Access regulation and the transition from copper to fiber networks in telecoms

Abstract

In this paper we study the impact of different forms of access obligations on firms’ incentives to migrate from the legacy copper network to next generation broadband infrastructures. We analyze geographically differential access prices of copper (that depend on whether or not an alternative fiber network has been deployed in the area) and ex-ante access obligations for fiber networks. We discuss how these regulatory schemes fare in addressing the tension among different objectives, such as the promotion of static efficiency, fostering investments in new infrastructures, and avoiding unnecessary duplication of (fiber) networks.

Appendix

1.1 Appendix A: Profits (per-area) for industry configurations (1) - (3)

Let $s_{i}$ denote firm $i$’s quality, with $s_{i}\in \left\{ s^{O},s^{N}\right\}$, and $i=1,2$.

(1) Service-based competition within the copper network.

We have $s_{1}=s_{2}=s^{O}$. The incumbent’s profit is $\pi _{1}^{O,O}=p_{1}q_{1}+aq_{2}$, and the entrant’s profit is $\pi _{2}^{O,O}=(p_{2}-a)q_{2}$. In the equilibrium of the quantity-setting game, we have $\pi _{1}^{O,O}\left( a\right) =\left( (1+s^{O})^{2}+5a\left( 1-a\right) +5as^{O}\right) /9$ and $\pi _{2}^{O,O}\left( a\right) =(1+s^{O}-2a)^{2}/9$. Note that $\pi _{2}^{O,O}\left( a\right) \ge 0$ if and only if $a\le \overline{a}^{O}=\left( 1+s^{O}\right) /2$. Besides, the incumbent’s gross profit is maximized at $\widehat{a}^{O}=\arg \underset{a}{ \max }\pi _{1}^{O,O}\left( a\right) =(1+s^{O})/2=\overline{a}^{O}$.

(2) Infrastructure-based competition between the copper and the fiber networks.

i. The incumbent uses its copper network and the entrant uses its own fiber network.

We have $s_{1}=s^{O}$ and $s_{2}=s^{N}$. The incumbent’s profit is $\pi _{1}^{O,N}=p_{1}q_{1}$, and the entrant’s profit is $\pi _{2}^{O,N}=p_{2}q_{2}$. In equilibrium, we have $\pi _{1}^{O,N}=(1+2s^{O}-s^{N})^{2}/9$ and $\pi _{2}^{O,N}=(1+2s^{N}-s^{O})^{2}/9$.

ii. The incumbent uses a fiber network and the entrant uses the incumbent’s copper network.

We have $s_{1}=s^{N}$ and $s_{2}=s^{O}$. The incumbent’s profit is $\pi _{1}^{N,O}=p_{1}q_{1}+aq_{2}$, and the entrant’s profit is $\pi _{2}^{N,O}=(p_{2}-a)q_{2}$. In equilibrium, we have $\pi _{1}^{N,O}\left( a\right) =((1+2s^{N}-s^{O})^{2}+5a\left( 1-a\right) +a(s^{N}+4s^{O}))/9$ and $\pi _{2}^{N,O}\left( a\right) =(1+2s^{O}-s^{N}-2a)^{2}/9$. Note that $\pi _{2}^{N,O}\left( a\right) \ge 0$ if and only if $a\le \overline{a} ^{N}=\left( 1+2s^{O}-s^{N}\right) /2$, and that $\overline{a}^{N}<\overline{a }^{O}$ as $s^{N}>s^{O}$. The incumbent’s gross profit is maximized at $\widehat{a}^{N}=\arg \underset{a}{\max }\pi _{1}^{N,O}\left( a\right) =(5+s^{N}+4s^{O})/10>\overline{a}^{N}$.

(3) Infrastructure-based competition between the fiber networks.

We have $s_{1}=s_{2}=s^{N}$. The incumbent’s profit is $\pi _{1}^{N,N}=p_{1}q_{1}$, and the entrant’s profit is $\pi _{2}^{N,N}=p_{2}q_{2}$. In equilibrium, we have $\pi _{1}^{N,N}=(1+s^{N})^{2}/9$ and $\pi _{2}^{N,N}=(1+s^{N})^{2}/9$

1.2 Appendix B: Properties of per-area profits

B1.

From the expressions of profits given in Appendix A, we have $\partial \pi _{2}^{k,O}\left( a\right) /\partial a\le 0$, for $k=O,N$. Furthermore, $\pi _{1}^{O,O}\left( a\right)$ increases with $a$, for all $a\le \widehat{a}^{O}=\overline{a}^{O}$. Similarly, $\pi _{1}^{N,O}\left( a\right)$ increases with $a$, for all $a\le \overline{a}^{N}<\widehat{a}^{N}$, as we have $\partial ^{2}\pi _{1}^{N,O}\left( a\right) /\partial a^{2}<0$ and $\partial \pi _{1}^{N,O}\left( a\right) /\partial a\left( a=\overline{a} ^{N}\right) =2\left( s^{N}-s^{O}\right) /3>0$.

B2.

