# Coase Theorem

## Coase in context

The Coase Theorem is a central result in economics. It shows how, under certain conditions, economic actors can arrive at an efficient solution to an externality without direct government involvement. Prior to the 1960 paper by Coase, economists thought that externalities, which are at the heart of environmental economics, necessitate government regulation, particularly taxation. Since then, Coase' result has sometimes been used to argue that environmental externalities do not necessitate government regulation beyond the establishment and enforcement of property rights. Skepticism remains, however, regarding the applicability of the Coase Theorem to real-world problems.

In its original form, the Coase Theorem is not a theorem in the conventional sense of the word. Coase did not formalize his theorem, let alone prove it. There is not a single equation or rigorous definition in the 1960 paper. Instead, Coase offers a detailed discussion of common law on liability and nuisance. Coase agrees with Pigou that externalities are a problem, but disagrees with Pigou's solution.

Coase's critique of the Pigovian framing of environmental problems focuses on the nature of the transfer payment required to internalize an externality. He argues that, because of the symmetry of the problem, a tax on producers of a negative externality is not the only possible solution:

"The traditional approach has tended to obscure the nature of the choice that has to be made. The question is commonly thought of as one in which A inflicts harm on B and what has to be decided is: how should we restrain A? But this is wrong. We are dealing with a problem of a reciprocal nature. To avoid the harm to B would inflict harm on A. The real question that has to be decided is: should A be allowed to harm B or should B be allowed to harm A?"

That is, Coase takes issue with Pigou's premise that the one who causes the externality should be the one who is rewarded (if the externality is positive) or penalized (if the externality is negative). Coase's criticism of Pigou's asymmetric treatment of pollutee and polluter is perhaps less well-known, but it was a key point in his seminal paper. To demonstrate the feasibility of alternative regimes, Coase discusses the different treatment under common law of escaped domesticated and wild animals. If a domesticated animal escapes and does damage, its owner is liable. If a wild animal escapes from captivity and does damage, the victim is liable rather than the former captor. Coase underlines the arbitrary nature of this distinction by discussing the rabbit, which many people would think of as \textit{domesticated} (and tame), but is actually a \textit{wild} animal under common law.

Coase' central example is cattle eating a neighbor's crops. Coase argues that, if the cattle-owner is liable for the damage done by her steers, she would limit the size of her herd to the point where the damage done by one additional steer equals the cattle's incremental profit. Coase then argues that, without such liability, the farmer would be willing to pay his neighbor to reduce the herd size, and that he would pay up to the point where the damage avoided by one fewer steer equals the marginal steer's value to the cattle-raiser. In other words, the final outcome is the same regardless of whether or not the cattle-owner has a duty to compensate for harm to her neighbor.

This example leads to the Coase Theorem: In the presence of externalities and well-defined property rights, and in the absence of transaction costs, agents can bargain their way to a Pareto optimum, and that Pareto optimum is the same regardless of who imposes an externality on whom.

Coase emphasizes that his conclusion only holds if there are no costs involved in the transaction and that it is easier to reach agreement if fewer parties are involved. He implicitly assumes that people are well-informed, act in their self-interest, that the money changing hands does not affect the demand or supply curves, and that the agreement reached by the bargaining parties will be enforced by courts if necessary. Another assumption is that the willingness to pay to avoid harm is equal to the willingness to accept compensation for harm. Coase himself did not seem to believe that these conditions were likely to be met in most situations, emphasizing the importance of considering the net value of alternative (imperfect) institutions that can be implemented in the presence of transaction costs. In his Nobel Prize lecture, he said that "the legal system will have a profound effect on the working of the economic system and may in certain respects be said to control it'".

Coase's Nobel Prize lecture continues: "[s]ince standard economic theory assumes transaction costs to be zero, the Coase Theorem demonstrates that the Pigovian solutions are unnecessary in these circumstances. Of course, it does not imply, when transaction costs are positive, that government actions (such as government operation, regulation or taxation, including subsidies) could not produce a better result than relying on negotiations between individuals in the market. Whether this would be so could be discovered not by studying imaginary governments but what real governments actually do. My conclusion; let us study the world of positive transaction costs." That is, in his Nobel Lecture, Coase does not take issue with Pigou, but rather with the assumption of zero transaction costs.

