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Federal Reserve Bank of Chicago

Duopoly Dynamics with a Barrier
to Entry
Jaap H. Abbring and Jeffrey R. Campbell

REVISED April, 2007
WP 2006-29

Duopoly Dynamics with a Barrier to Entry∗
Jaap H. Abbring† Jeffrey R. Campbell‡
April, 2007

Abstract
This paper considers the effects of raising the cost of entry for a potential competitor
on infinite-horizon Markov-perfect duopoly dynamics with ongoing demand uncertainty.
All entrants serving the model industry incur sunk costs, and exit avoids future fixed
costs. We focus on the unique equilibrium with last-in first-out expectations: A firm
never exits leaving behind an active younger rival. We prove that raising a second
producer’s sunk entry cost in an industry that supports at most two firms reduces the
probability of having a duopoly but increases the probability that some firm will serve
the industry. Numerical experiments indicate that a barrier to entry’s quantitative
relevance depends on demand shocks’ serial correlation. If they are not very persistent,
the direct entry-deterring effect of a barrier to a second firm’s entry greatly reduces
the average number of active firms. The indirect entry-encouraging effect does little
to offset this. With highly persistent demand shocks, the direct effect is small and the
barrier to entry has no substantial effect on the number of competitors. This confirms
Carlton’s (2004) assertion that the effects of a barrier depend crucially on industry
dynamics that two-stage “short run/long run” models capture poorly.

∗

We are grateful to Allan Collard-Wexler and Meredith Crowley for their insightful comments. The
National Science Foundation supported this research through Grant 0137042 to the NBER, and De Jonge
Akademie of the Royal Netherlands Academy of Arts and Sciences supported this research through a travel
grant. A replication file is available at http://www.nber.org/˜jrc/lifo
†
VU University Amsterdam and Tinbergen Institute. E-mail:jabbring@feweb.vu.nl
‡
Federal Reserve Bank of Chicago and NBER. E-mail: jcampbell@frbchi.org
JEL Codes L13, L41
Keywords: LIFO, FIFO, Sunk costs, Markov-perfect equilibrium, Competition policy

1

Introduction

This paper determines the effects of raising a late entrant’s sunk cost in a duopoly version of
the industry dynamics model presented in a companion paper (Abbring and Campbell, 2006).
Demand evolves stochastically; entry possibly requires paying a sunk cost; and continued
operation incurs fixed costs. The wish to avoid these per-period fixed costs in markets that
are no longer profitable motivates firms to exit. We assume that all participants rationally
expect exit decisions to follow a last-in first-out pattern. That is, no firm produces after the
exit of an older competitor. Two considerations motivate this focus. First, it embodies (in an
extreme way) the widespread observation that young firms exit more frequently than their
older counterparts. Second, the last-in first-out restriction favors incumbents; so we expect
them to deter entry and promote exit by shifting equilibrium expectations in this direction
with any available means.
Abbring and Campbell (2006) prove that there exists an essentially unique Markov-perfect
equilibrium with last-in first-out expectations, and they provide a simple procedure for its
calculation. These results allow us to compare duopoly dynamics with different sunk costs of
entry. We first provide an analytic characterization of the effects of barriers to entry. Raising
a second entrant’s sunk cost reduces the probability of two firms operating, but increases
the probability that some firm will serve the industry. The first effect is the expected entry
deterrence. The second reflects the positive influence of barriers to entry on the value of
being a monopolist, and it is familiar from the static analysis of patents and innovation. We
also show that raising the second entrant’s sunk costs reduces average entry and exit rates.
In this specific sense, barriers to entry stabilize industry structure.
These qualitative results say nothing about a barrier to entry’s quantitative importance,
so we complement them with a wide range of numerical examples. We find that when demand
shocks are not very persistent, a barrier to entry substantially lowers the average number of
firms serving the industry. The entry-encouraging effect is small and does little to offset the
direct deterrent effect. Raising the persistence of demand dramatically changes this. For a
wide range of specifications with persistent shocks, we find that a sizeable barrier to a second
firm’s entry has only trivial effects on the number of active firms. The entry-encouraging
effect remains quantitatively unimportant and does not explain this result. Rather, highly
persistent demand makes the direct deterrent effect small. We do not explore the welfare
implications of a barrier to entry in detail. However, our numerical results suggest that even
a large barrier to entry entails little welfare loss in the empirically-relevant case with highly
persistent demand shocks.
1

The term “barrier to entry” has a rich and confusing history. McAfee, Mialon, and
Williams (2004) review the many definitions proposed for it and conclude that none of them
are useful for antitrust analysis. They offer a new definition:
An economic barrier to entry is a cost that must be incurred by a new entrant
and that incumbents do not or have not had to incur.
The barrier to entry we consider satisfies this. Carlton (2004) asserts that the effects of
such a barrier depend crucially on the industry’s medium-run dynamics, so two-stage “short
run/long run” models capture them poorly. This paper’s results confirm and refine this point:
The positive effects of increasing a second entrant’s sunk costs substantially depend on the
stochastic rule governing the evolution of demand.
Most of this paper’s analysis assumes a last-in first-out (LIFO) pattern for entry and exit.
In the companion paper, we demonstrate that the Markov-perfect equilibrium satisfying this
restriction can be viewed as an approximation for a model with a learning curve. In the
learning-curve model, firms’ fixed costs decline as they age and no firm exits while leaving
behind a higher cost rival. Cabral (1993) uses this latter restriction to select a “natural”
equilibrium. This approximation result indicates that this paper’s analysis most naturally
applies to industries with technologies that favor experienced firms.
Models of Schumpetarian economic growth assume that technology favors younger firms,
so this situation also merits analysis. For this, we examine a version of our model in which the
younger firm’s continuation decision comes first. We prove the existence of an equilibrium
with first-in first-out expectations, and this is the unique equilibrium in which a higher
demand state never yields fewer active firms. In this sense, the equilibrium is also “natural”.
The qualitative effects of raising a barrier to entry in this environment are identical to those
from the case with LIFO expectations. Unfortunately, the FIFO analysis requires demand
to follow a particular stochastic process.
The remainder of this paper proceeds as follows. The next section presents the model’s
primitives and derives the effects of barriers to entry for a specific example with a penciland-paper equilibrium solution. Section 3 presents the analysis of the general model, and
Section 4 examines the model’s FIFO version. Section 5 explains the relationship between
our work and existing papers on entry deterrence, competitive industry dynamics, and the
computational analysis of competition policy. Section 6 contains some concluding remarks.

