CONNECTED SUBSTITUTES AND INVERTIBILITY OF DEMAND BY Steven Berry, Amit Gandhi ...

26 Pages·2013·0.29 MB·English

CONNECTED SUBSTITUTES
AND INVERTIBILITY OF DEMAND

BY

Steven Berry, Amit Gandhi, and Philip Haile

COWLES FOUNDATION PAPER NO. 1391

COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
YALE UNIVERSITY
Box 208281
New Haven, Connecticut 06520-8281

2013

http://cowles.econ.yale.edu/
Econometrica,Vol.81,No.5(September,2013),2087–2111
CONNECTEDSUBSTITUTESANDINVERTIBILITYOFDEMAND
BYSTEVENBERRY,AMITGANDHI,ANDPHILIPHAILE1
Weconsidertheinvertibility(injectivity)ofanonparametricnonseparabledemand
system.Invertibilityofdemandisimportantinseveralcontexts,includingidentification
ofdemand,estimationofdemand,testingofrevealedpreference,andeconomicthe-
oryexploitingexistenceofaninversedemandfunctionor(inanexchangeeconomy)
uniqueness of Walrasian equilibrium prices. We introduce the notion of “connected
substitutes” and show thatthis structure is sufficientfor invertibility. Theconnected
substitutesconditionsrequireweaksubstitutionbetweenallgoodsandsufficientstrict
substitutiontonecessitatetreatingtheminasingledemandsystem.Theconnectedsub-
stitutesconditionshavetransparenteconomicinterpretation,areeasilychecked,and
aresatisfiedinmanystandardmodels.Theyneedonlyholdundersometransformation
ofdemandandcanaccommodatemanymodelsinwhichgoodsarecomplements.They
allowonetoshowinvertibilitywithoutstrictgrosssubstitutes,functionalformrestric-
tions,smoothnessassumptions,orstrongdomainrestrictions.Whentherestrictionto
weaksubstitutesismaintained,oursufficientconditionsarealso“nearlynecessary”for
evenlocalinvertibility.
KEYWORDS:Univalence,injectivity,globalinverse,weaksubstitutes,complements.
1. INTRODUCTION
WECONSIDERTHEINVERTIBILITY(INJECTIVITY)ofanonparametricnonsep-
arabledemandsystem.Invertibilityofdemandisimportantinseveraltheoret-
icalandappliedcontexts,includingidentificationofdemand,estimationofde-
mandsystems,testingofrevealedpreference,andeconomictheoryexploiting
existenceofaninversedemandfunctionor(inanexchangeeconomy)unique-
ness of Walrasian equilibrium prices. We introduce the notion of “connected
substitutes”andshowthatthisstructureissufficientforinvertibility.
Weconsiderageneralsettinginwhichdemandforgoods1(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)J ischarac-
terizedby
(cid:2) (cid:3)
(1) σ(x)= σ (x)(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)σ (x) :X ⊆RJ →RJ(cid:2)
1 J
where x=(x (cid:2)(cid:3)(cid:3)(cid:3)(cid:2)x ) is a vector of demand shifters. All other arguments of
1 J
the demand system are held fixed. This setup nests many special cases of in-
terest. Points σ(x) might represent vectors of market shares, quantities de-
manded, choice probabilities, or expenditure shares. The demand shifters x
1ThispapercombinesandexpandsonselectedmaterialfirstexploredinGandhi(2008)and
BerryandHaile(2009).WehavebenefittedfromhelpfulcommentsfromJean-MarcRobin,Don
Brown,thereferees,andparticipantsinseminarsatLSE,UCL,Wisconsin,Yale,the2011UCL
workshopon“ConsumerBehaviorandWelfareMeasurement,”andthe2011“Econometricsof
Demand”conferenceatMIT.AdamKaporprovidedcapableresearchassistance.Financialsup-
portfromtheNationalScienceFoundationisgratefullyacknowledged.
©2013TheEconometricSociety DOI:10.3982/ECTA10135
2088 S.BERRY,A.GANDHI,ANDP.HAILE
might be prices, qualities, unobserved characteristics of the goods, or latent
preferenceshocks.SeveralexamplesinSection2illustrate.
