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International Comparisons of Welfare and Poverty: Dominance Orderings for Ten Countries Author(s): John A. Bishop, John P. Formby, W. James Smith Source: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 26, No. 3 (Aug., 1993), pp. 707-726 Published by: Blackwell Publishing on behalf of the Canadian Economics Association Stable URL: http://www.jstor.org/stable/135896 Accessed: 15/09/2010 05:10 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Canadian Economics Association and Blackwell Publishing are collaborating with JSTOR to digitize, preserve and extend access to The Canadian Journal of Economics / Revue canadienne d'Economique. http://www.jstor.org
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International Comparisons of Welfare and Poverty: Dominance Orderings for Ten CountriesAuthor(s): John A. Bishop, John P. Formby, W. James SmithSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 26, No. 3(Aug., 1993), pp. 707-726Published by: Blackwell Publishing on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/135896Accessed: 15/09/2010 05:10

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp://www.jstor.org/action/showPublisher?publisherCode=black .

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

Canadian Economics Association and Blackwell Publishing are collaborating with JSTOR to digitize, preserveand extend access to The Canadian Journal of Economics / Revue canadienne d'Economique.

http://www.jstor.org

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International omparisons of welfareand poverty: dominance orderings for tencountriesJ O H N A. B I S HO P East Carolina UniversityJO H N P. F O R M B Y University of AlabamaW. J A M E S S M I T H University of Colorado, Denver

Abstract. Inference-based tochastic dominance procedures are applied to Luxembourg In-come Study (LIS) data to rank ten western countries in terms of standards of living andpoverty. First-order dominance comparisons ranks more than 50 per cent of all pairwisecomparisons, and second-order generalized Lorenz) dominance adds another 25 per cent.Truncated dominance analysis is used to make inferences about poverty and up to 98 percent of the truncated distributions are ordered. The results differ substantially rom thoseobtained when only relative incomes and inequality are considered.

Comparaisons nternationales des niveaux de bien-etre et de pauvrete: classement de dixpays par la methode de la dominance. Les auteurs appliquent des procedures de dominancestochastique nf6ree des donnees disponibles sur les niveaux de vie et de pauvrete de dix paysoccidentaux selon le Luxembourg ncome Study. Les comparaisons de dominance de premierordre classent plus de cinquante pour cent des comparaisons par paire, et avec la dominancede second ordre on peut y ajouter un autre 25 pour cent. On utilise l'analyse de la dominancetronquee pour suggerer des inferences quant 'a la pauvrete et on peut ainsi classer quatre-vingt-dix-huit pourcent des distributions ronquees. Les resultats different substantiellementde ceux obtenus quand on examine seulement les revenus relatifs et l'inegalite relative.

I. INTRODUCTION

The dominance method for ranking Lorenz curves and entire distributions standsat the forefront of important advances in the theory and measurement of incomedistributions over the last decade. Important theoretical contributions by Atkinson(1970), Saposnik (1981), and Shorrocks (1983) establish a powerful relation be-tween the dominance of one income distribution over another and ordinal levelsof welfare. Foster and Shorrocks (1988) extend the theory to show an equallypowerful relation between dominance of income distributions and poverty. New

The authors express thanks to Paul Thistle for useful discussion and an anonymous referee forhelpful suggestions. The usual caveats apply.

Canadian Journal of Economics Revue canadienne d'Economique, XXVI,No. 3August aouit 1993. Printed n Canada Imprim6 au Canada

0008-4085 / 93 / 707-26 $1.50 ? Canadian Economics Association

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708 J.A. Bishop, J.P. Formby, W.J. Smith

statistical inference procedures pioneered by Beach and Davidson (1983) comple-ment the theoretical developments by permitting the dominance relation between

income distributions o be subjected to rigorous hypothesis testing. This paper ap-plies recent developments n the theory and measurement of income distributionsand welfare by testing for differences across ten countries.

We build upon recent international comparisons of income distributions byBishop, Formby, and Thistle (1991) and Bishop, Formby, and Smith (1991b) andapply inference tests for first- and second-order stochastic dominance. The paperalso tests for differences in international poverty using the truncated dominancemethod proposed by Foster and Shorrocks (1988).

The paper is organized as follows. The next section briefly reviews the relevant

theory and discusses recent international omparisons of income distributions. Sec-tion IIIpresents the empirical results. The fourth section integrates he findings withearlier work and provides a unifying perspective on cross country comparisons ofincome distributions. The final section provides brief concluding remarks.

II. STOCHASTIC DOMINANCE, WELFARE, AND POVERTY

Atkinson (1970) demonstrates hat for distributions with equal means, strong in-ferences about social welfare can be made about comparative states when one

Lorenz curve dominates another. As pointed out by Sen (1973), when the meansof the distributions of interest are unequal, the Lorenz dominance principle is de-void of welfare content. Saposnik (1981) and Shorrocks (1983) demonstrate hatthe dominance method can be used to evaluate income distributions with unequalmeans. Shorrocks shows that, when Lorenz curves are scaled by the mean of thedistribution, dominance comparisons can be made in the same fashion as withordinary Lorenz curves. Shorrocks refers to this rescaled Lorenz curve as the gen-eralized Lorenz (GL) curve. Like dominance in terms of ordinary Lorenz curves,GL dominance incorporates a preference for equality; unlike the Lorenz curve, theGL curve also incorporates a preference for efficiency. Analytically, GL dominanceis equivalent to second-degree stochastic dominance SSD), and we use these termsinterchangeably n the remainder of this paper.

Saposnik adopts a more straightforward approach and applies first-degreestochastic dominance (FSD) techniques directly to income distributions. The cri-terion compares absolute incomes in ranked (ordered) positions in the income dis-tribution and is referred o as 'rank dominance.' As is now well known, FSD impliesSSD and, as a result, rank dominance implies GL dominance. Rank dominance is apure efficiency criterion and unlike GL dominance does not contain a preferencefor equity.

