Network interconnectivity with competition and regulation

Transportation Research Part A 47 (2013) 97–110

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Transportation Research Part A

journal homepage: www.elsevier .com/locate / t ra

Network interconnectivity with competition and regulation

0965-8564/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.tra.2012.10.026

⇑ Corresponding author. Tel.: +44 (0) 1274 23 4773.E-mail addresses: [email protected] (J. McHardy), [email protected] (M. Reynolds), [email protected] (S. Trotter).

1 The development of ‘hub and spoke’ systems in air travel provides one illustration of a widespread change in that operating decision.

Jolian McHardy a,b, Michael Reynolds c,⇑, Stephen Trotter d

a Department of Economics, University of Sheffield, S1 4DT, UKb The Rimini Centre for Economic Analysis, University of Bologna, Italyc School of Social and International Studies, University of Bradford, Bradford, UKd Centre for Economic Policy, University of Hull, HU6 7RX, UK

a r t i c l e i n f o

Article history:Received 29 May 2012Accepted 9 October 2012

Keywords:Network interconnectivityMonopolyCompetitionRegulation

a b s t r a c t

A simple theoretical network model is introduced to investigate the problem of networkinterconnection. Prices, profits and welfare are compared under welfare maximisation, net-work monopoly and a scenario with competition over one part of the network. Given thatinducing actual competition may bring disbenefits such as cost duplication and co-ordina-tion costs, we also explore the possibility of a regulator using the threat of entry on a sec-tion of the monopoly network in order to bring about the socially preferred level ofinterconnectivity. We show that there are feasible parameter values for which such athreat is plausible.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Transport networks typically offer travellers more than one way to get from A to B with the directness of the origin–destination route choices available depending upon the extent to which origin–destinations are joined up or interconnected.A fully interconnected transport network would have the advantage of offering direct origin–destination travel (e.g. A to Bnot via C) for all journeys with obvious benefits for travellers but it does not take much imagination to see that in most casessuch a scenario is unlikely to be optimal. The relevant question is then not so much whether a particular transport systemdelivers a fully interconnected network, but rather whether it provides a degree of connectivity close to the socially optimallevel. A regime offering a suboptimal level of interconnectivity may result in travellers experiencing overly long andcircuitous travel options and excessive congestion. Whether a transport system will provide a socially optimal network willdepend on a number of factors, including the operator’s objective function and the relative costs of, and demands for, thedifferent links in the network.1 From the consumers’ side, different routes may provide complementary services for sometravellers and substitutes for others.

In recent years – and especially since the development of computer systems and the internet – there has been a renewedinterest in issues of competition and integration within markets for complementary and substitute products, which hasmotivated research into networks with varying degrees of connectivity. The initial technological development saw the re-sults of Cournot (1838) concerning complementary and substitute goods being extended by Economides and Salop (1992)and Matutes and Regibeau (1992), amongst others. Since then a number of studies have applied the analysis to policy deci-sions, such as studies on computer operating systems (Gisser and Allen, 2001; McHardy, 2006) and video games (Clementsand Ohashi, 2005). Gabszewicz et al. (2001) consider price equilibria where products are each indivisible but their jointconsumption results in a higher utility than that achieved when the products are consumed in isolation. Denicolo (2000)

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98 J. McHardy et al. / Transportation Research Part A 47 (2013) 97–110

considers compatibility within a bundling framework and Zhou (2003) looks at the level of access to telecommunicationsmarkets. In the transport arena, Else and James (1994) have looked at railways and McHardy and Trotter (2006) at airlines.Additionally, Newbery (1999) explores a number of issues regarding network utilities and Baumol and Sidak (1994) considerinterconnection and how to encourage entry, amongst other things, in local telephony.

Much of the literature deals with incentives in the network set-ups and resulting pricing. In particular, there is a greatdeal of research that discusses access pricing under various structures – see Laffont and Tirole (1994, 1996) and Armstronget al. (1996, 1998). These papers tend to focus on networks traditionally seen as natural monopolies, and consider how, atwhat price and where it is best to introduce competition to ensure beneficial results. This paper focuses on when and whereto encourage competition to bring about a network set-up that would be preferred by society. The model has two possiblenetwork configurations that can be selected by the network operator(s). One is fully connected and therefore more expen-sive, but allows quicker access; the second is cheaper but incomplete, with greater use of indirect connections. The paperconsiders different types of network operator, and situations where ownership regimes provide a fully-connected networkare compared. Using a simple transport network as an example, the paper explores the effects of permitting entry by otheroperators when the network is operated by an incumbent monopolist subject to a regulator, and how the regulator can useentry to achieve the socially desirable provision of the network.

