Carbon Taxation When Climate Affects Productivity

Transcription

1 Carbo Taxatio Whe Climate Affects Productivity William K. Jaeger ABSTRACT. Based o a model where climate chage affects productivity, the secod-best optimal carbo tax is foud to exceed margial social damage by 53% ad margial private damage (aggregate households willigess to pay) by 73%. Aual welfare gais are estimated at $3.58 billio whe margial damage is $40 per to; employmet also icreases. A carbo tax set at the Pigouvia rate raises welfare by oly $3.17 billio. The seemigly cotradictory results from the tax iteractio literature are show to arise oly whe the optimal evirometal tax is compared to margial private damage, ad oly for a ameity exterality. (JEL Q4, H2) I. INTRODUCTION Global climate chage is thought to be amog the most serious evirometal threats to humaity, oe that presets eormous itellectual ad policy challeges i a complex settig where the stakes are high. The policy questios surroudig climate chage have attaied heighteed importace sice 1997, whe 160 atios siged the Kyoto Protocol ad agreed to sigificatly reduce emissios of greehouse gases. This situatio represets a importat opportuity for ecoomists to provide policymakers with soud aalysis ad clear guidace at a time whe there is growig recogitio of the advatages of market-based policies. I this cotext, there has bee reewed iterest ad cosiderable cotroversy surroudig optimal evirometal taxatio i a secod-best settig with preexistig taxes. Tullock (1967) was the first to suggest that reveues from evirometal taxes could be used to fiace reductios i preexistig reveue-motivated taxes as a way to improve the eviromet ad reduce the welfare costs associated with the overall tax program. This otio, ow widely referred to as the double divided hypothesis, also carries Lad Ecoomics August (3): ISSN by the Board of Regets of the Uiversity of Wiscosi System with it the ituitive iferece that the optimal evirometal tax would geerally exceed margial evirometal damages whe the reveues are used i this way (see Terkla 1984; Lee ad Misiolek 1986; Pearce 1991; ad Oates 1995). Despite the ituitive appeal of the double divided hypothesis, a more recet literature has raised questios about its validity ad its implicatios for settig optimal evirometal taxes. Although the authors of this literature cocur that the use of evirometal tax receipts to fiace reductios i preexistig taxes will ideed have a positive welfare effect, they fid that the optimal evirometal tax lies below, rather tha above, the margial social damage from pollutio eve whe evirometal tax reveues are used to fiace reductios i preexistig taxes (Boveberg ad de Mooij 1994; Parry 1995; Boveberg ad Goulder 1996). To explai this result, the authors postulate that a previously urecogized welfare cost or tax iteractio effect offsets the positive effect associated with the double divided. The uexpected implicatio of these results ad their iterpretatio is that the welfare gais from evirometal tax reform, ad the ability to cotrol pollutio efficietly, is much more limited tha previously thought. 1 If correct, these results are believed to have profoud implicatios ot just for climate chage policy but for may other evirometal ad o-evirometal policies as well (Parry ad Oates 2000). The author is associate professor, Departmet of Agricultural ad Resource Ecoomics, Orego State Uiversity. This paper has beefited from commets ad suggestios by Do Bovee, Ralph Bradburd, Neil Bruce, Hayes Goddard, Va Kolpi, Suzae Leoard, Kerry Smith, ad a aoymous reviewer. This article was origially published as Orego Extesio & Experimet Statio Techical Paper # This ow-extesive literature also icludes Goulder (1995), Fullerto (1997), Schöb (1997), Boveberg ad de Mooij (1997), Goulder, Parry, Burtraw (1997), Parry, Williams, Goulder (1999), Goulder et al. (1999), Parry ad Beto (2000), amog others.

2 78(3) Jaeger: Carbo Taxatio Whe Climate Affects Productivity 355 The approach take i much of this recet tax iteractio literature has bee to specify aalytical or umerical geeral equilibrium models where a evirometal ameity affects utility directly, where labor is the oly productive factor ad source of icome, ad where demads for pollutig ad o-pollutig goods are assumed to be similar (so that i the absece of a exterality, uiform taxatio would be optimal to raise reveues). The curret aalysis employs a similar model, but oe i which climate chage damage reduces welfare by lowerig factor productivity, rather tha as a ameity exterality. For a umerical model of the U.S. ecoomy, a welfare-maximizig algorithm is employed to choose optimal taxes which both raise reveues ad cotrol carbo emissios. The aalysis fids that the optimal carbo tax exceeds margial social damages from climate chage by about 53%, a result cosistet with the double divided hypothesis. These results are also show to be i harmoy with aalytically derived expressios for the optimal evirometal tax. A compariso of the curret aalysis with that from the tax iteractio literature reveals that two differet defiitios of margial social damage are beig used. I the curret aalysis, the defiitio of margial social damage is based o the social margial rate of substitutio betwee icome ad evirometal quality, so that the social margial utility of icome is the umeraire uit of value. By cotrast, the tax iteractio literature uses a defiitio based o the margial private damage for a idividual household, ad the summed over the populatio. The two defiitios will differ wheever there are reveue implicatios from chages i either icome or climate, except at the first-best optimum. Whe the defiitio used i the tax iteractio literature is applied to the curret model, the optimal carbo tax exceeds margial social damages by a eve greater amout, 73%. Ideed, oly for the case of a ameity exterality, ad whe margial damage is defied as the sum of households margial private damage, does the relatioship betwee the optimal carbo tax ad margial damage appear to be icogruous with the double divided hypothesis. The geeral model ad umerical simulatio results are preseted i Sectio 2. Sectio 4 evaluates the differeces with the tax iteractio literature usig both the umerical model ad aalytical results. Sectio 4 cocludes. II. THE MODEL We begi with a climate ecoomy model which cotais two edowmets, icome i the form of household time (T ), ad a global climate. There are m idetical households who allocate their time betwee leisure (V ) ad labor supply (L), so that T L V. Utility is a fuctio of cosumptio (C) ad leisure. Cosumer goods are produced with two itermediate iputs, oe usig fossil fuels (F), ad oe usig o-carbo iputs (N), such that C c(f, N). We ca therefore write utility as U u(c, (T L)). [1] Productio is assumed to be competitive, ad labor is assumed to be the oly iput used to produce the itermediate iputs F ad N. The model is assumed to be for oe period ad thus does ot attempt to solve a explicit dyamic problem. Expected carbo damages are therefore beig characterized as cotemporaeous damages rather tha future cosequeces of cumulative carbo emissios. The structure of this model is similar to the oe i Parry, Williams, ad Goulder (1999). The climate, E, is altered whe carbo is emitted i fixed proportios to the cosumptio of F, such that E mφf. A chage i the climate is assumed to harm the ecoomy due to its adverse effect o factor productivity. May types of expected climate damages ivolve reduced output or losses of productivity, such as losses i the productivity of agriculture ad forestry, losses of productive assets due to sea level rise, reduced productivity of fixed assets ad ifrastructure, the redirectio of productive factors toward defesive ad restorative efforts, or direct reductios i labor productivity from morbidity ad mortality. Ideed, Clie (1992) estimates that about three-fourths of climate chage damages will come from losses to agriculture, forestry, ad damage from sea level rise.