The connected substitutes structure involves two conditions. First, goods
must be “weak substitutes” in the sense that, all else equal, an increase in x
j
(e.g.,afallinj’sprice)weaklylowersdemandforallothergoods.Second,we
require “connected strict substitution”—roughly, sufficient strict substitution
between goods to require treating them in one demand system. These con-
ditions have transparent economic interpretation and are easily confirmed in
many standard models. They need only hold under some transformation of
thedemandsystemandcanaccommodatemanysettingswithcomplementary
goods.
The connected substitutes conditions allow us to show invertibility without
thefunctionalformrestrictions,smoothnessassumptions,orstrongdomainre-
strictionsreliedonpreviously.Wealsoprovideapartialnecessityresultbycon-
sidering the special case of differentiable demand. There we show that when
the weak substitutes condition is maintained, connected strict substitution is
necessary for nonsingularity of the Jacobian matrix. Thus, given weak substi-
tutes,connectedsubstitutesissufficientforglobalinvertibilityand“nearlynec-
essary” for even local invertibility. A corollary to our two theorems is a new
globalinversefunctiontheorem.
Important to our approach is explicit treatment of a “good 0” whose “de-
mand”isdefinedbytheidentity
(cid:4)J
(2) σ (x)=1− σ (x)(cid:3)
0 j
j=1
Formally,(2)issimplyadefinitionofthefunctionσ (x).Itsinterpretationwill
0
vary with the application. When demand is expressed in shares (e.g., choice
probabilitiesormarketshares),good 0 mightbea“real”good—forexample,
anumeraire good,an“outside good,”ora goodrelative to which utilities are
normalized.Theidentity(2)willthenfollowfromthefactthatsharessumto1.
Inotherapplications,good 0 willbeapurelyartificialnotion introducedonly
as a technical device (see the examples below). This can be useful even when
anoutsidegoodisalsomodeled(seeAppendixC).
Itisclearfrom(2)that(1)characterizesthefulldemandsystemevenwhen
good 0 is a real good. Nonetheless, explicitly accounting for the demand for
good0inthiscasesimplifiesimpositionoftheconnectedsubstitutesstructure
on all goods. When good 0 is an artificial good, including it in the connected
substitutesconditionsprovesusefulaswell.Aswillbeclearbelow,itstrength-
ens the weak substitutes requirement in a natural way while weakening the
requirementofconnectedstrictsubstitution.
AlsoimportanttoourapproachisapotentialdistinctionbetweenthesetX
andthesubsetofthisdomainonwhichinjectivityofσ isinquestion.Ademand
system generally will not be injective at points mapping to zero demand for
CONNECTEDSUBSTITUTES 2089
somegoodj.Forexample,raisinggoodj’sprice(loweringx )atsuchapoint
j
typically will not change any good’s demand. So when considering conditions
ensuringinjectivity,itisnaturaltorestrictattentiontotheset
(cid:5) (cid:6)
X˜ = x∈X :σ (x)>0∀j>0
j
or even to a strict subset of X˜—for example, considering only values present
in a given data set, or only positive prices even though σ is defined on all of
RJ (e.g., multinomial logit or probit). Allowing such possibilities, we consider
injectivityofσ onanyset
X∗⊆X
(typicallyX∗⊆X˜).Ratherthanstartingfromtherestrictionofσ toX∗,how-
ever, it proves helpful to allow X∗ (cid:7)=X. For example, one can impose useful
regularity conditions on X with little or no loss (we will assume it is a Carte-
sianproduct).Incontrast,assumptionsontheshapeortopologicalproperties
of X∗ implicitly restrict either the function σ or the subset of X˜ on which in-
jectivitycanbedemonstrated.Avoidingsuchrestrictionsisonesignificantway
inwhichwebreakfromthepriorliterature.
