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Measuring lifecycle inequality Blewett, Edwin 1982

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MEASURING L I F E C Y C L E I N E Q U A L I T Y by EDWIN BLEWETT B . A . ( H o n . ) , T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1974 M . A . , Q u e e n ' s U n i v e r s i t y , 1977 A T H E S I S S U M B I T T E D IN P A R T I A L F U L F I L M E N T OF T H E REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY i n T H E F A C U L T Y OF GRADUATE S T U D I E S ( D e p a r t m e n t o f E c o n o m i c s ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y OF B R I T I S H COLUMBIA M a r c h 1982 © E d w i n B l e w e t t , 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ^OfrAJSV^^LA^ The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date n r . C I -) /1Q \ i i Abstract In t h i s thesis, t h e o r e t i c a l l y sound and empirically tractable solutions are provided to problems inherent in the t r a d i t i o n a l practice of measuring inequality in the d i s t r i b u t i o n of annual income. Inequality i s taken throughout to mean the extent to which society f a l l s short of a situ a t i o n in which everyone i s equally well-off. The measurement of annual income inequality i s inappropriate in th i s regard because i t i s consumption, not income, that produces welfare. Furthermore, ind i v i d u a l , and therefore s o c i a l , welfare depends on consumption over the l i f e c y c l e , not just in a single year. There are also problems of a less t h e o r e t i c a l nature. Measured annual inequality includes an age-related component attributable to the shape of l i f e c y c l e income p r o f i l e s . Annual inequality indices also f a i l to account for the effects of income mobility. In response to these problems, two new approaches to the measurement of inequality are proposed. In the welfare approach, an improved index of inequality i s sought by replacing annual income with a summary s t a t i s t i c of l i f e c y c l e consumption. L i f e c y c l e inequality i s then decomposed within and among age-cohorts. Intercohort inequality captures the contribution of economic growth to t o t a l inequality, while intracohort inequality i s an index of pure interpersonal inequality. The decomposition approach is a compromise between the inadequacy of measuring annual income inequality and the imposs i b i l i t y of measuring l i f e c y c l e consumption inequality. Total inequality i s measured in panel consumption data treated as a single d i s t r i b u t i o n , and then decomposed into indices of age-related, mobility-related, and pure interpersonal inequality. Empirical implementation of the decomposition approach indicates that age-, and especially mobility-related, inequality account for substantial portions of t o t a l measured inequality. S e n s i t i v i t y tests of the decomposition approach indicate that i t i s a robust method of measuring inequality. F i n a l l y , the decomposition approach i s applied to the problem of measuring the trend of inequality, widely observed to have been remarkably constant in the post-War period. Although the trend of measured annual inequality i s constant, l i f e c y c l e inequality as measured using the decomposition approach declines over the sample period. The p r i n c i p a l finding of t h i s thesis i s that the decomposition approach to the measurement of inequality i s esse n t i a l for an accurate assessment of the l e v e l and trend of pure interpersonal inequality. TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES , v i ACKNOWLEDGEMENTS v i i CHAPTER ONE: Introduction .• • • 1 CHAPTER TWO: A Welfare Approach To The Measurement Of Inequality 28 CHAPTER THREE: A Decomposition Approach To The Measurement Of Inequality 52 CHAPTER FOUR: Empirical Implementation Of The Decomposition Approach • • 72 CHAPTER FIVE: The Trend Of L i f e c y c l e Inequality ..107 CHAPTER SIX: Summary And Conclusions 119 SELECTED BIBLIOGRAPHY .126 APPENDIX A: An Alternative Procedure For Decomposing Inequality 1 32 APPENDIX B: Evaluation Of A Method Of Approximating Long-Run Inequality 148 APPENDIX C: Data . .161 APPENDIX D: Inequality In The Origi n a l Sample 166 APPENDIX E: A Proof 169 V LIST OF TABLES TABLE 1: Annual Means and Standard Deviations of the Panel Data 94 TABLE 2: Indices of Absolute Inequality , 95 TABLE 3: Indices of Relative Inequality 96 TABLE 4: Indices of Relative Equality 97 TABLE 5: The Effect of Age-cohort Bracket Size on Intra-and Intercohort Absolute Inequality 98 TABLE 6: The Effect of Age-cohort Bracket Size on Intra-and Intercohort Relative Inequality 99 TABLE 7: The Effect of Age-cohort Bracket Size on Intra-and Intercohort Relative Equality .100 TABLE 8: The Effect of Number of Years of Data on Indices and Subindices Of Absolute Inequality 101 TABLE 9: The Effect of Number of Years of Data on Indices and Subindices Of Relative, Inequality 103 TABLE 10: The Effect of Number of Years of Data on Indices and Subindices Of Relative Equality 105 TABLE 11: The Trends of Annual and L i f e c y c l e Inequality ...118 TABLE 12: Additively Decomposable Indices of Inequality ...146 TABLE 13: Total, Age-Related and Paglin Inequality Indices 159 TABLE 14: Intra- and Inter-Age-Cohort Inequality Indices ..1.60 TABLE 15: Relative Inequality In The Original Sample 168 v i LIST OF FIGURES FIGURE I: Cumulative Income D i s t r i b u t i o n Showing Age-Related Inequality 27 FIGURE I I : Mean And Equally Distributed Equivalent Consumption 51 FIGURE I I I : Histogram of the Panel Data ...93 FIGURE IV: Cumulative Income D i s t r i b u t i o n Showing Age-Related Inequality 158 v i i ACKNOWLEDGEMENTS I would l i k e to thank my supervisor, David Donaldson, for the very valuable contributions that he has made to this thesis and my understanding of i t s subject. My task would have been immeasurably more d i f f i c u l t without his help. My i n t e l l e c t u a l debt to Chuck Blackorby i s also substantial. He f i r s t taught me the theory of inequality measurement and has aided and encouraged me throughout t h i s project. I would l i k e to thank Jon Kesselman for his careful reading of the thesis in dr a f t . He helped to c l a r i f y a number of important empirical issues and improve my exposition. My thanks go also to David Stapleton and Terry Wales, who provided helpful suggestions along the way. Frank Flynn and Jean Wu kindly assisted in solving several programming problems. For f i n a n c i a l support I am indebted to the Social Sciences and Humanities Research Council of Canada for a Doctoral Fellowship and to the Department of Economics at the University of B r i t i s h Columbia. 1 CHAPTER ONE Introduction Measuring inequality in the d i s t r i b u t i o n of annual income was early established as a t h e o r e t i c a l and empirical norm. This approach has survived despite considerable evidence that annual income inequality is a poor index of the extent to which society f a l l s short of a situation in which everyone i s equally well-o f f . In empirical work, for example, i t was discovered that the Gini c o e f f i c i e n t i s sensitive to the length of the income accounting period. Income mobility, the tendency for individuals' r e l a t i v e positions in a d i s t r i b u t i o n to change over time, works to reduce the dispersion of incomes cumulated over several years. Since the choice of an accounting period i s largely a r b i t r a r y , what e t h i c a l content might otherwise be imputed to the Gini c o e f f i c i e n t of annual incomes i s e f f e c t i v e l y destroyed. In a similar vein, i t was recognized that measured annual inequality r e f l e c t s not only income differences within a population, but also i t s age-structure. L i f e c y c l e p r o f i l e s estimated from cross-sectional data show that income varies systematically with age, tending to r i s e at a decreasing rate over the working years, eventually l e v e l i n g off and declining somewhat after retirement. Young people and seniors therefore predominate in the low income portions of the d i s t r i b u t i o n while individuals in the prime of working l i f e are among the majority of high income receivers. Measured annual income inequality thus 2 includes an age-related component; a change in i t s value may as e a s i l y be the result of a demographic change such as the maturing of a "baby boom" generation, as a tendency for the r i c h to get richer at the expense of the poor. Theoretical c r i t i c i s m s of the t r a d i t i o n a l practice of measuring inequality in the d i s t r i b u t i o n of income have also been raised. Ideally, inequality should be measured in the d i s t r i b u t i o n of welfare. A problem a r i s e s , however, because individual u t i l i t y functions are known only up to a monotonic transformation. Measured inequality in the d i s t r i b u t i o n of the images of individual u t i l i t y functions depends on the p a r t i c u l a r functional representation of preferences, and i s therefore not unique. 1 The imposs i b i l i t y of measuring inequality in the d i s t r i b u t i o n of welfare led to i t s being approximated by inequality measured in the d i s t r i b u t i o n of income. Only recently have objections been raised against t h i s practice on the grounds that the d i s t r i b u t i o n of welfare i s more clos e l y related to- the d i s t r i b u t i o n of consumption than of income. Inequality would be more accurately approximated, i t has been argued, i f i t were measured in the d i s t r i b u t i o n of consumption. Furthermore, the importance, for considerations of welfare, of l i f e c y c l e consumption has been stressed, r a i s i n g further questions. For example, what summary s t a t i s t i c of l i f e c y c l e consumption i s appropriate for use as one of the arguments in an inequality 1The situ a t i o n i s akin to the problem pointed out by Atkinson [1970] that Dalton's [1920] measure of inequality i s not invariant with respect to linear transformations of individual u t i l i t y functions. 3 index? And how should the effects of economic growth, which puts the l i f e c y c l e consumption prospects of young people considerably above those of their elders, be taken into account in the measurement of inequality? These are some of the problems with the theory and practice of inequality measurement which have stimulated the present research. In t h i s introductory chapter a detailed analysis of these problems i s provided within the context of a review of the l i t e r a t u r e on the measurement of inequality. This leads to the development of two new approaches to the measurement of inequality. In the welfare approach, presented in Chapter Two, summary s t a t i s t i c s of l i f e c y c l e consumption p r o f i l e s replace annual incomes as the arguments of an inequality index. While t h e o r e t i c a l l y sound, this approach turns out to be impractical, and an empirically tractable alternative i s presented in Chapter Three. In the decomposition approach t o t a l inequality i s measured in panel consumption data and decomposed into three components, one of which may be interpreted as an index of pure interpersonal inequality. Empirical results for decomposition approach indices of inequality are presented in Chapter Four. Annual inequality indices computed from the same data set are also reported in order to evaluate and compare the performances of these two types of inequality indices. In addition, the robustness of the decomposition approach is investigated by examining the s e n s i t i v i t y of the computed indices to changes in the s p e c i f i c a t i o n of certain important variables. F i n a l l y , in Chapter Five, I apply the decomposition approach to the problem of analysing the trend of l i f e c y c l e inequality. I summarize the 4 results of my research and draw some conclusions from i t in Chapter S i x . 2 The d i s t r i b u t i o n of annual income has long been the centre of attention in both the theory and practice of measuring inequality. Lorenz [1905] and Gini [1912], for example, proposed methods of portraying and measuring inequality in income d i s t r i b u t i o n s that are s t i l l the best known and most popular techniques of inequality measurement. Dalton's [1920] pioneering the o r e t i c a l work on "The Measurement of the Inequality of Incomes" provided the f i r s t insights into the s o c i a l welfare foundations of the subject. Based on these i n f l u e n t i a l precedents, empirical studies of inequality have concentrated on the d i s t r i b u t i o n of annual income. This t r a d i t i o n i s continued in the modern theory, of inequality measurement, due primarily to Atkinson [1970], Kolm [1969], and Sen [1973]. Theoretical interest in the personal d i s t r i b u t i o n of income stems in part from the c l a s s i c a l economists' interest in the d i s t r i b u t i o n of factor shares and the associated neoclassical marginal productivity theory of d i s t r i b u t i o n , which are concerned with the d i s t r i b u t i o n of t o t a l product or, in monetary terms, income. The predominance of the income d i s t r i b u t i o n in empirical studies of inequality i s largely the result of the r e l a t i v e a v a i l a b i l i t y of annual income data. The t r a d i t i o n of 2There are two appendices attached to th i s thesis. In Appendix A, an alternative procedure for decomposing inequality within and among population subgroups i s evaluated and compared to the one that I have employed in the decomposition approach to the measurement of inequality. A method of approximating the degree of l i f e t i m e inequality using annual data is compared to the decomposition approach in Appendix B. 5 measuring inequality in the d i s t r i b u t i o n of annual income may thus be said to have been born of a marriage of empirical pragmatism and th e o r e t i c a l rationale. Measuring inequality in the annual income d i s t r i b u t i o n , however, i s both t h e o r e t i c a l l y and methodologically incorrect. B r i e f l y , the methodological problems concern the f a i l u r e of measured annual income inequality to take account of intertemporal and intergenerational aspects of inequality that should be distinguished from purely interpersonal inequality. In th i s sense, simply measuring annual income inequality i s incomplete. A theoret i c a l problem arises because the correct interpretation of inequality i s the extent to which individuals in society are not equally well-off, which implies that measuring inequality in the d i s t r i b u t i o n of annual income is misspecified. I w i l l discuss these problems and what has been written about them in turn, before drawing some conclusions about how inequality should properly be measured. Early empirical work indicated that t r a d i t i o n a l indices of income inequality are not independent of the length of the accounting period. Hanna [1948], for example, found that the incomes of a sample of Wisconsin taxpayers became more equally d i s t r i b u t e d when measured over a longer accounting period ( i . e . the Lorenz curve of incomes measured over two years lay everywhere inside the average of the two Lorenz curves of annual incomes). Soltow [1965] and Kohen, Parnes, and Shea [1975] also report an inverse relationship' between the Gini c o e f f i c i e n t of incomes and the length of the accounting period. This phenomenon is the result of changes in individuals' r e l a t i v e positions in 6 the income d i s t r i b u t i o n over time, or income mobility, and has recently been studied in depth by Shorrocks [l978a,b]. Those occupying the highest and lowest positions in the income hierarchy rarely remain there forever. So the aggregation of incomes over time tends to improve the r e l a t i v e position of those temporarily found at the bottom of the d i s t r i b u t i o n , and the situation of those at the top tends to deteriorate. For t h i s reason i t i s commonly supposed that inequality f a l l s as the accounting period i s lengthened. . . . (T)he l i t t l e evidence available agrees with expectations (Shorrocks [1978a, p. 377]). Measured annual income inequality thus includes a mobility-related component which should be distinguished from pure interpersonal inequality. The s o c i a l significance of the degree of pure interpersonal inequality is thus overstated by measured annual income inequality. The severity of the error inherent in measured annual income inequality depends, of course, on the degree of income mobility. If the income structure exhibits l i t t l e mobility, r e l a t i v e incomes w i l l be l e f t more or less unaltered over time and there w i l l be no pronounced e g a l i t a r i a n trend as the measurement period increases. In < contrast, inequality may be expected to decrease s i g n i f i c a n t l y in a very (income) mobile society (Shorrocks [1978a, p. 377]). The available evidence indicates that income mobility and mobility-related inequality are substantial. S c h i l l e r [1977] found that the United States i s characterized by a very high degree of r e l a t i v e earnings m o b i l i t y . 3 Shorrocks [1978b] quantified the effect of income mobility by charting the inverse re l a t i o n s h i p between measured inequality and the length of the 3However, the subjective nature of his analysis and absence of quantitative results impair the v a l i d i t y of his conclusion. 7 accounting period. Although the results were found to be sensitive to the choice of inequality index and age, declines in measured inequality of 5 to 52 per cent compared to annual income occurred in inequality of family incomes aggregated over nine years. My own estimates, reported in Chapter Four, indicate that mobility-related inequality accounts for 21 to 39 per cent of the t o t a l . " It would thus seem that the f a i l u r e to account for inequality attributable to income mobility represents a serious problem that has not yet been adequately solved. Shorrocks [1978a, b] has provided the best attempt to date to deal q u a n t i t a t i v e l y with the ef f e c t of income mobility on measured inequality. His suggestion i s to exploit the relat i o n s h i p between income inequality and mobility to construct an index of mobility that r e f l e c t s the extent to which incomes are equalized as the accounting period i s lengthened. More s p e c i f i c a l l y , he f i r s t proves that, for a large class of inequality indices which are convex functions of r e l a t i v e incomes and mean independent, inequality of incomes aggregated over a number of years cannot exceed a weighted average of annual income inequality, where the weights equal the proportions of aggregate income received in each year (Shorrocks [1978a], Theorem 1). The r a t i o of aggregate income inequality to average annual inequality i s therefore bounded above by unity, which represents a situation of complete income immobility or constant r e l a t i v e incomes over time. Shorrocks c a l l s t h i s an "This range r e f l e c t s only the choice of inequality index (or, more s p e c i f i c a l l y , the degree of inequality aversion) and would be wider s t i l l i f . the results were disaggregated by age. 8 index of income r i g i d i t y . He then defines an index of income mobility as the difference between unity and the value of the r i g i d i t y index. Shorrocks suggests that the r i g i d i t y index be computed over a two year period, then a three year period, and so on up to the maximum number of years for which data are a v a i l a b l e . 5 R i g i d i t y curves, showing the relationship between the value of the r i g i d i t y index and the number of years of data used to compute i t , can then be plotted. The r i g i d i t y curve of a completely income immobile society w i l l be a horizontal l i n e , since the value of a mean independent inequality index is invariant with respect to the length of the income accounting period when r e l a t i v e incomes are constant (Shorrocks [1978a], Theorem 2). Income mobility w i l l cause the value of the r i g i d i t y index to decline as the accounting period i s lengthened, and the shape of the associated r i g i d i t y curve thus r e f l e c t s the degree of mobility. The r i g i d i t y curve of a society in which there i s l i t t l e income mobility w i l l decline only s l i g h t l y and l i e close to the horizontal reference l i n e , while a more income mobile society w i l l be characterized by a more sharply declining r i g i d i t y curve. The shape of a r i g i d i t y curve reveals not only the degree of income mobility, but may also indicate something of the nature of the fluctuations in individual incomes over time. For example, suppose we were to compare two groups, 5Since individual incomes must be aggregated to calculate long period inequality in the numerator of the r i g i d i t y index, longitudinal data are required. 9 one of which had large variations in transitory income, whilst the other experienced substantial changes in permanent incomes (but small transitory changes). Year-to-year income variations might appear to be rather sim i l a r . Yet their " r i g i d i t y curves" may be expected to be r a d i c a l l y d i f f e r e n t . If income changes are purely due to transitory e f f e c t s , r e l a t i v e incomes w i l l rapidly approach th e i r permanent values and there w i l l then be no substantial further equalization. The r i g i d i t y curve w i l l therefore tend to become horizontal after the f i r s t few years. This contrasts with the group with more mobility in permanent incomes, whose r i g i d i t y curves w i l l continue to decline as the aggregation period i s extended. . . . (C)alculating values of R (the r i g i d i t y index) over d i f f e r e n t aggregation periods may thus be a l l that i s required to make the important d i s t i n c t i o n between these alternative types of income var i a t i o n s . (Shorrocks [1978a, p. 389]) Shorrocks [1978b] exploits this feature of r i g i d i t y curves in his empirical analysis of income s t a b i l i t y in the United States to conclude that transitory income fluctuations predominate among the younger members of society (the 20 to 29, and espec i a l l y the under 20, age groups) and among low income earning females into the middle age groups (Shorrocks [1978b, pp. 19-21]). The continual decline, over the nine year sample period, of the r i g i d i t y curves of middle aged men (aged 30-59) and a l l seniors indicates that income mobility in these groups is of a longer run nature. An important feature of r i g i d i t y curves i s thi s a b i l i t y to portray graphically some of the interesting c h a r a c t e r i s t i c s of income mobility. While income mobility is doubtless of i n t r i n s i c interest, i t s study i s motivated primarily by the recognition of i t s effe c t s on inequality: "estimates of the welfare loss due to inequality . . . tend to be biassed upwards i f they are computed from short-run ( i . e . annual) data" (Shorrocks [1978a, p. 388]). Thus, despite the elegance and appeal of Shorrocks' approach, i t 1 0 does not provide what i s most needed, a method of measuring inequality free of the e f f e c t s of income mobility. He does suggest that, "short run estimates of welfare losses due to inequality can be made consistent with the true long run value by reducing the short run estimate by the factor R (the value of the r i g i d i t y index)" (Shorrocks [1978a, p. 388, n. 14]). This i s rather ad hoc, however, and results in as many estimates of long-run inequality as there are years of data in the sample. In addition, while such indices may account for intertemporal income differences and their e f f e c t on measured inequality, they ignore the equally important intergenerational income differences which should also be excluded from measured in e q u a l i t y . 6 The annual incomes which an individual receives over the course of his l i f e vary with age, giving r i s e to the c h a r a c t e r i s t i c a l l y humped shape of l i f e c y c l e income p r o f i l e s . The systematic variation of income with age implies that the income differences observed in an annual income d i s t r i b u t i o n are partly the result of the age-structure of the population. This intergenerational aspect of inequality i s captured by indices of annual inequality, which must therefore be taken as overestimates of the degree of pure interpersonal inequality. In the extreme, i f l i f e c y c l e income p r o f i l e s were i d e n t i c a l across the population, measured annual inequality w i l l be e n t i r e l y age-6The intertemporal and intergenerational aspects of inequality are both accounted for in my suggested approaches to the measurement of inequality presented in the following two chapters. 11 related. In general the problem w i l l not be this severe, of course, but the fact remains that inequality measured in the d i s t r i b u t i o n of annual income must exclude the age-related component i f i t i s to be a r e l i a b l e estimate of pure interpersonal inequality. This intergenerational aspect of inequality has been recognized by many, including Paglin [1975] who argued that indices of annual income inequality, "combine and hence confuse intrafamily variation of income over the l i f e c y c l e with the more pertinent concept of interfamily income variation which underlies our idea of inequality . . . " (p. 598, emphasis in o r i g i n a l ) . Eschewing the use of estimated l i f e c y c l e income data or age-specific inequality indices, Paglin suggests that l i f e t i m e inequality may be approximated simply by redefining the standard of e q u a l i t y 7 as equality within age-cohorts rather than as equality across the population as a whole. 8 Paglin's method i s f i r s t to estimate the mean age-income p r o f i l e of the population from cross-sectional (annual) data. A Lorenz curve of this d i s t r i b u t i o n r e f l e c t s the inequality of annual incomes that would ex i s t , given the population age-structure, i f everyone traversed the same l i f e c y c l e income p r o f i l e . Paglin employs this "P-reference l i n e " as a new standard of equality to replace the t r a d i t i o n a l 45° l i n e of 7The d i s t r i b u t i o n with respect to which the s o c i a l significance of inequality in the actual d i s t r i b u t i o n is measured. 8An immediate problem i s that the method has been applied only to the Gini c o e f f i c i e n t . In Appendix B I have generalized Paglin's technique and compared the results to my own approach and to the use of age-specific indices of annual inequality. 