Advertising Competition in Retail Markets

Transcription

1 Advertising Competition in Retail Markets KyleBagwellandGeaM.Lee April 25, 2010 Abstract We consider non-price advertising by retail firms that are privately informed as to their respective production costs. We construct an advertising equilibrium, in which informed consumers use an advertising search rule whereby they buy from the highest-advertising firm. Consumers are rational in using the advertising search rule, since the lowest-cost firm advertises the most and also selects the lowest price. Even though the advertising equilibrium facilitates productive efficiency, we establish conditions under which firms enjoy higher expected profit when advertising is banned. Consumer welfare falls in this case, however. Under free entry, social surplus is higher when advertising is allowed. In addition, we consider a benchmark model of price competition; we provide comparative-statics results with respect to the number of informed consumers, the number of firms and the distribution of costs; and we consider the possibility of sequential search. Bagwell: Donald L. Lucas Professor of Economics, Stanford University; Senior Fellow at SIEPR and SCID; and NBER Research Associate. Lee: School of Economics, Singapore Management University. We thank two anonymous referees, Yeon-Koo Che, Pauline Ippolito, Jeong-Yoo Kim, Massimiliano Landi, Laurent Linnemer, Albert Ma and John Vickers for helpful comments. We also thank seminar participants at EARIE, ENPC-CERAS, EUI, the Far Eastern Meeting of the Econometric Society, Singapore Management University, and Southern Methodist University for helpful discussions. The material in this paper was originally contained in our earlier paper entitled Advertising and Collusion in Retail Markets, dated June The static analysis from the earlier paper is now found here, and the dynamic analysis from the earlier paper is now found in our companion paper (Bagwell and Lee, 2010). 1

2 1 Introduction The typical retail firm sells a large number of products. When a retail firm advertises using traditional media, such as radio, TV, billboards and magazines, the firm may thus be unable to communicate substantial hard information to its potential consumers. In particular, a retail firm that advertises using such media may be able to communicate only limited information regarding the prices of the large range of products that it carries. Retail advertising in such media is often thus best described as non-price advertising. The prevalence of non-price advertising in retail markets raises a number of interesting questions. First, is non-price retail advertising consistent with equilibrium behavior? Non-price advertising is costly, so firms use it only if consumers respond. Since non-price advertising may have little or no hard information, however, it is not clear why rational consumers would respond. Second, would banning non-price retail advertising raise or reduce consumer welfare? An answer to this question is necessary in order to evaluate the appropriate public policy toward non-price advertising. Finally, do retail firms have an incentive to encourage regulatory restrictions on non-price advertising? An affirmative answer to this question, for example, would support the view that restrictions on retail advertising may be a form of regulatory capture. These questions are of particular interest in light of striking empirical relationships between advertising and prices that have been identified in some retail industries. The classic study is by Benham (1972). Examining transaction prices in the retail eyeglass industry in the U.S. in the 1960s, he reports that prices were higher in states that prohibited all advertising than in states that had no restrictions on advertising; moreover, prices were only slightly higher in states that allowed just non-price advertising than in states that also allowed price advertising. Evidently, the ability to advertise, even if only in a non-price form, is sometimes associated with lower prices. Cady (1976) documents similar relationships in the U.S. retail market for prescription drugs in At a broad level, this work raises the possibility that retail firms might gain if they were able to limit advertising. An agreement to limit advertising is directly achieved in the presence of a state law that prohibits advertising; alternatively, retail firms might achieve such an agreement by forming a professional association that imposes advertising restrictions on its members. 2 Bagwell and Ramey (1994a) develop a complete-information model of retail competition with which to interpret Benham s findings. In their model, some consumers can identify the highestadvertising firm, while other consumers do not observe advertising levels. The former (latter) consumers are referred to as informed (uninformed) consumers. Each consumer possesses a downward- 1 Related findings are reported in other studies which focus on different retail markets, including the markets for gasoline, routine legal services and liquor. See Bagwell (2007, Section 3.2.4) for a survey. Rizzo and Zeckhauser (1992) consider the market for physicians and report evidence that advertising is associated with higher prices (and quality). As they note, however, this market has several novel features: quality is difficult to assess, consumers are highly sensitive to perceived quality, and consumers are relatively insensitive to price (given third-party reimbursement). 2 The FTC has argued that anti-competitive effects may be associated with price and non-price advertising restrictions imposed by a professional association. See California Dental Association v. Federal Trade Commission (1999). A further possibility is that firms interact repeatedly and achieve a self-enforcing agreement to limit advertising. We consider this latter possibility in our companion paper (Bagwell and Lee, 2010). 1

