@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Arts, Faculty of"@en, "Vancouver School of Economics"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Hill, Robert J."@en ; dcterms:issued "2009-04-16T18:46:45Z"@en, "1994"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The objective of this dissertation is to improve our understanding of the various Purchasing Power Parity (PPP) methods that have been advocated in the literature on international comparisons. The first of three essays builds on the pioneering work of Van Yzeren(1987) to rationalize the literature, by constructing a taxonomy of PPP methods. In particular, the taxonomy reinterprets PPP methods in a graph theoretic context. This reinterpretation yields many useful insights. The second essay was motivated by the realization that virtually all PPP methods have the same underlying graph theoretic structure. This essay develops a new PPP method which allows the data to choose the underlying structure by using Kruskal’s “Minimum Spanning Tree” Graph Theory algorithm to chain PPPs across countries rather than imposing the structure ex ante. The Minimum Spanning Tree (MST) method may potentially dramatically simplify the procedure for constructing PPPs. The MST method also has important implications for time series comparisons. The essay concludes with an empirical comparison using 1990 OECD data between the MST method and the three 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 of generating quantity indices that literally add up over different levels of aggregation when measured in value terms. However, it is widely claimed that Average Price methods overestimate the output shares of any outlier countries in a comparison. This is the so-called Gerschenkron effect. In spite of its significant implications, evidence for the Gerschenkron effect remains largely anecdotal. This essay explains the reasoning behind the Gerschenkron effect. As part of this explanation it is necessary to give a precise interpretation to the hitherto vague notion of an “outlier” country. Also frameworks are developed for empirically verifying and measuring the Gerschenkron effect, which are then applied to 1990 OECD data, with some surprising results."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/7235?expand=metadata"@en ; dcterms:extent "2203267 bytes"@en ; dc:format "application/pdf"@en ; skos:note "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 Methods70]. (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) 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. 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C 0’ C U) C —(NU)’00’U)’ a)’ct—(N C’) — (N — (“4 —— U) ‘0——— 0’—— —— a) (N U)C I I I I I I I I I I I I I I I I I I I I I I I00B—0Cd00ci)CdSL0CdE—1171)CCl)4- — — — — — — —D—r CN a- U) C’4 a) C’J O -O ‘0 a) ‘ to ‘0 a- a) r- (\\1 ‘0 — a) a) r- O C’J a) C’)C C’4 C’4 0 0 C”J (N 0 (N — a- (N CN— 0 a- 0 a) a- — 0 (N 0 0 — — — — C’) C’) C’) a)4- (N 0 (N (N (N (N (N (N (N — (N CN (N (N — (N — — (N (N (N (N (N (N— (N (N (N (N (N —Dqqqqqqqqqqoqqoqqqqqqqqqqqqqqqqq0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0C,)u).DI—U) C’) (N ‘0 C’) (N a) ——‘0 a) ‘0 a) C’) c’ C’) N. 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"@en ; edm:hasType "Thesis/Dissertation"@en ; vivo:dateIssued "1995-11"@en ; edm:isShownAt "10.14288/1.0088095"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Economics"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Purchasing power parity methods of making international comparisons"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/7235"@en .