Productivity Measurement in the Public Sector: Theory and Practice

Transcription

1 1 Produciviy Measuremen in he Public Secor: Theory and Pracice W. Erwin Diewer. 1 January 26, Discussion Paper 17-01, Vancouver School of Economics, Universiy of Briish Columbia, Vancouver, B.C., Canada, V6T 1L4. erwin.diewer@ubc.ca Absrac In many secors of he economy, governmens eiher provide various goods and services a no cos or a highly subsidized prices. I is usually possible o measure he quaniies of hese governmen secor oupus and inpus as well as inpu prices bu he problem is how o esimae he corresponding oupu prices. Once meaningful oupu prices have been esimaed, he measuremen of produciviy growh using index numbers can proceed in he usual manner. This chaper suggess hree possible general mehods for measuring public secor oupu prices and quaniies. If lile or no informaion on he quaniy of nonmarke oupus produced is available, hen he mehod recommended in he Sysem of Naional Accouns 1993 mus be used, where aggregae oupu growh is se equal o aggregae inpu growh. If informaion on nonmarke public secor oupus is available hen he second general mehod ses he missing oupu prices equal o he uni coss of producing each oupu while he hird general mehod uses purchaser s valuaions o deermine he missing oupu prices. Specific measuremen issues in he healh and educaion secors are discussed. Similar oupu and produciviy measuremen issues arise in he regulaed secors of an economy since regulaed producers are forced o provide services a prices ha are no equal o marginal or average uni coss. Finally, he problems associaed wih measuring capial services are discussed. The focus of he chaper is on he use of index number mehods o measure he Toal Facor Produciviy of producion unis in he public and regulaed secors. Keywords Measuremen of oupu, inpu and produciviy, nonmarke secor, healh, educaion, regulaed indusries, cos funcions, marginal cos prices, echnical progress, qualiy adjusmen, hedonic regressions, index number heory. JEL Classificaion Numbers C43, D24, D57, E23, H40, H51, H52, I10, I12, I20, O47 1 W. Erwin Diewer: School of Economics, Universiy of Briish Columbia, Vancouver B.C., Canada, V6T 1Z1 and he School of Economics, UNSW Ausralia, Sydney erwin.diewer@ubc.ca. The auhor hanks Knox Lovell, Kevin Fox, Paul Schreyer and Robin Sickles for helpful commens and he SSHRC of Canada and he Ausralian Research Council (DP ) for financial suppor. None of he above are responsible for any opinions expressed in he paper.

2 2 1. Inroducion: How Should Public Secor Nonmarke Oupus be Valued? In order o measure he Toal Facor Produciviy of a public secor producion uni (or esablishmen) using index number echniques, i is necessary o measure he prices and quaniies of he oupus produced and he inpus used by ha uni for wo periods of ime. Then Toal Facor Produciviy growh can be defined as a quaniy index of oupus produced divided by a quaniy index of inpus used by he esablishmen. 2 I is usually possible o measure he price and quaniy of inpus in a fairly saisfacory manner 3 bu here are problems in measuring he prices and quaniies of public secor nonmarke oupus. Thus in his chaper, we will ake a sysemaic look a possible mehods for he valuaion of nonmarke oupus produced by public secor producion unis. In many cases, i is difficul o deermine exacly wha i is ha a public secor producion uni produces. In his case, we may have neiher quaniies or prices for he oupus of he governmen service provider. However, in many cases, we can measure a leas he quaniies of he oupus produced by he public secor uni bu no he corresponding prices. Finally, in some cases, i may be possible o measure oupu quaniies produced and o obain esimaes of purchaser s valuaions for he missing nonmarke oupu prices. Thus he bes pracice mehodology ha can be used o form produciviy esimaes for he public secor uni will depend o a large exen on wha informaion on prices and quaniies is available. From he perspecive of measuring he effecs on he welfare of households of public secor producion, we sugges he following mehods for valuing governmen oupus in he order of heir desirabiliy: Firs bes: valuaion a marke prices or purchaser s valuaions; Second bes: valuaions a producer s uni coss of producion; Third bes: oupu growh of he public secor producion uni is se equal o real inpu growh and he corresponding oupu price growh is se equal o an index of inpu price growh. Obviously, he hird bes opion is he leas desirable opion. If i is used, hen produciviy growh for he public secor producion uni will be nonexisen by consrucion. If here are compeiive markes wih no economies of scope and consan reurns o scale in producion, hen he firs and second bes opions will be roughly equivalen; i.e., he purchaser s price will be approximaely equal o he long run marginal cos of producing a uni of he commodiy. However, in a nonmarke seing, even wih no economies of scope and consan reurns o scale in producion, here is 2 This follows he approach pioneered by Jorgenson and Griliches (1967) (1972). For furher developmens of he index number approach o measuring TFP growh, see Diewer (1976) (1980) (1983) (1992a) (1992b) (2014), Caves, Chrisensen and Diewer (1982a) (1982b), Diewer and Morrison (1986), Kohli (1990), Fox and Kohli (1998), Balk (1998) (2003), Schreyer (2001), Diewer and Nakamura (2003) and Inklaar and Diewer (2016). 3 However, here are some significan problems associaed wih he measuremen of capial services inpus and we will spend some ime dealing wih hese difficulies.

