Download TRUE MONOPOLISTIC COMPETITION AS A RESULT OF

TRUE MONOPOLISTIC COMPETITION AS A RESULT OF
IMPERFECT INFORMATION*
ASHER WOLINSKY
I. INTRODUCTION
The industrial organization literature does not make a clear
distinction between oligopolistic competition and monopolistic
competition. In some vague sense, the latter term often refers to
imperfect competition with "numerous" firms and free entry, while
the former term often describes competition among "fewer" firms
sometimes with, and sometimes without, free entry. In two recent
papers Hart [1985a, 1985b1 makes a clear distinction between the
two forms of competition by suggesting a certain definition of
large group Chamberlinian monopolistic competition. He mentions four properties that characterize this form of competition:
(1) there are many firms producing differentiated commodities;
(2)each firm is negligible in the sense that it can ignore its impact
on, and hence reactions from, other firms; (3) each firm faces a
downward sloping demand curve and hence the equilibrium price
exceeds marginal cost; (4) free entry results in zero-profit of operating firms (or, at least, of marginal firms).
Obviously, when there are a finite number of firms, this definition cannot be expected to hold accurately. Yet, it suggests the
following criterion. If in the limit, as the number of firms is made
arbitrarily large, the model satisfies the above properties, then
when there are sufficiently many firms the model can be viewed
as monopolistically competitive. In particular, monopolistic competition requires that, while each firm has negligible impact on
its rivals, it still maintains significant market power.
According to this approach, much of the recent literature on
imperfect competition in differentiated products markets (see, e.g.,
*I would like to thank Ariel Rubinstein for making some useful comments.
©1986 by the President and Fellows of Harvard College. Published by John Wiley & Sons, Inc.
CCC 0033-5533/86/030493-19804.00
The Quarterly Journal of Economics, August 1986
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The paper restates Hart's [1985a, 1985b] definition of large group monopolistic
competition, distinguishes it from oligopolistic competition, and raises the question of whether there are reasonable circumstances that give rise to true monopolistic competition as defined. The purpose is to show that consumers' imperfect
information may create conditions that will turn an otherwise oligopolistic market
into a truly monopolistically competitive one.
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behavior of the utility functions. He adopts the basic model of
Perloff-Salop [1983] and Sattinger [1984] in which there are many
consumers with different preferences for the many potential brands,
and adds to it an assumption that each consumer is interested in
only a fixed finite subset of the potential brands. This latter assumption is what limits the substitutability among brands and
is the key to the result that, at the limiting market where each
firm is negligibly small, it still faces a demand curve with finite
elasticity. The main weakness of this key assumption is that it
is somewhat arbitrary. It is not clear why from among the substitute brands the consumer likes only a fixed number and will
never pay a positive price for any other of the substitute brands.
The assumption is particularly disturbing if one thinks of the
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Spence [1976a], Lancaster [1979], and Salop [1979]) does not capture monopolistic competition. These models describe markets in
which a finite number of firms operate, and therefore firms are
not negligible in the sense of property (2). Moreover, even as the
setup costs in these models are made arbitrarily small, and hence
the number of firms becomes arbitrarily large, they still do not
approximate monopolistic competition. This is because, when the
number of firms is large and each firm is negligible, the demand
for each product becomes increasingly elastic, and the model approximates perfect rather than monopolistic competition. That is,
as the model comes close to satisfying property (2), it hardly satisfies property (3) anymore.
Exceptions to this can be found in Dixit-Stiglitz [1977] and
in some specifications of Perloff-Salop [1985] and Spence [1976b]
in which even as the number of brands is made arbitrarily large,
the demand for each brand does not approach a perfectly elastic
demand. However, in these models this result is due to peculiar
limiting behavior of the postulated consumers' utility functions.
In the Dixit-Stiglitz model, for example, the demand side is derived from a representative consumer's CES utility function, and
as a result the elasticity of demand, is approximately constant
regardless of the number of brands. But the limiting behavior of
this demand structure, as the number of brands is made arbitrarily large, involves a discontinuity in the MRS between a typical brand and the numeraire good at the limiting economy, and
unbounded utility at the limiting bundle.
Motivated by this, Hart [1985a, 198513] presents a true model
of monopolistic competition in the sense that the fulfillment of
the above requirements (1)—(4) does not owe to irregular limiting
TRUE MONOPOLISTIC COMPETITION
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substitute brands as points in bounded characteristics space and
hence expects that, as the number of produced brands increases,
each consumer can find increasingly closer substitutes for each
particular brand.
