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

Suppose there are fixed supplies of capital and labour to be allocated between the production of good x and good y. The production functions for x and y are given

(respectively) by x = k

αl

β

and y = k

γl

δ, where the parameters α, β, γ and δ will take on different values throughout this problem.

Using either intuition, a computer, or a formal mathematical approach, derive the production possibility frontier for x and y in the following cases.

a.

α = β = γ = δ = 1/2.

b.

c.

d.

e.

f.

α = β = 1/2, γ = 1/3, δ = 2/3.

α = β = 1/2, γ = δ = 2/3.

α = β = γ = δ = 2/3.

α = β = 0.6, γ = 0.2, δ = 1.0.

α = β = 0.7, γ = 0.6, δ = 0.8.

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