An emission fee levied against polluters will tend to

E M I S S I O N P R I C E S A N D F E E S

IN CHAPTERS 4 AND 5, WE SAW THAT MARKETS CANNOT BE RELIED ON TO PROVIDE THE

right amount of pollution—markets fail when it comes to public bads. Normally, consumer preferences are communicated to producers through the price system. The basic problem in the case of pollution is that without a price system, polluters do not "see" the damage caused by the pollution they emit. The simplest way to correct this lack of a signal to polluters is to establish a price for pollution. Just as the price of labor signals to a firm to conserve labor inputs, a price on pollution will signal to the firm to cut back on pollution as much as is eco- nomically feasible. The theoretical idea of placing a price on pollution is from the English economist Arthur C. Pigou and dates to the early twentieth century.1 The more practical idea of regulating pollution by charging polluters an emission fee per unit of pollution has its modern origins in the pathbreaking environmental economist, Alan Kneese, who spent virtually his entire professional career at Resources for the Future in Washington, DC.2

I. H O W D O P O L L U T E R S R E S P O N D T O E M I S S I O N F E E S ?

Before turning to how one determines how large or small an emission fee should be, it is important to understand how polluters will respond when charged a specific fee for their emissions. We first define an emission fee:

DEFINITION An emission fee is a price or fee paid by a polluter to a regulatory entity for every unit of emissions the polluter emits.

Consider a firm producing some good (for example, electricity) along with pollution, x. If the emission fee is p and the polluter emits x units of pollution, the payment from the polluter to the regulator is px. To determine how much the polluter will emit, let C(x) be the production costs associated with emitting x units of pollution, holding goods output fixed. One would expect C to decline as x increases—the polluter saves money by emitting pollution. Total costs for the firm (abatement costs and costs of the fee) are given by

TC(x) = C(x) + px (12.1)

241

242 CHAPTER 12 EMISSION PRICES AND FEES

We know something is minimized when its marginal is zero; furthermore, the marginal of a sum is the sum of the marginals (the derivative of a sum is the sum of the derivatives). Therefore, total cost in Eq. (12.1) is minimized when x is chosen so that

p = -MC(x) = MS(x) (12.2)

The interpretation of Eq. (12.2) is that the firm chooses to operate where the marginal savings from emitting one more unit of pollution (which happens to be equal to the mar- ginal cost of reducing pollution by one unit) is equal to the price of pollution. Emitting a smaller amount of pollution would involve abatement costs that are higher than the pol- lution price—lowering emissions raises total costs. Emitting a larger amount of pollution than called for in Eq. (12.2) would mean that abatement costs are lower than the emission fee; thus increasing emissions increases fee payments more rapidly than abatement costs are lowered and is also a bad idea.

Equation (12.2) indicates that when faced with an emissions fee, firms will abate pol- lution up to the point where the marginal cost of abatement (or equivalently, the marginal savings from emitting) is equal to the emissions fee. This point is where the firm's total costs are lowest.

With multiple firms, each firm will abate to the point where the marginal cost of abatement is equal to the emissions fee. Thus the equimarginal principal will automati- cally hold—each firm sets the marginal cost of abatement equal to the same number (the fee), and thus the marginal cost is the same for all firms.

This is illustrated in Figure 12.1 for two firms—Riley Wreckers and Tucker Tools. In the left-hand panel of the figure, marginal savings to the firm are shown as a function of emissions: by emitting one more unit, there is a cost savings to each of the firms. The right panel shows exactly the same thing, except from the point of view of abatement and the marginal cost of abatement. With no pollution regulations, there will be no abatement and emissions will be at the point where marginal savings is zero: £T for Tucker Tools and ER for Riley Wreckers.

to ra Dl ro z

p

— Tucker Tools

\ V<— MST

\ \ MSR

V

\ \ \ \ \ : V : \ • \

+-• o

c

A3 Z

/ /

/-<- /

/ /

/ /

S .'

-MCT

/**—MCR

ET ET ER ER Emissions AT AR

(a)

Abatement

(b)

FIGURE 12.1 Firm choice of emissions/abatement when faced with emission fee, p.

—*

Pigovian Fees 243

With an emission fee equal top, the two firms cut back on emissions (increase abate- ment) to the points £.,. and ER (respectively, A,, and AR) for the two firms. Firms operate where the marginal cost of abatement/marginal savings from pollution is exactly equal to the emissions fee.

