Uses of Ramsey-Cass- Koopmans Model
The Ramsey-Cass-Koopmans model is used to analyse the performance of the economy as the rational behaviour of utility maximising individuals. It is popular in modern economics because it is a very flexible, simple model that can be readily adapted to a wide range of different assumptions.
The model represents the economy as an optimal control problem for a single representative household, who has time (labour potential) and an endowment of capital from a previous time period. The person must choose between labour and leisure, and between consuming current output, and saving it to generate future wealth.
In economics, K always represents the total stock of capital; k (small) represents the supply available to our sample household at any moment in time. The symbol rho stands for the discounting of future consumption; economists assume that consumers value future consumption less than present consumption (time discounting).
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This is not as farfetched as it might seem. That’s because the “average” path taken by millions of households groping towards an optimal allocation may well fit this description. Groping comes in the form of endless brushes with frustration and lost opportunities.
Errors or eccentric decisions made by this or that individual may be expected to average out over very large numbers and over great lengths of time.
The RCK model represents many stylised facts about the economy; specifically, that the population can be represented by a single individual; that the individual has no liquidity constraints; relevant information, particularly about the implications of public policy, is cost-free; that technology is isotropic, and allows ready switching between capital and labour or different kinds of capital/labour; and others.
Proponents of dynamic general equilibrium (DGE) theory have argued that all analysis involves massive simplifications. No economic model can consist of uniformly realistic propositions (unless it was unmanageably complicated); what matters is how reliably the DGE model predicts.
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This is the function for constant relative risk aversion (CRRA); it is also known as the continuous intertemporal elasticity of substitution (CIES) function. Since we do not impose a time horizon, there’s a risk of what is called a “corner solution,” which is where the maximum point of a function lies at one limit or the other of its domain.
At the end of time, k8 would be extremely large, but the who affair would be utterly pointless since our whole effort to simulate the economy with an average household would lead to that household acting in accordance with totally arbitrary equations.
Such a scenario is unreasonable; people have to consume something even when their incomes are so low they can save scarcely anything, so we have limits to the value of infinitely postponed consumption.