Symbolic Constrained Optimization

Question

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:

$\max\limits_{x,y} U(x,y)$

$s.t.\ p_xx+p_yy=I$

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:

$U_x-\lambda$$p_x=0$
$U_y-\lambda$$p_y=0$
$I-p_xx-p_yy=0$

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.

Answer 1

You can adapt the answer to the question, How can I implement the method of Lagrange multipliers to find constrained extrema?, to obtain the first-order system.

Clear[U, px, py, x, y];
f[x_, y_] := U[x, y];
g1[x_, y_] := budget - {px, py}.{x, y}

h[x_, y_, λ_] := f[x, y] - λ g1[x, y]

Thread[
 D[h[x, y, λ], {{x, y, λ}}] == {0, 0, 0}
 ]
(*
  {px*λ + Derivative[1, 0][U][x, y] == 0, 
   py*λ + Derivative[0, 1][U][x, y] == 0, 
   -budget + px*x + py*y == 0}
*)

About the best I can do with an unspecified utility function U[x, y] is to eliminate λ:

Eliminate[
    {px*λ + Derivative[1, 0][U][x, y] == 0, 
     py*λ + Derivative[0, 1][U][x, y] == 0, 
     -budget + px*x + py*y == 0},
    λ
]
(*
  budget == px*x + py*y && 
   px*Derivative[0, 1][U][x, y] == py*Derivative[1, 0][U][x, y]
*)