THE EFFECT OF EXIT ON ENTRY DETERRENCE STRATEGIES

Abraham L. Wickelgren*

Federal Trade Commission

601 Pennsylvania Ave. NW, Room 5006

Washington, DC 20580

awickelgren@ftc.gov

12/04/2000

Keywords: Entry Deterrence, Exit

JEL Numbers: L10, L12

Abstract: Recent analyses of entry deterrence strategies have required an incumbent’s 

post-entry output or pricing strategy to be profit maximizing. However, most papers 

have continued to assume that either an incumbent can commit not to exit after entry or 

that exit is never optimal. When there are avoidable fixed costs of operating in any 

period, however, exit can be the optimal strategy. In this situation, entry deterrence 

strategies operate very differently than when exit is never optimal. In fact, the possibility 

of exit can make some, previously effective, strategies completely ineffective while 

improving the effectiveness of others.

 

*

 I thank Jeremy Bulow, Ezra Friedman, Jae Nahm and seminar participants at the Federal Trade 

Commission and Georgetown University for helpful comments. The views expressed in this paper are 

those of the author and do not necessarily represent the views of the Federal Trade Commission or any 

individual commissioner. All errors are my own.
1

“Because technology such as software requires huge fixed investment up-front, but 

involves trivial marginal costs, it is highly likely that competition will result in ‘fragile 

monopolies’ being created, with single companies dominating segments for a time, until 

they are toppled by rivals.” The Economist, “The New Enforcers.” October 7, 2000.

I. INTRODUCTION

The ability of incumbent firms to maintain their market power by deterring entry 

has been a topic of considerable interest for some time. With the advent of game theory, 

economists have recognized that entry deterrence strategies must be credible. Imposing 

this credibility requirement on an incumbent’s strategy in a price or quantity setting game 

with the entrant has had a profound effect on the analysis of several types of entry 

deterrence strategies, such as limit pricing (e.g., Milgrom and Roberts 1982) or capacity 

expansion (e.g., Dixit 1980). Typically, however, these analyses have not considered the 

credibility of the incumbent’s decision to remain in the market or to exit. They have 

assumed parameter values such that exit is never optimal (the no exit threat is always 

credible) or that the incumbent can commit not to exit the market. In industries with 

large fixed, but not sunk, costs, however, exit can be the profit maximizing strategy when 

faced with an entrant who may be much more efficient or have a far superior product. 

American Airlines, for example, decided to abandon its San Jose hub and many routes 

within California shortly after Southwest Airlines entered the San Jose market (San Jose 

Mercury News 1993; Washington Post 1993). In fact, as the above quote indicates, the 

credibility of the no exit threat could become increasingly important as rapidly advancing 

technology provides opportunities for entrants to topple dominant firms in many 

industries. Thus, it is more important than ever to analyze the effect of the exit option on 

the ability of incumbents to deter entry.
2

This is especially true given that this paper shows that allowing for the possibility 

that the incumbent may want to exit the market (and, thus, requiring that the threat to 

remain in the market after entry be credible) can have as profound effects on entry 

deterrence strategies as does imposing the credibility constraint on the incumbent’s 

pricing or output strategy. This is true both when the entry deterrence strategy in 

question is of the type that affects the entrant’s perceptions about profitability given entry 

(e.g., limit pricing), and when they are of the type that actually do affect future 

profitability (e.g., investments in capacity expansion or cost reduction or advertising).

In the case of limit pricing, while one might think that the possibility of exit1

always makes limit pricing less effective at deterring entry, this is not necessarily the 

case. In separating equilibria, of course, since the incumbent’s type is perfectly revealed, 

the possibility of exit does not change the fact that limit pricing does not affect the entry 

probability. For pooling equilibria, however, introducing the possibility of exit does 

matter. In particular, when exit is allowed, the high cost incumbent is less likely to pool 

(in a partial pooling equilibrium2), i.e., choose the same first period price as a low cost 

incumbent, than if it could not exit (because, say, fixed costs were sunk). Not pooling, 

and thus inducing entry, is now more attractive to the incumbent because it isn’t forced to 

remain in the market when this results in negative profits. This effect tends to make entry 

more likely when exit is possible.

 

1

 I will often speak of comparisons between cases where exit is possible and where it is not. This can be 

thought of as either comparing models where the incumbent can not commit not to exit to models where it 

can or as comparing models where levels of avoidable fixed costs are such that exit is sometimes optimal 

versus models where these costs are so small (almost all fixed costs are sunk) that it is never optimal.

2

 I define a partial pooling equilibrium as one where the high cost incumbent chooses the pooling action 

with positive probability less than one and separates with the complementary probability.
3

Because of this, however, the entrant is actually less likely to enter after observing 

the pooling price when exit is possible than when it isn’t (because the entrant is more 

likely to be up against a low cost incumbent). This more than makes up for the fact that 

if it is up against a high cost incumbent, the entrant will earn monopoly rather than 

duopoly profits due to exit. The net effect, then, of allowing exit on the total probability 

of entry is ambiguous. I show, however, that which effect dominates depends on the cost 

of the high cost incumbent. In fact, when the high cost incumbent’s costs are especially 

high, entry can actually be less probable when one allows for exit.

In the limit pricing model, I assume that while the incumbent’s type is private 

information, the entrant’s type is common knowledge. When analyzing the effect of exit 

on entry deterrence strategies that directly affect future profitability, I examine the 

reverse case: the incumbent’s type is common knowledge while only the entrant knows 

its type. I do this to show how entry-deterring strategies can affect the credibility of the 

incumbent’s threat to remain in the market after entry. The crucial insight here is 

somewhat similar to Nalebuff’s (1987) insight about the effect of credibility in pretrial 

settlement negotiations. The more the incumbent invests in decreasing the profitability of 

entry, the stronger a competitor it is likely to face given that entry occurs. This reduces 

the credibility of the incumbent’s threat not to exit.

In fact, I show that when exit is possible, but not inevitable, the only way the 

incumbent can deter entry is by making the no exit threat more credible; that is, by 

increasing its expected duopoly profits. Reducing the profitability of entry into a duopoly 

market will have no independent effect on the probability of entry. While investments in 

cost reduction still have entry deterring properties, it is only because they make the 
4

profits of the incumbent, given entry, larger, not because they also reduce the profits of 

an entrant. Increases in the fixed cost of entry, however, have no effect on entry 

probability because they don’t affect the incumbent’s profits given entry, and thus don’t 

affect its incentives to exit. Thus, strategies such as lobbying for tighter environmental 

standards for new plants will not be effective in deterring entry when exit is possible (but 

not certain). This will even be true for regulations that raise the marginal costs of a new 

entrant, thus reducing its expected profit from entry, so long as they also raise the 

incumbent’s marginal cost enough to reduce its profitability in the event of entry. As 

surprising as this may seem, it follows because the higher the cost of entry or the less 

profitable duopoly competition is for the entrant, the more effective a competitor an 

entrant must be to warrant entry. This reduces the expected profits of the incumbent, 

making it more likely to exit. This increase in the exit probability then induces more 

entry, counteracting the increase in the entry cost.

On the other hand, advertising that expands the market for both the incumbent and 

the entrant can actually deter entry when exit is possible, even though it would encourage 

entry were exit not possible. Again, the reason is that by increasing the demand for the 

product, the incumbent increases its expected profits if entry occurs.3 This makes it less 

likely it will exit, making entry less attractive. This effect more than compensates for the 

direct effect of the larger market on the profitability of entry.

While most of the literature on entry deterrence has assumed that exit by the 

incumbent firm is never optimal, there are a few exceptions. Judd (1985), in a model of 

 

3

 One can imagine some types of demand curves (such as constant elasticity demand) where parallel shifts 

of the demand curve might actually reduce duopoly profits by inducing each firm to act compete more 

aggressively. In these unusual circumstances, advertising that induced such a parallel shift in the demand 

curve would not deter entry since it would not increase the incumbent’s duopoly profits. I thank Jeremy 

Bulow for pointing this out.
5

spatial competition, shows that, when the incumbent can exit, its ability to pre-empt entry 

into a nearby product by entering first is limited. Eaton and Lipsey (1980) allow for exit 

to discuss the optimal durability and replacement of sunk capital. Neither of these 

papers, however, considers the impact of exit on the type of entry deterrence strategies 

discussed here.4 Moreover, both use complete information models, so the interaction of 

asymmetric information and exit, which is the essence of this paper, is not present in 

either of those papers.

In the limit pricing literature, LeBlanc (1992) allows the incumbent firm to exit 

and recover some of its fixed costs. Thus, his model contains some exit in some 

equilibria. His focus, however, is not on the effect of exit on the effectiveness of limit 

pricing. In fact, since he only examines separating equilibria, and pooling equilibria are 

the only ones where exit affects the ability of limit pricing to deter entry, he does not 

derive any results about the role of exit in limit pricing strategies.

