BART HOBIJN Federal Reserve Bank of New York *

Transcription

1 O BOH SIDES OF HE QUALIY BIAS I PRICE IDEXES BAR HOBIJ Federal Reserve Bank of ew York Verson 2, December 2002 Absrac I s ofen argued ha prce ndexes do no fully capure he qualy mprovemens of new goods n he marke. Because of hs shorcomng, prce ndexes are perceved o overesmae he acual prce ncreases ha occur. In hs paper, I argue ha he qualy bas n prce ndexes s jus as lkely o be upward as s o be downward. I show how boh he sgn and he magnude of he qualy bas n he mos commonly appled prce ndex mehods are deermned by he cross-seconal varaon of prces per qualy un across he produc models sold n he marke. I do so by smulang a model of a marke ha ncludes monopolscally compeng supplers of he varous produc models and a represenave consumer wh CES (consan elascy of subsuon) preferences. I llusrae he bas n he commonly appled prce ndex mehods by comparng her esmaes of nflaon wh he heorecal nflaon rae mpled by he daa-generang process. Keywords: Prce ndex heory, hedonc prce mehods, qualy bas, monopolsc compeon. JEL-codes: C43, D, D43, L3. Correspondence: Bar Hobjn, Domesc Research Funcon, Federal Reserve Bank of ew York, 33 Lbery Sree, 3 rd floor, ew York, Y E-mal: bar.hobjn@ny.frb.org. he vews expressed n hs paper are hose of he auhor and do no necessarly reflec he poson of he Federal Reserve Bank of ew York or he Federal Reserve Sysem as a whole. I would lke o hank Alexs Anonades for hs excellen research asssance.

2 . Inroducon here s a wdespread consensus among economss ha prce ndex mehods end o overesmae acual nflaon n markes where here s a rapd urnover of goods due o echnologcal progress. he Boskn (996) commsson made hs pon wh respec o he U.S. Consumer Prce Index, whle Gordon (990) used hedonc prce ndexes o correc for hs bas n equpmen prce ndexes. here s, however, also a small number of sudes ha challenge hs convenonal wsdom. Sudes by by rple (972,2002), Feensra (995), as well as Hobjn (200) have each made he pon ha qualy adjusmens n prce ndex mehods mgh acually lead o an undersaemen of nflaon. hs paper follows up on he above papers by nroducng a parsmonous heorecal model ha can generae boh a posve as well as a negave qualy bas n he mos commonly appled prce ndexes. he value added of hs approach s ha allows for he sudy of he facors ha deermne he sgn and magnude of he qualy bas n a sylzed framework. hs conrass srongly wh he mehodology ha s radonally appled n he prce ndex leraure. A large par of he leraure on prce ndexes compares varous prce ndexes calculaed for he same daase. hs s for example he approach of Azcorbe and Jackman (993), Manser and McDonald (988), and Brahwa (980) when assessng he magnude of subsuon bas as well as of Azcorbe, Corrado, and Doms (2000) and Slver and Herav (2002) n he comparson of hedonc and mached model prce ndexes. Such an approach allows us o consder he sensvy of prce ndexes o he choce of mehod appled. I does no, however, enable us o make any normave saemens abou whch ndex mehod s beer han anoher. Such normave saemens on prce ndexes are all based on he exensve heorecal prce ndex leraure, whch focuses on properes lke dealness, exacness, and superlavy of prce ndex formula. I urns ou ha he heorecal resuls derved n hs paper conradc some of he properes of prce ndexes ha are presumed n hs appled srand of he leraure. hree resuls sand ou n parcular. he mos mporan s ha he heorecal model n hs paper confrms he clams by rple (972,2002), Feensra (995), and Hobjn (200) ha he qualy bas n prce ndexes s no by defnon upward. Moreover, he sgn and magnude of he bas urn ou no o depend on he overall level of nflaon. Insead hey depend on he cross-seconal behavor of prces per qualy un across models sold n he marke durng he same perod. Secondly, he exsence and sgn of hs bas does no depend on he specfc prce ndex formula appled. I show how he applcaon of he mos popular prce ndex formulas, lke Laspeyres, Paasche, Geomerc Mean, Fsher Ideal, and ornqvs, all lead o smlar magnudes of he qualy bas. Fnally, hedonc prce ndexes suffer from he same qualy bas as mached model ndexes. Hence, he heorecal resuls here seem o dsagree wh he presumpon ha hedonc prce ndexes do a beer job a correcng prces for qualy mprovemens, as made n, among many, Pakes (2002) and Hulen (2002). he parcular heorecal model ha I use for my analyss n hs paper s ha of a marke wh a represenave consumer wh CES preferences over a se of models sold. hs seup s very smlar o Dx and Sglz (977) and Hornsen (993). he man dfference s ha he marke ha I consder has a 2

3 counably fne number of models and supplers. he advanage of hs choce of model s ha prce ndex heory for CES preferences s exremely well developed. Sao (976) derved he deal exac prce ndex for CES preferences when he same models are sold n boh he base- and measuremen perods. Feensra (994) exended Sao s ndex o an exac mached model ndex ha can be used when he unverses of models sold n boh perods do no concde. he resulng mehodology n hs paper s closely relaed o he Mone Carlo mehodology n economercs. In hs sense, I follow Lloyd (975) who also used numercal smulaon mehods o derve and quanfy ceran properes of prce ndexes. In Lloyd s (975) sudy he focus was on he subsuon bas n prce ndexes whle here he focus s on her qualy bas. he srucure of he paper s as follows. In he nex secon I nroduce he form of he CES preferences ha I consder n he res of he paper and derve he heorecal prce level ha prce ndexes are mean o measure. In Secon 3 I hen llusrae graphcally how convenonal prce ndex mehods mgh yeld a downward qualy bas for hese preferences. hs graphcal descrpon s essenally an nformal verson of he resuls ha are derved formally n he conex of he heorecal model. I nroduce hs heorecal model n Secon 4. I consder s demand and supply sde and show how s Pure Sraegy ash equlbrum exss and s unque. In Secon 5 I hen proceed by dervng some general resuls for he sgn of he qualy bas n mached model and hedonc prce ndexes calculaed for a specfc parameerzaon of he model. Fnally, n Secon 6 I presen he resuls of a se of numercal smulaons of he model and llusrae how hese smulaons confrm he resuls shown n Secons 3 hrough 5. Secon 7 concludes. 2. CES-preferences and he heorecal prce level he am of hs paper s o be able o make normave saemens abou prce ndex mehods and o say whch ones perform beer, n ceran suaons, han ohers. In order o make hese normave saemens we need o defne wha s we would lke our prce ndex mehods o measure. Snce Konüs (939) he man focus of prce ndex heory has been on consrucng a cos-of-lvng ndex (COLI). he am of a COLI s o rack he (percenage) changes n he mnmum expendures requred o reach a ceran base-level of uly over me. he mnmum amoun of expendures ha s necessary o reach a ceran uly level crucally depends on he underlyng preferences of he consumer. Hence, he heorecal prce level ha prce ndex mehods are afer depends on he preferences of he consumer. In realy, a marke consss of a specrum of consumers wh dfferen preferences. I urns ou ha s no always possble, n such cases, o specfy he heorecal prce level because aggregae demand does no always behave as f s generaed by a wellbehaved aggregae uly represenaon. he focus of hs paper s no on he condons for he exsence of an aggregae uly represenaon for aggregae demand. Wha I wll do s smply use one of he bes developed aggregae uly represenaons for whch has been proven ha can be nerpreed as he aggregae uly funcon of a marke wh a connuum of heerogeneous agens. hs aggregae uly represenaon s Consan Elascy of I wll focus on consumer prce ndexes hroughou hs paper. he heory presened n hs paper s also applcable o producer prce ndexes, whch are amed a rackng he mnmum cos requred o oban a base quany of oupu over me. 3

