On Advertising and Price Competition 1Ninghua Du January, 2005 Abstract If advertising lowers consumer search costs, it can affect competition. Previous studies by Butters (1977) and Robert and Stahl (1993) show that giving sellers the option of price advertising can significantly lower equilibrium market prices. These models assume that sellers make two bundled decisions: sellers determine the proportion of buyers that receive advertisements (ads) and reveal the price that they intend to charge in such ads. However, the vast majority of advertising does not reveal product pricing. In this paper, I argue that certain types of advertising may reduce consumer search costs without actually mentioning the price in the message. This leads me to propose a model in which firms first decide whether to advertise, and then set prices. In this model, the equilibrium price with advertising returns to the monopoly level. Keywords: Advertising, Consumer Search, Price Competition JEL Codes: C72, D21, D43, D83, L11, L13 Acknowledgements: I am grateful to Jim Cox, Martin Dufwenberg, Stan Reynolds and John Wooders for their guidance. I am also grateful to Greg Crawford, Price Fishback, David Reiley, Mark Walker, and theory workshop participants at the University of Arizona for their constructive comments. I retain responsibility for any errors. 1 School of Economics, Shanghai University of Finance and Economics, Shanghai, China. Email: ndu@ 1
I. Introduction Consumer search costs and the availability of sellers' advertising choices are key factors that influence how a market works. The classical model of Bertrand competition may be viewed as a simplified case where neither consumer search cost nor advertising plays a role. Under Bertrand competition, buyers get perfect information and take the lowest price in the market. Since the seller who offers the lowest price takes the whole market, sellers have an incentive to undercut their rivals and the market outcome is marginal cost pricing. However, if one relaxes the assumption that consumers have perfect information about prices, price competition will be imperfect and the market share of the high-price firm may be nonzero. One extreme example is the price adjustment procedure studied by Peter Diamond (1971). Given positive consumer search costs (uniformly bounded below by an arbitrary small positive number), Diamond shows that the unique market equilibrium is monopoly pricing. The intuition is that when all buyers have positive search costs for prices, buyers will not switch to other sellers when they see a price that is slightly higher than other prices; when other sellers are charging a price below the monopoly price, choosing a slightly higher price will strictly improve one seller's profit. One way to undermine the monopoly equilibrium derived by Diamond is to introduce price advertising. When advertising is available to sellers, sellers have an incentive to advertise their prices in order to increase their market shares. This way, buyers observe the advertised prices for free and buyers can switch to the advertised offers without paying a search cost. The studies by Butters (1977) and Robert and Stahl (1993) show that as the advertising costs go up, the degree of price dispersion (., the variance of prices) in 2
the market will go up. When advertising is free, the market outcome reverts to marginal cost pricing. Both Butters and Robert & Stahl assume that sellers make two simultaneous decisions: sellers determine the proportion of buyers that receive advertisements (ads) and reveal the price that they intend to charge in such ads. Price advertising is modeled in a way that presumes that prices are contained in the ads. However, this is not necessarily the most reasonable way to model advertising. In fact, the vast majority of advertising does not reveal product pricing. Examples include TV advertising, newspaper advertising and ads in the Yellow Pages. Internet advertising is another example. Merchants use banners, pop-up windows and sponsored links to advertise their products. However, in most cases, the banners, pop-up windows and links themselves do not directly convey product prices to buyers. Even in the circumstance that ads contain prices, the nominal prices shown in the ads may not be informative. In many cases, prices are multi-dimensional. When people shop for cars and houses, they not only care about the nominal price, but also care about the warrantee, insurance and related fees. In order to obtain this important information, people need to visit car dealers or real estate agents instead of sitting at home reading newspaper ads column. Without actually mentioning the price in the message, certain types of advertising still provide a way to reduce consumer search costs for advertisers' prices. An advertiser may leave its location and contact method in the ads. It is very common to see examples like the following in the Yellow Pages: "Insurance problems? Hassle free quotes! Call us first. We make house calls. Open seven days a week. Out of area call 1-800-xxx-xxxx." For internet advertising, by clicking the banners, pop-up windows and sponsored links, 3
