INDEX NUMBER THEORY AND MEASUREMENT ECONOMICS. By W.E. Diewert. February CHAPTER 9: Two Stage Aggregation and Homogeneous Weak Separability

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1 IDEX UMBER THEORY AD MEASUREMET ECOOMICS By W.E. Dewert. February 05. CHAPTER 9: Two Stage Aggregaton and Hoogeneous Weak Separablty. Introducton Most statstcal agences use the Laspeyres forula to aggregate prces n two stages. At the frst stage of aggregaton, the Laspeyres forula s used to aggregate coponents of the overall ndex (e.g., food, clothng, servces, etc.) and then at the second stage of aggregaton, these coponent subndexes are further cobned nto the overall ndex. The followng queston then naturally arses: does the ndex coputed n two stages concde wth the ndex coputed n a sngle stage? We wll address ths queston n secton 3 below. However, before answerng the above queston, we wll frst ask a ore fundaental queston: naely, what condtons on consuer s preferences justfy a two stage aggregaton procedure? We address ths secton n secton below.. The Assupton of Hoogeneous Separablty of Preferences It turns out that the assupton of hoogeneous separablty s one of the splest ways of justfyng aggregaton over coodtes n such a way that the coodty aggregate has an aggregate prce that behaves just as f t were a true croeconoc prce. Essentally, ths assupton allows us to apply croeconoc theory to aggregates! The assupton of hoogeneous separablty works n the followng anner. Suppose that a household or consuer has preferences over two groups of coodtes where there are coodtes q [ q,..., q ] n the frst group and coodtes q [ q,..., q ] n the second group. Let the consuer s preferences over all of the coodtes be represented by the nonnegatve, contnuous, quasconcave and ncreasng utlty functon U(q,q ) for q 0 and q 0. The consuer s preferences are hoogeneously separable n the two groups f there exsts a acro utlty functon F(Q,Q ) that s nonnegatve, contnuous, quasconcave and ncreasng n ts two Much of the ateral n secton 3 s adapted fro Dewert (978) and Alteran, Dewert and Feenstra (999). See also Balk (996) for a dscusson of alternatve defntons for the two stage aggregaton concept and references to the lterature on ths topc. The other sple way s through the use of Hcks Aggregaton Theore;.e., f the prces n a group of coodtes vary n strct proporton over te, then the factor of proportonalty can be taken as the prce of the group and the deflated group expendtures wll obey the usual propertes of a croeconoc coodty. Thus we have deonstrated atheatcally the very portant prncple, used extensvely n the text, that f the prces of a group of goods change n the sae proporton, that group of goods behaves just as f t were a sngle coodty. J.R. Hcks (946; 3-33).

2 nonnegatve arguents, Q and Q, and there exst two lnearly hoogeneous, quasconcave and nondecreasng cro utlty functons, f (q ) and f (q ), such that () U(q,q ) = F[f (q ),f (q )]. Let p >> 0 and p >> 0 be two postve vectors of coodty prces facng the consuer n a partcular perod. Snce the cro utlty functons f and f are lnearly hoogeneous, we know that ther correspondng cost functons have the followng for: () n q {p T q : f (q ) Q } = c (p )Q ; (3) n q {p T q : f (q ) Q } = c (p )Q where c (p ) s the unt cost functon that s dual to f and c (p ) s the unt cost functon that s dual to f. ow consder the cost nzaton proble for a consuer that has separable preferences of the for defned by () above: for p >> 0 and p >> 0 and u > 0, we defne the nu cost of achevng the utlty level u as follows: (4) C(u,p,p ) n q s { p T q + p T q : F[f (q ),f (q )] u} = n q s, Q s { p T q + p T q : F[Q,Q ] u, Q f (q ), Q f (q )} where we added two extra constrants to the cost nzaton proble by defnng Q f (q ) and Q f (q ) = n Q s {c (p )Q + c (p )Q : F[Q,Q ] u} usng () and (3) C*[u, c (p ),c (p )] where C*(u,P,P ) s the acro cost functon that s dual to the acro utlty functon F(Q,Q ). Lookng at (4), t can be seen that the unt cost functons, c (p ) and c (p ), act lke croeconoc prces for the quantty aggregates, Q = f (q ) and Q = f (q ). Ths s what s powerful about the assupton of hoogeneous separablty. 3 For certan functonal fors for c (p ) or f (q ) for =,, lsted n Pollak (983) or Dewert (976), we can use exact ndex nuber forulae to calculate these prce and quantty aggregates usng the assupton of optzng behavor along wth observed prce and quantty data for two perods. Proble. Repeat the analyss around equatons () to (4) assung 3 hoogeneous aggregates nstead of. 3 Ths odel of aggregaton dates back to Leontef (947) but t s ost clearly explaned by Shephard (953; 6-7) (970; 45-46). See also Arrow (974) (n the producer context), Sauelson and Sway (974), Dewert (980; 438-4) and Blackorby, Pront and Russell (978) for addtonal references and exposton.

