ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E15-8

A Dynamic Monopoly with Experience Goods andRisk-Averse Consumers
byMaría Martín-Rodríguez

December 2015

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

A Dynamic Monopoly with Experience Goods andRisk-Averse Consumers
María Martín-Rodríguez

∗Nagoya University
December 21, 2015

Abstract
I analyze the dynamic problem of a monopolist facing dierent masses of riskaverse consumers in two periods. All qualities are equally costly and the true quality
is uncertain for all the agents in the rst period. I nd that a semi-separating equilibrium is sustainable in the second period if the rst-period market share is strictly
positive due to social learning. Increasing patterns of prices may arise when low
qualities are revealed, but this outcome is less likely with less risk-averse consumers.
The welfare analysis suggests that the authority could only help when the consumers
were not very risk averse.
Keywords: asymmetric information, risk aversion, price pattern.
JEL classication: L12, L15

1

Introduction

From time to time, a truly innovative product appears into the market: the Walkman was1the rst portable cassette player that consumers could actually buy,

the Virtual Boy wasthe rst portable video game console capable of displaying true 3D graphics out of thebox, and several prototypes of wearable technology with an optical head-mounted display
are currently in progress.
∗

2

When the products have such innovative nature, both the rm

maria.martin.rdguez@gmail.com, maria.martin.rodriguez@e.mbox.nagoya-u.ac.jp.
Graduate School of Economics, Nagoya University. Furo-cho, Chikusa-ku, Nagoya 464-8601,

E-mail addresses:

Address:

Japan.
1 In fact, the rst portable cassette player was the Stereobelt, invented by Andreas Pavel. Even thoughhe led a patent in 1977, at that time there was no major vendor interested in manufacturing the device.
Sony began to commercialize the Walkman in 1979, with the subsequent legal battles that lasted until
2004.
2 The reader could be familiar with the Google Glass, although the idea does not belong to Google;
instead, the original device is the EyeTap, developed in the eighties by the father of wearable computing
Steve Mann.
1and the consumers ignore how well they will t the consumers' taste at the moment ofbeing introduced.
In this context of incomplete information, the monopolist may have incentives to
set a price low enough to induce experimentation, so that the parties involved in thetransaction learn how well the product ts the consumers' taste. But observe that the
aforesaid price might not be an introductory price: if consumers would nd that they
dislike the product, the monopolist could be forced to decrease the price in the period
next to the experimentation, depending on the severity of the dislike.
I study the dynamically optimal pricing strategy of a forward-looking monopolist that
introduces a novel product, when facing consumers who are risk-averse and non-strategic.

3

The product can be seen as an experience good at the moment of the introduction, whilst
signaling is possible afterwards due to non-idiosyncratic tastes if the monopolist induced
experimentation.

The cost of inducing experimentation increases as the risk aversion

increases, but the subsequent positive eect of removing the uncertainty could oset theeect derived from learning that the product does not t well the consumers' taste.
The rest of the paper is organized as follows:

Section 2 explains the model; theBayesian-Nash equilibrium is characterized in Sections 3 and 4; Section 5 discusses the
problem of the social planner; and Section 6 concludes.

 Literature Review:Dynamic pricing in the presence of uncertainty is an old question. A rst research line
considers forward-looking consumers and ignores signaling.

Shapiro (1983) studies the

dynamic pricing problem of a monopolist aware of its quality but unable to signal it;
thus, consumers only learn through experience. Introductory prices occur when consumers
underestimate quality, whereas in my model introductory prices may also happen when
consumers nd that they overestimated how well the product will t their taste, due tothe subsequent positive eect on utility of removing the initial uncertainty.
Another example is ?.

The authors study the dynamic monopoly pricing when all

the agents are initially uncertain about the number of high-type consumers, which inturn determines the extent of the network externality.

The rst-period price is below

the expected second-period price in order to induce the high-type consumers to buy inthe rst period, and so the externality becomes observable in the second period. In myframework the monopolist also sets a low price oriented to remove the initial uncertainty,
although it is due to the risk aversion of the consumers.
Bergemann and Valimaki (2006) and Villas-Boas (2006) consider frameworks in which
the perception of the quality is subjective. The former examine a monopoly who dynamically changes her pricing policy depending on the relative sizes of the segments of informed
and uninformed consumers, and nd that introductory prices happen in niche markets.
The latter examines a dynamic duopoly in which consumers have a relative preference for3 According to Nelson (1970), an experience good is a product such that some of its characteristics

cannot be observed in advance by the consumers, but instead are ascertained upon consumption.
2

the variety they learn about in rst place, and concludes that the anti-competitive eectof exploiting the informational advantage dominates the pro-competitive eects.
Empirical studies support the hypothesis that, for some goods, consumers can only
learn through experience.

4

For instance, Crawford and Shum (2005) use anti-ulcer drugs

data and conclude that, while there is substantial heterogeneity in the eectiveness acrosspatients, they and their doctors gradually reduce the costs of uncertainty through direct
trial of the dierent drugs. Israel (2005) uses car insurance data and nds that consumers
overestimate the quality when contracting the service, but that the impact of learning is
mitigated as the quality is discovered only after a road accident.
A second research line deals with signaling considerations.

Milgrom and Roberts(1986) consider a dynamic monopoly in which introductory prices and dissipative advertising are signals of the type, dened as the probability that a random consumer ndsthe product satisfactory.

