On the performance of endogenous access pricing

Abstract

Endogenous access pricing (ENAP) is an alternative to the traditional procedure of setting a fixed access price that reflects the regulator’s estimate of the supplier’s average cost of providing access. Under ENAP, the access price reflects the supplier’s actual average cost of providing access, which varies with realized industry output. We show that in addition to eliminating the need to estimate industry output accurately and avoiding a divergence between upstream revenues and costs, ENAP can enhance the incentive of a vertically integrated producer to minimize its upstream operating cost. However, ENAP can sometimes discourage surplus-enhancing investment.

Appendix

Proof of Proposition 1

From (1) and (5), the VIP’s equilibrium profit under ENAP is:

$$\begin{aligned} \pi _{0}^{*} = \left[ \frac{Q^{*}}{N+1}\right] P\left( Q^{*}\right) +\frac{F}{Q^{*}}\left[ \frac{NQ^{*}}{N+1 }\right] -F = \frac{1}{N+1}\left[ Q^{*}P\left( Q^{*}\right) -F\right] . \end{aligned}$$

(8)

From (3) and (5), equilibrium industry output under ENAP is given by:

$$\begin{aligned}&Q^{*}\left[ N+1\right] P\left( Q^{*}\right) +\left( Q^{*}\right) ^{2}P^{\prime }\left( Q^{*}\right) -NF = 0 \end{aligned}$$

(9)

$$\begin{aligned}&\Leftrightarrow Q^{*}P\left( Q^{*}\right) = \frac{NF-\left( Q^{*}\right) ^{2}P^{\prime }\left( Q^{*}\right) }{ N+1}. \end{aligned}$$

(10)

From (8) and (10):

$$\begin{aligned} \frac{d\pi _{0}^{*}}{dF}=-\frac{1}{\left[ N+1 \right] ^{2}}\left\{ \left[ \left( Q^{*}\right) ^{2}P^{\prime \prime }\left( Q^{*}\right) +2Q^{*}P^{\prime }\left( Q^{*}\right) \right] \frac{\partial Q^{*}}{\partial F}+1\right\} . \end{aligned}$$

(11)

(11) implies that if $P^{\prime \prime }(Q^{*})\le 0$ and $\frac{\partial Q^{*}}{\partial F}\le 0$, then $\frac{d\pi _{0}^{*}}{dF}<0$, and so the VIP will set $F= \underline{F}$  under ENAP. To determine when $\frac{\partial Q^{*} }{\partial F}\le 0$, let $h\left( Q^{*}\right) \equiv Q^{*}\left[ N+1\right] P\left( Q^{*}\right) +\left( Q^{*}\right) ^{2}P^{\prime }\left( Q^{*}\right)$. It is readily verified that $h^{\prime \prime }\left( \cdot \right) <0$, and so $h\left( \cdot \right)$ is a concave function of $Q^{*}$, under the maintained conditions. From (9), $Q^{*}$ is determined by $h(Q^{*})=NF$, and so (9) will have at least one real root when $F$ is sufficiently small. Furthermore, when (9) has two real roots, the larger root of (9) decreases as $F$ increases, and so $\frac{\partial Q^{*}}{\partial F}<0$, when $h\left( \cdot \right)$ is a concave function of $Q^{*}$. Fjell et al. (2013) prove that the larger root of (9) is the relevant root in cases where (9) has two roots. $\square$

The following Lemmas are instrumental in the proof of Proposition 2.

Lemma 1

Suppose Assumption 1 holds. Then given access price $\widehat{w}$, the equilibrium output of the VIP under EXAP is $\widehat{q}_{0}^{ *}= \frac{{a}+ \widehat{w} N}{b \left[ N+2\right] }$. The equilibrium output of each of the $N$ rivals under EXAP is $\widehat{q}_{i}^{ *}= \frac{a - 2 \widehat{w}}{{b}\left[ N+2\right] }\, { for} \,\,i = 1,\ldots ,N$.

Proof

Differentiating (1) and ( 2) provides:

$$\begin{aligned} \frac{\partial \pi _{0}}{\partial q_{0}}=a-2bq_{0}-b \sum _{j=1}^{N}q_{j} \quad \text {and} \quad \frac{\partial \pi _{i}}{\partial q_{i}}=a-bq_{i}-bq_{0}-b\sum _{j=1}^{N}q_{j}-w. \quad \quad \end{aligned}$$

(12)

In equilibrium, $\frac{\partial \pi _{0}}{\partial q_{0}}=\frac{\partial \pi _{i}}{\partial q_{i}}=0$. Therefore, from (12):

$$\begin{aligned} b\sum _{i=1}^{N}q_{i}=Nbq_{0}-w N. \end{aligned}$$

(13)

Since $\frac{\partial \pi _{0}}{\partial q_{0}}=0$ in equilibrium, (12) and (13) provide the identified expressions for $\widehat{q}_{0}^{ *}$ and $\widehat{q}_{i}^{ *}$. $\square$

Lemma 2

Suppose Assumption 1 holds. Then when the VIP’s fixed cost is $F$, the access price that will be set under EXAP is $\widehat{w} (F)= \frac{1}{2N}\left[ a\left( N+1\right) -\sqrt{ \widehat{G}(F)} \right]$ where $\widehat{G}(F) \equiv a^{2}\left[ N+1 \right] ^{2}-4bFN\left[ N+2 \right]$.

