The Agglomeration-Differentiation Tradeoff in ... - Yale University

The Agglomeration-Differentiation Tradeoff in Spatial Location Choice
Sumon Datta
Krannert School of Management
Purdue University
403 W. State Street
West Lafayette, IN 47907
Email: sdatta@purdue.edu
Phone: (765) 496-7747
Fax: (765) 494-9658
K. Sudhir
Yale School of Management
135 Prospect St, PO Box 208200
New Haven, CT 06520
Email: k.sudhir@yale.edu
Phone: (203) 432-3289
Fax: (203) 432-3003
June 2011

Abstract
Retailers often co-locate spatially to draw consumers, even though it increases price competition.
The paper develops a structural model of entry and location choice that isolates the
agglomeration benefit of co-location, after controlling for pure differentiation rationales for co-
location such as (1) high demand and/or low cost at the location; (2) zoning restrictions and (3)
format differentiation that minimizes the need for spatial differentiation. We augment entry and
location choice data used in the literature with revenue and price data to help identify the
agglomeration effect. We introduce a new approach to obtain zoning data across a large number
of markets that should be of general interest for a large stream of spatial location applications.
We find that agglomeration benefits explain a significant fraction of observed co-location. While
zoning restrictions have little direct impact on co-location, in combination with the
agglomeration benefit, they explain a surprisingly large fraction of observed co-location.
Keywords: Entry, Location Choice, Agglomeration, Differentiation, Zoning, Retail
Competition, Store Format, Discrete Games, Multiple Equilibria, Structural Modeling

1. Introduction
Spatial clustering is a common phenomenon in many types of retail markets such as
restaurants, automobile dealerships, electronics shops and bridal boutiques. That the
phenomenon is well recognized in the popular imagination is seen in the popular labels for such
retail clusters: hamburger alleys, restaurant rows, automobile malls etc. Consider for example,
the retail locations of competing grocery stores. Figure 1 shows the distribution of distance
between a grocery store and its nearest competitor in the three US states of New York,
Pennsylvania and Ohio. Somewhat surprisingly, over 45% of stores are located within 0.5 miles
of a competitor. What explains the high observed levels of co-location in grocery stores?
When a growing retailer embarks on a store expansion strategy it faces two key
questions: (1) Should it enter a particular market (entry decision), and if so, (2) Where within the
market should it locate the new store (location decision)? Economists have long recognized that
locating close to a competitor could increase profits by increasing aggregate demand at the
location even though the lack of spatial differentiation is likely to increase price competition
(Marshall, 1920). This agglomeration-differentiation tradeoff or volume-price tradeoff is a
central tradeoff in spatial location choice. Indeed, there is ample theoretical research (Varian
1980, Stahl 1982, Wolinsky 1983, Dudey 1990, Fischer and Harrington 1996, Bester 1998,
Arentze et al., 2005, Konishi 2005) to suggest that agglomeration benefits can act as an incentive
for firms to forego spatial differentiation. 1
But how can we measure the volume and price effects
due to competitors?
1 Some empirical evidence of the benefits of spatial co-location can be found in Fox et al. 2007 and Watson 2005.
Vitorino (2011) finds evidence for inter-store spillovers in a particular kind of retail cluster - Shopping Malls. But in
a mall setting, firms only make a strategic entry decision; they do not face the tradeoff of whether to co-locate or
spatially differentiate with rivals.
1

A retailer could use detailed household level data on consumer store choices across
several markets that vary in market characteristics (like population and income) and the number
of stores of different formats, and their locations, to estimate the benefit of agglomeration
(volume effect due to co-location) through a household level model of store choice. 2
Given these
household level estimates, one can then solve for a competitive pricing equilibrium to identify
the benefit of spatial differentiation (price effect due to differentiation). Such a method, however,
tends to be impractical because such detailed household level data across multiple retailers are
difficult to obtain and a household level analysis across markets is too onerous.
Another approach could be to use firm level data on revenues and prices of all stores
across several markets. Assuming store locations as given, one could develop a consumer
shopping behavior model to identify the benefit of agglomeration, coupled with a price
competition model to identify the benefit of differentiation. This approach, however, could suffer
from serious issues due to the endogeneity of market structure (i.e., number of firms that enter a
market and their locations). For instance, a location with a high unobserved demand shock is
likely to have higher revenues but the location is also likely to attract more firms. Not accounting
for the endogeneity of market structure can give biased estimates of the parameters capturing the
agglomeration benefit and the competitive interactions.
To infer the strategic interactions between firms, researchers in marketing and economics
have adopted an alternative empirical approach that uses readily observed entry and location
decisions of firms. The approach is built on the idea that firms take into account their
competitors’ actions when making their decisions. Thus, by solving for the location choice game
between firms we can infer the strategic interactions between firms. A vast majority of papers
2 For example, Fox et al., (2007) use data from a multi-outlet panel to study consumers’ shopping behavior and its
impact on store revenues. However their data is from a single major metropolitan market.
2

have used this approach to study firms’ entry decisions (e.g., Bresnahan and Reiss 1991; Berry
1992; Mazzeo 2002; Aguirregabiria and Mira 2007; Bajari et al., 2007; Vitorino 2011; Zhu,
Singh and Manuszak 2009; Ciliberto and Tamer 2009). 3
The reduced form approach is incapable of separating the ‘net effect’ of competitors into
a volume effect and a price competition effect which can independently describe the
agglomeration benefit from co-location and the benefit from spatial differentiation, respectively.
In particular, if consumers are in fact attracted to a location with multiple competing stores then
the demand at any location will be endogenous to a firm’s location choice decision and the
decisions of competitors. The reduced form approach cannot distinguish such endogenous
demand from the latent profit and is therefore unsuitable for studying firms’ agglomeration-
differentiation tradeoff.
3
A smaller literature has analyzed the
strategic location choice decisions (e.g., Seim 2006; Watson 2005; Orhun 2005; Zhu and Singh
2009), where firms not only decide whether to enter into a market but if they enter, where to
locate and how far to locate from a competitor. These structural models of location choice use a
reduced form profit function that allows latent profit in a location to depend on the number of
competitors, and their distances from that location. As we can only make the inference that a
firm’s chosen location must be more profitable in expectation than any alternative location, at
best, we can only estimate the average ‘net effect’ of competitors on firm profit. A net negative
effect is characterized as the competition effect and a decrease in the negative effect with the
distance of the competitor is highlighted to emphasize only the benefit of spatial differentiation.
Also, a crucial challenge in disentangling the agglomeration-differentiation tradeoff is
that observed co-location may be consistent with pure differentiation rationales. That is, even if
3 Some have addressed entry decisions of retail chains, considering how these chains build up their network (Jia
2008; Holmes 2008; Ellickson, Houghton, and Timmins 2008).

there are no agglomeration benefits firms may still locate close to each other when (a) there is
high demand at the location; (b) there is low cost at the location; (c) zoning regulations restrict
retailers to set-up stores in very concentrated areas and (d) the need for spatial differentiation is
lower when retailers can differentiate on other attributes or dimensions such as store formats.
Again, the existing structural models of firms’ entry and location choice decisions do not
incorporate most of these features. In this paper we develop a comprehensive structural model
that disentangles the agglomeration-differentiation tradeoff while simultaneously controlling for
the alternative explanations for co-location.
We make three crucial contributions to the literature. First, we introduce a novel
approach to obtain spatial zoning regulation data for any number of markets. Most towns and
cities in the U.S. practice single-use zoning wherein locations with high population and income
are often zoned as residential land where big-box retailers are not allowed to open stores.
Previous studies wrongly viewed the absence of stores in such locations as a strategic choice of
firms. Similarly, smaller and concentrated retail zones might force rivals to cluster together. But
previous models would infer such clustering by rivals to be the result of low competition. Datta
and Sudhir (2011) exhibit the role of zoning in firms’ entry location and store format choice
decisions, and the potential biases in inference that can result from ignoring zoning. Even though
the critical importance of spatial zoning is fairly well-known, extant research has completely
ignored this issue because of lack of availability of zoning data on a national scale across many
markets. To control for spatial zoning regulations, we use a publicly available, digital dataset
called National Land Cover Dataset (NLCD). In conjunction with Geographic Information
System (GIS) tools such as ArcGIS and Google Earth, we can recover zoning data in any number
of markets across the entire U.S. This is the first application of digital land cover data in
4

Marketing and the approach should be of general interest for a large stream of spatial location
applications.
Second, we decompose store profits into revenue and cost, and incorporate common
unobserved demand and cost shocks – Location specific demand (cost) characteristics such as,
say, traffic patterns (tax-breaks), which may be common knowledge for firms but which are
unobserved by the researcher. For this, we augment firms’ entry and location choice data with
store revenue data. 4 Extant structural models use a reduced form profit function that cannot
discern whether a location was chosen because of high demand or because of low costs.
Furthermore, when firms cluster in a location because the increased competition is more than
offset by an unobserved positive revenue (negative cost) shock at the location, existing models
would misattribute the co-location to low competition. 5
In our approach, the portion of observed
store revenue that is not explained by the observable demand factors or the observed market
structure is attributed to an unobserved revenue shock at the location which is a draw from a
distribution. Having accounted for revenue, we then identify the residual cost function through
the latent profit function and the data on observed entry and location choice decisions of firms.
Thus, we are able to infer how the observed market characteristics affect revenue and cost
differently which gives us better insights about the drivers of store location choice.
Third, we show how to disentangle the agglomeration-differentiation tradeoff by further
decomposing store revenue into its components of consumer shopping location choice based
volume and spatial competition based price. Specifically, for volume we model consumers’
4 Some recent research that has also used post-action performance data to gain richer insights about the drivers of
firms’ strategic decisions include Ellickson and Misra (2007) and Draganska et al., (2009).
5 Orhun (2005) attempts to control for location-specific common profit shocks. However, with only choice data, one
can only model latent profits whose errors have to be normalized for estimation. For instance, Orhun (2005)
assumed that the distribution of common profit shocks have a standard normal distribution.
5

