Solving the IndexNumber Problem in a Historical Perspective


 Eustacia Chandler
 2 years ago
 Views:
Transcription
1 Solvng the IndexNumber roblem n a Hstorcal erspectve Carlo Mlana * January 2009 "The fundamental and wellknown theorem for the exstence of a prce ndex that s nvarant under change n level of lvng s that each dollar of ncome be spent n the same way by rch or poor, wth all ncome elastctes exactly unty (the homothetc case). Otherwse, a prce change n luxures could affect only the prce ndex of the rch whle leavng that of the poor relatvely unchanged. Ths basc theorem was well known already n the 930's, but s often forgotten and s repeatedly beng redscovered". "* + Although most attenton n the lterature s devoted to prce ndexes, when you analze the use to whch prce ndexes are generally put, you realze that quantty ndexes are actually most mportant. Once somehow estmated, prce ndexes are n fact used, f at all, prmarly to 'deflate' nomnal or monetary totals n order to arrve at estmates of underlyng 'real magntudes' (whch s to say, quantty ndexes!)". "* + The fundamental pont about an economc quantty ndex, whch s too lttle stressed by wrters, Leontef and Afrat beng exceptons, s that t must tself be a cardnal ndcator of ordnal utlty"..a. Samuelson and S. Swamy (974, pp ) Introducton The ndexnumber problem s typcally a problem of aggregaton of changes n heterogeneous elements. Mathematcally, t conssts n reducng the relatve change of the elements of a vector nto changes n one sngle numercal value, a scalar. In hs famous Econometrca survey of general economc theory dedcated to the problem of ndex numbers, Ragnar Frsch (936, p. ) descrbed t n these terms: The ndexnumber problem arses whenever we want a * Isttuto d Stud e Anals Economca, azza dell Indpendenza, 4, 0085 Roma, Tel , emal: personal emal: paper has been prepared for the EUKLEMS project funded by the European Commsson, Research Drectorate General as part of the 6 th Framework rogramme.
2 quanttatve expresson for a complex that s made up of ndvdual measurements for whch non common physcal unt exsts. The desre to unte such measurements and the fact that ths cannot be done by usng physcal or techncal prncples of comparson only, consttute the essence of the ndexnumber problem and all the dffcultes center here. In economcs, the soluton of ths problem s necessary n every decomposton of changes of total nomnal values nto meanngful aggregate prce and quantty components. The natonal accountants are asked to provde a splt of the changes of nomnal economc aggregates nto a deflator and a volume component. Smlarly, montorng monetary polces usually entals a decomposton of the ndex of money supply nto an nflaton ndex and a volume representng the purchasng power of crculatng money. At frm level, changes n nomnal profts can be accounted for by decomposng them nto a productvty component (a volume ndex) and market prce condtons (a deflator or prce ndex). It turns out that ths s possble only under very restrctve condtons. In the general case, every attempt of forcng the applcaton of ndex number formulas s doomed to yeld msleadng results (see, e.g., McCusker, 200, Derks, 2004, Offcer and Wllamson, 2006 on ntertemporal comparsons of the purchasng power of money and Leontef, 936 and Samuelson, 947, p. 62, who warned us aganst the tendency to attach sgnfcance to the numercal value of the ndex computed ). Even when the aggregaton condtons are not rejected on the bass of the observed data, there stll remans a certan degree of uncertanty regardng the pont estmate of the ndex number. Followng the truly constructve method establshed by Afrat (98), we can bypass ths uncertanty by revertng the problem and askng: () whether the avalable data can be ratonalzed by wellbehaved true ndex functons, () f yes, what are the upper and lower bounds of the regon contanng the numercal values of possble ndex functons? () f the data cannot be ratonalzed by well behaved ndex functons, then ether the data are not generated by a ratonal behavour (and a correcton for neffcency may be attempted), or else the data are generated wthn a dfferent set of varables to be consdered n an alternatve or extended accountng framework. Snce wellbehaved true ndex functons respect, by constructon, all Fsher s tests (see Samuelson and Swamy, 974), also the reconstructed upper and lower bounds of the set of possble values of the true ndex respect those tests, and so does a geometrc mean of those bounds, whch may be requred for practcal needs of pont estmaton. Ths soluton s 2
3 purely constructve and s obtanable rrespectve of the actual exstence or nonexstence of the underlyng utlty of producton functons. The purpose of ths paper s to present a soluton of the ndex number problem n the perspectve of the theoretcal developments occurred durng the last century. It represents a further step forward wth respect to Afrat s (98)(2005) method used n Afrat and Mlana (2009) wth the defnton of approprate consstent tght bounds of the true ndex number. Further references to the current state of the theory and applcatons of ndex numbers can be found n Vogt and Barta (997), von der Lppe (200)(2007), Balk (2008), and the manuals on consumer prce ndces (CIs), producer prce ndces (Is), and mportexport prce ndces (XMIs) publshed jontly by ILO, IMF, OECD, UN, Eurostat, and The World Bank (2004a)(2004b)(2008). Although, for brevty reasons, we shall concentrate manly on the prce ndex, mportant mplcatons for the quantty ndex wll be also consdered. Irvng Fsher and the deal ndex number formula In Fsher's (9) book The urchasng ower of Money. Its Determnaton to Credt, Interest and Crss, the theory of the prce level