Competition Effects in a Liberalized Railway Market

Abstract

This paper presents a game-theoretic model of a liberalized railway market, in which train operation and ownership of infrastructure are vertically separated. We analyze how the regulatory agency will optimally set the charges that operators have to pay to the infrastructure manager for access to the tracks and how these charges change with increased competition in the railway market. Our analysis shows that an increased number of competitors in the freight and/or passenger segment reduces prices per kilometer and increases total output in train kilometers. The regulatory agency reacts to more competition with a reduction in access charges in the corresponding segment. Consumers benefit through lower prices, while individual profits of each operator decrease through a higher number of competitors. We further show that the welfare effect of increased competition in the freight and/or passenger segment is ambiguous and depends on the level of competition. Finally, social welfare is higher under two-part tariffs than under one-part tariffs if raising public funds is costly to society.

A Appendix

1.1 A.1 Proof of Lemma 2

The break-even condition for the infrastructure manager will be satisfied with equality in equilibrium because increasing governmental transfers above the break-even level is costly to society. The maximization problem of the regulatory agency can thus be rewritten as:

$$ \max\limits_{(a_{f},a_{p})\geq 0}\left\{ W^{\prime }=\Pi _{p}+\Pi _{f}+CS_{f}+CS_{p}-(1+\lambda )\left( F+(v-a_{f})Q_{f}+(v-a_{p})Q_{p}\right) \right\} , $$

(17)

with $Q_{k}=\sum_{i=1}^{n_{k}}q_{ik}$. The first-order conditions of the maximization problem (17) are derived as:

$$ \begin{array}{rll} \frac{\partial W^{\prime }}{\partial a_{k}}&=&n_{k}\left( -\frac{\left( \theta _{k}-a_{k}\right) ^{2}(2+c_{k})}{\left( n_{k}+1+c_{k}\right) ^{3}}\right) +n_{k}\left( \frac{n_{k}\left( \theta _{k}-\frac{n_{k}(\theta _{k}-a_{k})}{ n_{k}+1+c_{k}}\right) -n_{k}\theta _{k}}{n_{k}+1+c_{k}}\right) \\ &&-(1+\lambda )\left( \frac{n_{k}(2a_{k}-(\theta _{k}+v)}{n_{k}+1+c_{k}} \right) =0, \end{array} $$

with k ∈ {p, f}. Note that the second-order conditions for a maximum are satisfied if the number of competitors is sufficiently small with

$$ n_{k}<n_{k}^{SOC}\equiv 1+\lambda +\left[ 1+c_{k}+\lambda (4+2c_{k}+\lambda ) \right] ^{1/2}. $$

Solving the system of first-order conditions yields:

$$ a_{k}^{\ast }=\frac{1}{\varphi }\left[ (n_{k}+1+c_{k})(v(1+\lambda )+\lambda \theta _{k})-\theta _{k}(1+n_{k}(n_{k}-1))\right] , $$

(18)

with φ ≡ n _k (2 − n _k ) + 2λ(n _k  + 1 + c _k ) + c _k . To guarantee that access charges are not below marginal infrastructure costs v, we assume that the number of competitors is sufficiently small with

$$ n_{k}<n_{k}^{v}=\frac{1}{2}\left( 1+\lambda +\left[ \lambda (\lambda +6+4c_{k})-3\right] ^{1/2}\right) . $$

Thus, in the subsequent analysis, we assume that $n_{k}<n_{k,\max }\equiv \min \{n_{k}^{SOC},n_{k}^{v}\}$.

Finally, to show that access charges increase with higher costs c _k , we compute

$$ \frac{\partial a_{k}^{\ast }}{\partial c_{k}}=\frac{1}{\varphi ^{2}}\left[ (1+n_{k}(n_{k}-1))(\theta _{k}-v)(1+\lambda )\right] . $$

By noting that n _k  ≥ 1 and θ _k  > v, we derive that $\frac{ \partial a_{k}^{\ast }}{\partial c_{k}}>0$.¹⁹

1.2 A.2 Proof of Proposition 1

The partial derivatives of $a_{k}^{\ast }$ with respect to n _k is given by:

$$ \frac{\partial a_{k}^{\ast }}{\partial n_{k}}=-\frac{\left[ 2(n_{k}-1)+n_{k}+c_{k}(2n_{k}-1)\right] (1+\lambda )(\theta _{k}-v)}{\left[ n_{k}(2-n_{k})+2\lambda (n_{k}+1+c_{k})+c_{k}\right] ^{2}}. $$

By noting that n _k  ≥ 1 and θ _k  > v, we derive that $\frac{ \partial a_{k}^{\ast }}{\partial n_{k}}<0$.

