Exit Deterrence

Exit Deterrence∗ Martin C. Byford† Joshua S. Gans‡

Abstract This paper is the first to provide a general context whereby potential entry can lead incumbent firms to permanently reduce the intensity of competition in a market. All previous results found that potential entry would lead to lower prices and greater competition. Examining markets where entry occurs by the acquisition of access rights from an existing incumbent, we demonstrate that, where competitive choices are strategic complements, a more efficient entrant may be unable to acquire those rights from a less efficient incumbent due to the accommodating behavior of the efficient incumbent. Similarly, such accommodating behavior may deter efficient investment by an incumbent. These results have implications as to how economists view potential entry and its benefits.

This Version: June 2009

J.E.L. Classification: L13 Keywords: Entry, Dynamic Price Competition, Markov Perfect Equilibrium ∗ We are grateful to Yongmin Chen, Stephen King, Mark Armstrong (the editor) and two anonymous referees for comments on an earlier draft of this paper. The latest version of this paper is available at www.mbs.edu/jgans † Department of Economics, University of Colorado at Boulder. ‡ Melbourne Business School, University of Melbourne.

Economic models of the impact of potential entry on competition give rise to two distinct predictions. First, theories of contestable markets and limit pricing predict that credible entry threats will induce incumbents to price lower so as to deter that entry. Second, originating with Selten’s chain store paradox, potential entry will not induce a prior change in incumbent behavior as it either does not commit them to maintaining that behavior in the post-entry environment or their own forecasted behavior in that environment is sufficient to deter entry in of itself. Resolving the theoretical tension between the two predictions has relied on examination of the actions incumbents can take to commit themselves to tough post-entry competition or the possibility that pre-entry behavior might signal relevant information regarding anticipated post-entry behavior. Regardless, together, all of these theories predict that potential entry will not lead to less competition in a market.1 The existing models all have in common one feature: that entry, if it occurs, will be de novo; that is, the entrant adds to the pool of competitors in a market. In contrast, there are many situations where entry arises by acquisition of an incumbent. While, in some situations, this may be a choice aimed at eliminating incumbent competitors, in others, there may be market constraints on de novo entry. For instance, in a market with high sunk entry costs, a natural oligopoly might arise with a ceiling on the number of suppliers (Gilbert and Newbery, 1992). In this case, entry may only be credible if it is by acquisition. Consider a market in which there exists an asset that is required in order for a firm to gain access to the market. We term this asset as access right which has characterized by being (a) necessary for a firm to participate in a market; (b) potentially transferable and (c) rival in that it cannot be utilised by more than one firm at any point in time. The owner of such an asset may sell, but can never share, its access to a market. Transferability means that either the asset itself can be traded between firms 1

See Davis, Murphy and Topel (2004) for a recent treatment and Wilson (1992) for a review.

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or, alternatively, it can be traded as a pool of assets through a merger, acquisition or management buyout.2 Access rights can come in the form of government licensing arrangements — including broadcasting licenses for television and radio, and spectrum used by wireless carriers — as well as complementary assets that are uneconomic to replicate but critical for competition. For example, sea and air ports require significant amounts of land that satisfy the very specific requirements. Likewise, an asset such as “eyeballs” is necessary to compete in the the market for internet advertising; particularly targeted advertising connected to web searches and internet portals.3 The focus of this paper is on markets with restricted access: that is, where there is a strictly limited number of access rights. Entry into a restricted access market may only take place via acquisition of an incumbent’s access right: forcing the incumbent to exit. Simple intuition would suggest that, if an entrant emerges who is capable of utilizing the access right more efficiently than an incumbent, there would exist gains from trade in an exchange in which the access right is transferred to the entrant. Hence, it would normally be presumed that the emergence of a potential entrant would (a) not result in any change in pre-entry behaviour of incumbents, and (b) eventually result in entry. However, this simple intuition neglects the role that rival incumbents might play in influencing the terms of trade in the acquisition market for the access rights. Using an infinite horizon, dynamic model of competition, we show that, in a Markov perfect equilibrium, the threat of entry may result in reduced competition. In the model, an inefficient firm competes with an efficient firm in a restricted access market. 2

The rivalry assumption may seem questionable in a world in which regulators frequently require firms to grant their rivals access to privately owned infrastructure. However, this issue can be resolved by distinguishing between the market for infrastructure services, and the downstream market for which the infrastructure is a necessary input. 3 Moreover, the term access right could apply to situations in which firms need to access a constrained distribution system (Dana and Spier, 2007) or where limitations on physical capital restricts access.

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With the emergence of a potential entrant, the inefficient firm has an incentive to sell its rights. Nonetheless, we demonstrate that, where the actions of firms are strategic complements, when firms are sufficiently patient, there exists a Markov perfect equilibrium in which the efficient incumbent relaxes competition, directing a stream of profits to the inefficient firm sufficient to eliminate the gains from trade in the acquisition market. The efficient incumbent takes this action anticipating that the inefficient incumbent will respond by remaining in the market. In turn, the inefficient incumbent remains in the market anticipating that this accommodating behavior will continue into the future. Because the threat of entry allows the efficient firm to commit to this less aggressive stance, even in the absence of collusion, prices are higher. The higher prices, in turn, deter exit. Interestingly, compared to the situation where no potential entrant existed, it is possible for both incumbent firms enjoy higher profits. Hence, incumbents may be in favor of policies (e.g., spectrum or broadcast re-sale rights) that free-up entry into such markets. This result is significant because, to our knowledge, it is the first paper that provides a situation whereby potential entry could result in reduced competition in a market.4 Stiglitz (1981) develops a model whereby potential entry might temporarily cause prices to increase for a monopolist. He considers a situation whereby a monopolist chooses to extract a resource at a slower rate so as to forstall the entry of a competing substitute. Strictly speaking, in his model, it is the availability of the resource (akin to the incumbent holding additional capacity) that is entry deterring and the ability to sustain higher prices is, at best, temporary. The paper proceeds as follows: The general form of the model and the central 4

