SCHOOL OF ECONOMICS DISCUSSION PAPER

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1 An Economic Justification for the EKS Multilateral Index Kevin J Fox 2000/3 SCHOOL OF ECONOMICS DISCUSSION PAPER ISSN ISBN

2 An Economic Justification for the EKS Multilateral Index by Kevin J. Fox School of Economics The University of New South Wales Sydney, NSW 2052, Australia. Fax: Tel: August 2000 Abstract Solving a problem left by the seminal paper of Caves, Christensen and Diewert (1982), a justification for the use of the EKS multilateral index is given from the economic approach to index numbers. Key words: Multilateral index; Economic approach JEL classification: C43, O47 The insightful comments of Erwin Diewert are gratefully acknowledged.

3 1 Introduction Multilateral index numbers are used for output, input and productivity comparisons across economic entities, such as countries. They statisfy a circularity requirement so that comparisons between two countries are consistent with each country s comparison with, say, a third country. The EKS multilateral index (Eltetö and Köves, 1964; Szulc, 1964) is based on the use of the Fisher Ideal index formula (Fisher, 1922), and has been suggested for making international comparisons. 1 Caves, Christensen and Diewert (1982) (CCD), while introducing the translog multilateral index, made favourable comment about the attractive properties of the EKS method. However, they were unable to provide a theoretical justification for its use from the economic approach to index numbers (Diewert, 1976) along the lines of the justification they gave for the use of the translog multilateral index: It is not known whether the EKS can be derived directly from a flexible transformation function that is non-separable in inputs and outputs and permits nonneutral differences in productivity among countries. (Caves, Christensen and Diewert, 1982, page 83). Further, they note the following in footnote 1 on the same page: It is straightforward to derive the EKS index in the separable, neutral case, but we have not suceeded in deriving EKS in the general case. What they were able to do, however, was to set out in a systematic fashion an economic justification for the translog multilateral index that had already been proposed and implemented, (Caves, Christensen and Tretheway, 1981), based on results for the bilateral translog index from Diewert (1976). The CCD paper has popularized the use of the translog multilateral index to the extent that it is commonly referred to as the CCD index (see e.g. 1 Some authors use the term EKS is refer to the method of making any bilateral index number transitive, whereas here the more common usage is employed so that it refers to the multilateral index based on the bilateral Fisher index. 1

4 Selvanathan and Rao, 1992). More importantly, it has often relegated the EKS index to the relative obscurity of footnotes (e.g., Pilat and Rao, 1996; page 119), in spite of the favourable comments by CCD concerning this index. Coelli, Rao and Battese (1998; page 86) noted that in the multilateral context the CCD index is the form used in most empirical analyses of total factor productivity measurement conducted during the past decade. This seems to be due to the perception that the EKS method is not as well founded in economic theory as the CCD index (Neary, 1997; page 4). The aim of the current paper is of a similar nature to that of the CCD paper. Using results from Diewert (1992) for the bilateral Fisher index, it is shown that, as for the CCD index, the EKS index can be derived directly from a flexible transformation function that is non-separable in inputs and outputs and permits non-neutral differences in productivity among countries. In doing so, it is hoped that it will be clear that the EKS index should no longer be overlooked in favour of the CCD index on the basis of economic justification. An additional contribution is that this paper uses the same (transformation) function to derive the bilateral Fisher input, output and productivity indexes, whereas Diewert (1992) used different functions for each case. 2 This paper is organised as follows. Section 2 derives the EKS output and input indexes from a transformation function which has the Diewert form (Diewert, 1992; Balk, 1998). 3 Section 3 similarly derives the EKS productivity index for multilateral comparisons. 2 EKS Output and Input Comparisons We consider the following specification of a transformation function as a representation of some globally regular technology: F s (y s, x s ) = 1, (1) 2 Besides Diewert (1992), other authors who have recently strengthened the theoretical case for the use of the bilateral Fisher index formula for productivity measurement include Färe and Grosskopf (1990)(1992) and Balk (1993). 3 Balk (1998) dubbed this functional form the Diewert form. 2

