Open economies review 16: 23–31, 2005c© 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

A Theoretical Inefficiency in the InternationalMarketing of Tradable Global Warming EmissionPermits

ROBERT E. KOHN∗ Robert.Kohn.49@post.harvard.eduDepartment of Economics, Southern Illinois University at Edwardsville, Edwardsville, IL 62026, USA

Key words: non-uniform Pigouvian taxes, global pollutant, global warming, emission permits

JEL Classification Numbers: F02, Q4, Q25

Abstract

When the disutility of global pollution is measured in units of a numeraire good that is polluting inconsumption, the marginal rates of substitution in consumption are generally weighted differentlyin each country’s calculation of marginal damages. In that case, it is inefficient to control globalwarming by tradable emission permits, which in theory trade at a unique international equilibriumprice. This conclusion is derived from a model of Shibata (1996) in which consumption is pervasivelypolluting in global warming gases. Shibata dubiously based the inefficiency he discovered on thereciprocal external damages of global warming emissions rather than on the nullibicity he posedfor a non-polluting numeraire good.

The Kyoto Protocol on controlling global warming envisages an arbitrary cap ontotal emissions of carbon dioxide. After permits for this quantity of emissionsare allotted to existing sources, they may be bought and sold on an internationalmarket. Because of the “advantages of relatively cheap emission reductions indeveloping countries,’’ Harrison (2002, pp. 311, 312, 352) argues that there aresubstantial savings to be gained from emission trading over conventional com-mand and control regulations. However, the economic gains from trading arelimited to savings in total abatement costs and exclude the additional benefitsthat accrue when the total quantity of allotted permits is Pareto optimal ratherthan arbitrary. In the case of global warming control, Pareto optimality may intheory not be attainable with tradable emission permits. Hirofumi Shibata (1996,p. 298) contends that when permits to pollute are bought and sold in a singleworldwide market and therefore at a unique international equilibrium price, inorder to control “reciprocal external diseconomies such as global warming,’’ theequilibrium market price is not efficient. The problem with reciprocal external

∗Present address: 52 Broadview Drive, Clayton, MO 63105-3029.

24 KOHN

diseconomies was originally explained by Tietenberg (1973, p. 520) in the con-text of Pigouvian taxation:

“[F]or firms that are adversely affected by their own pollution, the efficient taxrate is equal to total marginal social damage minus the marginal social dam-age already internalized. Accordingly, any firm that has already internalizedpart of the damage caused by its waste products should, in an efficient allo-cation of resources, face a lower tax rate than firms that have not internalizedany of the damage their pollution is causing. Hence the use of a single taxrate in this case would not lead to an efficient allocation of resources.’’

That the efficient tax rate for the same pollutant would vary across industriesis mathematically simulated in Kohn (1998, pp. 58–59). However, economistshave generally agreed with Baumol and Oates that the difference in tax rates dueto reflexivity “will typically be miniscule, and since it complicates the analysisconsiderably, we have chosen to ignore it.’’ Shibata (1966, p. 314) not only doesnot ignore the possibility that “the polluter will adjust his own emission so asto reduce the damages that he otherwise inflicts upon himself,’’ but extendsthe reflexivity concept to entire countries that “voluntarily undertake at leastsome direct or indirect measures to reduce their emission of pollutants,’’ andthereby internalize, on behalf of their citizens the damages that in reflexivitytheory they would themselves internalize. Shibata’s interpretation is unorthodoxand largely unrealistic, because it is generally the case, though Shibata (1996,p. 315) dismisses it, that polluters “behave as if they disregarded the disutilitiesthat their own emissions impose upon themselves.’’ Nevertheless, Shibata’sconsequent insight about the inefficiency of a single rate Pigouvian tax or singleequilibrium price of permits is correct, and, apparently unaware of it himself,actually follows from his own special model, though for a very different reasonthan he offers.

