Railway restructuring and organizational choice: network quality and welfare impacts

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.

Appendix: Proofs and derivations

1.1 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. $\square$

1.2 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). $\square$

1.3 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 $CS_{H}^{*}>CS_{V}^{*}$. $\square$

1.4 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 $\psi (b_{V},\lambda )>2A(b_{V},\lambda )$. $\square$