the probability of market breakdown weakly increases.

Proof of Proposition 2: Suppose the introduction of a new potential entrant induces exit. The
number of firms in the maximal Nash equilibrium remains at NV, but one firm's entry cost has been
reduced. By Part (5) of the Corollary, expected total welfare strictly increases.

Suppose the introduction of a new potential entrant does not induce exit. There are two cases
to consider. First, suppose that there are still N firms in the maximal Nash equilibrium, but that
£. > Ey. In that case, the behavior of the N firms in the maximal Nash equilibrium of interest
is unchanged from their behavior before the introduction of the potential entrant. Consequently,
expected total welfare is unchanged.

Second, suppose that there are N + 1 firms in the maximal Nash equilibrium. Consider the

lowest possible entry cost for the new potential entrant that does not induce exit, namely

1
N Nat
1
Ee = (2 a) :
Tm py

If E, is any less, then exit will be induced. With the preceding entry cost, in the new maximal
Nash equilibrium the new potential entrant enters with probability zero, and the original N firms
enter with the same probability with which they entered the initial maximal Nash equilibrium.
Thus, expected total welfare in the maximal Nash equilibrium with N + 1 firms is the same as
in the maximal Nash equilibrium with N firms. As E, increases, expected total welfare strictly
decreases, by Part (5) of the Corollary. Thus, if exit is not induced, then expected total welfare
weakly decreases. Ml

Proof of Lemma 2: Suppose each firm i enters with probability aj. Map each af into a 6;
ordered such that 6; > 62 >--- > By, and let B= {1, B2,...,8nr}. I first show there are no
pure strategy equilibria. Suppose each entering firm i sets price p; with probability one. No firm
sets a price less than (legs q- Bx)) Tm, as firm i can guarantee itself that expected profit by
setting price pm. Thus, the minimum price set is strictly greater than zero. At least one firm scts
the minimum price. If no other firm sets the minimum price, then the firm setting the minimum
price has an incentive to raise its price to be arbitrarily close to the next highest price of the other
firms. If at least one other firm sets the minimum price, then a firm setting the minimum price has
an incentive to undercut that price slightly. Thus, there can be no pure strategy equilibria. For
the same reasons, there be no mass points on any price strictly less than pm, and at most one firm

can set price pj, with positive probability.

16


All other firms stay out. Upon entering, firms 1 and 2 set their prices according to the mized

strategy distributions
1— shy forp € [p.Pm)
1 for p=Dm

Fly) =

and

1 E:
Fale) = B [ - ial for p € |p, Pm];

respectively. Firm 1’s expected profit, net of entry costs, is Ey - E, > 0. Firm 2’s expected profit,

net of entry costs, is zero.

Firm 1’s expected profit is positive in this setting, in contrast to the unobserved entry setting with
identical entry costs across firms or to the observed entry setting. The reason that only the two
firms with the lowest entry costs enter with positive probability is that, upon entering, each firm
expects to earn the same profit, gross of entry costs. If firm 3 also entered with positive probability,
then its expected profit upon entering must be no less than E3. If that is the case, then both firms
1 and 2 expect to earn positive profits, net of entry costs. With that expectation, both firm 1 and

firm 2 would enter with probability one, in which case their expected profit upon entering would

be zero, a contradiction.

If multiple firms have the lowest entry cost, then the equilibrium is quite different.

Lemma 3 Suppose that the first k > 1 firms have identical entry cost E > 0, and that the remaining
N'—k firms have strictly higher entry costs. There exists a symmetric maximal Nash equilibrium

in which each of the first k firms enters with probability

a" (E,tm) =1- (2).

Tm,

while the remaining firms stay out. Upon entering, the firms set their prices according to the mized

strategy distribution

1
1 E RT
r0)~ fi (g8)"] tere eel
=~ aul DO) BP
Each firm’s expected profit, net of entry costs, is zero.

A result similar to Lemmas 2 and 3 obtains if firm 1 has the lowest entry cost and firms 2 through

10


Using the results from Lemma 2, H(p) may be written

E» J

Hp) = 1 loan

Let Gy(p) denote the distribution of the winning price in the N-firm symmetric case described by

LR, and assume that the “price” charged is pm in the event that zero firms enter. Using Result 2,

Gao(p) =1- soa] -

LR shows that Gy(p) is largest for N = 2, which implies that the expected price is lowest when

one can show

N =2. Note that H(p) > Go(p), with strict inequality for some p, which implies that the expected
price is strictly lower with EF, < Ey < E than with N > 2 firms each with entry cost E.

Proof of Proposition 4: Suppose that E, < E,. Denote the equilibrium mixed strategy pricing
distributions by F.(p) and (p), respectively. Denote the distribution of the winning price in the

new equilibrium by

Alp) =1- [1-F@)] [La Em) Aw],

where @} (E,7m) is firm 1's entry probability in the new equilibrium. Using Lemma 2, one can
write >

~ Ey

fio) = [si
Recalling the distribution of the winning price in the original equilibrium, calculated in the proof
of Proposition 3 and denoted H(p), one can show easily that H(p) > H(p), with strict inequality
for some p. Consequently, the expected price is strictly lower in the new equilibrium.