The applied welfare theory underpinning he dominance approach can be sum-marized by following Atkinson (1970) and assuming the relationship between thedistribution of income and standard of living is given by a social welfare function,which represents ethical judgements regarding ncome distributions. Both first- andsecond-degree stochastic dominance impose restrictions on the welfare function,which are detailed below.

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1. Rank (first-degree) dominanceFirst degree stochastic dominance (FSD) is underpinned by the strong Pareto prin-

ciple and anonymity. In addition, to facilitate comparisons of populations of dif-ferent sizes, we adopt the population principle (Sen 1976). Together these as-sumptions imply that statistical cumulative distribution functions (CDF) for in-come contain sufficient information for ranking social states. Fonnally, let Fdenote the income CDF. The inverse distribution function or quantile function,X(p) inf {x: F(x) ? p}, p E [0, 1], yields individuals' incomes in increasingorder.

We denote the class of anonymous, ncreasing welfare functions as Wp.Saposnik(1981) proves the following theorem on rank dominance.

THEOREM 1. X >R Y iff w(X) > w(Y) for all w E Wp. Thus distribution Xdominates distribution Y iff X(p) > Y(p) for all p E [0, 1]. Iffor all p E [0, 1],X(p) = Y(p), then X and Y have the same income distribution and standard ofliving. If X(p) > Y(p) for some p, and X(p) < Y(p) for some p (i.e., the quantilefunctions cross), the distributions are non-comparable and cannot be ordered usingthe rank dominance criterion.

Foster and Shorrocks 1988) provide an important orollary to theorem 1 linkingrank dominance (FSD) to the head-count poverty concept. To see this relationship,let H(z) represent he proportion of the population at or below any poverty line, z.Specifically, H(X; z) = q(x; z)/n(x), where q(x; z) is the number of incomes in Xthat do not exceed z.

COROLLARY . X >-R Y iffX H(z) Y for all z.

The corollary implies that an unambiguous decline in head-count poverty is suffi-

cient for rank dominance. Conversely, f distribution X1 rank dominates distributionX2, then headcount poverty in XI cannot exceed that in X2, regardless of the in-come cut-off, z, used. Thus, truncating distributions X1 and X2 above any arbitrarypoverty ine, z, and testing for rank dominance on the truncated distribution providesdominance ordering of head-count poverty. From an ordering of truncated quantilefunctions one can conclude that head-count poverty is unambiguously ower in XIthan X2 irrespective of where the poverty line is drawn.

These concepts are illustrated n figure 1 which depicts the quantile functions,Q(x0), Q(x2), and Q(X3),corresponding o hypothetical ncome distributions xl, x2,

and X3. In the figure income is measured on the ordinate and the population decileson the abscissa. In this example, Q(x2) and Q(X3) ie everywhere above Q(xl), andwe conclude that xl is rank dominated by both x2 and X3. In contrast, Q(x2) andQ(X3) cross and the conditions of theorem 1 are not satisfied with the result thatthese two income distributions cannot be ordered using the FSD criterion.

If the quantile functions cross in a pairwise comparison, he analyst can proceedin two ways. First, despite the crossing it may be possible to reach important

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710 J.A. Bishop, J.P. Formby, W.J. Smith

Max Q(x3)

I Max Q(x2)Q(x 3)

C.0/

Q(x) / 7 Max Q(x 1)

Q(x1)

0 .2 .4 .6 .8 1.0Quantiles anked by size of income

FIGURE 1 Hypothetical quantile functions

conclusions about poverty. For example, in figure 1,. if the poverty line is at orbelow point C (the crossing point), we can conclude that head-count poverty isunambiguously less in x2 than in X3. This is necessarily the case, since Q(x2)dominates Q(x3) at all income levels below C. Second, further restrictions can beplaced on the class of admissible welfare functions by assuming a social preferencefor equity. This leads to the application of the GL (SSD) criterion.

2. GL (second-degree) dominanceAs with rank dominance, the income distribution CDF) contains all the informationnecessary to apply the GL criterion. Also like rank dominance, t is more convenientto define the GL function in terms of the inverse function, F-1. Adapting Gastwith's(1971) definition of the Lorenz curve, we can write the GL curve as

Gx(p) = j Fl(x)dx = pxLx(p)

where Lx(p) is the ordinary Lorenz ordinate and GX(1) = ,x. The GL criterionrequires that the class of admissible welfare functions be restricted to only thosethat are equality preferring. Dasgupta, Sen, and Starrett (1973) demonstrate thatthis amounts to assuming that the welfare function is S-concave. We denote theclass of anonymous, ncreasing, and S-concave welfare functions as WE.Shorrocks(1983) demonstrates he relationship between GL dominance, WE,and second-degreestochastic dominance with the following theorem.

THEOREM . X >GL Y iff w(X) > w(Y) for all w E WE. Income vector X gener-alized Lorenz dominates Y, denoted X >GL Y, if, and only, if Gx(p) _ Gy(p) for

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all p E [O,1], with at least one strict inequality at some p. Like ordinary Lorenzcurves, the GL criterion provides only a partial ordering because crossing gen-eralized Lorenz curves cannot be ranked. Thus, GL curves can be compared inessentially the same manner as ordinary Lorenz curves.

As in the case of rank dominance and head-count poverty, Foster and Shorrocks(1988) provide a corollary that links the GL criterion to the income-gap povertyconcept. An income gap is defined as the weighted sum of the income shortfallsof the poor, or,

r

P(x; z) = [1 /n(x)] Z(z -xi)z,i=1

where xi is the ith individual's income and r is the order statistic corresponding othe poverty line income, z. For any given z, the income-gap criterion s X >p(z) Y,iff, (I/n)axi > (I/n)Xyi, for all i up to r.

COROLLARY 2. X ?GL Y iff X ?P(z) Y for all z.

Corollary 2 implies that an unambiguous decline in income gap poverty is suffi-cient for generalized Lorenz dominance. Conversely, if distribution X generalizedLorenz dominates distribution Y, then income gap poverty in X cannot exceed thecomparable poverty n Y, regardless of the income cut-off, z, used. Thus, truncatingthe distribution above any arbitrary poverty line, z, and testing for GL dominanceon the truncated distribution provides the income-gap poverty ordering.

