competition in a status goods market_jmr.11.0005

8/13/2019 Competition in a Status Goods Market_jmr.11.0005

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Abstract

Consumers value status goods due to the impression status-product ownership makes on other

consumers and this impression depends on the actual distribution of ownership in population.

Explicitly modeling consumer value of status products as coming from the information the

product ownership conveys to other consumers, this paper shows that a status-product manu-

facturer can benefit from a competitors cost reduction due to the competitors price reduction

associated with it. In other words, it shows that two status products which are (imperfect)

substitutes in the consumer utility function may be complements in the profit functions. As

a consequence, competition could lead to higher prices than the optimal ones under monopoly

ownership of both products. The authors confirm the assumptions that consumer value of a

status good depends positively on the proportion of desirable type among owners and negatively

on the proportion of the desirable type among non-owners in one experiment and find empirical

support for the positive effect of a price reduction of one product on the demand for the other

product from another experiment.

Keywords: Competitive Strategy, Pricing, Self-Expressive Goods, Status Goods, Fashion

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While consumers value some products for their utilitarian qualities, such as the taste and

nutritional value of food, warmth of clothes, or convenience a car provides, many products are

also valued for the status they are supposed to bestow on their owners. One can easily argue

that the value of precious stone and metal jewelry, designer handbags (e.g., Louis Vuitton, Prada

or Gucci), fine watches (e.g., Rolex), luxury cars (e.g., Mercedez, BMW), or even premium beer

(e.g., Heineken in the US) comes mainly from the perception that using these products would

elevate the person in the eyes of the onlookers.

Furthermore, competing status products differentiate from each other in a horizontal dimen-

sion. For another example, consider the luxury car market. BMW, Audi, Mercedes-Benz, and

Lexus cars competing in the luxury car segment all in some extent convey status, but consumers

differ in their preferences for these brands. Dan Pankraz, a senior planner at DDBO, describes

the differences between these brands by what the car says about the driver: for example, he

claims, an Audi says I improve with the times and a BMW says Im on the way to the top,

while Mercedes-Benz says I made it, and a Lexus says Im confident in my own skin. 1

Similar opinions are expressed about competing brands of Prada vs. Gucci and Omega vs.

Rolex. For example, Menkes (1999) begins her NY Times article about Gucci vs. Prada by

characterizing the differences between these two brands in the following way: They are the

fashion titans of the 1990s. In one corner, a streak of blood red across the sleek hair, scarlet

sequins dripping on the jacket and the crescendo from Psycho on the soundtrack. On the

other side, a khaki-clad figure, bag strapped to the chest, ready to stride out on biker boots

into the urban jungle.

Unlike the value of utilitarian products, the value of status products depends not only on

the physical attributes of the product, but also on the distribution of ownership of the product

in the consumer population (Veblen 1899). Specifically, the value of a status product to a

consumer comes from the comparison between the value of the perception other people will

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have of this consumer if she does not own a particular status product and the value of the

perception other people will have of this consumer if other people observe she owns this status

product. Therefore, the value of a status product is affected by its sales both due to the effect

of the change in the distribution of consumers who own it and due to the effect of the change

in the distribution of consumers who do not own it.

Furthermore, as the above examples illustrate, competition is ubiquitous in status goods

markets. The complexity of inter-dependence of value and price raises the question of whether

what we know about the effects of competition in regular goods markets would also apply in

status goods markets.

To better illustrate the issue of competitive implications of the endogenous value of a status

product, consider a market with two status products, A and B, competing for demand from

the high-class consumers who would like to use status goods to signal their identity to each

other. In this case, the consumer value of product A increases when either i) the proportion of

the high-class consumers in the total customer base of product A increases, or ii) the proportion

of the high-class consumers among the people not owning either product decreases. The first

effect is due to the increasing utility of the consumer if she buys product A, while the second

effect is due to the decreasing consumer utility if she does not buy either product. Consider

now what happens if the price of product B decreases and as a result, product B sells to more

high class consumers. Assume for the moment that product Bs price is high enough and the

lower price does not attract many consumers from outside the high-class segment. Then the

increased sales of product B have two effects on the demand for product A: a negative effect

of potentially taking away demand from product A by improving the value of product B, and

a positive effect of increasing consumer value of each of the products, since the proportion of

high class consumers among those who do not buy either product decreases. If the second effect

dominates the first, it could lead to the the counter-intuitive outcome that a decrease in price

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of one product would result in an increase in the demand for both products. In other words,

while the two status products could act as substitutes in the consumer utility function, they

can be compliments in the firms profit functions.

This paper illustrates the validity of the above intuition through a formal model of two

products competing in a market with two high-type consumer segments and one low-type con-

sumer segment. We assume that consumers value how they are perceived by other consumers

and thus their value of a product depends on how possession of this product will affect other

consumers beliefs about them.2 We then derive that one products price reduction could lead to

increased demand for both products and further validate this counter-intuitive result through

an experiment where participants act as consumers and choose which of the two products they

would buy given the prices and their social payoffs in the subsequent matching game. In this

experiment, by keeping the price of one product constant while changing the price of the other

product, we focus on examining the change in the demand of one product as a result of the price

change of the other one. Consistently with the model predictions, we find a significant increase

in the average demand for the product with constant price when the price of the other product

increased.

The above comparative statics can lead to the equilibrium result that a lower cost of one firm

benefits both competitors when the prices are set to optimally account for the demand and cost

factors. Another result we obtain is that a monopoly owner of both products should set both

of their prices lower than competitors who each own one product would. Therefore, in a status

goods market, horizontal integration may lead to lower rather than higher prices, as the case

with utilitarian products would be. Comparing the equilibrium outcomes in the market with

a single-product monopoly and in the market with a duopoly, we find that competition, i.e.,

entry of another firm, may either increase or decrease profits of the incumbent. Furthermore,

competitive entry may benefit the incumbent even when the entrant poaches away half of the

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customers who would buy from the incumbent in the absence of entry.

RELATION TO EXTANT LITERATURE

Consistently with Veblens idea that certain conspicuous consumption may signal belonging

to the high society, we define status products as products that help consumers identify them-

selves as members of a desirable type. This is a common definition used in literature. For

example, Grossman and Shapiro (1988) define status goods as goods for which the mere use or

display of a particular branded product confers prestige on their owners, apart from any utility

deriving from their function. An early treatment of social aspects of consumption belongs to

Leibenstein (1950), who speculated that either small or high market share may be valued by

consumers due to the value of uniqueness and conformity, as well as that some consumers could

value a high price. Becker (1991), Amaldoss and Jain (2005a, 2005b, 2010) and Balachander

and Stock (2009) also explore the implications of consumer value for uniqueness and conformity.

However, it is appealing to understand where the status meaning of a product or brand is

coming from in a framework where consumers rationally infer the signal from the environment.

In this regard, Muniz and OGuinn (2001) argue that consumers derive the symbolic meaning of

a brand from the type of consumers who buys that brand. Similarly, Escalas and Bettman (2005)

find experimental support for the hypothesis that the symbolic properties of reference groups

become associated with the brands those groups are perceived to use. In other words, they

argue that people derive brand associations from their use by the people they desire to appear

to be (the in group in Escalas and Bettmans terminology). They further show, through an

experiment, that the reverse holds as well: the brand desirability declines if more people from

undesirable group (the out group) use it. Similar arguments are also given by White and

Dahl (2006) and Han, Nunes and Dreze (2010). Applying these findings to status products,

one then expects that the meaning and value of a status product comes from the prevalence of

more desirable types of people among its owners. As Bagwell and Bernheim (1996) argue, such

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definition also directly follows from the classical and often cited essay by Veblen (1899).