3 sloping demand function and lacks direct information about firms prices: a consumer observes a firm s price only after choosing to visit that firm. Bagwell and Ramey compare two equilibria. In a random equilibrium, consumers ignore advertising and choose firmsatrandom. Firmsdonot advertise, and they enjoy symmetric market shares. By contrast, in an advertising equilibrium, the informed consumers go to the firm that advertises the most. Firms then use a symmetric mixed strategy, in which higher advertising choices are paired with greater investments in cost reduction and thus lower prices. Informed consumers are then rational in visiting the highest-advertising retailer, since this retailer also offers the lowest price. For a given number of firms, expected profit is higher in the random equilibrium, because advertising expenses are thereby avoided. Bagwell and Ramey then include an initial entry stage so as to endogenize the number of firms. Comparing freeentry advertising and random equilibria, they show that, in the advertising equilibrium, the market is more concentrated, prices are lower, and social welfare is higher. If the random equilibrium is associated with a setting in which advertising is banned, these findings are broadly consistent with the empirical patterns that Benham reports. In this paper, we modify the Bagwell-Ramey model by assuming that each firm has private information about its exogenous costs of production. Specifically, we consider a model with a continuum of possible cost types, where cost types are iid across firms. We then characterize an advertising equilibrium, in which firms use pure strategies and lower-cost firms advertise more than do higher-cost firms. In the incomplete-information model that we analyze here, informed consumers are rational in using the advertising search rule, since the lowest-cost firm advertises the most and also selects the lowest price. We also compare the advertising equilibrium with the random equilibrium in which no firm advertises, both when the number of firms is fixed and when the number of firms is endogenous. We thereby consider the short- and long-run implications of advertising competition for consumer surplus and firm profit. In addition, we analyze a benchmark model of price competition; we provide comparative-statics results with respect to the number of informed consumers, the number of firms and the distribution of costs; and we consider the possibility of sequential search. At a broad level, we can interpret our model as a purified version of the Bagwell-Ramey model. In fact, our analysis explores two notions of purification. Our first notion reflects the traditional view of purification under which the mixed-strategy equilibrium of a complete-information game can be interpreted in terms of the pure-strategy equilibrium of a nearby incomplete-information game. For the special case in which the support of possible costs is small, we report that the distribution of advertising levels in the pure-strategy advertising equilibrium of the incomplete-information game is approximately the same as the distribution of advertising levels in the mixed-strategy advertising equilibrium of the associated complete-information game. 3 Correspondingly, we show that the main predictions of Bagwell and Ramey directly extend to the private-information setting, if the support 3 Bagwell and Ramey (1994a) report a similar finding when the cost of advertising is private information and varies slightly across firms. We also note that the associated complete-information game that we consider here is closely related to but distinct from that considered by Bagwell and Ramey. An important difference is that we do not allow scale economies achieved through endogenous investments in cost reduction. 2

4 ofpossiblecostsissufficiently small. Under our second notion of purification, we consider whether the main predictions are robust across equilibria for the complete- and incomplete-information games, even when these games are not nearby to one another. For the general case in which the support of possible cost types may be large, we establish conditions under which the main predictions derived in the complete-information game emerge as well in the incomplete-information game. This second notion of purification sometimes requires additional structure on the distribution and demand functions. We develop our results while allowing for the general case of a large support of possible cost types. As mentioned, we establish that an advertising equilibrium exists, in which lower-cost firms advertise more and price lower than do higher-cost firms. We then establish four additional results. First, for any given number of firms, expected consumer surplus is higher in the advertising equilibrium than in the random equilibrium. Intuitively, in our model, the distribution of posted prices is not affected by firms ability to engage in non-price advertising; thus, uniformed consumers are indifferent between the advertising and random equilibria. 4 At the same time, non-price advertising enables informed consumers to locate the firm with the lowest cost and thus the lowest price; hence, the average transaction price is lower when non-price advertising is allowed. This first result is consistent with Benham s observation and holds independently of any assumption about the distribution of types or the elasticity of demand. Second, for any given number of firms, if the distribution of types is log-concave and demand is sufficiently inelastic, then firms earn higher expected profit in the random than in the advertising equilibrium. When the number of firms is fixed, therefore, firms and informed consumers rank the advertising and random equilibria in opposite fashion. Intuitively, in comparison to the advertising equilibrium, the random equilibrium offers firms an advantage as well as a disadvantage. The advantage of the random equilibrium is that firms avoid the expenses that are associated with advertising competition. The disadvantage of the random equilibrium is that it assigns the same market share to all firms, whereas the advertising equilibrium achieves greater productive efficiency by assigning higher expected market shares to lower-cost firms. Building on techniques used by Athey et al. (2004) in their study of price collusion, we show that the random equilibrium s advantage overwhelms its disadvantage, if the distribution of types is log-concave and demand is sufficiently inelastic. 5 Our second result thus gives conditions under which firms have an incentive, at least in the short run, to seek regulatory restrictions on advertising. The third result follows directly from the second: when the number of firms is endogenously determined by a free-entry condition, if the distribution of types is log-concave and demand is sufficiently inelastic, more firms enter when the random equilibrium is anticipated. Thus, advertising competition leads to a more concentrated market structure. 4 The prediction of an invariant distribution for posted prices is special to our model. It holds since we assume that costs are exogenous and exhibit constant returns to scale. 5 Athey et al. (2004) establish related conditions under which optimal collusion for sellers in a first-price procurement auction entails pooling at the buyer s reservation value. See also McAfee and McMillan (1992) for a related theory of identical bidding among collusive bidders in a first-price auction. Our model of advertising, by contrast, is analogous to an all-pay auction. 3