2

2

The Model

The model consists of a single duopolistic market in discrete time t ∈ {0, 1, . . .}. There is a
large number of firms that are potentially active in the market. We index these firms with N.
At time 0, N0 = 0 firms are active. Entry and subsequent exit of firms determines the number
of active firms Nt in each later period. At most two firms can be active in the industry at
any given time. The number of consumers in the market, Ct , evolves stochastically according
to a first-order Markov process on the interval [Ĉ, Č].
The sequence of events and actions within a period begins with the inherited values of Nt
and Ct−1 . First, all participants observe the realization of Ct . Then, all active firms receive
profits equal to (Ct /Nt ) × π(Nt ) − κ. Here, each firm serves Ct /Nt customers, and π(Nt ) is
the producer surplus earned from each one. We assume that π(1) > π(2)/2, so that entry
reduces an incumbent firm’s profits. The term κ > 0 represents fixed costs of production.
After serving the market, active firms decide whether they will remain so. These decisions
are sequential and begin with the oldest firm. In particular, if two firms are currently active
in the market (Nt = 2) the older duopolist will decide first on continuation and the younger
duopolist follows. For future reference, we introduce notation for the firm’s rank in this
sequence of exit decisions: If firm i is active in period t, then Rti = 1 if it is either a
monopolist or the older duopolist; and Rti = 2 if it is the younger duopolist. Exit is costless
but irreversible and allows the firm to avoid future periods’ fixed production costs.
If the continuation decisions do not commit two firms to serving the market, then the firm
with the smallest name (i) that has also not yet had an opportunity to enter makes an entry
decision. The cost of entry potentially depends on the number of firms already committed
to serving the market in the next period. We denote the entry cost for a firm that would
be the oldest among next period’s active firms with ϕ(1), and the same cost for a firm that
would be the youngest next period equals ϕ(2). The payoff to staying out of the industry
is always zero, because a firm with an entry opportunity cannot defer the option. If firm i
chooses to enter an otherwise empty industry, then firm i + 1 chooses between staying out of
i+1
= 2. Otherwise, the period ends. We specified the entry
the industry and entering with Rt+1
technology in this way to capture “free entry” within a game-theoretic environment, which
requires us to name every player and specify their possible actions and payoffs.1 Both active
firms’ and potential entrants’ decisions maximize their expected stream of profits discounted
1

The restriction to markets with at most two firms can be formalized in the more general model of Abbring
and Campbell (2006) by either setting the entry cost ϕ(R0 ) for a firm with potential next period’s rank R0 ≥ 3
to a very large value or by setting the producer’s surplus π(N 0 ) in a market with N 0 ≥ 3 firms to 0.

3

with β < 1. We assume that ϕ(2) ≥ ϕ(1) ≥ 0. If ϕ(2) > ϕ(1), we say that a second
entrant faces a barrier to entry (relative to a first entrant). We do not model the actions
that incumbents or policy makers take to impose these costs.

2.1

Markov-perfect equilibrium

We choose as our equilibrium concept symmetric Markov-perfect equilibrium. When firm
i decides whether to stay or exit, Nt − Rti (the number of active firms following it in the
i
(its rank in the next period’s sequence of active firms) are available
sequence), Ct , and Rt+1
i
). Similarly, the payoffand payoff-relevant. Collect these into Hit ≡ (Nt − Rti , Ct , Rt+1
i
). Note that Hit htakesiits values in
relevant state to
entrant is Hit ≡ (Ct , Rt+1
h a potential
i

HS ≡ {0, 1} × Ĉ, Č × {1, 2} for firms active in period t and in HE ≡ Ĉ, Č × {1, 2} for
potential entrants. Here and below, we use S and E to denote survivors and entrants.
A Markov strategy for firm i is a pair (AiS (HS ), AiE (HE )) for each HS ∈ HS and HE ∈
HE . These represent the probability of being active in the next period given that the firm
is currently active (AiS (·)) and given that the firm has an entry opportunity (AiE (·)). A
symmetric Markov-perfect equilibrium is a subgame-perfect equilibrium in which all firms
follow the same Markov strategy.
When firms use Markov strategies, the payoff-relevant state variables determine an active
firm’s expected discounted profits at any node of the game tree. We denote these at the
moment of the continuation decision with v(HS ). In a Markov-perfect equilibrium, this
function satisfies the Bellman equation
 0

C
0
0
v(HS ) = max aβE
π(N ) − κ + v(HS ) HS .
(1)
a∈[0,1]
N0

Here and throughout, we adopt conventional notation and denote the variable corresponding
to X in the next period with X 0 . In Equation (1), the expectation of N 0 is calculated using
all firms’ strategies conditional on the particular firm of interest choosing to be active.
Although firms make their continuation and entry decisions sequentially, the game’s infinite horizon removes the standard argument for uniqueness of subgame perfect equilibrium.
Abbring and Campbell (2006) show that uniqueness of equilibrium can be restored by focusing
only on Markov-perfect symmetric equilibria in which an older firm’s first-mover advantage
translates into longevity. By definition, firms in such an equilibrium follow a common last-in
first-out (LIFO) strategy.
Definition 1. A LIFO strategy is a pair of functions (AS , AE ) such that AS (HS ) ∈ {0, 1},
AE (HE ) ∈ {0, 1}, and AS (1, C, R0 ) ≥ AS (0, C, R0 ).
4

If all firms follow a common LIFO strategy and the market is served by two firms, then the
youngest of these two firms will decide to exit whenever the oldest does:
AS (1, C, 1) = 0 ⇒ AS (0, C, 1) = 0.
Constructing a Markov-perfect equilibrium in a LIFO strategy is straightforward. Consider the decisions of a firm entering with one firm already committed to produce in the next
period, that is with (potential) rank 2. This firm will produce no longer than the older firm
and will not experience further entry of competitors during its time in the market, so the
Bellman equation governing v(·, ·, 2) corresponds to a familiar non-strategic exit decision.
This Bellman equation obviously has a unique fixed point, and we choose its associated optimal decision rule to form AS (0, C, 2) and AE (C, 2). If the firm is indifferent between activity
and inactivity at some value of C, we choose the rule that defaults to inactivity. With this
complete, we can consider the problem of a firm with rank 1. Maximizing its profit requires
only expectations about the entry and exit of a firm with rank 2, which are already in hand.
Thus, standard dynamic programming arguments yield the unique value function satisfying
the Bellman equation. With this, we can set AS (0, C, 1), AS (1, C, 1), and AE (C, 1) to the
unique optimal decision rule that defaults to inactivity.
The construction of this equilibrium suggests that it is the only one in a LIFO strategy
that defaults to inactivity. We present a formal proof of this in Abbring and Campbell (2006,
Proposition 2). As we mentioned in this paper’s introduction, we find the restriction to
LIFO strategies natural for the problem at hand, because it embodies in an extreme form the
empirical regularity that young firms exit more frequently than their older counterparts and
because these expectations make entry less profitable by placing incumbents in a privileged
position. It is also powerful enough to select an essentially unique Markov-perfect equilibrium,
so we impose it on our analysis.

2.2

A Pencil-and-Paper Example

If we assume that Ct = Ct−1 with probability 1 − λ and that it equals a draw from a
uniform distribution on [Ĉ, Č] with the complementary probability, then we can calculate
the model’s equilibrium value functions and decision rules with pencil and paper. Before
proceeding, we examine this special case to illustrate the model’s moving parts. To ensure
that the equilibrium dynamics are not trivial, we assume that no firm will serve the industry

5

if demand is low enough and that two firms will serve the industry if it is sufficiently high.2
To begin, consider an incumbent firm with rank 2. In an equilibrium in a LIFO strategy,
its profit equals (C/2)π(2) − κ. It will earn this until the next time that Ct changes, at which
point demand will be statistically independent of its current value. These facts imply that
this firm’s value function is piecewise linear in C.
(
0
if C ≤ C 2 ,
v(0, C, 2) =
(1−λ)( C
π(2)−κ
+λe
v
(0,2)
)
2
β
if C > C 2 ,
1−β(1−λ)
where
1
ve(0, 2) =
2

Ĉ + Č
2

!

Z

Č

π(2) − κ +
Ĉ

v(0, C 0 , 2)
(Č − Ĉ)

dC 0 .

