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01 02 Chapter 2 03 04 PRODUCTIVITY AND EFFICIENCY 05 06 MEASUREMENT USING 07 08 PARAMETRIC ECONOMETRIC 09
10 METHODS
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13 Subal C. Kumbhakar
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18 19 Abstract 20 21 In this chapter we survey the econometric approaches to productivity and efficiency 22 measurement. Both primal and dual approaches are considered. More specifically, we
23 examine the production function and distance functions (multiple outputs), cost functions
24 (with single and multiple outputs), standard and alternative profit functions (with single and multiple outputs). Possible extensions of the traditional productivity and efficiency 25 measures in the light of non-traditional inputs are also discussed. 26
27 Keywords and Phrases: Partial and total factor productivity; technical change; returns 28 to scale; technical efficiency; technical efficiency change; production function; distance 29 function; cost function; profit function; alternative profit function; mixing models. 30 31 JEL Classification No.: C23, C33, D24, O30 32 33 “Productivity cannot be measured directly. Instead, it must be measured indi- 34 rectly as a relationship between physical outputs and inputs that can be 35 assembled.” 36 (John W. Kendrick 1984, p.9) 37 38 “To measure the productivity change of a firm or an industry, we first have to 39 define what we mean by a productivity change.”
40 (W. Erwin Diewert 1992, p. 163)
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01 1. INTRODUCTION1 02 03 The most commonly used definition of productivity is the amount of physical 04 output produced by one unit of a given factor of production at a stated period 05 of time. Thus, productivity indicators ordinarily relate output to a single factor 06 of production, creating measures such as labor productivity, capital productivity, 07 etc. These are also defined as partial factor productivity. In contrast, multifactor 08 (total factor) productivity measures output per unit of a set of combined factors 09 of production (such as labor capital, land, etc.) and gives a single overall measure 10 of productivity. These partial and multifactor productivity measures are based on 11 quantities of inputs and outputs and are called primal measures of productivity. 12 Instead of using input and output quantities, productivity can also be defined 13 using cost, profit and price information. For example, average cost can be used to 14 measure productivity. Note that these measures of productivity (average product 15 and average cost) can be measured directly from the data. 16 Productivity (no matter how it is defined) is likely to change over time. In simple 17 terms, productivity change/growth means more output is produced with a given 18 levelofinputs,whichoccurswhentechnologyimprovesovertime.Sinceinputsalso 19 change over time, productivity change is often defined as output growth net of input 20 growth rate. Accordingly, productivity change can take place without technical 21 change. In a single output single input case, productivity change is simply the rate 22 of change in average product. We will give a formal definition of productivity 23 change with single output and multiple inputs technology. This will be followed by 24 multiple outputs multiple inputs technology. In addition to measuring productivity 25 change, we also consider the sources of productivity growth. 26 Productivity and change in productivity can be measured using different tech- 27 niques. Diewert (1992) showed that productivity change can be calculated using 28 an index number approach (Fisher (1922) or Törnqvist (1936) productivity index. 29 Both indices require quantity and price information, as well as assumptions con- 30 cerning the structure of technology and the behavior of producers, but neither 31 requires estimation of anything econometrically. Productivity change can also 32 be calculated using the Divisia index, which is nonparametric. Finally, it can be 33 estimated using econometric techniques. 34 A disadvantage of index number techniques and the Divisia index is that they 35 do not provide sources of productivity growth, whereas nonparametric and econo- 36 metric techniques do. Although nonparametric and econometric techniques are 37 capable of measuring productivity change and its sources, only the econometric 38
39 40 1 Parts of this section are drawn from Chapter 8 of Kumbhakar and Lovell (2000). Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 23
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01 approach is capable of doing so in a stochastic environment. Here we show how 02 to use econometric techniques to estimate the magnitude of productivity change, 03 and then to decompose estimated productivity change into its various sources. 04 Productivity is affected in the presence of inefficiency. This is likely to affect 05 productivity growth as well, unless inefficiency is time-invariant. Traditional 06 econometric models of productivity change ignored the contribution of effi- 07 ciency change to productivity change. Productivity change was decomposed into 08 technical change and scale economies. However, if inefficiency exists, then effi- 09 ciency change provides an independent contribution to productivity change. If 10 efficiency change is omitted from the analysis, its omission leads to an erroneous 11 allocation of productivity change to its sources. Accordingly, it is desirable to 12 incorporate the possibility of efficiency into econometric models of productivity 13 and productivity change. 14 To get a flavor of the issues involved, we begin with a quantity-based (primal) 15 approach for the estimation and decomposition of productivity change. In doing 16 so we allow for the possibility that production plans can be technically inefficient. 17 The general structure of the primal approach is illustrated in Figure 2.1, in which 18 a single input is used to produce a single output. Assume that for a producer t t 19 at time t the production plan is given by the input-output combination x y 20 and the production frontier (the maximum possible output function given input 21 quantities) is f x t . Productivity for this input-output combination is defined t t 22 by the ratio of output to input, viz., y /x , which can be easily measured from t t 23 input and output data. Note that y < f x t , which means that the produc- t t 24 tion plan is technically inefficient. We define technical efficiency as y /f x t , 25 which is at most unity. Since the production frontier, f x t , is not known, 26
27 y f (x, t+1) 28 t+1 29 y f (x, t) 30
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37 t t+1 x x x 38 39 Fig. 2.1. The Primal Approach to the Estimation and Decomposition of 40 Productivity Change Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 24
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01 the measurement of technical efficiency requires estimation of the frontier. Fur- 02 thermore, productivity is reduced in the presence of technical inefficiency, i.e., t t t 03 y /x < f x t /xt. Alternatively, the more efficient a firm is, the higher is its 04 productivity, ceteris paribus. If we assume that the producer expands the pro- t t t+1 t+1 05 duction plan from x y to x y , and technical progress has occurred
06 between periods t and t +1, this would imply that f x t +1 > f x t . Ignoring
07 noise, it is clear that production is technically inefficient in both periods, since t t t+1 t+1 + 08 y < f x t and y < f x t 1 , and that technical efficiency has improved + t t t+1 t+1 + 09 from period t to period t 1, since y /f x t < y /f x t 1 . It is also t+1 t+1 t t 10 clear that productivity growth has occurred, since y /x > y/x . The
11 estimated rate of productivity growth can then be decomposed into returns to
12 scale, technical change and change in technical efficiency.
13 It is clear from the above discussion that in a single-output single-input case, productivity at a point in time is measured by yt/xt, which is nothing more 14 than the average product of x at time t. Productivity change (treating time as a 15 continuous variable) is measured by the rate of change in the average product of 16 x, i.e., ln yt/xt / t =˙yt −˙xt, where the dot over a variable indicates its rate of 17 change. Thus both productivity and productivity change can be measured from 18 observed data without estimating anything. If inefficiency is present, then the 19 productivity of an input is lower than its maximum possible value. The degree 20 to which productivity is lowered depends on the degree of inefficiency, which 21 has to be estimated. Similarly, the effect of inefficiency on productivity change 22 will depend on the temporal behavior of inefficiency. 23 If there are multiple inputs, one can obtain the partial factor productivity of 24 each input j, defined as y/xj and the partial factor productivity change defined 25 =˙−˙ as ln y/xj / t y xj. Observing that both the partial factor productivity and 26 its change (with or without technical inefficiency) depend on the usage of other 27 factors, which are likely to differ among inputs, a single measure of productivity 28 (i.e., total factor productivity, TFP) and productivity change is needed. In prac- 29 tice, labor productivity is often used as the measure of productivity, although its 30 level depends on the amount of other factors being employed, such as capital, 31 energy, and materials used, in the production process. 32 The objective of this paper is to survey some issues related to productivity 33 measurement and the decomposition of TFP change in the context of a panel 34 data framework. The focus is on the parametric econometric models based on 35 production, distance, cost, and profit functions. TFP change is decomposed into 36 technical change, scale economies, and technical efficiency components. Several 37 alternative strategies in modeling technical change and production technology 38 (in both primal and dual contexts) are considered. 39 We begin by using a production frontier approach to obtain estimates of 40 productivity change, and to decompose estimated productivity change into a Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 25
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01 technical change, a returns to scale, and a component associated with change 02 in technical efficiency. We also consider the factor augmenting approach to 03 technical change, which allows the researcher to decompose technical change into 04 contributions due to each individual input. To allow for the possibility of multiple 05 technologies and to measure the technology gap across regions, countries, etc., 06 we discuss mixing (latent class) models and estimation of a best practice (meta) 07 frontier. Furthermore, we use the input and output distance functions as an 08 alternative to the production function approach. We repeat this investigation 09 using a dual cost function approach, which has the advantage of accommodating 10 multiple outputs. With multiple outputs, an additional component of productivity 11 growth, viz., markup in output prices, is added. We also use a profit function 12 approach with both single and multiple outputs. The profit function approach 13 is extended to accommodate markups in output prices. Finally, we consider the 14 alternative profit function in which output prices (instead of output quantities)
15 are treated as endogenous.
