Transport tax reforms, to-part tariffs, and revenue reyling - A theoretial result Abstrat Jens Erik Nielsen Danish Transport Researh Institute We explore the interation beteen taxes on onership and on use of ars hen households fae a disrete hoie of purhasing a ar or not. We use a simple labor-leisure model ith a logit formulation for the disrete hoie of ar-onership to examine ho a tax reform, hih shifts taxes from onership to use of ars, affets elfare. Car transport is burdened ith negative externalities hih lead to feedbak effets on both the internal, and the external, margin. We sho that the elfare effet depends on hoies on both the internal margin and the external margin, and that effets on the external margin might affet the ongestion externality in ar transport signifiantly. Furthermore, the effet of suh a tax reform depends on the initial tax level on ar transport.. Introdution The importane of intervention in the transport setor has beome obvious in reent years ith externalities, and espeially ongestion externalities, inreasing rapidly in almost every major ity. The dilemma faing the transport authorities is that transport, hile ausing negative externalities is an essential part of soiety. This dilemma is pointed out in Parry & Bento (200), here it is shon, that the implementation of marginal ost priing to redue externalities, an ause negative elfare effets if the extra osts of transport disourage labor supply. This paper examines a tax reform in the transport setor. A fixed purhase (or onership) tax on ars is substituted ith a variable tax on the use of ars. This type of tax reform has, to our knoledge, not yet been analyzed onsistently, sine the onership deision has been left out of the previous analysis. We build upon several results from the eonomi literature. The expliit inlusion of time in eonomi models as first undertaken by Beker (965), DeSerpa (97), and others. They inlude time as a soure of utility for households, either diretly in the utility funtions (DeSerpa), or indiretly through a household prodution funtion (Beker). This approah has sine been used in numerous papers, among other: De Borger & Van Dender (2003) ho analyze the effet of a tax reform on the value of time. For the modeling of ar onership, e dra upon the results from Small & Rosen (98). They present a frameork for modeling elfare effets hen disrete onsumer hoies have to be taken into aount. De Borger (2000) demonstrates that the Small & Rosen approah an be implemented in a tax model, and he uses their frameork to derive the optimal to- Trafikdage på Aalborg Universitet 2006
part tariff in a model of disrete hoie hih he also extends to a situation ith externalities (De Borger 200). We sho that the elfare effet of a tax reform depends on a ombination of several fators, most of hih are identified in some of the papers mentioned above. The ne finding is that hanges on the external margin (e.g., hanges in ar-onership) ould be large and should probably be modeled expliit if orret estimates of the elfare effets are to be obtained. The paper has the folloing struture. Setion 2 presents the modeling frameork in hih household deisions on the internal margin (ho muh to onsume ontingent on aronership) is analyzed. The hoie on the external margin (the ar-onership deision) is also desribed in detail. The prodution setor and the government optimization problem are also presented. Setion 3 derives an expression for the elfare effet of a tax reform shifting from fixed to variable taxation of ars. The final setion onludes. 2. The model We onstrut a model in hih households deide on the onsumption of goods and leisure, ho muh to ork, and to ommute by either private transport ('ar') or publi transport ('bus'). Our model deviates from the standard tax model in that e do not assume households to be idential. Instead, e assume all soioeonomi harateristis of the households to be idential but e add a random term that aounts for unobservable heterogeneity beteen households. The prodution setor is fully ompetitive and operates under onstant returns to sale. All produer pries are thus onstant and equal to the marginal ost of prodution. The government tax goods, transport, and labor in order to generate revenue for some unspeified tasks (e.g., national defense or the health are system). We assume that ongestion externalities are present in 'ar transport'. 2. The households Households onsume to goods, 'pure leisure', L, and an aggregate onsumption ommodity, Z. Households supply labor, L, to the prodution setor and ommute to ork by either publi transportation ( bus ) or private transportation ( ar ). Households derive utility from onsumption of the aggregate good, leisure and transportation. Thus the utility funtion an be ritten as b U( L, Z) + u( Z, Z ) () b here Z is a trip by bus and Z is a trip ar defined in detail belo. The reparability assumption separates the transport mode hoie deision from the deision on ho muh to ork. It also allos the possibility for households to prefer one mode over the other and thus that publi and private transport are imperfet substitutes. We assume that H households exist and that H is large. Households have the same earning apaity and thus the same age rate,. Furthermore, households have the same endoment of time, L, and non-labor inome y. A trip to ork an be done by bus or by ar and eah trip Trafikdage på Aalborg Universitet 2006 2