The present paper provides an alternative explanation for the inefficiency oftradable global warming permits that economists appear to have overlooked.The basic concept is still Shibata’s for the model that demonstrates the ineffi-ciency is his own in that it focuses on the case in which the numeraire good,against which the marginal disutility of global warming is measured, is neces-sarily pollutive in consumption. Had Shibata worked through the mathemat-ics that his model dictates, he would surely have discovered this alternative,more substantive explanation for the inefficiency of tradable permits. Shibata’s(1996) model, which he restates in his 2003 paper, is an elegantly simple one-good model in which pollution adversely affects utility, and the source of thepollution is consumption rather than production. Because consumption real-istically pollutes at a different rate across countries, each country is likely tovalue marginal global pollution damage differently, and therefore the price fora permit to emit a unit of pollution would have to vary across countries. Thiswould not be possible in a single international market for permits.

THEORETICAL INEFFICIENCY IN THE INTERNATIONAL MARKETING 25

This problem of non-uniform marginal global damages across countries doesnot hold when the single good is polluting in production rather than in consump-tion, as in Kohn (2003), nor when, at the margin, the consumption good is equallypolluting in both countries, which is a very special case. Nor does this problemhold for models such as that of Kohn (2001) or of Caplan, Cornes, and Silva(2003) in which pollution adversely affects production functions rather than util-ity functions. Note that the problem of reflexive damages, on which Shibatabases his thesis, though it is generally ignored by economists, allows moregenerally for pollution in production as well as consumption and for pollutiondamage to firms as well as to consumers.

The non-uniformity of the efficient price of permits across countries criticallydepends on the explicit assumption that pollution affects utility and the as-sumption that the numeraire good is, inescapably, polluting in consumption.Although the latter is a strong assumption, and would not hold for a highlycompartmentalized utility function such as that of Parry and Williams (1999,p. 352), most household activities, including leisure, do depend upon on-sitecombustion for heating, cooking and private transportation. This is a significantsource of carbon dioxide for, based on study of the Netherlands by Biesiot andNoorman (1999, pp. 371, 373), approximately 20% of total energy used in thatcountry in 1990 was consumed in the direct combustion of heating and motorfuels by households.

Non-uniform damages across countries are not a problem when carbon diox-ide is controlled by Pigouvian taxes, for in that case each country can tax itsown emitters at the particular rate it deems globally efficient. To the extent,then, that consumption by households is pervasively and non-uniformly pollut-ing, the control of global warming by Pigouvian taxes is first-best, whereas thecontrol by internationally marketable emission permits is second-best. Whetheror not the latter, though second-best, is superior to the arbitrary emissions capapproach of the Kyoto Protocol, is beyond the scope of this paper. It may de-pend upon relative transaction costs of the alternative approaches, which is anissue that Pan and Regemorter (2004) raise. These in turn may be affected bythe fact that the initial allotment of permits increases the wealth of polluters,whereas emission taxes decreases it. Whereas the economics of the presentpaper are driven by principles different from those in Shibata’s paper, the creditstill goes to Shibata for conceiving the intriguing problem associated with aunique equilibrium permit price and for his brilliant insight that, for all practicalpurposes, when it comes to global warming there is no numeriaire good that isnot, to some extent, pollutive.

1. The model in which consumption is polluting

Based on Shibata’s (1996, 2003) model, there is a single good, good Y , anda single global pollutant, g, that is emitted when good Y is consumed. It isassumed here that the good, which may be thought of as household heating or

26 KOHN

automotive transportation, is produced with labor, L, and capital, K , accordingto a linear-homogeneous production function, Y = Y (L , K ), which may differacross countries. There are two countries in this model, one an industrializedcountry with a greater abundance of both inputs, called the North, the other adeveloping country, called the South. The model allows for transfers, Y T , of thegood from the industrial to the developing country, so that the quantity of goodY actually consumed by the North is y∗ = Y ∗ − Y T and the quantity consumedby the South is y = Y + Y T . This transfer can also represent, in whole or inpart, the purchase of emission permits by the North from the South. When bothcountries’ variables are simultaneously represented in this paper, asterisks areused to distinguish the North’s.

