GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 311

Waiting For Signalling Quality

Hikmet Gunay

August 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Waiting For Signalling Quality
Hikmet Gunay∗
University of Manitoba
July 26, 2013

Abstract
When a durable good of uncertain quality is introduced to the market, some consumers strategically delay their buying to the next period with the hope of learning
the unknown quality. We analyze the monopolist’s pricing and “waiting” strategies
when consumers have strategic delay incentives. We show when the monopolist oﬀers
introductory low prices in pooling equilibria. We also ﬁnd two types of separating equilibria: one where high type signals its quality by choosing a diﬀerent price than the low
type in the ﬁrst period, and another where the high-type monopolist announces the
product in the ﬁrst period and waits to sell only in the second period. Waiting createsa credible cost for signalling; hence, the monopolist uses it as a signalling device.

JEL Codes: C73, D42, D83, L12
Keywords: Strategic delay, Introductory prices, Penetration pricing, Cream-skimming,
Monopoly pricing, Coasian dynamics.

∗

I am greatful to M. Aoyagi and anonymous referees for their suggestions. I also thank I. Ishibashi,
S.Loertscher, M. Mitchell, R. Wang, S. M. Gorricho for their suggestions on an earlier version, and Osaka
ISER for hosting me during doing this research, and U of Manitoba for providing ﬁnancial assistance. I
thank Xin Meng for her superb research assistance. An earlier version of this paper is presented at Bilkent,
Ryerson, Osaka (ISER), York, Melbourne, Saskatchewan, Windsor, Ritsumeikan, Hiroshima, Kobe, and the
IIOC Conference held in Savannah, Midwest Economic Theory Conference held at the U. of Minnesota andDelta Marsh Conference held at the U. of Manitoba; I thank the participants for their suggestions. The
usual disclaimer applies.

11

Introduction

Microsoft ﬁnished Windows 7 before July, 2009; however, the launch date was set as October22, 2009.1 In the meantime, consumers heard about the quality of Windows 7 -e.g., comparison with Vista- through magazines such as Business Week. In this paper, we give one
possible explanation of why a monopolist announces a later launch date rather than sellingits product right away. We show that such a waiting strategy signals high quality.
In this paper, we not only analyze seller’s waiting incentives but also consumers’ waiting
incentives. Speciﬁcally, when a new product is introduced to the market, only a few consumers buy it right away. The others wait and then decide whether or not to buy. We showthat there are two reasons for this waiting. First, consumers wait to learn more about theunknown quality of the good (strategic delay eﬀect, an eﬀect neglected in the literature).
Second, they wait to take advantage of the lower price, if the price will decrease (price eﬀect,
a well-known reason in the literature). We analyze ﬁrms’ pricing strategies with these two
eﬀects and relate it to the practice of low introductory prices, and the strategy of waiting
before selling.
In our model, a continuum of consumers decide whether to buy one unit of a durable
good in one of the two periods. A consumer’s total willingness to pay for the good depends
on the consumer’s taste for quality (private value) and the good’s unknown intrinsic quality(common value). Consumers’ taste for quality is uniformly distributed. The quality of the
product is revealed in the second period. This uncertainty resolution makes the consumers
with moderate preferences delay their buying strategically. By delaying her purchase, theconsumer avoids buying a low quality good. However, she gives up the utility she could havederived from a high quality good in the ﬁrst period. The monopolist who knows its product’s
quality and decides how to price it and when to start selling it when there are both strategic
delay and price eﬀects.
We use Perfect Bayesian Equilibrium to solve our two period signalling model. We ﬁnd
two types of separating and one type of pooling equilibria depending on the parameters.
In all pooling equilibria, we ﬁnd the same suﬃcient condition for the monopolist to oﬀer
1

Business Week, October 26, 2009, page 71.

2

introductory low prices or to use a cream-skimming strategy.
The monopoly pricing literature is a huge one, and can be divided into durable goods(e.g. Coase (1972), Gul et. al. (1986), Stokey (1979)) and experience goods literature. The
experience goods literature focuses on introductory pricing (e.g., Shapiro (1983), Cremer(1984), Bagwell (1987), Schlee (2001), Bergemann and Valimaki (2006)) or price discrimination to new and repeat customers (Jing (2011)).2 In the aforementioned “experience goods”
papers, consumers learn their private valuation by “experiencing” the good; hence, they neither discuss signalling quality nor consumers’ strategic delay. (unlike this “durable goods”
paper).3 In other experience goods papers (e.g. Milgrom and Roberts (1986)), signalling via
pricing is discussed but consumers’ strategic delay incentive does not exist.
We have two kinds of separating equilibria. In the ﬁrst one, the high type monopolist
announces the product launch date in the ﬁrst period and sells its product only in the second
period although the product is ready from the start. Our paper’s innovation is to show that
waiting before selling serves as a signalling strategy, and that consumers have the option ofdelaying their buying. There are “vaporware” papers that show why some ﬁrms announce
launch dates before the product is 𝑟𝑒𝑎𝑑𝑦 (e.g., Bayus et al. (2001), Dranove and Gandal(2003), Kristiansen (2006), Choi et al. (2010)).4 Unlike these papers, our ﬁrm’s product isready in the ﬁrst period but the ﬁrm chooses not to sell at that time in order to signal its
quality.5 Waiting creates a credible cost for signalling; hence, the high-type monopolist uses
this as a signalling device.
In the other type of separating equilibrium, low and high types each chooses diﬀerent
prices in the ﬁrst period. Both low and high type use high and declining prices in such anequilibrium; in order to sell to the residual “low willingness to pay consumers” in the second
period, they have no choice but to lower the price. Our model is diﬀerent than Bagwell and
2

These papers generally discuss cream-skimming like our paper.
Van Ackere and Reyniers (1995) is a quasi-durable good paper that explains introductory oﬀers (andtrade-ins). Their model also does not have strategic delay.
4
Vaporware means falsely announcing product launch dates.
5
Some ﬁrms present their products in expos for the public; hence, experts and consumers can verify thatthe product is ready. For example, Sony showed its PlayStation 4 console (PS4) in E3-2013 Expo but Sonymade the announcement that it will release PS4 in Holiday 2013 season (which is November) according tovarious websites including www.pcadvisor.co.uk
33

Riordan (1991) (quality is learned in the second period, there are no expert consumers etc.)
so the diﬀerent result, in the sense that low type also uses high and declining price, is not
surprising.
The theme of this paper is also related to the literature on the durable good monopolist
in which buyers’ incentive to delay purchases causes the monopolist to reduce prices immediately (Coase (1972), Gul et. al. (1986)) and the intertemporal price discrimination literaturein which prices fall as time passes (Stokey (1979), Landsberger and Meilijson (1985), Tirole(1989)). Our paper extends these papers to allow for uncertainty on the common value
component (quality) of consumers’ total willingness to pay. We show that price increase ispossible in the short term, although “the higher willingness to pay” consumers, in terms ofpure private value, bought their unit demand in the ﬁrst period and exited the market. In
the second period, the residual “lower willingness to pay” consumers learn that the good’s
quality is much higher than their expectation. Then, the high-type monopolist can charge ahigher second period price in a pooling equilibrium.
Our paper also falls into the strategic delay literature (e.g., Aoyagi (1998), Heidhues andMelissas (2006), Gunay (2008a), Gunay (2008b)). To our best knowledge, there is no paperin this literature that shows how a monopolist will price its good when facing consumers who
delay their buying strategically. Bose et. al. (2008) discuss how a monopolist price its goods
when faced by one consumer at each period or Bhalla (2011) discusses how a monopolist
will price its good when faced with non-strategic consumers; however, their consumers do
not have strategic delay incentives.6
In what follows, we set up our model. We deﬁne our Perfect Bayesian Equilibrium in
the Appendix. We use this equilibrium concept to show what type of equilibria exist. We
discuss various assumptions and generalizations at the end. All proofs can be found in the
Appendix.
6

Lopomo and Squintani (2007) also discuss whether the seller will choose to sell in the spot market or
forward market. They assume all consumers have the same type, and the seller has a diﬀerent discountfactor unlike us. As a result, we ﬁnd diﬀerent separating equilibria.

4

2

The Model

A monopolist’s product is exogenously set to be either high quality 𝛾ℎ = 1 or low quality
𝛾𝑙 ∈ [0, 1). The monopolist has private information about his type, 𝛾𝑖 ∈ Ψ = {1, 𝛾𝑙 }. In
the ﬁrst period, he determines the price, 𝑝𝑖1 ∈ [0, ∞) (where 𝑖 = 𝐻, 𝐿 depending on quality)
or announces the launch date as the second period. That is, his action set in the ﬁrst∪
period is 𝐴1𝑀 = [0, ∞) {announce}. A continuum of consumers are indexed by their taste
parameters 𝜃 which is uniformly distributed on Θ = [0, 1]. Each consumer has a unit demandand believes that 𝛾𝑖 = 1 with probability 𝛼 > 0 and 𝛾𝑖 = 𝛾𝑙 , with probability 1-𝛼.7 Afterobserving the monopolist’s action 𝑎1𝑀 , each consumer updates her belief. We denote thisﬁrst period posterior by 𝜇1𝐵 (𝛾𝑖 = 1∣𝑎1𝑀 ). Given their beliefs, each consumer chooses anaction 𝑎1𝐵 ∈ 𝐴1𝐵 ={Buy, Delay}.8 The monopolist does not know the type of consumers
but knows their taste distribution; therefore, the monopolist cannot price discriminate inany given period.
SECOND PERIODFIRST PERIOD

Monopolist sets the price 𝑝𝑖2
Consumers observe quality and 𝑝𝑖2 .
Then decide whether to buy.

Monopolist observes
Consumers
quality andupdate priors.
sets the price 𝑝𝑖1
Decide whether
or announces launch date. to buy.