In the context of a status goods market, a number of game-theoretic papers have defined

the consumer value of a product as coming from the value of projecting the image consistent

with the products distribution of ownership (e.g., Karni and Schmeidler 1990, Wenerfelt 1990,

Pesendorfer 1995, Bagwell and Bernheim 1996, Yoganarasimhan 2012, Kuksov and Wang 2012).3

This is also the approach we adopt in this paper. Note that status goods in this definition are a

special case of self-expressive goods, with the distinctive characteristic being that they separate

high from low class, i.e., they signal a characteristic that is perceived as desirable by everybody.

The consumer use of status products we consider is akin to communication through costly

signaling. Although Spence (1973) introduced the idea of people signaling their types to firms,

the extensive marketing literature on costly signaling concentrated on studying the effects of

firms signalling to each other or to the consumers (e.g., Balachander and Srinivasan 1994, Desai

and Srinivasan 1995, Moorthy and Srinivasan 1995, Simester 1995, Anderson and Simester 1998,

Kalra et al. 1998, Desai 2000, Soberman 2003). In contrast, this paper analyzes costly signalling

of consumers to other consumers.

As we have noted above, in the economics literature, implications of consumer product use

to signal their types to other consumers were considered by Pesendorfer (1995) and Bagwell and

Bernheim (1996). In particular, Pesendorfer (1995) considers implications of a durable-good

monopoly selling status goods in a dynamic model and finds that innovation cycles would en-

dogenously occur. The idea is that when a product penetrates the high-type consumer segment,

the firm then has an incentive to sell to the low-type consumers, which degrades the signalling

value of the product and a new product (new fashion design) is then needed. In equilibrium, a

new product is therefore periodically introduced when the old design sufficiently penetrates the

low-type consumer segment. Bagwell and Bernheim (1996) ask a more general question of when

a market for status products may exist and derive general conditions. Since they concentrate

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on the demand side, they assume a perfectly competitive supply of the status goods.

The current paper extends this analysis to consider competitive implications of a product

being a status good when competition is non-trivial, i.e., when firms are differentiated. Note

that while Amaldoss and Jain (2005b) also model differentiated competition, they assume that

some consumers derive value from exclusivity and some from conformity. They show that the

demand from those who value exclusivity may increase with the price. However, in their model,

the total demand for a product always decreases in own price and increases in the competitors

price. We are able to show that demand for a product could decrease in the competitors price

because we derive the consumer value as endogenously coming from what the product possession

signals to other consumers relative to what others would assume if neither product would be

purchased. This competitive implication of our model is unique in the literature on status goods.

The intuition behind the above result in our model is that when either product penetrates

the market, it first penetrates the high-type segment. Therefore not owning either product

becomes more indicative of belonging to the low type and thus, the differential value of owning

either product increases. This effect is akin a category-wide network externality effect. In that,

our paper is related to the extensive literature on network externalities starting from Katz and

Shapiro (1985) and Farrell and Saloner (1986). Although that literature normally assumed that

network externality is product- rather than category-specific, similar results to ours could be

obtained in a market with category-wide network externality. When network externalities are

present, the value of a product also depends on the pattern of ownership and this can lead to

interesting effects (for example, Chen and Xie (2007) show how a larger size of loyal segment

can end up being detrimental to a firm in the presence of network externality). However, the

mechanism behind our results is different: in the case of network externality, the value of owning

a product increases when there are more products like it, while the value of not owning a product

is independent of the product penetration. In contrast, in a status goods market, the value of

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having a product does not change with the penetration of the competitors product, but the

value of not owning either product decreases.

An increasing body of marketing literature utilizes experiments to test theoretical predictions

of the consumer or firm behavior, including the predictions of interdependence of consumer

choices. For example, Amaldoss and Jain (2005a, 2005b) use laboratory experiments to show

empirical support for the key results of their theoretical models that the desire for uniqueness and

conformity can lead to an upward sloping demand curve from one of the consumer segments. In

other settings, experiments have been used by marketing researchers to validate their models of

consumer sweepstake design and contest promotions (Kalra and Shi, 2010), timeshare exchange

mechanisms (Wang and Krishna, 2006), sales contests (Lim et al., 2009), and contracting issues

between supply chain partners (Ho and Zhang, 2008). Following these researchers, we provide

empirical support from a lab experiment for one of the key results of the theoretical model.

MODEL

A unit mass of consumers consists of high (H) and low (L) consumer types. The high-

type consumer segment consists of two subsegments, which we will call H1 and H2 segments,

each of mass . Thus, the low-type consumer segment has mass = 12. Consumers

have preferences over what other consumers think their type is. Given the above three-type

consumer distribution, a belief about consumeris type can be defined by a triple (qH1, qH2 , qL)

of probabilities that this consumer is of type H1, H2, and L, respectively (qH1+qH2+qL = 1).

We will call this triple of probabilities by which a consumer is known to other consumers as the

consumers image (in the eyes of other consumers). Assume that consumeri values an image at

(1) Vi(qH1 , qH2, qL) =qH1vH1i + qH2v

H2i + qLv

Li,

wherev Ti is the weight consumeri places on being perceived as type T. In the next section, we

clarify how these consumer valuations for projecting images of themselves to other consumers

can be derived from a matching game. The conceptual meaning of high type is that all

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consumers prefer to be perceived as a high-type consumer than as a low-type consumer, i.e.,

vHji > v

Li for j = 1, 2. Without loss of generality, normalize the utility of each consumer so

that vLi = 0. Consumers would then have value for a product if it allows them to project an

image more desirable than the image represented by the population of consumers who do not

buy either product.

To obtain non-trivial (differentiated) competition, we assume thatHj-type consumers prefer

being perceived as Hj type rather than as H3j type, i.e., H1 and H2 type consumers are

horizontally differentiated in their preferences for the image to project. Namely, let vHji =

vH3ji +t for consumer i of type Hj. This assumption is similar to the in-group preference

argued for by some researchers (e.g., Escalas and Bettman 2005, White and Dahl 2006 and

Han, Nunes and Dreze 2010; for a theoretical argument in the context of a marriage market, see

also Becker 1973). Furthermore, to obtain a non-degenerate demand function, we assume that

consumer valuations are heterogeneous within each Hj segment. Specifically, assume that the

weight vhthat Hj-type consumer places on being perceived as Hjtype is uniformly distributed on

U[vh, vh]. Thus,Hj -type consumers value of perception (qH1, qH2 , qL) isV =qHjvh+qH3j(vht)

withvhU[vh, vh].

If low-type consumers derive higher value from projecting a high-type image than their

high-type counterparts, there may be no equilibrium in which a product can be valued solely for

the image it allows consumers to project of themselves. Intuitively, this is because if low-type

consumers would have higher value of projecting high-type image than high-type consumers do,

low-type consumers would be buying any product with desirable image in higher proportions

than the high-type consumers thus degrading the products image until it is no longer valuable.

