Suppose you have an income of $100 to spend on goods x1 and x2.
A. Suppose that you have homothetic tastes that happen to have the special property that indifference curves on one side of the 45 degree line are mirror images of indifference curves on the other side of the 45 degree line.
(a) Illustrate your optimal consumption bundle graphically when p1 = 1 = p2.
(b) Now suppose the price of the first 75 units of x1 you buy is 1/3 while the price for any additional units beyond that are 3. The price of x2 remains at 1 throughout. Illustrate your new budget and optimal bundle.
(c) Suppose instead that the price for the first 25 units of x1 is 3 but then falls to 1/3 for all units beyond 25 (with the price of x2 still at 1). Illustrate this budget constraint and indicate what would be optimal.
(d) If the homothetic tastes did not have the symmetry property, which of your answers might not change?
B. Suppose that your tastes can be summarized by the Cobb-Douglas utility function u(x1,x2) = x11/2 x21/2 .
(a) Does this utility function represent tastes that have the symmetry property described in A?
(b) Calculate the optimal consumption bundle when p1 = 1 = p2.
(c) Derive the two equations that make up the budget constraint you drew in part A(b) and use the method described in the appendix to this chapter to calculate the optimal bundle under that budget constraint.
(d) Repeat for the budget constraint you drew in A(c).
(e) Repeat (b) through (d) assuming instead u(x1,x2) = x13/4 x21/4 and illustrate your answers in graphs.