Marginal Utility of Income and value of time in urban transport. Abstract

Transcription

1 Margnal Utlty of Income and value of tme n urban transport Chrstelle Vauroux Unversty of Cncnnat Abstract We relax the assumpton of constancy of the margnal utlty of ncome nto a structural model of urban transportaton wth endogenous congeston. We examne the mpact of unobserved heterogenety n Margnal Utlty (MU) of ncome on the determnants of travel by estmatng the model usng household survey data. We show that the value of tme s no more statstcally dfferent across tme slots and that the model s robust to all other results. The author s grateful to the partcpants of the Mdwest Econometrcs Group for ther useful comments. The author's research was partally supported by Charles Phelps Taft grants. Ctaton: Vauroux, Chrstelle, (2008) "Margnal Utlty of Income and value of tme n urban transport." Economcs Bulletn, Vol. 4, No. 3 pp. 1-8 Submtted: January 26, Accepted: March 7, URL:

2 1. Introducton The obectve of the present note s to test the constancy of the MU of ncome n the estmaton of urban transportaton demand wth endogenous congeston. The decson of travelng s modeled as a Bayesan game, n whch travelers mpose externaltes on one another and possess prvate nformaton about ther own averson to traffc congeston. We assume that the traveler has a prvate knowledge of hs MU of ncome and that t s proportonal to hs tolerance to traffc. In agreement wth the lterature statng that margnal utlty of ncome decreases wth ncome, we assume that hgher ncome ndvduals are less constraned by travelng schedules than lower ncome ndvduals (See Frsch (1964), and Clark (1973) for emprcal measures of MU of ncome n transportaton). Under the assumpton of constant MU of ncome, Vauroux (2007) showed that the ndvdual value of tme n traffc s sgnfcantly dfferent between peak and offpeak perods wth an estmated valuaton of.769 durng peak perod 1 aganst.7213 durng off-peak perod (and an t-test statstc of ). We show that ntroducng unobserved heterogenety n the MU of ncome results n a constant average averson to traffc congeston across tme slots. Consequently, the dfference n averson between perods s only explaned by dfferences n the cost of transportaton usage. 2. Framework Ths secton ntroduces the transportaton demand model. The dea s that polces (such as tax or fare ncreases) affects ndvdual decsons of travelng and these decson n aggregate change economc varables such as traffc congeston, whch affect ndvdual decsons agan. Hence, the ndvdual preference for travelng depends on the antcpated level of congeston, whch n turn s determned by the travel decsons of all ndvduals. We let θ ndex ndvdual value of travelng, and refer to θ as ndvdual s type (for = 1,..., I). Here θ Θ = Θ = [θ, θ], where θ s a taste parameter ndexng the least tolerance (greatest averson) for congeston whle θ represents the greatest tolerance (lowest averson) for the externaltes. We denote by p the ex-ante probablty that ndvdual s type s θ, and we assume t has a probablty densty functon f(θ ). We also wrte θ = (θ, θ ) Θ Θ wth Θ := Θ. I Let p (θ θ ) denote the subectve probablty that would assgn to the event that θ Θ s the actual profle of types for the others, f hs own type were θ. We assume that the probabltes p are ndependent, so that the densty functon of p (θ θ ) s gven by f (θ ). Let q = (q c, qb ) Q denote the number of trps made by ndvdual where qc s the number of car trps creatng congeston, whle q b s the number of bus trps that does not create externaltes. 1 The peak perod refers to ndvduals departng between 7:00 a.m. and 9:00 a.m. or between 4:00 a.m. and 7:00 p.m. 1

3 2 Travelers utlty functons are of the form u (q, θ, ν ) = αq c [1 + ψ c + ln θ ln s ln q c ] + (1 α)q b [1 + ψ b + ln θ ln s ln q b ] + h(θ )ν of Hanemann (1984), where s s the average number of automoble trps made by ndvduals other than, ψ c (respectvely ψb ) denotes a measure of comfort of travelng by car (resp. by bus) for ndvdual, ν gves the amount of composte good consumed by, α (respectvely 1 α) represents the margnal utlty of usng the car (respectvely the bus) and the margnal utlty of ncome h(θ ) s a functon of θ. Indvdual faces the budget constrant a c + p c q c + a b + p b q b + p ν ν w, where p c (respectvely p b ) denotes the car and bus unt prce, ac (respectvely a b ) denotes the fxed charges assocated to car and bus use, p ν s the unt prce assocated wth the composte good ν (normalzed to 1) and w s the ndvdual endowment. Assumng that the number I of ndvduals s suffcently large, we have the followng Proposton 1. The maxmzaton of u ( ) under the budget constrant gves the optmal allocatons of trps for ndvdual : where we use the notaton s := 1 I Proof. : See Appendx 1. but q c (θ ) = θ q b (θ ) = θ I θ Θ c s eψ b s eψ Note that when θ s common knowledge, q c I h(θ )p c α, h(θ )p b 1 α, θ e ψc h(θ )p c α df (θ ) s := 1 θ e ψc h(θ )p c α I 1/2. (θ ) and q b (θ ) reman the same, Fnally, one can wrte the ndrect utlty functon of ndvdual as V (w a, p c, p b, θ) = αq c (θ) + (1 α)q b (θ) + h (θ ) (w a ), (1) where a = a c + ab. 3. Data and Estmaton We use a household survey n the greater Montpeller area (south of France; 229,055 nhabtants) recordng the transportaton actvty of 6341 ndvduals on a two days perod. A trp s seen as a more-than-300-meters drve or run between two places on a publc road. We focus on trps made for the purposes of work, school, shoppng, lesure, returns home are not accounted for. We use a maxmum lkelhood estmaton method. The observed number of trps s assumed to follow a 1/2.

