Retail Sales: A Study of Pricing Behavior in Supermarkets Author(s): MartinPesendorfer Reviewed work(s): Source: The Journal of Business, Vol.

75, No. 1 (January 2002), pp. 33-66 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/10.1086/323504 . Accessed: 26/06/2012 06:09
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Martin Pesendorfer
Yale University

Retail Sales: A Study of Pricing Behavior in Supermarkets*

I.

Introduction

Periodic price reductions, or sales, constitute a widely observed phenomenon in retailing. Sales occur on a regular basis, which suggests that they are not entirely due to random variations such as shocks to inventory holdings or demand. In recent years the frequency of periodic price reductions has increased, indicating that the sales phenomenon has become more important for retailers and consumers. This article examines a particular market in which price reductions occur, namely, the market for ketchup products in supermarkets in Springeld, Missouri, between 1986 and 1988. The data consist of daily shelf prices and quantities purchased for four product categories in 80% of all supermarkets in the region. The behavior of consumers and retailers involved in this market is described. The data analysis reveals the following about ketchup sales: ketchup prices stay at high levels for a number of time periods followed by a short time period
* I wish to thank the editor, Pradeep Chintagunta, and a referee for helpful comments. Dirk Bergemann, Ariel Pakes, Peter Rossi, and seminar audiences at a number of institutions provided helpful discussions. I am also grateful to the Institut dAnalisi Eco` nomica, Universitat Autonoma de Barcelona, for their hospitality ` ` during a visit and to the marketing group at the Graduate School of Business at the University of Chicago for making the data available. I wish to thank the National Science Foundation for nancial support under grant SBR-9811134. I can be contacted at martin.pesendorfer@yale.edu. The data can be accessed at http:// gsbwww.uchicago.edu/research/mkt/Databases/ERIM/ERIM.html. (Journal of Business, 2002, vol. 75, no. 1) 2002 by The University of Chicago. All rights reserved. 0021-9398/2002/7501-0002$10.00 33

This article examines temporary price reductions, or sales, on ketchup products in supermarkets in Springeld, Missouri, between 1986 and 1988. The descriptive data analysis indicates that intertemporal demand effects are present. A model of intertemporal pricing in which demand increases with the number of time periods since the last sale is considered and confronted with the data. The estimates indicate that demand increases in the time elapsed since the last sale. The timing of ketchup sales is well explained by the number of time periods since the last sale. Also, competition between retailers for accumulated shoppers inuences the sale decision.

34

Journal of Business

of low prices. Demand during periods of low prices is on average about seven times higher than it is during periods of high prices. Intertemporal effects in demand appear strong. Demand is signicantly higher if previous prices were high than if they were low. I consider a model of intertemporal pricing. The model assumes that demand at low prices accumulates in the time elapsed since the last sale. The optimal decision to conduct a sale involves randomization. The predicted probability of a sale increases in the time elapsed since the last sale. The data support the model. Demand during a sale period increases in the duration since the last sale in the store. The decision to conduct a sale is well explained by the time elapsed since the last sale in that and other stores. In addition, the sale decision is affected by competition between retailers for accumulated consumers who are shoppers. In the literature, different explanations for retail sales and price dispersion have been offered. Varian (1980) formulates a model in which there are consumers who are informed about prices and uninformed consumers. Retailers randomly choose prices every period, and informed consumers purchase from the retailer offering the lowest price. An implication of this model is that prices are not predictable and, thus, not correlated over time. Conlisk, Gerstner, and Sobel (1984) and Sobel (1991) study intertemporal pricing decisions of a durable goods monopolist with a constant inow of consumers. They consider strategic behavior by consumers who may purchase later if they expect the price to fall. The policy of the monopolist is to start at a high price, selling only to high-valuation consumers, then gradually lower the price over time until low-valuation consumers are willing to purchase. After a sale has occurred the cycle starts over. These gradual declines in prices differ from the observed price paths in my data, in which prices remain at high levels for extended periods of time followed by a sudden price cut. Sobel (1984) considers competition between retailers for lows. He assumes that only lowvaluation consumers behave strategically. High-valuation consumers do not discount the future. The resulting equilibrium price path consists of sudden price cuts similar to the observed price path in the data. There is little empirical literature on retail sales. Villas-Boas (1995) provides empirical evidence supporting the Varian model for the coffee and saltine crackers sold in supermarkets in Kansas City during the mid-1980s. Warner and Barsky (1995) document that sales for consumer appliances occur in periods of high demand, on weekends and holidays. Slade (1998, 1999) and Aguirregabiria (1999) present evidence showing that retail price reductions depend on intertemporal considerations. According to Slade, demand for crackers depends on a stock of goodwill that accumulates (or erodes) when a rm charges low (or high) prices. The stock of goodwill combined with menu costs explains the existence of price rigidities and temporary price reductions. In Aguirregabirias work, the effects of inventory decisions on price are studied. Price reductions arise more frequently immediately following a new inventory order. In contrast to the empirical literature, this article emphasizes the role of

Retail Sales

35

intertemporal demand effects in explaining the occurrence of sales. Ketchup is storable for extended periods of time, which suggests that consumers may be willing to wait for a price reduction. I examine the demand accumulation effect, in which demand at low prices accumulates in the duration since the last sale, and consider its inuence on pricing decisions. Section III describes the empirical evidence on sales. I document that sales do not arise at the same time across stores. This suggests that wholesale price variations do not cause the adoption of sales entirely. Intertemporal effects in demand appear important. Demand during periods of low prices depends positively on previous prices. If the past weeks prices were high, then demand is signicantly higher than if the past weeks prices were low. During periods of high prices, demand is also affected by previous prices but to a lesser extent. In addition, consumers who purchase at low prices visit competing retail stores frequently, which suggests that sales may be affected by competition between retail stores. Section IV describes a simple model of demand accumulation in which a xed number of consumers enter every period. In contrast to the durable goods literature, I do not consider strategic behavior by consumers. I assume that low-valuation consumers purchase the product as soon as it becomes affordable for them. I distinguish between two stocks of low-valuation consumers: store-loyal consumers and shoppers. Loyal consumers shop at one store only. Shoppers purchase at different stores. Retailers face different wholesale prices over time and decide when to hold a sale. Equilibria of the model are described for two special cases: rst, when there are no shoppers and the retailer is a monopolist; second, when there are no store-loyal lows and retailers compete for lows that are shoppers. The model predicts that the equilibrium decision to hold a sale is a function of the duration since the last sale in that store and the duration since the last sale in other stores. The implied price path consists of an extended period of high prices followed by a short period of low prices. The predicted price path is in accordance with the data. In Section V the implications of the theoretical model are confronted with the data. The stores decision to conduct a sale is estimated as a function of the duration since its last sale and the duration in other stores. The model ts the data well. The effects of the duration variables are signicant and have the predicted sign. The evidence suggests that demand accumulation is important in the decision to conduct a sale. In addition, competition between retailers for the stock of shoppers is a relevant factor for the timing of sales. The determinants of demand during a sale period are examined. I nd that demand increases in the duration since the last sale, which is in accordance with the model. The organization of the article is as follows. In Section II the market and the data are described. Demand and supply decisions are characterized in Section III, and it is documented that there are intertemporal linkages in both supply and demand. In Section IV, I present a model of intertemporal pricing

36

Journal of Business

decisions. The optimal sale decision is studied when demand accumulates in the duration since the last sale. Predictions of the model are discussed. In Section V, I examine the retailers decision to hold a sale and estimate demand during a sale period. In Section VI conclusions are given. II. Market

This article examines the pricing decisions of supermarkets in Springeld, Missouri, between 1986 and 1988. The data were collected by the Nielson marketing research company. The data consist of daily shelf prices of selected products for 80% of the grocery and drug retail stores in this region. Thus, the unique advantage of the data is the relatively complete coverage of competing chains in the same market. The list of products includes ketchup, detergent, yogurt, and soup products. In addition, the purchase behavior of a sample of 1,500 households is recorded. For each day the price and quantity purchased of the selected products is listed. The daily price data are constructed from data on the purchase behavior of households. On some days no purchase for a product is recorded and the price observation is missing. In the price data, missing observations have been lled by using adjacent purchase observations and by comparing price data across supermarkets within the same chain. Although a comparison between the actual and constructed daily price series revealed no apparent problem, this article uses mainly weekly price data. The market consists of consumers, retailers, wholesalers, and manufacturers. Manufacturers sell their products to retailers through wholesalers. The data do not contain information on the wholesale market. In particular, wholesale prices are not observed. After talking to several wholesale distributors and retailers in this area, I can describe the interaction among market participants as follows. There are a number of wholesalers supplying a variety of products to supermarkets. Retailers purchase directly from wholesalers at a prespecied price and distribute products to their retail outlets. There are no quantity restrictions in the wholesale market. Periodically, manufacturers offer their product at lower prices. Price reductions for ketchup, so-called trade promotions, are off invoice, meaning that the retailer can buy an unrestricted quantity at the lower price. According to conversations with wholesalers, the price in the wholesale market changes infrequently. After a price change, or trade promotion, occurs, the wholesale price remains at this level for about 3 months. A number of contributions to the marketing literature, including Blattberg and Neslin (1989), Lal (1990), and Blattberg, Briesch, and Fox (1995), point out that trade promotions provide weak incentives for retailers. Retailers need not pass the trade promotions on to the consumers. If a retailer decides to adopt a promotion he may amplify or mitigate the effect of the wholesale price reduction. Marketing studies nd that, on average, a smaller amount than the original discount is passed through to the consumer. An additional

Retail Sales TABLE 1 Variable Number of supermarkets per chain Market share of Heinz per chain Market share of Hunts per chain P-HEINZt ($) P-HUNTSt ($) P-MIN-HEINZt ($) P-MIN-HUNTSt ($) Summary Statistics for Selected Variables Mean 5.25 25.78 18.04 1.35 1.35 1.05 1.05 SD 2.63 3.10 6.10 .18 .18 .15 .11 Min 3 21.15 12.02 .79 .79 .79 .79