There are numerous examples of papers that show that increasing actual competition in a market can lead to less competition and increased prices (Satterthwaite, 1979; Perloff and Salop, 1985; Stiglitz, 1987; Schulz and Stahl, 1996; Janssen and Moraga-Gonz´alez, 2004; Gabaix, Laibson, and Li. 2005; Perloff, Suslow, and Seguin, 2006; and Chen and Riordan, 2008). These results are based on the interplay of product differentiation and/or consumer search. In contrast, in the model developed in this paper competition is greater post-entry than the pre-entry benchmark.

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results are set out in section 1. Section 2 presents a simple example on the Hotelling line, demonstrating that the emergence of a potential entrant can leave both incumbent firms better off. Section 3 presents a generalisation demonstrating that the equilibrium does not rely on knife edge assumptions. The paper concludes with a discussion of the results.

1

The Model

This section develops an infinite horizon, dynamic model of competition in a restricted access market. For the purposes of this paper it is assumed that there exist two access rights, and as a consequence there can be at most two firms competing in the market at any point in time. Initially, the access rights are owned by firms 1 and 2, with firm 1 assumed to be more efficient than firm 2. Firms 1 and 2 interact in a repeated game of duopoly competition. At some point an exogenous policy break occurs, permitting resale of access rights. For instance, the policy change may allow broadcast or spectrum license re-sale after some earlier allocation to firms 1 and 2. From this point forward firms external to the market become potential buyers for the access rights, and consequently potential entrants. Assumption 1. It is assumed that: (a) regardless of the history, at least one potential entrant is present in each period; (b) each potential entrant is as efficient as firm 1; (c) anti-trust regulators prevent any one firm acquiring both access rights. Intuitively, (a) can be understood as the analogue of free entry in a restricted access market. Capital seeks a profit and as a consequence there always exists a potential buyer for any under performing asset. Where trading of access rights has occurred in a previous period, a former incumbent may also be a potential entrant. (b) Can be interpreted as implying that both firm 1, and the potential entrant, are operating at the technological frontier. The efficiency of the potential entrants 4

suggests that the policy break permitting resale of access rights is desirable from a social perspective.5 Potential entrants are indexed in order of arrival. The first potential entrant to emerge is denoted as firm 3. If firm 3 ever succeeds in acquiring an access right and entering the market, firm 4 emerges as a potential entrant, and so on. After a potential entrant arrives, the game consists of two distinct phases. Each phase is itself a repeated game with the number of periods in each phase endogenously determined. 1. In the entry phase an efficient firm (initially firm 1) competes with firm 2, under the threat of entry by an efficient potential entrant (initially firm 3). Each period begins with the incumbent firms selecting actions and realizing instantaneous profits. The incumbent firms then have the opportunity to sell their access rights to an entrant. If no transaction takes place, or if the efficient incumbent sells out, then the game enters a new period of the entry phase. If the inefficient firm sells out, the game proceeds to the post-entry phase.6 2. In the post-entry phase, two efficient firms compete. In each period of that phase, the two firms simultaneously select actions before realizing instantaneous profits. As in the entry phase, either incumbent may trade its access rights to an equally efficient entrant. This game is repeated infinitely. All firms discount the future by the common discount factor δ ∈ (0, 1). Because the transition from the entry to post-entry phase only occurs with firm 2’s consent, the game may never reach the post-entry phase. Figures 1 and 2 illustrate the intraand inter-period timing of the game. 5

See Cramton (2000) on the effect of the FCC’s ”unjust enrichment” requirements on the ability to resell spectrum. 6 If the efficient incumbent exits the market, it is replaced by the entrant, and a new potential entrant emerges immediately. It is assumed that there always remains a potential entrant to take its place. Consequently, the replacement of an efficient incumbent, with an equally efficient entrant, does not alter the structure of the game.

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Competition Stage

Period t s

Exit Stage

s

s

firms select actions (ai , aj )

profits realised

Period t + 1

s

s

-

incumbents may exit

Figure 1: Intra-Period Timing Exogenous Policy Change

Firm 2 Exits

yes

yes -

Duopoly Phase

-

Entry Phase

6

Post-Entry Phase

6

6

no

no Figure 2: Inter-Period Timing

It will be useful to benchmark the outcomes here with a baseline duopoly phase involving competition between firms 1 and 2. Each period of the duopoly phase sees firms 1 and 2 simultaneously selecting actions and realising the consequent instantaneous profits. Exit is not possible by selling access rights to eachother. As will be demonstrated below, the duopoly and post-entry phases share the characteristic that, in a Markov perfect equilibrium, market outcomes involve firms considering only instantaneous profits. In contrast, as behavior in the entry phase might trigger a transition the post-entry phase, firms active in the entry phase consider outcomes beyond the current period.