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17 18 2. THE PRODUCTION FUNCTION APPROACH 19 20 2.1. Time Trend Representation of Technical Change 21 22 We summarize the technology of the firm in time period t by its period t t 23 production function f . Assuming that there is one output, the production = t 24 function can be represented as yit f x1it xJit , where yit is the output of th t 25 the i firm i = 1 N in period t t= 1 T f · is the production 26 technology, and x is a vector of J inputs. In order to estimate the parameters 27 of such a production function econometrically, it is necessary to relate the 28 production function in period t to the corresponding production function for 29 other periods. A common approach is to assume that the production function 30 is atemporal, meaning that its form and the parameters do not depend on the 31 time index t. Exploiting this fact and accommodating technical inefficiency, the 32 production function can be represented by 33 = − 34 yit f xit t exp uit (1) 35 ≥ 36 where uit 0 is output-oriented technical inefficiency. Technical inefficiency, 37 uit, measures the proportion by which actual output yit falls short of maximum = 38 possible output f x t . Technical efficiency is then defined by yit/f xit t − ≤ 39 exp uit 1. The time trend variable t in (1) represents technical change (a 40 shift in the production function over time, ceteris paribus). Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 26
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01 When input quantities change, productivity change is measured by what is 02 popularly known as TFP change (or the Divisia index of productivity change, ˙ 2 03 denoted by TFP) and is defined as 04 ˙ =˙−˙≡˙− a ˙ 05 TFP y x y Sj xj (2) j 06 07 a = a a = where Sj wjxj/C and C jwjxj wj being the price of input xj. Here the 08 rate of change in the composite input x is defined as the weighted average 09 of the rate of change on individual inputs. By using this definition, the TFP 10 = + ˙ = index can be constructed from TFPt TFPt−1 1 TFPt t 2 T when, 11 for example, TFP for the base year t = 1 is 100, i.e., TFP = 100. ˙ 1 12 To decompose TFP we differentiate (1) totally and use the definition of TFP 13 change in (2) to obtain 14 u f x 15 ˙ = − + j j − a ˙ TFP TC Sj xj 16 t j f 17 = − ˙ + + + − a ˙ RTS 1 jxj TC TEC j S xj (3) 18 j j j 19 20 where TC = ln f x t TEC =− u and RTS = ln y = ln f = f x / t t j ln xj j ln xj j j j 21 ≡ = = f j j is the measure of returns to scale. Finally, j fjxj/ kfkxk 22 j/RTS when fj is the marginal product of input xj. The relationship in 23 − ˙ (3) decomposes TFP change into scale RTS 1 j jxj , technical change 24 ln f x t − − a ˙ , technical efficiency change u/ t , and price effect j j Sj xj 25 t components. This last component captures either deviations of input prices from 26 the value of their marginal products, i.e., w = pf , or the departure of the 27 j j marginal rate of technical substitution from the ratio of input prices f /f = 28 j k w /w . Thus, the last component can be dropped from the analysis if one 29 j k assumes that firms are allocatively efficient.3 30 If technical inefficiency is time-invariant (i.e., − u/ t = 0), the decomposition 31 in (3) shows that TEC does not affect TFP change. Under the assumption of 32
33 34 2 Subscripts i and t are omitted to avoid notational clutter. 35 3 TFP growth is computed using the Divisia index. Therefore, the sum of the TFP growth components 36 obtained from the parametric model (for example, from (3)) has to be equal to the TFP growth obtained using the Divisia index. But in practice, a wide gap between the two measures is often 37 observed, especially when TFP growth (which is the sum of the TFP growth components) is estimated 38 using a parametric model. This problem can be avoided by using the TFP growth equation as an 39 additional equation in the system. See Kumbhakar and Vivas (2004a) for details on this issue in the 40 context of a dual cost function estimation. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 27
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01 constant returns to scale, the TFP change formula in (3) is identical to the one 02 derived in Nishimizu and Page (1982), viz.: 03 u 04 ˙ = − + − a ˙ TFP TC j Sj xj (4) 05 t j 06
07 The TFP growth formula with input-oriented technical inefficiency (for which the = − ≥ 08 production function is written as y f x exp t 0 being input-oriented
09 technical inefficiency) is presented in Appendix A. Technical inefficiency can
10 also be non-neutral. The TFP growth formulae for non-neutral (input- and output-
11 oriented) technical inefficiency are presented in Appendices B and C.
12 13 2.2. The Factor-Augmenting Model of Technical Change 14 15 Since technical progress is defined in terms of shifts in the production function 16 or the isoquants, it means that either a given level of output can be produced 17 with fewer inputs, or more output can be produced using the same amount of 18 inputs. This, in turn, implies that technical progress increases the productivity of 19 at least one input. Does technical progress increase productivity of all or some 20 of the inputs? Is it possible that the productivity of some inputs increases while 21 those of other inputs remain constant or decline? What is the contribution of 22 a particular input to overall technical change? The standard time trend model 23 (discussed in Section 2.1) gives an estimate of the overall effect of technical 24 change on output but it cannot answer the above questions. Answers to these 25 questions can be obtained by specifying technical change in factor augmenting 26 (FA) form (Beckmann and Sato (1969), Sato and Beckmann (1969), Kumbhakar 27 (2002, 2003)), viz.: 28 29 = − = − y f Ax exp u f A1 t x1 AJ t xj exp u 30 ≡ f x˜ x˜ exp −u = f x ˜ exp −u (5) 31 1 J 32 ˜ = th where xj Aj t xj is the j variable input measured in efficiency units, and 33 · f is the production technology. Aj t > 0 is the efficiency factor associated 34 with input j j= 1 J .4 It can also be viewed as an input-specific pro- 35 ductivity/efficiency index. If Aj t increases over time, then the productivity of 36 − − input j rises. Thus, Aj t Aj t 1 measures the productivity change of input 37 j from period t − 1 to period t. Consequently, productivity growth in input 38
39 40 4 Although we are making A a function of time, in principle, it can depend on other exogenous variables. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 28
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− − 01 j (i.e., Aj t Aj t 1 >0) implies an increase in partial factor productivity 02 y/xj but not vice-versa. 03 Using the same definition of technical change as before, TC in the FA model 04 can be expressed as 05 ˜ ˜ ˜ ln f x ln xj ln f x ˙ j 06 TC = = A ≡ TC (6) p ˜ ˜ j p ln xj t ln xj 07 j j j
08 j th TCp represents the contribution of the j input to the aggregate (overall) primal 09 j technical change TCp. It is clear from (6) that TCp depends on the rate of 10 ˙ ln f x ˜ change of input productivity A and ˜ , which under competitive market j ln xj 11 conditions, is the cost share of input j in total revenue. 12 The essential difference between the specifications of the production technol- 13 ogy in (1) and (5) is that technical progress in (1) shifts the production function 14 over time, ceteris paribus, whereas in (5) it enhances input quantities in effi- 15 ciency units x ˜ , thereby causing a movement along the production function 16 y = f x ˜ . Such a movement may be caused by factors other than time. 17 The input-specific productivity indices Aj can be functions of time as well 18 as variable and quasi-fixed inputs.5 Thus, depending on the specification of 19 Aj, technical change can be purely exogenous (functions of only time), purely 20 endogenous (functions of choice/decision variables), or a mixture of the two. 21 However, in the context of estimating a single equation production function, 22 endogeneticity of inputs is typically not taken into account. In such a case the 23 distinction between endogenous and exogenous technical change is not clear. 24 TFP growth in this setup is 25 TFP˙ = RTS − 1 x˙ + TC + − Sa x˙ − u/ t (7) 26 j j P j j j j j 27
28 To examine these components in detail, we assume a translog functional form
29 to represent the underlying production technology, viz.:
30 = + ˜ + 1 ˜ ˜ − 31 ln y 0 j ln xj jk ln xj ln xk u (8) j 2 j k 32 ˜ = 33 where xj Aj t xj. It is necessary to specify Aj t in order to estimate the above 34 model. To illustrate this, we consider the following two forms for Aj t . First, 35 we specify Aj t as a function of time with the following parameterization 36 = + 2 ln Aj t ajt bjt (9) 37
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39 5 Variables such as R&D expenditure that is considered a choice variable in the dynamic model also 40 affects technical change. Technical change caused by choice variables is often labeled endogenous technical change. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 29
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01 where aj and bj are parameters which are to be estimated along with the parame- 02 ters of the production function. In the second specification, the Ajs are functions 03 of time as well as other x variables, viz.: 04 05 = + ln Aj t aj bjk ln xk (10) 06 k 07 where a and b are parameters to be estimated. 08 j jk From the above specifications one can easily test whether the rate of change 09 in efficiency factors are constant or not, by restricting b = 0 ∀ j in (9) and 10 j b = 0 in (10). Similarly, the hypothesis of no change in productivity of inputs 11 jk can be tested from the restrictions a = b = 0 ∀ j in (9) and a = b = 0 ∀ j k 12 j j j jk in (10). Specification (10) assumes that productivity of an input at a point in 13 time depends not only on time but also quantities of inputs being used at that 14 point of time. That is, the productivity gain associated with an input is not purely 15 exogenous. For example, productivity of labor in an environment with computers 16 (higher level of capital) might be higher than a similar person working in an 17 environment with fewer computers. Thus, labor productivity might depend on the 18 stock of capital, and efficiency of capital might depend on the quantity of labor. 19 Using this specification we can test whether one input affects the productivity 20 of another input (i.e., whether some inputs are complementary to others from an 21 efficiency point of view). 22 The measure of overall technical change TCp using the above production 23 function is 24 25 ln y = = ˙ ≡ j TCp RjAj TC (11) 26 p t j j 27 28 where 29 ln y 30 R = = + x˜ j ˜ j jk ln j (12) 31 ln xj k 32 j = ˙ TCp RjAj t (13) 33
34 35 2.3. Latent Class Model 36 37 So far it has been assumed that there is a unique technology employed by every 38 firm. However, firms in particular industries may use different technologies. In 39 such a case, estimating a single frontier function encompassing every sample 40 observation may not be appropriate in the sense that the estimated technology is Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 30
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01 not likely to represent the “true” technolgy. That is, the estimate of the under- 02 lying technology may be biased. Furthermore, if the unobserved technological 03 differences are not taken into account during estimation, the effects of these 04 omitted unobserved technological differences might be inappropriately labeled 05 as inefficiency. 06 To reduce the likelihood of such misspecification, the sample observations 07 are often classified into certain categories using exogenous sample separation 08 information, and a separate technology is estimated for each group. For example,
09 Mester (1993) and Grifell and Lovell (1997) grouped banks into private and
10 savings banks. Kolari and Zardkoohi (1995) estimated separate costs functions
11 for banks grouped in terms of their output mix. Mester (1997) grouped sample
12 banks in terms of their location. In the above studies, estimation of the technology
13 using a sample of firms is carried out in two stages. First, the sample observations
14 are classified into several groups. This classification is based on either some
15 a priori sample separation information (e.g., ownership of firms (private, public
16 and foreign), location of firms, etc.) or applying cluster analysis to variables
17 such as output and input ratios. In the second stage, separate analyses are carried
18 out for each class/sub-sample.
19 To exploit the information contained in the data more efficiently, a latent
20 class model approach (hereafter LCM) is often used, in which technological
21 heterogeneity can easily be incorporated by estimating a mixture of production 6 22 functions. In the standard finite mixture model, the proportion of firms (obser-
23 vations) in a group is assumed to be fixed (a parameter to be estimated), see, for
24 example, Beard and Gropper (1991) and Caudill (2003). However, the probabil-
25 ity of a firm using a particular technology can be explained by some covariates
26 and may vary over time, and these technologies along with the (prior) probability
27 of using them (that might vary across firms) can be estimated simultaneously.
28 Once the technologies are estimated, one can define the best practice technology
29 (metafrontier) by taking the outer envelope of the individual technologies (see
30 Battese et al. (2004)). The advantage of defining the best practice technology
31 is that it can be compared to the individual technologies to measure technology
32 gaps among countries, industries, firms, etc.
33 Assume that there are j technologies that can be represented by the density f v j = J 34 functions j j 1 . Each producer in the sample uses one of these technologies (that is each firm at a point in time belongs to a particular class). 35 The analyst does not know who is using what technology. Sometimes the analyst 36
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38 6 Finite mixture/LCMs are widely used in marketing, biology, medicine, sociology, psychology, and 39 many other disciplines. For applications in social sciences, see, Hagenaars, J.A. and McCutcheon 40 (2002). Statistical aspects of the mixing models are dicussed in detail in Mclachlan and Peel (2000).