3 356 Lad Ecoomics August 2002 For simplicity, ad because labor is the oly productive factor i the model, we characterize climate damage i terms of its affect o output per uit of labor. Specifically, labor productivity is defied as a fuctio of carbo emissios, or h(e), where dh/de 0. For a give E, the margial product of labor is assumed to be costat so that output is a fuctio of labor supply ad climate. Exogeously determied reveues, mg, are collected usig excise taxes o F ad N. These reveues are retured to households i the form of lump-sum trasfers, rather tha itroducig a explicit ad detailed represetatio of the public sector. Taxes ca be iterpreted as paymets out of labor icome. A tax o F is also a tax o carbo emissios give the fixed proportios relatioship beig assumed. We defie output usig a liear productio fuctio such that for a ecoomy cosistig of m households, aggregate output is mh(e)l mf mn mg. Uits are chose so that margial rates of trasformatio amog F, N, ad G all equal oe. Assumig competitive productio ad with o direct taxes o C, the tax problem will be uchaged if we substitute the productio fuctio ito the utility fuctio to reflect derived demads for F ad N. We ca therefore write the represetative household s maximizatio problem as Max: F,N,V u(c(f, N), V ) s.t.(λ) (1 t F )F (1 t N )N h(e)l G, [2] where λ deotes the private margial utility of icome correspodig to the value of the implied Lagrage multiplier. To maximize social welfare, W, defied as the sum of idividuals utility, the plaer s problem ca ow be writte as Max: t F,t N m Max: F,N,V u(c(f, N), V) s.t.(λ) (1 t F )F (1 t N )N h(e)l G s.t.(µ) mt F F mt N N mg. [3] Let µ deote the shadow value of a uit of public reveue correspodig to the shadow value of the implied Lagrage multiplier for the reveue costrait. For [3] we ca express the social margial utility of carbo emissios as π where π dw dh mλ de de L mµ t F F (hl) (hl) E t N N (hl) (hl) E. [4] Each term icludes m due to the o-rival characteristic of E. (I a explicitly dyamic model we would wat to write this as the preset discouted value of expected losses i icome due to iduced chages i global climate over years). Note that this expressio icludes both the direct effect o utility from reductios i private cosumptio (the first term o the right-had side), plus the idirect effect due to lost reveues (the remaiig terms o the right-had side). Let Y deote a uit of icome correspodig to the household budget costrait where uits of time, T, are paid a wage h(e). The social margial utility of a uit of icome, α, ca the be expressed as α W Y λ µ t F F Y t N F N πmφ Y Y. [5] This expressio reflects the stadard defiitio of the social margial utility of icome from Diamod (1985), but with the additio of the evirometal damage term: the fial term o the right-had side reflects the disutility from additioal carbo damages arisig as a fuctio of the margial propesity to cosume fossil fuels where de mφdf. The margial social damage (MSD) from cosumptio of F is thus defied as the product of margial emissios, mφ, ad the margial rate of substitutio (MRS) betwee icome ad emissios, or MSD MRS de df W/ E de W/ Y df π mφ. [6] α To examie the relatioship betwee the optimal carbo tax ad margial social dam-

