MULTILATERAL APPROACHES TO THE THEORY OF INTERNATIONAL COMPARISONS by KEIR G. ARMSTRONG B.Sc., The University of Toronto, 1987 M.A., The University of British Columbia, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Economics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1995 © Keir G. Armstrong, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Department of COkO*)Icc The University of British Columbia Vancouver, Canada Date DE-6 (2/88) OCfO1€V 11-i Ic11 11 ABSTRACT The present thesis provides a definite answer to the question of how comparisons of certain aggregate quantities and price levels should be made across two or more geographic regions. It does so from the viewpoint of both economic theory and the “test” (or “axiomatic”) approach to index—number theory. Chapter 1 gives an overview of the problem of multilateral interspatial comparisons and introduces the rest of the thesis. Chapter 2 focuses on a particular domain of comparison involving consumer goods and services, countries and households in developing a theory of international comparisons in terms of the the (Kontis—type) cost—of—living index. To this end, two new classes of purchasing power parity measures are set out and the relationship between them is explored. The first is the many—household analogue of the (single—household) cost—of—living index and, as such, is rooted in the theory of group cost—of—living indexes. The second Consists of sets of (nominal) expenditure—share deflators, each corresponding to a system of (real) consumption shares for a group of countries. Using this framework, a rigorous exact index— number interpretation for Diewert’s “own—share” system of multilateral quantity indexes is provided. Chapter 3 develops a novel multilateral test approach to the problem at hand by generalizing Eichhorn and Voeller’s bilateral counterpart in a sensible manner. The equivalence of this approach to an extended version of Diewert’s multilateral test approach is exploited in an assessment of the relative merits of several alternative multilateral comparison formulae motivated outside the test—approach framework. 111 Chapter 4 undertakes an empirical comparison of the formulae examined on theoretical grounds in Chapter 3 using an appropriate cross—sectional data set constructed by the Eurostat—OECD Purchasing Power Parity Programme. The principal aim of this comparison is to ascertain the magnitude of the effect of choosing one formula over another. In aid of this, a new indicator is proposed which facilitates the measurement of the difference between two sets of purchasing power parities, each computed using a different multilateral index—number formula. iv TABLE OF CONTENTS Abstract Table of Contents iv List of Tables vi List of Figures vii Acknowledgements viii Chapter 1 Introduction 1 Chapter 2 Microeconomic Foundations 5 2.1 The Cost—of—Living Index 6 2.2 Country—Specific Indexes of Relative Purchasing Power 11 2.3 Bloc—Specific Indexes of Relative Purchasing Power 15 2.4 The Multilateral—KonUs PPP Index 18 2.5 The Multilateral—Allen Consumption—Share System 20 2.6 Plutocratic and Democratic PPP Indexes 25 2.7 Concluding Remarks 30 A Restricted—Domain Multilateral Test Approach 31 3.1 Definitions 32 3.2 Consumption—Share Equivalence 41 3.3 Some Examples 51 3.4 Concluding Remarks 60 Chapter 3 V Chapter 4 The Impact of Alternative Formulae 62 4.1 A Summary Measure of the Differences between Alternative Formulae 63 4.2 The Data 66 4.3 Empirical Results 68 4.4 Concluding Remarks 72 Bibliography 83 Appendix A Proofs of Theorems in Chapter 2 87 Appendix B Proofs of Theorems in Chapter 3 95 Appendix C Proofs of Theorems in Chapter 4 128 vi LIST OF TABLES Table 4.1 OECD—calculated PPPs for private final consumption expenditure in 1990 73 Table 4.2 Population data for the Member countries of the OECD in 1990 74 Table 4.3 Indexes of private fmal consumption in 1990 75 Table 4.4 Mean absolute log—percentage differences 75 Table 4.5 Indexes of per—household private final consumption in 1990 76 Table 4.6 PPPs for private final consumption expenditure in 1990 77 vii LIST OF FIGURES Figure 3.1 The hierarchy of multilateral comparison formulae under Diewert’s multilateral test approach Figure 4.1 HD, EKS, OK and ER indexes of per—household private final consumption in 1990 Figure 4.2 80 VH, EKS and OS indexes of per—household private final consumption in 1990 Figure 4.4 79 HD, AB and CD indexes of per—household private final consumption in 1990 Figure 4.3 61 81 AB, EKS and OK indexes of per—household private final consumption in 1990 82 viii ACKNOWLEDGEMENTS First and foremost, I owe a debt of gratitude to Erwin Diewert for his excellent supervision of my work. His insightful comments, helpful advice and friendly demeanour made the task at hand considerably easier to bring to fruition. My work in progress also benefited from the comments of Chuck Blackorby, John Helliwell, Ken White, Craig Brett and Ross McKitrick, and from the beautiful and stimulating environment of the University of British Columbia. Special thanks in this regard are due to Catherine Schittecatte. Finally, I acknowledge the unwavering moral support of my parents. Their continual encouragement and heart—felt belief in my ability to succeed have always emboldened me to undertake new and greater challenges. 1 CHAPTER 1 INTRODUCTION In a country like Canada, which covers a large territory and has a diverse economy, it is often necessary to compare regional real incomes or consumption levels so that federal transfers to poorer areas can be made according to some definite formula. Similarly, many international organizations require measures of real per—capita GDP which are comparable across their member countries in order to detennine desirable directions and magnitudes for aid flows. Measures of this sort are also essential in understanding economic growth, and in investigating the international distribution of world income and the extent of world poverty. Until as recently as fifteen years ago, no useful worldwide system of consistent international comparisons covering a substantial number of countries was available for general 1 The effort to come up with such a system has been led by various national statistical use. agencies collaborating under the auspices of the United Nations International Comparison Project (ICP). Established in the late 1960s, the ICP’s mandate was to bridge a gap in the world’s statistical system and thereby enable the conversion of GDP and other national— accounts aggregates of different countries to a common currency in such a way as to make them directly comparable. At that time, anyone requiring such data had to make do with inter—country comparisons calculated using exchange rates as conversion factors. According to Balassa (1964, p. 586), if “... international differences in productivity are greater in the sector of traded goods than in the non—traded goods sector,” the ratio of a currency’s relative purchasing power to its exchange rate will be an increasing function of relative per—capita real income. Since relative purchasing power and relative per—capita real income are not independent variables — their product being constrained to equal relative per— 1 The first such system, described by Summers et al. (1980), included estimates of real GDP and its components for 119 countries in 1950 and in each of the years between 1960 and 1977. 2 capita nominal income — this means that, although relative purchasing powers will tend to deviate from exchange rates in a systematic manner, there is no way to provide a general characterization of the former in terms of the latter. Thus, the relative purchasing power of currencies cannot be approximated using exchange rates. The logic of Balassa’s argument proceeds under the following assumptions: (i) capital flows and invisibles do not enter the balance of payments; (ii) there are no barriers to trade; (iii) transport costs are zero for traded goods; (iv) prices equal marginal costs; and (v) labour is mobile within countries but not between them. By assumptions (ii) and (iii), exchange rates will equate the prices of traded goods across countries. By assumption (iv), inter—country wage differences in the traded—goods sector will be positively related to productivity differentials. Assumption (v) implies that the wages of similar types of labour will be equalized within each country. Therefore, if international productivity differences among non—traded goods are smaller than those among traded goods, the former will be relatively more expensive in countries with greater productivity differentials in the production of the latter. Since relative purchasing powers are directly influenced by the prices of non—traded goods but exchange rates are not, the purchasing power of the currency of a low—productivity country relative to that of a high—productivity country will be lower than the corresponding equilibrium rate of exchange. The wider the productivity gap, the larger the differences in wages and non—traded--goods prices and, consequently, the larger the difference between relative purchasing power and equilibrium exchange rate. Presently, the United Nations, the World Bank, the International Monetary Fund (IMP), the Organization for Economic Co—operation and Development (OECD), and the Statistical Office of the European Communities (Eurostat) publish per—capita real income data for various countries which were calculated using one of two methods favoured by the ICP. Not withstanding the fact that either of these methods is an enormous improvement over the exchange—rate approach, all three suffer from the same apparent lack of grounding in 3 economic theory. This is also true of every other multilateral comparison formula developed to date. Nowhere in the relevant literature is there any strong justification for the use of one formula over another. The present thesis investigates the problem of multilateral international comparisons from the viewpoint of both economic theory and the test (or axiomatic) approach to index— number theory. It endeavours to give a definite answer to the question of how interspatial comparisons of aggregate quantities and price levels should be made. The usual objective of studies of this sort is to facilitate comparisons of real GDP among all the world’s countries based on all purchases of final—use commodities by economic agents during a single calendar year. Since there is often a need for quantity comparisons made on a restricted or altogether different basis, it is desirable that the recommended procedure be applicable in as many contexts as possible. A flexible basis for making place—to—place comparisons is established by the following definition: A domain of comparison is (i) a time period, (ii) a list of final—use commodities, (iii) a list of geographic regions, (iv) a matrix of positive regional commodity prices, (v) a matrix of regional quantities purchased, and (vi) a vector of regional purchaser populations. In its broadest interpretation, this definition could be used as the basis for comparing standards of living among groups of people. Leisure, family size, length of life, quality of life, etc. could be treated as final—use commodities with associated prices and quantities “purchased.” Moreover, any stratification of society could be treated as a list of “regions” over which levels of well—being are to be compared. In this way, systematic differences in the prices faced by different types of people in the same location could be recognized. 2 2 “Rich” versus “poor,” for example. 4 A natural way to begin thinking about international comparisons is in terms of the cost—of—living index. As shown in Chapter 2, the interspatial interpretation of this concept can be generalized to facilitate comparisons of real private consumption among groups of countries. It should be understood that, by necessarily limiting the domain of comparison to “private consumer goods and services,” “countries” and “households” instead of the more general “final—use commodities,” “geographic regions” and “purchasers,” this approach cannot offer a complete answer to the question posed above. It is, however, an important first step towards doing so. A novel multilateral test approach to the problem at hand is developed in Chapter 3 by generalizing Eichhorn and Voeller’s (1983) bilateral counterpart in a sensible manner. The equivalence of this approach to an extended version of Diewert’s (1986) multilateral test approach is exploited in an analysis of the major multilateral comparison formulae discussed in the literature. Two alternatives supported by the results in the second chapter are shown to be superior to the others in the sense of satisfying the most tests. The final chapter undertakes an empirical comparison of the methods compared on theoretical grounds in Chapter 3. In aid of this, a new indicator is proposed which facilitates the measurement of the difference between the results of two different formulae applied to the same data set. It is discovered that the formulae found to be best from the perspective of the test approach yield results which are substantially different from those generated by other methods. 5 CHAPTER 2 MICROECONOMIC FOUNDATIONS Any price index between two groups of countries is, by definition, a measure of “purchasing power parity.” The most commonly required indexes of this sort involve single— country groups in which commodities are valued using different currencies. As explained in Chapter 1, there is no definite relationship between such a measure and the corresponding exchange rate parity — the most readily observable and seemingly appropriate proxy. How, then, should suitable measures of purchasing power parity be constructed? In its current form, economic theory offers very little support for any of the practical procedures which have been developed to facilitate international comparisons of purchasing power. This is especially true of multilateral methods — i.e., those applicable to comparisons among two or more countries. The present chapter explores a novel consumer theory approach to the problem at hand with the object of finding an economic basis for one or more of the existing methods or for some heretofore undiscovered alternative. Three classes of purchasing power parity measures are set out below. The first, developed in Sections 1 and 2, is an interspatial interpretation of the single—household cost—of—living index. The second, developed in Sections 3 and 6, is the many—household analogue of the first and, as such, is rooted in the theory of group cost—of—living indexes. Members of the third class are sets of (nominal) expenditure—share deflators, each corresponding to a system of (real) consumption shares for a group of countries. Sections 4 and 5 establish reasonable generalizations of the cost—of—living index and the Allen consumption index as a consistent pair within this class. 6 2.1 The Cost—of—Living Index Consider a bloc comprising K this bloc there are N { 1,...., N) 2 countries indexed by the set K : = { 1,..., K). 2 well—defined types of consumer goods and services. denote the “general list” of these commodities and let Ik Within Let 1.1 := Al denote the subset which is available in country k E K. Every country—specific commodity list contains at least two items.’ In each country k € K, a representative household is assumed to purchase x of commodity ii E good or service n 0 units 2 For any Ilk at a price of p> 0 country—k currency units (k$, for short). € )l\Alk which is unavailable in country k, information to estimate a reservation price p> 0. Let representative country—k consumption bundle and let 4 0 and there is sufficient 4)’ € D denote the (pt,..., Pr)’ € denote the xk : = (x,..., k : vector of country—k commodity prices. 4 The matrix of all commodity prices in the bloc and the corresponding matrix of representative quantities are denoted by P := (p , 1 X := (x’,..., x9, , ... p1) and respectively. The tastes of the kth (representative) household are represented by a preference ordering ?tk defined over the commodity space indexed by 11. arbitrary consumption bundles, x Letting x E D and . E ll denote ? means that household k considers x to be at least as good as X. Assuming that ? is complete, reflexive, transitive, continuous and increasing, there 1 Formally, 1)41 2. 2 Representative households are assumed for the sake of expositional clarity. The need for their existence may be expunged by regarding the bloc as a collection of households rather than as a collection of countries. As such, K would index the constituent households, 14 would denote a household—specific commodity list, and x would be household k’s consumption of commodity n. Using a hedonic price index, for example. See Griliches (196 l)(1967) and Kravis and Lipsey (1971). Notation: The “prime” symbol denotes the transpose operator. 7 exists a continuous, increasing function Uk: D x >.k -‘ ll such that Uk(x) Uk() if and only if L Thus, Uk is a (direct) utility function corresponding to the preference ordering k•5 The regularity conditions for UC are summarized as Ri. continuity and strict positive monotonicity. Let (Uk) denote the range of Uk with its infimum value excluded. 6 expenditure function Ck: U÷ x 1(Uk) -* Household k’s IJ shows the minimum expenditure required to attain a given level of utility v at commodity prices p £ ll; i.e., Ck(p, v) := mm {p’x I Ukcx) v} (2.1) . Given that Uk satisfies Ri, Ck is non—decreasing, positively linearly homogeneous (PLH) and concave in p, increasing in v, jointly continuous in (p, v) and positive. 7 Whenever C” has these properties, it is said to satisfy R2. positivity, continuity in (p, v), strict positive monotonicity in v, and positive monotonicity, positive linear homogeneity and concavity in p. The (Konus—type) cost—of—living index for household k is the ratio of the minimum expenditure required to attain a particular utility level under two different price regimes: r Iv P ‘P P’ Vi C . C V kP, Ck(pr v) Thus, given household k’s indifference map (or preference ordering) , the household—k index Of course, any increasing transformation of Uk is also a utility function representing >.k 6 Since the commodity space excludes the origin, the expenditure minimization problem defined below does not have a solution when the utility level is at its infimum value. Proofs of these properties can be found in Diewert (1982). 8 depends on a base indifference curve v chosen from that map, a vector of reference prices p ’ 7 and a vector of comparison prices pC• There are five basic properties of the household—k cost—of—living index which follow directly from its definition and R2. pk is a positive function; i.e., for every (pf, pC, v) p2. (Uk), k(r pC, v)> 0. plc is non—decreasing in the comparison prices; i.e., for every (pr, pC pc v) IR x €(Uc) such that pC p• x € <PC pk(pr pC v) pk(pr, pC v). plc is PLH in the comparison prices; i.e., for every (pr, pC A, v) € pk(pr, ,k Apc, v) = Apk(pr pC, € x v). is transitive (or circular) with respect to the reference and comparison prices; i.e., for every (pV, p, pC, v) € x (Uk), pk(pf, p v)plc(p, pC, v) = pk(pr, pC, v). plc is concave in the comparison prices; i.e., for every (pr, pC p A, v) € (Uk), pl(pT, (1_A)pc + Ape, v) (1_A)pk(pr, pC v) + Ap’(p’, p x (0, 1) x v). Eight additional properties are implied by one or more of the preceding five. 1c P6. If the comparison prices are equal to the reference prices then for every (pr, v) € ÷ X (Uk), plc(pr, p ’, v) = 1. 