41
mation requires that preferences be directly observable. This is clearly unrealistic and trivializes the
complexities of retail trade pricing. In our baseline model, we assume that buyers’ preferences are
private information, so prices cannot depend on them. As we show, this is equivalent to assuming
that in equilibrium bilateral trades between buyers and sellers must be incentive compatible, in the
sense that buyers have no incentive to misrepresent their preferences.
3
The second way to implement the directed search equilibrium with full information is by in-
troducing ßat fees. In retail trade, we observe some instances in which ßat fees are charged to
buyers prior to their purchasing decision. For example, some warehouse clubs charge membership
fees. However, these fees are the exception rather than the rule. Moreover, even when they are
present, they cover only a small fraction of retail costs. Explaining why ßat fees are not common
practice is an open question of upmost importance. We do not give a Þnal answer to this question.
Rather we construct a stylized model of retail trade where ßat fees cannot arise as an equilibrium
outcome. In our baseline model, buyers learn their valuations as soon as they meet a seller and
observe the good the seller carries. This assumption, combined with the fact buyers’ valuations can
be arbitrarily low, rules out ßat fees. Buyers with small valuations will simply refuse to pay the
fee.
4
However, our environment does not rule out other forms of non-linear pricing. In fact, the
model predicts that per unit prices should decline with quantity. This may sound counterfactual.
However, retailers use both packaging and explicit quantity discounts to offer these decreasing per
unit prices. For instance, when we purchase a gallon of paint we pay much less than when we
purchase four separate quarters of a gallon of the same paint.
With the informational environment of our baseline model, the directed search equilibrium is
not efficient. In equilibrium the average lineup of buyers in front of a seller is inefficiently long.
Nevertheless, a policy maker who faces the same informational constraints as the sellers and whose
only policy instrument is to regulate market prices cannot improve upon the equilibrium allocation
when the production technology is linear (affine production function).
5
Thus, if the production
3Admittedly,sometransactionsinretailtradeinvolvebargaining,andthebargainingprocessmaypartiallyreveal
the willingness to pay of a buyer. Yet, the amount of information revealed in the bargaining process is also limited
by incentive compatibility constraints. Indeed, in the environment we study in this paper there is no exogenous
bargaining rule that attains the efficient outcome when buyers’ preferences are unobservable. Moreover, bargaining
in retail trade is rare and is inconsistent with ex-ante price posting.
4Using standard d terminology in the e mechanism
design literature, pricing mechanisms which include ßat fees
violate the individual rationality constraints of the buyers. On the other hand, the type dependent prices necessary
to implement an efficient allocation violate the buyers’ incentive compatibility constraints.
5Ingeneral equilibrium, efficiency in thecommercial sector is not enough. . We e also need production efficiency.
4
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37
technology is linear, the directed search equilibrium is second best efficient.
6
Assuming a linear
technology is certainly restrictive. However, as long as the production function is continuously
differentiable, we can approximate it with an affine production function. Therefore, as long as the
policy maker faces the same informational constraints as the sellers, the welfare gains of regulating
market prices are second order of magnitude. We use numerical simulations to check that these
potential welfare gains are minimal.
Our model can be easily incorporated in a Neoclassical growth framework with two sectors.
One of these sectors produces goods combining capital and labor as is typical in the Neoclassical
framework, while the other exchanges goods in retail markets. The combined model can be es-
timated using standard empirical data. In particular, the parameters of the retail sector can be
identiÞed and estimated using commercial margins and the average time allocation of households.
7
The tractability of the model makes it suitable for further extensions. For example, one can use
this framework to introduce money along the lines of the search theoretical approach of Kiyotaki
and Wright (1989 and 1993). Faig (2001) constitutes a Þrst attempt in this direction in a simpler
version of the present model. In this simpler version, sellers are restricted to make offers to the
buyers they are paired with which consist of a single quantity-payment pair. Our main improve-
ment with respect to Faig (2001) is the relaxation of this constraint by allowing sellers to make
offers that consist of a price schedule that maps the quantity chosen by a buyer into the payment
to be made to the seller. This price schedule serves as a mechanism to reveal the buyers’ private
information about their preferences. Also, in Faig (2001) search is undirected and sellers make
take-it-or-leave-it offers to buyers.