3 3 nohing o force he cos of producing a uni of public secor oupu o equal is value o recipiens of he commodiy. 4 In an exreme case, he producion uni could be producing nohing of value. Thus from he viewpoin of welfare economics, valuaion of public secor oupus a purchasers valuaions appears o be he preferred opion. 5 The ask of his chaper is o sugges mehods for measuring he Toal Facor Produciviy (TFP) of a public secor producion uni which produces a leas some oupus ha are allocaed o recipiens a nonmarke prices; i.e., a zero prices or highly subsidized prices ha do no cover heir uni coss of producion. In general, he TFP level of a producion uni is defined as he real oupu produced by he producion uni a a ime period divided by he real inpu uilized by he producion uni during he same ime period. There is TFP growh if real oupu grows more rapidly han real inpu. The main drivers of TFP growh are: (i) echnical progress; i.e., a ouward shif of he uni s producion possibiliies se; (ii) increasing reurns o scale combined wih inpu growh and (iii) improvemens in echnical and allocaive efficiency. 6 This volume oulines many mehods for measuring TFP bu in his chaper, we will concenrae on index number mehods. 7 The use of index number mehods means ha we will no be able o decompose he TFP growh of a public secor producion uni ino he conribuions of he above hree main componens of TFP growh; i.e., index number mehods do no allow us o measure separaely he effecs of he above explanaory facors; in general, all hree explanaory facors will be combined ino he overall TFP growh measure. However, he use of index number mehods requires a mehodological framework for he deerminaion of oupu prices for he nonmarke oupus produced by he public secor. If he public secor uni minimizes is cos of producion in producing is nonmarke oupus, hen from he viewpoin of he economic approach o produciviy measuremen, he righ prices o use o value nonmarke oupus are he (long run) marginal coss of producing he nonmarke oupus. Diewer (2012; ) provided a nonparameric mehodological jusificaion for his marginal cos mehod of oupu price valuaion for nonmarke oupus. 8 Diewer also jusified he use of he Fisher (1922) index number formula o aggregae inpus and oupus using his approach. We will summarize his approach in he Appendix o his chaper. Varians of his cos based mehod for valuing nonmarke oupus will be discussed in Secion 5. 4 For a more deailed discussion of valuaion principles based on purchaser versus supplier valuaions, see Hill (1975; 19-20), Akinson (2005; 88) and Schreyer (2012a; ). 5 However, from he perspecive of measuring he Toal Facor Produciviy growh of a public secor producion uni, valuaion of unpriced oupus by heir marginal or uni coss is bes as we shall see laer. 6 During recessions, oupus decrease more han inpus due o he shor run fixiy of many inpus. Thus producion unis may be in he inerior of heir producion possibiliies ses a imes and so movemens from a echnically inefficien allocaion of resources owards he producion fronier will improve TFP. 7 Oher mehods involve he esimaion of cos or producion funcions bu ypically, saisical agencies do no have he resources o underake he required economeric esimaion. Nonparameric produciviy measuremen mehods could also be used bu unless price informaion is used in addiion o quaniy informaion on he inpus used and oupus produced by a producion uni, hese mehods ofen do no generae reasonable esimaes in he ime series conex which is he focus of his chaper. For he applicaion of nonparameric mehods in he cross secional conex, see he pioneering papers by Charnes, Cooper and Rhodes (1978) and Charnes and Cooper (1985). 8 Akinson (2005; 88-90) advocaed he use of marginal coss o value oupus in he nonmarke secor. Hill (1975; 19-21) advocaed he use of (average) uni coss o value nonmarke oupus.

4 4 The hird bes opion oulined above is he only opion ha can be used when here is lile or no informaion on boh he prices and quaniies produced by a governmen esablishmen. This is he opion which is recommended in he Sysem of Naional Accouns 1993 o value governmen producion when direc informaion on he prices and quaniies of governmen oupus is no available. The quaniy or volume measure for esablishmen oupu ha resuls from using his mehodology can be inerpreed as a measure of real resources used by ha esablishmen and as such, i is an accepable indicaor of he oupu produced by a governmen uni. This hird bes opion will be discussed in more deail in he following hree secions. Secion 2 inroduces he mehod. Secion 3 discusses how user coss should be used o value public secor capial sock inpus. Secion 4 addresses he problems associaed wih choosing an index number formula and how TFP could be measured eiher as gross oupu TFP or as value added TFP. This secion also shows ha he choice of an index number formula o aggregae oupus and inpus does maer. The second bes opion for valuing public secor oupus will be discussed in secion 5 and he firs bes opion in secion 6. Secion 7 provides a numerical example due o Schreyer (2010; 21) ha illusraes how he firs bes opion (from he viewpoin of welfare economics) could be used in order o value nonmarke oupus. This secion also discusses some aspecs of he problem of adjusing oupu quaniies for qualiy change. Secion 8 discusses some of he pracical difficulies associaed wih he measuremen of public secor oupus in seleced indusries. Secion 9 concludes. 2. The Case where No Informaion on he Prices and Quaniies of Nonmarke Oupus is Available The hird bes opion oulined above is he only opion ha can be used when here is lile or no informaion on boh he prices and quaniies produced by a public secor producion uni (or here is no agreemen on how o measure he oupus of he uni). For example, usually here is lile informaion on he price and quaniy of educaional services produced by he public secor. 9 As was menioned earlier, he quaniy or volume measure for esablishmen oupu ha resuls from using his mehodology can be inerpreed as a measure of real resources used by he producion uni and as such, i is an accepable indicaor of he oupu produced by he uni. Alhough his hird bes opion is fairly sraighforward in principle (and has been exensively discussed in he naional income accouning lieraure), here are some aspecs of he mehod ha deserve some addiional discussion. The wo aspecs of he hird mehod ha we will discuss in more deail in he following wo secions are as follows: How exacly should he conribuion of durable inpus used in a public secor producion uni o curren period producion be valued? 9 In secion 8.1 below, we will discuss various suggesed mehods for esimaing oupu prices and quaniies for his secor. I will be seen ha here is no general agreemen on hese mehods.

