Question

An economy has a Cobb-Douglas production function: Y = K"(LE)!-a The economy has a capital share of 0.25, a saving rate of 40 percent, a depreciation rate of 3.00 percent, a rate of population growth of 0.75 percent, and a rate of labor- augmenting technological change of 2.0 percent. It is in steady state.

Answer 1

Answer

(b)

It is given that capital share of income = $\alpha$ = 0.25

Y = K^0.25(EL)^{1 - 0.25}

=> Y/(EL) = (K^0.25(EL)⁷⁵)/(EL) = (K/(EL))^0.25

Now Let y = Y/(EL) = Output per effective worker and k = K/(EL) = Capital per effective worker

=> y = k^0.25

Steady state occurs when $\Delta k = 0$, where $\Delta k$ means change in k and k = Capital per effective worker.

Also, $\Delta k$ = sy - (d +n + g)k.

where d = depreciation rate = 3% = 0.03, s = saving rate = 40% = 0.40, rate of labor augmenting technological change = 2% = 0.02 and n = population growth rate = 0.75% = 0.0075

Hence Steady state occurs when sy - (d +n + g)k = 0

Using above details we have :

sy - (d +n + g)k. = 0

=> 0.40k^0.25 - (0.03 + 0.0075 + 0.02)k = 0

=> 0.4k^0.25 = 0.0575k

=> k^0.75 = 0.4/0.0575

=> k = 13.28

Hence, Steady state level of capital per effective worker(k*) = 13.28

Steady state level of Output per effective worker(y*) = k^0.25 = 13.28^0.25 = 1.91

Marginal Product of capital = dy/dk = 0.25k^-0.75 and as k = 13.28

=> Marginal Product of capital = 0.25k^-0.75 = 0.25*13.28^-0.75 = 0.036