INDEX NUMBER THEORY AND MEASUREMENT ECONOMICS. By W.E. Diewert, March, 2016.

Transcription

1 1 INDEX NUMBER THEORY AND MEASUREMENT ECONOMICS By W.E. Diewer, March, CHAPTER 8: Fixed Base Versus Chained Indexes 1. Inroducion In his chaper 1, he meris of using he chain sysem for consrucing price indexes in he ime series conex versus using he fixed base sysem are discussed. The chain sysem 2 measures he change in prices going from one period o a subsequen period using a bilaeral index number formula involving he prices and quaniies peraining o he wo adjacen periods. These one period raes of change (he links in he chain) are hen cumulaed o yield he relaive levels of prices over he enire period under consideraion. Thus if he bilaeral price index is P, he chain sysem generaes he following paern of price levels for he firs hree periods: (1) 1, P(p 0,p 1,q 0,q 1 ), P(p 0,p 1,q 0,q 1 ) P(p 1,p 2,q 1,q 2 ). On he oher hand, he fixed base sysem of price levels using he same bilaeral index number formula P simply compues he level of prices in period relaive o he base period 0 as P(p 0,p,q 0,q ). Thus he fixed base paern of price levels for periods 0,1 and 2 is: (2) 1, P(p 0,p 1,q 0,q 1 ), P(p 0,p 2,q 0,q 2 ). Noe ha in boh he chain sysem and he fixed base sysem of price levels defined by (1) and (2) above, he base period price level is se equal o 1. The usual pracice in saisical agencies is o se he base period price level equal o 100. If his is done, hen i is necessary o muliply each of he numbers in (1) and (2) by 100. Due o he difficulies involved in obaining curren period informaion on quaniies (or equivalenly, on expendiures), many saisical agencies loosely base 3 heir Consumer Price Index on he use of he Laspeyres formula 4 and he fixed base sysem. Therefore, i is of some ineres o look a some of he possible problems associaed wih he use of fixed base Laspeyres indexes. 1 This secion is largely based on he work of Hill (1988) (1993; ). 2 The chain principle was inroduced independenly ino he economics lieraure by Lehr (1885; 45-46) and Marshall (1887; 373). Boh auhors observed ha he chain sysem would miigae he difficulies due o he inroducion of new commodiies ino he economy, a poin also menioned by Hill (1993; 388). Fisher (1911; 203) inroduced he erm chain sysem. 3 As we saw in chaper 7, Consumer Price Indexes are usually aken o be Lowe (1823) indexes, which have he formula P Lo (p 0,p,q b ) p q b /p 0 q b, where q b is he quaniy vecor of a base year and p 0 and p are monhly price vecors peraining o monhs 0 and. 4 The Laspeyres formula beween monhs 0 and is P L (p 0,p,q 0 ) p q 0 /p 0 q 0, where q 0 is he quaniy vecor of he base monh 0.

2 2 The main problem wih he use of he fixed base Laspeyres index is ha he period 0 fixed baske of commodiies ha is being priced ou in period can ofen be quie differen from he period baske. 5 Thus if here are sysemaic rends in a leas some of he prices and quaniies 6 in he index baske, he fixed base Laspeyres price index P L (p 0,p,q 0,q ) can be quie differen from he corresponding fixed base Paasche price index, P P (p 0,p,q 0,q ). 7 This means ha boh indexes are likely o be an inadequae represenaion of he movemen in average prices over he ime period under consideraion. The fixed base Laspeyres quaniy index canno be used forever: evenually, he base period quaniies q 0 are so far removed from he curren period quaniies q ha he base mus be changed. Chaining is merely he limiing case where he base is changed each period Discussion of he Advanages and Disadvanages of Chaining The main advanage of he chain sysem is ha under normal condiions, chaining will reduce he spread beween he Paasche and Laspeyres indexes. 9 These wo indexes each provide an asymmeric perspecive on he amoun of price change ha has occurred beween he wo periods under consideraion and i could be expeced ha a single poin esimae of he aggregae price change should lie beween hese wo esimaes. Thus he use of eiher a chained Paasche or Laspeyres index will usually lead o a smaller difference beween he wo and hence o esimaes ha are closer o he ruh. 10 Hill (1993; 388), drawing on he earlier research of Szulc (1983) and Hill (1988; ), noed ha i is no appropriae o use he chain sysem when prices oscillae (or bounce o use Szulc s (1983; 548) erm). This phenomenon can occur in he conex of regular seasonal flucuaions or in he conex of price wars. However, in he conex of roughly monoonically changing prices and quaniies, Hill (1993; 389) recommended he use of chained symmerically weighed indexes. 11 The Fisher and Walsh indexes are examples of symmerically weighed indices. 5 The Lowe index suffers from a similar problem: if he base year b is fairly disan from he base monh 0, he base year quaniy vecor q b can be unrepresenaive for boh monhs 0 and. 6 Examples of rapidly downward rending prices and upward rending quaniies are compuers, elecronic equipmen of all ypes, inerne access and elecommunicaion charges. 7 Noe ha P L (p 0,p,q 0,q ) will equal P P (p 0,p,q 0,q ) if eiher he wo quaniy vecors q 0 and q are proporional or he wo price vecors p 0 and p are proporional. Thus in order o obain a difference beween he Paasche and Laspeyres indexes, nonproporionaliy in boh prices and quaniies is required. 8 Regular seasonal flucuaions can cause monhly or quarerly daa o bounce using he erm due o Szulc (1983) and chaining bouncing daa can lead o a considerable amoun of index drif ; i.e., if afer 12 monhs, prices and quaniies reurn o heir levels of a year earlier, hen a chained monhly index will usually no reurn o uniy. Hence, he use of chained indices for noisy monhly or quarerly daa is no recommended wihou careful consideraion. 9 See Diewer (1978; 895) and Hill (1988) (1993; ). Laer in his chaper, we will examine more closely under wha condiions chaining will reduce he spread beween he Paasche and Laspeyres indexes. 10 This observaion will be illusraed wih an arificial daa se in a laer chaper. 11 Noe ha all known superlaive indexes are symmerically weighed.

