THE EFFECT OF EXIT ON ENTRY DETERRENCE STRATEGIES

Transcription

1 THE EFFECT OF EXT ON ENTRY DETERRENCE STRATEGES Abraham L. Wikelgren * Feeral Trae Commission 600 Pennsylvania Ave. NW, ail Drop 5016 Washington, DC 0580 awikelgren@ft.gov July 00 Keywors: Entry Deterrene, Exit JEL Numbers: L10, L1 * thank an anonymous oeitor, two anonymous referees, Jeremy Bulow, George Deltas, Ezra Frieman, Jae Nahm an seminar partiipants at the Feeral Trae Commission an Georgetown University for helpful omments. The views expresse in this paper are those of the author an o not neessarily represent the views of the Feeral Trae Commission or any iniviual ommissioner. All errors are my own.

2 ABSTRACT Reent analyses of entry eterrene strategies have reuire an inumbent s post-entry output or priing strategy to be profit maximizing. However, most papers have ontinue to assume that either an inumbent an ommit not to exit after entry or that exit is never optimal. When there are avoiable fixe osts of operating in any perio, however, exit an be the optimal strategy. n this situation, entry eterrene strategies operate very ifferently than when exit is never optimal. n fat, the possibility of exit an make some, previously effetive, strategies ompletely ineffetive while improving the effetiveness of others.

3 Beause tehnology suh as software reuires huge fixe investment up-front, but involves trivial marginal osts, it is highly likely that ompetition will result in fragile monopolies being reate, with single ompanies ominating segments for a time, until they are topple by rivals. The Eonomist, The New Enforers. Otober 7, NTRODUCTON With the avent of game theory, eonomists reognize that entry eterrene strategies must be reible. mposing this reibility reuirement in a prie or uantity setting game with the entrant has ha a profoun effet on the analysis of entry eterrene strategies suh as apaity expansion e.g., Dixit 1980 an Bulow et. al This literature has fouse on the ase where exit by the inumbent following entry is never optimal. n this paper, analyze the alternative situation where the inumbent may fin it ex post optimal to exit following entry an annot reibly ommit to remain in the market. While it is well known that the possibility of exit weakens an inumbent s ability to eter entry Bagwell an Ramey 1996, the literature has yet to analyze how the possibility of exit affets ifferent types of entry eterrene strategies. show that this effet an be uite profoun. n partiular, ereasing an entrant s uopoly profits often will not eter entry, but inreasing the inumbent s post-entry profitability often will. n inustries with large fixe, but not sunk, osts, exit may be the inumbent s profit maximizing strategy when faing an entrant that is muh more effiient or has a far superior prout. Amerian Airlines eie to abanon its San Jose hub an many routes within California shortly after Southwest Airlines entere the San Jose market beause it worrie it might not be able to ompete with Southwest s low fares San Jose erury News 1993; Washington Post n fat, as the above uote iniates, exit by ominant firms oul beome inreasingly ommon in many inustries as rapily avaning tehnology provies opportunities for entrants to evelop prouts that are greatly superior to those offere by inumbents. Thus, it is more important than ever to expan the analysis of entry eterrene strategies to over the ase where the inumbent might exit. 1

4 To show how entry-eterring strategies an affet the reibility of the inumbent s no exit threat, onsier a moel where the entrant s ability to ompete is private information. The ruial insight is similar to Nalebuff s 1987 insight about the effet of reibility in pretrial settlement negotiations. The more the inumbent invests in ereasing the profitability of entry, the more effiient is the entrant it is likely to fae given that entry ours. This reues the reibility of the inumbent s threat not to exit. As a result, in some irumstanes, inreases in entry osts have no effet on the probability of entry beause they o not affet the inumbent s inentive to exit. Strategies suh as lobbying for tighter environmental stanars for new plants, signing exlusive ontrats with suppliers, or buying up ritial resoures that are in limite supply will not be effetive in eterring entry when exit is possible but not ertain. This is even true for regulations that raise the marginal osts of a new entrant, thus reuing its expete profit from entry, so long as they also raise the inumbent s marginal ost enough to reue its profitability after entry. As surprising as this seems, it follows beause the less profitable entry is, the more effetive a ompetitor an entrant must be to warrant entry. This reues the expete profits of the inumbent, making it more likely to exit. This inrease in the exit probability inues more entry, ounterating the erease profitability of entry. On the other han, avertising that expans the market for both the inumbent an the entrant an atually eter entry in these irumstanes beause it inreases the inumbent s expete profits if entry ours. 1 While most of the literature on entry eterrene has fouse on the ase where exit by the inumbent is never optimal, there are a few exeptions. Ju 1985 shows that the inability to ommit not to exit limits an inumbent s ability to eter entry into a nearby market by preemptively entering that market first. Eaton an Lipsey 1980 allow for exit to isuss optimal urability an replaement of sunk apital. Neither of these papers, however, onsiers the impat of exit on the type of entry 1 One an imagine some types of eman urves suh as onstant elastiity eman where a parallel shift out of the eman urve might atually reue uopoly profits by inuing eah firm to ompete more aggressively. n these unusual irumstanes, avertising that shifte the eman urve out woul not eter entry sine it woul not inrease the inumbent s uopoly profits. thank Jeremy Bulow for pointing this out.