1.1

Primitives

All firms in the game share the instantaneous profit function π(ai , aj , θ); where for a closed bounded interval A ⊂ R, ai ∈ A is the firm’s own action, aj ∈ A is the rival firm’s action, and θ is a parameter describing the firm’s inefficiency or cost level. It is assumed that π is twice continuously differentiable in all its arguments and bounded from above. Moreover, π is assumed to be strictly concave in the firm’s own action

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(∂ 2 π/∂a2i < 0), while for all aj and θ there exists a function R(aj , θ) ∈ A such that
∂π R(aj , θ), aj , θ /∂ai = 0. It follows from the continuity and differentiability of π that R is likewise continuously differentiable. Moreover, if ∂ 2 π/∂ai ∂aj > 0 (< 0) the actions of firms are strategic complements (substitutes) and ∂R/∂aj < 0 (> 0). It is assumed that this derivative is bounded such that ∂R/∂aj ∈ (−1, 1).7 For the purposes of this paper, aggressive behavior by a firm is characterised by that firm selecting a high value of ai , while a low value of ai is interpreted as the firm adopting a passive position. Formally, this characterization requires the firm’s profit to be strictly decreasing in its rival’s action (∂π/∂aj < 0). The inefficiency parameter is assumed to have a negative impact on profits hence ∂π/∂θ < 0; as well as reducing the incentive for a firm to behave aggressively, ∂ 2 π/∂ai ∂θ < 0. This second condition implies ∂R/∂θ < 0. In words, increasing the inefficiency parameter causes the best response function to shift inward. In the baseline model it is assumed that firm 2 has an inefficiency of θ2 = c > 0, while θi = 0 for all remaining firms i ∈ {1, 3, 4, . . . }.

1.2

States

For the firms competing in this market, the state of the world is captured by the efficiencies of the two firms currently in possession of the access rights, as well as the existence and inefficiency of a potential entrant. Formally, the state space can be written as Ω = R2 × (R ∪ ∅) where elements in R represent the inefficiency of the three respective firms, while a value of ∅ for the third co-ordinate indicates that no potential entrant exists. Of course, at most three elements of this space may arise in any given specification of the model. In the duopoly phase, two firms are active with inefficiencies zero and c; in the entry phase a potential entrant emerges with inefficiency zero; while in the postentry phase both the two active firms, and the potential entrant share inefficiencies of 7

These bounds ensure the existence and uniqueness of an equilibrium in the static game.

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zero. It follows that we can utilize the discrete variables D = (0, c, ∅), E = (0, c, 0) and P = (0, 0, 0) to represent the state of the world in the duopoly, entry and postentry phases respectively. Note that while one efficient firm selling its access right to another efficient firm changes the identities of the firms active in the market, it does not alter the state.

1.3

Refining the Set of Equilibria

In accordance with the folk theorem, the model set out above possesses a large number sub-game perfect Nash equilibria. The purpose of this paper is to investigate the impact of potential entry on the oligopoly behaviour of the incumbent firms. In other words, the impact of the (potential) transition between the states D, E and P . The obvious refinement to employ is Markov perfection. The set of Markov perfect equilibria (MPE) is produced by refining the set of all sub-game perfect equilibria to only admit those sub-game perfect equilibria in which all firms play Markov strategies; that is, strategies that depend only on payoff relevant information in the game. However, while constraints are placed on the set of strategies that a firm can employ in equilibrium, every MPE must be robust against a unilateral deviation by a firm to any available strategy within the game; including non-Markov strategies. Markov perfection is a widely employed refinement where examining oligopoly (as opposed to collusive) behaviour in infinite horizon, dynamic oligopoly models.8 In the context of this paper the most significant feature of the MPE refinement is that in equilibrium firms may not condition their actions on the past behavior of any firm, where that behavior did not impact upon payoff relevant variables. This prevents firms from employing super-game punishment strategies in equilibrium; and as such 8

Maskin and Tirole (1988a&b) introduce MPE as means of examining oligopoly behavior over an infinite horizon, while Ericson and Pakes (1995) developed the foundation for employing MPE in empirical work. Maskin and Tirole (2001) provide a comprehensive discussion of the virtues of the Markov perfect refinement in applied work. See Vives (1999, Chapter 9) for a comprehensive treatment.

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an MPE cannot be regarded as a collusive outcome. It must be stated that the Markov perfect refinement has a dramatic effect in the model developed in this paper. As shown below, there exist at most two types of MPEs in the entry game. However, we do not regard the exclusion of non-Markov sub-game perfect equilibria as problematic. Quite the opposite. The Markov perfect refinement allows us to isolate a critical feature of our model. With super-game punishment strategies excluded in an MPE, any deviation by a firm from its static best response must be the consequence of at least one firm being able to induce a transition between states. Thus, in the present model, any discrepancy between MPE behavior, and the static oligopoly equilibrium, must be the consequence of firm 2’s ability to trade away its access rights.

1.4

Markov Strategies

In an MPE, firms respond to the prevailing payoff relevant variables in the market. It follows that each firm’s MPE strategy can be expressed as a functions of these variables. When selecting actions for the competition stage, the payoff relevant information is the prevailing state of the world ω ∈ {D, E, P }. It follows that the competition stage component of firm 1’s strategy can be written as the triplet P E (aD 1 , a1 , a1 ), where superscripts denote the state corresponding to the actions. Like-

wise, the competition stage component, of the entrant firm i’s strategy, is the pair P D E (aE i , ai ); while for firm 2 we have the pair (a2 , a2 ).

