SHADOW PRICE APPROACH TO PRODUCTIVITY MEASUREMENT

SHADOW PRICE APPROACH TO PRODUCTIVITY MEASUREMENT A Modified Malmquis Index TIMO KUOSMANEN Helsinki School of Economics and Business Adminisraion kuosmane@hkkk.fi THIERRY POST Erasmus Universiy Roerdam gpos@few.eur.nl ABSTRACT The Fisher ideal oal facor produciviy (TFP) index requires complee quaniy and price informaion. If inpu-oupu quaniies are allocaively efficien, he Fisher index can be approximaed by he Malmquis TFP index, which does no require price informaion. This paper elaboraes on he condiions under which he Malmquis and Fisher indexes are equivalen. Drawing from hese insighs, we develop a modificaion of he Malmquis approach o beer approximae he superlaive Fisher index wih quaniy daa only. We illusrae he new approach by an empirical applicaion o aggregae daa of 14 OECD counries. Key Words: Index Numbers and Aggregaion, Toal Facor Produciviy (TFP), Fisher ideal index, Malmquis index, Shadow prices JEL classificaion: C43, D24 1. INTRODUCTION Produciviy changes a he macro level of naions and indusries as well as he micro level of firms and plans is of obvious ineres o economiss. Produciviy change is defined as he raio of change in oupu o change in inpu, and i encompasses such facors as 1) echnological progress/regress, 2) operaional efficiency, and 3) uilizaion of economies of scale and specializaion. Measuring he degree of produciviy change, and explaining he measured changes by changes of hese underlying facors are he main concerns of he produciviy analysis. In he elemenary case where only a single oupu is produced by employing a single inpu, measuring he produciviy change is a rivially simple underaking. However, virually all producion unis produce muliple oupus and/or consume 1

muliple inpus. The fundamenal challenge in measuring Toal Facor Produciviy (hereafer TFP) changes comes from he need o aggregae he various inpus and oupus. Many alernaive approaches of aggregaing inpus and oupus are available (see e.g. Diewer, 1992, 2000, for discussion). However, here seems o be no quesion abou he need o accoun for he values of he inpus and oupus; expensive or imporan commodiies should be assigned a greaer weigh han inexpensive or unimporan ones. Convenional index heory ypically uses economic prices (or cos/revenue shares) as weighs. A prime example is he Fisher TFP index. Diewer (1992) considered 20 differen properies ha any produciviy index should have, and showed ha he Fisher ideal index has all suggesed properies, ouperforming any oher candidae index. Unforunaely, he Fisher ideal index canno be compued if he economically relevan prices (or cos/revenue shares) are no known. For many commodiies, moneary prices do no exis (free commodiies, inangible asses, new commodiies inroduced in he arge period), or he prices are biased due o marke failures (monopoly power, exernaliies) or governmenal inerference (ariffs, axaion, subsidizing, regulaion). Moreover, i can ofen be difficul o obain reliable price daa for research purposes even from commodiies ha do go wih he price ag. For example, pricing he services of durable inpus (capial goods) ha are used over several ime periods is a well-recognized problem. In hese circumsances, a frequenly employed alernaive is he Malmquis TFP index (inroduced by Caves, Chrisensen, and Diewer, 1982a,b), which does no require any price daa whasoever. Ineresingly, under some quie general condiions he Malmquis TFP index can approximae he Fisher ideal index, as poined ou by Diewer (1992), Färe and Grosskopf (1992), and Balk (1993). The approximaion essenially relies on recovering price informaion from he shadow prices of he observed quaniy choices, in he spri of he heory of revealed preference by Paul Samuelson (see e.g. Afria, 1972; and Varian, 1984; for insighful reamens of he revealed preference approach in he producion seing). Sill, he Malmquis TFP index only gives an inaccurae approximaion in he imporan case where prices and/or echnologies change over ime, as poined ou by Balk (1993). In his paper, we develop a modificaion of he Malmquis TFP index o eliminae he approximaion error in case of price and/or echnology change. Secion 2 formally inroduces he Malmquis and he Fisher ideal TFP indexes, and explores heir relaionship in more deail. In Secion 3 we use hese insighs o beer approximae he Fisher index by developing a modified Malmquis TFP index approach. An applicaion o he aggregae producion daa of 14 OECD counries illusraes our approach in Secion 5. Secion 6 concludes. 2. THE FISHER IDEAL INDEXES AND THE MALMQUIST TFP INDEXES In he elemenary case where only a single oupu y is produced by employing a single inpu x, measuring he produciviy change is rivially simple using he elemenary produciviy index 2

