Question

The next several questions refer to a case with the following production function Y = 12K^1/3L^2/3, where the level of capital in the economy is 1000 and the level of labor in the economy is 1000.

Compute the equilibrium real rental rate of capital.

Compute the equilibrium real wage

Does Eulers thoerem hold?

what fraction of output of this economy is paid to the owners of capital?

suppose new immigration laws reduce the labor supply. How will this affect
the real rental rate on capital?

how will the reduction in labor supply affect the REAL WAGE?

Answer 1

Equilibrium rental rate of capital=MPK=4(L/K)^(2/3)=4=r

Equilibrium real wage=MPL=dY/dL=8(K/L)^(1/3)=8=w

frsction of output to owner of capital=rK/Y=(4*1000)/(12(1000^(1/3))(1000^(2/3))=4000/12000=33.33%

If Labour supply reduces labour supply will shift leftwards and real wage would increase

Euler theorem says that (dF/dL)L+(dF/dK)K=nF

dF/dL=4(K/L)^1/3 and dF/dK=8(L/K)^(2/3)

(dF/dL)L+(dF/dK)K=4(K/L)^1/3 *L+8(L/K)^(2/3)*K=12K1/3L2/3=Y

Hence given function is homogenous degree of 1