There is a large literature on the injectivity of real functions, most of it de-
velopingconditionsensuringthatalocallyinvertiblefunctionisgloballyinvert-
ible.ThisliteraturegoesbackatleasttoHadamard(1906a,1906b).Although
wecannotattemptafullreviewhere,themonographofParthasarathy(1983)
provides an extensive treatment, and references to more recent work can be
found in, for example, Parthasarathy and Ravindran (2003) and Gowda and
Ravindran(2000).Localinvertibilityisitselfanopenquestioninmanyimpor-
tantdemandmodels,whereusefulconditionslikelocalstrictgrosssubstitutes
orlocalstrictdiagonaldominancefailbecauseeachgoodsubstitutesonlywith
“nearby”goodsintheproductspace(severalexamplesbelowillustrate).Even
when local invertibility is given, sufficient conditions for global invertibility in
this literature have proven either inadequate for our purpose (ruling out im-
portant models of demand) or problematic in the sense that transparent eco-
nomicassumptionsdeliveringthese conditions havebeenoverlyrestrictive or
evendifficulttoidentify.
Acentralresultinthisliteratureistheglobal“univalence”theoremofGale
and Nikaido (1965). Gale and Nikaido considered a differentiable real func-
tionσ withnonsingularJacobianonX∗=X.2 Theyshowedthatσ isglobally
injectiveifX∗ isarectangle(aproductofintervals)andtheJacobianisevery-
where a P-matrix (all principal minors are strictly positive). While this result
isoftenreliedontoensureinvertibilityofdemand,itsrequirementsareoften
2GaleandNikaido(1965)madenodistinctionbetweenX andX∗.Whenthesediffer,Gale
andNikaidoimplicitlyfocusedontherestrictionofσtoX∗.
2090 S.BERRY,A.GANDHI,ANDP.HAILE
problematic.Differentiabilityisessential,butfailsinsomeimportantmodels.
Examplesincludethosewithdemanddefinedonadiscretedomain(e.g.,agrid
ofprices),randomutilitymodelswithdiscretedistributions,orfinitemixtures
ofverticalmodels.Givendifferentiability,theP-matrixconditioncanbediffi-
cult to interpret, verify, or derive from widely applicable primitive conditions
(seetheexamplesbelow).Finally,thepremiseofnonsingularJacobianonrect-
angularX∗ isoftenasignificantlimitation.Forexample,ifxrepresentsprices
and σ (x)=0, an increase in j’s price has no effect on any good’s demand,
j
implying a singular Jacobian. However, the set X˜ (which avoids σ (x)=0) is
j
often not a rectangle. In a market with vertically differentiated goods (e.g.,
MussaandRosen(1978)),alowerqualitygoodhasnodemandunlessitsprice
is strictly below that of all higher quality goods. So if −x is the price vector,
X˜ generally will not be a rectangle. Other examples include linear demand
models, models of spatial differentiation (e.g., Salop (1979)), the “pure char-
acteristics” model of Berry and Pakes (2007), and the Lancasterian model in
AppendixC.WhenX˜ isnotarectangle,theGale–Nikaidoresultcandemon-
strateinvertibilityonlyonarectangularstrictsubsetofX˜.
Theliteratureonglobalinvertibilityhasnotbeenfocusedoninvertibilityof
demand, and we are unaware of any result that avoids the broad limitations
of the Gale–Nikaido result when applied to demand systems. This includes
results based on Hadamard’s classic theorem.3 All require some combination
of smoothness conditions, restrictions on the domain of interest, and restric-
tions on the function (or its Jacobian) that are violated by important exam-
plesand/oraredifficulttomotivatewithnaturaleconomicassumptions.4 The
connected substitutes conditions avoid these limitations. They have clear in-
terpretation and are easily checked based on qualitative features of the de-
mand system. They hold in a wide range of models studied in practice and
imply injectivity without any smoothness requirement or restriction on the
setX∗.
The plan of the paper is as follows. Section 2 provides several examples
that motivate our interest, tie our general formulation to more familiar spe-
3See,forexample,Palais(1959)orthereferencesinRadulescuandRadulescu(1980).