37

Max 9 27.78 26.46 1.79 1.69 1.39 1.39

Note.P-MIN-HEINZt denotes the lowest price for Heinz across all stores in a given time period. P-MINHUNTSt denotes the lowest price for Hunts across all stores in a given time period. Market share of Heinz and Hunts per chain refers to the shares held by 32 oz. bottles.

effect of trade promotions is that they induce retailers to purchase in excess of what they sell to consumers. Retailers may store the product. Neslin, Powell, and Schneider-Stone (1995) study optimal inventory policies and the adoption of the trade promotions by retailers. This study focuses on ketchup products.1 Ketchup is storable for fairly long periods of time. A particular ketchup product contained in the data set is homogeneous across time. And ketchup is offered in identical packages across stores. The data contain a total of 18 different ketchup products. Two products, Heinz and Hunts bottles weighing 32 ounces, are selected. There are several reasons for making this selection. First, the prices of these two products vary substantially over time. Second, these products have the highest market share in dollar revenues among ketchup products. Finally, the overall price distribution of the two products is very similar. Ketchup price reductions by supermarkets are typically advertised on yers enclosed in weekend newspaper editions. These advertisements are aimed at informing consumers about price reductions.2 Supermarkets offer a wide variety of products. The pricing decisions of one product may affect the pricing and purchase decisions for other products. Coordination of decisions across different product groups is not addressed in this article. I assume that decisions are separable. Table 1 provides summary statistics of selected variables. The market consists of a total of four supermarket chains, three national and one regional,
1. The data contain a number of other products. However, the other products exhibit characteristics that make them less attractive for this study. Prices of soup products exhibit little variation over time. Yogurt is perishable, and the decision to hold a sale may be strongly motivated by supply conditions. Detergent products exhibit dispersion in prices over time, but there is a large number of different detergent products, each with a small share of the market. On a given day, typically more than one product is on sale. In addition, the packages of the same detergent product may differ over time. This makes a comparison of behavior by consumers and retailers across different time periods more difcult. 2. National advertising is of less importance for ketchup sales. According to Advertising Age, March 3, 1997, the trade and consumer promotions have in the past decade played a much bigger role than advertising for the manufacturer of Heinz ketchup.

38

Journal of Business

Fig. 1.Prices for selected products in supermarket 1

with a total of 21 supermarket stores. The number of stores per chain differs, with an average number of ve, and ranges from three to nine. The market shares of the two products vary across chains. In three chains Heinz is the leading product, while in one chain Hunts has the highest sales volume. PHEINZ denotes the price of Heinz, P-HUNTS is the price of Hunts, P-MINHEINZ is the lowest price for Heinz across all stores in a given time period, and P-MIN-HUNTS is the lowest price for Hunts across all stores in a given time period. Both products exhibit substantial price dispersion on a given day, ranging from $0.79 to $1.79, with a sample mean of $1.35. For comparison on any given day, the lowest price in any of the 21 supermarkets is about 30% lower than the average price in a given store, with a sample mean of $1.05. Figure 1 illustrates the price over time of the two products in a typical supermarket. All prices in this study are nominal prices. The prices remain constant for long periods of time and occasionally drop for short periods. Other ketchup products in the sample typically exhibit less dispersion in prices over time. For example, gure 1 also illustrates the price over time of the third leading product, which is a store product. Observe that the price of this third product is substantially lower on a given day. In table 2 the distribution of daily prices for selected price levels in supermarkets is documented. The heading gives the price level, and the rst row gives the corresponding frequency in the distribution of prices. The 11 categories for Hunts and the nine categories for Heinz span 90% of all prices charged in the 21 supermarkets over the sample period. For Heinz, more than

Retail Sales

TABLE 2

Distribution of Prices P-HEINZ Price: $.79 .8 6.7 9.7 $.89 .5 5.3 6.7 $.99 13.2 33.5 24.6 $1.19 10.2 15.1 19.0 $1.29 4.1 3.6 19.9 $1.39 21.9 7.9 45.2 P-HUNTS Price: $.89 .9 4.1 10.5 $.99 6.4 32.1 22.8 $1.09 3.1 3.6 25.6 $1.19 15.4 18.5 35.3 $1.31 16.1 4.4 48.3 $1.39 8.6 3.0 56.7 $1.43 13.3 8.3 82.5 $1.44 5.1 2.1 59.2 $1.49 11.2 4.6 59.3 $1.53 4.8 3.3 96.3 $1.79 4.8 1.0 85.8 $1.45 4.1 2.0 53.4 $1.46 22.1 12.5 50.6 $1.49 13.1 5.4 20.3

Percentage Percentage of demand Average number of days

Percentage Percentage of demand Average number of days

39

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Journal of Business

60% of all daily prices fall in the range $1.39$1.49, and 23% of prices fall in the categories $0.99 or $1.19. I call the last two categories sale categories. Similarly for Hunts, 20% of prices fall into the sales categories of $0.99 and $1.19, and 54% of prices fall in the range $1.31 to $1.49. The second row in table 2 gives the percentage of all purchases undertaken at any given price level. About 45% of purchases of Heinz ketchup take place at prices below $1.00, and about 65% of all Heinz purchases take place at prices below $1.20. Similarly, approximately 36% of all Hunts purchases take place at prices below $1.00, and about 60% of all Hunts purchases take place at prices below $1.20. Adjusting for the relative frequencies of prices, it is apparent that there are substantial differences in the number of units sold between high and low price levels. At a price of $0.99, about seven times as many Heinz ketchup units are sold as at a price of $1.39. Similarly, about eight times as many Hunts ketchup units are sold at a price of $0.99 as at a price of $1.43. The third row in table 2 reports the average number of days during which the price stays at this level. For low price levels, the average number of days appears smaller than for high price levels. Moving from low to high prices, the average number of days increases for both Heinz and Hunts prices. To illustrate this relationship, I regress the number of days during which the price stays at this level on a constant and the price level. The relationship is signicant and explains roughly 10% of the variation for Heinz and about 15% in the variation for Hunts. The coefcient on the price level is signicant and positive for both Heinz and Hunts. Increasing the price of Heinz by 10 cents increases the number-of-days variable by about 6 days. Increasing the price of Hunts by 10 cents increases the number-of-days variable by about 9 days. Thus, nonsale periods are signicantly longer than sale periods. III. Description of Price Reductions

This section reports descriptive analysis of the data. I nd that prices (and sales) exhibit little correlation across stores. This suggests that variations in wholesale prices do not explain sales entirely. Second, I document that the distribution of prices depends on previous prices of the same and competing brands, which indicates intertemporal effects in supply. Intertemporal supply effects are not in accordance with the predictions of static models similar to Varian (1980). Third, I examine the dependence of demand on past prices. I nd that demand depends on past prices. Demand at low prices is signicantly higher if previous prices were high than if they were low. The intertemporal demand effects are not in accordance with assumptions made in static models of price reductions. Fourth, I examine how frequently consumers visit different stores. I nd that consumers who purchase ketchup at low prices are more likely to visit different stores. Table 3 gives correlation coefcients for prices of Heinz and Hunts over time. The correlation coefcients are given for prices within a chain and across

Retail Sales TABLE 3 Correlation between Heinz and Hunts Prices Within Chain P-HEINZt P-HEINZt P-HUNTSt .52 .14 P-HUNTSt .14 .47 Between Chains P-HEINZt .04 .07

41

P-HUNTSt .07 .21

Note.All correlation coefcients are signicant at the 1% level.

chains. The pricing behavior in supermarkets within a chain for the same product is very similarfor Heinz and for Hunts the correlation coefcient is about 0.5. For supermarkets in different chains the correlation in prices of the same product is positive, but with a smaller amount of correlation than appears within chains. The correlation between prices of different ketchup products is positive and, again, stronger within a chain than across chains. Since prices for supermarkets within a chain are highly correlated, it appears reasonable to assume that supermarkets coordinate their pricing behavior within a chain. The correlation of Heinz prices across chains is low (correlation coefcient of 0.04). This indicates that variations in wholesale prices do not explain price reductions entirely. Although the data do not provide wholesale prices paid by individual supermarkets, conversations with wholesalers indicate that supermarkets purchase ketchup products at the same wholesale price. Furthermore, unequal wholesale prices are not in accordance with the law.3 Static models of monopoly and Bertrand competition predict that retail prices follow a pattern similar to wholesale prices. According to these models, a strong variation in the wholesale price should yield a positive association between retail prices across chains. The small magnitude of the price correlation may suggest that the predictions of static models are not relevant and/or that changes in common cost factors are not the only factor determining price changes. Price changes mostly occur on particular weekdays. Sixty-eight percent of the price changes for Heinz and 77% of the price changes for Hunts occur on a Wednesday, while 20% of the price changes for Heinz and 15% for Hunts occur on a Thursday. The remaining price changes are evenly distributed across the remaining days of the week. An inspection of the number of price changes in a supermarket reveals that changes occur infrequently. The number of price changes, in the average supermarket, equals 24 during the sample period for Heinz and 18 for Hunts. This variable differs across supermarkets and ranges from 11 to 34. The absolute value of a price change is, on average, 30 cents for Heinz and 18 cents for Hunts. Figure 1 and table 2 suggest the possibility of serial correlation in prices
3. The Robinson-Patman Act, which amended section 2 of the Clayton Act in 1936, prohibits a manufacturer (or wholesaler) from price discriminating if it harms competition among the retail rms. The Robinson-Patman Act was passed in response to political pressure from small retail stores who complained that larger chains were able to purchase supplies on more favorable terms and thereby charge lower prices (Ross 1984).