In the exit stage of the entry phase the payoff relevant variable is the magnitude of the profits that a firm receives by staying the market. It follows that we can define the set πiω ⊂ R such that firm i will exit if, and only if, firm i’s instantaneous profit from the competition stage of the current period is an element of πiω . The superscript ω ∈ {E, P } represents the state to which the set πiω applies.

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1.5

Equilibrium Duopoly and Post-Entry Phase Profits

In an MPE, actions, and, therefore, instantaneous profits in the duopoly and postentry phases are independent of firm behavior in any other phase. Lemma 1. In a Markov perfect equilibrium firms play their static Nash equilibrium actions in both the duopoly phase and post-entry phase. Proof. Once in the post-entry phase, the exchange of access rights cannot alter the state, thus the post-entry phase repeats indefinitely. It follows that in a Markov perfect equilibrium the two firms active in the market both select the action aP that
maximises their instantaneous profits; that is to say aP = R R(aP , 0), 0 . While the game will eventually proceed beyond the duopoly phase, actions in the duopoly phase neither effect instantaneous payoffs in later phases nor the probability of entering the entry phase. It follows that in a Markov perfect equilibrium firms 1
D and 2 select actions to maximise instantaneous profits, thus aD 1 = R R(a1 , c), 0 and
D aD 2 = R R(a2 , 0), c . Lemma 1 establishes that both prior to the policy break that creates the possibility of a potential entrant, and following the displacement of the inefficient incumbent, the market behaves as in a one shot game. It is only during the entry phase that the strategic effect of the option to exit comes into play. Lemma 1 also defines the instantaneous profits for the duopoly and post-entry phases, and therefore both the viability of the potential entrant and the incentive for the efficient incumbent to resist entry. D P P Lemma 2. In a Markov perfect equilibrium π(aD 1 , a2 , 0) > π(a , a , 0). P Proof. First note that as R(·, c) lies strictly inside R(·, 0), aD 2 < a . It follows that,

P P D P D π(aD 1 , a2 , 0) > π(a , a2 , 0) > π(a , a , 0),

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where the first inequality results from the strict concavity of π, while the second is implied by assumption that π is strictly decreasing in a rival’s action. Lemma 2 unambiguously states that entry has a negative impact on the profits of firm 1, and therefore that firm 1 has an incentive to prevent entry. P P D Assumption 2. π(aD 2 , a1 , c) < π(a , a , 0) for all c > 0.

Assumption 2 is necessary for the replacement of firm 2 to be viable. Intuitively, an entrant and firm 2 will never be able to agree on a price for firm 2’s access rights if the access rights are more valuable to firm 2 than the entrant. Note that assumption 2 is an assumption over the form of the profit function.

1.6

Entry Phase

A potential entrant can only purchase an access right if there exists a fee that is both acceptable to the entrant, and sufficient to induce the incumbent to exit the market. For the fee to be acceptable to an incumbent in a sub-game perfect equilibrium, it must be greater than the stream of profits the incumbent would receive by remaining in the market. We make the following simplifying assumption: Assumption 3. All incumbent firms have a weak preference to remain in the market. Assumption 3 is necessary to avoid an openness problem. Intuitively, a weak preference to remain in the market can be understood as representing the existence of arbitrarily small transaction costs associated with the transfer of access rights. By purchasing firm 2’s access right, an entrant causes the game to transition to the post-entry phase. It follows that the return to purchasing firm 2’s access right in the entry phase, or any incumbent firm’s access right in the post-entry phase, is, Πi =

δ π(aP , aP , 0). 1−δ

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This is also the return that an incumbent firm receives by remaining in the market in the post-entry phase. The equality of these returns prevents entry in the postentry phase as a potential entrant is not willing to offer a fee that is greater than δ π(aP , aP , 0). 1−δ

This result generalizes to the entry phase as well.

To analyze the outcomes that occur, we first rule out acquisition by the entrant of the efficient incumbent. Lemma 3. In an MPE, an entrant will never purchase the access rights of an efficient incumbent. Proof. The exchange of access rights between two efficient firms does not change the state of the game, and therefore, in an MPE, does not alter the remaining incumbent’s behaviour. The entrant and efficient incumbent share identical profits functions and choice sets. It follows that in an MPE, an entrant cannot expect a stream of profits that is greater than the stream of profits that the original owner of the rights would be able to secure. In the proof, this is a direct consequence of the assumption that following the policy break, at least one potential entrant is always present. Consequently, acquisition of the efficient incumbent does not change the phase of the game and hence, there are no strictly positive gains from such an acquisition.9 The return that firm 2 receives by remaining in the market indefinitely — thereby preventing the game from proceeding to the post-entry phase — is, Π2 =

δ E π(aE 2 , a1 , c). 1−δ

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If there were only a single potential entrant, then acquisition of the efficient incumbent would eliminate the threat of entry and return the game to the duopoly phase. However, as is demonstrated in the Hotelling example of section 2, there exist parameters whereby the profits of the efficient incumbent are lower in the duopoly phase than in the subsequent entry phase. In such a case a variant of the lemma will continue to hold. In section 3 we also consider what happens when the potential entrants are less efficient than the efficient incumbent.

