Can Taxes on Cars and on Gasoline Mimic an Unavailable Tax on Emissions? 1

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1 Journal of Envronmental Economcs and Management 43, Ž do: jeem , avalable onlne at on Can Taxes on Cars and on Gasolne Mmc an Unavalable Tax on Emssons? 1 Don Fullerton Department of Economcs, Unersty of Texas at Austn, Austn, Texas and Sarah E. West Department of Economcs, Macalester College, St. Paul, Mnnesota Receved September 14, 1999; revsed November 24, 1999; publshed onlne February 13, 2001 An emssons tax s effcent, but measurement of every car s emssons would be naccurate and expensve. Wth dentcal consumers, we demonstrate the same effcency for: an emssons tax; a gas tax that depends on fuel type, engne sze, and polluton control equpment Ž PCE.; a vehcle tax that depends on mleage; or a combnaton of unform tax rates on gasolne and engne sze wth a subsdy to PCE. Wth heterogeneous consumers, effcency can be obtaned by a vehcle-specfc gas tax or mleage-specfc vehcle tax, but not by flat rates. We characterze second-best unform tax rates on gasolne and on car characterstcs Elsever Scence Contnued growth of ctes, ncreases n vehcle-mles traveled, and Amercans renewed love for large vehcles contrbute to ncreasng externaltes from vehcle emssons, ncludng worsened health, dmnshed vsblty, and possble global warmng. Technologcal advances n the measurement of car emssons renew hope that a tax can be leved drectly on these emssons ŽHarrngton et al If so, ndvduals would reduce polluton effcently ŽPgou 34.. At least for the tme beng, however, the emssons taxes or permts that work well for statonary sources such as electrc generatng plants are not consdered feasble for moble sources of polluton. The technology s not avalable to measure the emssons of each vehcle n a way that s cost-effectve and relable, that s resstant to tamperng by the vehcle s owner, and that satsfes legal restrctons aganst the search of a prvate vehcle. In ths paper, we nvestgate alternatve market ncentves for the reducton of car polluton. We focus on economc ncentves because these knds of polces tend to be less costly than approaches whch regulate the technology of the car or the fuel 1 Correspondence can be addressed to ether author at the addresses above, or by e-mal to dfullert@eco.utexas.edu or to wests@macalester.edu. We are grateful for helpful comments from Wnston Harrngton, Robert Innes, Raymond Robertson, Dan Slesnck, Kenneth Small, Margaret Walls, Pete Wlcoxen, Paul Wlson, Ann Wolverton, and anonymous referees. In addton, we thank the Publc Polcy Insttute of Calforna Ž PPIC. for fundng ths research. Ths paper s part of NBER s research program n Publc Economcs. Any opnons expressed are those of the authors and not those of the PPIC or the Natonal Bureau of Economc Research $ Elsever Scence All rghts reserved.

2 136 FULLERTON AND WEST ŽHarrngton et al. 23, p If a Pgovan tax were avalable, t would nduce households to drve fewer mles, to buy fuel-effcent cars, to nstall polluton control equpment Ž PCE., to purchase cleaner fuel, to avod cold start-ups, and perhaps to drve less aggressvely. 3 Dfferent households would choose dfferent amounts of each abatement method. Thus, any effcent polcy would need to nduce the same behavors. We nvestgate other polces that would nfluence people to drve fewer mles and to buy smaller cars, better polluton control equpment, and cleaner fuel. To determne the form of these effcent polces, we model the household choce of mles, vehcle attrbutes, PCE, fuel cleanlness, and other goods. Intally, we clarfy our basc framework n a model of homogeneous consumers. We then model heterogeneous consumers that dffer by ncome and two taste parameters Ž one for mles and one for vehcle sze.. Usng each model, we confrm the effcency of the emssons tax, and we evaluate other polces that dffer n terms of feasblty. For one case, we consder a tax on gasolne that can depend on the vehcle at the pump. Another alternatve s a car tax that depends on the car s characterstcs and mleage. Assumng those polces are not avalable ether, we then derve condtons that characterze second-best optmal unform tax rates on gas and on engne sze, and we dscuss how these rates depend on the jont dstrbuton of tastes for mles and engne sze. 4 We wsh to focus on the economc effcency of alternatve nstruments n the face of heterogeneous choces about gasolne and car characterstcs, but we wsh to abstract from purely dstrbutonal ssues. We therefore assume government can use ndvdual-specfc lump-sum taxes. We also gnore labor-lesure choces and assume government has no revenue constrant. Thus, rather than focus on Sandmo s second-best problem 36 of beng unable to tax lesure, we focus on Sandmo s 5 second-best problem 37 of beng unable to tax emssons. Because vehcle emssons cannot be montored at the source, Eskeland and Jmenez 14 analyze ndrect nstruments relatng to cars and fuels. Eskeland 12 expands ths analyss and bulds a smple general equlbrum model wth homoge- 2 For an estmate of the cost savngs from the use of ncentve nstruments rather than mandates, see Klng 28. For a revew of such studes, see Bohm and Russell 6. Many researchers evaluate costs of current ar polluton or the costs and benefts of abatement due to new vehcle and fuel technologes ŽFaz et al. 15, Hall et al. 20, Kahn 25, Kazm 27, Krupnck and Portney 30, Krupnck and Walls 31, Small and Kazm 40.. Others focus on the effects of command and control Ž CAC. polces such as emssons requrements and Corporate Average Fuel Economy Ž CAFE. standards ŽGoldberg 19, Harrngton 21, Kahn Because of cold start-up emssons, Burmch 8 fnds that a 5-mle trp has almost three tmes the emssons per mle as a 20-mle trp at the same speed. Serra Research 39 fnds that a car drven aggressvely has a carbon monoxde emssons rate that s almost 20 tmes hgher than when drven normally. 4 Optmal tax results are derved here analytcally, usng general functonal forms. In a later paper, Fullerton and West 17 assgn specfc functonal forms, use a large sample of households, smulate second-best polces, and compare welfare gans as a percentage of the gan from the deal-but-unavalable Pgovan tax. 5 See, also, Balcer 4 and Wjkander 43. Comprehensve treatments of optmal taxaton are n Auerbach 3 and Stern 41. Sandmo 37 has no revenue requrement, but assumes that government can tax nether the drty good nor lesure. Thus the second-best tax rates on other taxable goods depend on varous cross-prce elastctes. If all goods other than the drty good are taxable, then Fullerton and Wolverton 17 show that the frst-best allocaton can stll be attaned wth a tax on the endowment and a subsdy to all clean goods.