was related to the quantty theory of money. Let M = stock of money, V = the velocty of crculaton of money; p = prce level of the th transacton, T = volume of the th transacton carred out usng money. The startng (nfamous) equaton of exchange s () MV = p T + p 2 T p n T n, In order to make the foregong equaton workable, the followng verson s usually consdered (2) MV = T where s the aggregate prce level and T s the volume of all transactons, whch have been replaced wth the aggregaton Q of real outputs q, q2,..., q n, often measured by real GD, that s MV = Q (see Fsher, 9, Ch. 2). Equaton () does not necessarly mply equaton (2). Whle the former s n prncple based on observable varables, the latter contans nonobservable aggregates and reles on computaton technques n order to correctly construct them. It s n ths ven that Irvng Fsher dedcated energes and efforts n the search of hs deal ndex number formula satsfyng as many desred propertes as possble. Ths search culmnated n hs famous book The Makng of Index Numbers publshed n 922 (3 rd edton 3
4 927), where he recognzed that no ndex number would satsfy all the desred propertes, but he chose the geometrc mean of the Laspeyres and aasche ndces as hs deal ndex number formula. Appled to the prce ndex between the ponts of observaton 0 and, ths deal ndex number s gven by (3) 0, 0, 0, F L where 0, L 0 0 pq pq pq pq and 0, pq 0 0 pq pq pq t t t t t t t t where p [ pp 2... p n ] and q [ qq 2... q n ] are the prce and quantty vectors and, 0, L 0,,, and 0, F are the Laspeyres, aasche, and Fsher s deal prce ndces. Ths formula had been prevously consdered by Bowley and others before 899 (see Bowley, 923, p. 252) and recommended by Walsh and gou, although t does not generally satsfy the transtvty or crcularty property, that s 0,2 0,,2 F F F (whereas, any rato of aggregate prce levels, f any, s transtve by constructon: 2 / 0 = ( 2 / )( / 0 )). Surprsngly, Fsher dropped the requrement of ths property and deemed t as unmportant compared to other propertes whch hs deal formula always satsfes. In ther artcle dedcated to economc ndex numbers, Samuelson and Swamy (974) commented Fsher s choce n these terms: Indeed, so enamoured dd Fsher become wth hs socalled Ideal ndex that, when he dscovered t faled the crcularty test, he had the hubrs to declare, therefore, a perfect fulflment of ths socalled crcular test should really be taken as proof that the formula whch fulfls t s erroneous (922, p. 27). Alas, Homer has nodded; or, more accurately, a great scholar has been detoured on a trp whose purpose was obscure from the begnnng (p. 575). By contrast, n order to avod strong dscrepances n the results obtaned, the subsequent developments n ths feld have been devoted to satsfy, among the other tests, the transtvty property n multlateral comparsons. Constantutlty ndex numbers Bennet (920) ntroduced a method by whch a change of expendture can be analysed nto two parts, one correspondng to changes n cost of lvng and the other to changes n standard of lvng (p. 455). Ths decomposton was proposed n terms of absolute dfferences. Konüs (924) and Allen (949) have, respectvely, ntroduced the concepts of constantutlty ndexes 4
5 of prces and quanttes n terms of ratos. Konüs prce ndex s defned as K (, u) p q p 0 0, p q( p, u) whch takes nto account the prcenduced adjustments n quanttes for a gven level of utlty u. Settng u u yelds the Laspeyrestype Konüs prce ndex p q p q( p, u ), whle settng K pq 0 0, p q( p, u ) where p q p q( p, u ). 0 K 0 (, u ) p q p 0 0, pq where u u yelds the aaschetype Konüs prce ndex It must be noted that the constantutlty ndex numbers 0 K and K cannot be computed drectly snce the respectve compensated expendtures 0 p q( p, u ) and 0 0 p q( p, u ) cannot be usually observed. Unless the demand functons 0 qp (, u ) and qp (, u ) are somehow estmated and smulated wth prces p and 0 p respectvely (as n the econometrc approach), a way to proceed wth the concept of Konüs constantutlty ndex numbers s to establsh ther (upper and lower) lmts, when possble. In the general (nonhomothetc) case, Konüs had establshed the followng onesded bounds wth the prce ndex from the pont of vew of demand (on the supply sde, the algebrac sgns are reversed) 0 K 0 pq pq 0 0 L and pq pq 0 K snce 0 0 p q( p, u ) p q and p q( p, u ) p q, because the lefthand sdes of these last nequaltes are those actually consstent wth a costmmnzng behavour at the prces p and p 0 respectvely. Konüs (924) also consdered varous stuatons n relaton to the rankng between the Laspeyres and aasche ndces. In summary, from the pont of vew of demand, the followng alternatve cases are possble: Case : Laspeyres < aasche Case 2: Laspeyres aasche 0 K L K K L 5
6 or and or L K 0 K L 0 K L Konüs observed that t s always possble to fnd a reference utlty level, say u *, such that the cost of lvng ndex falls between the Laspeyres and aasche ndexes, that s or * K L n case * K L n case 2. Konus clamed that these results would suggest that we can work wth the Laspeyres and aasche bounds and take an average of the two to approxmate the true prce ndex. Allen (949) observed that the economc (utltyconstant) quantty ndex could be obtaned drectly, for gven reference prces p, as Q A p q( p, u ) 0 p q( p, u ) Settng 0 p p yelds the Laspeyrestype true Allen quantty ndex Q 0 A 0 0 (, u ) p q p 0 0, pq where p q p q( p, u ), and settng p p yelds the aaschetype true Allen quantty ndex Q A pq 0, p q( p, u ) 0 where p q p q p u (, ). The Laspeyres and aaschetype true Allen quantty ndex numbers can also be obtaned by deflatng the nomnal ncome rato between the two observaton ponts by the aasche and Laspeyrestype true Konüs prce ndex numbers, that s: Q Q p q( p, u ) p q p q p q A / / K p q p q p q( p, u ) p q 0 p q p q p q( p, u ) p q 0 A / / K p q( p, u ) p q p q p q The theory of bounds wth respect to the quantty ndex numbers s smlar to that of the prce ndex numbers. Followng Konüs suggeston, any pont of the numercal nterval between these two ndex numbers could correspond to the true quantty ndex wth a certan level of relatve prces. 6
