A General Blueprint for InternationalProductivity Comparison

Bert M. Balk∗

Rotterdam School of ManagementErasmus University

E-mail bbalk@rsm.nland

Statistics NetherlandsThe Hague

Draft, November 24, 2010

Abstract

The measurement of interspatial productivity differences (for examplebetween countries), also called the comparison of productivity levels,is usually based on models that make use of strong assumptions suchas competitive behaviour and constant returns to scale. This paperdevelops a blueprint for interspatial comparative productivity mea-surement. It will be shown that all the usual, neoclassical assumptionscan be avoided.

The key concept is a production unit’s profitability. In the KLEMS-Y framework this is defined as (output) revenue divided by (input)cost. Given detailed prices and quantities, revenue as well as cost iscomputed as sum of prices times quantities. For the input cost of

∗ The views expressed in this paper are those of the author and do not necessarily reflectany policy of Statistics Netherlands. Previous versions of this paper have been presentedat a conference on International Comparisons of Prices, Income and Productivity, Oxford,8-9 April 2010, and at a conference on Productivity and Internationalization, The Hague,2-3 September 2010.

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owned capital assets a number of decisions must be faced: 1) whichinterest rate must be used in the unit user cost expression?; 2) mustthe unanticipated revaluations (as measured ex post) be retained aspart of the user cost?; 3) must the productive capital stock be adjustedby a utilization rate? For the labour input cost imputations must bemade for self-employed persons.

A natural way of comparing two production units (during the sametime period) is to consider their profitability ratio. In principle, anyvalue ratio can be decomposed into a price index and a quantity index.The quantity component of the profitability ratio is called the (primal)Total Factor Productivity (TFP) index. The paper explores what arethe implications when a less than optimal decomposition is used.

When more than two production units enter the comparison exer-cise the transitivity issue arises. Also the concept of productivity gapis introduced.

Keywords: Productivity; international comparison; index numbertheory; transitivity.

JEL classification: C43, O47.

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1 Introduction

Productivity is about the efficiency of turning inputs into outputs.1 There is avivid interest in the productivity development over time of production units,of whatever level of aggregation. There is an equally vivid interest in com-paring such units cross-sectionally. There are subtle differences between thetwo viewpoints. Barring mergers, split-ups and the like, a certain productionunit can be observed repeatedly through time. However, the same produc-tion unit does not exist simultaneously at two or more locations. Thus, incross-sectional comparisons production units must be declared comparable.This is the tacit assumption in all, national or international, benchmarkingstudies.

In the case of international comparisons of industries an internationalclassification system must ensure that industry I in country A is comparableindeed to industry I in country B. The big question then becomes: Is I in Amore or less efficient than I in B? And, having established that, what are thedrivers behind such a difference?

This paper is about measurement. In two previous papers — Balk (2009a),(2010) — I discussed productivity measurement in the time domain. I wentstep-by-step through the whole measurement process to see whether the usualneo-classical behavioural and structural assumptions, such as the existence ofa production function, constant returns to scale, and competitive behaviour,are really necessary. The answer appeared to be that it is possible to avoidall those assumptions. At the same time this exercise made clear which deci-sions an analyst must make on the road to delivering outcomes. The impactof those decisions on the outcomes was the subject of a further study byVancauteren et al. (2009).

The present paper furthers this undertaking by looking at internationalcomparisons.

Two excellent studies served as orientation points for carving out theroute. The first, Griffith et al. (2004), looks at the way R&D influences pro-ductivity growth in a panel of industries across twelve OECD countries overthe period 1971-1990. The theoretical framework here consists of a Hicks-neutral production function for real value added, RV A = AF (L, K), whereA indexes technological change, F (.) is a linearly homogeneous translog func-

1The second sentence of Solomon Fabricant’s introduction to Kendrick (1961) reads“Productivity ... is a measure of the efficiency with which resources are converted into thecommodities and services that men want.”

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tion, L and K index labour and capital input respectively, and the standardmarket-clearing conditions prevail. Under these assumptions changes overtime and differences across countries in A can be measured by ratios of out-put and input quantity index numbers. The key interest of Griffith et al. isin finding the determinants of those changes.

In the second study, Inklaar and Timmer (2007) compared productivitylevels of 26 industries across seven countries in the year 1997, following “thestandard methodology developed by Jorgenson and associates”. In particularthe Caves et al. (1982) methodology was used, the important elements beinga Hicks-neutral, constant-returns-to-scale translog production function andthe necessary producer equilibrium conditions.2 A variation on this themewas recently provided by Inklaar and Timmer (2009). For the same yearthey compared productivity levels of 24 industries across twenty countries.