5 Finally, our fourth result concerns social welfare when the number of firms is endogenously determined. Under free entry, firms are indifferent between the advertising and random equilibria, since in each setting the number of firms adjusts until each firm earns zero profit. Whether social surplus is higher in the advertising or random equilibrium thus hinges entirely on the consumer surplus that is expected in each equilibrium. In fact, our first result may be easily extended to cover settings in which different numbers of firms enter in the advertising and random equilibria, respectively. Without making any assumption on the distribution of types or the elasticity of demand, we show that social surplus is weakly higher in the advertising than in the random equilibrium when the number of firms is endogenous; moreover, social surplus is strictly higher in the advertising equilibrium if at least two firms enter in that equilibrium. As noted, Bagwell and Ramey (1994a) also construct an advertising equilibrium, find that firms earn greater short-run profit in the random equilibrium, and find that the advertising equilibrium leads to a more concentrated market structure and greater social welfare in the long run. Our second, third and fourth results above thus establish a sense in which Bagwell and Ramey s main predictions extend to the private-information setting. We emphasize, however, that the second and third results now employ additional assumptions on the distribution of types and the elasticity of demand. These additional assumptions arise because of the productive-efficiency advantage that the advertising equilibrium offers in our incomplete-information setting. We also compare the advertising equilibrium with another benchmark. In particular, we follow Varian (1980) and suppose that informed consumers observe prices and buy from the lowest-priced firm while uninformed consumers pick a firm at random. Following Spulber (1995) and Bagwell and Wolinsky (2002), we modify Varian s model and allow that firms are privately informed about their production costs. 6 Let us refer to the (symmetric) equilibrium of this game as the pricing equilibrium. For any fixed number of firms, we show that firms earn higher expected profit inthe pricing equilibrium than in the advertising equilibrium. This is perhaps surprising, since competition in advertising is sometimes argued to be less aggressive than competition in prices. As we discuss, the key intuition is that price competition induces greater in-store demand from consumers and thus elevates the size of expected information rents for firms. With an analysis of the benchmark model in place, we are able to offer a more complete comparison across different advertising regulatory regimes. Provided that the market always has at least two firms, our results indicate that the average transaction price is lowest in the pricing equilibrium, somewhat higher in the advertising equilibrium, and higher yet in the random equilibrium. Likewise, when the number of firms is endogenous, social welfare is highest in the pricing equilibrium, somewhat lower in the advertising equilibrium, and lower yet in the random equilibrium. If we associate the pricing equilibrium with a setting in which price advertising is allowed, the advertising equilibrium with a setting in which only non-price advertising is allowed, and the random equilibrium with a setting in which all advertising is banned, then our results are broadly consistent 6 Bagwell and Wolinsky follow Varian and assume that each consumer possesses an inelastic demand function. We generalize this analysis slightly and allow for downward-sloping demand functions. Spulber considers a related model in which all consumers are informed. 4

6 with Benham s findings. We next examine the comparative-statics properties of the advertising equilibrium. When the number of informed consumers is increased, advertising increases for all types other than the highest type. Intuitively, firms advertise more heavily when the prize from advertising the most is increased. 7 Interestingly, the effect on advertising of an increase in the number of firms depends on a firm s cost type: lower-cost firms compete more aggressively and increase their advertising, but higher-cost firms perceive a reduced chance of winning the informed consumers and advertise less. An implication is that the support of observed advertising levels may be larger in markets with a greater number of firms. We also find that, for all types other than the lowest type, if the number of firmsissufficiently large, the equilibrium level of advertising is negligible. Finally, building on Hopkins and Korneinko s (2007) analysis of all-pay auctions, we show that, if the cost distribution shifts to make lower-cost types more likely in the sense of the monotone likelihood ratio order, then lower-cost firms advertise more while higher-cost firms become discouraged and advertise less. 8 We next modify the game to allow for sequential search. If demand is sufficiently inelastic or if the cost of sequential search is sufficiently high, then our results are maintained without modification. If these conditions do not hold, however, then higher-cost firmsmust limitprice (i.e., price below their monopoly prices), in order to deter sequential search. 9 An advertising equilibrium then continues to exist, if the support of possible cost types is not too large and the number of informed consumers is not too great. In this equilibrium, informed consumers use observed advertising behavior to locate the lowest price, and limit pricing by higher-cost firms ensures that uninformed consumers do not gain from actually undertaking sequential search. We argue as well that the possibility of sequential search mayevenstrengthenourresults,byraising the relative profitability of the random equilibrium. Other authors have also considered the effects of advertising regulations on the conduct of firms that are privately informed as to their respective costs of production. For example, Peters (1984) and LeBlanc (1998) consider the effects of a prohibition on price advertising in models with a fixed number of firms where each firm is privately informed about its production cost. By contrast, here we emphasize the effects of a prohibition on non-price advertising, and we utilize a free-entry condition to endogenize the number of firms. Also, Bagwell and Ramey (1994b) consider a duopoly model in which one firm has private information as to whether its costs are high or low. For settings in which non-price advertising is legal, they show that non-price advertising may be used to signal low costs and thus low prices. In the current paper, by contrast, we adopt a continuum-type model, assume that all firms are privately informed as to their costs, endogenize the number of firms using a free-entry condition, and report (non-monotone) comparative statics results. The paper is organized as follows. Section 2 defines the advertising game and contains our main 7 Bagwell and Ramey (1994a) report a related finding for their complete-information model. 8 This finding contrasts interestingly with the standard monotone comparative statics result for first-price auctions. See, for example, Athey (2002) and Lebrun (1998). 9 Our analysis here builds on Reinganum (1979) and Bagwell and Ramey (1996). Reinganum examines sequential search in a model with privately informed firms that are not allowed to advertise. Bagwell and Ramey examine sequential search when advertising is allowed but private information is absent. 5