5 where for the economic entity s uses the vector of M inputs x s to produce the vector of N outputs y s. 4 The specification of the transformation function in (1) indicates that the structure of production can differ in a non-neutral fashion across economic entities. In what follows, we adopt the convention of Caves, Christensen and Diewert (1982) in refering to these entities as countries, although they could be firms or any other kind of economic organisation. The corresponding Diewert form (Diewert, 1992) for the transformation function (1) can then be written as follows, for countries s = k, l: σ s [(y Ay) 1 x Cx + α s y 1 β s xy 1 B s x] 1/2, (2) where A and C are symmetric parameter matrices, a dot denotes the inner product, and σ s [(y s Ay s ) 1 x s Cx s ] 1/2 = 1 (3) and either B k x k = 0 M ; (y k ) 1 B k = 0 T N; α l (y l ) 1 = 0; β l x l = 0; α k (y l ) 1 β k x l = 0; (y k ) 1 B l x k = 0, (4) or B l x l = 0 M ; (y l ) 1 B l = 0 T N; α k (y k ) 1 = 0; β k x k = 0; α l (y k ) 1 β l x k = 0; (y l ) 1 B k x l = 0, (5) so that the transformation function is flexible at (x k, y k ) under restrictions (4), and flexible at (x l, y l ) under restrictions (5). 5 In addition, it can be verified that this transformation function is non-separable in inputs and outputs and permits non-neutral differences in productivity among countries. 4 See, e.g., Diewert (1973) for the properties of transformation functions. 5 The proof of this follows from the proofs in Diewert (1992) for the flexibility of the Diewert revenue function and the Diewert distance function. See also Balk (1998). A functional form is flexible if it can theoretically approximate an arbitrary twice continuously differentiable, linearly homogeneous function to the second order at a point, (Diewert, 1974; 113). 3

6 If we want to compare output between countries k and l we could use country k as the base country for comparisons, or country l. If we use country l as the base, then we can write the transformation function as: F k (δ l y l, x k ) = 1, (6) so that the output of k relative to l is the maximum possible proportional increase in the elements of the output vector y l (δ l ) given inputs and productivity levels of country k. Using (2), (3) and (4) or (5) we then have: σ k [(δ l y l Aδ l y l ) 1 x k Cx k ] 1/2 = 1, (7) and given that (2) exhibits constant returns to scale, we can solve for δ l : δ l = σ k [(y l Ay l ) 1 x k Cx k ] 1/2 = F k (y l, x k ). (8) Similarly, if we use country k as the base, then we can write: F l (y k /δ k, x l ) = 1, (9) so that the output of k relative to l is the minimum possible proportional decrease in the elements of the output vector y k (δ k ) given inputs and productivity levels of country l. Using (2), (3) and (4) or (5) we then have: σ l [(y k /δ k Ay k /δ k ) 1 x l Cx l ] 1/2 = 1, (10) and given that (2) exhibits constant returns to scale, we can solve for δ k : δ k = 1/σ l [(y k Ay k ) 1 x l Cx l ] 1/2 = 1/F l (y k, x l ). (11) Now, consider the square of the bilateral Fisher output index, Q F ( ), between countries k and l, and using (p s /p s y s ) = y F s (y s, x s ) from Caves, Christensen and Diewert (1982; 77) we proceed as follows: Q F (p l, p k, y l, y k ) 2 = [(p l /p l y l ) y k ]/[(p k /p k y k ) y l ] (12) 4

7 = [y k y F l (y l, x l )]/[y l y F k (y k, x k )] (13) = (σ l ) 2 y k Ay l x l Cx l (y l Ay l ) 2 /(σ k ) 2 y l Ay k x k Cx k (y k Ay k ) 2 (14) = (σ l ) 2 (y l Ay l ) 1 x l Cx l (y l Ay l ) 1 /(σ k ) 2 (y k Ay k ) 1 x k Cx k (y k Ay k ) 1 (15) = y k Ay k (y l Ay l ) 1 (16) = (σ k ) 2 (y l Ay l ) 1 x k Cx k /(σ k ) 2 (y k Ay k ) 1 x k Cx k (17) = F k (y l, x k ) 2 (18) = (δ l ) 2 (19) = y k Ay k (y l Ay l ) 1 (20) = (σ l ) 2 (y l Ay l ) 1 x l Cx l /(σ l ) 2 (y k Ay k ) 1 x l Cx l (21) = 1/F l (y k, x l ) 2 (22) = (δ k ) 2. (23) So, we see that the use of the bilateral Fisher output quantity index can be justified by the Diewert transformation function. This is similar to the Caves, Christensen and Diewert (1982) justification for the use of the translog output quantity index, except that they had to take the geometric mean of their δ l and δ k to prove their result. In the current case, as we can see from above, the Fisher output quantity index can be justified without taking the geometric mean of δ l and δ k, as either can be used by itself to derive the output quantity index. Now, consider the case where we want to make comparisons between countries k and l and a third country, m. It would be desirable for inter-country comparisons to be consistent, or transitive. That is, we would like the following circularity property to hold: δ kl = δ km /δ lm, (24) where δ kl = δ k = δ l from (23) and (19), which measures the difference between the output of country k compared with country l. Equation (24) will not necessarily hold if the bilateral 5