The model in this paper has a number of differences and similarities with other areas of network theory, not just the ac-cess pricing literature. Interconnection is applicable but there are no specific externalities attached to either of the networkconfigurations here. Nor are there any issues of compatibility. The network operator chooses the configuration of the net-work and the network will function correctly. This immediately differentiates this model from the network externalities lit-erature – such as the seminal paper by Katz and Shapiro (1985), which considers the presence of network externalities andhow this impacts upon competition and compatibility. However, their finding that firms may choose complete compatibilityto the detriment of consumer surplus is relevant when considering the conclusions of this paper. Additionally, Lambertiniand Orsini (2001), which includes network externalities and finds an oversupply of quality compared to the social optimum,is of interest when it comes to discussing the results in the final part of the present paper.

The following section introduces a simple three-sector network model. Section 3 considers the benchmark scenario of thewelfare-maximising social planner. Section 4 introduces a profit-maximising network monopolist, and considers the circum-stances under which the network monopolist and social planner’s choices over a complete or incomplete network eithercoincide (thus eliminating any need for regulation) or differ. Section 5 considers the possibility of entry on a sub-sectionof the network. Section 6 summarises the results and discusses their policy implications.

2. The network

Consider the simple network shown in Fig. 1. Let it represent (part of) the transport network in a circular city, and assumethere are three public transport services along routes 1, 2 and 3 between the three origins and destinations r, s and t. Thisnetwork can serve demands for radial travel between the city centre (s) and the perimeter of the city (points r and t) as wellas non-diameter chord travel (between r and t). This simple network, therefore, allows for direct travel between points onthe perimeter along route 1 as well as indirect travel between these points using routes 2 and 3. For passengers wishing totravel between r and t, the combined services along routes 2 and 3 (which are perfect complements) now provide a substi-tute for route 1.

We now make specific a key assumption of the model.

Assumption 1. Services on radial routes 2 and 3 are always provided.

In the next section we impose restrictions on key parameter values to ensure that Assumption 1 is justified within theappropriate maximising framework for each regime (so it is an equilibrium outcome of the model). However, the purposeof Assumption 1 is to frame the question of network interconnectivity in terms of whether or not the direct cross-city (chord)service (route 1) is provided. A key question is then whether or not a monopoly public transport provider would (would not)

Fig. 1. A simple network.

J. McHardy et al. / Transportation Research Part A 47 (2013) 97–110 99

provide the complete network when it is (is not) socially optimal to do so. In the situation where there is a mismatch be-tween the network monopoly decision and that of a social planner we consider how using entry or the threat of entry onthe network may help align the objectives of the public transport provider with society’s interests.

By definition, routes 2 and 3 are of equal length (the radius of the circular city), which for simplicity is normalised tounity. Therefore, route 1 is the chord joining r and t with length x. In this framework, x lies in the open interval 0 < x < 2.

Suppose the relevant public transport provider charges a fare, fi, for travel on route i (i = 1,2,3). Let the demand for directtravel along route i be given by:

2 It isgeneral

3 Thr4 Thi

Q 1 ¼ a� x� f1; ð1aÞQ j ¼ b� 1� fj; ðj ¼ 2;3Þ: ð1bÞ

where a and b are positive constants. As the city has a radius of one then to ensure that (1a) and (1b) are positive, at least atzero fares, and given 0 < x < 2, let:

a P 2; b > 1: ð2Þ

For simplicity we assume that the psychological passenger cost per unit distance is unity; thus for passengers on routes 2and 3 (which have unit length) the relevant cost is also unity, whilst for passengers on route 1 it is x.2 It follows that the gen-eralised cost of direct travel along mn (m – n = r,s, t), Gmn, is given by:

Grt ¼ f1 þ x; Grs ¼ f2 þ 1; Gst ¼ f3 þ 1: ð3Þ

Clearly each journey can be undertaken directly using one route or indirectly using the remaining two routes. This paperallows the provision of services along route 1 to be an option for the relevant service provider. In order to compare the gainsto the relevant operator with and without services on route 1 and to incorporate the fact that pricing decisions on route 1must be undertaken in the knowledge that too high a fare may divert passengers onto routes 2 and 3, it is necessary to con-sider the demands for the alternative journey rt via s. Assuming that there is no interchange penalty, the generalised cost fora passenger making indirect travel along mn via l (m – n – l = r,s, t), Gmln, is given by:

Grst ¼ 2þ f2 þ f3; ð4aÞGstr ¼ 1þ xþ f1 þ f3; ð4bÞGtrs ¼ 1þ xþ f1 þ f2: ð4cÞ

If f1 is prohibitively high or services on route 1 are not provided, then all rt travel is diverted through routes 2 and 3, andtravellers would face generalised cost (4a). Using the demand densities from (1) and the generalised cost (4a) the total de-mand for travel on route j (own route-specific demand and indirect rst demand), bQ j, is given by:3

bQ j ¼ aþ b� 3� 2f j � fk; ðj – k ¼ 2;3Þ: ð5Þ

In terms of the cost structure of the model, it is assumed that public transport provision on a route has a zero marginalcost per passenger; however, there is a non-zero operating cost per unit distance.4 Setting F as the operating cost per unitdistance, it follows that the operating cost for routes 2 and 3 (which have unit length) is F whilst for route 1 the operating costis Fx.

3. A first-best social planner

Before considering the conditions under which a first-best social planner would choose to provide services along route 1,we construct expressions for the consumer surplus associated with each individual route. With the social planner engagingin marginal-cost pricing, our assumption of zero marginal cost implies a fare of zero on each route:

f Si ¼ 0; ð8i ¼ 1;2;3Þ: ð6Þ

Consumer surplus on each route then becomes:

CS1 ¼

12ða� xÞ2; ð7aÞ

CS2 ¼ CS

3 ¼12ðb� 1Þ2: ð7bÞ

In order for the framework to be consistent with route 1 being the marginal route for the purposes of this paper, we im-pose constraints on the parameters of the model.

also assumed that all services operate at an equal (constant) speed; hence, there is no need to introduce a separate time–cost parameter in theised travel cost.oughout the paper, terms indicated with a ‘‘^’’ relate to the incomplete network – i.e. excluding route 1.s means that there are no capital costs in the model.


Lemma 1. 5 The following constraints on the values a, b, and x are sufficient to ensure that at the socially optimal prices the consumersurplus on the incomplete network with routes 2 and 3 provided cCSS

23 is always strictly greater than the level of consumer surplus onthe incomplete network with routes 1 and j ðj ¼ 2;3Þ cCSS

1j.

5 The6 In t

a > b� x� 1: ð8Þ

This justifies our assumption that routes 2 and 3 are always provided; implying that the demand on the radial routes isdenser.6 Given this assumption, the decision to provide route 1 is now of particular interest. Had route 1 been an importantroute to the traveller then finding it to be provided would be unremarkable.

We now consider the conditions under which a social planner would provide route 1. It follows from (6) that networkrevenue under the social planner is zero, so welfare is derived solely from consumer surplus. In the case where the socialplanner provides services on route 1, total consumer surplus across the system, CS, is then:

CS ¼X3

i¼1

CSSi ¼

12ða� 2Þ2 þ ðb� 1Þ2: ð9Þ

Welfare under the social planner with route 1 provided, WS, is consumer surplus minus the operating costs of the threeroutes:

WS ¼ CS � ð2þ xÞF ¼ 12ða� xÞ2 þ ðb� 1Þ2 � ð2þ xÞF: ð10Þ

If the social planner does not provide route 1 then the consumer surplus is measured in relation to the alternative demandsin (5). Welfare, cW S, is then:

cW S ¼ 12ða� xÞ2 þ ðb� 1Þ2 � 2F: ð11Þ

We are now able to consider the social planner’s optimal network provision.

Proposition 1. The welfare-maximising social planner would prefer to provide a complete network if the constant operating costper unit distance satisfies the inequality:

F <12x½ða� xÞ2 � ða� 2Þ2�: ð12Þ

Proposition 2. If the constant operating cost per unit distance is zero, the welfare maximising social planner will always provide acomplete network.

Setting (12) as an equality, we define the social planner’s threshold cost for providing a complete network.

Lemma 2. The level of constant operating costs (per unit distance) which makes the social planner indifferent between providing aservice on route 1 and not providing a service, eF S:

eF S ¼ 12x½ða� xÞ2 � ða� 2Þ2�: ð13Þ

Lemma 2 provides a useful benchmark level of operating cost. If for some level of F below eF S a regime does not provideroute 1 then the regime’s network is socially suboptimal.

