Estimating the Benefits and Costs of New and Disappearing Products December 18, 2017

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1 1 Esimaing he Benefis and Coss of New and Disappearing Producs December 18, 2017 Erwin Diewer and Rober Feensra, 1 Discussion Paper 17-10, Vancouver School of Economics, Universiy of Briish Columbia, Vancouver, B.C., Canada, V6T 1L4. Absrac A major challenge facing saisical agencies is he problem of adjusing price and quaniy indexes for changes in he availabiliy of commodiies. This problem arises in he scanner daa conex as producs in a commodiy sraum appear and disappear in reail oules. Hicks suggesed a reservaion price mehodology for dealing wih his problem in he conex of he economic approach o index number heory. Feensra and Hausman suggesed specific mehods for implemening he Hicksian approach. The presen paper evaluaes hese approaches and suggess some alernaive approaches o he esimaion of reservaion prices. The various approaches are implemened using some scanner daa on frozen juice producs ha are available online. Keywords Hicksian reservaion prices, virual prices, Laspeyres, Paasche, Fisher, Törnqvis and Sao-Varia price indexes, new goods, welfare measuremen, Consan Elasiciy of Subsiuion (CES) preferences, Konüs, Byushgens and Fisher (KBF) preferences, dualiy heory, consumer demand sysems, flexible funcional forms. JEL Classificaion Numbers C33, C43, C81, D11, D60, E31. 1 W. Erwin Diewer: Vancouver School of Economics, Universiy of Briish Columbia, Vancouver B.C., Canada, V6T 1Z1 and he School of Economics, UNSW Sydney, NSW 2052, Ausralia (erwin.diewer@ubc.ca); Rober Feensra, Universiy of California a Davis, Davis, Davis, CA, USA, ( rcfeensra@uc.davis.edu). The auhors hank Jan de Haan, Kevin Fox, Jerry Hausman, Alan Woodland and he paricipans a he EMG Workshop, December 1, 2017 for helpful commens. The auhors graefully acknowledge he financial suppor of a Trans Alanic Plaform Digging ino Daa gran. The projec ile is: Online Prices for Compuing Sandards of Living Across Counries (OPSLAC).

2 2 1. Inroducion One of he more pressing problems facing saisical agencies and economic analyss is he new goods (and services) problem; i.e., how should he inroducion of new producs and he disappearance of (possibly) obsolee producs be reaed in he conex of forming a consumer price index? Hicks (1940) suggesed a general approach o his measuremen problem in he conex of he economic approach o index number heory. His approach was o apply normal index number heory bu esimae hypoheical prices ha would induce uiliy maximizing purchasers of a relaed group of producs o demand 0 unis of unavailable producs. 2 Wih hese virual (or reservaion or impued) prices 3 in hand, one can jus apply normal index number heory using he augmened price daa and he observed quaniy daa. The pracical problem facing saisical agencies is: how exacly are hese virual prices o be esimaed? Economiss have been worrying abou he new goods problem a leas since he early conribuions of Lehr (1885; 45-46) and Marshall (1887; ), who independenly inroduced he concep of chained index numbers in order o deal wih his problem. 4 These auhors suggesed ha he bes way o deal wih he problem was o use he price and quaniy daa for adjacen periods and use a suiable index number formula on he se of producs ha were presen in boh periods. Keynes (1930; ) endorsed he idea of resricing index number comparisons o he se of producs ha were presen in boh periods being compared bu he preferred o use his maximum overlap mehod 5 in he conex of fixed base indexes. He rejeced he idea of using chained indexes because he fel ha chained indexes would suffer from a chain drif problem. 6 Indeed, we will find ha he problem of chain drif is a serious one when calculaing price indexes using scanner daa on he sales of a reail oule. Following up on he conribuion of Hicks, many auhors developed bounds or rough approximaions o he bias ha migh resul from omiing he conribuion of new goods in he consumer price index conex. Thus Rohbarh (1941) aemped o find some 2 The same kind of device can be used in anoher difficul case, ha in which new sors of goods are inroduced in he inerval beween he wo siuaions we are comparing. If cerain goods are available in he II siuaion which were no available in he I siuaion, he p 1 s corresponding o hese goods become indeerminae. The p 2 s and q 2 s are given by he daa and he q 1 s are zero. Neverheless, alhough he p 1 s canno be deermined from he daa, since he goods are no sold in he I siuaion, i is apparen from he preceding argumen wha p 1 s ough o be inroduced in order o make he index-number ess hold. They are hose prices which, in he I siuaion, would jus make he demands for hese commodiies (from he whole communiy) equal o zero. J.R. Hicks (1940; 114). Hofsen (1952; 95-97) exended Hicks mehodology o cover he case of disappearing goods as well. 3 Rohbarh inroduced he erm virual prices o describe hese hypoheical prices in he raioning conex: I shall call he price sysem which makes he quaniies acually consumed under raioning an opimum he virual price sysem.. E. Rohbarh (1941; 100). 4 See Diewer (1993a; 52-63) for addiional maerial on he early hisory of he new goods problem. 5 Keynes (1930; 94) called his he highes common facor mehod. 6 Keynes noed ha chained index numbers failed Walsh s (1901; 389) muliperiod ideniy es which is he following es: P(p 1.p 2,q 1,q 2 )P(p 2.p 3,q 2,q 3 )P(p 3.p 1,q 3,q 1 ) = 1 where P(p 1.p 2,q 1,q 2 ) is he bilaeral index number formula which is being used. The divergence of he produc of he 3 indexes from 1 serves as a measure of he amoun of chain drif.

