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Brand Loyalty, Volume of Trade and Leapfrogging:Consumer Behavior in Markets of Durable Experience Goods�

Martin Paredes y

Department of Economics,Universidad of Piura (UdeP)

Abstract

We study a dynamic model of a market for durable experience goods. In equilibrium, we �ndthat, despite obsolescence, (1) there are consumers who keep their durable goods after a goodexperience because they are unwilling to face the risk of replacing the good with a new one ofuncertain quality; (2) there are consumers who trade their durable goods, after a bad experience,with one of a di¤erent brand but with the same or lower quality, because of the uncertaintythey face when buying an unknown brand. One implication of such behavior is incompletetrade in secondary markets, a result that has been interpreted (in light of the adverse selectionliterature) as an indication of ine¢ ciency. However, as long as experience involves idiosyncratictastes, such lack of trade is e¢ cient in this model. A second implication is leapfrogging, withthe simultaneous presence of high-valuation consumers owning a low-quality good, and low-valuation consumers owning a high-quality good. As an extension we also examine the e¤ectof durability on loyalty and �nd that it reduces the amount of (i) consumers who switch brandwith every purchase, and (ii) consumers who stay loyal to the �rst brand they bought.

JEL Classi�cation: D82, D83, L15.Keywords: Durable goods, experience goods, consumer behavior.

�Originally entitled "Uncertainty and the Role of Experience in Durable Goods Markets". I am grateful toAlessandro Lizzeri and Luis Cabral for helpful suggestions at earlier versions of this papers. However, all errorsremain my own. Please do not cite, as the paper is currently under revision for submission.

yE-mail: [email protected]. Current address: Departamento de Economía, Universidad de Piura, MártirJosé Olaya # 162, Mira�ores, Lima 18, Perú.

1

1 Introduction

Most durable goods are also experience goods. Consider for instance the automobile industry. Cars

are not only durable goods but also experience goods since a consumer cannot ascertain all the

attributes of the car before buying it, either because some of those attributes are unobservable1,

or because the consumer is unsure about the e¤ect of such attributes on her own utility �evenif they are observed2�. She can get some information about a particular car before deciding topurchase it, but she will learn more about that car after driving it for some time �in particular,whether or not such car suits her needs�. But this uncertainty is not exclusive of the car market.Other industries where experience also plays an important role include appliances (such as washing

machines and microwave ovens), electronics (such as computers, game consoles and TV sets) and

transportation (such as airplanes and motorcycles).

Consumer brand loyalty is an indicator that highlights the importance of experience, since

the consumer�s choice of brand in her next purchase will clearly depend on her past experience.

Concretely, consumers usually stay loyal to a brand after a good experience, but switch brands

after a bad experience. In the car industry, less than half of returning customers repeat purchase of

the same brand they bought last time3. Table 1 shows a substantial dispersion of customer loyalty

across brands for this market between 2003 and 2009.

1An example from the automobile industry is repair and maintenance services o¤ered by the car dealer: consumerscannot know for certain whether they will be satis�ed with them until they bring the car for service.

2For example, consumers may not know a priori whether the rate of acceleration is adequate for their driving style,or whether they will feel comfortable with a car with leather seats, until after trying the car out.

3The index of customer loyalty, as well as the two indicators on satisfaction shown below, are published online byJ.D. Power & Associates.

2

Observe how American brands such as Chevrolet and Ford �not distinguished by their highquality� exhibit high loyalty, while some Japanese brands like Nissan and Mazda, or European

ones like Volvo and Volkswagen, do not. However, loyalty does not depend on the origin of the

brand, since there are Japanese and European brands (Toyota, Mercedes Benz) with high loy-

alty and American ones (Pontiac, Jeep, Mercury) with low loyalty. We also notice �uctuations in

customer loyalty per brand, but the index does not converge towards 0% or 100%,

To ascertain whether experience has an e¤ect on customer loyalty, we compare it with two in-

dices that measure consumers�satisfaction with their new vehicles during the �rst three years after

purchase. The Customer Satisfaction Index (CSI) evaluates satisfaction with dealer maintenance

and repair services, while the Automotive Performance, Execution and Layout index (APEAL)

quanti�es owners�satisfaction with the design, content, layout and performance of their new vehi-

cles. Table 2 shows there is in fact a signi�cant positive brand correlation between loyalty and both

satisfaction indices. Simply put, the higher the satisfaction of a consumer with her car �whichwould re�ect whether she has had a good experience with her previous car�, the larger the prob-ability that the customer will repeat the purchase of the same brand4. Thus, empirical evidence

from the automobile industry suggests that cars are experience goods.

Based on these observations, we construct a dynamic model for a good that has both charac-

teristics of durability and experience, and examine how consumers behave in that market. In the

model, a good lasts two periods, and the utility it provides to consumers can be separated into

two components: quality and experience. Quality refers to the utility obtained from characteris-

tics that are observable before the purchase, which shows obsolescence, so that a used good has

lower quality than a new good. Experience, instead, denotes the utility from attributes that are

unobservable before purchase, but can be learned after owning the good for one period. Consumers

have heterogeneous preferences regarding their valuation for both quality and experience, and buy

at most one unit of the good5.

4 It is possible that observable characteristics also a¤ect these indices. However, we can expect that, if cars werecompletely identi�able before a purchase, consumers will remain loyal to the brand.

5Then, in equilibrium, consumers with higher valuation for quality purchase new goods, while those with lowervaluation of quality buy used goods.

3

Given that the evidence from the car market shows �uctuations in customer loyalty, but no

convergence, we assume that experience is idiosyncratic and non-permanent. The dynamics of

both assumptions are better explained by means of an example. Suppose a consumer bought a new

Ford last period. Today, she owns a used Ford, and has already learned her match with that car.

Suppose she enjoyed a good experience. Since experience is idiosyncratic, however, there will be

some other consumers who bought a Ford last period and got a bad experience. Now, this Ford

owner has to decide whether she keeps her used car or sells it. If she chooses to sell, she also has to

decided whether she stays loyal to Ford or switches brands. Suppose the consumer decides to trade

the used good and buy a new Ford. But it is possible that the consumer gets a bad experience

with the new car because experience is not permanent: the fact that a consumer got a good match

today does not mean that she will get good matches in her next purchases of that brand.

In equilibrium we �nd that there is incomplete trade in secondary markets: there are always

some consumers who decide to keep a used good, despite its obsolescence, because they got a

good experience and do not want to face the risk of replacing the good with a new one of uncertain

quality. Further, incomplete trade is an e¢ cient outcome in my model, as long as experience involves

idiosyncratic tastes: the fact that a consumer got a good experience does not mean that the next

owner will get the same experience. This is in contrast to the literature on adverse selection, which

states that such lack of trade is ine¢ cient:

In the model �rms supply a constant output of new goods each period, and there is a secondhand

market where consumers can trade their used goods. But we can also interpret the model such that

�rms introduce an upgraded good every period, which provides higher quality than the previous

generation of the good. This alternative interpretation with technological obsolescence is the most

adequate to understand our second result: We �nd that some consumers who had a bad experience

with a higher-quality good will decide to switch brands but purchase a good of lower quality

because of the uncertainty they face when buying an unknown brand. Such trying behavior causes

leapfrogging within this subset, with some consumers with higher valuation of quality preferring to

buy a low-quality good while others with lower valuation of quality preferring a high-quality good.

In the model consumers always remains loyal to a brand after a good match, and switches brands

after a bad match. Hence brand loyalty is determined by the probabilities of getting a good or bad

experience, not by durability. But in the real world we observe consumers who switch brands even

after a good match, while others remain loyal to a brand even after a bad match. As an extension

we construct a model where those two behaviors are possible, and �nd that durability does have

an e¤ect on brand loyalty, by reducing (i) the fraction of consumers that switch brands with every

purchase, and (ii) the fraction of consumers who stay loyal to the �rst brand they bought.

There is a large literature that has studied durable goods markets. Recent developments have

4

focused on the �rm�s obsolescence decision, either through the choice of durability (Waldman, 1996;

Hendel and Lizzeri, 1999a) or the introduction of new products (Fudenberg and Tirole, 1998; Lee

and Lee 1998). Another important contribution has been the analysis of adverse selection in a

model of durable goods (Hendel and Lizzeri, 1999b, Johnson and Waldman, 2003). A survey by

Waldman (2003) provides a synthesis of the evolution of the microeconomics of durable goods.

But this literature has mostly ignored the role of experience and uncertainty regarding the

quality of a new good. When experience is introduced in a model of durable goods, it can have

a direct e¤ect on (i) the frequency consumers decide to replace their good, (ii) the type of good

they buy � new or used� , and (iii) the brand they choose. Through those channels experience

also has an e¤ect over prices. Herein lies the contribution of this paper. Although we focus only

on the consumer side, experience should also a¤ect decisions on the supply side, especially about

obsolescence and market coverage.

There is also a large literature on experience goods. Initially, it centered on the �rms�strategies

to overcome the asymmetric information problem, either by signaling high quality or by building

a reputation (Klein and Le er, 1981; Shapiro, 1982; Riordan, 1986; Milgrom and Roberts, 1986).

More recently, there has been a renewed interest on the consumer experimentation problem and

its relationship with oligopolistic competition (e.g. Bergemann and Valimaki, 1996, 1997, 2002).6

Other authors have shown the empirical relevance of experience in markets such as the anti-ulcer

drug market (Crawford and Shum, 2005), yogurts (Ackerberg, 2004), and laundry detergents (Er-

dem and Keane, 1996).

But durability introduces some dynamics in a model of experience goods that are absent in non-

durable goods markets, and we explore them here. For example, once purchased, a non-durable

goods has no value. Instead, consumers can continue to use a durable good for several periods.

Depending on the experience they got, consumers have to decide how long they will keep the good

they currently own. Further, if there is a secondhand market, they can sell the used product to

another consumer.

The paper is organized as follows. Section 2 introduces the model and presents some preliminary

results. Section 3 solves the model, while Section 4 discusses the main results of the paper. Section

5 extends the model to consider di¤erences in consumers�preferences for loyalty and switching.

Section 6 concludes.

6 In contrast, here I get away from such problem by limiting consumers�memory, similar to Villas-Boas (2001),although there he studied a non-durable good market with overlapping generations of consumers. Our papers alsodi¤er in the characterization of heterogeneity across consumers.

5

2 The Model

2.1 Setup

We consider an in�nite-horizon economy in discrete time, with a unit mass of consumers who

demand at most one unit of a durable good at any date. Consumers are born at the beginning of

time and live forever, and no new consumers enter the economy at any other time.