However, it is made clear by Hart's analysis that a true monopolistically competitive model must involve some type of limitation on the effective substitutability among brands, and the
question is whether there are reasonable circumstances under
which such a limitation would arise naturally and support monopolistic competition in the above sense. The purpose of the
present paper is to supplement Hart's analysis by suggesting that
consumers' imperfect information may create circumstances that
will restrict the effective substitutability among brands and hence
give rise to a true monopolistic competition.
Specifically, we adopt the Perloff-Salop model of product differentiation (which is similar to the model employed by Hart) and
modify it to allow for a certain form of consumers' imperfect information. The approach resembles that of Wolinsky [1984], and
the main differences are that there the product space is spatial
and the prices are exogenously fixed, while here the product space
is not spatial and the main focus is on the equilibrium prices that
are determined endogenously. After deriving the symmetric equilibrium of the model, we examine its limit as the per firm setup
cost is made small and the number of firms is made large so that
each firm becomes negligible. It turns out that the equilibrium
price does not approach the competitive price—the limiting value
of the price is bounded away from the marginal cost by a magnitude that depends on the cost of information. The conclusion
drawn from this is that consumers' imperfect information may
reduce the effective substitutability among brands in a way that
will result in a true monopolistically competitive model.
In the last section we discuss the relations between the model
of this paper and the well-known result by Diamond [1971]. We
distinguish between our result of prices exceeding marginal costs
in an equilibrium with many firms and the result by Diamond
that also describes such a phenomenon. We also show how the
present model can be used to rule out what is sometimes viewed
as an implausible result in Diamond's model.
Finally, although the presentation of the model was motivated by the issues described above, the reader can notice that
the model is potentially useful for considering other issues concerning imperfect competition in markets of differentiated prod-
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ucts. For example, one can investigate the familiar questions
regarding the provision of product diversity, or one may introduce
into the model the possibility of informative advertising and consider questions regarding its role.
II. THE MODEL
Firms
There are many potential firms. Each firm can produce a
single distinct brand. The firms have identical cost functions,
(1)
C(x) = F + ex.
Consumers
There are a large number L of consumers with different tastes.
Each consumer is interested in having one unit of the product. A
consumer's utility from having quantity x o of the numeraire good
and a unit of brand i is of the form,
(2)
u(xo;i) = xo +
where ( v i ,v 2 , . .) are the values attached by this consumer to the
different brands. It is assumed that consumers seek to maximize
the expected value of their utility as captured by (2). Thus, when
there are n available brands (i = 1,2, . . . , n) priced at rn1P21 • • • pa,
and the consumer knows their respective values (v i ,v 2 , .) for
him, he will prefer to buy the brand for which the net surplus
v, — p, is maximal, provided that v, — p, 0; if, for all i, v, — p, < 0,
this consumer will not buy at all.
Consumers differ with respect to their tastes. It is assumed
that for each consumer the valuations v, are realizations of independent and identically distributed random variables with distribution function G, which has finite support [v, v] and a differentiable density function g(v). Thus, the expected fraction of the
consumer population for which v, is less than 0, is equal to G(31)
—
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This section describes the Perloff-Salop model of product differentiation and adds to it the elements of consumer imperfect
information that are needed for the present framework. To facilitate the analysis, we make strong symmetry assumptions
throughout.
Consider a differentiated product that has an unlimited number of potential distinct brands denoted by i = 1,2, .. .
TRUE MONOPOLISTIC COMPETITION
497
and the expected fraction for which v 1 On is
G( 01)G( 52) . . . G( O n ). Note two implications of this assumption.
Information
Our main modification of the Perloff-Salop model concerns
consumers' information. Here it is assumed that initially a consumer knows the number of the available brands, but he does not
know the prices and the values v, of the available brands for him.
A consumer, however, can gather information by searching
among the available brands. We make the following assumptions
concerning consumers' search. First, at a cost s per brand a consumer can sample brands and find out about their prices and
values for him. Second, the consumer's search is without replacement and with recall. That is, each time the consumer incurs the
cost s he learns about a different brand, and he can proceed to
purchase any one of the brands he has already sampled without
incurring additional search costs. Third, the total number of observations that a consumer can make is bounded. Let k denote
the maximum number of observations that a consumer can make.
That is, if in the course of his search a consumer has already
sampled k brands, he may purchase one of the previously sampled
brands but cannot continue his search.