Figure 1: TIMING OF THE GAMEAt the beginning of the second period, the true quality of the product is revealed to
the “informed” consumers, and in this section we assume all consumers are informed.9 Letℎ = [𝑎1𝐵 (𝜃, 𝑎1𝑀 ), 𝑎1𝑀 ∈ 𝐴1𝑀 , 𝛾𝑖 ] ∈ 𝐻 denote the history that summarizes the actions taken
by each player in the ﬁrst period, the quality revealed in the second period and 𝐻 denote thehistory space. In period 2, the monopolist chooses its second period price 𝑝2 ∈ 𝐴2𝑀 = [0, ∞)7This is a durable good since consumers have a unit demand.
If “announce” is chosen by the monopolist, then all consumers delay in the ﬁrst period.
9
The justiﬁcation of this assumption is as follows. Consumer Report style magazines may reveal the
product once it is on the market or if some consumers bought the product, they may reveal the true quality
to the other consumers by using internet or other forms of media; they have no incentive to misrepresent
the true quality since re-selling is not modelled in this paper. We also relax this assumption later byassuming only a fraction 𝛽 of consumers learn the true quality in the second period. Informed
consumers, their type denoted by 𝜏 = 𝐼, learn the quality, and uninformed consumers 𝜏 = 𝑈
do not learn the quality (unless it is a separating equilibrium).
85and then each consumer chooses an action 𝑎2𝐵 ∈ 𝐴2𝐵 ={Buy, Do not Buy}. Consumers
beliefs in this period are 𝜇2𝐵 : 𝐻 ×𝐴2𝑀 ×{𝐼, 𝑈 } 7→ [0, 1]. Since we assume that all consumers
are informed (𝜏 = 𝐼) in this section, we have 𝜇2𝐵 = 1 if 𝛾𝑖 = 1; otherwise 𝜇2𝐵 = 0.
A (pure) behavioral strategy 𝜎𝑀 of the monopolist consists of two mappings: First,
𝜎1𝑀 : Ψ = {𝛾𝑙 , 1} 7→ 𝐴1𝑀 . Second, 𝜎2𝑀 : 𝐻 × Ψ 7→ 𝐴2𝑀 .
A (pure) behavioral strategy 𝜎𝐵 of the buyers/consumers consists of two mappings: First,
𝜎1𝐵 : Θ = [0, 1] × 𝐴1𝑀 7→ 𝐴1𝐵 .10 Second, 𝜎2𝐵 : 𝐻 × 𝐴2𝑀 × Θ 7→ 𝐴2𝐵 .
Both the consumers and the monopolist have the same discount factor 𝛿 ∈ (0, 1]. A type𝜃 who buys the good in the ﬁrst period derives an expected utility of (1 + 𝛿)[(𝜇1𝐵 + (1 −𝜇1𝐵 )𝛾𝑙 )𝜃 − 𝑝𝑖1 ] after observing the price 𝑝𝑖1 . Instead, if she delays buying to the second period,
then, looking from the ﬁrst period, she derives an expected utility of 𝛿[𝜇1𝐵 (𝜃1 − 𝑝𝐻2 ) + (1 −𝜇1𝐵 )(𝜃1 𝛾𝑙 − 𝑝𝐿2 )]. If she does not buy it at all, she derives a zero utility.
Both type of monopolist’s marginal and ﬁxed cost of production are equal to zero.
∫∫
The monopolist maximizes its payoﬀ 𝑢𝑀 (𝜎𝐵 , 𝜎𝑀 ) = 𝑝1 𝑉 𝑑𝑘 + 𝛿𝑝2 𝑌 𝑑𝑘. The sets 𝑉 =[𝜃∣𝜎1𝐵 (𝜃, 𝑝1 ) = 𝐵𝑢𝑦] and 𝑌 = [𝜃∣𝜎2𝐵 (𝜃, ℎ, 𝑝2 ) = 𝐵𝑢𝑦] are the sets of ﬁrst and second periodbuyers, respectively. The corresponding integrals give the mass of each set. Since consumers∩
have unit demand, we have 𝑉 𝑌 = ∅.

2.1

Perfect Bayesian Equilibrium

Our model is a two-period signalling game in which the uncertainty about the monopolist’s
type will be revealed in the second period. We deﬁne our Perfect Bayesian Equilibrium in
the Appendix (Deﬁnition 1). We ﬁrst deﬁne the sequential rationality conditions for themonopolist and the consumers (Condition 1 and 2). That is, when they reach the second
period (given the relevant history), they should act rationally. Condition 3 and 4 show how
the monopolist and each consumer choose their strategies in the ﬁrst period. Condition 5
is the usual Bayesian updating condition for the consumers. We only look for pure strategy
equilibria in which consumers use threshold strategies. Speciﬁcally, for any 𝑎1𝑀 , 𝑝2 , ℎ, there
exists 𝜃1 (𝑝1 ) and 𝜃2 (ℎ, 𝑝2 ) such that:
10

Note that buyers observe prices before making a decision. Their history is essentially (ℎ𝑡 , 𝑝𝑡+1 ), 𝑡 = 0, 1,
where ℎ0 denotes the null history.

6{
𝐵𝑢𝑦 if 𝜃 ∈ (𝜃1 , 1]
𝜎1𝐵 (𝜃, 𝑎1𝑀 ) =𝐷𝑒𝑙𝑎𝑦 if 𝜃 ∈ [0, 𝜃1 ) 𝑜𝑟 𝑎1𝑀 = {𝑎𝑛𝑛𝑜𝑢𝑛𝑐𝑒}{
𝐵𝑢𝑦 if 𝜃 ∈ [𝜃2 , 𝜃1 ]𝜎2𝐵 (𝜃, ℎ, 𝑝2 ) =𝐷𝑜 𝑁 𝑜𝑡 𝐵𝑢𝑦, if 𝜃 ∈ [0, 𝜃2 )In what follows, we will analyze the equilibrium strategies. Given condition 1 and 2 ofour equilibrium, we should start solving the problem backwards. The quality and 𝜃1 will be
revealed in the second period, then consumers will buy if their payoﬀ is non-negative; thatis, the second period buyers are 𝑌 = {𝜃 ≥ 𝜃2 ∣𝜃2 𝛾𝑖 − 𝑝𝑖2 = 0 𝑎𝑛𝑑 𝑎1𝐵 (𝜃) = 𝐷𝑒𝑙𝑎𝑦}. The 𝜃2
consumer is the threshold agent that is just indiﬀerent between buying and not buying in thesecond period. The monopolist’s maximization problem in this period gives the following
prices and the second period proﬁts, Π2 (𝛾𝑖 ):
𝑝𝑖2 (𝜃1 − 𝜃2 ) 𝑠.𝑡. 𝜃2 𝛾𝑖 − 𝑝𝑖2 = 0
max𝑖⇒ 𝑝𝑖2 =

𝑝2

𝛾𝑖 𝜃1
⇒
2

Π2 (𝛾𝑖 ) = 𝛾𝑖 (

𝜃1 2
)
2

(1)

Now moving back to the customers’ equilibrium strategies in period 1, each consumer
will form a belief 𝜇1𝐵 that the good is high quality after observing the ﬁrst period price, 𝑝1 .
The threshold agent 𝜃1 will receive the following payoﬀ from buying:(1 + 𝛿)𝜃1 [𝜇1𝐵 + (1 − 𝜇1𝐵 )𝛾𝑙 ] − 𝑝𝑖1
If she delays buying the good, she will receive a (discounted expected) payoﬀ of𝐿𝛿[𝜇1𝐵 (𝜃1 − 𝑝𝐻2 ) + (1 − 𝜇1𝐵 )(𝜃1 𝛾𝑙 − 𝑝2 )]where 𝑝𝑗2 is the second period price when the revealed quality is 𝑗 = 𝐻, 𝐿. By equatingthese two terms, we can calculate the set of ﬁrst period buyers 𝑉 = [𝜃1 , 1] and the set of
second period buyers 𝑌 = [𝜃2 , 𝜃1 ). Speciﬁcally, by plugging in for the second period priceand solving for the indiﬀerence condition gives us:
𝜃1 (𝑝𝑖1 , 𝜇1𝐵 ) =2𝑝𝑖1(2 + 𝛿)[𝜇1𝐵 + (1 − 𝜇1𝐵 )𝛾𝑙 ]7(2)

Now, we can write the monopolist’s proﬁt (payoﬀ) as a function of 𝑝𝑖1 and 𝜇1𝐵 with thehelp of equation 1:
𝑉𝛾𝑖 (𝑝𝑖1 , 𝜇1𝐵 ) = 𝑝𝑖1 (1 − 𝜃1 (𝑝𝑖1 , 𝜇1𝐵 )) + 𝛿𝛾𝑖 (2.2

𝜃1 (𝑝𝑖1 , 𝜇1𝐵 ) 2
)
2

(3)

Separating Equilibrium: Both Types Sell in Both Periods

In what follows, we will give diﬀerent types of separating equilibria. In the separating
equilibrium given in Proposition 1, consumers will be convinced of high quality if they see
a very high price in the ﬁrst period. The low type monopolist chooses its proﬁt maximizing
price rather than imitating the high-type. The high type will be better oﬀ by choosing theequilibrium price rather than waiting/announcing and selling only in the second period, or
posing as low type in the ﬁrst period. In this equilibrium, prices will decline for both types
in the second period. Therefore, the result is diﬀerent than Bagwell and Riordan (1991) inthe sense that low type also chooses high and declining prices.

Figure 2: A separation equilibrium example when high type separates itself by choosing
𝑝1 ∈ {1.253, 1.295}. The lines are equilibrium and deviation payoﬀs for high and low types.
While it is not shown on the graph, waiting to sell in the second period is not proﬁtable
when 0.34 < 𝑝1 < 1.3.

8

𝛿
Proposition 1 Assume that 𝛾𝑙 > 𝛿+4. There are separating equilibria in which both low andhigh type monopolist sell in both periods. There is an equilibrium that satisﬁes the Intuitive
Criterion.
The high type sets a (very) high price in the ﬁrst period and sells only to a small subset of
consumers, despite the fact that consumers correctly infer that this is a high quality product.
The low type cannot mimic this strategy of setting a high price in the ﬁrst period since selling
only a small subset of consumers in the ﬁrst period (and, as a result, having a low total sales
in both periods) is not proﬁtable. Instead, it wants to sell as much as possible both in theﬁrst and the second period. This can be done by setting a low enough price in the ﬁrst
period. As 𝛾𝑙 gets smaller, the low type’s proﬁt decreases in a separating equilibrium. If
it gets too small, then the low type would deviate by choosing the high type monopolists’s𝛿
price since this gives him a higher proﬁt. This is why we need the condition 𝛾𝑙 > 𝛿+4.

Figure 3 shows an example of such an equilibrium. It shows the price range in which thehigh-type and the low-type will not deviate. When the price is in the overlapping area, notypes will deviate and hence we have a separating equilibrium. The high type should also
not deviate to “waiting and only selling in the second period” equilibrium (this range is notshown in the ﬁgure).

