Question

Country A and Country B both have the production function:

Y = F(K,L) = K^(1/3)L^(2/3).

a) Does this production function have constant returns to scale? Explain.

b) What is the per-worker production function, y = f(k)

c) Assume that neither country experiences population growth or technological progress and that 20-percent of capital depreciates each year. Assume further that country A saves 10% of output each year and country B saves 30% of output each year. Using your answer from part (b) and the steady-state condition that investment equals depreciation, find the steady-state level of capital per worker for each country. Then find the stead-state levels of income per worker and consumption per worker.

d) Suppose that both countries start off with a capital stock per worker of 1. What are the levels of income per worker and consumption per worker?

Answer 1

Answer:

Explanation:

a) A production function has constant return to scale if the inputs and the outputs change by the same factor.

$Y = F(K,L) = K^{1/3} L^{2/3}$

Multiplying K and L by a constant C

$Y = F(CK,CL) =( CK)^{1/3} (CL)^{2/3}\\Y = F(CK,CL) = C^{1/3} K^{1/3} C^{2/3}L^{2/3} \\Y = F(CK,CL) = C K^{1/3}L^{2/3}$

Since Y = F(CK, CL) = CF(K,L), the production function have constant returns to scale

b) Per-worker production function, y = f(k)

$Y = K^{1/3} L^{2/3}\\Divide both sides by L\\\frac{Y}{L} = \frac{ K^{1/3} L^{2/3}}{L} \\\frac{Y}{L} = K^{1/3} L^{-1/3}\\\frac{Y}{L} = \frac{ K^{1/3}}{L^{1/3} }\\\frac{Y}{L} = (\frac{ K}{L})^{1/3}\\Since y = f(k)\\Per-worker production function, y = (k)^{1/3}$

c) % Capital depreciation per year, Δ = 0.2

Country A saves 10%, S = 0.1

The steady state level of income per worker and consumption per worker

$k = (\frac{s}{\delta} )^{3/2} \\k = (\frac{0.1}{0.2} )^{3/2} \\k = 0.354$

Country B saves 30%, S = 0.2

The steady state level of income per worker and consumption per worker

$k = (\frac{s}{\delta} )^{3/2} \\k = (\frac{0.3}{0.2} )^{3/2} \\k = 1.84$