A similar argument holds when Ey < E. < E>. In the remaining case, when E2 < E,, the

introduction of a new potential entrant does not change firms’ equilibrium behavior or the expected

price. Ml

References

{1} Elberfeld, W., and Wolfstetter, E. “A Dynamic Model of Bertrand Competition with Entry.”

International Journal of Industrial Organization, Vol. 17 (1999), pp. 513-525.

[2] Janssen, M., and Rasmusen, E. “Bertrand Competition Under Uncertainty.” unpublished man-

uscript (1999).

20


1 Introduction

When firms evaluate whether their cost of entering into a new market can be recouped, they
must consider both the entry decisions of their rivals and the nature of post-entry competition.
The firms’ entry decisions affect not only their own expected profits, but also the expected profits
of other market participants. Consequently, understanding how firms’ entry behavior responds to
different market conditions is quite important to every player in the market. For example, the
firms’ customers and suppliers can maximize their profits by optimally selecting from how many of
the firms to solicit bids when they let procurement contracts, while antitrust agencies can improve
social welfare by evaluating reductions in or restrictions on the number of potential entrants.

Two recent papers examine simultaneous and costly entry decisions by homogeneous product
Bertrand competitors with identical entry costs. The key difference between Elberfeld and Wolf-
stetter [1999] (hereafter EW) and Lang and Rosenthal {1991} (hereafter LR) is that in the EW
model the firms learn the identity of all entrants before they make their pricing decisions.! In
contrast, in the LR model the firms do not learn the number or identity of other entrants before
they make their pricing decisions.” Despite this important distinction, the two models’ results are
surprisingly similar. EW find that adding more identical potential entrants decreases expected
total welfare, while LR find that adding more identical potential entrants increases the expected
price.? Both results contradict the conventional wisdom that the degree of competition increases in
the number of potential entrants. The reasoning underlying both results is that an increase in the
number of potential entrants reduces the likelihood that any particular firm enters. In equilibrium,
the reduction in the original firms’ entry likelihood is sufficiently great that it outweighs the effect
of the new potential entrant’s probability of entry. Moreover, the lower likelihood of rivals’ entry
leads to on average less aggressive price-setting by firms that do enter.

If these models are to be considered accurate representations of the entry process, then the
robustness of their counterintuitive results must be tested. In this paper I extend the EW and LR

models by introducing the plausible assumption that the potential entrants have different entry

‘Levin and Smith (1994] examine the same problem in an auction context, in which firms see who has entered,
but are uninformed about rivals’ production costs. Their main results are qualitatively the same as those presented
in EW. I use the more recent paper as a starting point because its model is more closely related to the LR model.

7In a related paper, Janssen and Rasmusen [1999] analyze the pricing decisions of firms that have an equal and
exogenously specified likelihood of entering the market. Assuming that the firms do not learn the number or identity
of other entrants before they make their pricing decisions, the authors characterize post-entry prices and profits.
Though Janssen and Rasmusen [1999] exogenously specify the firms’ entry probabilities, those probabilities can be
determined endogenously by imposing entry costs, as is done in LR.

Sharkey and Sibley [1993] find the same result in an extension of the LR model that permits certain nonlinear
pricing schemes, in addition to the uniform prices used by LR.


Given that entering firms use a mixed strategy in equilibrium, I now show that the supports
of all entering firms’ mixed strategy distributions have the same minimum. Suppose firms i and j
have different lower supports for their mixed strategy distributions, with firm i setting the minimum
price, so that RP=PR,< B;- Price p; wins with probability one, while price P; wins with probability
less than one.

Denote firm i’s interim expected profit when it sets price p and firms enter with the proba-

bilities in @ by 7; (p|B). Now 7; (p18) = 1; (2,18). Also,

TM ) ~ IT [) ~ BF ,)] (2, ~ «)

kAi

and
= (8) =I b-a(2)](@-2),

where F,(p) denotes the mixed strategy price distribution employed by firm k. Comparing the two,

it is evident that
= (a8) > (8)
Now
TM (28) 2m (2,18) 275 (2,18) .

However, 7; (p18) = (2,18), which implies 7; (p18) > Tj (2, ia), which violates the equilib-
rium requirement 7; (2, i) Pas (2,18). Consequently, the supports of all entering firms’ mixed
strategy pricing distributions have the same minimum. As the minimum price in the support wins
with probability one, then each firm has the same expected payoff, gross of entry costs, upon
entering. Reverting to the ranking by a¥ rather than by §;, denote that expected profit by m (a).
Suppose that in equilibrium firm j > 3 enters with positive probability. For firm j to enter, it
must be that 7(a") > Ej. Consequently, 7 (a) > E2 > Ey, which implies that firm 1 and firm 2
each enter with probability one. Such entry behavior by firm 1 and firm 2 implies that 7 (a") =0,
which in turn contradicts 7 (a) > E;. Therefore, no firm j 2 3 enters with positive probability.
If in equilibrium only firm 1 enters with positive probability, then it will set a price of Pm upon
entering. In this case, any other firm will enter with probability one and slightly undercut firm
1, thus upsetting the purported equilibrium. Consequently, it cannot be the case that only firm 1

enters with positive probability. By similar reasoning, firm 2 cannot be the only firm that enters

17


Consider an equilibrium in which each of the first k firms enters with the same probability, a%.
Following the argument shown in the proof of Lemma 2, a firm’s expected profit upon entering,

gross of entry costs, is

a(a’) =(1-a")F ian.

