On Using Standard Values of Time in Project Appraisal: Income Equity vs. Preference Equity

Transcription

1 On Usng Standard Values of Tme n Project Apprasal: Income Equty vs. Preference Equty Lus Ignaco Rzz a Sej S.C. Stemetz b,* a Department of Engneerng and Transport Logstcs Pontfca Unversdad Católca de Chle Caslla 306, Códgo 105, Santago 22, Chle Emal: lrzz@ng.puc.cl Phone: b Calforna State Unversty, Long Beach 1250 Bellflower Boulevard Long Beach, CA 90840, Unted States of Amerca Emal: sej.stemetz@csulb.edu Phone: * Correspondng author Abstract We examne the practce of usng standard or equty values of tme to evaluate the benefts of transportaton mprovements, and we make explct the socal weghtng schemes that are consstent wth that practce. We demonstrate that f travel-tme preferences vary, then standard values of tme typcally mply socal weghtng schemes that favor those who gan the least utlty from travel-tme savngs. We further show how ths fndng perssts when mprovements are fnanced by user payments such as road tolls and taxes. We then descrbe a remedal weghtng scheme that generalzes several valuaton approaches developed n the lterature. JEL Codes: D61, D63, R41, R42 Keywords: ncome nequalty, cost-beneft analyss, value of tme, transportaton nvestment

2 I. Introducton The benefts of transportaton mprovements are typcally domnated by the travel-tme savngs that they offer. And so the subjectve value of travel-tme savngs, or smply the value of tme, s arguably the most mportant transportaton polcy parameter (Small and Verhoef, 2007). Intutvely, the economc benefts of a tme-savng mprovement would be calculated by multplyng each user s value of tme by the amount of tme saved. But dong so can result n a valuaton scheme that favors projects beneftng hgher ncome groups (Galvez and Jara-Daz, 1998; Macke et al., 2001; Macke et al., 2003). Specfcally, usng the value of tme for costbeneft analyss, n the context of a socal welfare functon, mples welfare weghts that are nversely proportonal to the margnal utlty of ncome. If the margnal utlty of ncome falls as ncome rses, then those wth hgher ncomes are assgned a hgher welfare weght. To address that equty concern, some natonal project apprasers (manly n the U.K. and European Unon) have adopted standard values of tme,.e. a sngle value appled to all users (Macke et al., 2001; Odgaard et al., 2005; Fowkes, 2010; MIDEPLAN, 2011). In the U.K., for example, t s the value of tme evaluated at the average ncome level sometmes referred to as an equty value. The use of standard values can also be justfed as a pragmatc approach, gven the dffculty n mappng heterogenety n the value of tme to correspondng members n the populaton. In the Unted States, for example, the value of personal travel tme s evaluated at the natonal medan of household ncome (U.S. Department of Transportaton, 2014). Whether standard values are adopted on the grounds of equty or parsmony, t s mportant to understand ther mplcatons for welfare analyss. The purpose of ths paper s to clarfy the welfare mplcatons of usng standard values of tme n transportaton project apprasal, and to demonstrate mplcatons that have been overlooked n the lterature. We show that there s more than one way to derve a standard value of tme from a socal welfare functon. And, more mportantly, we show that usng those standard values typcally results n apprasal schemes that gve more weght to those wth the lowest margnal utlty of travel tme. In other words, standard values of tme favor those who gan the least from travel-tme mprovements. Our dscusson proceeds as follows. From the relevant lterature n transportaton economcs, Secton II outlnes an analytcal framework for usng the value of tme n project apprasal n the context of a socal welfare functon. Wthn that framework, Secton III makes explct the socal weghtng schemes mpled by the practce of usng standard values for project apprasal, and demonstrates how that practce can requre a tradeoff between ncome equty and travel-tme preference equty. Secton IV proposes a smple and remedal socal weghtng scheme, whch generalzes other schemes developed n the lterature. Secton V offers concludng remarks. II. The Value of Tme n Project Apprasal Our analyss bulds upon the semnal work of Galvez and Jara-Daz (1998), and mportant dscussons and extensons offered by Macke et al. (2001), Macke et al. (2003), and Fowkes (2010). We focus on the value of non-work (or personal ) travel tme,.e. we do not consder commercal values of tme. We begn wth a socal welfare functon, W, defned as 1