4 Subsuon (CES) preferences. Anderson, de Palma, and hsse (993) nroduced he mcrofoundaons of CES preferences and showed how hey can be nerpreed as he aggregae uly represenaon of a marke conssng of a connuum of heerogeneous agens. Le X, be he quany consumed of good a me, where I wll assume ha good =0 s he numerare good. C s he unverse of goods sold a me. I wll assume ha aggregae demand n he marke n he heorecal model behaves as f resulng from he uly maxmzng decson of a represenave consumer wh he uly funcon U + α = ( a ) X, X 0, where >0 and 0<α< () C \{} 0 + hs s a relavely sandard CES uly funcon, where σ (+)/ s he consan elascy of subsuon. he only non-sandard feaures of () are ha he quanes for goods C \{0} are mulpled by a qualy parameer a and ha he numerare good, =0, s ncluded. Le p, be he prce of a un of good. Snce =0 s he numerare good, I wll assume ha p 0, = for all. In he res of hs paper, I wll focus on he consrucon of a prce ndex for he se of goods, whch I wll call models n he fuure, ha are conaned n he CES par of (). ha s, my focus s on he measuremen of he prce level of he se of models C where C =C \{0}. We are hus confroned wh wo ses of goods,.e. he numerare good and he models for whch we would lke o measure an aggregae prce level. Dewer (200) shows ha, because he preferences n () are separable beween X 0, and he oher goods, he aggregae prce level for he models C s well defned. In parcular, aggregae demand for he models C wll be as f was generaed by he represenave agen maxmzng he amoun of uly obaned from hese models for he expendures solely on hese models. hs mples ha he heorecal prce aggregae for he se of models C s he CES prce aggregae as appled n, among many, Dx and Sglz (977), Hornsen (993), and Feensra (994). hs aggregae, he value a me of whch I wll denoe by P, reads / P = ( p a ) (2) C I s a CES aggregae of he prces per qualy un for all models ha are raded n he marke. hs prce aggregae represens he money cos of a un of uly obaned from he consumpon of he compeng varees n he se of models C. hs money cos does no depend on he base-level of uly because he preferences are homohec. he am of prce ndex mehods s o consruc an ndex ha approxmaes, up o a consan, he pah of P. In parcular, he ndex mehods are mean o esmae he perod by perod percenage change n P. hroughou hs paper, I wll focus on he percenage change n P beween perods =0 and =. I wll refer o he percenage change n P beween hose wo perods as he heorecal nflaon rae and wll denoe as 4

5 P P 0 π = (3) P0 I represens he percenage change of he money cos of a un of uly beween perods 0 and. If one would know all he preference parameers n () hen would no be dffcul o calculae he heorecal nflaon rae n (3). In pracce, however, he preference parameers are no observed. ha s, we do no exacly know he elascy of subson,.e. σ (+)/. eher do we know he qualy emboded n each un sold for each model,.e. a. In fac, when we apply prce ndex mehods we do no even know by wha preference represenaon aggregae demand s generaed. here are bascally wo lnes of hough here, whch I wll boh pursue n hs paper. he frs lne assumes ha aggregae preferences belong o a ceran class and hen uses hs resrcon o oban an esmae of (3). For he CES preferences he ndex ha exacly measures he heorecal nflaon rae s he one derved by Sao (976). he deals of hs ndex are descrbed n able. Sao s ndex s vald under he assumpon ha he unverses of models sold n boh perods are he same, such ha C 0 = C. I s a proper prce ndex n he sense ha only depends on observables, namely expendure shares and prces. he requremen of concdng ses of models beng sold n boh perods renders he Sao (976) ndex napplcable a many lower levels of aggregaon. Many markes have a hgh rae of produc urnover, as llusraed n Azcorbe, Corrado, and Doms (2000) for he marke for Inel CPU uns and n Slver and Herav (2002) for he marke for laundry machnes. Hence, s hus essenal o develop prce ndex mehods ha allow for dynamc unverses of models ha change over me,.e. C 0 C. Feensra (994) exends Sao s resul o a quas-ndex ha s exac for CES preferences wh non-overlappng unverses of models. Feensra s s a quas-ndex because depends on he unobserved elascy of subsuon, whch has o be esmaed o mplemen he ndex. I s descrbed n able and I wll dscuss s nuon n more deal laer on. Prce ndex heory s hus very well developed for CES preferences. We know he form of he exac ndexes boh when he unverse of models s sac as well as when s dynamc. he problem s ha n many praccal cases s a bg leap o assume ha demand s generaed by aggregae CES preferences. hs brngs us o he second lne of hough. hs lne s o consruc prce ndexes ha do no exacly measure (3) bu nsead reasonably approxmae for a very broad class of preferences. hs s he approach mos commonly chosen for he calculaon of aggregae sascs. Classcal prce ndex heory, among ohers Konüs (939), Frsch (936) and Fsher (922), yelded many mporan resuls for he case n whch he unverse of models s sac. Konüs (939) nroduced he conceps of a cos-oflvng ndex and subsuon bas n prce ndexes ha prce a fxed baske of goods. Frsch (936) showed how Konüs s subsuon bas resul mpled ha for homohec preferences he change n he rue cos of lvng s bounded from above by he Laspeyres ndex and from below by he Paasche ndex. Fsher (922) showed how he geomerc mean of he Laspeyres and Paasche ndexes consues an deal ndex n he sense ha boh he prce and quany ndexes have he same funconal form. 5