3. Recent cross-country comparisonsIn two recent papers Bishop, Formby, and Smith (1991b) and Bishop, Formby, andThistle (1991) apply the dom:inance methodology to make cross-country compar-isons. The former paper uses US micro data to estimate decile Lorenz ordinates andtheir variances and conducts pairwise statistical tests for Lorenz dominance acrossnine countries. The result is an exceptionally clear picture of relative inequality.'The latter paper extends Shorrocks's (1983) analysis of generalized Lorenz domi-nance by comparing FSD and SSD rankings using simple (zero variance) numericalcomparisons.2 This paper goes beyond the earlier work by combining the statistical

1 Virtually all of the Lorenz curves are ranked. Sweden, followed by Norway and then West Ger-many, has the most equal income distribution, and Switzerland and the United States have thegreatest inequality. The remaining four countries - Australia, Canada, United Kindom and Nether-lands - fall in the middle of the inequality hierarchy. Three definitions of the income recipientunit, the family, the individuals within a family, and 'equivalent' individuals within a family areconsidered. For each definition of the recipient it is found that in only one of thirty-six pairwisecountry comparisons do Lorenz curves cross.

2 The method used by Bishop, Forinby, and Thistle (1991) was the standard n the literature untilthe development of Beach and Davidson's (1983) test procedure, which only recently has begunto be generally applied in income distribution tudies. The older approach reats each country'sincome distribution data essentially as descriptive statistics and ignores the fact that a sample isused to estimate the income shares and other distribution tatistics, which are widely reported bygovernments and international agencies.

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inference procedures with an analysis of FSD and SSD to test hypotheses concerningdifferences in welfare and poverty among the LIS countries.

III. EMPIRICAL ANALYSIS

We consider ten LIS countries Australia, Canada, France, he Netherlands, Norway,Sweden, Switzerland, United Kingdom, United States, and West Germany. TableAl of the appendix shows the original national surveys, sample sizes, and timeperiods to which the data apply.3 The income concept used is net cash income,defined as market income plus private and public transfers minus direct (incomeand payroll) taxes. The income recipient unit is per capita family income in which,

for example, a family of four with net cash income of $40,000 is reported as fourincomes of $10,000 each. The data are weighted by the LIS person weights.

Before stochastic dominance tests can be meaningfully applied, the levels ofincome in the different LIS data sets must be corrected or differences n purchasingpower of monetary units across countries and adjusted so that they refer to acommon time period. In a fashion analogous to Shorrocks (1983) and Bishop,Formby, and Thistle (1991), we make adjustments by converting incomes to acommon currency and standardizing n a single year. All LIS data sets are expressedin 1979 u.s. dollars by adjusting each individual country's mean income using theper capita gross domestic product GDP) estimates of Summers and Heston (1988),which are expressed in international prices.4 We note that there are two difficultiesassociated with standardizing n this manner. First, Summers and Heston's estimatesare not precise measures of real purchasing power. Second, income distributionsare known to contain an element of cyclicality, with inequality rising in recessionsand falling in recovery.5 As indicated by table Al of the appendix, some of LIS datasets are for the pre-recession year of 1979 (United Kingdom, Norway, and UnitedStates) and one (Netherlands) s for a year of recovery, 1983. The remainder are for

1981 and 1982, which were impacted by a serious recession in most countries. Weconsider the sensitivity of the results to these difficulties below, after we describethe conversion of the data, summarize the inference procedures, and present theoverall results.6

3 The LIS corrects the data for differences in the definition of income and income recipient units.With the exception of West Germany, he survey coverage is comprehensive and national inscope, with 95.5 to 99.2 per cent of the populations sampled. Detailed descriptions of the data areprovided by Buhmann et al. (1988).

4 By using GDP to measure the level and the LIS data to measure dispersion of incomes, we are

implicitly assuming that a dollar's worth of non-transfer xpenditures provides equivalent benefits,which are divided proportionally among income quantiles.5 The cyclicality of inequality in the United States is well established. See Beach (1977).6 While the LIS data are unquestionably he best available for making international omparisons sev-

eral cautionary notes concerning the data are in order. First, differing customs lead to somewhatdifferent definitions of the family across the sample of LIS countries considered. For a discussionof this point see O'Higgins, Schmaus, and Stephenson (1989). Second, national survey data oftentop code the highest incomes, and some countries follow different top coding practices. The LISstaff has made extensive efforts to make the data as comparable as possible, including corrections

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Table A2 of the appendix reports the raw LIS per capita family mean incomes(column 1), the Summers and Heston per capita GDP figures (column 2) and theGDP per capita relative to the United States (column 3). The data in columns 1 and3 of table A2 are used to transform he LIS incomes into constant 1979 u.s. dollars,as shown in column 4.

1. Estimates and statistical inferencesWe estimate the quantile functions as a step function of the sample decile condi-tional means, Gi, and the GL curves from a vector of decile sample GL ordinates,G, where Go = 0, Gi = fiLi, and Glo = i. To test for differences in quantilefunctions and GL curves we construct 95 per cent confidence bands around each

of the sample curves. The formulae for the asymptotically distribution-free vari-ance expressions used to construct confidence bands for the GL curves and quantilefunctions are given by Beach and Richmond (1985).

To provide a more intuitive understanding f the inference tests, we explain themwith reference to confidence bands around a sample quantile GL curve) function.To construct a confidence band, we use the information rom the decile conditionalmeans (GL ordinates). Because this requires drawing inferences from the unionof ten disjoint subhypotheses, simultaneous nference procedures are appropriate.Following Beach and Richmond's (1985) procedure for ordinary Lorenz curves

and Beach and Kaliski's (1986) results for weighted sample data, a joint confidenceinterval confidence band) around a quantile unction GL curve) is constructed usingthe Studentized Maximum Modulus sMM) variate.7 The 5 per cent SMM critical valuefor deciles is 2.80.

