Consider an economy that uses two factors of production, capital ( K ) and labor
ID: 1150620 • Letter: C
Question
Consider an economy that uses two factors of production, capital (K) and labor (L), to produce two goods, good X and good Y. In the good X sector, the production function is X = 4KX0.5 + 6LX0.5, so that in this sector the marginal productivity of capital is MPKX = 2KX-0.5 and the marginal productivity of labor is MPLX= 3LX-0.5. In the good Y sector, the production function is Y = 2KY0.5 + 4LY0.5, so that in this sector the marginal productivity of capital is MPK = KY-0.5 and the marginal productivity of labor is MPLY = 2LY-0.5. Finally, let the total endowment of capital in this economy be K = 800, the total endowment of labor be L = 1200, the price of good X be PX = 3 and the price of good Y be PY = 6.
How much capital will be allocated to producing good X?
Explanation / Answer
We have the following information
MPKX = 2KX^-0.5 and MPKY = KY^-0.5, Px = 3 and Py = 6. Total capital units KX + KY = 800
Now rental income to capital should be same for both industries
Px * MPKX = Py * MPKY
6KX^-0.5 = 6KY^-0.5
KY/KX = 1 or KY = KX. Hence we have KX + KX = 800 or K* = 400.
A tota of 400 units of capital will be allocated to producing good X.