3 MIMIC A TAX ON VEHICLE EMISSIONS? 137 neous consumers. These papers explore optmal combnatons of mandates and taxes that can mmc the unavalable emssons fee, wth homogeneous consumers. Eskeland and Devarajan 13 proceed to dscuss heterogenety, and they show how combnatons of polces can be used to approach the effect of a Pgovan tax. A key s whether emssons are as senstve to a gas tax as to an emssons fee. Other papers explore market ncentves that could be used n place of the 6 emssons tax. Harrngton et al. 22 consder the cost-effectveness of a mandated vehcle nspecton and mantenance Ž IM. program compared to ncentves, wth uncertanty. The ncentve s a fee that s based on the vehcle s emsson rate, assumng mles are not observable. Thus, motorsts can reduce ther fee by reparng ther vehcle, but not by drvng less. Sevgny 38 ncorporates the choce of mles wth a second-best emssons tax, but ths tax requres knowledge of each vehcle s average emssons per mle and the accurate measurement of mles traveled. 7 Snce our paper provdes a general equlbrum model wth heterogeneous consumers who can choose mles and car characterstcs, t s most smlar to the exstng paper by Innes 24. He also analyzes frst-best and second-best combnatons of feasble polcy nstruments. We state below where some of our results confrm those of Innes, but we also show how some other results dffer n ways that can be attrbuted to assumptons employed n each model. Because of the recent ncrease n consumers affnty for large vehcles, we focus on engne sze as one mportant determnant of emssons. 8 Thus our model dffers from Innes n three respects. Frst, we allow consumer heterogenety n terms of two dfferent taste parameters Ž for mles and for engne sze.. Second, we wrte explct expressons for mles per gallon Ž MPG. and emssons per mle Ž EPM. that are functons of engne sze and other car characterstcs. Thrd, we derve closed-form solutons for frst-best tax rates on gasolne, engne sze, and PCE. In both the homogeneous-consumer model and the heterogeneous-consumer model, we evaluate four knds of polces. Frst, we confrm that a sngle rate of tax on all emssons would acheve a frst-best allocaton of resources. Assumng that s not feasble, we then fnd a closed-form soluton for a frst-best gasolne tax where the rate depends on the vehcle at the pump. Ths second polcy may not be feasble ether, f consumers can sphon gas from one vehcle to another, so we 6 Plaut 35 compares nstruments one at a tme. Kohn 29 shows that any combnaton of a tax on emssons and subsdy to abatement are equvalent. For any such combnaton to be admnstered, however, emssons must be measurable. Tran et al. 42 analyze feebates, n whch rebates are provded to vehcles wth hgher-than-average fuel effcency and fees are leved on less effcent vehcles. These feebates are feasble ncentves because fuel effcency can be measured, but they are not perfectly effcent because they do not depend on mles drven. 7 All of these schemes are mperfect. Emssons per mle Ž EPM. cannot be measured perfectly, because t depends on how the car s drven. Mles cannot be measured perfectly, because drvers can roll back the odometer. Harrngton et al. 23 dscuss remote sensng at a selecton of locatons as a good approxmaton, but some drvers may dsproportonately mss or ntentonally avod those locatons. Our schemes are not perfect ether, as they mss some behavors mentoned above Žcold start-ups, aggressve drvng.. 8 Current Clean Ar Act regulatons mpose the same restrctons on all cars, so new-vehcle emssons vary by engne sze only because of weaker regulatons for trucks and sport utlty vehcles Ž SUV.. Lght trucks and SUVs are one out of every two famly vehcles sold, and wll be the fastest growng source of global warmng gases n the Unted States over the next decade ŽBradsher 7, p. 1.. In addton, actual subsequent emssons rates may vary by engne sze, even wthn a vehcle class.

4 138 FULLERTON AND WEST solve for a frst-best vehcle tax that depends on characterstcs of the vehcle and on mleage. Ths thrd polcy s not enforceable f consumers can roll back ther odometers. Last, we consder a three-part polcy wth a sngle rate of tax on gasolne, a sngle rate of tax per unt of engne sze, and a sngle rate of subsdy to PCE. Wth homogeneous consumers ths last polcy acheves frst best, but n the heterogeneous case the frst best would requre a dfferent rate of tax for each consumer. Assumng such rates are not feasble, we solve for condtons that characterze the second-best unform rates of tax on gasolne and engne sze. Ths model presents polcymakers wth complcated condtons for settng these second-best tax rates, so we then nvestgate easer approaches. In partcular, we nvestgate the bas from the smple but erroneous assumpton that consumers are homogeneous and that all drve the mean number of mles n the mean szed car Žusng the closed-form expressons for frst-best tax rates, evaluated at mean mles and engne sze.. Ths bas depends on convexty of MPG and EPM as functons of engne sze, and on the correlaton n consumer preferences for mles and engne sze. Prelmnary evdence suggests that both MPG and EPM are convex, and that therefore the erroneous assumpton of homogenety would lead to a tax rate less than the desred second-best unform tax on gasolne. On the other hand, prelmnary evdence suggests that mles and engne sze are negatvely correlated and that therefore the assumpton of homogenety would lead to a tax rate that exceeds the desred second-best unform tax on engne sze. I. HOMOGENEOUS CONSUMERS In ths secton, we use a smple model of homogeneous consumers to set up our notaton, to expost the basc model, and to evaluate all four knds of polces. In the sprt of Baumol and Oates 5, we assume perfect nformaton, perfect competton, and no market falures other than a negatve externalty from emssons. 9 We also assume that each household gets ts tax revenue back n a lump sum. Ths s a general equlbrum model, but a smple one where producer prces are fxed Ž and consumer prces vary wth tax rates.. The economy conssts of n dentcal households, each of whch owns one vehcle. Each vehcle s composed of some attrbutes that affect emssons Žsuch as engne sze, fuel effcency, and PCE. and other attrbutes that do not affect emssons Žsuch as leather seats or a sunroof.. Households buy gasolne n order to drve mles, and they choose among grades of fuel-cleanlness. Households gan utlty from drvng mles m, the sze of the vehcle s, and other goods and servces, x. Broadly nterpreted, s represents any vehcle characterstc that gves households utlty and that ncreases emssons per mle. More specfcally, we can defne s to be a measure of engne sze such as cubc nches of dsplacement Ž CID.. Also, consumers may gan or lose utlty from polluton-control equpment, c, and per-gallon fuel cleanlness, f. PCE ncludes catalytc converters and other emssons-reducng equpment drectly nstalled on a vehcle. In general, ths c should reflect the condton as well as the amount of PCE. Fuel 9 We gnore exstng mandates n the theoretcal model below, but we recognze that these mandates affect the estmated ways n whch actual emssons per mle depend on engne sze and other car characterstcs. Thus, ncentve polces may work because they encourage purchase of regulated cars.