The pollutant in this model is carbon dioxide, ambient air concentration unitsof which are emitted during the consumption of good y according to the func-tion, E = E(A, y), where A is the quantity of capital that consumers allocateto abatement. It is assumed that E A is less than zero and that E AA, Ey and Eyy

are greater than zero, where subscripts denote first and second partial deriva-tives with respect to those variables. Examples of abatement capital that canbe used by households are thermal solar collectors and gasoline-electric hy-brid automobile engines. Abatement possibilities such as these likewise figurein Shibata’s (1996) model.

The total emissions of the two countries, E∗ and E , are transboundary andadditive, and, assuming that the two countries comprise the entire world, theglobal concentration of carbon dioxide is the sum, g = E∗ + E . As in Shibata’smodel (1996), pollution is treated both as a flow variable and as an undesirableargument in the community utility function of each country, U = U (y, g), in whichUy is positive and Ug negative. Finally, the input endowments of each countyare Lo and K + A = Ko.

2. Marginal conditions for international economic efficiency

The marginal conditions for efficiency in this model are derived from theLagrangian expression,

£ = U (y∗, g) + λ[U (y, g) − Uo] + µ[y∗ − Y (L∗o, K ∗) + Y T ]

+ υ[y − Y (Lo, K ) − Y T ] + ρ[g − E(A∗, y∗) − E(A, y)]

+ τ [K ∗o − K ∗ − A∗] + δ[Ko − K − A]. (1)

Setting the derivatives of the Lagrangian with respect to K ∗, K , Y T , A∗ and Aequal to zero and solving the resulting equations simultaneously yields

Y ∗K

E∗A

= YK

E A, (2)

THEORETICAL INEFFICIENCY IN THE INTERNATIONAL MARKETING 27

which states that the efficient trade-off in production of good Y for pollution,g, achieved by shifting capital from the production of good Y to abatement,that is, the marginal rate of technical substitution of capital in production, byfirms, for capital in abatement by consumers, is the same in both countries. Incontrast, the marginal rate of technical substitution of capital for labor, that is,YL/YK , is not equal across countries in this model. It is tempting to call YK /E A

the marginal cost of abatement, measured in units of output per unit of pollution.However, abatement occurs during consumption, not production, so that outputY is forgone simply because capital is diverted from production of the good toabatement in the consumption-related activity. The marginal cost of abatementin consumption is actually YK /(−YL E A), which varies across countries.

Setting the derivatives of the Lagrangian with respect to y∗, y, and g equal tozero and solving all eight equations simultaneously yields

U ∗g

U ∗y

(E∗y − E∗

A/Y ∗K ) + Ug

Uy(Ey − E A/YK ) = −1, (3)

which sums the two countries’ efficiently weighted marginal rates of substitu-tion in consumption of good y for the amount of pollution emitted when themarginal unit of good y is consumed. It makes sense that economic efficiencyis achieved when, at the margin, the two countries together are willing to forgothe consumption of precisely one more unit of good y to avoid the pollutionfrom that one unit. It also makes sense that the North, the richer country thatconsumes more of the good, should, other things equal, have the higher rateof trade-off for pollution and should abate at a higher rate (so that the sum ofterms in parentheses are lower for it).