potential buyers are led to the advertiser's homepage that provide information about the product details, including prices. This way, consumers can easily find out product prices by several phone calls or several clicks. Ads substantially reduce consumers' time and effort during the search process. For these important reasons, in this paper I propose and examine the impact of an alternative way to model advertising. Given that ads reduce consumer search costs without mentioning prices, it is plausible to think of sellers as making two decisions sequentially. First, sellers decide whether to advertise. Second, after observing rivals' advertising choices, sellers choose their prices. The first stage is that advertising a firm's existence eliminates the search cost for a consumer to shop at that firm and the firms choose prices after observing all other firms' advertising decisions. I show that, in contrast to the results by Butters and Robert & Stahl, the market outcome under the proposed 2Advertise-then-Price assumption reverts to monopoly pricing. The paper is organized as follows. Section II describes two previous results of search and advertising. Section III discusses the market outcome under the Advertise-then-Price assumption. Section IV discusses three variations. Section V concludes. II. The Backdrop: Two Previous Results of Search and Advertising Diamond shows that in a market that consumers must search sequentially with a positive search cost, in finite time the prevailing price becomes that which maximizes firms' joint profits. However, one can avoid this surprising result by altering the 4
assumptions. When consumers have sources of free information, the monopoly equilibrium in the Diamond model may be undermined. Butters and Robert & Stahl study the interaction between advertising and consumer search and show that the market equilibrium goes back to marginal cost pricing when advertising is free. In this section I use two one-shot games to recapture the main results by Diamond and by Butters and Robert & Stahl. The search cost game is an extension of a simple Bertrand pricing game, and the price advertising game is an extension of the search cost game. First, these two games serve as the backdrop for the advertise-then-price game in the next section, and allow me to compare the market outcome of the advertise-then-price game with the market outcomes from the previous studies under the same structure of modeling. Secondly, reinterpreting the results from the previous studies under the same game theoretic framework itself enriches the literature. To make the games simple and illustrative, I assume there are two sellers, identified as 1 and 2, and each of them produces a homogeneous good with zero cost of production. Sellers' goal is to maximize profits. There is a population of identical buyers, who have reservation value v for one unit of the good. Buyers' goal is to maximize consumer surplus. I assume there is no resale among buyers. Therefore, buyers' optimal strategies are identical and I can further assume there is only one buyer that represents the population of buyers. Sellers' costs and the buyer's reservation value are common knowledge. All of the players are risk neutral. These assumptions are kept throughout this section. 2 In my model, I assume that ads do not contain prices. However, one may wonder what will happen if sellers can choose whether to put prices in the ads. In section IV I show that giving sellers the option of choosing the 5
1. The Search Cost Game When searching for new prices is costly for consumers, the market outcome moves from marginal cost pricing in Bertrand competition to monopoly pricing. Given positive search cost and the buyer's equilibrium belief that the unobserved price is not lower than the observed price, the buyer will not search for the unobserved price. In the search cost game, the buyer has search cost c > 0 and the value of this parameter is common knowledge. Let us assume c < v. The game proceeds in two stages. First, sellers 1 and 2 simultaneously choose their prices P and P. Then the buyer is randomly matched with one of the sellers and observes 12that seller's price for free. With probability of the buyer observes seller 1's price for free, 3and the buyer observes seller 2's price for free with the same probability. In the second stage, it is the buyer's turn to move. The buyer has three choices. She can take the observed offer, quit the market, or search for a new price. If the buyer takes the offer, then the buyer gets v - P and seller i gets P. If the buyer quits, all players get zero. If iithe buyer decides to search, then the buyer pays search cost c and observes the other seller's price. Now the buyer can either take the offer P or P, or quit the market. 12It is natural to restrict attention to price offers that do not give the player losses if accepted, so I will assume that sellers make price offers P∈[0, v], i=1, 2. It is further iassumed that the buyer accepts the lower price if the buyer knows both prices. Then the buyer's strategy can be characterized by decision d at the price offer observed for free. d 11is a function that maps the offer observed for free to an action, d: [0, v]→{A, S, Q}, where 1 format of advertising will not change the main conclusion of this paper. 3 The price observed for free can be interpreted as the buyer's local store price. The buyer observes each seller's price for free with probability of means that the population of buyers is evenly distributed between two sellers. 6