5 MIMIC A TAX ON VEHICLE EMISSIONS? 139 cleanlness s an attrbute of gasolne such as volatlty or oxygenaton. 10 As dscussed more below, c or f may rase utlty f the consumer enjoys provdng cleaner ar, or may reduce utlty f the car s performance s affected. Moreover, addtonal c or f s more expensve. Vehcle characterstcs not related to emssons are ncluded n x. Fnally, household utlty s affected by aggregate auto emssons, E. Thus the household s utlty functon s 11 u už m, s, c, f, x, E., Ž 1. where u s contnuous, dfferentable, and strctly quas-concave n ts frst fve arguments. We also assume that the technology s homogeneous and convex, so all dentcal consumers choose the same pont. We assume nternal solutons ntally but dscuss corner solutons later. 12 The emssons per mle Ž EPM. that a car dscharges depends postvely on sze 13 and negatvely on PCE and the clean-fuel characterstc. Thus EPM EPMŽs, c, f.. Snce each of the n households drves m mles, aggregate emssons E can be calculated by nmepmž s, c, f.. Next, fuel effcency s measured n mles per gallon Ž MPG. and depends on engne sze and the quantty of the clean-car good on the 14 vehcle, so MPG MPGŽ s, c.. Cars wth larger engnes get lower gas mleage, so MPGs MPG s s negatve. The addton of a clean-car good such as a catalytc converter adds weght to a vehcle, dmnshng fuel effcency, and therefore MPGc s also lkely to be negatve. 15 Consumers do not purchase m drectly, but through the combnaton they choose of gasolne Ž g., sze Ž s., and the clean-car good Ž c.. Gas demand s related to desred m by m g. Ž 2. MPGŽ s, c. 10 More volatle gasolne leads to more evaporatve emssons. The addton of oxygenates to gasolne alters the stochometrc arfuel rato. Provded the carburetor settng s unchanged, ths alteraton may reduce emssons of carbon monoxde Ž CO. and hydrocarbons Ž HC., but can also ncrease emssons of oxdes of ntrogen Ž NOX.. And, f the mxture becomes too lean Ž hgh arfuel., HC emssons can ncrease due to msfrng ŽOECD Drver utlty may also be affected by the age of the vehcle, and vntage s an mportant determnant of emssons. For smplcty, n ths paper, we gnore vntage and thus the possblty of a subsdy to newness. Addng vntage s straghtforward, and t yelds a newness subsdy analogous to the sze tax below Žsee Fullerton and West A referee ponts out that one could generalze ths model by allowng ncomes to vary whle assumng that consumers have dentcal quas-lnear preferences Ž lnear n the numerare, x.. 13 Prelmnary results ndcate that the effect of sze on EPM s mportant. We use data from the Calforna Ar Resources Board ŽCARB. 9 to estmate ths relatonshp for three pollutants ŽCO, HC, NOX. usng 342 cars and lght-duty trucks of model years 1962 through Emssons per mle frst decrease n CID, then ncrease, n convex U-shapes. Ths regresson for CO s typcal Žstandard errors n parentheses.: lnco lncid 1.36 lncid 2, R 2.14, n Ž. Ž. Ž. 14 Fuel effcency may also be a functon of the clean-fuel characterstc, f. Oxygenated fuel contans methyl tertary butyl ether Ž MTBE. or ethanol, each wth lower energy content per gallon than conventonal gasolne. For smplcty, we do not ncorporate f nto MPG, but we provde an ntutve explanaton of ths effect when warranted. 15 Accordng to Dunleep 11, the addton of one cylnder decreases fuel effcency by 3%. Also, the equpment mandated n U.S. ter 1 emssons regulatons lowers fuel effcency by 1%.

6 140 FULLERTON AND WEST Consumers ncorporate these relatonshps when they decde what sze car and how Ž. much gasolne wll maxmze ther utlty n Eq. 1 above. A. The Socal Planner s Problem The socal planner maxmzes utlty of the representatve household by choosng m, s, c, f, and x, recognzng that ndvdual amounts affect aggregate emssons. The socal planner s also constraned by exogenous prces and household ncome, y. Thus, the socal planner maxmzes utlty n Eq. Ž. 1 subject to Eq. Ž. 2 and a resource constrant ž / pg pf f u m, s, c, f, x, nmepmž s, c, f. y m ps s pc c x MPGŽ s, c. Ž 3. wth respect to m, s, c, f, and x. The prce per gallon of gas wthout any clean characterstc s p g, and the prce per unt of the clean-fuel characterstc per gallon s p. The total prce of a gallon of gasolne s Ž p p f. f g f, and the prvate cost of drvng a mle s Ž p p f. MPGŽ s, c. g f. The prce of s s p s, whch represents the prce of addng a cubc nch of dsplacement to an engne. The prce per unt of the clean-car good s p c, and the prce of x s normalzed to one. Ths maxmzaton yelds frst-order condtons n Eqs. Ž. 4, shown n Table I. Subscrpts on u, EPM, and MPG ndcate partal dervatves Ž u u m. m, and s the margnal socal value of ncome. The quantty n brackets n Ž 4a. s the total mplct prce of a mle, whle the quantty n brackets n Ž 4b. s the overall cost per unt of sze, ncludng the extra amount that must be pad for mles due to the lower MPG caused by the ncremental unt of s. Smlarly, the quantty n brackets n Ž 4c. s the overall cost of PCE, ncludng the extra amount that must be pad for mles due to the lower MPG. In Ž 4d., the term n brackets s the overall cost per unt of the clean-fuel characterstc. These frst-order condtons say that the margnal socal gan from drvng another mle, or from an addtonal unt of s, c, f, or x, s equal to the margnal socal cost of each. The u term on the left-hand sdes of Ž 4a. Ž 4d. E reflects the effect on utlty of the ncrement to aggregate emssons from drvng an addtonal mle, ncreasng vehcle sze, addng PCE, or cleaner gas. B. The Household Problem In contrast to the socal planner, a household does not recognze that ts own choces affect aggregate emssons. 16 However, t may face taxes or subsdes on ts consumpton of s, c, f, x, and g. If t were avalable, a tax on emssons would enter the budget constrant. The household optmzaton problem s to choose m, 16 Techncally, we could say that each household recognzes only ts own contrbuton to aggregate emssons, but that effect becomes nl as n becomes large. The key dstncton s that only the socal planner recognzes how the ndvdual s decson s multpled by n to determne aggregate emssons.

