The Role of Spatial Demand on Outlet Location and Pricingmela/bio/papers/Duan_Mela_2006.pdf · The Role of Spatial Demand on Outlet Location and Pricing 261 1By sales, we refer to

Journal of Marketing ResearchVol. XLVI (April 2009), 260–278260

© 2009, American Marketing AssociationISSN: 0022-2437 (print), 1547-7193 (electronic)

*Jason A. Duan is an assistant professor, Department of Marketing,McCombs School of Business, University of Texas at Austin (e-mail:[email protected]). Carl F. Mela is Professor of Marketing,Fuqua School of Business, Duke University (e-mail: [email protected]).The authors thank Bart J. Bronnenberg; J.P. Dube; Tom Nechyba; RonShachar; Christoper Timmins; and seminar participants at Berkeley Sum-mer Institute on Competitive Strategy, Duke University Marketing andEconomics, Georgia State University, IBM Watson Research Center, Indi-ana University–Bloomington, Massachusetts Institute of Technology, Pur-due University, Temple University, University of Connecticut, Universityof Georgia, University of Maryland, University of Minnesota, Universityof Pennsylvania, University of Texas at Austin, and the Yale School ofManagement for their helpful comments. Michel Wedel served as guesteditor for this article.

JASON A. DUAN and CARL F. MELA*

In this article, the authors consider the problem of outlet pricing andlocation in the context of unobserved spatial demand. The analysisconstitutes a scenario in which capacity-constrained firms set pricesconditional on their location, demand, and costs. This enables firms todevelop maps of latent demand patterns across the market in which theycompete. The analysis further suggests locations for additional outletsand the resultant equilibrium effect on profits and prices. Using Bayesianspatial statistics, the authors apply their model to seven years of data onapartment location and prices in Roanoke, Va. They find that spatialcovariation in demand is material in outlet choice; the 95% spatial decayin demand extends 3.6 miles in a region that measures slightly more than9.5 miles. They also find that capacity constraints can cost complexesupward of $100 per apartment. As they predict, price elasticities andcosts are biased downward when spatial covariance in demand isignored. Costs are biased upward when capacity constraints are ignored.Using the analysis to suggest locations for entry, the authors find thatproperly accounting for spatial effects and capacity constraints leads toan entry recommendation that improves profitability by 66% over a modelthat ignores these effects.

Keywords: outlet location, pricing, spatial statistics, structural models,competition

The Role of Spatial Demand on OutletLocation and Pricing

Outlet location and pricing are of central concern tomany firms. For example, the French retailer Carrefour SAadded 5930 stores between 1999 and 2003, and the U.S.company Dollar General added 2426 stores in the sameperiod (Euromonitor 2007). Paramount to expansion effi-cacy is the effect of outlet location on sales, prices, andprofits, which are moderated by the underlying demandacross regions in which a firm trades. Yet direct observationof demand across the areas in which firms trade is difficult;

not only is this demand apportioned among existing outletslocated at a small (relative to the trade space) number offixed points in space, but the latent demand also does notnecessarily comport with the observed density of popula-tion. For example, the presence of complementary stores ordesirable traffic patterns may elevate demand in a specificlocale.

Given the central role of spatial distribution of demand ina firm’s location decision, we focus on the inference oflatent spatial demand. Integrating spatial statistics with astructural model of firm conduct enables us to capture aflexible distribution of latent spatial demand and use it toengage policy simulations regarding the sequential effect oflocating an additional outlet on the demand, prices, andprofits for the new and existing outlets. This enables us toaddress questions such as the following:

•Can the distribution of category demand across a given marketor trade area be inferred from observations of retail sales atspecific points in space? The answer to this question leads to acontour map of latent spatial demand that can be used to iden-tify potential sites for new outlet entry or deletion.

•How do spatial demand and outlet location affect equilibriumprices and profits? We find that unobserved latent spatial

The Role of Spatial Demand on Outlet Location and Pricing 261

1By sales, we refer to the observed number of units sold at a particularoutlet. By demand, we refer to the distribution of product utility acrossvarious regions. It is the distribution of demand across space that drivessales at given locales.

demand leads to higher price variation over space and that aproper accounting of spatial demand effects improves the prof-itability of an entry recommendation by 66% over a model thatignores spatial covariance in demand and costs.

There has been considerable prior work in marketingregarding outlet sales dating back to the gravity model (see,e.g., Bucklin 1971; Ghose and Craig 1983). Our work dif-fers from this research in several respects. First, thisresearch typically assumes that prices are exogenous suchthat prices of competing outlets do not change with thelocation of an outlet. To the extent that firms react to thenew entry by changing prices, models that ignore this couldmisstate potential profits. In contrast, our model capturesspatial endogeneity in pricing through a structural linkbetween random spatial demand effects and equilibriumprices. Second, such models typically do not estimate latentdemand across regions (i.e., the demand apportioned to anoutlet arising from a particular point in space) but ratherassume that demand arises from observed differences inpopulation across regions. In contrast, our work can accom-modate unobserved sources of spatial demand. Third, wenote that sales data necessary to estimate these models areoften not observed, because firms sometimes keep theiroutlet sales data private (e.g., Wal-Mart). Our approachdoes not require information on outlet sales to infer latentspatial demand.1

In this sense, our model of spatial demand is relatedmore to analytical models of spatial location and demand ineconomics. Much of this work is theoretical, assumes a uni-form distribution for spatial demand, and focuses on theequilibrium location of outlet location and the corre-sponding prices (Ansari, Economides, and Ghosh 1994;D’Aspremont, Gabszewicz, and Thisse 1979; Hotelling1929). Recently, empirical models in economics and mar-keting have begun to appear that focus on solving the sub-game of equilibrium prices or sales conditional on outletlocation and capacity to infer latent spatial demand (Chan,Padmanabhan, and Seethuraman 2007; Davis 2001; Pinske,Slade, and Brett 2002; Thomadsen 2005, 2007; Venkatara-man and Kadiyali 2005). This subgame is a reasonablestarting point for the outlet location problem for severalreasons. First, it must be solved before the optimal outletlocation can be determined. Second, in a preponderance ofmarkets, outlet locations are extant and fixed at the time alate entrant decides to enter, so the relevant managerialdecision pertains to locating the next outlet and its capacity.Using the subgame, it is possible to explore the implica-tions of adding an outlet on equilibrium prices and demandfor the new and existing outlets. Third, competitiveresponse latencies in constructing outlets can be largebecause of land acquisition, zoning, permitting, and con-struction. Therefore, competitive response in location maybe impracticable over an intermediate duration, suggestingthat a focus on the subgame is appropriate. A central inno-vation in these empirical models of spatial demand is thatthey consider observed spatial demand effects arising from

2We differentiate between location-specific random effects (e.g., Bayerand Timmins 2007) that are commonly employed in sorting models andspatial random effects. Spatial models allow for spatial covariance in thelocation-specific effects. These covariances have implications for spatialprediction and outlet location and pricing. Our work further differs fromthis research stream insofar as we consider the supply-side problem (priceendogeneity) and capacity constraints in demand.

distance to some centroid, such as a population center or anairport. Our work complements the foregoing stream ofresearch in two key ways.

First, we supplement observed spatial demand factorsthrough spatially correlated unobserved demand effects.2Given that there are a plethora of potential spatial influ-ences, it is unlikely that a researcher can capture all or mostof them. For example, the presence of a particularemployer, hotel, traffic pattern, school, restaurant, shop,family member, or friend could affect the choice of outletand, therefore, equilibrium prices. Moreover, these influ-ences tend to induce spatial covariance in demand shocks.The existence of spatial covariance in demand and supplyshocks has several important implications for the outletlocation problem.

Inclusion of unobserved spatial effects can enhance theefficiency of model estimates in information-poor environ-ments in which covariates pertaining to spatial variation indemand are unobserved. We conjecture that such cases aremore common than those of complete information. Becausecomplete data are often prohibitively costly or unavailable,our approach affords a feasible alternative in information-poor environments.

The estimate of spatially correlated random effects inconjunction with spatial kriging yields a regional demandmap that can serve as a decision aid to (1) find new loca-tions and (2) afford insights into differences in demandacross the map (e.g., a peak in a certain area might suggestan important omitted variable).

We explicitly consider the case of apartments, in whichthe observed population distribution cannot be used to cap-ture latent spatial demand because the population locationitself is endogenous. By including latent spatial randomeffects, we can capture such phenomenon. In other con-texts, such as retail outlets, our model of random spatialeffects can be augmented using observed distances betweenconsumers and the outlets or demographics.

For the following four reasons, policy simulations per-taining to outlet location yield incorrect recommendationswhen spatial covariance is ignored. (In our simulation, thepolicy recommendation from a model that incorporates spa-tial covariance yields a locale that would increase profits by66% over a model that ignores spatial covariation.)

1. Spatial random effects are biased toward zero when spatialcovariance is ignored, leading to a downward bias in esti-mates of price parameter. The bias arises as a result of thesmall sample properties inherent in instrumental variable(IV) estimation (Altonji and Segal 1996; Buse and Moaz-zami 1991). We demonstrate these effects through simulationand a data application. Given that the number of outlets in ageographic trade area is often limited, the bias is consequen-tial for the outlet location problem.

2. The downward bias in the price parameter also leads to adownward bias in the estimate for marginal costs. Because

262 JOURNAL OF MARKETING RESEARCH, APRIL 2009

price is the sum of costs plus markups, for a given price, thelower estimated cost must be offset by a higher estimatedmarkup.

3. The inclusion of spatial covariance in demand implies substi-tution patterns that are a function of entry location (apartfrom observed spatial differences). This is one approach to addressing the spatial independence-from-irrelevant-alternatives problem, in which expected share loss as a resultof entry is apportioned not to proximal outlets but rather tothe largest ones.

4. In the absence of spatial covariance, predictions for demandare incumbent solely on observed spatial differences and arelikely to underestimate the true variation in demand. In ourdata, the omission of spatial covariance attenuates the stan-dard deviation of spatial demand by a factor of 11.

Second, we consider the issue of outlet capacity becausesupply is not limitless. Extant models do not accommodatethe possibility that capacity might be constrained (as wouldbe the case with categories such as restaurants, hotels, orapartments). This can be problematic for several reasons:

1. In the context of policy simulations, moving an outlet (oradding a new one) cannot increase sales among extant outletsbeyond their capacity.

2. Improperly accounting for this constraint leads to biasedestimates for marginal costs. When demand exceeds supply,firms can raise prices with no effect on sales. The high pricesassociated with at-capacity firms leads to inferences ofhigher costs when capacity constraints are ignored. The addi-tional error arising from poor predictions also leads to anincrease in the estimated variance of the marginal cost equa-tion in the supply-side model.

