Index Number Theory Using Differences Rather Than Ratios

Transcription

1 Idex Number Theory Usig Differeces Rather Tha Ratios By W. ERWIN DIEWERT ABSTRACT. Traditioal idex umber theory decomposes a value ratio ito the product of a price idex times a quatity idex. The price (quatity) idex is iterpreted as a aggregate price (quatity) ratio. The preset paper takes a alterative approach to idex umber theory, started by Beet ad Motgomery i the 92s, which decomposes a value differece ito the sum of a price differece plus a quatity differece. Axiomatic ad ecoomic approaches to this alterative brach of idex theory are cosidered i the preset paper. The aalysis preseted has some relevace to accoutig theory i which reveue, cost, or profit chages eed to be decomposed ito price quatity compoets or where stadard or budgeted performace is compared with actual performace (variace aalysis). The methodology preseted i the paper is also relevat for cosumer surplus aalysis. I Itroductio Whe, forty-two years ago, I wrote my doctor s thesis o certai mathematical ivestigatios i the theory of value ad prices, I was a studet *W. Erwi Diewert is a Professor of Ecoomics at the Uiversity of British Columbia. He has published over 7 papers i jourals ad over 7 chapters i books. His mai areas of research iclude duality theory, flexible fuctioal forms, idex umber theory (icludig the cocept of a superlative idex umber formula), the measuremet of productivity, ad the calculatio of excess burdes of taxatio. He has acted as a cosultat o measuremet ad regulatory issues for the Iteratioal Moetary Fud, the World Bak, the Bureau of Labor Statistics, the Bureau of Ecoomic Aalysis, the OECD, the New Zealad Treasury, the Busiess Roudtable i New Zealad, Bell Caada, B.C. Telephoe, the America Associatio of Railways, the Victoria Treasury, ad Idustry Caada. The research for this paper was partially supported by a Strategic Grat to a team of researchers at the Uiversities of British Columbia ad Alberta (the PEER Group). The author thaks Bert Balk ad PEER Group members Alice Nakamura, Alla Russell, ad Peter Lawrece for helpful commets. The America Joural of Ecoomics ad Sociology, Vol. 64, No. ( Jauary, 25). 25 America Joural of Ecoomics ad Sociology, Ic.

2 32 The America Joural of Ecoomics ad Sociology of mathematical physics ad, with youthful ethusiasm, dreamed dreams of seeig ecoomics, or oe brach of it, grow ito a true sciece by the same methods which had log sice built up physics ito a true ad majestic sciece. Irvig Fisher (933: 2) Rather tha review all of the may cotributios of Irvig Fisher to the idex umber literature, we will develop a alterative brach of idex umber theory that had its origis aroud the time that Fisher (9, 92, 922) developed his test approach. This alterative brach of idex umber theory was started by T. L. Beet (92) ad J. K. Motgomery (929, 937), but for various reasos, their approach ever prospered ad has bee mostly forgotte by presetday idex umber theorists. I order to explai this alterative approach to idex umber theory, we first eed to explai the traditioal approach. Suppose we have collected price ad quatity iformatio o N commodities for a base period ad a curret period. Deote the price ad quatity of commodity i period t as ad, respectively, for =, t t 2,..., N ad t =,. Defie the period t price ad quatity vectors t t t t as ( ) ad q t t t t p, p2,..., pn ( q, q2,..., qn) for t =,. The the value of the N commodities i period t is t t N p q  p t q t for t =,. = () Fisher (9, 922) framed the ow-traditioal test approach to idex umber theory as follows: fid two fuctios of the 4N price ad quatity variables that pertai to the two periods uder cosideratio, say P(p, p, q, q ) ad Q(p, p, q, q ), such that the value ratio for the two periods, p q /p q, is equal to the product of P ad Q, that is, p q p q Pp, p, q, q Qp, p, q, q, = ( ) ( ) (2) ad the fuctios P ad Q satisfy certai properties that allow us to idetify P(p, p, q, q ) as a aggregate measure of relative price chage ad Q(p, p, q, q ) as a aggregate measure of relative quatity chage. Fisher called these properties or axioms tests. The fuctio P(p, p, q, q ) is called the price idex ad the

3 Diewert o Idex Number Theory 33 fuctio Q(p, p, q, q ) is called the quatity idex. If the umber of commodities is oe (i.e., N = ), the the price idex P collapses dow to the sigle price ratio p / p ad the quatity idex Q collapses dow to the sigle quatity ratio q / q. The idex umber problem (i.e., the problem of determiig the fuctioal forms for P ad Q) is trivial i this N = case. However, i the multiple commodity case where N >, the problem of fidig fuctios P ad Q that satisfy Equatio (2) ad satisfy appropriate tests is far from trivial. Note that if we have somehow determied the appropriate fuctioal form for P, the Equatio (2) ca be used to defie the quatity idex Q that will be cosistet with it. Thus we ca cocetrate o fidig a fuctioal form for P that satisfies a appropriate set of tests. This is the traditioal test approach to idex umber theory i a utshell. The alterative brach of idex umber theory that we wish to study is the oe that uses a differece couterpart to the ratio Equatio (2). Thus we look for two fuctios of 4N variables, DP(p, p, q, q ) ad DQ(p, p, q, q ), which sum to the value differece betwee the two periods; i other words, we wat DP ad DQ to satisfy the followig equatio: p q p q DPp, p, q, q DQp, p, q, q - = ( )+ ( ) (3) ad the fuctios DP ad DQ are to satisfy certai properties or tests that will allow us to idetify DP as a measure of aggregate price chage ad DQ as a measure of aggregate quatity or volume chage betwee the two periods. As was the case with traditioal idex umber theory where P ad Q caot be determied idepedetly if Equatio (2) is to hold, the if Equatio (3) is to hold, DP ad DQ caot be determied idepedetly. Thus i what follows, we will postulate axioms or tests for DP ad oce DP has bee determied, we will defie DQ usig Equatio (3). As the otatio DP ad DQ is somewhat awkward, we will use the otatio I(p, p, q, q ) (for idicator of price chage) to replace DP(p, p, q, q ) ad we will use V(p, p, q, q ) (for idicator of volume chage) to replace DQ(p, p, q, q ). Usig our ew otatio, Equatio (3) ca be rewritte as follows:

