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DISCUSSION PAPERS IN
ECONOMICS
Working Paper No. 14-06
Why Branded Firms May
Benefit from Counterfeit Competition
Yucheng Ding
University of Colorado Boulder
October 2014
Department of Economics
University of Colorado at Boulder
Boulder, Colorado 80309
© October 2014 Yuchen Ding
Why Branded Firms May Benefit
from Counterfeit Competition
Yucheng Ding∗
October, 2014
Abstract
A durable-good monopolist sells its branded product over two periods. In period
2, when there is entry of a counterfeiter, the branded firm may charge a high price to
signal its quality. Counterfeit competition thus enables the branded firm to commit
to a high price in period 2, alleviating the classic time-inconsistency problem under a
durable-good monopoly. This can increase the branded firm’s profit by encouraging
consumer purchase without delay, despite the revenue loss to the counterfeiter. Total
welfare can also increase, because early purchase eliminates delay cost and consumers
enjoy the good for both periods.
JEL: D82, L11, L13
Keywords: Counterfeit, Durable Good, Quality Signaling
∗
University of Colorado Boulder, [email protected] I am very grateful for Yongmin
Chen’s advice and encouragement. I would like to thank Oleg Baranov, Jin-Hyuk Kim and Keith
Maskus for many helpful comments and discussions. All errors are mine.
1
1
Introduction
Counterfeits have become a fast growing multi-billion dollars business. In the 2007
OECD counterfeit report, the volume of counterfeits was around 200 billion dollars
in international trade, 2% of world trade.1 This figure does not include domestic
consumption of counterfeits or digital products distributed via internet. The U.S.
government estimated that counterfeit trade increased more than 17 fold in the past
decade (U.S. Customs and Border Protection 2008).
Counterfeits are generally viewed as harmful to both the authentic producers and
consumers, especially when they are deceptive, such as counterfeits of pharmaceutical products, eyeglasses, luxury goods or even normal textile products of famous
brands.2 . However, some recent empirical evidence suggests that (deceptive) counterfeits could actually benefit the branded firm. In particular, Qian (2008) finds that
the average profit for branded shoes in China is higher after counterfeit entry. Qian
(2011) provides further evidence that the impact of counterfeits on profit depends on
the quality gap between the authentic good and the counterfeit good; the branded
firm benefits from counterfeits when the quality gap is sufficiently large. In this
paper, I provide a theoretical explanation of why a branded firm can indeed benefit
from competition of a deceptive counterfeiter when the quality difference of their
1
The Economic Impact of Counterfeiting and Piracy http://www.oecd.org/industry/ind/
38707619.pdf
2
This does not mean consumers cannot distinguish products at all. It is just hard for buyers to
tell whether the good is authentic without any other information. For example, a consumer may
not be able to separate a genuine Chanel bag from a fake one only by appearance. However, if one
is priced at $3,000 and the other is sold for $50, she will know that the expensive one is more likely
to be authentic ex post. On the other hand, non-deceptive counterfeits are those that consumers
can easily recognize when purchasing, such as digital products.
2
products is large enough.
I consider a model with an authentic durable-good firm which sells in two periods.
Without counterfeits, the branded durable-good monopolist faces the classic timeinconsistency problem (Coase, 1972): after selling to high-value consumers in the
first period at a high price, it cannot resist cutting its price in the second period.
But then rational consumers will delay their purchase, forcing the monopolist to
reduce its price in the first period and lower the monopolist’s overall profit. Now
suppose that a counterfeiter will enter the market in the second period. In order to
separate its product from counterfeits, the branded firm needs to set a high price
to signal its quality. Thus the presence of counterfeits enables the branded firm to
commit to a high price in period 2, providing a solution to the time-inconsistency
problem. This then motivates more consumers to purchase in period 1 instead of
waiting to buy in period 2, even if the first-period price is high. When the quality
gap is sufficiently large, this “front-loading” effect will dominate the profit loss from
competition in the second period. In terms of total welfare, counterfeits are likely
to decrease surplus in the second period; however, first-period welfare increases due
to front loaded purchases. Early purchases contribute twice the surplus compared
to late purchases because consumers can use the good for two periods. Therefore, if
the quality gap is not too large, it is possible for counterfeits to increase welfare.
The results in this paper shed light on the policy towards counterfeits. Both
branded firms and consumers respond to counterfeits strategically. In the model, the
authentic firm separates itself from the counterfeiter through high price when the
quality gap is large enough. Therefore, consumers will not be fooled by counterfeits
with extremely low quality. Moreover, knowing the later counterfeit entry, consumers
are more inclined to purchase early, which benefits both the authentic firm and total
welfare in a dynamic context.
3
The existing literature has investigated varies strategies by the durable-good monopolist to resolve the commitment problem (see, e.g., Waldman, 2003 for an excellent survey). They include leasing rather than selling the durable good (Coase, 1972;
Bulow, 1982), special contracts between the monopoly and consumers (Butz, 1990),
offering an inferior version (Karp and Perloff, 1996; Hahn, 2006), and product-line
management (Huhn and Padilla, 1996). All of these involve tactics that the monopoly
adopts to alleviate the problem. The present paper suggests a novel commitment
mechanism through the competition from another firm.
Several other papers have discussed the counter-intuitive result of price- or profitincreasing competition (e.g., Chen and Riordan, 2008; Gaibaix et al., 2005; Perloff
et al., 2005; Thomadsen, 2007, 2012). In those papers, competition changes the demand curve of the incumbent firm. When the competitor attracts some price-elastic
consumers, the incumbent can concentrate on price-inelastic consumers by charging
a higher price. However, in my paper, quality signaling leads to the higher price.
In addition, in these static models, competition generally will not increase a firm’s
profit even if prices go up, because a monopoly will always earns higher profit than
a duopoly if the price is the same. However, in a dynamic model, price-increasing
competition helps the monopoly to overcome the time-inconsistency problem and
boosts profit.
There are other papers that discuss deceptive counterfeits. Grossman and Shapiro
(1988a), for example, discuss the problem in international trade; they show that
counterfeits will decrease the total welfare and the authentic firm’s profit. Qian
(2014) focuses on brand-protection strategies against counterfeits, including increasing price or upgrading quality, etc. She uses a vertical differentiation model similar
to my modeling of second-period competition. The main difference is that I investigate the counterfeit problem in a dynamic context. This new feature yields opposite
4
results from hers: in her paper, the authentic firm’s profit decreases with the threat
of counterfeits. Also, total welfare also drops when the ratio of uninformed consumer
is high. However, in the present paper, the branded firm’s profit and total surplus
might increase even if all consumers are uninformed.