The problem with the standard methodology is that, on the one hand,due to all the assumptions made, total factor productivity (TFP) change ordifference becomes synonymous with technological change or difference. Onthe other hand, TFP is measured residually as output quantity index overinput quantity index, and thus “includes a mix of both important policyrelevant information (...) as well as spurious information ...” (as voiced byMcMorrow et al., 2010, footnote 4)

This paper shows that for the definition and use of productivity indicesin a cross-sectional context no behavioural or structural assumptions areneeded. The discussion is framed in the KLEMS-Y model. The basic ac-counting relation of this model is explained in section 2. The capital inputcost deserves special attention because of a large number of measurementoptions. This is the subject of section 3. Section 4 then turns to the mea-surement of total factor productivity difference in the case of two countries.Section 5 considers what can be done when there are more than two countriesto compare and one wants transitive indices.

2 The basic KLEMS-Y model

Let us consider a certain industry, defined according to some internation-ally agreed system (such as SNA), in a certain country. For the purpose ofproductivity measurement, such an industry is considered as a consolidatedinput-output system. What does this mean?

2Hulten (2009) seems to consider this methodology as a sort of gold standard.

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For the output side as well as for the input side there is some list of com-modities, according to some internationally agreed classification scheme. Acommodity is thereby defined as a set of closely related items (goods or ser-vices) which, for the purpose of analysis, can be considered as “equivalent”,either in the static sense of their quantities being additive or in the dynamicsense of displaying equal relative price or quantity changes. Ideally, then,for any accounting period considered (ex post), say a year, each commoditycomes with a value (in monetary terms) and a price and/or a quantity. Ifvalue and price are available, then the quantity is obtained by dividing thevalue by the price. If value and quantity are available, then the price isobtained by dividing the value by the quantity. If both price and quantityare available, then value is defined as price times quantity. In any case, forevery commodity it must be so that value equals price times quantity, themagnitudes of which of course must pertain to the same accounting period.Technically speaking, the price concept used is the unit value, where the def-inition of the units is at the statistician’s discretion. At the output side, theprices must be those received by the industry, whereas at the input side, theprices must be those paid. Consolidation (sometimes also called ‘net-sectorapproach’ ) means that the industry does not deliver to itself. Put otherwise,all the intra-industry deliveries are netted out.

The inputs are customarily classified according to the KLEMS format.The letter K denotes the class of owned, reproducible capital assets.3 Thecommodities here are the asset-types, sub-classified by age category. Cohortsof assets are assumed to be available at the beginning of the accountingperiod and, in deteriorated form (due to ageing, wear and tear), still availableat the end of the period. Investment during the period adds entities tothese cohorts, while desinvestment, breakdown, or retirement remove entities.Examples include buildings and other structures, land, machinery, transportand ICT equipment, tools. As will be explained in the next section, thequantities sought are just the quantities of all these cohorts of assets (togetherrepresenting the productive capital stock), whereas the relevant prices aretheir unit user costs (per type-age combination), constructed from imputedinterest rates, depreciation profiles, (anticipated) revaluations, and tax rates.The sum of quantities times prices then provides the capital input cost Ct

K

of an industry.4

3Thus land and inventories are excluded, as well as intangibles.4The productive capital stock may be underutilized, which implies that not all the

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The letter L denotes the class of labour inputs; that is, all the types ofwork that are important to distinguish, cross-classified for instance accordingto educational attainment, gender, and experience (which is usually proxiedby age categories). Quantities are measured as hours worked (or paid), andprices are the corresponding wage rates per hour.5 Where applicable, im-putations must be made for the work executed by self-employed persons.The sum of quantities times prices provides the labour input cost Ct

L (or thelabour bill, or labour compensation, as it is sometimes called).6

The classes K and L concern so-called primary inputs. The letters E, M,and S denote three, disjunct classes of so-called intermediate inputs. First,E is the class of energy commodities consumed by an industry: oil, gas,electricity, and water. Second, M is the class of all the (physical) materialsconsumed in the production process, which could be sub-classified into rawmaterials, semi-fabricates, and auxiliary products. Third, S is the class ofall the business services which are consumed for maintaining the productionprocess. This includes the services of leased capital assets and outsourcedactivities. Though it is not at all a trivial task to define precisely all theintermediate inputs and to classify them, it can safely be assumed that atthe end of each accounting period there is a quantity and a price associatedwith each of those inputs. And intermediate inputs cost Ct

EMS is the sum ofall those quantities times prices.

Then, for each accounting period, production cost is defined as the sumof primary and intermediate input cost Ct ≡ Ct

K + CtL + Ct

EMS.At the output side, the letter Y denotes the class of commodities, goods

and/or services, produced by the industry. Though in some industries, suchas services industries or industries producing mainly unique goods, defini-tional problems are formidable, it can safely be assumed that at the end ofeach accounting period there are data on quantities produced. For industriesoperating on the market there are also prices. The sum of quantities timesprices then provides the production revenue Rt. Finally, profit (includingtaxes on production) Πt is defined as revenue minus cost: Πt ≡ Rt − Ct.Profit may be positive, negative, or zero.