5 5 How exacly should esimaes for he aggregae real oupu produced by a nonmarke producion uni be consruced? In paricular, which index number formula should be chosen o perform he aggregaion and does he choice of formula make a difference? 3. The Valuaion of Durable Inpus Used in he Public Secor The basic ool ha economiss use o value he conribuion of a durable inpu (or a capial inpu) o producion in an accouning period is he concep of a user cos. We will firs explain how o consruc he user cos of a capial inpu 10 for a durable inpu for which marke prices exis for he same inpu a differen ages. Consider a producion uni which purchases q unis of a durable inpu a he beginning of accouning period a he price P. Afer using he services of he capial inpu during period, he producion uni will have q u = (1 )q unis of used or depreciaed capial (in consan qualiy unis) on hand a he end of period where is he one period depreciaion rae for he capial good under consideraion. 11 Finally, we assume ha he producion uni has a one period financial opporuniy cos of capial a he beginning of period (i.e., a beginning of he period nominal ineres rae) equal o r. The gross cos of he capial inpu is he beginning of he period purchase cos of he capial inpus, P q. Bu his cos is offse by he revenue ha could be raised by selling he depreciaed capial sock value a is impued marke value a he end of he period, which is P +1 q u = P +1 (1 )q. Bu his impued revenue is no equivalen o he cos oulay made a he beginning of he period. To make i equivalen, we need o ake ino accoun ha money received a he end of he period is less valuable han money received a he beginning of he period and so he end of period marke value should be discouned by (1+r ). Thus he ne cos of using he services of he capial inpu during period is U defined as follows: 12 (1) U P (1+r ) 1 (1 )P +1. Define he consan qualiy asse inflaion rae over period, i, by he following equaion: (2) 1+ i P K +1 /P K. 10 A capial inpu is an inpu which conribues o producion for more han one accouning period. In pracice, naional income accounans rea capial inpus ha las less han hree years as nondurable inpus. 11 For simpliciy, we assume he geomeric model of depreciaion where he one period depreciaion rae remains consan no maer wha he age of he asse is a he beginning of he period. For more on he geomeric model of depreciaion, see Jorgenson (1989) (1996). 12 This simple discree ime derivaion of a user cos (as he ne cos of purchasing he durable good a he beginning of he period and selling he depreciaed good a an ineres rae discouned price a he end of he accouning period) was developed by Diewer (1974a; 504), (1980; ), (1992b; 194). Simplified user cos formulae (he relaionship beween he renal price of a durable inpu o is sock price) dae back o Babbage (1835; 287) and o Walras (1954; ). The original version of Walras in French was published in The early indusrial engineer, Church (1901; ) also developed a simplified user cos formula.

6 6 Subsiuing (2) ino (1) leads o he following expression for he ex pos user cos U defined by (1): (3) U P (1+r ) 1 (1 )(1+i )P = (1 r ) 1 [(1+r ) (1 )(1+i )]P. Raher han discouning he end of period value of he capial sock o he beginning of he period, i is more convenien o ani-discoun coss and benefis o he end of he period. Thus he ex pos end of period user cos of a capial inpu, u, is defined as (1+r ) imes U : 13 (4) u (1+r )U = [(1+r ) (1 )(1+i )]P = [r i + (1+i )]P. Using his formula, we add he opporuniy cos of ying up financial capial for one period, r P, o he beginning of he period purchase cos of he asse, P, o obain a oal cos of purchasing one uni of an asse and ying up financial capial for one period, which is (1+r )P. This oal asse cos is offse by he end of period (impued) benefi of he depreciaed value of one uni of he purchased capial sock, (1 )(1+i )P. Taking he difference of his cos and benefi leads o he user cos formula (4). This ex pos formula for he user cos of capial defined was obained by Chrisensen and Jorgenson (1969; 302) for he geomeric model of depreciaion. 14 However, he derivaion of he user cos of capial defined by (3) or (4) is no he end of he problems associaed wih valuing he cos of using a capial inpu over an accouning period. There are four addiional issues ha need o be discussed: Ex ane versus ex pos user coss; How o rea specific axes on some capial inpus such as propery axes; Wha o do if here are no marke prices for used capial socks and Wha should be done if inappropriae user coss of capial are used. We will address each of hese problems in urn. The problem wih he user cos formulae defined by (3) or (4) is ha hey use he ex pos or acual asse inflaion rae i defined by (2). For land asses in paricular, ex pos asse inflaion raes can a imes be so large ha i makes he user coss defined by (3) or (4) negaive. Basically, we would like he user cos of an asse o be approximaely equal o he renal price for he asse (if renal markes for he asse exis). Renal prices will rarely be negaive so obviously, a negaive user cos will be a poor approximaion o a renal 13 I should be noed ha he user coss ha are ani-discouned o he end of he period are more consisen wih commercial accouning convenions han he corresponding user coss ha are discouned o he beginning of he period; see Peasnell (1981). 14 Diewer (2005) (2010) and Diewer and Wei (2017) derive user cos formula for more general models of depreciaion. In paricular, one hoss shay or ligh bulb depreciaion may be a more appropriae model of depreciaion in valuing he conribuion of srucures and long lived infrasrucure asses. Diewer (2005) also discussed in more deail user coss ha are formed by discouning or ani-discouning coss and benefis o eiher he beginning or end of he accouning period.

