The Ladder of Investment in Telecoms: Stairway to Heaven or Highway to Hell?∗ Nicolas Schutz† Thomas Tr´egou¨et‡ Preliminary - Comments welcome

Abstract We analyze the investment decision of a potential entrant which can either build its own network, or get access to the incumbent’s network at a regulated access price. There are two periods of competition, and the entrant is uncertain about its retail marginal cost prior to entry. We derive the impact of the regulated access price on the ex ante probability of infrastructure investment, denoted by ρ. We show that, depending on the parameters’ values, three distinct investment regimes can arise. In the conventional wisdom regime, ρ is an increasing function of the access price. In the no-investment regime, the access price has no effect on the probability of investment. Last, in the ladder of investment regime, ρ is a hump-shaped function of the access price. We then provide some numerical computations to determine the range of parameters for which each regime arises. This analysis enables us to assess whether service-based and facility-based competition are substitutes or complements. Journal of Economic Literature Classification Number: D92, L51, L96. Keywords: Access pricing, regulation, investment, ladder of investment.

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Intellectual and financial support by France T´el´ecom and CEPREMAP is gratefully acknowledged. We also wish to thank Marc Bourreau, Johan Hombert and Jerome Pouyet for helpful comments and discussions. We are solely responsible for the analysis and conclusions. † Paris School of Economics. Address: 48 Boulevard Jourdan, 75014, Paris, France. E-mail: [email protected]. ‡ Ecole Polytechnique. Address: Department of Economics, Ecole Polytechnique, 91128 Palaiseau Cedex, France. E-mail: [email protected]

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1

Introduction

There is an old debate in the telecommunications industry on the relationship between access price regulation and incentives to invest in infrastructures. Until recently, the conventional wisdom could be summarized as follows. On the one hand, if the access price is low, operators can enter without having to build their own facilities. Low access prices can promote servicebased entry within a short time horizon, thereby eroding the incumbent’s market share on the retail market, and lowering retail prices. On the other hand, low access prices provide telecoms operators with little incentives to build their own networks. Therefore, the benefits from facility-based competition, e.g., more diversified offerings, quality improvements, larger scope for innovation, may fail to emerge in the long run. In a nutshell, there seems to be a conflict between short-run efficiency, favored by service-based competition, and long-run efficiency, favored by facility-based competition. The existence of this conflict was widely accepted by telecoms regulators, who admitted that they had to choose between servicebased and facility-based competition.1 This view was challenged recently by the so-called ladder of investment theory, elaborated by Cave and Vogelsang (2003) and further developed by Cave (2006). The proponents of this theory argue that there is actually no tradeoff between short-run and long-run efficiency. Attractive access tariffs promote service-based entry in the short run, which allows the entrants to become familiar with the retail market, to alleviate pre-entry uncertainty on the returns to investment, before choosing to become facility-based competitors. In other words, service-based competition in the short run favors the development of facility-based competition in the long run. This view was recently endorsed by the European Regulatory Group (ERG):2 “Service competition based on regulated access at cost-oriented prices can be (and in general is) the vehicle for long-term infrastructure competition. With this new entrants can decide on their investment in a step-by-step way and can establish a customer base (critical mass) before they go to the next step of deploying their own infrastructure.” 3 1

For instance, the UK regulator, Oftel (now Ofcom), chose to promote facility-based competition, while the majority of the other European regulators favored service-based entry. See Crandall and Waverman (2006) for an exposition of regulatory policies across countries from 1996, and their impact on the modes of entry. 2 See Oldale and Padilla (2004) for an exposition of the shift in the European regulatory paradigm after the adoption of the New Regulatory Framework in 2002. 3 “ERG Common Position on the approach to Appropriate remedies in the new regulatory framework”, 23 April 2004, p. 68.

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In this paper, we assess the theoretical validity of these two conflicting views. We consider a model in which an incumbent operator has already built its own network. A potential entrant, E, can enter either as a service-based or as a facility-based operator. If it chooses service-based entry, it gets access to the network of the incumbent at a regulated access tariff. After a period of service-based competition, it can decide to build its own network, or to remain a service-based operator. The entrant can also choose to invest in infrastructure immediately, without going through a spell of service-based competition. In this case, it has to pay immediately both the sunk cost of service-based entry and the sunk investment cost. Prior to entry, the payoffs are uncertain: before having operated on the final market, the entrant does not know its retail marginal cost. This creates a potential tradeoff between facility-based and service-based entry. On the one hand, if the entrant chooses to enter as a service-based operator, it has the opportunity to experiment and to learn its retail marginal cost before making its investment decision, but it has to pay the access charge as long as it remains a service-based competitor. On the other hand, if it invests right away, then, it saves on the access price, but it always pays the sunk cost of investment, even in ‘bad’ states of nature in which its retail marginal cost turns out to be high. To understand the impact of this uncertainty on the relationship between access regulation and investment, consider as a benchmark the situation in which firm E observes its cost before entry. In this case, the entrant’s investment decision are completely determined by the level of the regulated access price, and by the sunk investment cost. If the access price is low, the entrant has little incentives to invest in infrastructure, since it can operate on the retail market as a service-based competitor without incurring a significant cost disadvantage. By contrast, if the access price is high, the entrant can no longer compete on a level playing field with the incumbent if it remains service-based, and the incentives to invest strengthen. Putting these insights together, only two distinct investment regimes can emerge. In the no-investment regime, the sunk investment cost is so high that the entrant will never build its own infrastructure, whatever the access price regulation. The access price has no impact on investment decisions, since the investment cost is too large anyway. In the conventional wisdom regime, which arises if the investment cost is not prohibitive, the entrant invests if the access price is sufficiently high. Put differently, in this second regime, a larger access price always promotes infrastructure investment. When uncertainty is taken into account, the two investment regimes presented before should still be there. Again, if the sunk investment cost is high, the no-investment regime should emerge, whereas the conventional wisdom regime should emerge if the investment cost

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is low. More interestingly, due to the uncertainty, a third regime may appear for intermediate values of the sunk investment cost. This third regime, that we call the ladder of investment regime, can be described as follows. When the access price is low, again, the entrant chooses service-based entry, and remains a service-based competitor once it has learnt its marginal cost. When the access price is intermediate, firm E still chooses service-based entry, but it may invest subsequently if it discovers that its marginal cost is low. Service-based entry provides the entrant with the opportunity to experiment on the retail market, in order to learn its marginal cost, and to make a well-informed entry decision. Last, when the access price is high, service-based entry is no longer profitable, since the entrant would suffer from a significant cost disadvantage when competing with the incumbent. Facility-based entry is not profitable either, since the entrant anticipates that it will incur important losses if its marginal cost turns out to be large. Therefore, when the access price is high, neither servicebased entry, nor infrastructure investment take place. In the ladder of investment regime, infrastructure investment takes place only for intermediate values of the sunk investment cost. Formally, we show that, in this regime, there is a hump-shaped relationship between the access price and the ex ante probability of infrastructure investment. This behavior is in sharp contrast with the conventional wisdom. In the conventional wisdom regime, the ex ante probability of investment is increasing in the regulated access price. Put differently, in the conventional wisdom regime, service-based competition and facility-based competition are always substitutes. If the regulator increases the access price, it favors the development of facility-based competition, but it hampers the emergence of service-based competition. In the ladder of investment regime, service-based and facilitybased competition are substitutes for low values of the access price, and complements for high values of the access price. To see this, consider that the access price is initially low, so that infrastructure investment never takes place, and assume that the regulator increases this price. On the one hand, this tightens the conditions of service-based competition. On the other hand, this provides the entrant with incentives to invest once it has learnt its cost, thereby promoting facility-based competition. There is indeed a conflict between servicebased and facility-based competition when the access price is low. By contrast, assume that the regulator decreases the access price from a high to an intermediate value. Initially, servicebased entry is not profitable, and infrastructure investment never takes place. After the access price decreases, the entrant rationally decides to enter as a service-based competitor. If its marginal cost turns out to be low, it subsequently decides to invest. Service-based and facility-based competition are indeed complements when the access price is high.