2.3

Pooling Equilibrium and Introductory Low Price

In Proposition 2 below, we explain pooling equilibria.
Proposition 2 There are pooling equilibria in which both types choose the same price to sellin the ﬁrst period. When 𝛾𝑙 = 0, there is an equilibrium that satisﬁes the Intuitive Criterion.
Let us concentrate on the equilibrium that satisﬁes the Intuitive Criterion. Since 𝛾𝑙 = 0,
if the low type deviates, it cannot sell any product and end up with zero proﬁts. Therefore,
the low type does not deviate. The high type has no incentive to deviate to be seen as low
type but may have an incentive to deviate to waiting. When consumers believe that a ﬁrm
𝛿(2+𝛿)𝛼is a high type with probability 𝛼 if 𝑝1 in the range 4(2+𝛿)𝛼−2𝛿
< 𝑝1 < (2+𝛿)𝛼, then deviation2

to waiting is not proﬁtable, and a pooling equilibrium can be sustained.
91Given equation 2, we know that 𝑑𝜃
< 0. In other words, when consumers are more𝑑𝛼

pessimistic, more of them strategically delay their buying to the second period. By equation
2
1
1, we know that 𝑑𝑝= 𝛾2𝑖 𝑑𝜃< 0. That is, the more strategically delay their buying to𝑑𝛼𝑑𝛼

the second period, the higher the second period price of the monopolist. Not surprisingly,
the high type increases its price more, and this inevitably deters strategic delay incentives.
However, we admit that this may be due to modelling assumptions since as 𝜃1 increases (dueto a lower 𝛼), the remaining consumers have a higher willingness to pay (compared to alower 𝜃1 ). Hence, both type of monopolist can and do charge a higher price.
Our main interest is to analyze when the ﬁrm uses the introductory low price strategy andwhen they use the cream skimming strategy (that is, lower the prices in the second period).
The next proposition ﬁnds suﬃcient conditions for these strategies for a given equilibrium.
We note that this suﬃcient condition is exactly the same for all equilibria of Proposition 2.
The reason is that the consumers’ (oﬀ and on path) beliefs determine the ﬁrst period pricebut the monopolist’s proﬁt maximization problem determines the second period price.
Proposition 3 For any given pooling equilibrium: a) (introductory low price) the high-type𝑙 −𝛼𝛾𝑙 ). b) (cream-skimming) The
ﬁrm will set a higher second period price if 𝛿 < 1−2(𝛼+𝛾(𝛼+𝛾𝑙 −𝛼𝛾𝑙 )𝑙 −𝛼𝛾𝑙 ).
high-type ﬁrm will set a lower second period price if 𝛿 > 1−2(𝛼+𝛾(𝛼+𝛾𝑙 −𝛼𝛾𝑙 )As 𝑝2 is higher in equilibrium, the cost of delaying strategically will be higher for consumers. In addition, if the consumers are too pessimistic about the quality, ﬁrms are more
likely to oﬀer introductory low prices. The high-type monopolist sells to “the high willingness to pay” consumers in the ﬁrst period. In a pure private value durable good model,
the monopolist has no chance but to lower the price in the second period to sell to theremaining/residual lower types. In our model, the remaining lower willingness to pay consumers, in the second period, learn that the quality is much higher than their expectations.
Hence, their total willingness to pay increases. This will enable the monopolist to charge ahigher price. This result diﬀers from the intertemporal price discrimination literature (suchas Stokey (1979), Landsberger and Meilijison (1985), Tirole (1989)) which shows that theprice must decrease in all periods. Once the uncertainty is resolved (and if we had more
10

than two periods), we are essentially back in the pure private value model and the price will
decrease starting from period three onwards. Introducing uncertainty on quality is the key
factor that diﬀerentiates our results from the rest of the literature.11
In Shapiro (1983), myopic consumers have point estimates of the quality (biased expectations). If their estimate is lower than actual quality, the ﬁrm oﬀers a low introductory priceso that consumers can pay higher prices in repeat purchases. Since consumers are myopic,
they do not have the price expectation or strategic delay incentives that the consumers in
our model have.
Corollary 4 Let 𝛾𝑙 = 0. a) If 𝛼 < 13 , the high type will set a higher second period price. b)
If 𝛼 > 12 then the high type will set a lower second period price.
In the proofs, we show that only when 𝛾𝑙 = 0, then the pooling equilibrium satisﬁes theintuitive criterion. In this case, using proposition 3, the condition for oﬀering introductory. If 𝛼 ≤ 31 , this condition is always satisﬁed; hence, the high type
low price becomes 𝛿 < 1−2𝛼𝛼monopolist will always oﬀer introductory low price. Since consumers are pessimistic about
the quality, the high type monopolist has to oﬀer a very low price in the ﬁrst period (andeven then relatively a small percentage of consumers buy the product in the ﬁrst period).
When the quality is revealed, consumers are willing to pay a higher price for the high quality
good.
In the following example, for 𝛾𝑙 > 0, we give a parameter range in which the pooling
equilibrium exists while neither separating nor waiting equilibria exists.
Example 5 Let 𝜇1𝐵 = 𝛿 = 0.4 and 𝛾𝑙 = 0. Since 𝛾𝑙 = 0, there cannot be any separating
equilibrium (in which there is positive sale in the ﬁrst period). Low type will always mimic
the high type to guarantee a positive proﬁt rather than receiving zero separating equilibrium
proﬁt. In our pooling equilibrium, consumers will believe that any ﬁrm choosing 𝑝1 = 0.4
11There are two other papers that discuss Coasian dynamics with agents consuming two goods unlike ourpaper. Koh (2006) shows that a durable good monopolist who commits to its pricing path and faces inﬁnitely
lived heterogeneous agents consuming both perishable and durable good may oﬀer low prices initially. His
results depend on the consumption of perishable good in each period. Lee (2003) shows that Coasian
dynamics may lead to the sale of degraded goods that are as costly as producing high-quality goods. Sellingde-graded goods mitigate consumers’ waiting incentives.

11

is a high type with probability 0.4. In the proof of pooling equilibrium, we show that for𝛿any 𝑝1 ∈ {0.12, 0.48} (with belief 𝜇1𝐵 = 0.4) and 𝛾𝑙 < 2+𝛿
, waiting equilibrium payoﬀ islower than the pooling equilibrium payoﬀ for the high type and hence we have a pooling
equilibrium.12 In such an equilibrium, the second period price will be 𝑝2 ≃ 0.468 which isgreater than the ﬁrst period price.

2.4

Separating Equilibrium: High-Type Waits

In proposition 6 below, we give a waiting equilibrium. If the low type monopolist’s quality
is too low (and consumers believe that any ﬁrm waiting is a high type), the high-type ﬁnds
it proﬁtable to wait for the second period (i.e., announce) rather than selling as a low typein the ﬁrst period. The low type chooses its proﬁt maximizing price in the ﬁrst period.
𝛿Proposition 6 (Waiting equilibrium) Suppose that 𝛾𝑙 < 2+𝛿
holds. There are separating

equilibria in which low type sells in the ﬁrst period and the high type announces the productlaunch date in the ﬁrst period and waits to sell in the second period. If 𝛾𝑙 = 0, the equilibrium
satisﬁes the Intuitive Criterion.
If 𝛾𝑙 = 0, then the low type would like to mimic the high type’s price in the ﬁrst period
price. The high type responds by waiting to sell in the ﬁrst period, and hence, a “waiting
equilibrium” is sustained.
A monopolist may signal its quality through announcing the launch date of its productalthough its product is ready at the time. Hence, this is not a “vaporware” example (unlike
Dranove and Gandal (2003)) since the product is delivered on its launch date. In order tohave such a waiting equilibrium, 𝛾𝑙 should be low enough so that the high type will have agreat incentive to distinguish itself from the low type (depending on the oﬀ-path equilibrium
beliefs, it would not choose to sell in the ﬁrst period). Also, as discount factor increases, it
is more likely to have such an equilibrium since waiting is less costly.
When 𝛾𝑙 is zero, the intuitive criterion is satisﬁed since any deviation by the high typewill be followed by the low type who receives the zero payoﬀ in the waiting equilibrium. In
12

We impose the oﬀ-path equilibrium belief that for any other 𝑝1 , consumers believe that this is a low
type with probability 1.

12other words, there is no 𝑝1 choice by the high type that is equilibrium dominated for the low
type.

3

When only 𝛽 fraction of agents learn the true quality
in the second period

In this section, we will show that we can construct the “waiting equilibrium” when only a 𝛽
fraction of agents learn the true quality in the second period.
We assume that the monopolist cannot distinguish the informed and uninformed consumers in the second period so price discrimination is not possible.
The second period proﬁt maximization function of the 𝛾𝑖 > 0 type monopolist is asfollows:
𝑝𝑖2 [𝛽(𝜃1 −
max
𝑖
𝑝2

𝑝𝑖2
𝑝𝑖2
) + (1 − 𝛽)(𝜃1 −)]𝛾𝑖
𝜇2𝐵 + (1 − 𝜇2𝐵 )𝛾𝑙(4)

since informed agents buy the product as long as their type satisﬁes 𝜃𝛾𝑖 − 𝑝𝑖2 ≥ 0 and
uninformed agents buy as long as their type satisﬁes 𝜃(𝜇2𝐵 + (1 − 𝜇2𝐵 )𝛾𝑙 ) − 𝑝𝑖2 ≥ 0. The
uninformed agents use the expected value of 𝛾𝑖 given their second period beliefs 𝜇2𝐵 .
If we diﬀerentiate this function with respect to 𝑝2 and set it to zero, we can get the proﬁt
maximizing second period price 𝑝𝑖2 as a function of 𝜃1 which is𝑝𝑖2 =𝜃1 𝛾𝑖 (𝜇2𝐵 + 𝛾𝑙 − 𝛾𝑙 𝜇2𝐵 )2[(𝛽(𝜇2𝐵 + 𝛾𝑙 − 𝛾𝑙 𝜇2𝐵 − 𝛾𝑖 ) + 𝛾𝑖 )](5)Then, given 𝜃1 , the proﬁt maximization of the monopolist is:𝑝𝑖
𝑝2
max 𝑝1 (1−𝜃1 )+𝛿𝑝𝑖2 [𝛽(𝜃1 − 2 )+(1−𝛽)(𝜃1 −)]𝑝1𝛾𝑖
𝜇2𝐵 + (1 − 𝜇2𝐵 )𝛾𝑙𝑠.𝑡

𝑒𝑞 2 𝑎𝑛𝑑 𝑒𝑞 5 (6)To construct a waiting equilibrium, let us suppose that consumers believe any ﬁrm who
do not wait/announce is a low type; that is, 𝜇1𝐵 (𝑎1𝑀 = 𝑝1) = 0. First, let us ﬁnd the range
in which the low type monopolist will not imitate the high type who is waiting. The low type
monopolist’s proﬁt in the waiting equilibrium can be found from the equation 6 when 𝜇2𝐵 = 0
13since consumers can detect the type in a separating equilibrium.13 We need to calculate 𝜃1 .
Given the beliefs, we can calculate it from the equation (1 + 𝛿)𝜃1 𝛾𝑙 − 𝑝1 = 𝛿[𝜃1 𝛾𝑙 − 𝑝2 ]. By
2𝑝1
using equation 5, we derive that 𝜃1 = (2+𝛿)𝛾. Note that 𝛽 has no role since consumers know𝑙the type with certainty in a separating equilibrium. In this equilibrium, the proﬁt of low
type will be the same as in equation 6:

𝐿𝑜𝑤 𝑇 𝑦𝑝𝑒 𝑃 𝑟𝑜𝑓 𝑖𝑡 𝑖𝑛 𝑊 𝑎𝑖𝑡𝑖𝑛𝑔 𝐸𝑞 =𝛾𝑙 (2 + 𝛿)24(4 + 𝛿)If the low type deviates to waiting, then we can calculate the deviation proﬁt by setting
𝜇2𝐵 (., 𝜏 = 𝑈 ) = 1 but 𝜇2𝐵 (., 𝜏 = 𝐼) = 0. In this equilibrium, since low type does not sell
anything in the ﬁrst period, we have 𝜃1 = 1. By using equation 6, we get the deviation proﬁt
which is:𝐿𝑜𝑤 𝑇 𝑦𝑝𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑃 𝑟𝑜𝑓 𝑖𝑡 =𝛿𝛾𝑙4[(1 − 𝛾𝑙 )𝛽 + 𝛾𝑙 ]If the equilibrium proﬁt is higher than the deviation proﬁt, we know that low type would
not deviate to waiting. This condition is𝛾𝑙 (2 + 𝛿)2𝛿𝛾𝑙(4 + 𝛿)𝛿 − 𝛽(2 + 𝛿)2≥⇔ 𝛾𝑙 ≥
4(4 + 𝛿)4[(1 − 𝛾𝑙 )𝛽 + 𝛾𝑙 ](1 − 𝛽)(2 + 𝛿)2
If 𝛾𝑙 is big enough, then the low type does not have to deviate to waiting. Also, as 𝛽approaches to 1, the numerator becomes negative and the condition is always satisﬁed. The
more informed the consumers are, the more likely to have a waiting equilibrium in whichhigh type waits but the low type sells in the ﬁrst period. This is an intuitive result.
Now, we will derive the condition that the high type does not deviate.
It is easy to see that the proﬁt of the high type in a waiting equilibrium is the same asin the main model which is 4𝛿 .
If the high type deviates to selling in the ﬁrst period, then the beliefs will be 𝜇1𝐵 = 0,𝜇2𝐵 (., 𝜏 = 𝐼) = 1, and 𝜇2𝐵 (., 𝜏 = 𝑈 ) = 0. Then, the threshold agent 𝜃1 can be found from
13
Speciﬁcally, 𝜇2𝐵 (., 𝜏 = 𝐼) = 𝜇2𝐵 (., 𝜏 = 𝑈 ) = 0. That is, both informed and uninformed consumers knowthat this is a low type since this is a separating equilibrium.

14

the equation (1 + 𝛿)𝜃1 𝛾𝑙 − 𝑝1 = 𝛿(𝜃1 𝛾𝑙 − 𝑝2 ). By using this, and equation 6, we can calculate
the deviation proﬁt of the hight type:14(2 + 𝛿)2 𝛾𝑙 (1 − 𝛽(1 − 𝛾𝑙 ))4[4 − (4𝛽 + 2𝛿𝛽)(1 − 𝛾𝑙 ) + 𝛿]2(2+𝛿) 𝛾𝑙 (1−𝛽(1−𝛾𝑙 ))> 0 holds for someHence, the high type does not deviate if 4𝛿 − 4[4−(4𝛽+2𝛿𝛽)(1−𝛾𝑙 )+𝛿]

parameter range of (𝛽, 𝛾𝑙 , 𝛿). Intuitively, we expect that as 𝛽 → 1, there are waiting equilibria.
To show one equilibrium, we set 𝛾𝑙 = 0.25. Then, by using MAPLE, we showed that when𝛽 = 𝛿 = 0.9, there is a waiting equilibrium since both the low type and the high type would
not deviate. Of course, by continuity, one can see that there are inﬁnitely many waiting
equilibrium when we chose parameters close to the set ones.

4

Eﬃciency Loss and Welfare in Waiting Equilibrium
Case

In this section, we will calculate the eﬃciency loss for the equilibrium that satisﬁes theintuitive criterion (i.e., 𝛾𝑙 = 0) for the waiting equilibrium. In the Appendix, we analyze thewelfare loss for separating equilibrium case.
We will calculate the eﬃciency loss of waiting equilibrium when 𝛾𝑙 = 0 compared tothe perfect information case. The low type does not produce anything neither in the perfect information case nor in the waiting equilibrium so there is no eﬃciency loss when themonopolist is low type of 𝛾𝑙 = 0.
The high type will only sell in the second period and from equation 1, one can easily seethat 𝑝2 = 𝜃2 = 12 . Therefore, welfare under the waiting equilibrium, 𝑊𝑤 , given that 𝛼 is theprobability of having the high type, is:3
1 1
𝑊𝑤 = 𝛼𝛿( + ) = 𝛼𝛿4 8
8
The ﬁrst term in the parenthesis is the proﬁt and the second term is the consumer surplus∫1
which can be calculated from 𝛿 1 (𝜃 − 12 )𝑑𝜃.
214(2+𝛿)2 𝛾 2

If 𝛽 = 1, we get the same proﬁt as in the previous section which is 4[4𝛾𝑙 +2𝛿𝛾𝑙𝑙−𝛿]

15The welfare under the perfect information case, 𝑊𝑝 is as follows. In this case, 𝜇1𝐵 = 1
since there is perfect information. A quick calculation shows that second period maximization
2𝑝1
problem gives 𝑝2 = 𝜃2 = 𝜃21 . Since 𝜃1 = 2+𝛿
(from equation 2 when 𝜇1𝐵 = 1), the proﬁt

maximization problem becomes:

max 𝑝1 (1 −𝑝1

𝛿 2𝑝1 2
2𝑝1
)+ (
)
2+𝛿
4 (2 + 𝛿)

When we diﬀerentiate, we ﬁnd the ﬁrst order conditions:

1−

4𝑝1
𝛿2𝑝1
+
=0
2 + 𝛿 (2 + 𝛿)2

2

(2+𝛿)
2+𝛿
This gives 𝑝1 = 2(4+𝛿)and 𝜃1 = 4+𝛿
. We can calculate the proﬁt of high type (Fromequation 3):

𝑉𝛾ℎ (𝑝1 =(2 + 𝛿)2(2 + 𝛿)2, 𝜇1𝐵 = 1) =2(4 + 𝛿)4(4 + 𝛿)The consumer surplus is:∫ 1(2 + 𝛿)2(𝜃 −)𝑑𝜃 + 𝛿2+𝛿2(4 + 𝛿)
𝜃1 = (4+𝛿)∫ 𝜃2 = 2+𝛿

(4+𝛿)

2+𝛿
𝜃1 = 2(4+𝛿)

(𝜃 −

2+𝛿
)𝑑𝜃
2(4 + 𝛿)1
(2 + 𝛿)2(2 + 𝛿)2(2 + 𝛿)3(2 + 𝛿)2
=[ −
−
+
]
+
𝛿[
]
2 2(4 + 𝛿)2 2(4 + 𝛿) 2(4 + 𝛿)28(4 + 𝛿)2
where the ﬁrst bracket parenthesis is the ﬁrst period consumer surplus, the second oneis the second period consumer surplus (after simpliﬁcations). We summarize the eﬃciency
loss and its derivative with respect to 𝛿 in the following proposition:
Proposition 7 The eﬃciency loss in waiting equilibrium is:

𝐸𝐿𝑤 = 𝑊𝑝 − 𝑊𝑤 =𝛼[(3 + 𝛿)(4 − 3𝛿)]2(4 + 𝛿)2

Eﬃciency loss decreases as agents become more patient:𝛼(−44 − 19𝛿)𝑑(𝑊𝑝 − 𝑊𝑤 )=<0𝑑𝛿2(4 + 𝛿)316

As 𝛿 increases, eﬃciency loss decreases. Increase in 𝛿 has two (counter) eﬀects on 𝑊𝑝 .
The ﬁrst eﬀect causes price to increase. This makes fewer agents buy the good. Consumer
surplus decreases; hence, this lowers welfare. However, the second eﬀect causes proﬁts toincrease. The second eﬀect dominates and 𝑊𝑝 increases. It is easy to see that 𝑊𝑤 increases
with 𝛿. It must be the case that 𝑊𝑤 increases more than 𝑊𝑝 ; therefore, the eﬃciency loss
decreases.

5

Discussion and Conclusion

If we extend the model to three or more periods, we will ﬁnd that the price will decrease
from the third period onwards. Since the uncertain common value will be learned in thesecond period, this model will essentially be a pure private value durable good model. If
the monopolist wants to sell to the residual lower willingness to pay type consumers in thethird period, it has to decrease the price (as in Stokey (1979) or Landsberger and Meilijson(1985)).
The model produces many pooling equilibria; however, we show that the suﬃcient condition to oﬀer introductory low prices or use cream-skimming strategies is the same for any
given pooling equilibrium. In our pooling equilibria, the beliefs determine the ﬁrst period
price but the monopolist’s actual proﬁt maximization motive determines the second period
price. As long as the beliefs of consumers are known, the equilibrium that will be selected is
known by all parties; that is, there is no coordination problem. However, we do not attempt
to explain how these beliefs are formed.
We have used uniform distribution. We believe that using a general distribution would
not bring any new insights than the current paper but only complicate the model.

5.1

Conclusion

We show how a monopolist will price its good when consumers do not know the quality
but will learn it in the second period. This leads to strategic delay and price expectations
on the part of consumers. The monopolist will take these into account and price its good
dynamically.
17

One of our main contributions is showing that the high type monopolist can signal its
quality by announcing a future launch date even though the product is ready to sell today.
Microsoft’s launching of Windows 7 may be an example to this strategy.15
Signalling creates a welfare loss. We show that as agents become more patient or the low
type’s quality increases, this loss decreases.
One contribution of our paper is to show that consumers may delay their buying to learn
quality. This is in contrast with Coasian dynamics under a model in which there is nolearning.

6

AppendixDeﬁnition 1 (Perfect Bayesian Equilibrium): A proﬁle of (pure) strategies 𝜎 =(𝜎1𝐵 , 𝜎2𝐵 , 𝜎1𝑀 , 𝜎2𝑀 ) (where 𝜎1𝑀 : Ψ 7→ 𝐴1𝑀 and 𝜎2𝑀 : 𝐻 × Ψ 7→ 𝐴2𝑀 and 𝜎1𝐵 : Θ × 𝐴1𝑀 7→
𝐴1𝐵 , and 𝜎2𝐵 : 𝐻 × 𝐴2𝑀 × Θ 7→ 𝐴2𝐵 .) and the belief system 𝜇 = (𝜇1𝐵 , 𝜇2𝐵 ) (where
𝜇1𝐵 : 𝐴1𝑀 7→ [0, 1], and 𝜇2𝐵 : 𝐻 × 𝐴2𝑀 × {𝐼, 𝑈 } 7→ [0, 1] is a Perfect Bayesian Equilibrium
if
Condition 1. For any 𝛾𝑖 and ℎ, 𝜎2𝑀 (𝛾𝑖 , ℎ) solves∫∫
max𝑝2 𝑝2 [ 𝜎2𝐵 (𝜃,ℎ,𝑝2 ,𝜏 =𝐼)=𝐵𝑢𝑦 𝑑𝜃 + 𝜎2𝐵 (𝜃,ℎ,𝑝2 ,𝜏 =𝑈 )=𝐵𝑢𝑦 𝑑𝜃]Condition 2. For any 𝜃, 𝜏 , ℎ, and 𝑝2 , 𝜎2𝐵 (𝜃, 𝜏, ℎ, 𝑝2 ) = 𝐵𝑢𝑦 iﬀ 𝜃[𝜇2𝐵 (ℎ, 𝑝2 , 𝜏 ) + (1 −𝜇2𝐵 (ℎ, 𝑝2 , 𝜏 )𝛾𝑙 ] ≥ 𝑝2 .
Condition 3. For any 𝛾𝑖 , 𝜎1𝑀 (𝛾𝑖 ) solves∫∫

max{𝛿𝜎2𝑀 (𝛾𝑖 , ℎ(𝑎1𝐵 , 𝑎1𝑀 = 𝑎𝑛𝑛𝑜𝑢𝑛𝑐𝑒, 𝛾𝑖 ))𝑑𝜏 𝑑𝜃,𝑎1𝑀
𝜎2𝐵 (𝜃,𝜏,ℎ(.,𝑎1𝑀 =𝑎𝑛𝑛𝑜𝑢𝑛𝑐𝑒),𝜎2𝑀 (𝛾𝑖 ,ℎ(.,𝑎1𝑀 =𝑎𝑛𝑛𝑜𝑢𝑛𝑐𝑒)))=𝐵𝑢𝑦∫∫∫
max 𝑝1𝑑𝜃 + 𝛿𝜎2𝑀 (𝛾𝑖 , ℎ(𝑎1𝐵 , 𝑝1 , 𝛾𝑖 ))𝑑𝜏 𝑑𝜃}𝑝1

