rosen hedonics jpe 1974

8/13/2019 Rosen Hedonics JPE 1974

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Hedonic Prices and ImplicitMarkets:ProductDifferentiationn PureCompetition

SherwinRosenUniversityfRochesternd HarvardUniversity

A classofdifferentiatedroducts s completely escribed ya vector fobjectivelymeasured haracteristics. bservedproductprices and thespecific mounts fcharacteristicsssociatedwith ach good define setof mplicit r hedonic prices.A theory fhedonicprices sformulatedas a problemntheeconomics f patialequilibriumnwhich he entireset of implicitpricesguides both consumer nd producer ocationaldecisionsncharacteristicspace. Buyer nd seller hoices, s wellas themeaning and nature ofmarket quilibrium, re analyzed. Empiricalimplications or hedonic price regressions nd index number con-structionre pointed ut.

I. Introduction and SummaryThis paper sketches a model of product differentiationbased on thehedonic hypothesisthat goods are valued for theirutility-bearing ttrib-utes or characteristics.Hedonic prices are defined as the implicit pricesofattributes nd are revealed to economic agents fromobserved prices ofdifferentiated roducts and the specific amounts of characteristics asso-ciated with them. They constitute the empirical magnitudes explainedby the model. Econometrically, implicit prices are estimated by the first-step regression nalysis (product price regressed on characteristics) n theconstruction of hedonic price indexes. With few exceptions, structural

Thesubstancef his aper rose romonversationsithH. Gregg ewis everal earsago.Amultitudef ther eoplehave ontributeddvice nd criticism.mong hemreWilliam rock, tanley ngerman,obert .Gordon,viGriliches,obert . Lucas,Jr.,MichaelMussa, nd thereferee.emainingrrorsremy wnresponsibility.inancialsupportrom heCenter orNavalAnalysisndtheNational nstitutefEducationsgratefullycknowledged.34


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HEDONIC PRICES 35interpretationsf the hedonicmethod re not available.' Therefore, urprimary oal is to exhibit generatingmechanism orthe observationsin thecompetitive ase and to use that structure o clarify he meaningand interpretationfestimatedmplicit rices. t willbe shown hat hesedata generallyontain ess nformationhan scommonly upposed.How-ever, he modelsuggests method hatoften an identifyheunderlyingstructuralarametersf nterest. lso,as a generalmethodological oint,it is demonstratedhatconceptualizingheproblem fproductdifferen-tiationnterms f fewunderlyingharacteristicsnstead f largenum-berofcloselyrelatedgenericgoodsleads to an analysishavingmuch ncommon withthe economicsof spatial equilibrium nd the theory fequalizingdifferences.The model tself mountsto a description fcompetitive quilibriumin a plane of severaldimensions n whichbothbuyers nd sellers ocate.The classof goods underconsiderations described ynobjectivelymea-sured haracteristics.hus,any ocation n theplane, srepresented yavector of coordinatesz = (z,, Z2,. . . ) Zn), with zi measuring theamountofthe th characteristicontained n each good. Productsn theclass are completely escribed ynumerical aluesofz and offer uyersdistinct ackagesof characteristics.urthermore,xistence f productdifferentiationmpliesthat a wide varietyof alternative ackages areavailable. Hence, transactionsn products re equivalentto tied saleswhenthought f as bundlesofcharacteristics,uggestingpplicability ftheprinciple fequal advantagefor nalyzingmarket quilibrium.In particular, price (z) = P(Z1, Z2, . . . Zn) is defined t each pointon theplane and guidesboth consumer nd producer ocationalchoicesregardingpackages of characteristics ought and sold. Competitionprevailsbecause singleagentsadd zero weight o the market nd treatprices (z) as parametric o theirdecisions. n factthe function (z) isidenticalwith the set of hedonicprices- equalizing differences -asdefined bove, and is determined y some market learingconditions:Amounts f commodities ffered y sellers t everypointon the planemustequal amountsdemandedby consumers hoosing o locate there.Bothconsumers nd producers ase their ocational and quantitydeci-sionson maximizing ehavior, nd equilibrium rices re determined othatbuyersnd sellers re perfectly atched.No individual an improvehisposition,nd all optimum hoices refeasible.Asusual,market lear-ingprices, (z), fundamentallyre determined y the distributionsfconsumer astes nd producer osts.We showhow t spossible orecover,

1 Excellent summariesof the hedonic technique are available in Griliches (1971,chap. 1) and Gordon (1973). Major exceptions o the statement n the textare thosestudiesdealingwithdepreciation nd obsolescence see Griliches1971, chaps. 7 and 8)and some recentmodelsbased on markuppricing e.g., Ohta and Griliches1972).


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36 JOURNAL OF POLITICAL ECONOMYoridentify,ome of the parameters f theseunderlying istributionsy asuitable ransformationf the observations.Anearly ontributiono theproblem fquality ariation nd thetheoryofconsumer ehaviorhas been made by Houthakker1952). His analysisis designedto take account of the fact thatconsumers urchase trulynegligible ractionsf all goods available to themwithout aving o dealwith myriad f corner olutions equiredbyconventional heory. hatvirtue fHouthakker's reatments preservedn thepresentmodel. Morerecently ecker 1965), Lancaster (1966), and Muth (1966) have ex-tended Houthakker'smethods o moreexplicit onsideration f utility-bearing haracteristics.gain,theemphasis s on consumer ehavior ndproperties f market quilibriumhave not been workedout, a gap wehope to fill,n part,here. The spirit f theserecent ontributionss thatconsumers re also producers.Goods do not possessfinalconsumptionattributesut rather re purchased s inputs ntoself-productionunc-tionsforultimate haracteristics.onsumers ct as their wn middle-men, so to speak. In contrast,hemodelpresented elowinterposesmarketetween uyers nd sellers. roducershemselvesailor heir oodsto embodyfinal haracteristicsesiredby customers nd receive eturnsfor ervingconomic unctionss intermediaries.hese returnsrisefromeconomiesof specialized productionachieved by specializationanddivisionof labor throughmarkettransactions ot available outsideorganizedmarketswith elf-production.Section I discussesndividual hoices nthe market nd thenature fmarketequilibrium.Some simple examples of analyticsolutionsforgeneral equilibrium re givenin Section III. SectionIV presents nempiricalmethodfor identifyinghe underlying tructure rom theobservations, hile SectionV appliesthemodel to price ndex numberconstructionn thepresence f egislated estrictions.o highlightssen-tial features,hesimplest ossible pecificationsre chosenthroughout.As a furtherppeal tointuition,se ismade ofgeometricalonstructionswhereverpossible.II. Market EquilibriumConsidermarkets ora class of commodities hat are describedby nattributesr characteristics, = (z, Z2,. . . , Zn). The components fzare objectivelymeasured n the sensethat all consumers' erceptionsrreadingsof the amountof characteristicsmbodied n each good areidentical, hough fcourse onsumersmaydifferntheir ubjective alua-tionsof alternative ackages.The terms product, model, brand,and design are used interchangeablyto designate commodities ofgivenquality or specification. t is assumed that a sufficientlyarge number ofdifferentiated roducts are available so that choice among various com-


