{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":58449,"title":"Compute rational expectations in a static, linear NKM model","description":"Consider a static, linear approximation of the baseline New Keynesian macroeconomic model. This can be described by an IS equation of the form\r\n    ,\r\na PC equation of the form\r\n    ,\r\na (nominal) interest rate rule of Taylor type of the form\r\n    ,\r\nand a Fisher equation of the form\r\n    .\r\nIn these equations, , , ,  and  denote Okun's output gap, the inflation rate, the nominal interest rate, the real interest rate and the natural real interest rate respectively; a superscript  indicates expectations formed by agents, so  is the expected output gap and  the expected inflation rate. The terms ,  and  are white-noise shock terms; , , ,  and  are positive parameters (with ), and additionally, the Taylor principle holds so that .  is the central bank's (exogenously chosen) target inflation rate, and  is the implied target nominal interest rate.\r\nWe want to compute the rationally expected (model-consistent) inflation rate and output gap  and  that are implied by given values of the model's parameters and a given value of . To do this, we set  for any variable  for which agents form expectations (e.g.  and ), where  denotes the mathematical expectations operator and where the agents' information set  contains both the structure of the model equations and the values of all parameters (as well as  and ).\r\nSince  for any such  by the law of iterated expectations, and since  for the white-noise shocks (where  is ,  or ), this allows us to write down the IS equation as\r\n    ,\r\nthe PC equation as\r\n    ,\r\nand so on. Please solve the model in expectations (you may find it helpful to write the nominal Taylor rule in terms of the real interest rate gap  for this) and write a function that, for given parameter values and a given inflation rate target, computes the model-consistent expectations  and  for  and .\r\n\r\nBonus question 1: what can you say about the relationship of  and ? What happens if ?\r\nBonus question 2: why is there, in fact, always a unique solution for  and ?\r\nBonus question 3: what role does the parameter  play?","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 818px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 409px; transform-origin: 407.5px 409px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 21px; text-align: left; transform-origin: 384.5px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a static, linear approximation of the baseline New Keynesian macroeconomic model. This can be described by an IS equation of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.9167px; text-align: left; transform-origin: 384.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"140.5\" height=\"21\" style=\"width: 140.5px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ea PC equation of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.9167px; text-align: left; transform-origin: 384.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"112.5\" height=\"21\" style=\"width: 112.5px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ea (nominal) interest rate rule of Taylor type of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.9167px; text-align: left; transform-origin: 384.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"179\" height=\"21\" style=\"width: 179px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eand a Fisher equation of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"19\" style=\"width: 63px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 106.667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 53.3333px; text-align: left; transform-origin: 384.5px 53.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIn these equations, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ei\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"19\" style=\"width: 14.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e denote Okun's output gap, the inflation rate, the nominal interest rate, the real interest rate and the natural real interest rate respectively; a superscript \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAmCAYAAAAbdcG0AAAAAXNSR0IArs4c6QAAAPJJREFUSEvt0y1IBEEchvHfbBUvyGK0CGIVy4HYzVZRi+HAZFKDyQPBZrhwNj9AEJvJYtFmNlgEMYrZ5K4sq3Au3K5yZcNOnXmG//vMO8EIK4zAqiecTTWLJZFYYhz7ePuJOmzs7OAOUUyyi0lcood+GTyGQ8I0aQcv2AhspazjoQxewTlWcYFlbKKLWyTD4BhnhBnSK6IWyR1u8F581mLmecJ1DtrGR1kPivAC7nGAPXx+w5mHCbwOXlaEp3CKFtbwiDm5qCM8l8HZXju3bRFPOMbJXzL/q+r17HZlhGbsSkW/DzTCGmGVBpqSVCqqy6/6AmpmKCcgmFZrAAAAAElFTkSuQmCC\" width=\"7.5\" height=\"19\" style=\"width: 7.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e indicates expectations formed by agents, so \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the expected output gap and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e the expected inflation rate. The terms \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAoCAYAAAACJPERAAAAAXNSR0IArs4c6QAAAhtJREFUWEft1kuojGEcx/HPzIKFsGSjbFx2kp2SUqSTFB25bMhtJ5KFEuo4JZeTksRJWdig3LIRSmxkIZ2NIp3cdsqtXAoz+o9nOGcyM++ZM6+zMM9qpvd5/9/n/3t+z+95C8ZgFMaAqQPNVfUs8k5Cl6Ll+KHkKeYpGlRyDK9GusJG0CJWohePsQuDCbCYwk3KF7AVH0cCbgRdU6C//Au0Gk9S4WKRnhJ7CgyUhz/LxK4HnYwz6MZB7K1I+2d0K+hV1o/j+J6JlibVg85CSDcH+3BgJEWbza0HnYAT2IAH2IaHmJgKxh6WmxWv97zRns6k4s6umpffYgXutRM6HpvRgzc4hKv4gNjrGPG71C5odB7AMMgLrMfdVotnlXeoa8O92/H5X0KvYBPe1UDDZDuSsW60sqBaI8X/tTiVih3ByQQOFeZjI/pwv1UH/829EX8LU7zNRZzZSKNHuIbr+NRKh9V3sgR+1vpTsArLMDVF5/MhJ2F/Ovu5XOJxSVxKsHF4j6/pshiIDtrZaVWR2biI6DziM/wxLJvzgFaPXUDX4XXt/uQBjUQ7igU1V+Jvdh7QRQkazl+CW3l3Oh1bcBuncS7dxzPwDN/aZaRIqLjsX2JpMlHkdgC/4HIKkfPVD4F2yDsNZxEfcLtxBxEwO9MXx+GUYLGAymgHNGt45GqkpovodNpUotFM6Mg7GvWavtuRt6lEo5nw/8j7E8JMaCnv4ftYAAAAAElFTkSuQmCC\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAAAAXNSR0IArs4c6QAAAj5JREFUWEft1k+ITWEYx/HPvSkL+bOR1CwsRJL827FQymaSoikjipCd/AkbsRgTSZI/SZOEUv4skI1mR0oW0kgKNYUdIgqbca/e8U7M6d575px7xt3cs7qn9z7P93mf93d+z1vSwqfUQrY2vCXdH0vbp6BT2Rr8UvEaS5UNqjiF93krbwQvYx168RL7MBhBqyj1U72BHfiWp4BG8O4SfdU/wPV4FQHlMj0VDpYYqI5ey1RDPfhUXEQXjuHQcMv/Pl1KelX14QyGMlHjn+vB5yK0dCEO40ie5Gkx9eCTcA5b8AQ78RSTY8JwxtW05Gnrjc58DsNq7kwk+Yy1eJiWPG29FnwitqMHH3Ecd/AVQQvhCb8racnT1pPw8B7AQUhvsRkP0pLkXU/C/1V5UPsu/MibPC2uEfw2tuFLIkkQ4+4owPtpgEbrtdq+ARdi0AmcjwWErizDVpzE42YVX0twwVZXRNtcjPDNB3d7hru4h+/N7HgkdiyDJQtnAjowq07QC3waD/iSaLXL64AHEjOisJvMPKzG9Tj9RvwhOGMwqz34mSyq6LbPxxXsxRtcje/XaomzSHjIFXa4MhrVApxGN57XOooi4TPjToMjhplwFNPiUKp52SgKHj7P4Ib7sQnv4vn3x9kQ7gW3kjeeIuAzoulsjIZ0AIvwCEHh4Y53KQ6nUWO4CPhsXMaHeOZhIE3H2VhEuAnd/B9qz2JIhX3nmaDj4XCZCyjizDND2ztvtz23aJoJbLe9me7ljm1p238D9nFtKXHURRYAAAAASUVORK5CYII=\" width=\"15.5\" height=\"20\" style=\"width: 15.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"12.5\" height=\"20\" style=\"width: 12.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e are white-noise shock terms; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eβ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eκ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15.5\" height=\"20\" style=\"width: 15.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAoCAYAAAACJPERAAAAAXNSR0IArs4c6QAAAopJREFUWEft10mIVFcUxvHfK0UJDqHdGAyiLkRXBreKYISIoVXQKGLUbBIxWwfEgSBoRBFDQBQVFIV2tk2ICGIixI2uTJoILkJE4oAKggMu1ETryam+LU1L19RVrrpWVbzz7v/c757znVuZ8p8WtMm0yq3CjxXiq3qcVYiaiFMZY3M+x+WqVq0QVAk6F7/IXJJbgnvNhg7AFpn18sJ+iivxvNnQFpk2edZK/i32NwIYa5STdxLaZYbKzcEfmFRgSZF4VlDQoagd12tRoRw0zvAILuJLPEw7HYeD2IszKNaqQG/QQdiOOMdok3X4DyMUrFYswf6sFdYV3xt0FI5iOpam7x9hBQ7jVr3Acmc6VeZ8lnuQswCvk8Q/4FFfgOWgsaN9uICzWItdSeq8GdAP0uIB7v6JgvoK95sBHYNjmIINuIbjGIYv8FMzoJ/hV/yj8zzvpBaZh5/xNR73BdyzeuP3xpL9hTHwDZ4m0AE8S4lEUmGTn+IqntSSRE9oSLgHy/AdtiIK5+PSiOuEnE6tMzolEDEv+wKdgJP4BDPxW1osklucKjoSO5ecaE06hlqY73hv5yjjSurL7iYwEMvxPTpSG9XlSpXmaS07GImFmI1wr0X4N9XFZmzC7kpTphZg99j5aRBEEYaHR5G9wE381Sxo6YqD2Hl0QTjbq+5ZNVLernU/RLRXQGMk3u0pWTOgg7ET09K5/v0+oDMSdHKPtnvLbvROx6a2iuEQd6owlG0Yjxv4v1GFNCQ5023MSkUU/R3AuD3GgAhXO5HmctmLWbUtE3Z4CMPTteb30qVN6R9BWOkOxPB/e31ttLxVJdoPrUqmeoP65a1Xuare65e3KpnqDeqXt17lqnrvDVkVhCmbqy1EAAAAAElFTkSuQmCC\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e are positive parameters (with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"37.5\" height=\"18\" style=\"width: 37.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e), and additionally, the Taylor principle holds so that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42\" height=\"20\" style=\"width: 42px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACMAAAAmCAYAAABOFCLqAAAAAXNSR0IArs4c6QAAAnBJREFUWEft1UmozWEYx/HP/y4QkTHZmLK0UBaGEpEQMi0sDAssRBRKClESkSlThixcYYEMG2EjKRZs2CgLO0OGi6xw/3qu91zHzTn+5xpSzlunc86/9/883/d5fr/nzfxDK/uHWPyXMA0Yiufp88OG/K3KDCE7S34JWypJ40/CdMcM3MQbHMc5XMQkvMLtcrA/CTMMZzOaclYkgBsYi+3YhK34XAL6GUx/LCSbmslH5q3HiNe+/iv9yniaMx33yk7bAQE1E6NpeELzhVSR161B0guVYOL5LJnd8mxAlhLn3xhaKVoefX1+FQvwsi1MA3OaGdXAw2Yt+26l1n0nn0owUcolONLyUuaoPHtBvgrvMVymUW4PjqWIn/Cu7LQDcQq9sQzjcB3jsRlrsBfN1drUD/NxCB8wG+exmpbkcYD1WIspbUVYdtSumIb7eJYEHOKNNk3GYzyoRcBdcCCJr6SHgD2ZgizE0wJTfDBOI6y9rb3WHo4ruIPFqc8TcS0F3VjuhipQUc1BeJssXfPQiwChkV2IpGHDcMcOrEwz5HKBqhTeUs3apXaMKNNGnO4MOmEuHhXOVGBjNZh5yQ3llo1WxSRtxPLkrD74iKYC+apuqQQTl1okjaqEBcNFHVPLwqYhwmhdt/S9s6CQa4LphXVYhJ7JtmHzJ+iRKjIVd5P1w7oxOwL824BuZ4naVmYC4v6IFcNtadJIJOqMfWkYltKdSCKPYffLqy1MX+xHQG1I0zUma2mNwcHkqgA5nKB/GSQC/Oyi/C1Jigapw7T3Oiha4d+yr96meptqFVJdM3XN1DVTawXqmqm1YvU5U6liXwA/XocnAWgeeAAAAABJRU5ErkJggg==\" width=\"17.5\" height=\"19\" style=\"width: 17.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the central bank's (exogenously chosen) target inflation rate, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"74\" height=\"19\" style=\"width: 74px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the implied target nominal interest rate.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 52.5px; text-align: left; transform-origin: 384.5px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWe want to compute the rationally expected (model-consistent) inflation rate and output gap \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e that are implied by given values of the model's parameters and a given value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"17.5\" height=\"19\" style=\"width: 17.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. To do this, we set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"77.5\" height=\"19.5\" style=\"width: 77.5px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for any variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for which agents form expectations (e.g. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e), where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" style=\"width: 28.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e denotes the mathematical expectations operator and where the agents' information set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eI\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e contains both the structure of the model equations and the values of all parameters (as well as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"17.5\" height=\"19\" style=\"width: 17.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"19\" style=\"width: 14.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 21.4167px; text-align: left; transform-origin: 384.5px 21.4167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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width=\"226\" height=\"19.5\" style=\"width: 226px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for any such \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by the law of iterated expectations, and since \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"60.5\" height=\"20\" style=\"width: 60.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for the white-noise shocks (where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ei\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e), this allows us to write down the IS\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAmCAYAAAAbdcG0AAAAAXNSR0IArs4c6QAAAPJJREFUSEvt0y1IBEEchvHfbBUvyGK0CGIVy4HYzVZRi+HAZFKDyQPBZrhwNj9AEJvJYtFmNlgEMYrZ5K4sq3Au3K5yZcNOnXmG//vMO8EIK4zAqiecTTWLJZFYYhz7ePuJOmzs7OAOUUyyi0lcood+GTyGQ8I0aQcv2AhspazjoQxewTlWcYFlbKKLWyTD4BhnhBnSK6IWyR1u8F581mLmecJ1DtrGR1kPivAC7nGAPXx+w5mHCbwOXlaEp3CKFtbwiDm5qCM8l8HZXju3bRFPOMbJXzL/q+r17HZlhGbsSkW/DzTCGmGVBpqSVCqqy6/6AmpmKCcgmFZrAAAAAElFTkSuQmCC\" width=\"7.5\" height=\"19\" style=\"width: 7.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e equation as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"121\" height=\"19.5\" style=\"width: 121px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ethe PC\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAmCAYAAAAbdcG0AAAAAXNSR0IArs4c6QAAAPJJREFUSEvt0y1IBEEchvHfbBUvyGK0CGIVy4HYzVZRi+HAZFKDyQPBZrhwNj9AEJvJYtFmNlgEMYrZ5K4sq3Au3K5yZcNOnXmG//vMO8EIK4zAqiecTTWLJZFYYhz7ePuJOmzs7OAOUUyyi0lcood+GTyGQ8I0aQcv2AhspazjoQxewTlWcYFlbKKLWyTD4BhnhBnSK6IWyR1u8F581mLmecJ1DtrGR1kPivAC7nGAPXx+w5mHCbwOXlaEp3CKFtbwiDm5qCM8l8HZXju3bRFPOMbJXzL/q+r17HZlhGbsSkW/DzTCGmGVBpqSVCqqy6/6AmpmKCcgmFZrAAAAAElFTkSuQmCC\" width=\"7.5\" height=\"19\" style=\"width: 7.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e equation as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"93\" height=\"19\" style=\"width: 93px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eand so on. Please solve the model in expectations (you may find it helpful to write the nominal Taylor rule in terms of the real interest rate gap \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"50\" height=\"19.5\" style=\"width: 50px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for this) and write a function that, for given parameter values and a given inflation rate target, computes the model-consistent expectations \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBonus question 1:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e what can you say about the relationship of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"17.5\" height=\"19\" style=\"width: 17.