If Many Seek, Ye Shall Find: Search Externalities and
New Goods⇤
Maciej H. Kotowski†
Richard J. Zeckhauser‡
August 22, 2016
Abstract
Consumer search serves productive roles in an economy with multiple goods. In
equilibrium, search promotes the sorting of consumers among producers, thereby enabling the market for new goods, and potentially increasing welfare and profits above
the benchmark case (an economy with a single good, hence no search). When competitors are few, additional direct competitors may benefit a firm, as more sellers may
encourage more consumers to search. In return, consumer search entices producers of
new goods to enter. Neither of these externalities, nor the coordination problems faced
by consumers and producers, is appropriately recognized in the literature.
Keywords: Consumer Search, Product Differentiation, Price Dispersion, Imperfect Information, Search Externalities, Insurance
JEL: D43, D83
We are grateful to the editor and the referees for suggestions that greatly improved this study. We are
grateful to Daniel Hojman for stimulating discussions on this and related topics. Audiences at the 2014 CEA
meeting (Vancouver), the 2014 European Summer Meeting of the Econometric Society (Toulouse), and the
2015 AEA Meeting (Boston) offered appreciated constructive feedback. Part of the research was completed
when Maciej H. Kotowski was visiting the Stanford University Economics Department. He thanks Al Roth
and the department for their generous hospitality.
†
John F. Kennedy School of Government, Harvard University, 79 JFK Street, Cambridge MA 02138.
E-mail: <[email protected]>.
‡
John F. Kennedy School of Government, Harvard University, 79 JFK Street, Cambridge MA 02138.
E-mail: <[email protected]>.
⇤
1
“Too few competitors.” That is hardly a complaint of firms in the economics literature.
However, contrary to both the literature and standard intuition, a firm may suffer if there
are too few other firms producing its product. This surprising conclusion applies in the
not unusual case where two or more different goods compete, and consumers need to find
the producers of a product they desire. For a viable market to emerge and to be sustained
requires an appropriate interaction of consumers and producers. Sellers enter the market
confident that when their numbers are sufficient, well-matched buyers will search to find
them. Buyers will search only if the goods they want are available and sufficiently easy to
find.
The following analogue of a common situation illustrates our argument. Juan owns and
operates a taxi in a large city. Since taxis are alike in terms of quality and price, consumers
rationally hire the first available cab. Taking this random matching as given, for argument’s
sake assume that each driver makes about thirty trips per day. Suppose Juan invests in
a more comfortable car that will be appreciated by a third of the city’s taxi users. These
customers are willing to pay a premium for the added comfort of a luxury vehicle and Juan
charges them accordingly. For simplicity, say Juan needs to make only eighteen trips per
day at this higher price to recoup his investment.1 At first glance, Juan’s entrepreneurial
investment seems like a wise business move. He intuited a large untapped market, and he
went after it.
Despite the large latent demand for Juan’s unique service, we argue that he is unlikely
to be successful. Why? Although in aggregate many consumers want added comfort, Juan
is unlikely to encounter enough of them in a typical day. Indeed, unless comfort-conscious
consumers alter their behavior and purposefully wait or search for his car, Juan will complete
only ten trips per day and will fall short of his goal.2 When there are very few luxury taxis
in the city, even comfort-conscious consumers rationally take the first available cab. If the
chance of encountering a fancy taxi is very small, why incur the cost of waiting?
Perhaps paradoxically, Juan needs more direct competitors to stand a chance of being
successful with his new vehicle. If there were more luxury cabs, some consumers would wait
since they could expect that a deluxe taxi would be around shortly. If enough consumers
think it worthwhile to wait, Juan would be able to succeed. However, the market’s outcome
1
Although less common in North America, in many cities there is sizable variation in both the quality of
vehicles in the taxi fleet and in the fares they charge. For example, in Singapore (mid-2013) the flag rate for
a basic Toyota taxi is around S$3.00. This charge increases to around S$4.50 for a Mercedes taxicab.
2
Only one third of the thirty consumers whom Juan encounters randomly are willing to pay the higher
posted price.
2
is still far from obvious, as competition is a double-edged sword. While more luxury vehicles
may tempt consumers to seek out such taxis—effectively creating the market for this service
where none existed before—the increased competition from other drivers cuts into profits.
Equilibrium demands that these effects balance gracefully.
One might initially think that Juan’s problem is related to the excessive costs of effective
advertising,3 or with the difficulty consumers may have in contacting him specifically for a
ride. But, consider the extreme situation where Juan drives for Uber or a similar, internetenabled, ride-sharing platform. Would a typical consumer even consider the on-demand
luxury car service a realistic option if there was but one driver serving the entire city? Likely
not. Rather, Juan’s success requires the influx of other drivers offering a similar service and
who can also be found with similar ease. That is, he needs more competition.
The benefits of more competitors apply even when each supplier is providing a slightly
different good. Thus, a file-sharing service that seeks to compete with Dropbox may benefit if
a few other similar services enter the market. This makes it more likely both that consumers
will be aware of alternatives generally, and they will believe it worthwhile to seek a service
more appropriate to their needs in terms of capabilities and price. Online streaming music
and dating/match-making applications exhibit the same characteristics. In these cases, advertising is insufficient to adequately inform consumers about each product’s characteristics.
Consumers must search and experiment to find the product that works best for them.
This article studies large, almost competitive markets where consumers incur a cost to
find the producers of desired products. It addresses the conditions under which competition,
coupled with consumer search behavior, can create a market for new goods. When it does,
both consumer welfare and firms’ profits can rise. We argue that successful markets require
that both sides take costly actions. Firms must invest to enter a market while consumers must
undertake a proactive and costly search to seek out desired goods. The improved sorting of
consumers due to their search behavior yields an efficiency gain that producers and consumers
share. We have already sketched the essence of our argument in the market for transportation
and online applications. Similar conclusions apply, for example, to restaurants, professional
services, and a range of products capitalizing on 21st century technologies. An example at
the end of our essay applies our framework to insurance markets subject to adverse selection.
We assume that consumers must search to learn producers’ offers. As noted by Diamond
(1971), the existence of even a small search cost can seriously impede a market’s operation.
3
When a seller does advertise, consumers receiving and responding to the advertisement can go directly
to him. Others’ search is not directed and random as we assume.
3
In an outcome later dubbed the “Diamond paradox,” the market’s only equilibrium involves
all firms charging the monopoly price. Applied to our motivating discussion, in this outcome
all taxis charge the same high price for standard service. If consumers believe that all taxis
are the same, it is easy to appreciate the situation’s self-fulfilling logic.
This undesirable outcome is neither theoretically inevitable, nor often observed in the
real world. It can be avoided if there is consumer-level heterogeneity and the potential for
multiple goods. We take the simplest case where producers can supply one of two goods. For
example, a taxi may be either a standard or a luxury vehicle, as in our motivating vignette.
Initially, each consumer knows the product offered by a local, nearby supplier and he knows
the distribution of available products in the economy as a whole. If he finds the nearby offer
acceptable, he takes it. If it is unacceptable, he searches for a better deal.
We investigate two related questions. First, we consider the welfare and profit consequences of search and competition. When consumers search for a desired product, they
expand the demand faced by that product’s producers. Increased demand enhances the
incentives of firms to provide the product and, as the market expands, this virtuous circle
reinforces the effectiveness of consumers’ search efforts. By this process, search behavior generates a market-creation externality operating through the self-sorting of consumers
across producers of different goods. This externality, we believe, has been neglected in previous examinations of markets where consumers search for desired goods. (We discuss the
related literature below.) To highlight one consequence, even though consumer search is
costly, and their search tamps down producers’ market power, the resulting equilibrium can
Pareto-dominate the benchmark case. Compared to Diamond’s single-good, monopoly-price
equilibrium, multiple goods flourish in the market; consumers enjoy greater expected surplus;
and firms earn higher profits.
How might such an equilibrium come about? That is our second focus and it sounds
two recurrent themes—product differentiation and price dispersion. We construct equilibria
where multiple goods trade at multiple prices and we show how the combined incentives
offered by product differentiation and price dispersion support the appealing welfare consequences noted above. Two considerations render this conclusion non-trivial. First, consumers must have the incentive to search. Hence, some firms must promise consumers a
good deal. Second, and cutting against the first point, firms face a free-rider problem when
consumers are already searching. No firm wishes to be the one that promises the best deal
as it can often get away with leaving just a little less surplus to searching consumers than
other firms. However, if all firms succumb to this temptation the constellation of incentives
4
supporting the market unravels—consumers’ reason to search evaporates. In this regard, our
story goes beyond a simple coordination game. One side (firms) faces a steady incentive for
miscoordination among its members. Reconciling these opposing forces is the focus of our
technical analysis.
Consumer advocates often encourage people to “shop around” for a better deal. Many
fail to follow that advice.4 In some markets search offers little immediate gain. Hence,
as in the case of taxi services in Juan’s city when there are few luxury cabs, even luxuryseeking consumers may rationally eschew search. On the other hand, in some markets
extensive search appears to be consumers’ modus operandi. The market for air transport or
packaged vacations is one example, at least for many travelers.5 Our study emphasizes the
transition from one search regime to another. In our framework, it is a lost cause for one
or a few consumers alone to search for a better deal. Rather, welfare gains come only when
many consumers search simultaneously. Although a search-intensive equilibrium may appear
wasteful at first glance—many consumers are pursuing economic rents all the while incurring
direct search costs—we argue that by changing producers’ incentives Pareto-improving gains
can be realized. Producers’ incentives change on two margins. First, they face pressure on the
intensive margin as better-informed consumers curtail their market power. Second, producers
gain, now profitable, new opportunities on the extensive margin. When consumers search,
they self-sort among producers thereby creating an opening for new products and services.
When the second effect predominates, efficiency is enhanced, and welfare and profits can
both rise.
Three related claims constitute the primary economic contributions of our paper. First,
the benefits of costly search activity are intimately tied with the aggregate prevalence of
search behavior. Searching alone is often ineffective as a desired product or good is likely
not on offer. If many search, desired products will spring up. Resolving this coordination
problem and moving an economy from a low-search to a high-search equilibrium often yields
benefits for consumers and producers alike. To sustain such coordinated action, firms must
ensure that consumers’ costly activities are worthwhile. Thus, and second, firms often benefit
from direct competitors. When this is the case, each consumer is sure that his investment
4
For examples see Pratt et al. (1979). Even for goods that are standardized, relatively expensive, repeat
purchases, and for which search is in principle easy, prices can vary dramatically. For example, Medigap
insurance—a product to cover co-payments and deductibles in Medicare—is standardized by the government.
In 2003 in Colorado, the monthly premiums for a 75-year-old for Plan C coverage ranged from $58 to $271
per month. See Cutler and Zeckhauser (2004).
5
Actively searching in these markets may be worthwhile as price dispersion is common (Borenstein and
Rose, 1994; Gerardi and Shapiro, 2009). See also Sengupta and Wiggins (2014).
5
is not in vain. Third, intensive search behavior among consumers facilities the introduction
of new goods to the possible benefit of all. Despite the costs and inconvenience of search,
consumers gain from the new product’s presence. Suppliers of the novel good gain from
their new commercial success. And, most surprisingly, suppliers of old products can gain as
well. They can now take advantage of the enhanced sorting of consumers among products
offered in the economy. Our fourth contribution flows from a narrower technical perspective.
We identify necessary and sufficient conditions for the existence of multiple equilibria in our
model and, in particular, the existence of an equilibrium featuring price dispersion among
all goods in the economy and search by all types of consumers. As we explain below, it is
precisely in such cases that beneficial outcomes described above are most likely to occur.
After providing context for our study in Section 1, we introduce our model in Section
2. Sections 3 and 4 present the equilibrium analysis. We sketch extensions to our model
in Section 5. To highlight but one variation, we extend our model to allow for adverse
selection, which we then use to study insurance markets. We show how search activity can
sustain welfare- and profit-enhancing insurance products through better consumer sorting.
An appendix collects proofs and other technical remarks.
1
Related Literature
Since Stigler (1961) and Rothschild (1973), a large literature has examined consumer search
and market equilibrium. The findings can be quite surprising. As Diamond (1971) showed,
even a small search cost can squash any possible benefits from competition and lead to an allcharge-the-monopoly-price outcome. We take the monopoly-price outcome as the benchmark
case, and investigate how price dispersion and product differentiation can together support
Pareto-superior outcomes when consumers search. The equilibria we construct depend on
the combined presence of search externalities and complementarities between consumers and
firms. Our study relates to prior contributions on these three dimensions, as explained below.
Price Dispersion Price dispersion is a commonly observed phenomenon and many theories have been developed to explain its presence.6 Reinganum (1979) focuses on firm-level
heterogeneity. Stahl (1989, 1996) considers differences among consumers in their search
costs. Burdett and Judd (1983) propose a non-sequential/noisy search process. More gener6
We only cite a representative sample from this large literature. Stiglitz (1989) and Baye et al. (2006)
provide surveys. See also McCall and McCall (2008, Chapter 10).
6
ally, Burdett and Judd (1983) observe that the key driver behind price dispersion in nearly
all models (and likely in practice as well) are differences in consumer’s ex post information
concerning the prices charged by firms. This will be true in our setting as well. In our model,
as in Diamond (1987), differences in consumers’ willingness to pay for a product help to drive
dispersed prices. Some consumers will be more keen to search for cheap or new products
than others. Albrecht and Axell (1984) exploit the same type of heterogeneity to support
wage dispersion in their model of search unemployment. In contrast to these models, our
setting allows firms to further specialize in producing different products. This cross-cutting
dimension is critical for many of the beneficial welfare conclusions that we identify.
Though anecdotal evidence for price dispersion abounds, systematic documentation across
a variety of goods is less common. Recently, Kaplan and Menzio (2014) have documented
the patterns of price dispersion across a wide range of consumer products. Their data allow
for a decomposition of variance in prices and they find that search frictions account for 35
to 55 percent of the price differences observed across identical goods. They attribute the
bulk of the remainder to inter-temporal price discrimination. Our analysis focuses on search
frictions.
Product Differentiation Wolinsky (1986) examines how product differentiation motivates consumers to search. His framework represents a commonly encountered approach
integrating search and product differentiation.7 In his model, a consumer must search to
learn both the price and his own idiosyncratic valuation for each firm’s particular product.
In our setting, by contrast, each consumer has a persistent type and firms producing objectively similar products are viewed similarly. Hence we depart from the standard model
of product differentiation in a monopolistically-competitive market. Though we allow for
multiple goods, our environment also differs from a “multi-product” search setting in which
a consumer buys multiple products, perhaps from different suppliers (Burdett and Malueg,
1981; McAfee, 1995; Gatti, 2001; Rhodes, 2014; Zhou, 2012). In our model, a consumer
demands but one item; we relax this assumption in Section 5.
Search Externalities and Complementarities As noted above, our key economic conclusions stem from the presence of search externalities and complementarities. Several earlier
studies hint at similar implications to those that we identify. In line with our analysis, Pratt
et al. (1979) empirically document considerable price dispersion across a range of standard7
Wolinsky (1986) builds on Perloff and Salop (1985). See also Wolinsky (1984). Anderson and Renault
(1999, 2000), Armstrong et al. (2009), and Larson (2013) are more recent studies in this vein.
7
ized products.8 They also suggest a precursor to the market-creation/expansion implications
of search behavior that we examine. Their example of expensive and inexpensive motels along
a road (p. 209) could find a welcome home in this paper. Nevertheless, the environment analyzed here differs from their setting and accommodates conclusions outside of their model.