3 3 I is possible o be a bi more precise under wha condiions one should chain or no chain. Basically, one should chain if he prices and quaniies peraining o adjacen periods are more similar han he prices and quaniies of more disan periods, since his sraegy will lead o a narrowing of he spread beween he Paasche and Laspeyres indices a each link. 12 Of course, one needs a measure of how similar are he prices and quaniies peraining o wo periods. The similariy measures could be relaive ones or absolue ones. In he case of absolue comparisons, wo vecors of he same dimension are similar if hey are idenical and dissimilar oherwise. In he case of relaive comparisons, wo vecors are similar if hey are proporional and dissimilar if hey are nonproporional. 13 Once a similariy measure has been defined, he prices and quaniies of each period can be compared o each oher using his measure and a ree or pah ha links all of he observaions can be consruced where he mos similar observaions are compared wih each oher using a bilaeral index number formula. 14 Hill (1995) defined he price srucures beween he wo counries o be more dissimilar he bigger is he spread beween P L and P P ; i.e., he bigger is max {P L /P P, P P /P L }. The problem wih his measure of dissimilariy in he price srucures of he wo counries is ha i could be he case ha P L = P P (so ha he Hill measure would regiser a maximal degree of similariy) bu p 0 could be very differen han p. Thus here is a need for a more sysemaic sudy of similariy (or dissimilariy) measures in order o pick he bes one ha could be used as an inpu ino Hill s (1999a) (1999b) (2001) spanning ree algorihm for linking observaions. The Appendix o his chaper provides an inroducion o he sudy of dissimilariy indexes. For a more complee discussion, see Diewer (2002). 12 Walsh in discussing wheher fixed base or chained index numbers should be consruced, ook for graned ha he precision of all reasonable bilaeral index number formulae would improve, provided ha he wo periods or siuaions being compared were more similar and hence, for his reason, favored he use of chained indexes: The quesion is really, in which of he wo courses [fixed base or chained index numbers] are we likely o gain greaer exacness in he comparisons acually made? Here he probabiliy seems o incline in favor of he second course; for he condiions are likely o be less diverse beween wo coniguous periods han beween wo periods say fify years apar. Correa Moylan Walsh (1901; 206). Walsh (1921a; 84-85) laer reieraed his preference for chained index numbers. Fisher also made use of he idea ha he chain sysem would usually make bilaeral comparisons beween price and quaniy daa ha was more similar and hence he resuling comparisons would be more accurae: The index numbers for 1909 and 1910 (each calculaed in erms of ) are compared wih each oher. Bu direc comparison beween 1909 and 1910 would give a differen and more valuable resul. To use a common base is like comparing he relaive heighs of wo men by measuring he heigh of each above he floor, insead of puing hem back o back and direcly measuring he difference of level beween he ops of heir heads. Irving Fisher (1911; 204). I seems, herefore, advisable o compare each year wih he nex, or, in oher words, o make each year he base year for he nex. Such a procedure has been recommended by Marshall, Edgeworh and Flux. I largely mees he difficuly of non-uniform changes in he Q s, for any inequaliies for successive years are relaively small. Irving Fisher (1911; ). 13 Diewer (2002) akes an axiomaic approach o defining various indexes of absolue and relaive dissimilariy. 14 Fisher (1922; ) hined a he possibiliy of using spaial linking; i.e., of linking counries ha are similar in srucure. However, he modern lieraure has grown due o he pioneering effors of Rober Hill (1995) (1999a) (1999b) (2001). Hill (1995) used he spread beween he Paasche and Laspeyres price indexes as an indicaor of similariy and showed ha his crierion gives he same resuls as a crierion ha looks a he spread beween he Paasche and Laspeyres quaniy indexes.

4 4 The mehod of linking observaions explained in he previous paragraph based on he similariy of he price and quaniy srucures of any wo observaions may no be pracical in a saisical agency conex since he addiion of a new period may lead o a reordering of he previous links. However, he above scienific mehod for linking observaions may be useful in deciding wheher chaining is preferable or wheher fixed base indexes should be used while making monh o monh comparisons wihin a year. Some index number heoriss have objeced o he chain principle on he grounds ha i has no counerpar in he spaial conex: They [chain indexes] only apply o ineremporal comparisons, and in conras o direc indices hey are no applicable o cases in which no naural order or sequence exiss. Thus he idea of a chain index for example has no counerpar in inerregional or inernaional price comparisons, because counries canno be sequenced in a logical or naural way (here is no k+1 nor k 1counry o be compared wih counry k). Peer von der Lippe (2001; 12). 15 This is of course correc bu he approach of Rober Hill does lead o a naural se of spaial links. Applying he same approach o he ime series conex will lead o a se of links beween periods which may no be monh o monh bu i will in many cases jusify year over year linking of he daa peraining o he same monh. 3. When Will Chaining Give he Same Answer as Using a Fixed Base Index? I is of some ineres o deermine if here are index number formulae ha give he same answer when eiher he fixed base or chain sysem is used. Comparing he sequence of chain indexes defined by (1) above o he corresponding fixed base indexes defined by (2), i can be seen ha we will obain he same answer in all hree periods if he index number formula P saisfies he following funcional equaion for all price and quaniy vecors: (3) P(p 0,p 2,q 0,q 2 ) = P(p 0,p 1,q 0,q 1 ) P(p 1,p 2,q 1,q 2 ). If an index number formula P saisfies (3), hen P saisfies he circulariy es. 16 If i is assumed ha he index number formula P saisfies cerain properies or ess in addiion o he circulariy es above 17, hen Funke, Hacker and Voeller (1979) showed 15 I should be noed ha von der Lippe (2001; 56-58) is a vigorous criic of all index number ess based on symmery in he ime series conex alhough he is willing o accep symmery in he conex of making inernaional comparisons. Bu here are good reasons no o insis on such crieria in he ineremporal case. When no symmery exiss beween 0 and, here is no poin in inerchanging 0 and. Peer von der Lippe (2001; 58). 16 The es name is due o Fisher (1922; 413) and he concep was originally due o Wesergaard (1890; ). 17 The addiional ess are: (i) posiiviy and coninuiy of P(p 0,p 1,q 0,q 1 ) for all sricly posiive price and quaniy vecors p 0,p 1,q 0,q 1 ; (ii) he ideniy es; (iii) he commensurabiliy es; (iv) P(p 0,p 1,q 0,q 1 ) is

