About Envelope theorem

6.3 The envelope theorem
In economic theory we are often interested in how the maximal value of a function depends on some parameters.
Consider, for example, a firm that can produce output with a single input using the production function f . The standard theory is that the firm chooses the amount x of the input to maximize its profit p f (x) - wx, where p is the price of output and w is the price of the input. Denote by x*(w, p) the optimal amount of the input when the prices are w and p. An economically interesting question is: how does the firm's maximal profit p f (x*(w, p)) - wx*(w, p) depend upon p?

We have already answered this question in an earlier example. To do so, we used the chain rule to differentiate p f (x*(w, p)) - wx*(w, p) with respect to p, yielding

f (x*(w, p)) + x*¢p(w, p)[p f ¢(x*(w, p)) - w],
and then used the fact that x*(w, p) satisfies the first-order condition p f ¢(x*(w, p)) - w = 0 to conclude that the derivative is simply f (x*(w, p)).
That is, the fact that the value of the variable satisfies the first-order condition allows us to dramatically simplify the expression for the derivative of the firm's maximal profit. In this section I describe results that generalize this observation to an arbitrary maximization problem.

Unconstrained problems
Consider the unconstrained maximization problem
maxx f (x, r),
where x is a n-vector and r is a k-vector of parameters. Assume that for any vector r the problem has a unique solution; denote this solution x*(r). Denote the maximum value of f , for any given value of r, by f *(r):
f *(r) = f (x*(r), r).
We call f * the value function.

2007-05-25 21:46:30 補充：
由於字數太多，所以請到參考資料觀看更多的詳請！