Let us explain the assumptions just made. First, consider the
last assumption concerning the existence of a maximum number
k of possible observations. It should be emphasized that for the
purpose of deriving the results of this paper we need not place
any such constraint on the length of a consumer's search. Furthermore, throughout the analysis we shall regard k as large in
the sense that the constraint is, in fact, nonbinding for all but
possibly a negligible percentage of the consumers. However, since
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First, aggregate preferences for the different brands are symmetric in the sense that, for any two brands i and j, the expected
number of consumers who prefer i to j is equal to the expected
number who prefer j to i. Second, the value that a consumer
attaches to a particular brand is independent of the values that
he attaches to other brands.
Finally, note that the above assumptions imply that the value
that a consumer attributes to the best available brand is stochastically increasing with the number of the available brands
n. Of course, when n is made arbitrarily large, the expected value
attributed by the consumer to the best available brand approaches
the maximum possible value I).
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III. MARKET DEMAND
Suppose that there are n operating firms. The purpose is to
derive the demand facing each firm in a symmetric configuration
in which all other firms charge the same price p*.
Consumer Search
Consider first the decision problem of a consumer who expects
that all firms charge the price p* and can gather information at
the cost s per observation as described above. Define w*E[v,"v)
- to
be the (unique) solution for the equation,
(3
)
I. (u — w*)dG(v)
s,
1. This particular specification fits a situation in which the sampling cost is
required only to learn about the existence of a product and about its properties,
but it is not part of the costs associated with buying the product. That is, strictly
speaking, this specification does not fit the case in which s is a travel cost incurred
by a consumer who travels to stores to examine products, since this will violate
the assumed perfect recall property of the search. Of course, it is not that we are
especially interested in the particular situation that fits the specification, but
rather view it as an analytically convenient example.
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obviously there is some such upper bound on the number of observations (say, due to some wealth constraint or a physical constraint), and since we shall consider configurations with arbitrarily large numbers of available brands, this assumption is
imposed to ensure the reader that the results to be reported are
not obtained due to the violation of such a constraint.
Next consider the assumption concerning the recall and no
replacement properties of the search. Of course, this assumption
results in just one out of a number of possible specifications of
the consumer's search. 1 Note that since the search here is with
finite horizon, the different specifications (whether the search is
with or without replacement and with or without recall) will give
rise to different optimum search policies. However, when the horizon of the search is sufficiently long, the optimum search policies
and expected outcomes that correspond to the different specifications are arbitrarily close (with infinite horizon the optimum
policies for the different specifications coincide). Thus, the loss of
generality involved in focusing on a particular specification of the
search is not severe, especially as we shall be interested in situations where the horizon of search is long (i.e., both n and k are
large).
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499
and if (3) has no solution, define w* = v . Let T denote min[n,k],
where k is the postulated upper bound on the total number of
observations that a consumer can make. The following claim describes the search and purchase policy that maximizes the consumer's expected benefit, given a uniform price p* and the numbers n and k.
a. If w* p*, the consumer's optimum decision is not to participate in the market.
b. If w* > p*, the optimum policy is to follow sequential search
with a fixed reservation value w*. That is, the consumer stops
sampling and buys the first brand i such that v, w*; if no
such brand is found among the first T brands sampled by the
consumer, he buys the brand i such that v, is maximal among
these T valuations, provided that v, p*; otherwise he does
not buy.
The claim is proved in the Appendix. Note that if k > n, then
T is equal to the number of available brands n, and the constraint
on the sampling as captured by k does not affect the search.
Before we proceed, let us explain the reservation value w*
as defined by (3). Equation (3) states that once the consumer has
sampled a brand which he values at w*, the expected improvement from sampling another brand (captured by the left-hand
side of (3)) is exactly equal to the cost s of sampling it. Therefore,
given any brand for which the consumer's valuation is greater
(smaller) than w*, the expected improvement from sampling another brand is lower (higher) than s. Note that when s is large
relative to the distribution of valuations as captured by G, then
the reservation value w* is equal to v which means that, with a
uniform price p*, there is no incentive to search. In what follows,
we shall rule out the degenerate cases in which w* = v, and no
search takes place. Instead, we shall assume throughout that the
relations between s and G are such that w* > v so that some
search will take place at the symmetric configurations that will
be considered.
Suppose now that all operating firms but one charge the price
p* and that the remaining firm, say firm j, charges price p. If the
consumer expects that all firms charge the price p*, and if he
continues to hold this belief about other firms even after observing
the deviant firm's price, then upon sampling brand j he will stop
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CLAIM
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his search and buy it if and only if v, w(p), where w(p) is defined
by
w(p) p = w * p*.
(4)
If v , < w(p), the consumer will not buy brand j immediately, but
he may still buy it in the event that after observing a total of T
brands he realizes that v, > p and that v, — p is greater than the
surplus v, — p* associated with any other of the sampled brands.
