On Measuring the Contribution of Entering and Exiting Firms to Aggregate Productivity Growth

Transcription

1 1 On Measuring he Conribuion of Enering and Exiing Firms o Aggregae Produciviy Growh by W. Erwin Diewer and Kevin J. Fox* April 13, Absrac The problem of assessing he impac of firm enry and exi on aggregae produciviy growh is addressed. The proposed mehod overcomes some problems wih currenly proposed mehods. The paper also addresses some of he problems involved in aggregaing oupus and inpus when firms ener and exi so ha he one oupu and one inpu aggregae produciviy decomposiions can be applied. I urns ou ha mulilaeral index number heory is useful in performing he aggregaion of many oupus (inpus) ino a single oupu (inpu). Key words: Produciviy, index numbers, indusry dynamics, enry and exi of firms, mulilaeral index number heory. Journal of Economic Lieraure Classificaion Codes: C43, D24, E23. * The auhors hank Ber Balk and Alice Nakamura for helpful commens, he Universiy of Valencia for hospialiy, and he SSHRC of Canada, he Ausralian Research Council, he Minisry of Educaion and Science of Spain (Secrearía de Esado de Universidades e Invesigación, SAB ) and he School of Economics a he Universiy of New Souh Wales for financial suppor. None of he above are responsible for any opinions expressed in he paper. W. Erwin Diewer Kevin J. Fox Deparmen of Economics School of Economics The Universiy of Briish Columbia The Universiy of New Souh Wales Vancouver, B.C. V6T 1Z1 Sydney 2052 Canada. Ausralia. diewer@econ.ubc.ca K.Fox@unsw.edu.au

2 2 1. Inroducion A recen developmen in produciviy analysis is he increased focus on he impac of firm enry and exi ino an indusry on aggregae levels of produciviy growh. Haliwanger and Barelsman and Doms in heir survey papers make he following observaions: 1 There are large and persisen differences in produciviy across esablishmens in he same indusry (see Barelsman and Doms (2000) for an excellen discussion). The differences hemselves are large for oal facor produciviy he raio of he produciviy level for he plan a he 75 h percenile o he plan a he 5 h percenile in he same indusry is 2.4 (his is he average across indusries) he equivalen raio for labour produciviy is 3.5. John Haliwanger (2000; 9). The raio of average TFP for plans in he ninh decile of he produciviy disribuion relaive o he average in he second decile was abou 2 o 1 in 1972 and abou 2.75 o 1 in Eric J. Barelsman and Mark Doms (2000; 579). Thus he recen produciviy lieraure has demonsraed empirically ha increases in he produciviy of he economy can be obained by reallocaing resources 2 away from low produciviy firms in an indusry o he higher produciviy firms. 3 However, differen invesigaors have chosen differen mehods for measuring he conribuions o indusry produciviy growh of enering and exiing firms and he issue remains open as o which mehod is bes. We propose ye anoher mehod for accomplishing his decomposiion. I differs from exising mehods in ha i reas ime in a symmeric fashion so ha he indusry produciviy difference in levels beween wo periods reverses sign when he periods are inerchanged as do he various conribuion erms. 4 Our proposed produciviy decomposiion is explained in secions 2 and 3 below, assuming ha each 1 See heir papers for many addiional references o he lieraure. Some of he more imporan references are Baldwin and Gorecki (1991), Baily, Hulen and Campbell (1992), Griliches and Regev (1995), Baldwin (1995), Haliwanger (1997), Ahn (2001), Foser, Haliwanger and Krizan (2001), Aw, Chen and Robers (2001), Fox (2002), Baldwin and Gu (2002), Balk (2003), and Barelsman, Haliwanger and Scarpea (2004). 2 A more precise meaning for he erm reallocaing resources is changing inpu shares. 3 This conclusion has also emerged from he exensive lieraure on benchmarking and on Daa Envelopmen Analysis; e.g., see Coelli, Prasada Rao and Baese (1998). 4 Balk (2003; 29) also emphasized he imporance of a symmeric reamen of ime. A symmeric decomposiion was proposed earlier by Griliches and Regev (1995) and a modificaion of i was used by Aw, Chen and Robers (2001).

3 3 firm in he indusry produces only one homogeneous oupu and uses only one homogeneous inpu. Anoher problem wih he various produciviy decomposiions ha have been suggesed in he lieraure is ha hey ofen assume ha here is only one oupu and one inpu ha each producion uni in he indusry produces and uses. If he lis of oupus being produced and inpus being used by each firm is consan across firms, hen here is no problem in using normal index number heory o consruc oupu and inpu aggregaes for each coninuing firm ha is presen for he wo periods under consideraion. 5 However, his mehod for consrucing oupu and inpu aggregaes does no work for enering and exiing firms, since here is no naural base observaion o compare he single period daa for hese firms. This problem does no seem o have been widely recognized in he lieraure wih some noable excepions. 6 Hence in he remainder of his paper, we focus our aenion on soluions o his problem. Our suggesed soluion o his problem is o use mulilaeral index number heory so ha each firm s daa in each ime period is reaed as if i were he daa peraining o a counry. Unforunaely, here are many possible mulilaeral mehods ha could be used. In secion 5 below, we consruc an arificial daa se involving hree coninuing firms, one enering and one exiing firm and hen in he remaining secions of he paper, we use various mulilaeral aggregaion mehods in order o consruc firm oupu and inpu aggregaes, which we hen use in our suggesed produciviy growh decomposiion formula. The mulilaeral aggregaion mehods ha we consider are: he sar sysem (secion 6); he GEKS sysem (secion 7); he own share sysem (secion 8); he spaial linking mehod due o Rober Hill (secion 9) and a simple deflaion of value aggregaes mehod (secion 10). Secion 11 concludes. 5 An economic jusificaion for using a superlaive index o accomplish his aggregaion can be supplied under some separabiliy assumpions; see Diewer (1976). 6 Aw, Chen and Robers (2001) and Aw, Chung and Robers (2003) recognized he imporance of his problem and hey used a modificaion of a mulilaeral mehod originally proposed by Caves, Chrisensen and Diewer (1982). The modificaion ha hey used is due o Good (1985) and is explained in Good, Nadiri and Sickles (1997). The original Caves, Chrisensen and Diewer mehod was designed for use in a single cross secion and is no suiable for use in a panel daa conex if here is considerable inflaion beween he periods in he panel.