6 A large par of he leraure has focused on he queson whch prce ndex formula approxmaes (3) n he mos reasonable way. Lke Fsher (922), Dewer (976), as well as Lloyd (975), Brahwa (980), Manser and McDonald (988), and Azcorbe and Jackman (993). A much smaller par of he leraure has focused on he consrucon of reasonable approxmaons o (3) n case of dynamc unverses of models. he problem when he unverses of models are dynamc s ha he prces of new goods are no observed n he frs perod, whle he prces of obsolee goods are no observed n he second perod. I s hus no possble o measure he percenage change n he prces beween boh perods for new and obsolee goods. wo approaches are generally consdered when dealng wh hs problem. he frs, known as mached model ndexes, makes specfc assumpons abou he relave prce per qualy un of he new models versus he old models. hese assumpons are such ha hey mply ha he change n he overall prce level can be esmaed solely as a funcon of he prce changes of he models ha are sold n boh perods,.e. ha are mached. rple (2002) conans an overvew of he dfferen mached model mehods and he possble bases ha hey nduce. he second, known as hedonc prce ndexes, uses a regresson model ha relaes he prce of a model n a ceran perod o s characerscs o mpue he unobserved prces for he new and obsolee models. hs mpuaon complees he se of prces needed o apply convenonal prce ndex mehods developed for overlappng unverses of models. Afer he prce mpuaon of he mssng prce observaons, ndexes are hen consruced usng convenonal prce ndex mehods. 3. A graphcal llusraon of he man argumen he convenonal wsdom s ha he nroducon and obsolescence of goods n a marke would cause sandard prce ndex mehods o oversae he acual nflaon rae. he Boskn (996) commsson repor as well as s recen reassessmen by Lebow and Rudd (200) boh conan exensve descrpons of hs convenonal wsdom. here are hree man reasons why hs s argued o be he case. he frs reason, desgnaed qualy bas by he Boskn (996) commsson, s ha curren prce ndexes do no properly capure he qualy mprovemens emboded n new (or mproved) models. By underesmang hese qualy mprovemens, prce ndexes wll arbue oo much of changes n expendures o changes n prces raher han o changes n quanes. he second reason, desgnaed produc bas by he Boskn (996) commsson, argues ha prces of new goods end o drop faser han hose of esablshed models. Because new goods and models are only ncluded n he sample of goods used o calculae he prce ndex wh a ceran delay, he nal prce drops early n he produc cycle are no capured by curren prce ndexes. Fnally, here s he subsuon bas. hs bas s due o some prce ndexes, ncludng he CPI and mos prce ndexes calculaed n Europe, beng fxed weghed prce ndexes whch do no capure he ncreases n welfare from consumers beng able o subsue new goods for goods ha hey were prevously consumng. In he res of hs paper I wll manly focus on he qualy bas and gnore ssues relaed o he laer wo sources of bas. In general s hard o argue agans sascal agences ncludng new models and goods more mely n her samples and reducng he poenal sources of produc bas. Furhermore, he ssue of subsuon bas s currenly beng addressed, a leas for he U.S. CPI, by he jon publcaon of a fxed 6

7 weghed as well as a chan weghed prce ndex. he laer s mean o accoun for he subsuon bas. See Bureau of Labor Sascs (2002) for a dealed descrpon. he man pon of hs paper s ha he qualy bas n prce ndexes s no solely a source of upward bas. Insead, he qualy bas nduced by mos commonly appled prce ndex mehods can be boh upward as well as downward. Before I llusrae hs n a formal mahemacal economc framework, I frs descrbe he man nuon of he argumen graphcally n hs secon. he graphcal descrpon n hs secon s based on Fgure hrough Fgure 3. he op panel of Fgure depcs wo hypohecal prce schedules, for =0 and =, of a se of models ha dffer accordng o her qualy levels, a. I wll assume ha a s no drecly observed. herefore, he researcher observes he prce of each model,.e. p, bu does no know s relave poson on he x-axs. As explaned n he prevous secon, wha s mporan for he prce level assocaed wh he CES preferences ha I consder s no he acual prce levels, p, bu he prce per un of qualy, p /a, for each model. Panel (b) of Fgure depcs he assocaed schedule of prces per qualy un. Panel (c) conans he same prce per qualy un schedule and adds some of he noaon ha I wll use n he res of hs paper. Jus lke n he prevous secon C denoes he se of models sold n perod, whle P denoes he heorecal prce level a me. oe ha I have chosen o draw he example such ha P 0<P. ha s, n he graphcal example he acual prce level ncreases beween perods =0 and =, such ha here s posve nflaon. In each perod he se of models sold,.e. C, consss of a group of models ha are no sold n he oher perod,.e. he se A -B, as well as a group of models ha are mached n he sense ha hey are sold n boh perods,.e. he se B -D. Wha I wll now llusrae s ha, even hough he heorecal prce nflaon s posve for hese hypohecal prce schedules, mos commonly appled prce ndex mehods wll end o measure negave nflaon nsead. ha s, n hs graphcal example sandard prce ndex mehods wll end o underesmae acual nflaon raher han overesmae, as he consensus vew suggess. I wll llusrae hs for boh mached model as well as hedonc prce ndexes. here are several ways n whch mached model ndexes are calculaed. hey each make dfferen denfyng assumpons abou he relave prce per qualy un of he obsolee and new models n he marke. he frs mehod, ofen referred o as drec comparson, assumes ha he obsolee and new models can be drecly compared n he sense ha hey embody he same levels of qualy. Because hs mehod assumes ha here are no qualy mprovemens beween he old and new vnages of models, hs mehod s never appled n markes wh rapd produc urnover due o echnologcal progress, lke hose for compuers and oher elecronc producs for example. Because I wll focus on markes wh qualy mprovemens n he producs sold, I wll dsregard hs mehod n he res of hs paper. he second mehod, known as lnk-o-show-no-prce-change, assumes ha he prce per qualy un s he same for he obsolee and new models. In hs case, he relave prce of he obsolee and new models s assumed o be fully arbuable o qualy mprovemens. Azcorbe (200) uses hs assumpon for example o denfy he pars n semconducor prce changes arbuable o qualy changes and prce changes respecvely. oe ha, as rple (2002) descrbes n more deal, hs mehod overesmaes nflaon only 7