One of the bonuses of an emissions fee is that the equimarginal principle automati- cally holds. Because the two firms face the same emission fee, they will both set their mar- ginal cost of abatement to the same number and thus marginal costs will be equalized between the two firms (or among any number of firms facing the same emissions fee).

The London Congestion Charge

One of the most successful applications of an environmental charge is the charge on automobiles in the centers of congested cities. And one of the most prominent of these is the charge in London, England. Although not a charge on emissions, the congestion charge is still a good example of the use of a tax or fee to correct an externality.

By the 1990s the speed of travel in central London was slower than it was in 1900, before the advent of the automobile. In 2002 the average daytime speed was less than 15 km/hr, compared to more than twice that speed in the less congested night time. In order to address this problem, in 2003 a charge of £5/day (raised to £8/day in 2005) was levied on all vehicles traveling or parking in central London during the day, excluding weekends and holidays. The charge is collected by voluntary payment of the charge by motorists, accompanied by enforcement using cameras and computers that read license plates. Enforcement is only 85% or so effective, yet the £100 penalty for scofflaws is sufficient for the system to work.

Although paying the charge is not popular (a similar system has been proposed for New York but has faced political opposition), a significant reduction in congestion has been achieved. Private car travel in central London is down by a third and daytime travel speeds have increased 20%.

(Source: Leape, 2006.)

P I G O V I A N F E E S

As was mentioned at the beginning of the chapter, approximately a century ago the English economist Arthur C. Pigou argued for the imposition of taxes on generators of pollution. The Pigovian fee, as it has come to be called, is a special kind of emissions fee—an emission fee that is set at the marginal damage of pollution in an attempt to restore Pareto optimality to a situation of market failure:

DEFINITION A Pigovian fee is an emission fee exactly equal to the aggregate marginal damage caused by the emissions when evaluated at the efficient level of pollution.

We examine this definition in more detail, first for the case of a single polluter and then for the case of multiple polluters. In both cases, there are multiple victims of the pollution.

244 CHAPTER 12 EMISSION PRICES AND FEES

A . A S i n g l e P o l l u t e r

Suppose we have a factory generating pollution in the amount x at a cost C(x), with marginal costs, MC(x). Since costs decline as x is increased, marginal costs are actually negative. Another way of thinking about this is in terms of marginal savings—the savings from emitting one more unit of pollution. Of course the marginal savings is the negative of the marginal costs: MS(x) = -MC(x). As in the case of an emission fee, this is exactly the same as the marginal cost of abatement.

Further assume there are N people surrounding the factory and that pollution causes damage. For the time being, assume that people cannot use locational choice to change the amount of pollution they face. Thus there is nothing a person can do to reduce his or her exposure, short of getting the factory to cut back. For person i, the damage from pollution is D.(x), which is positive and increases in x. There are several other ways of interpreting this damage. We could also say that person i benefits from the pollution in the amount B.(x) with benefits negative and decreasing in x. Or, we could say that D.(x) is the willingness to pay to eliminate the pollution. Total damages are given by

D(x)=Z.Di(x) (12.3)

The efficient amount of pollution is the amount that minimizes total costs and damages:

x* minimizes {C(x) + D(x)} (12.4a)

We know that something is minimized when its marginal is zero. Further, the marginal of a sum is equal to the sum of the marginals. Thus we can set the marginal of the quantity in braces in Eq. (12.4a) to zero:

MC(x*) + MD(x*) = 0 (12.4b)

Substituting the marginal version of Eq. (12.3) into Eq. (12.4b) and recognizing that mar- ginal savings is the negative of marginal cost, we obtain

MS(x*) = E. MD.(x*) (12.4c)

In other words, we seek a level of pollution such that the marginal savings to the firm from pollution (-MC) is equal to the marginal damage from pollution over the entire population. Since pollution is a public bad, the aggregate marginal damage (MD) is the vertical sum of the individual marginal damages (MD.).

This situation is illustrated in Figure 12.2 for the case of one polluter and two victims of the pollution. Shown in the lower half of the figure is the marginal cost of pollution. Note that this is negative since every extra unit of pollution the factory is allowed to emit lowers total costs for the factory (up to a limit of course). The marginal savings to the factory is the negative of this and is shown in the first (upper) quadrant. As the factory increases pollution from no emissions at all, savings are initially quite high. When emis- sions are relatively large, the savings from emitting a little more are much smaller. Thus MS(x) is downward sloping.