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In the last part of this paper, we use numerical computations with a Cournot specification for the retail profits in order to identify more precisely the situations in which the different investment regimes arise. We show that the ladder of investment regime is more likely to arise when the sunk investment cost is intermediate, the probability for the entrant to be efficient is low, and the sunk cost for service-based entry is low. In any case, the set of parameters compatible with the ladder of investment regime is small. This suggests that, if the regulator is willing to promote infrastructure investment, a ladder of investment approach is not always relevant. In particular, a conventional wisdom approach should not be rejected without a strong presumption that the actual regime is indeed the ladder of investment regime. There is a growing body of literature on the impact of access regulation on infrastructure investment in telecoms. Bourreau and Dogan (2005 and 2006) analyze the optimal investment decision of an entrant when the sunk investment cost decreases over time, and the entrant can get access to the network of an incumbent if it does not invest. They assume that the access price is set once and for all at the beginning of the game (Bourreau and Dogan (2005)), or that the pattern of future access prices is decided upon at the beginning of the game (Bourreau and Dogan (2006)). In both cases, they show that, if the incumbent is free to set the access price, then, it tends to use it to delay facility-based entry. In these papers, the conventional wisdom always prevails, and a lower access charge always delays infrastructure investment. In a thought-provoking article, Sappington (2005) claims that the access price has actually no impact at all on the entrants’ investment decisions. To make this point, he considers a model in which an incumbent and a service-based entrant compete in prices on the Hotelling segment. When the incumbent increases its retail price, it internalizes the fact that any customers lost on the retail market can be recovered on the wholesale market. Therefore, it tends to behave on the retail market as if its total marginal cost were equal to its retail commercial cost plus the access price. In a Hotelling model, when the retail market is covered, this implies that the access price has no impact on the entrant’s profit, and hence, no impact on its incentives to invest. In this model, service-based and facility-based competition are neither friends nor enemies. However, this view has been challenged by Gayle and Weisman (2007) who tend to show that the conventional wisdom prevails once Hotelling competition with a covered market is assumed away. Hori and Mizuno (2006) and Vareda and Hoernig (2007) analyze the race for infrastructure investment between two entrants when the access tariff is regulated. Assuming that the total retail demand increases stochastically, Hori and Mizuno (2006) derive conditions under which

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an access-to-bypass equilibrium exists. In such an equilibrium, one of the two entrants, the leader, first builds its own network, when the retail demand is sufficiently large. Subsequently, the follower enters as a service-based competitor, and eventually builds its own network once the retail demand has become sufficiently high. When an access-to-bypass equilibrium exists, an increase in the regulated access price always induces the leader and the follower to invest earlier, and there is indeed a conflict between service-based and facility-based competition. This result is challenged by Vareda and Hoernig (2007), who solve a similar model, in which the retail demand is constant, and the sunk investment cost is decreasing over time, as in Bourreau and Dogan (2005 and 2006). They show that the follower’s investment always occur earlier when the access price raises. Yet, the impact on the leader’s investment date is ambiguous. Intuitively, when the access price increases, the leader expects to earn larger per-period profits, but it also anticipates that the duration of service-based competition will become shorter, since the follower will invest earlier. If this latter effect is strong enough, this can induce the leader to delay its investment. To the best of our knowledge, the only paper which explicitly mentions the ladder of investment concept is Vareda (2007). There are initially a facility-based incumbent and service-based entrant. There are two periods of competition, and the entrant can decide to build its network at the end of the first period. The entrant knows whether the potential demand is high or low, while the regulator cannot observe the potential demand. By assumption, it is profitable for the entrant to invest only if the demand is high. Vareda (2007) shows that, when the demand is high, the entrant has incentives to mimic the behavior of a lowtype entrant, in order to obtain a cheaper access charge. This implies that the access tariffs set by the regulator are distorted, with respect to the complete information benchmark. It may even be the case that the regulator does not try to induce the entrant to invest, since this would create significant welfare losses if the demand turns out to be low. It should be noted that Vareda (2007) assumes that it would not be profitable for the entrant to invest immediately without going through a spell of service-based competition. In the terminology of our paper, Vareda (2007) assumes that the industry is in the ladder of investment regime, and investigates the impact of asymmetric information on the optimal regulation. Our paper complements his analysis, by exhibiting conditions under which the ladder of investment regime actually arises. The remainder of this paper is organized as follows. In section 2, we lay out the framework of our model. Section 3 describes the optimal investment decision of the entrant. It also establishes necessary and sufficient conditions for the emergence of each investment regime.

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Section 4 presents our numerical computations. Section 5 concludes.

2

The model

There are two firms in this model: an incumbent I and a potential entrant E. We assume that the incumbent has already built a network at the beginning of the game. Put differently, the incumbent is already a facility-based firm, and we will not have to analyze its investment decisions. On the other hand, the potential entrant did not invest at all before the game starts. During the course of the game, it can sunk some investment cost to become a service-based or a facility-based operator. If it chooses service-based entry, it pays a sunk cost f and it has to obtain local access from the incumbent at some regulated access tariff a ≥ 0. In this case, its total marginal cost of operating on the final market can be decomposed into two terms: a, the access price, and c˜, the retail marginal cost. We denote by π(a + c˜) the operating profit flow earned by a service-based entrant. Function π(.) should be seen as the entrant’s equilibrium payoff resulting from the competition between the incumbent and the entrant on the final market. In section 4, we propose some numerical simulations, in which π(.) is computed as the outcome of Cournot competition between the incumbent and the entrant. We assume that there exists a threshold, cmax , such that π(c) = 0 if and only if c ≥ cmax . We assume that function π(.) is twice differentiable, strictly decreasing and strictly convex up to threshold cmax . If the entrant chooses to operate as a facility-based competitor, it also needs to sunk an investment cost. If it builds its infrastructure after a spell of service-based competition, it pays sunk cost fˆ. If it enters right away as a facility-based operator, it incurs a sunk cost F = f + fˆ.4 After investment has taken place, it can use its own network to provide telecoms services to its end-users. For simplicity, we normalize to zero the marginal cost of operating the network. In this case, the total marginal cost on the final market is equal to 0 + c˜, and the entrant earns an operating profit flow π(˜ c).5 We assume that, before having operated on the final market, the entrant does not know A priori, there is no reason to believe that F should be equal to f + fˆ. If we assumed that F 6= f + fˆ, this would introduce another trade-off between becoming facility-based right away on the one hand, and going through a spell of service-based competition before investing on the other hand. This trade-off may be interesting, but it is clearly orthogonal to the effects we emphasize in this article. 5 Notice that we have taken the same profit function when the entrant is service-based and facility-based: πS (.) = πF (.) = π(.). In particular, if the regulated access price were set at zero, a service-based entrant would earn as much profit as a facility-based entrant. We would obtain qualitatively similar results if we assumed that facility-based entry improves the quality of the final product, namely, πF (.) > πS (.). 4

