DISCUSSION PAPER. Should Urban Transit Subsidies Be Reduced? Ian W.H. Parry and Kenneth A. Small

Transcription

1 DISCUSSION PAPER JULY 2007 RFF DP Should Urban Transt Subsdes Be Reduced? Ian W.H. Parry and Kenneth A. Small 1616 P St. NW Washngton, DC

2 Should Urban Transt Subsdes Be Reduced? Ian W.H. Parry and Kenneth A. Small Abstract Ths paper derves ntutve and emprcally useful formulas for the optmal prcng of passenger transt and for the welfare effects of adjustng current fare subsdes, for peak and offpeak urban ral and bus systems. The formulas are mplemented based on a detaled estmaton of parameter values for the metropoltan areas of Washngton (D.C.), Los Angeles, and London. Our analyss accounts for congeston, polluton, and accdent externaltes from automobles and from transt vehcles; scale economes n transt supply; costs of accessng and watng for transt servce as well as servce crowdng costs; and agency adjustment of transt frequency, vehcle sze, and route network to nduced changes n demand for passenger mles. The results support the effcency case for the large fare subsdes currently appled across mode, perod, and cty. In almost all cases, fare subsdes of 50 percent or more of operatng costs are welfare mprovng at the margn, and ths fndng s robust to alternatve assumptons and parameters. Key Words: transt subsdes, scale economes, traffc congeston, welfare effects JEL Classfcaton Numbers: R48, H Resources for the Future. All rghts reserved. No porton of ths paper may be reproduced wthout permsson of the authors. Dscusson papers are research materals crculated by ther authors for purposes of nformaton and dscusson. They have not necessarly undergone formal peer revew.

3 Contents 1. Introducton Analytcal odel odel Assumptons Welfare and Optmal Subsdy Formulas Parameter alues Results Baselne Results Senstvty Analyss Relaton to Recent Studes Concluson...21 References...22 Appendx A: Analytcal Dervatons...27 Appendx B. Assessment of Parameter alues...35

4 Should Urban Transt Subsdes Be Reduced? Ian W.H. Parry and Kenneth A. Small 1. Introducton Passenger fares for publc transportaton are, for the most part, heavly subsdzed. Across the 20 largest transt systems n the Unted States (ranked by passenger mles), the fare subsdy, as measured by the dfference between operatng costs and passenger fare revenues, ranges from 29 to 89 percent of operatng costs for ral, and from 57 to 87 percent for bus (Table 1). Kenworthy and Laube (2001) document a smlar pattern of heavy fare subsdes across cty transt systems n other developed natons. Two classc ratonales for fare subsdes are often advanced (Glaster 1974; Henderson 1977; Jansson 1979). Frst, scale economes mply that the margnal socal cost of supplyng passenger mles s less than the average cost. These scale economes may arse from fxed costs, such as track and staton mantenance, but more mportantly they arse from the ohrng effect, whereby the user s costs of watng at transt stops or accessng transt declne as servce frequency or route densty s ncreased (ohrng 1972). A related pont s that hgher passenger densty allows vehcles to be operated wth hgher occupancy, thereby savng on agency costs. The second ratonale s that lower transt fares dscourage automoble use, thereby reducng external costs from traffc congeston, local and global ar polluton, and traffc accdents. Ths s a second-best argument, snce t assumes that these external costs cannot be nternalzed through approprate road prcng. Determnng whether current fare subsdes are warranted by these two arguments s complcated by several factors. Frst, the strength of both arguments may vary greatly by tme of day, mode, and locaton. Second, the approprate subsdy depends on how transt agences respond to ncreases n passenger demand at the margn whether by expandng servce through more vehcle mles (thus provdng hgher servce frequency and/or a denser route network) or by ncreasng vehcle occupancy (ether through hgher load factors and/or larger vehcles). Thrd, We are grateful to Rchard Arnott, Bruno De Borger, Stef Proost, George Tolley, and Kurt an Dender for helpful comments and suggestons and to Ellot Klen for research assstance. Kenneth Small thanks the Unversty of Calforna Energy Insttute for fnancal support. 1

5 transt vehcles themselves may contrbute to externaltes such as congeston and polluton, and ther passengers generate external costs on each other va crowdng (Kraus 1991) or ncreased boardng and alghtng tme. Fourth, automoble externaltes are partly nternalzed through fuel taxes. And fnally, alterng the subsdy for one mode wll cause substtuton effects across modes and tmes of day, wth secondary effects on economc effcency due to the many dstortons from optmalty condtons n the system. Several studes have estmated optmal transt prces, focusng on one or both of the prmary ratonales just mentoned and usually just one locaton. None of them encompass all of the addtonal complcatons dentfed above. In fact, exstng estmates of optmal transt prces (gven current road prces) vary enormously, from zero to more than 100 percent of operatng costs, provdng a confusng gude as to whether current fare subsdes should be preserved, expanded, or elmnated. 1 It s dffcult to dscern the reasons for the strkngly dverse results because the studes apply to dfferent regons and years, they account for dfferent factors, they make dfferent assumptons about transt agency response, not all of them dstngush tme of day, and some use smplfed analytcal models whle others use less transparent but more sophstcated computatonal models. Furthermore, assumptons about agency and travel responses that may be reasonable at current prces may not be at very dfferent prces; thus, for the purpose of drawng robust qualtatve results, t may be better to focus on the drecton of welfare effects from small changes to exstng prces rather than place too much emphass on fully optmzed prces. Ths paper provdes a general framework for evaluatng exstng fare subsdes and potental prcng reforms. It does so by developng a sngle analytcal model that ncorporates all the factors just descrbed, then derves formulas for optmal subsdes for bus and ral at peak and off-peak perods and calculates the welfare effects from ncrementally adjustng current fare subsdes. These formulas clarfy the contrbuton of all underlyng parameters and can be 1 For London, Glaster and Lews (1978, Table 4, lne 3b) estmate optmal ral and bus fares at about 50 to 60 percent of margnal operatng costs. For the San Francsco Bay Area and for Pttsburgh, ton (1983) fnds optmal fares to be vrtually zero. Wnston and Shrley (1998) fnd qute the opposte for the Unted States as a whole, wth optmal bus and ral fares coverng 84 percent and 97 percent of margnal operatng costs, respectvely. For a prototype Belgan cty, De Borger et al. (1996) estmate optmal transt fares are 50 to 114 percent of average agency costs, dependng on how servce frequency adjusts to passenger demand. For Brussels, an Dender and Proost (2004) estmate optmal transt fares to be nearly zero n peak perods and about double current fares n off-peak perods. Two recent studes of Washngton, D.C., by Wnston and aheshr (2007) and Nelson et al. (2007), estmate net total benefts from transt but wth conflctng results. Insofar as possble, we relate our fndngs to ths prevous lterature n Secton

6 emprcally mplemented n a spreadsheet. Followng an extensve complaton and estmaton of parameter values, we apply the formulas to three large but very dfferent metropoltan areas: Washngton, Los Angeles, and London. Our analyss ncludes vehcle captal costs (whch can be vared farly easly) but not nfrastructure nvestments; thus, followng prevous optmal transt prcng lterature, we explore how best to use exstng nfrastructure wthout worryng about recoverng sunk captal costs. The most strkng fndng s that, n almost all cases, extendng fare subsdes beyond 50 percent of operatng costs often well beyond s welfare mprovng at the margn across modes, perods, and ctes. And these fndngs are generally robust to plausble alternatve assumptons about parameters and agency behavor. The man reasons why large subsdes are welfare mprovng are the two classc ones. However, the relatve mportance of these two ratonales vares across dfferent cases and assumptons. We fnd bg gans from dvertng auto traffc, especally durng peak perods. Furthermore, to the extent that servce s ncreased n response to addtonal passenger demand, scale economes arsng from reduced user costs of wat and access are usually sgnfcant, especally for bus and for off-peak servce. And to the extent that transt vehcle occupancy s ncreased nstead, savngs n operatng costs typcally outwegh any extra costs from crowdng or ncreasng vehcle sze. Do our results mply that exstng operatng defcts should contnue to be fnanced through general taxaton, rather than beng reduced through substantal ncreases n passenger fares and reduced servce levels? One counterargument s that we gnore the broader effcency costs from fnancng operatng defcts through dstortonary taxes. However, as emphaszed n the lterature on envronmental tax shfts (e.g., Bovenberg and Goulder 2002; Parry and Bento 2001), there are mportant counteractng effects on tax dstortons elsewhere n the economy to the extent that lower transportaton costs encourage more economc actvty. We dscuss tax dstortons n Secton 5; based on a rough calculaton there, the net mpact of these dstortons on optmal subsdes appears to be moderate. Another ssue s to what extent our results may carry over to urban transt systems other than the three studed here. argnal congeston costs are lkely to be lower n most other ctes, but as dscussed n our senstvty analyss, optmal fare subsdes can stll be substantal because of other factors. A more defntve answer awats a detaled parameter assessment for other cases. 3

7 Probably the most mportant qualfcaton s that we do not explctly model the potentally lax ncentves for cost mnmzaton nherent n a publcly provded servce. As shown later, our general framework and results stll apply to more effcently managed transt systems wth lower operatng costs. However, there s evdence that subsdy programs themselves cause cost-nflaton through excessve compensaton, msuse of hgh-sklled labor n low-skll tasks, and neffcent use of labor and captal (Wnston and Shrley 1998; Small and Gomez-Ibanez 1999, sect. 3.3). One response to ths problem mght be to prvatze transt systems whle retanng some subsdes; but an alternatve would be to swtch to a fxed subsdy per passenger mle (by mode and perod) and requre the agency to cover the remander of ts operatng costs through ts prcng structure. The rest of the paper s organzed as follows. Secton 2 descrbes the analytcal model and derves key formulas; Secton 3 dscusses baselne data; Secton 4 presents the man quanttatve results and senstvty analyss; and Secton 5 concludes and elaborates on the qualfcatons. 2. Analytcal odel We develop a model of urban passenger travel by auto, ral, and bus at dfferent tmes of day, where transt user costs depend on congeston, transt frequency, route densty, and vehcle crowdng. Travel also produces polluton and accdent externaltes, some of whch are nternalzed by fuel taxes. The government chooses transt characterstcs and fares subject to a budget constrant, whle agents optmze over travel choces takng externaltes and transt characterstcs as gven odel Assumptons () User utlty. The representatve agent has preferences defned as follows: (1a) U = u( X,, Γ) Z (1b) = ({, = P, O; j = CAR, B, R}) (1c) Γ = Γ( T, W, A, C) (1d) T = t, W = w, = CAR A a, C = c CAR CAR where all varables are n per capta terms. In (1a), X s the quantty of a numerare or general consumpton good; s subutlty from passenger mles traveled; Γ s a generalzed (non- 4