•^

MD(x)

MDy(X~) MD2(,x)

x (emissions)

Pigovian Fees 245

FIGURE 12.2 Pigovian fee on pollutant emissions with two victims of pollution. MD^x), marginal damage to victim 1; MD2(x), marginal damage to victim 2; MD(x), aggregate marginal damage; MC(x), marginal cost of emitting for the polluter; MS(x), marginal savings from emitting for the polluter; X, pollution levels with no regulation; x*, efficient amount of emissions; p*, Pigovian fee.

Also shown in Figure 12.2 are the marginal damage functions for the two victims of the pollution: MDt(x). Marginal damage is the negative of the demand function for pollution for each of the individuals. Each of the marginal damage schedules is upward sloping. When pollution levels are small, one more unit of pollution causes little damage. When pollution levels are higher, that extra unit causes more damage. Since the pollution is a public bad, aggregate marginal damage, like aggregate demand, is the vertical sum of individual marginal damages. This is also shown in the figure [MD(x)]. The optimal amount of pollution is the x for which MD(x) = MS(x), shown as x* in Figure 12.2. Also shown in Figure 12.2 is the Pigovian fee, p*. If the polluter is charged p* per unit of pollu- tion, the polluter basically sees pollution as priced at p* and thus the polluter will pollute so that price equals marginal cost:

MC(x*) =-p* or MS(x*) = p' (12.5)

The total amount of money the firm pays for the pollution is p*x*. Note that the Pigovian fee is defined as the marginal savings from pollution genera-

tion at the optimal level of pollution. If we are not at the optimum, the Pigovian fee will be neither the current marginal cost of pollution control nor the marginal damage from pollution. Thus the Pigovian fee is not any emission fee; it is the marginal savings from pollution at the optimal pollution level.

B. M u l t i p l e Pol lu ters

We have seen how a Pigovian fee can generate the efficient amount of a public bad. The case we have looked at involves a single polluter. Suppose we have more than one polluter. For the time being, assume we have two polluters. Figure 12.3 illustrates the case for two polluters in which the marginal damage function (MD) is the aggregate damage to all consumers. Shown is the marginal savings to each of two firms from generating pollution. How much pollution should each firm generate and how should the Pigovian fee be set to support that level of pollution?

An aggregate marginal savings function for a group of polluters indicates what the marginal savings will be if the total amount of pollution increases by one unit. This

246 CHAPTER 12 EMISSION PRICES AND FEES

MD(x)

FIGURE 12.3 The case of two polluters. MSX (x), marginal savings from emitting firm 1; MS2(x), marginal savings from emitting firm 2; MS(x), aggregate marginal savings from emitting; MD(x), marginal damage from emitting; p*, Pigovian fee; x*, total amount of emissions with Pigovian fee; x*, emissions from firm 1 with Pigovian fee; x*. emissions from firm 2 with Pigovian fee.

Emissions(x)

depends, of course, on what assumptions are being made about how the total amot of pollution is distributed among the individual polluters. If one polluter is doing all i the pollution control and all other polluters are doing none, the marginal savings will I higher than if the pollution control were more "evenly" distributed among the polluters. One way of avoiding this ambiguity is to assume the polluters are sharing the obligation of pollution abatement in an efficient or least-cost manner. The logical (and cost-minimizing) way of apportioning the abatement responsibility is to equate marginal cost of abatement (marginal savings of emitting) among the firms. As we saw in Chapter 4, the equimar- ginal principle calls for the equalizing of marginal costs.

Numerical Example: Suppose the two marginal savings functions in Figure 12.3 are given by:

MS{xJ = 10 - 2 x, MS2(x2) = 5-3x2

(12.6a) (12.6b)

To construct an aggregate marginal savings function, first solve each of these for emissions:

x, = (10 - MS,)/2 x2 = (5 - MS2)/3

(12.7a) (12.7b)

We now ask what is the total emissions (X = x, + x2) for any given MS? By inserting MS into Eqs (12.7) and then summing the two equations one obtains:

X = (10 - MS)/2 + (5 - MS)/3

Which, with a little manipulation, becomes

MS = 8-1.2 X

(12.8a)

(12.8b)

The exact same approach would be taken in constructing an aggregate marginal abatement cost curve.