4 4 2. The Measuremen of Aggregae Produciviy Levels in he One Oupu One Inpu Case We begin by considering a very simple case where firms produce one oupu wih one inpu so ha i is very sraighforward o measure he produciviy of each firm by dividing is oupu by is inpu used. 7 We assume ha hese firms are all in he same indusry, producing he same oupu and using he same inpu, so ha i is also very sraighforward o measure indusry produciviy in each period by dividing aggregae indusry oupu by aggregae indusry inpu. Our measuremen problem is o accoun for he conribuions o indusry produciviy growh of enering and exiing firms. In wha follows, C denoes he se of coninuing producion unis ha are presen in periods 0 and 1, X denoes he se of exiing firms which are only presen in period 0, and N denoes he se of new firms ha are presen only in period 1. Le y Ci > 0 and x Ci > 0 respecively denoe he oupu produced and inpu uilized by coninuing uni i C during period = 0, 1. Le y 0 Xi > 0 and x 0 Xi > 0 respecively denoe 1 he oupu produced and inpu used by exiing firm i X during period 0. Finally, le y Ni > 0 and x 1 Ni > 0 respecively denoe he oupu produced and inpu used by he new firm i N during period 1. The produciviy level Ci of a coninuing firm i C in each period can be defined as oupu y Ci divided by inpu x Ci : (1) Ci y Ci / x Ci ; i C ; = 0,1. The produciviy levels of he exiing firms in period 0 and he enering firms in period 1 are defined in a similar fashion, as follows: 7 We will consider he case of many oupus and many inpus in secions 4-10 below.

5 5 (2) Xi 0 y Xi 0 / x Xi 0 ; i X ; (3) Ni 1 y Ni 1 / x Ni 1 ; i N. Since he producion unis are all producing he same oupu and are using he same inpu, a naural definiion for indusry produciviy 0 in period 0 is aggregae oupu divided by aggregae inpu: 8 (4) 0 [ i C y Ci 0 + i X y Xi 0 ] / [ i C x Ci 0 + i X x Xi 0 ] = S C 0 i C s Ci 0 Ci 0 + S X 0 i X s Xi 0 Xi 0 where he aggregae inpu shares of he coninuing and exiing firms in period 0, S C 0 and S X 0, are defined as follows: (5) S C 0 i C x Ci 0 / [ i C x Ci 0 + i X x Xi 0 ] ; (6) S X 0 i X x Xi 0 / [ i C x Ci 0 + i X x Xi 0 ]. In addiion, he period 0 micro inpu share, s Ci 0, for a coninuing firm i C is defined as follows: (7) s Ci 0 x Ci 0 / k C x Ck 0 ; i C. Thus s 0 Ci is he inpu of coninuing firm i in period 0, x 0 Ci, divided by he oal inpu used by all coninuing firms in period 0, k C x 0 Ck. Similarly, he period 0 micro inpu share for exiing firm i X, s 0 Xi, is defined he inpu of exiing firm i in period 0, x 0 Xi, divided by he oal inpu used by all exiing firms in period 0, k X x 0 Xk : (8) s Xi 0 x Xi 0 / k X x Xk 0 ; i X. 8 I is possible o rework our analysis by reversing he role of inpus and oupus so ha oupu shares replace inpu shares in he decomposiion formulae. Then a he end, we can ake he reciprocal of he aggregae inverse produciviy measure and obain an alernaive produciviy decomposiion. We owe his suggesion o Ber Balk.

6 6 Of course, he period 0 aggregae produciviies for coninuing and exiing firms, C 0, and X 0, can be defined in a similar manner o he definiion of 0 in (4), as follows: (9) 0 C i C y 0 0 Ci / i C x Ci = i C s 0 Ci 0 Ci ; (10) 0 X i X y 0 0 Xi / i X x Xi = i X s 0 Xi 0 Xi. Subsiuion of (9) and (10) back ino definiion (4) for he aggregae period 0 level of produciviy leads o he following decomposiion of aggregae period 0 produciviy ino is coninuing and exiing componens: (11) 0 = S C 0 C 0 + S X 0 X 0 (12) = C 0 + S X 0 ( X 0 C 0 ) where (12) follows from (11) using S C 0 = 1 S X 0. Expression (12) is a useful decomposiion of he period 0 aggregae produciviy level 0 ino wo componens. The firs componen, 0 C, represens he produciviy conribuion of coninuing producion unis while he second erm, S 0 X ( 0 X 0 C ), represens he conribuion of exiing firms relaive o coninuing firms o he overall period 0 produciviy level. Usually he exiing firm will have lower produciviy levels han he coninuing firms so ha 0 X will be less han 0 C and hus under normal condiions, he second erm on he righ-hand side of (12) will make a negaive conribuion o he overall level of period 0 produciviy. 9 9 Olley and Pakes (1996; 1290) have an alernaive covariance ype decomposiion of he overall level of produciviy in a given period ino firm effecs bu i is no suiable for our purpose, which is o highligh he differenial effecs on overall period 0 produciviy of he exiing firms compared o he coninuing firms.

7 7 Subsiuing (9) and (10) ino (12) leads o he following decomposiion of he period 0 produciviy level 0 ino is individual firm conribuions: (13) 0 = i C s Ci 0 Ci 0 + S X 0 i X s Xi 0 ( Xi 0 C 0 ) where we have also used he fac ha i X s Xi 0 sums o uniy. Obviously, he above maerial can be repeaed wih minimal modificaions o provide a decomposiion of he indusry period 1 produciviy level 1 ino is consiuen componens. Thus, 1 is defined as follows: (14) 1 [ i C y Ci 1 + i N y Ni 1 ] / [ i C x Ci 1 + i N x Ni 1 ] = S C 1 i C s Ci 1 Ci 1 + S N 1 i N s Ni 1 Ni 1 where he period 1 aggregae inpu shares of coninuing and new firms, S C 1 and S N 1, and individual coninuing and new firm shares, s Ci 1 and s Ni 1, are defined as follows: (15) S 1 C i C x 1 Ci / [ i C x 0 Ci + i X x 0 Xi ] ; (16) S 1 N i N x 1 Ni / [ i C x 0 Ci + i X x 0 Xi ] ; (17) s 1 Ci x 1 Ci / k C x 1 Ck ; i C ; (18) s 1 Ni x 1 Ni / k N x 1 Nk ; i N. The period 1 counerpars o C 0 and X 0 defined by (9) and (10) are he aggregae period one produciviy levels of coninuing firms C 1 and enering firms N 1, defined as follows: (19) 1 C i C y 1 1 Ci / i C x Ci = i C s 1 Ci 1 Ci ; (20) 1 N i N y 1 1 Ni / i N x Ni = i N s 1 Ni 1 Ni.

