1. John is a risk averse but dishonest bank clerk. His utility for wealth is u(x) = √x and he is an expected utility maximizer. His salary is W=$40,000 a year. By stealing small amounts at a time he can steal $4,100 a year without attracting too much attention. However there is routine monitoring of the accounts and there is a small chance that this monitoring will uncover the missing funds and that he will be caught. Let π be the probability of being caught stealing. If he is caught, he will have to reimburse the $4,100 that he has stolen and in addition he will have to pay a fine of $3,900 for stealing. (He will also be fired and will have to find another job but for the purpose of this exercise we will ignore this side of the story). We study whether John, who has no moral principles, will choose to steal, or whether he will be deterred by the risk of being caught and have to pay the fine.
(a) Draw a graph of John's utility for wealth. Insert the point (W, u(W)) which gives John's utility if he does not steal.
(b) Suppose π = 0.1. Compute the expected value E(X) and the expected utility E(u(X)) of the lottery X = Xb π Xg 1 - π! that he faces if he steals, where Xb is his income if he is caught and Xg his income if he is not caught. Plot the points (Xb, u(Xb)), (Xg, u(Xg)), (E(X), E(u(X)) in the graph of the previous question (the relation between these points must be shown).
(c) Will John choose to cheat? Explain your answer.
(d) The bank where John is working is loosing money and the management realizes that funds are disappearing because some employees are stealing. Thus they decide to monitor better their 1 employees. This stricter monitoring increases the probability of an employee being caught when stealing money. What is the minimum probability π m which will discourage John from stealing? Compute π m and the expected value E(Xm) of the lottery Xm = Xb π m Xg 1 - π m, where Xb and Xg are as in question (b).
(e) Draw again the graph of u and insert the points (W, u(W)),(Xb, u(Xb)), (Xg, u(Xg)), (E(Xm), E(u(Xm)).
(f) Interpret the difference E(Xm) - W.
2. Illegal parking is a severe problem in Paris and the mayor is studying the best way to reduce it. The reason why people use their cars to shop or go to work in Paris is that some people do not like taking the subway: it can be crowded, it does not smell good, it is not convenient with large or heavy shopping bags... Let c denote the amount that a given Parisian is ready to pay to have a comfortable ride in a car rather than taking the metro. c is thus the "cost" of taking the subway for a given resident of Paris. Parisians are expected utility maximizers and risk averse, i.e. their Bernouilli utility function u (the same for all Parisians) is increasing and concave. Currently the fine for illegal parking is (the equivalent of) $100. This is rather expensive and Paris's mayor does not think that he can increase the fine without provoking the anger of his constituents. But there are so many cars which are parked illegally and so few cops to give tickets that the probability of receiving a parking ticket when parked illegally is only 20%. This could be increased by hiring more personnel for parking enforcement.
(a) Draw a graph next page with consumption on the horizontal axis and utility on the vertical axis the typical shape of the utility index u. Then indicate the expected utility of a Parisian parking her car illegally. Argue (in words and with the help of the graph) that all the Parisians who choose to take their car and park it illegally must have a cost c of taking the metro which is at least 20 dollars. Show on the graph how you can determine the number ¯c such that all Parisians with a cost less than ¯c take the subway and all those with a cost larger than ¯c drive and park illegally.
(b) Knowing that the average monthly income in Paris is I = 3, 000 and that the utility index for the representative Parisian is u(x) = -1/x, what is the minimum cost ¯c such that all the 2 Parisians who have a cost of taking the subway larger than ¯c will choose to drive and take the risk of getting a parking ticket?
(c) You can interpret c as what the person would be ready to pay to take a taxi instead of the subway. Unfortunately ¯c is not sufficient to cover the true cost of a taxi, which is $30 on average. Calculate the minimum probability of being caught which will induce the people who do not like taking the subway to take a taxi rather than risking a parking ticket (the income and the utility are the same as in the previous question).
(d) If Parisians were risk neutral, what would be the answer to the previous question? Is it substantially different from the answer to the previous question?
3. Joe is invited to the annual party of the Fun Gamblers Society, a respected society in town. The big attraction of the party is a lottery, in which all the names of the guest are entered and half the names are drawn at random by the innocent hand of the young daughter of the Society's president-all accompanied by music and jokes. Those whose names are drawn receive $1000 while the unfortunate guests whose names are not drawn pay $1000 each to finance the lottery. Unfortunately Joe is risk averse-he is an expected utility maximizer with a concave utility function of wealth-and in addition his business is not doing very well so he does not feel rich.
(i) What is the maximum that he is ready to pay his brother Jim to replace him at the party if
a. his utility of wealth is u(W) = ln(W) and his current wealth is $50,000
b. his utility of wealth is v(W) = -13x-3 and his current wealth is $50,000
c. his utility of wealth is u(W) = ln(W) and his current wealth is $100,000
d. his utility of wealth is v(W) = -13x-3 and his current wealth is $100,000
Please do the calculations by two possible ways: first direct calculation, then using the formulathat I gave in class, so that you can see the error that you make when taking the approximation.
(ii) Explain the results of the cases a-d of the previous question (at least explain why the resultis smaller or larger in one case than in another) using the coefficients of risk aversion of the two functions u and v.