7 7 price. A way around he negaive user cos problem is o replace he ex pos asse inflaion rae by an anicipaed asse inflaion rae. Thus suppose ha a he beginning of period, he anicipaed end of period price for an asse of he same qualiy is P +1*. Use his price in order o define an anicipaed asse inflaion rae as i * (P +1* /P ) 1. Now replace he ex pos asse inflaion rae i which appears in he user cos formulae (3) and (4) by heir anicipaed counerpars i * and we obain he corresponding ex ane user coss, U * and u *. Jorgenson (1989) (1996) and his coworkers 15 endorsed he use of ex pos user coss, arguing ha producers can perfecly anicipae fuure asse prices. On he oher hand, Diewer (1980; 476) (2005; ), Schreyer (2001) (2009) and Hill and Hill (2003) endorsed he ex ane version for mos purposes, since hese ex ane user coss will end o be smooher han heir ex pos counerpars and hey will generally be closer o a renal or leasing price for he asse. 16 Diewer and Fox (2016) used secoral daa on he US corporae and noncorporae financial secor o compue capial services aggregaes and he resuling raes of TFP growh using boh Jorgensonian and smoohed user coss ha use prediced asse inflaion raes. 17 They found ha Jorgensonian ex ane user coss for land componens were indeed negaive for many years bu he use of ex ane or prediced asse inflaion raes cured he problem of negaive user coss and indeed, led o much smooher user coss as could be expeced. 18 There is anoher soluion o he problem of negaive user coss if renal prices for he asse are available: ake he maximum of he user cos and he corresponding marke renal price as he appropriae valuaion for he services of he asse during he period under consideraion. The user cos valuaion for he services of he asse is essenially a financial opporuniy cos of using he asse while he renal price is he opporuniy cos of using he services of he asse for producive purposes raher han rening i ou for he period. Boh valuaions are valid opporuniy coss so he rue opporuniy cos of using he asse should be he maximum of hese wo coss. Diewer (2008) called his opporuniy cos mehod for asse services valuaion. 19 We have ignored ax complicaions in deriving he user cos formulae (3) and (4). Any specific capial axes (such as propery axes on real esae asses) should be added o he user cos formula for he relevan asses. 20 Business income axes ha fall on he gross 15 See in paricular Jorgenson and Griliches (1967) (1972) and Chrisensen and Jorgenson (1969). 16 Of course, he problem wih using ex ane user coss is ha here are many mehods ha could be used o predic asse inflaion raes and hese differen mehods could generae very differen user coss. For empirical evidence on his poin, see Harper, Bernd and Wood (1989), Diewer (2005) and Schreyer (2012b). 17 The prediced asse inflaion rae was se equal o he ex pos geomeric average inflaion rae for he asse over he pas 25 years. 18 While Jorgensonian user coss may no be he bes for produciviy measuremen purposes, hey are he righ user cos concep o use when calculaing secoral ex pos raes of reurn. Moreover, Diewer and Fox (2016) found ha even hough many Jorgensonian land user coss urned ou o be negaive, he resuling raes of Toal Facor Produciviy growh did no differ much from he corresponding TFP growh raes using ex ane or smoohed asse inflaion raes. 19 See also Diewer, Nakamura and Nakamura (2009). 20 Thus he user cos formula (4) should be modified o u (1+r )U = [(1+r ) (1 )(1+i ) + ]P = [r i + (1+i ) + ]P where is he period specific ax rae on one uni of he asse. This modified user cos formula assumes ha he specific ax (such as a propery ax in he case of a srucure or a land plo) is paid a he end of he accouning period.

8 8 reurn o he asse base can be absorbed ino he cos of capial, r, so ha r can be inerpreed as he before income ax gross reurn o he asse used by he producion uni. 21 The user coss (3) and (4) were derived under he assumpion ha here are marke prices for used asses ha can be used o value he asse a he end of he accouning period. However, for many unique asses ha do no rade in each accouning period (such as real esae, inellecual propery and mining asses and cerain ypes of arisic asse like a movie), here are no end of period asse prices ha are available. In hese cases, esimaes of he fuure discouned cash flows ha he asse migh generae have o be used in order o value he asse as i ages. I will usually be difficul o form hese esimaes. 22 There is sill a cerain amoun of conroversy on how exacly o measure capial services in he inernaional Sysem of Naional Accouns. 23 A paricular problem wih he Sysem of Naional Accouns 1993 and 2008 is ha capial services in he general governmen secor are o be measured by depreciaion only; i.e., here is no allowance for he opporuniy cos of capial in governmen secor user coss 24 whereas as we have seen above, marke secor user coss of capial include boh depreciaion and he opporuniy cos of capial ha is ied up in holding producive asses. This omission of impued ineres cos will lead o a subsanial underesimae of public secor coss (from an opporuniy cos perspecive) and hence economy wide GDP will also be underesimaed. 25 The Office of Naional Saisics (ONS) in he UK has made a subsanial effor o measure produciviy in he public secor and i recognized ha he SNA recommended reamen of capial inpu in he public secor is no appropriae for produciviy measuremen purposes. Thus he ONS reamen of capial services coss in he public secor for produciviy measuremen purposes is differen from is Sysem of Naional Accouns reamen, which follows he inernaional guidelines. This is unforunae because ideally, we would like he official GDP measure o coincide wih he GDP measure ha is used for produciviy measuremen purposes. The above maerial should aler he reader o he fac ha he measuremen of capial services inpu in he public secor is no a compleely sraighforward exercise. Thus i is 21 For maerial on he consrucion of user coss for more complex sysems of business income axaion, see Diewer (1992b) and Jorgenson (1996). 22 For references o he lieraure on valuing fixed inpus as hey age, see Diewer (2009), Cairns (2013) and Diewer and Fox (2016). See Diewer and Huang (2011) for a discussion on how o value inellecual propery producs. Finally, for references o he lieraure on decomposing propery values ino heir land and srucure componens, see de Haan and Diewer (2011), Diewer, de Haan and Hendriks (2015) and Diewer and Shimizu (2015a) for residenial propery decomposiions, Diewer and Shimizu (2017) for condominium propery decomposiions and Diewer and Shimizu (2015b) (2016) and Diewer, Fox and Shimizu (2016) for commercial propery decomposiions. 23 See Diewer (1980; ) and Schreyer (2001) (2009). 24 The propery ax componen of he use cos of public secor propery inpu is also omied from he SNA user cos reamen. 25 Some counries wan o make heir GDP as small as possible in order o minimize inernaional ransfer paymens ha are based on heir per capia GDP. Thus he reamen of capial services for he general governmen secor in he Sysem of Naional Accouns is a leas parially a poliical issue raher han a pure measuremen issue.

