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The Role of Multiple Pollutants and Pollution Intensities in the Policy Reform of Taxes and Standards

Luis Gautier

Abstract

Countries with varying degrees of pollution intensities, facing increasing global competition and addressing emissions from multiple pollutants may undertake policy reforms inconsistent with cooperative outcomes, where global welfare is higher. Among others, this is because of the incentives to set laxer policy to be more cost competitive. A number of welfare-enhancing and emissions-reducing policy reforms consistent with the cooperative equilibrium, but also consistent with addressing concerns about global competitiveness are derived. The analysis indicates that the nature of multiple pollutants and asymmetries in pollution intensities are key in the design of policy reform and characterization of optimal policy. With complementarity and asymmetry in pollution intensities, laxer taxation and stricter standards are consistent with welfare gains. Laxer taxation arises with large asymmetry in pollution intensities regardless of whether pollutants are complements/substitutes. The policy reform of standards requires both complementarity and asymmetry in pollution intensities. Results are reversed if pollutants are substitutes.

JEL Classification: H23; Q5; D43

Appendix

Total differentiation of eqs. (2), (3), (5), (6) yields the following system (the “bar” is dropped from eˉ2h and eˉ2f for notational simplicity):

2Pqhh+qhPqhqhhCqhqhhPqfh+qhPqhqfhCqhe1hh0Ce1hqh0Ce1he1hh0Pqhf+qfPqfqhf2Pqff+qfPqfqffCqfqff0Cqfe1ff0Ce1fqff0Ce1fe1ffdqhdqfde1hde1f=Cqhe2hhde2hCe1he2hhde2h+dthCqfe2ffde2fCe1fe2ffde2f+dtf

where the determinant of the coefficient matrix is µ < 0.

In what follows the comparative statics effects on output and emissions for each country are derived. In particular, the change in output for the home and foreign countries are given by

(21)μCe1he1hhCe1fe1ffdqh=Cqhe1hhCe1he1hh2Pqff+qfPqfqff+Ce1fe1ffCqfqffCe1fqff2Ce1fe1ffdthCqfe1ffCe1fe1ffPqfh+qhPqhqfhdtf+Ce1he2hhCqhe2hhCe1he2hhCqhe1hhCe1he1hh2Pqff+qfPqfqff+Ce1fe1ffCqfqffCe1fqff2Ce1fe1ffde2h+Ce1fe2ffCqfe2ffCe1fe2ff+Cqfe1ffCe1fe1ffPqfh+qhPqhqfhde2f
(22)μCe1he1hhCe1fe1ffdqf=Cqfe1ffCe1fe1ff2Pqhh+qhPqhqhh+Ce1he1hhCqhqhhCe1hqhh2Ce1he1hhdtfCqhe1hhCe1he1hhPqhf+qfPqfqhfdth+Ce1fe2ffCqfe2ffCe1fe2ffCqfe1ffCe1fe1ff2Pqhh+qhPqhqhh+Ce1he1hhCqhqhhCe1hqhh2Ce1he1hhde2f+Ce1he2hhCqhe2hhCe1he2hh+Cqhe1hhCe1he1hhPqhf+qfPqfqhfde2h

where µ < 0:

(23)μ=Ce1he1hhCe1fe1ff(2Pqhh+qhPqhqh)(2Pqff+qfPqfqf)(Pqhf+qfPqfqh)(Pqfh+qhPqhqf)(Ce1he1hhCqhqhhCe1hqhh2)(Ce1fe1ffCqfqffCe1fqff2)+Ce1he1hh(2Pqh+qhPqhqh)(Ce1fe1ffCqfqffCe1fqff2)+Ce1fe1ff(2Pqf+qfPqfqf)(Ce1he1hhCqhqhhCe1hqhh2)<0

where CekekkCqkqkkCqkekk2>0, for k = h, f by the concavity assumption of the cost function.

Next, the effect of taxes on foreign and home emissions is given by

(24)μCe1he1hhCe1fe1ffde1h=Cqhe1hhCe1he1hhCqhe1hhCe1he1hh2Pqff+qfPqfqff+Ce1fe1ffCqfqffCe1fqff2Ce1fe1ff+μ/Cqhe1hhdth+Cqhe1hhCe1he1hhCqfe1ffCe1fe1ffPqfh+qhPqhqfhdtf
(25)μCe1he1hhCe1fe1ffde1f=Cqfe1ffCe1fe1ffCqfe1ffCe1fe1ff2Pqhh+qhPqhqhh+Ce1he1hhCqhqhhCe1hqhh2Ce1he1hh+μ/Cqfe1ffdtf+Cqfe1ffCe1fe1ffCqhe1hhCe1he1hhPqhf+qfPqfqhfdth

Next, the expressions in eqs. (13) and (14) are expanded; these are used to derive eq. (32): πh/th+e1h=qhPqfhqthfCe2hh(e2h/th), πh/e2h=qhPqfhqe2hfCe2hh, E/th=e1h/th+e2h/th+e1f/th+e2f/th, E/e2h=e1h/e2h+1+e1f/e2h+e2f/e2h, and e2f/th=0, e2f/e2h=0 since the home country takes the foreign country’s standard as given.

