Can Low Prices and Long Queues Deliver High
Profits?
Laurens Debo
Uday Rajan
Tuck School of Business
Stephen M. Ross School of Business
Dartmouth College
University of Michigan
Hanover, NH 03755
Ann Arbor, MI 48109
[email protected]
[email protected]
Senthil K. Veeraraghavan
The Wharton School
University of Pennsylvania
Philadelphia, PA 19104
[email protected]
August 2016
Abstract
Long queues are often observed during the introduction of new and innovative products. It is puzzling that firms do not increase their prices to clear such queues. To
explain this observation, we consider the informational role of queues when a firm can
also adjust its price to signal its quality to uninformed consumers. When the market
has many informed consumers, increasing the price of a high-quality good is a good
signaling strategy. However, when the proportion of informed consumers is low, as typically is for new innovative products, such separation is costly for a high-quality firm: to
prevent imitation by the low-quality firm, it has to raise the price beyond its monopoly
price. We show that, with low waiting costs, there may also exist a pooling equilibrium
in which both firms charge the same low price, but the queue length reveals the quality
almost perfectly. Further, the high-quality firm earns a higher profit in the pooling
equilibrium, as compared to the case of costly price separation. Therefore, when both
waiting costs and the proportion of informed consumers are low, a firm prefers to charge
a low price and signal its quality via the queue length.
1
Introduction
Long waiting lines are often observed when new products are introduced. For example,
when the iPhone 6 was being released, Time magazine advised its readers “You’ll Have to
Wait Till October to Get an iPhone 6 Plus” (Frizell, 2015). Similarly, ABC news reported
that many people (and a robot) waited in line for the product a day before it went on sale
(Utehs, 2015).
It is sometimes proposed that queues may increase the appeal for the product (for
example, see Lam, 2015), but it also has been documented that customers prefer shorter
lines. Queues and waiting times introduce inefficiencies in the market, by imposing waiting
costs on consumers that cannot be captured by the producer. Hence, when the demand for
a new product is strong, it is puzzling that producers allow these queues to develop, rather
than raise the price of the product.
In this paper, we demonstrate that a firm may prefer to have a low price and a long
queue for its product. We identify situations under which such a strategy allows the firm
to signal its quality to consumers and thereby improve its profit. Our model is therefore
a foray into understanding the informational benefits from long lines. To our knowledge,
ours is the first paper to consider the use of queue length as a signal when the firm can also
change the price of the product.
We devise a signaling game in which the type of the firm corresponds with the quality
of the good it sells, which can be high or low. Both products have the same production
cost; that is, the high-quality firm can use the same materials and labor in a way that
delivers more value to the customers. The quality of the product may be interpreted in
terms of aesthetic quality metrics, instead of just performance quality. Knowing its quality,
firm maximizes profit by choosing a price for the good. Consumers arrive at the market
according to a Poisson process. Both types of firm have the same exponential service
process. A consumer wishing to buy the good must join a queue if another consumer is
being served when she arrives at the market.
The consumers are heterogeneous in two ways. First, keeping the quality of the good
fixed, their consumption valuations differ. Second, a fraction of the consumers is informed;
that is, those customers know the quality of the product. The remaining fraction of the
population is uninformed, and knows the prior distribution over quality but does not observe
the realized quality itself. All consumers observe the price and the length of the queue when
they arrive at the market. On arrival, each consumer chooses whether to purchase the good
or leave the market.
1
We consider perfect Bayesian equilibria of the game. Each type of firm chooses its
optimal price, and consumers strategically join the service (and buy) or leave the market.
The decision of informed consumers is straightforward. Uninformed consumers update their
prior belief about the quality of the firm once they have observed the price and queue length,
and then determine whether to join the queue. We use the Intuitive Criterion of Cho and
Kreps (1987) to restrict off-equilibrium beliefs. We describe the model in Section 2. We
then show (in Section 3) that, when waiting costs are low, long queues can develop and the
informational dynamics due to queueing are most pertinent.
Throughout the paper, we consider a situation in which waiting costs are low relative to
the consumption value of the product. We provide analytic results for the limiting case as
the waiting cost goes to zero. In a separating equilibrium, the high- and low-quality firms
optimally choose different prices (high- and low- prices respectively). The price immediately
reveals the quality to uninformed consumers, so there is no informational role for the queue.
If the proportion of informed consumers is high, there is no cost to separation: The highquality firm can separate out even at its monopoly price. However, if the proportion of
informed consumers is low, separation is costly. In particular, it entails charging a price
beyond the monopoly level. In that case, in equilibrium the expected queue length is lower
for the high-quality firm than the low-quality one.
Critically, we also study pooling equilibria in which both types of firm charge the same
price. In this case, uninformed consumers learn nothing from the price of the good. However, informed consumers join at different rates for each type of firm, so the queue length
communicates information about quality.
We show that in equilibrium, as the cost of waiting per unit time diminishes, the queue
length may become almost perfectly informative. Uninformed consumers correctly infer
that the firm almost surely has low (high) quality if the queue is short (long). This result
holds when the pooling price is low, the service rate is at an intermediate level compared
to the arrival rate, and the proportion of informed consumers is low.
One question is whether a firm can attain higher profits in the pooling equilibrium, as
compared to a separating equilibrium. We demonstrate conditions under which the answer
is “yes.” Under our conditions, a pooling equilibrium and a costly separating equilibrium
are both supported. The pooling equilibrium features a price lower than, and the separating
equilibrium a price higher than, the monopoly price of the high-quality firm. We find that
the high-quality firm earns a higher profit in the pooling equilibrium. Hence, a firm may
choose to impose congestion on customers, rather than charge a higher price as a means of
signaling its quality.
2
Our model applies both to new goods and to existing goods which have been significantly
improved by firms. For such products, we expect that both informed and uninformed
customer bases exist. Since improvements cannot be easily communicated to the entire
consumer base, a firm may refrain from increasing its price, relying instead on long queues
to signal product quality.
1.1
Literature Review
Our work builds on both the service operations and signaling literatures. On price-signaling,
Bagwell and Riordan (1991) show that, in a multi-period game, a high-quality firm can
signal its quality by initially charging a high price that is subsequently reduced. To the
best of our knowledge, ours is the first model to consider a price-setting firm subject to
stochastic capacity (as in Naor, 1969) when the quality is not directly observed by some
consumers. Casual observation suggests that these features are present in many situations
that are notorious for long queues. Our analysis, therefore, is important in understanding
why firms allow long queues to persist.
We add to the emerging literature on signaling games in different operations management
environments. On the supply side, Cachon and Lariviere (2001), Gumus, Ray and Gurnani
(2012), Schmidt, et al. (2015) and Lai, et al. (2012) study contracts and capacity investment
as signals. On the demand side, Allon and Bassamboo (2011) consider a cheap talk game in
which delay announcements are signals of congestion. Long queues as signals of high quality
are considered by Debo et al. (2012) in a model in which a firm chooses the service rate
and by Veeraraghavan and Debo (2009, 2011) in models with and without waiting costs.
Hassin and Haviv (2003) provide an excellent introduction to queueing games.
A related marketing paper, Stock and Balachander (2005), shows that when the proportion of informed consumers is low, a high-quality firm may use scarcity rather than price
alone as a signaling device to separate itself from the low type. Our paper differs in two
fundamental ways. First, they consider firms that choose both a price and a capacity, with
scarcity resulting if consumer demand at that price exceeds the available capacity. We focus
on the price as the firm’s sole strategic decision variable as it can be changed in the short
run. Moreover price signals are more common, since capacity information is hard to signal
directly, or adjust in short-run. Second, they focus only on separating equilibria; that is,
equilibria in which the two types of firm choose different price-capacity pairs. Distinctively,
our focus is on equilibria which feature pooling on price. To summarize, we show that
even when the capacity cannot be directly used as a signaling instrument, the firm can
3
nevertheless signal with endogenously-created congestion.
In much of the aforementioned literature, there is one signaling instrument (such as
capacity or contract) that fully describes the information set of the uninformed agents. In
our model, the firm has a single signaling instrument, the price. However, the information set of uninformed consumers contains a second instrument, the queue length, which
is endogenously generated. This feature, while critical to analyze, introduces additional
analytical complexity into our model, and makes the demonstration of signaling benefits
from low prices technically challenging. We characterize the precise situations in which the
queue length may be informative about the firm’s quality even when the price is not.
2
Model
We consider a market in which a firm produces a product or an experience good. The
sequence of events in the game is as follows. At stage 0, nature chooses the type of the
firm (θ), which corresponds with the quality of the good produced by the firm. The type
is high, h, or low, `, each with probability 1/2. The unit cost of production is constant
across type; without loss of generality, we normalize the cost to zero. The firm privately
observes its own type. At stage 1, the firm chooses a price for its product, P . At stage
2, consumers arrive according to a Poisson process with rate Λ. Consumers who wish to
buy the product are serviced at a known exponential service rate µ. When a new consumer
arrives, if another consumer is being served, she must join a queue if she wishes to purchase
the good. Alternatively, the newly-arrived consumer balks and leaves the game. The service
follows FIFO discipline. The waiting cost for a consumer is c per unit of time.
2.1
Consumer and Firm Payoffs
On arrival, a consumer observes both the price chosen by the firm, P , and the length of
the queue, n. Each consumer has a consumption value tvθ for a good of quality θ. Here,
the parameter t is independent across consumers and is uniformly distributed over [0, 1].
Further, vh = v and v` = 12 v. A proportion q of the consumers are informed, and know the
type of the firm. The remaining 1 − q of the consumers are uninformed consumers. These
consumers know the prior probability the firm is the high type (1/2) and can update it
based on P and n.
Consider first an informed consumer who knows the firm has type θ and arrives when the
queue has length n. Her utility from purchasing the product at price P is tvθ −P −(n+1) µc .
She buys the good only if her utility is non-negative; that is, if t ≥
4
P +(n+1)c/µ
.
vθ
Denoting
x+ = max{x, 0}, the expected demand from informed consumers at that queue length and
+
price is then 1 − P +(n+1)c/µ
. Next, consider an uninformed consumer who sees the price
vθ
P and arrives when the queue has length n. Suppose this consumer has a posterior belief b
that the firm has high quality. Then, her posterior expectation of value is bv + (1 − b)v/2 =
(1+b)v/2. Therefore, her expected utility from joining is t(1+b)v/2−P −(n+1) µc . Therefore,
+
. Let
the expected demand from uninformed consumers given (P, n) is 1 − P +(n+1)c/µ
(1+b)v/2
sθ (P, n, b) ∈ [0, 1] denote the probability that a newly-arrived consumer will join when the
firm has type θ and charges a price P , the queue has n consumers and uninformed consumers
have posterior belief b that the quality is high. Then, for each θ = h, `,
P + (n + 1)c/µ +
P + (n + 1)c/µ +
+ (1 − q) 1 −
.
sθ (P, n, b) = q 1 −
vθ
(1 + b)v/2
(1)
Define nθ (P ) = b µvc θ (1 − P/vθ )c = b µc (vθ − P )c, where bxc is the largest integer smaller
than or equal to x. Then, if a firm of type θ charges a price P , there is zero demand
from informed consumers at any length greater than or equal to nθ (P ). It follows that for
n ≥ nh (P ), the demand from uninformed consumers is also zero. Let N̄ = nh (0) = bµv/cc;
then, at or above queue length N̄ , the overall demand is zero at any P ≥ 0.