takes up a ertain amount of time hih e ill denote L b and L. A trip is defined as traveling both to and from ork. This means that households either go bak and forth by ar or by bus. Folloing Parry & Bento (200) e assume that the number of orking hours in one day is fixed and equal to one. This means that the supply of labor hours given by L is equal to the total number of ommuting trips to ork b L = Z + Z (2) We let Z be total travel by ar and that this inrease the time requirement for ar transport and e thus have that L ( Z) ' 0 Z = > (3) Furthermore, e assume that households ignore their on influene on Z and, aordingly there is an externality problem in private transport. The bus servie is not affeted by ongestion. Even though this is not ompletely realisti it an be defended by several arguments. Cities ould have priority lanes for busses signs at rossings that give priority to publi transport, or the bus time shedules may simply be set to be so slo that the ongestion is part of the planned travel time. Letting P, b P, and P represent onsumer pries on goods, publi transport, and private transport, t be the tax on labor, P be the fixed ost of purhasing a ar, y non-labor inome and normalizing the age rate to one allo us to rite the onstraints hih the households fae as PZ + P b Z b + P Z + P = ( t ) L + y (4) b b L+ L Z + LZ + L = L (5) We ill label (4) the budget onstraint and label (5) the time resoure onstraint. They are interdependent through L, b Z and Z = then Z. As in De Jong (990) e have that if 0 P=0 and thus that households not using a ar ill also not buy a ar. Apart from the fixed ost P this part of the model is idential to the one used in Parry & Bento (200). 2..The household's hoie at the internal margin We no examine ho a household behaves dependent on its hoie of ar onership status. Sine households oning a ar and households not oning a ar an hoose beteen different onsumption bundles ('non-oners' an not hoose to travel by ar) and fae different budget onstraints ('oners' have to pay a fixed fee P) e analyze the to types of households separately. Trafikdage på Aalborg Universitet 2006 3
The hoie for ar oners Using the budget onstraints given in (4) and (5) and assuming that the households ignore their on influene on the level of Z e an speify the utility maximization problem for ar oners as s.t. max { (, ) ( b, U L Z + u Z Z )} b L, Z, Z, Z M PZ + P Z + P Z + P = ( t ) L + y ( λ ) b b T + + + = ( λ ) b b L L Z LZ L L (6) M T here λ is the marginal utility of inome, λ is the marginal utility of time as a resoure b and labor supply ill be given by L = Z + Z. With (6) being a standard maximization problem e an solve the system given by the first order onditions to obtain the folloing demand funtions L ( PP, b, P, Pt,, L b, L ( Z), yl, ) (7) Z ( PP, b, P, Pt,, L b, L ( Z), yl, ) (8) Z b ( PP, b, P, Pt,, L b, L ( Z), yl, ) (9) Z ( PP, b, P, Pt,, L b, L ( Z), yl, ) (0) and the indiret utility funtion V ( P, P, P, P, t, L, L ( Z), y, L) = max { U( L, Z ) + u( Z, Z ) b b b b L, Z, Z, Z λ ( PZ + P Z + P Z + P ( t ) L + y) M b b T b b λ + + + ( L L Z LZ L L)} () hih is no given as a funtion of variables exogenous to the household. The hoie for non-ar oners Using the budget onstraints given in (4) and (5) and assuming that the households ignore their on influene on the level of Z e an speify the utility maximization problem for ar oners as s.t. 0 0 b0 max { U( L, Z ) + u( Z,0)} 0 0 b0 L, Z, Z M 0 PZ + P Z = ( t ) L + y ( λ ) 0 b b 0 0 0 b b0 0 L L Z L L T 0 + + = ( λ ) (2) Trafikdage på Aalborg Universitet 2006 4
M 0 T 0 here λ is the marginal utility of inome, λ is the marginal utility of time as a resoure 0 b0 0 and labor supply ill be given by L = Z + Z. With (2) being a standard maximization problem e an solve the system given by the first order onditions to obtain the folloing demand funtions 0 b b L ( P, P, t, L, y, L ) (3) 0 b b Z ( PP,, t, L, yl, ) (4) Z b0 ( PP, b, t, L b, yl, ) (5) and the indiret utility funtion V ( P, P, t, L, y, L) = max { U( L, Z ) + u( Z,0) 0 b b 0 0 b0 0 0 b0 L, Z, Z λ ( PZ + P Z ( t ) L + y) M0 0 b b0 0 T0 0 b b0 0 λ + + ( L L Z L L)} (6) hih is no given as a funtion of variables exogenous to the household. 