4Afewresults,startingwithMas-Colell(1979),useadditionalsmoothnessconditionstoallow
thedomaintobeanyfulldimensioncompactconvexpolyhedron.However,inapplicationsto
demand,thenaturaldomainofinterestX˜ canbeopen,unbounded,and/ornonconvex.Exam-
plesincludestandardmodelsofverticalorhorizontaldifferentiationorthe“purecharacteris-
ticsmodel”ofBerryandPakes(2007).AndwhiletheboundariesofX˜ (iftheyexist)areoften
planeswhenutilitiesarelinearinx,thisisnotageneralfeature.AlesscitedresultinGaleand
Nikaido(1965)allowsarbitraryconvexdomainbutstrengthenstheJacobianconditiontorequire
positive quasidefiniteness. Positive quasidefiniteness (even weak quasidefiniteness, explored in
extensions)isviolatedonX˜ inprominentdemandmodels,includingthatofBerry,Levinsohn,
andPakes(1995)whenxis(minus)thepricevector.
CONNECTEDSUBSTITUTES 2091
cialcases,andprovideconnectionstorelatedwork.Wecompletethesetupin
Section 3 and present the connected substitutes conditions in Section 4. We
giveourmainresultinSection5.Section6presentsasecondtheoremthatun-
derliesourpartialexplorationofnecessity,enablesustoprovidetightlinksto
the classic results of Hadamard and of Gale and Nikaido, and has additional
implicationsofimportancetotheeconometricsofdifferentiatedproductsmar-
kets.
2. EXAMPLES
EstimationofDiscreteChoiceDemandModels
A large empirical literature uses random utility discrete choice models to
study demand for differentiated products, building on pioneering work of
McFadden (1974, 1981), Bresnahan (1981, 1987), and others. Conditional in-
directutilitiesarenormalizedrelativetothatofgood0,oftenanoutsidegood
representingpurchaseofgoodsnotexplicitlyunderstudy.Muchoftherecent
literature follows Berry (1994) in modeling price endogeneity through a vec-
tor of product-specific unobservables x, with each x shifting tastes for good
j
j monotonically. Holding observables fixed, σ(x) gives the vector of choice
probabilities(ormarketshares).Becauseeachσ isanonlinearfunctionofthe
j
entirevectorofunobservablesx,invertibilityisnontrivial.However,itisessen-
tial to standard estimation approaches, including those of Berry, Levinsohn,
and Pakes (1995), Berry and Pakes (2007), and Dube, Fox, and Su (2012).5
Berry(1994)providedsufficientconditionsforinvertibilitythatincludelinear-
ity of utilities, differentiability of σ (x), and strict gross substitutes.6 We re-
j
laxallthreeconditions,openingthepossibilityofdevelopingestimatorsbased
on inverse demand functions for new extensions of standard models, includ-
ing semiparametric or nonparametric models (e.g., Gandhi and Nevo (2011),
Souza-Rodrigues(2011)).
5TheBerry,Levinsohn,andPakes(1995)estimationalgorithmalsoexploitsthefactthat,in
themodelstheyconsidered,σissurjectiveatallparametervalues.Giveninjectivity,thisensures
thatevenatwrong(i.e.,trial)parametervalues,theobservedchoiceprobabilitiescanbeinverted.
Thispropertyisnotnecessaryforallestimationmethodsorforotherpurposesmotivatinginterest
intheinverse.However,Gandhi(2010)providedsufficientconditionsforanonparametricmodel
andalsodiscussedasolutionalgorithm.
6Although Berry (1994) assumed strict gross substitutes, his proof only required that each
insidegoodstrictlysubstitutetotheoutsidegood.HotzandMiller(1993)providedaninvertibility
theoremforasimilarclassofmodels,althoughtheyprovidedacompleteproofonlyforlocal,not
global, invertibility. Berry and Pakes (2007) stated an invertibility result for a discrete choice
model relaxing some assumptions in Berry (1994), while still assuming the linearity of utility
inx.Theirproofisincomplete,althoughaddingthesecondofourtwoconnectedsubstitutes
j
conditionswouldcorrectthisdeficiency.