Finally, the modeling of second-period counterfeit competition is related to the
literature of duopoly signaling games. Hertzendorf and Overgaard (2001), Fluet and
Garella (2002) and Yehezkel (2008) study similar games with advertising. These
papers focus on the role of dissipative advertising in expanding the separating equilibrium regime while I try to answer how counterfeits influence profit and welfare.
Like Qian (2014), these papers only investigate the static game while my paper
incorporates the signaling game into a durable-good model.
The paper is organized as follows. Section 2 presents the model and reviews the
monopoly benchmark. Section 3 investigates the effect of counterfeit competition on
profit and welfare in a specific equilibrium. Section 4 shows that the main results
continue to hold for other equilibria of the model under proper refinement. Section
5 concludes. All proofs are relegated to the Appendix.
2
The Model and the Monopoly Benchmark
I adapt the two-period durable-good model in Tirole (1988). A branded firm sells
a durable good that can be used in two periods. The quality of its product QA is
normalized to 1. In the second period, a counterfeiter producing a low-quality clone
QC = C < 1 will enter and compete with the branded firm.3 Firms have no marginal
cost to produce the good. Consumers know the quality of both products from the
3
This implicitly assumes that the authentic product has a lead time advantage. Many firms have
special designs on the new product so that imitators have to spend some time to learn and copy.
5
beginning of the game. However, they are not able to tell which good is produced
by the branded firm from its appearance before their purchase.4 This contrasts with
the standard assumption that consumers can trace the producer of the good.
There is a unit mass of heterogeneous consumer indexed by the taste parameter
θl ∼ U [0, 1]. Consumer’s utility has the linear function form:
Ul = θl Qi − pi , i ∈ {A, C}, where pi is the price of firm i
The discount factors of both firms and consumers are assumed to be 1.
Let µi (pA , pC ) be the probability that consumers believe the good from firm i
is the authentic good, given pA and pC . Unlike the traditional monopoly signaling
model, there are two signal senders here. Consumer belief is based on price and
the number of firms charging that price. Consumers are aware that two firms sell
the good and one of them is the counterfeiter. Thus, µA (pA , pC ) + µC (pA , pC ) = 1 in
equilibrium. In a pooling equilibrium, where pA = pC , consumers cannot separate two
products and µA = µC = 21 . In a separating equilibrium, where pA 6= pC , consumers
believe that the expensive good is authentic and the cheap one is counterfeit.
Given consumer’s belief, the firm’s profit is represented by
Πkit (pA , pC , µi ), t ∈ {1, 2}, k ∈ {P, S}
The subscript i, t stands for firm type and time respectively. We use the superscript
k to denote equilibrium values in the second period (P for Pooling Equilibrium and S
4
They are aware of the counterfeit quality in the first period. The assumption can be relaxed
such that consumers only know the distribution of the counterfeit quality, which will not change our
result qualitatively. The underling assumption is that counterfeits are deceptive and all consumers
are uninformed. An alternative assumption is that part of consumers are informed. As long as the
proportion of uninformed consumers are large enough, our qualitative conclusion will hold.
6
for Separating Equilibrium). Also, assume that the separating equilibrium is selected
when profits are the same for a separating and a pooling equilibrium.
The time-line of the game is as follows: the authentic firm sets the first-period
price p1 in t = 1. Consumers decide whether to buy or wait. The counterfeiter
enters in t = 2 and both firms set prices simultaneously. Then consumers observe
both prices and make a purchasing decision based on their beliefs.
Before analyzing the game with counterfeit competition, let’s first review the
benchmark monopoly model without entry.5
(i) When the monopoly lacks commitment power, it has an incentive to decrease
the price to reap the residual demand in t = 2. There is a marginal consumer θ1 who
is indifferent between buying in t = 1 and in t = 2. Therefore, the intertemporal
incentive compatibility constraint for her is:
2θ1 − p1 = θ1 − p2
The right (left) hand side is her surplus from buying in t = 1 (t = 2), given that her
expected second-period price is fulfilled in equilibrium (E(p2 ) = p2 ). In t = 2, the
1
optimal price pM
2 = 2 θ1 , combining with the intertemporal incentive compatibility
constraint, the monopoly’s aggregate profit can be written as:
Π = (2θ1 − θ1 + p2 )(1 − θ1 ) + p2 (θ1 − p2 )
The first (second) term is the profit from the first (second) period. Therefore, the
marginal buyer and the monopoly’s profit are θ1M =
3
5
and ΠM =
9
20
respectively.
(ii) When the monopoly can commit to the same price, there will be no sale in
t = 2 and p1 = 2θ. Henceforth, the monopoly’s profit is as follows:
Π = 2θ(1 − θ)
5
Since there is only one firm here, the subscript represents time and the superscript stands for
the equilibrium value in monopoly case.
7
This gives a optimal profit Π =
1
2
and θ1 = 21 . The profit in no commitment case is
lower because of the standard time-inconsistency problem: high valuation consumers
will anticipate the price reduction in the future and some of them postpone purchase
to the second period.
3
Equilibrium Analysis With Counterfeit Competition
In this section, I will first characterize the set of Perfect Bayesian Equilibrium (PBE)
under counterfeit competition. I then show that there exists an equilibrium at which
the counterfeit can increase the authentic firm’s profit and social welfare.6
Standard backward induction is applied to analyze the counterfeit game. As in
the benchmark, there is a marginal consumer θ1 , such that all consumers with taste
parameter above θ1 will purchase in the first period. The remaining consumers may
purchase in the second period. θ1 can be interpreted as the market size of the second
period.
3.1
Signaling Game in Second Period
In t=2, there is a signaling game played between a pair of vertically differentiated
firms and consumers. Consumers use market prices to update their beliefs. If both
firms have the same price, counterfeits are indistinguishable ex post and a pooling
equilibrium is sustained. If the counterfeiter sets a lower price than the branded firm
and reveals itself, there will be a separating equilibrium where consumers know for
6
In next section, I show all equilibria survive from the refinement have the desired result
8
sure which goods are counterfeits.7
In a pooling equilibrium, consumers are equally likely to pick a genuine product,
leading the expected quality of the product to be
1+C
.
2
The profit function is given
by the following equation.
1
1
2p2
1
)p2
ΠA2 (p2 , p2 , ) = ΠC2 (p2 , p2 , ) = (θ1 −
2
2
2
1+C
In a separating equilibrium, profit functions of both firms are the same as under
vertical price competition with complete information.
ΠA2 (pA2 , pC2 , 1) = (θ1 −
ΠC2 (pA2 , pC2 , 0) = (
pA2 − pC2
)pA2
1−C
pA2 − pC2 pC2
−
)pC2
1−C
C
The counterfeiter’s best response function is always pC2 =
C
p
2 A2
in a separating
equilibrium.
