Searching for the Holy Grail of Index Number Theory
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1 Searchig for the Holy Grail of Idex Number Theory Bert M. Balk Rotterdam School of Maagemet Erasmus Uiversity Rotterdam ad Statistics Netherlads Voorburg April 15, 28 Abstract The idex umber problem is kow as that of decomposig aggregate value chage, i ratio or differece form, ito two, ideally symmetric, factors. This ote commets o a recet cotributio of Casler [6]. Keywords: Idex umber theory; ideal idices; decompositio. JEL classificatio: C43. 1 Itroductio Time ad agai people are searchig for the Holy Grail of idex umber theory, here defied as beig a symmetric pair of price ad quatity idices that satisfy all kow requiremets. Sectio 2 more precisely describes the The author thaks Stephe Casler for his commets o a earlier versio. 1
2 objective of the search. Sectio 3 catalogues the fidigs. Sectio 4 discusses a recet fidig by Casler [6]. The coclusio ca be brief: the Holy Grail is a mirage! 2 The idex umber problem We cosider two time periods, a base period, deoted by the label, ad a compariso period, deoted by the label 1, ad a set of commodities, labeled from 1 to N. The vectors of (uit) prices ad quatities of these commodities will be deoted by p t (p t 1,..., p t N) ad q t (q1, t..., qn) t respectively (t =, 1). It is assumed that p t, q t R N ++. The value of a commodity at period t is the give by v t p t q t ( = 1,..., N; t =, 1), ad the aggregate value by V t N v t = N p t q t p t q t (t =, 1). The value share of a commodity is defied as s t v/v t t ( = 1,..., N; t =, 1). It is clear that s t = 1 (t =, 1); (1) that is, the base ad compariso period value shares add up to 1. I the classical idex umber problem oe wats to decompose the aggregate value ratio ito two parts, V = P (p1, q 1, p, q )Q(p 1, q 1, p, q ), (2) of which the first part, P (p 1, q 1, p, q ), measures the effect of differig prices ad the secod part, Q(p 1, q 1, p, q ), measures the effect of differig quatities. Both fuctios operate o the price ad quatity vectors of the two periods ad map these ito uitless scalars. Provided that certai reasoable requiremets are satisfied, the first part is called a price idex ad the secod part a quatity idex. The idices P (p 1, q 1, p, q ) ad Q(p 1, q 1, p, q ) should exhibit the basic properties of cotiuity, positivity, mootoicity i prices (quatities), liear homogeeity i compariso period prices (quatities), idetity, homogeeity of degree zero i prices (quatities), ad ivariace to the uits of measuremet (see Balk [1] for precise formulatios). The time reversal test stipulates that reversig the time periods yields a idex which is idetically equal to the reciprocal of the origial idex. The factor reversal test requires that (2) be satisfied whereby Q(p 1, q 1, p, q ) = P (q 1, p 1, q, p ); that is, price idex 2
3 ad quatity idex have the same fuctioal form except that prices ad quatities have bee iterchaged. A idex is called ideal if it satisfies the factor reversal test. The alterative problem, equally old but lesser kow, is to decompose the aggregate value differece ito two parts, V = P(p 1, q 1, p, q ) + Q(p 1, q 1, p, q ), (3) of which the first term, P(p 1, q 1, p, q ), measures the part of the value differece that is due to differig prices ad the secod term, Q(p 1, q 1, p, q ), measures the part of the value differece that is due to differig quatities. Both fuctios operate o the price ad quatity vectors of the two periods but map these ito moey amouts. Provided