To highlight but one difference, we show how an equilibrium with search can increase firms’
expected profits even above those with a monopoly price for a single good. This outcome
cannot occur in Pratt et al. (1979).
Formally, many of our conclusions rest on a strong feed-back effect between search activity
and production profitability. Diamond (1982, p. 882) has argued for the presence of such an
effect based on his analysis of a barter economy. His model assumes that goods are exchanged
on a one-for-one basis and therefore does not include price dispersion. In an important
paper, Burdett and Judd (1983) rely on the same complementarity between consumers’
search activity and firms’ pricing strategies to support multiple equilibria in a search model.
They also show that multiple price-dispersion equilibria can coexist with different levels of
consumer search intensity. In one equilibrium, a small fraction of consumers learns the prices
posted by more than one firm; in another, a large fraction of consumers sample more than
one firm.9 In their model, firms’ profits decline relative to the monopoly-price case whenever
more consumers search to learn the posted price of more than one firm. This effect is to be
expected and is present in some cases of our model as well. However, as we explain in Section
4.2, firms’ profits can also rise if they are also able to specialize in producing different goods.
With product differentiation, search activity also facilitates the sorting of consumers among
producers. Thus, firms can compensate for their reduced market-power with better-targeted
pricing.10
Cachon et al. (2008) identify a “market-expansion effect” tied to search activity. They
model an oligopolistic market where firms can expand product lines. Unlike our model,
Cachon et al.’s (2008) firms offer a continuum of firm-differentiated products. Each consumer
has an idiosyncratic taste for each good and must search to learn the price and appeal of each
good. The fine-grained idiosyncrasies in consumer tastes coupled with their cost specification
8
This observation continues to be salient despite the advent of new technologies that have, arguably,
lowered search costs (Baye and Morgan, 2001; Baye et al., 2006).
9
Burdett and Judd (1983) study a model with non-sequential search and a model with noisy search. We
are referring to their model with non-sequential search where the fraction of consumers learning one price is
endogenously determined in equilibrium.
10
Acemoglu and Shimer (2000) identify a similar phenomenon when studying labor markets. They argue
that better search-related sorting promotes “technology dispersion.” Their dispersion and the differentiation
that we study are roughly analogous.
8
produces a unique equilibrium without price dispersion. In contrast, our model generates
high-, moderate- and no-search equilibria, often with dispersed prices.
Market-expansion effects are also noted in the literature on agglomeration economies.
Its central idea is that concentrations of sellers attract increased numbers of consumers by
reducing consumers’ search costs (Dudey, 1990). New York’s Diamond District or the London
Silver Vaults are prime examples. The micro-level strategic interaction and competitive
incentives in our model are unrelated to those stemming from agglomeration effects. Our
consumers are not drawn to a particular locale.
While our model considers a non-monetary economy, some features of our environment
have been considered in recent search-theoretic models involving money, following Kiyotaki
and Wright (1989). Both Camera et al. (2003) and Fong and Szentes (2005), for example,
allow agents to endogenously specialize their production across an array of differentiated
products. Respectively, they show how money can increase specialization in production
or increase the production of costly, but higher-quality goods. A key component in our
framework is the strategic complementarity among producers of a similar product, which
requires a costly investment to produce. As noted by Lester et al. (2012), a similar strategic
complementarity exists for costly information acquisition when assets trade as currency: as
more agents recognize an asset as a safe medium of exchange, it becomes more worthwhile
for others to do the same. Finally, our main conclusions concerning welfare in equilibria with
intensive search and high price dispersion play off the counterbalancing of both intensive- and
extensive-margin effects. In a monetary model, such effects have been studied by Rocheteau
and Wright (2005) though they rely on a considerably different market mechanism than we
do.
Externalities of various sorts have always been a theme in the search literature. Effects
such as the thick-market externality, the congestion externality, and the sorting externally
have been identified and studied (Shimer and Smith, 2001; Burdett and Coles, 1997, 1999).
We relate our analysis to these findings in the conclusion, after our model’s details and
implications have been spelled out.
2
The Model
Our model features a large number of firms and consumers. Each firm chooses a good to
produce and its price. Consumers differ in their willingness to pay for goods. Each consumer
searches sequentially among the firms to satisfy his consumption needs. First, we describe
9
the two sides of the market. Subsequently, we introduce the market mechanism and search
process. We comment on our model’s interpretation before concluding this section.
Firms Firms move first in our economy. Each firm j 2 Z commits to a single contract—
a good-price pair, j = hxj , pj i—that each passing consumer may either accept or reject.
xj 2 {0, 1} is the good produced by firm j and pj is its per-unit price. For simplicity, assume
that good 0 can be produced at zero marginal cost and with no fixed cost. Good 1 can be
produced with a constant marginal cost c 0; moreover, if a firm produces good 1 it must
additionally incur a fixed cost of > 0. This fixed cost, for example, may represent an
investment in a product-specific technology. Thus, a taxi driver may invest in a fancier car
or a restaurant may opt for a more stunning decor.11
A firm’s profit depends on the number of consumers who accept its contract offer. This
number will be determined in equilibrium, but for now we observe that if n consumers accept
xj · (nc + ).12
j = hxj , pj i then firm j’s profit is ⇡(n| j ) = npj
Consumers Each consumer i 2 Z is willing to pay z > 0 for good 0. Consumers differ in
their willingness to pay for good 1. This variation is described by the random variable Vi ,
which we call the consumer’s type. A helpful interpretation is to consider good 0 to be an
established, standardized product and good 1 to be a novel product with more heterogenous
demand. For simplicity, suppose Vi takes on one of two values, v or v̄, where 0  c < v < v̄.
Consumers’ types are independently and identically distributed and := Pr[Vi = v̄]. This
distribution is common knowledge.
Goods 0 and 1 are substitutes. A consumer wishes to buy at most one unit of either
good, but not both. He needs but one cab ride, or one dinner. (Two cab rides would take
him away from his destination. Two dinners would give him indigestion.) Thus, if a type-vi
consumer accepts j = hxj , pj i—he buys one unit of good xj at price pj from firm j—his
immediate consumption payoff is u( j |vi ) = xj vi + (1 xj )z pj . A consumer’s purchase is
voluntary. He is free to exit the market for an immediate payoff of zero.
The Market Mechanism Although a consumer knows the aggregate distribution of contracts in the economy, he does not know the specific contract offers of all firms. We assume
11
Fishman and Levy (2012) also allow firms to make a costly investment in product quality, thus generating
product differentiation. In their model the outcome of this investment is stochastic.
12
Implicitly, we also allow a firm to forego production entirely. We do not model this decision directly as it
is a weakly dominated action. We thank an anonymous referee for suggesting this notational simplification.
10
that a consumer initially knows only the offer of one local firm; he must search to learn the
offers of others. Our market unfolds as follows:
1. Each firm j simultaneously commits to a single feasible contract,
feasible contracts is ⌃ = {hx, pi : x 2 {0, 1}, p 2 R+ }.
j
2 ⌃. The set of
2. Without cost, each consumer i learns the contract offer of firm j = i. Thus, at the
start of the market consumer i has access to contract i .
3. For t = 0, 1, . . . , consumer i may (a) decide to accept contract
better option, or (c) exit the market.
i+t ,
(b) search for a
(a) If the consumer accepts i+t he receives an immediate payoff of u( i+t |vi ) and he
exits the market. The firm offering the chosen contract supplies the good and
receives its quoted price as payment.
(b) If the consumer decides to search, he incurs a search cost of s > 0 and he learns
the contract offer of firm i + t + 1. Step 3 subsequently repeats with i+t+1 being
the available contract offer.
(c) If the consumer exits the market, he receives a payoff of zero and does not return.
Remark 1. If consumer i accepts j after sampling t (non-local) firms, his total payoff is
u( j |vi ) ts. If he exits the market without making a purchase, this total payoff is ts.
Remark 2. Our model assumes that consumers cannot return to a previously sampled firm.
That is, search is “without recall.” This assumption accords well with our motivating taxicab
example. Once a taxi drives by, it is usually gone for good. Anticipating our equilibrium
analysis, we emphasize that our argument also holds when search allows for recall (DeGroot,
1970).13 For example, when consumers search for a new cellphone plan, they can reconsider
previously encountered options without much hassle. In such applications, search costs often
stem from the challenge of evaluating a product’s qualities or fine-print specifics rather than
physically “finding” the good.14
13
If search allows for recall, at each date t consumer i will be able to pick his favorite option from the
contracts he has encountered on his search { i , i+1 , . . . , i+t } or to continue searching. Except for the
extension in Section 5.3 to increasing search costs, the analysis is unchanged if search with recall is allowed.
A unified exposition motivated our modeling choice.
14
Ellison and Wolitzky (2012) discuss how firms may manipulate consumers’ costs of learning product
attributes and prices.
11
Remark 3. Each consumer knows the offer of a “local” firm. This is equivalent to giving the
consumer one free sample before possibly embarking on a more extensive search. This is
a common assumption (Salop and Stiglitz, 1982; Stahl, 1989; Acemoglu and Shimer, 2000,
among others). Unlike studies following Varian (1980), we do not posit the existence of a
group of consumers who are perfectly informed about all firms’ offers.
Strategies and Equilibrium In many models generating price dispersion, firms adopt
mixed strategies. A similar conclusion applies here and we denote a mixed strategy for firm
j as j 2 (⌃).15
When firm j posts a contract, the number of consumers who eventually accept it is a
random variable, Nj . The distribution of Nj will depend on j , on other firms’ strategies,
j , and on consumers’ search strategies. Suppressing these latter elements in our notation,
firm j’s expected profit is ⇧j ( j ) := E [⇡(Nj | j )| j , j ].
A consumer’s strategy is a search-and-purchase plan. In each period, he must decide
whether to accept an available offer or to search further so as to maximize his expected utility
net of search costs. His beliefs regarding the distribution of contracts in the economy are a
critical input informing this plan. We represent these beliefs by ˜i 2 (⌃). When a consumer
searches, we assume that he views each encounter with a new firm as an independent draw
from ˜i .
With the above ideas in place, we can define an equilibrium in our market. We restrict
our analysis to symmetric equilibria and we will often drop the “symmetric” qualifier. The
definition conforms to the usual requirements. Firms’ strategies constitute a Nash equilibrium; each consumer’s search behavior maximizes his expected welfare given his beliefs; and
each consumer’s beliefs are correct.
Definition 1. A (symmetric) equilibrium is a tuple of production strategies, beliefs, and
search strategies such that:
1. Each firm adopts a common strategy j = ⇤ , which maximizes its expected profit
given the strategies of all other firms and consumers’ search strategies and beliefs.
2. Each consumer believes that the distribution of contracts in the economy is ˜i =
⇤
.
3. The search strategy of each consumer maximizes his expected utility given ˜i .
4. All consumers (of a particular type) adopt the same search strategy.
15
Equivalently, we can reframe our model so that firms follow pure strategies and equilibrium will be
described by the fraction of firms offering specific contracts. For example, Pratt et al. (1979) take this route.
12
Context and Interpretations Our model’s simplicity accommodates several interpretations and links easily with a range of models of market competition. First, we assume
that consumers acquire market information sequentially. Thus, we depart from a thoroughly
“classic” model where perfect information reigns supreme. If consumers enjoyed perfect information (and could immediately contract with their preferred provider), then our model
reduces to a case of Bertrand competition—each consumer will rush to the firm promising
the greatest surplus.
Second, our model features an orderly search process. Given our assumptions about
consumer beliefs, our model behaves as if a consumer samples from a large set of ex ante
identical firms, as is standard in the search literature. Our search process considerably
simplifies the analysis of consumer sorting. It lets us easily track the expected inflow of local
and non-local consumers to each firm.
Third, we are silent regarding the model’s time dimension. While the “t” superscript
suggests the passing of physical time and we often speak of “periods of search,” we stake no
claim on the model’s temporal calibration. We view “t” as an index describing the extent
of search undertaken by a consumer. Hence, our economy can unfold asynchronously with
different consumers searching at different paces or different times. The key restriction is that
firms cannot screen consumers based on how much previous search they have undertaken or
on their arrival “time.” All firms post a single price, which is a common practice.
Finally, we assume an equal number of consumers and firms. Our analysis is qualitatively
unchanged if each firm is initially matched with m consumers each of whom may accept its
offer or engage in search. The m = 1 case simplifies exposition.
3
Consumer Search Behavior
How should a typical consumer search? If he believes that all firms are offering contracts
according to ˜ 2 (⌃), his optimal search strategy involves a cutoff rule (Lemma A.1).
For each type of consumer vi , there exists a critical value u⇤ (vi ) such that if u( j |vi )
max{u⇤ (vi ), 0} then he stops searching and accepts j . If u⇤ (vi ) > max{u( j |vi ), 0} the
consumer takes a pass on j and samples his next available option. Otherwise, he exits
the market. Following DeGroot (1970), the cutoff value characterizing the search strategy
⇥
⇤
is the unique solution to the equation u⇤ (vi ) = E ˜ max{u⇤ (vi ), u( |vi ), 0}
s where is
distributed according to ˜.
This strategy has two noteworthy characteristics. First, it is stationary and independent
13
of the extent of prior search activity. This characteristic is a direct consequence of the
constant search costs and does not carry over to a more general setting (see Section 5.3).
Second, a consumer’s beliefs ˜ are constant and do not adjust in response to the observed
sample of contracts. Though in equilibrium a consumer’s beliefs are correct, he does not draw
any inferences from “off-equilibrium path” occurrences. Being surprised by an unexpectedly
good deal will not prime the consumer to anticipate even better offers elsewhere. Both
characteristics considerably simplify the analysis and are standard features of nearly all
search models.
Since consumer behavior is straightforward, we will typically identify equilibria by referring only to firms’ strategies. When we do so, our description implicitly assumes that
consumers hold correct beliefs, and that their search behavior is described by the above
cutoff strategy.
4
Market Equilibrium
We begin our analysis by introducing a maintained parameter restriction.
Assumption A-1. v
c
< (1
)z and (v̄
c)
< z.
When Assumption A-1 holds, producing good 1 in an economy without consumer search is
comparatively unprofitable. The assumption tilts our setting against the conclusions that
we are working toward and focuses our attention to (in our opinion) interesting cases.
The case with neither search, nor product differentiation, nor price dispersion will serve as
our benchmark. This equilibrium, which always exists, reproduces the well-known Diamond
(1971) paradox. A single product is sold at its monopoly price. As all consumers know this
to be true, search never pays. As Burdett and Judd (1983) show, this result is common
across search models whenever consumers learn the price posted by a single firm.
Theorem 1. There exists a unique equilibrium with no product differentiation. In this
equilibrium all firms offer good 0 at the monopoly price z.
From a consumer’s perspective, Theorem 1 describes this economy’s “bad equilibrium.”
Since consumers expect all firms to offer the same contract, they have no incentive to search.
Acting like a local monopolist, each firm extracts the surplus from its captive market and
earns a profit of z. If search costs are sufficiently high or there are too few consumers who
value good 1 highly, this will be the economy’s only equilibrium.
