Vol. 87 (2006), No. 1, pp. 73–87 DOI 10.1007/s00712-005-0160-8
Journal of Economics Printed in Austria
A Method for Transitive and Additive Multilateral Comparisons: A Transitive Bennet Indicator Kevin J. Fox Received March 18, 2005; revised version received September 30, 2005 Published online: December 29, 2005 Ó Springer-Verlag 2005
It is shown how the Bennet indicator (or ‘‘index’’) can be made transitive. This is in particular useful for making consistent (profit, cost, price, quantity) comparisons between firms when there are more than two firms and/or more than two periods. The method is given both statistical and economic justifications. Keywords: index numbers, indicators, firm comparisons. JEL Classification: C43, D24.
1 Introduction Modern index-number theory is based on comparisons in terms of ratios. Such index numbers are used to construct various measures of economic activity, such as Gross Domestic Product (GDP) and the Consumer Price Index (CPI). While economists typically prefer to think in terms of ratios (growth rates) in most contexts, in many business contexts it is more natural to think in terms of differences. Such cases include the determination of sources of profit change (from price and quantity changes), and ‘‘variance analysis’’ which determines the sources of deviations of actual values from budgeted or optimal values; see, e.g., Grifell-Tatje´ and Lovell (2003). Cost-benefit analysis is also usually performed in terms of differences (benefits less costs), rather than as ratios of benefits to costs. An advantage of working in differences is that the existence of zero quantities and/or prices in some periods do not cause the problems that
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exist when working with ratios (i.e., division by zero). This may be particularly useful in many business contexts where not all goods may be produced, purchased or transacted in every period. The difference approach to index numbers was developed by a number of authors early last century (e.g., Bennet, 1920; Montgomery, 1929), and was recently ‘‘unearthed’’ and re-invigorated by Diewert (1993; 2005). To distinguish index numbers in terms of differences from index numbers in terms of ratios, the former are referred to as ‘‘indicators." Since this reintroduction of indicators to the economics literature there has been increasing interest in their properties and theoretical applications in a variety of contexts (Chambers and Pope, 1996; Chambers, 1998; Balk, 1999; Grifell-Tatje´ and Lovell, 2000; Chambers, 2002; Balk, Fa¨re and Grosskopf, 2004). One, as yet, under-developed part of the theory on indicators relates to the issue of transitivity, sometimes referred to as the ‘‘circularity’’ property. This property is particularly important when there is no natural ordering (such as time) for data. This occurs in cases such as comparing firms in a single period, or comparing multiple firms over more than one period. In such cross-section and panel-data contexts there is often no obvious firm to use as the base for comparisons. If no such base firm exists then, depending on the properties of the indicator, comparisons between more than three firms may be inconsistent in the sense that inferring the relationship between firms A and B by their relationship with C may give a different answer than a direct comparison between A and B. This transitivity problem is well known in the index-number literature on international comparisons, and hence many methods have been developed to allow transitive comparisons. Commonly used methods include the EKS (Elteto¨ and Ko¨ves, 1964; Szulc, 1964; Gini, 1931), Geary-Khamis (Geary, 1958; Khamis, 1972) and CCD (Caves, Christensen and Diewert, 1982) methods. Transitive index numbers are reviewed and assessed by Balk (1996; 2001), Hill (1997), Diewert (1999), and van Veelen (2002). Such methods are used by agencies such as the World Bank, IMF, OECD, and the International Comparison Project (ICP). Chambers (1998) showed a method by which any indicator may be made transitive. In a new contribution, this paper shows that the method has an interpretation as the solution to a least squares problem; the proposed transformation of bilateral indicators minimizes the sum of squared deviations of the transitive indicators from their bilateral equivalents. The method is then applied to the Bennet indicator to explicitly derive for the
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first time a transitive Bennet indicator. It is shown that the proposed indicator can be used to decompose value differences into indicators of contributions from price and quantity differences, where all indicators are transitive and so provide consistent comparisons across multiple firms. A further justification for the transitive Bennet indicator is provided from economic theory. The proposed indicator will allow for consistent comparisons across firms in cross-section, panel, or even time-series contexts. This method has applications in comparing relative quantities, prices, productivity and efficiency. 2 Is the Bennet Indicator Transitive? Consider the following Bennet quantity indicator: kBkl ¼ ð1=2Þ