.
Suppose that p* < w* so that consumers choose to participate
in the market. Let D(p,p*,n) denote the expected demand for a
brand (say, brand j) as a function of its own price, p, the price of
all other brands p*, and the number of brands n. For prices p such
that w(p) > v ,
1
(5) D(p,p*,n) = L I [1
—
G(w(p))]4--, E G(w*)'
t=0
w(p)
G(p* — p
V)T- lg- (v)dv}.
p
The right-hand side of (5) is a product of the number of consumers
L multiplied by the probability that a randomly drawn consumer
will end up purchasing the considered brand j. The term Tin
captures the probability that brand j will be one of the first T
brands in a random order of all the n brands. That is, the probability that a consumer who samples sequentially T brands out
of the n available ones will sample the considered brand j. The
expression in the curly brackets captures the probability that a
consumer will purchase brand j, given that this brand is one of
the T potential observations of this consumer. The first expression
in the curly brackets captures the probability that the consumer
will sample brand j and immediately stop the search and buy it.
A typical component of this expression is a product of the following
probabilities: the probability 1 — G(w(p)), that the consumer would
indeed want to stop at brand j; the probability 1/T that brand j
is the ith brand in the random order in which the consumer samples; and the probability G( w*)i -1 , that the first i — 1 brands in
such a random sampling order are valued by the consumer at less
than w* (and hence are not purchased when sampled). The second
expression in the curly brackets captures the probability that the
consumer samples T brands and realizes that brand j is the best
buy out of them. Indeed, given a particular value v of brand j,
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Market Demand in Symmetric Configuration
TRUE MONOPOLISTIC COMPETITION
501
the probability that v — p v i — p* for all i out of the T — 1
other brands is equal to G(p* — p + v)T 1 . This is integrated over
all possible values v of brand j that are consistent with its purchase ( v -^ p) and are also consistent with brand j not being picked
up immediately (v w(p)).
To simplify (5), notice that the first expression in the curly
brackets is equal to
-
[1
-
G(w(p))](1 — G(w*) T )I(T[1 — G(w*)]).
Using (6), we have for p* < w*,
(7) D(p,p*,n) = L
T 1[1- G(w(p))1[ - G(w*) T
T[1 - G(w*)[
]
w(p)
[G(p* p
0]T-ig-(v)dul.
p
Substituting p = p* above, we get
(8)
D(p*,p*,n) = Lin [1 — G(p*) T].
The foregoing derivation of the demand is for prices p such
that w(p) > V. If p is such that w(p) v, then D(p,p*,n) is independent of p and is equal to the expected number of consumers
who sample this particular brand.
It can be shown that D(p,p*,n) is decreasing in p, and for
later reference we derive for p such that w(p) > v,
(9) Dp(p,p*,n) = L -
„ 1 - ( w* )T
G G(w*)]
w(p)
G(p* - p v)T-1g "
g(w(p))
-
p
Finally, let us remark on the role of the assumption concerning the constraint on the total number of observations that
a consumer can make. When k n, the constraint is not binding;
T = n; and obviously the demand is unaffected by the constraint.
When k < n, there is a positive probability that the constraint
will be binding; we have T = k; and the demand function (7) can
be written as
D(p,p*,n) = L 1[1 - G(w(p))]
[1 - G(w(p))]
k G(w*) k
n [1 - G(w*)]
[1 - G(w*)]
(10)
w(p)
+ k
[G(p* - p + vA k-ig(v)dv}.
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(6)
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IV. SYMMETRIC MARKET EQUILIBRIA
This section considers oligopolistic markets with and without
entry. A market without entry is modeled as a fixed number n of
operating firms that choose their prices simultaneously. A market
with entry is modeled as a two-stage process where in the first
stage the potential firms decide whether or not to enter (the entrants incur the cost F), and in the second stage the entering firms
choose their prices. It is assumed that the firms are maximizers
of expected profit and that they base their decisions on their expectations concerning the behavior of their rivals and on their
knowledge of consumers' demand.
The focus is on symmetric equilibria at which all operating
firms charge the same price p*, and all consumers correctly expect
the price to be p*. Let Tr(p,p*,n) denote the expected profit of a
firm that charges price p when all other firms charge p* and the
total number of operating firms is n:
(11)
Tr(AP*,n) = (p — c)D(p,p*,n) — F.
Following are the definitions of the symmetric oligopolistic equilibria for markets with and without entry.
DEFINITION 1. Given a fixed number of firms n, a symmetric oligopolistic equilibrium (SOE) is a price p* = p*(n) such that
p* < w* and
(12)
p = p* maximizes 7(p,p*,n).
