Optimal access regulation with downstream competition

Abstract

We analyze the setting of access prices for a bottleneck facility where the facility owner also competes in the deregulated downstream market. We consider a continuum of market structures from Cournot to Bertrand. These market structures are fully characterized by a single parameter representing the intensity of competition. We first show how the efficient component pricing rule should be modified as the downstream competitive intensity changes. We then analyse the optimal access price where a regulator trades off production efficiency and pro-competitive effects to maximize total surplus.

Appendix

Proof

(Proposition 2): We examine market outcomes for interior solutions ($c_{A}<A+\frac{Q^{c}+\left( 2+\beta \right) \left( c-c_{E}\right) }{2}$) in this proposition. For $c>c_{E}$ and $c_{A}<A+\frac{Q^{c}+\left( 2+\beta \right) \left( c-c_{E}\right) }{2}$, partially differentiating Eq. (18) gives$\frac{\partial q_{E}^{*}}{\partial \beta }\ge 0$. The efficient entrant’s output increases as $\beta$ increases. On the other hand, for $c<c_{E}$ and $c_{A}<A+\frac{Q^{c}+\left( 2+\beta \right) \left( c-c_{E}\right) }{2}$, partially differentiating Eq. (17) gives $\frac{\partial q_{I}^{*}}{\partial \beta }\ge 0$. The efficient incumbent’s output increases as $\beta$ increases. Thus, as $\beta$ increases, the efficient firm’s output increases.

For the market aggregate output, for $c<c_{E}$ and $c_{A}<A+\frac{ Q^{c}+\left( 2+\beta \right) \left( c-c_{E}\right) }{2}$, $\frac{\partial Q}{ \partial \beta }\ge 0$. For $c_{E}<c$, $\frac{\partial Q}{\partial \beta } \ge 0$ if $c_{A}$ is sufficiently small ($c_{A}\le A+\frac{2Q^{c}+c-c_{E}}{ 4}$). Although the market aggregate output is not always increasing in $\beta$ for any arbitrary $c_{A}$. Once the optimal setting of $c_{A}$ is considered, this condition is always satisfied. As shown in Proposition 3, for $c_{E}<c$, $c_{A}^{*}<A+\frac{2Q^{c}+c-c_{E}}{4}$. $\square$

Proof

(Proposition 3): Let $c_{B}$ denote the $c_{A}$ level such that $\pi _{I}\left[ c_{B}\right] =0$.

$$\begin{aligned} TS&= CS+\pi _{I}+\pi _{E} \nonumber \\&= \frac{1}{2}\left( a-P^{*}\right) Q^{*}{+}\left( P^{*}-\left( A+c\right) \right) q_{I}^{*}{+}\left( c_{A}-A\right) q_{E}^{*}{+}\left( P^{*}{-}\left( c_{A}+c_{E}\right) \right) q_{E}^{*} \nonumber \\&= \frac{1}{2}\left( Q^{*}\right) ^{2}+\left( P^{*}-\left( A+c\right) \right) Q^{*}+\left( c-c_{E}\right) q_{E}^{*}\nonumber \\ \frac{\partial TS}{\partial c_{A}}&= Q^{*}\frac{\partial Q^{*}}{\partial c_{A}}+\left( \frac{\partial P^{*}}{\partial c_{A}}\right) Q^{*}+\left( P^{*}-\left( A+c\right) \right) \frac{\partial Q^{*}}{\partial c_{A}}+\left( c-c_{E}\right) \frac{\partial q_{E}^{*}}{\partial c_{A}} \nonumber \\&= Q^{*}\left( -\frac{\partial P^{*}}{\partial c_{A}}\right) +\left( \frac{\partial P^{*}}{\partial c_{A}}\right) Q^{*}+\left( P^{*}-\left( A+c\right) \right) \frac{\partial Q^{*}}{\partial c_{A}}+\left( c-c_{E}\right) \frac{\partial q_{E}^{*}}{\partial c_{A}} \nonumber \\&= \left( P^{*}-\left( A+c\right) \right) \frac{\partial Q^{*}}{\partial c_{A}}+\left( c-c_{E}\right) \frac{\partial q_{E}^{*}}{\partial c_{A}}. \end{aligned}$$

(20)

For an interior solution, $\frac{\partial TS}{\partial c_{A}}=0$ gives

$$\begin{aligned} c_{A}=A-\frac{\left( \beta +1\right) \left( 2\beta +5\right) \left( c-c_{E}\right) +\left( 2\beta +1\right) Q^{c}}{\left( 2\beta +1\right) ^{2}} <A. \end{aligned}$$

(21)

Given this $c_{A}$, the downstream price is

$$\begin{aligned} P=A+c-\frac{2\left( c-c_{E}\right) \left( 1+\beta \right) }{2\beta +1}<A+c. \end{aligned}$$