The comparison of quantile unctions GL) confidence bands allows three possibleoutcomes. First, if the confidence bands overlap over the entire range, the quantilefunctions GL curves) are not significantly different and are ranked as equal. Second,if the quantile functions (GL curves) are not equal but intersect, the quantile (GL)

curves 'cross' and are non-comparable. Finally, if two quantile GL curves) neithercross nor are equal, then a rank (GL) dominance relation exits.

The inference procedure is illustrated in figure 2. Panel A of figure 2 plotsthe quantile function confidence bands for the United Kingdom and Australia.Australia's band lies below that for the United Kingdom from the first decile tothe fifth decile and lies above the British band from the eighth to tenth deciles.Thus, the quantile functions of Australia and the United Kingdom cross, and therank dominance criterion is unable to rank the two countries in terms of relativeeconomic well-being.

PanelB

of figure 2 plots theGL

confidence bands for Australia and the UnitedKingdom and depicts a statistical ranking of GL curves. Comparisons of panels

for differences due to variations n top coding practices. Nevertheless, differences remain and thisintroduces an element of noncomparability. Finally, the West German survey coverage (91.5) issignificantly below that of other countries because it excludes households headed by foreign-bornnationals. This too introduces an element of noncomparability.

7 Miller (1981) discusses simultaneous nference and the SMM distribution. Tables for the percentilesof the SMM distribution are provided by Stoline and Ury (1979).

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714 J.A. Bishop, J.P. Formby, W.J. Smith

Panel A Quantile confidence bands Panel B Generalized Lorenz confidence bands

$10 $4

8-

o 6

0S ~~~~~~~N2

0 U.K.-4 -

2-

t Australia Australia

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Quantiles Ranked by Size of Income Cumulative Fraction f Income Recipient Units

FIGURE 2 Confidence bands of quantile functions and generalized Lorenz curves

A and B of figure 2 illustrate the effects of the equity preference embodied inthe GLcriterion; he quantile functions cross, but the United Kingdom dominatesAustralia n terms of SSD. The British GL curve lies above Australia's at deciles onethrough eight, and the confidence bands overlap at deciles nine and ten. Equivalenceof the GL curves at the tenth decile implies that mean per capita incomes arenot significantly different. However, the greater equality of the British incomes isreflected in the fact that Britain's GL curve lies everywhere above Australia's up tothe eight decile, and this is sufficient for GL dominance. Thus, on the basis of FSD,

no conclusion can be drawn, but from the perspective of SSD, the level of economicwell-being in Britain is greater than Australia.

2. Stochastic dominance and economic well-beingTables A3 and A4 of the appendix provide the necessary information o apply FSD

and SSD to the ten LIS countries. Table A3 contains the decile means and standarderrors used to construct he confidence bands of quantile functions. Table A4 of theappendix reports the decile GL ordinates and standard errors, which are calculatedusing Beach and Kaliski's (1986) procedure or weighted data.

A summary of the results of the tests for stochastic dominance is provided intable 1. Rank dominance results are shown by the primary entries in table 1 with

marginal effects of GL dominance indicated by bold superscripts. A primary entryof '+' indicates that a country n a row rank dominates a country n a column. Forexample, a '+' in column 1, row 2, signifies that Canada rank dominates Australia.A '-' signifies that the country in a row is rank dominated by a country in acolumn. An entry of 'X' means that application of the statistical test reveals thatthe null and dominance hypotheses are rejected. This means that one country'squantile function lies below the other country's in one part of the distribution and

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TABLE 1Per capita welfare dominance

Aus. Can. Fr. Neth. Nor. Swed. Switz. W. Ger. U.K. U.S.

Australia *Canada + *France + -- *Netherlands + -- - *Norway + x + + *Sweden + - x x+ - *Switzerland + -- x- x+ x- x *West Germany + x + + + x+ *United Kingdom x - x - - *

United States + x x + x x + x x *NOTES

A '+' means that the statistical test reveals the country in the row rank dominates the country in thecolumn.

-' indicates that the statistical test reveals the country in the row is rank dominated by th}e ountry inthe column.

All 'x s' indicate the two distributions ross at the first degree. A superscript ndicates they are uncrossedusing the GL criterion.

An 'x+' means that the crossing at the first degree changes to '+' when the GL criterion s applied, inwhich case the country n the row GL dominates the country in the column.

An 'x -' means that the crossing at the first degree changes to '-' when the GL criterion is applied, in

which case the country in the row is GL dominated by the country n the column.

above it in another. We refer to this as a 'statistically significant crossing.' Forexample, an 'X' in column 2, row 5, indicates that application of the statisticalinference test reveals that Norway and Canada cannot be ordered using the rankdominance criterion.

Applying the tests for rank dominance to each of the forty-five pairwise com-parisons, we find twenty-seven cases (60 per cent) where unambiguous statements

about economic well-being can be made. In the remaining eighteen comparisonsthe rank dominance criterion is not sufficiently restrictive to provide a completepairwise ordering. Several results emerge. First, Canada, Norway, and the UnitedStates are not dominated by any other country, and West Germany s dominatedonly by Norway. Second, Australia dominates no other country and is dominatedby all except the United Kingdom. Third, the Netherlands and the United Kingdomdominate only Australia, are non-comparable o each other and to the United States,and are dominated by the other six countries. Fourth, Sweden and Switzerland arenon-comparable o each other and each dominates Australia, the United Kingdom,

and the Netherlands. Fifth, France dominates Switzerland, the Netherlands, Aus-tralia and the United Kingdom, and is non-comparable o the United States, Swedenand Switzerland. Finally, the United States is non-comparable o six countries andaccounts for one third of eighteen statistically significant crossings in table 1.

