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Purchasing power parity methods of making international comparisons Hill, Robert J. 1994

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PURCHASING POWER PARITY METHODS OF MAKINGINTERNATIONAL COMPARISONSByRobert J. HillB.A. (Economics and Econometrics) University of York (1990)M.A. (Economics) University of British Columbia (1991)A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESECONOMICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJune 1995©RobertJ. Hill, 1995In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.EconomicsThe University of British Columbia2075 Wesbrook PlaceVancouver, CanadaV6T 1Z1Date:T26 t95AbstractThe objective of this dissertation is to improve our understanding of the various Purchasing Power Parity (PPP) methods that have been advocated in the literature oninternational comparisons. The first of three essays builds on the pioneering work of VanYzeren(1987) to rationalize the literature, by constructing a taxonomy of PPP methods.In particular, the taxonomy reinterprets PPP methods in a graph theoretic context. Thisreinterpretation yields many useful insights.The second essay was motivated by the realization that virtually all PPP methodshave the same underlying graph theoretic structure. This essay develops a new PPPmethod which allows the data to choose the underlying structure by using Kruskal’s“Minimum Spanning Tree” Graph Theory algorithm to chain PPPs across countriesrather than imposing the structure ex ante. The Minimum Spanning Tree (MST) methodmay potentially dramatically simplify the procedure for constructing PPPs. The MSTmethod also has important implications for time series comparisons. The essay concludeswith an empirical comparison using 1990 OECD data between the MST method and thethree most widely used PPP methods.The third essay focuses specifically on the Average Price class of PPP methods identified in the taxonomy. Average Price methods have the very desirable property ofgenerating quantity indices that literally add up over different levels of aggregation whenmeasured in value terms. However, it is widely claimed that Average Price methodsoverestimate the output shares of any outlier countries in a comparison. This is theso-called Gerschenkron effect. In spite of its significant implications, evidence for theGerschenkron effect remains largely anecdotal. This essay explains the reasoning behind11the Gerschenkron effect. As part of this explanation it is necessary to give a preciseinterpretation to the hitherto vague notion of an “outlier” country. Also frameworks aredeveloped for empirically verifying and measuring the Gerschenkron effect, which arethen applied to 1990 OECD data, with some surprising results.111AbstractTable of ContentsIiList of TablesList of FiguresAcknowledgement1 Introductionviiviiiix12.1 Introduction2.2 Methodology and Notation2.3 The Exchange Rate Method2.4 Mean Value Share Methods2.5 Chain Methods2.6 Star Methods2.7 Symmetric Star Methods2.7.1 Average Basket Methods .2.7.2 Average Price Methods . .2.7.3 Törnqvist Star Methods .2.7.4 Fisher Star Methods2.7.5 The Van Yzeren Methods2.8 Asymmetric Star Methods10101214161718192022262729322 A Taxonomy of Multilateral PPP Methods.iv2.9 Mean Asymmetric Star Methods 332.9.1 Mean Quantity Index Methods 332.9.2 Mean Output Share Methods 362.10 Conclusion 373 Chained PPPs and Minimum Spanning Trees 393.1 Introduction 393.2 Methodology and Notation 423.3 Chain Methods 433.4 Malmquist and Konüs Indices 473.5 The Paasche-Laspeyres Spread (PLS) Index 493.6 The Minimum Spanning Tree Method 503.7 Applying the MST Method 533.8 Prior Restrictions 543.9 Sensitivity Analysis 563.10 A Comparison between the MST, Geary-Khamis, EKS and ECLAC Methods 593.11 Conclusion 614 Additive PPP Methods and the Gerschenkron Effect 654.1 Introduction 654.2 Notation 684.3 Average Price - (Additive) Methods 694.4 Average Basket Methods 724.5 The Intuition Behind the Gerschenkron Effect 744.6 Measuring the Gerschenkron Effect Directly from the Data 814.7 Gerschenkron Bias and Outlier Countries 834.8 Empirical Verification of the Gerschenkron Effect 88V4.9 Discriminating Between Additive PPP Methods 934.10 Conclusion 95Bibliography 97Appendix 102A Proofs to Chapter 2 102B Tables for Chapter 3 104C Tables for Chapter 4 112viList of Tables1.1 Spearman Rank Correlation Coefficients of Per Capita Income 4B.1 105B.2 106B.3 107B.4 108B.5 109B.6 110B.7 111C.1C.2C.3C.4C.5Paasche-Laspeyres Spread (PLS) IndicesFisher PPPsMinimum Spanning Tree PPPsMinimum Spanning Tree PPPs with Germany Deleted .Geary-Khamis PPPsEKS PPPsECLAC PPPsAllen-Diewert IndicesOutput SharesPaasche-Laspeyres Spread (PLS) IndicesRegression ResultsEstimates of Gerschenkron Bias113114115116117viiList of Figures1.1 Sensitivity of Per Capita Income to the Choice of Method 52.1 The Taxonomy 152.2 Examples of Spanning Trees 182.3 The Star Spanning Tree 193.1 Examples of Spanning Trees 453.2 Constructing Multilateral Quantity Indices using the Star Spanning Tree 463.3 The Minimum Spanning Tree for the OECD 553.4 The Minimum Spanning Tree for the OECD with Germany Deleted 584.1 The Consumer Substitution Effect for Average Price Methods 754.2 The Producer Substitution Effect for Average Price Methods 774.3 The Consumer Substitution Effect for Average Basket Methods 794.4 The Producer Substitution Effect for Average Basket Methods 814.5 The Gerschenkron Effect (Turkey) 904.6 The Gerchenkron Effect (USA) 91viiiAcknowledgementI would like to express my gratitude to my supervisory committee, Erwin Diewert, JohnCragg and Ken White, for their intellectual and emotional support at each stage of thisthesis. I am particularly grateful to Erwin Diewert for his questions and comments, whichled me to think more deeply about the subject and helped me improve my arguments.I would also like to thank my external examiner Bert Balk, for carefully reading thisdissertation and making numerous useful suggestions.This thesis would not have been possible without the assistance of the StatisticsDirectorate of the OECD, which sllpplied me with all my data. However, most of all Iwish to thank my father Peter Hill for stimulating my interest in Economics in generaland Purchasing Power Parities in particular.ixChapter 1IntroductionThe types of international comparisons addressed here are comparisons of output, percapita income and the purchasing power of currencies across countries. A PurchasingPower Parity (PPP) between two currencies A and B equals the number of units ofcurrency B that have the same purchasing power as one unit of currency A. Such comparisons are important in both the fields of international and development economics.For example, in international economics, comparisons of the purchasing power of currencies across countries are important to theories of exchange rate determination. Indevelopment economics, comparisons of per capita income across countries are relevantto discussions on global poverty, inequality and growth. Lucas(1988) goes further andargues that the central problem in development economics is to explain the observeddifferences in per capita income across couitries.By the problem of economic development I mean simply the problem of accounting for the observed pattern, across countries and across time, in levelsand rates of growth of per capita income. (Lucas, 1988)A natura.l way of making such comparisons is to use exchange rates to convert nationalGDP data into units of the same numeraire currency. However, there are problemswith this approach. In addition to being highly volatile, exchange rates also have aninherent bias. This is the so-called Balassa(1964)-Samuelson(1964) hypothesis, whichstates tha.t exchange rates have a systematic tendency to undervalue the purchasingpower of currencies in poorer countries relative to richer countries. This is because many1Chapter 1. Introduction 2services which tend to be relatively cheaper in poorer countries, generally are not tradedinternationally.1 The Balassa-Samiielson hypothesis implies that comparisons of realincome across countries that use exchange rates tend to underestimate real income levelsin poor countries. The World Bank Development Report(1983) is an example of such acomparison. Indeed the report prompted Lucas(1988) to make the following observation:The diversity across countries in measured per capita income levels is literallytoo great to be believed. Compared to the 1980 average for what the WorldBank calls the “industrialized market economies” of US $10 000. India’s percapita income is $240, Haiti’s is $270, and so on for the rest of the verypoorest countries. This is a difference of 40 in living standards! These latterfigures are too low to sustain life in say, England or the United States, sothey cannot be taken at face value.(Lucas, 1988)Alternatively, Purchasing Power Parities (PPPs) may be used instead of exchange ratesto convert national GDP data into the numeraire currency. The advantage of PPPs isthat they are not subject to the Balassa-Samuelson hypothesis, and are less volatile thanexchange rates.There is undoubtedly a growing interest in PPP based methods which are now beingused increasingly by the World Bank, IMF and OECD to make international comparisons.In particular, the IMF and OECD recently started using PPPs in their economic surveys.PPP based international comparisons are also being extended to the countries of theformer Soviet Union, with the help of the OECD and World Bank. The results of suchcomparisons influence international loan, aid and investment decisions.However, PPP based methods also have their problems. Firstly, there is the problemof collecting and harmonizing price and expenditure data over a large sample of goods1See for example Bhagwati(1984) or T.P.Hill(1986).chapter 1. Introduction 3and services across countries. For most PPP methods it is necessary that all countriessupply data on the same set of basic heading goods and services.2 This requirementcreates serious methodological problems, since a staple good in one country may be rareor even unobtainable in another country. This problem is reduced by allowing each basicheading category to cover a selection of similar goods and services. Two procedures havebeen advocated for aggregating up to the basic heading level. The Country ProductDummy (CPD) method was advocated by Kravis, Heston and Summers. It is discussedin Kravis, Heston and Summers(1982). Eurostat advocated a variant on the Eltetö,Köves and Szulc (EKS) method. It is discussed in Eurostat(1983). This dissertation forthe most part does not address aggregation problems below the basic heading level.Secondly there is the problem of how to proceed, once a common set of basic headinggoods and services has been agreed. There are a large number of competing methods,and there is still widespread disagreement over which is the best method. For example, the International Comparison Project (ICP), the World Bank and the IMF usethe Geary(1958)-Kharnis(1972) method. However, the World Bank is now consideringswitching to the Iklé(1972) method. The OECD currently uses both the Geary-Khamisand EKS methods. Eurostat has used the Geary-Khamis and Gerardi methods, but nowprefers the EKS method. Meanwhile the UN has used the ECLAC method to makecomparisons in Latin America.3 In addition, other methods such as the MultilateralTranslog, Own Share and Van Yzeren balanced method have been advocated by expertsin the literature.42The Minimum Spanning Tree method advocated in Chapter 3 is an exception to this rule.3The EKS method was first proposed by Gini(1931). It was later independently advocated by Eltetöand Köves(1964) and Szulc(1964) and popularized by Drechsler. The Gerardi method was first proposedby Walsh(1901). It was popularized by Gerardi(1982) and Eurostat(1983). ECLAC stands for UnitedNations Economic Commission for Latin America and the Caribbean. The ECLAC method was advocated by Walsh(1901), who called it Scrope’s method with arithmetic weights. For a detailed discussionon the evolution of the literature, see Diewert(1993b).4See Diewert(1986) and BaIk(1995).Chapter 1. Introduction 4ER ECLAC EKS GKER 1 0.777 0.805 0.832ECLAC 1 0.990 0.970EKS 1 0.988GK 1Table 1.1: Spearman Rank Correlation Coefficients of Per Capita IncomeIt must be emphasized that the results of international comparisons can differ dramatically depending on the method used to make the comparison. For example, as a resultof using the Geary-Khamis method rather than exchange rates, the May 1993 issue ofthe World Economic Outlook published by the IMF, found that the global GDP share ofthe industrialized countries fell from 73 to 54 per cent, while the share of poor countriesrose dramatically. In particular, China’s share rose from 2 to 6 per cent. Similar resultsare obtained from 1990 OECD data. The share of total output of the OECD’s poorestmember Turkey, rose from 0.57 to 1.53 per cent as a result of switching from exchangerates to Geary-Khamis. However as Figure 1.1 reveals, these results overstate the sensitivity of results to the choice of PPP method. Figure 1.1 depicts the ratio of per capitaincome of each OECD country relative to Turkey’s per capita income as measured byexchange rates and the ECLAC, EKS and Geary-Khamis PPP methods. The ECLAC,EKS and Geary-Kharnis methods are to date probably the three most widely used PPPmethods.5 There is a broad consensus between the three PPP methods in Figure 1.1.This consensus is borne out by the matrix of Spearman Rank Correlation Coefficients ofper capita income in Table 1.1. The divergence between exchange rate and PPP resultscan be attributed to a combination of the Balassa-Samuelson hypothesis and the inherentvolatility of exchange rates.5These methods are discussed in detail in Chapter 2. The ECLAC method is discussed in section 2.7.1on Average Basket methods, the EKS method in section 2.9.1 on Mean Quantity Index Methods, whilethe Geary-Khamis method in section 2.7.2 on Average Price methods.Chapter1Introduction14 12 10C) c8C) C C)6=- L)2 0-CC-ECD--E-L)t/)=L)CE-.-_L)L)-LZ‘C———c_CountriesOrderedbyPerCapitaIncomeasMeasuredbyGeary—KhamisFigure1.1: SensitivityofPerCapitaIncometotheChoiceofMethod——---—ER-ECLAC—.———EKS—0-———GKc-flChapter 1. Introduction 6In spite of the high correlation coefficients between PPP methods in Table 1.1, theresults are nevertheless sensitive to the choice of PPP formula. although admittedly notto the same extent as if PPPs are compared with exchange rates. For example, Turkey’sshare of total OECD output is 1.53 per cent according to Geary-Khamis as comparedwith 1.04 per cent according to ECLAC. Hence Turkey’s share of total OECD output is50 per cent higher according to Geary-Khamis than according to ECLAC. Moreover thedifference between Geary-Khamis and ECLAC results would in all likelihood rise if thecomparison was extended to a more heterogenous group of countries.These examples demonstrate the importance of the choice of method. Furthermore,such large discrepencies in the results of international comparisons clearly hinder theattempts of economists to empirically test the predictions of theoretical models in international and development economics.The objective of this dissertation is to improve the quality of international comparisons based on PPP methods. Firstly, the dissertation rationalizes the PPP literatureto make users aware of the alternative methods available for making international comparisons. Secondly, new perspectives are provided on these methods. Thirdly, newapproaches to the problem of making international comparisons are explored. Fourthly,some of the implications of the choice of method are analyzed. Finally the dissertationmakes some recommendations.Chapter 2 builds on the pioneering work of Van Yzeren(1987) to rationalize the literature. Chapter 2 develops a taxonomy of all the main PPP methods. The relationshipsbetween methods in the literature are frequently unnecessarily obscured, since the variousmethods were typically advocated by different authors from different perspectives usingdifferent notation. The taxonomy groups methods together if and only if they can beshown to be special cases of a more general method, thereby exposing generic similaritiesbetween apparently different methods. For example it is shown that the Geary-Khamis,Chapter 1. Introduction 7Iklé, Gerardi, Fixed Base Price, Ideal Price, and Van Yzeren homogenous group methodsare all special cases of a more general method called here the Average Price method. Thetaxonomy reinterprets PPP methods in a graph theoretic context. This reinterpretationyields many useful insights. In particular, the taxonomy reveals a rich underlying structure built on the one hand around Paasche, Laspeyres and Fisher indices, and on theother hand around arithmetic, geometric and harmonic means. Many of the characteristics of a method may be inferred directly from the taxonomy once the method’s genushas been identified.Chapter 3 was motivated by the realization that virtually all the PPP methods in thetaxonomy of Chapter 2 have the same underlying graph theoretic structure. However,this ‘star” structure is just one of an extremely large number of possible structures.This paper proposes a new PPP method which allows the data to choose the underlying structure by using Kruskal’s “Minimum Spanning Tree” Graph Theory algorithm tochain PPPs across countries rather than imposing the structure ex ante. By chainingis meant the procedure of linking together bilateral comparisons. The idea of chainingindex numbers dates back to Marshall(1887). The main advantage of chaining is thatit tends to reduce the spread between Paasche and Laspeyres indices and between allknown superlative indices. In other words chaining tends to reduce the sensitivity of theresults of a comparison to the choice of bilateral index number formula. Chaining hasbeen widely advocated in a time series context. However few attempts have been madeto construct chains across countries, since unlike time series there is no natural orderingof countries. One such attempt was made by Kravis, Heston and Summers(1982). Theyconsidered a number of different approaches ranging from cluster analysis to geographicalpropinquity. The Minimum Spanning Tree (MST) method advocated in this paper mayhe viewed as an extension of their pioneering work. The paper concludes with an empirical comparison between the Geary-Khamis, EKS, ECLAC and MST methods usingChapter 1. Introduction 81990 OECD data.. MST PPPs are found to lie most of the time between their corresponding Geary-Khamis aid EKS PPPs. The Minimum Spanning Tree (MST) methodmay potentially dramatically simplify the construction of PPPs by reducing the number of countries that must he compared directly. The MST method also has importantimplications for time series comparisons.Chapter 4 focuses specifically on the Average Price class of methods from the taxonomy of Chapter 2. Average Price methods have the very desirable property of generatingquantity indices that literally add up over different levels of aggregation when measuredin value terms. Additivity is extremely useful if international comparisons are requiredat various levels of aggregation as for example in national accounts comparisons. However, it is widely claimed in the PPP literature that Average Price methods are subjectto the Gerschenkron effect.6 The Gerschenkron effect refers to the purported tendencyof Average Price methods to overestimate the output shares of any outlier countries ina comparison. In spite of the significant implications of the Gerschenkron effect, theevidence for its existence remains largely anecdotal. That is to say that no one hasprovided a satisfactory explanation of why all Average Price methods are subject to theGerschenkron effect. Nevertheless, the Gerschenkron effect has recently prompted theWorld Bank to reconsider its use of the Geary-Khamis method. Instead the World Bankis considering using another Average Price method called the Iklé method on the groundsthat it is less sensitive than Geary-Khamis to the Gerschenkron effect.7 This paper explains the theoretical underpinnings of the Gerschenkron effect. Also frameworks aredeveloped for empirically verifying and measuring the Gerschenkron effect, which arethen tested using 1990 OECD data. Finally the chapter addresses Dikhanov’s assertionthat Iklé is less sensitive than Geary-Khamis to the Gerschenkron effect. The answer to6See for example Gerardi(1982) and Enrostat(1983).7See Dikhanov(1994).Chapter 1. Introduction 9this question seems to depend critically on how the Gerschenkron effect is measured. Alsothe Gerschenkron effect is shown to have important implications for the measurement ofinequality across countries and time.Chapter 2A Taxonomy of Multilateral PPP Methods.2.1 IntroductionMany multilateral Purchasing Power Parity (PPP) methods have been advocated in theinternational comparisons literature. A list of those that have received most attentionwould include the Geary-Khamis, Eltetö-Köves-Szulc (EKS), Van Yzeren, Iklé, Gerardi,Ideal Price (IP), Walsh, Fixed Base, ECLAC, Rao, Own Share, Multilateral Translogand Mean Output Share methods. Such an abundance of competing methods is potentially confusing for users. That there is no consensus can be deduced from inspecting themethods that have actually been used to make international comparisons. The International Comparison Project (ICP), World Bank and IMF use the Geary-Khamis method,although the World Bank is now considering switching to the Iklé method.’ Eurostat hasused the Gerardi and Geary-Khamis methods, but now prefers the EKS method. TheOECD currently uses both the Geary-Khamis and EKS methods. Meanwhile the UnitedNations Economic Commission for Latin America and the Caribbean (ECLAC) has usedthe Walsh and ECLAC methods. The Rao method has also been used to make comparisons in Latin America. In addition, other methods such as the Multilateral Translog,Own Share and Van Yzeren balanced methods have been advocated by experts in theliterature 2A key factor in the debate over the relative merits of competing methods which cuts‘See Dikhanov(1994).2See Diewert(1986) and Balk(1995).10Chapter 2. A Taxonomy of Multilateral PPP Methods. 11right across the taxonomy is the issue of whether a multilateral PPP method should giveall countries equal weights in the PPP formula, or whether larger countries should begiven larger weights. The Geary-Khamis and EKS methods are respectively the mostwidely used weighted and unweighted methods. However, the divisions in the literatureover the relative merits of these two methods run deeper than the issue of weighting. Foras the taxonomy reveals, the Geary-Khamis and EKS methods belong to fundamentallydifferent generic classes of methods with differing properties. Nevertheless the issue ofweighting is crucial to an understanding of the rationale behind and the relationshipsbetween the various competing multilateral methods. For example, the Gerardi andIklé methods were advocated as improvements on Geary-Khamis precisely because whilebelonging to the same class of methods as Geary-Khamis, unlike Geary-Khamis, theygive all countries in a comparison equal weight. Therefore the weighting properties ofmethods are discussed in some detail in this paper.The literature on multilateral PPP methods of making international comparisons isstill quite fragmented. The various methods were generally advocated by different authorsfrom different perspectives using different notation. Hence the relationships betweenmethods in the literature are frequently unnecessarily obscured. The taxonomy seeksto build on the work of Van Yzeren(1987), and rationalize the literature, by classifyingall the main multilateral PPP methods within a general framework. In particular, thetaxonomy reinterprets methods ill a graph theoretic context. This reinterpretation yieldsmany useful insights, and some new methods. The taxonomy reveals a rich underlyingstructure. The structure is built on the one hand around bilateral Paasche, Laspeyresand Fisher indices, and on the other hand around arithmetic, geometric and harmonicmeans.The taxonomy groups methods together if and only if they can be shown to be specialcases of a more general method thereby revealing the underlying generic similaritieschapter 2. A Taxonomy of Multilateral PPP Methods. 