8 when he prce per qualy un of he new models s lower han ha of he obsolee models. In ha case he mehod overesmaes he prce per qualy un for he new models and hus wll overesmae nflaon. he reverse s rue n our graphcal example here. In he example he prce per qualy un s hgher for he new models han for he old models. Consequenly, he mehod wll overesmae qualy changes and underesmae he acual level of nflaon. he fnal mached model mehod ha s frequenly appled s he Implc Prce Implc Quany 2 mehod (IP-IQ). hs mehod s based on he denfyng assumpon ha he overall prce change equals he prce change n he se of mached models. When one makes hs assumpon, he prce levels of he unmached models are no needed o measure nflaon. Hence, n hs case he unmached models are gnored,.e. deleed, and sandard prce ndex mehods are appled o he se of models ha s sold n boh perods,.e. he mached models. Fgure 2 llusraes he applcaon of he IP-IQ mehod n our graphcal example. he se of mached models n he example s he nersecon of C 0 and C. Consequenly, he IP-IQ mehod wll compare he B 0 -D 0 par of he perod 0 prce schedule wh he B -D par of ha of perod. For all models n hs range he prces are fallng. he IP-IQ mehod wll hus, ncorrecly, fnd a drop n he overall prce level. he smples way o see why he IP-IQ mehod underesmaes nflaon n hs case s o compare he relave prces of he deleed secons A 0 -B 0 and A -B wh he mached pars of he prce schedules. For he deleed par A 0 -B 0 n perod 0 we oban ha he prces per qualy un are lower han he prces per qualy un on he mached par of he schedule, B 0 -D 0. Consequenly, he deleon of he below average prces on he A 0 -B 0 par of he prce schedule wll lead o an nferred prce level n perod 0 ha s hgher han he acual level. Smlarly, when he above average prces of par A -B are deleed n perod, he prces of he mached models,.e. B -D, reflec a prce level ha s lower han he acual prce level. ha s, because he prce per qualy un s ncreasng n qualy and he wors models become obsolee whle he new models are of he hghes qualy, he prce level n perod 0 s overesmaed whle he prce level n perod s underesmaed. he combnaon of hese wo measuremen errors leads o an unambguous downward bas n he measured nflaon rae, ndependen of whch prce ndex formula s appled. One fnal hng s worh nong abou hs argumen. ha s ha he bas ncurred due o he applcaon of he IP-IQ mehod does no depend on he overall nflaon rae. Insead, compleely depends on he cross-seconal behavor of he prces per qualy un as a funcon of he qualy uns emboded n he models sold n he marke. I wll show hs n a numercal example laer on. hs resul conrass sharply wh he argumen n rple (2002) who argues ha he errors produced by he IP-IQ mehod are symmerc, n he sense ha when prces are fallng he IP-IQ mehod ends also o mss prce declnes. Prces have generally been fallng for elecronc producs, ncludng I producs. When he IP-IQ mehod s used o consruc prce ndexes for elecronc producs, he prce ndexes are based upward because hey do no adequaely measure prce declnes ha accompany new nroducons. he example here suggess ha wha maers for he IP-IQ bas n I produc nflaon s no wheher prces are declnng over me bu raher wheher prces per qualy un are declnng n he amoun of qualy emboded n he models. 2 hs s also ofen referred o as he deleon mehod. 8

9 Hedonc prce ndexes are, n some sense, he oppose of IP-IQ mached model ndexes. ha s, where he IP-IQ mehod delees he observed prces of he unmached models, hedonc mehods nser he unobserved prces of he unmached models. hs nseron, or more correcly mpuaon, s done by esmang a hedonc prce equaon ha relaes he prce of a model n a parcular perod o a se of s qualy characerscs and hen usng hs equaon o predc wha he unobserved prces of he unmached models would have been. Over he pas fve years, hedonc prce ndexes have been mplemened for an ncreasng number of goods for U.S. aggregae sascs. See Landefeld and Grmm (2000) as well as Moulon (200), for example, for a dscusson of he applcaon of hedonc prce ndexes n he U.S. naonal accouns. he man reason why hedonc prce ndexes are adoped for an ncreasng number of goods s he praccal problem ha he IP-IQ mehod ends up no usng a large par of he avalable prce quoes n markes where new and obsolee models make up he bulk of models raded. hs s parcularly a problem for compuers and relaed equpmen. he beleve s ha by akng he prce daa for he obsolee and new models no accoun and relang hem drecly o qualy characerscs, hedonc prce ndexes more properly adjus for qualy and are less subjec o qualy bas. hs seems o be confrmed by he fac ha hedonc prce ndexes end o fnd less nflaon for mos of he goods o whch hey are appled 3 han sandard mached model ndexes, whch are sad o overesmae nflaon. Is rue ha hedonc prce ndexes have a smaller qualy bas han mached model ndexes? o necessarly. In order o see why no, consder Fgure 3. Whch prces are mpued n a hedonc prce ndex depends on he prce ndex formula appled. he wo panels of Fgure 3 depc he wo mos common cases. he op panel consders a hedonc Laspeyres ndex, whch nends o measure he percenage change n he cos of he models sold n perod 0. he Laspeyres ndex requres he use of he prces of he models ha became obsolee n perod. herefore, a hedonc regresson model s used o mpue hese prces and he prce schedule n perod s exended by he mpued par D -E. he Laspeyres ndex hen bascally approxmaes he change n he overall prce levels mpled by he curves A 0 -D 0 and B -E. he overall prce level mpled by A 0 -D 0 concdes wh he acual prce level n perod 0,.e. P 0. he prce level mpled by B -D, denoed by P HL n he fgure, s lower han he acual prce level n perod. he reason s ha for he calculaon of he Laspeyres ndex he above average prces per qualy un n he par A -B are gnored. Moreover, he mpuaon adds below average prces per qualy un n he secon D -E. Hence, he nferred prce level n perod s below he acual prce level and nflaon s underesmaed. In fac, because he A 0 - D 0 schedule s above he B -E schedule everywhere, n hs example he hedonc prce ndex would fnd spurous prce deflaon. he boom panel depcs he calculaon of a hedonc Paasche ndex. I s mean o approxmae he change n he cos of he models sold n perod. herefore requres he mpuaon of he D 0 -E 0 par of he prce schedule n perod 0 and gnores he par A 0 -B 0 n s calculaon. he mpued par D 0 -E 0 consss of above average prces per qualy un and he gnored par A 0 -B 0 of below average prces per qualy un. 3 See for example Gordon s (990) hedonc prce ndexes. 9