4 78(3) Jaeger: Carbo Taxatio Whe Climate Affects Productivity 357 ages, we ote that the optimal taxes o F ad N will reflect both reveue-raisig taxes ad, i the case of F, a evirometal compoet. I order to isolate the evirometal compoet of the optimal tax o F, we will assume that both F ad N are average substitutes for leisure as has bee the stadard approach i the recet tax iteractio literature. With appropriate restrictios o prefereces, the optimal carbo tax ca the be iterpreted as t* F t* N. The relatioship betwee the optimal carbo tax ad margial social damage is estimated i the ext sectio based o a umerical model for the U.S. ecoomy. III. NUMERICAL MODEL AND RESULTS The umerical geeral-equilibrium model ivolves costat elasticity of substitutio fuctios for utility, u(c, V ), ad productio, c(f, N), ad is represeted as a sigle period model rather tha as a dyamic optimizatio problem. This approach is similar to the model of the U.S. ecoomy itroduced i Parry, Williams, ad Goulder (1999). Ideed, the structure ad parameters of the curret umerical model replicate their model, albeit with a more aggregated represetatio of productio, but calibrated to produce a similar margial abatemet cost curve. 2 Additioal details of the model s structure ad calibratio are preseted i Appedix A. For purposes of umerical estimatio, ested optimizatio models such as [3] are ofte represeted as a sigle-maximizatio problem by itroducig the household s firstorder coditios as costraits o social maximizatio, similar to the first-order approach i priciple-aget problems (Jewitt 1988). Lettig subscripts deote partial derivatives with respect to variable j (e.g., U j ad C j ), we write the social welfare maximizatio problem as Max: m[u(c(f, N), V)] t F,t N s.t. (α) (1 t F )F (1 t N )N h(e)l G (µ) t F F t N N G (η 1 ) U C C F (1 t N ) U C C N (1 t F ) (η 2 ) U V (1 t F ) U C C F h(e) (π) E mφf. [7] TABLE 1 Optimal Carbo Taxatio: Results for Numerical Model of the Uited States (Margial Social Damage (MSD) $40 per to) Optimal tax, F Optimal tax, N Social margial utility of emissios (π) Social margial utility of icome (α) Social margial utility of reveues (µ) Optimal carbo tax o F(t* F t* N ) (per to of carbo) $61.3 Private margial utility of icome (λ) Private margial utility of emissios (π P ) 9.03 Ratio of optimal carbo tax to MSD (π/α) 1.53 Ratio of optimal carbo tax to MPD(π P /λ) 1.73 Note: Resource allocatio at the optimum is: N 2,620,500, F 556,040, C 1,763,700, V 931,080, ad E 1, I this model, the shadow value of the Lagrage multiplier o icome, α, will reflect the social value of a uit of icome because all optima i this model represet Pareto efficiet states. For that reaso, the private margial utility of icome, λ, does ot appear directly i the model: it does ot reflect a Pareto efficiet use of a uit of icome. Rather, it reflects a movemet from a Pareto efficiet state to a o-pareto efficiet state. That is, to the extet that a uit icrease i icome causes a icrease i tax paymets (assumig a positive margial propesity to pay taxes), the value of λ does ot afford ay value to these added tax receipts. Prior to the itroductio of a carbo tax, the iitial coditio of the model reflects a reveue costrait requirig 67% excise taxes o F ad N, which is equivalet to a 40% icome tax rate. Parameter values for h(e) are chose so that the margial social damage from carbo emissios (π/α) is $40. The model is solved for the welfaremaximizig taxes o F ad N. The optimal carbo tax correspodig to t* F t* N is $61.3 per to of carbo, or about 53% higher tha margial social damage (see Table 1 for details). I a first-best settig with o bidig reveue-requiremets, the optimal carbo tax equals margial social damage. Hece, we ca ifer that the optimal 2 I the curret model the reveue costrait, G, is fixed i omial rather tha real terms.

5 358 Lad Ecoomics August 2002 FIGURE 1 Relatioship Betwee the Optimal Carbo Tax ad Margial Social Damage i the Uited States for Differet Levels of Reveue Requiremets carbo tax rises with icreased reveue requiremets, as idicated i Figure 1 based o results from the umerical model. For example, icreasig the reveue requiremet i the umerical model 25% above the iitial level produces a optimal carbo tax of $67/to, or 68% higher tha margial social damages. The welfare gai from the reveue-eutral itroductio of the optimal carbo tax is estimated at $3.58 billio per year. Employmet also icreases by early $900 millio i gross wages. These two chages broade the tax base by itroducig a tax o previously utaxed carbo emissios, ad by iducig higher labor supply. If the carbo tax is set equal to margial social damage, however, the welfare gai would be oly $3.17 billio ad employmet would rise by oly $640 millio. The ituitio for these results ca be compreheded i at least three ways. First, the otio uderlyig the double-divided hypothesis recogizes that whe pollutio tax receipts are used to lower reveue-motivated taxes that this will produce a welfare gai by lowerig the excess burde of the tax system. This additioal beefit from taxig pollutio lowers the et cost of emissios reductios, which i tur justifies a higher optimal carbo tax. A secod explaatio views carbo emissios as simply aother good i the ecoomy, oe that, like other similar goods, should be taxed above its social cost as part of a broadly based Ramsey tax program. As the reveue requiremet icreases, so too should the Ramsey tax premiums o all similar goods ad services. A third perspective poits out that the global climate ca be uderstood to be a utaxable edowmet i much the same way that leisure is ot directly taxable. From optimal tax theory we kow that reveuemotivated taxes are distortioary precisely because of the existece of utaxed edowmets, sice idividuals are able to avoid taxatio by cosumig the utaxed edowmet. Thus, a icrease i reveuemotivated taxes ca be expected to result i icreased cosumptio of the utaxable edowmets. I the curret model with two utaxed edowmets, this result is cofirmed with icreased cosumptio of climate quality. I the case of leisure, the icreased propesity to cosume leisure is offset by a substitutio effect as labor productivity rises. Fially, although these results may at first glace appear to cotradict the Samuelso Coditio (whereby the optimal provisio of

