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 of good X and good Y are produced?

Answer 1

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 KX* = 400. This also gives us KY = KX = 400.

Now use the same process to find LX and LY because wage rate should be same for all labor types

Px * MPLX = Py * MPLY

3*3LX^(-0.5) = 6*2LY^(-0.5)

(LY/LX)^0.5 = 4/3 or 9LY = 16LX. Use the fact that LX + LY = 1200 or LX = 1200 - LY

9LY = 16*(1200 - LY)

19200 = 25LY or LY* = 768 and LX* = 432

We have

X = 4KX0.5 + 6LX0.5 and   Y = 2KY0.5 + 4LY0.5

X = 4*(400)^0.5 + 6*(432)^0.5 and Y = 2*(400)^0.5 + 4*(768)^0.5

= 204.71 = 150.85

This shows that X = 204.71 units and Y = 150.85 units.