10 hs leads o he hedonc mehod overesmang he prce level n perod 0. hs esmae s denoed by P HP 0. Agan, he hedonc mehod wll fnd spurous deflaon. hs me because overesmaes he prce level n perod 0, raher han underesmaes he prce level n perod. hus, my enave graphcal example llusraes why mached model and hedonc mehods mgh acually resul n esmaes of nflaon ha are oo low raher han oo hgh. However, hs smple graphcal example can only be used for llusraon purposes, does no prove ha such bases mgh occur n he daa. In order o show ha hese bases are lkely o occur, I nroduce a farly sandard heorecal model n he nex secon and show how he equlbrum oucome of he model gves rse o bases of he same knd dscussed here. 4. heorecal model he am of hs secon s o nroduce a smple heorecal framework ha generaes he knd of bas ha I dscussed n he secon above. he heorecal framework nroduced here s based on he CES model frs consdered by Anderson, de Palma, and hsse (992). Feensra (995) appled hs model o hedonc prce ndexes. I wll nroduce he heorecal model n hree subsecons. he frs explans he demand sde of he marke, whle he second focuses on he supply sde of he marke. In he fnal subsecon, I wll prove exsence and unqueness of he Pure Sraegy ash equlbrum ha deermnes prces and quanes n he marke and wll derve some of he relevan comparave sacs for hs equlbrum. Demand sde of he marke he aggregae demand n hs marke can be represened as generaed by a represenave agen choosng he demand { X, o maxmze he aggregae uly funcon n equaon (). hs uly funcon s } C maxmzed subjec o he budge consran where Y denoes real ncome n erms of he numerare commody. he maxmzaon of hs uly funcon yelds he demand funcons X ~ Y p, Y = X + p X (4) 0, / ( p, / a ) ( p j, / a j ) j C C =, / for he non-numerare commody,.e. C. he varable ( ),, ~ / ( ) ( ) Y = p, / a (5) / p, P ~ Y = + α s he level of oal expendures Y on hese models. hese demand funcons are very smlar o he ones mpled by sandard CES preferences where he level of qualy for all goods s he same,.e. a = for all C. he man dfference s ha he relevan relave prce of each good ha deermnes s marke share s s prce per un of qualy, ha s p /a, raher han s un prce, p. 0

11 Supply sde of he marke he nex concern s he supply sde of he marke for C. I wll assume ha he producer of model a each pon n me,, faces a consan un producon cos c. I wll consder pure sraegy ash equlbra n prces for a marke wh a fxed se of models, a =. Such ash equlbra mply ha he suppler of model { } C a akes as gven he prces p j for j C \{} and chooses s prce p o maxmze s profs ( c ) X p (6) subjec o he demand funcon (5). he prof maxmzng choce of prce p n hs case sasfes he followng frs order condon. c p X = + X / p / p (7) hs condon mples ha he suppler of each model chooses s prce such ha s cos-prce rao equals one plus he nverse of he own prce elascy of demand for good. Snce he own prce elascy of demand for good s negave, hs mples ha c /p <. ha s, prce exceeds margnal and average cos and he frm charges a markup. For he prce elascy of demand, we oban ha X X ( p / a ) / / p = + = θ ( ), / / p p j a j / j C ( p a ) where θ ( p ) >0 s he negave of he prce elascy of demand for good and = { },a p C (8) p s he sequence of prces charged n he marke. Essenal for he resuls ha are o follow s ha hs elascy s specfc o good. hs s conrary o he seup of monopolsc compeon ha s ofen used o model mperfec compeon n models wh prce rgdes, lke n Hornsen (993). hese models generally consder a symmerc equlbrum n whch each monopolsc compeor s oo small o affec he aggregae prce level and s own prce elascy of demand. Usng he noaon above, he suppler of good wll se s prce such ha where ( p ) p c ( p, a ) ( p, a ) θ ( p a ) = = = µ ( ), (9) θ p, a θ µ,a > s he markup charged by he frm. Solvng for hs markup yelds ha (, ) = ( + ) ( p / a ) / + / µ p a (0) ( p j / a j ) j C \{}

12 hs mples ha he pure sraegy ash equlbrum n hs marke sasfes he followng sysem of equaons p = / + / ( + ) ( p / a ) c for all C ( p j a j ) () / j C \{} hs sysem of equaons wll be he cener of aenon n wha s o follow. Equlbrum ow ha I have derved he condons for a Pure Sraegy ash equlbrum n equaon (), he queson ha remans s wheher here exss a se of prces he sasfes hs equaon. In hs secon I wll no only show ha hs s he case, bu also prove he unqueness of hs equlbrum prce schedule. I wll hen proceed by dervng some of s comparave sacs ha are relevan for he prce ndex measuremen resuls ha I wll prove laer on. Frs and foremos hough, s mporan o realze ha he Pure Sraegy ash equlbrum n prces ha I consder acually exss and s unque. hs s wha I prove n he followng proposon. Proposon : Exsence and unqueness of equlbrum For any >0 and sequences a { } = a C and { } = c C unque pure sraegy ash equlbrum n prces. c where a,c >0 for all C here exss a Proof: (exsence) Exsence of he pure sraegy ash equlbrum follows from he applcaon of Brouwer s fxed pon heorem. In order o see how Brouwer s fxed pon heorem apples here, s mos convenen o defne c =c /a. Rewre he sysem of equaons ha defnes he equlbrum,.e. (), n he form p ca whch mples ha p c = + / = + / ( c j ) ( p j / c ja j ) j C / / / / ( c j ) ( p j / c ja j ) ( c ) ( p / ca ) j C / ( c j ) ( p j / c j ) / / / / / ( c j ) ( p j / c j ) ( c ) ( p / c ) j C j C / for all C (2) / for all C (3) Le be he number of elemens of C,.e. he number of compeng models n he marke a me. 0Defne 2

13 / v ( p / c ) and he space V { v } = { 0 v } = C R (4) hen (3) defnes a connuous mappng from V o V and hus, accordng o Brouwer s fxed pon heorem mus have a fxed pon. Hence, here mus exs an equlbrum. (unqueness) Defne ( c ) / = z, / p w = z, and W = w (5) c C hen (3) can we rewren as / W w = z + for all C (6) W w Gven W, for all C, here s one unque w R + ha solves (6). hs follows from a sraghforward applcaon of he nermedae value heorem o (6). Defne he funcon f: [0,W ] R + as / W f ( w ) = w z + (7) W w hen f(w ) s connuous and srcly ncreasng. Furhermore, f(0)=-z [+] -/ and f(w )=W. Hence, he nermedae value heorem mples ha here mus be a unque w, [0,W ] for whch f(w )=0. Suppose he equlbrum s no unque, hen here exs W and W such ha W >W =(+δ)w, where δ>0, such ha W = and W = w (8) w C and W and w for all C sasfy (6), whch s also rue for W and w for all C. oe ha he reason ha W and W can no be he same s because I have shown above ha he same W wll lead o he same bes response by he supplers of all models and hus o he same equlbrum. Wha I wll show n he followng s ha f (6) holds for W and w for all C, hen for all C mus be he case ha he w ha sasfes (6) gven W has o sasfy w <(+δ)w. However, hs would mply ha W <(+δ)w =W whch s a conradcon. In order o see hs, suppose ha w (+δ)w. In ha case, equaon (6) mples ha w = z < W + W w / = z ( + δ ) W z + ( + δ ) W ( + δ ) ( + δ ) w w + w / C ( + δ ) ( + δ ) W = z whch s a conradcon. Hence, here can only be one equlbrum. γ W w / W + W w / = w (9) 3