7 findings. In Section 3, we present a benchmark model of price competition. Our comparative-statics results are found in Section 4. We consider the possibility of sequential search in Section 5. Section 6 concludes. Remaining proofs are in the Appendix. 2 The Advertising Game In this section, we define an advertising game in which a fixed number of firms compete through advertising for market share. Firms are privately informed as to their respective costs, and each firm s advertising choice may signal its costs, and thus its price, to those consumers who are informed of advertising activities. We establish the existence of an advertising equilibrium, in which informed consumers visit the firm with the highest level of advertising. We compare the expected profit earned by firms in the advertising equilibrium with that which they earn in a random equilibrium, wherein all consumers pick firms at random. We also compare expected consumer surplus in the advertising and random equilibria. We then endogenize the number of firms and compare market concentration and social welfare across the two equilibria. 2.1 The Model We assume that N 2 ex ante identical firms compete for sales in a homogeneous-good market. Each firm i is privately informed of its unit cost level θ i. Cost levels are iid across firms, and cost type θ i is drawn from the support [θ, θ] according to the twice-continuously differentiable distribution function, F (θ), whereθ>θ 0. The density f(θ) F 0 (θ) is positive on [θ, θ]. As discussed in further detail below, after firms learn their respective cost types, they simultaneously choose their prices and levels of advertising. Following Bagwell and Ramey (1994a), we assume that advertising is a dissipative expense that does not directly affect consumer demand. The firms face a unit mass of consumers, where each consumer possesses a twice-continuously differentiable demand function D(p) that satisfies D(p) > 0 >D 0 (p) over the relevant range of prices p. We assume that price information cannot be directly communicated in this market; thus, consumers cannot observe prices prior to selecting a firm to visit and from which to purchase. Some consumers, however, do observe advertising activity prior to picking a firm. In particular, a fraction I of consumers are informed, in the sense that they observe firms advertising expenses. 10 Given this information, informed consumers form beliefs as to firms cost types and determine a visitation (search) strategy. For example, informed consumers may use an advertising search rule, whereby aconsumergoestothefirm that advertises the most. 11 The remaining fraction U =1 I are uninformed. Uninformed consumers do not observe advertising expenditures and thus may adopt a random search rule, whereby a consumer randomly chooses which firm to visit. 10 We assume that informed consumers observe advertising levels for simplicity. All of our results hold under the assumption that informed consumers observe only the identity of the highest-advertising firm(s). 11 If several firms tie for the highest advertising level, then the informed consumers divide up evenly over those firms. 6

8 The interaction between firms and consumers is represented by the following advertising game: (i) firms learn their own cost types, (ii) firms make simultaneous choices of advertising and price, and (iii) given any advertising information, each consumer chooses a firm to visit, observes that firm s price and makes desired purchases given this price. Note that a consumer is assumed to visit only one firm. 12 As we explain below, this assumption simplifies our analysis, by ensuring that each firm chooses the monopoly price that is associated with its cost type for any sales that it makes. We are interested in Perfect Bayesian Equilibria for the advertising game. Before formally defining our equilibrium concept, however, we impose two requirements. First, we focus on equilibria in which consumers do not condition their visitation decisions on firms names. Thus, uninformed consumers must use the random search rule; furthermore, for any given vector of advertising levels, informed consumers must treat symmetrically any two firms which advertise at the same level. Informed consumers hence satisfy this restriction if they use the random or advertising search rules. 13 Second, we focus on equilibria in which firms use symmetric advertising and pricing strategies. We note that the random search rule is indeed an optimal search strategy for uninformed consumers, when firms use symmetric pricing strategies. Givensymmetry,wecandefine a pure advertising strategy for firm i as a function A(θ i ) that maps from the set of cost types [θ, θ] to the set of possible advertising expenditures R + [0, ). Consider now firms other than firm i. Let A(θ i ) denote the vector of these firms selections when their cost types are given by the (N 1)-tuple θ i. Given the search rule used by informed consumers, the market share for firm i is determined by the vector of advertising levels selected by firm i and its rivals. The market share for firm i thus maps from R N + to [0, 1] andinequilibrium may be represented as m(a(θ i ), A(θ i )). 14 Note that, under our first requirement above, firm i s market share is not indexed by i and thus does not depend on firm i s name. If firm i has cost type θ i, its interim-stage market share under these strategies is E θ i [m(a(θ i ), A(θ i ))]. We next describe a firm s expected profit. A firm s net revenue is r(p, θ i ) (p θ i )D(p) (excluding advertising expenses) when it has cost type θ i, sets the price p and captures the entire unit mass of consumers. We assume that r(p, θ i ) is strictly concave in p with a unique maximizer p(θ i ) = arg max p r(p, θ i ). It follows that the monopoly price p(θ i ) strictly increases in θ i whereas r(p(θ i ),θ i ) strictly decreases in θ i. We further assume that the price at the top has a positive margin: p(θ) > θ. Given our requirement that all consumers, and particularly uninformed consumers, treat all firms symmetrically, we know that all firms receive positive expected market share. Therefore, in the equilibria upon which we focus, each firm must select the monopoly price given its cost type. Embedding the monopoly price into the revenue function, we may define the interim-stage 12 We extend the analysis to allow for sequential search in Section Under the random search rule, consumers randomly pick a firm from the set of all firms. Similarly, under the advertising search rule, informed consumers randomize over all firms that advertise at the highest level (if more than one such firm exists). 14 For example, if all consumers use the random search rule, then m(a(θ i), A(θ i)) = 1/N. If instead the uninformed consumers use the random search rule while the informed consumers use the advertising search rule, then m(a(θ i ), A(θ i )) = I + U/N if A(θ i ) >A(θ j ) for all j 6= i, whilem(a(θ i ), A(θ i )) = U/N if A(θ i ) <A(θ j ) for some j 6= i. For this latter set of consumer search strategies, if firm i ties with k 1 other firms for the highest advertising level, then m(a(θ i ), A(θ i )) = I/k + U/N. 7