3. Our approach yields estimates of the costs of these capacityconstraints (through estimates of the Kuhn–Tucker multi-plier), which can be used to assess the merits of expansion.

In summary, our goal is to develop a flexible structuralmodel of spatial demand to provide guidance to firms con-sidering the potential location of new outlets (or changingthe location of existing ones). In this sense, our work lies atthe intersection of two developing research streams in mar-keting: empirical economics (Chintagunta et al. 2006) andspatial statistics (e.g., Bronnenberg and Mahajan 2001;Bronnenberg and Mela 2004; Bronnenberg and Sismeiro2002). As such, we augment spatial statistics with structuralmodels of pricing. Then, we employ these spatial methodsin a different context: outlet location. Finally, given theemphasis on outlet location, we integrate capacity con-straints into a spatial model.

The article proceeds as follows: We begin by developingthe demand-side model. We then generalize the model toconsider the pricing problem firms face in the presence ofcapacity constraints, latent spatial demand, and the locationof other firms. Then, we outline how to estimate the modeland, using these estimates, how to forecast demand pricesand profits arising from the entry of a new outlet at anygiven location. We then describe the apartment data used tocalibrate the model and follow this with both simulated anddata-based results. We use the simulated data to show thebiases that arise when spatial covariance and capacity con-straints are omitted, and the real data show that these spatialeffects lead to considerable improvements in model per-formance. We conclude with the results of our data applica-tion, suggested locations for additional outlets, and direc-tions for further research.

3We considered other specifications for the temporal dependency in spa-tial shocks. In particular, we computed for each of the Jfirms for each of the draws of the in the Markov chain Monte Carlosampling chain described in the Appendix. The mean of ρ is .16, and thevariance is .23. Thus, autocorrelations appear statistically small. We con-jecture that the large dispersion in autocorrelations results from having alimited number of periods in our data (six periods) from which to computethem.

vts j

ρ( ),v vts t sj j, − 1

THE MODEL

Our model presentation proceeds as follows: First, wepresent the consumer demand model. Specifically, we applyan individual-level, random utility, discrete choice modelwith spatially correlated random effects and use this modelto infer the aggregate demand function. Second, we outlinethe equilibrium conditions of supply and demand in thecontext of firms that own outlets that are capacity con-strained. The equilibrium conditions are derived from theprofit-maximizing behavior of the firms and the utility-maximizing behaviors of the consumers. The market-clearing condition establishes the economic equilibrium ofdemand and supply, which leads to the structural estimationequations. These equilibrium conditions constitute a systemof equations that we estimate using advances in Bayesianspatial statistics.

Demand Model: Discrete Choice

Suppose that there are J outlets for a given type of goods(e.g., apartments, car dealers, hotels, bank branches) in agiven region. The location of the j ∈ J outlet is denoted assj. Let the number and the set of the potential customers fora set of outlets in a region be I, i = {1, ..., I}. Customer i’srandom indirect utility for choosing outlet j at location sj inperiod t is as follows:

In this equation, Xtj represents the attributes of the jth out-let, such as variety or amenities; Xtsj

represents theobserved attributes specific to the location (e.g., the dis-tance to a business center or a major highway); and Ptj is aprice index for an outlet. As is common in random utilitymodels of choice, εtij is assumed to be independently drawnfrom a Gumbel distribution. Consumers’ sensitivities toprice change are assumed to be normally distributed acrossthe population: Let γi = γ + ηi, where Wefocus on price response heterogeneity to reduce the model’sdimensionality. Gowrisankaran and Rysman (2007) findthat adding random effects to nonprice attributes essentiallyleads to no change in the model parameters and that therandom effects for these attributes do not differ from zero.

The θtsj, indexed by time t and location sj, is the time-

varying random demand shock. We decompose θtsj= θsj

+νtsj

, where θsjis a time-invariant spatial random effect that

is fixed over all the observed periods and νtsjis a spatial

random effect that is independent across time.3 Thisdecomposition implies that there is a systematic location-specific long-term effect (to capture unobserved spatialeffects that do not vary over the range of the data, such as the proximity to urban amenities, distance to majoremployers, and school zoning) and a short-term perturba-tion about this mean, νtsj

, to capture unobserved spatial fac-

η σγi N~ ( , ).0 2

( )1 1 2V X X Ptij tj ts tj i ts tijj j= + − + + .β β γ θ ε


tors that may vary over time (e.g., construction, traffic patterns). As a result, both θsj

and νtsjmay be spatially

correlated. We assume that θ = (θsj, j = 1, ..., J) ~ N[0,

RJ(φθ)] and νt = (νtsj, j = 1, ..., J) ~ N[0, RJ(φν)],

whose joint effect is as follows:

The entries of the correlation matrix RJ(φ) (φ = φθ or φν)can be constructed with an isotropic exponential decayfunction, exp(–φ × d), where d is the distance between anytwo locations, d = ||si – sj||; the Matérn class (Banerjee,Carlin, and Gelfand 2004); or a more general anisotropicnonparametric spatial Dirichlet process model (Duan,Guindani, and Gelfand 2007; Gelfand, Kottas, andMacEachern 2005). We note that these spatial structures arehighly flexible and admit many potential latent demandsurfaces.

Customer i may choose the outside good; that is, he orshe may not buy in the region at all or may consider expen-ditures on other types of goods sold in the region. The cus-tomer’s indirect utility for the outside good is given by

If the highest utility of choosing one outlet exceeds that ofthe outside good, the customer will select the highest-utilityoutlet.

The customer chooses the outlet with the highest utility.If we define the mean effects in the utility function as

the customer utility func-tion in Equation 1 can be rewritten as Vtij = ξtj – Ptjηi + εtij,where ηi and εtij capture consumer heterogeneity. Weassume that the distributions for the random effects εtij andηi are common knowledge to firms. This assumption is nec-essary for firms to form expectations regarding their marketshare. The Gumbel error assumption for εtij results in alogit choice likelihood that customer i chooses outlet j:

We obtain the expected market share for outlet j, Wtj, byintegrating over the remaining random effect, ηi, whichyields the following:

Multiplying the number of people in the market, I, byfirm j’s share leads to the total expected demand, Qtj = I ×Wtj. The second term in the denominator reflects thedemand for the outside good. Because I, Mt, and the inter-cept of the additive utility cannot be separately identified atthe same time, we make I and Mt constant, as we discuss inthe “Data” section.

Berry, Levinsohn, and Pakes (1995, hereinafter BLP)prove that Wtj and ξtj is a one-to-one mapping conditioningon the distribution of ηi by a contraction mapping theorem.The natural algorithm derived from the contracting map-ping to compute ξtj by setting the observed is asfollows:

W Wtj tj=

( )4

1

We

e edtj

P

JP M

tj tj i

tl tl i t

=+

−

=−∑

ξ η

ξ ηl

FF iη( )∫ .

We

e etij

P

JP M

tj tj i

tl tl i t

=+

−

=−∑

ξ η

ξ ηl 1

.

ξ β β γ θtj tj ts tj tsX X Pj j= + − +1 2 ,

(3) V = M + .ti0 t ti0ε

( ) ~ .2 0 2 2θ θ ν σ φ σ φθ θ ν νt t J JN R R= + , +⎡⎣ ⎤⎦( ) ( )

σν2σθ2where g is the index of iterations. This algorithm inverts thevector function in Equation 5 from the observed shares to the unobserved errors (asymptotically approached by

This inversion leads to a nonlinear transformation ofthe data, given that Note that is linearin the demand-side model parameters (β, γ, and θ), whichfacilitates the demand-side estimation.

Supply Model: Bertrand–Nash Game

In addition to consumers, the market comprises firmsthat compete on the basis of price, location, and capacity.As noted previously, we focus on the subgame of firm com-petition conditional on location choice and capacity. Weassume that a market is composed of F firms and that eachfirm f has a set Ff of outlets. Each firm maximizes the totalprofit of its outlets Ff. We first consider firm f’s strategy.Conditioning on the prices of the outlets not belonging tof’s chain, firm f faces the following profit maximizationproblem:

where ctj represents the variable cost of goods sold and Kjis outlet j’s capacity constraint, which is the maximumnumber of goods the outlet can produce, sell, or rent. Thecapacity is considered rigid over the short decision frame of a firm setting prices because the expansion or contrac-tion of an outlet’s physical plant involves construction/permitting time and considerable cost. This implies that thecapacity consideration is relegated to an “upper stage” ofthe game, in which a firm sets location and capacity beforechoosing price in the subgame. When a firm faces a differ-ent demand at time t as a result of setting the price Ptj, Kjbecomes an immutable short-term constraint to which theoutlet is subject.

Because the econometrician does not observe the firms’variable costs ctj, we model them as

where Ytj is a set of cost shifters (e.g., the type of inven-tory) and ζtsj

are spatially correlated cost shocks. We fur-ther decompose the time-varying cost shock as ζtsj

= ζsj+

where ζsjis a time-invariant random effect that is fixed

over the observed periods and is the cost shock that isindependent across time. The ζsj

effect captures the long-term cost factors that are not observed and controlled incost shifters Ytj. It can include the labor, maintenance,administrative costs, tax, insurance, and other unobservedfactors, whereas is the short-term changes in these fac-tors. Accordingly, ζsj

and are likely to be spatially cor-related as a result of these omitted variables in the costmodel. We assume the following:

ζ ζ σ ψζ ζ= ; = , ,( ) ,⎡⎣⎤⎦( )s J

t

jj … J N R

e

and1 0 2~ ,

cctsc

c J ce j … J N Rj

= ; = , ,⎛⎝⎜

⎞⎠⎟ ,⎡⎣ ⎤⎦( )1 0 2~ σ ψ ,,

e tsc

j

e tsc

j

e tsc

j

e tsc

j,

( ) ,7 3c Ytj tj ts j= +β ζ

( ) max6 Πtfj F

tj tj tj

t

f

P c Q

Q

= −∈

( )∑ ,

subject to jj jK≤ ,

ξ tF Ni( ) ( ).η σγ= ,0 2ξ t

g( )).ξ t

Wt

( ) ln ˆ ln51ξ ξ ξ ηt

gtg

t t tg

iW W F+( ) ( ) ( ) ( )= + − ,⎡

⎣⎢⎢⎤⎦⎥,


where RJ(ψ) (ψ = ψζ or ψc) can be of the same or differentfunctional form of the correlation in spatial demand RJ(φ).