We assume that the durable good lasts two periods and is produced by M �rms, with each �rm

supplying one brand. For every brand i = 1; :::M; the (gross) bene�t provided by each durable

good can be divided into two parts. We de�ne qi to represent the (gross) bene�t provided by the set

of characteristics that consumers can inspect before purchase, which will be referred as quality. In

turn, zi denotes the (gross) utility obtained from attributes that can be observed only after buying

and trying the product, which we call experience.

We assume that quality is deterministic and a¤ected by physical obsolescence, so that a new

good has quality qhi , while a used good has lower quality qli. We denote by �qi � qhi � qli the

depreciation of the good�s quality. We assume that experience represents the value of a consumer�s

match with a vintage of brand i; which can take two values: for a given vintage of brand i; some

consumers will get a good match zhi , while some others will get a bad match zli: Hence, experience

is idiosyncratic. We denote by �zi � zhi � zli the di¤erence in experience outcomes.A consumer who has never purchased a good of brand i will get a good experience with prob-

ability �0i . Since experience is associated with a particular vintage of the good, a consumer�s

match with brand i can change over time. Concretely, we assume that the probability of getting

a good experience depends on the experience with the consumer�s last purchase of that brand,

denoted by z0i: We thus de�ne �hi � P

�zi = z

0i = zhi

�and �li � P

�zi = z

0i = zli

�, and we assume

that 0 < �li < �0i < �hi < 1: Hence, experience is non-permanent7 and has the Markov property8.

Consumers�memory about past experiences lasts for T periods. We assume T < M; so that at any

date there are always some brands that the consumer has not experienced before.

Consumers�preferences are characterized by the pair (�; �) : Both � and � are independent of

each other, �xed throughout the consumer�s life, and known only by the consumer. � represents the

consumer�s valuation of quality, and is distributed according to the function F :��; ��! [0; 1]. We

assume F (�) is increasing and has a continuous density f (�) : In turn, � represents the consumer�svaluation of experience and has distribution function G :

��; �

��! [0; 1], which is also increasing

and has a continuous density g (�) :7However, if a consumer bought a new good last period and got experience z0j ; then she would enjoy that same

experience-related utility if she decides to keep the used good.8The Markov property is assumed for simplicity. It would be enough to assume that the last experience has a

weight larger than 50%.

6

Given a sequence of goods with total bene�tnfqit + zitgMi=1

o1t=s

and pricesnfpitgMi=1

o1t=s

at

time s for each brand i, the expected present value of the utility for a consumer with pair (�; �) is:

Es

" 1Xt=s

�t�sMPi=1ICit��qit + �E (zit=Hs+t)� IBit pit

�#

subject to the constraintPNi=1 I

Cit � 1: The indicator function ICit represents the consumer�s decision

at period t; with ICit = 1 if she consumes brand i; otherwise ICit = 0: The constraint allows for the

possibility that the consumer chooses not to consume any good. In turn, IBit is an indicator function

of whether or not the consumer has bought brand i in period t, while Hs+t denotes the consumer�s

history at time s+ t: The price can be either a that of new good, pNit; or a used good, pUit :

There is a constant �ow of yi new units of each brand coming into the market at every date,

so the number of units produced by the industry is Y = �iyi. Since the mass of consumers who

owns a durable good of brand i is at most 2yi at any date, we assume that 2Y < 1 so that there

are always some consumers who do not own a unit of the durable good. Hence, we analyze how a

given output �ow is allocated in equilibrium under conditions of uncertainty and experience, and

ignore how market structure and cost conditions may lead to this exogenous output.

Our objective is to characterize how the equilibrium is driven by consumers�expectations about

experience. As such, given the stationarity of the environment, we focus on symmetric equilibrium

outcomes at the steady state, where qit = q and zit = z for all i and t: Also, �ki = �k for k = 0; h; l

and all i. Further, we assume z =�zl; zh

; and q =

�ql; qh

: For convenience we also de�ne

pR � pN � �pU to be the good�s rental price, while �p � pR � pU :

2.2 Discussion and Benchmarks

Consumer behavior involves a sequence of decisions regarding what kind of good the consumer will

own every period, conditional on (i) the product she may own at the beginning of that period, and

(ii) her purchasing history, which includes the last experience with all brands owned in the last T

periods. If she buys the same brand she owned the previous period, we say that she stays loyal to

that brand; otherwise she switches brands.

We assume that experience is non-permanent and has the Markov property because producers of

durable goods introduce upgrades and modi�cations very frequently. As a result, (i) a consumer�s

match with a particular brand can change over time, (ii) the experience obtained with the last

purchase should provide better information for the next purchase than any other previous purchases

of that brand, but (iii) such last experience gives only an approximation of what to expect in their

next purchase. This explains why we assume 0 < �l < �0 < �h < 1:

7

The assumption of idiosyncratic experience means that the consumer�s private information

about a given vintage of a brand is useless to any other consumer. The consumer can only exploit

such information for her own purchasing decisions. With time, any experience becomes irrelevant,

which is why it is assumed that consumers have limited memory (i.e., T < M)9.

Although we assume that the quality of the durable good su¤ers from physical obsolescence, the

value loss can also be interpreted as technological obsolescence, where �rms produce an improved

good every period that render the previous generation obsolete. In that context �q denotes the

di¤erence between the upgraded and the obsolete good, in terms of observable quality From the

viewpoint of consumer behavior, both forms of obsolescence have the same e¤ects10. With some

minor adjustments, we can also extend the model to consider the absence of secondhand market.

At any point in time, a consumer can be an owner or a buyer, depending on whether or not she

purchased a new good last period. The consumer is a buyer if she owns no good at the beginning

of the period. She can decide to purchase one unit of the durable good, or nothing. If she decides

to buy the durable good, there are two levels of decision: (i) whether to buy a new or a used good,

and (ii) which brand to buy. If K denotes the set of brands, then J � K denotes the subset of

brands that any given buyer has owned in the last T periods. Then, from the viewpoint of such

buyer, any brand i =2 J is an unknown brand. Given her pair (�; �), her expected value function isde�ned by:

EV BUY (�; �) = maxi2K

�EV N (�; �) ; EV U (�; �) ; 0

(1)

where EV N (�; �) and EV U (�; �) denote respectively the expected value of purchasing a new good

and a used good. Both are de�ned as follows:

EV N (�; �) = maxi2K

� ��qhi + �E (zi=z

0i)� pNi + �EV OWN (�; �)

i2J ;�

�qhi + �E (zi)� pNi + �EV OWN (�; �)i=2J

�(2)

EV U (�; �) = maxi2K

� ��qli + �E (zi=z

0i)� pUi + �EV BUY (�; �)

i2J ;�

�qli + �E (zi)� pUi + �EV BUY (�; �)i=2J

�: (3)

where pN denotes the price of a new good, and pU the price of a used good, while z0i represents the

last experience she had with brand i, and E (zi=z0i) is the expected match with brand i; given her

last experience z0i. When a consumer decides to buy a new good, as shown in (2), she also has to

9 In other words, experience depreciates after T periods. The results do not change qualitatively if we assumeinstead that experience depreciates every period (e.g. at the same rate than quality).10There is a di¤erence between physical and technological obsolescence from the viewpoint of �rms. With physical

obsolescence, �rms invest to reduce the quality of the used good next period before they sell it today. Depreciationcannot be changed once the good is sold. Under technological obsolescence, they invest to increase today the qualityof the durable good they will sell next period. There is an issue of commitment in investment, which has beenanalyzed by Waldman (1996) and Nahm (2004).

8

decide whether she buys a brand she has already tried in the past, or an unknown brand. In the

�rst case, the consumer expects to get experience E (zi=z0i), while in the second case she expects

E (zi) : In any case, the consumer will be the owner of a used good next period, and her expected

value would be EV OWN (�; �) : If the consumer chooses to purchase a used good, she faces a similar

decision regarding which brand to buy, and the only di¤erence is that she will own no good at the

beginning of next period.

Consider now a consumer who owns a used good of brand j. As an owner, she has to decide

whether or not she keeps that good. If she chooses to keep it, she will enjoy the used good today,

and become a buyer next period. If she chooses not to keep it, she will sell it and become a buyer

today. If z0j denotes the experience she got with her used good, then the expected value function

of an owner is thus:

EV OWN (�; �) = maxi2K

n�qlj + �z

0j + �EV

BUY (�; �) ; pUj + EVBUY (�; �)

o(4)

In this model, experience causes ex-post di¤erentiation within the subset of consumers who have

bought brand i, and can thus explain why some consumers choose to stay loyal while others switch

brands. As a benchmark, suppose that there is no uncertainty, so that consumers are completely

informed about all attributes of the good before purchase. A consumer with pair (�; �) thus gets

a per-period utility �qi + �zi from the durable good, where qi =�qhi ; q

li

: Let pUi;d and p

Ni;d denote

respectively the price of a used good and a new good of brand i in this deterministic benchmark.

The utility of a buyer with pair (�; �) reduces to:

V BUY (�; �) = maxi2K

n�qhi + �zi � pNi;d + �V OWN (�; �) ; �qli + �zi � pUi;d + �V BUY (�; �) ; 0

o(5)

while the utility of an owner with pair (�; �) becomes:

V OWN (�; �) = maxi2K

n�qlj + �zj + �V

BUY (�; �) ; pUj;d + VBUY (�; �)

o(6)

In the symmetric stationary case we must have that pUi;d = pUd and pNi;d = pNd ; otherwise con-

sumers will prefer to buy the cheapest brand. Further, consumers will have no incentives to switch

brands: since all brands are identical, they would prefer to continue buying the brand they have

purchased before. Also, the opportunity cost of keeping a used good is the same as buying a used

good. A consumer can thus pursue one of three possible purchasing behaviors: (i) buy a new good

every period, (ii) buy a used good every period, or (iii) buy a new good every two periods and

keeping the good when used. For each of these cases, the discounted utility is respectively:

UNd (�; �) =�qh + �z � pRd

1� � ; UUd (�; �) =�ql + �z � pUd

1� � ; and

UKd (�; �) =�qh + �z � pNd + �

��ql + �z

�1� �2

:

9

In equilibrium only two of those behaviors are optimal: a mass Y of consumers with the highest

valuations of quality purchase new goods every period, while a mass Y of consumers with lower

valuations buy used goods. Hence, all used goods are traded in the secondhand market: if a

consumer bought a new good last period, then today she will sell the used good and buy a new

good again. Any consumer with quality valuation �N is indi¤erent between buying a new or a

used good, regardless of her valuation of experience �, where �N is de�ned by 1 � F��N�= Y .

