TI /3 Tinbergen Institute Discussion Paper Second-best Road Pricing Through Highway Franchising

Transcription

1 I /3 inbergen Institute Disussion Paper Seond-best Road Priing hrough Highway Franhising Erik Verhoef Department of Spatial Eonomis, Free niversity Amsterdam, and inbergen Institute.

2 inbergen Institute he inbergen Institute is the institute for eonomi researh of the Erasmus niversiteit Rotterdam, niversiteit van Amsterdam, and Vrije niversiteit Amsterdam. inbergen Institute Amsterdam Roetersstraat WB Amsterdam he etherlands el.: Fax: inbergen Institute Rotterdam Burg. Oudlaan PA Rotterdam he etherlands el.: Fax: Please send questions and/or remarks of nonsientifi nature to Most I disussion papers an be downloaded at

3 SECOD-BES ROAD PRICIG HROGH HIGHWAY FRACHISIG Erik. Verhoef * Department of Spatial Eonomis Free niversity Amsterdam De Boelelaan HV Amsterdam he etherlands Phone: Fax: everhoef@eon.vu.nl his version: 12/08/05 ey words: raffi ongestion, seond-best priing, highway franhising JEL odes: R41, R48, D62 Abstrat his paper onsiders the welfare impats of a range of franhising regimes for ongestible highways. For a single road in isolation, it is shown that a ompetitive aution with the level of road use as the deision riterion produes the soially optimal road in terms of apaity and toll level as the equilibrium outome, provided onstant returns to sale haraterize highway operations. he aution outperforms various alternatives, in whih the bidders are asked to minimize the toll level or toll revenues, or to maximize apaity or the bid for the franhise. When seond-best network aspets are taken into aount, the patronagemaximizing aution is no longer optimal. When unpried ongestion on parallel apaity dominates, the seond-best highway would generate losses and the zero-profit ondition beomes binding. he aution produes a below-optimal apaity. When unpried ongestion on serial apaity dominates, the aution produes an above-optimal apaity. * Affiliated to the inbergen Institute, Roetersstraat 31, 1018 WB Amsterdam.

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5 1. Introdution he private supply of highway apaity offers one way to deal with growing traffi ongestion in the fae of insuffiient publi funds to finane new apaity, and insuffiient support for publi road priing. Private involvement in highway supply is not exeptional. Around one-third of the Western European highway network is urrently under onession, with a strong onentration in the more Southern ountries of Frane, Spain, Italy and Portugal. Some of the value priing projets in the S involve private pay-lanes. And private toll roads are an inreasingly ommon phenomenon in developing ountries. Prolaimed potential advantages of private over publi highways inlude osteffiieny, innovativeness, and availability of funds. A main disadvantage is the divergene between the private objetive of profit maximization and the soial objetive of welfare maximization e.g. Edelson, 1971; Mills, 1981; Mohring, An important question is whether there are ways, partiularly through the design of autions for highway onessions, to make the private operator behave more losely in line with welfare maximizing prie and apaity setting. Suh strategies might preserve the advantages of private involvement, while limiting the potential disadvantages. Moreover, a properly designed aution would provide inentives to minimize the ost of supplying the apaity hosen, and would give the government an objetive way to selet a road operator among a larger set of andidates. And finally, the use of autions or omparable alloation mehanisms seems unavoidable in the awarding of onessions for highways, anyway. It is therefore important to have a proper understanding understand of the potential effiieny impats of the design of suh autions. his paper investigates one partiular aspet of autions for highway onessions, namely the extent to whih the hoie of the riterion used in the seletion of the winning bid affets the effiieny of the resulting highway. We will mainly onsider autions in whih private bidders are free to selet highway apaity and toll, but will briefly onsider more limited autions as well. o fous attention, a number of simplifying assumptions will be made. First, we onsider stationary traffi onditions with homogeneous users. Seond, we ignore speifi distortions that might arise from strategi interations between bidders in the aution by onsidering ompetitive autions only. here is no a priori reason to expet that these interations would systematially affet the ranking of the various riteria that we will onsider, although the welfare gains or losses from eah riterion would of ourse be different under non-ompetitive bidding. hird, we will not formally model demand unertainty and ontrat renegotiation. And fourth, we will assume that the government has suffiient power and redibility to enfore fulfilment of the bid, and an punish violations suh that the net profitability of winning the aution but not living up to it would be negative and hene below that of not winning. he set-up of the analysis is similar to that of bbels and Verhoef 2004, but extends it by onsidering the impats of an unpried omplement in addition to an unpried substitute; by onsidering as andidate riteria maximum patronage and, for pre-speified apaities, minimum toll revenues; and by using a different non-linear, BPR-type ongestion ost funtion in the numerial model.