First, since $\pi _{1}^{O,O}\left( a\right)$ increases with $a$ for $a\le \widehat{a}^{O}=\overline{a}^{O}$, we have $\pi _{1}^{O,O}\left( a\right) \ge \pi _{1}^{O,O}\left( 0\right) =\left( 1+s^{O}\right) ^{2}/9$. Since $s^{O}<s^{N}$, we have $\pi _{1}^{O,N}<\left( 1+s^{O}\right) ^{2}/9$, and hence, $\pi _{1}^{O,O}\left( a\right) >\pi _{1}^{O,N}$ for all $a$. Second, since $\pi _{1}^{N,O}\left( a\right)$ increases with $a$ for all $a\le \overline{a}^{N}$, we have $\pi _{1}^{N,O}\left( a\right) \ge \pi _{1}^{N,O}\left( 0\right) =(1+2s^{N}-s^{O})^{2}/9$. As $s^{N}>s^{O}$, we then have $\pi _{1}^{N,O}\left( a\right) >(1+s^{N})^{2}/9=\pi _{1}^{N,N}$.

1.3 Appendix C: Quality of the copper network and the fiber coverage

We study the effect of a higher quality for the copper network, $s^{O}$, on firms’ investment in fiber. We focus on the two asymmetric equilibria, $\left\{ z_{2}^{m},z_{1}^{c}\right\}$ if the entrant dominates in fiber investment, and $\left\{ z_{1}^{m},z_{2}^{c}\right\}$ if it is the incumbent that dominates. We find that $z_{1}^{c}$, $z_{1}^{m}$, $z_{2}^{m}$ , and $z_{2}^{c}$ decrease with the quality of the copper network, $s^{O}$. Indeed, $\partial z_{1}^{c}/\partial s^{O}=\partial (\pi _{1}^{N,N}-\pi _{1}^{O,N})/\partial s^{O}=-4(1+2s^{O}-s^{N\ })/9\le 0$, since $s^{N}<1+2s^{O}$ from our assumptions. We have $\partial z_{1}^{m}/\partial s^{O}=\partial (\pi _{1}^{N,O}-\pi _{1}^{O,O})/\partial s^{O}=-\left( a+4+4s^{N}\right) /9\le 0$, and $\partial z_{2}^{m}/\partial s^{O}=\partial (\pi _{2}^{O,N}-\pi _{2}^{O,O})/\partial s^{O}=-4\left( 1+s^{N}-a\right) /9\le 0$, as $a\le \overline{a}^{O}$ and $s^{N}>s^{O}$. Finally, we have $\partial z_{2}^{c}/\partial s^{O}=\partial (\pi _{2}^{N,N}-\pi _{2}^{N,O})/\partial s^{O}=-4\left( 1+2s^{O}-s^{N}-2a\right) /9\le 0$, since $a\le \overline{a}^{N}$.

1.4 Appendix D: Social welfare in local areas

D1. Expressions for social welfare.

Recall that $s_{i}$ is the quality offered by firm $i$. Consumer surplus is given by $CS=\int _{\widetilde{\tau }}^{1}\left( \tau - \widehat{p}^{*}\right) d\tau$, where $\widehat{p}^{*}=p_{1}^{ *}-s_{1}=p_{2}^{*}-s_{2}$ is the quality-adjusted price in the equilibrium of the quantity-setting subgame, and $\widetilde{\tau }=\widehat{p }^{*}$ is the marginal consumer. We find that $CS=\left( 2+s_{1}+s_{2}-a\right) ^{2}/18$. The local social welfare is $w=CS+\pi _{1}+\pi _{2}$, and $w=\left( 4+4s_{2}+a\right) \left( 2+2s_{2}-a\right) /18+11\left( s_{1}-s_{2}\right) ^{2}/18+4\left( a+1+s_{2}\right) \left( s_{1}-s_{2}\right) /9$.

D2. Variations of local welfare with the copper access price

When there is service-based competition within the copper network, we have $s_{1}=s_{2}=s^{O}$, and we find that $\partial w^{O,O}/\partial a=-\left( a+1+s^{O}\right) /9<0$. When firm 1 uses a fiber network and firm 2 uses the copper network, we have $s_{1}=s^{N}$ and $s_{2}=s^{O}$, and we find that $\partial w^{N,O}/\partial a=-\left( a+1+5s^{O}-4s^{N}\right) /9<0$, as $1+5s^{O}-4s^{N}>0$ under our assumptions on $s^{N}$.

1.5 Appendix E: Equilibrium coverage with differential access pricing

The entrant’s investment decisionAssume that firm 1 has covered the areas $\left[ 0,z_{1}\right]$. Firm 2’s profit is then given by

$$\begin{aligned} \Pi _{2}(z_{1},z_{2})=-C\left( z_{2}\right) +\left\{ \begin{array}{l@{\quad }c} z_{2}\pi _{2}^{N,N}+\left( z_{1}-z_{2}\right) \pi _{2}^{N,O}\left( a^{N}\right) +\left( \overline{z}-z_{1}\right) \pi _{2}^{O,O}\left( a^{O}\right) &{} \text {if } z_{2}\le z_{1} \\ z_{1}\pi _{2}^{N,N}+\left( z_{2}-z_{1}\right) \pi _{2}^{O,N}+\left( \overline{z }-z_{2}\right) \pi _{2}^{O,O}\left( a^{O}\right) &{} \text {if }z_{2}>z_{1} \end{array} \right. \text {.} \end{aligned}$$