DEFINITION 2. A symmetric free entry oligopolistic equilibrium
(SFEO) is a pair (p*,n*) such that p* < w* and
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Note that the part of the demand function which depends on k
(the two last terms in (10)) is of an order of magnitude of less
than 1/k of the total demand. That is, when k is sufficiently large,
the existence of that constraint has only a negligible effect on the
expected quantity demanded from a firm. This is not surprising,
since with the sequential search described here, the expected
number of observations made by a consumer is about 1/[1 — G( w*)]
and, when k is sufficiently large in comparison to 1/[1 — G(w*)],
the probability that a consumer will be affected by the constraint
is arbitrarily small.
TRUE MONOPOLISTIC COMPETITION
(13) p
503
p* maximizes Tr(p,p*,n*);
(14) IT(p * ,p * ,e)
0,
and Tr(p' ,p' ,n* + 1) < 0,
for any p' which is a SOE price with n* + 1 firms.
(15)
p* = c — D(p*,p*,n)ID p (p*,p*,n).
On substituting (8) and (9) into (15), we get
(16) p* = c
1 — G(p*)T
{g(w*)(1 — G(w*) T)I(1 — G(w*)) — T f G(v) T g' (U)C10
.
Pe
Thus, if there exist nontrivial equilibria of the types described by
definitions 1 and 2, the equilibrium price p* must satisfy (16).
Note that p* depends on the number of firms n through the number T = min[k,n]. Later on we shall derive the limiting equilibrium price as the number of firms n is made arbitrarily large,
and in particular, inquire whether the SFOE approximates a monopolistically competitive equilibrium in the sense described in
the introduction. However, before turning to that analysis it will
be useful to verify that the equilibrium concepts are not empty.
We shall not attempt general analysis, but rather point out cer-
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Note that since consumers' demand D(p,p*,n) is derived as described in Section III, the equilibrium configurations satisfy the
requirement that consumers' expectations are indeed consistent
with the equilibrium prices. Note also that condition (14) replaces
the customary "zero profit condition" which cannot be expected
to hold accurately when there are a discrete number of firms.
In the absence of the requirement that p* < w*, there will
always exist trivial equilibria in which all firms charge a sufficiently high price p* so that no consumer searches, and hence
demand is zero. The requirement p* < w* rules out these equilibria and admits only nontrivial equilibria in which trade takes
place.
The assumption made throughout that w* > v implies that
if p* < w* then, for p in the neighborhood of p*, the formulae (7)
and (9) are applicable. Therefore, the first-order condition for a
price p* to be a solution fir (12) (and. hence (13)) is Trp (p*,p*,n) = 0,
which using (11) can be written as
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tain sufficient conditions for existence and uniqueness of SOE and
SFOE.
PROPOSITION. If G( v)
is uniform on [v,v1 and w* + c)/2, then
(a) for any n and k there exists a unique SOE. (b) for all
values of the setup cost F below a certain level F there exists
a unique SFOE.
(17) p* = c + (ti — w*)
(
x [1
p*
il
12 T
- V
1
*
v
T
I1 - V
Equation (17) clarifies the purpose of the requirement that
w* (T) + c)/2, since it can be verified that, if that requirement
holds, the solution p* for (17) satisfies p* w*. Note that, since
w* is a decreasing function of the search cost s, the condition
w* (ti + c)I2 requires in fact that s is sufficiently small so as
to ensure that consumers participate in the market.
V. THE LIMITING EQUILIBRIA
In this section we show that, when the number of firms is
large, the above described oligopolistic equilibrium can be viewed
as a true monopolistically competitive equilibrium in the sense
explained in the introduction. Obviously, when the setup cost is
small and hence the number of firms at the SFOE is large, the
impact of each firm's actions on any one of its competitors is
2. Note that for sufficiently large T (i.e., sufficiently large n and k) equation
(16) always has a unique solution. There are two sufficient conditions that guarantee that this solution is an equilibrium. The first condition guarantees that the
solution p* for (16) is smaller than w*. The second condition requires that Tr(p ,p* ,n)
is a concave function of p so that the choice of p = p* indeed maximizes each
firm's profit. Now it is easy to verify that, given a sufficiently small s and a
sufficiently large T, there are many specifications of G for which the above two
conditions are satisfied. For example, this is the case for all G functions that
feature increasing hazard rate.
,
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The proposition is proved in the Appendix. The sufficient conditions of the proposition are admittedly quite special, and they can
no doubt be extended, 2 but this is outside our main interest in
this paper.