5 eterrene strategies isusse here. oreover, both use omplete information moels, so they on t onsier the interation of asymmetri information an exit, the essene of this paper. n the investment in entry eterrene literature, both Arvan 1986 an Bagwell an Ramey 1996 in apaity moels allow for the possibility of exit by the inumbent following entry. n Arvan s paper, however, beause the unertainty is about the inumbent s tehnology, not the entrant s, there is no interation between the entry eterring strategy an the exit eision. n Bagwell an Ramey, there is no asymmetri information; they link apaity expansion an exit via forwar inution. Due to forwar inution, the inumbent assumes that the entrant will proue a large enough uantity to over its entry osts assuming the inumbent takes this output level as given. As a result, the inumbent wants to onstrain its apaity so that the entrant an enter an make positive profit without the inumbent exiting. Like their paper, show that exit an make entry eterrene strategies less effetive. y paper, however, obtains this result with asymmetri information rather than forwar inution. This istintion is funamental beause it is both the possibility of exit an unertainty about the entrant s type that generate the result that, in some ases, entry an only be eterre by ereasing the inentive to exit, not by ereasing the profitability of entry. The plan of the paper is as follows. Setion outlines the moel. Setion esribes the euilibrium of the entry an exit subgame. Setion V works through a Cournot example that illustrates effets of the mixe strategy euilibrium. Setion V analyzes the mixe exit strategy euilibrium in a more general framework. Sub-setion V.1 analyzes entry eterrene in this euilibrium an ompares it to the no exit benhmark ase. Sub-setion V. isusses some robustness issues. Setion V onlues. The Appenix ontains proofs omitte from the text. The spatial preemption moel of Ju 1985 oul be seen as a speial ase of the strategies isuss below. However, sine there is no unertainty in his moel an the entry eterrene strategy is of fixe magnitue, the euilibrium result is exit with probability one. Therefore, the most interesting euilibrium, the mixe strategy euilibrium, oes not arise in Ju s moel. By the same token, sine this paper oes not onsier spatial markets, it oes not over the interesting aspets of Ju s moel. Thus, while his paper has similarities with this one, both the situations it overs an its results are uite istint. 3

6 . THE ODEL Consier a two perio moel where proution in any perio reuires the expeniture of fixe osts in that perio negative profits are possible. n perio 1, only the inumbent,, is in the market. ts osts are ommon knowlege. There is a potential entrant, E, that an enter in perio. ts marginal ost parameter,, is private information note, this oes not imply that the entrant s marginal osts are onstant, only that its marginal osts are a funtion only of its output an. The inumbent knows only that is istribute aoring to the ifferentiable umulative istribution funtion G, with ensity funtion g, an support [, ]. 3 n perio 1, hooses some observable ostly ation, a A, that affets either the entrant s or inumbent s net profits or both in perio. Perio is ivie into three stages. n.1, E eies whether to enter the market. n.,, having observe the entry eision but not E s ost parameter, eies whether or not to exit. n perio.3, an E if in the market earn profits. When analyzing the speial ase of Cournot ompetition in perio.3 setion V an Proposition, assume for simpliity that observes prior to hoosing its uantity in perio.3. o not make this assumption when analyzing the general game, nor oes the main result of the paper, Proposition 1, epen on this assumption. 4 employ the Perfet Bayesian Euilibrium onept for this game. Perio 1 Perio.1 Perio. Perio.3 hooses a earns profits E observes a an makes its entry eision eies whether to exit an E, if ative, earn profits s first perio monopoly profits are given by π 1 a. This inlues its fixe operating ost. 3 oul also be interprete as a parameter measuring the uality of the entrant s goo rather than its proution ost. n this interpretation, higher shoul be thought of as lower uality. 4 One an also generalize the proof of Proposition so that it applies to the ase where the entrant s osts remain unobservable when the inumbent hooses its uantity in the Cournot game. This proof is available from the author upon reuest. 4

7 assume π 1 a is ereasing an stritly onave in a: a is an ation that has immeiate osts but no immeiate benefits suh as an investment in future ost reution, avertising that expans the market in the future, or raising entry osts. Thus, the myopially optimal a is zero. f exits, it earns zero profits in perio.3. f oes not exit, then in.3 it earns uopoly profits of π a, if E entere in.1 an monopoly profits of π m a if E i not enter both net of fixe osts. π a, is inreasing in if the entrant s osts are higher, then the inumbent s uopoly profits are higher an both π a, an π m a are weakly inreasing in a this will be strit for some types of a, for example if a is investment in ost reution or avertising, an π m π a > a,,. f E enters in perio.1 an oes not oes exit in., then E s uopoly monopoly profits are given by π a, π a,, inlusive E Em of entry osts; π a, > π a,. Both are ereasing in an monotoni in a. The entrant s Em E profits an be monotonially ereasing in a, e.g., when a represents the inumbent s investment in ost reution or apaity expansion an oes not exit or when a is an investment in inreasing entry osts. The entrant s profits an also be monotonially inreasing in a, e.g., when a is avertising that expans the market. All profit funtions are twie ontinuously ifferentiable in both an a. At this point, an in most of the paper exept where speifie otherwise, the profit funtions for eah player in a given perio are reue form profit funtions. That is, o not expliitly moel how eah player hooses its ontrol variables that only affet urrent perio profits suh as prie or uantity. For example, onsier π a,. Whenever this general form is use, make no assumptions about whether perio.3 ompetition is, for example, Cournot or Bertran or perfet ollusion, or about whether E s ost parameter remains private information or beomes ommon knowlege before the inumbent hooses its ontrol variables. Any of these assumptions is permissible so long as π a, satisfies the properties assume above. f this is the ase, the partiulars of the perio.3 game o not affet the main result of the paper Proposition 1. Notie that, while have allowe the inumbent to save avoiable fixe osts by exiting the market after observing the entry eision, o not allow the entrant to exit, an save its entry osts, after 5