6 2 Seond-best road priing through highway franhising he plan of the paper is as follows. Setion 2 starts with some theoretial bakgrounds. Setion 3 onsiders a single road, while Setion 4 moves on to introdue network aspets. Setion 5 onludes. 2. heoretial bakgrounds his setion provides some theoretial bakgrounds for our analysis. Setion 2.1 identifies the onditions for surplus-maximizing and profit-maximizing road apaities and tolls, and thus identifies the diretion in and extent to whih an aution should ideally affet the private operator s hoies, ompared to unrestrited freedom in setting the toll and apaity. hree ases are disussed: the benhmark of an isolated road, and two seond-best ases allowing for simple network spill-overs, namely where either an unpried substitute or an unpried omplement is available. Setion 2.2 addresses the profitability of surplus-maximizing roads, and therewith identifies the desirability for autions designed so as to push bidding ompanies to a zero-profit bid Welfare maximizing and profit maximizing tolls and apaities Single road Consider a single road with apaity, whih is used under stationary traffi onditions by homogeneous drivers with an aggregate inverse demand funtion D, where denotes the equilibrium flow of traffi. he average user ost inludes all variable osts inurred by the user, inluding travel time, and depends, through ongestion, on and. It is denoted,. he generalized prie faed by road users, p,, is equal to the sum of, and a toll τ if levied. he per-unit-of-time apaity ost depends on the road s apaity and is denoted C. Ignoring external osts other than ongestion, the first-best optimal toll and apaity, defined so as to maximize soial surplus, an be determined by solving the following Lagrangian:, D Λ = D ndn, C λ τ 1 0 where the first three terms represent the objetive, and the fourth term the equilibrium onstraint λ is the Lagrangian multiplier. he set of first-order onditions w.r.t.,,τ and λ an be solved to yield two familiar onditions: τ = 2a = 2b C where subsripts denote partial derivatives. Equation 2a shows that the optimal toll should be set equal to the marginal external ongestion osts, while 2b shows that the marginal benefits of apaity expansion the l.h.s. should be equal to the marginal ost the r.h.s.. An unrestrited private road operator would maximize profits by solving:

7 Seond-best road priing through highway franhising 3, τ D Λ = τ C λ 3 he set of first-order onditions now yield: τ = 4a D = 4b C he profit-maximizing toll inludes the marginal external ongestion osts from 2a, but adds to this a standard monopolisti mark-up that inreases as demand beomes less elasti. he latter term has the onventional interpretation; the internalization of ongestion is motivated by the fat that any redution in ongestion osts an be turned into revenues for the operator by inreasing the toll aordingly. Internalizing the ongestion externality therefore ontributes to the profit. Interestingly, the profit-maximizing optimality ondition for apaity hoie 4b is the same as for the soial optimum 2b. he reason is that for a given, the operator an turn savings in average user osts into toll revenues on a Euro-by-Euro basis when inreasing apaity. he profit-maximizing trade-off is therefore idential to the surplus-maximizing trade-off. Of ourse, the differene between 2a and 2b will generally ause the profitmaximizer to evaluate 4b for a smaller than a surplus-maximizer would onsider, produing a lower optimal apaity. As a orollary, when demand is perfetly elasti so that 4a beomes equal to 2a, a profit-maximizing operator would set its instruments suh that the optimum is ahieved, and no further regulation is warranted. Beause estimates of demand elastiity for road transport usually indiate elastiities well below 1 in absolute terms, this observation is of very limited use for pratial poliy making. nder first-best priing and apaity hoie, equations 2a and 2b would hold for every single link in a network. And a private operator would attempt to add to this a monopolisti mark-up, as in 4a, for every origin-destination pair in the network. Indeed, network extensions beome more hallenging analytially when seond-best onditions apply elsewhere on the network. An important type of seond-best distortion would be the existene of untolled, ongested links, with apaities set arbitrarily. Solving the resulting seond-best optimal or revenue-maximizing tolls and apaities an yield tedious expressions see Verhoef, 2002ab that elude easy interpretation. More insightful expressions an be obtained by onsidering two partiular extensions of the one-link network onsidered above: one with an unpried parallel link a substitute, and one with an unpried serial link a omplement. npried substitute he ase of seond-best ongestion priing with an unpried substitute has been onsidered by various authors, inluding Lévy-Lambert he inlusion of apaity as a seond poliy instrument has been less ommon. sing supersripts and to denote the untolled and the tolled alternative, respetively, and assuming that is to be treated as given, the

8 Seond-best road priing through highway franhising 4 surplus-maximizing seond-best toll τ and apaity an be found by solving the following Lagrangian:,,,, d,, 0 D D C C n n D = Λ τ λ λ 5 he set of first-order onditions w.r.t.,,,τ, λ and λ an be solved to yield: = D D τ 6a C, = 6b where D denotes the slope of the single demand funtion. he seond-best optimal toll 6a is the same as the one reported by Lévy-Lambert 1968, and is therefore unaffeted by the possibility of also setting apaity for route. A more detailed interpretation of this toll expression an be found in Verhoef et al. 1996, but note that it is below the marginal external ongestion on route in order to optimize the ongestion spill-over onto route. he seond-best optimal apaity rule 6b is similar to the first-best rule 2b. Given the equilibrium level of use of route,, and given the assoiated generalized equilibrium prie τ, it is optimal to set apaity suh that the flow is served at a ombination of and τ that minimizes the soial ost of arrying. he optimality ondition for apaity is therefore the same as for a road without substitute. he profit-maximizing toll and apaity follow from:,,, D D C Λ = τ λ λ τ 7 he set of first-order onditions w.r.t.,,,τ, λ and λ now yields: = D D τ 8a C, = 8b he same tax rule an be found in Verhoef, ijkamp and Rietveld 1996 in whih is held fixed, and it is therefore again not affeted by the possibility to set apaity. ote that, in ontrast to the seond-best toll in 6a, this tax rule adds a positive term to the ommon firstterm that represents the marginal external ongestion ost on the tolled route. As we found for the single link, the optimality onditions for surplus-maximizing and profit-maximizing apaity, 6b and 8b, are the same. he hosen apaities will differ only beause the point of evaluation differs.

