Household heterogeneity, aggregation, and the distributional impacts of environmental taxes

Transcription

1 Houseold eterogeneity, aggregation, and te distributional impacts of environmental taxes Sebastian Rausc 1,2,, Giacomo Scwarz 1,2 June 8, 2015 Abstract Tis paper examines te incidence of an environmental tax using teoretical and numerical general equilibrium models tat allow for eterogeneous ouseolds, fully general forms of preferences, differential spending and income patterns, differential factor intensities in production, and fully general forms of substitution among inputs of capital, labor, and pollution. First, we focus on te ouseold aggregation problem and find tat te incidence of an environmental tax can be qualitatively affected by te level of consistency wit wic ouseold eterogeneity is integrated into te analysis. Distributional impacts of environmental taxes based on partial and general equilibrium analyses tat fail to consistently integrate ouseold eterogeneity are tus likely to be biased significantly. Second, we apply te eterogeneous ouseold model to analyze te distributional effects of a U.S. carbon tax. We find strong evidence tat suc a tax would be regressive. Wile tis result is robust wit respect to varying ouseolds and firms caracteristics, te regressivity is dampened considerably if labor is a good substitute for pollution relative to capital. Keywords: Environmental taxes, General equilibrium, Heterogeneous ouseolds, Distributional effects, Sources side, Uses side, Non-omotetic preferences JEL: H23, Q52 1. Introduction Te public acceptance for environmental taxes depends crucially on teir distributional consequences. A pletora of applied researc in public and environmental economics as investigated te incidence of environmental taxes in various policy settings. Not seldom, owever, te empirical evidence weter a specific tax is regressive or not is mixed even if te incidence of a given tax instrument is analyzed in a similar or identical policy context. Differences arise because te incidence analysis does not consider all relevant cannels troug wic an environmental tax affects market outcomes (see, e.g., Atkinson & Stiglitz (1980) and Fullerton & Metcalf (2002) for a discussion of incidence impacts in te public finance literature). Environmental taxes often appear to be regressive on te uses side of income as tey affect more eavily te welfare of te poorest ouseolds tan of te ricest ones, since poorer ouseolds spend a larger fraction of Corresponding autor: Department of Management, Tecnology, and Economics, ETH Zuric, Züricbergstrasse 18, ZUE E, 8032 Zuric, Switzerland, Pone: addresses: srausc@etz.c (Sebastian Rausc), giscwar@etz.c (Giacomo Scwarz) 1 Department of Management, Tecnology, and Economics and Center for Economic Researc at ETH Zuric. 2 Joint Program on te Science and Policy of Global Cange, Massacusetts Institute of Tecnology, Cambridge, USA.

2 teir income on polluting goods (e.g., energy or electricity). Sources side of income impacts can dampen or even offset te regressive incidence on te uses side to te extent tat environmental tax policies affect te returns to factors of production tat are disproportionately owned by ricer ouseolds and used intensively in te production of dirty relative to clean industries (e.g., capital). 3 Partial equilibrium assessments, comprising Input-Output analysis as a widely used metod, ignore general equilibrium effects, including sources side impacts. 4 Incidence analyses using a general equilibrium framework typically employ a single, representative ouseold model to derive factor and output price canges wic are used in an ex-post calculation to determine tax burdens across ouseolds. A problem wit suc an approac is tat it neglects te impact of ouseold eterogeneity on te equilibrium allocation, tus ignoring a potentially relevant cannel for tax incidence. Despite te ig policy relevance and academic interest for understanding te incidence of price-based pollution control policies, a teoretical analysis of te distributional impacts of environmental taxes in a general equilibrium setting wit eterogeneous ouseolds is lacking. 5 Tis paper develops a teoretical general equilibrium model in te spirit of Harberger (1962) of te incidence of an environmental tax tat features eterogeneous ouseolds, fully general forms of preferences, differential spending and income patterns, differential factor intensities in production, and fully general forms of substitution among inputs of capital, labor, and pollution. Its purpose is two-fold. First, we investigate te implication of te ouseold aggregation problem for te general equilibrium incidence of environmental taxes. In te absence of identical omotetic preferences for eac individual or omotetic preferences and collinear initial endowment vectors (i.e., identical income sares), aggregated preferences depend on te distribution of income (Polemarcakis, 1983). 6 Tus acknowledging eterogeneity in tastes undercuts te representative consumer framework tat is used to calculate te general equilibrium effects on output and factor prices (Kortum, 2010). How severe is te aggregation problem, i.e. to wat extent are incidence results derived from a general equilibrium analysis wic ignores ouseold eterogeneity and te potential for non-omotetic preferences biased? Second, we apply te eterogeneous ouseold model to assess te incidence of a tax on carbon dioxide (CO 2 ) emissions in te United States. Is a US carbon tax regressive? We assess te incidence on te sources and uses side, and explore ow sensitive incidence results are wit respect to key caracteristics governing ouseolds and firms beavior. Our paper builds on a small but growing literature tat uses analytical general equilibrium models to study te incidence of environmental taxes. Our model builds on a series of influential papers by Fullerton and oters (Fullerton & Heutel, 2007, 2010; Fullerton et al., 2012; Fullerton & Monti, 2013) tat extend te Harberger (1962) model and previous teoretical work by Rapanos (1992, 1995) to develop a model 3 Te regressivity of many environmental taxes on te uses side, including carbon pricing in te context of climate policy, constitutes a serious concern for policymakers and as been investigated extensively in te literature (Poterba, 1991; Metcalf, 1999; Fullerton et al., 2012). Gasoline taxes are generally found to be progressive on te uses side (Sterner, 2012). More recently, work by Fullerton & Heutel (2007), Araar et al. (2011), and Rausc et al. (2011) as also scrutinized te sources side impacts of carbon taxation. 4 Williams III & Goulder (2003) sows tat under typical conditions te simple excess-burden triangle formula substantially underestimates te excess burden of commodity taxes, in some cases by a factor of 10 or more, because it ignores general equilibrium interactions most important, interactions between te taxed commodity and te labor market. 5 Tis paper focuses on te incidence of an environmental tax itself, i.e., we do not consider te question of ow recycling te revenue from an environmental tax can sape distributional impacts across ouseolds. Wile a number of studies ave sown tat various revenue recycling options ave te potential to offset unintended distributional consequences (for example, Metcalf, 2007; Parry & Williams III, 2010; Rausc et al., 2010b), te coice and design of revenue recycling constitute a separate and distinct policy issue wic first requires an understanding of te incidence impacts per se. 6 On a more fundamental conceptual level, and not related to te incidence of (environmental) taxation, te aggregation problem for eterogeneous consumers in general equilibrium models as been studied by Ackermann (2002) based on prior work by Rizvi (1994) and Martel (1996). 2

