Railway restructuring and organizational choice: network quality and welfare impacts

An Erratum to this article was published on 01 February 2017

Abstract

This paper compares alternative ways of structuring competition in a railway system: vertical separation (VS) and horizontal separation (HS). We compare each structure in terms of its impact on network quality, consumer surplus and social welfare. To do so, we use a two-stage game under HS and a three-stage game under VS to derive Nash equilibrium network qualities, consumer surplus and social welfare, respectively. We highlight four distinct incentive effects that shape network quality under each structure, and we point out that, on balance, they tend to favor higher network quality under HS. However, intensity of transport service competition under each system also plays a critical role in shaping consumer surplus and social welfare. The best case for HS occurs when there is a moderate amount of price competition between the vertically integrated systems, while the best case for VS occurs when there is intense price competition between transport operators. Using computational analysis, we show that it is more likely that HS dominates VS on all three performance metrics.

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Notes

  1. 1.

    It should be noted that various forms of vertical separation exist, including accounting separation within a vertically integrated company, organizations separated into subsidiaries under an overall holding company, and complete vertical separation in which infrastructure management and transportation services are provided by distinct companies. In this paper, we consider the case of complete vertical separation.

  2. 2.

    Parallel competition (also known as end-to-end competition) refers to competition between two or more railroads that provide service between the same city pairs. Source competition between railoads (also known as geographic competition) arises when purchasers of a good that must be shipped by rail can purchase it from two or more different geographical sources and/or when suppliers of a rail-captive good can sell it in two or more geographic markets. For example, consider a purchaser of a commodity located in city A, and suppose there is a single railroad connecting A and region B and a different railroad connecting A and region C. If purchasers at A can source their products from either B or C, the two railroads can become implicit competitors with one another: if one railroad attempts to raise rates, some buyers may shift their purchases to the other region, transferring traffic to the other railroad.

  3. 3.

    There are papers that consider the welfare effects of vertically divesting a vertically integrated network provider. (See Sappington (2006) for an example, as well as references to other papers.) However, this literature does not contrast VS and HS.

  4. 4.

    Section 3.6 briefly considers the implications of incorporating subsidies into our analysis.

  5. 5.

    In Sect. 3.6, we discuss the implications of relaxing this assumption so that \(f>0\).

  6. 6.

    “Consumers” in our model can be thought of as business firms shipping good on freight trains, and the utility function in (1) can be interpreted as a reduced-form profit function for a representative firm.

  7. 7.

    Although, as we will see, given symmetry of networks, the equilibrium network qualities of the two systems will be the same.

  8. 8.

    Nothing in our analysis changes if q is the actual level of quality and a is simply a parameter in the consumer’s utility function.

  9. 9.

    In Sect. 3.5 we show that changes in the verifiability of quality that enable a higher minimum quality standard do not affect the comparison between HS and VS in terms of equilibrium network quality, consumer surplus, or social welfare.

  10. 10.

    We add the upper bound \(\overline{q}\) because for some parameter values, the firm’s profit function under HS and the regulator’s social welfare function under VS could be convex in q and c, respectively, giving rise in each case to a preference for an infinitely large network quality.

  11. 11.

    Our demand formulation corresponds to what is known in the industrial organization literature as the Shubik-Levitan specification of the linear demand, differentitated product model (Shubik and Levitan 1980). If we had not divided by \(1+b_{k}\) in (1), then at equal prices and qualities aggregate market demand would have decreased in \(b_{k}\), ceteris paribus. This would reflect (in this alternative specification) that the representative consumer places greater intrinsic value on the ability the choose among different products the less substitutable and the more complementary they are. In a railway context, the implications of this alternative specification would be implausible. For example, it would imply that holding prices and qualities fixed across organizational structures, a rail system consisting of a single line connecting two cities, A and B would generate more aggregate demand under HS—with one railroad serving A and a point halfway between A and B and another providing serivce the rest of the way to B—than under VS—with two competing operators providing service between A and B. Formally, by dividing by \(1+b_{k}\), we ensure that variations in \(b_{k}\) affect equilibrium quantities only through the (indirect) effect of \(b_{k}\) on equilibrium prices and qualities and not through a direct effect on market demand. In this way, we avoid biasing our welfare comparison between VS and HS solely due to differences in \(b_{V}\) and \(b_{H}\). We should note that, as with our maintained assumption that \(\delta =\frac{1}{2}\), our utility function specification removes a potential advantage that the analysis would otherwise confer on HS vis a vis VS due to the fact that \(b_{k}\) can be negative under HS but not under VS.