The Kuhn–Tucker conditions for this optimization prob-lem with inequalities constraints are as follows:

with the Kuhn–Tucker multipliers, , and

These multipliers reflect the marginal cost of the capacityconstraint on apartment profitability and reflect the value ofexpanding a particular outlet’s capacity by one unit.

Because Qtj = I × Wtj, the representation of Equation 8with market shares is as follows:

The Kuhn–Tucker conditions imply that firm f choosesoptimal prices for the outlets with expected vacancy andthe second-best price for outlets at capacity. The term

is the solution of the first-order condition in Equation10, where λtk = 0, whereas satisfies the binding capacityconstraint in Equation 9. We solve , and the nonzeromultiplier λtm simultaneously.

There are |Ff| (|Ff| denotes the number of outlets in Ff)equations arising from the first-order conditions for eachfirm. To facilitate explication, we define a |Ff| × |Ff| matrix,in which the (j, k)th element is

Let the vectors λtf = (λtm, m ∈ Ff), Ptf = (Ptj, j ∈ Ff), Wtf =(Wtj, j ∈ Ff), and and the submatrix Ytf =(Ytj, j ∈ Ff). Note that for the outlets with spare capacity,λtm = 0, and for the outlets with full capacity, Wtj = Kj/I.With these definitions, we can rewrite the first-order condi-tions in Equation 10 in matrix form as follows:

Likewise, for all the J outlets belonging to F firms in themarket, we define the following J × J matrix whose (j, k)thelement is as follows:

( )12 Ω jkt

tj tj t j t t

tk

W P P X

P( ) ,−( )=∂ , , ,

∂, for

θjj k Ff, ∈

⎧

⎨⎪

⎩⎪

same

otherwise0

.

Ω Ωt ftf tf tf tf

t ftfP Y W,( ) ( ) ,( )− − + = .β λ ζ3

ζ ζtf ts fjj F= , ∈( )

( )11

1

Ω jkt f tj

tk

i tij tij

W

P

W W dF

,( ) =∂∂

=− +( ) −( )γ η ηη

γ η η

i f

i tij tik i

j k F

W W dF j

( ) = ∈

+( ) ( )∫∫

, for

, for ≠≠ ∈

⎧

⎨⎪⎪

⎩⎪⎪ k Ff

.

�Ptm,Ptk∗

�Ptm

Ptk∗

�Ptm

Ptk∗

( )10 3j F

tj tj tjtj

tktk

mf

P YW

PW

∈( )

∈∑ − −

∂∂

+ −β ζFF

tmtm

tk

f

f

W

P

k F

∑ ∂∂

= ;

∈ .

λ 0

( ) , .9 0 0Q K Q Ktm m tm tm m tm= , > < , =iff and iffλ λ

λ tm ≥ 0

( )8 3∂∂

= − −∂∂

+

∈( )∑Πtf

tk j Ftj tj tj

tj

tkPP Y

Q

Pf

β ζ

QQQ

Pk Ftk

m Ftm

tm

tkf

f

−∂∂

= ; ∈∈∑ λ 0 ,

4The need for IV estimation in joint inference for γ arises because pricesin the supply-side model of Equation 14 are a function of Ω(t)[γ, P(ζt)] andWt[γ, P(ζt)], which depend on prices, which in turn depend on the costshocks. Thus, Ω(t) and Wt are endogenous to the errors in the cost equa-tion. This suggests the need for IV estimation when they are unknown. Ofnote, Ω and W enter the pricing model in a nonlinear fashion. The proper-ties of nonlinear IV estimators are not clear; as a result, we cannot be surethat an IV approach on the supply side leads to unbiased estimates for γ.

We can write the first-order conditions for all the outlets inmatrix form as

If Ω(t) is invertible, we have the following:

In Equation 14, Bt (λt, γ) ≡ λt – Ω(t)–1 Wt indicates thefirms’ markups because this expression represents the dif-ference between a firm’s prices and its costs. Note thatλtm = 0 for the outlets that have vacancy and λtm > 0 for theoutlets with full occupancy. Thus, markups increase whenfirms are capacity constrained. The economic interpretationof this equation is that the prices of the outlets with fullcapacity are higher than their optimal prices. Intuitively,this suggests that firms that face demand in excess of sup-ply will raise prices to the point at which demand is equalto capacity; lower prices serve only to decrease revenue.

Of interest, Equation 14 embeds all parameters in thesupply and demand system. This suggests that both thedemand-side and the cost-side parameters can be estimated,even in the absence of any demand-side data (Thomadsen2005). Similar to BLP and others, we assume that theseparameters, though unobserved to the econometrician, areknown to the firm and to the consumers.

ESTIMATION

We follow a two-stage estimation approach analogous totwo-stage least squares as in BLP and Nevo (2001). In thefirst stage, we estimate the demand-side model to obtainestimates for γ, Ω(t), and Wt, which we denote as , and . In the second stage, we treat these estimateddemand-side parameters as random variables in the supply-side model to obtain the estimates for costs and theLagrange multipliers.

Despite the loss of statistical efficiency that would begained by using a one-stage approach to estimate thedemand-side parameters (i.e., using information from boththe demand and the supply models to infer the demandparameters), the two-stage approach offers several advan-tages over a one-stage approach. First, a two-stageapproach mitigates the need to use IV estimation on thesupply side.4 Second, the Jacobian required for joint esti-mation is complex, rendering estimation substantially morecumbersome. Third, should the supply-side model be mis-specified, the resultant demand-side parameters will bebiased (Chintagunta, Kadiyali, and Vilcassim 2006).

Demand-Side Estimation

The demand-side estimation equations for all the outletsin the market are as follows:

Wt

ˆ ,( )Ω tγ

( )141

3P W Yt tt

t t t− + = + .−( )λ β ζΩ

( )13 3Ω Ωtt t t t

ttP Y W( ) ( ) ( )− − + = .β λ ζ


5Unlike the time-varying spatial effect νt, θ is a time-invariant spatialrandom effect. Accordingly, it need not be instrumented. Analogous toNevo’s (2001) use of brand-specific dummies, θs can be estimated usinglocation-specific dummy variables and the prior distribution N[0, RJ(φθ)].

6Inspection of the residual variograms provides evidence that the expo-nential function fits the data well. Furthermore, these residuals reveal littleevidence of any anisotropy in spatial decay.

σθ2

where Qtj = Kj if the outlet is at capacity (e.g., a full outlet)in that period. There is no closed form for the integral inEquation 15. However, given the distribution of randomprice effects, F(ηi), the integration involved to compute

can be approximated with a Monte Carlointegration. The integration proceeds by simulating Is indi-vidual ηi from the distribution F(ηi) and approximating theintegral in Equation 15, as follows:

Instead of using a maximum likelihood method to esti-mate model parameters, BLP propose an IV-basedapproach. An appealing aspect of this approach is thatunobserved effects affect both spatial demand and prices,and the IV approach controls for this possibility by forminginstruments for price. Stacking the ξtj in Equation 16 acrossj, we obtain the following:

The random spatial errors θ and νt in Equation 17 can bestructurally correlated with the prices Ptj as a result of theendogeneity of price to random spatial demand effects;prices tend to increase when demand is high.

Therefore, our goal is to compute ξt from Equation 16,so that we can use the orthogonality conditions between thespatial errors νt in Equation 17 and instruments Zt to esti-mate β1 and β2 (i.e., In their study, BLPprove that estimates are computed using the iterative pro-cedure, by showingthat this equation is a contraction mapping. When weobtain (approximated by as the sequential discrep-ancy between and approaches zero), we con-sider the following linear model:

which is mean independent of the instruments Zt and has aspatial covariance structure , given the trueβ1, β2, γ, and θ.5 We assume an exponential correlationfunction (Banerjee, Carlin, and Gelfand 2004) for the spa-tial random effects—that is, RJ(φ) = exp(–φ × ||s – s′||).This function implies an exponential decay in the covari-ance of spatial effects over distance.6

Using a likelihood approach (Chernozhukov and Hong2003; Kim 2002; Romeo 2007), we let ZT

tνt ~ N[0, ZTt RJ

(φν)Zt], or equivalently,σν2

N R J[ ( )]0 2, σ φν ν

( ) ˆ ( ) ,18 1 2ν ξ γ β β γ θt t t ts tX X P= − − + −

ξ tm( )+ 1ξ t

m( )ξ t

m( ),ξ t

ˆ ˆ ln ˆ ln (ˆ ),( ) ( ) ( )ξ ξ ξtm

tm

t t tmW W+ = + −1

ξ t

E Z tT

t[ ] ).ν = 0

( )17 1 2ξ β β γ θ νt t ts t tX X P= + − + + .

( )161

1

We

etj tj

i

IP

J

stj tj i

t

ξξ η

ξ( )

=

−

=

= ∑∑ l

ll tl i tP Me− +.

η

W Ft tg

i[ ( )]( )ξ η,

( )15

1

WQ

I

e

etj

tjP

JP

tj tj i

tl tl

= =−

=−∑

ξ η

ξ ηl

ii tedF

Mi

+∫ ( ),η

7Although the condition might sug-gest that ζ2 and are bounded from above, we note that P2t is set afterfirms observe ζ2 and , and P2t can be adjusted with ζ2 and Thus,this inequality does not place constraints on the cost shocks.

etc,2.et

c,2

etc,2

P W Y ett

t t tc

2 2 2 3 2 21+ > + +−

,[ ˆ ˆ ]( )Ω β ζ

Note that this likelihood, denoted as L(ξt), is specifiedover ξt, but the observed response variable is Wtj. There-fore, the likelihood for observed shares involves a nonlineartransformation of variables from Wt to ξt(Wt). Thus, werewrite the likelihood L(Wt) = L[ξt(Wt)]|∂ξt/∂Wt|, wherethe Jacobian is given by

Supply-Side EstimationIn the supply-side model, we attempt to infer both the

parameters in cost function and the nonzero Lagrangianmultipliers. As noted in the “Model” section, we useobserved demand market share to replace the expectedmarket share Wtj in Equation 14. For the outlets at capacityconstraint in period t, according to theBertrand–Nash equilibrium. Using the price parameter esti-mates and from the demand model, we compute

and impute it into the first-order conditions:

The markup for the outlet chain model is in which the subvector λ1t = 0 for the outlets under capac-ity. For these outlets, For the outlets at capacity, we have λ2t > 0, or equivalently,

7 To use the full infor-

mation available to infer the cost parameters, we supple-ment observed data from apartments below capacity withaugmented data λ2t for firms at or under capacity, formingthe following likelihood:

where we partition the covariance matrix into foursubmatrices. To draw the augmented parameter λ2t for usein Equation 22, we use the following truncated distribution,derived in the Appendix:

where k = 1, 2.�P P W Ykt kt kt

t k kt= + − −−[ ˆ ˆ ] ,( )Ω 1

3ζ β

( )23 2 2 21 111

12

22 21 111N P R R P R R Rt t tλ σζ|� �+ , −− − RR I t12 2 0( )⎡

⎣⎤⎦ >{ }λ ,

σ ψc cR2 ( )

( )

ˆ ˆ

ˆ

( )

( )22

1 1 1 3 1

2 2

1P W Y

P

tt

t t

tt

+ ⎡⎣ ⎤⎦ − −

+

−

−

Ω

Ω

β ζ

112 3 2 2

2 11 10

ˆ

~ ,

W Y

NR R

t t t

c

⎡⎣ ⎤⎦ − − −

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟β ζ λ

σ 22

21 22R R

⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜⎞

⎠⎟,

P W Y ett

t t tc

2 2 2 3 2 21+ > + +−

,[ ˆ ˆ ] .( )Ω β ζ

P W Y ett

t t tc

1 1 1 3 1 11+ = + +−

,[ ˆ ˆ ] .( )Ω β ζ

λ tt

tW− −ˆ ˆ ,( )Ω 1

( ) ˆ ˆ( )21 13P W Y

W P X

t tt

t t t

tj t t

− + = + ,

, ,

−λ β ζΩ and

ifθ λt j tjK I( ) = / > 0.

