Price Cycles and Booms:
Dynamic Search Equilibrium
Search theory has been extensicely and successfully applied to explain the
persistence of price dispersion. This paper presents an explicitly dynamic search
model which is able to account for cyclical patterns of prices and demand ocer
time. These cyclical features of the model are the consequence of the dynamic
strategic interaction between buyers and firms and do not require the presence of
extraneous factors such as shocks or heterogeneity of agents in order to obtain.
The model builds on earlier work by Kenneth Burdett and Kenneth L. Judd and
may be interpreted as a dynamic extension of their model. ( J E L D83, D40)
George J. Stigler's (1961) celebrated seminal article focused economists' attention on
the fact that, if prices are costly to observe,
buyers might consciously purchase at a price
exceeding the lowest obtainable price in the
market; with imperfect information, Stanley
Jevons's "law of one price" need not apply.
This theoretical insight is in accord with
empirical evidence substantiating the existence of significant price dispersion. In
Stigler's formulation, the buyer decides on
the number of prices to be sampled prior to
observing any price quotation. Subsequent
authors (e.g., Morris H. DeGroot, 1970;
John J. McCall, 1970; Meir Kohn and Steven
Shavell, 1974) pointed out that a sequential
strategy (i.e., a search procedure in which
the decision as to whether an additional
price is to be sampled depends upon the
realization of preceding samples) is generally superior to fixed-sample-size rules.
While the early search literature showed
that price dispersion is a viable prospect as
*Department of Economics, Tel-Aviv University,
Tel-Aviv 69978, Israel. We are grateful to seminar
participants at Tel-Aviv University, the Hebrew University of Jerusalem, the University of Toronto, the
University of Western Ontario, Columbia University,
the University of Pennsylvania, Northwestern University, Tilburg University, and Boston University and two
anonymous referees for valuable comments and suggestions. Financial assistance from the Foerder Institute of Economic Research is also gratefully acknowledged.
far as the demand side of the market is
concerned, its sustainability as an equilibrium phenomenon consistent with optimal
behavior on both sides of the market was
less clear-cut, as Michael Rothschild (1973)
argued forcefully. Moreover, Peter Diamond (1971) showed that when all consumers have positive search costs, however
small, all firms charge the monopoly price in
equilibrium. A large number of equilibrium
search models, which showed that price dispersion can be a viable market equilibrium,
arose in response to this challenge. These
include Bo Axell (19771, Steven Salop and
Joseph E. Stiglitz (19771, Jennifer F. Reinganum (1979), Louis L. Wilde and Alan
Schwartz (1979), Avishay Braverman (1980),
Richard D. MacMinn (19801, Hal R. Varian
(1980), Kenneth Burdett and Kenneth L.
Judd (1983), John A. Carlson and R. Preston McAfee (1983), James W. AIbrecht
and Axell (1984), Rafael Rob (19871, and
Joseph E. Stiglitz (1987).
The preceding literature shows that
search theory can successfully account for
price dispersion as an equilibrium phenomenon. The objective of this paper is to
demonstrate, in an explicitly dynamic setting, that search theory is also able to account for the existence of price and demand
cycles even in deterministic and stationary
environments.
On the demand side of the market, the
dynamics of buyers' search behavior in our
model follows recent developments in the
1222
THE AMERICAN ECONOMIC REVIEW
theory of optimal search. While the earlier
search literature imposed a dichotomy between fixed-sample-size and sequentialsearch procedures, several authors have
more recently pointed out that, in a dynamic framework, the optimal procedure is
generally a hybrid of the two (Shmuel Gal
et al., 1981; Richard Manning and Peter
Morgan, 1982; Jess Benhabib and Clive Bull
1983; Burdett and Judd, 1983; Morgan and
Manning, 1985).' Accordingly, we assume
the following search technology. At the beginning of each period, each buyer decides
the intensity of search (i.e., the number of
price quotations to be sampled). All prices
demanded at the beginning of the period
are simultaneously received at the end of
the period. At that point, each buyer decides whether to accept the best offer received or to continue sampling. A consumer's impatience to purchase her unit
might induce her to sample more than one
price at a time. Thus, the search procedure
is sequential but allows the number of prices
included in a sample to be determined endogenously.
