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Optimal Green Taxation With Both Emission and Commodity Taxes


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Basharat A.K. Pitafi and James Roumasset Working Paper No. 02-8


A Second-best Pollution Solution with Separate Taxation of

Commodities and Emissions

Basharat A. K. Pitafi and James A. Roumasset University of Hawaii at Manoaϒ

July 26, 2002

JEL Classification: D62, H21, H23

Keywords: Second-best environmental taxation, Pigouvian taxation, tax normalization, Revenue recycling, Tax interaction


Several authors have argued that the second-best environmental tax on a “dirty good” is less than marginal emission damage associated with its consumption. These studies limit their analysis to cases in which emissions can only be reduced by a reduction of the dirty good. With a more general specification that allows abatement through input substitution, we show that the direct emissions tax cannot be less than marginal emission damage, regardless of the normalization.

ϒ542 SSB, 2424 Maile Way, Honolulu, HI 96822. Fax: (808) 956-4347. basharat@hawaii.edu.

Authors gratefully acknowledge comments from Kwong Soo Cheong, Gerard Russo, Katerina Sherstyuk, and the participants of the World Congress of Environmental and Resource Economists, Monterey, CA,


1. Introduction

Recent interest in global warming and carbon taxes has spawned a vigorous discussion regarding the size of the optimal environmental tax in relation to marginal environmental damage. Early contributions stressed the additional benefits (denoted as the “revenue-recycling” effect by Parry, 1995) of using environmental tax revenue to reduce the excess burden of pre-existing taxes [Nichols 1984, Terkla 1984, Lee and Misiolek 1986, Pearce 1991, Repetto et. al. 1992]1. Nordhaus (1993), for example, using the DICE framework, empirically derived large carbon taxes when such revenue recycling was allowed, and much smaller taxes when it was not allowed.

While these studies ignored the tax interaction between environmental and other taxes, Bovenberg and de Mooij (1994)2 showed that such friction is not negligible. When other taxes are present in the system, environmental taxes may exacerbate pre-existing

distortions and this unfavorable “tax-interaction” effect may outweigh the favorable “revenue-recycling” effect, such that a second-best emissions tax is smaller than the marginal emissions damage (MED). This did not quite settle the issue, however, and soon Fullerton3 (1997) and Schöb (1997) showed that the second-best environmental taxes can be greater or smaller than MED depending on the choice of tax instruments. Further confusion was created in the literature by subsuming the size-of-emission-tax argument

1Tullock (1967) is often credited with pioneering the idea.

2 Also see Bovenberg and de Mooij (1997), and Bovenberg and Goulder (1996).

3 Fullerton’s treatment also clarifies how the results of Boveberg and de Mooij (1994) depend on the

complementarity between dirty good and labor. Fuest and Huber (1999) show that when there is gross substitutability between the dirty good and the other tax base (labor/clean good), the optimal tax on the dirty good always exceeds the Pigovian tax.


into the "double dividend" debate without a consensus on the definition of double dividend.

The purpose of all these analyses was to determine how the size of the environmental tax is affected by the presence of other taxes. At first blush, the problem seems

straightforward. One needs only to specify the nature of emissions in a fairly general way and then extend the theory of optimal commodity taxation to include emission taxes. Generality can be achieved by treating emissions as an input (e.g., Oates and Strassman 1984, and Fullerton and Metcalf 20014). Here we encounter a difficulty, however,

because the conventional method of solving for an optimal system of commodity taxes is not directly applicable in the presence of a non-market good (environment). In the

absence of an emissions price, the elasticities involving the emissions price, which would make a major component of a conventional tax solution, cannot be computed.

Sandmo (1975) and others using similar models (e.g., Bovenberg and de Mooij 1994, Fullerton 1997) avoid this difficulty by making emissions a function of a commodity called the dirty good and taxing the dirty good in order to tax emissions. While this approach renders the model tractable, it severely restricts pollution avoidance

possibilities. For instance, in this polar case, imposition of a labor tax does not result in the substitution of emissions for labor in production that would have increased the

demand for emissions input. This is the reason behind the Bovenberg and de Mooij result (emissions tax less than MED) and the Fullerton result (the tax-MED comparison


Fullerton and Metcalf use this technology in comparing the rent-generating and rent-capturing emissions control policies. Oates and Strassmann use it to compare the efficiency of effluent fees under different non-competitive market structures.


dependent on normalization or the choice of tax instruments).5 Moreover, the utility function is typically assumed to exhibit weak separability between emissions and other goods. This further restricts the applicability of the results from these models.

