The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Working Paper Series
Price Dynamics with Customer Markets
WP 14-17
Luigi Paciello
Einaudi Institute for Economics and
Finance and CEPR
Andrea Pozzi
Einaudi Institute for Economics and
Finance and CEPR
Nicholas Trachter
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Price Dynamics with
Customer Markets∗
Luigi Paciello
Einaudi Institute for Economics and Finance and CEPR
Andrea Pozzi
Einaudi Institute for Economics and Finance and CEPR
Nicholas Trachter
Federal Reserve Bank of Richmond
Working Paper No. 14-17
October 1, 2014
Abstract
We study a tractable model of firm price setting with customer markets and empirically
evaluate its predictions. Our framework captures the dynamics of customers in response to
a change in the price, describes the behavior of optimal prices in the presence of customer
acquisition and retention concerns, and delivers a general equilibrium model of price and customer dynamics. We exploit novel micro data on purchases from a panel of households from
a large U.S. retailer to quantify the model and compare it to the counterfactual benchmark of
the standard monopolistic competition setting. We show that a model with customer markets
has markedly different implications in terms of the equilibrium price distribution, which better
fit the available empirical evidence on retail prices. Moreover, the dynamic of the response
of demand to shocks that affects price dispersion is also distinctive. Our results suggest that
inertia in customer reallocation across firms increases the persistence in the response of demand
to these shocks.
JEL classification: E30, E12, L16
Keywords: customer markets, price setting, product market frictions
∗
Corresponding Author: luigi.paciello@eief.it. Previous drafts of this paper circulated under the title “Price Setting with
Customer Retention.” We benefited from comments on earlier drafts at the Minnesota Workshop in Macroeconomic Theory,
2nd Rome Junior Macroeconomics conference, 2nd Annual UTDT conference in Advances in Economics, 10th Philadelphia
Search and Matching conference, ESSET 2013, MBF-Bicocca conference, MaCCI Mannheim, the Collegio Carlo Alberto Pricing
Workshop, Goods Markets Paris Conference 2014, the Macroeconomy and Policy, NBER Summer Institute 2014, EARIE 2014
and seminars at Bank of France, Bank of Spain, Columbia University, Federal Reserve Bank of Richmond, the Ohio State
University, University of Pennsylvania, Macro Faculty Lunch at Stanford, and University of Tor Vergata. We thank Fernando
Alvarez, Lukasz Drozd, Huberto Ennis, Mike Golosov, Bob Hall, Hugo Hopenhayn, Eric Hurst, Pat Kehoe, Francesco Lippi,
Erzo Luttmer, Kiminori Matsuyama, Guido Menzio, Dale Mortensen, Ezra Oberfield, Facundo Piguillem, Valerie Ramey, and
Leena Rudanko. Luigi Paciello thanks Stanford University for its hospitality. The views expressed in this article are those of
the authors and do not necessarily represent the views of the Federal Reserve Bank of Richmond or the Federal Reserve System.
1
Introduction
The customer base of a firm, that is, the set of customers buying from it at a given point
in time, is an important determinant of firm performance and survival.1 Its effects are
long lasting, as customer-supplier relationships are subject to a certain degree of stickiness
(Hall (2008)).2 Therefore, firms actively seek to maintain and grow their customer base and
this effort impacts their pricing, as the price is an obvious instrument to attract and retain
customers (Phelps and Winter (1970)). Despite the centrality of price and demand dynamics
in informing several domains of public policy (e.g. monetary policy), our understanding of
how customer base concerns affect firms’ pricing is still scant.
In this paper we develop a model of firm price-setting with customer markets. We contrast its predictions with those of the standard monopolistic competition framework and
quantitatively evaluate them exploiting novel data. In the model, firms set prices responding to idiosyncratic productivity shocks taking into account the effects of a price change on
the dynamics of their customer base. Customers respond to price changes but face search
frictions that reduce their ability to reallocate across firms. This model is of interest in itself
because, while being tractable, it provides a rich laboratory to study how the relationship
between customer and price dynamics is shaped, in equilibrium, by idiosyncratic production
and search costs. Moreover, whereas the lack of appropriated data has so far limited the
possibility to measure the importance of customer markets, we exploit novel micro data to
estimate and quantify our model.
We combine the model and data to highlight two important implications of customer
markets. First, competition for customers creates a strong incentive for firms to cluster
prices. This results in a shape of the price distribution matching the one found in our data
and consistent with the findings of a recent literature documenting the pricing of homogeneous
goods (Kaplan and Menzio (forthcoming)). Second, with a counterfactual exercise, we show
that search frictions in customer reallocation shape in a sizeable and persistent fashion the
dynamic response of demand to shocks that affect price dispersion (e.g. exchange rate shocks,
changes in sales taxes, etc.). Inertia in customer reallocation tends to dampen short-term
effects and to magnify long-run ones.
The model features two types of agents, firms and customers. Firms produce a homogeneous good with a linear production function in a single variable input. Customers derive
utility from the consumption of the homogenous good. Customers perfectly observe the
variables characterizing the relevant state of the firm they buy from: its idiosyncratic productivity, which follows an exogenous Markov process with stationary normal distribution;
1
2
See Foster et al. (2013) for recent evidence.
See also Blinder et al. (1998) and Fabiani et al. (2007) for survey evidence.
1
and its customer base, the mass of customers who bought from it in the previous period.
Customers start each period matched to the same firm from which they bought in the previous period. Once firms have drawn the new productivity level and posted a price for the
period, customers have faculty to search for a new supplier. To search, customers must pay
an idiosyncratic search cost, drawn every period from an i.i.d. distribution. After the search
cost is paid, the customer gets randomly assigned another firm, she observes the state of the
new firm, compares it to that of her old supplier, and decides where to buy (extensive margin
of demand). After customers have made their search and matching decisions, each customer
decides her purchased quantity of the good (intensive margin of demand). In this framework,
firms face a trade-off between charging a higher price and extracting more surplus from high
search costs customers, versus posting a lower price to extract a lower surplus but from a
larger mass of customers. Since the customer base is sticky, changes in the customer base
have persistent effects on demand, making the price-setting problem of the firm a dynamic
one. The equilibrium of our model features both price dispersion and customer dynamics. In
particular, the equilibrium prices are decreasing in productivity whereas the growth in the
customer base is increasing in productivity, with more productive firms gaining customers at
the expenses of less productive ones.
We complement our modeling effort with an empirical analysis that relies on novel micro
data documenting pricing and customer base evolution for a large retail firm. We take
advantage of scanner data from a major U.S. retailer recording purchases for a large sample of
households between 2004 and 2006. Household-level scanner data are particularly well suited
to study customer base dynamics. First, we observe a wealth of details on all the shopping
trips each household makes to the chain (list of goods purchased, prices, quantities, etc.).
More importantly, we can infer when customers leave the retailer by looking at prolonged
spells without purchases at the chain. These data allow us to study the relation between a
customer’s decision to abandon the firm and the price of the good-or rather, in this case,
bundle of goods-she consumes there. We show that customer dynamics are indeed affected
by variation in the price: A 1% change in the price of the customer’s typical basket of grocery
goods would raise the firm yearly customer turnover from 14% to 21%.
We use the estimated price elasticity of the customer base, jointly with moments from
the distribution of prices posted by the chain, to identify the key objects of the model:
the distribution of search costs and the properties of the productivity process. We assess the
relevance of customer markets for price dynamics by comparing our model to a counterfactual
economy where only the intensive margin is present. This is an interesting benchmark, as the
pricing problem of the firm in such economy is similar to the one of standard macro models
where competition comes only from the downward-sloping demand of each customer, and the
2
customer base is constant. We design the experiment so that the counterfactual economy is
observationally equivalent with respect to average demand elasticity, price persistence, and
dispersion. The most obvious difference between our model and those that do not feature
an extensive margin of demand is that the former predicts equilibrium customer dynamics
whereas the latter imply no customer reallocation. Quantitatively, our model delivers a
yearly average turnover of 6%, nearly half of what we measure in the data (14%). This
implies that customer dynamics triggered by variation in prices due to idiosyncratic cost
shocks can explain a large fraction of the overall turnover.
We find that customer markets also have implications for firms’ pricing and document
that they are nontrivial in magnitude. First, we look at the implications for the shape of
the price distribution. The strategic complementarity effect leads firms facing high extensive
margin elasticity to set similar prices, generating a high mass of prices that cluster around the
mean that results in a distribution with high peakedness. A large fraction of prices bunched
around the mean also characterizes the price distribution in our data. In fact, the model
and data display a similar excess kurtosis even though the shape of the price distribution
was not targeted in our estimation procedure, and the underlying invariant distribution of
productivity is normal. The model without customer markets instead fails to capture the
distinctive characteristics of our pricing data as it delivers a nearly perfectly normal price
distribution.
Finally, we use our model to explore the effects of customer markets for the dynamics of
the response of demand to shocks. In particular, our model represents an interesting setting
to study the response of demand to shocks that, by hitting firms asymmetrically, affect
price dispersion. This, in fact, influences the incentives of customers to search, generating
persistent dynamics in demand, as it takes time for customers to reallocate across firms. In
our model demand responds to variation in prices through two channels. The first one is
the intensive margin: Customers can adjust the quantity of the good they purchase from
their supplier. This channel is static and its adjustment happens immediately. The second
channel is the extensive margin: Customers can decide to leave the firm. This channel is
dynamic because of the presence of a search friction. Therefore, the adjustment is delayed, as
it takes time for customers to relocate to firms with lower prices, and persistent because once
customers have moved, it is costly for them to change firm again. The shape of the response
of demand depends on the strength of these two components. This is a quantitative question
that we explore by simulating a persistent cost shock that affects a subset of the firms.3 We
3
This is far from an abstract setup: It captures the main features of an exchange rate shock (which affects
firms’ costs differently according to whether or not they buy inputs on the international market) or a change
in the state sales tax (which penalizes and benefits competing firms on the basis or where their headquarters
are established).
3
find that the extensive margin dominates: The long-run response of demand is substantially
larger than the short-run one. In the counterfactual economy, where the absence of customer
markets implies that only the intensive margin is at work, the response of demand on impact
of the same shock is much larger and decays monotonically as the effect of the shock on prices
vanishes.
Related Literature. Our paper relates to the seminal work by Phelps and Winter (1970)
who study the pricing problem of the firm facing customer retention concerns. In their paper,
the response of the firm’s customer base to a change in the firm’s price is modeled with an
ad hoc function. We instead endogenize customer dynamics in response to firms’ pricing as
the outcome of customers’ optimal search decisions. Fishman and Rob (2003), Alessandria
(2004), and Menzio (2007) also study the firm price-setting problem in models where search
costs prevent customers from freely moving to the lowest price supplier. These papers focus
on different issues. Fishman and Rob (2003) study the implications of customer markets for
firm dynamics. Alessandria (2004) shows that such a model can generate large and persistent
deviations from the law of one price, consistent with the empirical evidence on international
prices. Menzio (2007) looks at the role of asymmetric information and commitment in the
optimal pricing decision of the firm. Differently from our paper, in these papers customers
face an homogeneous search cost and, as a result, optimal pricing is such that no endogenous
customer dynamics occur in equilibrium.
Unlike the literature cited above, we exploit scanner data to discipline our model and
provide a quantitative assessment of the relevance of customer markets for pricing. In documenting the shape of the price distribution, our paper also relates to the recent empirical
work by Kaplan and Menzio (forthcoming). While their focus is on customers and the price
they pay for the same good (or bundle of goods), we are interested in the point of view of
sellers and the price they charge. Our finding on the dynamics of demand relates to studies
documenting the short-run and long-run elasticity of demand. In particular, it is consistent
with evidence on the short- vs. long-run Armington elasticity in the international economics
literature (Ruhl (2008)).
A related set of contributions use customer markets to address questions different from the
ones we study here. Gourio and Rudanko (2014) explore the relationship between the firm’s
effort to capture customers and its performance. They show that customer markets have
nontrivial implications for the relationship between investment and Tobin’s q. Drozd and
Nosal (2012) introduce in a standard international real business cycle model the notion that,
when they want to increase sales, producers must exert effort to find new customers. This
extension help to rationalize a number of empirical findings on the dynamics of international
4
prices and trade. Dinlersoz and Yorukoglu (2012) focus on the importance of customer markets for industry dynamics in a model where firms use advertising to disseminate information
to uninformed customers. Shi (2011) studies a model where firms cannot price discriminate
across customers and use sales to attract new customers. Kleshchelski and Vincent (2009)
examine the impact of customer markets on the pass-through of idiosyncratic cost shocks to
prices in a symmetric equilibrium that does not allow us to study the relationship between
customer dynamics and the price distribution. Burdett and Coles (1997) study the role of
firm size for pricing when firms use the price to attract new customers. Their work complements ours: Price and customer dynamics in their setting are shaped by the heterogeneity
in firm size (age). For us, the driving force is the heterogeneity in productivity.
Finally, a stream of studies analyzes the implications of product market frictions for business cycle fluctuations (Petrosky-Nadeau and Wasmer (2011), Bai et al. (2012) and Kaplan
and Menzio (2013)). In these papers, aggregate shocks alter the opportunity cost of searching, influencing markup and demand dynamics over the business cycle. While we abstain
from this type of analysis, our quantified model could be extended to allow for the search
opportunity-cost to vary with the aggregate state and be used as a laboratory to answer
similar questions.
The rest of the paper is organized as follows. In Section 2 we lay out the model, and
in Section 3 we characterize the equilibrium. Section 4 presents the data and descriptive
evidence of the relationship between customer dynamics and prices. In Section 5 we discuss identification and estimation of the model. In Section 6 we present some quantitative
predictions of the model, contrast them with the outcomes from a model without customer
markets, and compare them empirical evidence from our data. In Section 7 we introduce an
application of the model, which we use to study the implications of customer markets for the
dynamics of demand. Section 8 concludes.
2
The model
The economy is populated by a measure one of firms producing an homogeneous good and a
measure Γ of customers who consume it.
Customers. We use the index i to denote customers. Let d(p) and v(p) denote, respectively, the static demand and customer surplus as a function of the price p of the good. We
assume that: (i) d(p) is continuously differentiable with d0 (p) < 0, and bounded below with
limp→∞ d(p) = 0; and (ii) v(p) is continuously differentiable with v 0 (p) < 0, and bounded
above with limp→0+ v(p) < ∞. These properties are satisfied in standard models of consumer
5
demand.