8 observing the inumbent s exit eision. This is ritial, otherwise oul effetively ommit not to exit beause if E oul bak out of its entry eision at zero ost, the initial entry eision woul be meaningless, making a first mover. 5. PEROD EQULBRU Beause the entrant s osts are private information, the inumbent s response to entry annot be onitione on the entrant s type. Sine the inumbent observes the entry eision, however, its exit eision will epen on its beliefs about the entrant s type given the entry eision. Similarly, sine the entrant must enter before observing the inumbent s exit eision, its entry eision will epen on its beliefs about the probability of exit. A perio euilibrium ours when both the inumbent s an entrant s beliefs are orret an the entry an exit eisions are best responses to those beliefs about the other s strategy. f the inumbent exits with probability pa, then the entrant s profits from entry are: π a, = 1 p a π a, p a π a, 1 E E Em Euation 1 implies that there is a utoff level for E s ost parameter, ˆ a, suh that E enters if an only if < ˆ a if its osts are low enough, E an earn positive expete profits from entry. Of ourse, the entrant oes not observe pa when making its entry eision, although the entrant will orretly preit this exit probability in euilibrium. Thus, pa in 1 an below shoul be thought of as the entrant s onjeture of the inumbent s exit probability. This utoff, ˆ a, is efine impliitly by: 1 p a π a, ˆ a p a π a, ˆ a 0 E Em = t will not neessarily be the ase that ˆ a [, ]. f the ˆ a that solves is less than, then the entrant s osts will never be low enough to enter. On the other han, if the ˆ a that solves is 5 f, after the entrant sunk an entry ost in.1, the entrant an inumbent both ha avoiable fixe osts an playe a simultaneous move exit game in., the euilibrium in Setion remains an euilibrium of this moifie game, though it is no longer the uniue euilibrium. 6

9 greater than, then the entrant s osts will always be low enough that entry is profitable. further efine a an a m impliitly as follows: E = E m m = π a, a 0 anπ a, a 0 3 That is, a is the entrant s ost utoff level if the inumbent oes not exit pa=0 an a m is the utoff level when the inumbent always exits pa=1. t follows from that ˆ a is inreasing in pa, E s orret onjeture of the probability that exits, an a ˆ a a. This relationship is epite in the following figure. 6 m Figure 1 ˆ ĉ Given Conjeture Exit Probability p n this figure, the entrant will always enter if it believes the probability of exit is fairly high < ˆ a when p is large, an the entrant enters with positive probability even if it believes the inumbent will never exit < ˆ a even when p=0. As Figure 3, below, shows, neither of these will hol in general. What is generally true, as one an see from, is that the optimal entry utoff is inreasing in the entrant s onjeture of the probability of exit. 6 The exat urves in Figures 1,, an 3 are erive base on Cournot ompetition in perio.3, onstant marginal osts whih are ommon knowlege in perio.3, linear eman, an a uniform istribution for the entrant s marginal osts. The ualitative results from the figures, however, o not epen on these speifiations. 7

10 For any given onjeture about the entrant s entry strategy, enter if < ˆ a, the inumbent will exit with probability zero one whenever its expete seon perio uopoly profits are positive negative. f seon perio expete uopoly profits are zero, inumbent an play a mixe exit strategy. Thus, the inumbent s exit strategy is etermine by the sign of the following: ˆ a π a, g 4 Again, sine the inumbent oes not observe the entry strategy, ˆ a in 4 is the inumbent s onjeture orret in euilibrium of the entrant s atual entry utoff. Figure epits the inumbent s exit probability as a funtion of this onjeture of ˆ a. The point * a is the entry strategy that makes the * a inumbent inifferent between exiting an not, efine impliitly by π a, g 0. = Figure 1 p 's Exit Prob. Given Conjeture ĉ ˆ Combining these two plots on the same graph emonstrates that there is a uniue euilibrium to the perio entry an exit subgame in whih the entrant s beliefs about the inumbent s exit probability an the inumbent s beliefs about the entrant s entry strategy are orret. Figure 3 epits three possible euilibria the blak ots for three ifferent entrant profit funtions that iffer in entry osts, eah of whih generates a ifferent urve epiting the entrant s hoie of ĉ as a funtion of p. 8

11 Figure 3 p Entry & Exit Euilibria for 3 Different Entry Costs 1 Exit ixe NoExit 's Exit Prob ˆ wê High fe ˆ wê eium fe ˆ wê Low fe When entry osts are large the soli line, the entrant must have small marginal osts to enter for any given probability of exit. The perio euilibrium has exit with probability one onitional on entry beause when the inumbent observes entry it knows it will make negative expete profits ompeting with a low marginal ost entrant. This euilibrium ours whenever efine as follows: X a A, where X A is A X m a = { a A: π a, g 0} = { a A: a * a} m That is, X A is the set of ations that generate a seon perio market that is so iffiult for an entrant that in orer for an entrant to make positive monopoly profits its marginal osts must be so small that the inumbent annot make positive expete profits ompeting against this entrant. When entry osts are very small the ashe line, an entrant with high marginal osts an make positive uopoly profits. Given this, both the inumbent an entrant an simultaneously have positive 9

12 expete uopoly profits; the perio euilibrium has exit with probability zero. This euilibrium ours when N a A, where N A is the following set: A N a = { a A: π a, g 0} = { a A: a * a} That is, N A is the set of ations that generate a seon perio market that is profitable enough for an entrant that it an make positive uopoly profits with marginal osts that are high enough the inumbent an make positive expete profits ompeting against this entrant. For intermeiate entry osts epite by the otte line in the figure, the break-even ost urve intersets the exit probability urve exatly at the point where the inumbent earns zero expete profits given entry. This mixe strategy euilibrium ours when a A an A is given by: A a m a m = { a A: π a, g < 0, π a, g > 0} = { a A: a < * a < a} That is, A is the set of ations that generate a seon perio market that is not so profitable that the inumbent an make positive expete profits ompeting against an entrant that an make positive uopoly profits; but it is profitable enough that the inumbent an make positive expete profits ompeting against an entrant that makes positive monopoly profits. One might woner whether the sets that efine the type of entry an exit euilibrium that ours for any given a are onnete that is, are they eah one interval? f the effet of a on s an E s profits is in the same iretion, then j a π a, g is monotoni in a j=,m. This ensures that eah of the above sets is onnete. This overs all the examples have isusse exept for marginal ost reution. n that ase, a inreases s uopoly profits an ereases E s uopoly profits. Thus, it is possible that N A an/or A is not onnete X A will remain onnete sine a oes not affet E s monopoly profits. f inreasing investment in ost reution at some levels reues a so muh that it atually reues the inumbent s expete profits given entry, while at other levels the reverse is true, 10