For a formal argument, see Bagwell and Bernheim (1996). We therefore assume that vHji is

lower for a low-type consumer than it is for a high-type consumer. Note that this assumption

can also be justified through an in-group preference. Formally, denoting vHji =vLfor a low-type

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consumer i, assume that this value is the same for all low-type consumers. For simplicity, we

assume that if consumer i is of low type, her weight vHji is lower than a firm could reasonably

consider, e.g., it is lower than marginal product cost. This assumption greatly simplifies the

technical derivations, but is not strictly necessary. Given this assumption, the assumptions of no

heterogeneity of low-type consumer valuations and that low-type consumers do not distinguish

between H1 and H2 segments are not essential.

Although we have virtually assumed an inert low-type consumer segment, the existence

of this segment is still essential. This is because the presence of this segment implies that a

consumer buying a product signals to other consumers that she is of one of the high types. In

other words, the existence of low-type consumers makes the product ownership imply status

rather than only a horizontally differentiated image. Our main results would not hold if the

low-type consumer segment does not exist or is not large enough.

IfHj-type consumers are not heterogeneous in their valuations, either all or none of them

would buy a product and thus the demand curve would be degenerate. Furthermore, a small

degree of consumer heterogeneity does not resolve this issue since as we will see in the following

section, when the consumer demand from the high type consumer segment increases, the value

increases as well. Therefore, with small degree of consumer heterogeneity, one would obtain

that all high-type consumers buying and all high-type consumers not buying could both be

equilibrium outcomes at the same time. Therefore, to ensure unique equilibrium in consumer

choice we need sufficient heterogeneity in high-type consumer valuations. Formally, we assume

(2) vh

t > vL, and vh+ t > vh,

where the first inequality is the condition on low-type consumer value of being perceived as a

high type being lower than that of a high-type consumer, and the second inequality is the above

discussed condition on sufficient heterogeneity of high-type consumers and sufficient mass of

low-type consumers. It turns out that the second assumption is equivalent to assuming that

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when prices are equal, it is possible to have in equilibrium some but not allH1 andH2 segment

consumers buy. If this condition is not satisfied, reducing price from a level that has non-zero

sales never increases revenue and thus price would not be continuously changing in costs, which

would make the analysis of the pricing decision not interesting.

Two firms, named Firm 1 and Firm 2, produce one product each at marginal costs c1 and

c2, respectively, and simultaneously set prices p1 and p2. Then, consumers make purchase

decisions observed by other consumers. Then consumers receive payoffs based on the other

consumers beliefs about their type as discussed above. Note that in equilibrium, these beliefs

have to be consistent with the buying behavior and are derived according to the Bayes rule

from the prior beliefs determined by the relative segment sizes and the equilibrium buying

behavior. If in equilibrium, no consumers buy product k, we assume that consumer belief

about someone who deviates and buys this product is an arbitrary probability mix over Hj

types. This assumption can be justified by the ability of firms to target a few select (high-type)

consumers at product introduction to establish an image and is needed to rule out the trivial

equilibrium where no consumers buy either product because the belief is that whoever buys a

product is of the low class. Furthermore, whenpk > vL, as we will consider, such beliefs also

follow from the Intuitive Criterion. Although, technically, it could be important to postulate

how a consumer payoff depends on which consumers think what about her, since all consumers

have the same information, they must all have the same beliefs about a given consumer. Also

note that consumers do not need to observe all other consumer choices to make an inference

about a given consumer, since consumer can fully derive their beliefs about aggregate consumer

behavior from the equilibrium.

Before turning to the model analysis, in the following section, we discuss how the above-

defined consumer valuations may be derived from a more primitive assumptions on consumer

interaction. This analysis shows, in particular, that the utility function defined in Equation (1)

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may be a reduced-form representation of how product choices affect a customers image and how

this image value can be justified through more primitive values. We also further relate some of

the assumptions to the relevant literature.

A MATCHING GAME MICRO-FOUNDATION FOR IMAGE VALUE

The main departing point of our model from the standard models of competition is that we

consider products whose value is endogenously determined by the value consumers derive from

what they can convey to others through their product use. It is therefore important to under-

stand the assumptions made on this value. In this section, we discuss how the consumer value

of projecting a certain image may be justified through the interaction consumers have with

each other. The idea is that each consumer is ultimately interested in interacting with another

consumer and the payoffs of this interaction depend on the type of the consumer she interacts

with. This justification is not novel to our paper: in the case of two consumer segments, it has

been offered by Pesendorfer (1995) and later also used by Kuksov (2007) and Yoganarasimhan

(2010). Although previous research examined such matching game in the context of marriage

market, one can see that it could be applicable in other situations, such as when consumers are

interested in forming teams to perform a certain task and the consumer ability to contribute

depends on her type.

To justify the consumer value of projecting a certain image of themselves, let us assume

that consumers are interested in social interaction with other consumers. To be specific, assume

that each consumer is interested in pairing up with another consumer with the payoff from

the pairing depending on the consumer types (this assumption can be extended to consumers

interested in forming a group of certain size Nwhere the payoff from forming a group to an

individual depends on the individuals type and the group composition). Specifically, normalize

each consumers payoff from pairing with a low-type consumer to zero and denote the low-type

consumers payoff of pairing with a high-type consumer byvL. Let theHj -type consumers payoff

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of pairing with an Hj-type consumer be vh and that of pairing with an H3j type consumer be

vh t, where vhU[vh, vh], and t >0 is a parameter.

Further, assume consumers are risk neutral in their search for a partner and pairing occurs

when two consumers agree to form a pair. While consumers know their own type, they only

observe the purchase decisions but not the types of other consumers. Therefore, they have to

make the pairing decision based on the product ownership alone. The consumers can then be

segmented into groups of consumers who made the same purchase decision. From the point

of view of an Hj-type consumer, these groups can be ordered by the expected utility to this

consumer of pairing with a random member of each group. Moreover, all Hj-type consumers

order the groups in exactly the same way. Thus, Hj-type consumers who belong to the highest

group will desire to be matched within the same group, and they will be preferred for matching

by members of their group. This means thatHj-type consumers in a lower group do not have

a chance of matching with the higher group and repeating the argument across the two lower

groups, we conclude that they end up pairing within their own group. The Hj-type consumers

of the lowest group are then only able to match within their own group as well. The same

consideration applies to L-type consumers as well. Thus, all consumers end up pairing within

their own group, either by choice or by availability, and the pairing outcome is equivalent to

consumers using the same product randomly matching with each other. 4

To complete the matching model formulation, it remains to define what happens if only a

single consumer buys a product. This is important for consideration of the purchase when in

equilibrium nobody may end up buying this product. Similarly to the model described in the

previous section, to avoid this ambiguity and the possibility that a consumer is in a group of

her own, we can assume that a zero-mass but infinitely many high-type consumers own each

product j .

One can easily see that the above defined game leads to the utility of projecting an image

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as we defined in the previous section.

MODEL ANALYSIS

Solving the model through backward induction, we first discuss the consumer choice sub-

game, which is the stage of consumer product choice taking prices as given, and then the price

setting stage. By solving the consumer choice subgame, we obtain the comparative statics re-

sults about the response of demand for one product to the price change of the other product.

Based on this result, we then derive the optimal firms response to a change in one of the firms

costs and the equilibrium results of the full game.

