Measuring Productivity using the Index Number Approach: An Introduction

McLellan, Nathan Working Paper Measuring Productivity using the Index Number Approach: An Introduction New Zealand Treasury Working Paper, No. 04/05 Provided in Cooperation with: The Treasury, New Zealand Government Suggested Citation: McLellan, Nathan (2004) : Measuring Productivity using the Index Number Approach: An Introduction, New Zealand Treasury Working Paper, No. 04/05, New Zealand Government, The Treasury, Wellington This Version is available at: https://hdl.handle.net/10419/205545 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Measuring Productivity using the Index Number Approach: An Introduction Nathan McLellan N EW Z EALAND T REASURY W ORKING P APER 04/05 J UNE 2004 Treasury:529027v4 NZ TREASURY WORKING PAPER 04/05 Measuring Productivity using the Index Number Approach: An Introduction MONTH / YEAR June 2004 AUTHOR Nathan McLellan New Zealand Treasury PO Box 3724 Wellington 6015 NEW ZEALAND Email Telephone Fax [email protected] +64-4-471-5130 +64-4-473-1151 ACKNOWLEDGEMENTS I would like to thank Tony Booth, John Creedy, Kevin Fox, David Law, Peter Mawson, John Morris and Grant Scobie for comments on an earlier draft of this paper. NZ TREASURY New Zealand Treasury PO Box 3724 Wellington 6015 NEW ZEALAND Email Telephone Website [email protected] 64-4-472 2733 www.treasury.govt.nz DISCLAIMER The views, opinions, findings, and conclusions or recommendations expressed in this Working Paper are strictly those of the author(s). They do not necessarily reflect the views of the New Zealand Treasury. The New Zealand Treasury takes no responsibility for any errors or omissions in, or for the correctness of, the information contained in this Working Paper. The paper is presented not as policy, but to inform and stimulate wider debate. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION i Abstract This paper provides an introduction to productivity measurement using index number techniques. Attention is given to the construction of productivity series using common index number formulae, the economic and axiomatic approaches to selecting an index number formula, and the use of chaining. Special attention is also given to measuring physical capital inputs and quality adjusted labour inputs. Numerical examples are used throughout the paper to illustrate the analysis. JEL CLASSIFICATION C43 – Index numbers and aggregation D24 – Production; capital and total factor productivity; capacity E23 – Production KEYWORDS Productivity measurement; index numbers; capital, quality-adjusted labour inputs WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION ii Table of Contents Abstract ...............................................................................................................................i Table of Contents ..............................................................................................................ii List of Tables......................................................................................................................ii List of Figures....................................................................................................................ii 1 Introduction ..............................................................................................................1 2 Index number approach...........................................................................................2 2.1 Productivity index............................................................................................................2 2.2 Index number formulae...................................................................................................3 2.3 Selecting an index number formula: The economic and axiomatic (test) approaches ....................................................................................................................4 2.4 Numerical example using different index number formulae ...........................................6 3 Chained indices........................................................................................................8 4 Physical capital stock and the user cost of capital.............................................10 4.1 Productive capital stock................................................................................................10 4.2 Numerical example of the productive capital stock ......................................................11 4.3 Age-efficiency and age-price schedules, economic depreciation, and the user cost of capital ...............................................................................................................12 4.4 Numerical example of the net capital stock and economic depreciation......................17 5 Measuring quality-adjusted labour input .............................................................17 