13.5  Maximum-Value Functions and the Envelope Theorem

A maximum-value function is an objective function where the choice variables have been assigned their optimal values. The optimal values are, in turn, functions of the exogenous variables and parameters of the problem. Once the optimal values of the choice variables have been substituted into the original objective function, the function indirectly becomes a function of the parameters only. Thus the maximum-value function is also called the indirect objective function

The Envelope Theorem for Unconstrained Optimization

The indirect traces out all the maximum values of the objective function as the parameters vary, hence, working as an "envelope" of the set of optimized objective functions generated by the parameters of the model. This usually comes up in economics as a comparison of short-term and long-term cost curves. For example, the long-run curve is often taught as an envelope of all the short-run average cost curves. 
To illustrate, consider the following unconstrained maximization problem with two choice variables (x, y) and one parameter (\(\phi\)):
Max \(U = f(x, y, \phi)\)
The first-order necessary condition is
\(f_x (x, y, \phi) = f_y(x, y, \phi) = 0\)
If second-order conditions are met, these two equations implicitly define the solutions
\(x^* = x^*(\phi)\) and \(y^* = y^*(\phi)\)
If we substitute these solutions into the objective function, we obtain a new function
\(V(\phi) = f(x^*(\phi), y^*(\phi), \phi)\)
Therefore, this is the maximum-value function. If we differentiate with respect to \(\phi\), we have
\(\frac{dV}{d \phi} = f_x \frac{\partial x^*}{\partial \phi} + f_y \frac{\partial y^*}{\partial \phi} + f_\phi\)
However, from the first-order conditions we know that the first two terms equal zero and will disappear, resulting in
\(\frac{dV}{d \phi} = f_\phi\)
This result says, n words, that at the optimum, as \(\phi\) varies with x* and y* allowed to adjust, the derivative gives the same result as if x* and y* were treated as constants. Note that although \(\phi\) enters the function in three places (directly but twice indirectly through x and y), our last equation shows that only its direct effect matters at optimum. This is the essence of the envelope theorem; only direct effects of a change in an exogenous variable needs to be considered. 

The Profit Function

Now applying this to the profit function of a competitive firm, consider the case where
\(\pi = Pf(K, L) - wL - rK\)
The first-order conditions are
\(\pi _L = Pf_L(K, L) - w = 0\)
\(\pi _K = Pf_K(K, L) - r = 0\)
which respectively define the input-demand equations as
\(L^* = L^* (w, r, P)\)
\(K^* = K^* (w, r, P)\)
Substituting the solutions of K* and L* into the objective function gives us
\(\pi ^* (w, r, P) = Pf(K^*, L^*) - wL^* - rK^*\)
where \(\pi ^* (w, r, P)\) is the profit function (indirect objective function). 
Now consider the effect of a change in wages on a firm's profits. Differentiating the original profit function with respect to w, holding all other variables constant, 
\(\frac{\partial \pi}{\partial w} = -L\)
However, this doesn't take into account the profit-maximizing firm's ability to make a substitution of capital for labor and adjust the level of output in accordance with profit-maximizing behavior. So, we'll differentiate the optimized profit function with respect to w,
 \(\frac{\partial \pi^*}{\partial w} = (Pf_L - w) \frac{\partial L^*}{\partial w} + (Pf_K - r) \frac{\partial K^*}{\partial w} - L^*\)
From the first-order conditions, the two parenthesized terms equal zero. Thus the equation becomes
\(\frac{\partial \pi^*}{\partial w} = - L^* (w, r, P)\)
This result says that, at the profit-maximizing position, a change in profits with respect to a change in the wage rate is the same wether or not the factors are held constant. 

Reciprocity Condition

Consider again the original question
Max \(U = f(x, y, \phi)\)
We found the maximum-value function to be
\(V(\phi) = f(x^*(\phi), y^*(\phi), \phi)\)
By definition, \(V(\phi)\) gives the maximum value of f for any given \(\phi\). Now consider a new function that depicts the difference between the actual value and the maximum value of U: 
\(\Omega (x, y, \phi) = f(x, y, \phi) - V(\phi)\)
This new function has a maximum value of zero when \(x = x^*\) and \(y = y^*\). The first-order conditions of this equation are
\(\Omega _x(x, y, \phi) = f_x = 0\)
\(\Omega _y(x, y, \phi) = f_y = 0\)
\(\Omega _\phi(x, y, \phi) = f_\phi - V_\phi = 0\)
We can see that these are nothing but the original conditions of the function f. Differentiating both sides with respect to \(\phi\) gives us
\(V_{\phi \phi} = f_{\phi x} \frac{\partial x^*}{\partial \phi} + f_{\phi y} \frac{\partial y^*}{\partial \phi} + f_{\phi \phi}\)
Using Young's theorem, we can write
\(V_{\phi \phi} - f_{\phi \phi} = f_{x \phi} \frac{\partial x^*}{\partial \phi} + f_{\phi y} \frac{\partial y^*}{\partial \phi} > 0\)
Supposing the first-order condition such that \(f_{y \phi} = 0\), the above reduces to
\(f_{x \phi} \frac{\partial x^*}{\partial \phi} > 0\)
Applying this then to the profit-maximization model, we end with
\(\frac{\partial L^*}{\partial r} = \frac{\partial K^*}{\partial w}\)
Which is referred to as the reciprocity condition, since it shows the symmetry between the comparative-static cross effect produced by the price of one input on the demand for the other input. Specifically, in the comparative-static sense, the effect of the rental rate for capital (r) on the optimal demand for labor (L) is the same as the effect of the wage rate for labor (w) on the optimal demand for capital. 