1 0 y / y (1) P( y, x ) 1 0 x / x where he superscrips 0 and 1 refer o he base and he arge period of he index respecively. The major challenge of produciviy measuremen is he fac ha almos all producion aciviies involve more han jus a single inpu and oupu. A simple and frequenly employed soluion is o look a he parial produciviy indexes such as labor produciviy (i.e. oupu per labor inpu). Unforunaely, such parial indexes offer an incomplee picure of produciviy, since all he ineresing radeoffs beween differen facors are ignored. For example, high labor produciviy saisics can be a signal of genuinely high level of echnical efficiency, bu i migh equally well reflec low capial produciviy. A more ambiious sraegy is o aggregae he various inpus and oupus in one way or anoher o consruc a more comprehensive TFP index. The convenional index heory ypically uses he prices (or cos/revenue) shares as he weighs of quaniy indexes. Of course, he prices ofen change from one period o anoher, so we ineviably face he quesion of which se of prices provide he mos appropriae weighs (consider e.g. difference beween he classic Paashe and Laspayers indexes). Fisher (1922) solved his quesion in an ingenious way by compuing wo quaniy indices, one using he weighs of he base period and anoher using he weighs of he arge period, and aking he geomeric average of he wo. Specifically, denoing he oupu vecor of period s = 0,1 by y R + and he associaed price vecor by p, he Fisher ideal oupu quaniy index is defined as 1 1 (2) F (, ) j o p y ( p y ) 1 2. j= 0 = 0 m Similarly, denoing he inpu vecor of period by x R + and he associaed facor prices by w, he Fisher ideal inpu quaniy index is defined as (3) 1 1 j Fi ( w, x ) ( wx) 1 2 j= 0 = 0. Using he Fisher ideal indexes o aggregae boh inpus and oupus, he Fisher ideal TFP index is obained simply as he raio of he aggregaed oupu o he aggregaed inpu, i.e. (4) TFP p w y x F p y F w x F(,,, ) o(, )/ i(, ) 1 1 j p y = j j= 0 = 0 wx 1 2. Unforunaely, he Fisher ideal index canno be compued if he economically relevan prices (or cos/revenue shares) are no known. As discussed in he Inroducion, obaining reliable price daa is problemaic in mos research siuaions. In hese circumsances, he frequenly employed alernaive is he Malmquis TFP index (inroduced by Caves, Chrisensen, and Diewer, 1982a,b), which does no require any price daa whasoever. Le he producion echnology be characerized by he closed and non-empy producion se 3

(5) ( ) s+ m { + } T y, x R x can produce y a period, = 0,1. We define he Malmquis TFP index as 1 1 1 (6) (, ) (, ) j D y x 2 M y x ( ), j= 0 = 0 where (7) D ( y, x) Inf { θ ( y/ θ, x) cmc( T )}, = 0,1 is he Shephard oupu disance funcion. The disance funcion is convenionally defined relaive o he producion se T, which is ypically assumed o exhibi monooniciy, convexiy and consan reurns-o-scale (CRS). 1 By conras, we define i relaive o he convex monoone conical hull, cmc ( T ), i.e. he smalles monoone convex cone ha conains T. This deparure allows us o alleviae he sandard (bu noneheless raher resricive) assumpions ha he rue echnology exhibis monooniciy, convexiy and CRS. We need he 'benchmark echnology' cmc ( T ) (which exhibis hese properies by definiion) o esablish dualiy beween he disance measure and he economic objecive funcion (see below). However, he rue echnology may be non-monoone, non-convex, and may exhibi variable reurns-o-scale (VRS). Observe ha he oupu disance funcion (6) is he dual of he following equivalen 'primal' formulaion: ρy ρy' (8) D ( y, x) = Sup 1 ( y', x') T, = 0,1. s+ m ( ρω, ) R + ωx ωx' In conras o he dual formulaion (7), he primal formulaion (8) does no require he monoone convex conical hull (alhough i can be used wihou harm). This observaion is he key o undersanding why Theorem 1 does no require he rue echnology o exhibi monooniciy, convexiy and CRS. Heurisically, he oupu disance funcion (8) can be inerpreed as he reurn-ohe-dollar, 2 a he mos favorable prices (as represened by he weighing vecor ( ρ, ω), subjec o a normalizing condiion ha no feasible inpu-oupu vecors yields he reurn-o-he-dollar higher han uniy a hose prices. We will henceforh refer o he ρ, ω of (8) as he shadow prices of (y,x) wih respec o echnology opimal weighs ( ) T. By he supporing hyperplane heorem, here exiss a vecor of shadow prices for any arbirary inpu-oupu vecor, bu clearly, ha vecor need no be unique. The se of shadow price vecors is henceforh denoed by 1 In empirical applicaions, he rue echnology T is ypically unknown and mus be approximaed using some empirical benchmark echnology. In pracice, Malmquis indices are ofen compued using he convex monoone conical hull of he observed daa poins as he benchmark echnology (see e.g. Färe and Grosskopf, 1996, pp. 61-62). 2 The noion of reurn-o-he-dollar was inroduced by Georgescu-Roegen (1951). I equals one plus he profi margin (Jorgenson and Griliches, 1972). 4