7 The ndetermnacy of the numercal value of true ndex wthn the Laspeyres aasche bounds seemed to be elmnated by another fndng that s descrbed n the followng secton. Exact and superlatve ndex numbers Byushgens (924) and Konüs and Byushgens (926) have ntroduced the concept of exact ndex numbers for the true aggregator functon by showng that the Fsher deal ndex formula (the geometrc mean of the Laspeyres and aasche ndex numbers) may yeld the same numercal value of the ratos of values taken by a quadratc aggregator functon. If the observed data were generated by a demand governed by such functon, then the transtvty o crcularty property would be satsfed by Fsher deal ndex formula. Followng the modern generalzaton of ther proposton, let us assume a utlty functon such that the correspondng mnmum expendture functon has the quadratc meanoforderr functonal r / 2 r / 2 / r form C( p, u) c r ( p, u) u, where c r ( p, u) ( p A( u) p ) wth r 0, 0 < r, and the Q Q matrx A ( u) s a normalzed symmetrc matrx of postve coeffcents a ( u) a ( u) satsfyng the restrcton a j ( u ), so that c r ( p, u) f p [...]. j Q j j The functonal form c r can be seen as a generalzaton of a CES functonal form, to Q whch t collapses f all aj 0 for j (see McCarthy, 967 and Kadyala, 972), and t reduces to the Generalzed Leontef functonal form wth r (Denny, 972, 974) and the Konüs Byushgens (926) functonal form wth r 2 (Dewert, 976, p. 30). Snce the quadratc functonal forms can be seen also as secondorder approxmatons to any arbtrary functonal form, they have been called flexble by Dewert (976). We have, n fact, r/ 2 r/ 2 c ( (4) p r, u) ( u ) Q p A p c ( / 2 / 2 r p0, u0) r r Q p0 A( u0) p0 / r / r r / 2 r / 2 r / 2 r / 2 / r r / 2 r / 2 / r ( u) ( u0) 0 0 ( u ) r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 0 ( u) 0 ( u0) 0 0 ( u0) p A p p A p p A p p A p p A p p A p r / 2 r / 2 r / 2 r / 2 snce p A( ut) p0 p0 A( ut) p wth a symmetrc Au ( t ) r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 / r r / 2 r / 2 / r p A( u) p p pˆ0 pˆ0 A( u0) p0 p0 A( u) p r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 r / 2 0 ˆ ˆ ( u) 0 ( u0) 0 0 ( u0) p p p A p p A p p A p 7
8 where ^ denotes a dagonal matrx formed wth the elements of a vector where p a ( u ) p r/ 2 r/ 2 pq t t t j j t tj t r/ 2 r/ 2 ptjqtj pt A( ut ) p j t s r /2 p r s0 / 2 / 2 / r r /2 r r p0 p0 A( u) p r /2 r/ 2 r/ 2 p0 p0 A( u0) p s r /2 p, whch s the observed value share of the th quantty r 2 r /2 c( p, ) ( ) t u p t t aj ut p j tj t ( pt, t ) / t t t pt r/ 2 r/ 2 r ( pt A( ut ) pt ) q C u p u u by Shephard s lemma, wth a j beng the (,j) element of matrx A. Thus, the ndex number yelds exactly (s exact for) the same numercal value that would be obtaned as a rato of the values of the underlyng functon n the two compared stuatons. Dewert (976) called superlatve the ndex numbers that are exact for flexble functonal forms and descrbed them as approxmatng each other up to the second order. By contrast, t has been noted that these ndex numbers are far from beng secondorder approxmatons to each other (see Mlana, 2005 and Hll, 2006a) and that ths termnology dverges n meanng from that used by Fsher (922), who has defned superlatve those ndex numbers that smply performed very closely to hs deal ndex formula wth hs dataset. Snce all the prce varables and utlty are consdered here at ther current levels, the shares s t are those actually observed. As we shall see also below, n the homothetc case, we have C( p, u) c( p ) u and, consequently, the observed shares s t are equal to the theoretcal weghts that are functons only of prces (wth A( u0) A( u) A ). The frst multplcatve bracketed element of the last lne of (4) can be consdered as a canddate prce ndex number (5) r /2 r /2 p0 0 0 r /2 p0 r /2 p r ( p, p, q, q ) Q p s 0 s r whch corresponds to Dewert s (976, p.3) quadratc meanoforderr prce ndex number. As r tends to 0, the prce ndex r tends to the Törnqvst ndex number: Q lm exp[ ( )(ln ln )] 2 (6) r0 r T s Q 0 s p p0 whch s exact for the translog cost functon 8
9 (7) ct ( p, u) u exp( 0 u ln u lnp u ln p ln u j ln p ln p j ) 2 j If r = 2, then the prce ndex r s to the deal Fsher ndex. Q We note that, f the observed data were generated by a demand consstent wth a mnmum quadratc cost functon c r ( p, u) wth specfc parameters values, at least locally, Q then we would have r ( p0, p2, q0, q2) r ( p0, p, q0, q) r ( p, p2, q, q2), that s the exact ndex number Q Q Q r would satsfy the transtvty property as well as all the other Fsher s tests between Q the three observaton ponts. If the transtvty property s not satsfed, then ether the demand s not governed by a ratonal behavour or the ndex number formula s not exact for the actual cost functon or utlty functon consstent wth the data. At tme of the dscovery of Konüs and Byushgens (926), the concept of homothetcty of ndfference curves and ts relatonshp wth exstence of a pure prce (and quantty) ndex was not wdely known. The concept of homothetcty was explctly spelled out by Shephard (953) and Malmqust (953) n the feld of producton technology and ndependently by Afrat (972) under the termnology of concal functons n the feld of consumer utlty. Earler contrbutons datng back at least from Antonell (886) and ncludng Frsch (936, p. 