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its retail marginal cost c˜. This enables us to capture the fact that, prior to entry, the entrant is uncertain about the amount of profits it will be able to make on the market. Formally, c˜ is equal to 0 with probability ν, and c¯ > 0 with probability 1 − ν.6 After firm E has entered, as a service-based or as a facility-based operator, the uncertainty is resolved, and the entrant learns the value of c˜. The chronology of events runs as follows: 1. Firm E chooses whether to enter as a facility-based competitor, or as a service-based competitor, or not to enter at all. If it does enter, then, it pays the entry costs, and its retail marginal cost is revealed. 2. If entry has taken place, the entrant competes with the incumbent on the final market, and earns a payoff determined by function π(.). 3. If firm E entered as a service-based competitor in stage 1, and if it obtained a strictly positive operating profit flow, then it can pay investment cost fˆ to build its own network.7 4. If entry has taken place, the entrant competes with the incumbent on the final market, and earns a payoff determined by function π(.). We assume that the potential entrant maximizes the expected value of its total profits, net of the sunk costs. We adopt the following tie-breaking rule: if firm E is indifferent between entering and remaining out of the market, it stays out; if it is indifferent between investing and remaining a service-based operator, it does not invest. For the sake of simplicity, we choose not to discount future profit flows. We would also like to comment on the assumption that firm E can invest in period 3 if and only if it received some positive profit flows in period 2. If we did not make this assumption, then, for certain parameters’ values, the following unrealistic situation could be an outcome of our model. Consider that the regulated access price is so high that a service-based entrant will never be able to serve any customers whatever its retail marginal cost. This would be the case if a were set above cmax . Then, it could be optimal for firm E to choose service-based entry in stage 1 in order to learn its cost, even though it would not earn any operating profit 6

Our results would extend if c˜ were distributed over a continuous set. Notice that we do not enable firm E to enter in stage 3 if it did not enter in stage 1. Even if we did allow for this possibility, it would never be optimal for the entrant to use it. Indeed, if it is profitable to enter in stage 3, then, it is even more profitable to enter in stage 1. 7

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in stage 2. Subsequently, the entrant would invest in period 3 if and only if its marginal cost were low. It does not seem desirable to obtain this situation as the outcome of our model. Indeed, it seems unlikely that the entrant will be able to learn the retail market characteristics, i.e., in this model, to discover c˜, without actually operating on that market. If the entrant’s total marginal cost, c˜ + a, is so high that it cannot supply any customers, this basically means that service-based entry does not enable the entrant to experiment on the retail market, hence, that the entrant should not be able to learn its marginal cost. Assuming that the entrant has to serve a positive quantity in period 2 in order to invest in period 3 allows us to rule out these unrealistic situations.

3

Optimal investment decision

3.1

Investment decisions in stage 3

In this section, we analyze the investment decision in stage 3. If the entrant chose facilitybased entry in stage 1, by assumption, it remains a facility-based competitor subsequently. If it did not enter previously, again by assumption, it cannot invest in period 3. Assume now that firm E became a service-based operator in period 1, and consider its incentives to build its own network at stage 3. If the entrant remains service-based, it earns π(˜ c + a).8 If it invests in infrastructure, it earns a net profit flow π(˜ c) − fˆ. Therefore, the entrant invests if and only if π(˜ c) − fˆ > π(˜ c + a).9 Assume that the entrant has a low marginal cost: c˜ = 0. Then, it invests if and only if π(0) − fˆ > π(a) > 0. If π(0) > fˆ, then, since function π(.) is continuous and strictly decreasing, there exists a threshold a, such that 0 < π(a) < π(0) − fˆ if and only if a < a < cmax . On the other hand, if the sunk cost to become a facility-based operator is too high, namely, if fˆ > π(0), there exists no such a. If fˆ is too high, a service-based entrant never builds its own network, whatever the regulation. In this case, a does not exist. Intuitively, if the sunk cost of investment is too large, the entrant will never find it profitable to invest whatever the level of the regulated access price. On the other hand, when the sunk cost is sufficiently low, so that the entrant would decide to invest if it could not get access to the incumbent’s network, the regulation can have an important impact on the investment decision. If the regulated access price is too low, the entrant can remain a servicebased competitor without incurring a prohibitive cost disadvantage: it has therefore little 8 9

Recall that c˜ is known at date 3. This is true as long as π(˜ c + a) > 0, i.e., as long as a < cmax − c˜.

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incentives to build its own infrastructure. This is the conventional wisdom on the impact of access regulation on infrastructure investment: a low access price favors the emergence of service-based competition, yet, it is detrimental to facility-based entry. In the following sections, we will see that this statement may no longer be true once uncertainty is taken into account. If the entrant has a high marginal cost, c˜ = c¯, it invests if and only if π(¯ c) − fˆ > π(¯ c + a) > 0. Again, if π(¯ c) > fˆ, then, there exists a ¯, such that the entrant invests if and only if a ¯ < a < cmax − c¯. On the other hand, if π(¯ c) ≤ fˆ, there is not such a ¯, and the entrant chooses to remain a service-based competitor whatever the access price. Now that we have understood the impact of regulation on investment incentives for a given value of c˜, it remains to investigate whether a low marginal cost entrant has more incentives to invest than a high marginal cost entrant. This is done in the following lemma, which gives a complete characterization of the entrant’s optimal investment decision in period 3 as a function of the access price. Lemma 1. If a ¯ exists, then a exists, and a < a ¯: a low cost entrant has more incentives to invest than a high cost entrant. Proof. See Appendix A.1.

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π(0)

π(¯ c)

π(0) − fˆ

π(a) π(¯ c + a) π(¯ c) − fˆ

0

c¯

a

cmax − c¯ cmax

a ¯

a

Figure 1: Profit functions of the low cost and high cost entrant, when both a and a ¯ exist The above lemma, illustrated by Figure 1, states that an efficient entrant has more incentives to invest than an inefficient entrant. This result derives directly from the convexity of the profit function with respect to the total marginal cost. To see this, fix some a > 0, ¯ both exist. If the entrant invests, its operating profit increases by and assume that a and a π(˜ c) − π(˜ c + a), but it needs to pay the sunk cost fˆ. Since the profit function is strictly convex, π(0) − π(a) > π(¯ c) − π(¯ c + a): an efficient entrant benefits more from a decrease in its total marginal cost than an inefficient entrant. Intuitively, an efficient entrant receives more demand on the final market. Therefore, its total cost decreases more, and its profit increases more when its marginal cost decreases, which strengthens its incentives to invest.

3.2

Stage 1 profits

In this section, we derive the total expected profit of the entrant in stage 1, when it chooses to invest right away in infrastructure, and when it chooses service-based entry. If the entrant 11

becomes a service-based competitor, it anticipates that it will behave optimally in stage 3, once it will have learnt its cost. If the entrant builds its own network in stage 1, its expected profit is given by the following expression: EΠ(F ) = 2 (νπ(0) + (1 − ν)π(¯ c)) − (f + fˆ).

(1)

We have to be careful when writing the expected profit from service-based entry, since, as pointed out in the previous section, the entrant may or may not choose to invest in infrastructure in stage 3, depending on its cost and on the access regulation. Since the analytical expression of this expected profit, denoted by EΠ(S, a), is not illuminating, we relegate it to the appendix. The following lemma summarizes what we know about EΠ(S, a): Lemma 2. EΠ(S, a) is strictly decreasing in a for 0 ≤ a ≤ cmax . Besides, EΠ(S, 0) = EΠ(F ) + fˆ, and EΠ(S, a) = −f , for all a ≥ cmax . Proof. See Appendix A.2. Last, if the entrant stays out of the market, it obtains zero profit: EΠ(∅) = 0.