shopping location choice, which incorporates the spatial configuration of firms around
consumers, and we model price as a function of the spatial configuration of rivals around a store.
Hence, the benefits of agglomeration are realized through increased volume potential while the
benefits of spatial differentiation emerge from acquiring a greater share of that potential as a
result of decreased price competition. As competitors affect both volume and price, a non-
parametric identification of the two effects would require additional data on sales or prices.
Without this data, one would have to rely on suitable functional form assumptions so that the
locations of rivals affect store volume differently from the way they affect store prices. Hence we
further augment our data with price data for a set of stores that belong to one store chain.
Finally, when different store formats specialize in different product categories or pricing
strategies or services, the need for spatial differentiation may be lower. In the context of grocery
stores, format types include Supermarkets, Superstores, Limited Assortment and Warehouse
stores, Natural Foods stores, Food and Drug stores and Supercenters and Wholesale Clubs.
Datta and Sudhir (2011) show that when zoning restrictions increase in a market the entering
grocery retailers are more likely to exhibit greater diversity in their store formats as a means to
mitigate the reduced the scope for spatial differentiation. Hence, we control for format
differentiation by accounting for the different store formats.
The empirical strategy to investigate firms’ entry and location decisions involves solving
a choice game where firms’ strategies are interrelated. We estimate a static, structural
simultaneous move game for firms’ entry and location choice decisions with incomplete
information between firms. 6
We use maximum likelihood estimation (MLE) for estimation of the
6 We do not have store entry dates which are required to solve a dynamic choice game. However, our model can be
extended to a dynamic set-up similar to Aguirregabiria and Vicentini (2006) who have proposed a dynamic model of
an oligopoly industry characterized by spatial competition.
6

static discrete game. 7 Estimation challenges include the possibility of multiple equilibria in the
model, multiple equilibria in the data, and slow convergence or potential non-convergence of the
MLE algorithm. 8
We build on recent developments in the empirical literature to address each of
these challenges and these are explained in detail in the estimation section.
Our estimates and counterfactual analysis show that the agglomeration effect is strong
and explains a significant fraction of observed co-location of grocery stores across several
markets. Surprisingly, zoning has little direct effect on co-location. But tighter zoning
restrictions interact with the agglomeration effect to explain a surprisingly large fraction of
observed collocation. We find that a small change in zoning can cause a discontinuous impact on
the location pattern. The finding that zoning regulations and the agglomeration effect interact to
shape market structure has important policy implications for local government bodies that make
zoning decisions. It also highlights the value of a structural model in understanding how a small
perturbation of market characteristics can cause strategic firms to respond in complex and
nonlinear ways.
The rest of the discussion is organized as follows: Section 2 describes the model and
estimation strategy. Section 3 describes the data and the approach for recovering spatial zoning
data. Section 4 describes the estimates of the model. Section 5 presents the results of
counterfactual simulations. Section 6 concludes with a summary of the findings and the
limitations of this research.
2. Model and Estimation Strategy
7
Alternatives to likelihood based approaches include method of moments (Thomadsen 2005; Draganska et al.,
2009), minimum distance or asymptotic least square estimators (Pakes et al., 2007; Bajari et al., 2007; Pesendorfer
and Schmidt-Dengler 2008) and maximum score estimators (Fox and Bajari 2010; Fox 2007; Ellickson et al., 2010).
8
See Aguirregabiria et. al., (2008) for a discussion on the distinction between multiple equilibria in model and
multiple equilibria in data.
7

2.1. A Comprehensive Model of Strategic Entry and Location Choice
The entry and location choice game involves a nested framework with two stages. In the
first stage, each firm, i, decides whether or not to enter a market m (m = 1, 2,…, M). In the
second stage, the entering firms simultaneously choose their respective store type or format, f (f
= 1, 2,…, F) and the store location within the market.
For the purposes of illustration, imagine a square city with a grid of L m discrete blocks or
‘locations’ (Figure 2(a)). In extant models, firm i's payoff at each location, l (l = 1, 2,…, L m ), is
modeled as a reduced-form function of the market characteristics at the location, xl, the actions
(entry and location choices) of all firms, a = (ai, a-i), and an idiosyncratic profit shock, ε il , which
is the firm’s private information and is known to rivals (and the researcher) only in distribution:
m m
π ( a ) =Π ( x , a)
+ ε
(1)
ifl i f l il
In this incomplete information setup, a firm cannot predict rivals’ discrete actions but it
has rational beliefs about their strategies. For example, suppose firms are homogeneous, then
each firm will make its decision based on its belief about the number of firms that would enter
the market ,
m
N , and its belief that an entering rival will choose a particular location as
represented by a vector of conditional location choice probabilities,
8
( )
m m
P P { p1, p2,..., p m
L }
= .
For instance, the firm may have a belief that a rival, conditional on entry, will choose location ‘j’
with probability p j . Hence, for homogeneous firms the expected profit at location l can be
written as (after dropping subscript ‘f’ for format):

9
( ( ⎡ ⎤ ) )
m m m m
E[ πil( ai)] =Π xl, E⎣N ⎦ , P + εil(2)
We build on this popular modeling approach and introduce several new features. First, in
the extant models, firms are allowed to consider all L m locations in the market so that each
location has some positive probability of being chosen by a firm. However, since firms are not
allowed to set up stores in residential locations, we use our zoning data to exclude such locations
and concentrate only on a subset of potential retail locations, l = {1, 2,…, lm} (Figure 2(b)).
Second, we break down the reduced-form profit into revenue and a cost multiplier. 9
We
allow both revenue and cost to include observed and unobserved (to the researcher) components.
Third, instead of an idiosyncratic profit shock, we assume an idiosyncratic cost shock. Formally,
we revise Equation (1) as follows:
( ) ( )
m r m c
π ( a ) = R x , a, υ * C x , υ , ς
(3)
ifl i fl l l ifl l l il
where, revenue has the following multiplicative form:
( ) ˆ ( )
r r
R x , a, υ = R x , a * υ
(4)
fl l l fl l l
R ˆ
fl is the observed component of store revenue that is a function of the store format, f,
the market characteristics at the location, xl, and the actions (entry, location and format choices)
r
of all firms, a. The unobserved component of revenue, υ l , is a common location-specific shock
that is common knowledge for all firms at the time of entry. It accounts for location-specific
demand characteristics such as traffic density that are unobserved by the researcher.
The cost multiplier in Equation (3) has the following multiplicative form:
( , υ , ε ) = ˆ ( ) * υ *exp ( ξ ) *exp ( ε )
C x C x
m c c m
ifl l l il fl l l il
9
As described later (Equation 15), we will consider the log transformation of Equation (3) which will yield the
familiar form: Profit = Revenue – Cost.
(5)

where, the observed component, ˆ C fl , is a function of the store format, f, and the market
characteristics at the location, xl,. The unobserved component of cost consists of three elements:
c
(a) A common location-specific shock, υ l , that is common knowledge for all firms at the time of
entry. It accounts for location-specific cost characteristics such as commercial taxes that are
unobserved by the researcher. Now, the common unobserved cost shock at a location is likely to
be correlated with the common unobserved revenue shock at the location. We empirically check
for this correlation through the following assumption about the distribution of the two shocks:
( )
c ( υl
)
r ⎛ln υ ⎞ r
2
l ⎛ω⎞ ⎛ l 0 ⎡ σ r ρσ rσ⎤⎞ ⎜ ⎟=
⎛ ⎞
c
⎜ ⎟ N ⎜ ,
c ⎜ ⎟ ⎢ 2 ⎥⎟
⎜ln ⎟ ⎜ω⎟ ⎜ 0 ρσ l ⎝ ⎠ rσc σ ⎟
⎝ ⎠ ⎝ ⎣ c
⎝ ⎠
⎦⎠
m
(b) An overall market-specific attractiveness parameter, exp(
)
all firms but is unobserved by the researcher.
10
(6)
ξ , that is common knowledge for
(c) The firm’s idiosyncratic cost shock at the location, exp( ε il ) , that is the firm’s private
information and known to rivals and the researcher only in distribution.
Finally, to separate the agglomeration-differentiation effect, we decompose the observed
component of revenue, ˆ fl R , into a consumer shopping location choice based volume ( v fl ) and a
competition effect based price index ( pr fl ).
( ) ( )
Rˆ = v * pr
fl fl fl
This decomposition of revenue will enable us to separate the volume and price effects of
competitors and thus distinguish the benefits of agglomeration that increase volume, from the
benefits of spatial differentiation that reduce price competition. We now describe the volume and
price components of revenue.
2.1.1. Consumers’ Shopping Location Choice Based Volume
(7)

We have detailed information about consumers up to the Census Block Group (CBG)
level. Hence, in what follows, we use demographic data at the CBG level and assume that
consumers are located at the population density weighted center of their respective CBG.
However, the model can easily be extended to a household level.
Consumers in each CBG, g, choose the store format and the retail location where they
want to shop. They incur a travel cost (Tgl) to go to a retail location l. This travel cost could be a
non-linear function of the distance, dgl, between the consumer’s location and the retail location.
We also allow the travel cost to differ by the median household income of the CBG (med_hhI),
the median age (med_age), and the minimum distance consumers have to travel before they can
get to the nearest retail location (min_d). For instance, a consumer, who is located deep within a
residential zone and is far from the nearest retail location, may be more willing to go to a store
that is farther away, than say, a consumer who is close to several retail locations. Formally, the
travel cost is given by:
( )
Tvl = α d + α d + α med _ hhI + α med _ age + α min_ d * d
gl 1 gl 2
2
gl 3 g 4 g 5
g gl
A consumer who wants to buy, let’s say, groceries, may be attracted to a particular
grocery store in location l if the location also consists of other commercial activities that cater to
the consumer’s non-grocery needs (e.g., electronics and apparel stores). That is, there could also
be economies of scope from one-stop shopping or multipurpose shopping ( α MS ). Hence, we
account for the extent of commercial activity in the location ( comm l ). In addition, if consumers
expect low prices at the store then they may be even more likely to visit the store. To control for
this price effect ( α pr ), we account for the price index of the store format, pr fl . The price index
specification is described in the next sub-section.
11
(8)

