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| The '''Ramsey–Cass–Koopmans model''', or '''Ramsey growth model''', is a [[Neoclassical economics|neo-classical]] model of [[economic growth]] based primarily on the work of [[Frank P. Ramsey]],<ref>{{cite journal |first=Frank P. |last=Ramsey |title=A Mathematical Theory of Saving |journal=[[Economic Journal]] |volume=38 |issue=152 |year=1928 |pages=543–559 |doi= |jstor=2224098 }}</ref> with significant extensions by [[David Cass]] and [[Tjalling Koopmans]].<ref>{{cite journal |first=David |last=Cass |title=Optimum Growth in an Aggregative Model of Capital Accumulation |journal=[[Review of Economic Studies]] |volume=32 |issue=3 |year=1965 |pages=233–240 |jstor=2295827 }}</ref><ref>{{cite book |last=Koopmans |first=T. C. |year=1965 |chapter=On the Concept of Optimal Economic Growth |title=The Economic Approach to Development Planning |location=Chicago |publisher=Rand McNally |pages=225–287 |isbn= }}</ref> The Ramsey–Cass–Koopmans model differs from the [[Solow–Swan model]] in that the choice of [[Consumption (economics)|consumption]] is explicitly [[Microfoundations|microfounded]] at a point in time and so endogenizes the [[saving|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 efficiency|Pareto optimal]] or [[Pareto efficiency|Pareto efficient]].<ref group="note">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 [[Paul Samuelson|Samuelson's]] or [[Peter Diamond|Diamond's]] [[overlapping generations model]]s.</ref>
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| 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 [[Shock (economics)|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]].
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| ==Key equations of the Ramsey–Cass–Koopmans model==
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| [[File:Ramseypic.svg|thumb|300px|[[Phase space]] graph (or phase diagram) of the Ramsey model. The blue line represents the dynamic adjustment (or saddle) path of the economy in which all the constraints present in the model are satisfied. It is a stable path of the dynamic system. The red lines represent dynamic paths which are ruled out by the transversality condition.]]
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| 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, <math>F(K, AL)</math>, with factors capital <math>K</math>, labour <math>L</math>, and labour-augmenting technology <math>A</math>. The amount of labour is equal to the population in the economy, and grows at a constant rate <math>n</math>. Likewise, the level of technology grows at a constant rate <math>g</math>. The first key equation of the Ramsey–Cass–Koopmans model is the law of motion for capital accumulation:
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| :<math>\dot{k} =f(k) - c - (n + g + \delta)k</math>
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| where k is [[capital intensity]] (capital per worker), <math>\dot{k}</math> is change in [[capital (economics)|capital]] per worker over time (<math>\frac{dk}{dt}</math>), c is consumption per worker, f(k) is output per worker, and <math>\delta\,</math> is the [[Depreciation (economics)|depreciation]] rate of capital. Under the simplifying assumption that there neither population growth nor an increase in technology level, this equation states that [[investment]], or increase in [[capital (economics)|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 [[saving]]s.
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| It also yields a potentially optimal steady-state of the growth model, in which <math>\dot{k} = 0</math>, i.e. no (further) change in capital intensity. Now, an has to determine the steady-state which maximizes consumption <math>c</math>, and yields an optimal savings rate <math>s = 1 - c</math>. This is the “[[Golden Rule savings rate|golden rule]]” optimality condition proposed by [[Edmund Phelps]] in 1961.<ref>{{cite journal |last=Phelps |first=Edmund |title=The Golden Rule of Accumulation: A Fable for Growthmen |journal=[[American Economic Review]] |volume=51 |issue=4 |year=1961 |pages=638–643 |jstor=1812790 }}</ref>
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| :<math>I=sY=(1-c)Y</math>
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| 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.
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| 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:
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| rate of return on savings = rate at which consumption is discounted − percent change in marginal utility times the growth of consumption.
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| Mathematically:
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| :<math>r = \rho\ - %dMU*\dot c \,</math>
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| A class of utility functions which are consistent with a steady state of this model are the [[Isoelastic utility|isoelastic or constant relative risk aversion (CRRA) utility functions]], given by:
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| :<math>u(c) = \frac{c^{1-\theta}-1} {1-\theta} \,</math>
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| In this case we have:
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| :<math>%dMU = \frac{\frac{d^2u}{dc^2}}{\frac{du}{dc}} = -\frac{\theta}{c}</math>
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| Then solving the above dynamic equation for consumption growth we get:
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| :<math>\frac{\dot c} {c} = \frac{r - \rho} {\theta} \,</math>
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| which is the second key dynamic equation of the model and is usually called the "[[Euler equation]]".
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| 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]]
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| :<math>y=k^\alpha \,</math>
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| which implies that the gross interest rate is
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| :<math>R = \alpha k^{\alpha-1} \,</math>
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| hence the net interest rate r
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| :<math>r = R - \delta = \alpha k^{\alpha-1} - \delta \,</math>
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| Setting <math>\dot k</math> and <math>\dot c</math> equal to zero we can find the steady state of this model.
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| ==Notes==
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| {{Reflist|group="note"}}
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *{{cite book |first1= Robert J. |last1= Barro |authorlink1=Robert J. Barro |first2= Xavier |authorlink2= Xavier Sala-i-Martin |last2= Sala-i-Martin |chapter=Growth Models with Consumer Optimization |title=Economic Growth |location=New York |publisher=McGraw-Hill |year=2004 |edition=Second |isbn=0-262-02553-1 |pages=85–142 }}
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| *{{cite book |first=Olivier Jean |last=Blanchard |first2=Stanley |last2=Fischer |chapter=Consumption and Investment: Basic Infinite Horizon Models |title=Lectures on Macroeconomics |location=Cambridge |publisher=MIT Press |year=1989 |isbn=0-262-02283-4 |pages=37–89 }}
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| *{{cite book |first=Partha S. |last=Dasgupta |first2=Geoffrey M. |last2=Heal |title=Economic Theory and Exhaustible Resources |location=Cambridge, UK |publisher=Cambridge University Press |year=1979 |isbn=0-7202-0312-0 }}
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| *{{cite book |first=David |last=Romer |authorlink=David Romer |chapter=Infinite-Horizon and Overlapping-Generations Models |title=Advanced Macroeconomics |edition=Fourth |location=New York |publisher=McGraw-Hill |year=2011 |pages=49–77 |isbn=978-0-07-351137-5 }}
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| {{Macroeconomics}}
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| {{DEFAULTSORT:Ramsey-Cass-Koopmans model}}
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| [[Category:Economics models]]
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Hi there, I am Felicidad Oquendo. Interviewing is what she does but soon she'll be on her personal. Bottle tops gathering is the only hobby his wife doesn't approve of. Delaware has always been my living place and will by no means transfer.
Feel free to surf to my blog - Www.donorcall.com