25) and Samuelson (950, p. 24) have dealt wth t mplctly. When the true prce ndex defned by Konüs s not ndependent of the utlty level, as n the general nonhomothetc case, the correspondng Allen true quantty ndex fals to be lnearly homogeneous (f all the elementary quanttes are multpled by a factor λ, then the ndex number fals to be proportonal by the same factor λ). In Allen s (949, p. 99) words, *t+he ndex has no meanng unless we make the assumpton that the preference map s the same n the two stuatons. Ths affects, n a way, also the prce ndex: although ths ndex s always lnearly homogeneous by constructon n the nonhomothetc case t results to be a spurous prce ndex whose weghts are functons not only of prces but also of the utlty level and, then, of the demanded relatve quanttes. Ths has been usually overlooked also n the current lterature on economc ndex numbers. Wth the quadratc functon consdered above, only f A( u) A( u0) A would the weghts be functons only of prces. In the applcaton of ndexes defned by Dvsa (925), ths s called path ndependence snce the ndex s ndependent of the path taken wth respect to the reference quantty varables. Hulten (973) has shown that the Dvsa ndex s path 9
10 ndependent f and only f the underlyng functon s homothetc (tastes do not change). Ths can be seen mmedately related to the Törnqvst ndex number n the lmt of nfntesmal changes: hence ( ) (8) ln p ln p dln p d ln Dv lm t0 ( st s( t t) ) st 2 t dt (9) 0, t dln pt Dv exp( s ) t 0 t dt dt t t t t whch s the Dvsa prce ndex. If the weghts s t are not functons of the prces alone (as n the homothetc case), but depend also on relatve levels of the reference quanttes, then the Dvsa prce ndex s not a pure prce ndex. These consderatons were already mplct n the analyss of contrbutors n the early part of last century, who were well aware of the mportance of homothetc tastes for the exstence of economc aggregate ndex numbers. A. L. Bowley, for example, n search of a constantutlty prce ndex had been among the frst proponent of the geometrc mean of the Laspeyres and aasche ndexes (whch had later become famous as Fsher deal ndex). He also devsed another ndex as an approxmaton to the constantutlty prce ndex gven by the followng formula, prevously proposed by Edgeworth: (0) E 0 p ( q q ) 0 0 p ( q q ) to be appled under the hypothess of no changes n tastes. He, n fact, wrote: Assume that our records represent the expendture of an average man, and that the satsfacton he derves from hs purchases s a functon of the quanttes bought only, say u(q ), are the numbers of unts bought of the n commodtes. Further, suppose that the form and constants of ths functon are unchanged over the perod consdered. The last condton lmts the measurement to an nterval of tme n whch customs and desres have not changed and to a not very wde range of real ncome. The analyss and conclusons do not apply to comparsons between ctzens of two countres, nor over, say, 60 years n one country (Bowley, 928, pp ). Identcal preferences, mplyng a homothetc utlty functon, have been noted as early as the work of Antonell (886) as a necessary and suffcent condton for aggregaton. Condtons for aggregaton holdng only locally and allowng global preference heterogenety have been studed by Afrat (95356)(959) and Gorman (953)(96). 0
11 It s remarkable, however, that also the foregong BowleyEdgeworth ndex number does not satsfy the requrement of transtvty. In general, the lack of transtvty would sgnal the poor approxmaton gven by the formulas chosen. Ths s the stuaton encountered partcularly n nterspatal comparsons, where the alternatve measures could dffer more than 00% even wth superlatve ndex numbers (see, e.g., Hll, 2006a, 2006b). Gven the dscouragng results obtaned wth specfc ndex number formulas, we now turn to the method of lmts by consderng the exercse of testng the data for consstency wth hypothetcal homothetc changes, followng Keynes, Hcks, Samuelson, and Afrat. John Maynard Keynes method of lmts In hs Treatse on Money, Keynes (930, Vol.I, ch. 8) made no explct reference to the dea of a prce ndex. Rather, he compared the purchasng power of money n two stuatons of consumpton dfferng n relatve prces. The comparson was made by usng the socalled method of lmts (p. 98). No change n taste and proportonalty of composte quanttes (and prces) wth respect to total real expendture are assumed. These hypotheses mply monotoncty along a beam lne where, at gven relatve prces, all ndvdual quanttes change proportonally. Two alternatve ratos of real expendtures can be calculated at constant relatve prces of the base and the current stuatons, respectvely. It turns out that these ratos are the upper and lower lmts (bounds) of the ndex of the real expendture. (As shown by Leontef, 936, pp and Afrat, 977, pp. 085, 2005, these lmts correspond, respectvely, to the Laspeyres and aasche ndex numbers of real expendture.) Smlar methods were used by other authors. In hs famous revew artcle, Ragnar Frsch (936, p. 727) mentoned gou, Haberler, Keynes, Gn, Konüs, Bortkewcz, Bowley, Allen, and Staehle and dscussed them brefly. Keynes (930, p. 99) hmself observed: Ths concluson s not unfamlar * +. It s reached, for example, by rofessor gou (Economcs of Welfare, part I, chapter VI). The matter s also very well treated by Harberler (Der Snn der Indexzahlen, pp ). The dependence of the argument, however, on the assumpton of unformty of tastes, etc., s not always suffcently emphassed (talcs added). He wrtes, here, the followng footnote: Dr. Bowley n hs Notes on Index Numbers publshed n the Economc Journal, June 928, may be mentoned amongst those who have expressly ntroduced ths necessary condton.