The key question is when a separating equilibrium can be sustained. In the
standard monopoly signaling game, the separation is attained if the single-crossing
condition is satisfied: the firm with high marginal cost is willing to distort price
further than the low-cost firm because the profit depends only on its own price and
consumer belief. However, in a duopoly case, a firm’s profit is also affected by the
other firm’s price. When one sets a high price, the other one faces a trade off between
favorable consumer belief and demand: if the counterfeiter decides to pool with the
authentic firm, which tries to signal by pricing high, its product has 50% chance to
be treated as authentic. However, the demand is low because of the uniform high
7
Because of the asymmetric information, consumers only infer the quality of the firm from
its price. Henceforth, there is another symmetric separating equilibrium where the counterfeiter
charges a higher price than the branded firm. However, I will ignore that one since in this equilibrium
all consumers are paying a higher price for the fake product, which is unrealistic in real life.
9
price in the market. Alternatively, the counterfeiter can reveal itself with a lower
price, which may be better because the upward distorted price of the branded firm
mitigates competition and leaves a large market for the counterfeiter. Two incentive
compatibility constraints must be satisfied to support a separating equilibrium. The
first equation assures that the counterfeiter does not deviate to the authentic price
and the second one implies the branded firm wants to maintain the high price.
1
ΠC2 (pA2 , pC2 , 0) ≥ ΠC2 (pA2 , pA2 , )
2
(1)
1
ΠA2 (pA2 , pC2 , 1) ≥ ΠA2 (pC2 , pC2 , )
2
(2)
Lemma 1. (i) When the quality of the counterfeit is low (C ≤ C1 ≈ 0.604), a set
of separating equilibria can be sustained: pSA2 ∈ [p2 (θ1 , C), p2 (θ1 , C)]; pSC2 =
where p2 (θ1 , C) =
2(1−C 2 )
θ
C 2 −3C+4 1
and p2 (θ1 , C) =
(4−C)(1−C 2 )
θ.
2(2−C)(1+C)−C 2 (1−C) 1
C S
p ,
2 A2
(ii) For any
quality C, there exists a set of pooling equilibria where both firms price at
pP2 ∈ [0, p2 (θ1 , C)).
All equilibria listed in Lemma 1 can be supported by a system of beliefs off the
equilibrium path, such as the most pessimistic belief. For any separating equilibrium
with peA2 ∈ [p2 , p2 ] and peC2 =
C
p ,
2 A2
if the out of equilibrium belief is that any
0
deviating price p 6= {e
pA2 , peC2 } is conceived as a sign of counterfeits, then no firm
would deviate and that particular separating equilibrium is stable. Similarly, the
0
0
belief that µ(p , pe2 ) = 0, ∀p 6= pe2 can support all pooling equilibria.
The result is very intuitive: when the quality gap is large, the profit in a pooling
equilibrium is low because of the low expected quality. The authentic firm just needs
to slightly distort the price upward, which will reduce price competition and leave
the counterfeiter enough profit under separating regime. For the branded firm, since
price distortion is moderate, the cost of signaling is not too high. However, if two
10
products are close substitutes, the cost of signaling for the branded firm is so high
that it would rather pool with the counterfeiter.
As in other signaling games, this model also has multiple equilibria. In some pooling equilibria with low price, counterfeit competition is detrimental to the branded
firm’s profit. In this section, I will show that there exists an equilibrium in which
both the authentic firm and the society benefit from counterfeit entry under certain
conditions. In the next section, it is proved that all equilibria surviving from the
Competitive Intuitive Criterion refinement have similar properties.
The equilibrium I will focus on here is the one with the highest second-period
profit for authentic firm, which is defined as the profit-maximizing equilibrium. It
seems reasonable that consumers will believe that the authentic firm will choose the
price that maximizes its second-period profit. Therefore, consumers believe the firm
charging that price is the authentic firm. If both firms set that price, the good has
50% probability to be genuine. Any other price indicates a fake product. This is
the pessimistic belief that supports the profit-maximizing price in t=2. Formally,
consumer belief is defined as follow.
1
µi (p∗A2 , p∗A2 ) = ; µA (p∗A2 , p2 ) = 1, ∀p2 6= p∗A2 ;
2
µA (p2 , ·) = µC (·, p2 ) = 0, ∀p2 6= p∗A2
In this section, an extra asterisk is used in superscript to denote variables in the
P∗
profit-maximizing equilibrium. Let pS∗
be the authentic price in the opA2 and p2
P∗
timal separating and pooling equilibrium respectively. p∗A2 = arg max[ΠS∗
A2 , Π2 ] ∈
P∗
{pS∗
A2 , p2 } is the price that maximizes the branded firm’s second-period profit, which
is illustrated in the following lemma.
Lemma 2. In the profit-maximizing equilibrium: (i) if the counterfeit’s quality is
low enough (C ≤ C3 ≈ 0.512), the separating equilibrium is supported as the PBE
11
∗
S∗
of signaling game in t=2. p∗A2 = pS∗
A2 = p2 (θ1 , C), ΠA2 = ΠA2 =
4(1−C)2 (1−C 2 ) 2
θ1 .
C 2 −3C+4
(ii)
If the counterfeit’s quality is high (C > C3 ), the pooling equilibrium will be
selected. (a) For C3 < C ≤ C2 ≈ 0.702, p∗2 = pP2 ∗ =
For C > C2 , p∗2 = pP2 ∗ = p2 (θ1 , C), Π∗2 = ΠPA2∗ =
1+C
θ1 , Π∗2 =
4
C(1+C)(1−C 2 ) 2
θ .
2(C 2 −3C+4)2 1
ΠPA2∗ =
1+C 2
θ ;
16 1
(b)
Figure 1 illustrates the second-period price scheme in the profit-maximizing equilibrium. For C ∈ [0, C3 ], the price p2 (θ1 , C), which is the minimum price that prevents the counterfeiter from mimicking the branded firm, has an inverted-U shape
with respect to C and is higher than the monopoly price in benchmark. The counterfeiter’s profit in the pooling equilibrium increases faster with C than its profit in the
separating equilibrium when C is close to 0.8 Therefore, the authentic firm is forced
to increase the price in order to reduce competition and increase the competitor’s
profit in the separating equilibrium. As C gets larger, the condition will be reversed
and the authentic firm has no need to incur a large distortion to support the separating equilibrium. Combining these two segments give us an inverted-U shape price in
the separating equilibrium. When C ∈ (C3 , C2 ], the price increases with C because
of higher expected quality. When C is close to 1, the game converges to Bertrand
Competition of homogeneous good, and the price goes down to 0.
3.2
The Dynamic Game
In this subsection, I will analyze the dynamic game and illustrate why the entry of
counterfeiter may generate higher profit for the incumbent. Given the second-period
consumer surplus and the first-period price, the marginal buyer in the first period
will be determined. The authentic firm’s decision is to choose this marginal consumer
to maximize total profit.
