Consider an economy that uses two factors of production, capital (K) and labor (L), to produce...

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.

Suppose now that the price of good X relative to the price of good Y increases. What will happen to the quantities produced? Are workers better off or are capital owners better off? Explain.

Solutions

Expert Solution

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

The results are KY = KX = 400, LY = 768 and LX = 432. We also see that Px * MPLX = Py * MPLY and Px * MPKX = Py * MPKY. If there are no increases in PY, MPLY, MPLX, MPKY, MPKX and PX increases, then wage rate of workers in production of X and rental price of capital engaged in production of X will increase. Since LX is a smaller proportion of total labor than capital, capital owners will be relatively better off from this event.

Rahul Sunny answered 1 year ago

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