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Working Paper Series Equilibrium Price Dispersion Across and Within Stores WP 15-01 Guido Menzio University of Pennsylvania and NBER Nicholas Trachter Federal Reserve Bank of Richmond This paper can be downloaded without charge from: http://www.richmondfed.org/publications/ Equilibrium Price Dispersion Across and Within Stores Guido Menzio Nicholas Trachter University of Pennsylvania and NBER Federal Reserve Bank of Richmond January 2015 Working Paper No. 15-01 Abstract We develop a search-theoretic model of the product market that generates price dispersion across and within stores. Buyers di¤er with respect to their ability to shop around, both at di¤erent stores and at di¤erent times. The fact that some buyers can shop from only one seller while others can shop from multiple sellers causes price dispersion across stores. The fact that the buyers who can shop from multiple sellers are more likely to be able to shop at inconvenient times induces price dispersion within stores. Speci…cally, it causes sellers to post di¤erent prices for the same good at di¤erent times in order to discriminate between di¤erent types of buyers. JEL Codes: D43. Keywords: Search, Price dispersion, Price discrimination, Bargain hunting. Menzio: Department of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA 19013 (email: gmenzio@sas.upenn.edu). Trachter: Federal Reserve Bank of Richmond, 701 E. Byrd Street, Richmond, VA 23219 (email: nicholas.trachter@rich.frb.org). The views expressed in this paper are those of the authors and do not necessarily represent the views of the Federal Reserve Bank of Richmond or the Federal Reserve System. 1 1 Introduction It is a well-known fact that the same product is sold at very di¤erent prices, even when one restricts attention to sales taking place in the same geographical area and in the same narrow period of time. For instance, Sorensen (2000) …nds that the average standard deviation of the price posted by di¤erent pharmacies for the same drug in the same town in upstate New York is 22%. In a more systematic study of price dispersion that covers 1.4 million goods in 54 geographical markets within the United States, Kaplan and Menzio (2014b) …nd that the average standard deviation of the price at which the same product is sold within the same geographical area and the same quarter is 19%. Moreover, it appears that price dispersion is caused by both di¤erence in prices across di¤erent stores and di¤erence in prices within each store. For instance, Kaplan and Menzio (2014b) …nd that on average roughly half of the overall variance of prices for the same good in the same market and in the same quarter is due to the fact that di¤erent stores sell the same good at a di¤erent price on average, and the other half is due to the fact that the same store sells the same good at di¤erent prices in di¤erent transactions taking place during the same quarter. In this paper, we develop a search-theoretic model of price dispersion across and within stores by combining the standard theory of price dispersion of, e.g., Butters (1977) and Burdett and Judd (1983), and the standard theory of intertemporal price discrimination of, e.g., Conslik, Gerstner and Sobel (1984) and Sobel (1984). The resulting model o¤ers a tractable and uni…ed framework to study the extent and shape of price dispersion and its di¤erent causes.1 Speci…cally, we build a model of the market for an indivisible good. On the demand side, there are buyers who di¤er with respect to their ability to shop at di¤erent stores, as well as with respect to their ability to shop at di¤erent times: Some buyers can shop from only one seller while others can shop from multiple sellers, and some buyers can shop only in the daytime while others can shop both in the daytime and in the nighttime. On the 1 Besides intertemporal price discrimination, there are other explanations for why a seller would charge di¤erent prices for the same good within the same quarter. First, as in Sheshinski and Weiss (1974), Benabou (1988) or Burdett and Menzio (2013), a seller may change his nominal price during a quarter in order to keep up with the movements in the aggregate price level (more or less frequently depending on in‡ation and menu costs). Although, this theory of within-store price dispersion seems unlikely to be relevant in a low-in‡ation environment like the U.S. economy in the 2000s. Second, as suggested by Aguirregabiria (1999), a seller may change his price during a quarter in response to changes in his inventories of the good. Finally, as suggested by Menzio and Trachter (2014), a large seller may change his price over time in order to occasionally price out of the market a fringe of small sellers. 2 supply side, there are identical sellers and each seller posts a (potentially di¤erent) price for the good in the daytime and in the nighttime. We prove the existence and uniqueness of equilibrium. The equilibrium always features price variation across stores. Moreover, if the buyers who are able to shop at both times are— on average— also able to shop from more stores than the buyers who can only shop in the daytime, the equilibrium features price variation within stores. In particular, the equilibrium is such that some sellers post a strictly lower price in the nighttime than in the daytime. On the other hand, if the buyers who are able to shop at both times of day are— on average— less able to shop at multiple stores, the equilibrium features no price variation within stores. That is, sellers do not vary their price over time. Intuitively, price dispersion across stores arises because sellers meet some buyers who cannot purchase from any other store and some other buyers