Accordingly, we proceed with a different solution strategy. We maintain the generality afforded by regarding emission as an input to production and solve the non-market problem by treating emissions analogously to the other input, labor. Just as the consumer is endowed with units of labor that can be sold or consumed, she is also endowed with emission rights that can be sold to polluters or retained (thereby reducing pollution). We derive the Ramsey price of emissions and then exploit institutional equivalency under zero transaction costs to replace the pollution market with an equivalent pollution tax. We also relax the assumption of weak separability.

Using this framework, we show that, when the revenue requirement warrants distortionary taxation, the second-best emissions tax exceeds MED, and quantity of emissions is less than in the first-best solution. If the revenue requirement is small enough to be financed by emissions tax alone, then a first-best optimum obtains with the emissions tax equal to MED. These results are independent of tax normalization.

The rest of the paper is organized as follows. The model and derivation of result are provided in section 2. Section 3 summarizes the results and concludes.

5The only abatement method allowed is reducing the consumption of the dirty good. It does not allow for

abatement by switching to a cleaner process or using end-of-pipe treatment. This implies that consumer cannot choose the amount of emissions independently of the amount of the dirty good, and the two cannot, therefore, be taxed separately. Because the dirty tax changes with normalization, the implied emissions tax also varies.


2. The Model

Following Oates and Strassmann (1984), emissions are modeled as a virtual input. This allows for all three abatement methods, i.e., a) decreasing output, b) switching to a cleaner process, and c) “end-of-pipe” treatment.

In consonance with, but simultaneously generalizing, the prototypical approach6, we have two goods, one of which is “dirty” (good d) in the sense that emissions are necessary for its production, while the other is a “clean” good (c). The production functions are:

( 1 ) c=c(lc), d =d(ld,e),

where lc and ld are labor inputs to sector c and d respectively and e is the emissions input. For convenience, we first assume a market for emissions and then replace the market with an emissions tax.

The producers’ profit-maximizing first-order conditions (FOCs) are now:

( 2 ) d e d d c c p e d p w l d p w l c p = ∂ ∂ = ∂ ∂ = ∂ ∂ , ,

where w is the wage rate, pe is the emissions market price, and pc and pd are the producer prices of the good c and d respectively.

The consumer is initially endowed with N pollution rights,7 sells e rights to the producers in an emissions market and consumes the rest, n, in the same way that she sells part (l) of the total labor endowment (L) and consumes the rest as leisure (v). Thus, lc + ld = l = (L -

6Henceforth, “prototypical” or “standard” model or approach refers to Bovenberg and de Mooij (1994),

Fullerton (1997) and similar models.

7N can be set arbitrarily at some number greater than or equal to the optimal quantity of emissions. To fix


v) and e = (N - n). The consumer’s utility is, therefore, u(c, d, v, n, G), where G is a government good financed through taxes, is weakly separable in the utility function, and is held constant (following the standard models).

For the government revenue constraint, G, to be met, the consumer pays ad valorem

commodity8 taxes, tcand td. The revenue constraint is then: ( 3 ) tcc+td pddG

where pc and pd are the producer prices of the good c and d respectively, and the former is normalized to unity, without loss of generality. The consumer maximizes the utility subject to the budget constraint:

( 4 ) wl+ pee=(1+tc )c+(1+td )pdd

The resulting first-order conditions (FOCs) are:

( 5 ) c d d pe n u w v u t p d u t c u . iv) , . iii) ), 1 ( . ii) , ) 1 ( i) λ λ λ =λ ∂ ∂ = ∂ ∂ + = ∂ ∂ + = ∂ ∂

where λ is the marginal utility of income. Re-writing (5), we get a more intuitive form:

( 5A ) λ λ λ λ n u p v u w d u t p c u tc d d e ∂ ∂ = ∂ ∂ = ∂ ∂ = + ∂ ∂ =

+ ) , ii) .(1 ) , iii) , iv) 1

( i)

price were zero.

