1. Suppose that Alice consumes apples (good 1) and bananas (good 2), and prefers consumption bundle x to bundle xt if the total of apples and bananas in x is higher (i.e. her preferences are perfect substitutes).
(a) Show that Alice's preferences are complete, transitive, monotonic, and convex.
(b) Verify that Alice's preferences are represented by the following utility function:
u(x1, x2) = x1 + x2
(c) Consider another utility function of the following form:
u(x1, x2) = 3√(x1 + x2)
Verify that this function is equivalent to the previous utility function (i.e. also captures Alice's preferences) by considering the marginal rate of substitution (MRS) for each function.
2. Consider a consumer with utility function U (x1, x2) = min{x1, ax2} (with a > 0). Solve for the Marshallian demands x1(p1, p2, m) and x1(p1, p2, m). That is, solve the problem:
max {min{x1; ax2} : p1x1 + p2x2 ≤ m}
(x1 ,x2)∈R2
3. The indirect utility function v(p1, p2, m) of a consumer is defined as the highest utility the consumer can get for each combination of prices and income. Find the indirect utility function for the consumer in problem 2.
4. Re-do Problems 1 and 2 assuming the government imposes and income tax of s; i.e., when the new income of the consumer is m-s. Compare v(p1, p2, m) with v(p1, p2, m-s).
5. Suppose you are now interested in a different problem. You want to know the minimum amount of money a consumer with a utility function as the one in Problem 2 needs to get a level of utility U = u. Is this amount of money increasing or decreasing in u?
6. Solve for the Marshallian demands when the consumer has utility function U (x1, x2) = max{x1, x2}. That is, solve the problem:
max {max{x1, x2} : p1x1 + p2x2 ≤ m}
(x1 ,x2)∈R+2