Application of SSD provides additional rankings. Table 1 uses superscripts todenote those cases that are non-comparable using rank dominance but can be or-dered using GL dominance. Superscript +' means that the rank dominance crossing

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716 J.A. Bishop, J.P. Formby, W.J. Smith

changes to GL dominance, with the country in the row dominating the country inthe column. Superscript -' indicates that the rank dominance crossing changes toGL dominance, with the country in the row being dominated by the country in thecolumn. Seven additional cases can now be ordered for a total of thirty-four andthe percentage of unambiguous comparisons ncreases from 60.0 to 75.6 per cent.

Two difficulties associated with our adjustment of the levels of income in theLlS data sets are noteworthy. The first relates to the fact that Summers and Heston'sestimates of per capita GDP in international rices do not provide perfect adjustmentsfor differences in purchasing power across countries. But there is wide agreementthat the Summers and Heston estimates are far superior to income comparisonsbased on GDP per capita converted at official exchange rates. As an alternative o

the Summers and Heston estimates, the OECD provides 1979 purchasing price parity(Ppp) igures for a subset consisting of seven of the ten countries considered.8 Usingthe OECD data leads to results very similar to those reported above. There is onlyone statistically significant change from the rank dominance results reported above.We find that Canada irst degree dominates the United States when the OECD'S PPPS

are used to adjust incomes. All other pairwise comparisons are the same as thosereported n table 1.

The second difficulty with the stochastic dominance comparisons s associatedwith the fact that the LlS data do not refer to the same year and the recession of theearly 1980s may differentially mpact the data of some countries. Standardizing helevels of income to 1979 dollars eliminates some of the problem, but both SSD andFSD comparisons may be affected by the cyclicality of the income shares, whichare basic building blocks used in constructing GL curves and quantile functions.Inspection of table 1 reveals that six of the eleven pairwise comparison (54.5 percent) that cannot be ranked at the second degree involve the United States, thecountry with the highest per capita income. None of the pairwise crossings in table1 involving the United States would likely be undone f each country's LIS data were

available for the same year, for example, 1979. The explanation of this lies in thefact that the u.s. data are for 1979, a pre-recession year, and, as shown by Bishop,Formby, and Smith (199la), the 1979 u.s. Lorenz curve is dominated by each ofthe LIS countries that the United States crosses (Canada, France, Norway, Sweden,West Germany, and United Kingdom) in table 1. All of the crossings in table 1involving the United States are traceable to high-income inequality and relativelylarge absolute incomes, these conditions would only be magnified if Lorenz datafor the same year were available.

However, some of the other pairwise comparisons n table 1 could be impactedby the cyclicality of the Lorenz ordinates. The unavailability of data limits ourability to test for differences, but several observations can be made. A recent paper(Bishop, Formby, and Smith 1991a) establishes that the u.s. Lorenz curve shiftedsignificantly toward's greater inequality between 1979 and 1982, which impliesthat the United States's dominance over Australia (1981-2 LIS data), Netherlands

8 The OECD does not provide 1979 PPPs for Australia, Switzerland, and Sweden.

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(1983 LIS data) and Switzerland 1982 LIS data) in table 1 may be due to the use of1979 Lorenz ordinates n the construction of u.s. quantile functions and GL curves.Thus, what appears to be u.s. dominance in table 1 could turn out to be additionalcrossings once the greater u.s. inequality in the early 1980s is taken into account.

The availability of u.s. micro data for 1982 that is comparable o LIS data permitsus to investigate the sensitivity of the finding of u.s. dominance over Australia,Netherlands, and Switzerland by using 1982 Lorenz data for the United States. Asindicated by table 1, the original survey for the LIS data for the United States isthe Current Population Survey (cps). In the Luxembourg Income Study databasethe standard ps data are augmented by estimates of direct taxes, including federalincome taxes, state and local income taxes, payroll and property taxes estimates,

and some non-cash benefits to make them comparable to other LIS datasets. Weuse the public use file of the March 1983 cps After-Tax Money Income Estimates(calendar year 1982 income data) and extract all primary families and unrelatedindividuals, which makes the file comparable o the LIS'S primary amilies. We takea random sample of 15,081 to make the 1982 u.s. sample size comparable o 1979(15,134) and repeat the test procedure described above for pairwise comparisonsof the United States, Australia, Netherlands, and Switzerland. While the 'cps AfterTax Money Income File' is not identical to the U.S. LlS data set, it is extremelyclose. The use of 1982 after tax money income Lorenz ordinates, rather than the1979 ordinates from the official LIS file, does make a difference in the GL curve.When the 1982 ordinates are used, the statistical comparisons indicate that ratherthan dominating Australia, Netherlands, and Switzerland, the GL curves cross. Allof the crossings occur in the bottom two deciles and are a result of the fact thatthe u.s. Lorenz ordinates were smaller in deciles one and two in 1982 than 1979.9Therefore, we conclude that international omparisons of income distributions maybe significantly affected when LlS data sets are for different years and one of theyears involved precedes and another ollows a major recession. 10Researchers must

exercise appropriate aution in interpreting esults under these conditions.

3. Stochastic dominance and poverty rankingsThe relationship between rank dominance (FSD) and headcount poverty, and GL

dominance (SSD) and income gap poverty (Foster and Shorrocks 1988) provides a

9 The cps After-Tax Money Income Estimates provides the major adjustments o the ordinary cpsAnnual Demographic File (March tapes) that are used in constructing he U.S.LIS data set. Butdirect taxes are not the only ajustments used. In addition, certain non-cash transfers hat we areunable to replicate from the public use file of the cPs After- Tax Money Income Estimates orthe Annual Demographic File are added in constructing he U.S.LIS data set. Therefore, using the1982 Lorenz ordinates from the After-Tax Money Income File is not absolutely comparable othe LIS data. However, a comparison of 1979 u.s. Lorenz ordinates using cPs After-Tax MoneyIncome Estimates and the LIS data set suggests the inclusion of non-cash benefits has only a veryminor influence in the bottom deciles. Nevertheless, it is conceivable that if we had the necessarynon-cash benefits and included them in our calculations, our findings of U.S. crossings withAustralia, Netherlands, and Switzerland using 1982 Lorenz ordinates could be affected.