that certai reasoable requiremets are satisfied, the first part is called a price idicator ad the secod part a quatity idicator. The idicators P(p 1, q 1, p, q ) ad Q(p 1, q 1, p, q ) should exhibit the basic properties of cotiuity, mootoicity i prices (quatities), idetity, liear homogeeity i prices (quatities), ad ivariace to the uits of measuremet (see Diewert [7] for precise formulatios). The time reversal test stipulates that reversig the time periods yields a idicator which is idetically equal to the egative of the origial idicator. The factor reversal test requires that (3) be satisfied whereby Q(p 1, q 1, p, q ) = P(q 1, p 1, q, p ); that is, price idicator ad quatity idicator have the same fuctioal form except that prices ad quatities have bee iterchaged. A idicator is called ideal if it satisfies the factor reversal test. The lik betwee additive ad multiplicative decompositios is provided by the logarithmic mea. 1 The additive decompositio derived from expressio (2) is V = L(, V ) l P (p 1, q 1, p, q ) + L(, V ) l Q(p 1, q 1, p, q ). (4) Recall that L(, V ) is a average of the period 1 value ad the period value V, ad otice that l P (.) ad l Q(.) are approximately equal to the percetage of aggregate price ad quatity chage respectively. 1 The logarithmic mea is, for ay two strictly positive real umbers a ad b, defied by L(a, b) (a b)/ l(a/b) ad L(a, a) a. It has the followig properties: (1) mi(a, b) L(a, b) max(a, b); (2) L(a, b) is cotiuous; (3) L(λa, λb) = λl(a, b) (λ > ); (4) L(a, b) = L(b, a); (5) (ab) 1/2 L(a, b) (a + b)/2; (6) L(a, 1) is cocave. 3
4 Reversely, the additive decompositio (3) leads to { P(p 1 V = exp, q 1, p, q } { ) Q(p 1, q 1, p, q } ) exp, (5) L(, V ) L(, V ) as multiplicative decompositio of the value ratio. It is good to otice that properties of idices do ot automatically carry over to idicators, ad vice versa. 3 Ideal idices ad idicators History has provided us with a umber of ideal idices. Fisher s [8] solutio to the ratio type idex umber problem was V = ( p1 q p 1 q 1 ) 1/2 ( p q 1 p 1 q 1 ) 1/2 p q p q 1 p q p 1 q P F (p 1, q 1, p, q )Q F (p 1, q 1, p, q ). (6) Fisher s idices exhibit all the basic properties, plus the time reversal test, ad the factor reversal test. Motgomery s [9], [1] solutio was ( ) N p 1 L(v 1,v)/L(V 1,V ) N ( ) = q 1 L(v 1,v)/L(V 1,V ) V p q P MV (p 1, q 1, p, q )Q MV (p 1, q 1, p, q ). (7) Sice Vartia [14], [15] idepedetly rediscovered this solutio to the idex umber problem, the fuctios P MV (.) ad Q MV (.) are called Motgomery- Vartia idices. These idices satisfy the time reversal test ad the factor reversal test. Of the basic properties, they fail to satisfy mootoicity globally. However, as show by Balk [2], such a failure ca hardly be expected to occur i practice. More importat is the fact that these idices do ot exhibit the basic property of liear homogeeity i compariso period prices (quatities), because of the fact that the weights do ot add up to 1 (which i tur depeds o the cocavity of L(a, 1)). Thus this solutio is ot completely satisfactory. 4