7 MIMIC A TAX ON VEHICLE EMISSIONS? 141 TABLE I Frst-Order Condtons wth Homogeneous Consumers Ž. Equatons 4 from the Socal Planner s Problem pg pf f um uenepmž s, c, f. Ž 4a. MPGŽ s, c. ž Ž. / ž Ž. / pg pf f MPGs us uenmepms ps m Ž 4b. 2 MPGŽ s, c. pg pf f MPGc uc uenmepmc pc m Ž 4c. 2 MPGŽ s, c. u u nmepm f E f u x pm f MPGŽ s, c. Ž. Equatons 6 from the Household Problem ž Ž. / ž Ž Ž.. / ž Ž Ž.. / Ž 4d. Ž 4e. pg tg pf tf f um teepmž s, c, f. Ž 6a. MPGŽ s, c. pg tg pf tf f MPGs us ps ts m teepmsm Ž 6b. 2 MPGŽ s, c. pg tg pf tf f MPGc uc pc tc m teepmcm Ž 6c. 2 MPGŽ s, c. Ž pf tf. m u f teepmcm Ž 6d. MPGŽ s, c. u 1 t x x Ž 6e. s, c, f, and x to maxmze už m, s, c, f, x, E. ž / Ž pg tg Ž pf tf. f. y m Ž ps ts. s MPGŽ s, c. Ž 5. Ž pc tc. c Ž 1 tx. x teepmž s, c, f. m. In ths budget constrant, y s ncome ncludng the lump sum tax rebate, tg s the tax per gallon of gas, t f s the tax per unt of clean-fuel characterstc, ts s the tax per unt of sze, tc s the tax per unt of PCE, t x s the tax per unt of x, and te s the tax per unt of emssons. The frst-order condtons for ths problem are Eqs. Ž. 6, also shown n Table I. Emssons can be made to enter the consumer problem mplctly through the

8 142 FULLERTON AND WEST polluton tax t. The prce per mle would then nclude the emssons tax per mle. e Smlarly, the mplct prces of s, c, and f nclude the emssons tax assocated wth the change n emssons. C. Solutons 1. The Pgoan Tax. The tax on emssons, t e, provdes the basc effcent polcy aganst whch alternatves can be compared. Suppose all other tax rates are set to zero Ž t t t t t 0.. In ths case, Ž 4e. and Ž 6e. g f s c x mply. To equate the frst order condtons for mles, Ž 4a. and Ž 6a., replace nto Ž 4a. and subtract t from Ž 6a. to obtan ue n te MED. Ž 7. We defne the rght-hand sde as margnal envronmental damages Ž MED. per unt of emssons. It s the sum of all n households dsutltes from emssons, u E, translated nto money terms when dvded by the margnal utlty of ncome. Ths s the usual Pgovan tax, and t s greater than zero so long as ue 0. Usng ths value of t, and, then the frst order condton Ž 4b. matches Ž 6b., Ž 4c. e matches Ž 6c., and Ž 4d. matches Ž 6d.. Thus the Pgovan tax on emssons by tself nduces households to make all the optmal choces about mles, car sze, fuel, and polluton control equpment. 2. A Complcated Gas Tax. If the measurement of emssons were mpossble, so that te 0, we can fnd a dfferent polcy that attans the exact same effcent outcome. Ths polcy s a complcated gas tax that depends upon fuel characterstcs, f, and on vehcle characterstcs, s and c. Ths tax s ue n tg EPMŽ s, c, f. MPGŽ s, c.. Ž 8. Ths tax represents the addtonal damage caused by an ncrease of one gallon of gas. It s the damage per unt of emssons Ž MED. tmes emssons per mle Ž EPM. tmes mles per gallon Ž MPG.. If we substtute Ž 8. nto t n Ž 5. g before dfferenta- ton, then addtonal terms n the frst order condtons nvolve dervatves of EPM and MPG wth respect to s, c, and f. These frst order condtons match the socal optmum n Ž. 4. Thus, ths polcy attans the frst best wthout a separate tax on sze or subsdy to fuel cleanlness or PCE ŽInnes 24.. A useful ntuton s that emssons are determned by nmepmž s, c, f., where m s determned by g, so the optmalty of an emssons tax can be replcated perfectly by any polcy that can correctly nfluence every emsson determnant Žs, c, f, and g.. In addton, each household must know how ts gas tax s affected by ts own choce of s, c, and f. It seems reasonable for a gas tax to depend on the fuel characterstc, f. But to acheve the frst best, the gas tax must also depend on characterstcs of the vehcle at the pump Ž s and c.. A gas tax that s fxed to reflect the average vehcle wll not

