Countercyclical Average Price in Customer Market Naohiko Wakutsu∗ Institute of Innovation Research Hitotsubashi University Kunitachi, Tokyo, Japan 186-8603 October 1, 2012

Abstract We examine a customer market of a homogeneous good and show that the equilibrium average price may be countercyclical for two reasons. The first reason is that in times of high demand, sellers’ reputational concerns may increase, causing them to offer extra discounts. The second reason is that buyers’ search incentives may increase, causing buyers to become more price sensitive. The two reasons are interdependent in a manner not yet considered. The model developed is a two-period extension of Reinganum’s model. Our explanation nests two previous hypotheses in the literature and raises the possibility of multiple forces jointly generating countercyclical price movement. Also, it is shown that while the average price falls, price decreases may differ across sellers according to their types. Specifically, high-cost sellers tend to reduce their prices by larger amounts than low-cost sellers during peak demand periods. This insight is new in the theoretical literature and accords with the evidence of a recent empirical study. Keywords: countercyclical price, price dispersion, search, repeated purchase JEL classification: D39, D43, D83, L16

1

Introduction

A large body of empirical literature suggests that, on average, sellers set lower prices in response to positive product-specific demand shocks. That is, the price of a good is countercyclical to positive product-specific demand shocks. In a study of scanner data from a supermarket chain in Chicago, Chevalier, Kashyap, and Rossi (2003) demonstrate that prices and markups of products such as beer, cracker and tuna tend to fall in response to positive demand shocks. Warner and Barsky (1995) show that toys, bicycles and consumer ∗

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appliances fall in price during Christmas. MacDonald (2000) finds similar countercyclical patterns in prices of groceries. This is interesting for two reasons. First, a standard model of perfect competition is not consistent with this price movement. In a standard model of perfect competition, positive demand shocks result in either no change to or an increase in prices. Second, this might explain why prices are sticky at the macroeconomic level since individual demand peaks may not coincide with aggregate demand peaks. Hence, countercyclical pricing at the micro level does not necessarily imply that macro price indices should fall in times of high demand. Instead, it may lead to sticky prices. The theoretical literature on imperfect competition offers several hypotheses that potentially account for this price movement. These are: procyclical seller reputational concerns (Bagwell (2004)), procyclical price sensitivity in demand (Bils (1989) and Warner and Barsky (1995)), a loss-leader advertising model (Lal and Matutes (1994)), a procyclical price war in a collusive oligopoly (Rotemberg and Saloner (1986)) and procyclical seller entry (Plehn-Dujowich (2008)). This paper examines a customer market of a homogeneous good with many buyers and sellers. By customer market, we mean a market where buyers develop some attachment to sellers they have contacted previously. Papers developing related themes include Phelps and Winter (1970), Bils (1989) and Bagwell (2004). Here, this attachment is called reputation and arises since buyers are uncertain about sellers’ price selections, just like in the models due to Phelps and Winter (1970) and Bagwell (2004) and contrary to the Bils model (1989) where it arises because buyers are uncertain about the match of a seller’s product to their tastes. After characterizing an equilibrium, we will show that the equilibrium average price may be countercyclical for two reasons. The first reason is that in times of high demand, sellers’ reputational concerns may increase, causing them to offer extra discounts. The second reason is that buyers’ search incentives may increase, causing buyers to become more price sensitive. As explained below, these reasons are interdependent. Hence, multiple forces jointly generate countercyclical price behavior. The models due to Bagwell (2004) and Bils (1989) are two previous models on countercyclical seller pricing in a customer market and offer the hypotheses of procyclical seller reputational concerns and procyclical price sensitivity in demand, respectively. Here, both of their hypotheses are nested in our explanation. The model developed is a two-period extension of Reinganum’s model (1979) of a market for a homogeneous good with many identical buyers and differentiated sellers. While all sellers live for two periods, only a fraction of buyers do so, called “local consumers” rather 2

than “tourists”. Each period, sellers receive candidates for current-period selling costs in stage 1 and determine their costs. Given the realizations, sellers set prices in stage 2 and buyers shop in stage 3. In period 2, a buyer can choose repeated purchasing by returning to a seller he patronized in period 1 instead of searching for different sellers. However, repeated purchasing may or may not be beneficial to a buyer, since every seller has a chance to reduce his selling cost in period 2 and the seller he patronized in period 1 could be one that fails to reduce its price as much as other sellers. The countercyclicality of the equilibrium average price emerges as follows. An optimal rule for a buyer in period 2 is shown to be such that a buyer makes a repeated purchase if the price in period 1 of a seller he patronized in period 1 is low enough; and otherwise searches for a different seller. While setting his period-1 price, each seller correctly predicts that any buyer that finds his price low enough will then be willing to buy from him repeatedly, and that any buyer that finds his price merely acceptable will then buy from him this period only. The cutoff value between prices that are low enough and merely acceptable is called the reservation price of repeated purchase. Buyer behavior of this form confronts sellers with a tradeoff. While a higher price may raise short-run profits, it also may ruin the reputation for a “good” (i.e., low-price) seller, leading to less future profit. This tradeoff may induce sellers to offer extra discounts, since the short-term incentive to raise prices should be balanced against the ramifications that a favorable reputation has for long-run profit. The extra discounts to be offered by a seller is interpreted as the seller’s cost per buyer of establishing a good reputation. Since the cost is higher, the higher is a seller’s selling cost, sellers are divided into three subsets. The first set consists of low sellingcost sellers that can build low-price reputation without extra discounts. The second set is medium selling-cost sellers that must offer some extra discounts to establish favorable reputations and find it optimal to do so since the required values of the extra discounts are small. The third set contains high selling-cost sellers that must offer some extra discounts to establish low-price reputations and find it optimal not to do so since the required extra discounts are too large. So, the prices in period 1 of low- and medium-cost sellers are at least as low as the reservation price of repeated purchase, causing them to build good reputations, whereas those of high-cost sellers are higher than are the reservation price of repeated purchase. Now, if a positive product-specific demand shock is associated with an increase in the number of “local” buyers, then a positive demand shock is shown to affect neither the reservation price of repeated purchase nor on the number of low-cost sellers. Moreover, by turning relatively efficient high-cost sellers into medium-cost sellers, it increases the numbers of medium-cost sellers and decreases the number of high-cost 3

sellers. In sum, a positive demand shock increases the number of sellers that choose their period-1 prices to be at least as low as the reservation price of repeated purchase, causing more sellers to invest in building low-price reputations. Buyers predict this increase in sellers’ reputational concerns, causing them to anticipate a better price distribution in period 1. A better price distribution increases the value of search, so buyers tend to search longer, causing them to become more price sensitive in times of high demand. Again, while setting their prices in period 1, sellers predict this increase in price sensitivity in demand, causing the least efficient sellers to lower their prices. In this way, the equilibrium average price falls in response to a positive product-specific demand shock for two reasons that are interdependent. The source of procyclical seller reputational concerns in this model is that positive product-specific demand shocks cause the investment in future demands to be more profitable, especially for efficient high-cost sellers. A similar causality is provided in Bagwell’s model (2004). The procyclical price sensitivity in demand due to increased consumer search is also considered by Warner and Barsky (1995) and results from a different mechanism. In the Warner and Barsky model (1995), increased consumer search is due to a buyer’s longer shopping list and scale economies in search. Here, it is due to a buyer’s anticipation of a better price distribution, which is caused by sellers’ increased reputational considerations. Moreover, the model presented here predicts that while the average price falls, price decreases will differ across sellers according to their types. As we explained above, highcost sellers are more sensitive to positive product-specific demand shocks than are low- and medium-cost sellers and tend to reduce their prices by larger amounts during peak demand periods. Therefore, high-cost sellers may be more responsible than are low- and mediumcost sellers for countercyclical price movement. This causes the distribution of prices to be dispersed less in times of high demand, so both the average price and price dispersion are countercyclical. It merits emphasis that this insight accords with the recent finding of Nevo and Hatzitaskos (2005). Using the dataset of Chevalier, Kashyap, and Rossi (2003), they find that during peaks in demand for tuna, consumers become more price sensitive and substitute towards cheaper brands that do not lower their prices, while competitors respond by lowering their prices. The rest of this paper is organized as follows. The next section presents the model. In Sections 3 and 4 we consider the optimal buyer and seller behaviors in periods 1 and 2. Given these, in Section 5, we identify an equilibrium for the market. Section 6 shows the countercyclicality of the equilibrium average price. Section 7 concludes.