3 3 bounds for he bias while Hofsen (1952; 47-50) discussed a variey of approximae mehods o adjus for qualiy change in producs, which is essenially he same problem as adjusing an index for he conribuion of a new produc. Diewer (1980; ) developed some bounds for he bias in a maximum overlap Fisher (1922) index relaive o he bias ha would resul from using he Fisher formula where 0 prices and quaniies were used in he Fisher formula for he base period when a new produc was no available. 7 Addiional bias formulae were developed by Diewer (1987; 779) (1998; 51-54) and Hausman (2003; 26-28). These approximaions relied on informaion (or guesses) abou expendiure shares, elasiciies or raios of virual prices o acual prices. We will examine he Hausman approximae formula in more deail in secion 11 below. We urn now o mehods ha rely on some form of economeric esimaion in order o form esimaes of he welfare cos (or changes in he rue cos of living index) of changes in produc availabiliy. The wo main conribuors in his area are Feensra (1994) and Hausman (1996). 8 Economeric mehods for adjusing price and quaniy indexes will be he main focus of his sudy. We will apply various economeric mehods in order o adjus a consumer price index for changes in he availabiliy of producs. We will also obain economeric esimaes for he virual prices for unavailable producs for each period in our sample period. We will es ou our suggesed mehods on a scanner daa se ha is available on line. 9 The daa se is lised in Appendix A so ha researchers can use his daa se o es ou possible improvemens o our suggesed mehodology. Feensra s (1994) mehodology ress on he properies of he CES uni cos funcion. His mehodology is explained in secion 2. In secion 3, we adap his mehodology o he case of a CES uiliy funcion. Secion 4 inroduces our scanner daa se which we use o es ou Feensra s mehodology. Secion 5 esimaes a CES uni cos funcion using our daa se while secion 6 esimaes a CES direc uiliy funcion. Boh sysems of esimaing equaions use he sales shares of he 19 producs in our sample as he dependen variables in a sysems regression approach. If eiher he CES uni cos funcion model or he CES homogeneous uiliy model were o fi he sample daa perfecly, we would obain exacly he same resuls. However, neiher model fis he daa exacly. We find ha he CES uiliy funcion model fis he daa much beer han he CES uni cos funcion model. There are wo problems wih he CES uni cos funcion mehodology: The CES funcional form is no fully flexible 10 and The reservaion price ha induces a poenial purchaser o no purchase a produc is equal o plus infiniy, which seems high. Thus he CES mehodology may oversae he benefis of increases in produc availabiliy. 7 Diewer (1980; 501) concluded ha boh Fisher price indexes would probably have an upward bias bu he index which used zeros would definiely have a larger bias han he maximum overlap Fisher index. The similar ype of argumen appears in Diewer (1987; 779). 8 See also Hausman (1999) (2003) and Hausman and Leonard (2002) 9 The daa are described in secion 4 below. 10 See Diewer (1974) (1976) for he definiion of a flexible funcional form.

4 4 Thus in secion 7, we replace he CES uiliy funcion wih a flexible funcional form which was iniially due o Konüs and Byushgens (1926; 171). This uiliy funcion is u = f(q) (q T Aq) 1/2 where A is a symmeric marix of parameers and q T is he row vecor ranspose of he column vecor of quaniies purchased, q. Konüs and Byushgens showed ha if purchasers maximized his uiliy funcion in wo periods where hey faced he price vecors p 1 and p 2 and he uiliy maximizing vecors were q 1 and q 2, hen he uiliy raio, f(q 2 )/f(q 1 ), is equal o he Fisher (1922) quaniy index, QF(p 1,p 2,q 1,q 2 ) [p 1T q 2 p 2T q 2 / p 1T q 1 p 2T q 1 ]. 11 Thus we will call his funcional form for f he KBF funcional form. The advanage in working wih his flexible funcional form is ha when some componen of he q vecor is equal o 0, he resuling uiliy funcion is sill well defined and he corresponding reservaion price can be calculaed by parially differeniaing he esimaed uiliy funcion wih respec o he quaniy variable ha happens o equal 0 in he period under consideraion. In fac, Diewer (1980; ) suggesed exacly his mehodological approach o he esimaion of reservaion prices bu in he end, he suggesed ha i would be difficul o esimae all of he N(N+1)/2 unknown parameers in he A marix. In he presen paper, we solve his degrees of freedom problem by inroducing a semiflexible version of he flexible KBF funcional form. 12 This new mehodology is explained in secion 7. In secion 8, we aemp o esimae he KBF funcional form using he usual sysems approach o he esimaion of consumer demand funcions. However, he nonlineariy in our esimaing share equaions causes our nonlinear esimaing procedure o come o a premaure hal as we increase he rank of he A marix. Hence in secion 9, we drop he sysems approach o he esimaion of he unknown parameers in favour of he one big equaion approach. The laer approach has he advanage of being able o drop he observaions where a produc was missing. Alhough he implied fis in he produc share equaions were quie good using our one big equaion approach, when we moved from prediced shares generaed by our esimaes o prediced prices, we found ha prediced prices did no mach up well wih acual prices for he observaions where producs were presen. Thus in secion 10, we moved from shares as he dependen variables o using prices as he dependen variables. We coninued o esimae higher rank A marices using he one big equaion approach wih prices as he dependen variables unil we esimaed a rank 7 A marix wih 111 unknown parameers. We hen used our esimaed A marix in order o define virual or reservaion prices for he unavailable producs. We were also able o quanify he effecs of he changing availabiliy of producs and compare he resuls of he KBF esimaion wih he earlier CES benefi measures. We found ha he CES mehodology did indeed give much 11 Konüs and Byushgens (1926; ) also inroduced he KBF uni cos funcion, c(p) (p T Bp) 1/2 where B is a symmeric marix of parameers. They showed ha his uni cos funcion funcional form is exac for he Fisher price index. If A or B is of full rank, hen B = A 1. For a descripion of he conribuions of Konüs and Byushgens o index number heory and dualiy heory, see Diewer (1993a; 47-51). For a descripion of he regulariy condiions ha he marices A and B mus saisfy for he KBF f(q) or c(p) o be well behaved, see Diewer and Hill (2010). Diewer (1976) generalized he KB resuls o more general funcional forms for f and c. 12 Our new semiflexible funcional form has properies ha are similar o he semiflexible generalizaion of he Normalized Quadraic funcional form inroduced by Diewer and Wales (1987) (1988). In secion 7 below, we also show how he correc curvaure condiions can be imposed on our semiflexible KBF funcional form.

