Question

Production function Y = 4K^1/4L^3/4, where the level of capital in the economy is 1000 and the level of labor in the economy is 1000.

a) Compute the equilibrium real wage.

b) Compute the equilibrium real rental rate.

c) Does Euler's theorem hold?

d) What fraction of output of this economy is paid to the owners of capital?

e) Suppose tighter immigration policy reduces the labor supply. How will this affect the real rental rate on capital?

f) Suppose tighter immigration policy reduces the labor supply. How will this affect the real wage?

Answer 1

a) Real wage at full employment is equal to the marginal product of labor. Here MPL = dY/dL

Real wage = 4*K^(1/4) * (3/4)*L^(-1/4)

= 4*1000^(1/4) * (3/4)*(1000)^(-1/4)

= 3

Hence real wage rate is 3 units of output

b) Real rental rate at full employment is equal to the marginal product of capital. Here MPK = dY/dK

Real rental rate = 4*(1/4)*K^(-3/4)*L^(3/4)

= 1000^(-3/4) * (1000)^(3/4)

= 1

Hence real wage rental rate is 3 units of output

c) Yes. Total output is 4 * (1000)^(1/4) * (1000)^(3/4)

= 4 units

Capital's share = (1/4) x 4 = 1 and labor's share = (3/4) x 4 = 3. This shows that Capital's share + labor's share

= 4

d) The fraction of output goes to capital owners is (1/4)th

e) Real rental rate = 4*(1/4)*K^(-3/4)*L^(3/4). Now tighter immigration policy reduces L. This will reduce real rental rate since real rental rate is positively related with L.

f) Real wage = 4*K^(1/4) * (3/4)*L^(-1/4). Now tighter immigration policy reduces L. This will increase real wage rate since real wage rate is inversely related with L.