14
As noted above, the equilibrium described in Theorem 1 is neither inevitable nor particularly common. Goods usually differ both physically and in price. Similar products vary by
quality and the same product can vary in price from store to store. Both forms of variation
suggest alternative equilibrium arrangements that may dominate the benchmark case. More
precisely, we say that an equilibrium features product differentiation if there is a positive
probability that two firms produce different goods. Similarly, price dispersion exists if there
is a positive probability that two firms set different prices for the same good. Our analysis
below considers progressively more complex equilibria that differ on these two characteristics.
A coordination failure lurks behind Theorem 1. Given Assumption A-1, the sustainable
production of good 1 requires that some consumers make their purchase from a non-local
supplier. This is only possible after a costly search. We say that an equilibrium exhibits
search if there is a positive probability that at least one consumer searches for at least one
period. For search to be worthwhile, a consumer needs to be reasonably certain (i) that
it will yield a worthwhile prize, and (ii) that it will not be exceptionally costly. The first
criterion means that some consumers must anticipate receiving a positive surplus from some
contract(s) in the economy. The second criterion means that an adequate number of firms
must offer sufficiently attractive contracts.
Once multiple goods are traded simultaneously, a large set of equilibrium contracts may
emerge. This, however, does not happen and there is only a small set of relevant cases.
Theorem 2. Consider an equilibrium with product differentiation. There are at most two
distinct contracts offered by firms producing good 0 and there are at most two distinct contracts offered by firms producing good 1.
The intuition behind Theorem 2 is straightforward and has been noted before by Albrecht
and Axell (1984) and Diamond (1987), among others. Each posted equilibrium price must
correspond to the reservation price of some consumer. Suppose good 1 trades in equilibrium
at, say, three prices: p < p0 < p̄. Suppose that the cheapest contract is acceptable to both
type-v and type-v̄ consumers and the most expensive contract is acceptable only to type-v̄
consumers. But now, some firm has a profitable deviation. If the price p0 is acceptable to
both types of consumers, then any firm charging p can raise its price without hurting its sales
volume. Similarly, if p0 is accepted only by type-v̄ consumers, a firm charging p0 can boost
its profits by raising its price. While a full argument must consider additional scenarios, the
reasoning is similar.
Aspects of Theorem 2 are similar to those identified by Curtis and Wright (2004), who
propose a “law of two prices” within a monetary search model. The need to simultaneously
15
satisfy a family of incentive constraints supporting search behavior limits firms’ freedom to
set prices in equilibrium. Despite this similarly, as we show below, our analysis leads to
somewhat different conclusions concerning the incidence of price dispersion equilibria than
those proposed by Curtis and Wright (2004). For instance, they show that equilibria with
multiple prices occur for a wider range of parameters than equilibria with one price. Example
1, presented below, suggests the converse conclusion in our model.
Theorem 2 allows for a considerable simplification of our analysis, as there are at most
four classes of equilibria we need to consider. One class exhibits no price dispersion. Two
goods are available and each good sells at a uniform price. The other three classes feature
varying degrees of price dispersion. We focus on the two extreme cases in this quartet: (i)
product differentiation and no price dispersion, and (ii) product differentiation with both
goods selling at multiple prices. We comment briefly on the remaining cases—one good
trading at multiple prices—in this section’s conclusion. Neither of these cases generate
equilibria that Pareto-dominate the benchmark scenario. Thus, they are comparatively less
interesting than the other cases considered.
4.1
Product Differentiation without Price Dispersion
Starting with the first case, we begin with a lemma pinning-down the equilibrium prices.
Implicitly, it also defines the equilibrium search pattern, as explained below.
Lemma 1. Let 0 = h0, p0 i and 1 = h1, p1 i be the two contracts offered in an equilibrium
with product differentiation and no price dispersion. Then p0 = z and p1 = v.
Two immediate implications flow from Lemma 1. First, type-v consumers will never
search in an equilibrium absent price dispersion. All offered contracts guarantee them a
payoff of zero. Second, given Assumption A-1, a firm producing good 1 must in expectation
sell its product to more than one consumer. Hence, type-v̄ consumers must search and
accept 1 . This is a simple case of consumers self-sorting between products, a recurrent
theme in both our model and real-world markets. The implied pattern of search and the
flow of consumers is presented in Figure 1. After matching with a firm offering 0 = h0, p0 i,
a type-v̄ consumer is unsatisfied; he searches until he finds a firm offering 1 = h1, p1 i. This
flow is depicted by the solid arrows leading to 1 = h1, p1 i. In contrast, a type-v consumer
finds both 0 and 1 acceptable. We can summarize the resulting equilibrium as follows.
Theorem 3. Let
⇤
1
=
(v
(1
16
c)
)(v c
z)
.
(1)
1
Key
Type-v̄
Type-v
⇥
0
Figure 1: The flow of consumers in an equilibrium with product differentiation but no price
dispersion. Type-v consumers accept either contract while type-v̄ consumers search for 1 .
There exists an equilibrium with product differentiation and no price dispersion if and only
if
s
s
⇤
1
✓
v̄
( ⇤1 )2 (1
z
c
1
)(v
c)
+ /
⇤
1
◆
and
.
(2)
(3)
In such an equilibrium, a typical firm offers the contract 1 = h1, vi with probability ⇤1 and
⇤
⇤
0 = h0, zi with probability 0 = 1
1 . On the equilibrium path, a type-v̄ consumer searches
until he finds a firm offering the contract 1 , which he accepts. A type-v consumer does not
search, and accepts the first contract available.
As illustrated in Example 1 below, (2), (3), and Assumption A-1 can be simultaneously
satisfied for a wide range of parameters. By inspection, we observe that (2) and (3) hold
when search costs are sufficiently low (s ! 0). Conversely, if there are too few type-v̄ agents
( ! 0) such an equilibrium will not exist.16
The equilibrium in Theorem 3 sorts consumers between goods. However, this sorting is
incomplete as good 1 is consumed by a mixed group of consumers. This outcome is crucial
lest the market unravel. If a firm charges a price slightly more than v for good 1, its demand
will fall. Hence producers of good 1 face a sharp incentive to maintain prices at a (relatively)
16 ⇤
1
is increasing in . Hence, as
! 0, then
⇤
1
! 0. Condition (2) is eventually violated.
17
low level. Coincidentally, that same low price motivates type-v̄ consumers to search and it
is their inflow that makes producing good 1 a profitable undertaking.
Several intriguing comparative static implications follow easily from Theorem 3. First,
equilibrium prices are independent of search costs and are driven by consumers’ valuations.
Second, the equilibrium distribution of contracts does not vary with search cost. Instead,
the propensity of a firm to produce good 1 increases ( ⇤1 ") as the fixed costs of production
falls ( #), or as consumers’ preferences shift toward good 1 ( ").
Since the firm-consumer ratio is one, a natural welfare criterion is the aggregate expected
surplus generated by a firm-consumer pair. In the equilibrium without product differentiation, this value was z and the firm appropriated it all. When there is product differentiation,
each firm’s profit declines to (1
)z; type-v̄ consumers enjoy a positive surplus; but, type-v
consumers experience no welfare improvement. Aggregate welfare may increase or decrease
depending on the model’s specifics, even when search costs are small.
Corollary 1. Suppose search costs become small, i.e. s ! 0. The equilibrium with product
differentiation and no price dispersion generates greater aggregate expected surplus than the
equilibrium with no product differentiation if and only if v̄ z + v.
4.2
Product Differentiation and Price Dispersion
The preceding subsection showed that when there is product differentiation but no price
dispersion, some consumers benefit, others receive their reservation payoff, and firms’ profits
decline. The aggregate outcome might be better or worse than had the local monopoly
situation prevailed. These conclusions lack universal appeal. Ideally, we would like to identify
cases where all consumers and all producers are better off relative to the benchmark case,
at least in an ex ante sense. Whether such a Pareto-improvement is a realistic aspiration or
a Pollyannish fantasy is not obvious a priori.
Two facts hint at a possibility. First, searching consumers gain access to a wider set of
producers allowing price comparisons. This trims firms’ market power. Thus, ensuring that
consumers gain from active search does not seem too demanding. Second, consumer search
has a cross-cutting implication for firm profits. Search generates a movement of consumers
and also produces a sorting of consumer types. The former effect increases sales volumes
while the latter allows for more targeted pricing. Both effects counteract firms’ weakened
market power and will prove critical in supporting a Pareto-superior equilibrium.
The first challenge is to ensure that consumers search actively in equilibrium. To justify
costly search there must exist equilibrium contracts that consumers view as particularly
18
appealing. Promising carrots in equilibrium, however, poses a challenge for producers. The
problem here is that firms have the incentive to free-ride whenever consumers search. We
have in mind the following phenomenon. When consumers search for a prized contract,
that contract must promise a sufficiently high reward and be offered by sufficiently many
firms in order to justify the search costs incurred. When many firms offer the appealing
contract, the consumer is confident his search will be worthwhile. Since search costs are
incurred incrementally, a firm promising a very appealing contract has an incentive to slightly
downgrade its offer. A searching consumer will still stop and buy rather than search further.
The “deviating” downgraded contract does not affect a consumer’s beliefs but does trap
him like flypaper before he finds the anticipated deal. This points to a subtle and underappreciated paradox present in search economies. A firm benefits from having a sufficient
number of direct competitors since that induces consumers to search for its type of product.
It therefore wants competitors to coordinate, but the firm itself seeks to miscoordinate with
its competition in order to profit. The implied tension for an equilibrium is obvious.
To add some formalism, from Theorem 2 we know that at most four contracts—say 0 ,
1 , ˆ0 , and ˆ1 —can coexist in equilibrium. For notation, we let k = hk, pk i and ˆk = hk, p̂k i,
with the convention that pk < p̂k . A typical firm follows a mixed strategy offering contract
ˆ
k (ˆk ) with probability k ( k ). With notation set, we can consider our first key lemma.
This lemma accomplishes two tasks. First, it identifies constraints neutralizing the freeriding deviation described above. Second, coupled with Lemma 3 to follow, it defines the
equilibrium pattern of consumer search.
Lemma 2. Consider an equilibrium with product differentiation.
1. If good 0 sells at prices p0 and p̂0 , p0 < p̂0 , then z
u⇤ (v̄).
p̂0 = max{u⇤ (v), 0} and z
p0 =
2. If good 1 sells at prices p1 and p̂1 , p1 < p̂1 , then v
u⇤ (v̄).
p1 = max{u⇤ (v), 0} and v̄
p̂1 =
We can paraphrase Lemma 2 as follows. If a particular good’s prices are disperse, the
prices at which the good trades must correspond to the agents’ reservation values. These
reservation values are determined in equilibrium and account for both the goods’ intrinsic
utility and the search-and-purchase strategy adopted by the agents. Generally, of course,
these reservation values will differ for different types of agents. An implication of the four
equalities in Lemma 2 is that in equilibrium no firm can increase its posted price even
19
slightly without alienating some consumers.17 This serves as a disciplinary force holding the
equilibrium together. The same intuition is present in a variety of search models (Albrecht
and Axell, 1984; Diamond, 1987).
As noted above, we are particularly interested in equilibria with desirable welfare implications for both producers and consumers. This added requirement necessarily leads us to
consider cases where both goods sell at multiple prices.
Lemma 3. In any equilibrium that Pareto-dominates the equilibrium of Theorem 1.
1. All goods must trade at multiple prices.
2. On the equilibrium path, a type-v̄ agent searches when ˆ0 is available to him; else, he
accepts 0 , 1 , or ˆ1 . A type-v agent searches when ˆ1 is available to him; else, he
accepts 0 , ˆ0 , or 1 .
Lemma 3 summarizes a complex pattern of consumer search, illustrated in Figure 2. First,
from Lemma 2 we see that two contracts endogenously emerge as the “deals” motivating
search. The preferred option of a type-v̄ consumer is 1 while a type-v consumer views 0
as best. Consumers hunt for their preferred contract, but may settle for a more expensive
option or the other good should they stumble upon it.
With the pattern of search determined, the equilibrium’s remaining necessary features
briskly fall into place. Lemmas 2 and 3 allow us to characterize a typical consumer’s (cutoff)
search strategy and the quantitative relationship among equilibrium prices.
Lemma 4. Consider an equilibrium that Pareto-dominates the equilibrium of Theorem 1.
1. The cutoff values characterizing a consumer’s search strategy are u⇤ (v̄) = v̄
and u⇤ (v) = z p0 s/ 0 .
p1
s/
1
2. The prices associated with the contracts satisfy
p̂0 = p0 +
p̂1 = p1 +
s
p0 = p1 +
p1 = p0 +
0
s
1
s
1
s
0
+z
+v
v̄
z
Noting the relationship among prices, an immediate corollary is that
v̄
v=
s
0
17
+
s
.
(4)
1
Intuitively, we might say that consumers feel a smidgen of indignation and refuse to purchase any good
whose price even marginally exceeds expectations.
20
High Price
Low Price
Good 1
⇥
1
ˆ1
Key
Type-v̄
Type-v
Good 0
⇥
0
ˆ0
Figure 2: The flow of consumers in an equilibrium with product differentiation where both
goods trade at multiple prices.
Expression (4) carries two economic implications. First, it can be satisfied only if search
costs are sufficiently small. Second, when search costs rise, the prevalence of at least one
inexpensive contract must also rise. This response is counterintuitive since higher search
costs suggest that firms’ market power should be enhanced, which favors a shift toward
higher-priced goods. In equilibrium, however, the observed response is necessary to maintain
consumers’ incentive to search.
The search pattern also determines the profits associated with each contract. The lowprice contracts generate high volume sales while catering to a diverse group of consumers.
High-priced contracts feature targeted prices and a homogenous clientele.
Lemma 5. Consider an equilibrium that Pareto-dominates the equilibrium of Theorem 1.
The expected profits associated with each contract are:
⇧j ( 1 ) =
⇧j ( 0 ) =
✓
1
1
✓
1
1
ˆ1
ˆ1
+
+
1
ˆ0
1
ˆ0
◆
◆
(p1
c)
,
⇧j (ˆ1 ) =
⇧j (ˆ0 ) =
p0 ,
1
1
ˆ0
1
ˆ1
(p̂1
c)
,
p̂0 .
In an equilibrium, of course, the expected profits must be the same across all offered contracts. Solving ⇧j ( 0 ) = ⇧j (ˆ0 ), ⇧j ( 1 ) = ⇧j (ˆ1 ), and using Lemma 4 allows us to express
21
equilibrium prices as a function of the distribution of contracts. That is,
p0 =
1
1
ˆ0 1
·
ˆ1
·
s
and
p1 = c +
0
1
ˆ1
1
ˆ0 1
·
·
s
Thus, we can characterize the equilibrium entirely as a distribution of contracts— 0 ,
satisfying (4), (5), and the following four conditions:
⇧j ( 0 ) = ⇧j ( 1 ),
s
p1 = p0 +
+v
(5)
.
1
1,
ˆ0, ˆ1—
(6)
z,
(7)
+ ˆ 0 + ˆ 1 , and
(8)
0
z
p0
1=
s
0
+
1
(9)
0
0
Expression (6) ensures the equality in profits; (7) ensures the remaining relationship among
equilibrium prices is satisfied; (8) ensures the ’s define a valid probability distribution; and,
(9) ensures type-v agents wish to search, i.e. u⇤ (v)
0. All remaining constraints have
either been eliminated or incorporated into the above expressions.