On the supply side of the markets, we
assume that firms are able to change prices
in response to the dynamics of consumer
search. This absence of commitment on the
part of firms is found to affect the equilibrium price path in remarkable ways. In particular, if consumers are sufficiently patient,
equilibria distinguished by the existence of a
stationary (nondegenerate) price distribution do not exist. Moreover, the typical
equilibrium price path is cyclical. In each
period during which a consumer is in the
market, she has the option of not soliciting
any prices. Thus, in principle, there could
exist periods during which consumers choose
to defer their search such that the accumulated unsatisfied demand generates future
demand booms. It is shown that the abovementioned cyclical price pattern may indeed
account for such nonstationary demand in
equilibrium. The model we present can be
' ~ m o n gthese references, only Burdett and Judd
(1983) describe a market equilibrium.
DECEMBER 1992
viewed as a dynamic extension of Burdett
and Judd's (1983) nonsequential model, and
our analysis draws heavily on their results.
I. A Model of Dynamic Nonstationary Search
There is a continuum of firms and buyers.
Each firm can costlessly supply an unlimited
quantity of a homogeneous product. Firms
compete in prices. In each period, a new
cohort of identical risk-neutral buyers of
finite measure p > 0 per firm enters the
market. Each consumer has inelastic demand for exactly one unit for which she is
willing to pay p* > 0 at the most. Once a
buyer has purchased a unit, she leaves the
market forever. At any period t > 1, the
measure of buyers in the market may exceed p if in preceding periods, buyers have
chosen to delay their purchases. Accordingly, we let p, L p be the total measure
of consumers per firm at the outset of period t .
Let F,( ) denote the (possibly degenerate) distribution of prices at period t . Each
buyer in the market is assumed to know the
sequence of future price distributions but to
be unaware of the price charged by any
specific firm. In order to buy from a firm, a
buyer must first solicit a price quotation
from that firm. Any number of prices may
be solicited at the beginning of period t at a
constant cost of c > 0 per price. All prices
demanded at the onset of period t are
simultaneously received at the end of that
period. We denote by n , the number of
prices demanded at t .
Each buyer seeks to minimize her cost of
purchase, including search costs and the
price paid, subject to her discount rate as
discussed below. Specifically, utility from future consumption is discounted at a constant rate. We assume that this impatience
to consume translates into the existence of
S > 1 such that at any period a consumer is
indifferent between buying at once at the
price S p or buying at the next period at the
price p, and the consumer strictly prefers to
delay her purchase if the current price exceeds Sp. We emphasize that S is not the
discount factor but is inversely proportional
to it and so is always greater than 1 and,
V O L . 82 NO. 5
FERSHTMAN A N D FISHMAN: PRICE CYCLES A N D BOOMS
moreover, is larger the smaller is the discount rate.,
We assume that firms are not committed
to any price offer for more than one period.
Thus, although consumers might have perfect recall, previously observed prices are
rendered obsolete by the fact that the firms
may have already changed their prices. This
assumption implies that if a consumer buys
the product at period t she may choose only
from the set of firms sampled at that period.
In deciding on the intensity of search at
period t (i.e., on the size of the sample) and
in making their purchasing choice, consumers take into account not only the current price distribution but also the future
evolution of prices in the market.
It proves to be methodologically useful to
suppose temporarily that the number of periods is finite, say T > l. The characterization of the equilibria derived for an arbitrarily large finite horizon is then found to carry
over directly to the case of an infinite horizon. At any period t, let q: denote the
probability that a randomly selected consumer observes M prices, and let fit be the
acceptance price associated with that period
such that a unit is purchased only if its price
does not exceed 13,. The acceptance price is
determined endogenously, depending on the
current and future price distributions and
on the consumers' dynamic search strategy.
' ~ e tS ( p ) denote the consumer surplus from buying
at the price p where S ( p ) is continuous and monotonically decreasing. We assume, following Burdett and
Judd (1983), that S ( p * ) > 0. This is the case if, for
example, the consumers' demand is inelastic for more
than one unit but is elastic for less than a unit such
that the optimal monopoly price corresponding to this
demand curve is p*. Let future utility be discounted
according to the factor y < 1 (i.e., the utility from
buying at p one period hence is y S ( p ) . Then for each
p 5 p* there exists S ( p ) > 1 such that
Given this formulation, S ( p ) is seen to be decreasing
in p. T o simplify exposition and notation we have
assumed S to be constant in the text as this has no
effect on our results. We also assume that S ( p * ) - c 2 0
so that a consumer is willing to enter the market to
purchase a unit when she expects to pay the monopoly
price.
1223
Each firm optimally chooses its price, taking the price distribution at that period and
the search behavior of buyers, summarized
by fi, and the sequence {q:)Z=,, as given.
An equilibrium is a sequence of price distributions, one for each period, and a dynamic
search strategy such that no firm or buyer
can benefit by altering behavior. This is
stated more formally in the following definition.
