San José State University Department of Economics |
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THE MALTHUSIAN RACE BETWEEN THE LABOR FORCE AND THE CAPITAL STOCK |
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In the 19th century Thomas Malthus raised the spectre of inevitable misery for the human race because population would necessarily outrun the natural resources. That spectre did not materialize because of technical developments that increased the productivity of the natural resources. But there is another race, namely that between the capital stock of an economy and its labor force. The labor force may increase from natural growth and from immigration.
This problem is examined by means of a simple economic growth model. In the model there are only two resources, labor and capital. All of the wage income is consumed and all of profit, the reward for the use of the capital stock.
Let the production function for the economy be
where F(L, K) is homogeneous of degree one in L and K.
This means that
where k is the capital/labor ratio.
Then the marginal productivity of capital is
Thus the marginal productivity of capital is a function only of the capital/labor ratio.
Likewise the production function can be represented as
The marginal productivity of labor is given by
Thus also the marginal productivity of labor is a function only of the capital/labor ratio.
In a competitive economy the wage rate w will be equal to the marginal productivity of labor and the profit rate of capital, r, will be equal to the marginal productivity of capital. Thus
The level of consumption per person is proportional to w and hence what happens to w depends upon (dk/dt).
The dependence can be made more specific by representing the relationship between the wage rate and the capital/labor ratio as
The capital/labor ratio can be represented as
The differentiation of the above relation with respect to time gives
In the model the increase in the capital is equal to the profit less the depreciation of capital; i.e.,
Therefore
where λ' is the proportional rate of growth of the labor force, (1/L)(dL/dt).
Therefore the condition of life in the economy will be improving if the net profit rate (r−δ) is greater than the proportional rate of growth of the labor force. It will be decreasing if the net profit rate falls below the proportional rate of growth of the labor force.
To simplify the equations in the further developments let λ=λ'+δ
If F(L,K)=AL^{α}K^{1-α} then
thus r=(1-α)Ak^{-α}. Likewise w=αAk^{1-α}.
Solving both of these relations for k gives
and hence if any one of the three variables {k,r,w}l are known then the other two are known.
The time paths of {k,r,wk} can be determined. The solution for r(t) appears to be the easiest to derive. Its value, assuming a constant value for λ is
where r_{0} is the value of r at time 0.
This means that if at time 0, λ/r_{0}>1 then the denominator of the fraction representing r(t) is greater than 1 and hence r(t) will be less than λ forever.
The functional relationship between r and k was abovek solved for k as a function of r; i.e.,
Differentiation of this last equation with respect to time gives
Since (1/k)(dk/dt) is equal to (r−λ)
This reduces to the differential equation
Integration from 0 to t gives
This equation can be put into the form
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