9 Seond-best road priing through highway franhising 5 npried omplements Seond-best priing with an unpried omplement suh as a serial link has reeived onsiderably less attention in the literature than the unpried substitute. Maintaining the assumptions of a single origin-destination pair and ontrol over instruments at one of the two links only, the seond-best optimum is easily found by adapting 5 to the new network onfiguration that implies that the same users use link and : Λ = 0 D ndn λ,, C,, τ D, C, 9 he set of first-order onditions w.r.t.,,τ, and λ an be solved to yield: τ = 10a = C 10b, Intuitively, the seond-best optimal toll perfetly internalizes the marginal external ongestion osts for both links jointly. he rule that defines optimal apaity has the by now familiar form. Again no effets on link are present in this rule, whih reflets that indiret effets of hanges in upon ongestion on link anel beause the toll in 10a already perfetly internalizes this ongestion. Finally, the profit-maximizing hoie of instruments an be found from solving:,, τ D, Λ = τ C λ 11 he set of first-order onditions w.r.t.,,τ, and λ now yields: D τ = 12a = C 12b, he profit-maximizing toll is a straightforward generalization of 4a, like the surplusmaximizing toll 10a was from 2a. And also in this final network, the rule ditating profitmaximizing apaity is the same as the one for surplus maximization. he hosen apaities will again differ only beause the point of evaluation differs Optimality, self-finaning and maximized profits A famous result from the transport eonomis literature, derived by Mohring and Harwitz 1962, states that an optimally designed road i.e., with an optimal apaity and an optimal toll will be exatly self-finaning, provided some tehnial onditions are satisfied. hese tehnial onditions an be summarized as follows: 1 road apaity should be a ontinuous variable; 2 there should be onstant returns to sale in ongestion tehnology equiproportional hanges in use and apaity leave average user ost unaffeted; and 3 there

10 6 Seond-best road priing through highway franhising should be onstant eonomies of sale in highway onstrution the ost per unit of apaity is independent of total apaity. his self-finaning theorem has been shown to extend to eah road individually in a full network, and therefore also to the network in aggregate, provided eah link is optimally pried and all apaities are optimized Yang and Meng, he theorem also extends to dynami models Arnott, De Palma and Lindsey, 1993; and in present-value terms when allowing for adjustment osts and depreiation Arnott and raus, Empirial evidene suggests that onditions 2 and 3 may hold at least approximately: empirial estimates of the ratio of long-run average and marginal osts are often relatively lose to unity Small, 1992, Setions 3.4, Profits or defiits under optimal design and priing of a road will then be relatively small. Condition 1 seems unrealisti for a single road beause the number of lanes is disrete. But apaity per lane an be varied by widening lanes, by resurfaing, or by re-grading or straightening a streth of road. And when this is not the ase, an optimally designed road might still be self-finaning over the longer run, when periods of underapaity and overapaity alternate as demand grows over time. We an illustrate the self-finaning theorem in our model by replaing the general ost funtion, by the somewhat more speifi / seuring onstant returns in ongestion tehnology, and the apaity ost funtion C by γ, with γ denoting a onstant ost per unit of apaity seuring onstant eonomies of sale in highway onstrution. Observe that: / / 13a All onditions for optimal apaity hoie enountered thus far were of the same type, whih an be rewritten using our assumption of a onstant ost per unit of apaity as: = γ 13b Multiplying both sides by yields: = C 13 or, using 13a: = C 13d he l.h.s. of 13d gives total apaity ost when apaity is set aording to 13b, whih turns out to be equal to total toll revenues under the first-best priing rule of 2a. his means that, whenever 13b is satisfied but the toll rule deviates from 2a, optimal apaity hoie 1 More reently, Levinson and Gillen 1998 report a point estimate for the ratio between long-run average and marginal ost of 0.92 for auto, but 1.45 for single truks and 1.96 for ombination truks, suggesting mild diseonomies for passenger ars but onsiderable eonomies for truks.

