Suppose that a household derives income from a business whose only input is capital. The production function of the household is y = AKb where b < 1, A is a constant, and K is capital.

a. Show that the marginal return to capital is decreasing in K. Draw a diagram for the case where A = 1, and b = 0.5.

b. Suppose that when K > 8, the production function becomes y = 4AKb, because at such levels of K household-level knowledge is enhanced, and this, in turn, helps the household be more productive. Is the marginal return to capital still decreasing in this case? Construct a numerical example to illustrate your answer.