The Differential Approach to Superlative Index Number Theory
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1 The Dffeeal Aoach o Suelae Ide Numbe Theoy by Wllam A. Bae K-Hog Cho ad Taa M. Scla Decembe 8 Fohcomg he Ha Thel Memoal Secal ue of he Joual of Agculual ad Aled Ecoomc Wllam A. Bae ofeo of Ecoomc a Wahgo Uey. K-Hog Cho a he Naoal eo Reeach Cee Seoul Koea. Taa M. Scla a Wahgo Uey. Coac Auho: Wllam A. Bae Ecoomc Deame Wahgo Uey Camu Bo 8 Oe Boog De Sa ou MO 63 hoe: Fa: Emal: bae@uecoc.ul.edu
2 Abac Dee (976) uelae de umbe defed o be eac fo ecod ode aggegao fuco ufy de umbe heoy h aggegao heoy bu hae bee dffcul o defy. We ee a e aoach o fdg eleme of h cla. Th e aoach elaed o ha adocaed by He Thel (973) afom caddae de umbe o goh ae fom ad eloe coegece ae o he Da de. Sce he Da de couou me eac fo ay aggegao fuco ay dcee me de umbe ha coege o he Da de ad ha ha a hd ode emade em uelae. Keyod: Da de umbe uelae dee.
3 . Ioduco Accodg o Thel ( ) The ubjec of de umbe oe of he olde ac ad alo a egad he moe ecalzed ubjec of co of lg de umbe a old oe ecoomc. Alhough a old ubjec ecoom hae log uggled o defy ueful de umbe. The mo flueal eleco ceo ha he de umbe be eac fo a aggegao fuco ha ca oduce a ecod ode aomao o ay ce cououly dffeeable lealy homogeeou fuco. Dee (976) defed uch de umbe o be uelae. Suelae de umbe hu hae o acg ably elae o he eac aggegao fuco of ecoomc aggegao heoy. The cla of uelae de umbe coa a fe umbe of de umbe ce a fe umbe of ecod ode aggegao fuco e bu oly a mall umbe of de umbe he uelae cla hae o fa bee foud. The each oce ha eouly oled fdg a de umbe hch eac fo a o ecod ode algebac aggegao fuco o eachg fo a ecod ode aggegao fuco fo hch a o de umbe eac. No mle ocedue ha bee foud fo ehe deco. Fo eamle he mfle aue aggegao fuco ogaed by Bae ad ee (985) oe ffee yea ago o o be ecod ode bu o oe ha ucceeded fdg he de umbe ha ca ac eacly. I he ohe deco Fhe (9) ooed may de umbe h famou boo bu o he ee day he aggegao fuco aced eacly by hem ema uo fo mo of hoe de umbe. The Da couou me de hold a ome lace he leaue becaue Vlle Hule Samuelo ad Samy ad Bae ad Sele (. -) hae ho ha he Da le egal oduce he uque eac de umbe fomula fo ay eoclacal 3
4 aggegao fuco. Smlaly he Da ce de he uque eac de umbe fomula couou me fo he eoclacal aggegao fuco dual u co fuco. Thee eul mly ha he Da egal de he ooye ecoomc de umbe. Fo geeal ue hoee he Da couou me de mu be adaed o aly o dcee daa. A log-chage fom de uual fo all ell o dcee me aomao o he Da de. Th obeao aloe o uffce o deeme he egh fuco. Thu hee hae bee ublhed lage umbe of oeal fe chage aomao o he Da de. Each log chage fom ad hey ae dffeeaed by he egh. We ho ha Thel dffeeal aoach hch he ued o uo he Töq de (973) ca be ued yemacally o deeme hch fe chage aomao o he Da de ae uelae. Sce he Da le egal couou me eac fo ay aggegao fuco ay uelae dcee me de umbe mu: ) coege o he Da de a he me eal ao ad ) hae a hd ode emade em fo fe-chage me eal.. Th ue egadle of hehe o o e ae caable of fdg he ecod ode aggegao fuco fo hch he dcee me de umbe eac dcee me. To ue h aoach eceay o be able o u caddae de umbe o goh ae fom dcee me (Thel (974) og-chage fom) o ha coegece ae o he Da de ca be eloed. Ug coegece heoem hch ae dely aalable mahemac become oble o defy lage umbe o comaably good de umbe ad eha ee o fd e de umbe h bee oee ha he cuely o de umbe. I h ae e ll ho ha he mo ell o de umbe ca be eeeed log-chage fom. I he e eco e ll code he mea of ode cla of de 4