1 P7. If the comparison prices are proportional to the reference prices then p is equal to the factor of proportionality; i.e., for every (pr, A, v) € P8. is equal to unity; i.e., x R(U’), pk(pr APr, v) If the reference prices and the comparison prices are switched, the new reciprocal of the old; l/pk(pr,pC, v). i.e., for every (pr, pC v) € x €(Uk), = k pk(pC pr A. is the v) = 9 k P9. is non—increasing in the reference prices; i.e., for every (pr such that pr <Pr, pk(pr, pC, pk(pr, pC, v) pr pC v) E x v). plc is positively homogeneous of degree minus one in the reference prices; i.e., for P10. every P11. (pr, pC, A, v) x (Uk), £ pC, pk(Apr v) Alpk(p, = pC, v). If the reference prices and the comparison prices are multiplied by the same positive (pr, pC A, v) £ scalar, p ’ is unaffected; i.e., for every 1 k(rc x .Z(Uk), pk(Apr Ape v) = v). plc is bounded from below and above by, respectively, the smallest and the largest price P12. n € I; i.e., for every (pr, pC, v) € relative I mm c I iPi pk(pf,pC, v) EI k P13. C iP max— flEX 1_fl x Lnr 1_fl is convex in the reference prices; i.e., for every (pf, pr, pC A, v) £ (Uk), pk((1_A)pr + Apr, p”, v) x (0, 1) x (1_A)pk(pr,pC, v) + Apk(PT,pc, v). The necessary and sufficient conditions for an arbitrary function plc: to be a cost—of—living index are provided by the following theorem. THEOREM 2.1. Let p’’: iR vpk(p?’, p’, v). -‘ P satisfy P1-P5 and, for some pr € ll÷, let Ck(pe, v; pr) := Then Cc is an expenditure function which satisfies R2 and the money metric utility scaling property Ck(pr, v) and k = v Vv E D, (2.3) is the cost-of-living index corresponding to the preferences that are dual to Cc. 10 Conversely, given an expenditure function Ck: pk(pr - D which satisfies R2 and (2.3), v) : = C(pc, V)/Ck(pr, v) satisfies P1-PS and Ck(p, v) pC = vp k(pr, pC, 8 v). The next theorem asserts that the cost—of—living index is invariant to changes in the dimensionality and/or ordering of prices. It follows from the fact that the introduction of such a change imposes no restrictions on the functional form of p’. 2.2. Let IN be a permutation of the columns of the N x N identity matrix and, for THEOREM some A (A , 1 ..., - .\N)’ , = A,for all n E JL Then x (Uk) -* v) , and k: II x R(Uk) v) = - ll defined by (2.2) with Cc : Uk, satisfies defined by (2.1) with Uk := k(5JNpr I 5 Np’, pC, (2.4) UINz) is a utility function representing for every (pf, be the diagonal matrix with defined by Uk(z) := tk: 5 let € plc(pr, pC, v) (2.5) € llU x A quantity counterpart to the cost—of—living index can be obtained by using p’ as a deflator for household k’s expenditure ratio between two different price—utility situations, (pr, U?.) and (pC, More precisely, the implicit household—k (Konus—type real) consumption va). index is defined as k(pr pC Vr, Vc, v) := C(pc, p, The result of substituting for 8 pk(pf, pC, v) v) yr / pk(pr pC (2.6) v). using (2.2) and setting v equal to either v or v is the A less concise version of this theorem was established by Diewert (1983, Theorem 1). A version of this theorem was established by Samuelson and Swamy (1974, pp. 57 1—572). 11 household—k Allen (1949, p. 199) consumption index at prices p Aki (fi D, Vr, Vc) Since Ck is increasing in representing Thus, . k 5 q rk ,,O, . v, pC or p := pr: V V, for p constant, C’(p, v) is a (money—metric) utility function is a measure of welfare change for household k in moving from the reference situation to the comparison situation.’ 0 2.2 Country-Specific Indexes of Relative Purchasing Power Setting pr := p . 2 j € K, and p’ := p, I E K, in (2.2) yields the household—k cost—of—living index associated with the class of international comparisons defined in Chapter 1: k P ‘P 2 P’ r’k 1.)) — I \‘ Ck(pi, V v) 28 The number pk(pf, pZ, v) is the factor by which household k’s purchasing power at country—i prices must be deflated in order to make it equal to the same household’s purchasing power at country—f prices. Thus, pC(p.9, p, v) is of the dimensionality i$/j$ the number of units of — country i’s currency per unit of country j’s. To distinguish interspatial interpretations of the cost—of—living index from intertemporal ones, it is common practice to refer to the former as purchasing power parities (PPPs). Since there are K countries in the bloc, (2.8) defines K 2 PPPs for each of the K representative households. The K of these (K ) PPPs for which p 2 because, by P6, they are always unity. Half of the K(K — = p’ can be ignored 1) PPPs for which p$ * p’ can be ignored because, by P8, they are the reciprocals of the other half. Each of the remaining 10 is “reference—free” (i.e., independent of p) if and only if the household’s preferences are homothetic. See Blackorby and Donaldson (1983, pp. 379—38 1) for a proof of this statement. 12 K(K — the K 1)/2 PPPs corresponds to a distinct bilateral intra—bloc price level comparison. By P4, — 1 country—k PPPs for which j Clearly, since N = k and i * k constitute a basis for these comparisons. 2, the functional form of the country—k PPP index k depends on the functional form of the country—k expenditure function Cc which, in turn, depends on the functional form of the representative country—k utility function is unknown then so is that of k• Uk.h1 If the functional form of Under the assumptions made so far, the upper and lower bounds on p” stated in P12 (with pr := p’ and pC := p) represent the best that can be done. One way to do better is to assume that the household—k consumption bundle xk is expenditure—minimizing; i.e., pklxk = Ck(p, Uk(xi)). (2.9) Equation (2.9) suggests what would seem to be the “natural” (although by no means the only) choice of an indifference curve on which to base the country—Ic PPP index: corresponding to the utility level Uk := Uk(xk) The one attained by the representative country—k household when facing prices p. The “natural” country—k PPP index pk(pi, p, Uk) is of special interest because it can be bounded more tightly than that of the general case. Prior to demonstrating this, it is necessary to introduce the concept of an axiomatic PPP index. Unlike the economic PPP indexes discussed above, axiomatic PPP indexes ignore household preferences and treat prices and quantities as independent variables. Examples of such indexes include the country—k Laspeyres PPP index and the country—k Paasche PPP 1 For N = 1, P7 implies that plC(p, p, v) = p/p. 13 index defined, respectively, by xk, x) := PL(P’ p, (2.10) and (pk p, xk, x) (2.11) . THEOREM 2.3 [Pollak (1971, p. 11)1. For all (k, i) E K x K, pm mm pk(pk, — nEff p, Uk) PL(P pZ, xk, XZ) (2.12) . Note that since k PL(P ‘P I ,X k N x) k Xn , ‘xk k 1 n= 1 V N P k’Jc I I — k’ X k LE — max nEff 11 k[ 1 - — N since p’ E , xk LI-’ nJ ki k x E IJ& and £ Pkm 1 k iP X = 1, (2.13) the upper bound in (2.12) is an improvement upon the one stated in P12 (with p’ := p, pC := p and v := Uk). The lower bound is the same in both cases. COROLLARY 2.3.1 [Pollak (1971, p. 12)]. For all (k, i) € K x K, k 1 pp(p k , 1 x ) pk(pI k Uk) max {I The preceding corollary follows directly from (2.12), the fact that I1 PL(P. p, xe), P8 and P12. (2.14) pp(p, k x xk) 14 The bounds on pk can be tightened still further by assuming that the representative households have identical preferences. 2.4 [KonUs (1924, pp. 20—21)]. Let U: ll? -* I1 satisfy Ri and let C: + x (U) 1F satisfy R2. Suppose that U” := U and (pk, xk) satisfies the expenditure minimization THEOREM property (2.9) with Cc := C. € (U) bounded by u 2 Then, for all (j, i) E K x K, there exists a base utility level := U(x’) and u := U(x) such that p(p’, p, v) := C(pz, i4)/C(pJ, is bounded by p := pL(p ’ p, x, x) and p : 2 min{u,, u} 14 pp(p’, ‘4) pZ, x , x); i.e., V(j, i) € K x K, z4 3 2 max{u,, u} and min{p, pJ , p, 2 p(p z4) max{p, p} . (2.15) A stronger version of Theorem 2.4 would assert the existence of a ‘4bounded by u and u such that p(p , p,’4) is equal to a particular average of p and p. Unfortunately, since 2 p(p’, p, v) could be close to either one of p5, and pç but not the other for all v between U, 2 ii and there is, in general, no such i4. If p and p are sufficiently close, however, there exists a ‘4between u, and u such that p(,p’, p, 12 ‘4) is approximately equal to any average of the two. Empirical studies have demonstrated that p5 and p are usually too far apart to be of much practical use in approximating the “true” PPP index p”. Ruggles (1967), for example, compared country—f Paasche and Laspeyres PPP indexes for a bloc consisting of nineteen Central and South American countries and found that the average difference between the two was between thirty—five and forty—eight percent. Similar large disparities between these indexes were found by Kravis et al. (1975) for price level comparisons among OECD countries. Even if this sort of outcome were not the norm, the use of an (unweighted) average of Paasche and Laspeyres indexes to approximate pYZ would be problematic because no such average is transitive with respect to its component price vectors. 12 E.g., Fisher’s (1927) “ideal” PPP index 15 2.3 Bloc—Specific Indexes of Relative Purchasing Power The primary application of a set of PPPs is in extending the usefulness of national accounts data by making economically meaningful cross—country comparisons or combinations of such data feasible. This is achieved by using the PPPs as nominal—value deflators as in equation (2.6). The results of such calculations are needed for policy purposes and for the purposes of economic analysis by international organizations which exist to further the collective interests of a bloc of countries. For example, policy decisions regarding desirable levels of intra—bloc aid from “have” to “have—not” countries require a measure of each country’s per—capita consumption level relative to some numéraire. None of the country—specific indexes discussed above provide an appropriate basis for such a measure. This is due to the fact that, in general, different country—specific indexes yield different sets of PPPs for the members of the same bloc, and there is no good reason to choose one country’s representative household over another’s to represent the bloc as a whole. What is required, then, is an index that somehow reflects the preferences of all representative households in the relevant bloc. Under such a requirement, the purchasing power of one national currency relative to another will depend on whether or not some third country is a member of the same bloc. In other words, it will be “bloc specific.” Another way in which the appropriate PPP index might be context dependent is with respect to the nature of the purpose to which it is applicable. Different international organizations involving identical groups of countries may require different sets of PPPs just because they have different objectives. “... To borrow an example from Pollak (1971, p. 7), suppose the U.S. government wants to compare prices in Paris with those in Tokyo to decide on appropriate salary differentials for its diplomats.” Quite clearly, such a comparison should be made using a U.S.—specific PPP index. Suppose instead that a certain multinational 16 corporation wants to make the same comparison to decide on appropriate salary differentials for its sales agents. if these sales agents are drawn from the U.S. and France, say, then some kind of bloc—specific PPP index is called for. One way to construct such an index is by aggregating over the K instances of the country—specific variety. Indexes defined under this approach in the intertemporal context are called group cost—of—living indexes because they measure the impact on a group of households of moving from one price regime to another and are constructed as weighted averages of single—household cost—of—living indexes. The theory of such index—number formulae was developed by Pollak (1980)(1981) and Diewert (1984). Since the country—specific index k is a measure of the relative purchasing power of a single household considered to be representative of all Hk k€ 1 households living in country K, it may be desirable that a given bloc—specific counterpart take account of size differences among countries by considering the population vector H : = (Hi,..., HK) l Formally, then, a bloc—specific PPP index for country i relative to country j is constructed by choosing a real number r and a set of weights a € SK_l fl Zk = 1) denotes the unit simplex of dimension K r Mr,a: - — , where S’ 1 := {z € ll 1, and then applying a mean of order D defined by ifrE D\{O} Mr,a(Z) := I (2.16) ifr=O k=1Zk to the K country—specific PPP indexes: p(p’, p, where 13 j, i : = (ni,.... iK)’ Note that k £ H) := Mr,a[pl(p2, 1k), K , 0 i ... P’ ILK)] , (2.17) (Uk) is the vector of representative base utility levels. 13 As a is not necessarily equal to Uk := Uk(xk). 17 weighted average of the p”s, p inherits the dimensionality i$Ij$ and depends on the preference orderings of all the representative households in the bloc.’ 4 If, for each k € K, ak is chosen to be the fraction of households living in country k, then (2.17) defines a bloc—specific PPP index which is “democratic” in the sense of assigning weights to the associated country—specific indexes which are increasing in the number of households they represent. Alternatively, if ak is chosen to be the bloc expenditure share of the country—k households H, say — — defined in terms of prices p’. base utility levels and populations then (2.17) is “plutocratic” in the sense of giving more weight to pC which represent higher spending. In general, a is a function of 3 (p pZ, , non—decreasing and PLH, and since, for every x ll - H). r € Since Mr,a is positive, continuous, I}, there exists an a: x (U’) x ... x 1 fl I1{ such that the right—hand side of (2.17) is transitive with respect to S’ 2 and p, there is an uncountably—infinite number of ps satisfying the first four properties of p k (P1—P4). These are summarized as R3. positivity, continuity in (p’, p, I.L, H), positive monotonicity and positive linear homogeneity in pZ, and transitivity with respect to p 2 and pZlS Further investigation of the p—class of bloc—specific indexes defined by (2.17) is deferred until Section 6. The intervening sections that follow pursue alternative multilateral approaches to the construction of bloc—specific PPP indexes. 14 Consequently, a more accurate (and cumbersome) notation for the image of this function would be p(pi, pZ, H; k’,..., >.K) ‘ Since P1—P4 implies P9 and PlO, R3 implies negative monotonicity and positive homogeneity of degree minus one in pi. , 18 2.4 The Multilateral-Konus PPP Index In this section, a reasonable generalization of k is used to specify a PPP index which relates the general or average price level of each country to that of the bloc as a whole. The same outcome is derived indirectly in the next section by generalizing a different type of country— specific index. Using an exact index—number argument, both approaches are then shown to enable the justification of a particular system of axiomatic quantity indexes. To begin with, let denote the price of a unit of country k’s currency (1 k$) in k E terms of some numéraire. Consequently, the ratio with respect to country k. Ek/Ez (i$Ik$) is country i’s exchange rate In the rest of this chapter, ply Ekpk numéraire—denominated country—k commodity prices and P associated matrix. := denotes the vector of (pr,..., pK) denotes the Use of such a normalization allows the summation of household expenditure functions as in the following definition of country i’s share of (possibly hypothetical) bloc expenditure at utility levels : ) HC(p’, s’(P, , H) := (2.18) EiHk Ck(pk, /.Lk) Likewise, it enables the definition of the bloc expenditure function: (K C(P, u,H) := = miii k=l E Hkpk?zk 1 K Z,...,Z U=1 rninz klk(P {plcik Uk). 16 I ) U”(z’) Uk(zk) u, kE X J (2.19) Uk} (2.20) The substitution of C for Cc in (2.8) gives rise to a logical bloc—specific counterpart to the (Kontis—type) country—k PPP index. Specifically, the multilateral-Konüs (MK) PPP index 16 Note that C is not a Scitovsky expenditure function since prices are not, in general, equal across countries. 19 for country i relative to the bloc as a whole is defined as the ratio of the minimum expenditure required to attain utility levels u : = (Ui, ...., UK)’ when every representative household faces the prices of country i to the minimum expenditure required to attain the same utility levels when each household faces the prices of its home country: C(IY1K’,u, H) - öMK(P, u, H) := (2.21) - C(P, u, H) E= C’(]i, u,) H 1 = , by (2.20). (2.22) 1 Hk Cc(pk, Uk) E The number öK(P, u, H) may be interpreted as the factor by which cost—minimizing bloc expenditure at country—i prices and actual utility levels must be deflated in order to make it equal to nominal bloc expenditure. Thus, the numerator of (2.22) is the sum of (hypothetical) € K) faces the prices of country i, pz, and its utility Uk in conjunction with (2.22) reveals that the MK bloc expenditures when the jth household (j level is held constant at the actual value u . 2 Use of (2.18) and (2.8) with v : = PPP index is an expenditure—share—weighted sum of country—specific PPP indexes; i.e., u, H) = H).pk(pk, pZ, (2.23) Uk). k=1’ Following directly from this fact is a corollary to Theorem 2.3 establishing bounds on Skx. COROLLARY 2.3.2. For all i K E Si mm nEff where s, : = p”(H,xi)/ € 1p1 1 — [n.iJ rn K, K - kKP, u, H) x x) , E jPL(P” p, 2 , (2.24) plc? (Hk XC) denotes the actual bloc expenditure share for country j. Since ôj is not defined over an arbitrary vector of base utility levels, there is no corollary to 20 Theorem 2.4 establishing tighter bounds on this index when the representative households have identical tastes. 2.5 The Multilateral—Allen Consumption—Share System Setting k := i, v := u and Vr := ij, i € K, in (2.7) yields the country—i Allen consumption index: u) := The number q(p, j, to that at utility level for some country j. j€ (2.25) . u) is a measure of household i’s consumption at utility level u relative pj using reference prices p. If K, then is chosen so that Cz(p, t) C(p, u,) is a per—household consumption index for country i relative to A natural way to generalize this bilateral country—specific measure into a multilateral bloc—specific one is to use it as the basic building block of a system of consumption shares: H q (p,p, u) E i.Hk. k (.p, /Lk, Uk) — — HCz (p, u) /C( p E= lHk.Ck (p, Uk) I ) Ck ( HC!(p, uj/C’(p, u,) (2.26) = Uk)! since Ck(p, /Lk) = ) 2 C’(p, u C’(p, u) for all k € K. Thus, the multilateral-Allen (MA) consumption share for country i is defined as the ratio of the minimum country—i expenditure required to attain per—household utility level u at reference prices p to the minimum bloc expenditure required to attain per—household utility levels u at the same prices: 21 HC(p, H) MA(P’ u, The number ii, OkA(P, u) := (2.27) . ElHkCk(p, Uk) H) is the fraction of total bloc expenditure which would be attributable to country—i households at reference prices p. The data set P admits K possible choices for the reference—price vector p in (2.27). 11 the country—i price vector pZ is chosen, O’VEA(P, u, H) is called the MA own—price consumption MA(P’ u, share for country i. lii general, UMA(P, u, H) , 1 [cA(P u, H),..., H)]’ is only a quasi—consumption—share system since its components do not necessarily sum to unity. The MA own-price expenditure—share deflator for country i is a bloc—specific PPP index i5 defined implicitly by MAQ’, u, The number t5kA(P, u, H)01,).A(pz, u, H) = s(P, u, H). (2.28) H) is the amount by which country i’s actual expenditure share must be deflated in order to make it equal to the same country’s MA own—price consumption share. Since s(P, u, H) := [s P 1 , u, - E 1=1 6 M AQ, u, H),..., H)Jj(pz, u, Using the definitions of index MK: H)]’ € u, SkA, M 0 A H) = 1 and s, (2.29) . ‘ k 5 A can be shown to be equal to the MK PPP From (2.28), sz(P,u,H) - 6 M A(’, ii, H) = MA(P, u, HC = H) HC(.p, u,) 1 ( p, u) E , ElHkCl(pk, Uk) HC(p, u) by (2.18) and (2.27) 22 1 H C2(pz, u ) 2 — — k, Uk) 15 Hk Ck( =: Skx(P, u, H), by (2.22). (2.30) Thus, the MA own—price consumption— share system and the MK PPP index are completely consistent with one another. This fact together with (2.23) implies a corollary to Theorem 2.3 establishing bounds on COROLLARY r K I IIIA• 2.3.3. For all i £ K, 1—1 Hz.xz, H,x’) I q(j5, Lj1 J where qL(p, p. , H x, H,x’) := 7 K u, H) . 