A refers to accept, S refers to search and Q refers to quit. In order to guarantee the existence of Nash equilibrium, I will finally assume that the buyer always chooses A when the buyer is indifferent between A and S or the buyer is indifferent between A and Q. The following diagram illustrates the buyer's moves in the second stage given the buyer is matched with seller 1 and observes P in the first stage. 1 Buyer ( d) 1 A (Accept) Q (Quit) S (Search) Buyer ( d) 2 Seller 1 P 0 1 Seller 2 0 0 A A 12 Buyer v - P 0 (Accept P) (Accept P) Q (Quit) 112 P 0 0 1 0 P 0 2 v-c-P v-c-P -c 12 Figure 1: The Buyer's Actions and the Terminal Payoffs Since the buyer cannot observe the sellers' identities, at the decision node d the 1buyer must have a unique cutoff price r for both seller 1 and 2's offers. At d, the buyer 1accepts the observed offer p when p < r and the buyer searches for the unobserved offer when p > r. When p = r, the buyer is indifferent between A and S and the buyer chooses A according to the assumption stated previously. A Nash equilibrium in this game must be of the form "P=p, P =p, the buyer chooses cutoff pricep." It can be easily verified that in 1 2this game there are infinitely many Nash equilibria: for any p∈[0, v], the strategy profile "both sellers offerp, the buyer chooses cutoff pricep" is a Nash equilibrium. 7
However, any cutoff price p< v is not a credible threat, since it is not consistent with the buyer's belief along the equilibrium path. The reasoning is as follows. Consider the Nash equilibrium "The buyer chooses cutoff price p< v, both sellers offerp." Suppose the buyer is matched with seller 1 and observes P for free. On the equilibrium 1path, the buyer believes P=p. Then for any ex-post observation p< P< min(v, p+c), 2 1 the option 'search for P' is not optimal for the buyer, since the buyer's payoff by accepting 2P is v - P, and the expected payoff by search is v - p - c, which is less than v - P. 111In order to cater to this anomaly, one needs to employ a refinement of Nash equilibrium. Burdett and Judd (1983) were the first to provide a game theoretic interpretation of Diamond's model and they defined "search equilibrium," which has become the standard solution concept in the literature of consumer search and price 4dispersion. Before introducing the concept of search equilibrium, it is helpful to first characterize the buyer's cutoff price that is consistent with her belief about the unobserved offer. Observation 1 Suppose the buyer observes offer p for free and suppose the buyer's 5belief about the unobserved offer is a distribution of the form F(q), q∈[0, v]. Then the buyer's unique cutoff price r that is consistent with F(q) is given by the equation rc=F(p)dp. ∫0 4 Examples include Stahl (1989), Cason and Friedman (2003) and Morgan, Orzen and Sefton (2001). 5 In this section, the sellers do not mix their price choices. However, the notion of mixed strategy is useful since observation 1 will be applied to later sections, which involve mixed strategies. 8
Proof: Since the buyer takes the lower offer when she observes both 1 and 2's prices, the pvbuyer's expected payment conditional on search is E=qdF(q)+pdF(q). Therefore, ∫∫0pthe buyer accepts the observed offer when buyer's payoff by accepting offer p, which is v - p, is higher than buyer's expected payoff from search, which is v - c - E. This condition can be written as follows: A−>−−pvifvpvcE⎧d= where E=qdF(q)+pdF(q) or ⎨1∫∫0pSifv−c−E>v−p⎩pv⎧Aifp−qdF(q)−pdF(q)<c∫∫⎪0pd= ⎨1pvSifp−qdF(q)−pdF(q)>c⎪∫∫0p⎩pvppNotice thatp−qdF(q)−pdF(q)=(p−q)dF(q)=F(q)dq. ∫∫∫∫0p00pThe expressionF(q)dq is continuous in p, and strictly increasing on the support of F. ∫0pThe expressionF(q)dq is zero at p = 0, and is unbounded as p → ∞. According to the ∫0rintermediate value theorem, the equation c=F(p)dp has unique solution r, since the ∫0search cost c is positive. Now we can rewrite the buyer's decision at the offer observed for free in the following way: Aifp<rr⎧d= r is determined by c=F(p)dp. ⎨1∫0Sifp>r⎩ 9
Notice that r = q + c when F(p) = 0 for p < q and F(p) = 1 for p ≥ q. Since P∈[0, v] for i=1, i2, setting the cutoff price at min(v, r) represents the same buyer strategy as setting the cutoff price at r. Q. E. D. Given the buyer's cutoff price that is consistent with her belief on the unobserved offer, the search equilibrium that applies to the search cost game is stated as follows. ***Definition The strategy profile (P, P, r) is a search equilibrium if: 12 *****(1) For sellers, π (P, P, r) ≥ π (P, P, r) for ∀P, i = 1, 2, i ≠ j, where π is seller i's iijiijiipayoff. (2) For the buyer, *⎧Aifp<r* d= r= min(v, r), ⎨1*Sifp>r⎩rWhere p is the price observed for free, r is determined byc=F(q)dq and F is ∫0buyer's belief about the unobserved price. (3) In equilibrium, the buyer's belief is consistent with the equilibrium prices. *If seller i's price is not observed by the buyer, then F(q) =0 for q < P and F(q) =1 i*for q ≥ P. i In the definition, conditions (1) and (2) state that a search equilibrium must be a Nash equilibrium. Condition (3) imposes a restriction on the buyer's belief on the 10
unobserved price and this restriction is not required in Nash equilibrium. Therefore, the search equilibrium is a refinement. * **Observation 2 The unique search equilibrium is (P= P= v, r = v). Both sellers get 12expected payoff , and the buyer gets zero payoff. Proof: 1) The Nash equilibrium "The buyer chooses cutoff price p< v, both sellers offerp" does not satisfy the definition of search equilibrium. According to the definition of search equilibrium, given both sellers offerpin equilibrium, the buyer's equilibrium cutoff price is min(v, p+c) >p. This contradicts that the buyer chooses cutoff price p in equilibrium. 2) The strategy profile "The buyer chooses cutoff price p= v, both sellers offerp" satisfies the definition of search equilibrium. * **By 1) and 2), the unique search equilibrium is (P= P= v, r = v). Q. E. D. 12 Observation 2 indicates that the unique market outcome that satisfies the definition of search equilibrium is monopoly pricing. Two sellers split the market. The intuition is that choosing a slightly higher price strictly improves one seller's profit when its rival's price is less than v, since the buyer always accepts a slightly higher price offer given her positive search cost and market prices less than v. 11
