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Working Paper Series
Relative Price Dispersion: Evidence and
Theory
WP 16-02
Greg Kaplan
Princeton University and NBER
Guido Menzio
University of Pennsylvania and NBER
Leena Rudanko
Federal Reserve Bank of Philadelphia
Nicholas Trachter
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Relative Price Dispersion:
Evidence and Theory∗
Greg Kaplan
Guido Menzio
Princeton University and NBER
University of Pennsylvania and NBER
Leena Rudanko
Nicholas Trachter
FRB Philadelphia
FRB Richmond
January 2016
Working Paper No. 16-02
Abstract
We use a large dataset on retail pricing to document that a sizable portion of the
cross-sectional variation in the price at which the same good trades in the same period
and in the same market is due to the fact that stores that are, on average, equally
expensive set persistently different prices for the same good. We refer to this
phenomenon as relative price dispersion. We argue that relative price dispersion stems
from sellers’ attempts to discriminate between high-valuation buyers who need to make
all of their purchases in the same store and low-valuation buyers who are willing to purchase different items from different stores. We calibrate our theory and show that it is
not only consistent with the extent and sources of dispersion in the price that different
sellers charge for the same good, but also with the extent and sources of dispersion in
the prices that different households pay for the same basket of goods and with the relationship between prices paid and the number of stores visited by different households.
JEL Codes: L11, D40, D83, E31.
Keywords: Price Dispersion, Equilibrium Product Market Search.
We are grateful to Bob Hall for his comments on the previous work of some of the authors, which inspired
the current project. We are grateful to Jim Albrecht, George Alessandria, Fernando Alvarez, Ken Burdett,
Jeff Campbell, Glenn Ellison, Sarah Ellison, Doireann Fitzgerald, Gaetano Gaballo, Matthew Gentzkow, Erik
Hurst, Nir Jaimovich, Boyan Jovanovic, Jose Moraga-Gonzales, Stephan Seiler, Rob Shimer, and Randy
Wright for useful comments. We are grateful to our audiences at the NBER Summer Institute (IO and
Price Dynamics groups); Minnesota Macro Workshop; Canadian Macro Study Group; Philadelphia Search
and Matching Workshop; Econometric Society; Chicago FRB/St.Louis FRB Workshop on Money, Banking,
Payments and Finance; SED Annual Meeting; Bank of France Price Setting and Inflation Conference; EIEF
Junior Conference in Macroeconomics; University of Pennsylvania (Wharton); University of Wisconsin at
Madison; FRB Minneapolis; George Washington University; Universidad Torcuato di Tella; Universidad
San Andres; and UVA/Richmond FRB Research Jamboree. We thank the Kilts Center for Marketing and
the Nielsen Company for sharing their data. The views expressed herein are those of the authors and not
necessarily those of the Federal Reserve Bank of Philadelphia, Federal Reserve Bank of Richmond, or the
Federal Reserve System.
∗
1
1
Introduction
Using a large dataset on retail pricing, we document that a significant fraction of the crosssectional variation in the price at which the same good is sold in the same period of time and
in the same market is due to the fact that retailers that are, on average, equally expensive set
persistently different prices for that particular good. We refer to this phenomenon as relative
price dispersion. We propose a theory of relative price dispersion in an equilibrium model of
the retail market, where buyers and sellers, respectively, demand and supply multiple goods.
We argue that relative price dispersion stems from the sellers’ incentive to discriminate
between high-valuation buyers who need to make all of their purchases in the same store and
low-valuation buyers who are willing to purchase different items from different stores. We
calibrate our theory of relative price dispersion and show that it captures well the differences
between the extent and cause of variation in the price at which the same good is sold by
different sellers, and the extent and cause of variation in the prices that different households
pay for the same basket of goods.
In the first part of the paper, we measure the extent and sources of dispersion in the
price at which the same good is sold in the same week and in the same geographical area by
different retailers. We carry out the analysis using the Kilts-Nielsen Retail Scanner (KNRS)
dataset, which provides weekly price and quantity information for 1.5 million goods (defined
by their universal product code or UPC) at around 40,000 stores in more than 2,500 counties
across 205 designated market areas (DMA), which are geographical areas of roughly the same
size as metropolitan statistical areas.
Using the KNRS, we compute the average price of a particular good in a given week
and in a given geographical area. We normalize the price of the good at each different
store by expressing it as a percentage deviation from the average price in the relevant week
and market. We break down the normalized price of the good at each different store into
a store component—defined as the average of the normalized price of all the goods sold
by the store in the week—and a store-good component—defined residually as the difference
between the price of the good at the store and the average price of the store. We then
compute the variance of the price of the good at different stores, and we break down this
variance into a store component and a store-good component. We find that, on average, the
standard deviation of prices for the same good in the same week and market is 15.3%, and
the variance is 2.34%. We also find that only 15% of the variance is due to the variance in
the store component—i.e., due to the fact that the same good is sold at stores that have
a different average price—while 85% is due to the variance of the store-good component—
2
i.e., due to the fact that the same good is sold at different prices at stores that are equally
expensive on average.
We then use the time dimension in KNRS to identify the persistent component and the
transitory component of the price of a particular good at a particular store. To this aim, we
follow an approach that is commonly used in the literature on labor economics to analyze
wage inequality (see, e.g, Gottschalk and Moffitt 1994 and Blundell and Preston 1998) but
that had never been applied before to study price dispersion. Specifically, we estimate a
statistical model for prices, in which both the store and the store-good component of prices
are given by the sum of a fixed effect, an AR process and an MA process. When estimated,
the statistical model fits the auto-covariances of prices very well. The estimated model
implies that almost all of the cross-sectional variance of the store component of prices stems
from persistent differences in average store prices. Also, the estimated model implies that
roughly one-third of the variance of the cross-sectional variance of the store-good component
of prices stems from persistent differences in the price of the good at equally expensive stores,
while roughly two-thirds of the variance stems from transitory differences. Overall, a sizable
fraction of the variation in the price of the same good across different stores in the same week
and in the same market is due to the fact that equally expensive stores charge persistently
different prices for the same good. We refer to the persistent differences in the price of a
particular good relative to the average price of the store as relative price dispersion.
The literature offers compelling theories of short-term differences in the price of the same
good across equally expensive stores. For instance, according to the theory of intertemporal
price discrimination (see, e.g., Conlisk, Gerstner and Sobel 1984; Sobel 1984; and Menzio
and Trachter 2015a), sellers find it optimal to occasionally lower the price of a particular
good in order to discriminate between low-valuation customers who are willing to do their
shopping at any time during the month and high-valuation customers who need to make their
purchases on a specific day of the month. As different sellers implement these occasional
price reductions at different times, the equilibrium may feature short-term differences in
the price of the same good across equally expensive stores. According to the inventory
management theory (see, e.g., Aguirregabiria 1999), a seller finds it optimal to increase the
price of a good as the inventory of the good falls, and to lower the price when the inventory
of the good is replenished. As different sellers have different inventory cycles, the equilibrium
may feature short-term differences in the price of the same good across equally expensive
stores. In contrast, the literature does not offer much theoretical guidance in the way of
understanding relative price dispersion, i.e., long-term differences in the price of the same
good across equally expensive stores.
3
In the second part of the paper, we develop a theory of relative price dispersion. We
consider a retail market in which sellers and buyers, respectively, supply and demand two
goods. Sellers are ex-ante homogeneous, both with respect to their cost of producing the
goods and in terms of the type of buyers they meet. Buyers are ex-ante heterogeneous. One
type of buyers, which we call busy, has a relatively high valuation for goods and needs to
make all purchases in the same store. The other type of buyers, which we call cool, has
a relatively low valuation and can purchase different items at different stores. The retail
market is imperfectly competitive. As in Butters (1977) and Burdett and Judd (1983), we
assume that buyers do not have access to all the sellers but only to a subset of them. In
particular, a fraction of buyers can access a single seller, and a fraction of buyers can access
multiple sellers.
We find that, for some parameter values, the equilibrium of the retail market must display
relative price dispersion. This result follows from two properties of the equilibrium: (i) some
sellers find it optimal to post different prices for the two goods; and (ii) for every seller that
posts a lower price for one good, there is another seller posting a lower price for the other
good. There is a simple intuition behind the first property of the equilibrium. Consider a
seller that sets the same price for the two goods and suppose that this common price lies
in between the valuation of the cool buyers and the valuation of the busy buyers. Given
this common price, the seller trades with some of the busy buyers but never trades with the
cool buyers. Now, suppose that the seller lowers the price of the first good and increases the
price of the second good, so as to keep the average price constant. Since the busy buyers
must purchase both goods in the same location, they are indifferent to the change in prices.
Hence, the seller trades the same quantity of goods to this type of buyer. However, as the
seller keeps lowering the price of the first good, at some point it reaches the valuation of the
cool buyers. Hence, at some point, the seller trades the good to this type of buyer. Overall,
the seller is strictly better off setting different prices for the two goods, rather than setting
a common price. The second property of the equilibrium follows from the fact that, if there
were more sellers charging a lower price for one good than for the other, then there would
be some unexploited profit opportunities. When taken together, the two properties of the
equilibrium imply that there is dispersion in the price of the same good among sellers that
are, on average, equally expensive.
According to our theory, relative price dispersion is the general equilibrium consequence
of a new type of price discrimination. The difference in valuation between the busy buyers
and the cool buyers gives sellers a reason to try to price discriminate. The difference in
the ability of the busy and the cool buyers to make their purchases at different locations
4
gives the seller a way to price discriminate. Specifically, price discrimination is achieved by
setting asymmetric prices for the two goods, so as to charge a high price for the basket of
goods to the high-valuation buyers who need to purchase all items from the same retailer,
and to charge a low price (on one item) to the low-valuation buyers who have the ability to
purchase different items from different retailers.
In the last part of the paper, we calibrate and validate our theory of relative price
dispersion. We calibrate the theory so as to match the extent of dispersion in the persistent
component of prices at which different stores sell the same good, the contribution of the
(persistent) store component and of the (persistent) store-good component to the dispersion
of prices, and the elasticity of the price paid by a household for a given basket of goods with
respect to the number of stores from which the household shops. We find that our theory can
match these features of the data very well. Dispersion in the (persistent) store component of
prices is obtained for the same reason as in the single product retail market models of Butters
(1977) and Burdett and Judd (1983). Dispersion in the (persistent) store-good component
of prices is a general equilibrium consequence of price discrimination between low-valuation
buyers who can shop at multiple locations and high-valuation buyers who must shop at a
single location. The negative elasticity of the price paid by a household for a given basket of
goods with respect to the number of stores from which it shops is obtained because, when
there is relative price dispersion, buyers can achieve lower prices by purchasing different
items at different stores.
We validate the theory using the Kilts-Nielsen Consumer Panel (KNCP), which tracks
the shopping behavior of 50,000 households drawn from 54 different markets. We compute
the extent and sources of dispersion in the prices that different households pay for the same
basket of goods in the same quarter and in the same market. After removing the dispersion
due to transitory variation in the store-good component of prices, we find that the standard
deviation of prices paid by different households for the same basket of goods is 7.8%, and
the variance is 0.61%. We also find that 55% of the variance is due to persistent differences
in the store component of prices, while 45% is due to persistent differences in the storegood component of prices (and to the covariance term). We then compute the extent and
the sources of dispersion in the prices paid by different households that are implied by our
theory. We find that the predictions of the theory line up very well with the data. The
theory correctly predicts that the dispersion of prices paid by different households for the
same basket of goods is lower than the dispersion of prices posted by different stores for
the same good. Also, the theory correctly predicts that variation in the store component of
prices contributes to a larger share of the dispersion of prices paid by households than to the
5
dispersion of prices posted by stores. Intuitively, the predictions of the theory are correct
because the variation of the store-good component of prices of individual goods washes out
in the basket price of those households who need to purchase all goods from the same store.
Overall, our model of the retail market offers a parsimonious explanation for the extent
and sources of dispersion in the price that different sellers charge for the same good, and for
the extent and sources of dispersion in the price that different households pay for the same
basket of goods.
The paper contributes to the large empirical literature documenting price dispersion.
Ours is the first paper to use a large-scale dataset that covers multiple products, each sold
at multiple stores and each observed over a long period of time. The data allows us to use
techniques borrowed from labor economics to estimate the overall variance of prices, the
variance due to temporary and persistent differences in the store component of prices, and
the variance due to temporary and persistent differences in the store-good component of
prices. The variance decomposition identifies a novel feature of retail prices: relative price
dispersion. Part of the previous empirical literature has documented the extent of price
dispersion for particular products: cars and anthracite coal in Stigler (1961); 39 products
in Pratt, Wise, and Zeckhauser (1979); several books and CDs in Brynjolfsson and Smith
(2000); four academic textbooks in Hong and Shum (2006); illegal drugs in Galenianos,
Pacula, and Persico (2012); and mortgage brokerage services in Woodward and Hall (2012).
There are also studies that have used small-scale data that covers multiple products, each
sold at multiple stores and each observed over time: car insurance policies in Alberta in
Dahlby and West (1986), prescription drugs in upstate New York in Sorensen (2000), and
four products in Israel in Lach (2002). However, none of these papers has attempted to
decompose price dispersion as in our paper and, hence, identified relative price dispersion.
The paper also contributes to the search-theoretic literature on price dispersion. In
particular, we develop a version of the search-theoretic model of equilibrium price dispersion
of Burdett and Judd (1983) in which buyers and sellers trade multiple goods and buyers are
heterogeneous with respect to both their valuation and their ability to purchase different
items at different stores. Our version of the model leads to relative price dispersion and fits
well both data on the dispersion of prices across stores and data on the dispersion of prices
across households. There are many search-theoretic models of equilibrium price dispersion in
the retail market for a single good: Butters (1977), Varian (1980), Burdett and Judd (1983),
and Stahl (1989) in which price dispersion emerges because some buyers are in contact with
a single seller and others are in contact with multiple sellers; Reinganum (1979) and Albrecht
6
and Axell (1984) in which price dispersion emerges because of heterogeneity among buyers
or sellers; Rob (1985), Stiglitz (1987), and Menzio and Trachter (2015b) in which price
dispersion is obtained because sellers are large and have an impact on reservation prices.
There are fewer search-theoretic models of equilibrium price dispersion in the retail market
for multiple goods, as these models are famously difficult to analyze. The main exceptions
are McAfee (1995), Zhou (2014), Baughman and Burdett (2015), Rhodes (2015), and Rhodes
and Zhou (2015). Yet, none of these papers considers the type of buyers’ heterogeneity on
which our theory of relative price dispersion is built.
Outside of search theory, there are multiproduct models of the retail market in which
a retailer finds it optimal to charge different prices for goods that are equally valued by
buyers and equally costly to sellers (see, e.g., Lal and Matutes 1994). Asymmetric pricing is
optimal because it is assumed that buyers are aware of the price of only a subset of goods
when deciding where to shop. However, these theories typically imply that different retailers
should all charge lower prices for the same subset of goods. Hence, these theories do not
generate relative price dispersion. There are also some multiproduct models of the retail
market in which, as in our model, buyers differ with respect to their ability to purchase
different items in different locations (see, e.g., Lal and Matutes 1989, and Chen and Rey
2012). These models, even though different from ours in many dimensions, can also generate
relative price dispersion. Unlike ours, these models have not been confronted with the data.
2
Relative Price Dispersion: Evidence
In this section, we jointly analyze the dispersion and dynamics of the prices of identical
goods in the same geographical area and over the same period of time. We use a detailed
dataset on prices that includes the time series of the price of a large number of goods at
each of a large number of stores. We use these data to estimate a rich stochastic process
for the average price level of a store and for the price of a good at a store relative to the
average price level of the store. We then use the estimated stochastic process to decompose
the variance of the price of the same good in the same period of time and same geographic
area. The new finding that we emphasize is that a substantial fraction of the cross-sectional
variance of prices is due to the fact that stores that are, on average, equally expensive set
persistently different prices for the same good. We refer to this phenomenon as relative price
dispersion.
7
2.1
Framework and Estimation Strategy
Let pjst denote the quantity-weighted average price of good j at store s in time period t.
In our application, a time period is one week and a good is defined by its UPC (barcode).
We first decompose the log of each price pjst into three additively separable components: a
component that reflects the average price of the good in period t, µjt ; a component that
reflects the expensiveness of the store selling the good, yst ; and a component that reflects
factors that are unique to the combination of store and good, zjst .1 Formally, we decompose
the log of pjst as
log pjst = µjt + yst + zjst .
(1)
We model both the store component of the price, yst , and the store-good component of the
price, zjst , as the sum of a fixed effect, a persistent part and a transitory part. This statistical
model is motivated by the empirical shape of the auto-correlation functions of yst and zjst ,
which are illustrated in Figure 1. The auto-correlation functions of yst and zjst display
a sharp drop at short lags, followed by a smoothly declining profile that remains strictly
positive even at very long lags. The initial drop in the auto-correlation suggests the presence
of a transitory component in both yst and zjst . We model the transitory components as an
MA(q) process, rather than an IID process, to allow for the possibility that the transitory
component may reflect temporary sales. Indeed, since sales may last longer than one week
and since the timing of sales may not correspond to the weekly reporting periods, they
are better captured by a process with some limited persistence than with a weekly IID
process.2 The smoothly declining portion of the auto-correlation function is consistent with
the presence of an AR(1) component. Finally, the fact that the auto-correlation function
remains positive even after 100 weeks suggests the presence of a fixed effect.
Formally, the statistical model for yst and zjst is given by
P
T
yst = ysF + yst
+ yst
,
F
P
T
zjst = zjs
+ zjst
+ zjst
,
y
P
P
yst
= ρy ys,t−1
+ ηs,t
,
q
X
T
θy,i εys,t−i ,
= εys,t +
yst
P
P
z
zjst
= ρz zjs,t−1
+ ηjs,t
,
q
X
T
θz,i εzjs,t−i ,
zjst
= εzjs,t +
i=1
i=1
ysF
=
(2)
F
z
zjs
= αjs
,
αsy ,
1
We work with the natural logarithm of quantity-weighted average prices. This reflects an assumption
that innovations to prices enter multiplicatively, which is convenient when jointly analyzing prices of many
different goods.
2
Later in this section, we show that our findings are robust to alternative specifications for the transitory
component.
8
Figure 1: Auto-correlation function of prices
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
20
40
60
Lag (weeks)
80
0
0
100
(a) Store component
20
40
60
Lag (weeks)
80
100
(b) Store-good component
Notes: The figure plots the empirical auto-correlation functions of the store and store-good components, ŷs t
and ẑjst , together with their counterparts from the fitted statistical model.
P
F
denote the fixed effects of the store and the store-good components, yst
where ysF and zjs
P
T
and zjst
denote the persistent parts of the store and the store-good components, and yst
and
T
zjst
denote the transitory parts of the store and the store-good components. The parameters
z
αsy and αjs
are random variables with mean zero and variance σα2 y and σα2 z . The parameters
ρy and ρz are the auto-regressive parameters of the AR(1) part of the store and store-good
y
z
components, while ηs,t
and ηjs,t
are the innovations to the AR(1) part and are assumed to
be random variables with mean zero and variance ση2y and ση2z . Finally, the parameters θy,i
and θz,i are the coefficients of the MA(q) part of the store and store-good components, while
εys,t and εzjs,t are the innovations to the MA(q) part and are assumed to be normal random
variables with mean zero and variance σε2y and σε2z . All random variables are independent
across goods, stores, and times. In our baseline model, we set q = 1.
We estimate the parameters of the statistical model in (2) using data on quantity-weighted
average prices, pjst , for a large number of goods j = 1 . . . J, at a large number of stores
s = 1 . . . S, in a single geographic market m at a weekly frequency t = 1 . . . T . Given the
large number of goods, stores, and time periods, and the presence of unobserved components
in prices, estimating this model via maximum likelihood (or with panel data instrumental
variables regressions) is not feasible. Instead, we estimate the model using a multistage
generalized method of moments approach that is analogous to techniques that are commonly
used when estimating models of labor earnings dynamics (see, e.g., Gottschalk and Moffitt
9
1994, Blundell and Preston 1998, Kaplan 2012).
The estimation procedure involves four steps.
Step 1. We estimate the good-time mean, µjt , as the average of the log price, log pjst , across
all stores s in the market of interest, i.e.,
S
1X
log pjst .
µ̂jt =
S s=1
(3)
p̃jst = log pjst − µ̂jt .
(4)
We construct normalized prices as
Step 2. We estimate the store component, ŷst , by taking sample means of the normalized
prices across all goods in store s, i.e.,
njst
1 X
ŷst =
p̃jst ,
njst j=1
(5)
where njst is the number of goods for which we have data for store s in period t. In some
instances, njst < J because not every store-good combination will meet our sample selection
requirements in every week. We estimate the store-good component, zjst , as
ẑjst = p̃jst − ŷst .