The uniform distribution postulated in the proposition is such
that G(v) = (v — v)I(i — v), g(v) = 11(i.) — v), g'(v) = 0, and (16)
takes the particularly simple form,
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TRUE MONOPOLISTIC COMPETITION
(18)
lim p* = c + [1 — G(w*)]1g(w*).
For the special case in which G(v) is uniform over [v,i)] this limit
is
(19)
lim p* = c +
Note that when we refer in this ongoing discussion to an increasing sequence of n, we mean in fact a sequence of markets such
that the numbers of firms at the free entry equilibria form an
increasing sequence. Observe from (9) and (11) that the profit of
a firm at a SFOE (p*,n*) is
(20) Tr(p*,p*,n*) = L{(p* — c)[1 — G(p*) T]In* —
Thus, a sequence of markets over which n* is increasing requires
that F/L is decreasing. This can be achieved by looking at a sequence over which the size of the setup cost F approaches zero,
or at a sequence of markets over which the size of consumer
population L increases indefinitely. In the first case the extent of
the market is fixed, and as F —> 0 and n —> 00, the size of each firm
approaches zero. In the second case, as L and n —>x the size
of a firm remains roughly of the same magnitude, but the extent
of the market increases indefinitely.
Now, it follows immediately from (18) that even when the
number of firms n is arbitrarily large and when the constraint k
is so large that it is practically nonbinding, the equilibrium price
is still bounded away from the marginal cost. That is, even when
n and k are arbitrarily large, each firm maintains significant
market power in the sense that the demand for its product does
not approach a perfectly elastic demand. This is so despite the
,
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negligible, and the free entry condition closely approximates the
zero-profit condition. What has to be shown is that, even when
the equilibrium number of firms n is large so as to render each
firm negligible, firms still maintain significant market power. To
see this, suppose that G is such that, for any sufficiently small
values of the setup cost F and sufficiently large values of k, there
exists a unique SFOE. Observe that, for all values of n greater
than the maximum number of observations that a consumer can
make k, we have T = k, and the equilibrium price is fixed at the
level obtained by substituting T = k into (16). When both n and
k are large, the equilibrium price is in the neighborhood of
p*, where
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QUARTERLY JOURNAL OF ECONOMICS
fact that as n 00 each brand i has arbitrarily close substitutes
in the sense that for any consumer h and any number E > 0:
(21) prob {there is sold a brand j such that I v 1,1 — v,71 < El ---> 1.
n__..
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It should be emphasized that the discrepancy between the equilibrium price and the marginal cost is not due to the existence of
the constraint k. As is made clear by (18), even when k is arbitrarily large, p* is bounded away from c.
As noted in the introduction, there are other models of product differentiation in which the limiting value of the equilibrium
price is bounded away from the marginal cost (e.g., Dixit-Stiglitz
[1977], and certain specifications of Spence [1976b] and PerloffSalop [1985]). However, in those models the result is due to peculiar limiting properties of consumers' utility functions. For example, in the Perloff-Salop model, which is the perfect information
version of the present model, this result may obtain only if —v = 0c,
and it is made possible due to the fact that, as the number of
brands increases, consumers' willingness to pay for the best available brand increases without a bound.
When the models mentioned above are corrected to rule out
the peculiar limiting behavior of consumers' utility functions, the
monopolistically competitive result of lim p* > c is replaced by
lim p* = c. This is not surprising, since on considering, for example, the Perloff-Salop model with bounded consumer valuations (I) < it is easy to verify that as n—> oc, each brand has
arbitrarily close substitutes in the sense of (21) and therefore the
demand for each brand becomes perfectly elastic.
It is therefore obvious that if a model in which consumers'
utilities are well behaved in the limit is to have the monopolistically competitive feature of lim p* > c, there must be some limitation on the substitutability among brands. Indeed, this is the
case both in Hart [1985a, 1985b] and in the present model. In
Hart's model the substitution among brands is limited due to a
property of consumer tastes—each consumer is interested in only
a fixed number of brands. Hence, even as the market variety
increases indefinitely, the consumer will not have arbitrarily close
substitutes for his best buy. In the present model the availability
of substitutes is not limited, and as noted above, when n oc for
each consumer, there will be arbitrarily close substitutes for the
brand that he buys. The important limitation here on the effective
substitutability is not due to the existence of the upper bound k
TRUE MONOPOLISTIC COMPETITION 507
but rather due to the fact that it is suboptimal for consumers to
examine too many of the available substitutes. The costly information induces consumers to search the market sequentially with
a fixed reservation value w*, and therefore, even as n --> x and
even when k is large, each consumer samples on the average only
about 1/[1 — G(w*)] brands that might be just a small fraction
of the existing variety.