Next, a consumer shopping for groceries likely frequents locations where multiple
grocery stores are collocated (store agglomeration effect). Hence, we consider the effect of the
total number of competing stores at the location, Nl. We also consider any scope economies of
shopping within the grocery sector when grocery stores with different formats collocate (format
agglomeration effect). For instance, consumers may be more likely to visit a particular Food and
Drug store when it is located close to a Supercenter. Hence, we use an indicator, OF
I fl , for the
presence of store formats other than the focal format f, and we also allow the format
agglomeration effect to be format-specific. Finally, consumers may simply have a strong
intrinsic preference ( α f , Pref ) for the store format f and there could also have an unobserved
preference for the location, η gl . Formally, for a consumer in CBG g, the utility of shopping in
stores with format f in location l is:
U = Uˆ+ η
gfl gfl gl
ˆ =− + + ln + + + (9)
OF
and U Tvl α comm α ( pr ) α N α , I α ,
gfl gl MS l pr fl SA l f FA fl f Pref
We assume i.i.d. Type 1 extreme value distribution for the preference shock so that the
probability that a consumer in CBG g will shop in stores with format f in location l is given by
the standard logit form:
p
=
csr
gfl F
exp
∑ ∑
f '= 1 csr
j∈Lg ( Uˆ
gfl )
exp(
Uˆ
gf ' j )
where, the superscript ‘csr’ for the probability denotes that this is the choice probability of
consumers. We put a cap on consumers’ choice set by specifying that consumers may shop at
12
(10)

any retail location within a radius, Rad. 10
That is, we assume that Rad is the maximum distance
that a consumer will travel for shopping, and so in Equation (10), L is the set of retail locations
within the radius, Rad, from CBG g. Eventually, we estimate our model with different
specifications for Rad in order to empirically infer the maximum distance that consumers are
willing to travel.
Note that this maximum travel distance automatically implies that the trade radius of a
store (catchment area from where the store gets its customers) is Rad. Next, using consumers’
per capita income as a proxy for their consumption capacity or their purchasing ability, we
construct a metric called Customer Value ( CV fl ) for measuring the net worth of the consumers
who are attracted towards stores with format f in location l. For this, note that Equation (10) is
also the share of consumers located in CBG g, who will shop in stores with format f in location l.
We weigh consumers’ choice probability by the number of such consumers (CBG population,
Popg) and their per capita income (PCIg). 11
13
csr
g
Then the customer value metric, CVfl
, is obtained by
aggregating the influx of consumers from different CBGs around the location:
where,
location l:
csr ( )
CV = ∑ p Pop PCI . (11)
fl gfl g g
ret
g∈Ll ret
L l is the set of CBGs that lie within the trade radius, Rad, of location l.
We then transform this customer value metric into volume for stores with format f in
v fl CVfl α
= ⎡
⎣
⎤
⎦
10 If we do not impose such a cap on the maximum distance then the estimation becomes very cumbersome and slow
as our dataset consists of several large markets that consist of large number locations and CBGs.
11 We use per capita income for convenience. Alternatively, one could, of course, use other better variables such as
per capita expenditure on grocery.
fV ,
(12)

OF
Hence, in our framework, volume is endogenous to firms’ actions (through N l and I fl )
and it also depends on the market characteristics and consumer preferences.
2.1.2. Competitive Effect Based Price Index
Firms would like to differentiate spatially from rivals to reduce price competition. We
model the effect of competition on the price index of stores with format f that are in location l. 12
We use a flexible, semi-parametric approach so that the competition effect is split differentially
as a function of the store formats and distances of rivals from the location.
Similar to Seim (2006), we divide the area around a location (up to the trade radius, Rad)
into concentric circles or distance bands.
13
All rivals of a particular format type that are on a
distance band b (b = 1, 2,…, B) around location l are assumed to have the same effect on price.
Formally, the price index of a store with format f that is in location l is given by:
and
, * ( ) *exp ' ' *
'
pr
βx
⎛ ⎞
prfl = β f pr xl ⎜∑β f −fbNfbl+ ∑∑ β f −fbNfbl⎟
υl
⎝ b b f ≠ f ⎠
pr pr
2
ln( υl) ωlN(0, σ pr )
= (13)
where, β f , pr is a format-specific parameter which allows the intrinsic pricing ability of a store to
differ by the store format. This intrinsic pricing ability of a store format could be due to format-
specific differences in cost, efficiency, product mix, and service quality. However, we remain
agnostic about the specific reasons. The second component on the right-hand side allows the
pricing ability of firms to depend on exogenous observable location characteristics, xl. In our
application we use the per capita income of consumers within a 2 mile radius of the location to
allow for price discrimination or the ability to sell premium products in affluent neighborhoods.
12 We use sales-weighted prices across all categories in a store as the price index of the store.
13 Alternatively, one could employ a continuous distance weighting approach as in Orhun (2005).
14

The third component on the right-hand side of Equation 13 includes the intraformat
competition effect and the interformat competition effect. For intraformat competition we
consider the number of rivals that have the same format, f, as the focal firm and that are located
in distance band b around location l ( N fbl ). Here, β f − fb
15
is the competitive effect of one such
rival. For interformat competition, we consider the number of rivals that have a different format,
f’ ( f ' f
≠ ), and f ' fb
β − is the competitive effect of one such f’-format rival. If the estimates
reveal a weakening of the competitive effects at greater distance bands then that will indicate the
benefits of spatial differentiation whereas, estimates of weaker interformat competitive effects
relative to intraformat competitive effects within the same distance band will indicate the
benefits of format differentiation.
Finally, we introduce a common, location-specific price shock ( υ ) that is common
knowledge for all firms at the time of entry but is unknown to the researcher. We assume that
this price shock has a log-normal distribution.
To summarize, like volume, the price index also depends on market characteristics and is
endogenous to firms’ actions. As firms’ actions affect both volume and price, a non-parametric
identification of the competition effects would require price data for all stores. But we only have
price data for one store chain. Fortunately, this chain operates more than one store format and is
present in most markets in our dataset, and, therefore, experiences large variations in the spatial
distribution of market characteristics and rivals. Hence, despite its incomplete nature, the data
partly assists identification. However, we also rely on our functional form assumption of how
locations of rivals affect volume in a different way from how they affect prices.
pr
l

2.1.3. The Profit Function
Conforming to the multiplicative specifications so far, the observed component of the cost
multiplier, ˆ C fl (Equation 5), is specified as:
B
ˆ ⎛ ⎞
Cfl( xl) = exp⎜∑
γ fbxxbl ⎟
(14)
⎝ b=
1 ⎠
where, bl x are the observed cost shifters at distance band b around location l and γ fbx are format
and band specific cost parameters.
Substituting the expressions for revenue and cost into our profit specification, Equation
(3), then taking the log transformation, and after making some trivial sign reversals, we have a
equation for the transformed profit function that is very similar to equation (1):
( v pr ) ( Cˆ
)
( ) ( ) ( )
16
( )
r c m
π = ln π = ln + ln + ω − ln + ω + ξ + ε
ifl ifl fl fl l fl l il
2.4 Equilibrium Choice Probabilities:
Recall that the idiosyncratic cost shock, ε il , is known to rivals only in distribution. Due
to such incomplete information about rivals’ profits, a firm cannot exactly predict rivals’ discrete
actions but it can have rational expectations about rivals’ strategies. Hence, for a given set of
pr r c
vectors of price, revenue and cost shocks across all locations ( ω , ω , ω ), firm i can form
rational expectations about the number of firms that will enter the market, N m , and the location
and format choices of the (N m m m m m
-1) entering rivals, P P1 , P2 ,... PF
(15)
= ⎡
⎣
⎤
⎦
. That is, corresponding
to each format f (f’) firms we will have a vector of lm conditional choice probabilities (CCPs),
{ 1, 2,...,
}
P = p p p
m
f f f flm
m ( Pf ' { pf '1, pf '2 ,..., pf
'l
} ) m
= . For instance, fj
p ( f ' j)
p is a CCP of a f-

format (f’-format) rival and it represents the focal firm’s belief that a f-format (f’-format) rival
will choose location j when a total of
m
N firms enter the market.
Based on these beliefs, we can obtain expressions for the total number of competing
stores in a location ( [ l ] )
OF ( E⎡I ⎤ fl )
E N , the chance that there will be rivals with other formats in a location,
⎣ ⎦ and the number of f-format (f’-format) rivals in distance band b, E ⎡N⎤ fbl
17
⎣ ⎦ ( E ⎡
⎣N⎤ f 'bl
⎦ )
(Expressions for these expectations are shown in Appendix A). Consequently, given model
parameters and the vectors of location-specific shocks, firms can derive the expected values of
volume and price index which would then lead to the following expression for expected profit:
( ( ) ( ) ) ( ˆ )
( )
pr r c r c m
E ⎡ πiflω , ω , ω ⎤ E ⎡ln v ⎤ fl E ⎡ln pr ⎤
⎣ ⎦
=
⎣ ⎦
+
⎣ fl ⎦
+ ωl − ln c fl + ωl+ ξ + εil
(16)
Since fl v is a highly non-linear function of OF
N l and I , we will make the following
simplifying assumption:
OF
( ( ⎡ ⎤ ) )
( ) [ ]
E[ln vkl ] = E⎡ln vkl X, E Nl , E I fl ; α ⎤
⎣ ⎣ ⎦ ⎦ (17)
This expected volume can be calculated based on firms’ prediction of the outcome of
consumers’ shopping behavior as Equation (9) transforms into:
( ) [ ]
Uˆ =− Tvl + α comm + α E ⎡ln pr ⎤ α E N α E ⎡I⎤ ⎣ ⎦
+ + ⎣ ⎦+
α
gfl gl MS l pr fl SA l f , FA
OF
fl f ,Pref
We also have:
⎣
ln ( ) ⎦
ln ( ) ln ( ) ∑ ∑∑
E⎡ pr ⎤ = β + β x + β E⎡N ⎤+ β E⎡N ⎤+
ω
pr
fl f , pr x l f −fb⎣ fbl ⎦ f '−fb⎣ f 'bl⎦
l
b b f '≠
f
fl
. (18)
Thus, the expected profit in equation (16) can be rewritten as a function of the
equilibrium number of entrants in the market,
(19)
m
N , the equilibrium location choice probabilities

in the market for firms of all formats,
pr r c
ω , ω , ω ), and a set of model parameters, θ { αβγσρ , , , , }
m
P , the specific draws of price, revenue and cost shocks (
( )
= :
( )
pr r c m m pr r c m
E⎡ π , , ˆ
ifl ω ω ω ⎤
⎣ ⎦
= π fl N , P , ω , ω , ω , θ + ξ + εil
Note that m
ξ is common for all locations in the market and therefore does not influence
the location choice after firm i has decided to enter the market. Thus, if we assume that the
idiosyncratic component, ε il , has a Type 1 extreme value distribution that is independent across
locations and firms then the conditional probability (conditional on entry) that a f-format firm
chooses location l is given by the logit form:
m m pr r c
( N , P , , , , )
ψ ω ω ω θ =
fl F lm
∑∑
f '= 1 j=
1
18
m ( ˆ π fl ( N
m pr r c
P ω ω ω θ)
)
m
ˆ π f ' j(
N
m pr
P ω
r c
ω ω θ)
exp , , , , ,
( )
exp , , , , ,
Integrating over the distributions of the common unobserved shocks, we have the
location choice probability, conditional only on entry:
( ) ( )
m m m m pr r c pr r c pr r c
Ψ N , θ = ∫ ∫∫ Ψ N , ω , ω , ω , θ g( ω ) f( ω , ω ) dω dω dω
(22)
In equilibrium firms’ beliefs must match with rivals’ strategies. So:
( ; θ) ( , ; θ)
(20)
(21)
m m m m
P N = Ψ N P
(23)
This represents a system of equations that describes firms’ CCPs as the fixed point of a
continuous mapping between firms’ strategies and their beliefs about rivals’ strategies. As the
CCPs within market m must add up to 1, by Brouwer’s fixed point theorem, this system of
equations has at least one solution or fixed point.
Next, we normalize the profit from not entering a market to one so that the log of profit is
normalized to zero. Then the entry probability for a firm is given by the nested logit form:

pr r c
( ω , ω , ω , θ)
p Entry
F
lm
∑∑
f = 1 l=
1
19
( ˆ fl ( N P
) )
m m m pr r c
exp( ξ )* exp π , , ω , ω , ω , θ
f = 1 l=
1
=
F lm
m m m pr r c
1+ exp( ξ )* exp π , , ω , ω , ω , θ
∑∑
( ˆ fl ( N P
) )
Hence, if there are, say, E potential retail entrants then the expected total number of
entrants in market m is given by:
(24)
m
N = E * p( Entry)
(25)
By exogenously fixing E, and by observing the actual number of entrants,
the market specific cost parameter is:
( )
F m
( N ) ( E N ) ∑∑ ˆ fl ( N P
)
m
N , the estimate for
l
m pr r c m m ⎛ m m pr r c ⎞
ξ ω , ω , ω , θ = ln −ln − −ln⎜ exp π , , ω , ω , ω , θ ⎟
⎝ f = 1 l=
1
⎠
Again integrating over the distributions of the common unobserved shocks, we have:
( )
m m pr r c pr r c pr r c
ξ θ = ∫ ∫∫ ξ ω , ω , ω , θ g( ω ) f(
ω , ω ) dω dω dω
A simultaneous solution for Equations (23) and (27) gives the joint equilibrium
predictions for the number of entrants, and the format and location decisions of those entrants.
We assume that m
ξ is i.i.d. across markets, and follows a normal distribution,
Thus the probability that a total of
(26)
(27)
2
N ( µσ , ) .
m
N firms enter the market is given by the p.d.f. of this normal
distribution at the value obtained in Equation (27). Note that the value of m
ξ adjusts to the size
of E in relation to the outside option of no entry. Hence, although the size of E is not observed by

the researcher, varying the size will have only a miniscule effect on our inferences about firms’
strategies (See discussion in Seim (2006)).
Next, note that for a given θ and
m
N , we can get estimates of price and revenue when
firms’ locations are set to be identical to the observed spatial configuration of stores in the data.
We can compare these estimates with our price and revenue data and thus obtain the price and
pr r
revenue shocks, ( obv , obv )
ω θ ω θ , for the set of chosen locations that correspond to the observed
spatial configuration of stores in the data. These price and revenue shocks are included in the
likelihood function:
L
( )
Θ =
M
∏
m=
1
l ⎧ F m
I ( fl)
⎫
m m pr 2
r
⎨∏∏( ψ fl ( N , P ; θ) ) ⎬*
∏φ( ωobv θ,0, σ pr ) * ∏φ( ωobv θ,0,
Σ)
⎩ f = 1 l=
1
⎭
Price Data
Revenue Data
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ Location Choice
⎥
⎢ ⎥
⎢
m 2 * φξ ( ; µσ , ) ⎥
⎢ ⎥
⎢⎣ Entry Choice ⎥⎦
( θ) ( θ)
m m m m
s.t. P N ; =Ψ N , P ; , ∀ m
(28)
( )
2
where, Θ is the set of all model parameters { θ , µσ , }
Θ= , and I( fl ) is an indicator that
equals one if location l is chosen by a f-format firm, and is zero otherwise. φ is the pdf of a
normal distribution whereas φ has been used to indicate the pdf of the marginal distribution of
revenue shocks.
2.2 Estimation Strategy
2.2.1. Simplifying Restrictions
20

In the generalized model specification the number of model parameters increases
exponentially with the number of format types (F) due to the interformat and intraformat
competition effects (Equation 13). The number of distance bands (B) around each location
further explodes the number of parameters. For instance, in our empirical application in this
paper, we have six format types (F = 6). When we consider five 1-mile width distance bands
around each location (B = 5), the number of competition effect parameters is 180 (F 2 *B =
6*6*5). Also, the number of parameters for the observable component of cost (Equation 14) is
proportional to F*B. Furthermore, we are also constrained by data for only a limited set of
sample markets (small M). Hence, we make two restrictions in the model specification to reduce
the model parameters to a manageable number.
First, we assume that the competition effect between a pair of rivals is symmetric. That is,
for any distance band, b, and for two rivals with formats f and f’, we assume β f ' fb = β f f 'b.
In
21
− −
our empirical application for the grocery industry, this restriction implies that we treat the
competition effect of a Supermarket on a Superstore to be the same as the competition effect of a
Superstore on a Supermarket. Note however that we allow intraformat and inter-format effects to
be heterogeneous. Therefore, (1) the competition effect between two Supermarkets can be
different from that between two Superstores; and (2) the competition effect between, say, a
Supermarket and a Supercenter can be different from that between a Superstore and a
Supercenter.
Second, we assume that the ratio between the competition effect from a rival at a
particular distance band, b (b ≠ 1) and the competition effect from that rival in the first 0-1 mile
distance band is a constant value ( κ b ) . That is, we have:

β = β κ ; β = β κ ; ...; β = β κ
f −f 2 f −f1 2 f −f 3 f −f1 3 f −fB f −f1
B
β = β κ ; β = β κ ; ...; β = β κ
(29.1)
f '−f 2 f '−f1 2 f '−f 3 f '−f1 3 f '−fB f '−f1 B
( # competition effect parameters = ( F*( F + 1) / 2 ) + ( B-1)
)
Similarly, the impact of market characteristics on cost ( γ fbx ) are allowed to be format-
specific but we assume a constant ratio between the impact of a variable at a particular distance
band to the impact in the first 0-1 mile distance band. The constant is specific to the variable and
the particular distance band. For instance, suppose for cost (Equation 14) the coefficients of
population and per capita income in different distance bands are denoted by γ 1fb and γ 2 fb ,
respectively; then the restriction implies:
γ1= γ1 λ ; γ1= γ1 λ ; ... ; γ1= γ1 λ
f 2 f 1 2 f 3 f 1 3 fB f 1 B
γ2= γ2 ζ ; γ2= γ2 ζ ; ... ; γ2= γ2ζ f 2 f 1 2 f 3 f 1 3 fB f 1 B
( # observable cost component parameters ∝ ( F + B)
)
22
(29.2)
Note that if we allow the ratios or the multipliers, κb, λb and ζ b to be format-specific
then that is equivalent to directly estimating the format-specific coefficients, such as β f − fb , γ 1fb and γ 2 fb . In our estimation, we do not impose any restrictions on the values that the multipliers
can take at different distance bands. If these multipliers turn out to be decreasing with distance
and less than one then that would imply that the impact of the variable weakens with distance. In
particular, weakening of the competitive effects at greater distances would indicate the benefits
of spatial differentiation.
2.2.2. Multiple Equilibria in the Model

* *
Estimation involves finding the equilibrium solution, ( MLE , MLE )
23
P Θ , which is the global
optimum of Equation (28) where, *
Θ MLE are the Maximum Likelihood Estimates (MLE) and
*
P MLE are the corresponding equilibrium CCPs. Using a nested fixed-point (NFXP) approach for
estimation is computationally demanding as it involves solving for the fixed-point of Equation
pr r c
(22) for each draw of ⎡
⎣ω , ω , ω ⎤
⎦
and at each step of the likelihood maximization. More
importantly, NFXP suffers from the possibility of multiple equilibria in the model. Specifically,
for a value of θ , if Equation (22) has multiple solutions for CCPs then the likelihood is not well
defined. 14
A recursive extension of the PML, called the Nested Pseudo Likelihood (NPL) approach
addresses this problem at a relatively small additional computational cost (Aguirregabiria and
Mira, 2007).
Researchers have, therefore, developed two-step estimation approaches that avoid
these problems. In a two-step Pseudo Maximum Likelihood (PML) approach, the CCPs are
estimated in a parametric or nonparametric first step and the parameter estimates are obtained by
maximizing the resulting likelihood in the second step (Bajari et. al., 2007). However, in most
empirical contexts, consistent and precise first-stage estimates of CCPs are infeasible.
15
The standard NPL approach starts with an initial guess of the CCPs, and
converges to an equilibrium solution in the limit. For example, in our case, we would start with
initial guess values for firms’ beliefs about rivals’ CCPs, P . Then, using Equations (21) through
(28) we would obtain the likelihood, ( 0 ) ,
parameter estimates, 1
Θ , and new CCPs, 1
0
L P Θ . Maximizing the likelihood would give the
P . This would constitute one iteration, and the new
14 One way to deal with this problem is to provide sufficient conditions that the parameters, θ, must satisfy to ensure
a unique equilibrium (e.g., Seim, 2006; Zhu and Singh, 2009).
15 Another application of the NPL approach for a static game can be found in Ellickson and Misra (2008).