(6)
The above process leads to a S × T panel of store components {ŷst } and a (J × S) × T panel
of store-good components {ẑjst } (where there may be missing data for some combinations
of (j, s, t)).
Step 3. We construct the auto-covariance matrix of each of these panels up to L lags.
Step 4. We minimize the distance between the theoretical auto-covariance matrices implied
by the model and the empirical auto-covariance function from step three. We use a diagonal
weighting matrix that weights each moment by n0.5
jst . However, the main results are not
sensitive to using an identity weighting matrix instead.
2.2
Kilts-Nielsen Retail Scanner Dataset
We estimate the statistical model in (2) using the KNRS dataset. The KNRS dataset contains
store-level weekly sales and unit average price data at the UPC level. The dataset covers
10
the period 2006 to 2012. The full dataset contains weekly price and quantity information
for more than 1.5 million UPCs at around 40,000 stores in more than 2,500 counties across
205 DMAs. A DMA is a geographic area defined by Nielsen that is roughly the same size
as a metropolitan statistical area. Since our estimation procedure requires computing a full
auto-covariance matrix at the store-good-week level, it is not feasible to estimate the model
using anywhere near the full set of UPCs. For example, in the Minneapolis-St Paul DMA
alone, the full data set would consist of more than 200 million observations of pjst per year.
Thus, in order to keep the size of the analysis manageable, we restrict attention to a subset
of the data.
We focus our analysis on a single DMA: Minneapolis-St Paul. However, there is nothing
particularly special about Minneapolis-St-Paul; in Section 2.4, we show that our findings are
robust to extending the analysis to cover a broad set of geographically dispersed markets.
For the Minneapolis-St Paul DMA, we focus on the 1,000 UPCs with the largest quantities
of sales in the state of Minnesota in the first quarter of 2010. These 1,000 products span
50 product groups, as shown in Table 7 in Appendix A.3 Table 8 in Appendix A shows how
these 1,000 sample UPCs are distributed across goods departments. To give a sense of how
frequently these products are purchased in the Minneapolis market areas in the first quarter
of 2010, the product with the largest quantity of units sold (2.9 million units) was in the
“Fresh Eggs” product module. The least frequently sold of these 1,000 products (with just
under 50,000 units sold) was in the “Liquid Cocktail Mixes” product module. Even after
restricting attention to these 1,000 products, the dataset is extremely large. Over the seven
year period from 2006 to 2012, we have more than 40 million observations of prices pjst . To
ensure that our findings are not specific to this particular bundle of goods, in Section 2.4 we
re-estimate the model using a number of alternative sets of UPCs, chosen in various ways.
We always estimate the model separately for each geographic area that we consider. For
a given set of UPCs and a given geographic area, we select stores, goods, and weeks that
satisfy two criteria:
1. For each store/week combination, we have quantity and price data for at least N1 of the
UPCs in the given set. In our baseline estimation, we set N1 = 250, and we report results
for N1 ∈ {50, 500}.
3
Nielsen divides the full set of UPCs in the product database into 10 “departments,” which are subdivided
into around 125 “product groups,” that are further subdivided into around 1,075 “product modules.” For
example, different sized bottles of Heinz Tomato Ketchup have distinct UPCs in the “Catsup” product
module, which is one of 34 product modules in the “Condiments, Gravies and Sauces” product group, one
of 38 product groups in the “Dry Grocery” department.
11
2. For each good/week combination, we have quantity and price data for at least N2 stores.
In our baseline estimation, we set N2 = 50, and we report results for N2 ∈ {25, 100}.
These selection criteria ensure that we focus only on store/goods/weeks where we have
sufficient data to reliably estimate the good-time means and store-time means in the first
and second stages of the estimation procedure. In addition, to avoid the influence of large
outliers when computing the empirical auto-covariance function, we drop observations of the
store components and store-good components whose absolute value is greater than one.
2.3
Baseline Decomposition
We first present results for the baseline set of 1,000 goods in the Minneapolis-St Paul DMA.
We then consider the robustness of these results to a range of alternative specifications,
including whether they vary significantly across markets in the United States.
Figure 1 displays the fit of the auto-correlation function for the store component (Panel A)
and the store-good component of prices (Panel B) out to 100 lags. The parameter estimates
that correspond to this model are reported in Table 9 in Appendix A. For both components,
the statistical model provides an excellent fit to the shape of the auto-correlation function.
Several features of the auto-correlation functions are worthy of mention. First, the autocorrelation of the store component is very high even at long lags, foreshadowing our finding
that almost all of the store component is persistent in nature. The auto-correlation of the
store component asymptotes to around 0.8, equivalent to an auto-covariance of 0.0029, which
is what identifies the variance of the fixed effect (whose standard deviation is estimated to
be 5.3%). Second, the sharp drop in the auto-correlation of the store-good component after
one lag suggests the presence of a large transitory component in prices. Third, the slow
exponential decay and then flattening out of the store-good component suggest the presence
also of a substantial persistent part of the store-good component. The auto-correlation of the
store-good component asymptotes to around 0.17, equivalent to an auto-covariance of 0.0033,
which is what identifies the variance of the fixed effect (whose standard deviation is estimated
to be 5.7%). Fourth, the spike at 52 weeks reflects the fact that some products display annual
regularities in their prices. Finally, the zig-zag pattern of the auto-correlation of the storegood component is due to regularities in the patterns of sales that are not captured by our
statistical model.
Overall, the estimated model fits the data very well, and, for this reason, we are comfortable using it to decompose the cross-sectional variance of the price at which the same good
12
Table 1: Dispersion in prices: persistent and transitory
Variance
Store component
Transitory
Fixed plus Pers.
Total Store
Store-good component
Transitory
Fixed plus Pers.
Total Store-good
Total
0.000
0.004
0.013
0.007
Percent
3.2
96.8
0.004 100.0
Std. Dev.
15.5
0.011
0.059
0.060
64.1
35.9
0.020 100.0 84.5
0.023
100.0
0.113
0.084
0.141
0.153
Notes: The left panel presents the cross-sectional variances of UPC prices, as well as the store and store-good
components separately. The middle panel presents the decomposition of this variance into persistent and
transitory components. The right panel presents the cross-sectional standard deviations.
is sold in the same week and in the same market. The variance decomposition is reported in
Table 1. The overall variance of prices is 0.023, equivalent to a standard deviation of 15.3%.
The variance of the store component accounts for 15% of the overall variance of the price,
while the variance of the store-good component accounts for the remaining 85%. That is,
most of the variation in the price at which a good is sold is not due to the fact that the good
is sold at stores that are, on average, more or less expensive. Rather, most of the variation
in the price at which a good is sold is due to the fact that the good is sold at different prices
at stores that are, on average, equally expensive.
The variation in prices associated with the store and store-good components could be
due to either the transitory or persistent components. The statistical model (2) is designed
to distinguish between these two sources of variation. Since the estimated persistence of
the AR(1) component of prices is extremely close to unity for both the store and storegood components (the estimates of ρz and ρy for the baseline model are 0.965 and 0.983,
respectively), we group the fixed effect and AR(1) components together and refer to these as
the persistent part of the price, and we refer to the MA components as the transitory part
of the price.4
The decomposition in Table 1 reveals that nearly all of the price variance that is due to
variation in the store component comes from persistent differences in the average price of
different stores. In contrast, 64% of the price variance that is due to the variation in the
4
The estimation results remain very similar if we adopt an identity weighting matrix for the GMM
objective, instead of a diagonal weighting matrix based on sample sizes.
13
store-good component comes from transitory differences across stores in the price of the good
relative to the average price of the store. Yet, over one-third of the price variation that is due
to the variation in the store-good component comes from persistent differences across stores
in the relative price of the good. This is what we call relative price dispersion. Relative price
dispersion is a feature of the data that had not been documented before, and, at first blush,
it seems hard to rationalize. Why do stores that are, on average, equally expensive choose
to systematically charge different prices for the very same good?
Finally, notice that while variance decompositions are a convenient tool for breaking
down dispersion into orthogonal elements, the fact that variances are measured in squared
prices makes the comparison of the various elements somewhat hard to interpret. For this
reason, the last column of Table 1 reports the standard deviation of each of the orthogonal
components of prices implied by the model. The overall standard deviation of prices is 15%,
and the standard deviation due to persistent differences in relative prices is 8%. These figures
further emphasize the point that persistent differences in relative prices are an important
feature of the retail market.
2.4
Robustness
The estimates in Table 1 highlight two important features of price dispersion, both of which
turn out to be extremely robust. First, the vast majority of price dispersion is due to
variation in the store-good component of prices rather than the store component of prices.
Second, of the variation in the store-good component, at least one-third is due to highly
persistent differences across stores in the price of the good relative to the price of the store.5
In this section, we present results from a series of estimates under alternative assumptions
about the sample selection criteria, the statistical model for prices, for different types of
products, and different geographic areas.
2.4.1
Sample Selection
Our baseline selection criteria required that a minimum of N1 = 250 of the 1,000 goods in
our sample be sold at a given store in a given week for that store/week to be included in the
estimation sample. Table 2 reports variance decompositions for N1 ∈ {50, 500} and shows
that the results are not sensitive to this particular threshold. Our baseline selection criteria
5
An immediate implication of these two features of the data is that the price ranking of stores for a
particular good differs a lot from one good to another, and that the correlation over time of these rankings
is non-negligible.
14
Table 2: Robustness to sample criteria
N1 = 50
Sd
Dec/%
Store
Transitory
Fixed plus Pers.
Total Store
Store-good
Transitory
Fixed plus Pers.
Total Store-good
Total
N1 = 500
Sd
Dec/%
N2 = 25
Sd
Dec/%
N2 = 100
Sd
Dec/%
0.017
0.075
0.077
5.0
95.0
19.0
0.011
0.055
0.056
3.6
96.4
13.6
0.011
0.058
0.059
3.2
96.8
15.3
0.011
0.063
0.064
3.3
96.7
16.8
0.126
0.096
0.158
0.176
63.3
36.7
81.0
100.0
0.113
0.082
0.140
0.151
65.4
34.6
86.4
100.0
0.111
0.084
0.139
0.151
63.8
36.2
84.7
100.0
0.114
0.084
0.141
0.155
65.0
35.0
83.2
100.0
Notes: This table presents a robustness exercise comparing our baseline results to results obtained using
alternative cutoffs for required numbers of observations.
also required that a minimum of N2 = 50 stores have positive sales for a given good in a
given week for that good/week to be included in the sample. Table 2 also reports variance
decompositions for N2 ∈ {25, 100} and shows that the results are also not sensitive to this
threshold. Thus, it is unlikely that relative price dispersion is a statistical artifact of small
samples and/or insufficient overlap of goods across stores. This is important because for
some of the sets of UPCs considered below, we are required to set N1 = 50 and N2 = 25 in
order to have sufficient overlap for reliable estimation.
2.4.2
Statistical Model
In our baseline specification, we modeled the transitory part of the store-good component as
an MA(1), which implicitly assigns all price changes with a duration greater that one week
to the persistent component. Since the transitory component is intended to capture the
effects of temporary sales, the reader may be concerned that, if some sales last more than
one week, our baseline specification may be interpreting some sales-induced price variation
as relative price dispersion. To show that this is not the case, Table 3 reports the variance
decomposition when we model the temporary component as an MA(5) or MA(10), thus
allowing the transitory component to capture sales that potentially last up to 10 weeks. The
decomposition is barely affected — the persistent parts of the store-good component still
account for at least one-third of the variance of the store-good component.
The reader may also be concerned that modeling temporary sales as an MA process
of any order may fail to capture salient features of sales dynamics and that this may lead
15
Table 3: Robustness to statistical model
Sd
Store-good
Transitory
Fixed plus Pers.
Total Store-good
MA(5)
Dec/%
0.114
0.082
0.141
65.6
34.4
100.0
MA(10)
Sd
Dec/%
0.114
0.082
0.141
66.1
33.9
100.0
Skewed MA(1)
Sd
Dec/%
0.113
0.084
0.141
64.1
35.9
100.0
Sd
Sales
Dec/%
0.098
0.084
0.141
57.7
42.3
100.0
Notes: This table shows variance decompositions for the store-good component after replacing the MA(1)
process with: (i) an MA(5); (ii) an MA(10); (iii) an MA(1) that allows skewness in the disturbances; and
(iv) an explicit model of sales, as described in the main text.
us to understate the importance of sales. To show that this is not the case, we consider
two alternative approaches to modeling sales. First, we recognize that the cross-sectional
distribution of prices induced by periodic sales is likely to be negatively skewed, which
may help in identifying the component of price dispersion that is due to sales. To account
for this possibility we allow that the transitory innovation to the store-good component,
εzjs,t, is drawn from a skewed distribution, whose skewness we estimate alongside the other
parameters of the model.6 As expected, the estimated skewness is mildly negatively skewed
(coefficient of skewness = −0.6), consistent with sales. However, the decomposition of the
variance of prices, shown in Table 3, is not affected.
Second, we depart from the assumption of an MA process for the transitory store-good
component and replace it with a process that more explicitly resembles temporary sales.
In this model, the transitory store-good component is modeled as a 2-point process. In a
given week, there is a probability φ that each good is on sale, and sales are independently
distributed across goods and over time. If a good is on sale, then it is discounted from its
regular price by a fraction δ. We assume that all sales last exactly one week, and the day that
a good goes on sale is uniformly distributed within the week. This means that each sale will
affect the observed price of the good in two adjacent weeks, so the auto-covariance of prices
is impacted at both the zero and first lag (as with an MA(1) process). Since we work with
normalized prices, there is an additional restriction that the mean value of the transitory
component, after accounting for sales, is zero. The estimated weekly sales probability is
φ = 4.65%, and the corresponding average discount is δ = 52%.7 The associated variance
6
Note that our estimation procedure does not require distributional assumptions on the innovations. Our
baseline specification and procedure uses only second moments of the price data and only estimates the
distribution of price innovations up to their second moments. In order to achieve identification of third
moments, we include joint third moments of prices.
7
As with the skewed MA process, the sales process requires joint third moments of prices to be included
in the GMM objective in order to achieve identification. We have also explored richer specifications in which
16
decomposition, show in Table 3, reveals that the persistent components of the store-good
variance are even larger than in the baseline. We conclude that relative price dispersion is
not driven by temporary sales and is a distinct feature of price distributions.
2.4.3
Products
Our baseline set of goods is the 1,000 most-commonly purchased products in Minnesota in
the first quarter of 2010. We now show that relative price dispersion is not specific to this
set of goods but rather is a robust phenomenon that is present among samples of products
chosen in a broad variety of ways. The analysis also serves the purpose of rejecting some
simple explanations for relative price dispersion, such as managerial inattention, store-good
cost differentials, and different styles of shelf management.
Frequency of purchase Our baseline procedure weights each good equally when constructing the good-time means and the store components. In Table 4, we report the variance
decomposition when we use quantity weights to construct the good-time means and store
components. The decomposition is barely affected by this change.
Our baseline sample comprises only goods that are purchased very frequently. We examine whether relative price dispersion is a feature of the data for less frequently purchased
goods. To do this, we select a sample of the 1,000 goods ranked 9,001 to 10,000 in terms of
their frequency of purchase in Minnesota in the first quarter of 2010. This choice is motivated
by our desire to select substantially less-commonly purchased goods than in our baseline sample, while still satisfying the requirement that the goods are sufficiently commonly purchased
so that there is enough overlap across stores and enough continuity in weekly sales to meet
our two inclusion criteria. The types of goods in this alternative sample, shown in Table 10
in Appendix A, are quite different from those in the baseline sample. However, the variance
decomposition for this set of goods, shown in Table 4 (labelled “UPC-alt”), is extremely
similar to the baseline.
Lastly, we selected a different sample of goods based on frequently purchased goods
nationwide in the first quarter of 2010 rather than frequently purchased goods in Minnesota.
Selecting a sample in this way is useful for when we extend our analysis to other parts of
the country below. To construct this sample, we created two lists of the most commonly
the sales discount, δ, is itself a (possibly negatively skewed) random variable, and none of our main findings
are affected.
17
Table 4: Robustness
Baseline
Sd
Dec/%
Store
Transitory
Fixed plus Pers.
Total Store
Store-good
Transitory
Fixed plus Pers.
Total Store-good
UPC-national
Sd
Dec/%
3.2
96.8
15.5
0.019
0.037
0.041
21.0
79.0
6.5
0.006
0.054
0.055
1.4
98.6
18.7
0.011
0.072
0.073
2.2
97.8
18.9
0.113
0.084
0.141
64.1
35.9
84.5
0.124
0.096
0.156
62.6
37.4
93.5
0.082
0.080
0.114
51.1
48.9
81.3
0.119
0.095
0.152
61.0
39.0
81.1
High price
Sd
Dec/%
Low durability
Sd
Dec/%
High durability
Sd
Dec/%
0.024
0.078
0.082
8.7
91.3
20.6
0.025
0.059
0.065
15.6
84.4
15.9
0.013
0.062
0.063
4.0
96.0
19.3
0.027
0.043
0.051
27.9
72.1
19.4
0.122
0.105
0.161
57.4
42.6
79.4
0.130
0.071
0.148
77.0
23.0
84.1
0.103
0.077
0.129
64.0
36.0
80.7
0.077
0.069
0.103
55.8
44.2
80.6
Unilever
Sd
Dec/%
Store
Transitory
Fixed plus Pers.
Total Store
Store-good
Transitory
Fixed plus Pers.
Total Store-good
UPC-alt
Sd
Dec/%
0.011
0.059
0.060
Low price
Sd
Dec/%
Store
Transitory
Fixed plus Pers.
Total Store
Store-good
Transitory
Fixed plus Pers.
Total Store-good
Weighted
Sd
Dec/%
Coca-Cola
Sd
Dec/%
State: MN
Sd
Dec/%
County: Hennepin
Sd
Dec/%
0.035
0.058
0.068
27.4
72.6
21.3
0.030
0.070
0.076
15.5
84.5
26.2
0.011
0.070
0.071
2.5
97.5
17.6
0.015
0.058
0.060
6.2
93.8
12.5
0.101
0.081
0.130
60.9
39.1
78.7
0.106
0.071
0.127
68.9
31.1
73.8
0.120
0.096
0.154
60.9
39.1
82.4
0.128
0.095
0.159
64.4
35.6
87.5
Notes: This table presents a robustness exercise comparing our baseline results to results obtained using
alternative specifications: quantity weighting in constructing the store and store-good components, alternative samples of UPCs (UPC-alternative and UPC-national), the low- and high-price samples, the low- and
high-durability samples, the Unilever and Coca-Cola samples, and alternative definitions of a market (state
of Minnesota and Hennepin County). Tables 7, and 10-17 in Appendix A illustrate the diversity in product
groups across the alternative samples.
18
purchased UPCs, one based on quantity and one based on revenue. We then selected the
1,463 goods that appear in either list. The decomposition for this set of 1,463 goods is also
shown in Table 4 (labelled “UPC-national”). For this set of goods, the store component
accounts for slightly more of the overall price variation, but the persistent components of
prices account for even more of the variance of the store-good component. Hence, relative
price dispersion is larger in this set of goods than in the baseline.
High-price and low-price goods A simple explanation for relative price dispersion is
managerial inattention (see, e.g., Ellison, Snyder, and Zhang 2015). Equally expensive stores
may set persistently different prices for the same good because managers choose to not pay
much attention to the price of low-ticket items.8 This potential explanation for relative
price dispersion motivates us to decompose price dispersion for low- and high-price goods
separately. We divide our baseline sample of 1000 UPCs according to their average unit price.
The low-price subsample of 430 UPCs has a median unit average price of 99 cents, a 5th
percentile of 39 cents and a 95th percentile of $1.79; the high-price subsample of 315 UPCs
has a median unit average price of $3.59 cents, a 5th percentile of $2.39 and a 95th percentile
of $6.99. The variance decompositions for these two subsamples are shown in Table 4. The
low-price subsample features more relative price dispersion than the full sample: The storegood component accounts for 79% of the overall variance of prices, of which the persistent
components account for 43%. The high-price subsample features less relative price dispersion
than the full sample, but relative price dispersion is still a substantial fraction of overall price
dispersion. Hence, relative price dispersion is not only a feature of low-price, low-revenue
goods and thus is unlikely to be entirely due to managerial inattention.