N X ðpkn þ pln Þðqkn qln Þ;
ð1Þ
n¼1
where pjn is the price of good n for a firm (or country) j ¼ k; l, and qjn is the corresponding quantity. In a production context, if the N goods include both inputs and outputs then inputs can be treated as negative outputs to give, e.g., the change in real value added. We can consider N to be the full set of goods, but do not require each firm to produce/use each good; an advantage of using differences rather than ratios is that we do not run into problems of dividing by zero when a good is not produced/ used by firms. Through the additive nature of (1), it is possible to drop the sum over the N goods without loss of generality. That is, what holds for a one-good context holds for a many-good context, as the many-good context is simply the sum of many one-good contexts. For transitivity, we require, for some indicator kkl, that the following condition holds: kkl ¼ kkm klm :
ð2Þ
If the indicators have the Bennet form, then kBkm ¼ ð1=2Þðpk þ pm Þðqk qm Þ ¼ ð1=2Þðpk þ pm Þqk ð1=2Þðpk þ pm Þqm
ð3Þ
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and kBlm ¼ ð1=2Þðpl þ pm Þðql qm Þ ¼ ð1=2Þðpl þ pm Þql ð1=2Þðpl þ pm Þqm :
ð4Þ
It is clear that kBkm kBlm 6¼ kBkl , so the Bennet indicator fails to satisfy the transitivity condition given in (2). 3 A Transformation of the Bennet Indicator A popular approach to the transitivity problem in the context of index numbers has been to transform the bilateral indexes by the ‘‘EKS’’ method (Elteto¨ and Ko¨ves, 1964; Szulc, 1964; Gini, 1931; Fox, 2003).1 This method takes the geometric mean of all bilateral comparisons where each firm in turn is taken as the reference firm. The authors who originally (and independently) proposed the EKS method used the Fisher Ideal index (Fisher, 1922) as the bilateral index in their method. Caves et al. (1982) (CCD), used the EKS method, but with the To¨rnqvist bilateral index (To¨rnqvist, 1936). This latter approach (known as the CCD index) has an interesting algebraic outcome in that the application of the EKS method to the To¨rnqvist bilateral index is equivalent to constructing an average firm through which all other firms are compared; see Eq. (17) in Sect. 6 below. We propose a similar transformation method in the context of indicators in order to achieve transitive multilateral comparisons. Let there be a total of S firms, and consider the (arithmetic) average of comparisons between firm k and all S firms: kk ¼ ð1=SÞ
S X
kks ;
ð5Þ
kls :
ð6Þ
s¼1
and similarly for firm l: kl ¼ ð1=SÞ
S X s¼1
1 A simple method of ensuring transitivity exists if there is an obvious firm to which all bilateral comparisons should be made. More accurately, if there is a natural ‘‘base’’ firm for comparisons, the transitivity issue does not even arise. More generally, if a natural (or preferred) ordering for comparisons to take place exists, then again the transitivity problem does not exist. See, e.g., Hill (2001).
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Take the following as our indicator between firms k and l: kkl ¼ kk kl :
ð7Þ
Now, transitivity of this modified indicator holds if the following condition is satisfied: kkl ¼ kkm klm ;
ð8Þ
where kkm and klm are defined similarly to kkl . Hence, we have the following: kkm klm ¼ kk km ðkl km Þ ¼ kk kl ¼ kkl ;
ð9Þ
which satisfies the transitivity condition (8), so that the indicator in (7) is indeed transitive. Note that we have not specified a particular form for the bilateral indicators kks and kls in Eqs. (5) and (6). Hence, the transformation represented by Eqs. (5) to (7) can make any indicator transitive.2 Elteto¨ and Ko¨ves (1964) and Szulc (1964) derived the equivalent of this transformation in the case of index numbers by demonstrating that it is the solution to a least squares problem. This required taking a log-linear approximation to a bilateral index-number formula. In the context of indicators, a similar method of derivation can be used by considering a simple linear approximation to a bilateral indicator. Specifically, we use the following equation to approximate the structure of the bilateral indicator: kkl ¼ ak al þ kl ;
ð10Þ
where the aj , j ¼ k; l, are unknown parameters, and the kl are unknown error terms. Using least squares (with errors uncorrelated and of equal variance), then it is a standard result that the estimates ^aj satisfy: ^ak ^al ¼ ð1=SÞ
S X ðkks kls Þ
ð11Þ
s¼1
¼ kk kl ¼ kkl ; 2 This was also noted by Chambers (1998).