In equilibrium, the expected payoff from entering must equal the expected payoff from not entering.

Hence,

(1-0)! tm = B,

from which one can determine that

—
a" (Ey tm) = 1— (4) .

Tm

As a firm’s expected profit upon entering, gross of entry costs, must equal 7 (a) for any price p

in the support, [p, Pm], of the mixed strategy distribution, F(p), it must be the case that
[1 =a" (E, tm) Fp)" pD(p) = (1 ~ a (By tm) atm.

Using the previously derived expression for a“ (E, 7,,), one can manipulate the preceding expression

to show that

F(p) = — it f - (om) | for p € [p, Pm),

a
—
1-(#)

which is the desired expression. Finally, each firm’s expected profit, upon entering, is
(1 ~ a" (E,tm))* tm.

Using the expression for a“ (E, 7m), each firm’s expected profit from the two-stage game is zero. lll
Proof of Proposition 3: Suppose all firms have entry cost E > 0. Consider a perturbation to
cach firm’s entry cost so that firms 1 and 2 have entry costs E, and Ey strictly less than E, and
such that all other firms’ entry costs strictly exceed Ey. The equilibrium entry probabilities and
mixed strategy pricing distributions are presented in Lemma 2. The distribution of the winning
price is

H(p) = 1~- [1 — Fi(p)| (1 - 3 (E, tn) Fa(p)].

19


Taking the ratio of the two preceding expressions yields

l-a_ &E ~

l-ag EB;

The preceding relationship can be used in equation (1) to yield

_oyw-1 {_EN7t _ 7
(1~ a?) Ce *) Tm = Ej. (2)

Solving (2) for a? yields

a
Ly N-T
af (E, tm) =1—- Gers)"
which is the desired result.
Using the expression for a2, simple algebra yields the expressions for Pb (@°), pc (@?), and
Pm (a). The expression for total welfare then follows. il

Proof of Corollary 1: (1) The derivative of a? (E, 7m) with respect to E; is

da (E,tm) _ (N-2\ [1
dE, \N-1 an LL

which is strictly positive for N > 2. The derivative is zero is for N = 2.
(2) The derivative of a? (E, 7m) with respect to Ej, j #i, is
1

N NT
do? (E,tm) (1 (AT: Be)
dE; \N=1 E,E; ,

which is strictly negative.
(3) Follows from simple algebra.

(4) The derivative of p, (a?) with respect to E; is

anlar) _ (2) (2) (1B)

dE; N-1 Ej ,

which is strictly positive.

14


its entry costs. Hence, firm 1 would enter with probability one, which would unravel the purported
equilibrium and would imply that firm 1 and at most N — 2 other firms are in the maximal Nash

equilibrium.

Corollary 1 All else equal, a firm’s entry probability strictly increases as its entry cost increases
if tt has at least two rivals, and does not change otherwise. In contrast, a firm’s entry probability
strictly decreases as a rival’s entry cost increases. A firm with a higher entry cost is strictly more
likely to enter than is a firm with a lower entry cost. Moreover, the probability of market breakdown
strictly increases, and total welfare strictly decreases, as any firm’s entry cost increases. Formally,

(1) Stem) & (=)0 when N > (=)2, for all i.

(2) Gm) <0, for j #i.

(3) Ej > 5, => a? (E, 1m) > a2 (E, tm)

(4) S2" > 0, for all i.

(5) Pe Bre) <0, for all i.

Parts (1) and (2) can be explained by noting that when firm i’s entry cost increases, the remaining
firms must enter less frequently in order for firm i to remain indifferent between entering and not
entering. In order to keep any other firm k indifferent between entering and not entering, firm i
must enter more frequently to make up for the reduction in the frequency of entry of firms other
than i and k.6 Part (4) follows because the decrease in firm i’s probability of not entering just
offsets the increase in the probability of not entering by all firms other than i and k, and because
of the increase in firm k’s probability of not entering. Part (3) follows directly from the mixed
strategy nature of the equilibrium. A firm with a low entry cost must expect to receive lower
profit, gross of entry costs, than does a firm with a higher entry cost. The difference in the two
firms’ expected gross profits is driven by the difference in the probabilities that each of them does
not enter. Hence, the firm with a lower entry cost must be less likely to enter than the firm with
a higher entry cost. Finally, Part (5) follows because CS, > C'S; and because the decrease in the
probability of competition outweighs any possible increase in the probability of monopoly.

EW’s results are obtained from Lemma 1 by assuming that the entry costs are identical across

firms. The maximal Nash equilibrium includes all N’ firms.