CCPs would be used for firms’ beliefs about rivals’ actions in the next iteration. The n th iteration
of the standard NPL approach can be denoted by the following contraction mapping, M:
( Pn, n) ( Pn−1) where, n arg max L( Pn−1, ) ; Pn ( Pn−1,
n)
Θ =Μ Θ = Θ =Ψ Θ (30)
Θ
For a graphical illustration of the NPL iterations, suppose that the set ( P, Θ ) could be
‘collapsed’ onto one axis. In Figure 2(a) the X-axis corresponds to the vector Pn − 1,
the Y-axis
corresponds to the set ( Pn, Θ n)
, and the solid curve represents the contraction mapping ( P)
24
Μ .
The dotted lines represent the ‘track’ followed by the NPL iterations corresponding to a
particular starting value, P 0 . Note that a different starting value,
'
P 0 , would result in a different
track for the NPL iterations. With multiple iterations, if there is convergence, the contraction
* *
mapping would converge to an equilibrium solution or a NPL fixed point, ( , )
P Θ . In Figure
3(a), this is the point where Μ ( P)
intersects the 45 o line. Furthermore, if the fixed point is
* *
unique then it is, in fact, the global optimum, ( MLE , MLE )
2.2.2. Multiple Equilibria in the Data
P Θ .
The standard NPL approach, however, does not address the possibility of multiple
equilibria in the data which is when the contraction mapping in Equation (31) does not have a
unique NPL fixed point. The multiple eqilibria or the multiple NPL fixed points are essentially
the different ‘local optima’ of Equation (29). This is illustrated in Figure 3(b) where Μ ( P)
intersects the 45 o line at multiple points. Consequently, the NPL iterations may potentially
converge to a ‘local optima’ and not the global optimum. Further, as the track followed by the

NPL iterations depends on the starting value, P 0 , different starting values would result in distinct
tracks which could potentially converge to different ‘local optima’. One option is to spread the
search for the global optimum over a wide range of the contraction mapping, Μ ( P)
, by using
parallel-NPL where a large number of NPL algorithms, say, T, are run in parallel with different
starting values. By thus following T distinct tracks for the NPL iterations, this approach, upon
1* 1* 2* 2* T* T*
convergence, would give us a set of T fixed points, ( P , Θ ) ; ( P , Θ ) ;...; ( P , Θ )
25
⎡ ⎤
⎣ ⎦ 16
.
* * ( PMLE , ΘMLE
)
However, it does not guarantee that this set will contain the global optimum, .
For a more efficient search of the global optimum, Aguirregabiria and Mira (2005)
propose combining the parallel-NPL with a Genetic Algorithm (GA). GA is a search heuristic
that mimics natural evolution processes such as ‘selection’, ‘crossover’ or ‘reproduction’ and
‘mutation’, and can be used to obtain the global optimum of complex optimization problems.
Combining the parallel-NPL with GA has two advantages – (1) The crossover and mutation
steps spread the search for the global optimum over a much wider range of the contraction
mapping than what is feasible with just the parallel-NPL, and (2) The selection step steers the
tracks of the parallel-NPL iterations towards those regions of the contraction mapping that are
more likely to contain the global optimum. 17
In our estimation, we insert two GA steps after each iteration of the parallel-NPL. Note
that after the n th iteration of the parallel-NPL, we will have T vectors of CCPs,
⎡ ⎤
1 2 T
⎣Pn; Pn ;...; Pn
⎦ .
First, in a selection step, we evaluate each vector of CCPs by using a ‘fitness criterion’ where the
16 Many of the fixed points may be identical.
17 Su and Judd (2010) suggest using a Mathematical Programming with Equilibrium Constraints approach that finds
the parameter estimates and the equilibrium CCPs simultaneously. However, like the parallel-NPL, this approach
also relies on multiple runs with different starting values to find different equilibria. Hence, its ability to find the
global optimum in problems that have a large action space (as in our entry and location choice problem) is unclear.

CCPs that are likely to be closer to the global optimum are considered to be more fit. Analogous
to the natural selection process in nature, the more fit CCPs are given a greater chance of
survival and reproduction so that future search for the global optimum is concentrated in their
neighborhood. This is done by drawing, with replacement, T ‘mother’ CCPs,
and T ‘father’ CCPs,
1'' 2'' T ''
⎡
⎣Pn ; Pn ;...; P ⎤ n ⎦
, from the original set,
more fit CCPs have a greater chance of getting selected.
1' 1'' 2' 2'' T'T'' Next, each of the T ‘couples’, ( Pn , Pn ) ; ( Pn , Pn ) ;...; ( Pn , Pn
)
26
⎡ ⎤
1' 2' T '
⎣Pn ; Pn ;...; Pn
⎦ ,
1 2 T
⎡
⎣Pn; Pn ;...; P ⎤ n ⎦
, such that the
⎡ ⎤
⎣ ⎦
, go through a
crossover step to produce an ‘offspring’ that inherits the traits of both its parent CCPs. To the
extent that both parents are likely to be fit, the resulting offspring also has a high chance of being
fit. Hence, we obtain a new generation of T vectors of CCPs that are likely to be quite close to
the global optimum. To further reduce the chances of missing the global optimum, some
mutations may be implanted into the offsprings so that the search continues to span a wide range
of the contraction mapping. With multiple iterations of the parallel-NPL and GA steps, if there is
convergence, we would obtain a set of T fixed points which almost certainly would contain the
global optimum.
2.2.3. Convergence
The algorithm may not converge to the global optimum if the contraction mapping does
not have good local convergence properties around the global optimum. Intuitively, as shown in
Figure 3(c), convergence to a fixed point depends on the concavity or the convexity of the
mapping in the neighborhood of that fixed point. Kasahara and Shimotsu (2008) recommend
transforming the mapping by replacing ( P,
)
( P,
)
Ψ Θ and P :
Ψ Θ with the following log-linear combination of

( ) ( )
δ 1−δ
Λ P, Θ = ⎡ P, ⎤
⎣
Ψ Θ
⎦
⎡⎣P⎤⎦ ; δ ∈[0,1]
Note that P =Λ( P,
Θ ) and P ( P,
)
27
(31)
=Ψ Θ have the same fixed-point solution(s). An
appropriate value of δ can modify the concavity or convexity of the mapping such that the
transformed mapping is Locally Contractive around the fixed point and will converge even if the
original mapping does not. 18
Finally, even when the mapping does converge, the rate of
convergence could be extremely slow and may require a large number of iterations. To avoid
this, Kasahara and Shimotsu (2008) propose the following q-stage operator called q-NPL:
( ( ( ( ) ) ) )
q
Λ ( P, Θ ) =Λ Λ ... Λ P,
Θ , Θ ,..., Θ , Θ
q times
q
Again, P =Λ ( P,
Θ ) and P ( P,
)
(32)
=Ψ Θ have the same fixed-point solution(s). In
q
addition, Λ ( P,
Θ ) also has the locally contractive property of ( P,
)
Λ Θ . Hence, in our
estimation, we replace the standard NPL operator, Ψ , with the Locally Contractive, q-NPL
operator,
q
Λ . The resulting parallel NPL iterations are then combined with GA as described
above. This procedure searches efficiently over the space of possible equilibria and converges
fast to a set of equilibria which almost certainly contains the global optimum. Details of the
sequence of steps involved in estimation are provided in Appendix A2.
2.3 Identification
18 Kasahara and Shimotsu (2008) suggest the following procedure for selecting the value of δ : Simulate a sequence
N
{ P n} n=
0
by iterating the transformed mapping for different values of δ , say for δ ∈ { 0.1,0.2,...,0.9}
. Then
pick the value of δ that leads to the smallest value of the mean of
P P
P − P
n+ 1 n −
n N
across n = 1,…, N.

Extant models of location choice use only entry and spatial location choice data. They
exploit the variation in exogenous market characteristics around a location and the number and
geographical locations of rivals, in order to identify the effects of market characteristics and the
nature of competition. Given the entry and location choice data, they can only obtain make
inferences that the level of profits where a firm locates is greater than in locations where they do
not locate, conditional on what they expect competitors to do.
However, for identifying the agglomeration effect, we need to go beyond the profits and
decompose the quantity (demand) enhancing effects of agglomeration and the margin effects of
differentiation. We augment extant models with revenue data and price data. The revenue data
now helps isolate the cost impact from profits. The price data helps separate revenues into its
quantity and price components.
Identifying the quantity component helps to isolate the agglomeration benefit. We note
that the price data we have is only from one chain which has stores of different formats. Hence
the competitive effect on price is identified non-parametrically only in areas where this chain
locates its stores. We make the assumption that the price effect is identical across all stores of the
same format to facilitate identification in other locations.
3 Data
3.1 Store Data and Sample Markets
We investigate the spatial configuration of big-box grocery stores. We have store location
(latitude and longitude), store format and weekly revenue data at the national level for the period
2007-08 from Nielsen’s ‘Trade Dimensions’. For our analysis, we use average weekly store
revenue data. In a different dataset, we have store location and store format data (but no revenue
data) for the period 2000-01 and for a sample of local markets in the three states of New York,
28

Pennsylvania and Ohio. This second dataset also has price index data for stores belonging to one
store chain. 19
In our price model, price data from a different time period can be used to estimate
the competition parameters and the distribution of price shocks as long as we use the market
configuration and market characteristics corresponding to that period, and if we assume that the
price shocks do not change over the seven year period. Hence, we combine the two data sets so
that for a sample of markets we have the market configuration and revenue data for all stores in
one period (2008), and the market configuration and price data for one of the stores in many
markets, but for a different period (2001). The data constraint of having prices for only one store
chain may appear as a serious weakness. However, as discussed above, it aids our identification.
Also, it is interesting from a managerial perspective as it mimics a more realistic situation where
firms are likely to have more information about themselves (own prices and own revenue) and
relatively less information about rivals (only revenue information about rivals but no price index
information).
Among the markets for which we have price data, we select a sample of 98 fairly
isolated, small and medium sized towns to avoid the problems associated with large markets and
suburbs such as unclear market boundaries, cannibalization due to multiple stores of a firm in the
same market, and complex sub-zoning regulations. In 2008, our 98 sample markets had
19
We have weekly product category-level price index data for a one year period for 27 grocery product categories
and for each store that belongs to the store chain ( pr = ∑∑ w * pr ; where, w ciuts is the revenue share
cts ciuts ciuts
∀∈ i c∀u∈i of UPC, u, of item, i, within product category, c, for week t in store s). To construct store-level price indices we
adopt an approach similar to Chevalier et al., (2003; p. 22). That is, we aggregate over the product categories and
27 52
weeks to form a store-level price index ( pr = ∑∑ w * pr ; where w cts is the dollar share of category c in
week t in store s).
s cts cts
c= 1 t=
1
29

altogether 438 big-box grocery stores. 20
These stores have been classified into six format types
(i.e., F = 6): Supermarkets (SM), Superstores (SS), Supercenters and Wholesale Clubs (SC),
Limited Assortment and Warehouse stores (LA), Natural Foods stores (NF) and Food and Drug
stores (FD). Table 1 provides a description of these store formats.
3.2. Consumer and Retail Locations
Data on market characteristics are obtained from the U.S. Census. Although detailed
demographic data at a Census Block Group (CBG) level are available only for the year 2000, the
U.S. Census provides annual census projections for the county level. Hence, we project the CBG
level census data to their 2008 values by the proportion of change in the respective counties
between 2000 and 2008. As we do not have information about consumers beyond the CBG level,
we follow the convention in the literature and place consumers in a CBG at the population
weighted center of the CBG. These are our consumer locations.
For the location choice game, we divide a market into a uniform grid of discrete 1 sq.
mile blocks or market locations. Our 98 sample markets have a total of 4,792 such locations. But
zoning regulations dictate which of these locations are available for big-box retailers. Below, we
discuss our approach for identifying these potential retail locations and their commercial
centers. Just as consumers are placed at the population weighted center of CBGs, we place
retailers within a retail location at the commercial center of the location.
Our concept of market locations deviates from the standard approach in earlier research
that treats census divisions as market locations and places retail stores at the population weighted
center along with consumers. The standard approach simplifies the data setup process but it has
severe drawbacks: (1) The population weighted center of a census division is likely to be a
20 A comparison of the market configurations between 2001 and 2008 showed that the number of stores in these
markets increased less than 10% from 399 to 438.
30

