The power function Another function we will encounter often in this book is the power function: y 5 x , where 0 # #

The power function Another function we will encounter often in this book is the power function:

y 5 xδ

, where 0 # δ # 1 (at times we will also examine this function for cases where δ can be negative, too, in which case we will use the form y 5 xδ

/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 5 1 is a special case and that the function is “strictly”

concave only for δ , 1.

b. Show that the multivariate form of the power function y 5 f 1x1, x22 5 1x12 δ 1 1x22 δ

is also concave (and quasi-concave). Explain why, in this case, the fact that f12 5 f21 5 0 makes the determination of concavity especially simple.

c. One way to incorporate “scale” effects into the function described in part

(b) is to use the monotonic transformation g 1x1, x22 5 yγ 5 3 1x12 δ 1 1x22 δ4

γ

, where g is a positive constant. Does this transformation preserve the concavity of the function? Is g quasi-concave?