Erik T. Verhoef. VU University Amsterdam, and Tinbergen Institute.

Transcription

1 TI /3 Tinbergen Institute Discussion Paper Private Roads Auctions and Competition in Networks Erik T. Verhoef VU University Amsterdam, and Tinbergen Institute.

2 Tinbergen Institute The Tinbergen Institute is the institute for economic research of the Erasmus Universiteit Rotterdam, Universiteit van Amsterdam, and Vrije Universiteit Amsterdam. Tinbergen Institute Amsterdam Roetersstraat WB Amsterdam The Netherands Te.: +31(0) Fax: +31(0) Tinbergen Institute Rotterdam Burg. Oudaan PA Rotterdam The Netherands Te.: +31(0) Fax: +31(0) Most TI discussion papers can be downoaded at

3 PRIVATE ROADS Auctions and Competition in Networks Erik T. Verhoef * Department of Spatia Economics VU University Amsterdam De Boeeaan HV Amsterdam The Netherands Phone: / Fax: Emai: everhoef@feweb.vu.n This version: 25/06/08 Key words: Traffic congestion, second-best pricing, highway franchising, road networks JEL codes: R41, R48, D62 Abstract This paper studies the efficiency impacts of private to roads in initiay untoed networks. The anaysis aows for capacity and to choice by private operators, and endogenizes entry and therewith the degree of competition, distinguishing and aowing for both parae and seria competition. Two institutiona arrangements are considered, namey one in which entry is free and one in which it is aowed ony after winning an auction in which patronage is to be maximized. Both regimes have the second-best zero-profit equiibrium as the end-state of the equiibrium sequence of investments. But the auctions regime approaches this end-state more rapidy: tos are set equa to their second-best zero-profit eves immediatey, and capacity additions for the earier investments are bigger. When discreteness of capacity is reevant and imits the number of investments that can practicay be accommodated, the auctions regime may therefore sti resut in a more efficient end-state, with a higher socia surpus, athough the theoretica end-state is the same as under free entry. * Affiiated to the Tinbergen Institute, Roetersstraat 31, 1018 WB Amsterdam.

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5 1. Introduction Over the ast decades, there has been an increasing interest in private invovement in road infrastructure suppy. One important reason is that decining government budgets motivate the search for aternative funds for financing desired road capacity expansions. In addition, there is a rather wide-spread beief that the private sector woud be inherenty more efficient and innovative than their pubic counterparts, so that private roads may be buit and operated at ower costs than pubic ones. Another consideration coud be that the pubic at arge may accept the imposition of tos, generay beieved to be important in curbing traffic congestion, more easiy from private than from pubic operators. There are, however, aso potentia economic hazards in the private suppy of road capacity. Particuary, private to-road operators woud typicay be interested in maximizing profit rather than socia surpus, and sociay optima first-best pricing cannot be expected from them especiay not because the contro of a road (section) wi usuay impy a certain degree of market power. The impacts on the private operator s price setting has been studied, for instance, by Edeson (1971); Mis (1981); Mohring (1985); Verhoef, Nijkamp and Rietved (1996); Verhoef and Sma (2004); and De Pama and Lindsey (2000). One recurring and probaby not so surprising concusion from such studies is that profit-maximizing private road operators typicay set congestion tos above the optima eve: the profit-maximizing to not ony internaizes margina externa congestion costs, ike the efficient to does, but adds to this a monopoistic demand-reated mark-up that rises as demand becomes ess eastic. In addition, even though a profit maximizer has an incentive to offer the sociay optima amount of capacity given the prevaiing eve of demand, overpricing reduces demand, and hence the private suppy of capacity is generay beow the optima eve (for some further discussion, see for exampe Sma and Verhoef, 2007). Most studies of private road suppy take the number of private suppiers as given. Usuay ony one operator is considered, sometimes a duopoy (e.g. as in De Pama and Lindsey, 2000), but ony sedom more. This may ead to a somewhat pessimistic picture of the efficiency of private to roads: DeVany and Saving (1980) and Enge, Fisher and Gaetovic (2004) show how profit-maximizing tos fa as the number of parae competitors increases, approaching the optima vaue as firms become infinitesimay sma and competition becomes perfect. The imited attention for this theoretica benchmark resut can probaby be expained by the fact that perfect competition, with many parae competitors, seems a rather theoretica option, due to the umpiness of road infrastructure in practice. It is not ony just the number of competing private road suppiers that determine overa efficiency; it is aso their distribution over the network. Sma and Verhoef (2007, Ch. 6) iustrate this in a simpe exampe, by studying how tos and socia surpus wi vary if a road of a given capacity and ength is spit up and divided over an increasing number of symmetric private competitors in two contrasting cases: when they compete in parae as substitutes versus when they compete in series as compements. In accordance with the two studies just mentioned, they find that the tos approach the optima eve, which just