(9) s+ m ρy ρy' V ( y, x) ( ρω, ) R + = D ( y, x); 1 ( y', x') T, = 0,1. ωx ωx' In he spiri of he heory of revealed preference, he observed allocaion of inpus and oupus can indirecly reveal he economic prices underlying he producion decision. In line wih he definiion of produciviy, we assume he producion vecors are se o maximize he reurn-o-he-dollar, and define he concep of allocaive efficiency as: Definiion ( Allocaive Efficiency ): Producion vecor ( y, x) is allocaively efficien wih respec o echnology T and prices ( p, w ) iff ( p, w ) V ( y, x). Noe ha allocaive efficiency is obviously a necessary condiion for maximizaion of he reurn-o-he-dollar. However, i is no a sufficien condiion. For example, allocaively efficiency allows for echnical inefficiency in he sense of Farrell (1957) (i.e. producion in he inerior of he producion possibiliy se). We are now equipped o prove he following equivalence beween he Malmquis and he Fisher ideal indexes: THEOREM 1 ( The Equivalence Theorem ): The following condiions are equivalen: 0 0 1 1 1) Producion vecors ( y, x ) and ( y, x ) are allocaively efficien wih respec o prices 0 0 ( p, w ) and echnology 0 T, and wih respec o prices 1 1 ( p, w ) and echnology 2) The Malmquis TFP index M ( y, x ) and he Fisher ideal index TFP ( p, w, y, x ) are equivalen. F 0 0 1 1 Proof 1: If ( y, x ) and ( y, x ) are allocaively efficien wih respec o boh prices 0 0 1 1 0 1 ( p, w ), ( p, w ) and echnologies T, T, hen (by definiion) we find j p y j (ii) D ( y, x ) j, 0,1. j wx = { } Subsiuing he disance funcions in (6) by he revenue o cos raios (ii) gives (4). Q.E.D. In he earlier lieraure, Diewer (1992) poined ou he equivalence beween he Fisher ideal index and he Malmquis TFP index in case he disance funcion has a paricular flexible funcion form. A a more general level, Färe and Grosskopf (1992) derived an analogous equivalence resul o ha of Theorem 1 by resoring o he following se of assumpions: 1) he producion echnology exhibis monooniciy, convexiy, and CRS, 2) inpu-oupu vecors are allocaively efficien in erms of profi maximizaion. Ineresingly, Theorem 1 does no require monooniciy, convexiy and/or CRS. These properies do play a crucial role; he mahemaics of dualiy imply ha he 'dual' disance measure (7) uses he 'benchmark echnology' raher han he rue producion se. 1 T. 5

Sill, i is imporan o sress ha he rue echnology need no exhibi hese properies. This resul parly depends on assuming reurn-o-he-dollar maximizaion raher han profi maximizaion; he producion echnology mus ruly exhibi CRS under profi maximizaion, i.e. i does no suffice o use a benchmark echnology ha exhibis CRS. 3 However, he objecives of profi maximizaion and reurn-o-he-dollar maximizaion are in fac equivalen if he echnology exhibis CRS. In addiion, convexiy and monooniciy are no required even in case of profi maximizaion (alhough he 'dual' disance measure will use he convex monoone hull raher han he rue echnology as a benchmark). 4 Brief, CRS is no required in case of reurn-o-he dollar maximizaion, and monooniciy and convexiy are no required in boh cases. Hence, he above heorem is a rue generalizaion of he equivalence resul by Färe and Grosskopf. As noed by Balk (1993), i seems reasonable o assume ha he inpu-oupu vecor of any period is allocaively efficien wih respec o he echnology and prices of ha same period, i.e. (10) ( p, w ) V ( y, x ), = 0,1 However, he Equivalence Theorem also necessiaes he inpu-oupu quaniies of he base (arge) period o be efficienly allocaed wih respec o he echnology and prices of he arge (base) period, i.e. j j (11) ( p, w ) V ( y, x ), = 01;, j = 1. Balk (1993, pp. 680) has convincingly argued ha his condiion is compleely arbirary. For insance, if he prices and/or he echnology changes from he base period o he arge period, hen he opimal vecor of he base (arge) period is generally no allocaively efficien wih respec o he prices and echnology of he arge (base) period. In ha case, he Equivalen Theorem does no apply, and he Malmquis TFP index generally gives an inaccurae approximaion for he ideal index. 3. A MODIFIED MALMQUIST INDEX APPROACH Resoring o some general properies of he disance funcions, Balk (1993) showed ha he condiion of allocaive efficiency suffices for using he Malmquis and Fisher indexes as reasonable firs-order approximaions of each oher, even under price or echnology changes. However, if we are ineresed of finding he bes possible approximaion for he Fisher ideal index, hen we can use he above insighs o derive a beer approximaion han he convenional Malmquis TFP index. Specifically, we focus on an exac approximaion in case of unique shadow prices, and on an inerval approximaion in case of muliple opimal price soluions. 3 Noe ha he benchmark echnology generally needs o exhibi CRS. If he benchmark se exhibis VRS, he Malmquis index need no conain he elemenary produciviy index (1) as is special case. Moreover, under VRS he inpu-based and he oupu-based disance funcions can yield differen values of he Malmquis index. Under CRS, he inpu and oupu disance funcions are reciprocals o each oher (Färe and Lovell, 1978), and hence he orienaion of he (primal) disance funcion formulaion does no maer for he Malmquis index. 4 To rule ou some anomalies, we need he benchmark echnology o be monoonous. Specifically, in case of a non-monoonous benchmark echnology, he Malmquis index can decrease even if all oupus (inpus) increase (decrease); and conversely, he Malmquis index can increase even hough all oupus (inpus) decrease (increase). 6

