PRICING BEHAVIOR OF MULTIPRODUCT RETAILERS

Daniel Hosken and David Reiffen

Federal Trade Commission

Revised: May, 2001

keywords: Retailing, Pricing Dynamics, Sales

JEL codes: L81, L4, L11, D21

We would like to thank Cindy Alexander, Keith Anderson, Mark Armstrong, Jeremy Bulow,

John Howell, John Simpson, and Chuck Thomas for their helpful comments, and Elizabeth Autry,

Sara Harkavy and Cindy McMullen for providing us with excellent research assistance. The views

expressed in this paper are those of the authors and should not be viewed as those of the Federal

Trade Commission nor any individual Commissioner.
ABSTRACT

While retail sales are an important economic phenomenon, previous research has not explored how

the multiproduct nature of consumers’ purchasing decisions affects the pricing dynamics of

multiproduct retailers. In this paper, we first document the extent to which “sales”, defined as

temporary discounts in retail price, are a pervasive aspect of retailing and an important source of

retail price variation. Using a large data set for 20 categories of grocery products across 30 U.S.

metropolitan areas, we find that the majority of price changes and 25%-50% of retail price variation

is the result of retail sales. We then develop a model describing the pricing dynamics of

multiproduct retailers that is consistent with these empirical pricing regularities. Specifically,

because consumers prefer to buy a bundle of goods from the same retailer, a given discount on any

one good in the bundle will have a similar effect on consumers’ likelihood of visiting that retailer.

This implies that discounts on goods sold by a single retailer are substitutes instruments for retailers,

and factors that influence one good’s price will affect the pricing of other goods. Hence, if

intertemporal price changes are a means of price discriminating (as suggested in the literature), the

impact of these changes will be reflected in the prices of many goods, including even those for which

discrimination is not feasible. 
1This concept of a sale contrasts with other kinds of systematic price reductions that have

been documented. One such pattern is that prices for goods with a “fashion” element often

systematically decline over a fashion season (see, e.g., Pashigian (1988), Pashigian and Bowen

(1991), Warner and Barsky (1995)), as retailers learn which styles are popular with consumers. 

We view this type of sale as a fundamentally different phenomenon. 

1

I. Introduction

As consumers, we all have some familiarity with the complex pricing strategies employed

by supermarkets. For instance, among the more than 20,000 items they carry, supermarkets choose

to offer only a small fraction of the items at a low “sale” price each week. Despite the high

administrative costs of changing retail prices (Levy, et al.(1997)), retailers clearly find it profit

maximizing to put different items on sale each week. The goal of this paper is to provide some

insight into the extent of, and the reason for, this aspect of retailer behavior. First, we document the

extent to which sales - defined as temporary price reductions - occur, and the relative importance of

temporary versus permanent price changes.1 Second, we develop a model to explain retailers’

rationale for temporarily lowering prices. Specifically, we determine equilibrium pricing behavior

in a dynamic model in which competing retailers each sell two goods, a non-perishable good which

can be inventoried by consumers, and a perishable good which cannot. 

To document the extent to which sales occur, we make use of a novel data set. The data,

provided to us by the U.S. Bureau of Labor Statistics, consists of more than 300,000 monthly price

quotes in 20 categories of grocery items collected from retailers in 30 metropolitan areas for the

period 1988-1997. A desirable feature of this data set for studying sales is that it covers a wider

range of products and geographic areas than the data used in previous studies of retail pricing. We

observe multiple price series in each of the twenty categories of goods, where each price series

represents a particular grocery item (e.g. an 18 ounce container of brand x’s creamy peanut butter

from retailer y) for up to 5 years.

Our data show that temporary price reductions are empirically-important phenomena that

pervade retail pricing in the U.S. We find that the typical grocery product has a set “regular” price,

and most deviations from that regular price are downward. The data also demonstrate that sales are

an important source of retail price variation. Roughly 60% of price decreases are the result of sale

behavior. Further, between 25% and 50% of the observed annual variation in retail prices is the
2

result of retailers placing individual items on sale. 

To understand these empirical findings, we develop a model of pricing dynamics for

multiproduct retailers. The underlying logic of the model is that because consumers prefer buying

a bundle of goods from the same retailer, a retailer’s offer of a discount on any good in a bundle will

have similar effects on a retailer’s likelihood of attracting a given consumer. This implies that when

a retailer lowers the price of one good, it will likely raise the price of another good its sells. At the

same time, retailers can price discriminate against impatient, inelastic consumers by periodically

reducing prices on non-perishable goods (those that can be inventoried by consumers -see, e.g.,

Conlisk, Gerstner and Sobel (1984)). In combination, these two factors imply that prices for many

goods, including those for which price discrimination through intertemporal price changes is not

feasible (which we call perishables), will change periodically. In addition, we show that in

equilibrium, price movements will be quite different for perishable and non-perishable goods; nonperishable pricing will feature long periods of stable prices, following by significant but short-lived

price reductions, whereas perishable prices will move more frequently, but by smaller amounts. 

In the model, the desire of retailers to take advantage of differences in consumer’s inventory

costs is an important source of retail price variation. From this perspective, our empirical findings

suggests that consumer inventorying is a sufficiently important phenomenon to motivate a large

proportion of price movements. As has been noted elsewhere (e.g., Feenstra and Shapiro (2000)),

consumer inventorying means that the response to temporary price changes (which lead to inventory

behavior) may be substantially different than the response to permanent price changes (which do

not). The recent empirical studies we are aware of which estimate demand elasticities do not control

for the fact that much of the observed variation in retail prices is temporary, e.g. Hausman, Leonard,

and Zona (1994) or Cotterill, Putsis, and Dhar (2000). Hence, if consumer inventorying is important,

then the estimated demand elasticities do not correctly measure how consumers’ purchases would

change in response to permanent price changes.

II. Evidence Regarding Sales in the U.S.

This section provides systematic evidence regarding retail pricing regularities. While there

is no single metric by which to measure (or even define) sales, the pricing patterns we find are
2A category is a fairly narrow classification of consumer goods, e.g. cola drinks, eggs, and

white bread are BLS categories.

3

consistent with our notion of sale behavior. Our results suggest that sales are a widespread feature

of retail pricing in the U.S. Empirically, most grocery products have a “regular” price which is

charged most of the time. When a price is not at its regular level, it is much more likely to be below

the regular price than above it. The evidence suggests that sales do not appear to result primarily

from unexpected changes in demand or supply. For example, they are not due to unexpectedly high

inventories of perishable merchandise or short term changes in wholesale prices. Instead, retailers

appear to systematically place the same items on sale over time, suggesting that sales are not a

response to unexpected changes in inventory. Further, price changes across retailers within a market

on any particular item are not highly correlated, suggesting that sales are not simply the result of

short term changes in wholesale prices. Together, this evidence suggests that sales at least partially

represent changes in retailer margins rather than changes in wholesale prices. Section IV presents

a model of retailer sale behavior consistent with these empirical findings. The next two subsections

discuss the data used and the specifics analyses performed. 

Data Description

Our analysis makes use of two data sets. The first is a non-public data set we obtained from

the U.S. Bureau of Labor Statistics (BLS), which to our knowledge has not been used to study retail

pricing behavior. The second is a public data set provided by A.C. Nielsen. 

The data provided to us by the BLS are collected for use in calculating the Consumer Price

Index. Each month, the BLS samples food retailers in 88 geographic areas, collecting prices of

specific items in up to 94 categories of goods.2 Within each category, the BLS samples the price of

a specific item at the same store for up to 5 years. For example, a price series for cola may consist

of monthly prices of a 2-liter bottle of Coke in a particular retail outlet in Boston for a 5 year period.

The data we use in this study consists of these individual price series for specific products. While

most product categories have multiple price series in each geographic area, the price series provided

to us do not contain information that identifies the specific product and package size sampled within

each category. We only know that all of the prices within a price series correspond to prices for a
3For this reason, we cannot use the BLS data to examine any implications regarding the

relationship of prices movements on multiple products within a store.

4The geographic areas are: Atlanta, Boston, Buffalo, Chicago, Cleveland, Dallas, Dayton,

Denver, Detroit, El Paso, Greater Los Angeles, Jacksonville, Kansas City, Los Angeles, Miami,

Minneapolis, New Orleans, New York and Connecticut suburbs of New York City, Philadelphia,

Portland, Richmond, St. Louis, San Diego, San Francisco, Scranton, Seattle, Syracuse, Tampa,

Tucson, and Washington D.C. See Hosken, Matsa, and Reiffen (1999) for more details on this

data set.

4

specific product at a specific store within a category. For example, we do not know whether that

specific cola product in a price series is a 12-pack of Coke or a 2-liter bottle of Pepsi. We also

cannot identify the store or chain associated with each price series. Hence, we cannot determine

when two series are taken from the same store or chain.3

 The data we received from the BLS

contains all of the price series the BLS collected on 20 categories of goods (see Table 1 for specific

products) from 30 geographic areas for the period 1988-1997.4

To analyze the relationship between specific product prices across retailers at a point in time,

e.g. the price of a 28 ounce bottle of Heinz ketchup across retailers within a city, we use a public use

data set from A.C. Nielsen. This data set contains specific product and price characteristics for eight

categories of goods at the individual store level for Springfield, MO and Sioux Falls, SD for the 124

week period beginning January 23, 1985.