(2)

Now that these expected profits have been computed, finding the entrant’s optimal decision in stage 1 is just a matter of comparing expressions EΠ(F ), EΠ(S, a) and EΠ(∅) for given values of f , fˆ, c¯, ν and a. If EΠ(∅) > EΠ(F ) and EΠ(∅) > EΠ(S, a), which occurs for instance when fˆ and f are high enough, or when fˆ and a are high enough, then firm E prefers to stay out of the market. If EΠ(S, a) > max {EΠ(∅), EΠ(F )}, which would be the case if the regulated access price were low enough, or if fˆ were high enough, then firm E chooses to enter as a service-based competitor. Last, if EΠ(F ) > max {EΠ(∅), EΠ(S, a)}, which happens when a is high enough and fˆ is low enough, then firm E chooses facility-based entry. Yet, we are not only interested in the optimal investment decision of the potential entrant in period 1. Our main objective is to investigate the impact of access regulation on the emergence of facility-based competition at the end of the day. We claim that, in this model, the right way to deal with this issue is to analyze function ρ : a ∈ [0, ∞) 7→ P r1 (Firm E will be facility-based in stage 4) ,

(3)

where P r1 denotes the probability operator taken at date 1. Function ρ associates to each 12

value of the access price, a ∈ [0, ∞), the ex ante probability that the entrant will eventually build its own infrastructure, ρ(a). We will say that a change in the regulated access charge favors facility-based competition if and only if it increases the probability that infrastructure investment eventually takes place. In the following sections, we show that, depending on the parameters’ values, three regimes can be distinguished, when it comes to analyzing the impact of regulation on investment: • In the “no-investment” regime, ρ(.) is flat, and regulation has no impact on infrastructure investment. • In the “conventional wisdom” regime, ρ(.) is monotone: an increase in the access price always favors infrastructure investment. • In the “ladder of investment” regime, ρ(.) is hump-shaped : an increase in the access price fosters (hampers) investment for low (high) values of a.

3.3

No-investment regime: ρ is flat

The following proposition characterizes the no-investment regime: Proposition 1 (No-Investment Regime). Assume EΠ(F ) ≤ 0. If fˆ ≥ π(0), or if fˆ < π(0) and EΠ(S, a) ≤ 0, then ρ(a) = 0 for all a ≥ 0. Proof. See Appendix A.3. In the no-investment regime, the entrant will never build its own network whatever the regulation. For this kind of regime to arise, two ingredients are necessary. First, it must not be profitable to invest right away, otherwise, it would be possible to find a high enough a, so that EΠ(S, a) would be negative, and firm E would invest in period 1. For this a, ρ(a) would be equal to 1. Second, situations in which firm E chooses service-based entry in stage 1, and invests in stage 3 must not be optimal for the entrant, since they would imply a positive ρ(a). If π(0) ≤ fˆ, these situations cannot arise, since it is not profitable to invest in period 3, whatever the marginal cost and the regulation. If π(0) > fˆ, these situations may arise when a > a, but they are ruled out by the assumption EΠ(S, a) ≤ 0. In this regime, the best thing the regulator can do is set the access price to zero in order to maximize the benefits from service-based competition, since facility-based competition is definitely out of reach.

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3.4

Conventional wisdom regime: ρ is monotone

The following lemma defines threshold a ˜, which will be useful to describe the conventional wisdom regime: Lemma 3. If EΠ(F ) > 0, then, there exists a unique 0 < a ˜ ≤ cmax , such that EΠ(F ) > EΠ(S, a) if a > a ˜, and EΠ(F ) < EΠ(S, a) if a < a ˜. Besides, if π(¯ c) < fˆ, then a ˜ 0. Then, • If π(0) ≤ fˆ, or if π(0) > fˆ and EΠ(F ) ≥ EΠ(S, a), then, ρ(a) = 1{a>˜a} . • If π(0) > fˆ and EΠ(F ) < EΠ(S, a), then, a ˜ > a, and

ρ(a) =

    0 if a ≤ a,

ν if a < a ≤ a ˜,    1 if a ˜ < a.

Proof. See Appendix A.5. As pointed out in Proposition 2, the ex ante probability of investment can have two distinct shapes within the conventional wisdom regime. In the first subregime, this probability is equal to 0 for a ≤ a ˜, and 1 for a > a ˜. If a ≤ a ˜, the regulated access price is low, relative to the sunk investment cost. Therefore, the optimal choice of firm E is to become a servicebased competitor in period 1, and not to invest in stage 3. On the other hand, if a > a ˜, the access price is too high, so that firm E prefers to enter right away as a facility-based operator. This is the conventional wisdom on the impact of access regulation on infrastructure investment: a low access price favors service-based entry and hampers facility-based entry. Within this subregime, service-based competition and facility-based competition are substitutes. A policy that favors the emergence of service-based competition is necessarily detrimental to the development of facility-based competition. The second subregime is similar to the first one. For a ≤ a, the ex ante probability of investment is equal 0, while it is equal to 1 for a > a ˜: firm E remains service-based whatever its cost if the access price is low, while it chooses facility-based entry if a is high. However, 14

for a < a ≤ a ˜, ρ(a) = ν. For intermediate values of the access price, firm E enters as a service-based competitor in stage 1. In stage 3, it invests only if its retail marginal cost is low. Put differently, service-based entry allows firm E to learn its cost before making its investment decision. The entrant will use this option only for intermediate values of the access price. Indeed, if the access price is too low, the entrant prefers to remain service-based whatever its cost. On the other hand, if the access price is too high, service-based entry would be a bad choice, since the entrant would earn little profits in period 2. At the end of the day, this subregime reflects also the conventional wisdom: the ex ante probability is still increasing in the access price, and service-based and facility-based competition are still substitutes. Within the conventional wisdom regime, a regulator willing to promote infrastructure investment should set a high enough access price, namely, any a above a ˜.

3.5

Ladder of investment regime: ρ is hump-shaped

The following lemma defines threshold a ˆ, which will be useful to describe the ladder of investment regime: Lemma 4. If EΠ(S, 0) > 0, then, there exists a unique 0 < a ˆ ≤ cmax , such that EΠ(S, a) > 0 if a < a ˆ, and EΠ(S, a) < 0 if a > a ˆ. Besides, if EΠ(F ) ≤ 0, and if π(¯ c) > fˆ, then, a ˆ fˆ and EΠ(F ) ≤ 0 < EΠ(S, a), then, a < a ˆ, and, for all a ≥ 0,

ρ(a) =

    0 if 0 ≤ a ≤ a, ν if a < a < a ˆ,    0 if a ˆ ≤ a.

Proof. See Appendix A.7. In the ladder of investment regime, the ex ante probability of infrastructure investment is strictly positive only for intermediate values of the access price. Intuitively, if a is too low, then, it is profitable for firm E to enter as a service-based competitor. Since a is low, it 15