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E Let F π(aE 2 , a1 , c) represent the fee that an entrant pays to firm 2 in exchange for firm 2’s access right where, F

E π(aE 2 , a1 , c)




 ∈

 δ δ E E P P π(a2 , a1 , c), π(a , a , 0) , 1−δ 1−δ

and F 0 ≥ 0.10 In words, F is in the range that is acceptable to both firms, and is weakly increasing in the current profitability of firm 2. Given firm 2’s weak preference E to remain in the market, any fee that would induce firm 2 to exit where π(aE 2 , a1 , c) ≥

π(aP , aP , 0) would not be acceptable to the entrant. The following lemma summarises firm 2’s strategy in the exit stage of the entry phase. Lemma 4. In an MPE, firm 2 will exit in the entry phase if, and only if, its instantaneous profit is less than the entrant’s equilibrium instantaneous profit in the
post-entry phase; π2 = −∞, π(aP , aP , 0) . Firm 2’s response to a potential entrant’s offer of F is purely mechanical; an optimal response to the offer at the point in time that the offer is made. This must be the case in a Markov perfect equilibrium as Markov perfect equilibria are sequential. It turns out that this mechanical behavior extends to the competition stage of the entry phase. In other words, firm 2’s strategy in the entry phase, does not encompass any strategic objective beyond maximizing instantaneous profits. Lemma 5. In an MPE, firm 2 always plays its static best response in the competition E stage of the entry phase; aE 2 = R(a1 , c).

Proof. There two possibilities to consider. If, given firm 1’s choice of aE 1 , firm 2 E E P P E can select aE 2 such that π(a2 , a1 , c) ≥ π(a , a , 0) then selecting a2 to maximise π

delivers firm 2 an outcome that is strictly preferred to any fee that an entrant is willing to offer. 10

This formulation for F is satisfied by both the generalised Nash bargaining solution — a reasonable assumption where firms 2 and 3 are engaged in bilateral bargaining over the price of the δ access rights — and the case F = 1−δ π(aP , aP , 0) which would be expected to arise where two or more equally efficient potential entrants are in competition for the access rights.

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E E P P Where firm 2’s choice of aE 2 cannot result in π(a2 , a1 , c) ≥ π(a , a , 0) firm 2 will

exit at the end of the current period of the entry phase. Nevertheless, maximising π remains in firm 2’s best interests as it maximises firm 2’s instantaneous payoff in the current period, as well as weakly increasing the fee firm 2 receives from the entrant.

1.7

Markov Perfect Equilibria

It follows from lemma 5 that there exist at most two classes of Markov perfect equilibria to this game: A competitive equilibrium in which firm 2 exits in the first period of the entry phase, and an accommodating equilibrium in which firm 2 remains in the market indefinitely. The following lemmas characterise firm 1’s strategies in each of these equilibria. Lemma 6. In an MPE in which firm 2 remains in the market indefinitely — an accommodating equilibrium — aE 1 solves,
E π R(aE , c), a , c = π(aP , aP , 0). 1 1 Proof. From lemmas 4 and 5 it is straightforward to see that aE 1 must satisfy,
E P P π R(aE 1 , c), a1 , c ≥ π(a , a , 0).

(1.1)


E E To exclude the inequality note that π R(aE 1 , c), a1 , c is strictly decreasing in a1 . D P P E Given that π(aD 2 , a1 , c) < π(a , a , 0) it follows that in order to satisfy (1.1) a1 < E aD 1 and thus from the single crossing property of the best response functions a1 <
R R(aE 1 , c), 0 .
E P P Suppose to the contrary that π R(aE 1 , c), a1 , c > π(a , a , 0) in a Markov perfect

equilibrium. Given the continuity and concavity of π, firm 1 can increase its instantaneous profit in each period of the exit phase by marginally increasing aE 1 . For a small enough increase (1.1) will continue to hold and firm 2 will remain in the market. A contradiction. 14

Lemma 7. In an MPE in which firm 2 exits in the first period of the exit phase — a competitive equilibrium — firm actions and instantaneous profits in the entry phase are identical to their actions and instantaneous profits in the duopoly phase. Proof. It follows from lemmas 2, 4 and 5 that it is only necessary to show that D aE 1 = a1 . To see that this must be the case note that firm 1’s only concern in the

exit phase is to maximise its instantaneous profits. It follows from lemma 6 that a necessary condition for the existence of an accommodating equilibrium is,  

1 δ E E E π aE π(aP , aP , 0). (1.2) , R(a , c), 0 ≥ π R R(a , c), 0 , R(a , c), 0 + 1 1 1 1 1−δ 1−δ That is, the stream of profits firm 1 receives accommodating firm 2 must be at least as great as the profit firm 1 receives from deviating to its static best response in the first period of the entry phase, followed by infinite repetition of the post-entry game. Note that there exists a δ ∈ (0, 1) such that (1.2) holds if, and only if,
E P P π aE 1 , R(a1 , c), 0 > π(a , a , 0).

(1.3)

Similarly, from lemma 7 it follows that a necessary condition for the existence of a competitive equilibrium is, D π(aD 1 , a2 , 0) +

δ 1 π(aP , aP , 0) ≥ π(a1 , aD 2 , 0), 1−δ 1−δ

(1.4)

where a1 solves π(a2 , a1 , c) = π(a1 , aD 2 , 0). Firm 1 will not attempt to accommodate firm 2 so long as the return from accommodating firm 2 is weakly less than the return D from playing static best responses in every phase of the game. As π(aD 1 , a2 , 0) >

π(a1 , aD 2 , 0), condition (1.4) is always satisfied for δ sufficiently close to zero. Proposition 1. Where the actions of firms are strategic complements there exist ¯ 1) an accommodating c¯ > 0 and δ¯ ∈ (0, 1), such that for all c ∈ (0, c¯) and δ ∈ (δ, equilibrium exists. 15

Proof. Define a(c) as the action that solves, 
 π R a(c), c , a(c), c = π(aP , aP , 0), and note that a(c) is continuous, strictly decreasing in c, and that as c → 0, a(c) → aP . The continuity of all functions in c guarantees that there exists numbers cˆ > 0 and P P a ˆE ˆ), a(c) ∈ (ˆ aE 1 < a such that for all c ∈ (0, c 1 , a ).

