Dynamic Structural Models of Store Location: Methods and ...aguirregabiria.net/wpapers/slides_earie_storelocation...Methods and Applications Victor Aguirregabiria (University of Toronto)

Dynamic Structural Models of Store Location:Methods and Applications

Victor Aguirregabiria (University of Toronto)

34th CONFERENCE OF THE EARIE

Valencia. Sept. 6-9, 2007

Victor Aguirregabiria () Structural Models of Store Location Valencia. Sept. 6-9, 2007 1 / 50

Introduction: Importance of multi-store retailers

Multi-store retailers (retail chains) represent a large fraction of salesin retail industries such as supermarkets, gas stations, commercialbanking, restaurants, bookstores, co¤ee shops, drugstores, etc.

In the last years, Retail chains like Wal-Mart, Starbucks, McDonalds,Zara, etc, are among the most successful businesses.

Some authors claim that these �rms represent most of theproductivity growth in the US economy during the last decade.












Introduction: Geographic location of stores

Informal evidence seems to indicate that for retail chains thegeographic location of stores:

1 is the most important strategic variable ("Location, Location,Location");

2 is a forward-looking decision with signi�cant sunk costs due toinvestments which are both �rm- and location-speci�c.

3 is an important source of market power;

4 it accounts for a signi�cant fraction of their value;






























Introduction: Store location and market power

There are di¤erent ways in which the network of stores can be asource of market power

1 Consumer transportation costs: some locations are better than other;

2 Consumer transportation costs: distance to competitors�stores;

3 Multi-store (store substitution e¤ects);

4 Spatial preemption and entry deterrence;

5 Economies of density;

6 Sunk costs of setting-up a store;
























































Introduction: Recent empirical researchToday, we know very little about the relative contribution of thesefactors to explain the pro�tability of multi-store retailers;

Recent developments have generated an increasing interest in theempirical analysis on these questions:

1 Increasing availability of spatio-temporal panel datasets withinformation at the level of individual stores;

2 Advances in spatial econometrics and in tools for dealing withgeographic data (GIS);

3 Advances in methods for the estimation of structural models ofmarket entry (static and dynamic);

4 Increasing importance of multi-store retailers in today�s economies:mergers, anti-trust cases.





































Introduction: Dynamic game of store location

The speci�cation and estimation of a structural dynamic game can bea very useful tool to understand how store location contribute to thepro�tability of multi-store retailers.

1 Data on prices and quantities at the store level is rarelyavailable. BLP or Slade-Pinske approaches are not feasible

2 Important parameters do not appear in demand system andpricing equations: Fixed costs; Economies of density and scope;Entry costs; Store setup costs; Investment costs.

3 Endogenous mergers: Mergers are prevalent in some theseindustries. A merger can be the best strategy to increase the networkof stores.




















Introduction: The principle of Revealed Preference

The estimation of structural models of market entry or/and storelocation is based on the principle of Revealed Preference.

This principle establishes that if a �rm has a store in a particularlocation this is because the value of being active is greater than thevalue of shutting down and putting its assets in alternative uses.

Under this principle, �rms�location decisions reveal information aboutthe underlying latent pro�t parameters.












OUTLINE

1 A Dynamic Game of location by multi-store retailers.

2 Econometric Issues.

3 An Application to supermarkets.

4 Conclusions and further questions


OUTLINE






OUTLINE






OUTLINE






Dynamic Game: Previous related studies

The model builds on and extends the models in recent studies onstore location such as:

1 Seim (RAND, 2006) on video-rental stores

2 Toivanen and Waterson (RAND, 2005) on fast-food

3 Thomadsen (RAND, 2005) on gas stations

4 Thomadsen (MS, 2007) on fast food

5 Holmes (2007) on Wal-Mart

6 Aguirregabiria and Vicentini (2006)
























































Dynamic Game: Some general features

1 Dynamic: spatial preemption; sunk costs; �rst-mover advantages;

2 Game: important strategic considerations;

3 Multi-store

4 Consumer preferences and demand is explicitly modeled;

5 There is not an ad-hoc geographic de�nition of a market: this isendogenously given by the model (and varies over time and acrossmarkets);

6 Endogenous mergers: important in some retail industries.





3 Multi-store








3 Multi-store








3 Multi-store








3 Multi-store








3 Multi-store





A Dynamic Game: The Market

Market of a di¤erentiated retail product: e.g., food retailing

From a geographic point of view, the market is a set in the2-dimension Euclidean space.