6 78(3) Jaeger: Carbo Taxatio Whe Climate Affects Productivity 359 a public good is uderstood to vary iversely with reveue requiremets ad the cost of public fuds), i this case there is o cotradictio. The global climate is a edowed public good that is either directly taxed or publicly provided, so the cost ad expediture cosideratios uderlyig the Samuelso Coditio do ot directly apply. However, the public good of atmospheric waste disposal services for greehouse gases ca be directly taxed i this case: households ca be charged for emissios. Ideed, the model results demostrate that a rise i the reveue requiremet is accompaied by a icrease i the carbo tax ad a reductio i the cosumptio of these carbo disposal services, a chage that is fully cosistet with the Samuelso Coditio for this public good. Amelioratio of climate damages comes idirectly, as a byproduct of raisig taxes o carbo disposal services as well as o other taxable goods. IV. THE TAX INTERACTION LITERATURE The aalyses cotaied i the tax iteractio literature differ from the curret aalysis i three respects. First, they have geerally modeled evirometal exteralities as ameities that affect utility directly rather tha as productio exteralities that affect output. For a give value of margial social damage, however, there is o obvious reaso why the type of exterality beig modeled should affect the theoretical relatioship betwee the optimal evirometal tax ad margial social damage. Secod, the tax iteractio literature uses a tax rule that icludes a tax o labor supply (icome) rather tha by itroducig all taxes o expeditures. For this icome tax ormalizatio, N becomes the utaxed good (or itermediate iput) rather tha leisure, ad the tax o labor icome serves as the reveue-motivated tax. It is uderstood i these models that a labor tax is equivalet to uiform taxes o all expeditures, ad thus it ca serve as a optimal reveue-raisig tax if demads are symmetrical, that is if all goods are average substitutes for leisure (see Boveberg ad de Mooij 1994; Boveberg ad Goulder 1996). This symmetric demad coditio has ofte bee satisfied by assumig that all goods are homothetic ad separable from leisure ad evirometal quality i the utility fuctio. Like Parry, Williams, ad Goulder (1999), the curret model satisfies this coditio o demads for F ad N by assumig that both c(f, N) ad u(c, V ) are homothetic. Give this assumptio of symmetric demads, a labor tax will represet a optimal reveue raisig tax so that the etire remaiig tax imposed directly o the pollutig good ca be iterpreted as the evirometal tax. This is a coveiet attribute for the icome tax ormalizatio, however it also may cause cofusio because the uits i which icome is measured will vary with ay chage i the icome tax rate while at the same time, the uit for govermet reveues remais uchaged. Ideed, havig reveues measured i gross icome uits ad household icome measured i et icome uits may create cofusio for iterpretig the relatioship betwee the margial utility of icome ad the margial cost of public fuds. I order to miimize the sources of potetial cofusio o this topic, the curret model employs the stadard ormalizatio used i the optimal tax literature, where taxes are itroduced o expeditures oly. Although this itroduces a differece betwee the preset model ad the models i the tax iteractio literature, it is a differece that will have o effect o the results of the model or o ay real variables, ad thus should ot affect their iterpretatio (Schöb 1996; Fullerto 1997). Ideed, models ad results employig either of these tax ormalizatios ca be easily reormalized, or coverted, ito the other ormalizatio: it is a matter of redefiig uits. The curret model ca be reormalized to reflect a labor tax simply by dividig the household budget costrait i [3] by 1 t L, where t L is the tax o labor supply ad where 1 t N 1/(1 t L ). Whether the umeraire uit of icome is defied i gross or et uits will ot affect the aalysis so log as it is used cosistetly throughout. The third ad pivotal differece betwee the curret aalysis ad the tax iteractio

7 360 Lad Ecoomics August 2002 literature is the defiitio of the margial social damage from pollutio. I the curret aalysis, the defiitio of MSD reflects shadow prices, ad has bee derived based o the margial rate of substitutio betwee icome ad evirometal quality as expressed i [6]. By cotrast, the defiitio of evirometal damage used i the tax iteractio literature takes the margial damage for a represetative household, ad the sums these across all households. Thus, their defiitio reflects margial private damages (MPD), or the aggregate margial willigess to pay to avoid climate damage. The differeces betwee these two defiitios have crucial ad direct bearig o these results ad their iterpretatio. I a model with preexistig taxes ad a reveue requiremet, the social margial utility of a give resource (either icome or evirometal quality) will differ from its private margial utility to the extet that there are taxes preset ad reveue cosequeces. Diamod (1985) defied the social margial utility of icome as the gai i social welfare from provisio of additioal icome i umeraire uits, which is the sum of gais from idividual cosumptio ad from the margial propesity to pay taxes out of icome (p. 336). As a result, the social margial utility of icome will exceed the private margial utility of icome so log as the margial propesity to pay taxes is positive. I the curret model, the social margial utility of a chage i climate similarly has two compoets, oe reflectig the chages i private cosumptio ad the other reflectig the chages i tax receipts. I cotrast to this defiitio, the tax iteractio literature has defied the margial utility of icome as λ, the Lagragia multiplier for the household budget costrait. This reflects oly the private cosumptio compoet of the social margial utility for a uit icrease i icome, ad omits the secod ad third terms i [5] which reflect the utility from the reveues geerated by icremetal tax paymets, ad the evirometal cosequeces of chages i cosumptio correspodig to a icrease i icome. The differece betwee the social ad private margial utility of icome ca be uderstood as beig similar to the distictio i beefitcost aalysis betwee market prices ad shadow prices. I the tax iteractio literature, a evirometal ameity is assumed to be additively separable i the utility fuctio so that a chage i evirometal quality will ot affect households allocatio choices, ad therefore it will have o effect o tax paymets or reveues. Thus, due to the absece of ay iteractio betwee chages i evirometal quality ad chages i reveue paymets, the social margial utility of evirometal quality (i utility uits) will be the same as the private margial utility of evirometal quality (whe summed across households). I the curret model, however, climate chage damages affect icome directly by lowerig labor productivity, which reduces labor icome ad causes households to modify their cosumptio choices. As a result, the social margial utility of a chage i climate will differ from the private margial utility of a chage i climate due to its reveue cosequeces. Because climate damages reduce icome directly, the differece betwee the social ad private margial utility of climate will be similar (but ot idetical) to the differeces betwee the social ad private margial utility of icome. Thus, for the curret model MPD will differ i both its umerator ad deomiator from MSD. Let π P deote the private margial utility of evirometal quality. We ca write MPD as MPD π mλ dh P de L mφ mφ. [8] λ λ The umerator ad deomiator i [8] will be smaller i magitude tha i [6] give the explicit expressios i [4] ad [5], so log as the evirometal term i [5] is small relative to the reveue term. Private agets will igore the value of additioal reveues geerated whe their expeditures chage as a result of a chage i icome, ad this will be true for exogeous icome chages as well as those due to climate chage. Because both the umerator ad de-