Consumer Choice Subgame and Comparative Statics with Exogenous Price Changes

Letpk be the price of product k, and letdjk be the proportion ofHj -type consumers among

buyers of productk . Then Hj-type consumer prefers product 1 to product 2 if and only if

(3) dj1v+ (1 dj1)(v t) p1 > dj2v+ (1 dj2)(v t) p2,

wherev is the consumers payoff from projecting Hj-type image. The above inequality simplifies

to (dj1dj2)t > p1p2.Sincev does not enter this condition, we have that either all consumers

of the same subtype prefer product 1 to product 2 or vice versa, or all of them are indifferent.Consider first a potential equilibrium where the demand for each product comes from both

high-type segments. Then, indifference equation (dj1 dj2)t = p1 p2 must hold for each

j= 1, 2. Usingd2k = 1 d1k, we obtain (d11 d12)t= p1 p2 and (d12 d11)t= p1 p2, which

is only possible whenp1= p2 and d11= d12= 1/2. It will be clear from the consideration below

that in this case at least one of the firms should strictly prefer to change its price. Also, this

equilibrium is not asymptotically stable in the sense of t atonnement (see Fudenberg and Tirole

1991, pp. 23-25), because if an arbitrarily small mass of consumers of either high-type segment

deviates to either not buying a product or buying the other product, then all consumers from

that segment would prefer the other product to the first one and all consumers of the other high-

type segment would prefer the first product to the other one. Therefore, a one-step iteration of

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optimal response from this deviation would result in each product receiving demand only from

one of the high-type segments. The demands would then adjust to satisfy Equation (5) below,

since as shown in the appendix (see proof of Proposition 1), the left and right hand sides of each

equation diverge as the demands diverge from the unique solution to the system.

Consider next a potential equilibrium where demand for one of the products comes from

both Hj segments, but the demand for the other comes only from one Hj segment. Without

loss of generality, let the latter segment be H2 and the product they prefer be Product 2. Then

d11= 1 and the system of equations on the equilibrium similar to Equation (8) of the monopoly

case yields d2kd1k. Therefore, such equilibrium is only possible when p1 p2t/2.

Thus, when prices are not equal and not too different (|p1p2| < t/2) and both productsreceive positive demand, the only possible equilibrium is for H1-type consumers to only buy

one of the products and for H2-type consumers to only buy the other one. In the case one of

the products, say product 2, receives no demand, the other products demand is then derived

similarly to the single-product monopoly case (see below). As established there, we have d11=

d12 = 1/2. It is then optimal for consumers in both of the high-type subsegments not prefer

product 2 to product 1 only if the price of product 1 is higher (regardless of the beliefs about

the consumer who deviates and buys product 1 as far as the belief is that this consumer is from

a high-type subsegment). This outcome is not asymptotically stable when|p1p2| < t/2, as

the tatonnement with deviation of arbitrarily small mass from one of the high-type subsegments

to buy product 1 would converge to the separating equilibrium discussed below (similarly to

the argument in the case p1 = p2). Also, this can only be an equilibrium of the full game if

p2 c2. With this rationale, let us concentrate on the possibility for both products to have

positive demand.

Without loss of generality, let us assume that H1-type consumers buy Product 1, and let

Dk be the demand for product k . Then the mass D0= 1 D1 D2 of consumers who are not

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buying consists ofH1, H2, and L type consumers in proportions ( D1)/D0, ( D2)/D0,

and/D0, respectively, while d11 = d22 = 1. Consider anHj-type consumer who faces payoffv

from pairing with anHj -type consumer. This consumer then finds it optimal to buy product j

rather than not buy either product if

(4) v pj > ( Dj )v+ ( D3j )(v t)1 D1 D2 .

Thus, the equilibrium conditions on the consumer demands are

(5)

v1 p1= (D1)v1+(D2)(v1t)1D1D2 ,v2 p2= (D2)v2+(D1)(v2t)1D1D2 ,

wherevj denotes vh of the marginal Hj-type consumer in the decision to buy product j and

(6) Dj = max

0,

vh max{vj, vh}vh vh

, j = 1, 2.

This system of equations turns out to be linear invj(the quadratic terms cancel). For its solution

to represent an equilibrium consumer choice, it is also necessary that incentive compatibility

constraints are met: none of the consumers of type Hj should prefer buying product 3 j to

buying product j, and consumers of type L should not find it optimal to buy either of the

products. The former condition requirespjp3j < t,and the latter is satisfied ifvj > vL. We

summarize the equilibrium conditions and the important comparative statics they imply as the

following proposition.

Proposition 1. Equations (5) and (6) define a (unique asymptotically stable) consumer choice

equilibrium when the solution to these equations satisfies vj

[vL, vh] and

|pj

p3j

| < t/2

for j = 1, 2.5 Under the above conditions, a reduction in price of either product results in the

increased demand for both products as far asvj > vh for each j = 1, 2, i.e., as far as neither

high-type subsegment is fully penetrated.

Proof. See Appendix.

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The comparative statics result in the above proposition are essential to make possible the

equilibrium result we show in the next subsection. The first part of the comparative statics

results is that if a price reduction of one product would not lead to an increase in its own

demand, decreasing price could never be optimal. This part of the comparative statics result is

intuitive but important, since it ensures that the own-price elasticity for a status good behaves

as one would expect in the case of utilitarian products. The second part of the comparative

statics results is more interesting and counter-intuitive since it says that the cross-price elasticity

can be negative as well, i.e., the products may act as complements. The intuition for this result

is that when the price of one product decreases, this product converts some of the high type

population from non-buyers to buyers. Although this does not change the consumer payoffs

when buying either product, it decreases the payoffs when not buying either product. Therefore,

the willingness to pay of all consumers for either product increases. Since product images are

differentiated in the equilibrium and the price difference is not high enough, consumers still

prefer to buy the product with a matching image to the other product, and thus, the demand

for each product increases.

Note that the above mechanism of sales of a product benefiting from the sales of the other

one is not the same as that for products that are complementary in the usual sense. In the

usual case of complementary products, a consumers valuation of a product increases if the

consumer also owns the other product. In our case, the marginal consumer in the decision to

buy or not to buy a product views the two products as (imperfect) substitutes even though

the incentive compatibility constraints are not binding. Note that if one would like to have a

model with the same results but where incentive compatibility is binding for some consumers,

so that a price reduction of one product would draw some demand from the other product, one

can easily do this as follows. To the setup of the model we have, add a small fraction of high

type consumers of each subtype whose vh part of the payoff is distributed the same as for the

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rest of the high type consumers, but whose t part of the preference is replaced by uniformly

distributed on [0, t]. Keeping the mass of these added consumers small would ensure that they

have very little effect on the equilibrium demands from the other consumers, but some of these

new consumers would be switching products no matter what the price change is.

Moreover, ifvh > 112 t, all consumers view the two products as (imperfect) substitutes. To

see this, first note that product 2 has no value for H1-type consumer if she already possesses

product 1. Further, consider an H1-type consumer with payoff vh of matching with another

H1-type consumer. If she does not own either product, obtaining product 1 would increase her

utility by

(7) U(buy) U(not buy) =vh

D1D0

vh+ D2

D0(vh t)

.

However, if she owned product 2, her replacing it with product 1 would increase her utility by

vh (vh t) = t. Substituting D1 = (vh vh)/(vh vh), which holds if the consumer with

the payoffvh is a marginal consumer, andD0 = 1 D1 D2 in Equation (7), one obtains that

owning product 2 decreases marginalH1-type consumers value of product 1. To see when this

holds for all consumers, note that Equation (7) is smallest when D1 = D2 = 0 and D0 = 1.Substituting these, one obtains that owning product 2 decreases the H1-type consumers value

of product 1 as far as vh> 112 t.Therefore, Proposition 1 can be summarized as implying that

two status goods which are substitutes in the consumer utility functions can act as compliments

in the profit function.