1 2 e q u a l i t y ; i t e m b o d i e s , " e q u a l l i f e t i m e i n c o m e s , w i t h o u t t h e a d d e d c o n s t r a i n t o f a f l a t a g e - i n c o m e p r o f i l e " ( P a g l i n [ 1 9 7 5 , p p . 5 9 9 - 6 0 0 ] ) . T h e a c t u a l d i s t r i b u t i o n o f a n n u a l i n c o m e i s r e p r e s e n t e d by t h e u s u a l L o r e n z c u r v e . T h e s i t u a t i o n i s i l l u s t r a t e d i n F i g u r e I . 9 T h e t r a d i t i o n a l G i n i c o e f f i c i e n t i s e q u a l t o t w i c e t h e a r e a b e t w e e n t h e L o r e n z c u r v e a n d t h e d i a g o n a l , 1 0 a n d c a n be s e e n t o be c o m p r i s e d o f t h e sum o f two p a r t s . T h e s h a d e d a r e a b e t w e e n t h e P - l i n e a n d t h e d i a g o n a l r e p r e s e n t s a n n u a l i n c o m e i n e q u a l i t y a t t r i b u t a b l e t o t h e mean v a r i a t i o n o f i n c o m e w i t h a g e o v e r t h e l i f e c y c l e . T h e a g e - G i n i c o e f f i c i e n t i s e q u a l t o t w i c e t h i s a r e a . T h e u n s h a d e d a r e a b e t w e e n t h e P - l i n e a n d t h e L o r e n z c u r v e r e f l e c t s a n n u a l i n c o m e i n e q u a l i t y e x c l u d i n g a g e - r e l a t e d i n e q u a l i t y ; i t i s m e a s u r e d by t h e P a g l i n - G i n i w h i c h i s e q u a l t o t h e d i f f e r e n c e b e t w e e n t h e L o r e n z - G i n i a n d t h e a g e - G i n i . T h e L o r e n z - G i n i was f o u n d t o o v e r s t a t e l o n g - r u n i n t e r f a m i l y i n e q u a l i t y a s m e a s u r e d by t h e P a g l i n - G i n i by a s much a s 50 p e r c e n t i n s a m p l e d a t a . F u r t h e r m o r e , t h e t r e n d o f t h e P a g l i n - G i n i r e v e a l e d a 23 p e r c e n t d e c l i n e i n i n e q u a l i t y i n t h e p o s t - w a r p e r i o d , i n s h a r p c o n t r a s t t o , " t h e w i d e l y a c c e p t e d c o n c l u s i o n t h a t t h e r e h a s b e e n no s i g n i f i c a n t r e d u c t i o n o f i n e q u a l i t y f r o m 1947 t o 1972" ( P a g l i n [ 1 9 7 5 , p . 6 0 3 ] ) . I t w i l l s u r e l y be a g r e e d t h a t P a g l i n h a s a d d r e s s e d a n 9 A 1 1 F i g u r e s a n d T a b l e s a p p e a r a t t h e e n d o f t h e c h a p t e r . 1 ° T h e G i n i c o e f f i c i e n t i s d e f i n e d a s t h e r a t i o o f t h e a r e a b e t w e e n t h e L o r e n z c u r v e a n d t h e d i a g o n a l t o t h e t o t a l a r e a b e l o w t h e d i a g o n a l , t o w h i c h t h e d e f i n i t i o n i n t h e t e x t i s e q u i v a l e n t s i n c e t h e a r e a o f t h e s q u a r e i s u n i t y . 1 3 important and d i f f i c u l t problem. Given the lack of observed lifecycle•income data, the especial importance of Paglin's contribution l i e s in "reconstructing the reference l i n e of equality to match the excellent annual income data at our disposal" (p. 599). The Paglin-Gini has, nevertheless, been subject to considerable c r i t i c i s m on a number of counts. At least two authors have argued that Paglin's disaggregation of the Gini c o e f f i c i e n t i s incorrect. The usual Gini c o e f f i c i e n t measures inequality with respect to an optimal si t u a t i o n in which everyone receives the population-wide mean income. Wertz [1979], accepting Paglin's argument that the optimal income should be the age-group mean, proposed an adjusted Gini c o e f f i c i e n t which, unlike the Paglin-Gini, follows the logic of the Gini c o e f f i c i e n t in i t s construction. The adjusted Gini c o e f f i c i e n t suggested by Wertz measures non-age-related inequality with reference to the 45° l i n e of equality. The Paglin-Gini, on the other hand, compares the Lorenz curve of the actual d i s t r i b u t i o n to a redefined reference l i n e of equality, the P-line. Paglin concedes that neither method i s i n t r i n s i c a l l y superior but argues in favour of the Paglin-Gini on the grounds: (1) that the Lorenz curve corresponding to the adjusted Gini c o e f f i c i e n t can dip below the base l i n e of the Lorenz diagram into the negative income quadrant, and (2) that Wertz's adjusted Gini c o e f f i c i e n t i m p l i c i t l y assumes zero intracohort income mobility, and thus tends to overestimate l i f e t i m e inequality. The Paglin-Gini, though not e x p l i c i t l y accounting for the effects of income mobility, does a better job than the adjusted Gini c o e f f i c i e n t because the Paglin-Gini 1 4 varies inversely with the mean income difference between cohorts, which is p o s i t i v e l y correlated with income mobility (Paglin [ 1 9 7 9 , p. 6 7 6 ] ) . A second c r i t i c i s m of the Paglin-Gini along similar l i n e s was made by Nelson [ 1 9 7 7 ] , who argued that Paglin i m p l i c i t l y assumed that age-group income d i s t r i b u t i o n s do not overlap (as would be true, for example, i f families were grouped by income bracket). The difference between the Lorenz-Gini and the age-Gini calculated under this assumption, is an index of pure interpersonal inequality plus an interaction term. The degree of non-age-related inequality i s thus overestimated by the Paglin-Gini according to Nelson. Paglin supports his inclusion of the interaction effect in the intra-age-group component of inequality with an argument of Battacharya and Mahalonobis [ 1 9 6 7 , p. 1 5 0 ] : "(a)ssuming that the means of the groups are given, i t is reasonable to postulate that the between-groups component should not change simply because of the degree of within group v a r i a t i o n . " It follows that the between groups component in the general case i s the same as the between groups component in the special case where within group var i a t i o n is zero for every group. Battacharya and Mahalonobis conclude that while one cannot d i r e c t l y draw up a concentration curve of ov e r a l l within group inequality, as one can for the between group differences, the area between the l a t t e r curve and the L-curve 'indicates the eff e c t of within groups d i s p a r i t i e s . ' (Paglin [ 1 9 7 7 , pp. 520-21]) Paglin would seem to be secure on these quite defensible grounds. Nelson also argues, however, that the Paglin-Gini i s not a pure intracohort inequality measure because i t depends on cohort population and income shares as well as on inequality within 15 cohorts. Danziger, Haveman, and Smolensky [1977] also advanced thi s argument in their c r i t i q u e of the Paglin-Gini. They investigated the contributions of intracohort inequality, cohort population shares, and cohort income shares to the trend of pure interpersonal inequality, and found that, "while a l l three sources contributed to the increase in inequality from 1965 to 1972, two of the sources operated to decrease the Paglin-Gini. I r o n i c a l l y only the changes in cohort-specific Gini c o e f f i c i e n t s contributed to the increase in Paglin-inequality over this period" (Danziger, Haveman, and Smolensky [1977, p. 508]). The problem is that the Paglin-Gini is computed by subtracting the age-Gini, which i s not independent of cohort population and income shares, from the Lorenz-Gini; i t i s therefore sensitive to changes in these variables. More importantly, Paglin's major finding that the Paglin-Gini declines over time is seen to be the result of the trends of cohort population and income shares. Inequality within groups operated to increase Paglin inequality. This and related problems of the Paglin-Gini stem primarily from P a g l i n 1 s use of the actual cross-sectional l i f e c y c l e income p r o f i l e as the basis for correcting the Gini c o e f f i c i e n t for age-related income differences. "An inequality measure which allows for l i f e c y c l e variations i s appealing. However, such a standard requires an e x p l i c i t judgement on the optimum l i f e c y c l e pattern, and relying on annual observations of an a r b i t r a r i l y observed pattern is unsatisfactory" (Danziger, Haveman, and Smolensky [1977, p. 512]). The Paglin-Gini's lack of any normative underpinnings is i t s most serious drawback. I wish b r i e f l y to discuss other c r i t i c i s m s of the Paglin-Gini before 16 returning to th i s point. Several writers have argued that the Paglin-Gini estimates of inequality are too low. Johnson [1977] used a simple model of income d i s t r i b u t i o n to demonstrate t h i s r e s u l t . Nelson [1977] and Formby and Seaks [1980] have argued that the Paglin-Gini's underestimation of intracohort inequality results from the fact that i t i s not normalized to range over a [0,1] i n t e r v a l . Paglin has also been faulted by Danziger, Haveman, and Smolensky [1977] for his use of f u l l family money income in the computation of the Lorenz-, age-, and Paglin-Gini c o e f f i c i e n t s . (I)mplicit in Paglin's framework is a c r i t e r i o n for judging the effectiveness of income transfers, i f the objective i s to reduce inequality. An income transfer i s ' P a g l i n - e f f i c i e n t ' only i f i t reduces the variation of incomes within an age-cohort; transfers which involve intercohort r e d i s t r i b u t i o n are by d e f i n i t i o n ' P a g l i n - i n e f f i c i e n t . ' • . . . In th i s context, i t should be noted that, as calculated, the age- and Paglin-Gini c o e f f i c i e n t s incorporate transfers which are by d e f i n i t i o n P a g l i n - i n e f f i c i e n t , since Paglin's income concept i s census money income. . . . Consequently, Paglin's income p r o f i l e s are based on an inappropriate d e f i n i t i o n of income which biases his conclusions on the trend of functional inequality in the post war period. (pp. 510-11) In t h i s vein Minarik [1977] reports that the trend of earned income inequality i s considerably d i f f e r e n t from that of t o t a l family money income. He finds that the Lorenz-Gini r i s e s by 8 per cent and the Paglin-Gini by 2 per cent over the period 1967-1974. It has also been suggested that measured annual income inequality should be corrected for other factors in addition to the age-structure of the population. Minarik [1977], for example, found that, "while the Paglin-Gini, using the age-income p r o f i l e for a base, finds a 2 per cent, decrease in 17 inequality, the adjusted Paglin-Gini, based on separate age-income p r o f i l e s for groups with d i f f e r e n t schooling attainments finds a 2 per cent increase in inequality" (p. 515). The question here i s which factors should be included i n , and which excluded from, an inequality index. Paglin's purpose i s to p a r t i t i o n the area between the 45° l i n e and the Lorenz curve into two parts: that inequality which to him i s economically functional and, hence, of no concern for public policy, and the remaining . . . non-functional or policy-relevant inequality. Functional inequality in this instance r e f l e c t s society's needs for varying income over the l i f e c y c l e as well as other basic facts r e l a t i n g to productivity, investment in human resources, and the work-leisure preferences of households, but only in an average way, insofar as these factors express themselves through the age variable. (Danziger, Haveman, and Smolensky [1977, pp. 505-6]) Kurien [1977] c r i t i c i z e s the Paglin-Gini along similar l i n e s , and concludes that, "an ideal measure of income d i s t r i b u t i o n w i l l eliminate a l l choice-related variation (in incomes), but none of the d i f f e r e n t i a l opportunity-related v a r i a t i o n " (p. 518). The Paglin-Gini does not f a i l i r r e t r i e v a b l y as a result of any of the arguments just reviewed. The correct disaggregation could be derived, the resulting index could be normalized to range over a [0,1] i n t e r v a l , an appropriate income variable d e f i n i t i o n could be chosen, and decisions could be reached on the s o c i a l significance of various sources of inequality. A more serious problem, however, which was mentioned e a r l i e r but deferred momentarily, remains. It involves Paglin's approach to the problem of separating age-related income differences from, measured inequality. He has chosen to redefine the standard of equality to r e f l e c t , "equal l i f e t i m e incomes, but not the added 18 constraint of a f l a t age-income p r o f i l e " (Paglin [1975, p. 600]). This in i t s e l f is perfectly acceptable; i t could be j u s t i f i e d , and would probably be widely accepted, on v e r t i c a l equity grounds alone. Equality "across-the-board" may not, indeed, be the answer to how income ought to be d i s t r i b u t e d . But c l e a r l y the question is a normative one, and herein l i e s the problem with Paglin's formulation. The P-reference l i n e . . . i s a normatively empty box, devoid of any e t h i c a l content. (It) confuses the peaked age-income p r o f i l e thrown out by the market with the normative question of how income ought to be d i s t r i b u t e d . There i s no e t h i c a l content to the prescription that the young and elderly ought to have low incomes because, on average, they do have low incomes. A meaningless age-Gini subtracted from the Lorenz-Gini results in a meaningless Paglin-Gini. ( G i l l e s p i e [1979, p. 563]) An obvious solution to this problem i s to replace the P-reference l i n e with a normatively-based standard of equality which incorporates e t h i c a l consideration of how l i f e c y c l e income ought to be d i s t r i b u t e d . This may not be an easy task, however, and i t would seem simpler to retain the usual e t h i c a l standard of equality, but to disaggregate inequality by source. In the case of age-related inequality, t h i s would involve decomposing inequality within and among age-cohorts. Paglin in fact considered t h i s alternative but found the use of age-specific inequality indices unsatisfactory because the empirical c o e f f i c i e n t s available are not r e a l l y s p e c i f i c by age of family head but in fact represent broad age groups. This introduces spurious income variance by not f u l l y eliminating the effect of the age-income p r o f i l e . However, even i f we had t r u l y age-s p e c i f i c G i n i , we would have the problem of weighting and combining fifty-some measures into one c o e f f i c i e n t . (Paglin [1975, p. 602]) However, procedures for decomposing inequality within and among 19 population subgroups have been proposed by Blackorby, Donaldson, and Auersperg [1981] and Shorrocks [1980], among others, and there i s no reason to prevent age-groups being defined on an annual basis rather than by brackets including more than a single year. There is a genuine problem with t h i s suggestion, however, but i t i s not s p e c i f i c to the decomposition of inequality. It applies equally to Paglin's method, and i s in fact inherent in every measure of annual income inequality because they a l l f a i l to account for the e f f e c t s of income mobility. Mobility reduces the dispersion of l i f e t i m e incomes much below the annual income estimate. . . . While the P-Gini adjusts for average age-related inequality i t also f a i l s to catch the accompanying intracohort mobility. U n t i l we are able to modify our s t a t i c inequality c o e f f i c i e n t s by an index of mobility or c o l l e c t more longitudinal household income data for an extended period of time, our estimate of inequality of l i f e t i m e incomes (or the more d i f f i c u l t trend of the inequality of l i f e t i m e incomes) w i l l remain crude. (Paglin [1977, p. 527]) It would seem then that even Paglin agrees that the Paglin-Gini i s a stop-gap measure for use when panel data are not available. But Paglin finds fault with this practice too, on the grounds that economic growth renders l i f e t i m e income equality an unreasonable and unattainable goal. Paglin did not suggest a Gini c o e f f i c i e n t of l i f e t i m e income inequality based on the observed growth of real income over time. But since economic growth causes the l i f e t i m e incomes of currently young members of society to exceed the l i f e t i m e incomes of their elders, the appropriate solution would again seem to be the decomposition of l i f e t i m e income inequality within and among age-cohorts. To recapitulate, inequality attributable to intertemporal 20 and intergenerational income differences is confused with interpersonal inequality in indices computed from annual data. Paglin has proposed a method of excluding age-related income differences from the Gini c o e f f i c i e n t of annual incomes, but the Paglin-Gini has no e t h i c a l foundations as a measure of non-age-related inequality. A superior method to distinguish age-related inequality from pure interpersonal inequality is to decompose t o t a l inequality within and among age-cohorts. Indices of annual inequality, however, cannot account for the eff e c t s of income mobility. Shorrocks has suggested adjusting indices of annual inequality to approximate long-run inequality. It i s preferable, however, to compute long-run inequality d i r e c t l y from longitudinal data. To capture f u l l y the effects of income mobility and the shape of age-income p r o f i l e s requires that inequality be measured in the d i s t r i b u t i o n of l i f e t i m e income. It i s important to recognize that the intertemporal and intergenerational aspects of inequality do not disappear when a l i f e c y c l e perspective i s adopted; they each appear in a di f f e r e n t guise. The intergenerational problem, as was noted by Paglin, i s the result of economic growth which causes the li f e t i m e incomes of younger members of the current population to exceed those of elder members. There i s thus reason to decompose l i f e t i m e income inequality within and among age-cohorts., The intertemporal problem in measuring inequality of l i f e t i m e incomes i s to choose an appropriate summary s t a t i s t i c of l i f e c y c l e income for the purpose of measuring inequality. This i s intimately related to, and w i l l be discussed in the context of, the theoreti c a l d i f f i c u l t y with measuring inequality in the 21 d i s t r i b u t i o n of income, which I take up next. The d i s t r i b u t i o n of income monopolized the attention of economists interested in d i s t r i b u t i o n a l issues u n t i l very recently. As I suggested at the outset of t h i s chapter, this was l i k e l y the result of a superabundance of income data and a view of inequality as the degree to which the t o t a l product of the economy i s not equally shared among the population. Although not unreasonable grounds on which to j u s t i f y measuring inequality in the d i s t r i b u t i o n of income, i t s dominance in theory and practice seems curious in l i g h t of the welfare foundations of inequality measurement, o r i g i n a l l y established by Dalton [1920]. An American writer has expressed the view that "the s t a t i s t i c a l problem before the economist in determining upon a measure of the inequality in the d i s t r i b u t i o n of wealth i s i d e n t i c a l with that of the b i o l o g i s t in determining upon a measure of the • inequality in the d i s t r i b u t i o n of any physical c h a r a c t e r i s t i c . " But t h i s i s c l e a r l y wrong. For the economist i s primarily interested, not in the d i s t r i b u t i o n of income as such, but in the e f f e c t s of the d i s t r i b u t i o n of income upon the d i s t r i b u t i o n and t o t a l amount of economic welfare, which may be derived from income, (p. 348) In t h i s view, inequality i s interpreted as the degree to which individuals in society are not equally well-off. The measurement of inequality thus involves a s o c i a l evaluation of the d i s t r i b u t i o n of individual welfare or u t i l i t y . Dalton suggested that inequality be defined as the r a t i o of t o t a l welfare attainable under an equal d i s t r i b u t i o n to t o t a l welfare attained under the actual d i s t r i b u t i o n . Recognizing the d i f f i c u l t i e s of measuring welfare, however, Dalton argued that, "inequality, . . . though i t may be defined in terms of economic welfare, must be measured in terms of income" (Dalton [1920, p. 349], emphasis in o r i g i n a l ) . But welfare i s derived from income only 22 through consumption, so that there would seem to be something missing from Dalton's analysis of the problem. 1 1 Inequality, though defined in terms of welfare, must be measured in terms of consumpt ion. The significance for the measurement of inequality of the link from income through consumption to welfare was not f u l l y appreciated u n t i l quite recently. Bentzel [1970] was the f i r s t to argue that " i t is . . . th i s income-consumption-welfare nexus which i s the reason for the great interest in the income d i s t r i b u t i o n " (p. 254). That the observed inequality of incomes is not so much of i n t r i n s i c interest as i t i s an estimate of inequality in the d i s t r i b u t i o n of well-being raises the question of how accurately the former can be expected to approximate the l a t t e r . For i f i t is . . . the d i s t r i b u t i o n of welfare that is the relevant concept in p o l i t i c a l discussion, the economists' empirical analyses of income d i s t r i b u t i o n s w i l l be of interest only on the assumption that there is a f a i r l y close-connection between th i s d i s t r i b u t i o n and the corresponding welfare d i s t r i b u t i o n . (Bentzel [1970, p. 254]) With th i s in mind, Bentzel examined the relationships between the d i s t r i b u t i o n s of income, consumption, and welfare with regard to the measurement of inequality. He i d e n t i f i e d three reasons for d i s s i m i l a r i t i e s between the d i s t r i b u t i o n s of income 1 1As he must have recognized and, indeed, hinted at: "We have to deal, therefore, not merely with one variable, but with two, or  possibly more, between which certain functional relations may be presumed to e x i s t " (p. 348, emphasis added). Dalton's injunction to measure inequality in terms of income i s correct only i f the functional r e l a t i o n between welfare and income incorporates the relationships of welfare to consumption and consumption to income. 23 and consumption: saving and dissaving, consumption expenditure not out of own income, and the fact that the purchasing power of incomes varies with the price l e v e l . However s i g n i f i c a n t such eff e c t s might be — the consumption d i s t r i b u t i o n generally displays considerably less inequality than the d i s t r i b u t i o n of income — they pale in l i g h t of the d i f f i c u l t y of translating changes in the consumption d i s t r i b u t i o n into their e f f e c t s on the d i s t r i b u t i o n of welfare. Recently observed demographic phenomena such as the "graying of society" and the increasing number of working women cause income, and to a lesser extent consumption, inequality to r i s e , but i t i s considerably more d i f f i c u l t to say what are their e f f e c t s on the d i s t r i b u t i o n of welfare. Perhaps the most d i f f i c u l t problem of a l l i s accounting for, and determining the welfare e f f e c t s of, public consumption. Based on his analysis of the d i s t r i b u t i o n s of income, consumption, and welfare, Bentzel i s forced to a pessimistic conclusion regarding the prospects for learning much about the d i s t r i b u t i o n of well-being from an examination of the income d i s t r i b u t i o n . The situation could be improved s i g n i f i c a n t l y by measuring inequality in the d i s t r i b u t i o n of consumption. Interestingly, the importance of s h i f t i n g attention from income to consumption for the purpose of measuring inequality is t i e d in with the need to extend the temporal dimension of the analysis in order to estimate inequality more accurately. Nowhere has th i s point been made more c l e a r l y than in the theory of consumer behaviour. Both Friedman's [1957] theory of permanent income and the l i f e c y c l e hypothesis of Modigliani and . Brumberg [1954] view individual welfare as a function of 24 l i f e c y c l e consumption which depends in turn on l i f e t i m e income. CT)here need not be any close and simple r e l a t i o n between consumption in a given short period and income in the same period. The rate of consumption in any given period is a facet of a plan which extends over . . . the individual's l i f e , while the income accruing within the same period i s but one element which contributes to the shaping of such a plan. (Modigliani and Brumberg [1954, p. 391]) The implications for the measurement of inequality have been emphasized by Friedman: "the existence of large negative savings is a symptom that the observed inequality of measured income overstates substantially the inequality of permanent income" ([1954, p. 40]). Recent studies of inequality have thus focussed on the d i s t r i b u t i o n of l i f e c y c l e consumption rather than annual income (e.g. Nordhaus [1973], Blinder [1975], and Irvine [1980]). The intertemporal and intergenerational aspects of annual income inequality, which were e a r l i e r discussed at length, reappear in di f f e r e n t forms in the measurement of l i f e c y c l e consumption inequality. For example, the intertemporal problem i s to decide upon a summary s t a t i s t i c of l i f e c y c l e consumption suitable for the purpose of measuring inequality. Several have been suggested in the context of measuring l i f e t i m e income inequality, analogues of which might be considered as possible candidates. Summers [1956] estimated individual l i f e t i m e earnings and found average l i f e t i m e income to be more equally d i s t r i b u t e d than annual income. Weisbrod and Hansen [1968] suggested an income- net worth measure of economic welfare, equal to current income plus the l i f e t i m e annuity equivalent of current net worth. L i l l a r d [1977] measured inequality in the d i s t r i b u t i o n of human  wealth defined as the discounted present value of l i f e t i m e 25 earnings. A l l of these overlook Dalton's injunction, however, that i t is the welfare e f f e c t s of income which are of interest in the measurement of inequality. The discounted present value of l i f e t i m e income, or i t s annuity equivalent, r e f l e c t the magnitude and timing of the income an individual receives over the course of his l i f e , but not i t s significance in terms of economic w e l f a r e . 