8 Fisher output quantity index is used for comparisons. However, as in the translog case of Caves, Christensen and Diewert (1982), it is possible to modify the definition of output comparisons so that the transitivity requirement in (24) is satisfied. This modification yields the EKS multilateral output quantity index. Let the output of k relative to all countries s = 1,, S, be given by the geometric mean of the bilateral comparisons between k and each country s: δ k = (Π S s=1δ ks ) 1/s = (Π S s=1f k (y s, y k )) 1/s = (Π S s=1f S (y k, x s )) 1/s, (25) where δ ks can also be written as the Fisher output quantity index, as indicated by (12), (19) and (23). The EKS multilateral output quantity index between countries k and l, δkl, is then given by: δkl = δ k / δ l. (26) It can be easily verified that δkl satisfies (24), so that it satisfies the transitivity requirement. Hence, we have shown that the EKS multilateral output quantity index can be derived from the Diewert transformation function, defined by (2), (3) and (4) or (5), which is flexible, non-separable in inputs and outputs and permits non-neutral differences in productivity among countries. We can proceed in a similar fashion in order to make input comparisons. If we want to compare input levels between countries k and l we could use country k as the base country for comparisons, or country l. If we use country k as the base, then we can write: F k (y k, ρ k x l ) = 1. (27) The input of k relative to l is the minimum proportional increase in the elements of the input vector x l (ρ k ) so that y k can be produced using productivity levels of country k. Using (2), (3) and (4) or (5) we then have: σ k [(y k Ay k ) 1 ρ k x l Cρ k x l ] 1/2 = 1, (28) and given that (2) exhibits constant returns to scale, we can solve for ρ k : ρ k = 1/σ k [(y k Ay k ) 1 x l Cx l ] 1/2 = F k (y k, x l ). (29) 6

9 Similarly, if we use country l as the base, then we can write: F l (y l, x k /ρ l ) = 1, (30) so that the input of k relative to l is the maximum proportional decrease in the elements of the input vector x k (ρ l ) given output level y l and productivity levels of country l. Using (2), (3) and (4) or (5) we then have: σ l [(y l Ay l ) 1 x k /ρ l Cx k /ρ l ] 1/2 = 1, (31) and given that (2) exhibits constant returns to scale, we can solve for ρ l : ρ l = σ l [(y l Ay l ) 1 x k Cx k ] 1/2 = F l (y l, x k ). (32) Consider the square of the bilateral Fisher input index, Q F ( ), between countries k and l, and using (w s /w s x s ) = x F s (y s, x s ) from Caves, Christensen and Diewert (1982; 79) we proceed as follows:. Q F (w l, w k, x l, x k ) 2 = [(w l /w l x l ) x k ]/[(w k /w k x k ) x l ] (33) = x k x F l (y l, x l )/x l x F k (y k, x k ) (34) = (σ l ) 2 (y l Ay l ) 1 x k Cx l /(σ k ) 2 (y k Ay k ) 1 x l Cx k (35) = (σ l ) 2 (y l Ay l ) 1 /(σ k ) 2 (y k Ay k ) 1 (36) = (σ l ) 2 (y l Ay l ) 1 x k x k /(σ k ) 2 (y k Ay k ) 1 x k Cx k (37) = (σ l ) 2 (y l Ay l ) 1 x k Cx k (38) = F l (y l, x k ) 2 (39) = (ρ l ) 2 (40) = σ l (y l Ay l ) 1 x l Cx l /σ k (y k Ay k ) 1 x l Cx l (41) = 1/σ k (y k Ay k ) 1 x l Cx l (42) = 1/F k (y k, x l ) 2 (43) = (ρ k ) 2. (44) 7