5

In equilibrium, introductory prices are used if the good is not

perceived as surely satisfactory. Bagwell and Riordan (1991) study the signaling problem
of a static monopoly when some consumers are informed, nding that the price distortion
necessary to signal a high type decreases as the number of informed consumers increases.
In my model, the problem of the second period is very similar to the one analyzed in
Bagwell and Riordan (1991), although the amount of informed consumers was strategicallydetermined by the monopolist in the rst period. In contrast to Milgrom and Roberts(1986), the price plays no signaling role when introducing the product, and to spendmoney in dissipative advertising is not allowed.
Judd and Riordan (1994) analyze a dynamic monopoly in which both the buyers andthe seller acquire private, noisy information about the quality of the good after the rstpurchase. Higher prices signal higher qualities in the equilibrium of the second period,

6

and the equilibrium expected

second-period price exceeds the rst-period price. On thecontrary, I allow for the possibility of not buying in the rst period, so some consumers
may not get information.

Also, I consider that the information derived from the rst

purchase is accurate instead of noisy.
It is possible to consider signals other than price and advertising.

For instance, in

Bar-Isaac (2003) the signal is the decision of the monopolist of producing or not; in Boseet al. (2006) the signal for a consumer at a certain period is the history of previous
purchases by other buyers.

In the duopoly market considered by Caminal and Vives(1996), the previous market share also plays a crucial signaling role, leading to the use of
introductory prices.

4 Although not focused on the learning process, Farina (2012) studies a typical case of subjective

quality: the market of ready-to-drink orange juice. Using data from Brazil, concludes that the rmsshould give their juices for free to convince the uninformed consumers to taste them.
5 Ackerberg (2003) tests the eect of advertising in the introduction of a low-fat yogurt in the American
market, and nds that it only has a signaling eect on inexperienced consumers.
6 The realized second-period price is proportional to the private signal of the seller, so it can be below
the rst-period price if the signal is low enough.
3

2

The Model

A monopolist sells a non-storable experience good in a game of two periods (t = 1, 2). He
is located at the point 0 of the unit interval and his discount factor is equal to 1. It is
possible to charge dierent prices in each period, but discrimination across consumers wholive in the same period is not allowed. Price commitment across periods is not allowed
either.
In period 1, it is uncertain how well the product will t the consumers' needs (in a wide
sense, this concept would be identied with the quality). Since sometimes it is reasonable
to think that this adjustment depends more on some unobservable characteristics of the
population than on the intrinsic characteristics of the product, I will assume that all thetypes are equally costly without loss of generality, I normalize all the production costs
to zero. Therefore, in the rst period, all the agents assume that the quality is a random

2variable such thatq ∼ N (µ, σ), with µ > 0. In any period the consumers buy either one

unit or zero. In the rst period, only the consumers who acquire the product learn thequality after consumption. The monopolist can undertake some market research after therst-period consumption choices at a negligible cost, so he also observes the quality.
There is a mass 1 of risk-averse consumers uniformly distributed in each period. The
consumers have a unitary demand (they can buy either one unit or zero) and, since they
only live for one period, they maximize their instantaneous utility conditional on theirinformation sets. Although the consumers in each period are dierent, the information
is disseminated from the rst period to the second period in the following way:each

consumer i located at xi in the rst period truthfully communicates his gathered information about the quality to the consumer located at xi in the second period. This canbe interpreted as if consumers would only have access to the information gathered bysomeone with the same horizontal preferences: in most cases, individuals make friends
among other individuals with similar tastes, or they may join specialized clubs in which
they meet other people with criteria and preferences which are alike. In any case, thiscommunication structure captures two facts supported by evidence: the consumers learn
from the experience of others and they cannot access every single piece of information
available in the market. In a broad sense, it can be viewed as a very specic type of social
learning.

I also assume that, in the rst period, the monopolist launches an extensive

advertising campaign at a negligible cost to announce the existence of his product to theconsumers. This campaign cannot be taken as a signal, since the monopolist also ignores
how well the product that he supplies will t the necessities of the consumers. However,the campaign aects the consumers' utility (for instance, it may be oensive for the local
standards) but the monopolist cannot observe this before the rst-period purchases. In
the second period, no campaign is needed because the consumers are already aware of theexistence of the product. Then, the demand of the rst period will be random for themonopolist whereas the demand of the second period will be deterministic.
The timing is graphically explained in Figure 1. In words, is as follows: in the rst

2 A minimum degree of uncertainty σ ≥ 0.01 is necessary for the model to work.
4

period, no agent knows the quality. The monopolist launches the campaign and sets therst-period price, the consumers make their consumption choices and the rst-period payos are realized. Those who acquired the product observe the quality after consumption,and so does the monopolist. In the second period, the information dissemination takes
place: this divides the market into informed (those consumers who learn the quality fromthe previous experience of the others) and uninformed consumers. All the second-period
consumers observe the market share of the rst period. The monopolist sets the secondperiod price, and the uninformed consumers use the rst-period market share and thesecond-period price to update their beliefs. Since the informed consumers already know
the true quality, they do not make any inference process. Finally, the consumption choices
are made, the second-period payos are realized and the game ends.
I consider that the consumers exhibit a CARA (Constant Absolute Risk Aversion) utility

Figure 1: Timing of the game.

function. In the rst period, because of this utility form and the normal distribution of
the quality, maximizing the expected utility of the consumer i is equivalent to maximize
the following linear certainty equivalent:1
CEi = µ − ρσ 2 − p1 − xi + z1
2

(1)

where ρ > 0 is the risk-aversion coecient, p1 is the price charged by the monopolist inperiod 1 and xi is the location of the consumer.
and it aects all consumers in the same way.