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HEDONIC PRICES 37binations of z is continuous for all practical purposes. That is, there is aspectrum of products among which choices can be made. As will beapparent, this assumption represents an enormous simplification of theproblem. It is obviouslybetterapproximated in some marketsthan others,and there is no need to belabor its realism.2 To avoid complications ofcapital theory,possibilitiesforresale of used items n secondhand marketsare ignored, either by assuming that secondhand markets do not exist, oralternatively, hat goods representpure consumption.Each product has a quoted market price and is also associated with afixed value of the vector z, so that products marketsimplicitly reveal afunctionp(z) = p(z1,. . ., zn) relating prices and characteristics.Thisfunction s the buyer's (and seller's) equivalent ofa hedonic price regres-sion, obtained fromshopping around and comparing prices of brandswithdifferentharacteristics. t gives the minimumprice ofany packageofcharacteristics. f two brands offer he same bundle, but sell fordifferentprices, consumers only consider the less expensive one, and the identityof sellers is irrelevantto theirpurchase decisions. Adopt the conventionofmeasuring each zi so that theyall may be treated as goods (i.e., sothat consumers place positive rather than negative marginal valuationson them) in the neighborhood of their minimum technically feasibleamounts. Then firms an alter their products and increase z only by useof additional resources, and p(z1,. . ., z.) must be increasing in all itsarguments.Assumep(z) possesses continuous second derivatives. Since amajor goal of theanalysisis to presenta pictureofhowp(z) is determined,it is inappropriate to place too many restrictions n it at the outset. How-ever,note that there s no reason for t to be linear as is typicallythe case.The reason is that the differentiated roductsare sold in separate, thoughof course highlyinterrelated,markets. This point is spelled out in somedetail below.A buyer can forcep(z) to be linear ifcertain typesofarbitrage activi-ties are allowed. Let Za, Zb, and zc be particular values of the vector z.(i) Suppose Za = (1 It)Zb, and P(Za) < ( 1t)P(Zb), where t is a scalar andt > 1. Then t units of a model offering a yield the same amount ofcharacteristics s a model offeringb, but at less cost, rulingout transac-tionsin convex portionsofp(z). (ii) Suppose Za < Zb < zc and P(Zb) >6P(Za) + (1 - 6)p(zc), where 0 < 5 < 1 and Zb is defined by Zb =bza + (1 - 6)zc. Then characteristics n amount ofZb could be achievedby purchasing 5 units of a model containing Za and (1 - 5) units of amodel containing zc at lower cost than by direct purchase of a brandcontaining Zb, and products in concave portions ofp(z) would be un-economical. Arbitrage s assumed impossiblein what follows at thispoint

2 This assumptionwas first mployed by L. M. Court (1941) and allows the use ofmarginal nalysisrather han the programmingmethods equiredby Lancaster's 1966)formulation. ollowingthegeneral rule, t is notwithout tscosts,however see below).


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38 JOURNAL OF POLITICAL ECONOMYwe depart fromLancaster [1966]) on the assumption f indivisibility.This amounts to an assumption hat packages cannot be untied.Forexample,n terms fone characteristic,wo 6-foot ars are not equivalentto one 12 feet n length, ince they annotbe driven imultaneouslycase[i]); whilea 12-foot ar forhalf year and a 6-foot ar for he otherhalfisnot the ame as 9 feet ll yearround case [ii]). Similarly, ssume ellerscannotrepackage existingproducts n this manner or do not find teconomical o do so, as mightnot be the case withperfect entalmarketsand zero transactionsnd reassemblyosts.A. The ConsumptionecisionTo begin, uppose consumers urchaseonly one unit of a brandwithaparticular alue of z. Write theutility unction s U(x, zj, Z2, ... ) Zn)assumed trictlyoncave, n addition othe otherusual properties, herex is all othergoodsconsumed. t would not be difficulto treat as inter-mediategoodsand relate hem oyetmoreultimate ommodities hroughself-productionunctions,ut thatcomplications ignored. et thepriceofx equal to unity nd measure ncome, , in terms funitsof x: y =x + p(z). Maximizationofutility ubjectto thenonlinearbudgetcon-straint equires hoosing and (z1, ., Zn) tosatisfyhebudget nd thefirst-orderonditionsapl'z1= Pi= UzIUx, i = 1 .. ., n. Optimalityis achievedby purchasing brandofferinghedesired combination fcharacteristics.econd-order onditionsre fulfilledn theusual assump-tions egarding , so longasp (z) isnot ufficientlyoncave for generalstatementfthese onditions nder nonlinear onstraintee Intriligator[1971]).