8 Because demand function has the property of being homogeneous of degree zero in consumer prices, i.e.,

only relative prices matter, one of the goods can be chosen as numeraire (see, e.g., Keller 1980). The

numeraire’s price can be set to unity without changing the relative prices. Then all the others prices are stated in terms of units of the numeraire good. The same is true of supply function in terms of supplier prices. However, in the presence of taxes, the consumer and supplier prices are differentiated. For convenience and ease of interpretation, we often set the consumer and supplier prices of the same good to unity, causing the tax wedge between the two to disappear, i.e., we perform “tax normalization”. Since the choice of the numeraire is arbitrary, many alternate normalizations are possible. Here we are normalizing the labor tax to zero. Later we consider other normalizations and show that they yield the same results.


From (5)(i), the marginal utility of income, λ, is ( 6 ) (1 tc) c u + ∂ ∂ = λ

Note the relationship of λ to the tax on the clean good.

Now, we are ready to compare the emissions price with the marginal emissions damage. Plugging (6) in (5)(iv) gives:

( 7 ) c u e u c e c e t p c u n u t p ∂ ∂ ∂ ∂ − = + ⇒ ∂ ∂ ∂ ∂ = + ) (1 ) 1 ( Q n=Ne

where the R.H.S. is just the marginal emissions damage (MED). Re-arranging (7), we get: ( 8 ) pe ( 1 tc) (MED)( 1 tc) c u e u + = +      − = ∂ ∂ ∂ ∂ ( 9 )     = = > >

0 0 c c t when MED t when MED


Thus, when there are no commodity taxes, the emissions price and MED are equal.

Positive commodity taxation (warranted by a revenue requirement in excess of emissions tax revenue) must increase the efficiency price of emissions above MED. This is because the taxes on commodities raise consumer prices and encourage the consumer to substitute leisure and environment for consumption of commodities. This reduces the supply of emissions permits and raises the emissions price to the producers. When the emissions price is implemented as an emissions tax on producers, the emissions tax exceeds MED.


For purposes of illustration, consider a downward sloping demand for permits (value of marginal product of emissions) curve and an upward sloping supply of permits (marginal cost of damage from emissions) curve, as shown in Figure 1. Imposing the commodity tax, shifts the permits supply curve to the left, representing a substitution toward the consumption of untaxed environmental amenities and away from the consumption of the taxed commodity.

Figure 1: Second-best Emissions Price/Tax (induced by commodity taxation)

) 1 .( tc c u e u +       ∂ ∂ ∂ ∂ −       ∂ ∂ ∂ ∂ − c u e u       ∂ ∂ d d e d p.

In contrast, consider the case where G = 0. In the absence of distortionary tax, tc, the permit supply equals permit demand at the emissions level e1 and price p1. Since the government expenditures are zero, the tax revenue, p1.e1, is simply disbursed to the consumer in lump-sum fashion. Even as G increases, we remain at the first-best equilibrium so long as G = p1.e1.


Once the revenue constraint becomes binding, i.e., G =p1.e1, then distortionary taxes become necessary to raise any more revenue. As shown in Figure 1, this shifts the permit supply curve to the left, raises the emissions price to p2, and reduces the emissions quantity to e2. The value of MED at the new emissions level is MED2.9 The emissions price, p2, is larger than this MED level. Since this emissions price is implemented as a tax on producers, the resulting emissions tax is larger than MED. The emissions tax is also larger than the first-best MED level, i.e., p1.

2.1. The Normalization Problem

The conventional method of normalization in tax models is to use the same good as numeraire for both consumer and producer prices. As a result, without loss of generality, one of the taxes is zero. When we set the labor tax to zero, commodity taxes are used to meet any excess revenue requirement. We have shown above that, in this case, the emissions tax is greater than MED. This is because leisure and environmental amenities are substituted for other consumption and the emissions supply schedule shifts leftward. Similarly, when we use the labor tax to raise revenue and set the clean good tax to zero, again we get the result that the emissions tax is greater than MED. This is because labor tax reduces the consumer’s income and induces the consumer to substitute leisure and environmental amenities for other consumption, which, in turn, reduces the supply of emissions, as in Fig. 1. Any combination of the clean good tax and labor tax also yields the same results because they both affect emissions supply in the same way. 10Therefore,

9In the comparison between emissions tax and MED commonly used in the literature (e.g., Bovenberg and

de Mooij 1994, Fullerton 1997), MED is implicitly defined as its value at the second-best optimum.