5 5 higher esimaes for he gains from increases in produc availabiliy as compared o our KBF mehodology. However, due o he fac ha our esimaed KBF preferences did no fi he daa exacly, we found ha occasionally our esimaed gain from having an addiional produc had he wrong sign. Thus in secion 11, we developed an alernaive mehodological approach based on our esimaed KBF uiliy funcion (which is well behaved by consrucion) ha was free from anomalous resuls. This uiliy funcion based approach is an alernaive o Hausman s (1996) expendiure or cos funcion approach o measuring he gains from increases in produc availabiliy. In secion 12, we compare Hausman s approximae approach o a varian of our approach where we use a second order approximaion o he esimaed uiliy funcion. To keep hings simple, we consider only he case of wo producs in his secion. We obain a raher surprising equivalence resul. Secion 13 concludes. Appendix B ries ou Feensra s double differencing mehod for esimaing he elasiciy of subsiuion bu we apply i o he direc uiliy esimaion of he CES funcional form raher han esimaing he dual CES uni cos funcion. We find ha his mehod for esimaing he elasiciy of subsiuion worked very well on our scanner daa se. 2. Feensra s CES Uni Cos Funcion Mehodology In his secion, we will explain Feensra s (1994) CES cos funcion mehodology ha he proposed o measure he benefis and coss o consumers due o he appearance of new producs and he disappearance of exising producs. The mehodology assumes ha purchasers of a group of N producs all have he same linearly homogeneous, concave and nondecreasing uiliy funcion f(q), where he nonnegaive vecor of purchased producs is q (q1,...,qn) 0N and u = f(q) 0 is he uiliy ha he vecor of purchases q generaes. Given ha purchasers face he posiive vecor of prices p (p1,...,pn) a an oule, he uni cos funcion c(p) ha is dual o he uiliy funcion f is defined as he minimum cos of aaining he uiliy level ha is equal o one: (1) c(p) min q{f(q) 1; q 0N}. If he uni cos funcion c(p) is known, hen using dualiy heory, i is possible o recover he underlying uiliy funcion f(q). 13 Feensra assumed ha he uni cos funcion has he following CES funcional form: 13 I can be shown ha for q >> 0 N, f(q) = 1/max p {c(p): n=1 N p nq n 1 ; p 0 N}; see Diewer (1974; ) (1993b; 129) on he dualiy beween linearly homogeneous aggregaor funcions f(q) and uni cos funcions c(p).

6 6 (2) c(p) 0 [ n=1 N npn 1 ] 1/(1 ) if 1; 0 n=1 N p if = 1 n n where he i and are nonnegaive parameers wih i=1 N i = 1. The uni cos funcion defined by (2) is a Consan Elasiciy of Subsiuion (CES) uiliy funcion which was inroduced ino he economics lieraure by Arrow, Chenery, Minhas and Solow (1961) 14. The parameer is he elasiciy of subsiuion; 15 when = 0, he uni cos funcion defined by (2) becomes linear in prices and hence corresponds o a fixed coefficiens aggregaor funcion which exhibis 0 subsiuabiliy beween all commodiies. When = 1, he corresponding aggregaor or uiliy funcion is a Cobb-Douglas funcion. When approaches +, he corresponding aggregaor funcion f approaches a linear aggregaor funcion which exhibis infinie subsiuabiliy beween each pair of inpus. The CES uni cos funcion defined by (2) is of course no a fully flexible funcional form (unless he number of commodiies N being aggregaed is 2) bu i is considerably more flexible han he zero subsiuabiliy aggregaor funcion (his is he special case of (2) where is se equal o zero) ha is exac for he Laspeyres and Paasche price indexes. In order o simplify he noaion, we se r 1. Under he assumpion of cos minimizing behavior on he par of purchasers of he N producs for periods = 1,...,T, Shephard s (1953; 11) Lemma ells us ha he observed period consumpion of commodiy i, qi, will be equal o u c(p )/ pi where c(p )/ pi is he firs order parial derivaive of he uni cos funcion wih respec o he ih commodiy price evaluaed a he period prices and u = f(q ) is he aggregae (unobservable) level of period uiliy. Denoe he share of produc i in oal sales of he N producs during period as si pi qi /p q for i = 1,...,N and = 1,...,T where p q n=1 N pn qn. Noe ha he assumpion of cos minimizing behavior during each period implies ha he following equaions will hold: (3) p q = u c(p ) ; = 1,...,T where c is he CES uni cos funcion defined by (2). Using he CES funcional form defined by (2) and assuming ha 1 (or r 0), 16 he following equaions are obained using Shephard s Lemma: (4) qi = u 0 [ n=1 N n (pn ) r ] (1/r) 1 i (pi ) r 1 ; i = 1,,N; =1,...,T 14 In he mahemaics lieraure, his aggregaor funcion or uiliy funcion is known as a mean of order r 1 ; see Hardy, Lilewood and Polyá (1934; 12-13). 15 Le c(p) be an arbirary uni cos funcion ha is wice coninuously differeniable. The Allen (1938; 504) Uzawa (1962) elasiciy of subsiuion nk(p) beween producs n and k is defined as c(p)c nk(p)/c n(p)c k(p) for n k where he firs and second order parial derivaives of c(p) are defined as c n(p) c(p)/ p n and c nk(p) 2 c(p)/ p n p k. For he CES uni cos funcion defined by (2), nk(p) = for all pairs of producs; i.e., he elasiciy of subsiuion beween all pairs of producs is a consan for he CES uni cos funcion. 16 When = 1, we have he case of Cobb-Douglas preferences. In he remainder of his paper, we will assume ha > 1 (or equivalenly, ha r < 0).

7 7 = u c(p ) i (pi ) r 1 / n=1 N n (pn ) r. Premuliply equaion i for period in (4) by pi /p q. Using (2) and (3), he resuling equaions can be rewrien as follows: (5) si = i (pi ) r / n=1 N n (pn ) r ; i = 1,,N; = 1,...,T. The NT share equaions defined by (5) can be used as esimaing equaions using a nonlinear regression approach. We will implemen his approach laer in he paper. Noe ha he posiive scale parameer 0 canno be idenified using equaions (5), which of course is normal: uiliy can only be esimaed up o an arbirary scaling facor. Henceforh, we will assume 0 = 1. The share equaions (5) are homogeneous of degree one in he parameers 1,..., N and hus he idenifying resricion on hese parameers, i=1 N i = 1, can be replaced wih an equivalen resricion such as N = 1. Suppose ha all N producs are available in all T periods in our sample and we have esimaed he unknown parameers which appear in equaions (5). Then he period CES price index (relaive o he level of prices for period 1), PCES, can be defined as he following raio of uni coss in period relaive o period 1: (6) PCES [ n=1 N n (pn ) r ] (1/r) / [ n=1 N n (pn 1 ) r ] (1/r) ; = 1,...,T. Suppose furher ha he observed price and quaniy daa vecors, p and q for = 1,...,T, saisfy equaions (3) where c(p) is defined by (2) and he quaniy daa vecors q saisfy he Shephard s Lemma equaions (4). Thus he observed price and quaniy daa are assumed o be consisen wih cos minimizing behavior on he par of purchasers where all purchasers have CES preferences ha are dual o he CES uni cos funcion defined by (2). Then Sao (1976) and Varia (1976) showed ha he sequence of CES price indexes defined by (6) could be numerically calculaed jus using he observed price and quaniy daa; i.e., i would no be necessary o esimae he unknown n and (or r) parameers in equaions (6). The logarihm of he period fixed base Sao-Varia Index PSV is defined by he following equaion: (7) lnpsv n=1 N wn ln(pn /pn 1 ) ; = 1,...,T. The weighs wn ha appear in equaions (7) are calculaed in wo sages. The firs sage se of weighs is defined as wn * (sn sn 1 )/(lnsn lnsn 1 ) for n = 1,...,N and = 1,...,T provided ha sn sn 1. If sn = sn 1, hen define wn * sn = sn 1. The second sage weighs are defined as wn wn * / i=1 N wi * for n = 1,...,N and = 1,...,T. Noe ha in order for lnpces o be well defined, we require ha sn > 0, sn 1 > 0, pn > 0 and pn 1 > 0 for all n = 1,...,N and = 1,...,T; i.e., all prices and quaniies mus be posiive for all producs and for all periods. Now we can explain Feensra s (1994) model where new commodiies can appear and old commodiies can disappear from period o period.

