Level of access and infrastructure investment in network industries

Abstract

In this paper, we study how the terms of access to an incumbent’s infrastructure (i.e., the level of access and price) affect an entrant’s incentives to build its own infrastructure. Setting a high level of access (e.g., a resale arrangement), which requires relatively small up-front investment for entry, accelerates market entry, but at the same time delays the deployment of the entrant’s infrastructure. This is also true for a lower access price. We show that the socially optimal access price can vary non-monotonically with the level of access. We also study the case where access is provided at two different levels, and show that access provision at multiple levels can delay infrastructure building. Finally, we modify our baseline model to allow for experience and/or market share acquisition via access-based entry, and show that high levels of access may accelerate facility-based entry.

Appendix

1.1 Appendix 1: Proof of lemma 1

We start by determining the optimal facility-based entry date $t_{{F}}$, for a given service-based entry date $t_{{S}}$. For the moment, assume that there is an interior solution (i.e., $t_{{F}}>t_{{S}}$). From (3), the first-order condition is $\left( \pi ^{{F}}-\pi ^{{S}}(r)\right) e^{-\rho t_{{F}}}=-\lambda C^{\prime }\left( t_{{F}}\right)$, which yields the following optimal interior investment date:

$$\begin{aligned} t_{{F}}^{*}(\lambda ,r)=Z^{-1}\left( \frac{\pi ^{{F}}-\pi ^{{S}}\left( r\right) }{\lambda }\right) \text {.} \end{aligned}$$

The second-order condition, $\rho e^{-\rho t_{{F}}}(\pi ^{{F}}-\pi ^{{S}}(r))-\lambda C^{\prime \prime }\left( t_{{F}}\right) \le 0$, is satisfied, since $\rho e^{-\rho t_{{F}}}\left( \pi ^{{F}}-\pi ^{{S}}(r)\right) -\lambda C^{\prime \prime }\left( t_{{F}}\right) =\lambda Z^{\prime }(t_{{F}})e^{-\rho t_{{F}}}$, and $Z^{\prime }(t)\le 0$.

If there is no interior solution to the profit maximization problem, the entrant enters on the basis of facilities at date $t_{{S}}$. Therefore, for a given service-based entry date $t_{{S}}$, the optimal facility-based entry date is defined as $\max \left\{ t_{{S}},t_{{F}}^{*}(\lambda ,r)\right\}$.

We now determine the optimal service-based entry date, $t_{{S}}$. The optimal service-based entry date is interior if $t_{{S}}<t_{{F}}^{*}(\lambda ,r)$. If this is the case, as $t_{{F}}^{*}(\lambda ,r)$ does not depend on $t_{{S}}$, from (3), the first-order condition with respect to $t_{{S}}$ can be written as $\pi ^{{S}}(r)e^{-\rho t_{{S}}}=-(1-\lambda )C^{\prime }\left( t_{{S}}\right)$, which yields the following optimal service-based entry date

$$\begin{aligned} t_{{S}}^{*}(\lambda ,r)=Z^{-1}\left( \frac{\pi ^{{S}}(r)}{1-\lambda }\right) \text {.} \end{aligned}$$

The second-order condition, $\rho \pi ^{{S}}(r)e^{-\rho t_{{S}}}-(1-\lambda )C^{\prime \prime }\left( t_{{S}}\right) \le 0$, is satisfied as we have

$$\begin{aligned} \rho \pi ^{{S}}(r)e^{-\rho t_{{S}}}-(1-\lambda )C^{\prime \prime }\left( t_{{S}}\right)&= -(1-\lambda )\left[ \rho C^{\prime }\left( t_{{S}}\right) +C^{\prime \prime }\left( t_{{S}}\right) \right] \\&= (1-\lambda )e^{-\rho t_{{S}}}Z^{\prime }(t_{{S}})\le 0\text {.} \end{aligned}$$

If $t_{{S}}\ge t_{{F}}^{*}(\lambda ,r)$, there is no phase of service-based competition, as the entrant enters on the basis of facilities immediately at date $t_{{S}}$. Therefore, the entrant’s discounted profit is maximized at $t_{{S}}=t_{{F}}=\min \left\{ t_{{F}}^{*}(\lambda ,r),t^{*}\right\} =t^{*}$.

1.2 Appendix 2: Proof of proposition 2

If there is a phase of service-based competition in equilibrium, service-based entry occurs at $t_{{S}}^{*}=Z^{-1}\left( \pi ^{{S}}(r)/(1-\lambda )\right)$, while facility-based entry occurs at $t_{{F}}^{*}=Z^{-1}\left( (\pi ^{{F}}-\pi ^{{S}}\left( r\right) )/\lambda \right)$. Note that, if $\lambda =1-\pi ^{{S}}(r)/\pi ^{{F}}$, then $t_{{S}}^{*}=t_{{F}}^{*}=t^{*}$. However, from Lemma 2, there is a phase of service-based competition in equilibrium if and only if $\lambda >1-\pi ^{{S}}(r)/\pi ^{{F}}$. From Proposition 1, we have $\partial t_{{S}}^{*}/\partial \lambda <0$ and $\partial t_{{F}}^{*}/\partial \lambda >0$. Therefore, starting from $\lambda =1-\pi ^{{S}}(r)/\pi ^{{F}}$, and increasing $\lambda$ to have $\lambda >1-\pi ^{{S}}(r)/\pi ^{{F}}$, yields $t_{{S}}^{*}<t^{*}$ and $t_{{F}}^{*}>t^{*}$.