prefers to remain service-based in period 3 whatever its retail marginal cost. On the other hand, if a is too high, then, it is not profitable to enter as a service-based operator in stage 1. Since EΠ(F ) ≤ 0, it is not profitable to invest right away either. Therefore, entry does not take place at all, and the potential entrant will never invest. Last, if a is intermediate, then, service-based entry is profitable. However, a is sufficiently large, so that firm E has incentives to invest in stage 3 only if its retail marginal cost is low. As we said before, the conventional wisdom states that a larger regulated access price should always promote infrastructure investment. Proposition 3 tells us that, once uncertainty is taken into account, this statement may be flawed. In the ladder of investment regime, investing right away is not profitable: the entrant anticipates that, if its cost turns out to be large, then, its losses will be important; therefore, it is not ready to pay the sunk investment cost in this uncertain environment. By proposing a sufficiently attractive access price, the regulator provides the entrant with a way to experiment and learn its retail marginal cost, without incurring an important sunk cost, thereby partly alleviating the uncertainty. It is worth noticing that service-based and facility-based competition can be substitutes or complements, depending on the regulated access price. To see this, consider that the regulator increases the access price from a < a to a ∈ (a, a ˆ). This is clearly detrimental to service-based competition, as this reduces the expected profit from service-based entry. On the other hand, this favors the emergence of facility-based competition, by providing the low cost entrant with incentives to invest in period 3. Therefore, in the ladder of investment regime, service-based and facility-based competition are substitutes when the access price is low. Conversely, consider that the regulator increases the access price from a ∈ (a, a ˆ) to a > a ˆ. Again, this hampers service-based competition. Yet, this is also detrimental to facility-based competition, since the potential entrant no longer has the opportunity to experiment on the retail market as a service-based competitor: the ex ante investment probability drops from ν to 0. We can thus conclude that, in the ladder of investment regime, service-based and facility-based competition are complements when the access price is high. Notice that Propositions 1, 2 and 3 fully characterize the outcome of the model in terms of investment probability. If EΠ(F ) ≤ 0, then, Proposition 1 or 3 applies, depending on whether π(0) > fˆ and EΠ(S, a) ≤ 0. If on the other hand, EΠ(F ) > 0, then, Proposition 2 applies: the first conventional wisdom subregime arises if π(0) ≤ fˆ, or if π(0) > fˆ and EΠ(F ) ≥ EΠ(S, a), while the second one arises if π(0) > fˆ and EΠ(F ) < EΠ(S, a). Figure 2 summarizes Propositions 2 and 3. It depicts the ex-ante probability of investment as a function of the regulated access price in the conventional wisdom and in the ladder of investment regimes. 16

Conventional wisdom regime

ρ(a)

ρ(a)

1

1

ν

0

a ˜

a

a

a

0

a ˜

EΠ(F ) > 0

EΠ(F ) > 0

π(0) ≤ fˆ or EΠ(F ) > EΠ(S, a)

π(0) > fˆ and EΠ(F ) ≤ EΠ(S, a)

Ladder of investment regime

ρ(a) 1

ν

0

a

a ˆ

a

EΠ(F ) ≤ 0 π(0) > fˆ and EΠ(S, a) > 0 Figure 2: The conventional wisdom vs. the ladder of investment 17

a

Since the ladder of investment regime is the novelty of this paper, we would like to know for which parameters’ values this regime is more likely to arise. In the following proposition, we show that the ladder of investment regime can arise only for low enough values of ν: Proposition 4. A necessary condition for the ladder of investment regime to arise is ν < 1/2. Proof. See Appendix A.8. To grasp the intuition, consider the tradeoff between facility-based and service-based entry in period 1, when the access price is set at a. If firm E chooses service-based entry, it pays the sunk cost fˆ only with probability ν, at the cost of earning less operating profits. On the other hand, if it invests right away, it earns more operating profits since it saves on the access price, but it always pays the sunk investment cost. If ν is high enough, the probability that firm E will eventually pay fˆ after service-based entry is large, and the operating profit effect dominates the saving on sunk cost effect. The fact that ν has to be low enough for the ladder of investment regime to arise is the more general condition that we can derive if we stick to reduced form profit functions. In the next section, we solve the model using a Cournot specification of the profit functions. This enables us to understand more fully the impact of the parameters of the model on the investment regime.

4

Numerical computations

In order to go further and to identify more precisely the situations in which the different investment regimes arise, we specify the profit functions, and we resort to numerical computations.

4.1

The Cournot model

We assume that the incumbent and the entrant offer a homogenous product on the final market. They face a linear inverse demand function P = α − βQ, where Q = qI + qE denotes the total quantity proposed by both competitors. The intercept of this function, α, can be interpreted as the retail price above which no customer would consume the final product. Parameter β relates to the slope of the inverse demand function: a higher β implies that the retail price decreases more when the total quantity supplied increases. The incumbent’s retail marginal cost is equal to cI , and we normalize to zero its marginal cost of operating its network. 18

We assume that firms I and E compete `a la Cournot on the retail market. They set their quantities, qI and qE , simultaneously, internalizing the fact that an increase in the quantity supplied lowers the price of the final product. For the sake of simplicity, we assume that, once firm E has entered, its cost becomes public information. Assuming that firm E’s cost is private information this would introduce signaling issues into our framework, which would significantly complicate the analysis.10 With this assumption in mind, let us compute the entrant’s payoff at the Nash equilibrium of this game. Consider first that the entrant has not build its own infrastructure. Firms’ payoffs are given by the following expressions: πI = (P − cI )qI + aqE = (α − β(qI + qE ) − cI ) qI + aqE , (P − c˜)qE

πE =

= (α − β(qI + qE ) − c˜) qE .

(4)

Assume first that both firms supply a positive quantity in equilibrium. Then, since the payoff are strictly concave, the equilibrium quantities solve the following system of equations: ∂πI ∂qI ∂πE ∂qE

= 0 = (α − β(qI + qE ) − cI ) − βqI , = 0 = (α − β(qI + qE ) − c˜) − βqE .

(5)

Therefore, α + a + c˜ − 2cI , 3β α + cI − 2(a + c˜) = . 3β

qI =

(6)

qE

(7)

Notice that these quantities may well be negative, which would contradict our initial assumption. In order to rule out unrealistic situations, in which the incumbent would not be able to serve a positive quantity on the final market, we assume that, for all a ≥ 0, for all c˜ ∈ {0, c¯}, 2cI < α + a + c˜. The right-hand side of this inequality is minimal when c˜ = a = 0, therefore, we have to assume that cI < α/2. On the other hand, it may well be that the entrant is not able to supply a positive quantity 10

For instance, suppose that the entrant has chosen facility-based entry in stage 1. Then, the incumbent may try to infer the value of c˜ through the entrant’s choice of quantity. Anticipating this, a high cost entrant may have incentives to mimic the behavior of a low cost entrant, namely, to set a high quantity in period 2. If these effects were taken into account, there would be no reason to believe that the profit function π(.) should be the same in stages 2 and 4. This signaling mechanism may be interesting, then again, it is orthogonal to our point. In order to get rid of it, we make the assumption that the value of c˜ becomes publicly known as soon as firm E enters. For an analysis of these signaling issues in an entry model, see Milgrom and Roberts (1982).

19

at the Nash equilibrium. This situation occurs when 2(a + c˜) ≥ α + cI . At the end of the day, plugging the equilibrium quantities into the entrant’s payoff function, we obtain: ( πS (˜ c + a) =

(α+cI −2(a+˜ c))2 9β

if 2(a + c˜) < α + cI ,

0

if 2(a + c˜) ≥ α + cI .

(8)

Assume now that the entrant has built its own network. Then, with a similar argument, it is straightforward to show that: ( πF (˜ c) =

(α+cI −2˜ c)2 9β

if 2˜ c < α + cI ,

0

if 2˜ c ≥ α + cI .

(9)

In order to avoid undesirable situations, in which a firm invests in a network, and yet, is not able to attract any final consumer, we assume in the following that c¯ < (α + cI )/2, so that a facility-based entrant always earns a positive operating profit. Define π(x) =

(α+cI −2x)2 9β

if 2x < α + cI and 0 otherwise. Notice that π(.) = πF (.) = πS (.).