who can and— as explained in Butters (1977) and Burdett and Judd (1983)— this heterogeneity induces identical sellers to post di¤erent prices for the same good. Price dispersion within stores arises because— if the buyers who are more likely to be able to shop at nighttime are also more likely to be able to shop from multiple stores— a seller can compete more …ercely for these buyers without losing revenues on the other customers by charging a lower price at night than during the day. The paper’s main contribution is to combine in a uni…ed and simple search-theoretic framework the insights of the literature on price dispersion and the insights of the literature on intertemporal price discrimination. Compared to the classic papers on price dispersion, our model is richer because it introduces a time dimension and characterizing the equilibrium distribution of multidimensional price vectors across stores poses new technical challenges that are …rst tackled in this paper.2 Compared to the classic papers on intertemporal price discrimination, our model is richer as the analysis is carried out in an equilibrium framework that generates price dispersion across stores. Moreover, our model shows that the same element is su¢ cient to generate both equilibrium price dispersion and intertemporal price discrimination: Heterogeneity across buyers in their ability 2 In Butters (1977), Varian (1980), and Burdett and Judd (1983), each seller is indi¤erent between posting any price on the support of the equilibrium price distribution. Therefore, if sellers choose di¤erent prices on di¤erent days, these models would generate price dispersion both across and within stores. However, this result would not be robust to the introduction of menu costs (which would discourage sellers from resetting their prices if there is not a strictly positive bene…t from doing so) or to the introduction of heterogeneity in the sellers’cost of production (which would break the seller’s indi¤erence between any price on the support of the equilibrium distribution). 3 to shop around, at di¤erent locations and at di¤erent times of the day.3 The empirical evidence in Aguiar and Hurst (2007) and Kaplan and Menzio (2014a, 2014b) suggests that these traits are common to individuals with a relatively low value of time, such as the elderly and the unemployed. 2 Environment We consider the market for an indivisible good that operates in two periods: daytime and nighttime.4 On one side of the market, there is a measure 1 of identical sellers who can produce the good on demand at a constant marginal cost, which we normalize to zero. Each seller simultaneously and independently posts a pair of prices (pd ; pn ), where pd 2 [0; u] is the price of the good during the day, pn 2 [0; u] is the price of the good during the night, and u > 0 is the buyers’ valuation of the good. We denote as G(pd ; pn ) the distribution of prices across sellers. Similarly, we denote as Fd the marginal distribution of daytime prices and as Fn the marginal distribution of nighttime prices. Finally, we denote as Fm the marginal distribution of the lowest price of each seller. On the other side of the market, there is a measure a measure y x > 0 of buyers of type x and > 0 of buyers of type y. The two types of buyers di¤er with respect to their ability to shop from di¤erent sellers, as well as with respect of their ability to shop at di¤erent times of the day. In particular, a buyer of type x is in contact with only one seller with probability x sellers with probability 1 x. 2 (0; 1) and with multiple (for the sake of simplicity, two) The buyer observes both the morning and afternoon price of the sellers with whom he is contact. However, the buyer is able to shop from these sellers during both the daytime and the nighttime only with probability 1 3 x. With Existing theories of intertemporal price discrimination assume that buyers di¤er in their valuation of the good. In Conslik, Gerstner, and Sobel, (1984) there are high and low valuation buyers. In Sobel (1984), one type of buyer has a higher valuation and a higher discount factor than the other type. In Albrecht, Postel-Vinay and Vroman (2012), one type of buyer has a higher valuation and consumes the good faster than the other type. Generally, the elements that are needed for intertemporal price discrimination to be pro…table and feasible are that: (a) some buyers are willing to pay more for the good than others, and (b) these buyers are also less ‡exible in the timing of their purchases. In this paper we show that both of these elements follow from one common di¤erence: Some buyers are worse than others at shopping at di¤erent stores (which increases their expected willingness to pay) and at di¤erent times (which makes them less ‡exible in the timing of their purchases). 4 The reader should not interpret day and night literally. The key assumption is that some buyers are ‡exible with respect to their shopping time and others are not. Therefore, the reader can interpret the nighttime as Monday morning, as a particular day every week, or even as one particular day in any given month. Similarly, the reader can interpret the daytime as every time other than Monday morning or as every day of the week/month except for the one where sales are scheduled. 