# THE COASETHEOREM FOR NEGATIVE EXTERNALITIES

Allowing injuring and injured parties to negotiate compensation for a negative externality generates an efficient outcome (assuming negligible bargaining costs^{[1]}) regardless, in a sense (see below), of the legal regime.

The classic example used in Coase's landmark paper^{[2]} is that of a doctor disturbed in his work by noise caused by the machinery of a confectioner in a neighbouring building. The traditional economic opinion was restrictive: the noise is harming the doctor, so the confectioner is obliged to eliminate the source of the noise. Coase noticed a seemingly obvious fact: eliminating the source of noise might help the doctor, but it will harm the confectioner. Either way, someone will always suffer damage. From the overall perspective it would clearly be better to implement the option involving less damage. Coase added that if we allow bargaining between the injured party and the "guilty party", it is advantageous to both parties to implement the option involving the less serious harm.

In other words, when the theorem's assumptions are satisfied, negotiations between the parties will always lead to bargains that are advantageous to all the parties and simultaneously to optimal allocation of resources to environmental investments. Producers will, in their own interest in the given conditions, achieve what the state is trying in vain to achieve owing to a shortage of information. The negative externality will reach a level that is optimal from the point of view of society, for example from the point of view of a common owner of two factories.

In the following section we will for mulate the standard Coase theorem for the most common case of a negative externality affecting two producers.

## THE COASE THEOREM FOR NEGATIVE EXTERNALITIES: THE CASE OF TWO PRODUCERS

We solve the problem of two producers (a polluter and an injured party). We assume zero (negligible) costs associated with negotiating compensation.

We use the following notation: *x* is the amount of pollution,

*П(x)* is the polluter's profit at the level of activity causing pollution *x,*

*L(x*) is the financial loss of the party harmed by pollution *x.*

The optimum (the efficient level of pollution) *x** is the solution to the equation

*П'(x) = L'(x).*

If *П(x)* is a concave function and *L(x)* is a convex function, we have a necessary and sufficient condition for efficiency.

**Figure 53: The efficient level of a negative externality**

Figure 53 illustrates the polluter's decreasing marginal profit and the injured party's increasing marginal loss L'(x). Let's compare two possible legal regimes:

a) The law allows the polluter to damage the injured party at his will. Then, if the parties fail to agree on compensation, the outcome is the situation illustrated by point A, where the polluter ignores the loss of the injured party. At the efficient pollution level *x** [optimum compensation) the polluter will — by comparison with situation A — lose profit corresponding to area Px and the loss of the injured party will be reduced by the equivalent of area P1 + P2 . If the parties strike a bargain and the injured party pays the polluter compensation of

P1 + *a* • P2, where 0 < *a < 1,*

to voluntarily lower its output and thereby also cut pollution, both parties will be better off at the efficient pollution level *x** relative to situation A, because

• the polluter gains P1 + *a* • P2 - P1 = *a •* P2, and

• the injured party gains P1 + P2 - (P1 + *a* • P2) = (1 - *a)* • P2.

So, compared to situation A both parties are better off at the efficient pollution level *x**: they split the profit corresponding to area P2 .

b) The law bans pollution and the injured party can stop the polluter producing. Without compensation bargaining, the outcome is situation B, where the polluter is forced to halt production. A bargain struck between the two parties involving compensation of

P4 + *a* • P3, where 0 < *a <* 1

is again beneficial to both parties. Compared to situation B

• the polluter gains P3 + P4 - (P4 + a • P3) = (1 - a) • P3, and

• the injured party gains *a* • P3.

So, even in legal regime b), both parties will be better off at the efficient pollution level *x* relative to situation A: they split the profit corresponding to area P3 .

Bargaining therefore leads to a Pareto-efficient solution regardless of the legal regime. However, the law does significantly influence the distribution of profit for example, the polluter has profit (including compensation) of

• *П =* P1 + *a • P*2 *+* P3 + P4 when pollution is permitted, whereas

• when the injured party has the right to stop it producing, it has lower profit: П = P3 + P4 - (P4 + *a* • P3) = (1 - *a*) • P3.

In other words, from the overall perspective the legal regime is irrelevant, but from the parties' point of view it is significant.

Coasian bargaining is one of the possible ways of internalizing externalities. However, it is associated with fundamental problems. Besides the fact that bargaining costs and legal costs are not negligible in reality, there is the free rider problem. Free riders benefit from contracts to which they are not party and cannot be excluded from doing so. Another problem is the often dominant position of polluters in the emissions permit market. This can result in the polluter receiving the lion's share of the gain from bargaining.

Another internalization method is to levy on the polluter a Pigovian tax equal to the marginal damage imposed by the efficient pollution level:

constructed in such a way as to lead the polluter to choose the efficient pollution level *x* (see Figure 54).

**Figure 54: A Pigovian tax**

Even this approach is not free from fundamental problems. In particular, it is difficult to obtain the information necessary to calculate and determine the correct Pigovian tax. The polluter has a vested interest in distorting the information it provides to the state. There is another problem as well: if the polluter and the injured party are able to bargain, the amount of pollution will be deflected to x**. This new situation is Pareto-inefficient in the sense that a shift to point x* would increase tax revenue by *t* • (x* - x**), which is more than the two agents' gain from the bargain — the dark-shaded area P in Figure 54 is necessarily a part of the rectangle with sides t, x** - x*.

In this sense, the combination of taxation and bargaining space is disadvantageous. The state can avoid this by using the Pigovian tax to pay the injured party a compensatory subsidy equal to the amount of damage it suffers. This will remove the injured party's incentive to negotiate compensation. The outcome is still an efficient level of pollution.^{[3]}

The state should be interested in ensuring that environmental damage is cleaned up and environmental investment is financed by agents that can clean it up at the lowest possible cost.^{[4]} To achieve this, however, the state does not have information about the necessary cleanup costs. What is more, there is a morally questionable principle at stake here: if the injured party (e.g. a doctor) is able to remove the problem at lower cost (e.g. by installing a noise barrier) than the originator of the negative externality (e.g. by halting confectionery production), the legal regime that permits noise pollution is surprisingly more efficient.

- [1] The assumption of zero (negligible] transaction costs makes the Coase theorem applicable only to situations with a relatively small number of parties, since the number of possible combinations of actors of the interaction between the parties is subject to the "curse of dimensionality". If the number of parties is 10, the number of subsets created from this set of parties is less than 1000, and for n = 15 it exceeds 32,000. But for n = 20 there are more than 7 million "combinations” of parties to the Coase negotiation process. The transaction costs therefore exceed the positive effect of bargaining and the conclusion that bargaining is efficient no longer applies.
- [2] Coase, R. H.: The Problem of Social Cost. Journal of Law and Economics 3, Oct. (1960): 1-44.
- [3] This does not apply if the injured party is able to mitigate the damage caused by the polluter and the compensation varies with the amount of damage suffered. In such case the level of expenditure on miti-gating the damage is lower than the efficient level. See Gravelle, H., Rees, R.: Microeconomics. London: Longman, pp. 516–33.
- [4] See Frank, R. H.: Microeconomics and Behavior. New York: McGraw-Hill, pp. 663–700.