Here, ve(0, 2) is the firm’s average continuation value given a new draw of Ct and C 2 is the
largest value of C that satisfies v(0, C, 2) = 0. Optimality requires the firm to exit if C < C 2 .
This value function is monotonic in C, so there is a unique entry threshold C 2 that equates
the continuation value with the entry cost. Thus, a second duopolist enters whenever Ct
exceeds C 2 and exits if it subsequently falls at or below C 2 .
Next, consider the problem of an incumbent with rank 1. If this firm is currently a
monopolist, it expects to remain so until Ct > C 2 ; and if it is currently a duopolist, it
expects to become a monopolist when Ct falls below C 2 . This firm’s value function is also
piecewise linear. If the firm begins the period as the sole incumbent, it is


if C ≤ C 1

 0 (1−λ)(Cπ(1)−κ)+λev(0,1)
β
if C 1 < C ≤ C 2 ,
v(0, C, 1) =
1−β(1−λ)

C

 β (1−λ)( 2 π(2)−κ)+λev(1,1) if C > C ;
2

1−β(1−λ)

and if it begins as one of two incumbents it



 0 (1−λ)(Cπ(1)−κ)+λev(0,1)
β
v(1, C, 1) =
1−β(1−λ)

C

π(2)−κ)+λe
v (1,1)
(1−λ)
(
 β
2
1−β(1−λ)

equals
if C ≤ C 1 ,
if C 1 < C ≤ C 2 ,
if C > C 2 .

The exit threshold C 1 is the greatest value of C such that v(0, C, 1) = 0, and the average
continuation values following a change in Ct for a monopolist and a duopolist are
h
i
Ĉ+Č
π(1)−κ
< 0 and
Sufficient conditions for these two properties are (1 − λ) Ĉπ(1) − κ + λ 2 1−β



Č
λ
1 Č+Ĉ
(1−λ)( 2 π(2)−κ)+ 1−β 2 2 π(2)−κ
β
> ϕ(2).
1−β(1−λ)
2

6

Figure 1: Equilibrium in the Pencil-and-Paper Example

Ĉ + Č
2

ve(0, 1) =
ve(1, 1) =

1
2

!

Ĉ + Č
2

Z

Č

π(1) − κ +

v(0, C 0 , 1)
(Č − Ĉ)

Ĉ

!

Z

Č

π(2) − κ +

dC 0 ,

v(1, C 0 , 1)
(Č − Ĉ)

Ĉ

dC 0 .

This value function does not always increase with C, because slightly raising C from C 2
induces entry by the second firm and causes both current profits and the continuation value
to discretely drop. Nevertheless we know that they drop to a value above ϕ(1), because at
this point the second firm chooses to enter. Hence, it is still possible to find a unique entry
threshold C 1 that equates the value of entering with rank 1 to the cost of doing so.
Figure 1 visually represents the equilibrium. In each panel, C runs along the horizontal
axis. The vertical axis gives the value of a firm at the time that entry and exit decisions
are made. The top panel plots the value of a firm with rank 1, while the bottom plots the
7

Figure 2: Effects of Raising ϕ(2) in the Pencil-and-Paper Example
value for a competitor with rank 2. Both panels use the same vertical scale. The value of
a duopolist with rank 2 equals zero for C < C 2 , and thereafter increases linearly with C.
The entry threshold C 2 equates this value with ϕ(2). The value of an older firm with rank
1 has two branches. The upper monopoly branch gives the value of a monopolist expecting
no further entry. If C increases above C 2 and thus induces entry, the firm’s value drops to
the lower duopoly branch. This has the same slope as the value function in the lower panel.
Its intercept is higher, because the incumbent expects eventually to become a monopolist
the first time that C passes below C 2 . When this occurs, the firm’s value returns to the
monopoly branch. The entry and exit thresholds for this firm occur where the monopoly
branch intersects ϕ(1) and 0.
Now turn to the effects of increasing ϕ(2) on this duopoly’s dynamics. Figure 2 illustrates
this exercise. If a second firm actually enters the market, then the entry cost has no impact
on that firm’s optimal exit decision or value function. The value function in the bottom
panel is monotonic in C, so raising ϕ(2) directly increases C 2 . In this sense, raising ϕ(2)

8

directly deters entry. The impact of this on an incumbent’s value is clear. Both branches of
the value function shift up, from the grey to the solid black, and this lowers both C 1 and
C 1 . Intuitively, these changes lower the probability that either zero or two firms will serve
the industry.

3

Barriers to Entry

Incumbents’ attempts to deter contestants’ entry occupy a central place in dynamic industrial
organization. In this section, we characterize the effects of raising ϕ(2) for given ϕ(1). Actions
available to a real-world incumbent for increasing ϕ(2) include bidding-up the price of fixed
inputs, litigation, and purchases of legal entry restrictions. We do not explicitly model the
incumbent’s choice of ϕ(2). Instead, we examine the impact of exogenously varying it for
observable aspects of industry dynamics. The relative simplicity of our model allows us to
characterize this experiment analytically. We complement these qualitative results with a
numerical analysis of several model parameterizations.

3.1

Qualitative Characterization of a Barrier’s Impact

Suppose that two firms serve the market in equilibrium with positive probability. Consider
the graphical results from Figure 2. These clearly depend on the fact that entry and exit
thresholds characterize the equilibrium strategy. This property seems both natural and
intuitively desirable. However, it does not generally hold good. In Abbring and Campbell
(2006), we calculate the equilibrium for an example in which, given two initial values CL <
CH , the distribution of Ct+1 given Ct = CH stochastically dominates the same distribution
given Ct = CL . Although this condition guarantees that thresholds give a monopolist’s entry
and exit decisions in a market that supports at most one firm, the equilibrium entry rule for
a firm entering an empty industry that can hold two firms has no threshold representation.
That is, there is a “hole” in the set of Ct for which entry into an otherwise empty industry is
rational. This difficulty obviously reflects the non-monotonic value function: Increasing Ct
can make future entry likely enough to reverse a firm’s current entry decision. This difficulty
does not manifest itself in the pencil-and-paper example, because there increasing Ct has no
impact on the likelihood of future entry.
In Abbring and Campbell (2006), we address this issue with a condition on the stochastic
process for Ct (weaker than that from the pencil-and-paper example) which guarantees a
threshold representation for the equilibrium strategy. It ensures that increasing C cannot
9

move a “substantial” probability mass over another firm’s entry threshold. Rather than
repeat that analysis here, we make the following high-level assumption and refer the reader
to the companion paper for a more primitive analysis.
Assumption 1. Given the stochastic process for Ct , all other model parameters, and any
value of ϕ(2) ≥ ϕ(1), there exist four thresholds C 1 ≤ C 1 and C 2 ≤ C 2 such that AE (C, 1) =
I{C > C 1 }, AE (C, 2) = I{C > C 2 }, AS (0, C, 1) = AS (1, C, 1) = I{C > C 1 }, and
AS (0, C, 2) = I{C > C 2 }.
With this assumption in place, the graphical results from Figure 2 generalize immediately.3
Proposition 1. Increasing ϕ(2) weakly increases C 2 , leaves C 2 unchanged, and weakly decreases both C 1 and C 1 .
The inequalities here and below are weak only because the threshold entry and exit rules can
be trivial, prescribing activity or inactivity for all C.
Changes in the thresholds directly affect the evolution of the equilibrium number of firms.
To see how, suppose that ϕ(2) increases unexpectedly and permanently when the industry
begins with a particular value of C and no active producers. Entry of at least one firm occurs
if and only if C > C 1 . So, increasing a contestant’s entry cost increases the probability that
some firm will service the industry immediately. Suppose instead that the industry begins
with a single active incumbent. Because C 1 decreases and C 2 increases, any value of C that
induced neither entry nor exit will continue to do so. Hence, the probability of the industry
remaining with exactly one firm increases. Finally, an industry that begins with two firms
will remain unchanged in exactly the same conditions as before. The following proposition
shows that this intuition extends to the probability for a change in the number of firms that
conditions on the given initial values of C and N .
Proposition 2. Let Sj (C, N ) ≡ Pr{Nt+1 = · · · = Nt+j = N |Ct−1 = C, Nt = N } denote the
probability that the number of firms remains unchanged over j periods given the initial numbers of consumers and firms. Then, raising ϕ(2) weakly decreases Sj (C, 0), weakly increases
Sj (C, 1), and leaves Sj (C, 2) unchanged.
An immediate implication of Proposition 2 is that imposing a barrier to entry makes the
structure of an active industry more stable in the short and medium run. We now turn to
consider the effects of the entry barrier on that structure itself. The impact of ϕ(2) on the
3

Appendix A contains the proofs of this section’s propositions.

