A common utility function used to illustrate economic examples is the Cobb-Douglas function where U(X, Y) = X Y where and are decimal

A common utility function used to illustrate economic examples is the Cobb-Douglas function where U(X, Y) = Xα Yβ where α and β are decimal exponents that sum to 1.0 (that is, for example, 0.3 and 0.7).

a. Explain why the utility function used in problem 2 and problem 3 is a special case of this function.

b. For this utility function, the MRS is given by MRS = MUX=MUY = αY / βX. Use this fact together with the utility-maximizing condition (and that a α + β = 1) to show that this person will spend the fraction of his or her income on good X and the fraction of income on good Y—that is, show PXX = I = α, PYY / I = β.

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.