6 2 Private roads: auctions and competition in networks internaizes margina externa congestion costs, when the number of parae competitors approaches infinity. Efficiency thus rises with the degree of competition. In contrast, when the number of seria competitors increases, so does a road user s aggregate to (for using a seria road segments), and efficiency then fas with the number of competitors. These findings are in accordance with insights that Economides and Saop (1992) provide into the efficiency effects of mergers between seria and parae firms in network markets. When ooking at competition in network markets, it is therefore important to expicity consider the configuration of the network and the distribution of competitors over that network and in particuar to distinguish between seria and parae competition; i.e., competition between substitutes and compements. Besides competition, aso auctions for the right to operate a to road can be designed so as to improve the overa economic efficiency from private to roads. Enge, Fisher and Gaetovic (1996) for exampe argue how a Net Present Vaue auction may be used to circumvent probems of renegotiation under demand uncertainty. Verhoef (2007) studies how the criterion used for seecting the winning bid in an auction can affect the efficiency of the resuting equiibrium. The cassic criterion of the maximum bid pushes bidders towards the monopoistic profit-maximizing to and capacity, with the associated negative impacts on efficiency, and therefore does not seem to be very attractive from the socia viewpoint. Perhaps surprisingy, when the winning bid is defined as the one that maximizes the use or patronage of the new road, the resut wi correspond with the second-best zero-profit combination of to and capacity for the new ink. And that is the most efficient outcome that one coud reasonaby hope for when there is unpriced congestion esewhere on the network (which is why it is second-best), and no subsidies are granted to private road operators bidding competitivey (which is why a zero-profit constraint appies). Verhoef (2007) derives these resuts for a first toed ink at an exogenous ocation in an otherwise untoed network. A natura foow-up question, addressed in this paper, is whether this patronage auction retains its attractive properties in a more generaized setting. A first generaization is that aso the ocation of the ink to be suppied wi now be part of the auction, because the franchise wi be granted to the bidder that can attract the argest number of users to a new ink, irrespective of its ocation in the network. A second generaization is that we wi now consider a sequence of auctions, each of which can be won by incumbents or entrants, so that entry into the network is introduced endogenousy when new firms make the best bids. There are two natura benchmarks against which we can judge the performance of such a sequence of auctions. A first is a free-entry sequence, for which we assume that at each stage, a new ink is added by the firm who reaizes the highest profits from doing so, and who sets the profit-maximizing capacity and to. 1 A second one is the sequence where at each 1 An anonymous reviewer remarked that this free-entry sequence coud therefore aso be interpreted as a sequence of cassic bid-maximizing auctions at each stage. It is true that at each stage, the resuting capacity addition and to woud be the same for both sequences. But a main difference woud be that under competitive bid-based auctions, the payment of the bid immediatey exhausts the profits, so that a firm woud typicay run into osses after additiona parae capacity is auctioned off at a ater moment (this wi in fact be iustrated ater

7 Private roads: auctions and competition in networks 3 stage, the sociay most desirabe ink is added, with the second-best optima capacity and to. Both benchmarks wi be considered in this paper. It brings us in the ream of sequentia modeing of network evoution, a topic that has recenty been addressed aso by Levinson and Yerra (2006) and Zhang and Levinson (2006). This paper thus studies the efficiency impacts of private roads in initiay unpriced (hence pubic) networks. We aow for capacity and to choice by private operators, and endogenize entry and therewith the degree of competition, aowing for both parae and seria competition. Two institutiona arrangements are considered, namey one in which entry is free and one in which it is aowed ony after winning an auction. A number of simpifying assumptions are made for the dua purposes of keeping the anaysis manageabe and keeping the mode transparent, so that an economic interpretation of the resuts is more easiy given. The main assumptions are the foowing. The congestion externaity forms the ony reevant market faiure. We consider identica road users, and firms that are equipped with identica cost functions for providing road capacity. There are neutra economies of scae in road construction and the congestion technoogy exhibits constant returns to scae (i.e., the trave time functions are homogeneous of degree zero in traffic voume and capacity). Capacity is a continuous variabe, but we wi address quaitativey the question of how resuts might change when capacity woud become discrete, as it is usuay thought to be in reaity. Auctions are perfecty competitive: there are no strategic interactions between bidders during the bidding phase and the winner wi reaize a zero profit from carrying out the bid. Evidenty, each of these assumptions is debatabe empiricay, and may thus offer worthwhie extensions for further research. The present paper deiberatey focuses on this simpified environment, in the hope to derive transparent resuts that are indicative of the main economic forces in this type of probem, which wi remain reevant aso in a more compex setting that aows for some of the compications just mentioned. The paper is organized as foows. Section 2 introduces the mode and the main assumptions underying it, and discusses some anaytica backgrounds. Section 3 describes the numerica version to be used in this paper. Section 4 contains the simuation resuts, and Section 5 concudes. 2. Mode set-up 2.1. Network configuration We wi consider what is probaby the simpest possibe network configuration that aows us to incorporate interactions between both seria and parae roads in a network. This configuration is portrayed in Figure 1. There is a singe market for trips between one origin (A) and one destination (B). The road corridor between these ocations consists of two seria segments a and b, which are connected through an uncongested crossing X. in Figures 6 and 7). For that reason, we wi maintain the terminoogy and interpretation of a free-entry sequence versus a sequence of (patronage) auctions.