To gain some addiional insigh, consider firs a siuaion where he relevan (relaive) shadow price vecors are unique. As explained in he previous secion, he problem of he Malmquis TFP index is ha i uses shadow prices of he base (arge) period quaniies wih respec o he arge (base) period echnology. The approximaion obviously improves if we replace hese (generally) meaningless shadow prices wih he meaningful shadow prices wih respec o he base (arge) period echnology. This observaion moivaes us o define a enaive modificaion of he Malmquis TFP index as 1 1 1 (12) M ( y, x ) (, ) j D y x 2, ( ) j= 0 = 0 where (13) j y D ρ ( y, x ) sup, j j j ( ρω, ) V ( y, x ) ω x, j { 0,1} is he disance relaive o he shadow-price augmened benchmark se. 5 Observe he modified disance funcions (13) are equivalen o he sandard disance funcions (8) if = j ; differences occur for j. Ineresingly, if he relaive shadow prices are unique, he assumpion of allocaive efficiency suffices o recover he exac value of he Fisher ideal TFP index by using he modified version of he Malmquis TFP index. THEOREM 2 ( The 2 nd Equivalence Theorem ): The following condiions are equivalen if he shadow price vecors deermine unique relaive prices: ( y, x ), 0,1 is allocaively efficien wih respec o prices 1) Producion vecor { } (, ) p w and echnology T. 2) The modified Malmquis TFP index TFP ( p, w, y, x ) are equivalen. F 0 0 Proof 2: If (, ) 0 echnology T, we find 0 p y 0 (i) D ( y, x ), 0,1 0 wx = =. 1 1 Similarly, if (, ) 1 echnology T, hen 1 py 1 (ii) D ( y, x ), 0,1 1 wx = =. (, x ) and he Fisher ideal index M y 0 0 y x is allocaively efficien wih respec o prices (, ) p w and 1 1 y x is allocaively efficien wih respec o boh prices (, ) p w and 5 Using erminology of Kuosmanen and Pos (2001). 7

Subsiuing he disance funcions in (12) by he revenue o cos raios (i) and (ii) gives (4). Q.E.D. Figure 1 illusraes he difference o he convenional Malmquis TFP index for oupu based disance funcions relaive o a wo-oupu echnology. The wo dos labeled Y 0 and Y 1 represen he oupu bundles of he base year 0 and he arge year 1 respecively. The curves represen he froniers of he producion ses T 0 and T 1 respecively. The doed lines illusrae he pahs of equiproporionae scaling of Y 0 and Y 1 respecively, while he solid lines represen he iso-revenue surfaces a he shadow prices of Y 0 and Y 1. y 2 A Y 1 T 1 T 0 C B E D Y 0 F O y 1 Figure 1: The convenional Malmquis index compares Y 0 o he poins F and E on he echnology froniers, and Y 1 o C and A. By conras, he modified Malmquis index compares Y 0 o he poins F and D on he iso-revenue surfaces, and similarly Y 1 o B and A. The convenional Malmquis TFP index compares Y 0 o he poins F and E on he froniers, and similarly, Y 1 o C and A; he Malmquis TFP index is given by 1 1 1/2 OY / OA OY / OC (14) M( y, x ) = 0 0. OY / OE OY / OF By conras, he modified Malmquis TFP index compares Y 0 o he poins F and D on he iso-revenue surface, and similarly, Y 1 o B and A, i.e. he modified Malmquis TFP index is given by 8