Definition 1: Given the time horizon T, we
define {F,(P), r , , ?,, {q:);=,, fit):=, as a dynamic search equilibrium if:
(i) at every period t, given (I*,, {q:l:=,,
kt),
the profit of a firm is T,2 0 if it asks a
price in the support of F,(p) and is less
than or equal to r, if its price lies
outside the support; and
(ii) at every period t, ({q:):=,,
6,) represents the optimal search behavior of
consumers given the current and the
future sequence of price distributions.
The following definitions characterize
specific types of dynamic search equilibria.
Definition 2: A stationary equilibrium is
a dynamic search equilibrium in which
Ftl(p)= F,z(p) for every t,, t, I T.
Definition 3: A cyclic equilibrium is a nonstationary dynamic search equilibrium for
which there is an integer z , such that for
every t 4 T - z , F,(p) = F,+,(p).
Note from Definition 1 that qf may be
positive (i.e., consumers may abstain from
market participation at any period).
Definition 4: We say that active demand at
period t (AD,) is positive if a positive measure of buyers sample at least one price.
Note that in our model zero active demand cannot be due to a lack of potential
consumers since a new cohort enters at each
period. Rather, zero active demand characterizes a period at which all consumers,
although impatient, choose not to sample
and, as a consequence, choose not to buy.
DECEMBER I992
THE AMERICAN E( ZONOMIC REVIEW
1224
To be part of an equilibrium, this decision
must, of course, be consistent with their
objective of obtaining the product at the
lowest possible discounted cost.
where
Definition 5: A dynamic search equilibrium
is characterized by endogenous booms if
there are periods at which active demand is
zero and other periods at which active demand is greater than k .
In this equilibrium, firms charging higher
prices make fewer sales to consumers who
sample twice than do firms charging lower
prices, such that the profits associated with
the different prices in the price distribution's
support are equal.
In turn, q is determined by the consumers' optimal search strategy. Specifically, let V(p*,q) be the expected difference between the price paid by consumers
who sample twice and those who sample
once. The dependence of V( ., .) on p* and
q follows from the fact that the latter fully
determine the price distribution which, in
turn, determines V. Consumers who sample
twice pay less on average than those who
sample once, but consumers who sample
twice incur an additional sampling cost. In
equilibrium, consumers must be indifferent
between the two options. Thus it is required
that
The model of Burdett and Judd (1983) is
a static, one-period version of this model.
Their results for the one-period model are
important for our analysis and are briefly
summarized in what follows. The reader is
referred to their paper for further details.
First, in any dispersed price equilibrium,
there must be a proportion q, 0 < q < 1, of
consumers who observe only one price, and
a proportion 1 - q who observe two prices3
Intuitively, if all consumers sample only one
price, each seller will optimally charge the
monopoly price p*, while if all consumers
observe more than one price, prices will be
driven down to the Bertrand (zero-profit)
level as firms compete for each consumer.
Given q, and the monopoly price p*, the
equilibrium equal-profit condition [(i) of
Definition 11 determines the equilibrium
price distribution, which is continuous over
its support, as a function of q:
Using (1) and simplifying, it can be shown
that
The technical properties of this function are
important for our analysis (see Fig. 1).
Specifically, V( , ) attains a unique maximum at some q*, 0 < q* < 1, is strictly increasing (decreasing) if q* > q > 0 (if q* <
q < 1) and V(p*,q)+ 0 as q + 0 or q + 1.
Given the acceptance price p*, there exists a unique c* > 0 satisfying U p * , q*) =
c*. If c > c*, only a single price equilibrium
exists since for every q, V(p*,q) < c. In that
case, each firm charges the reservation price,
p*. If c I c*, dispersed price equilibria exist
as well. Specifically, if c = c*, there exists a
..
"n equilibrium no consumer observes more than
two prices because, given the price distribution, the
expected cost of purchasing when n prices are sampled
is a convex function which assumes a unique minimum
when n is allowed to be any positive number. Hence,
there are at most two integers, n* and n* + 1 that
minimize the cost of purchasing. Since n* must be 1,
as discussed in the text, n* 1 must be 2 (see Burdett
and Judd, 1983 p. 961).
+
VOL. 82 NO. 5
FERSHTMAN AND F I S H M N : PRICE CYCLES AND BOOMS
1225
the course of future price distributions.
Therefore it could in principle vary over
time. Accordingly, let V(fi,q) be defined
similarly to V(p*, q ) but for the acceptance
price fi [i.e., substituting fi for p* in (411.
Equation (4) implies that changing fi induces a parallel shift of the function V(fi, q).
