Theoretical Foundations for International Comparisons of Living Standards

Transcription

1 Theoretical Foundations for International Comparisons of Living Standards J. Peter Neary Oxford, CEPR, and CESifo Conference on International Comparisons Beijing Normal University October 29, 2018 J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

2 Introduction The ICP at 50 快乐 Happy Birthday! Lá breithe shona duit! J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

3 Introduction International Comparisons of Living Standards 1 Data ICP, Penn World Table, etc. Yield empirical index numbers An essential prerequisite; BUT: answers without questions (Afriat) 2 Parametric Theory Utility functions Yield theoretical/benchmark/ true index numbers 3 Non-Parametric Theory Revealed preference Can inform the links between 1 and 2 J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

4 Introduction Motivation International Comparisons of Living Standards: How to correct for departures from PPP? How to compare living standards when tastes differ internationally? 1 Assume Either: The same preferences in all countries 2 Or: Heterogeneous preferences, compared via a reference consumer Problems: Identical preferences are inherently implausible Who is the reference consumer? A real individual or reference group/country? The average employee of a multinational or a diplomatic corps? Estimated aggregate world preferences? [Neary (2004)] Data may be inconsistent with any maximizing consumer J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

5 Introduction This Lecture Substantive: Use revealed preference techniques to identify comparisons that violate either taste homogeneity or homotheticity Show how to quantify the importance of such weak links in the chain of international comparisons Methodological: New way of visualizing hypothetical and real data sets New results on GEKS index New comprehensive data set, drawn from all years of ICP J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

6 Introduction Background Almås, I., I. Crawford, and J. P. Neary (no date): International Comparisons of Living Standards with Heterogeneous Tastes, in preparation. Crawford, I., and J. P. Neary (2008): Testing for a Reference Consumer in International Comparisons of Living Standards, American Economic Review, 98(4), Feenstra, R. C., H. Ma, J. P. Neary, and D. P. Rao (2013): Who Shrunk China? Puzzles in the Measurement of Real GDP, Economic Journal, 123(573), Neary, J. P. (2004): Rationalizing the Penn World Table: True Multilateral Indices for International Comparisons of Real Income, American Economic Review, 94(5), J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

7 Introduction Outline 1 Bilateral Empirical Indexes 2 Bilateral Indexes: Parametric Theory 3 Bilateral Comparisons: Revealed Preference 4 Multilateral Comparisons 5 The Data 6 Conclusion 7 Additional Material J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

8 Bilateral Empirical Indexes 7 Additional Material J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51 Outline 1 Bilateral Empirical Indexes The Setting Log-Laspeyres Space The Fisher Index 2 Bilateral Indexes: Parametric Theory 3 Bilateral Comparisons: Revealed Preference 4 Multilateral Comparisons 5 The Data 6 Conclusion

9 Bilateral Empirical Indexes The Setting The Setting Countries: j = 1,..., J Commodities: i = 1,..., N Perfectly measured data: Quantities consumed: q j = {q 1j,... q ij,... q Nj } Local prices in a common currency: p j = {p 1j,... p ij,... p Nj } National expenditures: x j = p j q j J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

10 Bilateral Empirical Indexes The Setting The Research Agenda Goal: A set of index numbers Q jk, which give the real income/living standards of each country k relative to every other country j. Ignore: Data quality issues Within-country inequality: Almås and Sørensen (2012) Within-country taste heterogeneity Non-income components of living standards: Fleurbaey and Gaulier (2009) International differences in quality: Crawford and Neary (2018) Comparisons of national income/productive potential: Feenstra, Ma, Neary, and Rao (2013) J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

11 Bilateral Empirical Indexes Log-Laspeyres Space Log-Laspeyres Space Laspeyres indices, taking each country in turn as base Real income of k relative to j: L log Q kj Q L jk = pj q k p j q j L log Q jk Real income of j relative to k: Q L kj = pk q j p k q k J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