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Journal of Business

as prices remain constant for extended periods of time. To determine how current prices depend on past prices, weekly price changes can be studied. Although I do not report the test results in detail, I briey summarize the ndings: I examined whether the distribution of prices conditional on a price change depends on past prices. The null is that conditional distributions are identical for different past weeks prices. For example, under the null, the distribution of prices over the categories of $1.19, $1.39, and $1.46 is the same whether the past weeks price was $0.99 or $1.49. Pairwise comparisons of conditional distributions for the largest ve price categories for each product were considered. The null of identical price distributions can be rejected in 19 of 20 cases. I also considered tests of independence of previous prices of competing brands. The null of independence was again rejected. Chosen price levels depend also on the past weeks prices for the competing brand. I next examine the dependence of demand on lagged prices. Consider the following equation:
i Dk, t p l

( pt 7 Bl

i Xk, t bl )1{pt

Il}

i k, t

(1)

i where Dk, t denotes the number of units of product i purchased in supermarket k, pt 7 denotes a vector of lagged prices, X denotes additional explanatory variables, and 1{A} denotes an indicator function that equals one if condition A is true and zero otherwise. I permit the coefcients in the demand equation to vary with current period prices. Because of the discreteness of prices, I partition the set of current prices in a number of disjoint intervals, Il. I let the coefcients, Bl and bl, differ across intervals. Thus, the coefcients in equation (1) capture variations in the number of units purchased at a given price level. The data are obtained from a sample of households and contain a number of observations with zero units purchased. I account explicitly for the truncation of the dependent variable and estimate a Tobit model. Tables 4 and 5 report parameter estimates of Tobit regressions for Heinz and for Hunts. The dependent variable is constructed by adding one unit to the number of units purchased and taking the logarithm of the resulting sum. By construction, the dependent variable is truncated at zero. Explanatory variables include lagged supermarket-specic prices, lagged minimum prices across all stores, a set of advertising dummies that indicate whether a supermarket advertisement for a ketchup product appeared in the weekly newspaper, a Fourth of July dummy variable that equals one between June 15 and July 15 of every year. The set of explanatory variables is interacted with a set of six dummy variables indicating time t price intervals for both products. I select the price intervals to reect the distribution of prices. The ranges of price intervals are reported in the headings of tables 4 and 5. Thus, a column in tables 4 and 5 reports the coefcient estimates associated with a particular price interval. In addition to the reported coefcients, a set of weekday dummies and a set of supermarket-specic dummy variables are included but not reported.

Retail Sales

TABLE 4

Tobit Regression: Demand for Heinz Price Interval Dummies P-HEINZt ($): P-HUNTSt ($):
!1.2

.99 1.2 1,773 2.298 (5.7) 7.024 (6.9) 1.915 (3.8) 2.587 (3.0) 1.439 (16.8) .620 (1.2) .139 (.8) .208 (1.8) 41.5 Reject 11.4 Reject
!1.2

[1.00, 1.19] 1.2 1,192 .780 (1.1) 1.351 (1.9) .544 (.5) .176 (.2) 1.730 (13.6) .132 (.5) .074 (.6) 2.0 .2
!1.2

11.20

1.2 9,255 .130 (.4) 1.040 (3.2) .506 (2.3) .416 (1.4) 2.151 (14.6) .651 (1.3) .190 (2.3) .036 (.1) 5.4 Reject 2.8 Reject (6%)

Number of observations in this price range Interaction term: P-HEINZt 7 P-HUNTSt

838 2.078 (3.1) 1.021 (.9) 1.162 (1.3) .873 (.8) 1.030 (8.1) .153 (.5) .083 (.5) 6.93 Reject 142.7 Reject

570 1.453 (1.4) .061 (.1) 3.312 (1.9) 3.062 (2.6) 1.586 (8.0) .804 (2.0) .283 (.9) .179 (.9) .9 3.6 Reject (2%)

3,129 .664 (1.0) .084 (.2) .701 (1.3) .019 (.0) 2.019 (10.5) .152 (.9) .029 (.2) .082 (.7) .54 .9

P-MIN-HEINZt

P-MIN-HUNTSt

HEINZ-ADVERTISING HUNTS-ADVERTISING OTHER-PRODUCT-ADVERTISING 4TH-OF-JULY F-test: H0: P- p 0 F-test: H0: P-MIN- p 0

Notes.Coefcient estimates are reported for explanatory variables interacted with a set of price interval dummies. Thus, the rst row reports the estimated coefcient for P-HEINZt for individual price interval dummies. Nonqualitative explanatory variables are in logarithm. A set of supermarket dummies and weekday dummies is included but not reported. Absolute values of t-statistics are displayed in parentheses. A dash instead of a coefcient estimate indicates that the variable is collinear with other variables. Unless another value is given in parentheses, Reject denotes a rejection of the null hypothesis at the 1% level. Dependent variable: logarithm of Heinz equilibrium demand at price levels. Number of observations: 16,757. Degrees of freedom: 16,556. Chi-squared: 4,208.5.

43

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TABLE 5

Tobit Regression: Demand for Hunts Price Interval Dummies P-HUNTSt ($): P-HEINZt ($):
!1.2

.99 1.2 753 1.472 (2.6) 3.560 (2.2) 1.483 (1.1) .942 (.9) 1.186 (7.1) .055 (.1) .782 (2.6) 9.897 (3.0) 5.9 Reject .9
!1.2

[1.00, 1.19] 1.2 2,376 2.485 (3.4) 1.227 (1.7) 1.059 (1.5) .269 (.5) 1.799 (11.6) .378 (1.3) .077 (.5) 4.740 (1.5) 7.7 Reject 1.3
!1.2

11.20

1.2 9,255 .746 (1.8) .559 (1.6) .332 (.9) .301 (1.1) 2.344 (10.5) .148 (.5) .047 (.5) 6.109 (2.1) 3.3 Reject (4%) 1.5

Number of observations in this price range Interaction term: P-HUNTSt 7 P-HEINZt

292 .036 (.0) 7.427 (5.8) 4.771 (2.3) 8.379 (3.1) 1.319 (3.8) .001 (.0) .277 (.5) 2.6 Reject (8%) 29.1 Reject

1,116 2.881 (2.0) .055 (.8) .267 (.3) .678 (.6) 2.246 (2.8) .155 (.9) .548 (1.4) 5.524 (1.7) 2.2 .2

2,965 .103 (.2) .519 (1.0) 3.040 (2.8) 1.839 (2.4) 1.850 (5.8) .188 (1.4) .018 (.1) 10.797 (3.5) .6 6.3 Reject

P-MIN-HUNTSt P-MIN-HEINZt
7

HUNTS-ADVERTISING HEINZ-ADVERTISING OTHER-PRODUCT-ADVERTISING 4TH-OF-JULY F-test: H0: P- p 0 F-test: H0: P-MIN- p 0

Journal of Business

Note.Coefcient estimates are reported for explanatory variables interacted with a set of price interval dummies. Thus, the rst row reports the estimated coefcient for P-HUNTSt for individual price interval dummies. Nonqualitative explanatory variables are in logarithm. A set of supermarket dummies and weekday dummies is included but not reported. Absolute values of t-statistics are displayed in parentheses. A dash instead of a coefcient estimate indicates that the variable is collinear with other variables. Unless another value is given in parentheses, Reject denotes a rejection of the null hypothesis at the 1% level. Dependent variable: logarithm of Hunts equilibrium demand at price levels. Number of observations: 16,757. Degrees of freedom: 16,563. Chi-squared: 2,845.9.

Retail Sales

45

Under the null of no signicant effects of lagged price variables, H0: Bl p 0 for all l, an F-test can be constructed. The null hypothesis of jointly no signicant effects can be rejected if the F-statistic exceeds the critical value. The F-statistic equals 9.1 for Heinz and 3.1 for Hunts. Thus, for both Heinz and Hunts, lagged prices interacted with price interval dummies are signicantly different from zero at the 1% level. To test whether the coefcients associated with lagged prices differ across price intervals, we may conduct a likelihood ratio test. The null hypothesis of equal coefcients across price intervals (across columns), H0: B1 p B2 p . . . p B6, can be rejected, if the chi-squared statistic exceeds the critical value. The test statistic equals 143.3 for Heinz and 36.2 for Hunts. We can reject the null of identical coefcients for both Heinz and Hunts. At the bottom of tables 4 and 5 test results for the signicance of lagged store and minimum prices are reported for individual price intervals. I nd that for both Heinz and Hunts lagged supermarket-specic prices are signicant at the lowest two price intervals and at the highest price interval. For Hunts, the null is also rejected at the fourth interval. At other price intervals the null cannot be rejected. For Heinz, lagged minimum prices charged in other stores are signicant at the 2% level at the lowest three price intervals. They are also signicant at the highest price interval at the 6% level. At other intervals they are not signicant. For Hunts, lagged minimum prices charged in other stores are signicant at the lowest interval and in the fth interval. Lagged minimum prices are not signicant at other price intervals. The important effects in table 4 are the following. At the lowest price interval for Heinz, the demand for Heinz is signicantly higher if the past weeks prices in the same supermarket are high. When the price of Hunts is below $1.20, holding other variables constant, a 1% increase in P-HEINZt 7 increases Heinz demand by 2.1%. Moreover, an increase of P-HUNTSt 7 by 1% increases Heinz demand by 1.0%. When the price of Hunts exceeds $1.20, then the effects become even stronger: a 1% increase in P-HEINZt 7 increases demand by 2.3%. A 1% increase in P-HUNTSt 7 , increases Heinz demand by 7%. As we move to the right in the table, the magnitude of the effect of past prices declines. The effect is stronger when P-HUNTSt exceeds $1.2 than when it is below $1.2. Prices charged in other supermarkets have a signicant effect on demand at the lowest price level. Demand is higher if past prices in other supermarkets are high. The effect of the minimum price of Heinz equals 1.2% and 1.9% at the lowest two categories. The ndings in table 5 for the lagged product-specic price are similar to those in table 4. With the exception of the rst price category, lagged prices of Hunts have a positive effect on Hunts demand. A 1% increase in PHUNTSt 7 increases demand by 1.5% in the second category. The effect increases to 2.9% and 2.5% in the third and fourth categories. At the highest two categories, where the current price of Hunts exceeds $1.20, the effect declines in magnitude. It is positive and equals 0.1% and 0.7%. For Heinz and also for Hunts (with the exception of the rst category),