8 8 Feensra (1994) assumed CES preferences wih > 1 (or equivalenly, r < 0). He applied he reservaion price mehodology firs inroduced by Hicks (1940); i.e., Hicks assumed ha he consumer had preferences over all goods, bu for he goods which had no ye appeared, here was a reservaion price ha would be jus high enough ha consumers would no wan o purchase he good in he period under consideraion. 17 This assumpion works raher well wih CES preferences, because we do no have o esimae hese reservaion prices; hey will all be equal o + when > 1. Feensra allowed for new producs o appear and for exising producs o disappear from period o period. 18 Feensra assumed ha he se of commodiies ha are available in period is I() for = 1,...,T. The (impued) prices for he unavailable commodiies in each period are se equal o + and hus if r < 0, an infinie price pn raised o a negaive power generaes a 0; i.e., if produc n is unavailable in period, hen (pn ) r = ( ) r = 0 if r is negaive. The CES period rue price level under hese condiions when r < 0 urns ou o be he following CES uni cos funcion ha is defined over only producs ha are available during period : (8) c(p ) [ n=1 N n (pn ) r ] (1/r) = [ i I() i (pi 1 ) r ] 1/r. Using equaions (4) for his new model and muliplying he period demand qi by he corresponding price pi for he iems ha are acually available leads o he following equaions which describe he purchasers nonzero expendiures on produc i in period : (9) pi qi = u [ n I() n (pn ) r ] (1/r) 1 i (pi ) r ; = 1,...,T; i I() = u c(p ) i (pi ) r / n I() n (pn ) r. In each period, he sum of observed expendiures, n I() pn qn, equals he period uiliy level, u, imes he CES uni cos c(p ) defined by (8): (10) n I() pn qn = u c(p ) = u [ i I() i (pi 1 ) r ] 1/r ; = 1,...,T. Recall ha he ih sales share of produc i in period was defined as si pi qi / n I() pn qn for = 1,...,T and i I(). Using hese share definiions and equaions (10), we can rewrie equaions (9) in he following form: (11) si = i (pi ) r / n I() n (pn ) r ; = 1,...,T; i I() = i (pi ) r /c(p ) r 17 The same logic is applied o disappearing producs. 18 In many cases, a new produc is no a genuinely new produc; i is jus a produc ha was no in sock in he previous period. Similarly, in many cases, a disappearing produc is no necessarily a ruly disappearing produc; i is simple a produc ha was no in sock for he period under consideraion. Many reail chains roae producs, emporarily disconinuing some producs in favour of compeing producs in order o ake advanage of manufacurer discouned prices for seleced producs.

9 9 where he second se of equaions follows using definiions (8). Now we can work ou Feensra s (1994) model for measuring he benefis and coss of new and disappearing commodiies. Sar ou wih he period CES exac price level defined by (8) and define he CES fixed base price index for period, PCES, as he raio of he period CES price level o he corresponding period 1 price level: 19 (12) PCES c(p )/c(p 1 ) ; = 1,...,T = [ i I() i (pi ) r ] 1/r / [ i I(1) i (pi 1 ) r ] 1/r = [ Index 1] [Index 2] [Index 3] where he hree indexes in equaions (12) are defined as follows: (13) Index 1 [ i I() I(1) i (pi ) r ] 1/r / [ i I(1) I() i (pi 1 ) r ] 1/r ; (14) Index 2 [ i I() i (pi ) r ] 1/r / [ i I(1) I() i (pi ) r ] 1/r ; (15) Index 3 [ i I(1) I() i (pi 1 ) r ] 1/r / [ i I(1) i (pi 1 ) r ] 1/r. Noe ha Index 1 defines a CES price index over he se of commodiies ha are available in boh periods and 1. Denoe he CES cos funcion c * ha has he same n parameers as before bu is now defined over only producs ha are available in periods 1 and : (16) c * (p) [ i I() I(1) i (pi) r ] 1/r ; = 1,2,...,T. The period expendiure share equaions ha correspond o equaions (11) using he uni cos funcion defined by (16) are he following ones: (17) si * pi qi / n I() ) I(1) pn qn = 1,...,T; i I(1) I() = i (pi ) r / n I() ) I(1) n (pn ) r = i (pi ) r /c * (p ) r where he hird equaliy follows using definiions (16). Noe ha Index 1 is equal o c * (p )/c * (p 1 ) and he Sao-Varia formula (7) ( resriced o commodiies n ha are presen in periods 1 and ) can be used o calculae his index using he observed price and quaniy daa for he producs ha are available in boh periods 1 and. We urn now o he evaluaion of Indexes 2 and 3. I urns ou ha we will need an esimae for he elasiciy of subsiuion (or equivalenly of r) in order o find empirical expressions for hese indexes. I is convenien o define he following observable expendiure or sales raios: 19 In he algebra which follows, he prices and quaniies of period 1 can be replaced wih he prices and quaniies of any period. Feensra (1994) developed his algebra for c(p )/c(p 1 ).

