the welfare effects of environmental taxation on a green market where consumers emit a pollutant
TRANSCRIPT
Environ Resource Econ (2012) 52:87–107DOI 10.1007/s10640-011-9521-7
The Welfare Effects of Environmental Taxationon a Green Market Where ConsumersEmit a Pollutant
Isamu Matsukawa
Accepted: 2 October 2011 / Published online: 18 October 2011© Springer Science+Business Media B.V. 2011
Abstract This paper examines the welfare impact of emission taxes and subsidies in agreen market where consumers emit a pollutant through their usage of products produced byduopolists. For this purpose, we employ a discrete–continuous model including both con-sumer choice and usage of an environmentally differentiated product in a utility-consistentframework. The findings indicate that an emission tax is always welfare dominant over a sub-sidy on consumer purchases of the clean product because of its contribution to a reductionin environmental damage. It does this by both inducing firms to improve the environmentalqualities of their products and by constraining consumer usage of these products.
Keywords Discrete–continuous model · Emission taxes · Environmental quality ·Green market · Subsidies · Vertical differentiation
JEL Classification H23 · L13 · Q58
1 Introduction
Increasing concern about environmental problems, such as global warming, has made con-sumers more aware than ever of product attributes related to pollution. In the presence ofconsumers that are aware of the environmental attributes of products, each firm competesin the market by differentiating its product in terms of environmental quality. By assuminga vertical differentiation model, the literature on green markets investigates how taxationand subsidization affect social welfare (Cremer and Thisse 1999; Moraga-Gonzalez andPadron-Fumero 2002; Bansal and Gangopadhyay 2003; Kuhn 2005; Lombardini-Riipinen2005; Bansal 2008). However, these studies only concern the choice of product quality byconsumers and not consumer usage of these products.
I. Matsukawa (B)Faculty of Economics, Musashi University, 1-26-1 Toyotama-kami, Nerima-ku, Tokyo 176-8534, Japane-mail: [email protected]
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Along with the choice of product quality, the usage of the product also affects envi-ronmental pollution. Examples include the operation of energy-using durables, such asautomobiles and electrical appliances, which have increasingly affected global warmingthrough the emission of greenhouse gases. All other things being equal, we expect theimposition of an emission tax on consumers using gasoline and electricity to reducethe emissions of carbon dioxide through its effects on both the ownership and opera-tion of energy-using durables. This is because environmentally aware consumers considerenergy-efficient products, which emit less carbon dioxide per use than other productsto be of higher quality and hence more attractive. Further, with an emission tax theseconsumers are more likely to choose energy-efficient automobiles and electrical appli-ances. This provides firms with an incentive to develop more energy-efficient prod-ucts. Finally, given the ownership of energy-using durables, we expect an emissiontax to discourage consumer usage of durables because of the increase in operatingcost.
Using a discrete–continuous model of consumer choice, this paper explicitly inves-tigates the impact of emission taxes and subsidies on the choice and usage of anenvironmentally differentiated product. The main contributions of this paper are two-fold. First, we employ a utility-consistent framework to examine the effects of emis-sion taxes and subsidies in a green market. A discrete–continuous model enables us toconsistently analyze both the ownership and usage of an environmentally differentiatedproduct (Hanemann 1984). Accordingly, we can use this model to conduct an empir-ical analysis of the effects of environmental policies on environmentally differentiatedproducts.
Second, and more importantly, we investigate the welfare effects of emission taxes andsubsidies, and provide a welfare comparison of these environmental policies. We find that anemission tax always dominates a subsidy because the emission tax contributes to a reductionin environmental damage by inducing firms to improve the environmental qualities of theirproducts and by constraining consumer usage of the product. In contrast, a subsidy on con-sumer purchases of the clean product reduces environmental damage only by inducing firmsto improve the environmental qualities of their products. The difference in social welfarebetween the second-best emission tax and the second-best subsidy then corresponds to areduction in environmental damage net of the deadweight loss associated with the emissiontax.
Of past studies of green markets, Lombardini-Riipinen (2005) and Bansal (2008) are themost closely related to the present analysis. This is because both investigate the welfareeffects of emission taxes and subsidies in a green market fully covered by duopolists. Whilethe literature on green markets often examines partially covered markets, we consider a mar-ket fully covered by duopolists because our principal focus is the emission of pollutantsarising from consumer usage of environmentally differentiated products.
Lombardini-Riipinen (2005) argues that both a second-best emission tax imposed onfirms and a second-best subsidy on consumer purchases of the clean product are equal to thesocial valuation of the marginal damage from pollution caused by the production activitiesof firms. Lombardini-Riipinen (2005) also suggests that social welfare under the second-bestemission tax equals that under the second-best subsidy. Bansal (2008) concludes that thesecond-best emission tax imposed on firms equals the social valuation of the marginal dam-age from pollution caused by the production activities of firms. Bansal (2008) also arguesthat a second-best ad valorem tax dominates a second-best emission tax in terms of socialwelfare if the marginal damage from pollution is sufficiently small. Although we also findthat the second-best emission tax equals the social valuation of the marginal damage from
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The Welfare Effects of Environmental Taxation on a Green Market 89
pollution, we demonstrate that (i) the impact of an emission tax on product quality differsfrom that of a subsidy, (ii) the second-best subsidy rate deviates from the social valuation ofthe marginal damage, and (iii) the second-best emission tax always dominates the second-bestsubsidy.
The organization of the paper is as follows. Section 2 describes an economic model ofduopolists offering environmentally differentiated products to consumers whose usage of theproducts leads to the emission of a pollutant. This model is used to investigate the effectsof emission taxes and subsidies on the market. Section 3 examines welfare effects of emis-sion taxes and subsidies, and conducts a welfare comparison of these environmental policyinstruments. Section 4 concludes this study. The appendix presents proofs of propositions,the derivation of social welfare, and the second-order conditions for profit maximization andwelfare maximization. Welfare effects of ad valorem taxes–subsidies are also discussed inthe appendix.
2 The Model
2.1 A Discrete–Continuous Model of an Environmentally Differentiated Product
Suppose that firms G and B compete to provide consumers with a differentiated productwhose usage requires consumers to use fuel. A pollutant subject to environmental regula-tion is then emitted only by the consumption of fuel and no pollutant is emitted from theproduction process of firms. Emissions of the pollutant associated with the usage of firmi’s product, denoted ei , are assumed to be the product of the emission factor and consumerusage. The emission factor is defined as ρ (D − ai ), where ρ denotes emissions per fuel con-sumption, D > 0 denotes fuel consumption per usage if no activity for quality improvementis undertaken by the firm, and ai denotes the quality of the product offered by firm i . Thechoice of the product quality by firm i , ai ∈ [0, D] , i = G, B, reduces fuel consumption perusage to a level D −ai . Thus, the emission factor differs between two firms unless both firmschoose the identical quality. Consumer usage of the product, denoted ki , depends on sucheconomic factors as fuel price and an emission tax. Consumer characteristics including thenumber of household members and residential location also affect the usage of the product.The level of fuel consumption associated with the usage of firm i’s product, denoted xi ,is the product of D − ai and ki . Thus, ei = ρ (D − ai ) ki = ρxi . For the rest of the paper,we treat xi as a function of variables affecting fuel demand, which is consistent with utilitymaximization.
