QUESTION 1 Uncle Leo has preferences U(x, y) = xev Px = 4, py = 8 and m = 200 A) Set up the maximisation problem. B) Show that Leo's optimal bundle (x*, y') = (2.24) given these preferences are convex. What is the maximum level of utility Leo can achieve? () Show that In(U(x, y)) = In(x) +y- DJ Let Ux(x, y) = In(x) + y. Is Ux a positive monotonic transformation of U? E) Hence, what kind of preferences does Uncle Leo have? QUESTION 2 Carmelita treats coffee and tea as perfect substitutes. Let her utility function be U(c, t) = Ac + Bt, where the c represents units of coffee, t represents units of tea and the letters A and B are constants. A) What is the MRS? B) Consider three cases: i) Price of coffee Price of tea ii) Price of coffee Price of tea = iii) Price of coffee Price of tea What will Carmelita's optimal consumption bundle be in each case? C) Do changes in income affect Carmelita's purchasing decisions? QUESTION 3 Li's utility function is Up(x, y) = ex+> A) Find the MRS. B) Based on your answer to A), explain why Li treats goods x and y as perfect substitutes. C) Find 2 of Li's indifference curves and graph them on Desmos. D) Show that Li's utility function is a positive monotonic transformation of the utility function Uc(x, y) = 2x + yE) Consider Li's purchasing behaviour, assuming he is a rational utility maximiser. If px = 2 and py = 3, which good will he buy? QUESTION 4 A) Dimitri chooses a bundle consisting of 5 oranges and 6 pears over a bundle consisting of 3 oranges and 8 pears. The second time he buys groceries he purchases a bundle containing 4 oranges and 7 pears. Is Dimitri irrational? B) Suppose that the second time Dimitri goes shopping he purchases 4 oranges and 9 pears, and that this occurs after his income has increased (but prices have not changed in the meantime). Assuming Dimitri is a rational utility maximiser with well-behaved preferences, show how this would play out using a diagram, with two budget lines and two indifference curves. What kind of good are the oranges? QUESTION 5 Imani has preferences of the form U(x, y) = min{x + 2, y} A) Draw three of her indifference curves. B) If the price of good x is px, the price of good y is double the price of good x, her income is $50, and her optimal bundle occurs when x* = 2, show Px = 5. C) A 20% sales tax is placed on good x. Find how much tax the government will raise from Sally's purchases of good x. Assume income and prices have not changed from B). QUESTION 6 Draw 3 diagrams for each of the following scenarios, showing two budget lines in each (one before the change and one after): A) The prices of both goods rise, but the price of good 1 increases more so than the price of good 2. B) The price of good 2 falls and a lump-sum income tax is placed on the consumer. C) A new scheme is introduced: the price of good 2 drops after k units have been purchased, where 0