Proposition 3. The welfare maximising social planner would provide a complete network for an increasing (decreasing) range ofconstant operating costs per unit distance as a (x) rises.

Propositions 2 and 3 show that we have a well-behaved model with intuitive results. As the length of the direct route, x,falls – recall that the length of the indirect route along 2 and 3 remains constant – then the benefit that society gets from theexistence of route 1 increases. Equally, an increase in a means that there are more travellers moving along route 1, so theabsolute gains from providing the direct route increase.

4. Network monopoly

The case of network monopoly is more complicated than that of the social planner, since the monopolist has to select thenetwork structure (whether or not to provide route 1) and the optimum level of fares which are interdependent, whilst theplanner’s price policy is independent of choice of routes (given zero marginal cost). However, matters are made morestraightforward by the symmetrical nature of routes 2 and 3. It follows that whatever the monopolist’s choice of network

proofs to all Lemmas and Propositions are reported in Appendix A.he context of the transport example, this is intuitively appealing as we would expect more travellers to pass through a city centre.


configuration and fare structure, the optimal fare for route 2 will be the same as that for route 3. Dealing with the scenario inwhich the network monopolist supplies a complete network, the general expression for network profit, pM, is:

7 Con

pM ¼ f1ða� x� f1Þ þ 2f jðb� 1� fjÞð2þ xÞF; ð14Þ

where fj is the common fare on routes 2 and 3. Profit maximisation yields the following optimal network monopoly faresunder a complete network:

f1 ¼a� x

2; ð15aÞ

fj ¼b� 1

2: ð15bÞ

Substituting (15) into (14) gives the reduced-form network monopoly profit:

pM ¼ 14ða� xÞ2 þ 1

2ðb� 1Þ2 � ð2þ xÞF: ð16Þ

If, however, the network monopolist supplies an incomplete network (omitting route 1), the general expression for networkprofit is p̂M:

p̂M ¼ 2f jðaþ b� 3� 3f jÞ � 2F: ð17Þ

In this case profit maximisation yields the following optimal network monopoly fares under an incomplete network:

f̂ j ¼16ðaþ b� 3Þ; ðj ¼ 2;3Þ: ð18Þ

Substituting (18) into (17) gives the reduced-form expression for maximum monopoly profit with an incomplete network:

p̂M ¼ 16ðaþ b� 3Þ2 � 2F: ð19Þ

Proposition 4. The network monopolist would strictly prefer a complete network if the constant operating cost per unit distancesatisfies the inequality:

F <a2 þ 12a� 6axþ 3x2 þ 4b2 � 12� 4ab

12x: ð20Þ

The expression on the right-hand side of (20) gives us the network monopolist’s threshold operating cost below which afull network will be supplied, analogous to eF S for the social planner in the previous section. Like the social planner, the net-work monopolist would always find it more profitable to supply a complete network if there was no operating cost:

Corollary 1. If the constant operating cost per unit distance is zero, the network monopolist will always provide a completenetwork.

We now consider the monopolist’s decision when it faces the social planner’s threshold operating cost (eF S).

Proposition 5. When faced with the social planner’s threshold operating cost then the monopolist’s benefits from providing acomplete network increase as b and x rise, and as a falls.

Proposition 6. The monopolist will provide route 1 when the social planner does; however, when the social planner does not pro-vide a complete network it is possible that the monopolist will.

Proposition 5 establishes that the parameters for which the monopolist and social planner provide a complete network donot always match. Figs. 2 and 3 show this graphically by plotting the social planner’s and the monopolist’s threshold oper-ating costs. The area on and below the lines shows values for which each would provide a complete network; any point thatis below both lines represents a situation where the monopolist and social planner agree in providing a complete network.7

As stated in Proposition 6, Fig. 2 shows the monopolist provides route 1 for some values that the social planner would notand as a increases the monopolist becomes more willing to provide route 1 for values for which the social planner wouldprefer an incomplete network. Fig. 3 further underlines this by showing what happens if a takes on its largest possible valuein relation to b, that is if (8) becomes an equality.

versely, any point above both lines shows when the monopolist and social planner agree in providing an incomplete network.

Fig. 2. Network with a = 5 and x = 0.5.

Fig. 3. Network with a = b � 1 � x and x = 0.5.