3 W = W[U 1 (T 1, P 1, Y 1 ) U n (T n, P n, Y n )] (1) comprsng N transportaton system users ndexed by = 1,, n, each wth an ndrect utlty functon, U, whch depends on travel tme, T, prces, P, and ncome, Y. A tme-savng transportaton project generates a welfare gan of dw = W U dt U T (2) For ease of expresson we defne s W U θ U T (3) U Y where s s the socal weght appled to each ndvdual welfare gan, θ s the margnal utlty (.e. negatve margnal dsutlty) of travel tme, and s the margnal utlty of ncome. Those defntons mply that each user s (subjectve) value of tme, V, s V dy dt du =0 = U T U Y and that the total welfare gan s dw = s = θ (4) θ dt = s V dt (5) whch s the socally-weghted sum of ndvdual utlty gans. Expressng ths welfare gan as a monetary beneft for project apprasal requres a converson factor that transforms socal utlty to socal money (analogous to how transforms θ to an ndvdual s monetary valuaton). Let W denote that converson factor, and let db denote the monetary beneft of the project, such that db dw W. 1 An ntutve measure of that beneft, along the lnes of Small and Rosen (1981), would be the summaton of ndvduals travel-tme savngs multpled by ther values of tme,.e. db = V dt (6) 1 We dscuss specfc forms of W below. 2

4 Ths valuaton approach follows that prescrbed by Harberger (1971), where the benefts accrung to each user are added wthout regard to the ndvduals to whom they accrue. 2 In the context of the socal welfare functon n (1), the approach s consstent wth applyng a socal weght of s = W to each ndvdual gan. Henceforth we shall refer to h W as Harberger weghts and the apprasal scheme n (6) as the Harberger approach (wth s = h ). As expertly llustrated by Galvez and Jara-Daz (1998), ths approach rases ncome-equty concerns. Because the Harberger weghts are nversely proportonal to the margnal utlty of ncome, those wth hgher ncomes receve more weght n the apprasal process (to the extent that there s dmnshng margnal utlty of ncome). An analogous nterpretaton s that those wth hgher values of tme receve greater weght, as llustrated by (6). Ths presents economc apprasers wth a quandary. On one hand, the Harberger approach s consstent wth how choces n markets are actually made, and t may not be the concern of transport polcy to repar ncome nequalty (Macke et al., 2003). On the other hand, equty concerns are legtmate polcy concerns, and t may be desrable to make equty adjustments gven the opportunty (Fowkes, 2010). In practce, partcularly n the U.K. and E.U., equty adjustments have been appled to the value of tme, gvng rse to standard values for use n project apprasal. A promnent example of such an adjustment s to average the value of tme over ncome and apply t unformly to all travel-tme savngs. III. Standard Values of Tme Consder an average value of tme (typcally by ncome) defned as V for use as a standard value. The mpled monetary beneft of a tme-savng mprovement s db S = V dt (7) where the S subscrpt denotes that a standard (or equty ) value of tme s employed, as s often the case n Chle, the E.U., the U.K., and the U.S. (Macke et al., 2003; Odgaard et al., 2005; Fowkes, 2010; MIDEPLAN, 2011; U.S. Department of Transportaton, 2014). Our objectve here s to reconcle the Harberger approach wth the standard-value approach so we can make explct the socal weghtng scheme that converts (6) to (7). We nterpret the standard-value approach as a reweghtng of the Harberger approach on the bass of equty (or, n some cases, for smplcty). Let e represent an equty weght makng that adjustment. When appled to the Harberger approach, the socal weght, s, appled to each user s welfare gan becomes s = h e (8) Because the purpose of ths equty weght s to undo the ncome-regressvty of gvng more weght to those wth hgher values of tme (and less weght to those wth lower values), a natural 2 Ths s Harberger s Postulate C see Harberger (1971), p