8 78(3) Jaeger: Carbo Taxatio Whe Climate Affects Productivity 361 omiator are higher i MSD compared to MPD, the ratios may ot differ greatly. Ideed, for the curret umerical model where MSD is $40, MPD is $35.4. Sice the optimal carbo tax is foud to be $61.3, this result implies that, had Parry, Williams, ad Goulder (1999) evaluated a model with productio exteralities, they would have foud the optimal carbo tax to be approximately 73% higher tha MPD. It is essetial to demostrate that these umerical results, ad the reasos they differ from results i the tax iteractio literature, ca be recociled usig expressios that have bee derived theoretically. The distictio betwee MSD ad MPD is crucial, ad the implicatios of usig MPD versus MSD as a bechmark comparator for the optimal carbo tax will be demostrated usig the aalytical expressios from the tax iteractio literature. The derivatio of the aalytical expressio for the optimal evirometal tax for a geeralized versio of the curret model is preseted i Appedix B. A expressio for the optimal evirometal tax is derived i Boveberg ad Goulder (1996) for their model with a ameity exterality (a equivalet expressio ca be derived from the optimal tax expressios i Sadmo s (1975) semial article). The expressio is t * F MPD λ µ π λ mφ λ µ, [9] where a tilde over a variable deotes et icome uits correspodig to the icome tax ormalizatio. For their model i which evirometal quality is separable i utility so that a chage i the ameity exterality has o affect o cosumptio, there will be o differece betwee the private margial utility of evirometal quality ad the social margial utility of evirometal quality. The expressio is correct sice π π P. The correspodece betwee the two ormalizatios implies that λ λ (1 t L ) ad t * F (t* F t* N )(1 t L ). With these idetities, ad otig that α/α λ /λ, we ca substitute these expressios ito [9] ad plug i umerical values from Table 1 to get t* F t* N α(1 t N) µ mφ π α 0.4(1.641).429 mφ π α 1.53 mφ π α. [10] Thus, the optimal carbo tax obtaied directly from the umerical estimatio is show here to be cosistet with oe calculated by pluggig the shadow values α ad µ ito the aalytical expressio. It also cofirms that this relatioship is idepedet of the tax ormalizatio. We ca also use the expressio i [9] to poit out those circumstaces uder which the optimal evirometal tax may appear to be less tha margial damages. Cosider a model like those used i the tax iteractio literature ivolvig a evirometal ameity that is separable i utility, ad where margial evirometal damage is defied by MPD rather tha MSD. As explaied above, these restrictios o prefereces esure that a chage i evirometal quality will ot affect households allocatio choices or tax paymets, so that the social ad private margial utility of evirometal quality will be equal (π π P ). Thus, the umerators i MSD ad MPD will be idetical. The oly differece i this case betwee MPD ad MSD will be due to havig λ i the deomiator of MPD rather tha α which is i the deomiator of MSD. We ca express [9] by substitutig λ(1 t N ) λ ad pluggig i values from Table 1 to get t * F λ(1 t N) π mφ µ λ 0.255(1.641) π mφ λ 0.97 mφ π λ. [11]

9 362 Lad Ecoomics August 2002 Whe defiig margial evirometal damage usig λ rather tha α i the deomiator, the relatioship betwee the optimal evirometal tax ad the bechmark (MPD i this case) is altered. Because the umerators for both MPD ad MSD are the same, but λ is less tha α, it is possible that MPD will exceed the optimal evirometal tax, eve while MSD is less tha the optimal evirometal tax. 3 This particular result, however, should ot cast doubt o the validity of the double divided hypothesis. Comparisos betwee MPD ad the optimal evirometal tax may appear to be icogruous with our ituitio, but this is because MPD does ot reflect the social value of a umeraire uit of icome: λ does ot reflect the value of a uit of icome allocated to a Pareto efficiet use. For optimizatio models such as the oe represeted by [3] where households ad plaers are uderstood to respod to each other s actios, λ oly reflects the icreased cosumptio by households from a uit icrease i icome, but with o respose or chage o the part of the social plaer to the chages i tax receipts. For a icrease i icome, this implies that the icremetal reveues geerated have o value. Ideed, for a decremet of icome, λ would appear to reflect a chage that is ifeasible: a uit decrease i icome will lower cosumptio ad tax paymets, but trasfers of G (or the provisio of public goods) remai uchaged. Thus, i the case of a evirometal ameity, MPD may exceed the optimal evirometal tax makig it appear as though the optimal tax lies below the the Pigouvia rate that fully iteralizes the margial evirometal damages (Boveberg ad de Mooij 1994). But this iterpretatio is misleadig. Had the authors of the tax iteractio literature used MSD rather tha MPD as their bechmark comparator, they would have foud that the optimal evirometal tax exceeded that level. Alteratively, had they modeled a productio exterality rather tha a ameity exterality, they also would have foud that the optimal evirometal tax exceeded margial evirometal damage (o matter whether they used MPD or MSD). Oly for the particular case of a ameity exterality, ad whe usig MPD as the measure of margial damages, does the optimal evirometal tax geerally appear to be lower tha margial damages i these models. 4 IV. CONCLUDING COMMENTS The welfare effects of evirometal tax reform such as the itroductio of a carbo tax have become a importat policy questio as well as a source of debate i the theoretical literature. It is accepted wisdom that usig carbo tax reveues to fiace reductios i preexistig taxes will lower the excess burde of the tax program. Ituitively this implies that the et welfare chage from itroducig a carbo tax will exceed that which would be iferred by simply comparig margial social damages to margial abatemet costs; ad it follows logically that the optimal carbo tax will exceed the margial social damage (Lee ad Misiolek 1986; Terkla 1984). The curret aalysis is cosistet with this ituitio ad the double divided hypothesis. I a secod-best settig with preexistig taxes, ad where carbo damages affect productio, the optimal carbo tax is foud to exceed margial social damages by 53% based o a umerical model of the U.S. ecoomy. Whe margial climate damages are defied or estimated as the sum of household s margial willigess to pay, the the optimal carbo tax will exceed this measure of margial damages by 73%. The welfare gai from the reveueeutral itroductio of the optimal carbo tax whe margial social damage is $40 per to is estimated at $3.57 billio per year for the U.S. ecoomy. Employmet also icreases by $900 millio i gross wages aually. Differeces betwee the curret fidigs 3 For the curret umerical model, π/α 40, π P /λ 35.4, ad α/λ 1.57, from which we ca ifer that π/π P This ratio is larger tha α/λ due to the substitutio effect of a chage i labor productivity, whereas α is strictly a icome effect. 4 Of course, these geeralizatios ca oly be assumed to hold uder the stadard assumptios that have bee made i this literature: that labor supply is upward slopig ad that all goods are average substitutes for leisure.