Firms. Firms produce a homogenous good and are indexed by j. The only choice firms
make is to set the price p of the good they produce each period. We assume a linear production technology y j = z j `j where ` is the production input, and z j is the firm-specific
productivity. Idiosyncratic productivity is distributed according to a conditional cumulative
distribution function F (z 0 |z) with bounded support [z, z̄]. We also assume that F (z 0 | zh )
first order stochastically dominates F (z 0 | zl ) for any zh > zl to induce persistence in firm
productivity. Heterogeneity in firm productivity will be the driver of price dispersion in the
type of equilibrium we will focus on.4 The profit per customer accrued to the firms are
π(p, z) ≡ d(p)(p − w/z), where the constant w > 0 denotes the marginal cost of the input
`. We assume that profits per customer are single-peaked in p. Finally, we denote by mjt−1
the customer base of firm j, consisting of the mass of customers who bought from firm j in
period t − 1. As we will show later, the state of the firm j in period t is the pair {ztj , mjt−1 }.
Search, matching, and exit from the customer base. Each customer starts period
t matched to the firm she bought from in period t − 1. The customer observes perfectly
the state of the firm she is matched to (i.e. ztj and mjt−1 ), which allows her to assess the
probability distribution of the path of prices of that firm. Later we will be more specific
about the mapping from the state of the firm to the path of prices. After observing the
state of her current match, the customer decides if she is incurring a search cost and draws
another firm. In particular, each customer i is characterized by an idiosyncratic random
search cost ψ i ≥ 0 measured in units of customer surplus, which is drawn each period from
the same distribution with density g(ψ), and associated cumulative distribution function
denoted by G(ψ). For tractability, we restrict our attention to density functions that are
continuous on all the support. Heterogeneity, albeit transitory, in search costs allows us
to study firms’ pricing decisions that are not necessarily knife-edge in the trade-off between
maximizing demand and markups. The customer can search at most once per period. Search
is random, with the probability of drawing a particular firm j 0 being proportional to its
0
customer base, i.e. mjt−1 /Γ. This assumption captures the idea that consumers search new
suppliers not by randomly sampling firms but by randomly sampling other consumers and
following their behavior.5 On the technical side, this assumption implies that firms will
gain customers proportionally to their customer base. This simplifies the characterization of
the firm problem and implies that firm growth is independent of firm size consistent with
4
For tractability we will abstract from (possible) equilibria where symmetric firms charge different prices
as in Burdett and Judd (1983).
5
This behavior is known as preferential attachment in the extensive literature on network formation.
6
Gibrat’s law (Steindl (1965)). Conditional on searching, the customer takes then another
decision concerning whether to exit the customer base of her initial firm and match to the
new firm. In particular, the customer compares the distribution of the path of current and
future prices at the two firms and buys from the firm offering higher expected value. Finally,
we assume for simplicity no recall in the sense that the customer cannot go back to a firm
she was matched with in the past or whose price offer she rejected unless she randomly draws
it again when searching.
Timing of events. The timing of events is as follows: (i) productivity shocks are realized
for all firms and each firm j posts a price pjt ; (ii) each customer draws her search cost ψti and
observes the price pjt as well as the relevant state of the firm she is matched with (ztj and
mjt−1 ); (iii) each customer decides whether to search for a new firm or remain matched to her
current one; (iv) if the customer decides to search, she pays the search cost and draws a new
0
supplier j 0 with probability mjt−1 /Γ. The customer perfectly observes not only the price but
also the productivity and customer base of the prospective match and decides whether to
exit the customer base of the current supplier to join that of the new match or to stay with
the current match. Finally, (v) customer surplus v(pjt ) and firm profits mjt π(pjt , ztj ) realize.
Equilibrium. A firm and its customers play an anonymous sequential game. We look
for a stationary Markov Perfect equilibrium where strategies are a function of the current
state. There are no aggregate shocks. Although the relevant state for the pricing decision
of the firm could include both the stock of customers and the idiosyncratic productivity, we
conjecture and later show the existence of an equilibrium where optimal prices only depend
on productivity, and we denote by P(z) the equilibrium pricing strategy of the firm.
The relevant state for the search decision of a customer includes the expectations about
the path of current and future prices of the firm she is matched to, as well as the idiosyncratic search cost. Given the Markovian equilibrium we study, the current realization of
idiosyncratic productivity is a sufficient statistic for the distribution of future prices. As a
result, the search strategy of the customer depends on the current price and productivity
of the firm she is matched to, and on her own search cost. We denote the search decision
as s(p, z, ψ) ∈ {0, 1}, where s = 1 means that the customer decides to engage in search.
Conditional on searching, the exit decision depends on the continuation value associated to
the firm the customer starts matched to (the outside option), which is fully characterized by
posted price and productivity, as well as on productivity of the firm she has drawn upon the
search, z new , which determines the continuation value associated to the new firm. We denote
the exit decision as e(p, z, z new ) ∈ {0, 1}, where e = 1 means that the customer decides to
7
exit the customer base of her original firm.
2.1
The problem of the customer
Let V (pjt , ztj , ψti ) denote the value function of a customer i who has drawn a search cost ψti
and is matched to firm j, which has current productivity ztj and posted price pjt . This value
function solves the following problem,
V
(pjt , ztj , ψti )
n
o
j
j
j
j
i
= max V̄ (pt , zt ) , Ṽ (pt , zt ) − ψt ,
(1)
where V̄ (p, z) is the customer’s value if she does not search, and Ṽ (p, z) − ψ is the value if
she does search. The value in the case of not searching is
V̄
(pjt , ztj )
=
v(pjt )
∞
Z
Z
+β
0
z̄
V (P(z 0 ), z 0 , ψ 0 ) dF (z 0 |ztj ) dG(ψ 0 ) .
(2)
z
We notice that the state of the firm problem depends on the on the productivity z because
the pricing function P(·) mapping future productivity into prices in the Markov equilibrium
makes productivity z a perfect statistic for the distribution of future prices at the firm.6 The
value when searching is given by
Ṽ
(pjt , ztj )
Z
+∞
=
max V̄ (pjt , ztj ) , x dH(x) ,
(3)
−∞
where the customer takes expectations over all possible draws of potential new firms, and
where H(·) is the equilibrium cumulative distribution of continuation values from which
the firm draws a new potential match when searching. For instance, H(V̄ (pjt , ztj )) is the
probability of drawing a potential match offering a continuation value smaller than or equal
to the current match. The following lemma describes the customer’s optimal search and exit
policy rules.
Lemma 1 The customer matched to a firm with productivity ztj charging price pjt : i) searches
R∞
if she draws a search cost ψt ≤ ψ̂(pjt , ztj ), where ψ̂(p, z) ≡ V̄ (p,z) x − V̄ (p, z) dH(x) ≥ 0 is
the threshold to search; ii) conditional on searching, exits if she draws a new firm promising
a continuation value V̄ new larger than the current match, i.e. V̄ new ≥ V̄ (pjt , ztj ).
6
We also notice that the state of the firm problem includes the current price p, despite in equilibrium
productivity is enough to determine the current price, as this notation is needed to study the game between
the firm and its customers where the firm could, in principle, deviate from the equilibrium price.
8
The proof of the lemma follows immediately from equations (1)-(3). The lemma states
that, as search is costly, not all customers currently matched to a given firm exercise the
search option, only those with a low search cost do so. Notice that the threshold ψ̂(p, z)
depends both on the price of the firm, p, and its productivity, z. The dependence on the
price is straightforward, following from its effect on the surplus v(p) that the customer can
attain in the current period. The intuition behind the dependence on the firm’s productivity
is that, as searching is a costly activity, the decision of which firm to patronize is a dynamic
one, and involves comparing the value of remaining in the customer base of the current firm
with the value of searching. Because of the Markovian structure of prices, the customer’s
expectation about future prices is completely determined by the firm’s current productivity.
The next lemma discusses some properties of the continuation value function V̄ (p, z) and,
as a consequence, of the threshold ψ̂(p, z).
Lemma 2 The value function V̄ (p, z) (the threshold ψ̂(p, z)) is strictly decreasing (increasing) in p. If V̂ (z) ≡ V̄ (P(z), z) is increasing in z, the value function V̄ (p, z) (the threshold
ψ̂(p, z)) is increasing (decreasing) in z.
The proof of Lemma 2 is in Appendix A.1. The lemma states that customers obtain
strictly higher value from firms offering a lower current price and, if V̂ (z) is increasing in z,
also from firms characterized by higher current productivity. Notice that, under persistence
in the productivity process, a sufficient condition for the latter is that equilibrium prices are
decreasing in productivity. As a result, not only are customers more likely to search and
exit from firms charging higher prices, but also they are more likely to do so from firms with
lower productivity if V̂ (z) is increasing in z.
2.2
The problem of the firm
In this section we describe the pricing problem of the firm. We start by discussing the
dynamics of the customer base as a function of price and productivity, given the optimal
search and exit strategy of the customers. Then, we move to the characterization of the firm
optimal pricing strategy.
The customer base of firm j evolves as follows:
mj
mjt = mjt−1 − mjt−1 G ψ̂(pjt , ztj ) 1 − H(V̄ (pjt , ztj )) + t−1 Q V̄ (pjt , ztj ) ,
|
{z
} |Γ
{z
}
customers outflow
(4)
customers inflow
where G(ψ̂(pjt , ztj )) is the fraction of customers searching firm j’ customer base, a fraction
1 − H(V̄ (pjt , ztj )) of which actually finds a better match and exits the customer base of
9
firm j. The ratio mjt−1 /Γ is the probability that searching customers in the whole economy
draw firm j as a potential match. The function Q(V̄ (pjt , ztj )) denotes the equilibrium mass
of searching customers currently matched to a firm with continuation value smaller than
V̄ (pjt , ztj ). Therefore, the product of the two amounts to the mass of new customers entering
the customer base of firm j. We can express the dynamics of the customer base as mjt =
mjt−1 ∆(pjt , ztj ), where the function ∆(·) denotes the growth of the customer base and is given
by
∆(p, z) ≡ 1 − G ψ̂(p, z) 1 − H(V̄ (p, z)) +
1
Γ
Q V̄ (p, z) .
(5)
Notice that the growth of a firm is independent of its customer base and, therefore, of its
size. This result is known as Gibrat’s Law and is consistent with existing empirical evidence
on the distribution of firms’ size (see Luttmer (2010)), and depends on our assumption that
customers draw new firms with probability proportional to their share of customers. The
next lemma discusses the properties of the customer base growth with respect to prices and
productivity.
Lemma 3 Let p̄(z) solve V̄ (p̄(z), z) = maxz {V̄ (P(z), z)}; ∆(p, z) is strictly decreasing in p
for all p > p̄(z), and constant for all p ≤ p̄(z). If V̂ (z) ≡ V̄ (P(z), z) is increasing in z, then
∆(p, z) is increasing in z.
The proof of Lemma 3 follows directly from Lemma 2. The growth of the customer base
is decreasing in the current price because a higher price reduces the current surplus and
therefore the value of staying matched to the firm. When the price is low enough that no
firm in the economy offers a higher value to the customer, the customer base is maximized and
a further decrease in the price has no impact on the customer growth. If V̂ (z) is increasing in
z, the growth of the customer base increases with firm productivity, as a larger z is associated
to higher continuation value which increases the value of staying matched to the firm.
We next discuss the pricing problem of the firm. The firm pricing problem in recursive
form solves
Z z̄
j
j
j
j
W̃ (z 0 , mjt ) dF (z 0 | zt ) ,
W̃ (zt , mt−1 ) = max mt π(p, zt ) + β
p
z
subject to equation (4), where W̃ (ztj , mjt−1 ) denotes the firm value at the optimal price and
π(p, ztj ) = d(p) (p − w/ztj ) are profits per customer. We study equilibria where the pricing
decision of the firm only depends on productivity. Thus, we conjecture that in this equilibrium
the value function of a firm is homogeneous of degree one in m, i.e., W̃ (z, m) = m W̃ (z, 1) ≡
10
m W (z). W (z) solves
Z
W (z) = max ∆(p, z)
π(p, z) + β
p
!
z̄
W (z 0 )dF (z 0 | z)
,
(6)
z
|
{z
}
present discounted value of a customer ≡ Π(p,z)
and where we used equation (4) and we dropped time and firm indexes to ease the notation.
We assume that the discount rate β is low enough so that the maximization operator in
equation (6) is a contraction. Therefore; by the contraction mapping theorem we can conclude
that our conjecture about homogeneity of W̃ (z, m) is verified.
We can express the objective of the firm maximization problem as the product of two
terms. The first term is the growth in the customer base, ∆(p, z), which according to Lemma 3
is strictly decreasing in the price for all p > p̄(z) and is maximized at any price p ≤ p̄(z). The
second term is the expected present discounted value of each customer to the firm, which we
denote by Π(p, z). The function Π(p, z) is maximized at the static profit maximizing price,
p∗ (z) ≡
εd (p) w
.
εd (p) − 1 z
(7)
It follows that setting a price above the static profit maximizing price is never optimal.
Moreover, if p̄(z) ≤ p∗ (z), the optimal price will not be below p̄(z), because in that region
profit per customer increase with the price but the customer base is unaffected. Hence,
p̂(z) ∈ [p̄(z), p∗ (z)]. If instead p̄(z) ≥ p∗ (z), then the optimal price is the static profit
maximizing price, p̂(z) = p∗ (z), as at this price both the customer base and the profits per
customer are maximized. The following proposition collects these results.
Proposition 1 Let p̄(z) solve V̄ (p̄(z), z) = maxz∈[z,z̄] {V̄ (P(z), z)}, and let p∗ (z) expressed
in equation (7) be the price that maximizes static profits. Denote by p̂(z) a price that solves the
firm problem in equation (6). We have p̂(z) ∈ [p̄(z), p∗ (z)) if p̄(z) < p∗ (z), and p̂(z) = p∗ (z)
otherwise.
A proof of the proposition can be found in Appendix A.2.
3
Equilibrium
In this section we define an equilibrium, discuss its existence, and characterize its general
properties. We start by defining the type of equilibrium we study.
11
Definition 1 Let V̂ (z) ≡ V̄ (P(z), z) and p∗ (z) be given by equation (7). We study stationary
Markovian equilibria where V̂ (z) is non-decreasing in z, and for all z ∈ [z, z̄] the firm pricing
strategy p̂(z) solves the first order condition to the firm problem in equation (6) given by
p
p ∂∆(p, z)
∂Π(p, z)
=−
≥0.
∂p
Π(p, z)
∆(p, z) ∂p
(8)
A stationary equilibrium is then
(i) a search and an exit strategy that solve the customer problem for given equilibrium
pricing strategy P(z), as defined in Lemma 1;
(ii) a firm pricing strategy p̂(z) that solves equation (8) for each z, given customers’ strategies and equilibrium pricing policy P(z), and is such that p̂(z) = P(z) for each z;
(iii) two distributions over the continuation values to the customers, H(x) and Q(x), that
R ẑ(x)
solve H(x) = K(ẑ(x)) and Q(x) = Γ z G(ψ̂(p̂(z), z)) dK(z) for each x ∈ [V̂ (z), V̂ (z̄)],
where ẑ(x) = max{z ∈ [z, z̄] : V̂ (z) ≤ x}, and K(z) solves
Z
z
Z
K(z) =
z̄
∆(p̂(x), x) dF (s|x) dK(x) ds ,
z
(9)
z
for each z ∈ [z, z̄] with boundary condition
R z̄
z
dK(x) = 1.