  12. 12.

    And under HS, it also parameterizes the toughness or softness of network quality competition among the integrated systems.

  13. 13.

    This is, of course, a minor abuse of terminology: if \(b_{H}<0\) under HS, the two systems act as complementors not competitors

  14. 14.

    More specifically, if \(\lambda \le \phi (b_{H})\) there is no pair of positive qualities at which the quality reaction functions intersect. The function \(\phi (b_{H})\) is strictly less than 1 for \(b_{H}\) \(\in (-1,1)\). Thus, the existence condition \(\lambda >\phi (b_{H})\) will be satisfied for all \(b_{H}\) \(\in (-1,1)\) as long as \(\lambda >1\). Because in principle \(\lambda \) can take on any positive value, this is not particularly restrictive.

  15. 15.

    The complicated characterization in (9) reflects the fact that when \(b_{H}\in [0,1)\), the non-negativity condition \(\lambda >\phi (b_{H})\) is satisfied as long as the second-order condition is satisfied, but when \(b_{H}\le 0\) the non-negativity condition is stricter than the second-order condition. In this case, the equilibrium quality will be less than the upper bound \(\overline{q}\).

  16. 16.

    The proof of these results is in the Appendix.

  17. 17.

    It is possible that (8) begins to be violated for \(b_{H}<0.88\). In this case, \(\frac{\partial q_{H}^{*}}{\partial b_{H}}>0\) for all \(b_{H}\) such that \(q_{H}^{*}<\overline{q}\).

  18. 18.

    Strictly speaking, the derived demand curve for network access is \(2X_{V}^{*}(c)\). We use the terminology in the text to simplify verbiage.

  19. 19.

    There is a large literature on setting access prices in network industries, e.g., Armstrong et al. (1996), Laffont and Tirole (1994, 1996), Laffont et al. (1998a, (1998b), and Lewis and Sappington (1999). Much of this literature focuses on the issue of access pricing when the network firm is vertically integrated and competes with the downstream suppliers that require access to the natural monopoly network. Because we consider complete vertical separation, assume a relatively simple cost structure that is known to the regulator, and allow for unconstrained welfare-neutral transfers through the fixed charge F, the key issues this literature focuses on—e.g., the desirability of the Efficient Components Pricing Rule, the possibility of network bypass, the disentanglement of network and downstream costs, and private knowledge of costs by the regulated firm—do not arise in our model. For a comprehensive discussions of the issue of access pricing specific to railway networks, see Bureau of Transport and Regional Economics (2003) and OECD (2008).

  20. 20.

    This presumes, of course, that overall system profits are positive at the regulator’s optimal tariff. The assumption that \(a>\gamma _{V}+\eta _{V}\) ensures that this will be the case.

  21. 21.

    The proof of these results is in the Appendix.

  22. 22.

    If we had carried around a general value \(\delta \) for the economies of network size under VS, the expressions containing \(\lambda \) would instead contain \(2(1-\delta )\lambda \). This implies that increasing economies of network size by increasing \(\delta \) is tantamount to decreasing \(\lambda \), so the sign of comparative statics with respect to \(\delta \) would be opposite those for \(\lambda \). Thus, increasing the economies of network size results in higher quality and a higher variable access charge under VS.

  23. 23.

    Because b is assumed to the same across structure for the purpose of this discussion, we drop the H and V subscripts on the demand function. Finally, in writing (27), we use the fact the two transport operators under VS are symmetric so

    $$\begin{aligned} \begin{array}{c} \left[ \frac{\partial X_{1V}^{D}}{\partial P_{1}}\frac{\partial P_{1V}^{*}}{\partial q_{V}}+\frac{\partial X_{1V}^{D}}{\partial P_{2}}\frac{\partial P_{2V}^{*}}{\partial q_{V}}+\frac{\partial X_{1V}^{D}}{\partial q_{V}}\right] \\ \qquad \,=\left[ \frac{\partial X_{2V}^{D}}{\partial P_{1}}\frac{\partial P_{1V}^{*}}{\partial q_{V}}+\frac{\partial X_{2V}^{D}}{\partial P_{2}}\frac{\partial P_{2V}^{*}}{\partial q_{V}}+\frac{\partial X_{2V}^{D}}{\partial q_{V}}\right] \end{array}. \end{aligned}$$
  24. 24.