2 mm 1 5 j—_1 nEff 11 1 [j — —1 , (2.31) (H ) x /pz ‘(Hr x) denotes the country-i Laspeyres consump p 2 tion index. In addition to satisfying Ri, suppose that Uk is independent of k and (positively) homothetic. Consequently, there exists a PLH function U: IIi increasing function : (U) Uk(x) Since is = continuous U(x) = -* - D and a continuously such that b(U(x)). (2.32) and increasing, b_l(Uk(x)). Thus, U is a utility function representing ?. (2.33) The associated expenditure function can be written as C(p, v) = vc(p), (2.34) 23 where c(p) mm {p’x U(x) 1} (2.35) is the unit expenditure function for U. A bilateral axiomatic per—household (real) consumption index for country i relative to country j is a real—valued function 5 of the observed price and (per—household) quantity data for the two countries. Such an index is defined to be exact for a PLH utility function U if, for every (p’, x ) and (pZ, x) satisfying (2.9) with C’ 2 C and Uk := U, k i}, £ ç,(pi, p, x’, x) = {j, (2.36) . Diewert (1981, p. 181) noted that the price and quantity vectors in this equation are not completely independent variables since (9, xlè)kE(j,i} is assumed to be consistent with expenditure—minimizing behaviour. A bilateral axiomatic PPP index for country i relative to country j is a real—valued function p of p’, p*, X2 and x. This type of index is defmed to be exact for a PLH utility function U (and its dual unit expenditure function c) if, for every (p’. x’) and (p’. x) satisfying (2.9) with Ck := C and UC := U, k € p(p’, pZ, x, x) = {j I} (2.37) . Since argmin {px U(x) U(Hkxk)} Hk arin {pki [] U[-] U(xk)} = Hkxc, equation (2.36) is equivalent to pZ, H,x’, H x) = . (2.38) 24 Substituting for C := C and C’ := C in (2.27) using (2.34) with, respectively, v := u and v : u,, and setting Uk ivIA(P’ := U(xP) for all k E K yields U(x ) 1 ,..., U(x”), H) = {, K = :} r lj=1L =: (2.39) i—n—i H x z) j ,1 2 (p’, p, H,x , J by (2.38) J c(P, XH), where H is the K x K diagonal matrix with Hkk (2.40) = Hk for all k E K. Thus, given a bilateral axiomatic per—household consumption index which is exact for a PLH utility function representing the (homothetic) preferences of the (identical) representative households, the un— normalized country—i own share of bloc consumption, u(P, 17 is a direct approximation XH), for the MA consumption index for country i. The significance of this result is that it provides a rigorous exact index—number interpretation for Diewert’s (1986, p. 25) own—share system of axiomatic quantity indexes. Substituting for C’ := C and (ps, u,) and (p, v) : = (pk, Uk), := C in (2.22) using (2.34) with, respectively, (p, v) := and setting u : = U(x’) for all I £ K yields - MKCP,U(X),...,U(),H)— I K K E U(H,x’) U(H,jxC) —1 c(pz) 11 . c(pk) (2.41) j Since p must be homogeneous of degree zero in each of its quantity arguments in order to satisfy (2.37), ( K ),..., U(x’), H) = 1 MK(P, U(x k= 1 ok(P, X J)[p(jik, HkX”, Hix)]J (2.42) follows from (2.41) by (2.40) and (2.37). Therefore, the own—share—weighted harmonic mean ‘ This number is measured in the “metric” of country i. Since real—world data are seldom in accord with the maintained assumptions about household preferences, the metric for one country is not, in general, compatible with that of another. Consequently, further adjustments are necessary in order to ensure that the shares sum to unity. 25 of the bilateral axiomatic PPP indexes for country i is a direct approximation for the MK PPP index for country i when preferences are homothetic and identical across households. 2.6 Plutocratic and Democratic PPP Indexes Prais (1959) was the first to note that official group cost—of—living indexes like the Consumer Price Index assign an implicit weight to each constituent household’s consumption pattern which is proportional to its total expenditure. He called such indexes “plutocratic” and suggested an alternative “democratic” variety which treats all households equally. Pollak (1980) formalized these concepts by extending the theory of the (single—household) cost—of—living index to groups. The present section applies this extended theory to the p—class of bloc—specific PPP indexes and compares the results with those obtained in the intertemporal context. Under the maintained international—comparisons interpretation, Pollak’s Scitovsky group cost—of—living index becomes the (Prais—Pollak) plutocratic PPP index 18 and is defined as the ratio of the minimum bloc expenditure required to attain per—household utility levels at country—i prices to that required at country—i prices: p(p’, pZ, , H) := =iHk ck(pi, u1 k) (2.43) . 1 C’(p’, lit) E’f,H Using (2.18) with P := (p’, ... ,p2) and (2.8) with v := /.Lk, (2.43) can be re—written as an expenditure—share—weighted average of the corresponding country—specific PPP indexes: ppp(pi, p , 1 , H) , = k-1’ p, , H)pk(pi, p , 1 lik). (2.44) ... Pollak’s democratic group cost—of—living index becomes the additive democratic PPP index’ 8 18 This term is due to Diewert (1984). 26 and is defined as a population—share—weighted average of the corresponding country—specific PPP indexes; i.e., K PAD(P” p, , H) E k=1 ok(H)pk(p, p, /1k) (2.45) , where O’(H) : Hk/ E’ H is the country—k bloc population share. 1 As shown in Diewert (1984), a second type of democratic index can be constructed by replacing the arithmetic average in (2.45) by its geometric counterpart. The result of doing so is called the multiplicative democratic PPP index’ 8 for country i relative to countryj: , 1 PMD(P” p , H) := k=1 [pkc, p’ 0 k (J]) /k)] (2.46) . In both sorts of democratic PPP index, the use of population—share weights has the effect of counting every household equally. In contrast, by implicitly weighting each household— specific PPP index by its total expenditure, the plutocratic variety counts every dollar of consumption spending equally. Of the three bloc—specific PPP indexes just defined, PAD 15 the only one that does not satisfy R3 as it is not, in general, transitive with respect to p’ and p. Worse still, it does not even satisfy the weaker property of “country reversal” PAD(P’, p, t, H) PMD(P’ p, = , — the multilateral analogue to P8: H) 1 l/PMD 0 ’,/L,H) (p, l/p,(p, p’, , H), (2.47) where each of the two inequalities follows by the Theorem of the Arithmetic and Geometric 27 19 and the equality follows by the transitivity and positivity properties of Means The PMD• existence of an alternative to PAD which is both “democratic” and transitive eliminates the need to consider this index any further. As suggested by the following theorem, any p—type PPP index satisfying R3 can be provided with an axiomatic characterization similar to that provided for k in Theorem 2.1. THEoREM 2.5. Let p : IJ x 11K)’ E O_1 , t pjp(p and for all ll satisfy R3 and, for some p t (pC, Ilk) E k € K, X E ll÷, it-k := t, x H) E (i) p(p, p, (ii) p(p’, p’, , H) , H) = = , E K, Ilk-i, IL—k) 1. Further, for let p satisfy , 1 k=lk IlkP(p’, p II I p(p , p’, 2 k= 1 L Then, for all k (lu,..., let C(pc, t , t I; p k ek), where ek is the K-dimensional unit column vector with e pC all (p , p, 2 -* j, , etc) ek) and Ilk =0 , 1 IlkP(p’, p , ek) 0 Vk E K; or ink is an expenditure function which satisfies R2 and the money-metric utility scaling property (2.3) with v Ilk and, depending upon whether it satisfies (i) or (ii), p is the plutocratic or multiplicative democratic PPP index for country i relative to country j corresponding to the preferences that are dual to functions Ck: Hk Ck(p, 1 - }k 1C I which satisfy R2 and (2.3) with v := 1C (p’, 1 1H 1.1k)! E C(pi, { /1k) = ILkp(P’, and (b) p(pi, p, Il H) := U Conversely, given expenditure 1-uk (k e K), (a) p(p’, p, .u, H) ) satisfies (i), R3 and p, IL’ e”); (2.48) [1 Ck(p$, Ilk)ICk(pi, ILk)1 satisfies (ii), R3 and (2.48).20 19 See Hardy, Littlewood and POlya (1952, pp. 16—21) for a general statement and proof of this result. 20 A less concise version of this theorem was established by Diewert (1984, Theorems 6 and 10). 28 By rearranging the terms of the definition of the MA own—price consumption shares multiplied and divided by their sum, the former can be re—expressed as the product of the latter and the harmonic mean of the associated national expenditure ratios, each deflated by the corresponding plutocratic PPP index: From (2.27), H C I (p UMA(P, 1 , ) 1 ii (p, Urn) m E= H 1 rn C I K [k=1 Hk Ck(Pk, Uk) 1 Cl(plc, / Ef H Uj) EflHkCk(p, Uk)/5f,lHlCl(pk, ) i u, H) = Hk C’(, 1 Hm C”(, E Uk) Um)1 —1 H C(p, u’) Ef H C1(pk, uj)j 1 IK 1 Lk (p u C 1 H , ) 1 HkCk(,pk, Uk) / p(pk, Hk Ck(pk, K Uk) E k1 k, 5 H Cl() 1 Ef, Ui) K p, u, H) E J MA(P, u, H), (2.49) by (2.43) and (2.27). Dividing both sides of equation (2.49) by E=i okzA(Pk, is, H) reveals that the system of consumption shares that is dual to the plutocratic PPP index is the normalized MA own—price consumption—share system: - ojp(P, U, H) cTMA(P, : u, H) (2.50) El MA(P, u,H) = I I K H C 1 (p, u ) 1 E =1 —1 -1 /p(pkpUH) (2.51) . Hk Cc(pc, Uk) by (2.49). THEOREM 2.6. For any p satisfying R3, the right-hand side of (2.51) with p, system of consumption shares (P, u, H) := := p defines a [o’(P, u, H),..., a”(P, u, H)]’ € S’ 1 which is continuous in (, u, H) and homogeneous of degree zero in each of p’, ... , p’ andp’. 29 COROLLARY 2.6.1. The ratio of per-household consumption shares for countries i and j is equal to the per-household consumption index for country i relative to country j obtained by using p to deflate the per-household expenditure ratio between the two countries: o4(P, u, H)/H cri(P, U, H)/H, Cz(p, uj Ci(pi, u,) , p p — U H) =: çb(p , p, u 3 , u, u, H). 2 (2.52) Thus, under R3, any PPP index p and its quantity counterpart consumption shares o• are dual to some system of and its PPP counterpart 8, the jth element of which is defined as the amount by which the actual expenditure share s must be deflated in order to make it equal to c-i. The fmal result of this section shows that the translog PPP index (defined below) is exact for the multiplicative democratic PPP index evaluated at a particular vector of base utility levels when preferences are dual to a translog expenditure function. THEOREM 2.7 [Diewert (1976), p. 122)]. Let C: I’ - ll be a general translog expenditure function defined by N in C(p, v) := o + E /3,jn Pn + /3oln V N ÷ N E m=1,=1 7mnlfl J), In p, N + aE and /3,,E RforallnE )/U{O} =:J, i7mn = VC(pk, Uk). 0 for all m E JI. v + -yoo(ln 1, 7mn (2.53) , 2 v) 7nm€ For all k € K, suppose that for all (m,n)E (uk, uk) € JsxJl, LJ and x Then PMD(P” where E 7o,jnpln n= 1 p, vji,., vj, H) = = PT(P p, x Xi), (2.54) 30 N p$, 2 x x) := II , PT(P” ,?=1 P Vji 3 u) : = (u (2.55) — 2 n is the country-f translog PPP index, w, := and (w+w)/2 px/pk1xk is the flth household-k expenditure share 1/2• An obvious weakness of this result is its dependence on preferences being identical across households — a weakness not shared by the intertemporal analogue as comparison with Diewert (1984, Theorem 9) reveals. A similar asymmetry of outcomes is evident in the relative applicability of certain bounds on the plutocratic and additive democratic indexes: The bounding theorems for these indexes in the intertemporal context 21 have no direct counterparts in the international one. 2.7 Concluding Remarks The present chapter has extended the theory of the cost—of—living index into the realm of multilateral international comparisons. Such comparisons can be made from the viewpoint of an individual or from that of a group. Those that reflect the perspective of a group which includes a representative from each country being compared are called “bloc specific.” Different classes of bloc—specific indexes can be distinguished by the types of comparisons they facilitate. The dual relationships among four of these — two comprising indexes of relative purchasing power and two comprising indexes of real consumption above. — were established Within this framework, the plutocratic PPP index and multilateral analogues to the Konus PPP index and the Allen consumption index were shown to be mutually consistent, and Diewert’s (1986) own—share system of axiomatic quantity indexes was shown to be justifiable. In the next chapter, the theory of bloc—specific indexes is used to provide a solid foundation for the development of an appropriate multilateral test approach. 21 See Diewert (1984, Theorems 5 and 7). 31 CHAPTER 3 A RESTRICTED-DOMAIN MULTILATERAL TEST APPROACH The economic approach to index number theory pursued in Chapter 2 has a number of limitations. First, in deriving empirically useful results, it relies heavily on separability assumptions about the underlying aggregator functions which are unlikely to be correct. The most objectionable of these is the requirement that tastes or technologies be identical or, at the very least, closely related across countries. Second, in some contexts, the key assumption that agents behave optimally in allocating their available resources may be inappropriate. Finally, implementation of the economic approach may require unobservable ex ante expectations about future prices to enable the calculation of rental prices of durable goods. The test (or axiomatic) approach gets around these problems by focusing exclusively on axiomatic indexes; i.e., those based on ex post accounting data which are observable and treated as independent variables. Its ultimate objective is to specify a set of “reasonable” tests (or axioms or requirements) which is sufficient to determine a unique functional form for the index in question. Failing this, the specified tests can provide a basis for assessing the relative merits of alternative formulae motivated outside the test approach framework. For the most part, the existing literature in this field is concerned with bilateral 1. Working under the auspices of the United Nations International Comparison comparisons Project (ICP), Kravis et a!. (1975, p. 54) were the first to develop a set of tests that is applicable in a multilateral context. The latest version of this set was described by Gerardi (1982). Diewert (1986) proposed a more comprehensive system of multilateral tests and then used it to evaluate a number of different methods for making real output comparisons 1 See, for example, Fisher (1927), Voeller (1981), and Eichhorn and Voeller (1983). 32 within a bloc of countries. Balk (1989) used Diewert’s system to evaluate an additional output—comparison formula. In the sections that follow, a new framework for making multilateral international comparisons is developed. The various tests that define this framework are set out in Section 1. Many of these tests can be justified as “reasonable” using the fact that they are direct analogues to properties of the cost—of—living index. Further support for the new approach is provided in Section 2 by showing that it is equivalent to an extended version of Diewert’s (1986) multilateral test approach. Section 3 analyzes a number of alternative multilateral comparison formulae and establishes the relative superiority of two of them. Section 4 offers some concluding remarks. 3.1 Definitions The maintained domain of comparison can be characterized as a bloc of countries K := { 1,..., K) with household populations H := (Hi,..., HK)’ € , a set of consumer goods and services X:= {1,..., N) with country—specific prices P := (p’,...,p”) £ per—household consumption bundles X (x , 1 ..., x’) € DK. and a vector of In this chapter, unlike the preceding one, the underlying preferences which generate X are ignored. Further, the elements of P, X and H are treated as independent variables. From the viewpoint of the typical country—k household, the vectors p , p and 2 (j k e K) constitute the only available information which is relevant to the calculation of the purchasing power parity (PPP) between countries i and j. Prices outside i and j have no bearing on the cost of a commodity bundle in one of these countries relative to the cost of the same bundle in the other. Consumption bundles other than xk are generated by preferences which may be very different from those of the typical country—k household. Thus, it would 33 appear that the best way to make use of the available data in calculating PPPs which are specific to country k is by means of the formula pk defined by pk(p2,p:,X) := (3.1) . If the typical country—k household has preferences which admit very little substitution among the commodity types in ii, or if the various price vectors are not very different from one another, then this index will be approximately exact. The most obvious way to think about PPPs which are relevant to the bloc as a whole is as an aggregate of the K country—specific PPPs. To reflect the democratic principle of “one person, one vote,” the available population data could be used to provide appropriate weights for the different countries in constructing such an aggregate. Following this logic, a bloc—specific PPP index for country i relative to country j is a function p: IEI x +1) defined over (i) the price vectors for the pair of countries being compared, (ii) all of the per— household consumption bundles and (iii) the vector of household populations. Since there are K —2 price vectors which are not arguments of this index (but could be, in principle), p is called a restricted—domain index. Examples of such indexes are presented in Section 3 below. The first and second vector of prices over which p is defmed can be thought of as reference and comparison prices, respectively. Given the results in Chapter 2, it seems reasonable to require that p(p2, pZ, X, H) depend on p’ and p in the same way as the (Konus— type) cost—of—living index pI(pr, pC, u) depends on p’ and pC Accordingly, the first four tests for p encompass the direct analogues to properties P1 —P4. 2 Corresponding to P1 is the requirement that the value of p be a positive number. The motivation for this test, called positivity, comes from the fact that the PPP between any two 2 See Section 1 of Chapter 2. 34 countries is the number of currency units of the first country needed to buy a commodity bundle equivalent to one that can be bought with a single currency unit of the second country. P. Positivity: For all (j, i) € K x K, p(p’, p, X, H)> 0. The analogue to P2, called positive monotonicity, requires that an increase in one or more of the comparison prices cause the value of p to increase or remain the same. M. Positive Monotonicity: p(p’, p, X, H) For all (j, i) € K x K and for all p € llL÷, if p <jY then p(p’, p, X, H). The P3—analogue, linear homogeneity, requires that a common proportional change in all comparison prices cause the same proportional change in the value of p. H. Linear Homogeneity: For all (j, i) € K x K and for all .A € p(p’, Ap, X, H) = Ap(p’, p, X, H). The test for p which corresponds to P4 is called transitivity or, more traditionally, circularity. It requires that the PPP between two countries be equal to the product of the PPP between the first country and any third country and the PPP between the same third country and the second country. T. Transitivity: For all (j, i) € K x K and for all 1 € K, p(pi, pi, X, H)p(p , p, X, H) 1 = p(p’, p, X, H). Seven additional tests for p follow from the preceding four in the same way that properties P6—P12 of the cost—of—living index follow from Pl—P4. The first of these additional tests, called identity, requires the value of p to be unity if the reference and comparison countries are one and the same. 35 I. Identity: For alijE K, p(p’,p’,X,H) = 1. The second implied test for p. called proportionality, asserts that if the result of applying a common proportional change to a country’s prices is compared with its original situation, the value of p is the factor of proportionality. Note that this requirement contains I as a special case. PP. Proportionality: For all j £ K and for all A £ p(p’, Api, X, H) = A. The third implied test, country reversal, asserts that if the reference and comparison countries are switched, the new value of p is the reciprocal of the old. CR. Country Reversal: For all p X, H) , (j, i) £ K x K, p(p$, 2 = l/p(pi, p1, X, H). The fourth implied test, negative monotonicity, is the reference—price counterpart to M. It requires that an increase in one or more of the reference prices cause the value of p to decrease or remain the same. NM. Negative Monotonicity: p(pi, p , X, H) 1 For all (j, i) £ K x K and for all p’ £ if p’ 2 then <p p(p’, p, X, H). The fifth implied test, homogeneity of degree minus one, is the reference—price counterpart to H. It requires that a common proportional change in all reference prices cause the value of p to change by the reciprocal of the factor of proportionality. HDM. Homogeneity of Degree Minus One: p(Ap’, p, X, H) = For all (j, i) E K x K and for all A £ , A’p(p, p , X, H). 