4.6 Suppose that a fast-food junkie derives utility from three goods—soft drinks (x), hamburgers (y), and ice cream sundaes (z)—

according to the Cobb–Douglas utility function Uðx, y, zÞ ¼ x0:5 y0:5

ð1 þ zÞ

0:5

:

Suppose also that the prices for these goods are given by px ¼ 1, py ¼ 4, and pz ¼ 8 and that this consumer’s income is given by I ¼ 8.

a. Show that, for z ¼ 0, maximization of utility results in the same optimal choices as in Example 4.1. Show also that any choice that results in z > 0 (even for a fractional z) reduces utility from this optimum.

b. How do you explain the fact that z ¼ 0 is optimal here?

c. How high would this individual’s income have to be for any z to be purchased?