6 Conclusion..............................................................................................................20 References .......................................................................................................................21 List of Tables Table 1 – Index axioms .......................................................................................................................6 Table 2 – Prices and quantities of outputs and inputs ........................................................................6 Table 3 – Laspeyres, Paasche, Fisher and Törnqvist output quantity index ......................................7 Table 4 – Input and productivity indices..............................................................................................8 Table 5 – Chaining using Laspeyres output indices ...........................................................................9 Table 6 – Productive capital stock ....................................................................................................11 Table 7 – Net capital stock and depreciation ....................................................................................17 List of Figures Figure 1 – Age-efficiency and age-price schedules..........................................................................15 Figure 2 – Age-efficiency schedule, age price schedule and economic depreciation ......................16 WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 1 Measuring Productivity using the Index Number Approach: An Introduction 1 Introduction Productivity measures are required in several areas of economic analysis. These range from exchange rate determination to examining the (proximate) sources of economic growth. A variety of methods are available to the productivity analyst in calculating productivity estimates, with the choice of method partly dependent on the objectives of the analysis. This paper provides an introduction to measuring productivity using the index number method. Consideration is given to this approach owing to the use of index numbers in constructing economic aggregates (such as GDP and the Consumers’ Price Index) and because index number techniques are used by statistical agencies that publish official productivity measures. Readers interested in alternative approaches to productivity measurement should see Mawson, Carlaw and McLellan (2003). There are two main approaches to choosing an index number formula: the economic and axiomatic approaches. The former approach bases the choice of index formula on a producer’s underlying production technology, and therefore has theoretical microeconomic underpinnings. The axiomatic approach bases the choice of index formula on desirable properties that indexes should exhibit. Once the index formula is chosen, consideration then needs to be given as to whether the productivity index should be chained to reduce substitution bias associated with fixed weight indexes. Good measures of outputs and inputs are needed in forming reliable productivity measures. This paper gives special attention to measuring physical capital and qualityadjusted labour inputs, as measuring these inputs present particular challenges for productivity analysts.1 Measuring physical capital inputs requires the construction of productive capital stocks from data on past investments and the formation of rental prices for different asset types. This is achieved using an integrated framework in which the loss in the productive capacity of an asset is linked to economic depreciation and its rental price (or user cost of capital). The measurement of quality-adjusted labour inputs requires rich information on labour market earnings and worker characteristics. 1 This is not to say the measurement of other inputs and outputs is not difficult. For a discussion of problems in measuring outputs and the implications for productivity measurement see Baily and Gordon (1988) and Diewert and Fox (1999). WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 2 The remainder of the paper is organised as follows. Section 2 discusses the index number approach to measuring productivity and choice of index number formula. Included in this section are numerical examples, based on hypothetical price and quantity data, illustrating the construction of productivity indices using various index formulae. Section 3 outlines the rationale and procedure for chaining fixed weight indices, illustrating this procedure with a simple numerical example. Measurement of capital inputs using an integrated framework that links the loss in productive capacity of an asset with economic depreciation and its rental price is discussed in Section 4. The measurement of qualityadjusted labour inputs is canvassed in Section 5. Section 6 provides a brief summary. 