We study equilibria where the continuation value to customers is non-decreasing in productivity, implying that customers’ rank of firms coincides with their productivity. This is
a natural outcome as more productive firms are better positioned to offer lower prices and
therefore higher values to customers. The first order condition in equation (8) illustrates
the trade-off the firm faces when setting the price in a region where customer retention is a
concern. When p̄(z) < p∗ (z), the optimal price balances the marginal benefit of an increase
in price (more profit per customer) with the cost (decrease in the customer base). The requirement that the solution to the firm problem must satisfy the first order condition implies
that we study equilibria where the firm objective, and in particular ∆(p, z), is differentiable
in p. The next proposition states conditions under which such an equilibrium exists and
characterizes its properties.
Proposition 2 Let productivity be i.i.d. with F (z 0 |z1 ) = F (z 0 |z2 ) continuous and differentiable for any z 0 and any pair (z1 , z2 ) ∈ [z, z̄], and let G(ψ) be differentiable for all ψ ∈ [0, ∞),
with G(·) differentiable and not degenerate at ψ = 0. There exists an equilibrium as defined
in Definition 1 where p̂(z) satisfies equation (8), and
12
(i) p̂(z) is strictly decreasing in z, with p̂(z̄) = p∗ (z̄) and p̂(z̄) < p̂(z) < p∗ (z) for z < z̄,
implying that V̂ (z) is strictly increasing. Moreover, the optimal markups are given by
µ(p, z) ≡
p
εd (p)
=
,
w/z
εd (p) − 1 + εm (p, z) Π(p, z)/(d(p) p)
(10)
where εd (p) ≡ ∂ log(d(p))/∂ log(p), εm (p, z) ≡ ∂ log(∆(p, z))/∂ log(p), and p = p̂(z) for
each z.
(ii) ψ̂(p̂(z), z) is strictly increasing in z, with ψ̂(p̂(z̄), z̄) = 0 and ψ̂(p̂(z), z) > 0 for z < z̄,
implying that ∆(p̂(z), z) is strictly increasing, with ∆(p̂(z̄), z̄) > 1 and ∆(p̂(z), z) < 1.
Differentiability of the distribution of productivity F is not needed for the existence of an
equilibrium. We assume it to ensure that H(·) and Q(·) are almost everywhere differentiable
so that equation (8) is a necessary condition for optimal prices. However, even when F is not
differentiable and the first order condition cannot be used to characterize the equilibrium,
an equilibrium with the properties of Proposition 2 exists where p̂(z) and ψ̂(p̂(z), z) are
monotonic in z but not necessarily strictly monotonic for all z. Monotonicity of optimal
prices follows from an application of Topkis’ theorem. In order to apply the theorem to the
firm problem in equation (6) we need to establish increasing differences of the firm objective
∆(p, z) Π(p, z) in (p, −z). Under the standard assumptions we stated on π(p, z), it is easy to
show that Π(p, z) satisfies this property. The customer base growth does not in general verify
the increasing difference property. However, under the assumption of i.i.d. productivity,
∆(p, z) is independent of z, which, together with Lemma 3, is sufficient to obtain the result.
Finally, while the results of Proposition 2 refer to the case of i.i.d. productivity shocks,
numerical results in Section 6 show the properties of Proposition 2 extend to the case of a
persistent productivity process. More details on the proof of the proposition can be found in
Appendix A.3.
We now comment on the properties of the equilibrium highlighted in the Proposition. The
equilibrium is characterized by price dispersion: More productive firms charge lower prices
and, therefore, offer higher continuation value to customers. This is important, as price
dispersion is what motivates customers to search for lower prices. As in Reinganum (1979),
price dispersion hinges on the fact that there is heterogeneity in productivity. If all the firms
had the same productivity, Proposition 2 would imply a unique equilibrium where the price
is that which maximizes static profits, p∗ (z̃), and as a result the customer base of every firm
would be constant.7 The equilibrium is also characterized by dispersion in customer base
7
This special case is useful to understand our relation to Diamond (1971), which shows in a simple
search model that the resulting equilibrium when search cost is positive exhibits no price dispersion and
13
growth: More productive firms grow faster, and there is a positive mass of lower productivity
firms that have a shrinking customer base and a positive mass of higher productivity firms
that are expanding their customer base.
Optimal markups in equation (10) depend on three distinct terms: εd (p), εm (p, z), and
x(p, z) ≡ Π(p, z)/(d(p) p). The terms εd (p) and εm (p, z) represent the price elasticities of
quantity purchased (per-customer) and of customer growth, respectively. An increase in
price reduces total current demand both because it reduces quantity per customer (intensive
margin effect) and because it reduces the number of customers (extensive margin effect).
Moreover, the optimal markup solves a dynamic problem as a loss in customers has persistent
consequences for future demand due to the inertia in the customer base. This dynamic effect is
captured by the term x(p, z), which measures the firm present discounted value of a customer
scaled by the current revenues. It follows that active customer markets are associated with
a strictly lower markup than the one that maximizes static profit; the lower, the larger the
product εm (p, z) x(p, z).
The importance of the dynamic effect on optimal markups can be better understood
performing the following thought experiment. Define the overall demand elasticity of an
economy as the sum of its quantity elasticity and its customer growth elasticity: εq (p, z) ≡
εd (p)+εm (p, z). Take two firms characterized by the same productivity z and the same overall
demand elasticity εq (·), but by different combinations of εd (·) and εm (·). In particular, one
firm has lower quantity elasticity but higher customer growth elasticity than the other. Then
the optimal markup for the former is strictly lower than that for the latter.8 Intuitively, a
loss in demand associated to a loss in customers is a persistent loss and, therefore, has a
larger impact on firm value, inducing it to charge lower markups with respect to a firm that
operates in a market where the customer base is less elastic.
Finally, the next remark explores two interesting limiting cases of our model and showcases
the effect of the search friction on price dispersion.
Remark 1 Let search costs be scaled as n ψ, where n > 0. That is, let the value function in
equation (1) be
n
o
V (p, z, ψ) = max V̄ (p, z) , Ṽ (p, z) − nψ .
Two limiting cases of the equilibrium stated in Definition 1:
(1) Let n → ∞. Then, in equilibrium: (i) the optimal price maximizes static profits, i.e.
p̂(z) = p∗ (z) for all z ∈ [z, z̄], and (ii) there is no search in equilibrium. Furthermore,
firms behaving as monopolists. Our model delivers a different outcome because we allow firms to differ in
idiosyncratic productivity but can generate the Diamond (1971) results if heterogeneity in producvtivity is
shut down.
8
More details are available in Appendix A.4.
14
the equilibrium is unique.
(2) Let π(p∗ (z̄), z) > 0 and let the assumptions of Proposition 2 be satisfied. Then, p̂(z̄) =
p∗ (z̄) and max{p̂(z)} = p̂(z) approaches p∗ (z̄) as n → 0. As a result, in the limit, there
is no price dispersion in equilibrium and customers do not search.
A proof of the remark can be found in Appendix A.5. The first limiting case explores the
resulting equilibrium when we let search costs diverge to infinity. The model then reduces to
one where customer base concerns are not present. Because the customer base is unresponsive
to prices, the firm problem reverts to the standard price-setting problem under monopolistic
competition widely explored in the macroeconomics literature: The firm sets the price p,
taking into account only its impact on static demand d(p). Not surprisingly, the equilibrium
is unique, optimal prices maximize static profits, i.e. p̂(z) = p∗ (z) for all z ∈ [z, z̄], there is
price dispersion, and there is no search in equilibrium. The second limiting case explores the
resulting equilibrium when search costs become arbitrarily small. We restrict attention to
the model that satisfies the assumptions of Proposition 2, so that the first order condition
presented in equation (8) is necessary for optimality.9 In this case, as the scale of search
costs becomes arbitrarily small, equilibrium prices approach the lowest price in the economy,
p∗ (z̄). As a result, there is no price dispersion and customers do not search.
4
Data and descriptive evidence
We complement the theoretical analysis with an empirical investigation that relies on cashier
register data from a large U.S. supermarket chain. The empirical analysis has two purposes.
In this section we provide descriptive evidence that the price posted by the firm influences
customers’ decision to exit the customer base and measure the size of this effect. In Section 5,
we will use the data to estimate our model and quantify the importance of customer markets
in shaping firm price setting.
4.1
Data sources and variable construction
The supermarket chain shared with us scanner data detailing purchases by a panel of households carrying a loyalty card of the chain.10 The chain operates over a thousand stores across
The assumption π(p∗ (z̄), z) > 0 is purely technical, and it ensures that the first derivative of the profit
function is bounded in the relevant range.
10
The chain is able to associate the loyalty cards belonging to different members of a same family to a
single household identifying number, which is the unit of observation in our data. Therefore, in the analysis
we use the terms “customer” and “household” interchangeably.
9
15
10 states, and the data reflect this geographical dispersion. For every trip made at the chain
by customers in the sample between June 2004 and June 2006, we have information on the
date of the trip, store visited, and list of goods (identified by their Universal Product Code,
UPC) purchased, as well as quantity and price paid.
The population of shoppers visiting a retail store in a given time period can be divided
into regular and occasional customers. Regular customers shop routinely, at least for some
period of time, at the same store. Occasional customers are instead agents who typically
shop elsewhere and visit the chain for convenience, for instance because they happened to be
in the vicinity. To obtain an estimate of the extensive margin elasticity, we want to restrict
attention to regular customers who are the ones the firm is trying to retain.11 Luckily, our
data are ideal to separate regular and occasional customers as they only include information
on households carrying a loyalty card of the chain. The willingness to sign up for the loyalty
card signals some form of commitment of these households to the chain, which random
shoppers are unlikely to bother with. The customers in our sample make an average of 150
shopping trips at the chain over the two years; if those trips were uniformly distributed, that
would imply visiting a store of the chain six times per month. The average expenditure per
trip is $69 for the average household. There is a great deal of variation (the 10th percentile
is $29; the 90th is $118) explained, among other things, by income and family size of the
different households.
In the theoretical model we studied the behavior of customers buying from firms producing
a single homogeneous good; our application documents the exit decisions of customers from
supermarket stores where they buy bundles of goods.12 This can be reconciled if we assume
that customers’ behavior depends on the price of the basket of goods they typically buy at
the supermarket.13 Although the multiproduct nature of the problem may have implications
for the pricing decision of the firm, we abstract from the price-setting process and focus only
on the resulting price index of the customer basket in order to retrieve the response of the
customer base to variation in prices. In particular, to measure the comovement between
the customer’s decision to exit the customer base and the price of her typical basket of
goods posted at the chain, we need to construct two key variables: (i) an indicator signaling
when the household is exiting the chain’s customer base, and (ii) the price of the household
11
Occasional customers’ behavior does not contribute to the estimation of the extensive margin elasticity
and may, in fact, confound it. These customers could instead be useful if we wanted to relax our assumptions
on the customer attraction process.
12
The choice of focusing on the customer base of the store rather than that of one of the branded product
it sells is data driven. With data from a single chain we cannot track the evolution of the customer base of a
single brand. In fact, if we observed customers not longer buying a particular brand we could only infer that
they are not buying it at the chain we analyze, but we could not exclude they are buying it elsewhere.
13
Note that since customers baskets are in large majority composed of package goods, which are standardized products, the assumption that the basket is a homogenous good is not unwarranted.
16
basket. Below we briefly describe the procedure followed to obtain them; the details are left
to Appendix B.
We consider every customer shopping at the retailer in a given week as belonging to the
chain’s customer base in that week. We assume that a household has exited the customer
base when she has not shopped at the chain for eight or more consecutive weeks, and we
assume that the decision to exit occurred the last time the customer visited the chain. The
eight-week window is a conservative choice given the shopping frequency of households in our
sample.14 Regular customers are unlikely to experience a eight-week spell without shopping
for reasons other than having switched to another chain (e.g. consuming their inventory). In
fact, on average, four days elapse between consecutive trips and the 99th percentile of this
statistic is 28 days, half the length of the absence we require before inferring that a household
is buying its groceries at a competing chain.
We construct the price of the basket of grocery goods usually purchased by the households
in the following fashion. We identify the goods belonging to a household’s basket using
scanner data on items the household purchased over the two years in the sample. In a
particular week t, the price paid by customer i, shopping at store j and for its basket,
represented by the collection of UPC’s in Ki is
pit =
X
P
ωik pkt ,
ωik = P
Eikt
tP
k∈Ki
k∈Ki
t
Eikt
,
(11)
where pkt is the price of UPC k at the store customer i is matched to in week t and Eikt
is the expenditure (in dollars) by customer i in UPC k in week t. Note that the price of
the basket is household specific because households differ in their choice of grocery products
(Ki ) and in the weight such goods have in their budget (ωik ). We face the common problem
that household scanner data only contain information on prices and quantities of UPCs when
they are actually purchased. Therefore, we complement them with store level data on weekly
revenues and quantities sold.15 This data allows us to back out weekly prices of each UPC in
the sample by dividing total revenues by total quantity sold as in Eichenbaum et al. (2011).
14
We experimented with four weeks and 12 weeks as alternative lengths of the period of absence required
to infer the exit from the customer base. In both cases the results are qualitatively similar. However, in the
12-week case the number of exit events becomes so small that we do not have the power to detect significant
effects.
15
The retailer changes the price of the UPCs at most once per week, hence we only need to construct
weekly prices to capture the entire time variation.
17
4.2
Evidence on customer base dynamics
The availability of individual level data allows us to study the determinants of a customer’s
decision to exit the customer base of the firm she is currently shopping from. We estimate
a linear probability model where the dependent variable is an indicator for whether the
household has left the customer base of the chain in a particular week. Our aim is to capture
the effect of the price posted by the chain for the basket of goods purchased by the customer
on her decision to exit. To isolate this magnitude in a way consistent with the mechanism
described in the theoretical model, we need to include a series of controls.
We are interested in the effect of price variation induced by cost shifts idiosyncratic to
a firm. Aggregate cost shocks do not change the relative price and, therefore, should not
trigger exit from the customer base. Furthermore, our retailer is a major player in the markets
included in our sample and it is likely that the competition takes its prices into consideration
when deciding on their own. This possibly introduces correlation between price variations at
the chain and price variations at the alternative outlets the customer may visit. To identify
idiosyncratic price variations, we control for the prices posted by the competitors of the chain
using the IRI Marketing data set. This source includes weekly UPC’s prices for 30 major
product categories for a representative sample of chain stores across 64 markets in the United
States.16 Thanks to this data, we can observe the weekly price of a specific UPC at every
chainstore sampled by IRI in the Metropolitan Statistical Area of residence of a customer.