3 wic represents pollution as an input along wit capital and labor and tat allows for general forms of substitution between inputs. We extend te model presented in Fullerton & Heutel (2007) wic assumes a single consumer to include eterogeneous ouseolds. We additionally allow for preferences to be nonomotetic, terefore going beyond te standard assumption of omotetic preferences employed by most analyses tat assess te incidence of environmental taxes. By fully integrating ouseold eterogeneity in te model, our paper also differs from te contributions in Fullerton & Heutel (2010) and Fullerton et al. (2012) tat use price impacts derived from te single-consumer model in Fullerton & Heutel (2007) to determine te burdens of a carbon tax on ouseolds using ouseold survey data from te Consumer Expenditure Survey. Fullerton & Monti (2013) do integrate two types of ouseolds into an analytical general equilibrium model and investigate te distributional impacts of a pollution tax swap were revenues are recycled troug a (pre-existing) wage tax of low-income workers; tey do not, owever, focus on te impact of ouseold eterogeneity on equilibrium outcomes. Te present paper is also related to te literature tat uses computational metods to investigate te distributional impacts of environmental taxes. A widespread approac is to use Input-Output analysis to derive price canges for different consumers goods wic are ten applied to calculate tax burdens for ouseolds based on micro-ouseold survey data. Examples include Robinson (1985) wo studies te distributional burden of industrial abatement in te U.S. economy or, related to energy and CO 2 emissions taxes, Poterba (1991) wo analyzes te incidence of gasoline taxes; Bull et al. (1994); Hassett & Metcalf (2009) compare a tax based on energy content and a tax based on carbon, and Metcalf (1999, 2009) analyze a revenue-neutral package of environmental taxes, including a carbon tax, an increase in motor fuel taxes, and taxes on various stationary source emissions. Dinan & Rogers (2002) assess te efficiency and distributional impacts of a U.S. cap-and-trade program for CO 2 emissions, and Matur & Morris (2014) investigate te distributional effects of a carbon tax in broader U.S. fiscal reform. Oter works study te incidence impacts of greenouse gas emissions pricing policies across ouseold income groups for different countries (e.g., Labandeira & Labeaga (1999) for Spain, Callan et al. (2009) for Ireland, and Jiang & Sao (2014) for Cina). Common to tese studies is tat tey adopt a partial equilibrium perspective tat does not consider beavioral canges, focuses on te uses sides of te incidence only, and employs welfare metrics tat are not consistent wit utility-maximizing micro-economic beavior. A few papers use numerical general equilibrium models to derive price impacts on commodity and factor prices. Metcalf et al. (2008) carry out an analysis of carbon tax proposals and find tat a carbon tax is igly regressive but tat te regressivity is reduced due to sources side effects to te extent tat resource and equity owners bear some fraction of te tax burden. Similarly, Araar et al. (2011); Dissou & Siddiqui (2014) use te prices effects generated to assess te distributional impact of a carbon tax on ouseolds. None of tese studies, owever, consider te impact of ouseold eterogeneity on equilibrium outcomes as teir analysis is based on a single, representative consumer general equilibrium model. Lastly, some papers do consistently incorporate eterogeneous ouseolds into a numerical general equilibrium framework tus capturing te impact of ouseold eterogeneity on equilibrium prices as well as te uses and sources side impacts. For example, Rausc et al. (2010a,b) investigate te incidence of a US carbon tax in a model wit nine ouseolds representing different income classes and find tat te overall impact is neutral to modestly progressive due to sources side effects (including inflation-indexed government transfers). Williams III et al. (2014, 2015) and Ciroleu-Assouline & Foda (2014) employ calibrated overlapping generations models to assess te distributional incidence across generations but tey do not model intra-coort eterogeneity. A major weakness of analyses based on numerical simulation models is tat tey ave to rely on specific functional forms wit limited forms of substitution. In contrast, te present paper studies te incidence of an environmental tax in a teoretical model wit general forms of 3