4 34 The America Joural of Ecoomics ad Sociology N Â [ - ] = Ip (, p, q, q)+ Vp (, p, q, q). (4) = I Sectios III ad IV below, we simplify the problem of fidig a suitable I ad V that satisfy Equatio (4) by postulatig that the overall idicators of price ad volume chage, I ad V, decompose ito a sum of N commodity-specific idicators of price ad volume chage, N N S = I (,,, ) ad S = V (,,, ), where each of the fuctios I ad V deped oly o the two prices that pertai to commodity,, ad, ad the correspodig commodity quatities ad. Thus i this simplified separable approach, the value chage for each commodity is postulated to have the followig compoets of price chage I ad quatity chage V : - = I( p, p, q, q)+ V( p, p, q, q) ; =,..., N. (5) Oce the commodity-specific idicators of price chage I have bee determied, the overall idicator of price chage I is defied as the sum of the specific idicators: N Ip (, p, q, q) I ( p, q, p, q ). (6) Â = Of course, oce I(p, p, q, q ) has bee defied, the correspodig aggregate volume chage idicator V(p, p, q, q ) ca be determied usig Equatio (4). I comparig the value differece decompositio of Equatio (3) or (4) with the value ratio decompositio of Equatio (2), it is iterestig to ote that the value ratio decompositio is trivial i the oecommodity case (N = ) but otrivial i the geeral case (N > ), while the value chage decompositio of Equatio (5) is otrivial i the case of oe commodity but oce the oe-commodity decompositios of the form i Equatio (5) have bee determied, the may-commodity case is a straightforward summatio of the oecommodity effects; i other words, see Equatio (6). We summarize the above itroductory material as follows: we are searchig for reasoable cadidates for the commodity-specific idicators of price chage I (,,, ) to isert ito the commod- ity-specific value differece Equatio (5) or, more geerally, we are lookig for reasoable cadidates for the overall idicator of price

5 Diewert o Idex Number Theory 35 chage I(p, p, q, q ) that we ca isert ito Equatio (4) above. Oce the reasoable I(p, p, q, q ) has bee foud, the correspodig idicator of volume chage V(p, p, q, q ) ca be determied by solvig Equatio (4) for V. I Sectios II ad III below, we preset the solutios of Beet (92) ad Motgomery (929, 937) to this search for a reasoable idicator of price chage. I Sectio IV, we follow the examples of Fisher (9, 922) ad Motgomery (929) ad pursue axiomatic approaches to the determiatio of a idicator of price chage. I Sectio VI, we show that may reasoable idicators of price chage umerically approximate each other to the secod order aroud equal price ad quatity vectors. Sectio VII cocludes. Proofs of various propositios may be foud i Appedix. The remaiig appedices pursue some more specialized topics. At this poit, it is useful to cosider a couple of prelimiary problems. The first prelimiary problem that eeds to be addressed is cocered with the iterpretatio of the period t prices ad quatities t t for commodity,, ad. I real-life applicatios of value decompositios ito price ad quatity parts, durig ay period t there will typically be may trasactios i commodity at a umber of differet prices. Hece, there is a eed to provide a more precise defiitio for the average or represetative price for commodity t i period t,. Irvig Fisher (922: 38) addressed this prelimiary aggregatio problem as follows: Essetially the same problem eters, however, wheever, as is usually the case, the data for prices ad quatities with which we start are averages istead of beig the origial market quotatios. Throughout this book, the price of ay commodity or the quatity of it for ay year was assumed give. But what is such a price or such a quatity?...the quatities sold will, of course, vary widely. What is eeded is their sum for the year...or, if it is worth while to put ay fier poit o it, we may take the weighted arithmetic average for the prices, the weights beig the quatities sold. Thus Fisher more or less advocated the use of the uit value (total value trasacted divided by total quatity) as the appropriate price t for commodity ad the total quatity trasacted durig period

6 36 The America Joural of Ecoomics ad Sociology t t as the appropriate quatity. As a aggregatio formula at the first stage of aggregatio, the uit value ad total quatity trasacted has bee proposed by may ecoomists ad statisticias, perhaps startig with Walsh (9: 96, 92a: 88) ad Davies (924: 83, 932: 59) ad icludig may other more recet writers. 2 t If we wat to equal the total quatity of commodity trasacted durig period t ad we t t also wat the product of the price times quatity to equal the value of period t trasactios i commodity, the we are forced to t defie the aggregate period t price to be the total value divided by the total quatity, or the uit value. The secod prelimiary problem that eeds to be addressed is also a fudametal oe: For what purpose do we wat to decompose a value chage ito price ad quatity compoets? It is quite possible that a give decompositio with certai properties would be appropriate for oe purpose but ot aother. 3 There are at least five geeral areas i which it would be useful to be able to decompose a value chage ito price ad quatity compoets. The first four areas of applicatio pertai to busiess uits, while the fifth pertais to households.. Reveue Chage Decompositios. Whe a compay reports its curret period reveue ad compares it to the reveue of the previous accoutig period, it is useful to be able to decompose this reveue chage ito a part that is due to the chage i prices that the firm faced i the two periods ad a part that is due to icreased (or decreased) productio; i other words, a volume chage. While this type of reveue decompositio is ot widespread i fiacial accoutig, oil producers frequetly make this type of decompositio i their aual reports so that shareholders ca determie the effects of chages i the world price for oil o the reveue chage. 2. Cost Chage Decompositios. Shareholders are ot oly iterested i reveue decompositios, they are also iterested i decomposig a compay s chage i costs ito a chage i prices part (a exogeous effect that is mostly beyod the cotrol of the firm) ad a chage i quatities part (a edogeous effect that is presumably uder the cotrol of the firm). This type of decompositio is ot yet commo i aual reports, presumably due to the difficulties ivolved i decomposig cost values ito idetifiable price ad quatity