Thus, q* is no? affected by changes of p'.
For every c, define >(c) as the acceptance price that solves
unique dispersed price equilibrium. If c < c*
there exist two dispersed price equilibria
corresponding to the two values of q satisfying (3), as illustrated in Figure 1. This completes our review of Burdett and Judd's
(1983) analysis.
Although two dispersed price equilibria
exist when c < c*, only one of these is stable. Consider q, in Figure 1. If q > q,, the
price distribution is such that the value of
sampling a second price is less than its cost
(i.e., V < c). Thus, q will increase away
from q,. An analogous argument obtains
for q < q,. On the other hand, q, is stable;
if q, < q < q,, then V > c, and the number
of consumers who wish to sample twice increases, moving q toward q,. Similarly, if
q < q,, then V < c, and the number of consumers who sample once increases, moving
q toward q1.4 Although our subsequent
analysis refers to the stable equilibria, the
existence of cyclical equilibria, as demonstrated in the next section, applies to the
unstable dispersed price equilibria as well.
While in the nonsequential case the acceptance price is the exogenous reservation
price, p*, this need not be the case in the
multiperiod problem. In that case, the acceptance price is determined strategically by
4 ~ are
e grateful to Dale Mortensen for pointing
this out to us.
Note that B(c) is well defined since q* does
not depend on the acceptance price and
since, by (4), for every c there exists a
sufficiently small fi > 0 satisfying V(fi,q*)
< c. The price j ( c ) is the critical price such
that, for an acceptance price fi < B(c>, only
a single price equilibrium exists, while for
fi 2 6 , a dispersed price equilibrium exists
as well. Our analysis of the dynamic priceformation process will be based on the endogenous variation in fi over time, which in
turn shifts the function V(fi, q ) periodically,
affecting the relative intensity with which
consumers search at different periods.
11. Cyclical Equilibria
To understand how cyclical equilibria
arise in the model, it is instructive first to
consider some specific examples. The more
general analysis is relegated to the next
section. Consider a dynamic search model
in which the consumers' reservation price is
p* = 10, the sampling cost is c = 1, 6 = 1.2,
and the horizon is T = 10.
We shall first derive the equilibrium at
the last period and proceed by backwards
induction. First observe that in the nonsequential case, the equilibrium price distribution, as described by (I), is independent
of the measure of buyers in the market, k T .
Thus the analysis of the last period is identical to that of the nonsequential model. Since
it is the last period, the acceptance price fil,
is simply the reservation price p* = 10.
Therefore, regardless of what has transpired up to the last period, since V(10, q*)
2 1, there exists a dispersed price equilibrium at period 10 as specified by (I), (21,
THE AMERICAN ECONOMIC REVIEW
1226
DECEMBER I992
t
Parameter
1
2
3
4
5
6
7
8
9
10
and (4). Calculating the (stable) equilibrium
price distribution yields that 26 percent of
the consumers sample one price, and the
average price at period 10 is A,, = 3.34 (see
Table 1).
Now consider the decision problem of a
consumer in period 9. By postponing her
purchase to period 10, her expected total
outlay, including the expected price A,, and
the additional search cost, 1, is 4.34. Therefore, since 8 = 1.2, the most she is willing to
pay in period 9 is 1.2(A,, + 1) = 5.21.
Therefore, the period-9 acceptance price is
dramatically reduced to 6,= 5.21. Calculation shows that V(5.21,q)< 1 for every q,
implying that at period 9 the unique market
equilibrium is the single price equilibrium
in which each firm's price is 5.21. At period
8 the consumer has the option of delaying
her purchase for either one or two periods.
Therefore,
which, in our case, gives 6,= 6.25. Since
V(6.25, q ) < 1 for every q, we again have a
single price equilibrium at period 8 in which
each firm's price is 6.25.
The equilibrium prices in periods 7 and 6
are calculated in the same manner. This
process continues until the acceptance price
rises to a level at which a dispersed price
equilibrium is again sustainable. This occurs
at period 5 [i.e., 6,> jXc)]. At period 4
the acceptance price is again below b ( c ) ,
and the process is repeated. Thus in this
example the equilibrium price cycle is of
length 5.