Products from a single distributor Another possible explanation for relative price
dispersion is that equally expensive stores set persistently different prices for the same good
because they have better or worse relationships (and, hence, are charged lower or higher
prices) with the wholesaler.9 This potential explanation motivates us to decompose price
dispersion for a subset of products produced and distributed by a single wholesaler. Indeed,
if relative price dispersion is caused by different retailer-wholesaler relationships, relative
price dispersion should be absorbed by the store component when we restrict attention to
products from a single wholesaler.
8
9
We thank Stephan Seiler for suggesting this hypothesis.
We thank Matthew Gentzkow for suggesting this hypothesis.
19
We consider two subsamples of goods. In the first subsample, there are only products
from Coca-Cola. In the second subsample, there are only products from Unilever. The 3,608
UPCs in our Coca-Cola subsample are primarily various types of beverages. The 10,866
UPCs in our Unilever subsample come from a variety of product groups; “Hair Care” is the
product group with the largest fraction of UPCs (32%), followed by “Personal Soap and
Bath Additives” (13%), “Deodorant” (12%) and “Skin Care Preparations” (10%).
The variance decompositions for these two subsamples of goods are shown in the bottom
row of Table 4. For both samples, the overall degree of price dispersion is very similar to the
degree of price dispersion in our baseline sample. However, the fraction of variation that is due
to the store component is somewhat larger – 21% for Unilever and 26% for Coca-Cola,
compared with 16% for the baseline. This is consistent with the hypothesis that some part of
price dispersion is due to different relationships between particular stores and particular
distributors. However, for both of these distributors, the vast majority of price dispersion is
due to the store-good component, and, of this, the persistent parts account for 39% (Unilever)
and 31% (Coca-Cola). Thus, relative price dispersion exists even when only considering goods
from the same distributor and so is not only driven by heterogeneity in distributional
relationships.
High-durability and low-durability products Another natural explanation for relative
price dispersion is shelf management. Some stores may keep perishable goods on their
shelves for longer and, for this reason, sell them at systematically lower prices, while other
stores may remove perishable goods sooner and, for this reason, sell them at systematically
higher prices. This observation motivates us to decompose price dispersion separately for
two subsamples of goods: low-durability goods (i.e., perishable goods) and high-durability
goods.10 The variance decompositions for these two subsamples are shown in Table 4. Even
though the two subsamples contain very different sets of products, the overall decomposition
of price dispersion is quite similar. For both subsamples, the store component accounts for
approximately 20% and the store-good component for 80% of the cross-sectional variance
of prices. For both subsamples, the transitory part accounts for roughly two-thirds and
the persistent part for roughly one-third of the cross-sectional variance of the store-good
10
We thank Boyan Jovanovic for suggesting we measure relative price dispersion for low- and highdurability products. We also thank George Alessandria for sharing the durability indexes constructed in
Alessandria, Kaboski, and Midrigan (2010). We merge this index with the Nielsen database at the product
module level by comparing descriptions of products. We define low-durability goods as those with a durability index of less than 2 months, and high-durability goods as those with a durability index of more than
140 months.
20
.4
.25
.1
0
0
40
60
80
100
0
.05
.1
.1
.2
.2
.15
.3
.3
.2
.4
.4
.3
.2
.1
20
0
20
40
60
80
100
0
20
40
60
80
100
0
0
.14
.16
.18
.2
Store−good
Persistent
(b) Store vs store-good
Transitory
Persistent
(c) Store
Transitory
(d) Store-good
.2
.2
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
0
0
0
.1
.1
0
0
.1
.1
.2
.2
.3
.3
.3
.4
.3
.4
(a) Sd of prices
Store
.4
.12
.15
.16
.17
.18
(e) Sd of prices
.19
Store
Store−good
Persistent
(f) Store vs store-good
Transitory
(g) Store
Persistent
Transitory
(h) Store-good
Notes: These histograms present a robustness exercise by looking at how our results for price dispersion
and the variance decompositions vary across geographic regions across the country. The top row considers
DMAs across the US, and the bottom counties across the US.
Figure 2: Price dispersion and variance decompositions across geographic areas
component of prices. These findings suggest that relative price dispersion is unlikely to be
a phenomenon caused by different styles of shelf management for perishable goods. Indeed,
relative price dispersion turns out to be slightly more important in the subsample of goods
that are less perishable.
2.4.4
Markets
So far our analysis has focused on a single geographic region — the Minneapolis-St Paul
DMA. Here, we show that none of our results are specific to this level of geographic aggregation or this part of the country. First, we consider alternative levels of geographic aggregation
for the definition of a market. In Table 4, we report the variance decomposition when we
use a broader definition of market (the state of Minnesota) and a narrower definition of a
market (Hennepin County, which is contained in the Minneapolis-St Paul DMA). All our
findings are robust to switching to either of these alternative levels of aggregation.
Second, we extend our analysis to the whole of the United States in order to verify that
our findings are not specific to Minneapolis-St Paul. We present results both at the level of
a DMA and the county level. For each level of geographic aggregation, we selected the 25
largest areas by revenue and repeated the estimation for each market, using the same set
of 1,463 UPCs. As described earlier, this set of UPCs was chosen to reflect UPCs that are
commonly purchased nationwide.
21
Figure 2(a) displays a histogram of the standard deviation of prices in each of the 25
DMAs. The corresponding variance decomposition between the store and store-good components is shown in Figure 2(b), and the fraction of the variance of each component that is
due to transitory versus persistent factors is shown in Figures 2(c) and 2(d), respectively.
The analogous statistics are displayed for the 25 counties in the bottom row of Figure 2.
Figure 2 shows very clearly that our findings are not unique to any one particular region
but instead are a general feature of price dynamics and distributions. For all geographic
areas, virtually all of the variance of prices is due to the store-good component rather than
the store component, and a substantial part of the variance of the store-good component
(between one-third and one-half) is very persistent in nature.
3
Relative Price Dispersion: Theory
In this section, we develop a theory of relative price dispersion. According to our theory,
relative price dispersion emerges as a strategy that retailers use in order to price discriminate
between high-valuation buyers who need to make all of their purchases in the same location,
and low-valuation buyers who have the time to purchase different items in different places. In
Section 3.1, we describe the model, which is an extension to multiple goods and multiple types
of buyers of the canonical theory of price dispersion of Burdett and Judd (1983). In Section
3.2, we establish some general properties of equilibrium. In Section 3.3, we characterize the
equilibrium when competition between sellers is weak. In this equilibrium, sellers post prices
that are attractive only to the high-valuation buyers. Since high-valuation buyers purchase
all goods in the same location, the model behaves like the one-good model of Burdett and
Judd (1983). In Section 3.4, we characterize the equilibrium when competition between
sellers is stronger. In this equilibrium, some sellers post some prices that are attractive to
low-valuation buyers. Since low-valuation buyers can purchase different goods from different
locations, the model does not behave like Burdett and Judd (1983), and, indeed, the model
features relative price dispersion. In Section 3.5, we briefly describe the equilibrium when
competition between sellers is strongest.
3.1
Environment
We consider a retail market populated by homogeneous sellers and heterogeneous buyers
who trade two goods (i.e., good 1 and good 2). Specifically, the market is populated by a
22
measure s > 0 of identical sellers.11 Every seller is able to produce each of the two goods at
the same, constant marginal cost, which we normalize to zero. Every seller chooses a price
for good 1, p1 , and a price for good 2, p2 , so as to maximize his profits, taking as given the
distribution H(p1 , p2 ) of the vector of prices across sellers. We find it useful to denote as
Fi (p) the fraction of sellers whose price for good i ∈ {1, 2} is smaller than p, and as λi (p)
the fraction of sellers whose price for good i ∈ {1, 2} is equal to p. We refer to Fi (p) as the
distribution of prices for good i ∈ {1, 2}. Similarly, we find it useful to denote as G(q) the
fraction of sellers whose prices p1 and p2 sum up to less than q, and as ν(q) the fraction of
sellers whose prices sum up to q. We refer to G(q) as the distribution of basket prices.
On the other side of the retail market, there is a measure 1 of buyers. A fraction µb ∈ (0, 1)
of buyers are of type b and a fraction µc = 1 − µb of buyers are of type c, where b stands
for busy and c stands for cool. A buyer of type b demands one unit of each good, for which
he has valuation ub > 0. A buyer of type c demands one unit of each good, for which he
has valuation uc , with ub > uc > 0. More specifically, if a buyer of type i ∈ {b, c} purchases
both goods at the prices p1 and p2 , he attains a utility of 2ui − p1 − p2 . If a buyer of type
i ∈ {b, c} purchases one of the two goods at the price p, he attains a utility of ui − p. If a
buyer of type i ∈ {b, c} does not purchase any of the goods, he attains a utility of zero.
In the retail market, trade is frictional. We assume that a buyer cannot purchase from
just any seller in the market, as each buyer only has access to a small network of sellers.
In particular, a buyer of type b can access only one seller with probability αb and multiple
(namely, two) sellers with probability 1 − αb , with αb ∈ (0, 1). Similarly, a buyer of type c
can access only one seller with probability αc and two sellers with probability 1 − αc , with
αc ∈ (0, 1). For the sake of simplicity, we let αb = αc = α.12 We refer to a buyer who
can only access one seller as a captive buyer, and to a buyer who can access multiple sellers
as a non-captive buyer. We interpret these restrictions on the buyers’ access to sellers as
physical constraints (i.e., sellers that the buyer can easily reach) rather than as informational
11
We assume that sellers are ex-ante identical. If sellers had different costs or faced different populations
of buyers, it would be easy to generate relative price dispersion. However, such explanations of relative price
dispersion would be basically unfalsifiable, as data on wholesale prices and demand curves faced by different
retailers is generally unavailable. Indeed, as noted by Stigler (1961), “It would be metaphysical, and fruitless,
to assert that all dispersion is due to heterogeneity.” To further strengthen this point, let us draw a parallel
with the literature on temporary sales. Clearly, one could explain temporary sales with temporary declines
in wholesale prices or with temporary increases in the elasticity of demand faced by retailers. Yet, because
such explanations are hard to falsify and seem empirically implausible, the literature has developed theories
in which sellers choose to have temporary sales in a stationary environment (see, e.g., Conlisk, Gerstner and
Sobel 1984; Sobel 1984; or Albrecht, Postel-Vinay and Vroman 2013).
12
It is tedious but straightforward to generalize our theory of relative price dispersion to the case of
αc > αb .
23
constraints
13
(i.e. sellers of which the buyer is aware). We also assume that a buyer of type
b must always make all of his purchases from just one of the sellers in his network. In
contrast, a buyer of type c can purchase different goods from different sellers in his network.
We interpret this assumption as heterogeneity in the buyer’s ability or willingness to visit
multiple stores when shopping.
A few comments about the environment are in order. We consider a version of the
canonical model of price dispersion of Butters (1977) and Burdett and Judd (1983). In this
model, the co-existence of buyers who have access to only one seller and of buyers who have
access to multiple sellers induces identical sellers to post different prices for the same good.
We depart from the canonical model by considering a market in which buyers and sellers
trade two goods. Obviously, to develop a theory of relative price dispersion, it is necessary
to consider multiproduct retailing. The simplest case of multiproduct retailing involves two
products. We also depart from the canonical model by considering a market in which buyers
are heterogeneous. In particular, we assume that buyers of type b are willing to pay higher
prices for the goods than buyers of type c and that they are less willing to purchase different
goods from different retailers than buyers of type c. It is natural to think about type-b
buyers as buyers whose time has high value in the labor market, and about type-c buyers
as buyers whose time has low value in the labor market. Hence, type-b buyers are willing
to pay higher prices, but they are hesitant to spend time shopping at different retailers.14
The negative correlation in the buyers’ willingness to pay and ability to shop from multiple
retailers is the key to our theory of relative price dispersion.
Our model of the retail market is static, as in Butters (1977) and Burdett and Judd
(1983). We interpret the equilibrium price distribution of our static model as a long-term
outcome. Indeed, in a repeated version of the model, it can be seen immediately that
sellers would have nothing to gain from changing their prices over time. Moreover, in
the presence of any type of adjustment costs, sellers would face a loss from changing their
prices over time. Thus, in a repeated version of the model, sellers would keep their prices
constant. Given our interpretation of the model, we compare the equilibrium price
distribution to the distribution of the persistent component of sellers’ prices.15
13
The reader may want to think of our model as a version of Hotelling (1929), in which buyers have a zero
transportation cost to access some sellers and an infinite transportation cost to access some other sellers.
14
As suggested by Kaplan and Menzio (2014), the reader may want to think of type-b buyers as employed
ones, and of type-c buyers as the unemployed and/or retirees.
15
It would be easy to extend our model to capture the high-frequency fluctuations in sellers’ prices. Menzio
and Trachter (2015a) use a version of Butters (1977) and Burdett and Judd (1983) to show that, if the buyers
who are more willing to intertemporally substitute their purchases are also in contact with more sellers (or
have a lower willingness to pay for goods), then each seller will post different prices at different times of the
24
3.2
General Properties of Equilibrium
We start the analysis of the model by establishing some general properties of equilibrium.
First, we argue that there are no sellers whose prices (p1 , p2 ) sum up to more than 2ub .
Second, we consider sellers whose prices (p1 , p2 ) sum up to strictly more than ub + uc . These
are sellers who cannot price an individual good below the willingness to pay of type-c buyers
without raising the price of the other good above the willingness to pay of type-b buyers.
We show that, in any equilibrium, the prices of these sellers must lie in a particular region
in R2+ . Third, we consider sellers whose prices sum up to less than ub + uc and strictly more
than 2uc . These are sellers who can price one of the goods below the willingness to pay
of type-c buyers, while keeping the price of the other good below the willingness to pay of
type-b buyers. We show that, in any equilibrium, the prices of these sellers must lie in one
of two particular regions in R2+ .
Lemma 1 contains two results. First, the lemma shows that a seller never finds it optimal
to post a price greater than the willingness to pay of type-b buyers. This result is intuitive,
as neither buyers of type b nor buyers of type c are willing to purchase goods at a price
greater than ub. Second, the lemma establishes that, if the sum q of the prices posted by
a seller is strictly greater than ub + uc , then the seller’s prices p1 and p2 are both strictly
greater than the willingness to pay of type-c buyers and smaller than the willingness to pay
of type-b buyers. That is, if the sum q of the prices posted by a seller is strictly greater than
ub + uc , the seller’s prices p1 and p2 must fall in the R1 region in Figure 3. This result is also
intuitive. If a seller has a basket price strictly greater than ub + uc , he must post p1 > uc
and p2 > uc in order to make sure that both of his prices are smaller than ub .
Lemma 1: (i) In any equilibrium, no seller posts prices (p1 , p2 ), such that p1 > ub and/or
p2 > ub . (ii) In any equilibrium, a seller with a basket price q > ub + uc posts prices
(p1 , p2 ) ∈ R1 , where R1 is defined as
R1 = {(p1 , p2 ) : p1 ∈ (uc , ub ], p2 ∈ (uc , ub ], p1 + p2 > uc + ub } .
(7)
Proof : Appendix B.
Lemma 2 states that, if the sum q of the prices posted by a seller is smaller than ub + uc
(but strictly greater than 2uc ), then the seller’s prices p1 and p2 are such that one of them
is smaller than the willingness to pay of type-c buyers, and the other one is greater than the
week in order to discriminate between different types of buyers.
25
Figure 3: Pricing decision of sellers
45o
ub
q = 2ub
R1
p2
R2
uc
q = ub + uc
R2
Region where profit not maximized
uc
q = 2uc
ub
p1
Notes: This figure considers the pricing decision of sellers discussed in the text, illustrating which regions of
the (p1 , p2 ) space will not be profit-maximizing. Conditional on a basket price in the interval ub + uc ≤ q ≤
2ub , sellers will not price outside R1 . Conditional on a basket price in the interval 2uc ≤ q ≤ ub + uc , sellers
will not price outside R2 .
willingness to pay of type-c buyers. That is, if the sum q of the prices posted by a seller is
smaller than ub + uc and strictly greater than 2uc , the seller’s prices p1 and p2 must fall in
one of the two regions marked R2 in Figure 3.
Let us provide some intuition for Lemma 2. Consider a seller with a basket price of
q ∈ (2uc , ub + uc ] who posts the same price p1 = p2 = q/2 for both goods. The seller will
trade with some type-b buyers. However, the seller will not trade with any type-c buyers,
as the price of each good is above their willingness to pay, uc . Now suppose that the seller
starts lowering the price of one good and, at the same, starts increasing the price of the
other good in order to keep the price of the basket constant. The seller will trade with the
same number of type-b buyers as before because these buyers, having to make all of their
purchases in the same place, only care about the price of the basket. However, when the
seller brings the price of the cheaper good below uc , he will also start to trade this good to
some type-c buyers. Therefore, a seller with a basket price of q ∈ (2uc , ub + uc ] will never
find it optimal to post the same price for both goods. For this seller, it is optimal to price
26
the two goods asymmetrically.
Lemma 2: In any equilibrium, a seller with a basket price q ∈ (2uc , ub + uc ] posts prices
(p1 , p2 ) ∈ R2 , where R2 is defined as
R2 = {(p1 , p2 ) : p1 ∈ [0, uc ], p2 ∈ (uc , ub ], p1 + p2 ∈ (2uc , ub + uc ]}
∪{(p1 , p2 ) : p2 ∈ [0, uc], p1 ∈ (uc , ub], p1 + p2 ∈ (2uc , ub + uc ]}.
(8)
Proof : Suppose that there is an equilibrium where a seller posts prices (p1 , p2 ), with q ≡
p1 + p2 ∈ (2uc , ub + uc ], p1 ∈ (uc , ub) and p2 ∈ (uc , ub). The seller attains a profit of
S(p1 , p2 ) = µb [α + 2(1 − α) (1 − G(q) + ν(q)/2)] q.
(9)
Let us explain (9) in detail. The seller is in the network of µb α captive buyers of type b. A
captive buyer of type b will purchase both goods from the seller, as p1 ≤ ub and p2 ≤ ub .
The seller is in the network of µb 2(1 − α) non-captive buyers of type b. If the second retailer
in the network of such a buyer has a basket price q ′ > q, the buyer will purchase both
goods from the seller. In fact, if the buyer purchases both goods from the seller, he attains
a utility of 2ub − q. If the buyer purchases both goods from the second retailer, he attains
a utility of 2ub − q ′ (recall that all sellers post prices below ub ). If the buyer purchases
only one good, he attains a utility that is strictly smaller than ub − uc , which is smaller
than 2ub − q. If the second retailer in the network of the buyer has a basket price q ′ = q,
the buyer is indifferent between purchasing both goods from the seller or from the second
retailer. We assume that, in this case, the buyer will randomize. If the second retailer in
the network of the buyer has a basket price q ′ < q, the buyer will not purchase anything
from the seller. Overall, a non-captive buyer of type b will purchase both goods from the
seller with probability 1 − G(q) + ν(q)/2, and will purchase nothing from the seller with
complementary probability. The seller is also in the network of µc α captive buyers of type c,
and of µc 2(1 − α) non-captive buyers of type c. However, a buyer of type c does not purchase
any good from the seller, as both p1 and p2 are strictly greater than uc .
If the seller deviates from the equilibrium and posts prices (p′1 , p′2 ) with p′1 = uc , p′2 =
q − p′1 , he attains a profit of
S(p′1 , p′2 )
= µb [α + 2(1 − α)1 − G(q) + ν(q)/2] q
+µc [α + 2(1 − α)1 − F1 (p′1 ) + λ1 (p′1 )/2] p′1 .
(10)
The expression for S(p′1 , p′2 ) is easy to understand. The seller is in the network of µb α
captive buyers of type b, and he will sell both goods to each of them. The seller is in the
27
network of µb 2(1 − α) non-captive buyers of type b, and he will sell both goods to a fraction
1−G(q)+ν(q)/2 of them. The seller is in the network of µc α captive buyers of type c, and he
will sell good 1 to each of them. Finally, the seller is in the network of µc 2(1 −α) non-captive
buyers of type c. If the second retailer in the network of such a buyer has a price for good 1
greater than p′1 , the buyer will purchase good 1 from the seller. If the second retailer in the
network of such a buyer has a price for good 1 equal to p′1 , the buyer will randomize. If the
second retailer in the network of such a buyer has a price for good 1 smaller than p′1 , the
buyer will purchase good 1 from the second retailer. Overall, a non-captive buyer of type c
will purchase good 1 from the seller with probability 1 − F1 (p′1 ) + λ1 (p′1 )/2.
