Social indicators and comparisons of living standards

Journal of Development Economics 70 (2003) 501 – 529 www.elsevier.com/locate/econbase

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Social indicators and comparisons of living standards Steve Dowrick*, Yvonne Dunlop, John Quiggin School of Economics, Copland Building (no. 26), Australian National University, ACT 0200, Australia Received 1 April 2001; accepted 1 April 2002

Abstract Construction of an international index of standards of living, incorporating social indicators and economic output, typically involves scaling and weighting procedures that lack welfare-economic foundations. Revealed preference axioms can be used to make quality-of-life comparisons if we can estimate the representative household’s production technology for the social indicators. This method is applied to comparisons of gross domestic product (GDP) and life expectancy for 58 countries. Neither GDP rankings, nor the rankings of the Human Development Index (HDI), are consistent with the partial ordering of revealed preference. A method of constructing a utility-consistent index incorporating both consumption and life expectancy is suggested. D 2003 Elsevier Science B.V. All rights reserved. JEL classification: I31; O47 Keywords: International comparisons; Living standards; Revealed preference; Social indicators; Human Development Index; Ideal Afriat Index

International comparisons of living standards and development are inescapable. Public opinion, inasmuch as it is represented by the media, has an insatiable appetite for world or regional rankings. Politicians and policy analysts regularly use changes in these rankings as a basis for assessing the efficacy of national policies. Governments and international organizations use measures of development to allocate funding to countries or regions. The underlying indexes of national progress are used extensively in econometric testing of theories of growth and development. Here we propose a contribution to the method by which such international comparisons are made, based on the economic theory of revealed preference.

* Corresponding author. Tel.: +61-2-6125-4606. E-mail address: [email protected] (S. Dowrick). 0304-3878/03/$ - see front matter D 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-3878(02)00107-4

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Most commonly, international comparisons of living standards or development are made in terms of gross domestic product (GDP) per person. Such comparisons have been criticized on a number of grounds. The most frequent criticism is based on the observation that standard GDP indexes are more properly regarded as partial measures of aggregate output than as indicators of wellbeing. No allowance is made for environmental differences, domestic activities, or production and consumption externalities. For example, if polluting industries cause illnesses that require expensive medical treatment, both the output of the polluting industry and the expenditure on medical services will be counted as part of GDP. Thus, aggregate output may increase although wellbeing, as measured by various social indicators, declines. Here we develop a method of welfare comparison that incorporates both GDP and social indicators. Our approach treats the average bundle, consisting of life expectancy and average (per person) consumption of goods and services, as the choice of a representative agent facing the relative prices and budget constraints of that country. Following Kravis et al. (1982), revealed preference may then be used to compare average bundles and to test the hypothesis of common tastes amongst representative agents. The paper is organized as follows. In Section 1, we review the literature on extended measures of GDP, designed to take social indicators into account. In Section 2, we briefly summarize our application of the principle of revealed preference. In Section 3, we extend standard GDP comparisons by incorporating evidence on life expectancy across 58 countries. In our empirical application, we have chosen to concentrate on length of life as the single most important of the social indicators in common use. Marketed goods such as food, medical services and education are important determinants of mortality, so we are able to infer the opportunity cost in each country of an additional year of expected life. Other important determinants of average longevity include environmental conditions, social customs and income distribution. Although our approach does not directly value these distributional and environmental aspects of living conditions, they influence our welfare comparisons inasmuch as they impinge on observed longevity. We find that international data on life expectancy and GDP per person allow the construction of a partial ordering of the welfare of the ‘representative agent’ of each country. In Section 4, we test whether either of the two most common rankings of national welfare, by real GDP per person and by the Human Development Index (HDI), satisfy the partial ordering of our revealed preference analysis. Not surprisingly, we find that the ranking of real GDP per person fails to satisfy the revealed preference criteria in a number of cases where levels of GDP per person are fairly close but where life expectancy is substantially higher in the lower-ranked country (for example, comparing Belgium with the Netherlands). Perhaps more surprisingly, we find that the HDI ranking, amended to provide a fair comparison, also fails to satisfy the criterion of revealed preference. The HDI reverses the ‘mistakes’ of the GDP rankings, but, by our criteria, it overcompensates. For example, the HDI method places the Netherlands not only above Belgium but also above the United States. This HDI ranking contradicts our finding that the United States could have afforded the average Dutch consumption bundle and, at the same time, could have channeled sufficient resources into health and education to surpass Dutch longevity, whereas the Dutch could not have achieved US outcomes.

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Since neither GDP nor HDI rankings satisfy our revealed preference test, we present in Section 5 a method of constructing a utility-consistent index which does satisfy the test, based on the true Afriat index developed by Dowrick and Quiggin (1997). Finally, in Section 6, we consider possible extensions and modifications of the analysis.

1. Extended measures of GDP and multiple indicators The limitations of GDP measures as a basis for international and intertemporal comparisons of living standards have been discussed on many occasions. One response has been to extend and adjust the standard GDP measure in various ways, such as by constructing measures in which corrective or defensive expenditures are excluded and household activities are included. Eisner (1988) and Scott (1989) have reviewed such extended systems of national accounts. An alternative approach, particularly popular in the development literature, has been to report multiple indicators of social development including, life expectancy, literacy rates and, occasionally, measures of personal freedom and political democracy. Dasgupta (1988) has argued that indicators such as life expectancy and literacy rates are important indicators of development within a desire-fulfillment framework. Kakwani (1993) presents an axiomatic derivation of a strongly nonlinear achievement index that he applies to several social indicators. Comparison of multiple indicators is, however, problematic. Atkinson and Bourguignon (1982) have derived conditions under which general classes of social welfare functions will provide unanimous rankings of multiple indicators, but their application to indicators of GDP and life expectancy is incomplete. Multiple indicators have sometimes been aggregated into alternative indicators based on the statistical properties of the data. For example, Slottje et al. (1991) form indexes using principal components and other measures of statistical association within their data sets, and Noorbakhsh (1998) uses principal component analysis. In other work, such as that of Anand and Sen (1994), the aggregation is based on a scaling procedure designed to reflect human needs and a declining marginal utility of consumption, with an a priori assignment of weights. The United Nations Development Program takes such an approach in its annual publication of the Human Development Index (HDI), which is based on life expectancy, literacy and GDP per person. But its scaling and weighting procedures change from year to year, reflecting a degree of arbitrariness that is criticized by Srinivasan (1994). Whilst we may lack confidence in the manner in which these alternative measures extend the notion of GDP, there is considerable merit in the objective of deriving measures that more closely approximate a broad concept of welfare. Such measures should improve attempts to evaluate policies where benefits are diffused widely. Barro and Lee (1993, 1994) attempt to measure the social value of investment in human capital through schooling. Others, such as Aschauer (1988) and Easterly and Rebelo (1993), have tried to quantify the social returns to public investment in the core infrastructure of roads, water and sewerage. These studies are typically based on the premise that there may be diffuse but significant spillover benefits from particular

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investments, benefits that will not be picked up in an analysis of private returns or of the productivity of individual firms. These studies test for external effects in broad aggregates such as GDP and try to quantify social rates of return. Implicit in these tests is the notion that the GDP aggregates are inclusive and exhaustive of those benefits. The very nature of the potential spillovers in these examples highlight the inadequacies of aggregate measures which ignore health and environmental outcomes; hence, our concern to investigate methods of aggregation. The weighting procedures used to derive the HDI are based on judgement rather than on welfare theory. For Desai (1991, pp. 353– 354), this does not constitute a problem since: ‘‘Social reciprocity, participation, interdependence are put into the wellbeing function. It is perhaps possible though hardly worthwhile to translate back many of the things listed above into commodity terms since the externality/interdependence valuation so difficult in consumer theory remains. Thus the concept of human development deliberately goes beyond the utilitarian calculus.’’ This argument carries weight at the level of individual welfare measurement. However, composite indicators such as the HDI have, in practice, been developed and publicized at a national aggregate level. Ravallion (1997) discusses the implicit tradeoffs between income and life expectancy in the HDI, commenting that ‘‘The value judgements underlying these tradeoffs built into the HDI are not made explicit, and they are questionable.’’ We suggest that it is feasible to use welfare theory to measure the tradeoffs between indicators of social capability and consumption of commodities, and thereby to estimate the weights implicit in current social choices. So it may be worthwhile to attempt the ‘utilitarian calculus’, if only to determine how the relative weights of the calculus relate to the explicit, but essentially arbitrary, weights used in current composite measures such as the HDI.