12between apparently different methods. For example, it is shown that the Geary-Khamis,Iklé, Gerardi, IP, Fixed Base Price and Van Yzeren homogenous group methods areall special cases of a more general method, called here the Average Price method. TheAverage Price method in turn is shown to be a special case of the Symmetric Star method,etc. The underlying taxonomic structure is depicted by the tree diagram in Figure 2.1.It is likely that methods of the same type will exhibit similar behaviour. This property ofthe taxonomy may prove particularly useful for assessing the relative merits of competingmethods since it allows certain behavioural characteristics of a method to be illferreddirectly from the taxonomy, once the method’s genus has been identified.2.2 Methodology and NotationConsider the problem of calculating multilateral PPPs (Pk) and quantity indices (Qk)over a set of K countries. This paper does not address aggregation problems below thebasic heading level. It is assumed that each country indexed by k = 1,. . . , K, suppliesprice and quantity data (pkj, qk), defined over the same set of basic heading goods andservices, indexed by i = 1,. . . , N.It is useful to distinguish between two different notions of a base country. The “numeraire base” may be defined as the country whose currency is used as the numerairefor the comparison and hence whose PPP and quantity index are set equal to unity. Incontrast a “weighting base” exists if one and only one country’s price and/or quantityvectors are used as weights in the comparison. Not all PPP methods have a weightingbase, and for those that do, it need not necessarily coincide with the numeraire base.Changing the numeraire base serves only to rescale multilateral PPPs and quantityindices. Hence it is also useful to define multilateral output shares (5k). Output sharesare a set of quantity indices that have been rescaled to sum to unity. More precisely,Chapter 2. A Taxonomy of Multilateral PPP Methods. 13output shares must satisfy the following three conditions:(i) 8k > 0, (ii) s = 1, (iii) Sk = , V k = 1,.. . , K. (2.1)kz1Multilateral PPPs and quantity indices may be related using the Weak Factor ReversalTest stated below in (2.2):Weak Factor Reversal Test: = V b, k = 1,..., K. (2.2)Pb Qb =1pbqbMultilateral PPPs may be derived implicitly from multilateral quantity indices via theWeak Factor Reversal Test. The same holds in reverse. This result is useful since itallows some flexibility in the description of multilateral methods.In the remainder of the paper, the following bilateral formulae will be referred torepeatedly. It should be noted that these bilateral indices are intransitive. In otherwords, unlike their multilateral counterparts, bilateral PPPs and quantity indices aredependent on the choice of the numeraire base country b.Laspeyres Quantity Index: Q = pbqk (2.3)=i pbiqbiLaspeyres PPP: P = _i pkiqbi (2.4)i=1 pbiqbiPaasche Quantity Index: = ZiPk, (2.5)pkiqbiPaasche PPP: P = _ipkiqki (2.6)ii pbiqkiFisher Quantity Index:= (QQ)”2, (2.7)Fisher PPP: P=1/2, (2.8)-‘bi+kiN / .\ 2Törnqvist Quantity Index: = J] (-) (2.9)\qbijChapter 2. A Taxonomy of Multilateral PPP Methods. 14Törnqvist PPP: P=(i)2(2.10)Pbpkiqkiwhere Vk= Ni=1 pkiqki2.3 The Exchange Rate MethodPerhaps the simplest and most widely used multilateral method for making internationalcomparisons exploits the fact that in general due to arbitrage, exchange rates are transitive. Hence multilateral PPPs are obtained by simply setting each PPP equal to itscorresponding exchange rate.Vb,k=1,...,K (2.11)Pb ebIn equation (2.11), ek/eb denotes the exchange rate between countries b and k. ek/ebequals the number of units of currency in country k that may be exchanged for oneunit of currency in country b. Multilateral quantity indices are obtained implicitly fromexchange rates via the Weak Factor Reversal Test in (2.2). The two main criticismsof this method are firstly that exchange rates can be highly volatile making the resultspotentially very sensitive to the exact timing of the comparison. Secondly, exchange ratessystematically tend to undervalue the purchasing power of currencies in poorer countriesrelative to richer countries. This is the so-called Balassa-Samuelson hypothesis.3 Thesystematic bias is attributable to the fact that many labour intensive goods and servicesthat are not traded internationally are relatively cheaper in poorer countries. Henceinternational comparisons using the exchange rate method also tend to underestimatethe output shares and per capita incomes of poorer countries relative to richer countries.3See Balassa(1964) and Samuelson(1964).Chapter 2. A’ Tonomy of Multilateral PPP Methods.Multilateral Methods15Chain MethodsChain Methodswithout aWeighting BaseSymmetricStar MethodsNMean AsymmetricStar Methods7<Exchange RateMethodMean ValueShare MethodsStar MethodsAsymmetricStar MethodsAverage AverageBasket PriceMethods MethodsTörnqvistStarMethodsFisher Mean Quantity Mean OutputStar Index ShareMethods Methods MethodsFigure 2.1: The TaxonomyChapter 2. A Taxonomy of Multilateral PPP Methods. 162.4 Mean Value Share MethodsMean Value Share methods calculate the quantity index (or PPP) between countries band k directly by taking a weighted geometric mean of the quantity (or price) relativesof b and k. Transitivity is assured by the use of a common set of weights, based on meanvalue shares for every pair of countries.Let Vki denote the share of total expenditure in country k spent on commodity i.pkiqki= N (2.12)Z1pkiqkiAlso let M1(v) denote a symmetric mean of the value shares of commodity i acrossall countries in the set.4 The Mean Value Share method is unweighted ifM1(v) is asymmetric mean. If on the other hand MJ1(v) is a weighted mean, then the Mean ValueShare method is weighted in favour of the countries with larger weights in the weightedmean formula. However, all the Mean Value Share methods that have been advocatedin the literature are unweighted. Mean Value Share methods calculate quantity indicesas follows:r__________________= n (.) __1[M=ijjj1 (2.13)Qb qbMean Value Share methods differ only in the choice of the symmetric mean formulaMJ1(v). The two best known methods of this type use the geometric and arithmeticsymmetric mean formulae respectively.5The Walsh Method= fl(v) (2.14)The Walsh Method with Arithmetic WeightsMJ1(vj) = (2.15)4For a detailed discussion on the properties of symmetric means, see Diewert(1993a).5For a more detailed discussion on these two methods, see Ruggles(1967).Chapter 2. A Taxonomy of Multilateral PPP Methods. 17It should be noted that when MJ1(vj) denotes an arithmetic mean, the exponent in(2.13) reduces to MJ1(v). The Walsh quantity index with arithmetic weights is in facta multilateral generalization of the bilateral Törnqvist quantity index formula, as can beseen from (2.16), which places the two formulae side by side.—K‘bi+’kjA / \ K L3i N 2k1J(kz), Q=fl(-’) (2.16)Qb j=1 \qbij i1 \qbijOne problem with Mean Value Share methods is that the PPP formula obtained implicitlyfrom (2.2) is complicated and cannot be identified with any of the standard indices definedin the index number literature. Hence the implicit PPPs are difficult to interpret.2.5 Chain MethodsMultilateral quantity indices can be obtained from K— 1 intransitive bilateral quantityindices by chaining over the set of K countries if and only if the chain is connectedand spans the set.6 In the Graph Theory literature, such a chain is called a SpanningTree.7 However, over a set of K vertices (countries), KK_2 Spanning Trees exist, andeach Spanning Tree generates a different set of multilateral quantity indices. One of theproblems with Chain methods is deciding which Spanning Tree to use. Some examplesof Spanning Trees over the set of K 5 vertices (countries) are given in Figure 2.2.Star methods are special cases of Chain methods that use the “star” Spanning Treedepicted in Figure 2.2(a). The country at the centre of the star is the weighting base.Almost all the Chain methods that have been advocated in the literature are Starmethods. One exception is the International Comparison Project (ICP) in which regionalStar comparisons have been chained together to obtain global comparisons.8Technically,6By chaining is meant the procednre of linking together bilateral comparisons.7For an introduction to Graph Theory see Foulds(1991) or Wilson(1985).8See Kravis, Heston and Summers(1982).Chapter 2. A Taxonomy of Multilateral PPP Methods. 18(a)Figure 2.2: Examples of Spanning Treeswhen Stars are chained together, the overall comparison no longer has a Star structure.A second exception is the Minimum Spanning Tree method advocated in Chapter 3 ofthis thesis.2.6 Star MethodsAll the remaining methods discussed in this paper belong to the class of Star methods.The essential feature of such methods is that a country, say country X which is notnecessarily one of the countries in the set, is chosen a priori as the weighting base. Inother words comparisons between each pair of countries in the set are made indirectly viacountry X. The underlying Star structure is depicted in Figure 2.3. The vertices denotethe countries in the comparison.The Star method quantity index between countries b and k is calculated as follows:Qk_Qxk-b XbIn the taxonomy a crucial distinction is drawn between Asymmetric and Symmetric Starmethods. An Asymmetric Star method is one in which one of the countries in the set isarbitrarily chosen as the weighting base. Hence Asymmetric Star methods a priori do not(b) (c)Chapter 2. A Taxonomy of Multilateral PPP Methods. 19Figure 2.3: The Star Spanning Treetreat all countries in the set symmetrically. In contrast, for Symmetric Star methods, theweighting base X is an artificially constructed country that represents some symmetricaverage of the countries in the set. Hence Symmetric Star methods a priori treat allcountries in the set symmetrically.2.7 Symmetric Star MethodsSymmetric Star methods may differ in two respects. Firstly they may differ with respectto the bilateral quantity index formula ilsed in (2.17). Secondly they may differ in theformula used to calculate the average price and/or quantity vectors (px, qx) that definecountry X. Methods that use the bilateral Paasche quantity index formilla are consideredfirst.Chapter 2. A Taxonomy of Multilateral PPP Methods. 202.7.1 Average Basket MethodsAll Average Basket methods calculate the quantity index and PPP between countries band k as follows:fP 17 DLk ‘Xk 1c 1Xkr riP’ L’b Xb b Xbwhere Qck is the Paasche quantity index defined in (2.5), while Pkk is the LaspeyresPPP defined in (2.4). Alternatively, (2.18) may be written thus:c-’N‘‘k L1pkjqk? Ld=1pbqXr -‘N‘b L.jj pkiqxi 2ii pbiqbiN N Nk— 1pkqX Pxqx — 1pkqX—— N N— Nb j—1 pxiqxi ji pbqx ji pbqxAverage Basket methods satisfy the Average Test for PPPs stated below in (2.20), butfail the Average Test for quantity indices stated below in (2.19).Multilateral quantity indices satisfy the Average Test if:mini(<maxi(, Vb,k=1,...,K. (2.19)\qbij Qb \qbijIViultilateral PPPs satisfy the Average Test if:mini < <maxj (i), Vb,k= 1,...,K. (2.20)PbiJ Pb \PbiNeither Qk/Qb nor Pk/Pb as defined in (2.18) depends on the price vector Px of countryX. Therefore Average Basket methods need only define an average quantity vector q.Van Yzeren(1987) calls methods of this type, “q-combining” methods. Three natura.1formulae for qx are respectively, the arithmetic, geometric and harmonic means of thequantity vectors of all the countries in the set. These formulae are given below:9KVi=1,...,N, (2.21)k=19Rescaling the average basket qx has no effect on tile resulting output shares. Hence the presence ofan arbitrary positive constant c in each average basket formula.Chapter 2. A Taxonomy of Multilateral PPP Methods. 21q.=fiqK Vi=1,...,N, (2.22)/K i\qx=c(— Vi=1,...,N. (2.23)\k1 qkijThe Average Basket method that calculates qx using (2.21) is called the ECLAC method.ECLAC stands for United Nations Economic Commission for Latin America and theCaribbean. This method was advocated by Walsh(1901), who called it Scrope’s method.It is also discussed in Ruggles(1967) a.nd Diewert(1993h). The Average Basket methodbased on (2.22) was advocated by Walsh(1901) who called it Scrope’s method with geometric weights.1°However, it is referred to here as the Geometric Average Basket method.The Average Basket method based on (2.23) does not seem to have been advocated inthe literature. A logical name for it would be the Harmonic Average Basket method.The ECLAC method (2.21) gives countries with larger baskets greater weight in theqx formula. In contrast, the Harmonic Average Basket method (2.23) gives countrieswith smaller baskets larger weight in the qx formula. Hence both the ECLAC and Harmonic Average Basket methods are weighted methods, although in opposite directions.However, the Geometric Average Basket method (2.22) does not seem to have any obvious weighting bias. Equal weights in (2.21), (2.22) and (2.23) are obtained by dividingeach quantity vector qj by its corresponding quantity index.11 These formulae are givenbelow. The average baskets qx and Paasche quantity indices Qck are obtained by solvingthe system of N + K simultaneous equations given by (2.5) and the respective formulafor qx.K/qx = — Vz = 1,... ,N (2.24)k=1 \XkK / \1/IVi=1,...,N (2.25)k—1 \‘XkJ‘°Again see Diewert(1993b).“This point is made by Van Yzeren(1987). He calls this procedure “de-weighting”.chapter 2. A Taxonomy of Multilateral PPP Methods. 22(K 7Xi=Qi(P_) ) Vi=1,...,N (2.26)\k=1 Qxk JThe Average Basket method that calculates qx using (2.24) is known as the unweightedVan Yzeren(1956) heterogenous group method. This method is discussed in greater detailin section 8.5. Interestingly, the Average Basket methods based on (2.22) and (2.25) areequivalent, in the sense that they generate the same set of output shares.The final Average Basket method considered here is the Fixed Base Basket method.It uses country k as a weighting base. This method is an extreme example of a weightedmethod since it gives the quantity vector of country k a weight of one, and the quantityvectors of all other countries a weight of zero.qxz = cqkj Vi 1,...,N (2.27)2.7.2 Average Price MethodsAll Average Price methods calculate the quantity index and PPP between countries band k as follows:r r’L r— “‘Xk rj— 19r riL ‘ D DP’‘b SXb lb LXbwhere Q%k is the Laspeyres quantity index defined in (2.3), while xk is the PaaschePPP defined in (2.6).12 Alternatively, (2.28) may be written thus:r -‘ N -s N s—’ N— Lipxiqki21pXiqXi— Lipxiqki—— N, N— Nb 1pxqx =1px•qbN Nk 1pkqk 4pXqbN Nb 1pxqk 4pbqbAverage Price methods satisfy the Average Test for quantity indices stated in (2.19), butfail the Average Test for PPPs stated in (2.20). In addition to satisfying the Average Test‘2Comparing (2.18) with (2.28) it can be seen that a symmetrical relationship exists between AverageBasket and Average Price methods. Average Basket methods have a Paasche Star structure in theirquantity indices and a Laspeyres Star structure in their PPPs. For Average Price methods, this patternis reversed.Chapter 2. A Taxonomy of Multilateral PPP Methods. 23(2.19), Average Price quantity indices are also additive. Additive quantity indices havethe appealing property of literally adding up over different levels of aggregation whenmeasured in value terms. Additivity is also sometimes referred to as matrix consistencyin the literature.’3Neither Qk/Qb nor as defined in (2.28) depends on the quantity vector q ofcountry X. Therefore Average Price methods need only define an average price vectorPx Van Yzeren(1987) calls methods of this type, “p-combining” methods. The GearyKhamis, Iklé, Gerardi, Fixed Base Price, IP and Van Yzeren homogenous group methodsare all examples of Average Price methods.Unweighted Average PricesThe unweighted (or more precisely equally weighted) average price counterparts tothe unweighted average basket formulae in (2.24), (2.25) and (2.26) are given below.’4The average prices PXi and Paasche PPPs P are obtained by solving the system ofN + K simultaneous equations given by (2.6) and the respective formula for px.PXi =(k)Vi = 1 ,7\T (2.29)K / \1/Kpx=ofl(--) Vi=1,...,N (2.30)k=1 \‘xkJ1K /i IPxi=j—j j Vm=1 ,N (2.31)\i \ XkJ JThe Average Price methods based on (2.29) and (2.30) are better known respectively asthe unweighted Van Yzeren(1956) homogenous group method and the Gerardi method.’5An intriguing feature of the Gerardi method is that deflating each national price by its‘3See for example Kravis, Heston and Summers(1982).‘4Again rescaling the average price vector has no effect on the resulting output shares. Hence thepresence of an arbitrary positive constant in each average price formula.‘5The Van Yzeren homogenous group method is discussed in greater detail in section 8.5. The Gerardimethod was first proposed by Walsh(1901). It was later popularized by Gerardi(1982). Again for a moredetailed discussion on the evolution of the literature, see Diewert(1993b).Chapter 2. A Taxonomy of Multilateral PPP Methods. 24corresponding Paasche PPP, before calculating average prices has no effect on theGerardi output shares. Hence the two formulae in (2.32) are equivalent.K K / \1/Ki—r i/K rrlPki ‘1 TPxi=11Pki , Pxi=11jJ , Vz=1,...,A. (2.32)k1 k=i \‘XkJThe equivalence of these two geometric average price methods is analogous to the equivalence of the two average basket methods given in (2.22) and (2.25). The method basedon (2.31) has not been advocated in the literature.Weighted Average PricesThe most widely used Average Price method is the Geary(1958)-Khamis(1972) method.The Geary-Khamis method is a weighted version of (2.29). Geary-Kharnis average pricesare calculated using the following formula:’6KfPkiPXiZLvK pP Vi=1,A. (2.33)k=i XkA second weighted variant on (2.29) calculates average prices as follows:Pxi=a.Z( Q%k ) =cf(sk) vz=i,...,iv. (2.34)k=i j=1 Q3 Xk k=i XkThe method based on (2.34) is the weighted Van Yzeren homogenous group method. Afurther variant on (2.29) is the Iklé(1972) method. The Iklé method calculates PXi asfollows:PXi ( qk/Q4k ) Vi = 1,... ,N. (2.35), =i qji/Qxj xLike the Geary-Khamis and weighted Van Yzeren homogenous group methods, the Ikléaverage price formula is a weighted arithmetic mean of deflated prices across the collntries in the comparison. However, unlike the Geary-Khamis and weighted Van Yzerenmethods, the Iklé method is unweighted in the sense that it does not weight countries‘°Khamis(1972) proves existence and uniqueness for the Geary-Khamis system.Chapter 2. A Taxonomy of Multilateral PPP Methods. 25according to size.17 In fact Iklé average prices are obtained by applying Van Yzeren’sde-weighting procedure to the Geary-Khamis average price formula. The Iklé methodcan also be expressed as a weighted harmonic mean of deflated prices. Dikhanov(1994)demonstrates that Iklé average prices can be calculated as follows:’8PXi = { [( Pki] }‘ V i = 1,... ,N. (2.36)k=1 Zj1Vj XkVan Yzeren(1987) outlines a weighted version of the Gerardi method (2.30). The formulais given below:L K LPXi = H (pxk) =iQxJ = flp V i 1,... ,N. (2.37)The Fixed Base Price method uses country k as a weighting base. This method is anextreme example of a weighted method since it gives the price vector of country k aweight of one, and the price vectors of all other countries a weight of zero.PXi=Pki Vi=1,...,N (2.38)The final Average Price method considered here approaches the problem of calculatingaverage prices in a radically different way. The “Ideal Price” (IP) method was proposedby Gerardi.’9 Average prices are obtained as the solution to the following minimizationproblem:K K / N \2Min (io pxiqki — log Q)k=lb=1 lpxqbK K / rLor alternatively, Min iog — log Q) (2.39)k=lb=1 Qxb7Hence one must be careful to distinguish between a weighted mean and a weighted multilateralmethod. The Iklé method is unweighted, but calculates average prices using a weighted mean formula.‘8t’ki in (2.36) denotes the value share of commodity i in the basket of country k. Vki is defined in(2.12). It should be noted that Vki = 1, V k. Hence again the Iklé method as defined in (2.36)does not weight countries according to size.‘°See Eurostat(1983).Chapter 2. A Taxonomy of Multilateral PPP Methods. 26where Q is the Fisher quantity index defined in (2.7). The rationale behind the IPmethod sterns from the Gerschenkron effect. The Gerschenkron effect refers to the purported tendency of all Average Price methods to overestimate the output shares of outliercountries in a comparison.20 The IP method is an attempt to find the Average Price (additive) method that is least sensitive to the Gerschenkron effect. However, the IP methoddoes not generate economically meaningful results, since in general even assuming theexistence and uniqueness of a strictly positive solution, the IP average price vector is unlikely to bear any resemblance to the actual national price vectors of the countries in thecomparison.2’Therefore the IP quantity indices obtained at lower levels of aggregationwill be meaningless, even though they sum to something close to their correspondingFisher quantity indices at the highest level of aggregation. In short this cure for theGerschenkron effect is worse than the illness.2.7.3 Törnqvist Star MethodsAs far as the author is aware, the only Törnqvist Star method that has been advocatedin the literature is the Rao method.22 Törnqvist Star multilateral quantity indices havethe following structure:Qk_Q3kr rT’ L‘b ‘Xbwhere the Törnqvist quantity index formula is defined in (2.9). Törnqvist Star methodsin contrast to Average Basket and Average Price methods, must define both Px and qx.However, as with Mean Value Share methods, the Törnqvist Star PPP formula derivedimplicitly from (2.40) via the Weak Factor Reversal Test is difficult to interpret since it20The Gerschenkrori effect is discussed in detail in Chapter 4.211n a 1993 Eurostat mimeo, Cuthbert claims that the IP method does not always have a uniquesolution, and that sometimes one or more of the average prices may equal zero.22See PrasadaRao and Salazar-Carrillo(1988). The Multilateral Translog method advocated by Caves,Christensen and Diewert(1982), although also based on the Törnqvist index is not a Törnqvist Starmethod. It belongs to the class of Mean Asymmetric Star methods discussed in section 10.Chapter 2. A Taxonomy of Multilateral PPP Methods. 27is complicated and cannot be identified with any of the standard indices defined in theindex number literature.The Rao method in fact uses a weighted Törnqvist Star PPP. It calculates PPPsdirectly and quantity indices implicitly. Hence it is the Rao quantity indices rather thanthe PPPs that are difficult to interpret.where‘Xk = fl (2.41)b Xb j PXtand Vkj is the value share of good i in country k, as defined in (2.12). The Rao methoduses a weighted Törnqvist Star PPP, since in (2.41) it gives a weight of one to the valueshares Vk of country k and a weight of zero to the value shares vx of country X.23 TheRao method defines the vector Px as follows:K / \V.i Pkz I VkzPXI=J]j---, where v= -*K (2.42)k=1 \‘xkJ !_sk1VkiBy giving country X a zero weight in the Törnqvist formula, and defining Px as shownin (2.42), the Rao method avoids having to define the vector qx.2.7.4 Fisher Star MethodsFisher Star methods differ only in the way they calculate the price and quantity vectorsPx and q of country X. All Fisher Star methods calculate the quantity index and PIPbetween countries b and k as follows:24r rF n“dk “‘Xk ‘k ‘xkri riF’— F’“db “‘Xb b Xb231t should be noted that although the Rao method uses a weighted Törnqvist PPP, it neverthelessweights all countries equally.24Unlike Average Basket, Average Price and Törnqvist Star methods, Fisher Star methods satisfy theStrong Factor Reversal Test. A method satisfies the Strong Factor Reversal Test if firstly it satisfies theWeak Factor Reversal Test, and secondly if its PPP formula can be obtained from its quantity indexformula by simply interchanging the price and quantity vectors.Chapter 2. A Taxonomy of Multilateral PPP Methods. 28where Qck and denote the Fisher quantity index and PPP defined in equations (2.7)and (2.8) respectively. Alternatively, (2.43) may be written thus:,- ,mF /rP \ /rL \ 1/2 , / DP \ / ijL \ 1/2“k Xk I Xk \ I “Xk \ I k Xk f £ Xk I I L Xk= irP llrL I ‘ ppF pPIpL“db “‘Xb \‘5xbJ \“‘xb/ b Xb \ XbJ \ XbA comparison of (2.44) with (2.18) and (2.28) reveals that any Fisher Star quantity index(PPP) may be expressed as the geometric mean of a pair of Average Basket and AveragePrice quantity indices (PPPs).A plethora of Fisher Star methods may be generated by combining pairs of AverageBasket and Average Price methods. However for a Fisher Star method to be meaningful,its respective Average Basket and Average Price methods must have the same weightingstructure. As an example of a “bad” Fisher Star method consider combining the Gerardimethod (2.30) with the ECLAC method (2.21). The Gerardi method gives all countriesin the comparison equal weight. In contrast the ECLAC method gives larger countrieslarger weights. The Fisher Star method formed by combining these two methods is aconfused mixture of weighting schemes. On the other hand, the Fisher Star methodformed by combining the Gerardi and Geometric Average Basket (2.22) methods has aconsistent weighting structure since both of its constituent methods give all countriesequal weights. Similarly the Fisher Star method formed by combining the Geary-Khamis(2.33) and ECLAC methods also has a consistent weighting structure as demonstratedbelow in (2.45). Both its constituent methods have the same weighting structure withlarger countries given larger weights. This method could be called the GKEC method.qxi=cqkj, Pxi=c( qk ), Vi=l ,N. (2.45)k1 k1From (2.44) it follows that all Fisher Star methods can be decomposed into a pair ofAverage Basket and Average Price methods. However, this process of decompositionis not always straightforward. If the average basket formula of a Fisher Star methodChapter 2. A Taxonomy of Multilateral PPP Methods. 