8 8 Subsiuion of (19) and (20) back ino definiion (14) for he aggregae period 1 level of produciviy leads o he following decomposiion of aggregae period 1 produciviy ino is coninuing and new componens: (21) 1 = S C 1 C 1 + S N 1 N 1 (22) = C 1 + S N 1 ( N 1 C 1 ) where (22) follows from (21) using S 1 C = 1 S 1 N. Thus he aggregae period 1 produciviy level 1 is equal o he aggregae period 1 produciviy level of coninuing firms, 1 C, plus a second erm, S 1 N ( 1 N 1 C ), which represens he conribuion of he new enrans produciviy levels, 1 N, relaive o ha of he coninuing firms, 1 C. 10 Subsiuing (19) and (20) ino (22) leads o he following decomposiion of he aggregae period 1 produciviy level P 1 ino is individual firm conribuions: (23) 1 = i C s Ci 1 Ci 1 + S N 1 i N s Ni 1 ( Ni 1 C 1 ). This complees our discussion of how he levels of produciviy in periods 0 and 1 can be decomposed ino individual conribuion effecs for each firm. In he following secion, we sudy he much more difficul problem of decomposing he aggregae produciviy change, 1 / 0, ino individual firm growh effecs, aking ino accoun ha no all firms are presen in boh periods and hence, here is a problem in calculaing growh effecs for hose firms presen in only one period. 10 Baldwin (1995) in his sudy of he Canadian manufacuring secor showed ha on average, he produciviy levels of new enrans was below ha of coninuing firms bu if he new enran survived, hen hey reach he average produciviy level of coninuing firms in abou a decade. For addiional empirical evidence on he relaive produciviy levels of enering and exiing firms, see Barelsman and Doms (2000; 581), Aw, Chen and Robers (2001) (who also find ha he produciviy level of new enrans is below ha of incumbens) and secion 5 of Barelsman, Haliwanger and Scarpea (2004).

9 9 3. The Measuremen of Produciviy Change Beween he Two Periods I is radiional o define he produciviy change of a producion uni going from period 0 o period 1 as a raio of he produciviy levels in he wo periods raher han as a difference beween he wo levels. This is because he raio measure will be independen of he unis of measuremen while he difference measure will depend on he unis of measuremen (unless some normalizaion is performed). However, in he presen conex, as we are aemping o calculae he conribuion of new and disappearing producion unis o overall produciviy change, i is more convenien o work wih he difference concep, a leas iniially. Using formula (13) for he period 0 produciviy level 0 and formula (23) for he period 1 produciviy level 1, we obain he following decomposiion of he produciviy difference: (24) 1 0 = i C s Ci 1 Ci 1 i C s Ci 0 Ci 0 + S N 1 i N s Ni 1 ( Ni 1 C 1 ) S X 0 i X s Xi 0 ( Xi 0 C 0 ) (25) = C 1 C 0 + S N 1 ( N 1 C 1 ) S X 0 ( X 0 C 0 ) where (25) follows from (24) using (12) and (22). Thus he overall indusry produciviy change, 1 0, is equal o he produciviy change of he coninuing firms, 1 C 0 C, plus a erm ha reflecs he conribuion o overall produciviy change of new enrans, S 1 N ( 1 N 1 C ), 11 plus a erm ha reflecs he conribuion o overall produciviy change of exiing firms, S 0 X ( 0 X 0 C ). 12 Noe ha he reference produciviy levels ha he produciviy levels of he enering and exiing firms are compared wih, 1 C and 0 C respecively, are differen in general, so even if he average produciviy levels of enering and exiing firms are he same (so ha 1 N equals 0 X ), he conribuions o 11 This erm is posiive if and only if he average level of produciviy of he new enrans in period 1, N 1, is greaer han he average produciviy level of coninuing firms in period 1, C This erm is posiive if and only if he average level of produciviy of he firms who exi in period 0, X 0, is less han he average produciviy level of coninuing firms in period 0, C 0.

10 10 overall indusry produciviy growh of enering and exiing firms can sill be nonzero, provided ha N 1 C 1 and X 0 C The firs wo erms on he righ-hand side of (24) give he aggregae effecs of he changes in produciviy levels of he coninuing firms. I is useful o furher decompose his aggregae change in he produciviy levels of coninuing firms ino wo ses of componens; he firs se of erms measures he produciviy change of each coninuing producion uni, 1 Ci 0 Ci, and he second se of erms reflecs he shifs in he share of resources used by each coninuing producion uni, s 1 Ci s 0 Ci. As Balk (2003; 26) noed, here are wo naural decomposiions for he difference in he produciviy levels of he coninuing firms, (27) and (29) below, ha are he difference counerpars o he decomposiion of a value raio ino he produc of a Laspeyres (or Paasche) price index imes a Paasche (or Laspeyres) quaniy index: (26) C 1 C 0 = i C s Ci 1 Ci 1 i C s Ci 0 Ci 0 (27) = i C s Ci 0 ( Ci 1 Ci 0 ) + i C Ci 1 (s Ci 1 s Ci 0 ) ; (28) C 1 C 0 = i C s Ci 1 Ci 1 i C s Ci 0 Ci 0 (29) = i C s Ci 1 ( Ci 1 Ci 0 ) + i C Ci 0 (s Ci 1 s Ci 0 ). We now noe a severe disadvanage associaed wih he use of eiher (27) 14 or (29): hese decomposiions are no invarian wih respec o he reamen of ime. Thus if we 13 Haliwanger (1997) (2000; 10) argues ha if he produciviy levels of enering and exiing firms or esablishmens are exacly he same, hen he sum of he conribuion erms of enering and exiing firms should be zero. However, our perspecive is differen: we wan o measure he differenial effecs on produciviy growh of enering and exiing firms and so wha couns in our framework are he produciviy levels of enering firms relaive o coninuing firms in period 1 and he produciviy levels of exiing firms relaive o coninuing firms in period 0. Thus if coninuing firms show produciviy growh over he wo periods, hen if he enering and exiing firms have he same produciviy levels, he effecs of enry and exi will be o decrease produciviy growh compared o he coninuing firms. Balk (2003; 28) follows he example of Haliwanger (1997) in choosing a common reference level of produciviy o compare he produciviy levels of enering and exiing firms bu Balk chooses he arihmeic average of he indusry produciviy levels in periods 0 and 1 (which is a leas a symmeric choice) whereas Haliwanger chooses he indusry produciviy level of period 0 (which is no a symmeric choice). In any case, our approach seems o be differen from oher approaches suggesed in he lieraure.