Two recent papers, Soller-Curtis and Wright (2000) and Camera and Delacroix (2001) also study
search-theoretic models where the buyers’ willingness to pay for a good is private information. In
both of these papers, goods are indivisible and search is undirected. Soller-Curtis and Wright
assume that sellers make take-it-or-leave-it offers to buyers and they focus on the coexistence of
two prices for the same good in equilibrium. Camera and Delacroix focus on the endogenous
determination of the trade mechanism - sellers can choose if they want to commit to a posted
With a linear technology, the marginal return to labor in the production sector is Þxed. Thus, the global allocation
of labor is optimal as long as buyers and sellers are given the right incentives in the commercial sector.
6Withfullinformation,therestrictiontolineartechnologiesisnotnecessary.
7Shi(1999and2001a) alsoincorporates asearch modelin aNeoclassicalgrowth framework. . Themodelin n Shi
does not differentiate between producers and sellers as we do. Also, his model assumes full information and random
(undirected) search.
5
31
price or to bargain once they meet a buyer. When buyers have identical preferences and search
is undirected, pre-committing to a price is preferable from the sellers point of view because it
allows them to extract the whole trading surplus. However, when the preferences of the buyers
are heterogeneous, the bargaining process allows the seller to infer information about the buyers’
preferences and hence to price discriminate. In our paper, goods are divisible, so sellers can commit
to a non-linear price schedule which allows a restricted form of price discrimination even without
bargaining. Also, we incorporate the equilibrium concept of directed search which endogeneizes the
market power of sellers in a reasonable fashion.
Peters and Severinov (1997) also study a model of directed search where the preferences of the
buyers are private information. However, in their model the price setting mechanism is an auction
among the buyers that meet a seller. Moreover, because a single unit is sold in each auction, their
paper does not deal with the price incentives to endogenously determine the size of each purchase.
The paper is organized as follows. Section 2 presents an overview of the model and our equi-
librium concept. Section 3 characterizes the optimal behavior of a representative household for
given prices. Section 4 studies how prices are endogenously determined under directed search by
analyzing the interaction between buyers and sellers in the market place. Section 5 combines the
analyses of the previous two sections into a general equilibrium model where both the behavior of
households and prices in retail trade are endogenous. Section 6 studies the welfare properties of a
directed search equilibrium. Section 7 incorporates a generalized version of our model in a dynamic
Neoclassical growth framework and discusses how to identify the parameters using standard data.
Section 8 brießy discusses some of the issues one must confront when extending the present model
and concludes. The proofs are gathered in the Appendix.
2 Overview of the Model
The economy consists of a continuum of households with measure one who produce and consume
differentiated goods. Households do not consume the goods they produce so they need to trade.
Trading activities involve some degree of idiosyncratic uncertainty to be speciÞed below. To avoid
the ex-post heterogeneity inducedby idiosyncratic uncertainty,which severely limits the tractability
of the model, we follow Shi(1997) in assuming that each household is composed of a large number of
individuals. These individuals independently perform the production and exchange activities in the
household. Thanks to the large household assumption, each household faces no uncertainty, even
6
41
though the members of the household who perform trading activities are subject to idiosyncratic
risks. All members of the household equally share the consumption of goods which is their only
source of utility, so there is no conßict of interests among them.
8
To construct a simple environment where households buy the goods they consume in a retail
trade sector, we make the following assumptions. Each household produces a single divisible good
and likes to consume the goods produced by other households. Because of physical constraints, the
members of the household who perform manufacturing activities (producers) cannot simultaneously
sell the good they produce. Likewise, sellers must remain intheir retailoutlets toselltheir products.
Therefore, for trade to take place the buyers of a household must go around visiting sellers of other
households. Since sellers never meet each other, direct barter is ruled out.
All payments are denominated in an abstract numeraire. In the version of the model analyzed
in this paper, buyers do not need to carry money with them. Instead, all traders have access to a
central clearing-house that records the credits (payments received by sellers) and debits (payments
made by buyers) of all households and ensures that their budget constraint is satisÞed.
Trading activities are subject to two kinds of uncertainty. First, buyers sometimes Þnd goods
that, because of idiosyncratic factors, Þt well the needs of their households while in other occasions
they do not. Second, because of matching frictions, trading meetings between buyers and sellers
are partially random. Thus, a trader may or may not be able to perform a transaction during a
given period.