9 9 no ha simple o measure he nonmarke oupu produced by a public secor producion uni by is corresponding inpu measure due o he fac ha i is no compleely sraighforward o measure capial services in boh he privae and public secor. 4. Measuring Oupu Growh by Inpu Growh: Gross Oupu versus Value Added If i proves o be difficul or impossible o measure nonmarke oupu quaniies, hen as indicaed above, economic saisicians have generally measured he value of nonmarke oupus by he value of inpus used and implicily or explicily se he price of nonmarke oupu equal o he corresponding inpu price index. Akinson (2005; 12) describes he siuaion in he UK prior o 1998 as follows: In many counries, and in he Unied Kingdom from he early 1960 s o 1998, he oupu of he governmen secor has been measured by convenion as he value equal o he oal value of inpus; by exension he volume of oupu has been measured by he volume of inpus. This convenion regarding he volume of governmen oupu is referred o below as he (oupu = inpu) convenion, and is conrased wih direc measures of governmen oupu. The inpus aken ino accoun in recen years in he Unied Kingdom are he compensaion of employees, he procuremen coss of goods and services and a charge for he consumpion of fixed capial. In earlier years and in oher counries, including he Unied Saes, he inpus were limied o employmen. As was noed in he previous secion, he above convenions imply ha capial services inpu for governmen owned capial will generally be less han he corresponding capial services inpu if he capial services were rened or leased. In he owned case, he governmen user cos of capial consiss only of depreciaion bu in he leased case, he renal rae would cover he cos of depreciaion plus he opporuniy cos of he financial capial ied up in he capial inpu. Akinson (2005; 49) makes he following recommendaion on his issue (and we concur wih his recommendaion): We recommend ha he appropriae measure of capial inpu for producion and produciviy analysis is he flow of capial services of an asse ype. This involves adding o he capial consumpion an ineres charge, wih an agreed ineres rae, on he enire owned capial. In addiion o he above problem, here are some suble problems associaed wih measuring public secor oupus by heir real inpu uilizaion: How exacly should real inpu be measured; i.e., which index number formula should be used o aggregae inpus and does he choice of formula maer? Should he oupu aggregae for a public secor producion uni be measured by is real primary inpu or by is real gross inpu; i.e., by primary plus inermediae inpus? In order o address hese quesions, we will calculae alernaive inpu aggregaes for an arificial daa se. In Table 1 below, we lis he prices (w 1,w 2 ) and he corresponding quaniies (x 1,x 2 ) for wo primary inpus and he prices (p 1,p 2 ) and quaniies (z 1,z 2 ) for wo inermediae inpus for five periods, = 1,...,5. Table 1: Primary and Inermediae Inpu Prices and Quaniies

10 10 Period w 1 w 2 p 1 p 2 x 1 x 2 z 1 I can be seen ha he price of he firs primary inpu (labour) is slowly rending upwards while he price of he second primary inpu (capial services) is rending downwards a a faser rae. The quaniy of he firs primary inpu is slowly rending downwards while he quaniy of he second primary inpu is rending upward a a rapid rae. Thus normal subsiuion effecs are aking place over ime. The price of he firs inermediae inpu (general impors) is rending down while he quaniy rends up. Finally, he price of he second inermediae inpu (energy impors) is rising rapidly while he corresponding quaniy falls. The rends in hese prices and quaniies are fairly smooh. The mos commonly used index number formula ha are used o aggregae prices and quaniies are he Laspeyres, Paasche, Fisher and Törnqvis price indexes. Once he sequence of aggregae prices has been formed using hese formulae, he corresponding quaniy indexes are formed by dividing he period value for he aggregae by he corresponding price indexes. Le w and x denoe he price and quaniy vecors for primary inpus for = 1,...,5. Then he period fixed base Laspeyres, Paasche, Fisher and Törnqvis price indexes are defined as follows for = 1,...,5: 26 (5) P L P L (w 1,w,x 1,x ) w x 1 /w 1 x 1 ; 27 (6) P P P P (w 1,w,x 1,x ) w x /w 1 x ; (7) P F P F (w 1,w,x 1,x ) [P L (w 1,w,x 1,x )P F (w 1,w,x 1,x )] 1/2 ; (8) P T P T (w 1,w,x 1,x )] exp[ n=1 N (1/2)(s n 1 +s n )ln(w n /w n 1 )] where s n w n x n /w x is he nh primary inpu cos share in period. The four fixed base primary inpu price indexes defined by (5)-(8) are lised in columns 2-5 of Table 2 below using he primary inpu daa lised in Table 1 above. These period fixed base indexes are denoed by P LX, P PX, P FX and P TX. I can be seen ha P LX is always greaer han P PX for = 2,...,5 and he gap beween he fixed base Laspeyres and Paasche price indexes gradually becomes greaer as ime marches on. Since he Fisher index P FX is he geomeric mean of P LX and P PX, i lies beween hese wo equally plausible fixed baske z 2 26 For furher discussion on all of hese indexes and heir properies, see Fisher (1922) and Diewer (1978) (1992a). The US Bureau of Economic Analysis uses chained Fisher indexes o aggregae over inpus and oupus. The use of Törnqvis price and quaniy indexes in produciviy analysis can be raced back o Jorgenson and Griliches (1967) (1972). Jusificaions for he use of Törnqvis indexes based on he economic approach o index number heory can be found in Diewer (1976) (1980), Caves, Chrisensen and Diewer (1982a) (1982b), Diewer and Morrison (1986), Kohli (1990) and Inklaar and Diewer (2016). Jusificaions for he use of Fisher indexes based on he economic approach o index number heory can be found in Diewer (1992a) (2012; ). 27 Noaion: w 1 x 1 n=1 N w n 1 x n 1 denoes he inner produc of he vecors w 1 and x 1.