9 MIMIC A TAX ON VEHICLE EMISSIONS? 143 nfluence each household to modfy ts vehcle. However, the complcated gas tax would be costly to admnster A Complcated Vehcle Tax. If the gas tax cannot depend on characterstcs of the vehcle, the effcent outcome can stll be attaned by a vehcle tax that depends on mleage. Suppose that the consumer s budget constrant n Ž. 5 s modfed to subtract a tax t per vehcle Ž stll assumng that each household owns one vehcle.. Suppose all other tax rates are zero, and the vehcle tax s ue n t mepmž s, c, f.. Ž 9. To set ths tax for each car, authortes must know the car s characterstcs Žs and c., and ts mleage Ž m.. If drvers know how ther tax depends on these choces, 18 they can be nfluenced n these choces. Substtuton of Ž. 9 nto Ž. 5 before dfferentaton yelds all the socal planner s frst order condtons n Ž. 4. Note, however, that ths soluton s essentally an emssons tax! Authortes know the car s characterstcs and mleage, so Ž. 9 just calculates the car s emssons mepmž s, c, f. and multples by the Pgovan tax rate u n E. Lke an emssons tax, ths vehcle tax may not be feasble. It requres much nformaton, and t can be crcumvented by drvers who roll back ther odometers. 4. Separate Fxed Tax Rates. We now suppose that none of the polces above are avalable, but that government can set separate tax rates on gasolne, engne sze, and PCE. We assume that the gas tax can be made to depend on characterstcs of the fuel, f, but not characterstcs of the car. Smlarly, nether the sze tax nor PCE subsdy can depend on mles or other choces. As t turns out, the gas tax looks exactly lke Ž. 8 above, but t does not vary wth s or c. In other words, the authortes must calculate the fxed rate of tax from Ž. 8 usng the optmal values of s and c, but that rate does not vary wth one s own choces. Then the sze tax s uen MPGs ts m EPMs EPMŽ s, c, f.. Ž 10. MPGŽ s, c. The sze tax has two components. The frst term s u n Ž that s, MED. E tmes the change n emssons per mle from a change n sze Ž EPM s., tmes mles Ž m.. Ths gves the drect damage caused by an ncrease of one unt of sze. Ths term s postve as long as emssons affect utlty negatvely Ž u 0. E and sze affects emssons postvely Ž EPM 0.. The second term s an ndrect effect from an s 17 For example, a tamper-resstant computer code would lkely be requred on each automoble; smlarly, gasolne pumps would have to be equpped to automatcally tack the approprate tax onto any gasolne that s dspensed to a partcular automoble. Moreover, snce a smple sphonng of gas wll permt consumers to bypass taxes on hgh-emsson vehcles, the scope for abuse, partcularly among those hgh-emttng consumers who are arguably the most mportant targets of the tax, would be tremendous ŽInnes 24, p It would be hard for ths vehcle tax to vary wth f, to nduce the rght choce of fuel, but that one margn could be nfluenced separately by a subsdy to fuel producers to make cleaner fuel or a mandate to make cleaner fuel.

10 144 FULLERTON AND WEST addtonal unt of sze through ts effect on fuel effcency. 19 As long as MPGs 0, ths term s negatve and s thus a rebate. Specfcally, t s a rebate of part of the gas tax n Ž. 8. Because an addtonal unt of sze decreases fuel effcency, the household knows that an ncrease n the sze of ts vehcle s engne wll cost addtonal gas tax. Thus part of the external cost of sze s already nternalzed by the gas tax. 20 Because the two components of the sze tax are opposte n sgn, ths theory does not predct the sgn of t s. Snce the rght-hand term before the brackets s postve, the sgn of t s determned by the sgn wthn the brackets. Thus the sze tax s postve whenever s EPMs MPGs. Ž 11. EPMŽ s, c, f. MPGŽ s, c. These two terms are proportonal effects of sze on emssons per mle Ž EPM. and on mles per gallon Ž MPG.. When an addtonal unt of sze brngs about a larger percentage change n emssons per mle than n fuel effcency, the sze tax s postve. If fuel effcency deterorates proportonally more than emssons ncrease, then sze s subsdzed! In ths latter case, the gasolne tax more than completely nternalzes the mpact of sze on emssons. Emprcal exploraton of the relatve effects n Ž 11. wll uncover the sgn of the sze tax. Then t s uen uen MPGc tc EPMcm EPMŽ s, c, f. m. Ž 12. MPGŽ s, c. Ths tax s perfectly analogous to the sze tax. The frst term s negatve to reflect the effect on damages of an added unt of PCE. The second term s a rebate due to the effect that PCE has on fuel effcency Ž already nternalzed by the gas tax.. Snce the second term s also negatve, the sgn of the clean-car tax s always negatve. That s, tc s necessarly a subsdy. 21 All four polces n the homogeneous-consumer model nduce households to make socally optmal trade-offs at the margn, so they are vald only for nternal solutons. A more complete analyss s requred to deal wth corner solutons. 22 If households dslke polluton control equpment enough Ž u 0. c, then the subsdy wthn the gas tax or n Ž 12. may not nduce them to buy any of t. In ths case, the c 19 Both of these terms contan m, the baselne number of mles. Authortes calculate the fxed t s from Ž 10. usng the optmum m, but ths rate does not vary wth the ndvdual s own choce of mles. Of course, n ths model wth homogeneous consumers, all households drve the same type of vehcle the same number of mles per year, and the sze tax s the same for everyone. When we ntroduce heterogenety n Secton II, the frst-best soluton requres that each household pay a sze tax that reflects ts own choce of mles. 20 Ths rebate also appears n Innes second-best vehcle tax, whch equals the predcted socal costs Ž. of emssons, less the porton of these costs that are nternalzed by the gasolne tax Innes 24, p If c measures the amount of PCE nstalled, ths subsdy could be pad upon purchase of the vehcle. More generally, f c reflects the condton of the equpment as well as the amount, then ths subsdy could reward testng, mantenance, and repar of PCE. 22 We derved KuhnTucker condtons from a model wth non-negatvty constrants on clean-car Ž. Ž. Ž. and clean-fuel characterstcs. The results nclude Eqs. 8, 10, and 12 for nternal solutons and an nequalty for each corner soluton. The addtonal ntuton s mnmal, however, so we just outlne these results n the text.