Proof

From Lemma 1, $\widehat{Q}^{ *}=q_{0}^{*}+\sum _{i=1}^{N}\widehat{q}_{i}^{ *}= \frac{a\left[ N+1 \right] -wN}{b\left[ N+2 \right] }$.Therefore, when $Q^{e}= \widehat{Q}^{ *}, \ w= \frac{F}{\widehat{Q}^{ *}}= \frac{bF\left[ N+2\right] }{a \left[ N+1\right] -wN}$, which ensures that $\widehat{w}(F)$ is as specified in the lemma. $\square$

Lemma 3

Suppose Assumption 1 holds. Then for a given fixed cost, $F$, the VIP’s equilibrium profit under EXAP is:

$$\begin{aligned} \widehat{\pi }_{0}^{*}(F)&= \frac{1}{ 4bN^{2}\left[ N+2 \right] ^{2}} \left\{ 2aN\left[ N+4 \right] \sqrt{\widehat{G}(F)}+4bFN^{2}\left[ N+4 \right] \left[ N+2 \right] \right. \\&\quad \left. -2a^{2}N\left[ N^{2}+3N+4 \right] \right\} -F. \end{aligned}$$

Proof

From Lemmas 1 and 2:

$$\begin{aligned} \widehat{\pi }_{0}^{ *} = \widehat{q}_{0}^{ *}\left[ a-b \widehat{Q}^{ *}\right] +\widehat{w} \sum _{i=1}^{N}\widehat{q}_{i}^{ *}-F = \frac{H}{b\left[ N+2 \right] ^{2}}-F, \end{aligned}$$

where $H \equiv \left[ a+\widehat{w}N \right] ^{2}+\left[ N+2 \right] \widehat{w}N\left[ a-2 \widehat{w} \right]$. Substituting for $\widehat{w}(F) \$from Lemma 2 provides the identified expression for $\widehat{\pi }_{0}^{*}(F)$.$\square$

Proof of Proposition 2

Differentiating $\widehat{\pi }_{0}^{*}(F)$ provides $\ \widehat{ \pi }_{0}^{*\prime }(F)=\frac{N+4}{N+2}\left[ -\frac{a}{\sqrt{\widehat{G}}}+1\right] -1$, which implies $\widehat{\pi }_{0}^{*\prime }(F)\gtreqless 0$ $\Leftrightarrow$ $F \lesseqgtr \frac{3a^{2}\left[ N-2 \right] }{16bN}$. Therefore, $\frac{\partial \pi _{0}^{*}}{ \partial F}<0$ (and so $\widehat{F}^{*}=\underline{F}$) if $N\le 2$. In contrast, if $N\ge 3$, then $\widehat{F}^{ *}=\min \left\{ \max \left( \underline{F},\frac{ 3a^{2}\left[ N-2\right] }{16bN}\right) ,\overline{F}\right\}$. Consequently, $\widehat{F}^{*}> \underline{F}$ if $\underline{F}<\frac{ 3a^{2}\left[ N-2\right] }{16bN}$. This will be the case if $\underline{F} <\frac{a^{2}}{16b}$, since $z(N)\equiv \frac{N-2}{N}$ is an increasing function of $N$ with $z(3)= \frac{1}{3}$. $\square$

Proof of Proposition 3

The incumbent’s profit in the setting with variable access costs is:

$$\begin{aligned} \pi _{0} = q_{0}P\!\left( Q\right) +w\sum _{i=1}^{n}q_{i}-F-c(F)Q. \end{aligned}$$

(14)

Differentiating (14) provides:

$$\begin{aligned} \frac{\partial \pi _{0}}{\partial F}=\frac{ \partial }{\partial F}\left\{ q_{0}P\!\left( Q\right) +w\sum _{i=1}^{n}q_{i}-F\right\} -\frac{\partial }{\partial F}\left\{ c(F)Q\right\} \end{aligned}$$

(15)

where:

$$\begin{aligned} \frac{\partial }{\partial F}\left\{ c(F)Q\right\} =Q \left[ \frac{\partial c(\cdot )}{\partial F}\right] +c(F)\left[ \frac{ \partial Q}{\partial F}\right] =-Qr^{\prime }(F)+c(F) \left[ \frac{\partial Q}{\partial F}\right] . \end{aligned}$$

(16)

From Proposition 3 in Fjell et al. (2010), the equilibrium value of $Q$ is the same under EXAP and ENAP for a given $F$. Therefore, it must be the case that both $Q$ and the rate at which $Q$ varies with $F$ are the same at each $F$ under EXAP and ENAP. Consequently, (16) implies that for any given $F, \frac{\partial }{\partial F}\left\{ c(F)Q\right\}$ is the same under exogenous access pricing and endogenous access pricing.

Under the conditions specified in Proposition 2, $\frac{\partial }{\partial F}\left\{ q_{0}P(Q)+w\sum _{i=1}^{n}q_{i}-F\right\}$ is strictly positive under EXAP for $F\in [0, \widehat{F}^{*})$ (where $\widehat{F}^{*}>0$) and strictly negative under ENAP for all $F\ge 0$. Therefore, (15) implies that for each $F, \frac{ \partial \pi _{0}}{\partial F}$ is larger under EXAP than under ENAP, and so the VIP will implement a larger level of $F$ under EXAP than under ENAP in the setting with variable access costs. $\square$