46 TABLE 6

Journal of Business Distribution of Preferred Chains by Average Price Paid for Repeat Purchasers N First Chain 58.20 (14.8) 60.13 (17.7) 64.23 (17.2) 68.68 (18.8) 72.51 (18.7) 65.4 (18.3) 66.0 (18.2) Second Chain 28.08 (11.4) 25.39 (11.4) 24.38 (10.8) 21.72 (12.6) 19.48 (13.7) 23.4 (12.3) 23.1 (12.3) Third Chain 11.18 (7.0) 9.92 (7.7) 8.64 (7.3) 7.80 (7.2) 6.20 (6.4) 8.5 (7.3) 8.4 (7.4)

Average price paid: $0$.99 $.99$1.09 $1.09$1.19 $1.19$1.29 $1.29$1.79 All repeat purchasers* All consumers

43 80 130 95 81 429 1,453

* Data consist of 429 consumers with at least four purchases of Heinz or Hunts ketchup. Standard errors are displayed in parentheses.

the effect of lagged store prices of the same product are stronger when current prices are low than when they are high. This difference suggests that consumers who buy at low price levels are more sensitive to past prices in the same supermarket than consumers who buy at high price levels. Weekly advertisements in the newspaper have the expected sign. Advertisements have a positive effect on the demand for the product and a negative effect on the demand for competing brands. The 4TH-OF-JULY variable has a positive and signicant effect on Hunts demand and a mostly negative, but not signicant, effect on the demand for Heinz. In tables 4 and 5, the demand for Heinz and Hunts appears to depend on past prices charged in other supermarkets. This raises the question of how often consumers shop at different supermarket chains. The data contain information on the purchase behavior of a sample of households. Notice that an observation in these data is a household and not an individual. Table 6 reports the percentage of occasions in which households go shopping at their most preferred chain, second most preferred chain, and third most preferred chain. In the table, First Chain denotes the supermarket chain the consumer visits most frequently, Second Chain denotes the supermarket chain the consumer visits the second most frequently, and so on. The shopping frequencies are given for households with at least 70 shopping trips during the sample period. For consumers with at least four purchases of Heinz or Hunts ketchup, whom I call repeat buyers, the shopping frequencies are reported separately and also grouped by the average price of Hunts or Heinz ketchup paid per household. The percentages in the table sum to less than 100 since a (small) fraction of the store visits are for the fourth choice. On average (across consumers) two out of three times the rst chain is chosen, about one out of four times the second chain is chosen, and about

Retail Sales

47

every tenth visit the third chain is chosen. These numbers are roughly the same for repeat buyers as for other households. The standard deviations in table 6 are high. This suggests that some households shop around very little, while other consumers may shop around a lot. Indeed about 20% of all consumers go almost exclusively to one store (19 of 20 times). On the other hand, about 20% of all consumers go only around 40% of times to their most preferred chain; they go on every third shopping trip to the second most preferred chain, every fth trip to the third most preferred chain, and once every 15 shopping trips to the least preferred chain. Table 6 also suggests a dispersion in the average price paid by repeat buyers of Heinz and Hunts ketchup. A total of 429 consumers in the sample, or about one-third of all households, qualify as repeat buyers. The rst two elements in table 6 illustrate that about 30% of repeat buyers pay an average price of less than $1.09 per ketchup bottle. Calculating the mean average purchase price for this group yields an estimate of $1.02. At the other extreme, about 20% of consumers pay an average price of more than $1.29, with a mean average purchase price of $1.35. For comparison, a price-insensitive consumer, or random buyer, would pay an average price of $1.35, and a consumer buying only at the store offering the lowest price on a given day would pay, on average, $1.05. Assuming that my measure of the average purchase price for repeat buyers reects the true average price paid per household implies the following. About 80% of consumers buy at prices lower than the average daily shelf price. About 30% of consumers buy at prices lower than the lowest price offered across all stores on a given day. Unfortunately, in the data the number of purchases per consumer is relatively small, ranging between 4 and 10 purchases for 85% of consumers. This weakens the implication and makes a more careful analysis impossible. However, a narrower denition of repeat purchasers does not change the distribution of average price paid per consumer substantially. A total of 62 households satisfy the stronger requirement of at least 10 purchases of Heinz or Hunts ketchup. From these, about 21% buy at a price below $1.09, and about 21% buy at a price above $1.29, which yields a distribution similar to that given in table 6. Consumers who buy at low prices appear to shop around more frequently. In table 6 the fraction of visits to the most preferred chain appears to increase as we move down the table. To determine whether this relationship is signicant I regressed the percentage of visits to the rst chain on the average price paid and a constant. About 8% of the variation in the dependent variable is explained. Moreover, the regression coefcient for the average price is signicant and positive. Ketchup sales may be intended to increase the inow of consumers who otherwise do not shop in that store. These consumers may purchase other products and enhance the gains from conducting a sale. Lal and Matutes (1994) consider loss leader pricing in which one product is offered at marginal cost or below in order to increase store trafc. Using the data, I examine

48

Journal of Business

whether the demand for other products increases when ketchup is on sale. The evidence indicates only small effects if any. If consumers do not substitute between ketchup and other products and there are no budget constraints, then demand for other products may increase. I examined this hypothesis with the data on soup, detergent, and yogurt products. The logarithm of demand for the leading product in each category, at a given price level of the product, is regressed on explanatory variables (similar to tables 4 and 5). Price levels that account for at least 10% in the price distribution are used. This yields a total of 12 regressions. Explanatory variables include one indicator variable that equals one if Heinz or Hunts ketchup is on sale during that day, a set of supermarket-specic dummy variables, and a set of weekday dummy variables. In these 12 regressions, four sale dummies are signicant. Seven of the sale dummies are negative, and ve are positive. This evidence does not indicate positive externalities across groups of products. Summarizing, I nd that both demand and supply depend on past price choices. The distribution of prices depends on previous price choices of competing ketchup brands within the store. Across supermarket chains there is little correlation in prices. Sales do not arise at the same time across supermarket chains. Demand at low price levels is higher if past prices are high than if past prices are low, holding other variables constant. Demand at high prices is also affected by past prices, but to a lesser extent. Individual households differ substantially in the average price they pay and in their willingness to shop around. Consumers who buy on average at low prices are more likely to visit competing chains. This may indicate that competition between retailers affects sale decisions. Varian-style sales models in which prices are independently and identically distributed (i.i.d.) have received considerable attention in the literature. The data appear to rule out this type of static model. Static models fail to explain that current prices depend on past prices and that current demand depends on past prices. The empirical evidence suggests intertemporal effects. Based on the descriptive evidence, I formulate a model of intertemporal pricing. In the next section, the model is described, and predictions of the model are provided that can be confronted with the data. IV. Demand Accumulation

In this section I describe the implications of a model of intertemporal pricing for a single product in which the stock of low-valuation consumers accumulates in the duration since the last sale. Retailers face uncertain future wholesale prices and decide when to adopt a sale. Time is discrete. Agents are fully informed and risk neutral. There are m retailers, each of which sells the same product. Retailers choose retail prices to maximize discounted present value, calculated with discount factor d (with 0 ! d ! 1 ), taking the prices of other retailers as given. Retailers cannot store

Retail Sales

49

the product. The cost of holding inventories exceeds the gains from storing the product.4 The product is supplied to the retailer by a manufacturer. The retail market is small compared with the wholesale market, and I assume that the wholesale price (paid from the retailer to the manufacturer) in period t is exogenous to the retail market. The wholesale price equals c1 with probability r and c2 with probability 1 r, with c 1 ! c 2. One consumer enters the market each period. Each consumer wishes to purchase one unit of a product. Consumers differ in their valuation for the products. A fraction of the consumers have high valuations and buy only from a particular retailer. Specically, a fraction a/m of consumers value the product at v1 and buy only from retailer k, with 0 ! a ! 1 . A fraction 1 a of consumers value the product at v2, with v1 1 v2 1 c 1 1 0. Consumers who value a product at v1 are sometimes referred to as high-valuation consumers, and consumers who value the product at v2 are called low-valuation consumers. High-valuation consumers stay in the market for one period. Independent of whether they actually purchase the product, they leave the market at the end of the period they entered. In contrast low-valuation consumers who do not purchase a product stay in the market. Low-valuation consumers leave the market only after purchase of a product. They have discount factor zero. This distinction reects that some consumers must have the product immediately, provided they can afford it, while other consumers have low search cost and stay around until a product becomes affordable to them, at which point they buy the product immediately and leave the market. I distinguish two types of low-valuation consumers: store-loyal consumers and shoppers. This distinction may arise because of differences in traveling costs. A fraction g/m of low-valuation consumers have high traveling costs and buy only from retailer k. A fraction 1 g of low-valuation consumers are shoppers who purchase from the store offering the lowest price (provided the price is below v2). To illustrate the assumptions, I describe the demand for retailer k. During a period in which the price is above v2, demand consists of highs and equals a/m. Let Tk denote the number of time periods since the last sale in retail store k. During a sale period (when the price does not exceed v2) demand is given by a m (1 a) g m Tk (1 a)(1 g) min (T1 , T2 , . . . , Tm ). (2)

It consists of a/m highs, (1 a)(g/m)Tk store-loyal lows, and (1 a)(1 g) min (T1 , T2 , . . . , Tm ) lows that are shoppers. Hence, retailers have monopoly power over high-valuation consumers and store-loyal lows. Retailers compete with other retailers for the fraction of consumers who are shoppers.
4. Conversations with retailers indicate that ketchup is stored for a short time period at the retail store. Thus, ignoring retailer inventory holdings appears to be a reasonable assumption for this market.

50

Journal of Business

I assume that retail pricing strategies can depend only on the number of time periods since the last sale in individual stores (T1 , T2 , . . . , Tm ) and the current period cost realizations c, with c {c 1 , c 2} . This strategy space rules out behavior in which rms condition their pricing strategies on observed histories of prices. We rst examine a situation in which there are no shoppers, g p 1 (or m p 1). Proposition 1 characterizes the optimal decision to hold a sale for a monopolist in the retail market. Later we examine equilibrium sales behavior under competition among retailers. Before stating the proposition, I illustrate the intuition. For any realization of costs sufciently below v2, as the number of low-valuation consumers accumulates, the revenues of a price cut increase. Eventually the gains from a price cut exceed the expected gains from waiting one more period, and the retailer holds a sale. Proposition 1. Suppose g p 1 (or m p 1). Then there exist integers T(c 1 ), T(c 2 ) such that the price in a period with state (T, c) is given by p(T, c) p

v2 if T T(c); v1 otherwise.