10 So, we see that the Diewert transformation function can also provide a justification for the use of the bilateral Fisher input quantity index. Now, consider the case where we want to make comparisons between countries k and l and a third country, m. It would be desirable for inter-country comparisons to be consistent, or transitive. That is, we would like the following to hold: ρ kl = ρ km /ρ lm, (45) where ρ kl = ρ k = ρ l from (44) and (40), which measures the difference between the input of country k compared with country l. Equation (45) will not necessarily hold if the bilateral Fisher input quantity index is used for comparisons. However, as in the translog case of Caves, Christensen and Diewert (1982), it is possible to modify the definition of input comparisons so that the transitivity requirement in (45) is satisfied. This modification yields the EKS multilateral input quantity index. Let the input of k relative to all countries s = 1,, S, be given by the geometric mean of the bilateral comparisons between k and each country s: ρ k = (Π S s=1ρ ks ) 1/s = (Π S s=1f k (y k, x s )) 1/s = (Π S s=1f s (y s, x k )) 1/s, (46) where ρ ks also can be written as the Fisher input quantity index, as in (33). multilateral input quantity index between countries k and l, ρ kl, is then given by: The EKS ρ kl = ρ k / ρ l. (47) It can be easily verified that ρ kl satisfies (45), so that it satisfies the transitivity requirement. Hence, we have shown that the EKS multilateral input quantity index can be derived from the Diewert transformation function, defined by (2), (3) and (4) or (5), which is flexible, nonseparable in inputs and outputs and permits non-neutral differences in productivity among countries. 8

11 3 EKS Productivity Comparisons If we use country k as the basis for a comparison of productivity levels between countries k and l, then we can write the transformation function as F l (y k /λ k, x k ) = 1, (48) where λ k is the minimum proportional decrease in the output vector y k so that the resulting output vector is producible with inputs x k and the productivity level of country l. Using (2), (3) and (4) or (5), we can solve (48) for λ k : λ k = 1/σ l [(y k Ay k ) 1 x k Cx k ] 1/2 = 1/F l (y k, x k ). (49) If we use country l as the basis for a comparison of productivity levels between countries k and l, then F k (λ l y l, x l ) = 1 (50) where λ l is the maximum proportional increase in the output vector y l so that the resulting output vector is producible with inputs x l and the productivity level of country k. Using (2), (3) and (4) or (5), we can solve (50) for λ l : λ l = σ k [(y l Ay l ) 1 x l Cx l ] 1/2 = F k (y l, x l ). (51) Now, consider the square of the bilateral Fisher productivity index, which is the ratio of the bilateral Fisher output and input indexes, (i.e., the growth in outputs divided by the growth in inputs): (Q F ) 2 /(Q F ) 2 = [ [(p l /p l y l ) y k ]/[(p k /p k y k ) y l ] ] / [ [(w l /w l x l ) x k ]/[(w k /w k x k ) x l ] ] (52) = [ [y k y F l (y l, x l )]/[y l y F k (y k, x k )] ] / [ [x k x F l (y l, x l )]/[x l x F k (y k, x k )] ] (53) = [ [(σ l ) 2 y k Ay l x l x l (y l Ay l ) 2 ]/[(σ k ) 2 y l Ay k x k x k (y k Ay k ) 2 ] ] 9