5 umbe. The e ll code he quadac mea of ode cla of de umbe. I he cocluo e ll dcu he eeao of he egh log-chage fom ad ugge fuhe eeach h aea.. og-chage Reeeao of he Mea of Ode Cla of Ide Numbe e be he quay of good dug eod ad le be ce. The mea of ode de of aggegae ce chage beee eod ad a defed by Alle ad Dee (98) chaacezed by eleco of he eoe ad uec. If e defe he eod co hae a fo... N he mea of ode de of aggegae ce chage fo beee eod ad ug eod hae () hee he ce chage de h beg he ce de leel eod. beg he ce de leel eod ad Membe of h cla clude he aeye de ( ) ad he aache de ( - ). The mea of ode quay de defed aalogouly by echagg he ole of ce ad quae he defo. Theoem : The mea of ode ce de ca be afomed o log-chage fom ad he um of egh log-chage fom le ha o equal o uy. 5
6 6 oof: If e ae he aual logahm of boh de of he mea of ode ce de e fd: () l l l hee ad hee. l l l We o aly he coce of he log-mea hch a oduced by Vaa ad Sao o he ecoomc leaue fo y > : (3) ( ) y y l y y y Fo ou equao e le ad e le y. We he hae (4) l l
7 7 l. eg (5) e hae ha (6) l l hch he log-chage fom of he mea of ode ce de. The egh ca equalely be e a: (7). Fom h fom e ca ho ha um le ha o equal o uy a follo:
8 8 (8) Bu l ) ( cocae a oe he Aed. Hece by Jee equaly h equaly oly f o fo all. Sce he log-chage aomao o he Da de baed o he Weghed Mea Value Theoem (Ful(978). 6) e hae he equeme ha he dcee me eal hould be a ao a daa allo. A fo all he um of he egh aoache uy. Q.E.D. Noe ha h ecouage he ue of he cha mehod oduced by Alfed Mahall ad decbed by Fch. Dee (978) alo ague fo he ue of h mehod. Eamle: We ca e he log-chage fom of he aeye de of aggegae ce chage beee eod ad ( ) a follo: (9) aeye l l hee aeye aeye aeye l l l ad hee he aeye egh ae
9 9 aeye. Smlaly e ca e he log-chage fom of he aache ce de ( - ) a follo: () aache l l hee aache aache aache l l l ad hee he aache egh ae aache. 3. og-chage Reeeao of he Quadac Mea of Ode Cla of Ide Numbe The quadac mea of ode de cloely elaed o he mea of ode de fom aboe. Dee (976) defe he quadac mea of ode ce chage de em of he mea of ode ce chage de a follo: ()
10 fo hee he ce chage de h beg he ce de leel eod ad beg he ce de leel eod. Dee (976) ha ho ha he quadac mea of ode cla of dee uelae fo all. Tha cla he mo geeal uelae de umbe ecfcao o ad clude he Fhe deal de ad he Töq de. Coollay o Theoem : The quadac mea of ode ce de ca be afomed o log-chage fom ad he um of egh log-chage fom le ha o equal o uy. oof: The oof a mle algebac maulao alyg he elaoh beee he quadac mea of ode ce de he mea of ode ce de ad ou eul fom Theoem. Noe ha () l l l l l o ha
11 (3) l l l hee (4). We ca ead o deeme he alue of um: (5). If e ae he ummao of boh de e oba: (6). A he cae of he mea of ode de each of he o em h he oue aehee o he gh had de le ha o equal o oe. Hece he um of hoe em dded by o alo le ha o equal o oe ad aoache oe a fo all. Q.E.D. Eamle: We ca e he log-chage fom of he Fhe deal ce de ( ) a follo:
12 (7) ( ) aache aeye Fhe l l hee Fhe Fhe Fhe l l l. We alo o ha he Töq de a lmg membe of h cla (fo ). Sce e dd o code he cae of aboe e o ho ha h de alo ca be eeeed log-chage fom. Fo Alle ad Dee (98) defed he quadac mea of ode ce chage de a: (8) hee. Reul: ca be eeeed log-chage fom ad he egh he um o. oof: (9) ( ) l l l l hee () ( ) We ca ee fom he defo of he egh (... N) ha he um h ecal cae become eacly uy. Q.E.D.