4 probability x 2 (0; 1), the buyer is able to shop from the contacted sellers only during the daytime. Similarly, a buyer of type y is in contact with one seller with probability and with multiple (two) sellers with probability 1 y. y A buyer of type y is able to shop from the contacted sellers during both the daytime and the nighttime with probability 1 y and only during the daytime with probability utility of u y. Both types of buyers enjoy a p if they purchase the good at the price p and a utility of zero if they do not purchase the good. Without loss in generality, we assume that buyers of type x are in contact with fewer sellers than buyers of type y, i.e. we assume x y. The de…nition of equilibrium for this model market is standard (see, e.g., Burdett and Judd, 1983 or Head et al 2012). De…nition 1: An equilibrium is a price distribution G such that the seller’s pro…t is maximized everywhere on the support of G. 3 Characterization of equilibrium In this section, we characterize the equilibrium set. We carry out the analysis in four steps. In Subsection 3.1, we show that we can restrict attention to equilibria in which every seller chooses a price for the good in the night that is non-greater than the price for the good in the day. In Subsection 3.2, we consider an equilibrium in which the pro…t of a seller attains its maximum for all daytime prices pd on the support of the marginal distribution Fd , as well as for all nighttime prices pn on the support of the marginal distribution Fn . We show that— if and only if buyers of type x are in contact with fewer sellers than buyers of type y, and they are less likely to be able to shop in the nighttime— this equilibrium exists and features price dispersion across and within sellers. In Subsection 3.3, we consider an equilibrium in which the pro…t of a seller attains its maximum for all prices (pd ; pn ) with pd = pn . We show that— if and only if buyers of type x are in contact with fewer sellers than buyers of type y, and they are more likely to shop in the nighttime— this equilibrium exists and features price dispersion across sellers but not within sellers. Finally, in Subsection 3.4, we rule out other types of equilibria. 3.1 A general property of equilibrium As a preliminary step, we show that we can restrict attention to equilibria in which sellers post prices (pd ; pn ) such that pn pd . This is the case because— by assumption— those 5 buyers who can shop in the nighttime can also shop in the daytime and, hence, a seller posting a higher price in the nighttime than in the daytime enjoys the same pro…t and exerts the same competition on other seller as if he were to post the same price at both times of day. To formalize the above argument, consider an equilibrium in which the marginal price distributions are continuous functions5 Fd , Fn , and Fm . A seller who posts prices (pd ; pn ) 2 [0; u]2 enjoys a pro…t V (pd ; pn ) where the constants and the constants 1d 2d =[ 1d + 2d (1 Fd (pd ))] pd +[ 1n + 2n (1 Fm (minfpd ; pn g))] minfpd ; pn g, 1n are de…ned as and and 1d = x x x 1n = x x (1 2n are de…ned as 2d = 2 x (1 x) x 2n = 2 x (1 x )(1 + y y x) + + 2 y (1 x) Let us brie‡y explain (1). The seller meets y; y y (1 1d (2) y ); y) y; + 2 y (1 (1) y )(1 y ): (3) buyers who are not in contact with any other seller and who can only shop in the daytime. Each one of these buyers will purchase the good from the seller at the price pd . The seller meets 1n buyers who are not in contact with any other seller and who can shop both in the daytime and in the nighttime. Each one of these buyers will purchase the good from the seller at the price minfpd ; pn g. The seller meets 2d buyers who are in contact with a second seller and who can only shop in the daytime. Each one of these buyers will purchase the good from the seller if pd is lower than the afternoon price posted by the second seller they contacted, an event that occurs with probability 1 Fd (pd ). Finally, the seller meets 2n buyers who are in contact with a second seller and can shop both in the daytime and in the nighttime. Each one of these buyers will purchase the good from the seller if minfpd ; pn g is lower than the lowest price posted by the second seller they meet, an event that occurs with probability Fm (minfpd ; pm g). 1 5 The assumption that the distribution functions Fd , Fn , and Fm are continuous is for the sake of exposition only. It is straightforward to generalize Lemma 1 to the case in which these distribution functions have mass points. 6 A seller posting the prices (pd ; pn ) 2 [0; u]2 with pn > pd enjoys a pro…t V (pd ; pn ) =[ 1d + 2d (1 Fd (pd ))] pd +[ 1n + 2n (1 Fm (pd ))] pd : (4) Notice that the seller’s pro…t does not depend on the nighttime price. Indeed, if pn > pd , the seller never makes a sale in the nighttime. The customers who can only shop in the daytime will purchase at the price pd . The customers who can shop both in the daytime and in the nighttime will choose to purchase during the day at the price pd . Therefore, the seller enjoys the same pro…t if he were to post the prices (pd ; pd ) rather than (pd ; pn ). Now, suppose that there is an equilibrium G in which some sellers post (pd ; pn ) with ^ in which the sellers posting (pd ; pn ) pn > pd . Consider an alternative price distribution G with pn > pd change their prices to (pd ; pd ), while the sellers posting (pd ; pn ) with pn pd keep their prices unchanged. Clearly, the marginal price distributions F^d and F^m ^ are the same as the marginal price distributions Fd and Fm associated associated with G with G. Since the prices (pd ; pn ) with pn pd maximize the pro…t of the seller given G, ^ as the pro…t function (1) only depends they also maximize the pro…t of the seller given G, on the marginals Fd and Fm . Moreover, since the prices (pd ; pn ) with pn > pd maximize ^ the pro…t of the seller given G, the prices (pd ; pd ) maximize the pro…t of the seller given G, as the seller’s pro…t function (1) only depends on the marginal Fd and Fm and, as shown in (4), the seller is indi¤erent between posting (pd ; pn ) and (pd ; pd ). Thus, the joint price ^ is an equilibrium and it is— along all relevant dimensions— equivalent to distribution G the equilibrium joint price distribution G: We have therefore established the following Lemma. Lemma 1. Without loss in generality, we can restrict attention to equilibria G in which every seller posts a price(pd ; pn ) 2 [0; u]2 with pn pd , and the marginal distribution of lowest prices, Fm , is equal to the marginal distribution of night prices, Fn . 