11 11 reverse he roles of periods 0 and 1, we would like he decomposiion of he aggregae produciviy difference for coninuing firms, 0 C 1 0 C = i C s Ci 0 1 Ci i C s Ci 1 Ci, ino erms involving he individual produciviy differences 0 Ci 1 Ci and he individual share differences s 0 Ci s 1 Ci ha are he negaives of he original difference erms. 15 I can be seen ha he decomposiions defined by (26) and (28) do no have his desirable symmery or invariance propery. A soluion o his lack of symmery is o simply ake an arihmeic average of (26) and (28), leading o he following Benne (1920) ype decomposiion of he produciviy change of he coninuing firms: (30) C 1 C 0 = i C (1/2)(s Ci 0 + s Ci 1 )( Ci 1 Ci 0 ) + i C (1/2)( Ci 0 + Ci 1 )(s Ci 1 s Ci 0 ). The use of his decomposiion for coninuing firms daes back o Griliches and Regev (1995; 185). 16 Balk (2003; 29) also endorsed he use of his symmeric decomposiion. 17 We endorse he use of his decomposiion since i is symmeric and can also be given a srong axiomaic jusificaion. 18 Subsiuion of (30) ino (24) gives our final bes decomposiion of he aggregae produciviy difference 1 0 ino micro firm effecs: 14 The decomposiion defined by (26) is he one used by Baily, Hulen and Campbell (1992; 193) for coninuing firms excep ha hey used logs of he TFP levels Ci insead of he levels hemselves. 15 In oher words, we wan he difference decomposiion o saisfy a differences counerpar o he usual ime reversal es ha occurs in index number heory. 16 Griliches and Regev (1995; 185) also have a symmeric reamen of he indusry difference in TFP levels bu firms ha exi and ener during he wo periods being compared are reaed as one firm and hey make a direc comparison of he change in produciviy of all enering and exiing firms on his basis. I can be seen ha here are problems in inerpreaion if here happen o be no enering (or exiing) firms in he sample or more generally, if here are big differences in he shares of enering and exiing firms. Aw, Chen and Robers (2001; 73) also use his symmeric mehodology, excep hey work wih logs of TFP. 17 In view of is symmery i should be he preferred one. Ber M. Balk (2003; 29). 18 Diewer (2005) showed ha he Benne decomposiion of a difference of he form i p i 1 q i 1 i p i 0 q i 0 ino a sum of erms reflecing price change and a sum of erms reflecing quaniy change can be given an axiomaic jusificaion ha is analogous o he axiomaic jusificaion for he use of he Fisher (1922) ideal index in index number heory. The adapaion of his axiomaic heory o provide a decomposiion of i p i 1 s i 1 i p i 0 s i 0 is sraighforward.

12 12 (31) 1 0 = i C (1/2)(s Ci 0 + s Ci 1 )( Ci 1 Ci 0 ) + i C (1/2)( Ci 0 + Ci 1 )(s Ci 1 s Ci 0 ) + S N 1 i N s Ni 1 ( Ni 1 C 1 ) S X 0 i X s Xi 0 ( Xi 0 C 0 ). The firs se of erms on he righ hand side of (31), i C (1/2)(s 0 Ci + s 1 Ci )( 1 Ci 0 Ci ), gives he conribuion of he produciviy growh of each coninuing firm o he aggregae produciviy difference beween periods 0 and 1, 1 0 ; he second se of erms, i C (1/2)( 0 Ci + 1 Ci )(s 1 Ci s 0 Ci ), gives he conribuion of he effecs of he reallocaion of resources beween coninuing firms going from period 0 o 1; he hird se of erms, S 1 1 N i N s Ni ( 1 Ni 1 C ), gives he conribuion of each new enering firm o 0 0 produciviy growh and he final se of erms, S X i X s Xi ( 0 Xi 0 C ), gives he conribuion of each exiing firm o produciviy growh. Noe ha he decomposiion (31) is symmeric: if we reverse he role of periods 0 and 1, hen he new aggregae produciviy difference will equal he negaive of he original produciviy difference and each individual firm conribuion erm of he new righ hand side will equal he negaive of he original firm conribuion effec. None of he conribuion decomposiions suggesed in he lieraure have his ime reversal propery, wih he excepion of he decomposiion (51) due o Balk (2003; 28) bu Balk s decomposiion compares he produciviy levels of enering and exiing firms o he arihmeic average of he indusry produciviy levels in periods 0 and 1 insead of o he average produciviy level of coninuing firms in period 1 (in he case of enering firms) and o he average produciviy level of coninuing firms in period 0 (in he case of exiing firms). We now make a final adjusmen o (31) in order o make i invarian o changes in he unis of measuremen of oupu and inpu: we divide boh sides of (31) by he base period produciviy level Wih his adjusmen, (31) becomes: 19 Insead of dividing by 0, we could divide by he logarihmic mean of 0 and 1. The lef hand side of he resuling counerpar o (32) reduces o ln( 0 / 1 ), which is compleely symmeric in he daa whereas he lef hand side of (32) is no. We owe his suggesion o Ber Balk.

13 13 (32) [ 1 / 0 ] 1 = [ i C (1/2)(s Ci 0 + s Ci 1 )( Ci 1 Ci 0 ) + i C (1/2)( Ci 0 + Ci 1 )(s Ci 1 s Ci 0 ) + S N 1 i N s Ni 1 ( Ni 1 C 1 ) S X 0 i X s Xi 0 ( Xi 0 C 0 )]/Π 0. In he following secions, we will illusrae he aggregae produciviy decomposiion (32) using an arificial daa se. Noe ha (32) is only valid for an indusry ha produces a single oupu and uses a single inpu. However, in pracice, firms in an indusry produce many oupus and use many inpus. Hence, before he decomposiion (32) can be implemened, i is necessary o aggregae he many oupus produced and inpus used by each firm ino aggregae firm oupu and inpu. This aggregaion problem is no sraighforward because some firms are enering and exiing he indusry. In he following secion, we address his unconvenional aggregaion problem How can he Inpus and Oupus of Enering and Exiing Firms be Aggregaed? The aggregae produciviy decomposiion defined by (32) above assumes ha each firm produces only one oupu and uses only one inpu. However, firms in he same indusry ypically produce many oupus and uilize many inpus. Thus in order o apply (32), we have o somehow aggregae all of he oupus produced by each firm ino an aggregae oupu ha is comparable across firms and across ime periods (and aggregae all of he inpus uilized by each firm ino an aggregae inpu ha is comparable across firms and across ime periods). I can be seen ha hese wo aggregaion problems are in fac mulilaeral aggregaion problems; 21 i.e., he oupu vecor of each firm in each period mus be compared wih he corresponding oupu vecors of all oher firms in he indusry over he wo ime periods involved in he aggregae produciviy comparison. 22 In he following secions of his paper, we will illusrae how hese firm oupu and inpu 20 As noed earlier, Aw, Chen and Robers (2001) and Aw, Chung and Robers (2003) addressed his aggregaion problem using he mulilaeral mehod explained in Good, Nadiri and Sickles (1997). 21 Bilaeral index number heory compares he price and quaniy vecors peraining o wo siuaions whereas mulilaeral index number heory aemps o consruc price and quaniy aggregaes when here are more han wo siuaions o be compared. See Balk (1996) (2001) and Diewer (1999) for recen surveys of mulilaeral mehods. 22 Fox (2002) seems o have been he firs o noice ha aggregaing firm oupus and inpus ino aggregae oupus and inpus should be reaed as a mulilaeral aggregaion problem in order o avoid paradoxical resuls.