Equation (3) rearranges to

U ∗g

U ∗y

+ Ug

Uy

[Ey − E A/YK

E∗y − E∗

A/Y ∗K

]= −1

(E∗y − E∗

A/Y ∗K )

, (4)

where the right-hand side is the North’s marginal rate of transformation ofgood y for pollution, attained by adjusting firms’ levels of production as wellas consumers’ levels of abatement, and the left-hand side is the sum of thecorresponding marginal rates of substitution in consumption, except that theSouth’s marginal rate of substitution is efficiently adjusted upward from theperspective of the North, whose net rate of emission, E∗

y − E∗A/Y ∗

K , is lower.From the North’s perspective, the South undervalues pollution because the nu-meraire good is more polluting when it is consumed in the South. The effect ofthe weighting is that the South’s marginal rate of substitution is measured inequivalently polluting units of good y to those in the North. Equation (3) alsorearranges to Equation (5), the corresponding marginal rate of transformationfor the South,

U ∗g

U ∗y

[E∗

y − E∗A/Y ∗

K

Ey − E A/YK

]+ Ug

Uy= −1

(Ey − E A/YK ), (5)

28 KOHN

in which the left-hand side marginal rate of substitution in consumption in theNorth is weighted downward, according to the South’s perspective, so that theunits of good y that it consumes and that measure its trade-off against pollutionare equivalent in pollutiveness to those of the South. The mathematics confirmwhat is behaviorally realistic, that a country’s valuation of global marginal pol-lution damage should be weighted downward from the perspective of anothercountry whose consumption is more pollutive or else weighted upward from theperspective of the other country whose consumption is less pollutive. In otherwords, the industrialized country believes that the developing country under-values pollution abatement, whereas the developing country believes that theindustrialized country overvalues pollution abatement. Note that there are twomarginal rates of transformation in this model: the ratios in Equation (2) denotethe efficient trade-off of output Y for pollution, which is the same in both coun-tries, whereas the right-hand-sides of Equations (4) and (5) are the respectivetradeoffs of consumption y for pollution in the two countries.

Note that the left-hand sides of Equations (4) and (5) become the standardSamuelsonian summation when the countries are identically polluting, which isthe case when the ratios of national emission rates, in square brackets, equalunity. Given Equation (2) in which efficient E A/YK necessarily equals efficientE∗

A/Y ∗K , the equation in square brackets is unity when the emission rates, E∗

y andEy , are the same in both countries. However, this would be a very special case.The standard Samuelson summation also holds when goods are polluting inproduction rather than consumption, in which case, E∗

y and Ey both equal zero.The assumption that all consumption contributes to global warming is crucialin this interpretation of the theoretical inefficiency in the international marketingof emission permits.

3. Pigouvian taxes versus internationally tradable emission permits

In contrast to the above case in which Pareto optimality is not attainable withtradable permits, when the economies of both countries are competitive andeach government imposes a Pigouvian tax on its consumers’ emissions, Paretooptimality obtains even though consumption is non-uniformly polluting acrosscountries. It is assumed here that the richer North may give a lump-sum transferof Y T to the South to induce it to participate in Pareto optimal global warmingcontrol. This corresponds to the cooperative solution in Hoel (1991, p. 60) that“allows for side payments between the two countries.’’ Letting the wage rateequal one unit of each country’s respective currency, it follows that the rentalprice of capital is r = YK /YL and that the Pigouvian tax T ∗ on emissions in theNorth equals their marginal global pollution damage as perceived by the North,

T ∗ = −p∗y

[U ∗

g

U ∗y

+(

Ug

Uy

)(Ey − E A/YK

E∗y − E∗

A/Y ∗K

)], (6)

THEORETICAL INEFFICIENCY IN THE INTERNATIONAL MARKETING 29

while that of the South equals

T = −py

[U ∗

g

U ∗y

(E∗

y − E∗A/Y ∗

K

Ey − E A/YK

)+ Ug

Uy

], (7)

where p∗y and py are the prices of good y in the respective countries. Because

the North weights the South’s valuation upward and the South weights theNorth’s valuation downward, the North’s real Pigouvian tax, T/py , is higher thanthe South’s. This non-uniformity across countries holds because the numerairegood, in which the trade-off in consumption is measured, is differentially pollu-tive in consumption across countries.