2. The Price Advertising Game Now let us introduce sellers' advertising options into the search cost game. When advertising is available to sellers, the monopoly outcome in the previous buyer search model will be undermined. Since the buyer observes the advertiser's price for free, the sellers have an incentive to advertise their prices in order to increase their market shares. In the price advertising game, both sellers can advertise their prices. The whole buyer population observes the advertisers' prices without paying a search cost. Advertising costs sellers nothing. The timing of the game is as follows. Stage 1: Each of the sellers makes two bundled decisions: 1) choose product price and 2) choose whether or not to reveal this price to the buyer in the ads. The sellers make their decisions simultaneously. Then the buyer population is evenly allocated between two sellers and the buyer observes one of the sellers' prices for free, as described in the search cost game. Stage 2: The buyer then observes all of the advertised prices, and makes her decision. To illustrate the game clearly, let us discuss the three possible cases that can occur. Case I: The buyer receives two ads. In this case the buyer gets full market price information: she observes both prices offered on the market, and she can either take any one of the offers without paying a search cost, or quit from the market. If the buyer takes seller i's offer, then the buyer gets v - P and seller i gets P. If the buyer quits, all iiplayers get zero. 12
Case II: The buyer receives exactly one ad. Without loss of generality, suppose it is from seller 1. If the buyer is matched with seller 2 in stage 1, then the buyer observes both prices and it degenerates to the full information case. If the buyer is matched with 1 in stage 1, then the buyer cannot observe 2's price. The buyer can take 1's offer, quit from the market, or pay a search cost c to search for 2's price. In this case, the description of the buyer's moves is the same as in the search cost model. Case III: The buyer receives no ad. The buyer's moves degenerates to that of the search cost game: no advertising, the buyer pays a search cost to observe a new price. For detailed descriptions see the search cost model, the timing of the game, Stage 2. The sellers' strategies are characterized by (P, a), P∈[0, v], a∈{Ad, No Ad}, iiiiwhere Ad means 'to advertise' and No Ad means 'not to advertise'. I assume that the buyer accepts the lower observed offer when she gets full price information, and the buyer accepts one of the two offers with probability of when she observes a tie. Then the buyer's strategy is given by decision d at the price offer observed for free. d: ((P, a), (P, 11112a)) → {A, S, Q}. A refers to accepting the lower observed offer, S refers to search and Q 2refers to quit. Given the specification on the player's strategies, we have the following observation. Observation 3 In Nash equilibrium, both sellers must choose (P = 0, Ad), i =1, 2. i Proof: Let us first show P > 0, i =1, 2 cannot be part of a strategy in Nash equilibrium. i 13
Suppose seller 1 chooses (P > 0, Ad). Then choosing P< P cannot be part of 12 1seller 2's best response. By choosing P < P, seller 2's payoff must be less than or equal to 21~~P. However, If seller 2 chooses (p, Ad), where P<p< P, then seller 2 will get payoff 22 1~pwhich is strictly higher than P. Choosing P= P cannot be part of seller 2's best 22 1response. By choosing P = P, seller 2's payoff must be less than or equal to . If seller 2122 chooses (, Ad), then seller 2 will get payoff , which is strictly higher than . Finally, choosing P> P cannot be part of seller 2's best response since choosing P22 12 > P gives seller 2 zero payoff. In summary, (P > 0, Ad) cannot be a strategy in Nash 11equilibrium. Similar arguments apply to (P > 0, No Ad). 1Secondly, (P = 0, No Ad), i =1, 2 cannot be a strategy in Nash equilibrium. iSuppose seller 1 chooses (P = 0, No Ad). Then choosing (0 < P< c, Ad) gives 12 seller 2 positive payoff, which is higher than zero payoff from P = 0. Therefore P > 0, and 22choosing P = 0 cannot be part of seller 1's best response. 1Finally, it can be easily verified that (P = 0, Ad), i =1, 2, and the buyer accepts one iof the offers with probability is a Nash equilibrium. Q. E. D. Observation 3 indicates that the market outcome becomes marginal cost pricing when ads effectively reach the whole buyer population and the ads cost nothing to advertisers. The intuition is that when other sellers are charging a price above the marginal cost, one seller has an incentive to advertise a slightly lower price in order to take the whole market. This result is consistent with the previous literature. 14
III. Advertise-then-Price Finally, it is the time to introduce the Advertise-then-Price game. Consider the following advertising format: ads do not directly contain prices; meanwhile, ads still remove the consumer search cost for prices. The ads are considered "contact method advertising" rather than direct price advertising. In pre-stage, sellers decide whether to advertise the search methods for their prices. In the second stage, the sellers' advertising strategies are identified to the public and the sellers choose their prices. Finally it is the buyer's turn to accept one of the observed prices, search for a new price or quit. The timing of the game is as follows. Pre-stage: Sellers simultaneously decide whether to send ads to the buyer. Stage 1: Both sellers observe their rival's pre-stage decision. Sellers simultaneously choose their prices P and P. The buyer population is evenly distributed between two sellers and 12the buyer observes one seller's price for free, as described in the search cost model. Stage 2: Now it is the buyer's turn to move. The description of the buyer's moves is exactly the same as that of the price advertising model. The restrictions on players' strategy sets are the same as that of the price advertising model. The game tree for Advertise-then-Price is as follows: Seller 1 15
