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

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The power function Another function we will encounter often in this book is the power function: y = xdelta, where 0 delta 1 (at times we will also examine this function for cases where delta can be negative, too, in which case we will use the form y = xdelta/delta to ensure that the derivatives have the proper sign). Show that this function is concave (and therefore also, by the result of Problem 2.9, quasi-concave). Notice that the delta = 1 is a special case and that the function is "strictly" concave only for delta 1. Show that the multivariate form of the power function y = f(x1, x2) = (x1)delta + (x2)delta 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 delta(x1, x2) = y gamma = [(x1)delta + (x2)delta]gamma, where gamma is a positive constant. Does this transformation preserve the concavity of the function? Is g quasi-concave?

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