ˆ ˆ( )Ω ttW

−1σ γγ

ˆ /W K Itj j=

Wtj

( )20

11

∂∂

=− , =

−

( ) ( )⎡

⎣⎢

⎤

⎦⎥∫

t

t

tij tij i

W

W W dF jξ

η if kk

W W dF j ktij tik i− , ≠

⎧

⎨⎪⎪

⎩⎪⎪

.−

( )⎡

⎣⎢

⎤

⎦⎥∫

1

η if

( ) ˆ ~19 01 22Z X X P N Z Rt

Tt t ts t t

TJξ β β γ θ σ φν− − + −( ) , νν( )⎡⎣ ⎤⎦.Zt


As in the demand model, we assume an exponential cor-relation function (Banerjee, Carlin, and Gelfand 2004) for the spatial random effects ζ and . That is, RJ(ψ) = exp(–ψ × ||s – s′||). We assume that the spatial randomerror νt in the indirect utility function is independent of theshort-term supply-side error . This does not mean thatdemand and supply are independent; these are linkedthrough the structural effect of observed effects and otherfactors on equilibrium prices. Note that IV estimation onthe supply side is not required, because none of costshifters in Yt are correlated with .

Model Inference

We use Bayesian inference to estimate the parameters inthis model. The Bayesian estimation is accomplished by aMetropolis-within-Gibbs sampler, in which is sampledusing the Metropolis–Hastings algorithm. The Appendixprovides the full conditional distributions for the samplingchain.

IVs

Selection of instruments. In addition to price, the spatialshocks νtsj

may correlate with the observed attributes Xtjand Xtsj

(e.g., amenities may be more prevalent in moredesirable areas). This potential correlation suggests that theuse of local attributes as instruments, as in BLP (i.e.,E[νtsj

|Xtj, Xtsj] = 0), does not hold in the context of the out-

let location problem. Instead, we consider as potentialinstruments Zt (1) the proximity to work Xtsl

(from thesame chain and from competitors), (2) attributes of distantoutlets Xtl (e.g., clubhouse, tennis, swimming pool, gymna-sium, and heat attributes from the same chain and fromcompetitors), (3) distant cost shifters Ytl (from the samechain and from competitors), and (4) lagged distant pricesPt – 1,l (from the same chain and from competitors). Theseare given, respectively, by the following:

where l, j ∉ Ff denotes that outlets l and j are not in thesame outlet chain; dist(l, j) is the distance between l and j;and r is the cutoff distance for a competitor’s inclusion inthe summation. The considered instruments should be inde-pendent of spatial shocks and correlated with the endoge-nous prices. In light of this, we discuss our rationale for thisset of instruments next.

Attributes as instruments. Similar to BLP and Nevo(2001), we use attributes in other locations as instruments.However, in our context, the unobserved spatial randomeffect νtsj

may be correlated with the local observed attrib-utes Xtj, Xtsj

, and cost shifter Ytj. This correlation willdecrease as outlet locations become far from one another.

( )

( )

24 Z

X

tj

l j l j Fdist l j r

tslf

l

=

,≠ ; , ∈, >

∑≠ ; , ∉, >

≠ ; ,∑ ,j l j F

dist l j r

tsl j l jf

lX

( )∈

, >

≠ ; , ∉

∑ ,F

dist l j r

tl

l j l j Fdist

f

f

X

( )

(ll j r

tll j l j Fdist l j r

X Y

f, >

≠ ; , ∈, >

∑ ∑,

) ( )

ttll j l j Fdist l j r

tl

l j

f

Y, ,≠ ; , ∉, >

≠ ;

∑( )

ll j Fdist l j r

t ll j l jf

P, ∈, >

− ,≠ ; , ∉

∑ ,

( )

1FF

dist l j r

t l

f

P

( ), >

− ,∑

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥

1

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

,

σγ2

etc

e tc

e tc

8Using lagged distant price as an instrument affords advantages overusing distant price as in Nevo (2001) when capacity constraints are pres-ent. Firms at capacity raise prices, leading other firms to raise prices,implying that current distant prices might not be good instruments.Because lagged demand shocks do not affect whether capacity binds in thecurrent period, they are not afflicted by this consideration.

Beyond a distance r (which is called the “range” effect inspatial statistics), the correlation between the instrumentsand the spatial shock will approach zero. Moreover, if thiscutoff distance is not sufficiently large, the attributes atother outlets will affect the local apartment’s choice proba-bility in the logit demand system and therefore can beexpected to correlate with current prices. In summary, withthe proper choice of r, the distant attributes are uncorrelatedwith the errors but correlated with prices. The selection ofthe cutoff distance r must consider several factors. Selectingtoo large a distance will cause some outlets to have noinstruments because no other outlets exist beyond a largedistance. Conversely, selecting too small a distance willlead to high spatial correlations between the competingapartment attributes and the local spatial shocks. Given thatthe nature of competition differs for other locations within achain and other locations within competing chains, wedivide these instruments into a same-ownership group and adifferent-ownership group. An analogous logic holds tojustify our choice of cost shifters and distance to points ofinterest as instruments.

Prices as instruments. In our data, the correlationbetween lagged distant own outlet prices and price is .12,and the correlation between lag distant competitor outletprices and price is –.54, indicating that lag distant pricesmay be good candidates for instruments. Our finding thatPtj correlates positively (negatively) with Ptk of outlet k inits own (a competing) chain is also confirmed in our simu-lation. These correlations can be explained as follows: First,lag distant prices are correlated with the long-term demandshock for these locales, θsj

. Second, the long-term demandshocks are correlated with prices. An increase in demandshocks for competing distant outlets makes those compet-ing outlets more attractive. As a result, local firms mustlower prices to compete. The price rule in Equation 10 sug-gests that a firm adjusts the prices of all its own outlets inthe same direction. Thus, there is a positive correlationbetween the own chain and the local outlet. In this sense,we argue that lagged distant prices for own and competingchains are correlated with current local price and are rea-sonable choices for instruments.

We further reason that lagged distant prices are indepen-dent of local demand shocks, νt, because (1) distant pricesare correlated with distant demand shocks and (2) distantdemand shocks are independent of local demand shocksand serially independent.8 This logic rests on the assump-tion that the correlation of the demand shock does notextend as far as the effect of price on competing outlets.This assumption is likely to be true when the spatial decayis large (i.e., no spatial correlation) or the distance betweenthe outlet and the competitive outlets used for instrument-ing is sufficiently large.

Test of instruments. To explore the quality of our instru-ments further, we explored the orthogonality of the instru-ments by inspecting the plots of the marginal posterior


distributions for . We compute the 95% posterior pre-dictive interval to assess whether it excludes 0 as a test ofthe orthogonality conditions. We find that all the intervalscontain 0. However, this ignores the joint distribution of theorthogonality conditions. Therefore, we conducted anoveridentifying test (J-test) of the quality of the instrumentsusing the demand-side model. This test exploits the ideathat the use of instruments that are correlated with the errorwill lead to biased parameter estimates, thus leading to poorfit for the moment conditions of instruments that are not(i.e., ). Table 1 presents the results of this analysis.In each period, the J-statistic does not differ significantlyfrom zero, indicating that, overall, the instruments areorthogonal to the model errors.

To ascertain whether the instruments are correlated withthe regressors, we computed the Shea’s statistic. Theresults appear in Table 1 and indicate a moderate correla-tion between the instruments and the regressors, compara-ble to or higher than that in Thomadsen (2005). In light ofthe range restrictions on many of our instruments, the sta-tistic indicates that the instruments are reasonably well cor-related with the regressors.

These tests also afford insights into our choice of the cut-off distance, r. Using a cutoff of 3.8 miles (which we choseto be slightly greater than the 95% spatial decay [3.6 miles]interval estimated in the “Results” section), Table 1 indi-cates mean independence between our instruments and νtsjusing the J-test. Moreover, we find that smaller distancesfor r do not increase the correlation between the instru-ments and prices (the Shea’s does not increase), sug-gesting that little is to be gained by choosing a smallerdistance.

SPATIAL PREDICTION AND OUTLET LOCATIONDECISION

After estimates of the demand and supply model areobtained, it is possible to forecast how entry and capacitydecisions at any given locale will affect the demand, prices,and profits for the existing firms and the new entrant. Suchinformation is useful to firms that want to assess the profitpotential of entering at various sites. The entering firm’sdecision problem consists of two steps: (1) selecting thelocation and (2) setting optimal price and capacity condi-tional on location. We can solve this decision problem bybackward induction: If the location has been selected, thefirm will set the price and capacity to maximize its profit.To achieve this aim, the firm must predict spatial randomeffects in demand and cost at the potential entry locationsand then compute the resultant equilibrium sales, prices,and profits at these locations. We discuss each step in turn.