®Note that the latter increase is unnecessary if there exist only two firms, which explains the N = 2 condition in
Part (1).


to zero, and the monopoly price, pm, uniquely maximizes pD(p), with tm = PmD(pm). In the first
stage of the game, the firms simultaneously decide whether to enter the market. If firm i enters,
then it pays an entry cost Ej; > 0. If firm i does not enter, then it pays nothing and earns zero
profit. Assume E; < mm for all 7, as a firm with an entry cost exceeding 7 will not enter, and so
has no effect on the analysis. I refer to the N’ firms as potential entrants. In the second stage
of the game, each entrant learns the number and identity of all other entrants. Each entrant then
simultaneously selects the price it offers to the buyer. The buyer purchases D(p) units from the
firm offering the lowest price at the price that firm offered, with ties among sellers resolved by a
fair lottery.

Due to the nature of the post-entry price competition, if two or more firms enter the market,
then at least two of them set a price of zero. This competitive outcome occurs in all Nash equilibria
of the one-shot Bertrand competition in the second stage of the game. If only one firm enters, then
that firm sets a price of pm, and earns tm. Firms are risk neutral, so firm i’s profit if it wins with
price p; is pjD(p;) — Ej. If the monopoly price of pm is set, then consumer surplus is CSin. If the
competitive price of zero is set, then consumer surplus is C’'S,, with CS, > CSmm.

This game has two types of equilibria. In the first type, firm i € {1,...,N"} enters with prob-
ability one and charges pm, while all other firms stay out. There are N’ of these equilibria. In
the second type, k > 1 firms enter with positive probability and the remaining firms stay out. To
facilitate comparison with EW’s results from their symmetric model, I define a maximal Nash
equilibrium to be a Nash equilibrium of the second type consisting of the largest number of firms,
which may or may not be the entire set of N’ firms.‘ I restrict attention to maximal Nash equilibria,
because EW focus on the Nash equilibrium in which each of the N’ identical firms enters with the
same probability. That equilibrium is their model’s unique maximal Nash equilibrium.

This two-stage game is solved through backward induction. Let the maximal Nash equilibrium
include exactly N of the N’ firms. I refer to the N firms as active potential entrants, and the
N'—N firms not in the maximal Nash equilibrium as inactive potential entrants. Order the
active firms so that Ly < Ey <---< Ey? In stage two, suppose that each firm k has entered
with probability ag, where “o” denotes the observed entry game. Let a = {af, . 9}, and let

E = {E), Fy,..., En}. Because firm i sets a price of py if it is the only entrant, and sets a price of

‘Note that there may exist several maximal Nash equilibria
‘Note that I am not requring the N firms to be those with the lowest entry costs. The results to follow hold for
any maximal Nash equilibrium.


costs. These differences might arise from scale or scope effects in the firms’ lines of business, or
from differences in the firms’ outside opportunities. I find that the central results in both models
depend critically on the symmetry assumption.

In the observed entry setting, expected total welfare can increase following the introduction of
a potential entrant, reversing the main result in EW. Welfare increases when the introduction of
a sufficiently low cost potential entrant induces one or more existing high cost potential entrants
to become inactive, in the sense that they choose to enter with probability zero. However, it is the
case that expected total welfare falls with an increase in the number of firms entering with positive
probability, which follows from the logic described previously.

In the unobserved entry setting, the equilibrium outcome generally does not involve all firms
entering with positive probability. For example, if all firms’ entry costs differ, then the unique
equilibrium involves strictly positive entry probabilities only for the two firms with the lowest and
second lowest entry costs. If multiple firms all have the lowest entry cost, then in equilibrium only
those firms enter with positive probability. In contrast, in the symmetric setting the symmetric
equilibrium involves all firms entering with the same probability. Thus, the symmetric model is a
very special case of the more general model, and introducing arbitrarily small differences in firms’
entry costs yields vastly different predictions about entry and pricing. Moreover, the expected price
typically decreases or remains unchanged following the introduction of a potential entrant in the
asymmetric model, in contrast to the unambiguous price increase found in the symmetric setting.

The essential feature of my findings is that simply increasing the number of potential entrants
does not dampen competition. Instead. competition decreases if the introduction of a potential
entrant increases the number of firms that enter with positive probability. However, if a new
potential entrant has a sufficiently low entry cost, then one or more previously active firms will now
enter with probability zero. In this instance,the introduction of a new potential entrant increases
competition. This finding is obscured in the symmetric models, because a new potential entrant
always can be accommodated without requiring a previously active firm to become inactive.

Section 2 examines the equilibria of the observed entry setting, and compares the symmetric
and asymmetric outcomes. Section 3 does the same for the unobserved entry setting, while Section

4 briefly concludes. An Appendix contains all proofs.

2 Bertrand Competition with Observed Entry
Consider a two-stage entry and pricing game with N’ > 2 firms producing perfect substitutes

for a consumer with demand D(p) for the good. The firms’ marginal production cost is normalized


(5) The derivative of W° (a°, E, 7m) with respect to E; is

—
dW? (a? Bytm) _ 1 |, N (= Tita Ex)* ‘ os
dE; = ie N-1 Ej “3m +
N wa
1 1
1 | (ae Mea Be
(AM Be) -11 CS.