Returning to Figure 12.3, how do we aggregate the two marginal savings functions to obtain an aggregate? The aggregate marginal savings function is found by horizontally

Fees Versus Subsidies 247

summing the marginal savings for the two firms.3 The aggregate marginal savings curve is the curve that indicates for a particular pollution price, how much pollution each firm would generate. This is the standard way of generating an industry marginal cost func- tion from firm-level marginal cost functions. Thus for any level of the fee, MS(x) tells us how much x in total will be emitted and each MS.(x) tells us how much each firm will contribute to that total. We have constructed MS(x) in such a way that the amount of pol- lution from each firm will sum to the total.

In Figure 12.3, to determine the efficient amount of pollution, we note where the marginal savings curve (MS) intersects the marginal damage curve (MD). That deter- mines the optimal amount of pollution (x*) and the marginal savings to polluters at x*: p*. Thusp* is the correct Pigovian fee. This is shown in Figure 12.3. At that fee level, firm 1 will generate x* and firm 2 will generate x*. Note that each firm operates so that mar- ginal savings from polluting is set equal to the Pigovian fee:

MSi(x*)=p*

Furthermore, by the way in which MS(x) was constructed,

(12.9)

MS(x*)=p' (12.10)

Equation (12.9) illustrates one of the primary virtues of the Pigovian fee: all firms will control pollution at the same level of marginal costs. Marginal costs of pollution control will be equated across all polluters. Firms with different pollution control costs receive the correct signals regarding how much pollution to generate. Those with high control costs will control relatively less than firms with lower control costs.

F E E S V E R S U S S U B S I D I E S

The notion that the source or cause of a pollution problem should pay for cleaning up the problem rings true to most people. In many parts of the world, it is institutionalized in the polluter pays principle. But a logical question arises: is it possible to obtain the same outcome by subsidizing firms to reduce pollution? In the "real" world, is there any danger in providing tax breaks and other subsidies for pollution control, rather than making polluters pay for the pollution they generate? Is it possible to obtain efficiency using a subsidy instead of a fee? This is an important question since subsidies are usually much more politically popular than taxes or fees.

The answer to this is that these two approaches yield different outcomes.4 The tax is efficient, whereas the subsidy can result in too many firms in the industry and thus an inefficient amount of both pollution and the good associated with the pollu- tion. We will consider two cases. The first is for the short run—there is no time for new firms to enter the industry. The second is for the long run—there is time for entry and exit (although exit in the sense of shut-down is always possible, even in the short run).

248 CHAPTER 12 EMISSION PRICES AND FEES

A . T h e S h o r t R u n

Let us consider a competitive industry, producing some good in conjunction with pollti tion. Initially, assume all of the firms in the industry are identical. Under a pollution tax (t), the production costs of a typical firm would be

CT(y,e) = V(y,e) + t e + FC (12.11)

where y is the amount of the good being produced, e is emissions, V(y,e) represents vari- able production costs for producing e and y, and FC is the fixed cost of production. To simplify things, suppose there is a fixed relation between output and emissions—the more you produce, the more you pollute. In particular, suppose emissions are related to output by e = ay where a is a constant. We can rewrite Eq. (12.11) as

C (y,ay) = V(y,ay) + tay + FC (12.12a)

Recognizing that V and CT are now functions of y only, we let TC(y) = Cfy,ay) and VC(y) = V(y,ay) and rewrite Eq. (12.12a) as

TC(y) = VC(y) + tay + FC

This means that marginal production costs are

MC(y) = MVC(y) + at

(12.12b)

(12.13)

So basically, marginal costs are increased by at. Now consider what happens with a subsidy, s. With no attention to pollution control,

a firm might pollute at the level e. With a subsidy, the firm will be paid to reduce emis- sions. If the firm reduces emissions to e, the subsidy payment will be s(e- e). This means that costs will be

TC(y) = VC(y) + FC - s(e - e)

= VC(y) + say + {FC - se)

(12.14)