There is, of course, discussion possible about what to include or exclude atthe input and output sides. We are here more or less tacitly assuming a broad

capital costs are incurred in actual production. We return to this issue later.5We are here following Kendrick (1961, 33).6The utilization rate of the labour input factors is assumed to be 1. Over- or under-

utilization from the point of view of jobs or persons is reflected in the wage rates.

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production viewpoint, where for instance marketing services are included inthe set S. A broader viewpoint would take into account sales and uses frominventories.

Profit is a wellknown and important financial performance measure. Analternative measure is ‘profitability’, defined as revenue divided by cost,Rt/Ct. Profitability gives, in monetary terms, the quantity of output perunit of input, and is thus a measure of return to aggregate input (and insome older literature called ‘return to the dollar’). Profitability is alwayspositive.

It is useful to remind the reader that the notions of profit and profitability,though conceptually rather clear, are difficult to operationalize. One of thereasons is that cost includes the cost of owned capital assets, the measurementof which exhibits a substantial number of degrees of freedom, as will bediscussed in the next section. Also, labour cost includes the cost of self-employed persons, for which wage rates and hours of work usually must beimputed. It will be clear that all these, and many other, uncertainties spillover to the operational definitions of the profit and profitability concepts.

The basic accounting model for a certain industry in a certain countrycan thus be expressed as

CtK + Ct

L + CtEMS + Πt = Rt, (1)

where t denotes the accounting period under consideration (say, a certainyear).

3 Capital input cost

Revenue, labour cost, and intermediate inputs cost concern the monetaryoutcomes of so-called flow variables, ex post measured. Capital input costneeds a different treatment, because capital is a stock variable. Basically,capital input cost is measured as the difference between the book values ofthe industry’s owned capital stock at beginning and end of the accountingperiod considered.

Our notation must reflect this. The beginning of period t is denoted byt−, and its end by t+. Thus a period is an interval of time t = [t−, t+], wheret− = (t−1)+ and t+ = (t+1)−. Occasionally, the variable t will also be usedto denote the midpoint of the period.

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All the assets are supposed to be economically born at midpoints of pe-riods, whether this has occurred inside or outside the industry under con-sideration. Thus the age of an asset of type i at (the midpoint of) period tis a non-negative integer number j = 0, ..., Ji. The age of this asset at thebeginning of the period is j − 0.5, and at the end j + 0.5. The economicallymaximal service life of asset type i is denoted by Ji.

The opening stock of capital assets is the inherited set of past investmentsand desinvestments; hence, the opening stock consists of cohorts of assets ofvarious types, each cohort comprising a number of assets of the same age.By convention, assets that are discarded (normally retired or prematurelyscrapped) or sold during a certain period t are supposed to be discarded orsold at the end of that period; that is, at t+. Second-hand assets that areacquired during period t from other industries are supposed to be acquiredat the beginning of the next period, (t+1)−. However, all other acquisitionsof second-hand assets and those of new assets are supposed to happen atthe midpoint of the period, and such assets are supposed to be immediatelyoperational.

Hence, all the assets that are part of the opening stock remain activethrough the entire period [t−, t+]. The period t investments are supposed tobe active through the second half of period t; that is, [t, t+]. Put otherwise,the stock of capital assets at t, the midpoint of the period, is the same asthe stock at t−, the beginning of the period, but 0.5 period older. At themidpoint of the period the investments, of various age, are added to thestock. Notice, however, that the closing stock at t+, the end of the period,is not necessarily identical to the opening stock at (t + 1)−, because of theconvention on sale, acquisition, and discard of assets.

Let Ktij denote the quantity (number) of asset type i (i = 1, ..., I) and

age j (j = 1, ..., Ji) at the midpoint of period t. These quantities are non-negative; a substantial number of them might be equal to 0. Further, let I t

ij

denote the (non-negative) quantity (number) of asset type i (i = 1, ..., I) andage j (j = 0, ..., Ji) that is added to the stock at the midpoint of period t; asubstantial number of these quantities might be equal to 0 too.

Total user cost over all asset types and ages, for period t, is then naturallydefined by

CtK ≡

I∑i=1

Ji∑j=1

utijK

tij +

I∑i=1

Ji∑j=0

vtijI

tij, (2)

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where utij is the unit user cost over period t of an opening stock asset of type

i that has age j at the midpoint of the period and vtij is the unit user cost

of an asset of type i and age j that is acquired at the midpoint of period t.The definition of unit user cost will be discussed in a moment.