Tm Ei

Premultiplying by 7, and using the expression for a? (E, 7m) yields

dW? (0°, E, tm) N ° ° °
rm ( dE; )= f= (Fa) a-e9] C8m — [a2] CS. < a? [CSm — CSc] < 0,

which gives the desired result.

Proof of Proposition 1: Suppose the introduction of a new potential entrant induces exit. The
number of firms in the maximal Nash equilibrium remains at N, but one participating firm’s entry
cost has been reduced. By Part (4) of the Corollary, the probability of market breakdown strictly
decreases.

Suppose the introduction of a new potential entrant does not induce exit. There are two cases
to consider. First, suppose that there are still N firms in the maximal Nash equilibrium, but that
E, > En. In that case, the behavior of the N firms in the maximal Nash equilibrium of interest is
unchanged from their behavior before the introduction of the potential entrant. Consequently, the
probability of market breakdown is unchanged.

Second, suppose that there are N +1 firms in the maximal Nash equilibrium. Consider the

lowest possible entry cost for the new potential entrant that does not induce exit, namely

1a who
Ee=|—|[& .
Fm jy

If E, is any less, then exit will be induced. With the preceding entry cost, in the new maximal
Nash equilibrium the new potential entrant enters with probability zero, and the original N firms
enter with the same probability with which they entered in the initial maximal Nash equilibrium.
Thus, the probability of market breakdown in the maximal Nash equilibrium with N + 1 firms is
the same as in the maximal Nash equilibrium with N firms. As Eg increases, the probability of

market breakdown strictly increases, by Part (4) of the Corollary. Thus, if exit is not induced, then


and expected total welfare is the sum of expected consumer and producer surplus. Hence,

N
W? (a°, Eytim) = CS? (0°) + $a? [n?(a°, tm) — Ei).
i=l
As 19(@°, tm) = E;, in equilibrium, expected total welfare in equilibrium is determined entirely by

expected consumer surplus.

wh
Lemma 1 Suppose that all firms have strictly positive entry costs, and that E, > ‘ee Th, Ex) uo
There exists a Nash equilibrium in which each of the N firms enters the market with non-negative

probability. Firm i enters with probability

(Ae) "7

a2 (E,tm) = 1— E

The probability of market breakdown is

while the probability of competition is

wo N N= N
pe(a®) =1+(N-1) (=) (II) -s (sa).
TM mTM \k=1

Therefore, expected total welfare is

N

N N wh
1 1\34 =
W?(a?,E,tm) = [— Ex -w(z) [] 2 CS +
Tm Tm
k=1 k=1
1
we Nn Na N
1\Fa 1
1+(N-1)(— I I E; -— y E; ‘So.
* (55) (i ) 7m (x ) ce

The condition on £) in Lemma 1 ensures that all N firms enter with non-negative probability. If
the condition did not hold, then there would be an internal inconsistency, as the solution would
imply that firm 1’s entry probability is negative, which cannot occur. In such an instance, if firm 1’s
entry probability were constrained to be zero and if all other firms adjusted their entry probabilities

accordingly, then one can show that firm 1 would have a strictly positive payoff from entry, net of


{3] Lang, K., and Rosenthal, R. “The Contractors’ Game.” RAND Journal of Economics, Vol. 22,
No. 3 (1991), pp. 329-338.

[4] Levin, D., and Smith, J. “Equilibrium in Auctions with Entry.” The American Economic Re-

view, Vol. 84, No. 3 (1994), pp. 585-599.

{5] Sharkey, W., and Sibley, D. “A Bertrand Model of Pricing and Entry.” Economics Letters, Vol.
41 (1993), pp. 199-206.

{6] Thomas, C. “The Competitive Effect of Mergers Between Asymmetric Firms.” Federal Trade
Commission Working Paper # 220 (1998).

21


The Effect of Asymmetric Entry Costs on Bertrand Competition

Charles J. Thomas!

This revision: September 26, 1999

‘Harvard Law School and Federal Trade Commission. 1 thank Dave Reiffen for many helpful discussions
and Mike Vita and Bobby Willig for providing valuable comments. Send correspondence to Lewis Interna~
tional Law Center, Room 137. Harvard Law School, Cambridge, MA 02138, or to cjthomas@law. harvard.edu.
This work represents the opinions of the author and is not intended to represent the position of the Federal
Trade Commission or any individual Commissioner.

JEL: D44, L10
Keywords: oligopoly, entry, asymmetric firms


As the new potential entrant’s entry cost is less than firm 3’s entry cost, it seems most reasonable
to focus attention on the new maximal Nash equilibrium in which firm 3 becomes inactive. In
contrast, it seems less reasonable to consider an equilibrium in which firm 2 does not enter, though
one could construct such an equilibrium. Similarly, if E. = 4.5, then it seems unreasonable to
consider equilibria in which any of the original firms become inactive.”

With the preceding definition and with the characterization in Lemma 1, it is straightforward

to prove the following results.