We model the Þrst type of uncertainty by assuming a preference shock ε that scales the utility
that a good brings to the household. Preference shocks are realized once a buyer of the household
meets a seller. The realized value of ε is the buyer’s own private information and cannot be
observed by the seller. Therefore, in a trading meeting the seller ignores the willingness to pay of
the buyer. We believe that this ignorance is key for understanding retail pricing in the same way
that unobservable characteristics of taxpayers are key for understanding income tax schedules.
In order to model both the matching frictions and the price competition that characterize retail
markets, we assume the following form of directed search.
9
Prior to the trading period, each
seller j simultaneously posts and commits to a price schedule Z
j
(q). This schedule speciÞes the
payment required in a transaction as a function of the quantity exchanged.
10
In the next stage,
8Theabsenceofaconßictofinterestsbetweenthehouseholdanditsmembersisanimportantdifferencebetween
this paper and Shi (1997).
9OurdescriptionofthisconceptparallelstheformulationinAcemogluandShimer(1999).
10Equivalently,becauseeachsellerjmaymeetbuyerswithdifferentprivatevaluationsexpost,thesellerannounces
7
128
buyers observe the menu of price schedules posted by all sellers and simultaneously choose where
to shop. That is, each buyer i decides to trade at a particular price schedule Z
i
(q) in the set of
posted schedules {Z
j
(q) : for all j}.
11
If buyer i meets a seller posting Z
i
(q) during the trading
period, the buyer privately observes his valuation ε for the seller’s good, chooses the quantity to be
transacted and pays for it according to the posted price schedule. Buyers can always choose not to
buy anything in which case they pay nothing. If no meeting takes place, then there is no trade.
12
We refer to the set of sellers posting Z
j
(q) and the set of buyers that direct their search to this
price schedule as submarket j.
Depending on buyers’s search decisions, there may be longer lineups in some submarkets than
in some others. To capture this, we let θ
j
∈[0,∞] be the ratio of buyers to sellers in the submarket
with a posted price schedule Z
j
(q) :
θ
j
=
B
j
S
j
,
(1)
where S
j
is the measure of sellers posting Z
j
(q) andB
j
is the measure of buyers who decide to trade
at this priceschedule. We refer toθ
j
as the congestionin submarket j,or the averagequeue of buyers
in front of a seller in this submarket. For the time being and to facilitate the exposition, we assume
that buyers and sellers may perform at most one trade during the trading period (Subsection
5.1 relaxes this assumption). In a submarket with congestion θ
j
, the probability that a seller
meets a buyer is m
s
(θ
j
), where m
s
:[0,∞] → [0,1] is continuously differentiable, decreasing, and
concave. Symmetrically, a buyer meets a seller with probability m
b
(θ
j
), where m
b
:[0,∞] → [0,1]
is continuously differentiable, increasing, and convex. If many buyers seek a few sellers (θ
j
is high),
then it is easy for a seller to Þnd a buyer and hard for a buyer to Þnd a seller. By having m
s
and
m
b
depend only on θ
j
,we implicitly assume constant returns in matching,
13
so
M(B
j
,S
j
)= B
j
m
b
(θ
j
)= S
j
m
s
(θ
j
),
(2)
alist of quantity-payment pairs {q
jv
,z
jv
}
v∈V
. It is not restrictive to assume that the number of items in this list
is equal to the number of ex post buyer types. A seller’s strategy is summarized by a price schedule Z
j
:Q
j
→<
where Q
j
={q
jv
}
v∈V
and z
jv
=Z
j
(q
jv
)for all v ∈ V. See Maskin and Riley (1984) which analyzes the problem of a
monopolistic seller who faces no price competition. In their model, however, there are no matching frictions.
11Buyerscanalsoplaymixedstrategiesandrandomizeoverpriceschedulesforwhichtheyareindifferent.
12Bytherevelation principle,theformulationasellerannouncesapricescheduleand buyerswith differentreal-
izations of ε self-select along this schedule by choosing their most preferred price quantity combination is equivalent
to a formulation where the seller announces a direct revelation mechanism which induces truthful revelation by the
buyers. See Section 4.