2092 S.BERRY,A.GANDHI,ANDP.HAILE
NonparametricIdentificationofDemand
Separate from practical estimation issues, there has been growing interest
in the question of whether demand models in the spirit of Berry, Levinsohn,
andPakes(1995)areidentifiedwithoutthestrongfunctionalformanddistri-
butional assumptions typically used in applications. Berry and Haile (2010a,
2010b) have recently provided affirmative answers for nonparametric mod-
els in which x is a vector of unobservables reflecting latent tastes in a market
and/orunobservedcharacteristicsofthegoodsinamarket.Conditioningonall
observables,oneobtainschoiceprobabilitiesoftheform(1).Theinvertibility
resultbelowprovidesanessentiallemmaformanyofBerryandHaile’sidenti-
ficationresults,manyofwhichextendimmediatelytoanydemandsystemsat-
isfyingtheconnectedsubstitutesconditions.RelatedworkincludesChiappori
andKomunjer(2009),whichimposedfunctionalformandsupportconditions
ensuringthatinvertibilityfollowsfromtheresultsofGaleandNikaido(1965)
orHadamard(Palais(1959)).
InvertingforPreferenceShocksinContinuousDemandSystems
Beckert and Blundell (2008) recently considered a model in which utility
fromabundleofconsumptionquantities q=(q (cid:2)(cid:3)(cid:3)(cid:3)(cid:2)q )isgivenbyastrictly
0 J
increasing C2 function u(q(cid:2)x), with x∈RJ denoting latent demand shocks.
The price of good 0 is normalized to 1. Given total expenditure m and prices
p=(p (cid:2)(cid:3)(cid:3)(cid:3)(cid:2)p ) for the remaining goods, qu(cid:7)antities demanded are given by
1 J
q =h (p(cid:2)m(cid:2)x) j =1(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)J, with q =m− p q . Beckert and Blundell
j j 0 j>0 j j
(2008) considered invertibility of this demand system in the latent vector x,
pointing out that this is a necessary step toward identification of demand or
testingofstochasticrevealedpreferencerestrictions(e.g.,BlockandMarschak
(1960), McFadden and Richter (1971, 1990), Falmagne (1978), McFadden
(2004)).Theyprovidedseveralinvertibilityresults.Onerequiresmarginalrates
of substitution between good 0 and goods j>0 to be multiplicatively separa-
ble in x, with an invertible matrix of coefficients. Alternatively, they provided
conditions(onfunctionalformand/orderivativematricesofmarginalratesof
substitution)implyingtheGale–NikaidoJacobianrequirement.Weprovidean
alternativetosuchrestrictions.
One way to translate their model to ours is through expenditure shares. To
dothis,fixpandm,letσ (x)=p h(p(cid:2)m(cid:2)x)/mforj>0.Expenditureshares
j j
sumto1,implyingtheidentity(2).Othertransformationsarealsopossible(see
Example 1 below). And although Beckert and Blundell represented all goods
intheeconomybyj=0(cid:2)1(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)J,acommonalternativeistoconsiderdemand
foramorelimitedsetofgoods—forexample,thoseinaparticularproductcat-
egory.Inthatcase,therewillnolongerbeagoodwhosedemandisdetermined
from the others’ through the budget constraint, and it will be natural to have
a demand shock x for every good j. This situation is also easily accommo-
j
dated. Holding prices and all other demand shifters fixed, let σ (x) now give
j
CONNECTEDSUBSTITUTES 2093
thequantityofgoodj demandedforj=1(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)J.Tocompletethemappingto
ourmodel,let(2)definetheobjectσ (x).Ahintattherolethisartificialgood
0
0playsbelowcanbeseenbyobservingthatarisein σ (x) representsafallin
0
thedemandforgoodsj>0asawhole.
ExistenceofInverseDemand,UniquenessofWalrasianEquilibrium
Let −x be the price of good j. Conditional on all other demand shifters,
j
letσ (x)givethequantitydemandedofgoodj.Aneedforinvertibledemand
j
arises in several contexts. In an exchange economy, invertibility of aggregate
WalrasiandemandisequivalenttouniquenessofWalrasianequilibriumprices.