5 choce s one that gves less weght to above-average values (and more weght to below-average values). In that sprt, consder e V V (9) mplyng that db S = V dt as n (7). Equatons (7) (9) thus demonstrate that the standardvalue approach s consstent wth a socal weghtng scheme that apples an equty adjustment gven by (9) to the Harberger weghtng scheme. We are now ready to state a key fndng of ths paper: f a standard value of tme s used to correct the ncome-regressvty of the Harberger approach, then t wll also favor those wth the least averson to travel delays. In other words, attemptng to correct an nequty n the ncome dstrbuton comes at the expense of ntroducng an nequty n the dstrbuton of travel-tme preferences. To see ths, note that the socal weghtng scheme mpled by the standard-value approach s s = h e = V W θ (10) whch shows how those wth the lowest margnal utlty of travel tme receve the hghest weght. The ntuton behnd ths result s smple: because the value of tme s the rato of margnal utltes between tme and money, gvng more weght to those wth hgher ncome utlty mples gvng less weght to those wth hgher travel-tme utlty. Put dfferently, (10) can be expressed as s = V SPT, where V SPT V θ s an ndvdual s socal prce of tme as dscussed n Macke W et al. (2001), showng that ndvduals wth hgher socal opportunty costs of tme receve less weght. It follows that usng standard values of tme for project apprasal may not be as equtable as desred. A correspondng fndng s that f standard values are requred, ether for equty or smplcty, they can be obtaned wthout relyng on the (possbly unrealstc) assumpton that the margnal utlty of travel tme s equal for all users. 3 For example, Macke et al. (2003) show that (7) can be derved by mposng a sngle margnal utlty of travel tme wth the normalzaton θ = 1 and then rescalng the mpled welfare gan by V. Galvez and Jara-Daz (1998) and Macke et al. (2001) show that the assumpton of θ = θ yelds a standard value of θ. The two approaches W are consstent f = W or =, where represents an average of margnal utltes. In contrast, our analyss demonstrates that f heterogenety n travel-tme preferences exsts, then standard values of tme can be obtaned, but only n a manner that favors those wth the lowest travel-tme margnal utlty. 4 Our more general fndng s that employng standard values of tme s consstent wth a socal weghtng scheme n whch s 1. θ 3 The value-of-tme dervaton n DeSerpa (1971), for example, theoretcally motvates heterogenety n the margnal utlty of tme. Moreover, modern travel-demand estmaton methods frequently specfy random coeffcents for travel tme, necessarly mplyng heterogenety n travel-tme utlty. 4 An excepton s when standard values are calculated n a manner that explctly compensates for that effect, such as the equty value proposed by Fowkes (2010). We dscuss ths further n Secton IV. 4

6 Thus far we have not consdered any specfc form of W. We now extend our analyss to consder the form derved by Galvez and Jara-Daz (1998) for when a project s funded by user payments such as tax contrbutons. 5 Suppose that each user contrbutes dm toward a project that requres a total fundng of dm. The total welfare gan from the project s stll gven by (5) and converted to a monetary beneft when dvded by W. Galvez and Jara-Daz (1998) refer to W as the socal [margnal] utlty of money and defne t as where π gan s thus W = s π (11) dm dm db = dw s each user s share of the total payments. 6 The monetary value of the welfare = s θ dt W s π (12) Under the Harberger approach wth s = h = W, the value of the mprovement s agan db = V dt. 7 Under the standard-value approach, wth e = V, that beneft s measured as V db S = [ π ] 1 dt V = V HM dt (13) where V HM [ π ] 1 s the weghted harmonc mean of ndvdual values of tme, wth V payment shares servng weghts. Note that (13) holds for any ntal standard value of tme chosen for the equty adjustment, e. 8 The consequences are strkng: f the welfare gans of tme savngs and the welfare costs of fnancng are to be equally weghted, and f the margnal utlty of tme s allowed to vary, then the only standard value of tme avalable for project apprasal s the weghted harmonc mean value of tme. 9 And we see that ths standard value requres a socal weghtng scheme such that s 1. θ 5 For ease of expresson and comparson wth prevous studes, and followng Galvez and Jara-Daz (1998), we mplctly assume that the margnal cost of publc funds equals one. 6 See Galvez and Jara-Daz (1998), p. 210, equatons (13) (16). 7 An alternatve treatment of user payments s gven by Macke et al. (2003), where dw = s [θ dt + dm ]. The Harberger approach mples db = V dt + dm, whereas the standard-value approach, wth equty adjustment e = V, mples db V S = V dt + V dm V. Macke et al. (2003) derve ths latter expresson by assumng that the margnal utlty of travel tme s the same for all users. Our analyss shows that t can be obtaned whle allowng for heterogenety n travel-tme utlty. 8 Ths s due to the fact that f a constant s ncluded n the socal weght, s, t appears n both the numerator and denomnator of (12) and thus cancels. 9 If the margnal utlty of tme s assumed to be equal for all users,.e. θ = θ, then the standard value of tme s V θ = θ, as shown by Galvez and Jara-Daz (1998) and Macke et al. (2001). W 5