3 3 3. Two Stage Aggregaton of Index uber Forulae We return to the queston that was rased n the ntroducton; naely, does a Laspeyres ndex coputed n two stages concde wth a Laspeyres ndex coputed n a sngle stage? Suppose that the prce and quantty data for perod t, p t and q t, can be wrtten n ters of M subvectors as follows: (5) p t = [p t,p t,...,p tm ] ; q t = [q t,q t,...,q tm ] ; t = 0, where the densonalty of the subvectors p t and q t s for =,,,M wth the su of the densons equal to. These subvectors correspond to the prce and quantty data for subcoponents of the consuer prce ndex for perod t. ow construct subndces for each of these coponents gong fro perod 0 to. For the base perod, set the prce for each of these subcoponents, say P 0 for =,, M, equal to and set the correspondng base perod subcoponent quanttes, say Q 0 for =,,,M, equal to the base perod value of consupton for that subcoponent for =,,,M: 0 0 (6) P ; Q p q for,,..., M. ow use the Laspeyres forula n order to construct a perod prce for each subcoponent, say P for =,,,M, of the consuer prce ndex. Snce the densonalty of the subcoponent vectors, p t and q t, dffers fro the densonalty of the coplete perod t vectors of prces and quanttes, p t and q t, t s necessary to use dfferent sybols for these subcoponent Laspeyres ndexes, say P L for =,, M. Thus the perod subcoponent prces are defned as follows: p q (7) P PL ( p, p, q, q ) for,,..., M. p q Once the perod prces for the M subndexes have been defned by (7), then correspondng subcoponent perod quanttes Q for =,,,M can be defned by deflatng the perod subcoponent values = p q by the prces P : p q (8) Q P for,,..., M.