5 5 ha P mus have he following funcional form due originally o Konüs and Byushgens 18 (1926; ): 19 (4) P KB (p 0,p 1,q 0,q 1 N 1 p i ) 0 i 1 pi where he N consans i saisfy he following resricions: (5) i=1 N i = 1 and i > 0 for i = 1,,N. i Thus under very weak regulariy condiions, he only price index saisfying he circulariy es is a weighed geomeric average of all he individual price raios, he weighs being consan hrough ime. An ineresing special case of he family of indexes defined by (4) occurs when he weighs i are all equal. In his case, P KB reduces o he Jevons (1865) index: (6) P J (p 0,p 1,q 0,q 1 ) n=1 N (p n 1 /p n 0 ) 1/N. The problem wih he indexes defined by Konüs and Byushgens and Jevons is ha he individual price raios, p n 1 /p n 0, have weighs (eiher n or 1/n ) ha are independen of he economic imporance of commodiy n in he wo periods under consideraion. Pu anoher way, hese price weighs are independen of he quaniies of commodiy n consumed or he expendiures on commodiy n during he wo periods. Hence, hese indexes are no really suiable for use by saisical agencies a higher levels of aggregaion when expendiure share informaion is available. The above resuls indicae ha i is no useful o ask ha he price index P saisfy he circulariy es exacly. However, i is of some ineres o find index number formulae ha saisfy he circulariy es o some degree of approximaion since he use of such an index number formula will lead o measures of aggregae price change ha are more or less he same no maer wheher we use he chain or fixed base sysems. Irving Fisher (1922; 284) found ha deviaions from circulariy using his daa se and he Fisher ideal posiively homogeneous of degree one in he componens of p 1 and (v) P(p 0,p 1,q 0,q 1 ) is posiively homogeneous of degree zero in he componens of q Konüs and Byushgens show ha he index defined by (4) is exac for Cobb-Douglas (1928) preferences; see also Pollak (1983; ). The concep of an exac index number formula was explained in an earlier chaper. 19 This resul can be derived using resuls in Eichhorn (1978; ) and Vog and Bara (1997; 47). A simple proof can be found in Balk (1995). This resul vindicaes Irving Fisher s (1922; 274) inuiion who assered ha he only formulae which conform perfecly o he circular es are index numbers which have consan weighs Fisher (1922; 275) wen on o asser: Bu, clearly, consan weighing is no heoreically correc. If we compare 1913 wih 1914, we need one se of weighs; if we compare 1913 wih 1915, we need, heoreically a leas, anoher se of weighs. Similarly, urning from ime o space, an index number for comparing he Unied Saes and England requires one se of weighs, and an index number for comparing he Unied Saes and France requires, heoreically a leas, anoher.

6 6 price index P F were quie small. This relaively high degree of correspondence beween fixed base and chain indexes has been found o hold for oher symmerically weighed formulae like he Walsh index P W defined in earlier chapers. 20 Thus in mos ime series applicaions of index number heory where he base year in fixed base indexes is changed every 5 years or so, i will no maer very much wheher he saisical agency uses a fixed base price index or a chain index, provided ha a symmerically weighed formula is used. 21 This of course depends on he lengh of he ime series considered and he degree of variaion in he prices and quaniies as we go from period o period. The more prices and quaniies are subjec o large flucuaions (raher han smooh rends), he less he correspondence. 22 I is possible o give a heoreical explanaion for he approximae saisfacion of he circulariy es for symmerically weighed index number formulae. Anoher symmerically weighed formula is he Törnqvis index P T. 23 The naural logarihm of his index is defined as follows: (7) ln P T (p 0,p 1,q 0,q 1 ) n=1 N (1/2)(s n 0 +s n 1 )ln (p n 1 /p n 0 ) where he period expendiure shares s n are defined in he usual way. Alerman, Diewer and Feensra (1999; 61) show ha if he logarihmic price raios ln (p n /p -1 n ) rend linearly wih ime and he expendiure shares s n also rend linearly wih ime, hen he Törnqvis index P T will saisfy he circulariy es exacly. 24 Since many economic ime series on prices and quaniies saisfy hese assumpions approximaely, hen he Törnqvis index P T will saisfy he circulariy es approximaely. As was seen in an earlier chaper, he Törnqvis index generally closely approximaes he symmerically weighed Fisher and Walsh indexes, so ha for many economic ime series (wih smooh rends), all hree of hese symmerically weighed indexes will saisfy he circulariy es o a high enough degree of approximaion so ha i will no maer wheher we use he fixed base or chain principle. Walsh (1901; 401) (1921a; 98) (1921b; 540) inroduced he following useful varian of he circulariy es: 20 See for example Diewer (1978; 894). Walsh (1901; 424 and 429) found ha his 3 preferred formulae all approximaed each oher very well as did he Fisher ideal for his arificial daa se. 21 More specifically, mos superlaive indexes (which are symmerically weighed) will saisfy he circulariy es o a high degree of approximaion in he ime series conex. See chaper 4 for he definiion of a superlaive index. I is worh sressing ha fixed base Paasche and Laspeyres indices are very likely o diverge considerably over a 5 year period if compuers (or any oher commodiy which has price and quaniy rends ha are quie differen from he rends in he oher commodiies) are included in he value aggregae under consideraion. 22 Again, see Szulc (1983) and Hill (1988). 23 This formula was implicily inroduced in Törnqvis (1936) and explicily defined in Törnqvis and Törnqvis (1937). 24 This exacness resul can be exended o cover he case when here are monhly proporional variaions in prices and he expendiure shares have consan seasonal effecs in addiion o linear rends; see Alerman, Diewer and Feensra (1999; 65).

7 7 (8) 1 = P(p 0,p 1,q 0,q 1 ) P(p 1,p 2,q 1,q 2 ) P(p T 1,p T,q T 1,q T ) P(p T,p 0,q T,q 0 ). The moivaion for his es is he following one. Use he bilaeral index formula P(p 0,p 1,q 0,q 1 ) o calculae he change in prices going from period 0 o 1, use he same formula evaluaed a he daa corresponding o periods 1 and 2, P(p 1,p 2,q 1,q 2 ), o calculae he change in prices going from period 1 o 2,, use P(p T 1,p T,q T 1,q T ) o calculae he change in prices going from period T 1 o T, inroduce an arificial period T+1 ha has exacly he price and quaniy of he iniial period 0 and use P(p T,p 0,q T,q 0 ) o calculae he change in prices going from period T o 0. Finally, muliply all of hese indexes ogeher and since we end up where we sared, hen he produc of all of hese indexes should ideally be one. Diewer (1993a; 40) called his es a muliperiod ideniy es. 25 Noe ha if T = 2 (so ha he number of periods is 3 in oal), hen Walsh s es reduces o Fisher s (1921; 534) (1922; 64) ime reversal es. 26 Walsh (1901; ) showed how his circulariy es could be used in order o evaluae how good any bilaeral index number formula was. Wha he did was inven arificial price and quaniy daa for 5 periods and he added a sixh period ha had he daa of he firs period. He hen evaluaed he righ hand side of (8) for various formula, P(p 0,p 1,q 0,q 1 ), and deermined how far from uniy he resuls were. His bes formulae had producs ha were close o one. 27 This same framework is ofen used o evaluae he efficacy of chained indexes versus heir direc counerpars. Thus if he righ hand side of (8) urns ou o be differen han uniy, he chained indexes are said o suffer from chain drif. If a formula does suffer from chain drif, i is someimes recommended ha fixed base indexes be used in place of chained ones. However, his advice, if acceped would always lead o he adopion of fixed base indexes, provided ha he bilaeral index formula saisfies he ideniy es, P(p 0,p 0,q 0,q 0 ) = 1. Thus i is no recommended ha Walsh s circulariy es be used o decide wheher fixed base or chained indexes should be calculaed. However, i is fair o use Walsh s circulariy es as he originally used i i.e., as an approximae mehod for deciding how good a paricular index number formula is. In order o decide wheher o chain or use fixed base indexes, one should decide on he basis of how similar are he observaions being compared and choose he mehod which will bes link up he mos similar observaions. Appendix 1: An Inroducion o Indexes of Absolue Dissimilariy A.1 Inroducion 25 Walsh (1921a; 98) called his es he circular es bu since Fisher also used his erm o describe his ransiiviy es defined earlier by (3), i seems bes o sick o Fisher s erminology since i is well esablished in he lieraure. 26 Walsh (1921b; ) noed ha he ime reversal es was a special case of his circulariy es. 27 This is essenially a varian of he mehodology ha Fisher (1922; 284) used o check how well various formulae corresponded o his version of he circulariy es.