43

z1 represents the eect of the campaign,
The monopolist assumes z1 ∼ U[−Z, Z]

3 The transportation cost parameter has been normalized to τ = 1.

4 Although I have provided an interpretation in terms of an advertising campaign, the framework

works for any pair of independent random shocks on preferences with 0 mean across the periods 1 and 2.
5with suciently large value of Z

> 0: on average the campaign has no eect, but theprobability that the campaign does not aect the utility is 0. Normalizing the utility of
not acquiring the good to 0, the marginal consumer in the rst period, that determines
the demand bounded by 0 and 1, is
1x̂1 (µ, ρ, σ, p1 , z1 ) = (2µ − ρσ 2 − 2p1 + 2z1 )2(2)

Remember that in the second period there is no advertising campaign. Denoting the true
quality by q̄ , the demand from the informed consumers, bounded by 0 and x̂1 , isφ(q̄, p2 ) = q̄ − p2

(3)

The demand from the uninformed consumers, bounded by x̂1 and 1, would be referredas ω(ρ, p2 , x̂1 ).

The price here has two eects: rst, the direct eect of decreasing thedemand; second, the signaling eect. The rst-period market share only has a signalingeect. From now on, the total demand of the second period will be denoted by x2 .

3

Equilibrium

Since the monopolist is forward looking, the backward induction applies.

Notice thatnegative prices are weakly dominated strategies in the second period, but not in the rst.

3.1 Equilibrium of the second period
We may have three dierent situations, depending on the amount of informed consumers.

3.1.1

Full information

1(2+2p1 −2µ+ρσ 2 ),5 that is, when all second-period
2
From the prot-maximization problem making x2 = φ(q̄, p2 )This situation takes place when z1 ≥consumers are informed.

subject to 0 ≤ x2 ≤ 1, the equilibrium is


 0
I
q̄/2
p2 =

q̄ − 1


 0
I
q̄/2
x2 =

1

if q̄ ≤ 0
if q̄ ∈ (0, 2)
if q̄ ≥ 2

if q̄ ≤ 0
if q̄ ∈ (0, 2)(4)

if q̄ ≥ 2

In the full-information equilibrium, the higher the quality, the higher the charged priceand the larger the share of served consumers.

5z

1
2
1 ≥ 2 (2 + 2p1 − 2µ + ρσ )when acquiring the good.

means that the most distant consumer obtained a strictly positive surplus

63.1.2

No information1(2p1 − 2µ + ρσ 2 ),6 that is, when no agent knows the2true quality of the good, including the rm. In this case, the consumers correctly infer

This situation takes place when z1 ≤7that the price does not convey any information and the uncertainty cannot be ruled out.
2Then, ω(ρ, p2 , 0) = (2µ − ρσ − 2p2 )/2. Solving the prot-maximization problem making

x2 = ω(ρ, p2 , 0) subject to 0 ≤ x2 ≤ 1, the solution is20
if ρ ≥ 2µ/σ



2
2
(2µ − ρσ )/4
if µ ≤ 2and ρ < 2µ/σ ;
or
pU2 =
2
2
if µ > 2 and (−4 + 2µ)/σ < ρ < 2µ/σ



2
2
(2µ − ρσ − 2)/2 if µ > 2 and ρ ≤ (−4 + 2µ)/σ

0



(2µ − ρσ 2 )/4
xU2 =



1

if ρ ≥ 2µ/σ

(5)2if µ ≤ 2

andif µ > 2andif µ > 2andρ < 2µ/σ 2 ; or(−4 + 2µ)/σ 2 < ρ < 2µ/σ 2ρ ≤ (−4 + 2µ)/σ 2(6)Here, both the price and the share of served consumers increase as the expected quality
increases, and they decrease as the risk-aversion coecient increases.

3.1.3

Intermediate1(2p1 − 2µ + ρσ 2 < z1 < 21 (2 + 2p1 − 2µ + ρσ 2 ): the2monopolist and some consumers are informed, whereas the rest of the consumers use theThis situation takes place when

price p2 and the market share x̂1 as signals to update their beliefs.

8

As this is a game of asymmetric information, the equilibrium concept used is the
Perfect Bayesian Nash Equilibrium (PBNE): (i) given beliefs, each player chooses strategies that are sequentially rational; (ii) beliefs are established consistently with players'
strategies, in the sense that they do not contradict Bayes' rule.
Proposition 1.

that

For a given x̂1 ∈ (0, 1), there exists a semi-separating equilibrium such

· the qualities q̄ ≤ 0 pool at a price ps2 = 0;
· the rest of qualities charge a price ps2 = q̄ − x̂1 if q̄ ≥ 2x̂1 , and ps2 = q̄/2 otherwise.
6z

means that the closest consumer would have obtained a strictly negative
surplus if she had acquired the good.
7 Notice that, even if the monopolist would have learned his quality after the rst period, the secondperiod price cannot be a valid signal in this case because there are no costs derived from charging a pricenot corresponding to the true quality: there are no future penalties because the game ends after thesecond-period purchases and, since there are no informed consumers, the monopolist is not losing moneyfrom the informed sector if he charges a price higher than the one corresponding to the actual quality.
8 The signaling problem follows the spirit of Spence (1973). An analysis of signaling games with acontinuum of types can be found in Mailath (1987). See Athey (2001) for a study of games of imperfect
information with a continuum of actions. For a discussion of signaling equilibria with multiple signals
and a continuum of types, check Ramey (1996).
12
1 ≤ 2 (2p1 − 2µ + ρσ )7

Proof: in the Appendix.