4 substitution among inputs for bot firms and ouseolds. Moreover, none of tese studies investigates te impact of ouseold eterogeneity on equilibrium outcomes and te ouseold aggregation problem. Te key result of te present paper is tat te ouseold aggregation problem can ave severe implications for analyzing te incidence of environmental taxes: basing te analysis on a single, representative ouseold model can yield bot qualitative and quantitative conclusions tat differ from a general equilibrium model wic consistently integrates ouseold eterogeneity. Te important implication for analyzing environmental tax policy is tat bot partial equilibrium metods, including Input-Output analysis, and general equilibrium analyses tat fail to consistently integrate ouseold eterogeneity sould be used wit care or best be avoided altogeter. We use teoretical and numerical analyses to derive and support our results. In te case of omotetic preferences we sow tat te impact of ouseold eterogeneity on te equilibrium allocation can be caracterized by two statistical quantities wic reflect te degree of ouseold eterogeneity in terms of expenditure and income sares and te second-order properties of ouseolds preferences (i.e., te utility function). Tese metrics provide an intuitive way to express te deviation between te realistic case wit eterogeneous ouseolds and a ypotetical case of identical ouseolds. We provide examples of conditions for ouseolds and firms caracteristics determining teir respective equilibrium coices in response to an environmental tax under wic te aggregation bias does or does not matter. For example, for cases wit limited substitutability between inputs of capital, labor, and pollution in production, factor and output price canges can be reversed in turn yielding qualitatively different incidence results across groups of poor and ric ouseolds. Moreover, we find tat tere exist for (almost) any bencmark economy, specified by production data and te distribution of preferences and endowments, plausible values for production elasticities suc tat factor price canges derived from a single, representative ouseold model are of opposite sign relative to te model wit eterogeneous ouseolds. We furtermore find tat allowing preferences to be non-omotetic can qualitatively affect te burden of te environmental tax on factors of production, tus igligting te potential importance of relaxing te assumption of omotetic preferences in order to fully capture te general equilibrium incidence of an environmental tax. Using an empirically calibrated version of te teoretical model, based on istoric data for te U.S. economy, we find strong evidence tat a U.S. carbon tax would be regressive. Tis result is relatively robust to varying caracteristics of bot ouseolds and firms. Our analysis, owever, points to te importance of including sources of income impacts for tax incidence analysis. As we find tat sources side effects tend to be progressive, basing te incidence analysis on te uses side effects only may overestimate te regressivity of te environmental tax and can lead to qualitatively false conclusions. On te oter and, our finding tat sources side impacts drive most of te variation in welfare impacts calls for a careful model specification of ow bot polluting and non-polluting firms respond to an environmental tax. Te remainder of tis paper is organized as follows. Section 2 presents te model. Section 3 derives closed-form expressions to assess te incidence of an environmental tax, and presents and interprets our teoretical results. Section 4 uses an empirically calibrated version of te teoretical model to derive additional results by means of numerical analysis. Section 5 concludes. 2. Model We consider a static and closed economy wit two sectors and two factors of production. A clean good is produced using capital and labor, and a dirty good is produced using capital, labor and pollution. Capital and labor are supplied inelastically and are mobile across sectors. Te government taxes pollution, returning te revenue lump-sum to ouseolds. Our general equilibrium model follows closely Harberger (1962) and Fullerton & Heutel (2007) but differs in two important aspects. First, we introduce eterogeneous 4

5 ouseolds tat differ in terms of teir preferences and income patterns derived from endowments of capital and labor. Second, we generalize te representation of ouseold beavior by allowing for non-omotetic preferences. Using log-linearization, we analytically solve for first-order canges in equilibrium prices and quantities following an exogenous cange in te pollution tax rate. Our model allows us to quantify te general equilibrium incidence of te environmental tax in te context of an economy wit no a-priori restrictions placed on te number and caracteristics of ouseolds. Te clean sector production function X = X(K X, L X ) and te dirty sector production function Y = Y(K Y, L Y, Z) are assumed to exibit constant returns to scale, were K X, K Y, L X, and L Y are te quantities of capital and labor used in eac sector. 7 Let indexes n and m denote sectors and factors of production, respectively. Te total amounts of factors of production in te economy are exogenously given and fixed: K X + K Y = K and L X + L Y = L. Totally differentiating te resource constraints yields: ˆK X K X K + ˆK Y K Y K = 0 (1) ˆL X L X L + ˆL Y L Y L = 0, (2) were a at denotes a proportional cange, e.g., ˆK X dk X /K X. Pollution (Z) as no equivalent resource constraint and is a coice of te dirty sector. To ensure a finite use of pollution in equilibrium, we assume a pre-existing positive tax on pollution, τ Z. Firms in sector X can substitute between factors in response to canges in te wage rate (w) and capital rental rate (r) according to an elasticity of substitution in production, σ X. Differentiating te definition for σ X yields: ˆK X ˆL X = σ X (ŵ ˆr). (3) Te production decision of firms in sector Y depends additionally on te pollution price tey face, wic is given by te pollution tax rate τ Z. We model te coice between te tree inputs of capital, labor and pollution by means of te Allen elasticities e i j between inputs i and j (Allen, 1938). Te 3 3 matrix of Allen elasticities is symmetric (i.e., e i j = e ji ), its diagonal entries are less or equal to zero (i.e., e ii 0), and at most one of te tree independent off-diagonal elements can be negative. Furtermore, e i j is positive wenever inputs i and j are substitutes, and negative wenever tey are complements. Totally differentiating input demand functions for sector Y, wic describe te dirty sector s cost minimization problem, and dividing by te appropriate input level, yields: 8 ˆK Y Ẑ = θ YK (e KK e ZK )ˆr + θ YL (e KL e ZL )ŵ + θ YZ (e KZ e ZZ )ˆτ Z (4) ˆL Y Ẑ = θ YK (e LK e ZK )ˆr + θ YL (e LL e ZL )ŵ + θ YZ (e LZ e ZZ )ˆτ Z, (5) were θ mn is te sare of sector m s revenue paid to factor n, e.g. θ XK = rk X p X X. Let p X and p Y denote output prices for X and Y, respectively. Under te assumption of perfect competition, te following expressions old: ˆp X + ˆX = θ XK (ˆr + ˆK X ) + θ XL (ŵ + ˆL X ) (6) ˆp Y + Ŷ = θ YK (ˆr + ˆK Y ) + θ YL (ŵ + ˆL Y ) + θ YZ (ˆτ Z + Ẑ) (7) ˆX = θ XK ˆK X + θ XL ˆL X (8) Ŷ = θ YK ˆK Y + θ YL ˆL Y + θ YZ Ẑ. (9) 7 Note tat te production side of our model is te same as for te single-consumer model of Fullerton & Heutel (2007). In describing production we tus follow closely te model description in Fullerton & Heutel (2007, pp ). 8 Appendix Appendix A in Fullerton & Heutel (2007) derives equations (4)-(9). 5

