Question: iii a common utility function used to illustrate economic examples...
III. A common utility function used to illustrate economic examples is the Cobb-Douglas function where U(X, Y)= XαYβ, where α and β are decimal exponents that sum to 1.0 (for example, 0.3 and 0.7).
a. 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 α+ β =1) to show that this person will spend the fraction of his other income on good X and the fraction of income on good Y— that is, show PXX/I = α, PYY/I = β.
b. Use the results from part a to show that total spending on good X will not change as the price of X changes so long as income stays constant.
c. Use the results from part a to show that a change in the price of Y will not affect the quantity of X purchased.
d. 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.