The beliefs that support this equilibrium predict a higher quality from a higher price,
but weighting by the share of informed consumers:

this share is used to measure themoney that the monopolist will stop earning if he charges a price not corresponding to
the true quality. Intuitively, the equilibrium works because the beliefs are such that thedemand from the uninformed consumers is always zero. Regarding the separating partof the equilibrium, given a strictly positive quality and a share of informed consumersx̂1 ∈ (0, 1), it turns out that the corner pricing strategy9 is the most protable option
when q̄ ≥ 2x̂1 otherwise, the interior solution in which only a fraction of the informed
consumers acquire the good is optimal.

That is, if the quality is suciently low withrespect to the share of informed consumers, distortion on prices is not necessary because
the monopolist maximizes his prots just by charging the full information price; however,if the quality is suciently high with respect to the share of informed consumers, distortionon prices appears: given that the demand from the uninformed sector is zero, it is optimalfor the monopolist to x a price such that the marginal consumer of the rst period receives
exactly no surplus. This optimal price is higher than the full information price, but thesdistortion decreases as x̂1 increases. In summary, by using p2 and x̂1 , the uninformed
consumers can infer the true value q̄ > 0 with no error, although no consumer belonging
to this group will buy the product.

On the contrary, if the quality is negative, themonopolist is indierent among all the pricing strategies because the prots will always

10

be null, so he pools at a zero price without loss of generality.

The characterization of the equilibrium shows some similarities with Milgrom andRoberts (1986) and Tirole (1988): the marginal buyer of the rst period does not obtaina positive surplus in the second period.

Although only the former involves a quality-

signaling problem in the rst period, the problem of the second period is essentiallythe same in the two models:the monopolist can choose to milk the consumers who

previously purchased the good, or he can choose to attract new consumers.

In thesemodels, any consumer i may be satised with the product (implying a gross utility of ri )or not (implying a gross utility of 0). The consumer can only know whether she likes the
product after consumption. In this framework, the quality is dened as the probability
with which the product will satisfy any random consumer. Then, in the second perioda share q of the rst-period consumers are satised and have a higher willingness to pay
than: (i) those who are not satised, and (ii) those who have not purchased the good andwhose willingness to pay is determined by the expected utility. The strategy of expanding
the customer base in the second period is dominated because new consumers can only bereached by a substantial price decrease (price discrimination is not allowed). Although in
my model the quality is not subjective and the signaling problem may only arise in thesecond period, the problem of the second period is more extreme: in order to sustain thesemi-separating equilibrium, the beliefs held by the uninformed consumers are such that

9 The monopolist sets a price such that all the informed consumers acquire the good and the most

distant informed consumer obtains zero surplus: ps2 = q̄ − x̂1 .
10 The only prices that would be out-of-equilibrium are the negative prices. The out-of-equilibrium
beliefs assign any negative quality (the value does not really matter) to a negative price with no error.
8

new customers cannot be reached through any price change, so expanding the customer
base is simply impossible for the monopolist.
Notice that the equilibrium established in Proposition 1 collapses to the full-information
equilibrium when x̂1 = 1. However, this continuity breaks down when x̂1 = 0. The reason is that, given a certain qualityq̄ > 0, the distortion needed for maintaining thesemi-separating equilibrium decreases as the amount of informed consumers increases.
However, when the information about the quality cannot be credibly disclosed, only therisk-aversion parameter ρ and the expected quality µ matter for the consumers' utility.
It is worth explaining the dierence between the signaling role played by the market
share in Caminal and Vives (1996) and here. In the former it is assumed that some rstperiod consumers receive a signal about the quality dierential between two competingbrands before buying, whereas the rms are totally ignorant about this dierential. Despite no direct communication among consumers of dierent periods, the dierence in the11rst-period sold quantities is a noisy

signal of the quality dierential: consumers always

prefer higher qualities, and some of them made informed purchases. On the contrary, alarger rst-period market share does not signal a higher quality in this framework, because both the monopolist and the consumers have no information before the rst-period
purchases. Instead, due to the communication between the consumers of the two periods,
the market share is a measure of the amount of people who know the true quality in thesecond period and that, therefore, do not need any signal to update their beliefs. If thebeliefs of the uninformed consumers are such that the demand from this group is always
zero, the monopolist optimally maximizes the prots obtained from the informed groupof consumers. Since the size of the informed group is publicly observed and equal to therst-period market share, the uninformed consumers can use this information and the12second-period price to infer

the true quality of the product.

It is also remarkable that, given the share of informed consumers, the price distortionis unnecessary for some types. This result is a generalization of the conclusion obtainedin Bagwell and Riordan (1991): with two possible types (high and low) such that the hightype is more costly to produce and also more valued by consumers, the price distortion
decreases as the amount of informed consumers increases in the separating equilibrium(and, eventually, the high type does not need to distort). Instead, I consider a continuumof types equally costly to produce such that the consumers' valuation increases as thetype increases.

Given the share of informed consumers, the types above 2x̂1 optimally

distort the price whereas the rest charge the full-information price. Consider the followingq̄ = 0.4, q̄ = 1 and q̄ = 2.5, with full-information prices
0.2, 0.5 and 1.5. Given x̂1 = 0.3, the equilibrium prices are 0.2, 0.7 and 2.2: the typeq̄ = 0.4 charges its full-information price, whereas the other two types optimally distort. If,
instead, x̂1 = 0.6, the equilibrium prices are 0.2, 0.5 and 1.9: now the type q̄ = 1 charges its
full-information price and the type q̄ = 2.5 distorts, but less than when x̂1 = 0.3. However,example with three qualities:

11 The signal is noisy because, in addition to the informed consumers, there are some other consumers

whose purchasing decisions are made at random.
12 The inference is perfect for strictly positive prices. If the price is zero, the uninformed consumers
can only infer that the quality is negative.
9

the information sets that yield to the separating equilibrium in Bagwell and Riordan(1991) and here are dierent: in the former, the consumers know the cost structure of
the industry and ignore the share of informed agents; here, all the production costs arenormalized to zero and the consumers know the share of informed agents. Specially witha continuum of types, it seems more reasonable to assume that the consumers know thepast market share instead of the cost structure of the industry.