13 4. Cocluo We hae ho ha o lage clae of ecoomc de umbe ca be eeeed log-chage fom. Th ode a e mehod o deeme hehe de umbe ae uelae. Iead of eachg fo he ecod ode aggegao fuco fo hch he dcee me de umbe eac dcee me e ca e he uggeed de umbe coegece o he Da de log-chage fom. If a de coege o he Da de a he me eal ao ad ha a hd ode emade em fo fe-chage me eal he he de uelae. The log-chage fom ode a ufed e of de umbe fomula ad he coegece oee. Thee alo a ueful eeao of de umbe h fom. Recall he fom: () l l hee l l l. Equao () a adde decomoo (Töq Vaa ad Vaa (985)) of he global ae of goh l o each cobug faco l. Thee eul ugge he oeal oducy of fuhe eeach egag e de umbe by h mehod o ee f hey ae uelae o bee ha uelae h emade em of ode hghe ha 3. Aed e f() () hee ( ) defed a he oof of Theoem. We o l oe he follog lemma. 3
14 emma : f() cocae fo all >. oof: The ecod deae of f l l f ( ). 3 (l ) eg 3 d ( ) (l ) ad h ( ) l l e oba h( ) f ( ). d( ) Clealy d() egae fo < < ad oe fo >. Obee ha h ( ) h ( ) ad h ( ) ce h ( ) l ad h ( ) ( ). Thu e o ha a fleco o (ccal o) of he cue h (). Bu h ( ) < fo > ad h ( ) > fo << o ha h() cly cocae fo > ad cly coe fo <<. Hece h() egae fo > ad oe fo < <. I follo mmedaely ha f ( ) < fo all oe. Q.E.D. Th eul ca be geealzed o he full logahmc mea fuco y ( y). l l y We ae debed o W. E Dee fo he follog oof oded o u hough ae coeodece. 4
15 Coollay o emma : (y) cocae fo all y> y. oof: Code he Hea ma H(y) of ecod ode aal deae of (y). By a agume aalogou o ha ued he oof of emma follo ha he dagoal eleme of H(y) ae cly egae fo y. Bu ce (y) lealy homogeeou he deema of H(y) zeo. See Hada (97 eq ). A all o off of he ay y he eceay ad uffce codo ae heefoe afed fo H(y) o be egae emdefe. Q.E.D. 5
16 Refeece Alle R. ad W.E. Dee. Dec Veu Imlc Suelae Ide Numbe Fomulae. Ree of Ecoomc ad Sac 63(Augu 98): Bae W.A. ad A. Sele. The Theoy of Moeay Aggegao. Amedam: Noh Hollad. Bae W.A. ad Y. ee. The Global oee of he Mfle aue Geealzed eoef ad Talog Fleble Fucoal Fom. Ecoomeca 53(Noembe 985): Dee W.E. Eac ad Suelae Ide Numbe. Joual of Ecoomec 4(May 976):5-45. Dee W.E. Suelae Ide Numbe ad Coecy I Aggegao. Ecoomeca 46(July 978): Fhe I. The uchag oe of Moey. odo: Macmlla 9. Fch R. Aual Suey of Geeal Ecoomc Theoy: The oblem of Ide Numbe. Ecoomeca 4(Jauay 936):-38. Ful W. Adaced Calculu. Ne Yo: Joh Wley & So 978. Hada J. Mahemacal Theoy of Ecoomc Behao. Readg MA: Addo-Weley 97. Hule C. Da Ide Numbe. Ecoomeca 4(Noembe 973):7-6. Samuelo.A. ad S. Samy. Iaa Ecoomc Ide Numbe ad Caocal Dualy: Suey ad Syhe. Ameca Ecoomc Ree 64(Seembe 974): Sao K. The Ideal og-chage Ide Numbe. The Ree of Ecoomc ad Sac 58(May 976):3-8. 6
17 Thel H. A Ne Ide Numbe Fomula. The Ree of Ecoomc ad Sac 55(Noembe 973): Thel H. Be ea Ide Numbe of ce ad Quae. Ecoomeca 8(Al 96): Thel H. Moe o og-chage Ide Numbe. The Ree of Ecoomc ad Sac 56(Noembe 974): Töq.. Vaa ad Y.O. Vaa. Ho Should Relae Chage Be Meaued? The Ameca Saca 39(985): Vaa Y.O. Ideal og-chage Ide Numbe. Scadaa Joual of Sac 3(Seembe 976):-6. Nema.K. ad J. Vlle. The Eece Codo of A Toal Uly Fuco. The Ree of Ecoomc Sude 9(95-95):3-8. 7
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