11 11 ype price indexes. Noe ha he fixed base Törnqvis price index, P TX, is quie close o is fixed base Fisher counerpar, P TX. 28 Table 2: Fixed Base and Chained Laspeyres, Paasche, Fisher and Törnqvis Primary Inpu Price Indexes Period P LX P PX P FX P TX An alernaive o he use of fixed base indexes is o use chained indexes. Consider how a chained Laspeyres price index, say P LX *, is formed. For periods 1 and 2, he chained indexes coincide wih heir fixed base counerpars; i.e., P LX 1* 1 and P LX 2* P LX 2 = P L (w 1,w 2,x 1,x 2 ). The period 3 chained Laspeyres price index is defined as he period 2 chained index level, P LX 2*, muliplied by P L (w 2,w 3,x 2,x 3 ), which is (one plus) he Laspeyres rae of change of inpu prices going from period 2 o period 3. In general, he chained Laspeyres price level in period +1 is equal o he corresponding Laspeyres price level in period imes he Laspeyres chain link going from period o +1; i.e., P LX +1* P LX * P L (w,w +1,x,x +1 ). The chained Paasche, Fisher and Törnqvis inpu price indexes, P PX *, P FX * and P TX * are formed in a similar fashion, excep ha he Paasche, Fisher and Törnqvis chain link formulae P P (w,w +1,x,x +1 ), P F (w,w +1,x,x +1 ) and P T (w,w +1,x,x +1 ) are used insead of he Laspeyres chain link formula P L (w,w +1,x,x +1 ). The chained Laspeyres, Paasche, Fisher and Törnqvis inpu price indexes, P LX *, P PX *, P FX * and P TX * are lised in he las four columns of Table 2. I can be seen ha he use of he chained indexes dramaically reduces he spread beween hese four ypes of indexes as compared o heir fixed base counerpars. In period 5, he percenage difference beween he Laspeyres and Paasche fixed base indexes is 4.8% bu he difference beween he chained Laspeyres and Paasche indexes is only 1.6%. In period 5, he percenage difference beween he superlaive Fisher and Törnqvis fixed base indexes is 0.047% bu he difference beween he chained Fisher and Törnqvis indexes is only 0.003%, which is negligible. Thus chaining has subsanially reduced he spread beween he four mos popular index number formula ha are used in empirical applicaions. This will generally happen if annual daa are used so ha rends in prices and quaniies are fairly smooh. 29 P LX * P PX * P FX * P TX * 28 Boh of hese indexes are superlaive indexes; i.e., hey are exac for flexible funcional forms for an underlying flexible funcional form for an economic aggregaor funcion as defined by Diewer (1976). Diewer (1978) showed ha hese wo funcional forms for an index number formula approximaed each oher o he accuracy of a second order Taylor series approximaion when he derivaives are evaluaed a a poin where he wo price vecors are equal and where he wo quaniy vecors are equal. 29 Diewer (1978) poined his ou many years ago. However, chaining does no always work well if he daa are available on a subannual basis: seasonal flucuaions and price bouncing behavior can creae a chain drif problem, which was poined ou by Szulc (1983). Ivancic, Diewer and Fox (2011) and de Haan and van der Grien (2011) suggesed adaping mulilaeral indexes (used o make cross secional

12 12 The period primary inpu implici quaniy indexes ha correspond o he inpu price indexes lised in Table 2 can be obained by dividing he period aggregae inpu value by he corresponding period price index. 30 The resuling implici quaniy indexes are lised in Table 3. Table 3: Fixed Base and Chained Implici Laspeyres, Paasche, Fisher and Törnqvis Primary Inpu Quaniy Indexes Period Q LX Q PX Q FX Q TX Q LX * Q PX * Q FX * Q TX * Exacly he same mehodology can be used o form eigh alernaive price indexes for inermediae inpus using he daa lised in Table 1. Denoe he Laspeyres, Paasche, Fisher and Törnqvis inermediae inpu price indexes for period by P LZ, P PZ, P FZ and P TZ and heir chained counerpars by P * LZ, P * PZ, P * FZ and P * TZ respecively. These indexes are lised in Table 4. Table 4: Fixed Base and Chained Laspeyres, Paasche, Fisher and Törnqvis Inermediae Inpu Price Indexes Period P LZ P PZ P FZ P TZ Viewing he enries in Table 4, i can be seen ha he fixed base Laspeyres and Paasche inermediae inpu price indexes differ by almos 5% in period 5 while he Fisher and Törnqvis fixed base indexes are virually he same. The chained Laspeyres and Paasche inermediae inpu price indexes differ by abou 1% in period 5 while he corresponding chained Fisher and Törnqvis indexes are exacly he same. Thus again, chaining reduces he spread beween indexes for his daa se. P LZ * P PZ * P FZ * P TZ * comparisons) o he ime series conex in order o deal wih he chain drif problem ha generally arises when monhly or weekly daa are aggregaed. See also de Haan and Krsinich (2014). 30 Thus he period fixed base implici quaniy indexes ha mach up wih he period fixed base Laspeyres and Paasche price indexes are Q LX w x /P LX and Q PX w x /P PX and so on. I should be noed ha our Q LX is normally called he Paasche quaniy index and our Q PX is normally called he Laspeyres quaniy index. Our noaion is simplified if we label he implici quaniy index ha maches up wih a paricular price index formula in he same way. Noe ha our Q LX is less han our Q PX for = 2,...,5. This is a ypical relaionship beween Laspeyres and Paasche price and quaniy indexes (bu i does no always hold).

13 13 Use he same mehodology o form eigh alernaive aggregae inpu price indexes for boh primary and inermediae inpus using he daa lised in Table 1. Denoe he Laspeyres, Paasche, Fisher and Törnqvis aggregae inpu price indexes for period by P LY, P PY, P FY and P TY and heir chained counerpars by P * LY, P * PY, P * * FY and P TY respecively. These indexes are lised in Table 5. Table 5: Fixed Base and Chained Implici Laspeyres, Paasche, Fisher and Törnqvis Aggregae Inpu Price Indexes Period P LY P PY P FY P TY Viewing he enries in Table 5, i can be seen ha he fixed base Laspeyres and Paasche aggregae inpu price indexes differ by approximaely 5% in period 5 while he Fisher and Törnqvis fixed base indexes differ by The chained Laspeyres and Paasche inermediae inpu price indexes differ by abou 2% in period 5 while he corresponding chained Fisher and Törnqvis indexes differ by Thus again, chaining reduces he spread beween indexes. Also, he superlaive indexes (wheher hey are fixed base or chained) are much closer o each oher han are he corresponding Laspeyres and Paasche indexes. 31 Once he aggregae inpu price indexes have been calculaed, corresponding aggregae implici inpu quaniy or volume measures can be calculaed by deflaing each period s oal inpu coss by he appropriae aggregae inpu price index. Thus he period fixed base implici aggregae inpu quaniy ha corresponds o fixed base Laspeyres price aggregaion, Q LY, is defined as follows for = 1,...,5: 32 (9) Q LY (w x + p z )/P LY. The oher seven aggregae implici inpu quaniy levels, Q PY, Q FY, Q TY, Q ly *, Q PY *, Q FY *, Q TY *, are defined in an analogous manner. The eigh aggregae implici inpu quaniy indexes are lised in Table 6. Noe ha he value of gross oupu produced during period, say V Y, is equal o he value of aggregae inpu used during period ; i.e., we have for = 1,...,5: P LY * P PY * P FY * P TY * 31 Diewer (1978) found he same resuls using Canadian annual naional accouns daa and hus he advocaed he use of chained superlaive indexes for naional accouns purposes. This is probably good advice if he daa are a an annual frequency bu here is he possibiliy of some chain drif if quarerly daa are used. 32 As menioned earlier, our implici Laspeyres quaniy index Q LY corresponds o wha is normally called a fixed base Paasche quaniy index. Cos weighed inpu quaniy indexes of he ype defined by (9) have been used widely in he UK in recen years when consrucing measures of nonmarke oupu quaniy growh; see Akinson (2005; 88).