10 10 (18) n I() pn qn / n I(1) I() pn qn ; = 1,,...,T. (19) n I(1) I() pn 1 qn 1 / n I(1) pn 1 qn 1 ; =1,,...,T. We assume ha here is a leas one produc ha is presen in periods 1 and for each. Le produc i be any one of hese common producs for a given. Then he share equaions (11) and (17) hold for his produc. These share equaions can be rearranged o give us he following wo equaions: (20) i(pi ) r = [ n I() n (pn ) r ] pi qi /[ n I() pn qn ] ; (21) i(pi ) r = [ n I(1) I() n (pn ) r ] pi qi /[ n I(1) I() pn qn ]. Equaing (20) o (21) leads o he following equaions: (22) n I() n (pn ) r / n I(1) I() n (pn ) r = n I() pn qn / n I(1) I() pn qn = where he las equaliy follows using definiion (18). Now ake he 1/r roo of boh sides of (22) and use definiion (14) in order o obain he following equaliy: (23) Index 2 = [ ] 1/r = [ i I() pi qi / i I(1) I() pi qi ] 1/r. 20 Again assume ha produc i is available in periods 1 and. Rearrange he share equaions (11) and (17) for = 1 and produc i and we obain he following wo equaions: (23) i(pi 1 ) r = [ n I(1) n (pn 1 ) r ] pi 1 qi 1 /[ n I(1) pn 1 qn 1 ] ; (24) i(pi 1 ) r = [ n I(1) I() n (pn 1 ) r ] pi 1 qi 1 /[ n I(1) I() pn 1 qn 1 ]. Equaing (23) o (24) leads o he following equaions: (25) n I(1) I() n (pn 1 ) r / n I(1) n (pn 1 ) r = n I(1) I() pn 1 qn 1 / n I(1) pn 1 qn 1 = where he las equaliy follows using definiion (19). Now ake he 1/r roo of boh sides of (25) and use definiion (15) in order o obain he following equaliy: If new producs become available in period ha were no available in period 1, hen > 1. Recall ha r = 1 and r < 0. Index 2 evaluaed a period prices equals ( ) 1/r = ( ) 1/(1 ) and hus is an increasing funcion of for 1 < < +. Wih > 1, he limi of ( ) 1/(1 ) as approaches 1 is 0 and he limi of ( ) 1/(1 ) as approaches + is 1. Thus he gains in uiliy from increased produc variey are huge if is slighly greaer han 1 and diminish o no gains a all as becomes very large. Suppose ha =1.05 and = 1.01, 1.1, 1.5, 2, 3, 5, 10 and 100. Then Index 2 will equal , 0.614, 0.907, 0.952, 0.976, 0.988, and respecively. Thus he gains from increased produc variey are very sensiive o he esimae for he elasiciy of subsiuion. The gains are giganic if is close o If some producs ha were available in period 1 become unavailable in period, hen < 1. Index 3 evaluaed a period 1 prices equals ( ) 1/r = ( ) 1/(1 ) and is an decreasing funcion of for 1 < < +. Wih < 1, he limi of ( ) 1/(1 ) as approaches 1 is + and he limi of ( ) 1/(1 ) as approaches + is 1. Thus he losses in uiliy from decreased produc variey are huge if is slighly greaer han 1 and

11 11 (26) Index 3 = [ ] 1/r = [ n I(1) I() pn 1 qn 1 / n I(1) pn 1 qn 1 ] 1/r. Thus if r is known or has been esimaed, hen Index 2 and Index 3 can readily be calculaed as simple raios of sums of observable expendiures raised o he power1/r. Noe ha [ i I() pi qi / i I(1) I() pi qi ] 1. If period has producs ha were no available in period 1, hen he sric inequaliy will hold and since 1/r < 0, i can be seen ha Index 2 will be less han uniy. Thus Index 2 is a measure of how much he rue cos of living index is reduced in period due o he inroducion of producs ha were no available in period 1. Similarly, [ i I(1) I() pi 1 qi 1 / i I(1) pi 1 qi 1 ] 1. If period 1 has producs ha are no available in period, hen he sric inequaliy will hold and since1/r < 0, i can be seen ha Index 3 will be greaer han uniy, Thus Index 3 is a measure of how much he rue cos of living index is increased in period due o he disappearance of producs ha were available in period 1 bu are no available in period. Turning briefly o he problems associaed wih esimaing r (and he n) when no all producs are available in all periods, i can be seen ha he iniial esimaing share equaions (5) are now replaced by he following equaions: (27) sn = n (pn ) r / k=1 N k (pk ) r ; = 1,...,T; n I(). In he nex secion, we obain an alernaive se of share equaions ha could be used in order o esimae he elasiciy of subsiuion. 3. The Primal Approach o he Esimaion of CES Preferences I urns ou ha esimaing he purchaser s uiliy funcion direcly (raher han esimaing he dual uni cos funcion) is advanageous when esimaes of reservaion prices for producs ha are no available are required. In he case of CES preferences, his advanage is no apparen since he CES reservaion prices are auomaically se equal o infiniy. Bu i urns ou ha here may be advanages in esimaing he CES uiliy funcion direcly because of economeric consideraions as we shall see laer. Thus in his secion, we will derive he purchaser demand funcions ha are consisen wih he maximizaion of a CES uiliy funcion. Using he same noaion ha was used in he beginning of he previous secion, we assume ha he purchaser uiliy funcion f(q) is defined as he following CES uiliy funcion: (28) f(q1,...,qn) [ n=1 N nqn s ] 1/s diminish o no gains a all as becomes very large. Suppose ha =0.95 and akes on he same values as in he previous foonoe. Then Index 3 will equal 168.9, 1.670, 1.108, 1.053, 1.026, 1.013, and respecively. Thus he losses are giganic if is close o 1 and negligible if is very large.

12 12 where he parameers n are posiive and sum o 1 and s is a parameer which saisfies he inequaliies 0 < s 1. Thus f(q) is a mean of order s. Assume ha all producs are available in a period and purchasers face he posiive prices p (p1,...,pn) >> 0N. The firs order necessary (and sufficien) condiions (provided ha s 1) ha can be used o solve he uni cos minimizaion problem defined by (1) are he following condiions: (29) pn = nqn s 1 ; n = 1,...,N; (30) 1 = [ n=1 N nqn s ] 1/s. Muliply boh sides of equaion n in (29) by qn and sum he resuling N equaions. This leads o he equaion n=1 N pnqn = n=1 N nqn s. Solve his equaion for and use his soluion o eliminae he in equaions (29). The resuling equaions (where equaion n is muliplied by qn) are he following ones: (31) pnqn/ i=1 N piqi = nqn s / i=1 N iqi s ; n = 1,...,N. The equaions (29) and (30) can be used o obain an explici soluion for q1,...,qn and as funcions of he price vecor p. 22 Use hese soluion funcions o form he uni cos funcion, c(p) equal o n=1 N pnqn(p). This funcion urns ou o be he following one: 23 (32) c(p) = [ n=1 N n 1/(1 s) pn s/(s 1) ] (s 1)/s. I can be seen ha c(p) is proporional o a mean of order r where r = s/(s 1). Thus if f(q) is he CES uiliy funcion defined by (28), hen he corresponding elasiciy of subsiuion is = 1 r = 1 [s/(s 1)] = 1/( s 1) = 1/(1 s). Noe ha our assumpion ha s saisfies 0 < s 1 implies ha saisfies 1 <. If purchasers maximize he CES uiliy funcion defined by (28) when hey face he posiive price vecor p, he uiliy maximizing q will saisfy he share equaions (31). If we evaluae equaions (31) using he period price and quaniy daa, we obain he following sysem of esimaing equaions, assuming ha all producs are available in all periods: (33) sn pn qn / i=1 N pi qi = n(qn ) s / i=1 N i(qi ) s ; = 1,...,T; n = 1,...,N. I can be seen ha he righ hand sides of equaions (33) are homogeneous of degree 0 in he parameers 1,..., N so a normalizaion of hese parameers is required for he idenificaion of he parameers. The normalizaion n=1 N n = 1 can be replaced by an equivalen normalizaion such as N = Under hese condiions, he firs order necessary condiions (29) and (30) for solving he uni cos minimizaion problem are also sufficien condiions. 23 Explici soluions for he q n(p) can be obained by using Shephard s Lemma; i.e., q n(p) = c(p)/ p n for n = 1,...,N where c(p) is defined by (32).