8
When C is close to 0,
dΠP
C2
dC
=
1
2
(1+C)2 pA2
≥
dΠS
C2
dC
12
=
1
2
4(1−C)2 pA2 .
Figure 1: Equilibrium Price in t=2
HPA2 L*
C
0.512
0.702
1
Pooling Equilibrium
In the first segment of the pooling equilibrium (C3 < C ≤ C2 ), consumer surplus in period 2 decreases because the market is flooded with counterfeits. This
pushes more consumers to buy in the first period since the authentic good can be
guaranteed. However, the market price is lower than the benchmark, which makes
late purchase more attractive (pP2 ∗ =
1+C
θ1
4
≤ 21 θ1 ). Overall, consumer surplus falls
below the benchmark case and the time-inconsistency problem is mitigated. We call
this effect of making consumers buy early as the Front-Loading Effect. On the other
hand, counterfeit competition will decrease the branded firm’s revenue in the second
period, which is the Competition Effect. The change of the authentic firm’s profit is
determined by the magnitude of these two effects.
The marginal consumer who purchases at t=1 in the pooling equilibrium is determined by the binding incentive compatibility constraint:
2θ1 − p1 =
1+C
θ1 − pP2 ∗
2
13
The authentic firm’s maximization problem is:
max ΠPA∗ (θ1 ) = (1 − θ1 )(2θ1 −
θ1
1
2pP2 ∗ P ∗
1+C
θ1 + pP2 ∗ ) + (θ1 −
)p
2
2
1+C 2
The marginal buyer θ1P ∗ and equilibrium profit ΠPA∗ are:


1+ 3−C
4

C ∈ (C3 , C2 ]
 2(1+ 11−5C
)
16
P∗
θ1 =
2(1−C 2 )
3−C
[ 2 + 2
]

C −3C+4

 2[ 3−C + (1−C 2 ) + 4(1−C 2 )(1−C) ] C ∈ (C2 , 1)
2
ΠPA∗ =







C 2 −3C+4
(C 2 −3C+4)2
(1+ 3−C
)2
4
C ∈ (C3 , C2 ]
)
4(1+ 11−5C
16
2(1−C 2 ) 2
+ 2
[ 3−C
]
2
C −3C+4
2
2 )(1−C)
(1−C
)
4(1−C
4[ 3−C
+ 2
+
]
2
C −3C+4
(C 2 −3C+4)2
C ∈ (C2 , 1)
As Figure 2 shows, when C ∈ (C3 , C2 ], θ1P ∗ increases with C for two reasons.
Individual surplus in the second period increases as the counterfeit quality rises and
more customers tend to wait, which decreases the wedge between p1 and θ1 . On the
other hand, the branded firm balances the profit in each period to maximize total
profit by properly choosing θ1 . It is optimal to leave more customers in the second
period (increase second-period market size) because the higher second-period profit
increases with C.
When C ∈ (C2 , 1), θ1P ∗ first increases and then decreases in this range. When
C gets close to 1, the front-loading effect disappears because p2 is close to 0. The
branded firm decreases the market size in period 2 due to fierce competition. It can
be inferred that the incumbent does not benefit from counterfeit competition in this
range.
Separating Equilibrium
In the separating equilibrium, the competition effect is not as strong as in the
pooling equilibrium since the counterfeit quality is low. Also, as the high-quality
14
producer, the branded firm takes a larger share of the total profit compared to
the head-to-head competition in the pooling equilibrium. The mechanism of the
front-loading effect is slightly different. Consumers will not be fooled ex post but
face a super monopoly price in the second period as Lemma 2 indicated. Now, the
marginal buyer θ1P faces two options in the second period—buy the authentic good
or the counterfeit.
2θ1 − p1 = max{θ1 − p2 (C, θ1 ), Cθ1 −
C
p2 (C, θ1 )}
2
However, the buyer who is indifferent between a genuine product and a counterfeit
in the second period must below θ1 . Therefore, the outside option is purchasing the
authentic good in t=2. The incumbent’s profit maximization is as follow.
max ΠA (θ1 ) = (1 − θ1 )(θ1 + p2 (C, θ1 )) + ΠS∗
A2 (θ1 )
θ1
In equilibrium,
θ1S∗
=
ΠS∗
A =
2(1−C 2 )
C 2 −3C+4
2(1−C 2 )(−C 2 +C+2)
]
(C 2 −3C+4)2
1+
2[1 +
2(1−C 2 ) 2
]
C 2 −3C+4
2(1−C 2 )(−C 2 +C+2)
]
(C 2 −3C+4)2
[1 +
4[1 +
The left segment of lower curve in Figure 2 informs that θ1S∗ monotonically decreases with C. As the quality gap closes, the branded firm’s profit in the second
period decreases. It would be better to assign less weight in the second period by
decreasing θ1S∗ .
Profit Comparison
Proposition 1. In the profit-maximizing equilibrium, the authentic firm’s profit
will be higher than the monopoly benchmark if the counterfeit quality is sufficiently
15
Figure 2: Marginal Buyer in t=1
Θ1
0.6
C
0.512
0.702
1
low (C < C4 ≈ 0.188). When the counterfeit quality is above that threshold, at any
equilibrium in the second period, competition always decreases the incumbent’s
profit.
Figure 3 illustrates Proposition 1: when the pooling equilibrium emerges in the
second period, the competition effect is too strong and always dominates the frontloading effect. The authentic firm suffers from the counterfeit entry. In the first
segment of the pooling equilibrium, the front-loading effect gets weaker when the
quality increases (θ1P ∗ increases with C) and the time-inconsistency problem is reinforced. However, the high-quality counterfeit also weakens the competition effect
and raises the second-period profit. In the second segment, the competition effects
gets too strong and the front-loading effect disappears.
However, if the separating equilibrium is sustained, the branded firm’s profit has
an inverted-U shape and can be higher than the monopoly benchmark. When the
counterfeit quality is 0, the result with counterfeit competition is the same as the
monopoly benchmark. In the first period, since the high second-period price makes
consumers less likely to wait, the front-loading effect will be stronger when p2 (θ1 , C)
16
is high. Recall that p2 (θ1 , C) has an inverted-U shape, which implies the branded
firm’s profit will has the same curvature. On the other hand, the magnitude of
negative competition effect monotonically increases with C. Therefore, when the
quality of counterfeits is low, the combination of a strong front-loading effect and a
weak competition effect raises the branded firm’s profit above the benchmark. As C
increases, this condition will be reversed and the incumbent’s profit falls below the
monopoly case.