(15) 1 1 1/2 OY / OA OY / OB (, x ) = 0 0. M y OY / OD OY / OF If he shadow prices correcly represen he economic prices, i.e. he condiion of allocaive efficiency holds, hen (by Theorem 2) he modified Malmquis TFP index equals he Fisher ideal index. In ha case, he convenional Malmquis TFP index is associaed wih he approximaion error according o he formula 1/2 OC / OB (16) M ( y, x ) = M( y, x ). OE / OD Balk s argumen of he reasonable approximaion abiliy of he sandard Malmquis TFP index is based on he fac ha boh he nominaor (OC/OB) and he denominaor (OE/OD) of he error erm are always less han or equal o uniy. Sill, eliminaing his source of error can yield a considerable improvemen, especially when here are subsanial price changes leading o allocaive shifs, and/or biased echnology change, i.e. when (OC/OB) is subsanially differen from (OE/OD). However, as noed already in he previous secion, he shadow prices need no be unique. In many empirical sudies, he rue producion se is approximaed using he aciviy analysis (or Daa Envelopmen Analysis, DEA) approach. In ha approach, he empirical producion se has a piece-wise linear fronier ha generally involves muliple shadow price vecors for he echnically efficien producion vecors. Unforunaely, when he shadow prices are non-unique, he assumpion of allocaive efficiency does no suffice for recovering he rue economic prices from he quaniy daa. We propose o exend our enaive modificaion of he Malmquis TFP index in (12) o he cases wih non-unique shadow prices by deriving an upper bound and a lower bound for he Fisher ideal index, derived from 'mos favorable' and 'leas favorable' prices. The modified disance funcion (13) represens he 'mos favorable' prices. Similarly, we represen he 'leas favorable' prices by he following disance measure: y (17) E ρ ( y, x) inf, 0,1 =. ( ρω, ) V ( y, x) ω x Using hese disance funcions, we define he following upper bound for he Fisher ideal index: 1/2 1 1 1 0 1 1 D ( y, x ) D ( y, x ) (18) UF ( y, x ) 0 0 0 1 0 0. D ( y, x ) E ( y, x ) Similarly, we define he following lower bound for he Fisher index: 1/2 1 1 1 1 1 D ( y, x ) E ( y, x ) (19) LF ( y, x ) 0 0 0 1 0 0. D ( y, x ) D ( y, x ) THEOREM 3 ( THE INTERVAL THEOREM ): The following condiions are equivalen: 1) Producion vecor ( y, x ), = 0,1 is allocaively efficien wih respec o prices (, ) p w and echnology T. 9

2) The Fisher index TFP ( p, w, y, x ) is conained wihin he inerval F F(, ), F(, ) L y x U y x. Proof 3: 1) 2). By definiion, allocaive efficiency guaranees ha he economic prices are conained in he ses of shadow prices, i.e. ( p, w ) V ( y, x ), = 0,1. Under allocaive efficiency, he upper and he lower bounds saisfy following inequaliies by consrucion: L ( y, x ) TFP ( p, w, y, x ) U ( y, x ). F F F 2) 1). TFPF ( p, w, y, x ) LF( y, x ), UF( y, x ) immediaely implies ( p, w ) V ( y, x ), = 0,1, i.e. he allocaive efficiency. Q.E.D. The inerval approximaion of he Fisher index already allows for logical inference. For example if UF ( x, y ) 1, hen produciviy decline mus have occurred. Similarly, LF ( x, y ) 1 implies produciviy improvemen. In general, he smaller he se of shadow prices, he more narrow he inerval. Moreover, he inerval ends o widen when allocaive shifs and echnology change are considerable, i.e. in hose cases where he convenional Malmquis TFP index ends o be mos inaccurae. y 2 Y 1 A T 0 B T 1 D C Y 0 O y 1 Figure 2: The case of non-unique shadow prices. The lower bound is obained by comparing Y 0 o he poin D and Y 1 o A associaed wih he leas favorable shadow prices. The upper bound compares Y 0 o C and Y 1 o B in ligh of he mos favorable shadow prices. 10