14 14 aggregaes can be formed using several mehods ha have been suggesed in he mulilaeral aggregaion lieraure. In order o make he comparison of alernaive mulilaeral mehods of aggregaion more concree, we will uilize an arificial daa se. In he following secion, we able our daa se and calculae he aggregae produciviy of he indusry using normal index number mehods. 5. Indusry Produciviy Aggregaes Using an Arificial Daa Se We consider an indusry over wo periods, 0 and 1, ha consiss of five firms. Each firm f produces varying amouns of he same wo oupus and uses varying amouns of he same wo inpus. The oupu vecor of firm f in period is defined as y f [y f1,y f2 ] and he corresponding inpu vecor is defined as x f [x f1,x f2 ] for = 0,1 and f = 1,2,,5. Firms 1,2 and 3 are coninuing firms, firm 4 is presen in period 0 bu no in period 1 (and hence is he exiing firm) and firm 5 is no presen in period 0 bu is presen in period 1 (and hence is he enering firm). Firm 1 is a medium sized firm, firm 2 is a iny firm and firm 3 is a very large firm. The oupu price vecor of firm f in period is defined as p f [p f1,p f2 ] and he corresponding inpu price vecor is defined as w f [w f1,w f2 ] for = 0,1 and f = 1,2,,5. The indusry price and quaniy daa are lised in Table 1. Table 1: Firm Price and Quaniy Daa for Periods 0 and 1 Firm 1 Firm 2 Firm 3 Firm 4 Firm 5 Oupu prices p 11 p 12 p 21 p 22 p 31 = = Oupu quaniies y 11 y 12 y 21 y 22 y 31 = = Inpu prices w 11 w 12 w 21 w 22 w 31 = = p 32 y 32 w 32 p 41 y 41 w 41 p 42 y 42 w 42 p 51 y 51 w 51 p 52 y 52 w 52

15 15 Inpu quaniies x 11 x 12 x 21 x 22 x 31 = = x 32 Thus he period 0 oupu price vecor for firm 1 is p 1 0 = [1,1], he period 1 oupu price vecor for firm 1 is p 1 1 = [15,7] and so on. Noe ha here has been a grea deal of general price level change going from period 0 o x 41 x 42 x 51 x 52 In he following secions, we will look a various mehods for forming oupu and inpu aggregaes for each firm and each period bu before we do his, i is useful o compue oal indusry supplies of he wo oupus, y [y 1,y 2 ] for each period and oal indusry demands for each of he wo inpus x [x 1,x 2 ] for each period as well as he corresponding uni value prices, p [p 1,p 2 ] and w [w 1,w 2 ]. 24 This informaion is lised in (32) below. (33) p 0 = [0.946, 0.869]; p 1 = [14.468, 7.159]; w 0 = [0.968, 1.057]; w 1 = [9.308, ]; y 0 = [70, 68]; y 1 = [94, 63]; x 0 = [69, 58]; x 1 = [52, 44]. Noe ha indusry oupu 1 has increased from 70 o 94 bu indusry oupu 2 decreased slighly from 68 o 63. However, boh indusry inpu demands dropped markedly; inpu 1 decreased from 69 o 52 and inpu 2 decreased from 58 o 44. Thus overall, indusry produciviy improved markedly going from period 0 o 1. In order o benchmark he reasonableness of he various produciviy decomposiions given by (32) above for differen mulilaeral mehods o be discussed in he following four secions, i is useful o use he indusry daa in (33) in order o consruc normal index number esimaes of indusry Toal Facor Produciviy Growh (TFPG). Following 23 In some applicaions of he lieraure on he conribuion of enry and exi o aggregae produciviy growh, he comparison periods are a decade apar and so in high inflaion counries, he period 0 and 1 price levels can differ considerably. 24 The uni value price of oupu n in period is defined as p n f=1 5 p fn y fn / f=1 5 y fn for n = 1,2 and = 0,1. The uni value price of inpu n in period is defined as w n f=1 5 w fn x fn / f=1 5 x fn for n = 1,2 and = 0,1.

16 16 Jorgenson and Griliches (1967) (1972), 25 TFPG can be defined as a quaniy index of oupu growh, Q(p 0,p 1,q 0,q 1 ), divided by a quaniy index of inpu growh, Q*(w 0,w 1,x 0,x 1 ): (34) TFPG Q(p 0,p 1,q 0,q 1 )/Q*(w 0,w 1,x 0,x 1 ). In order o implemen (34), one needs o choose an index number formula for Q and Q*. From an axiomaic perspecive, he bes choices suggesed in he lieraure seem o be he Fisher (1922) ideal formula 26 or he Törnqvis (1936) Theil (1967) formula. 27 Wih hese wo choices of index number formula, he resuling TFP growh raes 28 for he daa lised in (33) are as follows: (35) TFPG F = ; TFPG T = If we subrac 1 from he above TFPG raes, we obain indusry aggregae counerpars o he lef hand side of (32), [ 1 / 0 ] 1. Thus using he Fisher formula, indusry produciviy improved 55.53% and using he Törnqvis Theil formula, indusry produciviy improved 55.73%. These produciviy growh raes should be kep in mind as we look a alernaive mulilaeral mehods for consrucing oupu and inpu aggregaes for each firm in each period so ha we can implemen he decomposiion formula (32). In oher words, a mulilaeral mehod ha leads o an aggregae produciviy growh rae [ 1 / 0 ] 1 ha is very differen from he range.5553 o.5573 is probably no very reliable. We now urn o our firs mulilaeral mehod for consrucing oupu and inpu aggregaes for each firm in each period. 25 For recen surveys on how o measure TFPG, see Balk (2003) and Diewer and Nakamura (2003). 26 See Diewer (1992). The Fisher oupu quaniy index is defined as Q F (p 0,p 1,q 0,q 1 ) [p 0 q 1 p 1 q 1 /p 0 q 0 p 1 q 0 ] 1/2 where p q denoes he inner produc of he vecors p and q. 27 See Diewer (2004). Boh of hese formulae can be given economic jusificaions as well; see Diewer (1976). 28 Acually hese raes are 1 plus he oal facor produciviy growh raes.

