2.12 Proof of the envelope theorem in constrained optimization problems Because we use the envelope theorem in constrained optimization problems often in the text, proving this theorem in a simple case may help develop some intuition. Thus, suppose we wish to maximize a function of two variables and that the value of this function also depends on a parameter, a: f(x1, x2, a). This maximization problem is subject to a constraint that can be written as: g (x1, x2,

a) ¼ 0.

a. Write out the Lagrangian expression and the first-order conditions for this problem.

b. Sum the two first-order conditions involving the x’s.

c. Now differentiate the above sum with respect to a—this shows how the x’s must change as a changes while requiring that the first-order conditions continue to hold.