    Specifically, under HS an integrated firm recognizes that quality affects profits indirectly through its impact on “own” equilibrium price in the pricing subgame, also indirectly through its impact on the other firm’s price in the pricing subgame, and finally, directly through the impact of quality on quantity demanded. Due to the envelope theorem, the first of these three terms—given by \(\left[ (P_{H}^{*}-\gamma -\eta )\frac{\partial X_{1}^{D}}{\partial P_{1}}+X_{1}^{D}\right] \frac{\partial P_{1H}^{*}}{\partial q_{1}}\)—is zero because profit maximization implies \((P_{H}^{*}-\gamma -\eta )\frac{\partial X_{1}^{D}}{\partial P_{1}}+X_{1}^{D}=0\). This, then, eliminates the term \(\frac{\partial X_{1}^{D}}{\partial P_{1}}\frac{\partial P_{1H}^{*}}{\partial q_{1}}\) from the first-order condition for quality under HS.

  25. 25.

    We could further “unpack” the horizontal coordination on quality effect by separately highlighting the difference between \(\frac{\partial X_{1}^{D}}{\partial P_{2}}\frac{\partial P_{2H}^{*}}{\partial q_{1}}=\frac{b}{1-b}\frac{-b}{\left( 2-b\right) (2+b)}\) and \(\frac{\partial X_{1}^{D}}{\partial P_{2}}\frac{\partial P_{2V}^{*}}{\partial q_{V}}=\frac{b}{1-b}\frac{1-b}{2-b}\). The former term is a strategic effect that arises in the second-stage quality competition game between the two integrated systems under HS. It is unambiguously negative (and thus “quality supressing” ) for the following reason. From (5), when \(b_{H}>0\) and one integrated firm increases its quality, it lowers the equilibrium transport price of the other integrated firm. (In essence, the firm’s quality improvement induces its rival to compete harder on the transport price.) This is a “bad” effect because it works to reduce the demand for the firm’s own transport services. When \(b_{H}<0\), the opposite happens, but the end result is also “bad” : the increase in quality increases the rival’s equilibrium transport price which, when \(b_{H}<0\), reduces the firm’s own transport services demand. The upshot is that, under HS, this strategic effect works, on the margin, to suppress quality. By contrast, the corresponding term under VS, \(\frac{\partial X_{1}^{D}}{\partial P_{2}}\frac{\partial P_{2V}^{*}}{\partial q_{V}}=\frac{b}{1-b}\frac{1-b}{2-b}\), is postive. Thus, the difference in strategic effects is, in isolation, a force that tends to reduce quality under HS relative to VS. However, the difference in these strategic effects is more than compensated by the difference between \(\frac{\partial X_{1}^{D}}{\partial q_{1}}\) under HS and \(\frac{\partial X_{1}^{D}}{\partial q_{V}}\) under VS, with the result that, as noted above, \(\left[ \frac{\partial X_{1}^{D}}{\partial P_{2}}\frac{\partial P_{2V}^{*}}{\partial q_{V}}+\frac{\partial X_{1}^{D}}{\partial q_{V}}\right] <\left[ \frac{\partial X_{1}^{D}}{\partial P_{2}}\frac{\partial P_{2H}^{*}}{\partial q_{1}}+\frac{\partial X_{1}^{D}}{\partial q_{1}}\right] .\)

  26. 26.

    Here is the proof. Recall that \(A(b_{V},\lambda )\equiv \frac{2\left[ 3-2b_{V}\right] -(1-b_{V})\Psi (b_{V},\lambda )}{2\left[ 3-2b_{V}\right] +\Psi (b_{V},\lambda )}\). If \(A(b_{V},\lambda )\le 0\), then \(\Psi (b_{V},\lambda )\ge \frac{2(3-2b_{V})}{(1-b_{V})}>2\) since \(\frac{3-2b_{V}}{1-b_{V}}>1.\) Now from (18) \(\frac{dX_{V}^{*}(c)}{dc}<0\) iff \(\frac{dq_{V}^{*}(c)}{dc}<1\). Since \(\frac{dq_{V}^{*}(c)}{dc}=\frac{2}{\lambda (2-bV)}=\frac{2}{\Psi (b_{V},\lambda )}\) the result follows immediately.

  27. 27.

    In particular, if \(\lambda >2\), it holds for all \(b_{V}\in [0,1)\).

  28. 28.

    Under HS where we imagine the upstream network unit selling network access at marginal costs, network profits are, of course, the negative of quality investment costs.

  29. 29.

    This is on top of the “stacking of the deck” in favor of VS due to (1) the assumed economies of network size under VS, i.e., \(\delta =\frac{1}{2}\) and (2) our representative consumer specification which rules out the possibility that aggregrate transport demand at equal prices and qualities goes up as transport services become weaker substitutes and/or stronger complements.