7 Diewert o Idex Number Theory 37 compoets. However, as computerizatio of all trasactios proceeds, i the future it should be possible to costruct detailed uit values for cost compoets. 3. Profit Chage Decompositios. As was the case with areas ad 2 above, shareholders will be very iterested i the decompositio of profit chage over two accoutig periods ito a price chage compoet ad a quatity or volume chage compoet. Ivestor iterest will focus o the volume chage idicator because this is a idicator of firm efficiecy improvemet: the more positive is the volume chage, the greater is the ex post efficiecy gai for the firm. Note that profit chage is equal to reveue chage less cost chage, so if reveue ad cost decompositios have bee calculated, the the profit chage decompositio is simply equal to the reveue decompositio less the cost decompositio. Profit chage decompositios ca also be used by the firm for iteral cotrol ad performace evaluatio purposes Variace Aalysis. For this use of the value decompositio, the period prices ad quatities are iterpreted as period budgeted or forecasted or stadard prices ad quatities that are supposed to prevail i period. The the value differece is the differece betwee the period actual value, p q, ad budgeted performace, p q. The idicator of price chage, I(p, p, q, q ), is ow iterpreted as the cotributio of price chage (betwee actual ad budgeted prices) to the ex post differece betwee actual ad stadard values, ad the idicator of volume chage, V(p, p, q, q ), is similarly iterpreted as the cotributio of quatity chage (betwee actual ad budgeted quatities). Variace aalysis ca be traced back to the early accoutig ad idustrial egieerig literature; see Whitmore (98, 93), Harriso (98), 5 ad Solomos (968: 46 47) for the early history. 5. Chages i Cosumer Surplus. I this applicatio, the values are the expeditures of a cosumer or household o N commodities durig two periods. The task is to decompose the chage i cosumer expeditures over the two periods ito a price chage compoet I(p, p, q, q ) ad a quatity chage compoet V(p, p, q, q ), which ca be iterpreted as a costat dollar measure of real utility chage. This lie of research was started by Marshall (89) ad Beet (92). Other early cotributors iclude Hotellig

8 38 The America Joural of Ecoomics ad Sociology (938: ) ad Hicks (94 942: 34, : 73, 946: ). 6 With the above prelimiary material disposed of, we tur our attetio to the cotributios of Beet. 7 II Beet s Idicator of Price Chage The fudametal idea is that i a short period the rate of icrease of expediture of a family ca be divided ito two parts, x ad, where x measures the icrease due to chage of prices ad measures the icrease due to icrease of cosumptio... T. L. Beet (92: 455) Beet (92: 457) proposed the followig decompositio of a value chage: p q - p q = ( 2) ( q + q ) ( p - p ) + ( 2) ( p + p ) ( q - q ) N = Â ( 2) ( q q p p + ) ( - ) = N + Â ( 2) ( p p q q (7) + ) ( - ). = That Equatio (7) is true follows simply by multiplicatio ad cacellatio of terms. Thus the Beet idicators of price ad volume chage are defied as follows: ( ) ( ) + B I p, p, q, q 2( q q ) ( p - p ); ( ) ( ) + B V p, p, q, q 2( p p ) ( q - q ). Beet (92: ) justified his volume idicator as a liear approximatio to the area uder a demad curve ad his price idicator as a liear approximatio to a area uder a iverse demad curve. Hece, Beet was followig i Marshall s (89) partial equilibrium cosumer surplus footsteps. However, it is possible to derive Beet s idicators by a alterative lie of reasoig, which we ow explai. I the early idustrial egieerig literature, Harriso (98: 393) made the followig decompositio of a cost chage ito a price variatio plus a efficiecy variatio: (8) (9)

9 Diewert o Idex Number Theory 39 p q - p q = q ( p - p )+ p ( q - q ). () Agai, the proof that Equatio () is true follows by straightforward arithmetic. The reader familiar with idex umber theory will recogize that Harriso s idicator of price chage, q (p - p ), is the differece couterpart to the Paasche price idex p q /p q. Similarly, Harriso s idicator of quatity or efficiecy chage, p (q - q ), is the differece couterpart to the Laspeyres quatity idex p q / p q. Thus we defie the Paasche idicator of price chage I P ad the Laspeyres idicator of quatity chage V L as follows: ( ) - P I p, p, q, q q ( p p ); ( ) - L V p, p, q, q p ( q q ). () (2) More recetly i the accoutig literature, 8 it was recogized that the traditioal variace aalysis decompositio of a value chage, Equatio () above, may ot be as appropriate as the followig decompositio: p q - p q = q ( p - p )+ p ( q - q ). (3) The reaso why the decompositio i Equatio (3) may be preferable to Equatio () i the cotext of comparig actual performace to stadard performace is that i the case of exogeous prices, the firm maager will have a icetive to maximize profits p q with respect to q ad thus the efficiecy chage term, p (q - q ) i Equatio (3), is cosistet with profit-maximizig behavior. Agai, the reader will recogize that the idicator of price chage i Equatio (3), q (p - p ), is the differece aalogue to the Laspeyres price idex, p q /p q, ad that the idicator of quatity chage i Equatio (3), p (q q ), is the differece couterpart to the Paasche quatity idex, p q /p q. Thus we defie the Laspeyres ad Paasche idicators of price ad quatity chage, respectively, as follows: ( ) - L I p, p, q, q q ( p p ); ( ) - P V p, p, q, q p ( q q ). (4) (5)

10 32 The America Joural of Ecoomics ad Sociology Now we ca explai our alterative derivatio of the Beet idicators: the Beet idicator of price (quatity) chage is simply the arithmetic average of the Paasche ad Laspeyres idicators of price (quatity) chage; that is, ( ) = ( ) ( )+ ( ) ( ) I B p p q q I L p p q q I P,,, 2,,, 2 p, p, q, q ; ( ) = ( ) ( )+ ( ) ( ) V B p p q q V L p p q q V P,,, 2,,, 2 p, p, q, q. (6) (7) We will preset a geometric iterpretatio due to Beet (92: 456) of his idicators i the oe-commodity case i the ext sectio, i which we discuss the work of Motgomery. III Motgomery s Idicator of Price Chage Oe of the reasos why there has bee so much cotroversy o the subject of the price idex of a group of commodities is that writers have ever agreed o a defiitio. J. K. Motgomery (937: ) I a rather obscure paper, Motgomery (929) 9 defied some iterestig idicators of price ad quatity chage. We revert to the separable framework explaied i Sectio I above where we first fid idicators of price ad volume chage, I ad V, for commodity (recall Equatio (5) above) ad the the overall idicators of price ad volume chage are obtaied by summig over the commodityspecific idicators (see Equatio (6) above). The Paasche, Laspeyres, ad Beet idicators of price ad quatity chage were well defied, irrespective of the sigs of the idividual prices ad quatities ad. However, i the preset t t sectio, we shall restrict all prices ad quatities to be positive sice it will be ecessary to take atural logarithms of the idividual prices ad quatities. The restrictio that all prices ad quatities be positive is ot restrictive i the cotext of computig reveue ad cost idicators. Obviously, a profit idicator ca be defied as the differece betwee the reveue ad cost idicators ad so, eve i this cotext, the postivity restrictios are ot too restrictive.