Figure 2 illusraes he upper and lower bounds. Again, he wo dos labeled Y 0 and Y 1 represen he oupu bundles of he base year and he arge year respecively. The solid piece-wise linear curves represen he froniers of he producion ses T 0 and T 1 respecively. Noe ha in his example, he evaluaed uni consiues an exreme poin of he echnology se in boh periods. The broken lines illusrae he pahs of equiproporionae scaling of Y 0 and Y 1 respecively. Finally he wo solid lines associaed wih he boh observaions represen he iso-revenue surfaces a he exreme shadow prices of Y 0 and Y 1. The minimum value of he modified Malmquis TFP index is obained by comparing Y 0 o he poin D on he iso-revenue surface and Y 1 o A, i.e. 1 1/2 OY / OA (20) LF ( y, x ) = 0. OY / OD Clearly, his index exceeds he uniy, which signals produciviy improvemen. The maximum value of he modified Malmquis TFP index is obained by comparing Y 0 o he poin C on he iso-revenue surface and Y 1 o B, i.e. 1 1/2 OY / OB (21) LF ( y, x ) = 0. OY / OC Finally, if one prefers o consider a poin esimae insead of he inerval, an obvious alernaive is o ake he geomeric average of he upper and he lower bounds (in he spiri of Fisher), i.e. (22) Mˆ ( y, x ) ( U (, ) (, ) ) 1/2 F y x LF y x. Again, in case he shadow prices are unique, he modified Malmquis TFP indexes (12) and (22) coincide. 4. EXAMPLE APPLICATION: TFP GROWTH IN OECD COUNTRIES To es he modified Malmquis TFP index approach oulined in he previous secions, we underook an applicaion o aggregae producion daa of OECD counries, in he spiri of Färe, Grosskopf, Norris, and Zhang (1994). Like Färe e al. (1994), we measured aggregae oupu by Gross Domesic Produc (GDP: measured in Mill. U.S. dollars a 1990 prices and purchasing power pariy), and considered wo inpus: Labor (in Thousands of employees) and Capial (Gross Capial Sock; in Mill. U.S. dollars a 1990 prices and purchasing power pariy). Cross-secional daa of years 1970, 1975, 1980, 1985, 1990, and 1994 were obained from Research Insiue for Finnish Economy (ETLA) for 14 counries: Ausralia (AUS), Belgium (BEL), Canada (CAN), Denmark (DEN), Finland (FIN), France (FRA), Grea Briain (GBR), Ialy (ITA), Japan (JPN), Neherlands (NLD), Norway (NOR), Sweden (SWE), Unied Saes (USA), and Wes Germany (WGR). Table 1 summarizes he daa se. 11

Table 1: Summary saisics of he oupu and inpu variables GDP Labor Capial Mean S. Dev. Growh * Mean S. Dev. Growh * Mean S. Dev. Growh * AUS 228109 62561 0,16 5740 796 0,07 902801 275414 0,19 BEL 132311 26986 0,12 3022 74 0,00 493663 134296 0,17 CAN 365282 95909 0,16 9842 1909 0,11 1623141 541238 0,20 DEN 63803 13032 0,11 2093 177 0,04 369467 73453 0,12 FIN 56515 13601 0,13 1845 157 0,00 336694 100584 0,18 FRA 775771 173145 0,13 18360 1129 0,03 2406982 620520 0,16 GBR 702660 128081 0,10 22670 702-0,01 2390752 537426 0,13 ITA 744208 171158 0,13 15125 761 0,02 2867872 785554 0,16 JPN 1719163 587729 0,21 45628 7247 0,09 4864027 2635125 0,42 NLD 189767 40869 0,13 4302 265 0,03 696085 148919 0,13 NOR 57215 16347 0,18 1681 218 0,07 284115 94309 0,21 SWE 112546 19381 0,09 3833 261 0,01 509968 121769 0,14 USA 4694372 1140329 0,13 90105 15083 0,09 17235742 4382503 0,15 WGR 932372 204628 0,12 23802 1467 0,03 3596828 932064 0,16 Enire Sample 769578 1231081 0,14 17718 23861 0,06 2755581 4465362 0,18 * Growh = Average growh rae of he variable from he base period o he arge period We used a CRS Cobb-Douglas producion funcion fied o he daa se by he Correced Ordinary Leas-Squares (COLS) echnique (Aigner and Chu, 1968; Richmond, 1974) as an empirical producion echnology. The poenial specificaion error associaed wih specifying a parameric form is a disadvanage of his approach. However, he advanages of his approach (relaive o non-parameric approaches) for he presen applicaion (which involves relaively small cross-secional samples) include a relaively high robusness wih respec o sampling variaion, and he exisence of unique and posiive shadow prices. These consideraions were confirmed by he oucomes of he applicaion of he nonparameric Daa Envelopmen Analysis (DEA) approach (see e.g. Färe e al., 1994; Färe and Grosskopf, 1996); he DEA shadow prices for many observaions equaled zero, or were non-unique 6. For hese reasons, we found he COLS approach preferable in he presen applicaion. Table 2 summarizes he parameer esimaes of he COLS regression. Table 2: Regression resuls (sandard errors in parenhesis) 1970 1975 1980 1985 1990 1994 R 2 0,806 0,817 0,820 0,822 0,827 0,862 Coefficiens: Consan 12.354 9.368 7.831 5.904 9.609 13.087 6 Sill, he disance measures esimaed by he COLS and he DEA echniques were found o be highly correlaed (he correlaion coefficien equaled 0.958). In our inerpreaion, his suppors he parameric specificaion of he COLS fronier. 12