4 4 ow defne subcoponent prce and quantty vectors for each perod t = 0, usng equatons (6) to (8) above. Thus defne the perod 0 and subcoponent prce vectors P 0 and P as follows: (9) P ( P, P,..., P ) ; P ( P, P,..., P ) M M where M denotes a vector of ones of denson M and the coponents of P are defned by (7). The perod 0 and subcoponent quantty vectors Q 0 and Q are defned as follows: (0) Q 0 [Q 0,Q 0,,Q M 0 ] ; Q [Q,Q,,Q M ] where the coponents of Q 0 are defned n (6) and the coponents of Q are defned by (8). The prce and quantty vectors n (9) and (0) represent the results of the frst stage aggregaton. ow use these vectors as nputs nto the second stage aggregaton proble;.e., apply the Laspeyres prce ndex forula usng the nforaton n (9) and (0) as nputs nto the ndex nuber forula. Snce the prce and quantty vectors that are nputs nto ths second stage aggregaton proble have denson M nstead of the sngle stage forula whch utlzed vectors of denson, a dfferent sybol s requred for the new Laspeyres ndex whch we choose to be P L *. Thus the Laspeyres prce ndex coputed n two stages s denoted as P L *(P 0,P,Q 0,Q ). ow ask whether ths two stage Laspeyres ndex equals the correspondng sngle stage ndex P L ;.e., ask whether () P L *(P 0,P,Q 0,Q ) = P L (p 0,p,q 0,q ). If the Laspeyres forula s used at each stage of each aggregaton, the answer to the above queston s yes: straghtforward calculatons show that the Laspeyres ndex calculated n two stages equals the Laspeyres ndex calculated n one stage. Probles. Verfy that the Laspeyres ndex calculated n two stages equals the Laspeyres ndex calculated n one stage. 3. Is t true that the Paasche ndex calculated n two stages equals the Paasche ndex calculated n one stage? ow suppose that the Fsher or Törnqvst forula s used at each stage of the aggregaton;.e., n equatons (7), suppose that the Laspeyres forula P L (p,p,q,q ) s replaced by the Fsher forula P F (p,p,q,q ) (or by the Törnqvst forula P T (p,p,q,q )) and n equaton (), P L *(P 0,P,Q 0,Q ) s replaced by P F * (or by P T *) and P L (p 0,p,q 0,q ) s replaced by P F (or by P T ). Then s t the case that counterparts are obtaned to the two stage aggregaton result for the Laspeyres forula, ()? The answer s no; t can be shown that, n general, M

5 5 () P F *(P 0,P,Q 0,Q ) P F (p 0,p,q 0,q ) and P T *(P 0,P,Q 0,Q ) P T (p 0,p,q 0,q ). Slarly, t can be shown that the quadratc ean of order r ndex nuber forula P r defned and the plct quadratc ean of order r ndex nuber forula P r * defned n chapter 5 are also not consstent n aggregaton. However, even though the Fsher and Törnqvst forulae are not exactly consstent n aggregaton, t can be shown that these forulae are approxately consstent n aggregaton. More specfcally, t can be shown that the two stage Fsher forula P F * and the sngle stage Fsher forula P F n (), both regarded as functons of the 4 varables n the vectors p 0,p,q 0,q, approxate each other to the second order around a pont where the two prce vectors are equal (so that p 0 = p ) and where the two quantty vectors are equal (so that q 0 = q ) and a slar result holds for the two stage and sngle stage Törnqvst ndexes n (). 4 As was seen n chapter 5, the sngle stage Fsher and Törnqvst ndexes have a slar approxaton property so all four ndexes n () approxate each other to the second order around an equal (or proportonal) prce and quantty pont. Thus for noral te seres data, sngle stage and two stage Fsher and Törnqvst ndexes wll usually be nuercally very close. 5 Ths result wll be llustrated n chapter for an artfcal data set. Slar approxate consstency n aggregaton results (to the results for the Fsher and Törnqvst forulae explaned n the prevous paragraph) can be derved for the quadratc ean of order r ndexes, P r, and for the plct quadratc ean of order r ndexes, P r *; see Dewert (978; 889). However, the results of Hll (006) agan ply that the second order approxaton property of the sngle stage quadratc ean of order r ndex P r to ts two stage counterpart wll break down as r approaches ether plus or nus nfnty. To see ths, consder a sple exaple where there are only four coodtes n total. Let the frst prce rato p /p 0 be equal to the postve nuber a, let the second two prce ratos p /p 0 equal the b and let the last prce rato p 4 /p 4 equal c where we assue a < c and a b c. Usng the propertes of eans of order r, the ltng value of the sngle stage ndex s: (3) l r 0 0 r 0 0 p p P ( p, p, q, q ) l P ( p, p, q, q ) n ax ac. 0 r r p p 0 ow aggregate coodtes and nto a subaggregate and coodtes 3 and 4 nto another subaggregate. Usng the propertes of eans of order r agan, t s found that the ltng prce ndex for the frst subaggregate s [ab] / and the ltng prce ndex for the second subaggregate s [bc] /. ow apply the second stage of aggregaton and use the propertes of eans of order r once agan to conclude that the ltng value of the two 4 See Dewert (978; 889). In fact, these dervatve equaltes are stll true provded that p = p 0 and q = q 0 for any nubers > 0 and > 0. 5 For an eprcal coparson of the four ndexes, see Dewert (978; ). For the Canadan consuer data consdered there, the chaned two stage Fsher n 97 was.38 and the correspondng chaned two stage Törnqvst was.330, the sae values as for the correspondng sngle stage ndexes.