In the investment in entry deterrence literature, both Arvan (1986) and Bagwell 

and Ramey (1996) (in capacity models) allow for the possibility of exit by the incumbent 

following entry. In Arvan’s paper, however, the only uncertainty is about the 

incumbent’s technology, not the entrant’s. So the effect I describe about the interaction 

of the entry deterring strategy with the credibility of the no exit threat does not exist in 

Arvan’s paper. In Bagwell and Ramey there is no asymmetric information, but they do 

derive a link between capacity expansion and exit by assuming forward induction. This, 

 

4

 The spatial preemption model of Judd (1985) could be seen as a special case of the strategies I discuss in 

Section III. However, since there is no uncertainty in his model and the entry deterrence strategy is of fixed 

magnitude, the equilibrium result is exit with probability one. Therefore, the most interesting equilibrium 

in Section III, the mixed strategy equilibrium, doesn’t arise in Judd’s model. By the same token, since this 

paper doesn’t consider spatial markets, the model in Section III doesn’t cover the interesting aspects of 

Judd’s model. Thus, while his paper has similarities with this one, the situations it covers and its results are 

quite distinct.
6

essentially, gives the entrant a limited ability to commit to its output level. The way this 

works is that, under forward induction, the incumbent assumes that the entrant will 

produce a large enough quantity to cover its entry costs assuming the incumbent takes 

this output level as given. A consequence of giving the entrant this commitment ability is 

that the incumbent then wants to constrain its capacity so that the entrant can enter and 

make positive profit without the incumbent exiting. Like their paper, I also show that 

exit can make entry deterrence strategies less effective. My paper, however, gets this 

result by introducing asymmetric information rather than limited entrant commitment 

made possible by assuming forward induction. This distinction is fundamental because it 

is both the possibility of exit and uncertainty about the entrant’s type that generate the 

result that entry deterrence is accomplished solely by decreasing the incentive to exit, not 

by decreasing the profitability of entry.

The plan of the paper is as follows. In Section II, I discuss the effect of exit on 

strategies, such as limit pricing, that only affect the entrant’s perceptions about the 

profitability of entry. Section III discusses entry deterrence strategies that affect the 

actual profitability of entry. Section IV concludes. The Appendix contains the proof of 

the first and third propositions.

II. STRATEGIES THAT AFFECT PERCEPTIONS: LIMIT PRICING

To show how the possibility of exit changes the effectiveness of limit pricing, I 

consider a simple model where there are fixed, but not sunk, costs of operating in any 

period. These costs, which can differ between the entrant and the incumbent, are 

common knowledge. The entrant’s marginal cost is also common knowledge, but the 

incumbent’s marginal cost is private information. The set up of the game is as follows. 
7

In period 1, only the incumbent, I, is in the market. It chooses some action, a ∈ A, that 

affects its profits in period 1 but has no effect on its profits, or the entrant’s, in period 2 

conditional on the market structure that exists in period 2. The potential entrant, E, 

observes a. Period 2 is divided into three stages. In period 2.1, the entrant decides 

whether or not to enter the market. In period 2.2, the incumbent, having observed the 

entry decision, decides whether or not to exit the market. In period 2.3, the incumbent 

and entrant (if in the market) earn profits, which vary according to the entry and exit 

decisions. The following timeline depicts the sequence of events.

Period 1 Period 2.1 Period 2.2 Period 2.3

I chooses a.

I earns profits 

E observes a and makes 

its entry decision

I decides 

whether to exit

I and E, if active, 

earn profits

Notice that while I have allowed the incumbent to save avoidable fixed costs by 

exiting the market after observing the entrant’s entry decision, I do not allow the entrant 

to exit, and save its entry costs, after observing the incumbent’s exit decision. This is the 

essence of the difference between the model in this paper and prior models of limit 

pricing. If the entrant could back out of its entry decision at zero cost, the initial entry 

decision would be meaningless. Thus, the model would be identical to those where the 

incumbent can commit to remaining in the market even after the observing entry 

behavior. Relaxing this assumption is the essence of the paper.

One might think that the timing assumption is still peculiar in that, after the 

entrant pays a sunk entry cost, the incumbent and entrant are still not on equal footing. 

The incumbent has the option to exit, saving some avoidable fixed costs, while the 
8

entrant does not. This assumption, however, is not essential. If I modified period 2.2 to 

allow the incumbent and the entrant to make simultaneous exit decisions (and divided up 

the entry costs into a sunk portion, paid in 2.1, and an avoidable portion, paid in 2.3) the 

equilibrium of the old game is still an equilibrium of the new game. If the incumbent 

thinks that after the entrant sinks its entry costs in period 2.1 it will never exit in period 

2.2, its exit strategy in 2.2 will be unchanged. Thus, anytime the entrant wants to enter in 

the original game it will enter in the modified game and not exit. So, as long as there is 

an equilibrium of the game outlined in the above figure where the entrant enters, the same 

equilibrium will obtain in this modified game.5

Thus, the incumbent’s inter-temporal profit function can be written as follows:

(1)

{ ( )(1 ( , ))( ( ) ) (1 ( ))( ( ) )}

( , ) ( , )

2 2

1

p a p a c c f p a c f

a c a c f

e x I d e I m

I I

− − + − −

Π = − +

b p p

p

In each period that the incumbent operates, it incurs an avoidable fixed cost, f. This cost 

is avoidable in that if it exits the market, it doesn’t pay f. In period 1, the firm’s gross 

profit is a function of its action, a (in the limit pricing interpretation a is the first period 

price), and its marginal cost parameter, c, (which only I knows). Profits in period 2, 

discounted by ß, depend on the entry and exit decisions. With probability pe(a) E enters. 

If that happens, I will exit with probability px(a,c). If I exits, it earns zero profits. If E

enters and I does not exit, I earns the period 2 gross duopoly profit level ( ) 2c p I d. If E

does not enter, then I earns the period 2 gross monopoly profit level ( ) 2m c p I. I assume 

 

5

 There could be another equilibrium (at least for some parameter values) of this modified game where a 

high cost incumbent does not exit in period 2.2 with positive probability, inducing the entrant to exit in 

period 2.2, which means the entrant will never enter in period 2.1. Not only will this alternate equilibrium 

not exist for all parameter values, unlike the equilibrium described in the text, the alternate equilibrium is 

inconsistent with forward induction (see Bagwell and Ramey 1996): In period 2.2 if a high cost incumbent 

observes entry it will have to assume the entrant will not exit or else the entrant would have been better off 

not entering in 2.1.
9

that m c f p I 2( ) > , so that when E does not enter, I never exits. Given its cost parameter, 

c, I chooses a to maximize (1).

From here on, I will examine the simple case where there are just two possible 

cost parameters for I, H Lc > c . The ex ante probability that I’s cost parameter is cH is q. 

E’s cost parameter, known to all, is E

c . E’s expected period 2 profits from entry are as 

follows:

(2)

H x L E dL x L E m E

E H x H E dH x H E m

p a p a c p a c f

a p a p a c p a c

+ − − + −

= − +

(1 ( ))[(1 ( , )) ( , ) ]

( ) ( )[(1 ( , )) ( , ) ]

2 2

2 2

p p

p p p

With probability pH (a) , I has high costs (notice that this is the posterior probability that 

the incumbent has high costs given the action observed in period 1). Given that, if I does 

not exit, then E earns gross duopoly profits p E2dH . If I does exit, then E earns gross 

monopoly profits of p E2m . With probability (1− pH (a)), I has low costs. In this case, 

I’s exit probability and E’s duopoly profits are adjusted accordingly. To enter the market, 

E must pay the fixed cost fE.

Because I want to examine the perfect Bayesian equilibrium of this model, I 

require that pH (a) be determined in accordance with Bayes’ rule. That is:

(3)

( ) (1 ) ( )

( )

( )

qL a q L a

qL a

p a

H L

H

H

+ −

=

In (3), Li represents the likelihood that an incumbent with cost level ci would choose 

action a.

In this simple model, exit only matters if the high cost incumbent will exit if the 

entrant enters and the low cost incumbent will not. Thus, I assume that 

( ) ( )

I 2d H I 2d L p c < f < p c . It is well known that in models of this type both pooling and 
10

separating equilibria are possible. In separating equilibria, the incumbent’s type is 

perfectly revealed, so limit pricing never deters entry whether exit is possible or not. 