10

equilibrium entry and exit thresholds strongly suggests that the probabilities of having 0 or
2 firms declines with ϕ(2). This is indeed the case
Proposition 3. Let Pj (N 0 , C, N ) ≡ Pr{Nt+j = N 0 |Ct−1 = C, Nt = N }. Then raising ϕ(2)
weakly lowers Pj (0, C, N ) and Pj (2, C, N ).
This proposition’s content is straightforward: Imposing a barrier to a contestant’s entry increases the probability that the industry is serviced at the expense of decreasing the
probability of a “competitive” industry structure. The literature on entry barriers does not
ordinarily associate them with increasing the number of firms, but this effect is familiar from
the analysis of patents: Increasing the cost of a imitator’s entry encourages entry by an
innovator. The model of this paper differs from one with patents in one substantial respect:
The sunk cost of entry is weakly higher for a second entrant instead of lower. Nevertheless,
this difference does not change the qualitative effects of a barrier to entry on the incentive
for “innovation” (broadly construed).
Proposition 3 characterizes the number of firms at any given date in the future. In this
sense, it characterizes the short- and medium-run responses to creating a barrier to entry.
The following corollary extends its conclusions to the industry’s ergodic distribution, if it
exists.
Corollary 1. Suppose that (Ct−1 , Nt ) has a unique ergodic distribution and let P (N ) denote
the probability of N firms serving the industry in this distribution. Then raising ϕ(2) weakly
decreases P (0) and P (2).
The omitted proof is an simple application of the bounded convergence theorem. By definition, the distribution of industry outcomes converges to the ergodic distribution as the time
horizon grows large. However, this long run differs substantially from the two-stage analysis
of industry structure which goes by the same label; because it gives a distribution of industry
outcomes given ongoing uncertainty and sunk costs rather than a single outcome at the end
of time.
We conclude this analytic characterization of the model’s dynamics by considering the
most familiar observable indicators of an industry’s propensity to change its structure, its
entry and exit rates. We define the net growth rate of the number of firms in the industry
with Gt ≡ 2 × (Nt − Nt−1 )/(Nt + Nt−1 ). With this, the industry’s entry and exit rates are
ERt ≡ min{0, Gt } and XRt ≡ min{0, −Gt }. Because these are only well defined if either
Nt or Nt−1 is positive, we assume that P (0) = 0. With this, the following proposition shows
that raising ϕ(2) has the expected effect of reducing average entry and exit rates.
11

Proposition 4. Suppose that (Ct−1 , Nt ) has a unique ergodic distribution with P (0) = 0.
Then increasing ϕ(2) weakly decreases E[ERt ] = E[XRt ], where the expectations are calculated using the ergodic distribution for (Ct−1 , Nt ).
The proof relies on the fact that raising C 2 reduces P (2) and thereby all future dates’ exit
rates.
This completes the analytic characterization of the equilibrium impact of erecting barriers to a duopolist’s entry. Analytic results from an infinite-horizon Markov-perfect equilibrium come infrequently, so their mere existence is somewhat novel. Nevertheless, Judd
(1997) cautions against focusing only on surprising qualitative results without examining
their quantitative importance. For this, we turn to a numerical exploration.

3.2

Quantitative Assessment of a Barrier’s Impact

Calculation of the effects of barriers to entry requires us to assign values to the model’s
parameters. Estimating or calibrating them necessitates a much richer exploration of a particular industry’s institutional background and technological constraints than is appropriate
for this paper. Instead, we follow the strategy advocated by Judd (1997) and calculate the
model’s equilibrium dynamics at a wide range of parameter choices. This reveals which
features of the model amplify or mute the above qualitative results.
3.2.1

Model Specification

Calculation of the model’s equilibrium uses values for β, κ, π(1), π(2), ϕ(1), ϕ(2), and
the parameters of the stochastic process governing C. We suppose that a model period
corresponds to five years and that the real interest rate is 5%, so we set β = 1.05−5 for
all of the model’s parameterizations. We set the support of ln C to a grid of 301 evenly
spaced points on [−1.5, 1.5] and we choose the stochastic process to approximate the linear
autoregression ln C 0 = ρ ln C + u, where u has a normal distribution with mean zero and
variance σ 2 . The approximation is a mixture over 51 conditional distributions for ln C 0 with
a uniform distribution with support width σk and and conditional mean


 ln Ĉ + σk /2 if ρ ln C < ln Ĉ + σk /2
µk (C) =
ρ ln C
if ln Ĉ + σk /2 ≤ ρ ln C ≤ ln Č − σk /2


ln Č − σk /2 if ρ ln C > ln Č − σk /2.

12

Parameter
β
ρ
σ
κ
ϕ
π

Possible Values
1.05−5
0.80, 0.90, 1.00
0.10, 0.20, 0.30
1
0, 2/5, 4/5
2

Table 1: Baseline Parameter Values for Numerical Experiments

This paper’s replication file contains the details of their specification. Our calculations use
three different values of ρ— 0.80, 0.90, and 1.0— and three values of σ— 0.10, 0.20, and
0.30. Only the specification with ρ = 1 satisfies Abbring and Campbell’s (2006) condition
for threshold equilibrium entry and exit strategies, so our computational procedure does not
require firms to employ such threshold rules.
We normalize the per-period fixed cost κ to one. To begin, we suppose that the sunk
costs of entry in an market that is not yet served and in a monopolistic market are the same,
so that ϕ(1) = ϕ(2) = ϕ. We set ϕ to either 0, 2/5, or 4/5.4 Because our model period is five
years, these sunk costs correspond to 0, 2 and 4 years of fixed costs, respectively. Conversely,
their five-year rental equivalent values ϕ(1 − β)/β are 0, 0.11, and 0.22.
Throughout, we assume that entry by a second producer has no effect on the producer’s
surplus per consumer served, π(1) = π(2) = π. This arises, for example, if producers are
monopolistic price setters serving equal market shares. Sutton (1991) refers to the slope of
π(·) as the “toughness” of competition. In this terminology, our specification exhibits low
toughness of competition.5 We set π = 2, so that the duopoly producer’s surplus when
C = 1 equals κ. For reference, Table 1 reports all of the possible values we give to the
model’s parameters in the baseline experiments without barriers to entry.
4

The case with ϕ(N ) = 0 is not equivalent to the infinite repetition of the static free-entry game, because
a firm’s operation in an earlier period gives it “priority” for serving the industry later.
5
We have experimented with an alternative specification in which duopolists compete in quantities. This
example of Cournot duopoly gives π(2) = π(1)/2 and thus exhibits high toughness of competition relative to
our baseline specification. We do not report numerical results for this specification, because they are very
close to the results for our baseline specification, given appropriate normalizations of the level of demand.