Rp2

Rp2

Z tT

tν ≠ 0

Z tT

tν Spatial Prediction of Demand and Cost Effects: BayesianKriging

The first goal is to forecast random effects for long-termdemand shock, θs, and cost shock, ζs, over space for anynew location s in the future period T. This yields a map oflatent spatial demand and cost that can be used to obtaininsights into the nature of the market in which the outletscompete. To conserve space, we illustrate the spatial predic-tion for θs because the same procedure applies to predict ζs.Using Bayesian kriging (Banerjee, Carlin, and Gelfand2004), we assume that firms can estimate the distribution ofthe spatial random effects at the potential entry locationsconditional on the observed prices and sales at existingoutlets. Suppose that an entering firm considers building asingle outlet at a new location in a futureperiod T. To select a preferred location from potential loca-tions the firm wants to estimate the demandfunction at each of the considered locations , k = 1, …, n:

where

where Θ represents all the parameters and random effects(e.g., β1, β2, γ, , , , , and , l = 1, ..., J) in thisdemand system. The Θ are sampled when we fit the model,and the integrals can be approximated using Monte Carlosimulations for ηi. However, k = 1, …, n, for the con-sidered location are unknown as of yet. The estimation ofthe is spatial prediction. The entering firm is assumed touse Bayesian kriging to obtain the posterior distribution forlatent demand and cost at location and the

demand function .We assume that the location preference error θs follows a

Gaussian random field. Thus, for any two locations s1 ands2, θs1

and θs2have a bivariate normal distribution with the

covariance calculated from the covariance function of theGaussian process. With our choice of an exponential corre-lation function for the spatial random effects, exp(–φθ ×||s – s′||), the covariance of and is given by

exp(–φθ × ||s1 – s2||). Likewise, if there are J randomvariables associated with location s1, …, sJ, thepairwise covariance of θsk

and θsjis calculated as

exp(–φθ × ||sk – sj||), which is also the (k, j)th entry inthe covariance matrix of the multivariate normal distribu-tion for We denote this matrix as .

For the spatial prediction problem in our model, theobserved firms occupy the locations (s1, …, sJ) and the

σθ2R Jθ θs s…J1

, , .

σθ2

θ θs s…J1

, ,σθ2

θs2θs1

TkQ�ks�ζ �sk

θks�

θks�

θks� ,

θslφθσθ2σν2σγ2

( ) ˆ26 W W dF dF data

W

Tk Tik i

Tik

= ( ) ( )∫∫ �

�

η | |Θ Θ and

== + − + + +eX X PTk T ks Tk i ksβ β γ η θ ν1 2� �( ) TTsk

Tl Tsl TlJ X X Pe

( )⎡

⎣⎢

⎛⎝⎜ =

+ −∑ l 11 2β β γ( + + +

=++∑

η θ ν

β β

i sl Tsl

Tm T msm

n X Xe

)

11 � 22

e

− + + +

+⎞⎠⎟⎤

⎦⎥

P

M

Tm i ms Tsm

T

( )

,

γ η θ ν�

Tk TkQ I W( ) ,25 � �= ×

�sk

( , ..., ),� �s sn1

� � �s s sk n∈( , ..., )1

Table 1TEST OF IVs

p-Value (Under the Null

Period J-Statistic Hypothesis) Shea’s R2

2 7.38 .39 .253 6.86 .44 .264 6.93 .44 .335 7.25 .40 .306 8.02 .33 .317 8.31 .31 .29

χ72


9Note that there is a marginal construction cost associated with the addi-tion of each unit of capacity. However, all the construction costs becomesunk after the apartments are built. Our analysis proceeds on the assump-tion that the net present value of discounted profits exceeds the total con-struction cost.

entering firms select The corresponding spatialrandom effects are (θs1

, ..., θsJ) and (θs1

, ..., θsn), in which

(θs1, ..., θsJ

) can be computed from Equation 18 and theparameter draws from the Gibbs sampler. From theassumption of the spatial Gaussian process, (θs1

, ..., θsJ) and

(θs1, ..., θsn

) jointly have the following multivariate normaldistribution:

in which σθ2Rj + n is a (J + n) × (J + n) covariance matrix, inwhich the covariance of θsJ

(for current locations) and θsk(for future locations) is, by definition, σθ2 exp(–φθ × ||sk –sj||).

To sample (θs1, ..., θsn

) conditioning on the already-sampled (θs1

, ..., θsJ), we partition σθ2RJ + n into four sub-

matrices as follows:

where σθ2RJ,n includes the covariances of θsJand θsk

andσθ2Rn is the covariance matrix of (θs1

, ..., θ sn).

Conditioning on θ = (θs1, ..., θsJ

), θ = (θs1, ..., θsn

) has thefollowing distribution:

This is a n-variate conditional normal distribution withmean vector and covariance matrix σθ2[Rn –

Sampling (θs1, ..., θsn

) from this distribution iscalled Bayesian spatial prediction or kriging.

Equation 28 also affords insights into the effect of spatialcovariance on estimated demand. When spatial covarianceis ignored [RJ + n(φ) = IJ + n], the conditional spatial randomeffects are attenuated in expectation to zero [E(θ|θ) =

if RJ + n(φ) ≠ IJ + n].

Predicting Price and Profit

After we obtain predictions of demand and cost effects,the second step of the outlet location problem is to predictprofits and prices in new locales. This problem can be for-mulated as follows:

where

There is no capacity constraint because the firm canbuild to demand.9 Thus, the entering firm maximizes itsprofit with respect to PTk and selects the capacity

( ) (30 3 3� � �c Y e dFTk k s Tsc

k k= + + ,

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪β ζ β ζ

kk ks Tsce data� �, .∫ | )

( ) max ,29 ΠTk Tk Tk TkP c Q= −( )� �

R RJ nT

J,− ≠1 0θ ,

R R RJ nT

J J n,−

,1 ].

R RJ nT

J,−1θ

( ) ~28 1 2 1�θ θ σθN R R R R R Rn J nT

J n J nT

J J n,−

,−

,, −⎡⎣ ⎤⎦⎦( ).

σ σ

σ σθ θ

θ θ

2 2

2 2

R R

R R

J J n

J nT

n

,

,

⎡

⎣⎢⎢

⎤

⎦⎥⎥,

( ) ~271 1

θ θ θ θs s s s J n… … NJ n

, , , , ,⎛⎝⎜

⎞⎠⎟ +� � 00 2, +( )σθRJ n ,

( , ..., ).� �s sn1 The first-order condition is asfollows:

The existing outlets adjust their prices accordingly, know-ing the location selection of the entering firm. Theirstrategic pricing problem is as follows:

Ideally, the optimal price of firm f is determined by the fol-lowing Kuhn–Tucker condition:

The corresponding demand may exceed the capacity K1. The implementation of theoptimization problem involves two steps: With a certain setof interim prices PTl, the demand is calculated; if

, the corresponding PTl is solved from the bindingconstraint in Equation 34 in the next step of the optimiza-tion, and if , the corresponding PTl is solved fromEquation 33 in the next step.

An important consideration is the uniqueness of prices inthis equilibrium because multiple price equilibria wouldimply different profit outcomes for each equilibrium. InWeb Appendix A (http://www.marketingpower.com/jmrapril09), we prove the existence and uniqueness of theequilibrium prices in Equation 33 by constructing a con-traction mapping on a bounded set.

Equation 31 for the entering firm and Equation 33 for theexisting firms constitute the strategic optimization problemthat solves the new prices , , and the profit , ,where is a function of location . Optimizing withrespect to the location solves the optimal market entryproblem. We simulate demand for the entrant using a modalconfiguration of attributes (across the existing outlets).

DATA

To estimate our model, we use panel data on apartmentdemand and prices. Apartments are a desirable category forillustrating the model because the number of outlets is suf-ficiently small to make estimation feasible but is suffi-ciently large to obtain relatively reliable estimates of spatialeffects. Moreover, unlike many previous applications ofspatial demand models in economics, latent demand forapartments cannot be estimated as a function of the under-

�sk

ΠTk∗�skΠTk

∗ΠTk∗ΠTf

∗PTk∗PTl

∗

�Q KTl l<

�Q KTl l≥�QTl

Tl Tl Tl Tk T lQ Q P P P∗ ∗ ∗,−∗= , ,� ( )

( )33∂∂

= −( )∂∂ +∈∑ΠTf

Tl j FTj Tj

Tj

TlTP

P cW

PW

f

��

�ll

m Fm

Tm

Tl

f

Tm m

f

W

P

l F

I W K

−∂∂

=

∈

× = ,

∈∑ λ

�

�

0

34

;

;

( ) iiff and iffλ λm Tm m mI W K> × < , =0 0, .�

( ) max32 ΠTfj F

Tj Tj Tj

f

P c Q= −( )∈∑ ,

subject to

� �

�Q KTj j≤ .

�sk

( )31∂∂

= −( )∂ ,∂

∗∗( )�

��ΠTk

TkTk Tk

Tk Tk T

TPP c

W P P

P kkTk Tk T

Tk Tk i Tik

W P P

P c W

+ , =

⇒ −( ) − +( )

∗( )

∗

�

� �

0

γ η 11

0

−( )( ) ( ) + = .

∫∫ �

�

W

dF dF data W

Tik

i Tkη | |Θ Θ

� �K Q P XTk Tk T T T= , ,( ).Θ

http://www.marketingpower.com/jmrapril09


lying observed spatial distribution of the populationbecause the population itself is endogenous. We use datafrom Roanoke, Va., for our analysis. This market showsgood spatial coverage of apartments and little variability insupply, and a long time series of prices and vacancy ratesare available for these markets. These characteristics makethem ideal for our analysis.

The data for this study are provided by Real Data ofCharlotte, N.C., and are detailed at www.aptindex.com.Real Data conducts an annual survey of apartments in agiven market. The survey data consist of apartment attrib-utes (e.g., whether the apartment has tennis courts, whetherit has a pool, the age of the complex), addresses (which weconvert to latitude and longitude), and prices. The Roanokedata cover 60 apartment complexes managed by 25 differ-ent firms and covering 7 years (1999–2005). Figure 1depicts the location and occupancy levels of the apartmentsin 2005; the shortest bar corresponds to 50 rented units, andthe largest corresponds to 426 rented units. The averageapartment capacity of these apartments is 152 units. Eightapartments are missing several years of data, and thus weexclude them from subsequent analysis.