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01 follows some ad hoc criteria such as the size of firm, region in which the firm is 02 located, etc., to group the sample firms. In the LCM one assumes that each firm 03 has a probability of belonging to any group. Reintroducing the firm subscript i, ≥ = ∀ = ∀ 04 these probabilities ij 0 j 1 J i and j ij 1 i can be either th 05 constants or functions of covariates. Thus, in a LCM the technology for the j 06 class is specified as 07 = + 08 ln yi ln f xi t j vi j (14) 09 10 where j = 1 J stand for class. For each class, the stochastic nature of the 11 frontier is modeled by adding a two-sided random error term vi j. The noise 12 term for class j is assumed to follow a normal distribution with mean zero and 13 2 constant variance vj . Thus, the conditional likelihood function for firm i given 14 that it belongs to class j is 15 16 v j f ij = i (15) 17 vj 18 19 where · is the pdf of a standard normal variable. Consequently, the uncondi- 20 tional likelihood function for firm i is 21 v j 22 = i l i ij (16) 23 j vj 24
25 where ij is the prior probability assigned by the analyst on firm i to be using 26 the technology of type j. Sometimes firm-specific covariates are used to explain 27
28 29 Output Metafrontier [AU1] 30
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38 Input 39
40 Fig. 2.2. The Metafrontier Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 32
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01 the prior probabilities. To impose the constraint that these probabilites are non- 02 negative and sum to unity, the multinomial form is often used. That is 03 exp z 04 = j j = ij J 0 (17) 05 exp zj j 06 j
07 where z are covariates explaining the prior probabilities and are the associated 08 j j parameters. 09 The log likelihood function is then 10
11 n 12 log L = log l i (18) 13 i=1 14 15 which is maximized with respect to j (which is the vector of parameters = 16 associated with the class j technology), j and vj j 1 J . Maxi- 17 mization of this likelihood function can be done with conventional gradient
18 methods. The main problem is the possibility of multiple optima. Thus, one
19 should use different starting values to make sure that the global maximum is 7 20 achieved. The EM algorithm is a particularly useful device in this setting.
21 Furthermore, the EM algorithm is simple and intuitive (see Caudill (2003)
22 for details). The above approach can easily be extended to incorporate techni-
23 cal inefficiency (Caudill (2003), Orea and Kumbhakar (2004), Kumbhakar and
24 Tsionas (2003)).
25 Given that there are J groups (industries, regions, countries, etc.), the LCM 8 26 estimates J different technologies. The estimated parameters can be used to com- 27 pute the conditional posterior class probabilites. Using Bayes’ theorem, the poste- = · J · 28 rior group probabilities can be obtained from P j i l i j ij/ j=1 l i j ij. 29 The highest posterior probability can be used to assign a group for each producer,
30 i.e., firm i is classified into group k = 1 J if P ki = max P ji . The j 31 technology of class k is then used as the reference technology to estimate techni- 32 cal efficiency of firm i. The outer envelope of these group technologies (frontier 33
34 35 7 See Dempster, Laird and Rubin (1977) and McLachlan and Peel (2000). 8 36 In estimating a latent class model one has to address the problem of determining the number of groups/classes. The AIC and BIC (Schwartz’s criterion) are the most widely used criterion in 37 standard latent class models to determine the appropriate number of classes. Both statistics measure 38 the model’s goodness of fit but penalize by the complexity (number of parameters) of the model. 39 Hence, they can be used to compare models with different numbers of classes. The best model is 40 the one with lowest AIC or highest BIC. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 33
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01 for each group) is used to define the metafrontier (best practice technology), = ∀ 02 f x maxj f x j j . Finally, the group technologies can be compared with 03 the best practice technology to obtain measures of technology gap, defined as 04 the ratio of f x j to f x . 05
06 07 2.4. Multiple Outputs 08 09 One can express the multiple output production technology either in input or out- 10 put augmenting form, viz., (i) F y x ˜ = a and (ii) F y x ˜ = a, where y˜ = D t y 11 is a vector of M outputs and D t is the corresponding vector of efficiency 12 factors while a is a constant. These models can be further extended to accom- 13 modate technical inefficiency by attaching the one-sided inefficiency term to 14 either inputs or outputs. One way to interpret the formulation of technical change 15 in (i) is that technical progress shifts the isoquants inward to produce a given 16 level of output (thereby meaning that less inputs are needed) because input 17 ˙ quantities in efficiency units are higher Aj t > 0 . In formulation (ii) technical 18 progress shifts the production possibility frontier (PPF) outward, thereby mean- 19 ˙ ing that more outputs are produced (which implies that Dm < 0 , given the input 20 quantities. 21 If the specification in (ii) is chosen, then the production function can be 22 written as 23
24 y = f D y D y x x exp −u (19) 25 1 2 2 M M 1 J
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27 where output 1 plays an asymmetric role. This is a major disadvantage because
28 estimation of the production technology is not invariant to the choice of output 1.
29 Furthermore, there is the endogeneticity problem. If output 1 is endogenous,
30 why not output 2, etc? Because of these problems, multiple output production
31 functions are not estimated econometrically. These functions will be considered
32 again in the dual set-up where behavioral assumptions (cost minimization and
33 profit maximization behaviors) are introduced explicitly into the model.
34 A convenient way of modeling multiple outputs in a primal framework is to use
35 distance functions. In this framework the homogeneity property (degree one) is 36 used to solve the asymmetry problem that was encountered in the multiple output 37 production functions. Both input and output distance functions are specified as 38 F x y = a. The homogeneity property separates one from the other. The input 39 (output) distance function is homogeneous of degree one in inputs (outputs). We 40 now discuss the distance function approach. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 34
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01 3. DISTANCE FUNCTIONS 02 03 When many inputs are used to produce many outputs, Shephard’s (1953, 1970) 04 distance functions provide a functional characterization of the structure of the 05 production technology. Input distance functions characterize input sets, and out- 06 put distance functions characterize output sets. Not only do distance functions 07 characterize the structure of production technology, but also they are intimately 08 related to the measurement of technical efficiency. 09 = ∈ An input distance function is a function DI y x t max x/ L y , 10 where L(y) describes the sets of input vectors that are feasible for each output 11 vector. It adopts an input-saving approach to the measurement of the distance 12 from a producer to the boundary of production possibilities. It gives the maximum 13 amount by which an input vector can be radially contracted while still being 14 able to produce the same output vector. In Figure 2.3, the scalar input x is ∗ 15 feasible for output y, but y can be produced with smaller input x/ , and so ∗ 16 = DI y x t > 1. 17 = ∈ An output distance function is a function DO x y t min y/ P x , 18 where P x describes the sets of output vectors that are feasible for each input 19 vector x. It takes an output-augmenting approach to the measurement of the 20 distance from an observed input bundle to the boundary of production possibil- 21 ities. It gives the minimum amount by which an output vector can be deflated 22 and still remain producible for the same vector of inputs. In Figure 2.4 scalar ∗ 23 output y can be produced with input x, but so can larger output y/ , and so ∗ 24 = DO y x t < 1. 25
26
27
28 x2 29
30 L(y)
31
32
33 34 x 35 x/λ* 36
37
38 x1 39
40 Fig. 2.3. An Input Distance Function N = 2 Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 35
Productivity and Efficiency Measurement Using Parametric Econometric Methods 35
01 y2 02 P(x) 03
04
05 y/µ* 06 y 07
08
09
10
11 y1 12 Fig. 2.4. An Output Distance Function J = 2 13
14 15 Fare and Primont (1995) discussed properties of both output and input distance 16 functions. They show how to derive returns to scale (RTS) using these distance 17 functions. These results are used next in the derivation of the TFP growth 18 formulae. 19
20 3.1. Output Distance Function 21
22 In the presence of technical inefficiency, the output distance function is written 23 as, D x y t ≤ 1 ⇒ D x y t exp u = 1, where u ≥ 0 is output-oriented 24 O O technical inefficiency. Differentiating totally, we get 25 26 D x y t D x y t D x y t u ln O ˙ + ln O ˙ + ln O + = 27 ym xj 0 m ln ym j ln xj t t 28 29 ⇒ ˙ − ˙ + + = Rmym Sjxj RTS ln DO/ t u/ t 0 30 m j 31 ⇒ ˙ = − ˙ − − TFP RTS 1 Sjxj ln DO/ t u/ t (20) 32 j 33
34 where 35 J ˙ ln DO x y t 36 TFP = R y˙ − S x˙ RTS =− m m j j ln x 37 m j j=1 j 38 w x ln D / ln x p y D / y D = j j = O j = m m = ln O ln m = ln O 39 Sj Rm wjxj ln DO/ ln xj pmym ln DO/ ln ym ln ym 40 j j m m Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 36
36 Transparency, Governance and Markets
= 01 since m ln DO/ ln ym 1 (which follows from the homogeneity prop- 9 02 erty of DO). To estimate the above model, we use the homogeneity prop- = ˜ 03 erty and write the output distance functions as DO x y t /y1 DO x y t ˜ = − = 04 where y y2/y1 yJ /y1 . Finally, we rewrite it as ln DO x y t ln y1 ˜ ⇒− = ˜ + =− ≤ 05 ln DO x y t ln y1 ln DO x y t u where ln DO u 0. After this, a ˜ 06 parametric function is assumed for ln DO x y t and a stochastic noise term is 07 added prior to estimation. The standard frontier technique of maximum likelihood 08 method can then be used to estimate the output distance function.