Thus, the effect of exit on separating equilibria is not very interesting. I will confine the 

analysis to equilibria with at least some pooling. That is, if aP is the (pooling) action 

chosen by the low cost incumbent, a high cost incumbent chooses aP with positive 

probability. For this to be a perfect Bayesian equilibrium the following conditions must 

hold:

(4a) ( ) ≤ 0 E P p a

(4b) ΠI(aP,ci) ≥ ΠI(a,ci) ∀a ∈ A,i = L, H

Condition (4a) requires that in any pooling equilibrium entry must be deterred 

with positive probability. If this condition doesn’t hold, then entry is inevitable and thus 

both types of incumbents should choose the myopically optimal a, which I assume differs 

across types. Condition (4b) is just the condition that the pooling equilibrium action, aP, 

is weakly optimal (given the beliefs of the entrant) for both types of incumbents. Given 

the assumption about the duopoly profit functions of the high and low cost incumbent, 

condition (4a) can be written as follows:

(4a?) pH (aP)p E2m + (1− pH (aP))p E2dL − fE ≤ 0

Notice that if (4a?) holds for p a q H ( P

) = (the posterior probability that an incumbent has 

high costs equals the prior probability), then there exists a complete pooling equilibrium 

(a pooling equilibrium where the high cost incumbent chooses aP with probability 1). If 

(4a?) holds strictly for p a q H ( P

) = , then entry and exit never occur in the pooling 

equilibrium, so exit does not affect this pooling equilibrium. If (4a?) holds at equality for 

p a q H ( P

) = , then there will be a complete pooling equilibrium with the incumbent 
11

choosing action aP both if exit is possible and if it isn’t. Because the entrant is indifferent 

about entry when there is exit, however, the exit pooling equilibrium (unlike the same 

equilibrium when exit is precluded) will involve entry, exit, and duopoly with positive 

probability. Below, I consider what happens when (4a?) does not hold for p a q H ( P

) = , 

so a complete pooling equilibrium is impossible.

When complete pooling is impossible, the high cost incumbent must be 

indifferent between mimicking the low cost incumbent and choosing its myopically 

optimal a, aH. Thus, the high cost incumbent plays the mixed strategy of choosing aP

with probability rp and aH with probability 1- rp. Then, rp must be small enough so that 

(4a?) holds when ( ) H P

p a is given by (3). The following proposition compares 

incomplete pooling equilibria in this model to similar equilibria when the incumbent can 

commit not to exit.6

Proposition 1. Assume a complete pooling equilibrium does not exist. Consider any 

partial pooling equilibrium with the pooling action aP. (a) There is less pooling with exit, 

that is, the high cost incumbent chooses aP with strictly lower probability when exit is an 

option than when it is not. (b) If aP is not a complete pooling equilibrium when exit is 

impossible, then, conditional on observing the pooling action, aP, the entrant enters less 

often when exit is an option than when it is not. (c) If aP is not a complete pooling 

equilibrium when exit is impossible, then, unconditionally, entry occurs with greater 

 

66

The comparison in this proposition between the exit case and the no exit case is NOT equivalent to a 

comparison of pooling equilibria where a high cost incumbent’s duopoly profits are zero and when they are 

negative. While it is true that introducing the possibility of exit changes the high cost incumbent’s duopoly 

profit in the second period from some negative amount to zero, it has an additional effect. That is, when 

the high cost incumbent can exit, this also increases the entrant’s profit from entry when she is facing a 

high cost incumbent. Hence, this proposition is not simply a special case of the standard limit pricing 

results.
12

probability when exit is an option if and only if cH < cH*. cH * is implicitly defined by 

the equation 

( ( *) )( ( ))( ( *) ( *))

( ( *) ( ))( )( ( *) )

2 2 2 2 2

2 2 2 2

E d H E E m E d L I m H I d H

E d H E d L E m E I m H

c f c c c

c c f c f

p p p p p

p p p p

− − −

− − − =

if aP completely deters entry in the exit case. If aP does not completely deter entry in the 

exit case then cH * is implicitly defined by the equation 

( ( ))( ( *))( ( *) ( *))

( ( *) ( ))( )( ( *))

2 2 2 2 2

2 2 2 2

E E d L E m E d H I m H I d H

E d H E d L E m E I d H

f c c c c

c c f f c

p p p p p

p p p p

− − −

− − − =

Proof. See Appendix.

At first glance, Proposition 1 (especially part (b)) may seem somewhat counterintuitive: the entrant enters less often (with smaller probability) after observing the 

pooling action aP when the incumbent can exit than when it can’t. This is explained as 

follows. Because aP represents a partial pooling equilibrium, the high cost incumbent is 

indifferent between it and choosing its myopically optimal action aH. Also, because aP is 

a pooling equilibrium, the entrant must make non-positive profits (when exit is possible) 

if it enters after observing aP. Now, suppose exit is not possible. A high cost incumbent 

will find the separating action, aH, strictly less profitable than it did when it could exit, 

since it now must earn negative profits after entry. To restore indifference, the pooling 

action, aP must be less profitable also. This requires that the entrant enter with a larger 

probability when exit is impossible (hence, with a smaller probability when exit is 

possible) after observing aP. Similarly, now that the high cost incumbent does not exit 

after entry, the entrant earns a strictly lower profit from entry when exit is impossible, 

making no entry strictly preferable. To make entry more profitable, restoring the 
13

entrant’s indifference, it must face high cost incumbent with greater probability after 

observing the pooling action. Thus, the high cost incumbent must pool more often when 

exit is impossible (or less often when exit is possible). 

The comparison of the total amount of entry, then, depends on the strength of the 

two competing effects. With exit, a high cost incumbent tries to deter entry by pooling 

less often, but pooling is more successful at deterring entry. When a high cost 

incumbent’s costs are very high, the second effect dominates; there is less entry when 

exit is possible. Otherwise, there is more total entry when the incumbent can exit after 

observing entry than when it can’t. The reason for this is that the exit option is relatively 

more valuable to the high cost incumbent the higher its costs are (the higher its costs the 

more negative its profits are after entry if it cannot exit). Thus, the amount by which aH

is more profitable when exit is possible relative to when it isn’t is increasing in the high 

cost incumbent’s costs. So, to maintain indifference, the amount by which the probability 

of entry falls (when exit is possible relative to when it isn’t), again, given the pooling 

action, must be greater the higher the high cost incumbent’s costs.

To see that it is possible to have either more or less entry when exit is possible, 

consider two extreme values for cH. First, if cH is very large, then the entrant’s profit in 

duopoly will only be marginally lower than its monopoly profit. Thus, the fraction of 

high cost incumbents that choose the pooling action need only be marginally higher when 

exit is not possible to ensure that the entrant remains indifferent about entry when exit is 

not possible. When the high cost incumbent cannot exit, however, choosing aH becomes 

much less profitable because its losses after entry are much larger due to its very high 

costs. Thus, to ensure that aP is not more profitable than aH now that exit is not possible, 
14

the probability of entry, conditional on aP, must be much larger when exit is not possible. 

So, when exit is not possible and cH is very large a high cost incumbent chooses the 

pooling (entry-deterring) action with only a marginally larger probability as when exit is 

possible, but that action is much less effective at deterring entry. So there will be, on 

balance, more entry when exit is impossible than when it is possible. 

On the other hand, if cH is small enough that the high cost incumbent is only 

barely better of by exiting after entry, then the myopically optimal action, aH, is only 

slightly better for it when exit is possible than when it isn’t. As a result, the entry will 

have to occur only slightly more often (given the pooling action) when exit is impossible 

compared to when it is possible. High cost incumbents, however, will still need to 

choose the pooling action with a much larger probability when exit is not possible to 

ensure that the ex post probability that an entrant is facing a high cost incumbent is large 

enough so that the entrant still makes zero expected profits from entry even though it will 

only earn duopoly profits. In this case, then, the probability of entry conditional on 

pooling is only marginally larger (when exit is not possible) while the probability of 

pooling itself (which is the action that can deter entry) is much larger when exit is 

impossible. So, on balance, when cH is very small, entry is less probable when exit is not 

possible than when it is.

Of course, the proposition doesn’t say anything about separating equilibria. From 

Milgrom and Roberts (1982), however, we already know that in separating equilibria 

limit pricing is completely ineffective at deterring entry. Thus, in analyzing the 

effectiveness of limit pricing strategies, pooling equilibria are the important ones to focus 

on. For those equilibria, the message of Proposition 1 is that assuming incumbents can’t 
15

exit the market is not a benign assumption when judging the effectiveness of limit pricing 

strategies. When the pooling equilibrium is complete, the possibility of exit 

unambiguously reduces the effectiveness of limit pricing at deterring entry. When the 

pooling equilibrium is incomplete, the option to exit will limit the effectiveness of 

deterring entry if the high cost incumbent’s costs are not too high. Otherwise, the option 

to exit will actually increase the effectiveness of this entry deterrence strategy.

III. STRATEGIES THAT AFFECT FUTURE PROFITABILITY

In this section, I examine the effect of exit on entry deterrence strategies that 

operate by affecting actual future profitability rather than the entrant’s perceptions of the 

profitability of entry. In order to make the distinction clean, in this section the entrant 

will have the private information, not the incumbent. Therefore, any entry deterrence 

strategy by the incumbent will not influence the entrant’s perception of the incumbent’s 

type. Entry deterrence can only work by affecting the actual profitability in period 2. 