8 8 An absolue index of price dissimilariy regards he vecors p 1 and p 2 as being dissimilar if p 1 p 2 whereas a relaive index of price dissimilariy regards p 1 and p 2 as being dissimilar if p 1 p 2 where > 0 is an arbirary posiive number. Thus he relaive index regards he wo price vecors as being dissimilar only if relaive prices differ in he wo siuaions. The relaive index concep seems o be he mos useful for judging wheher he srucure of prices is similar or dissimilar across wo counries. However, assuming ha he quaniy vecors being compared are per capia quaniy vecors, hen he absolue concep seems o be more appropriae for judging he degree of similariy across counries. If per capia quaniy vecors are quie differen, hen i is quie likely ha he rich counry is consuming (or producing) a very differen bundle of goods and services han he poorer counry and hence big dispariies in he absolue level of q 1 versus q 2 are likely o indicae ha he componens of hese wo vecors are really no very comparable. In any case, i is of some ineres o develop he heory for boh he absolue and relaive conceps. Relaive indexes of price and quaniy similariy or dissimilariy are very useful in deciding how o aggregae up a large number of price and quaniy series ino a smaller number of aggregaes. 28 Finally, absolue indexes of dissimilariy can be useful in deciding when an observaion in a large cross secional daa se is an oulier. 29 In his appendix, we provide an inroducion o his opic by sudying absolue dissimilariy indexes when he number of commodiies is only one. We offer wha we hink are a fairly fundamenal se of axioms or properies ha such an absolue dissimilariy index should saisfy and characerize he se of indexes which saisfy hese axioms. A.2 A Firs Approach o Indexes of Absolue Dissimilariy We denoe our absolue dissimilariy index as a funcion of wo variables, d(x,y), where x and y are resriced o be posiive scalars. The wo variables x and y could be he wo prices of he firs commodiy in he wo counries, p 1 1 and p 1 2, or hey could be he wo per capia quaniies of he firs commodiy in he wo counries, q 1 1 and q 1 2. I is obvious ha d(x,y) could be considered o be a disance funcion of he ype ha occurs in he mahemaics lieraure. However, he axioms ha we impose on d(x,y) are somewha unconvenional as we shall see. The 6 fundamenal axioms or properies ha we hink an absolue dissimilariy index should saisfy are he following ones 30 (noe ha he domain of definiion for d(x,y) is x > 0 and y > 0): 28 For applicaions along hese lines, see Allen and Diewer (1981). 29 Rober Hill poined ou his use for a dissimilariy index. 30 Counerpars o Axioms A2-A6 in he conex of relaive dissimilariy indexes were proposed by Allen and Diewer (1981; 433). Sergueev (2001; 4) also proposed counerpars o A2, A4 and A6 in he conex of similariy indexes (as opposed o dissimilariy indexes).

9 9 A1: Coninuiy: d(x,y) is a coninuous funcion. A2: Ideniy: d(x,x) = 0 for all x > 0. A3: Posiiviy: d(x,y) > 0 for all x y. A4: Symmery: d(x,y) = d(y,x) for all x and y. A5: Invariance o Changes in Unis of Measuremen: d( x, y) = d(x,y) for all > 0, x > 0, y > 0. A6: Monooniciy: d(x,y) is increasing in y if y x. Some commens on he axioms are in order. The coninuiy assumpion is generally made in order o rule ou indexes ha behave erraically. The ideniy assumpion is a sandard one in he mahemaics lieraure; i.e., he absolue disance beween wo poins x and y is zero if x equals y. A3 ells us ha here is a posiive amoun of dissimilariy beween x and y if x and y are differen. The symmery propery is very imporan: i says ha he degree of dissimilariy beween x and y is independen of he ordering of x and y. A5 is anoher imporan propery from he viewpoin of economics: since unis of measuremen for commodiies are essenially arbirary, we would like our dissimilariy measure o be independen of he unis of measuremen. Finally, A6 says ha as y ges bigger han x, he degree of dissimilariy beween x and y grows. This is a very sensible propery. Problem 1. Show ha axiom A3 is implied by he oher axioms. I urns ou ha here is a fairly simple characerizaion of he class of dissimilariy indexes d(x,y) ha saisfy he above axioms; i.e., we have he following Proposiion: Proposiion 1: Le d(x,y) be a funcion of wo variables ha saisfies he axioms A1-A6. Then d(x,y) has he following represenaion: (1) d(x,y) = f[max{x/y, y/x}] where f(u) is a coninuous, monoonically increasing funcion of one variable, defined for u 1 wih he following addiional propery: (2) f(1) = 0. Conversely, if f(u) has he above properies, hen d(x,y) defined by (1) has he properies A1-A6.

10 10 Proof: Using A5 wih = x 1, we have: (3) d(x,y) = d(1,y/x). Now use A5 wih = y 1 and we find: (4) d(x,y) = d(x/y,1) = d(1,x/y) using A4. For u 1, define he coninuous funcion of one variable, f(u) as (5) f(u) d(1,u) ; u 1. Using A2 and definiion (5), we have (6) f(1) = d(1,1) = 0. Using A6, we deduce ha f(u) is an increasing funcion of u for u 1. Now if x y, hen from (4) and definiion (5), we deduce ha d(x,y) = f(x/y). If however, y x, hen from (3) and definiion (5), we deduce ha d(x,y) = f(y/x). These wo resuls can be combined ino he following resul: (7) d(x,y) = f[max{x/y, y/x}] which complees he firs par of he Proposiion. Going he oher way, if f(u) is an increasing, coninuous funcion for u 1 wih f(1) = 0, hen if we define d(x,y) using (1), i is easy o verify ha d(x,y) saisfies he axioms A1-A6. Q.E.D. Problem 2. Show ha if f(u) is an increasing, coninuous funcion for u 1 wih f(1) = 0, hen d(x,y) defined by (1) saisfies he axioms A1-A6. Example 1: The asympoically linear dissimilariy index: Le f(u) u + u 1 2 for u 1. Noe ha f (u) = 1 u 2 > 0 for u > 1, which shows ha f(u) is increasing for u 1. Noe ha as u ends o infiniy, f(u) approaches he linear funcion u 2. Hence f(u) is asympoically linear. Since f(1) = 0, we see ha f(u) saisfies he required regulariy condiions and he associaed absolue dissimilariy index is 31 (8) d(x,y) = (x/y) + (y/x) 2 = [(x/y) 1] + [(y/x) 1] ; x > 0 ; y > 0 31 If x y, hen max {x/y, y/x} is x/y and d(x,y) f[max {x/y, y/x}] = f[x/y] = (x/y) + (y/x) 2. If y x, hen max {x/y, y/x} is y/x and d(x,y) f[max {x/y, y/x}] = f[y/x] = (y/x) + (x/y) 2 = (x/y) + (y/x) 2.