Moreover, for c 1 ! c 2, T(c 1 ) T(c 2 ); for c 2 [(1 d)v2 drc 1 ]/[1 d(1 )p . r)], T(c 2 All proofs are given in the appendix. The optimal strategy of the retailer for a given cost realization has the following qualitative form. For T periods the retailer charges v1 for the product. After this point, there is a sale in which the retailer charges v2, and all the low-valuation consumers buy the product on sale. The process then repeats. Sales occur earlier for lower cost realizations than for higher cost realizations. The assumption that wholesale prices are random implies that the probability of a sale equals the probability that the manufacturer offers the product at a wholesale price below a certain threshold. This threshold increases over time and therefore implies that the probability of a sale increases in the number of periods since the last sale. What is observed is a price cut on the product followed by an interval of at least T(c 1 ) of high prices. The assumption that high-valuation consumers leave the market at the end of the period they entered ensures that retailers sell to all high-valuation consumers in every period. If high-valuation consumers do not leave the market at the end of the period, then a retailer, facing a high wholesale price, may prefer not to sell to high-valuation consumers but to sell later at a lower wholesale price. Competition between retailers is described next. Consider the situation with no store-loyal lows, g p 0 . Equilibrium prices in a similar model are examined in Sobel (1984). The difference to Sobel is that he assumes that low-valuation consumers do not necessarily buy in the rst period at which the price is below v2 but that they may also wait if they expect the price to fall even further. As a result, Sobels equilibrium distribution of prices below v2 differs

Retail Sales

51

from that of this model. Moreover, as Sobel shows, there are possibly multiple symmetric equilibria. In contrast, in this model there is only one symmetric equilibrium. I do not consider strategic behavior by consumers and assume that lows purchase immediately provided that the product is affordable to them. Nonstrategic behavior is also emphasized in the marketing literature (Blattberg and Neslin 1989). The literature on reference prices illustrates that consumers remember a small set of reference prices for supermarket products. Thus, in the context of supermarket products it appears reasonable to rule out strategic purchase behavior. The following proposition describes equilibrium prices under competition. Proposition 2. Suppose g p 0, and let c denote the cost realization and T p min (T1 , T2 , . . . , Tm ). A symmetric equilibrium is given by the following prices: if T ! [(a/m)(v1 v2 )]/[(1 a)(v2 c)] or c v2, then the price in period T equals v1 in all retail stores. If T [(a/m)(v1 v2 )]/[(1 a)(v2 c)] and c ! v2, then the price in every retail store is drawn from the distribution,

G( p, T, c) p

if p ! c

(a/m)(v1 T(1 a)

c)

{ {

(a/m)(v1 T(1

c) a)]( p c) a)](v2

[(a/m)

c)

(a/m)(v1 T(1

[(a/m)

c)

} }

1/(m 1)

if p
1/(m 1)

(a/m)(v1 (a/m) T(1

c) a)

, v2 ;

if p

[v2 , v1 );

if p v1.

The equilibrium has the following qualitative form. For [(a/m)(v1 v2 )]/[(1 a)(v2 c 1 )] periods, retail stores charge the high price. After this time, stores hold a sale with positive probability in any period. If a sale occurs, then the sale price is drawn randomly from a distribution with support below v2, and all low-valuation consumers purchase from the store offering the lowest price. After this sale period the cycle starts over. The implied price path consists of a price cut in one store followed by an interval of high prices. Comparative statics for the probability of a sale are easily obtained. An increase in the number of low-valuation consumers increases the probability of a sale in any retail store. As the number of retailers gets large, the probability that a sale occurs in some retail store approaches one.5 The reason is that, holding the total number of shoppers per period constant, an increase in the number of retailers is equivalent to a decrease in the number of loyal consumers per store. In the proposition it is assumed that retailers face the same wholesale price and that retailers have the same share of high-valuation consumers. If one store can buy the product at a lower wholesale price, then this store offers
5. Examples can be constructed in which the convergence is not monotone.

52

Journal of Business

the product on sale with a higher probability. Similarly, if one store has a smaller share of high-valuation consumers, then this store may hold a sale earlier than other stores. Expectations about future wholesale prices do not affect the equilibrium in proposition 2. In particular, serial correlation in wholesale prices does not alter the equilibrium outcome. Proposition 2 characterizes the symmetric equilibrium. If the number of retailers exceeds two, then there also exist asymmetric equilibria. In asymmetric equilibria one or more retailers may completely refrain from holding a sale. Supermarkets label themselves as HiLo price setters and Every Day Low price setters. Asymmetric equilibria may explain this distinction. HiLo supermarkets charge high and low prices according to proposition 2 while constant prices are obtained by completely refraining from holding a sale. An example is given in the appendix. Propositions 1 and 2 describe the sales decision for two extreme cases: all lows are store loyal, or all lows are shoppers. In the general case we expect that the sales decision is a function of the stock of lows in all retail stores. This decision rule can be studied with the data. Prediction. To summarize, the predicted price path consists of a number of periods of high prices followed by a short period of low prices. After the price cut, the sale cycle starts over. The equilibrium prices are described by a draw from a distribution function, G, that depends on the current state: p G( pFT1 , T2 , . . . , Tm , c). (3)

The equilibrium distribution function in equation (3) has a mass point at v1 and, if g does not vanish, also at v2. When there are no shoppers, g p 1; then sales are entirely determined by the duration since the last sale in the store. When there are no store-loyal lows, g p 0 , then the minimum time since the last sale across stores, min (T1 , T2 , . . . , Tm ) , determines sales. In general, the model implies that a stores decision to conduct a sale is a function of the wholesale price and the duration since the last sale in that store and other stores. The model is presented with one product per store. The formulation can easily be extended to permit multiple products. Related products can be incorporated by distinguishing between two additional types of lows: productloyal lows and consumers who perceive products as perfect substitutes. Observe that competition between products is analogous to competition between stores. Although I do not model multiple products explicitly, the equilibrium decision to conduct a sale for a particular product is a function of stocks of product-loyal lows for all products in all stores. In addition, differences in wholesale prices between products can inuence the decision to conduct a sale.

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V.

The Sale Decision and Demand for Ketchup

This section examines the predictions outlined in Section IV. First, in Section V.A the determinants of the decision to hold a sale are examined. I nd that both the store-specic sale duration and the sale duration across stores explain the decision to hold a sale. Second, in Section V.B, the determinants of demand conditional on a sale period are studied. I examine whether variables measuring the time elapsed since the last sale have a positive effect on demand. A. The Sale Decision

This section examines the decision to conduct a sale. Equation (3) in Section IV implies that the distribution of prices is a function of wholesale prices and the time elapsed since the last sale in the store and other stores. In addition, the distribution of prices has a mass point at the nonsale price (or regular price) and mass contained in an interval below the sales price. The price decision problem can be modeled as a discrete choice problem with two prices: sale and no sale. The probability of holding a sale will be a function of the time elapsed since the last sale on the product in the store, the time elapsed since the last sale of competing products in the store, duration variables in other stores, the wholesale price, and idiosyncratic randomness. Idiosyncratic effects may include shocks to inventory holdings. The retailer decides not only whether to conduct a sale but also which product to offer on sale. It may be expected that the decision to hold a sale is correlated across products. Unfortunately, the data contain only a very few observations in which both products are on sale at the same time. An estimation of the bivariate decision model was not successful. The likelihood of the bivariate model is very at, and signicant estimates for the correlation in the decision variables were not obtained. I therefore report only estimates of the univariate case. i The decision to offer product i on sale in supermarket k, ykt p 1, can be described by the following rule:
i ykt p gZ k, t

b Xkt

kt

and
i ykt p

{1 if yy 0 if

i kt i kt

1 0;

0;

(4)

where Z is a set of variables that measure the time elapsed since the last sale in the store and in competing stores, Xkt are additional explanatory variables, and kt is idiosyncratic randomness. Under the assumption that kt is independent of past sale decisions, this model can be consistently estimated. Before proceeding to the estimation results, I discuss two choices that are made in estimating relation (4). First, the data do not include wholesale prices. According to distributors in this market, the wholesale price did not change for extended periods of time. After a trade promotion is announced, the whole-

54

Journal of Business

sale price stays at this level for an interval of about 3 months. As is discussed in Section IV, the predicted equilibrium outcome in proposition 2 extends to the case of serially correlated wholesale prices. Since I estimate the model using weekly prices,6 the persistence of wholesale prices will lead to serially correlated errors and, as a result, the estimators will be inconsistent. To resolve this problem I include a set of time dummy variables that equal one during a period of xed length. Variations across supermarkets are used to identify this parameter. Specically, I select a set of time dummies with interval length of 8 weeks. Eight weeks is used to account for possible variations in wholesale prices of the two competing products. I also estimated the model using dummies measuring shorter and longer time periods. The results were qualitatively very similar. The second issue concerns the selection of data. The theory in Section IV implies that a sale lasts one time period. From gure 1 and table 2 we see that sales may last for more than 1 week. If consumers only periodically go shopping, this may be intended to reach a large fraction of customers. The theory in Section IV does not provide an explanation for this behavior. To resolve this issue, I omit all sale observations except the rst week of the sale. An additional problem is that the initial values for lagged values are missing. In the estimation I omit observations stemming from the rst 12 weeks. The rst observation for supermarket k begins in week 13. Tables 7 and 8 report probit estimates for the decision to conduct a sale for Heinz and Hunts using weekly data.7 The dependent variable equals one if the price falls below a certain threshold. As discussed in Section II, I interpret prices below $1.20 as sale prices. Since this denition may seem somewhat arbitrary, I report estimates for different threshold levels. Specically, results are reported for thresholds of $0.99, $1.09, and $1.19. Observe that the definition of explanatory variables changes as the threshold varies. Explanatory variables include MIN-T-HEINZ, the minimum time elapsed since the last sale for Heinz ketchup across all supermarkets, MIN-T-HUNTS, the minimum time elapsed since the last sale for Hunts ketchup across all supermarkets, T-HEINZ, the time elapsed since the last sale for Heinz ketchup in the supermarket under consideration, T-HUNTS, the time elapsed since the last sale for Hunts ketchup in the supermarket under consideration, 4THOF-JULY, a dummy variable that equals one between June 15 and July 15, and a set of supermarket-specic dummy variables, which are not reported. In addition, a set of time-specic dummy variables is included but not reported. All nonqualitative variables are in logarithm. To account for possible nonlinearities, squared terms of variables are included in table 8.8 In table 7, the null hypothesis that the duration variables are not jointly
6. As described before, price changes occur typically on a Wednesday. Following a price change, the price stays at this level for at least 1 week. I select Wednesday observations in the estimation and omit observations during other weekdays. 7. An estimation using daily prices yielded qualitatively similar results. 8. I omit squared terms of variables in table 9 because of the small number of sales observations.