2..2 The household's hoie at the external margin b Faing the prie struture ( PP,, P, P), age tax t, having non-labor inome y, faing time requirements b L and L together ith externality level Z the household hoose beteen the utility level V⁰ and V¹. Sine households are utility maximizing they hoose i {0,} suh i 0 that V = max{ V, V }. Using the random utility approah the households behave as if the i indiret utility funtion is omposed of an observable deterministi part, V together ith stohasti error term ε i i i. We rite this as V + ε. The error term apture the unobservable harateristis hih made the household hoie seem random to the government. For simpliity and to ensure a losed form solutions e assume that these error terms are independently and identially distributed folloing a double exponentially distribution ith the sale parameter normalized to. This gives us a logit model for disrete hoie 0 We kno (Ben-Akiva & Lerman (985) that the probability of hoosing not to buy a ar, π, and the probability of hoosing to buy a ar, π, are given by 0 b b V e = V 0 V e + e π 0 ( PP,, P, Pt,, L, L( Z), yl, ) b b V e = V 0 V e + e π ( PP,, P, Pt,, L, L( Z), yl, ) (7) (8) It is orth noting that the probabilities shon in (7) and (8) depend on all the parameters in the model. This means that even though households not oning a ar do not affet the total Trafikdage på Aalborg Universitet 2006 5
level of ongestion by hanging behavior on the internal margin they still affet the level of ongestion by hanging behavior on the external margin. The expeted maximum utility W for a representative household is given by 0 V V = ln( + ) (9) W e e hih is also knon as the log-sum. The demand for goods and ommodities for a representative (or average) household as ell as the supply of labor an no be ritten as Z Z Z b 0 0 = π + π (20) Z = π Z + π Z Z 0 b0 b = π Z L = π L + π L 0 0 (2) (22) (23) hih is a eighted average of the demand for the to types of households in the eonomy. 2.2 The prodution setor and the publi transport setor We assume that all prodution setors are fully ompetitive and operate under onstant b returns to sale. No profits thus exist and the produer pries p, p and p for ommodities, publi transport (a 'tiket') and private transport ('fuel') beome onstant and equal to the marginal ost of prodution. The government an tax both private and publi transport. Letting t b and t b represent the tax on publi and private transport e an rite b b b P = p + t (24) P = p + t (25) We assume that the fixed fee P for the purhase of a 'ar' is paid diretly to the government. This assumption might seem a bit strange but it does not affet the analysis. 2.3 The government The government has to raise revenue G for some unspeified purposes using the taxes defined in (24) and (25) together ith the labor tax t and the fixed fee, P. We rite the soial elfare funtion for a representative household as 0 b b V V W( P, P, P, P, t, L, L ( Z), y, L) = ln( e + e ) (26) hih the government seeks to maximize. We define the governments' revenue funtion R as Trafikdage på Aalborg Universitet 2006 6
b b (,,, ) $ b R Pt t t = π P+ t L + t Z + t Z (27) here the first term is the fixed fee olleted from ar users, the seond term is the total labor tax and the last to terms represent taxes on bus and ar respetively. The government s budget onstraint is no given by b R( Pt,, t, t) = G (28) Taking a loser look at (26) e see that the effet of hanges in parameters are a eighted sum of hanges in the indiret utility funtions for households oning a ar and households not oning a ar. Letting Θ represent some poliy parameter that is hanged, the hange in maximum expeted utility ill be given by = π + π (29) W 0 0 V V Θ Θ Θ here e an interpret the probabilities as frations of households not oning and oning a ar. Sine for households being at the border beteen having and not having a ar e have V⁰ =V¹ the hange in probability at the margin does not hange the overall elfare. A hange in probabilities does therefore not enter the expression above diretly. 3. Tax reform analysis In this setion e examine ho the elfare hanges hen the government implements a tax reform reduing the purhase tax on ars to variable taxes on the use of ars. 3. Helpful derivations We kno that W M = πλ Z t (30) W M = π λ P (3) W T = πλ Z (32) Commenting on these effets affets e se that (30) and (32) resembles the results from Parry & Bento (200) exept for the probability eighting inluded here. Note that the effet of the fixed fee derived in (3) is idential to the effet of a lump-sum transfer in the Parry and Bento paper (again exept for the probability eighting here). This is a onsequene of P only having an inome effet on the internal margin for ar oners. Trafikdage på Aalborg Universitet 2006 7
3.2 Feedbak effets It ill be advantageous to kno ho the demand for private transport hanges as a funtion of the fixed fee. Sine e have externalities in the model e expet feedbak effets to be present both on the internal margin and on the external margin in the demand for private transport. To simplify notation e ill assume that Z = Z ignoring the number of households H in the derivations. This has no effet on the analysis sine e an inlude the number of households H in the definition of L. Evaluating the hange in demand hen the fixed fee hanges and the revenue is reyled through t e find that ( π + π dt ) Z + ( Z + Z dt ) π Z P t dp P t d P = P ( L'( Z π π + Z ) (33) The numerator apture the effet of the hange in fixed fee had there been no externalities present in the model sine there is a diret response to the inrease in P and a response from the revenue reyling given by Z t dt dp. With externalities present in the model a feedbak effet is present hih is aptured by the denominator. By assumption L >0 and sine e expet private transport to be a normal good the inrease in L ill ause the generalized Z π prie to inrease. We therefore have that < 0 and < 0 thus making the denominator exeeds. Sine e normally also expet that ' L '' Z = to be larger than zero the feedbak effet beomes larger hen the ongestion externality inrease. Furthermore e see that the size of the feedbak effet is determined by both the hange on the internal margin and the hange on the external margin. 