A End-of-pipe type cost function

Next, the case of end-of-pipe type cost function as in Definition 6.1 is examined, where for the home country:

Cqhh=c˜h+(1+ω)δhδhqhe1h+δhqhe2hCqhe1hh=(1+ω)δhCqhe2hh=(1+ω)δhCe1hh=(δhqhe1h)ω(δhqhe2h)Ce2hh=(δhqhe2h)ω(δhqhe1h)Ce1he1hh=1,Ce2he2hh=1Ce1he2hh=ωCe1he1hhCqhqhhCe1hqhhCqhe1hh=δh2(1+ω)(1ω)

where ω(1,1). Assume also a linear demand function Pk=αβ(qh+qf), where Pqhh=Pqfh=Pqff=Pqhf=β. Using first-order conditions eqs. (2), (3), (5) and (6) gives the following system, which implicitly determines the equilibrium qh, qf, e1h, e1f:

(26)qh2+β+2δf2(1+ω)+qfβe1h(1+ω)δh=αc˜h+(1+ω)δhe2h
(27)qh(1+ω)δhe1h=th+ωe2h
(28)qhβ+qf2+β+2δh2(1+ω)e1f(1+ω)δh=αc˜f+(1+ω)δfe2f
(29)qf(1+ω)δfe1f=tf+ωe2f

whence,

(30)κqh=2β+δf2(1+ω)(1ω)αc˜h(1+ω)δhth+(1+ω)(1ω)δhe2h+βαc˜f(1+ω)δftf+(1+ω)(1ω)δfe2f
(31)κe1h=(1+ω)δh2β+δf2(1+ω)(1ω)αc˜hδhth+δhe2hβαc˜f(1+ω)δftf+(1+ω)(1ω)δfe2f+2β+δf2(1+ω)(1ω)2β+δh2(1+ω)β2(th+ωe2h)

where κ=2β+δf2(1+ω)(1ω)2β+δh2(1+ω)(1ω)β2<0 is the determinant of the coefficient matrix, which is equal to µ in eq. (23) under end-of-pipe as defined in Definition 6.1. Analogous expressions apply to qf, e1f because of the symmetric nature of the model. From eqs. (30) and (31) the following are obtained:

κqhth=2β+δf2(1+ω)(1ω)(1+ω)δh<0κqhtf=(1+ω)δh>0κe1hth=2β+δf2(1+ω)(1ω)2β+2δh2(1+ω)β2<0=2β+δf2(1+ω)(1ω)2β+2δh2(1+ω)δh2(1+ω)2+δh2(1+ω)2β2=2β+δf2(1+ω)(1ω)2β+δh2(1+ω)(1ω)β22β+δf2(1+ω)(1ω)δh2(1+ω)2κe1htf=(1+ω)2δhδfβ>0

And

κqhe2h=2β+δf2(1+ω)(1ω)(1+ω)(1ω)δh>0κqhe2f=(1+ω)(1ω)δfβ<0κe1he2h=2β+δf2(1+ω)(1ω)(1+ω)2δh2ω2β+δf2(1+ω)(1ω)2β+2δh2(1+ω)β2=2β+δf2(1+ω)(1ω)(1+ω)2δh2ω2β+δf2(1+ω)(1ω)2β+2δh2(1+ω)δh2(1+ω)2+δh2(1+ω)2β2=2β+δf2(1+ω)(1ω)(1+ω)2(1ω)δh2ω2β+δf2(1+ω)(1ω)2β+δh2(1+ω)(1ω)β2κe1he2f=(1+ω)2(1ω)δhδfβ<0

B Derivation of eq. (19)

Next, eq. (19) is derived. Consider the first-order conditions Wthh(th,e2h,ω)=0, We2hh(th,e2h,ω)=0. Differentiation gives a system of two unknowns dth/dω, de2h/dω. Equation (19) is obtained by evaluating dth/dω at ω = 0, while keeping in mind Wthe2hh=0 because of the linear demand and end-of-pipe cost function assumption. Also,

qhω=0=(2β+δf2)(αc˜hδhth+δhe2h)+β(αc˜fδftf+δfe2f)>0κqhωω=0=(2β+δf2)δhthδftfβκe1hωω=0=δh(2β+δf2)(αc˜h+2δhe2h2δhth)+β(αc˜f2δftf+δfe2f)+e2h(2β+δf2)(2β+2δh2)β2κe1fωω=0=δf(2β+δh2)(αc˜f+2δfe2f2δftf)+β(αc˜h2δhth+δhe2h)+e2f(2β+δh2)(2β+2δf2)β2

C Strategic use of standards and taxation

Assume governments still choose standards and taxes, but now standards are chosen first using eq. (14) which gives, in the case of the home country, say, a standard as a function of the tax i. e. e2h(th). The home government then takes this standard into account and chooses the tax. This case can be thought of as one where a country sets a policy first (e. g. environmental standards to tackle one pollutant), but uses the other policy strategically. That is, I look at the interplay between the tax and standard. The analysis does not rely on Definition 6.1.