We can, therefore, construct the price of the basket bought by the customer at each store
in the MSA (at least for the part of it falling into product categories sampled by IRI) in
the same fashion described for the price of the basket at our chain. We take the average
of such prices across all stores, weighted by market share, to compute the average market
price of the basket (pmkt ). To further control for sources of aggregate variation, we include
in the regression year-week fixed effects to account for time-varying drivers of the decision
of exiting the customer base common across households (e.g., disappearances due to travel
during holiday season).
The coefficient on the retailer price of the basket is identified by UPC-chain specific
shocks as those triggered, for example, by the expiration of a contract between the chain
and a manufacturer of a UPC. Within the chain, the price of a same good moves differently
in different stores. This can be due, for instance, to variation in the cost of supplying the
store due to logistics (e.g. distance from the warehouse), which will hit differently goods with
different intensity in delivery cost (e.g. refrigerated vs. nonrefrigerated goods). Therefore,
16
A detailed description of the data can be found in Bronnenberg et al. (2008). All estimates and analyses
in this paper based on Information Resources Inc. data are by the authors and not by Information Resources
Inc.
18
we can also exploit variation in our data from UPC-store specific shocks for identification. It
is worth stressing that to trigger an exit from the customer base we do not need to observe
shocks that make a supermarket uniformly more expensive than the competition. Rather, it
is enough that they change the comparative advantage of a chain with respect to a subset
of goods on sale. If a chain becomes more expensive in some goods (even though it may
become cheaper for other goods), this is enough to induce the customers who particularly
care about those goods to leave. Kaplan and Menzio (forthcoming) use independent scanner
data to provide ample evidence for this source of variation. They report that the bulk of
price dispersion arises not from the difference from high-price and low-price stores but from
dispersion in the price of a particular good (or product category) even among stores with
similar overall price level.
Another set of regressors is meant to acknowledge that, unlike posited in the model, customers are heterogenous in more dimensions than their cost to search. We start by controlling
for customer heterogeneity, including observable characteristics (age, income, and education)
matched from Census 2000. We also consider location as a potential driver of the decision to
exit. We control for the size of the set of potential store choices by including the number of
supermarket stores in the zip code of residence of the customer and factor her convenience
in shopping by calculating the distance in miles between her residence and the closest store
of the chain and the closest alternative supermarket. To pick up the heterogeneity in the
types of goods different customers include in their basket, we control for the price volatility
of the customer-specific basket and for its price in the first week in the sample, as a scaling
factor. Finally, we calculate customer tenure, defined as the number of consecutive weeks
the customer has spent in the customer base of the chain, and include it in the regression
to account for the fact that long-term customers of the chain may be less willing to leave it
ceteris paribus.
In Table 1, we report results of regressions of the following form,
0
Exitit = b0 + b1 log(pit ) + b2 log(pmkt
it ) + b3 tenureit + Xi c + εit .
(12)
The retailer price in equations (12) would be endogenous if the chain conditioned the price to
unobserved (to the econometrician) variables that also influence the customer’s decision to
leave. Since the prices we include in the regression are customer specific, the firm would need
to acquire relevant information on each customer (market-level variables would not suffice)
and be willing to tweak the prices of their baskets in response to that.17 Nevertheless,
17
Note that if blockbuster goods, i.e. goods that account for a significant share of many consumers’ baskets,
exist, the chain may face a combinatorial problem and be unable to tune the prices of the individual baskets
at will. For instance, the chain may want to have the price of the basket rise for some customers and fall for
19
the retailer provided, along with the store price data, a measure of replacement cost as
in Eichenbaum et al. (2011). This represents a natural instrument so that we can estimate
equations (12) through instrumental variables and be protected against potential endogeneity
of price. We use the replacement cost measure to construct, for each customer in each week,
the cost of their basket, following the same procedure described in equation (11) to obtain
the price of the basket. The cost of the basket is then used as an instrument for the price of
the basket in all the specifications.
Table 1: Effect of the price of the basket on the probability of exiting the customer base
Exiting: Missing at least 8 consecutive weeks
(1)
(2)
(3)
(4)
log(p)
0.14**
-0.01
0.16*
0.15**
(0.071)
(0.030)
(0.089)
(0.070)
Walmart entry
0.019*
(0.011)
log(pmkt )
Tenure
Observations
0.001
0.000
0.001
(0.001)
(0.001)
(0.001)
-0.002***
-0.003***
-0.004***
-0.002***
(0.001)
(0.000)
(0.000)
(0.000)
52,670
52,670
66,182
52,101
Notes: An observation is a household-week pair. The results reported are calculated through two-stages least squares where
we use the logarithm of the cost of the basket (constructed based on the replacement cost provided for each UPC by the
retailer) as instrument for the logarithm of the price of the basket. In column (2), the price of the household basket is
substituted with a price index for the store overall. In column (4), the exit of the customer is attributed to the first week of
absence in the eight (or more) weeks spell without purchase at the chain instead that to the week of the last shopping trip
before the hiatus. We trim from the sample households in the top and bottom 1% in the distribution of the number of trips
over the two years. A series of variables are not reported for brevity: demographic controls, that rely on a random subsample
of households for which information on the block-group of residence was provided, including ethnicity, family status, age,
income, education, and time spent commuting (all matched from Census 2000) as well as distance from the closest outlet of
the supermarket chain and distance from the closest competing supermarket (provided by the retailer). The logarithm of the
price of the household basket in the first week in the sample and the standard deviation of changes in the log-price of the
household basket over the sample period are included as a controls in all specifications. Week-year fixed effects are also always
included. Robust standard errors are in parenthesis. ***: Significant at 1% **: Significant at 5% *: Significant at 10%.
The results are reported in Table 1. The main specification in column (1) shows that
the basket price posted by the retailer significantly impacts the probability of leaving. The
effect is also quantitatively important. The average probability of exiting the customer base
(0.3% weekly) implies a yearly turnover of 14%; if the retailer’s prices were 1% higher, its
others. However, if it raises the price of a blockbuster good, it may prove hard to undo this with “reasonable”
price decreases in other goods and obtain lower basket prices for some customers.
20
yearly turnover would jump to 21%. The coefficient on the competitors’ price, which we
would expect to enter with a negative sign, is not significant. This may be due to the fact
that the IRI data only allow us to imperfectly capture competitors’ behavior. First, the IRI
dataset contains price information only on a subset of the goods included in a customer’s
basket, although it arguably covers all the major product categories. Furthermore, we can
only match a customer with the stores in her Metropolitan Statistical Area of residence. The
set of stores the customer considers as alternative to the chain are probably located in a
much smaller area, which introduces measurement error. The negative coefficient on tenure
confirms the intuition that the longer the relationship between a firm and a customer, the
less likely they are to be interrupted. Among the several individual characteristics we control
for it is worth mentioning that distance from stores of the chain and distance from the closest
competing store enter with the expected sign. Customers living in proximity of a store of
the chain are less likely to leave it, and those living closer to competitors’ stores are more
inclined to do so.
In columns (2)-(4) we assess the robustness of these findings. We start by replacing the
price of the individual basket with a price index for the store basket, defined as the average
of the prices of the UPC’s sold by the store where the customer buys, weighted for their sales.
This price is, by construction, equal for all the customers shopping in the same store. Column
(2) shows that this results in a price coefficient that is negative and not significant. We take
this as evidence that the customer-specific basket price used in our main specification is a
meaningful object, whose significance stems from being able to capture the set of prices each
customer cares about and not just reflecting some aggregate trend in pricing. The fact that
the competitor price comes up as not significant may raise suspicion that the variable is too
noisy to control for the effect of competition. This is important because we are interested in
price movements driven by idiosyncratic firm shocks. To assess whether this is the case, we
experiment in column (3) with an alternative way to control for the effect of competition:
We exploit episodes of entry by Walmart, a major retailer with which our chain is in direct
competition. We use data from Holmes (2011) to identify the date of entry by a Walmart
supercenter-i.e. a store selling groceries on top of general discount goods-in a zip code where
our retailer also operates a supermarket. The resulting event study allows us to measure the
effect of the retailer price on the probability of exit controlling for the most relevant change
in the competitive environment. The estimated coefficient falls in the same ballpark as that
estimated in the main specification, which reassures on the effectiveness of the IRI price
in measuring the competitors’ behavior. In column (4), we change the assumption on the
imputation of the date of exit. Rather than assuming that the customer left on the occasion
21
of her last trip to the store, we posit that the exit occurred in the first week of her absence.18
Even in this case, the main result stays unaffected. Finally, we performed a placebo test
running 1,000 times the main specification, randomly assigning exits from the customer base
but keeping their total number the same as in the actual data. This exercise is meant to
assess the likelihood that the significance of our result is only due to a lucky occurrence. We
find that only in 2.8% of the cases the simulation yields and price coefficient are positive and
significant at 5%.
5
Parametrization and analysis of the model
In this section we discuss the procedure followed to estimate the model. We need to choose the
discount factor β as well as four functions: the demand function, d(p), the surplus function
v(p), the distribution of search costs G(ψ), and the conditional distribution of productivity
F (z 0 |z). Below we discuss the parametrization of the model in detail.
We assume that a period in the model corresponds to a week to mirror the frequency
of our data. We fix the firm discount rate to β = 0.995. In the set of parameters that we
consider, this level of β ensures that the max-operator in equation (6) is a contraction.19
We assume that customers have logarithmic utility in consumption. Consumption is
θ
θ−1
θ−1
θ−1
θ
θ
+n
, with θ > 1.20 The first
defined as a composite of two types of goods c ≡ d
good (that we label d) is supplied by firms facing product market frictions as described in
Section 2.2; the other good (n) acts as a numeraire and it is sold in a frictionless centralized
market. The sole purpose of good n is to microfound a downward sloping demand d(p) and,
therefore, allowing for an intensive margin of demand. The parameter θ is chosen so that
the implied average intensive margin elasticity of demand (εd (p)) is 7, a value in the range of
those used in the macro literature. The customer budget constraint is given by p d + n = I,
where I is the agent’s nominal income, which we normalize to one.21
While we fix the parameters listed above using external sources, our data allows us to
estimate those characterizing the idiosyncratic productivity process and the search cost distribution using a minimum-distance estimator. These are the key parameters of the model:
18
This alternative assumption matches more closely our model where the customer leaves after having seen
the prices of her current supplier and decided not to buy there.
19
This level of β reflects that the effective discount rate faced by the firm is the product of the usual time
preference discount factor and a rescaling element which takes into account the time horizon of the decision
maker, as for instance the average tenure of CEOs in the retail food industry reported in Henderson et al.
(2006).
20
Moving from these assumptions we can derive a demand function (d(p)) and a customer surplus function
(v(p)) consistent with the assumptions made in Section 2.
21
In Appendix C we show that I can be derived based on a model of the labor markets.
22
The productivity process influences the variability of prices, which is necessary for customers
to obtain any benefit from search. The parameters of the search cost distribution, on the
other hand, directly determine how costly it is to search. Below, we pick functional forms
for these objects and explain which moments we select in our data helps to identify them.22
We assume that the productivity evolves according to a process of the following form:
(
log(ztj ) =
j
log(zt−1
)
with probability ρ,
0
log(z ) ∼ N (0, σ) with probability 1 − ρ
When solving the model numerically, we approximate the normal distribution on a finite
grid, using the procedure described in Tauchen (1986). Finally, we normalize the nominal
wage equal to the price of the numeraire good, so that w = 1.23
The model specifies how the parameters of the productivity process impact on the autocorrelation and volatility of firm prices. We therefore estimate persistence and volatility
of productivity by matching the autocorrelation and the volatility of the logarithm of firm
prices to those measured in the data using a store-level price index.24
We assume that the search cost is drawn from a Gamma distribution with shape parameter
ζ, and scale parameter λ. The Gamma is a flexible distribution and fits the assumptions we
made over the G function in the specification of the model. In particular, for ζ > 1, we
obtain that the distribution of search costs is differentiable at ψ = 0.25
To identify the parameters of the search cost distribution we exploit the estimates of the
relationship between price and probability of exiting the customer base discussed in Section 4.
We identify the scale parameter λ by matching the average effect of log-prices on the exit
probability predicted by the model in equilibrium to its counterpart in the data measured by
the parameter b1 in equation (12). 26 The parameter ζ measures the inverse of the coefficient
22
The discussion on identification is provided only for the sake of intuition; given the nonlinearity of the
model, all the moments contribute to the identification of all the parameters.
23
This is equivalent to assume that the numeraire good n is produced by a competitive representative firm
with linear production function and unitary labor productivity. See Appendix C for details.
24
The store price index (pjt ) is computed in a fashion analogous to the customer price index. It is the
average of the price of all the UPCs sold in store j, weighted for the share of revenues they represent. To
estimate persistent and volatility of prices in the data, we exploit the first year of the sample span to obtain
the store-level average of the price index and use it to demean the variables so to remove store fixed effects. We
then estimate on the second year of data the equation (log(pjt )−log(pj )) = k0 +k1 (log(pjt−1 )−log(pj ))+τt +jt
pooling all stores. The time fixed effects are included to purge the data from aggregate effects and isolate
the variation in price driven by the idiosyncratic component.
25
In our estimation procedure we do not impose any constraints on the values the parameter ζ > 1 can
take. Our unconstrained point estimate lies in the desired region.
26
In particular, the equilibrium probability of exiting the customer base of a firm charging p and with
productivity z is E(z) ≡ G(ψ̂(p, z))(1 − H(V̄ (p, z))) where p = p̂(z) for all z, so that the model counterpart
to b1 is given by Cov(E(z), log(p̂(z)))/V ar(log(p̂(z))). The exit probability is decreasing in λ for given
equilibrium prices. This is exactly the case we are considering since we are targeting the persistence and
volatility of the empirical price distribution, which is indeed fixed in our analysis.
23
of variation of the search cost distribution. In the model, higher dispersion of search costs
(i.e., lower ζ) implies more mass on the tails of the distribution of search costs. The latter is
associated with larger variation in the sensitivity of exit probability to price. In the data, we
measure this variation by fitting a spline to equation (12), allowing for the marginal effect
of price on the probability of exit to vary for different terciles of price levels. We find that
higher prices imply a higher value of b1 as predicted by the model, and the dispersion in
the estimates of b1 is 0.03. The parameter ζ is estimated by matching this number to an
equivalent statistic generated by the model.27
Table 2: Parameter estimates
Value
Target
Persistence of productivity process, ρ
0.60
Log-price autocorrelation: 0.58
Volatility of productivity innovations, σ
0.11
Log-price dispersion: 0.02
Scale parameter of search cost distribution, λ
0.033
Average marginal effect, b1 : 0.14
Shape parameter of search cost distribution, ζ
3.00
Dispersion of b1 : 0.03
We define Ω ≡ [ζ λ ρ σ]0 as the vector of parameters to be estimated and estimate it with
a minimum-distance estimator. Denote by v(Ω) the vector of the moments predicted by the
model as a function of parameters in Ω, and by vd the vector of their empirical counterparts.