The types of strategies I have in mind here are incumbent investment in cost reduction (or 

capacity expansion) in period 1 that has the effect of reducing the incumbent’s marginal 

cost in period 2 or advertising by the incumbent in period 1 that affects the demand for 

the product in period 2. The incumbent could also sign exclusive contracts with suppliers 

(or buy up critical resources that are in limited supply) that increase either the marginal or 

the fixed cost of the entrant in period 2 (notice that some of these strategies will affect the 

entrant’s profits even if the incumbent exits).

The model in this section differs from the model in Section II in the following 

ways. First, the incumbent’s marginal cost parameter, c, is common knowledge. Second, 

the entrant’s marginal cost parameter, cE, is private information to the entrant. The 
16

incumbent knows only that it is distributed according to the cumulative distribution 

function G, with associated density function g, with support [ , ] E Ec c . In period 1, the 

incumbent takes an action, a ∈ A, that affects both its profits in period 1 and (potentially) 

its own profits and the profits of the potential entrant (should it enter) in period 2. The 

potential entrant, E, observes a. Period 2 proceeds exactly as in Section II. For this 

model, c and cE could also be interpreted as parameters measuring the quality of the 

goods rather than their production cost. For this interpretation, higher c and cE should be 

thought of as lower quality. I employ the Bayesian Nash equilibrium concept for this 

game.

Because the entrant’s costs are private information, the incumbent’s response to 

entry cannot be conditioned on the entrant’s type. The entrant’s profits from entry are:

(5) E E x E d E x E m E E(a,c ) = (1 − p (a)) (a,c ) + p (a) (a,c ) − f p p 2 p 2

Notice that this is strictly decreasing in its marginal cost parameter, cE. I assume that the 

support for cE is such that, for any a the incumbent might choose, there exists a cutoff 

level for the entrant’s cost parameter ˆ ( ) [ , ] E E E

c a ∈ c c such that if the entrant’s cost are 

cˆ (a) E it makes exactly zero profits from entry. Thus, if the entrant has a smaller cost 

parameter than this cutoff value, c cˆ (a) E < E

, the entrant enters, otherwise it does not. Of 

course, the exact value of cˆ (a) E will depend on the exit strategy of the incumbent, but, as 

I show below, this will also be determined by a, its period 1 action. In the figure below, 

the incumbent can move the cutoff line at cˆ (a) E

 to the left by its choice of a.
17

E

c cˆ (a) E E

c

Entry Region No Entry Region

Given this entry strategy, the incumbent’s inter-temporal profit function can be 

written as follows:

(6)

{(1 ( )) ( ( , ) ) ( ) (1 ( ˆ ( )))( ( ) )}

( ) ( )

2

ˆ ( )

2

1

p a a c f g c dc G c a a f

a a f

E E E I m

c a

c

x I d E

I I

E

E

− − + − −

Π = − +

∫

b p p

p

In period 1, the incumbent is a monopolist and earns the associated profit, ( ) 1

a p I, 

less its fixed operating cost of f. This profit is a function of a in that the monopolist’s 

first period profit will be reduced if it invests in second period cost reduction or 

advertising to increase second period demand. Thus, I will think of ( ) 1a p I

′ as being 

negative. Its second period profits are discounted by ß. When the entrant enters, which 

happens when its costs are between [c ,cˆ (a)) E E

, if the incumbent exits it earns zero. If 

the incumbent doesn’t exit, which happens with probability (1 p (a)) − x, the incumbent 

pays its fixed cost f and earns gross duopoly profits of ( , )

I 2d E p a c . These are increasing 

in cE, and I will often assume they are increasing in a (though they need not always be, 

depending on the type of action the incumbent takes), as they would be if a is cost 

reduction investment or advertising. If the entrant’s costs are high enough that it doesn’t 

enter, the incumbent earns monopoly profits in period 2, again generally (though not 

always) increasing in a, and pays its fixed cost f.
18

There can be equilibria where the incumbent always exits upon entry. For 

example, consider an entrant of type ( ) 1c a E such that it earns zero profits as a 

monopolist, that is, p E2m (a,cE1(a)) − fE = 0 . If ( ( , ) ) ( ) 0

( )

2

1

− ≤ ∫ E E

c a

c

I d E

a c f g c dc E

E

p

then the incumbent’s expected profits are non-positive if it does not exit. Hence, it will 

always exit if the entrant enters, and all entrants with ( ) 1c c a E < Eenter. In such cases, 

however, a can still have value to the incumbent since it may reduce the probability of 

entry and may increase the incumbent’s profits in period 2.3 when there is no entry. I 

define the set A1 as follows:

{ : ( , ( )) 0, ( ( , ) ) ( ) 0}

( )

1 2 1 2

1

= ∈ − = − ≤ ∫ E E

c a

c

E m E E I d E A a A a c a f a c f g c dc E

E

p p

If A1a ∈ then the incumbent exits with probability one if there is entry.

It is also possible for the incumbent to never want to exit upon entry. Consider 

( ) 2c a E such that p E2d(a,cE2(a)) − fE = 0 . If ( ( , ) ) ( ) 0

( )

2

2

− > ∫ E E

c a

c

I d E

a c f g c dc E

E

p

then all entrants with ( ) 2c c a E < E enter, and the incumbent never exits.7 If it is optimal 

for the incumbent to choose a such that this equilibrium obtains, then the option to exit 

makes no difference. I define the set A2 as follows:

{ : ( , ( )) 0, ( ( , ) ) ( ) 0}

( )

2 2 2 2

2

= ∈ − = − > ∫ E E

c a

c

E d E E I d E A a A a c a f a c f g c dc E

E

p p

 

7

 Even in this case, however, there could be equilibria where the incumbent exits with positive probability 

and entrants with higher costs than ( ) 2

c a E enter. If so, the analysis that follows will apply to these 

equilibria as well.
19

The set A2 is the set of all actions that for which there is an equilibrium where the 

incumbent does not exit in period 2.2. Thus, if A1 A2a ∉ ∪ then the incumbent will exit 

with some probability p (a) ∈(0,1)

x

. The following analysis is for this case.

Since p (a) ∈(0,1)

x

, the incumbent must earn zero profits when it remains in the 

market after exit. That is, the entrant’s entry strategy, to enter if its costs are less than 

cˆ (a) E, must be such that:

(7) ( ( , ) ) ( ) 0

ˆ ( )

2 − = ∫ E E

c a

c

I d E

a c f g c dc E

E

p

Equation (7) implicitly defines the entry cutoff cˆ (a) E

. Given this cutoff, the probability 

that the incumbent exits after entry is determined to ensure that the entrant whose costs 

are cˆ (a) E

 breaks even by entering:

(8) (1− px(a))p E2d(a,cˆE(a)) + px(a)p E2m (a,cˆE(a)) − fE = 0

Thus, the probability of exit is the following:

(9)

( , ˆ ( )) ( , ˆ ( ))

( , ˆ ( )) ( )

2 2

2

a c a a c a

a c a f

p a

E d E E m E

E d E E

x

p p

p

−

−

=

By differentiating (7) with respect to a, one can see how entry cutoff varies with 

a; that is, how effective an action a is at deterring entry. Doing so gives the following:

(10)

(ˆ ( ))( ( , ˆ ( )) )

( )

( ( , ) )

ˆ ( )

2

ˆ ( )

2

g c a a c a f

g c dc

a

a c f

c a

E I d E

E E

c a

c

I d E

E

E

E

−

∂

∂ −

′ = −

∫

p

p

As (10) shows, increasing a (when A1 A2a ∉ ∪ ) reduces entry if and only if increasing a

increases the incumbent’s gross duopoly profit. This follows because if the incumbent is 

to break even in expectation when the entrant enters, it must make positive net profits 

when the entrant is of the highest possible cost type. That is, p I 2d(a,cˆE(a)) − f > 0 .
20

By contrast, if exit is not possible, the cutoff level for entry, which I’ll call 

cˆ (a) En , will be determined implicitly by the following equation:

(11) p E2d(a,cˆEn(a)) − fE = 0

Differentiating this with respect to a and solving for the effect of a on entry gives the 

following:

(12)

E

E d En E E d En

En dc

d a c a

da

d a c a f

c a

( ( , ˆ ( )) ) ( , ˆ ( )) ˆ ( )

p 2 − p 2

′ = −

Since the denominator is negative (increasing its cost parameter reduces the entrant’s 

profits), when exit is not possible increasing a decreases the cutoff level for entry if and 

only if increasing a decreases the entrant’s profits. Thus, I have proved the following 

proposition.

Proposition 2. For all A1 A2

a ∉ ∪ , when exit is possible, a deters entry if and only if it 

increases the incumbent’s duopoly profit, whereas, when exit is not possible, a deters 

entry if and only if it reduces the entrant’s duopoly profit.