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2

The Model

I consider a market of a homogeneous good with a unit-measure of sellers and a finite measure λ of buyers. While all sellers live for two periods, only a fraction θ of the buyers do so, where 0 < θ < 1. A proposed interpretation of the fraction θ of the buyers is that they are local residents rather than tourists that are dropping in for a short visit. Each period i = 1, 2, the model is organized in three stages. In stage 1, sellers receive firm-specific productivity shocks zi from some distribution function G, and then determine their current-period marginal costs ci . In particular, I assume c2 = min{z1 , z2 }.

c1 = z1 ,

In words, the productivity shock is a candidate for a seller’s current-period marginal cost in both periods. In period 1, the shock becomes his selling cost. In period 2, he can choose the minimum of the two shocks. A proposed interpretation may be the following: Sellers are retailers and face a number of wholesalers. Each wholesaler charges a different wholesale price zi , and they collectively form a wholesale price distribution G(z) in period i = 1, 2. Each period, every retailer is assigned a wholesaler and receives zi . In period 2, if a retailer is assigned a better lowerpriced wholesaler, then he can choose this wholesaler and obtain a lower current-period marginal cost c2 = z2 < z1 . Otherwise, he uses the same wholesaler and c2 = z1 ≤ z2 . Then the resulting cost distribution across sellers is G in period 1, and 1 − (1 − G)2 in period 2. Here G is exogenous and its finite support [cmin , cmax ] is assumed to satisfy 0 < cmin < cmax ≤ cmin e/(1 + e). Given the realization of ci , sellers set prices pi in stage 2, and buyers do shopping in stage 3. All buyers have the same isoelastic demand function D(p) with elasticity e < −1. Before purchasing, each buyer searches sequentially among sellers at a constant marginal and average cost k > 0 to decide whether and where he wishes to purchase the good. These pieces of information are common knowledge. Buyers use these pieces of information to determine their optimal search strategy, and sellers use this information to maximize their individual discounted profits with the common discount factor β.

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Buyer Search

For a given exogenous price distribution, this section describes buyers’ optimal search rule, beginning with period 1. I will show that the optimal strategies are characterized by reservation prices and repeated purchasing. 5

3.1

Period 1

Let F1 be a known price distribution in period 1 and [pmin , pmax ] be its non-degenerate finite support. Then, the search problem facing a buyer is described as follows. By paying k, a buyer randomly draws a selling price p1 from a known distribution F1 . After each draw, he can choose whether to stop searching and buy at that price, or pay another k to obtain another price quotation. As is well known, a buyer’s optimal strategy here is a reservation price rule. That is, there is a reservation price s1 such that he continues to search until a selling price p ≤ s1 is discovered. Any such price p is accepted and search stops. Formally, the consumer’s surplus gained from discovering a price p′1 to a buyer who has already observed another price p1 is { ∫ } CS(p′1 ; p1 )

p1

= max 0,

D(p)dp . p′1

Then, the reservation price of search s1 in period 1 is defined as V S(s1 , F1 ) = k. Here, the left-hand side V S(p1 , F1 ) of the equation is the expected value from continuing to search to this buyer facing a price distribution F1 :1 ∫ p1 V S(p1 , F1 ) = CS(p′1 ; p1 )dF1 (p′1 ).

(1)

pmin

Examples are provided in Figure 1. Figure 1a illustrates two price distributions on [3, 4.5]. The dotted line is the graph of the continuous cumulative density function F1′ (p) = 1 − (1 − (p − 3)/1.5)2 . The solid line is the graph of the piecewise continuous cumulative density function  (p − 3)/1.5, p < 3.6    F1′′ (p) = (3.9 − 3)/1.5, 3.6 ≤ p ≤ 3.9    (p − 3)/1.5, p > 3.9. Note F1′ > F1′′ for all p ∈ (3, 4.5). That is, F1′ is strictly first-order stochastically dominated by F1′′ .2 What does the graph of V S(p, F1 ) look like when F1 is continuous, is piecewise continuous and makes a first-order stochastic dominant change? This is the point I am making with these examples. Assume that each buyer’s demand function is D(p) = 10 p−3 . 1

I note that the integral in (1) is Riemann-Stieltjes, since F1 may not be absolutely continuous. A cumulative distribution function F on R is said to strictly first-order stochastically dominate another cumulative distribution function G on R, if F (x) < G(x) for all x ∈ R. 2

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(b) Function V S

Figure 1: Price distribution F and function V S

The values of V S(p, F1 ) are displayed in Figure 1b. As can be seen, V S is continuous and strictly increasing with respect to p for both distributions. That is, the value from continuing to search is higher, the higher is the buyer’s current price at hand. Moreover, V S(p, F1′′ ) is strictly lower than V S(p, F1′ ) for all p, meaning that a first-order stochastic dominant change to F1 reduces V S. It is shown below that these properties hold in general.3 Lemma 1. V S is continuous and strictly increasing with respect to p1 . Since V S is continuous and strictly increasing with respect to p1 , the buyer continues to search until a selling price p ≤ s1 is discovered. The buyer then accepts it and stops searching. This is a reservation price rule and shown by McCall (1970). Lemma 2. Let F1′ and F1′′ be price distributions on [pmin , pmax ] and let F1′ strictly first-order stochastically dominate F1′′ . Then, V S(p, F1′ ) < V S(p, F1′′ ) for all p ∈ (pmin , pmax ]. Lemma 2, originally shown by Kohn and Shavell (1974), plays a role in establishing the countercyclicality of the equilibrium average price. A condition that is necessary for such an s1 to exist is the following.4 Assumption 1. V S(pmax , F1 ) > k. If V S(pmax , F1 ) < k then the expected value to a buyer from continuing to search is less than the search cost, even when the buyer’s price at hand is the highest price in the market. 3 4

Proofs of all results are in Appendix. The existence is assured by the Intermediate Value Theorem under Assumption 1.

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Clearly it is then never optimal for a buyer to search. This is introduced in Reinganum’s model (1979) as well. A buyer starts searching if the expected net value of search is positive, ∫ ∞ V S(pmax , F1 ) + D(p′1 )dp′1 > k. pmax

This always holds under Assumption 1.