Each iteration n of the estimation procedure unfolds according to the following steps:
1. Pick values for the parameters ρn , σn , λn and ζn from a given grid,
2. Solve the model and obtain the vector v(Ωn ),
3. Evaluate the objective function (vd − v(Ωn ))0 Ξ (vd − v(Ωn )). Where Ξ is a weighting
matrix that we assume to be the identity matrix.
We select as estimates the parameter values from the proposed grid that minimize the objective function.
27
In the model the marginal effect of prices on the exit probability is proportional to
G (ψ̂(p, z))/G(ψ̂(p, z))(1 − H(V̄ (p, z)))2 , where p = p̂(z) for all z. We compute the dispersion in this measure
across the different firms.
0
24
Implementing step 2 requires solving a fixed point problem in equilibrium prices. In
particular, given our definition of equilibrium and the results of Proposition 2, we look for
equilibria where prices are in the interval [p∗ (z̄), p∗ (z)]. In principle, our model could have
multiple equilibria; however, numerically we always converge to the same equilibrium despite
starting from different initial conditions. In Appendix D we provide more details on the
numerical solution of the model. The estimation results are summarized in Table 2.
6
Price and customer dynamics
In this section we use the parameter estimates reported in Table 2 to solve for the equilibrium
pricing and searching policies predicted by our model (henceforth “baseline economy”). To
appreciate the importance of customer markets for equilibrium dynamics we contrast these
results with those implied by the limiting case where competition for customers is shut down
by raising the scale of search costs (λ) to infinity. We refer to this benchmark as “counterfactual economy.” To make the comparison meaningful, we fix θ in this counterfactual economy
so that the resulting average total elasticity of demand is the same as in our baseline economy
(i.e. εq = εd ) and choose σ and ρ targeting the same volatility and autocorrelation of prices
as in the baseline estimates. Hence the two economies are observationally equivalent with
respect to the (average) price elasticity of demand and the equilibrium price process.
This exercise allows us to compare qualitatively the baseline and the counterfactual economy and also to quantitatively assess the properties of the model. We start by analyzing
customer dynamics, plotted in Figure 1. In fact, a first order difference between our model
and the counterfactual economy is the presence of customer dynamics in equilibrium. In
particular, firms with high productivity experience positive net growth of their customers
base, whereas lower productivity firms are net losers of customers. Quantitatively, the model
predicts a yearly customer turnover of about 6%. The unconditional frequency of exit from
the customer base in our data is 14% on a yearly basis. Thus, price variation arising from
idiosyncratic cost shocks explains almost half of customer dynamics observed in the data.28
Our baseline model also features distinctive implications with respect to the shape of the
price distribution. This is interesting for two reasons. First, we argue that the shape of the
price distribution is an indicator of the relevance of customer markets. Therefore, comparing
the predicted distribution with the one from our data we provide an external validation of
our model. Second, it allows us to relate to a recent literature documenting features of the
28
It can be noticed that net customer base growth increases in productivity at a decreasing pace. This
is dictated by the asymmetry between the retention and attraction margins in our model. For instance,
allowing for an advertising technology would enable the firm to affect the mass of customers arriving and
reinforce the link between the arrival rate of customers and firm productivity.
25
Figure 1: Customer base growth and productivity
Weekly Customer Base Net Growth: log(∆), %
2
0
−2
−4
−6
−8
−10
−12
−14
−16
Baseline Economy
Counterfactual Economy
−18
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Log-Prod u ctivity: log (z )
Notes: The figure plots net customer base growth as a function of a firm’s idiosyncratic productivity, for the baseline and the
counterfactual economy. The baseline economy is simulated using the parameter estimates in Table 2. The counterfactual
economy’s productivity process is obtained matching the same moments (autocorrelation and volatility of the prices of store
baskets) as in the baseline estimation but search is shut down (λ → ∞). Therefore, in the counterfactual economy there is no
extensive margin of demand. The parameter governing the intensive elasticity of demand is chosen for the counterfactual
economy so that it matches the same overall elasticity of demand (intensive plus extensive margin) featured by the baseline
economy.
price distribution (Kaplan and Menzio (forthcoming)).
Figure 2 plots the distribution of prices in the baseline and counterfactual economies and
compares it with the empirical distribution emerging from the data. The figure plots the
distribution of the standardized (log-) ratio between the price set by a firm and the average
price in the market, p̃jt ≡ log(pjt /pj,mkt
).29 The price distributions from our baseline model and
t
the counterfactual are markedly different. Even though we impose that the two economies
have the same dispersion in prices, the baseline model price distribution displays a larger mass
In the stationary equilibrium of the model, the market price (pj,mkt
) does not vary over time, and there
t
is only one market. However, we construct the ratio to make the output of the model comparable with the
data, where the market price can vary through time and controlling for it is necessary to isolate the price
variation driven by idiosyncratic shocks.
29
26
Figure 2: The distribution of standardized prices: model and data
Data
Baseline Economy
Counterfactual Economy
Normal Distribution
0.6
Density
0.5
0.4
0.3
0.2
0.1
0
−6
−4
−2
0
2
4
6
Log-Price Ratio: log (p j /p j,m k t ), standardized
Notes: The figure plots the distribution of the ratio between the price set by a firm and the average price in the market,
log(pj /pmkt ). The price ratio is standardized (i.e. reported in deviations from its mean and divided by its standard deviation)
to make the outcomes from the baseline and the counterfactual models comparable. The blue solid line refers to our baseline
economy with customer markets at parameters estimated in Section 5. The counterfactual economy (dashed red line in the
plot) features parameters of the productivity process chosen targeting the same moments (autocorrelation and volatility of the
prices, and the intensive margin demand elasticity) as in the baseline estimation, but search is shut down (λ → ∞). The green
histogram portrays the empirical distribution of the ratio of the price index of each store of the retail chain to the average price
index in the Metropolitan Statistical Area where the store is located. Both the numerator and the denominator of this ratio are
normalized by their respective averages. Stores whose coefficient of variation for the ratio exceeds 1 are trimmed.
of prices clustered together around the mean that reflects into a more pronounced peakedness
of the price distribution relative to the counterfactual economy, whose price distribution is
instead nearly normal. Quantitatively the fraction of prices within half a standard deviation
from the mean is 46% in the baseline model and 38% in the counterfactual. Another way
of summarizing this is to note that only the baseline model displays excess kurtosis (around
4.2).
The increase in the clustering of prices around the mean, with respect to a normal distribution, derives from the strategic complementarity in pricing introduced by customer markets.
To illustrate this point we refer to Figure 3, which plots the results from a comparative static
exercise where we simulate the model varying only the scale of the search cost λ. Unlike in
Figure 2, we are not forcing dispersion and persistence of the price process to be the same
27
Figure 3: Equilibrium prices as a function of the scale of search costs, λ
Optimal prices
Price distribution
20
0.5
Low λ
Intermediate λ
High λ
Low λ
Intermediate λ
High λ
18
0.4
16
14
D en sity
Log-Price: log (p j )
0.3
0.2
0.1
12
10
8
6
0
4
−0.1
2
−0.2
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Log-Productivity: log (z )
0
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Log-Price: log (p j )
Notes: The left panel plots the optimal log-prices as a function of productivity. The right panel figure plots the distribution of
the log-price set by a firm. The blue solid line refers to our baseline economy with customer markets at parameters estimated
in Section 5 (λ = 0.03). The black dotted line refers to our baseline economy where however we set λ = 0.08. The red dashed
line refers to our baseline economy where however we set λ = 0.12.
across the different simulations, so that the other parameters are kept constant at the values
of Table 2. More competition for customers (lower λ) increases incentives of each firm to
price closer to firms with higher productivity, resulting in a shift of mass from the right to
the left of the price distribution. This incentive to reduce prices weakens as productivity
grows. In fact, the firm with the highest productivity always charges the same price (the
one maximizing profits per customer), independently of λ. It follows that the stronger the
competition for customers, the more prices cluster together.30
We compare the model prediction with the data by computing the price index for store j
in each week t as the average of the prices of the UPCs sold by the outlet, weighted for the
share of total revenues they generate in the all sample. We then use the IRI data to obtain
the average market price, pj,mkt
, given by the period t average price index for the same basket
t
of goods posted by retailers operating in the same Metropolitan Statistical Area where store
30
Notice that the shape of the price distribution in our model significantly departs from that of other
models featuring equilibrium price dispersion like, for instance, Burdett and Judd (1983). The difference is
not only due to the presence of customer retention concerns but also to other divergences in the modeling
approach, as for instance allowing for idiosyncratic variation in productivity.
28
j is located.31 The statistics plotted in the histogram in Figure 2 is the (standardized) ratio
of the store and the market price index. To remove permanent cross-market heterogeneity,
we normalize the numerator and the denominator of the ratio by their averages computed
over time. The decision to consider the distribution of the store price index, as opposed, for
instance, to that of the price of individual UPC’s, implicitly assumes that a manager cares
about the overall price level at the store for the “average” customer.32
The distribution obtained from our data also features high mass around the mean and
is well fitted by our baseline model: The fraction of prices within half a standard deviation
is 45.5%, and excess kurtosis is 4.6. This is remarkable, given that we did not target at all
the shape of the price distribution in our estimation and that our underlying productivity
innovations in the model are drawn from a Normal distribution. It follows that the leptokurtic
distribution of prices delivered from the model is not the product of ad-hoc assumptions but
derives from the strategic interaction that is at its core.33
A final difference between our setup and a model without customer markets relates the extent of pass-through. Our calculation for the baseline model implies an average pass-through
of idiosyncratic cost shocks equal to 13%, well below the 79% predicted in the counterfactual
economy.34 In the presence of competition for customers, a price increase leads to a persistent loss of customers; this dynamic incentive is not present in the counterfactual economy.
This implies that, when experiencing an increase in production cost, the firm has a strong
incentive to reduce the pass-through to the price by compressing its margin. Furthermore,
as we showed in equation (10), customer markets compress firms’ margins with respect to
the margin that would maximize profits per customer. As a consequence, when the firm
31
The procedure requires the price index of the store to be computed on the subset of UPCs for which we
have price information both in the retailer’s data and for each store in the IRI data for every week in the
sample. This substantially reduces the size of the store basket: In our data, the average store price index is
computed using about 1,000 UPCs. On the upside, the procedure naturally selects the best-selling products
(for which price information is more likely to appear continuously for all stores).
32
Notice that in the case of Section 4 we used the individual customer’s basket price as our main regressor.
In that context we were analyzing customers’ behavior and it was therefore more appropriate to consider the
price the customer cares about, that of its own basket.
33
Kaplan and Menzio (forthcoming) use Nielsen data to document the features of the price distribution
of both single UPCs and of bundles of goods bought by the consumers. They find that the former displays
high kurtosis; whereas the latter is nearly normal. Our finding does not contrast with theirs. They analyze
the ratio of the grocery expenditure by a household and the expenditure she would have incurred had she
purchased each item at the average market price. As such, the figure at the numerator can derive from a
bundle of goods bought in different stores. Our leptokurtik distribution refers instead to the ratio between a
bundle of goods sold at a given store and the average market price of that same bundle. In Appendix E, we
show that we can replicate their findings on the distribution of UPC prices using our data.
34
Note that the pass-through is incomplete even in the counterfactual economy because, with CES preferences, the demand of good i depends on the relative price pi /P . With a finite number of goods in the basket
of the customer, an increase in pi also increases the price of the basket, P , thus reducing the overall increase
in pi /P and effect on demand. The effect on P is larger, the higher the weight of good i in the basket, that
is the lower the price pi and the higher its demand. Therefore, the elasticity of demand εd (p) increases in p.
29
experiences a decrease in production cost, it can increase profits per customer without losing
more customers. Therefore, the pass-through is incomplete also in this direction. The store
data provided by the retailer contain a measure of cost. Although this measure is imperfect
and does not precisely capture the marginal cost, we can use it to check the model’s predictions regarding pass-through. The results from this exercise are reported in Appendix F:
The pass-through measured in the data is in line with the predictions of a customer markets
model and much lower than what the counterfactual economy would imply.35
7
The dynamics of demand: short vs. long run
So far we have analyzed the dynamics of prices and customers in response to idiosyncratic
shocks in the presence of customer markets. In this section we broaden our scope, exploring
the relevance of customer markets for the propagation of a shock affecting a subset of the
firms in the economy with an emphasis of the dynamics of demand. A shock to the effective
cost of a subset of the players in the industry is the salient characteristic of a number of real
world scenarios. For instance, variations in state sales tax would affect local sellers but not
online ones located out-of-state, as they cannot be compelled to collect it. A similar effect
is generated by a shock to the real exchange rate affecting the competitiveness of foreign
producers, or by the introduction of size-contingent employment protection legislation.36
Exploring cost shocks that affect firms asymmetrically is useful to highlight one of the main
differences between our setup and the standard monopolistic competition models. In fact,
such shocks create price dispersion and, therefore, incentivize customers to search for a new
supplier. In the presence of customer markets, this leaves scope for customer reallocation
to impact persistently the response of demand and output, creating a distinctive difference
between long- and short-run response.
The specifics of the experiment we perform are as follows. We consider the economy
calibrated in Section 5 in steady state at period t0 . We assume that 10% of the firms in the
economy are hit by an unexpected and unforeseen shock to production cost in the form of
a scaling factor τt , so that the marginal cost of production of a firm hit by the shock goes
∗
∗
from wt /ztj to τt wt /ztj , where j ∗ denotes a firm being hit by the shock. The productivity
35
This result is not inconsistent with evidence of complete pass-through presented by Eichenbaum et al.
(2011) using the same data. First, they measure pass-through conditional on price adjustment; whereas
we look at the unconditional correlation between prices and costs. Second, they deal with UPC-level passthrough while we measure pass-through of a basket of goods. If retailers play strategically with the pricing
of different products, for example lowering margins on some UPC to compensate the cost increase they
experienced on others, we can obtain both high UPC-level pass-through and low basket-level pass-through.
36
Introducing nominal rigidities to our framework, a similar exercise could also be used to study the effects
of nominal shocks.