(22)

The break-even constraint for the incumbent is violated. For an interior solution in the final good market, the optimal access charge is the constrained optimum with the break-even constraint binding:

$$\begin{aligned}&c_{B}\left[ c_{E}\right] \\&\quad =A+\frac{\left( 4\beta ^{2}+8\beta +5\right) Q^{c}+4\left( c-c_{E}\right) \left( \beta +1\right) ^{2}}{2\left( 4\beta ^{2}+8\beta +5\right) } \\&\qquad -\frac{\left( 2\beta +3\right) \sqrt{\left( 4\beta ^{2}+8\beta +5\right) \left( Q^{c}\right) ^{2}+4\left( c-c_{E}\right) ^{2}\left( \beta +1\right) ^{3}}}{2\left( 4\beta ^{2}+8\beta +5\right) } \\&\quad <A. \end{aligned}$$

For the corner solutions: It is never optimal for the regulator to set $c_{A}>A+\frac{1}{2}\left( Q^{c}+\left( c-c_{E}\right) \left( 2+\beta \right) \right)$ and force the equilibrium $q_{I}>0$ and $q_{E}=0$. For $c_{A}<A-\left( 1+\beta \right) \left( Q^{c}-\left( 1+\beta \right) \left( c-c_{E}\right) \right)$ with $q_{E}>0$ and $q_{I}=0$, the $c_{A}$ required for break–even is at least $A$. $A<A-\left( 1+\beta \right) \left( Q^{c}-\left( 1+\beta \right) \left( c-c_{E}\right) \right)$ if $\beta > \frac{Q^{c}}{\left( c-c_{E}\right) }-1.$ $\square$

Proof

(Proposition 4): In an interior solution, the ECPR is given by

$$\begin{aligned} c_{A}=A+\frac{a+\left( 1+\beta \right) \left( C_{I}+C_{E}\right) -\beta \left( A-c_{A}\right) }{2\beta +3}-C_{I}, \end{aligned}$$

or

$$\begin{aligned} c_{A}=A+\frac{Q^{c}-\left( 1+\beta \right) \left( c-c_{E}\right) }{2}. \end{aligned}$$

(23)

For $c\ge c_{E}$, we have an interior solution and this access price is relevant if $\beta$ is sufficiently small, $\beta \le \frac{Q^{c}}{\left( c-c_{E}\right) }-1$. The access price implied by the ECPR is always above $A$ except for in the limit $\beta \rightarrow \infty$ and $c=c_{E}$. For sufficiently large $\beta$, we have a corner solution with $q_{E}>0$ and $q_{I}=0$, in which case the ECPR is given by

$$\begin{aligned} A+c+\frac{\beta \left( c_{A}-A\right) }{\beta +1}-A-c=c_{A}-A, \end{aligned}$$

or

$$\begin{aligned} c_{A}=A. \end{aligned}$$

For $c<c_{E}$, with $c_{A}=A+\frac{Q^{c}-\left( 1+\beta \right) \left( c-c_{E}\right) }{2}$, we always have a corner solution with $q_{I}>0$ and $q_{E}=0$. The equilibrium price is $P=c_{A}+c_{E}$. The minimum $c_{A}$ required is $c_{A}=A+\frac{Q^{c}+\left( 2+\beta \right) \left( c-c_{E}\right) }{2}$. $\square$

Proof

(Proposition ): Let $\varepsilon \equiv c_{E}-c>0$. The optimal access pricing rule is summarized in the following table, where $c_{AI}=A+\frac{\left( \beta +1\right) \left( 5+2\beta \right) \varepsilon -\left( 1+2\beta \right) Q^{c}}{\left( 2\beta +1\right) ^{2}}$, $\widetilde{ c_{A}}=A+\frac{1}{2}\left( Q^{c}{-}\varepsilon \left( 2+\beta \right) \right)$ , $\beta _{1}=\frac{2Q^{c}-5\varepsilon {-}\sqrt{\left( 2Q^{c}-7\varepsilon \right) \left( 2Q^{c}+\varepsilon \right) }}{4\varepsilon }$, $\beta _{2}= \frac{2Q^{c}-7\varepsilon +\sqrt{\left( 2Q^{c}-\varepsilon \right) \left( 2Q^{c}-9\varepsilon \right) }}{4\varepsilon }$, and $\beta _{3}=\frac{ 2Q^{c}-5\varepsilon +\sqrt{\left( 2Q^{c}-7\varepsilon \right) \left( 2Q^{c}+\varepsilon \right) }}{4\varepsilon }$. As discussed in Sect. 4.1, for the corner solution $q_{I}>0$ and $q_{E}=0$, the ECPR price is not defined. We argue that any access price keeping the inefficient entrant out of the market is consistent with the reasoning behind the ECPR. In such case, to facilitate presentation of the results, we abuse the language and simply write $c_{A}^{*}=c_{A}^{ECPR}$ in Table .