5 Sato [12] ad Vartia [14], [15] idepedetly discovered a ew pair of ideal price ad quatity idices. They are give by ( ) N p 1 L(s 1,s )/ N L(s1,s ) N ( ) = q 1 L(s 1,s )/ N L(s1,s ) V p q P SV (p 1, q 1, p, q )Q SV (p 1, q 1, p, q ). (8) These fuctios exhibit all the basic idex properties except global mootoicity (as show by Reisdorf ad Dorfma [11]). But, as show by Balk [2], the failure of mootoicity will oly materialize i rather exceptioal circumstaces. Moreover, the Sato-Vartia idices satisfy the time reversal test as well as the factor reversal test. The fourth pair of ideal idices was developed by Stuvel [13]. They are ot liearly homogeeous i compariso prices (quatities). See Balk [4] for more details o these idices. We ow tur to the differece type idex umber problem. Beet s [5] solutio was V = 1 2 (q + q 1 ) (p 1 p ) (p + p 1 ) (q 1 q ) = q + q 1 (p 1 p 2 ) + p + p 1 (q 1 q 2 ) P B (p 1, q 1, p, q ) + Q B (p 1, q 1, p, q ). (9) The Beet idicators exhibit all the basic properties, plus the time reversal test, ad the factor reversal test. The correpodig multiplicative decompositio is V = exp (p 1 2 p ) L(, V ) exp N q +q1 N p +p1 2 (q 1 q ) L(, V ), (1) Of the basic idex properties these Beet idices fail global mootoicity, as oe easily checks, as well as liear homogeeity i compariso period prices (quatities). The two reversal tests remai satisfied. Motgomery s [9], [1] solutio to the differece type idex umber problem was 5
6 V = L(v 1, v ) L(p 1, p ) (p1 p ) + L(v 1, v ) L(q 1, q ) (q1 q ) P M (p 1, q 1, p, q ) + Q M (p 1, q 1, p, q ). (11) The Motgomery idicators satisfy the time reversal test as well as the factor reversal test. Of the basic properties, they oly fail to exhibit mootoicity globally, but, as argued by Balk [2], this problem is ulikely to be of practical importace. Applyig the trasformatio give i equatio (5) to the Motgomery idicators brigs us back to the Motgomery-Vartia idices. 4 Casler s cotributio Agaist the backgroud sketched i the previous two sectios I ow tur to Casler s [6] cotributio. Basically, Casler either cosiders the value ratio /V or the value differece V, but the asymmetric, forward-lookig growth rate /V 1. It is simple to check that the followig expressio is a idetity, V 1 = N s p + p s q + q s p p q, (12) q where p p 1 p ad q q 1 q ( = 1,..., N). Sice p /p + q /q = p /p + q /q ( = 1,..., N), expressio (12) ca be replaced by V 1 = N s p p + which ca be decomposed as s q q + s p /p + q /q p p /p + q /q p q, (13) q V 1 = N + s p + p s q + q s p /p p /p + q /q s q /q p /p + q /q 6 p p p p q q q. (14) q
7 This expressio ca be simplified to V 1 = N s + s ( 1 + (1/2)H( p /p, q /q ) ) p p ( 1 + (1/2)H( p /p, q /q) ) q, (15) q where H(a, b) 2ab/(a + b) deotes the harmoic mea of a ad b. This appears to be the decompositio favoured by Casler. The first term at the right-had side of expressio (15) gives the cotributio of price chage, ad the secod term gives the cotributio of quatity chage to /V 1. Multiplyig both sides of this expressio by V delivers V = ( q 1 + (1/2)H( p /p, q /q) ) p ( + p 1 + (1/2)H( p /p, q /q) ) q. (16) As Casler observed, these idicators do ot satisfy global mootoicity (because of the iteractio terms), though i practice that might be a mior problem. Also, the time reversal test is ot satisfied. However, to compare this decompositio to Fisher s, oe should proceed oe step further ad tur expressio (16) ito a multiplicative decompositio, by usig the logarithmic mea trasformatio. This leads to { N q (1 + (1/2)H( p /p = exp, q /q } )) p V L(, V ) { N p (1 + (1/2)H( p /p exp, q /q)) } q. (17) L(, V ) These Casler idices have several defects. They are ot globally mootoic i prices (quatities). They are ot liearly homogeeous i compariso period prices (quatities). Ad they do ot satisfy the time reversal test. It is iterestig to observe that the Casler idicators, as i expressio (16), are actually members of a family. This family emerges whe i expressio (13) the ratio p/p + q /q p /p, which is idetically equal to 1, is replaced + q/q 7