30
shock is realized after the firm has learned about idiosyncratic productivity zt , but before
pricing and customer’s exit decisions are taken, and dies out according to an AR(1) process,
τt = ρτ τt−1 for t > t0 .37
Figure 4: The response of market share and relative price to an aggregate shock
Market share of firms hit by the shock
Relative price of firms hit by the shock
0
5
Baseline Economy
Counterfactual Economy
Baseline Economy
Counterfactual Economy
4.5
−0.1
4
−0.2
in % f rom s.s.
in % f rom s.s.
3.5
3
2.5
2
−0.3
−0.4
−0.5
1.5
−0.6
1
−0.7
0.5
0
0
2
4
6
8
10
12
14
16
18
20
weeks
−0.8
0
2
4
6
8
10
12
14
16
18
20
weeks
Notes: The figure plots the impulse response of market share of those firms being hit by the aggregate shock and relative
price of affected to unaffected firm, for both our baseline economy and a counterfactual one where customer markets are not
present. The baseline economy is simulated using the parameter estimates in Table 2. In the counterfactual economy the
parameter governing the intensive elasticity of demand is chosen so that it matches the same overall elasticity of demand
(intensive plus extensive margin) featured by the baseline economy.
In Figure 4 we plot the responses of the market shares of firms hit by the shock (left
panel) and relative average price (right panel) of firms hit by the aggregate shock to that
of unaffected firms.38 The plot refers to a parameterization where the shock leads to a 1%
increase in productivity with a persistence parameter ρτ = 0.9, so that the half-life of the
aggregate shock is approximately a quarter. The solid line plots the responses in the economy
with customer markets, while the dotted line refers to the response in the alternative economy
37
Since the shock implies aggregate dynamics, we augment our economy with a simple equilibrium model
of the labor market to capture the general equilibrium effects of the shock on wages and income. More details
on this extension are provided in Section C.
R
R
∗
∗
38
The market share of the firms hit by the shock is defined as mjt djt dj ∗ /(Γ djt dj), where j ∗ denotes a
firm being hit by the shock, and j denote any firm.
31
where customer markets are shut down. As usual, we make sure that the two economies are
comparable, fixing the total average elasticity of demand to be the same.
The difference in the propagation of the same shock in the two economies is striking. In
the baseline economy, the effect of the shock is persistent and the long-run impact is larger
than the short-run one. In the counterfactual economy, the short-run effect is larger than in
a world with customer markets but the response is purely transitory. The reaction of prices
is also quite different: relatively little movement in the baseline economy and much more
marked changes in the alternative one.
The response pattern in the alternative economy is easy to understand and similar to
what would happen in static models of monopolistic competition. Both the response of the
market share and relative price follow closely the process for the productivity shock, and
thus are transitory and their order of magnitude depends on the size of the shock. The
baseline economy has two distinctive characteristics that deliver the very different response
path. First, as we have shown, lower levels of search cost imply less pass-through. Therefore,
the immediate effect on prices is smaller in the baseline economy. This, in turn, leads to more
contained effect on demand in the short run. The general equilibrium effect compounds this
mechanism: even firms not hit by the shock have an incentive to decrease their prices, as
they experience an increase in the extensive margin elasticity. The second force at play is
connected to the process of customer reallocation. After the unexpected shock has hit, firms
that were affected are better than those that were not and customers would want to shop
there. Because of the search friction, the reallocation process takes time and therefore does
not immediately boost the market share of the firms hit by the shock. However, as time
goes by, the firms made more productive by the shock gain customers that they will not
lose even when the effect of the shock has completely faded away. In fact, at that point the
productivity processes of all firms will be the same and there will be no extra incentive for
customers to search.
This example showcases the flexibility of our setup: Different levels of search cost can
originate stark differences in the response to aggregate shocks. This implies that the possibility of empirically assessing the relevance of customer markets in an economy is key to
predict the response of demand to shocks. At our parameter estimates, the shock generates a
stronger impact in the long than in the short run and the effect will be persistent. Evidence
from the international macro literature provides some support for these predictions: Ruhl
(2008) documents that the elasticity of the share of imports to exchange rate shocks is indeed
larger in the long run than in the short run.
32
8
Conclusions
Across a broad range of industries, being able to retain customers and attract new ones is
key to firms’ success and survival. This is increasingly recognized by scholars, as witnessed
by the growing number of studies attempting to incorporate this feature and exploring its
implications. In this paper, we combined a formal model and novel data to study and quantify
the consequences of the presence of competition for customers on firms’ pricing.
Our first contribution is the introduction of a rich yet tractable model of customer markets. We generate stickiness in the customer base of a firm by positing that customers must
pay a search cost when they wish to look for a new supplier. We allow this cost to be heterogeneous across customers, and we also let firms differ in their idiosyncratic productivity.
We show the existence of a Markov Perfect Equilibrium where optimal pricing are decreasing
in productivity, whereas the customer base growth is increasing in it. This implies that our
equilibrium features both price and customer dynamics. Another important result from the
model states that the fiercer the competition for customers the lower optimal markups will
be. This implies that markups in our model will be lower than in a classic static demand
model, which does not contemplate customer markets.
Empirically, we exploited detail data from a retail chain to document a relationship
between a firm’s pricing and its customers’ decision to leave its customer base. Retrieving
this elasticity and using the data to form other appropriate empirical moments, we were
able to estimate the key parameters of the model: those pinpointing the distribution of the
search costs and those governing the productivity process. This allowed us to highlight two
important implications of our model. First, the incentive to attract and retain customers
introduces strategic complementarities pushing firms to set similar prices. This results in a
price distribution where a large fraction of prices lies close to the mean, generating a shape in
line with the one documented by the available empirical evidence. Second, we presented an
application that highlights how a customer market model may have important implications
for the response of demand to shocks. In particular, our counterfactual exercise showed that
frictions in customer reallocation contribute to magnify the size and the persistence of the
long-run effects of shocks.
References
Alessandria, G. (2004). International Deviations From The Law Of One Price: The Role Of
Search Frictions And Market Share, International Economic Review 45(4): 1263–1291.
33
Bai, Y., Rull, J.-V. R. and Storesletten, K. (2012). Demand Shocks that Look Like Productivity Shocks, mimeo.
Blinder, A. S., Canetti, E., Lebow, D. and Rudd, J. (1998). Asking About Prices: A New
Approach to Understanding Price Stickiness, New York: Russell Sage Foundation.
Bronnenberg, B. J., W., K. M. and Mela, C. F. (2008). The IRI Marketing data set, Marketing
Science 27(4): 745–748.
Burdett, K. and Coles, M. G. (1997). Steady state price distributions in a noisy search
equilibrium, Journal of Economic Theory 72(1): 1 – 32.
Burdett, K. and Judd, K. L. (1983). Equilibrium price dispersion, Econometrica 51(4): 955–
69.
Diamond, P. A. (1971). A model of price adjustment, Journal of Economic Theory 3(2): 156–
168.
Dinlersoz, E. M. and Yorukoglu, M. (2012). Information and Industry Dynamics, American
Economic Review 102(2): 884–913.
Drozd, L. A. and Nosal, J. B. (2012). Understanding international prices: Customers as
capital, American Economic Review 102(1): 364–95.
Eichenbaum, M., Jaimovich, N. and Rebelo, S. (2011). Reference prices and nominal rigidities, American Economic Review 101(1): 234–262.
Einav, L., Leibtag, E. and Nevo, A. (2010). Recording discrepancies in Nielsen Homescan data: Are they present and do they matter?, Quantitative Marketing and Economics
8(2): 207–239.
Fabiani, S., Loupias, C., Martins, F. and Sabbatini, R. (2007). Pricing decisions in the euro
area: how firms set prices and why, Oxford University Press, USA.
Fishman, A. and Rob, R. (2003). Consumer inertia, firm growth and industry dynamics,
Journal of Economic Theory 109(1): 24 – 38.
Foster, L., Haltiwanger, J. and Syverson, C. (2013). The slow growth of new plants: Learning
about demand?, Working Papers 12-06, Center for Economic Studies, U.S. Census Bureau.
Gourio, F. and Rudanko, L. (2014). Customer capital, The Review of Economic Studies
81: 1102–1136.
34
Hall, R. E. (2008). General equilibrium with customer relationships: A dynamic analysis of
rent-seeking, Working paper, Stanford University.
Henderson, A. D., Miller, D. and Hambrick, D. C. (2006). How quickly do CEOs become
obsolete? Industry dynamism, CEO tenure, and company performance, Strategic Management Journal 27: 447–460.
Holmes, T. (2011). The diffusion of Wal-Mart and economies of density, Econometrica
79: 253–302.
Kaplan, G. and Menzio, G. (2013). Shopping externalities and self-fulfilling unemployment
fluctuations, NBER Working Papers 18777, National Bureau of Economic Research, Inc.
Kaplan, G. and Menzio, G. (forthcoming). The morphology of price dispersion, International
Economic Review .
Kleshchelski, I. and Vincent, N. (2009). Market share and price rigidity, Journal of Monetary
Economics 56(3): 344–352.
Luttmer, E. G. (2010). Models of growth and firm heterogeneity, Annual Review of Economics
2(1): 547–576.
Menzio, G. (2007). A search theory of rigid prices, PIER Working Paper Archive 07-031, Penn
Institute for Economic Research, Department of Economics, University of Pennsylvania.
Petrosky-Nadeau, N. and Wasmer, E. (2011). Macroeconomic Dynamics in a Model of Goods,
Labor and Credit Market Frictions, IZA Discussion Papers 5763, Institute for the Study
of Labor (IZA).
Phelps, E. S. and Winter, S. G. (1970). Optimal price policy under atomistic competition, in
G. Archibald, A. A. A. and E. S. Phelps (eds), Microeconomic Foundations of Employment
and Inflation Theory, New York: Norton.
Reinganum, J. F. (1979). A simple model of equilibrium price dispersion, Journal of Political
Economy 87(4): 851–58.
Ruhl, K. J. (2008). The International Elasticity Puzzle, Working Paper 08-30, New York
University, Leonard N. Stern School of Business, Department of Economics.
Shi, S. (2011). Customer relationship and sales, Working Papers tecipa-429, University of
Toronto, Department of Economics.
35
Steindl, J. (1965). Random processes and the growth of firms: A study of the Pareto law,
Griffin London.
Tauchen, G. (1986). Finite state markov-chain approximations to univariate and vector
autoregressions, Economics Letters 20(2): 177–181.
36
A
A.1
Proofs
Proof of Lemma 2
The proof of Lemma 2 follows from the assumption of v(p) being strictly decreasing in
p so that V̄ (p, z) is decreasing in p. If V̂ (z) is increasing in z, V̄ (p, z) increases with z
because of the assumptions that the productivity process is persistent. Finally, notice that
ψ̂(p,z)
(p,z)
= −v 0 (p) (1 − H(V (p, z))) ≥ 0 and that ∂ ψ̂(p,z)
= − ∂ V̄∂z
(1 − H(V (p, z))) ≤ 0 .
∂p
∂z
A.2
Proof of Proposition 1
Let p̄(z) be the level of price at which no customer matched with a firm with productivity z
searches. Then p̄(z) satisfies V̄ (p̄(z), z) = maxz∈[z,z̄] {V̄ (P(z), z)}. First, given the definition
of p∗ (z) and the fact that ∆(p, z) is strictly decreasing in p for all p > p̄(z), and constant
otherwise, it immediately follows that p̂(z) ∈ [p̄(z), p∗ (z)] if p̄(z) < p∗ (z), and p̂(z) = p∗ (z)
otherwise. Next, we show that p̂(z) < p∗ (z) if p̄(z) < p∗ (z). Given the definition of p̄(z) and
the assumptions made on G, at p = p̂(z): i) W (p, z) is strictly decreasing in p; ii) ∆(p, z) is
strictly decreasing in p. Therefore; p = p̂(z) cannot be a maximum and the result follows.
A.3
Proof of Proposition 2
Monotonicity of prices. We first show that optimal prices p̂(z) are non-increasing in z. Given,
that productivity is i.i.d. and that we look for equilibria where p̂(z) ≥ p∗ (z̄), we have that
p̄(z) = p∗ (z̄) for each z. From Proposition 1 we know that, for a given z, the optimal price
p̂(z) belongs to the set [p∗ (z̄), p∗ (z)]. Over this set, the objective function of the firm,
W (p, z) = ∆(p, z) (π(p, z) + β constant) ,
(13)
is supermodular in (p, −z). Notice the i.i.d. assumption implies that future profits of the
firm do not depend on current productivity as future productivity, and therefore profits, are
independent from it. Similarly, ∆(p, z) does not depend on z, as the expected future value to
the customer does not depend on the productivity of the current match as future productivity
is independent from it. Abusing notation, we replace ∆(p, z) by ∆(p). To show that W (p, z) is
supermodular in (p, −z) consider two generic prices p1 , p2 with p2 > p1 > 0 and productivities
z1 , z2 ∈ [z, z̄] with −z2 > −z1 . We have that W (p2 , z2 ) − W (p1 , z2 ) ≤ W (p2 , z1 ) − W (p1 , z1 )
if and only if
∆(p2 )d(p2 )(p2 −w/z2 )−∆(p1 )d(p1 )(p1 −w/z2 ) ≤ ∆(p2 )d(p2 )(p2 −w/z1 )−∆(p1 )d(p1 )(p1 −w/z1 ),
37
which, using ∆(p2 )d(p2 ) < ∆(p1 )d(p1 ) as d(p) is strictly decreasing and ∆(p) is non-increasing,
is indeed satisfied if and only if z2 < z1 . Thus, W (p, z) is supermodular in (p, −z). By application of the Topkis Theorem we readily obtain that p̂(z) is non-increasing in z.
Existence of equilibrium. Next we prove existence of an equilibrium. The fixed point problem is a mapping from candidate functions of equilibrium prices, P(z), to the firm’s optimal
pricing strategy, p̂(z), where an equilibrium is one where p̂(z) = P(z) for each z. Differentiability of v(p) implies that W (p, z) is continuously differentiable in p, so that the operator
that maps P(·) into p̂(·) is given by the first order condition in equation (8). Moreover, notice
that W (p, z) in equation (13) is continuous in (p, z). By the theorem of maximum, p̂(z) is
upper hemi-continuos and W (p̂(z), z) is continuos in z. Given that p̂(z) is non-increasing in
z it follows that p̂(z) has a countably many discontinuity points. We thus proceed as follows.
Let P̂(z) be the set of prices that maximize the firm problem. Whenever a discontinuity
arises at some z̃ (so that P̂(z̃) is not a singleton), we modify the optimal pricing rule of the
firm and consider the convex hull of the P̂(z̃) as the set of possible prices chosen by the firm
with productivity z̃. The constructed mapping from z to P̂(z) is then upper-hemicontinous,
compact and convex valued. We then apply Kakutani’s fixed point theorem to this operator
and obtain a fixed point. Finally, notice that since the convexification procedure described
above has to be applied only a countable number of times, the set of convexified prices has
measure zero with respect to the density of z. Hence, they do not affect the fixed point.