Empirical Findings

Sales are a common feature of retail pricing. Table 1 provides summary statistics about the

extent of “sales” in the BLS data set. The first column of Table 1 shows the number of price series

in each category. The next two columns describe the sales phenomenon. Specifically, our concept

of the phenomenon is that prices are at their “regular” level most of the time, and are significantly

lower for brief periods. To examine whether this pattern exists, we begin by developing a measure

of a product’s regular price. We do this by dividing each price series in the data set into individual

price series for each calendar year (e.g. the tenth price series for crackers in Chicago for 1996). From

each calender year, we calculate the modal price for each annual price series, which we view as the

regular price. Given this measure, we then determine the frequency with which prices in each
5We considered five definitions of a sale in performing this calculation: a price decrease

in month t of at least 5, 10, 15, 20, or 25 percent followed by a price increase in month of t+1 of

at least the same amount. In the interest of brevity, only the results for the 10% definition are

presented here. The results for other sale definitions are quite similar.

6Specifically, we perform the following calculation. First, for every price series with at

least 24 observations, we record whether that series experienced a sale during the first 12 months

for which we have data. Next, we divide the sample into two parts: The first contains price series

5

individual price series are equal to their annual modal values. Finally, we calculate the average

frequency at the modal value over all of the series in each product category. For example, the

average baby food price is at its modal value 72.05% of the time. With the exception of eggs and

lettuce, the average product is at its modal price at least 50% of the time. We conclude that one

feature of retail pricing is that most products have a “regular” price.

The second aspect of the sale phenomenon is evaluating what happens when price is not at

its regular level. If sales are important, then when prices are not at their regular level, they should

be more likely to be below the regular price than above it. Hence, we test for sales by comparing the

percentage of deviations from the modal price that are below versus above the mode for each product

category in our sample. In the third column of Table 1 we test this hypothesis by calculating the ratio

of the number of observations below the regular price to the number above it. For every category

of goods, the ratio is well above 1. Testing the hypothesis that the number of prices below the mode

is larger than the number above the mode, we find that for every category we can reject the

hypothesis of equality at the 1% confidence level. 

While retailers certainly have incentives to place items on sale when their inventories

unexpectedly increase, the evidence also suggests that is not the primary cause of retail sales. This

conclusion follows from the observation that, unless retailers’ excess inventories (e.g., due to

forecasting errors) are systematically more common on specific products within categories, we

would expect that sales due to excess inventories would be equally common on all items within a

category. If instead retailers systematically place certain items within a category on sale quite often,

while others rarely (if ever) go on sale, it would suggest other motivations for sales. We examine

whether there are predictable patterns to which products go on sale by comparing the probability an

item goes on sale5 in year t conditional on the product going on sale or not going on sale in year t-1.6
that have at least one sale in the first 12 months and the second contains those price series that do

not have a sale. Within each product category we then calculate two conditional probabilities;

the probability that a price series would experience a sale during the second year of the sample

conditional on the product being in the first group (i.e., having a sale within the first 12 months),

and the probability of a sale in the second year conditional on being in the second group.

7The corresponding number of z-statistics over 2.5 using all 5 sale definitions was 91 out

of 100. For some of the comparisons of conditional probabilities, the number of price series is

very small. In these cases it is incorrect to assume that the difference in proportions is

approximately normal, and instead we simply interpret the computed z-statistics as measures of

the size of the difference between conditional probabilities. Tables showing these tests for all

five sale definitions are available from the authors on request.

8The assumption that in each city all retailers’ wholesale prices move together is based on

our understanding of industry practices, along with legal restrictions on differential pricing due to

the Robinson-Patman Act. The Robinson-Patman Act makes it illegal for a firm in the U.S. to

6

If products are put on sale randomly (e.g., because of unexpectedly high inventories), then the two

conditional probabilities should be the same. The results of these calculations appear in Table 2. For

every product category in our sample, the conditional probability of observing a sale in year t is

larger, often substantially larger, if the price series experienced a sale in year t-1. In fact, for all 20

comparisons shown in Table 2, we reject the null hypothesis with a z-statistic greater than 2.5.7

 This

result is robust across 20 large categories of goods, over time, across the U.S. and for five different

definitions of sales. 

This finding that within categories, goods differ in their probability of going on sale is

consistent with some recent findings. Hosken, Matsa and Reiffen (2000) show that within 

categories, goods that are more “popular” (e.g., those with higher market shares) are more likely to

go on sale. They also present evidence consistent with that result. Hence, it appears to be the case

that the sale pattern portrayed in Table 1 is more likely to be found for popular goods.

We also examined the extent to which retail price changes represent changes in retail

margins, rather than in wholesale prices. Under the assumptions that (1) prices paid by retailers

(wholesale prices) move together in each city, and (2) wholesale price changes are reflected in retail

prices with a lag that is common across all retailers, we would expect that retail price changes for

a given product would be highly correlated across retailers if sales were primarily driven by

wholesale price changes.8

 Using the data from A.C. Nielsen, we calculate the correlations of price
charge different prices to two buyers of a good, unless those buyers are end users (subject to

some exceptions and defenses). 

9Several of the correlation matrices are reported in Hosken and Reiffen (1999). The

entire set of correlation matrices are available from authors.

10Some direct evidence on this point can be found in Chevalier, Kashyap, and Rossi

(2000). Their data set includes both retail and wholesale prices for one U.S. supermarket chain. 

They find that most of the temporary reductions in retail prices for the goods they examine reflect

retail margin changes, rather than wholesale price changes. Evidence reported in Levy, et al.

(1997) also shows that most retail prices changes are actually changes in retail margins.

7

changes across stores for the top three products (ranked by market share) in the eight categories of

goods contained in the data set for two markets. We find that nearly half of the correlations are

negative, and only 7% are greater than 0.3 (see summary in Table 3).9

 This finding suggests that

retail price changes were not primarily driven by changes in wholesale prices.10

The evidence presented above conforms with our experience as consumers, as it shows that

most goods have a “regular” price, and that most deviations from those regular prices are downward.

A question that is perhaps less clear is the importance of sales as a source of price variation. That

is, are sales an empirically important phenomenon? To evaluate this question, we present several

empirical means of distinguishing between temporary and permanent changes in retail prices. 

One way to examine the extent of temporary changes is to analyze the time series of first

differences in price. Specifically, we examine the price changes between month t and t+1,

conditional on price falling between months t-1 and t. If a price reduction is temporary, rather than

permanent, then price would rise between month t and t+1. In contrast, if the price change between

months t and t+1 is zero (or negative), it would suggest that the retail price movement reflects a

permanent change in the retailer’s cost (and/or the manufacturer’s cost). In fact, as Table 4 indicates,

across all categories of goods, 60% of price reductions are followed by a price increase, while only

23.3% remain at the new, lower level. This suggests that the majority of retail price reductions are

temporary.

A more formal way to evaluate price movements is to statistically decompose the price

variation. We compare the price variation due to temporary reductions to the price variation caused

by other sources. In particular, for many products, there are predictable price changes due to harvest
11Where a product is defined to go on sale if its price falls by at least 5% between period

t-1 and t, and then rises by at least 5% between period t and t+1.

12There are three additional details involved in estimating equations (1) and (2). First,

because each price series within a product category corresponds to a unique product (e.g. within

the cola category one price series could correspond to a 2 liter bottle and another to a case of

cans), comparing the price level across products is not meaningful. For this reason we have

scaled each price series by its mean value, so that the mean of every product’s scaled price will

be 1, and every price change can be interpreted as a proportional change. Second, because we are

interested in describing the importance of relatively short term changes in price, we are

restricting attention to one year time periods. In particular, when we scale price by a price series’

mean, that mean is calculated over a 12 month period. Hence, our results only correspond to the

8

periods, or year-to-year changes in costs or demand. Thus, we first run a regression to determine the

extent to which national shocks in each time period explain retail price variation. Specifically, for

each product i, we regress its price in city j at time t on separate dummy variables for each of the 120

months in our data as depicted in equation (1) below.

The r-squared from these regressions tells us how much of the variation in retail price for each

category is explained by national, time-specific shocks, such as supply changes. If all supply

changes are national, and effect all products in a category, then the k will pick up the price changes

due to changes in supply. 

To determine the proportion of price variation accounted for by sales, we wish to design a

model which will capture the variation in retail price caused by sales. We do this by adding a

separate dummy variable to equation (1) which is equal to 1 each time a product goes on sale

(equation (2) below).11 The addition of a separate dummy variable for each observation that has a

sale effectively controls for all of the price variation associated with observations that have sales.

Thus, the residual from equation (2) is the result of “permanent” changes in retail prices, other than

those associated with nation-wide time-specific shocks (e.g., seasonality) and sales. Hence, the

difference in the r-squared between equations (1) and (2) has the interpretation of the additional

portion of the retail price variation explained by sales (i.e., beyond that explained by time

dummies).12 
importance of sales in explaining annual variation in price. Finally, because sale behavior differs

across product categories, equations (1) and (2) are estimated separately by product category.

9

Figure 1 shows the r-squareds from equation (1) for each product category (the left-hand bars), and

the difference between the r-squareds from equations (2) and (1). As one would expect, these

calculations show that the price variation due to nation-wide shocks (such as supply changes) are

largest for the three products - bananas, eggs, and lettuce - that are relatively unprocessed agricultural

products. The time effects have little explanatory power for the remaining products. In contrast, for

every product category, the incremental contribution of sales to explaining variance is at least 25%,

and as much as 50%. 