Similar to Bourreau et al. (2012), we define $z_{2}^{c}\left( a^{N}\right) =\left( c\right) ^{-1}\left( \pi _{2}^{N,N}-\pi _{2}^{N,O}\left( a^{N}\right) \right)$ and $z_{2}^{m}\left( a^{O}\right) =\left( c\right) ^{-1}\left( \pi _{2}^{O,N}-\pi _{2}^{O,O}\left( a^{O}\right) \right)$. Note that depending on the values of $a^{O}$ and $a^{N}$ we can have either $z_{2}^{m}(a^{O})>z_{2}^{c}(a^{N})$ or the opposite. If $z_{2}^{m}>z_{2}^{c}$ , the entrant’s best-response is

$$\begin{aligned} z_{2}^{\text {BR}}\left( z_{1}\right) =\left\{ \begin{array}{l@{\quad }ll} z_{2}^{m}\left( a^{O}\right) &{} \text {if} &{} z_{1}\le \widehat{z}_{1} \\ z_{2}^{c}\left( a^{N}\right) &{} \text {if} &{} z_{1}>\widehat{z}_{1} \end{array} \right. \text {,} \end{aligned}$$

where $\widehat{z}_{1}\left( a^{O},a^{N}\right) \in \left[ z_{2}^{c},z_{2}^{m} \right]$ is the lowest $z_{1}$ such that $\Pi _{2}\left( z_{1},z_{2}^{c}\left( a^{N}\right) \right) \ge \Pi _{2}\left( z_{1},z_{2}^{m}\left( a^{O}\right) \right)$. If $z_{2}^{c}\ge z_{2}^{m}$, the entrant’s best-response is

$$\begin{aligned} z_{2}^{\text {BR}}\left( z_{1}\right) =\left\{ \begin{array}{l@{\quad }l@{\quad }l} z_{2}^{m}\left( a^{O}\right) &{} \text {if} &{} z_{1}\le z_{2}^{m} \\ z_{1} &{} \text {if} &{} z_{2}^{m}<z_{1}\le z_{2}^{c} \\ z_{2}^{c}\left( a^{N}\right) &{} \text {if} &{} z_{1}>z_{2}^{c}\end{array} \right. \text {.} \end{aligned}$$

The incumbent’s investment decisionThe profits of the incumbent in both cases are as follows. If $z_{2}^{c}<z_{2}^{m}$, firm 1’s profit is given by:

$$\begin{aligned}&\Pi _{1}(z_{1},z_{2}^{\text {BR}}\left( z_{1}\right) )\\&\quad =-C\left( z_{1}\right) +\left\{ \begin{array}{l@{\quad }l} z_{1}\pi _{1}^{N,N}+\left( z_{2}^{m}-z_{1}\right) \pi _{1}^{O,N}+\left( \overline{z}-z_{2}^{m}\right) \pi _{1}^{O,O}\left( a^{O}\right) &{} \text {if } z_{1}\in \left[ 0,\widehat{z}_{1}\right] \\ z_{2}^{c}\pi _{1}^{N,N}+\left( z_{1}-z_{2}^{c}\right) \pi _{1}^{N,O}\left( a^{N}\right) +\left( \overline{z}-z_{1}\right) \pi _{1}^{O,O}\left( a^{O}\right) &{} \text {if }z_{1}\in \left[ \widehat{z}_{1},\overline{z}\right] \end{array} \right. \text {.} \end{aligned}$$

If $z_{2}^{c}\ge z_{2}^{m}$, firm 1’s profit is given by

$$\begin{aligned}&\Pi _{1}(z_{1},z_{2}^{\text {BR}}\left( z_{1}\right) )\\&\quad =-C\left( z_{1}\right) +\left\{ \begin{array}{l@{\quad }l} z_{1}\pi _{1}^{N,N}+\left( z_{2}^{m}-z_{1}\right) \pi _{1}^{O,N}+\left( \overline{z}-z_{2}^{m}\right) \pi _{1}^{O,O}\left( a^{O}\right) &{} \text {if } z_{1}\in \left[ 0,z_{2}^{m}\right] \\ z_{1}\pi _{1}^{N,N}+\left( \overline{z}-z_{1}\right) \pi _{1}^{O,O}\left( a^{O}\right) &{} \text {if }z_{1}\in \left[ z_{2}^{m},z_{2}^{c}\right] \\ z_{2}^{c}\pi _{1}^{N,N}+\left( z_{1}-z_{2}^{c}\right) \pi _{1}^{N,O}\left( a^{N}\right) +\left( \overline{z}-z_{1}\right) \pi _{1}^{O,O}\left( a^{O}\right) &{} \text {if }z_{1}\in \left[ z_{2}^{c},\overline{z}\right] \end{array} \right. \text {.} \end{aligned}$$

Let $z_{1}^{c}=\left( c\right) ^{-1}(\pi _{1}^{N,N}-\pi _{1}^{O,N})$, and $z_{1}^{m}\left( a^{O},a^{N}\right) =\left( c\right) ^{-1}(\pi _{1}^{N,O}\left( a^{N}\right) -\pi _{1}^{O,O}\left( a^{O}\right) )$. When $z_{2}^{c}<z_{2}^{m}$ and when $z_{2}^{c}\ge z_{2}^{m}$, we have the same two potential asymmetric equilibria as in the case with uniform access pricing, and either the incumbent or the entrant dominates fiber investments. That is, the equilibrium coverage is either $\left\{ z_{1}^{m}\left( a^{O},a^{N}\right) ,z_{2}^{c}\left( a^{N}\right) \right\}$ or $\left\{ z_{1}^{c},z_{2}^{m}(a^{O})\right\}$).⁵¹

1.6 Appendix F: Sign of first-order derivatives of welfare

• (i) $\partial z_{1}^{m}/\partial a^{O}<0$, $\partial z_{1}^{m}/\partial a^{N}>0$ and $dz_{2}^{c}/da^{N}>0$. Since $\pi _{1}^{N,O}\left( a^{N}\right)$ increases with $a^{N}$ and $\pi _{1}^{O,O}\left( a^{O}\right)$ increases with $a^{O}$, we have $\partial z_{1}^{m}/\partial a^{O}<0$ and $\partial z_{1}^{m}/\partial a^{N}>0$. Since $\pi _{2}^{N,O}\left( a^{N}\right)$ decreases with $a^{N}$, then $dz_{2}^{c}/da^{N}>0$.