The combination of the arrival rate Λ, the consumer joining process and the service rate µ
induce a birth-death process over queue lengths between 0 and N̄ . Let s = (s(0), · · · , s(N̄ −
1)) ∈ [0, 1]N̄ be a specific vector of joining probabilities, and π(n, s) be the probability of
queue length n ∈ {0, · · · , N̄ − 1} under the stationary distribution of this birth-death
process. As is standard from the PASTA property (Wolff, 1982), it follows that
π(0, s) =
1+
PN̄
n=1
1
n Q
n−1
Λ
j=0
µ
,
(2)
s(j)
n n−1
Y
Λ
π(n, s) = π(0, s)
s(j) for n = 1, · · · , N̄ .
µ
(3)
j=0
In the steady state, the throughput per unit of time for the firm is µ(1 − π(0, s)). Given a
price P , its profit is then P µ(1 − π(0, s)).
To define the payoff for a firm, its throughput must be defined for all values of P ,
including prices that are not chosen in equilibrium. The best response of informed consumers
at any P depends only on the type of the firm and the queue length. The best response of
uninformed consumers, however, depends on their beliefs over the type of the firm. To ensure
consistency of beliefs across different queue lengths, we update the beliefs of uninformed
consumers as follows. Let ψ(P ) denote the posterior probability an uninformed consumer
5
places on the firm having the high type if she only observes the price of the good, and has
not yet seen the queue length. We require that for each n, the posterior beliefs be updated
from a common ψ. Effectively, we consider a two-stage Bayesian updating process for each
(P, n), with the prior belief the firm has the high type, 1/2, being updated to ψ(P ) once P
has been observed, and then further updated to a belief γ(P, n) at a given n.
2.2
Equilibrium
We consider a perfect Bayesian equilibrium of the game. The equilibrium must describe an
optimal price for each type of firm and consumers’ joining strategies as well as uninformed
consumers’ beliefs for every pair of price and queue length, (P, n). Formally, we define a
pure strategy equilibrium as follows:
Definition 1 A pure strategy equilibrium of the game consists of prices for each type of
firm, Ph , P` , a joining strategy profile for consumers for each type of firm θ, αθ : [0, 1] →
[0, 1]N̄ , an intermediate belief for uninformed consumers that the firm has high quality,
ψ : [0, 1] → [0, 1], and a posterior belief for uninformed consumers that the firm has high
quality, γ : [0, 1] → [0, 1]N̄ such that:
(i) For each P ∈ [0, 1], αθ (P ) = (sθ (P, 0, γ(P, 0)), · · · , sθ (P, N̄ − 1, γ(P, N̄ − 1))), where
sθ (P, n, b) is as specified in equation (1).
(ii) For each P ∈ [0, 1], γ(P ) = (γ(P, 0), · · · , γ(P, N̄ − 1)), with
γ(P, n) =
ψ(P )π(n,αh (P ))
ψ(P )π(n,αh (P ))+(1−ψ(P ))π(n,α` (P ))
whenever the denominator is positive, and
where π(n, s) is as specified in equations (2) and (3).
(iii) For each P = Ph , P` , the value ψ(P ) is updated from the prior belief 1/2 using Bayes’
rule.
(iv) For each θ = h, `, Pθ ∈ arg maxP ∈[0,1] P µ(1 − π(0, αθ (P ))).
In equilibrium, on arrival each consumer observes a price P ∈ {Ph , P` } and the number
of consumers already in the queue, n. Condition (i) requires that all consumers (informed
and uninformed) play an optimal strategy; that is, they join if and only if they have a weakly
positive payoff. Condition (ii) requires that for any price a firm may choose, the beliefs of
uninformed consumers at different queue lengths are consistent with each other and depend
on the actual joining rate given that price. Given the intermediate belief ψ, Bayes’ rule is
used to update the posterior belief at each queue length. Consistency across different queue
6
lengths is important: For example, given some P , it cannot be that uninformed consumers
at queue length n believe the firm has high quality with probability 1 while those at length
n + 1 believe the firm has low quality with probability 1. Condition (iii) requires that the
intermediate belief ψ is consistent with the actual strategies of firms. Finally, condition (iv)
requires that each type of firm maximizes its profit, given the consumer joining strategy.
2.3
Off-equilibrium Refinement for ψ: Intuitive Criterion
We restrict ψ off the equilibrium path based on the widely-applied Intuitive Criterion of
Cho and Kreps (1987). Suppose the observed price is P and uninformed consumers have an
intermediate belief ψ. For a fixed ψ, we can solve for joining strategies for each type of firm,
αh , α` and posterior beliefs at each queue length γ(P, n) that mutually satisfy conditions
(i) and (ii) of Definition 1. Recall that we normalized the production cost of the good to
zero. We can therefore write the profit of firm θ as Π(P, ψ) = µ(1 − π(0, αθ ))P , where
both αθ and the overall profit depend on ψ. Fix a pair of equilibrium prices (Ph∗ , P`∗ ) and
an associated pair of profits, (Π∗h , Π∗` ). For any off-equilibrium price P 6∈ {Ph∗ , P`∗ ), define
the maximal profit a firm of type θ can make by deviating to P as maxψ∈[0,1] Πθ (P, ψ). It
is immediate that the maximal deviation profit at price P for either type of firm will be
obtained when ψ = 1; i.e., when all uninformed consumers believe with probability one that
the firm has high quality — at any given P , this belief induces the greatest demand from
uninformed consumers, and also weakly increases the maximal queue length at which they
join.
An equilibrium will fail the Intuitive Criterion unless the following condition is satisfied:
If only one type θ gains by deviating from Pθ∗ to P , uninformed consumers believe that
price P was posted by type θ. That is,
(i) If Π` (P, 1) > Π∗` and Πh (P, 1) < Π∗h , then it must be that ψ(P ) = 0.
(ii) If Π` (P, 1) < Π∗` and Πh (P, 1) > Π∗h , then it must be that ψ(P ) = 1.
(iii) In all other cases, no restriction is placed on ψ(P ).
Observe that there cannot be an off-equilibrium price P such that Πh (P, 1) > Π∗h but
Π` (P, 1) < Π∗` . If such a price existed, firm h should charge P rather than Ph , breaking the
conjectured equilibrium. Therefore, in all our equilibria, we set ψ(P ) = 0 for P 6= Ph , P` .
That is, at any price not chosen in equilibrium, uninformed consumers believe the firm has
low quality. In each kind of equilibrium we identify, we confirm that this belief satisfies the
Intuitive Criterion.
7
2.4
Separating and Pooling Equilibria
Both price and queue lengths may communicate information about quality to uninformed
consumers. However, the firm cannot directly fix the queue length, which is stochastic
in nature. Hence, the firm directly chooses only the price, which in turn affecting the
queue length distribution. As is conventional, we use the terms “separating equilibrium”
and “pooling equilibrium” to refer to different types of firm choosing the same or different
strategies (i.e., prices), respectively.
In a separating equilibrium, each type of firm chooses a different price, so the price fully
reveals the type. Let (Ph∗ , P`∗ ) be the equilibrium prices. Condition (iv) in the Definition 1
then requires that ψ(Ph∗ ) = 1 and ψ(P`∗ ) = 0. As mentioned earlier, at any P 6∈ {Ph∗ , P`∗ }
we assign the off-equilibrium belief ψ(P ) = 0. In a pooling equilibrium with price P ∗ ,
condition (iii) of Definition 1 implies that ψ(P ∗ ) = 1/2. For all P 6= P ∗ , we set ψ(P ) = 0.
This is consistent with the off-equilibrium belief restriction discussed above.
2.5
Assumptions on Service Rate
We assume that µ is smaller than Λ (to allow for the possibility of congestion) but greater
than Λ/2. Typically, the service capacities in stores for popular products are lower than
arrivals of foot traffic. In Lemma 1 below, we show that under full information and when
the waiting costs vanish, the throughput for firm θ is
Λvθ
2 .
Part (i) of Assumption 1 below
ensures that at the full-information price for either firm, the throughput is limited by
consumer demand rather than capacity. In part (ii) of the assumption, we require that
v` = (1/2)v >
c
µ.
This assumption merely ensures that there exists a price at which the
low-quality firm has a non-zero customer traffic.
Assumption 1 (i)
3
µ
Λ
∈ (1/2, 1) (ii)
c
µ
< v2 .
Long Queues
Consider the case in which the customer waiting cost per unit of time is small relative to
the value from consuming the product, so that 0 < c v. Throughout the paper, we
derive analytic results for the limiting case in which the waiting cost becomes negligible;
i.e, c/v → 0+ . We first consider a separating equilibrium, and then a pooling equilibrium.
We find many parameter values at which both separating and pooling equilibrium exist.
In the pooling equilibrium, both types of firm charge the same low price but there is near-
8
separation via the queue length, with long queues signaling high quality. Firm h earns a
higher profit in the pooling equilibrium.
3.1
Separating Equilibrium
First, consider a separating equilibrium. When the product value is significantly higher
than the waiting cost, the demand from informed consumers flattens across different queue
lengths. In a separating equilibrium, Ph 6= P` , so that consumer beliefs at the equilibrium
prices are degenerate. As mentioned earlier, we assign the off-equilibrium belief that the
firm has low quality. Suppose the true type of the firm is θ and uninformed consumers
believe the type is θ0 (possibly
equal to θ). Then, at any price P , in the limit as c/v → 0+ ,
+
+
+ (1 − q) 1 − vP0
. Observe that in the limit
the overall joining rate is Λ q 1 − vPθ
θ
the consumer demand is independent of queue length.
If the throughput is below the service rate µ, then the throughput must equal the
demand, as consumers are serviced as they arrive. If µ is less than the demand, at any
point of time some consumer is being serviced, and the throughput must equal the service
rate. Overall, therefore, the throughput must be the smaller of the service rate µ and
the joining rate. Recall that the profit of firm θ is denoted Πθ (P, ψ), where ψ represents
uninformed consumers’ posterior belief that the firm has high quality. In a separating
equilibrium, for any P the belief ψ is either equal to 0 or 1. Let 1{θ0 =h} be an indicator
function set to 1 if θ0 = h and zero otherwise. Then, in the limit as c/v → 0+ ,
n
h P + io
P +
+ (1 − q) 1 −
.