6 Houseolds, indexed by = {1,..., H}, maximize utility by coosing optimal consumption of goods X and Y subject to an income constraint. 9 Eac ouseold inelastically supplies fixed factor endowments K and L wic satisfy te following relations: K = K and L = L. Income for ouseold is terefore given by M = w L + r K + ξ τ Z Z, were ξ is te sare of te pollution tax revenue redistributed lump-sum to ouseold. Since te tax revenue is returned entirely to ouseolds, it follows tat ξ = 1. Before proceeding, we empasize tat te focus of our analysis in tis paper is on understanding te pure incidence of a pollution tax, i.e., witout te impact resulting from ow te tax revenue is redistributed. To isolate te pure incidence impacts, we terefore assume tat te redistribution of tax revenues occurs in a way tat does not influence te relative burdens across ouseolds. 10 Following Hicks & Allen (1934), we parameterize non-omotetic consumer preferences for te two goods using te elasticity of substitution between goods X and Y in utility σ, and te income elasticities of demand for goods X and Y, denoted by E X,M and E Y,M respectively.11 Appendix A derives te following expressions for canges in demand by ouseold in response to output and factor price canges: wit ˆM = ŵ w L M ˆX Ŷ = σ ( ˆp Y ˆp X ) + (E Y,M E X,M )(α ˆp X + (1 α ) ˆp Y ˆM ) (10) ˆX = (α E X,M + (1 α )σ ) ˆp X [(1 α )E X,M (1 α )σ ] ˆp Y + E X,M ˆM, (11) + ˆr r K M + τ Z Z p X X+p Y Y (ˆτ Z + Ẑ). Finally, totally differentiating te market clearing conditions for te two consumption goods, X = X and Y = Y, yields: ˆX = Ŷ = X X ˆX (12) Y Y Ŷ. (13) Equations (1) (13) are H equations in H unknowns ( ˆK X, ˆK Y, ˆL X, ˆL Y, ŵ, ˆr, ˆp X, ˆX, ˆp Y, Ŷ, Ẑ, H ˆX, H Ŷ ). Following Walras Law, one of te equilibrium conditions is redundant, tus te effective number of equations is H. We coose X as te numéraire good, wic implies ˆp X = 0. Te square system of model equations ten endogenously determines all te above unknowns as functions of bencmark parameters (caracterizing te equilibrium before te tax cange), beavioral parameters (elasticities of production and consumption), and te exogenous positive cange in te pollution tax (ˆτ Z > 0). 3. Analytical results and interpretations Wen solving for te model unknowns as functions of te exogenous tax cange, we are ultimately interested in te distributional incidence of te environmental tax. Let v denote te indirect utility function 9 We assume tat pollution, or environmental quality, is separable in utility, tus not influencing te optimal consumption coice. Te incidence analysis carried out in tis paper tus focuses on utility derived from market consumption only. 10 Te redistribution sceme wic ensures tat all ouseolds are impacted in te same proportion relative to teir income is ξ given by: = 1 M 1. M p X X+p Y Y 11 Homotetic preferences are represented by te special case EX,M = E Y,M = 1. In tis case te first-order beavior of ouseolds can be sufficiently described by σ, as for example in Fullerton & Heutel (2007). 6

7 of ouseold, and dv te cange in utility caused by an increase in te pollution tax rate by dτ Z. To compare impacts across ouseolds, we express utility canges in monetary terms relative to income: dv M M v measures te amount of income wic would cause a cange in utility equal to dv at prices prior to te tax cange, expressed relative to te income of ouseold. To isolate te distributional dimension from te economy-wide cost of te tax, we focus on te welfare impact of eac ouseold relative to te average welfare cange. Tis ensures tat results do not depend on te coice of numéraire. We can ten write te welfare impact of ouseold relative to te average economy-wide monetary loss per unit of income as: 12 Φ dv M M v 1 M dv = (γ α ) ˆp Y M v } {{ } =Uses of income impact + (θ L θ L)ŵ + (θ K θ K)ˆr, (14) } {{ } =Sources of income impacts were θ K r K and θ M L w L r K are te capital and labor income sares of ouseold, and θ M K p X X+p Y Y, w L θ L p X X+p Y Y and γ p X X p X X+p Y Y are te value sares of capital, labor and te clean sector in te economy. Te welfare decomposition underlying equation (14) enables an intuitive economic interpretation of te various cannels troug wic ouseold caracteristics determine incidence. On te one and, for given canges in te prices of goods and factors, variation in impacts across ouseolds arises for two reasons. First, ouseolds differ in ow tey spend teir income. For a given increase in te price of te dirty good ( ˆp Y > 0), consumers of te dirty good are more negatively impacted as compared to consumers of te clean good. Tis impact is referred to as te uses of income impact. Second, in a general equilibrium setting, a pollution tax also impacts factor prices. Houseolds wic rely eavily on income from te factor wose price falls relative to te oter will be adversely impacted compared to te average ouseold. Tese impacts are referred to as sources of income impacts. Since output and factor price canges are not independent of ouseolds caracteristics, two additional and more indirect determinants of incidence emerge from te expression (14). First, in an economy wit eterogeneous ouseolds, output and factor prices are not independent of te distribution of ouseolds consumption profiles and factor endowments across te population; welfare canges for a given ouseold type do not only depend on its own caracteristics but also on tose of oter ouseolds in te economy. Second, even in an economy wit identical ouseolds, te specifics of te ouseold s beavioural response to price and income canges can affect equilibrium outcomes. To analytically study incidence in our model, we now proceed by deriving expressions tat caracterize ow output and factor prices canges depend on model parameters. Appendix B derives te following general solutions for ˆp Y, ŵ and ˆr following a cange in τ Z : ˆp Y = (θ YLθ XK θ YK θ XL )θ YZ D A(e φ Z ZZ e KZ ) B(e ZZ e LZ ) + (γ K γ L )(δ ) θ YZ ˆτ Z +θ YZ ˆτ Z (15a) ŵ = θ XKθ YZ D A(e φ Z ZZ e KZ ) B(e ZZ e LZ ) + (γ K γ L )(δ ) θ YZ ˆτ Z (15b) 12 Recall tat p X is te numéraire. Ten dv = py v dp Y + M v dm = py v p Y ˆp Y + ( M v ŵw L + ˆrr K + ξ τ Z Z[ˆτ Z + Ẑ] ), and use Roy s identity (i.e., py v = Y M v ). Recall furtermore tat te incidence-neutral tax re-distribution sceme is caracterised by ξ = M. M 7

