Efficient deterrence for price fixers and monopolists
The previous section argued that agreements to fix prices or quantities between Cournot duopolists (or oligopolists, for that matter) may be inherently unstable and therefore difficult to observe and enforce. Nevertheless, price fixing cartels do exist. How, then, should price fixing be punished? This section applies the framework of Chapter 9 to address this question.
To illustrate the main idea, suppose again that the demand curve is:
We again assume that marginal costs are constant and equal to c > 0. The efficient quantity is Q* = a - c. Now consider a price-fixing cartel, which somehow manages to reach a stable, enforceable agreement between its members to fix its price and quantity at the monopoly level. Thus, we assume that the members of the cartel basically act as a single, profit-maximising monopolist.
Under these circumstances, the cartel's profit is:
At a price P, the consumer surplus is:
Following the results from Chapter 9, the marginal damage or harm to consumers from a price increase is simply the change in consumer surplus as the price rises, which is equal to:
Thus we have a situation that is similar to the general set-up in Chapter 9: there is a party (a cartel) whose actions are inflicting 'harm' on another group, consumers. It is important to point out, however, that the analogy with the results in Chapter 9 only goes so far. Price fixing is not 'theft', since the transactions between consumers and producers are entirely voluntary. The 'harm' that is created when a cartel increases its price only becomes an issue for efficiency analysis when price increases above marginal cost, in which case there is an efficiency loss.
Nevertheless, the analytical approach in Chapter 9 is very useful here. In Chapter 9, the optimal punishment strategy involved setting the marginal expected fine pf equal to the marginal social harm at the optimum. In this example, the marginal harm to consumers is the loss of consumer surplus, which is Q = a - P. At the optimum, P = c, so let us set the marginal expected fine equal to pf = Q* = a - c. That is, suppose that the marginal fine is equal to
Equation (10.38) states that the efficient marginal fine is equal to the efficient quantity, discounted by the probability of detection.
The first obvious question to address is: does this fine actually deter a cartel from price fixing? Remember that the fine here is a marginal fine: it is levied per unit of the harmful activity (which is the increase in the market price). Therefore, with this marginal fine in place, the cartel's expected profit is:
The cartel chooses P to maximise this expression. The result is:
Solving this expression, we obtain the result that the cartel chooses P = c, which is the efficient outcome: the fine effectively deters price fixing.
Under this legal rule, the cartel's expected profit is:
and expected revenue from the fine is:
With this fine in place, equation (10.41) indicates that the cartel earns negative profits, and so may be better off shutting down completely. This is inefficient. To avoid this occurring, we alter the marginal fine slightly so that it effectively also pays the cartel a total amount equal to ac - c2. If this is done, the cartel's expected profit would become:
The optimal punishment scheme now involves imposing a total fine on the cartel that is equal to:
Equation (10.43) is very intuitive. It states that the total fine should be equal to the efficient quantity Q* multiplied by the cartel's actual profit margin, discounted by the probability of detection. Note that the total fine is zero if the cartel prices at marginal cost. The rule in (10.43) therefore states that the total fine should be proportional to the cartel's 'overcharge' P - c, where the factor of proportionality is the
Figure 10.7.1 The cartel's profit function with and without the optimal fine efficient quantity discounted by the probability of detection. With this rule in place, the cartel's expected profit is E[n] = -(P - c )2, which again induces the cartel to choose P = c, so that its expected profits are zero. The expected fine revenue is:
and so there is no expected monetary gain to the enforcement authority from this system of fines.
Note that this rule does not set the fine equal to the actual profit of the cartel; nor does it consider the actual total harm to consumers when the cartel chooses a price of P. This happens for two reasons. First, since the probability of detection is lower than one, it is the expected fine that has the deterrence effect, not the fine itself. Second, a rule which focused on the actual harm to consumers (that is, the loss of consumer surplus at the cartel's price P, compared to the efficient outcome) would result in underdeterrence, because with a higher price, quantity is lower. Since the harm to consumers is the change in consumer surplus, and since the marginal change in consumer surplus is -QdP, setting a fine proportional to the actual quantity that is chosen will result in under deterrence, since when price exceeds marginal cost, Q < Q.