5. The impact of an entrant

This section considers the effect on a network monopolist’s fare structure and network configuration when a regulatorallows rival firms to enter the network.8 To emphasise that the network monopolist now faces the possibility of entry on partof its network we shall refer to it as an incumbent. Initially we look at the effect on network provision and the incumbent’sbehaviour when entry is allowed on to route 1. Then we move on to look at entry on route 2 – due to the symmetry of the model,the analysis would, of course, apply equally to route 3 – and pose the question ‘‘will the introduction of a rival on route 2 causethe incumbent to provide a complete network (when it was previously incomplete)?’’, by considering the case in which entry onroute 2 leads to a Cournot quantity game on that route.9 We investigate whether the incumbent can use its provision of route 1as a strategic tool to reduce profitability on route 2 and hence exclude the entrant(s).

The general expression for the profit of an incumbent providing a complete network, using (1), is:

8 Thi9 In b

pI ¼ qI1ða� x� Q 1Þ þ f2ðb� 1� f2Þ þ f3ðb� 1� f3Þ � Fð2þ xÞ: ð21Þ

s is not a free-entry model, instead the regulator is able to set the number of entrants.oth cases the analysis is restricted to when the operating costs of operation are low enough to accommodate n-firm entry under the Cournot regime.


The profit function for an entrant k on route 1 is:

pEk ¼ qEk1 ða� x� Q 1Þ � Fx; ð22Þ

where qEk1 is the entrant’s output. This assumes that the entrant faces the same operating cost as the incumbent.

If nE1 is the total number of entrants on route 1 then total output on route 1, Q1, will be the sum of the quantities produced

by all firms and can be defined by:

Q 1 ¼ qI1 þ nE

1qEk1 : ð23Þ

Profit maximisation implies the fares given in (15b) and with the Cournot assumption on route 1 the equilibrium fare is:

f1 ¼ða� xÞð1þ n1Þ

; ð24Þ

where n1 ¼ 1þ nE1. Calculating firm outputs and using (23) before substituting (24) into (21) and (22) gives profits:

pI ¼ ða� xÞ2

ð1þ n1Þ2þ ðb� 1Þ2

2� Fð2þ xÞ; ð25aÞ

pEk ¼ ða� xÞ2

ð1þ n1Þ2� Fx: ð25bÞ

If, on the other hand, the incumbent chooses not to provide services on route 1, but the entrant(s) do, then we do not get theswitch in demands from Eqs. (1)–(5) that we saw previously; the rival firms offer route 1 so the incumbent’s profit becomes:

p̂I ¼ ðb� 1Þ2

2� 2F: ð26Þ

Profit for an entrant i on route 2 is:

p̂Ek ¼ qEk1 ða� x� Q 1Þ � Fx: ð27Þ

As the incumbent is no longer supplying route 1 then qI1 ¼ 0, so bQ 1 can be defined as:

bQ 1 ¼ nE1qEk

1 : ð28Þ

Profit maximisation gives the equilibrium fare of the entrant:

f̂ 1 ¼a� x

1þ nE1

: ð29Þ

Calculating entrant output, then using this and (29) in (27) gives profit:

p̂Ek ¼ ða� xÞ2

ð1þ nE1Þ

2 � Fx: ð30Þ

Proposition 7. The entrant(s) will provide route 1 for higher levels of the constant operating cost per unit distance than theincumbent.

As the entrant always provides route 1 when the incumbent does then we can concentrate on the route 1 entrant’sthreshold operating cost as it determines when a complete network is provided, and we can compare this with the socialplanner’s preferred provision of route 1.

Proposition 8. Entry on route 1 can lead to the provision of a complete network that more closely matches the provision of thesocial planner than the monopolist’s provision.

Fig. 4 shows the entrant’s threshold operating cost just below the social planner’s threshold operating cost. This meansthat, with entry on route 1, there is the possibility that an incomplete network may be provided when a social planner wouldprefer a complete network; contrasting with the previous result (in Proposition 6) where the monopolist would provideroute 1 for a larger range of parameter values than the social planner would. Additionally, Fig. 4 highlights the possibilitythat the entrant’s threshold operating cost is very close in value to the social planner’s threshold operating cost, so thereis a greater range of parameter values for which the entrant and social planner agree on the decision to provide a completenetwork when compared to the monopolist. Fig. 5 looks at a higher level of a and shows that, again, for the most part, theentrant would provide a level of network provision in agreement with the social planner for a larger range of parameter

Fig. 4. Network with a = 5, x = 0.5 and n = 1.


values that the monopolist. However, now the entrant would provide a complete network when the social planner would notand it is also possible at lower levels of b that the entrant would provide a complete network for values that the monopolistwould not.

