# Ramsey Cass Koopmans Model

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- Additional Contributors: None
Categories: CategoryModel CategoryMacro CategoryGrowth

Related Models: Solow_Growth_Model

Created: 2008-07-25

Suggested Citation: See license page.

## Introduction

A growth model which extends the basic Solow_Growth_Model by the introduction of consumers -- formally represented by a single optimizing agent -- who provide labour to firms and consume output using the wages thereby earned. This allows the explicit introduction of welfare as well as the endogenization of the interest rate and saving. This basic model is the basis for much Dynamic General Equilibrium work and continues to be used in slightly modified form for much policy modelling, for example in relation to climate change. It is a consolidation of the work in:

David Cass (1965),

*Optimum Growth in an Aggregative Model of Capital Accumulation*, Review of Economic Studies.Tjalling Koopmans (1965),

*On the Concept of Optimal Economic Growth*(The Economic Approach to Development Planning).F.P. Ramsey (1928),

*A Mathematical Theory of Saving*, Economic Journal.

## Model

Based on the basic Solow_Growth_Model in that there an identical number of firms each with (neo-classical) production function $$Y=F(K,AL)$$. Markets are competitive so firms obtain their marginal product and $$A$$ (technological progress) grows at rate $$g$$. The main change compared to Solow is the introduction of a (representative) consumer who supplies labour (competitively) to firms and maximizes total utility:

$$ \[ U = \int_{0}^{\infty} e^{-\rhot} u(C_{t} L(t) dt \] $$

**Remark:** the utility function is additively time separable and future consumption is discounted at rate $$\rho$$ which for this reason is known as the rate of time preference or the intertemporal discount rate. $$L(t)$$ is the total population of the economy (and since each member is endowed with one unit of labour this also equals the labour force). Often one would divide the economy in $$H$$ households and thus $$L(t)$$ would be replaced in this equation by the labour per household $$L(t)/H$$ however as it makes no difference to the analysis WLOG assume $$H = 1$$. Just as with Solow the population grows at an exponential rate: $$L(t) = e^{nt} L(0$$.

The instantaneous utility function $$u(t)$$ is taken to be of a CES form in order to ensure convergence to a balanced growth path:

$$ \[ u(C(t)) = C(t)^{1-\alpha} / (1-\alpha) \] $$

Note that this utility function display constant relative risk aversion (CRRA): $$ -Cu' '(C)/u'(C) = \alpha$$.

## Solving

First move to the intensive forms for the production function and consumption (i.e. dividing by effective labour force $$AL$$: $$y=Y/AL, k=K/AL, c=C/AL, w=W/AL$$ etc. Capital is paid (via the interest rate) its marginal product as is labour (via the wage W) (NB: for notational convenience time argument $$t$$ is being dropped):

$$ \[ r = f'(k), w(t) = f(k) - kf'(k) \] $$

**Remark:** capital gets its marginal product $$f'(k)$$ and labour gets what is left.

Next step is to work through the household optimization problem. Doing this one derives:

$$ \[ \frac{\dot{c}}{c} = \frac{r - \rho - \alpha g}{\alpha} = \frac{f'(k) - \rho - \alpha g}{\alpha} \] $$

and

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

The "golden-rule" level of capital is $$k$$ such that $$f'(k) = n+g$$. The balanced growth path is the point where $$\dot{k} = 0, \dot{c} = 0$$. Using the equation for $$\dot{c}$$ this occurs when equilibrium rate of return $$r$$ is as follows (this in turn defines some optimal capital level $$k^{*}$$ via $$f'(k^{*}) = r$$):

$$ \[ r = \rho + \alpha g \] $$

**Remark:** This is useful as it associates the real rate of interest (observable) with other fundamental parameters: $$\rho = $$ the rate of time preference, $$\alpha$$ the elasticity of marginal utility of consumption, and $$g$$ the rate of technological growth in the economy.

**Remark:** The resulting balanced growth path is pareto optimal from a welfare point of view -- markets are complete and competitive and there are no externalities so the result follows from the first theorem of welfare economics.