Notice that the first term on the right-hand side of (10) is the same as S(p1 , p2 ). This
is because buyers of type b base their purchasing decisions only on the price of the whole
basket of goods, not on the price of individual goods. The second term on the right-hand
side of (10) is strictly positive as it is bounded below by µc αp′1 . Overall, the right-hand side
of (10) is strictly greater than S(p1 , p2 ). Therefore, there is no equilibrium in which a seller
with a basket price of q ∈ (2uc , ub + uc ] finds it optimal to post p1 ∈ (uc , ub) and p2 ∈ (uc , ub ).
This observation, combined with part (i) of Lemma 1, implies that, if a seller posts prices
(p1 , p2 ) with p1 + p2 ∈ (2uc , ub + uc ], he must be setting one of the two prices below uc , and
the other price in between uc and ub .
In Lemma 3, we establish two additional results. First, we show that the distribution of
basket prices across sellers, G(q), does not have any mass points. This property of equilibrium
is obtained for the same reason as in Butters (1977), Varian (1980) or Burdett and Judd
(1983). Specifically, if there is a mass point at q0 > 0, a seller with a basket price of q0 could
lower one of his two prices by an arbitrarily small amount, and, instead of selling to half of
the non-captive buyers of type b who are in touch with another seller charging q0 , he could
sell to all of them. Moreover, if there is a mass point at q0 = 0, a seller with a basket price
of q0 could raise his prices to p1 = p2 = ub and, instead of attaining a profit of zero, he could
attain a strictly positive profit.
Second, we show that the distribution of prices for individual good, Fi (p), does not have
any mass points for p ∈ (0, uc ]. The logic behind this result is also similar to the one in
Butters (1977), Varian (1980), and Burdett and Judd (1983). However, in this case, we
cannot rule out the possibility of a mass point at p0 > uc or at p0 = 0. There may be a mass
point at p0 > uc because, when the price of an individual good is higher than the willingness
to pay of type-c buyers, the seller never trades the good in isolation and his price does not
play any allocative role. There may be a mass point at p0 = 0 because the fact that a seller
28
trades one good at a price of zero does not imply that the seller’s profit is zero.
Lemma 3: (i) In any equilibrium, G(q) does not have any mass points. That is, ν(q) = 0
for all q. (ii) In any equilibrium, F1 (p) and F2 (p) do not have any mass points for p ∈ (0, uc ].
That is, λ1 (p) = 0 and λ2 (p) = 0 for all p ∈ (0, uc ].
Proof : Appendix B.
3.3
Bundled Equilibrium
In this section, we consider an equilibrium in which every seller in the market sets a basket
price q strictly greater than ub + uc . We have already established that sellers with a basket
price q strictly greater than ub + uc post prices for the individual goods that are strictly
greater than the willingness to pay of type-c buyers, and smaller than the willingness to
pay of type-b buyers. Hence, sellers with a basket price q strictly greater than ub + uc sell
the whole bundle of goods to buyers of type b, and do not sell anything to buyers of type
c. For this reason, we refer to this type of equilibrium as a Bundled Equilibrium. In the
first part of this section, we characterize the price distribution in a Bundled Equilibrium. In
the second part, we identify necessary and sufficient conditions for the existence of this type
of equilibrium. Even though a Bundled Equilibrium is equivalent to the equilibrium of the
one-good model of Burdett and Judd (1983) and does not necessarily feature relative price
dispersion, the analysis in this section is useful as a stepping stone in the characterization
of more complex equilibria.
In a Bundled Equilibrium, all sellers set a basket price q greater than ub + uc . In Lemma
1, we showed that a seller with a basket price q greater than ub + uc posts prices p1 and p2
that are strictly greater than uc and smaller than ub . That is, he post prices (p1 , p2 ) ∈ R1 .
In Lemma 4,we show that a seller who posts a pair of prices (p1 , p2 ) ∈ R1 would attain the
same profit by posting any other pair of prices (p′1 , p′2 ) ∈ R1 as long as p′1 + p′2 = p1 + p2 .
Lemma 4: The seller attains the same profit by posting the prices (p1 , p2 ) ∈ R1 and the
prices (p′1 , p′2 ) ∈ R1 as long as p′1 + p′2 = p1 + p2 .
Proof : The profit of a seller posting (p1 , p2 ) ∈ R1 is given by
S(p1 , p2 ) = µb [α + 2(1 − α)(1 − G(p1 + p2 ))] (p1 + p2 ).
(11)
The seller is in the network of µb α captive buyers of type b. A captive buyer of type b
purchases both goods from the seller with probability 1, since both p1 and p2 are smaller
29
than ub . The seller is also in the network of µb 2(1 − α) non-captive buyers of type b. A noncaptive buyer of type b purchases both goods from the seller with probability 1 − G(p1 + p2 ),
which is the probability that the second seller in the buyer’s network has a basket price
greater than p1 + p2 . Finally, the seller is in the network of some buyers of type c. However,
a buyer of type c never purchases from the seller, since both p1 and p2 are strictly greater than
uc . Notice that the seller’s profit in (11) only depends on the sum of p1 and p2 . Therefore,
the seller would attain the same profit by posting any other pair of prices (p′1 , p′2 ) ∈ R1 such
that p′1 + p′2 = p1 + p2 .
Lemma 4 is intuitive. Buyers of type b must make all of their purchases in the same
location. Hence, they are indifferent between visiting a seller with prices (p1 , p2 ) ∈ R1 or
a seller with prices (p′1 , p′2 ) ∈ R1 as long as the two sellers charge the same price for the
whole basket of goods, i.e. p1 + p2 = p′1 + p′2 . Buyers of type c never purchase from a seller
with prices (p1 , p2 ) nor from a seller with prices (p′1 , p′2 ), as both sellers charge prices that
are above their willingness to pay. Since both buyers of type b and buyers of type c are
indifferent between (p1 , p2 ) and (p′1 , p′2 ), so is a seller. In turn, this implies that the profit of
a seller with prices (p1 , p2 ) ∈ R1 can be written as a function of q = p1 + p2 , i.e.,
S1 (q) = µb [α + 2(1 − α)(1 − G(q))] q.
(12)
Now we are in the position to establish two properties of the distribution G of basket
prices in a Bundled Equilibrium. First, the highest basket price, qh , on the support of G
equals 2ub. To see why, suppose that qh is strictly smaller than 2ub . In this case, the profit
for a seller with a basket price of qh is then equal to µb αqh , as this seller is the one with
the highest basket price in the economy and, hence, only sells to captive buyers of type b.
However, if the seller sets a basket price of 2ub , he attains a profit of µb α2ub , as the seller still
only sells to captive buyers of type b. Since µb αqh < µb α2ub, it follows that the seller with
a basket price of qh is not maximizing his profit, and, hence, this cannot be an equilibrium.
Next, suppose that qh is strictly greater than 2ub. In this case, Lemma 1 is violated. Hence,
this cannot be an equilibrium either. Overall, in a Bundled Equilibrium, qh = 2ub.
Second, the support of G is an interval [qℓ , qh ]. To see why, suppose that the support of
G has a gap between the basket price q0 and the basket price q1 . In this case, a seller with
a basket price of q0 attains a profit of µb [α + 2(1 − α)(1 − G(q0 ))] q0 . A seller with a basket
price of q1 attains a profit of µb [α + 2(1 − α)(1 − G(q1 ))] q1 . Since G has a gap between q0
and q1 , G(q0 ) = G(q1 ) and the seller with a basket price of q0 makes the same number of
trades as a seller with a basket price of q1 but enjoys a lower profit per trade. Therefore, the
30
45o
1
ub
p2
G(q)
q = qh
q = ql
uc
q = 2uc
Region where profit maximized
Example of support of H
0
0
uc
ub
q = ub + uc
0
0 2uc
p1
ql
qh
q
Notes: This figure shows the possible range of the support of the joint distribution H(p1 , p2 ) and the shape
of the cumulative distributions G(q) in the Bundled Equilibrium.
Figure 4: Bundled Equilibrium
seller with a basket price of q0 does not maximize his profit, and, hence, this cannot be an
equilibrium.
Next, we can solve for the distribution G of basket prices. In a Bundle Equilibrium, the
seller’s profit must attain its maximum for any q on the support of G. That is, S1 (q) = S ∗
for all q ∈ [qℓ , qh ]. Since S1 (qh ) = S ∗ and qh = 2ub, the maximized profit S ∗ is given by
µb α2ub. Since S1 (q) = S ∗ for all q ∈ [qℓ , qh ] and S ∗ = µb α2ub, it follows that
µb [α + 2(1 − α)(1 − G(q))] q = µb α2ub.
(13)
Solving (13) with respect to G(q), we obtain the equilibrium distribution of basket prices
G(q) = 1 −
α
2ub − q
.
2(1 − α)
q
(14)
Solving G(qℓ ) = 0 with respect to qℓ , we obtain the lower bound of the equilibrium distribution of basket prices
qℓ =
α
2ub .
2−α
(15)
Figure 4 illustrates the equilibrium distribution of basket prices (14). Notice that the distribution of basket prices in (14) is the exactly the same as the equilibrium price distribution
of Burdett and Judd (1983). This result is not surprising. In a Bundled Equilibrium, a seller
and a buyer either trade the entire basket of goods or they do not trade at all. Hence, in a
31
Bundled Equilibrium, our model of multiproduct retailing boils down to the single-product
model of retailing of Burdett and Judd (1983), where the single product being traded in the
market is the entire basket of goods.
In a Bundled Equilibrium, the distribution G of basket prices across stores is uniquely
pinned down. The distribution of basket prices is non-degenerate, as there is a positive
measure of sellers with any basket price q in the non-empty interval between qℓ and qh .
The fact that the distribution of basket prices is non-degenerate means that the equilibrium
features price dispersion across stores, in the sense that some sellers are, on average, expensive
and other sellers are, on average, cheap.
In a Bundled Equilibrium, the distribution H of price vectors across stores is not uniquely
pinned down. In particular, any distribution of price vectors H(p1 , p2 ) that has support inside
R1 and that generates the distribution of basket prices G(q) in (14) is such that every price
on the support of H maximizes the profit of the seller and, hence, is an equilibrium. For
instance, there is an equilibrium in which there are G′ (q) sellers with a basket price of q,
and each one of them posts the price q/2 for both good 1 and good 2. This equilibrium does
not feature relative price dispersion, as there is no dispersion across sellers in the price of
a particular good at a particular seller relative to the average price charged by that seller.
However, there always also exists an equilibrium in which there are G′ (q) sellers with a basket
price of q, and each one of them posts prices (p1 , q − p1 ) where p1 is randomly drawn from
a uniform distribution with support (uc , q − uc ]. In this equilibrium, there is relative price
dispersion. Yet, such relative price dispersion is a matter of indifference, as neither buyers nor
sellers care about the price of any individual good. In this sense, a Bundled Equilibrium—
and, more generally, any model of multiproduct retailing in which buyers always purchase
the entire basket of goods in the same location—does not offer a particularly compelling
theory of relative price dispersion.
We conclude the analysis by identifying necessary and sufficient conditions for the existence of a Bundled Equilibrium. We find that this type of equilibrium exists if and only
if
µc
3α − 2
≤
− 1.
µb
(2 − α)uc /ub
(16)
Condition (16) is satisfied if: (i) the market is not too competitive, in the sense that the
fraction α of buyers who are in contact with only one seller is greater than 2/3; and (ii)
the relative number of type-c buyers, µc /µb , and/or the relative willingness to pay of type-c
buyers, uc /ub, is not too large. Intuitively, if the market is too competitive, some sellers
would want to set a price q for the basket of goods that is smaller than ub + uc . Hence, a
32
Bundled Equilibrium would not exist. If the relative number of type-c buyers and/or the
relative willingness to pay of type-c buyers is too high, some sellers would want to trade
with type-c buyers and set a price p for one or both goods that is smaller than uc . Hence, a
Bundled Equilibrium would not exist.
The following proposition summarizes the characterization of a Bundled Equilibrium.
Proposition 1: (Bundled Equilibrium) (i) In a Bundled Equilibrium, the distribution of
basket prices, G, is continuous over the support [qℓ , qh ] and given by (14); the distribution
of price vectors across sellers, H, is not uniquely pinned down. (ii) A Bundle Equilibrium
exists if and only if (16) is satisfied.
Proof : Appendix B.
3.4
Discrimination Equilibrium
In this section, we consider an equilibrium in which some sellers have a basket price q greater
than ub + uc , and some sellers have a basket price smaller than ub + uc and greater than 2uc .
We refer to this type of equilibrium as a Discrimination Equilibrium, as in this equilibrium
some sellers set their prices so as to discriminate between the high-valuation buyers who must
purchase all the goods in the same location and the low-valuation buyers who can purchase
different goods in different locations. In the first part of the section, we characterize the price
distribution in a Discrimination Equilibrium and show that it necessarily features relative
price dispersion. In the second part, we identify necessary and sufficient conditions for the
existence of this type of equilibrium.
We start the characterization of a Discrimination Equilibrium by focusing on the sellers
with a basket price q strictly greater than ub + uc . Following the same arguments as in the
previous section, we can easily show that, among sellers with q > ub + uc , the distribution
of basket prices G(q) has support over the interval [q ∗ , qh ], with ub + uc < q ∗ < qh = 2ub .
Moreover, using the fact that all of the sellers with q ∈ [q ∗ , qh ] must attain the maximized
profit S ∗ , we can easily show that the maximized profit S ∗ is equal to µb α2ub and that, for
all q ∈ [q ∗ , qh ], the distribution of basket prices G(q) is equal to
G(q) = 1 −
α
2ub − q
.
2(1 − α)
q
(17)
Next, we focus on sellers with a basket price q smaller than ub + uc and strictly greater
than 2uc . In Lemma 2, we established that a seller with a basket price q ∈ (2uc , ub + uc ] sets
33
the price of one good below the willingness to pay of type-c buyers, and sets the price of the
other good in between the willingness to pay of type-c buyers and the willingness to pay of
type-b buyers. That is, a seller with a basket price q ∈ (2uc , ub + uc ] sets prices (p1 , p2 ) ∈ R2 .
Now, consider a seller who posts prices (p1 , p2 ) ∈ R2 . The seller trades the basket of goods
to buyers of type c at the price q = p1 + p2 , and he trades the cheaper of the two goods (say
good i) to buyers of type b at the price pi . Thus, the seller attains a profit of
S2i (q, pi ) = µb [α + 2(1 − α)1 − G(q)] q
+µc [α + 2(1 − α)1 − Fi (pi )] pi .
(18)
Note that (18) makes use of the fact that G(q) does not have any mass points, and Fi (p)
does not have any mass points over the interval (0, uc ].
The next lemma shows that, for all p ∈ [0, uc ], the fraction of sellers charging less than p
for good 1 is exactly the same as the fraction of sellers charging less than p for good 2. That
is, F1 (p) = F2 (p) = F (p) for all p ∈ [0, uc ]. The lemma implies that the profit of a seller
in region R2 is symmetric in the two goods. That is, S21 (q, p) = S22 (q, p) = S2 (q, p). The
lemma is intuitive. If F1 (p) > F2 (p) for p ∈ (p0 , p1 ), with 0 ≤ p0 < p1 ≤ uc , then a seller
posting the prices (p, q − p) ∈ R2 would be better off posting the prices (q − p, p) instead. In
fact, the seller trades the basket of goods to the same number of type-b buyers and at the
same price by posting either (q − p, p) or (p, q − p). However, by posting (q − p, p) rather
than (p, q − p), the seller trades the cheaper good to more type-c buyers even though he
charges the same price for it. Hence, if F1 (p) > F2 (p) for p ∈ (p0 , p1 ), all sellers posting the
prices (p, q − p) ∈ R2 would be better off switching the price tags of the two goods until
F1 (p) = F2 (p).
Lemma 5: In a Discrimination Equilibrium, F1 (p) = F2 (p) for all p ∈ [0, uc].
Proof : Appendix B.
The next lemma shows that the profit of a seller in region R2 attains the maximum
S for all prices of the basket q and prices of the cheaper good p such that q is in the
∗
interval [qℓ , ub + uc ] and p is in the interval [pℓ , uc ], where qℓ denotes the lower bound on the
support of the price distribution of baskets and pℓ denotes the lower bound on the support
of the price distribution of an individual good. That is, S2 (q, p) = S ∗ for all (q, p) such
that q ∈ [qℓ , ub + uc ] and p ∈ [pℓ , uc ]. The proof of the lemma follows the same strategy as
Proposition 3 in Menzio and Trachter (2015b). The gist of the proof is to show that, if profits
are not constant for all (q, p) such that q ∈ [qℓ , ub + uc ] and p ∈ [pℓ , uc ], there are either gaps
on the support of the distribution of G over the interval [qℓ , ub + uc ] or gaps on the support
34
of the distribution F over the interval [pℓ , uc ]. In turn, if there are gaps on the support of
one of the two distributions, there are some sellers that could increase their profits by either
increasing the price of the basket or by increasing the price of one of the cheaper good.
Lemma 6: In a Discrimination Equilibrium, S2 (q, p) = S ∗ for all (q, p) such that q ∈
[qℓ , ub + uc ] and p ∈ [pℓ , uc ].
Proof : Appendix B.
We are now in the position to solve for the lowest basket price q ∗ posted by sellers in
region R1 , for the marginal distribution G(q) of basket prices among sellers in region R2 , and
for the marginal distribution F (p) of prices among sellers in region R2 . Lemma 6 implies
that a seller posting prices (uc , ub ) attains the maximized profit S ∗ , i.e.,
µb [α + 2(1 − α) (1 − G(ub + uc ))] (ub + uc ) + µc [α + 2(1 − α) (1 − F (uc ))] uc = S ∗ .
(19)
Similarly, a seller posting prices (p1 , p2 ) such that p1 ∈ (uc , ub], p2 ∈ (uc , ub], and p1 + p2 = q ∗
attains the maximized profit S ∗ , i.e.,
µb [α + 2(1 − α) (1 − G(q ∗ ))] q ∗ = S ∗ .
(20)
Notice that the fraction of sellers with a basket price smaller than q ∗ is the same as the
fraction of sellers with a basket price smaller than ub + uc , i.e., G(q ∗ ) = G(ub + ub ). Also,
notice that the fraction of sellers that charge less than uc for good 1 is half of the fraction
of sellers with a basket price smaller than q ∗ , i.e., F (uc ) = G(q ∗ )/2. Using these two
observations and the fact that the left-hand side of (19) is equal to the left-hand side of (20),
we obtain
µb [α + 2(1 − α) (1 − G(q ∗ ))] (ub + uc ) + µc [α + 2(1 − α) (1 − G(q ∗ )/2)] uc
= µb {α + 2(1 − α)[1 − G(q ∗ )]} q ∗ .
(21)
Equation (21) can be solved with respect to q ∗ to obtain
q∗ =
2α(1 + uc /ub) + α(µc /µb )(uc /ub )
2ub.
4α − (2 − α)(µc/µb )(uc /ub )
(22)
Lemma 6 implies that a seller posting any prices (p1 , p2 ) such that p2 ∈ (uc , ub] and
q = p1 + p2 ∈ [qℓ , ub + uc ] attains the same profit as a seller posting prices (uc , ub ), i.e.,
µb [α + 2(1 − α) (1 − G(q))] q + µc [α + 2(1 − α) (1 − F (uc ))] uc
= µb [α + 2(1 − α) (1 − G(ub + uc ))] (ub + uc ) + µc [α + 2(1 − α) (1 − F (uc ))] uc .
35
(23)
45o
1
q = qh
G(q)
ub
0
p2
0
2uc
ql
uc + ub q ∗
qh
ql − pl q ∗ /2
qh /2
q
q = q∗
1
q = ub + uc
Region where profit maximized
Example of support of H
q = 2uc
pl uc
F (p)
uc
pl
q = ql
ub
0
0
pl uc
p1
p
Notes: This figure shows the possible range of the support of the joint distribution H(p1 , p2 ), the shape of
the cumulative distributions G(q), and an example of the shape of the cumulative distribution F (p) in the
Discrimination Equilibrium.
Figure 5: Discrimination Equilibrium
Using the fact that G(ub + uc ) = G(q ∗ ) and solving (23) with respect to G(q), we find that
the distribution of basket prices for q ∈ [qℓ , ub + uc ] is given by
G(q) = G(q ∗ ) −
α + 2(1 − α) (1 − G(q ∗ )) ub + uc − q
.
2(1 − α)
q
(24)
Solving the equation G(qℓ ) = 0 with respect to qℓ , we find that the lowest price on the
support of the distribution of basket prices is given by
qℓ =
2αub ub + uc
.
2 − α q∗
(25)
Lemma 6 also implies that a seller posting prices (p1 , p2 ) such that p1 ∈ [pℓ, uc ], p2 ∈
(uc , ub ] and p1 + p2 = qℓ attains the same profit as a seller posting prices (uc , qℓ − uc ), i.e.