Labor 0.739 (0.0017) Capial 0.261 (0.0016) 0.682 (0.0016) 0.318 (0.0018) 0.648 (0.0018) 0.352 (0.0019) 0.585 (0.0016) 0.415 (0.0018) 0.671 (0.0017) 0.329 (0.0018) 0.721 (0.0013) 0.279 (0.0014) Figure 3 displays he observaions and he COLS froniers for 1990 and 1994. Noice he biased echnology change driven solely by he capial inpus. The figure also illusraes how counries almos invariably subsiued labor by capial. Similar biased fronier shifs as well as subsanial increases in he capial per labor were clearly idenifiable in each year under sudy. In addiion, i seems obvious ha he labor compensaions relaive o he capial rens have increased subsanially over he sample period. These consideraions immediaely call ino quesion he assumpion ha he 1990 (1994) observaions are allocaively efficien wih respec o he 1994 (1990) echnology, and plead for correcing he Malmquis TFP index. FIN'94 DEN'94 FIN'90 DEN'90 NOR'94 NOR'90 SWE'94 CAN'94 CAN'90 BEL'94 WGR'94 ITA'94 AUS'94 AUS'90 ITA'90 BEL'90 WGR'90 USA'94 NLD'90 GBR'94 USA'90 NLD'94 JPN'94 FRA'94 FRA'90 JPN'90 GBR'90 SWE'90 1990 fronier 1994 fronier 1990 daa 1994 daa Labor Figure 3: Esimaed inpu isoquans for 1990 and 1994 For comparison, we compued boh he convenional and he modified Malmquis indices relaive o he empirical Cobb-Douglas producion fronier. Table 3 summarizes he average resuls by counry. The wo indexes were found o be quie highly correlaed as expeced (he correlaion coefficien equaled 0.82). Sill, he sandard Malmquis TFP 13

index yielded invariably higher values han he modified counerpar, and hence sysemaically overesimaed TFP growh. In fac, here were 2 cases (Japan, 1970-75; Canada, 1980-85) where he alernaive indexes lead o a qualiaively opposie conclusion; i.e. he modified index suggesed produciviy decline, while he sandard index indicaed growh. On he average, he absolue values of differences varied beween 0.03 0.06 index poins. The larges difference occurred for Japan: The difference beween he indexes was greaer han 10 percen in 2 cases. Moreover, in oal of 22 cases he difference of he wo indexes exceeded 5 percen, and in 65 cases ou of 70 cases he difference was greaer han or equal o 1 percen. The average difference beween he indexes was found o be 4.2 percen. I is worh o recall ha any difference immediaely consiues an improvemen for he purpose of approximaing he ideal index. Figure 1 furher illusraes he disribuion of he differences. Table 3: Summary of he Sandard vs. Modified Malmquis indexes by Counry Geomeric average: Difference: Sandard Modified Minimum Mean Maximum AUS 1,0819 1,0431 0,0123 0,0356 0,0545 BEL 1,1234 1,0657 0,0333 0,0557 0,0721 CAN 1,0409 1,0143 0,0162 0,0282 0,0428 DEN 1,0698 1,0456 0,0074 0,0254 0,0414 FIN 1,1225 1,0687 0,0484 0,0612 0,0908 FRA 1,1100 1,0655 0,0262 0,0411 0,0559 GBR 1,1089 1,0618 0,0323 0,0481 0,0733 ITA 1,1131 1,0655 0,0380 0,0467 0,0652 JPN 1,1370 1,0336 0,0365 0,0900 0,1443 NLD 1,1033 1,0672 0,0017 0,0334 0,0595 NOR 1,0995 1,0476 0,0064 0,0428 0,0579 SWE 1,0668 1,0327 0,0304 0,0419 0,0732 USA 1,0406 1,0230 0,0048 0,0187 0,0343 WGR 1,0966 1,0525 0,0173 0,0413 0,0629 14

0,16 0,14 0,12 0,1 0,08 0,06 0,04 0,02 0 1 70 Figure 4: Disribuion of he difference of he Sandard and he Modified Malmquis index. The horizonal axis presens he observaions sored in ascending order (N=70). The primary objecive of his applicaion was o see how well he convenional Malmquis TFP index approximaes our modificaion in a ypical applicaion. The empirical resuls suppor our conjecure ha he approximaion error of he sandard Malmquis TFP index ends o increase when he echnology or prices change rapidly. I is worh o noe ha we used aggregae daa wih cross-secions from represenaive years. In a monhly or quarerly daa he echnology and price changes would end be smaller, and he sandard Malmquis TFP index could yield a more accurae approximaion. On he oher hand, we migh expec much more dramaic allocaive shifs and echnical progress o occur in he firm or shop-floor level produciviy analyses. Therefore, we believe he presen applicaion is represenaive of many research siuaions, and ha he proposed modificaion can make a subsanial difference in reallife applicaions. 7. CONCLUSIONS In his paper, we have revisied he condiions for equivalence beween he Malmquis TFP index and Fisher ideal TFP index. This quesion is of considerable ineres due o he problem of incomplee price or value informaion, which frequenly occurs in pracical applicaions; in conras o he Fisher ideal index, he Malmquis TFP index can be compued wihou any price (or share) daa. I was earlier argued by Färe and Grosskopf (1992) and Balk (1993) (and elaboraed above) ha he Malmquis TFP index is a reasonable approximaion of he Fisher ideal TFP index, provided ha a relaively general assumpion of allocaive efficiency holds. This resul enables one o 15