9 net revenue for firm i by R(A(θ i ),θ i ; A) r(p(θ i ),θ i )E θ i [m(a(θ i ), A(θ i ))]. Thus,R(A(θ i ),θ i ; A) is the interim-stage net revenue for firm i when firm i has cost type θ i, advertises at level A(θ i ), and anticipates that other firms also use the advertising function A to determine their respective advertising levels upon observing their cost types. Firm i s expected revenue is E θi R(A i (θ i ),θ i ; A), and firm i s expected profit isthuse θi [R(A(θ i ),θ i ; A) A(θ i )]. Given our embedded requirements, we may now define an equilibrium as an advertising strategy A, a belief function and search rules for consumers that collectively satisfy three remaining conditions. First, given the market share function, m, that is induced by consumers search rules, the advertising strategy A is such that, for all i and θ i,a(θ i ) arg max ai [R(a i,θ i ; A) a i ]. 15 Second, given an observed advertising level a i by firm i, informed consumers use Bayes Rule whenever possible (i.e., whenever a i = A(θ i ) for some θ i [θ, θ]) in forming their beliefs as to firm i 0 scost type θ i and thus price p(θ i ). Third, for any observed vector of advertising levels [a 1,...,a N ] R N +, given their beliefs, informed consumers search rule directs them to the firm or firms with the lowest expected price. We may now simplify our notation for equilibrium variables somewhat further. We may define firm i s interim-stage market share as M(A(θ i ); A) E θ i [m(a(θ i ), A(θ i ))]. Similarly, we can define firm i s interim-stage profit and net revenue as follows: Π(A(θ i ),θ i ; A) r(p(θ i ),θ i )M(A(θ i ); A) A(θ i ). R(A(θ i ),θ i ; A) A(θ i ). We note that the interim-stage profit function satisfies a single-crossing property: higher types are less willing to engage in higher advertising to increase expected market share. 16 For here and later use, we now write interim-stage profit in direct-form notation, ignoring subscript i: ifafirm of type θ picks an advertising level A( b θ) whenitsrivalsemploythestrategya, then we define Π( b θ, θ; A) Π(A( b θ),θ; A), M( b θ; A) M(A( b θ); A) and R( b θ, θ; A) R(A( b θ),θ; A). We are primarily interested in two kinds of equilibria. In an advertising equilibrium, informed consumers use the advertising search rule. Since p(θ) is strictly increasing, such equilibria can exist only if the advertising schedule A is nonincreasing, so that higher-advertising firms have lower costs and thus offer lower prices. In a random equilibrium, informed consumers ignore advertising and use the random search rule. A random equilibrium thus can exist only if firms maximize expected profits and do not advertise (i.e., A 0). 15 Notice that A(θ i) must be an optimal choice for firm i with cost type θ i in comparison to advertising deviations that are on-schedule (i.e., a i such that a i = A(θ) 6= A(θ i ) for some θ [θ, θ]) aswellas off-schedule (i.e., a i such that a i 6= A(θ) for any θ [θ, θ]). 16 When a firm increases its advertising level, it may confront a trade off between the larger advertising expense, a i, and the consequent higher expected market share, M(a i ; A). When the interim-stage profit is held constant, the slope da i/dm(a i; A) is given by r(p(θ i),θ i), which is strictly decreasing in θ i. 8

10 2.2 Advertising Equilibrium In an advertising equilibrium, informed consumers use the advertising search rule while uninformed consumers are randomly distributed across all N firms. We now report the following existence and uniqueness result. Proposition 1. There exists a unique advertising equilibrium, and in this equilibrium A(θ) is strictly decreasing and differentiable and satisfies A(θ) =0. Proof. We first derive the necessary features of an advertising equilibrium. The following incentive constraints are necessary: For any θ [θ, θ] and any b θ [θ, θ], r(p( b θ), b θ)m( b θ; A) A( b θ) r(p( b θ), b θ)m(θ; A) A(θ) r(p(θ),θ)m(θ; A) A(θ) r(p(θ),θ)m( b θ; A) A( b θ). Adding yields [r(p( b θ), b θ) r(p(θ),θ)][m( b θ; A) M(θ; A)] 0. Since r(p(θ),θ) is strictly decreasing in θ, it is thus necessary that M(θ; A) is nonincreasing. It follows from the incentive constraints that A(θ) is nonincreasing. Further, given the advertising search rule, it is clear that A(θ) cannot be constant over any interval of types: by increasing its advertising an infinitesimal amount, a firm with a type on this interval would experience a discrete gain in its expected market share. Thus, A(θ) must be strictly decreasing, and consequently it is necessary that M(θ; A) = U N +[1 F (θ)]n 1 I. It follows that M(θ; A) = U N. A firm with type θ thus cannot be deterred from selecting zero advertising, and hence A(θ) =0is also necessary. We next establish that A(θ) must be differentiable, and we also derive the necessary expression for A 0 (θ). Consider any b θ<θ.rearranging the incentive constraints presented above, we find that r(p(θ),θ)[m( b θ; A) M(θ; A)] b θ θ A(b θ) A(θ) b θ θ r(p(b θ), b θ)[m( b θ; A) M(θ; A)] b θ θ. Similarly, consider any b θ>θ. The incentive constraints may now be rearranged to yield r(p(θ),θ)[m( b θ; A) M(θ; A)] b θ θ A(b θ) A(θ) b θ θ r(p(b θ), b θ)[m( b θ; A) M(θ; A)] b θ θ. Allowing that b θ may approach θ from the right or the left, we may now take limits as b θ θ, use the differentiability of the function M(θ; A) = U N +[1 F (θ)]n 1 I, and conclude that A 0 (θ) =r(p(θ),θ) M(θ; A). θ When combined with the boundary condition A(θ) =0, this differential equation may be solved to yield Z θ A(θ) = r(p(x),x)[ M(x; A)/ x]dx, θ 9

