JU tj F ritvij a 0 f2x Ai a XtA x2 a

If we assume that the functions / and g possess continuous first and second derivatives, and that the determinant

then we can apply the implicit function theorem. This amounts to saying we can solve for the endogenous variables as differentiable functions of the exogenous variable in the neighborhood of the optimal point, so the value of the function / in the same neighborhood is

and V is known as the value function for the maximization problem. The value function expresses directly the idea that the maximized value of the function / depends, via the maximization procedure, only on the exogenous variable in the problem.

In the same way we can write the Lagrange function as a function of the parameter

We now notice an interesting fact. Consider the total derivative of the Lagrange function with respect to a. This is dC _ . dx | „ dx-> dk

da da da da

Now, al the optimal point, we have /, + kgi = 0. i ¡= 1. 2. and g = 0, and so if we evaluate dC/da at the optimal point, the first three terms vanish, and we are left with dC , , ,, d€

That is, although a change in a induces changes in the values of the endogenous variables, for small enough changes at the optimal point, the effects of these changes on the Lagrange function can be ignored because the partial derivatives of the Lagrange function with respect to the endogenous variables are zero at that point.

The envelope theorem establishes a connection between the derivatives of the value function and the derivatives of the Lagrange function, with respect to the parameter or. at the optimal point. Thus for the value function, using the chain rule of differentiation, we have

Substituting for J) and /j from the first-order conditions gives dV f dxx dx2 \

at the optimal point. We can write the constraint as

where, because the x, are optimal solutions to the constrained problem and therefore satisfy the constraint, this equality holds identically. Then differentiating with respect to a we have dx 1 dx2

Substituting into the expression for dV/da gives dV c BC

~r = 1« + 8a = -5-da da which is the form the envelope theorem takes for this example.

The envelope theorem tells us that we can find the effect of a change in the exogenous variable on the optimized value of the objective function simply by taking the partial derivative of the Lagrange function with respect to the exogenous variable at the optimal solution to the problem.

Theorem 14.2

(Envelope theorem) Given the problem

0 0

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