Retail Sales TABLE 7 Probit: Probability of a Sale on Heinz P-CRITICAL $.99 Number of sales Number of observations Degrees of freedom x2 T-HEINZt T-HEINZ-SQt T-HUNTSt T-HUNTS-SQt MIN-T-HEINZt MIN-T-HEINZ-SQt MIN-T-HUNTSt MIN-T-HUNTS-SQt 4TH-OF-JULY x2 test: H0: MIN-T- p 0 x2 test: H0: T- p 0 100 1,388 1,351 136.9 .7695 (2.2) .1772 (2.4) .6842 (2.6) .1181 (2.5) 1.0939 (1.9) .7335 (2.0) .2413 (.6) .0818 (.2) 1.7819 (6.3) 14.8 Reject 8.03 Reject (10%) $1.09 108 1,521 1,483 133.8 .6557 (2.1) .1617 (2.4) .4854 (2.9) .1091 (3.1) .9925 (1.8) .6823 (1.9) .1322 (.3) .7306 (2.5) 1.6852 (6.1) 12.9 Reject (2%) 8.6 Reject (10%) $1.19 109 1,484 1,447 110.1

55

.4455 (1.3) .0777 (.9) .3574 (2.5) .1613 (3.7) .5960 (2.3) .8215 (2.8) 1.0558 (4.3) 11.7 Reject 6.34

Note.All variables are in logarithms. A set of time-specic dummy variables is included but not reported. Absolute values of t-statistics are displayed in parentheses. T-HEINZt denotes the time elapsed since the last sale for Heinz in that supermarket measured in weeks. T-HEINZ-SQt denotes the square of T-HEINZt. THUNTSt denotes the time elapsed since the last sale for Hunts in that supermarket measured in weeks. MINT-HEINZt denotes the minimum time elapsed since the last sale for Heinz across all supermarkets. MIN-THUNTSt denotes the minimum time elapsed since the last sale for Hunts across all supermarkets. Chi-squared statistics are reported for two tests. The rst test, MIN-T- p 0 , reports the joint signicance of the parameters MIN-T-HEINZt, MIN-T-HEINZ-SQt, MIN-T-HUNTSt, and MIN-T-HUNTS-SQt. The second test, T- p 0, reports the joint signicance of T-HEINZt, T-HEINZ-SQt, T-HUNTSt, and T-HUNTS-SQt. Results of the tests are reported at the bottom of the table. Reject indicates that the null is rejected at the 1% level unless another level is indicated in parentheses. A dash indicates that the null cannot be rejected. Dependent variable: SALE p 1 if P-HEINZt P-CRITICAL; SALE p 0 otherwise.

signicant can be rejected for the rst two thresholds at the 1% level and for the threshold of $1.19 at the 2% level. In table 7, the null hypothesis of having no signicant effects can be rejected for all three thresholds at the 1% level. If sales are not affected by the interaction between retailers, then this should be detected by examining whether the minimum time elapsed since the last sale across stores has a signicant effect on the decision to hold a sale in a particular supermarket. Under the null hypothesis, T-H-MIN p 0 ; that is, the parameters measuring the effect of the elapsed time in other chains are zero. The null corresponds to the restriction g p 1 in Section IV. The test statistic is distributed as a chi-squared random variable. The test results are reported in tables 6 and 7. In table 7 the null can be rejected at the 2% level. In table

56 TABLE 8 Probit: Probability of a Sale on Hunts P-CRITICAL $.99 Number of sales Observations Degrees of freedom x2 T-HUNTSt T-HEINZt MIN-T-HUNTSt MIN-T-HEINZt 4TH-OF-JULY x2 test: H0: MIN-T- p 0 x2 test: H0: T- p 0 40 1,181 1,152 106.7 .5153 (4.3) .5190 (3.8) .7534 (2.2) .3094 (1.1) .8623 (2.8) 4.5 Reject (10%) 21.2 Reject $1.09 58 1,337 1,306 102.3 .5356 (4.8) .0846 (1.2) .5836 (2.2) .5061 (2.0) 1.0392 (3.7) 3.5 15.2 Reject

Journal of Business

$1.19 99 1,471 1,435 149.2 .3549 (3.9) .1299 (2.1) .0204 (.1) 1.0485 (3.0) .1311 (.6) 4.7 Reject (10%) 10.6 Reject

Note.All variables are in logarithms. A set of time-specic dummy variables is included but not reported. Absolute values of t-statistics are displayed in parentheses. T-HUNTSt denotes the time elapsed since the last sale for Hunts in that supermarket measured in weeks. T-HEINZt denotes the time elapsed since the last sale for Heinz in that supermarket measured in weeks. MIN-T-HUNTSt denotes the minimum time elapsed since the last sale for Hunts across all supermarkets. MIN-T-HEINZt denotes the minimum time elapsed since the last sale for Heinz across all supermarkets. Chi-squared statistics are reported for two tests. The rst test, MIN-T- p 0, reports the joint signicance of the parameters T-MIN-HEINZt and T-MIN-HUNTSt. The second test, T- p 0, reports the joint signicance of T-HEINZt and T-HUNTSt. Results of the tests are reported at the bottom of the table. Reject indicates that the null is rejected at the 1% level unless another level is indicated in parentheses. A dash indicates that the null cannot be rejected. Dependent variable: SALE p 1 if P-HUNTSt P-CRITICAL; SALE p 0 otherwise.

8 the null can be rejected in two of three cases at the 10% level. At the threshold level of $1.19 the null cannot be rejected. The main implications in table 7 are the following. An inspection of the linear and quadratic coefcients of time variables reveals that, with two exceptions, all linear coefcients are positive and all quadratic coefcients are negative. This implies that, as a particular explanatory variable increases, the probability of a sale increases initially and then decreases. The exceptions occur at the threshold of $1.09 for the variable MIN-T-HUNTS and at the threshold of $1.19 for the variable MIN-T-HEINZ. In the rst exception an increase in the variable reduces the probability for a small range and then increases it. In the second case, the variable has a negative effect on the probability of a sale. To evaluate and compare the effects of different variables, I calculate their effects at the sample mean of the variable and contrast it to their effects when the variable equals zero, holding other variables constant. This reveals that store-specic variables have stronger effects than variables measuring duration in competing stores. The relative effects are the following. I rst report effects for the threshold

Retail Sales

57

$0.99 and then comment on other thresholds. Supermarket-specic variables have strong effects, with an average effect of 0.81 for T-HEINZ and 0.98 for T-HUNTS. The effect of T-HEINZ-MIN equals 0.32. For T-HUNTS-MIN the effect equals 0.10 at the sample mean relative to the effect of zero. As we move to higher thresholds the effects of these variables become the following: T-HEINZ has effects of 0.62 and 0.06. The effect of T-HUNTS becomes 0.50 and then equals 0.10 at the threshold of $1.19. The effect of T-HEINZ-MIN declines to 0.28 and 0.24. The effect of T-HUNTS-MIN becomes 0.05 at the threshold at $1.09 and 0.30 at the threshold at $1.19. Table 8 reports the results for Hunts. There are a total of only 40 sales observations at the threshold of $0.99, and, therefore, only linear coefcients of duration variables are included. An examination of coefcients reveals that, with one exception, duration variables have positive effects. As the time since the last sale in the store or in competing stores increases, sales are more likely to occur. The exception occurs at the threshold of $0.99 for the variable MINT-HEINZ. There the effect is negative. Warner and Barsky (1995) document that sales for consumer appliances occur on weekends and holidays. Table 5 reveals that, at least for Hunts, the peak period of demand is during the early summer. A Fourth of July dummy variable is included in tables 7 and 8 to examine this effect. Both in table 7 and in table 8, the dummy variable is positive. It is signicant in all but one specication. The probability of a sale is higher during periods of high demand, which conrms the nding of Warner and Barsky. To determine whether the competition for accumulated shoppers is less important than the monopoly sales decision, I construct the following test. I estimate the sales decision of the monopolist, using only supermarket-specic variables measuring the time since the last sale, within the supermarket under consideration, for each product. I include variables for Heinz and Hunts separately to permit product-specic effects. The competitive sales decision is estimated using only the minimum time since the last sale across all supermarkets. The monopoly sale decision is estimated using only store-specic duration variables. The two alternative models correspond to the extreme cases of g p 0 and g p 1 in Section II. To determine which model ts the data best, I use Akaikes information criterion. Since the number of parameters is the same in both specications, this is equivalent to comparing the likelihood of the two specications. I nd that for Heinz the competitive model ts the data better than the monopoly model at the lowest two thresholds. For the highest thresholds the monopoly model ts the data better than the competitive model. For Hunts, the monopoly model ts the data better at all three thresholds. The test compares two extreme cases and suggests that competition for lows that are shoppers is important at least for the sale decision of Heinz. As is pointed out in Section IV, the model permits several equilibria: a symmetric equilibrium in which retailers behave in the same way in every time period and asymmetric equilibria in which one or more retailers refrain from holding a sale in some time periods. In the symmetric equilibrium the