  30. 30.

    Panels D and E indicate that the comparison between VS and HS in terms of each of the three metrics is also unaffected by variations in \(\gamma \) and \(\eta \).

  31. 31.

    Formally, as \(b_{H}\rightarrow 1\), we reach the point at which (8) does not hold, which (as discussed earlier) implies that each firm’s profit function eventually increases in quality without bound, irrespective of the quality of the other firm.

  32. 32.

    Pittman (2004a) points out that instead of subsidies, the fixed costs of network operation could be covered by revenue-enhancing price discrimination. Under VS, this could take the form of two-part access tariffs, while under HS it could take the form of third-degree price discrimination by customer segment or commodity. Our model does include a two-part access tariff paid by operators to the network firm under VS, but as we pointed out in the discussion of the regulator’s problem, the fixed access charge F can be chosen to guarantee both the network firm and transport operators a non-negative profit only if total system profits under VS are non-negative. As just noted, this might not necessarily be the case if the quality-independent fixed cost f is sufficiently large. (In which case, subsidies would be needed.) With our representative consumer model, it is not possible to analyze customer-based or commodity-based price discrimination in the context of HS in a meaningful way, so examination of this form of price discrimination is beyond the scope of this study. More fundamentally, though, Pittman (2004a) points out that opportunities to price discriminate in practice are likely to be more feasible under HS than under VS. This is because under VS attempts to use two-part access tariffs (or menus of such tariffs) may run afoul of policies intended to prevent the network firm from providing discriminatory access to some transport operators. (For example, under a two-part access tariff, smaller operators would pay a higher average price than larger operators.)

  33. 33.

    In 2011, two of these regional firms merged, so there are now two major freight rail firms in Mexico.

  34. 34.

    “A key part of the success of the other railways in North America is the existence of an economic regulator with a more clearly defined responsibility, adequate resources and expertise, and access to all necessary information needed to carry out its role. SCT [Secretaría de Comunicaciones y Transportes] in principle has a role vis-à-vis establishing the terms of the trackage rights but would need to develop its capabilities to be more effective in regulatory functions.” (OECD 2014, p. 25.)

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Acknowledgments

We thank Qiqin Dai, Robert Gallamore, Russell Pittman, Hanyu Shi, and two anonymous referees for their helpful comments and suggestions.

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Correspondence to David Besanko.

Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s11149-017-9318-x.

Appendix: Proofs and derivations

Appendix: Proofs and derivations

Proof of comparative statics results for network quality \(q_{H}^{*}\) under HS

The first results follow immediately from (9). The second result follows because \(q_{H}^{*}\) strictly increases in \(\phi (b_{H})\), and on the interval \([-1,1]\), \(\phi (b_{H})\) is a concave function that increases between \(-1\) and about 0.88 and decreases thereafter. The third result follows from differentiating (10) with respect to \(b_{H}\) to get \(\frac{\partial P_{H}^{*}}{\partial b_{H}}=-\frac{\left( a+q_{H}^{*}-\gamma _{H}-\eta _{H}\right) }{\left( 2-b_{H}\right) ^{2}}+\left( \frac{1-b_{H}}{2-b_{H}}\right) \frac{\partial q_{H}^{*}}{\partial b_{H}}\). Since \(\frac{\partial q_{H}^{*}}{\partial b_{H}}<0\) for \(b_{H}\) sufficiently large, the result then follows. \(\square \)

Derivation of solution to regulator’s problem

The regulator’s problem may have an interior solution (\(\eta _{V}<c<\overline{c}\)) or a corner solution (\(c=\eta _{V}\) or \(c=\overline{c}\)). The first-order condition for an interior solution is

$$\begin{aligned} \frac{dSW(c)}{dc}= & {} \sum _{i=1}^{2}\left[ \frac{\partial U_{V}(\mathbf {X}_{V}^{*}(c),\mathbf {q}_{V}^{\mathbf {*}}(c))}{\partial X_{i}}-(\gamma _{V}+\eta _{V})\right] \frac{dX_{V}^{*}(c)}{dc} \nonumber \\&+\sum _{i=1}^{2}\left[ \frac{\partial U_{V}(\mathbf {X}_{V},\mathbf {q}_{V})}{\partial q_{i}}\right] \frac{dq_{V}^{*}(c)}{dc}-\lambda q_{V}^{*}(c)\frac{dq_{V}^{*}(c)}{dc} \nonumber \\= & {} 0. \end{aligned}$$
(39)