11 Seond-best road priing through highway franhising 7 will result in an unbalaned budget. here will be a finanial surplus if the toll exeeds the marginal external ongestion ost as for all instanes of private priing we have onsidered, as well as for the ase of optimal priing with an unpried ongested omplement. here will be a defiit when the reverse holds as for optimal priing with an unpried ongested substitute. For a road in isolation or, indeed, a road in an otherwise optimally managed network equation 13d onfirms the self-finaning theorem. hese results have impliations for the potential of ompetitive autions to ahieve effiieny. At least when no subsidies are part of the aution, a ompetitive aution would drive profits to zero. For a road in isolation, and for whih the onstant-returns-to-sale assumptions are fulfilled, the optimal profit is zero, too. his means that there ould be ompetitive autions that would have the optimum road as an outome. For a road with unpried omplements or substitutes, this would not generally be the ase. 3. Competitive autions for an isolated road A ompetitive aution an be defined as one in whih a suffiiently large number of nonooperative bidders are ative, so that there is no sope for strategi behaviour and the bidders do not believe they will stand a hane of winning the aution when submitting a bid with a lower than their best sore on the seletion riterion used in the aution, beause the bidders are suffiiently omparable in terms of effiieny. he assumption of a ompetitive aution allows a lean analysis of the performane of different seletion riteria, without imperfetions in the bidding proess ompliating the piture. Suh imperfetions would in reality of ourse be of potentially deisive importane, but the questions of to exatly what extent the performane of aution would be affeted, and of whether the ranking of the different seletion riteria would be systematially affeted, are left for future researh. For a ompetitive aution thus defined, any seletion riterion that an be improved upon as long as profits are positive would ause bidders to be pushed towards a zero-profit bid. We will all suh riteria profit-exhausting. All riteria we will onsider in the sequel will be profit-exhausting. A somewhat unrealisti example of a riterion that would not be profit-exhaustive would be the riterion of soial surplus when an unpried ongested omplement is available; ompare equations 13d and 10a above. Indeed, if information would be so omplete that a meaningful aution with soial surplus as the riterion were possible, it would not be hard to define the optimal riterion for an aution. In pratie, more easily observable riteria will however have to be used. he set that we will onsider in our analysis is based on pratial examples, earlier proposals in the literature, and on an attempt to find a relatively effiient riterion. he riteria to be onsidered are: a maximum bid for the right to build and operate the road Bid, a maximum apaity supplied Cap ; a minimum toll harged oll ; a minimum toll harged for a pre-defined apaity oll-ap ; and the maximization of patronage of the new apaity Pat. If an aution is profit-exhausting, the ourrene of a winner s urse is of ourse not inoneivable: the winning bid is from the party that holds the most optimisti expetations

12 8 Seond-best road priing through highway franhising about market opportunities, and that therefore stands a onsiderable risk of below-projeted patronage and hene net losses one operation ommenes. We will not formalize the existene of a distribution in expetations aross potential bidders in our analysis. But one way of dealing with this problem in reality would be to ask bidders to supply, along with their bid, a detailed aount of the predited use levels, travel times, toll levels, and road design. his would allow verifiation of the plausibility of the travel times as a funtion of road harateristis apaity and patronage, as well as patronage as a funtion of travel time and toll level. For the latter test, existing transport network models ould be used; and further insights an be obtained by omparing bids with eah other. With a deviation above a ertain threshold, the bid ould be ignored to protet the bidder from a winner s urse, or larifiation or revision ould be demanded. And finally, a redible and effetive penalty should of ourse exist for underperformane ompared to the bid, if wining. his penalty should be suh that the firm should make a net loss from winning and underperforming, and might be oupled to the government obtaining the right to set toll levels when underperforming remains persistent. A numerial model: a single road We will illustrate the disussion of the various possible seletion riteria using the results of a small numerial simulation model. he model is highly stylized, but nevertheless alibrated so as to be representative for a highway that is ongested during peak times. he average user ost funtion is modelled aording to the well-known BPR-formulation Small, 1992: χ / = α t β f 1 14 where α is a parameter refleting the value of time set at 7.5 in our model, aording to onventional Duth estimates, t f is a parameter refleting the free-flow travel time set at 0.5, implying 60 km for a 120 km/hr highway, and β and χ are parameters that are set at 0.15 and 4, respetively; onventional values for the BPR-funtion. he units of apaity are hosen suh that one onventional traffi lane would orrespond to =1500. his implies a doubling of travel times at a use level of around 2400 vehiles per hour. his is roughly in aordane to the flow at whih, empirially, travel times double for a single highway lane and the maximum flow on a lane is reahed e.g. Small, 1992, Fig. 3.4, p. 66. A maximum flow, however, is not defined for BPR funtions. he prie of apaity, γ, is set equal to 7. With a unit of time of one hour, this parameter ought to reflet the hourly apital osts. o derive a value from empirial onstrution ost estimates, an assumption has to be made on whether the model aims to represent stationary traffi onditions throughout a day, or during peak hours only. Our parameterization onerns the latter. he value of 7 was then derived by dividing the estimated average yearly apital ost of one highway lane kilometre in he etherlands 0.2 million by working days times 5 peak hours per working day; assuming two

13 Seond-best road priing through highway franhising 9 peaks and next by 1500 the number of units of apaity orresponding with a standard highway lane, and finally multiplying by 60 the number of kilometres orresponding with a free-flow travel time of half an hour. Only welfare effets in peak hours are therefore onsidered in the numerial exerise, and it is assumed that off-peak travel is so modest that both the optimal off-peak toll and the marginal benefits of apaity expansion would be negligible. o maintain onsisteny, all seletion riteria to be onsidered below, where relevant, would also apply to peak hours only. And finally, no relevant welfare effets arise outside the peak, and therefore no toll revenues are supposed to be raised. Finally, it is assumed that a linear inverse demand funtion applies: D δ δ 15 = 0 1 A hoie of δ 0 = and δ 1 = , together with = 3000, produed a desired benhmark equilibrium where an equilibrium road use of = 5000 auses equilibrium travel time t to be around two times the free-flow travel time t f, while equilibrium demand elastiity ε is equal to Beause there are no toll revenues, profit π is negative in the benhmark equilibrium. his benhmark equilibrium will not be interpreted as some initial situation in the single-road analysis; that is, apaity will be allowed to beome smaller than the benhmark level. Equilibrium Optimum Bid Cap oll Pat t / t f n.r. b 1.37 ε τ n.r. b n.r. b 5.14 p n.r. b D π a S ω a he figure shown in fat gives the bid. After making this bid, profit will beome equal to zero. b ot relevant. able 1. umerial results for a single road he optimum onfiguration is depited in the seond olumn of able 1. As expeted, profits are exatly zero in the optimum. Optimal apaity is higher and optimal road use is lower than in the benhmark. As a result, travel times are lower 1.37 times the free-flow travel time. Let us now turn to the various riteria for autions. he first of these, Bid, fores the private operator to set the profit-maximizing toll and apaity identified in 4a and 4b the net profit, after the sum promised in the bid has been paid, will of ourse be zero. his leads to a toll that is nearly three times as high as the optimal toll, and a apaity that is exatly half the optimal apaity as an be expeted with a linear demand funtion and onstant long-run