8 money) cost of travel; and Z s dsutlty from polluton and traffc accdent externaltes. 2 In (1b), s passenger mles traveled durng perod by mode j where the two tme perods are = P (peak) and O (off-peak), and the three modes are j = CAR (auto), B (bus), and R (ral). 3 In (1c), T s total n-vehcle travel tme, W s tme spent watng at transt stops, A s tme spent accessng transt, and C s crowdng experenced on transt; as shown n (1d), these non-money costs are an aggregaton over mles traveled, each multpled by the respectve per mle costs t, w, a, and c. We assume u( ) s ncreasng and quas-concave n X and and decreasng and quas-concave n Γ; ( ) s quas-concave, mplyng travel by dfferent modes and tme of day are mperfect substtutes; and Γ( ) s ncreasng and quas-concave n non-monetary travel nputs. () Travel characterstcs. Several characterstcs of transt vehcles affect user and operator costs. Frst s vehcle occupancy, o, the average number of passengers n a bus or tran: (2a) o = / where s total vehcle mles. Second s the load factor, l, defned as the fracton of a vehcle s passenger capacty n that s occuped: (2b) l = o / n Thrd s the average servce frequency, f, along each bus or ral transt lne: (2c) f = / D where D s route densty, measured as total route mles wthn the fxed servce area. 4 These varables determne the per-mle travel characterstcs n (1d) as follows: j CAR B j CAR (3a) t = t ( + α ) + θ o, j = CAR, B; θ = 0 ; B t R = t R R + θ o (3b) w = w (f ), a = a (D ), c = c (l ), j=b,r where α B > 1 s the contrbuton of a bus to congeston relatve to that of a car, or the passenger R 2 We exclude possble externaltes from ol dependence because they are dffcult to defne; nsofar as they have been quantfed (for example, NRC 2002 put them at 12 cents per gallon of gasolne), ncorporatng them would make very lttle dfference to our results, as can be seen from our dscusson of senstvty wth respect to global warmng damages n Secton 5. Also, some of the costs of ol dependence are domestc rather than worldwde, and so ncludng all of them would be nconsstent wth the worldwde perspectve adopted n estmatng global warmng costs. 3 We hold trp length constant, so varatons n arse from varatons n the number of trps. 4 As s normal n economc models, all varables are flows. Thus, for example, s defned as ral vehcles per hour averaged over the peak perod. Hence /D has unts (vehcle/hour)/route; that s, vehcles per hour along a gven route. 5

9 car equvalent. In (3a), n-vehcle tme has two potental components. Frst s the tme transt vehcles are statonary at transt stops, expressed per passenger mle; ths s equal to vehcle occupancy tmes j θ, whch s the average dwell tme per passenger (boardng plus alghtng) dvded by trp length. Second s the tme the vehcle spends n moton per mle of travel, t j ( ), whch s the nverse of vehcle speed. For autos and buses, whch share the roads, t j ( ) s a weakly convex functon of aggregate road traffc, wth bus traffc weghted by α B ; buses travel more slowly than autos, therefore t B ( )>t CAR ( ). For ral we assume t R s fxed; that s, an extra tran does not slow down the speed of other trans n the system. In (3b) the per mle wat tme for transt vares negatvely wth servce frequency; the per mle transt access tme vares negatvely wth route densty; and per mle crowdng vares postvely wth the load factor. () Polluton and accdent externaltes. The nature of these externaltes has been dscussed extensvely elsewhere (e.g., 2005); we smply summarze ther aggregate cost by (4) Z = Σ z where z s the combned polluton and accdent external costs per vehcle mle. 5 (v) Household optmzaton. The household budget constrant s (5) I TAX = X + p where I s (exogenous) prvate ncome, TAX s a lump-sum tax to help fnance transt defcts, p s the money cost per passenger mle of travel, and the prce of X s normalzed to one. For bus and ral, p s the per mle fare, whereas for auto, p CAR CAR CAR = p + τ, where CAR fuel costs and τ s fuel taxes, both expressed per passenger mle ( tme of day because congeston affects fuel economy). CAR p s pretax CAR CAR p and vary by Households choose passenger mles and the numerare good to maxmze utlty (1) 6 τ 5 Some of the socal costs of traffc accdents (e.g., njury rsk to oneself) are nternal and are mplctly taken nto account n the subutlty functon ( ) for travel. 6 Other money payments (e.g., car mantenance, parkng fees) are assumed constant and are treated as subtractons from the utlty of travelng by car rather than explctly as costs. 6

10 subject to (5), takng p, t, w, a, c, Z, and TAX as gven. Ths yelds frst-order condtons, summarzed by (6a) u u X = q p + ρ T W A t + ρ w + ρ a + ρ C c (6b) k ρ u Γ / u Γ k X, k=t, W, A, C The quanttes ρ k are the (margnal) dollar values of n-vehcle tme, watng tme, access tme, and crowdng, whch are taken as fxed (although t s not ndcated by the notaton, we allow these values to vary by tme of day.) Thus s a generalzed prce, ncludng both money and nonmoney costs per mle; agents equate the margnal beneft from passenger mles q ths generalzed prce for each mode and tme perod. From (5), (6), and (1) we obtan the demand functons and ndrect utlty (denoted by ~): u u / X xy (7) = ({ q }, TAX ), X X ( q xy ~ xy = { }, TAX ), U = u~ ( { q }, TAX ) Z where {q xy } denotes the set of q for, j. to (v) Transt agency constrants. The agency s total operatng cost, OC, n perod for mode j, s (8a) (8b) OC = F + K = k + 1 k2 K n t where, > 0 are parameters. In (8a), F s a fxed cost representng, for example, the cost of k 1 k 2 operatng ral statons; consstent wth emprcal evdence, we assume there are no scale economes or dseconomes n provdng bus vehcle mles, F B =0 (Small 1992, 57). Operatng costs also nclude varable costs equal to total vehcle hours of operaton t multpled by varable costs per vehcle hour, K, whch prmarly reflect drver labor and vehcle captal. In (8b), K s a lnear functon of vehcle capacty, wth scale economes to the extent k 1 >0. We assume Pj Oj k1 > k1 because peak servce does not convenently ft an eght-hour workday, so ts unt labor costs are hgher; and we assume Pj Oj k2 > k2 prmarly for peak use are also avalable off-peak at lttle or no extra cost. because larger vehcles that are purchased 7

11 The agency budget constrant s CAR CAR (9) TAX + τ = ( OC p ) j CAR That s, revenues from lump-sum taxes and fuel taxes fnance the transt defct. 7 (v) Agency adjustment of transt characterstcs. Because there s lttle emprcal bass for quantfyng access and crowdng costs, we elmnate the need to do so by assumng that, for gven vehcle mles, the transt agency optmzes over route densty and servce frequency, and that for a gven vehcle occupancy, t optmzes over vehcle sze and load factor. These assumptons mply the followng frst-order condtons (see Appendx A): (10a) W ρ w ηw = A ρ a η a (10b) ρ C c η c o = t k 2 n where,, denote wat, access, and crowdng cost elastctes, all defned postvely: for η w example, η a η w η c = dw /df (f /w ). (10a) states that route densty s ncreased untl the ncremental cost of extra watng, resultng from less frequent servce, equals the ncremental reducton n access cost. (10b) states that transt vehcle sze s ncreased untl the ncremental reducton n crowdng costs to ts occupants equals the ncremental agency cost of the larger vehcle. Although these assumptons represent a neutral case, 8 we dscuss later the mplcatons of relaxng them. From (6a) and (10) we can express the generalzed user prce as T W (10c) q = p + ρ t + ρ w 1+ η / η ) + t k n /( o η ) ( w a 2 c 7 In practce, fuel tax revenues are earmarked for road and transt nfrastructure projects; accountng for ths could affect our results very slghtly, to the extent that the socal beneft per dollar of nfrastructure spendng dffers from unty. 8 For example, f the agency overnvests n servce frequency relatve to route densty, then usng (10a) and data on wat costs wll underestmate (unobserved) margnal access costs, and vce versa f there s undernvestment n servce frequency. Wthout relable data on access costs, we cannot say whch of these two cases mght be the more lkely. Condton (10a) gnores fxed costs of addtonal routes and so overstates the optmal route densty for ral. On the other hand, there s evdence that ral lnes have been bult that are not economcally justfed, and thus current route densty may also exceed optmal route densty. Furthermore, off-peak route densty can be adjusted even n the short run by makng some lnes peak-only. To analyze ths more thoroughly, we could omt (10a) for peak servce and assume nstead that D Pj s fxed at ts current value; ths would requre emprcal estmates of crowdng costs ρ C c Pj. 8

12 Followng an ncrease n demand for passenger mles, we assume that a (constant) fracton ε of t s accommodated through ncreased vehcle mles (of whch the ncreases n servce frequency and route densty are chosen to satsfy (10a)) and fracton 1 ε through hgher occupancy of transt vehcles (wth the ncrease n vehcle sze and load factors chosen to satsfy (10b)) Welfare and Optmal Subsdy Formulas () argnal welfare effects. We frst consder welfare effects of margnal changes n exstng transt prces. The resultng formulas are useful n drawng robust conclusons about whether ncreasng current fare subsdes mproves or reduces economc effcency robust because the formulas depend only on margnal rather than global assumptons about demand functons and agency adjustments. We focus on peak-perod ral for exposton; the formulas for other transt modes and perods are analogous. We dfferentate ndrect utlty wth respect to p ; that s, we consder an ncremental reducton n the fare, accountng for nduced changes n travel, user, and external costs, and n the agency budget. The resultng margnal welfare effect, defned (n consumpton unts) as ~ W ( du / ) / u, can be expressed as the sum of four components (see Appendx A): (11) W X margnal cost/prce gap ( C p )( supply ) net scale economy ( B C )( scale occ ) 64 externalty C =, CAR other transt ( C + C + C B p ) = OR, PB, OB supply ext occ scale ext In (11), the quantty d / s the margnal demand shft for mode nduced by a peak-ral prce change. Our preference assumptons mply that < 0 and 0 for ; that s, peak-ral rdershp goes up followng a decrease n the fare, dvertng rdershp away from autos and other transt. The other expressons n (11) are defned as follows: (12a) C supply = ( ε / o ) K t W (12b) B = ε ρ w η, scale w C occ = ( 1 ε )t k 2 n / o 9

13 (12c) z CAR CAR CAR CAR C ext = + Ccong τ / u X o CAR z C ext / = ε + Ccong o u ( 1 ε ) C dwell X +, j=b,r CAR k T k B B B B CAR R (12d) C = t ρ + t K ; C = α C ; = 0 cong CAR k = CAR, B T ( o K ) j C dwell = θ ρ + CAR cong B cong C cong In (12a), C supply s the margnal cost to the transt agency of supplyng an extra passenger mle; t equals the product of the travel tme per mle, the varable operatng cost per unt of tme, and the response of vehcle mles to an extra passenger mle,. Compared wth (8a), the margnal supply cost s lkely to be below the average operatng cost per mle, to the extent that ε < 1 and/or there are fxed costs. In (12b), B scale ε / o s the margnal user beneft per extra passenger mle from scale economes. It s postve to the extent that vehcle mles respond to passenger mles, ε > 0 ; t ncludes the reducton n wat costs from ncreased servce frequency and the reducton n access costs from the ncrease n route densty, wth the latter ncluded as a wat cost equvalent usng (10a). C occ postve f s the margnal cost of ncreased vehcle occupancy per extra passenger mle and s 1 ε > 0. It ncorporates the ncrease n agency supply costs from ncreased vehcle sze and the ncrease n crowdng costs from hgher load factors, wth the latter expressed as an agency cost equvalent usng (10b). In (12c), C ext denotes net external costs per passenger mle. For autos, t equals the pervehcle-mle external cost of polluton, accdents, and congeston (the latter denoted C cong ), net of the fuel tax, and all dvded by occupancy to convert to passenger mles. For transt, C ext ncludes these same costs to the extent that vehcle mles respond to passenger mles ( ε > 0 ), except there are no congeston costs for ral; n addton, t ncludes the margnal cost of ncreased dwell tme, C dwell, applcable to the extent that vehcle occupancy ncreases ( 1 ε > 0). Fuel taxes for transt are excluded from supply costs and thus do not need to be netted out here. 10