2 Index number approach This section presents an introduction to measuring productivity using the index number method. Subsection 2.1 discusses various productivity measures and subsection 2.2 presents several index number formulae often used in constructing productivity indices. The economic and axiomatic approaches to choosing an index number formula are discussed in subsection 2.3. Finally, subsection 2.4 presents numerical examples using the index formulae from subsection 2.2 to illustrate the construction of productivity indices and the differences between different index formulations. Readers interested in a more detailed review of the index number approach to productivity measurement should consult Diewert and Nakumara (2004). 2.1 Productivity index Productivity measures attempt to capture the ability of inputs to produce output (usually over time). In general a productivity index is defined as the ratio of an output quantity index to an input quantity index, that is: ; t t t Q A X = (1) for 0,...,tT= and where t A is a productivity index, t Q is an output quantity index and t X is an input quantity index. Each index represents accumulated growth from period 0 to period t. When t X comprises a single input, for example labour or physical capital, t A is a partial productivity index. The two well known partial productivity measures are labour and capital productivity. A limitation of partial productivity measures is that changes in productivity may reflect the impact of omitted inputs. For example, increases in labour productivity may be due to increases in the available amount of physical capital (one of the omitted inputs in the measurement of labour productivity) per worker, rather than increases in the underlying productivity of labour. When t X comprises two or more inputs, t A is a multifactor productivity index.2 Most often multifactor productivity is formed using labour and physical capital, although some 2 Multifactor productivity and total factor productivity are often used synonymously. However, strictly total factor productivity is measured by dividing an output quantity index by an input quantity index that is constructed using all inputs in the production process. Rarely is this the case, hence the preference for the name multifactor productivity. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 3 productivity studies have included additional variables such as land and inventories (see, for example, Diewert and Lawrence, 1999). Productivity indices are usually constructed using disaggregate prices and quantities of outputs and inputs. Because outputs and inputs are heterogenous it is simply not possible to add all outputs to form an output quantity index or, likewise, to add all inputs to form an input quantity index. Disaggregate data on the volumes of outputs and inputs need to be weighted to form output and input quantity indices. Output and input prices, or nominal output and input shares, are typically used as representative weights when forming output and input quantity indices. 2.2 Index number formulae When constructing productivity indices it is not immediately apparent which weighting procedure should be used to weight output and input quantities when forming output quantity and input quantity indices and on what basis the weighting structure should be chosen. There are numerous index formulae that can be used to construct output and input indices. The Laspeyres, Paasche, Fisher and Törnqvist indexes are some of the more widely used index formulae. Suppose information on the price and quantity of M outputs is available for period 0,...,tT=. Denoting the output price and quantity vectors in period t as 1 ( ,..., ) M t p pp≡ and 1 ( ,..., ) M t qq q≡, the Laspeyres output quantity index ( L t Q) is defined as follows: 0 1 00 1 0 10 M mm t Lm tM mm m m M mt m m p q Q p q q wq = = = = = ∑ ∑ ∑ (2) where 1 mm mtt tM mm tt m p q w p q = =∑ is output m’s nominal output share. Note that equation (2) shows the Laspeyres output quantity index is the period 0 share-weighted sum of quantity ratios. The Paasche output quantity ( P t Q) index is defined as follows: 1 0 1 1 1 10 M mm tt Pm tM mm t m m M mt tm m pq Q pq q wq = = − − = =    =    ∑ ∑ ∑ (3) The Paasche output quantity index uses period t prices as the weights, in contrast to the Laspeyres output quantity index that uses period 0 prices as weights. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 4 The Fisher output quantity index ( F t Q) is found by taking the geometric average of the Laspeyres and Paasche output quantity indexes, that is: 1 2 () FLP ttt QQQ= (4) Finally, the Törnqvist output quantity ( T t Q) index is defined as follows: () 0 1 2 10 mm t ww m M Tt tm m q Qq + =  =  ∏ (5) Input quantity indexes are defined in a similar manner using input prices ( t c) and input quantities ( t x ). 