12 / [ [(σ l ) 2 x k Cx l (y l Ay l ) 1 ]/[(σ k ) 2 x l Cx k (y k Ay k ) 1 ] ] (54) = (σ k ) 2 /(σ l ) 2 (55) = (σ k ) 2 (y l Ay l ) 1 x l Cx l /(σ l ) 2 (y l y l ) 1 x l Cx l (56) = (σ k ) 2 (y l Ay l ) 1 x l Cx l (57) = F k (y l, x l ) 2 = (λ l ) 2 (58) = (σ k ) 2 /(σ l ) 2 (59) = (σ k ) 2 (y k Ay k ) 1 x k Cx k /(σ l ) 2 (y k y k ) 1 x k Cx k (60) = 1/(σ l ) 2 (y k Ay k ) 1 x k Cx k (61) = 1/F l (y k, x k ) 2 = (λ k ) 2. (62) So, we see that the bilateral Fisher productivity index can be justified through the Diewert transformation function. Now, consider the case where we want to make comparisons between countries k and l and a third country, m. As for output and input comparisons, it would be desirable for inter-country productivity comparisons to be consistent, or transitive. That is, we would like the following to hold: λ kl = λ km /λ lm, (63) where λ kl = λ k = λ l from (58) and (62), which measures the difference between the input of country k compared with country l. Equation (63) will not necessarily hold if the bilateral Fisher productivity quantity index is used for comparisons. However, as in the translog case of Caves, Christensen and Diewert (1982), it is possible to modify the definition of productivity comparisons so that the transitivity requirement in (63) is satisfied. This modification yields the EKS multilateral productivity index. Let the productivity of k relative to all countries s = 1,, S, be given by the geometric mean of the bilateral comparisons between k and each country s: λ k = (Π S s=1λ ks ) 1/s = (Π S s=1f s (y k, x k )) 1/s = (Π S s=1f k (y s, x s )) 1/s, (64) where λ ks can also be written as the Fisher productivity index, as in (52). The EKS multi- 10

13 lateral productivity index between countries k and l, λ kl, is then given by: λ kl = λ k / λ l. (65) It can be easily verified that λ kl satisfies (63), so that it satisfies the transitivity requirement. Hence, we have shown that the EKS multilateral productivity index can be derived from the Diewert transformation function, defined by (2), (3) and (4) or (5), which is flexible, nonseparable in inputs and outputs and permits non-neutral differences in productivity among countries. 11

14 References Balk, B.M. (1993), Malmquist Productivity Indexes and Fisher Ideal Indexes: Comment, The Economic Journal 103, Balk, B.M. (1998), Industrial Price, Quantity, and Productivity Indices: The Micro- Economic Theory and an Application, (Kluwer Academic Publishers). Caves, D.W., L.R. Christensen and W.E. Diewert (1982), Multilateral Comparisons of Output, Input, and Productivity Using Superlative Index Numbers, The Economic Journal 92, Caves, D.W., L.R. Christensen and M.W. Tretheway (1981), U.S. Trunk Air Carriers, : A Multilateral Comparison of Total Factor Productivity, in T.G. Cowing and R.E. Stevenson (eds.), Productivity Measurement in Regulated Industries, (Academic Press, New York). Coelli, T., D.S. Prasada Rao and G.E. Battese (1998), An Introduction to Efficiency and Productivity Analysis, (Kluwer Academic Publishers, Boston/Dordrecht/London). Diewert, W.E. (1973), Functional Forms for Profit and Transformation Functions, Journal of Economic Theory 6, Diewert, W.E. (1976), Exact and Superlative Index Numbers, Journal of Econometrics 4, Diewert, W.E. (1992), Fisher Ideal Output, Input and Productivity Indexes Revisited, Journal of Productivity Analysis 3, Eltetö, Ö. and P. Köves (1964), One Index Computation Problem of International Comparisons, (in Hungarian), Statisztikai Szemle 7. Färe, R. and S. Grosskopf (1990), Theory and Calculation of Productivity Indexes: revisited, (Mimeo). Färe, R. and S. Grosskopf (1992), Malmquist Productivity Indexes and Fisher Ideal Indexes, The Economic Journal 102,

15 Fisher, I. (1922), The Making of Index Numbers, (Houghton-Mifflin, Boston). Neary, J.P. (1997), True Multilateral Indexes for International Comparisons of Purchasing Power and Real Income, Centre for Economic Performance Discussion Paper No. 329, London School of Economics and Political Science. Pilat, D. and D.S. Prasada Rao (1996), Multilateral Comparisons of Output, Productivity, and Purchasing Power Parities in Manufacturing, Review of Income and Wealth 42, Selvanathan, E.A. and D.S. Prasada Rao (1992), An Econometric Approach to the Construction of Generalized Theil-Törnqvist Indices for Multilateral Comparisons, Journal of Econometrics 54, Szulc, B. (1964), Indices for Multiregional Comparisons, (in Polish), Przeglad Statystyczny 3. 13

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