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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 Ramsey–Cass–Koopmans model ： ウィキペディア英語版 Ramsey–Cass–Koopmans model The Ramsey–Cass–Koopmans model, or Ramsey growth model, is a neo-classical model of economic growth based primarily on the work of Frank P. Ramsey, with significant extensions by David Cass and Tjalling Koopmans. The Ramsey–Cass–Koopmans model differs from the Solow–Swan model in that the choice of consumption is explicitly microfounded at a point in time and so endogenizes the savings rate. As a result, unlike in the Solow–Swan model, the saving rate may not be constant along the transition to the long run steady state. Another implication of the model is that the outcome is Pareto optimal or Pareto efficient.〔This result is due not just to the endogeneity of the saving rate but also because of the infinite nature of the planning horizon of the agents in the model; it does not hold in other models with endogenous saving rates but more complex intergenerational dynamics, for example, in Samuelson's or Diamond's overlapping generations models.〕 Originally Ramsey set out the model as a central planner's problem of maximizing levels of consumption over successive generations. Only later was a model adopted by Cass and Koopmans as a description of a decentralized dynamic economy. The Ramsey–Cass–Koopmans model aims only at explaining long-run economic growth rather than business cycle fluctuations, and does not include any sources of disturbances like market imperfections, heterogeneity among households, or exogenous shocks. Subsequent researchers therefore extended the model, allowing for government-purchases shocks, variations in employment, and other sources of disturbances, which is known as real business cycle theory. ==Key equations of the Ramsey–Cass–Koopmans model== Like the Solow–Swan model, the Ramsey–Cass–Koopmans model starts with an aggregate production function that satisfies the Inada conditions, of Cobb–Douglas type, $F(K,\; AL)$, with factors capital $K$, labour $L$, and labour-augmenting technology $A$. The amount of labour is equal to the population in the economy, and grows at a constant rate $n$. Likewise, the level of technology grows at a constant rate $g$. The first key equation of the Ramsey–Cass–Koopmans model is the law of motion for capital accumulation: :$\backslash dot\; =f(k)\; -\; c\; -\; (n\; +\; g\; +\; \backslash delta)k$ where $k$ is capital intensity (capital per worker), $\backslash dot$ is change in capital per worker over time ($\backslash tfrac$), $c$ is consumption per worker, $f(k)$ is output per worker, and $\backslash delta\backslash ,$ is the depreciation rate of capital. Under the simplifying assumption that there is neither population growth nor an increase in technology level, this equation states that investment, or increase in capital per worker is that part of output which is not consumed, minus the rate of depreciation of capital. Investment is, therefore, the same as savings. It also yields a potentially optimal steady-state of the growth model, in which $\backslash dot\; =\; 0$, i.e. no (further) change in capital intensity. Now, one has to determine the steady-state which maximizes consumption $c$, and yields an optimal savings rate $s\; =\; 1\; -\; c$. This is the “golden rule” optimality condition proposed by Edmund Phelps in 1961. :$I=sY=(1-c)Y$ where $I$ is the level of investment, $Y$ is level of income and $s$ is the savings rate, or the proportion of income that is saved. The second equation concerns the saving behavior of households and is less intuitive. If households are maximizing their consumption intertemporally, at each point in time they equate the marginal benefit of consumption today with that of consumption in the future, or equivalently, the marginal benefit of consumption in the future with its marginal cost. Because this is an intertemporal problem this means an equalization of rates rather than levels. There are two reasons why households prefer to consume now rather than in the future. First, they discount future consumption. Second, because the utility function is concave, households prefer a smooth consumption path. An increasing or a decreasing consumption path lowers the utility of consumption in the future. Hence the following relationship characterizes the optimal relationship between the various rates: rate of return on savings = rate at which consumption is discounted − percent change in marginal utility times the growth of consumption. Mathematically: :$r\; =\; \backslash rho\backslash \; -\; \backslash \%dMU$ *\dot c \, A class of utility functions which are consistent with a steady state of this model are the isoelastic or constant relative risk aversion (CRRA) utility functions, given by: :$u(c)\; =\; \backslash frac\; \backslash ,$ In this case we have: :$\backslash \%dMU\; =\; \backslash frac\}\}\; =\; -\backslash frac$ Then solving the above dynamic equation for consumption growth we get: :$\backslash frac\; =\; \backslash frac\; \backslash ,$ which is the second key dynamic equation of the model and is usually called the "Euler equation". With a neoclassical production function with constant returns to scale, the interest rate, r, will equal the marginal product of capital per worker. One particular case is given by the Cobb–Douglas production function :$y=k^\backslash alpha\; \backslash ,$ which implies that the gross interest rate is :$R\; =\; \backslash alpha\; k^\; \backslash ,$ hence the net interest rate r :$r\; =\; R\; -\; \backslash delta\; =\; \backslash alpha\; k^\; -\; \backslash delta\; \backslash ,$ Setting $\backslash dot\; k$ and $\backslash dot\; c$ equal to zero we can find the steady state of this model. 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「Ramsey–Cass–Koopmans model」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.