𝜎1𝐵 (𝜃,𝑝1 )=𝐵𝑢𝑦

𝜎2𝐵 (𝜃,𝜏,ℎ(.,𝑝1 ),𝜎2𝑀 (𝛾𝑖 ,ℎ(.,𝑝1 )))=𝐵𝑢𝑦

where ℎ(𝑎1𝐵 , 𝑎1𝑀 , 𝛾𝑖 ) is the history that arises when the monopolist oﬀers price 𝑝1 orchooses announce, the buyers choose their actions according to 𝑎1𝐵 in period 1 and thequality 𝛾𝑖 is revealed but only learned by the informed buyers (𝜏 = 𝐼)
15

We emphasize that our paper is a theoretical exercise and there may be other reasons for this waiting.

18

Condition 4. For any 𝜃 and 𝑝1 , 𝜎1𝐵 (𝜃, 𝑝1 ) = 𝐵𝑢𝑦 if and only if:(1 + 𝛿)𝜃(𝜇1𝐵 (𝑝1 ) + (1 − 𝜇1𝐵 (𝑝1 )𝛾𝑙 ) − 𝑝1 ≥𝛿[𝜇1𝐵 (𝑝1 )(𝜃 − 𝜎2𝑀 (𝛾ℎ , ℎ′ )) + (1 − 𝜇1𝐵 (𝑝1 ))(𝜃𝛾𝑙 − 𝜎2𝑀 (𝛾𝑙 , ℎ′ ))]where ℎ′ = ℎ({𝜎1𝐵 (𝜃′ , 𝑝1 )}𝜃′ ∕=𝜃 , 𝑎1𝐵 (𝜃) = 𝐷𝑒𝑙𝑎𝑦, 𝑝1 , 𝛾𝑖 ) is the history that arises when inperiod 1, the type 𝜃 consumer delays his decision while the monopolist oﬀers price 𝑝1 andother consumers choose their actions according to 𝜎1𝐵 , and the revealed quality is 𝛾𝑖 .
For any 𝜃, 𝜎1𝐵 (𝜃, 𝑎𝑛𝑛𝑜𝑢𝑛𝑐𝑒) = 𝐷𝑒𝑙𝑎𝑦.
Condition 5. Consumers update their beliefs using Bayesian rule whenever possible. In
the second period, since the quality is revealed to informed consumers, we have 𝜇2𝐵 (ℎ′ , 𝜎2𝑀 , 𝜏 =𝐼) = 1 if 𝛾𝑖 = 1, and 𝜇2𝐵 (ℎ′ , 𝜎2𝑀 , 𝜏 = 𝐼) = 0 if 𝛾𝑖 = 𝛾𝑙 .
Proof of Proposition 1]} comes fromWe will show that if consumers believe 𝑝1 ∈ {𝑟𝑜𝑜𝑡𝑙𝑜𝑤2, min[𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ2, 2+𝛿2
a high type, then there will be separating equilibria. We will give the explanation of 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2
and 𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ2 below.
We have already calculated the threshold agent 𝜃1 and proﬁt 𝑉𝛾 (𝑝1 , 𝜇1𝐵 ) for a given 𝜇1𝐵
and 𝑝1 in the text in equations 2 and 3. They were;𝜃1 (𝑝1 , 𝜇1𝐵 ) =2𝑝1(2 + 𝛿)[𝜇1𝐵 + (1 − 𝜇1𝐵 )𝛾𝑙 ]𝑉𝛾𝑖 (𝑝1 , 𝜇1𝐵 ) = 𝑝1 (1 − 𝜃1 (𝑝1 , 𝜇1𝐵 )) + 𝛿𝛾𝑖 ((2)

𝜃1 (𝑝1 , 𝜇1𝐵 ) 2
)
2

(3)

First, we will ﬁnd the price range that the low type monopolist will not deviate. In a
separating equilibrium, the low type monopolist will maximize its proﬁts as follows:
max𝑝1 𝑉𝛾𝑙 (𝑝1 , 0) = 𝑝1 (1 − 𝜃1 (𝑝1 , 0)) + 𝛿𝛾𝑙 ( 𝜃1 (𝑝21 ,0) )2 .
2𝛾𝑙Solving this maximization problem yields 𝑝1 = (2+𝛿)
which in turn yields 𝜃1 = 2+𝛿
.
2(4+𝛿)
4+𝛿

Thus, the proﬁt of low quality type in a separating equilibrium is given by:

𝜋𝑆𝑒𝑝𝐿𝑜𝑤 = 𝑉𝛾𝑙 (𝑝1 =(2 + 𝛿)2 𝛾𝑙𝛾𝑙 (2 + 𝛿)2, 𝜇1𝐵 = 0) =2(4 + 𝛿)4(4 + 𝛿)19

We now calculate the low type producer’s deviation proﬁts if it deviates to some pricerange so that consumers believe he is a high type. This deviation proﬁt is:
2𝑝1
𝛿𝛾𝑙 𝑝21)+(2 + 𝛿)(2 + 𝛿)2
To compare the proﬁts of the separating equilibrium with the proﬁts of the deviation, we
𝜋𝑆𝑒𝑝𝐿𝑜𝑤𝐷𝑒𝑣 = 𝑉𝛾𝑙 (𝑝1 , 𝜇1𝐵 = 1) = 𝑝1 (1 −subtract the deviation proﬁts from the separating equilibrium proﬁts and set the resulting
2

2

2𝑝
𝛿𝛾𝑙 𝑝
𝑙 (2+𝛿)function equal to 0. i.e. 𝜋𝑆𝑒𝑝𝐿𝑜𝑤 − 𝜋𝑆𝑒𝑝𝐿𝑜𝑤𝐷𝑒𝑣 = 𝛾4(4+𝛿)− [𝑝(1 − (2+𝛿)) + (2+𝛿)
2 ] = 0 = 𝑓 (𝑝).

Since 𝑓 and 𝜋𝑆𝑒𝑝𝐿𝑜𝑤𝐷𝑒𝑣 is a polynomial of degree two, we can ﬁnd at most two roots,and later we will show that there are exactly two roots. These roots will be positive since′′𝜋𝑆𝑒𝑝𝐿𝑜𝑤𝐷𝑒𝑣 (𝑝 = 0) = 0, 𝜋𝑆𝑒𝑝𝐿𝑜𝑤𝐷𝑒𝑣 is increasing at 0, concave (because 𝑓 ′′ = 𝜋𝑆𝑒𝑝𝐿𝑜𝑤𝐷𝑒𝑣=
𝛿(−4+𝛾𝑙 )−4< 0), and 𝜋𝑆𝑒𝑝𝐿𝑜𝑤 > 0 for any positive 𝛾𝑙 (see ﬁgure 3). For any price p lower than(2+𝛿)2the smallest root and any price p greater than the biggest root, the proﬁt from deviating isless than the proﬁt from the separating equilibrium. We calculate the roots of this function;
𝑟𝑜𝑜𝑡𝑙𝑜𝑤1 is the smallest root, and 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2 is the greatest root. In the equations below, theterm inside the square roots are positive; hence, there are exactly two roots.
√[(2 + 𝛿)2 [4 + 𝛿 − (4 + 𝛿)(1 − 𝛾𝑙 )[𝛿(1 − 𝛾𝑙 ) + 4) ]𝑟𝑜𝑜𝑡𝑙𝑜𝑤1 =2(4 + 𝛿)(𝛿(2 − 𝛾𝑙 ) + 4)√
2[(2 + 𝛿) [4 + 𝛿 + (4 + 𝛿)(1 − 𝛾𝑙 )[𝛿(1 − 𝛾𝑙 ) + 4) ]𝑟𝑜𝑜𝑡𝑙𝑜𝑤2 =2(4 + 𝛿)(𝛿(2 − 𝛾𝑙 ) + 4)

(7)
(8)

In what follows, we will calculate the two roots in which the high type monopolist will
not deviate if the price is within these two roots.
Let us calculate the high type’s proﬁt in a separating equilibrium given 𝑝1 .
2𝑝1
𝛿𝑝21
)+
(2 + 𝛿)(2 + 𝛿)2
The proﬁts of the high ﬁrm posing as a low ﬁrm as a function of 𝑝1 is given by 𝑉 (𝑝1 , 𝜇1𝐵 =𝜋𝑆𝑒𝑝𝐻𝑖𝑔ℎ = 𝑉𝛾=1 (𝑝1 , 𝜇1𝐵 = 1) = 𝑝1 (1 −
𝑝2

2𝑝1
1
0, 𝛾 = 1) = 𝑝1 (1 − 𝛾𝑙(2+𝛿)) + 𝛿 𝛾 2 (2+𝛿)
2 . If we maximize this function with respect to 𝑝1 , we𝑙𝛾 2 (2+𝛿)2

ﬁnd price that will maximize the deviation proﬁt, 𝑝1 = 2(4𝛾𝑙𝑙 +2𝛿𝛾𝑙 −𝛿) . Since, we assume𝛿that 𝛾𝑙 > 4+𝛿
, this will guarantee that 𝑝1 is positive plus the roots below (roothigh1 androothigh2) exist.
20

Figure 3: A separation equilibrium example when high type separates itself by choosing
𝑝1 ∈ {1.253, 1.295}. The lines are equilibrium and deviation payoﬀs for high and low types.
While it is not shown on the graph, waiting to sell in the second period is not proﬁtable
when 0.34 < 𝑝1 < 1.3.
By inserting the expression for 𝑝1 into the proﬁt function, we get the maximum proﬁt𝛿from deviating (and our assumption 𝛾𝑙 > 4+𝛿
implies it is always positive):