• (ii) $w^{N,O}\left( a^{N}\right) -w^{O,O}\left( a^{O}\right) -c\left( z_{1}^{m}\right)$ can be either positive or negative. From the definition of $z_{1}^{m}$, we have $c\left( z_{1}^{m}\right) =\pi _{1}^{N,O}\left( a^{N}\right) -\pi _{1}^{O,O}\left( a^{O}\right)$. We find that

  $$\begin{aligned} \Delta _{1}(a^{O},a^{N})&=w^{N,O}\left( a^{N}\right) -w^{O,O}\left( a^{O}\right) -\left( \pi _{1}^{N,O}\left( a^{N}\right) -\pi _{1}^{O,O}\left( a^{O}\right) \right) \\&=\left( 3\left( a^{N}\right) ^{2}-3\left( a^{O}\right) ^{2}+2a^{N}\left( s^{N}-3s^{O}-2\right) \right. \\&\quad \left. +\left( s^{N}-s^{O}\right) ^{2}+4a^{O}\left( 1+s^{O}\right) \right) /6\text {.} \end{aligned}$$

  We have $\Delta _{1}(a,a)=(s^{N}-s^{O})(s^{N}-s^{O}+2a)/6>0$. However, we can have $\Delta _{1}(a^{O},a^{N})<0$ too. For example, assume that $s^{O}=1$, $s^{N}=1.2$, then $\Delta _{1}(0,0.2)<0$.

• (iii) $w^{N,N}-w^{N,O}\left( a^{N}\right) -c\left( z_{2}^{c}\right) <0$. From the definition of $z_{2}^{c}$, we have $c\left( z_{2}^{c}\right) =\pi _{2}^{N,N}-\pi _{2}^{N,O}\left( a^{N}\right)$. We find that $w^{N,N}-w^{N,O}\left( a^{N}\right) -\left( \pi _{2}^{N,N}-\pi _{2}^{N,O}\left( a^{N}\right) \right) =\left( 3(a^{N})^{2}-2a^{N}\left( 1+s^{O}\right) -\left( s^{N}-s^{O}\right) ^{2}\right) /6\equiv \Delta _{2}$. $\Delta _{2}$ is a second-degree polynomial with an inverted bell-shape, and we have $\left. \partial \Delta _{2}/\partial a^{N}\right| _{a^{N}=0}<0$ and $\Delta _{2}\left( a^{N}=0\right) <0$. Besides, we have $\Delta _{2}\left( a^{N}=\overline{a}^{N}\right) <0$. Therefore, $\Delta _{2}<0$ always holds, and hence, $w^{N,N}-w^{N,O}\left( a^{N}\right) -c\left( z_{2}^{c}\right) <0$.

1.7 Appendix G: Access to fiber

G1. Profits with access to fiber.

When one firm (firm 1 or firm 2) leases access to the fiber network of its rival, both firms offer services of quality $s^{N}$. Let firm $i$ be the access provider, and firm $j\ne i$ be the access seeker, with $i,j=1,2$. We find that $\widetilde{\pi }_{i}^{N,N}\left( \widetilde{a}\right) =\left( (1+s^{N})^{2}+5\widetilde{a}(1+s^{N}-\widetilde{a})\right) /9$ and $\pi _{j}^{N,N}\left( \widetilde{a}\right) =\left( 1+s^{N}-2\widetilde{a} \right) ^{2}/9$. Firm $j$ has a positive demand if $\widetilde{a} \le (1+s^{N})/2$. We find that $\partial \widetilde{\pi }_{i}^{N,N}\left( \widetilde{a}\right) /\partial \widetilde{a}\ge 0$ for $\widetilde{a} \le (1+s^{N})/2$, and that $\partial \pi _{j}^{N,N}\left( \widetilde{a}\right) /\partial \widetilde{a}\le 0$.

G2. Equilibrium of the coverage game.

The entrant’s investment decisionGiven firm 1’s coverage $z_{1}$, firm 2’s profit is

$$\begin{aligned}&\widetilde{\Pi }_{2}\left( z_{1},z_{2}\right) \\&\quad =\left\{ \begin{array}{ll} z_{2}\pi _{2}^{N,N}+\left( z_{1}-z_{2}\right) \pi _{2}^{N,N}\left( \widetilde{a }\right) +\left( \overline{z}-z_{1}\right) \pi _{2}^{O,O}\left( a\right) -C\left( z_{2}\right) &{} \text {if }z_{2}\le z_{1} \\ z_{1}\pi _{2}^{N,N}+\left( z_{2}-z_{1}\right) \widetilde{\pi }_{2}^{N,N}\left( \widetilde{a}\right) +\left( \overline{z}-z_{2}\right) \pi _{2}^{O,O}\left( a\right) -C\left( z_{2}\right) &{} \text {if }z_{2}>z_{1} \end{array} \right. \text {.} \end{aligned}$$