Proposition 1 The introduction of a potential entrant strictly decreases the probability of market

breakdown if and only if it induces exit.

The finding that entry can decrease the probability of market breakdown reverses the first of two
main results in EW. The more important finding is how the introduction of a potential entrant

relates to total welfare.

Proposition 2 The introduction of a potential entrant strictly increases expected total welfare if

and only if it induces exit.

By reversing the second main result in EW, Proposition 2 shows that the conclusion that additional
potential entry reduces total welfare is not robust to the consideration of potential entrants whose
entry costs differ. More importantly, Proposition 2 illustrates fundamentally how the introduction
of a potential entrant affects total welfare in the observed entry setting, the reasons for which are
masked in the symmetric model. Proposition 1 follows from a similar argument.

If the introduction of a potential entrant increases the number of active potential entrants, even
if the new firm has a low entry cost. then total welfare strictly decreases. To see why this occurs,
imagine £, is such that the new potential entrant enters with probability zero.8 In this instance the
number of firms in the maximal Nash equilibrium has increased by one, but total welfare remains
unchanged. As EF, increases. application of the Corollary shows that total welfare falls. Thus, it is
the case that an increase in the number of active potential entrants, in the sense of an increase in

the number of firms in the maximal Nash equilibrium, reduces total welfare. Therefore, if N > 2

7One might take a more agnostic view about equilibrium selection, and instead limit attention to situations in
which the condition on E. in Lemma 1 is violated, so that some firm other than the new potential entrant must
become inactive. Using that criterion, all of the results to follow still hold, though their necessary conditions apply
less frequently.

°£. cannot be any lower, else the new maximal Nash equilibrium will not contain all N original firms.


Result 1 (EW) Suppose that all firms have the same entry cost E > 0. In the maximal Nash

equilibrium, all firms enter with probability

a? (E, tm)

The probability of market breakdown is

Pp (@®) = (=) wr ,

while the probability of competition is

nt
E\N NE
a?) =14+(N-1 ( ) - :
Pe ( ) ( ) Tm Tm
Therefore, expected total welfare is
N'E if E£ Yo E aS N'E
W° (a, E, tm) = -N (=) CS8m + |1+(N'-1) (=) -~—| CS.
Tn Tm Tm Tm

The probability of market breakdown strictly increases, and expected total welfare strictly decreases,

as the number of firms increases.

The next two propositions follow directly from Lemma 1. To aid the discussion to follow, I
introduce the following definition. Suppose that the original maximal Nash equilibrium consists
of N firms, and that a new potential entrant is introduced, with entry cost Ee. It is either the
case that the new maximal Nash equilibrium consists of N + 1 firms, or that the new maximal
Nash equilibrium consists of V’ firms or less. The latter occurs if the solution with N + 1 firms
violates the condition on E; in Lemma 1 (or the corresponding condition on Ee, if E, < Ey). 1
say that the introduction of a potential entrant induces exit if and only if the new maximal Nash
equilibrium consists of N’ firms or less and the new potential entrant’s entry cost is strictly less than
Ey. 1 use this definition to capture the idea that the introduction of the new firm induces one of
the firms in the original maximal Nash equilibrium to become inactive, in the sense that it enters
with probability zero. For example. suppose that initially there were three potential entrants in the
maximal Nash equilibrium, with 7, = 10, £, = 2, Bp = 3, and F3 = 4. Ifa fourth potential entrant

is introduced with E, = 3.5, then any new maximal Nash equilibria consist of only three firms.

x


zero otherwise, its expected profit from entry, gross of entry costs, is

me(a?,t7m) = { T] (1 = a8) | tm.
ki
For firm 7 to be willing to mix between the pure actions of entering and not entering, the payoffs
from these two actions must be identical. Thus, in equilibrium firm i’s expected profit, net of entry

costs, must equal zero, the payoff from not entering. That is, for each firm 4,

[Ja — aR) | tm = Ej.
kei
From the requirements for each of the N firms, one can determine the equilibrium values of the
entry probabilities in terms of the entry costs, E, and the monopoly profit, tm. Denote the vector
of equilibrium entry probabilities by a° (E, tm).
The buyer’s surplus depends on how many firms enter. The probability that zero firms enter,

which EW refer to as market breakdown (b), is

i=l

The probability that only one firm enters, which EW refer to as monopoly (m), is
N
Pm (2°) = Sa? | TT - a8) },
i=l ki

where the term in the summation is the probability that firm 7 is the only firm that enters. Finally,

the probability that two or more firms enter, which EW refer to as competition (c), is

Pe (°) = 1 — py (a) — pm (a).

Expected consumer surplus is

CS? (@°) = pm (O°) CSm + pe (&?) CSc,


with positive probability.

If in equilibrium firm 2 enters with positive probability, then it must be the case that 7 (aX) >
E2. Consequently, 7 (a) > Ey, and firm 1 enters with probability one.