8 4 Private roads: auctions and competition in networks Initia network: A Segment a X Segment b B Network with some parae inks added: A Segment a X Segment b B Figure 1. The initia network with untoed inks on segment a and segment b (upper diagram), and a possibe ater configuration after some inks have been added (ower diagram) Initiay, there is ony untoed capacity on both segments; these are the initia inks that wi be denoted as inks a0 and b0. We wi study how new inks are added to these on both segments, under different institutiona regimes. Note that the initia capacities can be set to zero without probem, so that absence of initia untoed capacity is just a specia case of the proposed mode. Each new ink covers either segment a or segment b, and is connected to the same crossing X. Because road users consider parae inks to be perfect substitutes, Wardropian equiibrium conditions appy on both segments individuay. This means that the generaized price faced by users, to be defined beow, must be equaized on a inks on a segment that carry traffic, and cannot be ower on unused inks for that segment (Wardrop, 1952). The ower diagram shows a possibe network configuration after three inks have been added; two on segment a and one on b. The dashing in the drawing aims to refect that one firm has become active on both segments, and a second firm ony on segment a. The excusion, by assumption, of possibe direct roads between A and B serves to maintain the origina network structure with substitute and compement roads; aowing for more structura changes in the network configuration is an interesting generaization for future study. We consider stationary-state congestion and assume that road users are homogeneous. Their inverse demand for traveing between A and B is given by the inverse demand function D(N), in which N denotes the number of trips per unit of time (or traffic fow). The average user cost on a certain ink incudes a variabe costs incurred by the users, incuding trave time, and depends, through congestion, on the ink fow N and the ink capacity K. It is denoted c (N,K ). The generaized price faced by users of a ink, p (N,K ), is equa to the sum of c (N,K ) and a to τ (if evied). Every possibe route r uses two inks, one on each segment, so that the number of possibe routes is equa to the product of the numbers of inks on segments a and b. Equiibrium is characterized by the foowing Wardropian conditions: a, r a, r b, r b, r p N K + p N K D N (, ) (, ) ( ) 0 r r : N 0 r a, r a, r b, r b, r N ( p ( N, K ) + p ( N, K ) D( N) ) = 0 N = r N r (1)

9 Private roads: auctions and competition in networks 5 where N is the vector of route fows. Note that a certain ink may carry users from mutipe routes, so that the generaized price for a ink used by route r may depend on more route fows than just N r (but of course not necessariy on a route fows). The composite superscript s,r denotes the specific ink that route r has on segment s. With (1) satisfied, the generaized prices for a used routes are equaized in equiibrium, and are equa to margina benefits D(N). Because users can freey choose combinations of inks from segments a and b, the equiibrium conditions (1) impy that on both segments the generaized prices for a used inks must be equaized. Assuming that the socia objective is to maximize socia surpus, defined as user benefits minus user cost minus capacity cost, we can next find the sociay optima or firstbest vaues of K andτ by maximizing, subject to (1): N c, S = D( n)dn N c ( N, K ) C ( K ) 0 (2) where C c, is the capacity cost for ink. Because a ink fow N is the sum of a route fows N r for routes that use that ink, and aggregate fow N is the sum of a route fows together, objective (2) can be maximized with respect to a route fows to find the short-run optimaity conditions (these conditions aso appy in the ong-run optimum, in which capacity is aso optimized). This produces first-order Kuhn-Tucker conditions that wi not be written out, because they ook very simiar to conditions (1). The ony difference is that p s,r is repaced by mc s,r : the (short-run) margina user cost on ink s,r. Observe that mc, in turn, is the sum of the generaized average cost c and the margina externa cost mec = N c / N : N c ( N ) c mc = = c + N N N (3) Because p = c + τ, the foowing tos wi consequenty achieve short-run optimaity: 2 c τ = N (4) N These are conventiona Pigouvian to expressions, equa to mec, as can be found in neary every transport economics textbook (e.g. Sma and Verhoef, 2007). Optima investment rues are found by optimizing (2) with respect to ink capacities K, which gives: N c K C = K c, (5) 2 Optima to vectors need not be unique; in the current network, a constant coud for exampe be added to a tos on segment a and subtracted from a tos on segment b without changing the equiibrium, so that (4) does not hod for any ink but the optimum is nevertheess achieved. When demand is not perfecty ineastic, a optima to schedues produce the same aggregate route tos, so that the tota to paid (over the fu trip) by any individua is the same irrespective of which among the possibe optima to patterns is appied. The to rue of (4) is, of course, the most natura and intuitive among these to patterns.