We define $\widetilde{z}_{2}^{c}$ and $\widetilde{z}_{2}^{m}$ as the values of $z_{2}$ that maximize the first and second lines of $\widetilde{\Pi } _{2}\left( z_{1},z_{2}\right)$, respectively, for $z_{2}\in \left[ 0, \overline{z}\right]$. We have $\widetilde{z}_{2}^{c}\left( \widetilde{a} \right) =\left( c\right) ^{-1}(\pi _{2}^{N,N}-\pi _{2}^{N,N}\left( \widetilde{a }\right) )$ and $\widetilde{z}_{2}^{m}\left( a,\widetilde{a}\right) =\left( c\right) ^{-1}(\widetilde{\pi }_{2}^{N,N}\left( \widetilde{a}\right) -\pi _{2}^{O,O}\left( a\right) )$. Since the wholesale migration condition holds, we have $\pi _{2}^{N,N}\left( \widetilde{a}\right) \ge$ $\pi _{2}^{N,O}\left( a\right)$, which implies that $\widetilde{z} _{2}^{c}\left( \widetilde{a}\right) \le z_{2}^{c}\left( a\right)$. Additionally, we have $\widetilde{z}_{2}^{m}\left( a,\widetilde{a}\right) \le z_{2}^{m}\left( a\right)$ as $\widetilde{\pi }_{2}^{N,N}\left( \widetilde{a}\right) \le \pi _{2}^{O,N}$ for all $\widetilde{a}\le \widetilde{a}_{1}^{\max }$. Finally, we have $\partial \widetilde{z} _{2}^{c}/\partial \widetilde{a}$, $\partial \widetilde{z}_{2}^{m}/\partial \widetilde{a}$, $\partial \widetilde{z}_{2}^{m}/\partial a\ge 0$. That is, increasing the access price to the copper network or to the fiber network increases fiber coverage. The entrant’s best-response function is then

$$\begin{aligned} \widetilde{z}_{2}^{\text {BR}}\left( z_{1}\right) \!=\! \left\{ \begin{array}{l@{\quad }ll} \widetilde{z}_{2}^{m} &{} \text {if} &{} z_{1}\le \widetilde{z}_{2}^{m}\left( a, \widetilde{a}\right) \\ z_{1} &{} \text {if} &{} \widetilde{z}_{2}^{m}\left( a,\widetilde{a}\right) <z_{1} \!\le \!\widetilde{z}_{2}^{c}\left( \widetilde{a}\right) \\ \widetilde{z}_{2}^{c} &{} \text {if} &{} z_{1} \!>\! \widetilde{z}_{2}^{c}\left( \widetilde{a}\right) \end{array} \right. \quad \!\text {and}\quad \widetilde{z}_{2}^{\text {BR}}\left( z_{1}\right) \!=\!\left\{ \begin{array}{l@{\quad }ll} \widetilde{z}_{2}^{m} &{} \text {if} &{} z_{1}\!\le \!\widetilde{z}_{1}\left( a, \widetilde{a}\right) \\ \widetilde{z}_{2}^{c} &{} \text {if} &{} z_{1}\!>\!\widetilde{z}_{1}\left( a, \widetilde{a}\right) \end{array} \right. \end{aligned}$$

for $\widetilde{z}_{2}^{c}>\widetilde{z}_{2}^{m}$ and $\widetilde{z} _{2}^{c}\le \widetilde{z}_{2}^{m}$, respectively, where $\widetilde{z}_{1}$ is the lowest $z_{1}$ such that $\widetilde{\Pi }_{2}\left( z_{1},\widetilde{z }_{2}^{c}\right) \ge \widetilde{\Pi }_{2}\left( z_{1},\widetilde{z} _{2}^{m}\right)$ holds.

The incumbent’s investment decisionIn the case where $\widetilde{z}_{2}^{c}>\widetilde{z}_{2}^{m}$, firm 1’s profit is

$$\begin{aligned}&\widetilde{\Pi }_{1}(z_{1},\widetilde{z}_{2}^{\text {BR}}\left( z_{1}\right) )\\&\quad =\left\{ \begin{array}{l@{\quad }l} z_{1}\pi _{1}^{N,N}+\left( \widetilde{z}_{2}^{m}-z_{1}\right) \pi _{1}^{N,N}\left( \widetilde{a}\right) +\left( \overline{z}-\widetilde{z} _{2}^{m}\right) \pi _{1}^{O,O}\left( a\right) -C\left( z_{1}\right) &{} \text { if }z_{1}\in \left[ 0,\widetilde{z}_{2}^{m}\right] \\ z_{1}\pi _{1}^{N,N}+\left( \overline{z}-z_{1}\right) \pi _{1}^{O,O}\left( a\right) -C\left( z_{1}\right) &{} \text {if }z_{1}\in \left[ \widetilde{z} _{2}^{m},\widetilde{z}_{2}^{c}\right] \\ \widetilde{z}_{2}^{c}\pi _{1}^{N,N}+\left( z_{1}-\widetilde{z}_{2}^{c}\right) \widetilde{\pi }_{1}^{N,N}\left( \widetilde{a}\right) +\left( \overline{z} -z_{1}\right) \pi _{1}^{O,O}\left( a\right) -C\left( z_{1}\right) &{} \text {if }z_{1}\in \left[ \widetilde{z}_{2}^{c},\overline{z}\right] \end{array} \right. \text {.} \end{aligned}$$