This distinction made by Proposition 2 is quite fundamental. When exit is 

possible, unless entry deterrence is either very cheap and avoidable fixed costs are large, 

or entry deterrence is very expensive and avoidable fixed costs are small, its effectiveness 

depends only on how it affects the incumbent’s profits, not the entrant’s. 

So long as, for the set of entrants who find it profitable to enter as a duopolist, the 

incumbent does not want to exit, decreasing the entrant’s profits from entry will decrease 

the cutoff value of the cost parameter, cˆ (a) E

, reducing the probability of entry. This is 
21

when A2a ∈ . However, as the profitability of entry falls, it is only the more efficient 

entrants who can profitably enter. This reduces the incumbent’s profits in period 2.3. 

Once the profitability of entry is reduced to the point where the incumbent earns 

zero expected net profits from competing against the entrant, reducing the profitability of 

entry, conditional on market structure, no longer deters entry (over the region 

A1 A2a ∉ ∪ ). The reason is that if the cutoff level of the cost parameter for entry 

continues to fall, the incumbent will expect negative profits in period 2.3, so it will exit 

with probability one. Knowing this, however, less efficient entrants (those with cost 

parameters above cˆ (a) E) will enter, which will induce the incumbent to remain in the 

market. Thus, past the point where the incumbent earns zero expected profits in period 

2.3, reducing the profitability of entry will only increase the incumbent’s probability of 

exit so as to continue to ensure that entrants with a cost parameter of cˆ (a) E break even 

from entry. Of course, once the profitability of entry falls below the point where an 

entrant with the old break even cost parameter cˆ (a) E

 can break even as a monopolist, 

then decreasing the profitability of entry will again reduce the probability of entry and the 

incumbent will exit whenever there is entry (now, A1a ∈ ).

On the other hand, if the incumbent’s action increases its duopoly profits given 

the efficiency of its competitor, it will remain indifferent about whether or not to exit 

only if the expected efficiency of the entrant it faces has increased. Thus, the entry rule 

must change (when A1 A2a ∉ ∪ ) to have entry only when the entrant’s cost parameter is 

below some lower level than before. To ensure that a less efficient entrant does not want 

to enter (since its profitability, conditional on the market structure, has not changed), the 
22

incumbent must exit with a smaller probability than if had not taken this action to 

improve its profitability.

Investments in marginal cost reduction (or quality enhancement) are entrydeterring strategies in both cases, though their effectiveness may differ greatly. 

Similarly, raising the entrant’s marginal costs (or reducing its quality) are effective entrydeterring strategies with or without exit. Strategies that affect fixed costs, however, will 

operate quite differently in the two situations. When exit is possible, fixed costs of entry 

are not entry barriers, that is, they don’t affect the probability of entry (so long as they 

aren’t too high). Thus, increasing the entrant’s fixed entry cost is pointless for the 

incumbent. As explained above, all that will do is increase the equilibrium exit 

probability without affecting the entry probability. On the other hand, if the incumbent 

can pay some of its fixed costs of operating before the entrant makes its entry decision 

(say by signing long-term employment contracts or long-term leases), this will deter entry 

by making its threat to remain in the market more credible. (Of course, to the extent 

these contracts could be renegotiated, the effectiveness of these strategies may be 

limited.)

This distinction also is critical when one considers the operation of advertising as 

an entry barrier. If the goods of the incumbent and the entrant are not highly 

differentiated, one might think that the incumbent would want to refrain from advertising 

to increase the size of the market because that might induce entry. When exit is possible, 

however, the exact opposite is the case. The incumbent will want to advertise to increase 

the size of the market, not only because doing so increases its profit the period 2, but also 

because it will deter entry. In this case, entry deterrence occurs because advertising has 
23

made the incumbent’s threat to remain in the market more credible. A similar argument 

would apply to investments in quality enhancement that are non-rival (improve the 

entrant’s quality as much as the incumbent’s). This effect is similar to Nalebuff’s (1987) 

point about the importance of credibility in pretrial settlement negotiations.

One might wonder how sensitive the results in Proposition 2 are to the need for a 

mixed strategy equilibrium in this model. To shed some light on this question, consider 

tweaking the model as follows. Instead of the incumbent’s cost parameter being common 

knowledge, assume that the incumbent is subject to some small, mean zero, cost shock in 

between period 1 and period 2. Then, instead of an incumbent being indifferent as to 

whether or not to exit in period 2.2, if the incumbent gets a favorable cost shock (say 

s < s *) it remains in the market, otherwise it exits.8

 

Consider the effect of an action, a, that only reduces the profitability of entry. 

There must be some small effect on the entry cutoff level. If not, then an incumbent will 

still remain in the market if and only if s < s * (since the incumbent’s expected profit 

from competing with the same distribution of possible entrants has not changed). So, an 

entrant whose cost parameter is exactly at the old cutoff level will not longer break even. 

As a result, the cutoff level for entry must fall, and this will reduce s*. Since s* falls, 

however, the probability that a potential entrant will be a monopolist increases. Thus, the 

cutoff level for entry cannot fall as much as it would if there were no exit. That is, the 

possibility of exit still reduces the ability of profit reducing strategies to deter entry, 

though it doesn’t entirely eliminate their effectiveness. 

As the magnitude of the cost uncertainty goes to zero, however, the effectiveness 

of reducing the profitability of entry does approach zero. If the possible cost parameters 

 

8

 I thank Ezra Friedman for suggesting this case.
24

of the incumbent in period 2 are very tightly clustered then it only takes a very small 

reduction in the cutoff level of entry to cause a big shift in the probability of exit. Thus, 

the entrant must only be slightly more efficient to still break even after the incumbent has 

acted to reduce the entrant’s profits. Thus, while stark nature of the results in Proposition 

2 is dependent on the mixed strategy equilibrium, the results do not change in a knifeedge way when some uncertainty is introduced. 

In addition, introducing this uncertainty will not change the fact that actions that 

only increase the duopoly profit of the incumbent deter entry. (By increasing s*, they 

reduce the probability that the entrant will be a monopolist.) Whether or not a strategy, 

such as advertising, that increases profitability for both the incumbent and the entrant 

deters entry will depend on the magnitude of the uncertainty. The smaller the uncertainty 

about the incumbent’s costs the more likely advertising will deter entry rather than 

encourage it.

Notice that when the conditions of Proposition 2 hold, there will necessarily be 

more entry when exit is possible than when it isn’t. This follows because the condition 

for the entry cutoff when there is no exit, equation (11), says that all entrant types that 

can make non-negative duopoly profits, net of their fixed entry cost, enter. When exit is 

possible, however, equation (8) indicates the entrant’s profits from entry are a weighted 

average of duopoly and monopoly profits. Since monopoly profits exceed duopoly 

profits, higher cost entrants will enter when exit is possible. If the conditions of 

Proposition 2 do not hold, there will either be exit whenever there is entry, in which case 

there is even more entry, or there will never be exit, in which case entry is unaffected by 
25

the exit option. Thus, not surprisingly, for entry deterrence strategies that actually affect 

future profitability, the exit option reduces the ability of incumbent to deter entry.

This does not necessarily imply, however, that the incumbent will make less effort 

to deter entry when exit is possible than it would were it impossible (where more effort is 

defined as a greater level of a, and thus more first period profits foregone). In fact, there 

is no clean comparison of the incumbent’s incentives to forego profits to deter entry in 

the two cases because the entry deterrence mechanism is so different when exit is 

possible than when it isn’t. This can be seen by examining the first order conditions for 

the entry deterrence action, a, in the two cases. When exit is possible (and occurs with 

positive probability) the marginal profit from a is the following:

(13) ( ) ( ) {(1 ( ˆ ( ))) ( ) ˆ ( ) (ˆ ( )))( ( ) )} 1 2 2 a a G c a a c a g c a m a f Π′

I = p ′I + b − E p I

′ m − E′ E p I −

This follows from differentiating (6) with respect to a, while fixing the incumbent’s profit 

when there is entry at zero. By substituting in for cˆ (a) E

′ using equation (10), this 

becomes:

(14) ( ( , ) ) ( ) }

( , ˆ ( ))

( )

{(1 (ˆ ( ))) ( )

( ) ( )

ˆ ( )

2

2

2

2

1

E E

c a

c

I d E

I d E

I m

E I m

I I

d a c da g c dc

a c a f

a f

G c a a

a a

E

E

∫ −

−

− ′ +

Π′ = ′ +

p

p

p

b p

p

For simplicity, I have assumed in (13) and (14) (and (15) below) that a does not 

affect fixed costs, though this is not essential for this point. As (14) indicates, a increases 

second period profits in two ways. First, when the entrant won’t enter anyway, it 

increases monopoly profits. Second, when the entrant would have entered, a increases 

duopoly profits. That is valuable, however, not because of the added profits given entry 

(that has to remain at zero to maintain the equilibrium), but, rather, because by reducing

the incumbent’s incentive to exit, it deters entry. From (14), one can see that the entry 
26

deterrence value from a is strictly greater than the amount by which it increases duopoly 

profits (

a c a f

a f

I d E

I m

−

−

( , ˆ ( ))

( )

2

2

p

p

>1, since monopoly is more profitable than duopoly). Thus, 

when exit is possible, the incumbent will choose a strictly larger a than it would have 

chosen if the only reason to choose a was to increase the incumbent’s second period 

profits, even if the incumbent could not exit and, thus, benefited from a with probability 

one in period 2.