To stresshe essential patialcontext ftheproblem, efine value orbid functionO(zl,. .., zn; u,y) according toU(Y15 - Zn) U. (1)

The expenditure a consumer is willing to pay for alternative values of(Z, . ... 5 Zn) at a given utility ndex and income is represented yO(z; u,y). It defines a familyof indifference urfacesrelating the zi withmoney (i.e., with x foregone), and has been widely used in urbaneconomics (e.g., see Alonso 1964). Differentiate 1) to obtain

ozi- Uzi/Ux > 05 O= -l/Ux < 0, and Oy= 1, (2)_ (U2U - 2UxUUxz, + U2Uxx) /U < 0, (3)

where the inequality in (3) follows from the assumptions about the bor-dered Hessian matrix of U. Also, strictconcavity of U implies that 0 isconcave in z. Equations (2) and (3) show that the value function isincreasing in zi at a decreasing rate. Alternatively,Oz, is the marginalrate of substitution between zi and money, or the implicit marginal


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HEDONIC PRICES 39pie ~~~~~~~~~~~~P(Z, Z* XZnP,~~~~~~~~ 2Z1-21 .) u

e (z *)

- ---- -- ~ ~ lZ2,-tl '2FIG..,z; Uu

I ~~~~~~~~~~~~~~~~~ZIFIG. 1

valuation the consumer places on zi at a given utility ndex and income.It indicates his reservation demand price for an additional unit of zi,which is decreasing in zi.The amount the consumer is willing to pay for at a fixedutility ndexand income is 0(z; u,y), while p(z) is the minimum price he must payin themarket.Therefore, utility smaximized when 0(z *; u*, y) = p(z *)and 0z.(z*; u*,y) = pi(z*), i = 1, . . ., n, wherez* and u* are optimumquantities. In otherwords, optimum location on the z-plane occurswherethe two surfacesp(z) and 0(z; u*,y) are tangent to each other. Onedimension of consumer equilibrium is illustrated in figure 1, where thesurfaceshave been projected onto the 0 - z1 plane cut at (z*, . . . Z *).A family of indifferencecurves, of which only one member (at u*) isshown, is defined by 0(z1, * .... *; u,y). Two differentbuyers areshown in the figure, one with value function O' and the other with 02.The latter purchases a brand offeringmore z 3In general, far less can be said than in the standard analysis aboutcomparative statics,because the budget constraint s nonlinear. Differen-tiate O0,with respectto u,Oziu= ( U., z, - Uz, U, U 2 the numeratorofwhich is recognized as determiningthe sign of the income elasticityofdemand for good zi in standard theorywhen the other componentsof z are held constant. If all these derivatives are positive (zi is nor-mal in this restricted ense for all i), the gradient of 0 unambiguously

3 Lewis (1969) employsa similar construction n analyzing the problem of hours ofwork s a tiedsale. Jobs offer fixedwage-hourpackage, which varies from ob to ob.The rimarketstablishes function elatingwages and hours on which both workers ndemployers ase theirdecisions.


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40 JOURNAL OF POLITICAL ECONOMYincreases s u increases.Additional ncome always increasesmaximumattainableutility.Hence ifp(z) is convex nd sufficientlyegular very-where,we might xpect higher ncome consumers o purchasegreateramountsofall characteristics. nly in that case would it be truethatlarger ncome eadsto an unambiguousncrease n theoverall qualityconsumed, nd differentiatedroducts'marketswould tendto be strati-fiedby income.However, n generalthere s no compelling easonwhyoverallqualityshouldalwaysincreasewith ncome.Some componentsmay increase nd others ecrease cf.Lipsey and Rosenbluth1971). Bethat s itmay, clearconsequence fthe model s thatthere renaturaltendencies owardmarketegmentation,n thesense hatconsumers ithsimilar aluefunctionsurchase roducts ith imilarpecifications.his sa well-knownesult f patial quilibriummodels. n fact, he above speci-ficationsvery imilar n spirit oTiebout's 1956) analysis fthe mplicitmarket orneighborhoods,ocal public goods beingthe characteristicsin thiscase. He obtainedthe result hatneighborhoods end to be seg-mentedby distinctncome, nd tastegroups also, see Ellickson1971).That result oldstruefor therdifferentiatedroducts oo.Allowinga parameterizationf tastes across consumers, he utilityfunctionmay be writtenU(x1,z1, .. Zn; OK),where o is a parameterthat differs romperson to person. Equilibrium value functionsdependon bothy and cx.A joint distributionfunctionF(y, or)is given in thepopulation at large, and equilibrium ofall consumers s characterized bya familyofvalue functionswhose envelope is the market hedonic or im-plicit price function.The model is easily expanded to include several quantities, so long asconsumers are restricted to purchasing only one model. FollowingHouthakker (1952), the utility function becomes U(x1,z1 ... Zn), M))where m s the numberofunitsconsumed of a model with characteristics .The constraint sy = x + mp(z), and necessaryconditions become

u= (z) Ux + Um = 0, (4)amau = -mpi(z)UX + Uzi = 0. (5)azi

The value function s still defined as the amount a consumer is willing topay for at a fixedutility ndex but now with the proviso that m s optim-ally chosen.That is,O(z1,. .., Zn) is defined y eliminatingm fromu = U(y -mMO, * ** Z.5 m)

Um/Ux = 0.

Again, OzQs proportionalto Uzl/Ux.The logic underlyingfigure1 remainsintact, and it can just as well serve forthis case. However, second-order


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HEDONIC PRICES 4Iconditionsare now more complex. For example, convexity ofp(z) is nolonger sufficient ora maximum as it was in the case where m was re-stricted to be unity. Also, it is necessaryto employ stronger ssumptionsthan thoseused above ifthe value function0 is to be concave.Note there s no question of monopsony involved here. Consumers actcompetitivelyn spiteof the factthat marginal costof quality,pj(z), is notnecessarilyconstant-it is increasing in figure 1-because as many unitsas desired of any brand can be purchased without affectingprices. Thefunctionp(z) is the same for all buyers and independent ofm.B. TheProductionecisionHaving set up the formal apparatus above, we give a symmetricalandconsequently brief account of producers' locational decisions. Whatpackage of characteristicsis to be assembled? Let M(z) denote thenumber of units produced by a firmof designs offering pecificationz.The discussion s limitedto the case of nonjoint production, n which eachproduction establishmentwithin the firm pecializes in one design, andthere are no cost spillovers from plant to plant. Thus a firm is anarbitrary collection of atomistic production establishments, each oneacting independentlyof the others.Analytical difficultiesrisingfrom ruejoint production are noted in passing.Total costsin an establishment re C(M, z; fi), erived fromminimiz-ing factor costs subject to a joint production functionconstraintrelatingM, z, and factors fproduction.The shift arameterfi eflects nderlyingvariables in the cost minimization problem, namely, factor prices andproduction functionparameters. Assume C is convex with C(O, z) = 0and CM and Cz1 > 0. There are no production indivisibilities,andmarginal costs of producing more units of a model of given design arepositive and increasing. Similarly, marginal costs of increasing eachcomponentof thedesignare also positiveand nondecreasing. (Ordinarily,there will be some technological constraintsthat limit the set of feasiblelocations on the plane.) Each plant maximizes profitXr= Mp(z) -C(M, z1, . . ., z.) by choosing M and z optimally,where unit revenue ondesign z is given by the implicit price functionfor characteristics, (z).f