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e? What happens if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"37.5\" height=\"18\" style=\"width: 37.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBonus question 2: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhy is there, in fact, always a unique solution for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBonus question 3: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhat role does the parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e play?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function xe_and_pie = ratexp(b, beta, kappa, k_pi, k_x, pistar)\r\n\r\n    xe  = 0;\r\n    pie = pistar;\r\n    \r\n    xe_and_pie = [xe; pie];\r\n    \r\nend","test_suite":"%%\r\nassert(max(abs(ratexp(1, 0.99, 0.3433, 1.5, 0.5, 2) - [0.056609114067365; 1.943390885932635])) \u003c 1e10)\r\n\r\n%%\r\nassert(max(abs(ratexp(2, 0.97, 0.2, 2, 0.8, 3) - [0.401785714285715; 2.678571428571428])) \u003c 1e10)\r\n\r\n%%\r\nassert(max(abs(ratexp(0.5, 0.98, 0.5, 1.3, 1, 2.5) - [0.088235294117647; 2.205882352941177])) \u003c 1e10)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":332395,"edited_by":332395,"edited_at":"2023-06-28T08:06:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-06-22T11:37:22.000Z","updated_at":"2023-06-28T08:06:52.000Z","published_at":"2023-06-22T11:40:17.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a static, linear approximation of the baseline New Keynesian macroeconomic model. This can be described by an IS equation of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex = x^e - b (r - r^n) + \\\\varepsilon_x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ea PC equation of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi = \\\\beta \\\\pi^e + \\\\kappa x + \\\\varepsilon_\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ea (nominal) interest rate rule of Taylor type of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei = i^* + k_\\\\pi (\\\\pi - \\\\pi^*) + k_x x + \\\\varepsilon_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand a Fisher equation of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei = r + \\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn these equations, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er^n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e denote Okun's output gap, the inflation rate, the nominal interest rate, the real interest rate and the natural real interest rate respectively; a superscript \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{}^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e indicates expectations formed by agents, so \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the expected output gap and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the expected inflation rate. The terms \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varepsilon_x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varepsilon_\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varepsilon_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are white-noise shock terms; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\beta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\kappa\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are positive parameters (with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\beta \\\\le 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e), and additionally, the Taylor principle holds so that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_\\\\pi \u0026gt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the central bank's (exogenously chosen) target inflation rate, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei^* = r^n + \\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the implied target nominal interest rate.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe want to compute the rationally expected (model-consistent) inflation rate and output gap \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that are implied by given values of the model's parameters and a given value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. To do this, we set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez^e = \\\\mathbb{E}(z \\\\mid I)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for any variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for which agents form expectations (e.g. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e), where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbb{E}(\\\\cdot)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e denotes the mathematical expectations operator and where the agents' information set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e contains both the structure of the model equations and the values of all parameters (as well as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er^n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(z^e)^e = \\\\mathbb E(\\\\mathbb E(z \\\\mid I) \\\\mid I) = \\\\mathbb E(z \\\\mid I) = z^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for any such \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by the law of iterated expectations, and since \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbb E(\\\\varepsilon_z) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for the white-noise shocks (where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e), this allows us to write down the IS\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{}^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equation as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e = x^e - b (r^e - r^n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe PC\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{}^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equation as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e = \\\\beta \\\\pi^e + \\\\kappa x^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand so on. Please solve the model in expectations (you may find it helpful to write the nominal Taylor rule in terms of the real interest rate gap \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(r - r^n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for this) and write a function that, for given parameter values and a given inflation rate target, computes the model-consistent expectations \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBonus question 1:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e what can you say about the relationship of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e? What happens if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\beta = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBonus question 2: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ewhy is there, in fact, always a unique solution for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBonus question 3: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ewhat role does the parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e play?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":58478,"title":"Optimal saving in Solow's classical growth model","description":"Let us consider a simplified version of Solow's classical growth model. Let , , ,   and  denote production, the capital stock, labor, (gross) investment, savings and consumption at time  respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, , satisfying the following conditions:\r\nThe marginal product of capital and labor is positive: , and .\r\nThe marginal product of capital and labor is diminishing: , and .\r\nProduction exhibits constant returns to scale:  is homogenous of degree one, i.e.  for all .\r\n satisfies the Inada conditions: , and .\r\nCapital in the economy accumulates according to the law of motion , where  is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output,  for all , for some . Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that  for all .\r\nAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we are considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by  throughout and taking advantage of the fact that  is homogenous of degree one. We use lower-case letters for per-capita terms:  is the capital intensity,  is output per capita, and so on. We also write ;  is the intensive form of the production function .\r\nThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by . An expression implicitly characterizing  can be derived from the law of motion for capital by moving to per-capita variables and replacing  and  with  throughout.\r\nSince in the steady state,  is constant, so is output per capita  and hence consumption per capita .  depends on three things: the depreciation rate , the savings rate , and the macroeconomic production function  (equivalently, ). A social planner seeking to maximize steady-state per-capita consumption may not be able to change  or , but can maximize  by influencing . We will call the savings rate that maximizes per-capita consumption the golden rule savings rate and denote it ; similarly, we will denote steady-state values for ,  etc. implied by  as ,  and so forth.\r\nTo find , we proceed as follows:\r\nfind an expression for  by using the relationship , moving to per-capita terms, and using the expression characterizing  to replace the term  with ;\r\ntake the derivative w.r.t. , keeping in mind that  depends on ;\r\nset the resulting expression to zero, obtaining an equality identifying  with the marginal product of capital, in per-capita terms, when the economy follows the golden rule;\r\nsubstitute this expression back into the expression characterizing  and solving for .\r\n can then be found by again considering the relationship  in per-capita terms in the steady state, with .\r\nYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, ,  (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter , the capital elasticity of output  and the depreciation rate , please compute the golden rule savings rate , and the resulting steady-state capital intensity , per-capita output  and per-capita consumption .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 965.688px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 482.844px; transform-origin: 407px 482.844px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42.25px; text-align: left; transform-origin: 384px 42.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eLet us consider a simplified version of Solow's classical growth model. Let \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"11\" height=\"20\" style=\"width: 11px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"13.5\" height=\"20\" style=\"width: 13.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"20\" style=\"width: 16px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e denote production, the capital stock, labor, (gross) investment, savings and consumption at time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81.5\" height=\"19\" style=\"width: 81.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, satisfying the following conditions:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 143.438px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 71.7188px; transform-origin: 391px 71.7188px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 35px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 17.5px; text-align: left; transform-origin: 363px 17.5px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe marginal product of capital and labor is positive: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"83\" height=\"35\" style=\"width: 83px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81\" height=\"35\" style=\"width: 81px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 37px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 18.5px; text-align: left; transform-origin: 363px 18.5px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe marginal product of capital and labor is diminishing: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg 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width=\"95\" height=\"37\" style=\"width: 95px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg 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width=\"91.5\" height=\"37\" style=\"width: 91.5px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.9375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4688px; text-align: left; transform-origin: 363px 20.4688px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eProduction exhibits constant returns to scale: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is homogenous of degree one, i.e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"141\" height=\"19\" style=\"width: 141px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for all \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36\" height=\"18\" style=\"width: 36px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 30.5px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 15.25px; text-align: left; transform-origin: 363px 15.25px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e satisfies the Inada conditions: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg 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margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eCapital in the economy accumulates according to the law of motion \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"63\" height=\"18\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"61\" height=\"18\" style=\"width: 61px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for all \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 124.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 62.25px; text-align: left; transform-origin: 384px 62.