Lecture 22: Externalities

In our analysis of consumer theory, producer theory, and market equilibrium, we implicitly made an assumption about costs and benefits: that they were purely private. If a consumer pays €4 for an apple at a local orchard and consumes the apple, the only benefit is their enjoyment of the apple, and the only cost is the €4 they paid for it. Likewise, the only benefit for the orchard is the €4 of revenue they received, and the only costs were the costs they paid growing the apple. If the consumer’s marginal rate of substitution was €5 per apple, and firm’s cost of production was €2 per apple, then the consumer enjoyed a surplus of €5 - €4 = €1, and the orchard had a producer surplus of €4 - €2 = €2; so the total surplus generated by the transaction would have been €1 + €2 = €3.

However, economic activity often impacts others beyond the consumers and producers who are party to a transaction. The orchard, for example, might use a pesticide which leaches into nearby streams, causing negative health effects in the surrounding communities. We call this an external cost or negative externality If so, the total benefit to society from the transaction won’t have been €3, but rather something less than that. Indeed, if the benefit to a transaction is small enough, and the external cost is large enough, it would be better for some transactions not to occur at all!

In this lecture, we’ll analyze two different types of externalities:

• Market externalities, in which the production and consumption of a good has a general impact on society. For example, greenhouse gases from transportation and energy production don’t impact specific individuals, but they do have a massive global impact.
• Individual externalities, in which the actions of one agent impact the welfare of another agent. The canonical example of this is a steel mill which operates upstream from a fishery, and pollutes the water

In each case, we’ll take the same approach:

1. use the models we’ve developed over the course of the last nine weeks to determine what quantity of the good would be produced in (free market) equilibrium
2. use a model of externalities to determine the efficient or optimal quantity of the good; and
3. analyze ways to achieve the efficient quantity

Let’s start by looking at a situation in which competitive markets do not, in fact, produce the welfare-maximizing quantity, and how a carefully targeted tax policy might move the situation closer to an efficient outcome.

Market externalities in the supply and demand model

Let’s consider as our first model the competitive model of supply and demand, which we developed in Module 6. In particular, in Lecture 20, we analyzed the way in which the demand curve represented the marginal benefit ($MB$) to consumers from a good, and the supply curve represented the marginal cost ($MC$) to producers. If all consumers and producers are price takers, then each consumer buys up to the point where their marginal benefit is equal to the price ($MB = P$), and each firm produces as up to the point where the price equals their marginal cost ($P = MC$), and since they all face the same price, in equilibrium the quantity produced is the one which sets $MB = MC$.

As described above, implicit in this analysis was the assumption that the only relevant benefits were the ones accruing to consumers from their private consumption, and the only relevant costs were the ones borne by the firms as part of their production processes. However, suppose there are benefits or costs which are external to the market transaction…what then?

In such cases, we need to draw a distinction between private benefits and costs (the benefits to the consumers, the costs to the firms), and external benefits and costs to people outside the immediate transaction. The sum of private and external benefits and costs are called social benefits and costs.

The diagram below extends the kind of analysis we did in Lecture 20. At first, it looks just the same: in the left graph, we see the gross benefit to consumers if the market produces $Q$ units of a good, as the area under the marginal benefit curve; in the middle, it shows the total cost to firms, as the area under the marginal cost curve; and in the right graph, it brings these two together to show the total welfare (benefit to consumers minus cost to firms) as the area below the MB curve and above the MC curve, representing the net benefit of each unit of the good produced (i.e. its marginal benefit minus its marginal cost):

Now, if you check the box marked “show the case of a negative externality,” you can see how the existence of external costs affects the analysis. The firm’s private marginal cost is labeled $PMC$. A brown line appears; this is the marginal external cost ($MEC$). The sum of the $PMC$ and $MEC$ is the social marginal cost, $SMC$. This line represents all of the costs generated by firms producing this good, including the costs they face ($PMC$) and the costs they impose on others via pollution or some other negative externality ($MEC$). As you can see from the right-hand graph, the socially optimal quantity lies at the intersection of the $MB$ and $SMC$ curves, because we want to produce the good up until the point where the marginal benefit to consumers is just offset by all the marginal costs of production. If we were to ignore these costs and produce at the point where $MB = PMC$, the last units produced would have a benefit that was great enough to offset the private costs, but not the additional external costs.

Solution: Pigovian Taxes

The fundamental problem here is that because market actors only make their decisions based on their own private costs and benefits, the market results in an overproduction of a good with a negative externality. (Likewise, it also results in an underproduction of a good with a positive externality, like those beautiful flowers in the front yard.) One solution to this is to adjust the incentives faced by firms so that they make the “right” decision, and produce the socially optimal quantity.