11 11 and i saisfies he axioms A1-A6. Example 2: The asympoically quadraic dissimilariy index: Le f(u) [u 1] 2 + [u 1 1] 2 for u 1. Noe ha f (u) = 2[u 1] + 2[u 1 1]( 1)u 2 > 0 for u > 1, which shows ha f(u) is increasing for u 1. Since f(1) = 0, we see ha f(u) saisfies he required regulariy condiions and he associaed absolue dissimilariy index is (9) d(x,y) = [(x/y) 1] 2 + [(y/x) 1] 2 ; x > 0 ; y > 0 and i saisfies he axioms A1-A6. Noe ha for boh of hese examples, he resuling d(x,y) is infiniely differeniable. Problems 3. Show ha he d(x,y) defined by (8) saisfies he axioms A1-A6. 4. Show ha he d(x,y) defined by (9) saisfies he axioms A1-A6. In he following secion, we show how a large class of one variable dissimilariy indexes can be defined. Then in he following secion, we will add some addiional axioms in an aemp o narrow down he choice of a paricular index o be used in applicaions. A.3 An Alernaive Approach for Generaing Absolue Dissimilariy Indexes. Le g and h be coninuous monoonically increasing funcions of one variable wih g(0) = 0 and consider he following class of dissimilariy indexes: (10) d g,h (x,y) g{ h(y/x) h(1) }. Thus we firs ransform y/x and 1 by he funcion of one variable h, calculae he difference, h(y/x) h(1), ake he absolue value of his difference and hen ransform his difference by g. I is easy o verify ha he d defined by (10) saisfies all of he axioms A1-A6 wih he excepion of A4, he symmery axiom, d(x,y) = d(y,x). However, his defec can be readily overcome. Noe ha d g,h (y,x) g{ h(x/y) h(1) } also saisfies A1-A6 wih he excepion of A4. Thus, if we ake a symmeric mean 32 of hese wo indexes 33, we will obain a new index which saisfies axiom A4. Hence, le m be a symmeric mean 32 Diewer (1993b; 361) defined a symmeric mean of a and b as a funcion m(a,b) ha has he following properies: (1) m(a,a) = a for all a > 0 (mean propery); (2) m(a,b) =m(b,a) for all a > 0, b > 0 (symmery propery); (3) m(a,b) is a coninuous funcion for a > 0, b > 0 (coninuiy propery); (4) m(a,b) is a sricly increasing funcion in each of is variables (increasingness propery). 33 Our mehod for convering a measure ha is no symmeric ino a symmeric mehod is he counerpar o Irving Fisher s (1922) recificaion procedure, which is acually due o Walsh (1921).

12 12 funcion of wo variables and le g and h be coninuous monoonically increasing funcions of one variable wih g(0) = 0 and consider he following class of symmeric monoonic ransformaion dissimilariy indexes: (11) d g,h,m (x,y) m[g{ h(y/x) h(1) }, g{ h(x/y) h(1) }]. Proposiion 2: Le g and h be coninuous monoonically increasing funcions of one variable wih g(0) = 0 and le m(a,b) be a symmeric mean. Then each member of he class of symmeric monoonic ransformaion indexes d g,h,m (x,y) defined by (11) saisfies he axioms A1-A6. Proof: The proofs of A1-A5 are lef o a problem. We verify axiom A6. Le y > y x > 0. Then (12) d g,h,m (x,y ) m[g{ h(y /x) h(1) }, g{ h(x/y ) h(1) }] = m[g{h(y /x) h(1)}, g{h(1) h(x/y )}] using y > x and he monooniciy of h > m[g{h(y /x) h(1)}, g{h(1) h(x/y )}] using y > y, x > 0 and he monooniciy of h, g and m > m[g{h(y /x) h(1)}, g{h(1) h(x/y )}] using y > y, x > 0 and he monooniciy of h, g and m = m[g{ h(y /x) h(1) }, g{ h(x/y ) h(1) }] using y > x and he monooniciy of h d g,h,m (x,y ). Q.E.D. Problem 5. Show ha he index d g,h,m (x,y) defined by (11) saisfies he axioms A1-A5. Le us ry and specialize he class of funcional forms defined by (11). The simples symmeric mean m of wo numbers is he arihmeic mean and so le us se m(a,b) = (1/2)a + (1/2)b. I is also convenien o ge rid of he absolue value funcion in (11) (so ha he resuling dissimilariy index will be differeniable) and his can be done in he mos simple fashion by seing g(u) = u This leads us o following class of simple symmeric ransformaion dissimilariy indexes, which depends only on he coninuous monoonic funcion h: (13) d h (x,y) (1/2)[h(y/x) h(1)] 2 + (1/2)[h(x/y) h(1)] There is anoher good reason for his choice of g. In mos applicaions, we wan he slope of g(u) o be zero a u = 0 and hen increase as u increases. This means he amoun of dissimilariy beween x and y will be close o zero in a neighborhood of poins where x is close o y bu he degree of dissimilariy will grow a an increasing rae as x diverges from y. We will formalize hese properies as axioms A7 and A8 in he nex secion. Hence if we wan he slope of g(u) o increase a a consan rae as u increases, hen g(u) = u 2 is he simples funcion which will accomplish his ask.

13 13 The wo simples choices for h are h(u) u and h(u) ln u. 35 lead o he following concree dissimilariy indexes: These wo choices for h Example 3: The linear quadraic dissimilariy index: (14) d(x,y) (1/2)[(y/x) 1] 2 + (1/2)[(x/y) 1] 2. Noe ha his example is essenially he same as Example 2. Example 4: The log quadraic dissimilariy index: (15) d(x,y) (1/2)[ln(y/x) ln(1)] 2 + (1/2)[ln(x/y) ln(1)] 2 = (1/2)[lny lnx] 2 + (1/2)[lnx lny] 2 = [lny lnx] 2 = [ln(y/x)] 2. Our conclusion a his poin is ha even in he one variable case, here are a large number of possible measures of absolue dissimilariy ha could be chosen. Hence, in he following secion, we add some addiional axioms o our lis of axioms, A1-A6, in an aemp o narrow down his large number of possible choices. A.4 Addiional Axioms for One Variable Absolue Dissimilariy Indexes Consider he following axiom: A7: Convexiy: d(x,y) is a convex funcion of y for y x > 0. The meaning of his axiom is ha we wan he amoun of dissimilariy beween x and y o grow a a consan or increasing rae as y grows bigger han x. Pu anoher way, we do no wan he rae of increase in dissimilariy o decrease as y grows bigger han x. Alhough his propery seems o be a reasonable one for many purposes, i mus be conceded ha his propery is no as fundamenal as he previous 6 properies. Proposiion 3: The asympoically linear dissimilariy index defined by (8) and he linear quadraic dissimilariy index defined by (14) saisfy he convexiy axiom A7 bu he log quadraic dissimilariy index defined by (15) does no saisfy A7. Proof: Le y x > 0. For he d(x,y) defined by (8), we find ha 2 d(x,y)/ y 2 = 2x/y 3 > 0 and so he asympoically linear dissimilariy index defined by (8) is convex in y. For he d(x,y) defined by (15), we find ha 2 d(x,y)/ y 2 = 2(x/y) 2 [1 x 1 ln(y/x)] which is negaive for y large enough and hence he log quadraic dissimilariy index defined by (15) does no saisfy A7. 35 Ber Balk suggesed he following choice for h: h(u) u 1/2.