𝛾𝑙2 (2 + 𝛿)2(9)4(4𝛾𝑙 + 2𝛿𝛾𝑙 − 𝛿)We have to ﬁnd the range in which a deviation will not occur. By equating the 𝜋𝑆𝑒𝑝𝐻𝑖𝑔ℎ𝐷𝑒𝑣
𝜋𝑆𝑒𝑝𝐻𝑖𝑔ℎ𝐷𝑒𝑣 =and 𝜋𝑆𝑒𝑝𝐻𝑖𝑔ℎ , we ﬁnd two roots. The smallest root is roothigh1 and the greatest root isroothigh2:√(2+𝛿)2 [(2𝛾𝑙 (2+𝛿)−𝛿− (1−𝛾𝑙 )(2𝛾𝑙 (2+𝛿)−𝛿)(𝛾𝑙 (4+𝛿)−𝛿))𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ1 =2(4+𝛿)[2𝛾√ 𝑙 (2+𝛿)−𝛿](2+𝛿)2 [(2𝛾𝑙 (2+𝛿)−𝛿+ (1−𝛾𝑙 )(2𝛾𝑙 (2+𝛿)−𝛿)(𝛾𝑙 (4+𝛿)−𝛿))𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ2 =2(4+𝛿)[2𝛾𝑙 (2+𝛿)−𝛿]When 𝑝1 ∈ [𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ1, 𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ2], the high type does not deviate from the separating
equilibrium (see ﬁgure 3). We should also make sure that announcing/waiting in the ﬁrst
period and selling only in the second period is not proﬁtable for the high type.16 The waiting
proﬁt of high type is calculated as 𝜋𝐻𝑖𝑔ℎ𝑊 𝑎𝑖𝑡 = 4𝛿 . The 𝜋𝑆𝑒𝑝𝐻𝑖𝑔ℎ − 𝜋𝐻𝑖𝑔ℎ𝑊 𝑎𝑖𝑡 = 0 function has
two roots:𝛿(2+𝛿)
and 2+𝛿
(4+𝛿)2
16

The low type will never ﬁnd waiting proﬁtable.

21

When 𝑝1 is ∈ [ 𝛿(2+𝛿), 2+𝛿
], waiting will not be proﬁtable.
(4+𝛿)2
Therefore, when we have max{ 𝛿(2+𝛿), 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2} < 𝑝1 < min{𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ2, 2+𝛿
},17 there is a(4+𝛿)2
separating equilibrium since neither low type nor the high type deviates.18 . We can restrict
this range further since it is easy to see that 𝛿(2+𝛿)< 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2.19(4+𝛿)In ﬁgure 3, we show such a separating equilibrium. When consumers believe that any
𝑝 ∈ {1.253, 1.295} is chosen by the high-type monopolist, all conditions are satisﬁed, andthere is a separating equilibrium.
Now, we will show that there is only one price that satisﬁes the Intuitive Criterion. First,
we assume that 𝑝1 = 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2 is greater than the high type’s proﬁt maximizing price in aperfect information case. In ﬁgure 3, we have such a case. The equilibrium that satisﬁes
the Intuitive Criterion is the minimum price in the given range of separating equilibria; thatis, 𝑝1 = 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2. For any other equilibrium 𝑝1 , the high type monopolist will switch to aslightly lower 𝑝 in the range (since this will be proﬁt increasing for the high-type by the factthat the payoﬀ function is concave, and as we approach to the proﬁt maximizing price fromright, proﬁts will be increasing) but this is equilibrium dominated for the low type monopolist(otherwise this would not be part of separating equilibrium) and hence, consumers should
believe that this deviation comes from a high-type monopolist. But then this violates theIntuitive Criterion. When 𝑝1 is the minimum price in the range, any deviation to a higher
price is equilibrium dominated for high type, and any deviation to the right will be betterfor both low and high types (this would break the separating equilibrium); hence, Intuitive
Criterion is satisﬁed.20
𝛿
Proof of Proposition 2: We will show this for two diﬀerent cases; 1) 𝛾𝑙 < 2+𝛿
and 2)
𝛿𝛾𝑙 > 2+𝛿.
𝛿
When 𝛾𝑙 < 2+𝛿
, 2+𝛿
2 is less than roothigh2.
While we could not prove that 𝑟𝑜𝑜𝑡𝑙𝑜𝑤1 < 𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ1, all our numeric examples indicated this result.
Hence, we do not try to look for other separating equilibria in the range 𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ1 < 𝑝1 < 𝑟𝑜𝑜𝑡𝑙𝑜𝑤1.
19
To show that we assume that the square root term in rootlow2 is zero (otherwise, rootlow2 will be evena greater number). Then, after some algebraic manipulation, we get 2𝛿 2 (2 − 𝛿) + 8𝛿) < 8 + 2𝛿 + 4𝛿 + 𝛿 2 .
This is always true since 𝛿 and 𝛾𝑙 are positive and less than 1.
20
If we ever have a separating equilibrium in which 𝑝1 = 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2 is smaller than the high type’s proﬁt
maximizing price in a perfect information case, then the proﬁt maximizing price will be the only price thatsatisﬁes the Intuitive Criterion.
17

18

22

Pooling Equilibrium

0.2
Profit
0.1
0

K0.1

0.2

0.4

p

0.6

0.8

1.0

K0.2
Low Type equilibrium profit

High type equilibrium profit
High type waiting (deviation) profit
Low type deviation profit

Figure 4: Pooling equilibrium when .38 < 𝑝1 < .82, 𝛿 = .9, 𝛾𝑙 = .2, 𝛼 = .5
𝛿1) Assume 𝛾𝑙 < 2+𝛿
. We will show such equilibria exist if the pooling price 𝑝1 consumers

believe is in the range of:𝛿(2 + 𝛿)[𝛾𝑙 + 𝛼(1 − 𝛾𝑙 )](2 + 𝛿)[𝛾𝑙 + 𝛼(1 − 𝛾𝑙 )]< 𝑝1 <
4[(2 + 𝛿)(𝛼(1 − 𝛾𝑙 ) + 𝛾𝑙 )] − 2𝛿2and√[𝛼(1−𝛾𝑙 )+𝛾𝑙 ](𝛿+2)2 {(4+𝛿)(𝛼(1−𝛾𝑙 )+𝛾𝑙 )− 𝛼(1−𝛾𝑙 )(𝛿+4)[4𝛾𝑙 +𝛼(1−𝛾𝑙 )(4+𝛿)]}< 𝑝1 <
2(4+𝛿)[2(2+𝛿)(𝛼+𝛾𝑙 −𝛼𝛾𝑙 )−𝛿𝛾𝑙 ]√[𝛼(1−𝛾𝑙 )+𝛾𝑙 ](𝛿+2)2 {(4+𝛿)(𝛼(1−𝛾𝑙 )+𝛾𝑙 )+ 𝛼(1−𝛾𝑙 )(𝛿+4)[4𝛾𝑙 +𝛼(1−𝛾𝑙 )(4+𝛿)]}2(4+𝛿)[2(2+𝛿)(𝛼+𝛾𝑙 −𝛼𝛾𝑙 )−𝛿𝛾𝑙 ]Note that this range is not empty since ﬁgure 4 gives a pooling equilibrium example.
To prove this, ﬁrst, let us calculate the no-deviation range for the high-type monopolist.
In a pooling equilibrium, 𝜇1𝐵 = 𝛼; hence,
2𝑝1
2𝑝1
Π𝐻𝑖𝑔ℎ𝑃 𝑜𝑜𝑙 = 𝑉𝛾=1 (𝑝1 , 𝛼) = 𝑝1 (1 − (2+𝛿)(𝛼+𝛾) + 4𝛿 ( (2+𝛿)(𝛼+𝛾)2𝑙 −𝛼𝛾𝑙 )𝑙 −𝛼𝛾𝑙 )The high type monopolist can deviate in two diﬀerent ways. One way is posing as low
type in the ﬁrst period. The other way is announcing/waiting in the ﬁrst period and selling𝛾 2 (2+𝛿)2

in the second period. If it poses as a low type, its proﬁt is 𝜋𝐻𝑖𝑔ℎ𝑃 𝑜𝑜𝑙𝐷𝑒𝑣 = 4(4𝛾𝑙𝑙 +2𝛾𝑙 𝛿−𝛿) . Its
waiting equilibrium proﬁt is 4𝛿 .21 A quick calculation shows that when our assumption in the21

Note that the out of equilibrium path is not important for this equilibrium since the monopolist will notsell in the ﬁrst period by announcing to wait, and then consumers will learn the type in the second period.

23

𝛿
proposition, 𝛾𝑙 < 2+𝛿, holds, then waiting equilibrium is higher than posing as a low typeequilibrium. Therefore, the high-type will not deviate if its proﬁt in the pooling equilibrium
is greater than or equal to the proﬁt of waiting equilibrium which is 4𝛿 . Just as the previous
case, we deﬁne a function that shows the diﬀerence of equilibrium proﬁt and deviation proﬁt,
and ﬁnd its roots.
2𝑝1
1
𝑔(𝑝1 ) = 𝑝1 (1 − −2𝛼−𝛼𝛿−2𝛾𝑙 −𝛾) + 4𝛿 ( −2𝛼−𝛼𝛿−2𝛾𝑙−2𝑝)2 − 4𝛿−𝛾𝑙 𝛿+2𝛾𝑙 𝛼+𝛾𝑙 𝛼𝛿𝑙 𝛿+2𝛾𝑙 𝛼+𝛾𝑙 𝛼𝛿

The roots are:𝑙 )]𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ𝑡𝑦𝑝𝑒2 = 𝛼 + 12 𝛼𝛿 + 𝛾𝑙 + 21 𝛾𝑙 𝛿 − 𝛾𝑙 𝛼 − 12 𝛾𝑙 𝛼𝛿 = (2+𝛿)[𝛾𝑙 +𝛼(1−𝛾2𝛿(2+𝛿)[𝛾𝑙 +𝛼(1−𝛾𝑙 )]𝑙 −𝛾𝑙 𝛿+2𝛾𝑙 𝛼+𝛾𝑙 𝛼𝛿)𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ𝑡𝑦𝑝𝑒1 = (1/2)𝛿(−2𝛼−𝛼𝛿−2𝛾= 2[2(2+𝛿)(𝛼(1−𝛾(2𝛾𝑙 𝛼𝛿−2𝛼𝛿+4𝛾𝑙 𝛼−4𝛼−2𝛾𝑙 𝛿+𝛿−4𝛾𝑙 )𝑙 )+𝛾𝑙 )−𝛿]𝛿
If 𝑝1 is between the roots above and 𝛾𝑙 < 2+𝛿
, then we get the no-deviation range for thehigh-type monopolist. Note that 𝑝1 < 𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ𝑡𝑦𝑝𝑒2 ensures that 𝜃1 is less than 1 (see equation2). In other words, it guarantees that waiting equilibrium payoﬀ is not greater than pooling
equilibrium payoﬀ.
For the low type, proﬁt in a pooling equilibrium22 is:
2𝑝1
2𝑝1
Π𝐿𝑜𝑤𝑃 𝑜𝑜𝑙 = 𝑉𝛾𝑙 (𝑝1 , 𝛼) = 𝑝1 (1 − (2+𝛿)[𝛼+(1−𝛼)𝛾) + 𝛿𝛾4 𝑙 ( (2+𝛿)[𝛼+(1−𝛼)𝛾
)2
𝑙]
𝑙]