We also explored the question of whether the (n may be capturing sub-national (i.e., regional)

supply shocks. To examine this question, we estimated equations (3) and (4), which generalize

equations (1) and (2) by allowing the time effects ( k

) to vary across the 4 census regions. 

The idea behind equations (3) and (4) is to allow for region-specific supply shocks. Figure 2 presents

the percentage of retail price variation explained by the region- and time-specific shocks in equation

(3), and the difference between the r-squareds from equations (3) and (4). Comparing Figures 1 and

2, we note that as one would expect, allowing the time effects to vary across census regions increases

the percentage of variation explained by the time effects. However, for 18 of the 20 categories, the

sale effect still explains more of the variation in retail prices than do the region specific time effects,

while sales explain about the same amount as the region-specific time effects for the two remaining

categories. The fact that regional shocks are more important for eggs and lettuce conforms with

economic logic. These are the two products most likely to be produced within every region, and

hence supply shocks for these products may well vary across regions. This provides some support

for our premise that the Bk in equation (1) (and the Bxk in equation 3) are capturing supply shifts. 
13Lal and Matutes (1994) use a similar explanation for competing multiproduct retailers

using different (static) pricing strategies for their array of goods. In their model, each retailer

has a low price on a different good, which causes low transportation cost consumers to buy at

more than one store each period, but allows the retailers to charge high prices on some items to

high transportation cost/high reservation value consumers. Banks and Moorthy (1999) show

that coupons can be another way of offering low prices to low reservation price/low search cost

customers, while maintaining high prices to high reservation price/high search cost consumers.

10

For every category, the incremental contribution of the sales effect is at least 25%, and

exceeds 40% for six categories. While the regional supply effects have relatively greater explanatory

power than the national shocks, the conclusion remains that temporary price reductions sales account

for a substantial proportion of the annual variation in retail prices. This is consistent with our earlier

conclusion that most deviations from the regular price are downward. 

III. Existing Models of Sales 

The previous section demonstrated that sales, in the sense of periodic, temporary reductions

in specific product prices are a ubiquitous feature of retail competition. To understand this behavior

we develop a model which draws primarily from work by Conlisk, Gerstner and Sobel (1984). The

basic intuition in their model is that consumers differ in reservation values and in their willingness

to wait (which is analytically similar to differences in costs of inventorying). Low-value consumers

are more willing to wait for price reductions because the cost of waiting is higher for the high-value

consumers, and hence only low-value consumers wait for the periodic price reductions. As a result,

periodic price reductions allow a seller to charge a low price to all low-value customers, while most

high-value customers purchase at a higher price.13 

Sobel (1984) extends this model to the case of multiple retailers. High-value consumers are

not only willing to pay more for the good and less willing to wait (as in Conlisk, Gerstner, and

Sobel), but they also are store loyal. That is, each of these consumers buys from a specific preferred

retailer if his price is below their reservation values, and do not buy at all if that retailer’s price is

above their reservation values. In contrast, low-value consumers are shoppers, buying from the

retailer offering the lowest price (as long as that price is sufficiently low). The basic characteristic

of the equilibrium in Sobel’s model is that retailers charge a high price in periods in which the
14 In contrast, in Sobel’s model, low-value consumers may wait to buy, even if price is

below their reservation values, if they expect price to fall further. Sobel shows that the expected

price decline eventually dissipates, and that consumers rationally purchase the good. Sobel’s

equilibrium constitutes a perfect Nash equilibrium, while Pesendorfer’s does not. However, 

the qualitative predictions of the Pesendorfer model are similar to Sobel’s results in that both

predict a mixed strategy equilibrium characterized by periodic sales. Because the model in

Pesendorfer is more tractable and yields similar results, our approach follows Pesendorfer’s. 

15 This formal equivalence require that low-value consumers have some inventory at the

beginning of period 1, and that when price is below their reservation values, these consumers buy

a sufficient quantity for storage to replace the inventory consumed since the previous sale. These

assumptions are discussed further in Section IV.

16Given symmetry between retailers, Varian’s assumption that some consumer randomly

choose a retailer and do not compare prices across retailers is equivalent to Sobel’s assumption

that 1/J of these consumers are “loyal” to each of the J retailers. 

11

aggregate purchases of shoppers would be small. As time passes, shoppers’ potential aggregate

purchases increase and it eventually becomes profitable to reduce price to compete for their business.

Pesendorfer (2002) both simplifies and generalizes the Sobel model. The simplification is

that he assumes low-value customers do not behave strategically - which is to say that they buy

whenever the price is below their reservation values.14 The generalization is that Pesendorfer allows

some portion of low-value consumers to be store-loyal. The Pesendorfer model is formally

equivalent to a model in which both types of consumers consume one unit of the good in every

period (rather than exit the market as soon they purchase one unit), but the low-value consumers

consume from their own inventory whenever the price is above their reservation values.

15

While Pesendorfer’s model explains price discounts for goods that can be inventoried, or

goods that are infrequently purchased, it does not explain discounts for perishable goods that are

frequently purchased and not inventoried by consumers, such as dairy products and produce. 

However, the evidence (see Table 1) suggests that prices of these items also vary considerably over

time. Varian (1980) provides a related explanation for price movements on products that are not

typically inventoried (such as consumer durables or perishable food items). As in Sobel and

Pesendorfer, Varian assumes that some customers do not compare prices across stores, but rather

buy as long as the (randomly-chosen) retailer’s price is below the consumer’s reservation value, and

others buy from the store with the lowest price.16 Retailers then choose between obtaining a high
12

price, and selling only to those customers who do not compare prices, or charging a "low" price, and

potentially selling to shoppers as well. Varian shows that the only symmetric equilibrium features

mixed strategies, where all retailers choose their price from a continuous distribution, so that each

retailer changes his price each period. 

Note that the reason for price movement in the Varian model is quite different from the

reason in Conlisk, Gerstner, and Sobel. In Varian, these movements result from competition among

imperfectly-competing retailers; a monopoly retailer would not vary price, given his demand

assumptions. In contrast, the Conlisk, Gerstner, and Sobel model is a monopoly model, and sales

are a means of price discrimination. Sales in the Sobel and Pesendorfer models combine elements

of both explanations; price movements reflect both competition and a desire to price discriminate.

IV. A Model of Sales and Multiproduct Retailers

The models described in the previous section describe how and why a single-product retailer

would adjust his price over time, even with unchanged costs. The phenomenon these models seek

to explain is the pattern of prices illustrated in Figure 3. As shown there, peanut butter prices tend

to remain constant for long periods, followed by brief periods of lower prices, followed by a return

to their initial levels. Further, as Table 1 suggests, this pattern is common for many of the goods sold

in supermarkets. One potentially-important abstraction in these models is that they consider the

pricing behavior of retailers selling only one product. Real retailers, such as supermarkets, sell a

variety of goods, and consumers prefer purchasing bundles of goods from a single retailer. In

evaluating whether the existing models explain pricing behavior, it is important to consider how

retailers’ actions might change if they had a richer set of pricing alternatives available because of

these two facts. 

To evaluate this question, we analyze competition between retailers, all of whom sell the

same two products, and those products have different storage characteristics. These two products

correspond to the two types of products described in the previous literature on retailer sale behavior

and reflect characteristics of products that supermarkets actually sell. The first good is a nonperishable (for which there is a potential for price discrimination) and the second is a perishable (for

which there is no such potential). Selling two different types of goods allows retailers to use one
17As Table 1 shows, perishable goods such as lettuce, eggs, ground beef, and bananas all

have mass points in their pricing distributions. 

18 Salop and Stiglitz (1982) make a similar assumption about the cost of visiting multiple

retailers in their analysis of sales. As discussed in Section V, similar results can be obtained

with weaker assumptions regarding the transactions cost of visiting multiple retailers.

13

product to compete with rivals in every period, while reserving the other for discriminating between

high-value and low-value consumers in certain periods. We show that, as in the Sobel model, the

desire to price discriminate results in periodic sales on the non-perishable product. We also find that,

as in the Varian model, the desire to attract shoppers leads to price movements for perishable

products. Moreover, we demonstrate that for multiproduct retailers, perishable prices move in

response to changes in non-perishable prices. This implies that there are mass points in the pricing

distribution for perishable products, an empirically-relevant result which does not follow from any

of the models described in Section III.17 Hence, incorporating the multiproduct aspect of retail

competition helps explain a broader set of pricing relationships. In addition, modeling retailers as

multiproduct sellers generates additional implications for prices. For example, we show that prices

for perishable and non-perishable goods sold by a retailer should vary inversely over time. 

As Bliss (1988) noted, an important aspect of multiproduct retailing is that most consumers

buy an array of goods each time they visit certain kinds of retailers (especially supermarkets). Our

model incorporates this feature by assuming that consumers know all of the relevant prices before

visiting any store, and shop at no more than one store in each period. Thus, if a consumer

purchases both goods in the same period, she buys both of them from the same retailer.18 It follows

that retailers compete for customers by attempting to offer the most attractive bundle of prices. 

In analyzing what constitutes the most attractive bundle, it is necessary to consider the

number of units of a good that a consumer might purchase during each visit. Of particular interest

to us are non-perishable goods, which have the property that some consumers can practically buy

more units of the good than they plan to consume in that period, inventorying a portion for later

consumption. Hence, while consumers all have unit demand for consuming each good in each

period, they do not necessarily all purchase one unit of the non-perishable each period. 