14 14 (10) V Y w x + p z = P LY Q LY = P PY Q PY =... = P TY * Q TY * Thus for each period, he price of gross oupu imes he corresponding quaniy of gross oupu is equal o he nominal value of aggregae inpu for each of our eigh pairs of gross oupu price and quaniy indexes. Table 6: Fixed Base and Chained Laspeyres, Paasche, Fisher and Törnqvis Aggregae Implici Inpu Quaniy Indexes Period Q LY Q PY Q FY Q TY Q LY * Q PY * Q FY * Q TY * The differences beween he Laspeyres and Paasche fixed base and chained aggregae implici inpu indexes grow over ime and are fairly subsanial by period 5 bu he oher four superlaive indexes are reasonably close o each oher. Wih no acual oupu prices and quaniies available for our imaginary public secor producion uni, we se he aggregae oupu price and quaniy levels for period o equal he corresponding aggregae inpu price and quaniy levels ha are lised in Tables 5 and 6. Thus if we choose o use he fixed base Laspeyres formula o form aggregae inpu prices, he period oupu price is se equal o P LY lised in Table 5 and he period aggregae oupu quaniy is se equal o Q LY lised in Table 6. The period Gross Oupu Toal Facor Produciviy level using he fixed base Laspeyres price index formula is defined as TFP LY equal o period gross oupu using Laspeyres fixed base price indexes divided by he period aggregae inpu level using fixed base Laspeyres price indexes which leads o he resul TFP LY Q LY /Q LY = 1 for each. Using he oher seven mehods for aggregaing inpus similarly leads o gross oupu produciviy levels ha are idenically uniy; i.e., TFP PY Q PY /Q PY = 1, TFP FY Q FY /Q FY = 1,..., TFP * TY Q * TY /Q * TY = 1 for all. Insead of calculaing gross oupu TFP levels, i is also useful o consruc value added TFP levels. 33 Nominal value added for a producion uni for period is defined as he value of oupus produced during period less he value of inermediae inpus used by he producion uni during period, where an inermediae inpu is an inpu ha was produced by anoher domesic or foreign producion uni. Thus for our numerical example, period nominal value added V O is defined as follows for = 1,...,5: (11) V O V Y p z = (w x + p z ) p z = w x 33 Generally speaking, gross oupu TFP growh will be smaller han value added TFP growh; see Schreyer (2001) and Diewer (2015) for explanaions of his phenomenon.

15 15 where he second equaliy follows using (10). Thus period nominal value added for our public secor producion uni, V O, is equal o period value of primary inpus used in he uni, w x, for each period. Thus period nominal value added V O is unambiguously defined. Bu how exacly should he price of real value added and he corresponding quaniy or volume be defined? We will follow he mehodology ha was used in he Producer Price Index Manual 34 and simply apply normal index number heory o he componens of value added, reaing all prices and oupu quaniies as posiive numbers bu changing he sign of inermediae inpu quaniies from posiive o negaive. 35 Thus he fixed base Laspeyres, Paasche, Fisher and Törnqvis price indexes for period value added are defined as follows for = 1,...,5: (12) P LO [P LY Q 1 LY p z 1 ]/[P 1 LY Q 1 LY p 1 z 1 ] ; (13) P PO [P PY Q PY p z ]/[P 1 PY Q PY p 1 z ] ; (14) P FO [P LO P PO ] 1/2 ; (15) P TO exp[(½)(s 1 T1 +s T1 )ln(p TY /P 1 TY )+(½)(s 1 T2 +s T2 )ln(p 1 /p 1 1 )+(½)(s 1 T3 +s T3 )ln(p 2 /p 1 2 )] where s T1 V Y /V O, s T2 p 1 z 1 /V O and s T3 p 2 z 2 /V O for = 1,...,5. Noe ha s T1 > 0, s T2 < 0, s T3 < 0 bu hese value added shares sum up o one; i.e., s T1 + s T2 + s T3 = 1 for = 1,...,5. The value added price indexes defined by (12)-(15) are lised in Table 7 below. The fixed base price index formulae defined by (12)-(15) can be modified o provide he corresponding chain link price indexes and hese links can be chained ogeher o defined he corresponding Laspeyres, Paasche, Fisher and Törnqvis value added chained indexes P LO *, P PO *, P FO * and P TO *. These chained indexes are also lised in Table 7. Table 7: Fixed Base and Chained Laspeyres, Paasche, Fisher and Törnqvis Value Added Oupu Price Indexes Period P LO P PO P FO P TO As usual, he Laspeyres ype price indexes are higher (afer period 1) han he corresponding Paasche ype indexes and he spread beween he fixed base Laspeyres and P LO * P PO * P FO * P TO * 34 See he IMF/Eurosa/ILO/OECD/UNECE/The World Bank (2004). 35 Jusificaions for his procedure ha are based on he economic approach o index number heory can be found in Diewer (1976), Diewer and Morrison (1986) and Kohli (1990) who jusified he use of Törnqvis indexes using he economic approach o he measuremen of produciviy. Diewer (1992a) (2012) jusified he use of Fisher indexes using he economic approach. If he public secor producion uni sells some oupus a marke prices, hen hese sales should become a par of he uni s inermediae inpu subaggregae excep ha he posiive signs associaed wih he quaniies of such oupus should be replaced by negaive signs.