14 10 Seond-best road priing through highway franhising marginal ost. he final row in able 1 shows an effiieny indiator ω, whih is for a partiular equilibrium alulated as the soial surplus in that equilibrium minus that in the benhmark, divided by soial surplus in the optimum minus that in the benhmark. It therefore gives the share of first-best surplus gains relative to the benhmark that a partiular aution ahieves; a negative value denotes a surplus below the benhmark level. his is for example the ase for the aution Bid. he poor performane of this poliy is in aordane with the rather pessimisti preditions of effiieny impats of profit-maximizing ongestion priing in various earlier studies e.g. Verhoef and Small, A seond aution, Cap, asks bidders to offer a apaity as large as possible. Beause the toll is not restrited to be set optimally, the likely result is that apaity would exeed the optimal level: in the urrent numerial example it is nearly twie as large. he high apital osts are overed by a toll that is nearly as high as the profit maximizing toll, beause it maximizes revenues given the apaity hosen. he resulting relatively small level of road use, in ombination with the relatively large apaity, ause soial surplus to be even lower than under Bid. t zp w zp Figure 1. Zero-profit toll left panel and relative effiieny right panel as a funtion of apaity A third aution, oll, would award the onession to the bidder requiring the lowest toll. Although this riterion may seem reasonable at first sight, it will generally fail to produe an interior solution with a positive apaity. he left panel of Figure 1 illustrates why. It shows the zero-profit ontour in the -τ spae, whih an be denoted τ zp : the orrespondene between zero-profit tolls and apaity. First note that for all apaities below the maximum apaity that an be offered without a defiit i.e., below the solution from aution Cap, around 5807 in the numerial model, there are in fat two toll levels that produe zero profits. For any apaity hosen, it is the lower of these two toll levels that would result with the riterion toll. he area bounded by the ontour τ zp and the vertial axis orresponds with positive profits; the area outside the ontour with negative profits. A suffiient ondition for bakward-bending pattern to arise is that the inverses demand funtion should interset

15 Seond-best road priing through highway franhising 11 both axes and that the elastiity of demand with respet to toll 2, denoted ε τ, should monotonously derease in absolute value in. All revenue levels below the maximum revenue for a ertain apaity at ε τ = 1 for that apaity an then be realized as a higher-toll lower-demand ombination the upper segment of the ontour τ zp or as a lower-toll higher-demand ombination the lower segment of τ zp. Beause the profit-maximizing toll for a ertain apaity lies between the two zero-profit tolls, and beause of our assumption of ε τ dereasing monotonously in, profit is inreasing in τ for a given apaity on the lower segment, and dereasing in τ on the upper segment. ear a zero apaity, the minimum zero-profit toll would be a delining funtion of apaity under rather general onditions. nder the onstant-eonomies-of-sale assumptions, this would be true provided demand is not perfetly elasti. A sketh of a proof is as follows. Consider a ertain apaity and a zero-profit toll below the revenue-maximizing toll for that apaity. hen imagine a simultaneous equi-proportional redution of apaity and use. his would leave average ost unaltered. herefore, to support this onfiguration as an equilibrium while marginal benefits D have inreased, τ should rise. his means that a profit will be made: apaity and use have fallen in the same proportion, so that maintaining zeroprofits would require the toll to remain at its original level, instead of rising. Beause revenues are inreasing in the toll level in this range, the toll should fall in order to return to zero profits. Hene, the minimum zero-profit toll is a delining funtion of apaity. As a result, the aution oll will not result in a winning bid with a positive apaity: bidders will be pushed towards bidding a zero apaity. he right panel of Figure 1 depits ω zp : the orrespondene between apaity and ω under zero-profit tolling. he upper segment of ω zp orresponds with the lower segment of τ zp, and reversely. It shows that the outome of the aution oll is the least effiient among the possible zero-profit ombinations of and τ. his is a general result under notperfetly-elasti demand, and is aused by the fat that under zero-profit tolling, marginal benefit is equal to average total ost user ost and apital ost together. With a falling demand funtion, total benefit therefore exeeds total ost for any positive use level, and soial surplus is always higher than in absene of the road as resulting from oll. he first-best optimum is on the lower segment of the ontour τ zp. An aution that would pre-speify the optimal apaity and next use the minimum toll as the riterion therefore would in priniple be suessful in ahieving the first-best optimum. his aution oll-ap would be lose to the one proposed by Engel, Fisher and Galetovi 1996, who propose an aution with as the riterion the minimization of the net present value of toll revenues PR before the highway is to be transferred to the government. he setting of Engel et al is rather different from that in this paper. hey are primarily onerned with the promotion of ost-effetiveness in onstrution and the avoidane of renegotiation of ontrats under demand unertainty; but they treat the hoie of apaity as exogenous and 2 he elastiity of demand with respet to toll ε τ differs from the onventional demand elastiity ε beause: a the toll τ differs from the generalized prie p=τ, and b d/dp d/dτ beause d/d 0.

