Ramsey–Cass–Koopmans model

The Ramsey–Cass–Koopmans model or the Ramsey growth model is a neo-classical model of economic growth based primarily on the work of the economist and mathematician Frank P. Ramsey, with significant extensions by David Cass and Tjalling Koopmans. The Ramsey model differs from the Solow model in that it explicitly models the choice of consumption at a point in time and so endogenizes the saving rate. As a result, unlike in the Solow 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. 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 subsequent researchers as a description of a decentralized dynamic economy.

Phase space graph of the Ramsey model. The blue line represents the dynamic adjustment path of the economy. It is a stable path of the dynamic system. The red lines represent dynamic paths which are ruled out by the transversality condition.

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:

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

where k is capital per worker, $\dot{k}$ is change in capital per worker over time, c is consumption per worker, f(k) is output per worker, and $\delta\,$ 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 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:

$r = \rho\ - %dMU*\dot c \,$

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:

$u(t) = \frac{c^{1-\theta}-1} {1-\theta} \,$

In this case we have:

$%dMU = \frac{\frac{d^2u}{dc^2}}{\frac{du}{dc}} = -\frac{\theta}{c}$

Then solving the above dynamic equation for consumption growth we get:

$\frac{\dot c} {c} = \frac{r - \rho} {\theta} \,$

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

$R = \alpha k^{\alpha-1} \,$

hence the net interest rate r

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

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. [1]
• Partha S. Dasgupta and Geoffrey M. Heal. Economic Theory and Exhaustible Resources. Cambridge, UK: Cambridge University Press, 1979.