Figure 3: Profit Difference
DP
0.188
3.3
0.512
0.702
C
Welfare and Policy Implication
In terms of welfare, the conventional wisdom is that the deceptive counterfeit is
harmful, because it fools consumers into buying the low-quality product at a relatively high price; this rationale led to trademark policy aiming to prevent consumer
confusion. Grossman and Shapiro (1988a) show deceptive counterfeits decrease welfare with free entry in trade. However, this paper shows that the impact on welfare
can be quite different in a dynamic context.
17
In the monopoly benchmark, total surplus is given by the following equation.
TS
M
Z
1
=
Z
θ1M
2θdθ +
θ1M
θdθ
θ2M
The first (second) term represents the surplus created by first (second) period
transaction9 . The total surplus decreases with θ1 , which is implied by the fact that
early buyer enjoy double surplus. Given the marginal buyer in each period, T S M =
0.775.
The welfare in the presence of deceptive counterfeit competition is a piecewise
function.
T S(C) =

R
R 1S∗
R θS∗

2
T S S∗ (C) = 1S∗ 2θdθ + θS∗
θdθ + S∗
Cθdθ
if C ≤ C3

T S P ∗ (C) = R 1P ∗ 2θdθ + R θ1P ∗
if C > C3
θ1
θ2
θ2P ∗
θ1
θ2
1+C
θdθ
2
In the separating equilibrium, there are two marginal consumers in the second period.
S∗
θ2 denotes the marginal consumer who is indifferent between the genuine good
and the counterfeit. θS∗
2 stands for the one who is indifferent between buying the
counterfeit and buying nothing. Surplus is discounted by C if the counterfeit is
purchased. In the pooling equilibrium, expected surplus is discounted by
1+C
2
for all
consumers because of confusion. Comparing welfare under two cases yields the next
result.
Proposition 2. The entry of deceptive counterfeits increases total welfare if and
only if the counterfeit quality is not too low (C ≥ C5 ≈ 0.078).
Deceptive counterfeits have two effects on welfare. Firstly, the second-period
surplus decreases because of competition with incomplete information, which is the
9
Surplus is attributed to the trading period. First-period buyer enjoys surplus in both periods
but the purchase is made in the first period, therefore all surplus belongs to the first period.
18
Figure 4: Welfare Difference
DTS
0.078
0.512
0.702
C
typical criticism against counterfeits. However, if the first-period welfare is taken
into account, the result will be quite different. As Figure 2 shows, there are always
more sales in t = 1 once C > 0. The front-loading effect pushes consumers to buy
in t = 1 either because the higher price or lower expected quality in t = 2. The
competition effect also forces the incumbent to reduce the market size in t = 2 by
decreasing first-period price and expanding the market in t = 1. Consumers who
purchase in the first period provide “double” contribution to surplus since they are
guaranteed with high quality for two periods, which is the reason that total welfare
could be higher under bad competition.
In Figure 4, the middle segment demonstrates the welfare difference under the
pooling equilibrium with C ∈ (C3 , C2 ]. The downward pressure on welfare decreases
with C because consumer confusion problem is alleviated. Since θ1P ∗ increases with C
in this range, the positive effect also decreases with C. Overall, the social welfare is
higher for all quality levels that sustain the pooling equilibrium in the second period.
In the right segment of Figure 4, the second-period price decreases with C, which
implies more trade and higher welfare.
19
The left segment is the welfare in the separating equilibrium. In Figure 2, as the
counterfeit quality improves, the positive effect increases with C roughly at the same
speed (
d2 θ1S∗
dC 2
is close to 0). The second-period welfare decreases because of upward
distorted prices. Since the second-period price has an inverted-U shape, the welfare
in that period will be an U shape curve. Combining these two effects, it is clear
why total welfare also has a U shape. When the counterfeit quality is 0, the model
coincides with the benchmark. When C is small, unlike the pooling equilibrium, θ1
is close to the benchmark value and decreases slower compared to the second-period
welfare. Therefore, when the counterfeit quality is sufficiently low, the overall welfare
effect is negative.
This proposition implies that deceptive counterfeits may have a positive effect
on welfare in a dynamic context, which is contrary to the traditional argument.
What is more surprising is that welfare is significantly higher when counterfeits
are indistinguishable ex post. The result reminds us to think deeply about the
counterfeit problem. Firstly, branded firms actively adopt strategies against clones.
Although counterfeits are deceptive ex ante, whether they can be recognized ex post
is endogenized. If the quality of clones is low, in which case consumer confusion
induced by counterfeits has a strong negative effect on welfare, the authentic firm
will signal by price and rational consumer will not be fooled. If consumers cannot
distinguish counterfeits from authentic goods ex post, it must be that the quality
gap is close enough. Even if consumers are diverted to counterfeits in that case,
the welfare loss is relatively small. Secondly, consumers respond rationally to the
problem. In the present paper, they are aware that surplus associated with future
purchase is lowered by the counterfeit competition. Thus, more people buy earlier,
which is beneficial for both the branded firm and welfare. However, as I point out,
when the authentic firm decides to separate itself by distorted price, the counterfeiter
20
can also charge a higher price in the second period. This “price collusion” created
by quality signaling might decrease welfare.
4
Equilibrium Refinement and Robustness
The profit-maximizing equilibrium discussed above is only one of equlibria in our
model. In this section, the Intuitive Criterion (Cho and Kreps, 1987) is applied to
refine equilibria. Since there are two signal senders here, I will use a competitive
version as Bontems et al. (2005) and Yehezkel (2008). We will show that all pooling
equilibria are eliminated with a tiny adjustment. The refinement is not applicable
to separating equilibria because both firms’ prices are informative.10 However, it is
proved that our general conclusion that counterfeit competition may increase the
branded firm’s profit and social welfare holds in all separating equilibria.
In previous discussion, both firms are assumed to have zero marginal cost. Now,
let the authentic firm has a slightly higher marginal cost > 0 which is arbitrarily
close to 0. This is just a tie-breaker that helps us to eliminate all pooling equilibria.
By continuity of all functions in the paper, this modification will not alter any of
our results except for the existence of pooling equilibria. For convenience, I only
explicitly state this adjustment in the refinement.
10
The Intuitive Criterion requires unilateral deviation. However, since the other firm charges the
equilibrium price, consumers can use that information to construct the out of equilibrium belief.
Therefore, I cannot simply assume a belief towards the deviating firm while the other one prices
at the equilibrium path. Bester and Demuth (2011), Bontems et.al (2006) and Hertzendorf and
Overgaard (2001) have discussed this issue.
21
4.1
Equilibrium Refinement
Pooling Equilibrium
The basic logic of the Intuitive Criterion is equilibrium dominance: an equilibrium should be eliminated if there exists an out-of-equilibrium price such that given
consumer’s most favorite belief, one type of firm would be better off by deviating
from the equilibrium price to that out-of-equilibrium price, while the other type of
firm cannot benefit from such deviation.