Our connected substitutes assumptions relax the strict gross substitutes prop-
erty that is a standard sufficient condition for uniqueness. In a partial equi-
librium setting, invertibility of aggregate Marshallian demand is required for
competition in quantities to be well defined. The result of Gale and Nikaido
(1965) has often been employed to show uniqueness. Cheng (1985) provided
economically interpretable sufficient conditions, showing that the Gale and
Nikaido(1965)Jacobianconditionholdsunderthedominantdiagonalcondi-
tionofMcKenzie(1960)andarestrictiontostrictgrosssubstitutes.Inaddition
to the limitations of requiring differentiability and, especially, a rectangular
domain, the requirement of strict gross substitutes (here and in several other
resultscitedabove)rulesoutmanystandardmodelsofdifferentiatedproducts,
where substitution is only “local,” that is, between goods that are adjacent in
the product space (see, e.g., Figure 1 and Appendix A below). Our invertibil-
ityresultavoidstheselimitations.Herewewouldagainusetheidentity(2)to
introduceanartificialgood0asatechnicaldevice.
3. MODEL
Weconsiderademandsystemcharacterizedby(1).Recallthat x∈X ⊆RJ
is a vector of demand shifters and that all other determinants of demand are
held fixed.7 Although we refer to σ as “demand” (and to σ (x) as “demand
j
forgoodj”),σ maybeanytransformationofthedemandsystem,forexample,
σ(x)=g◦f(x),wheref(x)givesquantitiesdemandedandg:f(X)→RJ.In
thiscase, σ isinjective onlyif f is.Ourconnectedsubstitutesassumptionson
σ are postponed to the following section; however, one should think of x as
j
amonotonicshifterofdemandforgoodj.Intheexamplesabove, x iseither
j
(minus) the price of good j, the unobserved quality of good j, or a shock to
tasteforgoodj.Inalloftheseexamples,monotonicityisastandardproperty.
LetJ ={0(cid:2)1(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)J}.Recallthatweseekinjectivityofσ onX∗⊆X,andthat
wehavedefinedtheauxiliaryfunctionσ (x)in(2).
0
7Whengood0isarealgoodrelativetowhichpricesorutilitiesarenormalized,thisincludes
allcharacteristicsofthisgood.Forexample,wedonotruleoutthepossibilitythatgood0hasa
pricex ,butareholdingitfixed(e.g.,at1).
0
2094 S.BERRY,A.GANDHI,ANDP.HAILE
ASSUMPTION1: X isaCartesianproduct.
This assumption can be relaxed (see Berry, Gandhi, and Haile (2011) for
details) but appears to be innocuous in most applications. We contrast this
withGaleandNikaido’sassumptionthatX∗ isarectangle.Thereisasuperfi-
cialsimilarity,sincearectangleisaspecialcaseofaCartesianproduct.Buta
fundamental distinction is that we place no restriction on the set X∗ (see the
discussionintheIntroduction).Further,Assumption1playstheroleofareg-
ularityconditionhere,whereasrectangularityofX∗ isintegraltotheproofof
GaleandNikaido’sresult(seealsoMoréandRheinboldt(1973))andlimitsits
applicability.
4. CONNECTEDSUBSTITUTES
Ourmainrequirementforinvertibilityisapairofconditionscharacterizing
connected substitutes. The first is that the goods are weak substitutes in x in
thesensethatwhenx increases(e.g.,j’spricefalls),demandforgoodsk(cid:7)=j
j
doesnotincrease.
ASSUMPTION 2—Weak Substitutes: σj(x) is weakly decreasing in xk for all
j∈{0(cid:2)1(cid:2)(cid:3)(cid:3)(cid:3)(cid:2)J},k∈/{0(cid:2)j}.
We make three comments on this restriction. First, in a discrete choice
model, it is implied by the standard assumptions that x is excluded from the
j
conditional indirect utilities of goods k(cid:7)=j and that the conditional indirect
utilityofgoodj isincreasinginx .
j
Second, although Assumption 2 appears to rule out complements, it does
not. In the case of indivisible goods, demand for complements can be rep-
resented as arising from a discrete choice model in which every bundle is a
distinctchoice(e.g.,Gentzkow(2004)).Asalreadynoted,theweaksubstitutes
condition is mild in a discrete choice demand system. In the case of divisible
goods, the fact that σ may be any transformation of the demand system en-
ables Assumption 2 to admit some models of complements, including some
witharbitrarilystrongcomplementarity.Example1belowillustrates.