µb [α + 2(1 − α)] qℓ + µc [α + 2(1 − α) (1 − F (p))] p
= µb [α + 2(1 − α)] qℓ + µc [α + 2(1 − α) (1 − F (uc ))] uc .
(26)
Using the fact that F (uc ) = G(q ∗ )/2 and solving the equation (26) with respect to F (p), we
find that the distribution of good-1 prices for p ∈ [pℓ , uc ] is given by
G(q ∗ ) α + 2(1 − α) (1 − G(q ∗ )/2) uc − p
−
.
F (p) =
2
2(1 − α)
p
36
(27)
Solving the equation F (pℓ ) = 0 with respect to pℓ , we find that the lowest price on the
support of the distribution of good-1 prices is given by
pℓ =
α + 2(1 − α) [1 − G(q ∗ )/2]
uc .
2−α
This completes the characterization of a Discrimination Equilibrium. In this type of
equilibrium, there is a group of sellers who sets a basket price of q ∈ [q ∗ , qh ] and the prices
p1 and p2 in between uc and ub . These sellers trade (with some probability) the basket of
goods to buyers of type b and never trade with buyers of type c. There is also a group of
sellers who set a basket price of q ∈ [qℓ , ub + uc ]. Half of these sellers set p1 below uc and p2
between uc and ub . These sellers trade (with some probability) the whole basket of goods to
buyers of type b and good 1 to buyers of type c. The other half of the sellers sets p2 below
uc and p1 between uc and ub . These sellers trade (with some probability) the whole basket
of goods to buyers of type b and good 2 to buyers of type c. There are no sellers who set a
basket price of q in the interval (ub + uc , q ∗ ).
The distribution of basket prices G(q) is given by (17) for q ∈ [q ∗ , qh ] and by (24) for
q ∈ [qℓ , ub + uc ]. The distribution G(q) is such that the seller’s profit from trading the
basket of goods to buyers of type b is equal to S ∗ for all q ∈ [q ∗ , qh ], and it is equal to
S ∗ − µc [α + 2(1 − α)(1 − F (uc ))] uc for all q ∈ [qℓ , ub + uc ]. The distribution G(q) has a
gap between ub + uc and q ∗ . The gap exists because a seller with a basket price of ub + uc
trades with both buyers of type b and buyers of type c, while a seller with a basket price
greater than ub + uc only trades with buyers of type b. Therefore, a seller strictly prefers
setting a basket price of ub + uc rather than setting any basket price just above ub + uc .
The distribution of prices for an individual good F (p) is given by (27) for p ∈ [pℓ , uc ]. The
distribution F (p) is such that the seller’s profit from trading the cheaper good to buyers of
type c is equal to S ∗ − µb (2 − α)qℓ for all p ∈ [pℓ , uc ]. The distribution F (p) is not uniquely
pinned down for p ∈ (uc , ub ]. Intuitively, this is the case because a seller who charges a price
of p > uc for one good only trades that good to buyers of type b together with the other
good.
The distribution of price vectors H is not uniquely pinned down. For sellers with a
basket price q ∈ [q ∗ , qh ], any distribution H that has support inside R1 and that generates
the marginal distribution of basket prices G(q) in (17) is consistent with equilibrium. For
example, there is an equilibrium in which, for all q ∈ [q ∗ , qh ], there are G′ (q) sellers with
a basket price of q and each of them posts the prices (q/2, q/2). For sellers with a basket
price q ∈ [qℓ , ub + uc ], any distribution H that has support inside R2 and that generates the
37
marginal distribution of basket prices G(q) in (24) and the marginal distribution of individual
good prices F (p) in (27) is consistent with equilibrium. For example, there is an equilibrium
in which, for all p ∈ [pℓ , uc ], 2F ′ (p) sellers have a basket price of φ(p), F ′ (p) sellers post the
prices (p, φ(p) − p), and F ′ (p) sellers post the prices (φ(p) − p, p), where
φ(p) =
[α + 2(1 − α)(1 − G(q ∗ ))] (ub + uc )
.
[α + 2(1 − α)(1 − G(q ∗ ))] + 2 [α + 2(1 − α)(1 − G(q ∗ )/2)] (uc − p)/p
(28)
A Discrimination Equilibrium features price dispersion across sellers, in the sense that
some sellers are on average more expensive, while some sellers are on average cheaper. This
property of equilibrium follows immediately from the fact that the distribution of basket
prices is non-degenerate. A Discrimination Equilibrium always features relative price dispersion, in the sense that there is variation across sellers in the price of a particular good
at a particular seller relative to the average price charged by that seller. This property of
equilibrium follows immediately from the fact that half of the sellers with a basket price
q ∈ [qℓ , ub + uc ] have a relative price for good 1 that is strictly greater than 1, while the
other half of the sellers with a basket price q ∈ [qℓ , ub + uc ] have a relative price for good 1
that is strictly smaller than 1.
Let us briefly explain why relative price dispersion must emerge in equilibrium. Competition between sellers drives part of the distribution of basket prices to the region where
q is between 2uc and ub + uc . A seller with a basket price between 2uc and ub + uc never
finds it optimal to post the same price for both goods. Instead, the seller finds it optimal
to set the price of one good below and the price of the other good above the willingness to
pay of type-c buyers. That is, a seller with a basket price q between 2uc and ub + uc finds
it optimal to follow an asymmetric pricing strategy for the two goods. However, if some
sellers post a higher price for good 1 than for good 2, other sellers must post a higher price
for good 2 than for good 1, or else there would be some unexploited profit opportunities.
That is, the distribution of prices for the two goods must be symmetric across sellers with a
basket price q between 2uc and ub + uc . The asymmetric pricing strategy followed by each
individual seller combined with the symmetry of the price distribution across sellers implies
relative price dispersion.
Sellers follow an asymmetric pricing strategy to discriminate between the two types of
buyers. The difference in the willingness to pay of type-b and type-c buyers gives sellers
a desire to price discriminate. The difference in the ability of type-b buyers and type-c
buyers to purchase different items in different locations gives sellers the opportunity to price
discriminate. In fact, by pricing the two goods asymmetrically, a seller can charge a high
38
average price to the high-valuation buyers who need to purchase all the items together (the
buyers of type b) and charge a low price for one good to the low-valuation buyers who can
purchase different items at different locations (the buyers of type c).
It is interesting to contrast the type of price discrimination described above with intertemporal price discrimination (see, e.g., Conlisk, Gerstner and Sobel 1984 and Sobel 1984
or, in a search-theoretic context, Albrecht, Postel-Vinay and Vroman 2013 and Menzio and
Trachter (2015b). The key to intertemporal price discrimination is a negative correlation
between a buyer’s valuation and his ability to intertemporally substitute purchases. A seller
can exploit this negative correlation by having occasional sales. The low-valuation buyers,
who are better able to substitute purchases intertemporally, will take advantage of the sales
and will end up paying low prices. The high-valuation buyers, who are unable to substitute
purchases intertemporally, will not take advantage of the sales and will end up paying high
prices. In contrast, our theory of price discrimination is based on a negative correlation
between a buyer’s valuation and his ability to shop in multiple stores. Moreover, while intertemporal price discrimination takes the form of time variation in the price of the same
good, our theory of price discrimination takes the form of variation in the price of different
goods relative to the average store price.
We conclude the analysis by identifying necessary and sufficient conditions for the existence of a Discrimination Equilibrium. We find that this type of equilibrium exists if and
only if
and
3α − 2
µc
>
−1
µb
(2 − α)uc /ub
(29)
µc
α − (2 − α)uc /ub 1 + uc /ub
≤
.
µb
1 + (2 − α)uc/ub uc /ub
(30)
Condition (29) guarantees that some sellers find it optimal to post basket prices below ub +uc .
The condition is satisfied if: (i) the market is sufficiently competitive, in the sense that the
fraction α of buyers who are in contact with only one seller is smaller than 2/3; or (ii) the
relative number of type-c buyers, µc /µb , and/or the relative willingness to pay of type-c
buyers, uc /ub , is large enough. Condition (30) guarantees that no seller finds it optimal to
post prices below 2uc . The condition is satisfied if: (i) the market is not too competitive,
in the sense that the fraction α of buyers who are in contact with only one seller is greater
than 2(uc /ub)/(1 + 2(uc /ub )); (ii) the relative number of type-c buyers, µc /µb , and/or the
relative willingness to pay of type-c buyers, uc /ub , are low enough. As we shall see in the
next section, conditions (29) and (30) define a non-empty parameter space.
39
Figure 6: Equilibrium type depending on buyers in market
1
Relative valuation of cool uc /ub
Unbundled Equilibrium
Discrimination Equilibrium
Bundled Equilibrium
0
0
1
2
Relative measure of cool µc /µb
Notes: This figure illustrates how the type of equilibrium depends on the shares and relative valuations of
the two types of buyers in the market.
The following proposition summarizes the characterization of a Discrimination Equilibrium.
Proposition 2: (Discrimination Equilibrium) (i) In a Discrimination Equilibrium, the distribution of basket prices, G, is continuous over the support [qℓ , ub + uc ] ∪ [q ∗ , qh ], and it is
given by (17) for q ∈ [q ∗ , qh ] and by (24) for q ∈ [qℓ , ub + uc ]; the distribution of prices for
good i, Fi , is continuous over the interval [pℓ , uc ] and it is given by (27) for both i = 1 and
2; the distribution of price vectors, H, is not uniquely pinned down. (ii) A Discrimination
Equilibrium exists if and only if (29) and (30) are satisfied.
Proof : Appendix B.
3.5
Other Equilibria
A Bundled Equilibrium exists if and only if the relative measure of type-c buyers, µc /µb ,
is smaller than the right-hand side of (16). The lowest curve in Figure 6 is the plot of the
right-hand side of (16) in the {uc /ub, µc /µb } space. Hence, a Bundle Equilibrium exists if and
only if the parameter values lie below the bottom curve.16 In this type of equilibrium, every
seller sets a basket price q strictly greater than ub + uc and posts prices for the individual
16
It is easy to verify that, when condition (16) is satisfied, the only equilibrium is a Bundle Equilibrium.
40
goods that are strictly greater than the willingness to pay of type-c buyers and smaller than
the willingness to pay of type-b buyers. Every seller trades the entire basket of goods to
buyers of type b and trades nothing to buyers of type c. The equilibrium distribution of
basket prices is the same as in the single-product model of retailing by Burdett and Judd
(1983).
A Discrimination Equilibrium exists if and only if the relative measure of type-c buyers,
µc /µb , is greater than the right-hand side of (29) and smaller than the right-hand side of (30).
Notice that the right-hand side of (29) is equal to the right-hand side of (16), which is the
lowest curve in Figure 6. Also, notice that the right-hand side of (29) is strictly smaller than
the right-hand side of (30), which is the middle curve in Figure 6. Hence, a Discrimination
Equilibrium exists if and only if the parameter values fall in the non-empty region between
the bottom and the middle curves.17 In this type of equilibrium, there are some sellers with
a basket price q > ub + uc who post prices p1 and p2 ∈ (uc , ub ]. These sellers trade the basket
of goods to buyers of type b and trade nothing to buyers of type c. There are also some
sellers with a basket price q ∈ (2uc , uc + ub ] who set the price of one good above and the
price of the other good below uc . These sellers trade the basket of goods to buyers of type
b and the cheaper good to buyers of type c.
If the relative measure of type-c buyers is greater than the right-hand side of (30), other
more complex types of equilibria emerge. In this paper, we do not wish to analyze these
equilibria in detail, but we still find it instructive to describe some of their properties. Any
equilibrium is an Unbundled Equilibrium when and only when
µc
1 − uc /ub
>
.
µb
uc /ub
(31)
Notice that the right-hand side of (30) is strictly smaller than the right-hand side of (31),
which is the top curve in Figure 6. Hence, an equilibrium is unbundled if and only if the
parameter values fall in the region above the top curve. In this type of equilibrium, every
seller sets a basket price q smaller than 2uc and posts prices for the individual goods that
are smaller than uc . Every seller trades the basket of goods to buyers of type b and either
the basket of goods or one of the two individual goods to buyers of type c.
If the relative measure of type-c buyers lies between the top and the middle curves in
Figure 6, the equilibrium is a combination between a Discrimination and an Unbundled
Equilibrium. In this type of equilibrium, there is a first group of sellers with a basket price
17
When conditions (29) and (30) are satisfied, the only equilibrium is a Discrimination Equilibrium.
41
q > ub + uc, a second group of sellers with a basket price q ∈ (2uc, ub + uc], and a third group
of sellers with a basket price q ≤ 2uc . Sellers in the first group trade the basket of goods
to buyers of type b and nothing to buyers of type c. Sellers in the second group trade the
basket of goods to buyers of type b and the cheaper good to buyers of type c. Sellers in the
third group trade the basket of goods to buyers of type b and either both or either one of
the individual goods to buyers of type c.
4
Calibration and Validation
In this section, we calibrate and validate our theory of relative price dispersion. In Section
4.1, we calibrate the theory using data on the extent and sources of dispersion in the price
at which the same good is sold by different sellers in the same market and week, the number of stores visited by different households, the relationship between the prices paid by a
household for the same basket of goods in the same market and quarter, and the number
of stores from which the households shops in a given quarter. In Section 4.2, we compare
the predictions of the calibrated theory on the extent and sources of dispersion in the prices
paid by different households for the same basket of goods in the same market and quarter.
We find that our theory of relative price dispersion is consistent not only with the key facts
about price dispersion but also with the main differences between the extent and sources of
price dispersion across stores and price dispersion across households.
4.1
Calibration
For the purposes of our quantitative analysis, we need to consider a dynamic version of the
static model of Section 3. We assume that the type of the buyer remains constant over time.
However, a buyer may change his network of sellers from one period to the next because,
for example, he moves from one part of the city to another, he changes his job, and so on.
In particular, we assume that a buyer keeps the same network of sellers from one period to
the next with probability ρ, and he samples a new network of sellers with probability 1 − ρ.
Conditional on changing his network, the buyer contacts one randomly selected seller with
probability α and two randomly selected sellers with probability 1 − α. Sellers post prices
in every period. Since the probability that a buyer changes network is independent of his
current network, the pricing problem of the sellers in every period is the same as in the static
model of Section 3. Hence, in the presence of any positive menu cost, sellers find it strictly
optimal to keep their prices constant from one period to the next.
42
When quantifying the model, we have to deal with the fact that the theory does not
uniquely pin down the distribution H of price vectors across sellers. As explained in Section
3, the theory uniquely pins down the distribution G of basket prices and the distribution
F of prices of individual goods, but it does not uniquely pin down H. Yet, we find it
natural to assume that: (i) sellers with a basket price of q ∈ (ub + uc , 2ub] set the same
price for both goods; and (ii) sellers with a basket price of q ∈ (2uc , ub + uc ] set the prices
(φ−1 (q), q − φ−1 (q)) with probability 1/2 and the prices (q − φ−1 (q), φ−1 (q)) with probability
1/2. Under conditions (i) and (ii), the equilibrium is symmetric, in the sense that H(p1, p2 ) =
H(p2 , p1 ), and rank-preserving, in the sense that the rank of a seller in the distribution of
basket prices is the same as the rank of a seller in the distribution of the lowest prices of
an individual good.18 There are other equilibria that are symmetric and rank-preserving.
However, conditions (i) and (ii) select the one with the lowest relative price dispersion.19
The dynamic version of the model has six parameters: the measure µb of buyers of type
b, the measure µc of buyers of type c, the valuation ub of buyers of type b, the valuation uc
of buyers of type c, the fraction α of buyers that have only one seller in their network, and
the probability ρ that buyers keep the same network of sellers from one week to the next.
The equilibrium of the model depends on the ratio between the measure of type-c buyers
and the measure of type-b buyers, but not on the two measures separately. Hence, we can
normalize µb to 1. Similarly, the equilibrium of the model depends on the ratio between the
valuation of type-c buyers and the valuation of type-b buyers, but not on the two valuations
separately. Hence, we can normalize ub to 1. We calibrate the remaining four parameters so
as to match four moments of the data. First, we target a measure of the dispersion of prices
for the same good in the same market and in the same week. Second, we target a measure of
the fraction of price dispersion that is due to differences in the store component of the price
and the fraction of price dispersion that is due to differences in the store-good component
of the price. Third, we target a measure of the effect of shopping from an additional store
on the prices paid by a household. Finally, we target a measure of the number of different
18
It is immediate to see that the equilibrium is symmetric under (i) and (ii). To see that the equilibrium
is rank-preserving, notice that, for sellers with q ∈ [qℓ , ub + uc ], the function φ−1 (q) relating the price of the
seller’s basket to the price of the seller’s cheapest good is such that F (φ−1 (q)) = G(q)/2 and, hence, the
rank of the seller in the distribution of basket prices is G(q) and the rank of the seller in the distribution of
the lowest price for an individual good is 2F (φ−1 (q)) = G(q). For sellers with q ∈ [q ∗ , 2ub ], the rank of the
seller in the distribution of basket prices is G(q) and the rank of the seller in the distribution of the lowest
price for an individual good is 2F (uc ) + G(q) − G(q ∗ ) = G(q).
19
The other symmetric, rank-preserving equilibria are such that half of the sellers with a basket price of
q ∈ [q ∗ , 2ub ] post prices (ψ(q), q − ψ(q)) and the other half post prices (q − ψ(q), ψ(q)), with ψ(q) ∈ (uc , q/2].
Clearly, conditions (i) and (ii) select the symmetric, rank-preserving equilibrium with the lowest amount of
relative price dispersion.
43
stores visited by a given household.
As discussed in Section 3, we interpret the model as a theory of the persistent component
of prices at different stores. For this reason, we want the model to match the variance of
prices that is caused by persistent price differences across stores. As documented in Section
2, the standard deviation of prices for the same good in the same market and in the same
week is 15.3%, and the variance is 2.34%. The fraction of the variance of prices that is due
to differences in the store component of prices is 15.5%. Nearly all of the differences in the
store component of prices are persistent, in the sense that they are either due to differences
in the fixed effect or to differences in the AR part of the store component. The fraction
of the variance of prices that is due to differences in the store-good component of prices
is 84.5%. Approximately 36% of the differences in the store-good component of prices are
persistent, and 64% are transitory. Based on these observations, the variance of prices that
comes from persistent differences in prices is 1% (and the standard deviation is 10%), the
fraction of this variance due to persistent differences in the store component is 34% percent,
and the variance due to persistent differences in the store-good component is 66% percent.
In order to obtain a measure of the effect of shopping at more stores on the prices paid by
households, we use the Kilts Nielsen Consumer Panel (KNCP), which tracks the shopping
behavior of approximately 50,000 households over the period from 2004 to 2009. Households
are drawn from 54 geographically dispersed markets, known as Scantrack markets, each of
which roughly corresponds to a metropolitan statistical area. Demographic data on panelists
are collected at the time of entry into the panel and updated annually through a written
survey. Panelists provide information about each of their shopping trips using a Universal
Product Code (UPC) scanning device. More specifically, when a panelist returns from a
shopping trip, he uses the device to enter details about the trip, including the date and the
store where the purchases were made. The panelist then scans the barcode of the purchased
good and enters the number of units purchased. The price of the good is recorded either
automatically or manually depending on whether the store where the good was purchased
is covered by Nielsen or not.
We follow the methodology in Aguiar and Hurst (2007) to compute a price index for every
household in the KNCP. Specifically, we define the price index of household i in market m
and quarter t as the ratio between the dollar amount that the household paid to purchase
its basket of goods and the amount that the household would have spent had it paid, for
each good in its basket, the average price of that good in market m and quarter t. We refer
the reader to Kaplan and Menzio (2015) for further details about the construction of price
44
Table 5: Regression of household price indexes on indicators of multistop shopping
(a)
Stores/Expenditure
Log
Log
Level
Level
-0.01124** -0.01291** -0.09099** -0.10365**
(0.00041)
(0.00041)
(0.00363)
(0.00349)
FE
R2
No
0.01776
Yes
0.0074
No
0.02041
Yes
0.01523
(b)
Log
Log
Level
Level
Stores
Expenditure
FE
R2
-0.03424** -0.01577** -0.01196** -0.00367**
(0.00072)
(0.00051)
(0.00027)
(0.00018)
0.00964** 0.01261** 0.00008** 0.00007**
(0.00041)
(0.00044)
(0.00000)
(0.00000)
No
0.02875
Yes
0.00753
No
0.02707
Yes
0.00235
Notes: This table presents results for regressions of household price indexes on indicators of multistop
shopping: the average number of different stores visited per dollar spent in the quarter (Panel A) and
number of different stores visited per quarter, conditioning on dollars spent per quarter (Panel B). The level
models have expenditures in levels and log models have expenditures in logs. In all regressions: N=880104,
clusters=78758.
indexes and for the sample selection criteria.
