A consumer has the utility function U(x,y) = xα yβ , where x$0 and y$0 represent her consumption of goods X and Y, and α>0 and β>0 are exogenous parameters. The consumer has exogenous income I to spend on goods X and Y, which she can buy at exogenous prices of px>0 for each unit of X and py>0 for each unit of Y. The consumer’s problem is to divide her income between goods X and Y in the way that maximizes her utility.

(a) Calculate the consumer’s demand functions for good X and (separately) good Y. These demands may be functions of α, β, I, px, and py. They should not be functions of x or y. You should justify your work carefully.

(b) Calculate the consumer’s elasticities of demand for X and Y. These elasticities may be functions of the exogenous variables α, β, I, px, and py. They should not be functions of x or y. You should show your steps carefully.

(c) Under what conditions (i.e., for what values of α, β, I, px, and py) are goods X and Y substitutes or complements?