3.3 Shifting from fixed to variable tax on ars We no examine the effet of hanging the fixed tax P on the purhase of ars and finaning this by making hanges in the variable tax t on the use of ars. The elfare effet is given by = + + ' (34) dw W W dt W L dz dp P t dp dp The first term on the right hand side is the diret effet on elfare from the hange. The seond term aptures the revenue reyling effet that orks through t. The last term apture the elfare effet of the hanges in the level of ongestion. We no differentiate the dg government budget onstraint (28) ith regards to P using = 0 to get dp dt dp = $ dπ P+ π + t dz + t d L dp dp dp Z (35) Trafikdage på Aalborg Universitet 2006 8
Substituting this together ith (30), (3) and (32) into (34) and doing some simple manipulation gives Z $ + Z dt P d P T dw M dπ d L t λ λ M dp dp dp L' Z λ L = ( P+ [ t + { }{ Z L' t }]) (36) Taking a loser look at this expression e see that several effets affet the outome of the proposed tax reform. The expression in square brakets is omprised of to terms. The first term, dl$ dp t, aptures the labor market effet sine hanges in labor supply ill hange the tax revenue olleted. The seond term in the square brakets is a bit more omplex. The first part aptures hanges in the demand for private transport inluding both tax interation effets and feedbak effets. The seond part of this term desribes the differene beteen the marginal external ost of transport, λ λ T M Z L ', and the tax on transport t. The sign of this term depends on the level at hih t is set. If it is above the marginal ost the term is negative and if it is set belo marginal ost it is positive. In the speial ase here the tax on transport is equal to the marginal external osts e see that this term anels out. The term in square brakets is struturally idential to formula 0 in Parry & Bento (200). The remaining term, dπ, aptures revenue effets oming from hanges in the number of ar oners. dp P 4. Conluding remarks This paper shos that deisions on the external margin, e.g., of ar onership, are an important element in elfare evaluation of tax reforms. Omission of deisions on the external margin ould therefore be ritial. Using a simple model for household deisions, taxation, and disrete hoie, e sho ho the feedbak effet as ell as the elfare effet depends on the onership deision and on the interation ith the labor supply deision. The next step is to implement the results in a numerial model in order to examine the size of the effets. This is left for future researh. Bibliography Beker, G. (965) A Theory of the Alloation of Time, The Eonomi Journal 75, 493-57. Ben-Akiva, M. and Lerman, S. R (985) Disrete Choie Analysis: Theory and Appliation to Travel Demand, The MIT Press, Cambridge, Massahusetts. De Borger, B. (2000) Optimal to-part tariffs in a model of disrete hoie, Journal of Publi Eonomis 76, 27-50. De Borger, B. (200) Disrete hoie models and optimal to-part tariffs in the presene of externalities: optimal taxation of ars, Regional Siene and Urban Eonomis 3, 47-504. De Borger, B. & Van Dender, K. (2003) Transport tax reform, ommuting, and endogenous values of time, Journal of Urban Eonomis 53, 50-530. Trafikdage på Aalborg Universitet 2006 9
De Jong, G. C. (990) An Indiret Utility Model of Car Onership and Private Car Use, European Eonomi Revie 34, 97-985. DeSerpa, A. C. (97) A Theory of the Eonomis of Time, The Eonomi Journal 8, 828-846. DORS (2006) Dansk Økonomi Forår 2006 (in Danish ith English summary), Det Økonomiske Råd (Danish Eonomi Counil). Kleven, H. J. (2004) Optimum Taxation and the Alloation of Time, Journal of Publi Eonomis 88, 545-557. Mayeres, I. (2000) The Effiieny Effet of Transport Poliies in the Presene of Externalities and Distortionary Taxes, Journal of Transport Eonomis and Poliy 34:2, 233-260. Mayeres, I. and Proost, S. (200) Marginal tax reform, externalities and inome, Journal of Publi Eonomis 79, 343-363. Nielsen, J. E. (2005) Externalities, Taxation and Time Alloation, Thesis paper. Parry, I. W. H. & Bento, A. (200) Revenue Reyling and the Welfare Effets of Road Priing, Sandinavian Journal of Eonomis 03:4, 645-67. Sandmo, A. (975) Optimal Taxation in the Presene of Externalities, Sedish Journal of Eonomis.77, 86-97. Small, K. A. & Rosen, H. S. (98) Applied elfare eonomis ith disrete hoie models, Eonometria 49, 05-30. Trafikdage på Aalborg Universitet 2006 0