17 17 6. The Sar Sysem for Making Mulilaeral Comparisons Recall ha in he previous secion, we defined he firm f and period oupu and inpu vecors as y f [y f1,y f2 ] and x f [x f1,x f2 ] for = 0,1 and f = 1,2,,5. However, for = 0 and f = 5 and also for = 1 and f = 4, here are no daa, since hese wo firms are enering and exiing respecively. Thus here are acually a oal of 8 oupu and inpu quaniy vecors insead of 10. I will prove o be more convenien o relabel our daa so ha here are only 8 disinc oupu and inpu quaniy vecors. Thus define he oupu quaniy vecors y 1, y 2, y 3 and y 4 as he previously defined vecors y 0 1, y 0 2, y 0 3 and y 0 4 respecively (hese are he nonzero period 0 oupu quaniy vecors) and define he vecors y 5, y 6, y 7 and y 8 as he previously defined vecors y 1 1, y 1 2, y 1 3 and y 1 5 respecively (hese are he nonzero period 1 nonzero oupu quaniy vecors). Similarly, define he oupu price vecors p 1, p 2, p 3 and p 4 as he previously defined vecors p 0 1, p 0 2, p 0 3 and p 0 4 respecively and define he vecors p 5, p 6, p 7 and p 8 as he previously defined vecors p 1 1, p 1 2, p 1 3 and p 1 5 respecively. Underake he same reordering of he daa for inpus. Now we are in a posiion o apply mulilaeral mehods in order o consruc oupu and inpu aggregaes for each firm in each period. In effec, we rea each of he 8 oupu (or inpu) price and quaniy vecors as if hey corresponded o he daa ha perained o a counry and we choose a mulilaeral mehod in order o consruc an oupu (or inpu) aggregae for each of our 8 counries. 29 The firs mulilaeral mehod ha we will consider is he Sar Sysem. 30 In order o implemen his mehod, we choose our favorie bilaeral index number formula, say he Fisher formula Q F, and we choose one observaion as he base (or sar), say observaion k, and hen we compue he Fisher quaniy aggregae of each observaion relaive o he base k, Q F (p k,p 1,y k,y 1 ), Q F (p k,p 2,y k,y 2 ),, Q F (p k,p 8,y k,y 8 ). The resuling sequence of 8 numbers can serve as comparable oupu aggregaes for our 8 observaions. 29 Noe ha we need o make wo mulilaeral comparisons: one for oupus and one for inpus. 30 This erminology is due o Kravis (1984; 10).

18 18 Of course, he problem wih he Sar Sysem aggregaes is ha i is necessary o asymmerically choose one observaion as he sar and usually, i is no clear which observaion should be chosen o be he sar. 31 Thus in Tables 2 and 3 below, we lis each of he 8 oupu and inpu aggregaes respecively, choosing each observaion as he base in urn. In order o make hese oupu and inpu aggregaes comparable, we divide each se of pariies by he pariy for he firs observaion. Thus he oupu and inpu pariies lised in Tables 2 and 3 are he following normalized pariies for oupus and inpus respecively, for k = 1,,8: 32 (36) 1, Q F (p k,p 2,y k,y 2 )/Q F (p k,p 1,y k,y 1 ),, Q F (p k,p 8,y k,y 8 )/Q F (p k,p 1,y k,y 1 ); (37) 1, Q F *(w k,w 2,x k,x 2 )/Q F *(w k,w 1,x k,x 1 ),, Q F *(w k,w 8,x k,x 8 )/Q F *(w k,w 1,x k,x 1 ). Table 2: Fisher Sar Oupu Aggregaes Oupus y 1 y 2 y 3 y 4 y 5 y 6 y 7 y 8 Base= Base= Base= Base= Base= Base= Base= Base= Table 3: Fisher Sar Inpu Aggregaes Inpus x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 Base= Base= Base= Base= Base= Base= Base= Base= In our paricular example, a case could be made for choosing eiher observaion 3 or 7; i.e., he observaions ha correspond o he very large firm. However, here are sill wo choices and again, i is no clear which of hese wo should be chosen. 32 Recall ha our final decomposiion of he indusry produciviy change defined by (32) does no depend on our raher arbirary unis of measuremen for aggregae firm oupus and inpus.

19 19 Noe ha he inpu aggregaes for observaions 1 and 2 using any of he observaions as he base are always he same. This is due o he use of he Fisher formula and he fac he inpu vecors for observaions 1 and 2 are proporional 33 ; i urns ou ha if he quaniy vecors for he wo observaions being compared are proporional, hen he Fisher quaniy index ha compares hese wo observaions will reflec his facor of proporionaliy. 34 However, in general, i can be seen ha he choice of he base counry or observaion does affec he oupu and inpu pariies. Now go along each row of Table 2 and divide each oupu aggregae by he inpu aggregae ha corresponds o ha observaion ha is lised in he corresponding row of Table 3. This deermines he produciviy level of each observaion using each of he 8 observaions as he base in he index number comparisons in urn. These sar produciviy levels are lised in Table 4. Table 4: Fisher Sar Produciviy Levels Base= Base= Base= Base= Base= Base= Base= Base= There can be a considerable amoun of variaion in he produciviy levels for each observaion, depending on which observaion is chosen as he base in he sar sysem comparison. Thus if we choose he base o equal 1 (firm 1 in period 0), he produciviy level of firm 2 in period 0 is whereas if we choose he base o equal 7 (firm 3 in period 1), he produciviy level of firm 2 in period 0 is 0.980, which is a 4% variaion in 33 The inpu vecor for firm 1 in period 0 is x 1 = [10,10] and for firm 2 in period 0 is x 2 = [1,1]. 34 Similarly, if he wo price vecors are proporional, hen he Fisher price index beween he wo observaions will reflec his facor of proporionaliy. The Fisher formula seems o be he only superlaive formula ha is consisen wih boh Hicks and Leonief s aggregaion heorems; see Allen and Diewer (1981).