14 B CAR In (12d), each of and measures the ncrease n travel tme to all hghway users from an extra passenger mle by bus or auto, scaled by the value of travel tme, plus the ncrease n bus operatng costs because t takes longer (and therefore requres more labor and captal nput) to supply a passenger mle wth slower-movng traffc. Fnally, s the effect on other passengers tme costs, and on agency operatng costs, due to the addtonal boardng and alghtng tme when an extra passenger mle s accommodated through hgher occupancy. C cong C cong C dwell Revstng (11), each term shows a component of welfare change due to shftng from other modes and/or tme perods nto peak ral. The margnal cost/prce gap term shows that welfare from a prce reducton s reduced to the extent that the fare for peak ral already falls short of the correspondng margnal supply cost. The net scale term ndcates that welfare from a fare reducton s larger to the extent that scale economes from ncreased peak-ral use outwegh the extra user costs due to crowdng and the extra user and agency costs from ncreased occupancy of peak-ral vehcles. The externalty term shows that welfare ncreases nsofar as polluton, accdent, and congeston externaltes from auto travel are reduced, although ths s partly offset f there are smlar externaltes from peak ral tself. Fnally, the other transt term ndcates that welfare mproves to the extent that passengers are dverted from other transt modes or tmes of day whose fares fall short of the correspondng margnal socal cost; that margnal socal cost ncludes ncremental supply cost, occupancy cost, and externaltes, less ncremental benefts from scale economes. () Optmzed transt subsdes. Equaton (11) gves us all we need to compute margnal welfare change from ncreasng an exstng subsdy. If we want to go further and fnd the optmal subsdy, we can do so by settng (11) to zero, wth the qualfcaton that we have less confdence n measurng ts components when prces are far from current values. Dong so, we obtan the followng result for optmal fare subsdy per passenger mle, ŝ : (13) sˆ = OC / pˆ average/margnal cost gap net scale economy = OC / C + ( B C supply scale occ ) 6444 externalty CAR CAR + C m C ext ext other transt ( C + C + C B p ) m = OR, PB, OB supply ext occ scale 11

15 where pˆ s value of p that sets (11) to zero and s the modal dverson rato, or fracton of ncreased travel by peak ral that comes from reduced travel by model j n perod. Equaton (13) mples that the optmal subsdy per passenger mle s postve to the extent that (a) margnal supply cost s below average operatng cost; (b) scale economes from ncreasng passenger mles outwegh costs from ncreased occupancy; (c) externalty gans from dvertng auto travel exceed the margnal external costs of the ncreased peak-ral travel; and (d) travel s dverted from other transt for whch the overall socal cost per passenger mle exceeds the fare. m = / As already dscussed, equaton (13) may be relable only when condtons are not too far from those currently observed. However, we found that attempts to smultaneously optmze all transt fares sometmes led to drastc changes n rdershp and consequently n transt characterstcs. Therefore n the emprcal smulatons presented here, we optmze over a sngle transt prce whle settng prces of competng transt modes and perods at ther currently observed levels. In other words, we ask what a gven fare should be, gven the possbly nonoptmal levels of other fares. C cong B CAR () Functonal forms. We assume that margnal congeston costs and are constant because road traffc changes only moderately n our polcy smulatons; we also assume that z / u and are constant but that and vary as dscussed n Secton 3. X η c η w η a C cong Passenger travel demands are assumed to have constant elastctes wth respect to own generalzed prce, and to adjust to other prces accordng to the modal dverson elastctes already defned. Wrtng ths out for changes n the prce of peak ral, we obtan (14a) (14b) ηq q =, q p = m, p where a bar denotes an ntal (currently observed) value, and η q s the elastcty of demand for peak ral wth respect to ts generalzed prce. Dfferentatng (14a), we obtan explctly the dependence of peak-ral demand on ts money prce, holdng the generalzed prces of other 12

16 modes constant: (14c) d = η q dq q In (14c), dq / s the total effect of a one-cent-per-mle ncrease n the passenger fare on the generalzed cost of peak-ral travel, through equaton (10c); that effect s greater than one cent because the reducton n peak-ral vehcle mles ncreases wat and access costs per mle (assumng ε>0), whch magnfes the depressng effect on rdershp Parameter alues We focus on areas served by the Washngton etropoltan Area Transt Authorty (WATA), the Los Angeles County etropoltan Transt Authorty (TA), and Transport for London (TfL) for year Appendx B provdes an extensve dscusson of data sources and varous estmaton procedures for all parameters. Below we comment on selected baselne data summarzed n Table 2; alternatve assumptons wth possble sgnfcance for our results are dscussed later. () System aggregates and agency adjustment. The Washngton and Los Angeles transt systems each carry nearly 2 bllon passenger mles a year across all modes and tmes of day; ths transt usage represents 4.3 percent of total passenger mles (auto plus transt) n Washngton but only 1.3 percent n Los Angeles. In London, the transt system carres more than 8 bllon passenger mles a year, or 21.7 percent of all passenger travel. For Washngton, passenger mles by ral are more than three tmes those for bus, whle the opposte apples to Los Angeles, wth ts extensve bus but lmted ral network. For London, the two modes are closer n sze, wth passenger mles for ral exceedng those for bus by 29 percent. Average transt vehcle occupances are broadly comparable across the ctes but are 26 to 76 percent greater durng peak than durng off-peak perods. Tran occupancy s around 5 to 10 tmes that for bus. 9 We make ths pont especally because many of the studes emprcally measurng money-prce elastctes of transt demand have not held wat and access costs constant whle observng changes n money prce. Thus the elastcty they measure nvolves the total money-prce dervatve defned by (14c) rather than a partal dervatve that holds servce characterstcs constant. 13

17 We assume that transt agences meet a 1 percent ncrease n passenger demand through a 0.67 percent ncrease n vehcle mles and a 0.33 percent ncrease n vehcle occupancy, or ε = As explaned n Appendx A, ths rule would apply, under certan smplfcatons, f the agency optmally trades off vehcle mles and occupancy and f wat and access tmes are nversely proportonal to frequency and route densty, respectvely. 10 We consder other values for ε n our senstvty analyss. () Operatng costs, margnal supply costs, and fares. Our cost data enable us to compute the parameters n (8). They mply that average operatng costs per vehcle mle are around 60 to 100 percent larger n the peak than n the off-peak perod. Peak costs are greater because they nclude vehcle captal costs, hgher unt labor costs due to rregular work hours, and n the case of bus, addtonal costs ncurred because t takes longer to drve a mle on congested roads. However, the peak/off-peak dscrepances n the average operatng costs per passenger mle are much smaller (approxmately zero for ral) because of the dfferent vehcle occupances. The resultng fgures for average operatng costs vary from 30 to 103 cents per passenger mle across modes, perods, and tme of day. For the U.S. ctes, average operatng costs per passenger mle are generally hgher for bus than for ral, partcularly for Washngton, where bus occupances are lower than n Los Angeles. The opposte apples for London. There, the margnal cost of supplyng passenger mles, from equaton (12a), s two-thrds of the average costs n the case of bus and only 60 percent of average costs n the case of ral, because ε = 0.67 and 10 percent of average ral costs s assumed fxed. Passenger fares are 20 to 25 cents per mle for Washngton and London; n Los Angeles they are only 14 cents per mle for bus and 8 cents per mle for ral. 11 Fare subsdes, defned as ( OC p ), are substantal and exceed 50 percent or more of average operatng costs n almost all cases; subsdes are especally large for Los Angeles ral (82 to 83 percent), wth ts unusually low fares, and also for Washngton bus (73 to 81 percent), whch has typcal fares but relatvely low occupances. 10 See also Nash (1988), Jansson (1997), and Small (2004). The result s a modfcaton of the better-known squareroot rule (ohrng 1972) that apples when route densty, and hence access costs, are fxed. 11 The low ral fare n Los Angeles was so pronounced that t resulted n a sut by a bus rders group aganst the operatng agency n 1996; however, ths resulted n lowerng the bus fare rather than rasng the ral fares to levels comparable to those n other ctes. 14

18 () User costs. Average wat tmes at transt stops are estmated from servce frequency. We assume that when vehcles are less than 15 mnutes apart, travelers arrve at random, so the wattme elastcty s one; but that as the tme between vehcles exceeds 15 mnutes, an ncreasng fracton of travelers use a tmetable, thereby lowerng the elastcty (see Appendx B). Expressng wat tmes on a per mle bass and multplyng by the value of wat tme ρ W (assumed from the emprcal lterature to be 60 to 80 percent of the market wage, dependng on tme perod) gve ntal wat costs that vary from 5 to 72 cents per passenger mle. Wat tmes are much larger durng the off-peak perod; they are also larger for bus than for ral. There s less emprcal bass for gaugng crowdng and access tme elastctes; we have assumed locaton-specfc values as explaned n Appendx B. At least when equaton (10) apples, our results are not very senstve to alternatve assumptons about these elastctes. (v) argnal beneft from scale economes and margnal occupancy costs. These are computed from (12b) usng our parameters for wat costs and vehcle captal costs. Because of greater wat tmes at transt stops, margnal scale economes are larger for bus than for ral and for off-peak than for peak travel; however, for bus travel, wat tmes are less responsve to servce frequency n off-peak perods when some people are usng schedules, whch narrows the dscrepancy n scale economes across tme of day for that mode. Overall, margnal scale economes vary between 3 and 34 cents per mle across modes, perods, and ctes. Increased occupancy costs per addtonal passenger mle counteract some, though usually not all, scale economes at peak perod; however, they are zero n the off-peak perod because all vehcle captal costs (and hence crowdng costs) are attrbuted to the peak. (v) Externaltes. argnal external costs per passenger mle for autos are domnated by congeston; that s, the net mpact of polluton and accdent externaltes and fuel taxes s relatvely modest. Ths s partcularly the case for London, where margnal congeston costs are 103 and 37 cents per passenger mle n the peak and off-peak perods, respectvely, but global and local polluton and accdent costs are only about 4 cents per passenger mle, wth offsettng fuel taxes of 6 to 9 cents per mle. Perhaps the most contentous assumpton s that global warmng costs amount to less than half of one cent per mle, though that s what most manstream estmates mply; even ncreasng our estmate several-fold would stll leave polluton 15