10 We are indebted to an anonymous referee for pointing out that a recession combined with datafrom different years could affect some pairwise comparisons.

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718 J.A. Bishop, J.P. Formby, W.J. Smith

TABLE 2Per capita poverty dominance

Aus. Can. Fr. Neth. Nor. Swed. Switz. W. Ger. U.K. U.S.

Australia *Canada + *France + - *Netherlands + - - *Norway + + + + *Sweden + - + + - *Switzerland + - 0 - - - *West Germany + x+ + + - 0 + *United Kingdom + - 0 x+ - - - - *

United States + - x + - x + x x *NOTESA '+' means that the statistical test reveals the country in the row head-count dominates the country n

the column.-' indicates that the statistical test reveals the country in the row is head-count dominated by the

country in the column.All 'xs' indicate the truncated distributions ross at the first degree. A superscript ndicates the truncated

distributions are uncrossed when the poverty gap concept is applied.A 'O' means the truncated distributions are equivalent.An 'x+' means that the crossing at the first degree changes to '+' when the income gap poverty concept

is applied.

An 'x -' means that the crossing at the first degree changes to '-' when the income gap poverty conceptis applied.

method for constructing poverty orderings. If the income distribution s truncatedabove the highest reasonable ncome that can be considered poor, and the stochasticdominance ranking rules are applied, a poverty ordering can be constructed withoutspecifying a fixed income cut-off. In this section we explore the possibility of addi-tional rankings by assuming that the poverty group is contained n the bottom three

deciles. In making poverty comparisons we refer to this as a 'truncated' distribu-tion, which means that we are making comparisons over only these three deciles.We note that tables A3 and A4 of the appendix contain sufficient information hatinterested readers can specify alternative poverty cut-offs.

The poverty rankings are given in table 2. The notation s the same as that usedin table 1, with the addition of a '0', which indicates that the null hypothesis of nosignificant difference between two truncated quantile unctions cannot be rejected."As with table 1, we begin with the truncated FSD (head-count poverty) criterion andrank as many distributions as possible. Next, we apply the GL (income gap poverty)criterion to those cases that cannot be ranked at the first degree.

Not surprisingly, runcating he quantile functions to test for headcount povertydominance results in a significant improvement n the ability to order the distri-butions. While we were able to rank only 60 per cent of the comparisons usingthe entire quantile function, the truncated FSD criterion ranks 89 per cent of the

11 The 5 per cent SMM critical value for k = 3 is 2.38.

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Welfare and poverty 719

pairwise head-count poverty comparisons. For example, while the Australian andBritish quantile functions are non-comparable over their entire range, the UnitedKingdom clearly dominates in terms of head-count poverty (see panel B of figure2). In only five cases, four of which involve the United States, do the truncatedquantile functions cross. If we further restrict the analysis to the income-gappoverty measure (truncated GL, dominance), we find only one case (the UnitedStates vs. the United Kingdom) where relative poverty cannot be evaluated. Thisresults in an income gap poverty ranking of almost 98 per cent (forty-four out offorty-five cases).

The general pattern of poverty orderings within the ten countries can be sum-marized as follows. Norway dominates and Australia is dominated by all other

countries. West Germany and Canada rank just below Norway, while the Nether-lands and the United Kingdom rank ust above Australia. France has a poverty levelequivalent to the United Kingdom and Switzerland and is dominated by Canada,Norway, Sweden, and West Germany. Compared with its position in terms of therankings based on the entire GL curve, Sweden moves up in the poverty compar-isons of the LIS countries. Only Australia, he Netherlands, and Switzerland have agreater degree of poverty than that experienced in the United States.12

IV. PARTIAL ORDERS, STATISTICAL TESTS, AND CROSS-COUNT'RY

COMPARISONS

This section integrates our findings relating to inference-based dominance rankingsof income distributions, welfare, and poverty in LIS countries with earlier work ondominance and provides a unifying perspective on cross-country comparisons. Wefocus first on the different partial orders obtained n international omparisons whenLorenz and stochastic dominance ranking criteria are used. We then discuss howthe statistical rankings n terms of stochastic dominance compare with zero varianceranking and contrast he LIS rankings to the earlier cross-country comparisons.

1. Income inequality versus stochastic dominanceExamination of the Hesse diagrams presented n figure 3 strongly suggests that thereis no general relation between the Lorenz and stochastic dominance partial orders.Panel A of figure 3 shows the inequality rankings based on statistically significantdifferences in Lorenz curves and, except for the addition of France, is equivalentto Bishop, Formby, and Smith's (1991b) figure 2. Panel B of figure 3 is based onthe SSD partial order given in Table 1 above and shows the statistically significant

differences n generalized Lorenz curves. The comparison of the two panels revealsan almost complete Lorenz partial order (only one crossing in panel A) and a much

12 Using the OECD purchasing power parities rather han Summers and Heston's adjustment o thedata yields two changes in the poverty comparisons reported above and both involve the UnitedStates. Using the PPPs to adjust incomes results in West Germany having a smaller degree ofhead-count poverty and the United Kingdom a smaller degree of income gap poverty comparedwith the United States. All other pairwise poverty comparisons are the same as those reported ntable 6.

