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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 Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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. 494 QUARTERLY JOURNAL OF ECONOMICS 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 Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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 495 Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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- 496 QUARTERLY JOURNAL OF ECONOMICS 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) — Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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 Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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). 498 QUARTERLY JOURNAL OF ECONOMICS 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. Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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). TRUE MONOPOLISTIC COMPETITION 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 Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 CLAIM 500 QUARTERLY JOURNAL OF ECONOMICS 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, Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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}. Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 (6) 502 QUARTERLY JOURNAL OF ECONOMICS 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 Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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- Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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 504 QUARTERLY JOURNAL OF ECONOMICS 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. , Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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, 505 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 , Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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 506 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__.. Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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,—, Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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. Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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 Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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, Downloaded from http://qje.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016 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 Diamond, P., "A Model of Price Adjustment," Journal of Economic Theory, III (1971), 156-68. Dixit, A., and J. Stiglitz, "Monopolistic Competition and Optimum Product Diversity," American Economic Review, LXVII (1977), 297-308. Hart, 0., "Monopolistic Competition in the Spirit of Chamberlin: A General Model," Review of Economic Studies, LII (1985a), 529-46. , "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 Press,. 1979). Perloff, J., and S. Salop, "Equilibrium with Product Differentiation," Review of Economic Studies, LII (1985), 107-20. 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 of Economic Studies, LI (1984), 321-32. Spence, M., "Product Differentiation and Welfare," American Economic Review, LXVI (1976a), 407-14. , "Product Selection, Fixed Costs and Monopolistic Competition," Review of Economic Studies, XLIII (1976b), 217-36. Wolinsky, A., "Product Differentiation with Imperfect Information," Review of Economic Studies, LI (1984), 53-61. 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.