20 20 produciviy levels due o he choice of a differen base for our bilaeral index number comparisons. 35 Aggregae oupu prices ha correspond o he 8 oupu aggregaes ha are lised in Table 2 for each choice of base observaion can be obained by dividing he value of oupu produced by each firm in each period by he corresponding oupu lised for ha observaion in Table 2. Similarly, aggregae inpu prices ha correspond o he 8 inpu aggregaes ha are lised in Table 3 for each choice of base observaion can be obained by dividing he value of inpus used by each firm in each period by he corresponding inpu lised for ha observaion in Table 3. Once hese aggregae oupu and inpu prices have been consruced, hen we are in a posiion o apply he decomposiion analysis ha was discussed in secions 2 and 3 above. We define he various erms ha occur on he righ and lef hand sides of he aggregae produciviy growh decomposiion (31) as follows: (38) Γ [ 1 / 0 ] 1 (aggregae indusry produciviy growh); (39) Γ CD i C (1/2)(s 0 Ci + s 1 Ci )( 1 Ci 0 Ci )/Π 0 (direc produciviy growh conribuion of coninuing firms); (40) Γ CR i C (1/2)( 0 Ci + 1 Ci )(s 1 Ci s 0 Ci )/Π 0 (reallocaion conribuion of coninuing firms); 1 1 (41) Γ N S N i N s Ni ( 1 Ni 1 C )/Π 0 (conribuion of enering firms o TFPG); 0 0 (42) Γ X S X i X s Xi ( 0 Xi 0 C )]/Π 0 (conribuion of exiing firms o TFPG). In our example, here are hree coninuing firms in each of he summaions in (39) and (40) bu only one erm in each of he summaions in (41) and (42) since we have only one exiing and one enering firm. 35 Ideally, we would like all he enries in each column of Table 4 o be idenical so ha he produciviy levels of each firm observaion would no depend on he choice of index number base.

21 21 The erms defined by (38)-(42) are lised in Table 5 below for each choice of base; i.e., we use he daa lised in Tables 2-4 above (along wih he corresponding prices) in order o consruc an aggregae indusry produciviy growh decomposiion for each of our 8 bases. Table 5: Aggregae Produciviy Growh Decomposiions for Each Choice of Base Γ Γ CD Γ CR Γ N Γ X Base= Base= Base= Base= Base= Base= Base= Base= I can be seen ha he choice of base maers. Aggregae produciviy growh using observaion 5 (daa of firm 1 in period 1) as he index number formula base leads o indusry produciviy growh of 51.74% whereas if observaion 4 (daa of he disappearing firm 4 in period 0) is used as he base, hen indusry produciviy growh is much larger a 60.25%. 36 Looking a he las 4 columns in Table 5, i can be seen ha he direc produciviy growh of coninuing firms accouns for mos of he indusry produciviy growh (somewhere beween 37.39% and 47.04%), he conribuion of he exiing firm is beween 10% and 11%, he conribuion of he enering firm is beween 3.0% and 4.4% and he reallocaion of resources beween coninuing firms sums o a negligible conribuion o overall produciviy growh. I is of some ineres o look a he direc produciviy growh conribuion and he reallocaion conribuion of each coninuing firm. Thus define he hree erms on he righ hand side of (39) as Γ CD1, Γ CD2 and Γ CD3, he direc produciviy growh conribuions of coninuing firms 1, 2 and 3 respecively and define he hree erms on he righ hand side of (40) as Γ CR1, Γ CR2 and Γ CR3, he reallocaion conribuions of 36 The choice of observaions 2, 3, 6 and 7 as he index number base gives rise o indusry TFP growh raes ha are closes o our arge raes of around 55.53% and 55.73%; recall (35) above. Noe ha he average of he indusry produciviy growh raes for he large firm observaions (3 and 7) is 55.74%.

22 22 coninuing firms 1, 2 and 3 respecively. These erms are lised in Table 6 for each of our 8 choices of index number base. Table 6: Direc and Reallocaion Conribuions o Aggregae Produciviy Growh for Each Coninuing Firm and for Each Choice of Base Γ CD1 Γ CD2 Γ CD3 Γ CR1 Γ CR2 Γ CR3 Base= Base= Base= Base= Base= Base= Base= Base= Viewing Table 6, i can be seen ha he larges conribuion o indusry TFP growh is he direc TFP growh of firm 3 (he large firm); i conribues an amoun somewhere beween 24.29% (he index base 5 esimae) and 32.61% (he index base 4 esimae). The nex larges conribuion comes from he medium sized firm 1; i conribues an amoun beween 12.10% (he index base 1 esimae) and 13.72% (he index base 7 esimae). The oher conribuion erms are all less han 5%. Obviously, some form of averaging of he differen sar decomposiions is called for. Our nex mulilaeral mehod simply akes he geomeric averages of he oupu and inpu aggregaes lised in Tables 2 and 3 and hen implemens he decomposiion (31) using hese new oupu and inpu aggregaes. 7. The GEKS Mehod for Making Mulilaeral Comparisons The GEKS mehod for making mulilaeral comparisons daes back o Gini (1931), Eleö and Köves (1964) and Szulc (1964). As was indicaed in he previous secion, his mehod simply akes each of he sar oupu and inpu pariies and akes he geomeric mean of hem. 37 These GEKS relaive oupu and inpu aggregaes are lised in Table The GEKS aggregaes can be defined in a number of equivalen ways bu his is one way; see for example, Diewer (1999; 31-37).

23 23 Once he oupu and inpu aggregaes have been consruced, hen he GEKS produciviy levels can be consruced by dividing he oupu aggregae by he corresponding inpu aggregae. The resuling 8 produciviy levels are also lised in Table 7. Table 7: GEKS Oupu and Inpu Aggregaes and Produciviy Levels Oupus y 1 y 2 y 3 y 4 y 5 y 6 y 7 y Inpus x 1 x 2 x 3 x 4 x 5 x 6 x 7 x Prod Levels y 1 /x 1 y 2 /x 2 y 3 /x 3 y 4 /x 4 y 5 /x 5 y 6 /x 6 y 7 /x 7 y 8 /x Aggregae oupu prices ha correspond o he 8 oupu aggregaes ha are lised in Table 7 can be obained by dividing he value of oupu produced by each firm in each period by he corresponding oupu lised for ha observaion in Table 7. Similarly, aggregae inpu prices ha correspond o he 8 inpu aggregaes ha are lised in Table 7 can be obained by dividing he value of inpus used by each firm in each period by he corresponding inpu lised for ha observaion in Table 7. Once hese aggregae oupu and inpu prices have been consruced, hen we can repea he decomposiion analysis ha was implemened in he previous secion. The produciviy growh decomposiion erms defined by (38)-(42) are lised in Table 8 below. We also lis he direc and reallocaion conribuion erms defined by he individual erms in (39) and (40) for each coninuing firm in Table 8. Table 8: The GEKS Aggregae Produciviy Growh Decomposiion Γ Γ CD Γ CR Γ N Γ X Γ CD1 Γ CD2 Γ CD3 Γ CR1 Γ CR2 Γ CR From Table 8, he GEKS aggregae produciviy growh Γ is 55.21%, which is reasonably close o our arge raes of around 55.53% o 55.73%; recall (35) above. Thus we conclude ha he GEKS mehod for consrucing relaive oupu and inpu levels for each firm in each period is saisfacory, a leas for our paricular numerical example.