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720 J.A. Bishop, J.P. Formby, W.J. Smith

less complete GL partial order (eleven crossings in panel B). Several observationsare relevant in this regard. In the absence of a systematic relation between meanincome and the distribution of income, it can be fonnally demonstrated hat ingeneral no relationship necessarily exists between Lorenz and generalized Lorenzdominance.13 Second, the test procedure s conservative and constructing criticalregions with the SMM distribution guarantees ests no larger than the nominal size,which results in a larger Type-2 error. As a consequence, the SMM test is more likelyto accept the null hypothesis of no difference in Lorenz curves than in GL curves,when it is in fact false. It is also true that Type-2 errors are more likely when theparameter stimates being tested are similar in size. Thus, if differences in Lorenzcurves are less pronounced han in GL curves (i.e., LC'S differ in dispersion, while

GL curves vary in both dispersion and scale), then we are more likely to falselyaccept the null hypothesis of no difference in two vectors of ordinates. Finally,on a related point regarding he ability to order distributions, we point out that inpractice zero variance comparisons tend to result in a large number of crossings.Statistical nference tests reveal, however, that many of these apparent rossings arenot significant once sampling errors are taken into account. Thus, inference-baseddominance analysis typically orders more of the distributions of interest than zerovariance comparisons.14

Table 3 presents evidence indicating hat there is a less complete GL partial orderof LIS countries than is obtained using the Lorenz dominance criterion. Table 3aand 3b summarize the zero variance and statistical comparisons. Table 3a showsthat in thirty-seven of forty-five zero variance comparisons (82.2 per cent) areranked n terms of the Lorenz dominance, but the GL criterion orders only twenty-three (51.1 per cent). Table 3b shows that statistical comparisons also result in aless complete GL partial order compared with the Lorenz partial order.15 Thus, interms of both zero variance comparisons and statistical inferences concerning LIS

income distributions he GL criterion results in a much less complete partial order

of countries than does Lorenz dominance.

2. The power of FSD and SSD in cross-country comparisonsIt seems intuitive hat the power of FSD to order ncome distributions hould be muchgreater when extremely diverse countries are being compared. In Bishop, Formby,

13 Simple use of inequalities and the fact that generalized Lorenz ordinates are equal to ordinaryLorenz ordinates scaled by the mean of the distributions an be used to show that when themeans are different, Lorenz dominance of distribution A by B is compatible with either general-ized Lorenz dominance of A by B, B by A, or a crossing, depending upon the ratio of the mean

incomes of the distributions. n a similar fashion, generalized Lorenz dominance is compatiblewith Lorenz dominance, reversal of Lorenz dominance, or a crossing. A proof is available fromthe authors upon request.

14 See Bishop, Formby, and Smith (1991a) and references therein on this point.15 It should be noted that in table 3b there are six 'Os' in the statistical Lorenz comparisons, ndi-

cating that the null hypothesis of no difference cannot be rejected. However, there are no 'Os'in the statistical comparisons of GL curves. Thus, the number of strict dominance relations in thestatistical-partial rder of Lorenz curves (thirty-eight) s much closer to the number of dominancerelations in the statistical-partial rder of GL curves (thirty-four).

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Welfare and poverty 721

Panel Panel

Lorenz Curves Generalized orenzCurves

Sweden Canada xxx W.Germnany xxx

Norway oooooooooNorway IxxxxxxxxxxxxxxxxxxxNowa10000000(?S X X [ -.German]y

U.K. looooooooo Face xxx S-weden xx[US

! > ~~~~xxxxxxxxoAustralia C FranceS xxxxxxxxxxx

Netherlands

Swi zerland xxxxxxx Ut.Sr. ]ai1

Solid ines from above ndicate dominance. Countries onnected y xxx's indicate pairwisecomparisons hatare characterized y statistically ignificant rossings. Countries onnected y ooo'sindicate airwise omparisons n which he null hypothesis f no difference annot be rejected.

FIGURE 3 Statistical comparisons of Lorenz and generalized Lorenz curves

and Thistle's (1991) sample of twenty-six countries, zero variance comparisons areused to construct FSD and SSD partial orders and 269 of 325 pairwise comparisons(82.8 per cent) are ordered. FSD accounts for 91 per cent of the pairwise compar-isons in the SSD partial order. The marginal effect of GL dominance increases thenumber of countries ordered rom 245 to 269 (9.8 per cent).'6 Thus, among diversecountries at different stages of development, rank dominance accounts for most ofthe ranking power, and generalized Lorenz dominance adds relatively little to ourability to order countries. Inspection of table 3 reveals that the results for the LIScountries are quite different. T'able 3a shows that only twenty-three of forty-fivezero variance comparisons can be ranked n terms of SSD (51.1 per cent), and FSD

accounts for sixteen of them. Among LIS countries the marginal effect of SSD usingzero variance comparisons is seven cases (43.8 per cent), which is much greaterthan in earlier cross country studies. The implications of this are clear. For zerovariance comparisons, the marginal effect of SSD on the stochastic dominance par-tial order depends considerably on the diversity and differences in the level ofdevelopment of the countries being compared.

Of course, the power of FSD and SSD can and should be looked at in terms ofthe statistical-partial rders. Table 3b shows that for the ten LIS countries in thisstudy SSD ranks thirty-four of forty-five (75.6 per cent) of the pairwise comparisons.Among the countries ranked by generalized Lorenz curves, FSD orders twenty-seven(79.4 per cent) and the marginal effect of GL dominance is seven cases (0.256 percent). Thus, in the LIS cross country comparisons we find that when sampling

16 Essentially the same result is found for Shorrocks's (1983) original twenty countries, which is asubset of the twenty-six examined.

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722 J.A. Bishop, J.P. Formby, W.J. Smith

TABLE 3Summary of Lorenz, generalized Lorenz and rank dominancecomparisons

a. Zero variance comparisonsGeneralized Rank

Lorenz Lorenz dominance

Dominance 37 23 16Crossings 8 22 29Total 45 45 45

b. Statistical comparisonsDominance 38 34 27

Crossings 1 11 18Equivalence 6 0 0Total 45 45 45

variability s taken into account, rank dominance accounts for most of the power,but GL dominance adds about 25 per cent to our ability statistically o order ncomedistributions.

V. CONCLUSIONS

This paper uses stochastic dominance, statistical nference procedures and the inter-nationally comparable Luxembourg ncome Study (LIS) database o rank en westerncountries n terms of standards of living and poverty. Two important advantages ofthe stochastic dominance procedures ie in the fact that they do not require that aset of distributional weights be specified in order to draw welfare conclusions, nordo they require that a fixed poverty income cut-off be chosen in order to constructpoverty orderings.