14 4 he benchmark case, and as urns ou he only one n whch sandard prce ndex mehods do no generae a bas, s he case n whch each suppler faces he same un producon cos per qualy un. As I show n he proposon below, he prce per qualy un s he same for all models n he marke n ha case. Proposon 2: Symmerc equlbrum he marke has a symmerc equlbrum n whch he prce per qualy un s consan across models,.e. p =p a for all C, f and only f he producer of each model faces he same margnal un producon cos per qualy un,.e. c =c a. Proof: ( ) If c /a =c and does no depend on, hen () reduces o ( ) ( ) {} \ / / C j j j c a p a p a p + + = for all C (20) When we choose p /a =p for all C and subsue n he sysem of equaons (20) we oban ha for all C {} \ / / C j c c p p p + = + + = (2) whch does no depend on. Hence, f c /a =c, hen p =(+ /( -))c s he symmerc pure sraegy ash equlbrum n whch all supplers charge he same prce per qualy un and all have an equal marke share. ( ) If here s a symmerc equlbrum, hen for all C + = = a c p a p (22) whch mples ha c p a c = + = (23) and does no depend on C. γ In he prevous secon I argued ha he bas ha I llusraed graphcally was he resul of he prce per qualy un no beng consan across models sold n he marke. In fac, n he example, he prce per qualy

15 un was hgher for beer models. In he symmerc equlbrum derved above he prce per qualy un s consan and s hus unlkely ha hs equlbrum wll yeld a bas of he sor descrbed before. However, f he margnal producon cos per qualy un s no he same across models sold n he marke, hen neher s he prce charged per qualy un. In ha case he marke equlbrum wll be asymmerc n he sense he models wll have dfferen marke shares. As I show n he followng proposon, he supplers ha produce he models wh he hgher margnal producon cos per qualy un wll charge a hgher prce per qualy un and wll have a lower marke share. Proposon 3: Properes of asymmerc equlbrum In he asymmerc equlbrum, producers wh hgher margnal producon coss per effcency uns,.e. c /a, () charge a hgher prce per effcency un, p /a, and () a lower markup, p /c. Proof: () Equaon (20) mples ha, when we defne hen for all C - mus be n equlbrum ha p a W = ( p / a ) j C /, (24) ( ) = W c + / W p / a a (25) Applyng he mplc funcon heorem o he above equaon yelds ( p / a ) ( c / a ) > 0 (26) Furhermore, because (p /a ) s hgher, equaon (5) mples ha he marke share of he model mus be lower. () In order o prove hs par, s eases o reconsder (3), whch reads p c / = + / ( c j ) ( p j / c j ) / / / / ( c j ) ( p j / c j ) ( c ) ( p / c ) j C j C / / (27) Agan, redefnng / ~ v ( p / c ) and defnng V = ( c ) = C / v (28) equaon (27) bols down o 5

16 / ~ V v = + ~ / for all C V ( c ) (29) v I s sraghforward o show ha he v ha solves hs equaon s ncreasng n c. Snce he markup, p /c, s decreasng n v, hs mples ha he equlbrum markup s decreasng n c, whch s wha s clamed. γ he above resul s mporan because suggess ha any asymmerc equlbrum exhbs prces per qualy un ha are unequal across he models sold n he marke and hus has he poenal of generang he bas descrbed n he prevous secon. 5. Prce ndex bas n he heorecal model ow ha I have developed he heorecal model of hs marke, s me o consder wha convenonal prce ndex mehods would measure n hs marke. In order o llusrae he qualy bas s essenal o consder dynamc unverses of goods such ha C and C (30) 0 C 0 C In prncple, here are many ways n whch he se of models raded n he marke can change and each of hese changes mgh have a dfferen effec n he heorecal example consdered here. Because s smply mpossble o consder all of hese dfferen cases, I wll lm myself o one specfc example. In he frs subsecon, I wll descrbe he parameerzaon of hs example n deal. hen, n he second subsecon, I wll consder wha happens when sandard prce ndex mehods are appled n hs example. Parameerzaon of example he example ha I wll consder s one where he model a he boom of he lne n perod =0 becomes obsolee n perod = and n whch n perod = a new op of he lne model s nroduced. he boom of he lne model a =0 s he lowes-qualy model,.e. he one wh he smalles a among all C 0. he op of he lne model nroduced n perod = s such ha s qualy exceeds ha of all models raded n perod =0. Consequenly, n boh perods he same number of models s sold. I wll denoe hs number by = 0 =. I wll ndex he models as =,,+, where model s he boom of he lne model ha becomes obsolee a me = and model + s he new op of he lne model nroduced a =. hs ndexaon mples ha C 0={,,} and C ={2,,+}. + wo hngs are sll o be defned. he frs s he parameerzaon of he qualy levels { a }. I wll = assume ha qualy s ncreasng n such ha ( + g) where g>0 represens he qualy growh rae across models. a = (3) 6

17 + = he second s he parameerzaon of he un producon coss { } choose for hese un producon coss s +γ c. he parameerzaon ha I wll c = c a (32) hs parameerzaon s such ha f γ=0 hen he producon coss per qualy un are dencal across models and he equlbrum s symmerc. If γ<0 hen he producon coss per qualy un are lower for beer models and her supplers wll charge a lower prce per qualy un and a hgher markup n equlbrum, as shown n proposon 3. Smlarly, f γ>0 hen producon coss per qualy un are hgher for beer models and, as n he graphcal example, he prce per qualy un s hgher for beer models. Hence, γ represens he seepness of he cross-model producon coss per qualy un schedule. Because of proposon 3, hs mples ha γ also represens he seepness of he cross-seconal prce per qualy un schedule. I wll parameerze he change of c / over me as follows. Le a = hen I wll assume ha c c~ a = where ( ) 0 γ ~ + π a C c~ = c (33) he reason ha I parameerze c lke hs s because, n equlbrum, he srucural parameer π has a specfc nerpreaon. hs s proven n he followng proposon. Proposon 4: Inerpreaon of srucural parameer π In equlbrum, he srucural parameer π equals he heorecal nflaon rae,.e. π=π. Proof: he bass of hs proof s he equlbrum equaon p = / + / ( + ) ( p / a ) c for all C ( p j a j ) (34) / j C \{} when we defne p ~ p = ac~ (35) hen (34) can be rewren as γ ( ) ( ~ / ~ p ) ( ) a p = + + for all C ~ / (36) p j a j C \{} wha I wll show s ha 7