Table 1 Optimal access charge with $c_{E}=c+\epsilon$

For $c_{E}=c+\varepsilon$, it can be shown that it is never optimal for the regulator to set a $c_{A}$ so low that in equilibrium $q_{E}>0$ and $q_{I}=0$. We only need to consider the interior solution and the corner solution where $q_{E}=0$ and $q_{I}>0$. Denote the critical $c_{A}$ level above which $q_{I}>0$ and $q_{E}=0$ by $\widetilde{c_{A}}$, $\widetilde{c_{A}}\equiv A+\frac{1}{2}\left( Q^{c}-\varepsilon \left( 2+\beta \right) \right)$. This critical $\widetilde{c_{A}}$ decreases in both $\varepsilon$ and $\beta$ as more cost differential and a more competitive market both facilitate corner solution in equilibrium. In particular, $\widetilde{c_{A}}\ge A$ for $\beta <\frac{Q^{c}-2\varepsilon }{\varepsilon }$. For $c_{A}\le \widetilde{c_{A}}$ , as shown in the proof of Proposition 3, $c_{A}^{*}=A+\frac{\varepsilon \left( \beta +1\right) \left( 5+2\beta \right) -\left( 1+2\beta \right) Q^{c}}{\left( 2\beta +1\right) ^{2}}$ (Eq. ). Denote this optimal $c_{A}$ as $c_{AI}$.

For $c_{A}>\widetilde{c_{A}}$, the FOC gives $c_{A}^{*}=A-\varepsilon$. This solution is interior, $A-\varepsilon >\widetilde{c_{A}}$, if $\beta > \frac{Q^{c}}{\varepsilon }.$

For $c_{A}\le \widetilde{c_{A}}\,$, $c_{AI}\ge \widetilde{c_{A}}$ if $\varepsilon >\frac{2Q^{c}}{7}$. For $\varepsilon$ large enough, it is optimal to set the access price such that only the efficient incumbent produces. For $\varepsilon \le \frac{2Q^{c}}{7}$, $c_{AI}\le \widetilde{ c_{A}}$ if

$$\begin{aligned}&\frac{2Q^{c}-5\varepsilon -\sqrt{\left( 2Q^{c}-7\varepsilon \right) \left( 2Q^{c}+\varepsilon \right) }}{4\varepsilon } \nonumber \\&\quad \le \beta \le \frac{2Q^{c}-5\varepsilon +\sqrt{\left( 2Q^{c}-7\varepsilon \right) \left( 2Q^{c}+\varepsilon \right) }}{ 4\varepsilon }. \end{aligned}$$

(24)

Note that $\frac{2Q^{c}-5\varepsilon -\sqrt{\left( 2Q^{c}-7\varepsilon \right) \left( 2Q^{c}+\varepsilon \right) }}{4\varepsilon }<0$ if $\varepsilon <\frac{Q^{c}}{4}$.

$c_{AI}\ge A$ if $\varepsilon \ge \frac{2}{9}Q^{c}$. For $\varepsilon < \frac{2}{9}Q^{c}$, $c_{AI}>A$ if $\beta >\frac{2Q^{c}-7\varepsilon +\sqrt{ \left( 2Q^{c}-\varepsilon \right) \left( 2Q^{c}-9\varepsilon \right) }}{ 4\varepsilon }.$

For $\varepsilon \le \frac{2}{9}Q^{c}$ and $\beta \le \frac{ 2Q^{c}-7\varepsilon +\sqrt{\left( \varepsilon -2Q^{c}\right) \left( 9\varepsilon -2Q^{c}\right) }}{4\varepsilon }$, $c_{AI}<A$, $P^{*}\left[ c_{AI}\right] >A+c$, and

$$\begin{aligned}&c_{B}\left[ c+\varepsilon \right] \\&\quad =A+\frac{\left( 4\beta ^{2}+8\beta +5\right) Q^{c}-4\varepsilon \left( \beta +1\right) ^{2}-\left( 2\beta +3\right) \sqrt{\left( 4\beta ^{2}+8\beta +5\right) \left( Q^{c}\right) ^{2}+4\varepsilon ^{2}\left( \beta +1\right) ^{3}}}{2\left( 4\beta ^{2}+8\beta +5\right) } \\&\quad <A. \end{aligned}$$

For this parameter range, $c_{A}^{*}=\max \left\{ c_{AI},c_{B}\left[ c+\varepsilon \right] \right\} <A$. $\square$