As an example, consider the fuel economy of a passenger vehicle, which indicates howlong a consumer can drive a car for a unit usage of gasoline. Here, given consumer usageof the car (ki , kilometers per year) and emissions per gasoline consumption (ρ, emissionsof carbon dioxide per liter), the larger the value of the measure of fuel economy, the smallerthe values of D − ai (liters per kilometer), xi (liters per year), and ei (emissions of carbondioxide per year).
Without loss of generality, the quality of firm G’s product is assumed to be equal to orhigher than that of firm B’s product. Thus, aG ≥ aB . Following Lombardini-Riipinen (2005)and Bansal (2008), the production cost of firm i is assumed to be given by
C (ai , qi ) = 0.5ba2i qi , (1)
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where C (ai , qi ) , qi , and b respectively denote firm i’s cost function, the amount of the prod-uct of firm i , and a positive parameter.1 In (1), production cost is proportional to the amountof the product, and the marginal cost with respect to quality is increasing.
Assume that each consumer purchases one unit of the product offered by firms so as tomaximize the net benefits of the ownership and usage of the product. A consumer’s max-imum willingness to pay for the product, denoted θ , is distributed according to a functionF(θ ) over the interval [θ, θ̄ ]. The willingness to pay of each consumer is private informationand firms only know the distribution function of the willingness to pay for the product priorto production.
An emission tax is imposed on consumers using fuel for their products or consumers pur-chasing products produced by firm G receive a subsidy that is proportional to the differencein environmental quality between the product of firm G and that of firm B.2 The emissiontax has then a direct impact on consumer usage of the product, while the subsidy has a directimpact on consumer choice of the product.
Assuming that each consumer’s income net of the purchase cost of the product is allo-cated between the expenditure on fuel for the usage of the product and the expenditure onthe composite good, her utility maximization conditional on the choice of firm i’s product isdescribed as:
max.U (xi , zi ; D − ai ) + θai − γ Exi , zi (2)
subject to
y − pi + d · s(aG − aB) ≥ (ω + teρ)xi + pzzi ,
where U is utility obtained from consuming fuel and the composite good, zi is compositegood consumption conditional on the choice of firm i’s product, y is income, pi is the priceof firm i’s product, ω is the price of fuel, pz is the price of the composite good, γ is the socialvaluation of the marginal damage from pollution, E is aggregate emissions, te is an emissiontax rate (te ≥ 0), s is a subsidy rate (s ≥ 0), and d is a binary variable that takes one if aconsumer chooses firm G’s product and zero otherwise.
In (2), the consumption of each good is conditional on each consumer’s choice of theproduct. Importantly, although consumers incur damage from pollution, they do not takeaccount of the effect of their product usage on aggregate emissions when undertaking utilitymaximization.3For the rest of our analysis, emissions per fuel consumption, ρ, are assumedto be one.4
To obtain a discrete–continuous model that is consistent with utility maximization in (2),we start from a parametric specification of a fuel demand function and treat Roy’s identity asa partial differential equation whose solution defines a conditional indirect utility function(Hausman 1981, p. 667; Dubin and McFadden 1984, p. 348). The fuel demand conditionalon the choice of firm i’s product is
1 Bansal (2008) assumes a more general form of the cost function, i.e., C (ai , qi ) = c (ai ) qi , where c′ (ai ) >
0, c′′ (ai ) > 0, c (0) = 0 and c′ (0) = 0, in the basic model, but applies such a specific functional form as in(1) to the welfare analysis of an ad valorem tax–subsidy and an emission tax.2 The form of a subsidy is the same as that assumed in Lombardini-Riipinen (2005). See Kuhn (2005, pp.111–119) for the examples of emission taxes and subsidies in practice.3 The assumption of the constant marginal damage from pollution is in line with Moraga-Gonzalez andPadron-Fumero (2002), Lombardini-Riipinen (2005) and Bansal (2008).4 Note that if ρ = 1, emission factors are equal to D − ai . Thus, emission factors differ between two firms.
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The Welfare Effects of Environmental Taxation on a Green Market 91
xi = x0 (D − ai ) − x1 (ω + te) /pz, i = G, B, (3)
where x0 > 0 and x1 > 0 denote parameters. Following the empirical literature that appliesa discrete–continuous model to consumer choice and usage of energy-using durables (Dubinand McFadden 1984; Hensher et al. 1990), the fuel demand conditional on the choice of firmi’s product is assumed to be the sum of two terms: Xi , the typical fuel consumption associatedwith firm i’s product, and f (·), a function of fuel price, an emission tax rate, and the priceof the composite good. Xi is assumed to be the product of two terms: x0, the typical usageof the product which is independent of product choice, and D − ai , the fuel consumptionper usage associated with product i . The variable x0 depends on consumer characteristicsincluding the number of household members and residential location. f (·) is assumed to be−x1 (ω + te) /pz in which fuel price and a tax rate are deflated by the price of the compositegood.
The assumption that the fuel demand function in (3) does not include any income effectneeds some clarification. The empirical literature on a discrete–continuous model reportsstatistically insignificant coefficients associated with income variables. The literature alsoreports extremely small income elasticities. In the analysis of household choice and usageof vehicles, Hensher et al. (1990) found that income elasticities ranged from 0.05 to 0.14and that they were statistically insignificant for most cases. In the analysis of householdchoice and usage of electric water heaters, Dubin and McFadden (1984) found that incomeelasticities ranged from 0.008 to 0.079 and that they were statistically insignificant for allmodels. These statistically insignificant and small estimates of income effects on fuel demandindicate the validity of the fuel demand function that does not explicitly take account of anyincome effect. In fact, Train and Mehrez (1994) assume a linear demand function withoutany income effect in their analysis of a discrete–continuous model of residential electricitydemand.
The demand function (3) facilitates analysis of the effects of alternative policy options onproduct qualities and social welfare. The inclusion of the income effect in the fuel demandfunction makes a welfare analysis of these policy options hard to conduct because of non-linearity in the indirect utility function. To see this, suppose the following functional form ofthe fuel demand function conditional on the choice of the product:
xi = x0 (D − ai ) − x1 (ω + te) /pz + β (Yi/pz) , i = G, B, (3′)
where Yi ≡ y − pi +d ·s(aG −aB). The fuel demand function in (3′) accounts for an incomeeffect on fuel demand in the parameter β. A demand function similar to (3′) is applied to theempirical analysis of household ownership and usage of energy-using durables in Dubin andMcFadden (1984) and Hensher et al. (1990). The conditional indirect utility function thatcorresponds to (3′) is (Hausman 1981, p. 668):5
ui = exp[−β (ω + te) /pz]{β (Yi/pz) − x1/β + x0 (D − ai ) − x1 (ω + te) /pz}/β+θai − γ E,
i = G, B,
5 Another functional form that includes an income effect is
ui = −exp[x0 (D − ai )
] [(ω + te) /pz
](1−x1)/ (1 − x1) + (Yi /pz)
(1−β) /(1 − β) + θai − γ E,
which leads to a log-linear demand function (Hanemann 1984, p. 551). Non-linearity in the first-order conditionwith respect to te makes it hard to obtain the analytical solution of the socially optimal emission tax.