8 ˆr = θ XLθ YZ D A(e φ Z ZZ e KZ ) B(e ZZ e LZ ) + (γ K γ L )(δ ) θ YZ ˆτ Z, (15c) were γ K K Y K X, γ L L Y L X, β L θ XL γ L + θ YL, β K θ XK γ K + θ YK, A γ L β K + γ K (β L + θ YZ φ Z ), B γ K β L +γ L (β K +θ YZ φ Z ), C β K +β L +θ YZ φ Z, D Cσ X +A[θ XK θ YL (e KL e ZL ) θ XL θ YK (e KK e ZK )] B[θ XK θ YL (e LL e ZL ) θ XL θ YK (e LK e ZK )] (γ K γ L )(θ XK (θ YL δ φ L ) θ XL(θ YK δ φ K )). Te remaining expressions depend explicitly on ouseold caracteristics: φ α L (1 γ )E w L X,M p Y Y + Y Y (E Y,M E X,M ) w L M, φ K and δ Y Y α (1 γ )E r K X,M p Y Y + Y Y (E Y,M E X,M ) r K, φ ( M Z σ + ( α γ 1)(σ EX,M ) + (E Y,M E X,M )(1 α ) α (1 ). γ )E X,M ξ τ Z Z p Y Y + Y Y (E Y,M E X,M ) ξ τ Z Z M Note tat in general ŵ = θ XK θ XL ˆr. Tus, in order to understand te burden of te cange in te pollution tax on te returns to factors, it is sufficient to study te cange in te returns to capital, keeping in mind tat given our coice of te numéraire good ŵ always as te opposite sign as ˆr. Wile te interpretation of te general solution is limited by its complexity, it is apparent from te analytical expressions above tat going beyond a single consumer and introducing multiple, eterogeneous ouseolds wit non-omotetic preferences into te model in general as a first-order impact on te market equilibrium, and tus on te incidence results. By considering expressions (15a) (15c) one can identify te following two effects, wic ave also previously been identified in te context of te Harberger (1962) model. Te (γ K γ L )(δ φ Z θ YZ ) term in equations (15b) and (15c) represents te output effect: te tax on sector Y reduces output, and consequently depresses te returns to te factor used intensively in te dirty sector. Te sign of te output effect follows tis intuition only if te denominator D is positive, wic in general is not te case, even for identical ouseolds (or equivalently a single ouseold) and omotetic preferences (Fullerton & Heutel, 2007). Introducing multiple, eterogeneous ouseolds and non-omotetic preferences adds anoter layer of complexity to tis indeterminacy, since δ cannot in general be signed, wereas tis expression is positive for identical ouseolds wit omotetic preferences. Te oter terms in equations (15b) and (15c) embody te substitution effects, wic reflect te reaction of firms to factor price canges. Again, wile for te case wit identical ouseolds and omotetic preferences te constants A and B can be signed as positive, tis is not te case in our more general model. Te substitution effect tus also bears a greater degree of indeterminacy as compared to te Fullerton & Heutel (2007) model. To better understand te various effects at work, it is necessary to depart from te generality of te above expressions. We terefore consider a series of special cases in wic we impose restrictions on ouseold and production caracteristics in order to seek definitive results for te canges in prices and returns to factors, and terefore better understand te implications for incidence. First, we present a special case for production under wic ouseold caracteristics ave no impact on price canges. Second, we consider cases wic allow for full ouseold eterogeneity in terms of preferences and income patterns but were preferences are assumed to be omotetic. Tird, te role of non-omotetic preferences is investigated for cases wit identical ouseolds. Comparing tese special cases will be useful to understand te interaction of production and ouseold caracteristics in determining te canges in output and factor prices, and consequently incidence. Lastly, te most general case described by te combination of non-omotetic preferences and eterogeneous ouseolds is studied by means of numerical analysis. φ Z θ YZ 8

9 3.1. Equal factor intensities in production Consider first te case in wic bot industries ave te same factor intensities, i.e., bot are equally capital and labor intensive. Under tis assumption, te price canges derived from a model wit eterogeneous ouseolds are identical to tose derived from a single ouseold model. Proposition 1. Assume bot sectors ave te same factor intensities, i.e., γ K = γ L. Ten, ˆp Y, ŵ and ˆr are independent of ouseold caracteristics and depend only on production parameters. Proof. If γ K = γ L, ten A = B = γ K C. It ten follows from (15a) (15c) tat all terms containing ouseold caracteristics in te expressions for ˆp Y, ŵ and ˆr cancel out. Proposition 1 implies tat in te case of equal factor intensities across industries, price canges derived from a single ouseold model wit omotetic preferences are sufficient to determine incidence of an environmental tax, even in an economy wit different ouseold types. Intuitively, as long as factor intensities are equal, canges in demands for X and Y do not affect relative demands for capital and labor tus implying tat relative factor prices are unaffected. Factor price canges are tus determined by te first-order response of firms alone, as accounting for first-order ouseold beavioral responses in combination wit first-order firm responses would capture a second-order effect. Te sign of factor price canges terefore depends only on production caracteristics. Incidence remains in general undetermined, since it depends on ow tese price canges affect individual ouseolds, as determined by teir income and expenditure sares Heterogeneous ouseolds wit omotetic preferences How does ouseold eterogeneity affect equilibrium factor and output price canges following a cange in te pollution tax? We focus on te conditions under wic ouseold eterogeneity can qualitatively reverse price canges as compared to a single-ouseold model. Tese cases tus illustrate conditions wic would give rise to significantly biased incidence results wen using as is often done in te literature a simplified general equilibrium tat abstracts from ouseold eterogeneity. To provide a clear intuition, we first restrict our attention to te case wit omotetic preferences. For omotetic preferences, te eterogeneity of ouseolds can be described by te ouseolds population distribution of te tree following ouseold caracteristics: (i) expenditure sares α, (ii) income sares θ L, and (iii) elasticities of substitution in utility σ. 13 Accordingly, we can summarize ouseold eterogeneity by te following two quantities. First, we measure te degree in wic expenditure and income patterns are correlated. To tis end, we define te covariance between te expenditure sare of te clean good and te labor income sare as: cov(α, θ L ) (α γ)m (θ L θ L). Te covariance is, for example, positive if ouseolds wo earn an above average sare of teir income from labor (i.e., θ L > θ L) spend an above average sare of teir income on te clean good (i.e., α > γ). Second, we quantify te interaction between expenditure sares α and substitution elasticities σ by defining te effective elasticity of substitution between clean and dirty goods in utility as: ρ 1 ( ) α (1 α )M p Y Y γ (σ 1) Note tat, for given ξ, a given θ L uniquely determines θ K. 9