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eare\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e throughout and taking advantage of the fact that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is homogenous of degree one. We use lower-case letters for per-capita terms: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44.5\" height=\"40\" style=\"width: 44.5px; height: 40px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the capital intensity, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"43\" height=\"40\" style=\"width: 43px; height: 40px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is output per capita, and so on. We also write \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"99\" height=\"20\" style=\"width: 99px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the intensive form of the production function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. An expression implicitly characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e can be derived from the law of motion for capital by moving to per-capita variables and replacing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADMAAAAoCAYAAABTsMJyAAAAAXNSR0IArs4c6QAAAvxJREFUaEPt2FvoVFUUx/GPJUakhD2FIVoR6Yu99ZA+6IPhBYMyEDVFSMlKBC+IF8IHC0TwQpApFAaVoZWiFGL1EFjhgxoEIRIkIVgomFERijcWrAPT8J8z/8Ez/5mB2S9z27P2+q6191q/fYYpH6PxIWZjNXY2md/Rn4c1WX0CDmI8ZuL7jnrbZPFmMM/hCL7FQlzsVZh7sQUbsBer8F+vwtSel+UJ1M0syrbZJHyGkZiD04jvYrvF6z34Mef83A1ZK4MJpz/CN1iAy5mWR/E+3sXnuNUt6WoEMwJb85xEOV6P63gIaxLiTLdAFH40ghmDjzEVL+X7h/EKPsBv3QYS/jSCmYxj+AMv4mZute240o0gZTCRgT04jqNYh7dTAdzuJZj70+kAqh1RCBbj916CGYf9eAYb8RM+wSjMxaEOwTyAJXgcmwZqBQOdmen4Cr/kebmQpfh5HMbL+HMIge7LPhfrzihTI/Uw8TmoQ8ZEw1yKvxLgPfydgAEbcmcaTuFqi3BR+iPTYS9KftkYnoXqMRzAyUbSqh4mFngHi/AG3kIc+EfyKhDOf5olemyCxZxr6U3Ym4+I5r4SD6NafocpLSjxJ1uFKf7wFJ7F13VORoUL4C+y86/N7VhUxtgGkdUVGcFGPEMCU0j+H7Kv1DbHSPcyvJmaLMp1oQImYmXqtn8yi9Gn4uow0BgSmBa3/v+mF1n9Eptxo9Pb7G5giqy+kFWvzFZXZ6a4yM3CPJyrISnU92ADVWjB+vktF4DBLlg/70FE6Y6q9nqW82JONOG4NtSOOGO78RrO1v12voGQHTKY4iIX5XhbCtOe3WaxleKyFgr7Uka2TCV07ZkpLnKh5XZlw4y7UNXVrHjsFW1jwIcrzR41DeYMBUw0ypA+OxLo3yZ/bCUzoSaeTmXxamY9Sv8J/Fq7ThUwgwGun/MEooSHAg9BW8noFEwlztcb6cO0JawVGO1npoIgtsVEPzNtCWsFRvuZqSCIbTHRz0xbwlqB0X5mKghiW0zcAdC9tyk09Uw1AAAAAElFTkSuQmCC\" width=\"25.5\" height=\"20\" style=\"width: 25.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"13\" height=\"20\" style=\"width: 13px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e throughout.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.75px; text-align: left; transform-origin: 384px 52.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSince in the steady state, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is constant, so is output per capita \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"19\" style=\"width: 16px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and hence consumption per capita \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e depends on three things: the depreciation rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the savings rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and the macroeconomic production function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (equivalently, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e). A social planner seeking to maximize steady-state per-capita consumption may not be able to change \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, but can maximize \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by influencing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. We will call the savings rate that maximizes per-capita consumption the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e savings rate and denote it \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; similarly, we will denote steady-state values for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ec\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e etc. implied by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"38.5\" height=\"20\" style=\"width: 38.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and so forth.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTo find \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we proceed as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 123.75px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 61.875px; transform-origin: 391px 61.875px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 42px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 21px; text-align: left; transform-origin: 363px 21px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003efind an expression for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAmCAYAAADTGStiAAAAAXNSR0IArs4c6QAAAf1JREFUWEft1UuoTWEYBuDnEKWIYqQcuUyUgTJwGaBToiiEklBM5FYkJo4oREZSBkoZuBaJkbuSKAMMkIkBmTBwSxLl0l/frm3lnPMva532ZO9atVd9//t+3/t/77s6tOjX0SJe/UE8AJPxLp5/ztYfxBNxHlewrydF6yIegUW4i484gYu4jHl4j/vNTdRFPCWm/IQtQXYLs3EIe3AAPxvkdREnvMFIDSzGTLzCpZj0A36XnTgtSwJchk68jPe0PPvxJgAbxEsxA89wDfdC/r+uu6+JxwT4BGzFo+h8Lm7gAtZjOE5jFDZiDm6iC3uxHUfwK0fqqTgWhWvxIv4PivvaEY2sCNssxGO8jeVKi5Wknh8qPc2RehJOhrSrcKdgi9TUGlzF9cL9jcfZsNPBMnYagsPYjKPYie8lEi5d3zh8DhtlB8issMZQLMHtEqTZpcXlSu+7InEeYCVeZ6OVKCwSD4uFWo1T2IQvJfCyS4vEI3Emkuc4tuFbNlqJwiJx8mPK2RQWvU2c/J3S6GsJrl4DZGDk6u7w7XI8bzqRGk2+TAGSamojThwND0/DOXTHgo3FBvyI4K909z1F5miswwJMx5PI3PSdfdgcfXVJ/b84pc/19ZEoDZh7oE2cq1TlurbUlSXMBWhLnatU5bq21JUlzAVoS52rVOW6lkn9B7+JXifHusByAAAAAElFTkSuQmCC\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by using the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, moving to per-capita terms, and using the expression characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to replace the term \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"20\" style=\"width: 40px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23\" height=\"19\" style=\"width: 23px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2188px; text-align: left; transform-origin: 363px 10.2188px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003etake the derivative w.r.t. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, keeping in mind that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e depends on \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.875px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4375px; text-align: left; transform-origin: 363px 20.4375px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eset the resulting expression to zero, obtaining an equality identifying \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with the marginal product of capital, in per-capita terms, when the economy follows the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2188px; text-align: left; transform-origin: 363px 10.2188px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003esubstitute this expression back into the expression characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and solving for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e can then be found by again considering the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in per-capita terms in the steady state, with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"38.5\" height=\"20\" style=\"width: 38.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 87px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43.5px; text-align: left; transform-origin: 384px 43.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"88.5\" height=\"22\" style=\"width: 88.5px; height: 22px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63.5\" height=\"18\" style=\"width: 63.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eA\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the capital elasticity of output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eα\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and the depreciation rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, please compute the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e savings rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and the resulting steady-state capital intensity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, per-capita output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and per-capita consumption \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [sg, kg, yg, cg] = Solow(A, alpha, delta)\r\n    \r\nend","test_suite":"%%\r\n[s, k, y, c] = Solow(1, 0.3, 0.05);\r\nassert(max(abs([s k y c] - [0.3 12.931373133239163 2.155228855539861 1.508660198877902])) \u003c 1e-12)\r\n\r\n%%\r\n[s, k, y, c] = Solow(2, 0.4, 0.025);\r\nassert(max(abs([s k y c] - [0.4 322.5397887730876 20.158736798317975 12.095242078990784])) \u003c 1e-12)\r\n\r\n%%\r\n[s, k, y, c] = Solow(3, 0.5, 0.1);\r\nassert(max(abs([s k y c] - [0.5 225 45 22.5])) \u003c 1e-12)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":332395,"edited_by":332395,"edited_at":"2023-07-02T18:27:24.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-07-01T21:55:49.000Z","updated_at":"2023-07-02T18:27:24.000Z","published_at":"2023-07-01T21:55:49.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet us consider a simplified version of Solow's classical growth model. Let \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eC_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e denote production, the capital stock, labor, (gross) investment, savings and consumption at time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY = F(K, L)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, satisfying the following conditions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe marginal product of capital and labor is positive: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_K = \\\\frac{\\\\partial F}{\\\\partial K} \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_L = \\\\frac{ \\\\partial F}{ \\\\partial L} \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe marginal product of capital and labor is diminishing: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_{KK} = \\\\frac{ \\\\partial^2 F }{ \\\\partial K^2 } \u0026lt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_{LL} = \\\\frac{ \\\\partial^2 F }{ \\\\partial L^2 } \u0026lt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eProduction exhibits constant returns to scale: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is homogenous of degree one, i.e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF(cK, cL) = c F(K, L)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e satisfies the Inada conditions: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lim_{K\\\\to 0} F_K = \\\\lim_{L\\\\to 0} F_L = \\\\infty\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lim_{K\\\\to\\\\infty} = \\\\lim_{L\\\\to\\\\infty} F_L = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCapital in the economy accumulates according to the law of motion \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_{t+1} = (1 - \\\\delta) K_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; \\\\delta \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_t = sY_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, for some \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; s \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eare\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e throughout and taking advantage of the fact that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is homogenous of degree one. We use lower-case letters for per-capita terms: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_t = \\\\frac{ K_t }{ L_t }\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the capital intensity, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey_t = \\\\frac{ Y_t }{ L_t }\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is output per capita, and so on. We also write \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(k_t) = F(k_t, 1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the intensive form of the production function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. An expression implicitly characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can be derived from the law of motion for capital by moving to per-capita variables and replacing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_{t+1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e throughout.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince in the steady state, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is constant, so is output per capita \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and hence consumption per capita \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on three things: the depreciation rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the savings rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the macroeconomic production function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (equivalently, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e). A social planner seeking to maximize steady-state per-capita consumption may not be able to change \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, but can maximize \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by influencing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. We will call the savings rate that maximizes per-capita consumption the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e savings rate and denote it \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; similarly, we will denote steady-state values for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc. implied by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = s_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and so forth.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo find \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we proceed as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efind an expression for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by using the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, moving to per-capita terms, and using the expression characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to replace the term \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003esf(k^*)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta k^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003etake the derivative w.r.t. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, keeping in mind that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eset the resulting expression to zero, obtaining an equality identifying \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with the marginal product of capital, in per-capita terms, when the economy follows the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003esubstitute this expression back into the expression characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and solving for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can then be found by again considering the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in per-capita terms in the steady state, with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = s_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = AK_t^\\\\alpha L_t^{1-\\\\alpha}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; \\\\alpha \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the capital elasticity of output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\alpha\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the depreciation rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, please compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e savings rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the resulting steady-state capital intensity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, per-capita output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and per-capita consumption \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":586,"title":"All Humans are Created Equal - Pareto Equality","description":"One way or the other two sets of identical types can come out ahead of the other by idea of Pareto equality. Pareto equality between two sets, or a group of sets, requires that atleast one element of each set is ranked higher than its corresponding element of the other set. Please see: http://en.wikipedia.org/wiki/Pareto_optimal for more information.\r\nBuild a function to take two cell-args, and return a boolean value true/false, to indicate their pareto equality.\r\nEx. \u003e\u003e ispareto( {1,'foo',40}, {0,'bar',30}) = true\r\n    \u003e\u003e ispareto( {2,-10,'z'},{0,-9,'t'}) = false\r\nCell-array can have only numbers and strings. Use natural comparison functions for numbers (a\u003eb), while strings have 'a' \u003e 'z' kind of comparison.\r\nNext, generalize this function to work with varargs, and see if the entire set is pareto optimal.\r\nTwo or more arguments will always be supplied.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 265.433px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.717px; transform-origin: 407px 132.717px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 363px 8px; transform-origin: 363px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne way or the other two sets of identical types can come out ahead of the other by idea of Pareto equality. Pareto equality between two sets, or a group of sets, requires that atleast one element of each set is ranked higher than its corresponding element of the other set. Please see:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehttp://en.wikipedia.org/wiki/Pareto_optimal\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.5px 8px; transform-origin: 67.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for more information.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 337.5px 8px; transform-origin: 337.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBuild a function to take two cell-args, and return a boolean value true/false, to indicate their pareto equality.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147px 8px; transform-origin: 147px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEx. \u0026gt;\u0026gt; ispareto( {1,'foo',40}, {0,'bar',30}) = true\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 20.4333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 192px 8.5px; tab-size: 4; transform-origin: 192px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 96px 8.5px; transform-origin: 96px 8.5px; \"\u003e    \u0026gt;\u0026gt; ispareto( {2,-10,\u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 12px 8.5px; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 12px 8.5px; \"\u003e'z'\u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 32px 8.5px; transform-origin: 32px 8.5px; \"\u003e},{0,-9,\u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 12px 8.5px; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 12px 8.5px; \"\u003e't'\u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 40px 8.5px; transform-origin: 40px 8.5px; \"\u003e}) = false\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCell-array can have only numbers and strings. Use natural comparison functions for numbers (a\u0026gt;b), while strings have 'a' \u0026gt; 'z' kind of comparison.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 289px 8px; transform-origin: 289px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNext, generalize this function to work with varargs, and see if the entire set is pareto optimal.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 149px 8px; transform-origin: 149px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eTwo or more arguments will always be supplied.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function z = ispareto(varargin)\r\n  x = varargin{1};\r\n  y = varargin{2};\r\nend","test_suite":"%%\r\ny_correct = true;\r\nassert(isequal(ispareto({1,'foo',40},{0,'bar',30}),y_correct))\r\n\r\n%%\r\ny_correct = false;\r\nassert(isequal(ispareto({-2,-10,'z'},{0,-9,'t'}),y_correct))\r\n\r\n%%\r\ny_correct = false;\r\nassert(isequal(ispareto({-2,-10,-2},{0,5,6},{0,2,30}),y_correct))\r\n\r\n%%\r\ny_correct = true;\r\nassert(isequal(ispareto({0,'bar',30},{1,'foo',40},{10,'zoo',-1}),y_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":8,"created_by":3378,"edited_by":223089,"edited_at":"2023-03-07T11:27:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":11,"test_suite_updated_at":"2023-03-07T11:27:08.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-04-14T16:48:30.000Z","updated_at":"2023-03-07T11:27:08.000Z","published_at":"2012-04-14T16:48:37.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne way or the other two sets of identical types can come out ahead of the other by idea of Pareto equality. Pareto equality between two sets, or a group of sets, requires that atleast one element of each set is ranked higher than its corresponding element of the other set. Please see:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttp://en.wikipedia.org/wiki/Pareto_optimal\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for more information.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBuild a function to take two cell-args, and return a boolean value true/false, to indicate their pareto equality.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEx. \u0026gt;\u0026gt; ispareto( {1,'foo',40}, {0,'bar',30}) = true\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e ispareto( {2,-10,'z'},{0,-9,'t'}) = false]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCell-array can have only numbers and strings. Use natural comparison functions for numbers (a\u0026gt;b), while strings have 'a' \u0026gt; 'z' kind of comparison.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNext, generalize this function to work with varargs, and see if the entire set is pareto optimal.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTwo or more arguments will always be supplied.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":58449,"title":"Compute rational expectations in a static, linear NKM model","description":"Consider a static, linear approximation of the baseline New Keynesian macroeconomic model. This can be described by an IS equation of the form\r\n    ,\r\na PC equation of the form\r\n    ,\r\na (nominal) interest rate rule of Taylor type of the form\r\n    ,\r\nand a Fisher equation of the form\r\n    .\r\nIn these equations, , , ,  and  denote Okun's output gap, the inflation rate, the nominal interest rate, the real interest rate and the natural real interest rate respectively; a superscript  indicates expectations formed by agents, so  is the expected output gap and  the expected inflation rate. The terms ,  and  are white-noise shock terms; , , ,  and  are positive parameters (with ), and additionally, the Taylor principle holds so that .  is the central bank's (exogenously chosen) target inflation rate, and  is the implied target nominal interest rate.\r\nWe want to compute the rationally expected (model-consistent) inflation rate and output gap  and  that are implied by given values of the model's parameters and a given value of . To do this, we set  for any variable  for which agents form expectations (e.g.  and ), where  denotes the mathematical expectations operator and where the agents' information set  contains both the structure of the model equations and the values of all parameters (as well as  and ).\r\nSince  for any such  by the law of iterated expectations, and since  for the white-noise shocks (where  is ,  or ), this allows us to write down the IS equation as\r\n    ,\r\nthe PC equation as\r\n    ,\r\nand so on. Please solve the model in expectations (you may find it helpful to write the nominal Taylor rule in terms of the real interest rate gap  for this) and write a function that, for given parameter values and a given inflation rate target, computes the model-consistent expectations  and  for  and .\r\n\r\nBonus question 1: what can you say about the relationship of  and ? What happens if ?\r\nBonus question 2: why is there, in fact, always a unique solution for  and ?\r\nBonus question 3: what role does the parameter  play?","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 818px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 409px; transform-origin: 407.5px 409px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 21px; text-align: left; transform-origin: 384.5px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eConsider a static, linear approximation of the baseline New Keynesian macroeconomic model. This can be described by an IS equation of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.9167px; text-align: left; transform-origin: 384.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"140.5\" height=\"21\" style=\"width: 140.5px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ea PC equation of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.9167px; text-align: left; transform-origin: 384.5px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"112.5\" height=\"21\" style=\"width: 112.5px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 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\" width=\"179\" height=\"21\" style=\"width: 179px; height: 21px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eand a Fisher equation of the form\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63\" height=\"19\" style=\"width: 63px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 106.667px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 53.3333px; text-align: left; transform-origin: 384.5px 53.3333px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIn these equations, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ei\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003er\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAmCAYAAAA4LpBhAAAAAXNSR0IArs4c6QAAAf5JREFUWEft1UuIjlEcx/HPGRILlyQRNStlJUWJhYVyW9EQogZFrrmumIwFucZYWFCUW2Mjwo7NSEmKha1LUS4pVmSB99F5naemMfN65u0Zs3nP5lk8//P//v+/8zv/EwzCCoPA1ID2R/UhmI251W+TtyrasDiwL2MY1uBpnrRMeVsFl2XWYSxeYjwuYAMulg2NxbcFVmTcx0Pcxi5sQSse14LGBJMxFdOamFkJJshsxgvMQ3tgTMa2BBidOlqOwzgSAU10VIKJMhvx6V/QUZiE04KFMlexEy2CjpAZmf3J0I5DsTjcwEesxys0CzoFXSoO4mcReYfjFGEr2R68xgKcwUlMx1o8SEa5hu04h1/JWPFfC251d2YtI42j2uEiROgs7MW7HtaO7jyW4lbhOdX7vzud5UrEXM/wPe6tBZ2Bu/iaQEdxr5e7lBf3GTvwBfkZ/0g5huJ6LnEt6KYkVeScT5VXK+2x8uKO42ySdgROYDX240reZa1O46YORPAHLMWT/kyMWrF9ddqMTszBpWSQbwMNnd/t/JbhZlnAvuStTpd0/7rSdXg/0NDu0yU69kAyR2nc3s40ny5TsAR3SqOlRL1B4zMUp8ujZPk3/wNaNuOvfGW+p4WLbUALS1VPYEPeelQrvKchb2Gp6glsyFuPaoX3NOQtLFU9gYMi72/BFmYnajK+ogAAAABJRU5ErkJggg==\" width=\"14.5\" height=\"19\" style=\"width: 14.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e denote Okun's output gap, the inflation rate, the nominal interest rate, the real interest rate and the natural real interest rate respectively; a superscript \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAmCAYAAAAbdcG0AAAAAXNSR0IArs4c6QAAAPJJREFUSEvt0y1IBEEchvHfbBUvyGK0CGIVy4HYzVZRi+HAZFKDyQPBZrhwNj9AEJvJYtFmNlgEMYrZ5K4sq3Au3K5yZcNOnXmG//vMO8EIK4zAqiecTTWLJZFYYhz7ePuJOmzs7OAOUUyyi0lcood+GTyGQ8I0aQcv2AhspazjoQxewTlWcYFlbKKLWyTD4BhnhBnSK6IWyR1u8F581mLmecJ1DtrGR1kPivAC7nGAPXx+w5mHCbwOXlaEp3CKFtbwiDm5qCM8l8HZXju3bRFPOMbJXzL/q+r17HZlhGbsSkW/DzTCGmGVBpqSVCqqy6/6AmpmKCcgmFZrAAAAAElFTkSuQmCC\" width=\"7.5\" height=\"19\" style=\"width: 7.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e indicates expectations formed by agents, so \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the expected output gap and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e the expected inflation rate. The terms \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15.5\" height=\"20\" style=\"width: 15.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"12.5\" height=\"20\" style=\"width: 12.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e are white-noise shock terms; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eβ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eκ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15.5\" height=\"20\" style=\"width: 15.