29is a function of average prices as well as of the national price and quantity vectors,while simultaneously average prices are a function of the average basket as well as of thenational price and quantity vectors, then it may be difficult to disentangle the AverageBasket and Average Price methods from each other. An example of such a Fisher Starmethod is Van Yzeren’s balanced method. The Van Yzeren average basket and averageprice formulae are given below in (2.54). qx in (2.54) is a function of p. Similarly, Pxis a function of qx.2.7.5 The Van Yzeren MethodsVan Yzeren(1956) proposed three multilateral methods. The rationale behind the VanYzeren methods proceeds as follows. The heterogenous group method finds the set ofPPPs that equate the ratio of the cost of buying the basket of country k in all countries inthe set, to the cost of buying the basket of country kin country k, across all countries k =1, ... K in the set. The equations defining the heterogenous group PPPs,.. ,may he written as follows:(pp) = Vj = 1,... ,K, (2.46)where Wk denote country weights and \ is the Perron-Frobenius root of the system.25Similarly the equations defining the heterogenous group quantity indices, Qxi. . . , QxKmay be written as follows:(wkQk__)=-_,Vj = 1,... ,K. (2.47)As stated earlier, the heterogenous group method belongs to the class of Average Basketmethods. The general formula for the heterogenous group average basket is given below:qxi=(wk) Vi=1,...,N. (2.48)2For a discussion on Perron-Frobenius roots, see Debreu and Herstein(1953).Chapter 2. A Taxonomy of Multilateral PPP Methods. 30If wj = 1, Vk = 1, . . . , K, then the average basket formula reduces to (2.24). If on theother hand the weights Wk are set equal to the output shares Sk = Qck/(l Q), theaverage basket formula reduces to (2.21).The homogenous group method finds the set of PPPs that equate the ratio of thecost of buying the baskets of all the countries in the set in country k, to the cost ofbuying the basket of country k in country k, across all countries k = 1,. .. , K in the set.The equations defining the homogenous group PPPs, Px1,... , Px-, may be written asfollows:(wkp)=,Vj=1,...,K, (2.49)where again, wk denote country weights and A is the Perron-Frobenius root of the system.Similarly the equations defining the homogenous group quantity indices, Qxi,. . . , QxK,may he written as follows:(wQQx) = Qxj, Vj = 1,...,K. (2.50)Van Yzeren(1956) proves existence and uniqueness for the heterogenous and homogenousgroup systems. As stated earlier, the homogenous group method belongs to the class ofAverage Price methods. The general formula for heterogenous group average prices isgiven below:i Pk.iPX(WkTj) Vi=1,...,Pv. (2.til)k=1 \ 1XicJIf wk = 1,Vk = 1,. . . , K, then the average price formula reduces to (2.29). If on theother hand the weights wk are set equal to the output shares Sk = Q%k/(Zl Q4), theaverage price formula reduces to (2.34).Van Yzeren’s balanced method exploits the fact that the Perron-Frobenius roots in(2.46) and (2.49) are the same. This is because the matrix of Laspeyres indices in (2.49)is the transpose of the matrix of Laspeyres indices in (2.46) and hence has the sameChapter 2. A Taxonomy of Multilateral PPP Methods. 31eigenvalues. Therefore (2.46) and (2.49) may be combined as follows:Vj=l,...,K. (2.52)Similarly, (2.48) and (2.50) may be combined as follows:(wkQk”)=(wkQ) Vj = i,...,K. (2.53)The general formulae for the balanced method average basket and average prices aregiven below:26K/qk1 I Pk --,qxi=cwk—), Pxi=crZ1Wk-p- Vz=1,...,]\’. (2.4)k=1 \ ‘dxkJ k=1 \ XkThe two natural choices of weights are either wk = 1 or Wk= Qk/(1 Q) =sk,Vk = 1, . . . , K. It is important to note that if ‘Wk = 1,Vk = 1,..., K, the averagebasket and average price formula in (2.54) do not reduce to the iinweighted heterogenousand homogenous group formulae respectively.Two rather peculiar examples of the weighted Van Yzeren balanced method wereproposed by Kurabayashi and Sakuma. They call these methods the YKS and YKS-Qmethods.27 The equations defining the YKS PPPs, P1,. . . , Px, are given below:=()2Vk = 1,...,K. (2.55)A comparison of (2.55) with (2.52) reveals that the YKS method sets the weights asfollows: Wk = 1/Pk. Meanwhile, the equations defining the YKS-Q method are givenbelow:K K /,.-. \2, Vk=1,...,K. (2.56)k=1 k1 \QXkJA comparison of (2..56) with (2.53) reveals that the YKS-Q method sets the weights asfollows: wk = 1/Qk. The weighting structures of both the YKS and YKS-Q methodsare a bit puzzling.26This result is proved in Appendix A. (Proof 1)27yK5 stands for Van Yzeren-Kurabayashi-Sakunia. See Kurabayashi and Sakuma(1990).Chapter 2. A Taxonomy of Multilateral PPP Methods. 322.8 Asymmetric Star MethodsAsymmetric Star methods select a priori one of the countries in the set (denoted by X)as the weighting base. Asymmetric Star quantity indices are defined as follows:Vk=1 K, (2.57)where Qxk denotes a generic bilateral quantity index between countries X and k. TheFixed Base Basket and Fixed Base Price methods defined in (2.27) and (2.38) are exam-pies of Asymmetric Star methods. They use the Paasche and Laspeyres bilateral quantityindex respectively. If multilateral comparisons are made specifically from the perspectiveof one country, then it is quite reasonable to use an Asymmetric Star method with thatcountry as the weighting base. However, if the comparison is not linked to a specificcountry, Asymmetric Star methods are considered less reasonable, since the obtained setof quantity indices and output shares depend critically on which country is selected asthe weighting base. Nevertheless, an Asymmetric Star method has been used for manyyears to make comparisons in Eastern Europe, using Austria as the weighting base. Itis still being used. The reasons have more to do with resources, costs and politics thanwith methodology. Austria can be seen as a link between Eastern and Western Europeand the Austrian Statistical Office has invested large resources in these comparisons.Asymmetric Star methods are in fact the international equivalent of fixed base timeseries methods. For time series comparisons the numeraire and weighting base yeartypically coincide. However, the weighting base year becomes progressively less and lessacceptable as time goes on. Hence the need to rebase periodically.chapter 2. A Taxonomy of Multilateral PPP Methods. 332.9 Mean Asymmetric Star MethodsIn general, multilateral methods are required to treat all countries symmetrically a priori.One way of resolving this problem is to create an artificial country which constitutes someaverage of the countries in the set and use it as the weighting base. This is the rationalebehind Symmetric Star methods. In contrast, Mean Asymmetric Star methods proceedby using sequentially each country as the weighting base. Hence K sets of quantity indicesare obtained. Each set is associated with a different weighting base. Mean Quantity Indexmethods obtain multilateral quantity indices by averaging over these K sets of quantityindices. In contrast, Mean Output Share methods first generate the set of output sharescorresponding to each of the K sets of quantity indices and then obtain multilateraloutput shares by taking an average over the K sets of output shares.2.9.1 Mean Quantity Index MethodsFour possible general forms for a Mean Quantity Index multilateral method are givenbelow in equations (2.58), (2.59), (2.60) and (2.61).1iK fri— JV1j1IiJjk2 58Qb — MJl(Qjb)ii,rK fri— lVlJ1bj2 59— MJ1(Qk)Qk — MJ1 [m(Qjk, j)]Qb Mjl[m(QJb,-)]IVIJl{m(—,Qbj)](261)Qb—iwJ1 [m(—,Qkj)]In the above equations, Qk again denotes a generic bilateral quantity index, whileMJl(Qk) denotes a symmetric or weighted mean of the bilateral quantity indices between country k and each of the countries in the set. m denotes a symmetric mean overChapter 2. A Taxonomy of Multilateral PPP Methods. 34its two arguments. All four formulae generate multilateral quantity indices, irrespectiveof the choice of symmetric mean formula for M and m. If Qk satisfies the CountryReversal Test,28 then (2.60) reduces to (2.58), while (2.61) reduces to (2.59).The Eltetö-Köves-Szulc (EKS) and Related MethodsSuppose both M and m denote geometric means. Under this scenario, equations (2.60)and (2.61) are equivalent. In addition suppose that the bilateral quantity index satisfiesthe Country Reversal Test. Under this scenario, all four general forms are equivalent.EKS quantity indices are obtained by using the Fisher quantity index. The EKS quantityindex and PPP formulae may be written thus:K frF 1/K i, K / DF\ 1/K— 11 1 rF I ‘ p — 11 pF‘b \‘4jbl Lb j1 \L jbThe EKS method was first proposed by Gini(1931), although credited to Eltetö andKöves(1964) and Szulc(1964). A weighted version of the EKS method due to Van Yzeren(1983) calculates quantity indices and PPPs as follows:_______QjK fr)F\ K n K /PF\ K‘4k rr ( ‘4jk \ D1= Q1 Fk— J—- ( ‘k \ =1In IIlriFI D i..L15“db=i \‘4jbJ ) j=i \ jbA variant on the EKS method was proposed by Caves, Christensen and Diewert(1982).They proposed the Multilateral Translog method, which uses the Törnqvist as opposed tothe Fisher quantity index.29 A Multilateral Translog quantity index is defined as follows:K /rT\1”k=(‘4ik). (2.64)Qb j=1 Qb28A bilateral quantity index satisfies the Country Reversal Test if and only if:1Qik =6kjThe Fisher and Törnqvist formulae satisfy this test, while the Paasche and Laspeyres formulae do not.29The Törnqvist quantity index is exact for a translog cost function.Chapter 2. A Taxonomy of Multilateral PPP Methods. 35Meall Asymmetric Star methods may be further classified according to the bilateralquantity index formula they use. According to such a classification, the EKS methodis a Mean Asymmetric Fisher Star method, while the Multilateral Translog method is aMean Asymmetric Törnqvist Star method. Mean Asymmetric Törnqvist Star methodslike Mean Value Share and Törnqvist Star methods, are subject to the criticism thatthe PPP formula derived implicitly from the Weak Factor Reversal Test is difficult tointerpret.Consider the following logarithmic least squares problem.4in10gg { [o — log Qba)2] } (2.65)b a=1b1 bThe solution to this problem is given below in (2.66). This result is proved in PrasadaRao and Banerjee(1986).1Qk— j (QbjQk (266)QbQkQb)In other words the solution to (2.65) is the geometric mean version of (2.60). If thebilateral quantity index Qba in (2.65) is set equal to any of the Paasche, Laspeyres orFisher quantity index formulae Q, Q or Q, then (2.66) reduces to the EKS quantityindex formula in (2.62). This result is due originally to Van Yzeren(1987). ThereforeEKS quantity indices deviate from Fisher quantity indices by the logarithmic least squaresamount necessary to obtain transitivity. If on the other hand the bilateral quantity indexQba in (2.65) is set equal to the Törnqvist quantity index formula Q, then (2.66) reducesto the Multilateral Translog quantity index formula in (2.64). Analogous results applyto PPPs.The Own Share MethodSuppose now that both M and m denote harmonic means. In addition suppose thatthe bilateral index satisfies the Country Reversal Test. Under this scenario, in contrast tothe geometric mean case, equations (2.58) and (2.59) are not equivalent. The Own ShareChapter 2. A Taxonomy of Multilateral PPP Methods. 36method, which was proposed by Diewert(1986), uses the Fisher index in equation (2.58).Hence like the EKS method, the Own Share method is a Mean Asymmetric Fisher Starmethod. Own Share quantity indices may be written as follows:/ \ —1_K 1 1 \Qk — j=1 — 2 67It is interesting to note from (2.67) that the Own Share method could equally well bedefined using an arithmetic mean in equation (2.59). Similarly, using a harmonic meanin equation (2.59) is equivalent to using an arithmetic mean in (2.58). This variant onthe Own Share method may be written as follows:—1= 1 (*) = (2.68)Qb (*) Z2.9.2 Mean Output Share MethodsDiewert(1986) also proposed the class of Mean Output Share methods. Mean OutputShare methods start by calculating K sets of Asymmetric Star output shares using sequentially each country as the weighting base. Multilateral output shares are then obtainedby averaging over the K sets of output shares using the following formula:KWjS, (2.69)j=1where w denote the country weights and 8k is the multilateral output share of country k,while s is the output share of country k calculated using country j as a weighting base.Diewert proposes three different formulae for the weights w1. The weights are democraticif wj = 1/K, for all countries j = 1,.. . , K. The weights are plutocratic if:=ipjiqji Vj=l.... K.Z(i(ipuqu)Chapter 2. A Taxonomy of Multilateral PPP Methods. 37Finally, the weights are called quantity weights if: w = s Vj = 1,. . , K, wheredenotes the output own share of country j.2.10 ConclusionThis paper develops a taxonomy of multilateral methods for making international comparisons. The taxonomy is based on the principle of grouping methods together if andonly if they ca be showi to be special cases of the same more general class of method.Most of the multilateral methods examined here fall into one or other of the followingfour groups.3°1. Average Price Methods Geary-Khamis, Iklé, Gerardi, Ideal Price(IP), FixedBase Price, Van Yzeren homogenous group, Harmonic Average Price.2. Average Basket Methods — ECLAC, Geometric Average Basket, Fixed BaseBasket, Van Yzeren heterogenous group, Harmonic Average Basket.3. Fisher Star Methods — Van Yzeren balanced, Van Yzeren-KurabayashiSakuma(YKS), YKS-Q, Geary-Khamis-ECLAC (GKEC).4. Mean Asymmetric Star Methods — Eltetö-Köves-Szulc(EKS), MultilateralTranslog, Own Share, Mean Output Share.The relationships between the four groups are depicted in Figure 2.1. For example,the first three groups — Average Price, Average Basket and Fisher Star methods— are allspecial cases of Symmetric Star methods. Symmetric Star methods themselves form onlyone class within the more general category of Star methods, as explained in section 2.6.It is hoped that the taxonomy will provide the debate over the relative merits of competing multilateral methods with more structure. In particular, the taxonomy should30As far as the author is aware, the Harmonic Average Price, Harmonic Average Basket and GearyKhamis-ECLAC (GKEC) methods and the variant on the Own Share method defined in (2.68) have notpreviously been discussed in the literature.chapter 2. A Taxonomy of Multilateral PPP Methods. 38yield useful insights into the generic relationships between methods. It may be conjectured that methods of the same type will exhibit similar behaviour. Further conjecturescould be made regarding methods of differing types. Notably, Fisher Star and MeanAsymmetric Fisher Star methods are likely to exhibit similar behaviour, as will MeanValue Share, Törnqvist Star and Mean Asymmetric Törnqvist Star methods. Finally,Average Basket and Average Price methods are in some sense opposites. For example,Average Price methods satisfy the Average Test for quantity indices stated in equation(2.19) while Average Basket methods do not. In contrast, Average Basket methods satisfy the Average Test for PPPs stated in equation (2.20) while Average Price methods donot. In addition, Average Price methods tend to overestimate the output shares of any“outlier” countries in a comparison, while Average Basket methods tend to do exactlythe opposite, namely to underestimate these same shares.3 These conjectures may betested using the multilateral test approach developed by Diewert(1986) and Balk(1995).This remains as a topic for future research.31This is the so-called Gerschenkron effect discussed in Chapter 4.Chapter 3Chained PPPs and Minimum Spanning Trees3.1 IntroductionThis paper proposes a new method for making multilateral international comparisonsthat uses Kruskal’s “Minimum Spanning Tree” Graph Theory algorithm to chain PPPsacross countries. The Minimum Spanning Tree (MST) method may dramatically simplifythe construction of PPPs by reducing the number of countries that must be compareddirectly. The method also has important implications for multilateral time series comparisons especially if the data are seasonal.By chaining is meant the procedure of linking together bilateral comparisons. Theidea of chaining index numbers dates back to Marshall(1887). In a time series context,chained comparisons were seen to have two advantages over fixed base comparisons.Firstly, in the words of Irving Fisher:It may be said that the cardinal virtue of the successive base or chain systemis the facility it affords for the introduction of new commodities, the droppingout of obsolete commodities, and the continued readjustment of the systemof weighting to new commodities. (Fisher- 1911)Secondly, Diewert(1978) advocates chaining since it tends to reduce the spread betweenPaasche and Laspeyres indices and between all known superlative indices.1 In other1Paasche and Laspeyres indices are defined in section 2. Superlative indices are discussed in Diewert(1976) and Diewert(1978).39Chapter 3. Chained PPPs and Minimum Spanning Trees 40words, chaining tends to reduce the sensitivity of the results of a comparison to the choiceof bilateral index number formula. However, chaining sometimes causes the Paasche—Laspeyres spread to rise rather than fall. Szulc(1983) and Forsyth and Fowler(1981)found that a sufficient condition to ensure that chaining reduces the Paasche—Laspeyresspread is for price and quantity vectors to change monotonically along the chain. If onthe other hand price and quantity vectors cycle (or using Szulc’s term “bounce”), thena situation akin to resonance may arise.2 In the presence of “bouncing”, the Paasche—Laspeyres spread instead of falling, may become arbitrarily large.Chaining has a third use which although widely appreciated is rarely stated explicitly.By a multilateral method, is meant a method that generates transitive quantity indicesand price indices (PPPs). Multilateral (transitive) indices can be obtained by chainingbilateral (intransitive) indices across any Spanning Tree.3All these advantages of chaining are also applicable to international comparisons.However, there are two important differences between multilateral time series and multilateral international comparisons. Firstly, there is a natural chronological orderingin time series data. No such natural ordering exists in cross-section data. Secondly, inmost multilateral international comparisons, it is important that all countries are treatedsymmetrically ex ante.4 The same does not apply to time series comparisons.All multilateral time series comparisons, and virtually all multilateral internationalcomparisons proceed by chaining intransitive bilateral indices across a Spanning Tree.Fixed Base time series comparisons use the “star” Spanning Tree depicted in Figure 3.1(a).an example of “bouncing” consider the following chain: Norway—Portugal—Sweden—Italy—Finland—Spain. The price and quantity vectors of the countries in this chain will almost certainlycycle along the chain.3Spanning Trees are discussed in greater detail in section 3. However, loosely speaking, a SpanningTree connects a set of vertices (countries or time periods) in such a way that there is exactly one pathbetween any pair of vertices. Examples of Spanning Trees are given in Figure 3.1.4Note this does not imply that a multilateral method for making international comparisons musttreat all countries equally, but rather that unequal treatment if it occurs, must arise from the datarather than being imposed ex ante.Chapter 3. Chained PPPs and Minimum Spanning Trees 41In contrast, “chained” time series comparisons are based on the “linear” Spanning Treedepicted in Figure 3.1(b). It is also possible to combine the two, for example by rebasingannual data at five year intervals as is now done by many national Statistical Offices tomeasure growth and inflation.Most multilateral methods of making international comparisons including the GearyKhamis, Eltetö-Köves-Szulc (EKS), and ECLAC methods, use the “star” Spanning Treedepicted in Figure 3.1(a). However, in the case of the Geary-Khamis and ECLAC methods, the country at the center of the star is an artificially constructed country obtainedby averaging over the price and quantity vectors of the countries in the comparison.5The EKS method instead places each country in turn at the center of the star and thenaverages over the indices obtained from these fixed base comparisons. The problem withusing the “star” Spanning Tree, whether in a time series or cross-section comparison isthat it is chosen arbitrarily. The “star” is just one of a vast number of possible SpanningTrees, each of which gives different results. In contrast, using the “linear” Spanning Treecan be justified in a time series context by the natural chronological ordering of the data.We can be fairly confident that price and quantity vectors will not cycle along a timeseries chain (assuming the data are not seasonal) and hence such chains should be wellbehaved.The multilateral method proposed in this paper differs from all other multilateralmethods in that it lets the data decide the Spanning Tree structure rather than imposingit ex ante. The Minimum Spanning Tree (MST) method uses Kruskal’s Minimum Spanning Tree algorithm to choose between the K2possible Spanning Trees in a comparisonbetween K countries.Section 2 discusses in more detail the properties of multilateral methods and introduces the notation used in this paper. Section 3 considers the general properties of Chain5For a taxonomy of multilateral methods for making international comparisons, see Chapter 2.Chapter 3. Chained PPPs and Minimum Spanning Trees 42methods. Section 4 defines the Malmquist and Konüs indices, and states some theoremsrelating to these indices. These theorems provide the rationale behind the MST method.The Paasche-Laspeyres Spread (PLS) index for a bilateral comparison is defined in Section 5. PLS indices form the input data for Kruskal’s Minimum Spanning Tree algorithm.The Minimum Spanning Tree (MST) method is outlined in section 6. In section 7, 1990OECD data are used to estimate MST Purchasing Power Parities (PPPs) for the countries of the QECD at the level of GDP. In section 8 it is shown how prior beliefs can beincorporated into the MST method. Section 9 does a sensitivity analysis on the MSTmethod while section 10 compares the MST method empirically with the Geary-Khamis,EKS and ECLAC methods. Section 11 concludes the paper.3.2 Methodology and NotationConsider the problem of calculating multilateral PPPs (Pk) and quantity indices (Qk)over a set of K countries. It is assumed that each country indexed by k = 1,. . . , K,supplies price and quantity data (Pkj, qkD, defined over the same set of basic headinggoods and services, indexed by i = 1 ,...,N.It is useful to distinguish between two different notions of a base country. The “nilmeraire base” may be defined as the country whose currency is used as the numerairefor the comparison and hence whose PPP and quantity index are set equal to unity. Incontrast a “weighting base” exists if one and only one country’s price and/or quantityvectors are used as weights in the comparison. Not all PPP methods have a weightingbase, and for those that do, it need not necessarily coincide with the numeraire base.Changing the numeraire base serves only to rescale multilateral PPPs and quantityindices. Hence it is also useful to define multilateral output shares (sk). Output sharesare a set of quantity indices that have been rescaled to sum to unity.Chapter 3. Chained PPPs and Minimum Spanning Trees 43Multilateral PPPs and quantity indices may be related using the Weak Factor Reversal Test stated below in (2.2):Weak Factor Reversal Test: = V b, k = 1,... , K. (3.1)P6 Qb 1pbqbIn the remainder of the paper, the following bilateral formulae will be referred to repeatedly. It should be noted that these bilateral formulae are