The importance given to experience is only relevant for the consumer�s whether or not to buy

a used good: Concretely, a consumer with pair (�; �) is indi¤erent between buying a used good

or not buying at all when �ql + �z = pUd : In equilibrium, the price of a used good11 is de�ned

by 1 �Z �

�F��Ud (�)

�� dG (�) = 2Y; where �Ud (�) �

pUd � �zql

; while the price of a new good is

determined by pNd = �N�q + (1 + �) pUd :

As a second benchmark, suppose now that experience takes two values, but is permanent. This

means that a consumer who gets a match zki with brand i (k = h; l) will always get such match in

future purchases of that brand. If in addition experience were idiosyncratic, then a consumer stays

forever loyal to the �rst brand that gives her a good experience, and switches brands until she gets

it. If the match probabilities are the same for all brands, then the steady-state equilibrium would

be similar to the previous benchmark: there is again full trade of used goods, with consumers with

� � �N who buy new goods every period, while those with � 2��U2 (�) ; �n

�purchase used goods,

where �U2 (�) �pU2 � �zh

ql. Also, pN2 = �N�q + (1 + �) pU2 : If, for comparison purposes, we assume

zi = E (zi) ; then in this benchmark prices of both goods are higher12:

3 Durability and Experience

We assume now that experience is idiosyncratic but non-permanent: a consumer who buys a good

of brand i at the beginning of the period learns her match only at the end of that period, but she

may not get the same experience in her next purchase of that brand. All proofs of the propositions

in this section can be found in Appendix A.

11 In the special case where z = 0; then we obtain the same results as in Hendel and Lizzeri (1999b)�s �rst benchmark,with �u given by F (�u) = 1� 2Y:12To see this, notice that

�hZ�l

F��Ud (�)

�� dG (�) =

�hZ�l

F��U2 (�)

�� dG (�) :

Since both �Ud (�) and �U2 (�) are linear in �, then there exists a �

0 2��l; �h

�such that �U1 (�

0) = �U2 (�0) : Then

pUd < pU2 ; and p

Nd < p

N2 :

10

In equilibrium we need that, in both markets of new and used goods, and for all brands,

(a) supply equals demand, and (b) consumers� expectations are correct. The last condition is

automatically satis�ed because of the assumption that consumers have rational expectations (i.e.,

E��k�= �k for k = 0; h; l). But the demand for each brand and type of good may include (i)

buyers who have never tried that brand, (ii) buyers who have already tried that brand and got a

good experience last time, and (iii) buyers who have already tried that brand but got a bad match.

In turn, the supply of used goods depends on which owners decide to sell its used good. Hence, we

need �rst to analyze consumers�optimal behavior to identify who wants to buy a new or a used

good, and of which brand.

3.1 Consumer Behavior

All consumers take their own characteristics (�; �) ; match probabilities��0; �h; �l

�and prices�

pN ; pU�as given. We characterize their optimal behavior from the viewpoint of buyers, i.e., when

they do not own a used good at the beginning of the period. In the benchmarks we found that

there were at least two types of buyers in equilibrium: those who buy a new good every period, and

those who buy a used good every period. When experience is idiosyncratic and non-permanent,

however, it may be optimal for some consumers (i) to keep the good for two periods, and (ii) to

choose the quality of the good (new or used) based on their previous experience.

In this subsection we proceed as follows. First, we examine the optimal choice for buyers of

used goods. Then, we analyze the behavior of buyers of new goods. Finally, we explore utility

maximization for buyers whose choice of quality (new or used) depends on their past experience.

In the next subsection we determine who buys used goods, who buys new goods, and who chooses

to buy a new or a used good based on her past experience.

3.1.1 Buyers of Used Goods

Consider �rst the set of consumers who decide to purchase a used good every period. Their expected

utility in (1) reduces to EV BUY (�; �) = max�EV U (�; �) ; 0

: Thus, their behavior is similar to

that of consumers of non-durable goods. The following proposition describes the optimal behavior

for this type of consumers.

Proposition 1 Optimal behavior for consumers who buy used goods every period depends on theirexperience with the good owned last period: they stay loyal after a good match, but switch to an

unknown brand after a bad match:

11

One important implication of this result is that the only relevant information for a consumer�s

purchasing decision is z0; the experience she got with the good owned last period. Her memory may

recall good matches with some other brands, but it must be the case that her last purchase of those

brands was a bad experience, otherwise she would have continued to buy that brand. Regarding

the brand of her last purchase, her memory must include only good experiences. When z0 = zh; it

is optimal for the consumer to stay loyal to that brand, but if z0 = zl; she will switch brands. As

shown in the proof, the expected present value of the buyer�s utility, denoted by UUU (�; �=z0) ; is

simply a weighted average of staying loyal to a brand and switching to an unknown brand, with a

larger weight for staying loyal when z0 = zh:

We also �nd that, when purchasing a used good, the consumer�s valuation of experience does

not have an e¤ect on her decision of which brand to buy. However, � does a¤ect the decision of

whether or not to buy a used good. Since there are not enough goods for all the population, there

will always be a mass of consumers who never buy a unit of the durable good. Consider a consumer

who has never purchased a good. Her expected utility for buying a used good for the �rst time is

equivalent to that of a consumer who has gotten a bad match with every purchase in the last T

periods. Following Proposition 1, such unexperienced consumer would purchase a used good of an

unknown brand, but would decide to buy nothing if UUU��; �=z0 = zl

�< 0; that is, when:

pU > �ql + �bz; (7)

where bz � ��0

1� � (�h � �0)E�z=zh

�+

1� ��h1� � (�h � �0)E (z) (8)

is a weighted average of experience between staying loyal and switching brands. Given that con-

sumers�utility increases with both � and � then, as expected, consumers with the lowest pair (�; �)

must be the ones who stay out of the secondhand market in equilibrium. For a given valuation

of experience �; we de�ne �UU (�) to be the marginal consumer for whom UUU��UU (�) ; �=z

l�= 0;

i.e. who is indi¤erent between buying a used good or not buying at all. �UU (�) is a decreasing,

continuous function de�ned by:

�UU (�) �pU � �bz

ql: (9)

It is also worth noticing that the results obtained here would also apply to a model of non-

durable goods where it is also assumed that experience is idiosyncratic and non-permanent. In

particular, if we assume that �rms produce two qualities of the non-durable good, qh and ql; and

that the quantity produced for each quality is the same and equal to y; then in the symmetric

stationary equilibrium we should have that consumers with � � �Y buy the high-quality good,

and those with � 2��UU (�) ; �Y

�purchase the low-quality good. Any consumer who has a good

experience with a brand stays loyal to it. Any consumer who has a bad experience switches brands..

12

3.1.2 Buyers of New Goods

In this subsection and the next we examine the combined e¤ects of experience and durability on

the consumer�s purchasing decision. Consider �rst the case of a consumer who, as a buyer, always

purchases a new good. As an owner, she has to decide when to keep her used good, taking into

account (i) the experience she got with that good and (ii) her expected experience with her next

purchase. Such comparison depends also on her type (�; �) : Hence, her behavior is more complex

than that of a buyer of a used good. The next result characterizes her behavior:

Proposition 2 For consumers who only buy new goods, optimal behavior is characterized by a

continuous, strictly increasing function �KN (�) that satis�es:

�KN (�) =�p

�q+

�1� �h

�(1� � (�h � �0))�

�z

�q(10)

such that a consumer with valuation of experience � keeps her used good only (i) after a good match

and (ii) when her valuation of quality is smaller than �KN (�) : Otherwise, she sells her used good

regardless of her type (�; �) ; and she switches brands if she got a bad match, but stays loyal if she

got a good match.

Thus, �KN (�) de�nes a cuto¤ rule for buyers of new goods regarding their behavior when they

own a used good that is a good match. If instead such good was a bad match, they will always

choose to trade it and switch to a brand they have not previously purchased. Since the cuto¤ is an

increasing function of �; then for an owner of a used good that had a good match, the larger the

importance given to experience, the more likely she is going to keep the used good.

This behavior means that, when goods are durable, not only the consumer�s previous experience

has an e¤ect on her purchasing decision, but also the consumer�s importance given to experience.

Even though the consumer already knows the realization of her used good, when experience is

idiosyncratic and non-permanent she still faces uncertainty regarding her match with a new good

of the same brand, which may induce her to keep a used good that was a good match.

3.1.3 Buyers who choose Quality based on Experience

The previous subsections focused on the e¤ect that the consumer�s previous experience can have

on (i) her decision as a buyer on whether to stay loyal or switch brands, and (ii) her decision as an

owner on whether to keep or trade a used good. However, experience can also a¤ect her decision

as a buyer on whether to purchase a new or a used good. Given that the consumer�s utility is

increasing in both quality an experience, intuition suggests that a purchasing behavior based on

13

experience would exist with consumers deciding to buy a used good after a bad match, and to buy

a new good after a good match. The next result discusses the feasibility of such behavior:

Proposition 3 De�ne the following two continuous, strictly decreasing functions:

�NN (�) =�p

�q�

��0�1� �h

�1� �2 (1� �0) (�h � �0)

��z

�q(11)

�NU (�) =�p

�q�

��h�1� �h

�1� � (�h � �0)�

�z

�q(12)

For any given valuation of experience �; the optimal behavior for a buyer whose valuation of quality

satis�es � 2��NU (�) ; �

NN (�)

�is to purchase a used good after a bad match, and a new good after a

good match. As an owner, she keeps her used good if she got a good match, but trades it and buys

a used good of an unknown brand if she got a bad match.

Hence, there exists a set of buyers who choose the quality of the good (new or used) based on

their last experience. Such consumers are very reluctant to try a new brand when their previous

experience has been bad, so they choose to buy a good of lower quality. After a good experience,

however, the uncertainty they face about that brand is reduced, and choose to buy a good of higher

quality, but still prefer to keep it when used rather than trade it.

For any �; the function in (11) de�nes a cuto¤ rule for buyers regarding what to do after a bad

match: those with � � �NN (�) decide to buy a new good, while those with � < �NN (�) choose a

used good. Given that �NN (�) decreases with �; then buyers with higher valuation of experience are

more likely to always purchase new goods, regardless of their last experience. In turn, (12) is cuto¤

rule after a good match: those with � � �NU (�) prefer a new good, while those with � < �NN (�) go

for a used one. Since �NU (�) also decreases with �; then buyers with lower valuation of experience

are more likely to always purchase used goods, regardless of their last experience.

