# Exercises

1. (The Unscrupulous Diners) Another example of overlapping usage rights occurs when a group of diners jointly enjoys a meal at a restaurant, and there is an explicit restaurant policy that the bill cannot be split. Alternatively, there is often an unspoken agreement or convention (or even an explicit agreement among diners) that the table's bill will be split equally.

Gneezy et al. (2004) perform a series of actual restaurant experiments to examine this problem. They observed and manipulated conditions for several groups of six diners at a popular dining establishment. When asked to choose, prior to ordering, whether to split the bill or pay individually, 80 per cent choose the latter. That is, individuals preferred the environment with private property rights and without externalities. However, in the presence of externalities, they nevertheless take advantage of others. When the bill was split, the group spent more.

Consider a simple example of the split bill problem. Suppose that there are two diners at a lunch (A and B). When A is making his order, he knows that each additional dollar of spending will be subsidised by B, and vice versa. Note here that the benefit of an additional dollar of spending on individual item is privatised, but the cost is socialised: the private cost to each individual diner of their last dollar of spending is only \$0.50, but the joint cost is \$1.00. Thus, each additional dollar of spending by A imposes a negative external cost on B. Alternatively, each reduction in spending by A creates a positive external benefit on B (his portion of the bill is now reduced by \$0.50 without having to do anything).

Suppose that each diner can order a main meal and a dessert. The price of a main meal is \$10, and the individual benefit is \$12. The price of a dessert is also \$10, but the individual benefit is only \$6.

• (a) Is it efficient for the diners to order dessert?
• (b) Write down the payoff matrix for the unscrupulous diner game when the bill is split. Find the Nash equilibrium. Show that the diners will overeat. If the diners get a zero benefit from not going to dinner together, show that if the bill is split they are better off not going to dinner at all.
• (c) Now assume that the diners agree not to split the bill. Repeat part (b). Show that the diners do not overeat and that there are positive individual and aggregate benefits from dining.
• (d) Why would a restaurant insist on a rule of 'no split bills'?
• 2. This question considers the framework introduced in Chapter 3 in a world with insecure property rights, and follows Robson and Skaperdas (2008). Consider an economy with two agents, 1 and 2. Agents own an initial amount of money (m) and can make payments to each other, but the total amount of money available in the economy remains constant. Each agent likes money, so utility functions are increasing in m.

Agent 1 undertakes an activity, x, which pleases agent 1, but makes his neighbour, agent 2, unhappy (for example, playing loud music). In other words, agent 1's utility function increases with x but agent 2's utility function decreases with x. There is a limit on x (say, 24 hours per day). Denote this limit by x.

There are two legal regimes, labelled I and II. In regime I, agent 1 is legally entitled to play loud music all day and agent 2 must pay agent 1 to get a specified number of hours without noise. In regime II, agent 1 must obtain, in exchange for a money payment, agent 2's permission to play a specified positive number of hours of loud music.

• (a) Represent this economy in an Edgeworth box diagram under two legal regimes. Illustrate the competitive equilibrium allocations of money and playing of loud music, and the competitive equilibrium price ratio in the two legal regimes. Are the competitive equilibrium allocations Pareto optimal? Explain. Is it necessarily the case that in the competitive equilibrium, agent 1 always chooses to play the same amount of loud music? Explain, with reference to the efficiency version and the invariance version of the Coase Theorem.
• (b) Now suppose that small transactions costs drive a wedge between the marginal rates of substitution of the agents. In other words, in a competitive equilibrium, marginal rates of substitution need not be equal, even though no gains from trade may exist. What effect will these small transactions costs have on the competitive equilibrium allocations? Explain, with reference to the Coase Theorem.
• (c) Now suppose that transactions costs are again zero, but suppose legal rights regarding the production of loud noise must are insecure, and must be clarified by a court or some other recognised tribunal which makes binding decisions. The agents can use their initial allocations of money to influence the court's decision on which legal regime prevails.

Suppose that the court's decision depends on the relative of quantity of money devoted to influence activities by each agent. For any pair (m1, m2) of expenditures on influence activities, let p(m1, m2) be the probability that the court will decide that regime I prevails, and let 1 - p(my m2) be the probability that the court will decide that regime II prevails, with 0 < p(m1, dp d2p dp Э2p

m2) < 1, and > 0, 2 < 0, < 0, 2 > 0. What implications

dm1 dm{ dm2 dm22

do these insecure legal rights have for the final production of loud music? Can the invariance version of the Coase Theorem still hold, even though initial legal rights are insecure or not well defined?