1.3 A.3 Proof of Proposition 2

• Part (i) Let φ ≡ n _k (2 − n _k ) + 2λ(n _k  + 1 + c _k ) + c _k . To prove that more competition in segment k reduces the price $p_{k}^{\ast }$ per kilometer and increases total output $Q_{k}^{\ast }$ in train kilometers, we derive the partial derivatives with respect to n _k as:

  $$ \begin{array}{rll} \frac{\partial p_{k}^{\ast }}{\partial n_{k}} &=&-\frac{1}{\varphi ^{2}} (\theta _{k}-v)(1+\lambda )\left(c_{k}+n_{k}^{2}+2\lambda (1+c_{k})\right)<0, \\ \frac{\partial Q_{k}^{\ast }}{\partial n_{k}} &=&\frac{1}{\varphi ^{2}} (\theta _{k}-v)(1+\lambda )\left(c_{k}+n_{k}^{2}+2\lambda (1+c_{k})\right)>0. \end{array} $$

  Furthermore, we compute:

  $$ \frac{\partial q_{ik}^{\ast }}{\partial n_{k}}=\frac{2}{\varphi ^{2}}(\theta _{k}-v)(1+\lambda )(n_{k}-(1+\lambda ))<0\Leftrightarrow n_{k}<n_{k}^{\prime } $$

  Thus, the effect of increased competition on individual output $q_{ik}^{\ast }$ of operator i is negative if $n_{k}<n_{k}^{\prime }$.

• Part (ii) To prove the claim, we substitute equilibrium access charges $a_{k}^{\ast }$ in the operator’s profit function (10) and derive that $\pi _{ik}^{\ast }=\left( 1+c_{k}/2\right) \left( q_{ik}^{\ast }\right) ^{2}-f_{ik}$, where $q_{ik}^{\ast }$ are the Stage 1 equilibrium outputs (Eq. 13) of operator i in segment k. From Proposition 2, we know that $\frac{\partial q_{ik}^{\ast }}{\partial n_{k}}<0$ if $n_{k}<n_{k}^{\prime }\equiv 1+\lambda$. Thus, $\frac{\partial \pi _{ik}^{\ast }}{\partial n_{k}}<0$ if $n_{k}<n_{k}^{\prime }$. We deduce that individual profits $\pi _{ik}^{\ast }\$of operator i in segment k decrease with a higher number of competitors, until the minimum is reached for $n_{k}=n_{k}^{\prime }$.

1.4 A.4 Proof of Proposition 3

• Part (i) To prove the claim, we have to show that $\frac{\partial T^{\ast }}{\partial n_{k}} <0\Leftrightarrow n_{k}<n_{k}^{T}$ and $\frac{\partial T^{\ast }}{\partial n_{k}}>0\Leftrightarrow n_{k}>n_{k}^{T}$. Remember that $T^{\ast }=F+(v-a_{f}^{\ast })Q_{f}^{\ast }+(v-a_{p}^{\ast })Q_{p}^{\ast }$. We derive:

  $$ \frac{\partial T^{\ast }}{\partial n_{k}}=\underset{<0}{\underbrace{ (v-a_{k}^{\ast })}}\underset{>0}{\underbrace{\frac{\partial Q_{k}^{\ast }}{ \partial n_{k}}}}-\underset{<0}{\underbrace{\frac{\partial a_{k}^{\ast }}{ \partial n_{k}}}}\underset{>0}{\underbrace{Q_{k}^{\ast }}} $$

  We define $z(n_{k}):=\frac{\partial T}{\partial n_{k}}$ and note that z(n _k ) is a continuous function in the range of feasible n _k . From the discussion of Proposition 2, we know that $a_{k}^{\ast }\geq v\Leftrightarrow n_{k}\leq n_{k}^{v}$. Thus, $z(n_{k}^{v})>0$. It follows that $n_{k}<n_{k}^{v}$ is a necessary condition for z(n _k ) = 0. We compute:

  $$ z(0)=\frac{(\theta _{k}-v)^{2}(1+\lambda )(1-\lambda (1+c_{k}))}{\left[ c_{k}+2\lambda (1+c_{k})\right] ^{2}}<0\Leftrightarrow \lambda >\lambda ^{\prime }\equiv \frac{1}{1+c_{k}}. $$

  • (a) Suppose that λ > λ′. According to the intermediate value theorem, there exists a number of competitors $n_{k}^{T}<n_{k}^{v}$, such that $z(n_{k}^{T})=0$. This proves the claim because T is a convex function in n _k .

  • (b) Suppose that λ < λ′. In this case, it holds that z(0) > 0. It follows that there does not exist a number of competitors $n_{k}^{T}\in (0,n_{k}^{v})$, such that $z(n_{k}^{T})=0$. Thus, z(n _k ) > 0 for all feasible n _k . This completes the proof of part (i).