10 6 Private roads: auctions and competition in networks The economic interpretation of (5) is straightforward: the margina benefits of capacity expansion (the eft-hand side), consisting of reduced aggregate user cost, shoud be equa to the margina cost (the right-hand side). A few reativey straightforward manipuations are sufficient to confirm the we-known sef-financing resut of Mohring and Harwitz (1962). 3 This resut impies that when (i) capacity is continuous, (ii) there are neutra economies of scae in road construction (i.e., the margina cost on the right-hand side of (5) is constant), and (iii) the congestion technoogy exhibits constant returns to scae (i.e., the trave time functions are homogeneous of degree zero in traffic fow and capacity), the tota to revenues equas the tota cost of capacity when (4) and (5) are both satisfied for a inks. The optima road network is then exacty sef-financing: the profits π on each ink are a zero. This resut is especiay significant in the context of the present paper, because it means that if free entry of firms, and competition between them, eventuay eads to a zero-profit outcome, this need not be inherenty inconsistent with a first-best equiibrium. The same hods for a sequence of competitive auctions that drives down profits to zero. Nevertheess, because there are many possibe combinations of tos and capacities that produce a zero profit, zero profits are of course not a sufficient condition for optimaity. Throughout this paper, it wi be assumed that the above conditions (i)-(iii), underying the exact sef-financing resut, are satisfied. However, because we wi aow for the continuing existence of unpriced and congestibe initia capacity, the first-best outcome wi generay be unattainabe. The existence of unpriced congestion wi for a parae toed ink cause a downward adjustment on second-best tos compared to Pigouvian tos, so as to reduce congestion spi-overs (e.g., Lévy-Lambert, 1968). In contrast, it typicay creates an upward bias on the to on a seria ink, which is adjusted in an attempt to aso (partiay) internaize downstream or upstream congestion. As shown in Verhoef (2007), the existence of unpriced congestion on parae or seria inks does, however, not affect the second-best investment rues for newy added priced capacity. Consequenty, the investment rue (5) remains vaid for a toed ink with unpriced parae or seria congestion, whie the to rue of (4) does not. The consequence is that second-best investments and pricing then do not generay resut in exact sef-financing of newy added toed capacity, simpy because this woud require (4) and (5) to be both satisfied. Equivaenty, when free entry or auctions cause ongrun profits on toed roads to become zero, whie unpriced initia capacity remains avaiabe, the resuting equiibrium cannot be second-best (which has a non-zero profit or oss). At best, it woud correspond to the second-best zero-profit configuration ( second-best because there is unpriced congestion esewhere in the network; zero-profit because the new capacity 3 The first step is to mutipy both sides of (5) by K. Because, by Euer s theorem, K c / K = N c / N when c is homogenous of degree zero as we assume, the eft-hand side of (5) is then equa to tota to revenues. And, after the said mutipication by K, the right-hand side of (5) gives tota capacity cost when the margina cost of capacity C c, / K is constant. Exact sef-financing thus appies. The Mohring-Harwitz resut is in fact more genera than this, and states that the degree of sef-financing (the ratio of tota revenues and tota capacity cost) is equa to the easticity of the capita cost function with respect to capacity (see aso Sma and Verhoef, 2007).

11 Private roads: auctions and competition in networks 7 is restricted to produce a zero surpus). In our anaysis beow, we wi therefore use both the first-best and the second-best zero-profit configuration as benchmarks for assessing the performance of the free-entry regime and the auctions regime Game-theoretic set-up in the free-entry regime Let us next turn to the assumed game-theoretic set-up for the free-entry regime, for which it is assumed that operators are free to add capacity to the network, and are free in setting tos. Before discussing various aspects of this regime in greater detai, it is usefu to sketch the more genera structure. A sequence of two-stage games is considered, where each two-stage game defines a round in the sequence (the initia equiibrium is round 0 ). The second stage in such a game invoves Bertrand to competition between road operators. The first stage invoves capacity choice for a singe added ink: it is assumed that in each round, ony one ink can be added to the network. Of course there are mutipe candidate road operators and mutipe candidate ocations (i.e., segment a or b) for such an added ink. For each candidate operator-ink combination, the described two stage game wi be soved, and it is assumed that the operator-ink combination that impies the highest profit gain for the associated operator is the one that wi materiaize. We then move to the next round; i.e., the next two-stage game. Note that there is thus fu rationaity within each two-stage game, whie we assume myopia between two-stage games. Let us now turn to the more detaied assumptions and, where needed, their justification. First, we assume that a firms have access to the same technoogy, and face the same user cost functions c (N,K ) and capacity cost functions C c,. Next, to account for the sequentia character of network deveopment in reaity, we choose to consider a sequentia game, with ony one capacity addition in each round, instead of a game where a potentia firms simutaneousy decide how much capacity to add on which segment. The moments at which investments are made are exogenousy determined in our mode. We thus ignore the optima timing of investments; we do this for simpicity and acknowedge it offers an important possibe extension of the present mode. Exogenous timing coud be reevant in reaity when the government woud not aow mutipe road construction projects to be carried out simutaneousy. Between capacity additions, the network configuration is given and the firms then present pay a Nash-Bertrand game when setting their tos. This means that they set their tos τ so as to maximize their profit π, treating as given any other operators tos, as we as a ink capacities. Note that this means that a firm operating more than one ink sets a his tos simutaneousy, so as to maximize his aggregate profit, summed over a his inks together. Bertrand to-setting behaviour of road operators, as assumed here, seems intuitivey more pausibe than a Cournot mode, where payers woud assume that the fows on the competitors inks are fixed. Bertrand competition is therefore common in network modes of competing operators (e.g., De Pama and Lindsey, 2000; De Borger, Proost and Van Dender, 2006). Furthermore, Nash behaviour seems a more neutra starting point than having a Stackeberg eader on the network, but this is another issue that may warrant further study (for