In Panel (a) of Table 5, we report the results of a regression of the price index of household
i in quarter t on the log (or the level) of the ratio between the number of different stores
in which household i shopped during quarter t and the dollar expenditures of household i
in quarter t. In all specifications, we control for household size, the age and education of
household members, and we add market dummies. In some specifications, we control for
household fixed effects, while in other specifications we do not. Column 1 reports the results
of the regression on the log of stores-per-dollar without household fixed effects. It shows
that a household that visits twice as many stores per dollar spent has a price index that is
1.12% lower. Column 2 reports the results of the regression on the log of stores-per-dollar
with household fixed effects. It shows that, in quarters when a household visits twice as
many stores per dollar spent, it enjoys a 1.29% lower price index. Columns 3 and 4 report
the results of the regressions on the level of stores-per-dollar when household fixed effects
are respectively excluded and included. For all specifications, the regression coefficient on
stores per dollar is negative and significant. In Panel (b), we carry out the same regression
using the log (or the level) of the number of different stores in which the households shopped
during quarter t. We find that the regression coefficient on the number of stores is always
45
negative and significant. Since our model has only limited heterogeneity across buyers, we
choose as a calibration target the regression coefficient on the log of stores-per-dollar.
In Figure 7, we display the distribution of the number of stores visited in a quarter by
households in KNCP. Approximately 32% of households do all of their shopping in a quarter
at one store. Approximately 28% of households do all of their shopping in a quarter at
two stores. Only 40% of households shop from more than two stores in a quarter. As a
calibration target for the model, we choose the average number of stores, 2.48, from which
a household shops in a quarter.
Figure 7: Number of different stores visited by households
Notes: This figure displays a histogram of the number of different stores visited by households within a
quarter.
We calibrate the parameters of the model by minimizing the sum of the absolute value of
the percentage deviation between each of the targeted moments and its counterpart in the
model. Table 6 reports the targeted moments, the value of these moments in the calibrated
model, and the calibrated value of the parameters.
46
Table 6: Calibration
Targets
Data
Model
Standard deviation of prices
Share of variance of prices due to store component
Average number of stores visited
Regression coefficient of price index on store number
10%
34%
2.48
-1.3%
10%
37%
2.41
-1.9%
7.8%
55%
6.6%
71%
Additional moments
Standard deviation of price index
Share of variance of price index due to store component
Parameter
Value
4.2
µc /µb
uc /ub
0.68
0.041
α
ρ
0.89 0.87
Validation
Our theory of shopping and pricing in the retail market—albeit rather stylized—matches
the extent and sources of dispersion in the price of the same good across different sellers.
Indeed, the calibrated model matches the standard deviation of prices posted by different
sellers for the same good in the same market and in the same week. Also, the calibrated
model matches quite well the fraction of the variance of prices posted by different sellers
that is due to differences in the store component of the price (37% in the model, 34% in
the data) and the fraction that is due to differences in the store-good component of the
price (63% in the model, 66% in the data). The theory’s explanation for this decomposition
is simple. There is dispersion in the store component of prices for the same reasons as in
Burdett and Judd (1983). Specifically, as some buyers have only one seller in their network
and other buyers have multiple sellers in their network, sellers must post different basket
prices in equilibrium. There is dispersion in the store-good component of prices (i.e., relative
price dispersion) because of price discrimination. Specifically, as high-valuation buyers need
to purchase all the goods from the same retailer and low-valuation buyers can purchase
different goods from different retailers, an individual seller wants to set asymmetric prices
for the two goods. And, in equilibrium, for every seller posting a higher price for the first
good than for the second, there must be another seller doing the opposite.
Our theory matches the negative relationship between the price index of a household and
the number of stores from which the household shops during a quarter (normalized by the
47
household’s expenditures). This is intuitive. According to the model, households who shop
from more sellers are more likely to be buyers who are willing to go through the stores in
their network to purchase different goods at the lowest price. Households who shop from
fewer stores are more likely to be buyers who need to purchase everything from one of the
stores in their network. Because of relative price dispersion, i.e., because equally expensive
stores posts different prices, a buyer who is willing to purchase different goods at different
stores in his network can pay less for the same basket of goods than a buyer who needs to
purchase everything from just one of the stores in his network. Thus, households who shop
from more stores tend to have a lower price index.
The theory makes other predictions about the dispersion in the prices paid by different
households for the same basket of goods. Before reviewing these predictions, we look at the
extent and sources of households’ price index dispersion in the KNCP data. We normalize
the price pijmt at which good j is traded in transaction i taking place in market m during
quarter t by the average price of that good in that market and in that quarter. We then
decompose the price pijmt into a store component, a store-good component, and a transaction
component. The store component is defined as the average of the (normalized) price of the
goods sold by the store si where transaction i took place. The store-good component is
defined as the difference between the average of the (normalized) price at which good j is
sold at store si during quarter t and the store component. The transaction component is
defined as the difference between the (normalized) price at which good j is sold in transaction
i and the average of the (normalized) price at which good j is sold at store si during quarter t
and the store component. We refer the reader to Kaplan and Menzio (2015) for details about
this decomposition. For our purposes, it is important to notice that, roughly speaking, the
store component and the store-good component of pijmt are “persistent” parts of the price,
as they are measured as quarterly averages. The transaction component is a “transitory”
part of the price, as it is measured as a deviation from a quarterly average.
We can decompose the households’ price indexes using the decomposition of the prices
pijmt . Specifically, we express the price index of a household as the sum of a store component, a store-good component, and a transaction component. The store component of the
price index is the (expenditure-weighted) average of the store component of all household
transactions. The store-good component of the price index is the (expenditure-weighted)
average of the store-good component of all household transactions. The transaction component of the price index is the (expenditure-weighted) average of the transaction component
of all household transactions. Then, we decompose the variance of the households’ price
indexes in market m and quarter t into the variance of the store component, the variance of
48
the store-good component, the variance of the transaction component, and two covariance
terms.
The average standard deviation of price indexes is 9%, and the average variance is 0.81%.
The fraction of the variance due to differences in the store component is 42%, the fraction
due to differences in the store-good component is 58%, and the fraction due to differences in
the transitory store-good component is 19%. The covariance between the store component
and the store-good component is −24%, while the covariance between the store component
and the transaction component is 5%. Since ours is a theory of the persistent component
of prices, we need to purge the price indexes from the transaction component. After doing
so, we find that the average standard deviation of price indexes is 7.8%, and the average
variance is 0.61%. The fraction of the variance due to differences in the store component is
55%, and the fraction due to differences in the persistent store-good component (plus the
covariance term) is 45%.
Two features of price index dispersion are remarkable. First, the standard deviation of
price indexes (7.8%) is smaller than the standard deviation of prices (10%). Second, the
fraction of the variance of price indexes due to differences in the store component (55%) is
higher than the fraction of the variance of prices due to differences in the store component
(35%). Conversely, the fraction of the variance of price indexes due to differences in the
store-good component (together with the covariance term) is smaller than the variance of
prices due to differences in the persistent part of the store-good component (45% vs 65%).
Our theory accounts surprisingly well for both features of price index dispersion, even
though it was not designed or calibrated to do so. First, in the calibrated model, the standard
deviation of price indexes is 6.6%, while the standard deviation of prices is 10%. Thus, the
model can account, both qualitatively and quantitatively, for the fact that the dispersion of
prices paid by households for the same basket of goods is lower than the dispersion of prices
posted by sellers for the same good. Second, in the calibrated model, the fraction of the
variance of price indexes due to differences in the store component is 71%, while the fraction
of the variance of prices due to differences in the store component is 37%. Conversely, the
fraction of the variance of price indexes due to the store-good component (together with the
covariance) is 29%, while the fraction of the variance of prices due to the persistent part of
the store-good component is 63%. Thus, the model can account qualitatively for the fact
that differences in the store component are a larger source of dispersion of prices paid by
households for the same basket of goods than they are a source of dispersion of prices posted
by sellers for the same good.
49
The explanation for these phenomena provided by our theory is simple. Some sellers
post a relatively high price for the first good and a relatively low price for the second good.
Other equally expensive sellers do the opposite, as they post a relatively low price for the
first good and a relatively high price for the second good. The variation across stores in
the store-good component of prices contributes to a large fraction of the overall variance of
posted prices. Now, let us turn to the buyers. Recall that buyers of type b have to purchase
both goods from the same seller. In the price index of this type of buyer, the variation in
the store-good component of prices washes out, as one of the prices they pay has a positive
store-good component and the other has a negative store-good component. For this reason,
the dispersion in price indexes across buyers is smaller than the dispersion in prices across
sellers. Also for this reason, the variation in the store component of prices is more important
for the dispersion of price indexes across buyers than for the dispersion of prices across sellers.
To summarize, our simple theory of pricing and shopping in the retail market provides
an explanation for the extent and sources of dispersion in the prices posted by different
sellers for the same good, for the buyers’ return from shopping at an additional store, and
for the extent and sources of dispersion in the prices paid by different households for the
same basket of goods. Thus, our theory responds to the call in Kaplan and Menzio (2015)
for a model of the retail market that is consistent not only with the key facts about price
dispersion but also with the key differences between the extent and sources of price and price
index dispersion.
5
Conclusions
In this paper, we used the KNRS to measure the extent and sources of price dispersion, i.e.
the dispersion in the price at which the same good is sold by different stores in the same
market and in the same week. We found that a significant fraction of price dispersion is due to
the fact that stores that are, on average, equally expensive choose to set persistently different
prices for the same good. We labelled this phenomenon relative price dispersion. We then
developed a theory of relative price dispersion in the context of the canonical model of price
dispersion of Burdett and Judd (1983). According to our theory, relative price dispersion is
an equilibrium manifestation of the sellers’ attempt to discriminate between different buyers.
In particular, an individual seller finds it optimal to charge asymmetric prices for different
goods to discriminate between high-valuation buyers who need to purchase everything in
the same location, and low-valuation buyers who are willing to purchase different goods at
50
different locations. In equilibrium, for every seller that charges a relatively high price for
one good, there must be another seller that is equally expensive on average but charges a
relatively low price for the same good. We showed that our theory of relative price dispersion
can not only account for the extent and sources of price dispersion, but also for the extent
and sources of variation in the prices paid by different households for the same basket of
goods. Thus, our theory offers a response to the call in Kaplan and Menzio (2015) for a model
of the retail market that is consistent not only with the key facts about price dispersion but
also with the key facts about the dispersion of price indexes across households.
Several extensions of our theory seem worthwhile. On the descriptive side, it would
be interesting to combine our model of the retail market with a model of temporary price
reductions (as Sobel 1984 or Aguirregabiria 1999). The resulting model would offer a truly
comprehensive theory of price dispersion, in which price dispersion occurs because some
stores are cheap and some are expensive, because equally expensive stores have different
average prices for the same good, and because the same store has a different price for the
same good on different days. As in this paper, the resulting theory could be tested using
data on the dispersion in prices paid by different households. On the normative side, it
would be interesting to use the calibrated model to measure the extent of inefficiency in
the retail market, identify which policies might be welfare improving, and which ones might
exacerbate inefficiencies. Finally, it might be worthwhile extending the model presented in
this paper to include a richer pattern of heterogeneity among buyers (i.e., more than two
types of buyers), a richer set of goods (i.e., more than two goods), to endogenize the buyers’
network of sellers, the buyers’ decision to shop from multiple sellers or not, and the sellers’
entry decision.
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54
Appendix
A
Empirical Appendix
Table 7: Share of UPCs across Product Groups: Baseline
Product Group
YOGURT
CARBONATED BEVERAGES
FRESH PRODUCE
BREAD AND BAKED GOODS
PIZZA/SNACKS/HORS DOEURVES-FRZN
MILK
VEGETABLES - CANNED
SOFT DRINKS-NON-CARBONATED
SOUP
CANDY
CEREAL
FRESH MEAT
SNACKS
CHEESE
PAPER PRODUCTS
BREAKFAST FOOD
CRACKERS
DRESSINGS/SALADS/PREP FOODS-DELI
PREPARED FOOD-DRY MIXES
PASTA
EGGS
JUICE, DRINKS - CANNED, BOTTLED
COOKIES
COT CHEESE, SOUR CREAM, TOPPINGS
BUTTER AND MARGARINE
PREPARED FOODS-FROZEN
CONDIMENTS, GRAVIES, AND SAUCES
PREPARED FOOD-READY-TO-SERVE
VEGETABLES-FROZEN
PACKAGED MEATS-DELI
GUM
SEAFOOD - CANNED
TOBACCO AND ACCESSORIES
Percent
10.7
9.3
6
5.4
4.4
3.6
3.4
3.3
3.3
3.2
3
3
3
2.9
2.8
2.3
2.1
1.8
1.8
1.7
1.6
1.6
1.3
1.3
1.2
1.2
1.1
1.1
1.1
1
0.8
0.7
0.6
55
DOUGH PRODUCTS
FRUIT - DRIED
JUICES, DRINKS-FROZEN
SALAD DRESSINGS, MAYO, TOPPINGS
SUGAR, SWEETENERS
BOOKS AND MAGAZINES
DESSERTS, GELATINS, SYRUP
ICE CREAM, NOVELTIES
ORAL HYGIENE
BAKING SUPPLIES
BREAKFAST FOODS-FROZEN
FRESHENERS AND DEODORIZERS
ICE
JAMS, JELLIES, SPREADS
BABY FOOD
BAKING MIXES
DESSERTS/FRUITS/TOPPINGS-FROZEN
FRUIT - CANNED
HOUSEHOLD CLEANERS
KITCHEN GADGETS
PACKAGED MILK AND MODIFIERS
WRAPPING MATERIALS AND BAGS
AUTOMOTIVE
COFFEE
DISPOSABLE DIAPERS
FLOUR
HARDWARE, TOOLS
HOUSEHOLD SUPPLIES
LIGHT BULBS, ELECTRIC GOODS
PICKLES, OLIVES, AND RELISH
SHORTENING, OIL
SNACKS, SPREADS, DIPS-DAIRY
SPICES, SEASONING, EXTRACTS
UNPREP MEAT/POULTRY/SEAFOOD-FRZN
0.5
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
Table 8: Share of UPCs across Departments
56
ALCOHOLIC BEVERAGES
DAIRY
DELI
DRY GROCERY
FRESH PRODUCE
FROZEN FOODS
GENERAL MERCHANDISE
HEALTH AND BEAUTY
MEAT
NON-FOOD
TOTAL
Baseline
UPC Alt
UPC National
Coca-Cola
Unilever
Low durab.
High durab.
Low price
High price
0%
22%
2%
52%
6%
9%
1%
0%
4%
4%
1000
0%
7%
1%
49%
1%
12%
4%
12%
3%
11%
1000
6%
12%
2%
53%
5%
4%
2%
2%
3%
11%
1463
0%
13%
0%
87%
0%
0%
0%
0%
0%
0%
3608
0%
2%
0%
12%
0%
8%
1%
62%
0%
14%
10917
0%
35%
0%
4%
9%
32%
0%
0%
20%
0%
12301
13%
0%
0%
0%
0%
0%
82%
4%
0%
0%
32989
0%
25%
1%
58%
4%
7%
1%
0%
1%
3%
430
0%
11%
3%
49%
6%
11%
1%
1%
10%
8%
315
Table 9: Parameter estimates
ρy
θy,1
0.982613573
0.983490975
0.991166659
0.987028926
0.990499883
0.983067304
0.983882217
0.98573063
0.989435059
0.868614266
0.980758409
0.980959918
0.962261334
0.985078092
0.984127103
0.995571178
0
0.038776
0.217979
0.086678
0
0
0.006013
0.154354
0.039211
0
0.131075
0.06653
0
0.408077
0.07472
0.0427
Models
Baseline
State
County
N1 = 50
N1 = 500
N2 = 25
N2 = 100
Quant weighted
UPC alt
UPC national
Low price
High price
Low durability
High durability
Unilever
Coca-cola
Models
Baseline
State
County
N1 = 50
N1 = 500
N2 = 25
N2 = 100
Identity weights
MA(5)
MA(10)
Skewed MA(1)
Quant weighted
UPC alt
UPC national
Low price
High price
Low durability
High durability
Unilever
Coca-cola
Sales
ρz
Store component
V ar(αy )
V ar(η y )
0.002793056
0.002663621
0.001667647
0.005004708
0.002280235
0.002687951
0.003159055
0.00103578
0.002554465
0.005151485
0.00555965
0.003284731
0.003505668
0.000928235
0.002913701
0.000230254
Store-good component
θz,1
V ar(αz )
2.46E-05
2.25E-05
3.04E-05
1.47E-05
1.32E-05
2.39E-05
2.40E-05
8.43E-06
8.57E-06
2.49E-05
2.16E-05
8.60E-06
2.45E-05
2.73E-05
1.28E-05
4.10E-05
V ar(εy )
0.000116634
0.000130635
0.000214859
0.00028866
0.000110802
0.000112249
0.000131709
0.000346942
4.07E-05
0.000119115
0.000576657
0.000645019
0.000160791
0.000614076
0.001244086
0.000892025
V ar(η z )
V ar(εz )
0.965159943
0.965000278
0.964487279
0.969583498
0.964557595
0.965167539
0.964220785
0.96724252
0.970311168
0.980000019
0.965159943
0.966668046
0.967386315
0.967495404
0.968290143
0.97228473
0.966802325
0.954017962
0.973958285
0.952179616
0.025611
0.042116
0.034764
0.03929
0.016314
0.026807
0.032336
0.030042
1
0.115018
0.025611
0.046261
0.241865
0.073928
0.058805
0
0.056946
0.071649
0.151937
0.004329
0.003273669
0.003592662
0.00267737
0.004748385
0.003151334
0.003236118
0.003191119
0.003211072
0.003151381
0.003150801
0.003273669
0.004811795
0.003009445
0.004770164
0.005416631
0.002602982
0.002670222
0.001431966
0.002962665
0.002045257
0.000262241
0.00027266
0.000245359
0.000264688
0.000252595
0.000258127
0.000266847
0.000246603
0.000213223
0.000204187
0.000262241
0.000285567
0.000216338
0.000269885
0.000350237
0.000133687
0.000216992
0.000294292
0.000187877
0.000280396
0.012660499
0.013102838
0.012897066
0.015791481
0.012829199
0.012361954
0.012973907
0.012723392
0.006327444
0.006453931
0.012660499
0.015285712
0.006292217
0.013976282
0.014827101
0.01691263
0.010632362
0.005906032
0.01006858
0.01120506
ρz
V ar(αz )
V ar(η z )
sale prob
discount
0.966006219
0.003260204
0.000251805
0.046535283
-0.520418332
Notes: The baseline model is estimated on a baseline sample of UPCs using data for the Minneapolis-St
Paul designated market area. The following rows present results for alternative specifications discussed in
the text: defining the market as the state or county, alternative cutoffs for constructing the samples, quantity
weighting in constructing the store components, the alternative selections of goods “UPC alt” and “UPC
nationwide,” the low- and high-price samples, the low- and high-durability samples, and the Unilever and
Coca-cola samples. The top table considers the store component and the bottom the store-good component.
For the latter we further investigate an alternative GMM
57 weighting matrix (identity weighting), as well as
alternative specifications for the transitory variation: MA(5), MA(10), MA(1) with skewness in disturbances,
and an explicit model of sales.