14 14 For he d(x,y) defined by (14), we find ha: (16) 2 d(x,y)/ y 2 = x 2 + 3x 2 y 4 2xy 3 g(y). Le us aemp o minimize g(y) defined in (16) over y x. We have: (17) g (y) = 12x 2 y 5 + 6xy 4 = 0. The posiive roos of (17) are y* = 2x and y** = +. We find ha g(y) aains a sric local minimum a y = 2x and his urns ou o be he global minimum of g(y) for y x. Thus we have for y x: (18) f (y) f (2x) = x 2 + 3x 2 (2x) 4 2x (2x) 3 > 0 and hence he linear quadraic dissimilariy index defined by (14) saisfies A7. Q.E.D. How can we choose beween he asympoically linear dissimilariy index defined by (8) and he asympoically quadraic dissimilariy index defined by (9) or (14)? Boh indexes behave similarly for x close o y bu as y diverges from x, he amoun of dissimilariy beween x and y will grow roughly quadraically in y for he index defined by (14) whereas for he index defined by (8), he amoun of dissimilariy will end owards a linear in y rae. Hence he choice beween he wo indexes depends on how fas one wans he amoun of dissimilariy beween x and y o grow as y grows bigger han x. I should be noed ha he index defined by (14) will be much more sensiive o ouliers in he daa so perhaps for his reason, he index defined by (8) should be used when here is he possibiliy of errors in he daa. Anoher axiom which is also no fundamenal bu does seem reasonable is he following one: A8: Differeniabiliy: d(x,y) is a once differeniable funcion of wo variables. The real impac of he axiom A8 is along he ray where x = y. If we look a he proof of Proposiion 1, we see ha if we add A8 o he lis of axioms, he effec of he differeniabiliy axiom is o force he derivaive of f(u) a u = 1 o be 0; i.e., under A8, we mus have f (1) = 0. In many applicaions, his will be a very reasonable resricion on f since i implies ha he amoun of dissimilariy beween x and y will be very small when x is very close o y. All of our examples 1 o 4 above saisfy he differeniabiliy axiom. We now consider anoher axiom for d(x,y), which is perhaps more difficul o jusify, bu i does deermine he funcional form for d: A9: Addiiviy: d(x,x + y + z) = d(x,x + y) + d(x,x + z) for all x > 0, y 0 and z 0.

15 15 Proposiion 4: Suppose d(x,y) saisfies he axioms A1-A6 and A9. Then d has he following funcional form: 36 (19) d(x,y) = [max{x/y, y/x} 1] where > 0. Proof: If d(x,y) saisfies A1-A6, hen by Proposiion 1, d(x,y) = f[max{x/y, y/x}] where f(u) is coninuous, increasing for u 1 wih f(1) = 0. Subsiue his represenaion for d(x,y) ino A9 and leing x > 0, y 0 and z 0, we find ha f saisfies he following funcional equaion: (20) f[1 + (y/x) + (z/x)] = f[1 + (y/x)] + f[1 + (z/x)] ; x > 0, y 0 and z 0. Define he variables u and v as follows: (21) u y/x ; v z/x. Subsiuing (21) ino (20), we find ha f saisfies he following funcional equaion: (22) f(1 + u + v) = f(1 + u) + f(1 + v) ; u 0, v 0. Define he funcion g as follows: (23) g(u) f(1 + u). Using (23), (22) can be rewrien as follows: (24) g(u + v) = g(u) + g(v) ; u 0, v 0. Bu (24) is Cauchy s firs funcional equaion or a special case of Pexider s (1903) firs funcional equaion 37 and has he following soluion: (25) g(x) = x ; x 0 where is a consan. Using (23) and (25), (26) f(u) = (u 1) ; u 1. Equaion (26) implies ha d is equal o he righ hand side of (19). However, in order ha f(u) be increasing for u 1, we require ha > 0, which complees he proof. Q.E.D. 36 The f(u) ha corresponds o his funcional form is f(u) [u 1] where > 0. The d(x,y) defined by (12) also saisfies he convexiy axiom A7 bu i does no saisfy he differeniabiliy axiom A8. 37 See chaper 2 or Eichhorn (1978; 49) for a more accessible reference.

16 16 Le us se = 1 in (19) and call he resuling d(x,y), example 5, he linear dissimilariy index. I can be seen ha for large y, he dissimilariy indexes defined by examples 1 and 5 will approach each oher. The big difference beween he wo indexes is along he ray where x = y: he linear dissimilariy index will no be differeniable along his ray, whereas he asympoically linear dissimilariy index will be differeniable everywhere. Also for x close o y, he linear dissimilariy index will be greaer han he corresponding asympoically linear dissimilariy measure. We conclude his secion by indicaing a simple way for deermining he exac funcional form for d(x,y): we need only consider he behavior of d(1,y) for y 1. This behavior of he funcion d deermines he underlying generaor funcion f(u) ha appeared in he Proposiion 1. Hence consider he following axioms for d: A10: d(1,y) = (y 1) y 1, where > 0; A11: d(1,y) = ln y ; y 1; A12: d(1,y) = e y e ; y 1. I is sraighforward o show ha if d(x,y) saisfies A1-A6 and A10, hen d is equal o he following funcion: (example 6): (27) d(x,y) = [max{x/y, y/x} 1] ; > 0. Of course, if = 1, hen Example 6 reduces o Example Similarly, i is sraighforward o show ha if d(x,y) saisfies A1-A6 and A11, hen d is equal o he following funcion: (example 7): 39 (28) d(x,y) = ln [max{x/y, y/x}]. Finally, if d(x,y) saisfies A1-A6 and A12, hen d is equal o he following funcion: (example 8): 40 (29) d(x,y) = e max{x/y,y/x} e. The funcional forms for he dissimilariy indexes defined by (27)-(29) are all relaively simple bu hey all have a disadvanage: namely, hey are no differeniable along he ray where x = y. Hence, hey are probably no suiable for many economic applicaions. We urn now o N variable measures of absolue dissimilariy. 38 The d(x,y) defined by (27) saisfies he convexiy axiom A7 if and only if This d(x,y) does no saisfy A7. 40 This d(x,y) does saisfy he convexiy axiom A7.