3.2 Equilibrium

In the past subsection we have characterized the optimal behavior of an individual consumer with

pair (�; �) for given prices and match probabilities. Now we need to aggregate such consumer

behavior to determine demand and supply for both markets of used and new goods. To determine

the demand for both types of goods (new and used), de�ne �Ni;t (�; �) to be the proportion of

consumers of type (�; �) who would like to buy a new good of brand i in period t, while �Ui;t (�; �)

is the proportion of those consumers who are present in the market of used goods at period t as

14

buyers of brand i. In a stationary symmetric equilibrium we have for each brand that:

DN =

Z �

�

Z �

��N (�; �) � dF (�)

!� dG (�) :

DU =

Z �

�

Z �

��U (�; �) � dF (�)

!� dG (�) :

A consumer who is not a buyer in period t must be an owner13, i.e., a consumer who bought

a new good last period and decided to keep the used good. Let �Ki;t (�; �) denote the proportion

of consumers of type (�; �) who decided to keep their used good in period t. For any type (�; �)

we requirePi

��Ni;t (�; �) + �

Ki;t (�; �) + �

Ui;t (�; �)

�= 1. In a stationary symmetric equilibrium, the

supply of used goods can be written as:

SU =

Z �

�

Z �

�

��N (�; �)� �K (�; �)

�� dF (�)

!� dG (�) :

Finally, recall that the supply of new goods was assumed to be constant and given by SN = y:

We have identi�ed at most four possible purchasing behaviors that are optimal: one for buyers

of used goods, one for buyers whose purchasing decision depends on their last experience, and two

for buyers of new goods. For the last group, the two optimal strategies di¤er on what the consumer

should do when she owns a used good that was a good match. All of those behaviors depend on

the consumer�s vector of characteristics (�; �) : For a given valuation of experience �; then �UU (�) is

the marginal consumer who is indi¤erent between buying a used good or not buying at all. Then,

consumers with � < �UU (�) never buy a good, while those with � 2��UU (�) ; �

NU (�)

�would prefer

to purchase used goods every period, and those with � 2��UU (�) ; �

NU (�)

�would choose the type of

good (new or used) depending on their last experience. Also, all buyers with � 2��NN (�) ; �

�prefer

new goods, but only those with � 2��NN (�) ; �

KN (�)

�would choose to keep their used good after a

good match. In any case, consumers switch brands if z0 = zl but stay loyal if z0 = zh: In Figure 1

we show the behavior for any consumer with type (�; �) for given prices and match probabilities,

and �14.

13 I exlcude from this analysis those consumers who stay out of the market because of the scarcity of goods, whichis given by those consumers with type � < �UU (�) :14However, it is possible that, for non-equilibrium values of pN and pU ; no types in the support of the distributions

satisfy the conditions outlined in the last Proposition: For example, if the price of a new good is lower than that ofa used good, then no one will buy used goods. In those cases some of the cuto¤s will be at one of the boundaries.

15

A symmetric equilibrium15 is then a pair of prices pN ; pU ; such that, for all brands in the market,

DN = y and DU = SU :

Proposition 4 There exists a stationary symmetric equilibrium.

To prove this proposition we need �rst to characterized �N (�; �), �K (�; �) and �U (�; �) for

each possible purchasing behavior. After that, we construct a map such that existence of a �xed

point of this map implies the existence of an equilibrium, and then we show that this map has a

�xed point.

4 Discussion

In this section we examine some implications of the stationary symmetric equilibrium we have just

characterized.15 In equilibrium we also need that consumers�expectations about the experience they would get when purchasing

a unit of the durable good are correct but, as we mentioned before, the assumption of rational expectations meansthat this condition is automatically satis�ed.

16

4.1 Volume of Trade and Welfare

One of the most important aspects analyzed in models of durable goods is the volume of trade in

the secondary market. Since SU denotes the amount of consumers who buy new goods and choose

to sell their used good regardless of the experience they got, then we can de�ne the volume of trade

as a percentage given by V oT =SU

y: The following proposition discusses its behavior.

Proposition 5 In a stationary symmetric equilibrium with durable experience goods, the volume oftrade in the secondhand market is always positive but incomplete: the market does not shut down,

but the volume is always less than 100%.

In other words, there is always a positive mass of buyers of new goods who decide not to

trade their used goods. The intuition for the absence of full trade is as follows. To have full

trade in equilibrium, any buyer of a new good must be willing to sell her used good, even after a

good experience. From (7), any used good is priced as if it provides an experience above average.

However, experience is idiosyncratic and not permanent, so for consumers who got a good experience

with their used good, the market price of a used good is not high enough to compensate them from

the risk of buying a new good of uncertain quality.

The intuition for the second part of the result is much simpler. To have no trade in equilibrium,

any buyer of a new good must prefer to keep her used good, even after a bad experience, but we

have shown that in such case consumers will always prefer to switch brands, so there will always

be some trade.

The result stated by Proposition 5 has some important implications. The most important

one is that the absence of full trade is an e¢ cient result. This is in contrast to the adverse

selection literature (e.g. Hendel and Lizzeri, 1999b, or Johnson and Waldman, 2003), where such

lack of full trade has been interpreted as an indication of ine¢ ciency. In both models, social

welfare is maximized when consumers with higher valuation consume higher quality. But as long

as experience involves idiosyncratic tastes for buyers of both new and used goods, there is no

asymmetric information in models of experience goods:

(1) In models with adverse selection, buyers in the secondhand market do not know the quality

of the used goods, but sellers do. As a result, any used good is priced as providing the same

average quality. Hence, the asymmetric information has two e¤ects: (i) it causes an ine¢ cient

allocation among the many types of buyers of used goods, and (ii) it forces owners of a used

good of high quality to keep such good because the price is based on the average quality of a

used good.

17

(2) In the model with durable experience goods, there are no informational gains that can be

earned by prospective buyers in the secondhand market because experience is idiosyncratic:

the fact that the �rst owner obtained a good match does not mean that the next owner will

get the exact same match. As a result, (i) the allocation among buyers of used goods is

e¢ cient because they all get the same observable quality, while the expectation of a good or

bad match depends on their previous experience, and (ii) some owners prefer to keep their

used good after a good match because they give great importance to experience and they face

uncertainty when trading it for a new good who may give them a bad match.

Hence, in this model a tax on keeping a used good, or a subsidy to trade it, will actually reduce

welfare, since the objective of both policies is to achieve full trade.

Another consequence from the fact that there are always some consumers who keep a used good

is that the mass of consumers that actually gets to buy a new good is larger than when there is

full trade.

4.2 Leapfrogging and Technological Obsolescence

Although we assumed that the quality of the durable good su¤ers from physical obsolescence, this

loss in value can also be interpreted as technological obsolescence, where �rms produce an improved

good every period that renders the previous generation obsolete. In that context, qh denotes the

quality of any new generation, which reduces to ql after the introduction of the next generation.

Then �q de�nes the di¤erence between the upgraded and the obsolete good From the viewpoint of

consumers, both forms of obsolescence have the same e¤ects16. This re-interpretation of the model

parameters from the viewpoint of technological obsolescence can help us understand the following

result.

Proposition 6 In a stationary symmetric equilibrium with durable experience goods there is alwaysleapfrogging.

In their paper on technological obsolescence17, Fudenberg and Tirole (1998) de�ned "leapfrog-

ging" as a situation where some high-valuation consumers stay with a low-quality good, while

16There is a di¤erence between physical and technological obsolescence from the viewpoint of �rms. With physicalobsolescence, �rms invest to reduce the quality of the used good next period before they sell it today. Depreciationcannot be changed once the good is sold. Under technological obsolescence, they invest to increase today the qualityof the durable good they will sell next period. There is an issue of commitment in investment, which has beenanalyzed by Waldman (1996) and Nahm (2004).17Fudenberg and Tirole assume that consumers were di¤erentiated according to their valuation of quality, which

was observable before purchase..

18

simultaneously some low-valuation consumers get a high-quality good. They consider a two-period

model of a durable goods market where a monopolist �rm can sell an upgrade of its product to

consumers who are di¤erentiated according to their valuation of quality. Assuming no secondhand

market, the authors found that, if the monopolist can segment the market between former and new

patrons (and price-discriminate accordingly), then it has an incentive to sell the upgraded good

to both groups, but some former patrons (who have a higher valuation for quality than any new

patron) will decide to keep the �rst-generation good.

In our dynamic model, we found an alternative explanation to such consumer behavior. The

introduction of uncertainty regarding the characteristics of the durable good results in the existence

of a positive mass of consumers who choose quality based on their past experience. After a bad

experience, these consumers are forced to face the risk of trying an unknown brand whose bene�t is

uncertain, so they prefer to acquire a good of lower quality rather than getting the higher quality.

As a result, we can �nd a consumer of type��0; �0

�within this set who purchases a low-quality

good, while a consumer with pair��00; �00

�buys a high-quality good, such that �0 � �00 and �0 � �00:

Notice that leapfrogging occurs without making any assumption about the market structure or the

information �rms may have about consumers�past behavior.

4.3 Consumers�Expectations and Loyalty

We turn our attention now to characterize consumers� expectations about the experience they

would get when purchasing a unit of the durable good, and to determine the total proportion of

consumers who buy a good of the same brand of their last purchase, which de�nes customer loyalty.

In equilibrium, consumers�expectations are always correct because of the assumption of rational

expectations. De�ne zN (�; �) to be the average expected experience by consumers of type (�; �)

who buy new goods. If ZN denotes the average expected experience by all buyers of new goods,

then:

ZN =

Z �

�

Z �

�zN (�; �) � �N (�; �) � dF (�)

!� dG (�)

Z �

�

Z �

��N (�; �) � dF (�)

!� dG (�)

By analogy, the average expected experience by all buyers of used goods is:

ZU =

Z �

�

Z �

�zU (�; �) � �U (�; �) � dF (�)

!� dG (�)

Z �

�

Z �

��U (�; �) � dF (�)

!� dG (�)

;

19

where zU (�; �) is the average expected experience by consumers of type (�; �) who buy used goods.

Similarly, we de�ne brand loyalty as:

L =

Z �

�

Z �

�` (�; �) �

��U (�; �) + �N (�; �)

�� dF (�)

!� dG (�)

Z �

�

Z �

�[�U (�; �) + �N (�; �)] � dF (�)

!� dG (�)

where ` (�; �) represents the fraction of consumers who are buying a good today (new or used) and

are purchasing the same brand that they owned last period.