13 then either N A or analysis that follows. A oul onsist of multiple intervals. This, however, oes not affet any of the Of ourse, there is no guarantee that all of these sets are non-empty for all possible parameter values. For example, if the market is large relative to entry osts, then the inumbent might be in the no exit set for all a. On the other han, if uopoly ompetition takes the Bertran form an m a is less than the inumbent s marginal ost, then the inumbent will always exit whenever there is entry. f both X A an N A are non-empty, however, then m that a < a an π a, is inreasing in. A must be non-empty also. This follows from the fat Given the euilibrium in this subgame, the inumbent s inter-temporal profit funtion is: β{1 p a ˆ a π Π a = π a 1 a, g 1 G ˆ a π m a} 5 n perio 1, is a monopolist, earning π. Seon perio profits are isounte by β. f 1 a exits, then earns zero. f, when E enters, i.e., when [, ˆ a, oes not exit this happens with probability 1 p a, then earns uopoly profits of π a,. f E s oes not enter, earns monopoly profits in perio. V. A COURNOT EXAPLE Before proeeing to a general analysis of the effets of the mixe exit strategy euilibrium, will illustrate the main results using a simple linear eman, Cournot example. Setion V shows that these results hol more generally. Say eman in eah perio is given by = z p, the inumbent has onstant marginal ost of an, the entrant s marginal ost, is uniformly istribute between 0 an 1. Further, to make etermining the euilibrium uantities simpler, assume that prior to perio.3, the inumbent learns the entrant s marginal ost. Then, in.3 firm j=,e where E = hooses j to maximize: z 6 j j j j 11

14 The first orer onition for j is: The Cournot-Nash euilibrium uantities are: z = 0 7 j j j j j z j = 8 3 The reue form uopoly profit funtions for j=,e are where f j is firm j s fixe ost: Similarly, the reue form monopoly profit funtions are: z j j π j = f j 9 9 z j π jm = f j 10 4 f the entrant expets the inumbent to exit with probability zero, then it will enter whenever π 0. From 9, this means the entrant enters if an only if: E < = 1/ z 3 f 11 E f the entrant expets the inumbent to exit with probability one, then it will enter whenever π 0. From 10, this means the entrant enters if an only if: m Em < = z f 1 E The inumbent s exit eision is epens on whether or not its expete uopoly profits are positive. Using 9, the inumbent exits, given entry, with probability one zero if an only if: ˆ 0 z 9 f < > 0 or 13 ˆ < > * = 1/ [6 3z 336 f z ] 14 Of ourse, the inumbent plays a mixe exit strategy if an only if ˆ = *. A mixe exit strategy euilibrium must our if an only if: 1/ z 3 f < 1/ [6 3z 336 f z ] < z f 15 E E 1

15 That is, 15 efines the set A whenever ĉ, z, f E an/or f are funtions of a. When 15 hols, if the inumbent exits with probability one, then the entrant will enter whenever = z f, m E making it optimal for the inumbent to exit with probability zero. While if the inumbent exits with probability zero, then the entrant will enter only if = 1/ z 3 f, making it optimal for E the inumbent to exit with probability one. Thus, the euilibrium must have ˆ = * = 1/ [6 3z 336 f z ] an exit with positive probability less than one. To see how various ations affet the probability of entry in the mixe strategy euilibrium, one nee only look at how * = 1/ [6 3z 336 f z ] varies with these ations. f the inumbent never exits following entry, however, then the effet of some ation on the probability of entry epens on how = 1/ z 3 f varies with this ation. For example, say the inumbent s E perio 1 ation inreases the entrant s fixe entry ost, that is, f E a. f 15 hols for f E = f E 0 the inumbent takes no ation to inrease entry osts, then, unless the inumbent inreases entry osts enough that 1/ [6 3z 336 f z ] z f, the inumbent must still play a mixe exit strategy. As a result, the probability of entry, the probability that < * = 1/ [6 3z 336 f z ], oes not hange, sine * is inepenent of f E. nreases in entry ost only reue the probability of entry when entry osts are low enough or high E enough that 15 oes not hol, in whih ase the utoff level for, ĉ, is given by or m. As 11 an 1 show, inreasing f E oes erease both z an m. Figure 4 illustrates this for 1, = 1/, f =.01. n the next four figures, the region where the inumbent oes not exit after entry = is enote by a soli thin line; the mixe exit strategy region is enote by a thik soli line; an the exit with probability one region is enote by a ashe line. 13

16 Figure 4 ˆ Effet of Entry Costs on Entry Cutoff NoExit ixe 0.45 Exit f E f the inumbent s ation, a, ereases its avoiable fixe osts, that is, f a, then it is lear from 11, 1, an 14 that this will eter entry if an only if 15 hols, i.e., if an only if we are in the mixe exit strategy euilibrium an illustrates this for z 1, = 1/, f =. 03 : = E m are inepenent of f, but * is inreasing in f. Figure 5 Figure ˆ Effet of 's Fixe Costs on Entry Cutoff ixe Exit NoExit f Now onsier the ase where a represents eman-enhaning avertising, that is, za. From 11, 1, an 14, one an see that suh avertising will eter entry in the mixe strategy euilibrium, 14