09
10 3.2. Input Distance Function 11
12 In the presence of technical inefficiency the input distance function is written 13 ≥ ⇒ − = ≥ as, DI x y t 1 DO x y t exp 1, where 0 is input-oriented 14 technical inefficiency. Total differentiation of the input distance function yields 15 16 ln D x y t ln D x y t ln D x y t I y˙ + I x˙ + I − = 17 m j 0 m ln ym j ln xj t t 18 ⇒ − −1 ˙ + ˙ + − = 19 RTS Rmym Sjxj ln DI / t / t 0 m j 20 21 ⇒ ˙ = − −1 ˙ + − TFP 1 RTS Rmym ln DI / t / t (21) 22 m 23 where 24 25 M ln D x y t ˙ = ˙ − ˙ −1 =− I 26 TFP Rmym Sjxj RTS m j m=1 ln ym 27 28 p y D / y w x ln D / ln x D = m m = ln I ln m = j j = I j = ln Rm Sj 29 pmym m ln DI / ln ym wjxj ln DI / ln xj ln xj 30 m j j
31 = since j ln DI / ln xj 1 (which follows from the homogeneity property of 32 = DI ). Using the same property, we can express DI x y t as DI x y t /x1 33 ˜ ˜ = − = ˜ DI x y t where x x2/x1 xJ /x1 . Thus, ln DI ln x1 ln DI x y t 34 − = ˜ − = ≥ which in turn implies that ln x1 ln DI x y t where ln DI 0. 35 Therefore, the second component in (21) (i.e., ln DI x y t / t) encapsulates 36 = ˜ ≥ ≤ technical change, where ln DI x y t / t ln DI x y t / t 0 0 implies 37 technical progress (regress). The last component of (21) (i.e., − / t) repre- 38 sents the effects of technical efficiency change. Once is estimated from the 39
40 9 See Brummer et al. (2002) for details on the decomposition. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 37
Productivity and Efficiency Measurement Using Parametric Econometric Methods 37
− = ˜ − 01 model ln x1 ln DI x y t , after a parametric function is assumed for ˜ 02 ln DI x y t and a stochastic noise term is added, / t can be easily esti- 03 mated. Thus, by estimating an input distance function, all three components of 04 TFP growth can be obtained. For example, if a translog form is assumed for ˜ 05 ln DI x y t , viz.: 06 J m J J 07 1 − ln x = + ln x /x + ln y + ln x /x ln x /x 08 1 0 j j 1 m m jh j 1 h 1 j=2 m=1 2 j=2 h=1 09 M M J J 10 + 1 + ml ln ym ln yl jk ln xj/x1 ln yk 11 2 m=1 l=1 j=1 k=1 12 1 J M 13 + + 2 + + + − 0t 00t jtt ln xj/x1 mtt ln ym v (22) 14 2 j=2 m=1 15 16 Standard frontier techniques can be used to estimate the above model from
17 which RTS Rm Sj and / t can be computed. These are then used to obtain 18 the components of TFP growth in (21).10 19
20 21 4. THE COST FUNCTION APPROACH11 22 23 4.1. Single Output 24 25 While modeling inefficiency in a cost model, we can argue that inefficiency 26 increases cost and therefore the cost function can be written as Ca = C0 × 27 CE where Ca is the observed cost, C0 is the minimum (frontier) cost, and 28 1/CE ≤ 1 is cost efficiency. While this argument is true for input-oriented 29 technical inefficiency, it is worthwhile to derive the above result from the firm’s 30 optimization problem, which is 31 32 min wx subject to y = f xe− t 33 x
34 where ≥ 0 is the input-oriented technical inefficiency. Assuming that firms are 35 = = allocatively efficient, fj/f1 wj/w1 j 2 J, where fj, is the marginal 36 product (MP) of input j. The solution of the above problem gives inefficiency 37
38 39 10 See Karagiannis et al. (2004) for details. 40 11 See Kumbhakar (2000) for details on this model with allocative inefficiency. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 38
38 Transparency, Governance and Markets
− 01 adjusted input demand functions, xj e as a function of (w, y, t). These input a 02 demand functions can be used to define actual (observed) cost, C , which is 03 Ca = w x ⇒ Cae− = w x e− = C0 x y t (23) 04 j j j j j j 05 0 06 where C (which is the minimum cost function without technical inefficiency). 07 The above cost function can be written as 08 ln Ca = ln C0 w y t + (24) 09 10 where, ≥ 0 can be interpreted as the percentage increase in cost due to technical 11 inefficiency.12 12 Following Denny and Waverman (1981), it can be shown that the TFP growth 13 formula in a cost minimizing set up is 14 ln C 15 ˙ ≡˙− a ˙ =˙ − − ˙ − TFP y Sj xj y 1 Ct 16 j ln y t 17 = 1 − 1/RTS y˙ − C˙ − (25) 18 t t 19
20 which decomposes TFP growth into scale, TC, and TEC components.
21 The cost function, which is dual to the factor augmenting production function, augmented to accommodate input-oriented technical inefficiency ( ) can be 22 written as 23 24 Ca = C w y ˜ exp (26) 25 ˜ = = ∀ 26 where wj Bj t wj.IfAj depends only on time, then Bj t 1/Aj t j. 27 Thus, an increase in efficiency of an input is equivalent to a decrease in its ˜ 28 effective price w . The overall technical change in a cost model is expressed =− ln C =− ln C ˙ = j 29 as TC ˜ B t , which can be expressed as TC TC c t j ln wj j c j c ln C 30 j =− ˙ where TC ˜ B t . Thus, TC is a weighted average of input productivity c ln wj j c 31 ˙ =−˙ a change Aj t Bj t , where the weights are the cost shares Sj . 32 Thus, the TFP growth formula can be written as 33 ln C 34 ˙ ≡˙− a ˙ =˙ − + − TFP y Sj xj y 1 TCc 35 j ln y t 36 37 = 1 − 1/RTS y˙ + TC − (27) c t 38
39 40 12 See Kumbhakar (1996) and Kumbhakar and Wang (2005) for the derivation of the cost function with output-oriented technical inefficiency. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 39
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01 The TFP growth formulae with output-oriented technical (along with and without 02 allocativity) inefficiency are given in Appendix D. In Appendix E, we derive the 03 TFP growth formulae with non-neutral output-oriented technical inefficiency. 04
05 06 4.2. Multiple Output Cost Function 07
08 ˙ = ˙ − If there are multiple outputs, TFP growth is defined as TFP m Rmym 09 a ˙ = j Sj xj, where Rm pmym/ l plyl with pm representing the price of output 10 = ym m 1 M . It can be shown (Denny et al. (1981)) that the components 11 of TFP are 12 13 TFP˙ = TC + 1 − RTS−1 Y˙ + Y˙ − Y˙ − 14 C P C t 15 = TC + Scale + Markup + TEC (28) 16
17 ˙ = cyl ˙ ˙ = plyl ˙ = 18 where Y y Y y , and ln C/ ln y . TC and RTS C l l cy l P l R l cyl l l = 19 in (28) can be obtained from a parametric cost function C C w y t when −1 = 20 RTS is defined as RTS l ln C/ ln yl. Thus, the first component of TFP 21 growth is TC and the second component is the Scale component (related to 22 RTS), which is zero if RTS is unity. The third component is non-zero if output 23 markets are non-competitive. That is, if output prices depart from their respective ˙ = ˙ 24 marginal costs then YP YC . Finally, the last component is due to technical effi- 25 ciency change TEC =− / t . It is positive (negative) if efficiency improves 26 (deteriorates), i.e., declines (increases) over time. 27
28 29 4.3. The Factor-Augmenting Model of Technical Change 30 31 Here we consider the multiple output production technology F y x ˜ = a with 32 input-oriented technical inefficiency, for which the cost function is Ca = 33 C w y ˜ exp . Following the procedure for the time trend model, the TFP 34 growth formula for the FA model can be written as 35 36 TFP˙ = TC + 1 − RTS−1 Y˙ + Y˙ − Y˙ − 37 c C P C t 38 = TC + Scale + Markup + TEC (29) 39 c 40 =− ln C =− ln C ˙ where as before TC ˜ B t . c t j ln wj j Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 40
40 Transparency, Governance and Markets
01 To pursue this decomposition in detail, we assume a translog form for C w y ˜ 02 and write it as 03 a = + ˜ + + ˜ 04 ln C 0 j ln wj m ln ym jm ln wj ln ym j m j m 05 06 1 07 + ln w˜ ln w˜ + ln y ln y + (30) 2 jk j k lm l m 08 j k l m
09 where w˜ = B t w. The cost function in (30) is assumed to satisfy symmetry and 10 j j linear homogeneity (in w˜ ) restrictions. We specify the B as quadratic functions 11 j of time, i.e.: 12
13 ln B = t a + b t 14 j j j 15 =− ln C ˙ We then use formula TC ˜ B t to estimate technical change where 16 c j ln wj j 17 ln C 18 = + w˜ + y ˜ j jk ln k jm ln m 19 ln wj k m 20 ˙ 21 with the appropriate forms for Bj derived from ln Bj t . Furthermore, technical 22 change as defined above can be decomposed into pure, scale and input price 23 components. 24 In banking applications, multiple outputs are almost always used. One com- 25 mon problem with multiple outputs is the presence of zero values for some 26 outputs (meaning that some banks do not produce all the outputs). Zero out- 27 puts create problems for the Cobb Douglas and translog cost functions, since 28 the log of zero is not defined. To avoid this problem, researchers often replace 29 the zero values with a small positive number. Some replace them by unity 30 so that log values are zero, while others add the small positive number to all
31 observations. For example, if output m has zero values for some banks, ln ym = + 32 is redefined as ln ym ln ym c , where c is a positive number supplied by 33 the user. This procedure, although widely used in practice, has two problems. 34 First, zero values for output(s) for a bank might be due to the fact that the 35 bank specializes in a few outputs, and adding small numbers for outputs that 36 are never produced puts the specialized banks in the same group as others. In 37 fact, this procedure does not recognize the fact that some banks can be spe- 38 cialized in certain outputs. Thus, no matter how small c is, this procedure is 39 not innocuous. Second, the output values are changed (either from zero to a + 40 positive constant c or from ym to ym c ) without changing the cost. That is, Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 41
Productivity and Efficiency Measurement Using Parametric Econometric Methods 41
01 the cost of the extra output c is zero. Looking at it from a purely mathematical = = 02 point of view, either C C w y1 ym yM t ,orC C w y1 ym + + 03 ym+1 c yM c t is true, but not both at the same time. 04 Replacing the zero values by unity makes sense if the technology of the banks 05 that are specialized in fewer outputs is the same as those that produce all the 06 outputs. For example, if the specialized banks produce Q outputs while the other
07 banks produce M M>Q outputs and the production technology for both types
08 of banks is Cobb-Douglas, i.e.: 09 J Q 10 = + + + = ln Ci 0 j ln wji m ln ymi vi i1 N1 11 j=1 m=1 12 J M = + + + = + 13 ln Ci 0 j ln wji m ln ymi vi iN1 1 N 14 j=1 m=1 15 then we can simply combine them and write the technology as 16 J M 17 = + + + = ln Ci 0 j ln wji m ln ymi vi i1 N 18 j=1 m=1 19 = = + = 20 where ymi 1 for m Q 1 M and i 1 N1. Since the hypothesis, 21 that the technology for the specialized and non-specialized banks is the same, is
22 testable, there is no point in making the assumption a priori. Thus the problem
23 is not as innocuous as it seems.
24 This zero value problem is not endemic to the multi-output cost functions. It
25 applies equally to output and input distance functions, as well as the latent class
26 models discussed before.