Πθ (P, 1{θ0 =h} ) = P min µ, Λ q 1 −
vθ
vθ 0
(4)
Before analyzing the separating equilibrium, we briefly present the optimal price and
resultant profit for each firm if consumers have complete information about its type; that
is, when q = 1. As expected, under full information, price and profit increase in quality.
All proofs are in the Appendix in §5.
We use the superscript
FI
to denote the full-information case. For notational conve-
nience, for any quantity x, we denote x = lim c →0+ x.
v
Lemma 1 When q = 1, in the limit the optimal prices are PhF I =
v
2
and P`F I =
v
4
and the
full-information profits are ΠFh I = Λ v4 and ΠF` I = Λ v8 .
Next, we consider the profit each firm θ can anticipate if it deviates from its separating
equilibrium price Pθ∗ . To ensure that firm ` has no incentive to deviate to the price Ph∗ ,
we need to determine its profit if uninformed consumers believe it has high quality; that is,
9
Π` (Ph∗ , 1). For completeness, we compute Π` (P, 1) for all P . Similarly, to ensure that firm
h does not wish to deviate to some other price P , we need to determine its maximum profit
if uninformed consumers believe it has low quality, Πmax
= maxP ∈[0,1] Πh (P, 0).
h
Lemma 2 (i) For any P ∈ [0, 1] the limiting profit of firm ` if uninformed consumers
believe it has high quality is given by:
(
P min{Λ(1 − P (1 + q)/v), µ} if 0 ≤ P < v2
Π` (P, 1) =
P min{Λ(1 − q)(1 − P/v), µ} if v2 ≤ P ≤ v.
(ii) The maximal profit of firm h if uninformed consumers believe it has low quality is
Πmax
h
=
Λv
4(2−q) .
Example 1:
Let Λ = 1, µ = 0.6, v = 1 and q = 0.1.
h-firm
h-firm
l-firm
Π*h
Π*l
Π*l
l-firm
Ph*
Pl*
The left panel shows the Full Information profits for the h and `-firm. The right panel shows the
deviation profits for the `-firm, as well as the separating price.
Figure 1: Separating equilibrium.
In Figure 1, we illustrate the Full Information profits in the right panel. The h-firm
(`-firm) profits are indicated with a solid (dashed) line. Notice that with Assumption 1,
at the optimal price for both firm types, the service rate exceeds the demand rate. The
Full Information optimal prices, v/4 and v/2 for the ` and h-firm respectively, are indicated
with dotted lines. The `-firm’s Full Information profits are Λv/8 = 0.125. In the left panel,
we illustrate Π` (P, 1) with a dashed line. The solid line is the h-firm profit. Between the
prices indicated by the vertical dotted lines, the `-firm obtains higher profits than its Full
10
Information profits via imitating the h-firm’s price. Outside the vertical dotted lines, the
highest profits the h-firm makes is at the right dotted line.
In a separating equilibrium, the price charged by each firm reveals its type to uninformed
consumers. Therefore, it must be that Π∗` = maxP ∈[0,1] Π` (P, 0). By Lemma 1, it follows
that in the limit as c/v → 0+ , P`∗ =
v
4
and its equilibrium profit is Π∗` =
Λv
8 .
The conditions
for a pair of prices (Ph∗ , P`∗ ) to comprise a separating equilibrium can now
follows: (i) Π∗h ≥ Πh max , which is the maximum profit firm h can obtain
believe it has low quality, and (ii) Π∗` ≥ Π` (Ph∗ , 1).
be stated as
if consumers
The intuition of separation is that by charging a sufficiently high price, the high-quality
firm can make it too costly for the low-quality firm to imitate. If the low-quality firm charges
the same high price Ph∗ , it loses some informed consumers (who know the quality is low)
while potentially gaining a higher profit from uninformed consumers (who now mistakenly
think that the quality is high). The ability of firm ` to successfully mimic the price of firm
h is therefore greater when the proportion of informed consumers is low.
Potentially, firm h may also be able to separate out by charging a sufficiently low price.
However, all demand at the low price cannot be met, since capacity will become an important constraint. As the throughput cannot exceed µ, the firm instead prefers to distinguish
itself by signaling at a high price. Congestion therefore rules out a deviation by firm h to
low prices, playing a similar role in our model as production cost plays in other models of
signaling by price (see Riley, 2001).
We first show that if the proportion of uninformed consumers is sufficiently high, i.e.
above 50%, the high-quality firm can obtain separation simply by charging its own fullinformation price, v/2. Separation is therefore costless. In describing the separating equilibria in the next two propositions, we focus on equilibrium prices and profits. As mentioned
in Section 2.4, intermediate beliefs ψ and posterior beliefs γ are trivial in a separating equilibrium, so the optimal consumer strategy for each P also follows immediately.
Proposition 1 Suppose q ≥ 1/2. Then, there is a separating equilibrium in which each
type of firm charges its full information price, so that Ph∗ =
profit Π∗` =
Λv
8
and Π∗h =
v
2
and P`∗ = v4 , with associated
Λv
4 .
If the fully informed customers are a minority in the population, i.e, q < 1/2, separation
is costly for firm h: it must distort its full information price in order to distinguish itself
from the firm `. We refer to the equilibria in Proposition 2 as costly separating equilibria.
11
Proposition 2 If q < 1/2, there is a separating equilibrium with limiting prices Ph∗ =
q
1−2q
v
∗
1
+
and P`∗ = v4 . Firm profits in the limit are given by Π∗` = Λv
2
8 and Πh =
2(1−q)
Λv
8(1−q) .
It is easy to see that, when q < 1/2, Ph∗ > PhF I . That is, to signal high quality, the firm
increases its price beyond the full information price. If firm ` adopts the same price as firm
h, no informed consumers join its queue. Therefore, the high quality firm’s separating price
increases in the fraction of uninformed customers, while the corresponding profits decrease.
If the price chosen by firm h strictly exceeds its optimal full-information price, on average
it has a shorter queue than firm `. The increase in price, therefore, comes at the cost
of reducing demand for its product. We show in the proof of Remark 2.1 that under
full information firms h and ` have identical queue distributions. Consider a separating
equilibrium in which the high-quality firm increases its price beyond v/2. Then, the joining
demand rate for the high-quality firm, and hence its expected queue length, is strictly
smaller.
Remark 2.1 In a separating equilibrium with Ph∗ >
v
2,
the expected queue length of the
high-quality firm is smaller than that of the low-quality firm.
As we show below, in a pooling equilibrium the high-quality firm has a strictly greater
expected queue length than the low-quality firm.
3.2
Pooling Equilibrium
In a pooling equilibrium, P`∗ = Ph∗ = (say) P ∗ , so that the posterior belief after observing the
price (and before observing the queue length) is equal to the prior belief, with ψ(P ∗ ) = 1/2.
That is, the equilibrium prices convey no additional information about the quality of the
firm. However, informed consumers know the quality of the firm, and join the queue for the
high- and low-quality firms at different rates. As a result, the queue length communicates
some information about quality to uninformed consumers.
In the limit as c/v → 0+ , firm ` can earn at least its full-information profit Π∗` =
by deviating to P ` =
v
4
Λv
8
and revealing itself to be the low type. At prices above v/2 − c/µ,
informed consumers will balk from an empty queue and firm ` earns zero profit. Therefore,
we focus on pooling equilibria at prices less than v/2 − c/µ.
We first show that for any P < v/2 − c/µ, there exist beliefs γ and consumer joining
strategies αh , α` that are consistent with each other in the sense of satisfying parts (i) and
(ii) of Definition 1.
12
Lemma 3 Suppose both firms charge some P ∈ (0, v/2 − c/µ). Then, there exist consumer joining strategies αh , α` and uninformed consumer beliefs γ such that αh , α` satisfy
Definition 1 (i) given γ, and γ satisfies Definition 1 (ii) given αh , α` and ψ(P ) = 1/2.
Obtaining the pair (αh , α` ) is non-trivial. In the proof of Lemma 3, we provide a
constructive fixed point method to compute (γ, αh , α` ) for a given price satisfying the
condition of Lemma 3. Next, we study the structure of the posterior belief after observing
the queue length, γ and obtain more insights when c/v → 0+ . In what follows, for notational
convenience, we suppress dependencies of different variables on P . Given P , choose a pair
of consumer joining strategies αh , α` that satisfy Lemma 3. These strategies give rise to
stationary probabilities over queue length as defined by equations (2) and (3). Consider the
stationary probabilities at each queue length. Recall that nh = b(v − P ) µc c is the highest
possible queue length at price P . Define N = (v − P ) µc ; then nh = bN c.
The role of queues for signaling quality will become apparent when we scale each queue
length n ∈ {0, . . . , nh } as follows. Let α =
n+1
N
denote the scaled queue length (i.e., a queue
length relative to nh ). Note that N → +∞ as c/v → 0+ . The condition that the highest
possible queue length (at some price) can be arbitrarily large will play an important role in
the subsequent discussion. Further, the scaled empty queue (where n = 0) tends to α = 0
and the scaled Naor threshold for firm h (i.e., nh ) tends to 1. For any c > 0, all queue
lengths in {1, . . . , nh − 1} are scaled into the range (0, 1). At any price P < v/2 − c/µ,
let α` =
v/2−P
v−P
α` = limc/v→0+
denote the limit of the relative queue length corresponding to n` ; that, is
n`
nh .
Given any pair of joining strategies αh , α` that are characterized in Lemma 3, we can
now compute via (2) and (3) the stationary distributions, π(bαnh c , αθ ) and corresponding
posterior belief b(bαnh c), where b(n) =
(1/2)π(n,αh )
(1/2)π(n,αh )+(1/2)π(n,α` ) ,
as a function of the scaled
queue length, α ∈ [0, 1].
We first introduce an example to illustrate our main point.
Example 2: We use the same parameters as Example 1. Let P = 3/16, which is less
than v/4 = 1/4. With these parameters, α` = 5/13 ≈ 0.385. We consider three values of
waiting cost, c = 0.001, c = 0.0001 and c = 0.00001. Consider a pair of joining strategies
characterized in Lemma 3. In Figure 2, we show the stationary distributions over (scaled)
queue length for each type of firm in the top line and the posterior belief of uninformed
consumers at different (scaled) queue lengths in the bottom line.
Notice in the top panel that the posterior belief increases in the (scaled) queue length,
confirming the intuition that the long lines signal quality.
13
γ∗(n,P)
γ∗(n,P)
γ∗(n,P)
β∗
β∗
β∗
π(n,α*(P))
π(n,α*(P))
π(n,α*(P))
π(n,α*h(P))
π(n,α*h(P))
π(n,α*h(P))
β∗
β∗
β∗
The waiting costs c = 0.001 (left panel), c = 0.0001 (middle panel) and c = 0.00001 (right panel).
The top line shows the posterior belief of uninformed consumers and the bottom panel the stationary
distributions over the scaled queue length.