17 17 A.5 Axioms for Absolue Dissimilariy Indexes in he N Variable Case We now le x [x 1,...,x N ] and y [y 1,...,y N ] be sricly posiive vecors (eiher price or quaniy) ha are o be compared in an absolue sense. Le D(x,y) be he absolue dissimilariy index, defined for all sricly posiive vecors x and y. The following 6 axioms or properies are fairly direc counerpars o he 6 fundamenal axioms ha were inroduced in secion A.2 above. B1: Coninuiy: D(x,y) is a coninuous funcion defined for all x >> 0 N and y >> 0 N. B2: Ideniy: D(x,x) = 0 for all x >> 0 N. B3: Posiiviy: D(x,y) > 0 for all x y. B4: Symmery: D(x,y) = D(y,x) for all x >> 0 N and y >> 0 N. B5: Invariance o Changes in Unis of Measuremen: D( 1 x 1,..., N x N ; 1 y 1,..., N y N ) = D(x 1,...,x N ;y 1,...,y N ) = D(x,y) for all n > 0, x n > 0, y n > 0 for n = 1,...,N. 41 B6: Monooniciy: D(x,y) is increasing in he componens of y if y x. The above axioms or properies can be regarded as fundamenal. However, hey are no sufficien o give a nice characerizaion Proposiion like Proposiion 1 in secion A.2. Hence we need o add addiional properies o deermine D. Possible addiional properies are he following ones: B7: Invariance o he ordering of commodiies: D(Px,Py) = D(x,y) where Px denoes a permuaion of he componens of he x vecor and Py denoes he same permuaion of he componens of he y vecor. B8: Addiive Separabiliy: D(x,y) = n=1 N d n (x n,y n ). The N funcions of wo variables, d n (x n,y n ), are obviously absolue dissimilariy measures ha give us he degree of dissimilariy beween he componens of he vecors x and y. Proposiion 5: Suppose D(x,y) saisfies B1-B8. Then here exiss a coninuous, increasing funcion of one variable, f(u), such ha f(1) = 0 and D(x,y) has he following represenaion in erms of f: (30) D(x,y) = n=1 N f[max{x n /y n, y n /x n }]. 41 Noe ha his axiom implies ha D has he homogeneiy propery D( x, y) = D(x,y). To see his, le each n =.

18 18 Conversely, if D(x,y) is defined by (30) where f is a coninuous, increasing funcion of one variable wih f(1) = 0, hen D saisfies B1-B8. Proof: Using B2 and B8, we have (31) D(1 N,1 N ) = n=1 N d n (1,1) = 0. Thus (32) D(x,y) = D(x,y) D(1 N,1 N ) using (31) = N n=1 d n (x n,y n ) N n=1 d n (1,1) using B8 = N n=1 d n *(x n,y n ) where he d n *(x n,y n ) are defined as: (33) d n *(x n,y n ) d n (x n,y n ) d n (1,1) ; n = 1,2,...,N. I is easy o check ha he d n * funcions saisfy he following resricions: (34) d n *(1,1) = 0 ; n = 1,2,...,N. Using (32) and (33), we have: (35) D(x 1,1 N 1,y 1,1 N 1 ) = d 1 *(x 1,y 1 ) + n=2 N d n *(1,1) = d 1 *(x 1,y 1 ) using (34). Properies B1-B6 on D imply ha d 1 *(x 1,y 1 ) will saisfy properies A1-A6 lised in secion A.2 of he Appendix above. Hence, we may apply Proposiion 1 and conclude ha d 1 *(x 1,y 1 ) has he following represenaion: (36) d 1 *(x 1,y 1 ) = f[max{x 1 /y 1, y 1 /x 1 }] for some coninuous, increasing funcion of one variable f(u) defined for u 1 wih f(1) = 0. Using B7, we deduce ha (37) d n *(x n,y n ) = d 1 *(x n,y n ) = f[max{x n /y n, y n /x n }] ; for n = 2,...,N using (36) and his esablishes (30). Q.E.D. The second half of he Proposiion is sraighforward. Thus adding he axioms B7 and B8 o he earlier axioms B1-B6 essenially reduces he N dimensional case down o he one dimensional case.

19 19 In applicaions, i is someimes useful o be able o compare he amoun of dissimilariy beween wo N dimensional vecors x and y o he amoun of dissimilariy beween wo M dimensional vecors u and v. If we decide o use he funcion of one variable f o generae he dissimilariy index defined by (30), hen we can achieve comparabiliy across vecors of differen dimensionaliy if we modify (30) and define he following family of dissimilariy indexes (which depend on N, he dimensionaliy of he vecors x and y): (38) D N (x,y) n=1 N (1/N)f[max{x n /y n, y n /x n }]. Recall examples 1 and 2 in secion 2. We use he generaing funcions f(u) for hese examples o consruc N variable measures of absolue dissimilariy beween he posiive vecors x and y. Using he generaing funcion f(u) [u 1] 2 + [u 1 1] 2 in (38) gives us he following N dimensional asympoically linear quadraic index of absolue dissimilariy, which is he N dimensional generalizaion of Example 1 above, which we now label as example 9: (39) D AL (x,y) (1/N) n=1 N [(y n /x n ) + (x n /y n ) 2]. Using he generaing funcion f(u) [u 1] 2 + [u 1 1] 2 in (38) gives us he following N dimensional asympoically quadraic index of absolue dissimilariy, which is he N dimensional generalizaion of Example 2 above, which we now label as example 10: (40) D AQ (x,y) (1/N) n=1 N [(y n /x n ) 1] 2 + (1/N) n=1 N [(x n /y n ) 1] 2. The indexes defined by (39) and (40) are our preferred indexes of absolue dissimilariy. The index defined by (40) is less sensiive o measuremen errors and ouliers so under mos circumsances, i seems o be a preferred choice. 42 We urn now o a discussion of relaive dissimilariy indexes in he case of N commodiy prices or quaniies ha mus be compared. 43 A.6 Axioms for Relaive Dissimilariy Indexes in he N Variable Case In making relaive comparisons, we regard x and y as being compleely similar if x is proporional o y or if y is proporional o x; i.e., if y = x for some scalar > 0. We denoe he relaive dissimilariy index beween wo vecors x and y by (x,y). The earlier axioms B1-B7 for absolue dissimilariy indexes are now replaced by he following axioms: C1: Coninuiy: (x,y) is a coninuous funcion defined for all x >> 0 N and y >> 0 N. 42 Hill (2004) has adoped he corresponding weighed relaive index of dissimilariy in his mos recen empirical work. 43 The case N = 1 is no relevan in he case of relaive dissimilariy indexes so we mus move righ away ino he N 2 dimensional case.