16 12 Seond-best road priing through highway franhising ignore the effet of toll setting on soial welfare in fat, they assume that the soial objetive is to minimize the expeted value of tolls paid. Demand unertainty and renegotiation are ignored in the present paper, but the impats on soial welfare are, in ontrast, entral. 3 From that perspetive, a number of observations an be made onerning Engel et al. s PR-aution. he first is that Figure 1 implies that over the relevant range of apaities, any target level of toll revenues below maximum revenues ould be ahieved by two toll levels, with strongly diverging welfare impliations. A riterion that is phrased in terms of toll revenues be it per-unit-of-time or in present value terms annot disriminate between these two tolls. As a onsequene, there is no guarantee that whihever toll revenue is raised per unit of time, it is raised using the more effiient toll. In the numerial example, even for the optimal apaity would the higher-toll equilibrium produe only very limited benefits ompared to the no-road situation ω is lose to 4. A seond observation is that the PRaution does not diret the operator towards an optimal toll revenue per unit of time: it is only the net present value that matters. If after transferring the road to the government, tolling is disontinued, the aution may ause priing to be non-optimal both before and after the transfer. It is therefore unertain whether an PR-aution would indeed produe an optimal outome, espeially when soial surplus would be the soial objetive. he final aution we onsider is Pat : the aution that awards the onession to the operator that offers the highest patronage. he final olumn in able 1 shows that the outome of this aution oinides with the first-best optimum. Again, this result an be expeted to arry over to more general settings, as long as the onstant-returns-to-saleassumptions are fulfilled. he intuition is as follows. First, observe that the maximization of demand requires the minimization of generalized prie, and hene of τ. ext, zero profits imply that τ is equal to apaity ost per user, so that the generalized prize beomes C /. Minimizing this with respet to, as the bidder would need to do to maximize, yields for any positive an expression that is equal to 4b: the optimality ondition for optimal apaity as a funtion of road use. Beause of the onstant-returns-to-sale assumptions, the generalized prie therefore oinides with the long-run marginal ost user ost and apaity ost jointly, so that also is optimized. And the joint ourrene of an optimal and an optimal seures ahievement of the first-best optimum. he perhaps ounter-intuitive onlusion is therefore that, provided the onstantreturns-to-sale onditions are fulfilled, the aution that maximizes soial surplus is the one that maximizes traffi flow. 4. Seond-best network effets An important simplifiation of the above analysis onerns the neglet of network effets. his is aeptable when studying a road in isolation, or under the hypothetial assumption of 3 Cost-effetiveness in onstrution is not onsidered expliitly in this paper, and firms are assumed to always operate on the apaity ost funtion C. A ompetitive profit-exhausting aution would, however, always seure ost-effetiveness in onstrution. Expliit onsideration of ost-effetiveness in onstrution would therefore not affet the onlusions.

17 Seond-best road priing through highway franhising 13 first-best priing throughout the rest of the network. It an also be onsidered instrutive to deliberately ignore network ompliations, beause doing so allows onentration on the primary effiieny impats of the various autions, independent of seond-best network spillovers. But network effets are likely to be important in reality, and may, as we shall see below, have signifiant impats on the performane of autions. o maintain fous and keep the exposition transparent, we will onsider two very simple networks in what follows, whih would represent the most important types of seond-best network issues that ould arise. Setion 4.1 onsiders the situation where an unpried perfet substitute for the new road is available i.e., a parallel road, while Setion 4.2 onsiders an unpried omplement i.e., a serial road npried substitute he existene of an unpried substitute road naturally redues the potential profitability of the new road. his effet an be substantial, whih is illustrated by the fat that when interpreting the rather heavily ongested benhmark road from the previous setion as pre-existing initial apaity, no profitable apaity-toll ombination for additional, pried apaity appears to be possible. Also the seond-best optimum, for whih the apaity of and toll on the tolled new parallel is optimized under the onstraint that initial apaity remains untolled, onsequently produes a finanial defiit. he results in able 2 show that only 7% of the apaity ost for the seond-best optimal toll road would be overed by the revenues from the seond-best toll. Equilibrium Optimum Seond-best τ τ π S ω Cost overage 0.07 able 2. umerial results for an unpried substitute: original parameterization nder this parameterization, no bids an be expeted in an aution if it does not inlude the possibility of subsidies. bbels and Verhoef 2004 explore the possibilities for and properties of autions with subsidies. In this paper, we do not onsider suh autions, motivated by the observation that if the required subsidy would be so large 93% of the onstrution osts in the numerial example, a government would most likely prefer to arry 100% of the