17 * is ereasing in z, but will enourage entry in either the no exit or the always exit euilibrium, both an m are inreasing in z. Figure 6 illustrates this for 1 /, f =.01, f =. 09 : = E ˆ Figure 6 Effet of arket Size on Entry Cutoff Exit 0.4 ixe 0.38 NoExit z f z=1 represents the market size with no perio 1 avertising, then small amounts of avertising may be ounterproutive sine this inreases the probability of entry. Larger amounts of avertising that push z 1.05,1.1 will atually erease the probability of entry beause it reues the probability of exit. Further avertising, however, pushes the inumbent into the never exit euilibrium where avertising begins to enourage entry again. Of ourse, the optimal amount of avertising will epen on more than just its affet on the probability of entry this is also true for any of the other ations the inumbent might take to eter entry. But the fat that avertising eters entry in the mixe exit euilibrium but enourages it otherwise suggests that the optimal amount of avertising may often reate a mixe exit strategy euilibrium. Lastly, onsier the ase where a is an investment in marginal ost reution, that is, a. f the inumbent reues its marginal ost, then this will obviously have no effet on the entrant s entry eision if the inumbent exits whenever there is entry. On the other han, it will eter entry reue the entrant s break-even marginal ost level if the inumbent never exits after entry sine the entrant s uopoly profits are inreasing in the inumbent s ost in 11 is inreasing in. n the mixe exit 15

18 strategy euilibrium, the inumbent s investment in marginal ost reution will also eter entry, though for a very ifferent reason. Beause lower marginal ost inreases the inumbent s uopoly profits, the mixe strategy euilibrium must have a lower probability of exit an a smaller marginal ost utoff for entry the more the inumbent reues its marginal ost * in 14 is inreasing in. Figure 7 emonstrates the effet of the inumbent s marginal ost on entry for z 1, f =.01, f =. 09 : = E ˆ Figure 7 Effet of 's C on Entry Cutoff Exit ixe NoExit As Figure 7 shows, the inumbent s marginal ost an have a greater effet on the probability of entry in the mixe exit strategy euilibrium than in no exit euilibrium. When this is the ase, if the ost of marginal ost reution inreases rapily, the marginal benefit of investment in ost reution may exee its marginal ost in the mixe exit strategy euilibrium, but not if the inumbent reues its ost so muh that it is in the no exit euilibrium. Of ourse, if ost reution is relatively heap, the inumbent may always want to reue its osts enough that it never exits following entry. To see that both outomes are possible, onsier the ase where the eman urve is stable aross perios; the inumbent s first perio profit funtion is iential to its monopoly profit funtion in the seon perio with no ost reution less the ost of any ost reuing investment it unertakes. assume ost reution osts are uarati. That is: z 1 π 1 a = f ka

19 an the inumbent s perio.3 marginal ost is = 1 a. The inumbent hooses its amount of ost reution, a, to maximize 5 where π a, π a,, an π are given by 16, 9, an 10 1 m a respetively. ĉ is given by if 1/ z 3 f > 1/ [6 3z 336 f z ], by E m if 1/ [6 3z 336 f z ] > z f an by E * = 1/ [6 3z 336 f z ] otherwise. Figure 8 illustrates how the euilibrium exit probability that results from this optimal hoie of a varies with k, the parameter that measures how ostly ost reution is, for z 1, 1 = 1/, f =.01, f =.06 an. 03: = E Figure 8 p 1 Effet of Cost Reution Costs on Exit Probability f E =.06 f E = k n this example, when investing in ost reution is not that ostly, k is small, an entry osts are not too large f =. 03, then the inumbent reues its osts enough so that it will never exit. As ost E reution gets more ostly, however, the inumbent will not fin it profitable to reue its ost so muh that it will never exit. When entry osts are larger, the average entrant is a tougher ompetitor, eteris paribus, so even when ost reution is relatively heap the inumbent still plays a mixe exit strategy. An when ost reution beomes uite expensive, the inumbent s osts, even after optimal ost reution, are so large that it always exits after entry. 17

20 V. THE XED EXT STRATEGY EQULBRU V.1 ANALYSS OF THE XED STRATEGY EQULBRU f plays a mixe exit strategy, then, as we saw in the above examples, what onstitutes effetive entry eterrene an be very ifferent from when exit is never optimal or always optimal. This setion shows that this result hols for arbitrary eman an ost funtions an arbitrary perio.3 ompetitive interation so long as the profit funtions satisfy the properties assume in Setion. n the no exit benhmark ase, the utoff level for entry is given by a ; the entrant s marginal osts must be low enough that it an over its fixe osts as a uopolist. etermine the effetiveness of a at eterring entry in this ase, N A a, by ifferentiating the first euation in 3 with respet to a an solving for a : π a a a a E, π E, a = 17 a Sine the enominator is negative, this has the sign of π E a, a. f the inumbent never exits, a then inreasing a ereases the probability of entry if an only if inreasing a ereases the entrant s uopoly profits. This is the stanar story about how an inumbent eters entry. To see how entry eterrene in the mixe strategy euilibrium a A iffers, reall that, beause earns zero expete uopoly profits, the utoff level for entry is given by the * that solves: * π a, g 0 18 = Beause ˆ a = * a must also satisfy whenever * a,, the probability of exit in the mixe strategy euilibrium must be: π E a, * a p a =. 7 π a, * a π a, * a E Em Now, ifferentiate 18 with respet to a an solve for * a : 7 f the entry utoff is not in the interior of the support, then this exit probability is only an upper or lower boun. 18