Proposition 7 In a stationary symmetric equilibrium with durable experience goods we always

have that E (z) � ZU � ZN � E�z=zh

�and �0 � L � �h:

The intuition for this proposition is the following. There are two kinds of consumers who

purchase a given brand, regardless of its quality: (i) those who enjoyed a good match with that

brand last period and decide to stay loyal, and (ii) those who got a bad match last period with

some other brand and decide to switch. Any consumer that stays loyal will get a good match

again with probability �h; so her expected experience is E�z=zh

�: Any consumer that switches

brands gets a good match with the new brand with probability �0; and as a result her expected

experience is E (z) : If at any purchasing behavior there is a positive mass of consumers who buy

a new good, then zN (�; �) must be a weighted average between E (z) and E�z=zh

�: The same is

true for zU (�; �).

The proposition also states that the expected experience of buyers of new goods is larger that

than of buyers of used goods. The reason for ZN � ZU relies on the existence of a positive mass of

consumers who choose quality based on experience. Those consumers choose to buy a used good

because they got a bad experience with the good they owned last period. As a result, the expected

experience for all buyers of used good in that subset is E (z) ; because they are all switching brands.

In contrast, they buy a new good only when they enjoyed a good match last period. Since they are

staying loyal to the last brand they owned, they expect E�z=zh

�in their next purchase.

Regarding loyalty, notice that the fraction of consumers who repeat purchase of a given brand

depends on the proportion of consumers who got a good match, which in a stationary symmetric

equilibrium is a weighted average between �0 and �h: In fact, we found that customer loyalty is a

constant given by:

L� = �0

1� (�h � �0) : (13)

20

4.4 Prices

We turn now to discuss the e¤ect of experience on prices, since such analysis allow us to gain some

insight regarding how experience might a¤ect the producers�behavior, given that we are assuming

a �xed output. First, notice that the price of a used good is de�ned by

1�Z �

�F��UU (�)

�� dG (�) = 2Y; where �UU (�) =

pU � �bzql

(14)

where bz was given in (8). To obtain the price of new goods, we use the fact that �NU (�) � as givenby (12)� de�nes the consumer with the lowest valuation of quality that purchases a new good.

Then:

pN = �NU (�) ��q +��h

�1� �h

�1� � (�h � �0)��z + (1 + �) p

U (15)

To understand the e¤ects of experience on prices, we compare them with the ones we obtained

in the deterministic benchmark without uncertainty that we characterized in Section 2.2. Recall

that the prices were determined by:

1�Z �

�F��Ud (�)

�� dG (�) = 2Y; where �Ud (�) �

pUd � �zql

and pNd = �N�q + (1 + �) pUd :

For comparison purposes, suppose z = E (z) : Then, the e¤ect of experience on the price of used

goods is unambiguous: buyers of used goods always pay a higher price. To see this, notice that:Z �

�F��UU (�)

�� dG (�) =

Z �

�F��Ud (�)

�� dG (�) :

Since both �UU (�) and �Ud (�) are linear in �, then there exists a �

0 2��; �

�such that �UU (�

0) =

�Ud (�0) : As a result pU > pUd : Intuitively, we know that any consumer who got a bad match can

reach a higher utility by switching brands rather than staying loyal. This choice e¤ect (which is

captured by the term bz � E (z)) increases the present value of the utility of any consumer of type

(�; �) who buys used goods every period, so the price of a used good must be higher to clear markets

out.

The e¤ect of experience on the price of new cars is more complex. Concretely, there are two

reasons for the di¤erence between pN and pNd . The �rst one is the marginal consumer e¤ect, since

in the equilibrium with uncertainty there are more consumers who have access to a new good.

Hence �NU (�) < �N for any � > 0: The second reason is the choice e¤ect we just mentioned above.

The choice e¤ect has a direct e¤ect, captured by the term��h

�1� �h

�1� � (�h � �0)��z (which was absent

21

in the case without uncertainty), and an indirect e¤ect, also called the resale value e¤ect, through

the price of a used good. While both choice e¤ects push the price of new goods up, the shift of

the marginal consumer pushes it down, so the net e¤ect is ambiguous, and will depend on the

distributions of � and � across the population

We can also compare the prices we characterized in (14) and (15) with those that would arise

from an equilibrium with uncertainty where a tax or subsidy succeeds to achieve full trade, as we

discussed in Section 4.1. As in any model with full trade, consumers of new goods trade their used

good every period. Then, a new good is priced according to:

pNft = �N�q + (1 + �) pUft: (16)

But the tax policy would not, in principle, have an e¤ect on the price of a used good. As a

result, pUft will also be de�ned by (14). Then, the di¤erence between pN and pNft is reduced to two

e¤ects: the marginal consumer e¤ect �which makes pNft to be larger� and the direct choice e¤ect

�which is absent from pNft; causing pN to be larger�. The net e¤ect is again ambiguous.

However, if the government implements a subsidy to trade that is accompanied by the scrapping

of the used good, then pU < pUft since there are less units in the market. As a result, a third di¤erence

�which we can call the scrapping e¤ect� will appear between pN and pNft; causing pNft to be larger.

5 Extension: Durability and Brand Loyalty

Until now, we have assumed that the consumer�s valuation of experience is the same for any brand,

regardless of whether or not she has tried that brand before. Then, one of the characteristics of

the equilibrium is that any optimal behavior involves to stay loyal to a brand after a good match,

and to switch brands after a bad match.

However, some consumers may be reluctant to switch brands and prefer to stay loyal to a brand

even after a bad match, either because they do not want to face the risk of trying an unknown brand,

or because they have become accustomed to a particular brand. Those consumers would then give

a higher valuation to previous experiences. On the other hand, there may be some consumers who

like variety and are thrilled at trying new brands. As such, they would give a higher valuation to

the experience they may get from brands they have never purchased before.

The point of this exercise is to show that durability does have an e¤ect on customers�brand

loyalty. To examine the dynamics generated by the two behaviors we just describe, suppose now

that consumers�preferences are characterized by the triple��; �L; �S

�; where �L represents how

a consumer values her experience with a brand she has bought in the last T periods, while �S

denotes the consumer�s valuation of the experience with an unknown brand. We assume that �L is

22

distributed according to the function GL :��; �

��! [0; 1] ; which is increasing and has continuous

density gL (�) : In turn, �S has distribution function GS :��; �

��! [0; 1] ; which is also increasing

and has continuous density gS (�) :Since we are interested in the interaction between �L and �S ; we make some simplifying assump-

tions to keep the exposition as concise as possible18. First, we assume that consumers�valuation of

quality takes only two values, � and �; with � < �; and the mass of high types, m��; is such that

m�� 1

2 : Second, we assume that the market is segmented, so that consumers with � buy only

new goods, while those with � buy only used goods19. To focus further on buyers of new goods,

all consumers of type � have the same valuation of experience, regardless of experience, so that

�L = �S = ��: Finally, we assume that zl = � = 0.

In this section we discuss results and intuition, while derivations are left for Appendix B. In the

environment we just described, we can classify buyers of new goods into three types, depending on

their choice of brand. These are shown in Figure 2.

18The main results, presented in Proposition 8, are not a¤ected qualitatively by these simpli�cations.19 In Appendix B we identify the conditions required for this market segmentation to occur in equilibrium.

23

The white area shows consumers whose choice of brand is based on their last experience, staying

loyal after a good match, but switching brands after a bad match. The light shaded area repre-

sents consumers who switch brands with every purchase, i.e., "brand-switching buyers". The dark

shaded area identi�es consumers who remain loyal to a brand even after a bad match, i.e., "brand-

loyal buyers". The 45� line constitutes the special case where �L = �S ; which is the equilibrium

characterized in Section 3.

As Figure 2 shows, there are two optimal purchasing behaviors for consumers who stay loyal

after a good match and switch after a bad match: (i) to buy a new good every period, and (ii)

to keep a used good after a good match, but trades it after a bad match. The cuto¤ rule �Se��L�

de�nes the set of owners who are indi¤erent between keeping the used good or trading it to buy

a new good when it was a good match. A consumer who gives a larger valuation to her past

experiences will be more likely to keep such used good. For both purchasing behaviors, a fraction

L� of consumers remain loyal to a given brand every period, with L� de�ned in (13).As we would expect, brand-switching buyers are characterized by having low �L and high �S :

There is only one purchasing behavior that is optimal for these consumers. As buyers, they always

purchase a new good from a brand they have no previous experience. As owners, they always trade

their used good. Hence, they purchase a new good every period from a di¤erent brand, with zero

loyalty. We �nd two cuto¤ rules in a comparison with the two purchasing behaviors from buyers

from who choose their brand based on their last experience. The �rst one, �S1��L�; characterizes

consumers who buy new cars every period and are indi¤erent between switching brands or remaining

loyal to the previous brand after after a good match. The second one, �S2��L�determines the cuto¤

rule between consumers who switch brands every period and those who keep their used good after

a good match �and buy a new good of the same brand next period�. A consumer at point A isindi¤erent between those three purchasing behaviors.

In turn, brand-loyal buyers would have high �L and high �S : In equilibrium, these consumers

pursue at most two purchasing behaviors, which are similar to ones followed by consumers who

choose their brand based on experience. Concretely, (i) there is a subset of brand-loyal buyers who

purchase a new good every period, and (ii) there is also a subset of such buyers who keep a used good

after a good match20. We �nd that, regardless of their type ��; consumers with �L � �L0 purchase

a new good every period, while those with �L > �L0 prefer to keep a used good after a good match.

In any case, consumers in this subset should have remain loyal to brand they bought �rst. When

we compare such behaviors with the ones for buyers whose choice of brand is experience-based, we

�nd that a consumer at point B is indi¤erent between the four behaviors The cuto¤ rule �S3��L�

20A purchasing behavior where all brand-loyal buyers keep their used car regardless of her last match is also possiblein equilibrium. However, it occurs when there are also some type-�H consumers who choose to buy a used good. Forthis reason it is not considered in the analysis.

24

indicates the threshold regarding the optimal decision after a bad match for consumers who buy a

new good every period. In turn, �S4��L�de�nes the decision rule after a bad match for consumers

who keep their used good after a good match.

