Consider the utility function U(x, y) = -8/x, -8/y. Find and expression for the marginal rate of substitution (MRS_x,y) at any point (x, y). Suppose x = 3 and y = 2, what is the MRS_x,y?

What about at (x, y) = (1,4)? Now consider another utility function W(x, y) = -(2/x)³ - (2/y)³ Does it represent the same preferences as the first one? Explain fully.

. The show that the following utility function is homothetic: U(x, y, z) = x^αy^βz^γ . Is it also homogeneous and if so what is the degree? Are all homothetic functions homogeneous? Show this or give a counterexample.

. Jon's favourite sports are golf and tennis, and his preferences are given by U(G, T) = G^1/3T^4/3. If the yearly prices for playing a half hour a week of each sport are PG and PT, and he has a budget of B, what are his demands for each sport as functions of prices and his budget?

. Remember Jon from question 4. Suppose that at the club where he plays, he pays P_G = 400 and P_T = 160, and suppose that Jon's budget is B = 1600, what is his utility? Now suppose he can buy a premium membership to the golf-tennis club for $500, and then he pays only half price for both sports. Will he be better off with the premium membership pass? Find the range of values for the cost of the membership pass where he will be worse off, and the range where he will be better off.