Besides, π(x) = 0 if and only if x ≥ cmax = (α + cI )/2. Last, π 0 (x) < 0 and π 00 (x) > 0 for all x < cmax . Therefore, all the assumptions we have made on π(.) in the previous sections are satisfied in this Cournot setting. The values of a and a ¯ can now be computed by solving the equations π(˜ c) = π(˜ c + a) + fˆ in a for c˜ ∈ {0, c¯}. We obtain:

a=

q α + cI − (α + cI )2 − 9β fˆ

and a ¯=

4.2

α + cI − 2¯ c−

2

(α + cI )2 if fˆ < , 9β

q (α + cI − 2¯ c)2 − 9β fˆ 2

(α + cI − 2¯ c) 2 if fˆ < . 9β

(10)

(11)

Numerical computations

These specifications allow us to estimate numerically the range of parameters for which the different investment regimes arise. In particular, we are interested in whether the ladder of investment regime arises for a wide range of parameters. Since the model involves many parameters – there are actually seven parameters: α, β, c¯, cI , ν, fˆ and f – we choose to focus the discussion on two of them: ν and fˆ, which seem to be the most relevant to our problematic. First, in our model, parameter ν is a proxy for pre-entry uncertainty in the market, which is the main point of this paper. The impact of ν on investment decisions has 20

to be clarified. We have already established that the ladder of investment regime does not arise if ν is greater than 1/2 (see section 3). However, no general conclusions can be drawn when ν is smaller than 1/2. Second, fˆ is the fixed cost of building a network. We have seen before that thresholds a, a ¯, a ˜ and a ˆ are all strongly affected by parameter fˆ. Therefore, it should have an important impact on investment decisions. Our methodology is as follows. First, we choose values for parameters α, β, c¯, cI and f . In particular, we normalize β to 1/9.11 Second, we pick a pair (fˆ, ν): ν is chosen in the interval [0, 1] and fˆ is chosen in the interval [0, (3/2 · α)2 ]. Both the lower and the upper bound of this last interval are taken so that a exists for low values of fˆ, while it does not exist for high values of fˆ.12 We then determine the investment regime for this pair (fˆ, ν). This is done by applying Propositions 1, 2 and 3. For instance, the pair (fˆ, ν) is compatible with the ladder of investment regime whenever inequalities EΠ(F ) ≤ 0, π(0) > fˆ and EΠ(S, a) > 0 hold. Third, we plot the range of parameters compatible with each regime of investment in the plane (fˆ, ν). Benchmark. Let us first present our benchmark. We choose α = 100, c¯ = 35, cI = 20 and f = 10. There are several good reasons for making this choice. First, cI is lower than α/2 so that the incumbent always serves a strictly positive quantity on the retail market. Second, even if ν = 0, service-based entry is profitable as long as the access price is not too large: 2π(¯ c) > f . The results are depicted in Figure 3. 11 12

This is equivalent to multiplying f and fˆ by 9β. 2 2 By equation (10), a exists if fˆ < (α + cI ) which is lower than (3/2 · α) since, by assumption, cI < α/2.

21

ν 1

1

EΠ(F ) = 0

0.5 EΠ(F ) = EΠ(S, a) 4 2

Ladder of investment

3

EΠ(S, a) = 0

0

fˆ

Figure 3: The benchmark case: α = 100, c = 35, cI = 20, f = 10. Investment regimes: 1 = Conventional wisdom 1, 2 = Conventional wisdom 2, 3 = Ladder of investment, 4 = No-investment. There are three regimes of investment for ν below 1/2, depending on whether fˆ is small, intermediate or large. If fˆ is small, the conventional wisdom prevails (area 1 or 2): since the cost of building the infrastructure is small, there exists an access price that induces the entrant to invest right away. If fˆ is intermediate, the ladder of investment theory prevails (area 3). In this case, fˆ is small enough so that a low marginal cost entrant wants to build its own network even though facility-based entry in stage 1 is not profitable. If fˆ is large, the infrastructure investment is not profitable. Therefore, the entrant never builds its own network (area 4). Result 1. The ladder of investment regime arises for ν below 1/2 and for intermediate values of fˆ. If ν is larger than 1/2, there are only two regimes of investment: the conventional wisdom regime (area 1) and the no-investment regime (area 4). For these values of ν, the model simply illustrates the standard cost-benefit trade-off: firm E builds its own network if and only if the benefits from investment outweigh the costs. Put differently, the uncertainty on retail marginal costs is no longer relevant if ν is too large. An important observation here is that the set of parameters compatible with the ladder of investment is relatively small (see the third area in Figure 3). This seems to indicate that the implicit assumptions behind the ladder of investment theory are strong. 22

Result 2. The set of parameters compatible with the ladder of investment regime is small. Thereafter, we investigate the robustness of this result to changes in cI , f and c¯. We first investigate what happens when the incumbent is (weakly) more efficient than the entrant (cI = 0) and when the incumbent is always less efficient than the entrant (cI = 40).13 The results are depicted in Figure 4. A couple of things are worth noting here. First, the ladder of investment area is small whatever the cost of the incumbent. Besides, the size of the area is not affected by the level of the incumbent’s marginal cost. Therefore, our Result 2 is robust to change in the incumbent cost. Second, the set of parameters for which investment arises with a positive probability (ladder of investment and conventional wisdom areas) is larger when cI is high. On Figure 4, areas 1, 2 and 3 shift to the right when cI increases. In particular, the ladder of investment regime arises for higher values of fˆ. To see this, assume that cI increases, and consider the impact on the frontiers between areas 2 and 3 and areas 3 and 4. These frontiers are defined by equations EΠ(F ) = 0 and EΠ(S, a) = 0 respectively. Fix some ν. When cI increases, both the expected profits from service-based and facility-based entry rise. Therefore, on the frontiers, fˆ has to increase in order to compensate the raise in cI . We also note that the ladder of investment regime is less likely to arise for ν close to 1/2 when cI is high. Figure 4 shows that when cI = 40 there is no value of fˆ such that the ladder of investment regime arises when ν is close to 1/2. The intuition is that the impact of uncertainty on investment is weaker when cI is high since the entrant is less likely to be less efficient than the incumbent. This effect is especially strong in our example since cI is chosen so that the entrant is always more efficient than the incumbent: c¯ = 35 < 40 = cI . Result 3. The level of cI does not impact the size of the ladder of investment area. An increase in cI has two effects: (i) the ladder of investment regime becomes more likely to arise for larger values of fˆ and (ii) the ladder of investment regime becomes more likely to arise for ν close to 1/2. 13

Notice that, in this model, comparative statics on cI and α are equivalent. The reason is that α and cI only appear in the sum α + cI in the expressions of the entrant’s expected profits (see equations (8) and (9)).

23

Figure 4: Impact of cI on the investment regimes. (Left: cI = 0, Right: cI = 40) We now discuss the impact of the fixed cost f on the ladder of investment. Our results are depicted in Figure 5. The graph on the left-hand side shows that an increase in f makes the ladder of investment regime less likely to arise. The intuition can be found in the equations that define the frontiers between each areas. The frontier between areas 1 and 2 is defined by EΠ(F ) = EΠ(S, a). It is therefore unaffected by a change in f since the equation compares two investment regimes in which entry takes place, i.e., in which f is paid in stage 1. The frontier between areas 2 and 3 is defined by EΠ(F ) = 0, i.e., by: 2 (νπ(0) + (1 − ν)π(¯ c)) − (f + fˆ) = 0 Therefore, keeping ν constant in this equation, an increase in f of ∆f must be compensated by a decrease in fˆ of −∆f . Following an increase in f , the frontier between areas 2 and 3 shifts to the left. Consider now the frontier between areas 3 and 4. It is defined by EΠ(S, a) = 0, where EΠ(S, a) is given by: ( EΠ(S, a) =

c + a) − (f + ν fˆ) iffˆ ≤ π(0) − π(cmax − c¯), ν(π(a) + π(0)) + 2(1 − ν)π(¯ ν(π(a) + π(0)) − (f + ν fˆ) iffˆ > π(0) − π(cmax − c¯).