4 Our inability o treat oint productionnontrivially etsimply temsfrom hespec-trum-of-commoditiesssumption. f a finitenumber say v) ofpackages is available, itwould be straightforwardormally o specify standard v-componentmultipleproductcostfunction or hefirm,nd proceedon that basis. In thepresent ase, firmsngage njoint production nly nsofars they wnestablishmentspecializingndifferentackages.However,genuineoint production equires ost dependencies etweenproduction nitswithin hefirm: he firmmust choose a functionM(z) describing n entire productline offeredn the market.The entirefunctionM(z) is an argument n each plant'scosts nd total costs n turn re thesum (or integral)over all production stablishmentcosts.A completetreatment equiresuse of functional nalysis nd is beyondthescopeof thispaper.


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42 JOURNAL OF POLITICAL ECONOMYAgain, firmsre competitorsnd not monopolists venthoughmarginalcosts fattributesi(z) are notnecessarilyonstant ecauseall establish-ments bserve he same prices nd cannot ffect hemby their ndividualproduction ecisions: (z) is independent fM.Optimalchoice ofM and z requires

Pi(Z) = CZi(M,Z1,.. . z)/M, i = 1 .. ., n (6)p(z) = CM(M, Z15 ... , Zn) (7)

At the optimumdesign, marginal revenuefrom additional attributesequals theirmarginalcost of productionper unit sold. Furthermore,quantities re producedup to thepointwhereunit revenue (z) equalsmarginalproduction ost,evaluatedat theoptimumbundleof charac-teristics. s above, convexityf C does not assure econd-orderonditionsdue tononlinearityfp z), and somestrongeronditions,ssumedto besatisfiedn whatfollows,re required see Intriligator 971).Symmetricallyith the treatment fdemand,define n offerunctionO(zl, . . ., Zn; i /3) ndicating unit prices (per model) the firm s willingtoaccepton variousdesigns t constant rofit hen quantities roducedof ach model re optimally hosen.A familyfproductionindifferencesurfacess defined y4. Then O(z1,. . ., Zn; 7),/3)sfound y eliminatingM from n = MO,0 C(Ml) Z1, -Z* on) /(8)and

CM(M, Z1,..., Zn) ,(9)and solvingfor bin terms fz, 7a, nd /3.Differentiate8) and (9) toobtainO, = CZ_/M 0 and /,, 1/M > 0.The marginal eservationupplypricefor ttributeat constant rofit,assumed ncreasing n Zi, is b Again convexity f C does not alwaysguarantee /ZZj> 0. Since b is the offer rice the seller s willingtoaccept on designz at profitevel 7a,whilep(z) is the maximumpriceobtainableforthose models n the market, rofits maximizedby anequivalentmaximizationftheofferrice ubject o theconstraint = b.Thus maximum profit and optimum design satisfypi(z*) =(b 4zl, . ., z*; j*, /3),for i = 1, . ., n, and p(z*) =(z7r*, /3).Producerequilibrium s characterizedby tangencybetweenaprofit-characteristicsndifferenceurface nd the market haracteristics-implicit rice urface.One dimension f the olution sdepictednfigure ,where

CZ , Zk I.. I Z \


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HEDONIC PRICES 43A,+ (f)2(ZI 2,........Zn; .772

/ (Z iZ*

..01Z )

it

FIG. 2defines family f curves on thez1 - q plane cut throughtheindifferencesurface at the optimum values of the other attributes.Only one memberis shown in the figure.The curve labeled /A efers o a production unitpossessingproductionand cost conditionsmakingit well suited to producelesseramounts ofz1, while the one labeled q2 refers o a firmwith a com-parative advantage at producing higher values of z1. That is, the twoplants have distinctvalues of the parameter /3.More generally, there s adistributionof/3cross all potential sellers. Let G(fl) representthat dis-tribution. Then producer equilibrium is characterized by a family ofoffer unctions hat envelop the market hedonic price functions.What is the empirical contentoffl? It is anythingthat shifts ost con-ditionsamong firms.Thus, differencesn factorprices are one possibility.For example, many products are produced in several countries and aretraded on national markets (forexamples, see Griliches [1971], chap. 5).There is no reason to assume equalization offactorprices in these cases.More generally, anything allowing identification of conventionalmulti-product production functions in cross-sectiondata serves to provokedifferencesn /1.Factor price differences cross states or regionswithinacountryoften ervethispurpose and do so here as well. Second, differencesin technology, as reflected by typically unmeasured, firm-specificfactors fproduction,also act as supply shifters cross firms.For example,agricultural production function research often treats education of thefarmoperator in thismanner. Firm-specificR&D expenditure as well asthe phenomena of progress-function-learninglso serve these purposes.