12 Keynes method of lmts has not been wdely used, probably because t has not been mmedately understood n ts fundamental reasonng. Frsch (936, p. 26), for example, whle concedng the correctness of Keynes proof, overlooked the real sense of hs proceedng by observng: If we know that q 0 and q are adapted and equvalent, the ndfferencedefned [prce] ndex can be computed exactly, namely, as the rato 0 0 p q / p q [snce t s assumed t t that q q( p, u) wth t = 0,]. In these crcumstances, to derve lmts for t s to play hdeandseek. It was Staehle who frst ponted ths out. In fact, Keynes dd not assume that q 0 and q were necessarly on the same ndfference curve, but on homothetc ndfference curves on the hypothess of unformty of tastes. Ths mples monotoncty along a beam (a lne where all ndvdual quanttes change proportonately) along whch the purchasng power of money can be compared at dfferent prces. Ths reasonng was later recovered and further developed by Afrat (977, pp. 085). Hcks Laspeyresaasche nequalty condton In a chapter enttled The IndexNumber Theorem of hs Revson of Demand Theory, John Hcks (956, pp ) establshed a proposton on the Laspeyresaasche nequalty on the demand sde () Laspeyres (L) aasche () (for both prce or quantty ndexes) (see also Hcks, 958 and the prevous prelmnary analyss contaned n Hcks, 940). The (nonnegatve) dfference between Laspeyres and aasche ndcates a substtuton effect (S) n the case the ponts of observaton are on the ndfference curve or the sum of substtuton effect and a certan ncome effect (I) n the case they are not on the same ndfference curve. In the more general case, we have (2) L = I + S where, L and are the Laspeyres and aasche ndexes (we use Hcks orgnal notaton denotng the aasche ndex as ). If the ncomeelastctes of all commodtes are the same (that s the preferences are homothetc), then I s equal to zero. In ths case, the proporton of demanded quanttes do not change as real ncome changes. We have the followng possble results: 2
13 Case : L < 0 (Hcks ndexnumber theorem breaks down) meanng ether that demand s not governed by ratonal behavour and/or the preferences are nonhomothetc wth a negatve and strong enough ncome effect so that realncome change nduces a relatve expanson n demand for those goods whose prces have relatvely rsen. A strong negatve ncome effect offsets a postve substtuton effect (I + S < 0) Case 2: L 0 (Hcks ndexnumber theorem holds), meanng ether that preferences are homothetc (so that I = 0 and S 0) or preferences are nonhomothetc (wth I 0 and I + S 0). If preferences are homothetc, mplyng that the ncomeelastctes of all commodtes are the same then the proporton of demanded quanttes do not change as real ncome changes. and I s equal to zero. The Hcks ndexnumber theorem pontng to a postve L dfference (case 2) s a necessary and suffcent condton for usng the observed data on prces and quanttes to reconstruct true ndex numbers based on hypothetcal homothetc preferences. These, however, do not necessarly concde wth the actual crtera governng the observed behavour. In other words, the L nequalty mght be the result of the concomtant nonproportonal effects of real ncome changes as well as substtuton effects under nonhomothetc preferences (f any), but the observed data could always be ratonalzed by a hypothetcal homothetc preference feld f L 0. Under ths condton we could always reconstruct true prce and quantty ndex numbers that are consstent wth those homothetc preferences and, as such, always respect all Fsher s requrement, ncludng transtvty. Ths s, n fact, (as Keynes had recalled) the only condton under whch t s possble to make such constructon. Samuelson s consderatons on the Laspeyresaasche nequalty Independently from Hcks (956) and consstently wth hs ndexnumber theorem, n ther surveys on the conclusons of the theory of bounds, Samuelson (974)(984), Samuelson and Swamy (974), and Swamy (984) have consdered the followng cases. Case : L < 0, so that the observed relatve prces are not negatvely correlated wth the observed relatve quanttes (as expected wth homothetc changes). In such an anomalous case, we mght obtan the followng rankng (wrtten n matrx notaton, where p t and q t are prce and quantty vectors at tme t): 3
14 0 0 p q( p, u ) p q p q p q p q p q p q p q( p, u ) Konüs Laspeyres aasche Konüs Laspeyrestype aaschetype wth 0 qp (, u ) and 0 qp (, u ) beng the vectors of nonobserved (theoretcal) quanttes that would have been demanded at the prceutlty combnatons 0 ( p, u ) and 0 ( p, u ), respectvely. Ths s a rather problematc case, where aggregaton s not possble. Even Fsher s deal ndex, whch conssts n the geometrc mean of the Laspeyres and aasche ndexes, fallng between these two ndexes, s farther than ths last ones from both true economc ndexes! (See the numercal example gven by Samuelson and Swamy, 974, where the Ideal ndex cannot gve hghpowered approxmaton to the true ndex n the general, nonhomothetc case, p. 585.) Case 2: L 0. If preferences are homothetc, then I = 0 and S 0. If preferences are nonhomothetc wth realncome changes nducng a relatve expanson n demand for those goods whose prces have relatvely fallen (a case consdered by Samuelson, 974, 984, Swamy, 984 and others under the name of EngelGerschenkron effect ), then I 0, whch renforces the postve substtuton effect S 0. In these two cases, we can rely on the followng rankng 0 p q p q p q p q p q( p, u ) p q aasche Konüs Laspeyres aaschetype and 0 0 p q (, u ) p q p p q p q p q p q aasche Konüs Laspeyres Laspeyrestype The Laspeyres and aasche ndex numbers correspond to alternatve fxed proportons utlty functons (wth zero commodty substtuton). Notng ths fact, Swamy (984, fn. 0) wrote: Ths s not to dssuade scholars from usng the Laspeyres and aasche ndexes, but merely to urge them to restrct the use of these ndexes to local changes n p. These ndexes can be used to determne bounds for the true ndex whch may not be known. 4