Figure 2: Queue length distributions and posterior beliefs of uninformed consumers
For c = 0.001 there is still substantial uncertainty about the quality for short lines. The
reason is that there is considerable overlap between the (scaled) queue length distributions
of the high- and low-quality firm. At short lines, more customers will balk, hence, keeping
lines short. That explains the mass at short lines (to the left of the vertical line). However,
when the lines become long enough, customers will believe that the quality is high and join.
Now, congestion costs will the counter-balancing force that makes them balk. Hence, the
mass to the left of the vertical line.
When the ratio of waiting cost over high-quality value, c/v, decreases, short lines are
more likely low quality, while long lines are more likely high quality. In particular, the
overlap between the (scaled) queue length distribution decreases: the low-quality firm’s
(scaled) queue length migrates further to the left of the vertical line. The mass at the left
of the vertical line of the high-quality firm disappears. The mass at the right of the vertical
line of the high-quality firm migrates further to the right.
For c = 0.00001 the posterior belief at the scaled queue length almost perfectly approx14
imates a step function going from 0 to 1 at some (scaled) queue length, which we refer
to as β ∗ . The queue length distributions are shown in the bottom panel. With low c/v
ratios, the queue length distributions are indeed more clearly separated from each other.
For c = 0.00001, the queue length distribution of the low (high) quality firm is almost surely
below (above) that threshold, hence, the sharp increase in posterior belief at β ∗ . Essentially,
the example shows that when the ratio of waiting cost over value, c/v falls, the (scaled)
queue length distributions will be almost perfectly separated by β ∗ . Long queues might
thus reveal quality when prices are uninformative. In Proposition 3 below, we characterize
β∗.
The distribution for firm ` has mass close to 1 at the left of β ∗ , whereas the distribution
for firm h has mass close to 1 at the right of β ∗ . At scaled queue lengths immediately below
and above β ∗ , the posterior belief for uninformed consumers increases rapidly from 0 to
1. This means that the errors of joining the wrong firm tend to zero when the queues can
become very long in principle (i.e. nh → +∞ for a low c/v ratio). The high-quality firm’s
queue most likely is above bβ ∗ nh c, queue lengths at which uninformed customers believe
that the quality is high, and join accordingly. Similarly, the low-quality firm’s queue most
likely is below bβ ∗ nh c, where uninformed customers believe that the quality is low. Thus,
learning about quality via long (or short) lines is almost perfect. Conversely, when the
waiting cost is high or the value is low, the (equilibrium) queue distributions have greater
overlap. There is still some learning (but less sharp), as the posterior belief increases in the
queue length. Thus, uninformed customers are less sure about the quality.
We now formalize our observations from Example 2. Let a denote a scaled queue
length. The belief about the quality, b(n), determines a joining strategy, which can be expressed as a function of the scaled queue length as sθ (P, banh c, b(banh c)). Denote s̃θ (a) =
sθ (P, banh c, b(banh c)), where for notational brevity we suppress the dependence of s̃θ on P .
The posterior expectation of value at a relative queue length a is given f (a) = (v/2)(1 +
b(banh c)). Now, for each θ = h, ` define
α
Z
Gθ (α) = exp
ln((Λ/µ) s̃θ (a))da .
(5)
0
We define the maximizer of Gθ (a) over [0, 1] as α̂θ . Rather than characterizing the stationary
probabilities as a function of the scaled queue length, π(bαnh c , αθ ), we are able to obtain
1
the limit of the expression (π(bαnh c , αθ )) N expressed in terms of Gθ (α):
Lemma 4 Suppose P < v/2 and for all c and all m < nh , the uninformed consumers’ posterior expectation is f ( m+1
N ), where f (a) > P + (1 − P )a for all a < 1. Then, the stationary
15
1
probabilities for each firm θ satisfy the following condition: limc/v→0+ (π(bαnh c , αθ )) N =
Gθ (α)
Gθ (α̂θ ) .
Lemma 4 is useful as follows: Consider any sequence of numbers x1 , x2 , · · · , xM such
1/M
that xm ∈ [0, 1] for each m and limM →∞ xM
be that limM →∞ xM > 0, whereas if
1/M
exists. Then, if limM →∞ xM
1/M
limM →∞ xM
= 1, it must
< 1, then limM →∞ xM = 0. Therefore,
it follows from Lemma 4 that, in the limit as c/v → 0+ , the probability at some queue
length nα is strictly positive only if α = α̂θ ; i.e., if the scaled queue length α is a maximizer
of Gθ (a).
We use this fact to characterize a pooling equilibrium with congestion when product
value is significantly higher than waiting costs. We first propose that there exists a β such
that the uninformed consumers believe the firm has low quality if α < β and high quality if
α > β. That is, the posterior expectation of uninformed consumers is f (α) = v/2 if α < β
and f (α) = v if α > β. Suppose uninformed consumers join optimally given these beliefs
(i.e., as in Definition 1 (i)). The joining strategy in turn implies a posterior expectation of
quality via the resulting queue length distributions (Definition 1 (ii)).
In Proposition 3, we show that in the limit this posterior expectation is also a step
function, equaling v/2 below some threshold and 1 above the threshold. We show that the
mapping from conjectured to posterior expectations has a fixed point α̂` = 0 < β ∗ < α̂h ,
where β ∗ satisfies the condition
Gh (β ∗ )
G` (β ∗ )
=
.
G` (α̂` )
Gh (α̂h )
(6)
From Lemma 4, it follows that at (scaled) queue length β ∗ the likelihood ratio of the queue
length is equal to one, or: π(n, αh ) ≈ π(n, α` ) for all queue lengths in a neighborhood
b(β ∗ ± c )nh c, where c > 0 is a (small) strictly positive number that depends on c > 0.
For shorter (longer) queue lengths, the likelihood ratio tends to zero (infinity) as c/v → 0+ .
Then, scaled queue lengths below and above β ∗ effectively separate the two firms even
though they pool on price.
In Proposition 3, we focus on prices below v/2. This choice is not restrictive, as we
show in Proposition 4 that, in the limit as c/v → 0+ , informative signaling through queues
occurs when both firms pool at the low-type firm’s full information price (v/4, which is well
within the interval of consideration).
The intuition behind the condition on
µ
Λ
in the statement of the proposition is as follows:
Suppose that uninformed customers believe that the quality is low. The condition
µ
Λ
>
1 − 2P/v ensures that the low-quality firm has a service rate high enough to exceed the
16
demand in the limiting case when c/v → 0+ ). As a consequence, α̂` = 0. The condition
µ
Λ
< 1 − P/v ensures that, when uninformed consumers believe the firm has high quality,
the service rate is less than the demand for the high-quality firm, so that α̂h > 0. These
conditions will induce the ‘sharpest’ difference in queue length distribution between the
high- and low-quality firm. We then show that there exists a β ∗ ∈ (α̂` , α̂h ) at which the
queues are perfectly separating.
Proposition 3 Suppose both firms charge the same price P < v/2 such that 1 − 2P/v <
µ
Λ
< 1 − P/v. Then, there exists a β ∗ ∈ (0, 1) and values c̄, δc , c > 0 such that for every
c ∈ (0, c̄), there is a best response strategy for uninformed consumers with corresponding
beliefs γ such that γ(P, n) < c if n < (β ∗ − δc )nh and γ(P, n) > 1 − c if n > (β ∗ + δc )nh .
Further, δc , c → 0 as c → 0.
Proposition 3 delivers a striking result. Both types of firm choose the same price for
the good. Nevertheless, when c/v → 0+ , there is near-perfect revelation of the firm quality
through the queue length. As informed consumers do not join the low-quality queue at
large n, the queue length distribution for firm ` has the bulk of its mass at low queue
lengths. When n bβ ∗ nh c, uninformed consumers correctly infer the firm has low quality.
Conversely, the queue length distribution for firm h has the bulk of its mass at long queue
lengths, and when n bβ ∗ nh c uninformed consumers infer the firm has high quality. The
threshold bβ ∗ nh c endogenously emerges as a queue length separating the low- and highquality firms.
At any strictly positive waiting cost, all queues are finite (and, in particular, no longer
than nh ). In this case, with some probability, the high-quality firm’s queue can be short
( bβ ∗ nh c) and the low-quality firm’s queue can be long ( bβ ∗ nh c). Therefore, uninformed customers may refrain from joining the (short) queue at firm h as they erroneously
believe the quality is low, making it impossible for the firm to grow its queue above bβ ∗ nh c.
Similarly, when the queue length for firm ` grows above bβ ∗ nh c, uninformed customers may
continue to join as they (erroneously) believe that the quality is high. We show in the proof
of Proposition 3 that these errors tend to zero when c/v → 0+ .
Formally, Gh (α) has a global maximum at αh∗ ∈ (β ∗ , 1). Uninformed consumers who
erroneously balk when the scaled queue length a is less than β ∗ will make the queue stall
below β ∗ , potentially creating a probability mass. Therefore, when c > 0, Gh (α) may also
have a strict local maximum at some queue length αh0 ∈ [0, β ∗ ]. In the limit, as αh∗ is the
17
global maximum over [0, 1], the mass at αh0 tends to zero, and the queue length is almost
surely greater than β ∗ .
Example 2 (Continued): Solving for β ∗ using equation (13) in the proof of Proposition
3, we find β ∗ = 0.07125. The scaled queue length β ∗ is marked by a dashed vertical line
in each sub-figure. It translates to an actual queue length bβ ∗ N c = 34 if c = 0.001, 347 if
c = 0.0001 and 3473 if c = 0.00001.
In any equilibrium, separating or pooling, firm ` must earn a profit at least equal to
Π` (P`F I , 0) (recall that PθF I is the optimal price of firm θ under full information). Therefore,
in the limit, the only pooling price that supports separation on queue length in equilibrium
is the low price P`F I . Hence, in order to separate via queues, the high quality firm must
(in the limit) pool on the low quality firm’s full information price. Recall from Lemma 1
that this price is lower than the high quality firm’s full information price. Hence, while in
a separating equilibrium with price, the high-quality firm’s price distortion is upward, in a
pooling equilibrium with informative queues, the price distortion is downward.
As c/v → 0+ , the price P`F I approaches v/4 (Lemma 1). When P = v/4, the conditions
in Proposition 3 identify restrictions on q, Λ and µ. Using Proposition 3 we obtain precise
conditions such that the P = v/4 can sustain a pooling equilibrium. In the proof of
Proposition 4 below, we show that the off-equilibrium belief ψ(P ) = 0 for P 6= v/4 satisfies
the Intuitive Criterion, and, in particular, satisfies the restrictions we discuss in Section 2.3.
Proposition 4 Suppose that 1/2 < µ/Λ < 3/4 and q < 1 −
pooling equilibrium in which both types of firm set price
on queue length. The equilibrium profits are Πh =
belief is ψ(P ) = 0 if P 6=
µv
4
P∗
=
v
4
Λ
2µ .
Then, there is a limiting
and there is separation based
and Π` =
Λv
8
and the off-equilibrium
P ∗.
