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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 Golden Rule savings rate ： ウィキペディア英語版 Golden Rule savings rate In economics, the Golden Rule savings rate is the rate of savings which maximizes steady state level or growth of consumption,〔 〕 as for example in the Solow growth model. Although the concept can be found earlier in John von Neumann and Maurice Allais's works, the term is generally attributed to Edmund Phelps who wrote in 1961 that the Golden Rule "do unto others as you would have them do unto you" could be applied inter-generationally inside the model to arrive at some form of "optimum", or put simply "do unto future generations as we hope previous generations did unto us."〔(Origin of the term described at newschool.edu )〕 In the Solow growth model, a steady state savings rate of 100% implies that all income is going to investment capital for future production, implying a steady state consumption level of zero. A savings rate of 0% implies that no new investment capital is being created, so that the capital stock depreciates without replacement. This makes a steady state unsustainable except at zero output, which again implies a consumption level of zero. Somewhere in between is the "Golden Rule" level of savings, where the savings propensity is such that per-capita consumption is at its maximum possible constant value. Put another way, the golden-rule capital stock relates to the highest level of permanent consumption which can be sustained. == Derivation of the Golden Rule savings rate == The following arguments are presented more completely in Chapter 1 of Barro and Sala-i-Martin 〔 〕 and in texts such as Abel ''et al.''.〔 〕 Let ''k'' be the capital/labour ratio (i.e., capital per capita), ''y'' be the resulting per capita output ( $y\; =\; f(k)$ ), and ''s'' be the savings rate. The steady state is defined as a situation in which per capita output is unchanging, which implies that ''k'' be constant. This requires that the amount of saved output be exactly what is needed to (1) equip any additional workers and (2) replace any worn out capital. In a steady state, therefore: $s\; f(k)\; =\; (n+d)k$, where ''n'' is the constant exogenous population growth rate, and ''d'' is the constant exogenous rate of depreciation of capital. Since ''n'' and ''d'' are constant and $f(k)$ satisfies the Inada conditions, this expression may be read as an equation connecting ''s'' and ''k'' in steady state: any choice of ''s'' implies a unique value for ''k'' (thus also for ''y'') in steady state. Since consumption is proportional to output ( $c\; =\; (1-s)f(k)$ ), then a choice of value for ''s'' implies a unique level of steady state per capita consumption. Out of all possible choices for ''s'', one will produce the highest possible steady state value for ''c'' and is called the ''golden rule'' savings rate. An important question for policy-makers is whether the economy is saving too much or too little. Given the interconnection of ''s'' and ''k'' in steady state, noted above, the question can be phrased: "How much capital per worker (k) is needed to achieve the maximum level of consumption per worker in the steady state?" To discover the optimal capital/labour ratio, and thus the golden rule savings rate, first note that consumption can be seen as the residual output that remains after providing for the investment that maintains steady state: $c\; =\; f(k)\; -\; (n+d)k$ Differential calculus methods can identify which steady state value for the capital/labour ratio maximises per capita consumption. The golden rule savings rate is then implied by the connection between ''s'' and ''k'' in steady state (see above). In detail, if $k^G$ is the golden rule steady state level of ''k'', then $k\; =\; k^G$ requires $dc/dk\; =\; 0$, i.e. $df/dk\; -\; (n+d)=\; 0$ $\backslash mbox\; \backslash frac\; =\; (n+d)$ The Inada conditions ensure that this rule is satisfied by a unique $k\; =\; k^G$ and thus produces a unique $y^G\; =\; f(k^G)$. Since steady state requires a particular level of investment, i.e., saved output: $i^G\; =\; (n+d)k^G$, then the ''golden rule'' savings rate must be whatever is required to generate this; $\backslash mbox\; s^G=\backslash frac$ Given the rule for optimal ''k'', this may also be expressed as $\backslash mbox\; s^G=\backslash frac$ in which $mpk^G$ is the marginal product of capital ( $df(k)/dk$ ) at the optimal value of ''k'' and $apk^G$ is the corresponding average product of capital ( $f(k)/k$ ) The actual values of $k^G$, $y^G$, $apk^G$, and $s^G$ depend upon the precise specification of the production function $f(k)$. For example, a Cobb-Douglas specification with constant returns to scale has $y=f(k)=k^a$, hence $apk=k^$ and $mpk=ak^$. This gives $s^G=a$ and hence $k^G=(a/(n+d))^$, $y^G=(a/(n+d))^$. 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「Golden Rule savings rate」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.