esidential zone so that placing retail stores there would confound the inclusion of zoning
regulations; (2) Stores are rarely present in the interior of a census division, rather, they are
present on roads that border these census divisions; (3) Census divisions vary extensively in size
so that, for large census divisions, stores may be located quite far from the center and also quite
far from each other. Such artificial distortions in distances between rivals can be very damaging
for our application as we are interested in explaining co-location of rivals through consumers’
willingness to travel to such locations. Our concept of market locations not only allows us to
incorporate spatial zoning regulations but it also avoids major distortions of the distances
between rivals and the distances of stores from population centers. 21
We next describe the National Land Cover Dataset (NLCD) and discuss how it is used in
conjunction with Geographical Information System tools such as ArcGIS and Google Earth to
recover the potential retail locations and their commercial centers.
3.3 Spatial Zoning Data
Multi-Resolution Land Characteristics Consortium, a conglomerate of several federal
agencies, has created two NLCD datasets that provide consistent and accurate digital land-cover
information for the coterminous U.S. The first national land-cover mapping project, NLCD 1992,
was derived from the early to mid-1990s Landsat Thematic Mapper satellite data. It applied a 21-
class, geo-referenced, land-cover classification (see Vogelmann et al., 2001). The second project,
NLCD 2001, updated the data for the year 2001 (see Homer et al., 2004). Both datasets have a
spatial resolution of 30 meters. That is, every 30 sq. meter area of land is classified as a specific
land type (e.g., deciduous forest, grassland, open water, etc.) and is allocated one pixel point with
21 In this paper, distance between two points always refers to the great-circle distance.
31

a distinct color code and the associated latitude and longitude. 22
Step 1: Constructing Market Boundaries and Market Locations
32
Interestingly, the land type
classifications include residential and commercial land. Residential land is further classified into
low and high intensity residential land, and commercial land comprises of highly developed
areas that do not include residential areas. We use the NLCD data in the following three steps to
identify the potential retail locations and their commercial centers.
We use the data in NLCD 2001 to construct the market boundaries of our sample markets.
The residential and commercial land area pixel points in each market are projected on a map by
using the ArcGIS software. This gives us the spatial area of interest for a market. A simple visual
inspection of the pixel density is used to construct the market boundaries where the pixels fade
away (See Figure 4(a)). As our sample markets are reasonably isolated from other towns and
cities, we can be flexible in choosing the shape of their boundaries. A rectangular shape is
preferred so that a market can be easily divided into a uniform grid of discrete blocks or market
locations. Thus, we construct imaginary rectangular borders (L miles X H miles where L and H
are integers that vary across markets) around the residential and commercial pixel points of each
market and then divide the market, specifically, into 1 sq. mile locations (See Figure 4(b)).
Step 2: Commercial Activity and Commercial Center in a Location
The extent of commercial activity in a location (as defined above) could affect firms’
profit in the location if consumers have a preference for multi-purpose shopping or one-stop
shopping. For instance, when shopping for groceries, consumers may like to combine their
shopping trip with non-grocery purchases such as clothing and electronics so that locations with
more retail businesses may be more attractive to firms. We isolate the NLCD 2001 pixel points
22
A pixel point is one of the individual dots that make up a graphical image. Each pixel point combines red, green,
and blue phosphors to create a specific color.

that correspond to commercial land with retail businesses (See Appendix C for technical details)
and use the number of pixel points in a location as a measure for the extent of commercial
activity in that location. The mean of the latitudes and longitudes of the commercial land pixel
points in a location gives us the commercial center of the location (See Figure 4(c)). We place all
retail stores within a location at the commercial center of that location.
Step 3: Discerning Potential Retail Locations from other Commercial Locations
The market locations which contain the commercial land pixel points are the commercial
locations and they constitute a very small share of all market locations. The locations without
any commercial activity are mostly residential locations and some barren land. Hence, we
account for residential zoning by excluding locations that do not have any commercial land pixel
points. But even within commercial locations, not all locations may be open to big-box retailers.
For instance, some commercial zones like, say, downtown areas, might only allow small
businesses such as banks and restaurants. An obvious candidate for a potential retail location for
big-box stores is any commercial location that has at least one big-box store which could be a
grocery store or a non-grocery store. Hence, we project the locations on to Google Earth and use
a tool called ‘Places Categories’ which shows the locations of various types of businesses in a
region (See Figure 4(d)). We carefully comb through the commercial locations, and specifically
check for the presence of major retail stores, major grocery stores and shopping centers to
identify the commercial locations that have at least one big-box store.
Now, the absence of big-box stores in a commercial location does not necessarily imply
that such stores are not allowed in that location. In particular, a commercial location that is open
to big-box stores may not have any such store if it is in an unfavorable or poor neighborhood and
33

cannot support a big store. 23
As we do not have a precise method for identifying such locations,
we use a stylized selection procedure. For each market, we find the minimum value of the total
income of consumers within a 2-mile radius of the commercial locations that have big-box
stores. We use this minimum as a benchmark for a commercial location in the market to be
attractive enough to support at least one big-box retail store. That is, if a commercial location
does not have any big-box store and the total income of consumers within a 2-mile radius of the
location is less than the market benchmark then we presume that the absence of a big-box store is
due to the unattractiveness of the location and not necessarily because of zoning restrictions.
Hence, a commercial location with no big-box store is still treated as a potential retail location
when the following condition is satisfied:
Income in 2-mile radius of a commercial
location that has no big-box store
≤
⎧Income in 2-mile radius of a commercial⎫
min ⎨ ⎬
⎩location that has a big-box store ⎭
To summarize, we use the NLCD data to construct market boundaries so that each market
can be divided into a grid of 1 sq. mile locations. Then the commercial land pixel points are used
to obtain the extent of commercial activity in a location and also to locate the commercial center
of the location. Extant models that do not account for zoning, assume that firms are allowed to
set up stores in any market location. In contrast, we account for residential zoning by excluding
locations that do not have any commercial land pixel points. Finally, we account for zoning
regulations particularly against big-box retailers, within commercial locations, by defining
potential retail locations as those commercial locations that (1) have at least one big-box store
which is either a grocery or a non-grocery store, and (2) do not have a big-box store and are in a
poor neighborhood which is below the market benchmark as described above.
23 Note that competition between stores in neighboring locations cannot explain the absence of big-box stores in a
location as we are considering big-box stores across any segment of the retail industry.
34

4. Results
The estimation results are presented in three parts. Table 2(a) presents the estimates for
the consumer shopping location choice or the demand side of the model. Table 2(b) presents the
results of the price index portion of the model. Finally, the estimates of cost and unobserved
shocks are presented in Table 2(c).
The demand side estimates indicate that consumers experience a negative travel cost that
is convex with respect to distance (The coefficient of 2
dgl is positive and significant). 24
Consumers who are far away from the nearest retail location (That is, when the value of
35
min_dg
is large), are more willing to travel long distances to get to a grocery store. Demographic
characteristics seem to have very little explanatory power for consumers’ travel costs.
The results show that consumers not only value economies of scope from the presence of
other, non-grocery businesses at a location but they also value the agglomeration of multiple
grocery stores at the location. The store agglomeration parameter ( α SA = 0.5342) is positive and
significant which suggests that consumers likely visit locations with multiple grocery stores. The
format agglomeration effect ( α f , FA ) is also positive and even significant for a few store formats
(Supercenters, Limited Assortment stores, and Food and Drug stores). Hence, consumers are
more likely to visit locations with multiple grocery stores when the cluster of stores consists of
different formats. Consequently, strategic store and format agglomeration increase consumers’
propensity to shop at a location, thus increasing volume at that location.
24 Comparing the results (not presented here) with different specifications for the maximum distance that consumers
may travel for shopping, Rad, suggested that a distance of 5 miles was sufficient. Rad values of 6 miles and above
did not change parameter estimates or increase the likelihood value significantly (vis-à-vis AIC and BIC criteria).
On the other hand, Rad values of 4 miles and below resulted in significantly different estimates for some model
parameters and also gave significantly smaller likelihood values.

Finally, consumers have a high preference for Supercenters and for Food and Drug stores
relative to the Supermarket format. Hence, consumers may be more willing to travel long
distances to get to such stores. Consumers have a relatively low preference for Limited
Assortment stores. This could be because Limited Assortment stores generally carry more name-
brand products and very few national brand products.
The results of the price index portion of the model (Table 2(b)) show that the format-
specific price constant, β f , pr , is lowest for Limited Assortment stores and Supercenters. This is
expected since the stores with these formats are typically EDLP stores or they offer relatively
more name-brand products that have low prices. For the effect of competition, recall that we
allowed for separate intra-format and inter-format competition, and we considered competition
from rivals in various 1-mile width distance bands (B = 5 when Rad = 5 miles). Our results show
that the competition effect decreases dramatically with distance.
Not surprisingly, intraformat competition is generally more severe than interformat
competition. The extent of intraformat competition is the highest between Food and Drug
combination stores, which is comparable to the competition between Supercenters. Superstores
are also found to compete quite heavily with each other. Interestingly, for some formats, the
interformat competition effect is found to be comparable to the intraformat competition effect.
For instance, the competition effect between Supermarkets and Superstores is quite comparable
to that between two Supermarkets. The competition effect between Superstores and Food and
Drug combination stores also seems to be quite high. The results highlight the importance of
accounting for format differentiation, in addition to spatial differentiation.
To explore the value of separating the agglomeration-differentiation effects of rivals, we
estimated a model that did not incorporate agglomeration benefits in the consumer model
36