11 MIMIC A TAX ON VEHICLE EMISSIONS? 145 corner soluton wth c 0 s ndeed part of the socal optmum, even though the margnal condtons Ž. 6 are not satsfed. If households care nothng about ths equpment, however, then a dfferent problem arses: when uc 0, the rght-hand sde of frst order condton Ž 6c. must equal zero at the optmum. Snce te 0, the subsdy to PCE Ž ether wthn the gas tax or t. c can only nduce consumers to buy any such equpment f t s equal to the entre prate cost of PCE, ncludng both the drect cost, p c, and the extra fuel costs ncurred due to the negatve effect that PCE has on fuel effcency. Wth a 100% subsdy, however, the choce of c s ndetermnate. Thus, f uc 0, then ncentves do not work. The optmal PCE can only be acheved by a mandate Žas n Innes 24.. The same analyss apples to the clean-fuel characterstc. When u f 0, the rght-hand sde of Ž 6d. must equal zero at the optmum. For households to choose cleaner gas, the subsdy wthn the gas tax must equal the entre cost of the attrbute, p f. We thnk that uc and u f are unlkely to be exactly zero. 23 In fact, these margnal utltes are lkely to fall wth the amount of c or f. Even f uc s negatve, a bg enough subsdy can nduce the household to buy more of ths good, untl uc on the left-hand sde of Ž 6c. falls to the level of the Ž negatve. prvate margnal cost on the rght-hand sde of Ž 6c.. II. HETEROGENEOUS CONSUMERS The tax rates derved n the prevous secton are unform across all consumers. In ths secton, we ntroduce heterogenety to see f and when the optmal tax rates need to dffer among consumers. If the emssons tax te were avalable, we confrm that a sngle te MED would acheve the frst-best socal optmum. If not, then ndvdual-specfc tax rates on other emssons-related goods can stll acheve the frst-best. If polcy s unable to apply ndvdual-specfc tax rates, then t cannot acheve the frst-best. We then characterze the second-best unform tax rates that best approxmate the unavalable tax on emssons. Whereas Innes 24 allows households to dffer n terms of ncome and one taste parameter, we use the parameter to represent the household s preference for mles and to represent the preference for sze of the car. Together wth ncome, these parameters are jontly dstrbuted accordng to the dstrbuton functon hž,, y. wth postve support on,, y, y. The ntegral of ths dstrbuton over,, and y s the populaton, n. In a CES or CobbDouglas specfcaton of utlty, for example, could be the weght on mles, would be the weght on sze, and Ž 1. would be the weght on x. Our analyss s not 23 Indvduals may get postve utlty from usng the latest technologes, or negatve utlty from nconvenence or decreased performance. Couton et al. 10 use data from France, where some vehcles do not have catalytc converters, to estmate a hedonc regresson of vehcle prce on vehcle characterstcs. They fnd that a catalytc converter ncreases prce by 8%. Ths hedonc prce was ncreasng sgnfcantly over the perod, whch would mply an ncreased demand of the equpment 10, p The OECD 33 fnds that oxygenated fuels perform better than hydrocarbon-only fuels; t hey gve better antknock performance and at hgh alttudes and n hot weather gve... better handlng performance compared wth wholly hydrocarbon fuels 33, p. 60. Also, Marell et al. 32 fnd that envronmental concern plays a role n the decson to replace an automoble.

12 146 FULLERTON AND WEST lmted to these specal cases, however, and t s not lmted by any partcular relatonshp between and. Those who lve far from ther place of work have a hgh demand for mles Ž., but they may prefer ether a small car Žfor better gas mleage. or a large car Ž for comfort and safety.. We show the mportance of the correlaton between and. To focus on the ssue of heterogenety, we now gnore the clean-car and clean-fuel characterstcs. Thus fuel effcency and emssons per mle depend only on sze, and each household generates mepmž s. unts of emssons. Aggregate polluton s thus HHH E mepmž s. hž,, y. y, Ž 13. y where choces of m and s are ndvdual-specfc. A household s utlty functon s U um, s, x;, E, Ž 14. where s the household s change n welfare from addtonal polluton Ž UE.. Whle we allow and to dffer among households, to analyze dfferent choces and abatement costs, we are not concerned wth dfferental benefts from envronmental protecton Ž.. 24 A. The Socal Planner s Problem The socal planner must maxmze a measure of socal welfare such as a weghted sum of n households utltes. To set up ths problem, we must specfy weghts that meet three crtera. Frst, we choose weghts so that a dollar gven to any household has the same effect on socal welfare. To acheve ths condton, we dvde each household s utlty by ts own margnal utlty of ncome Ž.. 25 Second, when te s avalable, we want the maxmzaton of our socal welfare functon to yeld the soluton of Pgou 34. Snce ths soluton s based on margnal condtons Žsuch as margnal envronmental damages. at the optmum, we use the values for that occur at the frst-best socal optmum Ž.. Thrd, when frst-best nstruments are not avalable, we want to be able to fnd second-best unform tax rates that maxmze the same socal welfare functon. Therefore we use prces at the Pgovan equlbrum to evaluate, and we use those to get the weghts Ž1. for all subsequent evaluatons of other polces. The result s a money-metrc measure of socal welfare. 24 Ths change restrcts utlty to a quas-lnear formlnear n emssons. If we kept the more general form, then frst order condtons and frst-best tax rates below would nclude complcated-lookng ntegrals over all dsutltes, u E, nstead of just n, but all else would reman the same. 25 To avod redstrbutons n the tax rate problem below, as n the homogeneous consumer model, we assume that each ndvdual s tax revenues are returned n a lump sum to the same ndvdual.

13 MIMIC A TAX ON VEHICLE EMISSIONS? 147 The socal planner s problem s to maxmze ths welfare functon subject to a Ž. resource constrant the ntegral over all ndvdual budget constrants, HHH y um, s, x HHH mepmž s. hž,, y. y hž,, y. y y Ž 15. HHH p g y m ps s x hž,, y. y y MPGŽ s. wth respect to each consumer s m, s, and x Ž gven ther ndvdual,, and y.. Income plus tax rebates s y, and the margnal socal value of ncome s. To maxmze Ž 15., we can gnore the outer ntegral to obtan the ndvdual margnal condtons and then ncorporate the mpact an ndvdual s choce of mles and sze has on emssons by dfferentatng the aggregate emssons term wth respect to the ndvdual s m and s. The resultng frst-order condtons for household are Eqs. Ž 16., shown n the top of Table II. The frst term n each equaton represents the ndvdual s money value of margnal utlty from each good. The second term n Ž 16a. represents the TABLE II Frst-Order Condtons wth Heterogeneous Consumers Equatons Ž 16. from the Socal Planner s Problem a u ž m / pg nepmž s. Ž 16a. MPGŽ s. u ž s / pgmpgs nmepms ps m Ž 16b. ž 2 MPGŽ s./ u ž x / Ž 16c. ž / Equatons Ž 18. from the Household Problem u pg t g teepmž s. Ž 18a. m MPGŽ s. ž / pg tg MPG s ps ts m teepmsm Ž 18b. 2 s MPGŽ s. u Ž. u 1 tx Ž 18c. x a These equatons represent n frst-order condtons, one for each ndvdual. a