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where ui≡ max.[U (xi , zi ; D − ai ) + θai − γ E | (ω + te)xi + pzzi ≤ Yi]. Social welfare
includes the term exp[−β(ω + te)/pz], which leads to non-linearity in the first-order condi-tion for welfare maximization with respect to te.6
The demand function in (3) indicates that the higher the quality of the chosen product, theless the demand for fuel. This implies, for instance, that a consumer choosing an automobilewith better fuel economy uses less gasoline, all other things being equal. The fuel demandalso depends on consumer characteristics. For example, age of a driver and residential loca-tion were found to affect vehicle use (Hensher et al. 1990), while the number of householdmembers was found to affect the usage of electrical appliances (Dubin and McFadden 1984).These effects are represented by x0 in (3). An emission tax directly affects fuel demand in(3) while a subsidy on the purchase of firm G’s product affects fuel demand only throughconsumer choice of the product and has no direct impact on fuel demand.
Given the choice of firm i’s product, each consumer has a conditional indirect utilityfunction
ui = Vi[(ω + te) /pz, Yi/pz; D − ai
] + θai − γ E, i = G, B. (4)
Applying Roy’s identity to the conditional indirect utility function yields
− (∂ui/∂ω) / (∂ui/∂y) = xi , i = G, B. (5)
Since pz (∂ui/∂ω) = ∂Vi/∂[(ω + te) /pz
]and pz (∂ui/∂y) = ∂Vi/∂ (Yi/pz), from (3), (4)
and (5),
−∂Vi/∂[(ω + te)/pz
]
∂Vi/∂(Yi/pz)= x0(D − ai ) − x1
(ω + te
pz
), i = G, B. (6)
Along a path of price change to stay on the indifference curve that corresponds to some levelof conditional utility, denoted ui0, we have
∂Vi
∂[(ω + te)/pz]d
(ω + te
pz
)+ ∂Vi
∂(Yi/pz)d
(Yi
pz
)= 0, i = G, B, (7)
which is derived from totally differentiating (4), given D, ai , θ, E and ui = ui0. Then, from(6) and (7), the variable Yi/pz can be expressed as a function of the variable (ω + te) /pz :
d(Yi/pz)
d[(ω + te)/pz] = x0(D − ai ) − x1
(ω + te
pz
), i = G, B, (8)
The differential equation (8) is solved to find
Yi/pz = c + x0 (D − ai )[(ω + te) /pz
] − 0.5x1[(ω + te) /pz
]2, i = G, B, (9)
where c, the constant of integration, depends on the utility level. By choosing c = ui −θai +γ E , we find the conditional indirect utility function
ui = Yi/pz−x0 (D−ai )[(ω + te) /pz
] + 0.5x1[(ω + te) /pz
]2 + θai−γ E, i = G, B.
(10)
For the conditional indirect utility function in (10) to be consistent with the utility maximi-zation problem in (2), it must be continuous and homogeneous of degree zero in income, fuel
6 The profit-maximizing quality of the product for each firm isam
G = exp[β(ω + te)/pz ](5θ̄ − θ)/(4b) + (βs − x0)/(bβ),
amB = exp[β(ω + te)/pz ](5θ − θ̄ )/ (4b) + (βs − x0)/(bβ).
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The Welfare Effects of Environmental Taxation on a Green Market 93
price, the price of the composite good, product price, a subsidy rate and an emission tax rate.This condition is automatically met by normalization using pz as numeraire (Hausman 1981,p. 668). The indirect utility function must also be non-decreasing in income and non-increas-ing in prices. It is non-decreasing in income because ∂ui/∂y = 1/pz > 0, and non-increasingin prices because
∂ui/∂ω = {−x0 (D − ai ) + x1[(ω + te) /pz
]}/pz = −xi/pz < 0,
∂ui/∂pz = {−pzYi + pz x0 (D − ai ) (ω + te) − x1 (ω + te)2} /p3
z = −zi/pz < 0,
as long as xi > 0 and zi > 0. Further, the indirect utility function must be quasi-convexin prices. The quasi-convexity condition indicates that the diagonal elements of the Slut-sky matrix must be non-positive. The diagonal elements of the Slutsky matrix, which areobtained from the expenditure function corresponding to the indirect utility function in (10),are −x1/pz < 0 for fuel price and −x1 (ω + te)2 /p3
z < 0 for the price of the compositegood. Thus, the quasi-convexity condition is met by the indirect utility function in (10).
As in Lombardini-Riipinen (2005), we assume that the market of the product is fullycovered, because we focus on externality caused by the usage of the product, which requiresthe ownership of the product. Moreover, the analysis of policy effects on product qualityis hard to conduct under the partially covered market because the analytical solution of theprofit-maximizing quality for each firm is difficult to obtain in the case of the partially cov-ered market. If both firms’ products are sold in the market, there exists some consumer whois indifferent between purchasing firm G’s product and firm B’s product. We denote thisconsumer’s willingness to pay for the product as θ1. Then, the following condition musthold:
θ1 = (pG − pB)/(aG − aB) − s − x0 (ω + te) . (11)
The price of the composite good is normalized to one for the rest of our analysis.Assuming that the market is fully covered and F (θ) is uniform, the demand functions for
firms G and B are written as
qG (pG , pB , aG , aB) =θ∫
θ1
d F(θ) = (θ − θ1)/(θ − θ), (12)
qB(pG , pB , aG , aB) =θ1∫
θ
d F(θ) = (θ1 − θ)/(θ − θ). (13)
Consumers with θ ≥ θ1 would purchase the product of firm G while those with θ ≤ θ1 wouldpurchase that of firm B. The demand function qi (pG , pB , aG , aB) implies the market shareof firm i , and the total output of the industry is constant because of the assumption of a fullycovered market. Substituting (11) into (12) and (13) yields the following demand functions:
qG (pG , pB , aG , aB) =[θ − pG − pB
(aG − aB)+ s + x0(ω + te)
](1
θ − θ
), (14)
qB(pG , pB , aG , aB) =[
pG − pB
(aG − aB)− θ − s − x0(ω + te)
] (1
θ − θ
). (15)
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94 I. Matsukawa
2.2 Market Equilibrium
We assume that a two-stage duopoly game derives the profit-maximizing product prices andqualities. In the first stage, firms simultaneously determine qualities to maximize their profits,given the environmental policy chosen by the government. The firms then simultaneouslydetermine their product prices to maximize their profits in the second stage after observingeach other’s product quality. This two-stage game is solved through backwards induction.First, the equilibrium product prices in the second stage are computed, given the qualities.Second, the equilibrium prices are described as functions of qualities and these functions areinserted into each firm’s profit, whose maximization leads to the equilibrium qualities in thefirst stage.