Two key assumptions in the literature on intertemporal price discrimination described in the
19These considerations also provide a justification for the assumption that reservation

values and shopping costs are correlated for the non-perishable; consumers who find shopping

particularly costly will have a high reservation value for branding.

14

previous section are that consumers are heterogeneous in both their valuations of the product and in

their cost of search, and that these two values are positively correlated across individuals. We view

such a positive correlation as plausible. For example, high-income consumers are likely to both have

higher reservation values for many goods, and due to a higher shadow value of time, lower

willingness to invest in learning about prices and taking steps to take advantage of that knowledge.

There is reason to believe this positive correlation is more likely for non-perishable goods than

perishables. For example, non-perishable goods typically have more value-added than perishable

goods (e.g., breakfast cereals or canned soup as compared to bananas or ground beef). Products with

considerable value-added by the manufacturer will typically be those for which brand names are

important. Economic theory suggests that brand names will be more valuable for consumers who

view search as particularly costly (see, e.g., Klein and Leffler, 1981, and Ward and Lee, 1999, for

recent empirical evidence). It follows that there will be greater heterogeneity in reservation values

for branded products than commodity products.

19 Relating this back to perishable and nonperishable products, we note that, consistent with the idea that brand names are more important for

non-perishables, supermarkets typically carry a single product in many perishable categories,

including several in our sample (bananas, lettuce, ground beef), while carrying multiple versions in

the non-perishable categories. 

Hence, following the literature described in Section III, we assume there are two kinds of

consumers; those who are store loyal, and do not compare prices across stores (i.e., they have high

search costs) and those who are shoppers, and evaluate stores on the basis of price. Following this

literature, we assume that store loyals have higher reservation values and storage costs for the nonperishable product than do the shoppers. Specifically, store loyals have reservation values of "H

,

which is higher than the value that shoppers ("L ) place on it. To isolate the incentive to price

discriminate, we assume that store-loyals have infinite storage costs for the non-perishable, and

shoppers have zero storage costs for it, although any significant difference in customers’ storage

costs would be sufficient for our purposes. 
15

For the reason discussed above, we assume that consumers are less heterogeneous with

respect to their reservation values for perishable goods than for non-perishables. Specifically, we

assume that all consumers have a common reservation value of $ for the perishable (which is

identical to the assumption in Varian). While this is a stronger assumption than needed for many

of our results, it allows for a simple closed-form pricing equilibrium. In section V, we discuss the

implications of more general assumptions regarding reservation values.

These assumptions imply the following about consumer behavior: Letting PP

 be the price

of the perishable and PN be the price of the non-perishable at her preferred store, a store-loyal

customer will make one of four choices in any period:

if PN > "H and PP > $ buy nothing

if PN < "H and PP > $ buy one unit of the non-perishable only 

if PN > "H and PP < $ buy one unit of the perishable only

if PN < "H and PP < $ buy one unit of each good

Shoppers also make one of 4 choices. Suppose there are J retailers and let the superscript j

index the specific store, then a shopper’s four choices in any period are: 

if minj (PN

j

) > "L and minj (PP

j

) > $ buy nothing

if minj (PN

j

) > "L and minj (PP

j

) < $ buy one unit of the perishable at lowest-priced

store 

if minj (PN

j

) < "L and minj

 (PP

j

) > $ buy multiple units of the non-perishable at the

lowest-priced store 

if minj (PN

j

) <"L and minj

 (PP

j

) < $ buy one unit of the perishable or multiple units

of the non-perishable (or both) at whatever store

offers the greatest consumer surplus. 

The difference between the fourth option in the two cases illustrates an important component

of shopping in our model. A store-loyal’s decision rules regarding her purchases of the two products
20 If PP

 > $ (or PN > "H), then retailer j makes no sales of the perishable (non-perishable). 

Hence, we restrict the analysis to values of PN < "H and PP < $. 

21Following Pesendorfer, we assume that the decision rule of low-value consumers is to

buy the non-perishable whenever PN < "L

. While the assumption that consumers exactly replace

their depleted inventory is not derived from a model of optimal consumer inventory behavior, 

this omission is not critical for the question we are interested in studying. The only property of

inventory behavior that is required for our results is that when a sale occurs, aggregate purchases

of the good by low-value consumers is increasing in the length of time since the previous sale. 

This property holds for some simple inventory models that we investigated. For this reason, our

model does not require identical inventorying behavior by all low-value consumers.

16

are independent; she buys one unit of good x at her preferred store if good x’s price is below her

reservation value for good x, without reference to good y’s price. In contrast, a shoppers’ decision

rules for the two goods are linked. Since by assumption consumers visit at most one store per

period, shoppers must consider the entire set of prices offered by each retailer, and determine the

consumer surplus offered by each store based on the observed prices, and choose the store that offers

the largest consumer surplus. Depending on the prices of the perishable and non-perishable items,

they may buy one unit of the perishable, multiple units of the non-perishable, or both goods. 

As long as PP

j < $ and PN

j < "H

, customers loyal to retailer j will buy both products at that

store. Indeed, if retailers only cared about selling to store-loyals, they would always charge PP =$

and PN = "H.

20 The reason that retailers might offer lower prices is that shoppers choose between

retailers on the basis of the consumer surplus they can obtain. 

Because shoppers purchase bundle of goods, they base decision on where to purchase on the

consumer surplus (summed over all goods) offered by each retailer. It follows that competition

between retailers can be described in terms of the consumer surplus offered and that retailers can

offer shoppers that surplus using either or both of two instruments: perishable prices (PP

) and nonperishable prices (PN

). Define *j to be the consumer surplus retailer j offers shoppers. For example,

if retailer j chooses to only offer the perishable good on “sale” (i.e., PN = H, PP < $) then *j = $ - Pp.

For the non-perishable, shoppers receive consumer surplus of ("L - PN) times the number of units

purchased. To conform with the models described in the previous section, we assume that shoppers

purchase a sufficient quantity of the non-perishable to replace the amount they consumed since the

previous sale.21 Letting M be the number of periods since the last sale, *j

 = max{[0,(M+1)("L

 - PN

j

)]
17

+ max[0, $- PP

j

]}. Whether retailer j makes any sales to shoppers depends on how *j compares to

the consumer surplus (*) offered by rival retailers.

To reduce notational complexity, we interpret "L

, "H and $ as the difference between the

consumer’s reservation value and the constant marginal cost of selling the good, so that we

normalize the retailers’ cost to zero. Additionally, we normalize the number of customers to one.

We also assume that the seller cannot determine an individual consumer’s type, so he cannot directly

price discriminate on that basis. Given these assumptions, we can derive retailer profits from

alternative pricing policies. Suppose that a portion, ( (where ( < 1) of customers are store-loyal, and

(1-() are shoppers. Retailers are assumed to be symmetric, so that (/J are loyal to each store. One

strategy for retailer j is to charge PN

j

 = "H and PP

j

 = $, which results in *j = 0 for all shoppers. This

yields profits of (("H + $)/J + (1-() $/J if all rival retailers choose these same prices, and (("H + $)/J

if any rival retailer chooses to offer * > 0. An alternative strategy is to have a sale on the perishable

only, so that PP

j

 < $ and PN

j

 = "H

. This yields profits of (("H + PP

j

 )/J + (1-() PP

j

 if retailer j offers

the highest *. Finally, he can only place the non-perishable on sale so that PN

j

 < "L and PP

j < $, and

retailer j will earn profits of (( PP

j+ PN

j

 )/J + (1-() [(M+1)PN

j

 + PP

j

)] if * j > * -j / maxij

 (* i) and

( (PN

j

 + PP

j

)/J otherwise. 

We now proceed to derive equilibrium pricing for the two goods, under the assumption that

retailers are all risk neutral. Our first result is that retailer j will at most, put one good on sale.

Proposition 1: It is more profitable for retailer j to place one good on sale than both goods (i.e., it

is not profitable to charge PN < "H and PP < $). 

Proof: Retailer j’s profit is 

If retailer j chooses PN

 and PP

 such that "H

 > PN

 > "L and PP

 < $ then * = $ - PP

, and a small increase

in PN

 increases profits without reducing *. Hence, "H

 > PN > "L

 and PP

 < $ is not a profitmaximizing strategy, as retailer j would raise PN to "H

. Conversely, if PN < "L and PP < $, * = $ -

PP + ("L - PN)(M+1), and an increase of , in PP accompanied by a decrease of ,/(M+1) in PN
22The implication that no more than one product will be on sale at any point in time

derives in part from the assumption that consumers necessarily visit no more than one retailer in

each period. As we discuss in Section V, in a model in which consumers can (at some cost) visit

multiple retailers, equilibrium might consist of multiple goods being on sale.

18

increases (strictly for M > 0) profits without changing *, so he would set PP = $. Hence, having

only one good on sale dominates having both PP < $ and PN < "H. 

The intuition behind Proposition 1 is that the cost of offering any given * to shoppers is the

foregone profits that could be obtained by selling to loyals only. It follows that for any given *,

retailer j wishes to offer it in a way that minimizes this cost. While it costs at least "H - "L to offer

any surplus through the non-perishable price, once PN is less than "L , the incremental cost of raising

* through a reduction in PN is small. Hence, the average cost (per unit of *) of offering surplus

through the non-perishable price is decreasing in the level of *, while the average cost of offering

* through lowering the perishable price is independent of the level of *. Thus, for low levels of 

the retailer will put the perishable on sale and for high the non-perishable will be put on sale. 