19 costs small relatve to congeston costs. 12 For the U.S. ctes, overall external costs for auto are 25 to 31 cents and 6 to 9 cents per passenger mle, respectvely, n the peak and off-peak perods; fgures for grdlocked London are much hgher, at 99 and 35 cents, respectvely. 13 Accdent and polluton costs for bus are mnmal per passenger mle because of the suffcently hgh vehcle occupances; the margnal costs of ncreased dwell tme are also not very large. However, margnal congeston costs are more substantal and amount to 15 to 29 cents per passenger mle for London (assumed passenger car equvalents for bus are between 4 and 5). argnal external costs for ral are neglgble, snce we assume no congeston. (v) Travel responses. Based on lterature surveys of transt demand elastctes (see Appendx B), we choose peak and off-peak own-fare elastctes of and for ral and -0.4 and -0.8 for bus. Elastctes wth respect to generalzed prces,, are then obtaned usng (10c), assumng that the emprcal estmates of own-fare elastctes ncorporate the ndrect effects dq / as emboded wthn (14c). odal dverson ratos are based on avalable evdence and our own judgment (Appendx B). We assume that 60 to 85 percent of ncreased passenger mleage n response to lower fares comes from dverted auto travel for U.S. ctes, and 40 to 50 percent for London, where autos account for a smaller share of passenger travel (Table 2). We assume that 10 percent of extra travel on one transt mode comes from the same mode n the other perod, and that the fracton from the other transt mode wthn the same tme perod s 5 percent for Los Angeles, 10 percent for Washngton, and 30 percent for London. η q 12 A gallon of gasolne contans tons of carbon, so even an extremely large carbon prce of, say, $200 per ton amounts to 48 cents per gallon, or about 3 cents per auto mle for peak perods n the Unted States and even less for other cases. 13 Our fgures are measured pror to the ntroducton of the London congeston toll n However, gven ts very lmted geographcal coverage, we would expect t to have only a modest effect on margnal congeston cost across the entre cty. If our study were more dsaggregated geographcally, congeston chargng mght have more sgnfcant effects on optmal transt fares wthn central London, both by reducng congeston and by ncreasng the value of τ CAR there. 16

20 4. Results 4.1. Baselne Results The upper part of Table 3 shows estmates of the margnal welfare effect of a one-centper-mle reducton n the passenger fare, startng ether at the current subsdy level or at a subsdy level equal to 50 percent of operatng costs. Results are expressed n U.S. cents per passenger mle (at 2004 prce levels). The most strkng result s that, wth the excepton of Washngton peak-perod bus, the margnal welfare effect of ncreasng the subsdy s postve across modes, perods, and ctes startng at subsdy levels of 50 percent. ost of these margnal welfare gans are between about 0.2 and 0.6 cents per passenger mle per one-cent ncrease n subsdy. Even startng at current subsdy levels, whch are typcally well above 50 percent, the margnal welfare effects from further lowerng transt fares are postve n 9 of 12 cases. The reasons for these results can be dscerned n the fgures for ndvdual components of margnal welfare at current subsdes (n the top part of Table 3). In all cases the margnal supply cost exceeds the fare at current prces, causng an ncremental welfare loss from ths source between 0.04 and 1.27 cents per passenger mle. However, n almost all cases ths loss s outweghed by ncremental welfare gans from the combnaton of net scale economes and externalty benefts. Washngton peak-perod bus s the excepton here because of ts especally hgh margnal supply cost. Welfare effects from nteractons among transt modes play a renforcng but generally more modest role, gven that most of the extra passengers on transt were prevously drvng. Although the contrbutons of net scale economes and externaltes to margnal welfare vary consderably, one or the other s mportant n almost every case. In 7 of the 12 cases, net scale economes are substantal between 0.3 and 1.5 cents per passenger mle. Net scale economes are larger for bus than for ral, and larger for off-peak than for peak travel; the reasons for ths, already mentoned, are amplfed by the greater prce-responsveness of passenger demand (and hence of servce frequency or route densty) n the cases of bus and of off-peak travel. Only for peak-ral servce n London are scale economes fully offset by hgher occupancy costs, presumably reflectng London s famous subway crowdng and ts already hgh servce frequency and densty. As for externaltes, most of the welfare gans come from reducng road 17

21 congeston; those gans are especally large n Los Angeles and London, except for off-peak bus servce. 14 The lower part of Table 3 shows the optmal subsdy level, expressed as a percentage of average operatng costs per passenger mle. We fnd that optmal fare subsdes are 68 to 90 percent or more of average operatng costs n 11 of 12 cases; Washngton peak bus s the excepton, where the optmal subsdy s only 44 percent. Agan, some combnaton of net scale economes and dverson of auto externaltes explans a major part of the optmal subsdy. The gap between the average and margnal cost of expandng passenger mles, due mostly to the savngs n agency costs when occupancy rather than servce frequency s ncreased, also plays a consstently mportant role Senstvty Analyss To explore how robust the above results are, we frst vary parameters wth potental sgnfcance for margnal welfare effects and then consder dfferent assumptons about agency adjustment. Table 4 reports results from varyng travel demand elastctes, congeston costs, the value of wat tme at transt stops, the costs of ncreasng vehcle sze, and per unt operatng costs. In all cases welfare effects change n the expected drecton. For example, when travel s more prce responsve, the sze of the welfare effect s magnfed but the sgn s unaffected, whle lower unt operatng costs lower the absolute value of the margnal cost/prce gap, mplyng a larger overall welfare effect. 15 For some perturbatons welfare effects change notceably. For example, off-peak bus s senstve to the value of wat tme through ts effect on scale economes, and peak ral n Los Angeles and London s senstve to congeston costs. Nonetheless, our basc qualtatve fndng that margnal welfare effects are postve at current subsdes n the majorty of cases s unaffected. Even startng at 50 percent subsdes (results not shown), our senstvty results show that Washngton peak bus s the only case wth negatve margnal welfare, wth two small exceptons when agency operatng costs are half agan as hgh as n our baselne. 14 In the case of Los Angeles, all components of our estmated welfare changes are compounded by exstng low fares because we assume constant-elastcty demand functons, such that a unt fare change s a larger percentage of the exstng fare. 15 For ths smulaton we adjust fares to keep subsdes constant as a proporton of operatng costs. 18

22 In Table 5 we compare two scenaros: an ncrease n demand for passenger mles s met entrely through ncreased vehcle supply (ε = 1.0), versus half of the ncrease s met through ncreased vehcle mles and half through ncreased occupancy (ε = 0.5). argnal welfare effects are generally lower n the former case than n our baselne, and larger n the latter. Ths follows because the agency costs of accommodatng extra passenger mles through more vehcle mles, net of scale economes, s greater than the agency and user costs of accommodatng extra passengers through hgher occupancy 16 (London off-peak bus s the lone excepton). But agan, our basc result that subsdes of at least 50 percent of operatng costs are welfare mprovng at the margn s robust. The man reason s that much of the change n agency supply costs per passenger mle, as we vary ε, s offset by changes n net scale economes, as can be seen from the decomposton of welfare effects n the table. 17 Fnally, suppose that the agency does not suboptmze over route densty/servce frequency and vehcle sze/load factor, and thus the condtons n (10) no longer hold. Suppose, for example, that servce frequency s excessve relatve to route densty. In ths case margnal wat costs wll underestmate margnal access costs, and correspondngly, the margnal beneft from scale economes B scale n (12b) wll be understated by a factor of A W { ρ a η / ρ w η ) 1} ( 1 ε f ) a w, where 1 ε f s the fracton of margnal changes n vehcle mles that are met through ncreased densty rather than ncreased servce frequency. If, for llustraton, ε f =0.5 and margnal access costs exceed margnal wat costs by 50 percent, then B scale wll be understated by 25 percent. However, we have already llustrated the effect of dfferent values for B scale when we vared the wat cost parameter n Table 4, and for the most part, results were only moderately senstve to dfferent values. Smlarly, relaxng suboptmzaton over vehcle sze and crowdng has essentally the same effect as varyng the cost of ncreasng vehcle sze, whch agan does not modfy our man qualtatve conclusons. 16 That s, the ncrease n the margnal cost/prce gap component of W, as ε s lowered from 1.0 to 0.5 n Table 5, s greater than the decrease n the net scale economy component. 17 The externalty component also changes n some cases, especally for London off-peak bus; ths s from the change n external costs from the transt vehcle tself as we alter the response of vehcle mles to passenger mles. 19

23 4.3. Relaton to Recent Studes Our fndngs on the margnal welfare of ncreasng current subsdes n London are consstent wth those of Glaster (1984): he estmates that a unform reducton n all London transt fares would produce net socal benefts of 0.41 per 1 of subsdy; our estmate s Our fndngs on optmal subsdes for London are somewhat hgher than those of Glaster and Lews (1978), who found them to be about 50 to 60 percent of operatng costs n ther preferred case. However, they obtan wdely dfferng results for dfferent parameter assumptons, and t s dffcult to pnpont the source of dfferences wth our results. Our results are also broadly consstent wth two studes of Chcago and several Australan ctes, whch fnd that although servce levels are sometmes neffcently hgh, fare subsdes could generally be ncreased wth postve effects on welfare (Savage 1997; Dodgson 1986). Our fndngs of very hgh optmal subsdes for peak-perod transt are smlar to those of an Dender and Proost (2004) for Brussels, n both cases because of the hgh value of reducng car traffc. However, we do not fnd a case, as they do, for much hgher peak fares; we are unsure of the reason for the dfference, but we suspect that scale economes are somewhat more mportant n our model than thers. Our results contrast wth those of Wnston and Shrley (1998), who fnd optmal subsdes to be very small; however, ths s not surprsng because they do not nclude an explct component for scale economes and they smultaneously optmze over transt prces and auto congeston tolls, effectvely elmnatng externalty benefts from dvertng drvers onto transt. Wnston and aheshr (2007) estmate that operatng the Washngton ral system produces a net annual aggregate welfare loss of $195 mllon, not countng ther adjustment for dstortonary tax fnance of agency defcts. Ths may appear to conflct wth our fndng that the current subsdy should be ncreased on welfare grounds. However, a closer look reconcles the apparent dfference. Frst, Wnston and aheshr (2007) nclude annualzed captal cost of $232 mllon as a subtracton from net benefts, whch are not relevant to the queston of whether operatons should contnue on the current system. Second, they use several assumptons that are dfferent from ours: they do not nclude scale economes or nteractons wth bus transt, they measure unt 18 Ths s calculated as the average of the W at current subsdy for the two modes and two tme perods as shown n Table 3, weghted by passenger mles as shown n Table 2. 20