ð12Þ
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which is the transitive indicator of Eq. (7). Thus, the proposed transformation of bilateral indicators minimizes the sum of squared deviations of the transitive indicators from their bilateral equivalents.3 This is a slightly neater result than the index-number case of Elteto¨ and Ko¨ves (1964), and Szulc (1964), as it has been derived in levels rather than logarithms.4 While any bilateral indicator can be used in Eqs. (5) to (7), the bilateral Bennet indicator has been shown to have a strong justification in terms of the mathematical properties that it satisfies (Diewert, 2005). Hence, form, consider the case where kks has the Bennet P the bilateral indicator P P and let pq ð1=SÞ ps qs , q ð1=SÞ qs and p ð1=SÞ ps . Then Eq. (5) can be written as kB ¼ ð1=2Þð1=SÞ k
S X ðpk þ ps Þðqk qs Þ s¼1
S X ðpk qk ps qs þ ps qk pk qs Þ ¼ ð1=2Þð1=SÞ s¼1
k pk qÞ ¼ ð1=2Þðpk qk pq þ pq k qÞ þ p q pq: ¼ ð1=2Þ½ðpk þ pÞðq
ð13Þ
Similarly, Eq. (6) becomes kB ¼ ð1=2Þ½ðpl þ pÞðq l qÞ þ p q pq: l
ð14Þ
Then, it follows from (7), (13) and (14) that the modified (transitive) Bennet indicator is B B kB kl ¼ kk kl k qÞ ð1=2Þðpl þ pÞðq l qÞ: ¼ ð1=2Þðpk þ pÞðq
ð15Þ
This shows that this transitive Bennet indicator has a nice intuition. It compares both k and l with an average firm, then compares the firms with each other. Thus the firms have a common firm as the reference firm through which comparisons are made. 3 A referee notes that the same result can be derived by a relatively straightforward application of projection theory. 4 For the (index number) case of errors that are correlated and of unequal variance, see Cuthbert (2003).
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For completeness, we note that in the multiple-good context the multilateral Bennet indicator in (15) is written as follows: kB kl ¼ ð1=2Þ
N N X X ðpkn þ pn Þðqkn qn Þ ð1=2Þ ðpln þ pn Þðqln qn Þ: n¼1
n¼1
ð16Þ This result for the Bennet indicator is analogous to the CCD index (Caves et al., 1982), where the To¨rnqvist index comparisons are also made to an average firm. The CCD index, QT kl , can be written as follows: ln QT kl ¼ ð1=2Þ
N X ðRkn þ Rn Þðln qkn ln qn Þ n¼1
ð1=2Þ
N X ðRln þ Rn Þðln qln ln qn Þ n¼1
¼
ln QT km
ln QT lm ;
ð17Þ
where, as before, a bar denotes an arithmetic mean over S firms, and Rkn is the revenue share of good n for firm k. The similar structure of the transitive Bennet indicator with the CCD index can be noted by comparing Eqs. (17) and (16). 4 Decomposing Value Differences into Price and Quantity Contributions Bennet (1920) originally proposed his indicator in order to decompose value changes. Hence, a change in values (profits, net revenue, costs) can be decomposed into contributions from price differences and quantity differences using a sum of a Bennet price indicator and a Bennet quantity indicator. In a comparison of, e.g., the profits of firms k and l, where the quantitiesP are negative if the good is an input, and using the notation pk qk ¼ Nn¼1 pkn qkn , we have pk qk pl ql ¼ ð1=2Þ
N X ðqkn þ qln Þðpkn pln Þ n¼1
þ ð1=2Þ
N X ðpkn þ pln Þðqkn qln Þ: n¼1
ð18Þ