• Part (ii) To prove the claim, we substitute equilibrium access charges $a_{k}^{\ast }$ in consumer surplus (Eq. 11) and derive that $CS_{k}^{\ast }=n_{k}^{3}/2\left( q_{ik}^{\ast }\right) ^{2}$. We compute the partial derivative of $CS_{k}^{\ast }$ with respect to n _k as:

  $$ \frac{\partial CS_{k}^{\ast }}{\partial n_{k}}=\frac{2n_{k}^{2}}{\varphi ^{3} }(\theta _{k}-v)^{2}(1+\lambda )^{2}\left[ n_{k}(2+n_{k})+2\lambda (3+n_{k})+3c_{k}(1+2\lambda )\right] $$

  with φ ≡ n _k (2 − n _k ) + 2λ(n _k  + 1 + c _k ) + c _k . Thus, $\frac{\partial CS_{k}^{\ast }}{\partial n_{k}}>0$ because φ is always positive for all $n_{k}<\min \{n_{k,\max }^{\prime },n_{k,\max }^{\prime \prime }\}$. This proves the claim that consumer surplus always increases with a higher number of competitors.

1.5 A.5 Proof of Proposition 4

Let φ ≡ n _k (2 − n _k ) + 2λ(n _k  + 1 + c _k ) + c _k . To prove the claim, we substitute equilibrium access charges $a_{k}^{\ast }$ in the welfare function (6) and derive the partial derivative of social welfare with respect to n _k as:

$$ \frac{\partial W}{\partial n_{k}}=\frac{\partial \Pi _{k}^{\ast }}{\partial n_{k}}\!+\!\frac{\partial CS_{k}^{\ast }}{\partial n_{k}}-(1\!+\!\lambda )\frac{ \partial T^{\ast }}{\partial n_{k}}=\frac{1}{2\varphi ^{2}}(\theta _{k}-v)^{2}(1\!+\!\lambda )^{2}(c_{k}(1\!+\!2\lambda )+n_{k}+2\lambda )\!-\!f_{ik}. $$

(19)

From Eq. 19, we derive that in a scenario without fixed costs, i.e., f _ik  = 0, social welfare would always increase with a higher number of competitors. However, in a scenario with fixed costs, i.e., f _ik  > 0, the effect on social welfare is ambiguous.

To prove this claim, we define $\mu (n_{k}):=\frac{\partial W}{\partial n_{k} }$ and note that μ(n _k ) is a continuous function in the range of feasible n _k . We compute $\mu (0)=\frac{(\theta _{k}-v)^{2}(1+\lambda )^{2}}{2c_{k}+4(1+c_{k})\lambda }-f_{ik}<0\Leftrightarrow f_{ik}>f_{ik}^{\ast }\equiv \frac{(\theta _{k}-v)^{2}(1+\lambda )^{2}}{ 2c_{k}+4(1+c_{k})\lambda }$. Moreover, we derive that $\lim_{n\rightarrow n_{k}^{SOC}}\mu (n_{k})=\infty$. According to the intermediate value theorem, there exists a number of competitors $n_{k}^{W}<n_{k}^{SOC}$, such that $w(n_{k}^{W})=0$. However, it is not guaranteed that $n_{k}^{W}<n_{k,\max }$.

We conclude that social welfare always decreases through more competition in segment k if $n_{k}^{W}>n_{k,\max }$. This is the case if the fixed costs f _ik of the train operators are sufficiently high because $n_{k}^{W}$ is an increasing function in f _ik . On the other hand, if the fixed costs of the train operators are sufficiently low then $n_{k}^{W}<n_{k,\max }$. In this case, social welfare initially decreases through more competition in segment k and reaches its minimum for $n_{k}=n_{k}^{W}$. For $n_{k}>n_{k}^{W}$, welfare increases through more competition in segment k.

1.6 A.6 Proof of Proposition 5

To prove that access charges under two-part tariffs $a_{k}^{\ast \ast }$ are lower than access charges $a_{k}^{\ast }$ under single tariffs if λ > 0, we compute:

$$ a_{k}^{\ast }-a_{k}^{\ast \ast }=\frac{\lambda (1+\lambda )(\theta _{k}-v)(2+c_{k})(n_{k}+1+c_{k})}{\varphi \cdot \tau } $$

with φ = n _k (2 − n _k ) + 2λ(n _k  + 1 + c _k ) + c _k and τ = n _k (2 − n _k ) + λ(2n _k  + c _k ) + c _k . It follows that $a_{k}^{\ast }>a_{k}^{\ast \ast }$ if λ > 0, while $a_{k}^{\ast }=a_{k}^{\ast \ast }$ if λ = 0.