Proposition 1 shows that no more than one product will be on sale at any point in time. We

now turn to the question of which, if either product, will be on sale. To characterize the equilibrium,

note that PN = "H

, PP = $ (or equivalently, * = 0) for all retailers is not an equilibrium, since any

individual retailer can profitably offer * > 0 (by setting PP slightly less than $), and make sales to all

the shoppers. Varian (1980) formally shows that the only symmetric equilibrium features a mixed

strategy in prices, or more generally, in *. Because the distribution of prices has no mass points, it

follows that at least one product will be on sale at all times. Varian’s result (which we refer to as

Lemma 1) combined with Proposition 1, implies that exactly one product will be on sale at each

point in time. This suggests an interesting empirical implication: price movements for the perishable

and the non-perishable goods should be negatively correlated. Specifically, in the symmetric

equilibrium, whenever the non-perishable price changes, the perishable price will move in the

opposite direction.22

Lemma 1: There are no point masses in the symmetric equilibrium distribution of *.
23Note that G(*) changes over time with changes in M, as detailed below. 

24The analysis here does not require PN

 > 0. Since we interpret PN

 as the margin on the

non-perishable, PN < 0 does not imply a negative price. Moreover, given our assumption that

shoppers buy both goods at the same store if ($ - PP) > 0 and ("L

 - PN) > 0, PN less than zero, but

more than

 might be profitable. Hence, using the non-perishable as a "loss leader" may

be profitable. 

19

Proof: See Varian (1980), Proposition 3.

This generalizes the earlier intuition that * = 0 for all retailers cannot be an equilibrium. The

idea is that if there were a specific which is offered with positive probability, a store would find

it profitable to deviate by offering a slightly higher *, say , with that same probability. Such

a store would expand its expected sales by a positive amount, since it would attract all of the

shoppers when all of its rivals tied (which occurs with positive probability), while the loss due to the

price reduction necessary to obtain the higher * is arbitrarily small. Lemma 1 says that in the

symmetric equilibrium, retailers do not offer any specific * with a positive probability; instead in

every period * is drawn from a common distribution function. That is, the only symmetric equilibria

involve mixed strategies in *j

; whereby retailers choose * according to some continuous distribution

G(*).23 The remainder of this section is devoted to explicitly deriving the pricing equilibrium. The

next two lemmas provide lower bounds for the pricing distributions for the two goods.24

Lemma 2: The lowest price any retailer will ever charge for the non-perishable is

Proof: Any sale price must yield profits at least as great as the profits from not having a sale. Note

that Proposition 1 implies that if PN < "L

, then PP = $, and the lowest price that retailer j will ever

find it profitable to charge for the non-perishable solves
20

(3)

where the left-hand side is the profit from not having a sale, and the right-hand side is the profits

from having a sale if he were certain that he would be offering the highest *. Simplification yields

 It follows from Lemma 1 that a necessary condition for retailer j to place the non-perishable on sale

is that 

Since the right-hand side of equation (3) is decreasing in M, a sale on the non-perishable becomes

profitable for a larger range of parameter values as M rises. The intuition is that as the number of

periods since the last sale (M) grows, the ratio of the quantity of the non-perishable bought by loyals

to the quantity bought by shoppers (who only buy the non-perishable during sales) falls. That is, the

profit from selling to new customers increases with M, while the lost profit from not charging "H

 to

loyal customers is independent of M. 

Lemma 3: The lowest price any retailer will ever charge for the perishable is 

Proof: Following the same logic as Lemma 1, the lowest PP that retailer j will ever choose solves

Solving for PP

 yields .

Proposition 1 along with Lemmas 2 and 3 implies that the maximum * any retailer will offer

is max{(M+1)("L

 - PN

), $ - PP }. As shown in Lemma 4, there is no pure strategy equilibrium in

pricing. Rather, in equilibrium, each retailer either chooses a * by either setting PN < "L or by

setting PP < $ in order to attract shoppers. In addition, for any given *, the retailer will choose the
21

combination of PN and PP that maximizes his profits. The next lemma develops the properties of the

profit-maximizing prices for any *. 

To determine which good is the more profitable means of offering any specific *, it is useful

to introduce some additional notation. Let BP

 (*) be the retailer’s profit from placing the perishable

on sale (i.e., setting PP

 = $ - *, which yields consumer surplus of *) and let BN

 (*) be the retailer’s

profit from placing the non-perisable on place (i.e., setting PN

 = "L

 - */(M+1), which also yields

consumer surplus of *). Recall that when consumer surplus is generated by lowering PN

 the average

cost (per unit of *) is decreasing in the level of *, while the average cost is constant if consumer

surplus is generated by PP

. This implies that BN

 (*) - BP

 (*) is increasing in *. Because of this

relationship, we can derive a , such that BN ( ) = BP ( ), which means that BN (*) > BP (*) for all

* > , with the reverse inequality for all * < . There are two kinds of solutions for the * at which

BN ( ) = BP( ); interior and corner. results in a corner solution if the * that equates BN and BP is

sufficiently large that it is not profitable to offer that * (i.e., a corner occurs if (M+1)("L - PN) < --

 in words, the * associated with the maximum profitable discount on the non-perishable is less

than ). In that case, BN

(*) <BP

(*) for all relevant *, and only the perishable will be discounted.

When there is a corner solution, pricing for the perishable is identical to the pricing that would occur

if retailers sold only that product. An interior solution arises if (M+1)("L - PN

) > . In that case,

either good may be offered for sale. Lemma 4 solves for and provides the basis for determining

the distribution function for *. 

Lemma 4: Let where Pr( ) is the probability that

retailer j attracts shoppers when it offers *j = (for M > 0). When M = 0, is implicitly defined

in if (("H - "L) < J (1 - () "L, and = ($ - PP) otherwise. Then 

a. > 0,
22

b. BP

 (*) > BN

 (*) for all * < ,

c. If (M+1)("L

 - PN

) > (i.e., an interior solution exists) then BP

 (*) < BN

 (*) for * such that (M+1)("L

- PN) > * > . 

Proof: See appendix.

Corollary: In the symmetric equilibrium,

a. = (M+1)/((M)[(("H- "L) - "L J(1-()(M+1)(G( ))J-1] > 0 for M > 0. is also positive for M =

0. 

b. As long as $ is greater than 0, BP (*) > BN (*) for * sufficiently small. That is, there will always

be a positive probability of a sale on the perishable for $ > 0.

Proof: See appendix

Lemma 4 and its corollary indicate that in the symmetric equilibrium, is always positive,

which means that there is always a range of * for which putting the perishable on sale is the most

profitable means of offering that * to shoppers. If there is a corner solution (for some M), then only

the perishable will be on sale for that M. However, because (M+1)("L - PN) is increasing in M, there

may be an interior solution for sufficiently large M. When there is an interior solution, then the

retailer will offer the non-perishable on sale when it offers a large amount of consumer surplus to

non-loyals (* > ) and place the perishable on sale for *< (Lemma 4, results b and c). The logic

for why sales on the non-perishable will be associated with large * is that it is costly (in terms of

foregone profit from store loyals) to offer any consumer surplus to shopper by setting PN below "L,

since the retailer has to sacrifice at least (("H - "L)/J. In order for the retailer to find such a price

reduction profitable, he must have a large probability of winning. In contrast, offering small * by

setting PP

 below $ entails only a small reduction in profits from loyals, and consequently small * is

always offered through discounts on the perishable. 

In terms of deriving equilibrium prices, Lemma 4 means that G(*) can be decomposed into

two cumulative distribution functions; G(*) = 1 - F1(PN ) for and G(*) = (1-D) (1 - F2(PP)) for
23

. Proposition 2 derives these two distribution functions. 

Proposition 2. Let F1(PN ) be the distribution of non-perishable prices and F2(PP ) be the distribution

of perishable prices in the symmetric equilibrium. 

a. If (M+1)("L - PN

) > , then retailer j puts the non-perishable on sale with probability D = 1 - G( ).

here PN ( ) = "L - /(M+1). When the non-perishable is on sale, PP = $.

b. If (M+1)("L - PN

) > , then the cumulative distribution function for PN is

c. If (M+1)("L - PN

) < , then D = 0 and F1(PN ) = 0 for PN < "H and F1("H ) = 1. 

d. With probability 1-D retailer j sets PN = "H , and chooses PP according to the distribution

 function . 