The profit function of firm i , denoted πi , is given by
max.πi = (pi − 0.5ba2i )qi
(pi, p j , ai , a j
), i, j = G, B; i �= j.
pi , ai (16)
The first-order conditions for maximizing the profits of firms in (16) with respect to theproduct prices in the second stage are given by
[θ + s + x0 (ω + te)
](aG − aB) − 2pG + pB + 0.5ba2
G = 0, (17)
pG − 2pB − [θ + s + x0 (ω + te)](aG − aB) + 0.5ba2B = 0. (18)
From (17) and (18), the product prices in the first stage of competition are
pG = {(aG − aB)[2θ − θ + s + x0 (ω + te)] + ba2G + 0.5ba2
B}/ 3, (19)
pB = {(aG − aB)[θ − 2θ − s − x0 (ω + te)] + 0.5ba2G + ba2
B}/ 3. (20)
Using (19) and (20), the product demands can be written as
qG = [2θ − θ + s + x0 (ω + te) − 0.5b (aG + aB)
]/[3(θ − θ)
], (21)
qB = [θ − 2θ − s − x0 (ω + te) + 0.5b (aG + aB)
]/[3(θ − θ)
]. (22)
Using the equilibrium product prices in (19) and (20) and the product demands in (21) and(22), firm i’s profit in the first stage of competition, denoted i , can be written as a functionof product qualities alone:
G (aG , aB) = (aG − aB)[2θ − θ + s + x0 (ω + te) − 0.5b (aG + aB)
] 2/[9(θ − θ)
],
(23)
B (aG , aB) = (aG − aB)[θ − 2θ − s − x0 (ω + te) + 0.5b (aG + aB)
] 2/[9(θ − θ)
].
(24)
The first-order conditions for maximizing the profits in (23) and (24) with respect to productqualities in the first stage of competition are given by
2θ − θ + s + x0 (ω + te) + 0.5b (aG + aB) − 2baG = 0, (25)
θ − 2θ − s − x0 (ω + te) − 0.5b (aG + aB) + 2baB = 0. (26)
As shown in Appendix A, the second-order conditions for profit maximization also hold atthe market equilibrium.
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The Welfare Effects of Environmental Taxation on a Green Market 95
The solution to the system of equations (25) and (26) yields the Nash equilibrium qualityof the product of firm i , denoted am
i :
amG = [5θ − θ + 4s + 4x0 (ω + te)]/(4b), (27)
amB = [5θ − θ + 4s + 4x0 (ω + te)]/(4b). (28)
The product quality for firm G is always strictly positive, because θ < θ by assumption. Theproduct quality for firm B at the equilibrium is also strictly positive, because the condition
θ >(
59
)θ must hold for full market coverage (Bansal 2008, footnote 15, p. 355).
Differentiating (27) and (28) with respect to te or s yields the effect of an emission tax ora subsidy on the quality of each firm’s product. While both firms’ qualities increase with theemission tax rate and the subsidy rate, thereby reducing aggregate emissions, the effect ofan emission tax is different from that of a subsidy.
Proposition 1 The improvement in the product quality of each firm by an increase in theemission tax rate, te, exceeds that by an increase in the subsidy rate, s, for x0 > 1. For x0 < 1, the quality improvement for each firm by an increase in the subsidy rate exceeds that by anincrease in the emission tax rate.
Proof From (27) and (28), ∂ami /∂s = 1/b and ∂am
i /∂te = x0/b, for i = G, B. Thus,
∂ami
∂te
>
<
∂ami
∂s⇔ x0
>
<1, i = G, B.
The effect of an emission tax on product quality in this study is different from that in
Lombardini-Riipinen (2005) and Bansal (2008). Both previous studies assume that the effectof an emission tax depends only on the cost parameter, b, and they do not take account ofthe impact of consumer usage on product choice. In contrast, the effect of an emission taxin this study depends on both the typical usage of the product, x0, and the cost parameter,because the typical usage of the product affects consumer choice of the product. The largerthe typical usage of the product, the larger the effect of an emission tax on product quality.
The difference in product quality between two firms at the market equilibrium, amG − am
B ,indicates the degree of product differentiation.7 As the degree of product differentiationdecreases, competition between firms intensifies. From (27) and (28), the degree of productdifferentiation is given by
amG − am
B = 1.5(θ − θ)/b. (29)
Equation (29) implies that neither an emission tax nor a subsidy affects the degree of productdifferentiation.
Substituting the product qualities in (27) and (28) into the demand functions in (21) and(22) yields the equilibrium output of firm i , denoted qm
i . This output corresponds to the mar-ket share of each firm, and qm
G = qmB = 0.5 at the market equilibrium. This equality of the
market share between two firms, which is due to the assumption of a quadratic form of thecost function, is also found in Lombardini-Riipinen (2005) and Bansal (2008). Substitutingthe equilibrium qualities into (11) yields the taste parameter θ1 at the market equilibrium,denoted θm
1 , which is equal to 0.5(θ + θ).
7 The degree of product differentiation can also be defined as the ratio of firm G’s product quality to firm B’sproduct quality (Moraga-Gonzalez and Padron-Fumero 2002; Kuhn 2005, p. 36).
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3 Welfare Analysis of Emission Taxes and Subsidies
3.1 Social Welfare
We define social welfare as the sum of five components: (i) the net benefits for consumersfrom ownership and usage of environmentally differentiated products and consumption ofthe composite good, (ii) the profits of the duopolists offering their products, (iii) the prof-its of firms supplying fuel associated with the usage of the environmentally differentiatedproducts, (iv) the profits of firms supplying the composite good, and (v) the government’sbudget surplus, which is positive with a tax and negative with a subsidy. Tax revenues areredistributed to consumers as a lump sum and subsidy expenditures are financed by a lump-sum tax imposed on consumers. We define the net benefit of each consumer as her utility lessexpenditure.