intransitive. In other words,unlike their multilateral counterparts, bilateral PPPs and quantity indices are dependenton the choice of the numeraire base country b.Laspeyres Quantity Index: = (3.2)Oi=1 pbiqbiLaspeyres PPP: pL =i Pkibi (3.3)Zj1pbiqbiPaasche Quantity Index: = (3.4)ii pkqbPaaschePPP: (3.5)i=1 pbiqkiFisher Quantity Index : (QQ)”2, (3.6)Fisher PPP: p =1/2, (37)3.3 Chain MethodsChain methods are an application of Graph Theory.6 A Graph c(V, ) consists of a set ofvertices denoted by V and a set of pairs of vertices denoted by. If the pairs are ordered,then the graph is directed. If the pairs are unordered, then the graph is undirected. Theelements of are called edges. In other words, each edge connects a pair of vertices. ForChain methods, each vertex in the set V corresponds to a country (time period) in the6For an introductory discussion on Graph Theory see Foulds(1991) or Wilson(1985).Chapter 3. Chained PPPs and Minimum Spanning Trees 44set ?C. Similarly, each edge in g connecting a pair of vertices j and k corresponds to abilateral comparison between countries (time periods) j and k.A connected graph is a graph such that a path of edges exists connecting each vertexto all other vertices in V. A special case of a connected graph is a complete graph.A complete graph is a graph without any missing edges. More precisely in a completegraph an edge is defined between each pair of vertices. A cycle exists, if it is possible totravel between two vertices j and k by two or more distinct paths. A Spanning Tree is aconnected graph which has no cycles. Hence over a set of K vertices, all Spanning Treeshave exactly K — 1 edges. If the Spanning Tree had less than K — 1 edges, then it couldnot be a connected graph. On the other hand, if it had more than K — 1 edges, then itmust have at least one cycle.Transitive quantity indices can be obtained from K — 1 intransitive bilateral quantityindices by chaining over the set of K countries, if and only if the chain forms a SpanningTree over the set of countries K. This result may be explained as follows. Multilateral(transitive) quantity indices can be obtained from bilateral (intransitive) quantity indicesby chaining if and only if the chain has no cycles. Furthermore, for the set of multilateralquantity indices to be complete, the chain must span the whole set of countries. In otherwords, the chain must be a connected graph. Since the chain must be a connected graphwithout any cycles, it constitutes a Spanning Tree over the set of countries .A K dimensional matrix of bilateral (intransitive) quantity indices defined on a set ofK countries, may be represented by a complete graph. A complete graph with K vertices,has K’2 different Spanning Trees contained within it. Each of these Spanning Treesmay be used to generate a different set of multilateral (transitive) quantity indices. Henceone of the problems with Chain methods is deciding which Spanning Tree to use. Someexamples of Spanning Trees over the set of K = 5 vertices (countries or time periods)are givell in Figure 3.1.Chapter 3. Chained PPPs and Minimum Spanning Trees 45(a)Figure 3.1: Examples of Spanning TreesStar methods are special cases of Chain methods that use the “star” Spanning Treedepicted in Figure 3.1(a). All Star methods have a weighting base, which is the country(time period) at the center of the star. For certain Star methods, such as the GearyKhamis and ECLAC methods, the weighting base is not one of the countries in the set,but rather an artificially constructed country obtained by averaging over the price and/orquantity vectors of the countries in the comparison. Using an artificially constructedcountry as the weighting base, ensures that ex ante all countries are treated symmetrically.To generate multilateral quantity indices from a Spanning Tree, it is first necessaryto specify a numeraire base. The procedure is demonstrated below for the star SpanningTree using two different numeraire bases. In Figure 3.2(a), country (2) is the numerairebase, while in 3.2(b), country (1) is the numeraire base.7 Let Qjk denote a generic bilateralquantity index while Qk/Q denotes a multilateral quantity index between countries j andk with j as the numeraire base.At this point a distinction must be drawn between directed and undirected graphs.The ordering of directed graphs is depicted by an arrow. Hence in contrast to Figure 3.1,the graphs in Figure 3.2 are directed. The reason for this difference will become apparentTIt should be noted that in both 3.2(a) and 3.2(b), country (1) is the weighting base.(b) (c)Chapter 3. Chained PPPs and Minimum Spanning Trees 46(3)= Q21Q13 (3) == 1 (1) = Q21Q14 =Q12 (1) =•(2) 14) = i 14)Q2 Qi(5)= Q21Q1.5 (5) =(a) (b)Figure 3.2: Constructing Multilateral Quantity Indices using the Star Spanning Treeshortly. For the multilateral quantity indices (Qk) in Figures 3.2(a) and 3.2(b) to beconsistent with each other, the bilateral indices (Qk) must satisfy the Country (Time)Reversal Test.8 Suppose the bilateral indices do not satisfy the Country Reversal Test.It turns out that it is still possible to generate multilateral quantity indices by imposingthe Country Reversal Test on the bilateral indices. Between each pair of countries, twobilateral indices are defined, namely Qjk and Qk3. For each pair of countries connectedby an edge in the Spanning Tree, one of these two indices must be redefined. Either Qkjmust be replaced with l/Qk, or Qjk must be replaced with l/Qk. This is exactly theprocedure followed by Star methods such as the Geary-Khamis and ECLAC methodswhich use the Lasperes and Paasche quantity index respectively. The exact procedureused may be described by a directed Spanning Tree. An arrow 011 the edge connectingj to k, travelling from j to k, implies that Q,j has been replaced by l/Qk, rather than8A bilateral quantity index satisfies the Country (Time) Reversal Test if and only if:Vj,k1C.QkThe Fisher formula satisfies this test while the Laspeyres and Paasche do not.Chapter 3. Chained PPPs and Minimum Spanning Trees 47vice versa. If the bilateral quantity index satisfies the Country Reversal Test, then thesearrows are superfluous.3.4 Malmquist and Konüs IndicesA Malmquist quantity index is defined as follows:DQ,qk)QM(q.qk,’u) =D(u,q)where D(’u,q) = rnaxh[h : F(q/h) n,h >0]. (3.8)F(q) is a continuous, increasing and quasiconcave aggregator function. D(u,q) in (3.8) isthe deflation factor h, that will just reduce the vector q proportionately so that F(q/h) =u. Similarly, a Konüs cost of living index is defined as follows:e(pk, u)., (3.9)ep3,U1where e(p, u) is an expenditure function. The Malmquist quantity index is dual to theKonfls PPP.9The Malmquist and Konüs indices may be viewed respectively as the true underlyingbut unobservable quantity index and PPP.’° Although the Malmquist and Konüs indicesare unobservable, the theorems that follow, place observable upper and lower bounds onthem. These theorems provide the rationale behind the Minimum Spanning Tree (MST)method.Neither the Malmquist quantity index nor the Konüs PPP is invariant to changes inu, unless preferences are hornothetic. When u is homothetic, D(u, q) = g(zt)(q), ande(p, n) = h(u)(p). Therefore, given this assumption, the Malmquist and Konüs indices9See Russell(1983).10The Malmquist and Konüs indices are defined here for a representative agent. However, they mayalso be extended to groups. See for example Diewert(1984).Chapter 3. Chained PPPs and Minimum Spanning Trees 48may be rewritten as follows:Dqi) ëpk)QM(q,qk,’u)= (qj)’ PK(p,pk,’u) = _()Malmqllist(1953,pp.231) proved the following theorem for the Malmquist quantity index.Theorem 1 — When u is homothetic,min(Q, Qk) Q(qj, q, u) <max(%, Qk) V •u.and in Theorem 1 denote respectively Paasche and Laspeyres quantity indices.These indices are defined in (3.4) and (3.2). Similarly, Frisch(1936,pp.25) proved thefollowing theorem for the Konüs PPP.Theorem 2 When u is homothetic,min(P,Pf,j P’(p,p,’u) <max(PJ,Pf) Vu.and Pfj in Theorem 2 denote respectively Paasche and Laspeyres PPPs. Theseindices are defined in (3.5) and (3.3). Theorems 1 and 2 imply that when preferencesare homothetic, the Laspeyres and Paasche quantity indices and PPPs act as upper andlower bounds on the true Malrnquist and Konüs indices respectively. However, a casecan also be made for these results even when preferences are not homothetic. Theorems3 and 4 below, generalize Theorems 1 and 2 respectively, by not assuming homotheticity.The importance of Theorems 3 and 4 is that they demonstrate that the assumption ofhomotheticity is not essential to the rationale behind the MST method.Theorem 3— There exists u = (q*) rhere n(qk) < n’< < uj(qj) such thatmin(Q, Q) QM(qJ, qk, u) <max(Q, Qk).Theorem 4— There exists u = (q*) where uk(qk) < u < zt(q) such thatmin(P,F) PK(p,pk,u ) max(P,Pfj).Theorems 3 and 4 are due respectively to Malmquist(1953) and Konfls(1924, pp.20-21). They imply that even when preferences are not homothetic, there exists a utility levelChapter 3. Chained PPPs and Minimum Spanning Trees 49between the two levels being compared for which the Laspeyres and Paasche quantityindices and PPPs act as upper and lower bounds on the Malmquist and Konfls indicesrespectively.113.5 The Paasche-Laspeyres Spread (PLS) IndexThe rationale behind the MST method is to let the data decide which Spanning Treeis used. To do this, it is necessary to have some means of discriminating between theKK_2 possible Spanning Trees. This is done using the PLS index. The PLS index PLSkbetween countries j and k is defined below:’2/ (DP ijLl\ / IriP çLmax1k,AJk1 I IPLSk(p,q,pk,qk)=1ogj,=logj.. (3.10)min jk’’ jk) / \ minj, ‘jk)Hence the PLS index of a bilateral comparison is simply equal to tile logarithm of thePaasche—Laspeyres spread. In other words it measures the dispersion between Paascheand Laspeyres indices. The dispersion between Paasche and Laspeyres quantity indicesis the same as between Paasche and Laspeyres PPPs. This property of the PLS index isstated below as Property 4.If changes in the price and quantity vectors across countries are negatively correlated, then a Laspeyres PPP (quantity index) exceeds its corresponding Paasche PPP(quantity index). Such negative correlation is consistent with consumer iltility maximization, On the other hand, if price and quantity changes are positively correlated, thena Paasche PPP (quantity index) exceeds its corresponding Laspeyres PPP (quantity index). Such positive correlation is consistent with producer profit maximization. Paascheand Laspeyres PPPs (quantity indices) are equal if and only if there is zero correlation11Diewert(1981,pp.168-176) has proofs of all four theorems.‘21n equation (3.10). P] and denote Paasche and Laspeyres PPPs between countries j and k.These indices are defined in equations (3.5) and (3.3) respectively. Similarly, Q and denotePaasche and Laspeyres quantity indices. These indices are defined in (3.4) and (3.2) respectively.Chapter 3. Chained PPPs and Minimum Spanning Trees 50between price and quantity changes across collntries. One desirable feature of the PLSindex as specified in equation (3.10) is its flexibility. It is equally applicable to worldswhere respectively consumer and producer behaviour dominate.PLSk has the following properties.Property 1 PLS = 0.Property 2— PLSjk 0.Property 3- PLSk = PLSk3.Property 4- PLSk(P) = PLSk(Q), since=Property 5 If p = then PLS3k = 0.Property 6 — If qj = 6q, then PLSk 0.The Laspeyres and Paasche PPPs (quantity indices) are both equal to the Konüs(Malmquist) index if N = 1. When N 2, the bounds in Theorem 2 (Theorem 1) areequal if the N goods satisfy either Hicks’(1946) or Leontief’s(1936) aggregation theorems.Properties 5 and 6 describe respectively the two situations when the set of goods overwhich countries j and k are being compared satisfy these aggregation theorems.3.6 The Minimum Spanning Tree MethodBefore discussing Minimum Spanning Trees it is first necessary to introduce the notionof a weighted graph. Each edge in a weighted graph has a numerical valile or weight associated with it. The Minimum Spanning Tree of a graph is the Spanning Tree containedin the graph which has the minimum sum of weights.The first step of the MST method is to construct a complete weighted graph definedon the set of K countries. The weight on an edge between countries j and k equals thePLS index PLSk between j and k. The graph is undirected because of Property 3 ofthe PLS index which states that PLSk = PLSk V j, k E . The MST method chainsChapter 3. Chained PPPs and Minimum Spanning Trees 51bilateral (intransitive) indices across the Minimum Spanning Tree of this weighted graphto obtain multilateral (transitive) indices.The natural choice of bilateral formula is the Fisher index.’3 The Fisher index hasa number of advantages over other bilateral formulae. Firstly, it satisfies the CountryReversal Test, and hence as explained in section 3, there is no need to impose thistest on the bilateral indices. As well as simplifying matters, this is important since theMinimum Spaniling Tree is undirected and hence the Country Reversal Test would haveto be imposed on the bilateral indices in an arbitrary fashion.Secondly, by construction, a Fisher PPP (quantity index) is constrained to lie betweenits corresponding Paasche and Laspeyres PPPs (quantity indices). Hence as the Paasche—Laspeyres spread shrinks, a Fisher quantity index must converge on its correspondingMalmquist index, while a Fisher PPP must converge on its corresponding Konüs index.Therefore the following Proposition may be surmised.Proposition 1 — A Fisher PPP (quantity index) between countries j and k tends toapproximate more closely its respective true underlying Konüs (Malmquist) index, thana Fisher PPP (quantity index) between countries m and n f PLSi <PLSmn.Proposition 1 provides a basis for discriminating between Fisher indices and henceindirectly also between Spanning Trees. The preferred Fisher indices are those betweencountries with the smallest PLS indices. Comparisons between countries with small PLSindices have the added advantage of being less sensitive to the choice of index numberformula.Thirdly, the Fisher index is superlative.’4Last but not least, the Fisher index satisfies‘3The Fisher quantity index and PPP are defined in equations (3.6) and (3.7) respectively.‘4For a definition and a discussion on the properties and advantages of superlative index numbers, seeDiewert (1976) or Diewert (1978).Chapter 3. Chained PPPs and Minimum Spanning Trees 52the Factor Reversal Test.’5 The Factor Reversal Test is important for the followingreason. Multilateral quantity indices may be obtained by chaining bilateral quantityindices across a Spanning Tree. Similarly, multilateral PPPs may be obtained by chainingbilateral PPPs across the same Spanning Tree. These two methods are dual to eachother in the sense of generating the same set of output shares, if and only if the bilateralquantity indices and PPPs used to chain across the Spanning Tree satisfy the FactorReversal Test. Unlike Paasche and Laspeyres indices, Fisher PPPs and quantity indicessatisfy the Factor Reversal Test. Hence the MST method applied to Fisher quantityindices is dual to the MST method applied to Fisher PPPs. In contrast, the MSTmethod applied to Paasche (Laspeyres) quantity indices is not dual to the MST methodapplied to Paasche (Laspeyres) PPPs.A number of algorithms exist in the Graph Theory literature for computing the Minimum Spanning Tree of a graph. In the empirical section that follows, Minimum Spanning Trees are computed using Kruskal’s algorithm run on Mathematica.16 Kruskal’salgorithm proceeds as follows. First of all, the algorithm ranks the edges according tothe size of their weights (PLS indices) .‘ Then the two edges with the smallest weightsare selected. Then the edge with the third smallest weight is selected, subject to theconstraint that it does not create a cycle. If selecting this edge creates a cycle, thenthe algorithm skips it and moves on to the edge with the next smallest weight. Thisprocedure for selecting edges is repeated until it is no longer possible to select any more‘5A bilateral index satisfies the Factor Reversal Test if:—1pkiqki Vj,k epfiqand P can be obtained from Q by interchanging prices and quantities. The Fisher index satisfies thistest while Laspeyres and Paasche do not.‘6See Skiena(1990).‘7The probability of encountering ties becomes negligible if the PLS indices are calculated to a sufficiently large number of decimal places.Ghapter 3. chained PPPs and Minimum Spanning Trees 53edges without creating a cycle, at which point the algorithm terminates. The set of vertices and selected edges constitutes the Minimum Spanning Tree. The rationale behindusing Kruskal’s algorithm to discriminate between the KK_2 competing Spanning Treesderives directly from Proposition 1. The Minimum Spanning Tree is constructed fromedges connecting pairs of countries with the lowest Paasche—Laspeyres spreads.A second justification for using Kruskal’s algorithm derives from the fact that theMinimum Spanning Tree is the Spanning Tree contained in a graph which has the minimum sum of weights.’8 Consider the problem of finding the chain path between a pairof countries with the minimum Paasche—Laspeyres spread. This problem is equivalent tofinding the chain path between these countries with the minimum sum of weights (PLSindices). This is because the sum of weights along a linear chain equals the logarithmof the Paasche—Laspeyres spread of the chain.19 Hence the chained Paasche—Laspeyresspread is a monotonic transform of the sum of weights. Minimizing the sum of weightsacross a Spanning Tree may be viewed merely as an extension of this problem to multilateral comparisons.3.7 Applying the MST MethodIn this section, MST PPPs are calculated for the countries of the OECD using 1990OECD data at the level of GDP.2°To construct MST PPPs the only data required are amatrix of bilateral Paasche and Laspeyres PPPs for the set of countries being compared.The matrix of PLS indices given in Appendix B, Table B.1 is derived from the Paasche‘8A proof of this result is given in Wilson(1985), pp.55.19Actually this result only holds if either P] PJ V j, k E )C, or if V j, k E K. However,at least at the level of GDP, Laspeyres PPPs always exceed their corresponding Paasche PPPs.20MST PPPs are calculated instead of quantity indices since the required Paasche and Laspeyres PPPdata are more readily available in the desired form. As discussed earlier, the MST method applied toPPPs is dual to the MST method applied to quantity indices if Fisher indices are used to chain acrossthe IVlinimum Spanning Tree.Chapter 3. Chained PPPs and Minimum Spanning Trees 54and Laspeyres matrices using equation (3.10).In the graph theory literature, Table B.1 is called an adjacency matrix. Any graph canbe represented as an adjacency matrix. The adjacency matrix of a graph with K verticeshas dimension K x K. The (x, y) element of the adjacency matrix equals the weight onthe edge connecting vertex x to vertex y. If there is no such edge in the graph, then thecorresponding element in the adjacency matrix is set equal to zero. Undirected graphshave symmetric adjacency matrices. Hence Table B.1 defines an undirected and completegraph.2 By feeding the adjacency matrix in Table B.1 into Kruskal’s Minimum SpanningTree algorithm, the Spanning Tree depicted in Figure 3.3 was produced as output.22MST PPPs are obtained by chaining Fisher PPPs across the Minimum SpanningTree in Figure 3.3. The matrix of Fisher and MST PPPs are presented respectively inAppendix B, Tables B.2 and B.3. It should be noted that the matrix of MST PPPs inTable B.3 has rank one. In fact a multilateral method must by construction generate amatrix of PPPs with rank one, irrespective of its dimension. Otherwise the output shareswould not be independent of the choice of numeraire base country. In contrast, a matrixof Fisher PPPs in general has full rank.3.8 Prior RestrictionsTwo types of prior restriction can easily be imposed on the MST method. The first typeinsists that one particular edge is included in the Spanning Tree. For example, supposeone wishes to insist that New Zealand is linked directly to Australia. This restriction isimposed by simply altering the PLS index between New Zealand and Australia in theadjacency matrix. If this PLS index is given a dummy value smaller than all the PLS21The graph is undirected since PLSk PLSk. It is complete since the only zero terms in theadjacency matrix are on the lead diagonal.22See Skiena(1990).Chapter 3. Chained PPPs and Minimum Spanning Trees 55IcelandFin andNew ea1and DemarkNor vayJapan SwedenSwitz riandLuxeiibourg A striaGerci yNethe1ands Fra ceUnited Kingdom Be1 iumIrel nd ItalCanada Portugal GreeceAustralia • SpainUnited tates Tur eyFigure 3.3: The Minimum Spanning Tree for the OECDChapter 3. Chained PPPs and Minimum Spanning Trees 56indices in the matrix, then Kruskal’s algorithm is certain to select its corresponding edge.When this restriction was imposed on the adjacency matrix in Table B.1, the obtainedMinimum Spanning Tree differed from Figure 3.3 only to the extent that New Zealand isow linked with Australia rather than Finland. The second type of prior restriction insiststhat one particular edge is excluded from the Spanning Tree. For example, suppose onewishes to insist that Japan is not linked directly to Norway. This restriction is imposedby giving the PLS index between Japan and Norway a large dummy value. When thisrestriction was imposed on the adjacency matrix in Table B.1, the obtained MinimumSpanning Tree differed from Figure 3.3 only to the extent that Japan now linked withSweden rather than Norway.It is also possible to measure the impact of prior restrictions by observing their influence on the sum of weights (PLS indices) in the Minimum Spanning Tree .As statedearlier, Kruskal’s algorithm selects the Spanning Tree with the minimum sum of weights.Hence the extent to which the sum of weights in the Minimum Spanning Tree rises as aresult of imposing a prior restriction is indicative of the extent to which the restrictioncontradicts the data. The impact of restrictions on the sum of weights may be usedto discriminate between alternative restrictions. For example, forcing New Zealand tobe linked to Australia caused the sum of weights to rise from 1.2117 to 1.2284, whileforbidding Japan from linking to Norway caused the sum of weights to rise from 1.2117to 1.2356. Hence we may conclude that the former restriction is more reasonable.3.9 Sensitivity AnalysisAll the multilateral methods that have been advocated in the literature fail to satisfyindependence of irrelevant alternatives in the following sense. For all these methods thereexists at least one country, that if removed from the comparison, causes the relativeChapter 3. Chained PPPs and Minimum Spanning Trees 57output shares of at least one pair of countries remaining in the comparison to change.The MST method is no exception to this rule.This sectioll assesses the sensitivity of the MST method to the deletion of a country.It would also be useful to analyze the sensitivity of the MST method over time. Unfortunately due to lack of appropriate OECD data in any year except 1990, it is not currentlypossible to assess the robustness of the MST method over time.Referring to the Minimum Spanning Tree in Figure 3.3, it is clear that the sensitivity of MST PPPs (quantity indices) to the deletion of a country from the comparisondepends crucially on which country is deleted. 