# Ramsey growth model

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Ramsey growth model

The Ramsey growth model is a neo-classical model of economic growth based primarily on the work of the economist and mathematician Frank Ramsey. The Solow growth model is similar to the Ramsey growth model, however without incorporating an endogenous saving rate. As a result, unlike the Solow model, the saving rate in general is not constant and the convergence of the economy to its steady state is not uniform. Another implication of the endogenous saving rate is that the outcome in the Ramsey model is necessarily pareto optimal or that the saving rate is at its golden rule level. It should be noted that 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 subsequent researchers as a description of a decentralized dynamic economy.

Key equations of the Ramsey model

There are two key equations of the Ramsey model. The first is the law of motion for capital accumulation:

$dot\left\{k\right\}=f\left(k\right) - delta,k - c$

where k is capital per worker, c is consumption per worker, f(k) is output per worker, $delta,$ is the depreciation rate of capital. This equation simply 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.

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 felicity 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:

$r = ho - %dMU*dot c$

A class of utility functions which are consistent with a steady state of this model are the CRRA utility functions, given by:

$u\left(t\right) =frac\left\{c^\left\{1- heta\right\}-1\right\} \left\{1- heta\right\}$

In this case we have:

$frac\left\{%dMU\right\} \left\{c\right\} = heta$

which is a constant. Then solving the above dynamic equation for consumption growth we get:

$frac\left\{dot c\right\} \left\{c\right\} =frac\left\{r - ho\right\} \left\{ heta\right\}$

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

$y=k^alpha$

which implies that the gross interest rate is

$R = alpha k^\left\{alpha-1\right\}$

hence the net interest rate r

$r = R - delta = alpha k^\left\{alpha-1\right\} - delta$

Setting $dot k$ and $dot c$ equal to zero we can find the steady state of this model.

Articles and Books

* Frank P. Ramsey. "A mathematical theory of saving". Economic Journal, vol. 38, no. 152, December 1928, pages 543–559. [http://links.jstor.org/sici?sici=0013-0133%28192812%2938%3A152%3C543%3AAMTOS%3E2.0.CO%3B2-R&origin=crossref]
* Partha S. Dasgupta and Geoffrey M. Heal. Economic Theory and Exhaustible Resources. Cambridge, UK: Cambridge University Press, 1979.

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