This section considers the behavior of the symmetric equilibrium price when the search cost s is small, and when both s
and the extent of the heterogeneity of the products are small. One
of the main reasons for examining these points is that they enable
us to understand the relations between the model of the present
paper and the well-known result presented by Diamond [1971].
It will be convenient to confine the discussion to the example
in which G(v) is uniform over [v, --u]. The reason is that we want
to make certain qualitative points rather than investigate the
theoretical bounds of the following statements. But it can be verified that here too the analysis can be extended to cover a wider
class of cases including, for example, the cases referred to in
footnote 2.
Notice from (3) that, for any T, lim o w* = 1.). It then follows
from (19) that
(22)
lim lim p* c.
That is, when the search cost s is sufficiently small and when the
equilibrium is such that both n and k are sufficiently large, the
equilibrium price is close to being competitive. This observation
is consistent with some natural intuition that leads us to expect
that intense competition (large n) and a negligible information
problem (s ---> 0) should result in a competitive price.
Next consider the nature of the limiting equilibrium price
when both s —' 0 and v V, where the latter means that the
product space under consideration degenerates into a homogeneous product valued by all consumers at V. It can be verified
from (19) that, if the rates at which s approaches zero and v
approaches V are such that w* is always greater than v, then
(23)
lim (lim p*) = c.
.9-0 7,—,
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VI. THE EQUILIBRIUM WHEN S IS SMALL
508
QUARTERLY JOURNAL OF ECONOMICS
(24) lim lim p* = R.
The two points concerning the relations between the Diamond
model and the present one are as follows. First, the contrast between (24) and (22), as well as the fact (from (19) and (3)) that
p*) — dw* 1
0.
ds ds 1 — G(w*) >
imply that the result of lim T x p* > c presented in the previous
section was not a repackaging of Diamond's result. This observation is important, since Diamond's result is often viewed as
suggesting a difficulty in modeling' rather than a positive prediction for the behavior of prices in a market with negligible
search costs and many competitors. Therefore, it is desirable to
verify that our result of lim p* > c is not subject to the same
difficulty.
Second, recall that the main objection to Diamond's result
arises because according to some natural economic intuition it is
felt that the equilibrium outcome in such a market should be
3. Note that the assumption made here that consumers always know the
correct price distribution (and not just at equilibrium) is different from our assumption that consumers' expectations concerning the price distribution are confirmed only at equilibrium. Without such a change of the assumption, the only
equilibrium in this version of Diamond's model is the degenerate equilibrium at
which no trade takes place.
4. The result is, of course, correct. It suggests a difficulty in modeling because
it is somewhat counterintuitive. In that sense, it belongs to a list of problems
such as the finitely repeated Prisoner's Dilemma and the Chain-Store Paradox.
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Thus, in a market where both n and k are large and where the
product is approximately homogeneous and the search cost is
negligible in the appropriate sense, the equilibrium price is approximately competitive.
Both the result captured by (22) and by (23) seem intuitively
plausible, but our main interest in them derives from their relations to the well-known result by Diamond [1971].
A particular version of Diamond [1971] describes a market
for a homogeneous good where each consumer wants to buy one
unit, provided that its cost does not exceed the reservation price
R. The identical firms set the prices. The consumers know the
price distribution, but can obtain a price quote from a particular
firm only by incurring a search cost' s. It turns out that, for any
s > 0, if the number of firms is sufficiently large, the unique
equilibrium price is p = R — s, which is the monopoly price. Letting n denote the number of firms, we thus have in Diamond's
model
TRUE MONOPOLISTIC COMPETITION
509
VII. CONCLUSION
The terms oligopolistic competition and monopolistic competition are often used interchangeably. Hart suggested the distinction according to which in monopolistic competition each firm
has both negligible impact on its rivals (a property that is usually
associated with perfect competition) and market power, while
oligopolistic competition lacks the former property. However, it
is difficult to think of a situation where products of the different
firms are substitutes for one another while still the required properties of monopolistic competition are satisfied, and the question
is whether the concept of monopolistic competition is meaningful.
This paper has suggested that this concept is meaningful by showing that a certain form of consumers' imperfect information can
give rise to monopolistic competition as defined.
APPENDIX
Proof of the Claim
The fact that, if the consumer decides to participate in the
market, the optimum policy is as described in (b) follows from a
result in search theory. The result states that the optimum policy
for a sampling process with finite horizon and perfect recall out
of a fixed distribution is sequential search with fixed reservation
value that is characterized by a marginal condition shown by
equation (3). This result, which is somewhat less known than the
similar result for sampling with infinite horizon, is mentioned by
Kohn and Shavell [1974] and proved for a special case in Wolinsky
[1984].