1 The sixth implied test, price dimensionality, requires that a common proportional change in all reference and comparison prices have no effect on the value of p. 36 PD. Price Dimensionality: For all (j, i) € K x K and for all A € D, p(Ap , Api, X, H) 2 = ,p(p 2 pz,X,H). The final implication of the four basic tests for p, the mean value test, asserts that the value of p lies between the smallest and the largest price relative p/p’, n MV. Mean Value Test: For all mm 4— , pZ, X, H) 2 p(p t [nj nEA I I max ,ZEN 1-’n THEOREM I. (j, i) € K x K, I i iPi € 4— 3.1. Suppose there exists a function p: If x p also satisfies (i) I; (ii) PP if H holds; (iii) CR; (iv) N+1) - ll satisfying P and T. Then NM if M holds; (v) HDM if H holds; (vi) PD ifH holds; (vii) MV if both H and M hold. The direct analogue to the invariance property of the cost—of—living index with respect to the dimensionality of each price and the position of each commodity in the “general list” is encompassed by a pair of tests. 3 The first of these, called commensurability, requires that a change in the unit of measure of each commodity 4 have no effect on the value of p. C. Commensurability: For all p(p’, t 5 p , .J’X, H) (j, i) € K x K and for all A := (Ai,..., AN)’ = € p(p’, p’, X, H), where A is the N x N diagonal matrix with L = A, for all n € ii. The second part of the invariance analogue is captured by commodity symmetry: a change in the ordering of the items in the general commodity list has no effect on the value of p. See Theorem 2.2. Such a change could include measuring the quantity of beer, say, in litres instead of gallons and the associated prices in currency units per litre instead of currency units per gallon. 37 CS. (j, i) £ K x K and for any permutation of the columns of Commodity Symmetry: For all the N X N identity matrix, denoted by IN, p(INp’, INPS, INX, H) = , pZ, X, H). 3 p(p The nature of the dependence of p on the matrix of per—household quantities and the vector of household populations cannot be established by analogy to properties of the cost—of— living index because neither set of variables is in the domain of this (latter) function. Consequently, from a theoretical economic standpoint, no pertinent test for p can be considered to be as desirable as those discussed above. From certain applied standpoints, however, this conclusion may not hold. Political or other non—economic considerations could lead to the prioritization of a particular requirement for p which is not grounded in the economic approach. One such requirement, weight symmetry, precludes the possibility that any country’s total consumption bundle (or weight) plays a special role in the determination of p. WS. (j, i) £ K x K and for any permutation of the columns of the Weight Symmetry: For all K x K identity matrix, denoted by p(p’, pZ, XIK, IK’H) = K, p(pi, pZ, X, H). Another “ungrounded” test lives up to the name population irrelevance by granting equal treatment to every country, regardless of size. P1. Population Irrelevance: , 2 p(p For all (j, i) £ K x K and for all H £ O, p(p’, pZ, X, R) = p, X, H). An obvious counterpart to the price dimensionality axiom discussed earlier, quantity dimensionality requires that a common proportional change in all per—household quantities 38 together with a possibly different proportional change in all household populations have no effect on the value of p. QD. Quantity Dimensionality: For all p(p.1, p, fiX, H 7 ) = (j, 1) E K x K and for all (fi, ‘y) E p(p’, p:, X, H). A stronger version of this requirement, strong quantity dimensionality, states that a common proportional change in the per—household quantities of any country has no effect on the value of p. SQD. Strong Quantity Dimensionality: For all , 1 p(p’, p, x ... , , 1 x 1 Axt, t x ’ , ... , (j, I) E K x K, for all 1 € K and for all A € ItH, , H) 1 x = , p, X, H) 2 p(p The importance of the distinction between total and per—household quantities implicit in the definition of p is assessed by the total quantities test. It demands that a change in per— household quantities and populations which is such that all total quantities remain the same have no effect on the value of p. TQ. Total Quantities Test: For all p(p’, p’, XH, iK) = (j, i) € K x K, p(p’, p, X, H), where H is the K x K diagonal matrix with Hkk = Hk for all k £ K and ‘K is the K—dimensional column vector of ones. Bilateral versions of the following test have been proposed by several authors, beginning with Fisher (1911). D. Determinateness: If any scalar argument in p tends to zero then the value of p tends to a unique positive real number. 39 Opinions on the desirability of this requirement are usually expressed in a categorically unequivocal manner. At one extreme is Frisch (1930, p. 405) who “feel[s] a great repugnance against any index which does not satisfy the determinateness test.” He justifies his position on practical grounds by adding that “...the withdrawal or entry of any [new] commodity will often have to be performed as a limiting case when either the quantity ... or the money value decreases toward zero, respectively increases from zero.” At the other extreme are Samuelson and Swamy (1974, p. 572) who consider the determinateness test to be “... odd ... and not at all desirable ... [because] it rules out the non—satiation assumptions often made in standard economic theory” thereby making it impossible for households to derive infinite utility when one or more prices vanish. Next, three tests are considered which require that the set of PPPs change in a consistent manner as the size of the bloc changes; i.e., they require consistency—in— aggregation. First up is the country partitioning test. It says that if some country 1 e K is partitioned into two new countries, each with the same per—household consumption bundle x , t then none of the PPPs among the rest of the countries are affected. If, in addition, the two new countries have the same price vector p . then each inherits the PPPs of the original 1 country 1. CP. Country Partitioning Test: For all 1 E K and for all A E (0, 1), ,H 1 •(p’, p, X, x , 1 ...., , (1—A)H — 1 H ,H 1 , ... HK, AH + 1 ) 1 , p(pi,p,X,H) if(j, i) E (K\{l}) x (K \{l}) ,p(p t p,X,H) if(j,i)€ {l,K+ 1} x (K “p1) p(p’,p , 1 X,H) if(j, I) € (K\{l}) x {l, K+ 11 1 , t p(p , X,H) p if(j,i)€ {l,K+ 1) x {l, K+ 1} under the addi— tional assumption that K+1 . 1 p A stronger version of this requirement is the strong country partitioning test. It says that if some country 1 € K is partitioned into two new countries, each with a per—household 40 consumption bundle which is possibly different from that of the other, then none of the PPPs among the rest of the countries are affected, if, in addition, the two new countries have the same price vector p . then each inherits the PPPs of the original country 1. 1 SCP. Strong Country Partitioning Test: For all 1 € K and for all (x’, 4 x A) € , such that (1A) 1 + Ax’’ 1 ,...., x 1 1 (p’, p ,x . = x t 1 x (0, 1) , t x ... , xt’, x+l, Hi, ... , , llt+i, 1 lit_i, (l—A)H ... , ) 1 Ilic, All p(p,p,X,H) if(j, i) € (K\{l}) x (K \{l}) p(p’, p$, X, H) if (j, i) € { 1, K + 1) x (K \{ 1)) p(p’,p , t X,H) if(j, i) € (K\{l}) x { 1, under the addi — tional assumption 1 tht a P K+ 11 p(p’,p , t X,H) if(j, i) € {l,K+ 1} x {l, K + 1} — The third consistency—in—aggregation requirement, tiny country irrelevance, states that if the population of some country 1 € K tends to zero, the PPPs among the remaining countries tend to those that would prevail if the bloc excluded country I altogether. TCI. Tiny Country Irrelevance: For all 1 E K, for all (j, i) € (K\{l}) x (K\{l}) and for all A € 1 im p(p’, p, X, Hi,..., H , All _ 1 ,1 1 H , + ..., HE) )- 0 = ,H 1 ). 1 p, X_ The next axiom is called the product test because it asks that the product of the values of p and a bloc—specific per—household consumption index b: II x N+K(N+i) -* D be equal to the corresponding per—household expenditure ratio. PT. Product Test: For all (j, i) € K x K, p(p’, p, X, H)(p’, p, x , xz, X, H) 2 = . (3.2) 41 Note that once a functional form is established for p. q can be defined implicitly by equation (3.2). Tn this case, PT is a tautology. The final axiom considered in this section is a strengthened version of PT. Factor reversal says that for any bilateral intra—bloc price level comparison given by : if x (N+1) - III, if the roles of prices and per—household quantities are reversed, the result can be regarded as the corresponding per—household consumption index. FR. FactorReversal: For any sub—bloc X K, )? = 2, and for all , x, H 3 , H)(x’, x, p’, p, H,, H) 3 (p’, p, x Note that FR is not a truly multilateral test since , (j, i) £ x = unlike p, is not defined over all per— household quantities and populations. Tn bilateral contexts, the validity of this requirement has occasionally come into question during the past seventy years because of its lack of intuitive appeal. This is unfortunate because, as the following theorem demonstrates, FR is of critical importance in establishing the axiomatic characterization of bilateral PPP indexes. THEOREM 3.2 [Funke and Voeller (1978)]. satisfies CR, FR, WS and P1 x 2(N+1) f and only if is the country-j Fisher PPP index; i.e., , xz, H, H) 2 p(p’, p, x 3.2 The bilateral PPP index i1: ‘x :‘x: /2 = =: , xz). 2 ’ p, x 2 PF(P (3.3) Consumption-Share Equivalence The focus of this section is the translation of Diewert’s (1986) multilateral test approach into the maintained domain of comparison. Following a detailed review and extension of the associated set of tests, a subset therefrom is shown to be equivalent to a subset 42 of the restricted—domain tests developed in the preceding section. This result serves to enhance the validity and usefulness of both approaches. In order to make it compatible with the test framework established above, Diewert’s multilateral system of output indexes is treated as a system of bloc—specific (real) consumption indexes. Any such system is characterized by a function x 0: (N+1) K defined over (i) all of the price vectors, (ii) all of the per—household consumption bundles and (iii) the vector of household populations. cr(P, X, H) := [g’(P, X, H),..., The 0 K (p, jth element (i € K) of the associated image vector X, H)]’ is to be interpreted as country i’s share of total bloc consumption. Desirable properties for a-, called share tests, are denoted by Si, S2, etc. The first such property is the fundamental share test — so named because it is essential to the interpretation of o- as a system of consumption shares. Si. Fundamental Share Test: a(P, X, H) > 0 for all i € K and E 1 a-’(P, X, H) = 1. The next share test is called weak proportionality. It says that if all of the price vectors are proportional to one another, all of the per—household quantity vectors are proportional to one another and all of the household populations are equal to one another, then country i’s share of total bloc consumption is equal to its (common) share in consumption of every item in the general commodity list. S2. Weak Proportionality: For all i € K, for all / E K, for all ‘y € 11, for all c:=(c1,...,aK)’ € D+andforall/3:= (fl1.,...,flK’E a(aip , 1 ..., KP ,...., 1 , flux 1 , 1 /3KX 7,...., 7) = +suchthat=l?9k= 1, f j 3 . A stronger version of this requirement is called proportionality. It says that if any country’s per—household quantity vector is multiplied by a positive scalar, then the ratio of the same country’s consumption share to that of any other country is equal to the original (pre— 43 multiplication) consumption—share ratio times the scalar; all other consumption—share ratios remain the same. S3. Proportionality: For all 1 E K and for all A € , 1 o(P, x — — ... , , 1 x 1 Ax , 1 ... , x’, H) o(P,X,H) {} 1+ (A—1)cri(P, X, H) 1•f•1€ K\l (P,X,H) 1 A ‘ 1+(A—1)ci’(P, X, H) if—l The fourth property, called the monetary unit test, states that multiplying each price vector, the matrix of per—household quantities and the population vector by (possibly different) positive scalars has no effect on the consumption share of any country. S4. Monetary Unit Test: For all i E K, for all a : = (ai,..., aK)’ € ÷ and for all (p3, 7) € aip , 1 ..., aKp’, fiX, 71]) = u(P, X, H). The fifth share test, commensurability, requires the consumption shares to be invariant to changes in the units of measure of commodities. S5. Commensurability: For all i € K and for all A := (Ai,..., AN)’ € i(p )JX, H) = Wj+, i(p X, H), where A is the N x N diagonal matrix with A, = A for all n € I. The sixth test is called country symmetry because it requires that quantities of every country in the same manner. i treat the prices and 44 S6. Country Symmetry: For any permutation of the columns of the K x K identity matrix, denoted by ‘K, 0(PIK,XIK, IK’H) = IK’J(P,X,H). The preceding axiom makes the names of countries irrelevant to the determination of consumption shares. Commodity symmetry does the same for commodity names. S7. Commodity Symmetry: For all i E K and for any permutation of the columns of the N x N identity matrix, denoted by IN, c?(INP, INX, H) o(P, X, H). All three of the following tests for 0 are consistency—in—aggregation requirements. The country partitioning test says that if some country 1 € K is partitioned into two new countries, each with the same per—household consumption bundle x’ and the same price vector , then none of the consumption shares among the rest of the countries are affected and the 1 p consumption—share ratio between the two new countries is equal to the corresponding population ratio. S8. Country Partitioning Test: For all 1 E K and for all A E (0, 1), i(p , 1 pi, X, x’, H ... , , (l—A)H _ 1 H ,1 1 H,÷ , ..., HK, All ) 1 o (P, X, H) = if 1 € K \{l} (l—A)o’(P, X,.H) if i= 1 Ao ( 1 P, X, H) if i=K+ 1 The second consistency—in--aggregation requirement for a-, tiny country irrelevance, states that if the population of some country 1 € K tends to zero, the consumption shares among 45 the remaining countries tend to those that would prevail if the bloc excluded country I altogether. S9. Tiny Country Irrelevance: For all 1€ K, for all i € K\{l} and for all A £ urn o(P, X, hi, -) ... 0 , , + 1 I-li—i, Ahlj, H ... , hhic) = —(p_i, X_ , H_t) 1 The last of the multilateral tests devised by Diewert (1986) is called strong dependence on a bilateral formula. Arguably the least compelling of the consistency—in—aggregation requirements, it asks that the consumption—share ratio between any two countries tend to the value given by some bilateral total—consumption index—number formula as the number of households in the rest of the bloc shrinks to zero. SlO. Strong Dependence on a Bilateral Formula: For all j E K, for all i £ K\{j} and for all A€ ++, 1 there exists a function : .0 i(p, x DN - U such that X, 1 All , 7 , ?J1 H , 1 _ , , 7 -- , ?JIi_i, his, 1 1 .Xh1 AhI+ , Allic) , ai(P, X, All ,..., AH_ 1 , H,, AHj+i,..., AH_ 1 , H, AH+ 1 ,..., AHx) 1 ... = ... ... q(p , 1 Hx’,Hx). Each of the next five tests is original. The first, monotonicity, says that if one or more of the prices in some country are increased, ceteris pan bus, then the percentage changes in the consumption shares of the other countries are at least as large as the difference between the percentage change in the inflated country’s consumption share and the percentage change in its per—household expenditure all divided by the latter percentage change incremented by one. Si 1. Monotonicity: For all — where (j, I) e K x K and for all p £ , if p < then sioijjp ainipudxo s&Iwnoo UO jj :sioJjp amiipudx jo CXICdH 1 X ,çd H — — sw.i an Os Uq ‘sioqiou si sa,ud oi &uu jo ‘ ‘d ‘d 1 ‘ _ :d 3uW1mbaL ‘ sq an soid ‘oq s ‘ d 1 )D &iipuodsiio oij zis ‘1) oti arnqs—uotdwnsuoo ji pq s.iss ‘ipou ‘Cmuapi )•z37/dun ‘si j UOI JO XpW s ‘ gums u prns—ai q u iuwaiinbai +:d ‘d ‘ 1 _ed ‘“‘ 1 d)c.o 1 (H ‘X ‘d ‘“‘ (H ‘X ‘Hd ‘Y x k’ 3 (! qi o jnb s w ijmp 02 snoojiuu (H ‘X ‘d (vs) ddd poqdx :Cmuapj ,n3iduq UO o amqs rn; jnb am suunoo puo3s MU qj iioiuoouow (Asod) sTq3 ‘sxpui ddd 2injdwi am siojjp ampudxo aui ‘1+: d ‘d ‘i:d (H ‘X ‘J) r 0 ‘ 5 xd n I:- :tus uiwai jo saioui 02 iowjjop am samou u qi 2uww 1pq2 usn pudx 521 sosno ! &uuno3 jo soud q3 jo amow io q2 uo UI s pidiw q uo £jnbu uTpaaid qi ‘suo2up A!pdsaI pu .q io; uminiusqns woij qM 2nsa1 uui.mai sw.i iC .o saiidwi [[S UV is .c _.: pU — — (FI’x’a):.o d 1 1 ‘, ‘“‘ ÷ d ‘x (H ‘d 1 _:d ‘“‘ 1 d)D 1 ‘“‘ + d ‘:d ‘ ‘x (H —. (iI’x’d)c.o ‘4 ‘ _ 1 ed 1 ‘“‘ d ).o —. 9t 47 The third new requirement for is the total quantities test. It asks that a change in per—household quantities and populations which is such that all total quantities remain the same have no effect on the consumption share of any country. S13. Total Quantities Test: For all I e K, ci(P, xfi, lK) = o(P, X, H), where Ii is the K x K diagonal matrix with fIkk = IIk for all k € K and ‘K is the K—dimensional colunm vector of ones. A strengthened version of S8, the strong country partitioning test says that if some country 1 E K is partitioned into two new countries, each with the same price vector p’ but possibly different per—household consumption bundles, then none of the consumption shares among the rest of the countries are affected and the consumption—share ratio between the two new countries is equal to the corresponding total—expenditure ratio. S14. Strong Country Partitioning Test: For all 1 € K and for all such that (l—A) + Ax 4 o(P, p , 1 ,x 1 ... A) e x (0, 1) = , x’, , x , , xK, 1 ... hi, ... , Hj—i, o’ (P. X, H) = , 1 (, x (1—A)Ht, hIi+i, ... , HK, Alhi) if i € K \{l} (l—A ’ ’ t ( 1 ] ’/p P x )[p , X, H) if i= 1 A[ ’ / 1 ] p” (P x p x , X, B) if i=K+ 1 The last axiom considered in this chapter is the ratio test. It provides a link between the two multilateral test approaches defined above by requiring that the ratio of any two 48 countries’ restricted—domain total consumption indexes 5 be equal to the corresponding consumption— share ratio. RatioTest: Forall(j,i)€XxXandforallkEX, RT. H ,(pic, 1 3 (pk, H , 1 pi, xk, x pi, xk, x, X, H) X, H) (P, X, H) 1 c o-i(P, X, H) — — (3 5) Using this axiom together with three others, it is possible to derive the precise mathematical relationship between the consumption—share system ci and the restricted—domain PPP index p. LEMMA ci: 3.1. Suppose there exists afunction p: K x satisfying Si. I (P X H — — (N+1) - P, and afunction satisfying N+K(N+1) Define the function : IR x implicitly by equation (3.2) and suppose that 0 x (, o-) satisfies RT. Then I p2’x’ p(pi, p 1 FI , X, H)1’ 1 Hp ’ 1 x p(pi, pi, X, H)J l,=i 36 If, in addition, p satisfies T then ’ 1 p(p’,p,X,H)= H iPXH’ x, H) (3.7) I-I,pi’xici(P, Equation (3.6) enables the derivation of each of the “non—fundamental” share tests (S2—S14) from one or more of the tests for p. LEMMA ci: 3.2. Suppose there exists afunction p: IRi x x jK(N+1) K satisfying Si. satisfies D satisfying Define the function : I} implicitly by equation (3.2) and suppose that satisfies H and HDM, (ii) S3 if p . (çi, a) satisfies x P, and a function N+K(N+1) . RT. Then ci satisfies (i) S2 if p SQD and T, (iii) S4 if p satisfies H, HDM and QD; (iv) 55 if p satisfies C; (v) S6 if p satisfies WS; (vi) S7 if p satisfies CS, (vii) S8 if p satisfies Recall that indexes of this sort can be defined implicitly in terms of a restricted—domain PPP index by equation (3.2). 49 CP; (viii) S9 if p satisfies TCI; (ix) SlO if p satisfies TCI, T and TQ, (x) Si] if p satisfies M; (xi) Si2 if p satisfies T, (xii) S13 if p satisfies TQ; (xiii) 514 if p satisfies SCP. The derivation of each of the tests for p — except P, T, P1, D, PT and FR — from one or more of the share tests is enabled by equation (3.7). 