The preceding discussion confirms that conditions (4)–(9) are necessary for the equilibrium we wish to construct. They encapsulate and simplify two sets of requirements. First,
the equilibrium distribution of contracts must ensure that the expected profits of each firm
are the same taking as given consumers’ search behavior. This requirement simultaneously
restricts the frequency with which different contracts are offered and the prices associated
with each good. Second, the equilibrium contract distribution must ensure that consumers’
payoffs respect the constraints in Lemma 2, thereby ensuring the required pattern of search
is consistent with equilibrium prices. Complementary activity of the two sides of the market
is therefore critical for the equilibrium to be sustained.
Satisfying conditions (4)–(9) is also sufficient to sustain an equilibrium where both goods
trade at multiple prices (Lemma A.15). Whereas the complexity of the required conditions
suggests that equilibria conforming to our desiderata are rare, this is not the case. The
following example offers some guidance when we may expect such equilibria to emerge and
when different types of equilibria coexist.
Example 1. Consider an economy where v̄ = 4, v = z = 1, = 1/3, and c = 0. Depending
on the magnitude of the search cost and the distribution of agents’ preferences, this economy may feature one equilibrium or multiple equilibria. Figure 3 summarizes the present
22
1 Good
Monopoly Outcome (
= 1)
2 Goods
No Price Dispersion (
< 1)
Price Dispersion (
< 1)
Price Dispersion (
> 1)
Figure 3: Multiple equilibria with product differentiation and price dispersion.
equilibria as a function of ( , s).18 The one-product equilibrium of Theorem 1 always exists.
It is the unique equilibrium when is sufficiently low. Given our parameterization, a typical
firm’s profit is equal to 1 in this equilibrium.
An equilibrium with product differentiation, but no price dispersion (Theorem 3), is
common once is sufficiently large (the gray region). In this equilibrium, the profits of a
typical firm are 1
, which is less than in the one-product benchmark scenario.
If is sufficiently large and search costs are sufficiently small, equilibria with both product
differentiation and price dispersion occur as well (the hatched region). Quite often such cases
lead to profits exceeding the benchmark case. For instance, when s = 0.03 and = 0.3, a
typical firms’ expected profit is approximately 1.18. This final case, and others like it in the
lower-right corner of Figure 3, Pareto-dominates the benchmark scenario as consumer also
enjoy a positive surplus as well.
The welfare implications associated with Example 1 are its key conclusion. The incentives
created by the economy’s multiple goods and dispersed prices ensure that consumers search
and sort among producers. While the resulting reshuffling of consumers is less than perfect
(some consumers do not purchase the item that they value the most), it sufficiently improves
matters to ensure that good 1 producers face adequate demand to render their production
18
To construct the figure we solved numerically for an equilibrium contract distribution satisfying conditions (4)–(9) over a fine grid of the parameter space.
23
profitable. Surprisingly, producers of good 0 also gain.
Intermediate Price Dispersion We have thus far focused on two extreme equilibria with
product differentiation. Either uniform prices prevail or all goods trade a multiple prices.
Intermediate cases are possible as well, though such equilibria do not enjoy the appealing
welfare and profit implications of the preceding case (see Lemma 3 above). For example,
it is straightforward to construct an equilibrium where good 0 trades at price p0 = z and
good 1 trades at prices p1 = v < p̂1 . A type-v consumer never engages in search and a
type-v̄ consumer searches for good 1, which he buys at both high and low prices. As in the
equilibrium of Theorem 3, firms’ profits are (1
)z.19 Similar logic lets us construct an
equilibrium where good 0 trades at multiple prices as well.
5
Interpretations and Extensions
To focus on important economic effects, we kept our model spare and simple. Naturally,
building on this streamlined formulation allows for many extensions and alternative interpretations. First, several technical amendments are possible. Our model extends to the case
of multiple consumer types and multiple goods.20 The importance of imperfect or incomplete
sorting of consumers as a mechanism that ties an equilibrium together continues to apply.
Second, with appropriate parameter restrictions, our model applies to a variety of problems. Markets with vertically-differentiated products, like our luxury/standard taxicab example, require the restrictions z  v < v̄ or v < v̄  z.21 When offered at the same
price, either good 1 or good 0 is universally preferred. To study a market with horizontallydifferentiated products, such as restaurants, it is sufficient to adopt the convention that
v < z < v̄. Now consumers disagree about which good is superior. Some prefer French
cuisine while others view Japanese food as more appetizing.
Our model’s ability to accommodate both vertical and horizontal differentiation spans
the ways in which new products surface. For example, Bower and Christensen (1995) and
19
+(1
)z v
Such equilibria are comparatively tractable. Notably: p̂1 = v + s/ 1 , 0 = (1 c+
)c+ +(1
)(z v) , 1 =
s
ˆ
0
1 . Given the parameters of Example 1, it can be shown that an
( +(1
)z) 0
(v c) , and 1 = 1
equilibrium in this class exists whenever there exists an equilibrium with product differentiation and no price
dispersion (the gray region in Figure 3). Examples exist showing that this inclusion is not true in general.
20
In a preliminary version of our analysis we considered a continuum of consumer types. In such a model,
there is additionally an endogenous cutoff consumer type delineating who seeks out good 1 versus good 0.
21
Wildenbeest (2011) develops a search model, along with an empirical analysis, focusing on vertically
differentiated products.
24
Christensen (1997) argue that new goods frequently enter at lower quality-tiers than existing
products. Nevertheless, with a careful pricing strategy targeting the appropriate market
segment such new goods can be very profitable (Christensen, 1997). The emergence of
large discount retailers and of no-frills airlines represent prime examples. Innovation along a
horizontal dimension rests on identifying products tailored to specific groups of consumers.
Many innovations in the technological realm, such as the file-sharing services mentioned
earlier, produce markets where multiple competitors offer modestly different capabilities
that appeal differentially to alternative consumers. Our model accommodates all of these
situations.
Third, labor markets are a prime example where search is paramount. A relabeling transposes our model to a labor-market context where firms search to recruit skilled workers.22
In this case a worker may choose whether to invest to upgrade his human capital or skill.
Skilled workers supply higher quality labor (good 1) while unskilled workers supply lower
quality labor (good 0). Type-v̄ employers are more eager to hire skilled workers than type-v
employers. Firms search for employees and incur a per-candidate interview cost, s. The
economy’s wage distribution, i.e. the distribution of prices, is the equilibrium outcome of
firms and workers interacting in this labor market with workers choosing whether to acquire
human capital.
Finally, many economically-motivated extensions are also easily accessible. We briefly
outline some of these below. For brevity and clarity, in each extension we confine our
attention to equilibria with product differentiation but no price dispersion (as in Theorem
3), and we suppress many technical caveats. More complex equilibria can also be constructed
with parallel implications to our preceding analysis.
5.1
Orphaned Consumers
We have assumed that each consumer knows the contract offer of a local producer. In practice
however, a local producer may not exist. For example, upon realizing the need to take a
taxi, a consumer may observe that no taxis are around. To accommodate this situation,
suppose that after firms have made their production decisions, but before consumers learn
any contract offers, with probability 1 q each firm experiences an independent shock that
prevents it from selling any goods. Consequently, an orphaned consumer’s initial set of
22
As documented by van Ommeren and Russo (2013), sequential search by firms appears to be an important
recruitment practice. However, they qualify this conclusion noting that it depends on the common role of
formal and informal search methods in the market under study.
25
available contracts is empty and, occasionally, a searching consumer may “find” a firm that
has gone out of business. Otherwise, the model remains unchanged.
The main conclusions of our original analysis continue to apply. In an equilibrium with
product differentiation but no price dispersion, firms would offer contracts 0 = h0, zi
and 1 = h1, vi. Of course, the shock impacts profitability; accordingly, the equilibrium
contract distribution adjusts. The probability that a firm offers good 1 now becomes
(v c)
⇤
. Also, and in common with the remaining extensions below, the suffi1 =
(1 )q(v c z)
cient and necessary conditions for this equilibrium to exist—analogous to those from Theorem
3—need to be amended appropriately. (We omit these calculations for brevity.) Formalities
notwithstanding, our model’s qualitative behavior remains unchanged.
5.2
Multi-Unit Purchases
Consider the problem faced by a visitor looking for lodging in an unfamiliar city. If the
traveler intends a long stay, he has a greater incentive to invest in an extensive search for
the right accommodations. The returns to a thorough search are far less for a one-time,
one-night visit. Within a broad range of prices, almost any hotel will do.
We address this situation by comparing one- and multi-unit purchasers. Some consumers
buy only one unit of a good while others acquire ⌧ > 1 units. The fraction of consumers
are multi-unit purchasers. We continue to assume that good 0 has uniform appeal. Each
consumer’s willingness to pay is z. Preferences for good 1 are heterogenous. Multi-unit
purchasers are willing to pay v̄ per unit. Its value to one-time buyers is v.
This formalization relates simply to our hotel example. Good 0 is a standard hotel located
in the city center that caters to short-term business travelers. Good 1 is a long-term hotel
that offers amenities that are more valuable to long-term guests (for example, kitchenettes),
but which are not important for one-night visitors.23
As usual, there exists an equilibrium where only h0, zi is provided. However, if type-v̄
consumers search, providers of good 1 will spring up. Again consider an equilibrium with
product differentiation but no price dispersion. Even though a fraction of consumers wishes
to purchase multiple copies of a good, a moment of reflection suggests that the usual two
contracts will be on offer: 0 = h0, zi and 1 = h1, vi. As before, the equilibrium distribution
of contracts must adjust. Now, ⇤1 = (1 ⌧ (v)(vc)c z) , which generalizes our original result.
23
As v̄ < z is possible, the long-term hotel may be located a couple of bus stops away from the city center.
Given the costs of understanding the bus system, it may be objectively inferior to the standard hotel for
one-night stayers.
26
5.3
Increasing Search Costs
Although a standard assumption in the literature, a consumer’s search costs are rarely constant in practice. Rather, search costs typically rise over time. The increasing marginal
cost of time, and the limiting case of deadlines, provide reasons why this is so. A more
subtle reason search costs drift upward is that a person’s search technology deteriorates as
he substitutes toward more costly (or less effective) search methods once easy options get
exhausted.24
In Appendix A.6 we generalize our model by assuming that search costs rise over time.
Under suitable conditions, we show that there exists an equilibrium paralleling our results
from Theorem 3. Firms offer 0 = h0, zi or 1 = h1, vi and a type-v̄ consumer searches for
1 . However, a searching consumer settles for 0 if his search fails to yield a positive result
sufficiently quickly. As above, type-v consumers forgo search entirely.
A qualitatively similar behavioral pattern emerges when a searching consumer has a
constant search cost but is learning about the distribution of contracts as he goes along.25
The associated intuition is simple. A consumer embarks on his quest with an air of optimism
believing his desired product is likely around the corner. Since he is uncertain regarding the
actual distribution of contracts, each unfavorable draw dulls his confidence. Eventually,
pessimism takes over and he settles for what is available.
5.4
Adverse Selection and Insurance
As a final extension, consider the case of adverse selection.26 As a specific example, consider
health insurance.27 Individuals who are willing to pay more for health coverage, tend to be
those who consume the most medical resources. Hence, insuring them is more costly. Insurance markets are also complex, and thoughtfully searching among alternative policies is often
difficult. Even in cases where the direct costs of search are low, such as Medigap insurance,
where policies are standardized and quotes are posted on the internet, the perception of
high search costs likely deters many consumers from seriously investigating their options.28
24
Osberg (1993) recognizes the substitution among search methods in an empirical study of job search.
See McCall (1970) or Rothschild (1974) for formal analyses.
26
Recent studies of search economies with adverse selection include Inderst (2005), Guerrieri et al. (2010),
and Lauermann and Wolinsky (2013). Related studies focusing on insurance markets include Mathewson
(1983), Schlesinger and von der Schulenburg (1991), and Seog (2002).
27
Cutler and Zeckhauser (2000) summarize this market’s key features. Our model does not address moral
hazard which, along with adverse selection, is an important caveat in studying insurance markets.
28
Lin and Wildenbeest (2013) and Kim (2013) both find the presence of substantial search fictions in the
market for Medigap insurance.
25
27
Search frictions facilitate higher insurance premiums by reinforcing insurers’ market power
(Cebul et al., 2011). Samuelson and Zeckhauser (1988) document the high persistence in
health insurance plan choice among Harvard employees. Search costs, of course, provide one
explanation for this observation.
Consider an insurance market and assume that both types of consumers have a willingness
to pay of z for a “standard” policy (good 0). The “premium” policy (good 1) offers enhanced
coverage and elicits a willingness to pay of v and v̄ from the two types of consumers. The
fixed cost of supplying the premium policy ( ) can be interpreted as an extra administrative
cost, or the cost of establishing a network of higher-quality health-care providers.
To accommodate adverse selection, we depart from our baseline model in two ways. The
first departure is substantive. Suppose that the cost of serving a customer depends on the
customer’s type. Type-v̄ consumers are at high health risk and the marginal cost of providing
them coverage is c0 when they enroll in a standard policy and it is c1 > c0 when they enroll
in a premium policy. Type-v consumers have a low health risk. The cost of insuring a type-v
consumer is normalized to zero, regardless of the policy that they purchase. Ex ante, an
insurer cannot distinguish between the two types of consumers.
The second departure is technical. For the sake of argument suppose that
v
(c1
c0 )
< (1
and
)z
(v̄
c1 + c0 )
< z.
(A-10 )
(A-10 ) gently modifies Assumption A-1 on account of the novel cost structure. Furthermore,
to focus on a case where adverse selection impedes the market’s operation, suppose z < c0 <
c1 < v < v̄.
An equilibrium exists where no one searches. The no-search equilibrium may take one of
two forms. First, if
⇡0 = (z c0 ) + (1
)z < 0,
then insurance provision is not profitable and no coverage will be available. In this case,
adverse selection effectively destroys the market. The second possibility is less ominous. It
occurs when ⇡0 = (z c0 ) + (1
)z 0. Now in equilibrium all firms offer the basic policy
0 = h0, zi and all consumers accept it immediately. Since z < c0 , the presence of high-risk
consumers reduces the profitability of a standard policy. In both cases, despite the presence
of many consumers who could be well-served by an enhanced policy, no firm will offer it.
Consider the case where consumers search for insurance. Now two types of insurance
contracts will be offered: 0 = h0, zi and 1 = h1, vi. Type-v̄ consumers will search to find
28
an enhanced policy while type-v consumers will default into either option without bothering
to search. The consequences of this behavior are surprisingly beneficial. A firm’s equilibrium
profit increases to (1
)z and consumers gain as well. Type-v consumers are no worse off,
while type-v̄ consumers now enjoy access to their preferred product. Search behavior and the
costly self-sorting of consumers has led to a Pareto improvement. In this regard, coordinating
on an outcome where it is common practice for consumers to “shop around,” even if the direct
costs of search remain relatively high, can prove beneficial.
Given the contrast between the no-search and search equilibria, it is instructive to investigate policies that can nudge a market toward the Pareto-superior outcome. As in the
original model, several conditions must be satisfied. First, type-v̄ consumers must consider
search worthwhile. A type-v̄ consumer begins to search when the fraction of firms offering
⇤
v s/ 1 0, or equivalently
1 , 1 , is large enough. More formally, u (v̄) = v̄
1
s
¯1 ⌘
v̄
v
(10)
.
