Externalities:  What are they?

Externalities are an important piece of the real-world optimization problem consumers and producers solve daily.  Most of the general public is aware of this concept but has not yet seen it formally discussed.  In short, an externality is present whenever the well-being of a consumer (or producer) is directly affected by the actions of another economic agent.  To tie an example to this, think pollution.  A factory produces paper, and through the process gives off harmful chemicals.  This expulsion of chemicals into the air could be considered an externality for those who are directly affected by the diminishing air quality (think neighborhood).  The easiest examples to think of in regards to externalities typically involve a harm imposed from one person to another.  However, positive externalities can exist as well.  Let’s suppose we are sitting in a classroom and the student sitting next to you is wearing cologne/perfume, and the scent provides you with some extra benefit of attending the lecture that day.  However, if that student goes too heavy on the cologne/perfume, this positive externality could quickly turn negative.  Examples of positive and negative externalities are plentiful, and thinking about them in daily occurrences can help you familiarize yourself with the concept and apply this thinking to how you evaluate certain political and economic scenarios.

What we have introduced so far is an effect on the overall utility (well-being) of another economic agent, these are considered true externalities.  However, there also exists another form of an externality, namely a pecuniary externality.  To introduce an example of this concept, let’s consider a residential complex that is near a local dump.  This area would produce a foul-smelling aroma that invades the noses of the people in and around the complex.  Upon building the development, the developer must consider the negative effect of poor-smelling air in the price of the homes.  What could have otherwise been a complex with houses averaging $500,000 can quickly turn into houses that average $400,000.  This 20% drop in housing prices is the externality being accounted for within the price mechanism.  Now, let’s suppose the dump ceases to exist and is replaced with a beautiful park.  This park would have the opposite effect through the price mechanism and could cause the property value in the complex to increase.  This increase in costs could be a positive pecuniary externality for those who already own the homes, but a negative pecuniary externality for someone who was considering moving into the area as it could effectively price them out of the market.

Externality Optimization

(If math scares you, we’re keeping it at a minimum here and only when completely necessary to get optimization outcomes across).  We can begin by analyzing the typical externality problem, in which there are two consumers (1,2) where consumer 1 imposes an externality on consumer 2.  A good illustrative case of this would be person 1 playing loud music while person 2 is trying to study.  The act of playing loud music imposes no extra cost on person 1 (the costs of electricity and instruments are considered in the consumption bundle separately from the externality), but negatively affects persons 2’s ability to focus, which is not considered within the price mechanism.  This lack of account into the price mechanism shows us that we are dealing with a true externality, as opposed to a pecuniary one.  Person 1, being the decider of how much of the externality to produce, chooses to produce this externality until his marginal utility is equal to zero (the benefit from producing the next unit of the externality would give him a negative amount of utility), call this value h.  In this basic example, even though person 2 is negatively affected, he has no say in the amount of loud music that person 1 plays.  Now, what would be the socially optimal level of production of the externality?  This question can be solved by summing up the utility functions of each individual and recalculating the optimization problem.  Here we find the optimal level of the externality is where the marginal utility of person 1 equals the negative of the marginal utility of person 2 (x1′ = -x2′).  This can be thought of as the marginal benefit of person 1 equaling the marginal damage to person 2.  If we consider the level of the externality produced when solving for the socially (Pareto) optimal example to be h*, we can now compare the two levels of the externality we’ve obtained.

Intuitively, we can deduce that the level of h produced when consumer 1 is optimizing will be higher than when we are optimizing social welfare.  Rearranging the socially optimal example would give us x1′ + x2′ = 0, compared with our original equation of x1′ = 0.  Intuition can lead us to the conclusion that the level of the externality produced in the x1′ = 0 case will be higher because he is not considering the disutility of person 2.  Thus, we have, in the case of a negative externality, an overproduction of harm that is not socially optimal.  Conversely, in the case of a positive externality, we see that the individual producing the externality would undersupply the benefit, and we would again have a non-optimal production of the externality.  So, how then do we solve this problem of optimally producing externalities?

Taxes and Subsidies

The traditional solution to problems of this nature typically revolves around the imposition of government force through either taxes (in the case of a negative externality) or subsidies (for positive ones).  Taxes can be imposed to deal with problems like this by causing the producer to bear some of the cost of producing the externality through a maximum allowable amount.  Conversely, a negative tax (subsidy) can be imposed to induce the production of more of the externality in the case that the socially optimal production is greater than what would be produced privately.  The exact value of these taxes is difficult for the government to calculate and enforce.  However, Pigouvian Taxation offers us some insight as to what these solutions would look like.

