Imperfect Price Information and Competition

Transcription

1 mperfect Price nformation and Competition Sneha Bakshi April 14, 2016 Abstract Price competition depends on prospective buyers information regarding market prices. This paper illustrates that if buyers are continuously and uniformly distributed between perfect information and perfect ignorance, then a symmetric pure strategy price equilibrium does not exist in a market of identical sellers. nstead there will always exist multiple prices at any point in time. For a finite number of sellers, each earns positive profit in the mixed strategy price equilibrium, with the lower support of equilibrium prices increasing if the number of sellers decreases. Marginal cost pricing results only in the limit if there are infinite number of sellers in the market. JEL Classification: D4, D8, L1. Keywords: price competition, price dispersion, imperfect information. wish to thank Kevin Siqueira and Bernhard Ganglmair for important comments, suggestions and criticisms that have helped improve this paper. All remaining errors are mine. University of Texas at Arlington. sneha.bakshi@uta.edu 1

2 1 ntroduction The popular way to model imperfect price-information among buyers in economic models has been via price-search. The results from search theory are, however, controversial. Sequential search models of price competition since Diamond (1971) have shown that prices in large markets can settle at monopoly levels. What is often referred to as the Diamond paradox challenges the prevalent view that the presence of rival sellers selling the same homogeneous good creates competition in the market. The literature is also characterized by the presence of a discontinuity such that as long as search cost is positive, buyers purchase with zero price comparison leading to monopoly price, but if search cost is zero, buyers know all prices and market price settles at marginal cost. Arbitrarily small search costs therefore, do not move the market any closer to marginal cost pricing. Also surprisingly, a costly search market is more likely to settle at monopoly price if the number of sellers in the market is large (Stiglitz, 1987; Stahl, 1989; Rosenthal, 1980) 1. This contradicts the economic intuition that as the number of sellers in a market increases, the market becomes more competitive, driving prices closer to cost. On the other hand, the illustration of a duopoly market in Bertrand (1883) shows a unique equilibrium at marginal cost, given perfect information for all buyers. The significance of this result is that there is no need for a large number of sellers in a market to achieve efficient competition that drives prices down to marginal cost. All that is needed is that each buyer have complete information of all prices in the market. Often referred to as the Bertrand paradox, this contradicts the notion that a larger number of sellers is needed to impose a higher degree of competition among existing sellers. 1 All these papers reach this result although Stiglitz (1987) models buyers as knowing price deviations by sellers before searching the market, Stahl (1989) models them as only knowing the Nash equilibrium distribution of prices, and Rosenthal (1980) models them as being exogenously informed/uninformed. 2

3 t is important to note that the essential difference between the two theories that drives seemingly opposing results is not the presence of search cost but the resulting information that it imposes upon buyers at the point of purchase. That is, the presence of non-zero search cost leads identical sellers to a single (monopoly) price equilibrium precisely because with zero variance in the price distribution, they expect buyers to purchase at the first price found, i.e. with zero price information. n other words, sellers incentive to reduce price is mitigated as doing so does not increase demand as long as the price difference is smaller than buyers search cost (even if buyers are assumed to ex ante see deviations in prices). Rather, by raising price by an amount less than the per unit search cost of buyers, sellers can charge higher prices without leading to price comparison and without deterring demand. t is therefore not surprising that monopoly price equilibrium results because with zero price comparison per buyer, every seller acts like a monopolist. The discontinuity that results from an arbitrarily small search cost is also not surprising because the expectation of buyers zero price information does not depend on the size of the search cost as long as it is strictly positive. However, zero price comparison or an ignorance of prices other than the one at the point of purchase, is unrealistic and fragile. Moreover, the assumption on which the sequential search theory thrives - that of ex ante buyer awareness of the underlying price distribution - is itself unrealistic (Rothschild, 1973), results in stationary and myopic search that is indifferent between permitting recall or not, and is used for the convenience it lends to the mathematical modeling of buyers sequential decision making (McMillan and Rothschild, 1994). Also, empirical evidence argues against the hypothesis that buyers search prices sequentially (Gastwirth, 1976; Kogut, 1990; Sonnemans, 1998; De los Santos et al., 3