In Proposition 4, the condition 1/2 < µ/Λ < 3/4 ensures that Proposition 3 applies at
price P ∗ = v/4. In addition, we require the proportion of informed consumers, q, to be
sufficiently low. The condition q < 1 − Λ/µ ensures that the equilibrium exhibited survives
the Intuitive Criterion. If q exceeds 1 −
P 6=
P∗
Λ
2µ ,
the off-equilibrium belief ψ(P ) = 0 for all
does not satisfy the restrictions imposed in Section 2.3, which are motivated by the
Intuitive Criterion. That is, we can then find a price P such that the maximal deviation
profit of the high-quality firm (across any belief for the uninformed consumers) exceeds its
equilibrium profit, whereas the maximal deviation profit for the low-quality firm is lower
than its equilibrium profit. In that case, firm h has a profitable deviation to P , breaking
the conjectured equilibrium.
18
Although there is pooling on price, the high-quality firm achieves near separation via
the queue length. For small queue lengths, uninformed consumers believe the firm is almost
surely firm `, and for large queue lengths uninformed consumers believe the firm is almost
surely firm h. An immediate observation is that:
Remark 4.1 In a pooling equilibrium, the expected queue length of the high-quality firm is
greater than that of the low-quality firm.
Observe the contrast with Remark 2.1: In a separating equilibrium, on average firm h
has a weakly shorter queue than firm `. However, in the pooling equilibrium, the expected
queue is longer at the higher quality firm.
3.3
Comparison of Pooling and Separating Profits for High Quality Firm
We now compare both the regions in which separating and pooling equilibria exist and the
profit of the high-quality firm in the region of overlap. Our main result is that whenever
a pooling equilibrium and a costly separating equilibria both exist, firm h earns a higher
profit in the pooling equilibrium. Because the low-quality firm’s profits are equal in both
equilibria, the pooling equilibrium Pareto dominates the costly separating equilibrium.
Proposition 5 Suppose the conditions of Proposition 4 are satisfied, then there is a pooling
equilibrium at price P ∗ =
firm h charges a price
v
4 . Then there is also a costly separating equilibrium in which
∗
Ph > v2 . Further, firm h earns a weakly higher profit in the pooling
equilibrium than in the costly separating equilibrium.
Recall from Proposition 4 that for the pooling equilibrium to exist, we need the proportion of informed consumers to be sufficiently low (in particular, q < 1 −
Λ
2µ ).
We show in
Proposition 5 that under those circumstances, if a separating equilibrium exists, the cost of
separation is significant for the high quality firm. The comparison of profits in separating
and pooling equilibria depends on the relative costs of separation and pooling. Separation
is costly when firm h must charge a higher price than PhF I , because of the resulting fall in
demand. The cost to firm h of pooling comes both from the low pooling price (v/4) and the
fixed capacity which implies congestion at low queue lengths. When q is sufficiently low,
the cost of separation is higher, so firm h prefers the pooling equilibrium.
Proposition 5 answers our main question: Rather than impose waiting times on consumers, why doesn’t a firm simply raise its price? Formally, our results have been shown
only for the limiting case in which c/v → 0+ , i.e., the case in which the waiting costs vanish
19
in the limit. Nevertheless, the intuition goes through when the waiting costs are sufficiently
small. We show that with small waiting costs, when the proportion of informed consumers
is low, it is cheaper for the high-quality firm to separate itself from the low-quality firm
by charging a low price and signaling its quality through the queue length. Conversely, we
expect that when waiting costs are high or the proportion of informed consumers is high,
separation is more cheaply achieved through a high price than through the queue length.
4
Conclusion
We have shown that, for the introduction of new and innovative products with uncertain
quality, queues can be a more efficient signaling instrument than prices. If the proportion
of informed consumers and the waiting costs are both low, and the service rate is an intermediate range, a firm with a high-quality product is willing to use congestion as a device to
communicate its quality to uninformed consumers. To credibly signal its quality by charging a high price, it needs to set a price so high that its profit is substantially reduced. As
a result, queues and congested services become the preferred signal of quality over price as
a signal.
We impose several restrictions on parameters in our model in the interest of parsimony.
For example, the prior places equal weight on both types of firms. Further, the consumption
value of the low-quality product is equal to half the consumption value of the high-quality
product. The parsimony allows us to demonstrate our results most cleanly.
The high-quality good in our model can be interpreted in terms of an improvement
to or a new version of a technology product such as a smart phone or app. Consider a
firm that sells an existing good with some quality `, priced at the full-information price
P`F I . Suppose the firm now develops a superior good with quality h, but not all consumers
understand that the quality has been improved. How can the firm communicate the quality
improvement to a large consumer base? It can increase price to PhF I (or higher), but the
consumers who are less informed about the product may balk from buying it, reducing
demand. Alternatively, it can continue to price at comparable price P`F I and allow the
increased demand from informed consumers to lead to congestion. We show the congestion
strategy can be beneficial even when the firm has the flexibility to raise its price.
Our model integrates elements of price signaling in economics and equilibrium queueing
in operations research. We hope that our research stimulates further interest in signaling
games in service operations environments.
20
5
Appendix: Proofs
Proof of Lemma 1
Consider a firm of type θ and let c/v → 0+ . In the limit, at any n, informed agents who
arrive join with probability 1− vPθ . That is, αθ (P, γ) approaches the vector 1 − vPθ , 1 − vPθ , · · · .
Since q = 1, informed agents arrive at the rate Λ. Therefore, the throughput approaches
o
n
R(αθ , P ) = min µ, Λ 1 − vPθ . If the throughput is unconstrained by µ, the profit of
firm θ is Πθ (P ) = Λ 1 − vPθ P . It is immediate that the optimal price is PθF I = v2θ ,
with associated profit Λ v4θ . The throughput in the limit at that price is R(αθ , v2θ ) =
Therefore, for µ ≥
Λ
2,
Λ
2.
the lemma stands.
Proof of Lemma 2
(i) Suppose θ = ` but uninformed consumersbelieve
firm has high quality. Then,
in
the
+
+ (1 − q) 1 − vPh
. If
the limit as c/v → 0+ , its throughput is min µ, Λ q 1 − vP`
n
h
io
P ≤ v` , the throughput is min µ, Λ 1 − P vq` + 1−q
, and if P > v` it reduces to
vh
min{µ, Λ(1 − q)(1 − P/vh )}. Substitute vh = v and v` =
v
2
to obtain part (i) of the Lemma.
(ii) Suppose θ = h but uninformed consumers believe the firm has low quality. Then,
io
n
h in the limit as c/v → 0+ , its throughput is min µ, Λ q 1 − vPh + (1 − q) 1 − vP`
o
n
if P ∈ [v` , vh ]. Suppose the throughput is unif P ≤ v` and min µ, Λq 1 − vPh
constrained by µ. Then, when P ≤ v` = v2 , the profit of firm h may be written as
h
i
Πh (P, 0) = Λ 1 − P vqh + 1−q
P , so that the optimal price is P ∗ = 12 q 11−q . At this
v`
price, the throughput is
Λ
2,
vh
+
v`
so it follows under Assumption 1 that µ exceeds the throughput.
Further, the profit is Π(P ∗ ) =
obtain P ∗ =
Λ
2P
=
Λ
4
1
q
+ 1−q
vh
v`
. Substituting in vh = v and v` =
v
2,
we
v
Λv
∗
2(2−q) and Π(P ) = 4(2−q) .
v When P ∈ 2 , v , the profit of firm h in the limit as c → 0+ is Λq 1 − Pv P , so the
optimal price is P ∗ = v2 . Now, the throughput is qΛ
µ, and the
2 , which is again less than n
o
max
Λv
1
firm’s profit is Π(P ∗ ) = Λqv
.
Comparing
both
cases,
we
obtain
Π
=
max
q,
h
4
4
2−q .
1
Now, for all q ∈ [0, 1], 2−q ≥ q. Therefore, if uninformed consumers believe firm h has
v
Λv
low quality, its optimal price is P ∗ = 2(2−q)
and its maximal profit is 4(2−q)
.
Proof of Proposition 1
From Lemma 1, when P ∗h = v2 , in the limit firm h earns its highest possible profit, so it
has no incentive to deviate.
21
Consider firm `. In the limit, if it charges a price P ` = v4 , its profit is Π∗` =
Λv
. Suppose
n8
o
v
v
.
it deviates to P = 2 . From Lemma 2 (i), at this price its profit is equal to 2 min µ, Λ(1−q)
2
As µ ≥
v
2,
Λ
2,
it follows that for all q ≤ 1, µ ≥
its profit is Π̃ =
Λ(1−q)
.
2
Therefore, if firm ` deviates to the price
Λ(1−q)v
.
4
Now, Π∗` ≥ Π̃ if and only if
1
1
2 ≥ 1 − q, or q ≥ 1/2. That is, when q > 2 , firm ` makes a
higher profit from setting P`∗ = v4 than by deviating to a price v2 .
Set the off-equilibrium belief that if P 6∈ {P ∗h , P ∗` }, then ψ(P ) = 0; that is, consumers
believe the firm has low quality. Note that at any price P 6= P ∗h = v2 , Πh (P, 1) < Π∗h .
Therefore, the off-equilibrium believe survives the restriction we discuss in Section 2.3, and
in turn the equilibrium satisfies the Intuitive Criterion.
Proof of Proposition 2
In any separating equilibrium, in the limit as c/v → 0+ , it must be that Π∗` =
Λv
4 ,
because
firm ` can always guarantee itself that profit even when uninformed consumers recognize
its type. Let Ph∗ be the price of the high firm. Then, Ph∗ = arg maxP ∈[0,1] Πh (P, 1) subject
to Π` (P, 1) ≤
Λv
4 ,
where the constraint ensures that firm ` has no incentive to deviate. At
equilibrium, the constraint must bind, else firm h can earn a higher profit by changing its
price by a small amount.
Let Dθ (P, 1) be the demand per unit of time for firm θ in the limit as c/v → 0+ (observe
that in the limit the demand does not depend on queue length n) and uninformed agents
believe the firm has high quality. Then, Dh (P, 1) = min{Λ(1 − P ), µ} and from Lemma 2
(i),
(
D` (P, 1) =
min{Λ[1 − P (1 + q)/v], µ} if 0 ≤ P <
min{Λ(1 − q)(1 − P/v), µ} if
v
2
≤P ≤v
Now, consider any two prices P 0 , P 00 such that each satisfy Π` (P, 1) =
It is straightforward to verify that
Πh
(P 00 , 1)
≥ Πh
(P 0 , 1).
D` (P,1)
Dh (P,1)
v
2
Λv
4
and P 00 > P 0 .
is weakly decreasing in P , so it must be that
Therefore, the profit of firm h at the price P 00 must exceed its profit
at P 0 .
Now, for notational simplicity, denote P = Ph∗ . Suppose that P ≥
v
2.
