A common utility function used to illustrate economic examples is the Cobb-Douglas function where U(X, Y) 5 XaYb, where a and b are positive decimal exponents that sum to 1.0

(for example, 0.3 and 0.7).

a. Explain why the utility function used in Problems 3.2 and 3.3 is a special case of this function.

b. For this utility function, the MRS is given by MRS 5 MUXyMUY5 aYybX. Use this fact together with the utility-maximizing condition (and that a 1 b 5 1) to show that the fraction of income that this person will spend on each good is given by PXXyI 5

a, PYYyI 5

b. That is, the fractions of income spent on each good is given by its exponent in the utility function.

c. Use the results from part b to show that total spending on good X will not change as the price of X changes so long as income stays constant.

d. Use the results from part b to show that a change in the price of Y will not affect the quantity of X purchased.

e. Show that with this utility function, a doubling of income with no change in prices of goods will cause a precise doubling of purchases of both X and Y