Advertise (Ad) Not to Advertise (No Ad) Seller 2 ----------------------------------- Seller 2 Ad No Ad Ad No Ad Bertrand Pricing One-Ad Pricing One-Ad Pricing Search Cost Subgame Subgame Subgame Subgame Note: the dotted line indicates seller 2's information set. Figure 2: Game Tree for Advertise-then-Price As shown above, advertising choices are realized before sellers choose prices. Therefore we can break the original game into an "advertising game" and a "pricing game"—given every seller's advertising choice, there is a corresponding separable pricing subgame. What we need to do is to discuss three possible cases. Case I: both of the sellers choose 'not to advertise.' In this case the pricing subgame degenerates to the search cost model. As shown in Figure 2, this scenario is the "search cost subgame." We use the unique search equilibrium as our solution—that is, P*=P*= v, the buyer sets the cutoff price at v. Both sellers get the 12expected payoff , and the buyer gets payoff zero. Case II: both of the sellers choose 'to advertise.' In this case, the pricing subgame degenerates to the Bertrand competition. As shown in Figure 2, this scenario is the "Bertrand pricing subgame." In equilibrium, 16
P*=P*=0, and the buyer takes the offer. Both sellers get payoff zero, and the buyer gets 12payoff v. Case III: One seller chooses 'to advertise,' but the other seller chooses 'not to advertise.' As shown in Figure 2, this scenario is the "one-ad pricing subgame." Without loss of generality, let us assume that seller 1 chooses 'to advertise,' but seller 2 chooses 'not to advertise.' In this case, the buyer always observes P. However, the buyer observes 2's 1price with probability , since with chance of 50/50 the buyer is matched with 2. The buyer makes choices after Pand P are chosen. Again, I restrict attention to price offers 1 2P∈[0, v], i=1, 2. Given price Pand the buyer maximizes her payoff conditional on the i1 price pair (P, P), seller 2's best response can be characterized as follows: 121⎧P−εif(P−ε)>(P−c−ε)111⎪P−εifP<2c⎧112 P= or P= ⎨⎨221P−c−εifP>2c⎩11⎪P−c−εif(P−ε)<(P−c−ε)111⎩2Since the buyer always observes P, seller 2 always has an incentive to undercut. If 1seller 2 chooses a price which is higher than P, seller 2 will never make a sale. If seller 2 1undercuts 1 by a sufficiently small amount of ε, then the fully informed buyer will buy 2's product and seller 2 gets payoff (P- ε). If seller 2 chooses to undercut by c + ε, then 1 both the fully informed buyer and the buyer who does not observe Pwill accept 2's offer 2 and seller 2 gets payoff P- c - ε. Therefore, when (P-ε) > (P-c -ε), or P< 2c, seller 2 1 1 1 1 chooses to undercut by ε. When (P- ε) < (P- c - ε), or P> 2c, seller 2 chooses to 1 1 1 undercut by c + ε. When P= 2c, seller 2 is indifferent. 1 Similarly, given price Pand the buyer maximizes her payoff conditional on the 2 price pair (P, P), seller 1's best response can be characterized as follows: 12 17
1⎧P−εif(P−ε)>(P+c−ε)222⎪P−εifP>c⎧222P= or P= ⎨⎨111P+c−εifP<c⎩22⎪P+c−εif(P−ε)<(P+c−ε)222⎩2Seller 1 faces tradeoff between sales volume and markup. If seller 1 chooses to undercut 2 by a sufficiently small amount of ε, then seller 1 will take the whole market. However, to increase the price by c - ε gives seller 1 higher profit margin without losing the market that the buyer does not observe P. Therefore, when (P- ε) > (P+ c - ε), or P22 2 2 > c, seller 1 chooses to undercut by ε. When (P- ε) < (P+ c - ε), or P< c, seller 1 2 2 2 chooses to raise the price by c - ε. When P= c, seller 1 is indifferent. 2 Since seller 1's best response to any P involves infinitely small quantity ε (same 2for seller 2's best response), there is no pure strategy Nash equilibrium in this pricing subgame. When P and P are higher than c, both seller 1 and seller 2 will undercut their 12rival. When the Pis at c, seller 2 will choose a price that is slightly lower than c. Given 1 seller 2's choice, seller 1 will increase the price by the amount of c - ε. Because of seller 1 and 2's asymmetric positions, price dispersion exists in the one-ad pricing subgame. I use search equilibrium defined in section II as the solution concept for the one-ad pricing subgame. The equilibrium is described by the strategy profile (F(P), F(P), r). 1122F(P) and F(P) are seller 1 and seller 2's price distributions. r is the buyer's cutoff price 1122iiiiwhen the buyer does not observe P. P∈[P, P], 0≤ P≤ P≤ v, i=1, 2. r = min(v, 2iLHL H r'r') where r' is determined by c=F(p)dp. The buyer always accepts the lower price 2∫0when she gets full price information. The unique search equilibrium in this one-ad pricing subgame is F(p)=1 - r/2p, p∈[, r), and Prob(P=r)= for seller 1; F(p) = 2 - r/p, 112 18
