# 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,[1] with significant extensions by David Cass and Tjalling Koopmans.[2][3] 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.[note 1]

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

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.

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:

$\dot{k} =f(k) - c - (n + g + \delta)k$

where k is capital intensity (capital per worker), $\dot{k}$ is change in capital per worker over time ($\frac{dk}{dt}$), c is consumption per worker, f(k) is output per worker, and $\delta\,$ is the 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 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 $\dot{k} = 0$, i.e. no (further) change in capital intensity. Now, an 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.[4]

$I=sY=(1-c)Y$

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:

$r = \rho\ - %dMU*\dot c \,$

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:

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

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.

## Notes

1. ^ 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.

## References

1. ^ Ramsey, Frank P. (1928). "A Mathematical Theory of Saving". Economic Journal 38 (152): 543–559. JSTOR 2224098.
2. ^ Cass, David (1965). "Optimum Growth in an Aggregative Model of Capital Accumulation". Review of Economic Studies 32 (3): 233–240. JSTOR 2295827.
3. ^ Koopmans, T. C. (1965). "On the Concept of Optimal Economic Growth". The Economic Approach to Development Planning. Chicago: Rand McNally. pp. 225–287.
4. ^ Phelps, Edmund (1961). "The Golden Rule of Accumulation: A Fable for Growthmen". American Economic Review 51 (4): 638–643. JSTOR 1812790.