10 ρ can be interpreted as a generalized weigted average of te σ s. 14 Proposition 2 proves tat te two quantities cov(α, θ L ) and ρ are indeed sufficient to fully caracterize te impact of ouseold eterogeneity on equilibrium prices and te level of pollution. For omotetic preferences, te system of equations (15a) (15c) caracterizing price canges in te general case simplifies to te following expressions, were te expression for ŵ as been omitted due to its simple relationsip to ˆr (see Appendix C.1 for te derivation): ˆp Y = (θ YLθ XK θ YK θ XL )θ YZ D H [ AH (e ZZ e KZ ) B H (e ZZ e LZ ) + (γ K γ L )ρ ] ˆτ Z + θ YZ ˆτ Z (16a) ˆr = θ XLθ YZ D H [ AH (e ZZ e KZ ) B H (e ZZ e LZ ) + (γ K γ L )ρ ] ˆτ Z, (16b) were A H γ L β K + γ K (β L + θ YZ ), B H γ K β L + γ L (β K + θ YZ ), C H β K + β L + θ YZ, D H C H σ X +A H (θ XK θ YL (e KL e ZL ) θ XL θ YK (e KK e ZK )) B H (θ XK θ YL (e LL e ZL ) θ XL θ YK (e LK e ZK )) (γ K γ L )ρ(θ XK θ YL θ XL θ YK ) (γ K γ L ) cov(α,θ L ) γp Y Y. Proposition 2 ten follows directly: Proposition 2. If preferences are omotetic, te impact of ouseold eterogeneity on output and factor price canges in equilibrium only depends on two quantities describing individual ouseolds caracteristics: (i) te covariance between te expenditure sare of te clean good and te labor income sare, cov(α, θ L ), and (ii) te effective elasticity of substitution between clean and dirty goods in utility, ρ. Proof. Equations (16a) (16b). Using te quantities cov(α, θ L ) and ρ, we are now in a position to investigate one key question of te paper: under wat conditions are price and pollution canges from an economy populated by eterogeneous ouseolds wit omotetic preferences identical to tose derived from an economy wit a single representative ouseold? Te next two propositions describe conditions in terms of ouseold preferences and income patterns under wic models wit and witout ouseold eterogeneity yield identical equilibrium outcomes. Proposition 3. Assume omotetic preferences and (i) identical expenditure sares (α = γ, ) or (ii) identical income sares (θ L = θ L, ). Ten: output and factor price canges are identical to tose for a single ouseold caracterised by omotetic preferences, clean good expenditure sare γ, and elasticity of substitution between clean and dirty goods in utility equal to te effective elasticity ρ. Proof. Eiter of te above assumptions (i) and (ii) implies cov(α, θ L ) = 0. From equations (16a) (16b) it is ten easy to see tat price canges are identical to tose derived for an economy wit a single consumer wit omotetic preferences, clean good expenditure sare γ, and elasticity of substitution in utility ρ. It follows tat in te case wit omotetic preferences and eiter identical expenditure sares or identical income sares (or bot), ouseolds beave in te aggregate as a single representative ouseold caracterized by an elasticity of substitution in utility given by ρ. In te case wit identical expenditure sares, te effective elasticity is equal to te weigted average of te individual ouseolds substitution 1 elasticities: ρ = M M σ. Te resulting aggregate beavior is tus completely independent of patterns of income from capital and labor, and does not depend on te number of ouseolds. Tis result, owever, breaks down if ouseolds ave identical income sares but exibit eterogeneity on te expenditure side. In te latter case, te value of ρ depends on te interaction between expenditure sares α and te substitution elasticities of individual ouseolds σ : if ouseolds wit an above average expenditure 14 To see tis, consider te case wit equal expenditure sares across ouseolds, i.e. α = γ,. Ten, ρ = M σ / M. 10