In summary, when consumers�preferences for experience are characterized by �L and �S ; it is

possible to have (a) consumers who switch brands with every purchase, and (b) consumers who

remain loyal to the �rst brand they bought. However, while in this setup there would always

exist brand-switching buyers, there may not be brand-loyal buyers. In particular, if we assume

that =�h � �01� �0 =

�0 � �l�0

determines the di¤erence between �h and both �h and �l; then no

consumer would remain loyal to the �rst brand they bought if >1

1 + �:

Suppose for a moment that there are enough new goods available for all consumers of type �,

that is, Y = m��: The equilibrium in that case will show all of those consumers purchasing a new

good every period. The only relevant cuto¤ rules will be �S1��L�and �S3

��L�such that consumers

with �S > �S1��L�are brand-switching buyers, those with �S 2

��S1��L�; �S3

��L��choose their

brand based on experience, while those with �S < �S3��L�are brand-loyal buyers. However, this

is also the equilibrium when the good is non-durable, where consumers of type � buy the good of

quality qh at a price pR: Hence, when Y < m��; it is possible that durability a¤ects brand loyalty.

Proposition 8 Suppose that consumers�preferences are characterized by the triple��; �L; �S

�: In

equilibrium, durability will lower:

(a) the mass of consumers who switch brands with every purchase.

(b) the mass of consumers who remain loyal to the �rst brand they bought.

Proposition 8 states that, if there are some consumers who keep their used good, then it must

be the case that slope��S1��L�

< slope��S2��L�and slope

��S3��L�

> slope��S4��L�. The

intuition for the �rst part is the following. We know that, in this setup, a consumer chooses to

switch brands with every purchase because she does not give enough value to the choice of staying

loyal after a good match. However, one of the bene�ts from keeping a used car after a good match

is that, on average, those consumers get to enjoy a good match more frequently than those who

trade their good every period. This di¤erence explains why, despite the lower quality, there will

be some additional mass of consumers choose to keep a used good after a good match rather than

trade it and switch brands.

For the second part of the proposition the intuition is as follows. A consumer chooses to stay

loyal to one brand because she does not give enough value to the choice of switching brands after a

bad experience but, as a result, gets to enjoy any bad match with that brand all too often. Then,

within the subset of consumers who decide to keep a used car after a good match, if we disregard

25

how they value their past experience, we observe that those who switch brands after a bad match

still get to enjoy a good match more frequently than those who remain loyal trade their good every

period, which leads then to more consumers choosing to do just that.

Although we conclude that durability can a¤ect customer loyalty, we cannot determine a priori

what the net e¤ect is. If GL (�) is negatively skewed, or GS (�) is positively skewed, and is

su¢ ciently low, then we will expect that durability reduces brand loyalty. If instead is large

enough so that there are no brand-loyal buyers, then durability has increases brand loyalty.

6 Concluding Remarks

We have presented a model for a market where goods exhibit both characteristics of durability and

experience. We found that the introduction of experience a¤ect some of results discussed in the

literature of durable goods. We have shown that the reason for consumers to keep a used cars

may not be an ine¢ cient problem as predicted by models of adverse selection, but rather a result

of experimentation. Also, we established a reason why some consumers with high valuation get

to hold a good of lower quality. On the other hand, we found that durability can a¤ect customer

loyalty.

There are at least two ways in which this paper can be extended. First, the empirical evidence

from the car market shows that, from the viewpoint of consumers, not all brands are the same.

As a result, di¤erences in the probability of a good match can explain other empirical facts. For

instance, we can expect that brands with higher loyalty are also the ones that (i) retain more value

(i.e., their price declines a slower pace) and (ii) exhibit longer ownership spells. In addition, more

realistic results can be obtained if instead we assume that experience takes more than two values.

A second extension involves the study of �rm behavior. If �rms have market power, there is the

possibility of strategic behavior of �rms in two fronts: market coverage (i.e., how much to produce)

and obsolescence (i.e., how fast to depreciate the quality of the good). The analysis of the e¤ect of

experience on prices gives us a few hints regarding production. In fact, the decision will depend on

the net e¤ect on the price of new cars and whether or not the choice e¤ect dominates the shift in

marginal consumer.

26

7 Appendix A

7.1 Proof of Proposition 1

Consider a buyer of used goods with pair (�; �) : Suppose she bought a used good of brand i last period and

got experience z0i. Today, the buyer can (i) stay loyal and buy a used good of brand i again; (ii) switch to

brand j =2 J; i.e., she buys an unknown brand, or (iii) switch to brand j 2 Jn fig ; so that she tries a brand(other than i) that she has experienced in the last T periods.

De�ne UUU��; �=z0i = zh

�to be the present value of the buyer�s utility under her optimal purchasing

policy when she got a good match last period, while 'h denotes the probability that she will get a good

match under such behavior. If the buyer stays loyal, then 'h = �h, but if she switches to brand j =2 J , then'h = �0; while a switch to brand j 2 Jn fig implies 'h = �l:

Similarly, we can de�ne UUU��; �=z0i = zl

�and 'l to be, respectively, the present value of the buyer�s

utility and the probability of a good match when she pursues her optimal purchasing behavior, given that

she got a bad match last period. When z0i = zl, a "loyal" consumer expects a good match with probability

'l = �l; a switch to brand i =2 J means 'l = �0 and a switch to brand i 2 Jn fjg implies 'l = �h:

Then, for k = h; l :

UUU��; �=zk

�= �ql � pU + �E

�z='k

�+ �EV BUYU

��; �='k

�where EV BUYU

��; �='k

�= 'kUUU

��; �=zh

�+ �

�1� 'k

�UUU

��; �=zl

�while E

�z='k

�= 'kzh +

�1� 'k

�zl:

Solving for UUU��=z0j = zh

�; we get:

UUU��=zh

�=

1

(1� �)

�ql � pU +

�1� �

�1� 'l

��(1� � ('h � 'l))�E

�z='h

�

+

1�

�1� �

�1� 'l

��(1� � ('h � 'l))

!�E�z='l

�!We also obtain that:

@UUU��=zh

�@'h

=

�1� �

�1� 'l

��(1� �) (1� � ('h � 'l))2

� ��z > 0 (A1)

Since �h = max�'h, then the optimal purchasing behavior for a buyer of a used good when z0i = zh is

to remain loyal to brand i, regardless of her choice when z0i = zl (i.e., regardless of the value for 'l). This

also means that the consumer�s last experience with any brand j 2 Jn fig must have been a bad match,otherwise the consumer would have stayed loyal to that brand.

When solving for UUU��=z0j = zl

�; we get:

UUU��=zl

�=

1

(1� �)

��ql � pU + �'l

(1� � ('h � 'l))�E�z='h

�+

�1� �'l

(1� � ('h � 'l))

��E�z='l

��

27

We also get:@UUU

��=zl

�@'l

=1� �'h

(1� �) (1� � ('h � 'l))2� ��z > 0 (A2)

From (A1); we determined that a consumer�s past history does not include good experiences with brand

i 2 Jn fjg : Hence the consumer only has two choices when z0j = zl: either she stays loyal to brand j or she

buys a used good of an unknown brand. From (A2), we conclude that the optimal choice when z0j = zl is to

switch to an unknown brand. Therefore, we get:

Therefore

UUU��=zh

�=

1

(1� �)��h1��ql + �E

�z=zh

�� pU

�+�1� �h1

� ��ql + �E (z)� pU

��(A3)

and UUU��=zl

�=

1

(1� �)��l1��ql + �E

�z=zh

�� pU

�+�1� �l1

� ��ql + �E (z)� pU

��(A4)

where

�h1 �1� �

�1� �0

�1� � (�h � �0) and �l1 �

��0

1� � (�h � �0)(A5)

Thus, the buyer�s expected utility is simply a weighted average of staying loyal to a brand after a good match

and switching to an unknown brand after a bad match. �


First, consider a consumer who always chooses to trade her used good with a new one regardless of her

experience. This is akin to the consumer "renting" a new good at a price pR = pN � �pU : Following

Proposition 1, we focus on the case where the consumer stays loyal to a brand when she gets a good match,

but switches to an unknown brand when she gets a bad match. Let UNN (�; �=z0) denote the expected utility

of a buyer of new goods under such optimal behavior. By analogy to the case of used-good buyer, then:

UNN��=zh

�=

1

(1� �)��h1��qh + �E

�z=zh

�� pR

�+�1� �h1

� ��qh + �E (z)� pR

��(A6)

and UNN��=zl

�=

1

(1� �)��l1��qh + �E

�z=zh

�� pR

�+�1� �l1

� ��qh + �E (z)� pR

��(A7)

where the constants �h1 and �l1 were de�ned in (A5).

Suppose now that the consumer, as an owner, still trades her used good after a bad match, but decides

to keep it after a good match. If she sells the used good, then she would buy a new good of an unknown

brand. If instead she keeps the used good, then she would purchase a new good of the same brand next

period. De�ne UKN (�; �=z0) to be the expected value of the consumer�s utility when she is a buyer and her

last experience was z0: When z0 = zk, k = h; l; her expected utility is:

UKN��=zk

�= �qh + �E

�z='k

�� pN + �

�1� 'k

� �pU + UKN

��=zl

��+ �'k

��ql + �zh + �UKN

��=zh

��

28

where E�z='k

�= 'kzh +

�1� 'k

�zl; while �h = �h and �l = �0: Solving for UKN

��; �=zh

�we obtain :

UKN��=zh

�=

1

(1� �)��h2��ql � pU + �zh

�+�1� �h2

� ��qh � pR + �

��h1E

�z=zh

�+�1� �h1

�E (z)

��(A8)

where �h2 ��h � �

��h � �0

��1 + ��0 � �2 (�h � �0)

: Hence, the expected utility can be written as the present value of a

weighted average of (i) keeping the used good, (ii) buying a new good after a good match, and (iii) buying

a new good after a bad match.

A comparison between the two purchasing behaviors we just analyzed gives us that a consumer who gets

a good match with her used good prefers trading it rather than keeping it when UNN (�; �=z0) � UKN (�; �=z

0) ;

i.e., when21 : �1� �

��h � �0

��[��q ��p] �

�1� �h

��z: (A9)

From (A9), we can determine the cuto¤ rule �KN (�), as de�ned by (10): Looking at (A9), consumers with a

large � and low � would favor keeping a used good after a good match, while those with a low � and a large

� would favor selling the good. �


Consider the set of buyers who purchase (i) a used good after a bad match, and (ii) a new good after a good

match. As owners, they must decide whether or not to keep the used good after a good match. Following

the results in previous proofs, we focus on consumers who switch brands after a bad experience, and remain

loyal after a good experience.