3.2

Period 2

Next consider the search problem facing a buyer in period 2. Suppose that (in some way) he has discovered a price p2 from a known price distribution F2 with support [pmin , pmax ]. Then, he can choose whether (i) to pay k and repurchase from the seller he patronized in period 1; (ii) to pay k and randomly draw another price from the distribution F2 ; or (iii) to stop searching and buy at the price p2 . The buyer’s choice between (ii) and (iii) is analogous to that in period 1. That is, he stops searching if V S(p2 , F2 ) < k, and draw another random sample if V S(p2 , F2 ) > k. On the other hand, the buyer’s choice between (i) and (iii) is characterized as follows. Suppose that the buyer’s purchase-price in period 1 is p1 . If his previous-period seller’s transition probability of current price is known as H(p′2 |p1 ) (indicating the probability that this previous-period seller’s current price is lower than p′2 ), then the expected value from revisiting the seller to this buyer is ∫ p2 V R(p2 , p1 , H) = CS(p′2 ; p2 )dH(p′2 |p1 ). pmin

Hence, the buyer’s optimal rule is to return to this seller if V R(p2 , p1 , H) > k, and to accept p2 and stop searching if V R(p2 , p1 , H) < k. In summary, given p2 and F2 , the buyer’s optimal search rule in period 2 is stated as follows. • If V R(p2 , p1 , H) > max{V S(p2 , F2 ), k}, then he pays k and continue to search by returning to the seller he patronized in period 1; • If V S(p2 , F2 ) > max{V R(p2 , p1 , H), k}, then he pays k and continues to search by sampling at random from the price distribution F2 ; • If k > max{V S(p2 , F2 ), V R(p2 , p1 , H)}, then he stops searching and buys at p2 . Apparently, the transition probability H and the price distribution F2 depend on the manner in which sellers set prices in period 2, given p1 . For instance, if each seller sets a

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price equal to his previous-period price or the monopoly price whichever is lower, then { G(p2 ), p2 < p1 , H(p2 |p1 ) = (2) 1, p 2 ≥ p1 . Moreover, if the market distribution of prices in period 2 is induced by the cost distribution in that period, then F2 (p2 ) = 1 − [1 − G(p2 )]2 .

(3)

As proved in a later section, these specifications of H and F2 turn out to be true for many sellers in an equilibrium. Thus, the rest of this subsection considers a buyer’s optimal repeat-purchasing strategy under the following assumption. Assumption 2. H(p2 |p1 ) and F2 (p2 ) are given by (2) and (3) respectively. An example of Assumption 2 is provided in Figure 2a below. Here, G = (p2 − 3)/1.5 and p1 = 4. Define r by V R(pmax , r, H) = V S(pmax , F2 ). Under the above assumption, the following lemma insists that it be always suboptimal for a buyer to return to his previous-period seller in period 2 if p1 > r. In this light, r so defined is called a reservation price of repeated purchase. Lemma 3. Under Assumption 2, V R has the following properties: (i) V R is continuous and non-increasing with respect to p1 . (ii) If V R(pmax , p1 , H) < V S(pmax , F2 ) then V R(p2 , p1 , H) < V S(p2 , F2 ) for all p2 . Part (i) of this lemma says that the value to a buyer from returning to the seller he patronized in period 1 is lower, the higher is the seller’s price in period 1. Part (ii) rules out the possibility of a buyer returning to the seller he patronized in period 1 for his ith price observation in period 2, where i = 2, 3, 4, . . . This is because a buyer who finds it beneficial to revisit his previous-period seller in period 2 should sample this seller before any others. The existence and uniqueness of r is ensured by the following lemma. Lemma 4. Under Assumptions 2, a unique r exists. Under Assumption 2 and with the demand function D(p) = 10 p−3 and the uniform distribution G(p2 ) = (p2 − 3)/1.5, the values of V R(p2 , p1 , H) are calculated for p2 = 4 and 4.5 and plotted in Figure 2 under Assumption 2. As Lemma 3 shows, V R is nonincreasing with respect to p1 . Specifically, V R is strictly decreasing with respect to p1 9

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Figure 2: Transition probability H and function V R

for p1 ≤ p2 and is constant for p1 > p2 , given p2 . The reason why V R is constant for p1 > p2 is obvious. ∫∞ If V S(pmax , F2 ) + pmax D(p′2 )dp′2 > k, then a buyer starts searching in period 2. This is always true under Assumption 3. Moreover, this assures the existence of s2 yielding V S(s2 , F2 ) = k. Assumption 3. V S(pmax , F2 ) > k. In summary, the optimal behavior in period 2 of any buyer whose previous-period purchasing equals p is as follows: Under Assumption 3, the buyer starts searching. Under Assumption 2, if p ≥ r then his initial search is random sampling. If p < r then he begins his search by returning to the seller he patronized in period 1. Either way, he continues his search until a selling price p′ ≤ s2 is discovered. Any such price p′ is accepted and search stops.

4

Seller Pricing

Given exogenous reservation prices of search and repeated purchase, this section considers a seller’s optimal price setting and the resulting distribution of prices in periods 1 and 2.

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4.1

Period 2

Given a reservation price s2 , suppose that q2 (p; s2 ) is the quantity demanded at a price p by a buyer with the reservation price s2 in period 2. Then, the seller’s profit per buyer in period 2 given c2 in [cmin , cmax ] and s2 is π(c2 ) ≡ max (p2 − c2 )q2 (p2 ; s2 ). p2 >0

From the previous discussion in Section 3, buyers’ optimal behavior in stage 3 suggests { D(p2 ), p 2 ≤ s2 , q2 (p2 ; s2 ) = 0, p2 > s2 . Together with the supposition that D(p) is isoelastic with elasticity e < −1, this implies that its optimal price in period 2 is { pm (c2 ), c2 < cs2 , p2 = s2 , c2 ≥ cs2 , where pm (c) ≡ ce/(1+e) and cs2 ≡ s2 (1+e)/e. In words, if the seller has a selling cost lower than cs2 , then his optimal price is a constant monopoly markup over the selling cost c2 , and otherwise is the reservation price s2 . This is the same pricing rule as in Reinganum’s model (1979). So, in summary, { (pm (c2 ) − c2 ) D(pm (c2 )), c2 < cs2 , π(c2 ) = (4) (s2 − c2 ) D(s2 ), c2 ≥ cs2 . Let n2 be the seller’s expected number of buyers in period 2. Once again, from the previous discussion in Section 3, buyers’ optimal behavior in stage 3 suggests that n2 depend only on his previous-period price p1 and not on his current price p2 as long as p2 is at least as low as s2 . So, n2 = n2 (p1 ) and the seller’s expected profit in period 2 is π(c2 )n2 (p1 ).

4.2

Period 1

Given reservation prices s1 , s2 and r in [pmin , pmax ], we now discuss the seller’s optimal price setting in period 1. As described in the previous subsection, its expected profit in period 2 depends on its selling price in period 1. Hence, its optimal price should maximize the present value ϕ(p1 ) of its expected profits in periods 1 and 2, which is the sum of its expected profit in period 1 and its discounted expected profit in period 2.

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Applying (4) and c2 = min{c1 , z2 } and recalling that z2 is distributed according to G, it is straightforward to see that its expected profit per buyer in period 2 does not depend on p1 . So, concisely it may be written as π e (c1 ).5 Given r, let µ be the expected measure of buyers in period 1, whose purchasing price this period is at least as high as r. Then, the previous discussion suggests { θ(λ + µ), p1 ≤ min{s1 , r} (5) n2 (p1 ) = θµ, p1 > min{s1 , r}. In words, the seller’s expected number of buyers in period 2 is the seller’s expected number of new buyers in period 2, plus the seller’s expected stock of customers in period 2. The seller’s expected number of new buyers is the expected ratio of buyers searching at random this period to sellers, while the seller’s expected stock of customers in period 2 depends on his price in period 1. If the seller’s price in period 1 is higher than min{s1 , r}, then he will have a zero stock of customers; otherwise, the stock is the expected number of his buyers in period 1 who can live for two periods. In summary, the seller’s discounted expected profit in period 2 given c1 is βπ e (c1 )n2 (p1 ). Here, β is the common discount factor with 0 < β < 1 and n2 (p2 ), a decreasing stepwise function as in (5). Its expected profit in period 1 is analogous to that in period 2. Suppose that q1 (p; s1 ) is the quantity demanded at a price p by a buyer whose reservation price of search is period 1 equals s1 and n1 , its expected number of buyers in period 1. From the previous discussion in Section 3, buyers’ optimal search in period 1 suggests that a seller’s expected number of buyers in period 1 be simply the ratio of buyers to sellers, as long as p1 is at least as low as s1 , i.e., n1 = λ. So, { (p1 − c1 )D(p1 )λ, p1 ≤ s1 , (p1 − c1 )q1 (p1 ; s1 )n1 = 0, p1 > s 1 . Hence, the present value ϕ(p1 ) to be maximized in period 1 is summarized as follows. { (p1 − c1 )D(p1 )λ + βπ e (c1 )θ(λ + µ), p1 ≤ s1 , ϕ(p1 ) = if s1 ≤ r, βπ e (c1 )θµ, p1 > s 1 , 5