24 operatng costs as 28 percent hgher than we do, and they mplctly assume that vehcle mles change n proporton to passenger mles (ε = 1). If we run our model under these assumptons, we fnd optmal subsdes of 50 percent for peak ral, about the same as current subsdes, but only 21 percent for off-peak ral, well below current subsdes; thus our model would call for curtalng, whle not fully elmnatng, ral operatons. Thrd, Wnston and aheshr use an ownfare demand elastcty for ral of -0.97; usng our smaller assumed ral demand elastctes would roughly trple ther estmated consumer surplus benefts from the ral system (by rasng the measured value of servce to current users), addng $564 mllon n benefts. akng just ths adjustment and excludng captal costs turns ther estmate nto a net aggregate welfare gan of $600 mllon. Ths latter fgure s roughly n lne wth the fndngs of Nelson et al. (2007), based on a transport network model; they estmate that the Washngton ral system produces an aggregate annual welfare gan of more than $700 mllon Concluson Our analyss suggests that today s substantal operatng subsdes for transt systems appear to be warranted on effcency grounds, at least for the three major metropoltan areas studed. The man caveat s that ths leaves asde the possblty that some of the subsdy may be lost to neffcency or hacked by labor unons. Thus, our analyss s most applcable to a transt agency wth ncentves to acheve the best-practce cost level; polces that mght promote such cost mnmzaton have been dscussed extensvely elsewhere (e.g., Wachs 1989; Wnston and Shrley 1998). Another caveat s that we have not ncorporated the burden on the broader tax system from the need to fnance agences operatng defcts. To the extent that dstortonary taxes elsewhere n the economy, such as ncome taxes, are hgher than they would otherwse be, ths causes effcency losses for example, by deterrng work effort. On the other hand, lower transportaton and commutng costs can stmulate more economc actvty and labor force partcpaton at the margn (Parry and Bento 2001). Suppose, for smplcty, that the rest of the tax system s collapsed nto a sngle tax rate on labor ncome. In ths case the optmal externalty tax or subsdy s gven by the Pgouvan 19 Nelson et al. (2007) exclude scale economes, but ther savngs n congeston costs are larger than both ours and Wnston and aheshr s because they nclude parkng search costs. 21

25 tax/subsdy dvded by the margnal cost of publc funds (Bovenberg and Goulder 2002), assumng the polcy s revenue neutral and the prced good or servce s an average substtute for lesure compared wth other goods. The latter assumpton seems a plausble approxmaton, at least for the bulk of transt travel, whch occurs durng a peak commutng perod. A typcal estmate for the margnal cost of publc funds (whch depends on the uncompensated labor supply elastcty) s around 1.1 to 1.2; makng ths adjustment for fscal consderatons would therefore mply some scalng back of the optmal subsdes calculated above, but not enough to overturn the basc fndng that large subsdes are stll warranted. We have also gnored dstrbutonal consderatons. Such concerns mght rase the optmal subsdy for hgh-densty bus servce, whch s heavly patronzed by lower-ncome people, and lower t for ral servce, whch typcally benefts wealther rders and owners of land near transt statons. Quantfyng these addtonal adjustments s contentous, as t brngs n value judgments about approprate dstrbutonal weghts; t also runs counter to the common vew that dstrbutonal concerns are most effcently addressed through the broader tax and beneft system. Fnally, we do not explore how optmal fares mght vary across dfferent routes for example, a route passng through bottlenecks n the central busness dstrct compared wth one servng reverse commutes or ntrasuburban trps. Analyzng ths ssue would requre a more dsaggregated model that accounted for substtuton effects among dfferent lnks n the road and transt network. References APTA Publc Transportaton Fact Book. Amercan Publc Transportaton Assocaton, Washngton, DC. Arnott, Rchard, André de Palma, and Robn Lndsey A Structural odel of Peak-Perod Congeston: A Traffc Bottleneck wth Elastc Demand. Amercan Economc Revew 83: Bovenberg, A. Lans, and Lawrence H. Goulder Envronmental Taxaton. In A. Auerbach and. Feldsten, eds., Handbook of Publc Economcs (second edton). New York: North-Holland, forthcomng. De Borger, Bruno, Inge ayeres, Stef Proost, and Sandra Wouters Optmal Prcng of Urban Passenger Transport: A Smulaton Exercse for Belgum. Journal of Transport Economcs and Polcy 30:

26 Dodgson, J.S Benefts of Changes n Urban Publc Transport Subsdes n the ajor Australan Ctes. Economc Record, 62: Dueker, Kenneth J., Thomas J. Kmpel, and James G. Strathman Determnants of Bus Dwell Tme. Journal of Publc Transportaton 7: FTA Natonal Transt Database. Federal Transt Admnstraton, Department of Transportaton, Washngton, DC. Avalable at Glbert, Chrstopher L., and Hossen Jallan The Demand for Travel and for Travelcards on London Regonal Transport. Journal of Transport Economcs and Polcy 25: Glaster, Stephen Generalsed Consumer Surplus and Publc Transport Prcng. Economc Journal 84: Glaster, Stephen The Allocaton of Urban Publc Transport Subsdy. In J. LeGrand and R. Robnson (eds.), Prvatsaton and the Welfare State, London: Allen and Unwn, Glaster, Stephen and Davd Lews An Integrated Fares Polcy for Transport n London. Journal of Publc Economcs 9: Goodwn, P.B., A Revew of New Demand Elastctes wth Specal Reference to Short and Long Run Effects of Prce Changes. Journal of Transport Economcs and Polcy: 26: Henderson, J. ernon Economc Theory and the Ctes. New York: Academc Press. Jansson, Jan Owen argnal Cost Prcng of Scheduled Transport Servces. Journal of Transport Economcs and Polcy 13: Jansson, Jan Owen Theory and Practce of Transport Infrastructure and Publc Transport Prcng. In G. de Rus and C. Nash (eds.), Recent Developments n Transport Economcs, Aldershot, UK: Ashgate, Kenworthy, Jeff, and Felx, Laube The llennum Ctes Database for Sustanable Transport. Internatonal Unon of Publc Transport, Brussels. Kraus, arvn Dscomfort Externaltes and argnal Cost Transt Fares. Journal of Urban Economcs 29: Lago, Armando., Patrck D. ayworm, and J. atthew cenroe Further Evdence on Aggregate and Dsaggregate Transt Fare Elastctes. Transportaton Research Record 799:

27 Lndberg, Gunnar Traffc Insurance and Accdent Externalty Charges. Journal of Transport Economcs and Polcy 35: ohrng, H Optmzaton and Scale Economes n Urban Bus Transportaton. Amercan Economc Revew 62: Nash, Chrs Integraton of Publc Transport: an Economc Assessment. In J.S. Dodgson and N. Topham (eds.), Bus Deregulaton and Prvatsaton, Aldershot, UK: Avebury, Nelson, Peter, Andrew Baglno, Wnston Harrngton, Elena Safrova, and Abram Lpman Transt n Washngton DC: Current Benefts and Optmal Level of Provson. Journal of Urban Economcs, forthcomng. NRC Effectveness and Impact of Corporate Average Fuel Economy (CAFE) Standards. Natonal Research Councl, Natonal Academy of Scences, Washngton, DC, Natonal Academy Press. Paulley, Nel, Rchard Balcombe, Roger ackett, Helena Ttherdge, John Preston, ark Wardman, Jeremy Shres, and Peter Whte The Demand for Publc Transport: The Effects of Fares, Qualty of Servce, Income and Car Ownershp. Transport Revews 13: Parry, Ian W.H., and Antono. Bento Revenue Recyclng and the Welfare Effects of Road Prcng. Scandnavan Journal of Economcs 103: Parry, Ian W.H., and Kenneth A. Small Does Brtan or the Unted States Have the Rght Gasolne Tax? Amercan Economc Revew 95: Pratt, Rchard H., Texas Transportaton Insttute, Cambrdge Systematcs, Parsons Brnkerhoff Quade & Douglas, SG Assocates, and ccollom anagement Consultng Traveler Response to Transportaton System Changes: Interm Handbook. Transportaton Research Board, Transt Cooperatve Research Program Web Document 12 (arch). Savage, Ian Evaluatng Transt Subsdes n Chcago. Journal of Publc Transportaton, 1: Schrank, Davd, and Tm Lomax The 2003 Urban oblty Report, Texas Transportaton Insttute, Texas A& Unversty System, College Staton, Texas. Small, Kenneth A Urban Transportaton Economcs. Fundamentals of Pure and Appled Economcs, olume 51, Harwood Academc Press, Chur, Swtzerland. Small, Kenneth A Road Prcng and Publc Transport. In Georgna Santos, ed., Road Prcng: Theory and Evdence, Elsever,

28 Small, Kenneth A., and Jose A. Gomez-Ibanez Urban Transportaton. In Paul Cheshre and Edwn S. lls, eds., Handbook of Regonal and Urban Economcs, olume 3: Appled Urban Economcs, Amsterdam, North-Holland, Talvte, Antt A Drect Demand odel for Downtown Worktrps. Transportaton 2: Tsato, Peter Optmal Bus Subsdy and Cross Subsdy wth a Logt Choce odel. Journal of Transport Economcs and Polcy 32: TfL London Travel Report London: Transport for London. accessed June 11, TfL. 2004a. About London Buses. accessed July 30, TfL. 2004b. Amazng Facts. accessed July 30, U.K. DfT Transport Statstcs Great Brtan. Department for Transport, London. Avalable at ctransportst5264. U.K. DOT, Transport Statstcs Great Brtan. London: Department for Transport. U.K. ONS Hours Worked: Growth Levels Off. In: Labour arket n 2002, Offce for Natonal Statstcs. accessed June 11, U.S. BLS etropoltan Area Occupatonal Employment and Wage Estmates. Bureau of Labor Statstcs, U.S. DOC Statstcal Abstract of the Unted States U.S. Department of Commerce, Census Bureau, Washngton, DC. U.S. FHWA Federal Hghway Cost Allocaton Study. U.S. Federal Hghway Admnstraton, Department of Transportaton, Washngton, DC. accessed July 30, U.S. OB Gudelnes and Dscount Rates for Beneft-Cost Analyss of Federal Programs, Crcular No. A-94, Revsed, Secton 8. U.S. Offce of anagement and Budget, Washngton, October. U.S. TRB Hghway Capacty anual. Transportaton Research Board, Natonal Research Councl, Washngton, DC. 25

29 an Dender, Kurt, and Stef Proost Optmal Urban Transport Prcng n the Presence of Congeston, Economes of Densty and Costly Publc Funds. Workng paper, Department of Economcs, Unversty of Calforna at Irvne. ton, Phlp A Pareto-Optmal Urban Transportaton Equlbra. In Theodore E. Keeler, ed., Research n Transportaton Economcs, olume 1. Greenwch, Connectcut: JAI Press, Wachs, artn U.S. Transt Subsdy Polcy: In Need of Reform, Scence 244: Wardman, ark A Revew of Brtsh Evdence on Tme and Servce Qualty aluatons. Transportaton Research E 36: Wnston, Clfford, and Chad Shrley Alternatve Route: Toward Effcent Urban Transportaton. Brookngs Insttuton, Washngton, DC. Wnston, Clfford, and kram aheshr On the Socal Desrablty of Urban Ral Transt Systems. Journal of Urban Economcs, forthcomng. 26