58

Journal of Business

variables measuring time since the last sale increase the probability of sale. Under asymmetric equilibria, these variables affect the probability of sale but not necessarily in a monotone way. Moreover, in the symmetric equilibrium there should be no differences across retailers. In asymmetric equilibria there may be differences across retailers. Of course, differences across retailers may also arise if demand differs across stores. The coefcients in tables 7 and 8 may indicate whether the observed behavior is more likely to have arisen from asymmetric or symmetric behavior. An examination of coefcients did not indicate asymmetric behavior. The estimates support the symmetric equilibrium. There are two variables that may be examined. First, the time since the last sale, and second, the supermarket-specic dummy. As described in table 7, the variables measuring the time since the last sale consistently increase the probability of a sale at a decreasing rate. This is in accordance with the predictions of the symmetric equilibrium. In table 8 the results are similar. The time since the last sale has a positive effect on the probability of a sale. The store-specic dummies may indicate differences across supermarkets. For Heinz we cannot reject the null, that supermarket-specic dummies have no effect. For Hunts, the null is rejected at the 1% level. A closer examination reveals that chain 4 offers Hunts on sale more frequently than other chains do. Between chains 1, 2, and 3 there are very few differences. To summarize, the estimates of the decision to hold a sale support the model in Section IV. Both store-specic and competitive-duration variables are signicant. Product-specic duration variables have stronger effects than duration variables for the competing brand. In the next section, the determinants of demand during a sale period are reported. B. Demand Effects

I next consider the determinants of demand. According to equation (2), demand i during a sale period in supermarket k for product i, Dk, t, equals the stock of low-valuation consumers plus high-valuation consumers. This can be estimated from the data using the time since the last sale as a measure for the stock of low-valuation consumers. Specically, demand conditional on a sale period is described by
i Dk, t p aTk, t

bXk, t

nk, t ,

(5)

where D denotes demand during the rst week of a sale, Tk, t denotes a vector of duration variables, X denotes additional explanatory variables, and nk, t is idiosyncratic randomness. To account for selection of sales periods, I include the Mills ratio in the estimation. It is constructed from the rst-stage probit estimates of equation (4). Tables 9 and 10 report the determinants of weekly demand in supermarkets during the rst week of a sale period. Explanatory variables include duration

Retail Sales TABLE 9

59 Ordinary Least Squares: Demand for Heinz during the First Week of a Sale Period P-CRITICAL $.99 $1.09 108 96 .60 .5874 (1.7) .1589 (2.0) .2349 (.9) .0190 (.5) .4555 (1.1) .0369 (.1) 1.7092 (2.5) 1.1463 (2.6) .9400 (7.1) .7404 (5.0) .0550 (.3) .61 1.4280 (3.8) .4076 (4.9) .2266 (1.3) .0152 (.5) .9210 (.4) .4913 (1.5) 1.1547 (1.8) .5163 (1.2) .8078 (6.0) .8214 (5.2) .2556 (1.2)

Number of sales Degrees of freedom R2 Variable: T-HEINZt T-HEINZ-SQt T-HUNTSt T-HUNTS-SQt MIN-T-HEINZt MIN-T-HEINZ-SQt MIN-T-HUNTSt MIN-T-HUNTS-SQt HEINZ-ADVERTISING AVERAGE-STORE-SALES MILLS-RATIO

100 88

Note.Dependent variable: Heinz demand. All nonqualitative variables are in logarithms. Absolute values of t-statistics are displayed in parentheses. T-HEINZt denotes the time elapsed since the last sale for Heinz in that supermarket measured in weeks. T-HEINZ-SQt denotes the square of T-HEINZt. T-HUNTSt denotes the time elapsed since the last sale for Hunts in that supermarket measured in weeks. MIN-T-HEINZt denotes the minimum time elapsed since the last sale for Heinz across all supermarkets. MIN-T-HUNTSt denotes the minimum time elapsed since the last sale for Hunts across all supermarkets.

variables that measure the time since the last sale in weeks,9 an advertising indicator that measures whether advertising appeared in the last edition of the newspaper, and a variable that measures average sales volume during a sale period in that store. In order to account for possible nonlinearities, squared terms for duration variables are included in table 9. In table 10, because of the smaller number of observations, only linear duration variables are used. In table 9 about 60% of the variation in demand is explained. In table 10 more than 49% in the variation in demand is explained. To determine whether duration variables are jointly signicant I construct an F-test. In table 9 the null of no signicant effects can be rejected at the threshold of $0.99 at the 5% level and at the threshold of $1.09 at the 1% level. In table 10, the null
9. Measuring the time since the last sale in weeks appears appropriate from table 2. However, I also used biweekly measurement, and the results were similar.

60 TABLE 10

Journal of Business Ordinary Least Squares: Demand for Hunts during the First Week of a Sale Period P-CRITICAL $.99 $1.09 58 50 .73 .0751 (1.0) .1583 (1.9) .5875 (.9) .6195 (.9) .5130 (2.4) .9379 (8.4) .3290 (1.8) .46 .0840 (.7) .3782 (2.9) .1962 (.2) .0418 (.0) .1618 (.5) .8720 (4.3) .2249 (.9)

Number of sales Degrees of freedom R2 Variable: T-HUNTSt T-HEINZt MIN-T-HUNTSt MIN-T-HEINZt HUNTS-ADVERTISING AVERAGE-STORESALES MILLS-RATIO

40 32

Note.Dependent variable: Hunts demand. All nonqualitative variables are in logarithms. Absolute values of t-statistics are displayed in parentheses. T-HUNTSt denotes the time elapsed since the last sale for Hunts in that supermarket measured in weeks. T-HEINZt denotes the time elapsed since the last sale for Heinz in that supermarket measured in weeks. MIN-T-HUNTSt denotes the minimum time elapsed since the last sale for Hunts across all supermarkets. MIN-T-HEINZt denotes the minimum time elapsed since the last sale for Heinz across all supermarkets.

cannot be rejected at the lowest threshold but can be rejected at the threshold of $1.09 at the 2% level. The effects of duration variables in table 9 are the following: an examination of coefcients reveals that linear coefcients have a positive sign and quadratic coefcients have a negative sign. The coefcients imply that, as a particular duration variable increases, demand is increased initially and then decreased. To illustrate the effect of variables, I report the marginal effect of duration variables evaluated at two values: 2 and 4 weeks. For comparison the sample mean of the variable T-HEINZ equals 3.8 weeks with a standard deviation of 8.1 weeks. I rst report effects at the lowest threshold and then comment on the effects at the threshold of $1.09. Two weeks after the last sale of Heinz, a 1% increase in T-HEINZ increases demand by 0.37%. This effect falls to 0.14% after 4 weeks. The marginal effect of T-HUNTS equals 0.21% after 2 weeks. This effect declines to 0.18% after 4 weeks. Two weeks after the last Heinz sale in any store, a 1% increase in MIN-T-HEINZ increases demand by 0.40%. After 4 weeks this effect declines to 0.35%. The marginal effect of MIN-T-HUNTS equals 0.12% after 2 weeks. At the threshold of $1.09 the effects in table 8 are the following: a 1% increase in T-HEINZ increases demand by 0.86% 2 weeks after the last sale and 0.30% after 4 weeks. The marginal effect of T-HUNTS equals 0.21% after 2 weeks and declines to

Retail Sales

61

0.18% after 4 weeks. MIN-T-HEINZ has an effect of 0.24% after 2 weeks, and the marginal effect of MIN-T-HUNTS equals 0.44 after 2 weeks. In table 10, duration variables enter only linearly. The effects of storespecic duration variables are the following: the effect of an increase of THUNTS by 1% on the demand for Hunts equals 0.08 at both thresholds. The effect of an increase of T-HEINZ by 1% on the demand for Hunts equals 0.14 at the threshold of $0.99, and it equals 0.36 at the threshold of $1.09. To assess the importance of the demand accumulation effects, I next report the cumulative effect of store-specic duration variables. For Heinz, the cumulative effect can be calculated from the linear and quadratic coefcients in table 9. I rst comment on the cumulative effects at the threshold of $0.99: T-HEINZ increases Heinz demand by 30% after 4 weeks. If all other stores do not hold a sale for Heinz during this time period, then Heinz demand increases additionally by 30%. T-HUNTS increases Heinz demand by 20% after 4 weeks. The effects are larger in magnitude at the threshold of $1.09: as T-HEINZ increases from 0 to 4 weeks, Heinz demand is increased by a total of 138%. T-HUNTS increases Heinz demand by 21% after 4 weeks. For Hunts the effects are, in general, smaller in magnitude. At the threshold of $0.99, T-HUNTS increases Hunts demand by 8% after 4 weeks. T-HEINZ increases Hunts demand by 19% after 4 weeks. The effect of T-HUNTS increases to 14% at the threshold of $1.09, and the effect of T-HEINZ increases to 200% at the threshold of $1.09. VI. Conclusion

This article examines supermarket prices for ketchup products to explain temporary price reductions. The descriptive data analysis appears to rule out a Varian-style sales model as an explanation of the temporary price reductions. I nd that demand at low price levels depends on past prices, suggesting intertemporal effects in demand. Based on the descriptive data analysis, I consider a model of intertemporal pricing in which a xed number of consumers arrive every period. Low-valuation consumers may wait for the occurrence of a sale, and, thus, the stock of low-valuation consumers accumulates in the duration since the last sale. I distinguish two stocks: a stock of storeloyal consumers and a stock of shoppers. I nd that the timing of ketchup sales is well explained by the model. Storeloyal consumers are important for the timing of sales. The probability of a sale is increasing in the stocks at a decreasing rate. In addition, competition between retailers for accumulated shoppers, as in Sobel (1984), inuences the sale decision. The demand accumulation effect implies that the number of units purchased at low prices increases in the duration since the last sale. At a sale price of $0.99, an increase in the time since the last sale of Heinz at the same store from 0 to 4 weeks increases Heinz demand by 31%. If all other stores do not hold a sale for Heinz during this time period, then Heinz demand increases

62

Journal of Business

additionally by 30%. The demand accumulation effect is asymmetric. It affects demand during low prices but not during high prices. Thus, estimates of the demand elasticity of ketchup based on a static model may be substantially overestimated.