13

6.45

0.80
−1.12

−0.79

Figure 3: Firms’ Value Functions in the Leading Example

3.2.2

A Specific Example

It is helpful to have a specific example in mind when considering the results of all of the
experiments. For this, we set ρ = 0.9, σ = 0.2, ϕ = 4/5, and π = 2. Figure 3 plots the value
functions of the two firms that might serve this industry. Its bottom panel displays the value
of a firm that has entered as a duopolist. This weakly increases with demand. The value of
continuation exceeds the cost of entry if ln Ct > 0.27, and it equals zero if ln Ct ≤ −0.10.
Figure 3’s top panel plots the value of a firm that has entered as a monopolist. This has
the same structure as the plot of the incumbent’s value function in Figure 2. The upper
branch corresponds to the value function when the industry has one firm, and the lower
branch gives the first firm’s value after the entry of a second firm. The arrows indicate that

14

the first firm’s value drops when the second firm enters and jumps after its exit. Unlike in
Figure 2, the monopoly and duopoly branches decrease near the second entrant’s entry and
exit threshold, respectively. This reflects the fact that here, unlike in the pencil-and-paper
example, a higher realization of Ct corresponds to a higher probability that a second firm is
active in the future. The stochastic process for ln Ct does not satisfy Abbring and Campbell’s
(2006) sufficient condition for threshold equilibrium entry and exit strategies. Nevertheless,
entry and exit thresholds, equal to −0.79 and −1.12, characterize this firm’s decisions.
To measure the impact of barriers to entry on equilibrium dynamics, we concentrate on the
number of firms’ average and standard deviation in the equilibrium’s ergodic distribution.
In this example, the average number of firms is 1.32 and the standard deviation is 0.47.
The barrier to entry we examine adds 4/5 to the sunk cost of a second firm’s entry. With
this modification, the second entrant’s entry threshold rises from 0.27 to 0.52. As Section
3.1’s results suggest, increasing the second entrant’s sunk costs encourages a first entrant.
However, even though the increase in these costs amount to a substantial four years of fixed
cost; the first entrant’s entry and exit thresholds drop very little. By construction, the second
entrant’s exit threshold remains unchanged. As a result of all this, the average and standard
deviation for the number of firms drop to 1.08 and 0.28. Thus, the intuition that raising a
barrier to entry stabilizes the number of firms holds good. Furthermore, the decrease in the
average number of firms conforms to the intuition from two-stage models that barriers to
entry increase concentration. In this particular example, imposing a barrier to entry hardly
changes the decision to enter or abandon an otherwise empty industry.
3.2.3

Experimental Results

Table 2 reports the average and standard deviation for Nt from the ergodic distribution for all
combinations of the parameter values we consider. The cost of entry influences the average
number of firms as expected. In all of the comparisons across parameterizations, raising ϕ
lowers the average number of firms. However, it lowers the number of firms much more if ρ
is low. For example, when σ = 0.1, raising ϕ from 0 to 4/5 decreases the number of firms
from 1.66 to 1.00 if ρ = 0.8. When ρ = 1.0, the same change lowers the average from 1.27 to
1.19.
A second notable pattern is that raising σ always increases the average number of firms.
This arises from the well-known effect of uncertainty on the real option embedded in a firm’s
value explained by Dixit and Pindyck (1994). Because firms can exit following an unprofitable
realization of Ct , uncertainty increases a firm’s value and in this way promotes entry.
15

ρ
0.80
0.90
1.00
0.80
0.90
1.00
0.80
0.90
1.00

σ
0.10
0.10
0.10
0.20
0.20
0.20
0.30
0.30
0.30

Average
Standard Deviation
ϕ = 0.0 ϕ = 0.4 ϕ = 0.8 ϕ = 0.0 ϕ = 0.4 ϕ = 0.8
1.66
1.04
1.00
0.47
0.19
0.00
1.65
1.30
1.04
0.48
0.46
0.21
1.27
1.22
1.19
0.83
0.84
0.85
1.70
1.35
1.07
0.46
0.48
0.26
1.66
1.48
1.32
0.47
0.50
0.47
1.33
1.27
1.23
0.80
0.82
0.83
1.72
1.50
1.28
0.45
0.50
0.45
1.66
1.54
1.44
0.47
0.50
0.50
1.39
1.32
1.28
0.77
0.80
0.81

Table 2: The Number of Firms in the Ergodic Distribution
The most notable feature of Table 2’s second panel is the relatively small impact of raising
ϕ on the standard deviation of Nt , unless both ρ and σ are small. In the four cases with the
lowest standard deviations, the industry is almost certainly a monopoly.6 The probability
of two firms serving the industry with all of the other specifications are substantial, but the
industry is likely to be inactive only if ρ = 1. These cases with substantial probabilities on
all three possible outcomes have the highest standard deviations.
Table 3 reports the changes in the number of firms’ average and standard deviation from
raising a barrier to entry. For each parameter combination, the cost of a second firm’s entry
was increased by 4/5 as in the leading example. In the reported experiments, raising a barrier
either lowers the average number of firms or leaves it unchanged. The largest decrease is 0.66
firms. In this case, the barrier completely deters a potential entrant. In the four cases in
which the market is unlikely to hold two active firms initially, raising ϕ(2) hardly changes
the average number of firms. A clear pattern emerges after eliminating these cases: Raising
a barrier to entry substantially lowers the average number of firms unless ρ = 1. With highly
persistent demand shocks, the average number of firms falls only trivially. The barrier’s effect
on the standard deviation also depends on the persistence of demand. Except in two cases,
the barrier decreases the standard deviation. This is the stabilization of market structure
that Proposition 3 leads us to expect. However, the standard deviation’s decline is always
trivial in the cases with ρ = 1.
To determine why the barrier to entry has such small effects with persistent demand
6

These cases are (ϕ, ρ, σ) = (2/5, 0.80, 0.10), (4/5, 0.80, 0.10),(4/5, 0.90, 0.10), and (4/5, 0.80, 0.20).

16

ρ
σ
0.80 0.10
0.90 0.10
1.00 0.10
0.80 0.20
0.90 0.20
1.00 0.20
0.80 0.30
0.90 0.30
1.00 0.30

Average
Standard Deviation
ϕ = 0.0 ϕ = 0.4 ϕ = 0.8 ϕ = 0.0 ϕ = 0.4 ϕ = 0.8
-0.66
-0.04
0.00
-0.47
-0.19
0.00
-0.61
-0.30
-0.04
-0.27
-0.42
-0.20
-0.04
-0.03
-0.03
-0.01
-0.01
-0.01
-0.62
-0.35
-0.07
-0.20
-0.42
-0.25
-0.34
-0.30
-0.23
-0.01
-0.11
-0.19
-0.05
-0.04
-0.03
-0.02
-0.01
-0.01
-0.44
-0.39
-0.26
0.00
-0.19
-0.30
-0.22
-0.20
-0.18
0.02
-0.02
-0.06
-0.05
-0.04
-0.03
-0.02
-0.02
-0.02