Assuming that the marginal costs of producing fuel and the composite good are constantand that both the fuel and composite good markets are perfectly competitive, social welfare,denoted W , can be written as
W = 0.5x1(ω2 − t2
e ) − x0ωD − γ E
+⎛
⎜⎝
θ∫
θ1
θaGd F +θ1∫
θ
θaBd F
⎞
⎟⎠ + x0ω
⎛
⎜⎝
θ∫
θ1
aGd F +θ1∫
θ
aBd F
⎞
⎟⎠
−θ∫
θ1
0.5ba2Gd F −
θ1∫
θ
0.5ba2Bd F, (30)
where aggregate emissions of the pollutant can be written as
E =θ∫
θ1
eGd F +θ1∫
θ
eBd F = x0 D − x1(ω + te) − x0
θ∫
θ1
aGd F − x0
θ1∫
θ
aBd F . (31)
Appendix B describes the derivation of social welfare in (30). Substituting (31) into (30)yields
W = W0 + x1te(γ − 0.5te) +⎛
⎜⎝
θ∫
θ1
θaGd F +θ1∫
θ
θaBd F
⎞
⎟⎠
+x0(ω + γ )
⎛
⎜⎝
θ∫
θ1
aGd F +θ1∫
θ
aBd F
⎞
⎟⎠ − 0.5b
⎛
⎜⎝
θ∫
θ1
a2Gd F +
θ1∫
θ
a2Bd F
⎞
⎟⎠ , (32)
where W0 ≡ x1ω (0.5ω + γ ) − x0 D (ω + γ ).Setting te = s = 0 and solving the system of the first-order conditions for maximizing
social welfare in (32) with respect to aG , aB , and θ1 gives the socially optimal values of thesevariables, denoted a∗
G , a∗B , and θ∗
1 :
a∗G = [3θ + θ + 4x0 (ω + γ )]/ (4b), (33)
a∗B = [θ + 3θ + 4x0 (ω + γ )]/ (4b), (34)
123
The Welfare Effects of Environmental Taxation on a Green Market 97
θ∗1 = 0.5(θ + θ). (35)
As shown in Appendix C, the second-order conditions for welfare maximization hold andthe corner solutions are excluded at the social optimum by assuming D to be sufficientlylarge. Comparing the socially optimal qualities of the products in (33) and (34) with themarket equilibrium qualities in (27) and (28) under no environmental regulation (i.e., settingte = s = 0) implies that quality competition alone in the product market, which leads to thesocially optimal allocation of consumers between qualities because θm
1 = θ∗1 , fails to achieve
the socially optimal provision of product qualities. Consequently, regulatory intervention isrequired to correct for this distortion in environmental quality.
3.2 Optimal Policy
Consider first an emission tax imposed on the consumption of the fuel associated withthe usage of the product. The socially optimal emission tax can be derived from the first-order condition of maximizing social welfare in (32) with respect to te. Inspection of thefirst-order condition for welfare maximization with respect to te indicates the equality ofthe second-best emission tax with the marginal damage from pollution (see the proof inAppendix D), and this is consistent with the literature (Lombardini-Riipinen 2005; Bansal2008). The imposition of a small emission tax is welfare improving because from (D1),(∂W/∂te)
∣∣te=0 = γ [x1 + (x20/b)] > 0 .
Next, consider a subsidy provided to consumers choosing the product of firm G. The first-order condition for maximizing social welfare in (32) with respect to s leads to the optimalsubsidy. The provision of a small subsidy to consumers purchasing G’s product raises socialwelfare because from (E1) in Appendix E, (∂W/∂s) |s=0 = γ (x0/b) > 0 . In contrast withthe second-best emission tax, the following proposition suggests that the second-best sub-sidy deviates from the social valuation of the marginal damage from pollution except wherex0 = 1:
Proposition 2 The socially optimal subsidy rate equals the social valuation of the marginaldamage from pollution if and only if x0 = 1.If x0 �= 1, the socially optimal subsidy ratedeviates from the social valuation of the marginal damage from pollution.
Proof See Appendix E.
The deviation of the optimal subsidy from the social valuation of the marginal damage frompollution found in this study contrasts with Lombardini-Riipinen (2005) argument that theoptimal subsidy rate must be equal to the social valuation of the marginal damage from pol-lution. To see the difference in the optimal subsidy between the previous literature and thisstudy, consider the following decomposition of welfare effects of a subsidy:
∂W
∂s= ∂
∂s
⎛
⎜⎝
θ∫
θ1
θaGd F +θ1∫
θ
θaBd F
⎞
⎟⎠ + x0(ω + γ )
∂
∂s
⎛
⎜⎝
θ∫
θ1
aGd F +θ1∫
θ
aBd F
⎞
⎟⎠
−0.5b∂
∂s
⎛
⎜⎝
θ∫
θ1
a2Gd F +
θ1∫
θ
a2Bd F
⎞
⎟⎠ . (36)
The first two terms on the right-hand side of (36) represent positive effects of a subsidyon social welfare through quality improvement. These positive effects are associated with
123
98 I. Matsukawa
three items: (i) the utility derived from the purchase of the product, which is equal to(θm
1 /b),
(ii) the damage from aggregate emissions, which is equal to[(x0γ ) /b
], and (iii) the utility
derived from product usage, which is equal to [(x0ω) /b]. The last term, which is equal to− [(
θm1 + s + x0ω
)/b
], represents adverse impacts of a subsidy on social welfare through
quality improvement. These negative effects are associated with production costs.The first two terms and the last term on the right-hand side of (36) offset each other, and
∂W/∂s = x0 (γ /b) − (s/b). Since Lombardini-Riipinen (2005) assumes that a reductionin the damage by one unit increase in product quality is independent of product usage, thepositive effect associated with the damage is equal to (γ /b), thereby leading to the equalityof the optimal subsidy with γ . In contrast, by assuming that a reduction in the damage byone unit increase in product quality depends on the typical product usage, this study findsthat the positive effect associated with the damage is equal to x0 (γ /b), thereby leading tothe deviation of the optimal subsidy from γ . The optimal subsidy rate increases along withthe typical usage of the product.
Note that the optimal emission tax rate is equal to the social valuation of the marginal dam-age from pollution. Consider the following decomposition of welfare effects of an emissiontax:
∂W
∂te= ∂
∂te
⎛
⎜⎝
θ∫
θ1
θaGd F +θ1∫
θ
θaBd F
⎞
⎟⎠ + x0(ω + γ )
∂
∂te
⎛
⎜⎝
θ∫
θ1
aGd F +θ1∫
θ
aBd F
⎞
⎟⎠
−0.5b∂
∂te
⎛
⎜⎝
θ∫
θ1
a2Gd F +
θ1∫
θ
a2Bd F
⎞
⎟⎠ + x1
∂
∂te
[te(γ − 0.5te)
]. (37)
The first three terms on the right-hand side of (37) represent welfare effects of an emissiontax through quality improvement. The fourth term represents an additional reduction in thedamage by the imposition of an emission tax on fuel consumption, net of a deadweight lossdue to an emission tax in the fuel market. The first two terms and the third term on theright-hand side of (37) offset each other, and ∂W/∂te = x2
0
[(γ /b) − (te/b)
] + x1 (γ − te).Thus, the optimal emission tax rate is equal to γ .
3.3 Welfare Comparison
If the government is restricted to the implementation of only a single environmental policyoption, which policy option should it choose from the viewpoint of social welfare? Thissection compares the social welfare of alternative policy options. The following propositionsupports the welfare dominance of the second-best emission tax over the second-best subsidy:
Proposition 3 The socially optimal emission tax is always welfare dominant over the sociallyoptimal subsidy.
Proof See Appendix F.
