Question

1a)For the next several questions refer to a case with the following production function Y = 648K^3/4L^1/4, where the level of capital in the economy is 16 and the level of labor on the economy is 81. Compute the equilibrium real rental rate of capital. (Depending on your computer, the production function might include "^", which indicates an exponent.)

1b)compute the equilibrium real WAGE.

1c) what fraction of total income of this economy is paid to the owners of capital?

1d) suppose a "shelter-in-place" order from the government reduces the labor supply without affecting the stock of capital. How should this affect the real rental rate on capital?

Answer 1

1a)

Y = 648K^3/4L^1/4

Real rental rate of capital = marginal product of capital

Marginal product of capital = dY / dK

= 648*0.75*K^{(3/4 - 1)}L^1/4

Real rental rate of capital = 486 K^-1/4L^1/4

Now substituting the values of L = 81 and K =16, we get

Real rental rate of capital = 486 * 16^-1/481^1/4

= 486*3/2 = 729

1b)

Y = 648K^3/4L^1/4

Real wage rate = marginal product of labour

Marginal product of capital = dY / dL

= 648*0.25*K^3/4L^{( 1/4 - 1 )}

Real wage rate = 162  K^3/4L^-3/4

Now substituting the values of L = 81 and K =16, we get

Real rental rate of capital = 162 * 16^3/481^-3/4

= 486*8/27 = 48

1c)

Total income being paid = K*( real rental rate of capital ) + L*( real wage rate )

= 16*729 + 81*48

= 15552

Fraction of income being paid to owners of capital = K*(real rental rate of capital) / total income

= 16*729 / 15552

= 0.75 ( or ) 75%

1d)

Suppose a "shelter-in-place" order from the government reduces the labor supply without affecting the stock of capital, it would mean that the amount of labour has decreased, without a decrease in the amount of capital. Therefore, in relative terms, labour has become more scarce and therefore becomes more valuable. Since labour is more valuable now, capital becomes less valuable. Therefore, the real rental rate of capital decreases.