11 Diewert o Idex Number Theory 32 The Motgomery (929: 5) idicators of price ad volume chage for the th commodity are defied as follows: ( ) { } [ ] M I p, p, q, q [ - ] [( l( )- l( ) ] l p p ; ( ) { } [ ] V M p, p, q, q [ - ] [( l( )- l( ) ] l p p. (8) (9) M Note that the fuctioal form for V is the same as the fuctioal M form for I except that the roles of prices ad quatities have bee iterchaged. Motgomery (929: 3 9) also showed that pq pq I p, p, q, q V p, p, q, q. - = ( )+ ( ) M M (2) I order to uderstad Equatios (8) ad (9) better, the reader should ote that L(a, b) [a - b]/[l a - lb] where a >, b > is kow i the ecoomics literature as the Vartia (976a, 976b) mea ad i the mathematics literature as the logarithmic mea. It ca be show that L(a, b) is a liearly homogeeous symmetric mea. 2 I Equatios (8) ad (9), a ad b p p. Motgomery (929: 7 9) derived his idicators by usig a very iterestig argumet (which parallels that of Beet 92) that we shall repeat sice it shows how a large umber of reasoable price ad quatity idicators ca be derived. Suppose that is fuctioally related to by a supply fuctio: q = s( p). (2) Note that the supply fuctio s is a partial equilibrium supply fuctio sice oly the price of the th good appears i Equatio (2) as a argumet of the fuctio. Motgomery (929: 7) assumed the followig fuctioal form for s : s ( p ) = a p b, a >, b π. (22) Now defie a theoretical price chage idicator as the area uder the supply curve goig from to :

12 322 The America Joural of Ecoomics ad Sociology p ( ) ( ) P* p, p, s s p dp p Ú Ú p b = a ( p ) dp usig Equatio ( 22) p - + b p = a ( + b ) p ) assumig b π- p b [ ] b = a ( + b ) ( p ) - ( p ) (23) (24) The ukow parameters a ad b that appear i Equatio (24) ca be determied by assumig that the two data poits (, ) ad (, ) are o the supply fuctio defied by Equatio (22). Thus we have b b q = a ( p ) ad q = a ( p ) or b b q a = ( p ) ad q a = ( p ) or b pq a = ( p) ad pq a = ( p) + + b. (25) (26) (27) By takig ratios i Equatio (25), we ca also deduce that b [ q q] = [ p p] or b = l[ q q] l[ p p] or + b = { l[ p p]+ l[ q q] } l[ p p] = { l[ ] = l[ ] } l [ p p]. (28) (29) Now substitute Equatio (27) ito Equatio (24) to get ( ) = + P* - p, p, s ( b) pq - = l[ p p] [ pq - ] { l[ ]- l[ ] } usig ( 29) M I p, p, q, q, [ ] = ( ) (3) M where the Motgomery idicator of price chage for good, I, was defied by Equatio (8). Motgomery (followig Beet 92: 456) also defied the theoretical idicator of quatity chage goig from to as the area uder the iverse supply curve S where = S ( ); that is, defied

13 Diewert o Idex Number Theory 323 q ( ) () Q* q, q, S S qdp. q Ú (3) If the supply fuctio s is defied by Equatio (22), the the correspodig iverse supply fuctio has the same fuctioal form; thus, we have (32) where d /b ; hece we require b π ad g (/a ) /b ; hece we require a π. Thus the same argumet that we used to derive Equatio (3) ca be adapted to prove that if S is defied by Equatio (32) or equivaletly if s is defied by Equatio (22), the V M b d p = S ( q ) = [ q a ] g q ( ) = ( ) Q* q, q, S V p, p, q, q, M (33) where is the Motgomery idicator of quatity chage defied by Equatio (9). Motgomery (929: 3) gave a ice geometric iterpretatio of his method for the case i which < < ad < < that we repeat i Figure., Figure Motgomery s decompositio of a value chage. q B Q M q A P M p p

14 324 The America Joural of Ecoomics ad Sociology b The lie AB is the curve defied by = a that passes through the observed poits A ad B. The period value, p, ca be iterpreted as the area of the big rectagle B, while the period value, p, ca be iterpreted as the area of the small rectagle A. The part of the value chage that is due to price M chage, I, is the shaded area below the curve AB (the area eclosed by AB ) ad the part of the value chage that is due to quatity M chage, V, is the shaded area to the left of AB (the area eclosed by q BA). It is easy to see how Motgomery s idea could be geeralized. Istead of usig the costat elasticity fuctioal form defied by Equatios (22) or (32) to joi the parts A ad B, ay mootoic curve could be used to joi A ad B ad the correspodig idicators of price ad quatity chage ca be defied by Equatios (23) ad (3). I fact, Motgomery (929: 9 ) cosidered a alterative curve. He repeated his aalysis assumig that the supply fuctio had the followig liear fuctioal form: s( p) a + bp. (34) For the liear supply fuctio defied by Equatio (34), the area uder the curve defiitio of the price chage idicator yields: p ( ) + P* p, p, s [ a b p] dp p Ú p p 2 = ap+ ( 2) bp] [ ] a p p b p 2 = [ - ]+ ( 2) ( ) - ( p 2 ). (35) As usual, we determie the ukow parameters a ad b i Equatio (35) by assumig that the observed poits (, ) ad (, ) lie o the curve defied by Equatio (34). Thus we have: Assumig that π, we fid that q = a + b p ; q = a + b p. [ ] [ - ] a = p q - p q p p ad [ ] [ - ] b = q - q p p. Substitutig Equatios (37) ad (38) ito Equatio (35) yields (36) (37) (38)

15 Diewert o Idex Number Theory 325 ( ) = ( ) + P* p p s q q p p I B,, 2( ) - p, p, q, q ; (39) i other words, the theoretical idicator of price chage P *(,, s ), uder the assumptio that s is the liear supply fuctio defied by Equatio (34), turs out to equal the Beet (92) idicator of price chage, I (,, y, y ), for commodity. B The iverse supply fuctio that correspods to Equatio (34) is: p = S( q) -( a b)+ ( b) q c + dq, (4) where c -a /b ad d /b (assumig b π ). The fuctioal form for the iverse supply fuctio defied by Equatio (4) is liear, ad so we ca simply adapt the above argumet that we used whe the direct supply fuctio s was liear. Thus if Equatio (34) or (4) holds, the p Q* q, q, S S qdq (4) ( ) () q Ú ( ) ( ) = ( 2) ( p + p) ( q - q) ( ) V B p, p, q, q, (42) B where V (,,, ) is the Beet quatity chage idicator for commodity. The geometry of the liear supply fuctio is give i Figure 2. Figure 2 Beet s decompositio of a value chage. q B Q B q A P B p p