## Coase formalized

Stigler coined the term Coase Theorem. Stigler did not, however, restate Coase' insight as a theorem. Let us do so. Consider two agents with an indirect utility function

$$v_i(p,w_i,h) = \max_{x_i \geq 0} u_i(x_i,h) \text{ s.t. } px_i \leq w_i \text{ for } i=1,2$$

where $$p$$ is the price vector for consumption bundle $$x_i$$ of agent $$i$$, $$w_i$$ is his budget constraint, $$u_i$$ is utility, $$v_i$$ is indirect utility and $$h$$ is the externality. Assuming a quasilinear utility function with respect to a numeraire, we can write $$v_i(p,w_i,h) = \phi_i(p,h)+w_i$$. If both agents are price-takers, we can write $$\phi_i(p,h)$$ as simply $$\phi_i(h)$$.

Suppose that agent 1 chooses $$h$$ to maximize $$\phi_1$$. Then $$\phi'_1(h^*) = 0$$. The social optimum maximizes $$\phi_1+\phi_2$$, so that $$\phi'_1(h^\circ) = -\phi'_2(h^\circ)$$. The equilibrium $$h^*$$ is suboptimal unless $$h^*=h^\circ=0$$. If $$\phi'_2(.)<0$$, the externality is negative and $$h^* > h^\circ$$, that is, agent 1 chooses too much $$h$$. If $$\phi'_2(.)>0$$, the externality is positive and $$h^* < h^\circ$$, that is, agent 1 chooses too little $$h$$.

Now suppose that agent 2 has the right to be free of externality $$h$$, but would be prepared to waive that right in return for compensation $$T>0$$. Then agent 2 would solve

$$\max_h \phi_2(h) + T \text{ s.t. } \phi_1(h) - T \geq \phi_1(0)$$

where the constraint comes about because agent 1 needs to agree to the bargain. As the constraint binds, this is equivalent to $$\max_h \phi_2(h) + \phi_1(h) - \phi_1(0)$$

The maximand is the social welfare function (shifted by a constant), and thus the equilibrium externality is the optimal one $$h^\circ$$.

If instead there are no restrictions on agent 1, agent 2 would need to compensate her with an amount $$T<0$$. Agent 1 would agree if $$\phi_1(h)-T \geq \phi_1(h^*)$$. Deciding on the offer made, agent 2 would solve

$$\max_h \phi_2(h) + \phi_1(h) - \phi_1(h^*)$$

The maximand is again the social welfare function (shifted by a different constant), and the equilibrium externality is the optimal one $$h^\circ$$.

This simple proof of the Coase Theorem also reveals key underlying assumptions:

• No wealth effect Quasi-linearity in the numeraire makes the externality $$h$$ independent of budgets $$w_i$$ and side-payment $$T$$.
• Perfect information The agents know each other's indirect utility functions.
• Rationality Agents maximize utility.
• No endowment effect The utility functions are smooth in the status quo, and economic agents behave the same whether or not they have the right to be free of externalities.
• Zero transaction costs The bargain can be struck without incurring costs.

The Coase Theorem can be split into three parts. The efficiency thesis states that, once property rights are assigned, a Pareto optimum is achieved. As the assignment of property rights completes the market, this result is equivalent to the First Fundamental Theorem of Welfare Economics. The invariance thesis states that the Pareto optimum is independent of the initial allocation, a result that is sharper than the Second Fundamental Theorem of Welfare Economics, which states that a Pareto optimum, rather that the same Pareto optimum, would be reached after reallocation.

Zero transaction costs is the third -- and most controversial -- part. Large parts of economic theory assume that transaction costs are negligible. If so, the Coase theorem illustrates that there is no need for direct government intervention to internalize externalities.

### Coase generalized

The Coase Theorem only holds for two economic agents -- one polluter and one pollutee, a $$1\times1$$ bargain. If there is more than one person involved on either side -- $$m\times1$$, $$1\times n$$ or $$m\times n$$ bargains -- then coordination problems between polluters or pollutees prevent the attainment of an efficient solution.