1 2 Measuring inequality of l i f e c y c l e consumption thus requires a welfare equivalent summary s t a t i s t i c of the l i f e c y c l e p r o f i l e , such as u t i l i t y equivalent annuity income, suggested and employed by Nordhaus [1973] or l i f e t i m e wealth, proposed by Pissarides [1978]. These are, respectively, the l i f e t i m e annuity and the corresponding discounted present value that provide the same u t i l i t y as the individual's chosen consumption plan. These ideas have been the subject of a recent paper by Cowell [1979], who was the f i r s t to recognize the importance of c a p i t a l market conditions. His welfare equivalent summary s t a t i s t i c s of l i f e c y c l e consumption, "wergild" and the associated "wergild annuity", are defined in terms of actual c a p i t a l market c o n d i t i o n s . 1 3 I follow t h i s practice in the welfare approach to the measurement of inequality presented in the next chapter. The intergenerational aspect of l i f e c y c l e inequality arises 12Two income p r o f i l e s with equal discounted present values, but d i f f e r e n t l y d i s t r i b u t e d over time, will,not y i e l d equal u t i l i t y to an individual without access to perfect c a p i t a l markets. 1 3Recent applied work on the measurement of l i f e c y c l e inequality (e.g. Nordhaus [1973] and Irvine [1980]) has focussed exclusively on consumption plans chosen by consumers facing perfect c a p i t a l markets. 26 because real economic growth causes consumption p r o f i l e s to s h i f t up over time. This w i l l be r e f l e c t e d in the values of the wergild annuities, which w i l l tend to be greater for l a t e r born ind i v i d u a l s . This intergenerational inequality should be distinguished from pure interpersonal inequality, and in the welfare approach this i s accomplished by decomposing l i f e c y c l e inequality within and among age-cohorts. 27 FIGURE I Cumulative Income D i s t r i b u t i o n Showing Age-Related Inequality 1 Income share 0 Population share 28 CHAPTER TWO A Welfare Approach to the Measurement of Inequality The modern theory of inequality measurement, attributable primarily to Atkinson [1970], Kolm [1969], and Sen [1973], attempts to provide a sound basis for evaluating the s o c i a l significance of inequality. Their work represents the most s i g n i f i c a n t theoretical contribution since Dalton's [1920] pioneering a r t i c l e and has kindled a burst of theore t i c a l and empirical work on inequality in the past decade. Nevertheless, their framework could be improved on a number of counts so as to strengthen i t s welfare, underpinnings. The major part of the work on inequality focusses on the annual income d i s t r i b u t i o n . A number of writers have argued that t h i s practice is e s s e n t i a l l y misguided, however, on the grounds that an individual's economic welfare i s r e f l e c t e d in his consumption rather than his income. Furthermore, an accurate assessment of economic position depends on consumption levels throughout l i f e ; r e s t r i c t i n g attention to a single year tends to produce a misleading indication of well-being. While some work on the theory and measurement of inequality has proceeded along these l i n e s of l a t e , i t has been plagued by an apparent confusion. Inequality, whether in the d i s t r i b u t i o n of annual income or l i f e c y c l e consumption, is measured in the actual d i s t r i b u t i o n with respect to an equally d i s t r i b u t e d a l t e r n a t i v e . Attempts to date to measure l i f e c y c l e consumption inequality have uniformly assumed, however, that a unique, 29 constant rate of interest p r e v a i l s in the market for saving and borrowing. Inequality is measured in t h i s d i s t r i b u t i o n of consumption plans chosen under optimal c a p i t a l market conditions with respect to an equal d i s t r i b u t i o n with the same mean. This situation c l e a r l y i s not representative of the actual c a p i t a l market conditions under which consumption plans are chosen. Individuals face a variety of means of real l o c a t i n g their income, with associated rates of interest, and d i f f e r e n t i a l rates for borrowing and lending. It is these actual c a p i t a l market conditions which underlie observed consumption over time and which should be i m p l i c i t in the measurement of l i f e c y c l e consumption inequality. The adoption of a l i f e c y c l e perspective on the measurement of inequality introduces a new factor contributing to measured inequality that i s absent when attention is limited to annual d i s t r i b u t i o n s . Real economic growth over time causes the l i f e c y c l e consumption opportunities of a young person to exceed those of someone older. Measured inequality in the d i s t r i b u t i o n of l i f e c y c l e consumption must therefore be decomposed within and among age-cohorts so as to di s t i n g u i s h pure interpersonal inequality from that due to growth. Two d i f f e r e n t decomposition procedures are available for t h i s purpose. One i s i n f e r i o r on both th e o r e t i c a l and empirical grounds, as i s argued in appendix A. The other is adopted for the decomposition of per capita inequality and Atkinson-Kolm-Sen (AKS) equality indices, while a new decomposition i s proposed and adopted for the decomposition 30 of AKS indices of i n e q u a l i t y . 1 " The approach I am proposing takes account of consumer choice exercised over a l i f e c y c l e planning horizon. It i s thus possible to measure the welfare loss attributable not only to the lack of equality among persons but also to the lack of perfect means of intertemporal r e d i s t r i b u t i o n . That i s , the welfare approach can be extended to measure the welfare loss implied by both interpersonal and intertemporal maldistribution. This i s done, and a decomposition of the t o t a l i s provided so that the two components and their interactive effect can be separately i d e n t i f i e d . Dalton was the f i r s t to point out .that the measurement of inequality i s a question of s o c i a l welfare. He, and la t e r Atkinson and-Kolm, suggested indices that measure inequality as the s o c i a l welfare loss implied by departures from e q u a l i t y . 1 5 In a l l of their work, however, u t i l i t y i s made a function of income. 1 6 Yet economic welfare i s generally taken to be a product of consumption. The p r i n c i p a l writers on the measurement 1'AKS indices measure the percentage of t o t a l consumption saved by moving from the actual d i s t r i b u t i o n to an equal d i s t r i b u t i o n that i s s o c i a l l y equivalent ( i . e . provides the same l e v e l of so c i a l welfare). Per capita indices measure the t o t a l saving from the same move on a per capita basis. 1 5Dalton's index i s not invariant with respect to linear transformations of individual u t i l i t y functions. The contribution of Atkinson and Kolm was to make measured inequality independent of monotonic transformations of individual u t i l i t y , functions through the use of the "equally d i s t r i b u t e d equivalent" in the construction of inequality indices. 1 6Dalton alone suggested that other variables might have to be taken into account. 31 of inequality thus seem to have stopped short of the goal of providing a welfare foundation for the theory of inequality measurement. The problems outlined b r i e f l y above are evidence of t h i s . In the new approach to the measurement of inequality presented in t h i s chapter a more accurate index of individual welfare i s substituted for annual income. Total inequality is decomposed within and among age-cohorts to obtain an index of pure interpersonal inequality. This chapter begins with a discussion of the desiderata of the modern theory of inequality measurement founded by Atkinson [1970], Kolm [1969], and Sen [1973]. I then present, in several steps, a thoroughly consistent welfare approach to the measurement of inequality. Beginning with individual u t i l i t y functions and l i f e c y c l e consumption p r o f i l e s , I define representative l i f e c y c l e consumption as the consumption annuity that provides the individual with the same l e v e l of u t i l i t y as the consumption plan that i t represents. The s o c i a l evaluation function i s defined over these representative l i f e c y c l e consumptions and is used to derive the (population-wide) equally d i s t r i b u t e d equivalent consumption. 1 7 The decomposition of inequality within and among population age-groups requires that an equally d i s t r i b u t e d equivalent consumption be defined for each age-cohort. This in turn requires some se p a r a b i l i t y in the s o c i a l evaluation function which r e s t r i c t s the class of 1 7The equally d i s t r i b u t e d equivalent of a given d i s t r i b u t i o n is defined by Atkinson [1970, p. 250] as, "the l e v e l of income^ (consumption) per head which i f equally d i s t r i b u t e d would give the same l e v e l of soc i a l welfare as the present d i s t r i b u t i o n . " 32 admissible s o c i a l evaluation functions and inequality indices to certain a d d i t i v e l y separable functions. Both AKS and per capita inequality indices bear interpretation as the s o c i a l saving which could be realized by moving from one d i s t r i b u t i o n (of representative l i f e c y c l e consumption) to a less unequal one which i s s o c i a l l y equivalent. The decomposition of inequality implies that inequality is eliminated in two stages: f i r s t within and then between cohorts. The i n d i v i d u a l , age-group, and population-wide equally d i s t r i b u t e d equivalents are used to define these hypothetical d i s t r i b u t i o n s from which AKS and per capita indices of inequality are computed. Inequality within age-groups i s taken as an index of pure interpersonal inequality, while the inter-age-cohort component of the decomposition represents inequality a t t r i b u t a b l e to economic growth. The decomposition procedure for indices of per capita inequality and AKS equality i s due to Blackorby, Donaldson, and Auersperg [1981]. Their decomposition of AKS inequality, however, suffers from several problems, and I suggest and employ a new procedure for decomposing AKS inequality indices that i s free of these problems. Having l a i d out the welfare approach to the measurement of interpersonal inequality, I then consider i t s extension to include measurement of the s o c i a l significance of intertemporal maldistribution. This involves the evaluation of actual l i f e c y c l e consumption p r o f i l e s with respect to a hypothetical s i t u a t i o n in which consumption plans are arranged through perfect c a p i t a l markets with a unique rate of i n t e r e s t . Inequality indices in the extended welfare approach can be decomposed into indices of interpersonal and "intertemporal" 33 inequality, plus a t h i r d term which r e f l e c t s the interdependency between them. F i n a l l y , the chapter closes with a discussion of the strengths and weaknesses of the welfare approach to the measurement of inequality. I begin with l i f e c y c l e consumption data on H individuals, and posit the existence of individual intertemporal u t i l i t y functions U :R -^>R1 with image ( 2 . 1 ) u w=U h(c h) (l<h<H) where c h = ( c h l , . . . ,c^ T^) i s the consumption plan of person h over the years of his l i f e . Each u t i l i t y function i s assumed to be continuous, increasing, and quasi-concave. An individual's observed consumption plan i s chosen to maximize ( 2 . 1 ) subject to an intertemporal budget constraint which r e f l e c t s his actual l i f e t i m e opportunities for saving and borrowing. My objective i s to evaluate the s o c i a l s i g n i f i c a n c e of differences in l i f e c y c l e consumption plans among individuals. This requires both a summary s t a t i s t i c of l i f e c y c l e consumption and a s o c i a l evaluation function defined in terms of that summary s t a t i s t i c . While individual u t i l i t y may seem the obvious candidate for t h i s purpose, i t i s in fact unacceptable since individual u t i l i t y functions are known only up to a monotonically increasing transformation. If the images of individual u t i l i t y functions were used as the arguments of an inequality index, measured inequality would depend upon the p a r t i c u l a r transformation which i s chosen. It would thus be possible to change the degree of inequality simply by applying a monotonically increasing transformation to i n d i v i d u a l u t i l i t y 34 f u n c t i o n s . 1 8 T h i s p r o b l e m i s s o l v e d b y t h e u s e o f C o w e l l ' s [ 1 9 7 9 ] e q u a l l y d i s t r i b u t e d e q u i v a l e n t summary s t a t i s t i c o f l i f e c y c l e c o n s u m p t i o n , w h i c h I c a l l r e p r e s e n t a t i v e l i f e c y c l e c o n s u m p t i o n . I t i s d e f i n e d a s t h e l i f e c y c l e c o n s u m p t i o n a n n u i t y w h i c h p r o v i d e s t h e same l e v e l o f u t i l i t y t o t h e i n d i v i d u a l a s t h e c o n s u m p t i o n p r o f i l e w h i c h i t r e p r e s e n t s . I t i s i n v a r i a n t w i t h r e s p e c t t o t r a n s f o r m a t i o n s o f t h e u t i l i t y f u n c t i o n , a n d i s i m p l i c i t l y d e f i n e d b y , ( 2 . 2 ) U h ( r h l T ^ ) = U W ( c K ) (1<h<H) w h e r e J_^ . i s a u n i t v e c t o r o f d i m e n s i o n T ^ . T h e p r o p e r t i e s o f uN..) e n s u r e t h a t r e p r e s e n t a t i v e c o n s u m p t i o n , r^ , i s u n i q u e a n d w e l l - d e f i n e d f o r e v e r y p o s s i b l e c o n s u m p t i o n p l a n , c ^ . ( 2 . 2 ) c a n t h e r e f o r e be w r i t t e n a s , ( 2 . 3 ) r h = R h ( c h ) 1<h<H) N o t e t h a t i s an e x a c t i n d e x o f p e r s o n h ' s w e l l - b e i n g ; t h a t i s , ( 2 . 4 ) r ^ r ^ • uNc^uNc,;) 0 < h < H ) T h e s o c i a l e v a l u a t i o n f u n c t i o n W:R^—•'R 1 ( w h e r e 'R* i s t h e n o n - n e g a t i v e E u c l i d e a n H - o r t h a n t ) h a s t h e i m a g e , ( 2 . 5 ) w=W(r) w h e r e r = ( r 1 ? . . . , ) i s a v e c t o r o f i n d i v i d u a l s ' r e p r e s e n t a t i v e l i f e c y c l e c o n s u m p t i o n s . W ( . ) i s a s s u m e d t o be 1 8 T h i s was t h e c r i t i c i s m o f D a l t o n ' s [ 1 9 2 0 ] m e a s u r e o f i n e q u a l i t y w h i c h l e d A t k i n s o n [ 1 9 7 0 ] t o p r o p o s e t h e u s e o f e q u a l l y d i s t r i b u t e d e q u i v a l e n t i n c o m e i n t h e c o n s t r u c t i o n o f i n e q u a l i t y i n d i c e s . AKS i n d i c e s a r e s e n s i t i v e t o t h e l e v e l f r o m w h i c h u t i l i t y i s m e a s u r e d ( i . e . a r e s c a l e i n d e p e n d e n t ) , w h i l e p e r c a p i t a i n d i c e s w i l l v a r y w i t h t h e u n i t s i n w h i c h u t i l i t y i s m e a s u r e d ( i . e . a r e o r i g i n i n d e p e n d e n t ) . 35 continuous, increasing, and S(chur)-concave. 1 9 The s o c i a l evaluation function (2.5) provides the e t h i c a l basis for the construction of AKS and per capita indices of inequality. AKS  indices measure the percentage of t o t a l consumption saved by  moving from the actual d i s t r i b u t i o n to an equal d i s t r i b u t i o n  that i s s o c i a l l y equivalent. Per capita indices measure the  t o t a l saving from the same move on a per capita basis. AKS and per capita inequality indices take the following forms respectively: (2.6) I=1-s/m (2.7) A=m-s These indices are constructed using m= 2L(1/H)r^, the mean of the vector r = ( r l f . . . ,r^) of individual representative l i f e c y c l e consumptions, and s, the equally d i s t r i b u t e d equivalent of r, defined i m p l i c i t l y by, (2.8) W(sl H)=W(r) An individual's representative l i f e c y c l e consumption, r^, r e f l e c t s both the position and shape of his l i f e c y c l e consumption p r o f i l e ( i . e . both the magnitude and d i s t r i b u t i o n of consumption during the course of his l i f e t i m e ) . Thus, even i f a l l consumption p r o f i l e s had exactly the same shape, the continual s h i f t i n g upward of their positions, because of real growth of the economy over time, would cause r^ to be larger for the younger members of the population. 19W(.) i s S-concave i f and only i f W(Br)>W(r) for a l l r in the domain of W(.) and for a l l bistochastic matrices B. W(.), i s s t r i c t l y S-concave i f and only i f W(Br)>W(r) whenever Br i s not a permutation of r. A bistochastic matrix i s a square matrix of nonnegative elements whose rows and columns each sum to unity. 36 While measured inequality captures both intergenerational and intragenerational aspects of differences in individual welfare, i t w i l l be desirable to distinguish the former economic-growth-related inequality from the l a t t e r pure interpersonal inequality. This requires that individuals be grouped by age-cohort. That i s , the population set N={1, . . . ,H} must be partitioned into subgroups by age, ft={N1, . . . ,NK} where_Nk is the subset of the population in the kth age-cohort. The s o c i a l evaluation function must be separable in the p a r t i t i o n ft, in which case i t can be written in the form, (2.9) w=^(W1(r1) W K(r K)) where W(.) i s increasing in W k(r^) and r k=(<r h> VhtN k) is the vector of representative consumption s t a t i s t i c s of a l l persons in the kth age-cohort. The conjunction of sepa r a b i l i t y and symmetry in W(.) imposes considerable structure on the s o c i a l evaluation function. As Blackorby, Donaldson, and Auersperg [1981, theorem 1] have shown, these conditions imply that W(.) i s addi t i v e l y separable; that i s , (2.10) w=W(Z.g(r h)') where W(.) i s increasing in i t s argument and g(.) i s i d e n t i c a l for a l l h because of the symmetry assumption. Furthermore, S-concavity of W(.) requires that g(.) be concave; s t r i c t S-concavity requires that i t be s t r i c t l y concave (Berge [1963]). Thus W(.) must be quasi-concave and symmetric. In t h i s case i t can e a s i l y be shown that the equally d i s t r i b u t e d equivalent representative consumption takes the form, 37 (2.11) s=S(r)=g- 1[d/H)21g(n)] h n The cohort s o c i a l evaluation function W^:RWk->R1 (where is the number of people in cohort k) has the image (2.12) w k=W W(r k) (l<k<K) and can be picked to have the properties of W(.). 2 0 (2.12) can be used to define the equally d i s t r i b u t e d equivalent of rk, (2.13) W H(s wl„ )=W k(r k) 0<k<K) The properties of W (.) ensure that representative cohort consumption, s^, i s unique and well defined for every vector r^; thus (2.14) s K = S U ( r l < ) = g - 1 [ ( l / n k ) h S k g ( r h ) ] ) d<k<K) The s o c i a l evaluation function defined over individual representative l i f e c y c l e consumptions must therefore be  continuous, increasing, symmetric, quasi-concave, and add i t i v e l y separable. Only then can i t provide the welfare basis for the construction of AKS and per capita indices of inequality which are decomposable within and among age-groups of the population. It i s generally desirable to go a step further, however, in order to derive r e l a t i v e indices (which are homogeneous of degree zero in their arguments) and absolute indices (which are invariant with respect to equal absolute changes in the values of their arguments). AKS indices are r e l a t i v e indices i f and only i f the so c i a l evaluation function i s homothetic. 2 1 Thus, r e l a t i v e inequality 2 0Blackorby, Primont, and Russell [1978]. 21W(.) is homothetic i f and only i f i t i s a monotonically increasing transform of a l i n e a r l y homogeneous s o c i a l evaluation function. 38 indices are based on so c i a l evaluation functions which are continuous, increasing, symmetric, quasi-concave, add i t i v e l y separable, and homothetic. The class of so c i a l evaluation functions which s a t i s f y these properties for positive representative consumptions 2 2 are the means of order R, (2.15) W R(r)=W[W R(r)] R<1 where W(.) is increasing and (2.16) WR(r)=< ( [ O / H j S l r * 0*R<1 TTr. 1'H R=0 K W R i s a free parameter determining the degree of r e l a t i v e inequality a v e r s i o n . 2 3 The corresponding r e l a t i v e inequality indices are members of the Atkinson family of indices, 1 - [ ( l / H ) X ( r , /m)*] 1/* 0#R<1 (2.17) I R ( r ) = 1- T T ( r , /m) w h 1 / H R=0 Per capita indices are absolute indices i f and only i f the  so c i a l evaluation function i s t r a n s l a t a b l e . 2 " Absolute indices are thus based on s o c i a l evaluation functions which are continuous, increasing, symmetric, quasi-concave, a d d i t i v e l y 2 2 I f the domain of W(.) i s the nonnegative orthant R" then we must have 0<R<1 in (2.16) and ( 2 . 1 7 ) . S-concavity, additive s e p a r a b i l i t y , and homotheticity are not possible over RRI except in the degenerate case R=1 (Blackorby and Donaldson [ 1 9 8 2 ] , theorem 4 ) . 2 3The degree of r e l a t i v e inequality aversion varies inversely with the _value of R. As R —>-<=©w^(r) and s=S(r) both go maximin; that i s , W R(r)=min^(r^}=S(r). Thus I R(r)=1-min^{r^}/m. 2 f lW(r) i s translatable i f and only i f W(rh=Vl[w(r) ] where W(.) is increasing in i t s argument and Vf(r+aJ_H) =W(r) +a, for a l l r, r+aj_ in the domain of W(.). 39 separable, and translatable, which r e s t r i c t s W(.) to the Kolm-Pollak (KP) family of s o c i a l evaluation functions, ( 2 . 1 8 ) W e(r)=-(1/G)ln{(l/H)Z.exp[(-G)r h]} G>0 and t h e i r corresponding absolute inequality indices, (2.19) A 6=(l/G)ln[(1/H)2Lexp{G(m-r h)}] G>0 The equally d i s t r i b u t e d equivalent consumptions for an i n d i v i d u a l , an age-cohort, and the o v e r a l l population are defined by (2.3), (2.14), and (2.11), respectively. By constructing reference vectors with these representative consumption s t a t i s t i c s as elements, e t h i c a l indices of inequality can be derived by computing the s o c i a l saving which could be r e a l i z e d by moving from one vector to another. AKS indices express t h i s saving as a percentage of the t o t a l and per capita indices express i t in per capita terms. I w i l l compute the per capita inequality indices and the AKS indices of inequality and equality. The corresponding absolute and r e l a t i v e indices can be found by duplicating t h i s procedure using the equally d i s t r i b u t e d equivalents corresponding to the s o c i a l evaluation functions (2.16) and (2.18) respectively. In order to measure intracohort inequality consider the replacement of the actual d i s t r i b u t i o n of representative consumpt ion, (2.20) (r, , . . . , r H ) by a s o c i a l l y equivalent one in which inequality i s eliminated within, but not between age-cohorts. In t h i s s i t u a t i o n , each ind i v i d u a l receives the equally d i s t r i b u t e d equivalent of the d i s t r i b u t i o n of representative l i f e c y c l e consumption within his cohort: 40 ( 2 . 2 1 ) (s. r , . . . , s k r ) The s o c i a l saving generated by the move from (2.20) to (2.21) r e f l e c t s intracohort inequality. If I now replace (2.21) by a s o c i a l l y equivalent equal d i s t r i b u t i o n , (2.22) (sJ[ H) inequality between cohorts w i l l have been eliminated and the soci a l saving accruing from the move from (2.21) to (2.22) can be used as a measure of intercohort inequality. Notice that the saving which could be realized by moving d i r e c t l y from (2.20) to (2.22) measures t o t a l inequality, indicating that i t w i l l be possible to aggregate the indices of i n t r a - and intercohort inequality into an index of t o t a l inequality. On a per capita basis, the savings generated by the move from (2.20) to (2.21) measure intracohort per capita inequality: \ = ( X r w - 2 n k s k ) / H (2.23) =21(nk/H) (m k-s k) where mk =(1/n^J^El^r^. 2 5 The mean s o c i a l saving which results from the move between (2.21) and (2.22) i s , (2.26) AR=X(nk/H)sk-s which measures intercohort inequality in per capita terms. It can e a s i l y be shown that (2.23) and (2.26) sum to (2.27) A=51(nk/H)mk-s =m-s 2 5 P e r capita inequality in cohort k i s defined, using (2.7), as, (2.24) Ak =mk-s so that intracohort inequality can be seen to be equal to a weighted average of inequality within cohorts, with the weights being cohort population shares. That i s , (2.25) A A=2L(n k/H)A k 41 which is the index of t o t a l inequality measured as the per capita s o c i a l saving to be realized by moving d i r e c t l y from (2.20) to (2.22). Before presenting the derivation of i n t r a - and intercohort AKS inequality indices, I wish to propose a new decomposition to replace the one suggested by Blackorby, Donaldson, and Auersperg [1981]. Their decomposition of AKS inequality indices, derived from their decomposition of AKS indices of e q u a l i t y , 2 6 has two serious drawbacks. The procedure gives d i f f e r e n t results depending on whether inequality within cohorts or inequality between cohorts is eliminated f i r s t . In either case, the decomposition lacks the simple additive aggregation of per capita inequality indices or m u l t i p l i c a t i v e aggregation of AKS indices of e q u a l i t y . 2 7 My decomposition of AKS inequality indices is derived from the decomposition of per capita inequality using the property that an AKS index is equal to the corresponding per capita index normalized on the mean (representative consumption). Thus, very simply, from the decomposition of per capita inequality, (2.28) A=AA+AR I obtain, by dividing through by m, (2.29) I = I A + I R where each index in (2.29) i s equal to the corresponding index in (2.28) divided by m. The decomposition (2.29) has, of course, 2 6 U s i n g the property that AKS indices of equality and inequality sum to unity. 2 7See (2.37) below. 42 the same simple additive structure of (2.28) and y i e l d s a unique decomposition of inequality within and among age-cohorts regardless of whether i n t r a - or intercohort inequality i s eliminated f i r s t . While t o t a l r e l a t i v e inequality in (2.29) and t o t a l r e l a t i v e equality in (2.37) below sum to unity, however, the subindices of r e l a t i v e inequality in my decomposition do not retain t h i s p roperty. 