When the fraction of firms offering 1 exceeds ¯ 1 , it is sufficiently easy for type-v̄ consumers
to find their desired contract. Second, it is straightforward to show that when 0 = h0, zi
and 1 = h1, vi are the available policies and type-v̄ consumers search for 1 , fraction
⇤
1
(v
(1
=
c1 )
)(v
z)
(11)
of firms must offer 1 in equilibrium. Expression (11) is the direct analogue of (1) from
Section 4.1. It ensures that 0 and 1 are equally profitable for the firms given the pattern
of consumer search.29 Together, (10) and (11) imply that
¯1
⇤
1
(12)
is a necessary condition for an equilibrium with product differentiation. Finally, in direct
parallel to (2) in Theorem 3 above, search costs cannot be too high, i.e.
s
( ⇤1 )2 (1
)v
.
(13)
If they were, a premium-policy firm would be tempted to slightly raise its price while still
catering to high-risk consumers who happen to find its offer first.
29
Using (A-10 ) it can be shown that 0 <
⇤
1
< 1.
29
The policy implications of (11), (12) and (13) are straightforward. First, if no consumers
are engaged in search, then policies that only reduce costs per search (s) fail to generate
benefits. Consumers have no incentive to utilize a superior or cheaper search technology if
there is nothing to find in the first place. By contrast, a policy that encourages the supply
of 1 to rise has the potential to be more effective in practice. For example, a lump-sum
subsidy of  paid to any firm that offers 1 = h1, vi can help offset its fixed cost. The
required magnitude if the subsidy depends on the market’s default state. If in the absence
of consumer search 0 is a profitable contract, i.e. ⇡0 0, a subsidy of  = ⇡0 ⇡1 , where
⇡1 = (v
c1 ) + (1
)v
,
is sufficient to ensure that a firm is indifferent between offering 0 or 1 .30 Crucially, the
subsidy need only be taken up by fraction ¯ 1 of firms. Once that threshold is met, consumers
become confident that their search activity will be rewarded, the market for 1 thickens
rapidly, and 1 becomes profitable on its own merits. Similarly, if ⇡0 < 0 and no insurance
is available, subsidizing fraction ¯ 1 of firms to offer the premium 1 policy would allow the
market to surface, giving all consumers a choice between deluxe and basic insurance policies.
Thus, the targeted support of one new product can encourage close substitutes to ride into
the market on its coattails. In this case, a subsidy of  = ⇡1 would be sufficient.31 A simple
cost-benefit calculation can determine if and when and if these policies are worthwhile.
Generally, their effectiveness is greater whenever search costs are relatively small and the
main obstacle to overcome is the coordination problem between consumers and firms.32
30
(A-10 ) ensures that  = ⇡0 ⇡1
0. The firm is indifferent between 0 and 1 conditional on no
consumer engaging in search. In expectations, this is the most pessimistic case from a firms’ perspective.
31
This subsidy level ensures that a firm is indifferent between offering 1 or no policy conditional on no
consumer searching.
32
Assuming the direct policy implementation costs are negligible, the subsidy is worthwhile from a costbenefit perspective if the gain in consumer welfare and firm profits exceeds the subsidy. For example, when
⇡0 > 0 then the proposed subsidy is worthwhile if
|
(v̄ + z
s(1 + 1/
{z
[1]
⇤
1 )) +
}
|
(c0 z)
{z }
[2]
¯1
|{z}
(14)
[3]
Term [1] is the per-capita change in ex ante consumer welfare. Term [2] is the per-firm change in firm profits.
And term [3] is the per-firm subsidy cost. Since ¯ 1 = s/(v̄ v), (14) will be satisfied when s is sufficiently
small. A similar analysis applies to the ⇡0 < 0 case.
30
6
Concluding Remarks
Our simple model highlights a market-creation externality that emerges when consumers
search. Search leads to both product differentiation and heightened competition. The effects
we emphasized operate through the enhanced sorting that costly search behavior necessarily
spawns. On many occasions, equilibria with active search produce greater welfare and higher
profits. This conclusion persists despite the deadweight of search costs (which consumers
must bear), and despite firms’ curtailed market power (which dampens their profits).
Before concluding, we would like to explain how our model contrasts with previous analyses that emphasize search externalities or that employ similar terminology. Shimer and Smith
(2001) discuss both the thick-market and congestion externalities tied to search behavior.
Our setting revolves around distinct, albeit related, effects.
First, regarding thick markets, we have argued that search supports market expansion
and creation. However, search behavior alone cannot foster this expansion. Rather, it
requires market entry by producers as well as consumer search. By coordinating around an
outcome with superior sorting, local markets for particular goods thicken. On the one hand,
this effect ensures that it is safe for producers to offer new products. On the other hand,
consumers become sufficiently confident in their prospects for a successful search to incur
the associated costs.
A congestion externality, as described by Shimer and Smith (2001), is not present in our
environment. In our setting, a firm can match with an unlimited number of consumers, and
thus consumers do not elbow out one another. That said, our environment exhibits a related
pecuniary externality. As more firms produce a specific good, they steal some consumers
from their direct competitors. This effect exists in any market, so is to be expected.
Our conclusions concerning consumer sorting are distinct from the sorting externalities
studied by Burdett and Coles (1997). Derived in the context of a dynamic “marriage market”
setting, their sorting externality refers to the change in the distribution of available partners
as agents match and subsequently exit the market (Burdett and Coles, 1999). These specific effects play no role in our setting. In practical terms, we allow producers to practice
polygamy. Thus, consumers experience no direct effect from the search activity of other
agents. The indirect effects can be substantial, as we have explained above.
Our model was kept spare to plainly exhibit the effects of interest. Many extensions
beyond those proposed here may be pursued. For example, our study focused on the case of
competitive firms. Allowing for oligopolistic firms or a monopolist, who can internalize some
31
of the highlighted externalities, would complement our model.33 Our conclusions should
carry over to these alternative market structures.
Similarly, we have abstracted from the role of advertising in our analysis (Robert and
Stahl, 1993).34 A natural way to introduce advertising into our model is to allow consumers
to direct their search toward producers offering products they like, without wasting effort
visiting suppliers of the other good.35 Thus, a consumer’s search would focus on a subset of
firms, but he still incurs a cost to learn the details of a firm’s offer. In this model, the added
information helps consumers, but it may fail to bring benefits to firms absent a robust search
effort among all consumers. For example, if only type-v̄ consumers search for good 1 and
type-v̄ consumers find both goods acceptable, as in the equilibrium of Section 4.1 above, all
firms’ equilibrium profits remain (1
)z. Though consumers benefit in comparison to the
single-good, monopoly price outcome, firms profits end up lower. Other forms of advertising,
particularly those that serve as a coordination device on the search-intensive equilibrium,
may help sustain more appealing market-wide outcomes.36
A natural interpretation of our model concerns comparisons across different markets,
with some markets exhibiting search-intensive equilibria, and others characterized by buyers’
ready acceptance of sellers. Alternatively, the model can also be viewed as a description of
the same market operating at different moments in time. The sale of existing homes offers
an illustration. While homes are bought and sold throughout the year, it is well known that
existing home sales cluster in the spring and summer. Patient sellers are often advised not
to list their house in the fall or winter, since few people are searching for homes then. (Stale
houses tend to lose their value, and it is also somewhat costly in terms of convenience to
have one’s house on the market.) In the spring many people are searching, so houses are
listed on the market in vastly disproportionate numbers. Throughout the year this market
moves between a period of collectively focused search among many types of buyers and a
period when search is far less intense. Despite the fact that disproportionately many other
sellers are listing their homes at the same time, a patient seller often gains by waiting for
the search-intensive period in the spring to sell. An extension examining the transition
among equilibria would naturally capture this dynamic. As suggested by our brief analysis
33
A monopolist could produce many versions of the same product or offer multiple goods.
Bagwell (2007) offers a review of models of advertising in economic analysis, including search economies.
35
In this sense search becomes partially directed (Menzio, 2007; Goldberg, 2007). For example, consumers
have a list of all firms offering good 0 and good 1 and can use this list to guide their store visits.
36
There is evidence that certain markets may display the qualities of a “search-intensive” equilibrium. For
example, Couture (2012) documents the distances travelled in search for restaurant meals in U.S. cities.
Consumers often forgo close options in favor of establishments further afield.
34
32
in Section 5.4, such transitions would undoubtably feature a critical thresholds of buyers and
sellers beyond which the market would tip—in the sense of Schelling (1969, 1971)—toward
an equilibrium with a significantly higher search intensity.
The economic effect we have emphasized—a search-generated and coordination-supported
market enhancement externality—is central in many markets. Capitalizing on this effect
should be a policy goal. While reducing search costs directly is certainly beneficial, as a policy it is unlikely to be as effective on the margin in markets where few consumers undertake
active search. The fundamental coordination problem between consumers and producers
may remain unresolved. Even if search costs are consequential, the welfare gains from a
search-intensive equilibrium can be substantial as new markets and new business opportunities arise due to this search activity. The recent emergence of ride-share and taxi-like
services like Uber and Lyft provides but one example where enhanced search (in this case
due to the enormous capabilities of the Internet) enables a multi-product equilibrium to supplant one with a single product. In many taxi markets, these platforms have added a dash of
price dispersion combined with a dollop of product differentiation to motivate the required
change in consumer behavior. Curiously, our analysis suggests that traditional taxi services
may ultimately benefit from these developments if consumers reduce their use of personal
vehicles sufficiently and substitute more cars for hire—sometimes taxis, sometimes car services. Better integrating this observation into applied studies in industrial organization and
market design would be a valuable next step. A successful market thrives on the symbiotic
relationship between producers and consumers. Absent a coordination mechanism beyond
personal incentives, appropriate complementary actions must emerge and be sustained on
the two sides of the market.
33
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38
All appendices are intended for online publication only.
A
Proofs
A.1
Preliminary Lemmas
Lemma A.1. Consider a type-vi consumer who has access to contract j . Suppose this
consumer believes that all firms not yet sampled are following the same independent production/pricing strategy, ˜ 2 (⌃). Define u⇤ (vi ) as the unique solution to37
⇥
⇤
u⇤ (vi ) = E ˜ max{u⇤ (vi ), u( |vi ), 0}
(A.1)
s.
1. If max{u( j |vi ), 0} u⇤ (vi ), then it is optimal for consumer i to accept
the market if 0 > u( j |vi )).
j
(or, to exit
2. If u⇤ (vi ) > max{u( j |vi ), 0}, then it is optimal for consumer i to learn the contract
offer of the next firm at a cost of s.
Proof. Consumer i’s decision problem is essentially one of sequentially sampling from a
stationary distribution of independently and identically distributed random variables. The
value of each sampled contract is the random variable [u( |vi ) _ 0]. (Each sample’s value is
bounded below by zero since a consumer may opt out of the market.) The distribution of
this value is derived from the distribution of , which the consumer believes to be ˜ 2 (⌃).
Since prices are non-negative, the expectation of [u( |vi ) _ 0] exists. By DeGroot (1970,
Sections 13.5 and 13.9), the optimal strategy for the consumer is the above cutoff rule.
Lemma A.2. Suppose there exists an equilibrium where all firms adopt the strategy ⇤ . Let
u⇤ (vi ) be the cutoff value describing a type-vi consumer’s equilibrium search strategy. Let ⇤
be a contract such that:
1. u( ⇤ |v̄) > max{u⇤ (v̄), 0} and u( ⇤ |v) > max{u⇤ (v), 0}; or,
2. u( ⇤ |v̄) > max{u⇤ (v̄), 0} and max{u⇤ (v), 0} > u( ⇤ |v); or,
37
The notation in (A.1) is self-explanatory. Here, “ ” is random. Its distribution is ˜.
39
3. max{u⇤ (v̄), 0} > u( ⇤ |v̄) and u( ⇤ |v) > max{u⇤ (v), 0}.
Then ⇤ is not in the support of ⇤ . That is, there exists an alternative contract that gives
firm j strictly greater profits than ⇤ given consumers’ search behavior.
Proof. We give the proof for case 1 with the remaining cases following analogously. Suppose
the conditions of the lemma are satisfied and that firm j posts ⇤ . This contract is accepted
by both types of consumers whenever they encounter firm j. Suppose firm j offers instead
the contract ✏⇤ , which is identical to ⇤ except that it sets a price that is ✏ > 0 greater.
Suppose ✏ is sufficiently small so that u( ⇤ |v̄) > u( ✏⇤ |v̄) > max{u⇤ (v̄), 0} and u( ⇤ |v) >
u( ✏⇤ |v) > max{u⇤ (v), 0}. Clearly, ✏⇤ is accepted by both types of consumers whenever ⇤
was. Hence ✏⇤ is accepted by the same expected number of consumers as ⇤ . Since this
number is strictly greater than zero and since ✏⇤ is associated with a strictly higher price,
it will generate greater expected profits than ⇤ . Therefore, ⇤ cannot be a contract that is
offered in equilibrium.
Lemma A.3. Let u⇤ (vi ) be the cutoff value characterizing a consumer’s search strategy. (1)
u⇤ (v̄)
u⇤ (v). (2) If u⇤ (vi )
0, then there exists some in the support of ˜ such that
u( |vi ) > u⇤ (vi ).
d
Proof. Suppose ⇠ ˜. To simplify notation, define the real-valued random variable Xvi =
[u( |vi ) _ 0]. Let Fvi (x) be the associated distribution function. Since u( |v̄) u( |v) for
all , Xv̄ first-order stochastically dominates Xv . Hence Fv (x) Fv̄ (x) for all x. Finally, we
R
can write u⇤ (vi ) = max{u⇤ (vi ), x}dFvi s. For this equality to hold, Fvi (u⇤ (vi )) < 1.
1. We argue by contradiction. Suppose u⇤ (v̄) < u⇤ (v). Since Xv̄ first-order stochastically
R
R
dominates Xv , u⇤ (v) = max{u⇤ (v), x}dFv s  max{u⇤ (v), x}dFv̄ s. Thus,
⇤
u (v)
⇤
u (v̄) 

Z
Z
⇤
max{u (v), x}dFv̄
(u⇤ (v)
 (u⇤ (v)
 (u⇤ (v)
Z
max{u⇤ (v̄), x}dFv̄
u⇤ (v̄))1(x  u⇤ (v̄))dFv̄
u⇤ (v̄))Fv̄ (u⇤ (v))
u⇤ (v̄))Fv (u⇤ (v))
Since Fv (u⇤ (v)) < 1, we arrive at a contradiction. Therefore, u⇤ (v̄)
u⇤ (v).
2. Again we proceed by contradiction. Suppose u( |vi )  u⇤ (vi ) for all in the support
of ˜. Then, u⇤ (vi ) = E ˜[max{u⇤ (vi ), u( |vi ), 0}] s  E ˜[max{u⇤ (vi ), u⇤ (vi ), 0}] s =
u⇤ (vi ) s, which is a contradiction since s > 0.
40
A.2
Proof of Theorem 1
Lemma A.4 is used in the proof of Theorem 1, which follows.
Lemma A.4. There does not exist an equilibrium where only good 1 is produced.
Proof. We divide the argument into three cases. Throughout we assume that if a firm
produces good 1, it sets a price in the interval [c, v̄]. Prices below c and above v̄ lead to
negative profits.
1. Suppose there exists an equilibrium where all firms offer ⇤ = h1, pi, p 2 [c, v̄]. Since
all firms are offering the same contract, u⇤ (vi ) = max{u( ⇤ |vi ) s, s} for vi 2 {v, v̄}.
There are two sub-cases, each generating a contradiction.
(a) Suppose p  v. If p < v, then u( ⇤ |v) > max{u⇤ (v), 0} and u( ⇤ |v̄) > max{u⇤ (v̄), 0}.