p∈[, r) for seller 2, and r = min(v, c/(1-ln2)) for the buyer. Seller 1's expected payoff is and seller 2's expected payoff is . The proof is in appendix. In the one-ad pricing subgame, both sellers get positive expected payoffs. The intuition is that when sellers choose different advertising strategies in the pre-stage, the advertiser keeps the market share from the buyer who does not observe the not advertised price and the seller who chooses not to advertise undercuts the advertiser in the fully informed market. Now we have solved for the equilibrium in every single pricing subgame. Therefore, we can use the equilibrium market outcomes in each pricing subgame to substitute for the pricing subgames and simplify the Advertise-then-Price game as follows: Seller 2 Not to Advertise To Advertise c (,v) Not to 1−ln2Advertise , (,v), Seller 1 1−ln2c (,v) 0 To 1−ln2 Advertise c0, (,v), 1−ln2 Table 1: Payoffs for the Sellers, Advertise-then-Price Game It is clear from the table that sellers' strategy 'not to advertise' weakly dominates the strategy 'to advertise.' When c/(1-ln2) < v, 'not to advertise' even strictly dominates the strategy 'to advertise.' The unique weakly dominant strategy equilibrium in this simplified 19
game is (1 not to advertise, 2 not to advertise) and both sellers get payoff . In contrast to the previous literature, the market outcome under Advertise-then-Price assumption reverts to monopoly pricing. IV. Variations In this section, we briefly discuss three extensions of the Advertise-then-Price model to check the robustness of the result. 1. What will happen if the sellers can choose the format of advertising? One may wonder what will happen if sellers can choose whether to put prices in the ads. In this subsection we show that giving sellers the option of choosing the format of advertising will not change the main conclusion of this paper. Suppose the procedure of the game is as follows. Pre-stage: Sellers simultaneously decide whether to send ads to the buyer. If seller i decides to send ads, she can choose whether to put her price P in the ads. iStage 1: Both sellers observe their rival's pre-stage decision. If seller i decides not to send ads or seller i decides to send ads without price in the pre-stage, then seller i will choose her price P in stage 1. The buyer population is evenly distributed between two sellers and the ibuyer observes one seller's price for free, as described in the search cost model in section II. Stage 2: Now it is the buyer's turn to move. The description of the buyer's moves is exactly the same as that of the price advertising model in section II. The restrictions and assumptions in the previous section apply to this game. Given every seller's pre-stage decision, there is a corresponding separable subgame in stage 1. 20
The structure of this game is similar to that of the advertise-then-price game in the previous section. What we need to do is to discuss the new cases that at least one of the sellers decides to put her price in the ads. Without loss of generality, suppose in the pre-stage seller 1 decides to put her price P in the ads. 1 Case I: Seller 2 decides not to advertise in the pre-stage. In this case, seller 2 observes P and undercuts seller 1. Following the discussion in 1the previous section, given positive P, seller 2's price is 1P−εifP<2c⎧11P= ⎨2P−c−εifP>2c⎩11In stage 2 the buyer observes P for sure and observes P with probability . 12When P> 2c, seller 1 gets payoff zero and seller 2 gets payoff P- c - ε. When P< 2c, 1 1 1 1 gets and 2 gets (P- ε). 11 Case II: Seller 2 decides to send ads without price in the pre-stage. In stage 1, seller 2 undercuts seller 1 by ε and takes the whole market. In stage 2 the buyer observes both P and P. Seller 1 gets zero and seller 2 gets P– ε. 121 Case III: In the pre-stage, seller 2 decides to put her price P in the ads. 2 The prices are already chosen in the pre-stage and the buyer observes both P and 1P. In this case no price will be chosen in stage 1. Play proceeds to stage 2 where the buyer 2will simply take the lower offer. 21
Combining with the analysis in the previous section, the market outcome in every subgame in stage 1 is unique. Now we can use this unique equilibrium market outcome in each subgame to substitute for the subgames in stage 1 and simplify the original game. In the simplified game, (1 not to advertise, 2 not to advertise) is a Nash equilibrium, which entails that both sellers subsequently choose the monopoly price in the search cost subgame as shown in section III. To see this, suppose seller 1 chooses 'not to advertise'. Then choosing 'not to advertise' gives seller 2 payoff . Sending ads without price gives seller 2 (v, c/(1-ln2)). Seller 2's payoff can never exceed (2c, v) when seller 2 decides to put P in the ads, since P ≤ v. Therefore, in the simplified game 'not to advertise' 22is seller 2's best response to seller 1's choice 'not to advertise'. The same argument applies 6to seller 1. 2. What will happen if advertising is not free? In the previous literature, Butters (1977) shows the comparative statics result that as the advertising becomes more expensive, buyers pay for it in terms of higher prices. Robert and Stahl (1993) show that as the advertising cost goes up, the probability that sellers choose monopoly price will go up. Now let us introduce advertising cost into Advertise-then-Price. A seller must pay fixed cost b if it chooses to advertise its price. Then the structure of the simplified advertising game is as follows: Seller 2 Not to Advertise To Advertise 6 One need to notice that in the simplified game, 'not to advertise' is not a dominant strategy for a seller. However, (seller 1 not to advertise, seller 2 not to advertise) is still a payoff dominant Equilibrium for sellers. 22