3.2 Equilibrium of the rst period
The discount factor of the monopolist is 1, so it maximizes the sum of current and futureprots.

The probability of the intermediate case is independent of the rst-period price.

Lemma 1.

Proof: in the Appendix.

Then, theexpected

13

intermediate case is irrelevant in the maximization problem. Denoting theIprot of the full-information case by Π2 and the prot of the no-information

Ucase by Π2 , the monopolist problem can be written as
maximize
p1

I
I
U
U
p 1 xE1 (p1 ) + P r (p1 )Π2 + P r (p1 )Π2

subject to

0 ≤ xE1 (p1 ) ≤ 1
0 ≤ P rI (p1 ) ≤ 1
0 ≤ P rU (p1 ) ≤ 1

where

1
2
xE
1 (p1 ) = (2µ − ρσ − 2p1 ) is the expectation of the rst-period market share21(2µ − ρσ 2 − 2 − 2p1 + 2Z) is the probability of the full-information case
P rI (p1 ) =4Z
1P rU (p1 ) =(−2µ + ρσ 2 + 2p1 + 2Z) is the probability of the no-information case4Zp
ΠI +2Zµ4
Let L1 = 2
ΠI2 − 4Z + 4Z 2 − 2(4Z−µ)and L2 = 2 2 . The equilibrium price is:
σ
σ2
Zσ

(2µ − ρσ 2 )/2
if ρ ≥ L2



2
I
U
(2Zµ
−
Zρσ
−
Π
+
Π
)/4Z
if L1 > 0and L1 < ρ < L2 ;
or
2
2
p∗1 =
(7)
if L1 < 0and ρ < L2

(2µ − ρσ 2 − 2)/2
if L1 > 0and ρ ≤ L1
This characterization would allow for an empirical test. Particularly, if dierent markets
are considered (say, dierent countries) in which the product is introduced simultaneously,
the price should decrease as the risk aversion of the consumers increases. Some measuresabout the risk aversion in dierent countries are available at the World Values Survey,

13 ΠI = P r(q < 2)0 + P r(q > 2)E[q − 1|q > 2] + P r(0 < q < 2)E[q 2 /4|0 < q < 2]2

10

but certain caveats should be kept in mind:

the most important one would refer tothe current possibility of the citizens of one country of buying the product in anothercountry through the Internet. In that case, it would be more correct to analyze rms who
work with experiences or services (for example, Smartbox) because these can only be
consumed at a certain location.
The previous characterization also gives rise to the next proposition:

The larger the degree of risk aversion, the smaller the probability ofobserving a decreasing pattern of prices.
Proposition 2.

Proof: in the Appendix.

Up to here, the model has been characterized for risk-averse consumers (ρ> 0).

However, the characterization for risk-neutral consumers is straightforward by making

ρ = 0. Then, I can state the following corollary:

It is less likely to observe a decreasing pattern of prices when the consumers
are risk averse than when they are risk neutral.
Corollary 1.

The intuition behind this result is simple: the risk-averse consumers need a compensation for the uncertainty faced in the rst period that the risk-neutral consumers do not,which is translated into a lower rst-period price when the consumers are risk-averse.
The patterns of prices and market shares predicted by the model have some similarities with those described in Shapiro (1983). In his model, Shapiro studies the optimal
price path of a monopolist that supplies an experience good over an innite number of
discrete periods.

There is a continuum of risk-neutral consumers of mass 1.

They areheterogeneous in the valuation of the quality q̄ , that is not idiosyncratic and can only
be learned after consumption. Although the monopolist knows his true quality, signaling
considerations are not included. Two cases are analyzed: pessimistic and optimistic.
In the pessimistic case, the consumers expect a quality lower than the true one: µ < q̄ .
The monopolist sets a low introductory price to spread the favorable information followed
by a higher price, that will be constant onwards. The market share is equal across all the
periods and less than the corresponding to the full information situation. In my model,
this outcome can be obtained even if µ > q̄ because of (i) the compensation required by
the consumers due to the risk of buying in the rst period, and (ii) the randomness ofthe rst-period market share for the monopolist. Consider for instance the following setµ = 3, σ = 1, ρ = 4, Z = 6, q̄ = 1.8. In this case, the full-information∗
market share is 0.9. The equilibrium price of the rst period is p1 = 0.426. Suppose
also that z1 = 0.126, so that x̂1 = 0.7. In this case, the monopolist sets a second-periodsprice p2 = 1.1 and supplies to x2 = 0.7; that is, with these parameters the model predictsof parameters:a price increase and a constant market share across the two periods. However, the real
quality is below the expected one: rst, the monopolist chose a low price to compensate
the risk; second, the market share remains constant because, given x̂1 , the true quality wassuciently high for the consumers to wish to repeat the purchase, once the uncertainty
was removed.

