Question

The annual productivity of certain country is Q(K, L) = 150(0.4K-1/2 +0.6L-1/2)^-2 units, where K is capital expenditure in millions of dollars and L measures the labor force in thousands of worker-hours.

1. Find the marginal productivity of capital and the marginal productivity of labor.
2. Currently, capital expenditure is 5.041 billion dollars (K =5,041) and 4,900,000 worker-hours (L =4,900) are being employed. Find the marginal productivities at these levels.
3. Should the government of the country encourage capital investment or additional labor employment to increase productivity as rapidly as possible?

Answer 1

The marginal productivity of capital (MPK) can be found by taking the derivative of Q with respect to K and the marginal productivity of labor (MPL) can be found by taking the derivative of Q with respect to L:
MPK = dQ/dK = 600(0.4K^-1/2 +0.6L^-1/2)^-3 * 0.4K^-3/2
MPL = dQ/dL = 600(0.4K^-1/2 +0.6L^-1/2)^-3 * 0.6L^-3/2

To find the marginal productivities at the current levels, plug in the values:
MPK = 600(0.4 * 5,041^-1/2 +0.6 * 4,900^-1/2)^-3 * 0.4 * 5,041^-3/2
MPL = 600(0.4 * 5,041^-1/2 +0.6 * 4,900^-1/2)^-3 * 0.6 * 4,900^-3/2

To determine whether the government should encourage capital investment or additional labor employment, compare the values of MPK and MPL. If MPK > MPL, then capital investment should be encouraged. If MPL > MPK, then additional labor employment should be encouraged. The larger the difference between the two, the more it should be encouraged