(Parameter estimates not shown). The results showed that the competition effects are biased
downwards for all format types. The bias was more severe for inter-format competition. Hence,
not modeling the agglomeration-differentiation tradeoff can highly underestimates the
competition intensity between stores with different formats. In retrospect, this is expected
because without an agglomeration effect the model would misattribute observed collocation of
stores in the data to low competition. Since the agglomeration benefit is higher when the cluster
of stores has different formats, it is understandable that the inter-format competition effect is
more biased.
Finally, the estimates of cost and unobserved shocks (Table 2(c)) also give some
interesting insights. Although the Supercenter format enjoys a high preference from consumers,
it also tends to incur high costs in densely populated neighborhoods. We find a strong negative
correlation between the location-specific cost shocks and demand shocks (-0.8932). This
conforms to the intuition that locations with high revenue potential are likely to be associated
with high costs.
5. Counterfactual Simulations
We report two counterfactual simulations which help assess the relative importance of
zoning and agglomeration effects. We consider three alternative scenarios: (1) There are no
zoning regulations in any market and consumers do not benefit from co-location of stores (i.e.,
‘Neither Zoning nor Agglomeration’), (2) Markets have zoning regulations but there are no
benefits from co-location (i.e., ‘Only Zoning; No Agglomeration’), (3) Agglomeration benefits
exist but there are no zoning regulations (i.e., ‘No Zoning; Only Agglomeration’), and (4) Both
zoning and agglomeration benefits exist. For the set of 98 sample markets we estimate the
equilibrium CCPs under these alternative market conditions, assuming that the equilibrium
37

number of entrants remains unchanged (An appropriate change in the market-specific terms, m
ξ ,
would ensure this). We use the estimated model parameters and find the fixed point of the
system of equations shown in Equation (23). For this, we use the NFXP approach.
Figure 5(a) shows the distribution of inter-store distance across the 98 markets, under the
first scenario of ‘Neither Zoning nor Agglomeration’. We see that only 28% of stores co-locate
within 1 mile of each other. 25
This level of co-location may be due to concentration of high
demand or low cost. When we turn on zoning (‘Only Zoning; No Agglomeration’), 32% of stores
are located within 1 mile of each other (Figure 5(b)). This suggests that zoning may force firms
to come a little closer to each other but it has very limited direct impact on their collocation
behavior. With only agglomeration turned on (Figure 5(c) - ‘No Zoning; Only Agglomeration’),
43% of stores located within 1 mile of each other, suggesting that agglomeration effects have a
substantial impact on collocation. Interestingly, the interaction between zoning and
agglomeration benefit is extremely high because when both effects coexist, co-location increases
to 60% (Figure 5(d) – ‘Both Zoning and Agglomeration’). This is quite close to the amount of
co-location that we observe in our sample data. Thus the impact of zoning on co-location is high
only in the presence of agglomeration benefits. Why is there an interaction effect between zoning
and agglomeration benefits?
To understand this interaction, we perform our second counterfactual analysis in a
hypothetical market where we gradually increase the zoning restriction which restricts the scope
for spatial differentiation. For a set of four grocery stores, the optimal locations are shown in
Figure 6. In the less restrictive zoning setting, we find that stores are located at the extremes of
25 For this counterfactual simulation, we are counting stores within 1 mi. of a rival as a co-located store. It is
plausible that the two stores belong to two neighboring 1 sq. mi. block retail location whose commercial centers
happen to be within 1 mi. of each other.
38

the commercial zone, suggesting that zoning restrictions constrain the extent of spatial
differentiation in this market. When zoning is made more stringent, one would expect that stores
would continue to be at the edges of the commercial zone. However, the optimal locations reveal
a surprising pattern. When zoning is more restrictive, we find that some stores actually
agglomerate. In retrospect, we can understand the logic of why this happens. When zoning is
relaxed, stores can be more spread out allowing for benefits of spatial differentiation to be large
enough. However when zoning is very restrictive, firms cannot differentiate enough; this leads to
a discontinuity where stores now recognize that by co-locating they can gain from agglomeration
benefits which may outweigh the relatively constrained benefits from differentiation because of
the tight zoning regulations. This explains the high interaction effect of zoning and
agglomeration that we find as we proceed from the scenario in Figure 5(a) to the scenario in
Figure 5(d).
6. Conclusion
The literature on retailer entry and location choices has thus far ignored the
agglomeration-differentiation tradeoff. We developed a comprehensive static, structural,
simultaneous move game model of firm entry and location choice that disentangles this tradeoff
while controlling for several alternative explanations for observed collocation. Taking advantage
of a publicly available, digital land cover database, NLCD, we are able to control for the effect of
zoning on entry and location choices. To control for demand and cost based explanations for
collocation, we decompose latent profits into revenue and cost and augment entry and location
data with store revenue data. To separate the benefits of agglomeration from the benefits of
spatial differentiation, we further decompose revenue into its components of consumer choice
based volume and competition based price. We use recent advances in the empirical estimation
39

literature of discrete games to address issues of multiple equilibria in the model and data as well
as problems due to slow convergence of the estimation algorithm.
The consumer and price model provided interesting insights about the differences in the
agglomeration and competition effects across store formats. These results and the subsequent
counterfactual analyses lead to the following takeaways: First, zoning, agglomeration effects,
spatial differentiation and format differentiation are all key drivers of observed store location
patterns. Second, zoning may force firms to locate closer than what they would like but it has
little direct effect on collocation of stores. Finally, zoning interacts with agglomeration to drive
observed collocation. The interaction between zoning and the agglomeration effect can have a
discontinuous impact on the location pattern of stores. This highlights the value of a structural
model in understanding how a small perturbation of market characteristics can cause strategic
firms to respond in complex and nonlinear ways.
We conclude with a discussion of some key limitations in this paper that warrant future
research. First, our identification of the volume and price effects is partially aided by functional
form assumptions for how locations of competitors affect volumes and prices differently. This is
because we only have price information for a set of stores belonging to one store chain.
Nonetheless, this is managerially interesting as it is closer to a realistic scenario where firms
usually have more information about themselves than about others. Second, we treat entry
decision in a static equilibrium framework, even though a dynamic model may be more
appropriate given that these decisions are made over time. Such a modeling approach requires
better data (timing of entry and exits) as well as richer modeling framework to solve the dynamic
game. Finally, we have treated store entry decisions across markets as independent, unlike recent
work by Jia (2008), who models the chain entry decision, taking into account the
40

interdependence across markets. However, her modeling approach is restricted to a small number
of competing chains and is hard to extend to our grocery market setting that involves a large
number of players. These important issues await future research.
41

Figure 1: Over 45% of big-box grocery stores are within 0.5 mi. of a rival store
1 2 3 …
Figure 2(a): An illustrative square market
with the geographical space discretized
into square blocks or ‘locations’.
(Data for 3 U.S. states of NY, OH and PA)
… L m
1 2
3 …
Figure 2(b): Due to zoning regulations,
firms can only choose among ‘potential
retail location’ (Area in white).
42
… lm

Figure 3(a): Graphical illustration of the standard NPL approach
Figure 3(b): With multiple equilibria
in the data, different starting values
may give different solutions
43
Figure 3(c): Depending on the local
convergence properties, the contraction
mapping may not converge to a fixed point

Figure 4(a): Constructing market boundaries based on visual
inspection of residential and commercial pixel density
44
Figure 4(b): Dividing a rectangular market into a grid of
1 sq. mile blocks or discrete locations

Figure 4(c): Using commercial land pixel data to obtain
extent of commercial activity within a location and the
commercial center of the location
45
Figure 4(d): Using ‘Places of Interest’ in Google
Earth to check for the presence of big-box stores in
commercial locations

120
80
40
0
Figure 5(a): Neither Zoning nor Agglomeration
120
80
40
0
28%
Figure 5(c): No Zoning; Only Agglomeration
46
120
80
40
0
Figure 5(b): Only Zoning; No Agglomeration
180
120
60
0
0.5
32%
43% 60%
1
1.5
2
2.5
3
Figure 5(d): With Zoning and Agglomeration
More

Notes: SM – Supermarket format; SS – Superstore format; LA – Limited Assortment format.
Figure 6: Equilibrium Store Locations in a Simulated Market – Shrinking Retail Zone
(Area in White represents retail locations)
47

Store Format
Examples of
Retailers 26
Total Number of
Stores in 98
Sample Markets
Maximum Number
of Stores in a
Market
Average Store
Area
(in sq. feet)
Average Annual
Store Revenue
from Grocery
Sales (in $ M)
Average Ratio of
Grocery Revenue
to Total Store
Revenue
Supermarket
(SM)
Hi-Low Food
Stores, Price
Chopper, Vons
Market
Superstore
(SS)
Jewel Food
Store, BI-LO,
Vons Market,
Albertsons,
Safeway,
Ltd. Assort.
(LA)
Save-A-Lot,
Price Rite,
Aldi, Smart &
Final
48
Natural Food
(NF)
Whole Foods,
Trader Joes
Food + Drug
(FD)
Jewel-Osco,
Kroger,
Albertsons,
Safeway
Supercenter
(SC)
Wal-Mart
Supercenter,
Super Target,
Meijer, Sams
Club, Costco
84 69 96 20 103 66
6 4 5 2 6 3
13,500 35,500 14,500 10,500 41,500 163,000
5.93 15.24 5.23 9.22 16.09 51.84
1 1 1 1 0.71 0.62
Table 1: Descriptive Statistics of Various Grocery Store Formats
26 Some retailers have more than one type of stores (e.g., Vons, Albertsons, and Safeway). We follow the format classification of individual stores
provided by AC Nielsen.

Variable 27
Travel Cost ( Tvl ) gl
Supermarket
(SM)
Superstore
(SS)
49
Ltd. Assort.
(LA)
Distance ( d gl ) - 0.1827
Natural
Food (NF)
Distance 2 ( d ) 0.9862***
2
gl
med _ hhI g * d 0.0436
gl
med _ ageg * d 0.0579
gl
min _ dg * d - 0.1482*
gl
Price ln ( pr fl )
- 0.4255
Economies of
Scope
Store
Agglomeration
Format
Agglomeration
Format
Preference
Customer Value
(Equation 12)
comm l
1.5468***
N l
0.5342**
Food + Drug
(FD)
Supercenter
(SC)
OF
I fl
0.6329 0.4917 0.9458** 1.2031 1.2974** 0.8337**
-- -0.3251 -0.2735** -0.5809 0.3683*** 0.2196**
CV fl 0.3932** 0.4885 0.3826* 0.5404 0.6391** 0.5673
27 Note: * : p < 0.1, ** p < 0.05, *** : p < 0.01; All significant estimates in bold.
Table 2(a): Consumers’ Shopping Location Choice Based Volume

