The Coase theorem states that in a situation in which a negative externality exists, putting a price on the externality will have the same effect on behavior regardless of which party bears the cost. It applies only if there are no transaction costs. The theorem is important in such areas as environmental policy because it suggests that there are multiple ways to address negative externalities such as pollution. Ronald Coase’s proposal of the theorem in his 1960 paper, *The Problem of Social Cost*, led the economic community to reevaluate its reliance on quantity regulation and Pigouvian taxes as the only tools to reduce negative externalities.

To understand the Coase theorem, it may be best to illustrate with an example. Consider two college roommates, Bob and Carl. Bob is in a difficult class and stays up late studying in their room. The bright light that Bob uses to read gives Carl a headache and prevents him from sleeping.

If Bob uses the light for x hours at night, he derives 24x – x^{2} units of utility from the higher grade that he receives in the class. It costs him 14x units—each hour he stays up does 14 units of damage, a quantity which represents his sleep deprivation, the actual cost of running the light and other factors. As long as Bob gets more utility from an hour of using the light than it costs him, he will not turn off the light.

The value that he gets from each additional hour of light is called the marginal value, and it is found by taking the derivative of the utility function. Bob’s marginal utility is 24-2x. This value diminishes with each additional hour of light, and he will only keep the light on until the marginal value of an hour of light is 14, which occurs after five hours.

Carl also has a utility function, but for him the light has a negative effect. If the light is on for x hours, he experiences 6x units of harm. He can deal with this in one of two ways.

One possibility is that Carl tells Bob that he dislikes the light and asks Bob to compensate him for the inconvenience of having it on. If Bob agrees, he will do extra chores that will give Carl 6 units of utility per hour that he uses the light, while Bob will lose 6 units per hour of light from doing them. This increases the cost to Bob of each hour of light from 14 to 20. His marginal value is now equal to his marginal cost after two hours, so he uses two hours of light.

The second possibility is that Carl decides that if he wants darkness in order to sleep, he must give up something to get it. He figures out that the maximum number of hours that Bob could have the light on per day is 12 hours, which is where Bob’s marginal utility is zero, and offers to pay him 6 units of utility for every hour of those 12 that the light is not on. If Bob uses x hours of light, he now gets an additional 6*(12 - x) units of utility. His new utility function is 24x – x^{2} + 6*(12 - x) = 72 + 18x – x^{2}, so his marginal utility is given by 18 – 2x. He still incurs a cost of 14 per hour, so he uses two hours of light.

From a purely mathematical standpoint, it does not matter whether Bob pays Carl compensation for his discomfort, or Carl pays Bob to turn off the light. This is the insight of the Coase theorem. It shattered the traditional theories of externality policy, which held that the only ways to curb negative externalities were to make laws against them or to force the creator of the externality to pay all of the costs associated with it.

In some cases, the Coase theorem does not apply because of transaction costs. For example, if the light were coming from outside, and Carl had to organize a group of students to ask the university to turn it off, then the effort that he put into organizing would be a transaction cost. He would be willing to offer less to turn off the light, so the effect on the externality would be smaller than if the university paid each student.

When there are no transaction costs, the Coase theorem introduces new possibilities and also new problems. The policy that is put into effect makes a statement about the values of the group. If Bob pays Carl, it implies that Carl has a right to darkness, but if Carl pays Bob, it means that Bob has a right to stay up studying. The prioritization of conflicting rights is a concern even though, as the Coase theorem shows, the numerical outcome is the same.