18 a (37) / / a = a+ a+ If hs s he case, hen mus be ha ~ p ~ = p( + )( + ). We know ha Gven hese ses, (3) mples ha = {, } and = { 2,, } C, 0 K C K (38) + ( ) = ( + ) ( ) ( ) g 0 =, whle a ( g) 2 ( g) ( g) = + = = + + = = ( + g) a 0 a + (39) Snce a + =(+g)a, hs mples ha herefore, ~ p ~ = p( + )( + ). hs mples ha ( + g) a /( + g) a a a a + / (40) / a+ = = P 0 / / 0 = ( p a ) = c~ ~ 0 0 p (4) = = and Hence, + + / / c P ( p a ) c~ ~ / p c~ ~ = = = p( + ) = P0 2 2 c~ = = = 0 ~ ~ P P0 c c0 π = = ~ = π P c 0 0 ~ (42) (43) hus, π represens he heorecal nflaon rae ha s supposed o be approxmaed by he emprcal prce ndex mehods. γ Gven hs parameerzaon, he queson s how esmaed nflaon on he bass of he varous prce ndex mehods depends on he underlyng srucural parameers,, g, π, γ, and and how compares o he acual level of nflaon, π. hs queson s addressed n he nex subsecon. Qualy bas Jus lke n he graphcal example of secon 3, I wll frs address he bas nduced by mached model ndexes and hen consder hedonc prce ndexes n hs heorecal model. For he mached model prce ndexes I wll solely consder he, mos frequenly used, IP-IQ mehod. he followng proposon saes he properes of he IP-IQ lnked mached model ndexes n hs example. Proposon 5: Mached model ndex properes An IP-IQ lnked mached model ndex yelds an esmae of nflaon, π M, ha has he followng properes: 8

19 () () π M =π f γ=0. π M >π f γ<0. () π M <π f γ>0. hs resul does no depend on whch of he prce ndex formulas (excep he Feensra (994) ndex whch s exac) s appled. Proof: (): oe ha he nflaon rae of good beween =0 and = s gven by p ~ ~ ~ ~ p0 c p c0 p0 π = = p ~ ~ 0 c ~ 0 p0 c~ ~ c0 c = [ ~ ~ ~ + ~ p p0] (44) c0 c0 = π + ( + π )( ~ p ~ p0 ) Hence, wha wll be essenal n he res of hs proof s he propery of ~ p ~ p 0. I urns ou ha p~ s ncreasng n ( a ) γ / a. In order o see why, s useful o rewre (36) as γ ~ ~ P ( ) a p = + ~ P ~ for all C / (45) p a where ~ P = ~ p C / (46) Applyng he mplc funcon heorem o he above wo equaons yelds n a sraghforward manner ha ~ p > 0 (47) γ a / a ( ) herefore a / a. Furhermore, noe ha f γ=0, hen he equlbrum s symmerc and p~ s equal for all C. Consequenly, f γ>0 hen ( a ) γ / a s ncreasng n and models of hgher qualy have a hgher p~. hs mples ha f γ>0 hen p~ s srcly ncreasng n ( ) γ ~ ~ ~ p ( + ) 0 > p0 = p( + ) (48) where he second equaly follows from he proof of proposon 4. Hence, f γ>0 hen for all C, 9

20 ( + π )( ~ p ~ p ) π π < (49) = π + 0 Because hs nequaly holds for all models sold n he marke n perods =0 and =, mached model prce ndexes calculaed usng he Laspeyres, Paasche, Geomerc mean, Fscher, ornqvs, and Sao formula wll all underesmae nflaon. he reason s ha all hese prce ndex formula have he propery ha measured nflaon s n he range of nflaon raes of he ndvdual models. Snce he acual nflaon rae s above he maxmum nflaon rae measured for he models mus be ha he acual nflaon rae s undersaed by he mached model ndexes. A reverse bu smlar argumen yelds ha he mached model ndexes oversae nflaon whenever γ<0. γ hs proposon s he formal mahemacal proof of he nformal argumen ha I saed wh respec o mached model prce ndexes for he graphcal example n secon 3. ha s, he sgn and magnude of he qualy bas n mached model prce ndexes does no depend on he sgn and magnude of he overall nflaon rae. Insead, depends on he cross-seconal behavor of prces per qualy un for he models sold n he marke. ha he bas does no depend on he sgn and magnude of he overall nflaon rae follows drecly from he fac ha he resul n proposon 5 does no depend on he srucural parameer π. he dependence of he bas on he seepness of he cross-seconal schedule of prces per qualy un across models s mpled by he bas n he mached model ndexes only dependng on he parameer γ. ha s, f γ>0 hen, accordng o proposon 3, he prce per qualy un s ncreasng n he level of qualy emboded n he model. hs s he case depced n he graphcal example of secon 3 and s he case ha yelds a downward bas n he measured nflaon rae. If γ<0 hen he reverse s rue. So, how do hedonc prce mehods behave n he heorecal model here? hs queson can only be answered condonal on he behavor of he mpued prce levels. I do so n he nex proposon. Proposon 6: Hedonc prce ndex properes A hedonc prce ndex yelds an esmae of nflaon, π H, ha has he followng properes: () π H =π f γ=0. () π H >π f γ<0, f he mpued prces sasfy he propery of he equlbrum prce schedule ha prces per qualy un are decreasng n he qualy emboded n he model. () π H <π f γ>0, f he mpued prces sasfy he propery of he equlbrum prce schedule ha prces per qualy un are ncreasng n he qualy emboded n he model. Jus lke n proposon 5, hs resul does no depend on whch of he prce ndex formulas (excep he Feensra (994) ndex whch s exac) s appled. Proof: (): If γ=0 hen he equlbrum prce schedule sasfes 20