2 8 Dividing through (2.23) and (2.26) by m y i e l d s , 2 9 (2.30) I A = (m -X(n w >/H)s k)/m and, Intra- and intercohort AKS inequality, (2.30) and (2.31), can e a s i l y be seen to sum to, (2.33) I=(m-s)/m F i n a l l y , the construction of AKS indices of equality proceeds as follows. When a move i s made from one s i t u a t i o n to another in which consumption i s less unequally d i s t r i b u t e d , the corresponding AKS equality index i s computed as the r a t i o of t o t a l representative consumption in the l a t t e r s i t u a t i o n to t o t a l representative consumption in the former. Thus the index 2 8Blackorby, Donaldson, and Auersperg [1981, pp. 673-4] have shown that no aggregation of subindices of r e l a t i v e equality measured as percentage savings of the o r i g i n a l d i s t r i b u t i o n e x i s t s . That i s , there i s no decomposition of r e l a t i v e equality corresponding to (2.29). 2 9Again, notice that (2.30), the AKS index of intracohort inequality, i s equal to a weighted average of AKS inequality within cohorts; that i s , with the weights being the cohort shares of t o t a l representative consumption. (2.31) I R = (2(n k/H)s k-s)/m (2.32) 43 of intracohort AKS equality i s , 3 0 (2.34) E A = X n k s k / r r w = 2(nkmk/Hm) (s k/m k) Intercohort AKS equality i s given by, (2.36) E R = s / Z ( n k / H ) s k In this case, the product of the i n t r a - and intercohort terms, (2.34) and (2.36), yields the index of t o t a l AKS equality: (2.37) E=s/m The welfare approach to the measurement of inequality presented above improves on the usual practice of measuring inequality in the d i s t r i b u t i o n of annual income by focussing on the d i s t r i b u t i o n of consumption and by adopting a l i f e c y c l e perspective. While some recent studies (e.g. Nordhaus [1973], Blinder' [ 1 97.5 ], Irvine [I980])have made the s h i f t from annual income to l i f e c y c l e consumption, they have measured inequality in the pote n t i a l d i s t r i b u t i o n of representative consumption computed under the assumption of perfect c a p i t a l markets. They should rather have examined the actual d i s t r i b u t i o n of representative consumption representing l i f e c y c l e consumption opportunities obtainable under exi s t i n g c a p i t a l market conditions. This i s accomplished in the welfare approach to the measurement of inequality by using data on actual consumption in the computation of representative l i f e c y c l e consumption defined 3 0(2.34) also demonstrates that intracohort AKS equality i s equal to a weighted average of AKS equality within each cohort, which i s given by, (2.35) E*=s./mk (l<k<K) with the weights being the shares of t o t a l representative consumption accruing to each cohort. 44 in (2.3). AKS and per capita indices can be interpreted as inequality measures which evaluate the actual d i s t r i b u t i o n with respect to a hypothetical, optimal a l t e r n a t i v e . When the s o c i a l evaluation function is continuous, increasing, symmetric, and quasi-concave, average representative consumption, r, i s the optimal ( i . e . s o c i a l welfare maximizing) d i s t r i b u t i o n . Social welfare in the actual situation is represented by the equally d i s t r i b u t e d equivalent of r. AKS and per capita indices measure inequality as a function of these two s t a t i s t i c s . Since the welfare approach incorporates the consumer choice problem, i t can be extended to measure the welfare loss a t t r i b u t a b l e not only to the degree of interpersonal inequality, but also to imperfections in the means of r e d i s t r i b u t i n g consumption over time. In the extended welfare approach, the actual s i t u a t i o n i s unchanged but the optimal si t u a t i o n becomes characterized by perfect c a p i t a l markets in addition to interpersonal equality. The optimal si t u a t i o n i s thus represented by the mean of a vector of potential individual representative l i f e c y c l e consumption s t a t i s t i c s representing consumption plans chosen under perfect c a p i t a l market conditions. The situation i s represented in Figure II, where variables representing the situation in which c a p i t a l markets are assumed free of imperfections are denoted by a prime. Consider f i r s t an index of per capita inequality. Total inequality i s , .(2.38) A=m'-s The welfare loss due to interpersonal inequality i s measured by 45 the difference between m, which r e f l e c t s s o c i a l welfare when interpersonal inequality i s eliminated, and s, which r e f l e c t s the s o c i a l evaluation of the actual d i s t r i b u t i o n of representative consumption. That i s , (2.39) AP=m-s This i s exactly the index of interpersonal inequality which was , derived in the welfare approach above, and which can be decomposed into i n t r a - and intercohort inequality as in (2.28); that i s , (2.40) A p=[Sl(n k/H) ( i \ - s k ) ] + [7L(r> k/H)s K"S] p p +A; An index of the welfare loss, measured in representative consumption d o l l a r s per capita, attributable to imperfections in c a p i t a l markets can analogously be defined as the difference between s', representing s o c i a l welfare when a l l "intertemporal" inequality has been eliminated, and s: (2.41) A T=s'-s F i n a l l y , account must be taken of the interaction between interpersonal and "intertemporal" inequality. Improved means of borrowing and lending may y i e l d greater benefits to some than others, a l t e r i n g the d i s t r i b u t i o n of representative l i f e c y c l e consumption and thus changing measured interpersonal inequality. S i m i l a r l y , r e d i s t r i b u t i o n among individuals w i l l a f f e c t the shape of their consumption p r o f i l e s and thus the s o c i a l s i gnificance of existing imperfections in c a p i t a l markets. Thus the i n t e r a c t i v e effect of interpersonal and intertemporal r e d i s t r i b u t i o n can be thought of either as the change in interpersonal inequality a t t r i b u t a b l e to the elimination of 46 c a p i t a l market imperfections, (2.42) A P T = (m'-s')-(m-s) or as the change in "intertemporal" inequality resulting from the elimination of interpersonal inequality, (2.43) APT=(m'-m)-(s'-s) These two interpretations of the interactive effect are c l e a r l y equivalent, as can be seen by comparing (2.42) and (2.43). The indices of per capita interpersonal and "intertemporal" inequality, (2.39) and (2.41), and their interactive e f f e c t , (2.42) or (2.43), can be aggregated into an index of t o t a l per capita inequality by adding them together: A P + A T + A f > T (2.44) =(m-s) + (s'-s) + [(m'-s')-(m-s)] =m'-s We may now use (2.44) to compute the corresponding AKS indices of inequality by di v i d i n g through by m1. This y i e l d s , (2.45) [(m-s)/m'] + [(s'-s)/m'] + {[(m'-s')/m']-[(m-s)/m']} P T PT =1 + I + I Recall that the per capita index of interpersonal inequality i s the same whether or not "intertemporal" inequality i s measured. The AKS index of interpersonal inequality, however, is d i f f e r e n t in the extended welfare approach because t o t a l representative consumption (the.basis on which AKS indices express inequality) i s greater when l i f e c y c l e consumption plans are chosen under perfect c a p i t a l market conditions. This i s ref l e c t e d in the denominator of the AKS index of interpersonal inequality, which is m' in the f i r s t term of (2.45) where i t had been m in (2.33). The index of interpersonal AKS inequality in this extended 47 welfare approach to the measurement of inequality can be decomposed within and among age-cohorts analogously to (2.29). This y i e l d s , (2.46) l[ =(m-Z:(nk/H)sk)/m' and, (2.47) l£ = (21(n k/H)s k-s)/m' AKS indices of equality are also e a s i l y extended to include the measurement of welfare losses due to differences between actual and potential intertemporal d i s t r i b u t i o n s . Total AKS equality in t h i s case i s equal to the r a t i o of s o c i a l welfare under the actual d i s t r i b u t i o n , measured in terms of representative consumption d o l l a r s , to s o c i a l welfare s i m i l a r l y measured in a p o t e n t i a l , optimal si t u a t i o n in which a l l inequality has been eliminated and c a p i t a l markets are free of imperfect ions: (2.48) E=s/m' Total inequality can be decomposed into the product of three terms as follows: (2.49) E=[s/m] [s/s'] [(s'/m*)/(s/m)] ' The f i r s t term in (2.49) is the AKS index of inequality (2.37) which can be m u l t i p l i c a t i v e l y decomposed into the two terms given in (2.34) and (2.36). The second term measures the s o c i a l cost of imperfections in the means of intertemporally r e a l l o c a t i n g income, and the t h i r d term r e f l e c t s the interaction of interpersonal and intertemporal r e d i s t r i b u t i o n . Indices of per capita inequality and the corresponding AKS indices of inequality and equality, derived in the welfare approach to the measurement of inequality and i t s extension to 48 include the measurement of "intertemporal" inequality, provide t h e o r e t i c a l l y sound measures of equality and inequality which have a number of very appealing c h a r a c t e r i s t i c s . F i r s t , they are defined in terms of, and are constructed with, summary s t a t i s t i c s of welfare which represent l i f e c y c l e consumption p r o f i l e s and are calibrated in units of real consumption d o l l a r s . Second, the indices d i s t i n g u i s h between actual and optimal, hypothetical d i s t r i b u t i o n s of well-being, and are c a r e f u l l y constructed to measure inequality in the actual d i s t r i b u t i o n with reference to the optimal a l t e r n a t i v e . To this extent the welfare approach f i t s within the framework suggested by Atkinson [1970], Kolm [1969], and Sen [1973] which i s now widely accepted as the foundation of the modern theory of inequality measurement..Third, welfare approach indices allow for the exercise of consumer choice to reallocate income streams to achieve desired consumption plans. While t h i s may seem an obvious point to anyone familiar with economic theory, i t has, in fact, largely been overlooked in the theory of inequality measurement to date. Fourth, the indices incorporate a l i f e c y c l e perspective on the measurement of inequality which i s both a necessary adjunct to the e x p l i c i t inclusion of consumer choice, and a great improvement on the predominant trend of measuring inequality in the d i s t r i b u t i o n of annual income or consumption. F i f t h , they incorporate a method of decomposing interpersonal inequality into intragenerational and intergenerational components so that they can be studied separately. And f i n a l l y , welfare approach indices can be constructed so as to include the e f f e c t s of both interpersonal and intertemporal inequality, and 49 can be disaggregated so as to id e n t i f y the r e l a t i v e magnitudes of these two sources of inequality. These many important advantages of a welfare approach to the measurement of inequality are, unfortunately, d i f f i c u l t to re a l i z e in practi c e . Empirical implementation of the welfare approach i s plagued by several problems. F i r s t , individual u t i l i t y functions are required for the construction of welfare approach indices of inequality. While actual consumption paths can be observed in the data (rather than derived by the maximization of u t i l i t y subject to actual market opportunities for r e a l l o c a t i n g income streams) u t i l i t y functions are required to compute representative l i f e c y c l e consumption. Furthermore, estimation of the indices which I have proposed requires l i f e c y c l e income data on a l l members of the population. Panel' data sets are rare and none covers the entire l i f e c y c l e of even one age-cohort in the population, l e t alone those of members of a l l cohorts represented in the population. To estimate welfare approach indices for the current population would require data c o l l e c t e d over a period of roughly one hundred f i f t y years; that i s , from the year of b i r t h of the oldest member of the current population, to the year of death of i t s o l d e s t - l i v i n g member. The prospects for empirically implementing the welfare approach thus appear bleak indeed. A situation in which the demands imposed by theory ou t s t r i p empirical resources and a b i l i t i e s i s not unfamiliar to economists. A solution i s to try to construct an alternative formulation which i s empirically tractable as well as th e o r e t i c a l l y sound and a t t r a c t i v e . This task i s taken up in the next chapter. 51 FIGURE II Mean and Equally Distributed. Equivalent Consumption Individual 1' s consumption (real $'s) 5 2 CHAPTER THREE A Decomposition Approach to the Measurement of Inequality There are two theoretical problems with measuring inequality in the d i s t r i b u t i o n of annual income. Their solution, I have argued, involves a s h i f t to measuring inequality in the d i s t r i b u t i o n of l i f e c y c l e consumption. But while t h i s solution, which i s the essence of the welfare approach to the measurement of inequality, may appear simple enough when put in such terms, i t s practice i s in fact e n t i r e l y precluded by lack of l i f e c y c l e consumption data and knowledge of individual u t i l i t y functions. A decomposition approach to the measurement of inequality i s an . attempt to tread the middle ground between the.theoretical rigour and empirical i n t r a c t a b i l i t y of a welfare approach and the empirically p r a c t i c a l but t h e o r e t i c a l l y misspecified . t r a d i t i o n a l approach. A decomposition approach also responds to c r i t i c i s m s leveled at the t r a d i t i o n a l approach to the measurement of inequality that indices of annual income inequality do not account for the ef f e c t s of income mobility and the age-structure of the population. I s h a l l b r i e f l y review here the methodological problems of measuring inequality in the d i s t r i b u t i o n of annual income. A number of empirical studies have assessed the s e n s i t i v i t y of popular and widely employed inequality indices, such as the Gini 31S-ee Chapter One for references. 53 c o e f f i c i e n t , to the length of the accounting p e r i o d . 3 1 A l l found a s i g n i f i c a n t equalizing effect associated with extending the period over which income (or consumption) i s cumulated. Furthermore, Shorrocks [1978a] has demonstrated that, for a large c l a s s of indices including those consistent with the approach i n i t i a t e d by Atkinson [1970], Kolm [1969], and Sen [1973], inequality of income aggregated over an extended accounting period cannot exceed a weighted average of measured annual inequality. In the best attempt to date to investigate the e f f e c t s of income mobility, Shorrocks [1978b] has constructed and computed indices which measure income mobility in terms of the extent to which inequality i s diminished by lengthening the accounting period. What his approach lacks i s an e x p l i c i t t h e o r e t i c a l l i n k between mobility and inequality indices which would allow measured annual inequality to be adjusted so as to account for the e f f e c t s of income mobility. The decomposition approach eschews the mobility index approach of Shorrocks in favour of inequality indices which can d i s t i n g u i s h pure interpersonal inequality from that at t r i b u t a b l e to income mobility. Another widely recognized source of bias in measured annual inequality i s due to the observed tendency of income streams and consumption paths to r i s e over the course of a l i f e t i m e . 3 2 Indices of annual inequality capture not only pure interpersonal inequality but also inequality related to the age-structure of 3 2 I n cross-sectional data t h i s pattern a declining t a i l after retirement. is often observed to have 54 the population when income and consumption p r o f i l e s display t h i s c h a r a c t e r i s t i c shape. Paglin [1975] has proposed a method for distinguishing age-related inequality from pure interpersonal inequality in the Gini c o e f f i c i e n t computed from annual data. The Paglin-Gini has been heavily c r i t i c i z e d 3 3 but no replacement has been suggested. Fortunately, however, this problem too can be solved by adopting the decomposition approach to the measurement of i n e q u a l i t y . 3 " The decomposition approach to the measurement of inequality starts from the premise that longitudinal data are necessary i f the intertemporal aspects of inequality are to be taken into account. A time-series of anonymous cross-sections i s of no use for t h i s purpose because the e f f e c t s of income mobility show up only when an individuals's income i s followed over time. 3 5 Panel data are thus required, which, even i f they do not cover entire l i f e c y c l e s , should at least allow the degree of mobility-related inequality to be approximated. The method of the decomposition approach i s to measure inequality in the panel data treated as a single d i s t r i b u t i o n . A part of t h i s t o t a l w i l l be due to the differences in income that an individual experiences from one 3 3 F o r a detailed, discussion of the Paglin-Gini and the c r i t i c i s m s of i t see Chapter One. 3"An evaluation of Paglin's method compared to the decomposition approach i s presented in Appendix B. 3 5 I n fact, the trend of annual inequality i s remarkably stable, but t h i s does not imply a low degree of income mobility because the symmetry of inequality indices means that permuting a p a r t i c u l a r d i s t r i b u t i o n w i l l not a l t e r the degree of measured inequality. Any amount of income mobility is consistent.with a stable trend of annual inequality. 55 year to another. By measuring the inequality attributable to these intrapersonal income differences, an index of mobility-related inequality is obtained which, in addition to i t s i n t r i n s i c interest, can be used to define a measure of interpersonal inequality net of the effects of income variation over time. 3 6 The remaining inequality i s not solely interpersonal, however, but also r e f l e c t s intergenerational income differences. The c h a r a c t e r i s t i c shape of l i f e c y c l e p r o f i l e s contributes to the variation of annual income among individuals even i f l i f e t i m e incomes are equally d i s t r i b u t e d . This source of inequality can also be distinguished from pure interpersonal inequality, y i e l d i n g an index of age-related inequality. This index i s of interest for i t s own sake and for use in deriving an index of pure interpersonal inequality net of the e f f e c t s of intertemporal and intergenerational income differences. The decomposition approach thus begins with panel consumption data. For each i n d i v i d u a l , h (l<h<H), in the sample, the panel data set has a time-series of annual consumptions over a T year period, c^ = ( c h 1 , . . . , c ^ ) . The entire data set can be arranged as a vector, c=(c^, . . . , c H ) , of dimension HT. In the decomposition approach to the measurement of inequality, t h i s vector i s treated as a single d i s t r i b u t i o n over a population of size HT in which t o t a l inequality i s measured. Inequality attributable to var i a t i o n in individual incomes over 3 6 I n t h i s sense intertemporal is a synonym for intrapersonal when speaking of inequality attributable to the time paths of individual incomes. 56 time can then be distinguished from that due to income variation among, individuals by decomposing inequality within and among population (of size HT) subgroups. The consumption vector of the members of subgroup h is c^. Inequality i s decomposed within and among population subgroups using a procedure suggested by Blackorby, Donaldson, and Auersperg [1981]. In t h i s procedure inequality within subgroups i s eliminated from the d i s t r i b u t i o n by assigning each individual the equally d i s t r i b u t e d equivalent of the d i s t r i b u t i o n within his own subgroup. 3 7 That i s , l e t t i n g the scalar be the equally d i s t r i b u t e d equivalent of the vector c^, intrapersonal (mobility-related) inequality i s eliminated by replacing the vector c^ = ( c ^ , . . . , c h T ) by the s o c i a l l y equivalent, equally d i s t r i b u t e d a l t e r n a t i v e , ( r ^ l T ^ ' 3 8 The equally d i s t r i b u t e d equivalent i s mean consumption adjusted for inequality in the d i s t r i b u t i o n i t represents. Thus the equally d i s t r i b u t e d equivalent of a d i s t r i b u t i o n cannot exceed i t s mean, and w i l l be s t r i c t l y less than the mean i f there is any inequality in the d i s t r i b u t i o n . Thus, t o t a l consumption in the o r i g i n a l d i s t r i b u t i o n c=(c^, . . . ,c^) cannot be less than t o t a l consumption in the d i s t r i b u t i o n in which intrapersonal 3 7An alte r n a t i v e procedure for decomposing inequality has recently been proposed independently by Bourguignon [1979], Cowell [1980], and Shorrocks [1980], in which subgroup mean incomes are used to eliminate intragroup inequality. In Appendix A Shorrocks' version of th i s alternative procedure i s compared to the Blackorby-Donaldson-Auersperg method, and their r e l a t i v e performances in the decomposition approach are evaluated. The conclusion reached i s that the Blackorby-Donaldson-Auersperg procedure i s superior for t h i s purpose because the alternative suffers from two theoreti c a l drawbacks which prove to be seriously damaging to i t s empirical performance. 3 8Where 1 i s a T-dimensional unit vector. 57 inequality i s eliminated, which i s characterized by the vector ( ri I T ' • • * f E*H J_T) • T n e saving generated by a move from the former to the l a t t e r represents the s o c i a l cost of mobility-related inequality. The replacement of the vector c^ by ( r h J _ T ) removes a l l intrapersonal inequality, leaving only inequality due to differences between the values of r^, (l<h<H). This remaining inequality i s attributable to differences related to the age-structure of the population as well as to pure interpersonal differences in consumption. To dis t i n g u i s h these two sources from one another, inequality may be further decomposed within and among age-subgroups of the population.. This i s accomplished by grouping together individuals of the same age, 3 9 and assigning members of the same cohort the equally d i s t r i b u t e d equivalent of their cohort d i s t r i b u t i o n . Letting s k be the equally d i s t r i b u t e d equivalent of the d i s t r i b u t i o n among the n^ members of cohort k, (l<k^K), the elimination of intracohort inequality results in the replacement of the d i s t r i b u t i o n (r1-L-r' • • • ' ' rH — b v ' s i , . . . , s K j _ r ( i ). Again, a move between these two vectors results in a saving which r e f l e c t s the so c i a l cost of inequality within cohorts, which I have c a l l e d pure interpersonal inequality. Intrapersonal and intracohort inequality having now been eliminated, only inequality between cohorts remains. This too can be eliminated by replacing the d i s t r i b u t i o n 3 9 0 r in the same age bracket i f age-cohorts are defined over a range of years. 58 (s , | J _ T ^ , . . . »sKJ_T.n^) by a d i s t r i b u t i o n in which everyone receives the population-wide equally d i s t r i b u t e d equivalent, s. This t h i r d move from the d i s t r i b u t i o n of cohort equally d i s t r i b u t e d equivalents to a s o c i a l l y equivalent equal d i s t r i b u t i o n over the entire population (of size HT) , ( s j _ H T ) , implies a further s o c i a l saving which is a measure of age-related inequality. The measurement and decomposition of inequality involve the repeated replacement of d i s t r i b u t i o n s (over subgroups of the population) by their equally d i s t r i b u t e d equivalents. In each case, since the equally d i s t r i b u t e d equivalent i s an inequality adjusted mean, there i s a s o c i a l saving created which r e f l e c t s the s o c i a l cost attributable to a p a r t i c u l a r source of t o t a l inequality. AKS indices express th i s saving as a proportion of t o t a l consumption, and per capita indices express i t in per capita terms. Thus e t h i c a l indices measure inequality as the amount wasted on inequality. This chapter begins with a formal discussion of the so c i a l welfare underpinnings of the decomposition approach to the measurement of inequality. The primary objective i s the d e f i n i t i o n of the equally d i s t r i b u t e d equivalents, r^ , s^, and s, which are fundamental elements of the decomposition indices that I wish to derive. This i s done by successively eliminating intrapersonal, intracohort, and intercohort inequality from the o r i g i n a l d i s t r i b u t i o n . The s o c i a l savings implied by moving through a succession of reference vectors measure the so c i a l costs associated with each source of inequality. AKS and per capita indices of inequality can then be constructed by 59 expressing the s o c i a l cost of inequality in percentage and per capita terms respectively. In the former case I again employ the decomposition presented in the preceding chapter which I have suggested to replace the Blackorby-Donaldson-Auersperg decomposition of AKS inequality indices. As mentioned before, the decomposition approach can be j u s t i f i e d on i t s own grounds--as a solution to the problems of measuring pure interpersonal inequality in an annual d i s t r i b u t i o n - - o r as an empirically tractable approximation to the welfare approach. The chapter concludes with a discussion of the common ground shared by the two approaches. I show that, by adopting two simplifying assumptions, an equivalence between the two approaches can be established. The measurement of inequality requires the existence of a so c i a l evaluation function W:R+ —*• R1 with image, (3.1) w=W(c) where c=(c^ , . . . , c 1 T , . . . , c H 1 , . . . , c H T ) i s a vector of the consumption paths of H people observed over T years. W(.) i s assumed to be continuous, increasing, and S - c o n c a v e . In the decomposition approach, c is treated as a d i s t r i b u t i o n of consumption over a population of size HT. I begin by decomposing this population into H exhaustive, mutually exclusive subgroups of size T, with the intention of measuring inequality in each subgroup independently of consumption of non-members. The s o c i a l evaluation function must therefore be separable in these subgroups, implying that W(.) can be written a s , (3.2) w=W(v!