21 This has the sign of π a a, * a π a, g * a = a 19 g * a π a, * a sine π a, * a 0 if the inumbent is to have zero expete > uopoly profits, then, beause π a, is inreasing in, it must make positive net profits when the entrant is of the highest possible ost type. nreasing a when a A reues the probability of entry if an only if inreasing a inreases the inumbent s net uopoly profit. This proves the following proposition. Proposition 1. For all a A, if the inumbent annot ommit not to exit, then it an reue the probability of entry if an only if it inreases its expete uopoly profit. When the inumbent an ommit not to exit or when N a A reues the uopoly profit of the break-even entrant., however, it an reue the probability of entry if an only if it Proposition 1 emonstrates that the results from the examples of the last setion hol more generally. To prove this proposition, i not make any assumptions about the istribution of the entrant s ost parameter. While have assume that the entrant s proution tehnology an be summarize by a single ost parameter, the proof oes not make any speifi assumption about how proution osts vary with this ost parameter other than the assumptions about how the entrant an inumbent s profits vary with. The proof oes use the assumptions that the entrant s profit is ereasing in an the inumbent s uopoly profit is inreasing in. These assumptions, however, are uite intuitive an shoul hol for almost all stanar moels of ompetitive interation. The istintion Proposition 1 makes between the mixe strategy ase an the benhmark ase is funamental. When exit is possible an the inumbent plays a mixe exit strategy, the effetiveness of an entry eterrene ation epens only on how it affets the inumbent s profits, not the entrant s. Beause inreasing the entrant s entry ost oes not affet the inumbent s uopoly profits, it has no affet on the 19

22 probability of entry in the mixe exit strategy euilibrium. By the same token, ereasing the inumbent s avoiable fixe ost will eter entry when the inumbent plays a mixe exit strategy, but not otherwise, sine it inreases the inumbent s uopoly profits but oes not affet the entrant s uopoly profits. Unlike strategies that affet fixe osts, investments in marginal ost reution or uality enhanement are entry-eterring strategies in both the mixe exit strategy euilibrium an in the no exit euilibrium. Similarly, both raising the entrant s marginal osts an reuing its uality are effetive entry-eterring strategies in either situation. Nonetheless, the possibility of exit is still relevant to these strategies. When exit is never optimal, the entry-eterring payoff from marginal ost reution omes exlusively from how this will reue the profitability of entry. f the inumbent might exit, over some regions its investments in marginal ost reution will eter entry beause they inrease its own profitability, reuing the probability that it will exit. Therefore, while the iretion of the effet of ost reution on entry is iential in either ase, the magnitue of this effet may iffer substantially, whih will affet the optimal investment in ost reution. This istintion is even more ritial when one onsiers the operation of avertising as an entry barrier. f the goos of the inumbent an the entrant are not highly ifferentiate, one might think that the inumbent woul want to refrain from avertising to inrease the size of the market beause that might inue entry. This is true when the inumbent will never exit. When the inumbent plays a mixe exit strategy, however, the exat opposite is the ase. The inumbent will want to avertise to inrease the size of the market, not only beause oing so inreases its profit the perio, but also beause it will eter entry. n this ase, entry eterrene ours beause avertising has mae the inumbent s threat to remain in the market more reible. A similar argument woul apply to investments in uality enhanement that are non-rival that improve the entrant s uality as muh as the inumbent s. This effet is similar to Nalebuff s 1987 point about the importane of reibility in pretrial settlement negotiations. Notie that in this moel, when a A, there will neessarily be more entry when exit is possible than when it is never optimal, as in Bagwell an Ramey This follows beause the 0

23 onition for the entry utoff without exit, the first euation in 3, says that all entrant types that an make non-negative uopoly profits will enter. When exit might our, however, the entrant s profits from entry are a weighte average of uopoly an monopoly profits. Sine monopoly profits exee uopoly profits, higher ost entrants will enter when exit happens with positive probability. This oes not neessarily imply that the inumbent will make less effort to eter entry when exit is possible than it woul if it oul ommit not to exit where more effort is efine as a greater level of a, an thus more first perio profits foregone. n fat, there is no lean omparison of the inumbent s inentives to forego profits to eter entry in these two ases beause the entry eterrene mehanism is so ifferent when the inumbent might exit versus when it will not. This an be seen by examining the marginal profit from the entry eterrene ation, a, when the inumbent might exit versus when it an ommit not to exit. 8 n the mixe exit strategy ase, the marginal profit from a is the following: Π a = π a β{1 G * a π a * a g * a π } 0 1 m m a This follows from ifferentiating 5 with respet to a, while fixing the inumbent s profit when there is entry at zero. By substituting in for * a using euation 19, this beomes: π m a * a Π a = π 1 a β{1 G * a π m a a, a g } a, * a π 1 π As 1 iniates, a inreases seon perio profits in two ways. First, when the entrant won t enter anyway, a inreases monopoly profits. Seon, when the entrant woul have entere, a inreases uopoly profits. That is valuable, however, not beause of the ae profits given entry that has to remain at zero to maintain the euilibrium, but beause, by reuing the inumbent s inentive to exit, it 8 prove the following laim in the Appenix. t establishes that omparing the marginal profit from a in the two ifferent ases the inumbent plays a mixe exit strategy versus the inumbent an ommit not to exit is suffiient to etermine in whih ase the optimal a is larger, even if the profit funtion is not single-peake. Claim: Consier two ifferent profit funtions Π, Y a an Π, Z a. f Π, Y a > Π, Z a for all a, then the a that maximizes Π is larger than the a that maximizes Π, even if neither Π nor Π is single-peake., Y a, Z a, Y a, Z a 1