In terms of pooling equilibria, the Competitive Intuitive Criterion requires that
0
there is no p , such that
1
0
ΠA2 (p , pP2 , 1) ≥ ΠA2 (pP2 , pP2 , )
2
1
0
ΠC2 (pP2 , p , 1) < ΠC2 (pP2 , pP2 , )
2
(3)
(4)
0
However, for every pooling equilibrium, there must exist a p such that both
equations hold, which means all pooling equilibria are eliminated. The reason is
similar to the refinement in the monopoly signaling game: The authentic firm with
a higher marginal cost , no matter how small it is, has a lower cost to signal its
quality. Since the the profit function satisfies single-crossing property, I can always
find an upward distorted price such that the authentic firm is willing to deviate to
that price if consumers believe its high quality, while the counterfeiter is not willing
to deviate even if people believe it produces genuine products at that price. The
detail can be found in Proof of Proposition 3.
Separating Equilibrium
Since the Intuitive Criterion cannot be applied to separating equilibria, Hertzendorf and Overgaard (2001) and Yehezkel (2008) use a stronger refinement named
22
Resistance to Equilibrium Defections (REDE) to select the unique and most intuitive separating equilibrium in the duopoly signaling game, which is similar to the
unprejudiced equilibrium in Bagwell and Ramey (1991).11 Only the least distorted
equilibrium survives that refinement, which is the profit-maximizing equilibrium investigated in the previous section. However, we don’t need to impose that extra
refinement since our main results hold in all separating equilibria, which is proved in
next subsection.
4.2
Robustness of Results
We have shown that separating equilibria survive the refinement and pooling equilibria are eliminated. The question is whether our main conclusions regarding incumbent’s profit and social welfare still hold in other separating equilibria. Let’s
first investigate the incumbent’s profit in all separating equilibria. In the previous
section, the profit-maximizing equilibrium is discussed in detail, which is the one
with lowest second-period price among all separating equilibria. For any C, it can be
proved that the branded firm’s profit increases with the second-period price among
equilibria because the front-loading effect grows faster than the competition effect.
Since the branded firm can benefit from counterfeits under the equilibrium with lowest second-period price, the result will hold under all other equilibria. Therefore, if
11
Basically, REDE assumes that consumers can still make reasonable inductions from the equi-
librium behavior of one sender even if they see out of equilibrium behavior from the other sender.
Mathematically, if consumers observe that one good is sold at a price pe ∈ [p2 (C, θ1 ), p2 (C, θ1 )] and
the other one is priced at p ∈ [0, p2 (C, θ1 )) but p 6=
C
e,
2p
then they will believe the one with pe is
genuine and the other one is counterfeit. This gives the authentic firm an incentive to unilaterally
deviate to the price that will maximize its profit within the separating equilibrium range. The
counterfeiter never deviates because any deviation cannot fool consumers.
23
the counterfeit quality is below C4 , the authentic firm’s profit is always higher with
the presence of counterfeits, no matter which separating equilibrium emerges in the
second period.
In terms of the impact on welfare, there is not such a nice monotonicity property
among equilibria because the welfare in t = 2 may be too low when the price is
high in that period. However, it is verified that if C is higher than a threshold, the
social welfare is higher with counterfeit competition in all equilibria. The economic
intuition is the same as the last section. All equilibria have higher second price (more
distortion) than the one that maximizes second-period profit. Thus, the incumbent’s
second-period profit in other equilibria is lower than that one. When the branded
firm maximizes the profit, it tends to reduce the weight on the second period (lower
θ1 ). Therefore, more consumers purchase in the first period and the welfare increases.
Proposition 3. All pooling equilibria are eliminated by the Competitive Intuitive
Criterion. In every separating equilibrium, when C ≤ C4 , the authentic firm’s
profit is higher with counterfeit competition. When C ≥ C6 ≈ 0.248, the social
welfare is higher in the presence of counterfeits.
4.3
Provision of a Damaged Good by the Branded Firm
This paper points out that the entry of a low-quality competitor can actually benefit
the incumbent. The downside to the brand is that it takes away part of the revenue.
An interesting question is whether the branded firm can overcome the competition
effect by offering an inferior version itself and earn higher profit? We do observe
many examples of damaged goods. Armani has a premium ready-to-wear line marketed as Giorgio Armani, relatively cheaper lines as Armani Collezioni and Emporio
Armani, as well as lines distributed in shopping malls like Armani Jeans and Armani
24
Exchange.
Firstly, the incumbent has no incentive to provide an inferior good in the second
period. Deneckere and MacAfee (1996) points out the linear utility function fails the
condition that damaged goods help to raise profit. In my model, no matter what
inferior quality the branded firm chooses, the optimal decision is to sell zero damaged
version in t=2. The second-period price and profit are the same as monopoly benchmark. Since the price is not higher than the monopoly price, the front-loading effect
does not exist. Therefore, the total profit can never be higher than the benchmark.
If there is any fixed cost associate with product line introduction, the profit is always
lower than the monopoly case.
Secondly, damaged goods introduced in the first period is not profitable as well.
Hahn (2006) discusses the benefit of introducing damaged goods in durable-good
model. In his paper, part of high (low) type consumers buy a high (low) quality
good in each period, which changes the ratio of consumer type. Since some low
types have purchased damaged goods in earlier period, the firm has less incentive
to decrease price sharply later, which relaxes the competition between two versions
and alleviates the time-inconsistency problem. However, with continuous consumer
type, it can be proved that if anyone buys a damaged good in the first period, then
all consumers with higher θ must purchase (a damaged or premium version) in that
period as well. Therefore, introducing damaged version in t = 1 only makes some
higher type consumers who would purchase the premium version select damaged
version, which decreases profit for sure.12 The mechanism that helps to solve Coase
Conjecture in Hahn (2006) disappears in my model and the firm would rather just
offer the original version.
12
The detailed proof is available upon request
25
5
Conclusion
While the conventional wisdom is that deceptive counterfeits are always harmful for
the authentic firm and social welfare, this paper argues that the opposite can hold
in durable-good markets. Despite the business-stealing effect, deceptive counterfeits
mitigate the time-inconsistency problem for the incumbent. It is demonstrated that
the effect of counterfeits crucially depends on their quality. When the quality gap is
sufficiently small, pooling equilibria are sustained in the second period. The frontloading effect cannot cover the loss from the competition and the authentic firm’s
profit decreases with counterfeit entry. However, if the quality gap is sufficiently
large, the low-quality counterfeiter only incurs a mild competition which is dominated
by increased sales in the first period, and the branded firm benefits from counterfeit
competition in this case. Moreover, the incumbent cannot earn higher profit by
offering a damaged good because then the front-loading effect then disappears. In
terms of welfare, contrary to traditional arguments, it is shown that in a large quality
range, the deceptive counterfeiter is actually beneficial to the society due to more
earlier purchases. Surprisingly, if counterfeits remain indistinguishable ex-post, total
surplus unambiguously increases.