The market may be a small town, a big city, a province, or even acountry.

We relax the assumption of "small and isolated local markets".

There is a �nite set of L pre-speci�ed locations where it is feasible tooperate stores.

The value of L can be very large (curse of dimensionality?).

We use the index variable ` for locations.

























































The market is populated by a continuum of consumers.

Each consumer is characterized by a pair (z , y) where z is his address,and y is a vector of attributes that a¤ects demand (e.g., income, age).

φt (z , y) is the density of consumers with attributes y at location z .

Consumers face transportation costs when visiting a store.

Transportation costs depend on the Euclidean distance kz � z`k(other metric? transportation time?)































There are F �rms in the market. We index �rms by f .

Some �rms are active or incumbents (they have stores in the market)and others are only potential entrants.

Every period, incumbent �rms:

1 Compete in prices;

2 Decide the number and location stores;

3 Decide to exit or not from the market;

4 Can merge









4 Can merge









4 Can merge









4 Can merge









4 Can merge









4 Can merge









4 Can merge


Consumer Behavior and Price CompetitionLogit model of demand with consumer transportation costs andconsumer heterogeneity in attributes y .

The utility of buying at store (f , `) for a consumer with attributes(z , y) is:

γ1(y) (ωf ` � pf `)� γ2(y) kz � z`k+ uf `

ωf ` is the quality of store (f , `);

pf ` is the price at (f , `);

γ1(y) and γ2(y) are parametric functions of the attributes y ;

uf ` are consumer idiosyncratic tastes which are iid extreme value




γ1(y) (ωf ` � pf `)� γ2(y) kz � z`k+ uf `








γ1(y) (ωf ` � pf `)� γ2(y) kz � z`k+ uf `








γ1(y) (ωf ` � pf `)� γ2(y) kz � z`k+ uf `








γ1(y) (ωf ` � pf `)� γ2(y) kz � z`k+ uf `








γ1(y) (ωf ` � pf `)� γ2(y) kz � z`k+ uf `






Consumer Behavior and Price Competition

Firms compete in prices a la Nash-Bertrand to maximize variablepro�ts:

L

∑`=1

nf `t (pf `t � cf `) qf `t

Firms may charge di¤erent prices at di¤erent stores.

The state of the market at period t is: (nt , mt , φt )

nt = fnf `t : f = 1, 2, ...,F ; ` = 1, 2, ..., Lg 2 f0, 1gFL represents thenetwork of stores for each (potential) �rm.

mt is a partition of the set f1, 2, ...,Fg that represents the ownershipstructure at period t;

φt is the state of the demand




L

∑`=1










L

∑`=1










L

∑`=1










L

∑`=1










L

∑`=1









For any given value (nt ,mt ,φt ), a Nash-Bertrand equilibrium exits.

However, the equilibrium is not necessarily unique.

Non-uniqueness is a potential problem for estimation and prediction.

To deal with non-uniqueness, we exploit some properties ofsupermodular games.

First, we establish su¢ cient conditions under which this pricing gameis smooth supermodular.

If the dispersion of the �logit error" is not too small, then the game issmooth supermodular.











































Supermodularity implies that the multiple equilibria in prices are"sorted": from the one with the lowest prices to the one with thelargest prices.

We can apply Topkis (1979) algorithm to obtain the two extremumequilibria.

Echenique (JET, 2007) has developed a simple and e¢ cient extensionof Topkis algorithm that provides all the equilibria of a supermodulargame.

We can apply Echenique�s algorithm to select a particular equilibriumor to obtain predictions that apply to all the equilibria.




















Consumer Behavior and Price CompetitionGiven a vector of equilibrium prices fp�f `tg associated with marketstructure (nt ,mt ,φt ), we can obtain the indirect variable pro�tfunction:

Rf (nt ,mt , φt ) = ∑L`=1 nf `t (p

�f `t � cf `) q�f `t

In principle, to obtain the equilibrium of the dynamic game we haveto obtain these variable pro�ts for every possible value of the marketstructure (nt ,mt , φt ).