27 28 4.4. Latent Class Models 29 30 The existence of multiple technologies can easily be accommodated in each of 31 the above cases. Orea and Kumbhakar (2004) used a multiple output cost function 32 approach and found four separate technologies used by the Spanish banks. Their 33 approach can easily be extended to construct the metacost function, which is the 34 inner envelope of the group cost frontiers. Such a metacost function can then be 35 used to measure technology gaps among banks using different technologies. The 36 Orea-Kumbhakar approach also accommodates technical inefficiency that varies 37 across banks and differs among technologies. Furthermore, these inefficiencies 38 can vary over time in a flexible manner. Because of the similarity of the approach 39 with the production function approach discussed earlier, no further detail is 40 given. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 42
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01 5. THE PROFIT FUNCTION APPROACH 02 03 5.1. Single Output 04 05 When modeling inefficiency by using a profit model, we start with the argument 06 that inefficiency increases cost and therefore the profit function can be written a 0 a 0 07 as = ×PE, where is the observed profit, is the maximum (frontier) 08 profit, and PE ≤ 1 is profit efficiency, which is modeled as a one-sided error 09 term. Furthermore, it is assumed to be independent of the regressors (input and 10 output prices and time). Here it is shown that this assumption is generally untrue. 11 The profit function, in the presence of technical inefficiency corresponding −u 12 to the production function in (1), can be written as p w t u = w pe t 13 (Lau (1978), Theorem II-3, p. 154). The rationale behind the above result is as = −u 14 follows. First, the first-order conditions of profit maximization are fj wj/pe . u 15 Second, the production function in (1) can be rewritten as ye = f x t Thus, if −u u 16 one substitutes pe for p and ye for y, the above first-order conditions and the
17 production function look like a standard neo-classical production function. Con-
18 sequently, the solutions of input demand and output supply functions (adjusted = −u = −u 19 for inefficiency) can be expressed as xj xj w pe t and y y w pe t . 20 Therefore, the profit function, conditional on u is defined as
21 −u −u w pe t = max u py − w x y = f x t e 22 ye x 23 which also equals the actual profit a = py − wx. It means that profit, when 24 price is p and output equals y, is the same as profit when output equals f x but 25 price equals pe−u. That is, a 10% reduction in output given inputs has the same 26 effect on profit as a 10% reduction in output price holding output constant. 27 Using ps ≡ pe−u, the profit function can be implicitly written as w ps t . 28 s = u =− Then the Hotelling’s lemma, / p ye and / wi xj can be applied 29 to obtain the input demand and technical inefficiency adjusted output supply 30 functions. Note that the argument of the profit function is ps, which in turn 31 implies that technical inefficiency may not appear additively in the log profit 32 function. For example, in the case of the translog profit function 33 = + + s + 34 ln 0 j ln wj p ln p tt j 35 36 1 + ln ps ln ps + ln w ln w + t2 37 pp jk j k tt 2 j k 38 + s + s + 39 jm ln wj ln p pt ln p t jt ln wjt (31) 40 j j Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 43
Productivity and Efficiency Measurement Using Parametric Econometric Methods 43
01 which can be rewritten as 02 ln = + ln w + ln p + t 03 0 j j p t j 04 1 05 + + + 2 + pp ln p ln p jk ln wj ln wk ttt jm ln wj ln p 06 2 j k j 07 1 2 08 + ln pt + ln w t − u + ln w + t + u (32) pt jt j p jm j pt 2 pp 09 j j 10 ≡ ln 0 − g u ln p ln w t 11 0 = + 12 where ln is the translog profit frontier, and g u ln p ln w t u p + − 1 2 ≥ 13 j jm ln wj ptt 2 ppu 0 is profit inefficiency (profit loss due to techni- 14 cal inefficiency). It is clear from the profit function above that g u ln p ln w t
15 is a non-linear function of u and it cannot assumed that g u ln p ln w t is
16 an independently and identically distributed random variable. As a result, the 13 17 standard tools used to estimate the production frontier cannot be used here. A
18 simple sign change is not enough to model technical inefficiency in the profit 14 19 function, unless the underlying production function is homogeneous.
20 This will also affect the TFP growth formulation. Differentiating the profit function = w ps t totally, we get 21 22 d ln ln s ln ln 1 s a ln 23 = p˙ + w˙ + = py p˙ − C S w˙ + s j s j j dt ln p ln wj t t 24 j j 25 = =− using / p y and / wj xj (Hotelling’s lemma). 26 a = − From the definition of profit py jwjxj we get 27 d ln py Ca 28 = ˙ +˙ − ˙ + ˙ y p Sjxj Sjwj 29 dt j 30 Equating the above two equations gives (after some algebraic manipulations), 31 32 ln = ˙ − a ˙ + a ˙ − a a ˙ + py y S xj py S xj C S xj py u/ t 33 j j j t j j j 34 = ˙ − s − a ˙ + 35 py TFP py RTS 1 Sj xj py u/ t j 36
37
38 13 See Kumbhakar (2001) and Kumbhakar and Tsionas (2005) for details on estimation issues. 39 14 There are numerous banking papers that use translog profit functions as ln a = ln 0 −u (for exam- 40 ple, see Berger and Mester (2003) and the references cited in there), which is clearly inappropriate. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 44
44 Transparency, Governance and Markets
01 The above equation can be rearranged to give 02 ˙ ln s a 03 TFP = + RTS − 1 S x˙ − u/ t (33) py t j j 04 j 05 where RTSs = 1 − ln / ln ps −1. Furthermore, it follows from the envelope 06 = f a = ln f = theorem that t p t , which in turn implies that py t t TC, which is 07 used in (3). There is one important difference between (3) and (33). In deriving 08 (3) we did not rely on any behavioral assumptions explicitly, whereas in (33) we 09 used the profit maximizing conditions to determine the allocation of inputs and 10 the production of output. Consequently, input quantities in (33) (and therefore 11 RTS) are affected by the presence of technical inefficiency.15 12 Using the following results, the components of TFP growth can be expressed 13 in terms of the profit function, viz.: 14 py ln 15 = s s 16 ln p 17 RTSs − 1 =− ln / ln ps −1 (34) 18 ln s/ ln w 19 a = j Sj 20 k ln / ln wk 21 2 −1 ln w ˙ = ln ln + ln − j 22 xj t ln wj ln wj t t 23 24 The problem with the above decomposition is that each component depends 25 on technical inefficiency. This can be avoided by using the relation ln = 26 ln 0 − g u ln p ln w t in the above formula. In doing so, we can capture both 27 direct and indirect effects (via input and output prices as well as time) of technical 28 inefficiency on TFP growth. However, if we specify (erroneously) the translog 0 29 profit function as ln = ln −u, then (i) the estimated parameters of the profit 30 function are likely to be biased, and (ii) the indirect effect (via input and output 31 prices and time) of technical inefficiency TFP growth would not be captured. 32
33 5.2. Multiple Output 34 35 ˙ ˙ = ˙ − a ˙ = As before, we define TFP as TFP mRmym jSj xj when Rm pmym/R 36 = and R mpmym. In the standard case, where firms are assumed to be both 37
38 39 15 See Kumbhakar (2000) for the TFP growth formula that takes into account both technical and 40 allocative inefficiency. The decomposition without any inefficiency is given in Kumbhakar (2000b). Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 45
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01 technically and allocatively efficient (so that the multiple output profit function 16 02 is = w p t ), the TFP change formula can be expressed as 03 ln 04 ˙ = + − ˙ TFP RTS 1 Sjxj (35) 05 R t j 06 − =− −1 07 where RTS 1 m ln / ln pm . s = −u 08 To derive the TFP change in the present case, we denote p e p1 pM = s 09 and start from the profit function w p t , which is defined as s = s u − 10 w p t m pmyme j wjxj. However, the actual (observed) profit is a = − = s u − = s 11 m pmym j wjxj m pmyme j wjxj w p t . Thus, the TFP 12 change formula is
13 s ln s u 14 ˙ = − + s − ˙ TFP RTS 1 Sjxj (36) 15 R t t j 16 = = 17 in which we used the results / pm ym (Hotelling’s lemma) and / wj − 18 xj. All the components in (36) can be expressed in terms of profit. For exam- = s s s s − =− s −1 19 ple, R m ln / ln pm RTS 1 m ln / ln pm , which gives s − =− a 20 RTS 1 R . Using these results, coupled with those in (34) for Sj , every- 21 thing in (36) can be expressed in terms of and rates of change in input and
22 output quantities.
23 The first term on the right-hand side of (36) can be expressed as
24 s s ln ln ym 25 = R (37) R t m t 26 m
27 where ym is the supply function of output ym. Since the expression in (37) is 28 a weighted average (weights being the revenue share of each output) of rates 29 of change of individual outputs ( ln ym/ t), holding everything else constant, it 30 can be viewed as a measure of output technical change. Note that this measure 31 of output technical change is different from profit technical change, defined as 32 ln s/ t. 33 It can be seen from (36) that contribution of technical change and returns 34 to scale depends on both technical inefficiency via ps. Since technical change, 35 returns to scale, etc., are usually defined in terms of the profit frontier, it is pos- 36 sible to rewrite (36) in terms of the profit frontier so that the effects of technical 37
38 39 16 See Karagiannis (2000) for TFP growth decomposition with multiple outputs in a profit function 40 framework with quasi-fixed inputs. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 46
46 Transparency, Governance and Markets
01 inefficiency on TFP change are separated. For example, in the translog case we 0 02 can express actual profit as ln = ln −g u ln p ln w t , where ln p and ln w 03 are vectors of output and input prices. This result can be used in the TFP growth 04 formula to separate technical inefficiency effects in the TFP growth components. 05
06 5.3. Alternative Profit Functions 07
08 Use of the profit function approach is based on the assumption that prices 09 are exogenous, and that producers seek to maximize profit (or variable profit) 10 by selecting outputs and inputs under their control. One justification for 11 exogeneticity of prices is that producers operate in competitive markets. If pro- 12 ducers have some degree of monopoly power in their product markets, then 13 demand would be exploited to determine output prices and quantities jointly, 14 and only input prices would be exogenous. 15 Recently Humphrey and Pulley (1997), among others, have introduced the 16 notion of an “alternative” profit frontier to bridge the gap between a cost fron- 17 tier and a profit frontier. For example, Berger and Mester (1997) suggest that 18 the alternative profit approach may be helpful when (i) there are substantial 19 unmeasured differences in the quality of banking services; (ii) outputs are not 20 completely variable; (iii) output markets are not perfectly competitive; and (iv) 21 output prices are not accurately measured. 22 An alternative profit frontier is defined as 23 24 A = max py − wxF x y t = 0 g p y x t = 0 25 p x 26 ⇒ A = y w t (38) 27 28 where the endogenous variables are p x and the exogenous variables are 29 y w t . F x y t = 0 is the production function (it can also be specified by 30 an output distance function), and g p y w t = 0 represents what Humphrey 31 and Pulley refer to as the producer’s “pricing opportunity set,” which captures 32 the producer’s ability to transform exogenous y w t into endogenous product 33 prices p. However, A is not dual to the production function because it 34 incorporates both the structure of production technology and the structure of the 35 pricing opportunity set. Moreover, without specifying the properties satisfied by 36 the function g p y w t = 0, it is not possible to specify the properties satisfied A 37 by . 38 The main problem with the alternative profit function is that it has no theo- 39 retical foundation such as with the standard profit and/or cost functions. Conse- 40 quently, the approach has no use other than measuring efficiency, which is also Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 47
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01 problematic, as will be shown later. Since the whole idea is based on the pricing 02 opportunity function g p y w t = 0, it is worth exploring the issue further. 03 First, the single output case is considered. The profit maximization problem 04 is partitioned into two steps. In step 1 the cost is minimized, given the pro- a 05 duction function, to obtain the cost function, C = w x = C w y t . In step 2 − = 06 the profit is maximized, viz., maxp py C w y t g p y w t 0 . Note that 07 g p y w t = 0 can be expressed implicitly, as p = p w y t , which is nothing
08 but the inverse demand function. Thus, there is no need to solve the optimization
09 problem in step 2 for p. Consequently, there is no meaning to the alternative
10 profit function because producers do not maximize profit to obtain p. The so-
11 called optimal price p is not related to the production technology (either the
12 production or the cost function).