4 2012) 2. This paper therefore takes a different approach to buyers imperfect information, untying it from both sequential search and the assumption of buyers ex ante knowledge of the underlying price distribution. t acknowledges that buyers price information is the result of various inter-related factors, for example: advertisement, history, buyer communication networks, and conscious price search with possible economies of scale. Buyers may also benefit from each or all of these factors to different extents. What is important then is that most buyers will purchase after utilizing some degree of price information and comparison, and that there will be a variety of buyers. therefore model a unit mass of buyers of different types, such that the type of a buyer represents the extent of market price information available to her at the time of purchase. Each buyer, given her information (type), will purchase at the lowest price known to her. assume that the distribution of buyer types is continuous and uniform, with the extremes being perfect ignorance and perfect information; regardless of the underlying price distribution 3. The uniform distribution is intuitively justifiable because lacking evidence regarding the true distribution of buyers by information, sellers would expect every buyer to be of any type with equal probability. Given the heterogeneity among buyers, sellers have incentives to undercut rivals and attract the more informed buyers, as well as to price high and sell to those relatively less informed and make large profits off them. n contrast 2 The empirical and laboratory data finds that buyers make search decisions based on total returns from search rather than marginal returns, contradicting the sequential search hypothesis (Kogut, 1990; Sonnemans, 1998). Also the decision to continue searching is found to be independent of the observed prices (De los Santos et al., 2012), again contradicting sequential search and favoring instead a predetermined search effort and sample size. 3 This implies for example that a highly informed buyer has information on many stores even if the underlying price distribution has zero variance. This seems apt because shoppers become aware of existing price variation only after they know the various prices in the market. Exogenous informational limitations of this sort have been modeled before (Rosenthal, 1980; Varian, 1980; Wilde and Schwartz, 1979), most commonly at a discrete level with two buyer types. 4

5 with the above literature, find that a duopoly does not suffice for marginal cost pricing and nor is a large market more likely to lead to monopoly pricing. As long as there are finitely many sellers in the market, competition cannot be perfect and a single price will not prevail. There will always be some price variation because sellers use mixed strategies in equilibrium. Moreover, each seller earns positive expected profit 4 in all such symmetric equilibria. The support of prices is bounded above by buyers value while the lower bound is a function of the number of sellers in the market. A larger number of sellers lowers this lower bound, thus making the market more competitive. A unique symmetric pure strategy equilibrium exists and occurs only in the limit when the number of sellers in the market tends to infinity. This is a zero profit equilibrium. Moreover, in the limit of an infinitely large market, the lower bound of the mixed strategy equilibrium also tends to marginal cost; thus there is indifference between the mixed strategy equilibrium and pricing at marginal cost when the number of sellers in the market tends to infinity. The intuition for the above results is that in a finite market the incentive to undercut rivals exists at any single price greater than cost. Therefore, if the market is finite, sellers keep undercutting each other until price is driven down to marginal cost. But at this price that gives zero profit, each seller prefers to deviate to a price at the monopoly level and sell to buyers unaware of other prices, thus triggering price undercutting again. This is why an equilibrium in a finite market can only exist in mixed strategies. Only if the market is infinitely large, do sellers no longer have the incentive to deviate from marginal cost pricing because the probability of being known by a completely ignorant buyer goes to zero. The nature of imperfect information used in this model is such that there 4 The persistence of positive profit differs from that found by Varian (1980) which has a similar (although discrete) information structure because of decreasing costs in that model and the imposition of a zero-profit equilibrium. 5

6 are always some (probabilistically) buyers who know only one price at the time of purchase. At the same time there are always some buyers who know two (or more) prices. Given such a structure, the result of price dispersion agrees with the results of Wilde (1977). This essay thus restores the correlation between market size and the level of price competition even in imperfectly informed markets, while at the same time illustrating the persistence of positive profits in finite markets. Also, instead of assuming any particular kind of search behavior by buyers that is refuted by evidence, the paper models buyers information as a continuous distribution. This has the benefit of lending continuity in the effect of imperfect information on market power 5, because collapsing the distribution of buyers to a single point either at perfect information or at perfect ignorance results in marginal cost pricing and monopoly pricing, respectively. 2 The Model The market consists of a unit mass of buyers and 2 identical sellers. Each buyer wants to buy at most one unit of the good. All buyers have the same value for the good, v, but have different degrees of information regarding market prices. A buyer s information type is private and given by k, where k [ 1, 1] represents the fraction of prices (sellers) known by the buyer in the market. Buyers are believed - by sellers - to be uniformly distributed in their information types, with cumulative distribution function G(k) = k 1 1. The range of k is limited by perfect ignorance (knowing only one price and purchasing at it) and perfect information (knowing all prices). The support of k is continuous, which 5 Kuksov (2006) also finds this continuity but as a result of the absence of sellers common knowledge regarding market demand. 6