12 Bilateral Empirical Indexes Log-Laspeyres Space Regions of Log-Laspeyres Space (i) Q L jk > 1, Q L kj > 1 Each country considers itself worse off than the other Super-Gerschenkron Effect (ii) Q L jk < 1, Q L kj > 1: Country j unambiguously better off. (iii) Q L jk < 1, Q L kj < 1: Each country considers itself better off than the other (iv) Q L jk > 1, Q L kj < 1: Country k unambiguously better off. (ii) (iii) L log Q kj (i) (iv) L log Q jk J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

13 Bilateral Empirical Indexes Log-Laspeyres Space Laspeyres and Paasche in Log-Laspeyres Space Real income of k relative to j: Laspeyres: Q L jk = pj q k p j q j Real income of j relative to k: A Paasche: Q P jk = pk q k p k q j Symmetry: Q L jk = pj q k = 1 p j q j Q P kj Q P jk = pk q k p k q = 1 j Q L kj B B A { log Q L jk = log Q P kj log Q P jk = log Q L kj Roles of j and k can be reversed Only 1 2 J(J 1) independent bilateral comparisons Laspeyres for tourists, Paasche for exiles J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

14 Bilateral Empirical Indexes Log-Laspeyres Space The Laspeyres-Paasche Spread The Laspeyres-Paasche Spread: Symmetry: S jk QL jk Q P jk S jk = Q L jkq L kj log S jk = log Q L jk + log Q L kj J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

15 Bilateral Empirical Indexes The Fisher Index The Fisher Index The Fisher Index: log F jk = 1 2 (log QL jk + log Q P jk) Rewrite as a difference of log Laspeyres: log F jk = 1 2 (log QL jk log Q L kj) Exact (i.e., equals true index) for homothetic quadratic preferences: Konüs (1939) Similar to Törnqvist: Exact for translog preferences [Diewert (1976)] J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

16 Bilateral Indexes: Parametric Theory Outline 1 Bilateral Empirical Indexes 2 Bilateral Indexes: Parametric Theory 3 Bilateral Comparisons: Revealed Preference 4 Multilateral Comparisons 5 The Data 6 Conclusion 7 Additional Material J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

17 Bilateral Indexes: Parametric Theory The Allen Index Let e(p, u) be the minimum expenditure to attain u facing prices p Then the true or Allen index of living standards is: Q A jk = e(p, uk ) e(p, u j ) Problems: Need to choose the preferences: Whose tastes are these? Need to choose the reference prices p Except when tastes are homothetic... J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

18 Bilateral Indexes: Parametric Theory The Allen Index with Homotheticity Homothetic tastes: L log Q kj e(p, u) = ē(p)ū(u) log S jk 0 Now, Q A jk independent of prices: Q A jk = e(p, uk ) e(p, u j ) = ēj (p)ū(u k ) ē j (p)ū(u j ) = ū(uk ) ū(u j ) A L log Q jk... and lies between Laspeyres and Paasche: A log Q jk Q L jk Q A jk Q P jk J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

19 Bilateral Comparisons: Revealed Preference Outline 1 Bilateral Empirical Indexes 2 Bilateral Indexes: Parametric Theory 3 Bilateral Comparisons: Revealed Preference WARP HARP 4 Multilateral Comparisons 5 The Data 6 Conclusion 7 Additional Material J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

20 Bilateral Comparisons: Revealed Preference WARP Implications of Rational Preferences WARP: Weak Axiom of Revealed Preference jpk jpk or kpj WARP is rejected when both: j chooses q j though q k is affordable p j q j > p j q k Q L jk < 1 WARP Rejected kpj and: k chooses q k though q j is affordable p k q k > p k q j Q L kj < 1 Note: The LP spread, log Q L jk log Q P jk, is not very informative log S jk > 0 sufficient but not necessary for taste homogeneity J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