30 Appendx A. Analytcal Dervatons Equaton (10): Agency optmzaton over route densty (D) and vehcle sze (n) Combnng (1), (4), and (5), the household s ndrect utlty functon n (7) s defned by ~ ~ { (A1) U = u ( p, t, w, a, c }, TAX ) Z = ax X,{ }, λ [ X, ({ }), Γ( Σ t, Σ w, Σ a, Σ c )] u + λ Σ [ I TAX X Σ p ] From the agency s pont of vew, (A1) can be transformed nto a socal utlty functon by substtutng the varous defntons and constrants of the system, namely, (2), (3), (8), and (9). In dong so, the revenues Σ p n the government s budget constrant (9) cancel those n the ndvdual s budget constrant n the last term of (A1); prces appear only nsofar as depends on them through the consumer s demand functons. The resultng socal utlty functon can be optmzed by settng λ=u X (the frst-order condton for X) and then by settng to zero ts partal dervatves wth respect to D, n,, and ether or p. (Henceforth we omt the superscrpts for smplcty and understand the precedng statement to apply to each and j.) Here t s convenent to use as the agency s choce varable; that s, we hold constant n takng the other three dervatves. We consder two of those n ths subsecton, deferrng the thrd () tll later. Each s a partal dervatve, holdng the other three varables constant. Thus, n optmzng route densty and vehcle sze, we hold constant and, whch mples also that occupancy o / s constant. Route densty affects user watng and access costs, and vehcle sze affects user crowdng costs and agency operatng costs OC. Thus each frst-order condton for optmzaton has two terms, and each term nvolves only the same and j, so we can contnue to omt the superscrpts wthout ambguty: (A2) ~ U ~ w ~ a W f A 0 = = U w + U a = λρ w f λρ ad D D D D (A3) ~ U ~ c ~ OC C l dk 0 = = U c U I = λρ c l λt n n n n dn z 27

31 where w f, a D, and c l are dervatves of the functons defned n (3b), and we have used the defntons of ρ k from (6b). The partal dervatves on the rght-hand sdes of (A2) and (A3) can be computed usng defntons (3) and (8), holdng,, and o constant. Ths yelds f / D = f / D, l / n = l / n, and dk/dn=k2. Insertng these and dvdng each equaton by λ yelds (10). Equaton (11). argnal welfare effects of reducton n peak-ral fare Partally dfferentatng (A1) and applyng (6b) gves ~ (A2a) U λ ; ~ T = λρ ; ~ W = λρ ; ~ A ~ C = λρ ; = λρ ; p U ~ (A2b) U = λ = ; TAX u X t ~ U Z = 1 U w Totally dfferentatng (A1) shows that when the agency changes peak-ral prce p, utlty changes accordng to ~ du ~ (A3) = U + p ~ U t dt ~ + U w dw ~ + U a da U a ~ + U c dc U c + U ~ TAX dtax z d From (A2) and (A3), we obtan ~ 1 du T dt W W = + ρ + ρ λ (A4a) dtax 1 d + + z λ dw A da + ρ C + ρ dc where d /d s a constant (1/o CAR ) for j=car and depends on the transt agency s operatng polcy for j=b, R. To keep track of ts parts, we wrte the components of (A4a) as (A4b) W = dtax + USERTI + WAITACC + CROWD + + POLLACC where WAITACC ncludes the terms nvolvng ρ W and ρ A and POLLACC, the last term n (A4a), represents changes n polluton and accdent externalty costs. We can compute dtax/ by rearrangng (9) wth only TAX on the left-hand sde, dfferentatng t, and usng (2a) and (8) to get 28

32 (A5a) dtax = + j CAR t A τ k CAR 2 dn CAR j CAR p + j CAR K t + dt where we hold constant τ A and all transt prces other than p. It s convenent to wrte the terms n (A5a) as changes n partcular fnancal flows: dtax (A5b) = FUELRE TRANSITRE + ( OPSUPPLY + OPCONG) + EHSIZE where the frst term s changes n peak-ral revenue from exstng passengers; the second s changes n fuel tax revenue; the thrd s changes n transt fare revenue due to mode and tme-ofday shfts; the fourth s changes n transt operatng cost related to travel tme (dvded nto two parts: changes due to shfts among dfferent modes and tmes of day wth dfferent average supply costs, and effects of congeston); and the last s changes n transt operatng cost related to vehcle sze. Note that new revenues reduce the lump-sum TAX that must be leved, whereas new costs ncrease t. Substtutng (A5b) nto (A4b), we see that the terms cancel, and we can rearrange the other parts nto a more convenent order for further calculaton, as follows: W = ( WAITACC TRANSITRE + OPSUPPLY ) (A6) + ( USERTI + OPCONG + POLLACC FUELRE ) + ( CROWD + EHSIZE) It s useful to summarze the defntons of elastctes of bus and ral travel characterstcs, recallng that all are defned so as to be postve: (A7a) f η w = w f, w D η a = a D, a l η c = c l,, j CAR c (A7b) ε = = o, o 1 ε = o = o,, j CAR We also defne how servce frequency and route densty change wth vehcle mles, and how vehcle sze and load factors change wth occupancy, as follows: (A7c) f ε f = f = D f, D 1 ε f = D = f D,, j CAR 29

33 (A7d) o n ε n = n o = l no, o 1 ε n = l o = n lo,, j CAR l We now proceed to compute key dervatves n (A4a) and (A5a) n terms of d /. The travel tme dervatve can be wrtten, usng (2a), (3a), and (A7b), as (A8a) dt = CAR CAR B B j ( tcar / o ) + ( tb / o ) ε + ( θ / )(1 ε ) j j R where t dt / d and t dt / d = α t. Note that t = R CAR t = 0 by our assumpton CAR CAR B B B CAR that ral speeds are unaffected by road traffc. Smlarly, the watng, access, and crowdng dervatves n (A4), whch apply only for j CAR, can be wrtten usng (2), (3), and (A7) as B (A8b) dw = w f df d = w η ε ε w f / (A8c) da = a D dd d = a η ( 1 ε ) ε a f / (A8d) dc = c l dl d = c η ( 1 ε )(1 ε ) c n / We now examne the terms n (A6) n groups. We begn by usng (A8b) and (A8c) to compute WAITACC as gven n (A4), usng (10a) to smplfy: WAITACC = (A9) = j CAR j CAR W dw ρ B scale A da + ρ = j CAR W ρ w η ε where the last equalty apples defnton (12b). Ths accounts for all the terms n (11) nvolvng B scale. As for the other terms n the frst group n (A6), we note that TRANSITRE, the thrd term n (A5a), accounts for all the terms n (11) nvolvng p. We also see that OPSUPPLY, as defned by the frst of the two terms nvolvng K n (A5a), can be wrtten usng (A7b) as (A10) OPSUPPLY = (ε / o ) K t j CAR = j CAR where the last equalty uses defnton (12a). Thus OPSUPPLY accounts for all the terms n (11) nvolvng C supply. C supply w 30

34 We now turn to the second group of terms n (A6). The terms USERTI and OPCONG, whch are the terms n (A4a) and (A5a) nvolvng dt /, can be combned and wrtten, usng (A8a), as USERTI + OPCONG ( T ρ + K dt ) = + + j= CAR, B j= CAR, B j= B, R T ( ρ T ε ( ρ T (1 ε )( ρ + K )( t + K CAR )( t + K / o B CAR / o ) B ) j )( θ / where we have adopted the notatonal conventon that K CAR =0. Usng the fact that t = α t, the defnton o = /, and defntons (12d), we obtan USERTI + OPCONG = (A11) + ( C j= B, R CAR cong / o (1 ε ) C CAR ) dwell CAR + ε ( C B cong / o B CAR ) B ) B B B CAR These terms are components of sums of C ext as defned n (12c). Next we obtan some other components of those same sums. Usng the defnton of ε and the fact that λ=u X, the change n external costs of polluton and accdents s (A12) 1 POLLACC λ z = z u CAR X 1 o CAR CAR + j CAR ( ε z / o ) u X Fnally, the fuel tax revenue term n (A5) s CAR CAR (A13) FUELRE = ( τ / o ) I CAR Addng equatons (A11)-(A13) and applyng defntons (12c) yelds C CAR ext + j= B, R C ext whch accounts for all the terms n (11) nvolvng C ext. Fnally, we consder the last group of terms n (A6), nvolvng crowdng and the costs undertaken to avod t. Usng (A7b), A7d), (A8d), and (10b), these terms add to 31

35 CROWD + EHSIZE = = = j CAR j CAR j CAR C (1 ε )(1 ε ) ρ c η C (1 ε )(1 ε )( t k occ n n 2 n c + / o ) j CAR + t k j CAR 2 n o o n (1 ε ) ε ( t k 2 n / o ) whch accounts for the terms n (11) nvolvng C occ. We have now accounted for all terms n (11), whch completes the proof. Transt agency optmzaton over vehcle mles of servce (). Now consder optmzng wth respect to, whch we consder only for some scenaros (ncludng the baselne scenaro). We compute the optmal value of ε under certan addtonal smplfyng assumptons, namely these three: Elastctes of watng and access tmes (defned postvely) are all equal to a common value ( η = η ζ ); w a The transt agency gnores ts own vehcles contrbutons to congeston ( = 0 ) and to other externaltes (z B =z R =0); Dwell tme for enterng and extng passengers s neglgble (θ B =θ R =0). The frst bullet s an assumpton common to the smpler models of ohrng effects for example, that of Small (2004). A specal case s when average watng tme s half the nterval between vehcles, and average access tme s proportonal to the dstance between parallel transt lnes; then ζ=1. Those assumptons enable us to derve a smple condton for maxmzng (A1) wth respect to the agency s choce varables, for gven travel demands { }. In what follows, we suppress superscrpts for smplcty. axmzng wth respect to D and n agan yelds (10). Gven t B 32

36 our frst smplfyng assumpton, we see mmedately from (10a) that average watng cost and access cost are equated: W A (A14) ρ w = ρ a Ths result s also n Jansson (1997). Snce D=/f, t can be wrtten as (A15) W ρ α f w ζ A = ρ α ( / f ) a ζ where we have substtuted n the constant-elastcty functons ζ w = α w f and a=αad -ζ descrbng watng and access tmes, respectvely. Solvng (A15) for f, we see that t s proportonal to the square root of. That s, f s adjusted when changes wth elastcty ε f =½. Therefore, (A16) ε f = ε f ε = ½ε We now consder maxmzng wth respect to. Gven our second assumpton, affects (A1) only through the terms nvolvng watng tme w, access a, crowdng c, and operator cost OC, the latter enterng through budget constrant (9). The frst-order condton s therefore ~ U ~ w ~ a ~ c ~ OC 0 = = U w + U a + U c + U TAX W A C = λρ w f λρ a D λρ c l ( do / d ) λkt λtk2n ( do / d ) f D where the last equalty uses the defntons n (6b) and (8) and the result λ=u X. Dvdng by λ and usng (A7), (2a), and (10), ths mples l o o W 0 = ρ wζε / f A C + ρ aζ (1 ε ) o + ρ cη (1 ε ) o k t k tn (1 ε ) f c n 1 2 n W = ρ wζ / k t 1 or w k1t (A17) = ρ W ζ Under our second assumpton, the rght-hand sde of (A17) s a constant as far as the agency s concerned. On the left-hand sde, w = α f w ζ. Therefore, (A18) f ζ 1 = constant Now let change parametrcally, wth all the servce varables f, n, and changng n response. Dfferentatng the logarthm of (A18) wth respect to log() yelds (A19) -ζε f + 1 ε = 0 33