As a corollary, if there is no coordination problem, the Coase Theorem does hold for more than two agents. For instance, a $$1\times n$$ bargain between 1 polluter and $$n$$ pollutees is equivalent to $$n$$ $$1\times 1$$ bargains if there is no fixed cost of emission reduction, the variable costs are linear in emission reduction, the environmental damage is linear in emissions, and the polluter cannot exert market power over the pollutees. Under these (stringent and unrealistic) assumptions, each pollutee would strike a separate bargain with the polluter and those bargains would be efficient as the pollutees do not affect each other.

In more realistic settings, the action of one pollutee does affect the other pollutees\textemdash or polluters may affect each other. This would be the case if, for instance, the impact of pollution is non-linear in emissions. Then, coordination problems arise, and a pollutee may choose to free-ride on the efforts of her fellow pollutees to bargain with the polluter.

Coordination problems are hard to solve. That said, while $$m\times n$$ bargains do not attain efficiency, they can still improve welfare.

## Coase in the lab

Since the assumptions underlying the Coase Theorem were first made explicit, many laboratory experiments have been designed to understand which of these assumptions are mathematically convenient but can be relaxed without overturning the practical implications of the Coase Theorem and which assumptions are crucial.

In the first experiment, subjects were assigned to groups of two or three. One or two subjects were randomly assigned to be controllers, who, analogously to being assigned initial property rights in the Coase Theorem, had the right to unilaterally choose the set of payoffs players would receive. The other participant(s) could attempt to influence the outcome via negotiations, including by offering to transfer some or all of her earnings to the controller. In each case there was a unique scenario that maximized total cash payments, but whether or not payments were known to all participants varied. Any contract between the players was enforced by the experimenter, and payments were made publicly. Under conditions where payoffs were known and there was only one controller, 89.5% of the 114 experimental decisions resulted in Pareto optimal outcomes. In experiments with limited information and joint controllers, success rates were substantially lower.

In another early experiment finds that a contract negotiated over an externality comes, on average, within 3% of the Pareto optimum, and that there is no statistically significant difference between cases where the polluter or pollutee holds the initial property rights. This experiment ends with an ultimatum, and players appear to be motivated by fairness as well as efficiency.

Many experimental studies of the Coase Theorem and its limitations have been conducted, yielding much insight about when property rights are sufficient to yield Pareto optimal outcomes. An experimental design that makes cooperation individually rational -- the original set-up, it was impossible to distinguish between a fair allocation and a Pareto optimal one -- find strong support for the Coase Theorem: The Pareto optimum is found in 97% of experiments. For zero transaction costs, complete information, and small incentives, subjects tend to opt for a fair allocation rather than a Pareto optimal one. Higher incentives lead to a shift to the Pareto optimum. The Pareto optimum becomes unattainable if transaction costs increase. Private (rather than public) information does not affect the ability of participants to attain the Pareto optimum. Asymmetric information does: Participants are less willing to trade in this case. Less secure property rights attenuate the effect of asymmetric information. Endowment effects would hamper Coasian bargaining.

The Coase Theorem holds also when stress-tested with larger numbers of participants, asymmetric payoffs, uncertain payoffs, and more complicated bargaining. Transaction costs and time-limits have a negative effect on the probability of attaining a Pareto optimum, while face-to-face bargaining and information have a positive effect. The Coase Theorem also holds if either party can block the transaction and have the experimenter take away the good that they are bargaining over with mininal compensation.

## Coase and the courts

To understand whether the experimental results discussed above have empirical counterparts, it is worth considering whether the world's legal institutions are conducive to Coasian bargaining.

Well-defined property rights (and, implicitly, enforceable contracts) are the key assumption underlying Coasian bargaining. In legal systems with strong protection of private property, such as the United States, clearly defining property rights may seem straightforward. However, specifying complete property rights requires attention to such details as mineral rights, wildlife harvesting rights, rights to make noise or emit noxious smells, and so on. As court cases demonstrate, there are many situations in which property rights are sufficiently vague to result in substantial disagreements between the affected parties about who holds a particular right. For simplicity, we refer to the party producing an environmental externality as the "polluter" and the party experiencing the environmental externality as the "pollutee".