Table 10: Share of UPCs across Product Groups: Alternative
Product Group
CANDY
PREPARED FOODS-FROZEN
SNACKS
JUICE, DRINKS - CANNED, BOTTLED
PAPER PRODUCTS
PACKAGED MEATS-DELI
BREAD AND BAKED GOODS
CONDIMENTS, GRAVIES, AND SAUCES
SOUP
ICE CREAM, NOVELTIES
PET FOOD
DEODORANT
CHEESE
CARBONATED BEVERAGES
PIZZA/SNACKS/HORS DOEURVES-FRZN
BABY FOOD
COOKIES
VEGETABLES-FROZEN
MEDICATIONS/REMEDIES/HEALTH AI
BREAKFAST FOOD
CEREAL
DESSERTS, GELATINS, SYRUP
DETERGENTS
PREPARED FOOD-DRY MIXES
SALAD DRESSINGS, MAYO, TOPPINGS
VEGETABLES - CANNED
CRACKERS
FRUIT - DRIED
SANITARY PROTECTION
DRESSINGS/SALADS/PREP FOODS-DELI
YOGURT
MILK
ORAL HYGIENE
PREPARED FOOD-READY-TO-SERVE
COFFEE
FRESH PRODUCE
LAUNDRY SUPPLIES
BAKING SUPPLIES
DISPOSABLE DIAPERS
BAKING MIXES
FIRST AID
SHAVING NEEDS
SKIN CARE PREPARATIONS
STATIONERY, SCHOOL SUPPLIES
GUM
NUTS
PERSONAL SOAP AND BATH ADDITIV
Percent
5.31
3.5
3.2
2.8
2.8
2.6
2.3
2.3
2.3
2.2
2.2
2.1
2
1.9
1.9
1.8
1.8
1.8
1.7
1.6
1.6
1.6
1.6
1.6
1.6
1.6
1.5
1.5
1.5
1.4
1.4
1.3
1.3
1.3
1.2
1.2
1.1
1
1
0.9
0.9
0.9
0.9
0.9
0.8
0.8
0.8
SOFT DRINKS-NON-CARBONATED
BAKED GOODS-FROZEN
COUGH AND COLD REMEDIES
FRESHENERS AND DEODORIZERS
FRUIT - CANNED
HARDWARE, TOOLS
HOUSEHOLD CLEANERS
HOUSEHOLD SUPPLIES
LIGHT BULBS, ELECTRIC GOODS
TEA
FRESH MEAT
HAIR CARE
TOBACCO AND ACCESSORIES
WRAPPING MATERIALS AND BAGS
COT CHEESE, SOUR CREAM, TOPPINGS
DOUGH PRODUCTS
GLASSWARE, TABLEWARE
JAMS, JELLIES, SPREADS
VITAMINS
DESSERTS/FRUITS/TOPPINGS-FROZEN
JUICES, DRINKS-FROZEN
SNACKS, SPREADS, DIPS-DAIRY
SPICES, SEASONING, EXTRACTS
UNPREP MEAT/POULTRY/SEAFOOD-FRZN
BREAKFAST FOODS-FROZEN
BUTTER AND MARGARINE
GROOMING AIDS
PASTA
PICKLES, OLIVES, AND RELISH
PUDDING, DESSERTS-DAIRY
SEAFOOD - CANNED
VEGETABLES AND GRAINS - DRIED
BATTERIES AND FLASHLIGHTS
BOOKS AND MAGAZINES
COOKWARE
PACKAGED MILK AND MODIFIERS
PET CARE
AUTOMOTIVE
ELECTRONICS, RECORDS, TAPES
FEMININE HYGIENE
FLOUR
GRT CARDS/PARTY NEEDS/NOVELTIE
KITCHEN GADGETS
SEASONAL
SHOE CARE
SHORTENING, OIL
SUGAR, SWEETENERS
TABLE SYRUPS, MOLASSES
WINE
58
0.8
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.5
0.4
0.4
0.4
0.4
0.4
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
Table 11: Share of UPCs across Product Groups: National
Product Group
CARBONATED BEVERAGES
FRESH PRODUCE
JUICE, DRINKS - CANNED, BOTTLED
CANDY
PAPER PRODUCTS
BEER
SNACKS
SOFT DRINKS-NON-CARBONATED
BREAD AND BAKED GOODS
YOGURT
CEREAL
TOBACCO AND ACCESSORIES
PACKAGED MEATS-DELI
PET FOOD
MILK
DRESSINGS/SALADS/PREP FOODS-DELI
SOUP
CHEESE
VEGETABLES - CANNED
CONDIMENTS, GRAVIES, AND SAUCES
DETERGENTS
PREPARED FOODS-FROZEN
BABY FOOD
BUTTER AND MARGARINE
CRACKERS
GUM
LIQUOR
PREPARED FOOD-READY-TO-SERVE
EGGS
MEDICATIONS/REMEDIES/HEALTH AI
PACKAGED MILK AND MODIFIERS
PIZZA/SNACKS/HORS DOEURVES-FRZN
PREPARED FOOD-DRY MIXES
COT CHEESE, SOUR CREAM, TOPPINGS
SEAFOOD - CANNED
BREAKFAST FOOD
COFFEE
COOKIES
BOOKS AND MAGAZINES
Percent
7.59
5.06
4.65
4.58
4.38
4.24
3.76
3.69
3.63
3.63
3.49
3.49
2.6
2.6
2.53
2.26
2.26
2.05
1.92
1.71
1.57
1.37
1.3
1.23
1.09
1.09
1.09
1.09
1.03
0.96
0.96
0.96
0.89
0.82
0.82
0.62
0.62
0.62
0.55
59
DISPOSABLE DIAPERS
TEA
VEGETABLES-FROZEN
WINE
ICE CREAM, NOVELTIES
PASTA
SALAD DRESSINGS, MAYO, TOPPINGS
SUGAR, SWEETENERS
COUGH AND COLD REMEDIES
FRUIT - CANNED
UNPREP MEAT/POULTRY/SEAFOOD-FRZN
BATTERIES AND FLASHLIGHTS
FRESH MEAT
JAMS, JELLIES, SPREADS
SHAVING NEEDS
WRAPPING MATERIALS AND BAGS
BAKING SUPPLIES
DESSERTS, GELATINS, SYRUP
HOUSEWARES, APPLIANCES
ICE
LAUNDRY SUPPLIES
SHORTENING, OIL
BAKING MIXES
BREAKFAST FOODS-FROZEN
DOUGH PRODUCTS
ELECTRONICS, RECORDS, TAPES
HOUSEHOLD SUPPLIES
PICKLES, OLIVES, AND RELISH
CHARCOAL, LOGS, ACCESSORIES
DESSERTS/FRUITS/TOPPINGS-FROZEN
NUTS
ORAL HYGIENE
SPICES, SEASONING, EXTRACTS
DIET AIDS
FLOUR
FRESHENERS AND DEODORIZERS
FRUIT - DRIED
HOUSEHOLD CLEANERS
PHOTOGRAPHIC SUPPLIES
SNACKS, SPREADS, DIPS-DAIRY
STATIONERY, SCHOOL SUPPLIES
VITAMINS
0.55
0.55
0.55
0.55
0.48
0.48
0.48
0.48
0.41
0.41
0.41
0.34
0.34
0.34
0.34
0.34
0.27
0.27
0.27
0.27
0.27
0.27
0.21
0.21
0.21
0.21
0.21
0.21
0.14
0.14
0.14
0.14
0.14
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
0.07
Table 12: Share of UPCs across Product Groups: Low Price
Product Group
Percent
YOGURT
CARBONATED BEVERAGES
VEGETABLES - CANNED
SOUP
SOFT DRINKS-NON-CARBONATED
CANDY
BREAD AND BAKED GOODS
FRESH PRODUCE
PASTA
PREPARED FOOD-DRY MIXES
PIZZA/SNACKS/HORS DOEURVES-FRZN
VEGETABLES-FROZEN
GUM
PAPER PRODUCTS
PREPARED FOOD-READY-TO-SERVE
EGGS
SEAFOOD - CANNED
JUICE, DRINKS - CANNED, BOTTLED
CHEESE
CONDIMENTS, GRAVIES, AND SAUCES
JUICES, DRINKS-FROZEN
PREPARED FOODS-FROZEN
DRESSINGS/SALADS/PREP FOODS-DELI
FRESHENERS AND DEODORIZERS
ICE
PACKAGED MEATS-DELI
BAKING MIXES
COT CHEESE, SOUR CREAM, TOPPINGS
DESSERTS/FRUITS/TOPPINGS-FROZEN
FRUIT - CANNED
KITCHEN GADGETS
ORAL HYGIENE
SUGAR, SWEETENERS
BAKING SUPPLIES
BUTTER AND MARGARINE
COOKIES
CRACKERS
DESSERTS, GELATINS, SYRUP
HARDWARE, TOOLS
LIGHT BULBS, ELECTRIC GOODS
MILK
PACKAGED MILK AND MODIFIERS
PICKLES, OLIVES, AND RELISH
SNACKS
SPICES, SEASONING, EXTRACTS
60
21.63
10.23
7.67
6.51
6.05
5.58
4.42
4.19
3.72
3.02
2.33
2.09
1.86
1.86
1.86
1.63
1.63
1.16
0.93
0.93
0.93
0.93
0.7
0.7
0.7
0.7
0.47
0.47
0.47
0.47
0.47
0.47
0.47
0.23
0.23
0.23
0.23
0.23
0.23
0.23
0.23
0.23
0.23
0.23
0.23
Table 13: Share of UPCs across Product Groups: High Price
Product Group
Percent
CARBONATED BEVERAGES
PIZZA/SNACKS/HORS DOEURVES-FRZN
FRESH MEAT
CEREAL
SNACKS
FRESH PRODUCE
PAPER PRODUCTS
BREAD AND BAKED GOODS
MILK
BREAKFAST FOOD
CRACKERS
COOKIES
DRESSINGS/SALADS/PREP FOODS-DELI
JUICE, DRINKS - CANNED, BOTTLED
SOFT DRINKS-NON-CARBONATED
TOBACCO AND ACCESSORIES
BUTTER AND MARGARINE
CANDY
COT CHEESE, SOUR CREAM, TOPPINGS
PACKAGED MEATS-DELI
CHEESE
EGGS
BOOKS AND MAGAZINES
ICE CREAM, NOVELTIES
SALAD DRESSINGS, MAYO, TOPPINGS
BABY FOOD
CONDIMENTS, GRAVIES, AND SAUCES
ORAL HYGIENE
PREPARED FOODS-FROZEN
YOGURT
BREAKFAST FOODS-FROZEN
COFFEE
DISPOSABLE DIAPERS
HOUSEHOLD CLEANERS
HOUSEHOLD SUPPLIES
SNACKS, SPREADS, DIPS-DAIRY
SUGAR, SWEETENERS
UNPREP MEAT/POULTRY/SEAFOOD-FRZN
VEGETABLES-FROZEN
WRAPPING MATERIALS AND BAGS
13.65
8.89
8.57
6.35
6.35
6.03
5.08
4.44
4.44
3.17
3.17
2.54
2.54
2.54
2.22
1.9
1.59
1.59
1.59
1.59
1.27
1.27
0.95
0.95
0.95
0.63
0.63
0.63
0.63
0.63
0.32
0.32
0.32
0.32
0.32
0.32
0.32
0.32
0.32
0.32
Table 14: Share of UPCs across Product Groups: Coca-cola
Product Group
Percent
CARBONATED BEVERAGES
JUICE, DRINKS - CANNED, BOTTLED
YOGURT
SOFT DRINKS-NON-CARBONATED
COFFEE
JUICES, DRINKS-FROZEN
VITAMINS
BUTTER AND MARGARINE
CHEESE
GLASSWARE, TABLEWARE
61
56.26
18.4
12.89
12.2
0.06
0.06
0.06
0.03
0.03
0.03
Table 15: Share of UPCs across Product Groups: Unilever
Product Group
Percent
HAIR CARE
PERSONAL SOAP AND BATH ADDITIV
DEODORANT
SKIN CARE PREPARATIONS
ICE CREAM, NOVELTIES
TEA
MEN’S TOILETRIES
SOUP
FRAGRANCES - WOMEN
GROOMING AIDS
YOGURT
SALAD DRESSINGS, MAYO, TOPPINGS
PREPARED FOOD-DRY MIXES
CONDIMENTS, GRAVIES, AND SAUCES
BUTTER AND MARGARINE
FIRST AID
SHAVING NEEDS
PAPER PRODUCTS
HOUSEWARES, APPLIANCES
DESSERTS/FRUITS/TOPPINGS-FROZEN
MEDICATIONS/REMEDIES/HEALTH AI
SPICES, SEASONING, EXTRACTS
COSMETICS
SANITARY PROTECTION
SHORTENING, OIL
SNACKS
SOFT DRINKS-NON-CARBONATED
JUICE, DRINKS - CANNED, BOTTLED
DRESSINGS/SALADS/PREP FOODS-DELI
DESSERTS, GELATINS, SYRUP
ETHNIC HABA
PREPARED FOOD-READY-TO-SERVE
BAKING SUPPLIES
VEGETABLES - CANNED
BABY NEEDS
BAKING MIXES
CANDY
AUTOMOTIVE
FRESHENERS AND DEODORIZERS
HOUSEHOLD CLEANERS
HOUSEHOLD SUPPLIES
JUICES, DRINKS-FROZEN
PACKAGED MILK AND MODIFIERS
SEWING NOTIONS
SNACKS, SPREADS, DIPS-DAIRY
VEGETABLES AND GRAINS - DRIED
62
31.83
13.55
11.87
9.51
7.82
5.25
2.78
2.39
2.1
1.95
1.39
1.36
1.22
1.03
0.98
0.95
0.67
0.62
0.58
0.45
0.35
0.32
0.23
0.08
0.08
0.08
0.08
0.07
0.06
0.06
0.04
0.04
0.03
0.03
0.02
0.02
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
Table 16: Share of UPCs across Product Groups: Low Durability
Product Group
Percent
ICE CREAM, NOVELTIES
PACKAGED MEATS-DELI
YOGURT
FRESH PRODUCE
MILK
CHEESE
BABY FOOD
COT CHEESE, SOUR CREAM, TOPPINGS
EGGS
BAKING SUPPLIES
DESSERTS/FRUITS/TOPPINGS-FROZEN
SNACKS, SPREADS, DIPS-DAIRY
SOFT DRINKS-NON-CARBONATED
CONDIMENTS, GRAVIES, AND SAUCES
DISPOSABLE DIAPERS
PAPER PRODUCTS
PET FOOD
PREPARED FOOD-READY-TO-SERVE
VITAMINS
CARBONATED BEVERAGES
CEREAL
GROOMING AIDS
HOUSEHOLD SUPPLIES
JAMS, JELLIES, SPREADS
31.52
20.29
18.56
9.28
8.59
4.11
3.16
2.15
1.78
0.28
0.09
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.01
0.01
0.01
0.01
0.01
Table 17: Share of UPCs across Product Groups: High Durability
Product Group
Percent
KITCHEN GADGETS
LIGHT BULBS, ELECTRIC GOODS
HOUSEWARES, APPLIANCES
LIQUOR
STATIONERY, SCHOOL SUPPLIES
PHOTOGRAPHIC SUPPLIES
BABY NEEDS
CHARCOAL, LOGS, ACCESSORIES
WINE
BEER
BATTERIES AND FLASHLIGHTS
JUICE, DRINKS - CANNED, BOTTLED
SOFT DRINKS-NON-CARBONATED
COOKWARE
DOUGH PRODUCTS
GRT CARDS/PARTY NEEDS/NOVELTIES
TOBACCO AND ACCESSORIES
63
35.83
18.04
13.32
13.01
10.08
4.73
4.4
0.4
0.12
0.03
0.01
0.01
0.01
0
0
0
0
B
Proofs for Section 3
B.1
Proof of Lemma 1
(i) First, consider a seller posting prices (p1 , p2 ) with p1 > ub and p2 > ub. The seller’s profit
is zero, as there are no buyers willing to purchase a good at a price strictly greater than
ub . If the seller instead posts the prices (p′1 , p′2 ) with p′1 = p′2 = ub, it sells both goods to all
captive buyers of type b and it attains a profit of at least 2µb αub > 0. Therefore, a seller
never finds it optimal to post prices (p1 , p2 ) such that p1 > ub and p2 > ub .
Consider a seller posting prices (p1 , p2 ) with p1 > ub and p2 ∈ (uc , ub]. The seller attains
a profit of
n
h
io
S(p1 , p2 ) = µb α + 2(1 − α) 1 − Ĝ(ub + p2 ) + ν̂(ub + p2 )/2 p2 .
(32)
The expression above is easy to understand. The seller is in the network of µb α captive buyers
of type b. A captive buyer of type b purchases good 2 from the seller with probability 1. The
seller is also in the network of 2µb (1 − α) non-captive buyers of type b. A non-captive buyer
of type b purchases good 2 from the seller with probability 1 − Ĝ(ub + p2 ) + ν̂(ub + p2 )/2,
where 1 − Ĝ(ub + p2 ) denotes the fraction of sellers that charge prices (p′1 , p′2 ) such that
min{p′1 , ub} + min{p′2 , ub} > ub + p2 , and ν̂(ub + p2 ) is the measure of sellers that charge
prices (p′1 , p′2 ) such that min{p′1 , ub} + min{p′2 , ub} = ub + p2 . The seller is in the network of
µc α captive buyers of type c and of 2µc (1 − α) non-captive buyers of type c, but it does not
trade with any of them as both of its prices are greater than uc . If the seller instead posts
the prices (p′1 , p′2 ) with p′1 = ub and p′2 = p2 , it attains a profit of
n
h
io
′
′
S(p1 , p2 ) = µb α + 2(1 − α) 1 − Ĝ(ub + p2 ) + ν̂(ub + p2 )/2 (ub + p2 )
(33)
> S(p1 , p2 ).
The strict inequality in (33) follows from the fact that, by lowering the price of good 1 to
ub , the seller trades both good 1 and good 2 to its customers of type b. Hence, a seller never
finds it optimal to post prices (p1 , p2 ) with p1 > ub and p2 ∈ (uc , ub ]. For the same reason,
a seller never finds it optimal to post prices (p1 , p2 ) with p1 ∈ (uc , ub ] and p2 > ub .
Finally, consider a seller posting prices (p1 , p2 ) with p1 > ub and p2 ∈ [0, uc ]. The seller
attains a profit of
S(p1 , p2 )
n
h
io
= µb α + 2(1 − α) 1 − Ĝ(ub + p2 ) + ν̂(ub + p2 )/2 p2
+µc {α + 2(1 − α) [1 − F2 (p2 ) + λ2 (p2 )/2]} p2 .
64
(34)
The expression in (34) is the same as in (32) with the addition of the term in the second
line. This term represents the profit that the seller makes from trading good 2 to buyers of
type c. If the seller instead posts the prices (p′1 , p′2 ) with p′1 = ub and p′2 = p2 , it attains a
profit of
S(p′1 , p′2 )
n
h
io
= µb α + 2(1 − α) 1 − Ĝ(ub + p2 ) + ν̂(ub + p2 )/2 (ub + p2 )
+µc {α + 2(1 − α) [1 − F2 (p2 ) − λ2 (p2 )/2]} p2
(35)
> S(p1 , p2 )
The strict inequality in (35) follows from the fact that, by lowering the price of good 1 to ub ,
the seller trades both goods 1 and good 2 to its type b customers. Hence, a seller never finds
it optimal to post prices (p1 , p2 ) such that p1 > ub and p2 ∈ [0, uc]. For the very same reason,
a seller never finds it optimal to post prices (p1 , p2 ) such that p1 ∈ [0, uc ] and p2 > ub .
(ii) Given part (i), it follows immediately that, if in equilibrium, a seller posts prices
(p1 , p2 ) with p1 + p2 > ub + uc , it must be the case that p1 ∈ (uc , ub ] and p2 ∈ (uc , ub].
B.2
Proof of Lemma 3
(i) On the way to a contradiction suppose there is an equilibrium where G has a mass point
at q0 , i.e. ν(q0 ) > 0. First, notice that no seller finds it optimal to post p1 = p2 = 0, and
hence the mass point cannot be at q0 = 0. Second, notice that if, in equilibrium, a seller
posts (p1 , p2 ) with p1 + p2 = q0 , it must be posting p1 ∈ [0, ub] and p2 ∈ [0, ub ]. Therefore,
this seller attains a profit of
S(p1 , p2 )
= µb {α + 2(1 − α) [1 − G(q0 ) + ν(q0 )/2]} q0
P
+ 2i=1 µc {α + 2(1 − α) [1 − Fi (pi ) + λi (pi )/2]} 1[pi ≤ uc ]pi ,
(36)
where 1[pi ≤ uc ] is the indicator function that takes the value 1 if pi ≤ uc and 0 otherwise.
Suppose that the seller deviates and posts prices (p′1 , p′2 ) with 0 ≤ p′1 = p1 − ǫ1 , 0 ≤ p′2 =
p2 − ǫ2 , ǫ = ǫ1 + ǫ2 , where ǫ1 ≥ 0, ǫ2 ≥ 0 and ǫ > 0 all arbitrarily small. Then, the seller
attains a profit of
S(p′1 , p′2 )
= µb {α + 2(1 − α) [1 − G(q0 − ǫ)]} (q0 − ǫ)
P
+ 2i=1 µc {α + 2(1 − α) [1 − Fi (pi − ǫi ) + λi (pi − ǫi )/2]} 1[pi ≤ uc ](pi − ǫi )
> S(p1 , p2 ),
(37)
where the inequality follows from the fact that G(q0 ) − ν(q0 )/2 − G(q0 − ǫ) ≥ ν(q0 )/2 while ǫ,
ǫ1 and ǫ2 are all arbitrarily small. Since S(p′1 , p′2 ) > S(p1 , p2 ), there cannot be a mass point
at q0 .