However, in any real application the number of possible marketstructures is extremely large.

Our method to approximate the equilibrium of the dynamic game onlyrequires to calculate Rf (nt ,mt , φt ) in a relatively small grid of marketstructures and then use interpolation methods (see below).



Rf (nt ,mt , φt ) = ∑L`=1 nf `t (p

�f `t � cf `) q�f `t






Rf (nt ,mt , φt ) = ∑L`=1 nf `t (p

�f `t � cf `) q�f `t






Rf (nt ,mt , φt ) = ∑L`=1 nf `t (p

�f `t � cf `) q�f `t





Dynamic Game of Store LocationEvery period �rms decide their networks of stores for the next period.

Firms maximize expected discounted pro�ts (taking into accountpossible future mergers).

A �rm can: open a new store; close an existing store; do nothing.

If time frequency is high enough, this is not really restrictive (data?).

Let aft be the decision of �rm f at period t such that:

1 aft = `+ OPENING a new store at location `

2 aft = `� CLOSING a store at location `

3 aft = 0 KEEP CURRENT NETWORK OF STORES

































































Dynamic Game of Store LocationFirm f total current pro�ts are:

πft = Rf (nt ,mt , φt )| {z } � FCft| {z } � SCft| {z }Variable pro�t Fixed costs Setup costs

Fixed Costs:

FCft = ∑L`=1 nf `t

�θFCf ` + εFCf `t � θEDf d(`, nft )

�

θFCf ` is the exogenous �xed cost: �rm and location speci�c.

εFCf `t is an iid zero mean private information shock.

d(`, nft ) measures the density of �rm f 0s stores around location `.

θEDf is a parameter that measures the degree of economies of density.




Fixed Costs:



�








Fixed Costs:



�








Fixed Costs:



�








Fixed Costs:



�








Fixed Costs:



�






Dynamic Game of Store Location

Setup Costs:

If �rm f opens a new store at location `:

SCft = θSC (+)f ` + ε

SC (+)f `t

If �rm f closes a store at location `:

SCft = θSC (�)f ` + ε

SC (�)f `t

εSC (+)f `t and ε

SC (�)f `t are iid zero mean private information shocks.



Setup Costs:



SC (+)f `t



SC (�)f `t

εSC (+)f `t and ε




Setup Costs:



SC (+)f `t



SC (�)f `t

εSC (+)f `t and ε




Setup Costs:



SC (+)f `t



SC (�)f `t

εSC (+)f `t and ε



Dynamic Game of Store LocationIt will be convenient to represent the current pro�t function ofchoosing alternative aft as:

πft (aft ) = zf (aft , nt ,mt , φt ) θf + εft (aft )

where:

θf is the column vector with structural parameters for �rm f :

θf =n1 , θFCf , θEDf , θSCf

o

zf (aft , nt ,mt , φt ) is a row vector with known functions:nRf (nt ,mt , φt ) , � nft , ∑L

`=1 nf `td(`, nft ) , 1(aft )o

And εft (aft ) is the private information shock that corresponds to thisaction and state.




where:



o


`=1 nf `td(`, nft ) , 1(aft )o





where:



o


`=1 nf `td(`, nft ) , 1(aft )o





where:



o


`=1 nf `td(`, nft ) , 1(aft )o

And εft (aft ) is the private information shock that corresponds to thisaction and state.Victor Aguirregabiria () Structural Models of Store Location Valencia. Sept. 6-9, 2007 22 / 50

Dynamic Game: Mergers

When a �rm decides opening/closing stores, it takes into accountthat mergers may occur in the future.

I present here a simple model of mergers between multi-store retailers.

This model implies that mt evolves according to a transitionprobability:

Pr (mt+1 j nt , mt , φt )

that is endogenously determined in the equilibrium of the model.

















