Only answer question 2

Question 1: [17 marks total] (+ 2 bonus) A lottery bond is a type of government bond in which investors receive reduced interest, or none at all, and instead receive a chance of a prize. Examples included the premium bond in the UK. Suppose that the government runs a lottery experiment in order to help them understand how to price the bonds. Suppose they offer small lotteries to consumers that are based on the outcome of a game in which the color red and black appear at random. Let R be the amount of money that the consumer receives if red occurs (the "red state of the world") and B be the amount of money the consumer receives if black occurs (the "black state of the world"). Suppose p is the probability of red, with 0 < p < 1. Suppose consumers have a utility of money given by u(m) = ln m, and that their preferences over these lotteries paying R with probability p and B with probability 1 p can be characterized by the expected value of their utility of money.

(a) [4 marks] Assume that p = .75 and that Ann is endowed with a lottery A that pays 50 dollars in the red state and 100 dollars in the black state. Letting the horizontal axis be the red state amount and the vertical axis be the black state amount, on the same graph (i) draw Ann's indifference curve passing through the point representing lottery A (label it "IC"), (ii) draw all positive payoff combinations of (R, B) that produce the same expected value as this lottery (label it "same EV"), (iii) draw all positive payoff combinations that yield the same payoff in each state (label "same payoff"),and indicate with "C" the lottery on the "same payoff" line that satisfies A C according to Ann, and indicate with "B" the lottery on the "same payoff" line that the risk neutral government would find equivalent to A (this B has nothing to do with the black state of the world). Mark the amount of money corresponding to A, B, C on the horizontal and vertical axis. The graph is enough, no need to justify.

(b) [2 marks] What is the least amount of money Ann would be willing to accept in exchange for lottery A from part (a)? Please justify briefly.

(c) [2 marks] Suppose the government offers Ann a lottery with p = .75, R = 100, and B undetermined . What is least B Ann would be willing to accept in exchange for this lottery instead of the A she is endowed with from part (a)? Please justify briefly.

(d) [3 marks] Suppose the government gives Ann a lottery in which p = .75 R = B = 100 dollars. Keeping p fixed, (i) what is the (instantaneous) rate at which Ann would be willing to give up money in the black state for more money in the red state, per dollar of extra money in the red state? (ii) what is the most amount money Ann would be willing to give up in the black state for an extra 25 dollars in the red state? Please justify briefly.

(e) [2 marks] Given p, R, B, Ann's value of a dollar in the red state is the rate at which Ann is willing to give up money in the black state for more money in the red state, per dollar of extra money in the red state. How does Ann's value of a dollar in the red state depend on p, R, B, i.e. for an increase in p, R, and,B respectively, is Ann's value of a dollar in the red state increasing, decreasing or constant? Please justify briefly.

(f) [4 marks] On a new graph with the red state on the horizontal axis, using Ann's indifference curve through certain lottery "basket" P = (R, B) = (10, 10) as a reference and assuming p = .75, plot the indifference curve over lotteries that passes through (R, B) = (10, 10) for each of four other individuals i = 1, 2, 3, 4 with utility functions of money given by ui(x), and with each of their preferences over lotteries characterized by the expected value of their utility of money. Assume that i = 1 is risk seeking, i = 2 is risk neutral, i = 3 is risk averse, but less so than Ann, wand i = 4 is risk averse, but more so than Ann. What this means is that: if i = 4 strictly prefers lottery "basket" E over P (written "E 4 P"), then so does Ann, (written "E A P"), also, if Ann strictly prefers D over P, i.e. "D A P", then so does i = 3, i.e. "D 3 P." In this way, this also means that if C 3 P then C 2 P and, finally that if B 2 P then B 1 P. Further assume the "basket" A is defined so that A 1 P but P 2 A. Along with the stylized indifference curves, plot the five lottery "baskets" A, B, C, D, E that satisfy the criteria just above under the assumption that they each pay strictly more in the red state than in the black state. Please briefly justify the relative location of the indifference curves, and the location of A, B, C, D, E.

(g) [2 *Bonus* marks] It just so happens that ln x = lim1 x 11 1 . Assume that the other people i = 1, 2, 3, 4 in part (f) have utility function ui(x) = x 1i1 1i , where i is different for each person i. Use a graphing software to plot these indifference curves along with Ann's and find the order of these values relative to each other, and relative to 0 and 1. You don't need to justify the ordering.

Question 2: [13 marks total] Ann has a 20 million dollar house. Suppose there is a p chance that house is safe from fire, and a 1 p chance of a fire, in which case the house is completely destroyed and Ann retains the value of the land, 5 million dollars. Assume Ann has utility of money given by u(m) = ln m, and that her preferences can be characterized by the expected value of her utility of money. A risk neutral investment company is considering offering Ann a contract. After the contract Ann's final wealth will be S in the "safe state of the world" and F in the "fire state of the world." Without a contract Ann's final wealth will be S0 = 20 and F0 = 5 respectively, this is the "lottery" Ann is endowed with. You may safely assume that Ann can make payments out of the value of her house in each state, but she cannot end up with negative wealth, i.e she cannot make an agreement in which S < 0 or F < 0.

(a) [4 marks] Plot the "lottery" that Ann is endowed with on a graph, labeling it H. Assume the safe state of the world is on the horizontal axis and the fire state of the world is on the vertical axis. Suppose the investment company is willing to offer any actuarially fair contract to Ann. Suppose p = .9. Plot all the final wealth (S, F) combinations available to Ann after contracting with this company. Plot Ann's indifference curve that passes through the lottery that she is endowed with. Plot the highest indifference curve Ann can reach by obtaining a contract with the investment company. What will Ann's final wealth be in each state of the world with this optimal contract? Show your work.

(b) [3 marks] On a new graph again plot the "lottery" that Ann is endowed with on a graph, labeling it H. Again have the safe state of the world on the horizontal axis and the fire state of the world on the vertical axis. Assume again that p = .9, but now Ann can choose to burn down her house without getting caught if she wants to (i.e. she can make p = 0 if it is to her advantage, but no one will know her choice, though they are aware it is a possibility). Suppose a rational and risk neutral investment company (with no costs) decides to charge an actuarially fair premium for contracts that pay a positive amount X in the fire state, and zero in the safe state. Plot the final wealth (S, F) combinations available to Ann after contracting with this company. Plot Ann's indifference curve through the lottery she is endowed with. Plot the highest indifference curve Ann can reach by obtaining a contract with this investment company. What contract will Ann purchase, i.e. what is the X that the company will pay her in the fire state? What premium will the company charge her in total? How does this compare to (a)? Show your work.

(c) [4 marks] Continue with the assumptions from (b) except now an insurance company instead decides to charge Ann .20 per unit paid to her in the fire state of the world, so that if Ann wants to be paid 10 in the fire state, she must pay 2 up front. On the same graph begun in part (b) plot the final wealth (S, F) combinations available to Ann after contracting with this insurance company, and indicate it as "unfair" with the one from (b) indicated as "fair." Plot the highest indifference curve Ann can reach by obtaining a contract with this insurance company. What contract will Ann purchase, i.e. what will be her payoff be in the fire state? What will her premium be? Show your work.

(d) [2 marks] What is the most Ann is willing to pay to eliminate all risk? Please justify.