What is important for us in this equation is that the total fixed cost is given by f + ν fˆ. Therefore, keeping ν constant on the frontier between areas 3 and 4, an increase in f of ∆f 24

must be compensated by a decrease in fˆ of − ν1 ∆f . Following an increase in f , the frontier between areas 3 and 4 thus shifts to the left. More interesting is the fact that this frontier shifts more to the left than the frontier between areas 2 and 3, since

1 ν

> 1. Put differently,

the size of the ladder of investment area decreases when f increases. Result 4. An increase in f makes the ladder of investment less likely to arise.

Figure 5: Impact of f on the investment regime. (Left: f = 1000, Right: f = 3500). Last, we discuss the impact of c¯ on the occurrence of the ladder of investment regime. Figures 6 and 3 together show that an increase in c¯ makes the ladder of investment regime more likely to occur. When c¯ is small (¯ c = 20 in our example), the uncertainty on the entrant’s marginal cost is no longer an issue: the entrant builds its own network whatever its cost if the access price is sufficiently high. This is the conventional wisdom. The reverse holds when c¯ is high (¯ c = 55 in our example): if it has already built its network, it is bad news for firm E to discover a high cost. The ladder of investment logic is at work in this case. Result 5. An increase in c¯ makes the ladder of investment more likely to arise.

25

Figure 6: Impact of c¯ on the investment decision. (Left: c¯ = 20, Right: c¯ = 55)

5

Concluding remarks

In this paper, we provide theoretical foundations to the ladder of investment concept. This allows us to derive necessary and sufficient conditions for the ladder of investment regime to arise. We show that these conditions are quite restrictive, which seems to indicate that a ladder of investment approach to access regulation is not always desirable. There are indeed situations, in which service-based and facility-based competition are complements. In these cases, the regulator can promote infrastructure investment by setting an intermediate access price. Yet, there also many situations, in which the conventional wisdom prevails, and the only way to favor the emergence of facility-based competition is to deter the conditions of service-based competition.

A

Appendix

A.1

Proof of Lemma 1

Proof. Assume that a ¯ exists. Then, π(¯ c) > fˆ. Since function π(.) is strictly decreasing, π(0) > π(¯ c) > fˆ, therefore, a exists as well. Let us now show that a < a ¯. Notice that function x 7→ π(x) − π(x + a) is strictly

26

decreasing, since π(.) is strictly convex. Therefore, fˆ = π(0) − π(a) > π(¯ c) − π(¯ c + a), and a
A.2

Proof of Lemma 2

Proof. Notice first that firm E never invests in stage 3 if a = 0. Therefore, EΠ(S, 0) = 2 (νπ(0) + (1 − ν)π(¯ c)) − f = EΠ(F ) + fˆ If firm E chooses service-based entry when a ≥ cmax , then, it makes zero operating profit in stage 2, whatever its retail marginal cost. Therefore, it cannot invest in stage 3, and it does not make any operating profit in stage 4 ever. At the end of the day, its total profit is: EΠ(S, a) = −f . We have to consider several cases to prove that EΠ(S, .) is strictly decreasing. Case 1: 0 < fˆ < π(¯ c). Then, a and a ¯ exist, and EΠ(S, a) can be written as:   2 (νπ(a) + (1 − ν)π(¯ c + a)) − f      c + a) − (f + ν fˆ)   ν (π(a) + π(0)) + 2(1 − ν)π(¯ EΠ(S, a) = ν (π(a) + π(0)) + (1 − ν) (π(¯ c + a) + π(¯ c)) − (f + fˆ)     ν (π(a) + π(0)) − (f + ν fˆ)     −f

if 0 ≤ a ≤ a, if a ≤ a ≤ a ¯, if a ¯ ≤ a < cmax − c¯, if cmax − c¯ ≤ a < cmax , if cmax ≤ a.

For all points in [0, cmax ) − {a, a ¯, cmax − c¯}, EΠ(S, a) is differentiable, with a strictly negative derivative, since π(.) is strictly decreasing. All we need to do now is check that EΠ(S, .) is also decreasing at the neighborhood of a, a ¯, and cmax − c¯: lim EΠ(S, a) = EΠ(S, a) = 2 (νπ(a) + (1 − ν)π(¯ c + a)) − f = lim+ EΠ(S, a),

a→a−

a→a

lim EΠ(S, a) = EΠ(S, a ¯) = ν(π(¯ a) + π(0)) + 2(1 − ν)π(¯ c+a ¯) − (f + ν fˆ) = lim+ EΠ(S, a),

a→¯ a−

lim

a→(cmax −¯ c)−

a→¯ a

EΠ(S, a) = ν(π(cmax − c¯) + π(0)) + (1 − ν)π(¯ c) − (f + fˆ) > ν(π(cmax − c¯) + π(0)) − (f + ν fˆ) = EΠ(S, cmax − c¯) =

27

lim

a→(cmax −¯ c)+

EΠ(S, a),

lim

a→cmax −

EΠ(S, a) = νπ(0) − (f + ν fˆ) > −f = EΠ(S, cmax ) =

lim

a→cmax +

EΠ(S, a).

Therefore, EΠ(S, .) is strictly decreasing. Case 2: π(¯ c) ≤ fˆ ≤ π(0) − π(cmax − c¯).14

¯ does not exist, and π(¯ c + a) > 0. Then, a exists, a

EΠ(S, a) can be written as:   2 (νπ(a) + (1 − ν)π(¯ c + a)) − f     ν (π(a) + π(0)) + 2(1 − ν)π(¯ c + a) − (f + ν fˆ) EΠ(S, a) =  ν (π(a) + π(0)) − (f + ν fˆ)     −f

if 0 ≤ a ≤ a, if a ≤ a ≤ cmax − c¯, if cmax − c¯ ≤ a < cmax , if cmax ≤ a.

For all points in [0, cmax ) − {a, cmax − c¯}, EΠ(S, a) is differentiable, with a strictly negative derivative, since π(.) is strictly decreasing. Besides, c + a)) − f = lim+ EΠ(S, a), lim EΠ(S, a) = EΠ(S, a) = 2 (νπ(a) + (1 − ν)π(¯

a→a−

lim

a→(cmax −¯ c)−

a→a

EΠ(S, a) = EΠ(S, cmax − c¯) = ν(π(cmax − c¯) + π(0)) − (f + ν fˆ) =

lim

a→cmax −

EΠ(S, a) = νπ(0) − (f + ν fˆ) > −f = EΠ(S, cmax ) =

lim

a→cmax +

lim

a→(cmax −¯ c)+

EΠ(S, a),

EΠ(S, a).

Therefore, EΠ(S, .) is strictly decreasing. ¯ does not exist, and π(¯ c + a) = 0. Case 3: π(0) − π(cmax − c¯) ≤ fˆ < π(0). Then, a exists, a EΠ(S, a) can be written as:   2 (νπ(a) + (1 − ν)π(¯ c + a)) − f     2νπ(a) − f EΠ(S, a) =  ν (π(a) + π(0)) − (f + ν fˆ)     −f 14

if 0 ≤ a ≤ cmax − c¯, if cmax − c¯ ≤ a ≤ a, if a ≤ a < cmax , if cmax ≤ a.

The inequality π(¯ c) < π(0)−π(cmax −¯ c) can be be rewritten as π(¯ c)−π(¯ c+(cmax −¯ c)) < π(0)−π(cmax −¯ c). Since π(.) is strictly convex and decreasing, this inequality is always satisfied.

28

For all points in [0, cmax ) − {a, cmax − c¯}, EΠ(S, a) is differentiable, with a strictly negative derivative, since π(.) is strictly decreasing. Besides, lim

a→(cmax −¯ c)−

EΠ(S, a) = EΠ(S, cmax − c¯) = 2νπ(cmax − c¯) − f =

lim

a→(cmax −¯ c)+

EΠ(S, a),

lim EΠ(S, a) = EΠ(S, a) = 2νπ(a) − f = lim+ EΠ(S, a),

a→a−

lim

a→cmax −

a→a

EΠ(S, a) = νπ(0) − (f + ν fˆ) > −f = EΠ(S, cmax ) =

lim

a→cmax +

EΠ(S, a).