There are at least four reasons for the continued existence of ambiguous property rights. First, the common law theory of nuisance makes it very difficult to fully and clearly assign the right to create or assign the externality to the polluter, particularly for new types of harms where precedent has not been established. Second, it is difficult to define terms used in legislation and regulation in a way that leaves no room for an alternative interpretation. Third, the existence of multiple levels of government and of multiple, related, laws sometimes creates ambiguity about which law applies to a particular situation. Fourth, new laws and regulations change property rights, and shifting social norms and legal principles change what is deemed permissible.

The (very old) common law principle of nuisance is the basic legal principle determining the allocation of property rights around externalities from private property -- who has the right to pollute and who has the right to be protected from pollution? In common law, the tort of nuisance goes back to the 13th century, in a case where King John of England (of Robin Hood and Magna Carta fame) ruled in favour of Simon of Merston after Jordan the Miller had flooded Simon's land in an attempt to expand the pond that powered Jordan's mill. Since the resolution of the Trail Smelter dispute, in which the smoke of a lead and zinc smelter in British Columbia affected farmers in Washington, the legal obligation to be a good neighbour also applies across country borders.

In modern legal theory, the nuisance principle allows for the "quiet enjoyment" of private property, while protecting other people from "unreasonable interference" as a result of that enjoyment. However, these are vague and general principles. In many situations, what constitutes "unreasonable interference" is unclear, or at least contested, resulting in both polluters and pollutees asserting that they hold the right to inflict the nuisance or to be free from it, respectively. These cases sometimes lead to costly nuisance lawsuits, requiring a judge to weigh in to resolve the ambiguous allocation of rights.

Similar issues surface in other environmental settings. There have been lawsuits over the exact definitions of "discharge", "fill material", "navigable waterway", "flood or flood waters", and "acceptable noise". In other cases, there is ambiguity over which laws apply and over who decides. The considerations above result in imperfectly defined property rights and therefore inhibit Coasian bargaining, at least before precedent has been established through the courts.

Furthermore, property rights are not immutable. Rewilding is one example. Large grazers were introduced in many nature reserves in Western Europe to keep landscapes open. Large predators are now being introduced to prevent overgrazing. As these wolves also kill the occasional sheep, the European Union now recommends full compensation for lost livestock. A customary privilege of safety for farm animals has been replaced by an explicit right to compensation.

Environmental standards, in particular, are generally tightened over time. This trend implies that rights to pollute tend to disappear and rights to be free of pollution tend to appear. Loosened regulations have the opposite effect. When governments tighten environmental regulations, compensation may be offered to the companies newly deemed to be polluters. Recent examples include more stringent standards for nitrate emissions and odour from farms, as well as pesticide bans, with politicians promising to make farmers whole. This is not Coasian bargaining -- which is bargaining given initial property rights\textemdash but rather bargaining over the assignment of initial property right -- meta-Coase bargaining, if you will.

Social norms can also play a role in defining the terms on which externalities are bargained over. Protests against and boycotts of large polluters have a long history. Like property rights, social norms can also shift over time with evolving standards of what constitutes a permissible nuisance as opposed to unacceptable behavior. Examples include shifting social norms around public littering or the disposal of dog waste. Decades ago, individuals had the "right" to dispose of waste in public spaces creating disamenities for others. But changing attitudes, sometimes driven by deliberate messaging campaigns and often codified in local laws and ordinances, shifted so that instead people now generally internalize at least some of the costs of responsible waste disposal while in public areas. As another example, the Stop the Child Murder movement in the Netherlands ensured that road safety standards were enforced. Similarly, China's Center for Legal Assistance to Pollution Victims focuses on the enforcement of existing environmental legislation. Lawsuits against emitters of carbon dioxide seek to establish a legal right to an unchanging climate.

At least in the lab, the Coase Theorem holds under its original assumptions -- and that it sometimes holds under conditions that are less strict.