Let $\widetilde{z}_{1}^{c}$, $\widetilde{z}_{1}^{d}$ and $\widetilde{z} _{1}^{m}$ denote the maxima of the first, second, and third lines of $\widetilde{\Pi }_{1}(z_{1},\widetilde{z}_{2}^{\text {BR}}\left( z_{1}\right) )$, respectively, for $z_{1}\in \left[ 0,\overline{z}\right]$. Firm 1’s profit can be written in a similar way for $\widetilde{z}_{2}^{c}\le \widetilde{z}_{2}^{m}$, which yields three maxima for different ranges of values for $z_{1}$. Similar to the baseline setting, we find two potential asymmetric equilibria, one in which the incumbent invests more than the entrant $\left( \left\{ \widetilde{z}_{1}^{m},\widetilde{z}_{2}^{c}\right\} \right)$, and one where it is the entrant that invests more $\left( \left\{ \widetilde{z} _{1}^{c},\widetilde{z}_{2}^{m}\right\} \right)$.

G3: Regulator’s choice of the access price of fiber.

Consider first the case where the incumbent invests more than the entrant. The equilibrium coverage are $z_{1}^{*}=\widetilde{z} _{1}^{m}\left( a,\widetilde{a}\right)$ and $z_{2}^{*}=\widetilde{z} _{2}^{c}\left( \widetilde{a}\right)$, and the social welfare is $W= \widetilde{z}_{2}^{c}\left( \widetilde{a}\right) w^{N,N}+\left( \widetilde{z} _{1}^{m}\left( a,\widetilde{a}\right) -\widetilde{z}_{2}^{c}\left( \widetilde{a}\right) \right) w^{N,N}\left( \widetilde{a}\right) +\left( \overline{z}-\widetilde{z}_{1}^{m}\left( a,\widetilde{a}\right) \right) w^{O,O}\left( a\right) -C\left( \widetilde{z}_{1}^{m}\right) -C\left( \widetilde{z}_{2}^{c}\right)$. Assuming an interior solution, the socially optimal access price of fiber solves

$$\begin{aligned} \frac{\partial W}{\partial \widetilde{a}}&=\frac{d\widetilde{z} _{2}^{c}\left( \widetilde{a}\right) }{d\widetilde{a}}\left( w^{N,N}-w^{N,N}\left( \widetilde{a}\right) -c\left( \widetilde{z} _{2}^{c}\right) \right) \\&\quad +\frac{\partial \widetilde{z}_{1}^{m}\left( a, \widetilde{a}\right) }{\partial \widetilde{a}}\left( w^{N,N}\left( \widetilde{ a}\right) -w^{O,O}\left( a\right) -c\left( \widetilde{z}_{1}^{m}\right) \right) \\&\quad +\left( \widetilde{z}_{1}^{m}-\widetilde{z}_{2}^{c}\right) \frac{ dw^{N,N}\left( \widetilde{a}\right) }{d\widetilde{a}}\equiv G\left( a, \widetilde{a}\right) =0\text {.} \end{aligned}$$

Let $\widetilde{a}^{w}$ denote the solution of $G\left( a,\widetilde{a} ^{w}\right) =0$. From the implicit function theorem, provided that the second-order condition holds, the sign of $\partial \widetilde{a} ^{w}/\partial a$ has the same sign as $\partial ^{2}W/\partial \widetilde{a} \partial a$. We find that

$$\begin{aligned} \text {sign}\left[ \frac{\partial \widetilde{a}^{w}}{\partial a}\right]&= \text {sign}\left[ \frac{\partial ^{2}W}{\partial \widetilde{a}\partial a} \right] \!=\!\text {sign}\left[ \frac{\partial ^{2}\widetilde{z}_{1}^{m}\left( a, \widetilde{a}\right) }{\partial \widetilde{a}\partial a}\left( w^{N,N}\left( \widetilde{a}\right) \!-\! w^{O,O}\left( a\right) -c\left( \widetilde{z} _{1}^{m}\right) \right) \right. \\&-\frac{\partial \widetilde{z}_{1}^{m}\left( a,\widetilde{a}\right) }{ \partial \widetilde{a}}\left( \frac{dw^{O,O}\left( a\right) }{da}+\frac{ \partial \widetilde{z}_{1}^{m}}{\partial a}c^{\prime }\left( \widetilde{z} _{1}^{m}\right) \right) +\left. \frac{\partial \widetilde{z}_{1}^{m}}{ \partial a}\frac{dw^{N,N}\left( \widetilde{a}\right) }{d\widetilde{a}}\right] \text {.} \end{aligned}$$