11 where M(x;A) x = (N 1)[1 F (x)] N 2 f(x)i <0 for all x<θ. We now integrate by parts and establish that A(θ) must take the following unique form: Z θ U A(θ) =R(θ, θ; A) R(θ, θ; A) D(p(x)) N +[1 F (x)]n 1 I dx, (1) θ where R(θ, θ; A) =r(p(θ), θ) N U. Rearranging, we note that interim-stage profit fortypeθ then must be given as Z θ U Π(θ, θ; A) =R(θ, θ; A)+ D(p(x)) θ N +[1 F (x)]n 1 I dx. (2) Observe that interim-stage profit is positive for all θ [θ, θ]. The second step in our proof is to construct an advertising equilibrium using the A(θ) function defined in (1). Observe that Π 1 (θ, θ; A) =r(p(θ),θ) M(θ;A) θ A 0 (θ) =0when this function is used. It follows that no type θ will deviate by mimicking some other type b θ, since for all b θ<θwe have Π(θ, θ; A) Π( b θ, θ; A) = = = Z θ θ Z θ θ Z θ Z θ θ x Π 1 (x, θ; A)dx [Π 1 (x, θ; A) Π 1 (x, x; A)] dx Π 12 (x, y; A)dydx > 0, where the inequality follows from Π 12 (x, y; A) =D(p(y))(N 1)[1 F (x)] N 2 f(x)i >0 for all x<θ. A similar argument ensures that Π(θ, θ; A) > Π( b θ, θ; A) for all b θ>θ.next, if no type θ>θ gains from deviating to A(θ), then a deviation to any advertising level a>a(θ) is also unattractive. Finally, since A 0 (θ) < 0, the advertising search rule is optimal for informed consumers. Proposition 1 thus establishes the existence and uniqueness of an advertising equilibrium. 17 The advertising equilibrium acts as a fully sorting (separating) mechanism: firms truthfully reveal their cost types along the downward-sloping advertising schedule. The informed consumers behave rationally in the advertising model: the lowest-cost firm advertises the most and offers the lowest price, and the informed consumers purchase from the highest-advertising firm. Thus, ostensibly uninformative advertising directs market share to the lowest-cost supplier and promotes productive efficiency. In the advertising equilibrium, the expected market share allocated to a firm of type θ takes the following form: M(θ; A) = N U +[1 F (θ)]n 1 I. A firm is sure to get its share of uninformed consumers; further, since the advertising schedule is downward sloping, a firm wins the informed consumers with the probability that the other N 1 firms draw higher types. The advertising 17 See Maskin and Riley (1984) for a related equilibrium characterization of bidding functions in the context of optimal auctions when buyers are risk averse. Our model also endogenizes the beliefs and strategies of informed consumers. For an advertising equilibrium, beliefs are uniquely definedontheequilibriumpath(bybayes rule)and off the equilibrium path (since the advertising search rule is optimal for informed consumers when they observe an advertising level in excess of A(θ) only if they believe that the deviating firm has cost type θ). 10