2. Revealed preference analysis Standard revealed preference analysis is based on the assumption that for each country i, we have a vector of relative prices Pi (where the nume´raire is the international average consumption bundle), and a vector of quantities Qi representing per person consumption of each group of goods. It is assumed that, for the representative individual, the budget set is given by: fQ:Pi  Q V Pi  Qi g;

ð1Þ

and that the consumption vector Qi is the most preferred element of this set. The usual revealed preference method cannot be applied to social indicators such as health, life expectancy and literacy rates because of the absence of a natural price measure. However, revealed preference principles may still be applied if the indicators are systematically related to the consumption of market goods: for example, if health

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outcomes can be regarded as outputs, with consumption of food and medical services as inputs. We assume that the vector of market goods may be partitioned into two sub-vectors Q1, 2 Q where the elements of Q1 are goods valued in their own right, and the elements of Q2 are valued only as inputs to the production of the final goods that are captured by the social indicators. Some elements of Q1 may also contribute to the production of the vector of social indicators which we denote Z. It is assumed that for each country i, there exists a technology represented by a set Ti: Ti ¼ fðQ1 ; Q2 ; ZÞ : Z is attainable given the inputs Q1 ; Q2 g:

ð2Þ

The feasible consumption set for country i is given by: Xi ¼ fðQ1 ; Q2 ; ZÞaTi : Pi  QVPi  Qi ; where Q ¼ ðQ1 ; Q2 Þg:

ð3Þ

Now consider two countries, A and B, with price-quantity vectors given by (Q1A, Q2A, Z , PA), (Q1B, Q2B, ZB, PB). Country A is revealed preferred to country B if: A

aQ2 ; ðQ1B ; Q2 ; ZB ÞaXA ;

ðiÞ

that is to say, if A could have afforded both B’s bundle of goods and the inputs required to match B’s social indicators. Similarly, B is revealed preferred to A if: aQ2 ; ðQ1A ; Q2 ; ZA ÞaXB :

ðiiÞ

If neither Eq. (i) nor Eq. (ii) holds, no ranking is possible in the absence of additional information on preferences. If both Eqs. (i) and (ii) hold, one of the maintained hypotheses, such as the hypothesis of common tastes, must be violated. The technology Ti defines the maximum indicator function, where L is one of the social indicators, an element of Z, and Z is the vector of other indicators: LA;B ðQ1B ; ZB ; PA Þ ¼ max:fL : Q z ðQ1B ; 0Þ; Z z ZB ; ðQ; ZÞaXA g: In the example we apply in the next section, LA,B is the maximum attainable life expectancy for country A, subject to the constraint that country A achieves at least the same consumption of goods and other social indicators as country B. The condition that country A is revealed preferred to country B is then: LA;B z LB :

ð4Þ

Revealed preference of this kind is sufficient to yield an ordinal ranking of consumption bundles, subject to a range of indeterminacy. To obtain a cardinal ranking, it is necessary to discover whether preferences can be represented by a homothetic utility function. This problem is discussed in Section 5.

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2.1. Life expectancy and the revealed preference approach The application of a welfare-theoretic approach to a comparison of international consumption levels and life expectancy poses a range of philosophical, theoretical and practical problems. Some difficulties that are inherent in the use of any measure of ‘average wellbeing’ become more salient when the criteria are extended to include social indicators. For example, the practice of ranking average national outcomes may seem distasteful or irrelevant to those whose prime interest lies in individual variety, particularly those concerned with inequality in social and economic outcomes. Nevertheless, given the frequent use of measures of average income and life expectancy in international comparisons, it is desirable to investigate whether such comparisons are open to welldefined economic interpretation. It is perhaps surprising to find that national average expenditure patterns happen to be consistent with the hypothesis of rational choice by a representative agent. Such findings have been reported by Manser and McDonald (1988), who examined US consumption between 1959 and 1985, and by Dowrick and Quiggin (1994) who analyzed international expenditure patterns. The use of a representative agent model to make comparisons of national living standards does not imply that individual preferences are necessarily capable of aggregation. What it does imply is that such comparisons of national averages can be undertaken using the standard principles of economic welfare analysis, and interpreted as if the average outcomes had been generated by a representative household. Fundamental theoretical issues do, however, arise from the fact that life expectancy is not simply an item of consumption. Rather, in the terminology of Sen (1998, p. 22), it is a capability, allowing individuals to engage in consumption or production, as well as having intrinsic importance. One way of combining the capabilities approach with the consumption approach has been put forward by Bleichrodt and Quiggin (1999) who propose an objective function of the multiplicatively separable form: W ¼ L  q  uðcÞ;

ð5Þ

where L is life expectancy, q is a quality-adjusted life-years measure with 1 corresponding to perfect health and 0 to death, and u(c) is a standard utility function for consumption of a bundle of goods and services. This formulation embodies the idea that life and health are not objectives of consumption, but provide the capability to engage in productive activity and consumption. Obviously, in order to consume and produce it is necessary to be alive. More generally, illhealth will reduce the benefits associated with any given level of consumption. Mathematically, this formulation is consistent with the revealed preference approach, which simply requires that individuals choose more health and more consumption of goods rather than less. The principal practical problem in applying revealed preference tests to social indicators is that we do not observe market valuations of these indicators. Varian (1988) has shown that the absence of even one price renders the standard revealed preference

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approach inapplicable. We argue that this problem may be overcome if it is possible to estimate a household production technology in which outcomes relevant to utility are determined, in part, by consumption of market goods such as food and health services. Given knowledge of such a technology and the relevant input prices, we demonstrate that it is possible to apply the theory of revealed preference to the resulting feasible consumption sets. Our procedure takes account of the possibility that some of the inputs such as food may be valued in their own right whilst other inputs such as medical expenditures may be purely instrumental. We also take account of the possibility that the implicit price of an indicator such as life expectancy is unlikely to be linear.