The second term is positive as $\partial \widetilde{z}_{1}^{m}/\partial \widetilde{a}\ge 0$, $dw^{O,O}\left( a\right) /da\le 0$, $\partial \widetilde{z}_{1}^{m}/\partial a\le 0$ and $c^{\prime }\left( z\right) \ge 0$ . The third term is also positive as $\partial \widetilde{z}_{1}^{m}/\partial a\le 0$ and $dw^{N,N}\left( \widetilde{a}\right) /d\widetilde{a}\le 0$. If $w^{N,N}\left( \widetilde{a}\right) -w^{O,O}\left( a\right) -c\left( \widetilde{z}_{1}^{m}\right) \ge 0$, the first term is positive if $\partial ^{2}\widetilde{z}_{1}^{m}\left( a,\widetilde{a}\right) /$ $\partial \widetilde{a}\partial a\ge 0$. We find that

$$\begin{aligned}&\frac{\partial ^{2}\widetilde{z}_{1}^{m}\left( a,\widetilde{a}\right) }{ \partial \widetilde{a}\partial a}\\&\quad \!=\!\frac{\partial \widetilde{\pi }_{1}^{N,N}}{ \partial \widetilde{a}}\frac{\partial \pi _{1}^{O,O}}{\partial a} c^{\prime \prime }\left[ \left( c\right) ^{-1}\left( \widetilde{\pi } _{1}^{N,N}\!-\!\pi _{1}^{O,O}\right) \right] \Big /\left( c^{\prime }\left[ \left( c\right) ^{-1}\left( \widetilde{\pi }_{1}^{N,N}\!-\!\pi _{1}^{O,O}\right) \right] \right) ^{3}\!\ge \! 0\text {,} \end{aligned}$$

as $\partial \widetilde{\pi }_{1}^{N,N}/\partial \widetilde{a}\ge 0$, $\partial \pi _{1}^{O,O}/\partial a\ge 0$, and $c^{\prime }\ge 0$, and provided that $c^{\prime \prime }\ge 0$ (i.e., the investment cost is convex). It follows that $\partial \widetilde{a}^{w}/\partial a\ge 0$. Finally, if $w^{N,N}\left( \widetilde{a}\right) -w^{O,O}\left( a\right) -c\left( \widetilde{z}_{1}^{m}\right) <0$, the first term is negative and therefore, the sign of $\partial \widetilde{a}^{w}/\partial a$ is ambiguous.

Now, we consider the case where the entrant invests more than the incumbent; the equilibrium coverage are $z_{1}^{*}=\widetilde{z} _{1}^{c}\left( \widetilde{a}\right)$ and $z_{2}^{*}=\widetilde{z} _{2}^{m}\left( a,\widetilde{a}\right)$. The social welfare is $W=\widetilde{ z}_{1}^{c}w^{N,N}+\left( \widetilde{z}_{2}^{m}-\widetilde{z}_{1}^{c}\right) w^{N,N}\left( \widetilde{a}\right) +\left( \overline{z}-\widetilde{z} _{2}^{m}\right) w^{O,O}\left( a\right) -C\left( \widetilde{z}_{1}^{c}\right) -C\left( \widetilde{z}_{2}^{m}\right)$. Assuming an interior solution, the socially optimal access price to fiber solves the first-order condition

$$\begin{aligned} \frac{\partial W}{\partial \widetilde{a}}&=\frac{d\widetilde{z} _{1}^{c}\left( \widetilde{a}\right) }{d\widetilde{a}}\left( w^{N,N}-w^{N,N}\left( \widetilde{a}\right) -c\left( \widetilde{z} _{1}^{c}\left( \widetilde{a}\right) \right) \right) \\&\quad +\frac{\partial \widetilde{z}_{2}^{m}\left( a,\widetilde{a}\right) }{\partial \widetilde{a}} \left( w^{N,N}\left( \widetilde{a}\right) -w^{O,O}\left( a\right) -c\left( \widetilde{z}_{2}^{m}\right) \right) \\&\quad +\left( \widetilde{z}_{2}^{m}-\widetilde{z}_{1}^{c}\right) \frac{ dw^{N,N}\left( \widetilde{a}\right) }{d\widetilde{a}}\equiv H\left( a, \widetilde{a}\right) =0\text {.} \end{aligned}$$

Let $\widetilde{a}^{w}$ denote the solution of $H\left( a,\widetilde{a} ^{w}\right) =0$. From the implicit function theorem, provided that the second-order condition holds, the sign of $\partial \widetilde{a} ^{w}/\partial a$ has the same sign as $\partial ^{2}W/\partial \widetilde{a} \partial a$. We find that

$$\begin{aligned} \text {sign}\left[ \frac{\partial \widetilde{a}^{w}}{\partial a}\right]&= \text {sign}\left[ \frac{\partial ^{2}W}{\partial \widetilde{a}\partial a} \right] \!=\!\text {sign}\left[ \frac{\partial ^{2}\widetilde{z}_{2}^{m}\left( a, \widetilde{a}\right) }{\partial \widetilde{a}\partial a}\left( w^{N,N}\left( \widetilde{a}\right) -w^{O,O}\left( a\right) -c\left( \widetilde{z} _{2}^{m}\right) \right) \right. \\&-\frac{\partial \widetilde{z}_{2}^{m}\left( a,\widetilde{a}\right) }{ \partial \widetilde{a}}\left( \frac{dw^{O,O}\left( a\right) }{da}+\frac{ \partial \widetilde{z}_{2}^{m}}{\partial a}c^{\prime }\left( \widetilde{z} _{2}^{m}\right) \right) +\left. \frac{\partial \widetilde{z}_{2}^{m}}{ \partial a}\frac{dw^{N,N}\left( \widetilde{a}\right) }{d\widetilde{a}}\right] \end{aligned}$$