In the preceding cyclic equilibrium, no
firm's price ever exceeds the consumer's
(current) acceptance price. Nevertheless,
there are periods during which there are no
sales. Consider a consumer in the market at
period 9. Once having paid for a price sample, she accepts the market price, 5.21. Buying at period 9 is not the consumer's optimal strategy, however. By sampling at this
period, her total outlay, including the search
cost and the purchasing price, is 5.21 + 1 =
6.21. Consider an alternative strategy of
postponing sampling to period 10. The expected total outlay from following this plan
is 1 + 3.34 = 4.34. Since 6.21 > 4.348, this
strategy is clearly superior to sampling and
VOL. 82 NO. 5
FERSHTMAN AND FISHMAN: PRICE CYCLES AND BOOMS
TABLE2-EXAMPLE OF CYCLICAL
EQUILIBRIUM
WITH CYCLE
OF LENGTH
3
(UNSTABLE
DISPERSED
PRICEEQUILIBRIUM)
t
Parameter
41
PI
41
PI
AD.
1
2
3
4
5
6
7
8
9
9.10
9.10
P
0
7.58
7.58
2F
0
5.32
10.00
0.48
3P
3u
9.10
9.10
F
0
7.58
7.58
2P
0
5.32
10.00
0.48
3P
9.10
9.10
P
0
7.58
7.58
2P
0
5.32
10.00
0.48
3P
3/1
3fi
the unstable dispersed price equilibrium
(note that, for convenience, in this example
T = 9).
111. General Properties of Price and
Demand Cycles
buying at period 9. So active demand at
period 9 is zero!
An analogous observation holds for periods 6, 7, and 8: active demand is zero at
each of them. Only in period 5 is the expected price sufficiently low to make market
participation attractive. This pattern of consumer behavior creates demand booms.
Consumers in the market during periods
6-9 accumulate until the boom in period
10. At this time the market is "cleared" as
all five cohorts sample prices and purchase
their units. The patterns of prices and booms
is illustrated in Figure 2.
The construction of the preceding equilibrium corresponds to the stable dispersed
price equilibrium. For the same parameters
one can generate a three-period cycle equilibrium in which a boom occurs every third
period. This equilibrium, given in Table 2
and illustrated in Figure 3, corresponds to
This section generalizes the intuition suggested by the above examples. As a temporary heuristic device, we assume a finite
horizon of length T > 1. As shown above,
the analysis of the last period is independent of the horizon, allowing us to proceed
through backwards induction.
If V(p*, q*) < c, only the monopoly price
equilibrium exists in the final period. Therefore this is the only equilibrium that exists
in any preceding period. Our concern is
with the nontrivial case, V(p*, q*) 2 c which
allows for a dispersed price equilibrium at
the last period. For concreteness, we shall
restrict attention to the stable dispersed
price equilibrium.
Let E , be the expected overall cost of
purchasing the product at period T. This
cost consists of the average price plus the
cost of sampling one firm:
where q, solves V(p*,q,) = c and p is
given by (2). From the equilibrium condition V(p*,q,) = c it is clear that a consumer who samples twice has the same ex-
1228
THE AMERICAN ECONOMIC REVIEW
pected overall purchasing cost. Thus E T is
well defined.
More generally, let E, be the expected
total outlay at any period t at which the
acceptance price is 3,. If there is a single
price equilibrium at period t, E, = fit c.
If there is a dispersed price equilibrium
at that period, E, is defined as in (61, with
fit replacing p*, p(fi,) = j,q, /(2 - q,) replacing p , and q, [& defined by V(fi,, g,) =
c] replacing q,.
The following Lemma is used extensively
in the subsequent analysis.
+
LEMMA 1: At any period at which there is
a dispersed price equilibrium, E, < fit.
(See the Appendix for a proof.)
Lemma 1 states, in particular, that if there
is a dispersed price equilibrium at period T,
then the sum of the average market price
and the cost of sampling one price is less
than the acceptance price (i.e., E , < p*).
Consider period T - 1. A consumer in the
market at this period buys at the lowest
price observed as long as this price does not
exceed 6ET. If the lowest price observed
exceeds this price, she prefers to defer consumption. Note that, although the consumer
is not certain to get E,, the assumption of
risk neutrality implies that she considers
only the expected price. Thus at period
T - 1 the new acceptance price is fiT-, =
min{p*,SE,}. Observe that, if there is a
dispersed price equilibrium at peroid T - 1,
then firm i's profits are
T h e equilibrium price distribution
FT-,(.) is derived from the equal-profit
~(p,), for every
condition: T&-l(pi) = d p, and pj in the support of F,-,( .). Thus,
from (7), p,-, plays no role in the derivation of FTPl(.).
Sincq E, is not a functionAof 6, there
exists 6 > 1 such that for 6 > 6, SET 2 p*.
In that case, the acceptance price at T - 1 is
DECEMBER 1992
p*, and F,-,(.) = F,(.). This is not the
case when 6 is small, however.