One way to achieve this would simply be to issue an edict that a certain quantity of the good (i.e. the socially optimal quantity) be produced. However, this would assume that some institutional authority knew exactly how much of a good was optimal; which is often not the case, as government’s don’t accurately know every consumer’s marginal benefit and every firm’s marginal cost.

Another method, suggested by the economist Arthur Pigou, was to levy a tax which would effectively force the firms to pay the external costs they would otherwise avoid: in other words, charge them for the marginal external cost they are imposing on others. If they have to pay that cost, the reasoning goes, they will stop producing when the price (plus tax) equals their private cost plus the external cost: \(P + \text{tax} = PMC + MEC\) You can see how this works in the graph below. The quantity $Q^E$ represents the market equilibrium quantity; if you check the box saying “show negative externality,” the quantity $Q^\star$ represents the socially optimal quantity. If you levy a tax, by adjusting the slider, you can reduce the equilibrium quantity bought and sold in this market. And if you levy just the right amount of tax, you can achieve the socially optimal quantity:

Note that, as in this case, the amount of external cost may depend on how much of the good is being produced. The trick, then, is to choose a tax scheme so that, as much as possible, the amount of tax paid per unit is equal to the marginal external cost at the optimal quantity. In this case, the MEC at the optimal quantity of $Q^\star = 50$ is $€20$ per unit; so a tax of that amount will result in the efficient quantity being produced.

Some important caveats, and some interesting potential solutions

The above analysis is temptingly simple: just figure out about how much damage in generated in the production process of a good, and tax the good accordingly. However, this “solution” raises some problems of its own.

For example, we know that driving internal combustion automobiles generates greenhouse gases, which in turn cause climate change. What should we tax, if we’re interested in reducing this pollution? Should we tax cars that run on gasoline, and/or subsidize electric cars? Should we tax gasoline itself? Should we subsidize carpools or public transportation, even if they run on gasoline, because the amount of greenhouse gases generated per person is less?

As a general rule, it’s better to tax the actual externality itself than something which is only correlated with the externality. So, for example, it’s better to tax gasoline than cars that run on gasoline, because the thing you want to make more expensive is the driving of cars, not the having of cars.

Furthermore, taxing the actual externality can provide an incentive to invest in less-polluting ways to do things. For example, cement is a huge source of greenhouse gases, estimated to account for as much as 9% of all human CO2 emissions. As described in this article, though, new, more sustainable techniques for creating cement are being developed, which could dramatically decrease the emissions from cement production. However, suppose these techniques are a bit more expensive than other, more polluting methods. If you tax cement itself, lowering the price cement manufacturers get for their product, you might make it cost prohibitive to invest in more sustainable manufacturing processes, ending up with less cement but more pollution! A particularly tricky homework problem will walk you through an analysis of how to solve this kind of problem.

Individual externalities: a steel mill and a fishery

Let’s now think about a less general kind of externality: one in which one agent’s actions directly affect another’s well-being.

The canonical example of this is a steel mill which lies upstream along a river from a fishery, and as part of its production process it dumps toxic waste in the river, harming the fishery. The more steel the mill produces, the more polluted the water becomes. Let’s say that each ton of steel produces waste which causes €2 of damage to the fishery.

The steel mill has a cost function given by \(c_S(S) = \frac{S^2}{200}\) and the fishery has a cost function given by \(c_F(F,S) = \frac{F^2}{10}+2S\) The $S^2/200$ is the total private cost to the steel mill from producing $S$ tons of steel. The $F^2/10$ is the total private cost to the fishery of producing fish. The $2S$ is the total external cost imposed by the steel mill on the fishery.

Both the steel mill and the fishery sell their products in competitive markets; steel sells at a price of $p_S = 10$, and fish sell at a price of $p_F = 50$. We can think of this meaning that the marginal benefit of steel to consumers is $€10$ per ton, and the marginal benefit of fish to consumers is $€50$ per fish.

Efficient Outcome

What is the optimal quantity of steel for this economy, if we define “optimal” as meaning the quantity that maximizes total combined profits? These profits are \(\begin{aligned} \Pi(S,F) &= \pi_S(S) & &+ \pi_F(F,S)\\\\ &= p_SS - c_S(S) & &+ p_FF - c_F(F,S)\\\\ &= 10S - \frac{S^2}{200} & &+ 50F - \left[\frac{F^2}{10} + 2S\right] \end{aligned}\) The first-order conditions of this are \(\begin{aligned} \frac{\partial \Pi(S,F)}{\partial S} &= 10 - \frac{S}{100} - 2 = 0 & \Rightarrow S^\star &= 800\\\\ \frac{\partial \Pi(S,F)}{\partial F} &= 50 - \frac{F}{5} = 0 & \Rightarrow F^\star &= 250 \end{aligned}\) With this level of production, the profits of the two fims are \(\begin{aligned} \pi_S &= 10 \times 800 - \frac{800^2}{200} = 4,800\\\\ \pi_F &= 50 \times 250 - \frac{250^2}{10} - 2 \times 800 = 4,650\\\\ \end{aligned}\) so total profits are 9,450.