For international comparisons formula (2) contains too much detail. Thusa number of simplifying assumptions are necessary. The first assumption isthat there are no transactions in second-hand assets. Then the number ofassets Kt

ij is equal to the number of new investments of j periods earlier,

I t−ji0 , adjusted for the probability of survival. Expression (2) then reduces to

CtK =

I∑i=1

Ji∑j=1

utijI

t−ji0 + vt

i0Iti0

. (3)

Second, it is assumed that new investments start their economic life at thebeginning of the first-next period. Then the last expression further reducesto

CtK =

I∑i=1

Ji∑j=1

utijI

t−ji0

, (4)

which is the classical formula for the value of capital services. This formulacan be rewritten as

CtK =

I∑i=1

uti1

Ji∑j=1

(utij/u

ti1)I

t−ji0

, (5)

provided that uti1 6= 0. If it is next assumed that, for each asset type, the

unit user cost ratios are independent of time, that is,

utij/u

ti1 = φij (i = 1, ..., I; j = 1, ..., Ji), (6)

then expression (5) reduces to

CtK =

I∑i=1

uti1

Ji∑j=1

φijIt−ji0

=I∑

i=1

uti1PCSt

i . (7)

Notice that φi1 = 1 (i = 1, ..., I). Moreover, common sense suggests that0 < φij ≤ 1 (i = 1, ..., I; j = 1, ..., Ji). For each asset i = 1, ..., I, thecoefficients φij serve to (linearly) transform assets of age 2 to Ji into assets of

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age 1. The set {φi1, ..., φiJi} is called the age-efficiency profile of asset type i.

The part between brackets in expression (7) is called the productive capitalstock of asset type i, measured in efficiency units, PCSt

i . Put otherwise,given the age-efficiency profile, the units of measurement become units ofage 1, and the only unit user cost that is needed for the computation of (7)is the user cost of a one year old asset.

It is good to recall here our convention on the measurement of age. Assetsof age 1 at the midpoint of period t were new at the midpoint of the previousperiod t−1, and they started their economic life at the beginning of period t;that is, at age 0.5. Hence, it is natural to value PCSt

i with the unit price (orvaluation) of an asset of type i and age 0.5 at t−, P t−

i,0.5. It is good to noticethat PCSt

i corresponds to the quantity of capital stock at the beginning ofperiod t as employed by Jorgenson and Nishimizu (1978).

Expression (7) is the point of departure for the aggregation of asset typesinto classes.

The ex post unit user cost over period t of an opening stock asset of typei that has age j at the midpoint of the period is defined as7

utij ≡ rtP t−

i,j−0.5 +(P t−

i,j−0.5 − P t+

i,j+0.5

)+ τ t

ij (j = 1, ..., Ji). (8)

There are three components here. We start with the second, P t−i,j−0.5−P t+

i,j+0.5.This is the value change of the asset between beginning and end of the ac-counting period. It is called (nominal) time-series depreciation, and combinesthe effect of the progress of time, from t− to t+, with the effect of ageing, fromj−0.5 to j+0.5. In general, the difference between the two prices (valuations)comprises the effect of exhaustion, deterioration, and obsolescence.

The third component, τ tij, denotes the specific tax(es) that must be paid

in relation to the use of an asset of type i and age j during period t.The first component, rtP t−

i,j−0.5, is the price (or valuation) of this asset atthe beginning of the period, when its age is j − 0.5, times an interest ratert. This component reflects the premium that must be paid to the ownerof the asset to prevent that it be sold, right at the beginning of the period,and the revenue used for immediate consumption; it is therefore also calledthe price of ‘waiting’. Another interpretation is to see this component asthe actual or imputed interest cost to finance the monetary capital that is

7In case of irreversible investments user cost must be replaced by user charge. It appearsthat the next formula still applies, with minor modifications. See Diewert, Lawrence andFallon (2009, Chapter 10).

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tied up in the asset; it is then called ‘opportunity cost’. Anyway, it is a sortof remuneration which, since there might be a risk component involved, isspecific for the industry.

The second component of expression (8) deserves closer attention. Thiscomponent can be decomposed as

P t−

i,j−0.5 − P t+

i,j+0.5 =(P t−

i,j−0.5 − E t−P t+

i,j+0.5

)+(E t−P t+

i,j+0.5 − P t+

i,j+0.5

), (9)

where E t− is an operator delivering the expected value at time t− of thevariable to which this operator is applied. In expression (9) the ex post time-series depreciation is decomposed in expected time-series depreciation anda remainder, called unexpected (or unanticipated) revaluation. Both partsneed to be operationalized. Ex post time-series depreciation is conventionallycalculated as

P t+

i,j+0.5

P t−i,j−0.5

=PPI t+

i

PPI t−i

(1− δi), (10)

where PPI ti denotes the Producer Price Index (or a kindred price index)

that is applicable to new assets of type i, and δi is the annual cross-sectiondepreciation rate that is applicable to an asset of type i and age j. Thisdepreciation rate comes from an empirically estimated geometric age-priceprofile. Expression (8) with (10) substituted corresponds to the formula forthe price of capital input as employed by Jorgenson and Nishimizu (1978).