Its deviation proﬁt is:2

𝑙 (𝛿+2)
Π𝐿𝑜𝑤𝑃 𝑜𝑜𝑙𝐷𝑒𝑣 = 𝛾4(4+𝛿)The roots of the diﬀerence function, Π𝐿𝑜𝑤𝑃 𝑜𝑜𝑙 − Π𝐿𝑜𝑤𝑃 𝑜𝑜𝑙𝐷𝑒𝑣 , is:
2

2𝑝1
2𝑝1
𝑙 (𝛿+2)
Π𝐿𝑜𝑤𝑃 𝑜𝑜𝑙 − Π𝐿𝑜𝑤𝑃 𝑜𝑜𝑙𝐷𝑒𝑣 = 𝑝1 (1 − (2+𝛿)[𝛼+(1−𝛼)𝛾) + 𝛿𝛾4 𝑙 ( (2+𝛿)[𝛼+(1−𝛼)𝛾)2 − 𝛾4(4+𝛿)𝑙]𝑙]⇔ 𝑝2 4(4 + 𝛿){2(2 + 𝛿)[𝛼 + (1 − 𝛼)𝛾𝑙 ] − 𝛿𝛾𝑙 } − 𝑝4(4 + 𝛿)(2 + 𝛿)2 [𝛼 + (1 − 𝛼)𝛾𝑙 ]2 + 𝛾𝑙 (2 +𝛿)4 [𝛼 + (1 − 𝛼)𝛾𝑙 ]2 = 0√[𝛼(1−𝛾𝑙 )+𝛾𝑙 ](𝛿+2)2 {(4+𝛿)(𝛼(1−𝛾𝑙 )+𝛾𝑙 )+ 𝛼(1−𝛾𝑙 )(𝛿+4)[4𝛾𝑙 +𝛼(1−𝛾𝑙 )(4+𝛿)]}2(4+𝛿)[2(2+𝛿)(𝛼+𝛾√𝑙 −𝛼𝛾𝑙 )−𝛿𝛾𝑙 ][𝛼(1−𝛾𝑙 )+𝛾𝑙 ](𝛿+2)2 {(4+𝛿)(𝛼(1−𝛾𝑙 )+𝛾𝑙 )− 𝛼(1−𝛾𝑙 )(𝛿+4)[4𝛾𝑙 +𝛼(1−𝛾𝑙 )(4+𝛿)]}𝑟𝑜𝑜𝑡𝑙𝑜𝑤𝑡𝑦𝑝𝑒1 =2(4+𝛿)[2(2+𝛿)(𝛼+𝛾𝑙 −𝛼𝛾𝑙 )−𝛿𝛾𝑙 ]𝑟𝑜𝑜𝑡𝑙𝑜𝑤𝑡𝑦𝑝𝑒2 =This gives the no-deviation range for the low-type monopolist. As we combine the nodeviation conditions for high and low type, we get pooling equilibria (see ﬁgure 4 for an
example).
𝛿
Case 2) Assume 𝛾𝑙 > 2+𝛿
. The only thing that will change is that if the high typedeviates, it will deviate to pose as a low type rather than to deviate to wait.
22

If the high type does not ﬁnd waiting proﬁtable, then the low type deﬁnitely does not ﬁnd it proﬁtable.

24

𝛾 2 (2+𝛿)2

2𝑝1
2𝑝1
Π𝐻𝑖𝑔ℎ𝑃 𝑜𝑜𝑙 − 𝜋𝐻𝑖𝑔ℎ𝑃 𝑜𝑜𝑙𝐷𝑒𝑣 = 𝑝1 (1 − (2+𝛿)(𝛼+𝛾) + 4𝛿 ( (2+𝛿)(𝛼+𝛾)2 − 4(4𝛾𝑙𝑙 +2𝛾𝑙 𝛿−𝛿) = 0𝑙 −𝛼𝛾𝑙 )𝑙 −𝛼𝛾𝑙 )⇔ 𝑝1 4(4𝛾𝑙 + 2𝛾𝑙 𝛿 − 𝛿)(2 + 𝛿)2 (𝛼 + 𝛾𝑙 − 𝛼𝛾𝑙 )2 − 2𝑝21 4(4𝛾𝑙 + 2𝛾𝑙 𝛿 − 𝛿)(2 + 𝛿)(𝛼 + 𝛾𝑙 − 𝛼𝛾𝑙 ) +4𝛿𝑝21 (4𝛾𝑙 + 2𝛾𝑙 𝛿 − 𝛿) − 𝛾𝑙2 (2 + 𝛿)4 (𝛼 + 𝛾𝑙 − 𝛼𝛾𝑙 )2 = 0√2𝑙 )+𝛾𝑙 )− 𝛼(1−𝛾𝑙 )(4𝛾𝑙 +2𝛾𝑙 𝛿−𝛿)[(4𝛾𝑙 +2𝛾𝑙 𝛿−𝛿)(𝛼+𝛾𝑙 −𝛼𝛾𝑙 )−𝛿𝛾𝑙 ]}⇔𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ𝑡𝑦𝑝𝑒2,1 = (2+𝛿) (𝛼+𝛾𝑙 −𝛼𝛾𝑙 ){(4𝛾𝑙 +2𝛾𝑙 𝛿−𝛿)(𝛼(1−𝛾
2(4𝛾𝑙 +2𝛾𝑙 𝛿−𝛿)[2(2+𝛿)(𝛼(1−𝛾𝑙 )+𝛾𝑙 )−𝛿]√(2+𝛿)2 [𝛼(1−𝛾𝑙 )+𝛾𝑙 ]{(4𝛾𝑙 +2𝛾𝑙 𝛿−𝛿)(𝛼(1−𝛾𝑙 )+𝛾𝑙 )+𝛼(1−𝛾𝑙 )[2𝛾𝑙 (2+𝛿)−𝛿][2(1−𝛼)(2+𝛿)𝛾 2 +[(3𝛼−2)𝛿+4𝛼]𝛾𝑙 −𝛿𝛼}𝑙⇔𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ𝑡𝑦𝑝𝑒2,2 =2(4𝛾𝑙 +2𝛾𝑙 𝛿−𝛿)[2(2+𝛿)(𝛼(1−𝛾𝑙 )+𝛾𝑙 )−𝛿]

For the low type, no-deviation range is still the same as Case 1. Hence, when price isbetween𝑀 𝑎𝑥{𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ𝑡𝑦𝑝𝑒2,1 , 𝑟𝑜𝑜𝑡𝑙𝑜𝑤𝑡𝑦𝑝𝑒1 } < 𝑝1 < 𝑀 𝑖𝑛{𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ𝑡𝑦𝑝𝑒2,2 , 𝑟𝑜𝑜𝑡𝑙𝑜𝑤𝑡𝑦𝑝𝑒2 }𝛿and 𝛾𝑙 > 2+𝛿
, we have a pooling equilibrium.

Figure 5: Indiﬀerence Curves at the Pooling equilibrium when 𝑝1 = .635, 𝛿 = .8, 𝛾𝑙 = .5,
𝛼 = .5 The high type’s (low type’s) is the one shifted further right (left). Preferable pairs of(𝑝1 , 𝜇1𝐵 ) lies inside for both types. High type may deviate to 𝑝1 = 1.15 which is equilibrium
dominated for low type (even when 𝜇1𝐵 = 1); hence, Intuitive Criterion does not hold.
When 𝛾𝑙 > 0, these pooling equilibria do not satisfy the Intuitive Criterion. For any
pooling equilibrium (𝑝1 , 𝜇1𝐵 ), we calculate the indiﬀerence curve, 𝑉𝛾 (𝑝1 , 𝜇1𝐵 ), (where 𝛾 ∈{1, 𝛾𝑙 }) for both types. We will show that these indiﬀerence curves satisfy the Spence-Mirrlees
condition (i.e., single crossing property). But then, we can show that high type monopolist
will deviate to a (very high) price that the low type will not deviate even if consumers’ belief
25change to 𝜇1𝐵 = 1 after observing such a deviation (See Figure 5). Therefore, the pooling
equilibria do not satisfy the intuitive criterion.
First, we calculate the following partial derivatives:

𝑑𝑉
4𝑝1
2𝛿𝛾𝑝1
(𝛾, 𝜇1𝐵 , 𝑝1 ) = 1 −+𝑑𝑝1((2 + 𝛿)(𝜇1𝐵 + (1 − 𝜇1𝐵 )𝛾𝑙 )) ((2 + 𝛿)2 (𝜇1𝐵 + (1 − 𝜇1𝐵 )𝛾𝑙 )2 )𝑑𝑉
2𝑝21 (1 − 𝛾𝑙 )2𝛿𝛾𝑝21 (1 − 𝛾𝑙 )=−𝑑𝜇1𝐵((2 + 𝛿)(𝜇1𝐵 + (1 − 𝜇1𝐵 )𝛾𝑙 )2 ) ((2 + 𝛿)2 (𝜇1𝐵 + (1 − 𝜇1𝐵 )𝛾𝑙 )3 )By using the equations above, we calculate the following derivative (and we can take thederivative since 𝛾𝑙 > 0:23

/𝑑𝑝1
−𝑑( 𝑑𝑉𝑑𝑉/𝑑𝜇
)
1𝐵

𝑑𝛾

= −(

(2 + 𝛿)(−𝛿𝜇1𝐵 − 𝛿𝛾𝑙 + 𝛿𝛾𝑙 𝜇1𝐵 − 2𝜇1𝐵 + 2𝛾𝑙 𝜇1𝐵 + 2𝑝1 − 2𝛾𝑙 )(−𝜇1𝐵 − 𝛾𝑙 + 𝛾𝑙 𝜇1𝐵 )2 𝛿2(𝑝21 (−2𝜇1𝐵 − 2𝛾𝑙 + 2𝛾𝑙 𝜇1𝐵 − 𝛿𝜇1𝐵 − 𝛿𝛾𝑙 + 𝛿𝛾𝑙 𝜇1𝐵 + 𝛿𝛾)2 (−1 + 𝛾𝑙 ))/𝑑𝑝1)/(𝑑𝛾) is always negative since the denominator is negative (−1+𝛾𝑙 <But then −𝑑( 𝑑𝑉𝑑𝑉/𝑑𝜇1𝐵0) and the numerator is also positive since(−𝛿𝜇1𝐵 − 𝛿𝛾𝑙 + 𝛿𝛾𝑙 𝜇1𝐵 − 2𝜇1𝐵 + 2𝛾𝑙 𝜇1𝐵 + 2𝑝1 − 2𝛾𝑙 ) ⇔ 𝑝1 <(2 + 𝛿)(𝛾𝑙 + 𝛼(1 − 𝛾𝑙 )2

The right hand side of the equation above should hold in any pooling equilibrium; otherwise, 𝜃1 > 1 holds and that means waiting equilibrium payoﬀ will be greater than pooling
equilibrium payoﬀ for the high type.
This is the Spence-Mirrlees condition so the single crossing holds. In addition, this showsthat the derivative of low type’s indiﬀerence curve (at the crossing point) is greater than thehigh type’s; hence, low type’s indiﬀerence curve touches 𝜇1𝐵 = 1 at a lower price (say 𝑝𝑙 )than the high type’s. If the high type deviates to a price slightly greater than this 𝑝𝑙 , thenthe low type would not follow high type even under the belief 𝜇1𝐵 = 1. Hence, intuitive
criterion will be violated.
23

1𝐵
Note that by dividing those derivatives, we get −𝑑𝜇which is the slope of the indiﬀerence curves.
𝑑𝑝1

26

However, when 𝛾𝑙 = 0, given the belief that 𝜇1𝐵 = 𝛼, now the low type will always mimic
the high type in the ﬁrst period, and no 𝑝1 can be equilibrium dominated for the low typesince it will be making zero proﬁts if consumers knew/learned that it is a low type. Hence,the pooling equilibrium satisﬁes the Intuitive Criterion for 𝛾𝑙 = 0 case.
This concludes our proof.