c (,v)- b Not to 1−ln2Advertise , (,v), Seller 1 1−ln2c (,v) To 1−ln2 - bAdvertise c - b, (,v)- b, 1−ln2 Table 2: Payoffs for the Sellers, Advertise-then-Price (advertising is not free) From the table we can see that seller i's strategy 'not to advertise' strictly dominates 'to advertise.' The unique strictly dominant strategy equilibrium in the simplified advertising game is (seller 1 not to advertise, seller 2 not to advertise). Not surprisingly, as advertising cost goes up, the monopoly price is more likely to be the market outcome. Therefore, analyzing the "boundary" case that advertising is free is enough to illustrate the characteristics of the Advertise-then-Price game. 3. What will happen if there are N sellers in the market? Let us assume that there are N sellers making decisions and the buyer population is evenly distributed among N sellers, otherwise the game is exactly the same as the Advertise-then-Price game described in the previous section. Given every seller's advertising choice in the pre-stage, there is a corresponding pricing subgame. We characterize the pricing subgames in three possible cases. Case I: All of the sellers choose 'not to advertise.' 23
In this case the pricing subgame degenerates to the search cost game. The buyer has unique cutoff price r. When the lowest observed price is less than or equal to r, the buyer accepts the lowest observed price offer. Otherwise the buyer searches for a new price. The *equilibrium cutoff price r depends on the equilibrium prices. 'P*=…=P*= v, the buyer 1Nsets the cutoff price at v' is the unique search equilibrium in this pricing subgame. Each of the sellers gets the expected payoff v/N, and the buyer gets payoff zero. Case II: At least two of the sellers choose 'to advertise.' The seller who advertises its price always has an incentive to undercut the other advertisers' prices in order to take the whole market. There is Bertrand competition among the advertisers. Finally, the advertisers set prices at zero and the nonadvertisers cannot make a sale. In Nash equilibrium, all sellers get payoff zero and the buyer gets payoff v. Case III: One seller chooses 'to advertise,' but all other sellers choose 'not to advertise.' The analysis of the one-ad pricing subgame is parallel to that of the two-seller case. Let the advertiser's price distribution be F and let the nonadvertiser's price distribution be G. r'The buyer's cutoff price is given by r = min(v, r'), where r' is determined by c=G(p)dp. ∫0The equilibrium payoff to the advertiser is r/N and the equilibrium payoff to the N−12nonadvertisers is r/N, where r = min(v, r') and r' is given by . The proof r'=cN−1−lnNis collected in the appendix. After the discussion of the equilibrium payoffs to sellers in all of the possible pricing subgames, we can easily verify the following observation. 24
Observation 4 In the N-seller Advertise-then-Price game, after substituting every pricing subgame by its solution, the unique weakly dominant strategy equilibrium in the simplified game is that every seller chooses not to advertise. V. Concluding Remarks The equilibrium prediction heavily depends on the way advertising is modeled. In this paper, changing the way of modeling is much more than playing around with assumptions, since this new way captures important features in the real business practice. There are some empirical studies that address the question whether price advertising reduces prices. Benham (1972) examines markets for eyeglasses in which advertising is prohibited and those in which advertising is allowed. Lower prices are found in markets that permit advertising, and there is no clear evidence to show that the quality of service is lowered in those markets, hence the null hypothesis that advertising restriction is a proxy of collusion cannot be rejected. Cady (1976) reports similar findings for retail prescription drug markets. Kwoka (1984) tests the claim that advertising lowers price and quality simultaneously, and hence others are forced to follow. The empirical evidence shows that the advertisers' prices and qualities are indeed lower, and while nonadvertisers' prices fall, their quality actually is greater. In Milyo and Waldfogel's "Rum and Vodka" paper (1999), they examine a single market with exogenous regulation change. They find advertising stores substantially cut prices of their advertised products, and follow their rivals when their rivals cut prices, while nonadvertising stores do not do so. 25
The theoretical study in this paper provides an alternative way to explain the empirical puzzle. Understanding the way to advertise is crucial to answer the question whether advertising reduces prices. If the format of advertising is to remove consumer search costs without directly revealing sellers' prices (for example, the "contact method advertising"), then it is possible for the sellers to choose not to advertise and to maintain a positive profit margin in the long run. 26
References Anderson, S., A. de Palma and J-F. Thisse, 1992. Discrete Choice Theory of Product Differentiation. Cambridge: MIT Press. Bagwell, K., and G. Ramey, 1994. Coordination Economies, Advertising and Search Behavior in Retail Markets. American Economic Review 84: 498-517 Benham, L., 1972. The Effects of Advertising on the Prices of Eyeglasses. Journal of Law and Economics 15: 337-52 Burdett, K., and K. Judd, 1983. Equilibrium Price Dispersion. Econometrica 51: 955-969 Butters, G. R., 1977. Equilibrium Distributions of Sales and Advertising Prices. Review of Economic Studies 44: 465-91 Cady, J. F., 1976. An Estimate of the Price Effects of Restrictions on Drug price Advertising. Economic Inquiry 14: 493-510 Capra, M., R. Gomez, C. Holt and J. Goeree, 2000. Learning and Noisy Equilibrium Behavior in an Experimental Study of Imperfect Price Competition. forthcoming in the International Economic Review. Cason, T., and D. Friedman, 2003. Buyer Search and Price Dispersion: A Laboratory Study. Journal of Economic Theory 112: 232-260 Diamond, P., 1971. A Model of Price Adjustment. Journal of Economic Theory 3: 158-168 Friedman, D., 1989. Producers' Markets: A Model of Oligopoly with Sales Costs. Journal Economic Behavior and Organization 11: 381-398 Grossman, G. M., and C. Shapiro, 1984. Informative Advertising with Differentiated Products. Review of Economic Studies 51: 63-81 27