(c1 ), . . . , V H ( c H ) ) 60 where W(. ) is increasing in V h(c^) and c h = ( c h 1 , . . . , c h T ) is a vector of the T consumptions observed for person h. The functions V^:RT^*R1 are id e n t i c a l for a l l h, and can be used to define individual representative consumption: (3.3) V ( r h l T ) = V ( c h ) (1<h<H) The properties of V(.), inherited from W(.), allow r^ to be uniquely determined for every c^ so that r^ can be written as, (3.4) r h=R(c^) 0<h<H) Substitution of (3.3) into (3.2) y i e l d s , w=W[V(r 1l T), . . . ,V(r„l T)] (3.5) =W( r^ , . . . , r^ ) This s o c i a l evalution.function defined over individual representative consumptions can be used as the basis of an index of interpersonal inequality free of bias due to consumption mobility. To separate intercohort e f f e c t s from interpersonal inequality within age-cohorts requires that individuals be grouped by age. Let "NWN1, . . . ,NK } be a p a r t i t i o n of the population set N={1, . . . ,H] into age cohorts, h e m e a n s person h i s a member of the kth age-cohort which has n k members, r* i s a vector of representative consumptions of individuals in cohort k. Each J>P must be separable from i t s complement in ft, in which case (3.5) can be written as, (3.6) w=w[WMr1), . . . ,W K(r K)] where W(.) i s increasing in Wk(r**). The conjunction of symmetry and sep a r a b i l i t y in W(.) implies that i t s structure i s additive (Blackorby, Donaldson, and Auersperg [1981, theorem 1]): (3.7) w=W[21g(rh ) ] 61 where W(.) i s increasing in i t s argument and g(.) is independent of h because of the symmetry assumption. S-concavity of W(.) requires that g(.) be concave; s t r i c t S-concavity requires that i t be s t r i c t l y concave (Berge [1963]). Thus W(.) must be quasi-concave and symmetric. The s o c i a l evaluation function  corresponding to decomposable AKS and per capita indices of  inequality must therefore be continuous, increasing, symmetric,  quasi-concave, and a d d i t i v e l y separable. Further properties are required of the s o c i a l evaluation function corresponding to r e l a t i v e indices (which are homogeneous of degree zero in their arguments) and absolute indices (which are invariant to equal absolute changes in their arguments). AKS indices are r e l a t i v e  indices i f and only i f the s o c i a l evaluation function i s  homothetic. In conjunction with the properties l i s t e d above, th i s r e s t r i c t s the class of admissible s o c i a l evaluation functions to the means of order R. Per capita indices are  absolute indices i f and only i f the s o c i a l evaluation function  is translatable. This r e s t r i c t s W(.) to the Kolm-Pollak family of s o c i a l evaluation functions. The properties of "W( . ) and wN . ) are again traced back to the o r i g i n a l s o c i a l evaluation function (3.1). Wk:Rn,< —••R1 can be used to define the representative consumption of cohort k, (3.8) Wk(s. 1 h )=W k(r k) (l<k<K) which i s the equally d i s t r i b u t e d equivalent of the vector of representative consumptions of a l l individuals in cohort k. The properties of W (.) ensure that s k can be e x p l i c i t l y defined as, (3.9) s k = S k ( r k ) (l<k<K) Since Wk (.) has the same additive structure as W(.), with the 62 s u m m a t i o n b e i n g o v e r members o f t h e k t h a g e - c o h o r t o n l y , r e p r e s e n t a t i v e c o h o r t c o n s u m p t i o n i s , ( 3 . 1 0 ) s k = g - 1 [ ( 1 / n k ) ^ k g ( r h ) ] 0 < k < K ) T h e i m p l i c i t d e f i n i t i o n o f c o h o r t r e p r e s e n t a t i v e c o n s u m p t i o n , ( 3 . 8 ) , c a n be s u b s t i t u t e d i n t o ( 3 . 6 ) y i e l d i n g , w = W[W 1 ( s 1 l n i ) , . . . , W K ( s K l n ^ ) ] ( 3 . 1 1 ) = W ( S l l n i s K l r , K ) F i n a l l y , t h e e l i m i n a t i o n o f i n t e r c o h o r t i n e q u a l i t y w i t h s o c i a l i n d i f f e r e n c e i s a c c o m p l i s h e d by a s s i g n i n g e a c h p e r s o n t h e p o p u l a t i o n r e p r e s e n t a t i v e c o n s u m p t i o n i m p l i c i t l y d e f i n e d by ( 3 . 1 2 ) w(s1 u ) = W ( s 1 1. , . . . , s K 1 n ) T h e p r o p e r t i e s o f W ( . ) . e n s u r e t h a t s i s u n i q u e l y d e f i n e d f o r a n y d i s t r i b u t i o n o f c o h o r t r e p r e s e n t a t i v e c o n s u m p t i o n s , a n d c a n be w r i t t e n ( 3 . 1 3 ) s = S ( S l l n i , . . . J.„K) W h i l e ( 3 . 1 3 ) d e f i n e s s a s t h e e q u a l l y d i s t r i b u t e d e q u i v a l e n t o f t h e v e c t o r ( s 1 J _ n , . . . , s KJ_ n ), i t s h o u l d be c l e a r f r o m ( 3 . 9 ) a n d ( 3 . 4 ) t h a t s c a n a l s o be e x p r e s s e d a s an e q u a l l y d i s t r i b u t e d e q u i v a l e n t o f e i t h e r i n d i v i d u a l r e p r e s e n t a t i v e c o n s u m p t i o n s o r t h e a c t u a l c o n s u m p t i o n p a t h s o f e a c h i n d i v i d u a l . AKS i n d i c e s m e a s u r e i n e q u a l i t y a s t h e p e r c e n t a g e o f t o t a l c o n s u m p t i o n s a v e d by m o v i n g f r o m t h e a c t u a l d i s t r i b u t i o n t o an e q u a l d i s t r i b u t i o n t h a t i s s o c i a l l y e q u i v a l e n t . P e r c a p i t a i n d i c e s e x p r e s s t h e same s a v i n g i n p e r c a p i t a t e r m s . B o t h i n v o l v e t h e s o c i a l e v a l u a t i o n o f two s i t u a t i o n s : o n e i n w h i c h i n d i v i d u a l s r e c e i v e t h e i r o b s e r v e d c o n s u m p t i o n p a t h s , a n d a n o t h e r i n w h i c h e v e r y o n e i s a s s i g n e d t h e r e p r e s e n t a t i v e , o r e q u a l l y d i s t r i b u t e d e q u i v a l e n t , c o n s u m p t i o n . 63 The decomposition of inequality can be thought of as a series of situations in which various sources of inequality are eliminated in succession, and the s o c i a l savings created by each move measure that part of t o t a l inequality attributable to a par t i c u l a r source. It is necessary f i r s t to eliminate inequality due to variation in consumption over time since summary s t a t i s t i c s of individual consumption paths are required in order to measure interpersonal inequality. Assigning each individual his representative consumption, r^, defined in (3.4), eliminates intrapersonal inequality and provides an exact welfare index of individual consumption paths. Interpersonal inequality indices based on the s o c i a l savings generated by a move from the d i s t r i b u t i o n of representative consumption to an equally d i s t r i b u t e d equivalent d i s t r i b u t i o n capture both real consumption differences among persons and inequality related to the age-structure of the population. Eliminating inequality within age-cohorts by next assigning individuals their representative cohort consumptions, s^ , leads to an index of intracohort i n e q u a l i t y . 4 0 F i n a l l y , an index of age-related inequality can be based on the s o c i a l saving r e s u l t i n g from the elimination of inequality between cohorts in the move to a situation in which everyone receives the population-wide equally d i s t r i b u t e d equivalent, s. *°Blackorby, Donaldson, and Auersperg [1981] have shown that subindices of i n t r a - and intercohort per capita inequality are invariant with respect to the order in which inequality is eliminated within and between age-cohorts. Although t h i s i s not true of their decomposition of AKS indices, i t does apply to the decomposition of AKS inequality that I proposed in Chapter Two and that I s h a l l employ here. 64 The equally d i s t r i b u t e d equivalent consumptions given by (3.4), (3.9), and (3.13) define the elements of three reference vectors which, with the vector of o r i g i n a l consumptions, represent the four situations which characterize the successive elimination of inequality, f i r s t within each individual's consumption path over time, then within age-cohorts, and f i n a l l y between age-cohorts: (3.14) ( c ^ , . . . ,c , . . . ,C|^^, . . • , c ) (3.15) (r-i IT * •• • • » R H IT * (3.16) (s. 1 - , . . . , s k 1 • ) (3.17) (SJ. H T) Movements between these reference vectors are made with s o c i a l indifference. The s o c i a l savings which accrue as a result of such movements can be used to construct AKS inequality indices when expressed as a proportion of t o t a l consumption, or per capita inequality indices when expressed in per capita terms. Consider, for example, the per capita saving which could be re a l i z e d in a move from (3.14) to (3.15), r e c a l l i n g that the consumptions of H individuals received over T years are being treated as the consumptions of a population of size HT. A. =( 1/HT) [ Z 2 c.. -ZTr, ] (3.18) =( 1/H) [ X ( m h - r h ) ] where m K = ( 1 / T ) ^ c k t i s the mean consumption of individual h during the time peri.od covered by the data. This intrapersonal per capita inequality index, A A P , is an average of A =(m^-r k), (l<h<H), the per capita inequality in each individual's consumption path. A move from (3.15) to (3.16) would produce a per capita 65 saving of: A A c = (l/HT)[ZTr K-ZlTn ks k] (3.19) =2L(nk/H) (m k-s k) where mk = 1/nk ) r k is the mean, and sk=S (r ) i s the equally d i s t r i b u t e d equivalent, of the vector of representative consumptions of individuals in the k_th age-cohort, r k = (<r w> VheN k). It can be seen from (3.19) that per capita intracohort inequality i s equal to the cohort-population-share weighted average of A =mk-sk, the per capita inequality within each age-cohort. This index measures pure interpersonal inequality, free of di s t o r t i o n s attributable to consumption mobility, the age-structure of the population, and economic growth. F i n a l l y , the per capita saving to be realized by moving from (3.16) to (3.17) i s : A R c = ( 1 /HT) [ T. Tn k s k -HTs ] (3.20) =Zl(n k/H)s k-s which measures per capita inequality between age-cohorts as the mean of the d i s t r i b u t i o n of cohort representative consumptions less i t s equally d i s t r i b u t e d equivalent. The sum of the three subindices of inequality can eas i l y be shown to be equal to the index of t o t a l per capita inequality, A=m-s,lt1 which i s the per capita saving that would result from a dir e c t move from (3.14) to (3.17). Thus the decomposition of to t a l per capita inequality i s , a iWhere m i s the mean and s i s the equally d i s t r i b u t e d equivalent of the o r i g i n a l consumption vector (3.14). 66 A=m-s (3.21) = [ ( l / H)I(Y^)] + [(l/H)(Zr H-In ks k)] + [I(n k/H)s k-s] = + + A f c AKS inequality indices can e a s i l y be computed from the decomposition of per capita inequality given in (3.21). The new decomposition of AKS inequality that I presented in Chapter Two to replace the Blackorby-Donaldson-Auersperg decomposition is calculated simply by dividing through (3.21) by m, the mean of (3.14). This y i e l d s : (3.22) I=I A p + I A c +I R C where the subindices of AKS inequality are defined as follows: (3.23) I A p =[ (1/H)2L (m L-r h ) ]/m (3.24) 1^ =[£ (n u/H)(m K-s k)]/m (3.25) 1^ = [ X ( n k / H ) s k - s ] / m . The AKS indices (3.23), (3.24), and (3.25) measure intrapersonal, intracohort, and intercohort inequality, respectively, as the s o c i a l cost associated with these sources of consumption differences expressed as a proportion of t o t a l consumption in the o r i g i n a l d i s t r i b u t i o n . The three indices may be interpreted as measures of mobility-related inequality, pure interpersonal inequality, and age-related inequality respectively. F i n a l l y , consider the construction of AKS indices of equality by a series of moves between successive pairs of the reference vectors (3.14) through (3.17). AKS equality indices constructed in this manner are,defined as the r a t i o of t o t a l consumption in one vector to t o t a l consumption in the preceding vector. For the move from (3.14) to (3.15) this y i e l d s , 67 E =ZTr, /Z 21 c,. (3.26) =( 1/H)2L rh/mh, where m= (1/HT)2. Z! c u+. i s the mean of (3.14). S i m i l a r l y , intracohort r e l a t i v e equality i s given as t o t a l consumption in (3.16) as a proportion of t o t a l consumption in (3.15). E * = £ T n k s k / ? T r h (3.27) = ? n k s k / | n k m k since m^ = (1/n^ K r k . And f i n a l l y , AKS intercohort equality i s E R t=HTs/^Tn ks k (3.28) =s/2t.(n k/H)s k The product of these three subindices of equality y i e l d the AKS index of t o t a l equality. Thus the decomposition of t o t a l AKS equality i s , E=s/m (3.29) =[ (1/H)Zlrh/m] [ Z n ^ ^ Z r J [ s / ( l / H ) | n k s k ] = EAP E A C ERc In the decomposition of per capita inequality, (3.21), AKS inequality, (3.22), and AKS equality, (3.29), t o t a l (in)equality i s expressed as a simple function of three subindices of (in)equality which measure the contributions of intrapersonal, intracohort, and intercohort (in)equality to the t o t a l . The motivation has been two-fold. The decompositions provide a solution to problems with the t r a d i t i o n a l approach of measuring inequality in annual d i s t r i b u t i o n s which, i t has been widely argued, confuses age- and mobility-related inequality with pure interpersonal inequality. Thus, the eff e c t s on measured inequality of consumption mobility and of the t y p i c a l l y non-68 constant time path of consumption have been i d e n t i f i e d and isolated , allowing an index of pure interpersonal inequality to be constructed. To be sure, mobility and the shape of consumption p r o f i l e s are sources of inequality which are of s i g n i f i c a n t i n t r i n s i c i n t e r e s t . " 2 "Inequality", however, as the phrase i s commonly used both by professionals and laymen, i s , I believe, meant to exclude inequality which arises from either of these two sources. And this i s precisely what the decomposition approach accomplishes. An alternative motivation for exploring the decomposition approach to the measurement of inequality is the hope that i t may offer a t h e o r e t i c a l l y sound and empirically tractable theory of inequality measurement. The t r a d i t i o n a l method of measuring inequality in the d i s t r i b u t i o n of annual income f a i l s on the • former count. The welfare approach presented in Chapter Two provides a sound theoretical basis for measuring inequality but is incapable of empirical implementation. The decomposition approach i s a successful method of measuring interpersonal inequality without confusing i t with inequality a r i s i n g from other sources. Dalton argued that, "the economist is primarily interested not in the d i s t r i b u t i o n of income as such, but in the ef f e c t s of the d i s t r i b u t i o n of income upon the d i s t r i b u t i o n and t o t a l amount of economic welfare" ([1921, p.348]). It should therefore be asked whether there are grounds for interpreting intrapersonal inequality in the decomposition approach as an * 2Witness the volume of l i t e r a t u r e written on these subjects, es p e c i a l l y the work of Shorrocks [l978a,b] and Paglin [1975]. 69 index of inequality in the d i s t r i b u t i o n of well-being. The answer i s a q u a l i f i e d yes. Take, for example, the decomposition of per capita inequality, (3.21). The index which I am arguing measures pure interpersonal inequality i s given by the second term, (3.19) A t o = ( l / H T ) [ l T r h - Z T n k s k ] In the welfare approach the corresponding index i s (2.23): A A = ( 1 / H ) [ r r h - ^ n k s k ] In the welfare approach, representative l i f e c y c l e consumption, r^, i s the l i f e c y c l e consumption annuity between which and his actual consumption p r o f i l e the individual is i n d i f f e r e n t . It is i m p l i c i t l y defined by, (2.2) U h ( r h l T J = U ( c M , . . . , c h T h ) In the decomposition approach, however, representative consumption i s defined in terms of a s o c i a l evaluation function rather than an individual u t i l i t y function. That i s , (3.3) V(r K l T ) = V ( c H 1 , . . . , c h T) (2.2) and (3.3) are sim i l a r , but d i f f e r in two important respects. One, already mentioned, i s that U (.) i s an individual u t i l i t y function while V(.) is a s o c i a l evaluation function which results from the separ a b i l i t y structure imposed on W(.), from which i t inherits i t s properties. The other difference i s that the domain of U (.) i s a T^-dimensional vector of consumption expenditures made by individual h during the course of his l i f e , while the dimension of the domain of V(.) is common to a l l individuals, being T observations on consumption. Despite these differences, there are grounds for arguing that r L in the decomposition approach i s a sat i s f a c t o r y summary 70 s t a t i s t i c of individual welfare, inequality is measured in terms of the s o c i a l cost implied by maintaining the actual d i s t r i b u t i o n rather than r e d i s t r i b u t i n g i t equally. A case might therefore by made that the arguments of an inequality index should be based on a s o c i a l , rather than private, evaluation of the individual welfare that results from a given consumption plan. In t h i s case V(.) may be viewed as a u t i l i t y function based on the preferences of a planner rather than on individual preferences. Second, and perhaps more importantly, r e c a l l that the welfare approach was found to be empirically impractical because i t requires that individual u t i l i t y functions be known and that data on the l i f e c y c l e consumption paths of a l l members of the population are avai l a b l e . When other writers have met these obstacles, they have invoked the simplifying assumptions of i d e n t i c a l u t i l i t y functions and length of l i f e across a l l individuals (Nordhaus [1973], Blinder [1975], Layard [1977], Irvine [1980]). Blinder [1975, pp. 31-2] has argued, furthermore, for the adoption of (the continuous-time version of) an i s o - e l a s t i c form for the u t i l i t y function. But thi s is precisely (the analogue of the discrete-time version of) the only functional form admissible as a s o c i a l evaluation function in the decomposition approach to the measurement of r e l a t i v e inequality. Faced with the task of implementing the welfare approach, Blinder would argue for substituting the known function V(.) defined over T years for the unknown U h(.) defined over years; precisely t h i s i s accomplished by adopting a decomposition approach to the measurement of inequality. For these reasons, in the next chapter I calculate 71 i n e q u a l i t y i n d i c e s b a s e d on t h e d e c o m p o s i t i o n a p p r o a c h t o t h e m e a s u r e m e n t o f i n e q u a l i t y . S e v e r a l f e a t u r e s o f t h e s e i n d i c e s w i l l be o f p a r t i c u l a r i n t e r e s t . Of t h e s e , p e r h a p s t h e m o s t i m p o r t a n t w i l l be t o c o m p a r e i n t r a c o h o r t i n e q u a l i t y i n t h e d e c o m p o s i t i o n ( w h i c h I t a k e a s a m e a s u r e o f p u r e i n t e r p e r s o n a l i n e q u a l i t y ) w i t h i n d i c e s o f i n e q u a l i t y i n t h e d i s t r i b u t i o n o f a n n u a l c o n s u m p t i o n . A l s o o f i n t e r e s t , h o w e v e r , w i l l be t h e r e l a t i v e i m p o r t a n c e o f t h e t h r e e s o u r c e s o f i n e q u a l i t y w h i c h h a v e b e e n i d e n t i f i e d i n t h e d e c o m p o s i t i o n a p p r o a c h . S h o r r o c k s [ 1 9 7 8 b ] h a s p r o d u c e d some i n t e r e s t i n g e m p i r i c a l r e s u l t s o n i n c o m e m o b i l i t y i n t h e U n i t e d S t a t e s , b u t n o t i n a f o r m t h a t a l l o w s t h e q u a n t i t a t i v e i m p a c t o f i n c o m e m o b i l i t y on m e a s u r e d i n e q u a l i t y t o be c a l c u l a t e d . T h i s i s p o s s i b l e w i t h t h e d e c o m p o s i t i o n a p p r o a c h t o t h e m e a s u r e m e n t o f i n e q u a l i t y , a s w i l l be s e e n i n t h e e m p i r i c a l r e s u l t s i n C h a p t e r F o u r . I s h a l l a l s o i n q u i r e i n t o t h e s e n s i t i v i t y o f t h e i n d i c e s d e v e l o p e d i n t h e d e c o m p o s i t i o n a p p r o a c h t o t h e c h o i c e s made r e g a r d i n g t h e s i z e o f a g e - c o h o r t s b r a c k e t s a n d t h e number o f y e a r s o f d a t a w h i c h a r e e m p l o y e d f o r t h e c o m p u t a t i o n o f t h e i n d i c e s . 72 CHAPTER FOUR Empirical Implementation of the Decomposition Approach Empirical implementation of the decomposition approach to the measurement of inequality requires consumption data observed over a number of years for a panel of individuals. This i s perhaps the heaviest requirement of the decomposition approach over and above those of the t r a d i t i o n a l approach of measuring annual income inequality. I have drawn upon the best source of panel data, the Panel Study on Income Dynamics conducted by the Survey Research Center [1968] of the Institute for Social Research at the University of Michigan." 3 Currently, there are eleven years of annual data available, running from 1968 to 1978, which report a wide variety of economic and demographic variables for 6154 families and. their almost 21,000 members. I have computed the consumption variable from the point of view of the family in keeping with the idea that the family acts as a unit in making private consumption decisions and i s treated as such by public transfer programs. The d i s t r i b u t i o n of consumption i s expressed on an individual basis, however, to exclude the effects of family size and so that inequality i s measured among individuals. In going from family to individual consumption i t is necessary to make use of adult equivalence scales because children do not require the same l e v e l of " 3Unfortunately no appropriate panel data exist for Canada. 73 consumption for their support as adults. There are a number of d i f f e r e n t ways to proceed. The usual practice has been to compute family size in terms of adult equivalents. For example, treating an adult as equivalent in terms of consumption to two children, a family of fi v e consisting of two adults and three children is equivalent to a family of three and one-half adults. If t o t a l family consumption i s $17,500, per capita adult equivalent consumption i s $5,000 (=$17,500/3.5). This family would then be counted, i f the usual practice were followed, as three and one-half individuals each with a consumption of $5,000. I prefer to keep the number of family members at i t s actual value, however, so that a l l individuals are represented in the d i s t r i b u t i o n of consumption. There are then two a l t e r n a t i v e . One i s to assign the adults in t h i s family consumptions of $5,000 and the children $2,500. This, however, creates intrafamily inequality which I do not wish to be included in measured inequality. I have therefore chosen to assign the per capita adult equivalent consumption to a l l family members.4'' This allows me to avoid introducing intrafamily inequality while ensuring that a l l individuals are represented in the consumption d i s t r i b u t i o n . 4 5 The consumption variable has been constructed by proceeding from market income through net income to consumption. Data 4 " T h i s practice is followed by Blackorby and Donaldson [1980b]. The idea i s due o r i g i n a l l y to A. Sen in a private communication to Blackorby and Donaldson. See also Sen [1979, pp. 292-3]. 4 5 T o t a l and therefore mean consumption are not, however, the same as in the o r i g i n a l d i s t r i b u t i o n . 74 lim i t a t i o n s have prevented this from being done exactly as i t should but I have attempted to compute the consumption variable to correspond as clo s e l y as possible with the theoret i c a l i d e a l . Beginning with family unit money income from market sources I added the rental value of free housing which represents one of the most important non-monetary components of market income. I then added transfers and subtracted taxes to arrive at net income, and added the amount saved on food stamps to incorporate an important non-monetary component of public transfers. My estimate of family consumption was then derived by adding income from private pensions and annuities and the rental value of owner-occupied housing.' 6 Since I am interested in welfare, and therefore in real rather than nominal consumption, the consumption variable has been deflated by the U. S. Consumer Price Index (1975=100) (International Monetary Fund [1980, p. 343]). The demographic data requirements include family composition (number of adults and children) for c a l c u l a t i n g the adult equivalent per capita consumption, and age of family head, according to which families are grouped into age-cohorts. I decided to drop families that experienced a change of family head during the sample period, to save on computing costs by reducing the sample si z e . By excluding families from the sample on th i s basis, the question of the age-cohort to which such 6The construction of the consumption variable i s described in d e t a i l in Appendix C. 75 f a m i l i e s s h o u l d be a s s i g n e d was a l s o a v o i d e d . " 7 S u r p r i s i n g l y , t h i s r e d u c e d t h e s a m p l e s i z e by 87 p e r c e n t . " 8 My e m p i r i c a l work i s b a s e d on t h i s s u b s a m p l e , b u t f o r c o m p a r a t i v e p u r p o s e s I a l s o c o m p u t e d r e l a t i v e i n e q u a l i t y i n t h e w h o l e s a m p l e . T h e r e s u l t s i n d i c a t e d t h a t m e a s u r e d i n e q u a l i t y i s 20 t o 25 p e r c e n t g r e a t e r i n t h e o r i g i n a l s a m p l e . T h e r e l a t i v e m a g n i t u d e s o f t h e s u b i n d i c e s o f i n e q u a l i t y , h o w e v e r , a r e v e r y c l o s e i n t h e two s a m p l e s . " 9 T h u s , w h i l e my r e s u l t s l i k e l y u n d e r e s t i m a t e t h e a c t u a l d e g r e e o f i n e q u a l i t y , t h e y a p p e a r t o be i n d i c a t i v e o f t h e d i f f e r e n c e s b e t w e e n t h e t r a d i t i o n a l a n d d e c o m p o s i t i o n a p p r o a c h e s t o t h e m e a s u r e m e n t o f i n e q u a l i t y . T h e f i n a l s a m p l e , t h e n , i n c l u d e s 797 f a m i l i e s on w h i c h d a t a a r e a v a i l a b l e o v e r a t e n y e a r p e r i o d , 1 9 6 8 - 1 9 7 7 . 