The Envelope Theorem for Constrained Optimization

Consider the problem
Max \(U = f(x, y; \phi)\)
subject to \(g(x, y; \phi) = 0\)
The Lagrangian for this problem is
\(Z = f(x, y; \phi) + \lambda [0 - g(x, y; \phi)]\)
With the first-order conditions
\(Z_x = f_x - \lambda g_x = 0\)
\(Z_y = f_y - \lambda g_y = 0\)
\(Z_\lambda = -g (x, y; \phi) = 0\)
Solving this system of equations yields
\(x = x^*(\phi) \)
\(y = y^*(\phi)\)
\(\lambda = \lambda^*(\phi)\)
Substituting the solutions into the objective function, we get
\(U^* = f(x^* (\phi), y^*(\phi), \phi) = V(\phi)\)
Where, once again, \(V(\phi)\) is the maximum-value function, or the maximum value of y for any \(\phi\) and \(x_i\)'s that satisfy the constraint. 
How does \(V(\phi)\)  change as \(\phi\) changes? First, differentiate V with respect to \(\phi\):
\(\frac{dV}{d \phi} = f_x \frac{\partial x^*}{\partial \phi} + f_y \frac{\partial y^*}{\partial \phi} + f_\phi\)
Since in constrained optimization it is not necessary to have \(f_x = f_y = 0\), we can substitute the solution to x and y into the constraint (producing an identity):
\(g(x^*(\phi), y^*(\phi), \phi) \equiv 0\)
And differentiating again with respect to \(\phi\) yields:
\(g_x \frac{\partial x^*}{\partial \phi} + g_y \frac{\partial y^*}{\partial \phi} + g_\phi \equiv 0\)
If we multiply by \(\lambda\), combine the results with our first differentiation, and rearrange terms we get 
\(\frac{dV}{d \phi} = (f_x - \lambda g_x) \frac{\partial x^*}{\partial \phi} + (f_y - \lambda g_y) \frac{\partial y^*}{\partial \phi} + f_\phi - \lambda g_\phi = Z_\phi\)
Where \(Z_\phi\) is the partial derivative of the Lagrangian function with respect to \(\phi\), holding all other variables constant. This result is the same as the others previous, and it reduces to:
\(\frac{dV}{d \phi} = Z_\phi\)
Which represents the envelope theorem in the framework of constrained optimization.

Interpretation of the Lagrange Multiplier

In the consumer choice problem of Ch. 12 we derived the result that the Lagrange multiplier represented the change in the value of the Lagrange function when the consumer's budget changed. We interpreted \(\lambda\) as the marginal utility of income. Now we'll derive a more general interpretation of the Lagrange multiplier with the assistance of the envelope theorem. Consider:
Max \(U = f(x, y)\)
subject to \(g(x, y) = c\)
Where c is a constant. The Lagrangian will be
\(Z = f(x, y) + \lambda [c - g(x, y)]\)
The first-order conditions are
\(Z_x = f_x(x, y) - \lambda g_x (x, y) = 0\)
\(Z_y = f_y(x, y) - \lambda g_y(x, y) = 0\)
\(Z_\lambda = c - g(x,y) = 0\)
From the first two equations, we get
\(\lambda = \frac{f_x}{g_x} = \frac{f_y}{g_y}\)
which gives us the condition that the slope of the level curve (indifference curve) of the objective function must equal the slope of the constraint at the optimum. The first-order conditions implicitly define solutions:
\(x^* = x^*(c)\)
\(y^* = y^*(c)\)
\(\lambda^* = \lambda^*(c)\)
Substituting these back into the Lagrangian yields the maximum-value function
\(V(c) = Z^*(c) = f(x^*(c), y^*(c)) + \lambda ^*(c) [c - g(x_1^*(c), y^*(c))] \)
Differentiating with respect to c and rearranging yields:
\(\frac{dZ^*}{dc} = [f_x - \lambda ^* g_x] \frac{\partial x^*}{\partial c} + [f_y - \lambda ^* g_y] \frac{\partial y^*}{\partial c} + [c - g(x^*, y^*)] \frac{\partial \lambda ^*}{\partial c} + \lambda ^*\)
We know again from the first-order conditions that the bracketed portions are equal to zero. Therefore this expression simplifies to 
\(\frac{dV}{dc} = \frac{dZ^*}{dc} = \lambda^*\)
Showing that the optimal value \(\lambda ^*\) measures the rate of change of the maximum value of the objective function when c changes, and is for this reason referred to as the "shadow price" of c. Note that, in this case, c enters the problem only through the constraint; it is not an argument of the original objective function. 

13.6  Duality and the Envelope Theorem

A consumer's expenditure function and his or her indirect utility function exemplify the minimum and maximum-value functions for dual problems. Duality in economic theory is the relationship between two constrained optimization problems. If one of the problems requires constrained maximization (i.e. utility) then the other  will require minimization (i.e. cost). The structure and solution of either problem can provide information about the structure and solution to the other problem. An indirect utility function specifies the maximum utility that can be obtained given prices, income, and the utility function.

The Primal Problem

Let \(U(x, y)\) be a utility function where x and y are consumption goods. The consumer has a budget B and faces market prices \(P_x\) and \(P_y\). This problem will be considered the primal problem:
Max \(U = U(x, y)\)
subject to \(P_x x + P_y y = B\)
For this problem, we have the familiar Lagrangian
\(Z = U(x, y) + \lambda (B - P_x x - P_y y)\)