Note that the term in braces is a fixed cost, consisting of the standard fixed cost plus a lump-sum transfer of se that is independent of the firm's choice of y or e. Thus the variable costs in both cases [Eqs. (12.12b) and (12.12)] are exactly the same. Only the fixed costs are different. Consequently the short-run marginal costs of production will be identical in the two cases and the firm will produce exactly the same amount of pollution and the good. In fact, the marginal costs from Eq. (12.14) are

MC(y) = MVC(y) + as (12.15)

which is exactly the same as Eq. (12.13) except we have an s here instead of a t. Consequently, our first result is that in the case of identical firms in the short run,

Pigovian fees and subsidies yield exactly the same outcome. We should note that this result applies even if there is a more complex relationship between output and emissions

Fees Versus Subsidies 249

than the fixed ratio assumed here. Showing that is more cumbersome mathematically so we omit it here.

We now turn to an industry with heterogeneous firms. This may be because of differ- ent technologies used by different firms due, for instance, to their different ages. This case is best understood graphically. Suppose we have an industry composed of two classes of firms, old firms and new firms. Newer firms may have higher fixed costs but lower vari- able costs. We are concerned with industry behavior in the short run under Pigovian fees and subsidies. Since this is the short run, no new firms may enter. Any firm may, however, choose to produce nothing, shutting down. If a firm produces nothing, the subsidy disap- pears. In other words, we only pay firms to produce less pollution. We do not continue to pay firms if they decide to go out of business.

Figure 12.4 shows the marginal cost curves and average variable costs for these two types of firms. Since this is a short-run analysis, we are not concerned with total costs. The issue is whether prices cover average variable costs and, if they do, production will be at marginal cost equals price. As we saw above, the effect of a tax on marginal cost is identical to the effect of a subsidy on marginal cost—both raise marginal costs relative to the unregulated case. The U subscripts in the figure correspond to average variable cost (AVC) and marginal cost (MC) in the unregulated, pretax, or presubsidy case. The T and S subscripts refer to the case of a Pigovian tax or subsidy, respectively.

i

Pj

Ps Pu

I MCj, MCS ,

* /A / / / / / / / /

" r " " I i I i I i /

i / i 1

/ v / ^ A V C u / f

/ / / / t / / / AVCS / / K * +* M / / • / • / X x t,s

>~ Goods output

?s Pu

MCj, MCS i lMQ

AVCS /

At/Cy

Goods output (a) (b)

FIGURE 12.4 Variable costs for heterogeneous industry, short run. (a) Old firms; (b) newer firms, MCU marginal cost unregulated case; MCS marginal cost, with emission control subsidy; MCV marginal cost, with emission fee; AVCU, average variable cost, unregulated case; AVCS, average variable cost, with emission control subsidy; AVCV average variable cost, with emission fee; pu, goods price, unregulated case; ps goods price, with emission control subsidy: pv goods price, with emission fee.

250 CHAPTER 12 EMISSION PRICES AND FEES

1

Pj

\ SupplyT ..,, /'. /bupplyu

/ „ Demand __/ .-4 Supplys

w Goods output

FIGURE 12.5 Short-run supply and demand, heterogeneous industry, with and without taxes and subsidies. Supply^ goods supply, unregulated case; Supplys, goods supply, with emission control subsidy; SupplyT, goods supply, with emission fee; demand, goods demand; pu, goods price, unregulated case; ps, goods price, with emission control subsidy; p , goods price, with emission fee.

Note in Figure 12.4 that although the tax and the subsidy raise the marginal costs by I same amount, the subsidy lowers average variable cost whereas the tax raises average vari; cost. The reason is that we have assumed the subsidy applies only if the firm is operating.' lump sum se in Eq. (12.14) goes away if the firm shuts down. Thus it counts as a variable cc Fixed costs (FC) are incurred whether or not the firm shuts down (in the short run).

Now we turn to determining what the market price of the good might be under the three regimes. Figure 12.5 traces out the short-run supply functions for this industr for the three cases, unregulated, Pigovian taxes, and subsidies. Also shown is a typic demand function for the good. Recall that firms will operate on the portion of their marginal cost curve that lies above the average variable cost. We can see that with bot the unregulated and subsidy cases, both types of firms are operating, yielding prodii prices of pu and ps, respectively. In the case of the Pigovian tax, only the newer firms operate, yielding product price px. These prices are shown on Figure 12.4. Note that all prices are above average variable costs for the newer firms, which is why they operate in all three cases. However, for the old firms, pT is below AVCT, the average variable cost for the Pigovian tax case. This is why the old firms shut down in this case.