By Lemma A.2 this is a contradiction. Hence, p = v. Therefore, the profits of a
typical firm must be v c
.
Suppose firm j offers instead the contract h0, zi. This contract is acceptable to all
type-v consumers and firm j ensures itself an expected profit of at least (1
)z.
By Assumption A-1, (1
)z > v c
. Hence, this alternative contract is more
profitable—a contradiction.
(b) Suppose instead that p 2 (v, v̄]. By reasoning analogous to the preceding case we
conclude that p = v̄. Thus, a typical firm’s profit (v̄ c)
. Now, the contract
h0, zi is acceptable by both types of consumer and will yield an expected profit
of z. Noting Assumption A-1, z > (v̄ c)
. Again, this is a contradiction.
2. Suppose there exists an equilibrium with price dispersion where all firms produce good
1 but consumers do not search. Thus, each firm sells its product to at most one
consumer.
Suppose = h1, pi and ˆ = h1, p̂i, c  p < p̂  v̄, are two contracts available in
equilibrium. Since a type-v̄ consumer does not search, max{u⇤ (v̄), 0}  u(ˆ |v̄) <
u( |v̄). If max{u⇤ (v), 0}  u(ˆ |v), then both contracts would be accepted by both
types of consumers on the equilibrium path. Such an outcome is not compatible with
profit equalization across contracts in equilibrium. Hence, u(ˆ |v) < max{u⇤ (v), 0} 
u( |v).
41
If max{u⇤ (v), 0} < u( |v), then Lemma A.2 implies a contradiction. Thus, max{u⇤ (v), 0} =
u( |v). Similarly, max{u⇤ (v̄), 0} = u(ˆ |v̄). Since the preceding reasoning applies to all
pairs of contracts there are at most two contracts available in this equilibrium.
If u⇤ (v)
0, then by Lemma A.3 there must exist some available contract, say ˜ =
h1, p̃i, such that u(˜ |v) > u⇤ (v). But this is a contradiction. Hence, u⇤ (v) < 0.
Therefore, u( |v) = 0, which implies p = v. Thus, the expected profit of a typical firm
in equilibrium is v c
< (1
)z. But then any firm offering can increase its
expected profit by posting h0, zi instead—a contradiction.
3. Suppose there exists an equilibrium with price dispersion where all firms produce good
1 but (some) consumers do search on the equilibrium path. If a consumer ever searches,
he will certainly also search whenever he matches with the most expensive contract
in the economy. Let ˆ = h1, p̂i be that contract. Note that both types of consumers
cannot search on the equilibrium path after matching with ˆ . Otherwise, any firm
offering it would earn strictly negative profits. Hence, there are two cases.
(a) Suppose a type-v consumer searches on the equilibrium path. Thus, u⇤ (v)
0
and u(ˆ |v)  u⇤ (v). Since ˆ must be acceptable to some type of consumer,
max{u⇤ (v̄), 0}  u(ˆ |v̄). Since u⇤ (v)
0, by Lemma A.3 there exists some
contract in the support of the firm’s strategy, = h1, pi, such that u⇤ (v) < u( |v).
But, without loss of generality, this also implies 0  u⇤ (v)  u⇤ (v̄) < u( |v̄), which
contradicts Lemma A.2.
(b) Suppose a type-v̄ consumer searches on the equilibrium path. Thus, u⇤ (v̄) 0,
u(ˆ |v̄)  u⇤ (v̄), and max{u⇤ (v), 0}  u(ˆ |v). By Lemma A.3 there exists some
contract in the support of the firm’s strategy, = h1, pi, such that u⇤ (v̄) < u( |v̄).
But this also implies max{u⇤ (v), 0} < u( |v). Again, we contradict Lemma A.2.
The three cases considered are exhaustive of the situations that may occur if only good 1 is
produced.
Proof of Theorem 1. We first argue that the proposed situation is an equilibrium. If every
firm offers ⇤ = h0, zi and consumers believe this to be the case, no consumer searches. Firm
profits are z. No firm can gain by offering h0, pi, p < z. Since no consumer is searching, this
alternative contract implies lower profits. If instead a firm offers h1, pi, its expected profits
are bounded above by max{v c
, (v̄ c)
} < z (Assumption A-1). Hence, this too
is not a profitable deviation. Therefore, the market is in equilibrium.
42
Lemma A.4 shows that there cannot exist an equilibrium where only good 1 is produced.
Thus, to establish uniqueness it is sufficient to show that there is no other equilibrium where
only good 0 is produced. Let ⇤ be the equilibrium strategy of a typical firm in a symmetric
equilibrium where only good 0 is produced. In this equilibrium, each firm must earn strictly
positive profits. To see this fact, note that firm j can offer the contract j = h0, ✏i where
0 < ✏ < s. Even if consumer i = j believed all other firms were offering good 0 at a price
of zero, he would still prefer to accept j without any search. This gives firm j a profit of
at least ✏. Since a firm’s equilibrium profit is strictly greater than zero, every contract in
the support of ⇤ must satisfy u( |vi )
max{u⇤ (vi ), 0}. (Note that u⇤ (v̄) = u⇤ (v) since
only good 0 is available.) By Lemma A.2, this implies u( |vi ) = max{u⇤ (vi ), 0}. Hence,
there is at most one contract offered in this equilibrium. But this implies u( |vi ) = 0 and so
z = p.
A.3
Proof of Theorem 2
Theorem 2 is proved by Lemmas A.5–A.12. In each lemma we assume that there exists a
symmetric equilibrium with product differentiation. ⇤ denotes the equilibrium strategy of
a typical firm. Ck is the set of contracts in the support of ⇤ that provide good k.
Lemma A.5. In an equilibrium with product differentiation, u⇤ (v̄)
0.
Proof. Suppose 0 > u⇤ (v̄). By Lemma A.3, 0 > u⇤ (v̄)
u⇤ (v). No consumer finds it
worthwhile search. Any firm offering h0, pi will have an incentive to increase p until p = z
since both types of consumers accept h0, pi for all p  z. Hence, in equilibrium each firm
earns z in profits. Since no consumers search, the maximal expected profits of a firm offering
good 1 are bounded above by max{ (v̄ c)
,v c
} < z, contradicting the equality
of expected profits across equilibrium contracts.
Lemma A.6. There exists
⇤
2 C1 such that u( ⇤ |v̄) > u⇤ (v̄).
Proof. Suppose the contrary. If u( |v̄)  u⇤ (v̄) for all 2 C1 , then by Lemma A.3 there
exists ˜ 2 C0 such that u(˜ |v̄) > u⇤ (v̄)
0. Since u(˜ |v̄) = u(˜ |v) and u⇤ (v̄)
u⇤ (v),
u(˜ |v) > max{u⇤ (v), 0}. But this contradicts Lemma A.2.
Lemma A.7. There exists
⇤
2 C1 such that u( ⇤ |v)
max{u⇤ (v), 0}.
Proof. Suppose max{u⇤ (v), 0} > u( |v) for all
2 C1 . By Lemmas A.5 and A.6 there
⇤
exists ˜ 2 C1 such that u(˜ |v̄) > max{u (v̄), 0}. But then max{u⇤ (v), 0} > u(˜ |v), which
contradicts Lemma A.2.
43
Lemma A.8. The expected profit of a firm in an equilibrium with product differentiation is
strictly positive.
Proof. Negative profits cannot be a feature of equilibrium. Thus, suppose they are zero.
This implies that every contract of the form h0, pi, 0 < p < z must not be acceptable to
either type of consumer. In particular, this implies that 0 < z  u⇤ (v).
By Lemma A.3, there exists a contract, ⇤ , offered in equilibrium such that u⇤ (v) <
u( ⇤ |v). If u⇤ (v̄) < u( ⇤ |v̄), then we would contradict Lemma A.2. Hence u⇤ (v̄) u( ⇤ |v̄).
There are now two possibilities.
1. If u⇤ (v̄) = u( ⇤ |v̄), then by Lemma A.3 there exists an alternative contract available
in equilibrium, ⇤⇤ , such that u( ⇤⇤ |v̄) > u⇤ (v̄). But in this case a type-v consumer
would also benefit from ⇤⇤ . Thus, u( ⇤⇤ |v) > u⇤ (v), contradicting Lemma A.2.
2. If u⇤ (v̄) > u( ⇤ |v̄), then we contradict Lemma A.2 immediately.
In either case we arrive at a contradiction.
Lemma A.9. For all
2 C0 , u( |v)
max{u⇤ (v), 0}.
Proof. Suppose there exists
2 C0 such that max{u⇤ (v), 0} > u( |v). Since u( |v̄) =
u( |v) < max{u⇤ (v), 0}  u⇤ (v̄), is not accepted by any consumers. Therefore, the profits
of a firm offering must be zero contradicting Lemma A.8.
Lemma A.10. |C0 |  2.
Proof. Let = h0, pi and ˆ = h0, p̂i and suppose , ˆ 2 C0 . Without loss of generality,
suppose 0 < p < p̂  z. By Lemma A.9, u( |v) > u(ˆ |v)
max{u⇤ (v), 0}. Since both
contracts must generate the same expected profits, ˆ must not be accepted by all type-v̄
consumers. Thus, u⇤ (v̄) u(ˆ |v̄).
Suppose u⇤ (v̄) = u(ˆ |v̄). If in equilibrium a type-v̄ consumer does not accept ˆ with
probability 1, then any firm offering ˆ can decrease its price by ✏ > 0, and induce all
type-v̄ consumers to accept. This infinitesimal decrease in price leads to a discrete increase
in demand and would be a profitable deviation. Therefore, u⇤ (v̄) > u(ˆ |v̄). If u(ˆ |v) >
max{u⇤ (v), 0}, then there would be a contradiction with Lemma A.2. Thus,
u(ˆ |v) = max{u⇤ (v), 0}.
44
(A.2)
Now consider . If u( |v̄) > u⇤ (v̄), then again Lemma A.2 is contradicted. If u( |v̄) <
u⇤ (v̄), then it is only accepted by type-v consumers—contradicting the equality of profits
between and ˆ . Therefore,
u( |v̄) = u⇤ (v̄).
(A.3)
Therefore, whenever there are at least two distinct contracts in C0 , they must satisfy (A.2)
and (A.3). Since these equations must hold for any pair of distinct contracts in C0 , there are
at most two elements in C0 .
Lemma A.11. For all
2 C1 , u( |v̄)
u⇤ (v̄).
Proof. We argue by contradiction. Suppose there exists ˆ 2 C1 such that u(ˆ |v̄) < u⇤ (v̄).
Without loss of generality we may assume that ˆ is the most expensive contract in C1 .
If type-v̄ consumers do not accept ˆ , type-v consumers must find it acceptable, u(ˆ |v)
max{u⇤ (v), 0}; else, a firm would earn negative profits. If the preceding inequality is strict,
Lemma A.2 would be violated. Hence, u(ˆ |v) = max{u⇤ (v), 0}. Since ˆ was the most
expensive contract in C1 , a type-v consumer must accept all contracts in C1 . By Lemma A.9,
a type-v consumer also accepts all contracts in C0 . Hence, a type-v consumer does search in
equilibrium.
Suppose u(ˆ |v) = 0. Then the profits of a firm offering ˆ are at most (1
)(v c)
.
Noting Assumption A-1, such a firm can strictly increase its profit by offering the contract
h0, zi instead since (1 )(v c)
 (1 )(v c )  (1 )z. Hence, u(ˆ |v) = u⇤ (v) > 0.
Next, suppose there exists ˜ 2 C0 such that u(˜ |v̄) > u⇤ (v̄). Since u⇤ (v̄)
u⇤ (v),
u(˜ |v̄) = u(˜ |v) > u⇤ (v). This, however, contradicts Lemma A.2. Therefore, u( |v̄)  u⇤ (v̄)
for all 2 C0 . Thus, noting Lemma A.3, we conclude that there exists ˇ 2 C1 such that
u(ˇ |v̄) > u⇤ (v̄) u⇤ (v) 0. But this also contradicts Lemma A.2.
Lemma A.12. |C1 |  2.
Proof. Let = h1, pi and ˆ = h1, p̂i and suppose , ˆ 2 C1 . Without loss of generality,
suppose 0 < p < p̂  v̄. Moreover, suppose is the least-expensive contract in C1 . By
Lemma A.11, both contracts are accepted by type-v̄ consumers:
u( |v̄) > u(ˆ |v̄)
max{u⇤ (v̄), 0}.
(A.4)
If u( |v) < max{u⇤ (v), 0}, Lemma A.2 is violated. Thus,
u( |v)
max{u⇤ (v), 0}.
45
(A.5)
If the inequality in (A.5) was strict, Lemma A.2 would be violated. Thus, u( |v) =
max{u⇤ (v), 0} > u(ˆ |v). But then Lemma A.2 implies that u(ˆ |v̄) = max{u⇤ (v̄), 0}. Since
ˆ was an arbitrary contract in C1 \ { }, we conclude that there are at most two distinct
elements in C1 .
A.4
Proof of Theorem 3
Theorem 3 draws on three preliminary results: Lemma 1, which is stated in the main text,
Lemma A.13, and Lemma A.14. The proof of Corollary 1 follows the proof of Theorem 3.
Proof of Lemma 1. Suppose a typical firm follows the mixed strategy ⇤ where it selects 1
with probability and 0 with probability 1
. By Lemma A.11, 1 must be accepted
by a type-v̄ consumer: u( 1 |v̄) max{u⇤ (v̄), 0}. By Lemma A.7, 1 must also be accepted
by a type-v consumer: u( 1 |v) max{u⇤ (v), 0}. By Lemma A.9, 0 must be accepted by a
type-v consumer: u( 0 |v) max{u⇤ (v), 0}.
Suppose u( 1 |v) > max{u⇤ (v), 0}. Then, u( 1 |v̄) = max{u⇤ (v̄), 0}, else Lemma A.2 is
violated. Since u( 1 |v̄) > 0, u⇤ (v̄) > 0. By Lemma A.3, u( 0 |v̄) > u⇤ (v̄), which implies
u( 0 |v) > max{u⇤ (v), 0}. However, this conclusion contradicts Lemma A.2. Therefore,
u( 1 |v) = max{u⇤ (v), 0}. Now there are two cases.
1. Suppose u( 0 |v) > max{u⇤ (v), 0}. Then if u( 0 |v̄) 6= max{u⇤ (v̄), 0}, Lemma A.2 is
contradicted. Hence, u( 0 |v̄) = max{u⇤ (v̄), 0} and so u( 1 |v̄) > max{u⇤ (v̄), 0} by
Lemma A.3. Therefore, on the equilibrium path no consumers search and the profit of
a typical firm is p0 = p1 c
.
Suppose a firm offered the contract ˜0 = h0, p̃0 i where p̃0 = p1 v + z. Since u(˜0 |v) =
z p̃0 = v p1 = u( 1 |v), this is a feasible contract and it is accepted by all type-v
consumers. Therefore, it gives the firm profits of at least (1
)p̃0 . However, from
Assumption A-1 and the fact that p1  v we conclude that
(1
)z > v
c
=) (1
)z > (1
)v + p1
() (1
)(z
() (1
)(p1 + z
v) > p1
() (1
)p̃0 > p1
c
v) + (1
c
)p1 > (1
)p1 + p1
c
c
Hence, a firm has a profitable deviation in offering ˜ . Hence, u( 0 |v) > max{u⇤ (v), 0}
is not compatible with an equilibrium.