Notice, however, that the marginal profit from a, when exit is possible, is 

independent of its effect on the entrant’s profit. This is very different from the 

incumbent’s marginal benefit from a when exit is impossible. To see this, differentiate 

(6) setting px(a) = 0 :

(15)

( ( , ) ) ( ) (ˆ ( ))[ ( ) ( , ˆ ( ))]ˆ ( )}

( ) ( ) {(1 (ˆ ( ))) ( )

2 2

ˆ ( )

2

1 2

d a c da g c dc g c a a a c a c a

a a G c a a

E E En I m I d En En

c a

c

I d E

I I En I m

En

E

− − ′

Π′ = ′ + − ′ +

∫

p p p

p b p

When exit is impossible, the incumbent’s incentive to increase a will come more from a’s 

effect on its monopoly profits in period 2 than from its effect on duopoly profits. This 

occurs not only because there is less entry when exit is impossible, but also because the 

effect on duopoly profits receives greater weight in (14) than (15). This is due to the fact 

that the value of duopoly profits in (14), in the exit case, is via entry deterrence, which is 

more valuable than the added increment to profit itself. Equation (15), however, has an 

added entry deterrence term of its own. From (12), we know that cˆ′En (a) < 0 if and only 

if increases in a reduces the profit of the entrant. Moreover, its magnitude is larger the 

more effective a is at reducing the profits of the entrant. Thus, whether an incumbent has 

a greater incentive to forego current profits by increasing a when exit is possible than 
27

when it isn’t will be heavily dependent on how strongly a affects the entrant’s profits. If 

the action a, however, has either no effect on the entrant’s profits or increases them (e.g., 

if it is advertising that expands the market), then it is much more likely that the 

incumbent will choose a larger a when exit is possible.

Focusing on the case where a is a cost reducing investment, one might also 

wonder if the threat of entry increases or reduces the incumbent’s incentive to reduce 

costs? In the exit case, one can look at (14) to answer this question. Notice that the 

marginal benefit from cost reduction, a, can be separated into two different terms:

(14?) ∫

− ′

−

−

Π′ = ′ + ′ +

ˆ ( )

2

2

2 2

1 2

( )] ( )

( , ˆ ( ))

( , ) ( )

[

( ) ( ) ( )

c a

c

I m E E

I d E

I d E I m

I I I m

E

E

a g c dc

a c a f

a f

da

d a c

a a a

p

p

p p

b

p bp

The first line of the right hand side is the marginal benefit from a when there is no threat 

of entry. The second line is adjustment in the marginal benefit due to the threat of entry. 

The threat of entry will increase the incumbent’s incentive to reduce costs, when exit is 

possible, if and only if the second line of (14?) is positive.

One can perform the same decomposition of the marginal incentive to reduce 

costs when exit is not possible:

(15?)

(ˆ ( ))[ ( ) ( , ˆ ( ))]ˆ ( )

( )} ( )

( , )

{

( ) ( ) ( )

2 2

ˆ ( )

2

2

1 2

g c a a a c a c a

a g c dc

da

d a c

a a a

En I m I d En En

c a

c

I m E E

I d E

I I I m

En

E

− − ′

− ′

Π′ = ′ + ′ +

∫

b p p

p

p

b

p bp

As before, the threat of entry increases the incumbents incentive to reduce costs, when 

exit is not possible, if and only if the last two lines of (15?) are, together, positive.

With completely abstract profit functions it is not possible, in either case, to say if 

the threat of entry increases or decreases cost reduction incentives. When exit is possible, 
28

however, Proposition 3 describes a leading case where one can say that the threat of entry 

always increases cost reduction incentives.

Proposition 3. Say profits in period 2.3 are determined by a one-shot Cournot quantitysetting game. If the reaction functions are downward sloping, then, if exit is possible, the 

threat of entry strictly increases the incumbent’s incentives to reduce costs. This is not 

necessarily the case when exit is not possible.

Proof. See Appendix.

This result is really driven by Proposition 2. There can be a reduced incentive to 

reduce costs in a duopoly market vis-à-vis a monopoly one because the incumbent 

produces less output. However, when exit is possible, the incumbent only cares about 

cost reduction in a duopoly market to make its threat of not exiting the market more 

credible. Thus, the benefit from cost reduction in a duopoly market is that by increasing 

the credibility of the incumbent’s threat not to exit, the incumbent is more likely to be a 

monopolist rather than a duopolist. So when exit is possible, cost reduction benefits still 

come from the entire monopoly output. Moreover, the benefit is not just the reduced cost 

but also the larger price the incumbent gets when it is a monopolist rather than a 

duopolist. As a result, the threat of entry strictly increases the incumbent’s incentive to 

reduce costs.

The above argument does not work for the Bertrand case due to the discontinuity 

of the effect of costs on profits. A simple example will illustrate how with Bertrand 

competition the threat of entry could actually reduce the incumbent’s incentive for cost 
29

reduction, even when exit is possible. Say the distribution of entrant types who enter are 

concentrated around two points: either the entrant has costs that are above the monopoly 

price for the incumbent (say entry costs are tiny), at its original cost level, or its costs are 

far enough below the incumbent’s that it isn’t profitable for the incumbent to reduce its 

costs to that level. Notice, the incumbent’s duopoly profits at cutoff level for entry are 

equal to its monopoly profits. So, the weight on the effect of a duopoly profits is just 

one; the entry deterrence value of cost reduction does not magnify the direct value of cost 

reduction on duopoly profits since the incumbent is earning monopoly profits as a 

duopolist. Thus, for the high cost entrant types, the incentive for cost reduction is 

unaffected by the threat of entry. If the entrant is of the low cost type, however, then the 

incumbent has no sales after entry, so cost reduction is of no benefit. Thus, the incentive 

for cost reduction must be strictly greater when there is no threat of entry.

When exit is not possible, the threat of entry can also either increase or decrease 

the incumbent’s cost reduction incentives, even under Cournot competition. The reason 

is that, when there is no exit, the entry deterrence benefit from cost reduction is (as 

Proposition 2 demonstrates) completely independent of the duopoly benefit from cost 

reduction. Thus, if the entry deterrence benefit is small (either because cost reduction has 

a small effect on the entrant’s profits or because monopoly and duopoly profits do not 

differ by much) then it is possible for the threat of entry to decrease the incumbent’s 

incentive to reduce costs.

IV. CONCLUSION

When there are fixed costs that can be avoided by exit, exit will sometimes be the 

incumbent’s ex post profit-maximizing strategy. This paper shows that considering this 
30

possibility can significantly alter the ability of entry deterring strategies to credibly deter 

entry. Thus, analyzing the effectiveness of entry deterrence strategies when exit is never 

optimal (or when the incumbent can credibly commit not to exit the market) can lead to 

as faulty conclusions about the effectiveness of entry deterrence strategies when exit is 

possible as assuming the incumbent can commit to a pricing or output policy. These 

credibility issues could be particularly important in industries where technology is rapidly 

advancing, making the risk that an entrant could supplant an incumbent quite large. Even 

where the exit by the incumbent does not happen immediately, as is depicted in this 

model, the qualitative effects of the possibility of later exit are likely to be similar, though 

a precise description of such effects is left for future research.
31

APPENDIX

Proof of Proposition 1. (a) Using (3), (4a?) can be written as follows (where pr is the 

probability the high cost incumbent chooses the pooling action):

(A1) 0

(1 )

1

(1 )

2 2 − ≤

+ −

−

+

+ −

E dL E

p

E m

p

p

f

qr q

q

qr q

qr

p p

When exit isn’t possible, the analogous condition is (here pn r is the probability the high 

cost incumbent chooses the pooling action when exit is impossible):

(A2) 0

(1 )

1

(1 )

2 2 − ≤

+ −

−

+

+ −

E dL E

pn

E dH

pn

pn f

qr q

q

qr q

qr

p p

If (A2) holds strictly, then aP must be a complete pooling equilibrium when exit is 

possible. Otherwise, there would be no entry when exit is impossible, making the 

pooling option is strictly better when exit is impossible than when it is possible. But the 

myopically optimal action, aH, is strictly worse without exit. Thus, if the incumbent is 

indifferent between the two actions when exit is possible, it will always choose the 

pooling option when exit is impossible. This ensures that rpn = 1. Since Proposition 1 

assumes there is no complete pooling equilibrium when exit is possible, p pn r < 1 = r .