Formatspecific
Pricing
Ability
Competition
Effect
Supermarket Superstore Ltd. Assort. Natural Food + Drug Supercenter
(SM) (SS) (LA) Food (NF) (FD) (SC)
Intrinsic Ability 1.4468* 1.7413* 0.9685* 1.2890 1.4963* 1.1372*
Variable 28
Per Capita Income in
2mi. radius ( x l )
SM; 0-1mi. coeff. - 1.4895***
SS; 0-1mi. coeff. -1.1043** -2.5038**
LA; 0-1mi. coeff. - 0.4712* - 0.5620* - 1.7921**
50
0.0861
NF; 0-1mi. coeff. - 0.8095 - 1.2597 - 0.5328 - 3.6044
FD; 0-1mi. coeff. - 0.5991** - 1.6469** - 0.8129** - 1.3031 -4.2357**
SC; 0-1mi. coeff. - 0.6344 - 0.3609* - 0.2315 - 0. 3988 -1.014* -3.9460*
1-2mi. multiplier ( κ 2 ) 0.5806***
2-3mi. multiplier ( κ 3 ) 0.3247***
3-4mi. multiplier ( κ 4 ) - 0.0521
4-5mi. multiplier ( κ 5 ) 0.0173
28 Note: * : p < 0.1, ** p < 0.05, *** : p < 0.01; All significant estimates in bold.
Table 2(b): Competition Based Price Index

Variable 29
Supermarket
(SM)
Superstore
(SS)
Ltd. Assort.
(LA)
Natural
Food (NF)
Food + Drug
(FD)
Supercenter
(SC)
Cost Intercept -- - 0.2451 0.0521 - 0.4489 0.3902 - 0.5813**
Commercial Activity ( comm ) l - 0.0774 0.2917* - 0.3265** 0.0592 - 0.1999* 0.3711**
0-1mi. coefficient - 0.5996 - 0.6233** - 0.2908* - 0.1370 - 0.4391** -1.2028**
Population
Per Capita
Income
Common
Unobserved
Location-Level
Shocks
Common
Unobserved
Market-level Cost
1-2mi. multiplier 0.7503**
2-3mi. multiplier 0.3132*
3-4mi. multiplier 0.0830
4-5mi. multiplier - 0.1114
0-1mi. coefficient - 0.7830 - 0.5548* - 1.2619* 0.3012 - 0.5693** - 0.8896*
1-2mi. multiplier 0.2815*
2-3mi. multiplier - 0.1002
3-4mi. multiplier - 0.0036
4-5mi. multiplier 0.0416
Std., Price Shock: σ 0.7783
p
Std., Revenue Shock: σ 1.0928**
r
Std., Cost Shock: σ 1.6041*
c
Revenue-Cost Corr.: ρ 0.8932**
µ ( ξ )
- 3.2962***
σ ( ξ )
1.3901***
29 Note: * : p < 0.1, ** p < 0.05, *** : p < 0.01; All significant estimates in bold.
Table 2(c): Cost and Common Unobserved Components
51

Appendix A
Expected total number of competing stores in a location:
[ ] m
E N = N∑ p
A.1
l fl
f
Expectation that a location will have stores with other formats besides format-f:
( )
OF
E ⎡
⎣I⎤ fl ⎦ = 1− prob location has only format- f stores
∏ ( ' )
= 1− p 1−
p
fl f l
f '≠
f
Expected number of format-f’ rivals in distance band b around a format-f store that is in location
l (interformat competition):
⎛ ⎞
m
E ⎡
⎣N ⎤ f 'bl ⎦ = ⎜N pf' j ⎟ ; f ' ≠ f
⎜ ∑ ⎟
⎝ j∈lb
⎠
where, is the set of locations in distance band b around location l.
lb
Expected number of format-f rivals in distance band b around a format-f store that is in location l
(intraformat competition):
When accounting for the number of rivals with the same format, we need to discount the
choice probability of the focal firm, conditional on its decision to enter the market:
⎛ ⎞ ⎛ ⎞
m
E⎡ ⎣N⎤ fbl ⎦ = ⎜N p fj ⎟−⎜1 ( p fj f Enters m)
⎟
⎜ ∑
⎟ ⎜ ∑
⎟
⎝ j∈lb ⎠ ⎝ j∈lb
⎠
Note that the probability that a f-format firm enters the market is simply
the probability ( p fj f Enters m ) is given by
fj fl
l=
1
52
lm
A.2
A.3
A.4
lm
∑ p fl . Hence,
l=
1
p ∑ p and Equation (A.4) can be rewritten as:
⎛ ⎞ ⎛ lm
⎞
m
E⎣ ⎡N fbl ⎦
⎤ = ⎜N pfj⎟−⎜1 pfjpfl⎟ ⎜ ∑
⎟ ⎜ ∑ ∑ ⎟
⎝ j∈lb ⎠ ⎝ j∈ lb
l=
1 ⎠
A.5

Step 0: Initial Population:
Appendix B
Generate a set of T vectors of starting values for retailers’ beliefs about rivals’ CCPs for
1 2 T
location choices, ⎡
⎣P0; P0 ;...; P ⎤ 0 ⎦ Also, create an initial guess for the parameter vector,
θ = αβγσρ , , , , .
( { } )
Step 1: Locally Contractive, q-NPL Iteration:
For the likelihood maximization, set up an internal loop to do the following for each of the T
CCP vectors:
Given the current parameter values, pick a large number of Halton draws of price,
revenue and cost shocks for all retail locations. Obtain the location choice probabilities
(Equations 20 - 22) and the market specific cost parameters (Equations 26 - 27). Next, calculate
the price indices of firms, sans the unobserved component, for the chosen locations and with the
observed configuration of stores. Compare the price estimates with the price data to obtain the
pr
price shocks at the chosen locations of the store chain for which we have price data, ( ωobv θ ) .
Also, calculate the revenues of stores, sans the unobserved component, for the chosen locations
and with the observed store configuration. Compare the revenue estimates with the revenue data
r
for all stores to obtain the revenue shocks of firms in their chosen locations, ( ωobv θ ) . We now
have all the components of the likelihood function. 30
Maximize the pseudo likelihood (Equation 28) to obtain a set of T vectors of parameter
t t
estimates: Θ n = arg max ( L( Pn−1, Θ ) ) , and a new population of CCPs using the q-NPL operator:
Θ
ˆ t q t t
P =Λ P −1,
Θ .
( )
n n n
Within each market, normalize the CCPs for each store format so that the CCPs of all formats
add up to one. Essentially, for each format f, and market location l, we have :
F lm
ˆ t q t t q t t
Pfln=Λ ( Pfln−1, Θn) ∑∑ Λ ( Pfln−1,
Θn)
B.1
53
f = 1 l=
1
30 We acknowledge that there is a potential selection bias because we only observe revenue data for locations that
were chosen. Ellickson and Misra (2007) propose a selection correction function in their application where
supermarkets choose from one of three pricing strategies. However, their approach suffers from a curse of
dimensionality in cases where the cardinality of firms’ action space is large, as is the case for firms choosing from
multiple locations within a market.

Step 2: Selection of Parents - Based on their fitness, draw, with replacement, T ‘mother’ CCP
1 2
vectors and T ‘father’ CCP vectors from the set, ⎡ ˆ
;
ˆ
;...;
ˆT
Pn Pn P ⎤
⎢ n and form couples or Parents.
⎣ ⎥⎦
ˆ t t
L P , Θ , and those closer to convergence (Absolute value
CCPs with high likelihood values, ( n n)
of
ˆ t ( Pn t
Pn−1) − closer to zero) are considered more fit to continue. In our problem, we use the
following fitness criterion:
( ) ln ( , )
h P
ˆ
= λ ⎡L P
ˆ
Θ ⎤−λ
P
ˆ
−P
−
⎢⎣ ⎥⎦
t t t t t
n 1 n n 2 n n 1
where, λ 1 and λ 2 are small positive constants. The t th CCP vector gets selected with the
probability:
T
ˆ ˆ
( ( n ) ) ( ( n ) )
54
j=
1
B.2
t t j
S = exp h P ∑ exp h P
B.3
Now, we have the set of couples:
ˆ1'ˆ1'' ˆ 2' ˆ 2'' ˆT'ˆT'' ( Pn , Pn ) ; ( Pn , Pn ) ;...; ( Pn , Pn
)
⎡ ⎤
⎢⎣ ⎥⎦
Step 3: Crossover and Mutation – Obtain an offspring from each couple as follows:
ˆ ' ' '' ''
( δ
ˆ ) ( 1
ˆ ˆ
) ( δ )
P = D• P + Z • • P + −D • P + Z • • P B.4
t t t t t
n n n n n n n n n
where, D is a vector of indicators for the identity of the parent who provides each element of the
CCPs. Its elements are i.i.d. with Pr ( D j = 1) = 0.5 for the j th element. Zn is another vector of
indicators for the identity of the elements of the CCPs which undergo mutation. Its elements are
also i.i.d. with Pr ( Z jn = 1) = 0.5 n.
Hence, with multiple iterations, as we get closer to the
global optimum, we allow the amount of mutations to reduce to zero. Finally, δn is a vector
whose elements represent the magnitude of a mutation. It is also defined such that its elements
δ ∈U − 0.5 n, 0.5 n
go to zero with multiple iterations. Specifically, we use: jn ( )
As with Step 1, within each market, again normalize the CCPs so that the CCPs of all
1 2 T
formats add up to one. Now, we have the new set of CCPs, ⎡
⎣Pn; Pn ;...; P ⎤ n ⎦ .
Iterate Steps 1-3 until the set of CCPs converges.

Appendix C
The following steps explain the technical operations involved in extracting commercial
land use pixel point data from NLCD. This is the authors’ original approach. However, a more
efficient approach may be plausible.
1. Open NLCD data in ArcGIS
2. Zoom in to the interested market area and select the data frame for further processing
3. Change coordinate system to WGS 1984
4. Reclassify the raster data to show only commercial land pixel points
5. Convert the reclassified raster data into Point Features and save as a Shapefile
6. Convert the saved Shapefile into a kml file using shp2kml software. The kml file can be
opened in Google Earth (GE), allowing us to see the pixel point data on GE
7. Make a copy of the saved kml file and rename the file from “.kml” to “.xml” This xml file
can be opened in Excel and the spreadsheet will show the coordinates (latitude and longitude)
of each pixel point, which may be used for further analysis
8. The count of these pixel points within each 1 sq. mi. block market location gives the measure
for the intensity of commercial activity in the location and the mean of the coordinates of
the pixel points within the location gives the commercial center of the location
In their classification of land types, NLCD 2001 combines high density residential land
and commercial land but NLCD 1992 separates them. Hence, we match the two data sets using
ArcGIS software to separate the pixel data for all residential land areas from land areas with
commercial activity in 2001. We are able to do this separation because land areas which were
high density residential in 1992 are unlikely to convert to commercial land areas by 2001, and
vice versa. In the rare instances where an area that was low-density residential in 1992 was
classified as commercial land in the 2001 data, we do a quick visual inspection of the
geographical area using Google Earth to confirm whether that area is truly commercial land or if
it has converted into a high density residential land.
55

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