Appendix
Proof of proposition 1. We consider the case in which m p 1. To obtain the case g p 1, demand in all time periods is rescaled by a factor of m. The analysis does not change. First observe that the retailer prefers selling to high-valuation consumers over not selling any goods at all. The optimal price in nonsale periods equals v1. During a sale period, the optimal price equals v2, since this is the highest price at which low-valuation consumers are willing to purchase. The policy of the retailer can thus take only two values, sale or no sale. Let V(T, c) denote the expected discounted future payoff to a retailer under the optimal pricing policy, with state of demand characterized by T and current cost realization c. We can write the functional equation of the problem in the following way: V(T, c) p max {[a a(v1 T(1 c) a)](v2 dEc V(T c) dEc V(1, c ), 1, c )}, (A1)

where Ec denotes the expectation operator. The rst term in the max is the gain from holding a sale, and the second term is the gain from not holding a sale. The retailer holds a sale if the rst term in the max exceeds the second term. Clearly a retailer will not hold a sale if c 1 v2 , since this results in a loss. If the cost is low, c ! v2, the retailer is willing to hold a sale if the gains from holding a sale exceed the expected future gains from not holding a sale in that period, [a T(1 a)](v2 c) dEc V(1, c ) a(v1 c) dEc V(T 1, c ).

Equivalently, this can be rewritten as T(1 a)(v2 c) a(v1 v2 ) dEc [V(T 1, c ) V(1, c )]. (A2)

Let Ti denote the smallest T such that equation (A2) is satised at cost realization ci. Observe that T1 ! T2, since the right-hand side in equation (A2) is independent of the current period cost realization and the left-hand side is decreasing in the cost. I next show that for T 1 Ti the retailer holds a sale if the cost realization is ci. I prove this rst for cost realization c1 and then for cost realization c2. Consider the following inequalities for cost realization c1: (T1 j)(1 a)(v2 c1) a(v1 c1) 1, c ) j v2 ) 1, c ) c1) V(1, c )] V(1, c )]

j(1

a)(v2

dEc [V(T1 j(1 a)(v2

1 dEc [V(T1 dEc [V(T1

1, c )

V(1, c )].

Retail Sales

63

The rst inequality is obtained by adding j(1 a)(v2 c 1 ) on both sides of inequality (A2). The second inequality is strict, since d ! 1 . To see the third inequality, observe that V(T 1, c ) V(T, c ) (v2 c 1 )(1 a). The policy in state (T, c ) that mimics the behavior of the policy at (T 1, c ) achieves this bound. Repeated application of this bound establishes the inequality. This completes the proof for cost realization c 1. To complete the proof I consider next the cost realization c 2 . I distinguish two cases, v2 c 2 dEc (v2 c) and v2 c 2 ! dEc (v2 c). First case: v2 c 2 dEc (v2 c). Consider the following inequalities. (T2 j)(1 a)(v2 c2) a(v1 c) v2 ) dEc [V(T2 j(1 a)(v2 1, c ) c) V(1, c )] V(1, c )]

j(1

a)dEc (v2

p dEc [V(T2 dEc [V(T2 j

1, c ) 1, c )

V(1, c )].

The rst inequality is obtained by adding j(1 a)(v2 c 2 ) on both sides of inequality (A2) evaluated at T2 and using v2 c 2 dEc (v2 c). The equality is obtained by rearranging terms. The last inequality repeatedly uses Ec [V(T 1, c ) V(T, c )] (1 a)Ec (v2 c). To see that this inequality is satised, observe that for T T2 a sale always occurs if the cost realization equals c1. The marginal gain in the value function is thus given by Ec [V(T 1, c ) V(T, c )] p (1 (1 a)r(v2 c 1) c 2 )(1 a), 2, c ) V(T 1, c )]}.

r) max {(v2

dEc [V(N

This equation is satised for any T 1 T2, and we can solve it. If (v2 c 2 )(1 a) dEc [V(N 2, c ) V(T 1, c )] for all T, then solving yields Ec [V(T 1, c ) V(T, c )] p (1 a)Ec (v2 c). Since v2 c 2 dEc (v2 c), this is indeed a solution. Suppose next that (v2 c 2 )(1 a) ! dEc [V(N 2, c ) V(T 1, c )] for some T. If (v2 c 2 )(1 a) dEc [V(N 3, c ) V(N 2, c )], then Ec [V(N 2, c ) V(T 1, c )] p (1 a)Ec (v2 c) a contradiction. Thus it has to be that (v2 c 2 )(1 a) ! dEc [V(N 2, c ) V(T 1, c )] for all T. Solving yields Ec [V(T 1, c ) V(T, c )] p {r(v2 c 1 )/[1 d(1 r)]}(1 a). Since v2 c 2 dEc (v2 c) this cannot be a solution and the rst guess is the only solution. This concludes the proof for the rst case. Second case: v2 c 2 ! dEc (v2 c). Since for T T2 a sale always occurs if the cost realization equals c1, the marginal gain in the value function is given by Ec [V(T 1, c ) V(T, c )] p r(v1 c1)(1 a) (1 r) max {(v2 c 2 )(1 a), dEc [V(N 2, c ) V(T 1, c )]}. Using the assumption v2 c 2 ! dEc (v2 c), we can solve this equation in the same way as before and obtain Ec [V(T 1, c ) V(T, c )] p {r(v2 c 1 )/[1 d(1 r)]}(1 a). Consider now equation (A2). Suppose it holds for the rst time at some T2. At T2 1 we have (T2 1)(1 a)(v2 c) a(v1 v2 ) ! dEc [V(T2 , c ) V(1, c )].

64

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Subtracting the right-hand side of this inequality from the left-hand side in equation (A2) and the left-hand side from the right-hand side in equation (A2) yields r(v1 c 1 ) V(T2 , c )] p (1 1 d(1 r)

(1

a)(v2

c 2 ) dEc [V(T2

1, c )

a).

The equality follows from the previous argument. Canceling and rearranging yields (v2 c 2 ) d[r(v2 c 1 ) (1 r)(v2 c 2 )]. The right-hand side equals dEc (v2 c), which is a contradiction. Thus, in the second case, T2 p . This completes the proof. Q.E.D. Before establishing the next proposition I state the following lemma, which was established by Sobel (1984). It shows that the per-period payoff during a sale period equals the payoff during a nonsale period. The proof is similar to a Bertrand argument. The lemma does not require a stationary wholesale price distribution, and it holds for any family of distribution functions. Lemma 3. For any retail store the prot in period T, with wholesale price c, is given by (a/m)(v1 c). Proof of proposition 2. Consider any state of demand T with cost realization c. Let G k ( p) denote the price distribution of the minimum price charged in retail stores other than retail store k. Let [p, p] be the set of points that contain the support of prices charged below v2 during a sale period for product i in period T. The prot from charging prices in this region is given by [(a/m) T(1 a)](p c)[1 G k ( p)] . From the previous lemma in equilibrium this equals (a/m)(v1 c). Rearranging this equality we obtain (a/m) [(a/m) (v1 c)

G k ( p) p 1

T(1

2a)](p

c).

Let G(p) denote the price distribution of retail store k. By symmetry assumption, the minimum price charged in retail stores other than retail store k equals the rstorder statistic of m 1 random draws from G. The distribution of the rst-order statistic equals G k ( p) p 1 [1 G( p)] m 1 . We can rewrite this expression to obtain the price distribution in retail store k as G( p) p 1 [1 G k ( p)] [1/(m 1)]. Substituting the above expression for G k ( p) yields the second line of G( p) in the proposition. To characterize the range [p, p], observe that the lower endpoint of the support is given by the equation G(p) p 0 or 1 [(a/m)/(v1 c)][(a/m) T(1 a)](p c) p 0. This can be rewritten as p c p [(a/m)(v1 c)]/[(a/m) T(1 a)]. To see that the upper endpoint of the support is given by p p v2, suppose not; that is, p ! v2. Then the profit of a seller charging v2 during a sale period is given by [(a/m) T(1 a)](v2 c){[(a/m)(v1 c)]/[(a/m) T(1 a)](p c)]} p [(v2 c)/(p c)](a/m)(v1 c) 1 (a/m)(v1 c), a contradiction to the lemma. Q.E.D. Example of an asymmetric equilibrium. Suppose all but the rst two retailers always charge price v1 and the rst two retailers charge a price according to the following strategy. If T ! a(v1 c)/(1 a)(v2 c), or if c 1 c j, then the price equals v1; otherwise the price is drawn from the distribution function

Retail Sales

65

G( p, T, c) p

0 1 1 1

if p ! c

(a/m)(v1 c) ; T(1 a) c (a/m)(v1 c) , v2 ; (a/m) T(1 a)

[ [

(a/m) (a/m)

(a/m)(v1 c) T(1 a)( p (a/m)(v1 c) T(1 a)(v2

c)

c)

] ]

if p if p

[v2 , v1 );

if p v1.

To see that this constitutes an equilibrium, consider either of the rst two retailers. If retailer 1 charges a price of v1, she makes a prot of (a/m)(v1 c), which is the best she can do by lemma 3. If she charges a price p [c (a/m)(v1 c)/[(a/m) T(1 a)], v2 ], she receives [(a/m) T(1 a)](p c)(a/m)(v1 c)/[(a/m) T(1 a)](p c), which equals (a/m)(v1 c). No other price yields higher payoffs, so the strategy is optimal for retailers 1 and 2. Now consider retailer 3. If he charges a price of v1, he makes a prot of (a/m)(v1 c), which is the best he can do by lemma 3. If he charges a price p [c (a/m)(v1 c)/[(a/m) T(1 a)], v2 ], he receives [(a/m) T(1 a)](p c){(a/m)(v1 c)/[(a/m) T(1 a)](p c)} 2, which equals (a/m)(v1 c){(a/m)(v1 c)/[(a/m) T(1 a)](p c)}; this is less than (a/m)(v1 c) since the term in square brackets is less than one. All other possible prices do not yield payoffs exceeding (a/m)(v1 c); thus the strategy is optimal for retailers k p 3, . . . , m. The strategies described above constitute an equilibrium.

References
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