3. Cross country comparisons based on disaggregated GDP and life expectancy Our primary sources of GDP data are the two parts of the report on Phase IV of the International Comparison Project (ICP), published by the United Nations and Commission of the European Community (1986, 1987). Part One presents the aggregated ‘league tables’ of GDP per person, evaluated at international prices for 60 countries in 1980. Part Two contains the details of prices and quantities disaggregated into private consumption, government consumption and capital formation. These broad expenditure categories are broken down into 38 more detailed categories. Life expectancy is the average for the period 1975– 1980, taken from United Nations (1989). This measure is based on the notional expected life of newborn infants if they were to face current age-specific mortality rates throughout their future lives. It is the single, most widely used measure of the health of a nation. The absence of demographic data for Botswana reduces the sample to 59 countries. We have also excluded Guatemala from our sample because we do not believe the data on medical expenditures can be correct; they suggest that Guatemala spent over 20% of GDP on health care, some three or four times as much as other countries at a similar stage of development. This reduces our sample size to 58. The data set is summarized in Table 1. GDP is converted into US dollars using the ICP’s aggregate purchasing power parities (PPP). Fig. 1 is a scatter plot of life expectancy against GDP per person. It shows that countries with higher GDP per person tend to have higher life expectancy. The relationship is strongly nonlinear. Whereas the rank correlation is 0.93, the linear correlation coefficient is only 0.82. The figure suggests a semilogarithmic relationship; when GDP per person is scaled logarithmically, the correlation coefficient with life expectancy rises to 0.93. The relationship between GDP per person and life expectancy is strong, but there are many interesting variations. The United States, for example, is the second richest country in the sample but its life expectancy of 73.2 years is 1 year below that of Canada and 2 years below that of Japan, Norway and the Netherlands. Luxembourg and Germany also rank significantly lower in life expectancy than in GDP per person. On the other hand, Sri Lanka, Spain and Greece stand out as countries that have much longer life expectancy than their GDP ranking might suggest. For these three long-lived populations, life expectancy is comparable to that typically found in countries with twice the level of GDP per person. These disparities highlight the nature of the problem of constructing a single measure or ranking of living standards. On standard comparisons of GDP per person, the United

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Table 1 Life expectancy and Real GDP per person in 1980 for 58 countries Rank

Rank

GDP

L

Country

GDP

L

GDP

L

Country

GDP

L

12 3 9 17 1 6 7 20 14 15 2 16 13 5 8 10 11 18 4 22 30 24 26 21 23 29 25 19 27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Japan Norway Netherlands Spain Canada Denmark France Greece Italy Hong Kong US Israel UK Germany Belgium Finland Austria Ireland Luxembourg Poland Costa Rica Yugoslavia Portugal Hungary Uruguay Panama Argentina Venezuela Chile

8413 11,324 9320 6352 11,619 9829 9780 5100 7787 7164 11,447 6803 8255 10,204 9435 8639 8628 5479 10,623 4321 3174 4044 3832 4631 4258 3186 3843 5429 3649

75.5 75.3 75.3 74.3 74.2 74.2 73.7 73.7 73.6 73.6 73.2 73.1 72.8 72.3 72.3 72.2 72.0 72.0 71.8 70.9 70.8 70.2 70.2 70.0 69.6 69.2 68.7 67.7 67.2

42 35 32 31 37 28 33 36 38 43 40 34 44 49 52 53 46 45 54 50 56 39 41 47 48 55 58 51 57

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58

Sri Lanka Paraguay Korea Colombia Dom. Rep. Brazil Ecuador Tunisia Philipp. Honduras El Salvador Peru Morocco Zimbabwe Kenya India Pakistan Indonesia Madagascar Zambia U. Rp. Tnz. Bolivia Ivory Coast Cameroon Nigeria Malawi Ethiopia Senegal Mali

1224 2131 2585 2840 1980 3347 2585 1993 1740 1212 1416 2507 1199 895 636 570 1097 1098 569 730 361 1633 1368 911 895 415 284 687 337

66.8 66.0 65.5 62.2 62.1 61.8 61.4 60.1 59.8 57.7 57.4 56.9 55.8 53.8 53.4 52.9 51.5 50.0 49.5 49.3 49.0 48.6 48.0 47.0 46.5 43.0 42.0 41.3 40.0

GDP per person is measured in 1980 international dollars. Life expectancy (L) is measured in years. Source: United Nations (1989) and United Nations and Commission of the European Communities (1986).

States appears to be much better off than Japan, and Brazil appears to be much better off than Sri Lanka. These rankings will be reversed if sufficient weight is attached to life expectancy. The choice of these weights is usually quite subjective, as is the choice of whether to enter GDP per person in linear or logarithmic form. Our revealed preference approach requires knowledge of the production technology for our single social indicator, life expectancy, denoted L. A number of approaches to the estimation of such a technology might be considered. First, there is a large and growing literature on the cost-effectiveness of particular medical interventions, most commonly assessed in terms of the marginal cost of gaining quality-adjusted lifeyears (QALY). Applying the results of this literature to health interventions that are marginal in a given country yields estimates of the marginal cost of increasing life expectancy.

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Fig. 1. Income and life expectancy in 58 countries, 1980.

Second, data on the compensating wage differentials associated with increases in the risk of work-related illness, death and injury can be used to estimate the social costs of lifesaving interventions. The assumption underlying this approach is that employers should be willing to pay at least as much for a given improvement in occupational health as they would otherwise pay in compensating differentials to workers for the associated health risk. In this paper, we adopt a third approach based on econometric estimates of the aggregate production technology for life expectancy. The main advantage of the crosscountry econometric approach is that it provides consistent estimates for a large number of countries of the opportunity cost of extending life expectancy. Dowrick et al. (1998) estimate the production function for life expectancy using 1980 data on real expenditures per person, disaggregated by 38 categories, for 58 countries. Employing general-to-specific hypothesis testing, we find three components of expenditure that systematically enhance life expectancy: health (H), education (E) and food ( F). Specification tests suggest the relationship is semilogarithmic. The implied relationship is: Li ¼ 7:45 þ 2:55logðHi Þ þ 3:47logðEi Þ þ 3:75logðFi Þ þ ei ;

ð6Þ

where Li is life expectancy measured in years in country i, real expenditures are calculated at world prices expressed in 1980 US dollars, and ei is a country-specific effect. Details of the regression results behind Eq. (6) are given in Table 2. Regression 1, estimated by Ordinary Least Squares, yields residuals that are strongly heteroscedastic. Fig. 2 shows that the variance of the residuals is negatively correlated with GDP per person, so this variable is used to weight the observations in regression 2. These results replicate those of Dowrick et al. (1998) who report that this semilogarithmic specification dominates both a linear model and Box – Cox nonlinear specifications. Regression 3 tests a log – log relationship, with life expectancy expressed as the shortfall from an assumed maximum of 80 years. This follows the formulation of Anand

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Table 2 Cross-country regression results for life expectancy (L) and comparison with the Anand – Ravallion specification Dependent variable Estimation method Explanatory variablesb Food Medical care Education Constant Summary and test statistics No. of countries R2 (adj) Standard error Heteroscedasticityc Resetd Davidson – McKinnone

L OLS 2.62 (1.9) 2.19 (3.1) 5.30 (4.5) 5.5 (1.1)

58 0.89 3.45 0.05 0.24

 Ln(80  L)

L a

WLS

3.75 2.55 3.47 7.45

OLS (2.8) (3.7) (3.4) (1.3)

58 0.85 2.56

0.13

0.129 (1.3) 0.129 (2.6) 0.349 (4.2)  6.07 (  16.8)

58 0.87 0.24 0.79 0.000 0.001

a

The weights are Real GDP per person. b Annual expenditure per person, log 1980 US dollars. c Minimum p-value from regressions of the squared residuals on the predicted value and predicted value squared. (Regression 1 displays significant heteroscedasticity.) d Minimum p-value from regressions augmented by predicted values to the powers of 2, 3 and 4, respectively. (Regression 3 displays significant functional form mis-specification.) e Variable addition p-value when regression 2 is augmented by the predicted value from regression 3, and vice versa. (The functional form of regression 2 is preferred on this test.)

and Ravallion (1993). We follow the method proposed by Davidson and MacKinnon (1984) for testing between non-nested model specifications. We augment regression 3 with the predicted values of the dependent variable calculated from regression 2, finding that this additional variable has strong additional explanatory power. Reversing the procedure, the added variable has no significant explanatory power. We conclude that the semilog specification of Eq. (6), corresponding to regression 2, is to be preferred.

Fig. 2. Squared residuals from regression 1.