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44 JOURNAL OF POLITICAL ECONOMYC. What o Hedonic ricesMean?An answer to the question s an immediate pplicationof the aboveanalysis. uperimpose igure onto figure . In equilibrium, buyer ndseller re perfectly atchedwhen heir espective alue andoffer unctionskiss each other,withthe commongradient t that pointgiven by thegradient f the market learing mplicitpricefunction(z). Therefore,observations(z) represent joint envelope fa familyfvalue functionsand anotherfamily f offer unctions. n envelope function y itselfrevealsnothingbout theunderlying embershatgeneratet; and theyin turn onstitutehegeneratingtructuref heobservations.ome quali-fications re necessary owever. a) Suppose there s no variance n fiand all firms re identical.Then thefamily foffer unctionsegeneratesto a single urface,ndp(z) mustbe everywheredenticalwith uniqueoffer unction. rice differences etweenvarious packages are exactlyequalizing among sellers because offerfunctions re constructed tconstantprofit.A variety f packages appear on productsmarkets osatisfy ifferencesn preferences mong consumers,nd the situationpersistsecause no firm indstadvantageous o alterthequalitycontentof ts products. b) Supposesellers iffer, ut buyers re identical.Thenthefamilyfvalue functionsollapses oa single unctionnd is identicalwith the hedonicprice function. bserved price differencesre exactlyequalizingacrossbuyers,ndp(z) identifieshestructurefdemand.III. Existence of Market EquilibriumAnalysisof consumerand producerdecisionshas proceeded on theassumptionfmarketquilibrium. hissection emonstratesomedetailsofequilibrium riceand quantitydetermination. arketquantityde-mandedforproductswithcharacteristicsis Qd(z), and Qs(z) ismarketquantity uppliedwiththose ttributes.t is necessaryofind functionp(z) such thatQd(z) = Qs(z)-for ll z, whenbuyers nd sellers ct in themanner escribed bove. The fundamentalifficultyosed bythisprob-lem is that Qd(z) and Qs(z) depend on the entirefunction (z). Forexample, suppose quantitiesdemanded and supplied at a particularlocationdo notmatch t prevailing rices.The effect f change npriceat thatpointis not confined o models with thoseparticular harac-teristicsut induces ubstitutionsnd locationalchangeseverywherentheplane. A verygeneraltreatmentf theproblem s found n Court(1941), and our discussion s devoted osomeexamples.These exampleshave been chosen for their implicity ut illuminate he problemandillustratemost of the basic issues. n contrast o the restof thepaper,discussions specialized o the case wheregoodsare described y exactlyone attribute i.e., n = 1). Therefore , representsn unambiguous


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HEDONIC PRICES 45measure of quality. When n = 1, the location surface degenerates to aline rather than a plane, and products are unequivocally ranked by theirz content.A. Short-RunquilibriumConsider a short-runequilibrium in which firmshave geared up for thequality (z1) of goods they can produce and are only capable of varyingquantities. The horizon is sufficientlyhort so that new entry is pre-cluded, and the distributionof firmsby quality is given as an initialcondition. The marketreveals an implicit price functionp(z1), and eachfirmdetermines the quantity it supplies to the market according to con-dition (7). Market supply in a small interval dz1 near quality z1 is foundby weightingfirm upply by the quality distributionfunction. Consumersdiffern tastes and income, but all determineoptimal quality and quantityas in (4) and (5). Market demand near any quality z1 is foundby usingthe conditions of consumer equilibrium to transform he distributionoftastes and income into a distribution fqualities demanded and weightingindividual quantities demanded by the resultingdistributionof qualities.Finally, settingdemand equal to supply yields a differential quation inp and z1 that must be satisfiedby market equilibrium, subject to someboundary conditions.To be specific,assume that C(N, z) = (a/2)M2z2 for all firms.Also,suppose firms are uniformly distributed by the characteristic z1:g(z1)dzl = kdz1 forz1, < z1 < z,1, where k is a constant and z1I andz s are exogenouslydeterminedupper and lower limitsoftheproduct line.Apply equation (7) to obtain firm upply: M(z1) = p/azf,since qualitiescannot be varied by assumption. Therefore,

Q5(zl)dzl = g(z1)M(z1)dz, = [(kla)p(z1)/Z2]dz1. (10)Assume a fixednumber of consumers in the population and that onlyone unitper customer oftheoptimal model is purchased. Consumers havethe same income, and utility s linear in x and z1, with the marginal rateof substitution,p, varyingfromperson to person. Maximize U(x, z,) =

x + pz1 subject to y = x + p(z1). Each consumer purchases a brandfor which dp/dz1 p'(z1) = p. In this case the value functions offigure1 are straight ines with a different lope, p, for each person. Themarginal condition characterizes consumer choice so long as p > 0,which will be shown to be true. Suppose p is distributed uniformly,f (p)dp = bdpfor , < p < Pmwhereb is a constant nd p, and p, are,respectively, he largestand smallestmarginal rates of substitution n thepopulation. Use the marginal conditionp' = p to transform (p)dp into


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46 JOURNAL OF POLITICAL ECONOMYp z)

0 Z/Zio A zle

FIG. 3

a distributionfz1. ThenQd(z)dz =f(z1) |- dzj = bp (zl)dzl. (11)dzj

Pricemust lear the market t every uality.Equating (10) and (11),p(z1) must atisfyhe differentialquation(k/ba)p/z= d2pldz'. (12)

Equation (12) is a specialcase of what s called Euler's equation andhas a known olution f the formP = c1z4 + C2ZS, (13)

where 1 and c2 re constants etermined ytheboundary onditionsndr and s are defined y r2 _ r - (a/bk)= 0: r = (1 + /I + 4a/bk)12and s = (1 - JI + 4a/bk)/2.he parameters and s are realnumbersandr > 0 and s < 0. Furthermore,'(z1) wouldnotbepositive hrough-out its rangeunlessc1 > 0 and c2 < 0, and consumers-could ot beinterior t thosepoints.Equation (13) is graphed n figure on thatassumption.Note thatp in (13) exhibits n inflection ointat z10 =(-c1/c2) /rs), and it so happens that &(z10) --0. Therefore > 0for 1 > z10.Boundaryonditions.-Competitionequires herebe no massesof con-sumers t any quality,for here re few ellers ocatedat any pointand


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HEDONIC PRICES 47theywould otherwise dd nonzeroweight othemarket.As seen n Sec-tion I, consumers ithhighvaluesof p buy higher-quality odels, ndit mustbe truethatthoseforwhomp = pi purchase he highest ualityavailable.Otherwise rices f quality 1 wouldfall, greatmass of con-sumerswould switch vertothem, riving he priceback up and causingthose buyersto relocateagain. Therefore, ne boundarycondition sp'(z11) p,, or p = rc z'-' + sc zS- (14)The other oundary ondition sfound yexamining he ower ndoftheline. The followinghree asescover ll relevant ossibilities:

1. zs= 0 and ps > 0. Firms hoose not to sellat negative rices seefig. 3) and all plantsgeared to produce qualities ess than z10 (to bedetermined) hut down. On theotherhand, all consumers alue z1 atleastas muchas its minimumupplyprice i.e., zero) and itmustbe truethat hey ll buysomevalue of 1. Individuals orwhomp = p, consumethe owestqualitiesappearingon themarket, or fthey hosequalitiesgreater han z1 0, pricesof models n the neighborhood f z10 would falltozero, nducingowp customersorelocate here nddrivingheir ricesbackup. Thus a secondboundary onditionsp'(z1o) = ps, or

ps =rc zr- l +

SC2Zs-i(15)

The parameters10,c1, ndc2aredetermined yequations 14) and (15)plus the definitionfz10. It can be shown that cl > 0 and c2 < 0, asrequired by the second-order onditionsof consumerequilibrium.Therefore,heequilibriumhedonicprice functionppears as a portionofthecurve n figure in the ntervalz10,z1 ).2. Ifps = 0 = z1s,all producersmustbe inthemarket,nd itfollowsthatz10 = 0. This onlyis possible fp'(O) = ps= 0 and c2 must bezero.In this ase price s a log-linear unction fquality.3. z1s > 0 and ps = 0. Now some consumers o not value z1 veryhighly, nd there s a definite imitto the smallest mountavailable.Clearly, (zls)must xceed zero and some consumersmustbe driven utofthemarket, indingtoptimalnotto consume heproduct t all. Ifnot,consumerswith mallvalues ofp wouldmasson z1, (therewouldbe acorner olution here), ddingfiniteweightto the market nd causingp(z1s) oexplode.Using hebudget onstraint,hemarket ateof xchangebetweennotbuying t all and buying js is [y - p(z1s)]/z1snd mustequal the lopeofthevaluefunctionor uyers t that extensive)margin.Thatis,theconditiony - p(zIs)]/zIs = p'(z1s)replaces quation 15)-after ubstitutingor andp' from13) in thedeterminationfc1 andc2. The hedonicprice function lso can be illustratedn figure as theportion fthe curvebetween hepoints uch as thosemarkedA(= z1)


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48 JOURNAL OF POLITICAL ECONOMYand zij. Again, c1 and c2 have the correct signs and the second-orderconditions are fulfilled.A second type of short-runequilibrium could be considered in whichexisting firmscan alter qualities as well as quantities of their products.W'Vhenhere is a distributionof cost functions, t is necessary to proceedanalogously to the treatmentof demand in the example above. For ex-ample, costs might be described by (a/2)N2ZI with A varying acrossfirms. hen (v./'2)= zlp'/p s used to transform he distribution f . into adistributionof qualities supplied. The resultingdistributionweightsfirmquantities supplied in the determination ofmarket supply at any quality.A littleexperimentationwill show that the differential quation resultingfrom setting Qd(z1) = Qs(zl) is nonlinear in most cases, and closedsolutions are not always feasible.

B. Long-RuniquilibriumFirms may vary qualities at will and also construct establishmentsofoptimum size. No entryrestrictionsmply the absence ofprofit mr* 0)and long-run offer price for each firm must satisfy O(z; /3)C(.M1, ; /)/IM. Plants are constructed to produce models of quality zat minimum cost. Hence scale economies are exhausted under competi-tion and the optimum production unit occurs where C(M, z, /3) islinear in Al, variations of quantity being achieved by changes in thenumber of establishments.Let h(z; /3) representminimum average costofZ foran establishmentofoptimum size. Then C(M, z; /3) = Mh(z; /3)in the long run. Therefore 0 = h(z; /3) nd p(z) = h(z; /3) s theequilibrium condition for maximum profit and p(z) is completelydetermined by supply, or by the envelope of the family h(z; /3)withrespect to /3.Generalization to n characteristics s obvious in this case.IV. An Identification ProblemSection III demonstrated that complete solutions for p(z) and thedistribution of qualities traded sometimes can be obtained if sufficienta priori structure s imposed on the problem. However, it is not alwayspossible to proceed in that manner. In general, the differential quationdefiningp(z) is nonlinear and it may not be possible to findclosed solu-tions.Moreover, a great deal of structuremust be imposed. For example,the distributionof income follows no simple law throughout its range,making it difficult o specify the problem completely. Finally, partial-differentialequations must be solved when there is more than onecharacteristic. This section sketches an alternative and more efficientprocedure, based on the analysis of Section II.


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HEDONIC PRICES 49p1701 I

2 3~~~~~~~

eztZI

FIG. 4

As shownabove, derivatives f a consumer's alue function,OZj, reproportional o marginalrates of substitution. hey are reservation-demandpricesfor dditionalamountsofzi at a constantutilityndex.Therefore zO(z)} are the inverses f a set of ordinary ompensateddemandfunctionsor hezip's. he marginal ostofzi tothe consumerspi(z), and optimal is determined heremarginal osts qual marginalvalues.One dimensionfthesemarginal onceptss llustratednfigure .The curves abeled OJ. re derivatives fOj in figure and reflect om-pensateddemandfunctionsorvarious buyers.The dashed line labeledp1(z) is the commonmarginalcost confrontingll buyers.Consumerchoice is given by the intersectionf demand and marginalcost. Itshouldbe emphasized hatthe functionsOz,(z) are compensated emandprices realincomeheld constant) nd can only be derived nce equilib-rium s determined,s in SectionII. For example,a new equilibriumresultingrom n exogenous hiftn p would not alwaysbe givenbytheintersectionfthe newmarginal osts, 1 z), and the nitial ompensateddemand pricefunctions. n exceptionoccurswhen Oziu= 0 and thefamily fsurfaces (z; u), such as depicted n figure , are all paralleltoeach other:0ZiU 0 is equivalent o constantmarginalutility fmoneyandOzi s uniqueand independent f uonly n thatcase. IfOza, 0, theshapeand location ftheOJ. unctionsre determined ytheequilibriumconditionsfSection I: tangency etween (z) and Oj(z, u*).A similar rocedure ppliestofirms: zj is thereservationupplypriceof incremental i and reflects profit-compensatedupplyfunctionor


