2.11 The power function Another function we will encounter often in this book is the power function: y xd , where 0 - d

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

y ¼ xd

, where 0 - d - 1 (at times we will also examine this function for cases where d can be negative, too, in which case we will use the form y ¼ xd

/d 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 d ¼ 1 is a special case and that the function is ‘‘strictly’’ concave only for d < 1.

b. Show that the multivariate form of the power function y ¼ fðx1, x2Þ¼ðx1Þ

d þ ðx2Þ

d is also concave (and quasi-concave). Explain why, in this case, the fact that f12 ¼ f21 ¼ 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 gðx1, x2Þ ¼ yg ¼ ½ðx1Þ

d þ ðx2Þ

d

(

g

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