1.3 Appendix 3: Second-order conditions

If the regulator’s optimal decision $\left( \lambda ^{w},r^{w}\right)$ is interior, that is, $\lambda ^{w}\in \left( 0,1\right)$ and $r^{w}\in \left( 0,\overline{r}\right)$, then $\left( \lambda ^{w},r^{w}\right)$ are solutions of the two first-order conditions, with respect to $\lambda$ and $r$. The local optimum defined by the first-order conditions corresponds to a maximum of the profit function if the Hessian matrix is negative definite. We have

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \lambda ^{2}}&= \alpha =-e^{-\rho t_{{S}}^{*}}\Delta _{{S}}\left( r\right) \left[ \frac{\partial ^{2}t_{{S}}^{*}}{\partial \lambda ^{2}}-\rho \left( \frac{\partial t_{{S}}^{*}}{\partial \lambda }\right) ^{2}\right] \\&-\,e^{-\rho t_{{F}}^{*}}\Delta _{{F}}\left( r\right) \left[ \frac{\partial ^{2}t_{{F}}^{*}}{\partial \lambda ^{2}}-\rho \left( \frac{\partial t_{{F}}^{*}}{\partial \lambda }\right) ^{2}\right] \\&+\,C^{\prime }(t_{{S}}^{*})\frac{\partial t_{{S}}^{*}}{\partial \lambda } -C^{\prime }(t_{{F}}^{*})\frac{\partial t_{{F}}^{*}}{\partial \lambda } \text {,} \\ \frac{\partial ^{2}W}{\partial r^{2}}&= \beta =e^{-\rho t_{{S}}^{*}}\left[ \frac{\partial t_{{S}}^{*}}{\partial r}\left( -\frac{\partial \Delta _{{S}}}{ \partial r}-\frac{\partial w_{{S}}}{\partial r}\right) -\Delta _{{S}}\left( r\right) \left( \frac{\partial ^{2}t_{{S}}^{*}}{\partial r^{2}}-\rho \left( \frac{\partial t_{{S}}^{*}}{\partial r}\right) ^{2}\right) \right] \\&+\,e^{-\rho t_{{F}}^{*}}\left[ \frac{\partial t_{{F}}^{*}}{\partial r} \left( -\frac{\partial \Delta _{{F}}}{\partial r}+\frac{\partial w_{{S}}}{ \partial r}\right) -\Delta _{{F}}\left( r\right) \left( \frac{\partial ^{2}t_{{F}}^{*}}{\partial r^{2}}-\rho \left( \frac{\partial t_{{F}}^{*}}{\partial r}\right) ^{2}\right) \right] \\&+\,\frac{e^{-\rho t_{{S}}^{*}}-e^{-\rho t_{{F}}^{*}}}{\rho }\frac{\partial ^{2}w_{{S}}}{\partial r^{2}}\text {,} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \lambda \partial r}&= \gamma =e^{-\rho t_{{S}}^{*}}\left[ -\frac{\partial t_{{S}}^{*}}{\partial \lambda }\frac{ \partial \Delta _{{S}}}{\partial r}-\Delta _{{S}}\left( r\right) \left( \frac{ \partial ^{2}t_{{S}}^{*}}{\partial \lambda \partial r}-\rho \left( \frac{ \partial t_{{S}}^{*}}{\partial r}\right) \left( \frac{\partial t_{{S}}^{*}}{\partial \lambda }\right) \right) \right] \\&+\,e^{-\rho t_{{F}}^{*}}\left[ -\frac{\partial t_{{F}}^{*}}{\partial \lambda }\frac{\partial \Delta _{{F}}}{\partial r}-\Delta _{{F}}\left( r\right) \left( \frac{\partial ^{2}t_{{F}}^{*}}{\partial \lambda \partial r}-\rho \left( \frac{\partial t_{{F}}^{*}}{\partial r}\right) \left( \frac{ \partial t_{{F}}^{*}}{\partial \lambda }\right) \right) \right] \\&+\,C^{\prime }(t_{{S}}^{*})\frac{\partial t_{{S}}^{*}}{\partial r} -C^{\prime }(t_{{F}}^{*})\frac{\partial t_{{F}}^{*}}{\partial r}\text {.} \end{aligned}$$

The first-order condition correspond to a local maximum if $\alpha <0$ and $\alpha \gamma -\beta ^{2}>0$, and we assume that this is the case.