13SeePissarides(1990).
8
54
where M is astandard matching functionthat maps the measures of buyers andsellers in submarket
jonto the measure of trading meetings in this submarket.
An interesting special case of m
s
is:
m
s
(θ
j
)= 1− exp(−θ
j
).
(3)
This case arises if buyers use identical mixed strategies to select a seller among those who post
equivalent price offers and, because selling is time consuming, each seller can serve at most one
customer (see Peters (2000)). This is the typical setup in the frictional assignment literature.
Another interesting special case is
m
s
(θ
j
)=
θ
j
1+θ
j
.
(4)
This special case arises if each buyer is randomly matched with a trader (buyer or seller) in the
submarket where search is directed. (In this case, m
s
(θ
j
) is equal to the fraction of buyers over
traders in submarket j).
We consider an environment where buyers and sellers are ex-ante symmetric. Given that buyers
are free to choose among different price schedules, in a directed search equilibrium, all buyers must
attain the same expected payoff. Furthermore, in equilibrium no seller should have an incentive
to deviate by posting a different price schedule. To attract buyers, the offer of the deviating seller
must yield at least the common expected payoff buyers attain in equilibrium.
14
Therefore, in a
directed search equilibrium the price schedules and the degree of congestion associated with them
must maximize the expected payoff of sellers subject to the constraint that the buyers get the
common expected payoff. As we show in Section 4, this implies that in our framework there is a
single price schedule and a single submarket in equilibrium.
We proceed to develop our model with the following steps. In Sections 3, 4, and 5, we construct
astatic version of the model. In Section 3, we analyze the optimal behavior of a household for a
given price schedule Z(q). Section 4 studies the endogenous determination of the price schedule
Z(q) under directed search. Section 5 collects the results of Sections 3 and 4 in a general equilibrium
model where both the behavior of the household and prices are endogenous.
14Sincebuyerscanplaymixedstrategies,theexpectednumberof buyersattracted byasingledeviatorneednot
be an integer.
9
46
3 A Representative Household
In this section, we describe the behavior of a household whose buyers and sellers trade in a retail
market with a given price schedule Z(q) and a given degree of congestion θ. Because there is a
continuum of households in the economy, the household takes the price schedule and the degree of
congestion as given. We assume Z(q) to be continuously differentiable and concave with Z(0) = 0.
The next section endogeneizes Z(q) and validates these assumptions. We adopt the following nota-
tion. Lower-case letters denote the decision variables of the household. Upper-case letters denote
the decisions of the other households and hence aggregate quantities, which are also taken as given
by our household. In a symmetric equilibrium, lower-case letters are equal to the corresponding
upper-case letters.
The household is composed of an inÞnite number of individuals. The individuals of the house-
hold are assigned to one of three different tasks: production (producers), purchase (buyers), and
sale of commodities (sellers). Each individual is endowed with one unit of labor and can perform
at most one task. The timing of the model is as follows. First, the members of the household are
divided into producers, buyers, and sellers. The measure of individuals assigned to each activity
is denoted by n, b, and s, respectively. Second, all members perform their assigned activities.
SpeciÞcally, producers use their labor n to generate output, which immediately becomes available
for sale. Buyers visit sellers from other households. Upon meeting a seller, they experience a ran-
dom shock ε that scales the incremental utility that the good brings to the household. Shocks are
uniformly distributed in [0,1] and are independent across trading meetings. While the distribution
of these shocks in public information, the realized value of ε is the buyer’s own private information.
Contingent on the realized value of ε, buyers choose the quantity q
ε
they want to acquire from the
seller, and pay z
ε
=Z(q
ε
).
15
This payment is immediately debited from the household’s account
in the clearing-house. Sellers go to retail outlets where they wait for buyers to visit them. When
aseller of the household is visited by a buyer who purchases a quantity Q
ε
,the seller fetches this
quantity from the current household production and collects a payment Z
ε
=Z(Q
ε
), which is im-
mediately transferred to the household’s account in the clearing-house. Finally, once all their tasks
are completed, all the individuals of the household get together and equally share the consumption
of the goods purchased.
The assumption that there is an inÞnite number of members in the household ensures full
15Weuseεsubscripttodenotethatqandzarefunctionsofε.
10
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