From Lemma
2 (i), if firm ` deviates to P , its profit is Π(P, 1) = Λ(1 − q)(1 − P/v)P . In a separating
2
v
equilibrium, this must equal Λ v8 . Consider the quadratic equation Pv − P + 8(1−q)
= 0. Its
q
q
1−2q
larger root is P 00 = v2 1 + 2(1−q)
, and the smaller root is P 0 = v2 1 − 1−2q
2−q . It is
immediate that P >
v
2
when q < 12 .
22
Now, as P > v2 , the throughput of firm h is strictly less than µ. Therefore, its limiting
q
1−2q
Λv
yields Πh = 8(1−q)
.
profit is Πh = ΛP (1 − P/v). Substituting in P = v2 1 + 2(1−q)
Set the off-equilibrium belief that ψ(P ) = 0 if P 6∈ {P`∗ , Ph∗ }. Suppose P < Ph∗ and
consumers believe the firm has high quality. Then, for any P in this range such that firm
h earns a higher profit from deviating, firm ` also earns a higher profit from deviating.
For any P > Ph∗ , neither firm h nor firm ` earn a higher profit from deviating, even if
consumers believe the firm has high quality. Therefore, this off-equilibrium belief satisfies
the restrictions we discuss in Section 2.3, so that the equilibrium survives the Intuitive
Criterion.
Finally, we show that given this belief, firm h has no incentive to deviate. In equilibrium,
firm h has a profit Πh =
maximal profit is Π̃ =
Λv
8(1−q) .
Λv
4(2−q) .
If it deviates to some other P , from Lemma 2 (ii), its
As 8(1 − q) < 4(2 − q), it follows that Πh > Π̃.
Proof of Remark 2.1
Let q = 1 and consider c/v → 0+ . In the limit, PθF I =
queue length n is
Λ
2,
vθ
2
and the joining rate for any
independent of θ. As µ is the same for both firms, the queue length
distributions must be the same. Now, consider a separating equilibrium in which Ph∗ > v2 .
The joining rate for firm h is now Λ(1 − Ph∗ ) < Λ/2, so the expected queue length must be
smaller than at PhF I , and hence smaller than the expected queue length for firm `.
Proof of Lemma 3
Consider any P ∈ (0, v/2 − c/µ). For any queue length n, let φ(n) denote the likelihood
that the firm has high quality. For notational simplicity, we suppress the dependence of
φ on P . Let x ∈ [0, 1] and define φ(0) = x. For any n ∈ {0, · · · , N̄ } such that φ(n)
is finite, let ψ(P ) = 1/2 and define the posterior belief that firm n has high quality as
γ(n) =
(1/2)φ(n)
(1/2)φ(n)+1/2
=
φ(n)
φ(n)+1 .
Given γ(n), for each θ determine the joining rate sθ from
(P,n,γ(n))
equation (1). Then, define φ(n + 1) = φ(n) ssh` (P,n,γ(n))
whenever s` > 0. If s` = 0, directly
set γ(n) = 1. For each θ = h, `, the strategy αθ = (sθ (P, 0, γ(0)), · · · , sθ (P, N̄ − 1, γ(N̄ −
1))) as defined induces a stationary distribution over the set {0, · · · , N̄ }, with stationary
probabilities given by equations (2) and (3). Define Φ(x, P ) =
π(0,αh )
π(0,α` ) .
Then, for any
x ∈ [0, 1] and any P > 0, the function Φ(x, P ) is well-defined.
Observe that when x = 1, γ(0) = 1/2. However, informed consumers join firm h at a
greater rate. Therefore, γ(1) < 1/2, so that αh (1, P, γ(1)) > α` (1, P, γ(n)). It then follows
that γ(2) ≤ γ(1), so that if αh (2, P, γ(2)) > 0, it must be that αh (2, P, γ(2)) > α` (2, P, γ(2)).
In general, it will follow that either αh (n, P, γ(n)) = α` (n, P, γ(n)) = 0 or αh (n, P, γ(n)) >
23
α` (n, P, γ(n)). As αh (·) weakly exceeds α` (·) for all n and strictly exceeds it for some n, it
follows that Φ(1, P ) > 1.
Next, consider x → ∞. Then γ(0) → 0, so that in the limit, uninformed consumers join
at the same rate as they would if the firm were known to have low quality. Hence, Φ(x, P )
remains finite as x → ∞. Now, by continuity of Φ in x, it follows that there exists at least
one x∗ (P ) such that Φ(x∗ (P ), P ) = x∗ (P ).
Choose any such x∗ (P ), set φ(0) = x∗ (P ), and construct γ and αh , α` as in the first
part of the proof. By construction, αh , α` satisfy Definition 1 (i) given γ, and γ satisfies
Definition 1 (ii) given αh , α` and ψ(P ) = 1/2.
Proof of Lemma 4
Throughout this proof, denote N = (v − P ) µc , so that nh = bN c and n` = b v/2−P
v−P N c.
Note that c/µ =
v−P
N .
Let h(n) denote the posterior expectation of an uninformed consumer
at price P and queue length n, so that h(n) = v/2 + γ(P, n)(v − v/2) = (1 + γ(P, n))v/2.
Thus, γ(P, n) =
h(n)−v/2
v/2
=
2h(n)−v
.
v
Then, the probability that a consumer joins the queue
for firm h at queue length n ∈ {0, 1, · · · , nh − 1} is
P + (n + 1)c/µ 2h(n) − v P + (n + 1)c/µ = q 1−
+ (1 − q) 1 −
sh P, n,
v
v
h(n)
n + 1 q 1 − q +
,
= 1 − P + (v − P )
N
v
h(n)
where the last equation uses
c
µ
=
v−P
N .
Similarly, the joining probability at queue length n for firm ` is

q
1−q
n+1

1 − (P + (v − P ) N ) v/2 + h(n) for n ∈ {0, 1, . . . , n` − 1}
2h(n) − v
s` P, n,
=
 (1 − q){1 − (P + (v − P ) n+1 ) 1 } for n ∈ {n` , . . . , nh − 1}.
v
N
h(n)
By assumption, for each n, the posterior expectation is h(n) = f
b(n)−v`
let s̃θ n+1
=
s
P,
n,
, to obtain equations (7) and (8):
θ
N
1−v`
n+1
N
. Suppressing P ,
1−q
s̃h (a) = 1 − (P + (v − P )a) q/v +
f (a)

 1 − (P + (v − P )a) q + 1−q
v/2
f (a) if a ≤ α`
s̃` (a) =
 (1 − q) 1 − (P + (v − P )a) 1
if a > α` .
f (a)
(7)
(8)
Denote ρ = Λ/µ. Observe that the stationary probability for firm θ at queue length k
24
is πθ (k) =
ρk Πk−1 s̃θ ( m+1
N )
PN m=0
m+1 .
s̃
1+ n=1 ρn Πn−1
m=0 θ ( N )
Therefore, we can write
k
ρ N exp
1
=
(πθ (k)) N
1
N
Pk−1
m+1
m=0 ln s̃θ ( N )
1 .
P
N N
PbN c n n−1
1
m+1
1 + n=1 ρ N exp N m=0 ln s̃θ ( N )
(9)
Consider the numerator of the RHS of equation (9) for some generic queue length k = αN ,
where α ∈ { N1 , N2 , · · · , nNh }. As c → 0+ , it follows that N → ∞. Therefore,
lim ρ
k
N
c/v→0+
= ρα exp
k−1
m + 1 1 X
1
bαN c
ln s̃θ
exp
= lim ρ N exp
N →+∞
N
N
N
m=0
1
N →+∞ N
bαN c−1
lim
X
m=0
m + 1 ln s̃θ (
)
= ρα exp
N
where the last step uses the fact that exp
R
Z
b
a ln(ρs̃θ (x))dx
α
bαN c−1
X
m=0
m + 1 ln s̃θ (
)
N
ln s̃θ a da = Gθ α ,
0
= ρb−a exp
R
b
a ln(s̃θ (x))dx
.
Applying similar algebra to the denominator, we obtain
1
lim (πθ (bαN c)) N
c/v→0+
Now, observe that
=
limN →∞
P
N
n=0
Gθ (α)
N 1 .
N
N
n=0 Gθ n/N
(10)
P
N 1
N 1
P
1
N
N
G
(n/N
)
Gθ (n/N )
= N N N1 N
. Beθ
n=0
1
cause limN →∞ N N = 1, it follows that
N 1 X
N 1
N 1
N
N
= lim
Gθ (n/N )
Gθ (n/N )
N →∞ N
N →∞
n=0
n=0
M Z 1
1 X
N 1
N
= lim lim
Gθ (n/M )
= lim
[G(a)]N da .
N →∞
N →∞ M →∞ M
0
lim
N X
n=0
Since G(·) is continuous and defined over the closed interval [0, 1], it follows that
R
1
N
1
limN →∞ 0 [G(a)]N da
= maxa∈[0,1] G(a) (this is the definition of the infinity norm).
1
Therefore, limc/v→0+ (πθ (bαN c)) N =
Gθ (α)
maxa∈[0,1] Gθ (a) .
Proof of Proposition 3
As in the proof of Lemma 4, define N = (v − P ) µc . Conjecture that there exists a β ∗
such that, in the limit as c/v → 0+ , the posterior belief of uninformed consumers is as
follows: If n < β ∗ N , then γ(P, n) = 1 and if n > β ∗ N , then γ(P, n) = 0. We will show that
the consumer best responses and the resulting stationary probabilities over queue lengths
lead to a posterior belief that satisfies the conjecture.
25
Let s̃θ
n+1
N
= sθ (P, n, 2h(n)/v − 1) denote the mass of consumers that join the queue
of firm θ at length n, where h(n) denotes the posterior expectation of value for uninformed
consumers. Given the conjectured beliefs, equations (7) and (8) in the proof of Lemma 4
imply that for each a =
n+1
N ,
in the limit as c/v → 0+ ,
(
1 − (P + (v − P )a)( vq +
s̃h (a) =
(1 − a)(1 − P/v)
(
s̃` (a) =
1−q
v/2 )
if a < β ∗
if a > β ∗
1
1 − (P + (v − P )a) v/2
if a < β ∗
(1 − q)(1 − a)(1 − P/v)
if a > β ∗
Now, define β ∗ to satisfy the following equation, where Gθ is defined equation (6) in the
text:
Gh (β)
G` (β)
=
.
maxa∈[0,1] G` (a)
maxa∈[0,1] Gh (a)
We proceed through a series of steps.
Step 1 : Characterizing β ∗ .
Denote α̂θ as the global maximizer of Gθ (α). Suppose that (i) α̂` = 0 and αˆh > β ∗ ,
(ii) β ∗ ∈ (0, α` ), where α` =
v/2−P
v−P
denotes the scaled queue length corresponding to n` .
These conjectures will be verified later. Because G` (0) = 1, β ∗ then satisfies the equation
G` (β) =
Gh (β)
Gh (α̂h ) .