13 of the 24 countries in Figure 3.3 areconnected to only one other country in the Minimum Spanning Tree. If the countrydeleted from the comparison is one of these countries, then independence of irrelevantalternatives will be satisfied, since removing that country and its connecting edge willstill leave a Minimum Spanning Tree now defined over the remai1ing 23 countries. If onthe other hand, the country deleted belongs to one of the 11 countries connected to twoor more other countries, then independence of irrelevant alternatives must be violated,since deleting that country and its connecting edges from the Minimum Spanning Treewill leave a disconnected graph. A disconnected graph cannot be a Spanning Tree letalone a Minimum Spanning Tree.The worst case scenario of country deletion in Figure 3.3 is to delete Germany. Germany is connected to six other countries. Figure 3.4 depicts the new Minimum SpanningTree obtained using the same OECD data but with Germany deleted from the comparison. Although comparisons between certain pairs of countries such as Norway andSweden are unaffected, comparisons between other pairs such as France and Belgium areaffected. The matrix of MST PPPs corresponding to Figure 3.4 is presented in AppendixB, Table B.4.Chapter 3. Chained PPPs and Minimum Spanning Trees 58IrelandUnited ‘kingdom NetTierlandsUSA Aus :raliaCaiadaFin ap/New ea1and DeimarkNor vayJapan SwedellSwitz riandLuxeiThourg Aus’hiaBel, iurnFrnceSpainItaly Gre cePor ugalTur eyFigure 3.4: The Minimum Spanning Tree for the OECD with Germany DeletedChapter 3. Chained PPPs and Minimum Spanning Trees 59The mean absolute percentage deviation between corresponding MST PPPs in Tables B.3 and B.4 is 0.47%. In other words, the average MST PPP in Table B.3 changedby about 0.47% purely as a result of omitting Germany from the comparison. In contrast,the average EKS PPP calculated using the same data set, changed by only about 0.073%as a result of omitting Germany. Similarly, the maximum absolute percentage deviationbetween corresponding MST PPPs is 2.6%, while between corresponding EKS PPPs itis 0.22%.These results suggest that MST PPPs (quantity indices) are more sensitive to thedeletion of countries than EKS PPPs (quantity indices). However, it should be noted thatthe MST PPP percentage deviations quoted above were calculated by deleting Germany,which plays a central role in the Spanning Tree in Figure 3.3. If on the other hand, Turkeywas deleted, the MST PPP mean percentage deviation would be zero. In contrast, EKSPPP percentage deviations remain fairly stable as the deleted country is varied.3.10 A Comparison between the MST, Geary-Khamis, EKS and ECLACMethodsThe Geary-Khamis, EKS and ECLAC methods are probably the most widely used multilateral methods. These three methods are also examples respectively of three of the maingeneric classes of multilateral methods.23 ECLAC stands for United Nations EconomicCommission for Latin America and the Caribbean. The ECLAC method was advocatedby Walsh(1901) who called it Scrope’s method with arithmetic weights. The GearyKharnis method was proposed by Geary(1958) and Khamis(1972). It is the preferredmethod of the International Comparison Project(ICP) and is also used by the OECD,IMF and World Bank. The EKS method was first proposed by Gini(1931). However,23See Chapter 2.Chapter 3. Chained PPPs and Minimum Spanning Trees 60it was credited to Eltetö and Köves(1964) and Szulc(1964). The EKS method is thepreferred method of Eiirostat, and is now also being used by the OECD.24Matrices of Geary-Khamis, EKS and ECLAC PPPs for the countries of the OECD,again calculated using 1990 OECD data are presented respectively in Appendix B, Tables B.5, B.6 and B.7. From the data it can be seen that EKS PPPs (quantity indices)generally lie between their corresponding Geary-Khamis and ECLAC PPPs (quantityindices) 25 For approximately 80% of the 276 possible bilateral comparisons betweenOECD countries, each EKS PPP lay between its corresponding Geary-Khamis andECLAC PPPs.26 The mean absolute percentage deviation between each EKS PPP andits corresponding Geary-Khamis PPP is about 4%. Similarly, the mean absolute percentage deviation between each EKS PPP and its corresponding ECLAC PPP is also about4%27A comparison of MST PPPs with their corresponding Geary-Khamis, EKS and ECLACPPPs reveals the following results. Using only the 223 comparisons for which each EKSPPP lies between its corresponding Geary-Khamis and ECLAC PPPs, approximately53% of the MST PPPs lay between their corresponding Geary-Khamis and EKS PPPs,while 22% lay between their corresponding EKS and ECLAC PPPs. Of the remaining25% of the MST PPPs that did not lie between their corresponding Geary-Khamis andECLAC PPPs, three quarters of these outliers were on the Geary-Khamis side. Theseresults suggest the following proposition regarding the MST method. MST PPPs (quantity indices) tend to lie in between Geary-Kharnis and EKS PPPs (quantity indices) and24For a discussion on the evolution of the literature, see Diewert(1993).2511 is not meaningful to ask whether in general a Geary-Khamis PPP is larger than its correspondingECLAC PPP, since the answer depends on the nilmeraire base.26This result is due largely to the Gerschenkron effect. The Gerschenkron effect is discussed in Chapter 4.271t should be noted that the size of these mean absolute percentage deviations depends on the setof countries being compared. In general, these percentage deviations increase as the set of countriesbeing compared becomes more heterogenous. For example, if the comparison was extended to includecountries from Asia, Africa and Latin America these percentage deviations could increase significantly.Chapter 3. Chained PPPs and Minimum Spanning Trees 61furthermore tend to approximate the latter more closely than the former.It is also informative to compare these multilateral PPPs with Fisher PPPs. TheFisher index is the generally preferred formula for making bilateral comparisons. Henceone could argue that the preferred multilateral method should be the one whose PPPs(quantity indices) most closely approximate Fisher PPPs (quantity indices). The meanabsolute percentage deviation between the Fisher and Geary-Khamis PPPs is 4.25%.Between Fisher and ElKS PPPs, it is 1.16%. Between Fisher and ECLAC PPPs it is4.21%. Finally between Fisher and MST PPPs, it is 1.78%. Not surprisingly, ElKS PPPsapproximate Fisher PPPs the most closely. This is not surprising since by constructionElKS PPPs are the solution to the problem of altering Fisher PPPs (quantity indices)by the logarithmic least squares amount necessary to obtain transitivity. Of course,there are many other criteria for discriminating between competing multilateral methods.In particular, Diewert(1986) and Balk(1995) develop a series of tests for multilateralmethods. To really discriminate between these methods it is necessary to subject themto a series of such tests. Nevertheless, it is informative to know that MST PPPs tendto approximate Fisher FPPs rather more closely than either Geary-Khamis or ECLACPPPs do.3.11 ConclusionThe Minimum Spanning Tree (MST) method proposed in this paper departs from othermultilateral methods of making international comparisons in that it lets the data determine the underlying Spanning Tree structure, rather than imposing it arbitrarily. Anordering of countries is constructed by applying Kruskal’s Minimum Spanning Tree algorithm to the data. Multilateral (transitive) PPPs and quantity indices are then obtainedby chaining bilateral (intransitive) Fisher indices across the Minimum Spanning Tree.Chapter 3. Chained PPPs and Minimum Spanning Trees 62Few attempts have been made in the international comparison literature to constructSpanning Trees across countries. One such attempt was made by Kravis, Heston andSummers(1982). They suggested an interesting variant on the “star” Spanning Tree. Using cluster analysis techniques the set of countries can be divided into more homogenoussubsets. Then star Spanning Trees defined over these subsets may be linked togetherto form a Spanning Tree defined over the whole set. The overall Spanning Tree doesnot have a star structure. Nevertheless, this method still imposes arbitrary constraintson the Spanning Tree. In particular, the weighting base for each cluster and the linkcountries between clusters are both chosen arbitrarily. Kravis et al considered variouscriteria for measuring similarity across countries to enable cluster formation, rangingfrom geographical propinquity and price correlation coefficients, to Paasche—Laspeyresspreads.The MST method may be viewed as an extension of the pioneering work of Kravis etal. in the field of chained international comparisons. Indeed the input into the MinimumSpanning Tree algorithm used by the MST method consists of a matrix of logged Paasche—Laspeyres spreads. However, rather than using these data to form clusters of countriesalong the lines of Kravis et al, and then constructing a Spanning Tree indirectly by linkingtogether star Spanning Trees defined over these clusters, the MST method obtains aSpanning Tree directly from the algorithm without imposing any restrictions.28Perhaps the most useful aspect of the MST method is that it reduces the number ofcountries that must be compared directly. Most PPP methods including Geary-Khamis,EKS aild ECLAC require all countries in a comparison to supply price and expendituredata over the same basket of goods and services. This requirement creates difficulties,since a staple good in one country may be rare or even unobtainable in another country.28However, as section 8 explains, prior restrictions may easily be imposed on the MST method ifdesired.Chapter 3. Chained PPPs and Minimum Spanning Trees 63This is particularly a problem for comparisons at a global level. In contrast, the MSTmethod requires each country to make comparisons with only a small number of othercountries, which moreover by construction have relatively similar price and expenditurepatterns. However, this is only true once a Minimum Spanning Tree has been constructed.This is not a problem if the Minimum Spanning Tree is stable over time, since onceconstructed, it would then he possible to use the same Spanning Tree for a number ofdifferent comparisons. Unfortunately the robustness of the MST method over time hasnot yet been verified, due to lack of data. Alternatively the MST method could be usedin a two step procedure. The first step would require all countries in the comparison tosupply price and expenditure data over the same basket of goods and services, so thatPaasche and Laspeyres indices between each pair of countries could be calculated. ThesePaasche and Laspeyres indices could then be used to generate a Minimum SpanningTree. The second step would require countries to supply more price and expendituredata. However, now each country would only have to harmonize its basket of goods andservices with its neighbours in the Minimum Spannillg Tree, thus allowing the basket tobe more representative, and hence improving the quality of the comparison.Empirically the MST method tends to generate PPPs (quantity indices) that liein between Geary-Khamis and EKS PPPs (quantity indices). Also, MST PPPs tendto approximate Fisher PPPs rather more closely than either Geary-Khamis or ECLACPPPs, although not as closely as EKS PPPs.Although this paper focuses on index number comparisons across countries, the MSTmethod also has important implications for time series comparisons. Unfortunately, dueto the lack of available matrices of bilateral Paasche and Laspeyres time series datafor any of the countries in the OECD, for the time being these implications can only beconjectured. It may be conjectured that in a time series context, the natural chronologicalordering of the data would cause the MST method to generate a “linear” SpanningChapter 3. Chained PPPs and Minimum Spanning Trees 64Tree as depicted in Figure 3.1(b). Many national Statistical Offices now use such linearchronological chains to measure growth and inflation over time. Therefore the MSTmethod may strengthen the arguments outlined by Fisher(1911) and Diewert(1978) forpreferring chained over fixed base multilateral time series comparisons, on the groundsthat a linear chronological chain is the most natural way of chaining over bilateral timeseries indices to obtain transitivity.A further intriguing application of the MST method would be to the problem ofconstructing index numbers for seasonal time series data. It may be conjectured that thestrllcture of the Minimum Spanning Tree would depend on how pronounced is the seasonalvariability in the data. If the seasonal variability is sufficiently weak, the MinimumSpanning Tree would again be a linear chronological chain. However, if the seasonalvariability is sufficiently strong, then a rather more complex structure could emerge. Itis perhaps for this reason that seasonal data have proved to be problematic in the indexnumber literature. The appropriate method for constructing index numbers for seasonaldata may depend critically on the relative strengths of the seasonal and annual trends.The MST method allows for this, by letting the data determine which of the vast numberof alternative chaining structures is most suitable.Chapter 4Additive PPP Methods and the Gerschenkron Effect4.1 IntroductionA PPP method is additive if its quantity indices literally add up over different levelsof aggregation when measllred in value terms. In the taxonomy of Chapter 2, additivemethods are referred to as Average Price methods, since algebraically additivity requiresthe set of countries to be compared using the same vector of average price weights.Additivity is extremely useful if international comparisons are required at various levels ofaggregation, as for example in a national accounts comparison. However, according to theGerschenkron effect, additive methods have a tendency to overestimate the output sharesof any oiitlier countries in a comparison.’ By implication the Gerschenkron effect alsopredicts that additive methods tend to overestimate the purchasing power of currenciesof outlier countries.The most widely used additive PPP method is the Geary-Khamis method, whichhas been extensively criticized on account of the Gerschenkron effect.2 According to theGerschenkron effect, the IMP World Economic Outlook of May 1993, which used theGeary-Khamis method, overestimated the output shares of China and other developing(outlier) countries. Remember in contrast, that according to the Balassa-Samuelsonhypothesis, the World Bank World Development Report of 1983 underestimated theoutput shares of these same countries.3‘By an outlier is meant a country whose price vector differs significantly from the average price vector.2See for example, Gerardi(1982) or Eurostat(1983).3See Chapter 1.65Chapter 4. Additive PPP Methods and the Gerschenkron Effect 66In spite of the significant implications of the Gerschenkron effect, the evidence for itsexistence remains largely anecdotal. That is to say that no one has provided a satisfactorytheoretical explanation as to why the Geary-Khamis method and additive methods ingeneral are subject to the Gerschenkron effect. In fact, recently Khamis(1993) wrote ashort paper arguing in defence of the Geary-Khamis method, saying that there was noclear evidence for the existence of the Gerschenkron effect.It is claimed that for a country whose price structure is very different from thestructure of the average prices of the multilateral Geary-Khamis method oneobtains a higher volume level than one obtains had the average prices used inthe aggregation been more characteristic of that country’s prices. Referenceto the overall results, e.g. those on page 96 of the Phase III of the ICP 1982report (Kravis, Heston and Sumrners,1982) do not justify a general statementalong these lines and no valid proof has been provided for such a statement.(Khamis, 1993)It should be noted that there is some confusion regarding the definition of the Gerschenkron effect. Gerschenkron’s original 1951 paper predates the Geary-Khamis method.The original Gerschenkron effect is aptly described by Samuelson and Swamy(1974).‘TIen a society’s frontier is augmented more in terms of one good (say, machines) than another (say, bread), the quantity mix of demand allegedly movesto favour the most augmented good whereas its price ratio moves against thatgood. (Samuelson and Swamy, 1974)However, in the international comparison literature, the Gerschenkron effect is used todescribe the very specific purported tendency of additive methods to overestimate theoutput shares of outlier countries. Nevertheless, these two definitions are consistent,Chapter 4. Additive PPP Methods and the Gerschenkron Effect 67since the latter definition implies that consumer theory substitution effects dominatethe producer theory effects. This is in fact a necessary condition for the Gerschenkroneffect. If the opposite were true, the output shares of outlier countries would instead beunderestimated by additive methods.The Gerschenkron effect has recently prompted the World Bank to reconsider its useof the Geary-Khamis method in international comparisons. Instead the World Bank isconsidering using another additive method called the Iklé(1972) method, on the groundsthat it is less sensitive than Geary-Khamis to the Gerschenkron efl’ect.4This paper takes up Khamis’s challenge and explains the theoretical underpinningsof the Gerschenkron effect. It derives from the fact that expenditure patterns change inresponse to changes in relative prices, i.e. the substitution effect. Hence the Gerschenkroneffect is inherent to the data irrespective of the choice of additive method. However, theextent of the Gerschenkron effect also depends on the choice of method. Frameworksare developed for measuring both the size of the substitution effect across data sets andthe extent to which the Gerschenkron effect differs across additive methods for a givendata set. As part of the process of measuring the Gerschenkron effect across additivemethods, it is necessary to give a precise interpretation to the hitherto vague notion of an“outlier” country. Also it is shown that a variant on the Gerschenkron effect is applicableto the class of Average Basket methods identified in the taxonomy of Chapter 2. Thelatter framework is then used to both empirically verify the existence of the Gerschenkroneffect using 1990 OECD data and to address Dikhanov’s claim that the Iklé method isless sensitive than the Geary-Khamis method to the Gerschenkron effect. The answerto this question is found to depend critically on how one measures the Gerschenkroneffect. The paper concludes by exploring the implications of the Gerschenkron effectfor the measurement of inequality across countries and across time. It is argued that4See Diklianov(1994).Chapter 4. Additive PPP Methods and the Gerschenkron Effect 68international comparisons based on the Kravis. Heston and Summers ICP data will tendto underestimate the level of convergence in per capita income across countries over time.4.2 NotationConsider the problem of calculating multilateral PPPs (Pk) and quantity indices (Qk)over a set of K countries. This paper does not address aggregation problems below thebasic heading level. It is assumed that each country indexed by k = 1,.. . , K, suppliesprice and quantity data (pkj, qk), defined over the same set of goods and services, indexedbyi=1,...,N.It is useful to distinguish between two different notions of a base country. The “nurneraire base” may be defined as the country whose currency is used as the numerairefor the comparison and hence whose PPP and quantity index are set equal to unity. Incontrast a “weighting base” exists if one and only one country’s price and/or quantityvectors are used as weights in the comparison. Not all PPP methods have a weightingbase, and for those that do, it need not necessarily coincide with the numeraire base.Changing the numeraire base serves only to rescale multilateral PPPs and quantityindices. Hence it is also useful to define multilateral output shares (8k). Output sharesare a set of quantity indices that have been rescaled to sum to unity.Multilateral PPPs and quantity indices may be related using the Weak Factor Reversal Test stated below in (4.1):Weak Factor Reversal Test: = Pkqk V b, k = 1 . . . . K. (4.1)Pb Qb Z—_1pbqb ‘In this paper the Paasche and Laspeyres bilateral formulae are referred to frequently.It should be noted that these bilateral indices are intransitive. In other words, unliketheir multilateral counterparts, bilateral PPPs and quantity indices are dependent on theChapter 4. Additive PPP Methods and the Gerschenkron Effect 69choice of the numeraire base country b.NL pbqkiLaspeyres Quantity Index:= N (4.2)i=i pbiqbiLaspeyresPPP: = (43)ii pbiqbPaasche Quantity Index: = (44)j=j pkqbPaasche PPP: = _ipkiqki (4.5)i=1 pbiqkiIn addition, it is useful to define the Malmquist(1953) index. The Malmquist index isthe true underlying but unobservable quantity index.5Malmquist Quantity Index: = D(u, q)D(n, qb)where D(u,q) = maxh{h : F(q/h) > u,h> O}. (4.6)F(q) is a continuous, increasing and quasiconcave aggregator function. D(n, q) in (4.6) isthe deflation factor h, that will just reduce the vector q proportionately so that F(q/h) =U.4.3 Average Price - (Additive) MethodsAll additive methods calculate the PPP and quantity index between countries b and k,and the output share of country k as follows:p pP L L•=‘ T’ SkKQL (4.7)5The Malmquist index is defined here oniy for a representative agent. However, it can also be extendedto groups, see Diewert(1984).Chapter 4. Additive PPP Methods arid the Gerschenkron Effect 70where Q4k is the Laspeyres qilantity index defined in (4.2), while Pk is the PaaschePPP defined in (4.5). Alternatively, (4.7) may be written thus:6N Nk — 1pkqk lpXqb—— N Nb i PbbQk = ;jpxiqki1pxiqxj = ipxiqki (4.8)Z1Pxb Zj=j pxqbAdditive methods differ only in how they define the average price vector Px. Eightalternative methods for calculating average prices are considered below. The averageprices PXi, Paasche PPPs P, and Laspeyres quantity indices Q%k are obtained bysolving the system of N + 2K simultaneous equations given by (4.5), (4.2) and therespective formula for PXi. Rescaling the average price vector has no effect on the resultingoutput shares. Hence the presence of an arbitrary positive constant c in each averageprice formula.Arithmetic Mean (Van Yzeren) method7K7i PkiPXi=O--, Vi=1...,N (4.9)k=1 \‘XkGeometric Mean (Gerardi) method8K / \1/RlPkiPXi , Vz=1,...,N (4.10)krrl \ XkJIklé method9PXi=( IL Vi=1,...,N (4.11)k=1 j=i qji/Qx Xk6From (4.8) it can be seen that Average Price PPPs and quantity indices satisfy the Weak FactorReversal Test stated in (4.1). Note however that the Qk are homogenous of degree one in the PXi whilethe Fk are homogenous of degree minus one in the PXi7This method is often referred to as Van Yzeren’s hornogenous group method.8The Geometric Mean average price formula may alternatively be written as follows:Ki/KPXi Pkk1It was advocated by Walsh(1901) and Gerardi(1982).9Iklé’s method may be viewed as an equally weighted variant on the Geary-Khamis method.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 71Harmonic Mean methodK / —1PXi= , Vi=1,...,N (4.12)k=1 XkGeary-Khamis methodL) Vi=1,...,N (4.13)k1 D1 qji XkWeighted Arithmetic Mean (Van Yzeren) methodPXi = ( , V i = 1 N (4.14)k=1 XlvWeighted Geometric Mean (Gerardi) method’°LPXi= k=l(PXk)Vi = l,...,N (4.15)Fixed Base Price methodPXi0ki, Vi=1,...,N (4.16)The Arithmetic (Van Yzeren), Geometric (Gerardi), Harmonic Mean and Iklé methodsgive all countries equal weight in determining the average price vector. In contrast, theWeighted Arithmetic (Van Yzeren), Weighted Geometric Mean (Gerardi) and GearyKhamis methods give more weight to countries with larger baskets. Finally the FixedBase Price method gives all weight to country k. In other words, Fixed Base Pricemethods use country k as a weighting base.10The Weighted Geometric Mean average price formula may alternatively be written as follows:K -K LVT Ldl XiPXiQ1IPkjk1Chapter 4. Additive PPP Methods and the Gerschenkron Effect 724.4 Average Basket MethodsAverage Basket methods calculate the PPP and quantity index between countries b andk, and output share of country k as follows:D IJL fl rP rP1k‘xk “k ‘‘Xk ‘Xkp pL’ r — rIP’ Sk — K rb Xb ‘b “6X6 j1 Xjwhere Q is the Paasche quantity index defined in (4.4), while P is the LaspeyresPPP defined in (4.3). Alternatively, (4.17) may be written thus:1’—N N N1k— ,pxqx— 1pkqx—— N N— Nb pxqx ,piqxzQk = Z,pkqk fLlpbqx (4.18)Qb =,pkqXAverage Basket methods differ only in how they define tile average basket qx. Fivealternative methods for calculating the average basket are considered below. The averagebasket q and Paasche quantity indices Qk are obtained by solving the system of N + Ksimultaneous equations given by (4.4) and the respective formula for qx. Rescaling theaverage basket has no effect on the resulting output shares. Hence the presence of anarbitrary positive constant in each average basket formula.Arithmetic Mean (Van Yzeren) method’2KfIqxj=a--- , Vz=1,...,N (4.19)k=1 \‘Xk“From (4.18) it can be seen that Average Basket PPPs and quantity indices satisfy the Weak FactorReversal Test stated in (4.1). Note however that the Qk are hornogenous of degree minus one in the qxjwhile the F,, are homogenous of degree one in the qx.‘2This method is often referred to as Van Yzeren’s heterogenous group method.