15 Afrat s ndex formula: Any pont n the Laspeyresaasche nterval, f any Along the lnes open by Hcks (956), the jont nformaton gven by the Laspeyres and aasche ndexes could provde us wth an alternatve nformaton concernng two lmtng functons allowng substtuton effects whose dfference s equal to S 0 consdered above. These two lmtng functons are pecewse lnear boundares of a set of possble homothetc utlty functons, whch can ratonalze the observed data. Even though these data have been actually generated under nonhomothetc preferences, the Hcks (956) Laspeyresaasche nequalty condton s necessary and suffcent for constructng true ndex homothetc functons that can also ratonalze the same data. It s n ths ven that Afrat (977, pp. 085) recovered Keynes (930) reasonng on the purchasng power of money under the hypothess of unchanged tastes and translated t nto the constructon of the bounds of a true prce ndex. As recalled by Samuelson and Swamy (974, p. 570), t s possble to nvoke the ShephardAfrat s factorzaton theorem under the hypothess of homothetcty to separate the expendture functon nto meanngful aggregates of prces and quanttes. As Samuelson and Swamy (974, p. 570) have recognzed, *t+he nvarance of the prce ndex [from the reference quantty base] s seen to mply and to be mpled by the nvarance of the quantty ndex from the reference prce base. Ths concluson was antcpated n Afrat (977, pp. 072). A pure prce ndex s consstent wth a concal (homothetc) utlty functon ratonalzng the observed prces and quanttes n dfferent stuatons. The concal (homothetc) utlty condton whch permts ths determnaton, for arbtrary p0 and p, s a nonobservatonal object, a purely hypothetcal metaphyscal concept. The correspondng dual mnmum expendture functon admts the factorzaton nto a product of the prce and quantty functons. C( p, u) c( p) u( q ) Defnng the amount of money devoted to total expendture (or ncome) as E, so that E C( p, u), we can obtan the cardnal measure of utlty as a deflated value of ncome, that s n the homothetc case where V( p, E) s the ndrect utlty functon. c ( p) u ( q) u V ( p, E) E c( p) c( p) 5
16 The observed (uncompensated) Marshallan demand functons for each elementary quantty s gven by Roy s dentty q V ( p, E) / p dv ( p, E) / de for =,2, whch, n the homothetc case, becomes q a ( p ) E where c( p) / p a ( p) c( p) for all s The ncome elastcty of the demanded th quantty s thus obtaned q E a ( p) E q a ( p) for all s that s, all ncome elastctes are equal to n the homothetc case. The problem s whether we can recover the prce ndex whch s expressed as a rato of prce levels / 0, 0 c( p ), c( p ) 0 0 whereas, the moneymetrc utlty ndex s measured by whch s the rato of volume levels E / E Q0, Q / Q0 0 0, Q E / c( p ), Q E / c( p ) The expendture ndex consstent wth the recovered homothetc utlty con be decomposed as follows: C( p, u) c( p) u c( p) E / c( p) 0, Q0, C( p, u ) c( p ) u c( p ) E / c( p ) Strctly speakng, the nverse of the prce ndex, / 0,, s the ndex of purchasng power of one unt of money and the quantty ndex Q0, ( E / E0) (/ 0, ) s the ndex of purchasng power of monetary ncome, or real ncome. Consstently wth the hypothess of homothetcty, ths last ndex corresponds to the ndex of utlty u( q) / u( q 0). In Afrat (977, p. 0) words: The concluson * + s that the prce ndex s bounded by the aasche and Laspeyres ndces. * + The aasche ndex does not exceed the Laspeyers ndex. * + The set of values *of the true ndex + s n any case dentcal wth the aasche 6
17 Laspeyres nterval. The true ponts are just the ponts n that nterval and no others; and none s more true than another. There s no sense to a pont n the nterval beng a better approxmaton to the true ndex than others. There s no proper dstncton of constant utlty ndces, snce all these ponts have that dstncton. The same concluson s replcated n Afrat (2005, p. xx): Let us call the L nterval the closed nterval wth L [Laspeyers ndex] and (aasche ndex] as upper and lower lmts, so the Lnequalty s the condton for ths to be nonempty. Whle every true ndex s recognzed to belong to ths nterval, t can stll be asked what ponts n ths nterval are true? The answer s all of them, all equally true, no one more true than another. When I submtted ths theorem to someone notorous n ths subject area t was receved wth complete dsbelef. Here s a formula to add to Fsher s collecton, a bt dfferent from the others. Index Formula: Any pont n the Lnterval, f any. In my revew artcle (Mlana, 2005), t s shown that any prce ndex number that s exact for a contnuous functon can be translated nto the followng form (3) ( ) s p 0 ( ) ( ) p0 N p q0 N p q 0, p 0 p0q 0 p0q ( ) s p where, for t = 0,, C( pt, ut ) C( pt, ut ) st pt / p j tj p p t tj C( pt, ut) pt qt / p j tjqtj usng Shephard s lemma ( q t = ) p and s an approprate parameter whose numercal value depends on the remander terms of the two frstorder approxmatons of C(p,u) around the base and current ponts of observatons. The ndex 0, s lnearly homogeneous n p (that s, f p p0, then 0, ). Wth 0, t reduces to a Laspeyres ndex number, whereas, wth, t reduces to a aasche ndex number. The true exact ndex number, f any, s numercally equvalent to 0,. If the functonal form of C( p, u ) s square root quadratc n p, then 0, can be transformed nto a Fsher t t t 7
18 deal ndex number. In ths case, the ndex 0, s numercally equvalent to a quadratc meanoforder2 ndex number. Here, agan, the prce ndex s nvarant wth respect to the reference utlty level f and only f C( p, u) s homothetcally separable and can be wrtten C( p, u) c( p ) u, so that s p q / p q t t t j tj tj p t c( pt) c( pt) / ptj. j p p t tj Moreover, Q0, [ C(, u) / C( 0, u0)]/ 0, p p s the quantty ndex measured mplctly by deflatng the ndex of the functonal value wth the prce ndex 0,. It has the meanng of a pure quantty ndex f and only f 0, s a pure prce ndex. The parameter, however, remans unknown and we cannot rely on the secondorder dfferental approxmaton paradgm. For ths reason, t s concluded that t would be more approprate to construct a range of alternatve ndex numbers (ncludng those that are not superlatve), whch are all equally vald canddates to represent the true ndex number, rather than follow the tradtonal search for only one optmal formula (Mlana, 2005, p. 44). revous attempts n ths drecton usng nonparametrc approaches based on revealed preference technques nclude Banker and Mandratta (988), Manser and McDonald (988), Chavas and Cox (990)(997), Dorwck and Quggn (994)(997), but these do not provde, n general, strngent tests for homothetcty and, more mportantly, the derved ndex numbers fal to satsfy the transtvty requrement. An alternatve approach to the Afrat methodology would be that of the econometrc estmaton of the functon c( p) n order to elmnate the ndetermnacy of the true ndex number (see, among the frst attempts, Goldberger and Gamaletsos, 970 and Lloyd (975), and, among the most recent contrbutons, Blundell et al. 2003, Neary, 2004, and Oulton, 2005), but ths mples the mposton of a subjectve choce of a pror functonal forms where stochastc components of the derved demand functons are also ncluded. The theory of bounds becomes more complex wth the addton of the stochastc term to each demand functon (see, e.g., hlps, 983). Crtcal remarks on ths approach could be made regardng the nondentfablty of the elastctes of substtuton and the bas n changes n technology or consumer tastes f no a pror nformaton s avalable (see, e.g., Damond, McFadden and Rodrguez, 978). 8