16 326 The America Joural of Ecoomics ad Sociology M M M Motgomery (929: 4) preferred the idicators I ad V V over B B the Beet (92) idicators I ad V because the costat elasticity supply curve defied by Equatio (22) passes through the origi, whereas the liear supply curve defied by Equatio (34) will ot geerally pass through the origi (uless / = / so that the price chage is equal to the quatity chage i proportioal terms). However, Beet (92: 457) argued that the straight lie joiig A ad B was the simplest hypothesis to use i the absece of other iformatio. Moreover, Motgomery s model, which has a supply curve goig through the origi, is ot appropriate i the cotext of a cosumer demad for commodities model or a producer s demad for iputs model i which a egatively sloped demad curve (which would ot pass through the origi) would be more appropriate. Beet s framework is perfectly valid i the demad cotext, whereas Motgomery s model breaks dow. 3 L Referrig to Figure 2, the Laspeyres I N idicator of price chage ca be represeted as the area of the rectagle ( - ), while P the Paasche idicator I ca be represeted as the area of the larger B rectagle ( - ). It ca be see that the Beet idicator I is the average of these two rectagles. The above aalysis shows that there are a large umber of possible measures of price ad quatity chage, I ad V, that satisfy Equatio (5). Which measure should we use i empirical applicatios? I the followig sectio, we suggest a axiomatic or test approach to the determiatio of the fuctioal form for I ad V while i Sectio V we suggest a ecoomic approach. IV The Test Approach for Idicators of Price ad Quatity Chage How is this (icrease i value) to be apportioed betwee the icrease i price...ad the icrease i value due to the icrease i quatity?... Agai we ote that if there is a icrease i price, but o icrease i quatity, the whole of the icrease i value is due to the icrease i price.... Coversely, if there is a icrease i quatity, but o icrease i price, the whole of the icrease i value is due to the icrease i quatity. J. K. Motgomery (929: 3)

17 Diewert o Idex Number Theory 327 We cosider axioms for the commodity price chage idicator, I (,,, ). We assume that the commodity volume chage idicator V (,,, ) is determied by Equatio (5) oce I has bee determied. We also assume that all of the scalar prices ad quatities,,, are positive. The tests ad axioms that we propose are, for the most part, aalogies to the tests that Diewert (992b) used to characterize the Fisher (922) ideal price idex. We assume that I ad V satisfy the idetity i Equatio (5). Thus if I is defied, the correspodig V is implicitly defied by: V( p, p, q, q) = pq - pq - I( p, p, q, q). (43) Thus tests or properties of the quatity chage idicator V ca be imposed o the price chage idicator I usig Equatio (43). The first property we wish to cosider for our price idicator I is the property of cotiuity. A I (,,, ) is defied for >, >, >, > ad is a cotiuous fuctio over this domai of defiitio. The followig two tests were first proposed by Motgomery (929: 3). If prices remai uchaged durig the two periods, the we wat our idicator of price chage to equal zero; if quatities remai uchaged durig the two periods, the we wat our idicator of quatity chage to equal zero. A2 Idetity Test for Prices: I (,,, ) =. A3 Idetity Test for Quatities: I(,,, ) = ( - ). We derived the test A3 usig Equatio (43). From Equatio (43), we have V( p, p, q, q) = pq - pq - I( p, p, q, q) = (44) ad the secod equatio i Equatio (44) is equivalet to the equatio i A3. Recall the Paasche ad Laspeyres price idicators defied i the P L previous sectio, I (,,, ) ( - ) ad I (,,, ) ( - ). From the perspective of the Motgomery approach preseted i the previous sectio, it ca be see that the Paasche ad Laspeyres price idicators give the most extreme results that could be

18 328 The America Joural of Ecoomics ad Sociology cosidered acceptable. Hece it seems reasoable to require that I (,,, ) lie betwee these two extreme idicators. A4 Boudig Test: mi{( - ), ( - ) } I(,,, ) max{( - ), ( - ) }. It ca be show that if I satisfies A4, the the correspodig V defied by Equatio (43) satisfies: { } ( ) { } mi ( q - q ) p, ( q - q ) p V p, p, q, q max ( q - q ) p, ( q - q ) p. (45) Our ext four tests are mootoicity properties for I ad V. A5 Mootoicity i Period Prices: I (,,, ) < I (,,, ) if <. If the period price icreases from price chage should also icrease. to, the the idicator of A6 Mootoicity i Period Prices: I (,,, ) > I (,,, ) if <. If the period price icreases from price chage should decrease. to, the the idicator of A7 Mootoicity i Period Quatities: - p p - I (,,, ) < - - I (,,, ) if <. Thus if the period quatity icreases from to, the implicit idicator of quatity chage V defied by Equatio (43) should also icrease; that is, A7 is equivalet to ( )< ( ) < V p, p, q, q V p, p, q, q if q q. (46) A8 Mootoicity i Period Quatities: - p p - I (,,, ) > p - - I (,,, ) if <. Thus, if period quatity icreases from to, the implicit idicator of quatity chage V defied by Equatio (43) should decrease; that is, A8 is equivalet to ( ) ( ) < V p, p, q, q V p, p, q, q if q q. (47)

19 The followig four tests are postivity (or egativity) tests that are somewhat weaker tha the previous four mootoicity tests. It ca be show that A2 ad A5 imply A9; A2 ad A6 imply A; A3 ad A7 imply A; ad A3 ad A8 imply A2. A9 A A A2 Diewert o Idex Number Theory 329 Positivity of Price Chage if the Period Price Exceeds the Period Price: I (,,, ) > if >. Negativity of Price Chage if the Period Price Exceeds the Period Price: I (,,, ) < if >. Positivity of Quatity Chage if the Period Quatity Exceeds the Period Quatity: p - p - I (,,, ) > if >. Of course, usig Equatio (43), the iequality i A is equivalet to ( ) > > V p, p, q, q if q q. (48) Negativity of Quatity Chage if the Period Quatity Exceeds the Period Quatity: p - p - I (,,, ) < if >. Usig Equatio (43), the iequality i A2 is equivalet to: V ( p, p, q, q )< if q > q. (49) I idex umber theory, we geerally require our price ad quatity idexes to be ivariat to chages i the uits of measuremet. It seems reasoable to impose this ivariace property i the preset cotext. A3 Ivariace to Chages i the Uits of Measuremet: I (,,, ) = I(l, l, l -, l - ) where l >. There are also the followig two liear homogeeity properties that we ca impose o P. A4 Liear Homogeeity i Prices: I (l, l,, ) = li (,,, ) for all l >. A5 Liear Homogeeity i Quatities: I (,, m, m ) = mi(,,, ) for all m >. Note that the price homogeeity axioms A4 ad A5 differ from their idex umber couterparts; i other words, the prices i both