I now derive the firms’ expected profit upon entering. Firm 1 always can earn an expected profit
of (1 - a$) mm by setting price pm. Hence, firm 1 will never set a price strictly less than (1 — a3) mm.
Moreover, the lowest price in the support of the mixed strategy distributions wins with probability
one. Consequently, p cannot strictly exceed (1 — a) 7m, or else firm 1 will not receive the same
expected profit from all prices in the support of the mixed strategy distribution. Therefore, p=
(1 — a3) mm. Consequently, each firm’s expected profit, upon entering, is (1 — a}) mm. Because firm
2 must be indifferent between entering and not entering, it must be the case that (1 — ay) tm = Ee,
which implies that

2)

a3 (E, tm) =1- oe
‘m

Any price p in the support of the mixed strategy distribution must yield an expected profit of

(1 — a3) 7m. For firm 1, it must be the case that
(1 — a3 (E, tm) Fe(p)] pD(p) = (1 — af (E, tn) tn

Using the previously derived expression for a} (E, 7m), one can manipulate the preceding expression

to show that

1

Po) = | -

Ey

A similar argument establishes that

1 = shy for p € [pyPm)

1 for p= Pm

Fi(p) =

Finally, as each firm’s expected profit, upon entering, is (1—a¥(E,7n)) 7m, firm 1’s expected
profit from the two-stage game is E2 — £), while firm 2’s expected profit is zero. I

Proof of Lemma 3: Using the argument in the proof of Lemma 2, suppose that in equilibrium
firm j > k enters with positive probability. For firm j to be willing to enter, it must be the case
that (a“) > E;. Consequently, 7 (a) > E, and each of the first k firms enters with probability
one, which contradicts 7 (a") > E;. Therefore, in equilibrium no firms other than the first k enter

with positive probability.

18




k have the second lowest.
As noted earlier, LR’s Theorem 1 can be extended to demand functions other than the inelastic

demand curve they consider. This is a consequence of Lemma 3, with k = N’.

Result 2 (LR) Suppose that all firms have identical entry cost E > 0. In the maximal Nash

equilibrium, each firm enters with probability

1
NTT
a® (E,m,) =1- (=) i .

Tm

Upon entering, the firms set their prices according to the mized strategy distribution

F(p) = | f - (am) | for p € [p, Pm].

Each firm’s expected profit, net of entry costs, is zero. The expected price strictly increases in the

number of potential entrants, N’.

As is evident from comparing the symmetric and asymmetric cases, even smal! perturbations
away from symmetry lead to dramatically different predictions about the firms’ behavior. In the
symmetric case, all firms enter with positive probability in the maximal Nash equilibrium. However,
once the firms’ entry costs differ by even an arbitrarily small amount, then only two firms enter
with positive probability, and the firm with the lowest entry cost enters with probability one. The
fact that the firm with the lowest entry cost is most likely to enter has more intuitive appeal than
the related result in Part (3) of Corollary 1. in which a firm is more likely to enter the higher is its
entry cost.

Even though the number of active firms differs once entry costs differ, it is not immediately
clear how the expected price changes. After all, the firms’ entry probabilities change when only
two firms are active, and the change in entry probabilities also changes firms’ price-setting behavior
upon entering. Fortunately, one can compare the equilibrium price distributions upon introducing
any type of asymmetry into the symmetric model. Proposition 3 illustrates one type of asymmetry

for which this comparison yields a strong result.

Proposition 3 Suppose that initially all firms have identical entry cost E > 0. If each firm i’s
entry cost is perturbed by a different amount A; to E+ A; > 0, with A; <0 for at least two of the

firms, then the expected price strictly decreases.

ll


model. In the general model, welfare decreases following an increase in the number of firms that
enter with positive probability. In the symmetric model, a new potential entrant always can be
accommodated by the incumbents without inducing exit. Consequently, expected total welfare
strictly decreases in the number of potential entrants in the symmetric model. However, in the
general model, welfare strictly increases following the introduction of a potential entrant, provided
that the introduction induces the exit of one or more of the original active potential entrants. Exit
is induced if the new potential entrant’s entry cost is sufficiently low. Thus, introducing the right
potential entrants increases welfare, a result which has greater intuitive appeal than the result from
the symmetric model.

Lang and Rosenthal [1991] assume that the firms make their price offers without knowing
the number of actual entrants. When firms have different entry costs, the equilibrium involves
positive entry probabilities only for the two firms with the lowest and second lowest entry costs.
In contrast, in the symmetric setting all firms enter with the same positive probability. Thus,
changing the symmetric setting by introducing arbitrarily small differences in firms? entry costs
leads to vastly different predictions about entry and pricing behavior. Moreover, given that the
expected price in the symmetric setting is lowest when potential entry is limited to only two firms,
introducing slight differences in potential entrants’ entry costs frequently causes the expected price
to decrease. Finally, introducing a new potential entrant in the asymmetric setting typically reduces
the expected price or has no price effect, in contrast to the unambiguous increase in the expected

price found in the symmetric model.
Appendix

Proof of Lemma 1: In equilibrium, firm i’s expected payoff, gross of its entry cost, must equal

its entry cost. Hence,

Il (1 — a2) | m = Ej. (1)
kei

The same relationship holds for any other firm j, so

[[ G = 02) | tm = Bj.