11

In the optimistic case, the consumers expect a quality higher than the true one: µ > q̄ .
The monopolist milks his reputation and decreases the price monotonically, reaching amarket share larger than would occur under full information. Subsequently, the price andmarket share go back to their full-information levels (the price increases and the market
share decreases) and remain constant onwards. The logic behind is that the monopolist
does not let the unfavorable information diuse too quickly: he can exploit the consumers
who still expect the quality to be larger than what actually is. In my model, when thetrue quality is lower than expected, it is possible to nd a similar pattern of market shares
jointly with a reversed pattern of prices. Consider now the following set of parameters:µ = 3, σ = 1, ρ = 4, Z = 6, q̄ = 0.5. As before, the equilibrium price of the rst period∗
is p1 = 0.426. If z1 = 0.126 again, then x̂1 = 0.7. In this case, the monopolist sets assecond-period price p2 = 0.25 and supplies to x2 = 0.25. Then, both the market shareand the price decrease in the second period with respect to their rst-period levels. The
reason is that in my framework the monopolist cannot really control how the information
about the quality diuses, because the rst-period market share depends on a component
that is unknown for him (and, therefore, considered as a random variable).

4

Welfare Analysis

In this section I discuss some options to improve the social welfare.
In view of the above results, it would be desirable to expand the consumer base in theintermediate case of the second period: qualities that would sell more units under perfect
information cannot reach new customers because of the signaling dynamics. However, to
set a price ceiling in the second period may not be a good solution: among others, thetypes q̄ < 0 will nd protable to deviate by charging the maximum price.

Then, theconsumers will have to adapt their beliefs to include this behavior, decreasing the inferred
quality from the price ceiling. Even more, the linearity of the certainty equivalent thatallows for a simple characterization derives from the normal distribution of the types: since
the separation of some positive and negative types will not be sustainable due to the priceceiling, the distribution of the pooling types will dramatically change with respect to theGaussian case, perhaps making the characterization impossible.
Nevertheless, there exists another possibility: a price ceiling in the rst period. Assuming that the realization of z1 is also unknown for the social planner, a limit on therst-period price can determine the probabilities of the full information event and of theno information event. The problem of the social planner is then to maximize the expected
total surplus.
The three graphs below show the rst-period price chosen by the monopolist to maximize his expected prots (dotted line) and the rst-period price chosen by the social
planner to maximize the expected total surplus, for three dierent values of the expected
2
quality µ with variance of σ = 1 and Z = 5. A more complete characterization of the
optimal policy of the social planner is provided in the Appendix.

12

Figure 2: Prices chosen by the monopolist and by the social planner; µ = 4.

Figure 3: Prices chosen by the monopolist and by the social planner; µ = 1.

Figure 4: Prices chosen by the monopolist and by the social planner; µ = 0.1.
There are two remarkable things. First, the price that maximizes the social welfare
does not strictly decrease as the risk-aversion parameter increases when µ is large enough,as can be noticed from the discontinuities in Figure 2 and Figure 3 (in Figure 4 there isno discontinuity because µ is small enough). The reason is that, for µ large enough andlow values of ρ, the option that maximizes the expected total welfare is to ensure the13diusion of the information (that is, to set a price such that the probability of the full
information event is 1). However, if the value of ρ is over a certain threshold, this option
becomes too costly and it is better for the society to allow for the no information eventto happen with a strictly positive probability. Second, the social planner weakly prefers
to diuse more information than the monopolist; that is, the social planner always sets alower or equal price than the one chosen by the monopolist. As can be observed in thethree previous gures, for values of ρ large enough the prices set by the social plannerand by the monopolist coincide: the best strategy is to set the expected market shareof the period 1 equal to 0, and to allow for both the full information event and the no
information event to happen with positive probability. This suggests that the authority
should only take part to improve the social welfare when the consumers are not very risk
averse.

5Conclusion

I have presented a model that describes the optimal strategy of a forward-looking monopolist in a game of two periods when a new product is introduced, facing risk-averse
consumers.

The dierence between the rst period and the second is the amount of

information spread in the market, that cannot be fully controlled by the monopolist.
In particular, the monopolist introduces a new, non-storable product. The strategic
variable of the monopolist is the price, and both intra-period discrimination and interperiod commitment are not allowed.
period.

There is a dierent mass of consumers in eachIn the rst period there is no information about the quality of the product,

although it is public knowledge that all the qualities are equally costly and that thequality is normally distributed with mean µ > 0 and standard deviation σ .

Thus, thepurchasing decisions of the rst-period consumers are based on the expected utility: theuncertainty is penalized whereas the expected quality is rewarded. Additionally, there isan element in the consumers' utility of the rst period that cannot be observed by therm, so I dierentiate among three potential outcomes: all consumers buy the product,
nobody buys the product, and only a share of consumers buy the product.
Consumers of the rst period who acquired the good learn the quality, and the monopolist learns the quality if some consumers do. The information is transmitted to thesecond-period consumers through a very specic channel: the consumer located at xi inthe rst period communicates her gathered information to the consumer located at xi
in the second period, so that the market is divided between informed and uninformed
consumers. The informed consumers do not penalize the uncertainty, and the uninformed
consumers use the rst-period market share and the second-period price to update theirbeliefs.
The model has two main predictions.

First, it is possible to observe an increasing

pattern of prices when relatively low qualities are revealed. The likelihood of observing adecreasing pattern of prices decreases as the degree of risk aversion increases: the reason
is that, the higher the degree of risk-aversion, the lower the rst-period price. Second,14

if only a share of consumers are informed, there exists a semi-separating equilibrium
in which the uninformed consumers can always infer with no error any strictly positive
quality. Interestingly, distortion is only necessary for qualities above a certain thresholdthat depends on the share of informed consumers. Even more, the distortion necessary
to sustain the semi-separating equilibrium is so intense that the uninformed consumers
never buy in equilibrium. It is important to remark the key signaling role played by therst-period market share: it is unrelated to the quality, but the uninformed consumers
know how many people are informed and they can understand the cost of charging a
deceiving price in terms of lost prots from the informed sector.