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50 JOURNAL OF POLITICAL ECONOMYcharacteristicz1; pi is the marginal revenue function for zi facing eachfirm. One dimension of producer equilibrium is shown in figure4 as theintersectionof a set of compensated supply curves for various firms oZ,with a common marginal revenue function,p1 z).Figure 4 reiterates the major conclusion of Section II in terms ofderivativesofp(z). Equlibrium is described by the intersectionofsupplyand demand functions.However, income effectshave been removed, indistinctionto the typical case. Observed marginal hedonic prices merelyconnect equilibrium reservation prices and characteristics and reveallittle about underlying supply and demand functions.However, figure 4 suggestsa method that can be used for estimation.In principle, data are available on designs purchased by buyersand alsoon theirincomes and taste variables such as age, education, etc. Denotethese empirical counterpartsofo by a vector Y1. Data are also potentiallyavailable on the characteristics' content of models produced by sellersand factor price and specific technological differences among them.Denote the empirical counterpartsof#by a vector Y2. Following figure4,let Fi(z, Yi) represent the marginal demand price forzi and Gi(z, Y2)represent the marginal supply price. Ignoring random terms,the modelto be estimated can be writtenas

pi(z) = F(zl, . . , z., Y1) (demand), (16)pi(z) = G (Zn. * . , z,, Y2) (supply), (17)

for = 1, .. n,wherepi and zi are all jointly dependent variables andY1 and Y2 are exogenous demand and supply shift variables. The 2nequations determine the 2n endogenous variables pi and zi. Estimationrequires a two-step procedure. First, estimate p(z) by the usual hedonicmethod,withoutregard to Y1 and Y2. That is, regressobserved differenti-ated products' prices, p, on all of their characteristics, , using the bestfitting unctionalform. This econometricallyduplicates the informationacquired by agents in the market,on the basis of which theymake theirdecisions. Denote the resulting estimate of the functionp(z) by p(z).Next, ompute setof mplicitmarginalrices, p(z)/8zi Pi(z) for achbuyer and seller, evaluated at the amounts of characteristics numericalvalues of z) actually bought or sold, as the case may be. Finally, useestimated-marginal prices Pi(z) as endogenous variables in the second-stage simultaneous estimation of equations (16) and (1 7). Estimation ofmarginal prices plays the same role here as do directobservations on pricesin the standard theory and converts the second-stage estimation into agarden variety identificationproblem. There are four cases to consider:1. There is no variance in # and cost conditions are identical acrossfirms.The variables Y2 drop out of equation (17) and p(z) identifies he


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HEDONIC PRICES 5 Tofferfunction. Similarly, the sample observations on Pi(z) and the ziidentify ompensated supply functions.Suppose several cross sectionsfordifferent ears are available and firms' production functionshave beensubject to technical change. Then within-yearhedonic price regressionsidentify upply conditionsforeach year. Changes in marginal prices andqualities induced by changing technology and cost conditions betweenyears approximately sweep out the structure of preferencesand com-pensated demand functions with due qualification forthe nonconstancyof the marginal utilityof money).2. If buyers are identical, but sellers differ,Y1 drops out of (16) andsingle cross-sectional observations trace out compensated demandfunctions.3. If buyers are identical and so are sellers,offer nd value functionsare tangentat a singlepoint, and only one quality appears on themarket.The observationsdegenerate to a singlepoint; there s no product differen-tiation and no problem.I. In general there is both a distributionof buyers and another dis-tributionofsellers.Both Y1 and Y2 have nonzero variance, and theusualidentifying ank and order conditions apply. A necessarypriorconditionforestimation s that p(z) be nonlinear at stage one. For ifp(z) happensto be linear, pi(z) are constants, independent of qualities traded, anddisplay zero variance across sample observations. As shown above,linearityof p(z) is unlikelyso long as thereis increasingmarginal cost ofattributesfor sellers and it is not possible to untie packages. But it isobvious that the model does not apply ifvery few distinctproducts areactually traded.V. Price Indexes, Economic Welfare, and Legislated RestrictionsThis sectionuses the model to analyze thewelfare consequences ofquality-standards legislation, a problem not easily handled by conventionalmethods. The discussion clarifies ssues in recent controversiesregardingtreatmentoflegislated standards in the constructionofprice indexes. Forexample, how should mandatory installation of seat belts and air bagsaffect he automobile price index? For expositoryconvenience, discussionis confinedto the case of one attribute. Generalization to several charac-teristics s immediate.A minimum quality standard means that z ? f, and brands con-taining less than f are prohibited from the market. Assume constantreturnsto quantities (as in Section III B). Then the law is irrelevantforall consumerspreviouslypurchasing packages containing more than thelegislated minimum. The situation fora buyer whose choice is affectedby the law is shown in figure5: z * was the original choice, whereas f ischosen afterthe law has been passed, since z* is no longeravailable. The


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52 JOURNAL OF POLITICAL ECONOMYp(z))

P2 ob

pi e~~~~~~~~~~~~~~~aP0

/t IZIZ, Z ZFIG. 5

minimum ttainablevalue function as shifted rom a to ob, and theconsumersworseoffsee eq. [2]).Choose the distanceAP = P2 - P1as a monetarymeasure f the ossin welfare. ince a0/ay= 1, AP is the bribenecessary or heconsumertopurchasef when * was available. Clearly, hismeasure snot unique(i.e., if compensations evaluated at a differentmount of z) unlessoziu= 0. The welfareosscan be estimated rom he mplicit rice andbid functions.The distance P2 -PO is given byJzi

or the area undermarginal ostfrom * tof,and is shown nfigure as4zkabf It representshe social opportunityost of additionalresourcesnecessaryo producef insteadofz*. The integral01 z)dz,Z*n

or the area undera compensated emandfunctioncompensatedt theoriginal evel of real income) between * and i in figure6 (z~ac 1)measures he amounttheconsumerwouldhave paid for heincrement(z1 - z) at the unrestrictedevel ofwelfare. t measuresP1 - PO infigure5 and represents he benefit f the restriction. he difference