10 78(3) Jaeger: Carbo Taxatio Whe Climate Affects Productivity 363 ad those i the tax iteractio literature are show to stem from differeces i the way margial social damage has bee defied. I the curret aalysis, margial social damage is defied based o the margial rate of substitutio betwee icome ad evirometal quality from society s perspective. I the tax iteractio literature, margial damage has bee defied i a way that does ot reflect the margial rate of substitutio betwee icome ad evirometal quality; their defiitio is a aggregatio across households of margial private damages, reflectig the private margial utility of icome rather tha the social margial utility of icome. The essetial poit for uderstadig the source of these ostesibly cotradictory results is that as reveue-motivated tax rates rise, the private margial utility of icome will declie relative to the social margial utility of icome, ad the divergece betwee the two defiitios will rise with icreasig reveue requiremets. If these distictios are overlooked, the expected ad ituitive correspodece betwee the optimal evirometal tax, margial social damage ad the validity of the double divided hypothesis will become obscured. Defiig the social value of a resource without takig accout of distortioary taxes or subsidies is cotrary to well-established cocepts ad methods of shadow pricig. This poit should ot be cotroversial. Cosider for example if climate chage damages were beig valued based o sea level rise ad the cosequet losses of agricultural lads. Ecoomists would be quick to poit out that market prices for farmlad should ot be used to measure these damages if farm subsidies have iflated their values. Obviously, the shadow value of agricultural lad is the appropriate measure rather tha the private value. The same logic applies to valuig other resources such as icome ad evirometal quality. Give the eed to provide policymakers with clear guidace o climate chage policy questios, these results, ad the reasos for the ostesibly icogruous fidigs i the tax iteractio literature, eed to be fully recogized ad uderstood i order to avoid cofusio by both policymakers ad the geeral public. For example, usig a model typical of the tax iteractio literature, Boveberg ad va der Ploeg (1994) cocluded that expadig evirometal policy will ted to slow the ecoomy ad lower employmet, forcig govermets to choose betwee a clea eviromet ad the provisio of other public goods. Evidece suggests that these iterpretatios have bee ifluetial i public ad political debates o evirometal policies such as gree tax reform i Europe ad the Uited States. The curret aalysis cocludes that evirometal policy ca actually icrease employmet ad ecoomic activity, ad that evirometal taxatio lowers, rather tha raises, the cost of raisig reveues to achieve other collective goals. APPENDICES Appedix A: Specificatio of the Numerical Climate-Ecoomy Model The umerical model represetig the U.S. ecoomy is represeted i [7]. The fuctioal forms ad parameter values are described here. These iclude a primary CES utility fuctio, U u(c, V) give as U [γc ρ (1 γ)v ρ ] 1/ρ, ad a secodary CES productio fuctio defiig substitutios betwee F ad N i C c(f, N) as C [βf δ (1 β)n δ ] 1/δ. This productio fuctio is a sigle CES fuctio rather tha the more disaggregated, ested CES structure of productio i the Parry, Williams, ad Goulder model (1999). The fuctio H h(e) ish (E ), ad φ The fuctios ad parameters have bee calibrated to correspod to the secodbest margial abatemet cost fuctio from Parry, Williams, ad Goulder (1999). Settig δ 0.5 implies that the elasticity of substitutio betwee carbo emittig ad o-carbo emittig cosumptio, σ NF, equals (1/1 δ) 2.0. The value of ρ 0.167, so that the elasticity of substitutio betwee cosumptio ad leisure, σ CV, equals (1/1 ρ) 1.2. I additio, γ 0.836, β 0.667, ad m 1. The desired labor supply elasticity (0.15) is

11 364 Lad Ecoomics August 2002 achieved by calibratig the share parameter γ. The iitial coditios are T , G 2,113,100, V 931,980 ad E These are idetical to the model i Parry, Williams, ad Goulder (1999). Appedix B: Derivatio of the Optimal Evirometal Tax for a Productio Exterality I the iterest of otatioal coveiece, ad to geeralize for may goods, detailed derivatios for a geeral model with goods are preseted, where X z is the otatioal equivalet to F. The essetial problem ca be stated as oe i which m idetical idividuals maximize utility U u(x 0, X 1,...,X Z,...,X ) for goods j 0,...,, where leisure is X 0 ad where labor supply is take out of a time edowmet ormalized to equal oe so that labor supply equals 1 X 0. Uits are chose for goods ad icome so that all pre-tax prices equal oe, ad where there are 1 o-pollutig X goods (excludig leisure) ad oe good, X z, which produces a evirometal exterality. For a productio exterality we assume that labor productivity, h, is a fuctio of evirometal quality such that h h(e). We defie aggregate output as mh(e)l mx i, ad where mg is fiaced through collectio of tax reveues. Each household s maximizatio problem ca be stated as Max: X 0...X u(x 0, X 1,...X ) s.t. hl G j1 (1 t j )X j, [A1] so that idividuals maximize utility subject to their budget costrait while igorig both the evirometal cosequeces of their ow cosumptio choices ad govermet behavior. The Lagragia expressio for each household is thus u(x 0, X 1,...X ) λ h(1 X 0) G for j 1,...,z,...,. j1 (1 t j )X j [A2] Cosumer prices are give as p j 1 t j, for j 1to, but where icome is utaxed, so that p 0 1. The first-order coditios for each household take the form U j λ(1 t j ), for j 1,..,, ad U o λh(e), for X 0. The social plaer s problem is the Max: t l...t m u(x 0,X 1,...X )s.t.h(1x 0 )G s.t. m j1 h h(e(mx Z )). t j X j G 1 (1t j )X j [A3] Takig the dual approach, we defie the household s idirect utility fuctio as v(p 0, p 1,...p ) u(x(p 0, p 1,...p ), so we ca state the social optimizatio problem as the Lagragia equatio mu(p 0, p 1,...p ) µ m j1 t j X j G. I the presece of evirometal effects o labor productivity, the first-order coditios for the social optimizatio problem are λx j µ j1 t i X i p j X j λ m(1 X 0) h µ j1 de E d(mx Z ) X t i (h(1 X 0 )) de i (h(1 X 0 )) E d(mx Z ) X Z p j 0 j 0, [A4] where λ dv/d(h(1 X 0 )) U*/ (h(1 X 0 )) is the household s margial utility of icome. Let π deote margial social damages i utility uits, or h de π ml λ E d(mx z ) µ j1 t i X i (h(1 X 0 )) (h(1 X 0 )) E de d(mx Z ), [A5] where margial social damage icludes both the direct loss of icome to households as well as the loss i reveues due to chages i labor icome, h(1 X 0 ), due to the chage i h as well as the resultig chage i labor supply.