Finally, consider the relation of this assumption to a requirement of strict
gross substitutes. If good zero is a real good, then our weak substitutes con-
dition is weaker, corresponding to the usual notion of weak gross substitutes.
When good 0 is an artificial good, Assumption 2 strengthens the weak gross
substitutesconditioninanaturalway:takingthecasewherexis(minus)price,
all else equal, a fall in the price of some good j >0 cannot cause the total
demand(overallgoods)tofall.
To state the second condition characterizing connected substitutes, we first
defineadirectionalnotionof(strict)substitution.
CONNECTEDSUBSTITUTES 2095
DEFINITION1: Goodjsubstitutestogoodkatxifσk(x)isstrictlydecreasing
inx .
j
Consideradeclineinx withallelseheldfixed.ByAssumption2,thisweakly
j
raises σ (x) for all k(cid:7)=j. The goods to which j substitutes are those whose
k
demands σ (x) strictly rise. When x is (minus) the price of good j, this is
k j
a standard notion. Definition 1 merely extends this notion to other demand
shifters that may play the role of x. Although this is a directional notion, in
most examples it is symmetric; that is, j substitutes to k if and only if (iff)
k substitutes to j.8 An exception is substitution to good 0: since any demand
shiftersforgood0areheldfixed,Definition1doesnotdefinesubstitutionfrom
good0toothergoods.9
Itwillbeusefultorepresentsubstitutionamongthegoodswiththedirected
graphofamatrixΣ(x)whoseelementsare
(cid:8)
1{goodj substitutestogoodkatx}(cid:2) j>0(cid:2)
Σ =
j+1(cid:2)k+1 0(cid:2) j=0(cid:3)
ThedirectedgraphofΣ(x)hasnodes(vertices)representingeachgoodanda
directed edge from node k to node (cid:6) whenever good k substitutes to good (cid:6)
atx.
ASSUMPTION 3—Connected Strict Substitution: For all x∈X∗, the directed
graphofΣ(x)has,fromeverynodek(cid:7)=0,adirectedpathtonode0.
Figure 1 illustrates the directed graphs of Σ(x) at generic x∈X˜ for some
standard models of differentiated products, letting −x be the price vector
and assuming (as usual) that each conditional indirect utility is strictly de-
creasing in price. The connected substitutes conditions hold for X∗ ⊆X˜ in
all of these models. As panel (e) illustrates, they hold even when J is com-
posedofindependentgoodsandeitheranoutsidegoodoranartificialgood0.
Eachoftheseexampleshasanextensiontomodelsofdiscrete/continuousde-
mand(e.g.,NovshekandSonnenschein(1979),Hanemann(1984),Dubinand
McFaden (1984)), models of multiple discrete choice (e.g., Hendel (1999),
Dube(2004)),andmodelsofdifferentiatedproductsdemand(e.g.,Deneckere
and Rothschild (1992), Perloff and Salop (1985)) that provide a foundation
forrepresentativeconsumermodelsofmonopolisticcompetition(e.g.,Spence
8See,forexample,Berry,Gandhi,andHaile(2011).Weemphasizethatthisreferstosymmetry
ofthebinarynotionofsubstitutiondefinedabove,nottosymmetryofanymagnitudes.
9Ifgood0isarealgooddesignatedtonormalizeutilitiesorprices,onecanimagineexpanding
xtoincludex anddefiningsubstitutionfromgood0toothergoodspriortothenormalization
0
thatfixesx .IfAssumption3,below,holdsundertheoriginaldesignationofgood0,itwillhold
0
foralldesignationsofgood0aslongassubstitution(usingtheexpandedvectorx=(x (cid:2)(cid:3)(cid:3)(cid:3)x ))
0 J
issymmetric(inthesensediscussedabove).