Consider Sunny who is committed to a life of crime. Sunny is risk averse, knows that he will enjoy consumption level x₁ if he does not get caught and consumption level of x₀ (very much below x₁) if he gets caught and goes to jail. He estimates the probability of getting caught as δ.

A. Suppose there are various mafia organizations that have connections in the District Attorney’s office and can affect the outcomes of court cases. Suppose initially that Sunny’s

tastes are state independent.

(a) First, consider a really powerful mafia that can insure that any of its members who is caught is immediately released. Can you illustrate how much such a mafia would be able to charge Sunny if Sunny is risk averse? What about if Sunny is risk loving?

(b) Next, suppose that the local mafia is not quite as powerful and can only get jail sentences reduced— thus in effect raising x0. It approaches Sunny to offer him a deal: Pay us p when you don’t get caught, and we’ll raise your consumption level if you do get caught by b. If the local mafia insurance business is perfectly competitive (and faces no costs other than paying for increased consumption in jail), what is the relationship between b and p? (Hint: Note that this is different than the insurance example in the text where my wife had to pay p regardless of whether she was in the good or bad outcome.)

(c) Suppose that Sunny can choose any combination of b and p that satisfies the relationship you derived in (b). What would he choose if he is risk averse? What if he is risk loving?

(d) Why does Sunny join the mafia in (a) but not in (c) if he is risk loving (and if “negative” insurance is not possible)?

(e) How much consumer surplus does Sunny get for buying his preferred (b,p) package when he is risk averse; i.e. how much more would Sunny be willing to pay to eliminate risk than he has to pay?

(f) Construct a graph with x_G —defined as consumption when not caught—on the horizontal and x_B —defined as consumption when caught—on the vertical axis. Illustrate, in the form of a bud

(g) Illustrate his optimal choice when he is risk averse and his tastes are still state-independent. How does this change if the corrupt jailer takes a fraction k of every dollar that the mafia makes available to Sunny in jail?

(h) Can you show in this type of graph where Sunny would optimize if he is risk-loving?

(i) Finally, suppose Sunny’s utility from consumption is different when he is forced to consume in jail than when he consumes on the outside. Can you tell an intuitive story for how this might cause Sunny to pick a (b,p) combination that either over- or under-insures him?

B. Suppose we express consumption in thousands of dollars per year and that x0 = 20 and x₁ = 80. Suppose further that δ = 0.25 and that the function u(x) = x^α is the utility function over consumption that allows us to express tastes over gambles through an expected utility function.

(a) Consider first the powerful mafia (from part A(a)) that can eliminate any penalties from getting caught. How much would Sunny be willing to pay to join this mafia if α = 0.5? What if α = 2?

(b) One of these cases represents risk averse tastes, the other risk loving. In light of this, can you explain your answer intuitively?

(c) Next, consider the weaker mafia that can raise consumption in jail. Suppose this mafia asks Sunny to pay p during times when he is not caught in exchange for getting an increase of b in consumption when he finds himself in jail. If you have not already done so in part A of the question, derive the relationship between p and b if the mafia insurance market is perfectly competitive (and faces no costs other than paying b to members who are in jail).

(d) Using the function u(x) = x^α, set up the optimization problem for Sunny who is considering which combination of b and p he should choose from all possible combinations that satisfy the relationship you derived in (c). Then derive his optimal insurance contract with the mafia.

(e) If α = 0.5, what is Sunny’s consumer surplus from participating in the mafia.

(f) Why does your solution to (d) give the wrong answer when α > 1? Explain using the example of α = 2.

(g) Suppose again that α = 0.5. What changes when the jailer takes a fraction k = 0.25 of every dollar that is smuggled into the jail?

(h) Finally, suppose that tastes are state dependent and that the functions u_B (x) = 0.47x^0.5 and u_G(x) = 0.53x^0.5 (where u_B applies in jail and u_G applies outside) allow us to represent

Sunny’s tastes over gambles using an expected utility function. Assuming that Sunny still chooses from the insurance contracts that satisfy the relationship between b and p you derived in (c), what contract will he pick? What if u_B (x) = 0.53x^0.5 and u_G(x) = 0.47x^0.5 instead? Can you make intuitive sense of your answers?