37 Substtutng (A16) nto (A19) and solvng yelds ε =2/(2+ζ). For the common case ζ=1, ths yelds ε =2/3, as n Small (2004) and a specal case of Nash (1988). The ntuton for ths result s somewhat subtle. If ζ s near zero, wat and access costs are relatvely unaffected by vehcle mles of servce, so vehcles are operated only as necessary to handle the passenger loads; thus ncreased passenger loads requre a proportonal ncrease n vehcle mles,.e. ε =1. If ζ s large, the operator accounts for the substantal effects on user costs by runnng extra vehcles for passengers convenence even when s small; n that case, when ncreases, the operator can absorb some of the ncrease through hgher occupancy, thereby reapng more of the advantages of scale; ths means choosng a smaller value of ε. We take ζ=1 as our base case (ε =2/3) and consder senstvty ζ [0,2] by treatng ε =1 and ε =½. 34

38 Appendx B. Assessment of Parameter alues Here we descrbe our methodology for estmatng parameter values along wth data sources; Table 2, whch s dscussed n the text, summarzes our key estmates. For some parameters, breakdowns by mode or tme of day are unavalable from statstcal sources; n these cases we use varous estmaton procedures or our own judgment. U.K. monetary numbers are converted to U.S. dollars usng the average exchange rate of 1.0 = US$1.6. System aggregates. Basc data are compled from the operatng agences and varous natonal statstcs. 20 For London, we allocate total passenger mles across tme of day usng the observed fracton of passenger trps occurrng at peak perod, 0.62 for ral and 0.48 for bus, and an assumed average trp length n the peak equal to 1.6 tmes that n the off-peak. 21 Passenger mles per hour are then computed assumng that the peak perod covers 6 hours per workday (30 hours per week) and the off-peak covers 10 hours every day (70 hours per week). We assume peak shares are each 0.05 hgher for Washngton (whch has a hgh proporton of government employment) and 0.05 lower for Los Angeles (whch has a smaller dscrepancy between peak and off-peak vehcles per hour). 22 To obtan vehcle mles per hour, we assume that observed total vehcle mles are allocated n proporton to ther respectve passenger mles per hour to the power ε =0.67, our baselne assumpton as dscussed n Appendx A. 23 For Washngton and Los Angeles, automoble vehcle mles by tme of day are from Shrank and Lomax (2003), and occupancy s from the 2001 Natonal Household Travel Survey on average occupancy per trp n large metropoltan areas. For London, auto passenger and vehcle mles by tme of day are from TfL (2003, Tables 1.2, 3.1 and 3.6), assumng the same rato of peak to off-peak occupancy as for Washngton. 20 For the Unted States, see the Natonal Transt Database (FTA 2003), and for the Unted Kngdom, see TfL (2003, Tables 1.1,1.2, 3.6), TfL (2004a, b), and U.K. DfT (2003, Tables 5.3, 5.16). Ral data encompass subways and lght ral but not commuter ral. 21 For the entre Unted Kngdom, commutng trps have around twce the length of trps for educaton, shoppng, or other personal busness (U.K. DOT, 2003, Table 10); however, we expect a smaller dscrepancy for transt trps because of the hgh fxed tme cost of accessng transt. 22 The Washngton adjustments are n lne wth unpublshed statstcs we obtaned from transt agency representatves; the Los Angeles transt authorty has no such data on trps by tme of day. 23 Total vehcle mles for ral were obtaned by multplyng vehcle-car mles by average cars per tran; for peak perods the latter s calculated by the rato of ral cars to trans. Off-peak tran length s assumed to be slghtly lower based on common observaton. 35

39 Operatng costs and fares. We assume that vehcle captal costs are proportonal to capacty n, whereas other operatng costs are ndependent of n. Thus n aggregate, vehcle captal costs consttute k 1 t and other operatng costs k 2 nt, usng (8). Operatng costs, asde from vehcle captal costs, are taken from the operatng statstcs of the transt agences. For ral, we assume R that 10 percent of these are the fxed cost of mantanng statons ( F n (8a)). When expressed per vehcle hour of servce, we assume that the rest of these costs are 25 percent greater durng peak than off-peak perods because of dffcultes n schedulng labor for splt shfts; hence we obtan k 1 n (8b). As for vehcle captal costs, we estmate them ourselves by annualzng the purchase cost of a ral or bus car, assumng lfetmes of 25 and 12 years, respectvely, and a real nterest rate of 7 percent. 24 We allocate vehcle captal costs entrely to the peak perod, on the assumpton that any ncrease n vehcle mles n that perod requres purchasng more vehcles, whereas an ncrease n the off-peak perod does not; hence we obtan Pj n Pj Oj and = 0 n (8b). ehcle captal costs are 27 to 52 percent of other peak varable operatng costs. Thus our assumpton that they are the porton of costs that s proportonal to n leads to results consstent wth several other studes of sze-related costs, as revewed by Small (2004, 156 and note 13). Fares were obtaned by dvdng agency passenger fare revenue by passenger mleage (for Washngton ral, peak fares were hgher than off-peak n 2002, but the dscrepancy was modest and we gnore t). k 2 k 2 Wat costs. Based on evdence summarzed n Small (1992, 44), we assume the value per mnute of n-vehcle tme ρ T s 0.5 and tmes the medan gross wage rate for peak and off-peak perods, respectvely; hourly wages are taken to be $16.93, $14.19, and $12.06 for Washngton, Los Angeles, and London, respectvely, and then expressed per mnute. 25 The standard consensus has been that the value of watng tme at transt stops, ρ W, s two to three tmes ρ T ; but a metaanalyss by Wardman (2001, 109, Table 2, frst column) on Brtsh studes produces a rato 24 We use U.S. natonwde fgures for all vehcle prces (from APTA 2002, Table 60) except for Los Angeles ral, for whch fgures were avalable from (where necessary, fgures are updated to 2002 usng the CPI for Transportaton Equpment). The vehcle lfetmes chosen are common n the transt cost lterature, and the nterest rate s that recommended for cost-beneft analyss by U.S. OB (1992). 25 From U.S. BLS (2002), TfL (2003, p. 49) and U.K. ONS (2004). 36

40 between 1.47 and To be conservatve, we set ρ W =1.6ρ T. We obtan ntal wat tmes and the wat tme elastcty as follows. Let H be average mnutes between transt vehcles at a gven stop, or headway (as calculated by the nverse of vehcles per hour). When H s small, t s reasonable to assume that travelers arrve randomly at a stop and ncur expected wat tme H/2. When headways are larger, at least some travelers wll use transt tmetables, whch, followng Tsato (1998), we assume nvolves three tme costs. The frst two are plannng and precautonary tme requred because the exact vehcle arrval tme s uncertan; we assume these are 1 and 5 mnutes, respectvely, and each s valued at rate ρ W. The thrd s the expected cost of early arrval at the destnaton, assumng the traveler chooses a transt vehcle arrvng pror to the desred tme to ensure aganst late arrval. Ths s ρ E H/2 where ρ E s the per mnute cost of early arrval, assumed to be 0.2ρ W (Arnott et al. 1993); that s, a mnute of early arrval s equvalent to 0.2 mnutes of extra plannng or precautonary tme. All these costs are therefore accounted for by settng w=6+0.2h/2 for those usng a tmetable. When H<15, all users wll arrve randomly, whereas when H>15, the average wat tme per trp s ( 1 λ ) H / 2 + λ (6 + H /10), where λ s the fracton of travelers followng a schedule, whch we assume rses lnearly from zero at H=15 to one at H=45. The elastcty then 1 for H<15 and declnes n magntude to 0.5 at H = 45. Wat tme, gven by the above expresson for headway, s dvded by trp length to express t on a per mle bass, and then multpled by ρ W to gve the ntal wat cost. η w s argnal benefts from scale economes and margnal cost from occupancy. These are easly W computed from (12b), usng above values for parameters ε, ρ, w,, and k n 2. η w argnal congeston costs. For automobles, CAR C cong s obtaned by multplyng estmates of average delay per passenger mle by 3.7 to convert them nto margnal hours of delay, and then T multplyng by the value of tme, ρ. 26 Average delay for the U.S. ctes s obtaned from dvdng total person hours of delay from Schrank and Lomax (2003) by passenger mles, and allocatng 85 percent of t to the peak perod (ths yelds an average peak delay of 0.33 mnutes 26 Ths s based on averagng over relatonshps ft by Small (1992, 70 71) whch suggest that total delay s well approxmated by a power functon of traffc volume, wth power 4.1 n Toronto and 3.3 n Boston. 37

41 per passenger mle for Washngton and 0.49 for Los Angeles). Our data provde drect estmates of average traffc speeds n Greater London durng the peak and daytme off-peak perods; we add 10 percent to the latter to account for evenngs and nghts. Average delay s then nferred assumng a free-flow speed of 30 mles per hour. We assume the passenger-car equvalent for buses, α B n (12d), s 4.0 for the U.S. ctes and 5.0 for London, where buses are larger and cars are smaller. 27 Polluton and accdent externaltes. We start wth natonwde average values from the assessment by (2005) of U.S. and U.K. automoble externaltes: namely, 2.0 cents per vehcle mle for local polluton; 6 cents per gallon of gasolne for global warmng; and 3.0 and 2.4 cents per vehcle mle for accdents n the Unted States and the Unted Kngdom, respectvely. To account for greater populaton exposure n urban areas, we double the local polluton fgure for Washngton and London, and we trple t for Los Angeles, whose topography causes pollutants to dsperse especally slowly. We do not adjust external accdent costs because the evdence suggests that, despte hgher traffc denstes n urban areas, external accdent rsks are not necessarly, gven the counteractng effect of slower-movng traffc (Lndberg 2001, ). Also from (2005), we assume fuel taxes of 40 cents per gallon for the U.S. ctes 28 and 280 cents per gallon n London. We use ther natonwde average fuel effcences of 20 and 30 mles per gallon for the off-peak perod (on the assumpton that most travel natonwde s n condtons smlar to off-peak travel n these very large metropoltan areas) but reduce them by 25 percent n the peak perod to adjust for the effect of congeston on fuel economy. For bus, accdents costs per vehcle mle are taken to be the same as for auto because buses move more slowly and are drven by professonals, offsettng ther much greater weght, but polluton s taken to be trple that for automobles. 29 When expressed per passenger mle, 27 U.S. FHWA (1997, Table -23) gves the passenger-car equvalent as only 2.0; however, ths s only for federal urban hghways where buses stop very nfrequently, and t excludes mleage on cty and suburban streets. 28 The federal tax was 18.4 cents per gallon; state-level taxes n Calforna, the Dstrct of Columba, rgna, and aryland were approxmately 20 cents per gallon (U.S. DOC 2003, Table 730). 29 These assumptons are consstent wth estmates of relatve external costs per vehcle mle for heavy trucks and autos n U.S. FHWA (1997, Table 13); separate estmates for bus are not avalable. 38