Proof: a and b. From Lemmas 1-3, we know that * is randomly drawn from a continuous distribution

with support (0, max{$ - PP, (M+1)("L- PN)}). In equilibrium, the profits from charging each price

for which the density function is positive must be equal to the profits from charging PN = "H and PP

= $, which are equal to ([$ + "H ]/J. To calculate G(*), note that by Proposition 1, retailer j will put
24

at most one good on sale. If (M+1)("L - PN) > , then retailer j will sometimes put the nonperishable on sale. Specifically, Lemma 4 implies that whether PN 

 or PP will be lowered in order

to generate consumer surplus of * depends on *. For * > , * is obtained by setting PN

 < "L

. Given

this result, when retailer j chooses a * > , the probability that a rival offers more consumer surplus

is equivalent to the probability the rival offers a lower PN

. Hence for * > , G(*) = 1 - F1

(PN ), where

F1(PN) is the common c.d.f. for PN. To determine F1(PN), note that any PN for which the density

function is positive must yield the same profits as can be obtained by not holding a sale. Hence, the

distribution function for PN, conditional on a sale occurring on the non-perishable must solve 

Solving for F1

(PN ) yields

The lower bound for the support is the lowest price the retailer could profitably charge for

the non-perishable item. As Lemma 2 shows, this price is

The highest PN for which G(*) = 1 - F1(PN ) corresponds to the * for which it is equally

profitable to have a sale on the perishable and non-perishable, or PN

 = "L - /(M+1). By Lemma

4, for any * < , it will be more profitable to lower PP rather than PN, so that letting D

, we know that F1

(PN ) = D on the open interval , and F1

("H ) =

1. By Proposition 1, when PN < "L, PP = $. 

c. If (M+1)("L - PN

) < , then it is more profitable to put the perishable on sale than the nonperishable for all relevant *. In that case, retailer j will set PN = "H

. 

d. From Lemma 1, we know that there is not a point mass at * = 0, so that the perishable must be
25

(4)

on sale whenever PN = "H. To solve for F2(PP), the c.d.f. of PP, first note that expected profits when

the perishable is on sale at PP = $ - * are (($- * + "H)/J + (1-() G(*)

J-1($ - *). In equilibrium, this

must equal the expected profits from not having a sale so that 

To relate F2

(PP

) to G(*), note that if retailer j puts the perishable on sale, a rival might offer

more consumer surplus either by putting the non-perishable on sale, or by offering a lower perishable

price. This means that the probability that any one rival offers more consumer surplus than retailer

j is 1 - G(*) = D + (1 - D)(F2

(PP

)) => G(*) = (1 - D)(1 - F2

(PP

)). Using (4) this implies

 

Proposition 2 shows that the profitability of alternative prices for the non-perishable depends

on M, the length of time since the previous sale by any retailer on that product. As long as "L> PN,

the probability of a sale and the cumulative distribution function for any PN

 < "L is strictly increasing

in M. Additionally, Lemma 2 implies that the lower bound for the support of the distribution of PN

declines as M rises. These results parallel the results in the Conlisk et al., Sobel and Pesendorfer

models. The primary difference is that sales on the non-perishable are more likely in the multiproduct environment (to see this, note that F1 is increasing in $, and the single-product models are

equivalent to $ = 0). The implications for perishable prices are similar to those in Varian; the

primary differences are that F2

 implies the distribution of prices will vary over time and may feature

a mass point at $. Example 1 presents an illustration of the equilibrium, and how the price

distributions change over time. 

Example 1: Suppose that "H = 5, "L = 2, $ = 1.5, ( = .75 and J = 2. This implies that the perishable

price will be at least .9 (i.e., PP

 = .9), while the lower bound on the support for the non-perishable

price distribution (PN ) depends on M. At M = 0 or 1, it turns out that there is a corner solution, so
26

that the non-perishable will never be discounted. For example, for M = 1, = 0.605 and PN = 1.71,

which means that the PN

 which generates (1.697) is below the lowest price a retailer could ever

profitably charge for the non-perishable when M= 1. Hence, for M = 0 or 1, * takes on a value

between 0 and 0.6, and * is always created by setting PP below $.

As M increases, the profitability of putting the non-perishable on sale rises. For example,

for M = 2, PN

 = 1.33, and a sale on the non-perishable would be profitable if < (M+1) ("L

 - PN

)

= 3 * .67 = 2. As shown in Table 5, for M = 2 = 0.46, so that this inequality is satisfied. That

is, for * between 0.46 and 2, * is created by lowering PN, and for * between 0 and 0.46, * is created

by lowering PP. The probability of a sale on the non-perishable (D) is .327 when M = 2. If there is

no sale on the non-perishable when M = 2, then since is decreasing in M, and (M+1) ("L - PN) is

increasing in M, will be less than (M+1) ("L - PN) for all M > 2. In fact, the probability of

holding a sale on the non-perishable is nearly 50% for M=3, almost 59% for M=4, and about 65%

for M=5. Table 5

M PN D PN

 ( )

2 1.333 .464 .327 1.845

3 1.091 .38 .491 1.905

4 .923 .323 .589 1.935

5 .8 .281 .654 1.953

Figure 4 portrays the c.d.f. for these four values of M. As the corollary to Lemma 4 implies,

D is strictly less than 1, so that for any M, there is a positive probability that the perishable will go

on sale. The c.d.f. for the perishable price changes with M though its effect on D. The c.d.f. for PP

for values of M between 1 and 5 is portrayed in Figure 5. 

The example illustrates several implications of the model. First, the probability of a sale on

a non-perishable is an increasing function of the elapsed time since the previous sale on that good

by any retailer. Second, F1

 implies that the probability and depth of a sale of the non-perishable is
27

increasing in J, the number of retailers. Similarly, F2 implies that the depth of sale on the perishable

is increasing in J. Hence, other things equal, markets with fewer retailers will have fewer and

shallower sales. Third, a price reduction on a perishable will be less likely in a period in which

there is a price reduction on the non-perishable. Finally, Proposition 2 implies that a non-perishable

is more likely to have the same price in consecutive periods than a perishable, and conditional on

a price reduction occurring, the average change will be larger for the non-perishable. 

V. Discussion

The model in Section IV explains some features of the observed pricing behavior of food

retailers. In this section, we discuss the robustness of these results to alternative assumptions, and

explore some of the empirical implications of the model. 

One important assumption in the model is that the degree of heterogeneity in consumer

valuation for the perishable is lower (or less correlated with type) than the heterogeneity for the nonperishable. To simplify the presentation in Section IV, we make the extreme assumption that

consumers are homogeneous in their valuations of the perishable. Similar results to those found in

Section IV can be obtained with less restrictive assumptions. One such alternative assumption is that

average reservation values for the perishable are the same as for the non-perishable, but unlike the

case for the non-perishable, the reservation values for the perishable are uncorrelated with whether

the consumer is a store-loyal or a shopper. Hence, for perishables, (% of both shoppers and storeloyals have a reservation value of "H

 and (1-()% of both groups have a reservation value of "L. A

second alternative assumption is that consumers are heterogeneous with respect to reservation values

for the perishable, and those reservation values are perfectly correlated with whether the consumer

is a store-loyal or a shopper, but the degree of heterogeneity is lower for the perishable (e.g., the

value that shoppers place on the perishable is of $L

, which is equal to "L

 but store loyal customers’

value of the perishable is $H < "H). 

The main change associated with using the second alternative assumption, rather than the

assumption in Section IV, is that Lemma 1 no longer holds. Instead, there is a mass point in the

distribution of * at * = 0, or equivalently at PN = "H, PP = $H. Lemma 1 does hold under the first

alternative assumption, but we cannot calculate a closed-form equilibrium price distribution under
28

that assumption. 

However, under either of these two alternative modeling assumptions, most of the important

results from section IV still hold. For example, Proposition 1 continues to hold under either of these

two assumptions. Significantly, under either of these assumptions, the basic finding that small * are

offered by placing the perishable on sale still holds, as does the finding that the probability of a sale

on the non-perishable increases over time.

Another important assumption in the model is that consumers necessarily visit no more than

one retailer in each period. This is an important assumption in establishing Proposition 1. If

consumers can (at some cost) visit multiple retailers, equilibrium might consist of multiple goods

being on sale. For example, in Lal and Matutes’ (1994) model, if rival retailers put different goods

on sale and transportation costs are low enough, then consumers will “cream skim” by buying some

items at each retailer. Under those circumstances, it may be profitable for a retailer to offer a given

* by placing several goods on sale to deter such cream skimming. 

Even if retailers place multiple products on sale (whether to prevent cream-skimming or

otherwise), the logic of our analysis suggests that for a retailer selling a large number of products,

the number of items on sale will be fairly stable, but the composition of the sale items will change

over time. In particular, the price-discrimination motive implies that specific non-perishables will

go on sale periodically. In some periods, it will be profitable to put a relatively large number of such

products on sale. In those periods, relatively few perishable products will be on sale. Conversely,

in periods in which there are relatively few non-perishables that can profitably be put on sale, a larger

number of perishables will be on sale. 

For this reason, we suspect that the model’s implication for the correlation of non-perishable

and perishable prices does generalize to a model in which multiple goods are placed on sale each

period. Because the BLS data do not identify the supermarket associated with each price series, we

cannot test this implication using the BLS data. However, a data set which consisted of a sufficiently

large number of items from individual supermarkets would allow for a test of this prediction. In

particular, such data would enable us to determine if price movements for perishable products were

negatively correlated with price movements for non-perishable products, as predicted by the theory.

Another implication of the model concerning perishable and non-perishable prices also
25By allowing separate time effects for each of the 120 months in our data set, we are

implicitly controlling for all national time-specific supply and demand shocks for each of the 20

product categories we study. Hence, the left bar for each product in Figure 1 shows an upper

bound on the proportion of retail price variation that can be accounted for by national changes in

wholesale prices. This is an upper bound because this technique will overstate the role of

wholesale price changes and understate the role of retailer behavior. For example, if there is a

29

generalizes to an environment with multiple goods of each kind. Because the motivation for sales

differs between perishable goods and non-perishables goods, pricing behavior will differ between

these types of goods; price movements would be more frequent, but perhaps smaller, for perishable

goods. To empirically examine that implication, one needs to operationalize the idea of

perishability. We view perishability as related to storage costs; goods with higher storage costs are

more perishable. Hence, although frozen orange juice will last months in a freezer, it is more costly

to store frozen orange juice than it is to store canned tuna fish, because the latter requires a lesscostly storage facility. For this reason, the notion of perishability is not as stark as the distinction

made in the model. 