Route 1 is not the only viable place for the introduction of entrants and it is possible to investigate what happens if com-petition is allowed on one of the radial routes – route 2 for convenience.

The general expression for the profit of the incumbent providing a complete network, using (1), is:

pI ¼ f1ða� x� f1Þ þ qI2ðb� 1� Q 3Þ þ f3ðb� 1� f3Þ � Fð2þ xÞ; ð31Þ

where qI2 is the demand supplied by the incumbent. If nE

3 is the total number of entrants then total output for route 3, Q2, canbe defined by:

Q2 ¼ qI2 þ nE

2qEk2 ; ð32Þ

where qEk2 is the demand supplied by each entrant. Profit for the kth entrant becomes:

pEk ¼ qEk2 ðb� 1� Q 2Þ � F: ð33Þ

Profit maximisation implies the first-order conditions (15a) and (15b) for j = 2 and given the Cournot assumption, on route 2the equilibrium fare is:




f2 ¼b� 11þ n2

; ð34Þ

where n2 ¼ 1þ nE2.

Calculating firm outputs then using this, (32) and (34) in (31) gives profits:

pI ¼ð1þ n2Þ2ða� xÞ2 þ n2

2 þ 2n2 þ 5� �

ðb� 1Þ2

4ð1þ n2Þ2� f ð2þ xÞ; ð35aÞ

pEk ¼ ðb� 1Þ2

ð1þ n2Þ2� F: ð35bÞ

If, on the other hand, the incumbent chooses not to provide services on route 1, demands on routes 2 and 3 become inter-linked by the diverted route 1 passengers. The profits for the incumbent and entrant, respectively, are:

p̂I ¼ qI2

2ðaþ b� 3� f2 � Q 2Þ þ f3ðaþ b� 3� f2 � 2f 3Þ � 2F; ð36aÞ


Fig. 8. Network with where n = 5, x = 1.6, and a = 8.


p̂Ek ¼ qEk2

2ðaþ b� 3� f2 � Q 2Þ � F: ð36bÞ

Maximising incumbent and entrant profits with respect to output on route 2, then substituting the values of qI2 and Q2

into (36a), before again maximising and solving gives the fare for route 2:

f3 ¼n2ð6n2 þ 5Þðaþ b� 3Þ

4ð4n22 þ 6n2 þ 3Þ : ð37Þ

Calculating firm outputs, and using (15), (32) and (37) in (36) yields profits for the incumbent and entrant, respectively:

p̂I ¼16n4

2 þ 96n32 þ 120n2

2 þ 64n2 þ 13� �

ðaþ b� 3Þ2

8 4n22 þ 6n2 þ 3

� �2 � 2F; ð38aÞ

p̂Ek ¼ n2ð6n2 þ 5Þ2ðaþ b� 3Þ2

8ð4n22 þ 6n2 þ 3Þ2

� F: ð38bÞ

Proposition 9. Entry on route 2 (or 3) can lead to an increased range of parameter values over which the provision of a completenetwork matches the choice of the social planner when compared to than the monopoly outcome.

At first glance, it might seem that the incumbent would provide route 1 as a strategic tool against the entry on route 2.What actually happens is that the entry on route 2 causes the price that the incumbent can charge on route 1 to fall and thisreduces the range of values over which the incumbent provides a complete network as shown in Figs. 6 and 7. However, itcan be seen that with entry on route 2 there is the possibility that the incumbent will not provide route 1 when the socialplanner would prefer it to.

Route 1 may be the preferable place to introduce entry as this is the route that completes the network; however this maynot always be the case.

Proposition 10. The welfare arising from complete network provision with entry on route 2 (or 3) can be larger than the welfarefrom complete network provision with entry on route 1, when all firms face the social planner’s threshold operating cost.

A regulator could introduce entry on route 3 in preference to entry on route 1.

6. Conclusions

This paper has explored the consequences for the services provided on a simple transport network of different objectiveson the part of the organisation operating that network. In particular we have compared the behaviour of a social plannerwith that of a (private) profit-maximising firm, and have investigated the circumstances under which different numbersof services over a simple circular city network would be provided. Although the services they provide are often the same,


even where the private operator is an unconstrained network monopolist, it is possible that there may be ‘over provision’ of acomplete network: whereas for some values of the parameters in the model both types of operator would provide a completenetwork, for other values the monopolist would provide this when the social planner would not (the converse never holds).This result is similar to those of Katz and Shapiro (1985) and Lambertini and Orsini (2001), although their models differ fromours.