7 does not restrict the number of prices known to be whole numbers, implying the possibility of a buyer knowing prices probabilistically 6. Each seller has the same marginal cost of production, s, and has no capacity constraints or fixed costs. For trade to be possible, assume s < v. Sellers can only post a price and allow prospective buyers the option of buying at the posted price or leaving (and possibly buying from another store). Because sellers are identical, the model will only look for symmetric Nash equilibria. f there is a single price in the market that results in positive profit, there will always exist price undercutting incentives. This is stated and proven as the following lemma. Lemma 1. At any common pure strategy price that exceeds the common marginal cost, every seller has the incentive to undercut rivals. For a market with 2 there will be no pure strategy Nash equilibrium wherein all sellers in the market post price p s.t. p > s. Proof. Let all sellers post price p such that p > s. Each seller has probability k of being known by a buyer of type k. For every buyer of type k, a seller then has a probability 1 k of making a sale because all sellers list the same price. Expected profit for any seller i from posting price p will be 1 1/ k p s (p s)dg(k) =. (1) k Whereas, with a small price cut - while all other sellers continue to post price, p - a seller can ensure making a sale to every buyer who knows its price. The probability of being known by a buyer of type k remains k, and therefore the 6 For example, say k = 5 for some buyer. This buyer therefore knows one store for sure, 4 and knows the second with probability 1/4. 7

8 expected profit from deviating is: 1 1/ k(p ɛ s)dg(k) = (p ɛ s) 2 1 (1 1 (p ɛ s) ) = (2) For a market with 2, there exists ɛ > 0 such that the right hand side (RHS) of (2) is larger 7 than the RHS of (1): (p ɛ s) ( + 1) 2 > p s. Therefore a slight price cut below rivals common price increases a seller s expected profit. The second statement of the lemma follows because each seller s incentive to deviate implies that no symmetric pure strategy Nash equilibrium higher than marginal cost is possible. The intuition is that if all sellers list the same price, the probability of a seller making a sale to a buyer of type k who knows its price, is 1 k. Whereas, by lowering price by an infinitesimally small amount, a seller ensures making a sale to such a buyer. Therefore, undercutting would drive the single price in the market down until it equals marginal cost and there exists no further incentive to undercut. However, because of imperfect information in the market, pricing at marginal cost and making zero profit need not be a Nash equilibrium as there may be an incentive to increase one s price to the monopoly level and sell only to consumers unaware of other prices. n the following proposition, show that this leads to an absence of a pure strategy symmetric Nash equilibrium in the finite market, and only in the limit when the number of sellers tends to infinity, can a pure strategy symmetric Nash equilibrium exist. Proposition 1. f there are a finite number of sellers, the market has no sym- 7 This becomes easier as the number of sellers,, increases. 8

9 metric pure strategy Nash equilibrium. There exists a unique symmetric pure strategy Nash equilibrium at p = s only in the limit as the number of sellers tends to infinity. Proof. From Lemma 4.1 we have that sellers want to keep undercutting rivals at any common price. Therefore, to prove Proposition 4.1 it suffices to show that when the number of sellers is finite, a common price at s is not an equilibrium; and that as the number of sellers tends to infinity, it becomes an equilibrium in the limit. Consider a seller raising price when all other sellers price at marginal cost. This seller thus makes successful sale only to buyers unaware of other prices, i.e. of types between 1 and 2. The best selling price therefore is the highest possible one, i.e. v. The probability of being known by a buyer of type k remains k, and the probability of making a successful sale in such a case is the same as the probability of the buyer not knowing a second seller, i.e. 1 (k 1 ). This gives the deviating seller an expected profit of: G( 2 ) 2/ 1/ k[1 (k 1 )](v s)dg(k) G( 2 ) = 2 (v s) 3 ( 1). (3) This is positive for finite. Therefore, each seller has incentive to raise price from marginal cost pricing and as a result, no symmetric pure strategy equilibrium exists in a finite market. For an infinitely large market, however, we have: p s lim 2 (v s) 3 ( 1), p s. That is, the deviation profit is no longer larger than the profit at a common price, even if that common price is at marginal cost. Therefore marginal cost pricing becomes a symmetric pure strategy equilibrium in the limit. The uniqueness 9

10 follows from Lemma 4.1. ntuitively, in an infinitely large market, the probability that a buyer is unaware of all other prices tends to zero, and thus all sellers pricing at marginal cost becomes an equilibrium. Note, however, that as long as the number of sellers is finite, this does not hold and being the highest priced seller is better than pricing at marginal cost. The model thus shows that an extension of Bertrand price competition to a continuum of imperfectly informed buyer types does not result in marginal cost pricing in finite markets, and certainly not in a duopoly. An infinitely large market is needed for a pure strategy equilibrium to settle, in the limit, to marginal cost pricing. This necessity for the market to be infinitely large to completely eliminate incentives to raise prices alone, contradicts the finding in Stiglitz (1987) and Stahl (1989) that a larger market is more likely to settle at monopoly price, given imperfect information. The absence of a symmetric pure strategy equilibrium in finite markets does not rule out a mixed strategy equilibrium. f such an equilibrium exists, all sellers must make equal profit in it as they are all identical. Therefore, let a representative seller in this equilibrium use the mixed strategy with (cumulative) probability function, F (p), such that it posts prices in the support, [p, p], with each price being posted with positive density. The following proposition establishes and defines the mixed strategy equilibrium. Proposition 2. A symmetric mixed strategy equilibrium exists on the support [p, v] such that in a finite market each seller earns positive profit. The lower support of the equilibrium mixture is inversely proportional to the number of sellers in the market and is given by p = s + 4(v s) 3( 2 1). Proof. Step 1: p = v. 10