The welfare outcomes of the second-best emission tax and the second-best subsidy foundin this study contrast with Lombardini-Riipinen (2005) argument that both policies lead to anidentical level of social welfare. The reason is that Lombardini-Riipinen (2005) focuses onthe emissions of the pollutant in the production process, such that both an emission tax anda subsidy mitigate the externality through the improvement in product quality. Thus, both
123
The Welfare Effects of Environmental Taxation on a Green Market 99
policies have the same impact on social welfare. In contrast, we focus on the emissions ofthe pollutant from consumer usage of the products, which is only affected by an emissiontax, given the product qualities. Accordingly, an emission tax reduces environmental dam-age by improving product qualities and discouraging product usage while a subsidy reducesenvironmental damage only by improving the product qualities.
The difference in social welfare between the second-best emission tax and the second-bestsubsidy corresponds to a reduction in environmental damage, given by x1t∗e γ , less the dead-weight loss associated with the emission tax, given by 0.5x1t∗2
e . As t∗e = γ , the emission taxleads to an additional gain in terms of social welfare, given by 0.5x1γ
2, through the reductionin environmental damage.8 However, if the pollutant subject to environmental regulation isonly emitted by the duopolists’ production activities, and an emission tax is imposed on theduopolists, the fuel market is irrelevant to an emission tax, and so an emission tax leadsto neither an additional reduction in environmental damage nor a deadweight loss in thefuel market. This is the case assumed in Lombardini-Riipinen (2005) who argues that boththe second-best emission tax and the second-best subsidy achieve the same level of socialwelfare.
The literature on green markets often investigates the effects of ad valorem taxes–subsi-dies that are applied to firms producing environmentally differentiated products. Under thead valorem tax–subsidy policy, the profit of firm i is written as [(1−t)pi −0.5ba2
i ]qi , where tdenotes the rate of the ad valorem tax–subsidy and t < 1. A negative value of t indicates an advalorem subsidy. Appendix G summarizes welfare effects of ad valorem taxes–subsidies. Assuggested by Lombardini-Riipinen (2005) and Bansal (2008), an ad valorem tax is imposedon firms if the marginal damage from pollution is relatively small, otherwise an ad valoremsubsidy is provided to the firms. As shown by Fig. 1 in Appendix G, if the marginal damagefrom pollution is relatively small, the second-best ad valorem tax dominates both the second-best emission tax and the second-best subsidy. However, the second-best ad valorem subsidy,which is provided to duopolists when the marginal damage from pollution is relatively large,is always dominated by both the emission tax and the subsidy in terms of social welfare.Although an ad valorem subsidy contributes to a reduction in aggregate emissions throughan improvement in environmental quality, it is the last policy chosen from the viewpoint ofsocial welfare.
4 Conclusion
Using a discrete–continuous model that treats consumer ownership and usage of durables in autility-consistent manner, this study conducts a welfare analysis of alternative policy optionsin the market for environmentally differentiated products supplied by duopolists when thepollutant subject to environmental regulation is emitted from consumer usage of products.This study focuses on two policy options that have been widely used as an economic instru-ment for controlling pollutant emissions: an emission tax imposed on consumers that reducesemissions of the pollutant by constraining consumer usage of the products, and a subsidyprovided to consumers choosing the ‘clean’ product that reduces emissions of the pollutantby providing consumers with an incentive to choose the product with a lower emission factor.
8 Assuming a linear demand function in (3) and a constant marginal cost of producing fuel in a perfectlycompetitive market, the deadweight loss caused by an emission tax can be written as 0.5 · (x1te) · te, whichequals a reduction in consumer surplus.
123
100 I. Matsukawa
We find that the second-best emission tax always dominates the second-best subsidy interms of social welfare. Although both an emission tax and a subsidy contribute to a reduc-tion in the environmental damage by inducing firms to improve their product qualities, anemission tax leads to an additional gain from reducing environmental damage by constrain-ing consumer usage of the same products. This additional gain exceeds the deadweight lossassociated with the distortion of the fuel market, thereby favoring the use of an emission taxover a subsidy in terms of social welfare. We also find that the second-best subsidy deviatesfrom the social valuation of the marginal damage from pollution, because of the impact ofthe typical product usage on a reduction in the damage through quality improvement.
Emission taxes and subsidies have been widely applied as environmental policy optionsin many countries (Kuhn 2005, pp. 111–119). The welfare dominance of an emission taxover a subsidy implies that an emission tax is more effective than a subsidy from a welfarepoint of view, especially where concern about the emission of the pollutant is closely relatedto consumer usage of energy-using durables, such as automobiles and electrical appliances.Future research in this area could apply the discrete–continuous model developed in thisstudy to an empirical analysis of the welfare improvements achieved by emission taxes andsubsidies in the market for energy-using durables.
Appendices
Appendix A: Second-Order Conditions for Profit Maximization
Using (23) and (24), we obtain
∂2G
∂a2G
=[ −2b
9(θ − θ)
] [2θ − θ + s + x0(ω + te) − b(3aG + aB)
4
], (A1)
∂2B
∂a2B
=[ −2b
9(θ − θ)
] [θ − 2θ − s − x0(ω + te) + b(aG + 3aB)
4
], (A2)
∂2G
∂aG∂aB= ∂2B
∂aB∂aG=
[b2
18(θ − θ)
](aG − aB). (A3)
Inserting the equilibrium quality ami in (A1), (A2) and (A3) yields
∂2G
∂a2G
= ∂2B
∂a2B
= −b
4< 0, (A4)
∂2G
∂aG∂aB= ∂2B
∂aB∂aG= b
12. (A5)
Inequality (A4) indicates that the second-order conditions for profit-maximization hold atthe market equilibrium. The sufficient condition for the stability of the market equilibriumis written as
(∂2G
∂a2G
) (∂2B
∂a2B
)
−(
∂2G
∂aG∂aB
) (∂2B
∂aB∂aG
)> 0. (A6)
Inequality (A6) holds at the market equilibrium because from (A4) and (A5),(
∂2G
∂a2G
) (∂2B
∂a2B
)
−(
∂2G
∂aG∂aB
)(∂2B
∂aB∂aG
)= b2
18> 0.
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The Welfare Effects of Environmental Taxation on a Green Market 101
Appendix B: Derivation of Social Welfare in (30)
Using (3) and (10) with pz = 1, the net benefits for consumers can be written as
θ∫
θ1
(uG − y)d F +θ1∫
θ
(u B − y)d F
=⎛
⎜⎝
θ∫
θ1
θaGd F +θ1∫
θ
θaBd F
⎞
⎟⎠−
θ∫
θ1
[pG − s(aG − aB) + x0 D(ω + te) − 0.5x1(ω + te)2]d F
−θ1∫
θ
[pB + x0 D(ω+te)−0.5x1(ω+te)2]d F +x0(ω+te)
⎛
⎜⎝
θ∫
θ1
aGd F +θ1∫
θ
aBd F
⎞
⎟⎠ − γ E
=⎛
⎜⎝
θ∫
θ1
θaGd F +θ1∫
θ
θaBd F
⎞
⎟⎠+x0(ω + te)
⎛
⎜⎝
θ∫
θ1
aGd F +θ1∫
θ
aBd F
⎞
⎟⎠+
⎛
⎜⎝
θ∫
θ1
s(aG − aB)d F
⎞
⎟⎠
+0.5x1 (ω + te)2 − x0 D (ω + te) −
⎛
⎜⎝
θ∫
θ1
pGd F +θ1∫
θ
pBd F
⎞
⎟⎠ − γ E, (B1)
where y = pi − d · s (aG − aB) + (ω + te) xi + zi for i = G, B at the equilibrium. Becauseof the constant marginal costs of producing fuel and the composite good, the profits of firmsin perfectly competitive markets for fuel and the composite good become zero. The sum ofthe net benefits for consumers in (B1), the duopolists’ profits, and the government’s budgetsurplus equals social welfare in (30).