Equilibrium Prices and the Effect of Costs on Profits

We now illustrate how the comparative statics derived in Proposition 1 may lead to one

firm benefiting from another firms equilibrium decision on price reduction due to a decrease

in its marginal costs. The first order conditions on prices of the two firms are derived from

the demand equations (5) and (6). To make sure that the solution of the first order conditions

constitutes a unique equilibrium, we must also check that a) a deviation by either firm j to set

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price at or below p3j t/2 to possibly capture the demand from both high subtypes leads to

a lower profit then the price determined by the first order conditions, and b) it is not optimal

to set price as to sell to the low type consumers. If condition a) is not satisfied, the equilibrium

will be in mixed strategies since a pure strategy equilibrium would have to satisfy the first order

conditions. The solution to the first-order conditions is algebraically complex. However, one

can easily see that the pure strategy equilibrium has to exist when costs are sufficiently high,

so that the first order conditions would lead to prices that exceed costs by no more than t/2.

Therefore, we have the following proposition.

Proposition 2. If the equilibrium is in pure-price strategies (which holds when it is not optimal

for either firm to set its pricet/2below the other price even if it then captures all demand)6 with

both firms having positive sales, then a cost reduction of one firm strictly benefits both firms if

the first firm did not fully penetrate either high-type subsegment.

Proof. See Appendix.

To illustrate the equilibrium price and profit changes due to one firms cost change, consider

20% decrease of Firm 2s cost starting from t= 1, vL = 1, vh [2, 4], c1 =c2 = 2. Given these

parameters, if = 1/4 we have the following equilibrium prices and profits:

Prices Profitsc2= 2 (2.171, 2.171) (.0205, .0205)c2 = 1.6 (2.247, 1.993) (.0354, .0781)

On the other hand, if = 1/10, we have the following equilibrium prices and profits:

Prices Profits

c2= 2 (2.723, 2.723) (.0361, .0361)c2 = 1.6 (2.741, 2.535) (.0373, .0584)

The above example illustrates a curious possibility that profits may increase when the size of

the high-type segment decreases. This is possible because while the potential demand for a

status product increases with the high-type segment size, the value of the product decreases in

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that size (the payoff of not owning the product increases in the high-type segment size while

the payoff of ownership does not change).

Proposition 2 is a fairly straightforward consequence of Proposition 1: as far as v2 > vh, a

cost reduction of Firm 2 makes it optimal for it to lower its price to increase its penetration

ofH2 consumer segment, and this increased penetration increases the value of both products

to both H1 and H2-type consumers. Here we considered the case when Firm 2 increases sales

without finding it optimal to take away sales from Firm 1. In the next section, we show that

a stronger result may also hold: in the case the optimal monopoly price of Firm 1 is such that

it equally penetrates both high-type segments, a competitor entry may still increase profits of

Firm 1 even as the entrant takes away all the demand from one of the high-type segments. We

also show another implication of Proposition 1 on equilibrium prices: the prices are lower when

the two firms are horizontally integrated.

Note that since the beneficial effect of one firms price reduction on the other firm relies

on changing the distribution of non buyers, the effect is weaker when is smaller. However,

it never disappears. Note also that in reality, should not be necessarily interpreted as a

proportion of Hj segment in the total consumer population, but rather as a proportion only

in the population of consumers that are not distinguishable from Hj -type consumers by other

consumers except through the use of products under consideration. On the other hand, because

one firm penetration reduces the contribution of image valued atvhtby the other subsegment,

the effect is stronger when t is smaller. The profit implications of smaller are not clear-cut:

although profits must tend to zero as tends to zero, as we have noted on the example above,

a smaller may lead to higher profits.

Comparison of Duopoly to Single- and Two-Product Monopoly

In this subsection, we complete the model analysis by considering two other market struc-

tures: the one-product monopoly case and the case of horizontal integration of the two firms.

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The first case is theoretically equivalent to one of the firms having cost above value of all con-

sumers. In this section, we also analyze the case when the price of the existing firm is low

enough so that both H1- and H2-type consumers buy the product, which adds the possibility

that the entrant takes away half of the incumbents consumers. As we show below, it is possible

that the incumbent firm benefits from the competitors entry even in this case.

Let us start with the analysis of one-product monopoly and the possibility that the product

sells to both Hj subsegments. In this case, equating the payoff of the marginal consumer of

each high-type segment when buying the product and not buying it, we have the following

equilibrium conditions on the demandsD1 andD2 from each of the two high-type segments:

(8)

D1v1+D2(v1t)

D1+D2p= (D1)v1+(D2)(v1t)

1D1D2,

D2v2+D1(v2t)D1+D2

p= (D2)v2+(D1)(v2t)1D1D2 ,

wherep is the products price and vj is still related to Dj according to Equation (6). It turns

out that the solution to the above system is always unique and symmetric:

(9) v1= v2 =(vh vh) + 2p(vh vh)

(vh

vh

)

2p , and D1+ D2 =

2p (2vh t)(vh

v

h

)

2p

When this solution implies positive demand and satisfiesvj > vL + t/2, it represents the unique

consumer choice equilibrium with demand coming from both high-type consumer segments. The

latter condition is necessary and sufficient to make it optimal for the L-type consumers not to

buy the product. Note that the highest price that results in positive sales in this equilibrium

is just below (vh t/2) and is lower than the maximal symmetric price that achieves positive

demand in the competitive case.

There may also be a consumer choice equilibrium where the demand is coming from oneHj

segment only. That potential equilibrium would not have to satisfy Equation (8), because no

consumers in the segment with zero demand are indifferent between buying and not buying.

The following are the equilibrium conditions for such asymmetric equilibrium with demand D

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are rewarded not according to the total profit impact of their pricing decisions but according to

the individual profit contributions of their respective products.

To see how a monopoly selling two products should modify its pricing relative to that of

a duopoly, consider the following. If monopoly increases one of the prices without decreasing

another, the monopoly profit would decrease because increasing price was not optimal for com-

petitors and cross-effect is also negative. Therefore, the monopoly should find it optimal to

either decrease one of the prices relative to a duopoly or keep them the same. Furthermore, if

one of the products did not fully penetrate its high-type segment, the profits would increase if

its price decreases because the effect on the profit from its sales is second order, but the effect

on the profit from the sales of the other product is first order. Also, since equal costs result in

equal equilibrium prices, the above implies that in the symmetric-cost case, both prices will be

optimally reduced in the case of horizontal integration, and thus, the total demand will increase.

We summarize these results in the following proposition.

Proposition 4. When the competitive equilibrium is in pure price strategies, a horizontal in-

tegration will result in the following: (1) both prices remain the same or the price of at least

one product decreases; (2) whenc1 = c2, the total category demand increases if not all high type

consumers purchase one of the products, and stays constant otherwise.8

While the above result could be counter-intuitive without the background of the prior anal-

ysis, it does not come as a surprise after Propositions 1 and 2 are understood. The intuition is

that the joint profit-maximization takes into account the positive externality of a price reduction

which increases the demand for both products. In addition to the managerial implications of

the proposition we described above, one should also note its public policy implication: while

competition may be promoting lower prices and higher consumption in most product categories,

competition may be promoting higher prices and lower consumption in the categories of status

goods.