Assuming that polluting consumers pay the Pigouvian tax on their emissions,the efficient price of good y is py = r/YK + EY T . Polluting consumers abateto the point that the marginal cost of abatement, which is r/(−E A), equals thePigouvian tax, and it follows therefore that py = r/YK +r Ey/(−E A). Because theleft-hand side of Equation (5), when multiplied by py , becomes marginal pollu-tion damage as perceived by the South, the right-hand side, when multipliedby r ((1/YK ) − (Ey/E A)), simplifies to the South’s marginal cost of abatement,r/(−E A).

Whereas Shibata (1996) assumes that an independent international agencyimposes the Pigouvian tax and collects the tax revenue, it is more realistically as-sumed here that each country imposes its unique tax rate on its own emissionsand collects and redistributes the revenue domestically, except for the portionof tax revenue that is transferred to other countries as lump-sum inducementsto cooperate. In the Pigouvian competitive equilibrium, gross national productin each country, which is GNP = py y, necessarily equals gross national income,which is GNI = Lo + r Ko + TE plus or minus the lump sum transfer, [r/YK ]Y T .(It is plus for the country that receives the lump-sum transfer and minus for thecountry that funds it, where [r/YK ]Y T is measured in the respective countries’own currencies.)

Although Pigouvian taxation achieves economic efficiency in this model, in-ternationally tradable emission permits do not, for they are based on buying andselling permits across countries at a unique equilibrium price. Even such a so-phisticated model as that of Caplan, Cornes, and Silva (2003), which envisionsan “ideal Kyoto protocol’’ in which the largest polluting country abstains fromparticipating, still retains the assumption of a unique international equilibriumprice for the permit. In the context of the present model, an equilibrium price ofpermits somewhere between the higher marginal global damages calculated bythe North and the lower marginal global damages calculated by the South wouldinduce the North to under-abate and to consume (and therefore, produce) toomuch of the good and would induce the South to over-abate and consume (andtherefore, produce) too little of the good. In other words, in countries where theequilibrium price of permits is higher than the domestically evaluated rate ofdamage, there would be excessive pollution abatement, whereas in countries

30 KOHN

where the equilibrium price is lower than the domestically evaluated rate ofdamages, there would be insufficient abatement.

4. Concluding remarks

Hirofumi Shibata (1996, p. 306) has usefully shown “that a tradable pollution per-mits system will not succeed in achieving a Pareto optimal solution because thedifferential pricing of permits . . . is not generally possible in a single competitivemarket of pollution permits.’’ According to Shibata’s (1996, p. 315) interpreta-tion, differential pricing of permits is efficient because polluters fully account for“the disutilities that their own emissions impose upon themselves.’’ Shibata didnot need to make that strong assumption on reflexivity, which Tietenberg (1973,p. 520) has judged to be “empirically [. . . ] one of lesser importance’’ and Bau-mol and Oates (1988, p. 42n) have viewed as a “miniscule’’ effect, because hisown model in which the numeraire good is polluting in consumption providesa simpler and more substantive explanation of the inefficiency in the interna-tional trading of global warming emission permits. Based on his assumption ofpollutive consumption, it follows straightforwardly that each country, summingover differently weighted marginal damage valuations, would have a differentrate of real marginal global pollution damage, and the efficient marginal cost ofabatement would vary accordingly across countries. Even if the Kyoto Protocolcould be implemented by capping allowable emissions at a Pareto optimal to-tal, as envisaged by Duggan and Roberts (2002), the equilibrium market priceof permits, being necessarily unique, would generally not be efficient.

Acknowledgments

For their helpful comments on earlier drafts of this paper, I am grateful to Brian R.Copeland, Michael Hoel and an anonymous referee. Despite my disagreementswith Professor Shibata, I consider the present paper to be more representativeof his contributions than my own, and am therefore in his debt. On this, theoccasion of my final publication in economics, I take the opportunity to thankProfessor Michele Fratianni for his good advice, his encouragement, and for hisgenerous publication of my papers over the past ten years.

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