Therefore, EΠ(S, .) is strictly decreasing. Case 4: π(0) > fˆ. Then, neither a, nor a ¯ exist. EΠ(S, a) can be written as:

EΠ(S, a) =

  c + a)) − f if 0 ≤ a ≤ cmax − c¯,   2 (νπ(a) + (1 − ν)π(¯ 2νπ(a) − f    −f

if cmax − c¯ ≤ a < cmax , if cmax ≤ a.

For all points in [0, cmax ) − {cmax − c¯}, EΠ(S, a) is differentiable, with a strictly negative derivative, since π(.) is strictly decreasing. Besides, lim

a→(cmax −¯ c)−

EΠ(S, a) = EΠ(S, cmax − c¯) = 2νπ(cmax − c¯) − f =

lim

a→cmax −

EΠ(S, a) = −f = EΠ(S, cmax ) =

lim

a→cmax +

lim

a→(cmax −¯ c)+

EΠ(S, a),

EΠ(S, a).

Therefore, EΠ(S, .) is strictly decreasing.

A.3

Proof of Proposition 1

Proof. Fix some a ≥ 0. To begin with, notice that firm E never chooses to invest in stage 1, since EΠ(F ) ≤ 0. Assume that fˆ ≥ π(0). If EΠ(S, a) ≤ 0, then, the entrant stays out of the market, and investment will never take place. If EΠ(S, a) > 0, firm E chooses service-based entry in 29

period 1, and we know that it will never invest in period 3, since fˆ ≥ π(0). In both cases, ρ(a) = 0. On the other hand, if fˆ < π(0), then, a exists. Assume first that a ≥ a. Given that EΠ(S, a) ≤ 0, and EΠ(S, a) is decreasing in a, the entrant does not want to become servicebased in stage 1. Therefore, for a ≥ a, the entrant optimally decides to stay out of the market, and ρ(a) = 0. Assume now that 0 ≤ a < a. If EΠ(S, a) > 0, then, the entrant chooses service-based entry in stage 1. Since a < a, it does not invest in stage 3, whatever its cost. If EΠ(S, a) ≤ 0, then, the entrant stays out of the market in stage 1. In both cases, ρ(a) = 0.

A.4

Proof of Lemma 3

Proof. Notice first that EΠ(S, 0) = EΠ(F ) + fˆ > EΠ(F ), and EΠ(S, cmax ) = −f < EΠ(F ), by Lemma 2. Since EΠ(S, .) is strictly decreasing, there exists a unique a ˜ ∈ (0, cmax ], such that EΠ(F ) > EΠ(S, a) if a > a ˜, and EΠ(F ) < EΠ(S, a) if a < a ˜. Assume π(¯ c) < fˆ. The expected profit from service-based entry when a = a ¯ is given by: EΠ(S, a ¯) = ν (π(¯ a) + π(0)) + (1 − ν) (π(¯ c+a ¯) + π(¯ c)) − (f + fˆ), which is strictly smaller than EΠ(F ) = 2 (νπ(0) + (1 − ν)π(¯ c)) − (f + fˆ). Therefore, a ˜
A.5

Proof of Proposition 2

Proof. Assume EΠ(F ) > 0 and fix some a ≥ 0. If π(0) ≤ fˆ, a service-based entrant never invests, whatever its cost and the access price. If a ≤ a ˜, then, by Lemma 3, EΠ(S, a) ≥ EΠ(F ). Therefore, firm E chooses service-based entry in period 1, and does not invest in period 3: ρ(a) = 0. If a > a ˜ then, EΠ(S, a) < EΠ(F ), and firm E invests in period 1: ρ(a) = 1. On the other hand, if π(0) > fˆ, then a exists. Assume first that EΠ(F ) ≥ EΠ(S, a). Then, by Lemma 3, a ˜ ≤ a. If 0 ≤ a ≤ a ˜, then, EΠ(S, a) ≥ EΠ(F ) > 0. As a result, firm E chooses service-based entry in stage 1. Since a ˜ ≤ a, we also have that a < a. Therefore, the entrant will never 30

choose to invest in stage 3. This implies that ρ(a) = 0. If a > a ˜, then, EΠ(F ) > EΠ(S, a). Since EΠ(F ) > 0, the entrant chooses facility-based entry in stage 1. Therefore, ρ(a) = 1. Assume now that EΠ(F ) < EΠ(S, a). Then, by Lemma 3, a ˜ > a. ˜, then, EΠ(S, a) > EΠ(F ) > 0. As a result, firm E chooses service-based If 0 ≤ a ≤ a < a entry in stage 1. Since a < a, the entrant will never choose to invest in stage 3. This implies that ρ(a) = 0. If a < a ≤ a ˜, then again, EΠ(S, a) ≥ EΠ(F ) > 0, and firm E chooses service-based entry in stage 1. If a ¯ does not exist, then, firm E invest only if its cost is low. On the other hand, if a ¯ exists, then by Lemma 3, a ˜ a ˜, then, EΠ(F ) > EΠ(S, a), and firm E chooses facility-based entry. This implies that ρ(a) = 1.

A.6

Proof of Lemma 4

Proof. The first result derives directly from the fact that EΠ(S, .) is strictly decreasing, EΠ(S, 0) > 0 and EΠ(S, cmax ) < 0. Assume also that EΠ(F ) ≤ 0 and π(¯ c) > fˆ. Then, EΠ(S, a ¯) = ν (π(¯ a) + π(0)) + (1 − ν) (π(¯ c+a ¯) + π(¯ c)) − (f + fˆ), < 2 (νπ(0) + (1 − ν)π(¯ c)) − (f + fˆ), < EΠ(F ). Therefore, EΠ(S, a ¯) < 0, and a ˆ
A.7

Proof of Proposition 3

Proof. By assumption, EΠ(S, a) > 0, therefore, by Lemma 4, a < a ˆ. If 0 ≤ a ≤ a, then, since EΠ(S, .) is strictly decreasing, EΠ(S, a) > 0 ≥ EΠ(F ), and firm E chooses service-based entry in stage 1. In stage 3, the entrant remains service-based whatever its cost, since a ≤ a. As a result, ρ(a) = 0. If a < a < a ˆ, then, again, EΠ(S, a) > 0 ≥ EΠ(F ), and firm E chooses service-based entry in stage 1. Since a > a, firm E invests in period 3 if its cost is low. If its cost is high, it may invest in period 3 only if a ¯ exists. Yet, we know from Lemma 4 that if a ¯ exists, and if

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EΠ(F ) ≤ 0, then, a ˆ
A.8

Proof of Proposition 4

Proof. By Proposition 3, we know that the ladder of investment regime arises if π(0) > fˆ and EΠ(F ) ≤ 0 < EΠ(S, a). By Propositions 1 and 2, we also know that, if any of these inequalities is not satisfied, then, either the no-investment or the conventional wisdom regime arise. Therefore, π(0) > fˆ and EΠ(F ) ≤ 0 < EΠ(S, a) are also necessary conditions for the ladder of investment regime. These conditions imply in particular that: 2 (νπ(0) + (1 − ν)π(¯ c)) − (f + fˆ) < 2 (νπ(0) + (1 − ν)π(¯ c + a)) − (f + 2ν fˆ). Rearranging terms, we obtain: (1 − 2ν)fˆ > 2(1 − ν) (π(¯ c) − π(¯ c + a)) . Since the term in the right-hand side is positive, the above inequality can be satisfied only if ν < 1/2.

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