7 In practce, however, the weghted harmonc mean s not employed (nor do we advocate t). It only arses as a standard value when the socal weghts on tme savngs and user payments are constraned to be equal. Instead, benefts are typcally measured as n equaton (7). We now demonstrate that the standard-value approach n (7) can arse from applyng dfferent socal weghts to tme savngs and user payments. 10 Defne s T as the tme-savngs socal weght and s M as the socal weght on money contrbutons. The correspondng benefts measure s then db = s Tθ dt s M π (14) Now consder s T = h e = V W θ, whch ncludes a value-of-tme equty adjustment, and s M = h = W, whch s an unadjusted Harberger weghtng scheme on money. What results s the standard-value approach where benefts are measured as n (7). Agan, whether equal or dsparate socal weghts are appled to tme savngs and user payments, we see that the standard-value approach s consstent wth socal weghtng schemes that correct for ncome regressvty at the cost of ntroducng travel-tme-preference regressvty. Ths may result n a msallocaton of resources, as dscussed by Pearce and Nash (1981), Sugden (1999), and Macke et al. (2003). For example, consder a project that prmarly affects low-ncome users who (at least for the sake of argument) also have low margnal utltes of travel-tme savngs. That project could be selected among competng projects based on the standard-value approach, even f those users would prefer longer commutes over a larger tax burden. IV. An Alternatve Socal Weghtng Scheme The cause of the tradeoff between ncome nequty and tme-preference nequty s n attemptng to make correctons through wllngness-to-pay values nstead of focusng on the culprt: the dmnshng margnal utlty of ncome. If a socal weghtng scheme wth equty adjustments s desred, we propose the followng: e (15) where s an average value of margnal utlty. For example, n a dscrete-choce model where travel cost s nteracted wth ncome, could be the margnal utlty of ncome evaluated at the average level of ncome. 11 Or n the absence of ncome nformaton, perhaps would be the estmated mean value of when specfed as a random parameter (to accommodate unobserved ncome heterogenety, nter ala). The role of e s to lend more weght to above-average 10 Macke et al. (2003) mplctly refer to ths requrement n ther dscusson of the mxture of apprasal practces n the U.K. that leads to a standard value of tme. 11 Consder the condtonal ndrect utlty specfcaton U = Xβ + γc + δc Y + ε such that the margnal utlty of ncome s measured by = U C = γ + δy. The average margnal utlty of ncome s then = γ + δy, where Y s the average level of ncome. If ncome categores are used n place ncome levels, then would be the average of margnal utltes weghted by the proporton of users n each ncome group. 6

8 margnal utltes of ncome (and less to below-average utltes). The resultng socal weghtng scheme s then s = h e = W (16) whch adjusts for ncome nequty wthout ntroducng tme-preference nequty. In general, the correspondng benefts measure becomes db = θ dt (17) where θ can be nterpreted as an ncome-adjusted value of tme that accommodates heterogenety n travel-tme preferences. Whle ths expresson s not as parsmonous as one comprsng a sngle value of tme, all of ts ngredents are readly obtaned from ordnary traveldemand analyss. When W s specfed as n Galvez and Jara-Daz (1998) and Macke et al. (2001), the benefts measure becomes db = θ dt, whch s equvalent to the neutral approach to apprasal π proposed n ther analyses. 12 Alternatvely, usng (14), f s T = h e = W If and π are uncorrelated, then db = θ dt as n (17). 13 and s M = h = W, then benefts are also measured as n (17). Ths latter nterpretaton may be pragmatc when π s dffcult to calculate but s beleved to be correlated wth. The benefts measure n (17) s also equvalent to the equty-value approach proposed by Fowkes (2010). That approach uses a dstance-weghted, average value of tme of the form V K (18) V K K where K s each user s travel dstance, and where db K V K dt. 14 The frst summand of (18) shows how the equty adjustment proposed n (15) s appled to that weghted average. In Fowkes s analyss t s assumed that each user s travel-tme savngs s proportonal to dstance such that dt = ρk, and summaton mples ρ = dt. It follows that K db K = V K K dt = θ K dt K = θ ρk = θ dt = db (19) 12 It s a neutral approach n the sense that t s equvalent to a socal weghtng scheme where s = If the covarance between and π s zero, then π =. 14 Usng our notaton, n (18) s actually expressed as W n Fowkes (2010). In dervng that expresson, however, a socal weghtng scheme s mposed such that W = 1 N =. 7