PROPOSITION 1: Corresponding to ecery
p* and c there exists S(p*,c) such that if
6 < 6( p*, c), no stationary dispersed price
equilibrium exists.
(See the Appendix for a proof.)
To better understand the role of 6 in
determining when a stationary dispersed
price equilibrium exists, return for a moment to the example from Section 11. There
it was shown that, for small 6 (e.g., 6 = 1.2),
the equilibrium price path is cyclic. Now
suppose that the stable dispersed price
equilibrium obtains at the final period, 10.
The average market price is then 3.34. A
consumer in the market at period 9 expects
a total cost of 3.34+ 1 = 4.34 if she delays
her purchase for one period and is therefore willing to pay up to 4.346 at period 9.
If 4.346 < 10 (i.e., if 6 < 2.3), then 5 , < fi,,,
and the dispersed price equilibrium is not
stationary. If 6 2 2.3, the acceptance price
(and therefore the dispersed price equilibrium) is the same at both of these periods.
The same argument extends back to any
period. Thus, in that case, the equilibrium
price distribution is stationary.
The possibility of zero active demand further complicates our analysis. Consumers
will not sample at period t if the expected
total purchasing cost at that period exceeds
the discounted expected cost at some later
period (not necessarily t + 1). Of course,
zero active demand can be introduced in a
trivial and artificial way if, at some periods,
all firms "happen to" charge more than the
reservation pricee6If this occurs, consumers
' ~ o t ethat Proposition 1 also holds for 6 = 1. This
follows directly from Lemma 1. The acceptance price
at T - 1 in this case is CT- 1 ET < p* and thus i T
< i T In this paper, however, we assume that 6 > 1,
since this condition is necessary for the existence of
cycles.
6 ~ o t that
e even in a Diamond-type equilibrium one
can define such zero-active-demand equilibria if all
firms charge more than p*.
l
VOL. 82 NO. 5
FERSHTMANAND FISHMAN: PRICE CYCLES AND BOOMS
will obviously defer their search to later
periods. This is not the type of effect we are
after. What we are interested in is the existence of periods in which, although it is
expected that firms will offer prices below
(or equal to) the acceptance price fi, if
contacted, consumers nevertheless prefer to
defer search to later periods. This may occur because the acceptance price defines
the most the consumer is willing to pay once
the search cost has been incurred. Prior to
incurring this cost, however, consumers
compare the current expected cost of purchase, including the expected price and c ,
with discounted future purchasing costs.
Thus a price that is acceptable once the
search cost has been paid might not be
acceptable before it has been paid.
The following propositions provide some
basic properties of the nonstationary equilibria.
PROPOSITION 2: Consider two consecutice periods t, t + 1 I
T, such that active demand is positice at each and such that at
period t 1 there is a dispersed price equilibrium. Then there is a dispersed price equilibrium at period t as well.
+
PROPOSITION 3: If at period t there is a
single price equilibrium in which the market
price is p, < p*, then:
(i) at t - 1 either there is a dispersed price
equilibrium or active demand is zero; and
(ii) if at period t - 1 there is Aa dispersed
price equilibrium, ;hen there is 6 such that,
for every 1 < 6 < 6, A , - , < p,. 7
(See the Appendix for proofs of these
propositions.)
As discussed above, at any period prices
are nondegenerately dispersed only if the
acceptance price at that period lies above
the critical price, p^(c). As the examples
presented in Section I1 suggest, cyclical behavior of prices and active demand are associated with transition periods such that
' ~ o t e that 6^=[g(q*)]-'
where g ( q * ) i l is as
specified by (A6) defines an upper bound for 6.
1229
immediately preceding such a period the
acceptance price is below the critical level,
$(c), while immediately following it the acceptance price lies above the critical level or
vice versa. It is natural to ask how general is
this phenomenon. The following proposition asserts that, if the time horizon is sufficiently long and 6 is sufficiently small,
then, if there is a dispersed price equilibrium at the final period, there exists a preceding period at which the only equilibrium
is a single price equilibrium and active demand is zero.
PROPOSITION 4: There exists 8 > 1 such
that to ecery 1 < 6 < 6 there corresponds T(6)
such that if the horizon is longer than T(6)
[i.e., 7' > T(S)], then there exists a period 5
1 5 t^ < T, such that at each t > t" there is a
dispersed price equilibrium and active demand is positice, while at t^ only a single pricc
equilibrium exists and actice demand is zero.
(See the Appendix for a proof.)
Using the preceding propositions, we can
use backwards induction to construct equilibria that exhibit cyclical patterns. Our construction is achieved by analyzing the evolution of the acceptance price 6, over time.