20 33 The America Joural of Ecoomics ad Sociology periods are scaled up by the multiplicative factor l i A4 ad A5. Note that if A4 holds, we have I (,,, ) = I (, /,, ) (let l = / ) ad if A5 holds, we have I (,,, ) = I(,,, / ) (let m = / ). If both A4 ad A5 hold, the we have A3 ad moreover: ( ) = ( ) I p, p, q, q p q I, p p,, q q. Our fial three tests are symmetry tests. (5) A6 Time Reversal: I (,,, ) = -I (,,, ): i other words, if we iterchage prices ad quatities for the two periods, the the price idicator fuctio should chage sig. Put aother way, the price chage goig from period to plus the price chage goig from period back to period should sum to zero; that is, I (,,, ) + I (,,, ) =. A7 Quatity Weights Symmetry: I (,,, ) = I (,,, ). Thus if (A7) holds, we ca iterchage the role of quatities i the two periods ad the idicator of price chage remais uchaged. This implies that the quatities ad eter ito the price idicator formula i a symmetric maer. A8 Price Weights Symmetry: p - p - I (,,, ) = - p p - I (,,, ). Makig use of Equatio (43), A8 is equivalet to: V p, p, q, q V p, p, q, q : (5) that is, if we iterchage prices i the idicator of quatity chage, this idicator of quatity chage remais uchaged. Thus ad eter ito the quatity idicator formula i a symmetric fashio. It turs out that the Beet price ad quatity idicators B I p, p, q, q 2( q q )( p - p ); B V p, p, q, q 2 p p q q ( ) = ( ) ( ) + ( ) ( + )( - ) (52) (53) satisfy all 8 of the above tests ad i fact are uiquely characterized by the above tests.

21 Diewert o Idex Number Theory 33 Propositio : If I satisfies the three symmetry tests A6, A7, ad B B A8, the I = I where I is defied by Equatio (52). See Appedix for proofs of propositios. Propositio 2: I B defied by Equatio (52) satisfies tests A A8. What tests does the Motgomery price chage idicator by Equatio (8) satisfy? defied M Propositio 3: I (,,, ) defied by Equatio (8) satisfies all of the tests except the mootoicity tests A5, A6, A7, ad A8 ad the symmetry tests A7 ad A8. It should be metioed that Motgomery (929: 3) also proposed the followig symmetry test, which is a couterpart to Fisher s (92: 53, 922: 72) factor reversal test i idex umber theory: A9 V (,,, ) = I (,,, ): that is, the volume idicator is equal to the price idicator whe the role of prices ad quatities is iterchaged i the latter fuctio. 4 Motgomery (929: ) oted that both the Beet ad Motgomery idicators satisfied the factor reversal axiom (A9) ad that he could ot fid ay other simple formula that satisfied this axiom. However, it is easy to adapt Fisher s (922: 42) 5 crossig of factor atitheses approach to rectify ay give idicator of price r chage I (,,, ) ito a idicator I (,,, ) that satisfies the factor reversal test A9. We show how this ca be doe. Let I (,,, ) be a arbitrary idicator of price chage that f perhaps does ot satisfy A9. Defie the factor atithesis I to I as follows: f I p, p, q, q p q p q I q, q, p, p. ( ) - - ( ) I M (54) Defie the factor rectified idicator I ad : I f I fr as the arithmetic average of ( ) ( ) ( )+ ( ) ( ) fr I p, p, q, q 2I p, p, q, q 2I f p, p, q, q. (55)

22 332 The America Joural of Ecoomics ad Sociology fr Propositio 4: The factor rectified idicator of price chage I defied by Equatio (55) satisfies the factor reversal test A9; that is, we have (56) It is also easy to modify Fisher s (922: 4) time rectificatio procedure, which will allow us to trasform a arbitrary idicator of price chage I (,,, ) that perhaps does ot satisfy the time reversal test A6 ito oe that does. Defie the time atithesis I to I as t follows: Defie the time rectified idicator I ad : I t fr I p, p, q, q I q, q, p, p p q p q. ( )+ ( ) = - fr t I p, p, q, q I p, p, q, q. (57) as the arithmetic mea of ( ) ( ) ( )+ ( ) ( ) tr I p, p, q, q 2I p, p, q, q 2I t p, p, q, q. (58) tr Propositio 5: The time rectified idicator of price chage I defied by Equatio (58) satisfies the time reversal test A6; that is, we have tr I p, p, q, q I p, p, q, q. ( ) - ( ) ( ) =- ( ) tr (59) The above propositios show that there are may parallels betwee the test approach to idex umbers ad the test approach to idicators of price chage. The Fisher (92, 922) ideal price idex (the geometric mea of the Paasche ad Laspeyres price idexes) appears to be best from the viewpoit of the test approach, 6 while the Beet idicator of price chage (the arithmetic average of the Paasche ad Laspeyres price idicators) appears to be best from the viewpoit of the test approach to idicators of price chage. We ote that the Paasche ad Laspeyres idicators of price chage, I P ad I L, defied by Equatios () ad (4), respectively, have rather good axiomatic properties, as is idicated i the followig result: Propositio 6: The Paasche ad Laspeyres idicators of price chage defied by Equatios () ad (4) satisfy all of the above axioms except the symmetry axioms, (A6), (A7), (A8), ad (A9). The reader may have oticed that we did ot strogly edorse the factor symmetry test A9; it is ot a particularly compellig test. 7 The I tr