In practice it is likely that a private monopoly operator would be subject to some form of regulation, and Section 5 of thepaper considered the case of an incumbent monopolist subject to regulation and the threat of entry. We find two ways inwhich a regulator can ensure that a complete network is offered when it would be socially beneficial to do so. Firstly, theregulator can take direct action by allowing entry on a route that the incumbent operator is not supplying (route 1 in ourexample); this may prompt the incumbent to provide a complete network, as the social planner would have done. Secondly,if allowing entry on the non-supplied route (route 1) is not plausible or desirable then the regulator can also induce the socialplanner’s preference of a complete network by allowing entry on a route already being provided (route 2 or 3 in our case).Indeed it is possible that this form of entry that leads to a complete network may provide a greater welfare than the firstmethod (route 1 entry).

Permitting entry is not risk-free, however. Section 5 has shown that it may lead to either too much or too little provisionin total (by the incumbent and the entrant combined), depending on the particular values of the parameters in the model.Thus a cautious approach is needed: the regulator must check that the appropriate conditions for the introduction of entryexist, and continue to monitor the industry if entry is allowed. None of this is likely to be easy – the regulator may also haveto decide where to permit entry and how many firms to allow in – but there are clear welfare gains to be made if the reg-ulator gets it right.

Acknowledgements

An earlier version of this paper was presented at the UTSG annual conference and we are grateful to participants there,and to Peter Else, for useful comments. The usual caveat applies.

Appendix A

Proof of Lemma 1. Given the socially optimal pricing (6), consumer surplus on the incomplete network excluding route 1 isgiven by:

cCSS23 ¼ ½b� 1þmaxð0;a� 2Þ�2; ðA:1aÞ

and consumer surplus on the incomplete network excluding route j (j = 2,3) is given by:

cCSS1j ¼

12½a� xþmaxð0;b� x� 1Þ�2 þ 1

2½b� 1þmaxð0;b� x� 1Þ�2: � ðA:1bÞ

Proof of Proposition 1. Subtracting (11) from (10) and rearranging in terms of F gives (12). h

Proof of Proposition 2. Given (2) and 0 < x < 2 we can see that all the elements of (12) are non-negative; that is (a � x)2 > 0,(a � 2)2 P 0, and 2x > 0. We can also see from (2) and from 0 < x < 2 that (a � x)2 > (a � 2)2. h

Proof of Proposition 3. Partially differentiating the bracketed part of the right-hand side of (13) with respect to x and a,respectively, gives:

@eF S

@x¼ x2 � 4aþ 4

2x2 ; ðA:2aÞ

@eF S

@a¼ 2� x

x: ðA:2bÞ

Using (2) it can be shown that (A.2a) is non-positive whilst (A.2b) is always strictly positive. h

Proof of Proposition 4. Subtracting (19) from (16) gives:

pM � p̂M ¼ a2 þ 12a� 6axþ 3x2 þ 4b2 � 4ab� 12� 12Fx: ðA:3Þ

It is straightforward to show that the right-hand side of this equation will be positive as long as F is less than the criticalvalue in (20). h


Proof of Proposition 5. The monopolist is indifferent between supplying a complete network and an incomplete networkwith operating cost, eF S, if #MðeF SÞ ¼ pMðeF SÞ � p̂MðeF SÞ ¼ 0. Using (13), (16) and (19) gives:

#MðeF SÞ ¼ a2 þ 6ax� 3x2 þ 4b2 þ 12� 12a� 4ab: ðA:4Þ

Partially differentiating (A.4) with respect to a, b and x gives, respectively:

@#MðeF SÞ@a

¼ 2ðaþ 3x� 6� 2bÞ; ðA:5aÞ

@#MðeF SÞ@b

¼ 4ð2b� aÞ; ðA:5bÞ

@#MðeF SÞ@x

¼ 6ða� xÞ: ðA:5cÞ

Using (2) it can seen that (A.5a) is always positive, so (A.4) is increasing in x. Given (8) then (A.5b) is negative and (A.5c) ispositive, so (A.4) is decreasing in a and increasing in b. h

Proof of Proposition 6. This proof is readily verified by using simulations in the relevant range of a, b, and x. Given (2) and(8) we can show that 6ax > 3x2 and 4b2 + a2 + 12 > 4a b + 12a so (A.4) is always positive. h