6 6 stage aggregaton usng P r as the ndex nuber forula s [ab c] /4. Thus the ltng value, as r tends to plus or nus nfnty, of the sngle stage aggregate over the two stage aggregate s [ac] / /[ab c] /4 = [ac/b ] /4. ow b can take on any value between a and c and so the rato of the sngle stage ltng P r to ts two stage counterpart can take on any value between [c/a] /4 and [a/c] /4. Snce c/a s less than and a/c s greater than, t can be seen that the rato of the sngle stage to the two stage ndex can be arbtrarly far fro as r becoes large n agntude wth an approprate choce of the nubers a, b and c. The results n the prevous paragraph show that soe cauton s requred n assung that all superlatve ndexes wll be approxately consstent n aggregaton. However, for the three ost coonly used superlatve ndexes (the Fsher deal P F, the Törnqvst-Thel P T and the Walsh P W ), the avalable eprcal evdence ndcates that these ndexes satsfy the consstency n aggregaton property to a suffcently hgh enough degree of approxaton that users wll not be unduly troubled by any nconsstences. 6 Proble 4. Use the propertes of eans of order r to establsh the results n (3) above. References Alteran, W.F., W.E. Dewert and R.C. Feenstra, (999), Internatonal Trade Prce Indexes and Seasonal Coodtes, Bureau of Labor Statstcs, Washngton D.C. Arrow, K.J. (974), The Measureent of Real Value Added, n atons and Households n Econoc Growth: Essays n Honor of Moses Abraovtz, P. Davd and M. Reder (eds.), ew York: Acadec Press. Balk, B.M. (996), Consstency n Aggregaton and Stuvel Indces, The Revew of Incoe and Wealth 4, Blackorby, C., D. Pront and R.R. Russell (978), Dualty, Separablty and Functonal Structure: Theory and Econoc Applcatons, ew York: orth-holland. Dewert, W.E. (976), Exact and Superlatve Index ubers, Journal of Econoetrcs 4, Dewert, W.E. (978), Superlatve Index ubers and Consstency n Aggregaton, Econoetrca 46, Dewert, W. E. (980), Aggregaton Probles n the Measureent of Captal, pp n The Measureent of Captal, Dan Usher (ed.), Unversty of Chcago Press, Chcago. 6 See chapter for soe addtonal evdence on ths topc.

7 7 Hcks, J.R. (946), Value and Captal, Second Edton, Oxford: Clarendon Press. Hll, R.J. (006), Superlatve Indexes: ot All of The are Super, Journal of Econoetrcs 30, Leontef, W. W. (947), Introducton to a Theory of the Internal Structure of Functonal Relatonshps, Econoetrca 5, Pollak, R.A. (983), The Theory of the Cost-of-Lvng Index, pp n Prce Level Measureent, W.E. Dewert and C. Montarquette (eds.), Ottawa: Statstcs Canada; reprnted as pp. 3-5 n R.A. Pollak, The Theory of the Cost-of-Lvng Index, Oxford: Oxford Unversty Press, 989; also reprnted as pp n Prce Level Measureent, W.E. Dewert (ed.), Asterda: orth-holland, 990. Sauelson, P. A. and S. Sway (974), Invarant Econoc Index ubers and Canoncal Dualty: Survey and Synthess, Aercan Econoc Revew 64, Shephard, R. W. (953), Cost and Producton Functons, Prnceton: Unversty Press. Prnceton Shephard, R. W. (970), Theory of Cost and Producton Functons, Prnceton: Prnceton Unversty Press.