Ramsey model of economic growth
The Ramsey growth model is a neoclassical model of economic growth based primarily on the work of the economist and mathematician Frank P. Ramsey. The Ramsey model differs from the Solow model in that it explicitly models the choice of consumption at a point in time and so endogenises 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 in that it corresponds to the Golden Rule savings rate.
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 maximising levels of consumption over successive generations. Only later was a model adopted by subsequent researchers as a description of a decentralised dynamic economy.
There are two key equations of the Ramsey model. The first is the law of motion for capital accumulation:
Where k is capital per worker, c is consumption per worker, f (k) is output per worker, 8 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 behaviour of households and is less intuitive. If households are maximising 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 equalisation 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 characterises the optimal relationship between the various rates:
Rate of return on savings = rate at which consumption is discounted – per cent change in marginal utility times the growth of consumption.
A class of utility functions which are consistent with a steady state of this model are the CRRA utility functions, given by:
This is a constant. Then solving the above dynamic equation for consumption growth we get:
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.
The Cass-Koopmans model is an extension of the Ramsey model. David Cass and Tjalling Koopmans both provided extensions to the Ramsey model. They used discounting, unlike in the Ramsey model. The underlying reasoning for imposing time preference is more a mathematical one than a logical one-necessary for solving an inter-temporal optimising programme.
They also extended the Phelps Golden Rule formulation by arguing that what the objective function should be maximising is utility and not consumption stream directly.
The argument here is that first, the society is not indifferent to the timing of utility receipts, and values and attaches greater value to utility today than to utility say, twenty years from mow. Hence society displays time preference.
Therefore total utility is a computed as the present discounted value of the utility stream, that is, here p is the rate of time preference or the discount rate. We can denoted (Hp) as p can call (the discount factor and denote total utility as
The question is what justifications can be given to put discounting and time preference into the picture. The first argument can be of course that it need not be justified at all and should be considered simply as taste or preference of the optimising agent or social planner.
It is an old dictum in economics that ‘the tastes are not to be disputed’. The other argument is to suggest that we no longer think of future generations but ‘dynasties’. Each person is concerned about his or her own children, they about their children, and so on.
Moreover, each person’s utility in the current period is a function also of the utility of his or her children, and their children, and so on. Some economists have proposed that we take a dynastic utility function that stays the same for that dynasty for all time.
All this necessitates some modification in the objective function to be maximised. The final argument to justify discounting future consumption is provided by economists like Robert Becker.
This argument says that the inter-temporal optimisation should not be seen as an exercise in welfare economics or normative economics, that is, it does not involve making ethical statements.
Supposing we look at the optimisation exercise as an analysis in positive economics that just describes the maximising action of a typical individual who maximises over time over an infinite horizon, who has perfect foresight and is supposed to be myopic. Then discounting necessarily enters the picture.
Now the optimisation exercise can be proceeded with. The Cass-Koopmans model begins is stating a simple representative consumer who maximises an intertemporal utility function: A characteristic of the above utility function is that it is additively separable over time.