3.2 Equilibrium with price dispersion across and within stores In this section, we look for an equilibrium G in which every seller posts prices (pd ; pn ) 2 [0; u]2 with pn pd and such that the marginal distribution of daytime prices, Fd , and the marginal distribution of nighttime prices, Fn , are respectively given by Fd (p) = 1 1d 2d u p p 7 , 8p 2 [pd` ; pdh ]; (5) and 1n Fn (p) = 1 u p p 2n , 8p 2 [pn` ; pnh ], (6) where the boundaries of the support of the distributions are pt` = 1t 1t + 2t u, for t = fd; ng (7) for t = fd; ng. pth = u; Given the marginal price distributions Fd and Fn in (5) and (6), we can identify the region where the pro…t of the seller attains its maximum. In general, a seller posting prices (pd ; pn ) 2 [0; u]2 with pn pd attains a pro…t of V (pd ; pn ) =[ +[ + 1n + 2d (1 1d 2n (1 Fd (pd ))] pd Fn (pn ))] pn . If the seller posts prices (pd; pn ) such that pd 2 [pd` ; pdh ], pn 2 [pn` ; pnh ] and pn (8) pd , his pro…t is given by V (pd ; pn ) = [ 1d + 1n ] u; (9) where (9) follows from (8) and from the expressions for the marginal price distributions Fd and Fn in (5) and (6). Notice that (9) is a constant, i.e., the seller’s pro…t attains the same value for all prices pd on the support of the marginal distribution Fd and for all prices pn on the support of the marginal price distribution Fn . Moreover, this pro…t is equal to the pro…t that the seller would attain if he were to charge the buyer’s reservation price u both in the daytime and in the nighttime and sell only to those buyers who are not in contact with any other seller. If the seller post prices (pd; pn ) such that pd 2 [pd` ; pdh ], pn 2 [0; pn` ) and pn pd , his pro…t is given by V (pd ; pn ) = <[ 1d u + ( 1n + 1d + 1n ] u; 2n )pn (10) where the …rst line on the right-hand side of (10) follows from (8) and the fact that Fn (pn ) = 0 for all pn pn < pn` and pn` = u pn 2 [0; pn` ), and pn pn` , and the second line on the right-hand side of (10) follows from 1n =( 1n + 2n ). Therefore, for any (pd ; pn ) such that pd 2 [pd` ; pdh ], pd , the pro…t of the seller is lower than in (9). This result is intuitive as lowering the price pn below pn` reduces the pro…t per sale without increasing the probability of making a sale to a night shopper. Similarly, for any (pd ; pn ) such that pd 2 [0; pd` ), pn 2 [0; pnh ], and pn pd , the pro…t of the seller is lower than in 8 (9), as lowering the price pd below pd` reduces the pro…t per sale without increasing the probability of making a sale to a daytime shopper. Finally, as established in section 3.1, the seller is indi¤erent between posting the prices (pd ; pn ) with pn > pd and the prices (pd ; pd ). Taken together, the above observations imply that the seller’s pro…t attains its maximum for all prices (pd ; pn ) such that pd 2 [pd` ; pdh ], pn 2 [pn` ; pnh ], and pn attains strictly less than the maximum for all other prices such that pn pd , and it pd . Therefore, an equilibrium such that all sellers post a nighttime price lower than the daytime price and where the marginal price distributions Fd and Fn are given as in (5) and (6) exists if and only if we can …nd a joint price distribution G such that: (a) the support of G lies in the region of prices (pd ; pn ) with pd 2 [pd` ; pdh ], pn 2 [pn` ; pnh ], and pn pd ; (b) the joint price distribution G generates the marginals Fd and Fn . Clearly, a necessary condition for the existence of the desired equilibrium is that the marginal distribution of prices in the daytime …rst order stochastically dominates the marginal distribution of prices in the nighttime, i.e. Fd (p) (5) and (6), it follows that Fd (p) the condition Fd (p) Fn (p) for all p 2 [0; u]. From Fn (p) is equivalent to Fn (p) or, equivalently, 1d = 2d 1d = 2d 1n = 2n 1n = 2n . Moreover, is also su¢ cient for the existence of the desired equilibrium. To see why this is the case, suppose that the distribution of sellers over night prices is the Fn in (6) and a seller who posts pn in the nighttime posts the price g(pn ) in the daytime, where 1 g(pn ) = 1n u+ 2n 1d 1n 2d 2n pn 1d (11) pn : 2d Given that sellers post (g(pn ); pn ), it is immediate to verify that the marginal distribution of prices in the afternoon is the Fd in (5). Moreover, if 1d = 2d 1n = 2n , it is easy to verify that the support of the joint price distribution G lies in the region pd 2 [pd` ; u], pn 2 [pn` ; u], and pn pd . Overall, the necessary and su¢ cient condition for the existence of the desired equilibrium is 1d 1n 2d 2n : (12) In words, the necessary and su¢ cient condition (12) states that the ratio of captive buyers— i.e. buyers who are in contact with a particular seller and nobody else— to noncaptive buyers— i.e. buyers who are in contact with a particular seller and a second 9 one— must be greater in the day than at night. In what follows, we vary the parameters of the model ( x; x; y; y) and verify whether condition (12) is satis…ed. Case 1: Buyers of type x are in contact with fewer sellers than buyers of type y and are less likely to shop at night, i.e. it is straightforward to verify that > x x > y y and and x > x y. > y Using (2)-(3) and (5)-(6), imply 1d = 2d > 1n = 2n and Fd < Fn . Since condition (12) is satis…ed, there exists a joint price distribution G whose support lies on the required region and that generates the marginals Fd and Fn in (5) and (6). Moreover, since