The apartment data are supplemented by two other datasources. First, we use the 2000 census data to attain thelocations of schools and the market size, I. The census dataat www.census.gov/geo/www/tiger contain the latitude andlongitude of the Roanoke schools. From these data, wecomputed the distance to the nearest school for use as anobserved spatial covariate in our analysis. Other censusdata (www.fedstats.gov/qf/states/51) report the number ofnon-home-owning households. We use this to determine I(for the Roanoke area, this was 29,489 households in

2000), but our results were not particularly sensitive to thischoice. Second, we determined the major employers in Roanoke from the Roanoke County Department ofEconomic Development (www.yesroanoke.com). For thelargest employers with a single central location, we com-puted the mean distance from each complex to the majoremployers. Table 2 summarizes the variables we use in ouranalysis:

To construct a price index from these prices for variousapartment sizes, we use the price of two-bedroom apart-

Figure 12005 LOCATIONS AND OCCUPANCY OF APARTMENTS IN ROANOKE, VA.

Table 2DATA DESCRIPTIVES

Attribute Observations M SD Maximum Minimum

Price (dollars) 364 491 102 753 290Capacity (units) 364 163 79 636 63Number of vacancies 364 10 17 150 0Clubhouse 52 35% 48% 1 0Tennis 52 32% 47% 1 0Swimming pool 52 77% 42% 1 0Gymnasium 52 20% 41% 1 0Heat included 52 30% 46% 1 0Year built 52 1976 11 2002 1952Average distance to

major employers (miles) 52 3.91 .86 5.7 2.28

Distance to nearest school (miles) 52 1.34 .74 3.05 .10

Maximum distance between apartments (miles) 1 9.6 — — —


Figure 2DISTRIBUTION OF 2005 RENTAL PRICES ($) IN ROANOKE, VA

900

800

700

600

500

400

300

.3

.25

.2

.15

.1

.05

0

–.5

Figure 3DISTRIBUTION OF 2005 VACANCY RATE (%) IN ROANAKE, VA.

.3

.25

.2

.15

.1

.05

0

–.5

ments (which we report in Table 2). We choose this simpleapproach for two reasons. First, to construct a measure thatconsiders all apartments, we would need to create an indexweight, and these weights may embed capacities, sales, orother factors that “contaminate” the pricing measure. Sec-ond, all the apartments have two-bedroom units, and thetwo-bedroom size is always the modal size (many apart-ments have only two-bedroom units). Thus, it is the mostrepresentative. When an apartment has multiple prices forthe two-bedroom units, we select the median value (no

information is available on the distribution of prices for agiven apartment size at a complex). During the interval ofthe data, average apartment prices in Roanoke increasedfrom $97 to $570 per unit, though they remained constantthe last three years. In the analysis, we adjust prices forinflation using Consumer Price Index figures from theBureau of Labor Statistics. Figure 2 presents a contour mapof 2005 rents in nominal dollars.

From Figure 2, we observe that the distribution of equi-librium prices is highly irregular, with the highest pricesobserved both near the downtown and on the periphery oftown. A band of lower prices snakes from northwest tosoutheast. Given that prices change nonmonotonically withdistance from the city center, it is desirable to capture ran-dom spatial effects rather than relying on the more com-monly used approach in which demand is a linear functionof distance to a population centroid, such as the center oftown. The proposed spatial random-effects approach canyield highly irregular demand and price surfaces. The spa-tial distribution of prices in Figure 2 also suggests that thereis spatial covariance in the data.

Vacancy rates averaged approximately 6%, with littlechange over time. In addition, total annual market capacitywas fairly constant over time, with a mean of 8985 unitsand a standard deviation of 337 units. No apartments werebeing constructed as of 2005. This suggests that it is appro-priate to model the subgame of prices in our data condi-tional on apartment locations and capacities. Figure 3depicts a contour map of vacancy rates. Vacancy rates tendto be highest on the west side of town.

RESULTS

Simulation Results

To show (1) how the omission of spatial covariance andcapacity constraints biases model estimates and (2) that ourmodel can recover the underlying parameters, we conduct asimulation using the approach described in Web AppendixB (http://www.marketingpower.com/jmrapril09). Usingthese simulated data, we estimate three models. In Model 1,we include spatial covariance and capacity constraints. InModel 2, we omit spatial covariance. In Model 3, we omitcapacity constraints. The Gibbs sampling chains proceedfor 10,000 iterations in each model, though they appear toconverge well before then (within 500 iterations). We dis-carded the first 20,000 draws to ensure convergence. Priorsare set to be as noninformative as possible. We use all avail-able attributes and prices from preceding periods as instru-ments. Subsequently, we report the parameter estimates andthe ability of three models to recover the parameters.

Full model. As Table 3 shows, the 95% posterior predic-tive interval for Model 1 contains the true parameter valuesfor all the parameters in our properly specified model, indi-cating that the model can recover the data-generating mech-anism. Of particular interest is the recovery of the estimatefor the highest Kuhn–Tucker multiplier. This parametercaptures the marginal cost of the constraint to the firm; inthis case, Apartment 39 in Year 8 would gain $131 inmonthly profit if it could add one apartment.

No spatial effects. Contrasting the full model to one inwhich we omit spatial correlation (Model 2), we observethe predicted downward bias in median price effects. The



Demand Side Posterior Median (95% Posterior Prediction Interval)

Parameter Variable Simulation Value Our Model No Spatial Correlation No Capacity Constraints

β1 Garage 1.00 1.01 (.87, 1.12) .94 (.85, 1.03) —β2 Distance to city –.30 –.29 (–.37, –.22) –.28 (–.36, –.19) —γ Price .01 .0096 (.0078, .012) .0077 (.0051, .0089) —σγ Standard deviation γ .002 .0017 (.0012, .0026) .0018 (.0013, .0026) —σθ Standard deviation θ .40 .38 (.33, .46) .42 (.32, .51) —φθ Spatial decay θ .40 .56 (.20, 1.02) — —σν Standard deviation ν .30 .29 (.25, .32) .35 (.30, .42) —φν Spatial decay ν .40 .44 (.20, .56) — —

Log-marginal likelihood –27.4 –43.9 —

Supply Side Posterior Median (95% Posterior Prediction Interval)

Parameter Variable Simulation Value Our Model No Spatial Correlation No Capacity Constraints

β3,0 Variable cost 150 147 (141, 154) 117 (103, 130) 173 (166, 180)β3,1 Apartment size 30 29.6 (29.0, 30.5) 29.5 (28.9, 30.6) 22.2 (19.8, 25.5)σζ Standard deviation ζ 10 10.5 (9.6, 11.3) 10.5 (9.6, 11.2) 20.6 (17.2, 25.7)ψζ Spatial decay ζ .40 .43 (.20, .74) — 1.31 (.86, 1.90)σc Standard deviation ec 10 10.1 (9.1, 11.3) 11.3 (9.9, 13.8) 17.3 (14.6, 20.1)ψc Spatial decay .40 .56 (.20, .87) — 1.00 (.74, 1.47)λmax Kuhn–Tucker 113 112 (105, 121) 115 (107, 129) —

Log-marginal likelihood –1093.000 –1130 –1276

Table 3SIMULATION RESULTS

10The ratio of the marginal likelihood (exponential of the log-marginallikelihood) is equivalent to a Bayes factor with noninformative priors, andtherefore we use this statistic for model comparison. We computed log-marginal likelihoods as in Gelfand and Dey (1994).

magnitude of this effect in this parameter is 20%. This biasis a consequence of how correlation in the spatial errors canexacerbate small-sample bias in generalized method ofmoments and IV estimation (Altonji and Segal 1996; Buseand Moazzami 1991). When these correlations are high,bias is present even in fairly large samples (Altonji andSegal 1996).

We also observe a downward bias of approximately 20%in the estimate for the intercept of marginal cost, β3,0. Thecause of this downward bias in estimated marginal costscan be observed by inspecting Equation 14. The downwardbias in the estimate for price elasticity γ lowers the expres-sion on the left-hand side of Equation 14. To maintain thisequality, estimates for the right-hand side must also belower, leading to a downward bias in the estimates for mar-ginal cost. Note also that there is a substantial decrease inthe demand-side log-marginal likelihood from –27.4 to–43.9.10 The log-marginal likelihood on the supply sidedecreases from –1093 to –1130.

No capacity constraints. Contrasting the full model toone in which we omit capacity constraints, we note thatcost estimates are biased upward. Firms raise prices in thepresence of capacity constraints to the point at whichdemand is equal to capacity because lower prices yieldslower per unit revenue with no attendant increase indemand. The observation of higher prices at a given level ofdemand leads to inferences of higher costs, consistent withEquation 14. Furthermore, the variance of the fixed and

time-varying spatial cost effects σζ and σc are over-estimated by nearly a factor of two as a result of the addi-tional error introduced by ignoring the capacity constraints.The spatial decay is overestimated by 150%. Reflectingthese biases, the log-marginal likelihood decreases consid-erably from –1093 to –1276.

In summary, the simulation data show that (1) our modelrecovers parameters well and (2) biases in parameter esti-mates that arise from ignoring spatial covariance andcapacity constraints are in the predicted direction. More-over, omitting spatial effects and capacity constraints has asubstantial impact on model fit.

Roanoke Results

Next, we apply the data detailed in the “Data” section toestimate our model of outlet demand and pricing. We ranthe sampling chain for 20,000 iterations. Inspection of thesequence of draws indicates good convergence afterapproximately 500 iterations, though we discarded the first2000 iterations. Moreover, little autocorrelation in thedraws was evident. In addition to estimating the full model,we provide two benchmarks: no spatial correlation and nocapacity constraints. Comparison of these models with thefull model yields insights into (1) the magnitude of parame-ter biases that can arise when capacity and spatial covari-ance in demand and prices are ignored and (2) the degree towhich spatial covariance improves model performance.