46
2. Suppose u( 0 |v) = max{u⇤ (v), 0}. Then u( 1 |v) = u( 0 |v). Hence, u⇤ (v)  0 and
0 = u( 1 |v) = u( 0 |v). But this implies p1 = v and p0 = z, as required.
Lemma A.13. Suppose all firms are following the strategy ⇤ where a firm selects 1
with probability
and 0 with probability 1
. Suppose all consumers adopt an optimal search strategy holding correct beliefs, ˜ = ⇤ . Suppose u( 0 |v)
max{u⇤ (v), 0},
u( 1 |v)
max{u⇤ (v), 0}, and u( 1 |v̄)
max{u⇤ (v̄), 0} > u( 0 |v̄). Suppose firm j deviates from ⇤ and offers j such that u( j |v̄) max{u⇤ (v̄), 0}. Then the expected number of
consumers with indices i < j who accept firm j’s contract is ( 1 1).
Proof. Given the contracts available, type-v consumers do not search. Type-v̄ consumers
search when they match to 0 . Otherwise, they accept 1 or j should they reach firm j.
First, we calculate the probability that exactly q consumers with indices i < j accept
firm j’s contract. Let k
q. Conditional on all firms j 0 2 {j k, . . . , j 1} offering the
contract 0 and firm j k 1 offering the contract 1 , for q consumers with indices i < j to
accept the contract offered by firm j, exactly q consumers with indices j k  i  j 1 must
be of type-v̄. In this case, no consumers with indices i  j k 1 will search long enough
to become acquainted with the contract offer of firm j. Thus, conditional on such a contract
offer by firms, the probability that q consumers with indices i < j accept the contract offered
by firm j is kq q (1
)k q . For each k, the events described above are disjoint. Thus, the
probability that exactly q consumers with indices i < j accept firm j’s contract is
1
X
k=q
(1
✓ ◆
k q
)
(1
q
k
)k
q
= (1
)q
q
( +
)
1 q
.
We can use this probability to compute the expected number of consumers with indices i < j
accepting j’s contract offer. This calculation simplifies conveniently:
P1
1
)q q ( +
) 1 q] =
1 .
q=1 q · [(1
Lemma A.14. There exists a unique
In fact, ⇤ = (1 (v)(vc) c z) .
⇤
2 (0, 1) such that 1
+
⇤
(v c)
= (1
)z.
Proof. The function f ( ) = 1
+ (v c)
is continuous on (0, 1] and strictly
decreasing. Moreover, lim !0+ f ( ) = 1 and f (1) = v c
< (1 )z. Hence, there exists
⇤
⇤
a unique
such that f ( ) = (1
)z. Solving this equation gives ⇤ = (1 (v)(vc) c z) .
47
Proof of Theorem 3. Suppose a typical firm follows the mixed strategy ⇤ where it selects
⇤
⇤
1 with probability 1 and 0 with probability 1
1 . When a consumer’s beliefs are that
⇤
˜ = , his search strategy is defined by the cutoff values u⇤ (v) = s and u⇤ (v̄) = v̄ v s/ ⇤ .
1
⇤
⇤
38
Condition (2) implies u (v̄) = v̄ v s/ 1
0. Hence, a type-v̄ consumer will search
for 1 in lieu of accepting 0 or opting out of the market. He will stop his search once he
encounters a contract ˜ such that u(˜ |v̄) u⇤ (v). A type-v consumer does not search.
We next compute a firm’s expected profits. If firm j offers ˜ , which is acceptable to
both types of consumers, the expected total number of consumers who accept firm j’s offer
is 1
+ ⇤ . Consumer i = j accepts the contract without engaging in search. To this
1
one consumer, we add / ⇤1
(Lemma A.13). This is the expected number of type-v̄
consumers who end up accepting firm
⇣ j’s contract
⌘ offer. In particular, when firm j offers 1 ,
its expected profits are ⇧j ( 1 ) = 1 + ⇤
(v c)
. If firm j offers 0 , its expected
1
profits are ⇧j ( 0 ) = (1
)z. An equilibrium requires ⇧j ( 0 ) = ⇧j ( 1 ). By Lemma A.14,
there exists a unique ⇤1 2 (0, 1) ensuring this equality: ⇤1 = (1 (v)(vc) c z) . Thus, both 0
and 1 give a firm the same expected profits.
The final step is to verify that no firm can profitably deviate. There are two relevant
classes of deviations. A firm may offer good 0 at a price less than z or good 1 at a price
more than v. Other deviations are clearly not optimal.
1. Suppose firm j offers ˜ = h0, p̃i, p̃ < z. If p̃ is very close to z, the firm’s expected
profits will decline. For ˜ to increase the firm’s profits, it must entice type-v̄ consumers
who otherwise would search to stop and to accept it. Thus, z p̃ v̄ v s/ ⇤1 . To
maximize its expected profit from this deviation, the firm would set p̃ = z v̄ + v +
s/ ⇤1 . Hence, the maximal expected profits associated with this alternative strategy
are ⇧j (˜ ) = (1 + / ⇤1
) (z v̄ + v + s/ ⇤1 ). Rearranging condition (2) we conclude
that ⇧j ( 1 ) ⇧j (˜ ). Hence, firm j cannot increase its profits through this deviation.
2. Suppose firm j offers ˜ = h1, p̃i where p̃ > v. In this case, no type-v consumers will
accept the contract. A type-v̄ consumer will accept ˜ if and only if v̄ p̃ v̄ v s/ ⇤1 .
The maximum price which ⇣will be accepted
is p̃ = v + s/ ⇣⇤1 . When p̃⌘= v + s/ ⇤1 , the
⌘
expected profit is ⇧j (˜ ) =
+ ⇤
(p̃ c)
= ⇤ v c + s⇤
. However,
1
1
1
s
⇧j (˜ ) ⇧j ( 1 ) = ( ⇤ )2 (1 )(v c)  0. Where the inequality follows from condition
1
(3).
38
To confirm this fact, it is sufficient to verify that z c
contrary and substitute for ⇤1 we observe that v > z + c + (v
a contradiction.
48
/(1
+ / ⇤1 )
v. If we suppose the
c)/( + z(1
)) > z + c + v c, which is
Proof of Corollary 1. From Theorem 3 it follows that if ⇤ is the equilibrium strategy of
a firm when the search costs are s̄, ⇤ continues to be an equilibrium if search costs are
s 2 (0, s̄).
The expected profit of a typical firm in the equilibrium of Theorem 3 is (1
)z. All
type-v consumers expect a payoff of zero. Type-v̄ consumers earn an expected payoff of
⇤
⇤
v) + (1
v s/ ⇤1 ). Hence, total expected surplus for a firm-consumer pair
1 (v̄
1 )(v̄
⇤
is (1
)z + ( ⇤1 (v̄ v) + (1
v s/ ⇤1 )). This expression must exceed z. Taking
1 )(v̄
the limit s ! 0+ , we see that such an outcome is possible if and only if v̄ v z.
A.5
Multiple Goods Trading at Multiple Prices
Proof of Lemma 2. The result follows from Lemmas A.5–A.12 in the proof of Theorem 2.
1. It is sufficient to establish that u( 0 |v) > u(ˆ0 |v) = max{u⇤ (v), 0} and u( 0 |v̄) = u⇤ (v̄).
By Lemma A.9, 0 and ˆ0 must be acceptable to a type-v consumer. Thus, u( 0 |v) >
u(ˆ0 |v) = max{u⇤ (v), 0}. The equality is due to (A.2) in the proof of Lemma A.10.
From (A.3) we have u( 0 |v̄) = u⇤ (v̄) as needed.
2. It is sufficient to establish that u( 1 |v) = max{u⇤ (v), 0} and u( 1 |v̄) > u⇤ (v̄) = u(ˆ1 |v̄).
By Lemma A.11, 1 and ˆ1 must be acceptable to a type-v̄ consumer. Thus, u( 1 |v̄) >
u(ˆ1 |v̄) = u⇤ (v̄) where the equality follows from the proof of Lemma A.12. Also from
the proof of Lemma A.12 we have u( 1 |v) = max{u⇤ (v), 0}.
Proof of Lemma 3. By Theorem 2, only four classes of equilibria may exist in our economy.
We analyzed the case of product differentiation and no price dispersion in section 4.1. In that
case we showed that a typical firm’s profits are (1
)z. Therefore, if a Pareto improvement
is to occur at least one good must trade at multiple prices.
1. Suppose there is an equilibrium where firms offer only 0 , 1 , and ˆ0 with positive
probability. That is, only good 0 trades at multiple prices. From Lemma 2 we know
that u( 0 |v) > u(ˆ0 |v) = max{u⇤ (v), 0} and u( 0 |v̄) = u⇤ (v̄) > u(ˆ0 |v̄). Hence ˆ0 is acceptable only to type-v consumers and is not accepted by type-v̄ consumers. Moreover,
type-v consumers always accept 0 .
49
Suppose that type-v consumers always accept 1 . Then the expected profit of a firm offering ˆ0 is at most (1 )z, which is not reflective of a Pareto improvement. Therefore,
a type-v consumer must search with positive probability if his only available contract
is 1 . That is, u( 1 |v)  u⇤ (v). There are two relevant sub cases:
(a) Suppose u( 1 |v) < u⇤ (v). Therefore, only type-v̄ consumers accept 1 . If u( 1 |v̄) =
v̄ p1 > max{u⇤ (v̄), 0}, then a typical firm offering 1 could slightly increase its
price without affecting its demand. Therefore, in equilibrium the price of good 1
must satisfy v̄ p1 = max{u⇤ (v̄), 0}. But this leads to a contradiction. Specifically, if v̄ p1 = u( 1 |v̄) = u⇤ (v̄) 0, then using Lemmas A.1 and 2 we see that
u⇤ (v̄) = 0 u⇤ (v̄) + ˆ 0 u⇤ (v̄) + 1 max{u⇤ (v̄), 0} s, which is impossible. On the
other hand, if u⇤ (v̄) < 0 then the profits generated by the contract 1 are at most
(v̄ c)
, which is less than z by Assumption A-1. Hence, this outcome cannot
lead to a Pareto improvement.
(b) Suppose u( 1 |v) = u⇤ (v). In this case, a type-v consumer is indifferent between
searching for an alternative offer and accepting 1 . Any firm offering 1 can
slightly decrease its price (which is strictly positive) and discretely increase its
expected profits by encouraging type-v consumers to accept its offer rather than
search. Hence, this case is not compatible with an equilibrium.
2. Suppose there is an equilibrium where firms offer only 0 , 1 , and ˆ1 with positive
probability. That is, only good 1 trades at multiple prices. From Lemma 2 we know
that u( 1 |v) = max{u⇤ (v), 0} > u(ˆ1 |v) and u( 1 |v̄) > u(ˆ1 |v̄) = u⇤ (v̄). Therefore, ˆ1
is only acceptable to type-v̄ consumers. If all type-v̄ consumers accept 0 , then the
expected profits of a typical firm offering ˆ1 are at most (v̄ c)
, which is less
than z by Assumption A-1. Hence, some type-v̄ consumers must search if 0 is their
only available contract. That is, u( 0 |v̄)  u⇤ (v̄). There are two sub cases to consider.
(a) If u( 0 |v̄) < u⇤ (v̄), then no type-v̄ consumers accept 0 . For 0 to generate an
expected profit greater than z, u( 0 |v) > max{u⇤ (v), 0}, so as to entice some
type-v consumers to search. (The strict inequality is necessary.) However, this
implies that any firm offering 0 can strictly increase its price slightly without
affecting its demand, contradicting the equilibrium condition.
(b) If u( 0 |v̄) = u⇤ (v̄), then any firm offering 0 can strictly increase its profit by
reducing p0 slightly thereby encouraging type-v̄ consumers to accept 0 in lieu of
searching for a good 1 contract. This contradicts the equilibrium condition.
50
Hence, three of the four possible classes of equilibria are not compatible with a Paretoimproving outcome. Therefore, if there is an equilibrium with the desired characteristics, it
must involve two goods trading at multiple prices.
Second, the requisite pattern of consumer search follows from Lemma 2. First, u⇤ (v̄) > 0
since u⇤ (v̄) = u( 0 |v̄) = z p0 > z p̂0 = u(ˆ |v) = max{u⇤ (v̄), 0} 0. Second, if u⇤ (v) < 0,
then ˆ0 is acceptable only to a type-v consumer. But this implies that ⇧(ˆ0 )) = (1
)z.
Thus, firms’ equilibrium profits fall below the benchmark case.
Remark A.1. In any equilibrium where a consumer is indifferent between accepting a contract
and continued search, he must accept the contract. Else, a firm offering the contract in
question can lower the price slightly prompting a discrete increase in expected profits.
Proof of Lemma 4. Since u⇤ (v̄) u⇤ (v) 0, Lemma 2 implies that z
and v̄ p1 > v̄ p̂1 = u⇤ (v̄). We can write (A.1) as
p0 = u⇤ (v̄) > z
p0 ) + ˆ 0 max{z p̂0 , 0, u⇤ (v̄)} + 1 (v̄ p1 ) + ˆ 1 (v̄
= 0 u⇤ (v̄) + ˆ 0 u⇤ (v̄) + 1 (v̄ p1 ) + ˆ 1 u⇤ (v̄) s
u⇤ (v̄) =
0 (z
= (1
1 )u
() u⇤ (v̄) = v̄
p1
⇤
(v̄) +
s/
1 (v̄
p1 )
p̂1 )
p̂0
s
s
1
The argument for u⇤ (v) is analogous.
To derive the relationships among prices we use equalities from Lemma 2 and the immediately preceding computations.
u(ˆ0 |v) = u⇤ (v) =) z
⇤
p̂0 = z
p0
s
=) p̂0 = p0 +
0
u( 1 |v) = u (v) =) v
p1 = z
u( 0 |v̄) = u⇤ (v̄) =) z
p0 = v̄
u(ˆ1 |v̄) = u⇤ (v̄) =) v̄
p̂1 = v̄
p0
s
0
=) p1 = p0 +
0
p1
s
s
1
s
+v
z
+z
v̄
0
=) p0 = p1 +
1
p1
s
s
1
=) p̂1 = p1 +
s
1
Proof of Lemma 5. From Section 4.2, we can easily describe consumer search behavior. A
type-v consumer searches for an alternative offer if ˆ1 is the only contract that is available.
He accepts 0 , ˆ0 , and 1 . A type-v̄ consumer searches for an alternative offer if ˆ0 is the
only contract that is available. He accepts 0 , 1 , and ˆ1 .
51
1. The contract 0 = h0, p0 i is acceptable to both types of consumers. We must compute
the expected number of consumers with indices i < j who accept firm j’s offer.
Suppose for the moment that at least one type-v̄ consumer with an index i < j accepts
1 must be offering the contract ˆ0 = h0, p̂0 i.
0 from firm j. This implies that firm j
But then no type-v consumers with indices i < j will accept firm j’s offer. Any type-v
consumer who is searching would accept ˆ0 from firm j 1 before learning the offer of
firm j. Likewise, if at least one type-v consumer with an index i < j accepts 0 from
firm j, then no type-v̄ consumers with indices i < j will accept the offer from firm j.
Thus, firm j sells its product to either q consumers of type-v̄ who are searching or to
q 0 consumers of type-v who are searching. Never does it sell its product to a mixed
group of buyers with indices i < j.
