# Exercises

1. Suppose that individuals are risk neutral and that the benefits from illegal

activity (denoted by x) are:

Suppose that the marginal social harm per unit of criminal activity is *h =* 2.

The resource cost of enforcement activity is:

where *p* is the probability of detection.

(a) For a given marginal fine *f* and a given probability of detection p, what level of illegal activity *x* will be chosen by individuals?

Suppose that the marginal fine *f* can be increased without limit.

- (b) What is the optimal marginal fine? What is the optimal enforcement probability? What is the efficient level of illegal activity, and what is the level of total welfare at this level? What is the government's expected revenue?
- (c) Now suppose that the fine cannot be increased without limit, and that the maximal fine is F = 3. Repeat part (a). What is the efficiency loss from not be able to increase the fine above F = 3?
- (d) Now suppose that the maximal fine is F = 4. Repeat part (a). What happens to the crime rate compared to your answer in part (b)?
- 2. Suppose that individuals are risk neutral and that the marginal gains from

illegal activity (denoted by *x*) are:

where A is a positive constant and e > 0.

- (a) What is the individual's 'demand curve' for illegal activity?
- (b) What is the point elasticity of the level of illegal activity with respect to changes in the expected per unit punishment?
- (c) Suppose that there is an imprisonment length of 1 year per unit of crime committed. Let the probability of detection equal 0.1, and suppose that each year in prison gives the individual disutility of $40,000. What is the expected 'price' of each unit of crime committed? If
*e =*0.5, what is the crime rate?

- (d) Use the results from this chapter to derive the optimal imprisonment term as a function of the number you computed in part (b), the marginal harm h, the probability of detection p, the disutility of imprisonment l, and the costs of imprisonment
*a.*How does the optimal imprisonment term*t*vary with each of these parameters? Explain. - 3. Consider the market for marijuana. Suppose that there is a constant marginal production cost of $50 per kilogramme for growing marijuana and selling it to buyers. Suppose that the private benefit of marijuana consumption is:

where *Q* is measured in kilogrammes. Suppose that the social external harm of marijuana consumption is:

(a) What is the efficient level of marijuana consumption and production? What level of consumption and production would be produced by a completely free and competitive market for marijuana? Is the free market outcome efficient? If so, explain why. If not, compute the welfare loss.

Now suppose that supplying marijuana is illegal. If authorities find marijuana growing or in the hands of dealers, they seize the product and fine the supplier.

Let *p* be the probability that a supplier is caught, and let *F* be the per kilogramme fine that is levied. The competitive 'street price' of marijuana is *P.* The total resource cost of enforcing laws against the production and supply of marijuana is:

- (b) If supplying marijuana is illegal, what is a marijuana supplier's expected marginal cost? What is a marijuana supplier's expected marginal revenue? Find an expression for the competitive street price
*P*in terms of the production cost, fine, and probability of detection. For any fine*F*and probability of detection*p,*what quantity will be consumed in the competitive equilibrium? - (c) If catching suppliers is costly and the fine
*F*could be increased without limit, what is the efficient fine? What is the efficient probability of detection? With this combination of*F*and*p*, what is the street price of marijuana, what is the quantity that is produced, and what is the government's expected revenue? - (d) Now suppose that the fine cannot be increased without limit, and that the maximal fine is
*F =*30. What is the marginal welfare gain from increasing*p*? What is the marginal welfare cost of increasing*p*? Use these results to compute the efficient probability of detection, and find the efficient quantity. Show that when detection is costly and there is a maximal fine, the quantity computed in part (b) is no longer efficient. What is the welfare loss from not being able to increase the fine above*F =*30?

- (e) Now suppose that the maximal fine is
*F =*40. Repeat part (d). What happens to the quantity of marijuana consumption compared to your answer in part (d)? Explain. - 4. This question follows on from Question 3 and uses the same model and parameters. Suppose that instead of there being a perfectly competitive market for marijuana, there is a monopoly marijuana firm, which is owned and run by organised crime bosses. All other assumptions remain the same.
- (a) Suppose that selling marijuana is not illegal and that the monopolist must charge a single price per kilogramme. What is the monopolist's profit- maximising price? Is the monopoly outcome efficient? If so, explain why. If not, compute the welfare loss from monopoly.
- (b) From an efficiency point of view, which is preferable: a free marijuana market or a monopoly supplier?

Now suppose that supplying marijuana is illegal. If authorities find marijuana growing or in the hands of dealers, they seize the product and fine the supplier.

Let *r* be the probability that a supplier is caught, and let *F* be the per kilogramme fine that is levied. The total resource cost of enforcing laws against the production and supply of marijuana is again:

- (c) If supplying marijuana is illegal, what is the monopolist's expected marginal cost? What is the monopolist's expected marginal revenue? Find an expression for the profit-maximising monopoly price in terms of its production cost, the fine, and the probability of detection. For any fine
*F*and probability of detection p, what quantity will be sold by the monopolist? - (d) If detecting sales is costly and the fine
*F*could be increased without limit, what is the efficient fine? What is the efficient probability of detection? With this combination of*F*and*p*, what is the street price of marijuana, what is the quantity that is produced, and what is the government's expected revenue? Is the expected fine with a monopoly supplier higher or lower than the expected fine under perfect competition?

5. Consider an economy in which all individuals have the following utility function over wealth:

(a) For what values of *b* are individuals risk-neutral? For what values of *b* are individuals risk averse? For what values of *b* are individuals risk-loving?

Suppose that the individual's initial wealth is w_{0}, and suppose that the individual can take two possible actions: he can either commit a crime, or not commit a crime. If he commits a crime and is caught, he faces a fine of *f* < w_{0}. The gain from the criminal act is *g* > 0. Suppose that the probability of getting caught is p, where 0 < *p* < 1.

- (b) What is the individual's expected utility if he commits the crime? Does the expected utility of committing crime rise or fall with increases in b? Write down the condition under which an individual will decide to commit a crime.
- (c) Given your answers in (a) and (b), if the government chooses a probability of detection
*p*and a fine f is this more likely to deter risk-averse individuals, risk-neutral individuals, or risk-loving individuals from committing crime?

Now suppose that the government can now alter *p* and f but can only do so in such a way as to keep the expected punishment *pf* constant.

- (d) If the government raises
*p*and lowers*f*(while keeping*pf*constant), what happens to the expected value of the individual's wealth if he commits a crime? What happens to the variance of the individual's wealth if he commits a crime? Is the change to*p*and*f*more likely to deter risk-averse individuals, risk-neutral individuals, or risk-loving individuals from committing crime? - (e) If the government lowers
*p*and raises*f*(while keeping*pf*constant), what happens to the expected value of the individual's wealth if he commits a crime? What happens to the variance of the individual's wealth if he commits a crime? Is the change to*p*and*f*more likely to deter risk-averse individuals, risk-neutral individuals, or risk-loving individuals from committing crime?