Fd < Fn , any such joint price distribution G must be such that a positive measure of sellers posts a strictly lower price at night than during the day. Hence, the equilibrium features price dispersion both across stores— in the sense that the marginal price distributions Fd and Fn are non-degenerate— and within stores— in the sense that a positive measure of sellers sells the good at di¤erent prices during di¤erent times of the day. As in Butters (1977) and Burdett and Judd (1983), price dispersion across stores emerges because the equilibrium price distribution makes sellers indi¤erent between posting a high price, enjoying a high pro…t margin, and selling a small quantity of the good and posting a low price, enjoying a low pro…t margin, and selling a large quantity of the good. As in Conslik, Gerstner, and Sobel (1984), price dispersion within stores emerges because, when x > y and x y, > sellers have the incentive and the opportunity to price discriminate between di¤erent types of buyers. Indeed, since the two types of buyers di¤er in their likelihood to shop at night, sellers face a di¤erent composition of buyers in the two times of the day. Moreover, since the type of buyer who is more likely to shop at night is also the type of buyer who is in contact with more sellers, sellers face more competition at night. As a result, sellers …nd it optimal to post lower prices— in the sense of …rst order stochastic dominance— at night than during the day. Case 2: Buyers of type x are in contact with fewer sellers than buyers of type y and they are more likely to shop at night, i.e., can verify that x > y and x < x y > y and imply x 1d = 2d < < y. Using (2)-(3) and (5)-(6), one 1n = 2n . Since condition (12) is violated, there exists no joint price distribution G whose support is on the required region and that generates the marginals Fd and Fn in (5) and (6). Let us explain this result. Since the two types of buyers di¤er with respect to their ability to shop at night, sellers face a di¤erent composition of buyers in the two times of the day. Moreover, since the type of buyer who is more likely to shop at night is in contact with fewer sellers, sellers 10 face less competition at night. Hence, sellers would like to post higher prices at night than during the day but this would induce. But this is not compatible with equilibrium, as it would induce buyers who can shop at night to visit the sellers during the day. In between cases 1 and 2, there are two knife-edge cases. Case 3: Buyers of type x are in contact with fewer sellers than buyers of type y, but are equally likely to shop at night, i.e. straightforward to verify that x y x and y and x = y turn, using (5) and (6), it is immediate to verify that Since 1d = 2d = 1n = 2n , x = y. Using (2) and (3), it is imply that 1d = 2d = 1d = 2d 1n = 2n = 1n = 2n . In implies Fd = Fn . we know that there exists a joint price distribution G whose support lies in the required region and that generates the marginals Fd and Fn . Moreover, since Fd = Fn and all sellers must post a nighttime price non-smaller than their daytime price, the only equilibrium G is the one where every seller posts the same price at both times of day. Hence, while the equilibrium features price dispersion across sellers, it does not feature price dispersion within sellers. This result is intuitive. Since the two types of buyers are equally likely to shop at night, sellers face the same composition of buyers and, hence, the same amount of competition in the daytime and in the nighttime. For this reason, the equilibrium marginal price distribution is the same in the two times of day. And since sellers post a lower price at night than during the day, this implies that every individual seller must always post the same price. Case 4: Buyers of type x are in contact with the same number of sellers as buyers of type y, but they are less likely to shop at night, i.e., x = is straightforward to verify that y imply x = y and x > y and x 1d = 2d > = y. Again, it 1n = 2n and Fd = Fn , which, in turn, implies that every seller posts the same price during the daytime and the nighttime. This result is also intuitive. Since the two types of buyers di¤er with respect to their ability to shop at night, a seller faces a di¤erent composition of buyers in the daytime and in the nighttime. However, since the two types of buyers are in contact with the same number of sellers, this di¤erence in composition does not translate into a di¤erence in competition. As a result, the equilibrium marginal price distribution during the day is the same as during the night, and every individual seller must always post the same price. The above analysis is summarized in Proposition 1. Proposition 1. An equilibrium G in which sellers post prices (pd ; pn ) 2 [0; u]2 with pn pd and such that the marginal price distributions Fd and Fn are as in (5) and (6) 11 exists if and only if x = y or y. x If x > y and price dispersion across and within sellers. If either x x = y, > or y the equilibrium features x = y, the equilibrium features price dispersion across sellers but not within sellers. 