2.3 Selecting an index number formula: The economic and axiomatic (test) approaches As discussed in the previous subsection, there are numerous index formulae that can be used to form aggregate productivity measures. This raises the question: are there any criteria that can be used to decide on the choice of index formula? The index number literature offers two main approaches: the economic approach and the axiomatic (test) approach.3 The economic approach bases the choice of index number formula on a producer’s underlying production technology (that is, the production, cost, revenue or profit function). This approach assumes competitive optimising behaviour by producers. In other words producers are assumed to maximise profit (minimise costs) for a given production technology.4 Consider the following production technology: (, , ) tttt QfAKH= (6) where t A is the level of multifactor productivity, t K aggregate physical capital services, and t H the aggregate labour input (in this case the total number of hours worked). An index expressed in terms of the above technology is known as a theoretic index. In continuous time the theoretic index is a Divisia index.5 As data are available in discrete time, rather than as continuous functions of time, it is necessary to use an index formula to approximate the Divisia index.6 A particular index is defined as an exact index when it corresponds directly to the theoretic index derived from the production technology (Diewert, 1976). For example, if production 3 A third approach, the stochastic approach, is less widely used. For a critical review of this approach see Diewert (1995). 4 Although, recent research by Diewert and Fox (2004) shows that an index of multifactor productivity can be derived from the economic approach without the need to assume competitive optimising behaviour. 5 Each of the index formulae in equations (2) to (5) use discrete data to measure quantity changes between period 0 and period t. The Divisia index is formed assuming data exists for all periods between period 0 and period t. 6 The Törnqvist index is often described as a Divisia index. It is actually a discrete approximation to the Divisia index. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 11 usually normalised so that 01 φ =. The age-efficiency schedule shows the decline in the productive capacity of an asset over its economic life.11 Three commonly used age-efficiency patterns are the linear, ‘one-hoss-shay’ and geometric age-efficiency schedules. The linear age-efficiency schedule assumes that the productive capacity of an asset depreciates linearly over the asset’s economic life. The ‘one-hoss-shay’ efficiency pattern assumes the productive capacity of an asset remains constant over its economic life but then falls to zero when the asset’s economic life ends.12 The geometric age-efficiency pattern assumes the productive capacity of an asset declines at a constant rate. Given real investment data, the functional form of the age-efficiency schedule, and assuming past vintages of a particular asset can be aggregated, the productive capital stock for a particular asset type in period t is calculated as follows: 0 S tsts s KI φ − = =∑ (10) Equation (10) is known as the perpetual inventory model of the productive capital stock.13 4.2 Numerical example of the productive capital stock Table 6 presents hypothetical data on the (new) price of purchasing a capital asset ( t p ) and the nominal investment in the asset ( t V). The volume of investment in the asset (t I ) is found by dividing the nominal investment series by the price series (that is, dividing column (2) by column (1)). The age-efficiency schedule, which is displayed in column (4), assumes the life of the asset is five years and the loss in productive capacity is linear (that is, the age-efficiency schedule ( t φ ) is linear). Table 6 – Productive capital stock (1) t p (2) t V (3) t I (4) s φ (5) 0 I (6) 1 I (7) 2 I (8) 3 I (9) 4 I (10) 5 I (11) t K ,0ts= 1.0 100.0 100.0 1.00 100.0 100.0 ,1ts= 1.1 120.0 109.1 0.80 80.0 109.1 189.1 ,2ts= 1.3 150.0 115.4 0.60 60.0 87.3 115.4 262.7 ,3ts= 1.5 160.0 106.7 0.40 40.0 65.5 92.3 106.7 304.4 ,4ts= 1.6 180.0 112.5 0.20 20.0 43.6 69.2 85.3 112.5 330.7 ,5ts= 1.7 195.0 114.7 0.00 0.0 21.8 46.2 64.0 90.0 114.7 336.7 To calculate the productive capital stock for the asset it is first necessary to account for the diminished productivity capacity of each investment over time. Columns (5) to (10) 11 This exposition ignores survival probabilities for capital assets to simplify the analysis. If survival probabilities are introduced into the analysis, the age-efficiency schedule represents the loss in the productive capacity of a capital asset conditional on survival. 12 The ‘one-hoss-shay’ efficiency schedule is also known as the ‘light bulb’ efficiency pattern because a light bulb delivers a constant flow of capital services before its life ends. 13 Equation (10) is often augmented to include the initial capital stock in period t-s. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 12 present the productive capacity of the asset for each investment. These series are calculated by multiplying the initial investment by the relevant value from the ageefficiency schedule. For example, to find the productive capacity of an investment in the asset in 3t= that was made in 0t=, the initial investment ( 0 I ) of 100 is multiplied by 0.4, the value of the age-efficiency schedule for a three period old asset ( 3 φ ). Likewise, the productive capacity of an investment in the asset in 4t= that was made in 2t= (69.2) is found by multiplying 115.4 ( 2 I ) by 0.6, the value of the age-efficiency schedule for a two period old asset ( 2 φ ). Finally, the productive capital stock of the asset is calculated using equation (10) by adding the efficiency adjusted investments for each period. The resulting productive capital stock for the asset is reported in column (11). 4.3 Age-efficiency and age-price schedules, economic depreciation, and the user cost of capital When forming an aggregate quantity index, prices are used to weight the different quantities. Hall (1968) showed that the rental price of capital (user cost of capital) is the relevant price when aggregating different types of capital For some assets the rental price for different asset vintages is observable because there is an active rental market (for example, residential and non-residential buildings). However, for other assets, rental markets do not exist or are very thin (for example, rental markets may not exist for certain types of specialised machinery). In this situation firms purchase capital assets and pay an implicit rental for their use. Although the user cost of capital is not directly observable, it can be imputed using information on the price of a new asset, the rate of economic depreciation, and asset price inflation. The assumption of perfect competition implies the price of a s-vintage asset in period t , () ts Pis equal to the discounted stream of future rentals , () ts U, that is: , ,1 0(1 ) V tvsv ts v v U Pr ++ + = =+ ∑ (11) where ris the discount rate and Vthe time the asset’s life ends. Noting that 1, 1 2, 2 , 1, 1 21 ... 1(1) (1) ts t s tVsV ts V UU U Prr r ++ ++ + + ++ + =+++ ++ + , equation (11) can be rewritten as: ,1,1 ,(1 ) (1 ) ts t s ts UP Prr ++ =+ ++ (12) If 0s= equation (12) states that the price of a new asset at the beginning of period t is equal to the discounted value of the rental for period t plus the discounted price of the one period old asset at the beginning of 1t+. When measuring the user cost of capital it is common to incorporate asset price inflation. When asset price inflation is incorporated the price of a s-vintage asset in period 1t+ is equal to the price of the s-vintage asset in period t multiplied by one plus that rate of asset price inflation: WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 13 1, , (1 ) ts ts t PP π +=+ (13) where t π is the asset price rate of inflation. Substituting the similar expression for 1, 1ts P++ from equation (13) into equation (12) and solving for the user cost of capital yields: ,, ,1 ,,,11,1,1 (1 ) (1 ) ()( ) ts ts ts t ts ts ts t s ts UP rP Pr P P P P π + ++++ =+− + =+− − − (14) Equation (14) is the basic user cost formula. The first term of equation (14) is the finance cost associated with purchasing the asset (or the opportunity cost from investing the funds used to purchase the asset elsewhere). The second term is the loss in the value of the asset due to ageing and represents economic depreciation. The final term represents the capital gains or losses associated with owning the asset. Equation (14) is often expressed in rate form as follows: ,, , ((1)) ts ts t t ts UPr d ππ =−++ (15) where ,ts dis the depreciation rate from an s-vintage asset in period t.14 In equation (15) the price of a new asset ( ,0t P) and ex post asset inflation ( t π ) are observable. The discount rate ( r) can be obtained from financial markets or an ex post internal rate of return can be computed using the approach suggested by Jorgenson and Griliches (1967). The latter approach involves equating capital income with the product of the user cost of capital and the productive capital stock and then solving for the discount rate. To calculate the economic depreciation rate ( ,ts d) information is used on the ageefficiency schedule that is used to calculate the productive capital stock. This recognises the fact that: One cannot select an efficiency pattern independently of the depreciation pattern and maintain the assumption of competitive equilibrium at the same time. And, one cannot arbitrarily select a depreciation pattern independently from the observed pattern of vintage asset prices t s P (suggesting a strategy for measuring depreciation and efficiency). Hulten 1990:129 To calculate economic depreciation using the age-efficiency schedule it is first necessary to calculate the age-price schedule from the age-efficiency schedule. The age-price schedule ( s θ ) gives the relative value of a s-vintage asset to the value of a new asset. Economic depreciation for a particular asset is then calculated by tracing the loss in value of an investment which is derived by multiplying the initial investment by the relevant value from the age-price schedule. The age-price schedule is usually normalised so that 01 θ =. 