4 3 Posson dstrbuton, the expectaton of whch s the equlbrum condtonal number of trps at the Nash equlbrum above (for a gven mode of transportaton). We use as average cost of the car, the prce per klometer tmes dstance Orgn-Destnaton. Bus fares vary by type and accordng to travelers socoeconomc characterstcs. They nclude the unt tcket: FF7 (1.07 euros), a booklet of three tckets: FF20 (3.05 euros), a booklet of 10 tckets (dscounted for handcapped or large famles), a 30 days lump sum (dscounted for students, non students-employed, unemployed, scholars dependng on dstrct subventons, retred wth and wthout no Carte Or subscrpton) an annual lump sum (dscounted for scholars and students, unemployed non students-scholars). Hence, each traveler possesses the followng mutually exclusve choces: he uses nether the car nor the bus (usng another mode of transportaton); V (θ) = h(θ)w ; he uses at least once the car but never the bus; e ψc hp c V c θ (θ) = α s α + h(θ)(w a c ); he does not use the car but he uses at least once the bus; then he can choose among J payment opton for the bus (J cases n total); V b (θ) = (1 α) θ s 1 α + h(θ)(θ)(w a b ), = 1,..., J; he uses both the car and the bus at least once; then he can choose agan among J e ψb hp payment opton for the bus (J cases n total). V cb (θ) = α θ s e ψc hp c α b + (1 α) θ s e ψb hp b 1 α + h(w a c ab ), = 1,..., J. The model s estmated by maxmum lkelhood where the lkelhood functon s the ont probablty of dong a number of trps and choosng a mode of payment. It s decomposed nto a probablty of makng a certan number of trps condtonal on makng trps wth that mode of transportaton and payment tmes the margnal probablty of choosng that mode of transportaton and mode of payment. The frst probablty s chosen to be a normalzed Posson dstrbuton whle the second s a multnomal logt. The lkelhood functons are reported n Appendx 2. We assume that h(θ ) = hθ, that s, the more tolerant to traffc congeston, the more an ndvdual uses transportaton despte traffc condtons and the hgher the margnal utlty of ncome. Vectors of comfort of travelng are specfed as ψ b = βb X b, and ψ c = βc X c where X b and X c are vectors characterzng the trp such as the tme between the Orgn and the Destnaton (O-D) as well as socoeconomc characterstcs of ndvdual. The respectve vectors of parameters to be estmated are denoted β c = { β c }=1,..J, βb = { } β b where J and J are the numbers of =1,..J varables ntroduced to respectvely characterze car and bus trps. Estmaton results are presented n Appendx 3. They show that the estmated 1 average averson to traffc congeston ( = ) s low and no more dfferent across tme slots as ths dfference s accounted for by the MU of ncome functon. Under the assumpton of a constant MU of ncome, Vauroux s (2007) estmaton results showed a sgnfcant dfference of the average averson to congeston between peak and off-peak perods. Allowng the MU of ncome to depend on averson to congeston results n a constant average averson. Consequently, the dfference n averson between perods s only explaned by dfferences n the cost of transportaton usage. The other results reman robust. The hgher the bus frequency, the more the ndvdual travels by bus and the effect s sgnfcantly stronger durng

5 4 off-peak tme. As the dstance from the Orgn to the Destnaton ncreases, ndvduals travel more by bus and less by car, whle both modes are used durng off-peak tmes. Scholars and unemployed travel more by bus for all tmes. 4. Concluson We show that the Bayesan approach used to model the endogenety of the congeston process n urban areas s robust to the relaxaton of the assumpton on the constancy of the margnal utlty of ncome. The only dfference s n the dfference n averson to traffc congeston found smlar across perods. References Clark, C. (1973): The margnal utlty of ncome, Oxford Economc Papers, 25 (2), Frsch, R. (1964): Dynamc Utlty, Econometrca, 32, Hanemann, W. M. (1984): Dscrete/contnuous models of consumer demand, Econometrca, 52, Vauroux, C. (2007): Structural estmaton of congeston costs, European Economc Revew 61, 1 25.

