2.10 The power function Another function we will encounter often in this book is the "power function": where 081 (at times we will also examine this function for cases where & can be negative, too, in which case we will use the form y=x/8 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 81 is a special case and that the function is "strictly" concave only for < 1.

b. Show that the multivariate form of the power function y= f(x, x)=(x)+(*)* = is also concave (and quasi-concave). Explain why, in this case, the fact that fiz f = 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 (x, z) = y = [(x)+(x)]. where y is a positive constant. Does this transformation preserve the concavity of the function? Is g quasi-concave?