65
(ii) On the way to a contradiction, suppose there is an equilibrium where F1 has a mass
point at p1,0 ∈ (0, uc ]. If in equilibrium a seller posts the price p1 = p1,0 for the first good, it
must post a price p2 ∈ [0, ub ] for the second good. This seller attains a profit of
S(p1 , p2 )
= µb {α + 2(1 − α)[1 − G(p1 + p2 )]} (p1 + p2 )
P
+ 2i=1 µc {α + 2(1 − α) [1 − Fi (pi ) + λi (pi )/2]} 1[pi ≤ uc ]pi .
(38)
If the seller deviates and posts prices (p′1 , p′2 ) with p′1 = p1,0 − ǫ, p′2 = p2 , for ǫ > 0 arbitrarily
small, it attains a profit of
S(p′1 , p′2 ) = µb {α + 2(1 − α)[1 − G(p1,0 + p2 − ǫ)]} (p1,0 + p2 − ǫ)
+µc {α + 2(1 − α) [1 − F1 (p1,0 − ǫ)]} (p1,0 − ǫ)
+µc {α + 2(1 − α) [1 − F2 (p2 ) + λ2 (p2 )/2]} 1[p2 ≤ uc ]pi
(39)
≥ S(p1 , p2 ),
where the first inequality follows from the fact that the F1 (p1,0 ) − λ1 (p1,0 )/2 − F1 (p1,0 − ǫ) ≥
λ1 (p1,0 )/2 and ǫ is arbitrarily small. Since S(p′1 , p′2 ) > S(p1 , p2 ), there cannot be a mass
point at p1,0 .
B.3
Proof of Proposition 1
We established part (i) in the main text. We have also established that, given the marginal
distribution G in (14), a seller attains a profit of S ∗ for all (p1 , p2 ) ∈ R1 such that p1 + p2 ∈
[qℓ , qh ]. In order to complete the proof of part (ii), all we need to do is find a condition under
which a seller cannot attain a profit strictly greater than S ∗ by posting some off-equilibrium
prices.
In Lemma 1, we proved that a seller never finds it optimal to post prices (p1 , p2 ) with
either p1 , p2 or both p1 and p2 strictly greater than ub . If the seller posts prices (p1 , p2 ) ∈ R1
with p1 + p2 ∈ (ub + uc , qℓ ), it attains a profit of
S1 (p1 + p2 ) = µb {α + 2(1 − α)} (p1 + p2 )
< µb {α + 2(1 − α)} qℓ = S ∗ ,
(40)
where the first line makes use of the fact that G(p1 + p2 ) = 0 and the second line makes use
of the fact that p1 + p2 < qℓ . Hence, a seller does not find it optimal to deviate from the
equilibrium and post prices (p1 , p2 ) ∈ R1 with p1 + p2 ∈ (ub + uc , qℓ ).
In Lemma 2, we proved that a seller never finds it optimal to post prices (p1 , p2 ) with
p1 + p2 ∈ (2uc, ub + uc ] and both p1 and p2 greater than uc and smaller than ub . If the seller
66
posts prices (p1 , p2 ) with p1 + p2 ∈ (2uc , ub + uc ], p1 ∈ [0, uc ] and p2 ∈ (uc , ub], it attains a
profit of
S(p1 , p2 ) = µb {α + 2(1 − α)} (p1 + p2 ) + µc {α + 2(1 − α)} p1
≤ µb {α + 2(1 − α)} (uc + ub ) + µc {α + 2(1 − α)} uc
(41)
= S(uc , ub),
where the first line makes use of G(p1 + p2 ) = 0 and F (p1 ) = 0, and the second line makes
use of G(uc + ub ) = 0, F (uc ) = 0, p1 + p2 ≤ ub + uc and p1 ≤ uc . The equilibrium profit S ∗
is greater than S(uc , ub ) if and only if
3α − 2
µc
≤
− 1.
µb
(2 − α)uc /ub
(42)
Hence, if and only if (42) holds a seller does not find it optimal to deviate from the equilibrium
and post any prices (p1 , p2 ) with p1 + p2 ∈ (2uc , ub + uc ], p1 ∈ [0, uc ] and p2 ∈ (uc , ub ].
Similarly, condition (42) guarantees that a seller does not find it optimal to deviate from
the equilibrium and post any prices (p1 , p2 ) with p1 + p2 ∈ (2uc , ub + uc ], p1 ∈ (uc , ub ] and
p2 ∈ [0, uc ].
If the seller posts prices (p1 , p2 ) with p1 ∈ [0, uc ] and p2 ∈ [0, uc ], it attains a profit of
S(p1 , p2 )
= (µb + µc ) {α + 2(1 − α)} (p1 + p2 )
≤ (µb + µc ) {α + 2(1 − α)} 2uc
(43)
= S(uc , uc ),
where the first line makes use of G(p1 + p2 ) = 0 and Fi (pi ) = 0 for i = {1, 2}, and the second
line makes use of G(2uc ) = 0, Fi (uc ) = 0 for i = {1, 2}, and p1 + p2 ≤ 2uc . The equilibrium
profit S ∗ is greater than S(uc , uc) if and only if
µc
α
≤
− 1.
µb
(2 − α)uc /ub
(44)
Hence, if and only if (44) holds a seller does not find it optimal to deviate from the equilibrium
and post any prices (p1 , p2 ) with p1 ∈ [0, uc ] and p2 ∈ [0, uc ]. Finally, notice that if condition
(42) holds, so does condition (44). Therefore, a seller does not want to deviate from the
equilibrium if and only if condition (42), which is the same as condition (16), is satisfied.
B.4
Proof of Lemma 5
It is easy to verify that in a Discrimination Equilibrium, Fi (p) does not have a mass point
at p = 0 for i = {1, 2}. Hence, Fi (0) = 0 and Fi (p) is continuous over the interval [0, uc ] for
67
i = {1, 2}. If F1 6= F2 for some p ∈ [0, uc ], there exist prices p′ and p′′ , with 0 ≤ p′ < p′′ ≤ uc ,
for which F1 (p′ ) = F2 (p′ ) and either F1 (p) > F2 (p) or F1 (p) < F2 (p) for all p ∈ (p′ , p′′ ).
Without loss in generality, assume F1 (p) > F2 (p) for all p ∈ (p′ , p′′ ). Any price p ∈ (p′ , p′′ )
for good 1 is posted by a seller in region R2 , i.e. a seller with a basket price of q ∈ (2uc , ub +uc ]
and a price for good 2 of p2 ∈ (uc , ub ]. This seller attains a profit of S21 (q, p). However, if
the seller inverts the prices of goods 1 and 2, it attains a profit of S22 (q, p) which is strictly
greater than S21 (q, p) because F1 (p) > F2 (p). In turn, S21 (q, p) < S22 (q, p) implies that in
equilibrium there are no sellers posting a price p ∈ (p′ , p′′ ) for good 1, i.e. F1 (p) = F1 (p′ )
for all p ∈ (p′ , p′′ ). Since F1 (p) = F1 (p′ ) for all p ∈ (p′ , p′′ ), F2 (p) ≥ F2 (p′ ) for all p ∈ (p′ , p′′ )
and F1 (p′ ) = F2 (p′ ), it follows that F2 (p) ≥ F1 (p) for all p ∈ (p′ , p′′ ). We have thus reached
a contradiction.
B.5
Proof of Lemma 6
We need to prove that the seller’s profit S2 (q, p) is constant for all (q, p) such that q ∈
[qℓ , ub + uc ] and p ∈ [pℓ , uc ]. To this aim, it is useful to write the seller’s profit S2 (q, p) as
S2 (q, p) = B(q) + C(p),
(45)
where B(q) and C(p) are defined as
B(q) = µb [α + 2(1 − α)(1 − G(q))] q,
C(p) = µc [α + 2(1 − α)(1 − F (p))] p.
In words, B(q) is the profit that the seller attains from trades with the buyers of type b, and
A(p) is the profit that the seller attains from trades with the buyers of type c.
First, we show that the distribution G(q) has full support over the interval [qℓ , ub + uc ].
On the way to a contradiction, suppose that G(q) is such that G′ (q0 ) > 0 and G(q0 ) = G(q1 )
for some q0 and q1 with qℓ < q0 < q1 ≤ ub + uc . Since G(q) = G(q0 ) for all q ∈ [q0 , q1 ], the
function B(q) is strictly increasing in q for all q ∈ [q0 , q1 ]. That is, the seller’s profit from
buyers of type b is strictly increasing in q for all q between q0 and q1 .
Now, consider a seller with a basket price of q̂ ∈ (q0 , q1 ). This seller may post any price
p for the cheaper good between q̂ − ub and uc . In fact, for p < q̂ − ub , the price of the more
expensive good would be greater than ub , which is never optimal. For p > uc , the seller
would be posting a price outside of the R2 region, which is also never optimal. Hence, the
profit of this seller is
S2∗ (q̂) = B(q̂) +
max
p∈[q̂−ub ,uc ]
68
C(p).
(46a)
Next, consider a seller with a basket price of q0 . This seller may post any price p for the
cheaper good between q0 − ub and uc . Hence, the profit of this seller is
S2∗ (q0 ) = B(q0 ) +
max
p∈[q0 −ub ,uc ]
C(p).
(47)
Since S2∗ (q0 ) ≥ S2∗ (q̂) and B(q0 ) < B(q̂), it must be that maxp∈[q0 −ub ,uc ] C(p) is strictly greater
than maxp∈[q̂−ub ,uc ] C(p). In turn, this implies that maxp∈[q0−ub ,q̂−ub ) C(p) is strictly greater
than maxp∈[q̂−ub ,uc ] C(p). Since q̂ was chosen arbitrarily in the interval (q0 , q1 ), C(q0 − ub ) is
strictly greater than C(p) for all p ∈ (q0 − ub , uc ].
A seller with a basket price of q ≤ q0 does not find it optimal to choose a price p ∈
(q0 − ub , q1 − ub] for the cheaper good, as C(q0 − ub ) > C(p) for all p ∈ (q0 − ub , q1 − ub ]. A
seller with a basket price of q ≥ q1 does not find it optimal to post a price p ∈ (q0 −ub , q1 −ub ]
for the cheaper good, as this would imply its prices are outside the R2 region. Finally, there
are no sellers with a basket price of q ∈ (q0 , q1 ), as G(q0 ) = G(q1 ). These observations imply
that the distribution F (p) has a gap between q0 − ub and q1 − ub . In turn, this implies that
the function C(p) is strictly increasing in p for all p ∈ [q0 − ub , q1 − ub ], which contradicts
C(q0 − ub ) > C(p) for all p ∈ (q0 − ub , uc ].
Second, we show that the distribution F (p) has full support over the interval [pℓ , uc ]. On
the way to a contradiction, suppose that F (p) is such that F ′ (p0 ) > 0 and F (p0 ) = F (p1 )
for some p0 and p1 such that pℓ < p0 < p1 ≤ uc . Since F (p) = F (p0 ) for all p ∈ [p0 , p1 ], the
function C(p) is strictly increasing in p for all p ∈ [p0 , p1 ]. By continuity of C(p), there is an
interval [p−1 , p0 ] such that C(p) < C(p1 ) for all p ∈ [p−1 , p0 ]. A seller with a basket prices
of q ≤ ub + p0 does not find it optimal to post a price p ∈ [p−1 , p0 ] for the cheaper good, as
C(p1 ) > C(p) for all p ∈ [p−1 , p0 ]. A seller with a basket price of q > ub + p0 does not find
it optimal to post a price p ∈ [p−1 , p0 ] for the cheaper good, as this would imply that its
prices are outside the R2 region. Therefore, there are no sellers posting p ∈ [p−1 , p0 ], which
contradicts F ′ (p0 ) > 0.
Finally, we show that C(p) is constant for all p ∈ [pℓ , uc ]. First, suppose that C(p) is
strictly increasing over some interval (p0 , p1 ) ⊂ [pℓ , uc ]. If that is the case, a seller with a
basket price of q ≤ ub + p1 does not find it optimal to post a price p ∈ (p0 , p1 ) for the cheaper
good, as it can post the price p1 instead and attain a higher profit. Similarly, a seller with
a basket price of q > ub + p1 cannot post a price p ∈ (p0 , p1 ), as this would imply that its
prices are outside of the R2 region. Hence the distribution F has a gap between p0 and
p1 . However, we have established that the distribution F has full support over the interval
[pℓ , uc ]. Therefore, C(p) must be weakly decreasing for all p ∈ [pℓ , uc ].
69
Now, suppose that C(p) is strictly decreasing over the interval p ∈ [pℓ , uc ]. In this case,
a seller with a basket price q ∈ [q, ub + uc ] chooses the lowest possible price p for the cheaper
good, i.e. ub + p. Hence, F (p) = G(ub + p)/2 for all p ∈ [p, uc ]. Moreover F (p) is such
that B(ub + p) + C(p) = S ∗ for all p ∈ [pℓ , uc ]. After solving this equal profit condition
with respect to F (p), we find that C(p) is strictly increasing in p over the interval [pℓ , uc ],
which contradicts the assumption that C(p) is strictly decreasing. The same argument
can be applied to rule out the case in which C(p) is strictly decreasing over some interval
(p0 , p1 ) ⊂ [pℓ , uc ]. Therefore, C(p) must be weakly increasing for all p ∈ [pℓ , uc ]. Since C(p)
is both weakly decreasing and weakly increasing, it must be constant for all p ∈ [pℓ , uc ]. In
turn, this implies that B(q) must be constant for all q ∈ [qℓ , ub + uc ].
B.6
Proof of Proposition 2
We established part (i) in the main text. Here we prove part (ii). To this aim, we need to
show that there exists a joint distribution H that generates the marginals G and F specified
in part (i), and such that, on every point on the support of H, the profit of the seller is
maximized.
We begin the analysis by identifying the region where the profit of the seller are maximized. In Lemma 1, we proved that a seller never finds it optimal to post prices (p1 , p2 )
with either p1 , p2 or both p1 and p2 strictly greater than ub . It is also straightforward to
show that a seller never finds it optimal to post prices (p1 , p2 ) with either p1 , p2 or both
strictly smaller than pℓ . Therefore, we only need to check the seller’s profit associated to
prices (p1 , p2 ) ∈ [pℓ , ub ] × [pℓ , ub].
First, we compute the seller’s profit for prices (p1 , p2 ) with p1 + p2 ∈ (ub + uc , qh ]. If the
seller posts prices (p1 + p2 ) with p1 + p2 ∈ [q ∗ , qh ], it attains a profit of S ∗ , as guaranteed by
the construction of G. If the seller posts prices (p1 , p2 ) with p1 + p2 ∈ (ub + uc , q ∗ ), it attains
a profit of
S1 (p1 + p2 ) = µb {α + 2(1 − α)[1 − G(p1 + p2 )]} (p1 + p2 )
< µb {α + 2(1 − α)[1 − G(q ∗ )]} q ∗ = S ∗ ,
(48)
where the second line uses the fact that G(p1 + p2 ) = G(q ∗ ) for p1 + p2 ∈ (ub + uc , q ∗ ).
Second, we compute the seller’s profit for prices (p1 , p2 ) with p1 + p2 ∈ (2uc, ub + uc ].
In Lemma 2, we showed that the seller never finds it optimal to post prices (p1 , p2 ) with
p1 + p2 ∈ (2uc , ub + uc ], p1 ∈ (uc , ub ] and p2 ∈ (uc , ub ]. If the seller posts prices (p1 , p2 )
with p1 + p2 ∈ (qℓ , ub + uc ], p1 ∈ (uc , ub ] and p2 ∈ [p, uc ], it attains a profit of S ∗ , as
70
guaranteed by Lemma 6. Similarly, if the seller posts prices (p1 , p2 ) with p1 +p2 ∈ (qℓ , ub +uc ],
p1 ∈ (uc , ub ] and p2 ∈ (uc , ub ], it attains a profit of S ∗ . If the seller posts prices (p1 , p2 ) with
p1 + p2 ∈ (2uc , qℓ ), p1 ∈ [pℓ , ub ] and p2 ∈ (uc , ub], it attains a profit of
S2 (p1 + p2 , p1 ) = µb {α + 2(1 − α)} (p1 + p2 ) + µc {α + 2(1 − α)[1 − F (p1 )]} p1
≤ µb {α + 2(1 − α)} qℓ + µc {α + 2(1 − α)[1 − F (p1 )]} p1 = S ∗ .
(49)
The first line uses the fact that the seller trades both goods with all of its potential customers
of type b. The second line uses the fact that the seller would also trade both goods with all
of its potential customers of type b at the basket price qℓ , and the last line uses Lemma 6.
Similarly, S2 (p1 + p2 , p2 ) ≤ S ∗ for all (p1 , p2 ) with p1 + p2 ∈ (2uc , qℓ ), p1 ∈ (uc , ub ], p2 ∈ [p, uc ].
Third, we compute the seller’s profit for prices (p1 , p2 ) ∈ [pℓ , uc ] × [pℓ , uc ]. If the seller
posts such prices, it attains a profit of
S(p1 , p2 ) = µb {α + 2(1 − α)} (p1 + p2 ) +
P2
i=1
µc {α + 2(1 − α)[1 − F (pi )]} pi
≤ µb {α + 2(1 − α)} 2uc + µc {α + 2(1 − α)[1 − F (uc )]} 2uc
(50)
= S(uc , uc ).
The first line uses the fact that the seller trades both goods to all its potential customers
of type b, it trades good 1 to µc {α + 2(1 − α)[1 − F (p1 )]} buyers of type c, and it trades
good 2 to µc {α + 2(1 − α)[1 − F (p2 )]} buyers of type c. The second line uses the fact that
the seller would also trade both goods to all its potential customers of type b at the prices
(uc , uc ), and that the profit that the seller makes off of buyers of type c by posting the price
pi ∈ [pℓ , uc ] for good i = {1, 2} is the same it would make by posting the price uc instead.
If and only if S(uc , uc ) ≤ S ∗ , the highest profit that the seller can attain is S ∗ . Using the
fact that S2 (ub + uc , uc ) = S ∗ , we can write the condition S(uc , uc ) ≤ S ∗ as
µb {α + 2(1 − α)} 2uc + µc {α + 2(1 − α)[1 − G(q ∗ )/2]} 2uc
≤ µb {α + 2(1 − α)[1 − G(q ∗ )]} (ub + uc ) + µc {α + 2(1 − α)[1 − G(q ∗ )/2]} uc .
(51)
After substituting out G(q ∗ ), we can write the inequality above as (30).
The functions G and F are proper distribution functions if and only if
and
3α − 2
µc
>
− 1,
µb
(2 − α)uc /ub
(52)
µc
α(1 − uc /ub )
<
.
µb
uc /ub
(53a)
71
Condition (52) is necessary and sufficient for G(q ∗ ) > 0, and it is condition (29). Condition
(53a) is necessary and sufficient for q ∗ < 2ub , and it holds whenever condition (30) is satisfied.
If and only if (52) and (53a) are satisfied, G and F are proper distribution functions. That
is, the interval [qℓ , ub + uc ] is non-empty and, over this interval, G(q) is strictly increasing
in q, and such that G(qℓ ) = 0 and G(ub + uc ) = G(q ∗ ), where G(q ∗ ) ∈ (0, 1). The interval
[q ∗ , qh ] is non-empty and, over this interval, G(q) is strictly increasing in q, and such that
G(q ∗ ) = G(ub + uc ) and G(qh ) = 1. Similarly, the interval [pℓ , uc ] is non-empty and, over
this interval, F (p) is strictly increasing in p and such that F (pℓ ) = 0 and F (uc ) = G(q ∗ )/2 ∈
(0, 1).
In the main text, we established that there exists a joint distribution H that generates the
marginal F for p ∈ [pℓ , uc ] and the marginal G for q ∈ [qℓ , ub + uc ] and that has support over
the region of prices (p1 , p2 ) such that p1 + p2 ∈ [qℓ , ub + uc ], and p1 ∈ [pℓ , uc ], p2 ∈ (uc , ub] or
p1 ∈ (uc , ub ], p2 ∈ [pℓ , uc ]. Over this region, the seller’s profit is S ∗ . Moreover, we established
that there exists a joint distribution H that generates the marginal G for q ∈ [q ∗ , qh ] and
that has support over the region of prices (p1 , p2 ) such that p1 + p2 ∈ [q ∗ , qh ] and p1 = p2 .
Over this region, the seller’s profit is S ∗ .
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@FedFRASER