A Neo-Classical Growth Model

N. LABIA *(1)CONSIDER a neo-classical growth model without technical progress and suppose that initially the capital stock is growing fasterthan the labour force, i.e. where K signifies capital stock, s the savings ratio, v the capital-output ratio and n the growth rate of the labour force or 'natural'rate.Initially, there will be a given stock of capital and a given labour force, which will jointly produce a certain output of goods andservices. The equilibrium rate of profit and real wage rate at any moment will be equal to the marginal productivity of the capitalstock and of the labour force, respectively, when fully employed.Suppose now that initially the rate of growth of the capital stock exceeds that of the labour force. As the wage and profit rates areassumed flexible in such a model the tendency will be for the rate of profit to fall and for the wage-rate to rise, thus inducing aswing to more capital-intensive and less labour-intensive techniques of production.We first investigate the behaviour of the wage rate and subsequently that of the profit rate.The Real Wage RateThe production function applicable to the economy is Y = F(K,L) and if we assume this to be linear and homogeneous, it can bewritten y = f(k) where is the output per worker and the capital per worker.Let w be the real wage rate, or , and r the rate of profit In an economy subject to constant returns to scale the marginal productivity of capital and of labour, i.e. the rate of profit and thewage rate, respectively, will depend purely on the capital-labour ratio , and will be independent of the scale of production. Hencewe can write w = w (k).

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Differentiating, we get

Let us define the elasticity of the wage rate with respect to the capital labour ratio as *(2)

Hence

In a neo-classical model of this kind the equilibrium condition is *(3)

Substituting, we get

The equilibrium, rate of increase of the wage rate is thus proportional:(a) to the elasticity of the wage rate with respect to the capital-labour ratio; and(b) to the gap between the warranted and the natural growth rates, i.e. to the gap between the rates of growth of capital and labour.Thus the more closely the economy approaches its equilibrium path, i.e. the closer the growth rate of capital becomes to that oflabour, the more slowly will the wage rate increase. Ultimately, of course, the wage rate will reach a steady state level when output,capital and labour are all increasing at the natural rate.The Rate of ProfitIn the same way the marginal productivity of capital, or rate of profit, will also be a function of the capital-labour ratio k, i.e., r = r(k). Assuming

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again that initially the capital stock of the economy is growing faster than the labour force, the rate of change of the profit rate willbe

As before, we define the elasticity of the profit rate with respect to the capital-labour ratio as

Then

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i.e. the rate of fall of the profit rate is proportional to e' and to the gap between the warranted and natural rates.ConclusionHistorically, the above process could describe the development of an economy starting with a labour force which is large relative toits capital stock. The real wage rate is initially low and the profit rate high. As capital is growing faster than labour (s/v > n) therate of profit declines, and the real wage rate rises, at the speeds given in the above expressions. The rates of change becomesmaller and smaller until eventually a 'golden age' is reached with capital, labour and output all growing at the natural rate and withthe profit and wage rates at steady state levels.School of Economic StudiesUniversity of the Witwatersrand

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Endnotes1 The author is Senior Lecturer in Economics, University of the Witwatersrand.

2 P. A. Neher, Economic Growth and Development: A Mathematical Introduction, John Wiley and Sons, New York, 1971, p. 90.

3 R. S. Solow, A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics, Feb. 1956.

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