Using the inference-based stochastic dominance methodology, we are able torank more than 75 per cent of the forty-five pairwise comparisons. In addition, weuse a truncated tochastic dominance analysis to draw inferences about poverty. Inthis case we are able to rank as many as 98 per cent of the forty-five truncateddistributions. The results of the statistical tests suggest that Canada and Norwayhave the highest levels of economic well-being and the lowest levels of poverty,while Australia has the lowest level of economic well-being and the highest level ofpoverty. The high per capita ncome and large degree of inequality n the U.S. resultsin non-comparability n six of the nine international comparisons of economic

well-being involving the United States. Restricting the analysis to poverty onlyby truncating the income distributions, however, results in unambiguous povertyorderings of the United States in eight of the nine international omparisons. Theresults for some pairwise comparisons are found to be sensitive to the fact that LIS

data for several countries are not for the same year. Income distribution data areimpacted by economic cycles, and interpretation f results that are based on datasets for pre- and post-recessions years should proceed with caution.

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Welfare and poverty 723

Two additional conclusions are worth emphasizing. We point out that a partialorder of countries based on generalized Lorenz dominance may be less completethan one based on ordinary Lorenz dominance, and this is shown to characterizethe ten LIS countries in this study. Finally, most of the power statistically to orderincome distributions across countries s contained n rank dominance, and general-ized Lorenz dominance adds about 25 per cent to the statistical partial order of LIS

countries.

REFERENCES

Atkinson, A.B. (1970) 'On the measurement of inequality.' Journal of Economic Theory2, 244-63

Beach, C.M. (1977) 'Cyclical sensitivity of aggregate income inequality.' Review ofEconomics and Statistics 59, 56-66

Beach, C.M., and R. Davidson (1983) 'Distribution-free tatistical inference with Lorenzcurves and income shares.' Review of Economic Studies 50, 723-35

Beach, C.M., and S.F. Kaliski (1986) 'Lorenz curve inference with sample weights: anapplication o the distribution of unemployment experience.' Applied Statistics, 35,439-50

Beach, C.M., and J. Richmond (1985) 'Joint confidence intervals for income shares andLorenz curves.' International Economic Review 26, 439-50

Bishop, J.A., J.P. Formby, and W.J. Smith (199la) 'Lorenz dominance and welfare:

changes in the u.s. distribution of income, 1967-1986.' Review of Economics andStatistics 73, 134-9(1991b) 'International omparisons of income inequality: tests for Lorenz dominanceacross nine countries.' Economia 58, 461-77

Bishop, J.A., J.P. Formby, and P.D. Thistle (1991) 'Rank dominance and internationalcomparisons of income distributions.' European Economic Review 35, 1399-410

Buhmann, B., L. Rainwater, G. Schmaus, and T. Smeeding (1988) 'Equivalence scales,well-being, inequality, and poverty: sensitivity estimates across ten countries using theLuxembourg Income Study (LIS) database.' Review of Income and Wealth 30, 115-42

Dasgupta, P., A.K. Sen, and D. Starrett 1973) 'Notes on the measurement of inequality.'Journal of Economic Theory 6, 180-7

Foster, J.E., and A.F. Shorrocks (1988) 'Poverty orderings.' Economica 56, 173-7Kakwani, N. (1984) 'Welfare ranking of income distributions.' In Advances in Economet-

rics: Vol. 3 ed. R.L. Basmann and G.F. Rhodes, Jr (Greenwich, CT: JAI Press)Miller, R.G. (1981) Simultaneous Statistical Inference (2e) (New York: Wiley)O'Higgins, M., G. Schmaus, G. Stephenson (1989) 'Income distribution and redistribution:

a microdata analysis for seven countries.' Review of Income and Wealth 35, 107-31Organization For Economic Cooperation and Development (OECD) (1982) National Ac-

counts (Paris: OECD)Saposnik, R. (1981) 'Rank dominance in income distribution.' Public Choice 36, 147-51Sen, A.K. (1973) On Economic Inequality (New York: Norton)- (1976) 'Real national income.' Review of Economic Studies 43, 19-39Shorrocks, A.F. (1983) 'Ranking ncome distributions.' Economica 50, 3-17Stoline, M.R., and H.K. Ury (1979) 'Tables of the studentized maximum modulus distri-

bution and an application o multiple comparisons among means.' Technometrics 21,87-93

Summers, R., and A. Heston, (1988) 'A new set of international omparisons of realproduct and price level estimates for 130 countries.' Review of Income and Wealth 34,1-25

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724 J.A. Bishop, J.P. Formby, W.J. Smith

APPENDIX

TABLE AlOverview of LIS data sets

Country Data set Sample sizea

Australia Income and HousingSurvey, 1981-82 15,985

Canada Survey of ConsumerFinances, 1981 15,136

France Survey of IndividualIncome Tax Returns, 1981 3,639

Netherlands Survey of Income ofProgram, 1983 4,833

Norway Norwegian TaxFiles, 1979 10,414

Sweden Swedish Income DistributionSurvey, 1981 9,625

Switzerland Income and WealthSurvey, 1982 7,036

United Kingdom Family ExpenditureSurvey, 1979 6,888

United States Current PopulationSurvey, 1979 15,134

West Germany Transfer Survey,1981 2,727

a Number of families in data set

TABLE A2Per

capita mean incomesLIS GDP GDP ? Adjusted LISincomea $1979b U.S. GDP incomeC

Country (1) (2) (3) (4)

Australia 5,542 8,152 0.703 3,907Canada 8,101 11,358 0.979 5,436France 23,016 9,625 0.830 4,602Netherlands 11,592 8,964 0.773 4,289Norway 27,137 10,708 0.923 5,129Sweden 34,268 8,750 0.754 4,180Switzerland 22,417 9,628 0.830 4,618West Germany 13,196 9,619 0.829 4,605United Kingdom 1,990 8,094 0.698 3,876United States 5,554 11,602 1.000 5,554

a LIS per capita incomes are reported n the country's own currency.b SOURCE: Summers and Heston (1989)c The incomes are adjusted to maintain the ratios of column 3 using

the LIS u.s. per capita income of $5,554 as the base.

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