12 78(3) Jaeger: Carbo Taxatio Whe Climate Affects Productivity 365 Simplifyig the otatio i [A4] usig [A5] we have λx j µ j1 t i X i p j X j π X Z p j 0 j 0. [A6] Derivatios of optimal tax rules ofte iclude substitutio of the Slutsky equatio i such a way that the social margial utility of icome, α, is represeted alog with the shadow cost of raisig a additioal dollar of reveue (Auerbach 1985). Divergig from the approach take by Sadmo (1975), we rearrage the plaer s first-order coditios ad use the Slutsky equatio to split the cross-price effects ito compesated effects (superscript U) ad effects o icome, Y, as X Z p i X U Z X X Z i p i Y. We substitute α to obtai αx j µ i t i X U i p j µx j π X U Z p j X X Z j Y 0 j 0. [A7] We defie Š as the determiat of the Slutsky matrix of compesated demads, so that S ij is the cofactor of the elemet for the jth row (price) ad ith colum (quatity). Usig Cramer s rule we ca solve for the optimal taxes t j X i S ij (µ α) i1 µ Ŝ π µ i1 X U Z X X Z i p i Š Y S ij, [A8] where the secod term o the right-had side is the evirometal compoet of the tax. From theorems about the expasio of determiats, we kow that X U 0 for j Z Z S ij p i1 i Š for j Z. Let R deote the Ramsey term for compesated demads or R i1 X i S ij /p j Š reflectig the reveue geeratig potetial for a margial chage i the tax o X i due to the direct ad idirect effects o cosumptio for all goods. Further simplify the otatio by defiig the icome effect o the eviromet as θ π i1 X i X Z Y. We ca thus rearrage terms ad simplify so that the optimal tax expressios ca the be writte as t j (1 t j ) ad t j (1 t j ) (µ α θ) R j z. [A9] µ (µ α θ) R µ π µ(1 t j ) j z. [A10] These implicit solutios are difficult to iterpret by ispectio, i part because of the lack of trasparecy i iterpretig R. Moreover, although the evirometal compoet of the tax i [A10] appears to be separable from the stadard formula, the idepedece is illusory both because the deomiator (1 t z ) is edogeous ad because the actual level of the exterality depeds o the actual equilibrium ad hece the optimal tax rates. The results differ from the expressios obtaied by Sadmo ivolvig ucompesated demads. Sadmo cocluded that the evirometal damages of X Z does ot eter the tax formulas for the other commodities, regardless of the patter of complemetarity ad substitutability (Sadmo 1975, 92). I this alterative derivatio, we see that the umerator i the first term o the right-had side icludes θ, a term ivolvig π, idicatig that the presece of a exterality raises the optimal tax o all goods due to their icome effect: by reducig real icome, all taxes discourage cosumptio of the exterality-producig good to some extet, ad these optimal tax rates will be higher as a result. These two versios of the optimal tax results are ot i coflict: i the

13 366 Lad Ecoomics August 2002 model ivolvig ordiary demads, the icome effects are implicit. The relatioship betwee the optimal evirometal tax ad margial evirometal damages ca be evaluated based o the optimal tax expressios derived above. Notig the idetity MSD π/α ad revertig to the otatio where F is the pollutig good ad N is the o-pollutig good, we ca write these as t* N (1 t* N ) µ α θ µ R [A11] ad t* F (1 t* F ) µ α θ µ R α µ MSD (1 t F ), [A12] where θ represets the direct ad idirect icome effects o the eviromet. These expressios, similar to those derived by Sadmo, are difficult to iterpret by ispectio because of the lack of trasparecy i R, ad the evirometal compoet of the optimal tax for F caot be evaluated separately by ispectio sice the left-had side deomiator is a fuctio of both terms. 5 The problem, however, ca be simplified by assumig that all goods are similar from a reveue raisig perspective. This has also bee doe i the recet literature by restrictig prefereces so that utility is homothetic i cosumptio, ad weakly separable i leisure, evirometal quality, ad govermet cosumptio (Boveberg ad de Mooij 1994; Boveberg ad Goulder 1996). Here, however, we assume oly that the Ramsey terms for both goods are equal, which allows us to derive a expressio for the evirometal compoet of the optimal tax. For the pollutig good, F, we rearrage [A12] to get t* F α θ 1 µ R α θ 1 1 µ R αmsd µ 1 α θ 1 µ. R [A13] From [A11] we ca express the Ramsey term as R t* N (1 t* N ) α θ 1 µ. [A14] To evaluate the optimal tax t* F relative to MSD, we substitute [A14] ito the secod term of [A13] ad rearrage to get t* F α θ 1 µ R α θ 1 1 µ R α(1 t N)MSD. µ [A15] We ca evaluate the relatioship betwee the optimal evirometal tax ad MSD i oe of two ways; either evaluatig the differece betwee the optimal tax expressios for N ad F or by differetiatig t F with respect to MSD. Takig the differetiatio approach, we obtai t* F (MSD) (1 t N)α, [A16] µ which idicates that the optimaltax may rise by more or less tha MSD depedig o the relatioship betwee the reveue-motivated tax rate t* N ad α/µ. To demostrate the similarity betwee these results ad those foud i Boveberg ad Goulder (1996), ote that a icome tax ormalizatio amouts to multiplyig the household budget costrait by (1 t L ) where t L is the icome tax rate ad where (1 t N ) 1/(1 t L ). This step implies that the household budget costrait ow represets uits of et icome. Let a tilde () deote et icome uits, so that the shadow value of et icome is λ, which will differ from the shadow value of gross icome, λ, such that λ 5 Although Fullerto (1997), Schöb (1997), ad Boveberg ad de Mooij (1997) have suggested that Sadmo s formula idicates that the optimal pollutio tax should be less tha margial social damages, this iterpretatio overlooks the edogeeity of the deomiator o the left-had side: oe caot ifer by ispectio that the differetial betwee the optimal tax o a pollutig good versus a similar o-pollutig good will simply equal the value of the secod term (sice the first term equals the tax o o-pollutig goods).