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The first term on the right-hand side of (18) is a Bennet price indicator, and the second term is the corresponding quantity indicator familiar from Eq. (1). As noted in Sect. 2, the price and quantity indicators in Eq. (18) are not transitive. Consider then the method of Sect. 3, where comparisons are initially made to an average firm. Then the value-difference decompositions for firms k and l become pk qk p q ¼ ð1=2Þ
N X ðqkn þ qn Þðpkn pn Þ n¼1
þ ð1=2Þ
N X ðpkn þ pn Þðqkn qn Þ;
ð19Þ
n¼1
and pl ql p q ¼ ð1=2Þ
N X ðqln þ qn Þðpln pn Þ n¼1
þ ð1=2Þ
N X ðpln þ pn Þðqln qn Þ:
ð20Þ
n¼1
A comparison of values between the firms can then be written as pk qk pl ql ¼ pk qk p q ðpl ql p qÞ ¼ ð1=2Þ
N X ðqkn þ qn Þðpkn pn Þ n¼1
ð1=2Þ
N X ðqln þ qn Þðpln pn Þ n¼1
N X ðpkn þ pn Þðqkn qn Þ þ ð1=2Þ n¼1
ð1=2Þ
N X ðpln þ pn Þðqln qn Þ:
ð21Þ
n¼1
The second and third lines of (21) make up the transitive Bennet price indicator, and the fourth and last lines are the transitive Bennet quantity indicator of Eq. (16).
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Hence, the proposed multilateral indicator can be used to decompose value differences into indicators of contributions from price and quantity differences, where all indicators are transitive and so provide consistent comparisons across multiple firms.
5 An Economic Assessment There are two main approaches to assessing the properties of index numbers – the economic approach and the axiomatic (or ‘‘test’’) approach. The economic approach to index numbers considers functions of economic concepts, (such as cost, production and profit functions), and for particular specifications of these functions derives the corresponding index number. Thus the To¨rnqvist index is ‘‘exact’’ for the translog functional form, as it can be derived from the geometric mean of two theoretical indexes specified as the ratio of translog functions. Because translog functional forms are ‘‘flexible’’ in terms of the technology that they can approximate, the To¨rnqvist index is an example of a ‘‘superlative’’ index (Diewert, 1976). This section follows the economic approach by examining whether or not the transitive Bennet indicator from the previous section can be given a justification from microeconomic theory.5 Specifically, we are interested in finding if there are any functional forms for which it has an ‘‘exact,’’ or ‘‘superlative,’’ relationship with some functional form. This approach requires making some assumptions about the underlying technology. The previous section derived the transitive Bennet indicator without relying on any such assumptions. However, if making such assumptions allows the establishment of a link with economic theory, then this is of use in assessing the properties of the indicator. For each firm j, the net revenue (or ‘‘restricted profit’’) function can be written as Rðpj ; xj Þ maxfpj qj : xj f ðqj Þg; qj
ð22Þ
5 The axiomatic justification for the bilateral Bennet indicator provided by Diewert (2005) extends to the transitive version in Eq. (16), which is just the difference between two bilateral indicators where comparisons are being made to an average firm.