At period T the acceptance price b, =
p*, and the expected purchasing cost is E ,
defined by equation (6). Now define the
following sequence.
For every t - 1 < T ,
where
and where q, is defined by V(b,, q , ) =-c.
According to Proposition 4, if 6 < 6 and
T > T(6), there exists a final transition period, t^, such that prices are dispersed and
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THE AMERICAN ECONOMIC REVIEW
active demand is positive at every period
5 T , while at t^ active demand is zero
and all firms charge the same price, 5;.
Since at period i there is a single price
equilibrium, fii-, = rnin[S($; + c), 6 ' ~ ; + , ]
= min[6(6E;+, + c),
= 62E;+1 =
65; > 5,. If fii-, L fi, then t - 1 is marked
by price dispersion and positive active demand. If not, t^ - 1 is also characterized by a
single price and zero active demand. More
generally, let 6 be the smallest positive integer such that 6%5;2 fi. Then price dispersion and positive active demand return at
period t^ - 6, which is the penultimate transition period. This marks the completion of
one cycle along the equilibrium price path.
Whereas prices are dispersed after t^ and
only single price equilibria obtain between
t^ - 6 and t", the reverse relation obtains for
t^ - 6. This construction can be rolled back
repeatedly until the first period.
It follows immediately from the construction of the cyclical equilibrium in Section I1
and the preceding analysis that an infinite
repetition of the cyclic price path is an
equilibrium for the infinite-horizon dynamic
search problem as well. Thus our contention that equilibrium search theory can
explain the occurrence of price cycles and
booms does not require the assumption of a
finite horizon. It is less obvious that Proposition 1 also holds for the infinite-horizon
case. Nevertheless, this turns out to be the
case.
t^ < t
PROPOSITION 5: Consider a dynamic
search pcoblem with an infinite horizqn. There
exists a 6 > 1 such that for 1 < 6 < 6 a stable
stationary dispersed price equilibrium does not
exist.
(See the Appendix for a proof.)
IV. Concluding Remarks
Search theory has been used successfully
to explain the persistence of price dispersion. In this paper we have used an explicitly dynamic search model to account for
cyclical patterns of prices and demand. From
the vantage point of firms, the dynamics of
the model refer to the absence of commit-
DECEMBER 1992
ment to previously announced prices; new
prices may be posted at any time. From the
point of view of consumers, the dynamics of
the model are expressed in two ways. First,
buyers are able to defer consumption to
future periods. Second, there is a lag between the time a mice is solicited and the
time it is received: If buyers are impatient
to consume, this lag may induce a demand
for the simultaneous solicitation of more
than one price. Interestingly, our analysis
shows that, even when this exogenous impatience tends to vanish (i.e., as 6 -+11, an
endogenous cause for impatience emerges:
future prices may be less favorable than
those presently available. Thus, even buyers
who are very patient simultaneously sample
several prices even if the horizon is infinite.
One unrealistic feature of our analysis is
that the equilibrium price cycle includes
periods in which active demand is zero. This
is a consequence of our assuming that consumers are homogeneous. If the more realistic assumption of consumer heterogeneity
were introduced, it is to be expected that
only consumers with low search costs would
accumulate until the boom period, while
those with high costs would buy at every
period. The assumption of homogeneity is
nevertheless maintained to deliver the paper's message in the simplest possible way.
More importantly, it serves to emphasize
how our approach can account for endogenous price and demand cycles without appeal to exogenous factors such as shocks or
heterogeneity of agents. This contrasts with
other microeconomic models of price cycles
(e.g., Eytan Sheshinski and Yoram Weiss,
1977; John Conlisk et al., 1984; Joel Sobel,
1984). Certainly one can expect that the
inclusion of shocks and differences between
agents would only tend to reinforce our
results. Indeed one may conceive of our
work as an exploration into the minimum
requirements for the generation of cyclical
effects. It is our hope that models of this
type will prove useful in improving our understanding of the micro foundations of
business cycles. The development of these
macroeconomic implications, while beyond
the scope of the present paper, is the subject of ongoing research.
VOL. 82 NO. 5
FERSHTMANAND FISHMAN: PRICE CYCLES AND BOOMS
APPENDIX
Differentiate 4 ( .) and $( .):
PROOF O F LEMMA 1:
Integrating El by parts and letting
F(p(fi,))
= 0 and F(fi,) = 1 yields
Solving the above integral yields
For 0 < q < 1, i,k(. ) and 4 ( . ) are monotonic
decreasing. Also, observe that, since
q(2 - q ) < 1 for 0 < q < 1, the absolute value
of 4'(.)'s numerator exceeds the absolute
value of $'(. 1's numerator while $'(. 1's denominator exceeds 4'(. )'s denominator.