Specifically, its expected profit per buyer in period 2 given c1 , denoted by π e (c1 ) ≡ E[π(c2 )|c1 ], is  ∫ c1  πm (z2 )dG(z2 ) + πm (c1 )[1 − G(c1 )], c1 ≤ cs2 ,   c e π (c1 ) = ∫ min ∫ cs2 c1    πm (z2 )dG(z2 ) + πs (z2 )dG(z2 ) + πs (c1 )[1 − G(c1 )], c1 > cs2 , cmin

cs2

where πm (c) ≡ (pm (c) − c) D(pm (c)) and πs (c) ≡ (s2 − c) D(s2 )

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and  (p − c1 )D(p1 )λ + βπ e (c1 )θ(λ + µ), p1 ≤ r,    1 r < p1 ≤ s1 , ϕ(p1 ) = (p1 − c1 )D(p1 )λ + βπ e (c1 )θµ,    βπ e (c )θµ, p1 > s1 , 1

if r < s1 .

(6)

Apparently, ϕ(p1 ) depends also among other things on c1 , s1 and r. Where appropriate in later discussion, this dependence will be made more explicit by writing ϕ(p1 ; c1 , s1 , r). Given these, its optimal price in period 1 is now specified as follows. Given s1 and r, there are two possible cases: s1 ≤ r and r < s1 . If s1 ≤ r, then the seller’s per-period profits in periods 1 and 2 are both maximized at min{pm (c1 ), s1 } (that is, pm (c1 ) if c1 < cs1 ≡ s1 (1+e)/e and s1 otherwise). Thus, the seller’s optimal price this period is min{pm (c1 ), s1 }. Since min{pm (c1 ), s1 } ≤ r for every c1 , he “captures” each one of his customers in period 1 until the next period and establishes a good reputation. Alternatively suppose r < s1 . If c1 ≤ cr ≡ r(1 + e)/e, then the seller’s optimal price in period 1 is the monopoly price pm (c1 ), since it maximizes both of his per-period profits in periods 1 and 2. However, if c1 > cr , then he faces a dynamic trade-off. While he can maximize his short-term profit in period 1 by choosing min{pm (c1 ), s1 }, this myopic price fails to maximize his long-term profit in period 2, since he can earn an extra profit E[π(c2 )|c1 ]θλ if underpricing to r to establish a good reputation. In this case, the seller must resolve this trade-off by choosing a price that balances his short-term incentives to raise prices against the ramifications such favorable reputation could have for his long-run profit. Specifically, suppose r < s1 and let x(c1 , s1 , r) denote the seller’s maximum discounted profit from the myopic price min{pm (c1 ), s1 }: x(c1 , s1 , r) ≡ max{ϕ(pm (c1 ); c1 , s1 , r), ϕ(s1 ; c1 , s1 , r)}. Then, his optimal price in period 1 is r if x(c1 , s1 , r) < ϕ(r; c1 , s1 , r) and min{pm (c1 ), s1 } otherwise. Which value is greater between x(c1 , s1 , r) and ϕ(r; c1 , s1 , r) depends on the value of c1 . With the demand function D(p) = 10 p−3 and the uniform distribution G(z) = z−2 on [2, 3], the values of x(c1 , s1 , r) and ϕ(r; c1 , s1 , r) are plotted in Figure 3. Here, the other parameter values used are s1 = 4.24, s2 = 3.92, r = 3.57, β = 0.025, θ = 0.6 and λ = 10. As can be seen, the value of ϕ(r; c1 , s1 , r) is higher if c1 is higher than but close to cr = 2.38, and the value of x(c1 , s1 , r) is higher if c1 is close to cmax = 3. Hence, in this example, the seller’s optimal price in period 1 is r if c1 is higher than but close to cr , and is min{pm (c1 ), s1 } if c1 is close to cmax . 13

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2.8

Figure 3: Functions x ≡ max{ϕ(pm (c1 )), ϕ(s1 )} and ϕ(r)

As shown below, this first observation holds in general. That is, the seller’s optimal price in period 1 is r if c1 is higher than but close to cr . The reason is that in this case the expected value of a good reputation per buyer to the seller is always greater than the associated expected cost, i.e., βE[π(c2 )|c1 ] θ > min{pm (c1 ), s1 } − r. Lemma 5. Let ϵ > 0 be very small. Given s1 and r with r < s1 , x(c1 , s1 , r) < ϕ(r; c1 , s1 , r) for any c1 ∈ (cr , cr + ϵ] For those with current costs c1 that are higher than and apart from cr , the optimal price in period 1 cannot be determined in general. However, if the highest-cost seller finds it optimal to set price r rather than s1 , then r is the optimal price for all sellers. Alternatively, if the highest-cost seller finds it optimal to pursue a maximized short-run profit by choosing price s1 , then the sellers with current costs c1 larger than cr are divided into two subsets. The first is the sellers with current costs lower than some critical value denoted by ca , which is determined shortly. For those sellers, the optimal price is r rather than min{pm (c1 ), s1 }. The rest of the sellers belongs to the other subset and find it optimal to set the price min{pm (c1 ), s1 } rather than r. These are the content of the next lemma. Given s1 and r, let ca be a selling cost that makes a seller indifferent between charging r and min{pm (c1 ), s1 } in period 1, i.e., x(ca , s1 , r) = ϕ(r; ca , s1 , r). Lemma 6. Given s1 and r with r < s1 , the following holds. (i) If x(cmax , s1 , r) ≤ ϕ(r; cmax , s1 , r) then x(c1 , s1 , r) < ϕ(r; c1 , s1 , r) for all c1 in [cr , cmax ]. (ii) If x(cmax , s1 , r) > ϕ(r; cmax , s1 , r) then there exists ca in [cr , cmax ] and is unique. Moreover, x(c1 , s1 , r) ≶ ϕ(r; c1 , s1 , r) if c1 ≶ a, for any c1 ∈ [cr , cmax ]. 14

The reason that this statement holds given s1 and r is as follows. Suppose c1 > cr . Then, the expected value of a good reputation per buyer to the seller is βE[π(c2 )|c1 ]θ, whereas the associated expected cost per buyer is min{pm (c1 ), s1 } − r. Although both increase with c1 , the increase is faster in the expected cost. Thus, the net value decreases with c1 . In Figure 3, this is illustrated by the decrease in the difference ϕ(r; c1 , s1 , r) − x(c1 , s1 , r) with respect to c1 in (cr , cmax ] = (2.38, 3]. The net value of reputation is the smallest to the highest-cost seller. So, if he finds it profitable to invest in reputation, then clearly it should be at least as profitable for any others. Otherwise, there should be some seller that is indifferent between setting min{pm (c1 ), s1 } to maximize his short-run profit and charging r to have a good reputation. In summary, given s1 and r with r < s1 , the seller’s optimal price in period 1 is described as follows. { pm (c1 ), c1 < cr , p1 = if x(cmax , s1 , r) > ϕ(r; cmax , s1 , r), r, cr < c1 , and  p (c ), c1 < c r ,    m 1 cr < c 1 < c a , p1 = r,    min{p (c ), s }, c < c m 1 1 a 1

if x(cmax , s1 , r) ≤ ϕ(r; cmax , s1 , r).