13 A similar result is obtained when multiple outputs are considered. The cost a = = 14 function from step 1 can still be written as C w x C w y t . The first- = = 15 order conditions from step 2 are: pm g / pm m 1 M where
16 is the Lagrange multiplier associated with the pricing opportunity function.
17 Thus, the output prices can be solved from the above first-order conditions and = 18 the pricing opportunity function g p y w t 0, without any reference to the
19 technology (production or cost functions). These solutions are, in implicit form, = 20 pm pm w y t . Thus, cost of production (namely, marginal cost) does not play
21 any role in determining optimal output prices. Now we examine the case with (input-oriented) technical inefficiency and 22 write the single output production function as y = f x exp − t , for which the 23 cost function is Ca = wx = C w y t exp . The first-order conditions from 24 the second step are exactly the same as before. Thus, the solution of p would 25 not be affected by the presence of technical inefficiency. Consequently, revenue 26 will be unaffected by the presence of inefficiency. Improvement in technical 27 inefficiency would not be transmitted to revenue through output prices. Using 28 this optimal price p = p w y t in the definition of profit we get 29 30 a = p w y t y − C w y t exp = A w y t exp − (39) 31 32 Thus, the above relationship cannot be expressed as ln a = ln 0 − ,asis 33 common in the literature (e.g., Berger and Mester (2003), DeYoung and Hasan 34 (1998), among others). That is, the observed profit is not necessarily reduced by 35 the same proportion by which cost is increased. For example, if profit without a = − = − = 36 inefficiency is 0 p w y t y C w y t 100 80 20 and the profit with a = − = − = 37 technical inefficiency is 0 p w y t y C w y t exp 100 80 1 10 38 12, profit is reduced by 40% although cost has increased by only 10%. a 39 In the presence of multiple outputs, the cost function is C = w x = 40 C w y t exp . Maximization of profit subject to g p y w t = 0 gives the Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 48
48 Transparency, Governance and Markets
= 01 solutions of output prices that can be implicitly expressed as pm pm w y t . 02 Note that these prices are not affected by the presence of technical inefficiency. 03 Thus, given output quantities, revenue is not affected by the presence of inef- 04 ficiency. Consequently, the argument that alternative profit function takes into 05 account the effect of inefficiency on both cost and revenue does not hold. Fur- 06 thermore, actual profit is 07 08 a = − = A − pm w y t ym C w y t exp w y t exp (40) 09 m
10
11 That is, actual profit is not reduced by the same percent by which cost is
12 increased. Therefore, the efficiency estimates based on the alternative profit
13 function cannot be related to technical inefficiency based on the production
14 technology. Since we cannot draw a meaningful interpretation of the inefficiency A = 0 − 15 term in the model ln ln w y t , and the whole idea of estimating the
16 alternative profit function is to estimate profit inefficiency, the alternative profit
17 function approach seems vacuous. However, this is not the case with the standard
18 profit function, although care has to be taken in modeling profit inefficiency (as
19 was shown in the previous section).
20 This brings us back to the standard profit function that can be extended to
21 accommodate non-competitive behavior in the output markets. This issue is
22 explored next (see Kumbhakar and Lozano-Vivas (2004) for details).
23
24
25 5.4. Modeling Markups in Variable Profit Functions
26
27 Assuming that the objective of the banks is to maximize profit, the bank’s
28 optimization problem can be formulated with the following two steps. In step 1,
29 a bank solves the following problem, given the output vector y, to determine the
30 least cost (variable) input quantities, i.e.:
31 = 32 Minx w x subject to F y x z t 0 33
34 where z is the vector of Q quasi-fixed inputs and output attributes. The solu- = 35 tion to the above problem gives the conditional input demand functions xj 36 xj w y z t that are then substituted into the objective function to obtain the 37 minimum cost function C w y z t . In step 2, the bank’s problem is to maximize
38 profit, i.e.:
39 = − 40 Maxy p y C w y z t Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 49
Productivity and Efficiency Measurement Using Parametric Econometric Methods 49
01 in which the choice variables are outputs. If the output markets are competitive, 02 the first-order conditions (FOC) for profit maximization can be expressed as 03 C 04 = ≡ = pm MCm m 1 M 05 ym 06 where MCm is the marginal cost associated with output ym. However, if the 07 output markets are not competitive and the banks possess monopoly power, the 08 FOC of the above problem become 09 10 ∗ = pm MCm (41) 11 12 ∗ = ≥ where pm pm m, with m 1 representing the markup factor. Thus, the presence 13 = ∀ of markups can be tested by restricting m 1 m. In the presence of markups, 14 ∗ = the relevant output prices are pm pm m and the output supply functions are 15 = ∗ ym ym w p z t . Consequently the variable profit function can be expressed 16 as ∗ = ∗ w p∗z t from which the input demand and output supply functions 17 can be obtained by using Hotelling’s lemma, viz.: 18 ∗ ∗ 19 = =− ym and xj 20 ∗ p wj 21 ∗ 22 The profit function is often labeled as the shadow variable profit function. ∗ ∗ = a a 23 However, it should be noted that is not observed and where a = − 24 is the actual profit, defined as m pmym j wjxj. Invoking Hotelling’s 25 lemma it can be shown that
26 27 a = ∗ ∗ + ∗ ≡ ∗ SRm/ m SCj H (42) 28 m j 29 ∗ ∗ ln ∗ ∗ ln ∗ ∗ 30 = + = =− when H m SRm/ m j SCj ln p∗ SRm and ln w SCj . The ∗ ∗ m j 31 shadow shares SCm and SCj are not observed. These shadow shares are related 32 to the actual (observed) shares as follows: 33 p y SR∗ 1 34 SRa = m m = m (43) m a 35 H m 36 w x SC∗ SCa = j j =− j 37 j a H 38 39 The above model can be estimated using either the variable profit function 40 in (42) along with the associated share equations in (43), or using the share Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 50
50 Transparency, Governance and Markets
01 equations strictly by themselves. Before proceeding we need to specify the
02 behavior of the markup factors m over time. First, m may be specified as 03 04 = + = m exp bm cmt m 1 M (44) 05
06 where bm and cm are parameters to be estimated. The exponential function is 07 ≥ often chosen to guarantee m 0. Depending on the signs of bm and cm m can 08 decrease or increase over time. We are primarily interested in testing whether 09 the s approach unity over time. 10 The system of equations in (42) and (43) can be operational only when a 11 parametric form of the shadow variable profit function ln ∗ is assumed. To 12 minimize a priori restrictions on the underlying production technology, we use 13 a translog form of the shadow variable profit function, ln ∗, and write it as 14 15 ∗ = + + + + 1 16 ln O j ln wj q ln zq tt jk ln wj ln wk j q 2 17 18 + + ∗ ∗ + 2 + 19 ql ln zq ln zl amn ln pm ln pn ttt bjq ln wj ln zq q l m n j q 20 + ∗ + ∗ + 21 cjm ln wj ln pm dmq ln pm ln zq jt ln wjt j m m q j 22 23 + ∗ + amt ln pmt bqt ln zqt (45) 24 m q 25 26 The above shadow profit function is assumed to satisfy the symmetry and 27 convexity conditions. Furthermore, it is homogeneous of degree one in input 28 and shadow output prices. The symmetry and homogeneity restrictions are 29 imposed in estimating the parameters of the model. The symmetry restrictions 30 on (45) are 31 32 = = = jk kj ql lq amn anm 33 34 and the homogeneity restrictions are 35
36 + a = 1 + c = 0 ∀ j a + c = 0 ∀ m 37 j m jk jm mn jm j m k m n j 38 + = ∀ + = ∀ 39 bjq dmq 0 q amt jt 0 t 40 j m m j Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 51
Productivity and Efficiency Measurement Using Parametric Econometric Methods 51
01 Normalizing the shadow profit function in (45) will impose the homogeneity 02 restrictions with respect to one price. Since the objective is to estimate markups th 03 for both outputs, we use the J input price to normalize the other prices. Then ∗ = ∗ = − 04 we calculate SRm and for m 1 2 M, and SCj for j 1 2 J 1. ∗ ∗ 05 From the shadow profit function in (45), SRm and SCj can be derived as 06 ∗ 07 ∗ = ln = + ∗ + + + SRm ∗ am amn ln pn cjm ln wj dmq ln zq amtt (46) 08 ln pm n j q 09 10 and 11 ∗ ∗ ln ∗ 12 − = = + + + + SCj j jk ln wk bjq ln zq cjm ln pm jtt (47) 13 ln wj k q m 14 = ∗ + ∗ 15 Finally, using (46) and (47) above, H mSRm/ m jSCj can be expressed 16 in terms of the unknown parameters and observed data. Consequently, the profit
17 system (after adding classical error terms to each equation) becomes
18 ln a = ln ∗ + ln H + v 19 20 p y SR∗ 1 SRa = m m = m + v (48) 21 m a m H m 22 w x SC∗ 23 SCa = j j =− j + j a j 24 H 25 which can be estimated using the iterative non-linear seemingly unrelated regres- 26 sion technique. One of the share equations has to be dropped because the shares 27 sum to unity.17 28 It should be noted here that the above approach does not recognize endoge- 29 neticity of output prices. If one assumes that producers have monopoly power 30 in the output market, then w have to add the demand (or inverse demand) func- 31 tions to take care of the endogeneticity of output prices. Adding M such inverse 32 demand functions (one for each output price) will make the above system consis- 33 tent in the sense that the number of equations equals the number of endogenous 34 variables (x y p). 35 One endemic problem in estimating profit functions (standard or alternative), 36 using a translog or Cobb-Douglas functional form, is that profit has to be pos- 37 itive. This is, however, not the case in reality. In many studies, especially in 38
39 40 17 Productivity decomposition formula for this problem is left to the readers. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 52
52 Transparency, Governance and Markets
01 banking, negative profits force researchers to use other functional forms that 02 allow negative profits. A majority of the banking studies, however, avoid this 03 problem by adding a positive number large enough to make profit for every 04 bank positive (see, for example, Berger and Mester (1997, 2003), Berger and 05 DeYoung (2001), and many others). This is clearly inadmissible, both econom- 06 ically and econometrically. Profit used in the profit function should be defined 07 as revenue minus cost (which may be different from accounting profit). So 08 any change in profit has to be reflected in the profit function. That is, if we a 09 assume no inefficiency and start from the profit function = p w t then a a 10 +c = p w t +c and, therefore, ln +c = ln p w t +c . However, a 11 what is used in practice is ln +c = ln p w t , i.e., no adjustment is made 12 on the right-hand side of the equation. This ad hoc procedure cannot be eco- 13 nomically justified. Adding something to the dependent variable and then taking 14 the log of it changes the intercept as well as the coefficients of all the right-
15 hand side variables (regressors) in the profit function. Thus, the estimates of
16 technological parameters can easily be manipulated (and therefore, estimates of
17 returns to scale, input substitutability, technical change, etc.) simply by changing
18 the positive constant that is added to profit. Since the alternative profit function
19 approach uses the same procedure to avoid taking log of negative profits, it is
20 subject to the same criticism.