18 14 Seond-best road priing through highway franhising onstrution osts and keep the road in publi hands altogether. An aution that raises only 7% of the onstrution ost while meaning loss of diret ontrol over the highway toll does not seem to be a very attrative option when soial surplus maximization is the overall objetive. In order to get an idea of the performane of zero-subsidy autions in the presene of an unpried substitute, the parameterization has to be adjusted, so as to reate the possibility of zero-profit bids with positive apaity. his was ahieved in the numerial model by reduing the initial apaity from 3000 to As a result, the benhmark equilibrium travel time beomes 4.3 times as high as the free-flow travel time. For suh a heavily ongested road, zero-profit bids for additional pried apaity do beome possible, and the results for the different riteria are shown in able 3. he first-best optimum for whih priing on both roads is allowed is of ourse equivalent to that for the road in isolation. Beause initial unpried apaity is relatively small, the seond-best equilibrium ahieves a relative effiieny of ω=0.91, whih is substantial. However, beause of the seond-best nature of this equilibrium, the toll is set aording to equation 6a, produing a toll that is only 16% of the first-best toll. As a result, a substantial defiit will our on the operation of the tolled road: the ost overage for the new road in the seond-best equilibrium is only 16%, despite the fat that tolled apaity makes up nearly two thirds of total apaity. Again, it would seem more attrative to keep the road in publi hands than to design an aution that would, if suessful in reproduing the seond-best optimum, require a subsidy of 84% of total onstrution ost. Equilibrium Optimum Seondbesbest Bid Cap oll Pat Seond- zp τ τ n.r. b n.r. b π a S ω Cost 0.16 overage a he figure shown in fat gives the bid. After making this bid, profit will beome equal to zero. b ot relevant. able 3. umerial results for an unpried substitute: adjusted parameterization If insuffiient publi funds are available to finane this investment, the question rises of how attrative zero-profit roads might be. he final olumn in able 3 shows the seond-best

19 Seond-best road priing through highway franhising 15 optimum under an additional zero-profit onstraint; so, the best ahievable benhmark outome for zero-profit autions. he toll is substantially higher and apaity lower than in the seond-best optimum. However, ω still reahes a level of 0.67 when the zero-profit onstraint is added to the seond-best problem. he levels of τ and are the same as for the first-best equilibrium, whih is aused by the fats that the aution indues the operator to minimize total ost for any given while keeping apaity self-finaning. his means seleting the same / ratio, and the same and τ, as for the first-best optimum. he aution Pat again ahieves the seond-best optimum under the zero-profit onstraint. Maximizing under a zero profit onstraint requires minimization of average user ost plus apital ost per user. Beause of the network equilibrium ondition, the minimization of the generalized prie on road implies that average user ost on road are also minimized., and, are therefore maximized and so is therefore total benefit. 4 For a given, and given τ = 0, these are the same onditions that define the seond-best zero-profit equilibrium. We an be brief on the other riteria. Bid still does not perform very good ω=0.38, whih is aused by the large disrepany between the revenue-maximizing and the surplusmaximizing seond-best toll; ompare 6a and 8a. Cap performs relatively good ω=0.59, whih is aused by the fat that there is not muh sope to expand apaity of the new road beyond the seond-best zero-profit level without running into losses. And oll, as before, will not result in the supply of a positive apaity. ote that all ω s are positive. he reason is that road users as a group an only benefit from the supply of additional apaity that is to be used voluntarily, while profits will be zero. Soial surplus, therefore, an only inrease. he absene of subsidization possibilities ombined with the absene of priing on initial apaity auses the maximum ahievable welfare gains to be around two thirds of those from first-best priing and apaity hoie. he size of the relative loss, around one third in this example, evidently depends on the assumed initial onditions, and may in some ases beome so large that the overall effiieny gain from the aution beomes unaeptably small. Would there, in suh ases, be a possibility to enhane the soial benefits from the aution by hanging its set up? One possible strategy, based on the observation that the soure of the redued effiieny gains is the existene of initial unpried apaity, would be to stipulate that the winning bidder will have to buy the existing road against the best estimate of the urrent onstrution osts for the same apaity, and to allow the winning bidder to apply a ongestion toll on this existing apaity. Provided the implied apital ost per unit of apaity for the initial apaity are the same as a bidder s ost per unit of new apaity, and provided the initial apaity is smaller than the apaity a bidder would hoose in an aution, he will in fat fae the same problem as for the road in isolation. he aution Pat would then again 4 he outome of the aution Pat appears to be independent of whether it is the patronage of the new apaity or of both roads together that is used as the riterion. Maximizing through minimizing the generalized prie on that road also maximizes, beause the generalized pries on both roads will be equalized in equilibrium.

20 16 Seond-best road priing through highway franhising ahieve the first-best optimum. herefore, there ertainly may be ways to avoid partiularly unattrative network spill-overs through autions, by making the ompulsory purhase of the assoiated links part of the onession npried omplement he logial ompanion problem to the existene of an unpried substitute is the existene of an unpried omplement. able 4 shows the numerial results, for whih in order to maintain omparability, the assumption was made that half the road s length would remain unpried and at the benhmark apaity 3000 so as to funtion as the unpried omplement, while the other half would be subjet to the aution. Equilibrium Optimum Seondbest Bid Cap oll Pat Seondbest zp τ n.r. b τ n.r. b τ n.r. b n.r. b n.r. b n.r. b π a S ω Cost overage 2.42 a he figure shown in fat gives the bid. After making this bid, profit will beome equal to zero. b ot relevant. able 4. umerial results for an unpried omplement Equations 10ab already showed that there will be a finanial surplus in the seond-best optimum, beause the seond-best optimal toll also internalizes the ongestion externality on the unpried omplement. he third olumn in able 4 shows that in the numerial example, the revenues will onsequently be more than twie as large as the apaity ost. he seondbest optimum with an additional zero-profit onstraint defines the best possible outome for profit-exhausting autions. he final olumn in able 4 shows that ω drops to 0.67 the similarity with the unpried-substitute ase in able 3 is a oinidene. Again, the aution Pat is the only aution that reprodues the seond-best zero-profit equilibrium: maximization of under a zero-profit onstraint apparently again implies maximization of the soial surplus under that same onstraint given the inability to adjust. he reason is that under zero-profit priing the generalized prie is equal to average total ost that is: average user ost and per-user apital ost for link jointly; apital ost for link are ignored but are fixed anyway. Soial surplus is therefore given by onsumer surplus, and this is maximized when is maximized by minimizing the generalized prie.