20 20 C2: Ideniy: (x, x) = 0 for all x >> 0 N and scalars > 0. C3: Posiiviy: (x,y) > 0 if y x for any > 0. C4: Symmery: (x,y) = (y,x) for all x >> 0 N and y >> 0 N. C5: Invariance o Changes in Unis of Measuremen: ( 1 x 1,..., N x N ; 1 y 1,..., N y N ) = (x 1,...,x N ;y 1,...,y N ) = (x,y) for all n > 0, x n > 0, y n > 0 for n = 1,...,N. C6: Invariance o he Ordering of Commodiies: (Px,Py) = (x,y) where Px is a permuaion or reordering of he componens of x and Py is he same permuaion of he componens of y. C7: Proporionaliy: (x, y) = (x,y) for all x >> 0 N, y >> 0 N and scalars > 0. The las axiom says ha he degree of relaive dissimilariy beween he vecors x and y remains he same if y is muliplied by he arbirary posiive number. The above axioms all seem o be fairly fundamenal in he relaive dissimilariy index conex. 44 We have no developed a counerpar o he absolue monooniciy axiom B6 for relaive indexes of dissimilariy because i is no clear wha he appropriae relaive axiom should be. This is a opic for furher research. Also, we do no have any nice characerizaion heorems for relaive dissimilariy indexes ha are analogous o Proposiion 5 in he previous secion. However, we do have a sraegy for adaping he absolue dissimilariy indexes o he relaive conex. Our suggesed sraegy is his. Firs, find a scale index S(x,y) ha is essenially a price or quaniy index beween he vecors x and y and ha has he propery S(x, x) =. Second, find a suiable absolue dissimilariy index, D(x,y). Finally, use he scale index S and he absolue dissimilariy index D in order o define he following relaive dissimilariy index : (41) (x,y) D(S(x,y)x, y). Thus in (23), we scale up he base vecor x by he index number S(x,y) which makes i comparable in an absolue sense o he vecor y. We hen apply an absolue index of dissimilariy D o he scaled up x vecor, S(x,y)x, and he vecor y. Naurally, in order for he defined by (41) o saisfy he axioms C1-C7, i will be necessary for D and S o saisfy cerain properies. We will assume ha he absolue dissimilariy index D saisfies B1-B5 and B7 in he previous secion. We will also impose he following properies on he scale index S(x,y): D1: Coninuiy: S(x,y) is a coninuous funcion defined for all x >> 0 N and y >> 0 N. 44 Axioms C2-C7 were proposed by Allen and Diewer (1981; 433).

21 21 D2: Ideniy: S(x,x) = 1 for all x >> 0 N. D3: Posiiviy: S(x,y) > 0 for all x >> 0 N and y >> 0 N. D4: Time or Place Reversal: S(x,y) = 1/S(y,x) for all x >> 0 N and y >> 0 N. D5: Invariance o Changes in Unis of Measuremen: S( 1 x 1,..., N x N ; 1 y 1,..., N y N ) = S(x 1,...,x N ;y 1,...,y N ) = S(x,y) for all n > 0, x n > 0, y n > 0 for n = 1,...,N. D6: Invariance o he Ordering of Commodiies: S(Px,Py) = S(x,y) where Px is a permuaion or reordering of he componens of x and Py is he same permuaion of he componens of y. D7: Proporionaliy: S(x, y) = S(x,y) for all x >> 0 N, y >> 0 N and scalars > 0. Proposiion 6: If he scale funcion S(x,y) saisfies D1-D7 and he absolue dissimilariy index D(x,y) saisfies B1-B5 and B7 lised in he previous secion, hen he relaive dissimilariy index (x,y) defined by (23) saisfies properies C1-C7. Proof: Properies C1 and C5 are obvious. Now check propery C2: (42) (x, x) D(S(x, x)x, x) using definiion (41) = D( S(x,x)x, x) using D7 = D( x, x) using D2 = 0 using B2. Now check propery C3. Given x and y, suppose ha y x for any > 0. Using definiion (41), we have: (43) (x,y) D(S(x,y)x, y) = D( x, y) where = S(x,y) > 0 using D3 > 0 using B3 since y x. Check propery C4: (44) (x,y) D(S(x,y)x, y) using definiion (41) = D(1 N, y 1 /x 1 S(x,y),...,y N /x N S(x,y)) using B5 = D(1 N, S(y,x)y 1 /x 1,...,S(y,x)y N /x N ) using D4 = D(x, S(y,x)y) using B5 again = D(S(y,x)y,x) using B4 (y,x) using definiion (23). Propery C6 follows from Properies B7 and D6.

22 22 Finally, check Propery C7. Le x >> 0 N, y >> 0 N and scalars > 0. Then by definiion (23), (45) (x, y) D(S(x, y)x, y) = D( S(x,y)x, y) using D7 = D(S(x,y)x, y) using B5 wih all n = = (x,y) using definiion (41). Q.E.D. The above Proposiion can be used in order o generae a wide class of relaive dissimilariy indexes. We conclude his secion by giving some examples of how Proposiion 6 could be applied in order o define some indexes of relaive dissimilariy. Example 11: Recall he N variable index of dissimilariy D AL (x,y) defined by (39) above. I can be verified ha his absolue index of dissimilariy saisfies axioms B1-B9. We need o choose a scale index S(x,y) ha saisfies he axioms D1-D7. The simples choice for such an S is: (46) S J (x,y) n=1 N (y n /x n ) 1/N. Thus S(x,y) is he geomeric mean of he y n divided by he geomeric mean of he x n. This funcional form (for a price index) is due o Jevons (1865) and i is sill used oday as a funcional form for an elemenary price index. I can be verified ha S J saisfies he axioms D1-D7. I should be noed ha he following scale indexes do no saisfy he ime reversal es, D4: (47) S A (x,y) n=1 N (1/N)(y n /x n ) ; (48) S H (x,y) [ n=1 N (1/N)(y n /x n ) 1 ] 1. Thus S A is he arihmeic mean 45 of he raios y n /x n and S H is he harmonic mean of he raios y n /x n. Insering S J defined by (46) ino formula (41) where D is defined by (39) leads o he following asympoically linear index of relaive dissimilariy (which saisfies C1-C7): (49) AL (x,y) D AL (S J (x,y)x, y) = n=1 N (1/N)[(S J (x,y)x n /y n ) + (y n /S J (x,y)x n ) 2]. 45 S A is known in he price index lieraure as he Carli (1764) index. Noe ha he geomeric mean of S A and S H does saisfy he axioms D1-D7 and hence could be used in place of he Jevons scale index S J. S AH (x,y) [S A (x,y)s H (x,y)] 1/2 has been suggesed as he funcional form for an elemenary price index by Carruhers, Sellwood and Ward (1980).