23 Diewert o Idex Number Theory 333 weight symmetry tests (A7) ad (A8) are also ot tests that simply must hold. However, the time reversal test is a importat test that we defiitely will wat to impose i ay cotext i which the data from the two periods is symmetric. With respect to the five applicatios (outlied i Sectio I) where idicators of price ad quatity chage might be useful, all of these applicatios would seem to require a symmetric treatmet of the data, with the exceptio of variace aalysis, where the data are ot symmetric. Thus the failure of P L the Paasche ad Laspeyres price idicators I ad I N to satisfy the time reversal test A6 meas that we should be cautious i usig these idicators. The failure of the Motgomery idicator of price M chage I to satisfy the mootoicity tests A5 A8 meas that we should be cautious i usig it as well. This leaves the Beet idicator of price chage, I, as the best from the viewpoit of the test B approach. We ow tur our attetio to ecoomic approaches to the measuremet of price ad quatity chage. V The Ecoomic Approach to Idicators of Price ad Quatity Chage Therefore the problem of costructig a true idex of the cost of livig is iseparably boud up with the geeral problem of establishig a fuctioal relatio betwee cosumptio ad prices. A. A. Köus ([924] 939: 2) I the ecoomic approach to idex umber theory, prices ad quatities are o loger regarded as beig completely idepedet variables: prices are regarded as exogeous, but quatities are determied as solutios to optimizatio problems. For example, i the cosumer cotext, it is assumed that a cosumer maximizes a utility fuctio f(q) subject to a budget costrait or, alteratively, the cosumer miimizes the cost of achievig a give utility level. Followig Samuelso ad Swamy (974) ad Diewert (976b), we assume that the utility fuctio is homogeeous of degree oe, i which case the cosumer s cost or expediture fuctio C, defied as ( ) { () } Cu, p mi p q: fq u, q (6)

24 334 The America Joural of Ecoomics ad Sociology has the followig decompositio: ( ) ( ) ( ) Cu, p uc, p uc p, (6) where c(p) C(, p) is the cosumer s uit cost fuctio. 8 I this case, where prefereces are homothetic, the observed price ad quatity data for period t, p t ad q t, satisfy: t t t t p q = f( q ) cp ( ); t =,. (62) Thus takig ratios, Equatio (62) implies (63) For certai specific fuctioal forms for the uit cost fuctio c, we ca fid a idex umber formula, P(p, p, q, q ), such that the uit cost ratio is equal to this price idex; that is, for such a idex, we have (64) Associated with such a exact price idex P is the quatity idex Q that satisfies Equatio (2) above. Usig Equatios (2), (63), ad (64), for such a Q, we have (65) This is a brief outlie of the theory of exact idex umber formulae that is discussed by Pollak (989: 22 32), Samuelso ad Swamy (974), ad Diewert (976b). A particular example of Equatio (64) that is used frequetly i empirical applicatios is the Fisher (922) ideal idex P F defied as: P p, p, q, q p q p q p q p q, F ( ) [ ] (66) which is exact for the homogeeous quadratic uit cost fuctio c(p) = (p Bp) /2 where B is a symmetric matrix of costats. 9 Aother example is the Törqvist 2 price idex P T defied as l P p, p, q, q 2 s s l p p, T p q p q = [ cp ( ) cp ( )][ fq ( ) fq ( )]. Pp (, p, q, q) = cp ( ) cp ( ). Qp (, p, q, q ) = f( q ) f( q ). 2 M Â = ( ) ( )( + ) ( ) (67) where the period t expediture share o commodity,, is defied t t as p /p t qt for =,..., N ad t =,. This idex is exact for a traslog uit cost fuctio. 2 Now retur to the case of a geeric exact price idex P satisfyig s t

25 Diewert o Idex Number Theory 335 Equatio (64), alog with its parter Q satisfyig Equatios (2) ad (65). We ca write the differece i cosumer expeditures betwee periods ad as follows: p q - p q = p q [( p q p q )- ] = p q [ PQ -] usig Equatio () 2 = p q ( 2) ( + Q) ( P- )+ ( 2) ( + P) ( Q- ) (68) [ ] where Equatio (68) follows from the lie above usig the Beet idetity: PQ - = ( 2) ( + Q) ( P - )+ ( 2 )( + P) ( Q -. ) (69) Now from Equatio (68) we see that the value chage has bee decomposed ito a ecoomic price chage (the term ivolvig P - ) ad a ecoomic quatity chage (the term ivolvig Q - ). Thus for our geeric exact ecoomic price ad quatity idexes P ad Q satisfyig Equatio (2), we defie the followig ecoomic idicators of price ad volume chage: E( ) I p, p, q, q ( ) [ + ( )][ ( )- ] 2p q Q p, p, q, q P p, p, q, q ; E( ) V p, p, q, q ( ) [ + ( )][ ( )- ] 2p q Pp, p, q, q Qp, p, q, q. (7) (7) Propositio 7: If the exact price ad quatity idexes P ad Q satisfy Equatios (2), (64), ad (65) ad there is cost miimizig behavior i the two periods uder cosideratio so that Equatio (62) holds, the the ecoomic price ad quatity idicators defied by Equatios (7) ad (7) satisfy the followig equatios: I ( p, p, q, q ) = ( 2) cp ( ) f( q ) + f( q ) f( q ) { cp cp } - E V ( p, p, q, q ) = ( 2) cp ( ) f( q ) + cp ( ) cp ( ) { f q f q } - E [ ][ ( ) ( ) ] = ( 2) fq ( )[ cp ( )- cp ( )]- ( 2) fq ( )[ cp ( )- cp ( )] = ( 2) [ fq ( )+ fq ( )][ cp ( )- cp ( )]; [ ][ ( ) ( ) ] = ( 2) cp ( )[ fq ( )- fq ( )]- ( 2) cp ( )[ fq ( )- fq ( )] = ( 2) [ cp ( )+ cp ( )][ fq ( )- fq ( )]. (72) (73) (74) (75) (76) (77)

26 336 The America Joural of Ecoomics ad Sociology Equatio (72) is a straightforward traslatio of Equatio (7) usig the exactess of the idexes P ad Q. Equatio (73) is a symmetric rearragemet of Equatio (72); it shows that the overall price chage ca be writte as the arithmetic average of the period to price chage c(p ) - c(p ) weighted by the period aggregate quatity f(q ) ad the egative of the period to price chage c(p ) - c(p ) weighted by the period aggregate quatity f(q ). Equatio (74) is the familiar Beet price chage decompositio, treatig c(p ) ad c(p ) as aggregate prices for periods ad ad f(q ) ad f(q ) as aggregate quatities for periods ad. Note that the left-had side of Equatios (72), (73), or (74) is I E (p, p, q, q ), which ca be readily calculated if P ad Q are give, whereas the right-had side of Equatios (72), (73), or (74) ivolves the uobserved aggregates c(p t ) ad f(q t ). It is useful to obtai a alterative formula for the ecoomic idicator of price chage I E defied by Equatio (7). Rearragig the right-had side of Equatio (7) yields IE( p, p, q, q ) = ( 2) p q [ P-]+ ( 2) p q QP [ -] = ( 2) p q [ P-]+ ( 2) p q [ p q p q P] [ P- ] usig () 2 = ( 2) p q [ Pp (, p, q, q )- ] - ( 2) p q [{ Pp (, p, q, q )} - ]. (78) Equatio (78) ca be used to establish the followig result. Propositio 8: If P satisfies the time reversal test for price idexes (i.e., P(p, p, q, q ) = /P(p, p, q, q )), the the correspodig ecoomic idicator of price chage I E defied by Equatios (7) or (78) satisfies a versio of the time reversal test for price idicators, A6. Obviously, there are other relatioships betwee the axiomatic properties of the price idex P ad the axiomatic properties of the resultig ecoomic price idicator I E defied i Equatio (78). For example, if P satisfies the idetity test for price idexes (P(p, p, q, q ) = ), the the correspodig I E will satisfy a versio of the idetity test A2; if P satisfies the ivariace to chages i the uits of