14 148 FULLERTON AND WEST external cost of an addtonal mle drven by ndvdual. Smlarly, the second term n Ž 16b. represents the external cost of an addtonal unt of sze purchased by ndvdual. The frst-order condtons Ž 16. say that the money-metrc socal margnal utlty of each good equals the socal margnal cost of that good. Also, lookng at Ž 16c., note that the left-hand sde s ndvdual s change n utlty from an addtonal unt of x, dvded by the margnal utlty of ncome. In other words, t s the dollar value of another unt of x Ž the prce of x.. Snce the prce of x equals one, Ž 16c. says that the socal margnal utlty of ncome,, also equals one. B. The Household Problem In contrast to the socal planner, a household does not recognze that ts own emssons add to aggregate emssons. The household problem s to maxmze Ž pg tg. už m, s, x. E y m Ž ps ts. s ž MPGŽ s./ Ž 17. Ž 1 tx. x teepmž s. m wth respect to m, s, and x. The frst-order condtons are shown n Eqs. Ž 18. of Table II. These equatons are heterogeneous counterparts to the frst-order condtons Ž. 6 of Table I, but wthout the clean-car and clean-fuel characterstcs. In addton, each consumer has a separate set of optmalty condtons. C. Solutons 1. The Pgoan Tax. To solve for a Pgovan tax, set all taxes except te equal to zero Ž t t t 0.. Then, usng 1, Ž 16c. and Ž 18c. s g x match each other. Also usng 1, set Ž 16a. and Ž 18a. equal to each other. The household-specfc varables drop out, leavng n te MED. Ž 19. Usng 1 and ths value of t, then Ž 16b. and Ž 18b. e also match each other. Thus, gven the weghts we have chosen, a unform Pgovan tax on emssons by tself nduces all households, no matter ther tastes for mles and sze, to drve the optmal number of mles n the rght-szed cars. Of course, polcymakers do not necessarly weght households so that ncome to one s the same as ncome to another. In ths paper, however, we weght households smply n a way that mples that the maxmzaton of socal welfare yelds the standard Pgovan formula Ž 19.. Ths frst-best unform Pgovan tax can be used as a benchmark to dentfy other frst-best polces, and more mportantly, aganst whch to evaluate other second-best polces. 2. A Complcated Gas Tax. As n the representatve-agent model, when a Pgovan tax s not possble Ž t 0. e, we can derve a frst-best gas tax. In the heterogeneous-consumer model, ths tax s t nepm s MPG s. 20 Ž. Ž. Ž. g

15 MIMIC A TAX ON VEHICLE EMISSIONS? 149 Ths formula s conceptually the same as n the homogeneous-consumer model Ž except that we dropped the c and f.. As n that pror model, a tax rate per gallon of gasolne that depends on the ndvdual s own choce of car characterstc Ž s. can optmally nfluence the determnants of emssons. The only dfference here s that heterogeneous consumers then optmally choose dfferent car szes and mleage. Thus each pays a dfferent rate per gallon. 3. A Complcated Vehcle Tax. Authortes mght be able to mpose a tax on each vehcle that depends on a drect measure of the emssons rate Ž EPM.,oron the determnants of emssons Ž s., and multply by a measure of mleage, t nepmž s. m. Ž 21. Ths formula agan matches that of the homogeneous-consumer model, and t agan acheves frst best, but n ths case the tax amount would dffer among heterogeneous households. Ths polcy would not acheve frst best n a model wth other less-measurable determnants of emssons Žlke cold start-ups and aggressve drvng.. Even the measure of mles s problematc, as annual odometer readngs would provde ncentve for ndvduals to roll back ther odometers Separate Fxed Tax Rates. Suppose now that the gas tax and sze tax can be set at dfferent rates for dfferent consumers, but that they must be fxed for each consumer. The gas tax cannot vary drectly wth one s own choce of vehcle, and the vehcle tax cannot vary drectly wth mles drven. We use the frst-order condtons n Table II to derve a gas tax that looks just lke Ž 20. and a sze tax: nepmž s. MPG s m ts nepmsm. Ž 22. MPGŽ s. Authortes could use Ž 20. and Ž 22. together to fx each household s tax rates based on that household s own optmal sze Ž s. and mles Ž m.. These tax rates acheve frst best, but the nformaton requrements are enormous. Because these frst-best rates must be ndvdual-specfc, any set of rates that are unform across all consumers cannot acheve frst best. Thus, we fnd that heterogenety matters. Suppose that the frst three polces above are not feasble, and polcy s lmted to a sngle unform rate of tax on gasolne and sngle unform rate of tax on engne sze Žor other vehcle characterstc.. Ths polcy acheves frst best n the homogeneous-consumer model, but not n the heterogeneous-consumer model. Moreover, a greater degree of heterogenety means greater dvergence from frst best. For these reasons, we now consder how to set the second-best unform tax rates on gasolne and engne sze. One possblty s that these rates could be calculated from Ž 20. and Ž 22. usng the mean sze and mles. How well these unform tax rates would perform depends on the technologcal relatonshps EPMŽ s. and MPGŽ s. and on the relatonshp n preferences between sze and mles. In the next secton, we fnd condtons that characterze second-best unform tax rates, and we compare them to the rates usng means n Ž 20. and Ž Even f only a small proporton of consumers cheat n ths way, those who cheat are lkely to be those who drve the most, who therefore have the greatest ncentve to cheat and who are arguably the most mportant targets of mleage taxaton ŽInnes 24, pp

16 150 FULLERTON AND WEST III. SECOND-BEST TAXES ON GASOLINE AND SIZE Ž. To fnd the second-best tax rates, we must fnd the sngle unform gas tax rate and sze tax rate that maxmze socal welfare, takng as gven households demand behavor for mles, sze, and other goods and servces. 27 Assumng producer prces are fxed, ths s equvalent to maxmzng ths weghted sum of ndrect utltes, HHH y Ž. V t, t, t ; y,, s g x E hž,, y. y, Ž 23. wth respect to ts and t g. As a normalzaton, the tax on x can be set to zero, as n the frst-best scenaro. 28 Usng Roy s Identty, ths maxmzaton results n the frst order condtons, s AŽ t. hž,, y. y hž,, y. y 0 HHH HHH s y y Ž 24a. g AŽ t. hž,, y. y hž,, y. y 0 HHH HHH g y y where Ž for j s, g. 29 Ž 24b. s s g AŽ t j. gmpgž s. EPM s gepmž s. MPG s EPMŽ s. MPGŽ s.. Ž 25. t t t j j j The chosen quanttes Ž s, g, and x. as well as the margnal utlty of ncome Ž. are functons of and. In Ž 24a., the frst term n the ntegral Žs. represents the change n welfare from a change n the sze tax, holdng aggregate emssons constant. The second term, nvolvng At Ž. s, s the change n utlty due to the effect 30 that a sze tax has on aggregate emssons. Smlarly, the frst term n Ž 24b. s the change n welfare from a change n the gas tax, holdng aggregate emssons constant. The second term ncorporates the change n welfare from the effect that the gas tax has on aggregate emssons. 27 For the sake of clarty, here we consder lnear second-best sze tax rates. Perhaps polcymakers could assess nonlnear sze tax rates farly easly. The use of nonlnear schedules could ncorporate heterogenety by accountng for convexty or concavty of EPMŽ s. and MPGŽ s., but not for the possble correlaton between sze and mles. 28 Income, y, s exogenous, so a Ž lump-sum. tax on ncome t y s equvalent to a tax on all commodtes at the same rate; any set of Ž t, t, t. s g x can be scaled up or down, wth commensurate changes n t y. Thus any one rate can be set to zero Žsee Fullerton Actually, each At Ž. also depends on the other tax rate Ž that s, both t and t. j s g, to the extent that s tj or g tj behavors depend on the level of the other tax rate. 30 When an emssons tax acheves the frst-best, where, then Ž 24a. says that the cost to the taxpayer of an ncrease n ts s the amount of s purchased, and that ths margnal cost should be equal to the margnal benefts n terms of reduced emssons.