Following the reasoning in the proof of Lemma A.13, the probability that exactly q
P
ˆk
ˆ 0 ) k q (1
type-v̄ consumers who are searching accept firm j’s offer is 1
k=q 0 (1
q
(1 ˆ 0 ) ˆ q0 q
.
(1 ˆ 0 (1 ))q+1
Similarly, the probability that exactly q type-v consumers who
P
(1 ˆ ) ˆ q (1 )q
ˆk
ˆ 1 ) k (1
are searching accept firm j’s offer is 1
)q k q = (1 1 ˆ 1 )q+1 .
k=q 1 (1
q
1
We can compute the expected number of consumers with indices i < j who accept firm
j’s offer of 0 to be
)k
q
=
1
X
q=1
"
(1
ˆ0)ˆq
0
ˆ 0 (1
(1
q
+
))q+1
(1
ˆ 1 ) ˆ q (1
1
(1
)q
ˆ 1 )q+1
#
q=
ˆ0
1
ˆ0
+
(1
1
)ˆ1
.
ˆ1
Hence, accounting
for consumer i⌘= j who
⇣
⇣ also accepts
⌘ 0 , the expected profit of firm
ˆ0
(1 ) ˆ 1
j is ⇧j ( 0 ) = 1 + 1 ˆ + 1 ˆ
p0 = 1 ˆ + 11 ˆ p0 .
0
1
0
1
2. The reasoning is identical to the preceding case.
3. The contract ˆ0 = h0, p̂0 i is accepted only by type-v consumers. Therefore, noting
the
the expected number of type-v consumers who accept h0, p̂0 i is
h above calculations,
i
(1 ) ˆ 1
1
(1
)+ 1 ˆ
= 1 ˆ . Hence, the expected profit is ⇧j (ˆ0 ) = 11 ˆ p̂0 .
1
1
1
4. The reasoning is identical to the preceding case.
Lemma A.15. There exists an equilibrium where both goods trade at multiple prices and
consumer search conforms to the pattern described in Lemma 3 if and only if there exists
52
strictly positive constants— 0 ,
1,
ˆ 0 , ˆ 1 —such that
v̄
s
v=
+
s
0
(A.6)
1
(A.7)
⇧j ( 0 ) = ⇧j ( 1 )
s
p1 = p0 +
+v
(A.8)
z
0
z
1=
s
p0
0
+
1
+ ˆ0 + ˆ1
(A.9)
(A.10)
0
0
where
p0 =
1
1
ˆ0 1
·
ˆ1
·
s
and
p1 = c +
0
1
ˆ1
1
ˆ0 1
·
·
s
.
(A.11)
1
Proof. Conditions (A.6)–(A.10) reproduce the conditions identified in Section 4.2. Thus,
necessity follows from the discussion in that section. To prove sufficiency we reverse the
argument. Let 0 , 1 , ˆ 0 , and ˆ 1 be the strictly positive values that satisfy the conditions
(A.6)–(A.10). We use these values to define four contracts, two for each good, as follows:
• Let
0
= h0, p0 i be a contract supplying good 0 at a price of
p0 =
ˆ0 1
·
ˆ1
1
1
·
s
(A.12)
.
0
• Let ˆ0 = h0, p̂0 i be a contract supplying good 0 at a price of p̂0 = p0 + s/ 0 , where p0
is given by (A.12).
• Let
1
= h1, p1 i be a contract supplying good 1 at a price of
p1 = c +
1
ˆ1
1
ˆ0 1
·
·
s
.
(A.13)
1
• Let ˆ1 = h1, p̂1 i be a contract supplying good 1 at a price of p̂1 = p1 + s/ 1 , where p1
is given by (A.13).
Suppose each firm adopts the strategy j⇤ where contract k (ˆk ) is offered with probability
ˆ
ˆ
ˆ
k ( k ). Since 1 = 0 + 1 + 0 + 1 , all prices defined above are strictly positive and the
strategy is a valid mixed strategy. To verify the equilibrium, we first characterize consumers’
search behavior given j⇤ . Subsequently we compute the expected profits associated with each
53
contract given the consumers’ search strategy. Finally, we verify that no firm has a profitable
deviation from the mixed strategy j⇤ .
The Optimal Search Strategy for a Type-v Consumer. Given the contracts defined above,
u( 0 |v) = z
u( 1 |v) = v
u(ˆ0 |v) = z
u(ˆ1 |v) = v
p0
p1
p̂0 = z
p̂1 = v
p0
p1
s/ 0
s/ 1 .
(A.10) implies that u( 0 |v) > u(ˆ0 |v) 0. Moreover, by (A.8), u( 1 |v) = v p1 = v
p0 s/ 0 v +z = z p0 s/ 0 = u(ˆ0 |v). Therefore, u( 0 |v) > u( 1 |v) = u(ˆ0 |v) > u(ˆ1 |v).
If a type-v consumer believes that all firms are following the mixed strategy j⇤ , his
optimal search strategy will be characterized by a constant cutoff value, u⇤ (v). From Lemma
A.1, u⇤ (v) solves
u⇤ (v) = E j⇤ [max{u⇤ (v), u( |v), 0}] s.
(A.14)
There are two possible cases:
1. Suppose that the value u⇤ (v) that solves (A.14) is such that u( 0 |v)
u⇤ (v)
⇤
u( 1 |v) = u(ˆ0 |v) > u(ˆ1 |v). Then, u⇤ (v) = 0 (z p0 ) + (1
s ()
0 )u (v)
⇤
u (v) = z p0 s/ 0 .
2. Suppose that the value u⇤ (v) that solves (A.14) is such that u( 0 |v) > u( 1 |v) =
u(ˆ0 |v) > u⇤ (v) · · · . Then,
⇤
u (v) =
p0 ) + ( ˆ 0 +
0 (z
1)
✓
z
p0
0
+ ˆ 1 max{u(ˆ1 |v), u (v), 0}
⇤
ˆ 1 )(z
=) u⇤ (v) = (1
(ˆ0 +
p0 )
s
1)
◆
s
s
+ ˆ 1 u⇤ (v)
s
0
=) u⇤ (v) = z
By assumption, however, z
ˆ0 +
1
But since 1
1
ˆ1
s
0
0
+
s
1
ˆ1
1
ˆ0
p0
p0
>
s
0
ˆ0 +
ˆ1
1
s/
()
1
0
s
0
s
1
ˆ1
> u⇤ (v). Therefore,
0 (1
ˆ 1 )(1
0
1
ˆ0
ˆ 1 )s < 0.
ˆ 1 = 0, this final condition is a contradiction.
Since case 2 cannot apply, we conclude that under the assumed values for the ’s, u⇤ (v) =
54
z p0 s/ 0 and u⇤ (v) > u(ˆ1 |v). Thus, when a type-v consumer believes all firms adopt
the strategy j⇤ , he finds acceptable the contracts 0 , ˆ0 , and 1 . If ˆ1 is the only contract
available, the consumer will search in lieu of accepting ˆ1 or exiting the market.
The Optimal Search Strategy for a Type-v̄ Consumer. Given the set of contracts defined
above, we can compute a type-v̄ consumer’s expected utility:
u( 0 |v̄) = z
u( 1 |v̄) = v̄
p0
p1
u(ˆ0 |v̄) = z
u(ˆ1 |v̄) = v̄
p̂0 = z
p̂1 = v̄
p0
p1
s/
s/
0
1
Again, u( 0 |v) > u(ˆ0 |v) 0 and substituting the expression from above yields u(ˆ1 |v̄) =
v̄ p1 s/ 1 = v̄ p0 s/ 0 v + z s/ 1 = z p0 . Hence, u( 1 |v̄) > u(ˆ1 |v̄) =
u( 0 |v̄) > u(ˆ0 |v̄). An argument analogous to the previous case establishes that under the
proposed conditions, the cutoff value charactering the search strategy of a type-v̄ consumer
is u⇤ (v̄) = v̄ p1 s/ 1 = z p0 . Thus, when a type-v̄ consumer believes all firms adopt
the strategy j⇤ , he finds acceptable the contracts 0 , 1 , and ˆ1 . If ˆ0 is the only contract
available, the consumer will happily search in lieu of accepting ˆ0 or exiting the market.
Expected Firm Profits. Noting the consumers’ search behavior we can compute the expected profits associated with each of the offered contracts. These are given by Lemma
5 above. It is straightforward to confirm that given the definition of prices and (A.7),
⇧j ( 0 ) = ⇧j (ˆ0 ) = ⇧j ( 1 ) = ⇧j (ˆ1 ). Therefore, each contract in the support of j⇤ yields
the same (strictly-positive) expected profit conditional on the consumers’ search strategy.
No Firm has a Profitable Deviation. It remains to verify that no alternative contract
can yield a greater expected profit to a typical firm given agents’ search behavior and the
strategy adopted by the other firms. Verification of this fact relies on the inequalities and
equalities summarized by the following conditions:
u( 0 |v) > u( 1 |v) = u(ˆ0 |v) = u⇤ (v)
max{u(ˆ1 |v), 0}, and
u( 1 |v̄) > u(ˆ1 |v̄) = u( 0 |v̄) = u⇤ (v̄) > u(ˆ0 |v̄).
(A.15)
(A.16)
(A.15) and (A.16) were derived above in relation to each type of consumers’ search strategy.
Suppose firm j posts the contract ˜ = h0, p̃i. Clearly, if p̃  p0 , the firm cannot improve
its expected profits. The lower price fails to attract more consumers than p0 and leads to
strictly lower profits. If p̂0 < p̃, then u⇤ (v̄)
u⇤ (v) = z p̂0 > z p̃. Thus, the more
expensive contract would not be accepted by any consumers. Finally if p0 < p̃ < p̂0 , then
z p̃ < u⇤ (v̄). Hence, the contract would only be acceptable to type-v̄ consumers. However,
55
ˆ0 was also acceptable to such consumers and it had a higher price. Hence, firm j could do
better by offering ˆ0 instead.
Suppose a firm posts ˜ = h1, p̃i. If p̃ < p1 , the firm can strictly improve its expected profit
by charging a slightly higher price since the contract would be accepted by all consumers.
Likewise if p̃ > p̂1 , the contract is not acceptable to any consumers—even type-v̄ consumers
will search in this case. If p1 < p̃ < p̂1 , the contract is acceptable only to type-v consumers.
But ˆ1 would yield a greater profit as it would be accepted by the same number of consumers
but charge them a higher price. Thus, no firm can deviate profitably from ⇤ .
A.6
Increasing Search Costs
In this section we elaborate on our proposed model extension with increasing search costs.
Suppose that the cost to learn the contract offer of the t-th additional firm is st where
0 < s1  s2  · · · . The sequence of search costs is common knowledge and limt!1 st > v̄.
Pt
If the consumer searches for t periods in total, his payoff will be u( i+t |vi )
k=1 sk if he
Pt
accepts i+t and
k=1 sk otherwise.
Let 0 = h0, zi and 1 = h1, vi. Suppose firms follow a strategy ( ⇤ ) where 1 is offered
⇤
with probability ⇤1 2 (0, 1) and 0 is offered with probability 1
1 . A type-v̄ consumer will
search for 1 , but only for a finite number of periods. If he searches for t⇤ periods at most,
t⇤ must satisfy
t⇤ = max{t :
⇤
1 u( 1 |v̄)
⇤
1 )u( 0 |v̄)
+ (1
st
u( 0 |v̄)}.
(A.17)
The expected payoff from taking the final sample must exceed the payoff of accepting the
available contract. We can characterize this consumer’s search strategy by a decreasing
sequence of cutoff values. As search costs rise over time, the consumer becomes less choosy.
In period t consumer i will accept i+t if u( i+t |v̄) u⇤t (v̄) where
u⇤t (v̄) = 1
(1
⇤
1)
t⇤
t
⇤ t
1 ) u( 0 |v̄)
u( 1 |v̄) + (1
t⇤X
t 1
st ⇤
k (1
⇤ t⇤ t 1 k
.
1)
k=0
These cutoff values can be computed via backward induction starting at t⇤ .
The preceding discussion has taken ⇤1 as given. However, it is determined endogenously
through the equilibration of firms’ profits. Given consumers’ search strategy, the expected
⇤ t⇤
⇤ t⇤
profit of a firm offering 0 is ⇧j ( 0 ) = 1
+ (1
z. The term (1
1)
1 ) accounts
for the small, but positive, probability that a type-v̄ consumer accepts 0 after a t⇤ -period
56
unsuccessful search for
1.
⇧j ( 1 ) =
Similarly,
✓
1+
✓
◆
1
1 (1
⇤
1
⇤ t⇤
1) )
(1
◆
(v
c)
.
In equilibrium, t⇤ 2 N and ⇤1 2 (0, 1) are such that (i) t⇤ satisfies condition (A.17),
and (ii) ⇧j ( 0 ) = ⇧j ( 1 ). Additionally, firms must not have an incentive to deviate from
⇤
. Hence a collection of additional constraints, analogous to those in Theorem 3 but now
accounting for the consumers’ new search strategy, must be satisfied.39 The following lemma
identifies sufficient conditions ensuring that t⇤ and ⇤1 exist in the defined environment.
Lemma A.16. Suppose search costs are non-decreasing. Define the functions
8
<max{t : (v̄ v) s
0} if {t : (v̄
t
⇢( ) =
:0
otherwise
⇣
⌘
v+z)
log 1 1 ( (c+
1)(c v)+ z
⌧( ) =
log(1
)
v)
st
0} =
6 ;
(A.18)
(A.19)
Let ¯ = (1 (v)(vc) c z) and suppose there exists ˆ 2 (0, ¯ ) such that ⇢( ˆ ) > ⌧ ( ˆ ). Then there
exists t⇤ 2 N and ⇤1 2 (0, 1) such that
⇤
1. t⇤ = max{t : ⇤1 u( 1 |v̄) + (1
1 )u( 0 |v̄)
⇣
⇣
⌘
⌘
1
⇤ t⇤
2. 1 +
1 (1 (1
c)
⇤
1 ) ) (v
st
u( 0 |v̄)}; and,
= 1
1
+ (1
⇤ t⇤
1)
z.
Proof. When 0 = h0, zi and 1 = h1, vi, u( 1 |v̄) = v̄ v and u( 0 |v̄) = 0. Now, ⇢( ) is a
non-decreasing step function taking on integer values. It is the blue curve is Figure A.1.
The function ⌧ ( ) is defined as the value of ⌧ that solves
✓
1+
✓
1
◆
1 (1
(1
⌧
◆
) ) (v
c)
= (1
+ (1
)⌧ ) z
(A.20)
When a solution exists, it is given by (A.19). ⌧ ( ) is strictly increasing and continuous.
v+ +c
Also, lim !0+ ⌧ ( ) = z (v
and ⌧ ( ) asymptotes to infinity as ! ¯ = (1 (v)(vc) c z) .
c)
⌧ ( ) is the red curve in Figure A.1.
If ⇢( ˆ ) > ⌧ ( ˆ ) there will exist some ⇤1 2 ( ˆ , ¯ ) where t⇤ = ⌧ ( ⇤1 ) and ( ⇤1 , t⇤ ) satisfies the
lemma’s conditions.
39
As in the case of the other extensions we analyze, we omit analyzing these conditions for brevity.
57
t
⌧( )
⇢( )
t⇤
0
⇤
1
¯
1
Figure A.1: An illustration of the situation in Lemma A.16.
58