27 Diewert o Idex Number Theory 337 measuremet test for price idexes, the the correspodig price idicator I E will satisfy a versio of the test A3, ad so o. Which I E defied by Equatio (78) is best from the viewpoit of the ecoomic approach? To aswer this questio, we must pick the best formula for the uderlyig ecoomic price idex P from the viewpoit of the ecoomic approach. Ufortuately, there is o sigle best price idex P from the viewpoit of the ecoomic approach. However, there is a class of price idexes P that are cosidered best from the viewpoit of the ecoomic approach: amely, those idexes P that are exact for a flexible fuctioal form for c or f. Such idexes P were called superlative by Diewert (976b: 7). 22 Hece if the price idex formula P i Equatio (78) is superlative, we will call the iduced idicator of price chage I E superlative as well. Sice the Fisher idex P F ad the Törqvist idex P T defied by Equatios (66) ad (67) above are superlative, the associated ecoomic idicators of F T price chage I E ad I E are also superlative idicators. Thus from the viewpoit of the ecoomic approach to price idicators, ay superlative price idicator could be cosidered best. VI Which Idicator of Price Chage Should Be Used i Practice? It is worth emphasizig that these theorems hold without the assumptio of optimizig behavior o the part of ecoomic agets; i.e., they are theorems i umerical aalysis rather tha ecoomics. W. E. Diewert (978: 889) From the viewpoit of the test or axiomatic approach to idicators of price chage, the results of Sectio IV above suggest that the Beet idicator I B defied by Equatio (8) is best. From the viewpoit of a ecoomic approach to idicators of price chage, the results of Sectio V suggest that a superlative idicator of price chage F such as I E (replace P i Equatio (78) by P F defied by Equatio (66) T above) or I E (replace P i Equatio (78) by P T defied by Equatio (67) above) is best. Which of these best idicators should we use i empirical applicatios? The result below suggest that it will ot matter much i practice which of these three idicators of price chage is used.

28 338 The America Joural of Ecoomics ad Sociology Propositio 9: The Beet idicator of price chage approximates ay superlative idicator (such as the Fisher or Törqvist idicators) to the secod order at ay poit where the two price vectors are equal (i.e., p = p ) ad where the two quatity vectors are equal (i.e., q = q ). The proof of the above propositio rests o aalogous results for superlative price idexes: every kow superlative price idex P(p, p, q, q ) has the same levels, vectors of first-order partial derivative, ad matrices of secod-order partial derivatives whe evaluated at a equal price (i.e., p = p ) ad quatity (i.e., q = q ) poit. 23 Somewhat surprisigly, the Motgomery idicator of price chage, I M (p, p, q, q N M M ) S = I (,,, ) where the I are defied by Equatio (8), also approximates the idicators described i Propositio 9 to the secod order aroud a equal price ad quatity poit. Propositio : The Motgomery idicator of price chage, I M (p, p, q, q ), approximates the Beet idicator I B (p, p, q, q ) to the secod order at ay poit where p = p ad q = q. We require all prices ad quatities to be positive i Propositio. All prices must be positive i Propositio 9, but the requiremet that the quatity vectors be strictly positive ca be relaxed provided that the superlative P(p, p, q, q ) is still well defied. Propositio appears to be the idicator couterpart to Theorem 3 i Diewert (978: 887), who showed that the Vartia I (976a: 24 25, 976b: 22) price idex approximated ay superlative price idex to the secod order aroud a equal price ad quatity poit. However, it should be oted that Motgomery (937: 35) actually came up with the formula for the Vartia I price idex may years before Vartia s derivatio. 24 The formula for the Motgomery-Vartia I idex is: N M Â = ( ) = ( ) l P p, p, q, q w l p p, (79) where the weights w are defied, usig the logarithmic mea L, as follows: w L( p q, p, q ) L p q, p q ;,..., N. ( ) = (8)

29 Diewert o Idex Number Theory 339 Thus the Motgomery price idicator fuctio I M plays the same role i Propositio above as the Motgomery-Vartia price idex P M played i Theorem of Diewert (978). I order to check the accuracy of Propositios 9 ad above, we calculated the Paasche ad Laspeyres idicators of price chage, I P ad I L defied by Equatios () ad (4) above, the Beet idicator I B defied by Equatio (8), the superlative Fisher ad Torqvist F T idicators I ad, ad the Motgomery idicator I M E I E usig the price ad quatity data o 36 primary commodities i the Uited States for the years that are tabled i Fisher (922: ). We calculated the chai liks for each year; that is, we calculated I(p t-, p t, q t-, q t ) for t = 94,..., 98. We also calculated the Fisher ad Törqvist chai liks P F (p t-, p t, q t-, q t ) ad P T (p t-, p t, q t-, q t ) to show the year-over-year iflatio rates. The results of these computatios ca be foud i Table. As ca be see from the table, there does appear to be a fairly close correspodece betwee the Beet, Fisher, Törqvist, ad Motgomery idicators of price chage, I B, I F, I T, ad I M, respectively. Note also that the Paasche ad Laspeyres idicators of price chage, I P ad I L, are rather differet from each other ad the other superlative ad pseudo-superlative idicators. 25 VII Coclusio Today s dollar is, the, a totally differet uit from the dollar of 897. As the geeral price level fluctuates, the dollar is boud to become a uit of differet magitude. To mix these uits is like mixig iches ad cetimeters or measurig a field with a rubber tape-lie. Livigsto Middleditch (98: 4 5) The above quotatio alerts us to a potetial problem with our treatmet of value chages; amely, if there is a great chage i the geeral purchasig power of moey betwee the two periods beig compared, the our idicators of volume chage may be excessively heavily weighted by the prices of the period with the highest geeral price level. Put aother way, the uits that quatities are measured i do ot require ay comparisos with other quatities, but the dollar