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e are positive parameters (with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"37.5\" height=\"18\" style=\"width: 37.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e), and additionally, the Taylor principle holds so that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42\" height=\"20\" style=\"width: 42px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"17.5\" height=\"19\" style=\"width: 17.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the central bank's (exogenously chosen) target inflation rate, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJQAAAAmCAYAAAAiPrKqAAAAAXNSR0IArs4c6QAABohJREFUeF7tmnmo1FUUxz93WsjCFpP2xUSxqPRBUfiKsqIssrKUMtupKEvNSiITS2wzwzSSSDBoMXtStv5RZJERRQQFLQTRgq22ZxtBy9z4/n73PsbXzLzz3vzGGeteGAZmzu/+7vme7z33e87v56g/NgemO5jpYRH5569erkl//48RcL34viPwELix4N8CzgDe/x/jlVzvBYHeCDWAEgtcmake7gWuAn5OqCYEaiHQG6F0XQkYCPxW5bjTfwcAX4dPQrr/CEzDMQ7P3cCT/Z+maVeaYm0hVL0VDsPRhc8AuLFprrR24s2A0cAR+XfpUyjPBk7AMQvPlsBZwBsNLnMOMA84O5cZbTeGOejyOdlrxro/hNoeOAV4CfgR3DLwjwJPAGOB74FX2g6Oxhd0LnA/cD4wCPgQ2MnBMg8XkUuCRsYccPPAtxOhNoh18LVurOsRSilOonxfYCjwVE4gOsB14fx6PNMCiZ4HjgTmAzcANwN/N4Jum10rnGY7ON3B6jK8HI6lGQ6meBDZXmtwze2YoTqALmA99Ii1Yz7+37GuRqitgcngJoE/JgOpxFLKXAn8HkBTmu+gxHjKdJZgbRkeC5npB8A3CG4zL5fPe4SNMhJKB+PKu+C5FPgAOBq4HtDuvJycPNsBy3BMxHETZW4JC1yEY1c8FwPfNLjoOQ7m+eKPvG10ojg40+M6wQ9CCPgY+hiq7Ee5oMJL7aE48ljDeAedHtZSJ9b1MpQmWeVgqCcDe2mVm0zAMRrPu8CzAXxlMesYAW4lMCpzptunist7+B2xcD7HJPvOCWDVcJpiW2B34I7QEnkQuAI4LYA5MCwlzjsSh1L9V3guAD4C9gZWQGkNlJWVG+3PNSNDDcexBM9xGwSkAtMe8P6SacMNJUsk1ARwo8HXjXU9Qp2stO5gnYeTKkTnEBzLnWewh8ugNAbKq7WzHcz1cDWwGCgbWDUCWIljVDefAkl0beUm6t5UYUNF7vUIvOGW3SZblWBhOfMh25UfQwb8Yge3e0cHnvOCVpToXg5MBe4Jx7lEunSkSPh4X25cw7ZoQu0M3AZIjrwIzAx9xMnAO8AOwJ3AH5CdPiKTYqbvuDmGBL8Hk+M0BshiDcylSqxrEUq/a9fpo8xzDvBdAEIthHHAm9mO1VGQC3IdeccHsaoFt/sQSMpMWrMIdWgA6IseC9cOlTaU3STgbfJcqiBIO6nZq7mER5QE/fG9SEJtAVwSgq9G9D7Aw8Bn5AXET8BhwDPAgqB5q8mUPse6FqFyzQATgVsBOVtNZEusrwgCVXab0jgIeBr4FRCJtP7nqjgQiafqdTogjRjx+TPMoUdUCli9Yy9muUYxOrwfVfSFIZ5RH6kVomQxAzgVeMGwKFOsaxFqJGSaYXhoEajCqzZ0vdgvxgvwTWloB+v40pA+rCw6Kv2IxNPxsSRsrAFhZ+v4uA54wJCdWkWoSH5lpPHA68BuFb0uretLQ+BMsa5FKKVxlYuqepSllOb/S0OEUCUjUq2rALqVPhZ55FX6cUiQJHoWG6WLdN+qgMG1QUcV4ns1QikdqmKaFbJUPHMLuWGPSXJRnlV5DY2+VHm6UajQ6ATuC2Jbj5ZaOZpBqHi0aW591B+UJpRu0vFdeBO1GqGk/iVWT6xYRLP6Sq0i1LEVemlCKChaSSbduxmEivhK/8R2wP7AI4B0Xzx9lLHVxP68URCqEapSP6mMVtkpJutxikrrTX1kXe+QhdeE53AWDdFsv4smlCo06b4poa0hYa4eYZQz0Xcd+WqHvGcU53VxqEaouHtfDWDvF7rKdxXQvGt2UCzzWytYy1xF2hRFKB1zSgDqBx4Y+krSTvENBkkZdfrVb9LDaDV59wyN3YZfTapGqFiNSMRJuO0VKplvi0SvhXNZK9iNvcSiCKWGploYRwUH9DrMNeH1I/0UH3JH/xRn/VZI4VWNUMOCUFVmUlm9MPReNjbAzbpf3DDKwCr7P2nWjfo4b1GEkjZSC0TzSSupiqtMBiKcYipNpayl/puq+UJGvUcvhdwgTWJGQG9u6AlEu75gZ3IkEcoEUzKyIpAIZUUq2ZkQSIQywZSMrAgkQlmRSnYmBBKhTDAlIysCiVBWpJKdCYFEKBNMyciKQCKUFalkZ0IgEcoEUzKyIpAIZUUq2ZkQSIQywZSMrAgkQlmRSnYmBBKhTDAlIysCiVBWpJKdCYFEKBNMyciKQCKUFalkZ0IgEcoEUzKyIvAP/USWNhGiU/kAAAAASUVORK5CYII=\" width=\"74\" height=\"19\" style=\"width: 74px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the implied target nominal interest rate.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 52.5px; text-align: left; transform-origin: 384.5px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWe want to compute the rationally expected (model-consistent) inflation rate and output gap \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAmCAYAAABK4fLXAAAAAXNSR0IArs4c6QAAAlxJREFUWEft1UuIjWEYB/DfdxZGFCHJwm0lxcqCjSwkaVwyG4RRTBFRKKlhNaEIyd3IwiUsbCiErUJZu2SBBSVFZOMyn54575nOHHNmjplGszjf7py+933+3//2ZIbAkw0BDOogSir8LybGYRkmYgZO4fH/AlFAU8amnD14juMKGnTYim8BZDCZiLtX4wCdA+9gVsb1vMjEWfyuBcRkNMs0ys0t0lfCnXf9zHJyPmApniWap2fcyHmKHZiEI3iI8/jelxwxaQWOykwpDslI07rFOt4s4rmHdfiUkLbKtMm1F/jRwVvcwquuE+mianLMp9BCxzl8LiLPPpLHF4WOs3EZx9Ce7vqFr2nAaFxIX78Wr3vro55AhIPj4OlEWRNuYmcaGmdasRuL8aiHAZGGq2jAGrxP74RRp+IdAnQ3gasBHYmTWFSmd4C8lA40K3qh8hmGtgR0Pa4ggG3DE9xFR60ggvbbKdMbkzQLcR8Hsa/k8B6AjMd+rMLPxGbI96JWT5RiEB4IR8ewuDC+8BC2Y3ky2oDXT289UaJ9Tpn203ANw7ESLweMoI+yCkOFluXRC0nC9ZGMUuMF7UH3l/4CqsbEzDQsWNiVUhFOD2m2lPlhVJLqcBWD1oSrEkQ4ODp+A8am+EVc32BMYqAxOTwivAQPEuBUoTXN/avvyv9YkGo1/otS2pw8EANGdC4fWsoOXEyVHCXV76eSiQk4gQCzN7VhV6lgXlrDkZIAcKa0CfuNYJC3aM24BnOV10HUzECtu+OfL+zPgbon6nJU+qbuibon6p6o1qb1dAypdPwB4XWCJzsSaV8AAAAASUVORK5CYII=\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e that are implied by given values of the model's parameters and a given value of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"17.5\" height=\"19\" style=\"width: 17.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. To do this, we set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"77.5\" height=\"19.5\" style=\"width: 77.5px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for any variable \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for which agents form expectations (e.g. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e), where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"28.5\" height=\"18.5\" style=\"width: 28.5px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e denotes the mathematical expectations operator and where the agents' information set \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eI\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e contains both the structure of the model equations and the values of all parameters (as well as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"17.5\" height=\"19\" style=\"width: 17.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"19\" style=\"width: 14.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.8333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 21.4167px; text-align: left; transform-origin: 384.5px 21.4167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSince \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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width=\"226\" height=\"19.5\" style=\"width: 226px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for any such \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by the law of iterated expectations, and since \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"60.5\" height=\"20\" style=\"width: 60.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for the white-noise shocks (where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ez\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ei\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e), this allows us to write down the IS\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAmCAYAAAAbdcG0AAAAAXNSR0IArs4c6QAAAPJJREFUSEvt0y1IBEEchvHfbBUvyGK0CGIVy4HYzVZRi+HAZFKDyQPBZrhwNj9AEJvJYtFmNlgEMYrZ5K4sq3Au3K5yZcNOnXmG//vMO8EIK4zAqiecTTWLJZFYYhz7ePuJOmzs7OAOUUyyi0lcood+GTyGQ8I0aQcv2AhspazjoQxewTlWcYFlbKKLWyTD4BhnhBnSK6IWyR1u8F581mLmecJ1DtrGR1kPivAC7nGAPXx+w5mHCbwOXlaEp3CKFtbwiDm5qCM8l8HZXju3bRFPOMbJXzL/q+r17HZlhGbsSkW/DzTCGmGVBpqSVCqqy6/6AmpmKCcgmFZrAAAAAElFTkSuQmCC\" width=\"7.5\" height=\"19\" style=\"width: 7.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e equation as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"121\" height=\"19.5\" style=\"width: 121px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ethe PC\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAmCAYAAAAbdcG0AAAAAXNSR0IArs4c6QAAAPJJREFUSEvt0y1IBEEchvHfbBUvyGK0CGIVy4HYzVZRi+HAZFKDyQPBZrhwNj9AEJvJYtFmNlgEMYrZ5K4sq3Au3K5yZcNOnXmG//vMO8EIK4zAqiecTTWLJZFYYhz7ePuJOmzs7OAOUUyyi0lcood+GTyGQ8I0aQcv2AhspazjoQxewTlWcYFlbKKLWyTD4BhnhBnSK6IWyR1u8F581mLmecJ1DtrGR1kPivAC7nGAPXx+w5mHCbwOXlaEp3CKFtbwiDm5qCM8l8HZXju3bRFPOMbJXzL/q+r17HZlhGbsSkW/DzTCGmGVBpqSVCqqy6/6AmpmKCcgmFZrAAAAAElFTkSuQmCC\" width=\"7.5\" height=\"19\" style=\"width: 7.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e equation as\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"93\" height=\"19\" style=\"width: 93px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e,\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eand so on. Please solve the model in expectations (you may find it helpful to write the nominal Taylor rule in terms of the real interest rate gap \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"50\" height=\"19.5\" style=\"width: 50px; height: 19.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for this) and write a function that, for given parameter values and a given inflation rate target, computes the model-consistent expectations \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eπ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBonus question 1:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e what can you say about the relationship of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"17.5\" height=\"19\" style=\"width: 17.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACEAAAAmCAYAAABK4fLXAAAAAXNSR0IArs4c6QAAAlxJREFUWEft1UuIjWEYB/DfdxZGFCHJwm0lxcqCjSwkaVwyG4RRTBFRKKlhNaEIyd3IwiUsbCiErUJZu2SBBSVFZOMyn54575nOHHNmjplGszjf7py+933+3//2ZIbAkw0BDOogSir8LybGYRkmYgZO4fH/AlFAU8amnD14juMKGnTYim8BZDCZiLtX4wCdA+9gVsb1vMjEWfyuBcRkNMs0ys0t0lfCnXf9zHJyPmApniWap2fcyHmKHZiEI3iI8/jelxwxaQWOykwpDslI07rFOt4s4rmHdfiUkLbKtMm1F/jRwVvcwquuE+mianLMp9BCxzl8LiLPPpLHF4WOs3EZx9Ce7vqFr2nAaFxIX78Wr3vro55AhIPj4OlEWRNuYmcaGmdasRuL8aiHAZGGq2jAGrxP74RRp+IdAnQ3gasBHYmTWFSmd4C8lA40K3qh8hmGtgR0Pa4ggG3DE9xFR60ggvbbKdMbkzQLcR8Hsa/k8B6AjMd+rMLPxGbI96JWT5RiEB4IR8ewuDC+8BC2Y3ky2oDXT289UaJ9Tpn203ANw7ESLweMoI+yCkOFluXRC0nC9ZGMUuMF7UH3l/4CqsbEzDQsWNiVUhFOD2m2lPlhVJLqcBWD1oSrEkQ4ODp+A8am+EVc32BMYqAxOTwivAQPEuBUoTXN/avvyv9YkGo1/otS2pw8EANGdC4fWsoOXEyVHCXV76eSiQk4gQCzN7VhV6lgXlrDkZIAcKa0CfuNYJC3aM24BnOV10HUzECtu+OfL+zPgbon6nJU+qbuibon6p6o1qb1dAypdPwB4XWCJzsSaV8AAAAASUVORK5CYII=\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e? What happens if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"37.5\" height=\"18\" style=\"width: 37.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBonus question 2: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhy is there, in fact, always a unique solution for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16.5\" height=\"19\" style=\"width: 16.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eBonus question 3: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhat role does the parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e play?\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function xe_and_pie = ratexp(b, beta, kappa, k_pi, k_x, pistar)\r\n\r\n    xe  = 0;\r\n    pie = pistar;\r\n    \r\n    xe_and_pie = [xe; pie];\r\n    \r\nend","test_suite":"%%\r\nassert(max(abs(ratexp(1, 0.99, 0.3433, 1.5, 0.5, 2) - [0.056609114067365; 1.943390885932635])) \u003c 1e10)\r\n\r\n%%\r\nassert(max(abs(ratexp(2, 0.97, 0.2, 2, 0.8, 3) - [0.401785714285715; 2.678571428571428])) \u003c 1e10)\r\n\r\n%%\r\nassert(max(abs(ratexp(0.5, 0.98, 0.5, 1.3, 1, 2.5) - [0.088235294117647; 2.205882352941177])) \u003c 1e10)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":332395,"edited_by":332395,"edited_at":"2023-06-28T08:06:52.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-06-22T11:37:22.000Z","updated_at":"2023-06-28T08:06:52.000Z","published_at":"2023-06-22T11:40:17.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eConsider a static, linear approximation of the baseline New Keynesian macroeconomic model. This can be described by an IS equation of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex = x^e - b (r - r^n) + \\\\varepsilon_x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ea PC equation of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi = \\\\beta \\\\pi^e + \\\\kappa x + \\\\varepsilon_\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ea (nominal) interest rate rule of Taylor type of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei = i^* + k_\\\\pi (\\\\pi - \\\\pi^*) + k_x x + \\\\varepsilon_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand a Fisher equation of the form\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei = r + \\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn these equations, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er^n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e denote Okun's output gap, the inflation rate, the nominal interest rate, the real interest rate and the natural real interest rate respectively; a superscript \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{}^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e indicates expectations formed by agents, so \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the expected output gap and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e the expected inflation rate. The terms \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varepsilon_x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varepsilon_\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\varepsilon_i\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are white-noise shock terms; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\beta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\kappa\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_x\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e are positive parameters (with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\beta \\\\le 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e), and additionally, the Taylor principle holds so that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_\\\\pi \u0026gt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the central bank's (exogenously chosen) target inflation rate, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei^* = r^n + \\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the implied target nominal interest rate.