For a given c > 0, there exists δ ∈ (0, 1) such that an accommodating equilibrium exists if, and only if, (1.3) holds. Note that in the limit, as c → 0, (1.3) holds with equality. Taking the derivative of the LHS of (1.3) yields,
 ∂π(ai , aj , 0) d  π a(c), R a(c), c , 0 = a0 dc ∂ai   ∂R(ai , c) ∂π(ai , aj , 0) 0 ∂R(ai , c) + a . (1.5) + ∂ai ∂c ∂aj (ai ,aj )=[a(c),R(a(c),c)]
The first term on the RHS of (1.5) is unambiguously negative as a(c) < R R(a(c), c), 0 , and approaches zero as c → 0. The second term is positive where the actions of firms are strategic complements. For arbitrarily small c > 0, the first term on the RHS of (1.5) is likewise arbitrarily close to zero and therefore (1.5) is positive. It follows that there exists c˜ > 0 such that (1.3) is satisfied for all c ∈ (0, c˜). Defining c¯ = min{ˆ c, c˜} completes the proof. Proposition 1 proves that strategic complementarity is a sufficient condition for the existence of an accommodating equilibrium for a non-degenerate interval of inefficiency parameters. Moreover, it is clear from the proof that the distortion away from static equilibrium behaviour, necessary to implement the accommodating equilibrium, is increasing in c. The following propositions provide a comparison of firm competition and payoffs under the competing equilibria. Proposition 2. For sufficiently high δ ∈ (0, 1), an accommodating equilibrium delivers firms 1 and 2 outcomes that are strictly preferred to a competitive equilibrium. 16

Proof. It is straightforward to see that firm 2 prefers the accommodating equilibrium. To see that firm 1 is likewise better off note that (1.3) must hold in an accommodating equilibrium. Proposition 3. In an accommodating equilibrium competition is at its lowest in the entry phase. D P Proof. Given the necessity of strategic complementarity aE and aE 1 < a1 < a 2 <

aD 2 . It is important to note that an accommodating equilibrium can only exit in an infinite horizon setting. Firm 1 takes an action that is less aggressive than its instantaneous best response, anticipating that firm 2 will respond by remaining in the market. In turn, firm 2 decides to remain in the market anticipating that firm 1 will continue its soft stance next period. The equilibrium depends upon the possibility of future interaction in every sub-game along the equilibrium path; unravelling in a finite horizon. In this way, the equilibrium developed in this paper is similar to the MPE of infinite horizon sequential moves price setting games developed by Maskin and Tirole (1988b) and Eaton and Engers (1990). All these equilibria involve firms taking actions that are less than competitive, and deliver firms profits that are in excess of competitive levels.

1.8

Costly Entry

Thus far we have assumed that the only cost of entry is cost of acquiring the access right. In reality, we would expect there to exist both transaction costs associated exchanging the access right and some expense required for firm 3 to establish a presence in the market. Here we show that the addition of an entry cost improves the ability of the efficient incumbent to accommodate its inefficient rival, while simultaneously reducing the impact of accommodating behavior on competition in the market. 17

Suppose that, in addition, to the fee paid for the access right, firm 3 incurs a cost C > 0 if it enters the market. The imposition of an entry cost reduces the return to firm 3 of purchasing the access rights. Going forward, the return to purchasing the access right in any given period is reduced by the amount of the entry cost to, Π3 =

δ π(aP , aP , 0) − C. 1−δ

It follows from lemma 4 that, in the presence of an entry cost, firm 2 will trade away its access rights if, and only if, E P P π(aE 2 , a1 , c) < π(a , a , 0) −

1−δ C. δ

From lemmas 5 and 6 it follows that in order to accommodate firm 2, firm 1 must select aE 1 such that,
1−δ E P P π R(aE C < π(aP , aP , 0). 1 , c), a1 , c = π(a , a , 0) − δ

(1.6)

The inequality in (1.6) indicates that it is easier for firm 1 to accommodate firm 2 where entry is costly. Given this structure the condition 1.3 is once again necessary for the existence of an accommodating equilibrium. Proposition 4. Suppose that the action of firms are strategic complements and that c ∈ (0, c¯), where c¯ is defined as in proposition 1. For all C > 0 there exists δ ∈ (0, 1) such that there exists an accommodating equilibrium to the entry game. Proof. The case in which, C≥

δ D π(aD 2 , a1 , c), 1−δ

D can be dismissed as aE 1 = a1 is sufficient to accommodate firm 2. For the remaining
E E cases note that by the monotonicity of π R(aE 1 , c), a1 , c in a1 , and the proof to
proposition 1, there exists aE c), aD that satisfies (1.6) and therefore, by the 1 ∈ a(¯ 1

proof to proposition 1, also satisfies (1.3).

18

Costly entry reduces firm 3’s willingness to pay for the access right. Consequently, the stream of profits that the efficient firm must direct to its rival — in order to prevent the inefficient firm from selling out to the entrant — is reduced, and the magnitude of the distortion that must be created in order facilitate accommodation is likewise smaller. In other words, the lower the cost of entry that the entrant faces, the greater will be the distortion away from a competitive outcome in an accommodating equilibrium.