21 Bilateral Comparisons: Revealed Preference WARP WARP with 2 Countries Recall: with two countries, there are four possible sign patterns of log L: [ ] [ ] [ ] [ ] [ ] 0 λ :,,, λ Define an adjacency matrix A(log L): a jk = 1 if j is revealed preferred to k By convention, set a jj = 0 So: [ ], [ ], [ ], [ Each of which corresponds to a particular configuration of choices... ] J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

22 Bilateral Comparisons: Revealed Preference WARP WARP with 2 Countries (cont.) [ ] [ ] [ ] [ ] x 2 x 2 x 2 x 2 J J K K K J J K x 1 x 1 x 1 x 1 (i) No Revealed Preference (ii) j Revealed Preferred to k (iii) Violation of WARP (iv) k Revealed Preferred to j J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

23 Bilateral Comparisons: Revealed Preference HARP Implications of Homothetic Preferences log Q L L jk logq kj L log Q kj HARP: Homothetic Axiom of Revealed Preference jpk jpk or kpj HARP rejected when: Details L log Q jk Q L jk < Q P jk pk q j p k q k < pj q j p j q k HARP Rejected kpj i.e., LP spread is less than one J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

24 Multilateral Comparisons Outline 1 Bilateral Empirical Indexes 2 Bilateral Indexes: Parametric Theory 3 Bilateral Comparisons: Revealed Preference 4 Multilateral Comparisons The Issues The GEKS Index Star Indexes WARP and GARP 5 The Data 6 Conclusion 7 J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

25 Multilateral Comparisons The Issues Multilateral Comparisons What are the new issues that arise? 1 Empirical multilateral index numbers: We want transitivity, which most bilateral index numbers do not satisfy How well do they approximate a true index? Candidates: The GEKS index (used by OECD) 2 True index numbers A star system; e.g., Geary-Khamis (used in Penn World Table) 3 Revealed preference Transitivity of preferences is more demanding with more countries GARP more general than WARP J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

26 Multilateral Comparisons The GEKS Index The GEKS Index Standard definition: Geometric mean of all possible ratios of Fishers: ( ) 1 n F jh G jk = F kh h Properties: Ensures transitivity/circularity; i.e., avoids path-dependence Given that, minimizes sum of squared log-deviations from Fishers i.e., maximizes characteristicity conditional on circularity Issues: 1 Not so much characteristicity: F jk gets a weight of J 1 Does it equal a true index? Only with quadratic preferences In which case it is identical to Fisher (Neary (2004)) Same applies to CCD (multilateral Törnqvist) and translog preferences All Fishers get equal weights in GEKS, whether they violate GARP, HARP or neither J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

27 Multilateral Comparisons The GEKS Index The GEKS Index Reinterpreted Restate the standard definition in logs: Γ jk log G jk and φ jk log F jk Log-GEKS is arithmetic mean of all possible differences of log-fishers: Alternative definition: Γ jk = 1 (φ jh φ kh ) J Arithmetic mean of log-fishers along all possible 3-country paths: h Γ jk = 1 (φ jh + φ hk ) J h J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

28 Multilateral Comparisons The GEKS Index The GEKS Index in Log-Laspeyres Space L log Q kj log Q L L jk log Q kj HK hk jk jk jh JK GEKS jk JH L log Q jk GEKS is a weighted average of the Fishers Bilateral Fisher, φ jk, gets a higher weight; here 50% J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

29 Multilateral Comparisons The GEKS Index The GEKS Index in Log-Laspeyres Space L log Q kj log Q L L jk log Q kj hk jk jk jh HK' JK' GEKS jk L log Q jk JH' Fishers, and hence GEKS, identical to previous example But now all Fishers violate either WARP or HARP i.e., purely empirical indices: Don t check for consistency with GARP etc. J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