A result from this price setting is as follows. If x(cmax , s1 , r) > ϕ(r; cmax , s1 , r), then all sellers successfully build a low-price reputation, because all set prices that are at least as low as the reservation price of repeated purchase. Alternatively, if x(cmax , s1 , r) ≤ ϕ(r; cmax , s1 , r), then only a fraction of the unit-measure of sellers can do so. The fraction is G(ca ) and they are the sellers with current costs at least as low as ca .

5

An Equilibrium

Given the derivations in Sections 3 and 4, we are now in the position to discuss an equilibrium, which is defined below. Definition 1. A set {s∗1 , s∗2 , r∗ , F1∗ (p1 ), F2∗ (p2 )} of reservation prices and price distributions is an equilibrium, if (i) given F1∗ and F2∗ , buyers choose s∗1 , s∗2 and r∗ to maximize their expected utility in period 1 and 2; and (ii) given s∗1 , s∗2 and r∗ , sellers collectively form F1∗ and F2∗ by each setting prices p1 and p2 that maximize his expected stream of future profits in periods 1 and 2.

15

Such an equilibrium is therefore a rational-expectation equilibrium in which both buyers and sellers correctly predict others’ behaviors. This definition is a natural extension of that characterized in Reinganum’s model (1979). In many markets, not all buyers may continue to patronize their sellers. Rephrased, in many markets, not all sellers may be viewed by their customers as “low-price” sellers. Rather, low-price sellers and high-price sellers often coexist. In what follows, I restrict my attention to characterize such a market in an equilibrium. ∫ cmax Assumption 4. cmin CS [z ′ e/(1 + e); cmax e/(1 + e)] dG1 (z ′ ) > k. Assumption 4 is merely a special case of Assumption 1 where the price distribution is that implied by the cost distribution G in period 1. Let c†a and r† and s†i , i = 1, 2 be the values defined implicitly by x(c†a , cmax e/(1 + e), r† ) = ϕ(r† ; c†a , cmax e/(1 + e), r† )

(7)

e), F2† )

(8)

V R(cmax e/(1 + e), r† , H) = V S(cmax e/(1 + V S(s†i , Fi′ ) = k

(9)

and let F1 , F1′ , F2′ , H and Fi† , i = 1, 2 be the cumulative distribution functions on [cmin e/(1+ e), cmax e(1 + e)] implicitly defined by F1 (p1 ) = G(p1 (1 + e)/e)   G(p1 (1 + e)/e), p1 < r†   F1′ (p1 ) = G(c†a ), r† ≤ p1 < c†a e/(1 + e)    G(p1 (1 + e)/e), p1 ≥ c†a e/(1 + e) F2′ (p2 ) = 1 − [1 − G(p2 (1 + e)/e)]2 { G2 (p2 (1 + e)/e), p2 < min{p1 , s†2 } H(p2 |p1 ) = 1, p2 ≥ min{p1 , s†2 } { ′ Fi (pi ), pi < s†i Fi† (p2 ) = 1, pi ≥ s†i

(10) (11)

(12) (13) (14)

Apparently, these definitions are interdependent, so the values of c†a and r† and s†i and the distributions F1 , F1′ , F2′ , H and Fi† , i = 1, 2, are determined simultaneously. How to determine these values and distributions is explained shortly. Proposition 1. Under Assumption 4, suppose that the following (15)–(17) hold: x(cmax , cmax e/(1 + e), r† ) > ϕ(r† ; cmax , cmax e/(1 + e), r† ) 16

(15)

V S(r† , F1 ) < k

(16)

c†a e/(1 + e) ≤ s†1

(17)

where c†a , r† , s†i , F1 , F1′ , F2′ , H and Fi† , i = 1, 2, are defined in (7)–(14). Then, an equilibrium exists and is given by (s∗1 , s∗2 , r∗ , F1∗ , F2∗ ) = (s†1 , s†2 , r† , F1† , F2† ). Proposition 1 says that if Assumption 4 and conditions (15)–(17) hold, then an equilibrium exists, which is represented as (s†1 , s†2 , r† , F1† , F2† ). Moreover, in this equilibrium, low-price sellers and high-price sellers coexist so that some buyers remain not “captive” in period 2. Assumption 4 ensures the existence of r† and s†i satisfying (8)–(9), while condition (15) ensures the existence of c†a satisfying (7). Condition (16) ensures r† < s† and also rules out the possibility of all sellers establishing good reputations. Condition (17) limits the equilibrium measure of sellers that establish good reputations. In particular, this prevents the equilibrium measure of sellers that establish good reputations from exceeding G(s†1 (1 + e)/e). In other words, it says that the equilibrium expected measure of buyers that are “captive” in period 2 is no greater than G(s†1 (1 + e)/e)θλ. Condition (17) implies min{pm (c†a ), s† } = pm (c†a ), so the equilibrium price of a seller that is indifferent between charging the reservation price r and the myopic price min{pm (c†a ), s† } is r† = pm (a† ) rather than r† = s†1 . All together they ensure the existence of such an equilibrium. Under Assumption 4, we can construct an equilibrium as follows by using backward induction. F2′ in (12) is a price distribution implied by the cost distributions 1 − (1 − G)2 in period 2. Given F2′ , we first obtain a candidate s†2 of the equilibrium reservation price of search in period 2 by using (9). Given s†2 , we construct a candidate F2† for the equilibrium price distribution in period 2 by using (14). Proceed backward to period 1. H in (13) is a transition probability implied by the exogenous distribution G that govern a probability shock in period 2, with a mass point at min{p1 , s†2 }. Given s†2 , F2† and H, we compute a candidate r† for the equilibrium reservation price of repeated purchase by using (8). Given r† , we can check whether conditions (15) and (16) hold. If they do, we can find c†a by using (7). Given s†2 , F2† , r† and c†a e/(1 + e), define F1′ by (14) and determine a candidate s†1 for the equilibrium reservation price of search in period 1 by using (9). Given c†a e/(1 + e) and s†1 , check whether condition (17) holds. If it does, all the conditions are satisfied and the candidate (s†1 , s†2 , r† , F1† , F2† ) turn out to be an equilibrium. As an example, the equilibrium price distributions F1∗ and F2∗ are illustrated in Figure 4 with the demand function D(p) = 10 p−3 , the uniform distribution G(z) = z − 2 on [2, 3]. For the parameter values θ = 0.6, λ = 10 and β = 0.025, each of Assumption 4 and conditions (15)–(17) turn out to be true. As can be seen, F1∗ exhibits two points of discontinuity at r∗ = 3.57 and s∗1 = 4.24. Since r∗ ≤ s∗1 holds, only a fraction of the sellers is expected to 17

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 3

3.2

3.4

3.6

3.8 price

4

4.2

3

4.4

(a) In period 1

3.2

3.4

3.6

3.8 price

4

4.2

4.4

(b) In period 2

Figure 4: Equilibrium price distributions

have low-price reputations. In other words, some buyers remain not captive in period 2.