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The analysis above also shows that as opposed to a product cannibalization concerns in

utilitarian markets, in status goods markets, a firm has an extra incentive to increase product

line. This could be a force towards increasing product line length in status goods. 9

At the same time, given that the differentiation is through image, product line extensions

may run a risk that consumers will confuse two products if they are coming from the same

manufacturer (or at least, if they have the same brand name). This raises another interesting

question: how imperfect observability of status good possession could change the dynamics of

a status goods market. This issue becomes especially relevant when brand counterfeits exist

for status goods. Applying the same line of reasoning about benefits of penetration, one could

imagine that if multiple consumer types are vertically heterogeneous, a counterfeit product

might even help the brand name one as it could separate medium-type subsegment from the

low-type one.

Product Differentiation in Physical Attributes

Although we have modeled differentiated product competition through high-type consumers

horizontal preference heterogeneity for the image to project, one could have alternatively as-

sumed that high-type consumers have heterogenous value for the functional product attributes.

For example, one could assume that a Hj-type consumer has value vh of projecting both the

image ofH1 type and that ofH2 type (i.e., t does not enter the image value), but in addition

derives functional value t from product j but zero from product 3 j. Our main results are

robust to this model modification. In fact, the benefit of one products price reduction on the

demand for the other product would be even stronger, because penetration of a product would

be taking out equally desirable consumers rather than less desirable ones from the population

of consumers who do not own a product. However, the model we developed shows that one does

not need status products to be functionally differentiated to obtain this and other results. It also

shows how in markets where self-expression is an important consideration, firms may become

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differentiated not due to firms actions but due to consumer choices. In similar vein, Kuksov,

Shachar, and Wang (2012) also show that the firm may in fact prefer to abstain from actively

creating image through advertising to facilitate the endogenous formation of image through

consumer communication. At the same time, interaction of functional and image differentiation

is also an interesting phenomenon to explore further. For example, in this regard, Muller and

Shachar (2006) argue that lack of functional differentiation could be a result of firms ability to

differentiate on a self-expressive attribute.

EXPERIMENTAL VALIDATION

This section describes several experiments supporting our theoretical model. First, we ex-

perimentally validate the key assumption of our model that the value of the product depends

on the type of consumers who own it and those who do not own it. Second, we experimentally

validate the result of Proposition 1 that as the price of one product declines, the demand for

both products increases.

Confirming Assumptions: Product Ownership Distribution and Perceived Product Value

One of the key assumptions in our theoretical model is that the value of a product used

by consumers to signal their type is endogenous and determined by the pattern of product

ownership: the proportion of product owners who are of the desirable type and the proportion

of non-owners who are of the desirable type. Specifically, the assumption of endogenous signal

value implied that the value should increase in the first proportion and decease in the second.

As we have discussed, this assumption is consistent with consistency of beliefs and outcomes

imbedded in the notion of Nash Equilibrium as well as reflected in some consumer behavior

and theoretical literature. To provide additional support for this assumption, we conducted a

consumer preference survey to examine the effect of proportion of different type of owners on

the perceived product value.

Eighty participants from a national Internet panel were randomly assigned to one of two

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conditions and completed a consumer preference survey in exchange for a $5 gift certificate.

Participants were asked to imagine that they are considering whether to purchase a luxury wrist

watch to wear at the final round interview for a high-paid management job at a company. They

were then informed about the percentage of managers and other employees at that company

who wear a luxury watch at work. Specifically, the following information was provided to

participants in the high-percentage of high-type owners (low-percentage of high-type owners)

condition, henceforth referred to as Condition 1 (Condition 2): You consulted a friend

of yours, who works at the company that you will interview with. The friend told you that

although almost never a choice by other employees, luxury watches are popular (somewhat

popular) among managers at the company with roughly four (one) out of ten of them seen

wearing one at work. Participants were then asked to indicate how much they desire to wear

a luxury watch at the interview on a 10-point Likert scale (1 = not at all desire; 10 = very

much desire). Note that although in either of these two conditions product ownership identifies

the employee as a manager, the absence of product ownership implies higher probability that

the employee is a manager under Condition 2 than it does under Condition 1. Therefore, our

prediction was that the desirability of product ownership would be higher under Condition 1.

The analysis of variance with the mean desirability evaluation as the dependent measure

indeed yielded a significant main effect such that participants in Condition 1 rated the luxury

watch as more desirable than the participants in Condition 2 did (MCond 1 = 5.88 vs. MCond 2=

4.50; F(1, 78) = 4.72, p < .05). This finding is consistent with our assumption that the perceived

value of the product increases when the proportion of the desirable consumers among non-owners

decreases.

To test whether the value of the product increases when the proportion of the undesirable

consumers in the total customer base of the product decreases (i.e., when owning a product

is stronger signal of being a high-type), we conducted the following manipulation: keeping

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constant the proportion of high-type employees (managers) wearing a luxury watch at one out

of ten, we varied the proportion of other employees wearing luxury watches from almost never

(henceforth, Condition I) to one out of ten (henceforth, Condition II). If our assumption

holds, the perceived value of product should be higher in Condition I when other employees

almost never wear a luxury watch. A separate set of 40 participants, randomly drawn from

the same Internet panel, went through the same procedure as participants in the first study

with the modification described above. The analysis of variance again yielded a significant

main effect: the participants in Condition I rated the luxury watch as more desirable than the

participants in Condition II (MCond I = 4.50 vs. MCond II = 3.19; F(1, 78) = 4.53, p < .05).

In summary, the data from the consumer survey is consistent with the assumptions of our

theoretical model.

Note that one explanation for the increased value in the first manipulation is the herding

effect: a products valuation may increase just because more people desire it (Becker 1991,

Zhang 2010). However, the second manipulation show that this is not exactly the full story:

when the product use increases due to the less desirable type of consumers, the valuation actually

decreases.

Testing Predictions: Experimental Validation of Proposition 1

The model implications derived in the previous section are crucially based on the consumer

choice predictions of Proposition 1 that as the price of one product declines, the demand for

the competing product increases. As this prediction might seem counter-intuitive, one may

ask: would consumers follow the optimal strategy, or would they tend to lower their choice

probability of a product if the price of the competing product declines as they would behave in

a regular goods market? In this section, we describe an experiment that lends support to the

consumer choice prediction of Proposition 1.

Model modification for an experiment. In the model, we assumed an infinite number of

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consumers of each type and a uniform distribution of payoffs for the high type consumers. Since

an experiment has to consist of finitely many consumers, we have to modify these assumptions

and re-derive the equilibrium. To adapt the uniform distribution of payoffs on [vh, vh] to a finite

consumer number, we use a discrete set of payoffs rounding them to an integer for simplicity:

vi= vh + i(vh vh)/N (i= 1,...,N) whereNis the number ofHj-type consumers. Specifically,

we used = 1/4, N= 7 (i.e., seven consumers per Hj segment and 14 low-type consumers) and

payoffs of pairing with high-type consumers defined as presented in the following table:

Consumer index 1 2 3 4 5 6 7Payoff when pairing with the same subtype 34 38 42 47 51 56 60Payoff when pairing with the other subtype 14 18 22 27 31 36 40

This distribution of payoffs across consumers approximates the uniform distribution from

vh = 30 to vh = 60 and t = 20. We numerically checked that for prices (p1, p2) = (33, 28) and

(33, 35.5), the unique consumer choice equilibrium demands are (D1, D2) = (6, 7) and (3, 2),

respectively. In other words, similarly to the main model setup, our theoretical prediction in

this model adaptation to finite consumer number is that when the price of one of the products

reduces from 35.5 to 28 and the price of the other product is held constant at 33, the demand

for both products increases.

