Issue of maximizing a profit function
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Zongo Amara Mohamed Isaac
on 23 Aug 2019
Answered: Raunak Gupta
on 26 Aug 2019
Morning everybody.
I've just become very familiar with Matlab, but I'm having trouble to maximization of a profit function:
we have profit function=
\pi_{I}= P_{I}*q_{I} - sqrt(X_{I})/2;
and demand function=
P_{I}=A+'\beta'*X_{I}+'\mu'*X_{E}-q_{E}-q_{I}
P_{E}=A+'\beta'*X_{E}+ '\mu'*X_{I}-q_{E}-q_{I}
with '\mu'<='\beta'
(0<='\beta')&('\beta'<= 1)
Please how to determine q_{I} who maximizes the profit function (\pi{I}) ?
Which syntax would be written in Matlab, please?
Best regards.
2 Comments
Ted Shultz
on 23 Aug 2019
One way to solve this class of problems is to make the function you want to maximize a minimization problem (multiply by -1) and then use a function like fminsearch.
Another way is to try to calculate the derivative, check locations where that is == 0 to see if it is a maximum.
Accepted Answer
Raunak Gupta
on 26 Aug 2019
Hi,
From the example given by you I can understand that the profit maximization problem is an unconstrained optimization problem which depends upon cost function and demand function.
I am assuming that X_{I} and X_{E} are the variables that does not depend on q_{E} and q_{I} and the optimization is with respect to q_{E} and q_{I} and constraints are on ‘\mu’ and ‘\beta’. In that case you can use fminsearch and define the profit function \pi_{I} in terms of q_{I} and q_{E} if the value of ‘\mu’ and ‘\beta’ are fixed.
The problem can be such that
\pi_{I} = (A + '\beta'*X_{I} + '\mu'*X_{E} - q_{E} - q_{I})*q_{I} - sqrt(X_{I})/2;
If ‘\mu’ and ‘\beta’ are actual constraint and can vary then you need to setup an optimization problem which will have total 4 variables that is
q_{E}, q_{I},'\mu','\beta'
Here since the constraints are not given on q_{E} and q_{I} therefore the solver can be positive as well as negative values, I am assuming these are demands so you may lower bound them to 0.
For more information on Optimization Setup you can refer to:
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