By assumption in the statement of the proposition, ρ(1−2P/v) < 1. Therefore, ρs̃` (0) <
1. Further, s̃` (a)ρ is decreasing for a < β ∗ . Hence, G` (α) is decreasing over the range [0, β ∗ ),
so has a local maximum at a = 0 (that is, at α̂` ). We verify in Step 4 that α̂` is a global
maximum of G` (α) over [0, 1].
Now, α̂h must satisfy G0h (a) = 0. Note that G0h (a) = Gh (a) ln(ρs̃h (a)), so α̂h satisfies
s̃(α̂h ) = ρ1 . For α > β, s̃h (α) = (1 − α)(1 − P/v), so that we have α̂h = 1 −
1
ρ(1−P/v) .
As
assumption ρ(1 − P/v) > 1 by assumption, we know that α̂h > 0. It is immediate to verify
that the second order condition G00 (α̂h ) < 0 holds. We verify in Step 5 that α̂h is a global
maximum of Gh (α) over [0, 1].
Observe that for α > β we can write Gh (α) = Gh (β)ρα−β exp
R
α
β ln s̃h (a)da
. Now,
Rα
ln[(1−P/v)(1−a)]da = −(1−α) ln[(1−P/v)(1−α)]−α+(1−β) ln[(1−P/v)(1−β)]+β, so
R
α
exp β ln[(1 − P/v)(1 − a)]da = [(1−P/v)(1−α)]−(1−α) [(1−P/v)(1−β)]1−β exp(−α+β).
β
Further, we have shown (1 − P/v)(1 − α̂h ) = 1/ρ, so
Gh (α̂h ) =
and
Gh (β)
Gh (α̂h )
Gh (β)
[ρ(1 − P/v)(1 − β)]1−β ,
exp(α̂h − β)
= exp(α̂h − β)ρ−(1−β) (1 − P/v)−(1−β) (1 − β)−(1−β) .
26
(11)
Now, consider G` (β) when β < α` . In this region, s̃` (a) = 1 − (P + (1 − P )a)/(v/2) =
1 − 2P/v − 2(1 − P/v)a. This expression is of the form y − za, where y = 1 − 2P/v and
R
y
z = 2(1−P )/v. Note that yz = v/2−P
v−P = α` . Further, ln(y−za)da = a − z ln(y−za)−a =
i
h (a − α` ) ln y 1 − αa` − a. Therefore,
Z
β
0
h
iβ
a ln(ρs̃` (a))da = β ln ρ + (a − α` ) ln y 1 −
−a
α`
0
= β ln ρ + β ln y + (β − α` ) ln(1 − β/α` ) − β.
Hence,
G` (β) = ρβ exp(−β)(1 − β/α` )β−α` (1 − 2P/v)β .
(12)
Putting together equations (11) and (12), we obtain that β ∗ is the solution to:
1
ρ[(1 − P/v)(1 − β)]1−β (1 − 2P/v)β (1 − β/α` )β−α` = exp 1 −
. (13)
ρ(1 − P/v)
Step 2 : Verifying that β ∗ ∈ (0, α` ).
Observe that
αˆh < α` ⇐⇒ 1 −
1
1 − 2P/v
1
1
1
1
<
⇐⇒ 1 − P/v − < − P/v ⇐⇒
> ,
ρ(1 − P/v)
2(1 − P/v)
ρ
2
ρ
2
which has been assumed in Assumption 1. We show that β ∗ ∈ (0, α̂h ), which then ensures
that β ∗ ∈ (0, α` ). Specifically, we evaluate equation (13) at β = 0 and β = α̂h . We show
that at β = 0, the LHS of equation (13) exceeds the RHS, whereas at β = α̂h , the RHS
exceeds the LHS.
Consider equation (13) when β = 0. The left-hand side is ρ(1 − P/v) > 1 by assumption
1
. Now, for any x > 1,
in the statement of the proposition. The RHS is exp 1 − ρ(1−P/v)
we have x > exp(1 − 1/x), so at β = 0, the LHS of equation (13) exceeds the RHS.
Consider equation (13) when β = α̂h . Recall that (1 − P/v)(1 − α̂h ) = 1/ρ. Therefore,
the LHS of equation (13) is equal to [ρ(1 − 2P/v)]α̂h (1 − α̂h /α` )α̂h −α` .
Note that ρ(1 − P/v) =
1
1−α̂h
and ρ(1 − 2P/v) = 2ρ(1 − P/v)α` =
2α`
1−α̂h .
Thus, for the
LHS of equation (13) to be strictly smaller than the RHS at β = α̂h , we want to show that:
2α α̂h
`
(1 − α̂h /α` )α̂h −α` < exp(α̂h )
1 − α̂h
or, (α̂h − α` ) ln(1 − α̂h /α` ) + α̂h [ln(2α` ) − ln(1 − α̂h ) − 1] < 0.
(14)
We show that inequality (14) holds for all α̂ ∈ (0, α` ). Define ξ(x) = (x − α` ) ln(1 − x/α` ) +
x[ln(2α` ) − ln(1 − x) − 1]. Observe that ξ(0) = 0. We show that ξ 0 (0) = 0 and ξ 0 (x) < 0 for
27
all x > 0 such that ξ(x) = 0. Note that
x − α` 1
x
ξ 0 (x) = ln(1 − x/α` ) +
−
+ ln(2α` ) − ln(1 − x) − 1 +
.
(15)
1 − x/α`
α`
1−x
When ξ(x) = 0, we have ln(2α` ) − ln(1 − x) − 1 = αx` − 1 ln(1 − x/α` ). Making this
substitution on the RHS of equation (15) and simplifying, ξ 0 (x) < 0 is equivalent to
α`
1
x
ln(1 − x/α` ) +
< 0, or, α` ln(1 − x/α` ) +
< 0.
x
1−x
1−x
Denote ζ(x) = α` ln(1 − x/α` ) +
`
Then, ζ(0) = 0. Further ζ 0 (x) = − α`α−x
+
x
1−x .
Therefore, when x > 0, ζ 02 > 1 − x/α` ⇐⇒
Therefore,
1
α`
(16)
1
α`
+ αh > 2. Now, α` =
1
2
×
1
.
(1−x)2
1−2P/v
1
1−P/v < 2 .
> 2, and the last inequality holds. That is, ζ(0) = 0 and ζ 0 (x) < 0 for all
x ∈ (0, α` ). Therefore, ζ(x) < 0 for all x ∈ (0, α` ), so that inequality (15) holds. That
is, we have shown that for any x ∈ (0, α` ), if ξ(x) = 0, then ξ 0 (x) < 0. As ξ(0) = 0 and
ξ 0 (0) = 0, this implies that ξ(x) < 0 for all x ∈ (0, α` ). That is, inequality (14) holds for all
α̂h ∈ (0, α` ), so that, at β = α̂h , the LHS of equation (13) is strictly smaller than the RHS.
Hence, we have β ∈ (0, α̂h ), so that β ∈ (0, α` ).
Step 3 : Uniqueness of β ∗ .
R
exp( 0α ln s̃` (α))
Rα
exp( 0 ln s̃h (α))
R
α
da
. Further, q > 0 implies that
= exp 0 ln s̃s̃h` (α)
(α)
R α s̃` (α)
G` (α)
s̃` (α) < s̃h (α) for all α < 1. Therefore, 0 ln s̃h (α) da is decreasing in α, so G
is strictly
h (α)
Observe that
G` (α)
Gh (α)
=
decreasing in α. Now, β ∗ is defined by the equation
G` (β ∗ )
Gh (β ∗ )
=
1
Gh (α̂h ) .
Because
G`
Gh
is strictly
monotone, this equation can have at most one solution.
Step 4 : G` (a) has a global maximum at a = α̂` = 0.
In Step 1, we have shown that G` (a) decreases over [0, β ∗ ), and G` (α̂` ) = 1. Now, we
argue that there cannot be a global maximum in (β ∗ , 1). In this region, s̃` (a) = (1 − q)(1 −
a)(1 − P/v). Recall that β ∗ satisfies the equation G` (β) =
∗
G` (β ) exp
Z
α̂h
β∗
Gh (β)
Gh (α̂h ) ,
ln(ρs̃h (a))da
= 1.
so that
(17)
Suppose that G` (a) has a local maximum at some α`0 > 0. To find α`0 , set G0` (a) = 0. This
R 0
α
1
yields α`0 = 1 − ρ(1−q)(1−P/v)
< α̂h . Observe that G` (α`0 ) = G` (β ∗ ) exp β ∗` ln(ρs̃` (a))da .
The second term in the latter expression is strictly smaller than the corresponding term in
equation (17), whereas the first terms are equal. Therefore, G` (α̂`0 ) < 1, so that G` (a) has
a global maximum at a = α̂` (= 0).
Step 5 : Gh (a) has a global maximum at a = α̂h .
28
The maximum value of Gh (a) in the region a ∈ [β ∗ , 1] is attained at α̂h . Consider the
maximum value in the region a ∈ [0, β ∗ ]. In this region,
q 1−q
s̃h (a) = 1 − (P + (v − P )a)( +
).
v
v/2
1−1/ρ
v 1+q −P
0
Define
to be the value of a at which ρs̃h (a) = 1, so that αh =
.
v−P
Now, there are three possibilities. First, suppose that αh0 < 0. Then, Gh (a) decreases
over [0, β ∗ ], so the maximum value in this region is attained at a = 0. Note that Gh (0) = 1.
Gh (β ∗ )
and G` (β ∗ ) < Gh (β ∗ ). Therefore, Gh (α̂h ) > 1, so
However, recall that G` (β ∗ ) = G
h (α̂h )
αh0
that α̂h is indeed a global maximizer of Gh .
Second, suppose that αh0 > β ∗ . Then, Gh (a) increases over [0, β ∗ ], and the maximum is
attained at a = β ∗ . As Gh (α̂h ) > Gh (β ∗ ), it follows that α̂h is a global maximizer of Gh .
Finally, suppose that αh0 ∈ (0, β ∗ ).
R
α̂
Gh (αh0 ) exp α0h ln(ρs̃h (a))da . Thus,
As α̂h > β ∗ > αh0 , we can write Gh (α̂h ) =
h
Z
Gh (α̂h )
=
exp
Gh (αh0 )
β∗
α0h
1+q
ln(ρ(1 − (P + (v − P )a)
))da +
v }
{z
|
Z
α̂h
β∗
ln(ρ(1 − a)(1 − P/v))da
|
{z
}
<0
>0
h (α̂h )
By definition of β ∗ , G` (β ∗ ) G
Gh (β ∗ ) = 1 = exp(0). This implies that:
Z
0
β∗
1
ln(ρ(1 − (P + (v − P )a)
))da +
v/2
Z
α̂h
ln(ρ(1 − a)(1 − P/v))da = 0.