Thus, time-series depreciation is modeled as a simple, multiplicative func-tion of two, independent factors. The first, PPI t+

i /PPI t−i , which is 1 plus

the annual rate of price change of new assets of type i, concerns the effectof the progress of time on the value of an asset of type i and age j. Thesecond, 1− δi > 0, concerns the effect of ageing by one year on the value ofan asset of type i and age j. Ageing by one year causes the value to declineby δi × 100 percent.

Anticipated time-series depreciation is measured as

E t−P t+

i,j+0.5

P t−i,j−0.5

= E t−(

PPI t+

i

PPI t−i

)(1− δi), (11)

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where the expected value of PPI t+

i /PPI t−i is usually based on past real-

izations. For example, a moving average of annual PPI changes. Then,combining expressions (10) and (11), unanticipated revaluation is measuredby

E t−P t+

i,j+0.5

P t−i,j−0.5

−P t+

i,j+0.5

P t−i,j−0.5

=

(E t−

(PPI t+

i

PPI t−i

)− PPI t+

i

PPI t−i

)(1− δi). (12)

Assembling all these bits and pieces it appears that expression (7) can bedecomposed as

CtK = Ct

K,w + CtK,e + Ct

K,u + CtK,tax, (13)

where the imputed interest cost is defined by

CtK,w ≡ rt

I∑i=1

P t−

i,0.5PCSti ; (14)

the cost of anticipated time-series depreciation is defined by

CtK,e ≡

I∑i=1

(1− E t−

(PPI t+

i

PPI t−i

)(1− δi)

)P t−

i,0.5PCSti ; (15)

the cost of unanticipated revaluation is defined by

CtK,u ≡

I∑i=1

(E t−

(PPI t+

i

PPI t−i

)− PPI t+

i

PPI t−i

)(1− δi)P

t−

i,0.5PCSti ; (16)

and the cost of tax is defined by

CtK,tax ≡

I∑i=1

τ ti1PCSt

i . (17)

Before proceeding to the issue of productivity comparison, a number of im-portant decisions must be faced.

First, we are interested in the efficiency of production. Put otherwise,we must relate outputs actually produced to inputs actually used. For allthe flow variables this is automatically accounted for via quantity measure-ment. For the productive capital stock this takes the form of a correction onthe quantities; that is, only the fraction of PCSt

i (i = 1, ..., I) that is actu-ally used in production should enter expression (7). However, such detailedinformation is usually not available.

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Let θtK (0 < θt

K ≤ 1) denote the fraction of the productive capital stock,averaged over all the types, that is actually used during period t. Assumingthat unit user costs are the same for used and unused assets, it appears thatexpression (1) can be written as

θtKCt

K + (1− θtK)Ct

K + CtL + Ct

EMS + Πt = Rt, (18)

where θtKCt

K and (1− θtK)Ct

K are the user costs of the used and unused partsof the capital stock respectively. The cost of unused capital must then beadded to profit.8

Second, it can be argued that unanticipated revaluations are not relatedto the ‘normal’ operations of the industry and must also be added to profit.Combining these two decisions, and using expression (13), expression (18)can be written as

θtK

(Ct

K,w + CtK,e + Ct

K,tax

)+ Ct

L + CtEMS + Π∗t = Rt, (19)

where Π∗t ≡ Πt + (1− θtK)Ct

K + θtKCt

K,u.Third, the imputed interest cost Ct

K,w depends on the interest rate rt.This can be a suitable exogenous magnitude. Or it can be the so-calledendogenous (or ‘internal’) rate, which is obtained by setting profit Π∗t = 0and then solving equation (19) for the remaining free variable rt. In a sensethe endogenous interest rate absorbs not only profit Πt, but also the cost ofunused capital (1 − θt

K)CtK and all the unanticipated revaluations θt

KCtK,u.

Expression (19) can be abbreviated as

C∗t + Π∗t = Rt, (20)

where C∗t ≡ θtK

(Ct

K,w + CtK,e + Ct

K,tax

)+ Ct

L + CtEMS. Notice that, if the

utilization rate θtK = 1 and unanticipated revaluations are retained in the

user cost of capital then equation (20) reduces to Ct +Πt = Rt. Moreover, ifan endogenous interest rate is applied then this equation further reduces toCt = Rt.