21
22 23 6. SOME NEW ISSUES AND CHALLENGES 24 25 6.1. Competitiveness, Deregulation and Efficiency 26 27 Economists since Adam Smith have argued in favor of the virtues of competitive 28 markets. The competitive framework is used as a benchmark because it leads to 29 socially efficient outcomes. Any departures from competitive input and/or output 30 markets lead to the inefficient allocation of resources and production of outputs, 31 resulting in a deadweight loss to society. There might be many reasons why 32 firms in some markets may not be competitive. For example, firms might posses 33 some degree of market power in selling their products and/or buying their inputs. 34 Consequently, the main driving force behind deregulatory effort is to increase 35 competition in the hope of reducing the deadweight loss to the society. Here we 36 examine the competitive and efficiency issues in the context of banking. 37 Like many other countries, the banking industry in Europe has faced important 38 changes. These changes were specifically designed to liberalize the provision 39 of services, bringing increased competition to the banking industry. Addition- 40 ally, the establishment of the economic and monetary union (EMU), along Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 53
Productivity and Efficiency Measurement Using Parametric Econometric Methods 53
01 with developments in information technology and removal of entry barriers are 02 other important changes faced by the European banking markets. Many of these 03 changes have important implications for the competitive structure of the banking 04 and financial sectors. Some of the techniques proposed in this survey can be used 05 to examine whether output markets are competitive or not, as well as the contri- 06 bution of non-competitive output markets on TFP growth. We can also estimate 07 the degree and the temporal behavior of non-competitiveness from the estimates 08 of markup factors. In a cross-country study the information on markups may be 09 useful to the policymakers (at the EU level), in the sense that they can examine 10 whether banks from the new EU member (or to-be EU member) countries can 11 survive if such markups are eliminated due to competition. More work is needed 12 in this area. 13 Given that the banking system plays a unique role in an economy, the effi- 14 ciency of that system can hardly be ignored. Studying the efficiency of the 15 individual banks, as well as the banking system, can provide feedback on ways to 16 improve policy-making decisions. Models of efficiency studies discussed in this 17 survey can answer many questions, some of which are: Does regulation increase 18 efficiency of banks? Are de novo banks more efficient than existing banks? Can 19 banks from the countries joining the EU survive in competition with the foreign 20 banks, if left without subsidy or other assistance? Can efficiency and/or scale 21 economies explain bank mergers? These questions can be addressed using micro 22 data from the EU countries. One can also use the aggregate industry level data 23 from the EU member countries to examine differences in efficiency, productiv- 24 ity, technology, and competitiveness of the banking industry. If the industry is 25 not competitive and one erroneously makes the assumption that it is competitive, 26 efficiency and productivity measures are likely to be biased (especially when an 27 econometric technique is used). 28 If one uses value added per worker as the measure of productivity and the 29 sole indicator of performance, the measure cannot separate the contribution of 30 competitiveness to productivity improvement. That is, if one compares value 31 added per worker from two different industries, the difference would not tell us 32 anything about the contribution of competitiveness, even if we know that one 33 industry is more competitive than the other. This is because difference in value 34 added per worker captures the effect of many other things such as difference 35 in technology, size and scale differences, price differences, etc. Contribution of 36 these factors to the overall productivity growth (whether using a partial or total 37 factor productivity index) cannot be separated without estimating the technology 38 directly or indirectly. For example, if cost-minimizing behavior is used, the 39 contribution of competitiveness on TFP growth can be computed once the cost 40 function is estimated. The benchmark is the competitive output markets. The Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 54
54 Transparency, Governance and Markets
01 main advantage of the cost function approach is that we do not have to assume 02 a particular form of market structure to estimate this effect. It can also handle 03 multiple outputs, and can disentangle the effect of non-competitive behavior in 04 each output market. This is especially useful if the industries under investigation
05 produce more than one output and the degree of competition varies across
06 outputs.
07 The cost function approach cannot test any hypotheses about competitive
08 output markets. For this, a profit function approach has to be used that can
09 allow distortions in output prices due to regulation, non-competitive behavior,
10 etc. Thus, if we want to model price distortion (and decomposes its effect on
11 productivity growth), it is necessary to use a profit function approach.
12 Another challenge is to incorporate quality differences in outputs that can
13 be confounded with non-competitive behavior, because quality difference is reflected in the prices. If output qualities differ across firms and these are not 14 taken into account, deviations in the p = MC rule might be thought of as non- 15 competitive behavior. Similar to quality differences, there might be heterogeneity 16 in products, behaviors, etc., among firms/industries operating in different coun- 17 tries. Such differences are to be taken into account. Furthermore, cross-country 18 data may not be compared directly, unless the outputs, inputs, behavior, etc., 19 are homogeneous. Another complication is that efficiency/performance of firms 20 is likely to change over time and a model should be used that allows for his 21 possibility. 22 Multi-dimensional indices of productivity and efficiency measures, such as 23 labor productivity, employment growth, competitiveness, globalization, foreign 24 investment, etc., are often constructed to examine performance. Some firms may 25 be champions based on, for example, one or some of these criteria. This does not, 26 however, mean that the same firms will be champions when the other criteria 27 are used. Thus, an overall productivity index has to be constructed from these 28 individual indices to make a meaningful comparison. That is, to get a macro 29 outlook, we have to dig down in to the microanalysis. 30 In all the approaches discussed so far, the effect of competition or lack of it 31 is examined from the producers’ point of view. However, it is well-known that 32 non-competitive markets contribute to deadweight loss to the society meaning 33 that the society would be better off under competitive markets. By focusing 34 only on the production side, we fail to see the compete picture and miss the 35 deadweight loss arising out of, for example, monopolistic markets. The dead- 36 weight loss can be measured from the markups in output prices and difference in 37 the outputs produced with and without competitive market conditions. Both the 38 output differential and markups can be obtained from the estimates of shadow 39 profit function. No one needs to look into the deadweight loss component in 40 TFP growth decomposition. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 55
Productivity and Efficiency Measurement Using Parametric Econometric Methods 55
01 6.2. Data Requirement for Improving the Efficiency 02 and Productivity Measures 03 04 To address the issues mentioned above, for example in banking, extensive data 05 is required, some of which are not reported in the balance sheets and income
06 statements. For a meaningful efficiency analysis, for example, data is needed
07 on different types of loans (consumer loans, business loans, real estate loans,
08 etc.), different types of deposits, assets, capital, foreign exchange reserves, labor,
09 wages and salaries, interest paid on different types of deposits, and rates charged
10 on different types of loans. Quality measure of loans might be important for dis-
11 entangling efficiency and productivity measures from output quality. Variables
12 on quality of inputs, R&D expenditure, expenditure on information technology,
13 etc., are important and useful in disentangling productivity differential. Depth of
14 the study will depend on the type and extent of data made available to do the analysis. 15 In this survey we discussed several methods, some of which can be used to 16 crosscheck results from competing models. Data requirements of these models 17 also vary, and so does behavioral assumptions. For example, the distance func- 18 tions require data on input and output quantities, while estimation of profit func- 19 tions require data on input and output prices as well as profit. While discussing 20 alternative profit functions in Section 5.3, we mentioned that profitability might 21 be affected through revenue and cost. For example, loan quality and non-interest 22 net income affect revenue, and cost inefficiency increases cost. Thus, excluding 23 quality of loans from the analysis is likely to mis-measure revenue, which is 24 likely to bias the results of the profit function approaches. If information on the 25 quality of output variables is obtained and these quality variables are used in the 26 cost function, estimates of the efficiency results would be much more reliable. If 27 the quality variables are not included (due to non-availability of data), estimates 28 of efficiency scores are likely to be contaminated. High efficiency score might 29 be due to bad output quality and vice versa. 30 To put some of these in a broader perspective, consider some of the simpler 31 non-econometric performance measures that are widely used in practice. For 32 example, partial factor productivity, such as value added per worker, is often 33 used as an indicator of performance. Another indicator is employment growth, 34 although comparison between growth rates of value added and employment 35 across industries, sectors, regions, etc., might not give similar performance mea- 36 sures. For example, value added per worker can increase without increasing 37 employment growth if workers become more productive by using more capital 38 (substituting labor with capital) and/or better technology. In this respect, growth 39 in value added per worker is a better measure than growth in employment. How- 40 ever, even with this simple measure, we would like to decompose growth in Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 56
56 Transparency, Governance and Markets
01 value added per worker into returns to scale and technical change components. 02 To do so, we have to estimate the underlying technology using a primal or dual 03 approach. In estimating the technology and using it to measure growth rates of 04 value added per worker, we can control for employment growth (indirectly in 05 the regression equation), instead of using it as a separate measure of produc- 06 tivity. Again, we can use the metafrontier approach to explain differences in 07 productivity growth across regions, industries, and over time. The advantage of 08 the metafrontier approach, as mentioned before, is that we can examine the role 09 of differences in technology (labeled as technology gap) and efficiency change 10 in the overall productivity change. This will be helpful in examining whether 11 the top performing firms (champions) are using the best practice technology 12 (metafrontier) and are also technically efficient. For some industries/regions, the 13 champion firms might be technically efficient and may also be using the best 14 practice technology. But this may not be true for every sector/region. The typical 15 growth accounting procedure (although useful and simple) cannot capture the 16 sources of productivity growth differentials across regions, industries, size of 17 firms in an industry, etc. Therefore, the simple value added approach cannot 18 separate the effects of technical efficiency from the technology. 19
20 21 6.3. Use of Aggregated and Micro Data 22 23 Since the number of EU member countries is relatively small, we have to pool 24 time series data on the member countries to estimate efficiency, productivity, 25 etc., especially if we want to use the aggregated macro data. This puts a limit on 26 allowing technological heterogeneity among EU member countries, especially in 27 econometric models. If we assume a single technology for all member countries, 28 the estimated inefficiency will be relative to the single frontier defined for all 29 member countries. Thus, we cannot address the question of technology gap. That 30 is, technology gap and inefficiency will be lumped together, yielding higher 31 estimates of inefficiency. Furthermore, the estimated technology might be biased, 32 if more than one technology is used in practice. This problem can be avoided 33 if we use either a mixing model approach or somehow group the countries 34 based on some a priori information, and estimate the technology for each group 35 separately. 36 Another potential problem in using cross-country data is consistency of the 37 data. That is, all the input and output variables, prices, and other control variables 38 (if any) need to be defined in the same way. Sometimes the variables are indeed 39 defined in the same way but certain components might be missing. This is hardly 40 the case in reality. For example, data on loans might have ten classifications in Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 57
Productivity and Efficiency Measurement Using Parametric Econometric Methods 57
01 one country but for another country there might be five categories. Thus, the 02 analyst has to know whether similar procedures are used in constructing the 03 variables that are used to estimate the technology. This problem is avoided if we 04 use bank level data and estimate a separate frontier function for each country. 05 The data definition/construction problem mentioned above is absent when we 06 uses cross-sectional (panel) data on banks from each country because the same 07 procedure is used to construct the variables for each bank. Thus, we can estimate 08 the frontier function for each country and measure efficiency and productivity 09 relative to the estimated frontier. However, sometimes we want to compare bank 10 efficiency across countries. Since each country has a frontier of its own, for a 11 valid comparison of efficiency we have to take into account the fact that the 12 technologies are different. Spanish banks might be efficient relative to their 13 own frontier, but they might not be using the best practice technology. Thus, 14 there might be a technology gap. To estimate the technology gap, we have to
15 construct a metafrontier using the country-specific frontiers. The deviation of
16 country frontiers from the metafrontier is then viewed as a technology gap. Thus,
17 for example, Swiss banks might be technically efficient (relative to their own
18 frontier) but when the technology gap is taken into account, they may not be as
19 so efficient. That is, if a technology gap exits, efficiency of Swiss banks relative
20 to the metafrontier (global frontier) would not be as efficient. In other words, to
21 get a full picture we have to take into account both the country-specific frontiers
22 and the metafrontier. This would be a better indicator of bank performance than
23 that which is currently used in the banking literature.