30 Multilateral Comparisons The GEKS Index The GEKS Index Extended to m-country Paths Why only 3-country paths? Extend to m-country Paths, m = 3,..., J: [Rao (1972)] Proposition Γ (m) jk Geometric mean of the Fishers along all possible m-country paths: Γ (m) jk 1 ( J m 2... φjh1 + φ h1h 2 + φ h2h φ hm 2k) h 3 h 1 h 2 h m 2 Γ (1) is degenerate: you must visit at least 2 countries Γ (2) is defined as just the Fisher: Γ (2) jk = φ jk (Only one path) Γ (3) is the standard GEKS: Γ (3) jk = Γ jk (J paths, J 2 via h j, k) = Γ jk for all m 3 and m J Proof relies on skew-symmetry of Φ Proof J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

31 Multilateral Comparisons The GEKS Index GEKS Extended to m-country Paths (cont.) Proposition gives an additional theoretical justification for GEKS Practical application? Result does not hold in general when steps are differentially weighted. J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

32 Multilateral Comparisons Star Indexes Star Indexes Add a new country (possibly one of 1,... J) as a reference An empirical star index takes some reference prices as base: Q jk = π.q k / π.q j π.q π.q = π.q k π.q j While a true star index takes reference prices and utility as base: Q jk = e(π, u k / ) e(π, u j ) e(π, u ) e(π, u ) = e(π, u k ) e(π, u j ) i.e., a reference consumer The Geary index is an example of the first, where the prices satisfy an aggregation condition The GAIA index (Neary (2004)) is an analogous example of the second J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

33 Multilateral Comparisons WARP and GARP WARP and GARP with 3 Countries GARP: Generalized Axiom of Revealed Preference All Laspeyres along a closed path are less than one WARP the special case with paths of length 2 How many configurations with 3 countries? With 2 countries, there are 2 different cross-comparisons, so 2 2 = 4 possible combinations With 3 countries, there are 6 different cross-comparisons, so 2 6 = 64 possible combinations! In general, n countries imply 2 n(n 1) possible combinations Over a million for n = 5, over a billion for n = 6,... So, no hope of understanding the issues by enumeration Instead, consider just three interesting configurations: J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

34 Multilateral Comparisons WARP and GARP WARP and GARP with 3 Countries (cont.) x 2 HK J JK JH H K x 1 (a) No Revealed Preference; WARP and GARP satisfied J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

35 Multilateral Comparisons WARP and GARP WARP and GARP with 3 Countries (cont.) x 2 K H JK HK JH J x 1 (b) All Possible Violations of WARP and GARP J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

36 Multilateral Comparisons WARP and GARP WARP and GARP with 3 Countries (cont.) In cases (a) and (b), GARP conveys no additional information to WARP Finally, consider: (c) GARP violated but WARP satisfied: q j revealed preferred to q h but not vice versa: Q L jh < 1, Q L hj > 1 q h revealed preferred to q k but not vice versa: Q L hk < 1, Q L kh > 1 q k revealed preferred to q j but not vice versa: Q L kj < 1, Q L jk > (c) GARP violated but WARP satisfied HK JH JK J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

37 Multilateral Comparisons WARP and GARP WARP and GARP with 3 Countries (cont.) GARP cannot be violated when WARP is satisfied with n = 2 But it can happen with n > 2 In general, with n goods, n < J, it is enough to check for GARP-violations along paths of length n or less [Deb and Pai (2014)] J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

38 The Data Outline 1 Bilateral Empirical Indexes 2 Bilateral Indexes: Parametric Theory 3 Bilateral Comparisons: Revealed Preference 4 Multilateral Comparisons 5 The Data 6 Conclusion 7 Additional Material J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

39 The Data The Data: All 6 ICP Rounds Number of Year Countries Comparisons , , , , ,576 ICP Rounds, Number of comparisons = 1 2 J(J 1) Consistently aggregated to 4 commodity groups J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

40 The Data Bilateral Comparisons: 2011 logq kj L logq jk L This shows log Laspeyres space for all 177 countries Very few bilateral comparisons violate WARP: only 14 out of 15,576 GARP: Adds 6 violations along paths of length 3, none of 4 J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