8Kendrick (1961, 32) argues that θtK should be set equal to 1 because “... capital is

counted as a cost when owned and thus available.” This was disputed in the same volume(page 226) by S. H. Ruttenberg, an NBER director appointed by the American Federationof Labor and Congress of Industrial Organizations, on the base that this would create aconceptual difference with labour input as measured by actual hours worked. Kendrick(1973, 26) repeats the earlier standpoint.

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Interesting empirical evidence on the relative importance of in- or exclud-ing unanticipated revaluations and the choice between an exogenous or anendogenous interest rate was provided, in a time-series context, by Inklaar(2010). On the impact of the utilization rate see Coremberg (2008).

4 Comparing two countries

4.1 General definitions

Consider now a certain industry in two countries c and c′. The two accountingequations read, conditional on utilization rate, treatment of unanticipatedrevaluations, and interest rate,

Cct + Πct = Rct (21)

Cc′t + Πc′t = Rc′t, (22)

where profits Πct and Πc′t may or may not be equal to 0. The ratio of thetwo profitabilities can be rewritten as

Rct/Cct

Rc′t/Cc′t=

Rct/Rc′t

Cct/Cc′t. (23)

Notice that, whereas each cost or revenue variable is expressed in a country’sown currency, expression (23) delivers a dimensionless magnitude.

Now, as we have seen in the previous two sections, revenue as well asthe various cost categories have the form of a sum of quantities times prices.Hence, for decomposing the revenue and cost ratios occurring in expression(23) into price and quantity components one can just look into the statisti-cian’s toolbox for good price and quantity indices. The basic requirement isthat these indices satisfy the Product Test, meaning that price index timesquantity index equals the value ratio. Then equation (23) transforms into

Rct/Cct

Rc′t/Cc′t=

PR(ct, c′t)

PC(ct, c′t)

QR(ct, c′t)

QC(ct, c′t), (24)

where PR(ct, c′t) is a price index and QR(ct, c′t) is a quantity index, bothacting on the price and quantity variables of the two situations for all thecommodities which are distinguished at the output (revenue) side. Simi-larly, PC(ct, c′t) and QC(ct, c′t) are price and quantity indices for the input

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(cost) side. Cross-sectional price indices such as PR(ct, c′t) and PC(ct, c′t)are conventionally called purchasing power parities. Notice, however, thatin cross-sectional comparions, such as between countries, there need not bedifferent currencies involved.

On the assumption that all the commodities are accounted for, the (grossoutput based) (primal) Total Factor Productivity (TFP) index for country crelative to country c′ is defined as

IPROD(ct, c′t) ≡ QR(ct, c′t)

QC(ct, c′t); (25)

thus, as the (gross revenue based) output quantity index divided by the (totalcost based) input quantity index.

The (gross output based) dual TFP index for country c relative to countryc′ is defined as

DIPROD(ct, c′t) ≡ PC(ct, c′t)

PR(ct, c′t); (26)

thus, as the (total cost based) input price index divided by the (total rev-enue based) output price index. Equation (24) tells us that dual and primalTFP index are the same if and only if the profitabilities of the two coun-trie are the same, Rct/Cct = Rc′t/Cc′t. When endogenous interest rates areemployed, then this condition is satisfied and, hence, primal and dual TFPindex coincide. One then says that the dual TFP index shows how the marketdistributes a productivity gain: by higher remunerations for the productionfactors and/or cheaper products.

4.2 Approximations

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4.3 Alternative models

The TFP index defined by expression (25) corresponds basically to the indexused by Inklaar and Timmer (2007). They did not adjust for utilization rates.All the unanticipated revaluations were retained in the user cost. Endogenousinterest rates (per country) were used.

Instead of the KLEMS-Y model one can opt for the KL-VA model. Thismeans that expressions (21) and (22) are replaced by

CctK + Cct

L + Πct = V Act (27)

Cc′tK + Cc′t

L + Πc′t = V Ac′t, (28)

where value added V At ≡ Rt − CtEMS. The TFP index then has the same

form as (25), but the gross revenue based output quantity index is replacedby a value added based output quantity index and the total cost based inputquantity index is replaced by a capital-plus-labour cost based input quantityindex. This basically corresponds to the approach of Griffith et al. (2004).For all countries they employed endogenous interest rates (which result fromsetting nominal value added equal to nominal capital-plus-labour cost), andadjusted capital input for utilization. Whether unanticipated revaluationswere retained in the user cost of capital or not cannot be decided from thetext of the paper.

Inklaar and Timmer (2009) also worked with the KL-VA model. As intheir previous study they did not adjust for utilization rates. The differenceis that for the treatment of capital they turned to the “hybrid approach”proposed by Oulton (2007). This approach can as well be explained in theframework of the KLEMS-Y model.