There are several directions for future research. For instance, I only investigates
the short-term effects of counterfeits. An interesting question is how the counterfeit
entry affects the incumbent’s incentive to innovate. Given that counterfeits may increase the branded firm’s profit, there is a possibility that they promote innovation
as well. Another extension is to endogenize the counterfeit entry by explicitly modeling public policies that affect its entry cost. Moreover, famous brands face many
counterfeiters with different qualities in reality. It would be interesting to study how
counterfeiters compete with each other and their impact on the branded firm.
26
References
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A
Appendix
Proof of Lemma 1 To sustain a separating equilibrium, the incentive compatibility
constraint for the counterfeiter requires that:
(
C
pA2 C
pA2 − C2 pA2
1
2pA2
− 2
) pA2 ≥ (θ1 −
)pA2
1−C
C 2
2
1+C
2(1 − C 2 )
pA2 ≥ 2
θ1 = p2 (θ1 , C)
C − 3C + 4
28
This equation is derived from Eq(1) by plugging the best response function of
the counterfeiter. Similarly, the incentive compatibility constraint for the authentic
firm requires that:
pA2 − C2 pA2
1
C
C
)pA2 ≥ (θ1 −
pA2 ) pA2
1−C
2
1+C
2
2
(4 − C)(1 − C )
θ1 = p2 (θ1 , C)
≤
2(2 − C)(1 + C) − C 2 (1 − C)
(θ1 −
pA2
Therefore, when p2 (θ1 , C) ≥ p2 (θ1 , C), a separating equilibrium exists. Otherwise, only pooling equilibria can be supported.
2(1 − C 2 )
(4 − C)(1 − C 2 )
θ
≥
θ1
1
2(2 − C)(1 + C) − C 2 (1 − C)
C 2 − 3C + 4
This implies that when C ≤ C1 ≈ 0.604, a separating equilibrium can be supported.
For pooling equilibria, as long as Eq(1) is violated, the counterfeiter is willing to
pool with the authentic firm. Therefore, ∀C, if pPA2 = pPC2 = pP2 ∈ [0, p2 (C, θ1 )), a
pooling equilibrium can be sustained by certain out of equilibrium beliefs. Q.E.D.
Proof of Lemma 2 In separating equilibria, it can be easily shown that all
authentic prices are higher than the unconstrained optimal price. Since the profit
function is a concave parabola, pS∗
A2 = p2 (θ1 , C). In any separating equilibrium, the
branded firm’s profit decreases with the counterfeit quality in the second period
because of intensified competition.
In pooling equilibria, when C < C2 ≈ 0.702, the unconstrained optimal price is
always less than p2 (θ1 , C). Therefore, the optimal price is the unconstrained optimal,
pP2 ∗ =
1+C
θ1 .
4
Fixing the market size, the authentic firm’s profit increases with C
within this range. That is because consumer confusion is alleviated, which enables
the firm to raise the price. However, when C > C2 , the quality gap is small and the
29
competition is intense. The unconstrained optimal is higher than p2 (θ1 , C). Since
the profit function is a concave parabola as well, pP2 ∗ = p2 (θ1 , C).
As Lemma 1 indicates, when C ≤ C1 , both types of equilibria exist and Π∗A2 =
P∗
M ax {ΠS∗
A2 , ΠA2 }. Given C1 < C2 , the price of the optimal pooling equilibrium is
pP2 ∗ =
1+C
θ1 .
4
Since
dΠS∗
A2
dC
< 0 and
∗
dΠP
A2
dC
> 0, there is a cut-off quality C3 ≈ 0.512
such that the optimal separating equilibrium is chosen if C ≤ C3 and the pooling
equilibrium would be selected for C3 < C ≤ C1 . When the counterfeit quality is
low, the profit in a separating equilibrium is high because of moderate distortion
while the profit in a pooling equilibrium is low due to low expected quality. As the
quality of the fake good increases, the profit associates with the pooling equilibrium
increases. For C > C1 , separating equilibria cannot exist and the only candidate is
pooling equilibria. When C1 < C ≤ C2 , the equilibrium price is the unconstrained
optimal price. If C > C2 , the price is the binding price p2 (C, θ1 ). Q.E.D
Proof of Proposition 1
(1) When C > C3 , the pooling equilibrium is sustained. There are two secondperiod prices given different C, both of which can be written as pP2 ∗ = F (C)θ1P ∗ .
Firstly, I will prove ∀C ∈ (C3 , 1),
∗
∂ΠP
A
∂F (C)
> 0.
(C)
+ F (C))[(3 − C)( 12 − F1+C
)+
( 3−C
∂ΠPA∗
2
=
2
(C) 2
∂F (C)
+ F (C)
+ F1+C
)
4( 3−C
2
2
Since 0 < F (C) ≤
P∗
1+C ∂ΠA
, ∂F (C)
4
F (C)
]
2
> 0.
Given this property, it can be shown that even with the larger second-period price
pP2 ∗ =
1+C
θ1 , ∀C
4
∈ (C3 , 1), counterfeit competition in the pooling equilibrium still
cannot increase the branded firm’s profit.
30
If pP2 ∗ =
1+C
,
4
(1 + 3−C
)(−2 + 35C−53
)
dΠPA∗
4
64
=
< 0, ∀C ∈ (C3 , 1)
11−5C 2
dC
(1 + 16 )
Therefore, ∀C ≥ C3 , ΠPA∗ (C) ≤ ΠPA∗ (C3 ). Since ΠPA∗ (C3 ) < ΠM , we have ΠPA∗ (C) <
ΠM , ∀C ≥ C3 .
(2) When C ≤ C3 , the separating equilibrium is supported in the second period.
If C = 0, the model is degenerated to the monopoly benchmark. ΠM = ΠS∗
A .
M
Let ∆Π(C) = ΠS∗
A − Π , then
d∆Π(C)
|C=0
dC
= 0.045 > 0. So there must exist some
C that is low enough such that the authentic firm’s profit would increase under the
competition.
On the other hand, the only root C4 ∈ (0, 1] of ∆Π(C) = 0 is C4 ≈ 0.188.
Henceforth, ∆Π(C) ≥ 0 if C ≤ C4 and vice versa. Q.E.D.
Proof of Proposition 2.
(1) In the pooling equilibrium, similar to proof of proposition 1, I can write
∂T S P ∗ (C)
∂F (C)
pP2 ∗ = F (C)θ1P ∗ . Firstly, I will show that
T S P ∗ (C) = (1 − (θ1P )2 ) −
< 0 for any C > C2 .
