Issue of maximizing a profit function

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.

[Ted Shultz]

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.

[Zongo Amara Mohamed Isaac]

I see and intend to use the first possibility. But can you help me with the syntax so that I can see clearly, please?

Answer 1

[Raunak Gupta]

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.

You can setup the optimization problem using the basic examples given in OptimizationProblem.

For more information on Optimization Setup you can refer to:

https://www.mathworks.com/help/optim/problem-based-approach.html