β∗
Thus, solving for the second expression and substituting into the RHS of
Gh (α̂h )
=
exp
Gh (αh0 )
Z
β∗
ln(ρ(1 − (P + (v − P )a)
α0h
1+q
)da −
v
Z
Gh (α̂h )
,
Gh (α0h )
we obtain:
β∗
ln(ρ(1 − (P + (v − P )a)
0
1
Now, in the region a ∈ [0, β ∗ ], we have ρ(1 − (P + (v − P )a) v/2
< 1, so that
R
∗
β
1
1
(P + (v − P )a) v/2 )da < α0 ln(ρ(1 − (P + (v − P )a) v/2 )da. Therefore,
R β∗
0
1
)da
v/2
ln(ρ(1 −
h
Gh (α̂h )
= exp
Gh (αh0 )
Z
Observe that, as
β∗
α0h
1+q
v
1+q
ln(ρ(1 − (P + (v − P )a)
)da −
v
Z
β∗
ln(ρ(1 − (P + (v − P )a)
α0h
1
)da
v/2
< v2 , the term inside the exponential term in the previous equation is
strictly positive. Therefore,
Gh (α̂h )
Gh (α0h )
> 1. That is, Gh (α̂h ) > Gh (α̂h0 ), and α̂h is the global
maximizer of Gh .
Step 6 : The best response strategy of the consumers leads to the conjectured belief for
uninformed consumers.
29
We have shown that
greater than 1 for α
G` (α)
Gh (α)
< β∗
π` (bαN c)
πh (bαN c)
1
G` (α)
=G
Gh (α̂h ).
h (α)
1
c) N
is strictly decreasing in α. Therefore, limc/v→0+ ππh` (bαN
is
(bαN c)
From Lemma 4, using G` (0) = 1, it follows that limc/v→0+
N
and less than 1 for α > β ∗ . It follows that limN →∞
π` (bαN c)
πh (bαN c)
is
infinite if α < β ∗ and 0 if α > β ∗ . Then, γ(P, n) = 0 for n < β ∗ N and γ(P, n) = 1 for
n > β ∗ N . That is, in the limit as c → 0+ , the consumer best response strategies indeed
generate the conjectured belief. By continuity of the G` and Gh functions, the result holds
approximately when c is close to zero. That is, there exist N̄ (alternatively c̄), c and δc such
that if N > N̄ (alternatively, c < c̄) then γ(n, P ) < c if α < β ∗ − δc and γ(n, P ) > 1 − c
if α > β ∗ + δc . Further, c , δc → 0+ as N → ∞ (or c/v → 0+ ).
Proof of Proposition 4
Observe that when P = P ∗ = v/4, the condition 1 − 2P/v < µ/Λ < 1 − P/v in the
statement of Proposition 3 reduces to 1/2 < µ/Λ < 3/4. Therefore, Proposition 3 applies
when the latter condition is satisfied. In the rest of the proof, we show that (a) no firm has
an incentive to deviate and (b) when q < 1 − 2Λ/µ, the off-equilibrium beliefs satisfy the
restrictions motivated by the Intuitive Criterion.
Define q̄ = 1 −
P∗
Λ
2µ .
For notational convenience, in the rest of the proof, we denote
∗
=P .
(a) Let the off-equilibrium belief be ψ(P ) = 0 for any P 6= P ∗ . Then, firm ` clearly has no
incentive to deviate, since its maximal deviation profit is also
Λv
8 .
From Lemma 2, given the
equilibrium belief, the maximal profit of firm h if it deviates is (in the limit as c/v → 0+ )
Λ
equal to Πmax
h
4(2−q) .
Recall that the length of the queue reveals the type of the firm. Thus, when P ∗ = v4 ,
the consumer demand for firm h in the limit as c/v → 0+ is Λ(1 − P ∗ /v) = 3Λ/4. By
assumption, this exceeds the service rate µ. Therefore, the throughput equals µ, and the
equilibrium profit of firm h in the limit is
deviation, it must be that
µv
4
≥
Λv
4(2−q) ,
µv
4 .
Now, for firm h to not have a profitable
or q ≤ 2 −
Λ
µ
= 2q̄.
(b) Suppose that, after a deviation, uninformed consumers believe the firm has high quality.
We show that, if firm h strictly gains by deviating, firm ` also strictly gains by deviating.
Define P = {P | Πh (P, 1) > Π∗h }. We proceed in two steps.
Step 1 : We show that P = (P ∗ , P̄ ) for some P̄ .
Suppose firm h deviates and uninformed consumers believe the firm has high quality.
The maximal throughput of firm h at any price is µ. Therefore, at any price P < P ∗ , the
30
maximal profit it can earn is µP <
µv
4 .
Hence, there is no price P < P ∗ in P.
Consider prices above P ∗ . Given that consumers believe the firm has hiqh quality, the
throughput of firm h at such prices is min{µ, Λ(1−P/v)}. Recall that ρ =
Λ
µ.
Then, for any
price P < v(1 − 1/ρ), the throughput is µ and for any price P > v(1 − 1/ρ) the throughput
is Λ(1 − P ). It is immediate that (P ∗ , v(1 − 1/ρ)] ⊂ P.
Consider prices P > v(1 − 1/ρ). The profit of firm h after the deviation is ΛP (1 −
P/v). The profit-maximizing price in the region [v(1 − 1/ρ), v] is P̃ =
v
2.
As Πh (v(1 −
1/ρ), 1) > Πh ∗ , and Πh (P, 1) is increasing in the region [v(1 − 1/ρ), v/2], it follows that
[v(1 − 1/ρ), v/2] ⊂ P. In the region [v/2, v], the condition ΛP (1 − P/v) ≥
p
d
P ≤ v2 1 + 1 − 1/ρ = P̄ . Therefore, we have P = (P ∗ , P̄ ).
µv
4
implies
Step 2 : We now need to show that Π` (P, 1) > Π` ∗ for all P ∈ P. Observe that P̄ > v/2.
As the throughput for firm ` changes at P = v/2, we break up this part of the proof into
two sub-steps. In Case (i) below, we consider prices in the range (P ∗ , v/2]. In Case (ii), we
consider prices in the range (v/2, P̄ ).
Case (i) First, consider prices in the range (P ∗ , v/2]. Suppose firm ` charges a price in this
range and uninformed consumers believe the firm has high quality. Then, the throughput
q
+
of firm ` is min{µ, Λ(1 − P ( v/2
1−q
v ))}
= min{µ, Λ(1 − P (1 + q)/v)}. The deviation
profit, in turn, is Π` (P, 1) = min{µP, ΛP (1 − P (1 + q)/v)}. We show that each of the two
components exceed Π∗` for all P ∈ (P ∗ , v/2].
First, consider the component µP . This component is strictly increasing in P . Further,
at P = P ∗ , this component is µv/4 > Π∗` = Λv/8, where the inequality follows from
1/ρ > 1/2. Therefore, the component µP strictly exceeds Π∗` for all P ∈ (P ∗ , v/2].
Next, consider the component ΛP (1 − P (1 + q)/v). Observe that this component is
strictly concave, so cannot have an interior minimum on the set P ∈ [P ∗ , v/2]. Therefore,
it is sufficient to show that it exceeds Π∗` at the two corner points, P ∗ and v/2.
Consider a sequence of prices P ∗ + where > 0, and take the limit of the component
ΛP (1 − P/v) as ↓ 0. In the limit, the value of the component is ΛP ∗ (1 − P ∗ (1 + q)/v) =
Λ(v/4)(1 − (1 + q)/4) = Π∗` ×
3−q
2
> Π∗` as q < 1.
Next, consider the price P = v/2. The value of the component is then Λ(v/2)(1 − (1 +
q)/2) = Π∗` × 2(1 − q). Now, by assumption in the statement of the proposition, q < 1 − ρ/2.
As ρ > 1, we have q < 1/2, so that 2(1 − q) > 1. Therefore, the value of the component at
P = v/2 exceeds Π` .
Hence, for all prices P ∈ (P ∗ , v/2], we have ΛP (1 − P (1 + q)/v > Π∗` . As we have
also shown that µP > Π∗` over this range, it follows that for all P ∈ (P ∗ , v2], we have
31
Π` (P, 1) > Π∗` .
Case (ii) Next, consider prices in the range (v/2, P̄ ). Suppose firm ` charges a price
in this range and uninformed consumers believe the firm has high quality. Then, the
throughput of firm ` is min{µ, Λ(1 − q)(1 − P/v)}.
The deviation profit, in turn, is
Π` (P, 1) = min{µP, Λ(1 − q)(1 − P/v)P }. We show that each of the two components
exceed Π∗` for all P ∈ (v/2, P̄ ).
First, consider the component µP . It is immediate that µv/2 > Π∗` = Λv/8, as
1
ρ
> 1/2.
Further, µP is strictly increasing in P . Therefore, µP > Π∗` for all P ∈ (v/2, P̄ ).
Next, consider the component Λ(1 − q)(1 − P/v)P . Observe that this component is
strictly concave in P , so cannot have an interior minimum over the range P ∈ (v/2, P̄ ).
Therefore, it is sufficient to show that it exceeds Π∗` at the two corner points P = v/2 and
P = P̄ .
At P = v/2, the value of the component is Λ(1 − q)v/4 = Π∗` × 2(1 − q). As argued in
Case (i) above, 2(1 − q) > 1 whenever q < 1 − ρ/2. Therefore, at P = v/2, this component
strictly exceeds Π∗` .
Consider the price P = P̄ = v2 (1 +
p
1 − 1/ρ). At this price, the value of the component
simplifies to Λ v4 (1 − q)/ρ = Π∗` × 2(1 − q)/ρ. Now, q < 1 − ρ/2 implies that 2(1 − q)/ρ > 1,
so that the value of the component strictly exceeds Π∗` at P = P̄ . Hence, the component
Λ(1 − q)(1 − P/v)P is strictly greater than Π∗` at all P ∈ (v/2, P̄ ).
Putting together Cases (i) and (ii), we have shown that for all P ∈ P, Π` (P, 1) > Π∗` .
Therefore, the belief ψ(P ) = 0 for all P 6= P ∗ satisfies the restrictions motivated by the
Intuitive Criterion.
Proof of Proposition 5
(a) Suppose the conditions in Proposition 4 are satisfied. Observe that µ < Λ implies that
Λ
2µ
> 12 , so that q̄ < 21 . Therefore, q ≤ q̄ implies that q <
It follows that there is a separating equilibrium with
1
2 so
Ph∗ > v2 .
that Proposition 2 applies.
In each of the separating and pooling equilibria considered, the profit of firm ` is
In the limit as c/v → 0+ , the profit of firm h is
µv
4
in the pooling equilibrium and
Λv
8 .
Λv
8(1−q)
in
the separating equilibrium. It follows that the profit in the separating equilibrium is weakly
higher whenever
Λ
2(1−q)
≥ µ, or q ≤ q̄ = 1 −
Λ
2µ .
32
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