Experiment setup. The objective of the experiment was to trace the change in the demand for

one product in response to the price change of the other product. Therefore, we kept the price

of one product constant at 33 francs, while we changed the price of the other product between

28 and 35.5 francs. The prices were manipulated within participants. In the experiment, we

define three types of consumers, Type 0, Type I and Type II, which corresponded to L,H1

and H2 type consumers, respectively, in our theoretical model. In each experimental session,

seven Type I buyers and seven Type II buyers are played by the experiment participants. The

purchase decision of the Type 0 buyers was straightforward because their values of matching is

defined to be so low that buying neither product is always optimal for them regardless of what

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they thought about the decisions of the other buyers. Therefore, we automated the 14 type 0

buyers to reduce the number of subjects needed.

We recruited 56 undergraduate students from a subject pool of a mid-western US university

to participate in four such experimental sessions and paid them a monetary award contingent

on individual performance. On average, the participants ended up earning approximately $15.

During the experiment, we used a hypothetical currency unit called franc convertible to the

dollar payoff at the rate of 100 francs per US $1 at the end of the experiment.

When participants arrived at the lab, they were given an information sheet describing the

purpose of the study. They were told that the study aims to explore how prices affect consumer

purchase decisions in a competitive status goods market where consumers use products to signal

their type to other consumers. They were then informed that during study they will be given the

payoff matrix in which their payoff from purchasing a status good depends on their and other

consumers purchase decisions, and their objective is to make a series of purchase decisions to

maximize their payoffs in the computerized study given the price information presented. After

that, participants were told that at the beginning of the study, they have 100 francs each which

they can use to purchase one of the two products or neither. Before they started the first trial

of the session, they were informed about the number of buyers of each type in the experiment,

their own type (randomized between Type I and Type II), and that the formula for their payoffs

is derived from the inference one can make of their type from their purchase decision given the

realized distribution of choices across participant types.

Further, the subjects were given written instructions, which included the following: There

are three types of buyers: Type 0, Type I and Type II. You are a Type I (II) buyer. There are 14

Type 0 buyers, 7 Type I buyers (including you [if applicable]), and 7 Type II buyers (including

you [if applicable]). The type-0 buyers are computerized and never buy either product, while the

type I and type II buyers are played by actual people like you. You will be repeatedly presented

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with prices of two products, and your task is to decide whether to purchase a product, and if

so, which one to purchase. In each period, after all buyers make their choice, you will be told

the choice of other buyers (the number of Type I and Type II buyers who purchased product 1

and product 2) and you will receive your payoff which will be calculated based on the following

formula (for this purpose, the computers of all experiment participants communicated with one

central computer with the master program):

(12) Payoff = w1M1+ w2M2M0+ M1+ M2

P,

where w1 is your value of matching with Type I buyer, w2 is your value of matching with

Type II buyer, M0, M1 and M2 are the numbers of Type 0, I and II buyers who have made

the same purchase decision as you, and P is the price of the product chosen. If you choose

not to purchase, P = 0. Type 0 buyers are computerized and always choose not to purchase

either product. Type I and Type II buyers are all actual players in this lab. We provided a

software calculator to each participant for calculating the payoff of Equation (12) conditional on

their predictions of purchase decisions of other consumers. After that, they were informed (on

the computer screen) about their individual values for w1 and w2, which were different acrossparticipants as reported above. They were then told that these values would remain the same

until a change was announced.

To allow participants to become familiar with the structure of the game, they were required

to play five practice trials. They were told that the purpose of the practice trials are to

familiarize themselves with the buttons and the visual presentation of the software and that

the results of these trials would not affect anything else. Then the game was restarted for the

actual trials that would be counted towards their payoffs. We kept the prices constant for 20

trials, before we changed one of the prices. After another 20 trials, we restarted the game again

switching the type of each participant from type I to type II, and vice versa, and switching the

participants values ofw1 and w2 from high to low, and vice versa. The participants were told

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that their weights and types will be re-assigned, but were not told the exact rules of how we

switched these values. We then ran another 40 trials under the new setup, again changing one

of the prices right in the middle of these 40 trials. The purpose of restarting the game was to

allow for more independence between trials: as one can see from the equilibrium predictions, it

is the consumers with the lower valuation who are supposed to switch their behavior. For the

exact price sequence manipulation in each experimental session, please refer to Table 1.

Insert Table 1 about here.

As Table 1 shows, the product which price was changed and the price trend whether it was

increasing, i.e., when the price of one product increased from 28 to 35.5 francs, or decreasing, i.e.,

when the price of one product decreased from 35.5 to 28 francs were counterbalanced across

the four experimental sessions so that we could test whether our prediction holds regardless of

these two factors. At the end of each session, the cumulative payoff was calculated in francs

and then converted to U.S. dollars, to be mailed to the participants in checks later. Then, the

participants were debriefed and dismissed.

Data analysis and results. Table 2 reports the average realized demand in each of the four

experimental sessions.

Insert Table 2 about here.

When the price changed from 35.5 to 28 francs for one product, its own demand increased on

average from 1.25 to 5.69 units. In other words, the own-price elasticity is negative, which is

intuitive and expected. The average demand for the other product with price held constant

throughout the experiment also increased from 2.51 to 3.61 units, as predicted by the model.

This change is also statistically significant (F(1,318) = 50.35, p < 0.0001) and consistent with

the prediction of Proposition 1. Moreover, as reported in Table 2, when examining the average

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demand within each experimental session for the product with constant price, we obtained

similar results. The significance level of the demand changes for the product with unchanged

price when the price of the other product changes is p < 0.0001, p < 0.001, p < 0.05, and

p 0.3993), is significant.

To further understand which consumer type the increased demand is coming from, in each of

the 20-trial experiment segment with constant prices, we define each products consumer type

to be the type of consumers predominently associated with this product, i.e., the consumer type

which results in the majority of demand for this product. We then split the demand for each

product into that coming from its consumer type and that coming from the other consumer

type and obtain the following. With a reduction of price from 35.5 to 28 francs, the average

demand for the product from its consumer type increased from 1.12 to 5.38 units, and from 0.13

to 0.31 from the other consumer type. At the same time, the average demand for the product

with constant price increased from 2.33 to 3.45 unit from its consumer type, while the demand

for it from the other consumer type decreased from 0.18 to 0.16 units. In other words, we find

that which consumer type generated extra (or less) demand for each product when one of the

products price changed is also consistent with our model prediction.

Although the equilibrium of the empirical model makes point prediction about consumer

demand, we observed that actual demand varies quite a bit across trials in the experiment.

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Figure 1 plots the frequency distribution of demand for the product with constant price over

320 trials across the four experimental sessions.

Insert Figure 1 about here.

In equilibrium, the demand for this product should be three units when the price of the other

product is high (i.e., at 35.5 francs). We observe that under this condition, the actual demand

for the product ranges from 0 to 7 units, with mean = 2.51 and median = 3. If the price of the

other product is low (i.e., at 28 francs), the demand for the product should be six units, based

on the prediction of our empirical model. The observed demand ranges from 0 to 6 units, with

mean = 3.6 and median = 4. Thus, the actual demand is lower than theoretically predicted.