11 sare on te dirty good ave iger substitution elasticities, te single representative ouseold responds in a more price-elastic manner as compared to a case wit te same σ s but α s tat are identical across ouseolds. Proposition 3 motivates te definition of ρ as well as its interpretation as te effective elasticity of substitution between clean and dirty goods: wen cov(α, θ L ) = 0 tat is wen eiter te ouseolds are identical on te expenditure or te income side (or bot) ten in te aggregate, ouseolds effectively beave like a single ouseold wit substitution elasticity ρ. As will become clear below, tis aggregation result does not old in te more general case for cov(α, θ L ) 0. Wile Proposition 3 describes te conditions for ouseold eterogeneity wic allow for consumer aggregation, it is clear tat in reality consumers differ in ways wic would violate tese conditions. A central question for our purpose of incidence analysis is ten to investigate to wat extent ouseold eterogeneity can reverse output and factor price canges, ence giving rise to qualitatively different outcomes on te uses and sources side of income for eterogeneous ouseolds. Proposition 4. Assume different factor intensities (i.e., γ K γ L ) and correlated income and consumption patterns (i.e., cov(α, θ L ) 0). Assume omotetic, unit-elastic preferences (i.e., σ = 1, ). Ten, for any observed consumption and production decisions before te tax cange, tere exist production elasticities (i.e., σ X and e i j ) suc tat te relative burden on factors of production is opposite compared to te model wit a single consumer, coupled to te same production side data. Proof. See Appendix C.2. Proposition 4 suggests tat for a trutful portrayal of incidence impacts among eterogeneous ouseolds it is pivotal to consider te impact of ouseold eterogeneity on equilibrium outcomes. It proves tat, in te presence of eterogeneous ouseolds, te sources of income impacts from a pollution tax not only differ quantitatively but can yield qualitatively different predictions wen relying on factor price canges derived from a single-ouseold general equilibrium model. Importantly, te possibility of reversed factor price canges does not depend on a particular distribution of ouseolds caracteristics as long as te covariance between income and expenditure patterns is non-zero. It seems to be indisputable tat cov(α, θ L ) 0 is te most relevant case wic describes reality. To furter illustrate te range of (differing) equilibrium outcomes wic depend on te nature and degree of ouseold eterogeneity, we provide an example for a special case of our simple economy. Consider unitelastic preferences (σ = 1) and Leontief production (σ X = e i j = 0). Under tese assumptions, equations (16a) and (16b) for price canges can be written as: ˆp Y = cov(α, θ L ) D H,1 γp Y Y (γ K γ L )θ YZ ˆτ Z (17a) ˆr = θ XLθ YZ D H,1 (γ K γ L )ˆτ Z, (17b) were D H,1 (γ K γ L )(θ XL θ YK θ XK θ YL ) (γ K γ L ) cov(α,θ L ) γp Y Y. Proposition 5 ten follows directly: Proposition 5. Assume omotetic, unit-elastic preferences (i.e., σ = 1), Leontief tecnologies in clean and dirty good production (i.e., σ X = e i j = 0), and tat te dirty sector is relatively capital-intensive (i.e., γ K > γ L ). Ten, te following olds: Note tat for te case were te dirty sector is relatively labor-intensive (i.e., γ K < γ L ), te sign of all te results in Proposition 5 is te opposite. 11

12 (i) if consumers are identical on te sources or uses side of income, or bot: ˆp Y = 0, ŵ > 0, and ˆr < 0. (ii) If labor ownersip and clean good consumption ave a negative covariance, ten ˆp Y > 0, ŵ > 0 and ˆr < 0. (iii) If labor ownersip and clean good consumption ave a positive covariance, ten ˆp Y < 0, ŵ > 0, ˆr < 0 if te covariance is low (i.e., D H,1 > 0), and ˆp Y > 0, ŵ < 0, ˆr > 0 if te covariance is ig (i.e., D H,1 < 0). Proof. Equations (17a) (17b). Proposition 5 illustrates tat, following a cange in te pollution tax, widely differing equilibrium outcomes are possible and depend on te type and degree of ouseold eterogeneity. Depending on assumptions about eterogeneity of ouseolds expenditure and income patterns, one can generate almost any combination of ˆp Y 0, ŵ 0, ˆr 0. Tis suggests tat even in a simple stylized general equilibrium model as ours, te incidence results of a pollution tax cange can bring about quite different qualitative conclusions as far as uses and sources of income side impacts across eterogeneous consumers are concerned. Note tat te results in Proposition 5 old under te assumption of unit-elastic preferences. One can easily sow on te oter and tat for a model wit a single ouseold and Leontief production, ˆp Y = 0. Hence, Proposition 5 provides an example of a case in wic patterns of output and factor price canges derived from an economy wit eterogeneous ouseolds cannot be generated in an economy wit te same production caracteristics, coupled to any single representative consumer wit omotetic preferences. Tis argument provides additional support for our ypotesis tat it is pivotal to consistently integrate ouseold eterogeneity in general equilibrium models wen assessing te incidence of environmental taxes across different ouseold groups Identical ouseolds wit non-omotetic preferences Our results ave so far proven tat ouseold eterogeneity can ave a qualitative result on te market equilibrium following an increase in a pollution tax, wit implications for incidence. We now abstract from ouseold eterogeneity in order to focus on te effect of non-omotetic preferences on te market equilibrium. To acieve tis, we assume tat ouseolds are identical. As te following special case illustrates, accounting for non-omotetic preferences can also qualitatively effect te market equilibrium. Assume tat all cross-price elasticities ave te same positive value c: σ = σ X = e KL = e KZ = e LZ c > 0. Price canges are ten of te following form: ˆp Y = θ XKθ XL γθ YZ D ID [(γ K γ L ) 2 (E Y,M E X,M )]ˆτ Z + θ YZ ˆτ Z (18a) ˆr = θ XLθ YZ D ID [(γ K γ L )(E Y,M E X,M )(1 γ)]ˆτ Z, (18b) were EX,M E X,M and EY,M E Y,M, D ID C ID + A ID θ XL + B ID θ XK + (γ K γ L ) 2 θ XK θ XL 1 γ, A ID τ γ L β K + γ K (β L + θ YZ + (E X,M E Y,M ) Z Z p X X+p Y Y ), B τ ID γ K β L + γ L (β K + θ YZ + (E X,M E Y,M ) Z Z p X X+p Y Y ), τ C ID β K + β L + θ YZ + (E X,M E Y,M ) Z Z p X X+p Y Y. In order to determine te sign of te price canges, we define te following Condition 1: D ID > 0. Condition 1 olds if te expenditure sare on te clean good increase wit income (E X,M > E Y,M ). It also olds wen te clean good expenditure sare decreases wit income (E Y,M > E X,M ), but te difference between te income elasticities is not too large. We can ten prove tat a wide range of possible combinations of output and factor price canges are possible in tis special case, depending on te parameters describing te non-omotetic preferences. 12 γ