16 16 Paasche price indexes is larger han he spread beween heir chained counerpars. As usual, he superlaive indexes are generally close o each oher. Noe also ha he value added price indexes lised in Table 7 end up below uniy in period 5 whereas he gross oupu price indexes lised in Table 5 end up above uniy in period 5 (excep for he fixed base Paasche index). The period implici real value added oupu quaniies or volumes ha correspond o he four fixed base value added price indexes defined by (12)-(15) are defined by (16) and he corresponding chained indexes are defined by (17) for = 1,...,5: (16) Q LO V O /P LO ; Q PO V O /P PO ; Q PO V O /P PO ; Q PO V O /P PO ; (17) Q LO * V O /P LO * ; Q PO * V O /P PO * ; Q PO * V O /P PO * ; Q PO * V O /P PO *. The above implici quaniy indexes of real value added are lised in Table 8 below. The subsanial difference beween he fixed base implici Laspeyres and Paasche indexes of real value added in period 5 is noeworhy. Table 8: Fixed Base and Chained Implici Laspeyres, Paasche, Fisher and Törnqvis Indexes of Real Value Added Period Q LO Q PO Q FO Q TO Q LO * Q PO * Q FO * Q TO * The primary inpu quaniy indexes lised in Table 3 above along wih he real value added oupu indexes lised in Table 8 can be used o form TFP indexes by dividing period real value added by he corresponding measure of real inpu. Thus he period Value Added Toal Facor Produciviy level using he fixed base Laspeyres price index formula is defined as TFP LO Q LO /Q LX for each. Using he oher seven mehods for aggregaing inpus similarly leads o he following alernaive measures of real value added TFP for period : TFP PO Q PO /Q PX ; TFP FO Q FO /Q Fx ; TFP TO Q TO /Q TX ; TFP * LO Q * LO /Q LX ; TFP * PO Q * PO /Q PX ; TFP * FO Q * FO /Q Fx ; TFP * TO Q * TO /Q TX. These alernaive measures of value added produciviy are lised in Table Table 9: Fixed Base and Chained Laspeyres, Paasche, Fisher and Törnqvis Indexes of Toal Facor Produciviy Based on Real Value Added Period TFP LO TFP PO TFP FO TFP TO TFP LO * TFP PO * TFP FO * TFP TO * 36 Noe ha he TFP levels lised in Table 9 can also be generaed by dividing he inpu price indexes lised in Table 2 by he corresponding real value added oupu prices lised in Table 7. This resul was firs esablished by Jorgenson and Griliches (1967; 252) and is a consequence of he fac ha for our example, he value of inpus is always exacly equal o he value of oupus for each period.

17 Since here are no independen measures of oupu for our producion uni, i should be he case ha all TFP levels for our public secor producion uni should equal uniy. Looking a Table 9, i can be seen ha his resul holds whenever Laspeyres or Paasche indexes are used. For he Fisher and Törnqvis mehods of aggregaion, i can be seen ha his resul does no hold bu i does hold o a high degree of approximaion. 37 This has been a raher lenghy discussion on how o consruc price and quaniy esimaes for a public secor producion uni when informaion on he price and quaniy of he uni s oupus is no available. In he end, we have seen ha i does no really maer very much wheher we measure he produciviy of he uni on he basis of is gross oupu or on is value added oupu: in he firs case, he level of TFP is always equal o uniy and in he second case, i is eiher equal o uniy if Laspeyres or Paasche aggregaion is used or almos equal o uniy if superlaive index aggregaion is used. Thus i would seem ha he choice of index number formula is no very relevan for public secor producion unis ha do no have independen oupu measures. While his is rue as far as he calculaion of TFP is concerned, i is definiely relevan when measuring he size of he real value added (and gross oupu) of public secor producion unis. From Table 6 above, i can be seen ha he fixed base Laspeyres and Paasche esimaes of real gross oupu in period 5 were versus while from Table 8 above, he fixed base Laspeyres and Paasche esimaes of real value added in period 5 were versus These are subsanial differences: he choice of index number formula does maer. 5. Cos Based Mehods for Valuing Public Secor Oupus when Informaion on he Quaniies of Nonmarke Oupus is Available In his secion, we assume ha here is quaniy informaion on he nonmarke oupus produced by a public secor producion uni bu here are no corresponding marke prices o value he nonmarke oupus. The main public secors a he naional or sae level where his siuaion arises are he healh, educaion and social services secors. 38 A he municipal or local level, he same siuaion arises for provision of some services such as wase disposal, waer and sewage services. 39 The provision of road and highway services 37 The reason why TFP = 1 when Laspeyres or Paasche indexes are used as he mehod of aggregaion over commodiies is due o he fac ha hese wo formulae are consisen in aggregaion; i.e., if a Laspeyres aggregae is formed by using he Laspeyres formula o aggregae wo or more subaggregaes and hen he resuling subaggregae price and quaniies are aggregaed in a second sage using he Laspeyres formula again, hen he resuling wo sage Laspeyres indexes of price and quaniy are exacly equal o he corresponding Laspeyres indexes of price and quaniy ha are consruced using a single sage of aggregaion. On he oher hand, he wo superlaive indexes are no exacly consisen in aggregaion bu hey are approximaely consisen in aggregaion; see Diewer (1978) for an explanaion of hese resuls. 38 We will discuss in more deail he problems associaed wih measuring healh and educaion oupus in secion 8 below. The Akinson (2005) Repor on measuring governmen oupus in he conex of he naional accouns has a much more deailed discussion of he associaed measuremen problems. 39 Daa Envelopmen Analysis or benchmarking producion unis for relaive performance was iniiaed by Farrell (1957) and Charnes, Cooper and Rhodes (1978) and hese mehods have been widely applied o