12 8 Private roads: auctions and competition in networks exampe, Ubbes and Verhoef, 2008, compare Nash versus Stackeberg behaviour in a twostage game-theoretic mode of two competing governments suppying toed infrastructure, and find that in their mode the difference between the two types of competition is much more pronounced in the to stage than in the capacity stage). We now turn to the firm s behaviour when considering whether or not to invest and add capacity to the network. In fact, it is not so straightforward to choose an appropriate specification. A strict adherence to Nash behaviour might ead to a mode in which it is assumed that a firm woud not expect other firms to change their tos in response to its own investment even though the addition wi make a non-margina change to the network. But this seems a rather naïve assumption, especiay if it is commony known from earier investments that firms do adjust their tos when the system moves from the one Nash equiibrium to the other. This is why we use the two-stage set-up in each round, which impies that a firm reaizes that after it wi have invested, a new Nash equiibrium in tos wi resut. Each firm, incumbent or entrant, is assumed to cacuate, for both segments of the network, for which eve of investment the increase in its profits between the current and the new Nash equiibrium is maximized. If the firm invests in a certain round, it wi then choose that segment and that capacity eve that produces the highest profit gain. However, ony one of these candidate investments wi be made in each round and we assume that it is the one by the firm that has the highest profit gain from investing in that particuar round. The motivation for this assumption coud be that in absence of entry barriers, the firm expecting the argest profit woud be the most ikey one to invest when ony one addition can be made. When deciding on capacity additions, firms are therefore neary-myopic : when investing, they optimize by ooking no further than the immediate post-investment Nash equiibrium but they do predict this equiibrium correcty. It is important to acknowedge that there is some inconsistency in assuming, on the one hand, that the firm reaizes that, in the second stage, other forms wi change their tos after it has made an investment, and, on the other, assuming that the firm wi nevertheess not set its to and capacity on the new ink as a Stackeberg eader. There are two reasons for accepting this inconsistency. One is that we prefer to eave the consideration of Stackeberg behaviour in investing and to setting for ater study, having Nash behaviour as the natura benchmark. The second is that it seems equay (or even more) inconsistent to assume that a firm behaves as a to-eader when panning an investment, but next vountariy moves to the roe of foower when a next investment is made, by another firm. Finay, note that the assumption of neary-myopic behaviour, rather than forward ooking behaviour, is again consistent with Nash behaviour, in the sense that it prevents firms in our mode from setting capacities strategicay i.e., so as to aso affect capacity choice by future entrants in the network. But it does eave open a question of regret. In particuar, an undesirabe property of a predicted equiibrium sequence of entries woud be that at some moment aong the sequence, one of the firms woud regret earier decisions, because it starts running into osses. We sha see that this does not occur in our mode: profits wi never fa beow zero, which is due to our neutra-scae-economies assumptions. Therefore, athough

13 Private roads: auctions and competition in networks 9 profits wi fa over time, there is never a reason to regret having entered the market. Moreover, we wi see that in the ong-run end-point equiibrium, a profits wi have faen to zero, so that a firms wi have become indifferent with respect to the capacity they chose when making their investments. The assumption we make on the sequentia process is therefore not too harmfu, in the sense that it does not ead to persisting different profitabiity eves for individua firms Auctions The second regime of interest is the auctions regime. For this regime, we assume that there is a sequence of auctions in which the right to buid a singe toed ink on either segment a or b is granted to the firm that makes the best offer. As in the free-entry regime, we thus have a sequence of equiibria, in successive rounds, at the beginning of which the network configuration is changed because a singe ink is added on either segment a or b. Foowing Verhoef (2007), we consider patronage-maximizing auctions, which in his mode reproduce the second-best zero-profit outcome. With this auction, the right to buid and operate a new addition to the network is granted to the firm that offers to carry the highest traffic fow on that new piece of capacity. We assume that bidding firms commit to a particuar combination of capacity and to, and that these impy a eve of patronage for the new capacity which the reguator can cacuate correcty, and use to determine the winning bid. We aso assume that neither the to nor the capacity can be changed when further capacity additions are made to the network ater on. There is, therefore, no direct to competition between firms. The patronage-maximizing auction is profit-exhausting, at east for firms who are not yet active in the network: under competitive auctioning, as we wi assume to appy, it pushes newy entering firms to make an offer that produces a zero profit on the new capacity (Verhoef, 2007). It is important to reaize that every traveer on the new ink carries, as a generaized price, the sum of average user cost and, through zero-profit toing, average capacity cost. When a bid successfuy maximizes the use of the ink, it must have minimized this sum of average user cost and average capacity cost. The auction therefore induces such newy entering operators to enter according to the ong-run cost function as we wi ca it. This means that the post-investment fow on the new ink is served against minimized socia cost. In other words, the first-best investment rue of (5) wi appy. Because neutra scae economies appy, the accompanying to that produces zero profits is aso the first-best to, given in (4), and its vaue is independent of the scae of operations. Therefore, the to eve is the same as it woud be in the ong-run first-best optimum, and the capacity is the one that minimizes socia cost for the post-investment fow eve on the ink. Because ony one right is granted in each round, the set-up induces firms to again behave neary-myopicay, in the sense that they are asked to ony maximize the immediate post-investment patronage of the new capacity. When further capacity additions foow ater on, they shoud keep the to to which they have committed unchanged, but the resuting changes in patronage are not considered to be a vioation of their earier bids.