13 Proposition 6. Assume identical ouseolds and equal cross-price elasticities (σ = σ X = e KL = e KZ = e LZ c > 0). Ten, te following olds: (i) If preferences are omotetic, ten ˆp Y = θ YZ ˆτ Z, and ŵ = ˆr = 0. (ii) Assume tat te dirty sector is relatively capital-intensive (i.e. γ K > γ L ). 16 (a) If Condition 1 olds, ten for E Y,M > E X,M : ˆp Y < θ YZ ˆτ Z, ŵ > 0 and ˆr < 0, and for E Y,M < E X,M : ˆp Y > θ YZ ˆτ Z, ŵ < 0 and ˆr > 0. (b) If Condition 1 does not old, ten for E Y,M > E X,M : ˆp Y > θ YZ ˆτ Z, ŵ < 0 and ˆr > 0, and for E Y,M < E X,M : ˆp Y < θ YZ ˆτ Z, ŵ > 0 and ˆr < 0. Proof. Equations (18a) (18b). For (i): use E Y,M = E X,M. We ave terefore illustrated tat tere exist cases were te relative burden on factors of production depends crucially on te interaction between production caracteristics and te income elasticities of demand for te clean and te dirty goods. It follows tat, by extending te Fullerton & Heutel (2007) model to incorporate ouseold eterogeneity and non-omotetic preferences, we ave added two dimensions tat can bot qualitatively alter te economy s reaction to an exogenous increase in te pollution tax. Bot features are terefore in general significant for incidence. 4. Numerical analysis We now assign plausible values to parameters to numerically examine te teoretical effects derived above. Given a version of our model wic is calibrated to current data for te U.S. economy and available estimates from te literature, we seek to answer te following questions. First, ow severe is te aggregation bias? In oter words, ow plausible is te assumption tat te overall effect of a pollution tax on factor and output prices wit eterogeneous consumers is te same as in te aggregate model wit a single representative ouseold? Employing te empirically calibrated model as a starting point, we explore ow sensitive te answer is wit respect to te degree of ouseold eterogeneity, te structure of preferences, and te interplay wit production parameters. In doing so, we also focus on te implications of te aggregation bias for incidence analysis, i.e., does using a simplified aggregate model produces qualitatively identical results in terms of a progressive or regressive incidence pattern. Second, we use te calibrated model to assess te incidence of a tax on carbon dioxide (CO 2 ) emissions in te U.S. Is a U.S. carbon tax regressive? Given te considerable uncertainty surrounding te parametrization of firm and ouseold beaviour, we explore te robustness of te incidence result troug sensitivity analysis by varying ouseold and production caracteristics and by identifying te relative importance of uses and sources effects of income Data and Calibration In order to situate our study in te context of te literature, we calibrate our model to data used previously for a two-sector general equilibrium environmental tax incidence analysis. For tis purpose, we cose te production and consumption data of Fullerton & Heutel (2010). Tey aggregate a data set of te U.S. economy to a dirty and a clean sector, were te dirty sector comprises te igly CO 2 -intensive industries (electricity generation, transportation and petroleum refining). As in Fullerton & Heutel (2010) 16 Note tat for te case wit γ K < γ L, te results for ŵ and ˆr are of opposite signs to te analogous expressions in Proposition 6 (ii). Te results for ˆp Y remain uncanged, as long as factor intensities differ (γ K γ L ). 13

14 Table 1: Houseold expenditures on clean and dirty goods and ouseold income by source for annual expenditure deciles (in % of total expenditure for a given ouseold group) Expenditure Income sources Expenditure by commodity decile Labor Capital Clean Dirty Notes: Houseold data is based on Consumer Expenditure Survey (CEX) data as sown in Fullerton & Heutel (2010). we assume an initial and pre-existing carbon tax of $15 per metric ton of CO 2. Our comparative-static analysis considers a 100% increase in te carbon tax. All prices in te bencmark are normalised to one, and quantities are normalised suc tat te total value of te economy is equal to one, i.e., p X X + p Y Y = 1. Calibrated values for outputs and inputs are as follows: X = 0.929, L X = 0.579, L Y = 0.029, K X = 0.350, K Y = 0.037, and Z = Houseolds are grouped by annual expenditure deciles, 17 and data for expenditures by clean and dirty goods as well as capital and labor income are sown in Table 1. Note tat we abstract from government transfers. 18 Incorporating eterogeneous ouseolds in a calibrated general equilibrium model of te U.S. economy requires tat at te aggregate level data describing ouseold consumption and income are consistent wit te production data on output by sector and aggregate, economy-wide factor income. To reconcile data sources, we adjust te ouseold data to be consistent wit aggregate production data wile preserving te relative caracteristics of ouseold expenditures across ouseold groups (expenditure deciles). More specifically, data adjustments for eac expenditure decile are as follows. First, we scale income to mac expenditure wile keeping fixed te decile s capital-to-labor ratio. Second, we scale te capital ownersip of all deciles by a common factor in order for aggregate ouseold income by factor to matc production side data, wilst preserving te relative capital ownersip amongst deciles. Tird, we perform an analogous scaling for consumption of te dirty good. Tis procedure yields a data wit consistent ouseold and production data wic is used to calibrate te general equilibrium model. 17 It is well-known in te literature on tax incidence tat absent a fully dynamic framework, categorizing ouseolds by expenditure deciles is a better proxy for lifetime income as compared to a ranking based on annual income deciles (see, for example, Poterba1991; Fullerton & Heutel, 2010). 18 Incorporating government transfers in te model would require including oter taxes in our analysis in order to finance tese transfer payments. As tax revenue would cange following a cange of te pollution tax, tis would imply tat oter taxes also would ave to adjusted simultaneously to ensure tat te government budget is balanced. We deliberately refrain from including tis aspect in our analysis as it would add significant complexity and pus te paper more towards te issue of (green) tax reform. In abstracting from government transfers, one sould bear in mind tat tey are to a large extent indexed to inflation, tus effectively protecting poorer ouseolds wic receive a iger proportion of transfers from increases in prices for final consumption goods. Adding government transfers to te picture would tus probably make our incidence results look somewat less regressive. 14