It remains to prove that it is optimal for the consumer to
participate in the market if and only if w* .-- p*. It is a well-known
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approximately competitive. The result captured by (23) suggests
that, in some sense, it is legitimate to view the equilibrium in
such a market as approximately competitive. This is in the sense
that if a market for a homogeneous product with many competitors and small search costs can be viewed as the limiting case of
such markets in which there is also some degree of product heterogeneity, then the counterintuitive character of the equilibrium
disappears, and it becomes approximately competitive. The explanation for this result is that the heterogeneity of the products
induces consumers to conduct some search, even when all prices
are the same, and the byproduct of this is downward pressure on
the price.
510
QUARTERLY JOURNAL OF ECONOMICS
J
* [v — pG(v) — s
P*
= i [v — plc1G(v) + [w* — p*][1— G(w*)].
P*
which is greater than zero iffp* < w*. Since the value of participation when T = 1 is smaller than the corresponding value for
T > 1, the condition p* --. w* is also sufficient for participation.
Q.E.D.
Proof of the Proposition
For G(v) uniform over [v,)] equation (16) takes the form,
(A.1) p* = c + (T — w*)
p* _ 1)
x [1
(T—v
T
1
w*
v
T
ii — v
To prove part (a), observe that, for a given T, the condition
w* _^.-- (V + c)I2 implies that (A.1) has a unique solution p*,
c p* .--- w*. This is because the right-hand side of (A.1) is a
decreasing function of p* over the interval p* E[c,w*[. For p* = c
the value of this function is greater than c, for p* = w* it is
smaller than w* and hence must have a unique fixed point p*E[c,w*].
By construction, the solution p* for (A.1) satisfies the first-order
condition irp (p*,p*,n) = 0. To see that p* is indeed a SOE, it is
enough to establish that Tr(p,p*,n) is concave in p over the interval
of prices such that w(p) > v. It follows from (9) that, with the
uniform G(v), we have D„(p,p*,n) = 0. Hence, Tr„(p,p*,n) =
2Dp (p,p*,n) < 0, and 7r(p,p*,n) is a concave function of p.
To prove part (b), let R (n) denote the firm's profit gross of
the setup cost when k is given, there are n operating firms, and
the price p* is given by (A.1). That is,
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result that, if the consumer faced an unbounded number of brands
and was not constrained with respect to the total number of samples, then the expected value of sampling optimally under the
conditions of the present model is equal to w*. Hence the expected
surplus associated with participation in the market is equal to
w* — p*. Therefore, the expected surplus associated with participating in a market in which the sampling is constrained to, at
most, T samples cannot exceed w*, and hence w* -^- p* is a necessary condition for participation in this market. Note from (3)
that when T = 1, the expected value of participation is equal to
TRUE MONOPOLISTIC COMPETITION
R(n)
511
H(p*,p*,n) + F
T2
= L(T) — w*)[1 (131* — 12 ) / n[1 (w
— v 1) 1
,
THE HEBREW UNIVERSITY
REFERENCES
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Hart, 0., "Monopolistic Competition in the Spirit of Chamberlin: A General Model,"
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, "Monopolistic Competition in the Spirit of Chamberlin: Special Results,"
Economic Journal, XCV (1985b), 889-908.
Lancaster, K., Variety, Equity and Efficiency (New York: Columbia University
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Salop, S., "Monopolistic Competition with Outside Goods," Bell Journal of Economics, X (1979), 141-56.
Sattinger, M., "Value of an Additional Firm in Monopolistic Competition," Review
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Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016
For values of n such that n k, we have T = k, and R(n) is
obviously a monotonically decreasing function of n. For values of
n such that n < k, we have T = n, and it can be shown that for
values of n greater than some number n the function R(n) is
monotonically decreasing in n (to verify this, set T = n in R(n),
differentiate it with reference to n, and notice that, for sufficiently
large n, this derivative is always negative). Thus, regardless of
the size of k, there exist a number n such that for values of n
greater than ii, R(n) is monotonically decreasing.
Define F = mink,„,,R(n). It follows from the above that given
k, for any value of the setup cost F < F, there exists a unique
nk(F) ii such that R(nk(F)) F > R(nk(F) + 1). Therefore, given
k, for any F < F the pair (p*,n*), where n* = nk (F) and p* solves
(A.1) for T = min[nk (F),k], is a unique SFOE.
Q.E.D.