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The country-specific component of the production technology in Eq. (6) is captured by the disturbance term ei. A positive value of ei indicates that, given levels of health and education services and food, life expectancy in country i is higher than average. These residuals might capture measurement errors, country-specific variables such as differences in climate, or omitted variables such as public health infrastructure, and, as discussed below, income distribution. The residuals, along with data on L, H, E and F, and the relative prices of the inputs, are listed in Appendix A. Using Eq. (6), Dowrick et al. (1998) estimate the cost of saving the life of a newborn individual in a wealthy country at between $US1 million and $US2 million. This estimate is comparable to estimates of the subjective value of life derived from studies of wage premiums (Hwang et al., 1992) and from applications of the contingent valuation method (Viscusi, 1990). Thus, for developed countries at least, the results of the cross-country econometric approach agree with those of intra-country approaches. Other cross-country studies have produced comparable results for the determination of life expectancy. Pritchett and Summers (1996) use instrumental variable estimation on a panel of countries. They find that life expectancy rises with GDP per person, but that there is no independent causal effect from years of schooling. Dasgupta and Weale (1992) find that life expectancy is strongly correlated with income, infant mortality, adult literacy and a composite indicator of wellbeing. They do not, however, analyze the quantitative determinants of life expectancy. Given the nonlinear relationship between life expectancy and health-promoting expenditures, we would expect that the distribution of health expenditure should have a significant impact on average life expectancy. With a declining marginal product of healthpromoting expenditure, as in Eq. (6), the transfer of a dollar of medical expenditure from a rich group to a poor group is predicted to decrease the longevity of the former by less than the increase in longevity to the latter. It follows that our estimates are conditional on the prevailing distribution of income and expenditures. The study by Bidani and Ravallion (1997) has the advantage that it introduces distributional variables into the cross-country regressions for average life expectancy. This paper, which draws on data from the World Development Report (World Bank, 1993), supersedes the earlier work of Anand and Ravallion (1993). They confirm that both aggregate spending on health and the targeting of expenditure on the poorer part of the population affect average life expectancy. Their evidence supports Sen’s (1998) argument that ‘support-led’ processes, particularly public provision of health care and basic education, are at least as important as ‘growth mediated’ processes in increasing life expectancy. The relevant part of their estimated relationship is: L ¼ 37 ½poor þ 58 ½1  poor þ 0:15 ½public health ½poor þ . . . ; R2 ¼ 0:74;

ð7Þ

where [poor] is the proportion of the population living below a poverty line of $US2 per day and [public health] is nominal expenditure, unadjusted for purchasing power, converted into US dollars at the market exchange rate.

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We note that the two empirical approaches described in Eqs. (6) and (7) yield broadly comparable estimates of the cost of achieving increases in life expectancy. Taking the regression coefficients in Eq. (7) and using an estimate from Chen and Ravallion (2001) that, for a sample of developing countries, the average proportion of people living below a poverty line of $US2 per day is 0.62, the impact of an additional 1990 US dollar of public health spending in a developing country is an additional 0.09 years of aggregate life expectancy. The prediction from the semilog specification in Eq. (6) is that an additional 1980 US dollar of total health spending increases aggregate life expectancy by 2.55/H, where H is real medical expenditure per person. For India, where 1980 real medical expenditures were $US33 per person, the predicted impact is 2.55/33 = 0.08 years. Translating into 1980 US dollars, Eq. (6) yields the prediction that an additional 1980 US dollar of total health spending increases aggregate life expectancy by 0.05 years. The revealed preference method that we are proposing asks whether country A could have afforded country B’s consumption bundle and could have achieved B’s life expectancy, given the price structure and budget constraint faced by A. To answer this question, we require data on relative prices and real quantities, measured at purchasing power parity. The World Development Report (World Bank, 1993) data, which underlies the estimates of Eq. (7), does not allow us to make such comparisons, because they report domestic nominal expenditures converted into US dollars at the prevailing market rate of exchange. Accordingly, we use the production function estimates (Eq. (6)), which are consistent with the ICP purchasing power parity estimates. 3.1. Adapting the revealed preference approach It is a straightforward exercise to determine for any given pair of countries, A and B, the maximum life expectancy attainable by country A subject to the constraint that consumption of all goods other than health services is greater than or equal to that of country B. Subject to a budget constraint, we choose H, E and F to maximize A’s predicted life expectancy as given by Eq. (6), including the country-specific component, eA. The maximization problem may be written as: LAB ¼ Max LA ðH; E; FÞ;

ð8Þ

subject to the constraints (i) E z EB; (ii) F z FB; (iii) H z 0; and A (iv) PEA(E  EB) + PFA( F  FB) + PH (H  HB) V PAQA  PAQB u SAB. The final constraint is simply that country A can spend more than country B on education, health and food only if there is a surplus, denoted SAB, after consuming country B’s goods. Given that the production function is strictly increasing in all its arguments, the budget constraint will always be binding. Further, the semilog form ensures that health expenditures must always be strictly positive. Thus, constraint (iii) will never bind, and constraint (iv) will always bind.

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It is possible, however, that the constraints on education and food may be either binding or slack. Predicted life expectancy might be increased by transferring resources from, say, food to health, but we cannot carry out a revealed preference comparison if food consumption falls below that in the reference country. Accordingly, we can limit the feasible solutions to Eq. (8) to the following four cases: (a) (b) (c) (d)

constraints (i) and (ii) bind; constraint (i) is binding, constraint (ii) is slack; constraint (ii) is binding, constraint (i) is slack; and constraints (i) and (ii) are slack.

Given the semilog form of the budget constraint, it comes as no surprise that the solution under each of these cases can be expressed as a simple sharing rule. First calculate ‘disposable income’, the maximum expenditure available after matching the other country’s consumption of goods for which the constraints are binding. Then allocate this disposable income amongst health, education and food (or the subset of these for which constraints are not binding). The budget shares for this allocation are in direct proportion to the regression coefficients, aH, aE, aF in the production function (Eq. (6)). In case (d) where neither E nor F is constrained, the optimal allocation of resources to health, for example, is given by: H d;AB ¼ ðS AB þ PHA H B þ PFA F B þ PEA E B Þ  ðaH =½aH þ aF þ aE Þ=PHA :

ð9Þ

The sharing rule for Ed and Fd is identical in form. Substitution of these optimal expenditures into Eq. (8) yields Ld,AB. A similar sharing rule applies to the division of disposable income between health and education alone, as in case (c), or between health and food, as in case (b). In case (a), all disposable income is spent on health. It is then straightforward to compute the maximum feasible life expectancy LAB = max{La,AB, Lb,AB, Lc,AB, Ld,AB}, allowing for the country-specific factor, ea. If this life expectancy is greater than or equal to that of country B, A is revealed preferred to B.

4. Revealed preference results Results of the revealed preference exercise are summarized in the matrix in Table 3 where element ij = ‘ + ’ indicates that i is revealed preferred to j, ‘  ’ indicates the reverse, and ‘nc’ indicates that the revealed preference test is inconclusive. There is one case where the Generalized Axiom of Revealed Preference, discussed by Varian (1982, 1983), enables us to resolve a comparison which is indeterminate using the Weak Axiom of Revealed Preference: Luxembourg and Japan are not directly comparable on the basis of the Weak Axiom, but Luxembourg is revealed preferred to France, and France is revealed preferred to Japan. In this case, the symbols G+ and G  indicate the rankings derived from the Generalized Axiom. The only instance where we reject the maintained hypothesis of common tastes is in the comparison between Finland and Austria, represented as ‘!!’.