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWe want to compute the rationally expected (model-consistent) inflation rate and output gap \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that are implied by given values of the model's parameters and a given value of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. To do this, we set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez^e = \\\\mathbb{E}(z \\\\mid I)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for any variable \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for which agents form expectations (e.g. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e), where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbb{E}(\\\\cdot)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e denotes the mathematical expectations operator and where the agents' information set \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e contains both the structure of the model equations and the values of all parameters (as well as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003er^n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(z^e)^e = \\\\mathbb E(\\\\mathbb E(z \\\\mid I) \\\\mid I) = \\\\mathbb E(z \\\\mid I) = z^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for any such \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by the law of iterated expectations, and since \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbb E(\\\\varepsilon_z) = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for the white-noise shocks (where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ez\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ei\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e), this allows us to write down the IS\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{}^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equation as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e = x^e - b (r^e - r^n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethe PC\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e{}^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e equation as\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e = \\\\beta \\\\pi^e + \\\\kappa x^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e,\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eand so on. Please solve the model in expectations (you may find it helpful to write the nominal Taylor rule in terms of the real interest rate gap \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(r - r^n)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for this) and write a function that, for given parameter values and a given inflation rate target, computes the model-consistent expectations \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBonus question 1:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e what can you say about the relationship of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e? What happens if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\beta = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBonus question 2: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ewhy is there, in fact, always a unique solution for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\pi^e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBonus question 3: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003ewhat role does the parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e play?\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":58478,"title":"Optimal saving in Solow's classical growth model","description":"Let us consider a simplified version of Solow's classical growth model. Let , , ,   and  denote production, the capital stock, labor, (gross) investment, savings and consumption at time  respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, , satisfying the following conditions:\r\nThe marginal product of capital and labor is positive: , and .\r\nThe marginal product of capital and labor is diminishing: , and .\r\nProduction exhibits constant returns to scale:  is homogenous of degree one, i.e.  for all .\r\n satisfies the Inada conditions: , and .\r\nCapital in the economy accumulates according to the law of motion , where  is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output,  for all , for some . Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that  for all .\r\nAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we are considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by  throughout and taking advantage of the fact that  is homogenous of degree one. We use lower-case letters for per-capita terms:  is the capital intensity,  is output per capita, and so on. We also write ;  is the intensive form of the production function .\r\nThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by . An expression implicitly characterizing  can be derived from the law of motion for capital by moving to per-capita variables and replacing  and  with  throughout.\r\nSince in the steady state,  is constant, so is output per capita  and hence consumption per capita .  depends on three things: the depreciation rate , the savings rate , and the macroeconomic production function  (equivalently, ). A social planner seeking to maximize steady-state per-capita consumption may not be able to change  or , but can maximize  by influencing . We will call the savings rate that maximizes per-capita consumption the golden rule savings rate and denote it ; similarly, we will denote steady-state values for ,  etc. implied by  as ,  and so forth.\r\nTo find , we proceed as follows:\r\nfind an expression for  by using the relationship , moving to per-capita terms, and using the expression characterizing  to replace the term  with ;\r\ntake the derivative w.r.t. , keeping in mind that  depends on ;\r\nset the resulting expression to zero, obtaining an equality identifying  with the marginal product of capital, in per-capita terms, when the economy follows the golden rule;\r\nsubstitute this expression back into the expression characterizing  and solving for .\r\n can then be found by again considering the relationship  in per-capita terms in the steady state, with .\r\nYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, ,  (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter , the capital elasticity of output  and the depreciation rate , please compute the golden rule savings rate , and the resulting steady-state capital intensity , per-capita output  and per-capita consumption .","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 965.688px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 482.844px; transform-origin: 407px 482.844px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42.25px; text-align: left; transform-origin: 384px 42.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eLet us consider a simplified version of Solow's classical growth model. Let \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"11\" height=\"20\" style=\"width: 11px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"13.5\" height=\"20\" style=\"width: 13.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"20\" style=\"width: 16px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e denote production, the capital stock, labor, (gross) investment, savings and consumption at time \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81.5\" height=\"19\" style=\"width: 81.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, satisfying the following conditions:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 143.438px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 71.7188px; transform-origin: 391px 71.7188px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 35px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 17.5px; text-align: left; transform-origin: 363px 17.5px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe marginal product of capital and labor is positive: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"83\" height=\"35\" style=\"width: 83px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"81\" height=\"35\" style=\"width: 81px; height: 35px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 37px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 18.5px; text-align: left; transform-origin: 363px 18.5px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe marginal product of capital and labor is diminishing: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg 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width=\"95\" height=\"37\" style=\"width: 95px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-16px\"\u003e\u003cimg 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width=\"91.5\" height=\"37\" style=\"width: 91.5px; height: 37px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.9375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4688px; text-align: left; transform-origin: 363px 20.4688px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eProduction exhibits constant returns to scale: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is homogenous of degree one, i.e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"141\" height=\"19\" style=\"width: 141px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for all \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36\" height=\"18\" style=\"width: 36px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 30.5px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 15.25px; text-align: left; transform-origin: 363px 15.25px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e satisfies the Inada conditions: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-15px\"\u003e\u003cimg 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margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eCapital in the economy accumulates according to the law of motion \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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width=\"63\" height=\"18\" style=\"width: 63px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" 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style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"61\" height=\"18\" style=\"width: 61px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e for all \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003et\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 124.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 62.25px; text-align: left; transform-origin: 384px 62.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eare\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e throughout and taking advantage of the fact that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is homogenous of degree one. We use lower-case letters for per-capita terms: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"44.5\" height=\"40\" style=\"width: 44.5px; height: 40px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the capital intensity, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-17px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"43\" height=\"40\" style=\"width: 43px; height: 40px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is output per capita, and so on. We also write \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"99\" height=\"20\" style=\"width: 99px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is the intensive form of the production function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAmCAYAAADTGStiAAAAAXNSR0IArs4c6QAAAn1JREFUWEft1lvIjWkUB/Df55jTxbiSQ4wUaqJcuHAoakxJY2aijGOEkCuH5JArprhgIhqKqBnDmBkzZHIsSuRiHKK5UmqaGlPKjFyIwrRqfbXb7W+/77tv7ad2+93PXuv5r/Vfa/2ft0Pj1RMr8XF+BuAWFuDPLnwqbXcUWPfFPqzAt1iLF5UQujAuAv4gAWdhPb4uCdoPY/EHXjbyKQIeg9MYgZm4WRJ4Hk5hSlc+RcCzcRbXsRB/NwGODCfgV0TAUaKleI5g7HKtfzPg7tiBLTiMdV3RlsGswe7MdC9W45vcm4q5uNYZeDPg2vrGIQFetMInQD7HqKT5FzysD7oZ8Dj8hP74FHcQe0F5fHfDvbTpbKJO4MhuKG7gN9zH69qomwEHwHe4mvP7NB0/xNGk8We8zf3PcgIuYheW4Qi+wkf4EreLqO6VzlHXGKHNGfFAbEAA3q3jPaidmBnG84Fsrpj76biAZ0XAg3EC07AonwdhFY6XUK85OIZParMsQ/XkjPCf7MY3Sfee2qibdFpvjMRjvKoiIJHZIVzCOWzC/qT9XVFrl/m/UXP1SYAAr13RZEvwpMzBRTaNgIfje0zCVjzAScQNFbU7U3Romf8bAc9IeXuU9f0rx+cLhBgsx79lDm9mUw8cv7elVIZ4xHUYWhtgMZMxGiEOobshqTEmv+O/qoHUAwedB7EY23P4o5mGpDgE0I85VsMyiBCIhp1bJePR+AHjcwavpHMEOD87PYI7n4q1EVGSyqs+485rsNFrTo98HdqZGh0jVq9epQMouo9LH1TVsA1clbGW7dtUt0xdVcc21VUZa9m+TXXL1FV1bFNdlbGW7d8/qv8Hq2N9JwVuavQAAAAASUVORK5CYII=\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. An expression implicitly characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e can be derived from the law of motion for capital by moving to per-capita variables and replacing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"25.5\" height=\"20\" style=\"width: 25.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"13\" height=\"20\" style=\"width: 13px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e throughout.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 105.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.75px; text-align: left; transform-origin: 384px 52.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSince in the steady state, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e is constant, so is output per capita \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"16\" height=\"19\" style=\"width: 16px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and hence consumption per capita \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e depends on three things: the depreciation rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the savings rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and the macroeconomic production function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (equivalently, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ef\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e). A social planner seeking to maximize steady-state per-capita consumption may not be able to change \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e or \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eF\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, but can maximize \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by influencing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. We will call the savings rate that maximizes per-capita consumption the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e savings rate and denote it \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e; similarly, we will denote steady-state values for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ec\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e etc. implied by \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"38.5\" height=\"20\" style=\"width: 38.