41 The Data Violations of WARP, 2011 Belarus Bulgaria Grenada Jordan Brazil Haiti Djibouti Iran Colombia Ecuador Armenia Barbados Lithuania El Salvador Norway St.Vincent & Grenadines United Arab Emirates Egypt South Africa Chile Latvia Seychelles Paraguay Fiji J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

42 The Data Violations of GARP, 2011 Jordan Grenada Paraguay Brazil Belarus Bulgaria Fiji United Arab Emirates Norway St.Vincent & Grenadines Egypt South Africa Chile Latvia Seychelles Haiti Djibouti Armenia El Salvador Ecuador Lithuania Barbados Colombia Iran Ukraine J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

43 The Data Extra Information from GARP Iran Colombia Ukraine Columbia: Iran: Ukraine: J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

44 The Data WARP Comparisons: All 6 ICP Rounds Number of WARP Rejections Year Countries Comparisons Number % , , , , , ICP Rounds: WARP rejections J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

45 The Data Other Results from the 2011 Data What else do we learn from the 2011 data? 65 comparisons exhibit super-gerschenkron 14 violations of WARP involve 24 countries All the rest (153) can be stars WARP adds only one (Ukraine): It can be a star 7,741 comparisons violate HARP: nearly 50%! J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

46 The Data Bilateral Comparisons: 2011 logq kj L logq jk L Very few bilateral comparisons violate WARP: only 14 out of 15,576 Many violate HARP: 7,741 which is nearly 50% J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

47 The Data WARP and HARP Comparisons: All 6 ICP Rounds Number of WARP Rejections HARP Rejections Year Countries Comparisons Number % Number % , , , , , , , , ICP Rounds: WARP and HARP rejections J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

48 Conclusion Outline 1 Bilateral Empirical Indexes 2 Bilateral Indexes: Parametric Theory 3 Bilateral Comparisons: Revealed Preference 4 Multilateral Comparisons 5 The Data 6 Conclusion 7 Additional Material J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

49 Conclusion Conclusion Comparing living standards internationally requires choosing a reference consumer Countries may not be comparable in principle Desirable to check consistency with GARP If not, weak links in the chain of international comparisons Fortunately, ICP data seem broadly consistent Problem much less acute for a star system Choice of reference consumer matters much more with non-homotheticity Homotheticity overwhelmingly rejected by the data J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

50 Conclusion Thanks and Acknowledgements Thank you for listening. Comments welcome! This research has received funding from the European Research Council under the European Union s Seventh Framework Programme (FP7/ ), ERC grant agreement no The contents reflect only the author s views and not the views of the ERC or the European Commission, and the European Union is not liable for any use that may be made of the information contained therein. J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

51 Additional Material 7 Additional Material More about HARP Afriat Efficiency The Log-Fisher Matrix J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51 Outline 1 Bilateral Empirical Indexes 2 Bilateral Indexes: Parametric Theory 3 Bilateral Comparisons: Revealed Preference 4 Multilateral Comparisons 5 The Data 6 Conclusion

52 Additional Material More about HARP HARP Let e(p, u) be the minimum expenditure to attain u facing prices p With homotheticity, this becomes: e(p, u) = uē(p) Expenditure minimization implies: e(p j, u k ) p j q k Dividing by e(p j, u j ) = p j q j yields: Q L jk = pj q k p j q j e(pj, u k ) e(p j, u j ) = uk ē j (p) u j ē j (p) = uk u j = QA jk Repeating for country k prices and combining yields: Q L jk Q A jk Q P jk So this is sufficient for homotheticity, and it can be shown that it is also necessary Corollary: With homotheticity, the Allen index lies within the Laspeyres-Paasche bounds Back to text J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

53 Additional Material Afriat Efficiency Quantifying Departures from WARP jpk jpk or kpj WARP Rejected kpj J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

54 Additional Material Afriat Efficiency Afriat Efficiency q 2 p k jpk jpk or kpj q j jk jk jk log 3 log 2 log 1 A j jk kj A k q k p j WARP Rejected kpj q 1 Inconsistent tastes: { pj q j > p j q k Afriat Efficiency: p k q k > p k q j } ε jk min{ε jk, ε kj } Dashed lines are iso-ε jk loci ε jk = pj q k = 1 < 1 p j q j Q P jk ε kj = pk q j p k q = Q L k jk < 1 J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