Appendix C: Second-Order Conditions for Welfare Maximization and Exclusion of CornerSolutions
Differentiating social welfare in (32) with respect to ai yields:
∂W
∂aG=
[θ − θ1
2(θ − θ)
][θ + θ1 − 2baG + 2x0(ω + γ )
], (C1)
∂W
∂aB=
[θ1 − θ
2(θ − θ)
] [θ1 + θ − 2baB + 2x0(ω + γ )
]. (C2)
The first-order condition for welfare maximization with respect to θ1 is given by
∂W
∂θ1=
[aB − aG
2(θ − θ)
] [2θ1 − b(aB + aG) + 2x0(ω + γ )
] = 0. (C3)
From (C3), the socially optimal value for θ1, denoted by θ∗1 , becomes 0.5b (aG + aB) −
x0 (ω + γ ). Substituting θ∗1 for θ1 in (C1) and (C2) and differentiating (C1) and (C2) with
respect to ai yields
123
102 I. Matsukawa
∂2W
∂a2G
=[ −b
4(θ − θ)
] [4θ + 4x0(ω + γ ) − b(3aG + aB)
], (C4)
∂2W
∂a2B
=[ −b
4(θ − θ)
][b(aG + 3aB) − 4θ − 4x0(ω + γ )
], (C5)
∂2W
∂aG∂aB= ∂2W
∂aB∂aG= b2(aG − aB)
4(θ − θ), (C6)
At the social optimum, these derivatives in (C4), (C5) and (C6) become
∂2W
∂a2G
= ∂2W
∂a2B
= −3b
8< 0, (C7)
∂2W
∂aG∂aB= ∂2W
∂aB∂aG= b
8. (C8)
From (C7) and (C8),
(∂2W
∂a2G
) (∂2W
∂a2B
)
−(
∂2W
∂aG∂aB
)(∂2W
∂aB∂aG
)= b2
8> 0. (C9)
Inequalities (C7) and (C9) indicate that the second-order conditions for welfare maximi-zation hold at the social optimum.
By assuming D to be sufficiently large so that D > [x0 (ω + γ ) + (3θ + θ)/4]/b,
(∂W/∂aG)∣∣aG=D < 0 . Setting ai to zero in (C1) and (C2) leads to (∂W/∂aG)
∣∣aG=0 > 0and (∂W/∂aB)
∣∣aB=0 > 0 . Thus, corner solutions for product qualities are excluded at thesocial optimum. Corner solutions for θ1 are excluded at the social optimum, because setting
θ1 to θ in (C3) with ai = a∗i leads to (∂W/∂θ1)
∣∣∣θ1=θ < 0 , and setting θ1 to θ in (C3) with
ai = a∗i leads to (∂W/∂θ1)
∣∣θ1=θ > 0 .
Appendix D: Proof of the Equality of the Second-Best Emission Tax Rate with the SocialValuation of the Marginal Damage from Pollution
Setting s to zero and substituting (27) and (28) into (32) yields
W = x1te(γ − 0.5te) + W0
+7(θm
1
)2 − 3θθ + 4θm1 x0(ω + te) + 4x0(ω + γ )
[θm
1 + x0(ω + te)]
4b
−13(θm
1
)2 − 9θθ + 4x0(ω + te)[2θm
1 + x0(ω + te)]
8b. (D1)
The first-order condition for maximizing social welfare in (D1) with respect to te yieldst∗e = γ , where t∗e denotes the welfare-maximizing rate of the emission tax.
123
The Welfare Effects of Environmental Taxation on a Green Market 103
Appendix E: Proof of Proposition 2
Setting te to zero and substituting (27) and (28) into (32) yields
W = W0 + 7(θm
1
)2 − 3θθ + 4θm1 (s + x0ω) + 4x0(ω + γ )
(θm
1 + s + x0ω)
4b
−13(θm
1
)2 − 9θθ + 4(s + x0ω)(2θm
1 + s + x0ω)
8b. (E1)
The first-order condition for maximizing social welfare in (E1) with respect to s yieldss∗ = x0γ , where s∗ denotes the welfare-maximizing subsidy rate.
Appendix F: Proof of Proposition 3
Substituting t∗e = γ into (D1) yields social welfare under the welfare-maximizing emissiontax rate, denoted W
(t∗e
). Similarly, substituting s∗ = x0γ into (E1) yields social welfare
under the welfare-maximizing subsidy rate, denoted W (s∗). The difference in social welfarebetween the optimal emission tax and the optimal subsidy can be written as:
W(t∗e
) − W(s∗) = x0γ
(x0t∗e − s∗)
b− x2
0 t∗2e − s∗2
2b+ x1t∗e
(γ − 0.5t∗e
). (F1)
Using t∗e = γ and s∗ = x0γ, the welfare difference in (F1) can be rewritten as
W(t∗e
) − W(s∗) = 0.5x1γ
2 > 0. (F2)
Appendix G: Welfare Effects of Ad Valorem Taxes–Subsidies
Effects on Product Quality
Under an ad valorem tax–subsidy, the Nash equilibrium quality of the product of each firmis:
amG = (1 − t)[5θ − θ + 4s + 4x0 (ω + te)]/(4b), (G1)
amB = (1 − t)[5θ − θ + 4s + 4x0 (ω + te)]/(4b). (G2)
An ad valorem tax (subsidy) reduces (raises) both firms’ qualities. This effect of an advalorem tax–subsidy is also found in Arora and Gangopadhyay (1995), Moraga-Gonzalez andPadron-Fumero (2002), Bansal and Gangopadhyay (2003), Lombardini-Riipinen (2005), andBansal (2008). Other things being equal, the imposition of an ad valorem tax on duopolistsraises aggregate emissions. The degree of product differentiation is given by
amG − am
B = 1.5(1 − t)(θ − θ)/b. (G3)
Equation (G3) implies that an ad valorem tax (subsidy) decreases (increases) the degreeof product differentiation.