24 eters entry. From 1, one an see that the entry eterrene value from a is stritly greater than the amount by whih it inreases uopoly profits sine monopoly is more profitable than uopoly. n the mixe strategy euilibrium, the marginal profit from a is inepenent of its effet on the entrant s profit. This is very ifferent from the inumbent s marginal benefit from a when the inumbent will not exit. The marginal return to a in this ase is: a π Π a = π 1 a β{1 G a, a g g a[ π m a π m a π a a, a] a} With no exit, the inumbent s inentive to inrease a will ome more relative to the mixe strategy ase from a s effet on its monopoly profits in perio than from its effet on uopoly profits. This ours not only beause there is less entry when the inumbent oes not exit, but also beause the effet on uopoly profits reeives greater weight in 1 than. The reason is that, in the mixe exit strategy ase, inreasing uopoly profits is valuable beause it eters entry, whih is worth more than the ae inrement to profit itself. Euation, however, has an ae entry eterrene term of its own. From euation 17, we know that a < 0 if an only if a reues the entrant s profits. Thus, whether an inumbent has a greater inentive to inrease a when it plays a mixe exit strategy than it oes when it never exits epens on how strongly a affets the entrant s profits. f the ation a has either no effet on the entrant s profits or inreases them e.g., if it is avertising that expans the market, then it is muh more likely that the inumbent will hoose a larger a when exit is possible. Fousing on the ase where a is a ost reuing investment, one might also woner if the threat of entry inreases or reues the inumbent s inentive to reue osts? n the mixe strategy ase, one an look at 1 to answer this uestion. The marginal benefit from a in 1 an be separate into two ifferent terms: * a π Π = a, π m a a π 1 a βπ m a β [ π ] m a g 1 a π a, * a

25 The first two terms of the right han sie give the marginal benefit from a if there is no threat of entry. The integral term is an ajustment in the marginal benefit ue to the threat of entry. The threat of entry will inrease the inumbent s inentive to reue osts if an only if this term is positive. One an perform the same eomposition of the inentive to reue osts in the no exit ase: β a π a, { π a m Π a = π 1 a βπ a} g βg m a[ π a m a π a, a] a As before, the threat of entry inreases the inumbent s inentive to reue osts if an only if the last line of is positive. With ompletely abstrat profit funtions, it is not possible in either ase to say if the threat of entry inreases or ereases ost reution inentives. f profits in perio.3 are etermine by a stanar one-shot Cournot uantity-setting game with onstant an known marginal osts, 9 however, then one an efinitively answer this uestion. Consier Cournot ompetition where the entrant s marginal ost is unknown to the inumbent in perio. but reveale in perio.3. The inumbent s marginal ost in perio 1 is, an its marginal ost in perio.3 is a. Perio.3 profit funtions are as follows: π a, = p 3 a π = p 4 E n 3 an 4, an Cournot-Nash euilibrium uantities given a an. With this setup, an now prove the following proposition. Proposition. Say p < 0 an p 0 an the threat of entry reates a mixe exit strategy euilibrium, a A. f perio.3 profit funtions are given by 3 an 4, then the threat of entry 9 Sine the inumbent oes not know the entrant s marginal ost in perio., this reuires that the inumbent learns the entrant s marginal ost between perio. an perio.3. While this assumption is implausible in many ases, as mentione above, it is not neessary for this result a proof of this is available upon reuest. 3

26 stritly inreases the inumbent s inentive to reue osts when the entrant s ost parameter is reveale prior to perio.3. Proof. See Appenix. Proposition 1 rives this result. A uopoly market an reue the inentive for ost reution visà-vis a monopoly one beause the inumbent proues less output. However, when the inumbent plays a mixe exit strategy, its benefit from ost reution in a uopoly market arises solely from the inrease in the reibility of its threat not to exit, whih inreases the probability that it will remain a monopolist. Beause of this, the magnitue of this benefit epens on the entire monopoly output. oreover, the benefit is not just the reue ost but also the larger prie the inumbent gets when it is a monopolist rather than a uopolist. As a result, the threat of entry stritly inreases the inumbent s inentive to reue osts so long as the effet on uopoly profits is large enough. f uopoly ompetition takes the Cournot form, then this is always the ase. When the inumbent never exits, the threat of entry an either inrease or erease the inumbent s ost reution inentives uner Cournot ompetition. The appenix provies an example that emonstrates this using a linear inverse eman urve an a uniform istribution of entrant types. The reason for the ambiguity is that the entry eterrene benefit from ost reution is as Proposition 1 emonstrates ompletely inepenent of the uopoly benefit from ost reution. Thus, if the entry eterrene benefit is small either beause ost reution has a small effet on the entrant s profits or beause monopoly an uopoly profits o not iffer by muh, then it is possible for the threat of entry to erease the inumbent s inentive to reue osts. V. ROBUSTNESS OF THE XED STRATEGY EQULBRU One might objet to the foregoing analysis by arguing that even though there are values of a for whih must play a mixe exit strategy, there may not be reasonable irumstanes where will inee hoose suh a value of a. This objetion is misguie on three ounts. First, Proposition 1 will often affet the inumbent s hoie of a even when that hoie oesn t lea to a mixe exit strategy. For example, onsier s eision about whether to lobby for new plant regulations that will inrease entry 4

27 osts. f exit were never optimal or, oul ommit not to exit, any level of lobbying effort oul be optimal. When this isn t the ase, Proposition 1 says that while may try to inrease entry osts up to the point where it has to play a mixe exit strategy, it has no inentive for further marginal inreases in entry osts. Only a isrete jump in the level entry osts that moves out of the mixe exit strategy euilibrium an into the always exit euilibrium oul be optimal. Seon, one an also think of Proposition 1 as a omparative statis result for exogenous parameter hanges. For example, Proposition 1 tells us that larger entry osts will not neessarily erease the probability of entry. On the other han, reutions in avoiable fixe osts an sometimes have a ramati impat on the probability of entry. Thir, the prior setion provie examples where hoosing a A was optimal for a range of parameter values for the ost reution ase. While these are only examples, sine m < monopoly is stritly more profitable than uopoly, there exists a ost funtion for any ation, a, that eters entry in the mixe strategy euilibrium suh that optimal whenever A a < inf A but rises very rapily when a mixe exit strategy euilibrium. a A is is non-empty. f the marginal ost of this ation is lose to zero for all A, then will always want to hoose an a that generates a One might also woner how sensitive the results in Proposition 1 are to the nee for a mixe strategy euilibrium. To she light on this uestion, onsier introuing some eman unertainty, so that the inumbent never plays a mixe exit strategy. Say that in perio., prior to making its exit eision, the inumbent observes a small, mean zero, shok, s, to its eman. For simpliity, assume this oes not affet the entrant s eman. Then, instea of an inumbent being inifferent about exit, if the inumbent gets a favorable ost shok say s > s * it remains in the market, otherwise it exits. 10 For an ation, a, that only reues the profitability of entry, there will be a small effet on the entry utoff level. To see this, suppose that the entry utoff is unaffete. Then the inumbent s expete uopoly profits are unhange, so it still remains in the market if an only if s > s *. Now, an entrant 10 thank Ezra Frieman for suggesting this ase. 5

