Externalities and the Numbers Problem

The purpose of this newsletter is to revitalize interest in price theory. Thus far, the newsletter has been well-received. Nonetheless, we do get criticism. One frequent criticism is related to our argument that the competitive model is potentially useful regardless of the number of firms in the market. A second criticism that we receive is that price theory is really just some sort of free market indoctrination scheme. As a result of these criticisms, I would like to use today’s post to discuss a scenario in which I think the number of firms matter and I would like to provide an example in which price theory might imply more policy intervention than might be conventionally argued.


Most introductory economics courses address the concept of externalities (I’ve discussed this before). Negative externalities occur when economic activity generates external costs that are not internalized by the decision-makers. In order to think through the implications of externalities and why the number of firms might matter, let’s think through a popular example.

Suppose that there is a railroad line constructed and that it runs alongside a farm. For simplicity, let’s assume that there is only one railroad and one farm. When trains run across the railroad tracks, they produce sparks. Since the railroad line runs alongside the farm, these sparks can possibly cause fires and ruin some of the farm’s crops, just from the ordinary operation of the trains. The railroad therefore imposes an externality on the farm since the farm’s output is effected by the output of the railroad and the railroad does not have an incentive to internalize this cost. Nonetheless, as Coase explained, there is potentially a contractual solution to this problem.

If the farm owns the property rights to the land immediately adjacent to the railroad tracks, the farm could make the railroad pay them not to farm on the adjacent land in order to avoid crop damage. Alternatively, the railroad could purchase the land next to the tracks and clear it to avoid damage. On the other hand, if the railroad owns the property rights to the land, the farmer might pay the railroad to install spark arresters on its trains to prevent sparks. In a world with zero transaction costs, the assignment of property rights are unimportant. Of course, the real world is full of transactions costs. Nonetheless, I am going to sweep that issue aside for now to emphasize that when there is one imposer and one victim of an externality, a contractual solution is possible.


Now consider a different example. Suppose that there are a large number of factories located along a river. In the process of production, these factories pollute the river. Downstream from these factories are a large number of firms that catch fish. The factories therefore impose an externality on these downstream firms.

Unlike the railroad spark example in which there is one party imposing the externality and one party that is the victim of the externality, a contractual solution is unlikely to occur. The reason is that there are a number of firms in the fishing industry and there are a large number of factories. The transaction costs associated with organizing the respective groups and negotiating is likely to be prohibitive. As a result, to induce the factories to internalize the costs of polluting the river, policymakers could impose a Pigouvian tax on the factories. In other words, the government could levy a tax on the factories that is equal to the marginal cost of the pollution.

Thus, when there is one imposer and one victim, a contractual solution can address the externality problem. However, when there are a large number of imposers and a large number of victims, a Pigouvian tax is likely needed to internalize the external costs.


To this point, this discussion represents a pretty standard approach to externalities. In fact, this is likely the sort of discussion that one would hear in an Econ 101 course. However, I want to depart from this standard approach and consider a different sort of example. In particular, I want to consider an example in which there is one imposer, but many victims.

Let’s stick with the previous example of factories and fishermen. However, now suppose that there is only one factory that pollutes the river. We will maintain the assumption of a large number of firms in the fishing industry. Furthermore, I will make two additional assumptions. The first assumption is that the marginal external cost imposed on the fishing industry is increasing in the production of the fishing industry. In other words, the more the fishing industry produces, the greater the harm imposed by the factory’s pollution at the margin. The second assumption that I will make is that fishing industry output is decreasing in the amount of factory output. Put differently, the more that the factory produces, the more pollution it creates. As a result, the fishing industry will produce less output as pollution rises. This could be because it is harder to fish in polluted waters or because there aren’t as many fish that survive in polluted waters or that the number of firms engaged in fishing declines as a result of the pollution.

Since there is one imposer and many victims, a contractual solution might still be prohibitive since the fishing industry still faces the same organizational costs and some of the same negotiation costs. Thus, one might be inclined to conclude that the optimal solution is a Pigouvian tax solution in which the externality imposer is taxed at a rate equal to the marginal external cost. However, as Ron Batchelder and Earl Thompson demonstrate, this is not the case. This reason is simple. Since (a) the marginal external cost is a function of the production of the fishing industry, and (b) the fishing industry produces less output when there are greater amounts of pollution, the factory could actually reduce its tax bill by polluting more! The reason is that when the factory increases production and therefore pollution, the fishing industry reduces its production. Since the marginal external cost is increasing in the production of the fishing industry, a decline in the production of the fishing industry produces a decline in the marginal external cost and therefore a decline in the tax imposed on the factory.

It is important to note that this result is entirely driven by the fact that there is one externality imposer. In a world with many externality imposers, one factory’s production would have no effect on the production of the fishing industry since it would only have a small effect on the total production of all factories. Thus, even with the same assumptions about the external costs, a standard Pigouvian tax would optimal when there are many imposers and many victims of an externality. It is the existence of single imposer that renders the standard Pigouvian tax suboptimal.

I should also note that when there is one imposer, there is an additional extreme scenario that one might want to consider. In the case of one imposer that I just described, the factory could reduce its tax bill by producing more output. However, if this is a possibility, the factory might want to behave strategically by producing so much output (and so much pollution) that it drives the fishing industry out of the river entirely. According to the tax scheme just described, this would eliminate the factory’s tax liability (at least as it pertains to pollution).

This is, of course, an extreme example and it is questionable whether any firm would be willing to pursue such a strategy. However, something similar to a hold-up problem might emerge, as I discussed in my post on the need for a theory of publicly-provided goods. A firm might buy land adjacent to the river and threaten to open a factory that pollutes the river unless the fishing industry buys back from the land from the firm at a premium. Having a standard Pigouvian tax in place will not solve this problem for the reasons I outlined above. Nonetheless, these sorts of costly interactions is what we as a society would typically want to avoid. As a result, as Thompson outlined here, it might be optimal in these sorts of scenarios to enact zoning laws that prevent factories from locating adjacent to the river. Or, in cases in which the factory needs to be located by the water, the government might impose direct regulation on the amount of pollution.

The purpose of this thought exercise was to demonstrate that (a) there are times when the number of firms matters for economic analysis, and (b) a serious price theorist might be able to explain interventionist policies — even those typically frowned upon by other economists. Nonetheless, one might wonder about the empirical relevance of this thought experiment. In lieu of empirical evidence, I will leave you with an anecdote.

When I first moved to Mississippi, I lived in a neighborhood that was outside the city limits. Located in the county, the neighborhood was not subject to any zoning restrictions enacted by the city. In fact, the neighborhood was governed by the county, which was fairly indifferent about zoning. There was a segment of my neighborhood that was undeveloped. One year, a developer purchased this land and announced his intention to build a very large, multi-story assisted living facility in this undeveloped part of the neighborhood. The residents of the neighborhood were not happy about this. They appealed to the county to prevent the construction, but to no avail. As a result, the homeowner’s association held a special meeting to vote on whether the HOA should buy back the land from the developer to prevent the construction. The plan was approved, the developer sold the land and built the facility in a more appropriate location on the other side of town.

Of course, we can debate whether this was truly an externality and how generalizable this experience is (or would be) in the absence of zoning. Nevertheless, consistent with the thought experiment above, this entire problem could have been avoided through zoning restrictions. Instead, a lot of time, energy, and money were spent with the end result being a redistribution from the neighborhood to the developer.

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