Now I consider the case where (A2) holds at equality. Solving for pn r yields:

(A3)

( )

(1 )( )

2

2

E dH E

E E dL

pn

q f

q f

r

−

− −

=

p

p

Solving (A1) for pr gives the following:

(A4)

( )

(1 )( )

2

2

E m E

E E dL

p

q f

q f

r

−

− −

≤

p

p

Since monopoly profits are greater than duopoly profits, p pn r < r .

(b) When aP is not a complete pooling equilibrium when exit is impossible then(A2) 

holds at equality, by the argument in (a). Since aP is not a complete pooling equilibrium, 

the high cost incumbent must be indifferent between aP and aH. That is:

(A5) ( , ) ( H , H )

I P H I p a c = p a c

When exit is possible entry always follows aH and the high type always exits following 

entry. Thus, (A5) becomes:

(A6) a c f p a c f aH cH f p I1( P, H ) − + b{(1− e( P))(p I 2m ( H ) − )} = p I1( , ) −

Solving for the probability of entry after observing aP gives the following:

(A7)

( ( ) )

( , ) ( , )

( ) 1

2

1 1

c f

a c a c

p a

I m H

I H H I P H

e P

−

−

= −

b p

p p

If exit is not possible, I can write (A5) as follows.

(A8)

( , ) ( ( ) )

( , ) { ( )( ( ) ) (1 ( ))( ( ) )}

1 2

1 2 2

a c f c f

a c f p a c f p a c f

I H H I d H

I P H en P I d H en P I m H

= − + −

− + − + − −

p b p

p b p p

Now ( ) pen aP represents the entry probability with no exit. Solving (A8) for ( ) pen aP

gives the following.
32

(A7)

( ( ) ( ))

( , ) ( , )

( ) 1

2 2

1 1

I m H I d H

I H H I P H

en P

c c

a c a c

p a

b p p

p p

−

−

= −

Since ( ) ( ) p I 2m cH > f > p I 2dcH , ( ) ( ) pen aP > pe aP.

(c) There are two cases to consider, where the pooling action in the exit case deters entry 

completely and when it doesn’t. The former, case (i), occurs if and only if 

( ( ) ) ( , ) ( , ) b p I 2m cH − f = p I1 aH cH − pI1 aPcH . Otherwise, in case (ii), 

( ( ) ) ( , ) ( , ) b p I 2m cH − f > pI1 aH cH −p I1 aPcH , so there is some entry.

Case (i). The total probability of entry in the exit case is (1 ) p q − r , since there is only 

entry when the incumbent has high costs and doesn’t pool. The total probability of entry 

in the no exit case is: ( )(1 ) (1 ) en P pn pn p a − q + qr + q − r . Using the equations above, 

subtracting the probability of entry in the no exit case from the probability of entry in the 

exit case is at least the following:

(A8)

( )( ){ ( ) ( ))}

{( )( )( ( , ) ( , ))

{ ( ) ( ))( )( )}

(1 )

2 2 2 2 2

2 2 2 1 1

2 2 2 2

E dH E E m E dL I m H I d H

E dH E dL E m E I H H I P H

I m H I d H E dH E E m E

f c c

f a c a c

c c f f

q

b p p p p p

p p p p p

b p p p p

− − − −

− − −

− − −

−

The likelihood of entry in the exit case less the likelihood in the no exit case exceeds 

(A8) because (A8) uses the value of pr from the right hand side of (A4). Now using the 

fact that ( ( ) ) ( , ) ( , ) b p I 2m cH − f = p I1 aH cH − pI1 aPcH , I can rewrite this as follows:

(A9)

( )( )( ( ) ( ))}

{( )( )( ( ) )

{ ( ) ( ))( )( )}

(1 )

2 2 2 2 2

2 2 2 2

2 2 2 2

E dH E E m E dL I m H I d H

E dH E dL E m E I m H

I m H I d H E dH E E m E

f c c

f c f

c c f f

q

p p p p p

p p p p

p p p p

− − − −

− − −

− − −

−

The denominator is positive since monopoly profits exceed duopoly profits and the 

entrant makes net profits both as a monopolist and when the incumbent’s costs are high 

as a duopolist. The numerator will be positive when cH is low enough that the entrant 

gets almost no profits from entry, (p E2dH − fE) → 0 , or the high cost incumbent would 

almost breaks even competing as a duopolist, cH f p I 2d( ) → . The numerator will be 

negative when cH gets so high that the entrant’s profit as a duopolist is almost as large as 

its profit as a monopolist, p E2dH →p E2m .

Now it only must be shown that the numerator is monotonically decreasing in cH. 

Differentiating the numerator of (A9), ignoring the 1-q term, gives the following:

 (A10)

H

E dH

E m E I m H I m H I d H E m E dL

E dH E E m E dL I m H I d H

E dH E dL E m E I m H

dc

d

f c f c c

f c c

f c

2

2 2 2 2 2 2

2 2 2 2 2

2 2 2 2

[( ){ ( ) ) ( ( ) ( ))( )]

( )( ){ ( ) ( ))

( )( ) ( )

p

p p p p p p

p p p p p

p p p p

− − − − −

− − ′ − ′ +

− − ′ −

Rearranging terms:
33

(A11)

H

E dH

E m E I m H I m H I d H E m E dL

E dH E E m E dL I d H

E m E dH E E dL I m H

dc

d

f c f c c

f c

f c

2

2 2 2 2 2 2

2 2 2 2

2 2 2 2

[( ){ ( ) ) ( ( ) ( ))( )]

( )( ) ( )

( )( ) ( )

p

p p p p p p

p p p p

p p p p

− − − − −

− − ′ +

− − ′ +

The first line is negative since E makes more profit as a monopolist than a duopolist, E

would lose money entering when I’s costs are low, and I’s profit is decreasing in its costs. 

The second line is negative because E makes positive net profit as a duopolist against a 

high cost I and makes more profit as a monopolist than a duopolist. The third line is 

negative since cH f p I 2d( ) < , I loses money as a high cost duopolist, E dL E < f p 2, E

loses money as a duopolist with a low cost incumbent, and because E’s profits are 

increasing in I’s costs.

Case (ii). The total probability of entry in the exit case is ( )(1 ) (1 ) e P p p p a − q + qr + q − r . 

The total probability of entry in the no exit case is: ( )(1 ) (1 ) en P pn pn p a − q + qr + q − r . 

Using the equations above, subtracting the probability of entry in the no exit case from 

the probability of entry in the exit case is the following:

(A12)

( )( )( ( ))}

{( )( )( ( ) ( ))

{ ( ) ( )){ ( ) )( )( )}

(1 )( ( , ) ( , ))

2 2 2 2

2 2 2 2 2

2 2 2 2 2

1 1

E dH E dL E m E I d H

E E dL E m E dH I m H I d H

I m H I d H I m H E dH E E m E

I H H I P H

f f c

f c c

c c c f f f

q a c a c

p p p p

p p p p p

b p p p p p

p p

− − − −

− − −

− − − −

− −

The denominator is positive since monopoly profits exceed both fixed costs and duopoly 

profits and the entrant makes net profits both as a monopolist and when the incumbent’s 

costs are high as a duopolist. The numerator will be positive when cH is low enough that 

the entrant gets almost no profits from entry, (p E2dH − fE) → 0 , or the high cost 

incumbent would almost breaks even competing as a duopolist, cH f p I 2d( ) → . The 

numerator will be negative when cH gets so high that the entrant’s profit as a duopolist is 

almost as large as its profit as a monopolist, p E2dH →p E2m . Now I show that the 

numerator term in curly brackets is monotonically decreasing in cH. Differentiating it 

gives the following:

(A13)

( )( ) ( )

( )( ){ ( ) ( ))

[( ( ))( ) ( )( ( ) ( ))]

2 2 2 2

2 2 2 2 2

2

2 2 2 2 2

E dH E dL E m E I d H

E E dL E m E dH I m H I d H

H

E dH

I d H E m E E E dL I m H I d H

f c

f c c

dc

d

f c f f c c

p p p p

p p p p p

p

p p p p p

− − − ′

− − − ′ − ′

− − − + − −

Rearranging terms:

(A14)

( )( ) ( )

( )( ) ( )

[( ( ))( ) ( )( ( ) ( ))]

2 2 2 2

2 2 2 2

2

2 2 2 2 2

E dH E E m E dL I d H

E E dL E m E dH I m H

H

E dH

I d H E m E E E dL I m H I d H

f c

f c

dc

d

f c f f c c

p p p p

p p p p

p

p p p p p

− − − ′

− − − ′

− − − + − −

Each line in (A14) is clearly negative because of the assumptions on the profit relations. 