Therefore l@(q)l> It,kf(q)l for 0 < q < 1. It
follows that 4 ( q ) > 1/49) for 0 < q < 1.
Substituting for -p ( j , ) yields
PROOF O F PROPOSITION 1:
Given p* and c, one can use (I), (2), and
(4) to calculate the equilibrium price distribution at period T, F,. Lemma 1 guarantees that E T < p*. Note also that ET does
not depend on 6. Thus let 6(p*,c) satisfy
6(p*, c)ET = p*. Since j,-, 1 6ET, for every 6 < S(p*, c), 6,-, < p* (i.e., the acceptance price at period T - 1 is below the
acceptance price at period T). Since the
derivation of FT- ,(. ) is analogous to that of
FT(.) with j,-, replacing p*, it is clear
that F,-, is different from F,.
Setting c = V(fi,, q,) implies that
Substituting (A4) into (A3) yields
where
Therefore, El - fit < 0 = g(q,) < 1.
It is enough to show that, for every 0 <
q < 1, g(q) < 1. Simplifying (A6), the claim
g(q) < 1 reduces to
Let 4(q) = 2(1- q)/q(2- q) and $(q) =
ln([2- ql/q). Observe that $(I) = $(I) = 0.
PROOF O F PROPOSITION 2:
Suppose that all firms charge the same
price at t . A buyer who has already paid for
a price offer is willing to pay at most fit =
SE,,,. Thus fit is the only possible equilibrium price at t . The expected cost of buying
at t is therefore c + 6E,+, > 6E,+ Since
the discounted cost of buying at t exceeds
that at t + 1, active demand at t must be
zero, which yields a contradiction.
,.
PROOF O F PROPOSITION 3:
(i) Suppose to the contrary that all firms
charge the same price at t - 1, say p,-,. By
an already familiar argument, p,-, = fit-, =
6(p, + c). Thus the total cost of purchasing
at t - 1 is p,-, + c > 6(p, + c), which implies that active demand at t - 1 is zero.
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THE AMERICAN ECONOMIC REVIEW
(ii) Consider the case in which there is a
(stable) dispersed price equilibrium at period t - l . Using (A51 yields E,-l =
g(q,-l)fiL-l, where q,-, is determined by
V(q,- p,- ,) = c. A tedious but routine calculation shows that g(q) is an increasing,
monotonic function and that g(q) < 1 for all
0 < q < 1. Specifically the monotonicity implies that for all 0 < q I q*, g(q) I g(q*) <
1. Thus there exists E ~ 0> such that for
every O < q s q * , g ( q ) < l - ~ l .
Since 3,-, I 6(p, + c) we can use Lemma
1 to obtain the following inequalities:
,,
Now let 8 = (1 - E,)-'. Thus for every 6 < 8,
A,-, + c < p, + c, which completes the
proof.
PROOF O F PROPOSITION 4:
Using (A5) yields that, at every period at
which prices are dispersed, E, = g(q,)fi,,
where q, is determined by V(q,,$,)= c.
Moreover, since we consider only the stable
equilibrium, we obtain from the monotonicity of the function g(q) that g(q) _< g(q*)
for every 0 < q I q*. Thus if at period t
there is a dispersed price equilibrium, then
DECEMBER 1992
PROOF O F PROPOSITION 5:
Suppose the contrary and let F"(.) be
the stationary dispersed price equilibrium.
Let j" be the (constant) acceptance price.
Then, at any period, the construction of the
dispersed price equilibrium must follow that
of the one-period (nonsequential) model of
Burdett and Judd (1983). Thus F"(p) must
be as described by (I), with j" substituted
for p*, 0 < q, < 1 substituted for q, and
pa = pffiqffi/(2- qm)substituted for p. Let A"
be the mean of F" and E" = A" T c. Then,
by already familiar arguments, j" =
min(p*, SE"). If p" = p*, then F" is precisely the stable dispersed price equilibrium
that obtains at the terminal period T in the
finite model. However, then Proposition 1
applies to any period t to the effect that the
acceptance price at t is strictly less than p*
if 6 is sufficiently near 1, a contradiction.
Thus, p" = 6E". Applying (A6) to substitute
for E" gives
However, since for the stable equilibrium
q" 5 q*, we obtain from the monotonicity of
g(q) that there exists E > 0 such that g(q")
< g(q*) < 1 - E. Thus, for 6 < [g(q*)l-l,
< 1, which yields a contradiction.
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*
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