14 10 Private roads: auctions and competition in networks We again face the question of whether firms may run into osses at a ater stage because of the capacities they chose and the and tos they have committed to. Because new entrants wi aways enter according to the ong-run cost function, and wi do so ony when demand is arge enough to prevent osses from doing so, and because Wardrop equiibrium conditions impy that equa tos on parae inks wi ead to equa trave times and hence equa fow/capacity ratios on these parae inks, earier investors need not fear osses as ong as they have committed to the ong-run first-best to eve. As expained, new entrants, with no capacity from earier investments, wi indeed choose that to eve. But what about a firm that aready has capacity on the seria segment? Because the firm is committed to the to set earier on its seria say, downstream ink, its profit on that downstream ink increases when the use of that ink increases after an investment on the upstream segment raises the equiibrium fow N. As a resut, the firm s aggregate profit, over both segments together, may be maximized at a patronage eve for the new upstream ink that exceeds the eve consistent with entry according to the ong-run cost function. This bid woud then win from bids according to the ong-run cost function, as new entrants woud make. But it may invove a to eve beow the ong-run first-best eve, and wi then ead to negative profits after further additions on both segments, by other firms, woud drive average cost and generaized prices to their ong-run first-best eves on the inks competing with those of the firm under consideration. The firm woud then regret its earier to bid. We assume that firms wi not make such bids, and appy a ower bound on the tos they bid that prevent the investment from becoming oss-generating in the future. 4 Effectivey, this means that a capacity additions wi be according to the ong-run cost function Second-best zero-profit entries Finay, we briefy characterize a sequence of ink additions that we ca the second-best zero-profit entries sequence. This sequence is a reevant benchmark for judging the auctions regime. It invoves a sequence of capacity additions that are chosen such that each addition has the maximum possibe contribution to socia surpus, under the constraint that the to and capacity produce a zero profit on the added capacity at east before any further capacity addition is made to the network. This sequence aows us to verify whether Verhoef s (2007) finding that the patronage-maximizing auction produces the second-best zero-profit road for a 4 In the numerica anaysis beow, this assumption is binding but has ony imited consequences. The ower imit becomes binding in round 4, but ony if inks a1 and a2 woud be operated by the same firm. That firm woud then offer a higher patronage for ink b2 than impied by entry according to the ong-run cost function, namey 132 versus 109, at a sighty ower to eve: versus In round 5, aso an operator with ony a singe ink, namey b1, woud offer a higher patronage for a3 than impied by entry according to the ong-run cost function: 77 versus 12, again at a sighty ower to eve: versus, again, These differences in tos are conceptuay significant, as they indicate that the auctions regime and the second-best zero-profit regime are not formay identica in terms of their resuts if we do not add the assumption that firms wi not bid tos that wi impy osses in a ater phase. But the size of the differences is negigibe in the present mode, especiay because the auctions sequence is aready very cose to the end-state.in round 4.

15 Private roads: auctions and competition in networks 11 first auctioned road, in an exogenousy specified network, remains vaid for a sequence of auctions in a network, which deveops in an endogenous manner through these auctions. 3. A numerica mode 5 We wi iustrate the reative performance of the three regimes of interest by using the resuts from a numerica simuation mode. The mode is very simiar to that used in Verhoef (2007), and the discussion in this section cosey foows his exposition. The mode is rather styized, but sti it is caibrated so as to be representative for peak-hour highway congestion. The average user cost function c takes on the we-known BPR (Bureau of Pubic Roads) form (see, for exampe, Sma and Verhoef, 2007): χ N c( N / K ) = α t + β f 1 (6) K The parameter α represents the vaue of time, and is set at 7.5 in our mode, cose to the vaue (in Euros) currenty used for officia Dutch poicy evauations. The parameter t f represents the free-fow trave time, and is set at 0.25 for both segments a and b, impying a tota trip ength of 60 km for a 120 km/hr highway. Finay, β and χ are parameters that are set at 0.15 and 4, respectivey, which are conventiona vaues for the BPR-function. The units of capacity are chosen such that a conventiona traffic ane corresponds to K =1500. This impies a doubing of trave times at a use eve of around 2400 vehices per ane per hour. This is roughy in accordance to the fow at which, empiricay, trave times are twice their free-fow vaues for a singe highway ane, and the maximum fow on a ane is reached (e.g. Sma and Verhoef, 2007, Fig. 3.3, p. 74). A maximum fow as such, however, is not defined for BPR functions. Note that the BPR function exhibits constant returns to scae in congestion technoogy: the underying trave time function is homogeneous of degree zero in N and K. Next, capacity cost is assumed to be proportiona with capacity, so as to secure neutra scae economies in road construction: C c, ( K ) = γ K (7) The unit price of capacity, γ, is set equa to 3.5 for both segments. Because our unit of time is one hour, this parameter refects the houry capita costs. To derive a vaue from empirica highway construction cost estimates, we have to make an assumption on whether the mode aims to represent stationary traffic conditions throughout a day, or during peak hours ony. Our parameterization concerns the atter. The vaue of 3.5 was derived by dividing the estimated average yeary capita cost of one highway ane kiometre in The Netherands ( 0.2 miion) 6 by 1100 (220 working days times 5 peak hours per working day; assuming two 5 The exposition in this section draws heaviy from Verhoef (2007). 6 With an infinitey-ived highway, without maintenance and an interest rate of 4%, this impies investment costs of 5 mn per ane-km, or 8 mn per ane-mie. This order of magnitude is reasonaby in accordance with