12 equilibrium thus induces a market share allocation that is strictly decreasing in a firm s type. The highest type does not advertise and sells only to its uninformed consumers: M(θ; A) = U N. We now characterize the expected profit forfirms in the advertising equilibrium. Using (2) and integrating by parts, we find that expected profit may be represented as: E θ [Π(θ, θ; A)] = r(p(θ), θ) U N + E θ D(p(θ)) F U f (θ) N +[1 F (θ)]n 1 I. (3) The first term on the RHS is the profit at the top. As noted, the fully sorting scheme allocates a market share of only N U to the highest type, θ. The second term represents the expected information rents. It is not immediately clear whether a strictly decreasing market share allocation enhances the magnitude of this term. The strength of the fully sorting scheme is based on downwardsloping demand. Lower-cost firms set lower prices and thus generate greater demand from visiting consumers; hence, D(p(θ)) is decreasing in θ. By directing more market share to lower-cost firms, the fully sorting scheme thus acts to expand the size of the market and increase expected information rents. The weakness of the fully sorting scheme is associated with the term F f (θ). For many popular distributions, F is log-concave ( F f (θ) is nondecreasing in θ).18 By allocating less market share to higher types, the fully sorting scheme works against the direct to which log-concavity of F appeals. Log-concavity of F plays a prominent role in our analysis below. It is thus important to develop some intuition for the role played by this property. 19 Market share must be allocated so as to satisfy incentive compatibility. When greater market share is directed to type θ, thistype earns greater profit and is thus less tempted to mimic lower types. Lower types can then lower advertising expenses and earn greater profit without inducing a violation of incentive compatibility. Intuitively, the ratio F f (θ) describes the contribution to expected profit thatismadewhentypeθ receives greater market share, since this ratio measures the proportion of types below θ conditional on the occurrence of type θ. When F f (θ) is nondecreasing in θ, an increase in market share to type θ contributes more to expected profit whentypeθ is higher. 20 The advertising equilibrium can be understood as a purification of Bagwell and Ramey (1994a). In their paper, advertising directs market share to the firm that offers the best deals, but equilibrium advertising takes the form of a mixed strategy. To see how our model constructs a purified version, consider a complete-information game, where production costs are fixed at a constant c>0. Then, as we establish in the Appendix, there exists a unique symmetric mixed-strategy equilibrium in this game as in Bagwell and Ramey (1994a) and Varian (1980). 21 Consider next an incomplete- 18 This assumption is common in the contract literature and is satisfied by many distribution functions. 19 For further discussion, see also Athey et al. (2004). 20 We build on this intuition below, when we compare expected information rents under the advertising and random equilibria. The random equilibrium is a limiting case in which firms pool at zero advertising. The resulting market share allocation is incentive compatible. 21 In the complete-information game considered here, all firms set the same price and informed consumers are indifferent when using the advertising search rule. By contrast, Bagwell and Ramey (1994a) allow firmstomake cost-reducing investments, and this ensures that higher-advertising firms offer strictly lower prices. In the analysis of advertising equilibria considered here, the advertising search rule is strictly optimal for informed consumers provided that incomplete information is present so that production costs vary (at least a little) with types. 11

13 information game, where production costs rise in types θ. As we show in the Appendix, if a firm of type θ uses the advertising strategy A(θ) in the unique advertising equilibrium of the incompleteinformation game, then the probability distribution induced by A is approximately the same as the distribution of advertising in the mixed-strategy equilibrium of the complete-information game, when the production costs for types θ (say, c(θ)) approximate the constant c. This purification result offers a useful link between the complete- and incomplete-information analyses; however, it does not establish whether the main predictions of Bagwell and Ramey carry over when, as seems plausible, production costs vary significantly with types. As we show below, when some additional structure is placed on the demand and distribution functions, the main predictions of the complete-information model can be captured in the general incomplete-information setting. 2.3 Random Equilibrium In this subsection, we analyze the random equilibrium, wherein all consumers use the random search 1 ruleandthusdivideupevenlyacrossfirms. Each firm then receives an equal share, N, of the unit mass of consumers. Given the random search rule, firms necessarily choose zero advertising, since even informed consumers are unresponsive to advertising; furthermore, when firms pool and do not advertise, the random search rule is a best response for each consumer. 22 The random equilibrium thus exists and takes the form of a pooling equilibrium. In the random equilibrium, the interim-stage profit forthefirm of type θ is given by r(p(θ),θ) 1 N. The random equilibrium sacrifices productive efficiency; however, all advertising expenses are avoided. Using dr(p(θ),θ) dθ = D(p(θ)), it is straightforward to confirm that the expected profit for a firm in the random equilibrium is E θ r(p(θ),θ) 1 = r(p(θ), θ) 1 N N + E θ D(p(θ)) F f (θ) 1. (4) N The RHS contains the profit at the top and the expected information rents, respectively. 2.4 Comparison of Advertising and Random Equilibria We now compare the advertising and random equilibria. We begin by comparing the expected consumer surplus in these equilibria. An uniformed consumer expects the same consumer surplus whether the advertising or random equilibrium is anticipated. For both equilibria, the uninformed consumers samples from the induced distribution of monopoly prices and expects to pay E θ [p(θ)]. 22 If informed consumers observe a deviation whereby some firm selects positive advertising, then random search remains optimal in the event that informed consumers believe that the deviating firm has an average type. Since such a deviation may be more attractive to a lower-cost type, the random equilibrium may fail to be a refined equilibrium in the static model. See Bagwell and Ramey (1994b) for an analysis of the refinedequilibriuminarelatedmodel of advertising in which one firm has two possible cost types. As noted in the Introduction, the random equilibrium can also be associated with a setting in which advertising is prohibited (in which case deviant positive advertising selections are not possible). Our analysis here of random equilibria is also useful for our companion paper (Bagwell and Lee, 2009), where we consider the repeated game and the possibility of a self-enforcing agreement among firms in which a deviation from zero advertising would cause a future advertising war. 12