Necessity of the first order condition. We show that Q and H are almost everywhere differentiable, so that Proposition 1 implies that equation (8) is necessary for an optimum. We guess
that p̂(z) is strictly decreasing and almost everywhere differentiable. It immediately follows
that V̂ (z) is strictly increasing in z and almost everywhere differentiable. Then, given the assumption that F is differentiable, we have that K is differentiable. From H(x) = K(V̂ −1 (x))
it follows that H is also almost everywhere differentiable. Given that G and H are differentiable, so is Q. Then the first order condition in equation (8) is necessary for an optimum,
which indeed implies that p̂(z) is strictly decreasing and differentiable in z in any neighborhood of the first order condition. Finally, given that p̂(z) has a countably many discontinuity
points, it has countably many points where it is not differentiable, and the first order condition does not apply at those points, but applies everywhere else. These points have measure
zero with respect to the density of z and therefore p̂(z) is almost everywhere differentiable.
Point (i). We already proved that p̂(z) is non-increasing in z. The proof that p̂(z) is
strictly decreasing follows by contradiction. Consider that p̂(z1 ) = p̂(z2 ) = p̃ for some
z1 , z2 ∈ [z, z̄]. Also, without loss of generality, assume that z1 < z2 . Given that we already
established the necessity of the first order condition presented in equation (8) when prices
38
are monotonic, suppose that it is satisfied at the duple {z2 , p̃}. Notice that, because of the
assumed i.i.d. structure of productivity shocks together with πz (p, z) < 0, it is not possible
that the first order condition is also satisfied at the duple {z1 , p̃}. Moreover, because the
first order condition is necessary and we already established that p̂(z) cannot be increasing
at any z, we conclude that the optimal price at z1 is strictly larger than at z2 . That is,
p̂(z1 ) > p̂(z2 ). Notice that this verifies the conjecture used to prove the necessity of the first
order condition, which in turn validates the use of equation (8) here.39
Notice that, because p̂(z) is strictly decreasing in z, the fact that v 0 (p) < 0 together with
i.i.d. productivity, implies, through an application of the contraction mapping theorem, that
V̂ (z) = V̄ (p̂(z), z) is increasing in z.
Point (ii). ψ̂(p, z) ≥ 0 immediately follows its definition. The fact that V̂ (z) is strictly
increasing in z, together with Lemma 2, immediately implies that ψ̂(p̂(z̄), z̄) = 0 and that
ψ̂(p̂(z), z) is strictly increasing in z. Finally, Lemma 3 implies that ∆(p̂(z), z) is increasing
in z. Because of price dispersion, some customers are searching, which guarantees that
∆(p̂(z̄), z̄) > 1. Likewise, ∆(p̂(z), z) < 1.
A.4
Thought experiment of Section 3
We show that µ(p, z) is increasing in εq (p, z). Notice that equation (10) can be rewritten as
µ(p, z) =
εq (p, z) + εm (p, z)x̃(p, z)
,
εq (p, z) − 1 + εm (p, z)x̃(p, z)
where x̃(p, z) ≡ Π(p, z)/π(p, z). From the equation above we obtain
∂µ(p, z)
x̃(p, z)
=
(1 − µ(p, z)).
∂εm (p, z)
εq (p, z) − 1 + εm (p, z)x̃(p, z)
A direct implication of nonnegative prices is that εq (p, z) − 1 + εm (p, z)x̃(p, z) ≥ 0, so that
sign [∂µ(p, z)/∂εm (p, z)] = sign[(x̃(p, z))(1 − µ(p, z))]. There are two cases to consider. The
first one is when π(p, z) > 0, which occurs if and only if µ(p, z) > 1. It implies x̃(p, z) > 0
39
If prices are not strictly decreasing, this argument cannot be used as the first order condition is not
necessary. However, it is possible to prove that p̂(z) is strictly decreasing in z for some region of z. The
argument follows by contradiction. Suppose that p̂(z) is everywhere constant in z at some p̃. Then p̄(z) = p̃
for all z. If p̃ > p∗ (z̄), then p̃ would not be optimal for firm with productivity z̄, which would choose a lower
price. If p̃ = p∗ (z̄), then continuous differentiability of G together with H = G = Q = 0 at the conjectured
constant equilibrium price imply that the first order condition is locally necessary for an optimum, and a
firm with productivity z < z̄ would have an incentive to deviate according to equation (8), and set a strictly
higher price than p̃. Finally, the result that p̂(z) < p∗ (z) for all z < z̄ and that p̂(z̄) = p∗ (z̄) follows from
applying Proposition 1, and using that p̂(z) ≥ p̂(z̄) and p̄(z) = p̂(z̄) for all z.
39
and, therefore, ∂µ(p, z)/∂εm (p, z) < 0. The second case is when π(p, z) < 0, which occurs
if and only if µ(p, z) < 1. It implies x̃(p, z) < 0 and, therefore, x̃(p, z) < 0. As a result,
∂µ(p, z)/∂εm (p, z) < 0.
A.5
Proof of Remark 1
Part (1). Start by noticing that, because the mean of G(ψ) is positive, the expected value of
searching diverges to −∞ as n diverges to infinity. Because prices are finite for all z ∈ [z, z̄],
the value of not searching is bounded. As a result, customers do not search so that firms
do not face customer base concerns. Formally, p̄(z) → ∞ for all z ∈ [z, z̄]. Because p∗ (z) is
finite for all z ∈ [z, z̄], it follows immediately that p∗ (z) < p̄(z) for all z ∈ [z, z̄]. Then, using
Proposition 1 we obtain that p̂(z) = p∗ (z) for all z ∈ [z, z̄].
Part (2). From Proposition 2 we have that, in equilibrium, the highest price is p̂(z). Moreover, under the assumptions of Proposition 2, the first order condition is a necessary condition
for optimality of prices. We use this to show that, as n approaches zero, p̂(z) has to approach
p̂(z̄) = p∗ (z̄).
In equilibrium, it is possible to rewrite equation (8), evaluated at {p̂(z), z}, as LHS(p̂(z), n) =
RHS(p̂(z), n), where
ψ̂p (p̂(z), z)
n
!
!
1
ψ̂(p̂(z), z)
+ G
H 0 (V̄ (p̂(z), z)) + Q0 (V̄ (p̂(z), z)) V̄p (p̂(z), z) ,
n
Γ
!!
πp (p̂(z), z)
ψ̂(p̂(z), z)
RHS(p̂(z), n) ≡ −
1−G
,
Π(p̂(z), z)
n
LHS(p̂(z), n) ≡ G0
ψ̂(p̂(z), z)
n
!
given that H(V̄ (p̂(z), z)) = Q(V̄ (p̂(z), z)) = 0.
ψ̂(p̂(z),z)
Suppose that as n ↓ 0, ψ̂(p̂(z), z) does not converge to zero. Then, G
↑ 1 as
n
n ↓ 0. This implies that limn↓0 RHS(p̂(z), n) > 0.
Consider now the function LHS(p̂(z), n). Again, suppose that as n ↓ 0, ψ̂(p̂(z), z) does
not converge to zero. Notice that the second term of the function approaches a finite number
as V̄p (p̂(z), z) is bounded by assumptions on v(p) and H 0 (V̄ (p̂(z), z)) and Q0 (V̄ (p̂(z), z)) being
bounded as a result of Proposition 2. Moreover, as long as p̂(z) > p̄(z) = p∗ (z̄), we have
that
ψ̂p (p̂(z),
z) > 0 so that ψ̂p (p̂(z), z)/n diverges as n approaches zero. This means that
0 ψ̂(p̂(z),z) ψ̂p (p̂(z),z)
G
is divergent, and therefore the first order condition cannot be satisfied.
n
n
This analysis concluded that, if ψ̂(p̂(z), z) does not converge to zero as n becomes ar40
bitrarily small, the first order condition, i.e. equation (8), cannot be satisfied. This occurs
because LHS(p̂(z), n) would diverge to infinity, while RHS(p̂(z), n) would remain finite. It
then follows that, as n approaches zero, a necessary condition is that ψ̂(p̂(z), z) also approaches zero. This condition can be restated as requiring that p̂(z) approaches p̄(z) as n
approaches zero. Moreover, given the assumptions of Proposition 2, p̄(z) = p̂(z̄) = p∗ (z̄).
In the end, if p̂(z) approaches p∗ (z̄) as n becomes arbitrarily small (so that ψ̂(p̂(z), z) → 0
and ψ̂p (p̂(z), z) → 0), we have that limn↓0 LHS(p̂(z), n) < ∞ and limn↓0 RHS(p̂(z), n) < ∞
as πp (p∗ (z̄), z) is bounded as π(p∗ (z̄), z) > 0. However, if p̂(z) does not approach p∗ (z̄) as n
becomes arbitrarily small, we have that LHS(p̂(z), n) diverges as n approaches zero, while
LHS(p̂(z), n) remains finite. As the first order condition has to be satisfied in equilibrium, a
necessary condition is that, as n approaches zero, the highest price in the economy, i.e. p̂(z),
has to approach the lowest price in the economy, i.e. p∗ (z̄).
B
B.1
Data sources and variables construction
Data and selection of the sample
The empirical evidence presented in Section 4 is based on two data sources provided by a
large supermarket chain that operates over 1500 stores across the United States. This implies
that we can observe our agents behavior only when they shop with the chain; on the other
hand, cash register data contain significantly less measurement error than databases relying
on home scanning (Einav et al. (2010)).
The main data source contains information on grocery purchases at the chain between
June 2004 and June 2006 for a panel of over 11,000 households. For each grocery trip made
by a household, we observe date and store where the trip occurred, the collection of all
the UPCs purchased with quantity and price paid. The data include information on the
presence and size of price discounts but do not generally report redemption of manufacturer
coupons. Data are collected through usage of the loyalty card; purchases made without
using the card are not recorded. However, the chain ensures that the loyalty card has a high
penetration by keeping to a minimum the effort needed to register for one. Furthermore,
nearly all promotional discount are tied to ownership of a loyalty card, which provides a
strong incentive to use it.
Household-level scanner data report information on the price paid conditional on a certain
item having been bought by the customer. Therefore, if we do not observe at least one
household in our sample buying a given item in a store in a week, we would not be able
to infer the price of the item in that store-week. This has important implications as our
41
definition of basket requires us to be able to attach a price to each of the item composing it
in every week, even when the customer does not shop. The issue can be solved using another
dataset with information on weekly store revenues and quantities between January 2004 and
December 2006 for a panel of over 200 stores. For each good (identified by its UPC) carried by
the stores in those weeks, the data report total amount grossed and quantity sold. Exploiting
this information, we can calculate unit value prices every week for every item in stock in a
given store, whether or not that particular UPC was bought by one of the households in our
main data. Unit value prices are computed using data on revenues and quantities sold as
U V Pstu =
T Rstu
,
Qstu
where TR represent total revenues and Q the total number of units sold of good u in week t
in store s.
As explained in Eichenbaum et al. (2011), this only allows us to recover an average price
for goods that were on promotion. In fact the same good will be sold to loyalty card carrying
customers at the promotional price and at full price to customers who do not have or use
a loyalty card. Without information on the fraction of these two types of customers it is
not possible to recover the two prices separately. Furthermore, since prices are constructed
based on information on sales, missing values can originate even in this case if no unit of a
specific item is sold in a given store in a week. This is, however, an unfrequent circumstance
and involves only rarely purchased UPCs, which are unlikely to represent important shares
of the basket for any of the households in the sample. For the analysis, we only retain UPCs
with at most two nonconsecutive missing price observations and impute price for the missing
observation interpolating the prices of the contiguous weeks.
On top of reporting revenues and quantities for each store-week-UPC triplet, the storelevel data also contain a measure of cost. This variable is constructed on the basis of the
estimated markup imputed by the retailer for each item and includes more than the simple
wholesale cost of the item (the share of transportation cost, etc.). Eichenbaum et al. (2011)
suggest to think about it as a measure of replacement cost, i.e. the cost of placing an item
on the shelf to replace an analogous one just grabbed by a consumer. We use this measure
to construct our instrument of the basket price.
It is important to notice that the retail chain sets different prices for the same UPC in
different geographic areas, called “price areas.” The retailer supplied store-level information
for 270 stores, ensuring that we have data for at least one store for each price area. In
order to use unit value prices calculated from store-level data to compute the price of the
basket of a specific household, we need to determine to which price area the store(s) at
42
which she regularly shops belong. This information is not supplied by the retailer that
kept the exact definition of the price areas confidential. A possible solution is to infer in
which price areas the store(s) visited by a household are located by comparing the prices
contained in the household panel with those in the store data. In principle the household
data should give information on enough UPC prices in a given week to identify the price
area representative store whose pricing they are matching. However, even though two stores
belonging in the same price area should have the same prices, they may not have the same
unit value prices if the share of shoppers using the loyalty card differs in the two stores.
Therefore, we choose to restrict our analysis to the set of customers shopping predominantly
(over 80% of their grocery expenditure at the chain) in one of the 270 stores for which the
chain provided complete store-level data. This choice is costly in terms of sample size: Only
1,336 households (or 12% of the original sample) shop at one of the 270 stores for which
we have store-level price data. However, since the 270 representative stores were randomly
chosen, the resulting subsample of households should not be subject to any selection bias.
A final piece of the data is represented by the IRI-Symphony database. We use storelevel data on quantities and revenues for each UPC in 30 major product categories for a
large sample of stores (including small and mom & pop ones) in 50 Metropolitan Statistical
Areas in the United States. The data allow to construct unit value prices for all the stores
competing with the chain who provided the main dataset. However, the coarse geographic
information prevents us from matching each customer with the stores closer to her location
(in the same zip code, for instance) and forces us to adopt the MSA as our definition of a
market.
B.2
Variables construction
Exit from customer base. The dependent variable in the regression presented in equation (12) is an indicator for whether a customer is exiting the customer base of the chain.
With data on grocery purchases at a single retail chain it is hard to definitively assess whether
a household has abandoned the retailer to shop elsewhere or is simply not purchasing groceries in a particular week, for instance because she is just consuming its inventory. In fact,
we observe households when they buy groceries at the chain but do not have any information
on their shopping at competing grocers. Our choice is to assume that a customer is shopping
at some other store when she has not visited any supermarket store of the chain for at least
eight consecutive weeks. The Exit dummy is then constructed so that it takes value of one
in correspondence to the last visit at the chain before a spell of eight or more weeks without
shopping there. Table 3 summarizes shopping behavior for households in our sample. It is
immediate to notice that an eight-week spell without purchase is unusual, as customers tend
43
to show up frequently at the stores. This strengthens our confidence that customers missing
for an eight-week period have indeed switched to a different retailer.
