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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 Atkinson–Stiglitz theorem ： ウィキペディア英語版 Atkinson–Stiglitz theorem The Atkinson–Stiglitz theorem is a theorem of public economics which states "that, where the utility function is separable between labor and all commodities, no indirect taxes need be employed" if non-linear income taxation can be used by the government and was developed in a seminal article by Joseph Stiglitz and Anthony Atkinson in 1976. The Atkinson–Stiglitz theorem is generally considered to be one of the most important theoretical results in public economics and spawned a broad literature which delimited the conditions under which the theorem holds, e.g. Saez (2002) which showed that the Atkinson–Stiglitz theorem does not hold if households have heterogeneous rather than homogeneous preferences. In practice the Atkinson–Stiglitz theorem has often been invoked in the debate on optimal capital income taxation: Because capital income taxation can be interpreted as the taxation of future consumption in excess of the taxation of present consumption, the theorem implies that governments should abstain from capital income taxation if non-linear income taxation is an option since capital income taxation would not improve equity by comparison to the non-linear income tax, while additionally distorting savings. ==Optimal taxation== For an individual whose wage is $w$, its budget constraint is given by :$\backslash sum\_\; q\_\; x\_\; =\; \backslash sum\_\; (x\_\; +\; t\_\; (x\_\; )\; )\; =\; wL\; -\; T(wL)\; \backslash ;\; ,$ where $q\_$ and $x\_$ are the price and the purchase of the i-th commodity, respectively. To maximise the utility function, the first order condition is: :$U\_\; =\; \backslash frac\; )\; \}\; \backslash ;\; (j=1,2,\; ...\; ,\; N).$ The government maximises the social welfare function, and so :$\backslash int^\_\; \backslash left(wL\; -\; \backslash sum\_\; x\_\; -\; \backslash overline\; \backslash right)\; d\; F\; =\; 0\; \backslash ;\; .$ Then we use a density function $f$ to express the Hamiltonian: :$H\; =\; \backslash left(G(U)\; -\; \backslash lambda\; \backslash left\backslash lbrace\; wL\; -\; \backslash sum\_\; x\_\; -\; \backslash overline\; \backslash right\backslash rbrace\; \backslash right)f\; -\; \backslash mu\; \backslash theta\; U\_\; \backslash ;\; .$ Taking its variation with regard to $x\_$, we use the condition for its maximum. :$-\; \backslash lambda\; \backslash left(\backslash left(\; \backslash frac\; \backslash right)\_\; +\; 1\; \backslash right)\; -\; \backslash frac\; \backslash left(\backslash frac\; \backslash left(\; \backslash frac\; \backslash right)\_\; +\; \backslash frac\; \backslash right)\; =\; 0\; \backslash ;\; .$ Then the following relation holds: :$\backslash left(\; \backslash frac\; \backslash right)\_\; =\; -\; \backslash frac\; =\; -\; \backslash frac\; \backslash ;\; .$ Substituting this relation into the above condition yields: :$\backslash lambda\; \backslash left(\backslash frac\; -\; 1\; \backslash right)\; =\; \backslash frac\; \backslash left(\backslash frac\; \backslash cdot\; \backslash frac\; -\; \backslash frac\; \}\; \backslash frac\; \backslash left(\; \backslash ln\}\; \backslash right)\; \backslash ;\; ,$ and we obtain :$\backslash lambda\; \backslash left(\backslash frac\; -\; 1\; \backslash right)\; =\; \backslash frac\; \backslash frac\; \backslash left(\; \backslash ln\; \backslash right)\; \backslash ;\; .$ Note that there is no loss of generality in setting $t\text{'}\_$ zero, therefore we put $t\text{'}\_\; =\; 0$. Since $U\_\; =\; (1\; +\; t\text{'}\_\; )\; \backslash alpha$, we have :$\backslash frac\; =\; \backslash frac\; \backslash frac\; \backslash left(\; \backslash ln\; \backslash right)\; \backslash ;\; .$ Thus it turns out that no indirect taxation need to be employed,〔 i.e. $t\_\; =\; 0$, provided that the utility function is weakly separable between labour and all consumption goods. 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「Atkinson–Stiglitz theorem」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.