55 Additional Material Afriat Efficiency Quantifying Departures from HARP jpk jpk or kpj HARP Rejected kpj Now the LP spread is crucial J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

56 Additional Material The Log-Fisher Matrix The Log-Fisher Matrix Implication of Laspeyres-Paasche symmetry: The log-fisher matrix is skew-symmetric: log F jk = 1 ( log 2 QP kj log Q L kj) = log F kj Φ log F Φ = Φ J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

57 Additional Material Proof of Proposition for m-country GEKS Proof of Proposition for m-country GEKS Rewrite Γ (m) jk, m = 3,..., J: Back to text Γ (m) jk = 1 J m 2 h 1 h 2 h 3... (φ jh1 }{{} h m 2 (A) +φ h1 h 2 + φ h2 h φ }{{} hm 2 k) }{{} (B) (C) (A) Sum of all m-country paths starting from j: This is unaffected by the order in which other countries are visited:... φ jh1 = J m 3 φ jh1 = J m 3 φ jh h 3 h m 2 h 1 h h 1 h 2 (B) m 4 sums of m-country paths that do not start with j or end with k: Each of these reduces to the sum of all terms in the Φ matrix. Since this matrix is skew-symmetric, all these sum to zero:... φ hh = J m 4 φ hh = 0 h 3 h m 2 h h (C) h 1 h 2 Sum of all m-country paths leading to k: Similar to (A)... φ hm 2k = J m 3 φ hm 2k = J m 3 h 3 h m 2 h m 2 h h 1 h 2 J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51 φ hk

58 Additional Material Proof of Proposition for m-country GEKS Proof of Proposition for m-country GEKS (cont.) Back to text Collecting terms: Γ (m) jk reduces to Γ (3) jk, i.e., the original GEKS index. J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

59 Bibliography References I Almås, I., I. Crawford, and J. P. Neary (no date): International Comparisons of Living Standards with Heterogeneous Tastes, in preparation. Almås, I., and E. O. Sørensen (2012): Global Income Inequality and Cost-of-Living Adjustment: The Geary-Allen World Accounts, Discussion Paper, 20/2012. Caves, D. W., L. R. Christensen, and W. E. Diewert (1982): The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity, Econometrica, 50(6), Crawford, I., and J. P. Neary (2008): Testing for a Reference Consumer in International Comparisons of Living Standards, American Economic Review, 98(4), (2018): Hedonic Models and New Characteristics, presented at the Econometric Society European Meetings 2018, Cologne. Deb, R., and M. M. Pai (2014): The Geometry of Revealed Preference, Journal of Mathematical Economics, 50, J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51

60 Bibliography References II Diewert, W. E. (1976): Exact and Superlative Index Numbers, Journal of Econometrics, 4(2), Feenstra, R. C., H. Ma, J. P. Neary, and D. P. Rao (2013): Who Shrunk China? Puzzles in the Measurement of Real GDP, Economic Journal, 123(573), Fleurbaey, M., and G. Gaulier (2009): International Comparisons of Living Standards by Equivalent Incomes, Scandinavian Journal of Economics, 111(3), Konüs, A. (1939): The Problem of the True Index of the Cost of Living, Econometrica, 7(1), Neary, J. P. (2004): Rationalizing the Penn World Table: True Multilateral Indices for International Comparisons of Real Income, American Economic Review, 94(5), Oulton, N. (2012): How to measure living standards and productivity, Review of Income and Wealth, 58(3), Rao, D. P. (1972): Contribution to Methodology of Construction of Consistent Index Numbers. Ph.D. Thesis, Indian Statistical Institute, Calcutta. J.P. Neary (Oxford) Theoretical Foundations: Intl. Comparisons Beijing: Oct. 29, / 51