16 12 Private roads: auctions and competition in networks peaks), and next by 1500 (the number of units of capacity corresponding with a standard highway ane), and finay mutipying by 30 (the number of kiometres corresponding with a free-fow trave time of 15 minutes). Ony wefare effects in peak hours are therefore considered in our mode, and it is assumed that off-peak trave is so modest that both the optima off-peak to and the margina benefits of capacity expansion woud be negigibe. To maintain consistency, no reevant wefare effects are assumed to arise outside the peak, and therefore aso no to revenues are supposed to be raised. Because firms are assumed to have access to the same technoogy, the cost functions of (6) and (7) appy, with equa parameters, to a firms incumbents and entrants. Finay, it is assumed that a inear inverse demand function appies: D( N) δ δ N (8) = 0 1 We choose δ 0 = and δ 1 = , together with initia capacities K a0 = K b0 = 1500, to obtain a sufficienty congested benchmark equiibrium, that aows a reasonabe number of further capacity additions in a sequence of investments. The initia equiibrium road use of N = 3500 causes equiibrium trave time t to be around 5.4 times the free-fow trave time t f, which is high but empiricay not unreasonabe (it corresponds to a speed of around 22 km/hr for a 120 km/hr road). The equiibrium demand easticity ε is equa to 0.5 in the initia equiibrium, whie it wi be equa to 0.21 in the second-best zero-profit outcome. Averaging over the extremes of the range of use eves considered in our anaysis, we therefore find a reasonabe Tabe 1 provides the vaues of some of the mode s key variabes in a few benchmark equiibria. First, as a matter of notation, note that in Tabe 1 we use a sighty different doube index to distinguish inks than before, the first character (a or b) sti indicates the segment, but now the second character identifies the individua ink on that segment. A singe index a or b refers to aggregates for a segment, summed over a its inks. The base equiibrium, in the first coumn, is as described above. Because no tos are charged, the operator s profits π on both segments are negative, refecting the capacity cost of the initia capacity. The generaized price in the first-best optimum is neary 50% ower, despite the imposition of a to. This is a consequence of the capacity expansion, which is in reative terms substantiay bigger than the increase in traffic fow. As anticipated, both segments have a zero profit in the first-best optimum. Next, the second-best equiibrium, in which the initia capacity remains unpriced but the new capacity is toed, has a remarkaby high socia surpus. The reative efficiency indicator ω, defined as the increase in socia surpus compared to the base equiibrium, reative to the increase obtained through first-best pricing and capacity, amounts to This is due to the fact that the initia capacity is very ow, so that the expansion of capacity brings figures that Litman (2006) presents for the US. He quotes diverging estimates that suggest that the median investment cost per ane mie woud be in the range of $ 5 10 mn; more than a third exceeds $ 10 mn.

17 Private roads: auctions and competition in networks 13 substantia net benefits. Because the second-best to is beow the margina externa cost on the new capacity, a deficit now resuts. Base equiibrium First-best Second-best Second-best zero-profits Maximum added capacity (zero-profits) S ω π a0, π b π a1, π b π a, π b K a0, K b K a1, K b K a, K b τ a0, τ b τ a1, τ b c a0, c b c a1, c b D = p N a0, N b N a1, N b N a, N b, N Note: Due to the assumed symmetry, equiibrium vaues for segments a and b are equa for a reevant variabes, and are therefore shown in a singe row. Tabe 1. Key characteristics of some benchmark equiibria Imposing a zero-profit condition, as in the second-best zero-profits poicy, 7 avoids such a deficit, but at the expense of a ower reative efficiency (ω = 0.783), and by setting a higher to. In fact, the numerica vaue of the to τ, as we as the fow/capacity ratio and therewith the average user cost c, are equa to their first-best counterparts. The intuition is that, under the constraint that the new capacity be sef-financing, the best thing to do is to set capacity at a eve that impies the minimization of average socia cost (user cost and capacity cost combined). Equiibrium route choice behaviour impies that this aso minimizes the average 7 It is perhaps important to note the difference between the second-best zero-profits entries regime introduced in the previous section, and the second-best zero-profits benchmark discussed here. The former imposes a zeroprofit constraint on newy added capacity on one of the two segments a or b, keeping capacity at the other segment fixed. This eads to a sequence of capacity additions aternatey on segments a and b, as we wi see shorty. The second-best zero-profits benchmark equiibrium in Tabe 1, in contrast, aows for a simutaneous optimization of newy added (toed) capacities on both segments. After optimization, there is of course no scope for further zero-profit capacity additions, so this benchmark invoves a singe static equiibrium, not a sequence.