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e as \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAoCAYAAAACJPERAAAAAXNSR0IArs4c6QAAAqpJREFUWEft10uszVcUx/HP1YYIHZSJaLwGDSYMJRhURUNEQ1oRigmiYeSRhjYmaGvSEInQpkTjFW8aIS0DEY9BvULEgAEhSKRUDETjlSXryL2He879u/8zu3tyXuu/v2utvdZvr9Ok9voYWzAeC7G6jn2bfm6qYzUIu9Af43CqTbvWMaoH/RIHcRzf4E6joR9gBZbiVyzAk0ZDm5/ntwkug6lWeodgD7pjAs4hvos0x2snXEibK0WyUAsam2/FMUzD/QxzADZiPfbiRdHwW4N2xqo8x2iTJfgfPbAoYeeLwir2rUF7Yxs+w/R83wtzsRk33xcYz7UGHYEjuIev8TxT/AsetAdYCxoRbcBf+BPfYW0q0stGQLvm5gFuvqKgZuJuI6D9sB3D8T0uYQc+wlfY1wjoGPyNa3met7JFJmE/ZuFhe8DVhRSff0j5C2GYjUcJ+h2P05FwKmRyFM7ivyJOVEMjheswA8vwI6JwPskrLiC7s3X6pANh87Q90IHYiaH4Akdzs3BualZ0OHYolWhxHkMR5lt9WrnKTmdfNheBDzEHK1Nzo42qVSn0eBjGZvR9M3OXm3tV7z4tEkGX7OeAzsftrI1ovQginLxaSxyKwCq2o7O656VsRi1EXUR9hKpF671eZUZaAUzJ9FeKMkadEJUbjYDGVbg8BeQiPsemTPEfeNYIaLecGCOyM1n9AQv5bNFSZaY32m1N9vbJWkVRJjQGuJ/wD37OCfKdclkWNNolNDlS+ymikmPFNBmTx5vzLKt6e+ZocwCHUzYDHlGPxMRqESkj0lCxmImrh/H4dxCjTYjCibIVKVL5GyZXRRTaHfdvnHWLEaeMSEOTQ3Ei0rgIrmNwzlXxP+jf6kouA1pYMjughVNW5IGO9BbJVmHbjvQWTlmRB14B+0qIKXSh+E4AAAAASUVORK5CYII=\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAoCAYAAAACJPERAAAAAXNSR0IArs4c6QAAAkVJREFUWEft1kuoTmEUBuDnuCQhAzPlUhJmyoCBDMillAiRW2EglwGKElLumZAyILkkEQMGIkJCJLlEMjEgUgYYSCK31un79bf5z9lfZ59M/j3au9Ze77fetd53fS3+w9PyHzA1QTuV9TL0dsFIzMJAvEjf77Adr3NP2B7ogJR4CFbjAX5hIq7gLJbhYw5wW6CjcCAlW4zn6b07dmBdOsTcVH1p3EagI3A00bkA1wsZ40CLcAmXU/UdAu2JPViF/ViPr6Uzlgj8V6XjcBq9MQPXSuTJCimCxvdGbMMdzMOrrIwlgougfdLwLMQJrMSnEnmyQoqg/XASk3EQa/AlK2OJ4CJoXxxORtBWpaHfD/hcAuOvkCJoV2zB5qTL2XhW91fET0nmEDGVgEb+mkZH4xQ2pWEahOX4ht0d6XUjc+iPJZiKMXiEW0lK9/CzAa3h03HYYCO0HV4drva0Pr49781pWY9kJAEaU/8mSS+8OeQXJtNqpVWCTsA5rEgKiMUQfQ/5xYZ6UqugStAawJzUjprmByefftkZoPOxFTPxGONxJFF8HN87A7QX1iIqu4tJCLCrxYVRJb3DsC/t2tttTWCVoBuwE/exCzca3SiqAg25LE3UDkVMcjyxrfbW97MqycSSCIc6j4vpFhF9jarHYjoeVm0O09I2iul9W5d8OI4lU7hZNWhQeQixHOoriukN+USvYyP9earoabfkOFHphXQzjKXxA2fwvjjJVYDm+HNrbBM0m7KcH5r05rCVHdukN5uynB9+AxIebClqF16sAAAAAElFTkSuQmCC\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and so forth.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTo find \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, we proceed as follows:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 123.75px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 61.875px; transform-origin: 391px 61.875px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 42px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 21px; text-align: left; transform-origin: 363px 21px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003efind an expression for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by using the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, moving to per-capita terms, and using the expression characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e to replace the term \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"20\" style=\"width: 40px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"23\" height=\"19\" style=\"width: 23px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2188px; text-align: left; transform-origin: 363px 10.2188px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003etake the derivative w.r.t. \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, keeping in mind that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e depends on \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 40.875px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4375px; text-align: left; transform-origin: 363px 20.4375px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eset the resulting expression to zero, obtaining an equality identifying \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e with the marginal product of capital, in per-capita terms, when the economy follows the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2188px; text-align: left; transform-origin: 363px 10.2188px; white-space: pre-wrap; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003esubstitute this expression back into the expression characterizing \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"15\" height=\"19\" style=\"width: 15px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and solving for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003es\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 21.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.75px; text-align: left; transform-origin: 384px 10.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB0AAAAoCAYAAAACJPERAAAAAXNSR0IArs4c6QAAAkVJREFUWEft1kuoTmEUBuDnuCQhAzPlUhJmyoCBDMillAiRW2EglwGKElLumZAyILkkEQMGIkJCJLlEMjEgUgYYSCK31un79bf5z9lfZ59M/j3au9Ze77fetd53fS3+w9PyHzA1QTuV9TL0dsFIzMJAvEjf77Adr3NP2B7ogJR4CFbjAX5hIq7gLJbhYw5wW6CjcCAlW4zn6b07dmBdOsTcVH1p3EagI3A00bkA1wsZ40CLcAmXU/UdAu2JPViF/ViPr6Uzlgj8V6XjcBq9MQPXSuTJCimCxvdGbMMdzMOrrIwlgougfdLwLMQJrMSnEnmyQoqg/XASk3EQa/AlK2OJ4CJoXxxORtBWpaHfD/hcAuOvkCJoV2zB5qTL2XhW91fET0nmEDGVgEb+mkZH4xQ2pWEahOX4ht0d6XUjc+iPJZiKMXiEW0lK9/CzAa3h03HYYCO0HV4drva0Pr49781pWY9kJAEaU/8mSS+8OeQXJtNqpVWCTsA5rEgKiMUQfQ/5xYZ6UqugStAawJzUjprmByefftkZoPOxFTPxGONxJFF8HN87A7QX1iIqu4tJCLCrxYVRJb3DsC/t2tttTWCVoBuwE/exCzca3SiqAg25LE3UDkVMcjyxrfbW97MqycSSCIc6j4vpFhF9jarHYjoeVm0O09I2iul9W5d8OI4lU7hZNWhQeQixHOoriukN+USvYyP9earoabfkOFHphXQzjKXxA2fwvjjJVYDm+HNrbBM0m7KcH5r05rCVHdukN5uynB9+AxIebClqF16sAAAAAElFTkSuQmCC\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e can then be found by again considering the relationship \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"71\" height=\"20\" style=\"width: 71px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e in per-capita terms in the steady state, with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"38.5\" height=\"20\" style=\"width: 38.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 87px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 43.5px; text-align: left; transform-origin: 384px 43.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-8px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"88.5\" height=\"22\" style=\"width: 88.5px; height: 22px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"63.5\" height=\"18\" style=\"width: 63.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eA\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, the capital elasticity of output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eα\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and the depreciation rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eδ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, please compute the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003egolden rule\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e savings rate \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAoCAYAAADt5povAAAAAXNSR0IArs4c6QAAAjpJREFUWEft1kuoTXEUBvDfjQiFQt4ZeIUSkSEjEiUpJSIiE0oieSaFkUchjyLJxDNG3hMDA4mByKMMmDBCeZZnS2vXdXTPPefe7Y7Onu32f69vr29937d2kw6+mjoYTwOwdMarURrPJmI9euIeBuAxJuMg7tf7RdUAo+gFPMByvENvHMBwLMTLMgHXYh+OYw0+Z/Hpeb84P6IuzGodFoBPsAAPs/JoLMUOfK0Ljaq2mILLGIjD2IBP9QJUnq/WYVfswrp8aSP24nt7QFszfqgyZjgbH7ASZ/4nYNSehlMYhltYgtdtBa3ssAtChbfxMYt2yvntzvsZuFkWYJ8UyHY8bVZ0fHpyZCr2bFmAI3JGR3CiWdGwQoAEA/MzbdqEWUlp0ckbLMOLtM4inEa7lVoJODU7iBnNwnsMQTccytn+bKG1GMe8zNv+eJRi+9L8fGu2qJW2CYgxhIVC0WNz5iG8sFH490cUKwOwR3bfDysQ4yhmfgXbCrCyAIvi17ElkyisdQOrcbRswL4pqAj3AIxI3I9BWFW5wsqgNJiKoI/cPYdxucqis1f1hHetgumMWGWDsQl/qfJ/ABZhMQrHcBLPW9oqZVA6JsURu3JOKvRO/pY8K7vDuZiJzXibgono24mLmUzfyjJ+rzR6rKygsriCtQj/2KUx29KSpjB8dBYZW3QSv5R7cAnX8KusDqPOUGzNxDqP7piEq7iLf3K3DNHUap8/5xqAddFVy+EGpbWwVNeZBqV10VXL4d9fFm8psCAQRAAAAABJRU5ErkJggg==\" width=\"14\" height=\"20\" style=\"width: 14px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, and the resulting steady-state capital intensity \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, per-capita output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e and per-capita consumption \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"14.5\" height=\"20\" style=\"width: 14.5px; height: 20px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [sg, kg, yg, cg] = Solow(A, alpha, delta)\r\n    \r\nend","test_suite":"%%\r\n[s, k, y, c] = Solow(1, 0.3, 0.05);\r\nassert(max(abs([s k y c] - [0.3 12.931373133239163 2.155228855539861 1.508660198877902])) \u003c 1e-12)\r\n\r\n%%\r\n[s, k, y, c] = Solow(2, 0.4, 0.025);\r\nassert(max(abs([s k y c] - [0.4 322.5397887730876 20.158736798317975 12.095242078990784])) \u003c 1e-12)\r\n\r\n%%\r\n[s, k, y, c] = Solow(3, 0.5, 0.1);\r\nassert(max(abs([s k y c] - [0.5 225 45 22.5])) \u003c 1e-12)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":332395,"edited_by":332395,"edited_at":"2023-07-02T18:27:24.000Z","deleted_by":null,"deleted_at":null,"solvers_count":2,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-07-01T21:55:49.000Z","updated_at":"2023-07-02T18:27:24.000Z","published_at":"2023-07-01T21:55:49.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eLet us consider a simplified version of Solow's classical growth model. Let \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eI_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eC_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e denote production, the capital stock, labor, (gross) investment, savings and consumption at time \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e respectively (all variables are in real rather than nominal terms), and assume that output is produced using a neoclassical production function using capital and labor as inputs, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY = F(K, L)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, satisfying the following conditions:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe marginal product of capital and labor is positive: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_K = \\\\frac{\\\\partial F}{\\\\partial K} \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_L = \\\\frac{ \\\\partial F}{ \\\\partial L} \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe marginal product of capital and labor is diminishing: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_{KK} = \\\\frac{ \\\\partial^2 F }{ \\\\partial K^2 } \u0026lt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF_{LL} = \\\\frac{ \\\\partial^2 F }{ \\\\partial L^2 } \u0026lt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eProduction exhibits constant returns to scale: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is homogenous of degree one, i.e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF(cK, cL) = c F(K, L)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec \u0026gt; 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e satisfies the Inada conditions: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lim_{K\\\\to 0} F_K = \\\\lim_{L\\\\to 0} F_L = \\\\infty\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\lim_{K\\\\to\\\\infty} = \\\\lim_{L\\\\to\\\\infty} F_L = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCapital in the economy accumulates according to the law of motion \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eK_{t+1} = (1 - \\\\delta) K_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; \\\\delta \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the rate of depreciation; investment equals savings, which are assumed to be a constant fraction of output, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eS_t = sY_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, for some \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; s \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Output that is not saved is consumed (in other words, we assume a closed economy with no government activity), so that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for all \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003et\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAssume that the population and hence the labor force is constant (this kind of defeats the purpose of a growth model, but we \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eare\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e considering a simplified version only). It is helpful to recast the model in per-capita (technically, per-laborer) terms by dividing by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e throughout and taking advantage of the fact that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is homogenous of degree one. We use lower-case letters for per-capita terms: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_t = \\\\frac{ K_t }{ L_t }\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the capital intensity, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey_t = \\\\frac{ Y_t }{ L_t }\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is output per capita, and so on. We also write \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef(k_t) = F(k_t, 1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is the intensive form of the production function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe model economy is in its steady state when the per-capita variables do not change; denote the steady-state capital intensity by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. An expression implicitly characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can be derived from the law of motion for capital by moving to per-capita variables and replacing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_{t+1}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e throughout.