21 p = p for all C a and for =0, (50) If he mpued prces, p ˆ +, 0 and ˆp,,from he hedonc regresson model also sasfy hs propery such ha pˆ +,0 a + = p 0 as well as hen we fnd ha he observed and mpued nflaon raes sasfy p ˆ, a = p (5) π p = p, +, p p,0 pˆ pˆ +,0 pˆ, p p,0,0 +,0,0 p p0 = = π p0 p p0 = = π p0 p p0 = = π p 0 for = 2, K, for = + for = (52) Consequenly, no maer wha ype of weghed average one akes of he observed and mpued nflaon raes across models o calculae π H, hs average wll always equal π. (): If γ<0 hen he equlbrum prce schedule sasfes p p a < for all,- C and for =0, (53) a If he mpued prces n he hedonc regresson model also sasfy hs propery, such ha pˆ +,0 p, 0 pˆ, p2, < and > (54) a a a a + hen, n erms of he noaon of proposon 5, he observed and mpued prces obey ~ p ~ = ~,0 < p,0 p, for =2,,, as well as +,0 < p,0 = p +, p ˆ~ ~ ~ and p ˆ~, > ~ p ~ p 2, =, 0 (55) hs means ha he observed and mpued nflaon raes sasfy ( + π )( ~ p ~ p ) π +,,0 > π = + ( + )( ~ ˆ~ π π π p +, p +,0 ) > π + ( + )( pˆ~ ~ π π, p,0 ) > π for = 2, K, for = + for = (56) Hence, no maer wha weghed average one akes of hese nflaon raes across models o calculae he hedonc nflaon rae π H, wll always yeld π H >π. 2

22 (): hs follows n he same way as he proof of par (). he only hng ha s dfferen s ha n hs case he equlbrum prce schedule s such ha he prces per qualy un are ncreasng n he qualy levels of he models, whch yelds a reversal of he nequaly sgns. γ he proof of he proposon above gves some neresng nsghs. Frs of all, he hedonc prce ndexes only yeld an unbased esmae of nflaon whenever he equlbrum s such ha he prce per qualy un s consan across he models raded n he marke. However, f he prce per qualy un s consan across models, hen mached model ndexes wll do jus fne. In fac, f he prce per qualy un s consan across he models sold n he marke, hen one can smply measure overall nflaon by consderng he percenage prce change of a sngle model. ha s, when he prce per qualy un s consan across he models sold n he marke qualy bas s no an ssue. hs self s an mporan observaon. Bls and Klenow (200) for example use mcrodaa from he Consumer Expendure Survey o esmae he qualy bas n he CPI for several durable consumpon goods. hey do so by esmang a srucural model of durable goods consumpon. In order o quanfy qualy growh n hs model, however, hey assume ha ndependen of each household s expendures on a parcular durable consumpon good, he prce pad per qualy un s consan for all households. Hence, no maer wha model he households are buyng, hey are assumed o pay he same prce per qualy un. hs means ha Bls and Klenow (200) mplcly assume ha he prce per qualy un s consan across models. However, f hs denfyng assumpon would be rue n he daa hen he BLS would have had no problem quanfyng qualy growh n he frs place. If he prce per qualy un s consan, hen relave prces represen relave qualy dfferences. In ha case he coeffcens n he hedonc regresson model wll represen he margnal qualy coeffcens of he qualy ndcaors. Feensra (995) shows ha when hese coeffcens represen hese margnal values, hedonc prce ndexes wll work properly. In fac, for ceran classes of preferences Feensra (995) derves exac hedonc ndexes. However, when he consders he exsence of markups he also observes ha when hs s no he case hen he esmaed hedonc regresson coeffcens mgh over- or underesmae he qualy dfference beween he models. hs s he case when γ>0 and γ<0. In hose cases hedonc regresson coeffcens do no only reflec he margnal qualy dfferences beween he models bu also reflec he slope of he prce per qualy un schedule. In order o llusrae hs pon n pracce, I presen he resuls of a numercal smulaon of he heorecal model n he nex secon. 6. A numercal smulaon So far, I have presened he properes of he prce ndexes n my heorecal model n he form of several formal proposons. hese proposons proved he exsence of he qualy bas ha can be boh upward as well as downward and exss for boh mached model as well as hedonc prce ndces. However, s worhwhle o see how bg he bas s n he model when we acually pu n some numbers. hs s wha I wll do n he numercal smulaon n hs secon. 22

23 he numbers o be pu n are he srucural parameers. hese are whch s he number of models sold n each perod, g he qualy growh rae across models, he preference parameer ha deermnes he elascy of subsuon, γ he slope of he average cos curve per qualy un, and π he heorecal nflaon level. Proposons 5 and 6 clam ha her resuls do no depend on he prce ndex formula appled, excep he exac mached model ndex for CES-preferences by Feensra (994). For ha reason, I wll apply all he mos commonly used prce ndex formulas. hese formulas are lsed n able. he able conans he names and defnon of he formulas as well as a bref descrpon where some of hem are appled n pracce. Fnally, n order o smulae he hedonc prce ndexes, I have o choose a parcular model specfcaon for he hedonc regressons. I wll assume ha he researcher observes a for each model bu does no realze ha s he acual qualy level of he model and uses as a qualy ndcaor n a hedonc regresson. he hedonc regresson model ha I apply s of he followng log-log form ln p = β a + ε (57) β ln a + β 2 ln I allow he coeffcens n he regresson o be changng over me 4. hs means ha I wll perform separae cross-seconal hedonc regressons for =0 and =. I wll presen my smulaon n wo pars. In he frs par, I wll presen a benchmark example ha urns ou o yeld resuls ha are smlar o he graphcal example ha I gave n secon 3. I dscuss hs example and hese smlares n deal. hen, n he second par, I wll presen he resuls for a se of oher examples ha each devae from he benchmark n he dfference n one parcular srucural parameer. Benchmark case he benchmark case ha I wll consder s ha of a marke wh en models,.e. =0. he qualy rankng of he models s such ha each model s 5% beer han he nex bes one,.e. g=0.05. he elascy of subsuon beween he varous models s consan and assumed o equal wo, such ha =. On he cos sde, average producon coss per qualy un are ncreasng n qualy 5, such ha s elascy wh respec o qualy s 0.. hs mples ha γ=0.. Fnally, he benchmark case s such ha overall nflaon s zero,.e. π =π=0. Hence, all nflaon or deflaon ha s measured by he prce ndexes s spurous. he op panel of Fgure 4 depcs he equlbrum prce schedules for =0 and =. Wha a researcher would observe n hs marke s ha he model wh he lowes prce a =0 drops ou of he marke, whle he prces of he oher models drop by abou half a percen. he model ha drops ou of he marke s replaced by a model wh a prce ha s hgher han ha of he oher models a =. Because of hs hgher prce of he new model he average prce pad per un sold n he marke ncreases beween =0 and =. However, he 4 In praccal applcaons of hedonc prce ndexes he regresson coeffcens urn ou o flucuae a lo over me. hs has been a opc of dscusson n he leraure for a whle. See Hulen (2002) for revew of hs dscusson. 5 One reason ha producon coss per qualy un mgh be ncreasng n he number of qualy uns s when he bes models are he newes models and learnng by dong reduces producon coss over me a a hgher rae han qualy per model grows. ha learnng by dong mgh be a sgnfcan source of prce declnes has been argued for semconducors by Irwn and Klenow (994). 23