Parameter estimates. Table 4 presents the results of ourestimation. The full model is the best-fitting model, and itsimprovement over the model with no spatial covariance issizable (the improvement in the log-marginal likelihood is18 on the demand side and 22 on the supply side). This pro-vides evidence that spatial covariance matters in practice.The improvement over a model with no capacity constraints


Demand Side Posterior Median (95% Posterior Prediction Interval)

Parameter Variable Full Model No Spatial Correlation No Capacity Constraints

β1,1 Clubhouse .31 (.06, .57) .30 (.03, .58) —β1,2 Tennis .55 (.31, .80) .58 (.35, .83) —β1,3 Pool .51 (.20, .78) .45 (.14, .76) —β1,4 Gym .15 (–.19, .50) –.02 (.34, .27) —β1,5 Heat .32 (.14, .51) .32 (.15, .50) —β2,1 Distance to school .03 (–.16, .23) .05 (–.18, .28) —γ Price .0035 (.0024, .0046) .0032 (.0022, .0042) —σγ Standard deviation γ .00057 (.00030, .00084) .00058 (.00033, .00085) —σθ Standard deviation θ .60 (.50, .72) .72 (.60, .84) —φθ Spatial decay θ .84 (.39, 1.38) — —σν Standard deviation νt .11 (.10, .12) .11 (.10, .13) —φν Spatial decay νt 2.8 (2.0, 3.73) — —Log-marginal likelihood –2.84 –20.9 —

Supply Side Posterior Median (95% Posterior Prediction Interval)

Parameter Variable Full Model No Spatial Correlation No Capacity Constraints

β3,0 Variable cost intercept 432 (368, 498) 408 (346, 472) 405 (340, 465)β3,1 Heat 70.2 (33.5, 107.8) 61.6 (25.2, 98.6) 57.1 (22.3, 90.3)β3,2 Age –7.3 (–9.3, –5.2) –7.2 (–9.3, –5.3) –6.7 (–8.7, –4.6)σζ Standard deviation ζ 88.2 (75.1, 106.0) 91.0 (78.7, 108.3) 102.0 (90.3, 117.8)ψζ Spatial decay ζ 2.8 (.46, 5.66) — 1.9 (1.0, 5.6)σc Standard deviation et

c 26.6 (24.8, 28.8) 27.0 (25.4, 29.2) 34.7 (30.2, 39.7)ψc Spatial decay et

c 3.2 (.46, 6.00) — 4.2 (1.6, 7.5)λmax Kuhn–Tucker 88.5 (48.2, 129.6) 65.2 (26.1, 118.4) —Log-marginal likelihood –1362 –1384 –1503

Table 4ROANOKE DATA RESULTS

11When estimating the model, we noted that the correlation betweendistance to schools and distance to employers was high, so we omitted thelatter because it had less explanatory power.

is even more substantial, leading to a 141-point gain in thelog-marginal likelihood.

Demand-side estimates. All attributes, except for theaddition of a gym, play a significant role in apartmentchoice. The low effect of a gym may arise from many com-peting options, including work and office gyms and homeexercise equipment. Among the attributes, pool and tenniscourts each play the greatest role in apartment choice. The95% posterior predictive interval for distance to schoolsincludes zero, suggesting that this effect is negligible.11 Thesmall effect may reflect few school-age children amongapartment dwellers or the local school zoning policies. Theeffective range of the median spatial decay, in which 95%of the spatial effect has decayed, is given by 3/φθ (Banerjee,Carlin, and Gelfand 2004), or 3.6 miles. This suggests thatthe demand-side spatial covariance is sizable because themaximum distance between apartments is 9.6 miles. Theprice parameter is positive, indicating that an increase inprice lowers the likelihood of apartment choice (becausethis enters our likelihood function with a negative sign).Consistent with our previous findings, the price parameteris biased toward zero when spatial covariance and capacityconstraints are ignored; the bias is approximately 10%.

Supply-side estimates. Table 4 indicates that heatincreased the variable costs of the apartment by approxi-

mately $70 per month per unit in 1999 dollars and thatnewer apartments had higher operating costs, perhapsbecause of greater amenities. The highest median Kuhn–Tucker multiplier was $88 per month per unit for Apart-ment 37 in 2004. Among apartments at capacity, this apart-ment had the second-lowest costs. Because this apartmentis so efficient, its capacity constraint was especially costly.In addition, apartments with high rents tended to have highcosts of capacity constraints.

According to a 1998 survey of apartment managers con-ducted by the Institute of Real Estate Management, annualoperating expenses for apartments average 42% of revenue;these costs include administrative expenses, operatingexpenses, maintenance expenses, tax/insurance, and payrolland amenities. For the average rent of $485 in our data (in1999 dollars), this implies that our estimates for variablecosts should be approximately $204 per apartment in 1999dollars. Averaging our marginal cost estimates across apart-ments yields $233 in 1999 dollars, close to the $204implied by the survey.

The spatial effects on the supply side (prices), with aneffective range of 1.1 miles, are smaller than those on thedemand side. We conjecture that costs are a more “global”variable in the sense that all apartments likely face a similarcost of capital, utilities, labor, and supplies. On the demandside, however, certain regions are more desirable than oth-ers, leading to higher spatial correlation.

As we expected, cost estimates are biased downwardwhen spatial covariance is ignored, and the cost error vari-ance increases when capacity constraints are ignored. We


12Figure 4 interpolates latent demand for locations in which there are noapartments, based on our observed estimates for θ as well as kriged θ in a10 × 10 grid.

do not observe an upward bias in costs for the no-capacity-constraint model, perhaps because a limited number ofapartments are constrained in our data. Consistent with thesimulation findings, the no-capacity-constraint-cost modelhas a poorer fit, as indicated by higher estimates for ζ andσc. These parameters are, respectively, 16% and 30% toohigh when capacity is ignored.

Latent spatial effects. Using the median of the samples ofthe spatial random effects, we create contour plots of spa-tial random demand θsj

and cost effects ζsjand offer a dis-

cussion of these results. Figure 4 plots the θsjand then

interpolates latent demand between these observationsthrough triangle-based cubic interpolation.12 This yields amap of latent demand.

Figure 4 indicates several modes of high latent demand,including just west of downtown and a more prominentmode southwest of downtown. We compare Figure 4, whichcomputes the latent spatial demand effects, with Figure 2,which depicts vacancy. The highest demand region south-west of town not only has high latent demand but also hashigher prices. Taken together, these factors imply that thismodal location for demand is an ideal spot to locate a newcomplex.

However, selecting an optimal outlet locale must alsoassess the effects of competition and costs. For example,Figure 3 indicates that the mode in latent demand south-west of downtown also corresponds to higher vacancy ratesbecause of a surfeit of apartments. Thus, the problem of

outlet location is a complex calculus involving an array ofcountervailing factors. We attempt to explore these trade-offs more precisely in the next section.

MANAGERIAL IMPLICATIONS

Our approach can be used to engage policy experimentspertaining to the selection of a new outlet location becauseit structurally links prices to outlet entry. For example, afirm could compare across available properties the effect oflocating an additional outlet on demand, prices, and profitsat (1) the additional outlet and (2) other outlets in the chain.In our analysis, we compare the desirability of these entryoptions subject to the caveat that no other outlets enter.However, even in the context of competitive response, theeffect of various competitive location responses to thefirm’s choices of next outlet location could be simulated.This suggests that many scenarios could be used to simu-late entry effects.

It might be conjectured that firms already occupy theoptimal locations, so that the policy simulation is of littlevalue. We believe that this is unlikely primarily because theebb and flow of people into the market and changes in theattributes of various locales (e.g., new roads) over the yearsare likely to render the extant distribution of existing outletssuboptimal. Therefore, it is likely to be useful to conjecturewhat the next best location may be.

Using the procedure discussed in the “Model” section,we assess the effect of an additional apartment on equilib-rium demand, prices, and profits. We begin by creating a10 × 10 grid of potential apartment entry locations withinthe convex hull defined by extant apartment locales inRoanoke. For each location on the 100-point grid, we com-pute equilibrium profits associated with an entry at thatlocation. This computation assumes a modal apartmentdesign for the new complex—that is, using the modal fea-ture set in the data. We execute this procedure twice, oncefor our full model and once for a model that omits spatialcovariance and capacity constraints. This enables us to con-trast the recommendation of each model. Figure 5 depictsthe predicted equilibrium profits at these locations.

The top half of Figure 5 portrays the equilibrium profitsfor the various entry options using the full model. The areasof the solid circles correspond to predicted monthly profits,with a maximum of $61,427, a median of $37,025, and aminimum of $20,356. The bottom half depicts predictedprofits for the various entry options using the model withno spatial effects. The predicted profits from this modelrange from $38,136 to $35,572. The median predictedprofit is $36,971. Most strikingly, the constrained modelshows a lack of variation in predicted profits because itignores unobserved information regarding latent demandand costs from extant apartments to forecast demand andcosts at new locales. Instead, it uses only observed spatialeffects, such as distance to the nearest school, and theseobserved effects appear to be small compared with theunobserved spatial variation. The standard deviation in pre-dicted profits is 10.8 times larger for the spatial covariancemodel. With this model, it might be inadvertently con-cluded that some of the lower-profit locations would yield agood profit. Furthermore, Figure 5 indicates several “falsemodes,” in which the full model predicts small profits.

Figure 4LATENT SPATIAL DEMAND FOR APARTMENTS IN ROANOKE,

VA.

1.5

1

.5

0

–.5

–1

–1.5

1.5

1

.5

0

–.5

–1

–1.5


Figure 5KRIGED PROFITS FOR POTENTIAL APARMENT LOCATIONS IN

ROANOKE, VA.

Kriged Profits in the Full Model

Kriged Profits Omitting Spatial Effects and Capacity

The open circles in the top and bottom halves of Figure 5enclose the locales with the highest predicted profit in eachof the respective models. For the full model, the highestpredicted profit corresponds to a lesser mode in latentdemand just west of downtown, as evidenced in Figure 4.The no-spatial-variation demand model recommends alocale in the far southeastern part of Roanoke. If the recom-mendation of the model without spatial covariance were tobe adopted, expected profits would be $37,062 (which rep-resents the predicted profits from the full model using therecommended locale from the constrained model). Using

the recommendation from the spatial model increasesexpected profits to $61,427, or 66%. Because these esti-mates are cash flows and because firms expand into manydifferent cities, the profit implications of our model couldbe considerable.

The second-highest mode in profits is colocated with thehighest mode in latent demand we discussed in the“Model” section. To provide more intuition regarding whythe lesser mode in latent demand west of town is optimaland the larger mode in latent demand southwest of town isnot, Figure 6 depicts the optimal locale superimposed onthe spatial distributions of historical rental prices, historicalvacancy rates, latent costs, and latent demands. In these dis-tributions, a darker shade corresponds to a lower number.Black squares indicate apartment locations, and the whitecircle denotes optimal entry locale. The optimal locationresults because (1) latent demand is high, (2) latent costsare low, and (3) vacancy rates are low. Although the histori-cal rental prices are low, this is in part offset by lower costs,suggesting that high margins are possible even in the faceof lower rents. Furthermore, there is only one competingcomplex in this area.

In contrast, the second-highest mode in profits and high-est mode in latent demand corresponds to (1) higher latentcosts, (2) higher vacancy rates, and (3) more competingchains. Thus, the presence of competition affects the rec-ommendation for optimal location. We contend that widelyemployed tools for site location often neglect this aspect ofthe site location problem (Buckner 2004).