6

Countercyclicality

A large body of empirical literature documented that on average, sellers set lower prices in response to positive product-specific demand shock. This section considers the cyclicality of the equilibrium average price to a positive product-specific demand shock. It is shown below that if a positive product-specific demand shock is associated with an increase in θ, then the equilibrium average price in period 1 is countercyclical for two reasons. The first reason is that as θ increases, sellers’ reputational concerns increase, causing them to offer extra discounts. The second reason is that buyers’ search incentives increase, causing them to become more price sensitive. The two reasons are interdependent, so they jointly generate countercyclical price movement. An increase in θ may be interpreted as a situation where the market consists of more local customers rather than tourists. Lemma 7. An increases in θ has no effects on r∗ , increases c†a and decreases s∗1 . The lemma says that an increased θ does not change the equilibrium reservation price of repeated purchase r∗ but decreases the equilibrium reservation price of search s∗1 . Moreover, increased θ raises the value of a selling cost in period 1 of a seller that is indifferent between charging the reservation price r∗ and the myopic price pm (c†a ) in an equilibrium. Therefore, an increase in θ causes a situation where more number of inefficient (i.e., relatively high-cost) sellers find it optimal to set price r∗ rather than pm (c1 ) in order to build 18

1

0.8

0.6

0.4

0.2

0 3

3.2

3.4

3.6

3.8 price

4

4.2

4.4

Figure 5: Change in equilibrium distribution in period 1

good reputations. Hence, as θ increases, the equilibrium measure G(c†a ) of sellers that price at least as low as r∗ increases while the highest equilibrium price s∗1 decreases. The next proposition is an immediate result from this lemma. It says that the equilibrium average price in period 1 is countercyclical to a positive product-specific demand shock. Proposition 2. An increases in θ lowers the equilibrium average price in period 1. Intuitively, the reason that more number of inefficient sellers find it optimal to set price r∗ as θ increases is that this is when winning customers’ continued patronage and establishing good reputations become more profitable. An inefficient seller whose cost in period 1 is higher than c∗r ≡ r∗ (1 + e)/e must pay the expected cost min{pm (c1 ), s∗1 } − r∗ to build an equilibrium good reputation. By doing so, it can secure all of its current stock of customers until the next period. Since a seller’s current stock of customers increases with θ, the investment in establishing good reputation is more profitable, as θ increases. The reason that buyers’ reservation price of search declines in response to an increase in θ is that increased θ raises sellers’ reputational concerns, causing buyers to anticipate a better price distribution in period 1 in the sense of the first-order stochastic dominance. Therefore, the two hypotheses of procyclical buyer search and procyclical seller reputational concern are nested in our explanation. While one interacting the other, they jointly generate the countercyclical price movement. On the basis of the possible interaction between sellers and buyers, this interpretation of ours is in start contrast to previous explanations in the literature. Hence it merits emphasis. With the demand function D(p) = 10 p−3 and the cumulative distribution G(z) = z − 2 19

on [2, 3], Figure 5 illustrates the change to an equilibrium price distribution F1∗ when θ increases from 0.3 to 0.6. Here, the other parameter values used are λ = 10 and β = 0.025, the same as before. The dotted line is the (original) equilibrium distribution for λ = 0.3, where r∗ = 3.57, c†a e/(1 + e) = 3.76 and s∗1 = 4.26. The solid line is the (new) equilibrium distribution for λ = 0.6, where r∗ = 3.57, c†a e/(1 + e) = 3.83 and s∗1 = 4.24. As can be seen, the new distribution lies above the original distribution, since the reservation price r∗ does not change while a† ≡ c†a e/(1 + e) increases and the reservation price s∗1 decreases. Hence, increased θ lowers the equilibrium average price. Lemma 7 also suggests that while the average price falls, price decreases vary among sellers according to their types. Specifically, as θ increases, while sellers with selling costs at least as low as the original value of c†a do not change their prices, sellers with selling costs higher than c†a tend to reduce their prices. Moreover, while the decrease in price by a seller with a selling cost close to c†a is driven by his increased reputational concerns, the decrease in price by a seller with a selling cost lower than but close to the original value of c∗s1 ≡ s∗1 (1 + e)/e is caused by increased search incentive of buyers. Since different sellers face different incentives, they react differently according to their types. Furthermore, since sellers with selling costs higher than the original value of c†a are more sensitive to an increase in θ than are the other sellers, they tend to reduce their prices by larger amounts. Thus, they may be more responsible for countercyclical price movement than are the other sellers. This can cause the distribution of prices to be dispersed less in times of high demand, so both the average price and price dispersion are countercyclical. These are two additional testable hypotheses that may be useful to differentiate further our explanation from those in the literature. Remark 1. An increase in λ has no effects on r∗ , c†a or s∗1 An increase in λ may be interpreted as an inflow of new buyers. According to Remark 1, an inflow of new buyers by itself does not change an equilibrium average price. However, this together with Lemma 7 implies that an inflow of new buyers also causes a countercyclical average pricing as long as an inflow of new buyers is associated with more local customers than tourists.

7

Conclusion

A large empirical literature suggests that, on average, sellers set lower prices in response to positive product-specific demand shocks. The theoretical literature offers several hypotheses that potentially account for this price movement. These are: procyclical seller 20

reputational concerns (Bagwell (2004)), procyclical price sensitivity in demand (Bils (1989) and Warner and Barsky (1995)), a loss-leader advertising model (Lal and Matutes (1994)), a procyclical price war in a collusive oligopoly (Rotemberg and Saloner (1986)) and procyclical entry (Plehn-Dujowich (2008)). This paper examined a customer market of a homogeneous good with many identical sellers and differentiated sellers and showed that the equilibrium average price may be countercyclical for two reasons. The first reason is that in response to positive productspecific demand shocks, sellers’ reputational concerns may increase, causing them to offer extra discounts. The second reason is that buyers’ search incentives may increase, causing them to become more price sensitive. These reasons were shown to be interdependent in a manner not yet considered. The model developed was a two-period extension of Reinganum’s model (1979). The models due to Bagwell (2004) and Bils (1989) are two previous models of countercyclical seller pricing in a customer market, and offer the two hypotheses of procyclical seller reputational concerns and procyclical price sensitivity in demand respectively. The explanation presented here nests these hypotheses and raises the possibility of multiple forces jointly generating countercyclical price movements.

Appendix: Proofs This appendix contains proofs of the results given in the text. Proof of Lemma 1. To show the stated monotonicity, consider any ϵ > 0. Then, ∫ p1 ∫ p1 +ϵ ′ ′ V S(p1 + ϵ, F1 ) = CS(p1 ; p1 + ϵ)dF1 (p1 ) + CS(p′1 ; p1 + ϵ)dF1 (p′1 ). pmin

limϵ→0 p1 +ϵ

Since CS(p′ ; p) is strictly increasing with respect to p for any p′ with p′ ≤ p, I have ∫ p1 +ϵ V S(p1 + ϵ, F1 ) > V S(p1 , F1 ) + CS(p′1 ; p1 + ϵ)dF1 (p′1 ) > V (p1 , F1 ). limϵ→0 p1 +ϵ

Therefore the claim holds. For proof of the continuity, I will establish the following: (i) V S is continuous at p1 when F1 is continuous, (ii) V S is continuous at p1 when F1 has a mass point at p1 , (iii) V S is continuous at p1 when F1 is right-continuous at p < p1 and cont. elsewhere. The initial claim is clear. To prove part (ii), suppose that F1 has a mass point at p1 . Let ϵ > 0. Then, ∫ p1 −ϵ CS(p′1 ; p1 − ϵ)dF1 (p′1 ) lim V S(p1 − ϵ, F1 ) = lim ϵ→0

ϵ→0

pmin

21



p1

CS(p′1 ; p1

= lim ϵ→0



ϵ)dF1 (p′1 )

pmin

∫ − lim ϵ→0

p1

CS(p′1 ; p1 − ϵ)dF1 (p′1 )

p1 −ϵ

= V S(p1 ; F1 ) − CS(p1 ; p1 ) lim (F1 (p1 ) − F1 (p1 − ϵ)) ϵ→0

= V S(p1 ; F1 ) and ∫ lim V S(p1 + ϵ; F1 ) = lim ϵ→0

ϵ→0

p1

CS(p′1 ; p1

+

ϵ)dF1 (p′1 )

pmin



p1 +ϵ

CS(p′1 ; p1 + ϵ)dF1 (p′1 )

