16.4 The purpose of this problem is to examine the relationship among returns to scale, factor intensity, and the shape of the production possibility frontier.

Suppose there are fixed supplies of capital and labor to be allocated between the production of good X and good Y. The production function for X is given by X= K aLP and for Fby Y= K^L 8

, where the parameters

a, (3, y, 8 will take on different values throughout this problem.

Using either intuition, a computer, or a formal mathematical approach, derive the production possibility frontier for Xand Fin the following cases:

a. a = /3 — y = 8 = ^.

d. a — p = y = 8 = j.

b. a=p=\,y=\, 5=f.

e. a = (3 = .6, y = .2, 5 = 1.0.

c. a = p = ^ , y = 8 = j.

f. a = /3 = .7, y = .6, 8 = .8.

Do increasing returns to scale always lead to a convex production possibility frontier?

Explain.