14 The basic user cost of capital formula shown in equation (15) can be augmented to include additional information such as the impact of taxation on the user cost of capital. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 14 The link between the age-efficiency schedule and age-price profile can be seen as follows. In a competitive market the ratio of the s-vintage rental price to the 1s+-vintage rental price asset is equal to the relative efficiency of the s-vintage asset to the 1s+- vintage asset: ,1 1 , ts s ts U U φ + + = (16) Substituting equation (16) into equation (11) yields: ,11, , ,21 , 1 0 ... (1 ) (1 ) (1 ) (1 ) s ts s t s s V t Vs ts V V sv tvs v v UU U Prr r U r φφ φ φ ++ + + + ++ + = =+ ++ ++ + =+ ∑ (17) Hence the vintage asset price sequence is a function of the age-efficiency profile. Assuming the user cost of capital grows at a constant nominal rate (that is, ,, (1 )v tvs ts UUg +=+ for 0,...,vV=), then equation (17) can be rewritten as: , ,1 0 (1 ) (1 ) v V sv ts ts v v Ug Pr φ + + = + =+ ∑ (18) Consider a new asset purchased at period t for ,0t P and its subsequent sequence of vintage prices. The rental price in period 0 is found by solving equation (17) for ,0t U (which is possible since ,0t P is observed). The subsequent sequence of vintage prices is also calculated by using equation (17) and the assumption that the rental price grows at a constant rate. Values of the age-price schedule are found by taking the price of the new asset to the price of the s-vintage asset: , ,0 ts s t P P θ = (19) where ,0t P is the price of the new asset in period t. Figure 1 shows the age-efficiency and age-price schedules for the linear, ‘one-hoss-shay’, and geometric efficiency patterns. The age-price schedules are calculated from the ageefficiency schedules assuming rental prices grow at a constant 2% and that the discount rate is 6%. When using the linear and ‘one-hoss-shay’ age-efficiency schedules the economic life of the asset is assumed to be five periods. Figure 1 illustrates the linear age-efficiency schedule produces a non-linear age price schedule and the ‘one-hoss- shay’ age-efficiency schedule produces a linear age-price profile. The geometric ageefficiency schedule produces an identical age-price schedule. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 15 Figure 1 – Age-efficiency and age-price schedules Linear age-efficiency schedule 0.0 0.2 0.4 0.6 0.8 1.0 1.2 012345 time age-efficiency/age-price schedule Age-efficiency schedule Age-price schedule 'One-hoss-shay' age-efficiency schedule 0.0 0.2 0.4 0.6 0.8 1.0 1.2 012345 time age-efficiency/age-price schedule Age-efficiency schedule Age-price schedule Geometric age-efficiency schedule 0.6 0.7 0.8 0.9 1.0 1.1 012345 time age-efficiency/age-price schedule Age-efficiency schedule Age-price schedule Once the age-price schedule is obtained from the age-efficiency schedule, economic depreciation ( t D) can then be derived as follows: 11 1 () S tstssts s DII θθ −−−− = =− ∑ (20) WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 16 Equation (20) shows that economic depreciation is calculated by summing the loss in value of the s-vintage investments between two consecutive periods. The real net capital stock, which is the logical base for the economic depreciation rate, is calculated by summing the value for each investment after adjusting for economic depreciation (using the age-price schedule): 0 S tsts s NI θ − = =∑ (21) The economic depreciation rate in period t is then calculated by taking the ratio of economic depreciation to the real net capital stock, that is: t t t D dN = (22) To summarise, Figure 2 shows the link between the age-efficiency schedule, the age price schedule and economic depreciation. It also serves to highlight the point that the loss in productive capacity of an asset cannot be determined independently of the depreciation pattern. Figure 2 also shows the relationship between productive capital stocks and real net capital stocks. Figure 2 – Age-efficiency schedule, age price schedule and economic depreciation Age-efficiency schedule Investment Productive capital stock Vintage prices Age-price schedule Depreciation and real net capital stock WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 17 4.4 Numerical example