Q.E.D.
34

Proof of Proposition 3. From equation (14?), it is easy to see that to show that the threat 

of entry increases the incumbent’s incentive to reduce costs it is enough to show that :

(A15) ( ) 0, [ , ˆ ( )]

( , ˆ ( ))

( , ) ( )

2

2

2 2

a c c c a

a c a f

a f

da

d a c

I m E E E

I d E

I d E I m − ′ > ∀ ∈

−

−

p

p

p p

When the incumbent is a monopolist or a duopolist competing with the highest cost 

entrant, its gross profits will exceed its fixed operating cost, f. Moreover, the monopoly 

profit will always exceed the duopoly profit. Thus, if the following is positive then (A15) 

will be positive:

(A16) ( ) 0, [ , ˆ ( )]

( , ˆ ( ))

( , ) ( )

2

2

2 2

a c c c a

a c a

a

da

d a c

I m E E E

I d E

I d E I m −p ′ > ∀ ∈

p

p p

Now I use the fact that duopoly profits are determined by a one-shot Cournot 

quantity setting game. Without loss of generality, I will say that the cost reducing 

investment is measured in the amount by which costs are reduced. Hence, the 

incumbent’s costs in period 2 are c-a. Thus, the left hand side of (A16) can be written as 

follows:

(A17) Im

Im Im

( ( ))

( ( ))

3 ( )

(4 ( 2 ) )

q

q p c a

q p c a

p q q p

q p q q p

Id Id Ed Id Id Id

Id Id Id Ed Id −

− −

− −

′ + + ′′

′ + + ′′

The arguments are suppressed in the above expression. All price terms and their 

derivatives are functions of the total quantity produced. Expression (A17) is positive if 

and only if the following inequality holds:

(A18) 1

( ( ))

( ( )) )

3 ( )

( )

(1

Im >

− −

− −

′ + + ′

′ + ′′

+

p c a

p c a

p q q p

p q p

Id Id Ed Id Id

Id Ed Id

The second fraction on the left hand side is greater than one since prices in a monopoly 

are higher than in a duopoly. Thus, (A18) will hold if 0

3 ( )

( )

>

′ + + ′

′ + ′′

Id Id Ed Id

Id Ed Id

p q q p

p q p

. But 

this is always positive since the numerator and denominator are both negative by the 

assumption that the reaction functions are downward sloping.

To show that if exit is not possible the threat of entry can either increase or 

decrease the incumbent’s incentive for cost reduction, under Cournot competition, 

consider a linear inverse demand function an a uniform distribution of entrant types. 

That is, let p q z z q 1 2

( ) = − and let ~ ( , ) cE U cEL cEH . The assumption of Cournout 

competition with these functional forms gives the following explicit forms for the profit 

functions:

(A19)

2

2

1

2

2

2

1

2

2

2

1

2

2

2

1

2

4

( )

; ( )

9

( (2 ( ))) ( , )

4

( ( ))

; ( )

9

( (2( ) )) ( , )

z

z c

c

z

z c c a

a c

z

z c a

c

z

z c a c

a c

E

E m E

E

E d E

I m E

E

I d E

−

=

− − −

=

− −

=

− − −

=

p p

p p

Using these profit functions and solving (11) for cˆ (a) En gives the following:

(A20) ( 3 )

2

1

ˆ ( ) 1 2

c a c a z f z En = − + − E

Using (A19) and (A20), I can write the last two lines of (15?), the added marginal benefit 

of cost reduction due to the threat of entry, as follows:
35

(A21)

72 ( )

{27 2( ( ) 2 )( 5( ) 4 )}

2

2 1 1

EH EL

E EL EL

z c c

f z z c a c z c a c

−

b + + − − − − +

Expression (A21) is positive if and only if:

(A22)

}

2

15 {5 7 2 3 ( )

5

1

}

2

15 {5 7 2 3 ( )

5

1

2 2

1 1

2 2

1 1

f z

c c z z c

a

f z

c c z z c

E

EL EL

E

EL EL

< − + + − +

− + − − + <

Thus, when exit is not possible, with these functional forms, the threat of entry will 

decrease the incumbent’s incentive to reduce costs when the optimal amount of cost 

reduction is either very large or very small. This will more likely be the case when the 

first period loss of profit from cost reduction is very small or very large. For example, 

when the parameters take on the following values:

3.5, 0, 7, 5, 1, 1, .95 c = cEL = cEH = z1 = z2 = fE = b =

the threat of entry reduces the incumbent’s incentive to reduce costs when the first period 

loss from cost reduction is 2− a , but increases the incumbent’s incentive to reduce costs 

when this loss is given by 2− .4a . Q.E.D.
36

REFERENCES

Albaek, Svend and Overgaard, Per Baltzer. “Advertising and Pricing to Deter or 

Accommodate Entry when Demand is Unknown: Comment.” The International Journal 

of Industrial Organization, March 1994, 12(1), pp.83-87.

Arvan, Lanny. “Sunk Capacity Costs, Long-Run Fixed Costs, and Entry Deterrence 

under Complete and Incomplete Information.” Rand Journal of Economics, Spring 1986, 

17(1), pp.105-21.

Bagwell, Kyle. “A Model of Competitive Limit Pricing.” Journal of Economics and 

Management Strategy, Winter 1992, 1(4), pp. 585-606. 

_____ and Ramey, Garey. “Advertising and Limit Pricing.” Rand Journal of 

Economics, Spring 1988, 19(1), pp.59-71.

_____. “Advertising and Pricing to Deter or Accommodate Entry when Demand is 

Unknown.” The International Journal of Industrial Organization, March 1990, 8(1), pp. 

93-113.

_____. “Oligopoly Limit Pricing.” Rand Journal of Economics, Summer 1991, 22(2), 

pp.155-72

_____. “Capacity, Entry, and Forward Induction.” Rand Journal of Economics, Winter 

1996, 27(4), pp.660-80.

Benoit, Jean-Pierre and Krishna, Vijay. “Entry Deterrence and Dynamic Competition.” 

The International Journal of Industrial Organization, 1991, 9(4), pp.477-95.

Bulow, Jeremy; Geanakoplos, John and Klemperer, Paul. “Holding Idle Capacity to 

Deter Entry.” The Economic Journal, March 1985, 95, pp.178-82.

Dixit, Avinash. “The Role of Investment in Entry-Deterrence.” The Economic Journal, 

March 1980, 90, pp.95-106.

Eaton, B. Curtis and Lindsey, Richard G. “Exit Barriers are Entry Barriers: The 

Durability of Capital as a Barrier to Entry.” The Bell Journal of Economics, Autumn 

1980, 11(2), pp.721-29.

The Economist, “The New Enforcers.” October 7, 2000. 

http://www.economist.com/business/displayStory.cfm?Story_ID=387757

Harrington, Joseph E. Jr. “Oligopolistic Entry Deterence under Incomplete Information.” 

Rand Journal of Economics, Summer 1987, 18(2), pp.211-31
37

Judd, Kenneth J. “Credible Spatial Preemption.” Rand Journal of Economics, Summer 

1985, 16(2), pp.153-166.

LeBlanc, Greg. “Signalling Strength: Limit Pricing and Predatory Pricing.” Rand 

Journal of Economics, Winter 1992, 23(4), pp.493-506.

Martin, Stephen. “Oligopoly Limit Pricing: Strategic Substitutes and Strategic 

Complements.” The International Journal of Industrial Organization, March 1995, 

13(1), pp.41-65.

Maskin, Eric S. “Uncertainty and Entry Deterrence.” Economic Theory, August 1999, 

14(2), pp.429-37.

Matthews, Steven A. and Mirman, Leonard J. “Equilibrium Limit Pricing: The Effects 

of Private Information and Stochastic Demand.” Econometrica, July 1983, 51(4), pp. 

981-96.

Milgrom, Paul and Robers, John. “Limit Pricing and Entry under Incomplete 

Information: An Equilibrium Analysis.” Econometrica, March 1982, 50(2), pp.443-59.

Nalebuff, Barry. “Credible Pretrial Negotiation.” Rand Journal of Economics, Summer 

1987, 18(2), pp. 198-210.

Salonen, Hannu. “Entry Detrrence and Limit Pricing under Asymmetric Information 

about Common Costs.” Games and Economic Behavior, March 1994, 6(2), pp.312-27.

San Jose Mercury News. “American Begins Notifying Workers Which Jobs Will Go.” 

June 24, 1993.

Schmalensee, Richard. “Advertising and Entry Deterrence: An Exploratory Model.” 

Journal of Political Economy, August 1983, 91(4), pp.636-53.

Spence, Michael. “Entry, Investment, and Oligopolistic Pricing.” Bell Journal of 

Economics, Autumn 1977, 8(2), pp.534-44.

Waldman, Michael. “The Simple Case of Entry Deterrence Reconsidered.” UCLA 

Working Paper No. 517, September 1988.

Washington Post. “The Southwest Revolution.” September 12, 1993.