Return to expression (25). The quantity index occurring in the denom-inator, QC(ct, c′t), can be seen, and is in practice usually calculated, as atwo-stage index9 of a quantity index of capital input and a quantity index ofthe remaining LEMS inputs; that is, formally, there exists a function Q∗(.)such that

QC(ct, c′t) = (29)

Q∗(QK(ct, c′t), QLEMS(ct, c′t); CctK , Cct

LEMS, Cc′tK , Cc′t

LEMS),

9For a brief survey of the theory of two-stage indices the reader is referred to Balk(2009a, Appendix A).

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where CctLEMS ≡ Cct

L + CctEMS, Cct

K + CctLEMS = Cct, and the two quantity

indices are supposed to satisfy the Product Test; that is, there exist priceindices PK(ct, c′t) and PLEMS(ct, c′t) such that

PK(ct, c′t)QK(ct, c′t) = CctK/Cc′t

K (30)

PLEMS(ct, c′t)QLEMS(ct, c′t) = CctLEMS/Cc′t

LEMS. (31)

Thus, expression (29) says that the (total cost based) input quantity indexQC(ct, c′t) is a function of the quantity index of capital input QK(ct, c′t)and the quantity index of all the other inputs QLEMS(ct, c′t), whereby thesetwo subindices are somehow weighted by country c values Cct

K , CctLEMS and

country c′ values Cc′tK , Cc′t

LEMS.The first step of the “hybrid approach” now consists in calculating the

quantity index QK(ct, c′t) subject to the condition that in the unit user costsan exogenous interest rate is used and unanticipated revaluations are ex-cluded. The calculation of the quantity index QLEMS(ct, c′t) proceeds asconventional. In the second step these two indices are aggregated by meansof

QhC(ct, c′t) ≡ (32)

Q∗(QK(ct, c′t), QLEMS(ct, c′t); CctK + Πct, Cct

LEMS, Cc′tK + Πc′t, Cc′t

LEMS).

There is a subtle difference between the right-hand sides of expression (32)and the former expression (29): the weight of the capital input quantity indexis increased or decreased from (Cct

K , Cc′tK ) to (Cct

K +Πct, Cc′tK +Πc′t) (dependent

on whether profits are positive or negative). Notice that CctK + Πct = V Act −

CctL , which is called cash flow (or ‘residual income’).

The “hybrid approach” can now be interpreted as follows. In addition tothe labour and intermediate inputs categories there is a category which couldbe called “capital plus entrepreneurship”. Its cost in countries c, c′ amountto Cct

K + Πct, Cc′tK + Πc′t, respectively. The quantity component of this cost

difference is given by the index QK(ct, c′t), and the price component by theremainder; that is,

CctK + Πct

Cc′tK + Πc′t

/QK(ct, c′t). (33)

Whether this is an adequate view on the production process remains to beseen.

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5 Comparing more countries

5.1 Getting transitive indices

Given a set C of countries we end up with a matrix of TFP index numbersfor any country relative to any other country. If the output and input quan-tity indices satisfy the Time Reversal Test, then the TFP indices too. ThenIPROD(ct, c′t) = 1/IPROD(c′t, ct). But this still leaves us with a triangu-lar matrix of TFP index numbers which are not transitive; that is, in generalIPROD(ct, c′t)IPROD(c′t, c′′t) will be unequal to IPROD(ct, c′′t).

The GEKS procedure10 can then be used to obtain transitive TFP indicesof the form

IPRODGEKS(ct, c′t) ≡∏

c′′∈C(IPROD(ct, c′′t)IPROD(c′′t, c′t))

1/card(C),

(34)where card(C) denotes the number of countries in C. These indices are tran-sitive; that is,

IPRODGEKS(ct, c′t)IPRODGEKS(c′t, c′′t) = IPRODGEKS(ct, c′′t), (35)

as can be checked easily. It is important to note that IPRODGEKS(ct, c′t)depends on the data of all the countries involved. Extending the set C there-fore implies recalculation of all the index numbers.

Notice that when Tornqvist quantity indices are employed for the bilateralTFP index numbers than the GEKS index numbers reduce to the indexnumbers developed by Caves et al. (1982).

5.2 The frontier concept

According to Griffith et al. (2004) the frontier is “the country with thehighest level of total factor productivity.” To obtain a precise definition ofthis concept it is useful to consider first the (imaginary) case of single-inputsingle-output production units. Let yct and xct denote the (positive) outputquantity and input quantity respectively of country c in period t. The pro-ductivity of c in period t is then given by the ratio yct/xct. The highest pro-ductivity level during this period is obviously given by max{yc′t/xc′t|c′ ∈ C}.

10See Balk (2008, Section 7.3.1) or Balk (2009b, 63-64) for details.

