San José State University
Department of Economics

applet-magic.com
Hjřrdis Bierman
& Thayer Watkins
Silicon Valley
& Tornao Alley
USA

THE MALTHUSIAN RACE BETWEEN
THE LABOR FORCE AND THE CAPITAL STOCK

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

#### Q = F(L, K)

where F(L, K) is homogeneous of degree one in L and K.

This means that

#### F(L, K) = LF(1,K/L) = LF(1,k) = Lf(k)

where k is the capital/labor ratio.

Then the marginal productivity of capital is

#### ∂Q/∂K = Lf'(k)(∂k/∂K) = Lf'(k)(1/L) = f'(k)

Thus the marginal productivity of capital is a function only of the capital/labor ratio.

Likewise the production function can be represented as

#### F(L, K) = KF(L/K, 1) = KF(1/k, 1) = Kg(k)

The marginal productivity of labor is given by

#### ∂Q/∂L = Kg'(k)(∂k/∂L) = Kg'(k)(−K/L²) = −g'(k)(K²/L²) = −g'(k)(K/L)² = −g'(k)k² = h(k)

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

#### r = f(k) w = h(k) and hence dr/dt = f'(k)(dk/dt) dw/dt = h'(k)(dk/dt)

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

#### ln(w) = ln(h(k)) which, upon differentiation by t (1/w)(dw/dt) = (h'(k)/h(k))(1/k)(dk/dt)

The capital/labor ratio can be represented as

#### ln(k) = ln(K) − ln(L)

The differentiation of the above relation with respect to time gives

#### (1/k)(dk/dt) = (1/K)(dK/dt) − (1/L)(dL/dt)

In the model the increase in the capital is equal to the profit less the depreciation of capital; i.e.,

Therefore

#### (1/k)(dk/dt) = [r − δ] − λ' or, equivalently (1/k)(dk/dt) = r − [δ + λ']

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

## The Special Case of a Cobb-Douglas Production Function

If F(L,K)=ALαK1-α then

#### ∂Q/∂K = (1-α)ALαK-α = (1-α)Ak-α

thus r=(1-α)Ak. Likewise w=αAk1-α.

Solving both of these relations for k gives

#### k = (r/(1-α)A))-1/α = (w/(αA))-1/(1-α)

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

#### r(t) = λ/[1 − (1−λ/r0)exp(−αλt)]

where r0 is the value of r at time 0.

This means that if at time 0, λ/r0>1 then the denominator of the fraction representing r(t) is greater than 1 and hence r(t) will be less than λ forever.

## Derivation of the Solution for r(t)

The functional relationship between r and k was abovek solved for k as a function of r; i.e.,

#### k = ((1-α)1/αr−or, equivalently ln(k) = ln(1-α)/α − ln(r)/α

Differentiation of this last equation with respect to time gives

#### (1/k)(dk/dt) = − (1/α)(1/r)(dr/dt)

Since (1/k)(dk/dt) is equal to (r−λ)

#### −(1/α)(1/r)(dr/dt) = r − λ

This reduces to the differential equation

#### (1/(r(r−λ))dr = −αdt which can be put into the form [1/(r−λ) − 1/r]dt = −αλdt

Integration from 0 to t gives

#### [ln(r−λ) − ln(r)] − [ln(r0−λ) − ln(r0)] = −αλt or, equivalently [ln(1−λ/r) − ln(1−λ/r0)] = −αλt

This equation can be put into the form