28 whose ost parameter is exatly at the ol utoff level will no longer break even sine the inumbent hose an ation a that reue the entrant s profitability an so will not enter. This reues the utoff level for entry, so s* must inrease the average entrant has lower osts, so the inumbent nees a better eman shok to make zero expete profits. The inrease in s*, however, inreases the probability that the entrant will be a monopolist, whih inreases the profitability of entry. Thus, the utoff level for entry annot fall by as muh as it woul if there were no exit. That is, the possibility of exit still reues the ability of profit reuing strategies to eter entry, though it oes not entirely eliminate their effetiveness. As the magnitue of the eman unertainty goes to zero, however, the effetiveness of reuing the profitability of entry oes approah zero. f the istribution of eman shoks is very tightly lustere, then it only takes a very small reution in the utoff level of entry to ause a big shift in the probability of exit beause the aompanying small hange in s* has a large effet on the probability of exit when eman shoks are very small. While stark nature of Proposition 1 epens on the enogenous unertainty of the mixe strategy euilibrium, ualitatively similar results arise with eman unertainty. 11 n aition, introuing this unertainty will not hange the fat that ations that only inrease the uopoly profit of the inumbent eter entry. When uopoly is more profitable for the inumbent, it oes not reuire as favorable a eman shok to remain in the market. This reues the probability that the entrant will be a monopolist, making entry less profitable. Whether or not a strategy, suh as avertising, that inreases profitability for both the inumbent an the entrant eters entry will epen on the magnitue of the unertainty. The smaller the eman unertainty, the more likely avertising will eter entry rather than enourage it. One oul also objet to the robustness of the results by arguing that if the inumbent an the entrant ompete for many perios following entry, the inumbent might never exit until it learne the entrant s osts. t is far from ertain, however, that the inumbent woul neessarily learn the entrant s 11 Profit reuing strategies will be less effetive at eterring entry when the entrant s ost parameter has a large effet relative to the eman shok on the inumbent s uopoly profits. 6

29 osts through uopoly ompetition. Just as there are pooling as well as separating euilibria in the ilgrom an Roberts 198 limit priing moel, there oul be pooling euilibria here where the entrant s proution an priing eisions are inepenent of its osts. oreover, even if there is a separating euilibrium, exogenous fators suh as ost or eman shoks might slow the inumbent s learning ramatially, making the benefits of auiring further information not worth the expete urrent proution losses. This is espeially true given that separating euilibria typially reuire low ost types to behave more aggressively than in a myopi setting, making information auisition that muh more ostly. Thus, while allowing for more perios of potential ompetition oul erease the range of parameters for whih there will be a mixe exit strategy euilibrium, it will by no means eliminate this euilibrium. The results from the two perio moel still provie valuable insights for the multi-perio ase. V. CONCLUSON When there are fixe osts that an be avoie by exit, exit will sometimes be the inumbent s ex post profit-maximizing strategy. This paper shows that onsiering this possibility an signifiantly alter the ability of entry eterring strategies to eter entry reibly. These reibility issues oul be partiularly important in inustries where tehnology is rapily avaning, making the risk that an entrant oul supplant an inumbent uite large. Even where the inumbent might ompete with the entrant for more than one perio following entry, as isusse briefly above, the ualitative effets of the possibility of exit will often be similar. A omplete esription of suh a multi-perio moel is left for future researh. 7

30 APPENDX Proof of Claim. Say, ontrary to the Claim, that * = arg max Π, < * = arg max Π, while a Y Y az, Z Π, Y a > Π, Z a for all a. an write Π, Y a Z * Π, Y ay * = Π, Y a a an Π az *, Z Z, Z Y *, a Z Y a * Π a * = Π a a. Using these, the fat that a < a * implies that: a * Z ay * az * ay * Y * Z Π, a * Π, a * Π, a * Π, a * = Π, a Π, a a > 0 Y Z Y Y Z Z Thus, Π a * Π a * < Π a * Π a *. But Π a * Π a * 0 sine Z, Y Y, Y Z, Z Y, Z Z Y Y Z, Z Y, Z Z < a Z * maximizes Π, Z an Π, Y a Y * Π, Y az * > 0 sine a Y * maximizes Π, Y, a ontraition. Q.E.D. Proof of Proposition. From euation 1, the threat of entry will inrease the inumbent s inentive to reue osts if: π a a, π m a π π a, * a m a > 0, [, ˆ a] A1 Without loss of generality, measure the ost reuing investment by the magnitue of the ost reution. Hene, the inumbent s perio marginal ost is a. Using the fat that uopoly profits are etermine by a one-shot Cournot uantity setting game, the left han sie of A1 is: m pm a 1 p m A p a The arguments are suppresse in the above expression. All prie terms an their erivatives are funtions of the total uantity proue an represents the entrant s uopoly uantity. Expression A is positive if an only if the following ineuality hols: m 1 p a p > 1 A3 p a The seon fration on the left han sie is greater than one sine monopoly pries are greater than uopoly pries. Thus, A3 will hol if < 0 an < 0 sine p < 0. t is well known that 8