Table 3: Effects of Imposing a Barrier to Entry on the Number of Firms
shocks, we began with an inspection of the (unreported) changes in the ergodic distributions
and thresholds underlying Table 3’s results. We found that the barrier had a negligible entryencouraging effect with all parameterizations. It changes neither the first entrant’s thresholds
nor the probability of an inactive market substantially. The barrier’s direct deterrent effects
explain the decreases in the average number of firms. The absence of a substantial effect in the
cases with ρ = 1 corresponds to small deterrent effects, not to offsetting entry-encouraging
effects. Even though the youngest firm’s entry threshold increases, the resulting change in
the probability of an active duopoly is small in these cases. Consequently, the changes in the
number of firms’ standard deviations are also small.7
The case with very persistent demand shocks obviously has empirical relevance, so it is
worth confirming that the very small effect of a barrier to entry we document is not specific
to the particular parameterizations used to create Table 3. To show this, we recalculated the
results of the same experiment for a variety of alternative parameter values. Increasing the
barrier from 4/5 to 8/5 leads to somewhat more pronounced effects without changing the
7

The pencil-and-paper example provides a qualitative understanding of how the effects of a barrier to
entry depend on the persistence of demand. There, the ergodic distribution of (Ct−1 , Nt ) only depends on
the persistence parameter λ through the thresholds. If we increase persistence by decreasing λ, then the slope
of the younger duopolist’s value function directly increases. This makes C 2 less sensitive to changes in the
sunk cost of entry. In the general model, the persistence parameter ρ affects the ergodic distribution not only
indirectly through the thresholds, but also directly. The numerical experiments confirm the pencil-and-paper
example’s intuition that the thresholds are less sensitive to a barrier to entry if ρ = 1. They also show that
the direct effect of ρ explains some of the fall in sensitivity to a barrier to entry.

17

main conclusions. The results for the cases with ρ = 1 also continue to hold for larger values
of σ than those reported in Table 3. In the case with ϕ = 0 and σ as high as 0.60, raising a
barrier to entry entices a first entrant to lower its entry and exit thresholds so that it becomes
and remains active at all demand levels. Consequently, the market is somewhat more likely
to be served by at least one firm, and the average number of firms increases slightly (by 0.02)
in this case. This shows that the entry-encouraging effect exists and can lead to an increase
in the number of firms without changing our conclusion that it is numerically unimportant.
Finally, we experimented with many alternative specifications of π(1) and π(2) to ensure
that our results did not depend on assuming that competition is not tough. In specifications
with π(2) < π(1) and ρ = 1, a barrier to entry continues to affect industry dynamics only
trivially.

4

A First-In First-Out Example

The above analysis heavily leverages the LIFO assumption. Although we find this assumption
natural for the reasons given above, confirmation that our results can be obtained without
it enhances confidence in their conclusions. Here, we present one such case which uses the
stochastic process for C from the paper-and-pencil example. We replace the assumption
that older firms make their continuation decisions first with its opposite, and we focus on
equilibria in which older firms exit before their younger competitors. We refer to this as the
first-in first-out (FIFO) example. As in the original model, de novo entry decisions follow
incumbents’ continuation decisions. We limit entry into the industry to at most one firm per
period.8
In Appendix B, we show that an equilibrium in a FIFO strategy that is characterized by
threshold rules exists, provided that ϕ(1) is sufficiently small and ϕ(2) is sufficiently large.9
Figure 4 plots this equilibrium’s value functions for a case in which ϕ(2) = ϕ(1).10 The solid
black line represents the value of a firm with rank 1. As before, it has monopoly and duopoly
8

This final assumption avoids complications that arise when there are no incumbents, the value of being
a duopolist with rank 2 exceeds ϕ(2), and the value of being a duopolist with rank 1 is less than ϕ(1).
9
Unlike in the LIFO case, we do not demonstrate that there is a unique Markov-perfect equilibrium in
a FIFO strategy. The equilibrium constructed in Appendix B is merely the unique equilibrium in which a
threshold rule governs a contestant’s entry.
10
To create Figure 4, and Figure 5 in Appendix B, we calculated the equilibrium for particular parameter
values so that the bounds on ϕ(1) and ϕ(2) are satisfied (Assumptions 2 and 3 in Appendix B), so the analysis
in Appendix B does not presume an impossible configuration of parameters.

18

Figure 4: Equilibrium in the FIFO Example

branches. They intersect the horizontal axis at C 1 (0) and C 1 (1). The grey line gives the
value of a firm with rank 2. This is slightly above the rank 1 duopoly branch and far below
the rank 1 monopoly branch. Suppose that initially N = 2. If C passes below C 1 (1), then
the older firm exits and the younger firm’s value jumps to the rank 1 monopoly branch.11 If
?
C then jumps above C 2 , another firm enters and the original firm’s value falls to the rank 1
duopoly branch.
Now consider the consequences of raising ϕ(2) while holding ϕ(1) constant. In Appendix
B, we prove
?

Proposition 5. Increasing ϕ(2) alone weakly increases C 2 , weakly decreases C 1 (0) and
C 1 (0), and leaves C 1 (1) unchanged.
This proposition is nearly identical to its predecessor from the general analysis of duopoly
11

The discount rate used to calculate this example was relatively high, so this possibility contributes little
to the value of a duopolist with rank 1.

19

with a symmetric LIFO strategy. There, C 1 and C 1 play the same role as C 1 (0) and C 1 (0)
here, while the FIFO analogue of C 2 is C 1 (1). Extending the propositions from the LIFO
duopoly to this FIFO case thus requires only relabeling the relevant thresholds. For the sake
of completeness, we state them here.12
Proposition 6. Define Sj (C, N ) as in Proposition 2. Raising ϕ(2) weakly decreases Sj (C, 0),
weakly increases Sj (C, 1), and leaves Sj (C, 2) unchanged.
Proposition 7. Define Pj (N 0 , C, N ) as in Proposition 3. Raising ϕ(2) weakly decreases
Pj (0, C, N ) and Pj (2, C, N ).
Corollary 2. Denote the probability of N firms serving the industry in the ergodic distribution for (C, N ) with P (N ). Raising ϕ(2) weakly decreases P (0) and P (2) and increases
P (1).
Proposition 8. Suppose that (Ct−1 , Nt ) has a unique ergodic distribution with P (0) = 0.
Then increasing ϕ(2) weakly decreases both E[ERt ] and E[XRt ], where the expectations are
calculated using the ergodic distribution for (Ct−1 , Nt ).
It is somewhat remarkable that the same results hold good for the opposite extreme
assumptions of LIFO and FIFO entry and exit for this particular stochastic process. This
reflects the robustness of two economic principles from finite-stage games. First, raising the
cost of late entry directly deters that entry. Second, this deterrence indirectly encourages
early entry. Of course, there are specifications for the dynamic game of entry and exit other
than LIFO and FIFO, so showing that the analytic duopoly results are robust to this change
does not demonstrate that they always hold good. Nevertheless, it does show that a LIFO
pattern for entry and exit is not necessary for this paper’s analytic results.

5

Related Literature

This paper implicitly relies upon a great deal of previous work. This section serves to
acknowledge this dependence explicitly. There are three areas of previous research that are
particularly important for us.
12

The corollary and final proposition take for granted the fact that (C, N ) does have an ergodic distribution.

20

5.1

Entry Deterrence

Barriers to entry play a prominent role in the structure-conduct-performance framework,
because no firm can maintain market power without an impediment to competition. Subsequent work using game-theoretic tools examined incumbents’ ability to deter entry. Dixit
(1980) pioneered this with a demonstration that an incumbent’s investment can deter entry
by committing the firm to tough post-entry price competition, and Milgrom and Roberts
(1982) continued this by showing how actions with no intrinsic dynamic consequences can
nevertheless deter entry by credibly signalling the incumbent’s payoff-relevant private information. Much work has followed this delineating the tools available to incumbents for entry
deterrence. Unlike that work, we take as given the existence of an economic entry barrier
(defined as in McAfee, Mialon, and Williams, 2004); and we focus instead on such a barrier’s
dynamic consequences.