As $\partial \widetilde{z}_{2}^{m}/\partial \widetilde{a}\ge 0$, the second term is negative if $dw^{O,O}\left( a\right) /da+$ $\partial \widetilde{z} _{2}^{m}/\partial a\times c^{\prime }\left( \widetilde{z}_{2}^{m}\right) \ge 0$, and we assume that this is the case (note that $dw^{O,O}\left( a\right) /da\le 0$, while $\partial \widetilde{z}_{2}^{m}/\partial a\times c^{\prime }\left( \widetilde{z}_{2}^{m}\right) \ge 0$). The third term is always negative as $\partial \widetilde{z}_{2}^{m}/\partial a\ge 0$ and $dw^{N,N}\left( \widetilde{a}\right) /d\widetilde{a}\le 0$. Finally, we find that $w^{N,N}\left( \widetilde{a}\right) -w^{O,O}\left( a\right) -c\left( \widetilde{z}_{2}^{m}\right) \le 0$, as

$$\begin{aligned}&w^{N,N}\left( \widetilde{a}\right) -w^{O,O}\left( a\right) -c\left( \widetilde{z}_{2}^{m}\right) \\&\quad =w^{N,N}\left( \widetilde{a}\right) -w^{O,O}\left( a\right) -\left( \widetilde{\pi }_{2}^{N,N}\left( \widetilde{a} \right) -\pi _{2}^{O,O}\left( a\right) \right) \\&\quad =\frac{1}{9}\left[ 5\widetilde{a}\underset{(-)}{\underbrace{\left( \widetilde{a}-\left( 1+s^{N}\right) \right) }}+\underset{(-)}{\underbrace{ \left( 1+s^{O}-2a\right) ^{2}-\left( 1+s^{N}\right) ^{2}}}\right] \le 0\text { .} \end{aligned}$$

The first term of sign$\left[ \partial \widetilde{a}^{w}/\partial a\right]$ is then positive as

$$\begin{aligned}&\frac{\partial ^{2}\widetilde{z}_{2}^{m}\left( a,\widetilde{a}\right) }{ \partial \widetilde{a}\partial a}\\&\quad \!=\!\frac{\partial \widetilde{\pi }_{2}^{N,N}}{ \partial \widetilde{a}}\frac{\partial \pi _{2}^{O,O}}{\partial a} c^{\prime \prime }\left[ \left( c\right) ^{-1}\left( \widetilde{\pi } _{2}^{N,N}\!-\!\pi _{2}^{O,O}\right) \right] \Big /\left( c^{\prime }\left[ \left( c\right) ^{-1}\left( \widetilde{\pi }_{2}^{N,N}-\pi _{2}^{O,O}\right) \right] \right) ^{3}\!\le \! 0\text {,} \end{aligned}$$

since $\partial \widetilde{\pi }_{2}^{N,N}/\partial \widetilde{a}\ge 0$, $\partial \pi _{2}^{O,O}/\partial a\le 0$, and $c^{\prime }\ge 0$, and provided that $c^{\prime \prime }\ge 0$. Though the sign of $\partial \widetilde{a} ^{w}/\partial a$ is ambiguous in general, we can have $\partial \widetilde{a} ^{w}/\partial a\le 0$ when the entrant dominates fiber investments in particular when $w^{N,N}\left( \widetilde{a}\right) -w^{O,O}\left( a\right) -c\left( \widetilde{z}_{2}^{m}\right)$ is high enough (provided that $dw^{O,O}\left( a\right) /da+$ $\partial \widetilde{z}_{2}^{m}/\partial a\times c^{\prime }\left( \widetilde{z}_{2}^{m}\right) \ge 0$ and that the investment cost is convex).

1.8 Appendix H: NGA regulation in selected OECD countries

In the following table we report the current state of regulatory interventions on NGA infrastructures in some selected OECD countries. Different types of both physical or “passive” (access to ducts or to dark fibre) and service-level or “active” (unbundling, bitstream) wholesale broadband remedies are considered. “Yes” means that the wholesale product/service is regulated; “no”, that it is not. The data are drawn from Cullen International (2013), Ovum (2013) as well as from direct interviews with national regulators.⁵²

Country	Geographically segmented remedies	Access to		In-building fiber	Fiber Unbundling	Bitstream on FFTH
		ducts	dark fibre
France	National	Yes	No	Yes (sym.)	No	No
Germany	National	Yes	Yes	Yes (sym.)	Yes $^{\ddag }$	Yes
Italy	Competitive areas	Yes	Yes	Yes (sym.)	No (VULA)	Yes
	Non-comp. areas	Yes	Yes	Yes (sym.)	No (VULA)	Yes
Netherlands	National	No	Yes$^{\lozenge }$	No	Yes $^{\ddag }$	No
Portugal	National	Yes	Yes	Yes (sym.)	No	No
Spain	National	Yes $^{*}$	Yes$^{\lozenge }$	Yes (sym.)	No	Yes$^{\dag }$
UK	Competitive areas	Yes	No	No	No (VULA)	No
	Non-comp. areas	Yes	No	Yes (sym.)	No (VULA)	Yes
Australia	National	Yes	Yes	Yes	No	Yes
Japan	National	Yes (asym.)	Yes (asym.)	Partially	Yes	No
New Zealand	National	No	Yes $^{\#}$	No	Yes $^{\#}$	Yes
USA	National	No	No	No	No	No

1. * Only in urban areas; $\lozenge$ When access to ducts is not possible; ‡ Only for residential customers; † Only for bandwidth under 30Mbit/s; # Point-to-point fibre only until 2019