Proof of Proposition 7. The incumbent’s threshold operating cost in the provision of route 1, calculated by subtracting (26)from (25a), is:

eF I ¼ ða� xÞ2

ð1þ n1Þ2x: ðA:6Þ

The incumbent would thus provide route 1 if:

eF I <ða� xÞ2

ð1þ n1Þ2x: ðA:7Þ

The entrant’s threshold operating cost if the incumbent did not provide route 1 would, from (30), be:

eF E ¼ ða� xÞ2

1þ nE1

� �2x: ðA:8Þ

The entrant would thus provide route 1 when the incumbent does not if:

eF E <ða� xÞ2

1þ nE1

� �2x: ðA:9Þ

As nE1 < n1 we can see that (A.6) would always be smaller than (A.8). h

Proof of Proposition 8. Figs. 4 and 5 depict simulations of (13), (20) and (A.8) indicating a non-empty parameter set underwhich entry on route 1 brings about the provision of a complete network, completing the proof. h

Proof of Proposition 9. Calculating the net gain to the incumbent from offering a complete network relative to an incom-plete network by subtracting (35a) from (38a) and solve for F, gives:

eF IZ ¼ð1þ n2Þ2ða� xÞ2 þ n2

2 þ 2n2 þ 5� �

ðb� 1Þ2

4xð1þ n2Þ2�

16n42 þ 96n3

2 þ 120n22 þ 64n2 þ 13

� �ðaþ b� 3Þ2

8x 4n22 þ 6n2 þ 3

� �2 : ðA:10Þ

Figs. 6 and 7 depict simulations of (13), (20) and (A.10) indicating a non-empty parameter set under which entry on route 2(or 3) increases profit for the incumbent by moving to a complete network from an incomplete network. h

Proof of Proposition 10. Using (15b) and (24) in (1) gives the following outputs:

Q11 ¼

n1ða� xÞð1þ n1Þ

; ðA:11aÞ

Q12 ¼ Q 1

3 ¼a� 1

2: ðA:11bÞ


Welfare for each route can be calculated using:

Wi ¼ fiQ i þ12ðc � fiÞQ i � TFi; ðA:12Þ

where c is the y-axis intercept on the demand curve and TFi is the total constant operating cost associated with running theroute. Substituting the relevant values for each route into (A.12) gives the welfare for each route:

W11 ¼

3n1ða� xÞ2

2ð1þ n1Þ2� n1Fx; ðA:13aÞ

W12 ¼W1

3 ¼3ða� 1Þ2

8� F: ðA:13bÞ

Summing the elements of (A13) together gives total welfare:

W1T ¼

3n1ða� xÞ2

2ð1þ n1Þ2þ 3ðb� 1Þ2

8� Fð2þ n1FÞ: ðA:14Þ

If the incumbent does not enter route 1 then welfare becomes:

W1T ¼

3nE1ða� xÞ2

2 1þ nE1

� �2 þ3ðb� 1Þ2

8� F 2þ nE

1F� �

: ðA:15Þ

Now let us consider when there is entry on route 3. Using (15) and (34) in (1) gives the following outputs:

Q 21 ¼ða� xÞ

2; ðA:16aÞ

Q 22 ¼ðb� 1Þ

2; ðA:16bÞ

Q 23 ¼

n2ðb� 1Þ1þ n2

: ðA:16cÞ

Substituting the relevant values for each route into (A.12) gives the welfare for each route:

W21 ¼

3ða� xÞ2

8� Fx; ðA:17aÞ

W22 ¼

3ðb� 1Þ2

8� F; ðA:17bÞ

W23 ¼

3n3ðb� 1Þ2

2ð1þ n3Þ2� nF: ðA:17cÞ

Summing (A17) together gives total welfare:

W2T ¼

3ða� xÞ2 þ 3ðb� 1Þ2

8þ 3n2ðb� 1Þ2

2ð1þ n2Þ2� Fðxþ 1þ n2Þ: ðA:18Þ

If the incumbent does not enter route 1 then welfare becomes:

W3T ¼ð3ða� xÞ2Þ þ 3ðb� 1Þ2

8þ 3nE

2ðb� 1Þ2

2 1þ nE2

� �2 � F xþ 1þ nE2

� �: ðA:19Þ

Producing simulations of (A.14) and (A.19) using eF S as the operating cost gives Fig. 8. h

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Network interconnectivity with competition and regulation