5.2

Competitive Industry Dynamics

We are not the first to analyze the influence of sunk costs on industry structure in an infinitehorizon framework. Hopenhayn (1992) does so in a model of competitive industry dynamics
with atomistic price taking firms. He shows that increasing the sunk cost of entry lowers
entry and exit rates in his model’s stationary equilibrium. This result is reminiscent of our
Proposition 4, but the two results reflect entirely different mechanisms. In our model, raising
a second entrant’s sunk cost raises that firm’s entry threshold and leaves the exit threshold
unchanged. The reduction in entry and exit rates arises from the greater distance that Ct
must fall before inducing that firm to exit. In Hopenhayn’s model, raising the sunk cost of
entry directly raises the competitive equilibrium price, which in turn lowers firms’ common
exit threshold.
Models of industry dynamics that assume either perfect competition (such as Jovanovic’s,
1982 or Hopenhayn’s, 1992) or monopolistic competition without strategic interaction (such
as Fishman and Rob’s, 2003) have proven invaluable in the empirical analysis of firm and
industry evolution, because they replicate the large simultaneous entry and exit rates documented by Dunne, Roberts, and Samuelson (1988). These models abstract from strategic
interaction, so they are very tractable. However, they imply nothing for the evolution of
oligopolistic markets. Campbell and Hopenhayn (2005) and Campbell (2006) have documented that observations from U.S. cities’ retail trade industries generally conflict with these
non-strategic models’ cross-market predictions, so their further analysis apparently requires

21

consideration of oligopoly. The appropriate extension of this paper’s model can serve as the
foundation of such work if we consider a given industry with many firms to be a collection
of many oligopolies.

5.3

Computational Analysis of Competition Policy

Our focus on the positive consequences of erecting a barrier to entry places our work close to
previous investigations of competition policy in dynamic settings. For example, Fershtman
and Pakes (2000) examine the effects of collusive pricing on the number and quality of
products offered by an oligopoly; and Cheong and Judd (2000) examine mergers’ profitability
and welfare consequences in an infinite-horizon setting with aspects of both Bertrand and
Cournot competition. Those papers’ results come from computed Markov-perfect equilibria.
This paper’s model considers only two forward-looking decisions (entry and exit); and this
simplicity allows us to obtain some of our results analytically. Judd (1997) opines that
analytical results from special cases of otherwise intractable dynamic models complement
computational results from richer parameterizations. This paper’s use of analytical and
numerical results to reinforce each other embodies this complementarity.

6

Conclusion

The phrase “barriers to entry” has a long history in industrial organization, much of which
revolves around its imprecise definition. We adopt a specific definition— sunk costs that
only late entrants pay— and examine their effects on duopoly dynamics in a particular
Markov-perfect equilibrium. In theory, exogenously imposing a barrier to entry can either
raise or lower the number of competitors, but we show that the barrier’s entry-encouraging
effects are quantitatively unimportant. If demand is not very persistent, then the barrier’s
direct deterrent effects are substantial. However, a barrier to entry effects the number of
competitors little in the empirically-relevant case with highly persistent demand shocks.
The numerical procedure that we have used to calculate the effects of barriers to entry in
a potential duopoly can be applied directly to industries that support more than two firms.
For an example of this, see Abbring and Campbell (2006). An extension of this paper’s
analysis to general oligopolies presents nontrivial economic difficulties, because it involves
several incumbents jointly erecting barriers to deter multiple contestants’ entry. Sorting this
out by modelling the actions that create barriers to entry lies beyond the scope of this paper.

22

References
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Reserve Bank of Chicago Working Paper 2006-28. 1, 3, 4, 5, 9, 13, 15, 22
Cabral, L. M. (1993). Experience advantages and entry dynamics. Journal of Economic
Theory 59, 403–416. 2
Campbell, J. R. (2006). Competition in large markets. Federal Reserve Bank of Chicago
Working Paper 2005-16. 21
Campbell, J. R. and H. A. Hopenhayn (2005). Market size matters. Journal of Industrial
Economics 53 (1), 1–25. 21
Carlton, D. W. (2004). Why barriers to entry are barriers to understanding. American
Economic Review 94 (2), 466–470. 0, 2
Cheong, K.-S. and K. L. Judd (2000). Mergers and dynamic oligopoly. Journal of Economic
Dynamics and Control Forthcoming. 22
Dixit, A. K. (1980, March). The role of investment in entry-deterrence. Economic Journal 90 (357), 95–106. 21
Dixit, A. K. and R. S. Pindyck (1994). Investment under Uncertainty. Princeton University
Press. 15
Dunne, T., M. J. Roberts, and L. Samuelson (1988). Patterns of firm entry and exit in U.S.
manufacturing industries. Rand Journal of Economics 19, 495–515. 21
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Rand Journal of Economics 31 (2), 207–236. 22
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Journal of Economic Theory 109 (1), 24–38. 21
Hopenhayn, H. A. (1992). Entry, exit, and firm dynamics in long run equilibrium. Econometrica 60 (5), 1127–1150. 21
Jovanovic, B. (1982). Selection and the evolution of industry. Econometrica 50 (3), 649–670.
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23

Judd, K. L. (1997). Computational economics and economic theory: Substitutes or complements? Journal of Economic Dynamics and Control 21, 907–942. 12, 22
McAfee, R. P., H. M. Mialon, and M. A. Williams (2004). What is a barrier to entry?
American Economic Review 94 (2), 461–465. 2, 21
Milgrom, P. and J. Roberts (1982). Limit pricing and entry under incomplete information:
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Sutton, J. (1991). Sunk Costs and Market Structure. Cambridge, MA: MIT Press. 13

24

Appendix A

Proofs of Results in Section 3

T
Proof of Proposition 2. We begin with Sj (C, 0). This equals Pr[ j−1
{Ct+i ≤ C 1 }|Ct−1 = C],
Ti=0
j−1
which is clearly increasing in C 1 . Next, note that Sj (C, 2) = Pr[ i=0 {Ct+i > C 2 }|Ct−1 = C],
T
which is decreasing in C 2 . Finally, Sj (C, 1) = Pr[ j−1
i=0 {C 1 < Ct+i ≤ C 2 }|Ct−1 = C],
which decreases with C 1 and increases with C 2 . The conclusion follows immediately from
Proposition 1.
Proof of Proposition 3. First, consider Pj (0, C, N ). Given Nt = N > 0, the event Nt+j = 0
equals
(
)
j−1
j−2
\
[
[
{Ct+k ≤ C 1 }
{Ct+i ≤ C 1 }
A0 (j − 1) ≡
{Ct+j−1 ≤ C 1 }
i=0

k=i+1

That is, the state must pass below C 1 at some date t + i and thereafter remain below C 1
S
until t + j. Given Nt = N = 0, the event Nt+j = 0 equals A0 (t, j − 1) B0 (t, j − 1), where
j−1

B0 (t, j − 1) ≡

\

{Ct+j ≤ C 1 }.

i=0

Similarly, given Nt = N < 2, the event Nt+j = 2 equals
)
(
j−1
j−2
[
\
[
{Ct+k ≥ C 2 }
{Ct+j−1 ≥ C 2 },
A2 (t, j − 1) ≡
{Ct+i ≥ C 2 }
i=0

k=i+1

and given Nt = N = 2, this event equals A2 (t, j − 1)

S

B2 (t, j − 1), where

j−1

B2 (t, j − 1) ≡

\

{Ct+j ≥ C 2 }.

