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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 Varian's theorems ： ウィキペディア英語版 Varian's theorems In welfare economics, Varian's theorems are several theorems related to fair allocation of homogeneous divisible resources. They describe conditions under which there exists a Pareto efficient (PE) envy-free (EF) allocation. They were published by Hal Varian in the 1970s. == Examples == All examples involve an economy with two goods, x and y, and two agents, Alice and Bob. A. Many PEEF allocations: Alice and Bob have linear utilities, representing substitute goods: :$u\_A(x,y)=2x+y$, :$u\_B(x,y)=x+2y$. The total endowment is (4,4). If Alice receives at least 3 units of x, then her utility is 6 and she does not envy Bob. Similarly, if Bob receives at least 3 units of y, he does not envy Alice. So the allocation () is PEEF with utilities (6,9). Similarly, the allocations () and () are PEEF. On the other hand, the allocation () is PE but not EF (Alice envies Bob); the allocation () is EF but not PE (the utilities are (6,6) but they can be improved e.g. to (8,8)). B. Essentially-single PEEF allocation: Alice and Bob have Leontief utilities, representing complementary goods: :$u\_A(x,y)=u\_B(x,y)=\backslash min(x,y)$. The total endowment is (4,2). The equal allocation () is PEEF with utility vector (1,1). EF is obvious (every equal allocation is EF). Regarding PE, note that both agents now want only y, so the only way to increase the utility of an agent is to take some y from the other agent, but this decreases the utility of the other agent. While there are other PEEF allocations, e.g. (), all have the same utility vector of (1,1), since it is not possible to give both agents more than 1. 〔Note that a similar economy appears in the 1974 paper as an example that a PEEF allocation does ''not'' exist. This is probably a typo - the "min" should be "max", as in example C below. See this (economics stack-exchange thread ).〕 C. No PEEF allocations: Alice and Bob have concave utilities: :$u\_A(x,y)=u\_B(x,y)=\backslash max(x,y)$. The total endowment is (4,2). The equal allocation () is EF with utility vector (2,2). Moreover, ''every'' EF allocation must give both agents equal utility (since they have the same utility function) and this utility can be at most 2. However, no such allocation is PE, since it is Pareto-dominated by the allocation () whose utility vector is (4,2). 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「Varian's theorems」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.