13 13 We now consider he case where no all producs are available in all periods. The parameer s is assumed o be greaer han 0 (and less han or equal o 1 so ha he resuling CES uiliy funcion is concave). If produc n is no available in period, we can se qn = 0 and (qn ) s = (0) s = 0 and hus produc n will drop ou of he uiliy funcion. Thus if we simply se quaniies equal o 0 when he corresponding producs are no available in a period, he overall CES uiliy funcion evaluaed a he period quaniy daa (wih he appropriae 0 values insered), f(q ), will be equal o [ n I() n (qn ) s ] 1/s, he uiliy funcion f which is defined over jus he producs ha are acually available during period ; i.e., he following equaions will be saisfied where we define uces as he period aggregae CES uiliy or quaniy (or volume) level: (34) uces = f(q ) [ n=1 N n (qn ) s ] 1/s = [ n I() n (qn ) s ] 1/s ; = 1,...,T where he las equaliy follows under he assumpion ha s > 0. Once he aggregae uiliy or quaniy levels uces have been defined by equaions (34), he corresponding CES fixed base quaniy index can be defined as follows: 24 (35) QCES uces /uces 1 ; = 1,...,T f(q )/f(q 1 ) ; = [ i I() i (qi ) s ] 1/s / [ i I(1) i (qi 1 ) s ] 1/s = [ Index 1 * ] [Index 2 * ] [Index 3 * ] where he above indexes are defined as follows: (36) Index 1 * [ i I() I(1) i (qi ) s ] 1/s / [ i I(1) I() i (qi 1 ) s ] 1/s ; (37) Index 2 * [ i I() i (qi ) s ] 1/s / [ i I(1) I() i (qi ) s ] 1/s ; (38) Index 3 * [ i I(1) I() i (qi 1 ) s ] 1/s / [ i I(1) i (qi 1 ) s ] 1/s. Noe ha Index 1 * defines a CES quaniy index over he se of commodiies ha are available in boh periods and 1. Denoe he CES uiliy funcion f * ha has he same n parameers as in definiion (28) bu is now defined over only producs ha are available in periods 1 and : (39) f * (q) [ i I() I(1) i (qi) s ] 1/s ; = 1,2,...,T. Uiliy maximizing behavior on he par of purchasers will imply ha he following counerpars o equaions (17) will hold: (40) si * pi qi / n I() ) I(1) pn qn = 1,...,T; i I(1) I() = i (qi ) s / n I() ) I(1) n (qn ) s = i (qi ) s /f * (q ) s 24 In he algebra which follows, he period 1 quaniy vecor can be replaced by he quaniy vecor of any period.

14 14 where he hird equaliy follows using definiions (39). Noe ha Index 1 * is equal o f * (q )/f * (q 1 ) and he modified Sao-Varia formula (7) (resriced o commodiies n ha are presen in periods 1 and and where quaniies and prices are inerchanged in he formula) can be used o calculae his index using he observed price and quaniy daa for he producs ha are available in boh periods 1 and. As usual, we assume ha here is a leas one produc ha is presen in periods 1 and for each. Le produc i be any one of hese common producs for a given. Then he ih share equaion in (33) and (40) for period can be rearranged o give us he following wo equaions: (41) i(qi ) s = [ n I() n (qn ) s ] pi qi /[ n I() pn qn ] ; (42) i(qi ) s = [ n I(1) I() n (qn ) s ] pi qi /[ n I(1) I() pn qn ]. Equaing (41) o (42) leads o he following equaions: (43) n I() n (qn ) s / n I(1) I() n (qn ) s = n I() pn qn / n I(1) I() pn qn = where he las equaliy follows using definiion (18). Now ake he 1/s roo of boh sides of (43) and use definiion (37) in order o obain he following equaliy: 25 (44) Index 2 * = [ ] 1/s = [ i I() pi qi / i I(1) I() pi qi ] 1/s. Again assume ha produc i is available in periods 1 and. Rearrange he share equaions (33) and (40) for = 1 and produc i and we obain he following wo equaions: (45) i(qi 1 ) s = [ n I(1) n (qn 1 ) s ] pi 1 qi 1 /[ n I(1) pn 1 qn 1 ] ; (46) i(qi 1 ) s = [ n I(1) I() n (qn 1 ) s ] pi 1 qi 1 /[ n I(1) I() pn 1 qn 1 ]. Equaing (45) o (46) leads o he following equaions: (47) n I(1) I() n (qn 1 ) s / n I(1) n (qn 1 ) s = n I(1) I() pn 1 qn 1 / n I() pn 1 qn 1 = where he las equaliy follows using definiion (19). Now ake he 1/s roo of boh sides of (47) and use definiion (38) in order o obain he following equaliy: If new producs become available in period ha were no available in period 1, hen > 1. Recall ha he elasiciy of subsiuion in erms of s is equal o 1/(1 s) where s saisfies 0 < s 1. Thus as s increases, also increases. Wih > 1, he limi of ( ) 1/s as s approaches 0 is + and he limi of ( ) 1/s as s approaches 1 is > 1. Thus Index 2 * is a decreasing funcion of s for 0 < s 1. Suppose ha =1.05 and s = 0.005, 0.01, 0.1, 0.3, 0.5, 0.9, 0.99 and Then = 1/(1 s) is equal o 2, 10, 100 and The corresponding Index 2 * values are , 131.5, 1.629, 1.177, 1.103, 1.056, and Thus he value of Index 2 * is very large when is close o 1 and is value declines o as approaches plus infiniy.

