Consider a person who is thinking about whether to engage in a life of crime. He knows that, if he gets caught, he will be in jail and will sustain a consumption level of x0 but if he does not get caught, he will be able to consume x1 considerably above x0.
A: Suppose that this person cares only about his consumption level (i.e. he has state-independent tastes).
(a) On a graph with consumption x on the horizontal axis and utility on the vertical, illustrate this person’s consumption/utility relationship assuming he is risk averse.
(b) Suppose the probability δ of getting caught is 0.25. Illustrate the expected utility of choosing a life of crime. What if δ = 0.75?
(c) Redraw the consumption/utility graph and suppose δ = 0.5. Let  indicate the income this person would need to be able to make honestly in order for him to be indifferent between an honest living and a life of crime.
(d) Senator C believes the criminal justice system spends too much effort on identifying criminals but not enough effort on punishing them harshly. He proposes an increased deterrence policy under which penalties for committing crimes are raised while less is spent on law enforcement. This implies a drop in both x0 as well as δ. Suppose the expected consumption level for a person engaged in a life of crime remains unchanged under this policy. Will the person who was previously indifferent between an honest living and a life of crime still be indifferent?
(e) Senator L believes we are treating criminals too harshly. He proposes an increased enforcement policy that devotes more resources toward catching criminals but then lowers the penalties that criminals face if caught. The policy thus increases x0 as well as δ. Suppose that the expected consumption level of a person engaged in a life of crime is again unchanged under this policy. Will the person who was previously indifferent between an honest living and a life of crime still be indifferent?
(f) True or False: If criminals are risk averse, the increased deterrence policy is more effective at reducing crime than the increased enforcement policy.
(g) How would your answers change if criminals were risk loving?
B. Suppose that x0 = 20 and x1 = 80 (where we can think of these values as being expressed in terms of thousands of dollars), and suppose the probability of getting caught is δ = 0.5.
(a) What is the expected consumption level if the life of crime is chosen?
Suppose the potential criminal’s tastes over gambles can be expressed using an expected utility function that evaluates the utility of consumption as u(x) = ln(x). What is the person’s expected utility from a life of crime?
(c) How does the expected utility compare to the utility of the expected value of consumption? Can you tell from this whether the criminal is risk averse?
(d) Consider the level of consumption this person could attain by not engaging in a life of crime. What level of consumption from an honest living would make the person be indifferent between a life of crime and an honest living? Denote this consumption level .
(e) Now consider the increased deterrence policy described in A (d). In particular, suppose that the policy increases penalties to the point where x0 falls to 5. How much can δ drop if the expected consumption level in a life of crime is to remain unchanged?
(f) What happens to the  as a result of this increased deterrence policy?
(g) Now consider the increased enforcement policy described in A (e). In particular, suppose that δ is increased to 0.6. How much can x0 increase in order for the expected consumption in a life of crime to remain unchanged?
(h) What happens to x as a result of this increased enforcement policy?
(i) Which policy is more effective at reducing crime assuming potential criminals are risk averse?
(j) Suppose that the function u(x) that allows us to represent this individual’s tastes over gambles with an expected utility function is u(x) = x2. How do your answers change?