2.11 the power function A function we will encounter often is the power function:

y = x

δ, where 0 ≤ δ ≤ 1 (at times we will also examine this func tion for cases where δ can be negative, too, in which case we will use the form y = x

δ/δ to ensure that the derivatives have the proper sign).

a.

Show that this function is concave (and therefore also, by the result of Problem 2.9, quasi-concave).

Notice that the δ = 1 is a special case and that the function is ‘strictly’ concave only for δ < 1.

b.

c.

Show that the multivariate form of the power function y = f

(x1, x2) = (x1)δ + (x2)δ

is also concave (and quasi-concave). Explain why, in this case, the fact that f12 = f21 = 0 makes the determination of concavity especially simple.

One way to incorporate ‘scale’ effects into the function described in part

(b) is to use the monotonic transformation g(x1, x2) = y

γ = [(x1)δ + (x2)δ]γ, where γ is a positive constant. Does this trans formation preserve the concavity of the function?

Is g quasi-concave?