of the net capital stock and economic depreciation Table 7 presents information to construct the economic depreciation rate for the asset (t d), which is needed to construct the user cost of capital ( t U). Based on the linear ageefficiency schedule presented in column (4) of Table 6, column (1) of Table 7 displays the corresponding age-price schedule. The age-price schedule is calculated assuming a discount rate of 6% and that the rental price grows at a constant 2%. To calculate economic depreciation for the asset ( t D) it is first necessary to trace the loss in the value of each investment in the asset over time. This is presented in columns (2) to (7) of Table 7. Economic depreciation ( t D) is calculated by summing the loss in value of each investment between two consecutive periods. For example depreciation in 2t= is equal to the loss in value of the initial investment ( 0 I ) between 1t= and 2t= (67.5 – 41 ≈ 26.5) plus the loss in value of the investment made in 1t= (109.1 – 73.6 ≈ 35.5), which equals 61.9. Table 7 – Net capital stock and depreciation (1) s θ (2) 0 I (3) 1 I (4) 2 I (5) 3 I (6) 4 I (7) 5 I (8) t D (9) t N (10) t d ,0ts= 1.00 100.0 100.0 ,1ts= 0.67 67.5 109.1 32.5 176.6 0.2 ,2ts= 0.41 41.0 73.6 115.4 61.9 230.0 0.3 ,3ts= 0.21 20.8 44.7 77.9 112.5 86.6 255.9 0.3 ,4ts= 0.07 7.0 22.7 47.3 75.9 112.5 103.0 265.4 0.4 ,5ts= 0.00 0.0 7.6 24.0 46.1 75.9 114.7 111.7 268.4 0.4 The base used to calculate the depreciation rate is the net capital stock. This is calculated by adding the age-price weighted investment for each period and is reported in column (9) of Table 7. The economic depreciation rate ( t d) is displayed in column (10) of Table 7. 5 Measuring quality-adjusted labour input Multifactor productivity is most often measured using labour and physical capital inputs. The number of hours worked rather than the number of people employed or the number of hours paid is usually the preferred labour input when measuring productivity. This is because the number of people employed does not capture changes in the number of hours worked by each worker nor changes in the composition of part-time versus full-time workers, while the number of hours paid may not accurately capture the number of hours actually worked by salaried workers. If the number of hours worked is used as the labour input when constructing productivity measures, differences in the human capital associated with each hour worked are not accounted for. Essentially the hours worked by different types of workers are treated as if they were all identical, with differences in the human capital, or the quality of workers WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 18 subsumed within the productivity measure. For example, the difference in the human capital embodied in the hours worked by a heart surgeon and a school teacher will be ascribed to the productivity measure. Moreover, changes in the human capital of workers owing to further education or greater work experience will be captured by changes in productivity over time. Productivity analysts are often interested in gauging the contribution to changes in aggregate output from changes in the human capital or the quality of labour inputs. This requires an adjustment for differences in the quality of hours worked by different types of workers. This is done by separately accounting for different types of labour inputs when forming productivity measures.15 In section 2, when discussing the economic approach to choosing an index number formula, the aggregate production function was denoted as follows: (, , ) tttt QfAKH= (23) where t H denoted the aggregate number of hours worked, and was calculated by summing over hours worked at the sub-aggregate level (for example industries). It was also outlined that when the production function was given the translog functional form, the continuous time (Divisia) index could be approximated using the Törnqvist index formula. An alternative specification to the production function presented in equation (23) is the following: 11 ( , ,..., , ,..., ) M N tttttt QgBk kh h= (24) In this specification each of the capital inputs ( 1,..., M tt kk) and each of the labour inputs (1,..., N tt hh) are accounted for separately. t B denotes the alternative measure of multifactor productivity (which is interpreted below). As discussed in section 3, when the productive capital stocks of various asset types are used, the aggregate capital stock measure is formed using the corresponding user cost of capital measures as weights in the index formula. Likewise, when different types of labour inputs are used, income shares for the different types of labour inputs are used as weights in the index formula. The difference between the multifactor productivity measure ( t A) corresponding to the underlying production function in equation (23) and the multifactor