3.3. Suppose there exists a function p: LEMMA function a: x 11 (N-I-i) - ( x (N+1) - fi satisfying P and T, and a satisfying Si. Define the function : II x implicitly by equation (3.2) and suppose that (, N÷K(N+i) a) satisfies RT. Then p satisfies (i) M if a satisfies Si]; (ii) H if a satisfies S4; (iii) C if a satisfies S5; (iv) CS if a satisfies S7; (v) WS if a satisfies S6; (vi) QD if a satisfies 54, (vii) SQD if a satisfies S3; (viii) TQ if a satisfies S13; (ix) CP if a satisfies S8; (x) SCP if a satisfies 514; (xi) TCI if a- satisfies S9. Under the hypothesis that a together with defined implicitly in terms of p satisfies the ratio test, the next theorem establishes the equivalence of Diewert’s (1986) multilateral test approach and that of Section 1 by combining the results presented in Lemmas 3.2 and 3.3. THEOREM 3.4. Suppose there exists a function p: If x function a: X j1 (N+i) -÷ W satisfying (N+1) -‘ O satisfying P and T, and a Si. Define the function : implicitly by equation (3.2) and suppose that (q, a) satisfies RT. Then (i) a- satisfies S3 if and only if p satisfies SQD; (ii) a- satisfies S4 if and only if p satisfies H and QD; (iii) a satisfies S5 if and only if p satisfies C; (iv) a- (v) a satisfies S7 if and only if p satisfies CS; (vi) a satisfies 58 if and only if p satisfies CP, (vii) a satisfies 59 if and only if p satisfies TCI; satisfies S6 if and only if p satisfies WS, x N+K(N+1) 50 satisfies Si] if and only if p satisfies M; (viii) ci (ix) o- satisfies 512,’ (x) ci (xi) a- satisfies S14 satisfies 513 if and only if p satisfies TQ; f and only f p satisfies SCP. By stating that two independently developed test approaches imply one another, this theorem reinforces the “reasonableness” of both. It should be understood, however, that such equivalence holds only for a particular class of PPP indexes and a particular class of consumption—share systems. The next lemma shows that the transitivity axiom restricts the admissible p indexes to ratios of national price levels which are independent of foreign prices. The theorem that follows shows that national expenditures deflated by these price levels and then normalized to sum to unity comprise the class of admissible consumption shares. This restriction on a’ is a direct consequence of the ratio test. LEMMA 3.4 [Eichhorn (1978, pp. 156—157)]. The function p: if and only if,for some 6’: ÷ x ‘ P\P ‘P’ X H’) - — ‘ THEOREM function — IR x (N+1) K 1 - IR satisfies T , X, H) 1 6’(p ô(pi, X, H) 38 x (N+1) - satisfying P and T, and a satisfying Si. Define the function : [R x implicitly by equation (3.2) and suppose that K(N+1) + N+1) ‘‘) 3.5. Suppose there exists a function p: 0-: x pN+K(N+1) (, a-) satisfies Ri’. Then, for some 6: I1 x ++, ci(P, X, H) = H 1 ‘x ö(p,X, H) K {j1 H- i ‘xi 6(pX, H)} (3.9) The practical value of consumption—share equivalence is that it enables the evaluation of indexes of the form (3.8) either directly via the axioms of Section 1 or indirectly via those of the present section. Consequently, any admissible restricted—domain PPP index can be 51 compared with any consumption—share system under the share—test approach. Such comparisons are undertaken at the end of the next section. 3.3 Some Examples There are many different ways in which the available price and quantity data can be aggregated into a bloc—specific index of relative purchasing power. In this section, twelve such alternatives are introduced and evaluated in light of the foregoing pair of test approaches. Patterned after the multiplicative democratic PPP index, 6 the household democratic PPP index for country i relative to country j is defined as the population—share—weighted geometric mean of the K country—specific PPP indexes given by (3.1): K PHD(P’,P,X,H) := 0 k (JJ’) i’xk k=l[pixk] (3.10) , where Oc(H) := Hk/E’f=iHj denotes the country—k bloc population share. By assigning each country—k PPP index a weight which is proportional to the number of households that it represents, PHD affords equal treatment to all households in the bloc. THEOREM 3.6. The household democratic PPP index PHD satisfies all of the restricted-domain tests except P1, TQ, SCP and FR. COROLLARY 3.6.1. The associated system of consumption p := PHD’ satisfies shares, H 0 D defined by (3.6) with all of the share tests except S13 and S14. A weaker democratic aggregation rule would treat countries as equals rather than households. Accordingly, define the country democratic PPP index for country i relative to 6 See Section 6 of Chapter 2. 52 country j as the unweighted geometric mean of the pbs: K ___ PCD(pPXH).=ll[pj1kj THEOREM 3.7. (3. 1) • The country democratic PPP index PCD satisfies all of the restricted-domain tests except CP, SCP and TCI. COROLLARY 3.7.1. The associated system of consumption shares, °CD’ defined by (3.6) with p := PcD satisfies all of the share tests except S8-S1O and S14. Although PCD fails one fewer test than PHD’ the former’s shortcomings can easily be seen to be much worse than the latter’s. If, for example, the size of the bloc is likely to change over time, the benefit of satisfying P1, TQ and FR will be more than offset by the cost of satisfying none of the consistency—in—aggregation requirements. The preceding PPP indexes can be regarded as examples of “external average” formulae. In each case, the per—household country—k basket xk is priced at both p$ and p) for all k € K, and then an average over the resulting K relative costs is calculated. An alternative methodology along similar lines would be to compute an average over the country—k baskets before doing the costing at pZ and p3. Such an “internal average” formula was once used by the United Nations Economic Commission for Latin America (ECLA) for measuring relative purchasing powers among the countries of Central and South America. Specifically, the ECLA or average basket PPP index for country i relative to country j is defined as the ratio of the cost of the bloc per—household consumption bundle in the two countries being compared: PAB(P” pZ, X, H) 1 ’ (XH) pi’X(X, H) (3.12) where X(X, H) : = E 1 U k(H)xk denotes the average household consumption bundle purchased in the bloc. By substituting for x and then 1C 0 using their respective definitions, (3.12) can be 53 re—written as the axiomatic analogue to the (Prais—Pollak) plutocratic PPP index : 6 ElHkpxk ’ p, X, H) 2 PAB(P THEOREM = (3.13) . H p”x 1 E’f t 3.8. The average basket PPP index PAB satisfies all of the restricted-domain tests except P1, SQD and FR. COROLLARY 3.8.1. The associated system of consumption shares, 0 A B’ defined by (3.6) with p := PAB’ satisfies all of the share tests except S3. Since TQ is arguably neither “desirable” nor “undesirable” as a requirement for p, comparison of Theorems 3.6 and 3.8 reveals that the relative merit of PAB and PHD depends on the relative desirability of SCP and SQD. However, due to the fact that weaker versions of these tests hold for both indexes, any preference for one over the other is unlikely to be very intense. Most of the multilateral PPP indexes considered in the literature are not independent of prices outside the countries being compared. In order to accommodate this fact, it is necessary to introduce a class of bloc— specific PPP indexes which is more general than that of Section 1. Accordingly, an unrestricted—domain bloc—specific PPP index for country i relative to country j is a function p’ : iii x 1 - D with image p(P, X, H). For a given system of p is defined by the right—hand side of equation Z consumption shares a, 2 (3.7). Clearly, such an index has a restricted domain if and only if there exists a function such that (, CT) satisfies RT. Kravis (1984, p. 10) pointed out that early multilateral comparison methods were based on bilateral index—number formulae. The simplest and most popular of these methods 54 involved the use of the Laspeyres formula in making binary comparisons between a pre selected base country and each of the other countries in the bloc. The first use of this sort of “star system” 7 was by the British Board of Trade (1908— 1911) in a series of inquiries into the costs of living of workers in the major industrial centres of the United Kingdom, Germany, France, Belgium and the United States. In general, bilateral—formula—based multilateral comparison methods can depend on any index—number formula of the form (p’, p, x , xi). 2 Thus, for a given base country k € K, the country—k star system of consumption shares is defmed by H (pk, p, ol*(P, X, H) := xk, x) (3.14) . EIf=iHiq(pk,pl, XC, x ) 1 Recall that a bilateral PPP index for country I relative to country j is a function : Ifl x N+1) - ll with image (p’, p, x’, equation (3.2) with p := and H, , x) := 2 q(p’, p$, x THEOREM 3.9. Suppose = H = X, H,, He). If satisfies P1 then, using 1, the associated consumption index is defined as x x, 1, 1) , / (p’, p, 2 (3.15) . satisfies P. M, PP, CR, SQD, C, CS, WS and P1. Then the country-k star system k* 0 with q5 defined by (3.15) satisfies all of the share tests except S6, S9, SlO, S12 and S14. Moreover, p defined by the right-hand side of (3.7) with o• := k* 0 is not a restricted-domain PPP index. A second multilateral comparison method based on a bilateral formula is known by the initials of its three independent re—discoverers, Eltetö and Köves (1964) and Szulc (1964). Named for the fact that its graph, constructed by associating nodes with countries and edges with admissible binary comparisons, looks like a star. ‘ 55 The (generalized) EKS system of consumption shares is defined by H ll. 1 [Hk 1 k p: xk x)] 1/K ( 5 , pi, p .H 1 1 ll, [ Hf 2 THEOREM system 514. 3.10. Suppose ciEKS (3.16) xl)]V” satisfies P, M, PP, CR, SQD, C, CS, WS and P1. Then the EKS with q5 defined by (3.15) satisfies all of the share tests except S8-S1O, 812 and Moreover, PAks defined by the right-hand side of (3.7) with ci := OEKS is not a restricted-domain PPP index. A third bilateral—formula—based multilateral comparison method is due to Diewert (1986, P. 25). His own—share system of consumption indexes is defined by H cibs(P, X, H) { E. 1 Hk [(pk, 3.11. Suppose own-share system o• with and 814. XIC, x)]’} (3.17) . H 1 = THEOREM :, [2,pl,x2,xl)]}l satisfies P. M, PP. CR, SQD, C, CS, WS and P1. qi Then the defined by (3.15) satisfies all of the share tests except 83, 812 Moreover, p defined by the right-hand side of (3.7) with ci o• is not a restricted-domain PPP index. The next three multilateral methods are based on weighted averages of the country—k star systems. Respectively, the democratic weights, plutocratic weights and quantity weights consumption—share systems are defined by 1 K cibwP, X, H) a(P, X, H), k=1 (3.18) 8 Tn the version of this index advanced by Eltetô and Köves (1964) and Szulc (1964), the Fisher formula was used in place of q. The more general version stated here is due to Gini (1931, p. 12). 56 K °1’w(, X, H) := E sk(Pe, X, H)o(Pc, X, H) (3.19) cTkos(P, X, H)cx(P, X, H), (3.20) k=1 and 4 w (’ X, H) : k=l where sk(P, X, H) := Hk (ekpk)fxk H 1 E= (3.21) 1 (ep’)’x is country k’s share of (nominal) bloc expenditure, is the K x K diagonal matrix with rates and THEOREM 3.12. Suppose democratic weights system e := (El,..., EK)’ = €i, is a vector of exchange for all k € K. satisfies P, M, PP, CR, SQD, C, CS, WS and P1. DW with c4 defined by (3.14) and q defined by (3.15) Then (i) the satisfies of the share tests except S3, S8-S1O, S12 and S14; (ii) the plutocratic weights system 4. defined by (3.14) and QW with ci defined by (3.17), a defined by defined by (3.15) satisfies all of the share tests except S3, S12 and S14. Moreover, p, defined by the right-hand side of (3.7) with o := right-hand side of (3.7) with r := QW with defined by (3.15) satisfies all of the share tests except S3, S4, S12 and S14; and (lii) the quantity weights system (3.14) and Ip all crp and pjr 0 D W’ , defined by the 1 p3 defined by the right-hand side of (3.7) with are not restricted-domain PPP indexes. Returning now to multilateral methods that are not based on a bilateral formula, two additional procedures deserve consideration. The first is a proposal by Geary (1958) which was later amplified by Khamis (1970)(1972); the second is van Ijzeren’s (1956) weighted balanced method. 57 The Geary—Khamis or GK consumption shares are found by solving the following system of equations: N = x3 1 E ir,,,[H ,= i , = 1,..., K, (3.22a) 1 n= 1,...,N, (3.22b) EiHkx where px/pxz := is the flth country—i per—household expenditure share. Equations (3.22b) define the “international price” of each commodity as the ratio of the per— household expenditure—share—weighted sum of the K consumption shares to the total quantity consumed. Equations (3.22a) define the share of bloc consumption for each country as the cost of its national basket at international prices. The N + K equations (3.22) N K ,=1 not independent since each constituent set implies K x 1 H E are (3.23) = j=1 and, consequently, at least one non—trivial solution exists. Khamis (1970, Section 3) showed that, subject to any normalization on the u s, the system consisting of equations (3.22) has a unique positive solution. solution is denoted by THEoREM and S14. OGK(P, any N+ K — Under the normalization E crj 1 of the = 1, this X, H) := [K(P, X, H),..., cK(P, X, H)]’. 3.13. The Geaiy-Khamis system T G K satisfies all of the share tests except 33, 312 Moreover, p defined by the right-hand side of (3.7) with a- := a-GK is not a restricted-domain PPP index. The consumption shares associated with van Ijzeren’s weighted balanced method are found by solving the following system of equations: 58 E pilxkJJjgj ak k p ‘x’ H E — Ok — k pk’xiJ-J ak pk ‘xk Hk Ui 1 ‘ K 324 — where ak is the country—k “weighting coefficient.” If i ’(Hix 1 p ) /o-i, ... , PKI(HKxK)Io.K are called “equivalents,” the left—hand side of (3.24) is the number of equivalents that would be required to buy, in country i, the quantities in the weighted national baskets that can be bought for one equivalent in countries 1,..., i — 1, i + 1,..., K. The right—hand side is the number of equivalents that would be required to buy, in each of countries 1,..., i — 1, 1+ 1,..., K, the weighted quantities purchased in country i for one equivalent. The balanced method asserts that, for i = 1,..., K, these two quantities of money are equal. Van Ijzeren (1956, pp. 25—27) showed that, subject to any normalization on the us, the system consisting of any K normalization 1 E . o ‘$H(’ = — 1 of equations (3.24) has a unique positive solution. Under the 1, this solution is denoted by vH(P, X, H) := 4 [ H (P, X, H),..., X, H)]’ if a := Hk and UVQ(P, X, H) := [c4Q(P, X, H),..., o5Q(P, X, H)]’ if c := Ok. The former weighting scheme originates with van Ijzeren (1956, p. 4); the latter with van Ijzeren (1983, p. 45). THEOREM 3.14. The population-weighted van Ijzeren system except S12-S14; the quantity-weighted van Ijzeren system except S3, S8-S1O and S12-S14. Moreover, := VH p4r VH satisfies C V T Q all of the share tests satisfies all of the share tests defined by the right-hand side of (3.7) with and p4 defined by the right-hand side of (3.7) with ci CTVQ are not restricted-domain PPP indexes. One multilateral comparison method is said to “dominate” another if, in addition to satisfying every potentially desirable share test satisfied by the second method, the first method satisfies at least one other such test. Using this criterion in conjunction with the three corollaries and the final six theorems of this section, a merit—based hierarchy can be 59 established among the associated methods. Since the EKS system satisfies S3 in addition to satisfying every share test satisfied by the democratic weights system, the former method dominates the latter. Due to the fact that the total quantities test Si 3 is value— neutral, the democratic weights method neither dominates nor is dominated by van Ijzeren’s quantity— weighted balanced method. Consequently, since neither of these methods satisfies S8—S 10, both are dominated by the GK, own—share and quantity weights methods in addition to being dominated by the EKS method. By virtue of satisfying S4, the GK, own—share and quantity weights methods dominate the plutocratic weights method as well. In turn, these methods are dominated by the average basket method which satisfies two further tests (S12 and S14). By virtue of satisfying S12, the country democratic method dominates the EKS method. Since S13 is value—neutral, van Ijzeren’s population—weighted balanced method dominates the EKS method (by S8—S10), the k—star method (by S6, S9 and Sb), and the GK, own—share and quantity weights methods (by S3). Similarly, the household democratic method dominates the country democratic method (by S8—S10) and the population—weighted balanced method (by S 12). Thus, only the average basket and household democratic methods are undominated. The hierarchy of multilateral comparison formulae is illustrated by Figure 3.1. Therein, the twelve methods under consideration are grouped in boxes according to the tests they satisfy: Methods satisfying the same tests are contained in the same box; methods satisfying different tests are contained in different boxes. These boxes are arranged so that the vertical distance between any pair of them is proportional to the difference in the number of tests satisfied by the methods inside. The higher up a given method is in the diagram, the more tests it satisfies. The dominance of one method over another is represented by a straight line connecting the boxes that hold them. Each of these lines is labelled with the names of the tests that are satisfied by the methods in the higher box but not by the methods in the lower one. 60 It is significant that the best multilateral methods from the test—theoretic perspective are those which have interpretations that are firmly rooted in the theory of group cost—of— living indexes. Even methods which are justifiable via an exact index—number argument are dominated by these restricted—domain formulae. justification are the own—share system — The two methods which have such a shown in Chapter 2 to be a direct approximation for the system of multilateral—Allen consumption indexes when based on a bilateral axiomatic per—household consumption index which is exact for a positively linearly homogeneous utility function — and the quantity—weighted balanced method — shown by Diewert (1995) to be exact for homogeneous quadratic utility functions. 3.4 Concluding Remarks The novel feature of the test approach developed in the early part of this chapter is the imposition of an economically sensible restriction on the price domain of admissible PPP indexes. Consequently, most of the multilateral comparison methods proposed in the literature are summarily ruled out. That this should be the case is reinforced by the fact that, under an extended version of Diewert’s (1986) test approach, the best methods are those associated with a restricted—domain PPP index. Kravis et a!. (1975, p. 66) stated that “[e]conomic theory gives no explicit procedure for ... [determining PPPs] in the the sense of providing a specific computing algorithm.” The present chapter in conjunction with the preceding one have demonstrated that this is not so. What remains to be established is whether or not the choice of one method over another is important from an empirical standpoint. Do different methods yield substantially different PPPs for a given bloc of countries? The next chapter endeavours to answer this question. C -n Cl) c) 0 Cl) cp C’) CA) Cl) 0 Cl) C,) Cl) F’) C,) F’, Cl) 62 CHAPTER 4 THE IMPACT OF ALTERNATIVE FORMULAE lii Chapter 3, twelve different methods for aggregating microeconomic price and quantity data into a bloc—specific index of relative household purchasing power were evaluated in light of a novel test approach. It was determined that ten of these methods are “dominated” by the other two since the tests satisfied by of any one of the former comprise a proper subset of the tests satisfied by one or both of the latter. What remains to be shown is that such dominance matters: that the choice of one formula over another can have a substantial impact on the resulting international comparisons. The question of how to compare multilateral purchasing—power—parity formulae from an empirical standpoint has received scant attention in the literature. If the formulae under consideration satisfy a certain minimal set of requirements, then the application of any one of them to a bloc consisting of K countries yields a vector of K — 1 numbers which can serve as a basis for all possible binary comparisons within the bloc. The universal means by which two such vectors have been compared in the past has been an assessment of the component—wise percentage differences between them. 1 This approach is unsatisfactory for a couple of reasons. First, the percentage difference between two numbers is an asymmetric indicator of the relative difference between them because it depends on which number is used as the point of comparison. To paraphrase an example from Tornqvist et at. (1985), 250 is twenty—five percent more than 200, or 200 is twenty percent less than 250. Second, component—wise comparisons between two vectors are unlikely to give rise to a very accurate assessment of the overall difference between them unless the components are few in number or the calculated differences exhibit little variation in size. 1 See, for example, Kravis et at. (1975, ch. 1 and 5) and Ruggles (1967). 