9 Hence the equty adjustment proposed n (15) yelds the dstance-weghted, equty value of tme derved by Fowkes (2010). However, we see that there s no need to tabulate trp dstances or formulate a standard (equty) value because (17) provdes an equvalent benefts measure. V. Conclusons Our precedng analyss s desgned to clarfy the roles of socal weghts n economc apprasals of transportaton mprovements, and to demonstrate the consequences of mplctly makng equty adjustments to those weghts by employng standard or equty values of tme. In partcular, we show that ths standard-value approach typcally favors those who derve the smallest utlty gan from a tme-savng mprovement, whch could lead to a msallocaton of resources. A correspondng fndng s that standard values of tme can be obtaned wthout relyng on the assumpton that everyone derves the same utlty gan from such mprovements. The mplcatons of our analyss extend beyond the valuaton of transportaton projects. Consder, for example, an mprovement n ar qualty that s valued by the margnal rate of substtuton between polluton and ncome, or any other margnal valuaton along the lnes of Small and Rosen (1981). Those wth hgher ncomes may have a greater wllngness to pay for clean ar, n whch case a Harberger valuaton approach would favor polluton-abatement projects n hgher-ncome regons. Although t may be socally desrable to correct for that nequty, applyng a standard margnal value of ar qualty to all projects could favor those who derve the least utlty from cleaner ar. In response, we propose a smple equty adjustment that targets the culprt of the ncome nequty: the dmnshng margnal utlty of ncome. Ths adjustment s feasble wth ordnary travel-demand analyss, s applcable beyond transportaton projects, and generalzes several apprasal schemes dscussed n the lterature. Acknowledgments: We are grateful to offcals from MIDEPLAN and SECTRA who were most wllng to dscuss these deas wth us, and we thank Anthony Fowkes for hs helpful comments and suggestons. Lus Rzz thanks the Insttuto Mleno Sstemas Complejos en Ingenería (ICM:P F; CONICYT: FBO16) for ts fnancal support. Sej Stemetz thanks the Calforna State Unversty, Long Beach for supportng ths research through a Research, Scholarly, and Creatve Actvtes award. References De Serpa, A. (1971). A Theory of the Economcs of Tme, Economc Journal, 81, Harberger, A. (1971). Three Basc Postulates for Appled Welfare Economcs: an Interpretve Essay, Journal of Economc Lterature, 9,

10 Fowkes, A. (2010). The Value of Travel Tme, n Van de Voorde, E., Vanelslander, T., eds., Appled Transport Economcs: a Management and Polcy Perspectve. Antwerp: De Boeck. Gálvez, T. and S. Jara-Díaz, (1998). On the Socal Valuaton of Travel Tme Savngs, Internatonal Journal of Transport Economcs, 25, Macke, P., S. Jara-Díaz, and T. Fowkes (2001). The Value of Travel Tme savngs n Evaluaton, Transportaton Research Part E, 37, Macke, P., M. Wardman, A. Fowkes, G. Whelan, J. Nellthorp, and J. Bates, (2003). Values of Travel Tme savngs UK, Insttute of Transport Studes, Unversty of Leeds, Workng Paper 567 MIDEPLAN (2011) Precos Socales para la Evaluacón Socal de Proyectos. Mnstero de Planfcacón, Dvsón de Planfcacón, Estudos e Inversón. Acceddo el 1/5/2011 en Odgaard, K., C. Kelly, and J. Lard (2006). Current Practce n Project Apprasal n Europe, Project Report, Developng Harmonsed European Approaches for Transport Costng and Project Assessment. Pearce, D. and C. Nash (1981). The Socal Apprasal of Projects: a Text n Cost-Beneft Analyss. London: MacMllan Press. Small, K. and H. Rosen (1981). Appled Welfare Economcs wth Dscrete Choce Models, Econometrca, 41, Small, K. and E. Verhoef (2007). The Economcs of Urban Transportaton. New York: Routledge. Sugden, R. (1999). Developng a Consstent Cost-Beneft Framework for Mult-modal Transport Apprasal, Unversty of East Angla, U.K., Report to Department of Transport. U.S. Department of Transportaton (2014). Revsed Departmental Gudance on Valuaton of Travel Tme n Economc Analyss, Memorandum from the Offce of the Secretary of Transportaton, U.S. Department of Transportaton, Washngton D.C. 9