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSince in the steady state, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e is constant, so is output per capita \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and hence consumption per capita \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on three things: the depreciation rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the savings rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the macroeconomic production function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (equivalently, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ef\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e). A social planner seeking to maximize steady-state per-capita consumption may not be able to change \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e or \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eF\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, but can maximize \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by influencing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. We will call the savings rate that maximizes per-capita consumption the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e savings rate and denote it \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; similarly, we will denote steady-state values for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e etc. implied by \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = s_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e as \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and so forth.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo find \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, we proceed as follows:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efind an expression for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e by using the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, moving to per-capita terms, and using the expression characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e to replace the term \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003esf(k^*)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta k^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003etake the derivative w.r.t. \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, keeping in mind that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e depends on \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eset the resulting expression to zero, obtaining an equality identifying \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with the marginal product of capital, in per-capita terms, when the economy follows the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003esubstitute this expression back into the expression characterizing \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and solving for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e can then be found by again considering the relationship \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = C_t + I_t\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e in per-capita terms in the steady state, with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es = s_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYour task is now simple (in principle): assume that macroeconomic production follows a Cobb-Douglas relationship, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eY_t = AK_t^\\\\alpha L_t^{1-\\\\alpha}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e0 \u0026lt; \\\\alpha \u0026lt; 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e (you may verify that this satisfies the conditions listed above). For given values of the (constant) technology parameter \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the capital elasticity of output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\alpha\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and the depreciation rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, please compute the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003egolden rule\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e savings rate \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003es_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, and the resulting steady-state capital intensity \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, per-capita output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ey_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and per-capita consumption \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ec_g\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":586,"title":"All Humans are Created Equal - Pareto Equality","description":"One way or the other two sets of identical types can come out ahead of the other by idea of Pareto equality. Pareto equality between two sets, or a group of sets, requires that atleast one element of each set is ranked higher than its corresponding element of the other set. Please see: http://en.wikipedia.org/wiki/Pareto_optimal for more information.\r\nBuild a function to take two cell-args, and return a boolean value true/false, to indicate their pareto equality.\r\nEx. \u003e\u003e ispareto( {1,'foo',40}, {0,'bar',30}) = true\r\n    \u003e\u003e ispareto( {2,-10,'z'},{0,-9,'t'}) = false\r\nCell-array can have only numbers and strings. Use natural comparison functions for numbers (a\u003eb), while strings have 'a' \u003e 'z' kind of comparison.\r\nNext, generalize this function to work with varargs, and see if the entire set is pareto optimal.\r\nTwo or more arguments will always be supplied.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 265.433px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 132.717px; transform-origin: 407px 132.717px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 363px 8px; transform-origin: 363px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne way or the other two sets of identical types can come out ahead of the other by idea of Pareto equality. Pareto equality between two sets, or a group of sets, requires that atleast one element of each set is ranked higher than its corresponding element of the other set. Please see:\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ehttp://en.wikipedia.org/wiki/Pareto_optimal\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.5px 8px; transform-origin: 67.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for more information.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 337.5px 8px; transform-origin: 337.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eBuild a function to take two cell-args, and return a boolean value true/false, to indicate their pareto equality.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 147px 8px; transform-origin: 147px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eEx. \u0026gt;\u0026gt; ispareto( {1,'foo',40}, {0,'bar',30}) = true\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 20.4333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 192px 8.5px; tab-size: 4; transform-origin: 192px 8.5px; unicode-bidi: normal; white-space: pre; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 96px 8.5px; transform-origin: 96px 8.5px; \"\u003e    \u0026gt;\u0026gt; ispareto( {2,-10,\u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 12px 8.5px; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 12px 8.5px; \"\u003e'z'\u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 32px 8.5px; transform-origin: 32px 8.5px; \"\u003e},{0,-9,\u003c/span\u003e\u003cspan style=\"border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 12px 8.5px; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 12px 8.5px; \"\u003e't'\u003c/span\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; perspective-origin: 40px 8.5px; transform-origin: 40px 8.5px; \"\u003e}) = false\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 382.5px 8px; transform-origin: 382.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eCell-array can have only numbers and strings. Use natural comparison functions for numbers (a\u0026gt;b), while strings have 'a' \u0026gt; 'z' kind of comparison.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 289px 8px; transform-origin: 289px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eNext, generalize this function to work with varargs, and see if the entire set is pareto optimal.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 149px 8px; transform-origin: 149px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eTwo or more arguments will always be supplied.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function z = ispareto(varargin)\r\n  x = varargin{1};\r\n  y = varargin{2};\r\nend","test_suite":"%%\r\ny_correct = true;\r\nassert(isequal(ispareto({1,'foo',40},{0,'bar',30}),y_correct))\r\n\r\n%%\r\ny_correct = false;\r\nassert(isequal(ispareto({-2,-10,'z'},{0,-9,'t'}),y_correct))\r\n\r\n%%\r\ny_correct = false;\r\nassert(isequal(ispareto({-2,-10,-2},{0,5,6},{0,2,30}),y_correct))\r\n\r\n%%\r\ny_correct = true;\r\nassert(isequal(ispareto({0,'bar',30},{1,'foo',40},{10,'zoo',-1}),y_correct))\r\n\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":8,"created_by":3378,"edited_by":223089,"edited_at":"2023-03-07T11:27:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":11,"test_suite_updated_at":"2023-03-07T11:27:08.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-04-14T16:48:30.000Z","updated_at":"2023-03-07T11:27:08.000Z","published_at":"2012-04-14T16:48:37.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne way or the other two sets of identical types can come out ahead of the other by idea of Pareto equality. Pareto equality between two sets, or a group of sets, requires that atleast one element of each set is ranked higher than its corresponding element of the other set. Please see:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ehttp://en.wikipedia.org/wiki/Pareto_optimal\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e for more information.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBuild a function to take two cell-args, and return a boolean value true/false, to indicate their pareto equality.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEx. \u0026gt;\u0026gt; ispareto( {1,'foo',40}, {0,'bar',30}) = true\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    \u003e\u003e ispareto( {2,-10,'z'},{0,-9,'t'}) = false]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eCell-array can have only numbers and strings. Use natural comparison functions for numbers (a\u0026gt;b), while strings have 'a' \u0026gt; 'z' kind of comparison.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNext, generalize this function to work with varargs, and see if the entire set is pareto optimal.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTwo or more arguments will always be supplied.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"term":"tag:\"economics\"","current_player_id":null,"fields":[{"name":"page","type":"integer","callback":null,"default":1,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"per_page","type":"integer","callback":null,"default":50,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"sort","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"body","type":"text","callback":null,"default":"*:*","directive":null,"facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":false},{"name":"group","type":"string","callback":null,"default":null,"directive":"group","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"difficulty_rating_bin","type":"string","callback":null,"default":null,"directive":"difficulty_rating_bin","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"id","type":"integer","callback":null,"default":null,"directive":"id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"tag","type":"string","callback":null,"default":null,"directive":"tag","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"product","type":"string","callback":null,"default":null,"directive":"product","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_at","type":"timeframe","callback":{},"default":null,"directive":"created_at","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"profile_id","type":"integer","callback":null,"default":null,"directive":"author_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_by","type":"string","callback":null,"default":null,"directive":"author","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player_id","type":"integer","callback":null,"default":null,"directive":"solver_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player","type":"string","callback":null,"default":null,"directive":"solver","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"solvers_count","type":"integer","callback":null,"default":null,"directive":"solvers_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"comments_count","type":"integer","callback":null,"default":null,"directive":"comments_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"likes_count","type":"integer","callback":null,"default":null,"directive":"likes_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leader_id","type":"integer","callback":null,"default":null,"directive":"leader_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leading_solution","type":"integer","callback":null,"default":null,"directive":"leading_solution","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true}],"filters":[{"name":"asset_type","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":"\"cody:problem\"","prepend":true},{"name":"profile_id","type":"integer","callback":{},"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":"author_id","static":null,"prepend":true}],"query":{"params":{"per_page":50,"term":"tag:\"economics\"","current_player":null,"sort":"map(difficulty_value,0,0,999) 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