With this definition in mind, we present the following suggestive evidence. Consider the 20

categories of goods discussed in Section II. Of these goods, we view four as clearly the most

difficult to store for long periods; eggs, lettuce, bananas and ground beef. At the other extreme,

several of the products can be readily stored for long periods of time. Such products include baby

food, soap and detergent, canned soup, peanut butter and paper towels. 

Table 6 presents the frequency of non-zero price changes between consecutive months for

each of the 20 categories in our sample. Consistent with our expectation, price changes fairly

frequently for the four highly-perishable categories of goods. For example, lettuce prices are

unchanged in consecutive months less than 10% of the time. To be sure, lettuce (and perhaps the

three other products as well ) is more likely to be subject to supply shifts than other products in our

sample. Such shifts change wholesale prices and hence retail prices, and for all of our products some

of the observed price movements are not “sale” behavior in the sense used here. 

Ideally, we would control for these shifts using wholesale price data. Because we lack that

data,we instead capture aggregate cost shifts using time-specific indicator variables. Consistentwith

our expectation (see Figure 1),25 bananas, eggs, and lettuce are the three products with the greatest
common national demand shift (e.g., hot dogs for the 4th of July holiday) and this shift leads to

more sales (consistent with the findings in Hosken, Matsa, and Reiffen (2000) and Chevalier,

Kashyap, and Rossi (2000)), our time dummies will attribute the reduction in price to changes in

wholesale price, rather than retail changes. 

26As an example, there was a recent NBER conference on the use of high-frequency data,

such as that from supermarket scanners.

30

proportion of their price variation explained by these time-specific effects (approximately13%, 26%,

and 22% respectively). In terms of testing the theory, what is noteworthy is that sales appear to

account for an additional 44%, 31%, 25%, and 44% of retail price variation for bananas, eggs,

lettuce, and ground beef. This suggests that for these products retail margin changes play at least as

important a role in price variation as do supply changes.

At the other extreme, the data are less consistent with our expectations. While baby food has

the greatest percentage of months with no change, some of the other products that we viewed as good

examples of non-perishables have more frequent price movement than goods that we view as more

perishable. For example, white bread’s (a fairly perishable product) price moves less often than

prices for products like canned soup, peanut butter, soap and detergent, and paper towels. To

examine this question more rigorously would require a more objective measure of perishability and

better data (especially data on wholesale prices). 

Of course, other factors are important in determining which specific grocery items go on sale.

For example, Hosken, Matsa and Reiffen (2000) develop a model, supported by empirical evidence,

which shows that if advertising a sale is costly, then retailers will choose to put on sale those

products that are popular with the largest fraction of the population. In terms of the model in

Section IV, this result implies that among perishables, more popular products (those that offer the

most * per advertising dollar) such as lettuce and yogurt will be put on sale, rather than less popular

ones like radishes.

VI. Conclusion

With the increasing availability of high-quality data on retail prices and quantities,

economists (as well as marketing professionals and others) have enthusiastically begun to estimate

economic magnitudes, such as demand elasticities.26 It is well understood that identifying these
27This is not simply an artifact of our assuming that consumers have unit demand for both

goods. More generally, a retailer having a very successful sale, i.e. attracting many shoppers,

will also sell more of the goods not on sale because consumers purchase bundles of goods.

31

magnitudes requires variation in some independent variable, such as price. What is perhaps less well

appreciated is the relevance of the source of this variation. This paper shows that the majority of

retail price changes are actually sales; that is, temporary reductions in retail prices. These sales

account for 25-50% of the annual price variation for the grocery products we study. Because these

temporary reductions are such an important source of price variation, understanding why these

changes occur is critical to interpreting econometric estimates which use this data.

This paper explores the pricing behavior of multiproduct retailers, and it provides an

explanation for some observed retail pricing regularities. An implication of our analysis is that an

important source of price variation is the desire of retailers to take advantage of differences in

inventory costs across consumers. As noted above, if differences in consumer inventory costs are

an important determinant of the consumers’ purchasing patterns of, then estimated elasticities may

not correspond to the experiment of interest to the analyst. In fact, given the multiproduct nature

of a retailer’s offerings, it is by no means clear that a retailer’s unit sales of a particular class of

products will vary inversely with its price. For example, in our model, unit sales of the perishable

at an individual store are highest when the non-perishable is on sale (because more consumers are

in the store); in that instance, however, the perishable price is high.27 

More generally, taking account of the multiproduct nature of a grocery retailer’s offerings

yields a richer set of implications than can be derived from models of single-product retailers. For

example, the model implies that individual perishable product prices vary inversely with individual

non-perishable product prices. The model also implies that perishable and non-perishable products

should have different pricing dynamics; perishable products will go on sale more often, but at less

dramatic discounts than non-perishables. This last point is supported by our empirical analysis

which shows that the most perishable products in our sample are the most likely to go on sale. 

 We view the multiproduct nature of consumers’ purchases as an important aspect of the

demand facing retailers. The model presented here shows how this aspect makes the two-product

retailer choose different prices than two single-product retailers. In this sense, the model helps
32

explain some observed pricing regularities. Of course, goods sold by a single retailer differ in ways

other than those modeled here, and consequently retailers have even richer pricing alternatives than

our model suggests. Future research that analyzes the impact of these differences across products

(e.g., differences in likelihood of purchase) would help develop a more complete understanding of

the observed pricing behavior of multiproduct retailers.
33

References

Banks, Jeffrey and Sridhar Moorthy (1999) “A Model of Price Promotions with Consumer Search,”

International Journal of Industrial Organization; 17, pp. 371-98. 

Bliss, Christopher (1988) “A Theory of Retail Pricing,” The Journal of Industrial Economics; 36,

pp. 375-91. 

Chevalier, Judith, Anil Kashyap, and Peter Rossi (2000) “Why Don’t Prices Rise During Periods of

Peak Demand? Evidence from Scanner Data” Mimeo. 

Conlisk, John, Eitan Gerstner, and Joel Sobel (1984) "Cyclic Pricing by a Durable Goods

Monopolist," Quarterly Journal of Economics; 99, pp. 489-505

Cotterill, Ronald W. Williams P. Putsis Jr., and Dhar Ravi (2000) “Assessing the Competitive

Interaction between Private Labels and National Brands” Journal of Business; 73, pp. 109-

37.

Feenstra, Robert and Matthew Shapiro (2000) "High-Frequency Substitution and the Measurement

of Price Indexes", Mimeo

Hausman, Jerry, Gregory Leonard, and J. Douglas Zona (1994) “Competitive Analysis with

Differentiated Products,” Annales D Economie Et De Statistique; 34, pp. 159-180.

Hosken, Daniel and David Reiffen (1999) “Pricing Behavior of Multiproduct Retailers” Federal

Trade Commission Bureau of Economics Working Paper 225.

Hosken, Daniel, David Matsa and David Reiffen (2000) “How Do Retailers Adjust Prices: Evidence

from Store-Level Data” Federal Trade Commission Bureau of Economics Working Paper

230.

Klein, Benjamin and Keith Leffler (1981) “The Role of Market Forces in Assuring Contractual

Performance ” Journal of Political Economy, 614-41.

Lal, Rajiv and Carmen Matutes (1994) "Retail Pricing and Advertising Strategies," Journal of

Business; 67, pp. 345-70.

Levy, Daniel, Mark Bergen, Shantanu Dutta, and Robert Venable (1997) “The Magnitude of Menu

Costs: Direct Evidence from Large U.S. Supermarket Chains,” Quarterly Journal of

Economics; 112, pp. 791-825

Pashigian, B. Peter (1988) "Demand Uncertainty and Sales: A Study of Fashion and Markdown
34

Pricing," American Economic Review; 78, pp. 936-53.

___ and Brian Bowen (1991) "Why are Products Sold on Sales?: Explanations of Pricing 

Regularities," Quarterly Journal of Economics; 106, pp.1014-1038. 

Pesendorfer, Martin (2002) "Retail Sales: A Study of Pricing Behavior in Super Markets" Journal

of Business (forthcoming)

Salop, Steven C. and Joseph Stiglitz (1982) "The Theory of Sales: A Simple Model of Equilibrium

Price Dispersion with Identical Agents", American Economic Review; 72, pp. 1112-30

Sobel, Joel (1984) "The Timing of Sales," Review of Economic Studies; 51, pp. 353-68. 

Varian, Hal R. (1980) "A Model of Sales", American Economic Review; 70, pp. 651-9.

Ward, Michael R. and Lee, Michael J. (2000). "Internet Shopping, Consumer Search 

and Product Branding," Journal of Product and Brand Management; 9, pp. 6-18.

Warner, Elizabeth J. and Robert B. Barsky (1995) “The Timing and Magnitude of Retail Store

Markdowns: Evidence from Weekends and Holiday,” Quarterly Journal of Economics;

110, pp. 321-52.
35

Appendix

Lemma 4: Let where Pr( ) is the probability that

retailer j attracts non-loyals when it offers *j = (for M > 0). When M = 0, is implicitly defined

in if (("H

 - "L

) < J (1 - () "L

, and = ($ - PP

) otherwise. Then 

a. > 0,

b. BP (*) > BN (*) for all * < (i.e., an interior solution exists).

c. If (M+1)("L - PN) > then BP (*) < BN (*) for * such that (M+1)("H - PN) > * > . 