19 Consstent prce ndces between several observaton ponts The approach outlned n the prevous secton can be enhanced by consderng more than two observaton ponts smultaneously. Ths dea had been advanced durng the debate on ndex numbers n the early part of last century. Frsch (936, p. 36), commentng the soexpendture method of Staehle (935), wrote: The comparson between two paths wll be more exact f made va an ntermedate path. The closer the ndvdual paths the better. Knowng a very close pathsystem s equvalent to knowng the ndfference surfaces themselves. In ths case the ndfference ndex can be computed exactly. Smlar statements were wrtten also by Samuelson (947, ch. VI). It s worth quotng Samuelson and Swamy s (974, p. 476) own words: * + Fsher mssed the pont made n Samuelson (947, p. 5) that knowledge of a thrd stuaton can add nformaton relevant to the comparson of two gven stuatons. Thus Fsher contemplates Georga, Egypt, and Norway, n whch the last two each have the same prce ndex relatve to Georga : We mght conclude, snce two thngs equal to the same thng are equal to each other, that, therefore, the prce levels of Egypt and Norway must equal, and ths would be the case f we compare Egypt and Norway va Georga. But, evdently, f we are ntent on gettng the very best comparson between Norway and Egypt, we shall not go to Georga for our weghts *whch are+, so to speak, none of Georga s busness. *922, p. 272]. Ths smply throws away the transtvty of ndfference and has been led astray by Fsher s unwarranted belef that only fxedweghts lead to the crcular s test s beng satsfed (an asserton contradcted by our / and Q / Q forms. j j One of Afrat s man contrbuton n ndex number theory has been the development an orgnal approach of constructng aggregatng ndex numbers usng all the data smultaneously (see Afrat, 967, 98, 984, 2005). He also has developed an effcent algorthm to fnd the mnmum path of chaned upper lmt ndex numbers (the chaned Laspeyres ndces on the demand sde). In the followng secton ths algorthm s brefly descrbed. From these chaned upper lmt ndex numbers can be derved drectly the chaned lower lmt ndex numbers (the chaned aasche ndces on the demand sde). 9
20 The proposed method In ths secton, for expostonal convenence, some notaton s changed wth respect to the prevous sectons. The matrces of blateral Laspeyres (L) and aasche (K) ndex numbers comparng aggregate prces at the pont of observaton relatve to those at pont j, for,j =, 2,, N, are respectvely L L2... L N L2 L22... L 2N L and LN LN 2... LNN K K... K K K... K K K K... K 2 N N N N 2 NN where j Lj pq j j pq, and Kj pq j pq L j. Obvously, K j and L K. L j The Laspeyres and aasche ndex numbers are usually consdered as two alternatve measures of the unknown true ndex number j whch can be seen as an aggregaton of r j r the elementary prce ratos p / p or, alternatvely, as a rato of aggregate prce levels,.e. /, where and j j j are true aggregate prce levels at the th and jth ponts of observaton. The prce level rato, always respects, by constructon, the base reversal test, that s /, and the crcularty test, that s. By contrast, n the general case j j 20 t tj j where the elementary prce ratos and the relatve quantty weghts change, the Laspeyres and aasche ndces fal to be base and chanconsstent, that s L / L K, j j j Lt Ltj Lj and Kt Ktj Kj. Even more unacceptable s wellknown falure of chaned ndexes to return on the prevous levels f all elementary prces go back to ther older levels (the socalled drft effect ): L L L. and K K K. These falures make the two t t t t ndex number formulas, lke all the other alternatve formulas, unsutable to represent a prce ndex. Nevertheless, as we shall see below, they are useful for testng the exstence of the true prce ndex and constructng ts consstent bounds. ( Lj Kj The socalled Lnequalty condton s that Lj Kj on the purchaser s sde ndex number on the suppler s sde) s necessary and suffcent for the exstence of a true prce j wth a numercal value fallng between the Laspeyres and aasche ndces. If ths condton s not satsfed for all pars of observaton, then a correcton of the data for
Recurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationQuality Adjustment of Secondhand Motor Vehicle Application of Hedonic Approach in Hong Kong s Consumer Price Index
Qualty Adustment of Secondhand Motor Vehcle Applcaton of Hedonc Approach n Hong Kong s Consumer Prce Index Prepared for the 14 th Meetng of the Ottawa Group on Prce Indces 20 22 May 2015, Tokyo, Japan
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationComparing Class Level Chain Drift for Different Elementary Aggregate Formulae Using Locally Collected CPI Data
Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Gareth Clews 1, Anselma DobsonMcKttrck 2 and Joseph Wnton Summary The Consumer Prces Index (CPI) s a measure of consumer
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationLecture 3: Force of Interest, Real Interest Rate, Annuity
Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annutymmedate, and ts present value Study annutydue, and
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationNasdaq Iceland Bond Indices 01 April 2015
Nasdaq Iceland Bond Indces 01 Aprl 2015 Fxed duraton Indces Introducton Nasdaq Iceland (the Exchange) began calculatng ts current bond ndces n the begnnng of 2005. They were a response to recent changes
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationWeek 6 Market Failure due to Externalities
Week 6 Market Falure due to Externaltes 1. Externaltes n externalty exsts when the acton of one agent unavodably affects the welfare of another agent. The affected agent may be a consumer, gvng rse to
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationProduction. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.
Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s nonempty If Y s empty, we have nothng to talk about 2. Y s closed A set
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationIDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS
IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationA Computer Technique for Solving LP Problems with Bounded Variables
Dhaka Unv. J. Sc. 60(2): 163168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationMultivariate EWMA Control Chart
Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 MultpleChoce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multplechoce questons. For each queston, only one of the answers s correct.
More informationOn the Optimal Control of a Cascade of HydroElectric Power Stations
On the Optmal Control of a Cascade of HydroElectrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationIMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1
Nov Sad J. Math. Vol. 36, No. 2, 2006, 009 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More information1.1 The University may award Higher Doctorate degrees as specified from timetotime in UPR AS11 1.
HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationAddendum to: Importing SkillBiased Technology
Addendum to: Importng SkllBased Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our
More informationIntroduction: Analysis of Electronic Circuits
/30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,
More informationSIMPLE LINEAR CORRELATION
SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationImplied (risk neutral) probabilities, betting odds and prediction markets
Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT  We show that the well known euvalence between the "fundamental theorem of
More informationStudy on CET4 Marks in China s Graded English Teaching
Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes
More informationHOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA*
HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* Luísa Farnha** 1. INTRODUCTION The rapd growth n Portuguese households ndebtedness n the past few years ncreased the concerns that debt
More informationWill the real inflation rate please stand up overlooked quirks of a favoured chainlinking technique
Wll the real nflaton rate please stand up overlooked qurks of a favoured chanlnkng technque Dr Jens Mehrhoff*, Head of Secton Busness Cycle, Prce and Property Market Statstcs * Jens Ths Mehrhoff, presentaton
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationTrade Adjustment and Productivity in Large Crises. Online Appendix May 2013. Appendix A: Derivation of Equations for Productivity
Trade Adjustment Productvty n Large Crses Gta Gopnath Department of Economcs Harvard Unversty NBER Brent Neman Booth School of Busness Unversty of Chcago NBER Onlne Appendx May 2013 Appendx A: Dervaton
More informationMAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date
Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPPATBDClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller
More informationProject Networks With MixedTime Constraints
Project Networs Wth MxedTme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationAnalysis of Premium Liabilities for Australian Lines of Business
Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton
More informationStructural Estimation of Variety Gains from Trade Integration in a Heterogeneous Firms Framework
Journal of Economcs and Econometrcs Vol. 55, No.2, 202 pp. 7893 SSN 20329652 ESSN 20329660 Structural Estmaton of Varety Gans from Trade ntegraton n a Heterogeneous Frms Framework VCTOR RVAS ABSTRACT
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationPassive Filters. References: Barbow (pp 265275), Hayes & Horowitz (pp 3260), Rizzoni (Chap. 6)
Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called
More informationThe covariance is the two variable analog to the variance. The formula for the covariance between two variables is
Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.
More informationChapter 4 Financial Markets
Chapter 4 Fnancal Markets ECON2123 (Sprng 2012) 14 & 15.3.2012 (Tutoral 5) The demand for money Assumptons: There are only two assets n the fnancal market: money and bonds Prce s fxed and s gven, that
More information10. XMPI Calculation in Practice
10. XMPI Calculaton n Practce A. Introducton 10.1 Ths chapter provdes a general descrpton wth examples of the ways n whch XMPIs are calculated n practce. The methods used n dfferent countres are not exactly
More informationSolutions to First Midterm
rofessor Chrstano Economcs 3, Wnter 2004 Solutons to Frst Mdterm. Multple Choce. 2. (a) v. (b). (c) v. (d) v. (e). (f). (g) v. (a) The goods market s n equlbrum when total demand equals total producton,.e.
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationSolution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.
Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces
More informationState function: eigenfunctions of hermitian operators> normalization, orthogonality completeness
Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators> normalzaton, orthogonalty completeness egenvalues and
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationGeneralizing the degree sequence problem
Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts
More informationTo manage leave, meeting institutional requirements and treating individual staff members fairly and consistently.
Corporate Polces & Procedures Human Resources  Document CPP216 Leave Management Frst Produced: Current Verson: Past Revsons: Revew Cycle: Apples From: 09/09/09 26/10/12 09/09/09 3 years Immedately Authorsaton:
More informationCourse outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy
Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton
More information9. PPI Calculation in Practice
9. PPI Calculaton n Practce A. Introducton 9.1 Ths chapter provdes a general descrpton of the ways PPIs are calculated n practce. The methods used n dfferent countres are not exactly the same, but they
More informationCHAPTER 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY. Themes of the chapter. Nominal rigidities, expectational errors and employment fluctuations
CHAPTER 18 INFLATION, UNEMPLOYMENT AND AGGREGATE SUPPLY Themes of the chapter Nomnal rgdtes, expectatonal errors and employment fluctuatons The shortrun tradeoff between nflaton and unemployment The
More informationWORKING PAPERS. The Impact of Technological Change and Lifestyles on the Energy Demand of Households
ÖSTERREICHISCHES INSTITUT FÜR WIRTSCHAFTSFORSCHUNG WORKING PAPERS The Impact of Technologcal Change and Lfestyles on the Energy Demand of Households A Combnaton of Aggregate and Indvdual Household Analyss
More informationI. SCOPE, APPLICABILITY AND PARAMETERS Scope
D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationSIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA
SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands Emal: e.lagendjk@tnw.tudelft.nl
More informationAPPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT
APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT Toshhko Oda (1), Kochro Iwaoka (2) (1), (2) Infrastructure Systems Busness Unt, Panasonc System Networks Co., Ltd. Saedocho
More informationNumber of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000
Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from
More informationQuestions that we may have about the variables
Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent
More informationOn Mean Squared Error of Hierarchical Estimator
S C H E D A E I N F O R M A T I C A E VOLUME 0 0 On Mean Squared Error of Herarchcal Estmator Stans law Brodowsk Faculty of Physcs, Astronomy, and Appled Computer Scence, Jagellonan Unversty, Reymonta
More informationNPAR TESTS. OneSample ChiSquare Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationBrigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
More informationAryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006
Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,
More informationCHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol
CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL
More informationThe program for the Bachelor degrees shall extend over three years of fulltime study or the parttime equivalent.
Bachel of Commerce Bachel of Commerce (Accountng) Bachel of Commerce (Cpate Fnance) Bachel of Commerce (Internatonal Busness) Bachel of Commerce (Management) Bachel of Commerce (Marketng) These Program
More informationDescribing Communities. Species Diversity Concepts. Species Richness. Species Richness. SpeciesArea Curve. SpeciesArea Curve
peces versty Concepts peces Rchness pecesarea Curves versty Indces  mpson's Index  hannonwener Index  rlloun Index peces Abundance Models escrbng Communtes There are two mportant descrptors of a communty:
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More informationFixed income risk attribution
5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two
More informationTime Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University
Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton
More informationSensitivity Analysis in a Generic MultiAttribute Decision Support System
Senstvty Analyss n a Generc MultAttrbute Decson Support System Sxto RíosInsua, Antono Jménez and Alfonso Mateos Department of Artfcal Intellgence, Madrd Techncal Unversty Campus de Montegancedo s/n,
More informationJoe Pimbley, unpublished, 2005. Yield Curve Calculations
Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More information