The imposition of a tax on the producer (person 1) would now be accounted for in their optimization problem where there is an interaction between the per unit tax and the amount of externality produced (t*h).  Solving the first order condition as we did previously will show us that the tax imposed should be equal to the disutility of person 2.  This will give us the Pareto Optimal outcome reached before (h*).  Thinking about this in the standard supply and demand framework allows us to consider the tax as a horizontal threshold equal to the disutility of person two, an upward sloping line.  With this, person 1 will not want to produce beyond the point of intersection of their two curves, for the tax would cause person 1’s marginal utility to fall below zero, by giving them utility that falls below the horizontal tax line.  If this remains unclear, let’s consider a real-world example of something like this.  Let’s assign a tax of $2 for the sake of simplicity, which then means the marginal utility of person 2, referred to as MU2, is also equal to 2 upon optimizing.  Because of the concavity of the utility function, we have the MU1 increasing at a decreasing rate.  Since we are working with the situation where the MU1 = MU2 = Tax, that implies that MU1 = 2.  Because MU1 is increasing at a decreasing rate, each unit of the externality produced beyond this point will increase person 1’s utility by a smaller amount than the unit before.  This concept is known as the Law of Diminishing Marginal Utility.  To keep the numbers easy, let’s say the level of harm (externality) produced at the socially optimal point is h* = 2.  If person 1 did not have the constraint of the tax, he would continue to produce the externality until his MU = 0, which would produce more of the externality than socially optimal (ex. h =4).  Therefore, the tax imposed has effectively reduced the amount of the externality produced to the socially optimal level by forcing the producer to incur the costs of his damage to society.  This same situation can be paralleled with a positive externality where we replace the tax with a subsidy to promote the socially optimal behavior.

While this is easy to see how a tax such as this would work in theory, it is much more difficult to implement in practice.  First off, we have more than just two individuals that make up society, and generally, there are a larger percentage of people affected by externalities.  This would require us to now sum up the marginal harm or benefit of each individual to discover where we should set the tax.  Discovering this point for one person would be near impossible (how much marginal harm do you receive from the paper factory down the street?).  Now imagine trying to expand this framework to consider thousands or millions of individuals, and we can see how quickly this problem becomes incalculable.  Even if we try to discover the average marginal harm to citizens, there is no way to find a credible interval within which the bounds of the tax should fall.  This is due to many reasons, namely the standard of exaggerating on surveys that try and capture this type of information.

Beyond this, the proper implementation of the tax is essential if it is going to function the way we have laid out theoretically in this section.  It is imperative the tax is applied directly to the externality-producing activity.  What exactly do I mean by this?  A quick analysis of taxing cars, rather than pollution itself, should provide some insight.  Let’s suppose the government imposes a tax on the car itself.  This tax on the production of cars could cause the company to produce fewer cars, but it’s hard to say that it would necessarily reduce the pollution itself.  If we suppose that efficient technology to produce x number of cars is expensive, and the only way the firm can afford to operate this technology is by increasing production to enjoy economies of scale, then this reduction in quantity forced by the tax on vehicles themselves could lead the firm to switch back to its less environmentally friendly production equipment, and thus could lead to more pollution than we started with.  A final thought on this subject will be covered in the concluding remarks.

Property Rights and Coase’s Theorem

In 1991 Ronald H. Coase of the University of Chicago received the Nobel Prize in Economics for one of the most elegant theories I have seen in my short time in graduate school.  The crux of Coase’s analysis falls on well-defined property rights in a society.  This directly addresses the issue in the first case presented in this post, where person 1 held the property rights to pollute and consideration for person 2 was absent.  If we suppose that person 1 held the property rights to the pollution, then we could apply Coase’s Theorem to solve the socially optimal allocation.  With this situation of well-defined property rights, Coase now supposed that person 1 and 2 could engage in negotiation so that both parties could be satisfied.

The bargaining mechanism followed by the parties would be:

  1. The consumer with the property rights to the pollution offers the other a take-it-or-leave-it contract specifying some transfer payment (T) and some level of externality.
  2. If the person accepts the offer, then we arrive at the contractual solution.  However, if the individual/firm decides that solution does not work in their favor then we reach a conditional situation.
    • If the person doing the polluting owns the property rights, then the same solution outlined in the beginning of this post will arise (the polluter will produce until his marginal utility equals zero).
    • If the person being affected by the pollution owns the property rights, then the level of pollution goes towards zero where the producer isn’t allowed to produce any.

