I have a question regarding evaluating constrained optimization problems in symbolic terms. I would like to perform How can I implement the method of Lagrange multipliers to find constrained extrema? without specifying the objective function.
As a simple example, suppose I have a constrained utility maximization problem:
where $U_x,U_y>0$, the Hessian matrix of the second-order derivatives is negative definite, and $p_x$ and $p_y$ are unspecified coefficients.
Using Lagrangian multipliers, the first-order conditions are:
Of course, the model that I want to use is a little bit more complicated than this (otherwise it would be no problem doing it by hand). Thus, given an unspecified objective function $U(x,y)$ and budget constraint(s) with unspecified coefficients, I was wondering if it was possible in Mathematica to:
1) produce the first-order conditions, and
2) given the system of equations in (1), produce symbolic rather than numerical solutions, which are functions of $U_x$, $U_y$, $p_x$, $p_y$, etc.
Again, this differs from How can I implement the method of Lagrange multipliers to find constrained extrema? because: that question specified that the objective function $f(x,y,z)=xy+yz$, the coefficients of the constraints are implicitly equal to 1, and thus, it produces numerical solutions for the constrained extrema.