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Table 3 Revealed preference over GDP and life expectancy (ranked by GDP) symbols explained in text GDP rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

GDP

L

1 2 3 4

Canada 11,619 74.2 US 11,447 73.2  Norway 11,324 75.3 nc Luxembourg 10,623 71.8  Germany FR 10,204 72.3  Denmark 9829 74.2  France 9780 73.7  Belgium 9435 72.3  Netherlands 9320 75.3  Finland 8639 72.2  Austria 8628 72.0  Japan 8413 75.5  UK 8255 72.8  Italy 7787 73.6  Hong Kong 7164 73.6  Israel 6803 73.1  Spain 6352 74.3  Ireland 5479 72.0  Venezuela 5429 67.7  Greece 5100 73.7  Hungary 4631 70.0  Poland 4321 70.9  Uruguay 4258 69.6  Yugoslavia 4044 70.2  Argentina 3843 68.7  Portugal 3832 70.2  Chile 3649 67.2  Brazil 3347 61.8  Panama 3186 69.2  Costa Rica 3174 70.8  Colombia 2840 62.2  Korea 2585 65.5  Ecuador 2585 61.4  Peru 2507 56.9  Paraguay 2131 66.0  Tunisia 1993 60.1  Dom. Rep. 1980 62.1  Philippines 1740 59.8  Bolivia 1633 48.6  El Salvador 1416 57.4  Ivory coast 1368 48.0  Sri Lanka 1224 66.8  Honduras 1212 57.7  Morocco 1199 55.8  Indonesia 1098 50.0  Pakistan 1097 51.5  Cameroon 911 47.0 

+ nc nc nc  nc                                                                                      

5 6 7 8 9

+ + + + + + + + nc + + + nc nc + nc nc + nc nc +           nc         G                                                                                                                                               

+ + + + + + +

10 11 12 13 14 15 16 17 18 19 20 21 22

+ + + + + + + + + + + + + + + + + + nc + + () + + (+) + +   !!!   !!! nc  (+) nc   (+) nc   (+)                                                                                                                                     

+ + + + + + + + + + + + G+ + + + + + + + + + + + + + + + nc + + + + + + + () () () + nc nc + + nc nc + nc nc + nc nc +       nc    nc                                                                                                                        

+ + + + + + + + + + + + + + nc nc                              

+ + + + + + + + + + + + + + nc nc                              

+ + + + + + + + + + + + + + + + +  nc                           

+ + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + nc + () nc (+) + nc  nc  nc nc  nc nc   nc                                                                    

+ + + + + + + + + + + + + + + + + + nc + nc nc  nc                      

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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + nc + + nc + + nc nc nc

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

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Table 3 (continued) GDP rank 48 49 50 51 52 53 54 55 56 57 58

Nigeria Zimbabwe Zambia Senegal Kenya India Madagascar Malawi U. Rp. Tanzania Mali Ethiopia

GDP

L

1 2 3 4

895 895 730 687 636 570 569 415 361

46.5  53.8  49.3  41.3  53.4  52.9  49.5  43.0  49.0 

337 284

40.0                       42.0                      

        

        

        

5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

GDP is ICP 1980 per capita GDP L is 1980 life expectancy.

Countries are ranked in Table 3 by PPP-adjusted GDP per person. There are six pairwise comparisons where the ranking contradicts the partial ordering suggested by revealed preference. These cases are marked ‘(+)’ and ‘(  )’. For instance, The Netherlands (GDP per person=$US9320) is revealed preferred to Belgium (GDP per person=$US9435). Japan, the United Kingdom and Italy are each revealed preferred to Finland, even though GDP per person in Finland is higher; similarly, Greece is revealed preferred to Venezuela, and Zimbabwe is revealed preferred to Nigeria. We can go further than these outright reversals of rankings to consider the ‘ties’— those cases where countries cannot be separated by revealed preference. The GDP rankings understate the relative standard of living in countries such as Norway, Belgium, Italy, Sri Lanka and Tanzania which rank higher on life expectancy than on GDP per person. Countries that are several places higher on the ICP rankings and are revealed preferred on the standard GDP data are not revealed preferred when life expectancy is included in the comparison. Most notably, Peru is not revealed preferred to Sri Lanka, although, on the ICP measure, Peru is more than twice as well off. Neither is Bolivia, with an ICP measure of $US1633, revealed preferred to Zimbabwe with an ICP measure of only $US895. Equally, the standard of living appears to be overstated by GDP measures that ignore the comparatively low life expectancy of both Peru and Bolivia and other countries such as Luxembourg, Belgium, Brazil, Senegal and Malawi. All of these countries are not revealed preferred to countries that are lower on the ICP rankings. Does the HDI procedure, which incorporates life expectancy and GDP per person in a composite index, yield rankings that are more consistent with our revealed preference ordering? To address this question, we follow the method of constructing the index described in the Human Development Report 1999 (UNDP 1999, technical note, pp. 159– 160), but use only the indexes for GDP per person and life expectancy, excluding the index of educational attainment. This gives a partial HDI measure, lying between zero and one: HDI ¼ ½logðGDPÞ  logð284Þ=2logð11619=284Þ þ ½L  40=2½75:5  40

ð10Þ

S. Dowrick et al. / Journal of Development Economics 70 (2003) 501–529

517

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58         

        

        

        

        

        

        

        

        

        

        

        

        

        

        

        

  nc               

        

nc  nc               

        

 nc nc nc              

nc () + nc (+) +       nc    nc    nc            

+ + + + nc nc nc nc nc nc nc     

+ + nc nc nc

+ + + + + + nc + + + + +  +     nc

+ + + + + + + nc

+ + + + + + + + + + + + + + nc + nc +

                                nc nc +                                   

where 284 and 11619 are the minimum and maximum observed levels of GDP per person and 40 and 75.5 are the limits observed on life expectancy in our data. This measure of a partial HDI captures the important features of scaling and weighting between GDP and life expectancy. It is listed in Table 4 and used as the ranking for the revealed preference matrix. The six mis-rankings found in the comparisons based on GDP are corrected by the inclusion of life expectancy in the HDI. For example, the Netherlands is now ranked above Belgium, in line with the revealed preference test. But the HDI ranking is biased in the opposite direction: the Netherlands is ranked above the United States, Denmark, Germany and Luxembourg although those four countries are each revealed preferred. There are 27 pairwise comparisons where the HDI rankings contradict revealed preference. Since neither GDP nor HDI rankings satisfy our revealed preference test, the challenge of the next section is to provide a method of constructing a utility-consistent index that does satisfy the test.

5. Construction of a true index of GDP per person incorporating life expectancy In order to make cardinal comparisons, we need to restrict the form of the notional preference relationship. One approach would be to estimate the parameters of a specific functional form for a utility or expenditure function, an approach surveyed by Diewert and Nakamura (1993). The alternative, which we prefer, is to require that the preference relationship be homothetic. This is a necessary and sufficient condition to yield a nontrivial money-metric utility that is independent of the reference price vector. Whilst homotheticity may appear to be a strong condition, it has been found that both time series data for US GDP and crosscountry GDP data can be so represented: see Manser and McDonald (1988) and Dowrick and Quiggin (1997) who develop a method for constructing true multilateral indexes. In order to incorporate social indicators into this index number framework, however, we need a linear approximation to the price of the indicator. We achieve this by

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Table 4 Revealed preference over GDP and life expectancy (ranked by HDI) GDP 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Norway Canada Netherlands US Denmark Japan France Germany FR Luxembourg Belgium Italy UK Finland Austria Hong Kong Spain Israel Greece Ireland Poland Hungary Venezuela Yugoslavia Uruguay Portugal Costa Rica Argentina Panama Chile Korea Brazil Paraguay Colombia Ecuador Sri Lanka Dom. Rep. Tunisia Peru Philippines El Salvador Honduras Morocco Bolivia Zimbabwe Pakistan Ivory Coast Indonesia Kenya

L

HDI GDP 1 rank

11,324 75.3 0.99 11,619 74.2 0.98 9320 75.3 0.97 11,447 9829 8413 9780 10,204

73.2 74.2 75.5 73.7 72.3

3 1 9

0.97 2 0.96 6 0.96 12 0.95 7 0.94 5

10,623 71.8 0.94

4

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

nc + nc +  

nc + + + + +   +

+ + nc + + + + + + + nc   (+) +

+ + +

+ + +

+ + +

+ + +

+ + +

+ + +

+ + +

+ + +

+ + +

    

+ +  +     +  nc +

+ + + nc    +

nc    

+ +  nc +

nc  +

 nc G+ +

nc

+ nc G  nc

+ + nc + +

+ + nc + +

+ + nc + +

+ + (+) + +

+ + nc + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+ + + + +

+

+

+

+

+

+

+

+

+

+

+

+ + + + + nc

+ + + + + nc

+ + + + + +

+ + + + + +

+ + + + + +

nc                             

nc + +   nc      ()                                                

9435 7787 8255 8639 8628 7164

72.3 73.6 72.8 72.2 72.0 73.6

0.93 0.92 0.92 0.91 0.91 0.91

8 14 13 10 11 15

     

     

()     

     

     

nc nc nc () nc 

     

     

     

+

    

+ nc

+ + (+)  nc (+) nc () () !!! + nc !!!    