15 15 (48) Index 3 * = [ ] 1/s = [ n I(1) I() pn 1 qn 1 / n I() pn 1 qn 1 ] 1/s. If s is known or has been esimaed, hen Index 2 * and Index 3 * can readily be calculaed as simple raios of sums of observable expendiures raised o he power1/s. Noe ha [ i I() pi qi / i I(1) I() pi qi ] 1. If period has producs ha were no available in period 1, hen he sric inequaliy will hold and since 1/s > 0, i can be seen ha Index 2 will be greaer han uniy. Similarly, [ i I(1) I() pi 1 qi 1 / i I(1) pi 1 qi 1 ] 1. If period 1 has producs ha are no available in period, hen he sric inequaliy will hold and since1/s > 0, i can be seen ha Index 3 will be less han uniy. The inerpreaions of Index 2 * and Index 3 * are no as simple as were he inerpreaions for Index 2 and Index 3. These indexes reflec he effecs of changes in he availabiliy of producs bu hey also reflec increases and decreases in uiliy ha are due o changes in oal expendiures ha vary across periods. In secion 6 below, we will explain how he mehodology developed in his secion can be modified o provide valid counerpars o Feensra s uni cos funcion mehodology ha was explained in secion 2 above. However, wha is rue is ha he uiliy raio decomposiion ha is defined by (35) above can be implemened using observable prices and quaniies provided an esimae for s or is available. In his respec, he uiliy funcion decomposiion (35) is similar o Feensra s uni cos funcion decomposiion defined earlier by (12). Noe ha he purchasers sysem of uiliy maximizing nonzero share equaions for each period ha is defined by equaions (33) can be rewrien as he following sysem of equaions: 27 (49) si = i (qi ) s / n I() n (qn ) s ; = 1,...,T; i I(). Recall ha he purchasers sysem of cos minimizing share equaions using he CES uni cos funcion defined by (2) was given by equaions (11); si = i (pi ) r / n I() n (pn ) r for = 1,...,T and i I(). Equaions (11) and (49) have exacly he same dependen variables bu hey have oally differen independen variables: period prices for equaions (11) and period quaniies for equaions (49). In Secions 5 and 6 below, we will use some scanner daa and esimae boh sysems of equaions and see which sysem fis he daa bes. In he following secion, we will explain our daa se. 4. Scanner Daa for Sales of Frozen Juice 26 If some producs ha were available in period 1 become unavailable in period, hen < 1. Index 3 * evaluaed a period 1 prices equals ( ) s = ( ) 1/(1 ) and is an increasing funcion of for 1 < < +. Wih < 1, he limi of ( ) 1/(1 ) as approaches 1 is 0 and he limi of ( ) 1/(1 ) as approaches + is < 1. Suppose ha = 0.95 and equals 1.005, 1.010, 1.111, 1.429, 2, 10, 100 and Then Index 3 * will equal , , , , , , and Because of he separabiliy properies of he CES uiliy funcion, he assumpion of uiliy maximizing behavior on he par of CES purchasers will imply ha he share equaions (40) and (49) will hold simulaneously.

16 16 We will use he daa from Sore Number 5 28 in he Dominick s Finer Foods Chain of 100 sores in he Greaer Chicago area on 19 varieies of frozen orange juice for 3 years in he period in order o es ou he CES models explained in he previous wo secions; see he Universiy of Chicago (2013) for he micro daa. The micro daa are weekly quaniies sold of each produc and he corresponding uni value price. However, our focus is on calculaing a monhly index and so he weekly price and quaniy daa need o be aggregaed ino monhly daa. Since monhs conain varying amouns of days, we are immediaely confroned wih he problem of convering he weekly daa ino monhly daa. We decided o side sep he problems associaed wih his conversion by aggregaing he weekly daa ino pseudo-monhs ha consis of 4 consecuive weeks. In he Appendix, he monhly daa for quaniies sold and he corresponding uni value prices for he 19 producs are lised in Tables A1 and A2. There were no sales of Producs 2 and 4 for monhs 1-8 and here were no sales of Produc 12 in monh 10 and in monhs Thus here is a new and disappearing produc problem for 20 observaions in his daa se. Laer in his paper, we will impue Hicksian reservaion prices for hese missing producs and hese esimaed prices are lised in Table A2 in ialics. The corresponding impued quaniy for a missing observaion is se equal o 0. Expendiure or sales shares, si pi qi / n=1 19 pn qn, were compued for producs i = 1,...,19 and monhs = 1,..., We compued he sample average expendiure shares for each produc. The bes selling producs were producs 1, 5, 11, 13, 14, 15, 16, 18 and 19. These producs had a sample average share which exceeded 4% or a sample maximum share ha exceeded 10%. These shares are lised in Table A3 in he Appendix. The remaining 10 producs are he lesser selling producs and hese shares are lised in Table A4 in he Appendix. See Chars 1 and 2 below for plos of hese shares. 28 This sore is locaed in a Norh-Eas suburb of Chicago. 29 In wha follows, we will describe our 4 week monhs as monhs.

17 17 Char 1: Sales Shares of Bes Selling Producs s1 s5 s11 s13 s14 s15 s16 s18 s19 Char 2: Sales Shares of Leas Popular Producs s2 s3 s4 s6 s7 s8 s9 s10 s12 s17 I can be seen ha here is remendous volailiy in he sales shares, boh for he bes selling and leas popular producs. In Chars 3 and 4 below, we plo he relaive prices for he bes selling and leas popular producs. The relaive price for produc i in period is defined as pri pi /pi 1 for i = 1,...,19 and = 1,..., For convenience, he impued reservaion prices for producs 2, 4 and 12 were used in Char 4.

18 18 Char 3: Normalized Prices for Bes Selling Producs pr1 pr5 pr11 pr13 pr14 pr15 pr16 pr18 pr19 Char 4: Normalized Prices for Leas Popular Producs p R2 p R3 p R4 p R6 p R7 p R8 p R9 p R10 p R12 p R17 I can be seen ha here is also remendous volailiy in produc prices for boh he bes selling and leas popular producs. Finally, in Chars 4 and 5 below, we plo he relaive quaniies for he bes selling and leas popular producs. The relaive quaniy for produc i in period is defined as qri qi /qi 1 for i = 1,...,19 and = 1,..., However, q 2 = 0 and q 4 = 0 for =1,...,8. Thus we define q R2 as q 2 /q 2 9 and q R4 as q 4 /q 4 9 for = 1,...,8.