One possible explanation is that not buying may be considered as a safer choice because buying

a product could generate negative payoff under the structure of the game. Another potential

reason for this is related to the question which consumer type should correspond to buying which

product. Due to symmetry, there are two equilibria (leading to the same demand predictions):

one in which Product 1 is bought only by Type I consumers and one in which Product 1 is

only bought by Type II consumers. Thus, in the first few trials of each experimental session,

we normally observed the demand for the lower priced product coming from both Type I and

Type II consumers. Such mixed demand decreases the value of the product and therefore the

demand for it. Remind that in each experimental session, we ran four sequences of 20 identical

back-to-back repeated trials for a given price combination.

In the sequences where one of the prices was low (28 francs), the perfect Type/Product

correspondence always emerged by the end of 20 repetitions with Product 1 receiving only

demand from Type I participants in 5 out of 8 sequences. In the sequences where one of the

prices was high (35.5 francs), the perfect Type/Product correspondence emerged in 7 out of 8

sequences with Product 1 receiving demand from Type I in 5 out of these 7 sequences. In two

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statics that one products price reduction could lead to increased demand for the competing

product. A lab experiment where participants act as consumers and make a sequence of pur-

chase decisions under a game structure similar to our model setup, empirically validates this

prediction. Second, based on this comparative statics, we show that in a status goods market,

a firm may benefit from the entry of a competitor. This result provides an additional justifica-

tion for co-location of different specialty stores. Furthermore, we show that a firm may benefit

from a competitors cost reduction, and the optimal price response to a price reduction by the

competitor is often to increase own price.

One of the uncertainties in a status goods market with multiple potential status products

available is whether consumers would choose a given product as means of signaling their status to

other consumers. Bagwell and Bernheim (1996) show that when the price of a product is reduced

below a certain limit, this product may result in consumers dropping it from being considered

as a status good. In this case, a price reduction by one manufacturer could benefit the other

manufacturer by essentially making it the monopolist in the status goods market. However, in

this case, the positive correlation between one products price and another products demand

would not be observed in equilibrium where firms choose prices because a firm would not choose

to reduce the price if doing so would lead to lower demand for its product. On the other hand,

in our model, a lower cost makes it optimal for the firm to reduce the price and increase its

penetration of the market, and this increased penetration increases consumer valuation for both

products, which in turn, helps both firms. In other words, in our model, an equilibrium price

reduction of one of the products increases value of each of the products rather than destroys

the value of the product with the lower price.

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FOOTNOTES

1 See www.slideshare.net/razzy/luxury-car-brand-positionings-by-dan-pankraz?from=share email.

Not surprisingly given this BMW brand image, BMW topped the 2010 ranking of the most desir-

able brands in China ahead of Apple and Rolex (see http://performancecarpartz.com/?p=222953).

2 Micro-modeling the dependence of the value of a status good possession as coming from its

social function allows us to endognenize the functional form of this value. In particular, such

value then exhibits both snob and bandwagon effects (Bagwell and Bernheim, 1996). We also

show that this value of other consumers beliefs can be endogenized through a matching game

where consumers decide whom to pair up with based on the information they have and receive

payoffs depending on the the type of consumer they have paired up with.

3 Kuksov (2007) also adopts the same definition to analyze the consumer value of brand image

in a horizontally heterogeneous consumer population, i.e., in the market of self-expressive/image

products that do not necessarily convey the message of status.

4 Burdett and Cole (1997) demonstrate how a similar argument works with a continuum of

types when potential partners arrive through a Poison process and consumers have a disutility

of waiting. In that case, consumers pair up not with exactly the same type but rather the

population splits into classes within which consumers end up randomly pairing. Alternatively,

Pesendorfer (1995) immediately starts with the assumption that after the consumer purchase

decisions, consumers who made the same purchase decision are randomly paired up.

5 The only other (not asymptotically stable) equilibria are: 1) when p1 = p2, both products

receive demands in equal proportion from both Hj segments and 2) one of the products (the

one with the higher price if the prices are not equal) receives zero demand while the other one

receives demand fromHj subsegments in equal proportions.

6 For example, this condition trivially holds when the solution of the first order conditions for

prices leads to prices at mostt/2 above the lower of the two firms costs.

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7 It is possible to show that the symmetric consumer choice equilibrium above is stable if and

only if this asymmetric one does not exist. The two asymmetric equilibria (mirror of each other),

whenever exist, are always stable. However, we will not need to use the asymptotic stability

refinement for the propositions that follow.

8 Numerical simulation shows that this result holds for asymmetric costs as well. However, we

are unable to prove this analytically.

9 There are many other issues affecting product line decisions, such as the expense the firms have

to incur to communicate product line with the consumer (Villas-Boas 2004), and distribution

channel issues (Villas-Boas 1998; Liu and Cui 2010).

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Table 1: The Sequence of Price Changes in the Experiment (p1, p2)

Trial 1-20 Trial 21-40 Trial 41-60 Trial 61-80

Session 1 (33, 35.5) (33, 28) (33, 28) (33, 35.5)Session 2 (28, 33) (35.5, 33) (35.5, 33) (28, 33)

Session 3 (35.5, 33) (28, 33) (28, 33) (35.5, 33)

Session 4 (33, 28) (33, 35.5) (33, 35.5) (33, 28)

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Table 2: Average Demand in the Experimental Sessions

Product with price change Product with constant price

Sessions SessionsPrice 1 2 3 4 All 1 2 3 4 All

28 5.75 5.33 5.68 6.00 5.69 3.98 3.78 3.13 3.53 3.61

35.5 1.25 1.10 0.90 1.73 1.25 2.25 2.43 2.63 2.73 2.51

Notes: the price listed here is the price of the product (in francs) which the price was changed in

the experiment. The price of the other product was kept constant at 33 francs.

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Figure 1: Distribution of Demand for Product with Unchanged Price

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7

Freqency

Demand

LOWPricefortheOtherProduct

HIGHPricefortheOtherProduct

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APPENDIX

Proof of Proposition 1. The main text already establishes that when Equations (5) and (6)

result in |p1p2|< tandvj < vLforj = 1, 2, the solution to the equations defines an equilibrium

E0 with demand for productj only coming fromHj segment. Furthermore, since the solution isunique, whenvj(vh, vh), the solution is a unique equilibrium with some but not all consumers

of eachHj segment buying productj .

To prove the comparative statics part of the proposition, we need to show that the derivative

of demand of one of the products, say product 1, with respect to the other price is negative

given Equation (2). The derivative ofD1 with respect top1 simplifies to

(13) D1

p2=

2(p1 t)[v h+ t vh][(vh vh) (p1 t) p2]2[(vh vh) t]

m,

which is negative because: p1 < t would result in v1 < vh; Equation (1) states vh+t > vh;

the first term in the denominator is squared, and the second term

(14) (vh vh) t= (v h+ t vh) + 2(vh t)> 0,

since both of the terms in the parentheses above are positive when Equation (2) is satisfied.

To prove that there are no other equilibria with demand for product j coming only fromHj

subsegment (i.e., no corner solutions), it suffices to prove that asvj changes andv3j is kept to

satisfy the solution of the (3j)-th equation of s

competition in a status goods market_jmr.11.0005

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