63 Section 1 proposes a new index of the difference between the results of two multilateral comparison methods applied to the same data set. Based on the symmetric and additive log(arithmic) difference indicator, this index overcomes the problems mentioned above to provide an appropriate summary measure of the differences between the purchasing power parities (PPPs) or consumption shares associated with the two methods. Section 2 describes the data used in Section 3 to undertake an empirical comparison of the methods compared on theoretical grounds in the preceding chapter. Section 4 concludes by explaining why different sources disagree on the values of the same PPPs. 4.1 A Summary Measure of the Differences between Alternative Formulae Recall that the maintained domain of comparison consists of a bloc of countries K { 1,..., K} with household populations services H = (Hi,..., HK)’ £ Q÷, a set of consumer goods and N := { 1,..., N} with country—specific prices per—household consumption bundles X := P := (p , 1 (x , 1 ..., xK) £ WNK. , ... p ’ 1 ) £ iiI and a vector of As in Chapter 3, the elements of P. X and H are treated as independent variables. An unrestricted—domain (axiomatic) bloc—specific PPP index for country i relative to country j is a function p: [I x N+i) - IR with image p’(P, X, H). It is assumed that, at the very least, this index is positive and transitive with respect to j and i. The positivity requirement enables the usual interpretation of p”(P, X, H) as the number of country—i currency units needed to buy a commodity bundle equivalent to one that can be bought with a single country—f currency unit. P. Positivity: For all (j, i) £ K x K, p-”(P, X, H)> 0. The transitivity requirement guarantees that the results of applying p” to a bloc comprising three or more countries are self—consistent. 64 T. Transitivity: Forall(j,i)EKxKandforalllEK, pfi(p, X, H)p(P, X, H) = pi(P, X, H). In addition to satisfying T, a self—consistent set of PPPs has two further properties. The first, called weak identity, requires the value of p1 to be unity when i WI. Weak Identity: For all j The second, CR. country € K, p”(P, X, H) j. 1. 2 is the reciprocal of the value reversal, asserts that the value of p of p. Country Reversal: For all (j, i) THEOREM 4.1. If p” = = satisfies € (P, X, H) 2 K x K, pz l/p z 2 (P, X, H). P and T then it also satisfies WI and CR. Consider two sets of bloc—specific PPPs, A and B, each computed using a different multilateral index—number formula satisfying P and T. For ease of exposition, let the K—dimensional square matrices (p’ ) and (p) represent the elements of A and B, respectively. 4 Since P together with T implies WI so that p’ = p’ = 1 for all j € K, there are up to K 2 — K possible differences between these matrices. Define the mean absolute log difference between (pj) and (p) as the sum of the absolute log differences between corresponding off—diagonal elements divided by their number: K E j=1 i*j in (p/p3) I LA,B= . K(K- 1) Since both A and B are transitive, for any h € K, (4.1) can be re—written as K E j=1 I.A B E i*j Iln[(pp)/(pp)]I = K(K- 1) (4.1) 65 = 2 K’K— 1) K—i K E E hi PB in in — j=li=j+1 hj PB — , by CR. (4.2) f3 pj’ For a particular bloc of countries, let P := {A, B, ... } denote the set of all sets of PPPs which satisfy P and T. THEOREM 4.2. defined by the right-hand side of (4.2) is a metric on 1; i.e., valued function on P x P which satisfies (i) (iii) A,B = B,A; and (iv) 1A,B &t,c + IA,B 1c,B 0; (ii) ZA,B = s.,. is a real- 0 if and only if A = B; (triangle inequality). Thus, z possesses the most important properties of ordinary distance in O making it a reasonable and intuitive measure of the difference between alternative sets of PPPs. Table 4.1 contains PPPs calculated by the Organization for Economic Cooperation and Development (OECD) using both the Eltetö—KUves—Szulc (EKS) method and the Geary— Khamis (GK) 2 method. For comparison, the corresponding exchange rates are also included. The differences among these three sets of numbers can be summarized by computing the values using equation (4.2): associated 1GK,ER = 1EKS,GK 0.04773, EKS,ER = 0.28832 and 0.30469. Thus, the OECD PPPs differ from one another by about 4.8 percent and from the corresponding exchange rates by roughly thirty percent. A system of bloc—specific (real) consumption indexes for countries 1,...., K is a function a: enable the jth x pI(N+1) - ll with image o(P, X, H) := [aL(P, X, H),..., a’(P, X, H)]’. To element (i E K) of this system to be interpreted as country i’s share of total bloc consumption, a is required to satisfy Si. 2 Fundamental Share Test: a(P, X, H)> 0 for all i E K and 1 E a (P, X, H) See Section 3 of Chapter 3. = 1. 66 THEOREM 4.3. If o satisfies Si then pZ defined implicitly by :(P p”(P, X, H) i(p’ X H) H) : = H ‘x 1 H;.pi’xi (4.3) satisfies P and T, and o(P, X, H) = { j=1 H P’,x’ p2i(P, X, H)} $P X (4.4) . p Z (P, X, H) is the amount by which Under the assumptions of this theorem, the number 2 the total bloc expenditure of country—i households relative to those of country j must be deflated in order to make it equal to the corresponding total consumption ratio. Substituting for p’ in (4.2) using (4.3) yields an equivalent expression for the mean absolute log difference between multilateral comparison methods A and B: &4,B Thus, ZA,B 2 K’K— ‘. K-1K I E E in — j1j+1 — in — . (4.5) 4 uf can be calculated from associated basis sets of PPPs using (4.2) or from the associated consumption—share systems using (4.5). 4.2 The Data The raw price and expenditure data used in the empirical work of the next section are those of the Eurostat—OECD PPP Programme. These data cover the bloc comprising the twenty—four OECD countries of 1990 and the general commodity list made up of the 158 basic headings 3 of the major aggregate called “Final Consumption of Resident Households.” In principle, a basic heading consists of a small group of similar well—defined goods or services. In practice, it is the lowest level of classification for which expenditures can be estimated. Consequently, an actual basic heading can cover a broader range of commodities than is theoretically desirable. 67 Let V := (v) denote the (158 x 24) matrix of national expenditures (in national currency units) at the basic heading level, and let P := (j5) denote the corresponding matrix of basic— heading PPPs in national currency units per U.S. dollar. Hence, for all n E A and for all k E K, v Hkpx (4.6) and — (47) Several different sources were employed in the determination of the 1990 household population data (H) presented in Table 4.2. For the United States and Japan, Turkey, and each of the Nordic countries excluding Iceland, 4 the corresponding datum was furnished by, respectively, the United Nations (1993, Table 3), the State Institute of Statistics (1993, 92), p. and the Nordic Statistical Secretariat (1994, p. 168). The raw data used to form estimates of Hk for Austria, Switzerland and Iceland were provided by the United Nations (1993, Table 3) and the OECD (1993a, pp. 156— 157) (1994, p. 210) whereas those used in the cases of Portugal and the Netherlands were provided by Eurostat (1993, pp. 121, 137) and the OECD (1994, p. 210). Eurostat (1993, pp. 121, 137), OECD (1993a, p. 157) (1994, p. 210) and United Nations (1993, Table 3) afforded a basis for the interpolation of Hk in each of the remaining countries of the European Community. 5 Similar estimations for Canada, Australia and New Zealand were carried out using data from Statistics Canada (1992, p. 125), the Australian Bureau of Statistics (1993, p. 120), the United Nations (1993, Table 3) and the OECD (1993a, p. 157) (1994, p. 210). Viz., Denmark, Finland, Norway and Sweden. Viz., Belgium, France, Germany, Greece, Ireland, Italy, Luxembourg, Spain and the United Kingdom. £ 68 For any scale factor /3 € Qt, a matrix X () of scaled per—household quantities consistent with (V, P, H) is defined by (3 v’ (4.8) PHk E /3pUSk,, by (4.6) and (4.7). (4.9) In the present chapter, a /3—value of 100,000 was used in the construction of 4.3 . Empirical Results The calculation of PPPs based on the data set (P. , H) can only be accomplished by means of formulae which, in addition to satisfying P and T, satisfy two other axioms. The first of these, called quantity dimensionality with respect to X, requires that a common proportional change in all per—household quantities have no effect on the value of p’s. QDX. Quantity Dimensionality with Respect to X: For all (j, I) p’(P, /3X, H) = E K x K and for all /3 € p(P, X, H). The second, commensurability, requires that a change in the unit of measure of each commodity have no effect on the value of pYZ• C. Commensurability: For all (j, i) p(AP, 1 5. X , H) = € K x K and for all A := (Ai,..., p ( 3 P, X, H), where A is the N x N diagonal matrix with THEOREM 4.4. p”(P, , H) = AK)’ € p(P, X, H) 5 = ) for all n f and only fp’ € 1. satisfies QDX and C. 69 Similarly, the calculation of consumption shares based on the data set (P, Z H) can only be facilitated by formulae which satisfy Si and two other axioms. The first of these, called the monetary unit test with respect to X, states that multiplying the matrix of per— household quantities by a positive scalar has no effect on the consumption share of any country. S4X. Monetary Unit Test with Respect to X: For all i E K and for all /9 € c(P, fiX, H) = o(P, X, H). The second, share commensurability, requires the consumption shares to be invariant to changes in the units of measure of commodities. S5. Share Commensurability: For all i € K and for all A := (Ai,..., AK)’ € uz(AP, 1 5J X , H) THEOREM 4.5. (0 P, Z H) = = a(P, X, H). o(P, X, H) f and only if o satisfies 54X and S5. All twelve of the multilateral comparison methods examined in Chapter 3 satisfy Si, 54X and S5 or, equivalently, by Theorem 4.3, P, T, QDX and C. Consequently, the computation of consumption shares was a straightforward exercise involving simple substitutions into the various formulae of Section 3.3. Table 4.3 contains a selection of the results of this exercise. included are the three restricted—domain methods democratic (HD), the average basket (AB) and the country democratic (CD) unrestricted—domain methods not based on a bilateral formula weighted van Ijzeren (VH) — — — — the household two of the three the GK and the population— and three of the six bilateral—formula—based methods EKS, the own—share (OS) and the k—star with k := US (US*). consumption index bF defined by — the The Fisher “ideal” 70 , x) := 2 F(P” p’, x i’x 1! i2 (4.10) [‘X was used as the basis for each of the bilateral—formula—based methods. Each of the other unrestricted—domain methods was calculated iteratively using the household—democratic consumption shares as initial values. The mean absolute log differences among the eight methods of Table 4.3 and the exchange—rate approach (ER). 6 are expressed as percentages in Table 4.4. If the cutoff between “substantial” and “insubstantial” is set at two percent, this table partitions the considered methods into five groups based on whether or not they are substantially different from one another. 7 HD, AB and CD are grouped together since all of the differences among them lie below the cutoff while all of the differences involving just one of them lie above. Similarly, VII, EKS and OS 8 form a group 9 as do each of GK, US* and ER. Thus, the choice of one method over another can have a substantial impact on international comparisons of consumption. More importantly, this is true of the theoretically justifiable restricted— domain methods (HD and AB) relative to those of the unrestricted—domain variety. Table 4.5 presents eight per—household consumption indexes derived from the results in Table 4.3 using the household population data in Table 4.2. Each of these indexes measures the consumption of the average household in each OECD country as a percentage of that in the 6 The exchange—rate—based consumption—share system was calculated by substituting the country—i exchange rate with respect to country j for piz(P, X, H) in equation (4.4). To get a feel for what this means, consider a hypothetical international project to be financed by reference to 1990 OECD consumption shares. Using the own—share system instead of the USK system (100 us*, os 2) would change the average national contribution by 1.58 percent. For some countries, however, this switch in methods would change their contribution by as much as 3.99 percent. In an era of government fiscal restraint, four dollars per hundred can easily be viewed as a substantial difference. Since it is so close to two, 100. US*, Os = 1.992 is treated as a substantial difference. An extended version of Table 4.4 would show that the quantity—weighted van Ijzeren, democratic weights, plutocratic weights and quantity weights methods also belong to this group. 71 United States. Figure 4.1 is a graphical representation of selected results in Table 4.5 along with those of the exchange—rate approach. Therein, the relevant countries’ 0 are arranged from left to right along the horizontal axis in order of decreasing per—household consumption calculated via the household democratic method. Constructed in the same manner as Figure 4.1, the next three figures serve to illustrate the “difference partition” established above. The close proximity of the per—household consumption lines in each of Figures 4.2 and 4.3 conveys the similarity of outcomes generated by methods belonging to the same group. By contrast, the relative separation of the corresponding lines in Figure 4.4 conveys the dissimilarity between methods belonging to different groups. In Table 4.6, each entry is a PPP associated with the corresponding consumption share in Table 4.3 by means of equation (4.3). Comparison of Tables 4.6 and 4.1 reveals small differences between the two sets of GK and EKS PPPs.’ 1 For both methods, these differences have arisen because the definition of private final consumption expenditure employed herein excludes expenditures by private non—profit institutions serving households whereas that of the OECD does not. For the EKS method, the differences are also due to the imposition of “fixity” by the OECD. Under this requirement, the “official” PPPs for the European Community (BC) must remain unchanged in any comparison involving a larger group of countries. The achievement of fixity is a two—step process. 12 First, each OECD—specific PPP comparing two BC countries is replaced with the corresponding BC—specific PPP. Second, 10 The United States (US),. Luxembourg (LUX), Switzerland (CHE), Canada (CAN), Italy (1TA), Japan (JAP), Australia (AUS), Iceland (ICE), France (FRA), the United Kingdom (UK), Germany (GER), Belgium (BEL), New Zealand (NZ), Austria (AUT), Spain (SPA), the Netherlands (NLD), Ireland (IRE), Finland (FIN), Denmark (DNK), Sweden (SWE), Norway (NOR), Greece (GRE), Portugal (PRT) and Turkey (TUR). The mean absolute log—percentage differences are 0.3 13 and 0.472, respectively. 12 Such a process is necessary since Pks is not invariant to changes in the size of the bloc. 72 each OECD—specific PPP comparing an EC country and a non—BC country is adjusted to restore transitivity. Thus, there are three distinct PPP concepts embedded within the OECD—EKS results. 4.4 Concluding Remarks Gordon (1995, P. 7) notes that the use of PPPs from alternative sources can lead to very different assessments of the relative standards of living of countries. In an expression of the widespread confusion that exists among users of PPP data about this seemingly “fragile state of international ... comparisons,” he then goes on to ask the obvious question: “[W]hy [do] the sources differ so much?” There are three essential reasons. First, different sources calculate the same PPPs in the context of different blocs of countries. An example of this was given above when the OECD—calculated EKS PPPs comparing two EC countries were contrasted with the corresponding EKS PPPs calculated by the author. The differences between them are due (in part) to the fact that the EKS index, like all other multilateral indexes, is bloc—specific: The comparison of two EC countries in the context of the BC is conceptually different from the comparison of the same two countries in the broader context of the OECD. Second, different sources build the same aggregates from different baskets of goods and services. For example, Final Consumption of Resident Households consists of 159 basic headings under the OECD’s classification and 215 under Eurostat’s. This fact reveals an additional dimension of conceptual disparity among the OECD—EKS PPPs since those which compare two BC countries were calculated on the basis of the latter classification while all the others were calculated on the basis of the former. Third, different sources calculate the same PPPs using different methods of aggregation. Using a new a new type of difference indicator, the preceding section showed 73 that the choice of one method over another can have a substantial impact on the results obtained. More importantly, it was demonstrated that the restricted—domain methods motivated in Chapter 2 and shown to be theoretically superior in Chapter 3 are sufficiently different that they cannot be approximated by any other method. 4.1. OECD—calculated PPPs for private final consumption expenditure in 1990 national currency per U.S. dollar. TABLE — Country Belgium Denmark France Germany Greece Ireland Italy Luxembourg Netherlands Portugal Spain United Kingdom Austria Switzerland Finland Iceland Norway Sweden Turkey OECDEKS OECDGK Exchange Rate 40.4 9.92 6.69 2.06 140. 0.688 1,380. 36.6 2.15 105.5 113.1 0.597 14.2 2.23 39.1 9.08 6.48 2.00 132. 0.679 1,332. 36.0 2.02 93.7 109.5 0.586 14.0 2.20 33.3 6.17 5.43 1.61 158. 0.603 1,195. 33.3 1.82 142.2 101.6 0.56 1 6.87 90.9 10.68 9.50 1,597. 6.66 85.4 10.13 9.02 1,232. 11.3 1.38 3.83 58.3 6.26 5.92 2,613. Australia New Zealand Japan 1.44 1.65 207. 1.39 1.58 186. 1.28 1.68 145. Canada United States 1.34 1.00 1.31 1.00 1.17 1.00 Sources: OECD (1992, Table 2.5) (1993b, Table 2.8). 74 4.2. Population data for the Member countries of the OECD in 1990. TABLE Country Persons (1,000) Belgium Denmark France Germany Greece Ireland Italy Luxembourg Netherlands Portugal Spain United Kingdom Austria Switzerland Finland Iceland Norway Sweden 9,967 5,141 56,735 63,253 10,089 3,503 56,737 382 14,951 9,808 38,959 57,411 House— holds (1,000) Ratio 2.6 2.3 2.6 2.3 2.9 3.3 2.8 2.7 2.5 3.0 3.4 2.6 7,718 6,712 3,853 2,229 21,818 27,412 3,425 1,047 19,970 140 6,055 3,300 11,439 21,731 2,968 2,685 4,986 255 4,241 8,566 2,037 85 1,751 3,830 2.4 3.0 2.4 2.2 Turkey 56,570 11,189 5.1 Australia New Zealand 17,045 3,363 5,679 1,148 3.0 2.9 Japan 123,540 40,670 3.0 Canada United States 26,610 249,924 9,782 93,347 2.7 2.7 OECD Total 836,466 297,591 2.8 2.6 2.5 75 TABLE 4.3. Indexes of private final consumption in 1990 (OECD Country HD AB CD GK VH Belgium Denmark France Germany Greece Ireland Italy Luxembourg Netherlands Portugal Spain U.K. 1.12 0.482 6.66 8.02 0.575 0.239 6.50 0.050 1.60 0.535 3.06 6.46 1.12 0.478 6.61 8.03 0.568 0.237 6.42 0.050 1.60 0.526 3.03 6.42 1.13 0.492 6.74 8.04 0.590 0.243 6.63 0.051 1.60 0.546 3.12 6.49 1.12 0.488 6.61 7.82 0.592 0.233 6.54 0.051 1.59 0.578 3.11 6.34 Austria Switzerland 0.799 0.938 0.806 0.944 Finland Iceland Norway Sweden 0.802 0.938 0.441 0.026 0.351 0.795 1.11 0.505 6.58 7.73 0.615 0.229 6.62 0.051 1.61 0.628 3.13 6.28 0.777 0.889 0.438 0.026 0.350 0.790 0.449 0.027 0.353 0.808 Turkey 1.46 1.43 Australia N.Z. 1.82 0.315 1.81 0.315 Japan Canada U.S. 100). = OS US* 1.11 0.484 6.58 7.75 0.596 0.236 6.58 0.051 1.58 0.577 3.13 6.37 1.12 0.488 6.61 7.85 0.590 0.230 6.51 0.051 1.60 0.575 3.09 6.31 1.14 0.483 6.63 8.04 0.579 0.221 6.42 0.051 1.62 0.561 3.00 6.14 0.787 0.908 0.438 0.027 0.354 0.806 0.784 0.916 0.435 0.027 0.351 0.801 0.772 0.923 0.436 0.027 0.360 0.8 19 0.787 0.915 0.436 0.027 0.353 0.803 0.427 0.026 0.344 0.789 1.50 2.14 1.74 1.71 1.74 1.73 1.83 0.318 1.80 0.314 1.79 0.312 1.79 0.311 1.80 0.313 1.80 0.315 EKS 13.1 13.1 13.0 14.3 13.5 13.3 13.6 13.7 3.40 41.2 3.42 41.5 3.37 40.9 3.28 39.8 3.32 41.0 3.31 41.2 3.33 40.9 3.38 40.9 TABLE 4.4. Mean absolute log—percentage differences. HO ER US* Os EKS VH GK CD AB AB 26.7 26.5 4.17 4.39 3.70 4.18 3.72 4.26 3.74 4.26 7.89 8.38 1.15 1.83 0.744 CD GK 26.8 4.40 3.42 3.29 3.39 7.37 30.6 5.37 4.31 4.61 4.26 VII EKS 28.7 28.7 2.39 3.07 0.423 1.23 0.859 OS US* 28.6 1.99 28.6 76 TABLE 4.5. Indexes of per—household private final consumption in 1990. Country HD AB CD GK VH EKS Os US Belgium Denmark France Germany Greece Ireland Italy Luxembourg Netherlands Portugal Spain U.K. 66.0 49.0 69.1 66.3 38.0 51.8 73.7 81.3 59.7 36.7 60.6 67.3 65.5 48.2 68.1 65.8 37.3 50.9 72.2 80.7 59.2 35.8 59.6 66.4 66.7 50.3 70.5 66.9 39.3 52.8 75.8 82.6 60.3 37.8 62.1 68.1 67.8 53.2 70.7 66.1 42.1 51.3 77.8 84.8 62.5 44.6 64.2 67.8 65.9 49.8 69.0 65.0 39.4 50.6 74.6 82.4 59.9 39.9 61.9 66.4 65.3 49.2 68.3 64.0 39.4 51.0 74.6 82.2 59.0 39.6 62.0 66.4 66.1 49.9 69.1 65.3 39.3 50.2 74.3 82.1 60.3 39.7 61.6 66.2 67.3 49.4 69.2 66.8 38.5 48.1 73.3 82.2 61.2 38.8 59.8 64.4 Austria Switzerland 61.2 79.1 62.0 80.2 61.4 77.7 60.4 77.6 60.1 76.6 60.3 77.8 59.3 78.4 Finland Iceland Norway Sweden 49.0 70.4 45.4 47.0 60.5 78.6 48.4 69.8 45.0 46.4 50.3 71.4 46.0 48.1 50.2 75.3 48.2 50.1 48.8 71.5 45.9 47.8 48.8 71.1 45.8 47.7 48.7 71.4 45.7 47.7 47.8 70.6 44.9 47.0 Turkey 29.6 28.8 30.5 44.8 35.5 34.6 35.5 35.3 Australia N.Z. 72.3 62.1 71.8 61.7 73.6 63.2 74.3 64.1 71.9 62.0 71.3 61.3 72.1 62.2 72.3 62.6 Japan 73.0 72.3 72.9 82.2 75.5 74.3 76.0 76.8 78.7 100. 