Given (2), we can simplify (39):

$$\begin{aligned} \frac{dSW(c)}{dc}=SMB_{X}\frac{dX_{V}^{*}(c)}{dc}+SMB_{q}\frac{dq_{V}^{*}(c)}{dc}=0, \end{aligned}$$
(40)

where

$$\begin{aligned} SMB_{X}\equiv 2\left[ P_{V}^{*}(c)-\gamma _{V}-\eta _{V}\right] \end{aligned}$$
(41)

is the (net) social marginal benefit of transport services, and

$$\begin{aligned} SMB_{q}\equiv 2X_{V}^{*}(c)-\lambda q_{V}^{*}(c) \end{aligned}$$
(42)

is the social marginal benefit of network quality. The marginal social benefit of increasing the access charge c is thus a weighted sum of the social marginal benefit of transport services and the social marginal benefit of network quality, where the weights are \(\frac{dX_{V}^{*}}{dc}\) and \(\frac{dq_{V}^{*}}{dc}\), respectively. Condition (40) indicates that the regulator chooses the variable access charge to balance the social marginal benefits of quantity and quality.

The second-order condition is

$$\begin{aligned} \frac{d^{2}SW(c)}{dc^{2}}=2\left[ \frac{dP_{V}^{*}(c)}{dc}\right] \frac{dX_{V}^{*}(c)}{dc}+\left[ 2\frac{dX_{V}^{*}(c)}{dc}-\lambda \frac{dq_{V}^{*}(c)}{dc}\right] \frac{dq_{V}^{*}(c)}{dc}<0, \end{aligned}$$

which can be shown to be equivalent to \(\psi (b_{V},\lambda )-2A(b_{V},\lambda )>0\), where \(\psi (b_{V},\lambda )\) and \(A(b_{V},\lambda ) \) are given in (21) and (22) above. Solving (40) gives us

$$\begin{aligned} c_{V}^{*}=\eta _{V}+\frac{\psi (b_{V},\lambda )A(b_{V},\lambda )\left( a-\gamma _{V}-\eta _{V}\right) }{\left[ \psi (b_{V},\lambda )-2A(b_{V},\lambda )\right] }. \end{aligned}$$
(43)

If the second-order condition holds and \(A(b_{V},\lambda )>0\), then the expression in (43) exceeds \(\eta _{V}\). If (as we’ve assumed), \(\overline{q}\) is very large, then \(\overline{c}\) will in fact exceed the expression in (43). Thus, when \(\psi (b_{V},\lambda )-2A(b_{V},\lambda )>0\), the expression in (43) does indeed represent an interior solution to the regulator’s problem.

When \(\left. \frac{dSW(c)}{dc}\right| _{c=\eta _{V}}\le 0\), we have a corner solution at which \(c=\eta _{V}\). This condition is equivalent to \(A(b_{V},\lambda )\le 0\). When the second-order condition is not satisfied, the regulator’s objective function will eventually increase without bound in c. In this case, welfare is maximized by setting \(c=\overline{c}\). Pulling all these cases together gives us the expression for \(c_{V}^{*}\) in (20). \(\square \)

Proof of comparative statics results for network quality \(q_{V}^{*}\) and variable access charge \(c_{V}^{*}\) under VS

\(\frac{\partial c^{*}_V}{\partial \lambda }<0,\frac{\partial q^{*}_V}{\partial \lambda }<0, \mathrm{and}\ \frac{\partial p^{*}_V}{\partial \lambda }< 0: \) Note that \(\psi \) increases in \(\lambda \) and A decreases in \(\psi \). Thus we have \(\frac{\partial \psi (b_{V},\lambda )}{\partial \lambda }>0\) and \(\frac{\partial A(b_{V},\lambda )}{\partial \lambda }<0\), and from (23), it follows that \(q_{V}^{*}\) decreases in \(\lambda \). By the same token, it is straightforward to show that \(c_{V}^{*}\) increases in A and decreases in \(\psi \) and thus \(q_{V}^{*}\) decreases in \(\lambda \).