i=0

The sets A0 (t, j − 1) and B0 (t, j − 1) are both increasing in C 1 and C 1 , and both A2 (t, j − 1)
and B2 (t, j −1) are decreasing in C 2 and C 2 . The conclusion that Pj (0, C, N ) and Pj (2, C, N )
decrease as ϕ(2) increases follows immediately from this and Proposition 1.
Proof of Proposition 4. Without loss of generality, take as given that (C0 , N1 ) = (C, 1). Then
T
XRt = 2/3 if the event A2 (1, t − 2) {Ct ≤ C 2 } occurs, and XRt = 0 otherwise. The set
T
A2 (1, t − 2) {Ct ≤ C 2 } is decreasing in C 2 , so we have that E[XRt |C0 = C, N1 = 1] is
decreasing in ϕ(2). Taking the limit as t → ∞ yields the result that the expected value
of XRt calculated using the ergodic distribution decreases with ϕ(2). The conclusion then
follows from the fact that E[ERt ] = E[XRt ] in the ergodic distribution.
25

Appendix B

Section 4’s First-In First-Out Example

We continue to denote a firm’s rank with R, but the FIFO assumption implies that a firm’s
rank can decrease during its lifetime. To begin, consider the continuation decision of an
duopolist with rank 1. This firm rationally expects its rival to not exit before it does, so the
corresponding dynamic programming problem is identical to the problem of a duopolist with
rank 2 under LIFO:
(
0
if C ≤ C 1 (1),
v(1, C, 1) =
(1−λ)( C
π(2)−κ
+λe
v
(1,1)
)
2
β
if C > C 1 (1).
1−β(1−λ)
Here, C 1 (1) is the largest value of C that sets v(1, C, 1) to zero and
!
Z Č
1 Ĉ + Č
v(1, C 0 , 1) 0
π(2) − κ +
dC .
ve(1, 1) =
2
2
(Č − Ĉ)
Ĉ
This value function is identical to v(0, C, 2) from the paper-and-pencil LIFO example.
The decisions of a potential or actual incumbent monopolist depend on their expectations
for further entry. To calculate an equilibrium we suppose that all future potential duopolists
actually enter if and only if C > C 2 . We then place conditions on the parameters that
guarantee that the best response by a current potential duopolist to such a strategy is to
0
enter if and only if C exceeds a threshold C 2 . With this in place, we demonstrate that there
is a unique value of this threshold that is its own best response. Finally, we characterize the
entry decisions of a potential monopolist given this equilibrium duopoly entry policy.
Begin with a given value for C 2 . With this, we can specify the closely related dynamic
programming problems of an incumbent monopolist and of an incumbent duopolist with rank
2. Their value functions satisfy


if C ≤ C 1 (0),

 0 (1−λ)(Cπ(1)−κ)+λev(0,1)
β
if C 1 (0) < C ≤ C 2 ,
v(0, C, 1) =
1−β(1−λ)

C

(1−λ)
π(2)−κ
+λe
v
(1,1)
(
)
 β
2
if C > C ,
2

1−β(1−λ)

and



 0 (1−λ)(Cπ(1)−κ)+λev(0,1)
β
v(0, C, 2) =
1−β(1−λ)

C

 β (1−λ)( 2 π(2)−κ)+λev(0,2)

if C ≤ C 1 (0),
if C 1 (0) < C ≤ C 1 (1),
if C > C 1 (1).

1−β(1−λ)

26

As above,
ve(0, 1) =

Ĉ + Č
2

!

Z

Č

π(1) − κ +

v(0, C 0 , 1)
(Č − Ĉ)

Ĉ

dC 0 ,

and
1
ve(0, 2) =
2

Ĉ + Č
2

!

Z

Č

π(2) − κ +
Ĉ

v(0, C 0 , 2)
(Č − Ĉ)

dC 0 .

The threshold C 1 (1) is that defined from the older incumbent duopolist’s problem, and the
threshold C 1 (0) is the greatest value of C such that v(0, C, 1) = 0.
We can now consider the entry decision of a potential duopolist given that all future
potential duopolists use the threshold C 2 . We simplify the analysis with the assumption that
such entry is never optimal if it immediately induces the incumbent to exit.
Assumption 2. If C 2 = Č, then v(0, C 1 (1), 2) < ϕ(2).
Because ϕ(2) played no role in the construction of v(0, C 1 (1), 2), Assumption 2 places a
(possibly trivial) lower bound on ϕ(2). With Assumption 2 in place, the incumbent chooses
to continue and the potential entrant stays out of the market when C 1 (0) < C < C 1 (1).
If C > C 1 (1), continuation is a dominant strategy for the incumbent; and the branch of
v(0, C, 2) to the right of C 1 (1) gives the payoff from entry. This is linear and increasing in
0
C, and Assumption 2 implies that it is below ϕ(2) for C close to C 1 (1). Define C 2 (C 2 ) to
equal the greatest value of C such that v(0, C, 2) ≤ ϕ(2). Creating a duopoly through entry
is optimal if C exceeds this threshold.
0
Figure 5 plots C 2 (C 2 ) for an example that satisfies Assumption 2. Increasing C 2 increases
0
the value of entering as a duopolist, so C 2 (·) is weakly decreasing. It is not difficult to
0
demonstrate that it is also continuous, so it has a unique fixed point if C (Č) < Č. In this
?
0
?
?
case, we set C 2 to that fixed point. If C (Č) ≥ Č, then we set C 2 to Č. In either case, C 2 is
the only possible equilibrium entry threshold for a second duopolist in an equilibrium with
a FIFO strategy.13
Determining the threshold for entry into the industry when it has no incumbents completes the determination of equilibrium entry and exit rules. By assumption, a firm entering
?
an empty industry will be a monopolist for one period. If C ≤ C 2 , the firm’s payoff in the
13

Unlike in the LIFO case, we have not demonstrated here that there is a unique Markov-perfect equilibrium
in a FIFO strategy. The equilibrium under construction is merely the unique equilibrium in which a threshold
rule governs a potential duopolist’s entry.

27

?

Figure 5: Equilibrium Determination of C 2 in the FIFO Example

next period is identical to a similarly situated incumbent monopolist. Hence, the value of
?
entering in this case is v(0, C, 1). If instead C > C 2 , then this firm expects to become a
duopolist following the next period if C remains unchanged. It is possible that the resulting
payoff could fall below ϕ(1), in which case equilibrium entry rule would not follow a threshold
rule. In the following, we assume that this does not occur.
Assumption 3. The following inequality holds.

 ?


?
β (1 − λ) C 2 π(1) − κ + v(1, C 2 , 1) + λe
v (0, 1) ≥ ϕ(1)
Because ϕ(1) played no role in the construction of the left-hand side of this inequality, it is a
positive upper bound on ϕ(1). It guarantees that there is a single value of C which equates
the payoff to entry as a monopolist with ϕ(1). This is the monopoly entry threshold, C 1 (0).
Finally, consider raising ϕ(2) while holding ϕ(1) constant. Increasing ϕ(2) directly in0
?
creases the function C 2 (·), so it also weakly increases C 2 . This in turn raises the rank 1
value function’s monopoly branch (vS (0, C, 1)) and leaves its duopoly branch (vS (1, C, 1))
unchanged. From this, Proposition 5 follows.
28

Working Paper Series
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Working Paper Series (continued)
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Working Paper Series (continued)
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Working Paper Series (continued)
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6

Working Paper Series (continued)
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7

Working Paper Series (continued)
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8