CONCLUSIONS

In this article, we address the problem of the firm’s outletlocation problem. A necessary step in this process is tosolve the subgame problem of firm pricing conditional onfirm locations. Firms can then use this model to simulatethe effect of locating an incremental outlet on equilibriumprices, demand, and profits. Our model applies these con-cepts in the context of apartment data and affords recom-mendations about the next best location for entry.

Our work extends prior research in several respects.Unlike gravity models, we explicitly and structurally con-sider competitive response in prices. Unlike analytical mod-els, our approach is data focused to enable insights intodata-driven decisions in several different marketing con-texts. We extend the structural model of the subgame ofoutlet location and prices to include, among other things,spatial correlation and capacity constraints. It is desirable toconsider spatial correlation for unobserved random demandand costs effects for several reasons. First, the omission ofspatial correlation in demand leads to downward biasedestimates of price effects and marginal costs in the smallsample sizes common in the outlet location problem. Sec-ond, spatial demand effects are noteworthy in their ownright, leading to a managerially informative map of latentdemand. Neglecting these effects obscures considerablespatial demand variation inherent in our data. It is alsodesirable to consider capacity constraints. Properlyaccounting for these effects mitigates an upward bias incosts, and the Kuhn–Tucker multipliers provide an estimateof the cost of the capacity constraint. Presumably, firmswith higher costs would be more inclined to consider addi-tional capacity. In addition, failure to consider capacity


Figure 6OPTIMAL LOCATION (WHITE CIRCLE) SUPPERIMPOSED ON RENT, VACANCY, COST, AND DEMAND

Rental Prices ($) in Roanoke, Va. Vacancy Rates (%) in Roanoke, Va.

Latent Costs in the Full Model Latent Demand in the Full Model

constraints and spatial effects leads to suboptimal recom-mendations for firm location.

To achieve these aims, we integrate Bayesian spatial sta-tistics with structural models of competition in a model ofoutlet demand and pricing. Simulations indicate that theresultant model recovers parameters well and that ignoringspatial effects and capacity constraints leads to biasedparameter estimates. We then apply this model to a novelapartment data set that includes prices, demand, and capac-ities over a six-year period. We find that the inclusion ofspatial covariance in demand improves model fit and that

omitting spatial effects and capacity constraints leads tobiased parameter estimates, as predicted.

Given that we attempt to assess the policy implicationsof outlet location, we use a spatial kriging approach cou-pled with our supply-side model to explore the impact oflocating an additional outlet at potential sites of interest to afirm (or more systematically on a grid). These simulationsreveal that (1) the best entry locations are related to highlatent spatial demand and a dearth of nearby outlets, (2)accommodating capacity and unobserved spatial effects inpolicy simulations improves the profitability of recom-


13We thank an anonymous reviewer for this observation.

mended entry locales by 66% in our data, and (3) ignoringspatial covariance leads to little variation in predicted prof-its, understating the actual standard deviation in predictedprofits across space by a factor of 11.

Several limitations exist that also represent opportunitiesfor further research. First, we consider the pricing subgameonly and do not model the outlet location game. We do thisbecause the subgame is an important problem in its ownright and is useful for policy simulations over the interme-diate term. An important extension would be to consider theentry problem as well. We believe that this would be a diffi-cult extension because it involves a dynamic program tosolve for sequential entry and prices, and it is likely thatsolutions to this problem would not be unique. Moreover,the distribution of latent demand can render such equilibriaobsolete after a few years.

Second, many outlets sell multiple goods or have prod-ucts that appeal to different markets (e.g., supermarkets,medical centers). It would be desirable to ascertain whethercompetition across goods leads to different outcomes forlocation and pricing than in the indexed single-good casewe model herein.

Third, we presume that attributes such as pool and tennisare exogenous. It would be desirable to model not only thelocation of an outlet but also its optimal design. We believethat this exercise in combinatorial optimization wouldprove challenging, especially with respect to documentingthe uniqueness of these equilibria.

Fourth, our model is of limited applicability when alter-native channels, such as the Internet, constitute a significantportion of demand. Multichannel models (Ansari, Mela,and Neslin 2008) could be overlaid with the spatial modelsto develop unique insights into this setting.

Fifth, in Web Appendix A (http://www.marketingpower.com/jmrapril09), we establish the uniqueness of prices onthe supply side under the condition that price sensitivity isnegative. Given that the supply-side model is therefore afunction, this result implies that it may be possible todevelop a full-information likelihood-based inferentialtechnique for supply and demand. Such an approach couldlead to gains in efficiency over extant estimation approachesand yield more desirable small-sample properties.

Finally, it would be worthwhile to consider capacity astrategic variable. Because we observe only one complexchange capacity in our data, it is an ideal setting to exploreBertrand–Nash competition; however, we believe that it isdesirable to extend this research to the context of Cournotcompetition. Of note, the Bertrand and Cournot pricing out-comes align in a two-stage game in which capacity is set inStage 1 and prices are realized conditional on capacity inStage 2 (Kreps and Scheinkman 1983). Given that wemodel prices in a subgame conditional on capacity con-straints, this indicates that the predicted prices in our modelmay be equivalent to the Cournot pricing outcomes.13

In summary, the approach we develop incorporates spa-tial effects into a model of outlet competition and affordsinsights into the role of unobserved spatial effects on outletpricing and demand. We hope that our research will sparksubsequent research related to these and other remainingissues.

APPENDIX: SAMPLER IN THE ESTIMATION SECTION

Demand Model

Step 1: Sample (θ, β1, β2, γ). The location-specific effectsfor an outlet are given by θ ~ N[Xα1 + Xsα2, d)],where X and Xs are time-invariant attributes. Note that

. Let and θA =(θ, β1, β2, γ), where θA has the multivariate normal priorN(μ0, Σ0), where

The prior is conjugate to the likelihood. The full condi-tional for θA is N(μA, ΣA), where

Step 2. Sample α1, α2, and θ. Let αA = (α1, α2) and XA =(X, Xs). Given , , , and the prior for thefull conditional distribution from the posterior for α1 andα2 is

In our Roanoke data analysis, we use independent and dis-perse priors for α1, α2, β1, and β2: N(0, 108).

Step 3. Sample σγ and Metropolis–Hastings algo-rithm. Suppose that σγ has a truncated normal prior N(σγ|0,

Use a truncated normal N(σγ|μp,

to generate . After recalculating conditional on using a contraction mapping as we indicated previously, theacceptance probability for σγ and is given as

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Step 4. Sample and We use the conjugate inversegamma (1, 1) priors for and . The posterior distribu-tions are also inverse gamma.

Step 5. Sample φθ and φν: discrete sampler. We assumethat φν (or φθ) can only take n values: (φ1, …, φn). For eachφν (or φθ), we calculate the posterior probability

We Sample φ with replacement from (φ1, …, φn).

Cost Model

Step 1. Sample β3 and ζ. Let YtA = (IJ, Yt), and let ζA =(ζ, β3), where ζA has the multivariate normal prior N(μ0,Σ0), where

Given λ2t and γ (γ is sampled from the demand model), thelikelihood is as follows:

where whereλ1t = 0 and λ2t > 0). The full conditional for ζA is N(μA,ΣA), where

In our Roanoke data analysis, we use independent anddisperse priors for β3: N(0, 108Ip).

( )A Y R YAt

T

tAT

J c tA61

1

21

10

1Σ Σ= +⎡

⎣⎢

−

=

− ( ) −∑σ ψζ⎢⎢

⎤

⎦⎥⎥

==

− ( )∑

and

( )A Y R PA At

T

tAT

J c712

1

1μσ

ψζ

Σ tt t tB− ,( )⎡⎣ ⎤⎦ +⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪.−λ γ μΣ0

10

( [ ] ,λ λ λt t tT= ,1 2B Wt t t

tt( ) ˆ ˆ( )λ γ λ, = − −Ω 1

( ) expA P B Yc t

TT

t t t tA A51

2 21

− − ,( ) −⎡⎣ ⎤⎦=∑σ λ γ ζ

⎧⎧⎨⎪

⎩⎪

− ,( ) −⎡⎣ ⎤⎦⎫⎬⎪

⎭⎪− ( )R P B YJ c t t t tA A

1 ψ λ γ ζ ,

μ μσ φβ θ θ

β0 0 0

2

0

0 3

3= , =

( )⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥( ) andT

JRΣ

Σ ⎥⎥.

−( )=

( )− −∑12

1

2 21

Z R Z X ZtT

J i tt

TT

t tA Aφσ ξ θν

exp ˆtt

t

T

tT

J i tZ R Z

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−

=

( )∑1

1

φtt

T

tT

t tA A

J i

Z X

Ror

=( )

−( )

∑ −⎫⎬⎪

⎭⎪

−

1

12

1

ˆ

exp

ξ θ

φ22 2

1

σθ φ θθ

TJ iR− ( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪.

σθ2σν2σν2.σθ2

tξ

2,N Z Z XtT

t tT

tAξ θ σν|ˆ ZZ R Z

N

tT

J tt

T

W

t

t

φ

σ

ν

ξγ

( )=

∂

∂

⎡⎣ ⎤⎦⎧⎨⎪

⎩⎪

,

∏1

0ˆ

ˆ| σσ σ μ σγ0

2 2( ) ( ) ,( )⎫⎬⎭⎞

⎠

⎟⎟⎟

Nn

p p| 1, .

Step 2. Sample λ2t: data augmentation. Let

Because is equivalent to it followsthat

or, equivalently,

making use of the properties of the conditional normal andsubtracting from both sides of the resultant conditionaldistribution for Noting that λ2t is structurallygreater than zero, we sample λ2t from the following trun-cated normal distribution:

Step 3. Sample and We use the conjugate inversegamma (1, 1) priors for and . The posterior distribu-tions are also inverse gamma.

Step 4. Sample ψζ and ψc. We use a discrete sampler. Weassume that ψc (or ψζ) can take only n values (ψ1, …, ψn).For each ψi, we calculate the posterior probability,

Then, we sample ψc (or ψζ) with replacement from (ψ1, …,ψn).

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−( )=

( ) − ( )− −∑12

1

2 21

1R P RJ ic t

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σλ ψexp � ��P

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�

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t tt

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t t

= ( )= + ( ) − −−

21

1 1 1 3 11

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ˆ ˆ( )Ω β ζ ,, + ( ) − −⎡⎣

⎤⎦.

−P W Ytt

t t2 2 2 3 21ˆ ˆ( )Ω β ζ


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The Role of Spatial Demand on Outlet Location and Pricingmela/bio/papers/Duan_Mela_2006.pdf · The Role of Spatial Demand on Outlet Location and Pricing 261 1By sales, we refer to