+ lim ϵ→0

limϵ→0 p1 +ϵ

= V S(p1 ; F1 ). Hence the claim holds as well. Similarly, to prove part (iii), suppose that F1 has a mass point at p < p1 . Then, for any ϵ > 0, ∫

limϵ→0 p−ϵ

V S(p1 , F1 ) =

CS(p′1 ; p1 )dF (p′1 ) ∫

pmin

+ CS(p; p1 ) lim (F (p) − F (p − ϵ)) + ϵ→0

p1

CS(p′1 ; p1 )dF (p′1 ).

limϵ→0 p+ϵ

As before, take the right-side and the left-side limit of V S at p1 and then equate them. The claim follows. These establish the continuity of V S with respect to p1 . Lemma A1. Suppose that T1 and T2 are bounded functions on [xmin , xmax ] with xmin < xmax and have the following properties for some b ∈ (xmin , xmax ): { = T2 (x), x ∈ {[xmin , b), xmax } T1 (x) > T2 (x), x ∈ [b, xmax ). Suppose f is a nonnegative and smooth function on [xmin , xmax ], and let ∫ x α(x, Ti ) ≡ f (x′′ )dTi (x′′ ), i = 1, 2. xmin

Then the following holds. (i) α(x, T1 ) = α(x, T2 ) for any x ∈ [xmin , b). (ii) α(xmax , T1 ) ⋛ α(xmax , T2 ) as f ′ (x) ⋚ 0 for all x ∈ (xmin , xmax ). (iii) α(x, T1 ) > α(x, T2 ) for any x ∈ (b, xmax ) if f ′ (x) ≤ 0 for all x ∈ (xmin , xmax ). Proof of Lemma A1. The initial claim is clear. To prove parts (ii) and (iii), suppose that f is monotonic. Also, let S ≡ T1 − T2 . Then, ∫ x f (x′′ )dS(x′′ ) α(x, T1 ) − α(x, T2 ) = xmin

22

′′

′′

= f (x )S(x

∫ x )|xmin



x

= f (x)S(x) −

x



f ′ (x′′ )S(x′′ )dx′′

xmin

f ′ (x′′ )S(x′′ )dx′′

b

where the second equality holds by integration by parts while the third, by S(x) = 0 for x ∈ [xmin , b]. Since S(x) > 0 for any x ∈ (b, xmax ], α(x, T1 ) − α(x, T2 ) is positive for any x ∈ (b, xmax ) if f ′ ≤ 0, and α(xmax , T1 ) − α(xmax , T2 ) coincides with −f ′ in sign. Hence, the claims hold. Proof of Lemma 2. This is simply a corollary to part (iii) of Lemma A1, since CS(p′1 ; p1 ) is a strictly decreasing function with respect to p′1 . Proof of Part (i) of Lemma 3. Under Assumption 2, an increase in p1 causes a firstorder stochastic dominant change to H(p2 |p1 ). That is, H(p2 |p′1 ) ≤ H(p2 |p1 ) for any p1 and p′1 with p1 < p′1 , given p2 . Recalling that CS(p′2 ; p2 ) strictly decreases with p′2 , the stated monotonicity is simply a corollary to part (iii) of Lemma A1. For proof of the continuity, consider any ϵ > 0 and let p1 < p′1 ≡ p1 + ϵ. Define S(p2 ; p1 , p′1 ) = H(p2 ; p1 ) − H(p2 ; p′1 ). Then, ∫ p2 ′ V R(p2 , p1 , H) − lim V R(p2 , p1 , H) = CS(p′2 ; p2 )dS(p′2 ; p1 , p′1 ) = 0 ϵ→0

pmin

since S(p2 ; p1 , p′1 ) tends to vanish at the limit. Similarly, by considering any p′′1 and p1 with p′′1 ≡ p1 − ϵ < p1 we can obtain limϵ→0 V R(p2 , p′′1 , H) − V R(p2 , p1 , H) = 0. Hence, V R is continuous with respect to p1 under the stated assumption. Lemma A2. Under Assumption 2, the following holds: (i) V R(p2 , p1 , H) − V S(p2 , F2 ) < 0 for p2 < p1 . (ii) V R(p2 , p1 , H) − V S(p2 , F2 ) is increasing with respect to p2 > p1 . Proof of Lemma A2. Let S(p2 |p1 ) ≡ H(p2 |p1 ) − F2 (p2 ). Then, S < 0 for pmin < p2 < p1 and S ≥ 0 for p1 ≤ p2 < pmax . Part (i) of Lemma A2 is simply a corollary of part (iii) of Lemma A1. Prove the latter part. Let 0 < ϵ < pmax − p1 . Then, for p2 > p1 , we have V R(p2 + ϵ, p1 , H) − V S(p2 + ϵ, F2 ) ∫ ∫ p2 ′ ′ CS(p2 ; p2 + ϵ)dS(p2 |p1 ) + = pmin



> V R(p2 , p1 , H) − V S(p2 , F2 ) +

p2 +ϵ

CS(p′2 ; p2 + ϵ)dS(p′2 |p1 )

limϵ→0 p2 +ϵ p2 +ϵ

CS(p′2 ; p2 + ϵ)dS(p′2 |p1 )

limϵ→0 p2 +ϵ

23

≥ V R(p2 , p1 , H) − V S(p2 , F2 ), where the first inequality holds since CS(p′ ; p) is strictly increasing with respect to p for any p′ < p, while the equality in the second (weak) inequality holds when S = 0 for all p′2 ∈ [p2 , p2 + ϵ]. Therefore, V R(p2 , p1 , H) − V S(p2 , F2 ) is increasing with respect to p2 > p1 . Proof of Part (ii) of Lemma 3. The claim is immediate from Lemma A2. Proof of Lemma 4. First, it is necessary under Assumption 2 that H(p2 |pmax ) < F2 (p2 ) < H(p2 |pmin ) holds for any p2 . So, V R(pmax , pmax , H) < V S(pmax , F2 ) < V R(pmax , pmin , H) As established in part (i) of Lemma 3, V R is continuous with respect to p1 under Assumption 2. Then, by the Intermediate Value Theorem the existence of r is assured. Recall that V R strictly decreases with p1 ≤ p2 and remains constant elsewhere, as can be seen in Figure 2b. Therefore r is unique. Proof of Lemma 5. Consider the right-side limit of ϕ(r; c1 , s1 , r) − x(c1 , s1 , r) at c1 = cr . Since this is strictly positive, the claim holds for sufficiently small ϵ > 0. Proof of Lemma 6. By construction, x(c1 , s1 , r) is the upper envelope of ϕ(pm (c1 ); c1 , s1 , r) and ϕ(s1 ; c1 , s1 , r). So, x(c1 , s1 , r) decreases at a diminishing rate until c1 = cs1 and at a constant rate beyond that. On the other hand, ϕ(r; c1 , s1 , r) declines at a constant rate. Therefore, the difference ϕ(r; c1 , s1 , r) − x(c1 , s1 , r) decreases with c1 . This proves the first claim. Since x(c1 , s1 , r) and ϕ(r; c1 , s1 , r) are continuous on (cr , cmax ], Lemma 5 implies that if x(cmax , s1 , r) > ϕ(r; cmax , s1 , r) holds, then there is a ca in (cr , cmax ) satisfying x(ca , s1 , r) = ϕ(r; ca , s1 , r). The rest of the statement is the result of the monotonicity of x(c1 , s1 , r) and ϕ(r; c1 , s1 , r). Proof of Proposition 1. Let F2 (p2 ) be given by (12). Then, since F2 (p) > G[p(1 + e)/e] holds for all p in (cmin e/(1 + e), cmax e/(1 + e)), part (iii) of Lemma A1 implies that Assumption 4 ensures Assumption 3. Let s∗2 be the reservation price of search for F2 . Then s∗2 = s†2 . Given s∗2 , a profit-seeking seller with c2 sets the monopoly price pm (c2 ) if c2 < cs2 ≡ s∗2 (1 + e)/e, and the reservation price s∗2 otherwise. Note cmax < cmin e/(1 + e) < s∗2 . So, each seller’s expected profit is positive and none exits. Hence, selling prices this period range from cmin e/(1 + e) to s∗2 < cmax e/(1 + e) and the induced price distribution, denoted by F2∗ , coincides with F2† in (14) since s∗2 = s†2 . Note F2∗ = F2′ almost everywhere (a.e.)