\(\frac{\partial c^{*}_V}{\partial b_V}>0\, \mathrm{and}\, \frac{\partial q^{*}_V}{\partial b_V}> 0:\) Recall

$$\begin{aligned} \psi (b_{V},\lambda )= & {} \lambda (2-b_{V}). \\ A(b_{V},\lambda )= & {} \frac{2(3-2b_{V})-(1-b_{V})\lambda (2-b_{V})}{2(3-2b_{V})+\lambda (2-b_{V})}. \end{aligned}$$

Thus

$$\begin{aligned} \frac{\partial A}{\partial b_{V}}= & {} -\lambda \left( b_{V}-2\right) \frac{2\lambda -4b_{V}-\lambda b_{V}+4}{\left( 2\lambda -4b_{V}-\lambda b_{V}+6\right) ^{2}} \\= & {} \lambda \left( 2-b_{V}\right) \frac{\lambda (2-b_{V})+4(1-b_{V})}{\left( 2\lambda -4b_{V}-\lambda b_{V}+6\right) ^{2}}. \end{aligned}$$

When \(b_{V}\in (0,1),\lambda >0\), we have \(\lambda (2-b_{V})+4(1-b_{V})>0\). This implies\(\frac{\partial A}{\partial b_{V}}>0,\)when \(b_{V}\in (0,1),\lambda >0\), and thus A increase in \(b_{V}\).

Now, \(\frac{\psi A}{\psi -2A}=\frac{A}{1-\frac{2A}{\psi }}\) decreases in \(\psi \) and increases in A. Moreover, \(\psi \) decreases in \(b_{V},\) and A increases in \(b_{V}\), so \(\frac{\psi A}{\psi -2A}\) increases in \(b_{V}\), i.e.,

$$\begin{aligned} \frac{\partial (\frac{\psi A}{\psi -2A})}{\partial b_{V}}>0\text { when }b_{V}\in (0,1). \end{aligned}$$

This implies

$$\begin{aligned} \frac{\partial (c_{V}^{*}-\eta _{V})}{\partial b_{V}}>0\;\text {when} ~ {b}_{V}\in (0,1). \end{aligned}$$

Also \(\frac{2A}{\psi -2A}\) decreases in \(\psi \) and increases in A. Moreover, \(\psi \) decreases in \(b_{V},\) and A increases in \(b_{V}\), so \(\frac{2A}{\psi -2A}\) increases in \(b_{V}\), i.e.,

$$\begin{aligned} \frac{\partial (\frac{2A}{\psi -2A})}{\partial b_{V}}>0\text { when }b_{V}\in (0,1), \end{aligned}$$

which implies

$$\begin{aligned} \frac{\partial (q_{V}^{*})}{\partial b_{V}}>0\;~\mathrm{when} ~ {b}_{V}\in (0,1). \end{aligned}$$

\(\frac{\partial A}{\partial \lambda }(b_V,\lambda ) <0\ \mathrm{and}\ \frac{\partial A^{*}}{\partial b_V}(b_V,\lambda )> 0:\) Implied by the derivations above.\(\square \)

Proof of Proposition 1

The condition for a corner solution at which \(c_{V}^{*}=\eta _{V}\) and thus \(q_{V}^{*}=0\) is, as noted in the text, \(A(b_{V},\lambda )\le 0\). It is straightforward to show that this is equivalent to (28). By contrast, as just noted, incremental network quality under HS is strictly positive for all \(\lambda \). Thus, when (28) holds, \(q_{H}^{*}>q_{V}^{*}=0.\) The function \(\frac{2\left[ 3-2b_{V}\right] }{(1-b_{V})(2-b_{V})}\) is strictly increasing on \(b_{V}\in [0,1)\), and thus (28) is more likely to hold the closer \(b_{V}\) is to zero. \(\square \)

Proof of Proposition 2

The first part of the proposition follows because \(X_{H}^{*}=X^{*}(q_{H}^{*},\gamma +\eta ,b)\ge X^{*}(q_{H}^{*},\gamma +c_{V}^{*},b)>X^{*}(q_{V}^{*},c_{V}^{*}+\eta ,b)=X_{V}^{*}\), where the first inequality follows because \(c_{V}^{*}\ge \eta \) and \(X^{*}(q,MC,b)\) is decreasing in MC, and the second inequality follows by the assumption that \(q_{H}^{*}>\) \(q_{V}^{*}\) and \(X^{*}(q,MC,b)\) is increasing in q. From (30) and (31), we can immediately conclude that \(CS_{H}^{*}>CS_{V}^{*}\). The second part of the proposition follows because \(X_{H}^{*}=X^{*}(q_{H}^{*},\gamma +\eta ,b)>X^{*}(q_{H}^{*},\gamma +c_{V}^{*},b)=X^{*}(q_{V}^{*},\gamma +c_{V}^{*},b)=X_{V}^{*}\), where the first inequality follows because \(c_{V}^{*}>\eta \), and the last equality follows because we assume \(q_{H}^{*}=\) \(q_{V}^{*}\). We can immediately conclude, as in the first part, that \(CS_{H}^{*}>CS_{V}^{*}\). \(\square \)