Assumptions on mergers:

1 Mergers are irreversible.

2 At a given period, mergers only of two �rms can occur.

3 After a merger, the new �rm inherits both networks of stores and acost structure that results from combining in the most e¢ cient waythe costs of the two �rms;





















More Assumptions on mergers:

4. There are exogenous "merging costs":

Merging Cost (A and B) = θMG (nAt , nBt ) + εMGABt

They represent actual costs and restrictions from the regulator.

5. Mergers occur sequentially, starting with the one with higher value. Amerger occurs i¤:

(i) it generates positive value (net of merging costs);

(ii) a previous merger does not make it infeasible (i.e., merger ACmakes AB infeasible).




















Dynamic Game: MergersDe�ne ∆AB as the added value of a merger between �rms A and B:

∆AB � VAB � VA � VB �MG Cost

Suppose that there are three four �rms: A, B, C and D.

And suppose that:

∆AB > ∆AC > ∆BC > 0 > ∆BD > ∆AD > ∆CD

The model predicts that the merger AB will occur, but AC and BCwill not.

This model of mergers provides a transition probability for theownership variable mt :






And suppose that:

∆AB > ∆AC > ∆BC > 0 > ∆BD > ∆AD > ∆CD








And suppose that:

∆AB > ∆AC > ∆BC > 0 > ∆BD > ∆AD > ∆CD








And suppose that:

∆AB > ∆AC > ∆BC > 0 > ∆BD > ∆AD > ∆CD








And suppose that:

∆AB > ∆AC > ∆BC > 0 > ∆BD > ∆AD > ∆CD





Markov Perfect Equilibrium

A �rm�s strategy depends only on its payo¤ relevant state variables:(nt ,mt ,φt ,εft ).

A MPE can be represented in terms of a set of choice probabilitiesconditional on common knowledge state variables:

Pf (aft j nt ,mt , φt )

for every �rm f , any action aft and any state (nt ,mt , φt ).

A MPE is a �xed point of a best response mapping that maps choiceprobabilities into choice probabilities.

P = Ψ(P)

where P is the vector with all the choice probabilities.








P = Ψ(P)









P = Ψ(P)



MPE: Equilibrium mapping

In our model, the equilibrium mapping Ψ(.) takes a simple form.

Ψ(P) = fΨf (af j n,m, φ;P) for any (f , af , n,m, φ)g such that:

Ψf (af j n,m, φ;P) =exp

�z̃Pf (af , n,m, φ) θf

∑j2Af

exp�z̃Pf (j , n,m, φ) θf

z̃P (af , n,m, φ) is the discounted and expected value of future

realizations of vectors z�af ,t+j , nt+j ,mt+j , φt+j

�given that the

current action and state is (af , n,m, φ) and that all the �rms behavein the future according to the choice probabilities P.

A MPE exits (simply by Brower�s Theorem).







∑j2Af




�given that the









∑j2Af




�given that the









∑j2Af




�given that the




MPE: Non-uniqueness

In general, the equilbrium is not unique.

This poses some estimation problems.

The recent literature on estimation of dynamic games has proposeduseful methods to deal with this problem.

Under the assumption that the data has been generated from onlyone MPE, say P0, it is possible to estimate structural parametersusing relatively simple methods.

More importantly, non-uniqueness implies serious problems when weuse the estimated model to make counterfactual experiments.

I�ll discuss di¤erent approaches which have been used/proposed todeal with this problem.


MPE: Non-uniqueness








MPE: Non-uniqueness








MPE: Non-uniqueness








MPE: Non-uniqueness








MPE: Non-uniqueness








MPE: ComputationLet�s use x to represent (n,m, φ). Then:

z̃Pf (af , x) = zf (af , x) + β pPf (af , x) WPf

where:

β is the discount factor;pPf (af , x) is a vector of transition probabilities (that depends on

competitors�choice probabilities);W Pf is a matrix that solves the �xed point problem:

W = ∑afPf (af ) � fzf (af )+β pf (af ) Wg

Given that the dimension of P and W Pf is extremely large, solving

exactly for W Pf is impractical.

