Constrained maximization of ....

Question

I have to maximize $U(x,y)= Min(ax+y, by+x)$ s.a $p_{1}x +p_{2}y =m$. I try the traditional solution for a leontieff $(ax_{1}+y= by_{1}+x)$ function but I'm not sure.. beacause exist regions where one plan is under the other and only one of them is a minimun...

Answer 1

I do not think an analytical solution exists for the problem.

For a numerical solution, you can use the simplex algorithm to solve the problem, once you have linearized it as follows: $$ \mbox{Maximize }\; Z= t $$ subject to $$ ax+y\ge t\\ by+x\ge t\\ p_1x+p_2y=m $$

Answer 2

It seems if you solve the system of equations [ p1x+p2y=m; ax+y=by+x] you get exactly the solution.