19 19 Char 5: Normalized Quaniies for Bes Selling Producs q R1 q R5 q R11 q R13 q R14 q R15 q R16 q R18 q R19 Char 6: Normalized Quaniies for Leas Popular Producs q R2 q R3 q R4 q R6 q R7 q R8 q R9 q R10 q R12 q R17 I can be seen ha he volailiy of quaniies (relaive o monh 1) grealy exceeds he volailiy of prices (relaive o monh 1). When a produc goes on sale a say ½ of is normal price, he volume sold of he produc can easily increase 10 fold or more. In he following secion, we will use his daa se in order o implemen Feensra s uni cos funcion mehodology for he reamen of new and disappearing producs.

20 20 5. The Esimaion of CES Preferences Using he Uni Cos Funcion Approach We assume ha purchaser preferences are defined by he uiliy funcion ha is dual o he CES uni cos funcion defined by (2) in Secion 2 above. Recall ha he sysem of esimaing equaions (5) was obained for his model. 32 In his secion, we compare wo mehods for esimaing he elasiciy of subsiuion for his model: he firs mehod uses he nonlinear equaions in (5), and he second mehod is a simplified version of he esimaor proposed by Feensra (1994). Using he firs mehod, we have a nonlinear sysem of 19 esimaing equaions where he ih equaion for period is si = i(pi ) r / n=1 N n(pn ) r for i = 1,,19 and = 1,...,39. We add error erms, i, o hese equaions where ( 1,..., 19 ) is assumed o be disribued as a mulivariae normal random variable wih mean vecor 019 and variance-covariance marix for = 1,...,39. In order o idenify he i, we impose he following normalizaion: (50) 19 = 1. Since he shares si sum o one for each period, all 19 error erms i for i = 1,...,19 canno be disribued independenly so we dropped he equaion for produc 19 from our lis of esimaing equaions. 33 We chose o esimae he key parameer r in wo sages. In he firs sage, we se r = 1, so ha we obained he following sysem of nonlinear esimaing equaions: (51) si = [ ipi / n=1 N npn ] + i ; = 1,...,39; i = 1,...,18. We used he nonlinear regression sofware package in Shazam 34 o esimae he unknown i in equaions (51). The final log likelihood urned ou o be The equaion by equaion R 2 values were as follows: , , , , , , , , , 0.047, , , , , , , and Thus he fis for his preliminary regression were no very good bu his is o be expeced: Model 1 defined by equaions (51) corresponds o preferences ha exhibi no subsiuion beween producs, which is implausible for closely relaed producs. 32 The acual esimaing equaions are defined by equaions (27), which ake ino accoun he prices and quaniies which are missing due o produc unavailabiliy. We will explain how we deal wih he problem of unavailable commodiies in subsequen foonoes. 33 There is a problem wih his sochasic specificaion: for 20 observaions, he price of a produc ha is no available is aken o be + in he nonlinear regression model ha corresponds o equaions (5). If r < 0, hen (+ ) r = 0 and he corresponding quaniy and expendiure share will also be 0 so in our regression model, here will be 20 observaions (ou of a oal of 702) ha will auomaically have 0 error erms. In order o apply sandard nonlinear regression sofware for sysems of equaions, we have emporarily ignored his problem. 34 See Whie (2004). For he 20 observaions where prices and quaniies were no available, we se he corresponding prices, quaniies and shares equal o 0 in he sysem of esimaing defined by equaions (51). 35 The R 2 concep ha we used is he square of he correlaion coefficien beween he dependen variables and he corresponding prediced variables.

21 21 The esimaed n coefficiens generaed by he above special case of CES preferences were used as saring coefficien values (along wih r = 1 as a saring value) in he uni cos CES model defined by he normalizaion (50) and equaions (52): (52) si = [ i (pi ) r / n=1 N n (pn ) r ] + i ; = 1,...,39; i = 1,...,18. Again, Shazam was used o esimae he 19 unknown parameers in equaions (52). 36 The final log likelihood for his Model 2 was , an increase of over he previous Model 1 regression for adding one parameer. The esimae for r was wih a sandard error equal o Hence he resuling poin esimae for he elasiciy of subsiuion is 1 r = Thus here is a considerable amoun of subsiuion beween he 19 frozen juice producs. The equaion by equaion R 2 values were as follows: , , , , , , , , , , , ,0.7419, , , , and These R 2 values are considerably higher han he corresponding ones from he firs regression model. However, he average R 2 was only equal o which is no very saisfacory. While he esimae of = for he elasiciy of subsiuion seems like a reasonable amoun of subsiuion, i is considerably lower han he esimae ha we shall obain in he nex secion from he uiliy funcion approach. Raher han proceed wih his iniial esimae, we now presen an alernaive mehod for esimaing he elasiciy ha is a simplified version of he esimaor in Feensra (1994). A key feaure of his esimaor is ha i akes ino accoun measuremen error in he prices, which can arise because we are aggregaing he prices over ime, i.e. over weeks in our iniial daa, and over monhs in he daase ha we use o consruc all of our price indexes. So he prices in he daa are acually uni values of each produc defined over hese ime inervals. We will find ha he alernaive esimaor of Feensra (1994), which conrols for his measuremen error, provides an esimae of ha is much higher han We begin by noing ha he prices pi ha are lised in Appendix A are no he rue, minue-by-minue selling prices in he sore. The prices pi are aggregaes over ime weekly or monhly. We refer o hese aggregaes over ime as uni values, and we assume ha hey are relaed o he rue prices i by: (53) lnpi = ln i + ui, 36 Recall ha we have 20 observed prices p i ha are enered as zeros. Bu (0) r is no well defined if r < 0. Thus for = 1,...,8, he 0 price erms (p 2 ) r and (p 4 ) r on he righ hand side of equaions (52) were replaced by (p 2 ) r and (p 4 ) r where we se p 2 = p 4 = 1 for = 1,...,8 and where is a dummy variable which akes on he value 0 for = 1,...,8 and is equal o 1 for = 9,10,...,39. Similarly, he 0 price erms (p 12 ) r for = 10 and = 20, 21 and 22 were replaced by 12 (1) r where 12 is a dummy variable which akes on he value 0 for = 10, 20, and is equal o 1 for oher periods. Thus we se he prices for he 20 missing observaions equal o 1 bu we nullified erms involving hese prices using our dummy variables. Wih hese modificaions, he Shazam sysem nonlinear regression package worked well. The saring log likelihood for he nonlinear regression model defined by his modificaion of (52) was equal o he final log likelihood for he model defined by (51) which is a check ha our modified model behaved in he desired manner.