42 these external costs are small. Polluton and accdent external costs per passenger mle for ral are taken to be zero. Dwell tmes. For bus, we adopt the mdrange values for typcal boardng and alghtng tmes from U.S. TRB (2000, Exhbt 27-9), assumng two doors for alghtng and boardng. We assume cash payment for the U.S. ctes and prepayment (whch allows rear-door boardng) for London. Ths yelds values of seconds for the U.S. ctes and sec for London (for comparson, Dueker et al estmate 5.18 sec n Portland). For ral, we use the estmate by Kraus (1991, 256) from observatons n Boston, whch s 1.0/N T sec where N T s the number of cars per tran. In each case we dvde by trp length to specfy parameter θ. The margnal cost of ncreased dwell tme s then calculated from (12d), usng parameters already descrbed. Generalzed prce of travel. The components of q are gven by (10c); besdes parameter values already descrbed, we need the tme per mle of transt vehcles t and access and crowdng elastctes and. (Ths s n fact the only place where we need an emprcal estmate of.) η c η a η c To calculate t, we dvde total vehcle mles by vehcle hours to gve average speeds, over the day, of 23 and 11 mles per hour for Washngton ral and bus, and 23 and 12 mles per hour for Los Angeles ral and bus. For London, we have a drect estmate of ral speed from the agency, of 20 mles per hour. For all three ctes we assume the rato of peak to off-peak speed s 1.0 for ral, whle for bus t s the same as that for autos: approxmately 0.8 for Washngton and London and 0.75 for Los Angeles. The access-tme elastcty depends on route densty n a manner smlar to how the wat-tme elastcty depends on servce frequency. It s one f people lve at unformly dstrbuted locatons and walk to the nearest transt stop, and smaller f people lvng farther away choose a faster access mode wth a fxed cost (e.g., park and rde). The less dense the transt network, the more mportant these other access modes, so the lower the elastcty. We assume other access modes have mnor mportance n London but more n Washngton and more stll n Los Angeles, and so choose η a η a = 0.8, 0.65 and 0.5 for these ctes, respectvely. 39

43 There s lttle emprcal bass for gaugng, whch s postve only for peak servce; we set t to 1.5 n the baselne, though our results are not senstve to dfferent assumptons (because crowdng costs are relatvely small). η c Own-prce travel demand elastctes. Our model calls for elastctes of each mode s passenger demand wth respect to ts own generalzed prce q, denoted as evdence s based on elastctes wth respect to fare p, whch we denote as η p the evdence on, then descrbe how we convert to. η q η q. However, most emprcal η p. We frst revew Based on Lago et al. (1981), Goodwn (1992), and Pratt et al. (2000), we assume that the own-fare demand elastcty, averaged over peak and off-peak tme perods, s -0.5 for bus and -0.3 for ral, 30 and that n each case the elastcty n the off-peak perod s twce that n the peak. Gven that about 70 percent of passenger mleage occurs durng the peak perod, the values just stated mply own-fare elastctes η p of approxmately and -0.8 for peak and off-peak bus, and and for peak and off-peak ral, respectvely. To convert these to generalzed-prce η q elastctes, we assume that the emprcal measurement of ncorporates the effects of p on w n (10c), as dscussed n the dervaton of (14c); that s, we assume (B1) p d dq p η p = = η q dq q dq where the rato and the dervatve on the rght-hand sde are both obtaned from (10c). Thus we smply nvert equaton (B1) to obtan our estmates of η q η p, whch we assume to be constants. odal dverson ratos, m kl. Pratt et al. (2000, ff.) provde several estmates for U.S. ctes of the proporton of ncremental transt trps that are dverted to or from other modes followng a change n transt prce; typcal numbers, averaged across tme of day, are about 65 percent and 80 percent for Atlanta and Los Angeles, respectvely. Nearly all of these shfts are to or from 30 A recent revew of mostly U.K. studes by Paulley et al. (2006) produces somewhat larger long-run elastctes, whch they suggest s because elastctes have rsen n magntude and are hgher n the Unted Kngdom than n other natons. any of the studes reled upon by Paulley et al. are unpublshed, and we do not feel the evdence s strong enough to apply these hgher elastctes to our U.K. smulatons. 40

44 cars. We assume that Washngton s lke Atlanta, and that peak values off-peak values Oj m OCAR Pj m PCAR Now consder the cross-elastctes between bus and ral transt. The few studes avalable typcally fnd them to be about half the drect elastctes n ctes wth good coverage by both bus and ral transt systems, such as London and Chcago (Glbert and Jallan 1991, Table 3b; Talvte 1973). Assumng equal travel volume by mode, ths would mply m R m B = R B 0.5 =P,O. However, we expect the substtutablty between modes to decrease as one expands beyond the cty to the metropoltan area, and to decrease more for ctes wth less and less well developed ral networks. We also expect them to have declned consderably from the 1970s or 1980s to the year 2000 because of ncreasng competton from the automoble. Fnally, n the newer U.S. transt systems the bus lnes are typcally reconfgured to serve as feeders to the ral system, wth compettve routes dscontnued. Therefore, we assume the cross-mode dverson B ratos to be just 10 percent for Washngton ( m = R R m = 0.1) and 5 percent for Los Angeles B ( m = R R m = 0.05 ). B 0.05 lower, than these average values. For London, we expect less dverson to automoble and more to the other transt mode because of the smaller ntal share of automobles and travelers more extensve transt choces. We therefore set London s dverson ratos to be lke those for Washngton, except 0.20 smaller for auto n the same tme perod, and 0.20 larger for other transt n the same tme perod. Lttle nformaton s avalable about shfts of transt rders across tme perods. We assume that n each case, 10 percent of the change n transt rdershp represents such shfts, and that the shfts occur entrely to the same transt mode. Those assumptons lead to the values shown n Table 2. The fracton of extra transt trps from ncreased travel demand s a resdual, equal to between zero and 20 percent. The revew by Pratt et al. (2000) suggests that 10 percent and 26 percent of new transt trps n Los Angeles and Atlanta, respectvely, represented some combnaton of changes n walkng, trp frequency, and destnaton durng the 1990s; gven the lkely further declne n ths fracton due to metropoltan decentralzaton, ths evdence s roughly consstent wth our assumed values. B are 0.05 hgher, and for 41

45 Table 1. Passenger Fare Subsdes for the 20 Largest U.S. Transt Authortes Fare subsdy, % of average operatng cost per passenger mle Passenger mles ral bus combned total, mllon % ral %bus TA New York Cty Transt, Brooklyn, NY , New Jersey Transt Corporaton, Newark, NJ , TA Long Island Ral Road/Bus, Jamaca, NY , etro-north Commuter Ralroad Co., New York, NY 40 n/a 60 2, Washngton etrop. Area Transt Authorty, Washngton, DC , assachusetts Bay Transportaton Authorty, Boston, A , Los Angeles County etrop. Transp. Authorty, Los Angeles, CA , Chcago Transt Authorty, Chcago, IL , Northeast Illnos Regonal Commuter Ralroad Corp., Chcago, IL 56 n/a 45 1, Southeastern Pennsylvana Transp. Authorty, Phladelpha, PA , San Francsco Bay Area Rapd Transt Dstrct, Oakland, CA 42 n/a 58 1, etropoltan Atlanta Rapd Transt Authorty, Atlanta, GA aryland Transt Admnstraton, Baltmore, D Kng County Dept. of Transp. - etro Transt Dvson, Seattle, WA etrop. Transt Authorty of Harrs County, Texas, Houston, TX n/a Tr-County etrop. Transp. Dstrct of Oregon, Portland, OR am-dade Transt, am, FL Dallas Area Rapd Transt, Dallas, TX Denver Regonal Transportaton Dstrct, Denver, CO Port Authorty of Allegheny County, Pttsburgh, PA Average (unweghted) , Average (weghted by passenger mles) , Source: FTA (2003). 42

46 Table 2. Selected Baselne Parameter alues Washngton Los Angeles London Ral Bus Ral Bus Ral Bus Peak Off- Peak Off- Peak Off- Peak Off- Peak Off- Peak Off- Peak Peak Peak Peak Peak Peak TRANSIT Annual passenger mles, mllons 1, ,302 1,265 2,115 1,432 Passenger mles per hour, thousands , , ehcle mles per hour, thousands ehcle occupancy Average operatng cost, $/ veh-m Avg operatng cost, c/pass-m argnal supply cost, c/pass-m Fare, c/pass-m Subsdy, % of average operatng cost Cost of n-vehcle travel tme, c/pass-m Wat cost, c/pass-m Wat tme elastcty argnal scale economy, c/pass-m argnal cost of occupancy, c/pass-m argnal external cost, c/pass-m arg. congeston cost, c/pass-m Polluton. & accdent, c/pass-m argnal dwell cost, c/pass-m Generalzed prce, c/pass-m Elastcty of passenger demand wrt fare Fracton of ncreased transt comng from auto--same perod same transt mode--other perod other transt mode--same perod ncreased overall travel demand AUTO Peak Off- Peak Off- Peak Off- Peak Peak Peak Annual passenger-mles, mllons 19,583 22,055 69,519 75,226 13,397 15,859 Occupancy argnal external cost, c/pass-m arg. congeston cost, c/pass-m Pol. & acc. less fuel tax, c/pass-m Source: See Appendx B. 43

47 Table 3. Baselne Welfare and Optmal Subsdy Estmates Washngton Los Angeles London Ral Bus Ral Bus Ral Bus Peak Off- Peak Off- Peak Off- Peak Off- Peak Off- Peak Off- Peak Peak Peak Peak Peak Peak Current subsdy, % of op. cost W at current subsdy a margnal cost/prce gap net scale economy externalty other transt W at 50% subsdy a Optmum subsdy, % of op. cost > > >90.0 > >90.0 >90.0 proporton of subsdy due to average-margnal cost gap net scale economy externalty other transt Note a Ths s the margnal welfare gan from a one-cent-per-mle reducton n the passenger fare, expressed n cents per passenger mle. 44

48 Table 4. Results wth Alternatve Parameter alues: argnal Welfare Effects at Current Subsdes Washngton Los Angeles London Ral Bus Ral Bus Ral Bus Peak Off- Peak Off- Peak Off- Peak Off- Peak Off- Peak Off- Peak Peak Peak Peak Peak Peak Baselne results Travel demand elastctes Increased by 30% Reduced by 30% argnal congeston costs Increased by 50% Reduced by 50% alue of wat tme at transt stops Increased by 50% Reduced by 50% ehcle sze costs Increased by 50% Reduced by 50% Agency operatng costs Increased by 50% Reduced by 50% Note All values are n cents per passenger mle per one-cent ncrease n subsdy. 45

49 Table 5. Results for argnal Welfare Effects wth Alternatve Assumptons for Agency Adjustment Washngton Los Angeles London Ral Bus Ral Bus Ral Bus Peak Off- Peak Off- Peak Off- Peak Off- Peak Off- Peak Off- Peak Peak Peak Peak Peak Peak ε = 1.0 W at current subsdy margnal cost/prce gap net scale economy externalty other transt W at 50% subsdy ε = 0.5 W at current subsdy margnal cost/prce gap net scale economy externalty other transt W at 50% subsdy Note All values are n cents per passenger mle per one-cent ncrease n subsdy. 46