Question

Suppose we know that output in the economy is given by the production function:

Y_t = A_t K_t^(1/3) L_t^(2/3)

A.  Use partial derivative techniques to solve for the marginal product of capital. (Remember, it is an equation and NOT a single number). You must show your work, don’t just simply copy the answer from the book or lectures.

B. Explain what happens to the MPK as K is increased. Make sure your explanation includes a behavioral rather than just mathematical explanation. For example, just stating that the function depends positively on K is not sufficient, explain in economic behavioral terms. Does MPK rise or fall as more capital is added and WHY? What do we call this in economics?

C.  Explain (again, in economic behavioral terms) what happens to the MPK as more LABOR (instead of capital) is added and WHY?

Answer 1

a)

b) We can see from MPk equation that an increase in capital allocation causes MPk to fall. As more and more capital is employed for a given labor input, due to the diminishing return of capital, the per unit contribution of capital to the output falls. Thus the capital will generate less return as more units of capital are employed successively.

c)

MPK rises in this cases as more labor is added. Since now more labor is available for per unit of capital, the productivity of capital and thus the MPk rises in this case. This is also evident from equation 1 where the MPK depends positively on L.