Proof of Proposition 3:
For a given equilibrium we compare the equilibrium prices we computed for any pooling
equilibrium. For part a), we have

𝑝1 < 𝑝2 =𝑝1
1 − 2(𝛼 + 𝛾𝑙 − 𝛼𝛾𝑙 )⇐⇒ 𝛿 <(2 + 𝛿)(𝛼 + 𝛾𝑙 + 𝛼𝛾𝑙 )(𝛼 + 𝛾𝑙 − 𝛼𝛾𝑙 )Note that 𝑝2 = 𝜃21 (from the second period maximization problem of the high type
monopolist).
Proof of part b follows from part a.
𝛿
Proof of Proposition 6: We will show that such an equilibrium exists when 𝛾𝑙 < 2+𝛿

and consumers believe that any ﬁrm selling in the ﬁrst period is a low type.24
The waiting proﬁt of the high-type ﬁrm is given by 𝜋𝑤𝑎𝑖𝑡 = 4𝛿 , since 𝜃1 = 1.
The proﬁts of the high ﬁrm posing as a low ﬁrm is the same as in equation 9, which is

𝜋𝐻𝑖𝑔ℎ𝐷𝑒𝑣 =(2 + 𝛿)2 𝛾𝑙24(4𝛾𝑙 + 2𝛿𝛾𝑙 − 𝛿)For the ﬁrm to wait, and not deviate and pose as a low type, the proﬁts of waiting must
be greater than or equal to the proﬁts of deviating as a low type. The root of the function
𝛿, the waiting proﬁt will be higher, and𝜋𝑤𝑎𝑖𝑡 − 𝜋𝐻𝑖𝑔ℎ𝐷𝑒𝑣 is 𝛾𝑙 = 𝛿/(2 + 𝛿). When 𝛾𝑙 < 2+𝛿

hence, the high type waits for the second period to sell.
If 𝛾𝑙 = 0 holds, then no 𝑝1 can be equilibrium dominated for the low type since it isalready making zero proﬁts. Hence, this satisﬁes the Intuitive Criterion. If 𝛾𝑙 is positive,
24

There may be other waiting equilibrium for diﬀerent parameter ranges with diﬀerent beliefs so we do
not claim that this is the only waiting equilibrium. This equilibrium, however, is suﬃcient to show “waiting
equilibrium” exists.

27

then assume that high type deviates to some 𝑝1 ∈ (𝑟𝑜𝑜𝑡𝑙𝑜𝑤2, 𝑚𝑖𝑛{𝑟𝑜𝑜𝑡ℎ𝑖𝑔ℎ2, 2+𝛿
}). we have2
already showed in the separating equilibrium proof that these price ranges are equilibrium
dominated for low type even under the belief 𝜇1𝐵 = 1 but is proﬁtable for the high typecompared to waiting. Therefore, such equilibria cannot satisfy intuitive criterion.

7

Appendix B

7.1

Separating Equilibrium Case

Here, we will calculate the welfare under the separating equilibrium that satisﬁes the intuitive
criterion. This is when25√[(2 + 𝛿)2 [4 + 𝛿 + (4 + 𝛿)(1 − 𝛾𝑙 )[𝛿(1 − 𝛾𝑙 ) + 4) ]𝑝1 = 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2 =2(4 + 𝛿)(𝛿(2 − 𝛾𝑙 ) + 4)
Let us ﬁrst calculate the welfare under the high type monopolist, then we will turn to
the low type monopolist. Finally, we will add these welfare calculations to solve for the total
ex-ante expected welfare.
The proﬁt of high type monopolist in a separating equilibrium is 𝑉𝛾=1 (𝑝1 = 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2, 𝜇1𝐵 =1). To calculate social welfare, we need 𝜃1 , and it can be calculated from equation 2.
√−(2 + 𝛿)(4 + 𝛿 + (4 + 𝛿)(−1 + 𝛾𝑙 )(−𝛿 + 𝛿𝛾𝑙 − 4))𝜃1 (𝑝1 = 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2, 𝜇1𝐵 = 1) =((4 + 𝛿)(−2𝛿 + 𝛿𝛾𝑙 − 4))In the equilibrium, we have 𝜃2 = 𝑝2 = 𝜃21 . Consumer surplus can be calculated by inserting𝜃1 , 𝜃2 , 𝑝1 and 𝑝2 to the following equation:
∫ 1
𝐶𝑆ℎ =

∫ 𝜃1
(𝜃 − 𝑝1 )𝑑𝜃 + 𝛿

𝜃1(𝜃 − 𝑝2 )𝑑𝜃𝜃2After taking the integral and simplifying, and adding the proﬁt, we calculate the welfare
under the high type monopolist in a separating equilibrium.26 Since, the expression is cumbersome, we provide its three dimensional graph in ﬁgure 6. While drawing the graph, we
assumed 𝛼 = 1.
25
26

In the proofs, we demonstrate that 𝑝1 = 𝑟𝑜𝑜𝑡𝑙𝑜𝑤2 satisﬁes the intuitive criterion.
MAPLE is used in making these calculations and drawing the graphs. It is available from the authors.

28

0.0

0.0
0.25
0.5

0.25

gammal

0.5

0.75

0.4

0.5

0.1
0.0 1.0

0.75

1.0
0.2

0.25

0.25

0.2

0.75

0.35

1.0
0.0

0.3

0.4
0.6

0.15

del

gammal0.05 1.0

0.8

delFigure 6: Welfare under a high-type monopolist (on the left) and its (always positive) derivative w. r. t. 𝛿 (on the right)

0.0

0.0
0.25
0.5

0.25

gammal

0.5

0.75
0.44
1.0
0.39
0.0
0.34
0.25
0.29
0.5
del
0.24
0.75
0.19 1.0

gammal
0.75

0.23

1.0
0.0

0.18
0.25

0.13
0.08
0.03 1.0

0.5
0.75

del

Figure 7: Welfare (on the left) and Derivative of Welfare w. r. t. 𝛿 (on the right)29

As expected, welfare increases with 𝛾𝑙 since this will decrease signalling cost for the hightype. This is clear from the ﬁrst graph in ﬁgure 6. It also increases with 𝛿. As delta increases,
price increases and hence less consumers buy the good. This decreases consumer surplus.
However, proﬁt always increases as delta increases. The net eﬀect is positive.
Now, we will calculate the welfare under a low type monopolist in a separating equilibrium. The low type monopolist actually produces the amounts it would produce in a perfect
information model in a separating equilibrium.
22𝛾𝑙𝑙 (2+𝛿)The proﬁt is 𝑉𝑙 (𝑝1 = (2+𝛿), 𝜇1𝐵 = 0) = 𝛾4(4+𝛿). To calculate the consumer surplus, we2(4+𝛿)2+𝛿
2+𝛿
need 𝜃1 = 2+𝛿, 𝜃2 = 2(4+𝛿), 𝑝2 = 𝛾𝑙 2(4+𝛿)
.
4+𝛿

∫ 1

(2 + 𝛿)2 𝛾𝑙
𝐶𝑆𝑙 =
(𝜃 −
)𝑑𝜃 + 𝛿2+𝛿2(4 + 𝛿)
𝜃1 = 4+𝛿

∫

2+𝛿
(4+𝛿)
2+𝛿
2(4+𝛿)(𝜃 − 𝛾𝑙2+𝛿)𝑑𝜃
2(4 + 𝛿)Taking the integral, we have,

𝐶𝑆𝑙 =

1
[3𝛿 3 + 12𝛿 2 + 28𝛿 + 48 − 2𝛿 3 𝛾𝑙 − 16𝛿 2 𝛾𝑙 − 40𝛿𝛾𝑙 − 32𝛾𝑙 ]28(4 + 𝛿)then,by adding the proﬁt, we have the welfare when we have a low type:𝑊𝑙 =3𝛿 3 + 12𝛿 2 + 28𝛿 + 48 − 2𝛿 3 𝛾𝑙 − 16𝛿 2 𝛾𝑙 − 40𝛿𝛾𝑙 − 32𝛾𝑙 𝛾𝑙 (2 + 𝛿)2+8(4 + 𝛿)24(4 + 𝛿)Total expected welfare is 𝛼𝑊ℎ + (1 − 𝛼𝑊𝑙 ). However, calculating total expected welfare
analytically is cumbersome so we give its three dimensional graph in ﬁgure 7. We assume
𝛼 = 12 while drawing the graph. The graph shows that welfare increases as 𝛾𝑙 increases; as
𝛾𝑙 increases, the cost of signalling for high-type decreases; hence, the welfare increases. The
derivative of welfare with respect to delta is positive.
Let us now calculate the eﬃciency loss for the separating equilibrium. The low typeproduces the same output in both perfect information and separating equilibrium. The
loss comes from the high type’s signalling eﬀort. We have already calculated welfare when
there is a high type monopolist under perfect information and under separating equilibrium.
By subtracting these welfare and multiplying with 𝛼, we calculate the eﬃciency loss. We
report its three dimensional graph in ﬁgure 8 (while drawing the graph, we used 𝛼 = 1. As
30

0.0

0.0
0.25
0.5

0.25

gammal

0.5

0.75
0.2
0.1
0.0 1.0

0.5

1.0
0.0

-0.15

0.25
0.75

0.75

-0.05

1.0
0.0

0.3

-0.25

del

-0.35 1.0

gammal

0.25
0.5
0.75

del

Figure 8: Eﬃciency Loss and its derivative w. r. t. 𝛿
the probability of having a high-type monopolist increases, eﬃciency loss decreases.) andshow that eﬃciency loss decreases as delta increases. Welfare under separating equilibriumincreases faster as delta increases. The cost of signalling must be decreasing as agents become
more patient.

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