123
104 I. Matsukawa
The Second-Best Ad Valorem Tax–Subsidy
Setting te and s to zero and substituting (G1) and (G2) into (32) yields
W = W0 +(1 − t)
[7
(θm
1
)2 − 3θθ + 4θm1 x0ω + 4x0(ω + γ )(θm
1 + x0ω)]
4b
−(1 − t)2
[13
(θm
1
)2 − 9θθ + 4x0ω(2θm1 + x0ω)
]
8b. (G4)
The first-order condition for maximizing social welfare in (G4) with respect to t yieldsthe welfare-maximizing rate of the ad valorem tax–subsidy, denoted by t∗:
t∗ =6
[(θm
1
)2 − θθ]
− 4γ x0(x0ω + θm
1
)
13(θm
1
)2 − 9θθ + 4x0ω(x0ω + 2θm
1
) . (G5)
From (G5), t∗ = γ if and only if γ = γt , where
γt ≡6
[(θm
1
)2 − θθ]
13(θm
1
)2 − 9θθ + 4x20ω(ω + 1) + 4θm
1 x0(1 + 2ω), (G6)
and
t∗ >
<0 ⇔ γ
<
>
1.5[(
θm1
)2 − θθ]
x0(x0ω + θm
1
) .
Thus, the socially optimal ad valorem tax–subsidy deviates from the marginal damage frompollution except where γ = γt . As long as the marginal damage from pollution is small, theadverse effect of the tax on the externality through the deterioration in quality is more thanoffset by a welfare improvement through increased competition. This leads to a decrease inthe degree of product differentiation. Thus, an ad valorem tax should be imposed on firmsif the marginal damage is relatively small. In contrast, an ad valorem subsidy should beprovided to firms if the marginal damage is relatively large, because a quality improvementthrough the ad valorem subsidy contributes to a reduction in environmental damage, and thisexceeds the decrease in welfare from the market imperfection. These results are consistentwith Lombardini-Riipinen (2005) and Bansal (2008). The provision of a small ad valoremsubsidy to the duopolists raises social welfare if and only if
γ > 1.5[(
θm1
)2 − θθ] / [
x0(x0ω + θm
1
)].
Welfare Comparison
The social welfare under the welfare-maximizing rate of the ad valorem tax–subsidy iscompared with that under the welfare-maximizing rate of the emission tax or that underthe welfare-maximizing rate of the subsidy. Substituting (G5) into (G4) yields social welfareunder the welfare-maximizing rate of the ad valorem tax–subsidy, denoted W (t∗). The differ-ence in social welfare between the optimal ad valorem tax–subsidy and the optimal emissiontax can be written as:
123
The Welfare Effects of Environmental Taxation on a Green Market 105
W(t∗
) − W(t∗e
)
=4
[4x2
0
(x0ω+θm
1
)2−(
x20 +bx1
)(B1+B3)
]γ 2−8x0(B1−B2)
(x0ω+θm
1
)γ +(B1−B2)2
8b(B1+B3),
(G7)
where B1 ≡ 13(θm1 )2 − 9θθ , B2 ≡ 7(θm
1 )2 − 3θθ , and B3 ≡ 4x0ω(x0ω + 2θm1 ). Note
that B1 > B2 > 0. The welfare difference, W (t∗) − W(t∗e
), in (G7) can be regarded as a
quadratic function of the marginal damage from pollution, which is concave because
4x20
(x0ω + θm
1
)2 − (x20 + bx1)(B1 + B3) = −bx1(B1 + B3) − 9x2
0
[(θm
1
)2 − θθ]
< 0.
The welfare difference, W (t∗)− W(t∗e
), intersects the vertical axis at
(0,
(B1−B2)2
8b(B1+B3)
), and
it has a negative slope at this point. Define
γ1 ≡ (B1 − B2)
2
[√(x2
0 + bx1)(B1 + B3) + 2x0(x0ω + θm
1
)] > 0. (G8)
Then, W (t∗) − W(t∗e
)><
0 ⇔ γ <>
γ1.As long as the marginal environmental damage is relatively large, the second-best emis-
sion tax dominates the second-best ad valorem subsidy in terms of social welfare. This isbecause an emission tax corrects for the distortion arising from the externality more than anad valorem subsidy. In contrast, with relatively small marginal damage, the second-best advalorem tax dominates the second-best emission tax. This is because the distortion arisingfrom imperfect competition, which is not mitigated by an emission tax but by an ad valoremtax, is relatively more important than the distortion arising from the externality.
Turning to a welfare comparison between the second-best subsidy and the second-bestad valorem tax–subsidy, the difference in social welfare between the optimal ad valoremtax–subsidy and the optimal subsidy can be written as:
W(t∗
) − W(s∗)
=4
[4x2
0
(x0ω+θm
1
)2−x20 (B1+B3)
]γ 2 − 8x0(B1 − B2)
(x0ω + θm
1
)γ + (B1 − B2)
2
8b(B1 + B3).
(G9)
The welfare difference, W (t∗) − W (s∗), in (G9) can be regarded as a quadratic functionof the marginal damage from pollution, which is concave because
4x20
(x0ω + θm
1
)2 − x20 (B1 + B3) = −9x2
0
[(θm
1
)2 − θθ]
< 0.
The welfare difference, W (t∗)− W (s∗), intersects the vertical axis at(
0,(B1−B2)2
8b(B1+B3)
), and
it has a negative slope at this point. Define
γ2 ≡ (B1 − B2)
2x0[√
B1 + B3 + 2(x0ω + θm
1
)] > 0. (G10)
Then, W (t∗) − W (s∗) ><
0 ⇔ γ <>
γ2.As long as the marginal environmental damage is relatively large, the second-best subsidy
on the purchase of firm G’s product dominates the second-best ad valorem subsidy in terms of
123
106 I. Matsukawa
Fig. 1 Welfare comparison of the three policy options
social welfare. In contrast, the second-best ad valorem tax dominates the second-best subsidywith relatively small marginal damage.
Figure 1 summarizes the results of the welfare comparison of the three second-best poli-cies. The inequality γ1 < γ2 holds because from (G8) and (G10),
γ2 − γ1 =(B1 − B2)
[√(x2
0 + bx1)(B1 + B3) − x0√
B1 + B3
]
2x0
[√(x2
0 + bx1)(B1 + B3) + 2x0(x0ω + θm
1
)]
[√B1 + B3 + 2
(x0ω + θm
1
)]
> 0. (G11)
As shown in Fig. 1, as long as the marginal damage from pollution is modest, the advalorem tax dominates the other policy options. However, as the marginal damage increases,the optimal policy shifts from the ad valorem tax to the emission tax. In fact, if the marginaldamage from pollution is substantial, both the emission tax and the subsidy dominate the advalorem subsidy in terms of social welfare, because B1−B2
4x0(x0ω+θm1 )
> γ2 > γ1. The outcomes
of the welfare comparison of an emission tax, a subsidy and an ad valorem tax–subsidy alsodepend on the price of fuel. Since ∂γ1/∂ω < 0 and ∂γ2/∂ω < 0, the results of the welfarecomparison in Fig. 1 indicate that an increase (decrease) in the fuel price weakens (strength-ens) the welfare dominance of the socially optimal ad valorem tax over the socially optimalemission tax and the socially optimal subsidy.
Acknowledgments The author gratefully acknowledges comments from the editor and two anonymousreviewers. The remaining errors are solely those of the author’s. This study was supported by the Grant-in-Aidfor Scientific Research (C) of the JSPS KAKENHI (20530217), and by the Musashi University ResearchProject.
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