24

on [cmin e/(1 + e), s∗2 ]. Part (i) of Lemma A1 then ensures V S(s∗2 , F2 ) = V S(s∗2 , F2∗ ), so s∗2 , defined to be the reservation price for F2 remains the reservation price for F2∗ . Let H be given by (13). Note that for these H and F2∗ , Assumption 2 and thereby Lemmata 3 and 4 hold. Let r∗ be the reservation price of repeated purchase for H and F2∗ , so that V R(cmax e/(1 + e), r∗ , H) = V S(cmax e/(1 + e), F2∗ ). By Lemmata 3 and 4, the existence and uniqueness of r∗ is assured. So r∗ = r† . Given r∗ = r† , suppose (15). Then, Lemma 6 assures the existence of c†a , which is given by (7). Let F1† be in (11) and let s†1 be the reservation price of search for F1† , so that V S(s†1 , F1† ) = k. Part (iii) of Lemma A1 implies that Assumption 4 assures Assumption 3, so s†1 exists. Note that c†a is in (c∗r , cmax ), while s†1 , in (cmin e/(1 + e), cmax e/(1 + e)). There are three possibilities: s†1 ≤ r∗ < a† , r∗ < s†1 < a† and r∗ < a† ≤ s†1 where a† ≡ c†a e/(1 + e). Suppose r∗ < a† ≤ s†1 as in (17). A profit-seeking seller with c1 then charges price pm (c1 ) if c1 ≤ c∗r , price r∗ if c∗r < c1 ≤ c†a , price pm (c1 ) if c†a < c1 ≤ c†s1 and price s†1 if c1 < c†s1 . All sellers are expected to have positive profits, so none exits. In this case, selling prices this period range from cmin e/(1 + e) to s†1 < cmax e/(1 + e) and the induced distribution is written in

  G(p1 (1 + e)/e), p1 < r∗      G(c† ), r∗ ≤ p1 < a† a  G(p1 (1 + e)/e), a† ≤ p1 < s†1      1, p1 ≥ s†1 .

Note this distribution corresponds to F1† a.e. on [cmin e/(1 + e), s†1 ]. Applying part (i) of Lemma A1, s†1 , defined to be the reservation price for F1† remains the reservation price for this induced distribution. Given F1∗ , buyers whose purchase-price in period 1 is at least as low as r∗ should have no identification problem in inferring sellers’ costs, excepts those whose purchase-price in period 1 have happened to be price r∗ , since F1∗ has a point of discontinuity at r∗ but is continuous elsewhere on [cmin e/(1 + e), r∗ ]. Although H may be different from the transition probability functions actually deduced by buyers whose purchase-price in period 1 have happened to be price r∗ , they are equivalent a.e. on [cmin e/(1 + e), r∗ ], since buyers know G. Using part (i) of Lemma A1, r∗ , defined to be the reservation price for F2∗ and H therefore remains the reservation price for F2∗ and the probability distribution buyers actually deduce. These establish Proposition 1 as its entity. Proof of Lemma 7. Both V R(pmax , p1 , H) and V S(pmax , F2 ) are independent of λ. Therefore, clearly an increase in λ has no effect on r∗ . 25

To establish the claim that an increase in θ raises c†a , consider any c1 in (cr , c†a ], Then, by condition (17), x(c1 , s1 , r) = max{ϕ(pm (c1 )), ϕ(s1 )} = ϕ(pm (c1 )) for such c1 . Let ∆ = ϕ(r)−ϕ(pm (c1 )). As can be seen in Figure 3, ∆ is strictly decreasing in c1 and c†a is a unique root of ∆. So, if I can show that an increase in θ raises the value of ∆ for any c1 in (cr , ca ], then the claim follows. The former condition holds if ϕ(r) increases more than ϕ(pm (c1 )) does by an increases in θ for any c1 in (c∗r , c†a ]. From (6), a direct calculation yields ∂ϕ(r) = βπ e (c1 )(λ + µ), ∂θ

∂ϕ(pm (c1 )) = βπ e (c1 )µ. ∂θ

Hence the stated condition holds, and the claim follows. These effects of an increase in λ on r∗ and a† causes a change in the price distribution F1′ in (11) so that the new distribution weakly lies above the original distribution for all p1 ; or F1′ (p1 ) ≤ F1′′ (p1 ) for all p1 where F1′′ indicates the new distribution. Applying Lemma 2, the other claim that increased λ decreases s∗1 instantly follows. Proof of Proposition 2. Lemma 7 implies F1∗∗ > F1∗ where F1∗∗ indicates the new equilibrium distribution for increased λ. Applying part (ii) of Lemma A1, the claim follows.

References Kyle Bagwell. Countercyclical Pricing in Customer Markets. Economica, 71:519–542, 2004. Mark Bils. Pricing in a Customer Market. Quarterly Journal of Economics, 104(4):699–717, 1989. Judith A. Chevalier, Anil K. Kashyap, and Peter E. Rossi. Why Don’t Prices Rise During Periods of Peak Demand? Evidence from Scanner Data. American Economic Review, 93(1):15–37, 2003. Meir G. Kohn and Steven Shavell. The Theory of Search. Journal of Economic Theory, 9 (2):93–123, 1974. Rajiv Lal and Carmen Matutes. Retail Pricing and Advertising Strategies. Journal of Business, 67(3):345–370, 1994. James M. MacDonald. Demand, Information and Competition: Why Do Food Prices Fall at Seasonal Demand Peaks? Journal of Industrial Economics, 48(1):27–45, 2000. J. J. McCall. Economics of Information and Job Search. Quarterly Journal of Economics, 84(1):113–126, 1970. 26

Aviv Nevo and Konstantinos Hatzitaskos. Why Does the Average Price of Tuna Fall During Lent? NBER Working Paper, 11572, 2005. Edmund S. Phelps and Sidney G. Winter, Jr. Optimal Price Policy under Atomistic Competition. In Edmund S. Phelps, editor, Microeconomic Foundations of Employment and Inflation Theory, pages 309–337. W. W. Norton, 1970. Jose M. Plehn-Dujowich. On the Counter-Cyclicality of Prices and Markups in a Cournot Model of Entry. Economics Letters, 99:310–313, 2008. Jennifer F. Reinganum. A Simple Model of Equilibrium Price Dispersion. Journal of Political Economy, 87(4):851–58, 1979. Julio J. Rotemberg and Garth Saloner. A Supergame-Theoretic Model of Price Wars During Booms. American Economic Review, 76(3):390–407, 1986. Elizabeth J. Warner and Robert B. Barsky. The Timing and Magnitude of Retail Store Markdowns: Evidence from Weekends and Holidays. Quarterly Journal of Economics, 110(2):321–52, 1995.

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Countercyclical Average Price in Customer Market

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