1.9

Endogenous Efficiency Enhancing Investment

By assumption, the efficient production technology is available to all potential entrants. It is therefore reasonable to ask: Why does the inefficient incumbent not simply implement the cost reduction itself? It turns out that, with or without the presence of a potential entrant, there exists an accommodating equilibrium in which firm 2 will not adopt an efficiency-enhancing technology, even where upgrading the technology is costless. To see this, suppose that as well as (or instead of) the option to exit the market, firm 2 has the option to reduce its inefficiency from c to zero. Doing so causes the game to transition to the post-entry phase. Once again it is firm 2 that controls the transition between states. The return to implementing the efficient technology is exactly the stream of post-entry profits

δ π(aP , aP , 0). 1−δ

It follows that if there exists

an accommodating equilibrium to the entry game where firm 2’s inefficiency is c, the same accommodating equilibrium will prevent firm 2 implementing the more efficient technology unilaterally.

2

Accommodation in Hotelling Competition

This section utilizes the example of price competition on the Hotelling line to illustrate propositions 1–3. Hotelling competition provides a clear example as not only is

19

competition reduced in the entry phase of an accommodating equilibrium in accordance with proposition 3, but the profits of both firms increase with the emergence of a potential entrant. The Hotelling example also provides a basis for a social welfare analysis of accommodating behavior.

2.1

The Example

Firm 1 is locates at the left end of a unit line and faces a marginal cost of zero, while firm 2 is located at the right end of the line and faces a marginal cost of c ∈ (0, 3t).11 A unit mass of consumers is uniformly distributed along the line with each consumer having unit demand and linear transport cost t > 0 per unit travelled. The market is assumed to be covered in each phase. If firm 3 replaces firm 2 in the entry phase it takes firm 2’s position at the right end of the line. Each period firms compete by simultaneously setting prices. Given that low prices represent aggressive behaviour, a firm’s action is interpreted as being the negative of its price. For a firm with marginal cost θ, instantaneous profit is given by the function π(pi , pj , θ) = (pi − θ)

pj − pi + t , 2t

where pi is the firm’s own price and pj is the price set by the rival firm. The corresponding best response is given by the function, R(pj , θ) =

pj + t + θ . 2

Employing the results of the previous section, this structure is sufficient to establish the Markov perfect equilibrium behaviour of firms in both the duopoly and post-entry 11

The upper-bound on c ensures that firm 2 will always produce a strictly positive quantity in a static equilibrium.

20

phases. Lemma 1 implies, pD 1

1 = t + c, 3

pD 2

2 = t + c, 3

 1 = t+ 2t  1 D D π(p2 , p1 , c) = t− 2t t π(pP , pP , 0) = . 2

D π(pD 1 , p2 , 0)

pP = t,

1 c 3

2

1 c 3

, 2 ,

P P D D D It is straightforward to see that π(pP , pP , 0) > π(pD 2 , p1 , c) and π(p1 , p2 , 0) > π(p , p , 0).

The value of the variables in the entry phase depend on the nature of the equilibrium. If an accommodating equilibrium exists pE 1 is defined implicitly by, E

pE t 1 − R(p1 , c) + t E E π R(pE = = π(pP , pP , 0). 1 , c), p1 , c = R(p1 , c) − c 2t 2 E E The unique solution to this equation is pE 1 = t + c. Moreover, p2 = R(p1 , c) = t + c

yielding instantaneous profits of, π(t + c, t + c, 0) =

t+c , 2

t π(t + c, t + c, c) = . 2

To prove the existence of an accommodating equilibrium it only remains to show that there exists δ ∈ (0, 1) such that (1.2) holds. R(t + c, c) = t + c/2 thus (1.2) becomes, 2      1 δ t+c 1 1 t ≥ t+ c + , 1−δ 2 2t 2 1−δ 2 c2 δ c≥ , 1−δ 4t  c 1 δ≥ ∈ 0, . 4t + c 7 In words, an accommodating equilibrium exists for all c ∈ (0, 3t) and δ ∈ (1/7, 1); a wide range of discount factors. Surprisingly, both incumbent firms benefit from the emergence of a potential entrant in an accommodating equilibrium. Given that c < 3t,  2 t+c 1 1 > t+ c , 2 2t 3 21

(2.1)

indicating that firm 1 enjoys higher profits in the entry phase than it does in either the post-entry or duopoly phases. Intuitively, with prices strategic complements in Hotelling competition, the inefficient incumbent responds to a relaxation of competition by itself reducing competition. In turn this reduces the cost to firm 1 of accommodating firm 2. Absent a potential entrant, firm 1 cannot credibly commit to
this reduction in competition as R R(p1 , c), 0 < p1 for p1 > t + c/3, implying that firm 1 has the incentive to unilaterally lower its price. This incentive is eliminated by the fact that firm 2 will sell out to an efficient entrant should the flow of profits to the firm fail to match the value of the access rights to the entrant.