Differentiation of the welfare function with respect to the tax and substituting eq. (14) into eq. (13) gives

(32)th=φh+qhPqfhqthf+qhPqfhqe2hfe2hth+φhe1fthe2hthe1fe2h1e1hth(1η)

where η=(e2h/th)(e1h/e2h)/(e1h/th), e1h/th<0, e1f/th>0, and Pqfh<0, qthf>0 (subscripts denote partial derivatives). The first term denotes the upward adjustment in the tax to tackle damages from home pollution. The second and third terms denote profit-shifting effects, and the fourth and fifth terms adjustments in the tax to address damages from transboundary pollution. The term 1η captures tax adjustments due to how emissions change, which has an upward pressure. But also the interplay between the tax and standard, which sign depends upon complementarity/substitutability.

The term e1h/e2h captures the degree of complementarity/substitutability of pollutants in the home country: if pollutants in the home country are complements (substitutes), then e1h/e2h>(<)0, meaning that a reduction in the second pollutant results in a reduction (increase) in the first pollutant, which is consistent with Ce1he2hh<(>)0. And the term e2h/th denotes the interplay between the tax and standard in the home country: if e2h/th>(<)0, then laxer taxation in the home country results in the home government setting a stricter (laxer) standard.

To illustrate the role of complementarity/substitutability of pollutants, as a benchmark case suppose these are independent (i. e. e1h/e2h0) and that adjustments in the home tax have a negligible effect on the standard i. e. e2h/th0. Then, the expression in eq. (32) reduces to

(33)e1htht1h=φhe1hthqhPqfhqthf+φhe1fth

where the first, second and third terms denote, respectively, damages from local pollution, profit-shifting effects and adjustments from transboundary pollution.

Now suppose home pollutants are complements (i. e. e1h/e2h>0) and to analyze the interplay between the home tax and standard assume, also, that the home country sets a stricter standard as a result of a laxer tax i. e. e2h/th>0. As a result of complementarity, in eq. (32) (i) qe2hf<0 and e1f/e2h<0: with complement pollutants a stricter standard raises foreign output and emissions of the first pollutant in that country via the strategic interaction of firms in the output market (the second pollutant in the foreign country is set by the binding agreement and so does not change). And (ii) 1η>0, (e1f/th)(e2h/th)(e1f/e2h)>0. Under (i) and (ii), then, the optimal tax is set below marginal damages due to two factors. First, profit shifting incentives via lower taxation, but also because the presence of the standard (third term in 32) renders home firms less cost competitive and so the tax is lowered to offset this. Second, a lower tax at home reduces damages coming from transboundary pollution, but also the presence of the standard raises foreign output (since home pollutants are assumed to be complements) and thus damages from transboundary pollution are addressed via lower taxation (fifth term in 32). Similar results are obtained if pollutants are substitutes and e2h/th<0. But interestingly, if pollutants are complements and, also, e2h/th<0 and large (i. e. 1η<0 in the last parenthesis in eq. (32) and (e1f/th)(e2h/th)(e1f/e2h)<0), then the tax exceeds marginal damages from home pollution if damages from transboundary pollution are small. This is because with a sufficiently large adjustment in the standard (i. e. very lax standard) damages from home pollution rise and so addressed via higher taxation; there is no need to control transboundary pollution via lower taxation. These cases point to the important role of complementarity/substitutability of pollutants in the interplay between taxes and standards.

Result A.1.

Suppose home chooses the standard first, followed by the choice of the tax. The non-cooperative tax in the home country is less than marginal damages from home pollution if (i) home pollutants are complements (substitutes) and (ii) a reduction in the tax is met by a stricter (laxer) standard in the home country i. e.e2h/th>(<)0.

Result A.2.

Suppose home chooses the standard first, followed by the choice of the tax. The non-cooperative tax in the home country exceeds marginal damages from home pollution if (i) home pollutants are complements, (ii) there is a sufficiently laxer standard in the home country (i. e.e2h/th<0and large), and (iii) damages from transboundary pollution are small.

Acknowledgments

I am truly grateful to two anonymous referees and Professor Frans de Vries, for very constructive comments and suggestions.

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Published Online: 2019-06-14

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