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

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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

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