11 Notice that the support of equilibrium prices must be such that p s, because posting price below cost with positive probability is dominated by a price at least equal to cost. On the other hand, posting price above buyers value is ruled out because no buyer would purchase at that high a price. Therefore, p v. Also, because the equilibrium mixture must be the representative seller s best response, each price in the support must give equal expected profit (given rival sellers equilibrium mixtures), which is: (p s) π(p; F (p)) = [ 1 ][ 2 1 k{1 (k 1 )}dk 2 + k(k 1){1 F (p)} k 1 dk / k{1 F (p)} k 1 dk]. The terms in the above expected profit are respectively, from selling to buyers unaware of all other prices, from selling to buyers who have a positive probability less than one of knowing a second price, and from selling to buyers who surely know other prices 8. Solving, gives : π(p; F (p)) = 2 (p s) 3 ( 1) + (p s) ( 1) [ (k 2 k){1 F (p)} k 1 dk + k{1 F (p)} k 1 dk]. 2/ (4) Note that because p v, it must be that p [ p, v], F (p) = 1. This implies that p [ p, v], π(p; F (p)) = 2 (p s) 3 ( 1). Because this is maximized at p = v, no seller would price at any p [ p, v). Therefore, it must be that p = v. 8 A buyer of ability k 2 has probability k 1 of knowing a second price; and [1 F (p)] k 1 is the probability that all other prices known by buyer type k are higher than p. 11

12 Step 2: f the market is finite, s / [p, p] and equilibrium expected profit for each seller is positive. Because all prices in the equilibrium mixture must give the same expected profit, we have π(p; F (p)) = 2 (v s), p [p, v]. (5) 3 ( 1) This implies that s / [p, p] because from (4) we have π(s; F (p)) = 0, resulting in a contradiction if the market is finite. t also implies that the expected profit from the mixed strategy equilibrium for each seller is 2 (v s) 3 ( 1), which is positive as long as is finite. Step 3: p = s + 4(v s) 3( 2 1). Because F (p) = 0, we have π(p; F (p)) = 2 (p s) (p s) + 3 ( 1) ( 1) [ ntegrating within limits, gives: (k 2 k)dk + kdk]. 2/ (p s)( + 1) π(p; F (p)) =. (6) 2 From (5), this implies that (p s)(+1) 2 = 2 (v s) 3 ( 1), which gives p as a function of : 4(v s) p = s + 3( 2 1) ; or equivalently, p = 4v + s(32 7) 3( 2. (7) 1) Therefore, the lower support of the equilibrium mixture is an inverse function of the number of sellers in the market. 12

13 The equilibrium probability function, F (p), is found by equating the expected profit from every price in the equilibrium support to the expected profit from pricing at v (using equations (4) and (5)). The substitution 1 F (p) = A is used to simplify notation. 2 (p s) (p s) + 3 ( 1) ( 1) [ (k 2 k)a k 1 dk + ka k 1 dk] = 2 (v s) 2/ 3 ( 1). ntegrating and simplifying, this gives the equation that defines the equilibrium probability function: 6(A 1 A ln A) + 3 ln A(1 + A 1 ln A A 1 ) 2(v p) (ln A) 3 = p s. t thus turns out that the lower support of the equilibrium mixture of prices depends on the size of the market, and so does the equilibrium probability function. The lower support of equilibrium prices decreases toward marginal cost as the number of sellers in the market increases. As long as the market is finite but larger than one seller, sellers make positive expected profit in the symmetric mixed strategy equilibrium. Moreover, when the number of sellers tends to infinity, the equilibrium support of prices tends to [s, v] and expected profit tends to zero. Equivalently, marginal cost pricing becomes an equilibrium in the limit as the market gets infinitely large. 3 Conclusion This paper is an illustration that a more realistic, finer distribution of buyers information in a market gives results in between the findings of marginal cost pricing under full information and monopoly pricing under perfect ignorance. 13

14 n imperfectly informed markets, a finite number of identical sellers will always make positive profit in mixed strategies. This also means that there will always exist multiple prices in a finite market. Equilibrium price settles at marginal cost only if the number of sellers tends to infinity. An important implication is that the number of sellers is critically important to the level of price competition and to the extent of profit margins in a market. Both the price variance in equilibrium, as well as the equilibrium probability distribution of prices depend on the size of the market. More sellers competing to sell the same good to imperfectly informed buyers leads to a lower support of equilibrium prices. Possible extensions of this work should examine the question with the distribution of buyers information being non-uniform. 14

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