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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 fundamental theorems of welfare economics ： ウィキペディア英語版 fundamental theorems of welfare economics There are two fundamental theorems of welfare economics. The first states that any competitive equilibrium or Walrasian equilibrium leads to a Pareto efficient allocation of resources. The second states the converse, that any efficient allocation can be sustainable by a competitive equilibrium. The first theorem is often taken to be an analytical confirmation of Adam Smith's "invisible hand" hypothesis, namely that ''competitive markets tend toward an efficient allocation of resources''. The theorem supports a case for non-intervention in ideal conditions: let the markets do the work and the outcome will be Pareto efficient. However, Pareto efficiency is not necessarily the same thing as desirability; it merely indicates that no one can be made better off without someone being made worse off. There can be many possible Pareto efficient allocations of resources and not all of them may be equally desirable by society. The second theorem states that ''out of all possible Pareto-efficient outcomes, one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take over''. This appears to make the case that intervention has a legitimate place in policy – redistributions can allow us to select from all efficient outcomes for one that has other desired features, such as distributional equity. The shortcoming is that for the theorem to hold, the transfers have to be lump-sum and the government needs to have perfect information on individual consumers' tastes as well as the production possibilities of firms. An additional mathematical condition is that preferences and production technologies have to be convex. Because of welfare economics' close ties to social choice theory, Arrow's impossibility theorem is sometimes listed as a third fundamental theorem.〔 * 〕 The ideal conditions of the theorems, however are an abstraction. The Greenwald-Stiglitz theorem, for example, states that in the presence of either imperfect information, or incomplete markets, markets are not Pareto efficient. Thus, in real world economies, the degree of these variations from ideal conditions must factor into policy choices. Further, even if these ideal conditions hold, the First Welfare Theorem fails in an overlapping generations model. ==Proof of the first fundamental theorem== The first fundamental theorem of welfare economics states that any competitive equilibrium is Pareto-efficient. This was first demonstrated graphically by economist Abba Lerner and mathematically by economists Harold Hotelling, Oskar Lange, Maurice Allais, Kenneth Arrow and Gérard Debreu. The theorem holds under general conditions.〔 The theorem relies only on three assumptions: (1) complete markets (i.e., no transaction costs and where each actor has perfect information), (2) price-taking behavior (i.e., no monopolists and easy entry and exit from a market), and (3) the relatively weak assumption of local nonsatiation of preferences (i.e., for every bundle of goods there is another similar bundle that would be preferred). The formal statement of the theorem is as follows: ''If preferences are locally nonsatiated, and if (x *, y *, p) is a price equilibrium with transfers, then the allocation (x *, y *) is Pareto optimal.'' An equilibrium in this sense either relates to an exchange economy only or presupposes that firms are allocatively and productively efficient, which can be shown to follow from perfectly competitive factor and production markets.〔 Suppose that consumer ''i'' has wealth $w\_i$ such that $\backslash Sigma\; \_i\; w\_i\; =\; p\; \backslash cdot\; \backslash omega\; +\; \backslash Sigma\; \_j\; p\; \backslash cdot\; y^$ *_j where $\backslash omega$ is the aggregate endowment of goods and $y^$ *_j is the production of firm ''j''. Preference maximization (from the definition of price equilibrium with transfers) implies: ::if $x\_i\; >\_i\; x^$ *_i then $p\; \backslash cdot\; x\_i\; >\; w\_i$ In other words, if a bundle of goods is strictly preferred to $x^$ *_i it must be unaffordable at price ''p''. Local nonsatiation additionally implies: ::if $x\_i\; \backslash geq\; \_i\; x^$ *_i then $p\; \backslash cdot\; x\_i\; \backslash geq\; w\_i$ To see why, imagine that $x\_i\; \backslash geq\; \_i\; x^$ *_i but . Then by local nonsatiation we could find $x\text{'}\_i$ arbitrarily close to $x\_i$ (and so still affordable) but which is strictly preferred to $x^$ *_i. But $x^$ *_i is the result of preference maximization, so this is a contradiction. Now consider an allocation $(x,\; y)$ that Pareto dominates $(x^$ *, y^ *). This means that $x\_i\; \backslash geq\; \_i\; x^$ *_i for all ''i'' and $x\_i\; >\_i\; x^$ *_i for some ''i''. By the above, we know $p\; \backslash cdot\; x\_i\; \backslash geq\; w\_i$ for all ''i'' and $p\; \backslash cdot\; x\_i\; >\; w\_i$ for some ''i''. Summing, we find: ::$\backslash Sigma\; \_i\; p\; \backslash cdot\; x\_i\; >\; \backslash Sigma\; \_i\; w\_i\; =\; p\; \backslash cdot\; \backslash omega\; +\; \backslash Sigma\; \_j\; p\; \backslash cdot\; y^$ *_j Because $y^$ * is profit maximizing, we know $\backslash Sigma\; \_j\; p\; \backslash cdot\; y^$ *_j \geq \Sigma _j p \cdot y_j , so $\backslash Sigma\; \_i\; p\; \backslash cdot\; x\_i\; >\; p\; \backslash cdot\; \backslash omega\; +\; \backslash Sigma\; \_j\; p\; \backslash cdot\; y\_j$. Hence, $(x,y)$ is not feasible. Since all Pareto-dominating allocations are not feasible, $(x^$ *,y^ *) must itself be Pareto optimal.〔 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「fundamental theorems of welfare economics」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.