24 24 One problem wih he (unweighed) GEKS mehod is ha each firm observaion is given an equal weighing when he oupu and inpu aggregaes are consruced. Since small firms may have differen daa han large firms and hence heir sar pariies could be quie differen from hose of large firms, i may no be wise o give hese small firms equal weighing in he consrucion of he oupu and inpu aggregaes. Thus in he following secion, we look a a mulilaeral mehod for consrucing oupu and inpu aggregaes ha gives large firms more weigh han small firms. 8. The Own Share Mehod for Making Mulilaeral Comparisons Recall our discussion in secion 6 when we described how he sar oupu aggregaes could be consruced using observaion k as he base. We noed ha he sequence of 8 numbers, Q F (p k,p 1,y k,y 1 ), Q F (p k,p 2,y k,y 2 ),, Q F (p k,p 8,y k,y 8 ), could serve as comparable oupu aggregaes for our 8 observaions. Hence, using observaion k as he base, he share of oal oupu of observaion k is: (43) s k * Q F (p k,p k,y k,y k )/[Q F (p k,p 1,y k,y 1 ) + Q F (p k,p 2,y k,y 2 ) + + Q F (p k,p 8,y k,y 8 )] = 1/[Q F (p k,p 1,y k,y 1 ) + Q F (p k,p 2,y k,y 2 ) + + Q F (p k,p 8,y k,y 8 )] ; k = 1,,8, where he las equaion in (43) follows from he fac ha he Fisher ideal quaniy index saisfies an ideniy es and hence Q F (p k,p k,y k,y k ) equals 1. Thus, using he meric of observaion k o make he index number comparisons, he share of observaion k in world oupu, s k *, is defined by (43) for k = 1,2,, 8. Thus each observaion s own share of world oupu is defined by (43). Pu anoher way, if we look a he enries in Table 2 above, he numbers lised in he Base=1 row deermine he share of observaion 1 in oal oupu over he wo periods, s 1 *; he numbers lised in he Base=2 row deermine he share of observaion 2 in oal oupu over he wo periods, s 2 *; ; and he numbers lised in he Base=8 row deermine he share of observaion 8 in oal oupu over he wo periods, s 8 *. Thus each row in Table 2 deermines only one share of world oupu and so he rows ha correspond o smaller shares of world oupu ge a smaller influence in

25 25 he overall mulilaeral comparison; i.e., he own share sysem does weigh he individual sar pariies according o heir economic imporance as opposed o he more democraic GEKS mehod where each sar pariy has he same imporance. Unforunaely, he own shares s k * defined by (43) do no sum up o uniy and so we renormalize hese shares o sum up o uniy as follows: 38 (44) y k s k */[ j=1 8 s j *] ; k = 1,,8. The oupu aggregaes y k defined by (44) are he own share aggregaes. 39 procedure can be used in order o define own share inpu aggregaes. The same These own share relaive oupu and inpu oupu and inpu aggregaes are lised in Table 9. Once he oupu and inpu aggregaes have been consruced, hen he own share produciviy levels can be consruced by dividing he oupu aggregae by he corresponding inpu aggregae. The resuling 8 produciviy levels are also lised in Table Table 9: Own Share Oupu and Inpu Aggregaes and Produciviy Levels Oupus y 1 y 2 y 3 y 4 y 5 y 6 y 7 y Inpus x 1 x 2 x 3 x 4 x 5 x 6 x 7 x Prod Levels y 1 /x 1 y 2 /x 2 y 3 /x 3 y 4 /x 4 y 5 /x 5 y 6 /x 6 y 7 /x 7 y 8 /x Aggregae oupu prices ha correspond o he 8 oupu aggregaes ha are lised in Table 9 can be obained by dividing he value of oupu produced by each firm in each period by he corresponding oupu lised for ha observaion in Table 9. Similarly, aggregae 38 In our empirical example, he s k * summed up o so ha he differences beween he y k and he s k * were negligible. The corresponding inpu shares summed up o The own share sysem was proposed by Diewer (1988; 69). For he axiomaic properies of his mehod, see Diewer (1999; 37-39). 40 Noe ha he unis of measuremen for he oupu and inpu aggregaes are quie differen in Tables 7 and 9. This illusraes he imporance of providing a produciviy growh decomposiion ha is independen of he unis of measuremen.

26 26 inpu prices ha correspond o he 8 inpu aggregaes ha are lised in Table 9 can be obained by dividing he value of inpus used by each firm in each period by he corresponding inpu lised for ha observaion in Table 9. Once hese aggregae oupu and inpu prices have been consruced, hen we can repea he decomposiion analysis ha was implemened in he previous secions. The produciviy growh decomposiion erms defined by (38)-(42) are lised in Table 10 below. We also lis he direc and reallocaion conribuion erms defined by he individual erms in (39) and (40) for each coninuing firm in Table 10. Table 10: The Own Share Aggregae Produciviy Growh Decomposiion Γ Γ CD Γ CR Γ N Γ X Γ CD1 Γ CD2 Γ CD3 Γ CR1 Γ CR2 Γ CR From Table 10, he own share aggregae produciviy growh Γ is 55.45%, which is very close o our arge rae of around 55.53% o 55.73%; recall (35) above. 41 Thus we conclude ha he own share mehod for consrucing relaive oupu and inpu levels for each firm in each period is very saisfacory, a leas for our paricular numerical example. 9. Hill s Mehod for Making Mulilaeral Comparisons Anoher mehod for finding oupu and inpu aggregaes can be based on he following idea: observaions which are mos similar in heir price srucures (i.e., heir oupu prices are closes o being proporional across iems) should be linked using a bilaeral index number formula firs. Then he observaion ouside of he firs wo observaions ha has he mos similar relaive prices o he firs wo observaions should be added o he chain 41 Noe ha he own share decomposiion is very close o he GEKS decomposiion lised in Table 8 above. Diewer (1988; 69) (1999; 38) showed ha he own share aggregaes and he GEKS aggregaes will usually approximae each oher fairly closely.