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where qj ½qj1 ; . . . ; qjN is a vector of ‘‘netputs,’’ with qjn > 0 if good n is an output and qjn < 0 if good n is a variable (in the short run) input. As before, pj is the vector of prices corresponding to the quantity vector qj. xj ¼ f ðqj Þ is the minimum amount of a fixed (in the short run) input needed given the vector qj.6 In addition to optimising behaviour (net revenue maximisation subject to an input constraint), we follow the other typical assumption in the economic approach to index number theory (Samuelson and Swamy, 1974; Diewert, 1976; Diewert, 2005, p. 21), of homogeneity of degree one so that Rðpj ; xj Þ xj Rðpj ; 1Þ xj rðpj Þ;
ð23Þ
where rðpj Þ is the firm’s unit input net revenue function. We consider an indicator which has the Bennet form, but which is expressed in terms of unit revenue functions (rðpk Þ and rðpl Þ) and the corresponding input requirements functions (f ðqk Þ and f ðql Þ), where pj qj ¼ f ðqj Þrðpj Þ, j ¼ k; l. Then an economic indicator of quantity (‘‘volume’’) change between l and k, VE ðpl ; pk ; ql ; qk Þkl , can be written as follows (Diewert, 2005): VE ðÞkl ¼ ð1=2Þ½rðpk Þ þ rðpl Þ½f ðqk Þ f ðql Þ;
ð24Þ
where rðpk Þ and rðpl Þ are treated as the aggregate prices for firms k and l, and f ðqk Þ and f ðql Þ are the corresponding quantities. The quantity indicator in (24) is clearly not transitive. Applying the same transformation as in Eqs. (5) and (6), we get: VE ðÞk ¼ ð1=SÞ
S X
VE ðÞks ;
ð25Þ
VE ðÞls :
ð26Þ
s¼1
and VE ðÞl ¼ ð1=SÞ
S X s¼1
6 This is usually taken to be a capital aggregate, but could be any input which is relatively fixed in the short run.
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Then consider the following modified economic indicator of quantity change: VE ðÞkl ¼ VE ðÞk VE ðÞl ¼ ð1=2Þð1=SÞ
S X ½rðpk Þ þ rðps Þ½f ðqk Þ f ðqs Þ s¼1
ð1=2Þð1=SÞ
S X ½rðpl Þ þ rðps Þ½f ðql Þ f ðqs Þ s¼1
¼ VE ðÞkm VE ðÞlm ;
ð27Þ
which satisfies transitivity. We now see if we can establish a relationship between the economic quantity indicator VE ðÞkl and the Bennet quantity indicator kB kl in Eq. (16). We re-express (27) as follows: VE ðÞkl ¼
¼
S 1 X rðpk Þ f ðqk Þ 1 rðps Þf ðqs Þ 1 þ 2S s¼1 rðps Þ f ðqs Þ S 1 X rðpl Þ f ðql Þ 1 rðps Þf ðqs Þ 1 þ 2S s¼1 rðps Þ f ðqs Þ S 1 X ps qs ½1 þ Pks ½Qks 1 2S s¼1
S 1 X ps qs ½1 þ Pls ½Qls 1; 2S s¼1
ð28Þ
where the last line follows if an ‘‘exact’’ relationship of the form Pks ¼ P ðps ; pk ; qs ; qk Þ
rðpk Þ rðps Þ
ð29Þ
exists for some price index-number formula Pks and corresponding functional form for rðpj Þ, and a similar exact relationship of the form Qks ¼ Qðps ; pk ; qs ; qk Þ
f ðqk Þ f ðqs Þ
ð30Þ
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exists for some quantity index-number formula Qks and corresponding functional form for f ðqj Þ. The well-known Fisher Ideal index has such an exact relationship with the homogeneous quadratic functional form (Diewert, 1976). As a quadratic function is a flexible functional form, the Fisher Ideal index is superlative. Equation (28) involves the difference of two indicators which are sums over functions of bilateral index numbers. If all of the bilateral index numbers in (28) are superlative, the transitive indicator VE ðÞkl is called a ‘‘superlative indicator,’’ following the terminology of Diewert (2005) for the bilateral indicator case. Proposition 9 of Diewert (2005) shows that the Bennet indicator can approximate any superlative indicator to the second order around an equal price (pk ¼ ps ) and quantity (qk ¼ qs ) point. Due to this ability to approximate a superlative indicator, Diewert labelled the Bennet indicator ‘‘pseudosuperlative.’’ As the transitive indicator in (28) involves the sum of superlative indicators, the bilateral Bennet indicator can approximate each, so that (16) is also pseudosuperlative.7 Thus, the transitive Bennet indicator has the desirable property of being able to approximate a function of some underlying general technology quite closely. 6 A Numerical Example We consider a simple example, where there are three firms (k, l, and m), and two commodities (n ¼ 1; 2), with prices and quantities as follows: ðpk1 ; pk2 Þ ð1; 1Þ;
ðpl1 ; pl2 Þ ð1; 1:5Þ;
ðqk1 ; qk2 Þ ð1; 2Þ;
ðql1 ; ql2 Þ ð1; 2Þ;
ðpm1 ; pm2 Þ ð1:5; 1Þ; ðqm1 ; qm2 Þ ð2; 5Þ:
ð31Þ
Using the bilateral Bennet indicator of Eq. (1), the bilateral comparisons are as follows: kBkl ¼ 0;
kBkm ¼ 4:25;
kBlm ¼ 5:
ð32Þ
From (32), it is clear that kBkl 6¼ kBkm kBlm ¼ 0:75 so that the transitivity condition in Eq. (2) is not satisfied. 7 A similar economic justification for the multilateral Bennet indicator could have been derived using the results of Balk, Fa¨re and Grosskopf (2004), that showed the Bennet indicator to be ‘‘exact’’ for a (normalized) quadratic function.