78.7 100. 78.6 100. 78.6 100. 77.3 100. 76.6 100. 77.6 100. 78.8 100. Canada U.S. 77 4.6a. PPPs for private final consumption expenditure in 1990 national currency per U.S. dollar. TABLE Country Belgium Denmark France Germany Greece Ireland Italy Luxembourg Netherlands Portugal Spain United Kingdom lID AB CD GK 40.1 9.85 6.64 2.00 146. 0.674 1,404. 37.5 2.12 113.2 115.7 0.591 40.5 10.01 6.73 2.02 148. 0.685 1,433. 37.8 2.14 116.1 117.7 0.599 39.7 9.59 6.50 1.98 141. 0.660 1,366. 36.9 2.10 110.1 112.9 0.584 39.1 9.08 6.48 2.01 131. 0.68 1 1,331. 35.9 2.03 93.1 109.2 0.587 Austria Switzerland 14.0 2.16 Finland Iceland Norway Sweden 6.84 91.7 10.80 9.64 Turkey Australia New Zealand Japan Canada United States 1,869. 1.43 1.63 211. 1.31 1.00 14.2 2.17 6.92 92.5 10.89 9.76 1,920. 1.44 1.64 213. 1.31 1.00 13.8 2.12 14.0 2.19 6.66 90.4 10.64 9.40 6.67 85.8 10.17 9.03 1,811. 1.40 1.61 211. 1.31 1.00 1,235. 1.39 1.58 187. 1.31 1.00 — 78 TABLE 4.6b. PPPs for private final consumption expenditure in 1990 national currency per U.S. dollar. Country Belgium Denmark France Germany Greece Ireland Italy Luxembourg Netherlands Portugal Spain United Kingdom Austria Switzerland Finland Iceland Norway Sweden Turkey Australia New Zealand Japan Canada United States VII EKS OS US* 40.2 9.68 6.64 2.04 140. 0.690 1,387. 37.0 2.11 104.2 113.2 0.599 40.6 9.80 6.71 2.07 140. 0.684 1,386. 37.1 2.15 104.9 113.1 0.599 14.3 2.22 40.1 9.67 6.63 2.03 141. 0.696 1,392. 37.1 2.10 104.6 113.8 0.601 14.2 2.19 39.4 9.77 6.62 1.98 144. 0.725 1,413. 37.1 2.07 107.2 117.4 0.617 14.5 2.17 6.88 90.4 10.71 9.50 7.01 91.6 10.92 9.64 14.2 2.20 6.86 90.3 10.67 9.47 1,559. 1.43 1.64 203. 1.33 1.00 6.87 90.9 10.69 9.50 1,598. 1.45 1.65 207. 1.35 1.00 1,559. 1.43 1.63 202. 1.33 1.00 1,565. 1.43 1.62 200. 1.31 1.00 — I 0 0 I C C,) Per-Household Consumption I I I 0 C3 0 F) 0 - I I I 0 0) 0 Cii 0 O 0 - (DO 0 0 _L - I I 0 L r) 0 . C) 0 I C Xc) rn 0 m C,) -1, (0 C CD G) N -1 -‘ 7 / fJ LI” 6L ‘1 I 0 0 Per-Household Consumption 0 0 0 - 0 0 CO 0 o 0 C (I) C I C) m z C C C) m C,) -I, :ij C > G) C -‘ CD m w m I z N F’) C H C,) -El z I Q ‘1 m z Cl) z mz 0 G) :ij m -1 H C 08 0 0 Per-Household Consumption 01 0 0) 0 0 0 CD 0 - 0 0 C C,) I C XC) C) m z C-. > > C C,) C, m m C -‘ CD C ‘1 -I, G) m I C) z N C -1 ci) -tJ z I ci 11 z C’) m ci z mz 0 G) m -o -I -I C 18 -n Co C -‘ CD I j.3 0 0) 0 Per-Household Consumption 0 (.71 0 0) 0 0 0) 0 (0 0 -L 0 0 C Cl) C XC-) I C-) m z C -ti C C,) C-) m ‘1 c 7c 0 m w m I z N C -I Cl) z I -I, m z C,) z mz 0 G) ::j m -o -1 -I C 83 BIBLIOGRAPHY Allen, R.G.D. (1949), “The Economic Theory of Index Numbers,” Economica, Vol. 16, pp. 197—203. Australian Bureau of Statistics (1993), Year Book: Australia, Number 76, Cat. No. 1301.0, Canberra. Balassa, Bela (1964), “The Purchasing—Power Parity Doctrine: A Reappraisal,” Journal of Political Economy, Vol. 72, pp. 584—596. Balk, Bert M. 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Voeller, Joachim (1981), “Purchasing Power Parities for International Comparisons,” Discussion Paper No. 161, Tnstitut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe. 87 APPENDIX A PROOFS OF THEOREMS IN CHAPTER 2 PROOF OF P1. Since C’ is positive, pk(pr,pc, PROOF OF v) Ck(pc, C(pr, v) 0 . I] v) v) C(p, v) — Cc(pr, Ckpc, v) : = Cktp r, v) v) p v) a . v) Ck(pc, v) Clc(pr, v) — — Apk(pr, pC, v) a . P4. pIC(pr,p, v)plcQ5,pc, Ck(p, v) v) := Ck(pr, v) PROOF OF =: k(r, P3. Since Cc is PLH in p, pk(pr, ), pC 1 PROOF OF Ck(pc, v) Cc(pr, v) > P2. Since Ck is non—decreasing in p. pk(pr PROOF OF _ P5. Since Ck is concave in p. p.” Cc(p, v) — Ck(p, p ’ 1 , v)pk(p?’, pr, v) Thus, by P1, pk(pr p ’ v) 1 = 1 a . = v) is concave in pC•• a PRoOF OF P6. By P4, for anypcE pIc(pc — pr, v). C(pc, v) Ck(pr, v) =: pC, v) a . 88 PROOF OF P7. k(r, )pV, = Apk(pr, pV, v) v), by P3 =A, byP6.D PROOF OF P8. pk(pc,pr, v)= — — pk(pr, pC,V)pk(pC, pk(pr, pk(pr, pr, pk(pr, pC, v) v) v) byPi byP4 1 PROOF OF pr, v) pC, = pIc(pr, pC, v) byP6.D = pk(pc, pr, v) by P8 P9. k(r v) 1 1 byP2 pk(pc, pr = PROOF OF pk(pf, pC, v) v), by P8. a PlO. pC, 1 v) pk(pc, Apr, = 1 Apk(p, = Apk(pr, PRooF OF by P8 v) pr, pC, ‘ v) by P3 v), by P8. a P11. pk(Apr Ape, v) A_lApk(pr, pk(pr, pC, pC, v), by PlO and P3 v) a .. 89 PROOF OF P12. Py — := mm nEil 11 F Li = k pf, P pr, — — pk(pf,pC, v by P7 , by P2 since pc v), p _pr; P. C C Pm := max nEA( Pm — iPni — F C = plc pr, LIJ Pm pr, — v , by P7 Pm C pk(pr, pC, v), by P2 since pC Pm _pr• o Pm PROOF OF P13. Define p k(C, p’s, v) (1—A)p’ + )j5’. By P5, (1_A)pk(pC, , v T p + Apk(pc, pr pf, v)]l_[pk(Pc, p v) v)] by the Theorem of the Arithmetic and Geometric Means. If P1 holds, this inequality is equivalent to 1 1 pk(pC, p, v) pk(pc, 1 pr, pk(pC, pr, v) v) 1 (1—A) pk(pC, pf, 1 +A v) pk(pc, pf, v) by the Theorem of the Arithmetic and Geometric Means. The required inequality follows by P8. u 90 PROOF OF THEOREM Vpk(pr, pr, v) 2.1. To prove necessity, recall that P4 implies P6. Hence, , ‘ v) vpk(pf, — pC, v) — v — Ck(pc, v) C)c(pT, v) To prove sufficiency, note that k P1—P5. Finally, by (2.3), v) := Ck(pr, V)p(pr, pC v) PROOF OF THEOREM Ck(pc, 2.2. satisfies (2.2). Since Cc satisfies R2, Since A € . Ck(AINp, v) := mm {(AINp)’z = mm {p’x I = k must also satisfy vplv(pr, pC, v). ci Uk is an increasing transformation of Uk and, , consequently, a utility function representing PRooF v) := (pand = V pk wherex:= Ck(pr, Equation (2.5) follows directly from I Uk(z) v) v} =: Ck(p, v), (P(x) A’N2D OF THEOREM f. 2.3. Ck(pi, Uk) Ck(pk, Uk) k( k j3 ,, Uki = Ck( k’ k px lclxk PROOF OF THEOREM 2.4. ço(A) := p(p’, p, Fix v) for all A , by (2.9) since the minimum expenditure required to attain prices pi cannot exceed p ci j and i, € k’ [0, 1]. and define x” := (1 — A)x’ + Axz, Uk at v := U(x) and Since both C and U are continuous over their respective domains, çü is continuous over [0, 1]. inequalities between the four numbers (0) = There are twenty—four p(p p, u,), ,(1) , 3 However, by Theorem 2.3 and Corollary 2.3.1, ço(O) = p5 and pj (= 4!) possible p(p’, pZ, ui), p5 and p. ço(l). Subject to these restrictions, there are just six possible inequalities between the four numbers: (i) (0) p 91 pj, ço(l); (ii) (O) p; (v) p ço(O) p (1); (iii) (O) p p; (vi) pj, ‘(l) ,(O) p i(1) p ,(l). Since p; (iv) pj, , (1) continuous on [0, 1] implies that for all p € O between ço(O) and ço(l) there exists a ). € [0, 1] such that is clear that there exists a p ) £ [0, 1] such that p5 p for cases (ii)— (vi). Thus, there exists a ço) between 3 u p and 1 U ,.kp(pr, pr, — — ek) jjp(pr, pr, Li, p(pr, ek)p(pr, pC, LI etc) ..i, ek) pC, by the positivity property of p , = by the transitivity property of p = ILk; p(p’, p, H) (ri = ElHkpkp(pr, p, I’. H pj p(pf, P” /L 1 Y2f —. IL H) by the transitivity property of p p(pr, p2: IL: H) —. p(p, P ’ 1 H) = , EflHkCI(p1, ILk) E= H 1 jC’(p, ILl) [p(p, = k=1 etc) ,by(i) e”) k(Jj) 0 ek)] by (ii) p, it which satisfies 2.5. Necessity: Clc(pr, uk) := = for case (i) or (2.15). 0 PROOF OF THEOREM ço(O) 92 — — p(pr, pi, ji, p(pf, k=1l. ek)1 e’)p(pi, pi, Li, pi, p., ek) U j by the positivity property of p p, f fp(pr, LukP(pr, pi, — — k= 1 , ek)1 0(H) p., e’)j by the transitivity property of p and since p. tk FCk(pi, Ilk) —. € c(p2, lAk) 0 Ic = iL —. Sufficiency: Straightforward. PROOF OF THEOREM Since both p and CIc (k 2.6. E K) are positive and continuous in their respective arguments, so is o’. Now, K E o(P, = 1 I K — U, H) := E i = 1 K 1 E1 = H Ci(pi, u) Hk Ck(pc, Uk) I K H C z(pc, E 1 = = u.).( 1 I p(pk, p, u, H) p(p’, K E = HkCk(ptc, Uk). 1 p(p’, _11_1 p, u, H)1 pk u, H) by the positivity and transitivity properties of p K H C( p, u) =E I K HIc Ck( pk, Uk) i=1p(pi,pi,U,H)[k=1p(i,pk,u,H) =1. Finally, - oP, u, H) := f I H Ci(pi, K E 1 Hk Ck(pk, u) / uk) p(pk, pZ, _11_1 u, H) 1’ 93 [Hi C(fIps, u) K = J E Hk Ck(f.pk, 1 = since pk := I 1 since both —il—i ] fkpk —1 —1 / Hk C(p’c, u, H) Uk) [Hi Cz(pz, u’) K [k I p(pk, .p: k i, u, H)] Uk) } is homogeneous of degree minus one in pk and and C are PLH in p p p =: cTz(P,u,H).D PROOF OF COROLLARY 2.6.1. Ci(pi, u) ElHkCk(pk, Ci(pi, u) ElHkCk(pk, ey ( 1 P u, 1 H)/H cri(P, u, H)/H, — — C ( 1 p, u:) k1 IIk c k Uk) Uk) k Ci(pi, u,) p(pk, p(pk, Uk) p’, u, H) , 5 p u, H) p(pk, Hk C(pc, p Uk) , u, H) /p(p i, pZ, p(pk p, u, H) by the positivity and transitivity properties of p — (p u) p(p, pZ, u, H) C , 1 a Ci(pi, u,) i . — PROOF OF THEOREM 2.7. in PMD(P’, p , 1 = = vj..., v, H) := C(p , 1 )j 1 k=1 [Cpi, vj rcc,, v)] since EKOk(JJ) v)j Lc(pi, in C(pi, v,) — in C(p’, v,) Oc(I-I) 1 U, H) 94 N = N fl, [in p n=1 E — in p] + N E 7mn. [in p,. in p in p in ph,] — ,n=ln=1 N ÷ E 7odlfl p — In p] in vp., by (2.53) m= 1 N = N E j3jln p in pg,] + E 70 ,jln p N ÷ = 1m = 7mn [in Nr N E 7m,Jfl p + 7onifl r N ,n=1 [in p , u) 1 ôinC(p — in p] + U + /3,?, + N 4 p 41 E I + I [1np pi’xij 1Lpi1xi N in N = =: P F12 in II mi[p] in pT(p” p, in p I xz) . ci E 7mnlfl p m=1 = ôlnC(pi, 1 u ) l j[Inp rp = — since in v from (2.53) = Vfi N + ‘yonln =E ‘p in pJ in p + in p, ] [in p 1 E I/3 + — N E m = — — inp] 1n u + in — inp] U 95 APPENDIX B PROOFS OF THEOREMS IN CHAPTER 3 PROOF OF THEOREM (i) 3.1. ByT,foranylEK, H),p(p p’, X, H) , p(p’, p’, X, 2 = p(p’, p , X, H). 2 Thus, by P, p(p’, p’. X, H) (ii) = , .\p’, X, H) 2 p(p 1 = p(p’, p’, X, H), by H =, (iii) p(p, p’. X, H) = — — (iv) byl. p(pi, p, X, H)p(pz, pi, X, H) p(pi, pi, X, H) p(pi, pi, X, H) p(pi, pi, X, H) by T 1 p(pi, p, X,H) byl. 1 p(p, 7 p X, H) , byCR For any p’ > p’. p(p, p, X, H) = 1 > = p(p, i 1 X, H) , 3 byM , p, X, H), by CR. 2 p(p , by P 96 (v) 1 p(pi, Api, X, H) p(Ap’, pZ, X, H) by CR , 1 = .Xp(p, pi, X, H) by H p(p’ pZ, X, H), by CR. 4 ) (vi) p(Ap’, .Xp, X, H) )Ap(p’, p, X, H), by H and HDM = = p(p ,ps,X,H). 2 I p 1p1 1 L = p I p’, p’, X, H I 7EJIp3J 1 p (vii) — : = mm p1 i — — L r4 by PP , J p(pJ,pz,X,H), byMsincep p _pi; pq — . I p, 1 P 1 I = p I p’, p’, X, H I = max ?lEIIniI p 2 L. u’mJ j — i Pm — , by PP i Pm p(p’,p,X,H), byMsincep p’.c PROOF OF THEOREM 3.2. Necessity: [(pi, p, x , x, 2 2 H)] pxz 21 p x.’ — XZ, p, H, H) byFR (x’, x, p, p, H,, Hj (p pZ, x , 2 x, H, 1 H ) (x, pi’xi p3, x, x , 2 pi’xi (p’, H, H)(x , 1 — — — — p’x pz’xi pi ‘Xi . by FR and PT. . p3, H, Hj, by CR 3 p x’, p’. p3, I-I,, Hj, by WS 97 Sufficiency: Straightforward. PROOF OF THEOREM 3.3. (&, — 1 E )/(1 + k(p, ) X, H) > 0 &.> Suppose K\{i}. Vj € = 1. Since p 1 . By <jV 1 wk(p, EL Si, 0, - •.. , , _ 1 p &.> , 1 p But &k> 0 Vk E K implies that E o(p1 , ... , p’, ... nii , 1- 3 ?r By Sil, 0. pK, X, H) , ,..., 1 p 11 p = p”, X,H)> PROOF OF LEMMA 3.1. By RT, X, H) cri(P, X, H) 1 (p’, H , 1 p x, x, X, H) p’, , 2 x — — - H, ‘(p , 2 1 pl’xi p(pi, H — H,pi’xip(pi, pi, — , 1 p 1 pi’xi (pi, 1H — — — p(pi, p1, X, H) Hph1x1 by (3.2) (B.1) X, H)p(p i, p, X, H) p(pi, pi, X, H) p.i, } , by T } , oi(P, X, H) H,pi’xi o4(P, X, H) Hp1’xl — (B.1), K =1 H pi’x 1 1 1p H 1 ‘x p(pi, X, H) u ( P X, H) X, H) 1 , 1 p p(p’, pi, a ( 1 P, X, H) = f I = PROOF OF LEMMA (i) X, H) X, H) pi’xi p(pi,pi,X,H)} 1 [H H p 1 1?xi — From x’, X, H) crl(aipl, 1 K = E u(P, X, H) =1 pi’xi p(pi, p, X, 1 1 H H) H p1 ‘xl p(pi, pi, X, H) f by Si. a 3.2. ... = , £LKp, I lj= , 1 ,6x ... , , 1 I3KX 7, ... ‘7) )’(f3x p(c,p 1 7(ajp ) , cjp, 1 1 flux , ..., p’)’(fl 7( x l) p(ajp , 1 1 1 jp’, KX 3 , 1 fluX’,..., flKX’, 7,..., 7) by (3.6) ( cq’ [(H ‘x ‘cd ‘çdd ,r,,dlH (H ‘X ‘).°(i—v)+i (H ‘X ‘J) c]( — — 1 L(H 1 ‘X ‘d ‘cd)dcx,cdH I — ) + (H ‘X = r(H ‘x ‘d ‘cd)d:x,dH’ ‘X ‘:d cp)tIcX,cp.tH _ _ 1 LCH 1 H ‘X d)d 7 (El ‘X ‘d ‘ d ‘:d (H ‘X ‘d ‘ d)d 1 ‘2 j (•) IxlldlHl(J —v) :X, :d H] x, 7 r(H ‘x ‘d ‘,d)d 7 dIH] ‘ ‘ d d)d 1 7 gX d H (I ‘X 1 _ _ 1 L(H 1 aOs iq x,dH ,x,dIH (I — + i }H ‘x ‘J)D = V) + 1 _[(H ‘X ‘J)1]} ‘ d )d I(H ‘x ‘,d 2 _l(H ‘X ‘d ‘,d)d 1 ‘ ‘cd ‘d)d + (H (H ‘X ‘d ‘d)d cX,cdH ‘‘ — , j (9E) £q f(H I. (H ‘“ ‘x ‘ ‘x x ‘ 1 ‘ x 7 x 7 1 ‘ ‘ x (H ‘x ‘ ‘1+ ‘x ‘‘ x 7 (H ‘ .... 1 ‘. x 7 ‘ ...x 1 ‘ ‘,d ‘ d)d 7 ‘:d ‘,d)d ‘x X 1 ‘ ‘ ‘_x ‘ x 7 _ 1 ‘71V ‘ xy 7 ‘ ‘x x 1 ‘ 1 x x’), 1 ( d2H + ‘d ‘d)d x,dj_j ‘d ‘td)d cX,cd H (H I ÷,X ‘ 1 ‘ _X 1 XV ‘ 2 ‘ ‘X ‘ ‘j).o ‘{ }\‘ 3 c:;/ ‘ = uis 7 iU JO (1!) = = IAKII1 Pu H q xX ‘ J I(L ‘“‘L. 1 ‘ x 7 X ‘,d ‘,d)d ‘ ‘,d)d d 2 ‘“,x’/ ‘‘ x 2 I ‘“.‘ dT } 86 (9) iCq ‘ (9) iCq ‘(H ‘X ‘i)’ = ‘X ‘cd ‘çd)d :I,:dH iç 1 1 j=( H 7 [(H ‘x ‘d ‘cd)cI ?(i_VV),:2 J 11 t (H ‘x ‘d ‘d)dxNjd (H ‘x ‘d ‘d)d _ 1 X( ),dcH 1(11 ‘K_’ ‘cdV ‘dv)d (xi_V),(:dV)H I _’ ‘d( ‘cdV)d(cxiV)j(cdV)cj-j 1 1 _ [(H ‘X “ 1 j i= = “ j (H ‘X 1 ‘c1v° (9E) pu ciO ‘NIH ‘H i(q ‘(H ‘X ‘j).o { - ( £) (j-j1 (HL. ‘xLI ‘cd’o ‘dc)d d H ‘xLI ‘gdV ‘cd2)d cccdLHT Gd) HL 1 ‘xI ‘dc ‘dft)d (x&/) ‘xLI ‘dv ‘d)I (cx€J)j(d)LjqA. I(HA-. 1 _ l(H’ (jjL ‘/ ‘= H J I ‘ 1(j — ‘ (Al) = = = ‘dxV ‘“‘idTV):D D(J—y)+J 7 (H ‘X ‘J) Dv 7 (H’x’J) (9)iq ciOs q (iii) — — _[(H’x’J) D ]}v {(i—v) + 1 1 ,Id7H (H ‘X ‘çd ‘cd)d 1 v) + (H d ‘cd)d CX,cdH 7 ‘X ‘ = = (9•) iq I(H 1 _ l(H (H ‘x (H ‘MX ‘ 1 H ‘ MX ‘lX ‘1+1 ‘ 1 T+? ‘ ‘ x 7 y ‘lIV ‘pc ‘V x 7 _ 1 ‘ ‘ ‘lAX ‘T....X ‘ x 1 ‘ ‘ ‘T ‘“‘ix (,xy), d lj_j x ‘,d ‘ 1 ‘ d)d 7 7 + d 7 7 ‘ ‘ d )d ( 1V),,dlJ_I 1 ‘TX ‘çd ‘cd)d (,xy) 1 d ‘ ‘cd)cf cr,cd H d 7 = ÷rr ‘V 1 (H ‘,X ‘“‘ ‘i ‘ D 1 ‘cI) 66 100 (v) Consider the bijective mapping ço: K COl,(l) ‘K = - K which satisfies (B.2) Coli ‘K for any 1 € K. For all k € K, let [pP(k) co(k) H(k)] := [F, X, H’Jco1k) = (B.3) ‘K [P,X,H’]COlkIK, by(B.2) =: (plC, xk, Hk). (B .4) Now, cT(PIK, XiK, IK’H) I [I,(y) K = 1 = = -. (vi) I I ‘‘ p(p, 1 IK’H)l’ H,( ) 1 p ‘(i) p(p0(i), p(i), XIK, IK’H)j Hp”x’p(p’, p, K E XIK, x, H)’ F ,by(3.6) by (B.4) and WS , 1 p(pi, pi, X, H)J H p ‘x c(P, X, H), by (3.6). I - o(INP, INX,H) = K Hj(lNp’)’(JNx’)p(lNp, INp, INX, E [j=1 - H)1 - Hj(INp)P(INxz) p(b.rp’, ‘NP , INX, H) 2 by (3.6) I K Hjp’(lN’iN)xp(p, pZ, X, H) ,byCS p’, I —1 K =E [=1 Hp ’ 2 INx’p(p’, p, X, X, H) H)1’ HzpZIp.rxZ p(p2, pS’, X, H) ‘ ‘ (‘MY “'Fl ‘“ (‘flY “H ‘ “H “H(YI) “‘H ‘ “‘H “H(V—I) “‘H “H ‘,x “H ‘,‘ ‘x ‘d ‘d)d x ‘X ‘,d ‘cd)d d ‘H(V—T) 1 x 7 = CH ( H 7 v “H ‘““‘H “H(V—T) “‘H ‘ “H X 7 ‘ o 1 ‘x ‘,d ‘j) (9) iq ‘(H ‘X ‘d)° JD q = ((I-I ‘X ‘cd ‘cd)d x :d11 1 ‘d ‘cd)cl cX cd’H ,, 1 ,_l(H ‘‘ (9E) q f (?gy 1.Qi-iv •1— “Fl ‘“•“‘H ‘)IfJ ‘ “ii(v—i) “—‘H “‘H “H(V—T) “‘H ‘ ‘ “I-I “H ‘,X X 7 ‘ ‘ ‘x gX, : ‘,d ‘,d)cI x ‘ d ‘,d) dH ,;,d ‘H[v+(v —1)1 (‘nv “H (‘nv ‘ “‘H “F[(v—L) “‘H ‘ “H ‘x ‘, + ‘d ‘d)d x “I-i ‘““‘H “H(V—T) “‘H ‘““H ‘,x ‘x ‘:d ‘d)d x d 1 j-j H (‘Hv “'I—I ‘ ‘T+pj .j 1 “Ff(V—T) “— ‘Ij_j ‘ x ‘x ‘,d ‘j) 7 ‘ i tU JO{ (9) cq ‘(H ‘X ‘d):0 (MA) = Jo’ ‘ ,_l(H ‘x 1(11 (9E) icq ‘(H ‘x ‘j).oy ‘d ‘cd)d,x,,dIH’ T_V} ‘ ‘d)d cx,cdH d 7 = = (9E) q ‘?d) ‘X ‘ d)d 7 ‘d ‘ ‘x x 7 IQHY “H ‘““-FJ 1 1‘ 1(Y—1) 1 “ i -i ‘““H ‘ T_IQHV “'H ‘““‘H “H(V—T) “ H ‘•••“H 7 1 dIHV x, + z t7 hH[V+(y , —1)1 ‘ (‘Hv “'H ‘ ( H 7 v “H ‘“ H 7 1 “ ‘ H (V—I) “‘H ‘ “‘H “H(Y—T) “‘H “H ‘9’ ‘x ‘c( 1 ‘d)d ‘ ‘d) d ‘9’ ‘X 7 x “H 7 x ,,dljqy ,cl X,dcH +IH “H(V—T) “‘u T i( 1 iv “'H ‘“‘ ‘ = “ii ‘x ‘x ‘,d ‘i),+ (9•) i(q ‘(H ‘X ‘d):°(V1) dD q ‘ 1(ii ,_l(H x 7 ‘cd)d d lH’’ , ‘x ‘d 1 ‘ d ‘cd)d cX,cd’j-i 7 ‘ } ‘I = = (9E) I(Hv “'H ‘““ H “H(V—T) “‘H 7 ‘I ‘XII ‘ “H “ii(v—T) “—H ‘ ‘ “H ‘x “H ‘x d)d 1 d ‘ 7 ‘ d ‘ 7 d) 1 ‘9’ ‘X ‘ ctlH(V— I) 1 9’, d ‘H[V+(V—I)l ; 7 + OT (“ ‘X ‘X CH ‘H ‘X ‘X QN ‘H ‘C gX ‘X H ‘d ‘d) (H ‘ X ‘.X H ‘:d ‘d) (H ‘C ‘d ‘d) CXICJcH x ‘d ‘d)d ‘? “N ‘X (N “H ‘x ‘x ‘d ‘d)dx,d’j-j + CN ‘H (N ‘J-J ‘x ‘x ‘d gd)dX,dJJ + (gjj ‘:X ‘çX CH ‘H ‘X ‘çX CH ‘H ‘X ‘X (N ‘H ‘X ‘çX CH ‘H ‘ ‘-d ‘cd)d:; d H ‘cx ‘:d ‘:d)cXdcH ‘d ‘d) 5 dH x, 5 ‘:d ‘d) ‘x (“ “HV ‘ d ‘: ‘ ‘ +1 1 d ‘H X ‘d 5 ‘ d ) d) 5 ‘d ‘ +1 5, X :d 5 H (9•;) t(q “NV ‘X ‘cd ‘cd)d x, sd !H ‘cd)dc c 1 dfj-j x +1+ (...“NV ‘x ‘ d 5 (••• “NV ‘X d t 5 d)dcX,cd 5 ‘ N d 1 ‘ (“ “NV ‘X ‘ ( “NV C” “HV ‘X ‘cd sd)d:X,:dzH +1+ (““HV ‘X ‘cd ‘ (“NV ‘ (“NV “ “HV “I-I “HV “HV “I-I “NV ‘ ‘ ‘ d)d 5 d H X, 5 s’C*y ‘d)dX, 1 3 jjy d ‘,d)dcX,cd.’N ‘,d)d,X,,d1HV “‘HV CN “‘NV “NV tH “NV (9) IaL’q ‘ ‘ “ ‘+H ‘‘IIV “‘II ‘1+ljj 1 ‘ 1 1Y ‘iI ‘ ‘ WIT = 0 ‘-V “NV ‘X ‘J)c.0 wij (xi) “NV ‘ ‘(‘—H “X “—i) = 5 ,d N *çl (‘—H ‘X ‘d ‘d) x • J d ‘d)d X-d’H 5 - f (‘H “X ‘ I (“N d‘ )dO’—Vmij “N ‘x ‘d 1 “N ‘X ‘d ‘,d)d0-’tuTj V 0 W + ,x,,d’H (“H ‘““‘N “NV ‘‘H ‘““N ‘X ‘ed ‘cd)d0-(wq d X 5 H , d ‘Cd)d0-VWIJ cXcdfH 5 (“H ‘““‘N “NV “‘N ‘•••“H ‘X ‘ (•) (q L J = dj_j 5 f(”N ‘““‘N “(-IV ‘Tjj ‘““H ‘X ‘,d ‘,d)d x d ‘,d)d ,x, ,d ‘j-j’<’ + 5 ‘111 ‘X ‘ ,_tQ’N ‘““H “I-IV “‘H ‘ ‘1ii 5 d H ‘cd ‘d)d x (“H “‘T+1 “IJY “H ‘X d ‘cd)d çx,d t 5 (“H ‘““‘H “NV “‘H ‘““H ‘X ‘ H ‘ ‘ i j WI I = 0 ‘-V (“H ‘ “‘iI “(-IV “‘lI “I-I ‘X ‘J) ° 5 ‘ WIT (WA) £01 104 — — — = (x) 1 (pi, pi, Hp x x, x, H, Hi), by T H,pi’x2 11 p ) 1 (Hx pi, H,xi, Hx , 1, 1) 1 pi’(H,xi) /(p2, q(pi, pi, H,xi, For any p > by TQ , H,xi, Hx , 1, 1), by (3.2). 1 :, p(pi, p, X, H) p(pi, pi, X, H) pz, X, H) p(pi, pJ, X, H) < p(pi, , by P and M Hpslxz i(p, X, H) Hplx a’(p 1 ,..., p 1—1 p, p i+1 ,..., p K X, H) H,p”x’ Hp”x’ oz(p 1 ,..., —i p i—i p, p i+1 ,..., p K X, H) X, H) — , , , by (B.1) ji(pl,..., i, pil, pi+l , .... , K I— oi(P,X, H) > (xi) rp’.1— [ps’xij u(p’,..., , i, 1 pi_ pK, X, H) (P,X, [1) Ifp=p’then p(p, p, X, H) = 1 by I (@ P A T) , H p’x ai(p 1 oi(p , ..., Hpi’xi 1 , (xli) X, H) I xk, lK) = pzl, pi, pi+l, pZ_l nY ,iI+1 1 — = 1 , •••, pK, pK, X, H) X, H) pu’(HxJ) p(pi, p, XH, lK) K E 1 — , ... p’ (H: x) p(p, p . XH, lK) 2 { H p 1 i’xi p(pi, pZ, X, H)1 = 1 H p2 ‘x p(pi, p2, X, H) j u(P, X, H), by (3.6). 1 = , by (3.7). 1 , by (3.6) by TQ , x 1 ‘ ‘ 11 ‘X ‘ (‘IJY “'H (lHy 1(V—T) “‘H 1‘ “-ii 2 ‘1+711 “ii(v ‘ 1 ‘“ ‘ x x 2 _ x 1 1 “‘ ‘ 1 ‘ x 7 ÷ 7 x 1 ‘ ‘ x ‘‘ 7 7 ‘“ ‘x x (‘ir’ “H ‘ ‘“ x 1 ‘ — j) ‘I—J ‘ ‘ “11 ‘111 d H 7 (V—J) c1 7 ‘d ‘d) , d ‘d) 1 ‘ 11 d c 1 x = H “H(V—I) “i-i ‘““H 7 “ •‘,÷ 1 ‘“ x I+MX ‘,x v, + 1 ,÷ x ,(v—j) x 7 _ 1 ‘ ‘ ‘x ‘(H ‘X ‘ci)r° (9•E) = f (H ,_1(H d ‘d),.o 7 ‘ = JDS us pu ‘= ç’ dH ‘X ‘d ‘cd)d : H ,, ‘X ‘:d ‘cd)OcX,cd (9E) cq —T) “H v “'H ‘““‘H “H(V ( H 1 (‘iiy “'H ‘ ‘1+111 ‘‘H(Y—I) “—H 7 ‘“‘,x x 1 x ‘ 1 x ‘ ÷ 1 ‘ ÷?c ‘ _ 1 ‘‘— , 1 ‘ x ‘“ X 1 1 ‘ X 1 ‘ “H 1+? ‘x ‘IfJ ‘ r, d H d ‘,d)d 7 ‘ + , ]-j ij 7 ) y + (7 ,, y— j d 1 [ ‘,d) x ‘ ‘ ‘ (‘i-iv “'H ‘ -iv “'H ( 1 7 ‘1+711 —i(Y i) “H i‘ 7 ‘ — H 7 ‘1+711 7 -1(V i) “ 1‘ ‘1j_j ‘ — T+? )1 T+l X _ 1 X ‘ 7 X 7 ‘1+ ‘ x 1 ‘ ‘,x _X ‘d ‘d) x, :d 11 * I ‘:d ‘d) x,dcH ‘‘“ ‘ (‘Hv “'H = “H “H “H(v—T) “H ‘“ ‘ J’c 1 + Mx )Ir 1 + _ ‘“‘,x 7 ‘, ‘ x ‘,d ‘j) ‘{ih 3 i AÜB 1O{ (mx) coi ‘....‘ ‘...‘ ‘I+ZH “H ‘1+711 IQHV “'H ,_IQHV x 7 ‘I_ x 7 I+ ix ‘,_,x x 7 + 1 ‘“‘,x ‘H@(—T) “‘FI “ii(v—i) )d ‘ d d 7 7 ‘ ‘1—Ijj ‘ ‘ (9) q “H “H ,,d ,÷, x 1 X 1 ‘,+, ‘X ‘,÷,X ‘X -jy 1 j + ‘“‘,x j i)], 1jd ‘,d)d[,+,xV + d 7 ‘ 7 ,a(y— ‘,‘• (lily ‘Xjj ‘ V ‘il ( H 7 x 1 ,Ix ‘, ‘ + 1 x 1 ‘I+, ‘,lx ‘ X ‘X 7 ‘, ‘1+711 ‘11(( I) “—‘ii ‘1+711 X 7 _ 1 ‘ ‘x T+ ‘ — i(V —1) “‘Il ‘ 1 7 X 1 ‘ ‘“‘,x ‘ (‘liv “'H ‘ ‘d ‘çd) ‘x ‘ ‘x + ,(y—j) ‘i1 ‘ X 11 d ‘d) 1 ‘ 7 ‘IIV d :* X,dcH i(V—T) “‘u i‘ “‘H 1 ÷x ‘ = “H x ‘‘,x 7 x ‘,_ 7 x ‘ ÷ 1 ‘“‘ ‘(H ‘X (9) = T+,lxv ous d11_j 17 ((H ‘X ‘cd ‘d)d ,x ,_(H ‘j),.o ‘,d ‘J),÷.o = d 7 ,, pu dDS (q I,x,,d ‘‘ ,_[,,,d _i)] d ‘cd)d cXcdH 7 ‘ } (9ç) q ‘I+HX ‘•••“H ‘,÷X I QHV “'H ‘““‘H ‘ H(V—T) “RH ‘““H 1 L’iiv , x 7 ÷ 1 ‘ “i-i “NH “i-i(v—i) ‘ ‘T1)J X 1 ‘ M d ‘H( v—i) ) ‘ d ‘“‘,x ‘ct 1 x 7 x ‘, 7 ‘ + ,( y— j )],,dZjj y + [,÷, x ‘,d ‘,d) 1 x 1 x ‘“‘ 1 ‘ x 7 ‘,÷ 107 = {{ ’x 1 p i = Hpi’xi p(pi, p . X, H)j 1 1 1 p’ ‘x H 1 p(pi, pi, X, H) f by SCP and since (1—)t + )pl = t x CT(P,X,H), by(3.6).D = PROOF OF (i) LEMMA 3.3. For any p > p, ’x oi(P, X, H) p H1 p(p’,p,X,H).