4
1+C
[1 −
F (C)2 ]
2
4
(1 + C)
∂T S P ∗ (C)
∂θP ∗ (C) 1 + C F (C)2
F (C) P ∗
= 2(θ1P ∗ )2 [ 1
(
−
− 1) −
θ ]
∂F (C)
∂F (C)
4
1+C
1+C 1
Plugging in θ1P ∗ and
∂θ1P ∗ (C)
,
∂F (C)
it is easy to verify that
∂T S P ∗ (C)
∂F (C)
< 0.
Given this property, it is proved that with the larger second-period price pP2 ∗ =
1+C
θ1 , ∀C
4
∈ (C3 , 1), counterfeit competition in pooling equilibrium still increases the
31
total welfare. When pP2 ∗ =
TS
P∗
1+C
θ1 ,
4
1
Z
θ2P ∗ = 12 θ1P ∗ .
Z
θ1P ∗
2θdθ +
(C) =
θ2P ∗
θ1P ∗
1+C
θdθ
2
3(1 + C)
)
16
5
3(1 + C)
∆T S P ∗ (C) = T S P ∗ (C) − T S M (C) = (θ1M )2 − (θ1P ∗ )2 (1 −
)
8
16
d∆T S P ∗ (C)
8(1 + C)
=
>0
dC
(27 − 5C)3
= 1 − (θ1P ∗ )2 (1 −
Therefore, ∆T S P ∗ (C) ≥ ∆T S P ∗ (C3 ), ∀C > C3 . Since, ∆T S P ∗ (C3 ) > 0, deceptive
counterfeits always yield a higher welfare under the pooling equilibrium.
(2) In the separating equilibrium,
S∗
θ2 =
S∗
Z
2−C
1
p2 (C, θ1 ), θS∗
2 = p2 (C, θ1 )
2(1 − C)
2
1
T S (C) =
Z
θ1S∗
Z
2θdθ +
θ1S∗
θdθ +
(2−C)(1+C) S∗
θ
C 2 −3C+4 1
1−C 2
θS∗
C 2 −3C+4 1
(2−C)(1+C) S∗
θ
C 2 −3C+4 1
Cθdθ
(1 + C)2 (4 − 3C)(1 − C)
1
]
= 1 − (θ1S∗ )2 [1 +
2
(C 2 − 3C + 4)2
Since
d∆T S S∗ (C)
|C=0
dC
< 0, if C is sufficiently low, T S S∗ (C) < T S M (C). Moreover,
there is only one root C5 ∈ (0, 1] such that ∆T S S∗ (C) = 0. Therefore, ∀C ≤ C5 ,
T S S∗ (C) ≤ T S M (C) and vice verse. Henceforth, the counterfeit entry increases total
welfare if its quality C ≥ C5 ≈ 0.078. Q.E.D.
Proof of Proposition 3.
(i) The elimination of all pooling equilibria.
0
Firstly, ∀p ∈ [0, p2 (C, θ1 )), there exists a p < p < p + (1 − C)θ1 , such that
(3) is binding. Choosing a δ that is arbitrarily close to 0. Then ΠA2 (p + δ, p, 1) >
32
ΠA2 (p, p, 12 ) and ΠA2 (p + (1 − C)θ1 , p, 1) = 0 < ΠA2 (p, p, 21 ). Therefore, by the
0
continuity of profit function, there must exist a p < p < p + (1 − C)θ1 that makes
0
ΠA2 (p , p, 1) = ΠA2 (p, p, 21 ).
0
Plug p and Eq(3) into Eq(4),
1
0
ΠC2 (p, p , 1) − ΠC2 (p, p, )
2
0
p −p 0 1
2p
=(θ1 −
)p − (θ1 −
)p
1−C
2
1+C
0
0
p −p p−p
=(θ1 −
)(
)<0
1−C p−
0
Hence, for every pooling equilibrium, there is a price p that the authentic firm wants
to deviate and the counterfeiter does not given consumer’s best belief.
Now let’s make some preliminary definition for separating equilibria
p2 (C, θ1 ) =
2(1 − C 2 )
(4 − C)(1 − C 2 )
θ
,
θ1
p
(C,
θ
)
=
1 2
1
C 2 − 3C + 4
2(2 − C)(1 + C) − C 2 (1 − C)
For convenience, let
K(C) =
2(1 − C 2 )
(4 − C)(1 − C 2 )
,
K(C)
=
C 2 − 3C + 4
2(2 − C)(1 + C) − C 2 (1 − C)
In any separating equilibrium, the authentic firm’s price is between K(C)θ1 and
K(C)θ1 .
(ii) For incumbent’s profit:
ΠSA
1 [1 + K(C)]2
=
2−C
4 [1 + 2(1−C)
K(C)2 ]
2−C
[1 + K(C)][1 − 2(1−C)
K(C)]
∂ΠSA
=
2−C
∂K(C)
2(1 + 2(1−C)
K(C)2 )2
33
Since 1 −
2−C
K(C)
2(1−C)
≥ 1−
2−C
K(C)
2(1−C)
> 0,
∂ΠS
A
∂K(C)
> 0. The profit-maximizing
equilibrium is the one that yields lowest total profit for the incumbent. In that
equilibrium, when C ≤ C4 , the profit with counterfeit entry is higher. Therefor no
matter which separating equilibrium is sustained in the second period, ∆ΠSA (C) ≥ 0
if C ≤ C4 .
(iii) For total welfare:
4−3C
1 [1 + K(C)]2 [1 + 4−4C K(C)2 ]
∆T S (C, K(C)) = 0.225 −
2−C
8
[1 + 2−2C
K(C)2 ]2
S
When C = 0,
∆T S S (0, K(0)) = 0.225 −
Since
[1+K(0)]2
1+K(0)2
1 [1 + K(0)]2
8 1 + K(0)2
increases with K(0),
∆T S S (0, K(0)) ≤ 0.225 −
1 [1 + K(0)]2
=0
8 1 + K(0)2
When C = C1 , ∀K(C1 ),
4−3C
2
2
1 [1 + K(C1 )] [1 + 4−4C11 K(C1 ) ]
∆T S (C1 , K(C1 )) = 0.225 −
2−C1
8
[1 + 2−2C
K(C1 )2 ]2
1
S
> 0.225 −
1 [1 + K(C1 )]2
2−C1
8 [1 + 2−2C
K(C1 )2 ]
1
≥ 0.225 −
1 [1 + K(C1 )]2
>0
2−C1
8 [1 + 2−2C
K(C1 )2 ]
1
Therefore, when C = 0, the welfare differences under all separating equilibria
are negative. However, when C = C1 , the welfare differences under all separating
equilibria are strictly positive. By continuity of the welfare difference function, there
must exist a threshold C6 such that as long as C ≥ C6 , the welfare is higher in
presence of deceptive counterfeits for any separating equilibrium. Numerically, I find
that C6 ≈ 0.248. Q.E.D.
34
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