Table 3: Descriptive statistics on customer shopping behavior
Mean Std.dev. 25th pctile 75th pctile
Number of trips
150
127
66
200
Days elapsed between consecutive trips
4.2
7.5
1
5
Expenditure per trip ($)
69
40
40
87
0.003
0.065
Frequency of exits
Composition of the household basket and basket price. The household scanner data
deliver information on all the UPCs a household has bought through the sample span. We
assume that all of them are part of the household basket and, therefore, the household
should care about all of those prices. Some of the items in the household’s basket are bought
regularly, whereas others are purchased less frequently. We take this into account when
constructing the price of the basket by weighting the price of each item by its expenditure
share in the household budget. The price of household i’s basket purchased at store j in
week t is computed as:
pijt =
X
P
t Eikt
,
ωik = P P
k
t Eikt
ωik pkjt ,
k∈Ki
where K i is the set of all the UPCs (k) purchased by household i during the sample period,
pkjt is the price of a given UPC k in week t at the store j where the customer shops. Eikt
represents expenditure by customer i in UPC k in week t and the ωik ’s are a set of householdUPC specific weights. There is the practical problem that the composition of the consumer
basket cannot vary through time; otherwise basket prices for the same customer in different
weeks would not be comparable. This requires that we drop from the basket all UPCs for
which we do not have price information for every week in the sample. However, the price
information is missing only in instances where the UPC registered no sales in a particular
week. It follows that only low market-share UPCs will have missing values and, therefore,
the UPCs entering the basket computation will represent the bulk of each customer’s grocery
expenditure. The construction of the cost of the basket follows the same procedure where we
substitute the unit value price with the measure of replacement cost provided by the retailer.
44
We choose to calculate the weights using the total expenditure in the UPC by the household over the two years in the sample. This can lead to some inaccuracy in identifying the
goods the customer cares for at a given point in time. For example, if a customer bought only
Coke during the first year and only Pepsi during the second year of data, our procedure would
have us give equal weight to the price of Coke and Pepsi throughout the sample period. If we
used a shorter time interval, for example using the expenditure share in the month, we would
correctly recognize that she only cares about Coke in the first twelve months and only about
Pepsi in the final 12 months. However, weights computed on short time intervals are more
prone to bias induced by pricing. For example, a two-weeks promotion of a particular UPC
may induce the customer to buy it just because of the temporary convenience; this would
give the UPC a high weight in the month. The effect of promotion is instead smoothed when
we compute weights using expenditure over the entire sample period.
The construction of the price of the competitors occurs in two steps. First, we use the
IRI data and the same procedure described above to obtain a price for the basket of each
consumer at every store located in her same MSA. Next, we average those prices across
stores to obtain the average market price of the consumer basket. In particular, the price is
computed as:
pmkt
it
=
X
z∈M i
sz
X
k∈Ki
ωik pkzt ,
P
t Eikt
,
ωik = P P
k
t Eikt
P
sz = P
Rzt
tP
z 0 ∈M
t
Rz0 t
where M i is the MSA of residence for customer i and Rzt represents revenues of store z
in week t. In other words, in the construction of the competitors’ price index, stores with
higher (revenue-based) market shares weight more.
Composition of the store basket and basket price. The construction of the price (and
cost) index for the store is conceptually analogous to that described above for the household
basket. In principle, we would want to compute the store price index including all the
UPCs sold at a store throughout the sample period, weighted by the share of revenues they
generated. However, to keep the composition of the store basket constant through time, we
must restrict ourselves to the UPCs for which we have no missing price information in any
of the weeks in the sample span. This severely reduces the size of the store basket. At the
same time, goods without missing information are the best sellers, which are those the store
is likely to care more about.
45
C
A simple model of the labor market
In this appendix we provide details on how the model can be extended to use it to evaluate
the role of aggregate shocks.
We assume that each household is divided into a mass Γ of shoppers/customers and a
representative worker. The preferences of the household are given by
Z
Γ
Et
Vt (pit , zti , ψti ) di
− Jt
,
(14)
0
where Vt (pit , zti , ψti ) is defined as in equation (1) and it is the value function that solves the
P
T −t
customer problem in Section 2.1. We denote as Jt ≡ φ ∞
`T with φ > 0 the disutility
T =t β
from the sequence of labor `T . The aggregate state for the household includes the distribution
of prices, the distribution of customers over the different firms and the level of income, the
wage, and their laws of motion. Given that we allow for aggregate shocks, we have to consider
the possibility that the aggregate state varies over time. We index dynamics in the aggregate
state through the time subscript t for the value function.
The worker chooses the path of `t that maximizes household preferences in equation (14).
The search problem of each customer is as described in Section 2.1. As for the consumption
decision, each customer allocates her income across consumption of the good sold in the local
market, the demand of which we denote by d, and another supplied in a centralized market
by a perfectly competitive firms, the demand of which we denote by n, to solve the following
problem
d
θ−1
θ
vt (pt ) = max
+n
θ−1
θ
θ
θ−1
(1−γ)
1−γ
d, n
s.t. pt d + qt n ≤ It ,
(15)
(16)
where θ > 1 and It ≡ (wt `t + Dt )/Γ is nominal income, which the customer takes as given.
Nominal income depends on the household labor income (wt `t ) and dividends from firms
ownership (Dt ). The first order condition to the problem in equations (15)-(16) delivers the
following standard downward sloping demand function for variety d
dt (pt ) = It
p−θ
t
.
pt1−θ + qt1−θ
(17)
Without loss of generality we use the price qt as the numeraire of the economy. From the first
order conditions for the household problem, we obtain that the stochastic discount factor is
46
−γ
RΓ
i
given by β Λt+1 /Λt , where Λt+s = 0 cit+s
/Pt+s
di is the household marginal increase
in utility with respect to nominal income; cit+s denotes customer i’s consumption basket in
1
i
= ((pit+s )1−θ + (qt+s )1−θ ) 1−θ is the associated price.
period t + s, and Pt+s
The production technology of the perfectly competitively sold good (good n) is linear
in labor, so that its supply is given by ytn = Zt `nt , where Zt is aggregate productivity, and
`nt is labor demand by this firm. The production technology of the other good (good d)
is also linear in labor, so that its supply is given by ytj = Zt ztj `jt , where Zt is aggregate
productivity, and `nt is labor demand by this firm, where j indexes one particular producer.
Perfect competition in the market for variety n and in the labor market implies that workers
are paid a wage equal to the marginal productivity of labor so that wt = qt Zt . Equilibrium
R1
in the labor markets requires `t = `nt + 0 `jt dj.
There are two exogenous driving processes in our economy: aggregate productivity Z and
the numeraire q. We consider an economy in steady state at period t0 where expectations
are such that Zt = 1 and qt = 1 for all t ≥ t0 . Notice that in this case the economy coincides
with the economy described in Section 2.
D
Numerical solution of the model
In order to solve the model, we start by setting the parameters. The parameters β, w, q, and I
are constant throughout the numerical exercises. For the set of estimated parameters Ωn =
[λn , ζn , ρn , σn ]0 , we set a search grid. The grid is different for each parameter, as they differ
both in their levels and in the sensitivity of the statistics of interest to their variation. We
consider a grid with an interval of 0.01 for σ, 0.05 for ρ, 0.5 for ζ, and 0.01 for λ. Each Ωn
corresponds to a particular combination of parameters among these grids. For each Ωn we
set θ to obtain E[εd (z)] = 7.
We next describe how we solve for the equilibrium of the model for a given combination
of parameters. We start by discretizing the AR(1) process for productivity to a Markov
chain featuring N = 25 different productivity values. We then conjecture an equilibrium
function P(z). Given our definition of equilibrium and the results of Proposition 2, we look
for equilibria where P(z) ∈ [p∗ (z̄), p∗ (z)] for each z, and P(z) is decreasing in z. Our initial
guess for P(z) is given by p∗ (z) for all z. We experiment with different initial guesses and
found that the algorithm always converges to the same equilibrium.
Given the guess for P(z), we can compute the continuation value of each customer as a
function of the current price and productivity, i.e. V̄ (p, z), and solve for the optimal search
and exit thresholds as described in Lemma 1. Given P(z) and the customers’ search and
exit thresholds we can solve for the distributions of customers Q(·) and H(·) as defined in
47
Definition 1. Notice that the latter also amounts to solve for a fixed point in the space of
functions. Here, standard arguments for the existence of a solution to invariant distribution
for Markov chains apply. Therefore, the assumption that F (z 0 |z) > 0 and ∆(p̂(z), z) > 0
ensure the existence of a unique K(z) that solves equation (9). Finally, given Q(·), H(·),
P(z) and V̄ (p, z), we solve the firm problem and the obtain optimal firm prices given by
the function p̂(z). We use p̂(z) to update our conjecture about equilibrium prices P(z), and
iterate this procedure until convergence to a fixed point where P(z) = p̂(z) for all z ∈ [z, z̄].
Once we have solved for the equilibrium of the model at given parameter values, we
construct the statistics to be matched to their data counterpart as follows.
• Log-price dispersion:
σ̂p ≡
sX
K(zj )(log(p̂(zj )) − Mp )2
j
P
where Mp = j K(zj ) log(p̂(zj )) and K(zj ) is the equilibrium fraction of customers
buying from firms with productivity zj .
• Average comovement between the probability of exiting the customer base and the
price:
b̂1 = Cov(E(z), log(p̂(z)))/(σp )2
P
where E(z) ≡ G(ψ̂(p̂(z), z))(1−H(V̄ (p̂(z), z))), and Cov(E(z), log(p̂(z))) = i K(zj )(log(p̂(zj ))−
P
Mp )(E(zj ) − ME ) and ME = j K(zj ) E(zj ).
• Dispersion in the marginal effect of the price on the probability of exiting the customer
base:
sX
K(zj )(b̂1 (zj ) − Mb̂1 )2
σ̂b1 =
j
where b̂1 (zj ) = G0 (ψ̂(p̂(z), z))/G(ψ̂(p̂(z), z))(1−H(V̄ (p̂(z), z)))2 and Mb̂1 =
P
i
K(zj )b̂1 (zj ).
The autocorrelation of prices, ρ̂p coincides with the parameter ρ governing the persistence
and autocorrelation of productivity. Thus the model-predicted statistics used to estimate
the parameters are given by the vector v(Ωn ) = [ρ̂p , σp , b̂1 , σ̂b1 ]0 . We then evaluate the
objective function (vd − v(Ωn ))0 (vd − v(Ωn )) at each iteration. We select as estimates the
parameter values from the proposed grid that minimize the objective function and check that
the optimum in the interior of the assumed grid.
E
Price distribution of individual UPCs
Kaplan and Menzio (forthcoming) perform a thorough study of the properties of the distri48
bution of prices in the grocery sector which is highly related to ours. However, our analysis
in section Section 6 focuses on a normalized store-level price index; whereas theirs considers
an index of dispersion of households’ expenditure in grocery stores (not necessarily at a same
store or chain). As such, our results on store baskets and their evidence on price distribution
and dispersion for bundles of goods cannot be directly compared.
However, Kaplan and Menzio (forthcoming) also present evidence at the single good
(UPC) level. Although this is not the relevant level of observation for our study, we use our
data to replicate their findings and establish that any difference between our and their results
on bundles of goods comes from the choice of a different object of interest, rather than from
some dishomogeneity in the underlying data.
In Figure 5 we plot a distribution comparable to the one Kaplan and Menzio (forthcoming)
report in their Figure 2, panel (a). In particular, we take the set of all the UPCs used to
compute the store-level price index whose distribution we depict in Figure 2. For each UPC
k sold in store j belonging to Metropolitan Statistical Area m, we take the price posted by
j(m)
the store in week t (Pkt ) and normalize it dividing it by the mean of the prices posted in
m
). In computing
the same week for the same UPC by the stores active in the same MSA (Pkt
the MSA average, we weight the different stores by their market shares. Formally, we define
the normalized price as follows
j(m)
j(m)
pkt
=
Pkt
m
Pkt
In Figure 5 we plot the distribution of the normalized price across UPCs, stores and
weeks. Just as the comparable figure in Kaplan and Menzio (forthcoming), the distribution
exhibits excess kurtosis. It is unimodal, has a peak close to the mean, and thicker tails than
a normal distribution with the same mean and variance.
F
Pass-through of idiosyncratic shocks
To measure pass-through of idiosyncratic shocks, we regress the log-price index of each store
in a given week on its log-cost index. The price index pj,mkt
is constructed as described in
t
Section 6 and the cost index is analogously computed using the data on replacement cost
provided by the retailer. To avoid inflating the short-term (weekly) pass-through due to the
persistence of both price and cost variables, we include in the specification lagged values of
the independent variable. We experiment with an alternative way to deal with the persistence
of the dependent variable by measuring the short-term pass-through using first differences.
Finally, we include time and market fixed effects to control for aggregate trend (e.g. demand
49
0
1
Density
2
3
Figure 5: Distribution of normalized UPC prices
0
.5
1
Normalized price
1.5
2
Notes: The histogram plots the distribution of normalized prices across UPCs, stores, and weeks. The normalized price of a
UPC in a week is defined as the ratio of the weekly price of the UPC at a store of the chain that shared data with us to the
average price of the same UPC in the Metropolitan Statistical Area where the store is located. The latter is computed using
the IRI Marketing database. The set of UPCs considered is that used to compute the store-level price index whose distribution
is presented in Figure 2; we discard UPCs whose coefficient of variation is larger than 1. The solid line plots the density of a
normal with the same mean and variance as the empirical distribution of the normalized prices.
shocks) that can move prices independently from cost shifts. The results are reported in
Table 4 and deliver a consistent picture. The weekly pass-through ranges between 13% and
24%, in line with the customer markets model predictions.
50
Table 4: Pass-through of idiosyncratic shocks
Dep. variable
(1)
log(pjt )
(2)
log(pjt )
log(costjt )
0.17***
0.24***
(0.04)
(0.09)
log(costjt−1 )
log(costjt−2 )
log(costjt−3 )
log(costjt−4 )
0.04
0.06
(0.03)
(0.04)
-0.01
0.02
(0.05)
(0.07)
0.06***
0.05*
(0.02)
(0.03)
0.07
0.07
(0.05)
(0.05)
∆ log(costjt )
(3)
∆ log(pjt )
0.13*
(0.07)
Observations
MSA f.e.
Time f.e.
12,915
No
No
8,295
Yes
Yes
8,295
Yes
Yes
Notes: An observation is a store(j )-week(t) pair. The dependent variable is the price index of the store and the independent
variables are the cost index of the store and its lags. Standard errors are in parenthesis and are clustered at the store level.
***: Significant at 1% **: Significant at 5% *: Significant at 10%.
51
@FedFRASER