Let UNU (�; �=z0) denote the present value of the buyer�s utility whose previous match was z0, such that,

as an owner, she always sells her used good even after a good match. Then:

UNU��=zh

�= �qh + �E

�z=zh

�� pN + �

�1� �h

� �pU + UNU

��=zl

��+ ��h

�pU + UNU

��=zh

��and UNU

��=zl

�= �ql + �E (z)� pU + �

�1� �0

�UNU

��=zl

�+ ��0UNU

��=zh

�Then, this consumer buys some good every period. Solving for UNU

��; �=zh

�we obtain:

UNU��=zh

�=

1

(1� �)��h1��qh + �E

�z=zh

�� pR

�+�1� �h1

� ��ql + �E (z)� pU

��(A10)

where �h1 was de�ned in (A5). However, a comparison with the present value of the utility when this consumer

always buys a new good, denoted by UNN��; �=zh

�in (A6), gives that UNN

��; �=zh

�� UNU

��; �=zh

�when

��q � �p: In turn, comparing (A10) with (A3), which de�nes the present value of the utility when the

consumer always buys a used good, we obtain that UUU��; �=zh

�� UNU

��; �=zh

�when ��q � �p: Therefore

21All the cuto¤ rules de�ned in this paper �except the ones involving buyers of used goods� can also be foundusing owners�utilities. For consistency, we choose to use buyers�utilities throughout the paper.

29

UNU��; �=zh

�� max

�UNN

��; �=zh

�; UUU

��; �=zh

�for any consumer with pair (�; �) : Hence, the purchasing

strategy determined by UNU��; �=zh

�is never optimal.

Suppose now that the owner keeps a used good after a good match (and buys a new good of the same

brand next period), but trades it after a bad match. Let UKU (�; �=z0) denote the present value of the utility

for this consumer when she is a buyer. When z0 = zh, then:

UKU��=zh

�= �qh + �E

�z=zh

�� pN + �

�1� �h

� �pU + UKU

��=zl

��+��h

��ql + �zh + �UKU

��=zh

��:

After a bad match, this buyer will choose to buy a used good for an unknown brand. Hence UKU��; �=zl

�=

UNU��; �=zl

�: Solving for UKU

��=zh

�we get:

UKU��=zh

�=

1

(1� �)��h3��qh � pR + �E

�z=zh

��+�1� �h3

� ��ql � pU + �

��h4E (z) +

�1� �h4

�zh��

(A11)

where:

�h3 =1� �

�1� �0

�1 + ��0 � �2�h (1� �0)

and �h4 =

�1� �h

�1� ��h (1� �0)

Hence, the present value of the utility of this buyer is a weighted average of the per-period utilities of (i)

buying a new good after a good match, (ii) buying a used good when after a bad match, and (iii) keeping a

used car after a good match.

We compare now the present value of utility in (A11) with that in (A8). The di¤erence between both

cases lies on the decision of which quality to buy after a bad match: the consumer in (A8) favors a new good

of an unknown brand, while the consumer�s choice in (A11) is a used good. We obtain that UKN��; �=zh

��

UKU��; �=zh

�whenever:�

1� �2�1� �0

� ��h � �0

��[��q ��p] � ��0

�1� �h

��z (17)

which de�nes the marginal consumer �NN (�) stated in (11). Looking at (17), consumers with a large � and

a large � would prefer to buy a new good after a bad match, while those with low � and low � would prefer

a used good.

Let us compare now the present value of utility in (A11) with that in (A3). In this case, the di¤erence rests

on the decision of which quality to buy (new or used) after a good match, since the consumer�s choice in (A11)

is to buy a new good, while the buyer in (A3) prefers a used good. We get that UKU��; �=zh

�� UUU

��; �=zh

�occurs when: �

1� ��h � �0

��[��q ��p] � ��h

�1� �h

��z (18)

From (18), we �nd the cuto¤ rule �NU (�) de�ned in (12): Looking at (18), consumers with high � and high

� would favor buying a new good after a good match, while those with low � and low � would favor a used

good. �

30


The �rst step is to �nd the values for �N (�; �) ; �U (�; �) and �K (�; �) for each of the four possible purchasing

behaviors that characterize consumers�optimal behavior. Within each interval, we also solve for ` (�; �) ;

the fraction of buyers who are repeating purchase of the same brand. We compute as well zN (�; �) and

zU (�; �) ; which gives the average expected experience by consumers of type (�; �) who buy new goods and

used goods, respectively. Fixing �; we have the following intervals:

1. � 2 [�KN (�);�] : All consumers within this interval buy new goods every period, and sell their usedcars accordingly. Hence �Ki;t (�; �) = �Ui;t (�; �) = 0 for all i and t. Consumers who want to buy a new

good of brand i include (i) consumers who bought a new good of brand i last period and got a good

match, and (ii) consumers who bought brand j 6= i and got a bad match; of those consumers, only a

fraction �Nj;i (j 6= i = i =2 J) will choose to buy a new good of brand i, taking into account every possibleconsumer history in which they have not purchased brand i in the last T periods. Let xNi;t (�; �) denote

the fraction of consumers who bought a new good of brand i in period t and got a good match. Then:

�Ni;t (�) = xNi;t�1 (�) � �Ni;t�1 (�) +Xj 6=i

�Nj;i (�) ��1� xNj;t�1 (�)

�� Nj;t�1 (�) :

Since consumers who got a good match with brand i in period t� 1 always decide to stay loyal, thenthe proportion of consumers who buy brand i again in period t is:

ì;t (�; �) =xNi;t�1 (�) � �Ni;t�1 (�)

�Ni;t (�)

To determine xNi;t (�; �) and zNi;t (�; �) ; we know that consumers that stay loyal to brand i will get a

good match again with probability �h; and their expected experience is E�z=zh

�. Consumers who

switch to brand i will get a good match with probability �0; and their expected experience is E (z) :

Hence, for any t:

xNi;t (�) = ì;t (�) � �h + (1� ì;t (�)) � �0:

and zNi;t (�; �) = ì;t (�) � E�z=zh

�+ (1� ì;t (�)) � E (z)

In a stationary symmetric equilibrium we would have that �Nj;i (j 6= i = i =2 J) = 1M�1 ; so:

xN (�; �) = ` (�; �) = �L ��0

1� (�h � �0) and �N (�; �) =1

M:

We also �nd that the average expected experience reduces to:

zN (�; �) = �Z � �L � E�z=zh

�+ (1� �L) � E (z)

2. � 2 [�NN (�);�KN (�)] : These consumers buy new goods every time they are in the market, so �Ui;t (�; �) =0: Consumers who want to buy a new good of brand i at period t include (i) all consumers who owned

a used car of brand i in period t� 1 (they bought it new in t� 2, got a good match and chose to keep

31

it as used), and (ii) a fraction �Nj;i (j 6= i = i =2 J) of the set of consumers who bought brand j 6= i in

period t� 1; got a bad match, and have not purchased brand i in the last T periods. Hence:

�Ni;t (�) = �Ki;t�1 (�) +Xj 6=i

�Nj;i (�) ��1� xNj;t�1 (�)

�� Nj;t�1 (�)

where �Ki;t�1 (�; �) = xNi;t�2 (�; �) � �Ni;t�2 (�; �) : Since all consumers who kept their used car of brand iin period t� 1 will buy the same brand in period t; then ì;t (�; �) is given by:

ì;t (�; �) =�Ki;t�1 (�)�Ni;t (�)

which in turn determines that:

xNi;t (�) = ì;t (�) � �h + (1� ì;t (�)) � �0

and zNi;t (�; �) = ì;t (�) � E�z=zh

�+ (1� ì;t (�)) � E (z)

As before, in a stationary symmetric equilibrium we should have �Nj;i (j 6= i = i =2 J) = 1M�1 , and we

obtain again:

xN (�; �) = ì;t (�; �) = �L and zN (�; �) = �Z

Since �N (�; �) + �K (�; �) = 1M ; then:

�N (�; �) =1

M� 1

1 + �Land �K (�; �) =

1

M� �

1 + �L

3. � 2 [�NU (�);�NN (�)] : Within this interval, buyers of a used good of brand i in period t includes aproportion of consumers who bought a good (either new or used) of brand j 6= i in period t � 1; gota bad experience with it, and do not have any previous experience with brand i in the last T periods.

Then:

�Ui;t (�) =Xj 6=i

�Nj;i (�) ��1� xNj;t�1 (�)

�� Nj;t�1 (�) +

Xj 6=i

�Uj;i (�) ��1� xUj;t�1 (�)

�� Uj;t�1 (�)

where �Nj;i (�; �) and xNj;t (�; �) are de�ned as before, while x

Uj;t (�; �) is the fraction of consumers of type

(�; �) who bought a used good of brand j in period t and got a good experience, and �Uj;i (j 6= i = i =2 J)is the fraction of consumers who bought a used good of brand j last period and chose to switch to

brand i since they got a bad experience, given that they have not tried brand i in the last T periods.

Notice that xUi;t (�; �) = �0 and zUi;t (�; �) = E (z) since all buyers of used goods are switching brands.

Buyers of a new good of brand i in this interval include (i) consumers who purchased a used good of

this brand in period t � 1 and got a good match, and (ii) consumers who bought a new good of thisbrand in period t� 2 and kept in period t� 1 because they got a good match. Then:

�Ni;t (�) = xUi;t�1 (�) � �Ui;t�1 (�) + �Ki;t�1 (�)

32

where again �Ki;t�1 (�; �) = xNi;t�2 (�; �) � �Ni;t�2 (�; �) : Since all buyers of new goods are remaining loyalto the brand they owned last period, then xNi;t (�; �) = �h and zNi;t (�; �) = E

�z=zh

�for any i and t:

In a stationary symmetric equilibrium where �Nj;i (j 6= i = i =2 J) = �Uj;i (j 6= i = i =2 J) = 1M�1 , we get

�U (�; �) =1� �h�0

�N (�; �) : Solving, we obtain:

�U (�; �) =1

M� 1� �

h

�; �K (�; �) =

1

M� �

0�h

�and �N (�; �) =

1

M� �

0

�;

where � ��1 + �0 � �h

�1� �0

��:

4. � 2 [�UU (�);�NU (�)] : All consumers within this interval buy used goods every period, so �Ki;t (�; �) =�Ni;t (�; �) = 0 for all i and t. By analogy with the results we got in the �rst interval, we would have in

a stationary symmetric equilibrium that:

xU (�; �) = ` (�; �) = �L; �U (�; �) =1

Mand zU (�; �) = �Z

The second step is to characterize the demand and supply in both markets. De�ne p =�pN ; pU

�to be

the vector of prices, and � =��0; �h

�to be the vector of probabilities. Given the de�nitions above, we can

rewrite the demand of goods in the used market as DU =

Z �

�

dU (p;�) � dG (�) , where:

dU (p;�) � 1

M

�1� �h�

hF��NN (p;�)

�� F

��NU (p;�)

�i+hF��NU (p;�)

�� F

��UU (p;�)

�i�

Demand in the market for new goods is DN =

Z �

�

dN (p;�) � dG (�) , where:

dN (p;�) � 1

M

�h1� F

��NK (p;�)

�i+

1

1 + �L

hF��NK (p;�)

�� F

��NN (p;�)

�i+�0

�

hF��NN (p;�)

�� F

��NU (p;�)

�i�

Supply in the used market is SU =Z �

�

sU (p;�) � dG (�) ; where:

sU (p;�) � 1

M

�h1� F

��NK (p;�)

�i+1� �L1 + �L

hF��NK (p;�)

�� F

��NN (p;�)

�i+�0�1� �h

��

hF��NN (p;�)

�� F

��NU (p;�)

�i!