1.4 Appendix 4: Proof of proposition 3

Assume that $W$ is concave with respect to $\lambda$ and $r$. From the implicit function theorem, the variations of $r^{w}(\lambda )$ are given by the sign of $\partial ^{2}W/\partial \lambda \partial r$. We have

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \lambda \partial r}&= e^{-\rho t_{{S}}^{*}} \left[ -\frac{\partial t_{{S}}^{*}}{\partial \lambda }\frac{\partial \Delta _{{S}}}{\partial r}\right] +e^{-\rho t_{{F}}^{*}}\left[ -\frac{ \partial t_{{F}}^{*}}{\partial \lambda }\frac{\partial \Delta _{{F}}}{ \partial r}\right] \\&-\,e^{-\rho t_{{S}}^{*}}\Delta _{{S}}\left( r\right) \left( \frac{\partial ^{2}t_{{S}}^{*}}{\partial \lambda \partial r}-\rho \left( \frac{\partial t_{{S}}^{*}}{\partial r}\right) \left( \frac{\partial t_{{S}}^{*}}{\partial \lambda }\right) \right) \\&-\,e^{-\rho t_{{F}}^{*}}\Delta _{{F}}\left( r\right) \left( \frac{\partial ^{2}t_{{F}}^{*}}{\partial \lambda \partial r}-\rho \left( \frac{\partial t_{{F}}^{*}}{\partial r}\right) \left( \frac{\partial t_{{F}}^{*}}{\partial \lambda }\right) \right) \\&+\,C^{\prime }(t_{{S}}^{*})\frac{\partial t_{{S}}^{*}}{\partial r} -C^{\prime }(t_{{F}}^{*})\frac{\partial t_{{F}}^{*}}{\partial r}. \end{aligned}$$

The first line of this expression has an ambiguous sign; we have $\partial t_{{S}}^{*}/\partial \lambda <0$ and $\partial t_{{F}}^{*}/\partial \lambda >0$, but we can have either $\partial \Delta _{\tau }/\partial r>0$ or the reverse, with $\tau =S,F$. If $Z^{\prime \prime }\le 0$, then $\partial ^{2}t_{{S}}^{*}/\partial \lambda \partial r\ge 0$. Since $\left( \partial t_{{S}}^{*}/\partial r\right) \left( \partial t_{{S}}^{*}/\partial \lambda \right) <0$, the second line has the sign of $-\Delta _{{S}}\left( r\right)$. Similarly, we find that the third line has the sign of $-\Delta _{{F}}\left( r\right)$. Finally, the fourth line is negative, as $C^{\prime }\le 0,\,\partial t_{{S}}^{*}/\partial r>0$, and $\partial t_{{F}}^{*}/\partial r<0$. Overall, the sign of $\partial ^{2}W/\partial \lambda \partial r$ is therefore ambiguous.

1.5 Appendix 5: Proof of proposition 4

Condition (16) introduces two additional constraints on the access scheme. First, the left part of (16) can be rewritten as

$$\begin{aligned} \pi ^{{S}}(r_{2})\le \frac{1-\lambda _{2}}{1-\lambda _{1}}\pi ^{{S}}(r_{1}) \text {.} \end{aligned}$$

Second, the right part of (16) can be rewritten as

$$\begin{aligned} \lambda _{2}\ge \lambda _{1}\frac{\pi ^{{F}}-\pi ^{{S}}(r_{2})}{\pi ^{{F}}-\pi ^{{S}}(r_{1})}\text {.} \end{aligned}$$

(17)

Equation (17) defines a minimum for $\lambda _{2}$. Since $t_{{F},2}^{*}$ increases with $\lambda _{2}$, the facility-based entry date $t_{{F},2}^{*}$ is lowest at the minimum for $\lambda _{2}$. Replacing for the minimum for $\lambda _{2}$ into $t_{{F},2}^{*}$ yields the following minimum facility-based entry date,

$$\begin{aligned} t_{{F},2}^{\min }=Z^{-1}\left( \frac{\pi ^{{F}}-\pi ^{{S}}(r_{1})}{\lambda _{1}} \right) \text {.} \end{aligned}$$

Therefore, with two levels of access, the entrant enters on the basis of services at $t_{{S},1}^{*}$ and then on the basis of facilities at earliest at $t_{{F},2}^{\min }$. In other words, in the best case (in terms of an early facility-based entry date), everything is as if there were a single level of access at $\lambda _{1}$ with access price $r_{1}$. The regulator can thus always replicate the best outcome with two access levels with a single access level. Since $\lambda _{2}$ should be strictly higher than the minimum defined by (17) for condition (16) to hold, facility-based entry occurs later than $t_{{F},2}^{\min }$ with two levels of access. Therefore, starting from two levels of access, the regulator could implement an earlier facility-based entry by reverting to a single access level at $\lambda _{1}$ with access price $r_{1}$.