productivity measure (t B ) corresponding to equation (24) is the latter measure accounts for changes in the composition or quality of labour inputs. This can be seen from the analysis that follows. Consider the case where the Törnqvist index is used to measure multifactor productivity. Assuming that the aggregate capital stock has been formed using rental prices for different asset types, the multifactor productivity indices can be written as follows: 15 Related to the measurement of changes in the quality or composition of labour inputs is measuring human capital stocks for inclusion in (comprehensive) national wealth accounts. Constructing human capital stocks using the lifetime-income method has recently been revived by Jorgenson and Fraumeni (1989 and 1992), although this approach has a long ancestry in economics dating back to work by Petty (1690). The lifetime-income approach to measuring human capital stocks has similarities with the method used to measure the productive capital stocks discussed in section 4, where the discounted stream of future rentals is used to value an asset at a point in time. The lifetime-income income approach values an individual’s human capital using the discounted stream of future labour market income. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 19 00 11 ()() 22 KK LL tt t tww ww tt Q A KH ++ = (25) 00 11 ()() 22 KK LL tt t tww ww tt Q B KL ++ = (26) where K t w is capital’s income share and L t w is labours income share, 1 0 1 N n t n tN n n h H h = = = ∑ ∑ the total number of hours worked, and nn 0 1() 2 10 t ww n N t tn n h Lh + =  =  ∏ a Törnqvist index of aggregate labour input. Furthermore, the labour quality index ( t LC ) can be written as: t t t L LC H = (27) This index is the ratio of the aggregate labour input index to an index of total hours worked. This labour quality index is akin to that adopted in work by Jorgenson, Gallop and Fraumeni (1987) and Jorgenson and Fraumeni (1989, 1992).16 Finally, substituting equations (25), (26) and (27) yields the following index for multifactor productivity: 0 1() 2 L L t ww ttt BALC + = (28) Equation (28) shows the alternative multifactor productivity index ( t B ) is simply the original multifactor productivity ( t A) adjusted for the quality composition of the labour input. In forming the alternative multifactor productivity measure ( t B ) it is necessary to have estimates of labour income shares for the various labour types. Labour shares for the various labour types can be estimated in one of two ways. One approach is to classify the labour inputs into different categories based on the characteristics of various workers and then use the average wage in forming labour shares for the various labour inputs. For example, workers could be classified into various categories based on their level of educational qualification. This approach has been adopted in work by Jorgenson, Gallop and Fraumeni (1987). An alternative approach is to estimate wage equations econometrically using worker characteristics, such as the number of years worked, as explanatory variables and then use the predicted values from the wage equations to form weights for the various types of labour inputs. This approach has been used by the Bureau of Labour Statistics (1993) 16 In Jorgenson and Griliches (1967) the interpretation given to their version of equation (27) was that owing to errors in aggregation of labour services. More recently Jorgenson, Gallop and Fraumeni (1987) and Jorgenson and Fraumeni (1989, 1992) have interpreted this as a labour quality index. WP 04/05 MEASURING PRODUCTIVITY USING THE INDEX NUMBER APPROACH: AN INTRODUCTION 20 when forming their multifactor productivity estimates that accounts for changes in the composition of labour over time (Bureau of Labour Statistics, 1993). 6 Conclusion This paper provides an introduction to productivity measurement using the index number approach. Consideration was given to this approach, rather than alternative approaches to productivity measurement, owing to the widespread use of index techniques in constructing economic aggregates and because the index number approach is used by statistical agencies in constructing official productivity measures. Attention was given to common index number formulae, whose application were illustrated using simple numerical examples, and approaches to choosing an index number formula. Special attention was also given to measuring physical capital inputs using an integrated framework that links productive capital stocks, economic depreciation, and rental prices; and measuring quality adjusted labour inputs. In regard to the latter, the alternative measure of multifactor productivity that incorporated quality-adjusted labour inputs was shown to be the original multifactor productivity measure adjusted for the composition of the aggregate labour input.