14 78(3) Jaeger: Carbo Taxatio Whe Climate Affects Productivity 367 λ (1 t L ). I the Boveberg ad Goulder model, household icome is expressed i uits of et icome whereas the govermet reveue costrait cotiues to be expressed i uits of gross icome (as i [3]), which meas that the shadow value o their reveue costrait, µ, reflects gross icome uits. To be cosistet, this shadow value ca also be expressed i et icome uits by substitutig the idetity µ µ (1 t L ). If we also substitute the idetity, λ /λ α /α, Boveberg ad Goulder s result i [9] for a ameity exterality ca be expressed as t * F α mφπ α µ (1 t L ). [A17] We ca differetiate [A17] with respect to MSD ( mφπ/α ) to get t * F (MSD ) α µ (1 t L ) (1 t N)α. [A18] µ This expressio is idetical to [A16] but for the differece i ormalizatio. Refereces Auerbach, A The Theory of Excess Burde ad Optimal Taxatio. I Hadbook of Public Ecoomics, Vol. 1, ed. A. Auerbach ad M. Feldstei. Amsterdam: North-Hollad Press. Boveberg, A. L., ad L. H. Goulder Optimal Evirometal Taxatio i the Presece of Other Taxes: Geeral Equilibrium Aalysis. America Ecoomic Review 86 (4): Boveberg,A. L., adr. A. demooij Evirometal Levies ad Distortioary Taxatio. America Ecoomic Review 94 (4): Evirometal Levies ad Distortioary Taxatio: Reply. America Ecoomic Review 87 (1): Boveberg, A. L., ad F. va der Ploeg Evirometal Policy, Public Fiace ad the Labour Market i a Secod-Best World. Joural of Public Ecoomics 55 (3): Clie, W The Ecoomics of Global Warmig. Washigto, D.C.: Istitute of Iteratioal Ecoomics. Diamod, P. A A May-Perso Ramsey Tax Rule. Joural of Public Ecoomics 4 (4): Fullerto, D Evirometal Levies ad Distortioary Taxatio: Commet. America Ecoomic Review 87 (1): Goulder, L. H Effects of Carbo Taxes i a Ecoomy with Prior Tax Distortios: A Itertemporal Geeral Equilibrium Aalysis. Joural of Evirometal Ecoomics ad Maagemet 29 (3): Goulder, L. H, I. W. H. Parry, D. Burtraw Reveue-Raisig vs. Other Approaches to Evirometal Protectio: The Critical Sigificace of Preexistig Tax Distortios. RAND Joural of Ecoomics 28: Goulder, L. H., I. W. H. Parry, R. C. Williams III, D. Burtraw The Cost-Effectiveess of Alterative Istrumets for Evirometal Protectio i a Secod-Best Settig. Joural of Public Ecoomics 78 (3): Jewitt, I Justifyig the First-Order Approach to Pricipal-Aget Problems. Ecoometrica 56 (5): Lee, D. R., ad W. S. Misiolek Substitutig Pollutio Taxatio for Geeral Taxatio: Some Implicatios for Efficiecy i Pollutio Taxatio. Joural of Evirometal Ecoomics ad Maagemet 13 (4): Oates, W. E Gree Taxes: Ca We Protect the Eviromet ad Improve the Tax System at the Same Time? Souther Ecoomic Joural 61 (4): Parry, I. W. H Pollutio Taxes ad Reveue Recyclig. Joural of Evirometal Ecoomics ad Maagemet 29 (3): Parry, I. W. H. ad A. M. Beto Tax Deductios, Evirometal Policy, ad the Double Divided Hypothesis. Joural of Evirometal Ecoomics ad Maagemet 39 (1): Parry, I. W. H., ad W. E. Oates Policy Aalysis i the Presece of Distortig Taxes. Joural of Policy Aalysis ad Maagemet 19 (4): Parry, I. W. H., R. C. Williams III, ad L. H. Goulder Whe Ca Carbo Abatemet Policies Icrease Welfare? The Fudametal Role of Distorted Factor Markets. Joural of Evirometal Ecoomics ad Maagemet 37 (1): Pearce, D The Role of Carbo Taxes i Adjustig to Global Warmig. Ecoomic Joural 101 (407): Sadmo, A Optimal Taxatio i the Presece of Exteralities. Swedish Joural of Ecoomics 77 (1): Schöb, R Evaluatig Tax Reforms i the Presece of Exteralities. Oxford Ecoomic Papers, 48 (4): Terkla, D The Efficiecy Value of Effluet Tax Reveues. Joural of Evirometal Ecoomics ad Maagemet 11 (2): Tullock, G Excess Beefit. Water Resources Research 3 (2):