Proof: a. To see that must be non-negative (for M > 0), note that since "H

 > "L

, < 0 would imply

that Pr( ) > 0; i.e., firm j can attract shoppers by offering negative consumer surplus. This violates

consumer rationality. For M = 0, must be greater than 0, since both (("H - "L) and J(1 - () "L are

positive

b. By Proposition 1, retailers will never put both products on sale. The PP

 required to generate

consumer surplus of * is $- *. Hence, the profits from putting the perishable on sale to generate *

are 

Where Pr(*j) is the probability that retailer j attracts non-loyals when it offers *j. Since the nonperishable price which yields consumer surplus of * equals "L

 - */(M+1), the profits from putting

the non-perishable on sale to generate * are 

First note that BN (*) - BP (*) is strictly increasing in *. Hence, if BN (*) - BP (*) is positive
36

for some , it will be positive for all * > , and if BN (*) - BP (*) is negative for some , it will be

negative for all * < . Solving for the * at which BN (*) = BP (*) allows us to divide the set of all

possible * into two mutually exclusive sets; one in which lowering PP is a more profitable way to

generate * and one in which lowering PN is more profitable . Specifically, BP (*) >BN (*) if 

 (for M > 0) For M = 0, the same calculation

yields the implicit solution to if (("H - "L) < J (1 - () "L .

c. By construction, BN >BP if * > . In addition, if (M+1)("H

 -PN

) > , then offering a sale on the

non-perishable yields higher profits to retailer j than having a sale on neither good, assuming no

other retailer offers more than in consumer surplus. 

Corollary: In the symmetric equilibrium,

a. = (M+1)/((M)[(("H - "L ) - "L J(1-()(M+1)(G( ))J-1] > 0 for M > 0. is also positive for M

= 0. 

b. As long as $ is greater than 0, BP (*) > BN (*) for * sufficiently small. That is, there will always

be a positive probability of a sale on the perishable for $ > 0.

Proof: a. In the symmetric equilibrium, Pr(*j

) = Prob(*j

 < *-j) = (G(*))J-1, where G( ) is a distribution

function common to all retailers. By Lemma 4.a, is non-negative. Using the definition of , 

= 0 would imply (("H - "L ) = "L J(1-()(M+1)(G(0))J-1, and hence a mass point at G(0). As Varian

(1980) showed, the equilibrium distribution cannot have a mass point (see Lemma 1). Hence, >

0 for M > 0. For M = 0, since both (("H - "L) and J (1 - () "L are positive, the implies Pr( ) is
37

strictly positive, and less than 1 if (("H - "L) < J (1 - () "L.

b. First note that $ is necessarily greater than PP. This means that there is always a profitable

discount that can be offered on the perishable, as long as the probability of winning is sufficiently

high. At the same time, part a. shows that > 0, which implies that BP (*) > BN (*) for * sufficiently

small.
38

Table 1: Proportion of Observations at the Mode and Proportion of 

Deviations above and Below Mode by Product Category

Product Number of Price

Series

Percent of Obs.

At Mode

Ratio of the # of Prices

Below Mode to the # of

Prices Above Mode*

Baby Food 299 72.05 1.74

Bananas 1142 56.19 2.01

Canned Soup 1310 67.06 1.93

Cereal 1631 65.16 1.74

Cheese 1233 65.35 1.53

Snacks 1288 74.05 2.46

Cola Drinks 1116 62.53 2.23

Cookies 750 71.47 2.37

Crackers 311 64.75 3.29

Eggs 905 39.07 1.26

Frozen Dinners 561 68.08 2.77

Frozen Orange Juice 491 57.78 2.23

Ground Beef 909 60.36 2.16

Hotdogs 471 63.06 2.39

Lettuce 672 20.98 3.57

Margarine 477 62.81 2.11

Paper Products 620 66.87 2.41

Peanut Butter 342 64.13 1.93

Soap and Detergents 820 68.04 2.40

White Bread 1043 69.05 1.69

All Products 16391 59.37 2.09

*In all cases the null hypothesis that the proportion of price increases equals the proportion of price

decreases can be rejected at a confidence level of =.0001.
39

Table 2 - Percent of Price Series Experiencing at Least One Sale in the Second Year of the 

Sample, Conditional on Whether there is a Sale within the First Year

Product Conditional on at

least one sale within

the First Year

(number of price

series)

Conditional on no 

Sale within the First

Year

(number of price

series)

Z-Statistic

(p-value)

Baby Food 26.7%

(15)

3.7%

(82)

3.17

(.0016)

Bananas 84.0%

(401)

52.9%

(87)

6.41

(0)

Canned Soup 51.8%

(110)

17.4%

(265)

6.81

(0)

Cereal 53.2%

(77)

22.0%

(259)

5.29

(0)

Cheese 56.1%

(139)

21.0%

(257)

7.07

(0)

Snacks 68.5%

(124)

25.8%

(151)

7.08

(0)

Cola Drinks 72.0%

(157)

25.4%

(122)

7.72

(0)

Cookies 66.7%

(63)

20.0%

(115)

6.18

(0)

Crackers 84.9%

(53)

25.5%

(51)

6.10

(0)

Eggs 63.5%

(244)

38.5%

(218)

5.37

(0)

Frozen Dinners 60.9%

(46)

34.2%

(38)

2.43

(.015)

 Frozen Orange Juice 64.6%

(113)

36.4%

(118)

4.28

(0)

Ground Beef 70.3%

(246)

36.1%

(216)

7.37

(0)
40

Hot Dogs 65.1%

(83)

37.5%

(56)

3.20

(.0014)

Lettuce 96.1%

(417)

70.0%

(40)

6.59

(0)

Margarine 66.2%

(74)

32.1%

(109)

4.54

(0)

Paper Products 76.5%

(17)

32.3%

(31)

2.93

(.0034)

Peanut Butter 49.0%

(51)

17.4%

(109)

4.17

(0)

Soap and Detergent 64.5%

(31)

21.2%

(33)

3.51

(.0004)

White Bread 60.9%

(151)

15.0%

(233)

9.34

(0)

Sale defined as a 10% or more reduction in price in month t, followed by an increase of similar

magnitude in month t+1.
41

Table 3: Summary of Correlations of First Differences in the Prices of Three Leading Brands

Across Grocery Chains in Springfield, Missouri and Sioux Falls, South Dakota

Sioux Falls Springfield Total

Number of Correlations 238 121 359

Percent Positive 51.2% 52.9% 51.8%

Percent Larger than .3 7.1% 6.6% 7.0%

Percentage Positive and

Significant at 10% level

14.3% 16.5% 15.0%

Percentage Positive and

Significant at 5% level

10.9% 13.2% 11.7%

The table is a summary of the correlations of first differences in price of specific brands across

chains. For example, we calculate the correlation of the first difference in price of Skippy Peanut

Butter across all the grocery chains in Springfield, Missouri. A similar calculation is done for each

of the three leading brands for eight categories of consumer goods in the data set (peanut butter, tub

margarine, stick margarine, ketchup, sugar, light tuna, dark tuna, and tissue). The difference in the

number of correlations between Sioux Falls and Springfield reflects differences in the products

carried across chains and differences in the number of chains.
42

Table 4: Proportion of Price Increases or Decreases Following a Price Decrease

Product Observations Price Increase Price Decrease

Baby Food 329 46.2% 4.9%

Bananas 5970 58.7% 16.8%

Canned Soup 2370 60.0% 6.6%

Cereal 2246 62.5% 6.5%

Cheese 3124 57.8% 9.8%

Snack Food 2254 66.6% 7.1%

Cola Drinks 3123 57.2% 13.4%

Cookies 1596 73.9% 6.1%

Crackers 1077 71.7% 7.1%

Eggs 7330 55.1% 21.8%

Frozen Dinners 946 65.4% 7.4%

Frozen Orange Juice 2306 52.1% 13.7%

Ground Beef 5006 60.8% 13.5%

Hot Dogs 1301 64.2% 10.1%

Lettuce 10328 60.7% 34.3%

Margarine 1752 58.6% 11.1%

Paper Towels 463 60.3% 8.6%

Peanut Butter 1303 53.5% 10.4%

Soap and Detergent 612 60.0% 7.8%

White Bread 2032 68.2% 6.3%

All Products 55468 60.0% 16.7%
43

Table 6: Proportion of Price Changes by Product

Product Observations Percentage of Price Changes

Baby Food 6058 17.8%

Bananas 24567 52.3%

Canned Soup 24254 26.1%

Cereal 24067 27.0%

Cheese 24871 31.2%

Snack Food 19310 27.4%

Cola Drinks 17754 39.0%

Cookies 12792 30.0%

Crackers 6404 39.3%

Eggs 26196 63.7%

Frozen Dinners 6531 33.6%

Frozen Orange Juice 12809 38.6%

Ground Beef 26013 42.4%

Hot Dogs 8692 35.7%

Lettuce 24158 90.0%

Margarine 11017 36.0%

Paper Towels 3225 34.5%

Peanut Butter 8602 33.3%

Soap and Detergent 4610 31.2%

White Bread 22674 24.9%

All Products 314604 40.6%
44
45
46
47