Our conclusions are that taxes and subsidies have different effects in the short run. A subsidy may allow firms to continue operating that would not continue in the case of a tax. Which is efficient? The subsidy requires a lump-sum transfer, which has to be obtained from somewhere. Even more important, the subsidy involves the operation of firms that are really losing money (negative profits). This is not efficient.

B. T h e L o n g R u n

We now turn to the case of the long-run effects of Pigovian taxes and subsidies. We saw in Eq. (12.12)-(12.15) that the effect of a tax or subsidy was to raise marginal costs, but that a subsidy lowered average costs while a tax raised average costs. This applies in the short run as well as in the long run. If we assume a constant-cost industry, all firms will operate at the bottom of their average total cost curve, in long-run equilibrium. Thus the supply sched- ules for the industry will be horizontal and as shown in Figure 12.6b. The result of this is that goods prices will be higher with a Pigovian tax than with a subsidy. Furthermore, there will be more firms in the industry with a subsidy than with a Pigovian tax.

Other than the fact that a subsidy has to come from somewhere, a subsidy is undesir- able because it does not allow the market to communicate the true costs of the product

Imperfect Competition 251

MCTc /

i

X i T

1 i \ 5u

1 1 i \ Ss

D

1 1 1 w

(a) Goods output (b)

QT 9U 9S

Goods output

FIGURE 12.6 Long-run supply and demand, constant cost industry, with and without taxes and subsidies. MCU, marginal cost, unregulated case; MCV marginal cost, with emission control subsidy or emission fee; ATCU, average total cost, unregulated case; ATCS, average total cost, with emission control subsidy; ATCV average total cost, with emisison fee; D, goods demand; Su, long-run supply of good, unregulated case; Ss, long-run supply of goods, with emission control subsidy; ST, long-run supply of goods, with emission fee; qu equilibrium goods output, unregulated case; qs, equilibrium goods output, with emission control subsidy; (c/T, equilibrium goods output, with emission fee.

being consumed to the consumer. To be quite concrete, suppose we are dealing with paper mills producing paper from trees and polluting rivers at the same time (virgin mills). Other mills produce paper from recycled products and we will assume they are pollution free (which is not actually the case). A pollution subsidy to clean up the virgin mills will make virgin paper more attractive (compared to recycled paper) than if the virgin paper manufacturer had to pay a pollution tax. The result is that a subsidy results in the overuse of trees and underuse of recycled paper, compared to the case of a Pigovian tax. If a prod- uct generates pollution, we want consumers to see the full costs associated with producing that product when the consumers decide what to buy and how much to buy.

IV. I M P E R F E C T C O M P E T I T I O N

When markets are not competitive, a host of efficiency problems generally arise, and controlling pollution is no exception. There are two cases we will consider. One concerns

252 CHAPTER 12 EMISSION PRICES AND FEES

a monopolist in some goods market who is also a polluter. For instance, the monopolist may dominate the steel market while also generating smoke. The second case is more likely: a firm that is the only producer of smoke in some appropriate region. In this case we have a monopoly, but a monopoly in the provision of the bad.

The general conclusion that results from the analysis is that when there is market power, an emmisions fee can make matters worse. This is a result in what economists call the theory of "second best." When there is a distortion in an economy, such as monopoly, levels of output and prices will be distorted—not at their efficient levels. In such a case, the best way to correct inefficiencies such as pollution will not be to blindly use prescriptions from theory developed for efficiency. Other, "second best" methods must be used to cor- rect the problems. This is a large area of study in economics. We will consider only the example of monopolists and monopsonists generating pollution.

A . M o n o p o l i s t in t h e G o o d s M a r k e t

Assume the steel industry is a monopoly and it produces smoke.5 There are many smoke producers but only one producer of steel. For simplicity we will assume, as in the previ- ous section, that smoke and steel output are proportional. We know from intermediate microeconomics what the steel mill should do. This is illustrated in Figure 12.7. Shown in Figure 12.7 is a typical demand curve for steel and the associated marginal revenue func- tion. Also shown is the marginal cost of producing steel, ignoring pollution, MCV, and the marginal social cost of producing steel, including pollution, MCr The producer will produce where