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This defines the period t frontier. The distance of c to the frontier is thendefined by the ratio of c’s productivity level and the level that correspondsto the frontier; that is, [yct/xct]/ max{yc′t/xc′t|c′ ∈ C}. It is straightforwardto check that this can be rewritten as min{[yct/yc′t]/[xct/xc′t]|c′ ∈ C}. No-tice that [yct/yc′t]/[xct/xc′t] is an output quantity index divided by an inputquantity index.

It is by now obvious how to generalize the distance-to-the-frontier con-cept. The distance of country c at period t to the period t frontier is definedby

PRODGAP (ct) ≡ min{IPROD(ct, c′t)|c′ ∈ C}. (36)

This is a radial measure of the distance of country c to the best performingcountry, both at period t.

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References

[1] Balk, B. M., 2008, Price and Quantity Index Numbers: Models for Mea-suring Aggregate Change and Difference (Cambridge University Press,New York).

[2] Balk, B. M., 2009a, Measuring Productivity Change Without NeoclassicalAssumptions, Discussion Paper 09023 (Statistics Netherlands, The Hague,www.cbs.nl).

[3] Balk, B. M., 2009b, “Aggregation methods in international comparisons:An evaluation”, in Purchasing Power Parities of Currencies; Recent Ad-vances in Methods and Applications, edited by D. S. Prasada Rao (EdwardElgar, Cheltenham, UK, Northampton, MA, USA).

[4] Balk, B. M., 2010, “An Assumption-free Framework for Measuring Pro-ductivity Change”, The Review of Income and Wealth 56, Special Issue 1,S224-256.

[5] Caves, D. W., L. R. Christensen and W. E. Diewert, 1982, “MultilateralComparisons of Output, Input and Productivity Using Superlative IndexNumbers”, The Economic Journal 92, 73-86.

[6] Coremberg, A., 2008, “The Measurement of TFP in Argentina, 1990-2004: A Case of the Tyranny of Numbers, Economic Cycles and Method-ology”, International Productivity Monitor 17, 52-74.

[7] Diewert, W. E., D. Lawrence and J. Fallon, 2009, The Theory of Net-work Regulation in the Presence of Sunk Costs, Technical Report preparedfor Commerce Commission (Economic Insights, Hawker, ACT 2614, Aus-tralia).

[8] Griffith, R., S. Redding and J. van Reenen, 2004, “Mapping the TwoFaces of R&D: Productivity Growth in a Panel of OECD Industries”, TheReview of Economics and Statistics 86, 883-895.

[9] Hulten, C. R., 2009, Growth Accounting, Working Paper 15341 (NationalBureau of Economic Research, Cambridge, MA).

[10] Inklaar, R., 2010, “The Sensitivity of Capital Services Measurement:Measure All Assets and the Cost of Capital”, The Review of Income andWealth 56, 389-412.

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[11] Inklaar, R. and M. P. Timmer, 2007, “International Comparisons ofIndustry Output, Inputs and Productivity Levels: Methodology and NewResults”, Economic Systems Research 19, 343-363.

[12] Inklaar, R. and M. P. Timmer, 2009, “Productivity Convergence AcrossIndustries and Countries: The Importance of Theory-Based Measure-ment”, Macroeconomic Dynamics 13, Supplement No.2, 218-240.

[13] Jorgenson, D. W. and M. Nishimizu, 1978, “U. S. and Japanese Eco-nomic Growth, 1952-1974: An International Comparison”, The EconomicJournal 88, 707-726.

[14] Kendrick, J. W., assisted by M. R. Pech, 1961, Productivity Trends inthe United States, National Bureau of Economic Research, General Series,Number 71 (Princeton University Press, Princeton).

[15] Kendrick, J. W., assisted by M. R. Pech, 1973, Postwar ProductivityTrends in the United States, 1948-1969, National Bureau of EconomicResearch, General Series, Number 98 (Columbia University Press, NewYork and London).

[16] McMorrow, K., W. Roger and A. Turrini, 2010, “Determinants of TFPGrowth: A Close Look at Industries Driving the EU-US TFP Gap”, Struc-tural Change and Economic Dynamics 21, 165-180.

[17] Oulton, N., 2007, “Ex Post versus Ex Ante Measures of the User Costof Capital”, The Review of Income and Wealth 53, 295-317.

[18] Vancauteren, M., D. van den Bergen, E. Veldhuizen and B. M. Balk,2009, Measures of Productivity Change: Which Outcome Do You Want?,Invited Paper for the 57th Session of the International Statistical Institute,16-22 August 2009, Durban, South Africa. Also presented at the EconomicMeasurement Group Workshop, University of New South Wales, Sydney,9-11 December 2009.

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