—. Hpi’xi u(P, X, H) by(3.7) ’x p’x u’(p 1 1 H• p ,..., 1 —i p, p, p i+1 ,..., p K X, H) Hjpx)p?x i(p I ,..., —i p i—i p, p i+1 ,..., p K X, H) , , , by Sli Hpxz 2 a ( p1 : 1 p, p. p 1+1 ,..., p K X, H) 1 Hp”x’o(p,..., p i—i p, p i+1 ,..., p K X, H) ,..., - , = (ii) p(p’, Apz, X, H) , , p(p, p, X, H), by (3.7). = 1 &(p ‘x , 1 H pi’xi 1 up ( 1 S,., ,..., +l,..., 1 , 1 p 1 1 )p p , 1 p 1+1 11 Ap p , by (3.7) = (P, H p’x 1 i0 H) A Hpi’xi gi(P, X, X, H) = Ap(p’, p , X, H), by (3.7). 1 by S4 , , ... , pK, pK , X, H) X, H) 108 Hj(Apz)(5x$) t72(5P, X 1 , H) (iii) X, H) 1 p(Ap’, Ap, ). = , Hj(Ap2)’Qx2) o(AP, 1 A X , H) by (3.7) H p 1 ”(AA)x o-i(P, X, H) ,byS5 = Hp’(A5’)x’ o(P, X, H) = (iv) p(INp’, INP, INX, H) p(p’, p, X, H), by (3.7) and since .Xj1 = IN. H(INp)’(INX) O’(iNP, INX, H) = , by (3.7) Hj(INp’)’(INx’) O(INP, INX, H) HIPI(INIIN)XZ j(p X, H) ,byS7 — Hjp2’(lp/IN)x2cr(P, = (v) X, H) p(p’,p,X,H), by (3.7) and since IN’IN= IN. Consider the bijective mapping : K - K which satisfies (B.2) for any 1 E K and let (B.3) hold for all k E K. Now, p(p’, p, X?K, IK’H) = p(pq’(i), ‘(i) XIK, IK’H), by (B.4) r(PIK, XIK, IK’H) H(j)p(i)1x4d(2> — — = (vi) by (3.7) ‘(PK, XIK, IK’H) H p’x o(P, X, H) by (B.4) and S6 Hpi’xi oi(P, X, H) p(p’, p, X, H), by (3.7). )p’(6x) i(P, f3X, 7H) 1 (7H ( H 7 ,)pi’(/3xi) o(P, /3X, ‘yH) p(p’, p, f3X, 7H) — — = 1 o(P, X, H) Hp’x Hpi’xii(P, X, H) p(p’, pZ, X, H), by (3.7). byS4 , by (3.7) £s’cq ‘ (LU) . US ‘ (H ‘X ‘ci)c” cx,cdcH (H ‘x ‘d)z° X,dH (Mj ‘Mi’ ‘Mi’ =)i ‘“ ‘ cq ‘i-x ‘ — ‘ci)” Cx H),d ‘HX ‘d ‘d)d tq (H (H ‘ MX ‘ ‘MX ‘ ‘V X’ 1 ‘ii’ X _ 1 ‘ ‘ Aq ‘(H ‘x ‘d ‘d)d = ‘ii’ ‘j).o ‘ii’ ‘ci),” q (,XV), 1 d 7 H x, d H ‘ ‘ii’ ‘ (H (H ‘Mi’ ‘“‘T+i’ ‘V X\’ 7 ‘i+,i’ ‘ ‘MX ‘ ‘ ‘ ‘ii’ ‘ i 1 ’ — — — ‘:d ‘,d)d ‘(H ‘X ‘,d ‘d)cI = (H ‘X ‘ci),°(I—V)+Tcx,c”H IXIdIH (H’x’cI)c”Y ‘PCV ‘i—P’ ‘,i’V ‘i_ui’ “‘ X ÷ 1 ‘j+ji’ ‘ — — cd H 1 ‘d) . 1 ° cx ‘ H = ‘ci)c0 (,xy , x ‘,xy ‘ 7 ÷ 1 “‘ x 7 _ 1 q ‘ x 1 ‘ ‘i—ui’ ‘“‘i ‘ci) X ‘j)cD 1 ‘ ‘_X ‘ X 7 ÷ 1 ‘“‘ ‘,d ‘d)d ‘(H ‘X ‘d ‘d)d = cdii 1 (H ‘X ‘ci) (H ‘X ‘cI)i°(T—V) + i cx :X,:dH (H ‘X ‘ci)i.°(1—V)+T (H’X’ci)c” cq (NA) = x 7 _ ‘ r ’,d’,d)d ’”’ x (H’, ÷ ’”’ 1 ,x’,xv ‘i = 1 ‘i+i’ ‘i+ii’ (H ‘X ‘ci),°(i—v)+i i(q — (“i ‘HX ‘ci)” (x ‘H),c = (Mj ‘ (L) d‘ 7 d)d 7 ‘(H ‘X ‘ (iv a (L) 2ç,d ‘),.0(T—y)+J H (H’X’d):°V (H ‘X 7 cdH (H ‘X ‘ci),”(T—v)+T (H’x’ci),”v (H (H ‘ x’’ ‘_i’ 7 +,x ‘ 1 (H Mi’ (L) (H’K’ci)°V q (L) x 1 (H ‘, (L€) Es (LU) x 1 (H ‘, cx,cd’H H ‘,xy ‘,x x ‘“‘ i 1 ’ ({l1\x) — — — — ‘:d ‘d)d (z ‘I) icu ioj (iA) 601 (‘HV “H ‘““‘H ‘ H(V—T) ‘‘H ‘““H 7 X 1 ‘ (‘iiv (L) i(q ‘x ‘,d ‘j),÷o x,d’j-j ‘X’,’ ’ 1 d)cQ ,x,,dIj_jy H “H(V—i) “‘H ‘“‘H ‘x ‘x ‘,+,.,d ‘d)d 7 (‘i-iv “'H “‘I+ ‘ utp ,d = ,÷,,d ; (c) Aq ‘(H ‘X ‘d ‘,d)d 8S q 7 ,,d1jj(—j) (H ‘x’).o x 12 ZH x H’X ‘j),’(v—j) d = — — (L) q d ‘j).o 7 1 x,,d ‘H(V—i) (‘liv “'ii ‘““‘H “[[(V—i) “‘ii ‘““H ‘,x ‘x ‘ (‘I-iy “I-I ‘ “‘H “H(V—i) “‘ii ‘‘““H ‘,x ‘x ‘,d ‘d),.o (‘i-iv “H ‘“‘“‘H “H(V—i) “‘H ‘ “• Iq x ‘x ‘d ‘,d)d 1 ‘TJ-J ‘ (L’E) cq ‘(H ‘X ‘d ‘cI)d (H ‘X ‘d)-° cX,cddli (H ‘X ‘j).o gX,d7-J 8S q = — — (L’) q (‘liv “H (jjy “'I-I ‘‘“ “ ‘ “‘H “H(V—i) “‘H ‘•‘•“H ‘,x ‘x ‘,d ‘j) cx,d’H “‘H “ti(v—i) ‘7j ‘ “ x d :H “H ‘r ‘x ‘,d ‘j). 2 (‘liv “'H ‘““‘H “H(V—i) “‘H ‘““H ‘({l}\Y) ‘(ce) cq ‘(H ‘x ‘d ‘d)d X ‘,‘ ({i)’) 3 ‘X ‘d ‘d)d ‘1) t cu oj (i (xi) = 011 111 x’ 1 p 1 )H 1 a’(P, X, H) Hpi’xika(P, x, = = (x) byS8 p(p’, p , X, H), by (3.7). 1 For any (j, i) € (K\{l}) x (K\{l}), , 1 , p, x 2 (p ... , x’’ ..,.K+1 ,H ,..., 1 x ’ 1 , 5c 1 1 p ‘x -i(P, p H 1 x1 , , H,p”x’ (P, p’. x , 1 1 Hi, x ... , IIj+i, 1 (1—)j11 , x11 x 1 x 1+1 , ... , , —l x 1 1 .x, r , ... , , , , ... , II_, (1— A)11 , I-Ii+i, 1 ... , , (1— A)11 — 1 H , + 1 , H 1 , ... HK, \.J-Ij) xK ... , FI-, All ) 1 ... , ) 1 HK, All by (3.7) , 1 x 1 — — = , 1 H (P, X, H) H px1 i0 o4(P, Hpi’xi X, H) by S14 p(p’, p , X, H), by (3.7); 1 , 1 ,x 1 ,p 1 (p ... , , 1 x ., , + 1 x 1 1 H x 1 p 1 ... , xK, iii, (P, p , 1 , x 1 1 p (1—A)11 t ’ 1 x 1 ( P, p, x , 1 x”’, H, , 1 x 1 H , 1 x 1 p 1 H 1 1 (1—A)Hjp = if K+l = , ... , , , (1—A)11 — 1 H , lJi÷i, 1 , , 1 x .x 1+1 , ... , , . .. , ... , HK, ) 1 All 1+1 1 , ... , , (1— A)H — 1 H , ÷ 1 , H 1 ... , , 1 lIii, (1— A)11 ... , HK, ) 1 ).J1 ... , HK, ) 1 Al-I by (3.7) +i, 11 (1—A)p ’ 1 . 1 r ( P, X, H) bySl4 p ’ 1 x’ ,p 1 p(p , X, H), by (3.7); 1 1 then p ... ... o ( 1 P, X, H) 112 (2 pK+i ,x 1 ..., Jipx+i — , o’(P, , 1 pt,x +l(P, p . t H p 5 i’x2 — xK x+i, ‘Ii,...., H (l—A)li Ili+i,.... lix, AH , , 1 — 1 ) 1 ., , 1 H ... ,x 11 ,x , 1 + x , 1 ..., , 1 x , , 1 x •• , H 1 , + 1 , (1—A)H — 1 H ... , H , 1 ..., Ii—i, (1—A)H , lit÷i, 1 ... , ..., H, Al-li) by (3.7) .xK+1 )JJllxK+i ’x t p o ( 2 P, X, H) H p 2 ”x’ Ap ’ 1 x’ ci’(P, X, H) = = (xi) , bySl4 p(p’, p , X, H), by (3.7). t lim p(p’, p, X, H , Al-I — 1 ,..., H 1 , + 1 ,H 1 ... , A-, 0 — ) 1 lix, All 1. Hx) H p’x o’i(P, p , X, x t , H t , 1 H (1—A)li , — 1 , ll+i, 1 ) 1 lix, All H p 2 i’xi o(P, t x H , ,...., H 1 , 1 — 1 (1—A)H H , p X, t , ,....., HK, 1 + 1 AH ) ... , , ... , by (3.7) Hp’x’ o(P_ , X1 ) 1 , H_ 1 ,byS9 = Hp”x’ (P_ , X1 ) 1 , H 1 = PROOF , H_ 1 ) ci 1 (p’, p, X_ OF THEOREM . 3.4. Part (i) follows from Lemmas 3.2(u) and 3.3(vii). Part (ii) follows from Lemma 3.2(iii), Theorem 3.1(v) and Lemmas Lemmas 3.2(iv) and 3.3(iii). 3.3(u) and 3.3(vi). Part (iii) follows from Part (iv) follows from Lemmas 3.2(v) and 3.3(v). Part (v) follows from Lemmas 3.2(vi) and 3.3(iv). Part (vi) follows from Lemmas 3.2(vii) and 3.3(ix). Part (vii) follows from Lemmas 3.2(viii) and 3.3(xi). Part (viii) follows from Lemmas 3.2(x) and 3.3(i). Part (ix) follows from Lemma 3.2(xi). Part (x) follows from Lemmas 3.2(xii) and 3.3(viii). Part (xi) follows from Lemmas 3.2(xiii) and 3.3(x). ci 113 PROOF OF LEMMA 3.4. Necessity: By T, p(p’ p, X, H) p(p’, pi, X, H) = p(p 1 pi, X, H) Since the left—hand side and, consequently, the right—hand side of this equation is independent of p, it can be rewritten as p(p, p . X, H) 1 where iN p(lN, p, p(lN, pi, = X, H) X, H) , X, H) 1 S(p S(pi, X, H) —. —. is the N—dimensional column vector of ones. Sufficiency: Straightforward. PRooF OF THEOREM 3.5. By (3.6) and I (= P A T), p’x H { j=1 1 PROOF OF THEOREM p(pi,pI, X, H)} = pi’xi 6(p, X, H)}_l 1 H p 1 H ’x t5(pi, X, H) { = Hp ’ 1 x , X, H) 1 (p I K lj1 , by Lemma 3.4 pi’xi 1 H 6(pi, X, H)J 3.6. Positivity: , X, H) 1 PHD(P’, p Fpz’xk Ok(H) 11 k=1LP . >0. Positive Monotonicity: For any p <ps, 0 k ( K h’xk p X, H) := . PHD(P” 1 k=1 [P’] Linear Homogeneity: For any A r(APi)1k1°W) k=lL pi’xk Aji, X, H) := II PHD(’ p X, H). , 1 £ K PHD(P” k=1[P”] ] K = All [Pi?xk]OAXH) =: APHD(P’,P,X,H). 114 Transitivity: For any 1 E K, pZ, X, H) , t ’p 3 PHD(P , X, H)pHD(p 1 K II := k= 1 K [ pl’xkpi’xk pj’k pl’xk] Ok(H) k(Jq) 0 rpi?xk =11 i[pilxk] k= =: PHD(P” p, X, H). Commensurability: For any A € K PHrXP” Apz, )JX, H) : II k 1(pi)?(xk) Ok(H) I ) 1 K rp’(’- .llI 1 0(H) k= 1[pi?(5?5j1)xk] K p’xk since k=l[Pj] =: = IN PHD(P’,P,X,H). Commodity Symmetry: For any permutation matrix K PHD(’NP’, INP, INX, H) 5_1 := II k=1 K IN, 1( p 1 )?(xk) I 9k(H) 1 LNP’YaNX] N 1 Ipul( N )xk ? =flI k= 1Lpi(N 1 Ok(ii) IN)Xkj K rpi’xkl = =: [_ 1 pixkj PHD(P , 2 P,A’,H). Since IN’IN IN 115 Weight Symmetry: Consider the bijective mapping ,: K x K which satisfies (B.2) for any 1 E K and let (B.3) hold for all k E K. Now, PHD(P’, (k)(Jq) Ip ‘x 0 K pZ X7K, IK’H) I := k =1 Lp’x Ok(ii) K = =: Quantity Dimensionality: For any (9 pZ, ,6X, H 7 ) PHD(P ’ 2 K r := II I k=1LP ] f i’xk ‘1 k =1 LP PHD(P, 1 P,X,H). )€ z’(f3xk) 7 k 0 ) 1 p ( K fpi’xk] =11 (flxk)] =: k-lLPj’i PHD(P” p, X, H). Strong Quantity Dimensionality: For any 1 e K and for any A e ll, PHD(P” pZ, 1 ,x ..., AX, x ,..., 1 , H) pilxkl ( 0 H) p’(Ax’) 0(H) [pi’xkj := K = [pi’(Ax)] Ok(ii) pi’Xk k=1[P”] =: PHD(P” p, X, H). Determinateness: For any n E urn X, H) pO urn p-0 I and for any 1 PHDO” pZ, X, H) := := E K, lim pO K lim p-4O k=1 rpi,xk10k() [pi’xkj rpi’xk] 0”(H) f1 [pi’xkj k= K = F pi?xk k= 1 L] K 1pI’x ilL] = k=1 > 0; O(H) >0; 116 urn p, X, H ):= urn PHD(P ’ 4-40 3 x,-0 K p’xk [pi’xk] k=1 1pxk =i 1 Ok(ii) k* 1[pi?xk] F__1 [p’xi,j p’x. >0; urn X, H) H0 PHD(P” ‘ urn H0 fj k= plixic i[pi’xk] K rpi’xkl =rH1 [pi’xkj , ... 1 1 Ok(H H _, 0, H , ... HK) + 1 , , k= >0. Country Partitioning Test: For any 1 E K and for any ). £ (0, 1), PHD(P” pS, X, x ,H 1 , ... 1 (l—))Hi, IIi+i, ... HK, All ) 1 , , Ipi’xk 0(1-I) klLPj”] II = Ipi’xk k(Jq) 0 [] (1—A)0’(H) [p’x1] (H) [$l]O 1 k =: O. 1PHD(P’,P,X, Tiny Country Irrelevance: For any 1 £ K, 1 irn PHD(P” p•, X, H ,..., H 1 , AH _ 1 ,H 1 , ... HK) ÷ 1 , -, 0 := urn -+ 0 ) 1 rpilxklHkI(Em*tHm+ AH k * iLpJlxk] I(Em*iHm. + AH 1 AH ) 1 p”xj 117 rpi’xk = klLP’] =: HD(P” PROOF OF COROLLARY kD(, p, X_ , H— 1 ) c 1 3.6.1. Since PHD X, H) := { . satisfies P, HIP”X’PHD(P’, p, Hp’x PHD(p’, pi, X, H) X, H)} > o. Now, K K H p 7 i’xi : UHDQ,X,fO= E i I_l[_lHIP$1Xk_l[P31Xk] j=1 v’K i =1 Hp’x — 1 H, p”x’ TI E i[pilxkf Ok(H) =1. Thus, aHD PROOF OF satisfies Si. By Lemma 3.2, THEOREM satisfies S2—S12. D 3.7. Straightforward. PROOF OF COROLLARY PROOF OF THEOREIvI aHD 3.7.1. Straightforward. 3.8. Positivity: PAB(P” pS, X, H.) := pElHkxlC p”EiHk xk > 0. Positive Monotomcity: For any p <ps, , 3 PAB(P p$, X, H) := p’E H 1 kx ‘EiHkX ‘iHk ?EHxk PAB(P’,P,X, H). 118 Linear Homogeneity: For any A E (Ap) ‘ 1 Hk x E PAB(P” Apt, X, H) := piIEKHxk = A 1 Hk X” p”E APAB(P’, p, p”E H 1 kxP Transitivity: For any 1 E K, PAB(P” p’, X, 1 H)pAB(p , pZ, X, H) := pl?Y2lHkxJc pElHkxk p ’ Ei.Hk .XC p” E Hkx’ 1 — — pElHkxk pilE Hkxk ,P,X,H). 2 =: PAB(P Commensurability: For any A E XIC) 1 ( p t 5 )’ Ef, 1 Hk(A_ PABQP’ X’X, H) := (pi)’ 1 Hk(5 E x k) p2?(,1)jl)E,Hkx’C pElHkxk = p”E’iHk xk =: PAB(P” 5?_1 = p, X, H). Commodity Symmetry: For any permutation matrix p(Np2, INP, INX, H) := since 7, N (INpzYEf.lHk (INXP) (iNp’Y E 1 Hk (IpjxC) IN X, H). 119 pZf(iNfIN)Y4=iHkx p’(N’ IN)EiHk Xk pElHkxk since = pi1EHkxk IN’IN IN =: PAB(P’,P,X,H). Weight Symmetry: Consider the bijective mapping çÜ: K x K which satisfies (B.2) for any 1 E K and let (B.3) hold for all k € K. Now, pS? PAB(P” p, X, IK’H) := 1 H(k) X E piiElHk)xk Quantity Dimensionality: For any (/9, ‘y) E 1 Hk x” p” E= PAB(P,P,X,H). = 2 Hj(/9xk) 7 pEl ( PAB(P” p, fiX, ‘yH) := 1 Hk Xk p’ E= Ps(P’ P’ X, H). = pilE fl(7Hk)(flxC) p’EiHk xk pUE(1)(Hxk) pHEHXk Total Quantities Test: p X, lK) := PAB(P” p’Ei.(1)(Hk xk) Determinateness: For any n € I and for any 1 € urn p, X, H) : PAB(P ’ 2 = pO = H 1 pilE =: PAB(P’,P,X,H). K, plElHkxk >0; , 1 Hk E pzlE Hxk urn urn X,H)— p-4O PAB(P” P’ p-0 ElHkxk 1 Hk X p’ E — p’EtHkx > — pilElfrJkxk 0; 120 urn 1 ) ilHkxk urn X, H) := x-0 PAB(P” p, %—O >0; p’E H 1 k Xk iyK Hkx’ urn := H-O ElHkxk urn — pEklI-IkxP >0. — P’EklHk Strong Country Partitioning Test: For any 1 € K and for any A £ (0, 1), PAB(P” p, , 1 x ... , , 1 x 1 X , 1 ... , .x, [l, ... , II_, (1—A)H , IIi+i, 1 pz[EHxlc + (1—A)H ’ + AH 1 ] x 1 p”[E . 1 Hk x’ + (1— A)Hi 1 .x API ] ’ 1 + plHkXlC p”EiHk Xk =: , 1 , 2 PAB(P X,H). P Tiny Country Irrelevance: For any 1 € K, urn PAB(P’. p ,..., H 1 , AH — 1 . X, H 1 ,H 1 , + 1 ... A-’ 0 , HK) pi[EHkxlc + AH ] x 1 urn ‘° — p?[Ek*lHkxk + 1 AH x ’J PEklHkX’ P2EklHkJk PAB(P” p, X_ . 1 , H— ) 1 . ci ... , HK, All ) 1 s.jdojd UAS (i ‘I ‘r - ‘d ‘dw 1 H ‘c / (XcH),Cd (:XH),gd (gX 11 ‘d ‘d)q ‘xc :iijuoi.iodoij dd ( ‘I ‘X ‘çX ‘d ‘d) / ‘ o < (I ‘ = ‘ cx,cd X ‘x ‘cx ‘d ‘d) —. (cd’() U çX = a iCq sq u!MoT1oJ —. (x ‘x ‘d ‘d)çb x pu dV cX, d / x,d =: = d jj :ipupj SJJSIS ?j ( ‘x ‘d ‘d) xpu’ uoidwnsuoo jijq 4J 6 t’IS—frS JOOJ do iO :iClTAi2TsoJ NIoHj, 1S sJsT3es UIW1 ‘i’ ‘snqj, •T= E T ,d/xddH X 1HT — Td/x 1 , 1 d H’JJE f_________ J H — = H (H ‘X ‘J)D ‘I ‘MOM 0 < 1 ‘= J ‘x ‘d ‘cd)8Yd x,dj-j (H ‘X ‘d ‘cd)EVd çxçd LII {(H — ‘J SJSIS (H ‘x ‘i)-° —. —. T8 3UTS AIVTIOIO3 dO dOOTJ Ill 122 /(pi,pi,x3,x,1,1), bySQD H,p2’x’ — — H =: H, ç5(p’, p, x , x. 2 — Q4. Strong Monetary Unit Test: (cp ) 1 ’(I3x) (x,pi)’(f3xi) / (a,p’, j3p f3x’, flx) : pi’x api’x — { (pi,pI, fiX’, /3XZ, 1, 1) , 1, 1)} 1 x by H, HDM (4= H A CR) and QD (4= SQD) =: q(p’, pZ, Q5. xi). Commensurability: (‘\p)’ )p: 51xi _1Xi) 5 (A_lxi) := (Ap’)’ p , ,\pZ, 2 / ()p 12 51 ‘()5J ) 1 x = /(pi,p,x , 2 x, 1, 1), byC piI(5._1)Xi cb(p’, p, Q6. x’, x) since 5.A-’ = IN. Country Reversal: q5(p, p, x, x’) := / (p$, p, x, x, 1, 1) —1 = ‘Ip”x2 / =: 1/ 1, 1) x) 1 (5J p, x, x, 1, 1)} c;b(p’, p, x, x) , by CR and WS 123 Q7. Commodity Symmetry: (INp’)’(INx) (iNp’, INP, INX’, INX) := I (I,p’, INP, INX’, INX, 1, 1) ‘(INX’) p(lN’ IN)X 5 1 1’1 by CS , pZ, x x 2 / (p = , , , p”(lN’ IN)X’ q’(p’, pZ, Since q satisfies Q1—Q7, k* 0 x) since IN’ IN = IN. satisfies S1—S5, S7 and S8 by Diewert (1986, Prop. 8). Forj * I * k, 1 +1 K ,p,p,...p,X,H) p,...,p _ o*(P, X, H) H 2m*iHm qXpk, çb(pc xk, x) pi. H çb( k ) + m tm xk, x p , 1 p’xH pIxz > — [] P’x pz, x’, x, Em*z H 1 _ PXZ Hm cb( k pm, xk, Xm) + [pilxz/pkIxk]/(pkpi H q ( 5 , xC, x) 1 , k, X, ( pi, k* satisfies Sil. x) XC, XZ) 1 1)/oL*(P, X, H) pi xk pk , Thus, p, by (3.15) 1 1—1 K ,p,p j+1 ,...p,X,H) ol*( p,...,p L(P, X, H) (k 1, 1)/o(P, X,H) k m,.k x) +H Em*iHm(P,P , H by M x) by (3.15) and (3.14). 124 Now, q5(pk p , Hk. XC, H x) 1 c4*(P, XH, lK) IYfib(pkpt ,Hkxk,H x ) 1 [pz(Hi X)/p’’(Hk X!c)]Ip(plc, p, Hxk, H x, 1, 1) — — 1 [p i=i ?Hl x’)/p Hkxk)j/(ppl,Hkxk,Hlxt, 1 xk, x, p H 1, 1) 1, 1) = , H [ph1xh/pk?x9/(pk,pl,xk,x1, 1, 1) 1 E bySQD 1 (pk p H . XC, x, 1, 1) 1 = , EhfiHi(plc,pl,xk,xt, 1, 1) —. Z —. Thus, 0 k * (D , by (3.15) V .11, satisfies S13. Since (k 4(P, X, H) 1 H o*(P, X, H) H (k p, xk , 1 p ) 1 x Xl) depends on prices other than p’ and p, there does not exist a restricted—domain consumption index satisfying RT with o PROOF OF THEOREM 0 k *. 3.10. Since Therefore, p is not a restricted—domain PPP index. o satisfies Q1—Q7, EKS satisfies S1—S7 by Diewert (1986, Prop. 8). The remaining parts are straightforward. PROOF OF THEOREM 3.11. Since satisfies Qi Q7, o satisfies Si, S2 and S4— Si 0 by — Diewert (1986, Prop. 8). The remaining parts are straightforward. 125 3.12. PROOF OF THEOREM Since satisfies Qi—Q7, satisfies Si, S2 and S4—S7, uw ‘JDW satisfies Si, S2 and S5—SiO, and uqw satisfies Si, S2 and S4--S1O by Diewert (1986, Prop. 8). The remaining parts are straightforward. 3.13. Since X £ PROOF OF THEOREvI Prop. 13). AH+ , 1 ..., For all k £ K, let To show that ?HK). K, 11 a satisfies Si, S2 and S4—S9 by Diewert (1986, GK 0 , H,, AHj+i,..., AH 1 ,..., .XH,_ 1 , H, _ 1 oK(P, X, AH := satisfies SlO, substitute for 7r,L in (3.22a) using (3.22b), GK replace Hk by AHk for all k € K\{j, 1) and take the limit as \ N K E E Hk x lim.) k*j,i)’O m=1 H N H, 4 + p, 4 — = •-, p1’ (H 1 2 H, x’) v K p, 0: X lim—=1 E 11 p t -‘O 1 ‘x H N X p lim—+E ° 1 H x + H4p”x’ lim +1 x 1 H x 1 H X -1. - = H 1 x + N Ep =1 H, 4) (H 1 4) N (H, 4) Wi x) 4 H 4 1 p’x N P 4 7Llp’x 14 H4 + H Ep, n=1 4 1 H4+H The remaining parts are straightforward. PROOF OF THEOREM 3.14. H) and let Proportionality: 0k := 4 H(, + Hk2 For all k € K, let ã := vH(P, x , 1 X, H). For i * 1, H p’(Ax) H2 = k*1 K 1’ Xk E Ilk 2P X 2 O k=1 P K = k’ I ‘ 2 pk 1 D H k=1 , 2k xk x + H? ...., , t x 1 1 x 126 0j cli — = A ak and — ‘Yk = — 0 0-i 0-i Country Partitioning: For all k € K, let HK, ) and let 1 AH clk := t4fi(P, p’, X, x ,H t ,..., H 1 , (1 — 1 0-k := — ,1 1 A)H H , + ..., 4 H (P, X, H). For i * 1, Hk2.h3tk E for all k € K\{l}. — — k*1 + [(1 — 2 A)H] — P!_ + (AH 2 1) 0-j k 0 0-K+1 pkfxi Ok = E H 2 k*1 K k=1 0-i — UK+1 + H 2 — + H 2 pklxkai 0pi’xk pi’x: ‘Yk 0_i — k — 0- (1—A)— =A 0-i 0— = (TK+1 0-i (ri and — 0-i — k 0- = — forallk€K\{l}. (Ik Thus, K+1 k 0- —= E K+1 since E (Ii = k=1 = 0-i k1ci0-i k=1. (1 — 1 and A)u K k = 0- E (Yk = 1 k=1 0K+1 = Aoi. Strong Dependence on a Bilateral Formula: For all k E K, let H,, 1 AH+ , ..., AH , H, 1 1 AH+ , ..., AHK). Now, o := H 4 (, X, 1 AH , ..., AH,_ , 1 127 Iofl urn k*i,j )‘ Pz 0 X urn 0 I— [J + lim 1o1 0 io’i 0 [uj 1a.1 = E k*i,j urn A-’ 0 p x xi)pi’(J-J pi’(H x 1 i) = The remaining parts are straightforward. p”xJ 0 1/2 =: 128 APPENDIX C PROOFS OF THEOREMS IN CHAPTER 4 PROOFOFTHEORBM4.1. ByT,foranylE K, ’p t (P, X, H)p (P , X, H) 22 = p”(P, X, H). Thus, by P, X, H) = 1 Now, p’(P, X, H) = — — = PROOF OF THEOREM 4.2. pit(P, X, H) pii(P, X, H) pii(P, X, H) pii(P, X, H) pii(P, X, H) , 1 p2(P, X, H) = by P by T byWI.D First, by (4.2), there is a real number associated with any two elements A and B of 1’. Clearly, AA,A , 2 K(K— 1) K—i K E E = 1 i = r hi’ h Next, AA,B = = = 2 K(K— 1) 2 K(K—1) AB,A. K—i K E E = = K—i K E E [ PB’ ’ 1 PB PAJ PA in _l_ln[_II hi I h] j+i =i i=j+i hi’ I I r hi’ PAI range A which is 0 if B * A. If B PA I 1 iini_i_in[_ii =0. I hi I hi] LPAJ PA I IPAI j+1 AA,B> AA,B £ hy 11 PA IinI_I_in[_li I I hil hj LPBJ PB I = A then (H ‘X (H ‘X o is cq ‘ (H ‘X ‘d)-° {(H X ‘j):d = gX,gdJ_J T=c } = (H ‘X ‘cI):D (H ‘X ‘cl)r0(H ‘X ‘d)d t,d ‘H (H’x’d)c.° = (H’X’d):D(H’X’c1):1’11 cd £H 1 cX ‘() wolJ ‘iirni (H ‘X ‘c1)° CX,CdcH (H ‘X ‘J)c.° sXdH cO.o ‘J).D ,x,,dIH (H ‘X ‘s),.o x, ‘H 7 zX,:d H (H ‘X ‘cI)c° d (H ‘X ‘d),d(H ‘X ‘cJ),cd ‘XN (H ‘X ‘c1):° CX,CdcH 0 < (H ‘X ‘J)c.° H —. —. ‘IS D rIrDd Iii. Ill_i LI L ffdJ — n 6 ro + I—I — I I I—i UTI} I I uijJ UJ — 1+ I (H ‘X ‘j)d ‘ I7 NOHJ IO 1OOUJ — =c i÷ç= = , 1= = c = I I..1 .J = i 1 I I i I I—I u + I—I uji IDdf I IfdI I “v + °“V = i I rvcfl fd1 UJ — ryp v [] I I—I 1 IM/J I 1?d1 IlI L4J II 1 i I I I—lull Iu,jI 4d ‘L 3 3 iU JOJ ‘i(JTu 6T 130 PROOF OF THEoREM 4.4. By (4.7) and (4.9), p’(P, X, H) where ii’ is the N x = pizcc N diagonal matrix with equivalence follows from setting PROOF OF THEOREM o(P, )‘P f3jUS H), X ( S 9 )_1• = p ci 4.5. By (4.7) and (4.9), , H) = ((pUS)-1p, I3PUSX H). The required equivalence follows from setting : (pus )_1 for all n E A’. The required
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Multilateral approaches to the theory of international comparisons Armstrong, Keir G. 1995
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Title | Multilateral approaches to the theory of international comparisons |
Creator |
Armstrong, Keir G. |
Date Issued | 1995 |
Description | The present thesis provides a definite answer to the question of how comparisons of certain aggregate quantities and price levels should be made across two or more geographic regions. It does so from the viewpoint of both economic theory and the “test” (or “axiomatic”) approach to index-number theory. Chapter 1 gives an overview of the problem of multilateral interspatial comparisons and introduces the rest of the thesis. Chapter 2 focuses on a particular domain of comparison involving consumer goods and services, countries and households in developing a theory of international comparisons in terms of the the (Kontis-type) cost-of-living index. To this end, two new classes of purchasing power parity measures are set out and the relationship between them is explored. The first is the many-household analogue of the (single-household) cost-of-living index and, as such, is rooted in the theory of group cost-of-living indexes. The second Consists of sets of (nominal) expenditure-share deflators, each corresponding to a system of (real) consumption shares for a group of countries. Using this framework, a rigorous exact index- number interpretation for Diewert’s “own-share” system of multilateral quantity indexes is provided. Chapter 3 develops a novel multilateral test approach to the problem at hand by generalizing Eichhorn and Voeller’s bilateral counterpart in a sensible manner. The equivalence of this approach to an extended version of Diewert’s multilateral test approach is exploited in an assessment of the relative merits of several alternative multilateral comparison formulae motivated outside the test-approach framework. Chapter 4 undertakes an empirical comparison of the formulae examined on theoretical grounds in Chapter 3 using an appropriate cross-sectional data set constructed by the Eurostat—OECD Purchasing Power Parity Programme. The principal aim of this comparison is to ascertain the magnitude of the effect of choosing one formula over another. In aid of this, a new indicator is proposed which facilitates the measurement of the difference between two sets of purchasing power parities, each computed using a different multilateral index-number formula. |
Extent | 2951176 bytes |
Subject |
Purchasing power parity -- Mathematical models Prices -- Mathematical models Econometric models |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-15 |
Provider | Vancouver : University of British Columbia Library |
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. |
DOI | 10.14288/1.0088103 |
URI | http://hdl.handle.net/2429/7180 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
Graduation Date | 1995-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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