2.2

Welfare in an Accommodating Equilibrium

A welfare analysis of the accommodating equilibrium in the Hotelling case is straightforward. The twin assumptions of market coverage and unit demand imply that social welfare is a function of productive efficiency alone. Productive efficiency, and therefore welfare, is highest in the post-entry phase where the efficiency of both firms gives rise to a zero cost of production. In the duopoly phase the inefficient firm serves a fraction q2D = (3t − c)/6t of the market at a marginal cost of c. In other words, going forward the welfare gain that results where the inefficient firm is replaced by an efficient entrant is,   δ c c2 ∆WGain = − . 1 − δ 2 6t By choosing to accommodate its rival in the entry phase the efficient incumbent not only prevents the efficiency gain ∆WGain but also concedes market share to its inefficient rival. In turn, this results in the cost of production rising to c/2 each period; higher than the cost of production in either the duopoly or post-entry phases. In fact, in an accommodating equilibrium, the emergence of a potential entrant results in the welfare loss, ∆WLoss

δ = 1−δ 22



c2 6t

 .

The c2 term represents the fact that as the inefficient firm’s cost disadvantage increases, so too does the share of the market that the efficient firm must sacrifice to the inefficient firm in an accommodating equilibrium. At the same time the higher value of c increases the welfare cost of any given market share that the efficient firm sacrifices. The order of prices — implied by proposition 3 and illustrated in this example — have implications for cases in which demand is downward sloping. In an accommodating equilibrium prices are highest in the entry phase and lowest in the post-entry phase. This suggest that in addition to the loss of productive efficiency, accommodation may also increase the magnitude of the dead-weight loss in the market.

3

Asymmetric Entrants

The model developed in section 1 assumes that both firm 1 and the potential entrants share equally efficient production technologies. Here we verify that the existence and nature of the MPEs is not an artefact of this knife edge assumption. This section develops a generalisation in which the efficiency of the potential entrants is less than that of firm 1 by an arbitrarily small amount ε > 0. Formally, θi = ε for all firms i ∈ {3, 4, . . . } that may emerge as potential entrants. This difference may represent that firm 1 possesses some market specific knowledge or advantage, or access to a patented technology, not available to an entrant. For the purposes of this section it is assumed that the actions of firms are strategic complements, and that ε < c and c ∈ (0, c¯), where c¯ is as defined in proposition 1.

3.1

States

Given the asymmetry between the efficiencies of incumbents and potential entrants, the state space is more complex than in the baseline model. Once the policy break permits trading in access rights, there are four states in the game may operate.

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Initially, firms 1 and 2 compete under threat of entry by firm 3; state (0, c, ε). From this state, an entrant can replace either of the incumbents giving rise to the states (ε, c, ε) or (0, ε, ε). If entry eventually results in the replacement of both of the original incumbents the state becomes (ε, ε, ε). From lemma 3 it follows that in an MPE there can be no entry in the states (0, ε, ε) and (ε, ε, ε). Consequently, in an MPE, firms play their instantaneous best (0,ε,ε)

responses in these states. We write these MPE actions as a1 (0,ε,ε)

and note that π(a1

3.2

(0,ε,ε)

, a3

(0,ε,ε)

, 0) > π(a(ε,ε,ε) , a(ε,ε,ε) , ε) > π(a3

(0,ε,ε)

, a3

(0,ε,ε)

, a1

and a(ε,ε,ε) ;

, ε).

Accommodating Equilibrium

For sufficiently small ε > 0, accommodation is possible in the states (0, c, ε) and (ε, c, ε). The case in which the entrant purchase access rights from the efficient incumbent, bringing about the state (ε, c, ε), is equivalent to the case analysed in section 1. Given the continuity assumptions, for sufficiently small ε > 0 an accommodating equilibrium (ε,c,ε)

exists and the corresponding MPE instantaneous profits satisfy π(a3 (ε,c,ε)

π(a2

(ε,c,ε)

, a3

(ε,c,ε)

, a2

, ε) >

, c) = π(a(ε,ε,ε) , a(ε,ε,ε) , ε).

Accommodation in the state (0, c, ε) requires firm 1 to take an action a1 such
(0,ε,ε) (0,ε,ε) , ε). Moreover, for sufficiently small ε > 0, , a1 that π R(a1 , c), a1 , c = π(a3
(0,ε,ε) (0,ε,ε) π a1 , R(a1 , c), 0 > π(a1 , a3 , 0) and therefore there exists δ ∈ (0, 1) such that playing the strategy a1 is rational. To see that an entrant would not be willing to purchase firm 1’s access right in (0,ε,ε)

the state (0, c, ε), given that π(a(ε,ε,ε) , a(ε,ε,ε) , ε) > π(a3

(0,ε,ε)

, a1

, ε), an entrant must

take a softer action in the states (ε, c, ε), than firm 1 must take in the state (0, c, ε), in order accommodate firm 2. Combined with the efficiency advantage of firm 1,
(ε,c,ε) (ε,c,ε) π a1 , R(a1 , c), 0 > π(a3 , a2 , ε).

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4

Conclusion

This paper provides a general context where potential entry can reduce the intensity of competition. The mechanism for this result is the combination of an environment in which entry requires the acquisition of critical incumbent assets and key competitive variables are strategic complements. In this situation, an inefficient incumbent may be deterred from selling assets to an efficient entrant by the accomodating actions of a more efficient incumbent. That incumbent sacrifices short-term profits to improve the profitability its rival to such an extent that there are no gains to trade in selling out to a more efficient entrant. As a novel result in economics, the result in this paper should make economists somewhat more cautious in advocating reforms that will improve potential entry in markets. In some circumstances, we have shown that those reforms may lead to reduced welfare. Of course, the extent to which the theoretical conclusions reached here are of relevance in policy-making is ultimately an empirical question that we leave for future researchers.

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