3.3 Equilibrium without within-store price dispersion In this section, we look for an equilibrium in which the joint price distribution, G, is such that every seller posts the same price for the good during the day and during the night and such that the marginal distribution of daytime and nighttime prices is given by + 2n + 1n Fd (p) = Fn (p) = 1 1d u p p 2d , 8p 2 [p` ; ph ], (13) where the boundaries of the support of the distributions are + 1n + 1d + 1n p` = 1d 2n + u; (14) ph = u: 2d First, consider a seller posting the prices (p; p) with p 2 [p` ; ph ]. This seller obtains a pro…t of V (p; p) = =[ 1d 1d + + 1n +( 2d + 2n ) + 2d + 1d 1n 2n u p p p (15) 1n ] u. The …rst line on the right-hand side of (15) follows from (8) and the expression for the marginal price distributions Fd and Fn in (13). The second line on the right-hand side of (15) follows from algebraic manipulation of the …rst. Notice that the second line on the right-hand side of (15) is a constant. That is, the seller attains the same pro…t by posting any prices (p; p) on the support of the joint distribution G. Moreover, this pro…t is equal to the pro…t that the seller would attain if he were to charge the buyer’s reservation price u both in the daytime and in the nighttime and sell only to those buyers who are not in contact with any other seller. pn Second, consider a seller posting prices (pd ; pn ), with pd 2 [p` ; ph ], pn 2 [p` ; ph ], and pd . This seller obtains a pro…t of V (pd ; pn ) = + 1d 1n + + + 2d + 1d 2d + 2d + 1d 2n 12 1n u pd 2n 1n 2n pd u pn pn pd (16) pn : Notice that the derivative of the seller’s pro…t with respect to the nighttime price, pn , is strictly positive if 1n = 2n and pn pn > 1d = 2d . 1n = 2n 1d = 2d ; < it is zero if 1n = 2n = 1d = 2d ; and it is negative if Hence, if the seller posts prices (pd ; pn ) with pd 2 [p` ; ph ], pn 2 [p` ; ph ) pd , he attains a pro…t non-greater than (15) if and only if 1n = 2n 1d = 2d . Third, consider a seller posting prices (pd ; pn ), with pd 2 [p` ; ph ], pn 2 [0; p` ], and pd . This seller’s pro…t is lower than what he could attain by posting the prices (pd ; p` ), as lowering the price pn below p` reduces the pro…t per sale without increasing the probability of making a sale to a night shopper. Similarly, for any (pd ; pn ) such that pd 2 [0; p` ), pn 2 [0; ph ], and pn pd , the seller’s pro…t is lower than what he could attain by posting the prices (p` ; pn ), as lowering the price pd below p` reduces the pro…t per sale without the probability of making a sale to a day shopper. Finally, as established in section 3.1, the seller is indi¤erent between posting the prices (pd ; pn ) with pn > pd and the prices (pd ; pd ). From the above observations, it follows that the seller’s pro…t is maximized everywhere on the support of the joint price distribution G if and only if 1n 1d 2n 2d (17) : In words, the necessary and su¢ cient condition (17) states that the ratio of captive buyers to non-captive buyers must be greater at night than during the day. Notice that condition (17) is the opposite as condition (12) and, hence, for any values of the parameters, there exists either the type of equilibrium studied in Subsection 3.2 or the type of equilibrium studied in this subsection. Moreover, the two types of equilibria coexist only when 1n = 2n = 1d = 2d , which is a knife-edge con…guration of parameters. In particular, we have the following cases. Case 1 Buyers of type x are in contact with fewer sellers than buyers of type y and are less likely to shop at night, i.e., x > y and x > y. When x > y and x > y, condition (17) is violated and, hence, there is no equilibrium in which all sellers post the same price at both times of day, and the marginal price distributions Fd and Fn are given by (13). Intuitively, when x > y and x > y, sellers face more competition at night than during the day. For this reason, sellers have an incentive to post lower prices— in the sense of …rst order stochastic dominance— at night than during the day. Case 2: Buyers of type x are in contact with fewer sellers than buyers of type y and they 13 are more likely to shop at night, i.e., x > y and x y. < When x > and y x < y, condition (17) is satis…ed and, hence, there exists an equilibrium in which all sellers post the same price at both times of day, and the marginal price distributions Fd and Fn are given by (13). In this equilibrium, there is price dispersion across stores— in the sense that di¤erent sellers post di¤erent prices— but no price dispersion within stores— in the sense that every seller posts the same price at all times. Intuitively, when x < y, > x y and sellers face more competition during the day than at night. For this reason, sellers want to post a nighttime price as high as possible. However, sellers cannot post a nighttime price higher than the daytime price or, else, buyers who can shop at night will purchase the good during the day. As a result, sellers post a nighttime price equal to the daytime price. In between cases 1 and 2, there are two knife-edge cases. In these cases, the type of equilibrium that we considered in Subsection 3.2 and the type of equilibrium that we are considering here coexist and coincide. Case 3: Buyers of type x are in contact with fewer sellers than buyers of type y, but are equally likely to shop at night, i.e., x y and x = y. In this case, condition (17) holds with equality. Therefore, there exists an equilibrium in which sellers post the same price during the day and during the night, and the marginal price distributions Fd and Fn are given as in (13). Intuitively, when x and y x y, = sellers face the same composition of buyers during the day and during the night and, hence, they have no incentive to vary their price over time. Notice that, when x y and x = y, condition (12) holds as well and, hence, there exists also an equilibrium in which the marginal price distributions Fd and Fn are given as in (5) and (6). However, as discussed in the previous subsection, this equilibrium is also such that sellers post the same price in the two periods. Moreover, it is immediate to see that the marginal price distributions Fd and Fn in (5) and (6) are the same as in (13). Hence, the two types of equilibria coexist and are identical. Case 4: Buyers of type x are in contact with the same number of sellers as buyers of type y, but they are