Let’s analyze the case where the person polluting owns the property rights first.  Person 2 needs to consider the optimization problem of the polluting person, otherwise, they will bargain for too low of a level of pollution or ask for too large of a transfer payment.   To solve this, person 2 will optimize their utility plus the transfer subject to the utility of person 1 less the transfer which needs to exceed the level of utility gained if they don’t accept the contract, in this case, U(h).  Due to the binding nature of the constraint, the optimization problem will represent the utility of person 1, person 2, with the base level of utility (U(h)), which is optimized at the Pareto Optimal solution, h*, which we solved for in the previous examples.  Rearranging our constraint, we can find that the transfer from person 1 to 2 would be the difference between the new level of utility U(h*) and the minimal level of utility they would accept, U(h).  What this implies is that the payment from person 1 to person 2 will be equal to the level of utility person 1 would receive if there was no bargaining U(h), minus the socially optimal level of utility, U(h*).

The second scenario of a rejected contract is one in which the person being harmed by the externality owns the property rights.  In this situation, we see the constraint change to U(h) – T ≥ U(0), because if the person polluting rejects the contract he will be forced to cease all production of the externality.  Going through and solving this problem will give us the same solution we derived in the Pareto Optimal situation of maximizing social welfare, and the same level of the externality produced in the previous example where the person doing the polluting held the rights.  However, due to the low level of utility received from the producer (person 1) if he rejects the contract, person 2 can require a higher transfer payment to deal with the externality.  With this, we can see the elegance of this solution.  The Coase Theorem shows that regardless of who holds the rights, as long as they are well defined, and the parties involved can bargain, we arrive at the socially optimal solution.  This showcases that individuals can bargain amongst themselves and reach the same social optimum as could be reached through much costlier government involvement.  Here, the government still has a very important role in defining these property rights and doing so clearly.

There are issues involved with the assumptions of the theory, namely that it is costless to bargain, and that each party has the same bargaining capabilities, which is obviously not the case when considering individuals being harmed by large corporations.  Overall, however, this theory gave the economic world a valuable insight into the study of transactions costs and property rights, and how defining these rights effectively can lead to individuals voluntarily engaging in a negotiation that can leave both parties walking away happy.

Final Thoughts

Externalities are crucial to understanding how actions of individuals can impose pain on other individuals, and the study of these affects can give us insight on how to solve these problems most efficiently.  Coase Theorem gives us a clean solution as to how individuals could solve problems for themselves, at least on the small scale.  For example, disputes between neighbors on noise pollution or trash could be solved to a socially optimal amount by allowing the parties to engage in discussion and bargaining with one another without involving the court system directly.

To my point earlier about the car situation:  Let’s consider a tax placed on cars to try and reduce carbon emissions.  For ease of discussion let’s say that every car on the road needs to average 25 miles per gallon.  This sounds great when considering the harm reduction occurring because people will either be driving more fuel-efficient cars, carpooling, or taking public transportation.  However, what are some of the unintended consequences?  Well, if we consider the business side of things, we all buy goods that are shipped at some point via ground transportation.  These trucking companies would now need to comply with the environmental regulation, causing them to buy new trucks and, for the sake of this example, reduce the size of the loads they carry with each trip.  This can harm consumers in multiple ways.  The price of the new trucks will not be borne entirely by the company; it will be passed onto consumers in the form of higher prices – in this case an increase in shipping costs.  Along with this, it would be reasonable to expect that shipping times would also increase because they may only be able to afford 3 regulation-compliant trucks whereas they previously had 8 now ineligible vehicles for transportation.  Couple that with less cargo on each trip and we can see wait times for items double.  Another quick side note about the corporate effects, who do you think is affected disproportionately by this regulation?  That’s right BIG BUSINESS!  Just kidding…big businesses are actually in favor of these types of regulation because they can afford to upend their current practices and adapt to the new regulation.  It’s the smaller companies that will struggle to find the capital necessary to restock their fleet and will be forced to go under, thus giving the bigger companies a larger share of the market.

Well, I hope you have enjoyed my first ever blog post.  I appreciate all comments, constructive, destructive, or otherwise.  I hope it was an insightful lesson into the world of externalities, with a side discussion of the pecuniary externality imposed on society by taxes and regulation.

“The law of property determines who owns something, but the market determines how it will be used.” – Ronald H. Coase