Proof that consumer surplus under both HS and VS increases in the competition intensity parameter, \(b_{k}\)

We first establish that \(\frac{\partial CS_{H}^{*}}{\partial b_{H}}>0\), when \(b_{H}\in (-1,1)\) and \(\lambda >\phi (b_{H}).\) Recall

$$\begin{aligned} q_{H}^{*}= & {} (a-\gamma _{H}-\eta _{H})\frac{\phi (b_{H})}{\lambda -\phi (b_{H})}. \\ \phi (b_{H})= & {} \frac{(2+b_{H})(2-b_{H})-b_{H}^{2}}{(2-b_{H})^{2}(2+b_{H})}. \end{aligned}$$

Substituting these into (11) and simplifying algebraically gives us

$$\begin{aligned} X_{H}^{*}=-\lambda \frac{b_{H}^{2}-4}{8\lambda +2b_{H}^{2}-4\lambda b_{H}-2\lambda b_{H}^{2}+\lambda b_{H}^{3}-4}(a-\gamma _{H}-\eta _{H}). \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\partial X_{H}^{*}}{\partial b_{H}}=\frac{\lambda b_{H}^{4}-8\lambda b_{H}^{2}-8b_{H}+16\lambda }{\left( 8\lambda +2b_{H}^{2}-4\lambda b_{H}-2\lambda b_{H}^{2}+\lambda b_{H}^{3}-4\right) ^{2}}\lambda (a-\gamma _{H}-\eta _{H}). \end{aligned}$$

Now,

$$\begin{aligned} \frac{\partial (\lambda b_{H}^{4}-8\lambda b_{H}^{2}-8b_{H}+16\lambda )}{\partial b_{H}}= & {} 4\lambda b_{H}^{3}-16\lambda b_{H}-8=4\lambda b_{H}(b_{H}^{2}-4)-8. \\ \frac{\partial (4\lambda b_{H}(b_{H}^{2}-4)-8)}{\partial b_{H}}= & {} 4\lambda \left( 3b_{H}^{2}-4\right) <0. \end{aligned}$$

Define

$$\begin{aligned} \begin{array}{c} f(b_{H})\equiv \lambda b_{H}^{4}-8\lambda b_{H}^{2}-8b_{H}+16\lambda \text {, } \\ g(b_{H})\equiv 4\lambda b_{H}(b_{H}^{2}-4)-8\text {, }\end{array} \end{aligned}$$

The function \(g(b_{H})\) decreases in \(b_{H}\), which implies

$$\begin{aligned} \max g(b_{H})= & {} g(-1)=12\lambda -8>0. \\ \min g(b_{H})= & {} g(1)=-12\lambda -8<0. \end{aligned}$$

Let \(b_{H}^{0}\) be the root of \(g(b_{H})=0\). This implies

$$\begin{aligned} \begin{array}{c} g(b_{H})<0\text { when }b_{H}\in (b_{H}^{0},1). \\ g(b_{H})>0\text { when }b_{H}\in (-1,b_{H}^{0}).\end{array} \end{aligned}$$

which in turn implies

$$\begin{aligned} \begin{array}{c} f(b_{H})\text { decrease in }b_{H}\text { when }b_{H}\in (b_{H}^{0},1). \\ f(b_{H})\text { increase in }b_{H}\text { when }b_{H}\in (-1,b_{H}^{0}).\end{array} \end{aligned}$$

Now,

$$\begin{aligned} f(1)= & {} 9\lambda -8>0. \\ f(-1)= & {} 9\lambda +8>0, \end{aligned}$$

which implies

$$\begin{aligned} \begin{array}{c} f(b_{H})>0\text { when }b_{H}\in (b_{H}^{0},1) \\ f(b_{H})>0\text { when }b_{H}\in (-1,b_{H}^{0})\end{array}, \end{aligned}$$

and thus \(f(b_{H})>0\) when \(b_{H}\in (-1,1)\). Because \(a-\gamma _{H}-\eta _{H}>0\), we have \(\frac{\partial X_{H}^{*}}{\partial b_{H}}>0\) when \(b_{H}\in (-1,1)\) and thus \(\frac{\partial CS_{H}^{*}}{\partial b_{H}}>0\) when \(b_{H}\in (-1,1)\).