RAG

13


Abstract

By permitting firms to have different entry costs, I generalize two previously studied models of
two-stage entry and pricing amongst Bertrand competitors. I find that the existing results depend
critically on the symmetry assumption. For example, if firms’ entry decisions are observed before
price-setting occurs, then total welfare can increase following the introduction of a potential entrant,
in contrast to the unambiguous welfare reduction found in the symmetric setting. If firms’ entry
decisions are unobserved before price-setting occurs, then the expected price typically decreases or
remains unchanged following the introduction of a potential entrant, in contrast to the unambiguous
price increase found in the symmetric setting. In both price-setting environments, competition
increases following the introduction of potential entrants with sufficiently low entry costs, a finding

that is obscured by focusing on the symmetric models.






firms are in the maximal Nash equilibrium, then total welfare always is increased by reducing the
number of active potential entrants to 2.

If the introduction of a potential entrant induces exit, then that is equivalent to holding the
number of active firms constant and reducing one firm’s entry cost. Part (2) of the Corollary shows
that the remaining original firms are more likely to enter than they were before the introduction.
This increase outweighs the reduction in the new firm’s probability of entry. relative to the entry
probability of the firm whose exit was induced, as is shown by applying Part (5) of the Corollary.
Consequently, expected total welfare strictly increases.

In the general setting in which costs differ, the proof of Proposition 2 shows why expected
total welfare falls as the number of active potential entrants increases. In the symmetric model,
the introduction of a new potential entrant always can be accommodated without inducing exit.

Consequently, expected total welfare strictly decreases in the number of potential entrants.

3 Bertrand Competition with Unobserved Entry

Consider a two-stage entry and pricing game identical to the one presented in Scction 2, with
the exception that entrants do not learn the number or identity of any other entrants before each
simultaneously selects the price it offers to the buyer. This game is identical to one variant studied
by LR, with a slight modification that permits more general demand functions than the inelastic
demand structure that they imposed. As will be evident, their results in the symmetric setting
continue to hold with more general assumptions about demand.

This two-stage game is solved through backward induction. In stage two, suppose each firm k has
entered with probability az, where “u” denotes the unobserved entry game. Let a% = {ay}, .., a) }.
Because entry is unobserved, the post-entry competition does not have the all-or-nothing character
seen in the observed entry setting. Instead, entrants set their prices using mixed strategies. This
pricing game is very similar to a first-price auction in which firms’ costs either are zero or exceed

Pm For technical reasons, assume that pD(p) is strictly increasing for p < pm.!°

Lemma 2 Suppose that all firms have different entry costs. There exists a unique Nash equilibrium

in which firm 1 enters with probability one and firm 2 enters with probability

Fy

Tm

03 (E, tm) = 1—

®See Thomas [1998] for a related model.
*°See Janssen and Rasmusen {1999} for details on this assumption, which is necessary to ensure that the equilibrium
mixed strategy price-setting distributions are strictly increasing in the price.

9


Proposition 3 shows that if each firm’s entry cost, E, is changed so that at least two of the firms
have entry costs strictly less than E, then the expected price falls. The expected price falls further
for additional reductions in the two lowest entry costs.

LR show that the expected price is lowest when there are only two potential entrants, under the
assumption that all potential entrants have the same entry cost. Consequently, if the entry costs in
the duopoly setting are perturbed in the manner described in Proposition 3, so that E, < E2 < E,
then the expected price falls further. With the introduction of mean zero random shocks to all
firms’ entry costs in the symmetric model, it is quite likely that the entry costs for at least two of
the firms will fall. Therefore, if initially there exist N’ > 2 potential entrants with identical entry
costs, then introducing differences in those costs frequently causes the expected price to fall.

One also can draw conclusions about the effect on the equilibrium price distribution of the

introduction of potential entrants, in the same manner as was done in LR.

Proposition 4 Suppose that all firms have different entry costs. The introduction of a potential
entrant strictly decreases the expected price if and only if the entrant has the lowest or second lowest

entry cost.

Introducing a potential entrant that has the third lowest entry cost, or higher, has no effect on
any firm’s entry probability. If the new potential entrant has the lowest or the second lowest entry
cost, then its introduction leads to more frequent entry and to more aggressive price-setting upon
entering. One can show that the expected price increases if and only if the new potential entrant
has the same entry cost as the firm initially with the second lowest entry cost. Consequently, in the
asymmetric model, introducing a potential entrant typically does not increase the expected price,

in sharp contrast to the symmetric model.

4 Conclusion

This paper analyzes a two-stage entry and pricing game amongst producers of homogeneous
products. By permitting firms to have different entry costs, the paper extends the analyses in two
recent papers that assume that firms’ entry costs are identical. I find that the existing results
depend critically on the symmetry assumption.

Elberfeld and Wolfstetter [1999] assume that the firms make their price offers after the number
of actual entrants is made known. Their finding that welfare falls as the number of potential
entrants increases is not an artifact of the assumption that entry costs are identical across firms.

However, in the general model the conclusion is not unambiguous, in contrast to the symmetric

12