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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Shephard's lemma ： ウィキペディア英語版 Shephard's lemma Shephard's lemma is a major result in microeconomics having applications in the theory of the firm and in consumer choice. The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost minimizing point of a given good ($i$) with price $p\_i$ is unique. The idea is that a consumer will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of utility given the price of goods in the market. The lemma is named after Ronald Shephard who gave a proof using the distance formula in his book ''Theory of Cost and Production Functions'' (Princeton University Press, 1953). The equivalent result in the context of consumer theory was first derived by Lionel W. McKenzie in 1957. It states that the partial derivatives of the expenditure function with respect to the prices of goods equal the Hicksian demand functions for the relevant goods. Similar results had already been derived by John Hicks (1939) and Paul Samuelson (1947). ==Definition== In consumer theory, Shephard's lemma states that the demand for a particular good ''i'' for a given level of utility ''u'' and given prices p, equals the derivative of the expenditure function with respect to the price of the relevant good: :$h\_i(\backslash mathbf,\; u)\; =\; \backslash frac$ where ''hi(p,u)'' is the Hicksian demand for good $i$, ''e(p,u)'' is the expenditure function, and both functions are in terms of prices (a vector ''p'') and utility $u$. Likewise, in the theory of the firm, the lemma gives a similar formulation for the conditional factor demand for each input factor: the derivative of the cost function ''c(w,y)'' with respect to the factor price: :$x\_i(\backslash mathbf,\; y)\; =\; \backslash frac$ where ''xi(w,y)'' is the conditional factor demand for input $i$, ''c(w,y)'' is the cost function, and both functions are in terms of factor prices (a vector ''w'') and output $y$. Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the envelope theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Shephard's lemma」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.