8 by f( p/p )+g( q/q ) f( p /p for arbitrary fuctios f(a) ad g(a), which is also )+g( q/q idetically equal to 1. ) The Beet idicators materialize i the case where f(a) = g(a) = 1 for all a. 5 Coclusio It is hard to beat the Fisher idices as decompositio of a value ratio. However, though ideal, they are ot perfect. For example, they are ot cosisteti-aggregatio, ad it is ot straightforward to represet them as weighted sums or products of idividual price (quatity) relatives. O the last poit see Balk [3]. The Motgomery-Vartia idices do exhibit cosistecy-i-aggregatio, ad are icely decomposable to the idividual commodity level, but they are ot liearly homogeeous i compariso prices (quatities). The Sato-Vartia idices share with Fisher the icosistecy-i-aggregatio, but are icely decomposable as appears from expressio (8). The failure of global mootoicity is practically ot very relevat. Though the Stuvel idices are cosistet-i-aggregatio, they are decompositio-resistat ad ot liearly homogeeous i compariso prices (quatities). Though performig perfectly as idicators, the Beet idices fail global mootoicity as well as liear homogeeity i compariso prices (quatities). Both the Casler idicators ad idices fail global mootoicity, ad time reversal. I additio the Casler idices fail liear homogeeity i compariso price (quatities). That the Holy Grail of idex umber theory has as yet ot bee foud, does ot come as a surprise. The thig does ot exist, as substatiated extesively by Balk [4]. But the quest cotiues to deliver iterestig fidigs. 8
9 Refereces [ 1 ] B. M. Balk, Axiomatic Price Idex Theory: A Survey, Iteratioal Statistical Review 63 (1995), [ 2 ] B. M. Balk, Ideal Idices ad Idicators for Two or More Factors, Joural of Ecoomic ad Social Measuremet 28 (23), [ 3 ] B. M. Balk, Decompositios of Fisher Idexes, Ecoomics Letters 82 (24), [ 4 ] B. M. Balk, Price ad Quatity Idex Numbers. Models for Measurig Aggregate Chage ad Differece, Cambridge Uiversity Press, New York, 28 (to appear). [ 5 ] T. L. Beet, The Theory of Measuremet of Chages i Cost of Livig, Joural of the Royal Statistical Society 83 (192), [ 6 ] S. D. Casler, Discrete Growth, Real Output, ad Iflatio: A Additive Perspective o the Idex Number Problem, Joural of Ecoomic ad Social Measuremet 31 (26), [ 7 ] W. E. Diewert, Idex Number Theory usig Differeces rather tha Ratios, Discussio Paper No. 98-1, Departmet of Ecoomics, The Uiversity of British Columbia, Published i Celebratig Irvig Fisher: The Legacy of a Great Ecoomist, edited by R. W. Dimad ad J. Geaakoplos, The America Joural of Ecoomics ad Sociology 64(2), Blackwell Publishig, 25. [ 8 ] I. Fisher, The Makig of Idex Numbers, Houghto Miffli, Bosto, [ 9 ] J. K. Motgomery, Is there a Theoretically Correct Price Idex of a Group of Commodities?, Iteratioal Istitute of Agriculture, Rome, [ 1 ] J. K. Motgomery, The Mathematical Problem of the Price Idex, P. S. Kig & So, Lodo, [ 11 ] M. B. Reisdorf, ad A. Dorfma, The Sato-Vartia Idex ad the Mootoicity Axiom, Joural of Ecoometrics 9 (1999), [ 12 ] K. Sato, The Ideal Log-Chage Idex Number, The Review of Ecoomics ad Statistics 58 (1976), [ 13 ] G. Stuvel, A New Idex Number Formula, Ecoometrica 26 (1957),
10 [ 14 ] Y. O. Vartia, Relative Chages ad Idex Numbers, Licetiate Thesis, Uiversity of Helsiki, 1974; The Research Istitute of the Fiish Ecoomy, Helsiki, [ 15 ] Y. O. Vartia, Ideal Log-Chage Idex Numbers, The Scadiavia Joural of Statistics 3 (1976),
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