+ + + + +

6352 6803 5100 5479 4321 4631 5429 4044 4258 3832 3174 3843 3186 3649 2585 3347 2131 2840 2585 1224 1980 1993 2507 1740 1416 1212 1199 1633 895 1097 1368

74.3 73.1 73.7 72.0 70.9 70.0 67.7 70.2 69.6 70.2 70.8 68.7 69.2 67.2 65.5 61.8 66.0 62.2 61.4 66.8 62.1 60.1 56.9 59.8 57.4 57.7 55.8 48.6 53.8 51.5 48.0

0.90 0.89 0.86 0.85 0.80 0.80 0.79 0.78 0.78 0.78 0.76 0.76 0.74 0.73 0.66 0.64 0.64 0.62 0.60 0.57 0.57 0.55 0.53 0.52 0.46 0.44 0.42 0.36 0.35 0.34 0.32

17 16 20 18 22 21 19 24 23 26 30 25 29 27 32 28 35 31 33 42 37 36 34 38 40 43 44 39 49 46 41

                              

                              

                              

                              

                              

                              

                              

                              

                              

                              

                              

                              

                              

nc nc                             

   

 

           

1098 50.0 0.32 45 636 53.4 0.30 52

     

     

     

                              

+ + + + nc + +   nc  nc    nc      nc                                          

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+ + + + +

+ + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + +

+ + + + + +

+ + + + + +

+ + + + nc

+ + + + (+) + + + nc + nc + nc nc nc nc     nc nc  nc  nc                                  

nc  nc                      

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

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+ + + + + +

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+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + + +

+ + + + nc nc nc nc

+ + + + + + + + nc

+ + + + + + + + + +

+ + + + nc + nc nc + nc 

+ + + + + + + nc + nc nc +

+ + + + + + + nc + nc nc + 

+ + + + + + + + + + nc + + +

+ + + + + + + + + nc nc + nc nc nc

+ + + + + + + + + + + + + + nc +

+ + + + + + + + + + + + nc + nc + 

+ + + + + + + + + + + + + + nc + nc +

+ + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + nc + + + + nc

+ + + + + + + + + + + + + + + + + + + nc nc

+ + + + + + + + + + + + + + nc + nc + + nc  

+ + + + + + + + + + + + + + + + + + + nc + + +

+ + + + + + + + + + + + + + + + + + + nc + + + +

+ + + + + + + + + + + + + + + + + + + nc + + + + +

+ + + + + + + + + + + + + + + + + + + nc + + + + + nc

+ + + + + + + + + + + + + + + + + + + nc + + + + nc nc nc

+ + + + + + + + + + + + + + + + + + + nc + + + + + + + nc

+ + + + + + + + + + + + + + + + + + + nc + + + + + + + + nc

+ + + + + + + + + + + + + + + + + + + nc + + + + + nc  nc  

+ + + + + + + + + + + + + + + + + + + nc + + + + + + nc + nc  +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + nc + + + + + + + + nc + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + nc + + + + + + + + + + + + + + + + + (+) + nc + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

nc                     

 nc nc nc  nc               

+ nc nc nc nc               

                  

+  nc  nc             

 nc               

nc nc nc nc  nc  nc        

              

+ nc    nc        

            

           

nc nc nc nc nc nc nc nc nc nc nc

nc +        

+        

       

      

  nc   

nc nc   nc

nc  nc   nc + nc + +

              nc       nc  nc +  + + + + + + + + + + +                            nc nc   + + nc + + +

(continued on next page)

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Table 4 (continued) GDP 49 50 51 52 53 54 55 56 57 58

India Zambia Cameroon Nigeria Madagascar U. Rp. Tanzania Senegal Malawi Ethiopia Mali

L

HDI GDP 1 rank

2

3

4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19 20

570 730 911 895 569 361

52.9 49.3 47.0 46.5 49.5 49.0

0.28 0.26 0.26 0.25 0.23 0.16

53 50 47 48 54 56

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

687 415 284 337

41.3 43.0 42.0 40.0

0.14 0.09 0.03 0.02

51 55 58 57

   

   

   

   

   

   

   

   

           

   

   

   

   

   

   

   

   

   

substituting into the standard GDP bundle the notional quantity of medical services that would be predicted by Eq. (6) to have delivered the observed longevity. This procedure has the effect of transforming the units of life expectancy, for which the cost is highly nonlinear, into units of equivalent medical services with a constant price. Use of this approximation allows us to test the GDP and life expectancy data for homotheticity and derive a set of ideal true index numbers for the countries that fit a common homothetic representation, following the procedure described by Dowrick and Quiggin (1997). The first step is to calculate the matrix, L, of log Laspeyres ratios between each pair of countries. If the generating preference relationship is homothetic, the true utility ratio, ui/uj, must lie below the Laspeyres ratio. We can test whether there exists a set of utility numbers, {u1, u2, . . .,. . ., un}, that satisfies all the bilateral conditions. Using the algorithm suggested by Varian (1983), we replace each element Lij by its minimum path, Mij, to construct the minimum path matrix, M, where: Mij ¼ min:ðk;...;mÞ ½Lij ; ðLik þ Lkl þ . . . þ Lmj Þ: The element Mij gives the upper homothetic bound for the logarithm of j’s utility index relative to i’s. Afriat (1981) shows that a homothetic representation is possible if and only if the (principal) diagonal elements of this matrix, namely Mjjj = 1. . .n, are nonnegative. We find that this condition is satisfied if we exclude five countries: Finland, Israel, Hungary, Poland and Paraguay. The row and (negative) column averages of the minimum path matrix give, respectively, the lower and upper bounds to the true index of country i relative to the mean of the index. These bounds are listed in Table 5 in logarithmic form, so that the number 0.22, for example, indicates that the bound is e0.22 = 1. 24 times the geometric mean of the sample. The sets of upper and lower bounds each constitute a true index. We also list the midpoints of the bounds, which constitute the Ideal Afriat Index, a true index which reduces to the Fisher ideal index in the case of pairwise comparison. This gives our preferred true index, A={A1,A2,. . .,Ah}, over the set of observations, h, for which a homothetic representation is possible.