Holt, C., 1994. Oligopoly Price Competition with Incomplete Information: Convergence to Mixed-Strategy Equilibria. Working Paper. Kreps, D., and J. Scheinkman, 1983. Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes. The Bell Journal of Economics 14: 326-337 Kwoka, J. E. Jr., 1984. Advertising and the Price and Quality of Optometric Services. American Economic Review 74: 211-16 Milyo, J., and J. Waldfogel, 1999. The Effect of Price Advertising on Prices: Evidence in the Wake of 44 Liquormart. American Economic Review 89: 1081-96 Morgan, J., H. Orzen and M. Sefton, 2003. An Experimental Study of Advertising and Price Competition. European Economic Review, forthcoming. Morgan, J., H. Orzen and M. Sefton, 2001. An Experimental Study of Price Dispersion. Games and Economic Behavior, forthcoming. Robert, J., and D. O. Stahl, II, 1993. Informative Price Advertising in a Sequential Search Model Econometrica 61: 657-86 Stahl, D., 1989. Oligopolistic Pricing with Sequential Consumer Search. American Economic Review 79: 700-712 Stigler, G. J., 1961. The Economics of Information. Journal of Political Economy 69: 213-225 Tirole, J., 1988. The Theory of Industrial Organization, Cambridge, MA: MIT Press. Weitzman, M., 1979. Optimal Search for the Best Alternative. Econometrica 47: 641-654 28
Appendix: MATHEMATICAL DETAILS Claim 1: Consider the one-ad pricing subgame in two-seller advertise-then-price game. Suppose seller 1 chooses to advertise and seller 2 chooses not to advertise. Then the unique search equilibrium in this one-ad pricing subgame is F(p)=1-r/2p, p∈[, r), 1and Prob(P=r)= for seller 1; F(p) = 2-r/p, p∈[, r) for seller 2, and r = 12min(v, c/(1-ln2)) for the buyer. Seller 1's expected payoff is and seller 2's expected payoff is . Proof: Seller 1's expected payoff from charging price p is ψ + [1 - F(p)]p. ψ = p 2r'when p≤ min(v, r'), otherwise ψ = 0. r' is given by c=F(p)dp. The first component is 2∫0from the buyer that does not observe Pand the second component is from the buyer with 2 full price information. 21First, it must be true that P< P, since seller 2 always undercuts seller 1. It is also H H211212true that P≤ P. Suppose P< P. Then charging a price p, P< p < P, gives seller 1 L LL LL L12payoff p. Therefore, seller 1's payoff is increasing in p when P< p < P. This contradicts L Lthat p is in the support of F(P). 11Then we show that the upper limit of seller 1's price distribution F(P) is the lesser 1111of the buyer reservation value v and r', P = min(v, r'). The reasoning is as follows. If P HH11< min (v, r'), then charging P gives seller 1 payoff , by charging a price p HH. 11slightly higher than P, P < p < min (v, r'), sellers 1 gets payoff which is strictly HH111higher than . This contradicts the fact that P is in the support of F(P). If r' < P HH11H111and r' < v, then seller 1's payoff from P is 0 (notice that 1- F(P) = 0, given P< P). H2H2H 29
However, charging price r' gives seller 1 strictly positive payoff. Again this contradicts the 11fact that P is in the support of F(P). Finally it is obvious that v < P and v ≤ r is H11Himpossible, since any price higher than the reservation value will be rejected by the buyer. 11Therefore, P =min(v, r'). It follows that seller 1's equilibrium payoff is , which HHequals to (v, r'). Now we can solve for the equilibrium. For seller 1's price p in the support of F(p), 1we must have +[1 – F(p)]p = (v, r'). Then F(p) = 2 - [min(v, r')]/p, 22p∈[(v, r'), min(v, r')). For seller 2's price p in the support of F(p), we have [1 2- F(p)]p = π, where π is seller 2's equilibrium payoff. Since p=(v, r') belongs to the 121support of F(p) and P≤ P, π = (v, r'). Therefore F(p)=1-[min(v,r)]/2p, 2L L1p∈[(v, r'), min(v, r')), and Prob(P=min(v, r'))=. 1r'Solving for r' fromc=F(p)dp, we have r' = c/(1-ln2). 2∫0Q. E. D. Claim 2: Consider the one-ad pricing subgame in N-seller advertise-then-price game. In search equilibrium, the advertiser's expected payoff is r/N and the expected payoff to the N−12nonadvertisers is r/N, where r = min(v, r'), r'=c. N−1−lnNProof: The proof is parallel to that of two-seller case. Let the advertiser's price distribution be F and let the nonadvertiser's price distribution be G. The buyer's cutoff price is given by r = min(v, r'), where r' is determined r'by c=G(p)dp. ∫0 30
In equilibrium, the buyer never searches for a new price: first, the upper limit of the support of F is min(v, r'); secondly, the upper limit of the support of G will not exceed min(v, r'), since the nonadvertiser always undercuts the advertiser. The advertiser's 1N−1expected payoff from charging price p isp+[1−G(p)]p. The first component is NNfrom the buyer that only observes the advertised price and the second component is from the buyer that observes the advertised price and one not advertised price. The advertiser's 1expected payoff from charging price p isp[1−F(p)]. Solving for the mixed strategy Nsearch equilibrium, we have min(v,r')F(p)=1− for p∈[min(v, r')/N, min(v, r')), and Prob(P=min(v, r'))=1/N. (1) 1NpNmin(v,r')G(p)=− p∈[min(v, r')/N, min(v, r')). (2) N−1(N−1)pN−1r'=c (3) N−1−lnNThe equilibrium payoff to the advertiser is min(v, r')/N, the equilibrium payoff to 2the nonadvertisers is min(v, r')/N, r' is given by equation (3). Q. E. D. 31