The consumer budget constraint with two commodity and one labor taxes is:

e l d d c d p t lw t ep t c.(1+ )+ . .(1+ )= . .(1− )+ .


under any normalization, the emissions tax is equal to or greater than MED at the

second-best optimum. Our results are the same under any choice of tax instruments. They agree with those of the prototypical model in case of a commodity tax but differ in case of a labor tax. The intuition behind this difference is explained next.

2.2 Comparison with the Standard Model

Why do we get the result that the emissions tax is equal to or larger than MED (under any choice of tax instruments), in contrast with the standard result that the dirty good tax is less than MED (at least in the presence of a labor tax)? As suggested in section 1, the answer lies in the input substitutability between emissions and labor. In the standard model (e.g., Bovenberg and de Mooij 1994, and Fullerton 1997) does not allow

substitution between emissions and labor as inputs in the production of a dirty good. By assuming that emissions are a fixed function of the dirty good, the elasticity of input substitution is implicitly assumed zero. This implies that the consumer cannot choose the amounts of environmental amenities to consume (or emissions to tolerate) independently of the amount of dirty good consumed. Therefore, taxing emissions or taxing dirty good is equivalent. More importantly, when a labor tax is imposed, producers cannot respond by substituting emissions for labor. If they could, the demand for emissions would

Multiplying through by a constant, Q, the consumer FOCs now become:

Q p n u Q t w v u Q t p d u Q t c u e l d d

c). ii) . .(1 ). , iii) . .(1 ). , iv) . . 1 .( i) λ , λ λ =λ ∂ ∂ − = ∂ ∂ + = ∂ ∂ + = ∂ ∂

Note that we can set to eliminate the clean good tax or to eliminate the

labor tax. Dividing (iv) by (i) gives:

) 1 ( 1 Q= −tl ) 1 ( 1 Q= +tc ). 1 ( . MED c u n u Q t Q p c e = ∂ ∂ ∂ ∂ = + e c .(1 tc) (MED).(1 tc) u n u p  + = +      ∂ ∂ ∂ ∂ = ⇒ MED pe≥ ⇒ when tc∈[0,1] 9


increase, as would the emissions price/tax, as in our model discussed above.

3. Conclusions

We examine the optimal emissions tax in a model that allows for multiple means of abatement. When there is no distortionary taxation to meet the revenue requirement, the emissions tax equals MED. When there is distortionary taxation, the emissions tax exceeds MED evaluated at the second-best optimum, and the emissions quantity is less than in the first-best solution. The second-best emissions tax also exceeds MED at the first-best optimum.

The assumption of no input substitution is the underlying reason for the results of the prototypical model. Without this restriction, our model provides the result that emissions tax is larger than MED and remains larger under any normalization. Thus, modeling the production function to allow for input substitution, and the resulting abatement

possibilities, avoids the troublesome normalization problem that has plagued the double dividend debate. When the elasticity of input substitution is positive, the emissions tax is greater than or equal to MED.



Bovenberg, A. L. and R. A. de Mooij, 1994, Environmental Levies and Distortionary Taxation, American Economic Review 84, 1085-1089.

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Bovenberg, A. L. and L. H. Goulder, 1996, Optimal Taxation in the presence of Other Taxes: General Equilibrium Analyses, American Economic Review 4, 985-1000.

Fuest, C. and B. Huber, 1999, Second-Best Pollution Taxes: An Analytical

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American Economic Review 1, 245-251.

Fullerton , D., G. E. Metcalf, 2001, Environmental controls, scarcity rents, and pre-existing distortions, Journal of Public Economics 80, 249–267.

Keller, W. J., 1980, Tax Incidence: A General Equilibrium Approach (North-Holland, New York).

Lee, D. R. and W. S. Misiolek, 1986, Substituting Pollution Taxation for General Taxation: Some Implications for Efficiency in Pollution Taxation, Journal of Environmental Economics and Management 13, 338-347.

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