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521

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58      

     

     

     

     

     

     

     

     

     

     

     

     

     

  nc nc  

     

     

     

     

     

     

     

     

     nc  ()    

   nc  

     

nc nc nc nc + + + + + +      

  +   + nc + nc +     

   

   

   

   

   

   

   

   

   

   

   

   

   

–   

   

   

   

   

   

   

   

   

   

   

   

 –  

nc   

   

   

nc   

nc   

   

nc   

+ + + + +

nc nc + + nc 

+ + + + + nc

+ + + + + +

+ + + + + nc

+ + + + nc  + nc     nc  nc +

These index numbers have a meaningful cardinal interpretation. If A’s index number is twice that of B’s, we infer that the representative household would be indifferent between, on the one hand, the actual GDP bundle and life expectancy of A, and, on the other hand, twice the GDP bundle of B with twice the notional health expenditures of B. Dealing with the five ‘non-homothetic’ countries is a problem if we want to establish a method that can be applied to all countries. One solution is to follow Varian (1983) in defining a utility function for any consumption bundle, xk: Pðxk Þ ¼ minðAi þ lnðpi xk =pi xi ÞÞ: iah

This utility function yields Ak for any observation within h and the minimum of the set of Paasche valuations (using i’s prices to value xk relative to each xi) for any observation k outside h. An obvious counterpart is to define the utility function: Lðxk Þ ¼ maxðAi þ lnðpk xk =ppk xi ÞÞ; iah

which yields Ai for any observation k within h and gives the maximum of the Laspeyres valuations (using country k’s own prices) for k outside h. A natural extension is to define the mid-point of these two functions, A*ðxÞ ¼ ½Pðxk Þ þ Lðxk Þ=2; which extends the true index A to include observations which do not fit the homothetic representation. The values Ak* can be given an economic interpretation even for such observations. They give the utility of the representative agent whose preferences are homothetic and consistent with observations in the set h, if they were allocated the consumption bundle xk, even though they would not have chosen the bundle xk at price pk. The extended Ideal Afriat index is listed in Table 5. An asterisk marks the five countries with non-homothetic observations. The countries are listed in descending order of GDP per person. We find that these rankings are fully consistent with the partial ordering of the

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Table 5 True index numbers and GDP True indices

Canada US Norway Luxembourg Germany FR Denmark France Belgium Netherlands Finland* Austria Japan UK Italy Hong Kong Israel* Spain Ireland Venezuela Greece Hungary* Poland* Uruguay Yugoslavia Argentina Portugal Chile Brazil Panama Costa Rica Colombia Korea Ecuador Peru Paraguay* Tunisia Dom. Rep. Philippines Bolivia El Salvador Ivory Coast Sri Lanka Honduras Morocco Indonesia Pakistan Cameroon Nigeria

Log GDP

Ideal Afriat (A)

Upper bound

Lower bound

1.58 1.54 1.54 1.46 1.43 1.42 1.40 1.36 1.40 1.22 1.26 1.28 1.23 1.22 1.06 0.93 1.05 0.83 0.60 0.85 0.73 0.69 0.56 0.56 0.52 0.44 0.37 0.24 0.25 0.33 0.10 0.03  0.06  0.13 0.05  0.29  0.24  0.41  0.71  0.59  0.70  0.67  0.68  0.78  0.89  0.98  1.12  1.14

1.65 1.61 1.65 1.55 1.50 1.48 1.47 1.43 1.48 1.27 1.33 1.39 1.30 1.29 1.18 0.99 1.13 0.90 0.70 0.96 0.80 0.70 0.66 0.66 0.64 0.54 0.51 0.35 0.39 0.42 0.22 0.21 0.04  0.04 0.16  0.21  0.13  0.33  0.54  0.51  0.62  0.36  0.61  0.70  0.79  0.89  1.04  1.06

1.51 1.47 1.43 1.36 1.36 1.35 1.34 1.29 1.33 1.18 1.19 1.16 1.15 1.16 0.94 0.88 0.97 0.77 0.50 0.73 0.67 0.68 0.46 0.46 0.40 0.35 0.23 0.13 0.10 0.24  0.02  0.15  0.16  0.21  0.06  0.36  0.34  0.49  0.88  0.66  0.78  0.97  0.75  0.87  0.99  1.08  1.21  1.22

1.52 1.50 1.49 1.43 1.39 1.35** 1.35 1.31 1.30** 1.22 1.22 1.19 1.18 1.12** 1.03 0.98 0.91** 0.77** 0.76** 0.69** 0.60 0.53 0.51 0.46 0.41 0.41 0.36 0.27 0.22 0.22** 0.11 0.01 0.02  0.02**  0.18  0.25  0.25  0.38  0.45**  0.59  0.62**  0.73  0.74  0.75  0.84  0.84**  1.03**  1.05**

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Table 5 (continued ) True indices

Zimbabwe Zambia Senegal Kenya India Madagascar Malawi U. Rp. Tanzania Mali Ethiopia

Log GDP

Ideal Afriat (A)

Upper bound

Lower bound

 1.00  1.31  1.50  1.36  1.40  1.56  1.95  1.96  2.20  2.28

 0.90  1.18  1.37  1.29  1.30  1.48  1.83  1.88  2.11  2.19

 1.11  1.43  1.64  1.44  1.50  1.64  2.07  2.04  2.29  2.37

 1.05  1.25  1.31**  1.39  1.50**  1.50  1.81**  1.95  2.02**  2.19

True index numbers and GDP are measured in logs relative to the 53 country-means. * Indicates countries outside the homothetic set, for which we use the index A*. ** Indicates that the ratio of GDP to the sample mean lies outside the true bounds.

revealed preference tests. For purposes of comparison, GDP per person is also listed in Table 5 in logarithmic form, normalized to a zero mean for the sample of 53 countries. We find that in one-third of the cases, indicated by ‘**’, the log GDP index lies outside the true bounds. For example, the Netherlands is only 1.30 above the mean on the log GDP index, but the true bounds are (1.33, 1.48). Fig. 3 displays GDP per person and the extended Ideal Afriat index A*, both measured on the left-hand axis, and the partial HDI index measured on the right-hand axis. The countries are ordered by A*, so non-monotonicity of the GDP or HDI line indicates a change in the ordering relative to A*. Fig. 3 illustrates the tendency for countries, like the Netherlands, which are ranked too low by the GDP index to be ranked too high by the HDI index, and vice versa.

6. Concluding comments Two different types of objection have been made to the use of measures of GDP per person as a basis for international comparisons of living standards. First, the set of consumption items that make up the GDP index omits important elements of wellbeing, such as health status. Second, standard procedures for combining quantities of individual goods into a GDP index at national or international prices do not typically reflect social opportunity costs, so that rankings derived from such indexes may be inconsistent with rankings inferred from revealed preferences. In this paper, it has been shown that these difficulties may be overcome using a generalization of the revealed preference approach pioneered by Samuelson (1947). This generalization, using an implicit cost function, provides a natural framework for the incorporation of information on social indicators into revealed preference analysis. We find that the international data on life expectancy and GDP per person allow the construction of a partial ordering of the welfare of a ‘representative agent’ facing the prices and outcome of each country.

S. Dowrick et al. / Journal of Development Economics 70 (2003) 501–529

Fig. 3. Comparison of indexes.