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 73Geometric Mean method’3K 1/Kqxi = a(ak)V i = 1,.. . , N (4.20)Weighted Arithmetic Mean method’4Kqx=cZqk, Vi=1,...,N (4.21)Weighted Geometric Mean method’5PXkqx = akl ($k)(4.22)Fixed Base Basket Method= aqkj, Vi = 1,... (4.23)The first two methods give all countries equal weight, while the next two methods givecountries with larger baskets larger weights. Finally, the Fixed Base Basket method givesall weight to country k. In other words, Fixed Base Basket methods use country k as aweighting base.13This method may also be written as follows:K1/Kqxi=a qk1This method was advocated by Walsh(1901), who called it Scrope’s method with geometric weights. SeeDiewert(1993:52-58) for references to the early history of multilateral methods.14This method was advocated by Walsh(1901), who called it Scrope’s method. It is also known as theECLAC method, since it has been used by the United Nations Economic Commission for Latin Americaand the Caribbean.‘5This method may alternatively be written as follows:XkK .ç-ic p1r x3qx2=cqqkjk= 1Chapter 4. Additive PPP Methods and the Gerschenkron Effect 744.5 The Intuition Behind the Gerschenkron EffectThe Gerschenkron effect derives from the fact that expenditure patterns change in response to changes in relative prices. Consumers tend to switch their consumption towardsrelatively cheaper goods and services. An implication of the substitution effect is thatas relative prices change, consumers can maintain the same level of utility with a lesscostly basket of goods and services given the new set of prices. Now suppose that thelevel of expenditure corresponding to the utility maximizing basket of goods and servicesis measured using the wrong price vector. The level of utility implied by this amountof expenditure overstates the actual level of utility. To see this, consider the followingexample. In Figure 4.1 countries A and B are on the same indifference curve. However,if per capita expenditures in the two countries are calculated using country A’s pricevector PA, then B has the higher measured expenditure (utility). Conversely if per capitaexpenditures are calculated using country B’s price vector PB, then A has the highermeasured expenditure (utility).Chapter 4. Additive PPP Methods and the Gerschenkron Effect 75YPBPBxPAFigure 4.1: The Consumer Substitution Effect for Average Price MethodsChapter 4. Additive PPP Methods and the Gerschenkron Effect 76The producer substitution effect acts in the opposite direction.16 In response to achange in relative prices, producers tend to switch their production towards relativelymore expensive goods and services. If the revenue corresponding to the actual productionof goods and services is measured using the wrong price vector, the production possibilities frontier implied by this level of revenue will underestimate the actual productionpossibilities frontier. For example, in Figure 4.2 countries A and B are on the sameproduction possibilities frontier. However, if revenue in the two countries is calculatedusing country A’s price vector PA, then A has the higher measured revenue (output).Conversely if revenue is calculated using country B’s price vector PB, then B has thehigher measured revenue (output).‘6llowever at least at the level of GDP, the consumer substitution effect dominates the producersubstitution effect. This is reflected in the fact that Laspeyres quantity indices (PPPs) invariably exceedtheir corresponding Paasche quantity indices (PPPs).Chapter 4. Additive PPP Methods and the Gerschenkron EffectYPAPAxFigure 4.2: The Producer Substitution Effect for Average Price Methods77PBChapter 4. Additive PPP Methods and the Gerschenkron Effect 78For Average Basket methods, the level of utility will he underestimated if the percapita expenditure corresponding to observed prices is estimated using the wrong basketof goods and services. For example, in Figure 4.3 countries A and B are again on the sameindifference curve. It should be noted that the indirect utility function is quasi-convexand decreasing in prices. If per capita expenditure in the two countries is calculated usingcountry A’s basket q, then A has the higher measured expenditure (utility). Converselyif per capita expenditure is calculated using country B’s price vector q, then B has thehigher measured expenditure (utility). This is analogous to the Gerschenkron effect inreverse.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 79pyqBqBPxqAFigure 4.3: The Consumer Substitution Effect for Average Basket MethodsChapter 4. Additive PPP Methods and the Gerschenkron Effect 80Again the producer substitution effect acts in the opposite direction to the consumersubstitution effect. A country’s production possibilities frontier will be overestimated ifit is measured using the wrong basket of goods and services. For example, in Figure 4.4countries A and B are on the same production possibilities frontier. However, if revenue iscalculated using country A’s basket q, then B has the higher measured revenue (output).Conversely if revenue is calculated using country B’s basket q, then A has the highermeasured revenue (output).Chapter 4. Additive PPP Methods and the Gerschenkron Effect 81pyqAqAPxFigure 4.4: The Producer Substitution Effect for Average Basket Methods4.6 Measuring the Gerschenkron Effect Directly from the DataIt is useful to view the Gerschenkron effect from two different perspectives. For AveragePrice and Average Basket methods the Gerschenkron effect is unavoidable because of thesubstitution effect. However, the magnitude of the Gerschenkron effect may neverthelessdiffer across methods. The problem of measuring the Gerschenkron effect across methodsfor a given data set is addressed later in this paper. This section focuses specifically onqBChapter 4. Additive PPP Methods and the Gerschenkron Effect 82the problem of measuring the magnitude of the Gerschenkron effect inherent in a dataset.Average Price and Average Basket methods avoid the Gerschenkron effect if and onlyif there is no substitution effect. Hence to avoid the Gerschenkron effect either preferencesmust be Leontief, or by chance the income and substitution effects across countries mustinteract in such a way that all countries end up with either the same set of relative pricesor multiples of the same basket of goods and services. The former scenario implies thatthe price data satisfy Hicks’s(l 946) aggregation theorem. The latter scenario implies thatthe quantity data satisfy Leontief’s(1936) aggregation theorem. If the price and quantitydata of a pair of countries satisfy either of these aggregation theorems, then the Paascheand Laspeyres quantity indices (PPPs) between these countries are equal. Hence the sizeof the Faasche-Laspeyres spreads across the countries in a comparison provide a measureof the magnitude of the substitution effect in the data. A metric for measuring the sizeof the Paasche-Laspeyres spread between countries j and k is defined below:’7(max(P, P) (max(Q,Q3k)PLSk(p,q,pk,qk) =logj. j =1og [rIP iLN (4.24)min jk’ jk) \ min, ‘jk) ,‘The PLS index equals zero if and only if the data satisfy either Hicks’s or Leontief’saggregation theorems. Otherwise the index is strictly positive. By taking the mean ofthe PLS indices across all pairs of countries, an overall measure of the substitution effectis obtained. The bigger the value of this mean, the more of a problem the Gerschenkroneffect becomes. In practice, the mean PLS index is likely to fall as the set of countriesbecomes more homogenous. For example the mean PLS index defined over the 24 OECDcountries at the level of GDP in 1990 was 0.152. By restricting the comparison to the18 European countries in the OECD, the mean PLS index fell to 0.123. By furtherrestricting the comparison to the five Scandinavian countries, the mean PLS index fell to‘7The PLS index is also used in Chapter 3. PLS indices between the 24 OECD countries based on1990 data are reported in Appendix C, Table C.3.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 830.049. In all likelihood if the OECD comparison was extended to include some developingcountries, the mean PLS index would exceed the overall OECD estimate of 0.152. Byimplication the price paid for obtaining additive quantity indices increases as the set ofcountries being compared becomes more heterogenous. Potentially the mean PLS indexcould be used to decide whether or not an additive PPP method should be used in acomparison. For a given data set, if the mean PLS index lies below a certain critical value,one would conclude that the gains of additivity outweigh the inaccuracies resulting fromthe Gerschenkron effect. If on the other hand the mean PLS index exceeds this criticalvalue, one concludes that the costs of additivity are too great, irrespective of the choiceof additive method. In such a situation a suitable alternative would be to use the EKSmethod.184.7 Gerschenkron Bias and Outlier CountriesThis section provides a more rigourous analysis of the Gerschenkron effect by explainingthe link between outlier countries and the inherent bias of Average Price and AverageBasket methods. To do this, it is necessary to give a precise interpretation to the word“oiitlier”. Also it is useful to refer to the true underlying but unobservable Malmquistquantity index as a pomt of reference against which to measure the bias of AverageBasket and Average Price methods.Consider first the class of Fixed Base methods defined in (4.16) and (4.23). Underhomothetic preferences, Laspeyres and Paasche quantity indices constitute upper andlower bounds on the true underlying Malrnquist quantity index.min(Qf, Qtk) Q max(Q, Qk (4.25)Without loss of generality, country 1 is assumed to be the weighting base. Q, Qf and18The EKS method is discussed in section 4.8.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 84Qfk denote respectively the Paasche, Malmquist and Laspeyres quantity indices definedin (4.4), (4.6) and (4.2). Malmquist(1953) showed that even when preferences are nothornothetic, there exists a utility level between the two levels being compared for whichthe Laspeyres and Paasche upper and lower bounds still apply.’9If differences in the price and quantity vectors across countries are negatively correlated, then Qfk > Qfk. Such correlation is consistent with consumer utility maximizingbehaviour and is observed empirically at the level of GDP. In this case, (4.25) reduces tothe following:P<rM<rL1k — ‘d1k—4.Assilming homothetic preferences and consumer utility maximizing behaviour, it followsfrom (4.7) and (4.26) that Fixed Base Price quantity indices Qfk overestimate their corresponding Malmqiiist quantity indices Q for all countries k = 2,. . . , K. If on the otherhand differences in the price and quantity vectors across countries are positively correlated, then Qfk < Qf. Such correlation is consistent with producer profit maximizingbehaviour. In this case, (4.25) reduces to the following:rL çM P— ‘1k — (Assuming homothetic preferences and producer profit maximizing behaviour, it followsfrom (4.7) and (4.27) that Fixed Base Price quantity indices Qf’k underestimate theircorresponding Ma.lmquist quantity indices Q for all countries k = 2,. . . , K.The output share Sk of country k may be obtained from the quantity indices as follows:= QikL (4.28)j=1 Q1Assuming consumer data, the fact that Qfk overestimates Qf for k 1, does not necessarily imply that Sk (k 1) will be overestimated. From (4.28) it can be seen that‘9This point is discussed in more detail in Chapter 3, section 4.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 85whether or not sk is overestimated depends on how much Qfk overestimates Q relative to the extent that the other Qf for j = 2,. . . , K overestimate their correspondingMalmquist illdices. The only certainty is that the output share of country 1 will beunderestimated.However, according to the Gerschenkron effect, the bias of the output shares dependssystematically on the similarity between each country and the weighting base country.A measure of the similarity between the price and quantity vectors of two countries isprovided by the Paasche-Laspeyres Spread (PLS) index defined in (4.24). The larger thevalue of PLS1k, the less similar the price and quantity vectors of countries 1 and k, oralternatively the more country k is an outlier from the perspective of country 1. ThePLS index equals zero if and only if the N goods and services over which the comparisonbetween the two countries is being made satisr either Hicks’s(1946) or Leontief’s(1936)aggregation theorems. These aggregation theorems provide the basis for measuring similarity between countries. The overall measure of similarity depends both on the similaritybetween the price vectors of countries 1 and k, and the similarity between the quantityvectors. Assuming consumer data, the Gerschenkron effect for Fixed Base Price methodsresults from the fact that as the PLS index of country k rises, from (4.26) 50 will theupward bias of Qfk relative to Q. Other things equal, this in turn causes the outputshare Sk of country k to rise. Hence as country k becomes more of an outlier, its outputshare tends to rise. However, assuming producer data, from (4.27) this result is reversed.As country k becomes more of an outlier, its output share tends to fall. In other wordsif producer data is used, Fixed Base Price methods tend to underestimate the outputshares of outlier countries. This is the Gerschenkron effect in reverse.For Fixed Base Basket methods, analogous arguments apply. Assuming homothetic preferences and consumer utility maximizing behaviour, it follows from (4.17) and(4.26) that Fixed Base Basket quantity indices Q underestimate their correspondingChapter 4. Additive PPP Methods and the Gerschenkron Effect 86Malmquist quantity indices Q, for all countries k = 2,. . . ,K. On the other hand,assuming homothetic preferences and producer profit maximizing behaviour, it followsfrom (4.17) and (4.27) that Fixed Base Basket quantity indices Q overestimate theircorresponding Malrnquist quantity indices Q, for all countries k = 2,.. . , K.Therefore assuming consumer data, it follows from (4.26) that as the PLS index ofcountry k rises, so will the downward bias of Qfk relative to Q. Other things equal,this in turn causes the output share 5k of country k to fall. In other words as countryk becomes more of an outlier, its output share tends to fall. Hence if consumer datais used, Fixed Base Basket methods tend to underestimate the output shares of outliercountries. This is analogous to the Gerschenkron effect in reverse. However, assumingproducer data, from (4.27) this result is in turn reversed. As country k becomes more ofan outlier, its output share tends to rise and therefore Fixed Base Basket methods tendto overestimate the output shares of outlier countries.The Gerschenkron effect generalizes to the whole class of additive methods. However,the analysis must be modified since in general no average basket vector is defined foran additive method. Hence a different measure of similarity must be used based onlyon Hicks’s aggregation theorem. Such a measure of similarity is provided by the AllenDiewert(1981) index.The Allen-Diewert distance index of prices between countries X and k is defined asthe Ordinary Least Squares sum of squared errors derived from the following regressionequation:log = + j, i = 1 . . . , N. (4.29)\PxiJIn (4.29), Pxi and Pki are the price vectors of countries X and k, is the parameter tobe estimated, and the j denote errors. The formula for the Alleii-Diewert index (ADXk)Chapter 4. Additive PPP Methods and the Gerschenkron Effect 87between countries X and k is given below:= = E [log () — log H]2 (4.30)The Allen-Diewert distance index provides a measure of similarity between each country’sprice vector Pk and the average price vector Px.20 The Allen-Diewert index equals zero ifand only if the prices of the N goods and services over which the two countries are beingcompared satisfy Hicks’s(1946) aggregation theorem. The index can be used to identifyoutlier countries in a comparison. These will be the countries with the largest AllenDiewert distance indices. Allen-Diewert indices between the 24 OECD countries basedon 1990 data for 31 different additive methods are reported in Appendix C, Table C.i.A second complication is that (4.25) is not applicable to the more general case wherethe artificial country X replaces country 1 as the weighting base. This is because wecannot assume cost minimizing behaviour on behalf of the artificial country X. Hence wecannot appeal to the true underlying Malmquist index. However, it is still possible toappeal to (4.25) in the limit as the Allen-Diewert indices of any of the countries in thecomparison tend to zero. This is beca.use in the limit as the Allen-Diewert index betweenPx and P1 tends to zero, all additive methods tend to the Fixed Base Price methodwith country 1 as the weighting base. Hence in the limit, the argument based on theMalmquist index may be invoked. Assuming consumer data, as the Allen-Diewert indexof country k rises, its output share tends to do likewise. This is the Gerschenkron effect.However, the only certainty is that in the limit assuming consumer data and homotheticpreferences, as a country’s Allen-Diewert index tends to zero, its output share will beunderestimated. For producer data, this result is reversed.For average basket methods, analogous arguments apply. The Allen-Diewert distanceindex of quantities between countries X and k is defined as the Ordinary Least Squares20A simple correlation coefficient between pk and px is an inappropriate measure of similarity, sinceit is not invariant to changes in the units of measurement.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 88sum of squared errors derived from the following regression equation:iog(L=+j, i=1,...,N. (4.31)‘qxjjIn (4.31), q, and qx are the quantity vectors of countries k and X, is the parameter to beestimated, and the j denote errors. The Allen-Diewert distance index provides a measureof similarity between each country’s basket of goods and services qj and the averagebasket qx. The Allen-Diewert index equals zero if and only if the prices of the N goodsand services over which the two countries are being compared satisfy Leontief’s(1936)aggregation theorem. The outlier countries will be those with the largest Allen-Diewertdistance indices.Again in the limit as the Allen-Diewert index between qx and q tends to zero, allAverage Basket methods tend to the Fixed Base Basket method with country 1 as theweighting base. Hence in the limit, the argument based on the Malmquist index maybe invoked. Assuming consumer data, as the Allen-Diewert index of country k rises,its output share tends to fall. This is analogous to the Gerschenkron effect in reverse.However, the only certainty is that assuming consumer data and homothetic preferences,as a country’s Allen-Diewert index tends to zero, its output share will be overestimated.For producer data this result is reversed.4.8 Empirical Verification of the Gerschenkron EffectThe framework developed in the last section can be used to empirically verify the existence of the Gerschenkron effect. Since Average Price (additive) methods have attractedrather more attention in the PPP literature than Average Basket methods due to thedesirability of additivity, the empirical comparisons in this section focus exclusively onadditive methods. However, the same framework could be used with slight modificationsto empirically verify the existence of the Gerschenkron effect for Average Basket methods.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 89The empirical comparisons all use 1990 OECD price and expenditure data defined over198 basic heading categories for the 24 member countries. Thirty-one different additivemethods are compared. These correspond to the first seven methods in section 3, and the24 Fixed Base Price methods obtained by using each of the countries in the comparisonas the weighting base.Figures 4.5 and 4.6 are examples of the Gerschenkron effect in action. Figure 4.5plots Turkey’s estimated output share against its corresponding Allen-Diewert index, foreach of the thirty-one additive methods. Similarly, Figure 4.6 plots the estimated outputshare of the USA against its corresponding Allen-Diewert index, for each of the thirty-one additive methods. The graphs reveal clear positive correlation between the size of acountry’s output share and its Allen-Diewert index. As the Gerschenkron effect predicts,the output share rises as the country becomes more of an outlier.EKS output shares are not subject to the Gerschenkron effect.2’ Hence they can beused to measure the bias of the output shares of additive methods. According to theGerschenkron effect, the difference between a country’s estimated output share and itscorresponding EKS output share (8k — 8EKS) should rise with its Allen-Diewert index(zk).22 Consider the following linear regression models:(sk—S )=1+1zk+E1k, k=1,...,K, (4.32)/ 5EKS\kEKS ) 2+/3Zk E2k, k=1,...,K. (4.33)\ 8k /Allen-Diewert indices for each of the 31 additive methods are reported in Appendix C,Table C.1. Output share estimates for each of the 31 additive methods and the EKS21The EKS method is named after Eltetö, Köves(1964) and Szulc(1964). However, it was first proposedby Gini(1931). The EKS method is discussed in detail in Chapter 2, section 9.1.22Actually this is not necessarily true for Fixed Base Price methods since the Allen-Diewert index ofprices ignores Leontief’s aggregation theorem, and hence similarity between the quantity vectors. Theappropriate measure of similarity for Fixed Base methods is the PLS index. In principle, the PLS andAllen-Diewert indices could move in opposite directions. However, the correlation between price andquantity changes across countries, makes such an occurrence unlikely.Chapter4.AdditivePPPMethodsandtheGerschenkronF1fect0.025—UUUUU •U.UIU0.02-.— (-)0,015--—) C C L.-0.01-Cl)-‘0.005-0—III020406080100120140Turkey’sAllen—DiewertIndexFigure4.5: TheGerschenkronEffect (Turkey)Chapter4.AdditivePPPMethodsandtheGerschcnkronFffect0.41—I0.405-0.4O.395-0.39--aaa I-0.385--a0.38--a•CI)y0.375-CI)I=0.37-0.365-0.360.355III020406080100120140TheUSA’s Allen—DiewertIndexFigure4.6: TheGerschenkronEffect(USA)Chapter 4. Additive PPP Methods and the Gerschenkron Effect 92method are reported in Table C.2. PLS indices for the 24 Fixed Base additive methodsare reported in Table C.3. The results obtained from running the regressions (4.32) and(4.33) on the 31 additive methods are reported in Table C.4.Assuming that the consumer substitution effect dominates the producer substitutioneffect, the Gerschenkron effect predicts that the parameter /3 in (4.32) should be positive,while should be negative. When this regression was run for the 31 additive methods, aspredicted the OLS estimate of was in all cases positive, while was always negative.However, the explanatory power of the Alleri-Diewert indices was rather low. On average,the R2 coefficient was 0.1907. The low R2 may be attributed partly to the fact that(4.32) fails to take account of differences in the size of the various countries in thesample, or alternatively to the fact that only three of the thirty-one additive methodsgive more weight to larger countries in the average price formula. In (4.33), the outputshare biases are adjusted for size by dividing by their corresponding EKS output shares(Sk — The Gerschenkron effect predicts that the parameter /32 in (4.33)should be positive, while 2 should be negative. Again for all 31 additive methods, theparameters have the expected sign. On average, the R2 coefficient was 0.4514.23 Theaverage can be increased for Fixed Base methods by regressing the Gerschenkron Biason the PLS indices rather than the Allen-Diewert indices. This is because Allen-Diewertindices ignore Leontief’s aggregation theorem and hence similarity in quantities. Whenthe Allen-Diewert indices were replaced in (4.33) by PLS indices for the 24 Fixed BasePrice methods, the average R2 rose to 0.5988. Finally, for both models the values ofthe parameters c and /3 were negatively correlated with the mean of the Allen-Diewertindices. Since the parameters are not stable across additive methods, this implies thatthe Allen-Diewert indices by themselves do not provide a good measure of Gerschenkron23The R2 coefficients cannot be compared across (4.32) and (4.33) because the regressions do not havethe same dependent variable.