approximae he inuiively and axiomaically aracive Fisher ideal TFP index even when he complee price informaion is no available. However, as emphasized by Balk, an accurae approximaion is obained only by acciden. In fac, approximaion error can be quie subsanial when allocaive shifs (reflecing price changes) or biased echnology changes occur. This observaion moivaed us o modify he sandard Malmquis TFP index o minimize approximaion error. Focusing solely on he meaningful shadow prices, we consruced upper and lower bounds for he Fisher ideal index. Whenever he shadow prices suppor unique relaive prices, he exac value of he Fisher index can be recovered wih full accuracy. To illusrae he pracical applicaion of he proposed mehod and o demonsrae he gains achievable by i, we applied our mehod o aggregae producion daa of 14 OECD counries. The empirical resuls demonsrae ha he proposed modified indexes can subsanially improve esimaion in real-life applicaions. REFERENCES Aigner, D.J., and S.F. Chu (1968): On Esimaing he Indusry Producion Funcion, American Economic Review 58, 226-239 Balk, B. (1993): Malmquis Produciviy Indexes and Fisher Ideas Indexes: Commen, The Economic Journal 103, 680-682 Caves, D.W., L.R. Chrisensen, and W.E. Diewer (1982): The Economic Theory of Index Numbers and he Measuremen of Inpu, Oupu and Produciviy, Economerica 50, 1393-1414 Diewer, W.E. (1992): Fisher Ideal Oupu, Inpu and Produciviy Indexes Revisied, Journal of Produciviy Analysis 3(3), 211-248 Diewer, W.E. (2000): Alernaive Approaches o Measuring Produciviy and Efficiency, paper presened a he Norh Americal Produciviy Workshop a Union College, Schenecady, N.Y., June 15-17 Färe, R., and S. Grosskopf (1992): Malmquis Produciviy Indexes and Fisher Ideas Indexes, The Economic Journal 102, 158-160 Färe, R., and S. Grosskopf (1996): Ineremporal Producion Froniers: Wih Dynamic DEA, Kluwer Academic Publishers, Färe, R., S. Grosskopf, M. Norris, and Z. Zhang (1994): Produciviy Growh, Technical Progress, and Efficiency Change in Indusrialized Counries, Americal Economic Review 84(1), 66-83 16

Färe, R., and C. A. K. Lovell (1978): Measuring he Technical Efficiency of Producion, Journal of Economic Theory 19, 150-162 Farrell, M. J. (1957): The Measuremen of Producive Efficiency, Journal of Royal Saisical Sociey, Series A, 120(3), 253 290 Fisher, I. (1922): The Making of he Index Numbers, Boson, Houghon Mifflin Georgescu-Roegen, N. (1951): The Aggregae Linear Producion Funcion and Is Applicaions o von Neumann's Economic Model", in T. Koopmans (Ed.), Aciviy Analysis of Allocaion and Producion. Jorgenson, D.W., and Z. Griliches (1967): The Explanaion of Produciviy Change, Review of Economic Sudies 34, 249-283 Kuosmanen, T., and G.T. Pos (2001): Measuring Economic Efficiency wih Incomplee Price Informaion: Wih an Applicaion o European Commercial Banks, European Journal of Operaional Research, o appear Richmond, J. (1974): Esimaing he Efficiency of Producion, Inernaional Economic Review 15, 515-521 Törnqvis, L. (1936): The Bank of Finland s Consumpion Price Index, Bank of Finland Monhly Bullein 10, 1-8 17

APPENDIX This appendix presens he Linear Programming formulaions for compuing he modified disance funcions relaive o a Daa Envelopmen Analysis (DEA) fronier.le he oupu daa of period be denoed by he marix Y = ( y 1 yn ), = 0,1; and use X = ( x 1 xn ) for he marix of inpu vecors respecively. We sar by compuing he sandard disance funcions D by solving he following wo Linear Programmin (LP) problems of he sandard form: D ( y, x ) = Max ρ y k k ρω, k ω xk = s.. ρy ωx 1 0 ρω, 0 = 0,1 We nex use he shadow prices associaed wih he opimal soluions of he above LP problems o obain he modified disance funcions. This boils down o solving he following LP problems D ( y, x ) = Max ρ y j k k ρω, k ω xk = j j s.. 1 ρy ωx 0 j j j j j ρyk = ωxkd ( yk, xk) ρω, 0 = 1 j j = 0,1 and E ( y, x ) = Min ρ y j k k k ρω, s.. ω x = 1 ρy j k ωx 0 j j j j j ρy = ωx D ( y, x ) ρω, 0 = 1 j j = 0,1 j k k k k 18