less likely to shop at night, i.e., x = y and x y. In this case, condition (17) holds with equality. Therefore, there exists an equilibrium in which sellers post the same price during the day and during the night, and the marginal price distributions Fd and Fn are given as in (13). Intuitively, when x = y and x y, sellers face a di¤erent composition of buyers during the day and during the night but this 14 di¤erence in composition does not translate into a di¤erence in competition because both types of buyers are in contact with the same number of sellers. For this reason, sellers have no incentive to vary their price over time. Notice that, also when y, x x = y and this equilibrium coexists and coincides with the one studied in Subsection 3.2. The above analysis is summarized in Proposition 2. Proposition 2. An equilibrium G in which all sellers post the same price in the morning and in the afternoon and in which the marginal price distributions Fd and Fn are given as in (13) exists if and only if 3.4 x = y or y. x Other equilibria The …nal step of the analysis is to rule out the existence of any type of equilibrium di¤erent from those studied in Subsections 3.2 and 3.3. To this aim, consider an equilibrium distribution of sellers over prices, G(pd ; pn ). Let Fd (pd ) denote the marginal distribution of sellers over daytime prices and as md (pd ) the measure of sellers who post a daytime price of pd , i.e., the mass point associated with the price pd . Similarly, let Fn (pn ) denote the marginal distribution of sellers over nighttime prices and as mn (pn ) the measure of sellers who post a morning price of pn . In light of Lemma 1, we can restrict attention to equilibria in which all sellers post a price pn pd and, consequently, such that the marginal distribution of sellers over their lowest price, Fm , is equal to Fn . In equilibrium, a seller posting prices (pd ; pn ) with pn pd attains a pro…t of (18) V (pd ; pn ) = Vd (pd ) + Vn (pn ); where Vd and Vn are respectively de…ned as Vd (pd ) = 1d + 2d 1 1 Fd (pd ) + md (pd ) 2 pd ; (19) Vn (pn ) = 1n + 2n 1 1 Fn (pn ) + mn (pn ) 2 pn : (20) and In words, Vd (pd ) denotes the seller’s pro…t from daytime trades. In fact, in the daytime, the seller meets 1d captive buyers and 2d non-captive buyers. A captive buyer purchases the good from the seller with probability one. A non-captive buyer purchases the good from the seller with probability one if he is in contact with a second seller whose price is strictly 15 greater than pd , an event that occurs with probability 1 Fd (pd ) or with probability 1=2 if he is in contact with a second seller whose price is equal to pd , an event that occurs with probability md (pd ). Similarly, Vn (pn ) denotes the seller’s pro…t from nighttime trades. Every price pair (pd ; pn ) on the support of the distribution G must maximize the pro…t V (pd ; pn ) of the seller. We use this property to establish several features of the equilibrium. Claim 1. The marginal price distributions Fd and Fn have no mass points. Proof : We begin by proving that Fd has no mass points. On the way to a contradiction, suppose that there exists an equilibrium G in which Fd has a mass point at pd . Consider a seller posting the prices (pd ; pn ) with pn < pd . From (18), it follows that this seller can attain a strictly higher pro…t by posting the prices (pd ; pn ) for some > 0 su¢ ciently small. Hence, no prices (pd ; pn ) with pn < pd can be on the support of G. Next, consider a seller posting prices (pd ; pd ). From (18), it follows that this seller can attain a strictly higher pro…t by choosing the prices (pd ; pd ) for some > 0 su¢ ciently small. Hence, the prices (pd ; pd ) cannot be on the support of G. Finally, since G is such that every seller posts a price pn smaller than pd , no prices (pd ; pn ) with pn > pd can be on the support of G. We have thus reached a contradiction. The proof that Fn has no mass points is analogous. Claim 2. The marginal price distribution Fd has no gaps and pdh = u. Proof : We …rst establish that Fd has no gaps. On the way to a contradiction, suppose that Fd has a gap between p0 and p1 with p1 > p0 . Since Fd (p1 ) = Fd (p0 ), a seller posting prices (p0 ; pn ) with pn p0 can attain a strictly higher pro…t by choosing the prices (p1 ; pn ) instead. Hence, the prices (p0 ; pn ) with pn p0 cannot be on the support of G. Similarly, since G is such that every seller posts a price pn smaller than pd , no prices (p0 ; pn ) with pn > p0 can be on the support of G. We have thus reached a contradiction. The proof that pdh = u is analogous. Claim 3. Let pn` be the lower bound of the support of the marginal price distribution Fn . The pro…t function Vn (pn ) is weakly increasing in pn over the interval [pn` ; u]. Proof : On the way to a contradiction, suppose Vn (pn ) is strictly decreasing over the interval (p0 ; p1 ), with pn` p0 < p 1 u. If this is the case, Vn (pn ) < Vn (p0 ) for all pn 2 (p0 ; p2 ) where p2 > p1 . Any seller with a daytime price pd nighttime price pn such that pn p2 , will choose a pd and pn 2 = (p0 ; p2 ). Any seller with a daytime price 16 pd 2 (p0 ; p2 ) will choose a nighttime price pn pd p0 . And any seller with a daytime price p0 will choose a nighttime price smaller than pd . Therefore, the marginal price distribution Fn has a gap between p0 and p2 , i.e., Fn (pn ) = Fn (p0 ) for all p 2 (p0 ; p2 ). From (20), it follows that if Fn is constant over the interval (p0 ; p2 ), then Vn (p) is strictly increasing over the interval (p0 ; p2 ), which contradicts the assumption that Vn (p) is strictly decreasing over the interval (p0 ; p1 ). Claim 4. The function Vn (pn ) is either strictly increasing for all pn 2 [pn` ; u], or it is constant for all pn 2 [pn` ; u]. Proof : Suppose Vn (pn ) is strictly increasing over some region (p0 ; p1 ), where pn` p1 u. This implies that a seller posting a daytime price pd price pn p0 < p1 chooses a nighttime p1 . A seller posting a daytime price pd 2 (p0 ; p1 ) chooses a nighttime price pn = pd . And a seller posting a daytime price pd p0 must post a nighttime price pn p0 . Therefore, for all p 2 (p0 ; p1 ), the fraction of sellers with a nighttime price smaller than p is equal to the fraction of sellers with a daytime price smaller than p, i.e., Fn (p) = Fd (p) for all p 2 (p0 ; p1 ). Using this fact and V (p; p) = V (p1 ; p1 ) for all p 2 (p0 ; p1 ), we obtain 1n Fn (p) = Fn (p1 ) + 1d +( + 2n + 2n 2d )(1 Fn (p1 )) p1 p p 2d , 8p 2 (p0 ; p1 ): (21) Given the expression for Fn in (21), we can compute the derivative of the function Vn (pn ), which is given by Vn0 (pn ) = + 2n + 1n 1n 2n The derivative is strictly positive if and only if 1d = 2d , 1d 2d , 8p 2 (p0 ; p1 ): 1n = 2n > 1d = 2d . (22) Thus, if 1n = 2n the function Vn (pn ) cannot be strictly increasing over the region (p0 ; p1 ) and, in light of Claim 3, it must be constant for all p 2 [pn` ; u]. Conversely, suppose Vn (pn ) is constant over some region (p0 ; p1 ) with p0 pn` . In this case, we can prove that the function 1n = 2n 1d = 2d . Thus, if 1n = 2n > 1d = 2d , Vn (pn ) cannot be constant over some region (p0 ; p1 ) and, in light of Claim 3, it must be strictly increasing for all p 2 [pn` ; u]. Now, suppose that the equilibrium is such that Vn (pn ) is constant over the interval [pn` ; u]. In this case, it is straightforward to verify that the marginal distribution of nighttime prices, Fn , is given as in (5). Moreover, since Vn (pn ) is constant, the function Vd (pd ) must also be constant over the interval [pd` ; u]. It is also straightforward to verify that this implies that the marginal distribution of daytime prices, Fd ; is given as in (6). 17 Thus, the only equilibrium with a constant Vn (pn ) is the one characterized in Subsection 3.2. Next, suppose that the equilibrium is such that the function Vn (pn ) is strictly increasing over the interval [pn` ; u]. In this case, every seller posts the same price in the morning and in the afternoon and the marginal price distributions Fd and Fn are identical. In turn, this implies that the marginal price distributions Fd and Fn are given as in (13). Thus, the only equilibrium with a strictly increasing Vn (pn ) is the one characterized in Subsection 3.3. Thus, we have established the following result. Proposition 3. Any equilibrium G is such that either: (i) the marginal price distributions Fd and Fn are given as in (6) and (7); or (ii) the marginal price distributions Fd and Fn are given as in (13). 4 Conclusions We developed a search-theoretic framework that generates equilibrium price dispersion across sellers and within sellers. Price dispersion across sellers is obtained because the buyers are heterogeneous in their ability to shop at di¤erent stores. Price dispersion within sellers is obtained when the buyers who are better at shopping at di¤erent stores are also better at shopping at less di¤erent times and, hence, sellers can discriminate between di¤erent types of buyers by varying their price over time. Our model is simpler and its predictions richer than standard models of intertemporal price discrimination. Our model could be estimated by extending the econometric techniques developed by Hong and Shum (2006) and Moraga-Gonzales and Wildenbeest (2009). References [1] Aguiar, M., and E. Hurst. 2007. “Life-Cycle Prices and Production.”American Economic Review, 97: 1533-1559. [2] Aguirregabiria, V. 1999. “The Dynamics of Markups and Inventories in Retailing Firms.”Review of Economic Studies, 66: 275-308. 18 [3] Albrecht, J., Postel-Vinay, F., and S. Vroman. 2013. “An Equilibrium Search Model of Synchronized Sales.”International Economic Review, 54: 472-493. [4] Benabou R. 1988. “Search, Price Setting and Inzation.”Review of Economic Studies, 55: 353-376. [5] Burdett, K., and K. Judd. 1983. “Equilibrium Price Dispersion.”Econometrica 955969. [6] Burdett, K., and G. Menzio. 2014. “The (Q; S; s) Pricing Rule.” NBER Working Paper 19094. [7] Butters, G. 1977. “Equilibrium Distributions of Sales and Advertising Prices.”Review of Economic Studies, 44: 465-491. [8] Conlisk, J, Gerstner, E., and J. Sobel. 1984. “Cyclical Pricing by a Durable Goods Monopolist.”Quarterly Journal of Economics, 99: 489-505. [9] Head, A., Liu, L., Menzio, G, and R. Wright. 2012. “Sticky Prices: A New Monetarist Approach.”Journal of the European Economic Association, 10: 939-973. [10] Hong, J., and M. Shum. 2006. “Using Price Distributions to Estimate Search Costs.” Rand Journal of Economics, 37: 257-275. [11] Kaplan, G., and G. Menzio. 2014a. “Shopping Externalities and Self-Ful…lling Unemployment Fluctuations.” Journal of Political Economy. Forthcoming. [12] Kaplan, G., and G. Menzio. 2014b. “The Morphology of Price Dispersion.”International Economic Review. Forthcoming. [13] Menzio, G., and N. Trachter. 2014. “Equilibrium Price Dispersion with Sequential Search.”NBER Working Paper 19990. [14] Moraga-Gonzales, J., and M. Wildenbeest. 2008. “Maximum Likelihood Estimation of Search Costs.”European Economic Review, 52: 820-848. [15] Sheshinski, E., and Y. Weiss. 1982. “In‡ation and Costs of Price Adjustment.”Review of Economic Studies, 44: 287-303. [16] Sobel, J. 1984. “The Timing of Sales.”Review of Economic Studies, 51: 353-368. 19 [17] Sorensen, A. 2000. “Equilibrium Price Dispersion in Retail Markets for Prescription Drugs.”Journal of Political Economy, 108: 833-850. [18] Varian, H. 1980. “A Model of Sales.”American Economic Review, 70: 651-659. 20
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