17 MIMIC A TAX ON VEHICLE EMISSIONS? 151 Thus the tax rates on sze and gasolne should each be set so that the aggregate margnal gan n prvate welfare equals the aggregate margnal loss from the effect on emssons. As shown n the At Ž. j term of each frst order condton, the extent to whch emssons are reduced depends on the degree of responsveness of mles and sze to taxes on sze and gasolne. Thus second-best optmal tax rates on sze and gasolne depend on the elastctes of demand for these goods. But the way n whch changes n sze affect emssons s through the technologcal relatonshps that sze has wth emssons per mle and fuel effcency. The functons EPMŽ s. and MPGŽ s. are therefore major determnants of the second-best tax rates. These frst order condtons cannot be used to solve for the second-best unform tax rates on sze and gasolne. To fnd closed-form solutons we would have to specfy the functonal forms of hž,, y., EPMŽ s., MPGŽ s. and the demands for sze, mles, and other goods and servces. Stll, these frst order condtons can be used to shed some lght on the nature of such taxes. Instead of tryng to rase revenue effcently ŽSandmo 36., these tax rates are tryng to tax somethng that approxmates emssons ŽSandmo 37.. In partcular, f consumers wth a hgh preference for mles Ž hgh. also happen to have a hgh preference for sze Žhgh., then that correlaton s lkely to affect the second-best optmal tg and t s. In addton, preferences for sze determne emssons through the relatonshp sze has wth MPG and EPM. Furthermore, frst-order condtons Ž 24. do not provde clear gudance about how to set unform tax rates n the face of heterogenety. Closed-form solutons for the second-best tax rates are not avalable, but two alternatves come to mnd. Frst, we can calculate the expected value or weghted average of the frst-best ndvdual-specfc tax rates n Eqs. Ž 20. and Ž 22.. These average rates mght then be appled unformly to all ndvduals. These rates are not the same as the second-best rates from Ž 24., but at least they ncorporate nformaton about the dstrbuton hž,, y. of heterogeneous ndvduals. Second, polcymakers mght smply gnore heterogenety, and just use the economy-wde means for mles and sze as f all ndvduals were the same. A comparson of these two alternatves wll not tell us how near or far we are from true second best rates, but t wll reveal somethng about the drecton of the bas We recognze that comparng a rate evaluated at the averages wth the average of the rates s not the same as comparng t wth the second-best unform rate. By nspecton of Ž 24., however, nonlnearty can be seen to affect second-best rates n the same drecton as t affects average rates. Consder, for example, the effect of convexty on the second-best gas tax rate n Ž 24b., usng Ž 25.. If ether EPMŽ s. or MPGŽ s. s convex, or both, then the thrd term n Ž 25. ncreases wth sze at an ncreasng rate; households wth large cars emt a dsproportonately hgh amount. The effect of the gas tax on the gasolne consumpton and emssons of these households s hgher than for those wth small cars. Households wth larger cars thus have larger At Ž. g terms, and larger socal margnal costs of emssons n Ž 24b.. The second-best unform gas tax rate ncreases to reflect these costs. As we shall see below, convexty has the same effect on the average of the ndvdual-specfc gas tax rates. The bas of rates evaluated at the averages s thus n the same drecton relatve to average rates as relatve to second-best rates.

18 152 FULLERTON AND WEST Ž. The average of all dfferent gas tax rates n 20 s HHHnEPMŽ s. MPGŽ s. hž,, y. y y HHH Ž. y t g h,, y y Ž 26. HHH EPMŽ s. MPGŽ s. hž,, y. y. y We ask how ths concept dffers from the smple calculaton of the gas tax rate for the person wth average choces: t s nepm s MPG s. 27 g Ž. Ž. Ž. Ž. Specfcally, we want to dentfy the crcumstances under whch the average of the gas tax n Ž 26. s greater than the gas tax rate for the person wth average choces n Ž 27.. We can thus dscover the condtons under whch unform taxes based on average choces would lkely fall short of attanng the second-best emssons reducton. Snce mles Ž m. do not appear n these two equatons, whether Ž 26. exceeds Ž 27. does not depend on correlaton between sze and mles. It does depend on the characterstcs of EPMŽ s. and MPGŽ s.. Convexty of EPMŽ s., for example, would mean that ncreases n sze ncrease emssons per mle at an ncreasng rate. Ths would rase the weghted average usng EPMŽ s. n Ž 26. relatve to the tax rate usng average sze n Ž 27.. Convexty n MPGŽ s. also rases Ž 26. relatve to Ž 27.. So, f ether functon s suffcently convex Ž or f both are convex., then the use of average sze to calculate the gas tax rate would result n a lower tax rate than Ž would understate. the second-best unform tax rate. Conversely, f ether EPMŽ s. or MPGŽ s. s suffcently concave, then usng the average value of sze to calculate the gas tax rate would result n a hgher rate than Ž would overstate. the second-best unform tax rate. To determne the lkely magntude of Ž 26. relatve to Ž 27., we need estmates of the possble nonlnear effect of engne sze on fuel effcency and emssons. Whle t s wdely known that fuel effcency decreases wth engne sze, a lterature search locates no statstcal estmates of the nonlnear nature of ths relatonshp. Nor could we fnd any estmates of the effect of sze on emssons per mle. 32 For these reasons, we use the CARB data to estmate EPM and MPG as polynomal functons of engne sze. These regressons omt other explanatory varables n order to capture the full effect of sze, the only taxed characterstc. These very prelmnary results suggest that EPM s ncreasng over most of the range of sze 32 Dunleep 11 provdes only rough estmates of the effect of sze on MPG. Kahn 26 examnes the effect of sze on emssons n parts per mllon rather than on emssons per mle.