5 0 T h e s a m p l e mean o f t h e d i s t r i b u t i o n o f a d u l t e q u i v a l e n t p e r c a p i t a c o n s u m p t i o n among a l l p e r s o n s i n 7 , 9 7 0 (=797x10) h o u s e h o l d s ( i . e . i n t h e p o o l e d s a m p l e ) i s $ 4 0 9 1 , w i t h a s t a n d a r d d e v i a t i o n o f $ 2 7 4 1 . A h i s t o g r a m o f t h e p o o l e d s a m p l e a p p e a r s a s F i g u r e I I I . T h e means a n d s t a n d a r d d e v i a t i o n s o f t h e a n n u a l d i s t r i b u t i o n s a n d o f t h e p o o l e d d i s t r i b u t i o n a r e g i v e n i n " 7 T h e r e a r e s e v e r a l p o s s i b l e s o l u t i o n s : ( l ) f a m i l i e s c o u l d be r e a s s i g n e d t o t h e a g e - c o h o r t o f t h e i r new h e a d when a c h a n g e o c c u r s ; ( 2 ) f a m i l y p e r c a p i t a a d u l t e q u i v a l e n t c o n s u m p t i o n c o u l d be c a l c u l a t e d a n d t h e n a s s i g n e d t o e a c h f a m i l y member i n h i s own a g e - c o h o r t ; o r ( 3 ) a g e - c o h o r t s c o u l d b e d e f i n e d o v e r l o n g e r t i m e s p a n s o f , s a y , f i v e o r t e n y e a r s . " 8 I t i s n o t c l e a r t h a t t h i s i s e n t i r e l y d u e t o c h a n g e s i n f a m i l y h e a d . E x a m i n a t i o n o f t h e r e p o r t e d " a g e o f f a m i l y h e a d " f o r some o f t h e f a m i l i e s d r o p p e d f r o m t h e s a m p l e i n d i c a t e d t h a t a p a r t o f t h e t o t a l m i g h t r e f l e c t r e p o r t i n g e r r o r s o n l y . " 9 S e e A p p e n d i x D f o r f u r t h e r d e t a i l s . 5 0 T h i s c o r r e s p o n d s t o t h e 1 9 6 9 - 1 9 7 8 w a v e s o f t h e P a n e l S t u d y on  I n c o m e D y n a m i c s s i n c e t h e d a t a a r e c o l l e c t e d on t h e p r e v i o u s y e a r ' s i n c o m e , t a x e s , t r a n s f e r s , e t c . T h e f i r s t wave was n o t u s e d b e c a u s e o f m i s s i n g d a t a . 76 Table 1 . Kolm-Pollak indices of absolute inequality based on the KP family of s o c i a l evaluation functions, and Atkinson indices of r e l a t i v e equality and inequality based on the means of order R s o c i a l evaluation functions, ([ (1/H )Zc* ] 0*R<1 have been computed. Both (4.1) and (4.2) represent families of so c i a l evaluation functions, whose members r e f l e c t varying degrees of inequality a v e r s i o n . 5 1 Since e t h i c a l indices of inequality measure the s o c i a l significance of inequality, measured inequality in a given d i s t r i b u t i o n w i l l depend on the degree of inequality aversion which the s o c i a l evaluation function e x h i b i t s . t A l l the indices reported in thi s chapter have been computed by f i r s t c a l c u l a t i n g i n d i v i d u a l , cohort, and population-wide equally d i s t r i b u t e d equivalent consumptions, the formulae for which are i d e n t i c a l to (4.1) and (4.2) (with the l i m i t s of the summation being appropriately changed). These representative consumptions have been used to construct reference vectors corresponding to a series of situations in which various sources of inequality are successively eliminated, and the inequality 5 1The degree of inequality aversion exhibited by members of the families of s o c i a l evaluation functions (4.1) and (4.2) i s ref l e c t e d in the curvature of the boundaries of their l e v e l sets. (4.1 ) G>0 (4.2) R=0 77 indices calculated as the per capita or percentage s o c i a l saving generated by a move from one vector to the next. AKS indices of equality are computed as t o t a l representative consumption in one situa t i o n as a proportion of t o t a l representative consumption in the preceding situation in which consumption is more unequally d i s t r i b u t e d . 5 2 Indices of (in)equality are reported for a number of values of the free parameters, G and R. Gini and Maximin indices have also been calculated for comparative purposes, although these indices are not addit i v e l y separable and thus are not guaranteed to aggregate to t o t a l measured inequality as do the Atkinson and KP indices. The absolute inequality results are reported in Table 2. For each value of G, the degree of absolute inequality aversion, t o t a l absolute inequality in the panel and i t s decomposition into subindices of intrapersonal, intracohort, and intercohort absolute inequality, are given in the f i r s t four rows of the table. The value of each subindex as a percentage of the t o t a l i s given in parentheses. Below these are shown, for comparative purposes, the minimum, maximum, and mean of the ten annual absolute inequality indices. The la s t i s a population share weighted average of the ten annual absolute inequality indices, which is the index of inequality within years that would result from a decomposition of t o t a l inequality within and 5 2 T h e r e p o r t e d v a l u e s o f t h e s e i n d i c e s s h o u l d be a c c o m p a n i e d by some m e a s u r e o f t h e i r s t a t i s t i c a l s i g n i f i c a n c e i f t h e y a r e t o be u s e d t o i n f e r t h e d e g r e e o f ( i n ) e q u a l i t y i n t h e p o p u l a t i o n . B e a c h a n d D a v i d s o n ( B e a c h [ 1 9 8 0 ] ) h a v e d e r i v e d a s y m p t o t i c s t a n d a r d e r r o r s f o r i n e q u a l i t y i n d i c e s w h i c h c o u l d be u s e d t o c o n s t r u c t i n t e r v a l e s t i m a t e s o f i n e q u a l i t y i n t h e p o p u l a t i o n d i s t r i b u t i o n . 78 among years. Mean annual absolute inequality is thus an appropriate index with which to compare the value of pure interpersonal inequality as measured by the index of absolute intracohort inequality. Absolute inequality indices measure the amount of consumption per capita that i s wasted on inequality. In a pure r e d i s t r i b u t i o n of consumption, everyone would receive the mean consumption, but when the so c i a l evaluation function exhibits a positi v e degree of inequality aversion, a s o c i a l l y equivalent d i s t r i b u t i o n results when individuals receive the equally d i s t r i b u t e d equivalent consumption (mean consumption adjusted for i n e q u a l i t y ) . The amount by which the equally d i s t r i b u t e d equivalent consumption f a l l s short of mean consumption i s the amount of consumption per capita wasted on inequality. This i s equal to the value of the absolute inequality index. Absolute inequality can thus range between zero and the mean. The lower bound i s attained when there i s no inequality in the o r i g i n a l d i s t r i b u t i o n or no aversion to inequality ( i . e . G=0). Absolute inequality equal to mean consumption implies that at least one individual receives zero consumption in the o r i g i n a l d i s t r i b u t i o n and that the s o c i a l evaluation function exhibits an i n f i n i t e degree of inequality aversion. Notice f i r s t of a l l that t o t a l inequality measured in the entire panel, A, given in the f i r s t row of Table 2, i s roughly equal to mean annual inequality given in row 6. This should be expected in l i g h t of the often-observed tendency of annual inequality to remain roughly constant over time. If the d i s t r i b u t i o n of annual consumption were i d e n t i c a l year after 79 year, then these two figures would be equal, since the KP indices s a t i s f y the p r i n c i p l e of population r e p l i c a t i o n . It should also be noted that the magnitudes of the decomposition approach subindices of absolute inequality reported in Table 2 display considerable s t a b i l i t y r e l a t i v e to one another. Each of these subindices measures inequality attributable to a particular source of consumption differences. The increase in t o t a l measured inequality as the value of G rises r e f l e c t s the greater s o c i a l significance of a fixed amount of consumption dispersion, objectively measured, at higher degrees of absolute inequality aversion. This i s as expected, and the r e l a t i v e s t a b i l i t y of the subindices may thus be interpreted as an indication that the decomposition approach i s robust. 5 3 The decomposition.of absolute inequality reveals that intrapersonal and intracohort inequality account for most of t o t a l measured inequality. As the degree of absolute inequality aversion (G) r i s e s , the magnitude of intrapersonal inequality r i s e s r e l a t i v e to intracohort inequality, from about 34 per cent when G=5X10" 6 to about 63 per cent when G = 5 X 1 0 " 3 . In a l l cases except maximin, intercohort inequality accounts for only 6 per cent of the t o t a l . The most important comparison to make i s of intracohort inequality with mean annual inequality, for the former, I have argued, measures pure interpersonal inequality, while the l a t t e r 5 3See .Appendix B for a decomposition of annual inequality in which the subindices exhibit extreme variation over a range of the degree of inequality aversion. 80 represents the t r a d i t i o n a l method of measuring inequality among individuals. Since mean annual inequality i s roughly equal to to t a l inequality, which i s equal to the sum of three nonnegative terms, i t should not be surprising to find that, taken i n d i v i d u a l l y , these terms are less than mean annual inequality. In p a r t i c u l a r , intracohort inequality i s 58 to 70 per cent of mean annual inequality. Thus, in th i s data set, consumption mobility and the variation of consumption over the l i f e c y c l e account for between 30 and 42 per cent of measured annual consumption inequality, which has t r a d i t i o n a l l y been interpreted as an index of interpersonal inequality. Annual inequality, that i s , overstates pure interpersonal inequality by 43 to 72 per cent. I turn now to the empirical r e s u l t s for indices of r e l a t i v e inequality presented in Table 3. Relative inequality indices range over a [0,1] i n t e r v a l , their value representing the proportion of t o t a l consumption wasted on inequality. Relative inequality equal to 0.5 means that one-half of t o t a l consumption could be thrown away, or, equivalently, that each individual would need to receive only one-half the mean consumption, in an equal d i s t r i b u t i o n that i s s o c i a l l y equivalent to the o r i g i n a l . Since r e l a t i v e inequality i s equal to absolute inequality normalized on the mean, the conditions for attaining the upper and lower bounds are the same for r e l a t i v e and absolute indices. Zero r e l a t i v e inequality implies an equal o r i g i n a l d i s t r i b u t i o n or zero inequality aversion in the s o c i a l evaluation function. Relative inequality equal to unity i s attained when someone receives zero consumption and the degree of inequality aversion 81 is i n f i n i t e . Total r e l a t i v e inequality, given in the f i r s t row of Table 3, i s decomposed into the sum of i t s three components which are given in rows 2 to 4. Relative magnitudes of the subindices, as percentages of t o t a l inequality, are shown in parentheses. Once again, minimum, maximum, and mean annual r e l a t i v e inequality are given for purposes of comparison. AKS indices are reported for eight values of the degree of r e l a t i v e inequality aversion parameter, and the Gini and Maximin indices have also been computed. Similar patterns emerge here as were seen in Table 2.5<t Total r e l a t i v e inequality i s , as expected, approximately equal to average annual inequality. Of the three components into which t o t a l inequality i s decomposed, intercohort inequality again accounts for the smallest part of t o t a l inequality — in most cases less than 10 per cent. For high degrees of r e l a t i v e inequality aversion (R<-5) intercohort inequality i s about 20 per cent of the t o t a l . Intracohort inequality is the largest of the three subindices of r e l a t i v e inequality for a l l cases. Intrapersonal inequality ranges from less than 28 per cent of intracohort inequality when R=0.9 (indicating a low degree of r e l a t i v e inequality aversion), to about 44 per cent when R=-10. In the case of maximin, intrapersonal inequality i s 61 per cent, 5 t tIn the case of the Gini c o e f f i c i e n t , the r e l a t i v e magnitudes of the per capita and AKS subindices of inequality are i d e n t i c a l because the Gini per capita and AKS inequality indices are both based on the same s o c i a l evaluation function. The same i s true of the maximin indices. The other inequality indices reported in Tables 2 and 3 are based on d i f f e r e n t s o c i a l evaluation functions (see (4.1) and (4.2)). 82 and in the Gini case, 70 per cent, of intracohort inequality. The most interesting comparison i s of intracohort inequality in the decomposition approach with mean annual inequality. It can be seen that, in a l l cases, the index of annual inequality overstates pure interpersonal inequality as measured by intracohort inequality in the decomposition approach. At lower degrees of inequality aversion (R>-1) the magnitude of t h i s overstatement is about one-third but thereafter i t rises monotonically. When R=-10 mean annual inequality exceeds intracohort inequality by 65 per cent. Notice that the problem is worse for the Gini index than any of the AKS indices, and that the maximin index i s the worst of a l l . Annual inequality overstates intracohort inequality by 76 per cent in the case of the Gini index, and by 106.per cent in the case of the maximin. These empirical results indicate the importance of adopting a decomposition approach to the measurement of inequality. To do otherwise -- that i s , t o measure inequality in the d i s t r i b u t i o n of annual consumption — is to ris k seriously overstating the degree of pure interpersonal inequality. Consider now the indices of r e l a t i v e equality reported in Table 4. One should expect certain patterns to emerge because of the fact that indices of r e l a t i v e equality and inequality sum to unity. This i s true of the annual indices (rows 5 to 7) and of the index of t o t a l inequality (row 1), but not of the subindices (rows 2 to 4). The reason for this i s that the bases on which the decomposition indices of inequality and equality express the s o c i a l cost of maldistribution are d i f f e r e n t . The subindices of r e l a t i v e inequality are calculated as the social saving which 83 results from eliminating some inequality, expressed as a proportion of the same base, namely t o t a l consumption in the o r i g i n a l s i t u a t i o n . The base on which subindices of r e l a t i v e equality are constructed, however, changes with each move between successive pairs of reference vectors: the base i s t o t a l consumption in the more unequally d i s t r i b u t e d reference vector. It i s thus necessary to examine separately the empirical results on indices of r e l a t i v e equality. Once again the fact that annual r e l a t i v e equality does not change much from one year to the next shows up in the s i m i l a r i t y between mean annual r e l a t i v e equality and t o t a l equality measured in the panel o v e r a l l . The increase in measured equality that i s expected when the decomposition approach i s employed rather than the t r a d i t i o n a l approach of measuring annual equality i s also borne out by the empirical r e s u l t s . For degrees of inequality aversion between 0.9 and -2 inclus i v e , mean annual equality i s roughly 80 to 100 per cent of intracohort equality. For higher degrees of r e l a t i v e inequality aversion, t h i s proportion f a l l s to less than one-half, and in the case of the maximin index, i t is a mere 22 per cent. What the indices reported in Table 4 lack i s s t a b i l i t y in the r e l a t i v e magnitudes of the decomposition approach subindices which characterized the subindices of absolute and r e l a t i v e inequality. The intrapersonal and intercohort aspects of the d i s t r i b u t i o n share the property of being least unequallly d i s t r i b u t e d . At low degrees of inequality aversion (R>-2), intercohort r e l a t i v e equality i s greater than intrapersonal r e l a t i v e equality. The reverse is true for higher degrees of 8 4 inequality aversion. The results reported in the preceding three tables indicate that concern about the inadequacy of measuring annual inequality i s well j u s t i f i e d . Both mobility- and age-related inequality account for s i g n i f i c a n t portions of t o t a l measured inequality and thus of annual i n e q u a l i t y . 5 5 Intercohort inequality, which r e f l e c t s consumption differences attributable to the age-structure of the population, runs in the neighbourhood of 5 to 10 per cent of the t o t a l . Intrapersonal inequality, which captures the effect of individual consumption mobility, averages about one-quarter to one-third of t o t a l absolute inequality, and about o n e - f i f t h of t o t a l r e l a t i v e inequality. Accounting for these sources of inequality and excluding them from the measurement of pure interpersonal inequality indicates that annually measured inequality overestimates pure interpersonal inequality by at least one-third, often as much as one-half, and in some cases by more than 70 per cent. Clearly, measured annual inequality cannot be r e l i e d upon to provide an accurate assessment of the s o c i a l significance of pure interpersonal inequali ty. There are several variables in the s p e c i f i c a t i o n of the decomposition approach to the measurement of inequality that may d i f f e r from one application of the method to another. It i s important that the decomposition approach be robust to such changes i f i t i s to be judged a r e l i a b l e method of measuring 5 5 T h i s arises because t o t a l and annual inequality are roughly equal, indicating a f a i r l y constant trend of annual inequality. 85 i n e q u a l i t y . I h a v e t h u s i n v e s t i g a t e d t h e s e n s i t i v i t y o f t h e d e c o m p o s i t i o n a p p r o a c h i n d i c e s t o c h a n g e s i n t h e s i z e o f a g e -c o h o r t b r a c k e t s a n d t h e number o f y e a r s o f d a t a u s e d i n t h e c o m p u t a t i o n o f t h e i n d i c e s . F i r s t , a g e - c o h o r t s h a v e b e e n d e f i n e d a n n u a l l y t h u s f a r . T h e n u m b e r o f y e a r s i n c l u d e d i n e a c h a g e - c o h o r t (B) s h o u l d a f f e c t t h e d i v i s i o n o f i n t e r p e r s o n a l i n e q u a l i t y w i t h i n a n d b e t w e e n a g e -c o h o r t s , an i n c r e a s e i n t h e s i z e o f t h e b r a c k e t c a u s i n g i n e q u a l i t y w i t h i n c o h o r t s t o r i s e a t t h e e x p e n s e o f i n t e r c o h o r t i n e q u a l i t y . T h e r e s u l t s r e p o r t e d i n T a b l e s 5 a n d 6 c o n f i r m t h i s ( o n l y i n t r a - a n d i n t e r c o h o r t i n d i c e s a r e r e p o r t e d s i n c e a g e -c o h o r t b r a c k e t s i z e d o e s n o t a f f e c t t o t a l o r i n t r a p e r s o n a l i n e q u a l i t y ) . T a b l e 7 i n d i c a t e s t h a t i n t r a c o h o r t e q u a l i t y v a r i e s i n v e r s e l y , c e t e r u s p a r i b u s , w i t h t h e s i z e o f a g e - c o h o r t b r a c k e t s . T h i s i s a s e x p e c t e d . S i n c e t h e i n d e x o f i n t e r p e r s o n a l r e l a t i v e e q u a l i t y i s e q u a l t o t h e p r o d u c t o f t h e i n t r a - a n d i n t e r c o h o r t i n d i c e s , a n d r e l a t i v e e q u a l i t y i n d i c e s r a n g e o v e r a [ 0 , 1 ] i n t e r v a l , t h e s u b i n d i c e s o f i n t r a - a n d i n t e r c o h o r t e q u a l i t y m u s t h a v e v a l u e s g r e a t e r t h a n t h a t o f t h e i n d e x o f i n t e r p e r s o n a l r e l a t i v e e q u a l i t y . A s t h e w i d t h o f a g e - c o h o r t b r a c k e t s i s i n c r e a s e d , g r e a t e r e m p h a s i s i s p l a c e d on t h e i n t r a c o h o r t c o m p o n e n t o f i n t e r p e r s o n a l r e l a t i v e e q u a l i t y a t t h e e x p e n s e o f t h e i n t e r c o h o r t c o m p o n e n t . When B i s s o l a r g e t h a t e v e r y o n e i s i n c l u d e d i n t h e same c o h o r t , i n t r a c o h o r t a n d i n t e r p e r s o n a l e q u a l i t y c o i n c i d e ( i . e . a l l i n t e r p e r s o n a l e q u a l i t y i s w i t h i n c o h o r t s s i n c e t h e r e i s o n l y one c o h o r t ) . T h u s i n t r a c o h o r t e q u a l i t y must f a l l a s B , t h e number o f y e a r s i n e a c h a g e - c o h o r t , r i s e s . 86 Although there are no r e s t r i c t i o n s on the number of years included in each age-cohort bracket, one, f i v e , and ten years seem the most natural d e f i n i t i o n s . Tables 5 and 6 reveal that the r e l a t i v e magnitudes of i n t r a - and intercohort inequality change f a i r l y smoothly as the size of age-cohort brackets increases, with most of the effect having been f e l t by the time age-brackets span five years. There is only a small ef f e c t when age bracket size i s increased from fiv e to ten years. Notice that t h i s increase in intracohort inequality as age-cohort brackets are widened reduces the discrepancy between the decomposition and t r a d i t i o n a l approaches to measuring inequality. With annual age-cohort brackets the r a t i o of mean annual absolute inequality to intracohort absolute inequality ranges from 1.40 to 1.72. The range of the same r a t i o with fiv e year age-cohort brackets i s 1.31 to 1.60, indicating that the mild quantitative e f f e c t s of widening age-cohort brackets are not s u f f i c i e n t to a l t e r the conclusion regarding the importance of adopting a decomposition approach to the measurement of inequality. I have also investigated the s e n s i t i v i t y of the decomposition indices of equality and inequality to the number of years of data employed in their c a l c u l a t i o n . There are both the o r e t i c a l and empirical reasons for doing t h i s . In the welfare approach, the equally d i s t r i b u t e d equivalent consumption i s a summary s t a t i s t i c of an individual's l i f e c y c l e p r o f i l e . *It is used in the computation of intrapersonal inequality, which r e f l e c t s that part of t o t a l inequality a t t r i b u t a b l e to the shape of consumption p r o f i l e s . But l i f e c y c l e 87 o data are not available, so the decomposition approach i s employed to compute indices and subindices of inequality from panel data. When the number of years of data i s small, however, i t is d i f f i c u l t to claim that an individual's equally d i s t r i b u t e d equivalent consumption r e f l e c t s the shape of his consumption p r o f i l e . In the short run, representative consumption accounts for the effects of mobility, and intrapersonal inequality may be interpreted as an index of mobility-related inequality. These are not dichotomous interpretations, but rather the extremes of a continuous s h i f t in the interpretations of individual representative consumption and intrapersonal inequality indices with the number of years of data. This i s important for my purpose because the behaviour of the decomposition indices may provide information on the r e l a t i v e importance of these two aspects of the intertemporal variation of individual consumption. With only one year of data, of course, representative consumption equals annual consumption and intrapersonal inequality i s zero. As the number of years of data r i s e s , the effects of consumption mobility are increasingly reflected in representative consumption and intrapersonal inequality. If short-run mobility were the only source of intertemporal variation i t should be expected that both representative consumption and the index of intrapersonal inequality would approach l i m i t s : the former to average consumption adjusted for 88 the degree of m o b i l i t y 5 6 and the l a t t e r to the degree of inequality that t h i s intertemporal d i s t r i b u t i o n displays. On the other hand, i f intertemporal v a r i a t i o n in consumption i s of a long-run nature, r e f l e c t i n g the shape of l i f e c y c l e consumption p r o f i l e s , representative consumption would be less l i k e l y to approach a l i m i t . Thus the s e n s i t i v i t y of the index of intrapersonal inequality to the number of years of data may indicate whether the variation in individual consumption over time i s a short-run or long-run phenomenon.57 It may well be that ten years of data are not enough to provide a clear indication of the nature of consumption va r i a t i o n over time, and in any event, i t is l i k e l y that both e f f e c t s are operative, and the d i s t i n c t i o n between them not clear cut. I leave i t open to an examination of the results of t h i s experiment to reveal what they may.. The second purpose for conducting t h i s experiment i s more for reasons of empirical p r a c t i c a l i t y . It may be that the decomposition indices approach l i m i t i n g values s u f f i c i e n t l y c l o s e l y when computed with less than ten years of data. For example, i f the trend of annual inequality is roughly constant, the decomposition approach index of t o t a l inequality w i l l show l i t t l e v a r i a t i o n as the number of years of data increases. If the index of intrapersonal inequality approaches a l i m i t i n g 5 6 T h i s i s exactly analogous to the idea that equally di s t r i b u t e d equivalent consumption is mean consumption adjusted for inequality. 5 7 S i m i l a r l y Shorrocks [1978a, p.389] argues that mobility may occur in either the transitory or permanent.component of t o t a l income. 89 value f a i r l y quickly, then the intracohort inequality index may also display considerable s t a b i l i t y since the index of intercohort inequality i s l i k e l y to show l i t t l e s e n s i t i v i t y to the number of years of d a t a . 5 8 In thi s case considerable data c o l l e c t i o n and computing cost savings could be realized because s u f f i c i e n t l y accurate indices could be produced with less than the f u l l ten years of data. While there is'no reason a p r i o r i to expect decomposition approach indices to approach l i m i t s or display such s t a b i l i t y , I am encouraged by Shorrocks' [1978b] finding that, in some population age-cohorts, his mobility i n dex 5 9 approached a l i m i t when computed with as few as fi v e years of data. The results of my experiment are reported in Tables 8, 9, and 10. Table 8 reveals that t o t a l absolute inequality varies "somewhat with the number of years of data, but within a f a i r l y r e s t r i c t e d range. Its changing value r e f l e c t s differences in annual inequality. Intrapersonal inequality, on the other hand, increases monotonically as the number of years of data i s increased. This i s due to the tendency for real consumption to grow over time and the fact that indices of absolute inequality are not mean independent. Both i n t r a - and intercohort absolute 5 81ntercohort inequality r e f l e c t s the contribution of the age-structure of the population to t o t a l inequality when computed with annual, or only a few years of data. With l i f e c y c l e data i t measures inequality due to economic growth. In either case i t i s not l i k e l y to depend much on the number of years of data used in i t s computation. 