標題:
About Envelope theorem
發問:
近日看書時本書提到這條定理 。有沒有人可以略述一下條定理的內容呢。 最好有例子說明。 謝謝。
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最佳解答:
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！
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