18 14 Private roads: auctions and competition in networks user cost on the initia capacity, given that new capacity shoud be sef-financing. Therefore, the sum of the average costs that can be affected (a socia cost components except capita cost of the initia capacity) is minimized, and because the generaized price faced by traveers is equa to the resuting average cost eve, the socia surpus is maximized. In this equiibrium, we are therefore adding new capacity according to the ong-run cost function using a socia cost-minimizing ratio of traffic fow and capacity, and a to and a generaized price that woud aso appy in the ong-run first-best optimum. The fina benchmark invoves maximum added capacity (zero-profits), where again the initia capacity is assumed to remain untoed. This equiibrium is incuded because it identifies the maximum eve of new capacity that one coud expect when a zero-profit constraint appies, either because profit-exhausting auctions are used or because free entry of road operators continues unti profits are exhausted. In this equiibrium, reative efficiency ω is, not surprisingy, beow the eve under second-best zero-profits : The to τ on the toed capacity, as we as the generaized price, exceeds the first-best eve because we are no onger operating aong the ong-run cost function. The vaues presented in Tabe 1 are the reevant benchmarks against which to assess the vaues of key variabes at various stages during the three regimes of interest, free entry, auctions and second-best zero-profit entries. This is what we wi do in the next section. 4. Simuation resuts 4.1. Patterns of entry and network growth A first property of interest of the three regimes concerns the pattern of entries, which is characterized not ony by the specific order of additions to segments a and b, respectivey, but aso by the identity of the firm that makes the investment. For the free-entry regime in our numerica mode, we find a very reguar pattern of entries, where in every odd round a new firm enters on segment a, whie in the even round that foows, an addition to segment b is made by that same firm (given the assumed symmetry in the network we can, without oss of generaity, assign the abe a to the segment to which the first firm makes the first addition). Athough this pattern is not the ony possibe equiibrium sequence under free entry, 8 there is a good economic intuition for why it shoud be a pausibe pattern. After the first investment on segment a, by a firm that we wi refer to as firm I (firms wi be numbered consecutivey by roman numbers in order of entry), it is pausibe that segment b is more attractive to enter for a new entrant (firm II) than segment a, because there wi be ess competition and a smaer aggregate capacity on segment b than on a. It is aso immediatey cear that segment b must be more attractive than segment a for firm I: we do not expect a possibe profitabe investment on segment a for firm I if it aready optimized the to and capacity of its added capacity on segment a in round 1. Finay, the incumbent firm I wi enjoy a higher profit increase from adding capacity to segment b than a new entrant does, 8 For a different parameterization of the numerica mode, we for exampe found three successive entries by the first firm entering, on segments a, b and then again a.

19 Private roads: auctions and competition in networks 15 because firm I can maximize the joint profits on both segments, whie a new entrant II wi end up in a situation of seria competition with the incumbent I. Therefore, in round 2, it is pausibe that firm I shoud add capacity to segment b. Next, when firm I optimizes its ink on segment b in round 2, the capacity on segment a is arger than was the capacity on segment b when that same firm I optimized its first investment, on segment a. As a resut, it is ikey that it chooses a arger capacity for its ink on segment b in round 2, than for its ink on segment a in round 1. A potentia new entrant wi, in round 3, therefore find segment a more attractive to enter than segment b. But aso the incumbent firm I woud prefer investing on segment a over segment b in round 3, as it has just optimized its ink on segment b. The question therefore is this: wi a new entrant II foresee greater profits from investing in segment a than the profit increase expected by the incumbent firm I? This cannot be said with certainty. The incumbent firm has the advantage that it can avoid competition between inks on segment a, so it is ikey to end up with higher tos. But the incumbent firm has the disadvantage that new capacity wi reduce demand for its earier capacity on segment a. It imposes, as it were, a pecuniary externaity upon the profitabiity of its own earier capacity. The incumbent firm wi take into account the impied fa in profits on its earier capacity, a oss that a new entrant wi not face. Depending on which of these two effects dominate, it may be the incumbent or a new entrant who invests on segment a in round 3. In our numerica mode, it is the new entrant, whom we wi refer to as firm II. Finay, in round 4, there are six possibe entries to consider: the incumbent firms I and II and a new entrant may each add capacity to segments a or b. Because the aggregate capacity is now arger on segment a whie the tos are ower, each of these firms woud prefer an investment on segment b. The comparison between the profit gains for firms I and II invoves the same trade-off as just described for round 3, and aso for round 4 it resuts in a net advantage for firm II in our numerica mode. The comparison between firm II and a new entrant, firm III, invoves the same trade-off as described above for round 2, and again it resuts in a net advantage for firm II. Firm II therefore invests in segment b in round 4. This pattern of new firms entering on segment a in an odd round and, after that, on segment b in the succeeding even round, is maintained in our numerica mode as far as we have tested it (4 firms; 8 rounds). As stated, this pattern is not the ony possibe equiibrium sequence, but it is a ikey pattern because of the considerations and trade-offs sketched above. Because it resuts in a configuration with parae competition on both segments, it suggests that the inefficient pattern of seria monopoists studied by Sma and Verhoef (2007), with a singe monopoist on each seria segment, wi not easiy arise spontaneousy, in its pure form, as the outcome of free entry of road providers. At the same time, because road users can switch between road providers at the intersection hafway the two segments in our mode, an operator that sets its to on the one segment is ikey to consider the tos on most of the inks on the other segment as given. So it remains to be seen whether the resuting tos tend towards the competitive eve, as suggested by the degree of parae competition on