21 Seond-best road priing through highway franhising 17 he relative ranking of Bid and Cap has reversed ompared to the unpriedsubstitute ase. Bid leads to the profit maximizing outome and therefore avoids the potentially substantial overinvestment in link s apaity that the revenues from impliit ongestion priing of link allow. Indeed, ould be expanded up to more than three times its seond-best level without running into losses; ompare Cap and Seond-best in able 4. Beause Cap aggravates this distortion, its effiieny is relatively low. oll, finally, again suffers from the problem of not produing a positive equilibrium apaity. Apart from making the ompulsory purhase of the unpried link part of the aution, as for the unpried-substitute ase, a simpler solution to the problem of over-investment appears possible in this ase, and that would be to inform the bidders that they will be harged a toll equal to the marginal external ongestion ost on link for every user passing that link. his would take away the exess profits and leave the private bidders faing the same onditions as in Setion 3, meaning that Pat would again reprodue the optimum. 5. Conlusion Shortage of funds for road expansion, politial unaeptability of publi road priing, and perhaps expetations of higher effiieny from private operations may all be fators that ause the private provision of toll roads to beome an attrative option to ope with growing traffi ongestion. Conessions for private road operation will typially be autioned. his paper showed that the seletion riterion used in suh autions may have a deisive impat on the effiieny of the resulting winning bid. A bid for the right to build and operate the road pushes the bidders towards a profit-maximizing design, whih will typially be quite different from a surplus-maximizing road. he maximization of apaity typially leads to exessive apaity, and a revenue-maximizing toll given that apaity; both reduing surplus below ahievable levels. he minimization of tolls pushes bidders towards a zero apaity, unless apaity is set a priori. he minimization of toll revenues does the same, and in addition suffers from the fat that there may be multiple toll levels that, given a apaity, yield the same revenue but differ strongly in welfare impats. However, an aution that asks to maximize patronage appeared to reprodue the first-best road in absene of network spillovers and under onstant-eonomies-of-sale assumptions, and the seond-best zero-profit road with network spill-overs. Many important questions that need further onsideration an be identified. A first one is whether a redible and effiient penalty system an be thought of that would guarantee the winning bidder to live up to the bid. A seond one is whether a mehanism an be developed to ope with demand unertainty and avoid renegotiation of ontrats. A third one involves extension of the urrent analysis to larger, ideally generalized networks, and multiple time periods notably peak off-peak. A fourth one involves strategi behaviour and interations during the bidding proess. he list ould probably be extended easily, and illustrates that there is still suffiient potential for future researh.

22 18 Seond-best road priing through highway franhising Referenes Arnott, R., A. de Palma and R. Lindsey 1993 A strutural model of peak- period ongestion: a traffi bottlenek with elasti demand Amerian Eonomi Review Arnott, R. and M. raus 1998 Self-finaning of Congestible Failities in a Growing Eonomy. In: D. Pines, E. Sadka and I. Zilha eds opis in Publi Eonomis: heoretial and Applied Analysis Cambridge niversity Press, Cambridge, pp Edelson,.E Congestion tolls under monopoly Amerian Eonomi Review Engel, E., R. Fisher and A. Galetovi 1996 Highway franhising: pitfalls and opportunities Amerian Eonomi Review, Papers and Proeedings Lévy-Lambert, H arifiation des servies à qualité variable: appliation aux péages de irulation Eonometria Mills, D.E Ownership arrangements and ongestion-prone failities Amerian Eonomi Review, Papers and Proeedings Mohring, H Profit maximization, ost minimization, and priing for ongestion-prone failities Logistis and ransportation Review Mohring, H. and M. Harwitz 1962 Highway Benefits: An Analytial Framework orthwestern niversity Press, Evanston Il. Small,.A rban ransportation Eonomis. Fundamentals of Pure and Applied Eonomis. Harwood, Chur. bbels, B. and E.. Verhoef 2004 Autioning onessions for private roads Disussion paper I /3, inbergen Institute, Amsterdam-Rotterdam. Verhoef, E a Seond-best ongestion priing in general stati transportation networks with elasti demands Regional Siene and rban Eonomis Verhoef, E b Seond-best ongestion priing in general networks: heuristi algorithms for finding seond-best optimal toll levels and toll points ransportation Researh 36B Verhoef, E.., P. ijkamp and P. Rietveld 1996 Seond-best ongestion priing: the ase of an untolled alternative Journal of rban Eonomis Verhoef, E.. and.a. Small 2004 Produt differentiation on roads: seond-best ongestion priing with heterogeneity under publi and private ownership Journal of ransport Eonomis and Poliy Yang, H. and Q. Meng 2002 A note on Highway priing and apaity hoie in a road network under a build-operate-transfer sheme ransportation Researh 36A