Next, we establish that \(\frac{\partial CS_{V}^{*}}{\partial b_{V}}>0.\) Recall

$$\begin{aligned} \psi (b_{V},\lambda )= & {} \lambda (2-b_{V}). \\ A(b_{V},\lambda )= & {} \frac{2(3-2b_{V})-(1-b_{V})\psi (b_{V},\lambda )}{2(3-2b_{V})\!+\!\psi (b_{V},\lambda )}=\frac{2(3-2b_{V})-\lambda (2-b_{V})(1-b_{V})}{2(3-2b_{V})+\lambda (2-b_{V})}. \end{aligned}$$

Thus,

$$\begin{aligned} q_{V}^{*}= & {} (a-\gamma _{V}-\eta _{V})\frac{2A(b_{V},\lambda )}{\psi (b_{V},\lambda )-2A(b_{V},\lambda )}. \\ c_{V}^{*}= & {} \eta _{V}+(a-\gamma _{V}-\eta _{V})\frac{\psi (b_{V},\lambda )A(b_{V},\lambda )}{\psi (b_{V},\lambda )-2A(b_{V},\lambda )}. \end{aligned}$$

If we substitute these expressions into (25) then with several straightforward steps of algebra we get

$$\begin{aligned} X_{V}^{*}=\frac{\left[ \frac{\lambda (2-b_{V})(1-\frac{2(3-2b_{V})-\lambda (2-b_{V})(1-b_{V})}{2(3-2b_{V})+\lambda (2-b_{V})})}{\lambda (2-b_{V})-2\times \frac{2(3-2b_{V})-\lambda (2-b_{V})(1-b_{V})}{2(3-2b_{V})+\lambda (2-b_{V})}}\right] }{2-b_{V}}(a-\gamma _{V}-\eta _{V}). \end{aligned}$$

From this it follows that

$$\begin{aligned} \frac{\partial X_{V}^{*}}{\partial b_{V}}&=\frac{\lambda (2-b_{V})+2(1-b_{V})}{\left( \lambda ^{2}b_{V}^{2}-4\lambda ^{2}b_{V}+4\lambda ^{2}+6\lambda b_{V}^{2}-20\lambda b_{V}+16\lambda +8b_{V}-12\right) ^{2}}\\&\quad \times \,4\lambda ^{2}\left( 2-b_{V}\right) (a-\gamma _{V}-\eta _{V}). \end{aligned}$$

Because \(b_{V}\in (0,1),\lambda>1,a-\gamma _{V}-\eta _{V}>0\), we have

$$\begin{aligned} (2-b_{V})+2(1-b_{V})>0,4\lambda ^{2}\left( 2-b_{V}\right) >0, \end{aligned}$$

which implies \(\frac{\partial X_{V}^{*}}{\partial b_{V}}>0\) when \(b_{V}\in (0,1)\) and thus implies \(\frac{\partial CS_{V}^{*}}{\partial b_{V}}>0\) when \(b_{V}\in (0,1). \) \(\square \)

Proof of Proposition 3

When \(b_{V}\rightarrow 1\),

$$\begin{aligned} \psi (b_{V},\lambda )\equiv & {} \lambda>0. \\ A(b_{V},\lambda )\equiv & {} \frac{2}{2+\lambda }>0, \end{aligned}$$

and so from (23), \(q_{V}^{*}>0.\) Now, using (32) (evaluated at \(b_{H}=-1\)) and (33) (evaluated at \(b_{V}=1\)) we have

$$\begin{aligned} \frac{CS_{H}^{*}}{CS_{V}^{*}}=\frac{1}{9}\frac{\left( \frac{9\lambda }{9\lambda -2}\right) ^{2}}{\left( \frac{\lambda ^{2}}{\lambda ^{2}+2\lambda -4}\right) ^{2}}\left[ \frac{a-\gamma _{H}-\eta _{H}}{a-\gamma _{V}-\eta _{V}}\right] ^{2}\le \frac{1}{9}\frac{\left( \frac{9\lambda }{9\lambda -2}\right) ^{2}}{\left( \frac{\lambda ^{2}}{\lambda ^{2}+2\lambda -4}\right) ^{2}}, \end{aligned}$$
(44)

where the inequality follows from the assumption that, \(\gamma _{V}+\eta _{V}\le \gamma _{H}+\eta _{H}\). Straightforward through tedious algebra establishes that the right-hand side of the above expression is less than 1,  provided \(\lambda >0\) and \(\psi (b_{V},\lambda )>2A(b_{V},\lambda )\). \(\square \)

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Besanko, D., Cui, S. Railway restructuring and organizational choice: network quality and welfare impacts. J Regul Econ 50, 164–206 (2016). https://doi.org/10.1007/s11149-016-9309-3

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Keywords

  • Railway restructuring
  • Infrastructure
  • Network quality

JEL Classification

  • L92
  • L51
  • D60