I use interpolation techniques: Rust (1997); Aguirregabiria (2007).




where:β is the discount factor;

pPf (af , x) is a vector of transition probabilities (that depends oncompetitors�choice probabilities);

W Pf is a matrix that solves the �xed point problem:








where:β is the discount factor;pPf (af , x) is a vector of transition probabilities (that depends on

competitors�choice probabilities);

W Pf is a matrix that solves the �xed point problem:

































DataWe have a panel data set where observe the location of stores over ageographic market.

Data : faft , nft : f = 1, 2, ...,F ; t = 1, 2, ...,Tg

We also observe the history of mergers:

Data : fmt : t = 1, 2, ...,Tg

And the distribution of consumer demographics:

Data : fφt (z , y) : for any (z , y); t = 1, 2, ...,Tg

Demographics are observed at the level of census tract. We assumethat they are uniformly distributed over the census tract.





Data : fmt : t = 1, 2, ...,Tg








Data : fmt : t = 1, 2, ...,Tg








Data : fmt : t = 1, 2, ...,Tg



Demographics are observed at the level of census tract. We assumethat they are uniformly distributed over the census tract.Victor Aguirregabiria () Structural Models of Store Location Valencia. Sept. 6-9, 2007 31 / 50

Structural parameters of the model

1 Cost adjusted qualities: fω̃f ` � ωf ` � cf `g

2 Consumer heterogeneity: γ1(y) and γ2(y).

3 Fixed costs: fθFCf ` g and θED

4 Setup costs; fθSC (+)f ` g and fθ

SC (�)f ` g

5 Merging costs: fθMG g







SC (�)f ` g








SC (�)f ` g








SC (�)f ` g








SC (�)f ` g



Unobserved location heterogeneity

Unobserved heterogeneity across locations is typically very importantin these applications: See Seim (2006), Thomadsen (2007), amongmany others.

Not accounting for this heterogeneity can imply serious biases in theestimates of some structural parameters. For instance:

1 Underestimation of competition e¤ects (to explain the agglomerationof stores of di¤erent �rms in some "good" locations/regions).

2 Overestimation of economies of density (to explain the agglomerationof stores of the same �rm in some "good" locations/regions)




















Unobserved location heterogeneityThe model can incorporate unobserved location heterogeneity in avery �exible way: qualities, �xed costs and setup costs.

ωf ` = ω̄f + σωξ ξω

`

θFCf ` = θ̄FCf + σFCξ ξFC`

θSCf ` = θ̄SCf + σSCξ ξSC`

θ̄FCf , ω̄f , θ̄

SCf , σω

ξ , σFCξ and σSCξ are parameters.

ξ` are random variables with zero mean, unit variance, knowndistribution, and discrete support.

De�ne ξ` � fξω` , ξ

FC` , ξ

SC` g as the market type, that follows a

discrete spatial stochastic process.




`




SCf , σω




FC` , ξ






`




SCf , σω




FC` , ξ






`




SCf , σω




FC` , ξ





Therefore, the structural parameters are:

1 Cost adjusted qualities: fω̄f g, σωξ


3 Fixed costs: fθ̄FCf g, σFCξ and θED

4 Setup costs; fθ̄SC (+)f g, fθ̄

SC (�)f g, σ

SC (+)ξ , and σ

SC (�)ξ









SC (�)f g, σ

SC (+)ξ , and σ

SC (�)ξ









SC (�)f g, σ

SC (+)ξ , and σ

SC (�)ξ









SC (�)f g, σ

SC (+)ξ , and σ

SC (�)ξ









SC (�)f g, σ

SC (+)ξ , and σ

SC (�)ξ









SC (�)f g, σ

SC (+)ξ , and σ

SC (�)ξ



Estimation method: NPL

Two-step estimation methods (as Hotz-Miller, orBajari-Benkard-Levin) cannot be applied to models with unobservedlocation heterogeneity.

Nested Pseudo likelihood (NPL) method

Let Q(θ,P) be the (pseudo) likelihood function based on bestresponse probabilities Ψf (af j n,m, φ;P) for an arbitrary value of P.

In our model Q(θ,P) is the likelihood function of a randomcoe¢ cients conditional logit model, and it is globally concave in θfor any possible value of P.




