6 5 Appendx 1. Proof of Proposton 1 Proof. Takng nto account the budget constrants, the utlty functons u are gven by the formula u (q, q, θ) = αq c [ 1 + ψ c + ln θ ln s ln q c ] + (1 α) [ 1 + ψ b + ln θ ln s ln q b ] + h (θ ) (w a c p c q c a b p b q b ). Ths defnton leads to a pure multstrategy equlbrum correspondng to the value of q (θ) whch maxmzes u (q, q, θ). In order to smplfy the computaton, let us admt that the varable q (θ) can be changed contnuously, and let us wrte down the frst order condtons assocated to the above maxmzaton. Both partal dervatves wth respect to q c and qb must vansh at the equlbrum pont, namely, Solvng for q c α [ ψ c + ln θ ln s ln q c ] h (θ ) p c = 0, (2) (1 α) [ ψ b + ln θ ln s ln q b ] h (θ ) p b = 0. (3) and q b we obtan the frst two nequaltes of the proposton : q c (θ ) = θ s e ψc h(θ c )p α, (4) q b (θ ) = θ s e ψb h(θ b )p 1 α. (5) In order to determne the value of s, ntegrate both parts of (4) wth respect to θ of densty f(θ ); we obtan s (q c ) = 1 I 1 = 1 I 1 I I θ Θ θ Θ s q c (θ )f(θ )dθ θ e ψc h(θ c )p α f(θ )dθ. Assumng that one ndvdual s neglgble n the contnuum of ndvduals so that s = s, we may denote ths common value by s. It follows that (s ) 2 1 I I θ e ψc h(θ c )p α f(θ )dθ. In the case of complete nformaton the proof remans the same, except the determnaton of s where we do not have to ntegrate over Θ. Then we obtan (s ) 2 1 I I θ e ψc h(θ )p c α.

7 6 To obtan the expresson of the ndrect utlty functon, let us rewrte the equaltes (2) and (3) n the form Then we obtan 1 + ψ c + ln θ ln s ln q c 1 + ψ b + ln θ ln s ln q b = 1 + h (θ ) p c, α = 1 + h (θ ) p b 1 α. as stated. V (w a c a b, p c, p b, θ) = [α + h (θ ) p c ]q c + h (θ ) [w a c p c q c (θ) + [1 α + h (θ ) p b ]q b (θ) (θ) a b p b q b (θ)] = αq c (θ) + (1 α)q b (θ) + h (θ ) (w a c a b ) Appendx 2. Expresson of the lkelhood functon Let kn denote the contrbuton to the lkelhood n case of k car trps and n bus trps by usng the mode of payment to take the bus f n > 0. It s the product of the probablty to observe q c car trps and/or q b bus trps by the logstc probablty to choose the car and/or bus modes of transportaton. The probablty of makng q b or q b trps follows a Posson dstrbuton. Then the uncondtonal lkelhood functon s gven by the product wth where 00 = k0 = 0n = kn = Θ Θ Θ Θ e V S df (θ ); exp ( q c ) (q c ) k k! (1 exp ( q c exp ( q b )) ) (q b ) n n! ( 1 exp ( q b exp ( qc )) L = N =1 kn e V c S df (θ ), k = 1, 2,... ; b ev ) exp ( q b k!n! (1 exp ( q c )) ( 1 exp ( q b k, n = 1, 2,..., = 1,..., J, S := e V + e V c + S df (θ ), n = 1, 2,..., = 1,..., J; ) (q c ) k (q b )n cb ev J =1 (e V b )) S ) + e V cb. df (θ ), Note that the structure of the lkelhood s hghly nonlnear n h(θ ). Our expresson of the lkelhood s sensbly dfferent from constant MU of ncome case.

8 7 Appendx 3. Estmaton results Table 1: Estmaton Results Varable Parameter θ Beta(1,µ) Comparson Peak Off-Peak t-test Margnal Utlty Transport α (0.0097) (0.0069) Prvate nformaton µ (0.0008) (0.0009) Slope of MU ncome h (0.1367) (0.1630) Bus 1 β1 b (0.1356) (0.1753) Bus Frequency β2 b (0.0143) (0.0219) Dstance O-D β3 b (0.0128) (0.0180) Tme O-D β4 b (0.0007) (0.0018) Student-Scholar β5 b (0.0769) (0.1313) Car 1 β1 c (0.1795) (0.2221) Dstance O-D β2 c (0.0084) (0.0127) Tme O-D β3 c (0.0004) (0.0018) Power of the car β4 c (0.0066) (0.0104) Unemployed β5 c (0.0963) (0.0765) Student-Scholar β6 c (0.0957) (0.1280) Age β7 c (0.0087) (0.0096) Age*age β8 c (0.0001) (0.0001) Ad. (Mean Log L.) Standard Devatons n parentheses.