5 9 T h i s index is the r a t i o of a weighted average of annual inequality indices to an inequality index of consumption accrued over the entire time period (number of years) for which data are available. 90 inequality tend to decline with increases in the number of years of data used in their computation. Neither appears to approach a l i m i t i n g value. Tables 9 and 10 report the results of the experiment with the ef f e c t of the time span of the data set on indices of r e l a t i v e inequality and equality. Since these indices are mean independent, t o t a l measured (in)equality r e f l e c t s the impact only of the d i s t r i b u t i o n , and not the mean, of each additional year of data. The remarkable s t a b i l i t y displayed by t o t a l measured r e l a t i v e (in)equality lends support to the widely observed tendency of annual (in)equality to remain f a i r l y constant over time. Intrapersonal r e l a t i v e inequality (equality) can be seen to r i s e ( f a l l ) monotonically as the number of years of data i s increased. It i s not possible to reach a conclusion on the nature of individual consumption variation over time on the basis of these r e s u l t s . It c e r t a i n l y supports the view, however, that measuring inequality in an annual d i s t r i b u t i o n i s inadequate. There i s some indication that the subindices of intracohort r e l a t i v e (in)equality do approach a l i m i t within the ten years of the data set. If this i s a general rather than data-specific property, then decomposition approach indices of pure interpersonal inequality could be computed accurately with fewer than ten years of panel d a t a . 6 0 Based only on the current evidence, however, th i s would have to be taken as a very °This property would also prove advantageous to the measurement of the trend of inequality in the decomposition approach since, as i s explained in Chapter Five, a cohort must be excluded from the population for every year of data used in the computation of the decomposition approach index of intracohort inequality. 91 tentative conclusion. The empirical results presented in this chapter w i l l , I hope, contribute to a better understanding of the decomposition approach to the measurement of inequality. The evidence indicates that the decomposition approach provides a considerably clearer picture of the dimensions of inequality by allowing indices of mobility- and age-related inequality to be computed. These indices are of i n t r i n s i c interest since they quantify the e f f e c t s of two sources of'consumption differences on measured inequality. In addition, of course,, they allow an index of pure interpersonal inequality to be calculated. In the d i s t r i b u t i o n taken from the Panel Study on Income Dynamics i t was found that annual inequality overstates pure interpersonal inequality by at least one-third, and in some cases up to 7 5 per cent. Such magnitudes underscore the theoretical arguments in favour of adopting the decomposition approach to the measurement of inequality. The decomposition of inequality within and among age-cohorts was shown, as expected, to depend on the width of cohort brackets. Nevertheless, the quantitative effects are not s u f f i c i e n t l y large to undermine the conclusion regarding the superiority of the decomposition approach over the t r a d i t i o n a l practice of measuring annual inequality. The investigation of the s e n s i t i v i t y of decomposition approach indices to the number of years of data suggested reasonably accurate results for r e l a t i v e (in)equality might be obtained with fewer than ten years of data. Further empirical work i s required to substantiate t h i s tentative conclusion. 92 The decomposition approach to the measurement of inequality thus seems to be an accurate, e f f i c i e n t , and robust method of measuring inequality that can successfully be employed to solve several long-standing problems of inequality measurement. The effects of consumption mobility and the age-structure of the population confuse not only the measurement of the le v e l of inequality, however, but also i t s trend. An accurate assessment of "trend" inequality c l e a r l y depends on careful and correct measurement of " s t a t i c " inequality. The decomposition approach index of intracohort inequality provides such a measure of pure interpersonal inequality. In the next chapter I apply the decomposition approach to the problem of determining the trend of inequality. 93 m i d d l e o f i n t e r v a l 5 0 0 . 1 5 0 0 . 2 5 0 0 . 3 5 0 0 . 4 5 0 0 . 5 5 0 0 . 6 5 0 0 . 7 5 0 0 . 8 5 0 0 . 9 5 0 0 . 1 0 5 0 0 . 1 1 5 0 0 . 1 2 5 0 0 . 1 3 5 0 0 . 1 4 5 0 0 . 1 5 5 0 0 . 1 6 5 0 0 . 1 7 5 0 0 . 1 8 5 0 0 . 1 9 5 0 0 . 2 0 5 0 0 . 21 5 0 0 . 2 2 5 0 0 . 2 3 5 0 0 . 2 4 5 0 0 . 2 5 5 0 0 . 2 6 5 0 0 . 27-500. 2 8 5 0 0 . 2 9 5 0 0 . 3 0 5 0 0 . 31 5 0 0 . 3 2 5 0 0 . 3 3 5 0 0 . 3 4 5 0 0 . 3 5 5 0 0 . 3 6 5 0 0 . 3 7 5 0 0 . 3 8 5 0 0 . 3 9 5 0 0 . 4 0 5 0 0 . 4 1 5 0 0 . 4 2 5 0 0 . 4 3 5 0 0 . 4 4 5 0 0 . 4 5 5 0 0 . 4 6 5 0 0 . 4 7 5 0 0 . 4 8 5 0 0 . 4 9 5 0 0 . 5 0 5 0 0 . 5 1 5 0 0 . number o f o b s . 972 4743 5803 5055 41 53 2699 1 633 1 230 71 1 452 316 217 1 22 71 53 39 35 21 8 1 5 22 10 1 1 5 1 4 3 0 3 0 0 2 1 0 0 2 0 2 0 0 0 0 0 0 0 1 F I G U R E I I I H i s t o g r a m . o f t h e P a n e l D a t a * * * * * * * * * **************************************** *********************************************** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *********************** ************** *********** ****** **** * * * ** ** * * * * * * * * * * * * * * * * * * * * * * e a c h * r e p r e s e n t s 120 o b s e r v a t i o n s 94 T A B L E 1 A n n u a l M e a n s a n d S t a n d a r d D e v i a t i o n s o f t h e P a n e l D a t a YEAR MEAN STANDARD D E V I A T I O N 1 9 6 8 3 4 8 0 . 4 2 3 1 7 . 3 1 9 6 9 3 6 2 6 . 2 2 3 7 0 . 6 1 9 7 0 3 7 6 1 . 1 2 4 8 7 . 5 1 971 3 9 2 2 . 3 2 6 2 0 . 1 1 9 7 2 4 1 9 1 . 2 2 7 1 4 . 3 1 9 7 3 4 4 0 2 . 2 2 8 1 9 . 2 1 9 7 4 4 3 4 1 .3 2 9 5 3 . 1 1 9 7 5 4 3 0 0 . 3 2 9 8 9 . 3 1 9 7 6 4 5 0 4 . 6 3 0 4 3 . 9 1 9 7 7 4 5 4 9 . 9 2 8 9 6 . 1 p o o l e d 4 0 9 1 .1 2741 .4 95 A A A F V m i n mean max G = 5 x 1 0 " 6 18.61 4 . 4 9 ( 2 4 . 1 ) 1 3 . 0 5 ( 7 0 . 1 ) 1 . 0 7 ( 5 . 7 ) 1 3 . 2 9 1 8 . 2 9 2 2 . 8 7 T A B L E 2 I n d i c e s o f A b s o l u t e I n e q u a l i t y D E C O M P O S I T I O N I N D I C E S G = 5 x 1 0 " 5 1 6 9 . 1 7 4 2 . 8 8 ( 2 5 . 3 ) 1 1 8 . 5 4 ( 7 0 . 1 ) G = 1 x 1 0 - 4 3 1 1 . 1 2 8 2 . 0 0 ( 2 6 . 4 ) 2 1 6 . 4 3 ( 6 9 . 6 ) 7 . 7 5 1 2 . 6 9 ( 4 . 6 ) ( 4 . 1 ) ANNUAL I N D I C E S 1 2 3 . 1 6 1 6 6 . 3 2 2 0 7 . 7 9 2 3 0 . 13 3 0 6 . 12 381 . 4 9 G = 5 x 1 0 " * 1 0 4 0 . 5 6 3 1 6 . 7 6 ( 3 0 . 4 ) 6 8 4 . 9 5 ( 6 5 . 8 ) 3 8 . 8 5 ( 3 . 7 ) 8 1 3 . 7 8 1 0 2 7 . 8 2 1 2 4 3 . 9 8 A P 4 A c G I N I 1 3 6 2 . 5 2 5 3 6 . 8 5 ( 3 9 . 4 ) 7 6 7 . 0 3 ( 5 6 . 3 ) 5 8 . 6 3 ( 4 . 3 ) m i n 1 1 7 1 . 4 3 mean 1 3 4 7 . 9 0 max 1 5 2 7 . 6 8 D E C O M P O S I T I O N I N D I C E S G = 5 x 1 0 " 3 G = 1 x 1 0 " 3 1 5 4 4 . 4 6 5 0 3 . 7 5 ( 3 2 . 6 ) 9 7 7 . 9 3 ( 6 3 . 3 ) 6 2 . 7 8 ( 4 . 1 ) 2 7 6 2 . 0 1 1 0 0 9 . 4 7 ( 3 6 . 5 ) 1 5 9 1 . 8 9 ( 5 7 . 6 ) 1 6 0 . 6 5 ( 5 . 8 ) ANNUAL I N D I C E S 1 2 3 5 . 9 7 1 5 2 8 . 9 7 1 8 1 1 . 7 0 2 2 5 2 . 4 6 2 7 4 5 . 1 5 3 2 4 2 . 13 MAXIMIN 3 9 3 4 . 2 2 1101.81 ( 2 8 . 0 ) 1 8 1 3 . 8 3 ( 4 6 . 1 ) 1 0 1 8 . 5 8 ( 2 5 . 9 ) 3 1 0 0 . 6 0 3 7 3 5 . 1 7 4 3 5 9 . 2 9 96 AP Rc R = . 9 . 0 1 8 5 . 0 0 3 8 ( 2 0 . 5 ) . 0 1 3 6 ( 7 3 . 5 ) .001 1 ( 5 . 9 ) m i n . 0 1 6 8 mean .0181 max . 0 1 9 0 T A B L E 3 I n d i c e s o f R e l a t i v e I n e q u a l i t y D E C O M P O S I T I O N I N D I C E S R = . 5 R=0 R = - . 5 . 0 8 9 9 . 1 7 4 7 . 2 5 5 3 . 0 1 8 7 . 0 3 7 2 . 0 5 5 4 ( 2 0 . 8 ) ( 2 1 . 3 ) ( 2 1 . 7 ) . 0 6 6 5 . 1 2 8 7 .1861 ( 7 4 . 0 ) ( 7 3 . 7 ) ( 7 2 . 9 ) . 0 0 4 8 . 0 0 8 8 . 0 1 3 7 ( 5 . 3 ) ( 5 . 0 ) ~ ( 5 . 4 ) ANNUAL I N D I C E S . 0 8 1 6 . 1 5 8 0 .2301 . 0 8 8 0 .1713 . 2 5 0 3 . 0 9 2 4 . 1 7 9 0 . 2 6 1 7 GINI . 3 3 3 0 .1312 ( 3 9 . 4 ) . 1 8 7 5 ( 5 6 . 3 ) .01 43 ( 4 . 3 ) . 3 1 7 0 . 3 2 9 5 .3391 AP Ac R c R=-1 ' .3321 . 0 7 2 9 ( 2 2 . 0 ) .2391 ( 7 2 . 0 ) . 0 2 0 2 ( 6 . 1 ) m i n . 2 9 7 8 mean . 3 2 6 0 max . 3 3 9 4 D E C O M P O S I T I O N I N D I C E S R=-2 R = - 5 R=-10 .4761 . 7 9 5 6 . 9 0 7 6 .1041 . 1 6 8 8 . 2 2 3 6 ( 2 1 . 9 ) ( 2 1 . 2 ) ( 2 4 . 6 ) . 3 3 1 0 . 4 6 4 0 . 5 0 7 9 ( 6 9 . 5 ) ( 5 8 . 3 ) ( 5 6 . 0 ) .0411 . 1 6 2 9 . 1 7 6 0 ( 8 . 6 ) ( 2 0 . 5 ) ( 1 9 . 4 ) ANNUAL I N D I C E S . 4 1 8 2 . 6 4 4 4 . 7 4 8 7 . 4 6 9 2 . 7 3 7 0 . 8 3 9 5 . 5 1 2 8 . 8 6 6 4 . 9 3 0 5 MAXIMIN . 9 6 1 6 . 2 6 9 3 ( 2 8 . 0 ) . 4 4 3 4 ( 4 6 . 1 ) . 2 4 9 0 ( 2 5 . 9 ) . 8 5 1 6 . 9 1 3 0 . 9 6 3 9 97 T A B L E 4 I n d i c e s o f R e l a t i v e E q u a l i t y DECOMPOSITION I N D I C E S R = . 9 R = . 5 R=0 R = - 0 . 5 G I N I E .981 5 .9101 . 8 2 5 3 . 7 4 4 7 . 6 6 7 0 EAP . 9 9 6 2 . 9 8 1 3 . 9 6 2 8 . 9 4 4 6 . 8 6 8 8 EAC . 9 8 6 4 . 9 3 2 3 . 8 6 6 3 . 8 0 2 9 . 7 8 4 2 E R C . 9 9 8 9 . 9 9 4 8 . 9 8 9 4 . 9 8 1 9 . 9 7 9 0 ANNUAL I N D I C E S m i n . 9 8 1 0 . 9 0 7 6 . 8 2 1 0 . 7 3 8 2 . 6 6 0 9 mean . 9 8 1 9 . 9 1 2 0 . 8 2 8 7 . . 7 4 9 7 . 6 7 0 5 max . 9 8 3 2 . 9 1 8 4 . 8 4 2 0 . 7 6 9 9 . 6 8 3 0 D E C O M P O S I T I O N I N D I C E S R=-1 R=-2 R=-5 R=-1 0 MAXIMI1 E . 6 6 7 9 . 5 2 3 9 . 2 0 4 4 . 0 9 2 4 . 0 3 8 4 EAP .9271 . 8 9 5 9 . 8 3 1 2 . 7 7 6 4 . 7 3 0 7 EAC . 7 4 2 2 . 6 3 0 6 .441 9 . 3 4 5 8 . 3 9 3 2 ERC . 9 7 0 9 • . 9 2 7 3 . 5 5 6 4 . 3 4 4 3 . 1 335 ANNUAL I N D I C E S m i n . 6 6 0 6 . 4 8 7 2 . 1 3 3 6 . 0 6 9 5 .0361 mean . 6 6 0 6 . 5 3 0 8 . 2 6 3 0 . 1 6 0 5 . 0 8 7 0 max . 7 0 2 2 . 5 8 1 8 . 3 5 5 6 . 2 5 1 3 . 1 484 98 T A B L E 5 T h e E f f e c t o f A g e - c o h o r t B r a c k e t S i z e on I n t r a - a n d I n t e r c o h o r t A b s o l u t e I n e q u a l i t y NUMBER OF YEARS IN EACH A G E - C O H O R T BRACKET B=1 B=3 B=5 B=10 5 x 1 0 " 6 A u 1 3 . 0 5 1 3 . 7 3 1 3 . 9 4 1 3 . 9 5 A ? ; 1 .07 . 3 8 .18 .1 6 5 x 1 0 ' 5 A A C 1 1 8 . 5 4 1 2 3 . 2 5 1 2 4 . 9 7 1 2 5 . 0 7 A R C 7 . 7 5 3 . 0 4 1.31 1.22 1 x 1 0 - * A A C 2 1 6 . 4 3 2 2 3 . 8 3 2 2 6 . 9 8 2 2 7 . 1 9 A R c 1 2 . 6 9 5 . 2 9 2 . 1 5 1.93 5 x 1 0 - * A A C 6 8 4 . 9 5 7 0 3 . 8 3 7 1 6 . 2 4 7 1 7 . 6 0 A R C 3 8 . 8 5 1 9 . 9 7 7 . 5 6 6 . 2 0 G I N I A ^ 7 6 7 . 0 3 7 9 6 . 2 3 8 0 8 . 3 8 8 1 1 . 1 4 A , ^ 5 8 . 6 3 2 9 . 4 4 1 7 . 2 8 1 4 . 5 3 1 x 1 0 " 3 AA c_ 9 7 7 . 9 3 1 0 0 5 . 6 3 1 0 2 6 . 1 6 1.028.78 A , ^ 6 2 . 7 8 3 5 . 0 8 1 4 . 5 5 1 1 . 9 4 5 X 1 0 " 3 A A c 1 5 9 1 . 8 9 1 6 7 6 . 1 9 1 7 1 7 . 3 2 1 7 2 3 . 7 6 A , ^ 1 6 0 . 6 5 7 6 . 3 6 3 5 . 2 3 2 8 . 7 9 MAXIMIN A ^ 1 8 1 3 . 8 3 2 0 8 1 . 1 9 2 2 8 8 . 4 9 2 3 4 8 . 1 9 A ^ 1 0 1 8 . 5 8 7 1 5 . 6 2 5 5 3 . 0 5 5 2 7 . 5 3 99 T A B L E 6 T h e E f f e c t o f A g e - c o h o r t B r a c k e t S i z e on I n t r a - a n d I n t e r c o h o r t R e l a t i v e I n e q u a l i t y R NUMBER OF YEARS IN E A C H A G E - C O H O R T BRACKET B=1 B=3 B=5 B=10 . 9 - . 5 G I N I -1 - 2 - 5 - 1 0 MAXIMIN Ac Rc Ac RC AC Rc Ac AC RC Ac RC AC RC Ac Rc Ac RC AC 0136 001 1 . 0 6 6 5 . 0 0 4 8 . 1 2 8 7 . 0 0 8 8 .1861 .01 37 . 1 8 7 5 . 0 1 4 3 .2391 . 0 2 0 2 .331 0 .041 1 . 4 6 4 0 . 1 6 2 9 . 5 0 7 9 . 1760 . 4 4 3 4 . 2 4 9 0 . 0 1 4 3 . 0 0 0 4 . 0 6 9 3 . 0 0 2 0 . 1 335 . 0 0 4 0 . 1 931 . 0 0 6 8 . 1 946 . 0 0 7 2 . 2 4 8 7 . 0 1 0 5 . 3 4 9 4 . 0 2 2 7 . 5 0 8 3 . 1 187 . 5 5 4 6 . 1293 . 5 0 8 7 . 1 7 4 9 01 45 0002 . 0 7 0 5 . 0 0 0 7 1362 001 4 1974 0024 1976 0042 2551. 0041 . 3 6 2 6 . 0 0 9 5 .5391 . 0 8 7 8 . 5 8 1 9 . 1 0 2 0 . 5 5 9 4 . 1 3 5 2 .01 45 . 0 0 0 2 . 0 7 0 6 . 0 0 0 7 . 1 364 .001 1 . 1 9 7 9 .001 9 . 1 9 8 3 . 0 0 3 6 .2561 .0031 . 3 6 5 2 . 0 0 6 8 . 5 4 5 2 . 0 8 1 7 . 5 8 9 0 . 0 9 5 0 . 5 7 4 0 . 1 2 8 9 100 T A B L E 7 T h e E f f e c t o f A g e - c o h o r t B r a c k e t S i z e on I n t r a - a n d I n t e r c o h o r t R e l a t i v e E q u a l i t y NUMBER OF YEARS IN E A C H A G E - C O H O R T BRACKET B=1 B=3 B=5 B=1 0 E A o . 9 8 6 4 E ^ . 9 9 8 9 9857 9995 9854 9998 . 9 8 5 4 . 9 9 9 8 J Ac 3 Rc . 9 3 2 3 . 9 9 4 8 . 9 2 9 4 . 9 9 7 9 9282 9992 .9281 . 9 9 9 3 2 R C . 8 6 6 3 . 9 8 9 4 . 8 6 1 3 . 9 9 5 2 8586 ,9984 . 8 5 8 3 . 9 9 8 6 - . 5 . 8 0 2 9 . 9 8 1 9 . 7 9 5 6 . 9 9 0 9 791 0 ,9967 . 7 9 0 4 . 9 9 7 5 G I N I E Ac 7842 9790 . 7 7 6 0 . 9 8 9 3 7726 9937 .7718 . 9 9 4 7 -1 7422 9707 . 73 18 . 9 8 4 5 7248 9939 . 7 2 3 8 . 9 9 5 3 - 2 J«c 6306 9273 . 6 1 0 0 . 9 8 5 6 5953 9823 . 5 9 2 4 .9871 - 5 E A C E * C 441 9 5564 . 3 8 8 6 . 6 2 3 8 351 5 6 9 9 5 .3441 . 7 1 4 5 - 1 0 ; A C he 3458 3443 . 2 8 5 7 .41 68 2505 4754 .241 4 . 4 9 3 3 MAXIMIN E A c . 3 9 3 2 E R c . 1 3 3 5 . 2 9 5 4 . 1 7 9 8 2368 2210 . 2 2 5 7 . 2 2 9 2 101 T A B L E 8 T h e E f f e c t o f Number o f Y e a r s o f D a t a on I n d i c e s a n d S u b i n d i c e s o f A b s o l u t e I n e q u a l i t y NUMBER OF YEARS OF DATA 5 x 1 0 - " 4 5 6 7 A 173. .25 1 6 7 . 16 1 7 1 . 6 4 165.91 A A P 26, (15, .83 . 5 ) 2 9 . ( 1 7 . 98 9) 3 4 . 5 0 ( 2 0 . 1 ) 3 6 . 6 5 ( 2 2 . 1 ) A A c 1 36, (78, .12 . 6 ) 1 2 7 . ( 7 6 . 30 2) 1 2 7 . 5 0 ( 7 4 . 3 ) 1 2 0 . 4 3 ( 7 2 . 6 ) kRc 10 (5 . 3 0 . 9 ) 9 . ( 5 . 88 9) 9 . 6 3 ( 5 . 6 ) 8 . 8 2 ( 5 . 3 ) mean 1 72 . 5 0 1 6 5 . 85 1 7 0 . 4 6 1 6 4 . 1 8 A 1 055 .74 1 0 2 3 . 71 1 0 4 0 . 8 4 1 0 1 6 . 1 6 A A F 203 (19 . 6 2 . 3 ) 2 2 4 . (21 . 33 9) 2 5 0 . 4 4 • ( 2 4 . 1 ) 2 6 6 . 5 2 ( 2 6 . 2 ) A Ac 801 (75 . 2 0 . 9 ) 7 5 2 . ( 7 3 . 30 5) 7 4 3 . 5 3 ( 7 1 . 4 ) 7 0 7 . 5 1 ( 6 9 . 6 ) A R c 50 (4 .91 . 8 ) 4 7 . ( 4 . 07 6) 4 6 . 8 7 ( 4 . 5 ) 4 2 . 13 ( 4 . 1 ) mean 1 052 . 2 6 1017. 70 1 0 3 5 . 6 0 1 0 0 8 . 1 0 A 2863 . 7 4 2 7 6 9 . 26 2 7 8 3 . 5 0 2 7 2 3 . 8 7 A A P 628 (21 . 3 7 . 9 ) 7 1 2 . ( 2 5 . 92 7) 7 9 5 . 2 6 ( 2 8 . 6 ) 8 4 2 . 9 6 ( 3 0 . 9 ) V 1 964 (68 . 2 2 . 6 ) 1 8 2 3 . ( 6 5 . ,33 8) 1 7 6 1 . 7 2 ( 6 3 . 3 ) 1 6 9 0 . 9 6 ( 6 2 . 1 ) A R C 271 (9 .15 . 5 ) 2 3 3 . ( 8 . ,01 ,4) 2 2 6 . 5 2 ( 8 . 1 ) 1 8 9 . 9 5 ( 7 . 0 ) mean 2845 . 3 4 2 7 5 4 . ,60 2 7 6 7 . 0 8 2 7 0 6 . 5 3 1 02 T A B L E 8 (CON'T) NUMBER OF Y E A R S OF DATA 8 9 10 171 . 21 1 6 6 . 46 1 6 9 . 1 7 3 9 . ( 2 3 . 42 0) 4 0 . ( 2 4 . 62 4) 4 2 . ( 2 5 . 88. 3) 1 2 3 . ( 7 2 . 34 0) 1 1 7 . ( 7 0 . 83 8) 1 1 8 . ( 7 0 . 54 1 ) 8. ( 4 . 45 9) 8. ( 4 . 02 8) 7. ( 4 . 7 5 6 ) 1 6 9 . 28 1 6 3 . 84 1 6 6 . 32 1 0 4 3 . 90 1 021 . 37 1 0 4 0 . 56 2 8 9 . (.27. 64 7) 2 9 8 . ( 2 9 . 08 2) 3 1 6 . ( 3 0 . 76 4 ) 7 1 3 . ( 6 8 . 62 4) 6 8 4 . ( 6 7 . 6.1 0) 6 8 4 . (6.5. 9 5 8 ) 4 0 . ( 3 . 64 9) 3 8 . ( 3 . 67 8) 3 8 . ( 3 . 8 5 7 ) 1 0 3 5 . 70 1 0 0 9 . 53 1 0 2 7 . 82 2 7 7 0 . 23 2 7 1 3 . 71 2 7 6 2 . 01 911 . ( 3 2 . 85 9) 9 5 0 . ( 3 5 . 68 0) 1 0 0 9 . ( 3 6 . 47 5) 1 6 7 5 . ( 6 0 . 94 5) 1 601 . ( 5 9 . 1 1 0) 1591 . ( 5 7 . 8 9 6 ) 1 8 2 . ( 6 . 45 6) 1 6 1 . ( 6 . 91 0) 1 6 0 . ( 5 . 65 8 ) 2 7 5 4 . .48 2 6 9 5 . ,28 2 7 4 5 . ,15 103 T A B L E 9 T h e E f f e c t o f Number o f Y e a r s o f D a t a on I n d i c e s a n d S u b i n d i c e s o f R e l a t i v e I n e q u a l i t y NUMBER OF YEARS OF DATA 2 3 4 5 6 I . 0 8 4 3 . 0 8 5 2 . 0 8 6 5 . 0 8 7 0 . 0 8 7 7 . 0 0 6 0 ( 7 . 1 ) . 0 0 9 6 ( 1 1 . 3 ) . 0 1 0 9 ( 1 2 . 6 ) . 0 1 2 8 ( 1 4 . 7 ) .01 40 ( 1 6 . 0 ) . 0 7 2 9 ( 8 6 . 5 ) . 0 6 9 6 ( 8 1 . 7 ) . 0 6 9 6 ( 8 0 . 5 ) . 0 6 8 4 ( 7 8 . 6 ) .0681 ( 7 7 . 7 ) . 0 0 5 4 ( 6 . 4 ) . 0 0 6 0 ( 7 . 0 ) . 0 0 5 9 ( 6 . 8 ) . 0 0 5 8 ( 6 . 7 ) . 0 0 5 7 ( 6 . 5 ) mean .0841 . 0 8 4 7 . 0 8 6 0 . 0 8 6 2 . 0 8 7 0 I .241 5 . 2 4 3 0 . 2 4 5 6 . 2 4 6 5 . 2 4 7 2 : A P . 0 1 7 5 ( 7 . 2 ) . 0 2 7 8 ( 1 1 . 4 ) . 0 3 1 9 ( 1 3 . 0 ) . 0 3 7 2 ( 1 5 . 1 ) . 0 4 0 8 ( 1 6 . 5 ) : A c . 2 0 7 8 ( 8 6 . 0 ) .1981 • ( 8 1 . 5 ) . 1 9 7 0 ( 8 0 . 2 ) . 1 935 ( 7 8 . 5 ) . 1 906 ( 7 7 . 1 ) . 0 1 6 3 ( 6 . 7 ) .0171 ( 7 . 0 ) .01 66 ( 6 . 8 ) . 0 1 5 8 ( 6 . 4 ) -.01 58 ( 6 . 4 ) mean . 2 4 0 6 .241 3 . 2 4 4 2 . 2442 . 2 4 5 3 I .7141 . 7 6 8 7 .8251 . 8 1 3 2 . 8 0 8 3 XAP . 0 5 0 6 ( 7 . 1 ) . 0 7 9 2 ( 1 0 . 3 ) . 0 9 4 9 ( 1 1 . 5 ) .1114 ( 1 3 . 7 ) . 1 2 4 2 ( 1 5 . 4 ) he . 5 0 7 2 ( 7 1 . 0 ) .4981 ( 6 4 . 8 ) .4961 ( 6 0 . 1 ) .481 9 ( 5 9 . 3 ) . 4 6 8 3 ( 5 7 . 9 ) . 1 5 6 2 ( 2 1 . 9 ) . . 1 9 1 5 ( 2 4 . 9 ) .2341 ( 2 8 . 4 ) . 2 1 9 7 ( 2 7 . 0 ) . 2 1 5 7 ( 2 6 . 7 ) mean . 6 9 3 3 . 7 2 5 2 . 7 6 0 7 . 7 3 8 8 . 7 2 5 0 1 04 T A B L E 9 ( C O N ' T ) NUMBER OF YEARS OF DATA R 7 8 9 10 I . 0 8 8 4 .0891 . 0 9 0 0 . 0 8 9 9 . 0 1 5 6 ( 1 7 . 6 ) . 0 1 6 5 ( 1 8 . 5 ) . 0 1 7 9 ( 1 9 . 9 ) . 0 1 8 7 ( 2 0 . 8 ) . 0 6 7 4 ( 7 6 . 2 ) . 0 6 7 5 ( 7 5 . 8 ) .0671 ( 7 4 . 6 ) . 0 6 6 5 ( 7 4 . 0 ) . 0 0 5 4 ( 6 . 1 ) .0051 ( 5 . 7 ) . 0 0 5 0 ( 5 . 6 ) . 0 0 4 8 ( 5 . 3 ) mean . 0 8 7 2 . 0 8 7 9 . 0 8 8 2 . 0 8 8 0 I . 2 4 9 8 .251 7 . 2 5 4 6 . 2 5 5 3 JAP . 0 4 5 2 ( 1 8 . 1 ) . 0 4 8 4 ( 1 9 . 2 ) . 0 5 2 9 ( 2 0 . 8 ) . 0 5 5 4 ( 2 1 . 7 ) - • 5 I Ac . 1 8 9 8 ( 7 6 . 0 ) .1891 ( 7 5 . 1 ) . 1878 ( 7 3 . 8 ) . 1861 ( 7 2 . 9 ) he . 0 1 4 8 ( 5 . 9 ) .0141 ( 5 . 6 ) . 0 1 3 9 ( 5 . 5 ) .01 37 ( 5 . 4 ) mean . 2 4 6 7 . 2 4 8 4 . 2 4 9 8 . 2 5 0 3 I . 7 9 9 4 . 8 0 1 0 . 7 9 3 7 . 7 9 5 6 IAP . 1373 ( 1 7 . 2 ) . 1 4 7 7 ( 1 8 . 4 ) . 1 598 ( 2 0 . 1 ) . 1 6 8 8 ( 2 1 . 2 ) . 4 6 7 5 ( 5 8 . 5 ) . 4 6 4 5 ( 5 8 . 0 ) .4601 ( 5 8 . 0 ) . 4 6 4 0 ( 5 8 . 3 ) . 1 9 4 6 ( 2 4 . 3 ) . 1 887 ( 2 3 . 6 ) . 1 7 3 8 ( 2 1 . 9 ) . 1 6 2 9 ( 2 0 . 5 ) mean . 7 2 0 5 . 7 3 2 0 . 7 2 8 4 . 7 3 7 0 105 T A B L E 10 T h e E f f e c t o f Number o f Y e a r s o f D a t a on I n d i c e s a n d S u b i n d i c e s o f R e l a t i v e E q u a l i t y NUMBER OF YEARS OF DATA 2 3 4 5 6 E . 9 1 5 7 . 9 1 4 8 . 9 1 3 5 .91 30 . 9 1 2 3 E A 0 . 9 9 4 0 . 9 9 0 4 . 9 8 9 1 . 9 8 7 2 . 9 8 6 0 E A c . 9 2 6 7 . 9 2 9 7 . 9 2 9 6 . 9 3 0 7 . 9 3 1 0 V .9941 . 9 9 3 5 . 9 9 3 6 . 9 9 3 7 . 9 9 3 8 mean ..9159 .91 53 .91 40 .91 38 . 9 1 3 0 E . 7 5 8 5 . 7 5 7 0 . 7 5 4 4 . 7 5 3 5 . 7 5 2 8 E A P . 9 8 2 5 . 9 7 2 2 . 9 6 8 1 . 9 6 2 8 . 9 5 9 2 E A c . 7 8 8 5 . 7 9 6 2 . 7 9 6 5 .7991 . 8 0 1 3 E R c . 9 7 9 0 . 9 7 7 9 . 9 7 8 4 . 9 7 9 4 . 9 7 9 4 mean . 7 5 9 4 . 7 5 8 7 . 7 5 5 8 . 7 5 5 8 . 7 5 4 7 E . 2 8 5 9 . 2 3 1 3 . 1 7 4 9 . 1 8 6 8 . 1 9 1 7 E A P . 9 4 9 4 . 9 2 0 8 . 9 0 5 1 . 8 8 8 6 . 8 7 5 8 E A c . 4 6 5 7 .4591 . 4 5 1 9 . 4 5 7 7 . 4 6 5 3 E R C . 6 4 6 7 .5471 . 4 2 7 7 . 4 5 9 2 . 4 7 0 6 mean . 3 0 6 7 . 2 7 4 8 . 2 3 9 3 . 2 6 1 2 . 2 7 5 0 1 06 T A B L E 10 N U M B E R O F ( C O N ' T ) Y E A R S O F D A T A 7 8 9 10 E . 9 1 1 6 . 9 1 0 9 . 9 1 0 0 .9101 EAP . 9 8 4 4 . 9 8 3 5 .9821 . 9 8 1 3 E A c .931 5 . 9 3 1 4 . 9 3 1 7 . 9 3 2 3 .9941 . 9 9 4 4 . 9 9 4 5 . 9 9 4 8 mean .9 1 2 8 .9121 . 9 1 1 8 . 9 1 2 0 E . 7 5 0 2 . 7 4 8 3 . 7 4 5 4 . 7 4 4 7 EAF . 9 5 4 8 . 9 5 1 6 .9471 . 9 4 4 6 . 8 0 1 3 . 8 0 1 2 . 8 0 1 7 . 8 0 2 9 . 9 8 0 6 . 9 8 1 5 . 9 8 1 7 . 9 8 1 9 mean . 7 5 3 3 .751 6 . 7 5 0 2 . 7 4 9 7 E . 2 0 0 6 . 1 9 9 0 . 2 0 6 3 . 2 0 4 4 EAp . 8 6 2 7 . 8 5 2 3 . 8 4 0 2 .831 2 E A c .4581 . 4 5 4 9 . 4 5 2 4 . 4 4 1 9 E R C . 5 0 7 5 . 5 1 3 3 . 5 4 2 8 . 5 5 6 4 mean . 2 7 9 5 . 2 6 8 0 .271 6 . 2 6 3 0 1 07 CHAPTER FIVE The Trend Of L i f e c y c l e Inequality One of the s t y l i z e d facts about inequality i s that i t has remained v i r t u a l l y constant in the post-World-War II period. A l l the c r i t i c i s m s that have been leveled at the practice of measuring inequality in the d i s t r i b u t i o n of annual income, however, apply to the determination of the trend of inequality as well. An accurate assessment of the trend of inequality presumes an accurate measure of " s t a t i c " 6 1 inequality. The development and i l l u s t r a t i o n of the l a t t e r have occupied the preceding three chapters. In t h i s chapter I consider the trend of l i f e c y c l e inequality. As the s t a t i c measurement of l i f e c y c l e inequality d i f f e r s from that of annual inequality, so does the trend of l i f e c y c l e inequality d i f f e r from i t s annual counterpart. When inequality is measured in the d i s t r i b u t i o n of annual income, i t s trend i s simply the time series of the annual values of the index; i t changes from year to year as individual incomes change. But the d i s t r i b u t i o n of representative l i f e c y c l e consumption, in which l i f e c y c l e inequality i s measured, i s independent of time; i t changes only as the population changes. Of course, changes in 6 1By " s t a t i c " I mean inequality measured in a given d i s t r i b u t i o n , without allowing for any change to occur. For example, annual inequality i s a " s t a t i c " measure, as are the indices reported in Chapter Four despite the fact that they measure l i f e c y c l e inequality. 108 the population occur over time, as individuals begin and end their (economic) l i v e s , but in a l i f e c y c l e context a calendar year serves only to identify a population. Inequality i s measured in the d i s t r i b u t i o n of representative l i f e c y c l e consumption over the members of thi s population. Thus, although l i f e c y c l e inequality may be said to evolve over time, i t s trend is an interpersonal or interpopulation, rather than an intertemporal, phenomenon. Measuring the trend of l i f e c y c l e inequality thus requires that measured inequality in s o c i a l states with d i f f e r e n t populations can meaningfully be compared. It w i l l therefore be necessary to replace a single s o c i a l evaluation function or inequality index with a family of such functions or indices having one member for each possible population s i z e . As a consequence, some means of li n k i n g together members of a family w i l l be required to ensure that a l l functions or indices in a family r e f l e c t the same set of e t h i c a l judgments. The chapter begins with a discussion of the nature of the trend of l i f e c y c l e inequality and the theoreti c a l implications for fixed population s o c i a l evaluation functions. I then consider the construction of variable population s o c i a l evaluation functions and the e t h i c a l judgments that can be b u i l t into them, p a r t i c u l a r l y as regards the evaluation of inequality in d i f f e r e n t s o c i a l states. There follows a th e o r e t i c a l discussion of the trend of inequality in the welfare and decomposition approaches to the measurement of inequality, and, in the l a t t e r case, presentation of empirical evidence on the trend of l i f e c y c l e inequality. The chapter closes with a 109 comparison of these results with the trend of annual inequality and some concluding comments. In addition to " s t a t i c " inequality, investigators may wish to examine the trend of inequality. When inequality is measured in an annual d i s t r i b u t i o n , i t s trend i s reflected in the time path of the inequality index, which changes year after year as i n d i v i d u a l incomes change. The trend of l i f e c y c l e inequality i s e s s e n t i a l l y d i f f e r e n t , however, because representative l i f e c y c l e consumption is a summary s t a t i s t i c of the l e v e l of well-being that an individual i s afforded by his l i f e c y c l e consumption p r o f i l e ; i t attaches to an individual independently of time and so does not change from one year to the next. The d i s t r i b u t i o n of representative l i f e c y c l e consumption changes only when the population under investigation changes. L i f e c y c l e inequality is time-dependent only in the sense that i t is over time that changes occur in the population. If the population were unchanged from one year to the next, so would be measured l i f e c y c l e i n e q u a l i t y . 6 2 T y p i c a l l y , of course, the population does change over time, and the trend of inequality in a l i f e c y c l e context r e f l e c t s the effect of this population change on inequality measured in the d i s t r i b u t i o n of representative l i f e c y c l e consumption. The trend of l i f e c y c l e inequality i s thus seen to be an interpersonal or interpopulation phenomenon rather 6 2The same is not true of annual inequality, unless the new d i s t r i b u t i o n i s simply a permutation of the old, because ind i v i d u a l incomes change from year to year.. 110 t h a n an i n t e r t e m p o r a l o n e . 6 3 S i n c e t h e p r o b l e m i s o n e o f m e a s u r i n g a n d c o m p a r i n g i n e q u a l i t y i n d i f f e r e n t p o p u l a t i o n s , a s o l u t i o n may n a t u r a l l y be s o u g h t i n t h e t h e o r y o f v a r i a b l e p o p u l a t i o n s o c i a l e v a l u a t i o n f u n c t i o n s . T h e f o l l o w i n g b r i e f o u t l i n e o f t h e t h e o r y i s d r a w n f r o m B l a c k o r b y a n d D o n a l d s o n [ 1 9 7 9 ] , a l t h o u g h I p r e s e n t i t i n t e r m s t h a t a r e r e l e v a n t t o t h e t r e n d o f l i f e c y c l e i n e q u a l i t y ( i . e . t h e a r g u m e n t s o f t h e f i x e d p o p u l a t i o n s o c i a l e v a l u a t i o n f u n c t i o n s a r e r e p r e s e n t a t i v e l i f e c y c l e c o n s u m p t i o n s ) . T h e c o n s t r u c t i o n o f a v a r i a b l e p o p u l a t i o n s o c i a l e v a l u a t i o n f u n c t i o n i n v o l v e s two s t e p s . F i r s t , i n p l a c e o f a s i n g l e s o c i a l e v a l u a t i o n f u n c t i o n a f a m i l y o f ( f i x e d p o p u l a t i o n ) s o c i a l e v a l u a t i o n f u n c t i o n s , o n e f o r e v e r y p o s s i b l e p o p u l a t i o n s i z e , i s r e q u i r e d . T h e p r o b l e m o f t h e r e b e i n g d i f f e r e n t p e o p l e i n two d i f f e r e n t s o c i a l s t a t e s i s r e d u c e d t o a p r o b l e m o f d i f f e r e n t p o p u l a t i o n s i z e s by t h e a d o p t i o n o f t h e a s s u m p t i o n s o f a n o n y m i t y 6 " a n d a v a r i a b l e p o p u l a t i o n a n a l o g u e o f w e l f a r i s m . 