33

An equilibrium exists if there is a vector of prices p that satis�es DN (p;�) = DS = y; and DU (p;�) =

SU (p;�) : De�ne eN (p;�) and eU (p;�) to be the excess demand functions in the market of new and used

goods, respectively. Then:

eN (p;�) = DN (p;�)� y and eU (p;�) = DU (p;�)� SU (p;�)

De�ne also the functions:

�N (p;�) = 1��eN (p;�)��

max fDN (p;�) ; yg

�U (p;�) = 1��eU (p;�)��

max fDU (p;�) ; SU (p;�)g

Finally, de�ne the compact set S =��qh + �zl; �qh + �zh

��qh + �zl; �qh + �zh

�; and the vector

valued function H =�hN ; hU

�, such that:

hN (p;�) = �N (p;�) � pN +�1� �N (p;�)

� �I�eN (p;�) � 0

�� qh + �zl

+ I�eN (p;�) < 0

�� qh + �zh

�hU (p;�) = �U (p;�) � pU +

�1� �U (p;�)

� �I�eU (p;�) � 0

�� qh + �zl

+ I�eU (p;�) < 0

�� qh + �zh

�It can be easily veri�ed that H maps S into itself, and is continuous. Hence H has a �xed point. We

can rewrite both hN and hU as follows:

hN (p;�) = pN +�1� �N (p;�)

� �I�eN ((p;�) �) � 0

�� qh + �zl

+ I�eN (p;�) < 0

�� qh + �zh � pN

�hU (p;�) = pU +

�1� �U (p;�)

� �I�eU (p;�) � 0

�� qh + �zl

+ I�eU (p;�) < 0

�� qh + �zh � pU

�Therefore, there is a �xed point ofH when E� (z) = E0 (z) ; and the following two conditions are satis�ed:�

1� �N (p;�)��

I�eN (p;�) � 0

�� qh + �zl + I

�eN (p;�) < 0

�� qh + �zh � pN

= 0�

1� �U (p;�)��

I�eU (p;�) � 0

�� qh + �zl + I

�eU (p;�) < 0

�� qh + �zh � pU

= 0

In both expressions, the term in braces is never equal to zero. Hence, it must be the case that �N (p;�) =

�U (p;�) = 1. This is only possible when eN (p;�) = eU (p;�) = 0. �

34


To show that the volume of trade is never 100%, suppose by way of contradiction that there is full trade:

Then, we must have in equilibrium that SU = DN = y; which means that every consumer who buys a new

good must choose to sell its used good regardless of the experience they got. For this to be true, then there

should be only two optimal purchasing behaviors in equilibrium: to buy every period either new goods or

used goods. Then UNN (�; �=z0) = UUU (�; �=z

0) de�nes those consumers with type (�; �) who are indi¤erent

between buying new and used. Solving, we get ��q = �p: This means that, regardless of their valuation of

experience, any consumer with � � �N would prefer to buy new goods every period, with �N � �p

�q: For this

behavior to be optimal, we must have that for any owner of type��N ; �

�:

�Nql + �zh + �UKN

��N ; �=z0 = zh

�� pU + UNN

��N ; �=z0 = zh

�where the left hand side denotes the utility for a consumer with �N from keeping used good after a good

match , and buying a new good next period, while the right hand side represents the utility from selling that

used good . Using (A6) and (A8), we obtain:

��zh � E (z) + �h

�E�z=zh

�� E (z)

�� N�q ��p = 0

Since the left hand side is positive for any � > 0; we get a contradiction that buying a new good every period

is an optimal purchasing policy for consumers with �N :

Let us now prove that the volume of trade is always positive. Suppose by way of contradiction that there

is zero trade, so that in equilibrium SU = 0: This means that the only optimal purchasing behavior occurs

when all buyers of new goods decide to keep their used good, even after a bad experience. Such behavior

implies that, for k = h; l :

UKK��; �=zk

�= �qh + �E

�z=zk

�� pN + �

�1� �k

� ��ql � pU + �zl + �UKK

��; �=zl

��+��k

��ql � pU + �zh + �UKK

��; �=zh

��Solving for UKK

��; �=z0 = zh

�; we obtain:

UKK��; �=zh

�=

1

1� �

�1

(1 + �)

��qh � pR + �

��ql � pU

��+

�

1� �2 (�h � �l)��1� �2

�1� �l

��E�z=zh

�+ �2

�1� �h

�E�z=zl

��(A12)

However, any consumer would prefer to switch brands after a bad match (and buy either a new or a used

good) rather than stay loyal and keep the used good. To see this, let us �rst compare (A12) with (A8), which

represents the consumers�s utility when switching to a new good. In that case, we get that UKN��=zh

�>

UKK��=zh

�when, for any pair (�; �):

��q ��p��z

> � (1 + �)�0 + �

��0 � �l

�1� �2 (�h � �l)

35

Let us now compare (A12) with (A11), which represents the present value of the utility when the consumer

switches to a used good after a bad match. In that case, we get that UKU��=zh

�> UKK

��=zh

�when, for any

pair (�; �) and any given prices and match probabilities:

��q ��p��z

<(1 + �)

� (1� �0)�0 + �

��0�h � �l

�1� �2 (�h � �l)

Hence UKK��=zh

�< max

�UKN

��=zh

�; UKU

��=zh

�for any consumer with pair (�; �) ; which means that the

purchasing strategy where buyers of new goods always keep her used good regardless of experience is never

optimal. �


The possibility of leapfrogging arises within the set of consumers who decide to buy quality based on experi-

ence. Recall that, as buyers, such consumers switch brands and purchase a used good after a bad match, but

stay loyal and get a new good after a good match. As owners, they keep the used good if they got a good

experience. In Proposition 3 we found that the present value of the utility from this purchasing behavior

was given by UKU (�; �=z0) : Suppose by way of contradiction that no consumer follows that behavior. For

this to be true, there should be only three optimal purchasing behaviors in equilibrium: (i) buy a new good

every period, getting utility UNN (�; �=z0) (ii) buying new goods whenever the consumer owns no good, but

keep the good after a good match, characterized by UNK (�; �=z0), and (iii) buy a used good every period

UUU (�; �=z0). When comparing UNK (�; �=z

0) with UKU (�; �=z0) we found that, after a bad match, a consumer

with pair (�; �) prefers to buy a new good of an unknown brand, rather than a used good, when:

��q ��p��z

� ��0

�1� �h

�1� �2 (1� �0) (�h � �0)

:

In a comparison of UUU (�; �=z0) and UKU (�; �=z

0) we also found that, after a good match, a consumer with

pair (�; �) prefers to buy a used good, rather than a new good, when:

��q ��p��z

� ��h

�1� �h

�1� � (�h � �0) :

Then there will be no consumer with pair (�; �) who chooses quality based on experience if:

��h

�1� �h

�1� � (�h � �0) � �

��0�1� �h

�1� �2 (1� �0) (�h � �0)

After some simpli�cation we get: ��h � �0

� �1 + ��0 � �2�h

�1� �0

��< 0

which is clearly a contradiction since both terms in the left hand side are positive. As a result, there

is a positive mass of consumer who buy quality based on experience, which means that there is always

leapfrogging. �

36


In the proof of Proposition 4 we found the values for ` (�; �) ; zN (�; �) and zU (�; �) for each of the optimal

purchasing behaviors. Concretely, we obtained that the fraction of buyers who decide to repeat purchase of

the same brand in each of the four intervals is equal to � � �0

1� (�h � �0) : As a result, then also:

L = �0

1� (�h � �0) :

It can be very easy to see that �0 � L � �h:

Regarding consumers�expectations, the average expected experience by all buyers of used goods is:

ZU =1

DU

"Z �

�

E (z)

�1

M

1� �h�

hF��NN (p;�)

�� F

��NU (p;�)

�i�� dG (�)

+

Z �

�

�Z

�1

M

hF��NU (p;�)

�� F

��UU (p;�)

�i�� dG (�)

#

Using the fact that �Z = E (z) + �L�E�z=zh

�� E (z)

�; we obtain::

ZU = E (z) + �L

�1� XU

DU

��E�z=zh

�� E (z)

�(19)

where

XU =

Z �

�

�1

M

1� �h�

�hF��NN (p;�)

�� F

��NU (p;�)

�i� dG (�)

is the mass of consumers with � 2h�NU (�) ; �

NN (�)

ithat buy used goods. It is immediate to see that

E (z) � ZU � E�z=zh

�:

In turn, the average expected experience in the new market is:

ZN =1

DN

"Z �

�

�Z

�h1� F

��NK (p;�)

�i+

1

1 + �L

hF��NK (p;�)

�� F

��NN (p;�)

�i�� dG (�)

+

Z �

�

E�z=zh

�� 1

M

�0

�

hF��NN (p;�)

�� F

��NU (p;�)

�i�� dG (�)

#Using again the de�nition of �Z , we can rewrite it as:

ZN = E (z) +

�1� (1� �L)

�1� XN

DN

��E�z=zh

�� E (z)

�(20)

where

XN =

Z �

�

�1

M

�0

�

�hF��NN (p;�)

�� F

��NU (p;�)

�i� dG (�)

is the mass of consumers with � 2h�NU (�) ; �

NN (�)

ithat buy new goods. Then E (z) � ZN � E

�z=zh

�: A

comparison of both (19) and (20) gives us that ZN > ZU since 1� XN

DN +XU

DU > 0: �

37

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39

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