14 By contrast, an informed consumer expects strictly higher consumer surplus in the advertising equilibrium than in the random equilibrium. The key point is that, in the advertising equilibrium, an informed consumer can infer the identity of the lowest-cost, and thus lowest-price, firm. Accordingly, while the distribution of prices is not altered across equilibria, the informed consumer in the advertising equilibrium transacts at the lowest available price. Formally, an informed consumer expects to pay E θ [p(θ)] in the random equilibrium and E θ [p(θ min )], whereθ min min{θ 1,...,θ N },inthe advertising equilibrium. Under our assumption that N 2, wehavethate θ [p(θ min )] <E θ [p(θ)]. The comparison of expected profit across equilibria is more subtle. As illustrated in (3) and (4), in both types of equilibria, expected profit consists of two terms: the profit atthetopandthe expected information rents. To increase the profit at the top, the random equilibrium (pooling) is strictly preferred to the advertising equilibrium (full sorting). Intuitively, the highest-cost firm is never out-advertised in the random equilibrium and thus sells to its share of all consumers, 1 N ; by contrast, in the advertising equilibrium, the highest-cost firm is always out-advertised and thus sells only to its share of uninformed consumers, N U. To increase expected information rents, however, it is not immediately clear whether the random or advertising equilibrium is preferred. On the one hand, if F f (θ) is nondecreasing, then the random equilibrium is attractive, since this equilibrium allocates more market share to higher-cost types. On the other hand, downward-sloping demand creates a force that favors the advertising equilibrium, which allocates more market share to lower-cost types, since these types price lower and thus generate larger demand D(p(θ)). For the special case in which the support of possible cost types is small, we can unambiguously rank expected profits under the advertising and random equilibria. As θ θ approaches zero, expected information rents also approach zero in both the advertising and random equilibria. Profit at the top remains strictly higher under the random equilibrium, however, since the highest-cost firm gets strictly more market share in the random than the advertising equilibrium. Thus, for θ θ sufficiently small, expected profit is strictly higher under the random equilibrium than under the advertising equilibrium. Given the purification result described above and established in the Appendix, this finding can be understood as a direct extension of Bagwell and Ramey s (1994a) analogous finding for the associated complete-information game. Consider next the general case in which the support of possible costs may be large. To go further in ranking expected profits, we must formally analyze the expected information rents. 23 Let A denote the advertising schedule used in the advertising equilibrium, in which the market share allocation, M(θ; A) = U N +[1 F (θ)]n 1 I, is strictly decreasing. Similarly, let A p 0 denote the advertising schedule used in the random (pooling) equilibrium, in which the market share allocation, M(θ; A p ) 1 N, is constant. We now define the distribution function G(θ; A) R θ θ M(x; A)f(x)dx R. θ θ M(x; A)f(x)dx 23 Our analysis here builds on arguments made by Athey et al. (2004) in their analysis of price collusion. 13

15 The distribution G(θ; A p ) is similarly defined. The denominator represents the (ex ante) expected market share, which equals 1 N. Since M(θ; A) is strictly decreasing, M(θ; Ap )= 1 N crosses M(θ; A) from below. This implies in turn that G(θ; A p ) first-order stochastically dominates G(θ; A): G(θ; A p ) G(θ; A). Thus, if D(p(θ)) F f (θ) is nondecreasing, then Z θ θ D(p(θ)) F f (θ)dg (θ; Ap ) Z θ θ D(p(θ)) F (θ)dg (θ; A). f The inequality can be rewritten as E θ D(p(θ)) F f (θ)m(θ; Ff Ap ) E θ D(p(θ)) (θ)m(θ; A). (5) Referring to (3)-(5), we conclude that, if D(p(θ)) F f (θ) is nondecreasing, then expected information rents are weakly higher in the random equilibrium than in the advertising equilibrium. In fact, given our assumption that f(θ) > 0, we can go further and establish that, if D(p(θ)) F f (θ) is nondecreasing, then the random equilibrium must generate strictly higher information rents than the advertising equilibrium. This follows since D(p(θ)) F f (θ) is strictly increasing at θ, givenf(θ) > 0. We may now summarize our findings reagarding the comparison of expected profit acrossequi- libria. Relative to the advertising equilibrium, the random equilibrium has strictly higher profit at the top and, if D(p(θ)) F f (θ) is nondecreasing, strictly higher expected information rents. As suggested above, D(p(θ)) F f (θ) is nondecreasing if the log-concavity of F is significant in comparison to the extent to which demand slopes down. Further insight is possible by considering the limiting case in which D(p(θ)) is perfectly inelastic, so that D(p(θ)) is constant for all prices up to a reservation value. In this case, if F (θ) is log-concave, then D(p(θ)) F f (θ) is nondecreasing. We may now state the following conclusion: Proposition 2. (i) Informed consumers enjoy strictly higher expected consumer surplus in the advertising equilibrium than in the random equilibrium, and uninformed consumers enjoy exactly the same expected consumer surplus in both equilibria. (ii) If F is log-concave and demand is sufficiently inelastic, or if the support of possible cost types is sufficiently small, then firms make a strictly higher expected profit in the random equilibrium than in the advertising equilibrium. Proposition 2 (i) captures the implications of advertising restrictions for consumer welfare. While the ability to advertise does not affect the distribution of posted prices, it does lead to a strict reduction in the average transaction price. This finding is consistent with Benham s observations. We discuss this point further in the next subsection, when we allow for endogenous entry. Proposition 2 (ii) indicates that important circumstances exist under which firms gain when the use of advertising is restricted. As our discussion of the random equilibrium confirms, advertising would not be used if informed consumers were to ignore it. If informed consumers were responsive to advertising, however, then firms might nevertheless achieve a restriction on the use of advertising if advertising were legally prohibited. For a fixed industry structure, Proposition 2 (ii) thus suggests 14