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525

We have then constructed a true index (with well-defined upper and lower bounds) which incorporates both the standard components of GDP and life expectancy. Although this method requires that we redefine the utility function on ‘notional health services’ rather than directly on life expectancy, the resulting ranking is fully consistent with the partial ordering of revealed preference. Where a unique set of cardinal index numbers is required, this method may be preferable to the relatively arbitrary weighting procedures of other composite indexes. These new index numbers incorporate life expectancy into measures of national income and welfare for the purposes of quantitative policy analysis. Although our empirical analysis has been undertaken for the case of a single social indicator, the general method set out above is applicable to a vector of indicators. A straightforward extension would be to include, for instance, literacy rates and other measures of educational attainment as a function of consumption of education services and possibly other goods. Another possible extension would be to replace life expectancy with the multiple measures of health outcomes employed to form the quality-adjusted life-years (QALYs) index. The main limitation of our revealed preference approach is that it relies on an assumption that welfare is based on the consumption of goods and services. It may be argued that social indicators are categorically different from the consumption items incorporated in GDP, so that no coherent index that aggregated them could be formed. We argue that it is not necessary to accept this argument in relation to literacy or life expectancy inasmuch as they are regarded as capabilities that are prerequisites for the enjoyment of consumption. A multiplicative utility function that captures the notion of capabilities is still consistent with the revealed preference. However, as argued by Desai (1991), there remains a class of indicators of human welfare, including notions of freedom and democracy and community, which are probably not amenable to the ‘utilitarian calculus’. Our application of the revealed preference method allows the incorporation of outputs such as life expectancy valued at social opportunity cost, where costs vary across countries according to both the level and distribution of income as well as climatic and institutional factors. Ours is a positive rather than normative approach. It may well be argued that life should be valued more highly. Indeed our results demonstrate that the Human Development Index does, implicitly, value life expectancy above its opportunity cost. The implication of our results is not to say that the HDI is ‘wrong’, rather it is to highlight the fact that life expectancy in many parts of the world could be extended at surprisingly low cost. Our index number approach should allow improved testing of models of economic development in which economic welfare is defined more widely than by conventional GDP. For example, our cross-country analysis confirms the findings of intra-country studies that education yields both direct and indirect benefits for health in general and life expectancy in particular. This finding suggests that estimates of the impact of human capital investment on economic growth may understate the welfare benefits if they measure growth solely in terms of GDP and ignore important indicators such as life expectancy. Equally, the index number approach may have important implications for the design of studies testing the relationship between measures of inequality and

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economic development, since reductions in inequality are likely to increase longevity. Application of the index number approach to a panel of observations on GDP, life expectancy and inequality would enable these hypotheses to be tested.

Acknowledgements We thank Nancy Wallace for helpful comments and criticism.

Appendix A . Regression data and residuals

Food

Medical care Actual

Canada US Norway Luxembourg Germany FR Denmark France Belgium Netherlands Finland Austria Japan UK Italy Hong Kong Israel Spain Ireland Venezuela Greece Hungary Poland Uruguay Yugoslavia Argentina Portugal Chile Brazil Panama Costa Rica Colombia Korea Ecuador Peru

Education

Life expectancy

Notional

Quantity

Price

Quantity

Price

Quantity

1070 939 825 900 940 848 1058 1040 943 770 818 606 769 1145 779 811 1205 665 585 962 599 569 868 461 807 921 534 816 575 494 441 433 621 447

1.018 1.000 1.539 1.345 1.448 1.569 1.363 1.330 1.253 1.401 1.372 1.603 1.239 1.054 0.840 0.931 0.969 1.124 1.110 0.975 0.554 0.781 0.973 1.135 1.552 0.825 0.993 0.570 0.770 1.015 0.678 0.979 0.668 0.607

892 889 1298 705 923 892 1154 876 998 1012 1022 995 725 544 455 675 574 561 294 354 740 445 280 523 105 395 214 212 230 280 255 132 177 125

0.258 1.000 0.668 0.769 0.962 0.986 0.821 0.824 0.847 0.568 0.645 0.509 0.606 0.560 0.464 0.479 0.613 0.715 0.522 0.446 0.085 0.212 0.622 0.273 1.457 0.300 0.468 0.368 0.395 0.358 0.215 0.192 0.297 0.221

681 458 2188 632 512 671 673 327 1147 847 646 2439 626 661 775 835 1687 843 207 2181 958 1799 183 2207 733 733 295 45 719 1204 109 369 35 11

Price 0.338 1.940 0.396 0.858 1.734 1.310 1.409 2.207 0.737 0.679 1.021 0.208 0.702 0.461 0.272 0.387 0.208 0.476 0.741 0.072 0.066 0.052 0.953 0.065 0.208 0.162 0.339 1.728 0.127 0.083 0.503 0.068 1.480 2.540

Actual

Residual (Years)

Quantity

Price

(Years)

997 1152 768 636 818 1295 882 1020 1069 681 735 1048 1012 797 1074 843 463 756 706 411 433 373 874 329 262 351 519 276 444 567 315 338 394 367

1.067 1.000 1.302 1.267 1.241 1.154 1.167 1.259 1.231 1.417 1.156 1.051 0.912 0.846 0.529 0.644 0.847 0.610 0.503 0.581 0.386 0.403 0.358 0.707 1.253 0.452 0.379 0.525 0.372 0.390 0.271 0.372 0.284 0.252

74.2 73.2 75.3 71.8 72.3 74.2 73.7 72.3 75.3 72.2 72.0 75.5 72.8 73.6 73.6 73.1 74.3 72.0 67.7 73.7 70.0 70.9 69.6 70.2 68.7 70.2 67.2 61.8 69.2 70.8 62.2 65.5 61.4 56.9

 0.689  1.692 1.334  0.280  1.504  0.725  1.378  2.515 0.356  0.455  1.172 2.290  0.374 0.497 1.361 0.544 2.754 1.038  0.895 4.645 0.657 3.565  1.090 3.677 4.968 1.576 0.825  3.948 2.906 3.724  2.172 2.633  4.099  6.233

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Appendix A (continued) Food

Medical care Actual

Quantity Paraguay Tunisia Dom. Rep. Philippines Bolivia El Salvador Ivory Coast Sri Lanka Honduras Morocco Indonesia Pakistan Cameroon Nigeria Zimbabwe Zambia Senegal Kenya India Madagascar Malawi U. Rp. Tanzania Mali Ethiopia

Price

Quantity

Education

Life expectancy

Notional Quantity

Actual

Residual

Price

Quantity

Price

Price

(Years)

(Years)

361 500 505 374 394 225 229 279 230 349 265 354 112 215 120 147 254 146 160 209 159 113

0.897 0.760 0.887 0.610 0.704 0.851 1.327 0.396 0.777 0.875 0.559 0.448 1.407 1.557 0.871 1.274 0.922 0.763 0.597 0.825 0.503 1.175

57 111 273 168 167 163 11 127 171 36 19 112 67 26 31 34 13 21 33 4 14 13

0.428 0.266 0.289 0.187 0.279 0.292 0.918 0.050 0.251 0.474 0.355 0.082 0.609 0.845 0.450 0.566 0.670 0.284 0.140 0.524 0.342 0.216

593 58 83 91 2 126 5 1959 325 36 9 10 8 3 116 45 0 52 79 14 5 35

0.041 0.513 0.947 0.345 23.366 0.379 2.154 0.003 0.132 0.471 0.744 0.943 5.202 6.644 0.120 0.427 21.214 0.116 0.058 0.147 0.988 0.080

334 240 322 215 134 147 108 232 78 144 101 108 120 97 109 48 84 143 81 81 37 70

0.201 0.309 0.276 0.162 0.243 0.293 0.562 0.071 0.386 0.342 0.248 0.154 0.359 0.415 0.413 0.474 0.255 0.248 0.109 0.267 0.168 0.177

66.0 60.1 62.1 59.8 48.6 57.4 48.0 66.8 57.7 55.8 50.0 51.5 47.0 46.5 53.8 49.3 41.3 53.4 52.9 49.5 43.0 49.0

5.999  1.676  3.030  1.564  11.303  0.663  2.178 6.984 1.632 0.019  1.887  6.233  5.475  5.264 3.367 0.717  8.820 2.284 2.260 3.246  2.708 2.548

126 61

0.772 0.597

5 13

0.451 0.261

3 6

0.805 0.604

30 70

0.163 0.124

40.0 42.0

 1.481  2.142

Quantities are valued in $1980 at US prices. Prices are normalised to unity for the US. The residuals are from the life expectancy regression (2) reported in Table 2. Notional Quantity and Price, H* and P*, are derived from Eq. (6) as H*=exp[L3.47log(E)3.75log(F)e7.45]/ 2.55; and P*=PH/H*. Source: UN and CEC (1987) and UN (1989).

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