Since the consumersof the two periods are dierent and all qualities are equally costly, the prices cannot becredible signals by themselves if the rst-period market share is not observed.
Also, I have explored some potential policies to improve the total welfare. I argue thatsetting a price cap in the second period breaks the signaling properties of the second-period
equilibrium. Instead, the social planner can set a price cap in the rst period. I show thatthe social planner weakly prefers to have more informed consumers than the monopolist.
However, when the risk aversion parameter is large enough, both the social planner andthe monopolist would implement the same price.
Finally, I discuss the role of some assumptions. First, the monopolist considers thatthe variable z1 , that he cannot observe, is uniformly distributed. The advantage of this
distribution is that it allows to nd the closed-form solution of the equilibrium.

The

distribution may be changed, but the equilibrium should be calculated through numerical methods. Also, this distribution makes irrelevant for the intertemporal problem thesituation of the second period in which only some consumers are informed.

The mainproperties of the equilibrium would not change if a dierent distribution was assumed, aslong as the probability of the event in which only some consumers are informed was smallenough. Second, it is possible to relax the assumption of perfect learning by consideringthat the consumers receive a noisy signal about the quality. Thus, the uncertainty would
still be penalized in the second period, although not so intensely as in the rst period.
Even more, it is possible to nd a semi-separating equilibrium analogous to the one existing with perfect learning, but instead of the true quality, the uninformed consumers
would infer the signal received by the informed consumers. Third, it is important to2
remark that, when ρ ≥ 2µ/σ , the equilibrium with forward-looking consumers works as
explained in the model, with a dierent mass of consumers in each period. The economic
logic is as follows: rst, no consumer with positive utility in the rst period wishes topostpone her consumption, because her overall utility would be zero (if nobody buys, thesecond-period utility is zero because of the degree of risk aversion, and if she is more distant to the monopolist than the marginal consumer, she will be out of the second-period
consumption); second, a consumer with negative utility may want to buy if consumers
behind her in the line also buy. However, the most distant consumer does not want to buy
in the rst period if she has a negative utility, because the utility of the second period iszero for sure. Then, the marginal consumer buys only if she is indierent between buying
2or not in the rst period. However, more complicated dynamics arise if ρ < 2µ/σ , inwhich the consumers will balance the information acquisition and the future prices. These15

considerations are left for future research.

Acknowledgments
I am in debt with Antonio Cabrales and Makoto Hanazono for their careful supervi-asion. I also acknowledge Natalia Fabra, M Ángeles de Frutos, Daniel García, Ángel Hernando, Diego Moreno, Sayaka Nakamura, Emmanuel Petrakis, Zeynep Özkök and Alvaro
Domínguez for useful discussion and suggestions. Any remaining errors are solely mine.
An earlier version of this paper was part of my dissertation at Universidad Carlos III de
Madrid. Financial support from FPU scholarship is kindly appreciated.

Appendix
Proof of Proposition 1

Proof. Let me set the beliefs of the uninformed consumers, q̃ , when they observe a strictlyspositive price p2 and a rst-period market share x̂1 ∈ (0, 1):


q̃ =

2ps2
ps2 + x̂1s
if p2 < x̂1s
if p2 ≥ x̂1s
Since these beliefs assign a single inferred quality to a given information set {p2 , x̂1 }, thesvariance is zero. Then, the demand would be ω(ρ, p2 , x̂1 ) = q̃ − p2 . Moreover, since it isbounded below by x̂1 , this sector in fact demands no quantity of the good for the previous
beliefs. This implies that the monopolist is constrained to maximize its prots within theinformed sector.

It turns out that if the type, q̄ , is large enough compared to the sizeof the informed sector, q̄ ≥ 2x̂1 , the only price that maximizes the monopolist's protss
is p2 = q̄ − x̂1 : it corresponds to a corner solution in which the most distant informed
consumer obtains zero surplus. If q̄ < 2x̂1 , the only price that maximizes the monopolist'ssprots is p2 = q̄/2, that corresponds to an interior solution within the informed sector.
This is then a separating PBNE equilibrium: the beliefs are conrmed (q̃ = q̄ ) and notype has incentives to deviate.
Regarding the pooling part of the equilibrium, all the negative types pool at a zero
price. If the uninformed consumers observe such a price, they know that the true type
belongs to the interval (−∞, 0]. The demand from this sector would be zero independentlyof the beliefs (that are required not to contradict Bayes' rule) and of the uncertainty
penalization

14, and the demand from the informed consumers is also zero. Notice thatthe monopolist obtains no gains from deviation given the beliefs specied above for strictly
positive prices. Then, this constitutes a pooling PBNE equilibrium.
Proof of Lemma 1

14 This happens because, even in the most favorable case of an inferred quality of 0 with no penalty,

the demand is zero.