24 While working with aggregate country level data, we often look at things
25 beyond the traditional production/cost/profit function approach. For example, a
26 new literature on productivity measurement (productivity and the new economy
27 (Nordhous, 2001)) has recently emerged. Following this approach, variables can
28 be included, such as technological capacity, human capital, financial capacity,
29 enterprise and innovation, openness, adaptability, etc. to enrich the productivity
30 measures. These variables (for which indices are to be constructed) can affect
31 productivity either indirectly by enhancing efficiency of traditional inputs such
32 as capital and labor, or directly through improving efficiency, i.e., increasing
33 outputs holding inputs constant.
34
35 36 7. CONCLUSIONS 37 38 In this survey we focused on the parametric models to estimate and decompose 39 TFP growth into scale, technical change, and technical efficiency change com- 40 ponents. For modeling inefficiency we concentrated on output-oriented technical Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 58
58 Transparency, Governance and Markets
01 inefficiency. The TFP growth formulas for input-oriented technical inefficiency 02 (for some selected models) are given in the appendix. Throughout the anal- 03 ysis, we assumed that firms are allocatively efficient to make things simple. 04 For modeling technical change we used both time trend and factor-augmenting
05 approaches. In the primal (quantity based) models we discussed the single output
06 production function approach, as well as input and output distance functions
07 (that accommodates multiple outputs). To allow for the possibility that sample
08 firms might use more than one technology and the analysts might not know who
09 is using which technology, we considered the latent class modeling approach.
10 In addition to estimating technical inefficiency for each firm, this approach is
11 capable of measuring technology gap (distance between the individual/group
12 frontiers from the metafrontier).
13 In the dual cost function approach, our TFP decomposition analysis was done for single and multiple output cost functions. We briefly discussed the modeling 14 issues for the latent class models. Finally, we considered the profit function 15 approach in both single and multiple output frameworks. We also examined the 16 alternative profit function and discuss problems associated with this approach. 17 Finally, we discussed extensions of the standard profit function to accommodate 18 markups in the output markets. 19 Estimation issues are not addressed in this survey. Since one standard tech- 20 nique would not fit all the models, and some of the techniques (especially when 21 the cost and profit function models that require use of a system approach) are 22 quite involved, we decided not to discuss estimation techniques in this sur- 23 vey. However, econometric techniques are available to estimate every model 24 discussed in this survey. 25
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29 Lee, Y. and P. Schmidt, (1993) A production frontier model with flexible temporal
30 variation in technical efficiency, in Fried, H., Lovell, C.A.K. and Schmidt, S. (eds). The Measurement of Productive Efficiency: Techniques and Applications 31 , Oxford: Oxford University Press, pp 3–67. 32 Malmquist, S. (1953) Index numbers and indifference surfaces. Trabajos de Estadistica 33 4: 209–42. 34 McLachlan, G.J. and D. Peel (2000) Finite Mixture Models. New York: John Wiley & Sons. 35 Mester, L. (1997), Measuring efficiency at US banks: Accounting for heterogeneity is 36 important. European Journal of Operational Research 98: 230–424. 37 Nishimizu, M., and J.M. Page Jr. (1982) Total factor productivity growth, technologi- 38 cal progress and technical efficiency change: Dimensions of productivity change in 39 Yugoslavia, 1965–78. Economic Journal 92: 920–36. 40 Nordhouse, W. (2001) Productivity and the new economy, NBER working paper. Elsevier AMS 0BTG02 22-3-2006 8:06a.m. Page: 61
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11 y = f x t ˆ ≡ f x exp − t 12 13 ⇒˙= 1 ˆ ˙ − d + ln f y fjxj xj (A.1) 14 f j dt t 15 16 ⇒ ˙ = − ˙ + − a ˙ − + ˙ TFP RTS 1 jxj j sj xj RTS j ft 17 j j j t 18 ln f 1 − 19 ˆ = − = = = where xj xj exp RTS ln x f fjxje j fjxj/ fkxk, and j j k 20 the dot over a variable represents its rate of change. 21 Since RTS depends on (so is j), the above formula can be rewritten as 22 23 d TFP˙ = RTS0 − 1 0x˙ + f˙ 0 − + misc (A.2) 24 j j t dt 25 26 0 = = 0 = = ˙ 0 = ˙ = where RTS RTS 0 TC TC 0 and ft ft 0 . Finally, 27 the miscellaneous component can be further decomposed into deviations of RTS 28 from RTS0, TC from TC0, etc. 29 Note: The OO and IO measures are identical if the production function is 30 homogeneous. In other words, one is a constant multiple of the other and there 31 is no difference in estimating these models. 32
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11 y = f x t exp −u z t where z does not include x 12 −u 13 ⇒˙= 1 ˙ + ln e ˙ − u + ˙ y fjxjxj zk ft 14 f k ln zk t 15 u ˙ 16 = RTS x˙ + I z − + f j j k k t t 17 j k
18 ln e−u = 19 where I is efficiency change induced by z ln zk k k 20 u 21 ˙ = − ˙ + ˙ − + ˙ + − a ˙ Thus TFP RTS 1 jxj ft Ikzk j sj xj (B.1) 22 j t k j 23 = Scale + TC + TEC + induced by the z variables + price/allocative 24
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11 y = f x t ˆ ≡ f x exp − z t t 12 13 RTS d ⇒˙y = f xˆ x˙ + I z˙ − + f˙ 14 ˆ j j j k k t fkxk j k dt 15 k 16 ln y 17 = = 1 ˆ where RTS f fjxj depends on 18 j ln xj 19 ˙ = − ˙ + ˙ + ˙ 20 Thus TFP RTS 1 jxj ft RTS Ik zk j j k j 21 22 − + − a ˙ RTS j / t j sj xj (C.1) 23 j j 24 = Scale + TC + Induced by z + TEC + Price/Allocative 25 26 Note: Each component of TFP growth depends on . The formula can be 27 rewritten so that the above components, except a residual component, are free 28 from . The residual term will contain , which can be further decomposed into 29 scale, TC, etc. 30
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11 Min wx s t y= f x t e−u 12 x 13 ⇒ Ca = C w yeu 14 ˙ ˙ ˙ u 15 ⇒ TFP = Y 1 − E − C − E (D.1) cy t cy t 16 17 = where Ecy ln C/ ln y. 18 19 With allocative inefficiency 20 21 Production function: yeu = f x t 22 j s 23 f w e w j = j ≡ j = 24 The first-order conditions are s 1 0 f1 w1 w1 25 ⇒ = s u xj xj w ye 26 27 ⇒ s = s = s s u C wjxj C w ye t 28
s wsx 29 ln C = s = j j Shephard’s Lemma: s Sj s 30 ln wj C
31 a = = s s s ≡ s · · 32 Thus C wjxj C wjSj /wj C G j 33 a s 34 ⇒ ln C = ln C + ln G · 35 · ˙ s a s ˙ s s u ln G 36 ⇒ TFP =˙y 1 − E + S − S w˙ − C − E − cy j j j t cy t t 37 j
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39 where Es = ln Cs/ ln y. 40 cy
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11 Production function: y = f x t exp −u z t 12
13 = j = s The first-order conditions fj/f1 wje /w1 wj/w1 14 a s s u z t 15 ⇒ ln C = ln C w ye t + ln G · 16 ˙ s a s ˙ s s u s ln G 17 TFP =˙y 1 − E + S − S w˙ − C − E + I z˙ E − (E.1) cy j j j t cy t k k cy t 18 19 = ln e−u z t where Ik 20 ln zk 21 Note: (i) As before, the above formula can be expressed in terms of scale, TC, 22 TEC, etc., defined at the frontier u = 0 . (ii) Cost function with non-neutral 23 IO technical inefficiency will be similar to the neutral case except for one extra 24 ˙ term involving Ikzk . (iii) Every component depends on u. 25 Rewrite TFP˙ as 26 u 27 ˙ =˙ − 0 − ˙ 0 − 0 + TFP y 1 Ecy Ct Ecy w u y (E.2) 28 t
29 where the miscellaneous component, w u y , can be further decomposed 30 into deviations of technical change, RTS, TEC, etc., from their respective values 31 at the frontier. 32
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