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Substituting the numbers in (31) into the transitive Bennet indicator of Eq. (16) we get: kB kl ¼ 0:25;
kB km ¼ 4:5;
kB lm ¼ 4:75:
ð33Þ
B B From (33), we see that kB kl ¼ kkm klm ¼ 0:25, which empirically demonstrates transitivity of the modified indicator. The above numbers are interpreted as weighted means of differences in quantities across firms. As in the bilateral Bennet indicator case, some normalisation of these transitive indicators can be made if that is required by the specific context; see, e.g., Chambers (2002). For comparison purposes, it is of interest to compare the results with a standard multilateral index-number method. Using the CCD index of Eq. (17), we get the following results:
QT kl ¼ 1:0031;
QT km ¼ 0:4316;
QT lm ¼ 0:4302:
ð34Þ
From the numbers in (34), it can be verified (with a slight discrepancy due T T to rounding error) that QT kl ¼ Qkm =Qlm ¼ 1:0031. Note that index numbers are centred around one, while indicators are centred around zero. A comparison of (33) and (34) reveals that both the indicator and index approaches give similar relative comparisons between the quantity vectors, the difference being the scale on which they are measured. 7 Conclusion A new, transitive, indicator function for multilateral comparisons has been proposed. It is based on the bilateral Bennet indicator for quantity and price comparisons, and can be derived from solving a simple least-squares problem. An additional justification for the transitive Bennet indicator was given from the economic approach to indicators, drawing on the results of Diewert (2005) for the bilateral case. While the results have been presented in terms of a quantity indicator, similar results can be shown for price and productivity indicators. In addition, if the ‘‘budgeted values’’ in the variance analysis are predicted values, then transitive indicators with efficiency-analysis applications can also be derived.
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Acknowledgements The author thanks Bert M. Balk and two referees for helpful comments, the Universitat Auto´noma de Barcelona and the University of Valencia for hospitality, and the Australian Research Council and the Ministry of Education and Science of Spain (Secretarı´a de Estado de Universidades e Investigacio´n, SAB2003-0234) for financial support. References Balk, B. M. (1996): ‘‘A Comparison of Ten Methods for Multilateral International Price and Volume Comparisons.’’ Journal of Official Statistics 12: 199– 222. Balk, B. M. (1999): ‘‘On the Decomposition of Cost Variation.’’ Presented at the 2000 North American Productivity Workshop, Union College. Balk, B. M. (2001): ‘‘Aggregation Methods in International Comparisons: What Have We Learned?’’ Report Series Research in Management ERS-2001-41MKT (Erasmus Research Institute of Management, Erasmus University Rotterdam). Balk, B. M., Fa¨re, R., and Grosskopf, S. (2004): ‘‘The Theory of Price and Quantity Indicators.’’ Economic Theory 23: 149–164. Bennet, T. L. (1920): ‘‘The Theory of Measurement of Changes in Cost of Living.’’ Journal of the Royal Statistics Society 83: 455–462. Caves, D. W., Christensen, L. R., and Diewert, W. E. (1982): ‘‘Multilateral Comparisons of Output, Input, and Producitivity Using Superlative Index Numbers.’’ The Economic Journal 92: 73–86. Chambers, R. G. (1998): ‘‘Input and Output Indicators.’’ In Index Numbers: Essays in Honor of Sten Malmquist, edited by R. Fa¨re, S. Grosskopf and R. R. Russell. Amsterdam: Kluwer Academic Publishers. Chambers, R. G. (2001): ‘‘Consumers’ Surplus as an Exact and Superlative Welfare Indicator.’’ International Economic Review 42: 105–120. Chambers, R. G. (2002): ‘‘Exact Nonradial Input, Output, and Productivity Measurement.’’ Economic Theory 20: 751–765. Chambers, R. G., and Pope, R. D. (1996): ‘‘Aggregate Productivity Measures.’’ American Journal of Agricultural Economics 78: 1360–1365. Cuthbert, J. R. (2003): ‘‘On the Variance/Covariance Structure of the Log Fisher Index, and Implications for Aggregation Techniques.’’ Review of Income and Wealth 49: 69–88. Diewert, W. E. (1976): ‘‘Exact and Superlative Index Numbers.’’ Journal of Econometrics 4: 114–145. Diewert, W. E. (1993): ‘‘The Early History of Price Index Research.’’ In Essays in Index Number Theory, edited by W. E. Diewert and A. O. Nakaura. Amsterdam: North-Holland. Diewert, W. E. (1999): ‘‘Axiomatic and Economic Approaches to International Comparisons.’’ In International and Interarea Comparisons of Income, Output
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and Prices, edited by A. Heston and R. E. Lipsey. Chicago: Chicago University Press, NBER. Diewert, W. E. (2005): ‘‘Index Number Theory Using Differences Rather Than Ratios.’’ American Journal of Economics and Sociology 64(1): 347–395. ¨ ., and Ko¨ves, P. (1964): ‘‘One Index Computation Problem of InternaElteto¨, O tional Comparisons’’ (in Hungarian). Statisztikai Szemle 7: 507–518. Fisher, I. (1922): The Making of Index Numbers. Boston: Houghton-Mifflin. Fox, K. J. (2003): ‘‘An Economic Justification for the EKS Multilateral Index.’’ Review of Income and Wealth 49: 407–413. Geary, R. G. (1958): ‘‘A Note on Comparisons of Exchange Rates and Purchasing Power between Countries.’’ Journal of the Royal Statistical Society A: 97–99. Gini, C. (1931): ‘‘On the Circular Test of Index Numbers.’’ Metron 4: 3–162. Grifell-Tatje´, E., and Lovell, C. A. K. (2000): ‘‘Cost and Productivity.’’ Managerial and Decision Economics 21: 19–30. Grifell-Tatje´, E., and Lovell, C. A. K. (2003): ‘‘The Managers Versus the Consultants.’’ Scandinavian Journal of Economics 105(1): 119–138. Hill, R. J. (1997): ‘‘A Taxonomy of Multilateral Methods for Making International Comparisons of Prices and Quantities.’’ Review of Income and Wealth 43(1): 49–69. Hill, R. J. (2001): ‘‘Measuring Inflation and Growth Using Spanning Trees.’’ International Economic Review 42: 167–185. Khamis, S. H. (1972): ‘‘A New System of Index Numbers for National and International Purposes.’’ Journal of the Royal Statistical Society A: 96–121. Montgomery, J. K. (1929): ‘‘Is There a Theoretically Correct Price Index of a Group of Commodities?’’ Rome: Roma L’Universale Tipogr. Poliglotta (privately printed paper, 16 pages). Samuelson, P. A., and Swamy, S. (1974): ‘‘Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis.’’ American Economic Review 64: 566–593. Szulc, B. (1964): ‘‘Indices for Multiregional Comparisons’’ (in Polish). Przeglad Statystyzcny 3: 239–254. To¨rnqvist, L. (1936): ‘‘The Bank of Finland’s Consumption Price Index.’’ Bank of Finland Monthly Bulletin 10: 1–8. van Veelen, M. (2002): ‘‘An Impossibility Theorem Concerning Multilateral International Comparisons of Volumes.’’ Econometrica 70: 369–375. Address of author: – Kevin J. Fox, The University of New South Wales, Centre for Applied Economic Research and School of Economics, Sydney, NSW 2052, Australia (e-mail:
[email protected])