16

Proof. There will be a rst-period demand x̂1 ∈ (0, 1) if the random shock

z1 ∈

1
1
p1 − µ + ρσ 2 , 1 + p1 − µ + ρσ 222By the uniform distribution of z1 , the probability of this event is




1 2
1 2
F 1 + p1 − µ + ρσ − F p1 − µ + ρσ22
2
2 + 2p1 − 2µ + ρσ + 2Z 2p1 − 2µ + ρσ 2 + 2Z
−
4Z
4Z
1
2Z


=
=

Proof of Proposition 2

Proof. By simple algebra it is possible to check that p∗1 < pU2 ∀ρ ≥ 0. The intuition
behind this result is that, in the rst period, the monopolist has incentives to spread theinformation: if the uncertainty is removed, the willingness to pay of the consumers will
increase in expected terms and he could charge a higher price in the second period. Then,
to observe a decreasing pattern of prices, a rst-period market share x̂1 > 0 is necessary.
sI
Depending on the value of x̂1 , ṗ2 will refer to p2 or to p2 .
∗∗Since I am looking for decreasing patterns of prices, I focus on p1 > 0 (if p1 < 0, thepattern of prices will be increasing because in the second period the negative prices are
Consider a set of parameters µ, σ and ρ such that the∗corresponding rst-period equilibrium price is p1 > 0. Given x̂1 , I denote by q̇ the quality∗
for which ṗ2 = p1 holds: all the qualities below q̇ yield a decreasing pattern of prices.
∗
Since p1 decreases as ρ increases, so does q̇ .
weakly dominated strategies).

Characterization of the social planer's problem

The problem of the social planner is as follows:

p 1 xE1 (p1 ) +maximizep1
subject toZ xE1 (p1 ) 0E0 ≤ x1 (p1 ) ≤ 1
0 ≤ P rI (p1 ) ≤ 1

1µ − ρσ 2 − p1 − xi2dxi + P rI (p1 )T W2I + P rU (p1 )T W2U

0 ≤ P rU (p1 ) ≤ 1
where

Z 1

T W2I =ΠI2 + P r(q < 2)0 + P r(q > 2)E[q − (q − 1) − xi dxi |q > 2]0

Z q/2
q − (q/2) − xi dxi |0 < q < 2]

+ P r(0 < q < 2)E[
0

17

and


0



3(2µ − ρσ 2 )2 /32
T W2U =
(2µ − ρσ 2 − 1)/2

if ρ ≥ 2µ/σ

2if µ ≤ 2

andif µ > 2andif µ > 2andρ < 2µ/σ 2 ; or(−4 + 2µ)/σ 2 < ρ < 2µ/σ 2ρ ≤ (−4 + 2µ)/σ 2

Also, notice that the expected total welfare of the rst period only depends on p1 if
2 2xE
1 (p1 ) ∈ (0, 1). In that case, its simplied expression is [(2p1 − 2µ + ρσ ) /8] + [p1 (2µ −2Eρσ − 2p1 )/2]. If x1 (p1 ) = 0, the expected total welfare of the rst period is also 0. If
2
xE
1 (p1 ) = 1, the expected total welfare of the rst period is 2µ − ρσ − 1)/2.
The welfare-maximizing price will correspond to one out of the following ve options:
1. Option 1 (O1):

1;15

I
U
p1 such that 0 < xE
1 (p1 ) < 1, 0 < P r (p1 ) < 1 and 0 < P r (p1 ) <E
IU
2. Option 2 (O2): p1 such that x1 (p1 ) = 0, 0 < P r (p1 ) < 1 and 0 < P r (p1 ) < 1;IUE
3. Option 3 (O3): p1 such that x1 (p1 ) = 1, 0 < P r (p1 ) < 1 and 0 < P r (p1 ) < 1;UIE
4. Option 4 (O4): p1 such that P r (p1 ) = 1 (mathematically, x1 (p1 ) > 1 and P r (p1 ) <U
0, so the corner solution xE
1 (p1 ) = 1 and P r (p1 ) = 0 applies);
U
E
5. Option 5 (O5): p1 such that P r (p1 ) = 1 (mathematically, x1 (p1 ) <I
P rI (p1 ) < 0, so the corner solution xE
1 (p1 ) = 0 and P r (p1 ) = 0 applies).

0 and

By simple algebra, it is possible to show that the options O3 and O5 always generateless welfare than the other alternatives for any µ > 0 and any ρ > 0.

Also, it is easy

to show that the option O2 maximizes the welfare for any µ > 0 and large values of ρ.
Intuitively, the option O2 generates the largest welfare when ρ is large enough because
in the rst period it provides no welfare in expected terms (the other options provide
negative expected levels of welfare if ρ is large), and in the second period it provides
no welfare if the information cannot be disclosed and a positive expected welfare in the
full-information case, what happens with a strictly positive probability (whereas the restof the options provide the same levels of expected welfare in the second period, but theprobabilities of the two events dier). Then, it remains to determine the chosen options
for lower values of ρ.
The characterization method works as follows: rst, compare the welfare provided by
the options O1 and O4 when ρ = 0 and nd the value µ that make them equal. If we make2
the variance equal to σ = 1, this threshold is approximately µ̄ = 0.442: for lower valuesof µ the starting welfare-maximizing option is O1, and for higher values of µ the starting
welfare-maximizing option is O4. Second, if the welfare-maximizing option for low valuesof ρ is O4, it is necessary to determine the µ̃ that divides the paths of the options chosen
by the social planner in two: (O4, O1, O2) and (O4, O2). To do it, equalize the total
welfare obtained with O1 and O4 and keep the obtained ρ(µ). Then, plug this value into

15 Notice that the structure of the price chosen under the option O1 changes depending on the corre-

sponding value of ΠU2 .

18

E
xE
1 and evaluate it in the optimal price chosen under O1. The value of µ that makes x1
2
equal to 0 is the threshold µ̃. In the case of the variance σ = 1, µ̃ = 1.461, approximately.
Then, for µ ∈ (0, µ̄) the social planner chooses (O1, O2) and there is no jump in the pricewhen moving from O1 to O2; for µ ∈ [µ̄, µ̃] the social planner chooses (O4, O1 and O2)
with a jump in the price when changing from O4 to O1; and for µ > µ̃, the social plannerchooses (O4, O2) with a jump in the price when changing from O4 to O2.

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