Equilibrium Outcome

Because the negative externality just affects the fishery’s fixed cost, its production decision remains the same; it will produce $F = 250$. The steel mill will maximize its own profits of \(\pi_S(S) = 10S - \frac{S^2}{200}\) by setting marginal profit equal to zero \(\pi^\prime_S(S) = 10 - \frac{S}{100} = 0 \Rightarrow S = 1000\) Therefore profits for the two firms will be \(\begin{aligned} \pi_S &= 10 \times 1000 - \frac{1000^2}{200} = 5,000\\\\ \pi_F &= 50 \times 250 - \frac{250^2}{10} - 2 \times 1000 = 4,250\\\\ \end{aligned}\) so total profits are 9,250.

Analysis of the Problem

Compared to the social optimum, the equilibrium outcome has 200 less in combined profits: specifically, the steel mill makes 200 more in profits, and the fishery makes 400 less. The steel mill produces up until the point where its marginal benefit (the price, 10) equals its marginal private cost. However, it imposes a cost of 2 for each unit of steel it produces; so once its marginal cost exceeds 8, it’s making less marginal profit than the cost it imposes on the fishery.

Solutions to the Problem

We could, as before, impose a “Pigouvian tax” on the steel mill; but it’s not clear that every steel mill imposes exactly 2 dollars of marginal external cost on a neighbor, so that may not be efficient.

However, there’s an alternative solution: assigning property rights to pollution. Let’s say we give the fishery the right to clean water, and that it can charge the steel mill a price $c$ per ton of steel. Then the two firms’ profit functions become \(\begin{aligned} \pi_S(S) &= 10S - \frac{S^2}{200} - cS\\\\ \pi_F(F,S) &= 50F - \left[\frac{F^2}{10} + 2S\right] + cS = 50F - \frac{F^2}{10} + (c-2)S \end{aligned}\) The fishery has the right to deny the steel mill any production at all; but if it charged any price $c \ge 2$, it would be happy to let the steel mill pollute as much as it liked, because each unit of pollution would earn the fishery an additional $c - 2$ in profits. Therefore the lowest price it would accept per ton of steel would be $c = 2$. This would result in the steel mill perfectly internalizing its externality, so it would produce $S = 800$. It would earn a profit of 3,200 and the fishery would earn a profit of 6,250.

Now let’s suppose we said the steel mill had the right to produce as much as it wanted to, but the steel mill could charge the fishery $c$ for every ton of steel it cut back production below its equilibrium level of 1000. Then the two firms’ profit functions become \(\begin{aligned} \pi_S(S) &= 10S - \frac{S^2}{200} + (1000-S)c\\\\ \pi_F(F,S) &= 50F - \left[\frac{F^2}{10} + 2S\right] - (1000-S)c = 50F - \frac{F^2}{10} - 1000c + (c-2)S \end{aligned}\) We can see that the most the fishery would be willing to pay would be $c = 2$; and if the steel mill could earn 2 for each ton of steel it didn’t produce, it would again choose the optimal quantity of $S = 800$. In this case, the steel mill would earn a profit of 5,200, and the fishery would earn a profit of 4,250.

Coase Theorem

In this case, we can note that the assignment of property rights results in the efficient quantity, regardless of who has the property rights. This is an illustration of the Coase Theorem, which states that, under certain circumstances (including low transaction costs), bargaining between parties can lead to the Pareto efficient outcome. However, it’s worth noting that the solutions weren’t symmetric: assigning the right to clean water to the fishery, while improving overall welfare, hurt the steel mill relative to the equilibrium outcome. Assiging property rights to the steel mill, on the other hand, made the steel mill better off and the fishery no worse off than in the equilibrium outcome. So even though assigning property rights can make both parties better off, it can also make one party much better off and the other slightly less worse off.

Concluding thoughts: the limits of policymaking

In each of these examples, we’ve applied the same general process: find what agents will do in equilibrium, determine the “optimal” outcome, and then try to design a policy that achieves it.

As we’ve seen, there are several places where the potential solution can be difficult to implement:

• Finding the optimal quantity. If there is private information about costs and benefits, it might be difficult to know what the true measure of the externality is, and therefore what the actual optimal quantity of each good is. We can know that driving causes climate change; but that doesn’t answer the important questions of what amount of driving is truly optimal, let alone the question of who should get to drive those miles.
• Implementing targeted policies that actually work. Even if we accept that we know the optimal quantity, it may be difficult or impossible to create a mechanism that achieves it; and even the most well-meaning policies can have unintended consequences.

We’ll talk more about this in class, and I’m planning on adding some fancy new graphs to the above analysis; but that’s enough reading to get you prepared. See you in class!

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