In the next step, we compare social welfare under single tariffs with social welfare under two-part tariffs. From the maximization problems (17) and (15), we know that social welfare under single tariffs is given by:

$$ W^{\ast }=\Pi _{f}^{\ast }+\Pi _{p}^{\ast }+CS_{f}^{\ast }+CS_{p}^{\ast }-(1+\lambda )\left[ F+\big(v-a_{f}^{\ast }\big)Q_{f}^{\ast }+\big(v-a_{p}^{\ast }\big)Q_{p}^{\ast }\right] \text{, } $$

while social welfare under two-part tariffs yields:²⁰

$$ \begin{array}{rll} W^{\ast \ast } &=&CS_{f}^{\ast \ast }+CS_{p}^{\ast \ast }-(1+\lambda )\left[ F-\big(T_{f}^{\ast \ast }+T_{p}^{\ast \ast }\big)+\big(v-a_{f}^{\ast \ast }\big)Q_{f}^{\ast \ast }+\big(v-a_{p}^{\ast \ast }\big)Q_{p}^{\ast \ast }\right] \\ &=&(1+\lambda )\big(T_{f}^{\ast \ast }\!+\!T_{p}^{\ast \ast }\big)\!+\!CS_{f}^{\ast \ast }+CS_{p}^{\ast \ast }\!-\!(1+\lambda )\left[ F+\big(v\!-\!a_{f}^{\ast \ast }\big)Q_{f}^{\ast \ast }+\big(v\!-\!a_{p}^{\ast \ast }\big)Q_{p}^{\ast \ast }\right] , \end{array} $$

with $T_{k}^{\ast \ast }=\sum_{i=1}^{n_{k}}T_{ik}^{\ast \ast }$.

Suppose that λ > 0: because $a_{k}^{\ast }>a_{k}^{\ast \ast }$, we derive that $CS_{k}^{\ast \ast }>CS_{k}^{\ast }$, $Q_{k}^{\ast \ast }>Q_{k}^{\ast }$ and $T_{k}^{\ast \ast }>\Pi _{k}^{\ast }$. One can show that the higher consumer surplus and operators’ lump sum fees under two-part tariffs compensate for the (eventually) higher value of $F+(v-a_{f}^{\ast \ast })Q_{f}^{\ast \ast }+(v-a_{p}^{\ast \ast })Q_{p}^{\ast \ast }$, such that W ^∗ ∗ > W ^∗ always holds.

Suppose that λ = 0: because $a_{k}^{\ast }=a_{k}^{\ast \ast }$, we derive that $CS_{k}^{\ast \ast }=CS_{k}^{\ast }$, $Q_{k}^{\ast \ast }=Q_{k}^{\ast }$ and $T_{k}^{\ast \ast }=\Pi _{k}^{\ast }$. It follows that W ^∗ = W ^∗ ∗.

1.7 A.7 Proof of Proposition 6

By computing the first-order conditions of the maximization problem (16) and solving the resulting equations systems, we derive the access charges in the passenger segment as:

$$ \begin{array}{rll} a_{p}^{A} &=&\frac{2v(1+\lambda )(1+c_{p}/2)+\theta _{p}(2\lambda (1+c_{p}/2)-1)}{c_{p}(1+2\lambda )+4\lambda +1}\text{ (Regime A),} \\ a_{p}^{B} &=&\frac{v+\theta _{p}}{2}+\frac{v-\theta _{p}}{2(c_{p}(1+\lambda )+4\lambda +3)}\text{ (Regime B).} \end{array} $$

The access charges in the freight segment are given in both regimes by:

$$ a_{f}^{A,B}=\frac{v+\theta _{f}}{2}+\frac{v-\theta _{f}}{c_{f}(1+\lambda )+3\lambda +1}\text{ (Regimes A and B).} $$

Let φ = (c _p (1 + 2λ) + 4λ + 1)(c _p (1 + λ) + 4λ + 3).

• ad (i) We compute $a_{p}^{A}-a_{p}^{B}=-\frac{1}{\varphi } (2+c_{p})^{2}(1+\lambda )(\theta _{p}-v)<0.$ Thus, access charges are higher in Regime B than in A.

• ad (ii) Note that governmental transfers are given by $T^{s}=F+(v-a_{p}) \widehat{q}_{p}+2(v-a_{f})\widehat{q}_{f}$ in Regime s ∈ {A, B}. Substituting equilibrium access charges from Regimes A and B in T ^s, we compute $T^{A}-T^{B}=\frac{1}{\varphi ^{2}}(2+c_{p})(1+\lambda )(\theta _{p}-v)^{2}\left[ 5+8\lambda +c_{p}(5+c_{p}+\lambda (6+c_{p}))\right] >0$. Thus, governmental transfers are higher in Regime A than in Regime B.

• ad (iii) Substituting equilibrium access charges from Regimes A and B in the welfare function, we compute $W^{A}-W^{B}=\frac{1}{2\varphi ^{2}} (2+c_{p})^{2}(1+\lambda )^{2}(\theta _{p}-v)^{2}>0$. Thus, social welfare is higher in Regime A than in B.