Estimation method: NPLThe NPL estimator is de�ned as a pair (θ̂, P̂) that satis�es twoconditions:

(1) θ̂ = argmaxθ Q(θ, P̂)

(2) P̂ = Ψ(θ̂, P̂)

A simple algorithm that can �nd the NPL estimator is: start with anarbitrary P̂0; then at iteration k obtain θ̂k as:

θ̂k = argmaxθ Q(θ, P̂k�1)

And update the probabilities using P̂k = Ψ(θ̂k , P̂k�1). That is:

P̂f ,k (af j n,m, φ) =exp

nz̃ P̂k�1f (af , n,m, φ) θ̂f ,k

o∑j2Af

expnz̃ P̂k�1f (j , n,m, φ) θ̂f ,k

o




(2) P̂ = Ψ(θ̂, P̂)






o∑j2Af


o




(2) P̂ = Ψ(θ̂, P̂)






o∑j2Af


oVictor Aguirregabiria () Structural Models of Store Location Valencia. Sept. 6-9, 2007 37 / 50


The main computational burden of this algorithm comes from thecomputation of the present values z̃ P̂f .

In fact,the exact computation of z̃ P̂f is not feasible.

I use interpolation techniques to approximate z̃ P̂f .

Advantages of interpolation versus simulation.




















Application: Supermarkets in North Carolina

Markets: Three most important metropolitan areas in NC:Greensboro, Charlotte and Raleigh-Durham.

Period: 1921 to 2002.

Firm Data: Annual information on entry, exit, ownership and addressof every supermarket and grocery store in these cities. Source: Citydirectories

Demographic Data: For each decennial census, we collected at thelevel of census tract: population, median household income, andmedian rent.

Estimation period: 1961 - 2002.




Period: 1921 to 2002.







Period: 1921 to 2002.







Period: 1921 to 2002.







Period: 1921 to 2002.





Descriptive evidence: Comparing periods 1960-79 and1980-2002

1 Number of stores has been declining since the 1960s.

2 Proportion of stores from multi-store retailers has been increasing.

3 Distance between stores has been increasing, but the nature di¤ers inthe periods 1960-79 and 1980-2002.

In 1960-79 average distance increases because new stores arecreated far from existing stores (in suburban areas). That is not thecase during 1980-2002.

In 1980-2002 average distance increases because store closings inrelatively crowded markets.

4. More clustering of stores of the same �rm (segmentation) in 1960-79than in 1980-2002.

5. More merging activity in 1980-2002.






















































5. More merging activity in 1980-2002.Victor Aguirregabiria () Structural Models of Store Location Valencia. Sept. 6-9, 2007 40 / 50


Stores Location: Year 1930130 �rms and 173 stores











Empirical Questions and approach

Can the model explain these patterns in the data?

I estimate the model for two separate periods. Any parameter canchange between the two periods.

Then, I study how the estimated changes in the structural parameterscan explain the observed di¤erences in number of stores, averagedistance within and between �rms, etc.

Counterfactual experiments using a Taylor approximation (based onthe di¤erentiability of the mapping Ψ with respect to (θ,P). Implicitassumption: no "jumps" in the equilibrium selection mechanism.




















A summary of estimation results

The reduction in the number of stores is explained partly by anincrease in �xed costs but very particularly by a reduction intransportation costs that makes price competition tougher.

Economies of density are statistically and economically signi�cant, butthey are very similar in the two periods. Therefore, they cannnotexplain the di¤erences in market segmentation.

The sunk component of setup cost is signi�cantly smaller in thesecond period (setup costs are less location or �rm speci�c?).

The very large sunk costs in the �rst period can explain part of themarket segmentation (spatial preemption?)




















Conclusions and further questions

Multi-store retailers are a very important part of our economies.

Little is known about their sources of market power and pro�tability.

The location of stores is potentially an important source of marketpower.

The estimation of dynamic games of store location can be a usefultool to understand the sources of market power of multi-store �rms.

Recent methodological contributions have made it possible toestimate and solve these models.

We need more empirical APPLICATIONS, for di¤erent retailindustries, regions, etc.










































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Dynamic Structural Models of Store Location: Methods and ...aguirregabiria.net/wpapers/slides_earie_storelocation...Methods and Applications Victor Aguirregabiria (University of Toronto)