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 93Bias across additive methods.244.9 Discriminating Between Additive PPP MethodsAlthough these results provide empirical support for the Gerschenkron effect, they donot indicate which additive methods have the lowest bias. Two alternative measures ofGerschenkron Bias across methods are given below.GB(1) = (s - EKS)2 (4.34)1K / — EKS\2EKS ) (4.35)‘lk=1 5kThese metrics provide a framework for discriminating between additive methods. GB(l)measures errors in output shares, while GB(2) measures percentage errors in outputshares.25 The two measures GB(1) and GB(2) generate very different results. UsingGB(1), the Weighted Geometric Mean (Gerardi) method has the lowest GerschenkronBias followed closely by the Geary-Khamis method. In contrast, using GB(2), the FixedBase Price method with Italy as base has the lowest Gerschenkron Bias, followed by the24If the parameters were stable, then the variance of each method’s Allen-Diewert indices would beindicative of the Gerschenkron Bias.25llowever, these framework should be used with caution. The mathematical solution is to choose theaverage price vector that minimizes the relevant Gerschenkron Bias measure. This approach is similar tothe one advocated by Gerardi in his Ideal Price (IP) method. (The IP method is discussed in section 7.2of Chapter 2). The average price vector in the case at hand contains 198 elements and there are 24countries. This corresponds to 198 unknowns in 24 equations. Clearly as it stands, multiple solutionsexist for which the Gerschenkron Bias equals zero. However, to avoid negative prices, it is necessaryto also impose an additional 198 inequality constraints. This minimization problem is computationallyextremely laborious. More importantly, the whole methodology is flawed from an economic perspective,since in general a mathematically optimal average price vector is unlikely to be economically meaningful.If the average price vector bears no resemblance to the actual national price vectors, then the resultsobtained at lower levels of aggregation will be meaningless, even though they add up to equal the EKSresults at the highest level of aggregation. In short, the cure is worse than the illness. Hence if GB(1)or GB(2) are used to discriminate between additive methods, they must be used subject to the provisothat the average price vectors under consideration bear some resemblance to the national price vectorsactually observed in the sample.Chapter 4. Additive PPP Methods and the Gerschenkron Effect 94Fixed Base Price methods with Spain, Ireland and Greece as base respectively. These arethen followed in turn by the equally weighted Iklé, Harmonic Mean, Arithmetic Mea.n(Van Yzeren) and Geometric Mean (Gerardi) methods. The two complete sets of resultsare given in Appendix C, Table C.5.26The last column of Table C.5 gives Turkey’s output share as calculated by each ofthe 31 additive methods and the EK5 method. The Gerschenkron Bias of each methodcannot be inferred directly from these estimates, since the extent to which Turkey is anoutlier differs across methods. However, these output shares serve as a useful illustrationof the Gerschenkron effect, since for 27 of the 31 additive methods, Turkey is the biggestoutlier in the sample as measured by the Allen-Diewert indices. The four exceptionsare the Fixed Base Price methods with Greece, Spain, Portugal and Turkey respectivelyas the weighting base.27 As expected for the 27 additive methods for which Turkey isthe biggest outlier, Turkey’s estimated output share exceeds the EKS estimate. Of theremaining four methods, Turkey’s estimated output share is less than the EKS estimateonly for the Fixed Base Price methods with Portugal and Turkey respectively as base.These results suggest that for the Fixed Base methods with either Greece or Spain asweighting base, although Turkey is not the biggest outlier, it is nevertheless still anoutlier, since its estimated output share exceeds the EKS estimate.The choice between the two measures GB(1) and GB(2) is essentially a matter oftaste. If one places more weight on obtaining an accurate output share estimate forthe United States than for Luxembourg, then it is perfectly reasonable to use GB(1) todiscriminate between additive methods. Hence the choice between GB(1) and GB(2) isequivalent to the choice between weighted and unweighted PPP methods. If one prefersGB(1) then the Geary-Kharnis method has the second lowest Gerschenkron Bias, and one261n Table C.5, GK denotes Geary-Khamis, GM Geometric Mean, AM Arithmetic Mean, WGMWeighted Geometric Mean, WAM Weighted Arithmetic Mean, and HM Harmonic Mean.27For these four methods Japan is the biggest outlier.chapter 4. Additive PPPMethods and the Gerschenkron Effect95must conclude that there isno basis for the World Bank abandoning it in favourof Iklé.If on the other hand one prefers GB(2), then Dikhanov’s result is obtained. AccordingtoGB(2) Iklé is less sensitive thanGeary-Khamis to the Gerschenkron effect. However, itis perhaps surprising to find that according to GB(2) a simple Fixed Base Price methodwith any of Italy, Spain, Ireland or Greece as the weighting base for the givendata setin turn seems to have a smaller Gerschenkron Bias than any of the more complicatedGeary-Khamis, Iklé, Arithmetic (Van Yzeren), Geometric(Gerardi) and Harmonic Meanmethods. Hence if one prefers GB(2), the World Bank may do better by switching fromGeary-Khamis to a Fixed Base Price method rather than to Iklé.4.10 ConclusionThe Gerschenkron effect has recently prompted the World Bank to reconsider its use ofthe Geary-Khamis methodin PPP comparisons, since the Gerschenkron effectimpliesthat the Geary-Khamis method tends to systematically overestimate the output sharesof outlier (poor) countries in international comparisons.Instead the World Bank isconsidering using the Iklé method. The Iklé method isalso subject to the Gerschenkroneffect. However, it has beenclaimed that it is less sensitive to the Gerschenkroneffectthan the Geary—Khamis method.28This paper explains the theoretical basis behind the Gerschenkron effect. It is shownthat the Gerschenkron effect is more pervasive thanit appears, since it is applicableto all Average Basket methods as well as all Average Price (additive) methods. TheGerschenkron effect is alsodeceptively complex. For example, the direction of theGerschenkron Bias depends onwhether producer or consumer substitution effects aredominant in the data. The Gerschenkron effect derivesfrom the substitution effect and28 Dikhanov( 1994).Chapter 4. Additive PPPMethods and the Gerschenkron Effect96is inherent to the datairrespective of the choice of additive method.However it alsodepends on the particular choice of additive method. Frameworks aredeveloped formeasuring both the magnitude of the substitution effect across data sets, and the extentto which the Gerschenkron effect differs acrossadditive methods for agiven data set.The latter framework may be used to discriminate between additive methods. Applyingit to the Geary-Khamisversus Iklé debate generates mixed results. Which method hasthe lower GerschenkronBias seems to depend critically on how one defines and measuresGerschenkron Bias. Nevertheless this is a usefulresult, since it makes explicit the issuesat stake in the choice between Geary-Khamis and Iklé.The Gerschenkron effecthas important implications for the measurement of inequality across countries andacross time.29 It seemsquite likely that the increased level ofinternational trade overthe last two decades has caused some convergence in relativeprices across countries reducing the magnitude ofthe substitution effect. Ifthis hypothe—sis is correct, then the size of the Gerschenkron Bias inherent in the datawill have fallenover time. Therefore theupward bias on the output shares of outlier (poor) countriesin international comparisois will also have fallen over time. This in turn implies thatstudies of international convergence in living standards based on the Kravis, Heston andSummers ICP data which used the Geary-Khamis PPP method, will have tended tounderestimate the rate of convergence in per capita income across countries over time.29See Nuxoll(1994).Bibliography[1] Allen, R.C. and W.E. Diewert, (1981), “Direct Versus Implicit Superlative IndexNumber Formulae,” Review of Economics and Statistics 63, 430-435.[2] Balassa, B. (1964), “The Purchasing Power Parity Doctrine: A Reappraisal,” Journal of Political Economy 72, 584-596.[3] Balk, B.M. (1995), “Multilateral Methods for International Price and Volume Comparison,” Journal of Official Statistics, forthcoming.[4] Bhagwati, J.N. (1984), “Why are Services Cheaper in the Poor Countries,” EconomicJournal 94, 279-286.[5] Caves, D.W., L.R. Christensen and W.E. Diewert, (1982), “Multilateral Comparisons of Output, Input and Productivity using Superlative Index Numbers,” Economic Journal 92, March, 73- 87.[6] Cuthbert, J.E. (1993), “IP and EKS Aggregation Methods: Preservation of PriceStructure under the IP Method,” Eurostat Mimeo, Luxembourg: Eurostat.[7] Debreu, G. and I.N. Herstein, (1953), “Nonnegative Square Matrices,” Econometrica21, October, 597-607.[8] Diewert, W.E. (1976), “Exact and Superlative Index Numbers,” Journal of Econometrics 4, 115-146.[9] Diewert, W.E. (1978), “Superlative Index Numbers and Consistency in Aggregation,” Econometrica 46, 883-900.[10] Diewert, W.E. (1981), “The Consumer Theory of Index Numbers, A Survey,” 163-208 of Essays in the Theory and Measurement of Consumer Behaviour in Honourof Sir Richard Stone, A. Deaton (ed.) London: Cambridge University Press.[11] Diewert, W.E. (1983), “The Theory of the Cost of Living Index and the Measurement of Welfare Change,” 163-234 of Price Level Measurement: Proceedings from aConference Sponsored by Statistics Canada, Diewert, W.E. and C. Montmarquette,(ed.) Statistics Canada.[12] Diewert, W.E. (1984), “Group Cost of Living Indexes: Approximations and Axiomatics,” Methods of Operations Research 48. 23-45.97Bibliography 98[13] Diewert, W.E. (1986), “Microeconomic Approaches to the Theory of InternationalComparisons,” NBER Technical Working Paper No. 53.[14] Diewert, W.E. (1993a), “Symmetric Means and Choice Under Uncertainty,” in Essays in Index Number Theory, Vol. 1, ed. Diewert, W.E. and A. Nakamura: NorthHolland.[15] Diewert, W.E. (1993b), “The Early History of Price Index Research,” in Essays inIndex Number Theory, Vol. 1, ed. Diewert, W.E. and A. Nakamura: North Holland.[16] Dikhanov, Y. 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(1991), “Graph Theory Applications,” New York: Springer-Verlag.[23] Frisch, R. (1936), “Annual Survey of General Economic Theory: The Problem ofIndex Numbers,” Econometrica 4, 1-39.[24] Geary, R.G. (1958), “A Note on the Comparison of Exchange Rates and PPPsbetween Countries,” Journal of the Royal Statistical Society, Series A, 121, 97-99.[25] Gerardi, D. (1982), “Selected Problems of Inter-Country Comparisons on the Basisof the Experience of the EEC,” Review of Income and Wealth 28, 38 1-405.[26] Gerschenkron, A. (1951), “A Dollar Index of Soviet Machinery Output,” Santa Monica, CA: Rand Corporation.[27] Gini, C. (1931), “On the Circular Test of Index Numbers,” International Review ofStatistics, Vol.9, No.2, 3-25.Bibliography 99[28] Hicks, J.R. (1946), “Value and Capita’, Second Edition, Oxford: Clarendon Press.[29] Hill, T.P. (1986), “International Price Levels and Purchasing Power Parities,” OEJDEconomic Studies 6, 133-159.[301 Iklé, D.M. (1972), “A New Approach to the Index Number Problem,” QuarterlyJournal of Economics, Vol.86, 188-211.[31] International Monetary Fund (May 1993), “World Economic Outlook,” Washington,D.C.: IMF.[32] Khamis, S.H. (1972), “A New System of Index Numbers for National and International Purposes,” Journal of the Royal Statistical Society, Series A, 135, 96-121.[33] Khamis, S.H. (1993), “On Some Aspects of the Measurement of Purchasing PowerParities,” Eurostat Mimeo, Luxembourg: Euorstat.[341 Konüs, A.A. (1924), “The Problem of the True Index of the Cost of Living,” translated in Econometrica 7 (1939) 10-29.[35] Kravis, I.K., A. Heston aild R. Summers, (1982), “World Product and Income: International Comparisons of Real Gross Product,” Baltimore: Johns Hopkins UniversityPress.[36] Kurabayashi, Y. and I. Sakuma, (1990), “Studies in International Comparisons ofReal Product and Prices,” Tokyo: Kinokuniya Company Ltd.[37] Leontief, W. (1936), “Composite Commodities and the Problem of Index Numbers,”Econometrica 4, 39-59.[38] Lucas, R.E. (1988), “On the Mechanics of Economic Development,” Journal of Monetary Economics, 22, 3-42.[39] Malmquist, 5. (1953), “Index Numbers and Indifference Surfaces,” Trabajos de Estadistica 4, 209-242.[40] Marshall, A. (1887), “Remedies for Fluctuations of General Prices,” ContemporaryReview 51, 355-375.[41] Nuxoll, D.A. (1994), “Differences in Relative Prices and International Differences inGrowth Rates,” American Economic Review 84, 1423-1436.[42] Organization for Economic Co-operation and Development, Statistics Directorate,(1992), “Purchasing Power Parities and Real Expenditures - EKS Results (1990)”.Bibliography 100[43] Organization for Economic Co-operation and Development, Statistics Directorate,(1993), “Purchasing Power Parities and Real Expenditures- Geary-Khamis Results(1990,),” Paris: OECD.[44] Prasada Rao, D.S. and K.S. Banerjee, (1986), “A Multilateral Index Number SystemBased on the Factorial Approach,” Statistische Hefte/Statistical Papers 27, 297-313.[45] Prasada Rao, D.S. and J. Salazar-Carrillo, (1988), “A General Equilibrium Approach to the Construction of Multilateral Index Numbers,” in World Comparisonsof Incomes, Prices and Product, ed. Salazar-Carillo, J. and D.S. Prasada Rao, Amsterdam: North-Holland.[46] Ruggles, R. (1967), “Price Indexes and International Price Comparisons,” in TenEconomic Studies in the Tradition of Irving Fisher, ed. Feliner, W., New York: JohnWiley.[47] Russell, R.R. (1983), Comments (on Diewert(1983)), 234-239 of Price Level Measurement: Proceedings from a Conference Sponsored by Statistics Canada, Diewert,W.E. and C. Montmarquette, (ed.) Statistics Canada.[48] Samuelson, P.A. (1964), “Theoretical Notes on Trade Problems,” Review of Economics and Statistics 46, 145-154.[49] Samuelson, P.A. and S. Swarny, (1974), “Invariant Economic Index Numbers andCanonical Duality: Survey and Synthesis,” The American Economic Review 64(4),566-593.[50] Skiena, S. (1990), “Implementing Discrete Mathematics: Combinatorics and GraphTheory with Mathematica,” Redwood City, California: Addison Wesley.[51] Szulc, B. (1964), “Indices for Multiregional Comparisons,” Przeglad Statystyczny 3,Statistical Review 3, 239-254.[52] Szulc, B. (1983), “Linking Price Index Numbers,” 537-567 of Price Level Measurement: Proceeding from a Conference Sponsored by Statistics Canada, Diewert, W.E.and C. Montmarquette, (ed.) Statistics Canada.[53] Van Yzeren, J. (1956), “Three Methods of Comparing the Purchasing Power ofCurrencies,” Statistical Studies No. 7, The Hague: The Netherlands Central Bureauof Statistics.[54] Van Yzeren, J. (1983), “Index Numbers for Binary and Multilateral Comparison,”Statistical Studies No. 34, The Hague: The Netherlands Central Bureau of Statistics.Bi bliograpliy 101[55] Van Yzeren, J. (1987), “Bias in International Index Numbers: A Mathematical Elucidation,” Dissertation submitted to the Committee of Scientific Qualifications of theHungarian Academy of Sciences for award of the degree of Candidate of Sciences(Ph.D) in Economics.[56] Walsh, C.M. (1901), “The Measurement of General Exchange Value,” New York:Macmillan and Co.[57] Wilson, R.J. (1985), “Introduction to Graph Theory,” Third Edition, New York:Longman.[58] World Bank (1993), “Purchasing Power of Currencies - Comparing National Incomesusing ICP Data,” Washington, D.C.: World Bank.Appendix AProofs to Chapter 2102Appendix A. Proofs to Chapter 2 103Proof 1 From (2.53), Van Yzeren balanced quantity indices Qxi,. . . , Qxi- are obtained as the solutions to the following system of K equations:K ( L Qxj’ K / L QxkjUkQk =(wkQk) Vj=1,...,K. (A.1)k=1 QxkJ k1 QxjjIt follows from (A.1) that:K /I W/ ‘lPiiki) =Qik1kv1pjqj) Vi 1 ,K. (A.2)Now by reversing the order of the summation signs and using the Weak Factor ReversalTest, (A.2) can be rewritten as follows:2 [Pu ()] — [ (wk) qi]= 1 K. (A.3)1pjiqji— -Z1pxqxNow define Px and qx as follows:K’ K(\ / PkNqx = w— j, PXi = V i = 1,. . . , N. (A.4)k=1 QxkJ k=1 PxkBy substituting for Px and qx from (A.4) into (A.3), the following expression is obtained:(z pxiqji 1/2) (QP L \1/2Qx = xjQxj) Vj = 1,...,K. (A.5)Using an analogous method, it can be shown that Px=F, Vj = 1,.. . , K. Howeverthis is not necessary, since from the Weak Factor Reversal Test, Qx = implies thatPx = Pf. Hence (A.4) may be rewritten as follows:K’ K(Pki’(\qx=Z Wkj, PXi Wk), Vi=1,...,N. (A.6)k4 QxkJ k=1 Xk/Appendix BTables for Chapter 3104AppendixB.TablesforChapter3GERFRATANETeaLUXUKDIREDNKGRCSPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGER00.05.40.08700360.0430.0460.0870.0170.0860.2140.0970.3030.0230.0020.0670.0880.060.0630.550.1020.1140.1080.1010.124FRA0.05400.05500970.0570.080.1050.0660.1150.160.0950.2040.0660.080.0770.1120.0870.0910.4040.0940.0910.1420.1050.202ITA0.0870.05500.1320.1090.1050.0890.080.1390.150.0780.1680.0970.1070.1250.1980.1760.1150.3480.1390.1320.1930.1490.216NET0.0360.0970,132Q0.0680.1210.0790.0670.0760.1830.1270.2930.0860.0690.0950.1120.0920.0850.4680.1350.1040.130.1470.18BEL0,0430.0570.1091106800.070.0820.0730.1150.2210.0960.2350.0910.0640.0890.1260.1210.1150.5080.1010.1010.120.1130.179LUX0.0460.080.1050.1210.0700.1730.1020.1810.2440.1380.3680.0660.040.1330.0910.1810.1140.6260.1350.1250.1360,1170.115UKD0.0870.1050.0890.0790.0820.17300.0240.130.1990.160.1950.170.1640.1660.1980.2030.1320.3820.1330.1180.1950.1690.176IRE0.0170.0660.080.0670,0730.1020.02400.0680.1880.1170.2140.0920.0560.0750.1440.1330.0790.3750.0320.0690.1220.0660.105DNK0.0860.1150.1390.0760,1150.1810.130.06800.1830.1380.2820.1110.1340.0630.080.0750.080.3460.1240.1050.1390.1930.277GRC0.2140.160.150.1830.2210.2440.1990.1880.18300.0870.1890.2120.2490.2260.2420.3290.2630.1760.1310.1650.3690.2640.306SPA0.0970.0950.0780.1270.0960.1380.160.1170.1380.08700.0990.1560.1520.1470.240.2620.2170.280.1320.1140.3110.1680.216PRT0.3030.2040.1680.2930.2350.3680.1950.2140.2820.1890.09900.3160.3470.2810.350.4650.3590,2060.2620.2340.4790.3210.398AUT0.0230.0660.0970.0860.0910.0660.170.0920.1110.2120.1560.31600.0390.0840.1260.0980.0780.450.0860.1160.1260.0640.114CHE0.0020.080.1070.0690.0640.040.1640.0560.1340.2490.1520.3470.03900.06.40.0860.0320.10.490.0850.0810.1430.0530.11FIN0.0670.0770.1250.0950.0890.1330.1660.0750.0630.2260.1470.2810.0840.06400.0570.0130.0550.3580.0470.0530.10.0670.165ICE0.0880.1120.1980.1120.1260.0910.1980.1440.080.2420.240.350.1260.0860.05700.080.0740.3980.0790.0820.0950.0940.175NOR0.060.0870.1760.0920.1210.1810.2030.1330.0750.3290.2620.4650.0980.0320.0130.0800.0370.570.0710.0860.0590.0590.199SWE0.0630.0910.1150.0850.1150.1140.1320.0790.080.2630.2170.3590.0780.10.0550.0740.03700.4490.10.0830.0830.1170.214TUR0.550.4040.3480.4680.5080.6260.3820.3750.3460.1760.280.2060.450.490.3580.3980.570.44900.4050.3810.6010.5370.634AUS0.1020.0940.1390.1350.1010.1350.1330.0320.1240.1310.1320.2620.0860.0850.0470.0790.0710.10.40500.0690.1240.0520.126NZL0.1140.0910.1320.1040.1010.1250.1180.0690.1050.1650.1140.2340.1160.0810.0530.0820.0860.0830.3810.06900.1170.0970.152JAP0.1080.1420.1930.130.120.1360.1950.1220.1390.3690.3110.4790.1260.1430.10.0950.0590.0830.6010.1240.11700.0960.222CAN0.1010.1050.1490.1470.1130.1170.1690.0660.1930.2640.1680.3210.0640.0530.0670.0940.0590,1170.5370.0520.0970.09600.078USA0.1240.2020.2160.180.1790.1150.1760.1050.2770.3060.2160.3980.1140.110.1650.1750.1990.2140.6340.1260.1520.2220.0780TableB.1: Paasche—LaspeyresSpread(PLS)IndicesC UiAppendixB.TablesforChapter3GERFRAITANETBELLUXUKDIREDNKGRCSPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGER13.134678.21.03318.9419,390.2880.3334.35567.9152.6550.086.7591.0392.98139.244.5984.328728.10.6610.76291.710.6220.488FRA0.3191214.90.3295.9526.1010.0910.1041.4121.1316.4715.592.1260.3360.96612.621.4741.412222.70.2090.24329.320.1980.152ITA0.0010.00510.0020.0280.0284E-045E-040.0070.0980.0770.0740.010.0020.0040.0590.0070.0071.0831E-030.0010.1369E-047E-04NET0.9683.041653,7118.3418.680.2780.3194.26665.6550.2747.176.4381.022.91737.734.5144.303671.30.6370.74190.240.6020.478EL0.0530.168360.05511.0360.0150.0180.2353.5642.7882.6350.3590.0560.1612.1050.2440.23737.710.0350.044,8270.0330.026LUX0.0520.16.435.560.05.40.96510.0150.0180.2343.622.7722.6810.3510.0530.1582.0320.2430.2340.010.0350.044.7910.0320.025UKD3.47511.0323863.59366.4665.0911.14915.59229.9179.9167.522.983.60410.57138.516.2615.724612.3032.68318.82.1621.6.46IRE2.9999.64520783.13457.1255.920.87113.6201.7160.9155.220.393.1189.21123.914.113.5523052.0042.337282.61.8731.409DNK0.230,709150.90.2344.2474.2720.0640.074114.5811.3210.821.5050.2380.6788.751.0471.008157.10.1520.17721.290.1440.109GRC0.0150.04710.180.0150.2810.2760.0040.0050.06910.7740.7390.0990.0160.0460.610.0710.06710.150.010.0111.3750.0090.007SPA0.0190.06112.930.020.3590.3610.0060.0060.0881.29310.9330.1280.020.0590.7680.0890.08813.310.0120.0151.8170.0120.009PRT0.020.06413.530.0210.380.3730.0060.0060.0921.3521.07210.1320.0220.0630.820.0930.09214.110.0130.0161,9340.0130.01AUT0.1480.471020.1552.7832.8470.0440.0490.66510.097.837.60210.1550.455.9320.6880.663109.60.0980.11414.020.090.07CHE0.9622.98647.50,9817.8918.750.2770.3214.20963.4749.3746.226.44212.86537.274.4684.242676.50.6280.71991.430.5820.457FIN0.3351.035222.90.3436.1976.3330.0950.1091.47521.6116.8915.892.2240.349112.721.5151.48230.30.2170.25330.690.2050.154ICE0.0250.07917.060.0270.4750,4920.0070.0080.1141.6.41.3021.220.1690.0270.07910.1190.11418.520.0170.022.3990.0160.012NOR0.2170.6791470.2224.1044.1110.0610.0710.95514.1211.2410.771.4530.2240.668.4110.953159.50.1420.16319.70.1350.102SWE0.2310.708152.60.2324.2284.3510.0640.0740.99314.8711.3510.821.5070.2360.6768.7771.0491161.30.150.17520.630.1420.108TUP0.0010.0040.9230.0010.0270.0254E-044E-040.0060.0990.0750.0710.0090.0010.0040.0540.0060.00611E-030.0010.1369E-047E-04AUS1.5124.79210351.57128.528.920.4340.4996.569103.180.1574.3510.231.5924.61258.777.0216.655104311.154140.40.9360.724NZL1.3134.117872.61.3525.0524.970.3730.4285.66488.1468.4563.578.7621.3913.95951.156.125.718891.20.8671120.30.8170.63JAP0.0110.0347.3790.0110.2070.2090.0030.0040.0470.7270.550.5170.0710.0110.0330.4170.0510.0487.3540.0070.00810.0070.005CAN1.6075.0410921.66130.4631.080.4630.5346.968106.784.8679,1711.131.724.87862.97.437.03911301.0681.224149.310.785USA2.056.58514382.12638.9140.020.6070.719.208141112.210414.282.196.48982.589.7959.24614811.3811.587197.21.2741TableB.2: FisherPPPsCAppendixB.TablesforChapter3GERFRAITANETBELLUXUKDIREDNKGRCSPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGER13.134673.41.03318.9419.490.290.3334.5267.3452.148.626.7591.0393.06538.994.6434.425683.60.6680.77491.450.6250.491FRA0.3191214.90.336.0436.2180.0930.1061.44221.4916.6215.512.1570.3320.97812.441.4811.412218.10.2130.24729.180.20.157ITA0.0010.00510.0020.0280.0294E-045E-040.0070.10.0770.0720.010,0020.0050.0580.0070.0071.0151E-030.0010.1369E-047E-04NET0.9683.035652.1118.3418.870.2810.3234.37765.2150.4447.076.5441.0062.96837.764.4954.2846620.6470.7588.550.6050.475BEL0.0530.16535.560.05511.0290.0150.0180.2393.5562.7512,5670.3570.0550.1622.0590.2450.23436.10.0350.0414.8290.0330.026LUX0.0510.16134.560.0530.97210.0150.0170.2323.4562.6732.4950.3470.0530.1572.0010.2380.22735.080.0340.044.6930.0320.025UKD3.44510.823203.55865.2667.1411.14915.57232179.5167.523.293.5810.56134.31615.2423552.3022.668315.12.1541.691IRE2.9999.39920203.09756.858.440.87113.56202156.2145.820.273.1179.193116.913.9213.2720502.0042.322274.31.8751.472DNK0.2210.6931490.2284.194.3110.0640.074114.911.5310.761.4950.230.6788.6261.0270.979151.20.1480.17120.230.1380.109GRC0.0150.047100.0150.2810.2890.0040.0050.06710.7740.7220.10.0150.0460.5790.0690.06610.150.010.0111.3580.0090.007SPA0.0190.0612.930.020.3640,3740.0060.0060.0871.29310.9330.130.020.0590.7480.0890.08513.120.0130.0151.7550.0120.009PRT0.0210.06413.850.0210.390.4010.0060.0070.0931.3851.07210.1390.0210.0630.8020.0950.09114.060.0140.0161.8810.0130.01AUT0.1480.46499.640.1532.8022.8830.0430.0490.6699.9647.7087.19310.15.40.4545.7690.6870.655101.20.0990.11513.530.0930,073CHE0.9623.0166480.99418.2318.750.2790.3214.3564.850.1346.786.50412.9537.524.4684.258657.90.6430.745880.6020.472FIN0.3261.022219.70.3376.1796.3580.0950.1091.47521.971715.862.2050.339112.721.5151.4432230.2180.25329.840,2040.lóICE0.0260.0817.270.0260.4860.50.0070.0090.1161.7271.3361.2470.1730.0270.07910.1190.11317.530.0170.022.3450,0160.013NOR0.2150,675145.10.2224.084.1980.0630.0720.97414.5111.2210.471.4560.2240.668.39910.953147.30.1440.16719.70.1350.106SWE0.2260.708152.20.2334.2814.4040.0660.0751.02215.2211.7710.991.5280.2350.6938.81310491154.50.1510.17520.670.1410.111TUR0.0010.0050.9850.0020.0280.0294E-045E-040.0070.0990.0760.0710.010.0020.0040.0570.0070.00611E-030.0010.1349E-047E-04AUS1.4974.69110081.54628.3529.170.4340.4996.766100.877.9872.7710.121.5554.58858,376.9496.623102311.159136.90.9360.735NZL1.2924.048869.71.33424.4o25.170.3750.4315.83886.9767.2862.798.7291.3423.95950.365.9965.714882.90.8631118.10.8080.o34JAP0.0110.0347.3640.0110.2070.2130.0030.0040.0490.7360.570.5320.0740.0110.0340.4260.0510.0487.4750.0070.00810.0070.005CAN1.5995.01210771.65230.2931.170.4640.5337.229107,783.3277.7510.811.6624.90262.367.4257.07610931.0681.238146.310.785USA2.0376.38.413722.10438.5839.70.5910.6799.208137.2106.199.0313.772.1176.24479.439.4579.01313931.3611.577186.31.2741TableB.3: MinimumSpanningTreePPPsAppendixB.Tablesfor Chapter3FRATANETBELLUXUKDIREDNKGRCSPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAFRA1214.90.3335.9526.2410,0920.1061A4821.4916.6315.512.1440.3330.98212.721.4871.417218.20.2130.24829.290.1990.156TA0.00510.0020.0280.0294E-045E-040.0070.10.0770.0720.010.0020.0050.0590.0070.0071.0151E-030.0010.1369E-047E-04NET3.003645.3117.8818.740.2780.3194.34864.5449.9346.596.4380.9992.94838.24.4654.256655.20.6390.74587.960.5980.47BEL0.16836.10.05611.0490.0160.0180.2433.612.7932.6060.360.0560.1652.1370.250.23836.650.0360.0424.9210.0330.026LUX0.1634.430.0530.95410.0150,0170.2323.4432.6642.4860.3440.0530.1572.0380.2380.22734.960.0340.044.6930.0320.025UKD10.8123243.60164.3767.4911.14915.65232.4179.8167.823.183.59910.62137.616.0815.3223592.3022.681316.72.1541.691IRE9.41220233.13456.0358.740.87113.63202.3156.514620.183.1339.24119.713.9913.3420532.0042.334275.71.8751.472DNK0.691148.40.234.1124.3110.0640.073114.8411.4810.721.4810.230,6788.7871.0270.979150.70.1470.17120.230.1380.108GRC0.0479.9990.0150.2770.290.0040.0050.06710.7740.7220.10.0150.0460.5920.0690.06610.150.010,0121.3630.0090.007SPA0.0612.930.020.3580.3750.0060.0060.0871.29310.9330.1290.020.0590.7650.0890.08513.120.0130.0151.7620.0120.009PRT0.06413.850.0210.3840.4020.0060.0070.0931.3851.07210.1380.0210.0630.820.0960.09114.060.0140.0161.8880.0130.01AUT0.466100.20.1552.7762.9110.0430.050.67510.027.7557.23610.1550,4585.9340,6940.661101.80.0990.11613.660.0930.073CHE3.005645.71.00117,8918.750.2780.3194.3564.5749.9546.626.44212.9538.224.4684.258655.50.640.745880.5990.47FIN1.019218.90.3396.0646.3580.0940.1081.47521.8916.9415.82.1840.339112.961.5151.443222.20.2170.25329.840.2030.159ICE0.07916.890.0260.4680.4910.0070.0080.1141.6891.3071.220.1690.0260.07710.1170.11117.150.0170.0192.3020.0160.012NOR0.673144.50.2244.0034.1980.0620,0710.97414.4511,1810.431.4420.2240.668.55610.953146.70.1430.16719.70.1340.105SWE0.706151.60.2354.2014.40.40.0650.0751.02215.1711.7310.951.5130.2350.6938.9771.04911540.150.17520.670.1410.11TUR0.0050.9850.0020.0270.0294E-045E-040.0070.0990.0760.0710.010.0020.0040.0580.0070.00611E-030.0010.1349E-047E-04AUS4.69810091.56427.9629.320.4340.4996.80110178.172.8810.071.5634.61259.766.9856.657102511.165137.60.9360.735NZL4.033866.51.3432425.170.3730.4285.83886.6667.0462.568.6461.3423.95951.35.9965.714879.80.8581118.10.8030.631JAP0.0347.3370.0110.2030.2130.0030.0040.0490.7340.5680.530.0730.0110.0340.4340,0510.0487.4490.0070.00810.0070.005CAN5.01910791.67129.8831.330.4640.5337.266107.983.4577.8710.761.674.92763.857.4637.11310951.0681.24514710.785USA6.39313742.12938.0539.90.5910.6799.255137,4106.399.1813.712.1286.27681.339.5059.05913951.3611.585187.21.2741TableB.4: MinimumSpanningTreePPPswithGermanyDeletedC ODAppendixB.TablesforChapter3GERFRATANETBELLUXUKDIREDNKGRCSPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGER13.144674,61.01618.719.120.2860.3354.2463.1451.5244.716.7741.0533.03138.614.4934.376573.30.6590.75990.290.6210.487FRA0.3181214.60.3235.9476.0840.0910.1071.34920.0816.3914.222.1550.3350.96412.281.4291.392182.40.210.24128.720.1980.155ITA0.0010.00510.0020.0280.0284E-045E-040.0060.0940.0760.0660.010.0020,0040.0570.0070.0060.851E-030.0010.1349E047E-04NET0.9853.095664.2118.4118.830.2820.334.17562.1650.7344.026.6691.0362.98438.014.4234.308564.40.6490.74788.90.6110.48BEL0.0530.16836.080.05.411.0230.0150.0180.2273.3772.7562.3910.3620.0560.1622.0650.240.23430.660.0350.0414.830.0330.026LUX0.0520.16435.270.0530.97810.0150.0180.2223.3012.6942.3380.35.40.0550.1582.0190.2350.22929.980.0340.044.7210.0320.025UKD3.49610.9923583.55165.3666.8611.17114.82220,7180.1156.323.683.6810.613515.715.320042.3032.653315.62.1711.704IRE2.9849.38220133.03155.857,080.854112.65188.4153.8133.420.223.1419.046115.213.4113.0617111.9662.265269.51.8541.455DNK0.2360.741159.10.244.4094.510,0670.079114.8912.1510.541.5980.2480.7159.1061.061.032135.20.1550.17921.290.1460.115GRC0.0160.0510.680.0160.2960.3030.0050.0050.06710.8160.7080.1070.0170.0480.6120.0710.0699.080.010.0121.430.010.008SPA0.0190.06113.090.020.3630.3710.0060.0070.0821.22510.8680.1310.020.0590.7490.0870.08511.130.0130.0151.7530.0120.009PRT0.0220.0715.090.0230.4180.4280.0060.0070.0951.4121.15210.1520.0240.0680.8640.10.09812.820.0150.0172.020.0140.011AUT0.1480.46499.580.152.762.8230,0420.0490.6269.3217.6066.610.1550.4475.70.6630.64684.630.0970.11213.330.0920.072CHE0.952.987640.90.96517.7618.170.2720.3184.02859.9848.9542.486.43612.8836.684.2684.157544.70.6260.72185.790.590.463FIN0.331.037222.60.3356.1686.310.0940.1111.39920.831714.752.2350.347112.741.4821.444189.10.2170.2529,790.2050.161ICE0.0260.08117.470.0260.4840.4950.0070.0090.111.6351.3341.1580.1750.0270.07910.1160.11314.850.0170.022.3390.0160.013NOR0.2230.7150.20.2264.1624.2570.0640.0750.94414.0511.479.9521.5080.2340.6758.59410.974127.60.1470.16920.10.1380.108SWE0.2290.718154.20.2324.2734,370.0650.0770.96914.4311.7710.221.5480.2410.6938.8231.02711310.1510.17320.630.1420.111TUR0.0020.0051.1770.0020.0330.0335E-046E-040.0070.110.090.0780.0120.0020.0050.0670.0080.00810.0010.0010.1580.0019E-04AUS1.5184.77110241,54228.3829.030.4340.5096.43695.8378.267.8610.281.5984.658.66.8196.642870.111.1521370.9430.74NZL1.3174.142888.71.33824.6325.20.3770.4415.58683.1867.8858.98,9241.3873.99350.875.9195.765755.30.86811190.8180.642JAP0.0110.0357.4710.0110.2070.2120.0030.0040.0470.6990.5710.4950.0750.0120.0340.4280.050.0486.3490.0070.00810.0070.005CAN1.615.06210861.63530.130.790.4610.546.827101.782.9671.9910.911.6954.8862.177.2347.046923,11.0611.222145.410.785USA2.0526.4513842.08438.3639.240.5870.6888.7129.5105.791.7413.92.166.21979.229.2188.97911761.3521.557185.31,2741TableB.5:Geary—KhamisPPPsCAppendixB.TablesforChapter3GERFRAITANETBELLUXUKDIREDNKGRCSPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGER13.1556781.03518.8519.050.2880.3294.45166.9752.1949.36.6981.0493.04639.424.6434.454711.50.6620.76893.190.6210.477FRA0.3171214.90.3285.9756.0380.0910.1041.41121.2316.5415.632.1230.3320.96512.491.4721.412225.50.210.24329.540.1970.151ITA0.0010.00510.0020.0280.0284E-045E-040.0070.0990.0770.0730.010.0020.0040.0580.0070.0071.0491E-030.0010.1379E-047E-04NET0.9663.049655.1118.2218.410.2780.3184.364.7150.4347.646.4721.0132.94338.094,4864.304687.50.6.40.74290,050.6010.461BEL0,0530.16735.960.05511.0110.0150.0170.2363.5522.7692.6150.3550.0560.1622.0910.2460.23637.740.0350.0414.9430.0330.025LUX0,0520.16635.590.05.40.9910.0150.0170.2343.5162.742.5880.3520.0550.162.0690.24.40.23437.350.0350.044.8920.0330.025UKD3.47110.9523533.59265.4466.1211.14115.45232.5181.2171.123.253.64110.57136.816.1215.4624702.2982.665323.52.1571.656IRE3.0439,60120633.14957.3757.970.877113.54203.8158.815020.383.1929.26912014.1313.5621652.0152.337283.61.8911.452DNK0.2250.709152.30.2334.2364.280.0650.074115.0511.7311.081.5050.2360.6848.8581.0431.001159.90.1490.17320.940.140.107GRC0.0150.04710.120.0150.2810.2840.0040.0050.06610.7790.7360.10.0160.0450.5890.0690.06710.620.010.0111.3910.0090.007SPA0.0190.0612.990.020.3610.3650.0060.0060.0851.28310.9450.1280.020.0580.7550.0890.08513.630.0130.0151.7860.0120.009PRT0.020.06413.750.0210.3820.3860.0060.0070.091.3581.05910.1360.0210.0620.80.0940.0914.430.0130.0161.890.0130.01AUT0.1490.471101.20.1552.8152,8440.0430.0490.6649.9997.7937,36110.1570.4555.8860.6930.665106.20.0990.11513.910.0930.071CHE0.9533.008646.40.98717.9718.160.2750.3134.24363.8549.76476.38612.90437.594,4274.247678.40.6310.73288.850.5930.455FIN0.3281,036222.60.346.1896.2550.0950.1081,46121.9917.1416.192.1990.344112.941.5241.462233.60.2170.25230.60.2040.157ICE0.0250.0817.20.0260.4780.4830.0070.0080.1131.6991.3241.2510.170.0270.07710.1180.11318.050.0170.0192.3640.0160.012NOR0.2150.681460.2234.064.1030.0620.0710.95914.4211.2410.621.4430.2260.6568.49110.959153.201430.16520.070.1340.103SWE0.2250.708152.20.2324.2324.2770.0650.0740.99915.0411.7211.071.5040.2350.6848.851.0421159.70.1490.17220.920.140.107TUR0.0010.0040.9530.0010.0260.0274E-045E-040.0060.0940.0730.0690.0090.0010.0040.0550.0070.00619E-040.0010.1319E-047E-04AUS1,5114.76610241.56328.4828.780.4350.4966.723101.278.8474.4710.121.5844.60159.557.0146.729107511.16140.80.9390.721NZL1.3024.1098831.34824.5524.810.3750.4285.79687.2267.9764.218.7231.3663.96751.346.0475.801926.60.8621121.40.8090.621JAP0.0110.0347.2750.0110.2020.2040.0030.0040.0480.7190.560.5290.0720.0110.0330.4230.050.0487.6350.0070.00810.0070.005CAN1.6095.07710911.66530.3430.650.4640.5297.162107.883.9879.3310.781.6884.90163.447.4717.16811451.0651.23615010.768USA2.0966.61314212.16939.5139.930.6040.6899.328140.4109.4103.314.042.1986.38482.639,7319.33614911.3871.609195.31.3031TableB.6:EKSPPPsCAppendixB.TablesforChapter3GERFRATANETBELLUXUKDIREDNKGRCSPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGER13.2266781.05418.8719.610.2960.3354.6371.4355.5655.566.7571.0543.11539.244.8084.587728.30.670.7811000.6220.488FRA0.3112000.3265.9526.0240.0920.1041.43522,7316.9517.242.1050.3260.96512.51.4881.4182500.2080.24230.30.1920.15ITA0.0010.00510.0010.0270.0284E-045E-040,0070.1040.0780.0790.010.0010.0040.0580.0070.0061.2441E-030.0010.1389E-O47E-04NET0.9493.065669.2118.1818.520.2810.3184.40571.4352.6352.636.45212.95938.464.5664.34810000.6360.74290.910.590.461BEL0.0530.16836,690.05511.0130.0150.0170.2413.8022.8652.8990.3540.0550.1622.110.250.23845.450.0350.0415.0510.0320.025LUX0.0510.16636.220.0540.98710.0150.0170.2383.7592.8332.8570.3490.0540.162.0830.2470.23545.450.0340.044.9750.0320.025UKD3.37710.923803.55764.8965.7211.13115.6325020020022.733.55910.53142.916.1315.3833332.2622.639333.32.0961.639IRE2.9869.63821043.14557.3658.10.884113.89200166.7166.720.413.1459.34612514.2913.7250022.331333.31.8551.449DNK0.2160.697152.20.2274.1494.2020.0640.072115.8711.912.051.4680.2270.6738.7721.0370.9882000.1450.16920.830.1340.105GRC0.0140.0449.6430.0140.2630.2660.0040.0050.06310.7540.7610.0930.0140.0430.5550.0660.06312.050.0090.0111.3260.0090.007SPA0.0180.05912.790.0190.3490.3530.0050.0060.0841.32611.0090.1230.0190.0570.7360.0870.08315.870.0120.0141.7610.0110.009PRT0.0180.05812.670.0190.3450.350.0050.0060.0831.3140.99110.1220.0190.0560.7290.0860.08215.870.0120.0141.7420.0110.009AUT0.1480.475103.60.1552.8252.8620.0440.0490.o8110.758.1048.18110.1550.4585.9520.7070.6731250.0990,11514.290.0910.071CHE0.9493.065669.2118.2418.470.2810.3184.39669.452.3252.826.45612.95938.464.5664.34810000.6360.74290.910.590,461FIN0.3211.036226.20.3386.1656.2450.0950.1071.48623.4617.6917.852.1820.338112.991.5431.4682500.2150.25131.250.1990.156ICE0.0250.0817.380.0260.4740.480.0070.0080.1141.8021.3591.3720.1680.0260.07710.1180.11321.740.0170.0192.3920.0150.012NOR0.2080.672146.70.2193.9984.0490.0620.070.96415.2111.4711.581.4150.2190.6488.43910.9522000.1390.163200.1290.101SWE0.2180.7051540.234.1974.2510.0650.0731.01215.9712.0412.151.4860.230.6818.861.0512000.1460.17121.280.1360.106TUR0.0010.0040.8040.0010.0220.0223E-044E-040.0050.0830.0630.0630.0080.0010.0040.0460.0050.00518E-049E-040.1117E-046E-04AUS1.4934.81910521.57228.6829.050.4420.56.913109.182.2883.0510.151.5724.65260.547.1746.833130911.167142.90,9280.725NZL1.284.13901.91.34824.5824.90.3790.4295.92593.5370.5271,198.7021.3483.98851.896.1495.85711220.85711250.7950.621JAP0.010.0337.2670.0110.1980.2010.0030.0030.0480.7540.5680.5740.070.0110.0320.4180.050.0479.0420.0070.00810.0060.005CAN1.o095.19511341.69530.9231.320.4770.5397.453117.688.789.5410.951.6955.01665.277.7347,36714111.0781.258156.110.781USA2.056.6514522.1739.5840.090.610.699.54150.6113.5114.614.012.176.4283.559.99.4318071.381.61199.81.281TableB.?:ECLACPPPsAppendix CTables for Chapter 4112AppendixC.‘rabiesforChapter4GERFRAITANLDBELLUXUKIREDNKGRESPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGK11.6813.9819.5313.9715.3614.3319.2619.9218.7532.0825.5543.8215.516.422.4328.7624.5720.2596.0216.422.3333.5513.7713.79IkIe10.349.6215.689.7469,949.57515.1613.2811.9422.3917.6633.3610.913.6315.1419.8917.516.1680,8614.4117.8935.3613.9925.58GM9.6649.15815.39.28.410.129.8314.3811.2610.1921.5215.9330.59.712.3512.1317.2615.1413.7777.5713.7817.2532.6313.8925.45AM10.449.81416.2410.0610.5410.7315.2811.5910.4721.3516.5230.4510.0612.7611.8316.9714.7413.9975.8313.9517.5632.9214.1925.52WGM10.2212.9619.7112.7915.3713.7518.0716.715.8530.9423.7641.5513.5613.6418.9224.0321.2817.3691.9114.1821.1529.8511.5911.57WAM10.6512.9920.2713.0915.214.2718.8316.8315.7630.6124.241.614.1813.618.1923.9820.8717.4390.4614.4421.1829.4512.0412.57HM9.9589.51215.059.31210.659.90614.1111.6610.5221.715.5430.3710.0813.1813.1918.216.114.4376.4614.4317.2932.6514.1125.81GER01Z3823.9212.1313.410.7819.8622.6419.6339.6832.0844.5312.617.0328.1736.4329.7524.48107.222.8131.9844.9523.9429.65FRA12.38017.9715.2212.3513.2121.323.3422.1431.9124.737.2615.6720.3726.4233.3929.326.9698.4623.7730.9844.527.8938.41ITA23.921797028.5227.0822.2533.2730.5429.8528.8516.3130.5820.7228.7133.8443.3740.0235.1891.2633.5242.2453.3937.9346.13NLD12.1315.2228.52012.8714.4717.1320.0314.2237.5231.345.2517.9322.3425.3730.9329.1724.25101.921.1126.6343.6124.2134.74BEL13.412.3527.0812.87011.1920.5923.121.5835.1827.3141.219.3322.0426.0935.4428.1326.8298.1524.5431.0547.226.3739.98LUX10.7813.2122.2514.4711.19023.7525.9621.8931.6.427.444.416.1615.8827.0534.131.9724.54100.724.4431.3948.8826.6635.28UKD19.8621.333.2717.1320.5923.75019.0624.544.4233.644.5522.93033.5141.132.5129.71100.727.2932.4147.429.6443.94IRE22.6423.3430.5420.0323.125.9619.06019.3632.7329.2939.524.0827.623.7224.6725.8928.8197.1923.0227.9745.5125.1939.13DNK19.6322.1429.8514.2221.5821.8924.519.36040.4729.4650.3119.7122.0818.6224.4920.2416.9397.9623.6827.344.325.7438.09GRC39.6831.9128.8537.5235.1831.6444.4232.7340.47023.3231.536.7145.739.1336.6346.6149.1464.6638.4143.5164.844.8157.26SPA32.0824.716.3131.327.3127.433.629.2929.4623.32025.4827.2336.3233.636.5938.6138.9274.6136.2540.6455.4139.5952.18PRT44.5337.2630.5845.2541.244.444.5539.550.3131.525.48045.8459.1752.3255.0956.7766.1762.7159.2857.4178.0259.5373.25AUT12.615.6720.7217.9319.3316.1622.924.0819.7136.7127.2345.84015.9724.6130.2228.422.598.623.7330.1546.6725.5335.61CHE17.0320.3728.7122.3422.0415.883027.622.0845.736.3259.1715.97019.2426.0624.3315.61119.426.7734.3338.7824.6432.18FIN28.1726.4233.8425.3726.0927.0533.5123.7218.6239.1333.652.3224.6119.24016.7612.0811.6499.8629.2227.3540.6828.3541.57ICE36.4333.3943.3730.9335.4434.141.124.6724.4936.6336.5955.0930.2226.0616.76023.2826.498.9833.4233.0148.8829.3743.74NOR29.7529.340.0229.1728.1331.9732.5125.8920.2446.6138.6156.7728.424.3312.0823.28016.83104.431.2829.7544.0824.7143.37SWE24.4826.9635.1824.2526.8224.5429.7128.8116.9349.1438.9266.1722.515.6111.6426.416.830113.127.7529.3838.8926.4137.99TUR107.298.4691.26101.998.15100.7100.797.1997.9664.6674.6162.7198.6119.499.8698.98104.4113.10108.3102.2118.5104.1127.4AUS22.8123.7733.5221.1124.5424.4427.2923.0223.6838.4136.2559.2823.7326.7729.2233.4231.2827.75108.3020.3954.7618.4926.47NZL31.9830.9842.2426.6331.0531.3932.4127.9727.343.5140.6457.4130.1534.3327.3533.0129.7529.38102.220.39053.0620.8738.15JAP44.9544.5533943.6147.248.8847.445.5144.364.855.4178.0246.6738.7840.6848.8844.0838.89118.554.7653.06045.8662.79CAN23.9427.8937.9324.2126.3726.6629.6425.1925.7444.8139.5959.5325.5324.6428.3529.3724.7126.41104.118.4920.8745.86021.59USA29.6538.4146.1334.7439.9835.2843.9439.1338.0957.2652.1873.2535.6132.1841.5743.7443.3737.99127.426.4738.1562.7921.590‘i’ableC.1:Alien—DiewertIndicesAppendixC.TablesforChapter4GERFRAITANLDBELLUXUKIREDNKGRESPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGK0.0740.0690.0640.0160.0115E-040.0650.0020.0060.0060.0340.0070.0090.010.0063E-040.0040.010.0230.0190.0030.1530.0360.373IkIe0.0730.0670.0630.0150,0115E-040.0650.0020.0060.0060.0330.0060.0090.0090.0063E-040.0040.010.0210.0190.0030.1520.0350.384GM0.0720.0670.0630.0150.0115E-040.0640.0020.0060.0060.0330.0060.0090.0090.0063E-040.0040.010.0210.0190.0030.150.0350.387AM0.0720.0670.0630.0150.0115E-040.0640.0020.0060.0060.0330.0060.0090.0090.0063E-040.0040.010,0210.0190.0030.150.0350.388WGM0.0730.0680.0640.0150.0115E-040.0650.0020.0060.0060.0340.0070.0090.010.0063E-040.0040.010.0220.0190.0030.1510.0350.377WAM0.0740.0680.0640.0150.0115E-040.0650.0020.0060.0060.0340.0070.0090.010.0063E-040.0040.010.0220.0190.0030.1510.0350.379HM0.0720.0670.0630.0150.0115E-040.0650.0020.0060.0060.0330.0060.0090.0090.0063E-040.0040.010.0210.0190.0030.150.0350.386GER0.0720.0680.0640.0150.0115E-040.0650.0020.0060.0060.0330.0070.0090.0090.0063E-040.0040.010.0220.0190.0030.1530.0360.379FRA0.0710.0640.0620.0150.0115E-040.0640.0020.0060.0060.0330.0060.0090.0090.0063E-040.0040.010.0210.0190.0030.1510.0350.393TA0.0720.0650.0590.0150.0115E-040.0630,0020.0060.0060.0320.0060.0090.0090.0063E-040.0040.010.0190.0190.0030.1530.0350.395NLD0.0710.0680.0650.0140.0115E-040.0630.0020.0060.0060.0340.0070.0090.0090.0063E-040.0040.010.0220.0190.0030.150.0360.385BEL0.0720.0660.0640.0150.015E-040.0640.0020.0060.0060.0330.0060.0090.0090.0063E-040.0040.010.0220.0190.0030.1520.0360.385LUX0.0730.0670.0630.0160.0115E-040.0650.0020.0060.0060.0330.0070.0090.010.0063E-040.0040.010.0220.0190.0030.1530.0360.379UK0.0720.0660.0610.0150.015E-040.060.0020.0060.0060.0340.0060.0090.010.0063E-040.0040.010.0210.0190.0030.1550.0360.388IRE0.0720.0670.0630.0150.0115E-040.0630.0020.0060.0060.0330.0060.0090.010.0063E-040.0040.010.020.0190.0030.150.0350.392DNK0.0710.0670.0640.0150.0115E-040.0640.0020.0050.0060.0340.0070.0090.0090.0053E-040.0040.010.020.0180.0030.1450.0350.396GRE0.0730.0650.060.0150.0115E-040.0620.0020.0050.0050.0310.0060.0090.010.0063E-040.0040.010.0180.0180.0030.1590.0350.392SPA0.0710.0650.0610.0150.015E-040.0630.0020.0050.0050.030.0060.0090.0090.0063E-040.0040.010.020.0190.0030.1550.0350.395PRT0.0730.0640.060.0150.015E-040.060.0020.0050.0060.030.0050.0090.010.0063E-040.0040.010.0180.0190.0030.1570.0350.399AUT0.0720.0670.0630.0150.0115E-040.0660.0020.0060.0060.0340.0060.0080.0090.0063E-040.0040.010.0210.0190.0030.1490.0360.386CHE0.0710.0690.0640.0150.0115E-040.0660.0020.0060.0060.0340.0070.0090.0090.0063E-040.0040.010.0220.0190.0030.1490.0350.381FIN0.0720.0670.0640.0150.0115E-040.0650.0020.0060.0060.0340.0070.0090.0090.0053E-040.0040.010.0210.0180.0030.1460.0340.392ICE0.0730.0680.0660.0150.0115E-040.0670.0020.0050.0060.0350.0070.0090.0090.0053E-040.0040.010.020.0180.0030.1430.0340.386NOR0.0710.0660.0640.0150.0115E-040.0670.0020.0060.0060.0350.0070.0090.0090.0053E-040.0040.010.0220.0180.0030.1450.0340.3915WE0.0710.0670.0630.0150.011SE-040.0650.0020.0060.0060.0350.0070.0090.010.0063E-040.0040.0090.0210.0190.0030.1470.0350.39TUR0.0750.0640.0610.0140.0116E-040.0620.0020.0050.0050.030.0050.0090.010.0053E-040.0040.010.0140.0180.0030.150.0360.406AUS0.0740.0680.0640.0150.0115E-040.0660.0020.0060.0060.0330.0070.0090.010.0063E-040.0040.010.0220.0180.0030.150.0350.382NZL0.0730.0670.0650.0150.0115E-040.0650.0020.0060.0060.0330.0070.0090.0090.0063E-040.0040.010.0220.0190.0030.150.0350.383JAP0.0720.0670.0640.0150.015E-040.0650.0020.0060.0060.0360.0070.0090.010.0063E-040.0040.0090.0240.0190.0030.1370.0340.394CAN0.0750.070.0660.0160.0115E-040.0680.0020.0060.0060.0340.0070.0090.0090.0063E-040.0040.010.0230.0190.0030.150.0340.371USA0.0760.0720.0660.0160.0115E-040.0660.0020.0060.0060.0340.0070.0090.010.0063E-040.0040.0110.0240.0190.0030.1550.0360.359EKS0.080.0680.0640.0160.0115E-040.0630.0030.0060,0050.0320.0060.0090.010.0063E-040.0050.010.0180.0190.0030.1520.0360.376lableC.2:OutputSharesAppendixC.TablesforChapter4GERFRAITANLDBELLUXUKIREDNKGRESPAPRTAUTCHEFINICENORSWETURAUSNZLJAPCANUSAGER000540.0870.0360.0430.0460.0870.0170.0860.2140.0970.3030.0230.0020.0670.0880.060.0630.550.1020.1140.1080,1010.124FRA0.05400.0550.0970.0570.080.1050.0660.1150.160.0950.2040.0660.080.0770.1120.0870.0910.4040.0940.0910.1420.1050.202ITA0.0870.05500.1320.1090.1050.0890.080.1390.150.0780.1680.0970.1070.1250.1980.1760.1150.3480.1390.1320.1930.1490.216NLD0.0360.0970.13200.0680.1210.0790.0670.0760.1830.1270.2930.0860.0690.0950.1120.0920.0850.4680.1350.1040.130.1470.18BEL0.0430.0570.1090.06800.070.0820.0730.1150.2210.0960.2350.0910.0640.0890.1260.1210.1150.5080.1010.1010.120.1130.179LUX0.0460.080.1050.1210.0700.1730.1020.1810.2440.1380.3680.0660.040.1330.0910.1810.114O.62o0.1350.1250.1360.1170.115UK0.0870.1050.0890.0790.0820.17300.0240.130.1990.160.1950.170.16.40.1660.1980.2030.1320.3820.1330.1180.1950.1690.176IRE0.0170.0660.080.0670.0730.1020.02400.0680.1880.1170.2140.0920.0560.0750.1440.1330.0790.3750.0320,0690.1220.0660.105DNK0.0860.1150.1390.0760.1150.1810.130.06800.1830.1380.2820.1110.1340.0630.080.0750.080.3460.1240.1050.1390.1930.277GRE0.2140.160.150.1830.2210.2440.1990.1880.18300.0870.1890.2120.2490.2260.2420.3290.2630.1760.1310.1650.3690.2640.306SPA0.0970.0950.0780.1270.0960.1380.160.1170.1380.08700.0990.1560.1520.1470.240.2620,2170.280.1320.1140.3110.1680.216PRT0.3030,2040.1680.2930.2350.3680.1950.2140.2820.1890.09900.3160.3470.2810.350.4650.3590.2060.2620.2340.4790.3210.398AUT0.0230.0660.0970.0860.0910.0660.170.0920.1110.2120.1560.31600.0390.0840.1260.0980.0780.450.0860.1160.1260.0640.114CHE0.0020.080.1070.0690.0640.040.1640.0560.1340.2490.1520.3470.03900.0640.0860.0320.10.490.0850.0810.1430.0530,11FIN0.0670.0770.1250.0950.0890.1330.1660.0750.0630.2260.1470.2810.0840.06400.0570.0130.0550.3580.0470.0530.10.0670.165ICE0.0880.1120.1980.1120.1260.0910.1980.1440.080.2420.240.350.1260.0860.05700.080.0740.3980.0790.0820.0950,0940.175NOR0.060.0870.1760.0920.1210.1810.2030.1330.0750.3290.2620.4650.0980.0320.0130.0800.0370.570.0710.0860.0590.0590.199SWE0.0630.0910.1150.0850.1150.1140.1320.0790.080.2630.2170.3590.0780.10.0550.0740.03700.4490.10.0830.0830.1170.214TUR0.550.4040.3480.4680.5080.6260.3820.3750.3460.1760.280.2060.450.490.3580.3980.570.44900.4050.3810.6010.5370.634AUS0.1020.0940.1390.1350.1010.1350.1330.0320.1240.1310.1320.2620.0860.0850.0470.0790.0710.10.40500.0690.1240.0520.126NZL0.1140.0910.1320.1040.1010.1250.1180.0690.1050.1650.1140.2340.1160.0810.0530.0820.0860.0830.3810.06900.1170.0970.152JAP0.1080.1420.1930.130.120.1360.1950.1220.1390.3690.3110.4790.1260.1430.10.0950.0590.0830.6010.1240.11700.0960.222CAN0.1010.1050.1490.1470.1130,1170.1690.0660.1930.2640.1680.3210.0640.0530.0670.0940.0590.1170.5370.0520.0970.09600.078USA0.1240.2020.2160.180.1790.1150.1760.1050.2770.3060.2160.3980.1140.110.1650.1750.1990.2140.6340.1260.1520.2220.0780TableC.3: Paasche—LaspeyresSpread(PLS)IndicesU]116ci)L)CU)ci)ccicjU)CU)U)nfla.)ci)cci— N. a) ‘0 CN 0’ U) ‘00’ U) C>—U) a) U) ‘0 CX) CO 0’ — CX) ‘0 C’) N. 0’ 0’ a)(I) N. C’.J C’4 U) LI) — U) az N. U) a) C C’) 0’. C’) C)) 0’ C’J 0’ ‘0 C’) N. CN 0’ ‘ci’ — — ccD C C C C C C C C C C C C C C C C C C C 0 C C C C C CCC.))&—0’ U) C’) N. U) C’) C” ‘0 C’) N. C’) ‘0 — C’) ‘ci’ N. C’) C’) ‘0 ‘0 cc C’) C’.J N. ‘0 ‘0 ‘0 — N. —qqco.o’. .o:!U) ‘ci’ U) U) C’) U) ‘ci’ C’.i U) LI) C’) ‘ci’ 0 (N — U) U) C’) ‘0 U) (N ‘ci’ U) U) U)C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’) C’)99 999 90 9c?00090 9LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU— (N CD N. (N (N (N a) ‘ci’ 0’ (N (N — cc C N. 0) C CD C’) ‘0 — U) ‘0 (N C ‘0 C’) ‘ci’qc’qq’coc0 C’) C’) C’) C’) C’) C’) C’) C’) (N — C’) C’) (N (N (N C’) a) — — CN C’) C’) (N ‘ci’ C’) — (N C’) ‘cl’ C’) C’).0‘0 C’) N. — N. (N 0’ N. ‘ci’ ‘0 — ‘ci’ ‘C3’ N. a) — (N ‘00’ ‘ci’ C’) U) (N C’) (N (N ‘0 C’)QcU-) cq1-!C7 C??9? C)U—C’) C’) C’) 0’ C’) U) U) — C’) ‘‘O a) — (X) 0’ N. C)’ ‘00’ N. U) ‘0 ‘0— CC) 0’ ‘00’ (NN.C—U)U)000’C’)U)(NN.N.N.N.N.a)N.a)0’N. ,—N.—a)—N.0’0’N.0’.-.--,- ‘—O’’-(N——0qqqqqqqqqq oqoqoqqqqq oqooo -.9999900 c?c?c?909 .9CN. U) U) (‘-I a) 0’ N. C’) N. (N 0’ (N C’) — CC ——0’ (‘4 N. N. — U) ‘0 (“4 cl’—U)O’Cra)’crU)’Cl’CN.C’)CU)Ca)CC0’ 0’U)CCC C’4CN.—C’J0Qd’ -qooo ——c’)cclD0000COC 0000COC CCC CC 000CCCU)C)R— 0’ ‘0 cc 0’ C’) ‘0 (N —0’ N. ‘0 ‘0 —‘0— C’) C’) N. a) N. N. (‘4 C’) 0’ N. (N — N. (N0’ C’0 — C’) 0’ N. — 0 0’ C’) CX) C’) C’) —0’ CN (N C’) U) (N —‘0 C’)—U) N. ‘0 ‘0— Co C’) (N — C’) (“I — ‘ (N — (N CN C’) (N — C (N (N (N — (‘4 — C — — — — C’) C’)U) U) U) LI) U) U) U) U) U) U) U) U) LX) U) U) ‘ci’ ‘cJ’ ‘ci’ U) U) U) U) LX) U) U) U) U) ‘ci’ U) ‘ci-C C C C C C C C C C C 9 C C C 9 0 0 C C 99 C 9 C C C C C C CUi UJ LU LU LU LU LU LU LU LU LU LU UI Ui LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU“‘O U) C (N — U) 0’ 0’ U) C’) 0’ ‘-0 U) C C N. — N. C’) C’0 a) cl’ ‘ci’ ‘0 C LI) N. N. C’)’OCqqro’crr-c’‘0 ‘O’O N. ‘0 U) —0 ‘0 a) —‘0 N. 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C’) a- (N U) ‘0 ‘0 U) a) U) (N (N‘0 — cr o o U) c’ r-.. — cr to — — r—. o r—. a- r-. a- tO ‘0 c’ a- 00— 0————00—CNQqoqqqqqqqqqoqqqqooqqqqqqqqqqqq0 000000000 0000 000000000000C(N‘-a)(- .cr ‘ C’) c C’) ‘ ‘ C’) ‘T0 9 0 9 9 00000000 9 9 00 9 9 0000000 9 0000UJ LU LJ LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LU LUU) ‘0 00 C’) ‘0 ‘0 N. a- 00 (N CO a- C’) ‘0 U) N. a) U) Ct) — C’) ‘0 (N a) — a)C’) f Ct) ‘0 a) Ct) U) a) a- U) U) a) ‘0 N. a- a) a) — U) a) ‘0 a) N.— —‘ a)—‘-a)(D

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