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HEDONIC PRICES 5dpdz,

ZZZr~~~~~~~~~~~ZFIG. 6

betweencostsand benefits s given byP2 -P1, or thedifference etweenthe areas under the marginal cost and compensated demand functions,the shaded area in figure6. In the general case of several attributes,APmust be measured by a line integral. Otherwise, everything else isunchanged.When the marginal utilityofmoney is constant,AP is unique and theprice restrictions equivalent to an additive increase in implicit prices inamount AP everywhere. n figure5, O,, = 0 means that all value func-tions are parallel, and if the budget constraint wasy = x + p(z) + APinstead ofy =x + p(z), the consumer would have arrived exactly at Oof hisown free hoice. The real priceof thecharacteristichas risenbecausechoices are restricted, nd the price index should rise to reflect hat fact.A natural measure of the real price increase imposed by the law is aweighted average of terms uch as AP (including buyersforwhom AP =0), where the weights are expenditure shares among all consumers.5This measure overstates the loss insofaras the restriction ctually forces5A complete ssessment f the aw and its effect n theprice ndexrequiresbalancingthe costs calculated above againstany externality-inducedocial benefits fthe restric-tion. n our udgment, eat belts nd air bags are in a differentategorythan emission-control evices. n regard o the atter, heapparatusabove can be usedeasilyto analyzethe effect f the European system f taxing engine displacement.An ad valorem taxincreases verageand marginal osts fpackageswith arger iter apacity, nd the usualincome and substitutionffectspply: packageswith smaller mounts of this and com-plementaryharacteristicssuchas size ofcar) are purchased.


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54 JOURNAL OF POLITICAL ECONOMYsome consumers completely out of the generic goods market since theyescape thefull oss AP. Also, standard index number problems arise whenthe marginal utilityof money is not constant.VI. ConclusionsThis paper has drawn out the observational consequences ofthe constructof mplicitmarketsforcharacteristics mbodied in differentiated roducts.When goods can be treated as tied packages of characteristics,observedmarket prices are also comparable on those terms.The economic contentof the relationshipbetween observed prices and observed characteristicsbecomes evident once price differences mong goods are recognized asequalizing differences or the alternative packages they embody. Here,as elsewhere, pricedifferences enerallyare equalizing only on the marginand not on the average. Hence, estimated hedonic price-characteristicsfunctionstypically identifyneither demand nor supply. In fact, thoseobservations are described by a joint-envelope function and cannot bythemselves identifythe structureof consumer preferences nd producertechnologiesthat generate them.The formal analysis is complicated by the fact that budget constraintsare nonlinear. Consequently, it is not surprising hat far weaker theoremsthan usual apply. However, a feasible econometric procedure for esti-mating the underlying generatingstructurehas been derived throughtheuse of derivative transformations. When constraints are nonlinear,marginal prices serve the same role as average prices do in the linearcase. Finally, the essential spatial context of the problem means thatsubstitution nd income effectsmustbe more carefullydistinguishedthanusual. Indeed, here is a major practical instance where compensateddemand and supply functionsbecome the relevant fundamental concepts.These compensated functions are estimated by the econometric methodand measures of consumer and producer surplus can be derived directlyfromthem. We anticipate that the basic conceptual frameworkoutlinedabove will have a variety of applications to many practical problemsinvolvingequilibrium in cross-sectiondata.The analysis has been simplifiedby assuming divisibility n production.Generalization has to incorporate nonconvexities, and discontinuitiesmust result. When nonconvexities are not small relative to the market, tis obvious that only isolated locations on the characteristicssurface willbe filled. In otherwords,such a generalization will naturally incorporatethe case ofmonopolisticcompetition,and observed distances (in termsof characteristics)between differentiated roducts will be endogenouslydetermined. The methods employed above do not carry throughbecausecertain nonmarginal decisions must be analyzed, and far more sophisti-cated techniques are required.


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HEDONIC PRICES 55ReferencesAlonzo,William.LocationndLand Use. Cambridge,Mass.: HarvardUniv. Press,1964.Becker,GaryS. A Theoryof theAllocation fTime. Econ.J. 75 (September1965): 493-517.Court,Louis M. Entrepreneurialnd ConsumerDemand Theoriesfor Com-moditySpectra. Econometrica, no. 1 (April 1941): 135-62; no. 2 (July-October 1941): 241-97.Ellickson, Bryan. JurisdictionalFragmentationand Residential Choice.A.E.R. 61 (May 1971): 334-39.Gordon,RobertJ. The MeasurementfDurableGoods Prices. Mimeographed.Nat. Bur.Econ. Res., 1973.Griliches, Zvi, ed. Price Indexes nd Quality Change.Cambridge, Mass.: Harvard

Univ. Press,1971.Houthakker,H. S. CompensatedChanges in Quantitiesand Qualities Con-sumed. Rev.Econ. tudies 9,no. 3 (1952): 155-64.Intriligator,Michael D. MathematicalOptimizationnd EconomicTheory. nglewoodCliffs, .J.: Prentice-Hall, 97 .Lancaster,KelvinJ. A New ApproachtoConsumerTheory. J.P.E. 74 (April1966): 132-56.Lewis,H. Gregg. Interesdel empleador n as horas de Trabajo del empleado[Employernterestsnemployee ours fwork].CuadernoseEconomia,atholicUniv. Chile, 1969.Lipsey,Richard G., and Rosenbluth, ideon. A Contributiono theNew De-mand Theory: A Rehabilitationof the GiffenGood. CanadianJ. Econ.4(May 1971): 131-63.Muth, RichardF. HouseholdProduction nd ConsumerDemand Functions.Econometrica4 (July 1966): 699-708.Ohta, Makoto, ndGriliches,vi. Makes and DepreciationntheU.S. PassengerCar Market. Mimeographed.HarvardUniv., 1972.Tiebout,CharlesM. A PureTheory fLocal Expenditure. J.P.E. 64 (October1956): 416-24.

rosen hedonics jpe 1974

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