6 5 W e l f a r i s m i m p l i e s t h a t s o c i a l s t a t e s c a n be f u l l y c h a r a c t e r i z e d by t h e v e c t o r o f r e p r e s e n t a t i v e l i f e c y c l e c o n s u m p t i o n s o f a l l i n d i v i d u a l s who e x i s t i n t h a t s o c i a l s t a t e . T h e p r o b l e m o f e v a l u a t i n g s o c i a l s t a t e s i s t h u s r e d u c e d t o t h e n e e d f o r a s o c i a l o r d e r i n g o v e r a l l f i n i t e l y - d i m e n s i o n e d v e c t o r s o f 6 3 I t s h o u l d a l s o be c l e a r t h a t t h e t r e n d o f a n n u a l i n e q u a l i t y i n v o l v e s b o t h i n t e r t e m p o r a l a n d i n t e r p e r s o n a l e f f e c t s s i n c e n o t o n l y a n n u a l i n c o m e s , b u t t h e p o p u l a t i o n t o o , c h a n g e f r o m y e a r t o y e a r . 6 " T h i s i s t h e a s s u m p t i o n t h a t who a p e r s o n i s d o e s n ' t m a t t e r . I t i s c a p t u r e d b y t h e s y m m e t r y o f t h e s o c i a l e v a l u a t i o n f u n c i o n s a n d i n e q u a l i t y i n d i c e s . 6 5 W e l f a r i s m i s i m p l i e d by t h e c o n j u n c t i o n o f U n l i m i t e d D o m a i n , I n d e p e n d e n c e o f I r r e l e v a n t A l t e r n a t i v e s , a n d P a r e t o I n d i f f e r e n c e . 111 r e p r e s e n t a t i v e c o n s u m p t i o n s . T h i s s o c i a l o r d e r i n g i s r e p r e s e n t e d by a f a m i l y o f ( f i x e d p o p u l a t i o n ) s o c i a l e v a l u a t i o n f u n c t i o n s , W n : R N — * R 1 , a t y p i c a l member o f w h i c h h a s t h e i m a g e , ( 5 . 1 ) w n = w " ( r n ) w h e r e r K = (r^ , . . . , r N ) i s t h e v e c t o r o f r e p r e s e n t a t i v e l i f e c y c l e c o n s u m p t i o n s i n s o c i a l s t a t e n . I a s s u m e t h a t r n c a n be r e p r e s e n t e d by i t s e q u a l l y d i s t r i b u t e d e q u i v a l e n t , i m p l i c i t l y d e f i n e d b y , ( 5 . 2 ) w " ( s n I N ) = W n ( r f t > w'V.) i s a s s u m e d t o be c o n t i n u o u s , i n c r e a s i n g , a n d S - c o n c a v e — p r o p e r t i e s w h i c h g u a r a n t e e t h e u n i q u e e x i s t e n c e o f t h e p o p u l a t i o n - w i d e r e p r e s e n t a t i v e c o n s u m p t i o n . ( 5 . 3 ) s n = S n ( r M ) M e m b e r s o f t h e f a m i l y ( 5 . 1 ) m u s t somehow be l i n k e d t o e n s u r e t h a t t h e y a l l r e f l e c t t h e same s e t o f e t h i c a l j u d g m e n t s . T h i s i s a c c o m p l i s h e d by t h e a d o p t i o n o f t h e P o p u l a t i o n  S u b s t i t u t i o n P r i n c i p l e ; 6 6 I f r = ( r n , r m ) , r n £ R N , r f^c R M , t h e n s M 4 r M = S  m(r)=S ( r n , r w ) = S ( s n l N , r w ) w h e r e s n =s" ( r n ) By r e q u i r i n g t h a t t h e e q u a l l y d i s t r i b u t e d e q u i v a l e n t c o n s u m p t i o n be u n c h a n g e d when a g r o u p ' s v e c t o r o f r e p r e s e n t a t i v e c o n s u m p t i o n s i s r e p l a c e d by i t s e q u a l l y d i s t r i b u t e d e q u i v a l e n t , t h e P o p u l a t i o n S u b s t i t u t i o n P r i n c i p l e e n s u r e s t h a t t h e s o c i a l e v a l u a t i o n o f t h e g r o u p ' s s i t u a t i o n i s c o n s i s t e n t w i t h t h a t o f 6 6 T h e P o p u l a t i o n S u b s t i t u t i o n P r i n c i p l e i m p l i e s some o t h e r , w e a k e r , p o p u l a t i o n p r i n c i p l e s , i n c l u d i n g D a l t o n ' s [ 1 9 2 0 ] P r i n c i p l e o f P o p u l a t i o n R e p l i c a t i o n . 1 1 2 the population. This implies that the members of the family of (fixed population) s o c i a l evaluation functions or equally d i s t r i b u t e d equivalent functions represent the same s o c i a l preferences. 6 7 The Population Substitution P r i n c i p l e also implies that the family of fixed population s o c i a l evalution functions must be ad d i t i v e l y separable (Blackorby and Donaldson [1979], theorem 3.2). This should not be too surprising given that the Population Substitution P r i n c i p l e views two so c i a l states with d i f f e r e n t sized populations as subgroups of a single population comprised of their sum, which i s exactly analogous to the structure imposed on the so c i a l evaluation function in the decomposition of inequality within and among subgroups. An add i t i v e l y separable functional structure results in both cases. Thus the families of indices that are admissible in the welfare and decomposition approaches to the measurement of inequality s a t i s f y the Pr i n c i p l e of Population Substitution. The second step in the construction of a variable population s o c i a l evaluation function i s the d e f i n i t i o n of the function i t s e l f . This requires an assumption that fixed population s o c i a l evaluation functions can be represented by their equally d i s t r i b u t e d e q u i v a l e n t s . 6 8 The variable population s o c i a l evaluation function f:R2~»-R1 (5.4) f(s,n) 6 7The members of such a family are characterized by the same degree of inequality aversion. For a proof of t h i s proposition see Appendix E. 6 8 T h i s i s equivalent to assuming that a vector of the u t i l i t i e s of individuals in a p a r t i c u l a r s o c i a l state can be represented by i t s equally d i s t r i b u t e d equivalent. 1 1 3 represents the s o c i a l ordering over possible trade-offs between representative consumption and population s i z e . The most basic dichotomy in c r i t e r i a for evaluating s o c i a l states with d i f f e r e n t sized populations i s whether the s o c i a l ordering r e f l e c t s a preference for t o t a l or per capita s o c i a l u t i l i t y . The adoption of additional population p r i n c i p l e s embodying either so-called c l a s s i c a l or average population rules further r e s t r i c t s the structure of the variable population so c i a l evaluation function (5.4) (Blackorby and Donaldson [1979], theorems 4.1 and 5.1). I am not interested, however, in the evaluation of d i f f e r e n t s o c i a l states in general, but rather in a p a r t i c u l a r aspect of these states, namely, their degree of inequality. S p e c i f i c a l l y , I am seeking a t h e o r e t i c a l foundation for the c a l c u l a t i o n of a meaningful trend of l i f e c y c l e inequality. The adoption of the P r i n c i p l e of Population Substitution i s s u f f i c i e n t for t h i s , as i t ensures that the same set of e t h i c a l judgments are being used in the measurement of the s o c i a l significance of inequality regardless of the size of the population under study. Since inequality indices are functions of representative consumption, but are independent of population si z e , i t should be clear that the measurement of inequality and i t s trend i m p l i c i t l y follows an average rule for evaluating s o c i a l states with variable sized populations. The variable population s o c i a l evaluation function i m p l i c i t in the trend of l i f e c y c l e inequality represents a s o c i a l ordering which is equivalent to ranking s o c i a l states according to their degree of measured inequality. In the welfare approach to the measurement of inequality, 114 inequality is measured in the d i s t r i b u t i o n of representative l i f e c y c l e consumptions. This d i s t r i b u t i o n changes only with changes in the population. L i f e c y c l e inequality therefore remains constant so long as the population remains unchanged. Ty p i c a l l y , of course, the population does change, and moreover i t s evolution occurs over time, so that the trend of l i f e c y c l e inequality i s presented as a time series of values of an inequality index. Over time, additions to and subtractions from the population ( i . e . the beginning and ending of individual economic l i f e c y c l e s ) result in changes to the vector of representative l i f e c y c l e consumptions which, of course, a l t e r measured l i f e c y c l e inequality. Although in fact the population is changing continuously i t i s s u f f i c i e n t to recompute l i f e c y c l e inequality so as to record i t s trend, not with every economic b i r t h or death, but rather at regular time i n t e r v a l s . Given annual data, the most l o g i c a l choice i s once a year. Approximating the trend of l i f e c y c l e inequality in the decomposition approach i s complicated by the fact that panel data c o l l e c t i o n , and the d e f i n i t i o n of the subset used in the computation of " s t a t i c " inequality indices, assume an unchanging po p u l a t i o n . 6 9 Since population changes over time are not e x p l i c i t l y incorporated into the panel data set, i t i s necessary to simulate the evolution of the population by a r t i f i c i a l l y 6 9The Panel Study on Income Dynamics allows for the introduction of new young members of society only through the formation of households by offspring of households currently in the sample. The data subset used to compute decomposition indices of inequality in Chapter Four include no households originating after 1968. 1 1 5 introducing each year a new cohort of individuals who are just beginning th e i r economic l i f e c y c l e s , while removing the eldest cohort whose members i t i s assumed have finished their economic l i v e s . Thus, in any given year, decomposition indices of inequality should be computed for age-cohorts in the range k_ to k (l<k_<k<K). The decomposition indices should then be recomputed for the following year using cohorts k-1 through k-1, to r e f l e c t ( a r t i f i c i a l l y ) the evolution of the population during that time. With ten years of panel data, t h i s method of simulating the ef f e c t s of a changing population on l i f e c y c l e inequality results in the loss of ten cohorts from the weighted average of inequality within cohorts which comprises the index of intracohort inequality. That i s , i f the data set includes households.with family heads aged 18 to 70 years, say, then intracohort inequality in the f i r s t year should be calculated, for the purposes of determining the trend of l i f e c y c l e inequality, as the weighted average of inequality within the 27 to 70 year old age cohorts. In the second year the age-range should be from 26 to 69 years, and so on u n t i l the f i n a l year in which the 18 to 61 year old age-cohorts should be used in the computation of intracohort inequality. Naturally, there is no the o r e t i c a l ground for t h i s exclusion of ten age-cohorts from each " s t a t i c " index of l i f e c y c l e inequality, but the reduced accuracy of measured inequality that results i s , in the absence of panel data that e x p l i c i t l y incorporates a new youngest cohort each year, an unavoidable cost associated with the cal c u l a t i o n 1 1 6 o f t h e t r e n d o f l i f e c y c l e i n e q u a l i t y . 7 0 I h a v e c o m p u t e d d e c o m p o s i t i o n i n d i c e s o f r e l a t i v e i n e q u a l i t y f o r a r e s t r i c t e d r a n g e o f a g e - c o h o r t s i n t e n y e a r s . T a b l e 11 c o n t a i n s t h e r e s u l t s o f c a l c u l a t i n g t h e t r e n d o f l i f e c y c l e i n e q u a l i t y i n t h e d e c o m p o s i t i o n a p p r o a c h , p l u s t h e t r e n d o f a n n u a l i n e q u a l i t y , f o r t h r e e v a l u e s o f t h e p a r a m e t e r o f r e l a t i v e i n e q u a l i t y a v e r s i o n , R. A l l f i g u r e s w h i c h r e f l e c t a n i n c r e a s e i n m e a s u r e d i n e q u a l i t y o v e r t h e p r e c e d i n g y e a r ' s f i g u r e a r e m a r k e d b y a n a s t e r i s k . T h e m o s t i m m e d i a t e l y n o t i c e a b l e c o n t r a s t b e t w e e n t h e t r e n d s o f l i f e c y c l e a n d a n n u a l i n e q u a l i t y i s t h a t a n n u a l i n e q u a l i t y i s f a r more c y c l i c a l , w h i l e l i f e c y c l e i n e q u a l i t y t e n d s t o f o l l o w a d e c l i n i n g t r e n d i n t e r r u p t e d o c c a s i o n a l l y by a one y e a r r i s e i n m e a s u r e d i n e q u a l i t y . T h e t o t a l d e c l i n e i n l i f e c y c l e i n e q u a l i t y o v e r t h e t e n y e a r p e r i o d r a n g e s f r o m 4.1 p e r c e n t ( R = - 5 ) t o 7 . 4 p e r c e n t ( R = - . 5 ) . T h i s i s e q u i v a l e n t t o a n a v e r a g e p e r annum r e d u c t i o n i n l i f e c y c l e i n e q u a l i t y o f f r o m 0 . 4 2 p e r c e n t t o 0 . 7 7 p e r c e n t f r o m t h e p r e c e d i n g y e a r . 7 1 A l t h o u g h t h e m e t h o d o f s i m u l a t i n g p o p u l a t i o n c h a n g e s o a s t o c o m p u t e t h e t r e n d o f i n e q u a l i t y c a s t s some d o u b t on t h e r e l i a b i l i t y o f t h e r e s u l t s , t h e y a r e , I t h i n k , i n d i c a t i v e o f t h e d i f f e r e n c e b e t w e e n t h e t r e n d s o f a n n u a l a n d l i f e c y c l e i n e q u a l i t y . A n n u a l i n e q u a l i t y c o m p u t e d f r o m t h e p a n e l d a t a 7 0 0 n t h e o t h e r h a n d , i f new c o h o r t s w e r e i n c o r p o r a t e d e a c h y e a r , t h e e n t i r e number o f y e a r s o f d a t a i n t h e p a n e l c o u l d n o t be u s e d i n t h e c o m p u t a t i o n o f t h e t r e n d o f l i f e c y c l e i n e q u a l i t y b e c a u s e o f i n s u f f i c i e n t d a t a on t h e m o s t r e c e n t l y a d d e d c o h o r t s . 7 1 T h i s may be c o n t r a s t e d w i t h t h e r e p o r t e d 23 p e r c e n t r e d u c t i o n • i n P a g l i n [ 1 9 7 5 ] - i n e q u a l i t y o v e r a 25 y e a r p e r i o d , w h i c h i s e q u i v a l e n t t o an 1 .04 p e r c e n t a v e r a g e p e r annum r e d u c t i o n i n i n e q u a l i t y . 1 1 7 conforms to the s t y l i z e d fact that the trend of annual inequality is constant. The moderately declining trend of l i f e c y c l e inequality, on the other hand, demonstrates that the trend of annual inequality i s not a r e l i a b l e indicator of the eff e c t of economic developments and s o c i a l policy on the d i s t r i b u t i o n of income. 118 T A B L E 11 T h e T r e n d s Of A n n u a l A n d L i f e c y c l e I n e q u a l i t y R= . 5 R=-- . 5 R= - 5 f e c y c l e a n n u a l l i f e c y c l e a n n u a l l i f e c y c l e a n n u a l . 0 7 1 2 . 0 9 1 5 . 2 0 0 8 . 2 6 1 8 . 4 8 6 3 . 6 9 6 4 .0701 . 0 8 8 7 . 1 983 . 2 5 5 6 * . 4 9 0 7 . 6 9 0 9 .0701 . 0 8 7 0 . 1 974 . 2 4 4 2 . 4 8 7 5 . 6 4 4 4 .0691 . 0 8 5 8 . 1 9 4 3 . 2 4 2 7 .481 2 * . 7 9 2 9 . 0 6 8 2 * . 0 8 6 7 . 1 9 1 8 * . 2 5 1 6 . 4 7 5 8 . 7 3 5 0 . 0 6 7 8 . 0 8 6 1 .1911 .2301 * . 4 8 0 4 .6531 .0671 * . 0 8 9 9 . 1883 * . 2 5 3 0 . 4 7 2 0 * . 8 6 6 4 . 0 6 6 5 * . 0910 . 1 8 6 8 . 2 5 0 6 . 4 6 8 7 . 6 5 5 2 . 0 6 6 3 * . 0 9 2 4 . 1 8 6 3 * . 2 6 0 6 . 4 6 7 7 * . 8 1 0 6 . 0 6 6 4 . 0 8 6 3 * . 1864 .2551 . 4 6 7 3 * . 8129 119 CHAPTER SIX Summary and Conclusions In this f i n a l chapter I w i l l b r i e f l y summarize the substance and main findings of my thesis and draw some conclusions from the th e o r e t i c a l and empirical results which have been presented. I have been concerned with problems that surround the t r a d i t i o n a l practice of measuring inequality in the d i s t r i b u t i o n of annual income. Throughout the thesis inequality i s taken to mean the extent to which society f a l l s short of a hypothetical sit u a t i o n in which everyone is equally well-off. The measurement of annual income inequality i s inappropriate in thi s regard because i t i s consumption, not income, that i s productive of welfare. Furthermore, welfare depends on consumption over the l i f e c y c l e , not just in a single year. Students and seniors may l i e f a i r l y far down in the d i s t r i b u t i o n of annual income, but th i s indicates l i t t l e about the extent to which they share in the f r u i t s of society during the course of their l i v e s . Neither does the negative annual income of a bankrupt businessman r e f l e c t his current consumption, l e t alone his probably enviable position in the d i s t r i b u t i o n of l i f e c y c l e consumption. Even were there not these th e o r e t i c a l objections to measuring inequality in the annual income d i s t r i b u t i o n , there are a number of methodological problems. Annual inequality indices, that i s , f a i l to take account of the source of income differences which results in their overstating the degree of 1 20 pure interpersonal inequality. For example, the receipt of income tends to follow a r i s i n g path over much of the l i f e c y c l e . Thus, even i f everyone traversed i d e n t i c a l l i f e c y c l e income paths, measured annual inequality would be posit i v e because of income differences attributable to the age-structure of the population. This source of inequality should properly be excluded from an index of pure interpersonal inequality. Additionally, i t has been observed that measured inequality i s sensitive to the length of the income accounting period. This i s explained by the fact that, over time, individuals tend to change their r e l a t i v e positions in the income d i s t r i b u t i o n . Empirical investigations have found evidence of substantial r e l a t i v e income mobility. The tendency for low income recipients to move up, and high income recipients to move down in the d i s t r i b u t i o n , has an averaging effect on incomes as the accounting period is lengthened. This explains the often observed tendency of measured inequality to vary inversely with the length of the income accounting period. This effect should also be taken into account when measuring inequality. In response to these problems I have proposed two new approaches to the measurement of inequality. In the welfare approach, presented in Chapter Two, inequality i s measured in the d i s t r i b u t i o n of a summary s t a t i s t i c of l i f e c y c l e consumption. Since the consumption p r o f i l e s of younger members of the current population w i l l tend to l i e above those of elder members because of real growth, i t i s necessary to decompose l i f e c y c l e inequality within and among age-cohorts of the population. Intracohort inequality i s an index of pure 121 interpersonal inequality while intercohort inequality captures the contribution of economic growth to t o t a l measured inequality. The welfare approach requires that individual u t i l i t y functions are known (so that representative l i f e c y c l e consumption can be computed), and that l i f e c y c l e consumption data are ava i l a b l e . Clearly neither of these conditions i s met. 7 2 I have therefore devised an alternative method, capable of empirical implementation, which i s the subject of Chapter Three. The decomposition approach i s a compromise between the inadequacy of measuring annual income inequality and the imp o s s i b i l i t y of measuring l i f e c y c l e consumption inequality. The method i s to treat panel consumption data on H individuals over T years as a d i s t r i b u t i o n of consumption among a single population of size HT. Total inequality i s then decomposed within and among various subgroups of that p o p u l a t i o n . 7 3 F i r s t , the T observations on each individual are treated as H separate subgroups of the population, and inequality thus decomposed within and among persons. Intrapersonal inequality (attributable 72Some recent work has attempted to measure l i f e c y c l e consumption inequality by assuming that individual u t i l i t y functions are i d e n t i c a l and of a p a r t i c u l a r functional form, and using estimated l i f e c y c l e consumption data computed by maximizing u t i l i t y subject to an (estimated) l i f e t i m e income constraint which assumes c a p i t a l markets are perfect. I have chosen a d i f f e r e n t exit from the impasse imposed by the requirements of the welfare approach to the measurement of inequality. 7 3 U s i n g a procedure suggested by Blackorby, Donaldson, and Auersperg [1981]. In Appendix A the performance of an alt e r n a t i v e procedure for decomposing inequality, suggested by Shorrocks [1980], i s evaluated. It was rejected for use in.the decomposition approach to the measurement of inequality on both th e o r e t i c a l and empirical grounds. 1 22 to variation in individuals' consumption over time) r e f l e c t s the effe c t on inequality of consumption mobility. Interpersonal inequality is further decomposed within and among age-cohorts. Age-related inequality i s captured by the index of intercohort inequality, leaving intracohort inequality as an index of pure interpersonal inequality. In Chapter Four I report the results of the empirical implementation of the decomposition approach to the measurement of inequality. Ten years of data drawn from the Panel Study on  Income Dynamics were used to compute decomposition approach indices of absolute inequality and r e l a t i v e equality and inequality. The results indicate that age- and es p e c i a l l y mobility-related consumption differences account for a substantial portion of t o t a l measured inequality in th i s d i s t r i b u t i o n . Annual inequality overstates pure interpersonal inequality by no less than one-third, and in some cases by as much as 7 5 per cent. Although the magnitude of t h i s overstatement depends in part on the width of age-cohort brackets, only a moderate reduction in the d i s p a r i t y between the t r a d i t i o n a l and decomposition approaches resulted from a widening of age-cohort brackets. There was also some indication that the decomposition approach index of intracohort inequality approached a l i m i t within the ten years of the data set. If th i s is a general rather than data-specific property, then decomposition approach indices may be computed s u f f i c i e n t l y accurately with as few as ten years of data. One of the most well-established empirical observations about inequality is that i t s trend in the post-World-War II 1 23 period has been remarkable constant. An accurate assessment of the trend of inequality, however, depends c r u c i a l l y on the measurement of " s t a t i c " inequality. In Chapter Five I investigate the trend of l i f e c y c l e inequality. The f i r s t step i s to recognize an essential difference between the trends of annual and l i f e c y c l e inequality. The former is simply the time-series of annual inequality, r e f l e c t i n g the fact that consumption changes from one year to the next. The trend of l i f e c y c l e inequality, however, depends not on the passage of time but on the evolution of the population. The equally d i s t r i b u t e d equivalent of an individual's l i f e c y c l e p r o f i l e remains constant, by d e f i n i t i o n , and the problem, therefore, is one of measuring and comparing inequality in d i f f e r e n t populations. This requires families of so c i a l evaluation functions and inequality indices with one member for every possible population s i z e . In addition, some means of li n k i n g together members of a family is required to ensure that they a l l represent the same set of e t h i c a l judgments. Having established the the o r e t i c a l basis for measuring and evaluating the trend of l i f e c y c l e inequality, I discuss how thi s would be accomplished in the welfare approach, and how i t can be approximated in the decomposition approach. Since the members of the panel do not change over time, the evolution of the population must be simulated by allowing each year for the demise of members of the eldest cohort and the " b i r t h " of a new cohort whose members are just beginning their economic l i f e c y c l e s . The empirical results proved very i n t e r e s t i n g . The trend of annual inequality measured in the d i s t r i b u t i o n i s 1 24 roughly constant, but l i f e c y c l e inequality declines over the ten year period. This i s taken as further evidence in support of the decomposition approach to the measurement of inequality. The p r i n c i p a l conclusion of t h i s thesis i s that a decomposition approach to the measurement of inequality i s essential for an accurate assessment of inequality. Measured annual inequality includes the ef f e c t s of mobility and the age-structure of the population. 7* In the decomposition approach, indices of age- and mobility-related inequality are computed and used to construct an index of pure interpersonal inequality. This requires panel data. The c o l l e c t i o n of such data, not currently available for Canada, would be extremely useful for many kinds of economic and s o c i a l research including the measurement of inequality. For t h i s purpose, i t would be useful to record both income and consumption expenditure (or assets and l i a b i l i t i e s , which would allow consumption expenditure to be computed from income). 7 5 Furthermore, i t would be useful for the purpose of measuring the trend of inequality i f a new cohort representing individuals just beginning their economic l i f e c y c l e s could be added to the panel each year. 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American Economic Review, 69(4), (September), pp. 670-72. 1 32 APPENDIX A An Alternative Procedure for Decomposing Inequality The procedure suggested by Blackorby, Donaldson, and Auersperg [19813 for decomposing inequality employs subgroup equally d i s t r i b u t e d equivalent incomes to eliminate inequality within groups. An alternative method has recently been proposed independently by at least three authors, Bourguignon [1979], Cowell [1980], and Shorrocks [1980], who have studied the class of a d d i t i v e l y decomposable inequality indices in which subgroup mean incomes are used to eliminate intragroup inequality. In this appendix, I describe t h i s decomposition procedure, drawing mainly on Shorrocks [1980], and compare i t with the Blackorby-Donaldson-Auersperg version. I then show how i t can be applied to the problem of constructing an index of pure interpersonal inequality by decomposing t o t a l inequality in panel income data within and among persons, and then further decomposing interpersonal inequality within and among age-cohorts, as was done with the Blackorby-Donaldson-Auersperg procedure in Chapter Three. Empirical results are provided to i l l u s t r a t e the performance of the class of indices defined by Shorrocks in both the t r a d i t i o n a l and decomposition approaches to the measurement of inequality. The l a t t e r i s also compared to the decomposition approach as implemented in Chapter Four in order to evaluate the two decomposition procedures on the empirical grounds most relevant to my objective of developing an index of pure interpersonal inequality. 1 33 Shorrocks [1980] has studied the class of continuous and symmetric inequality measures, bounded below by zero (when a l l members of the population have the same income), which are decomposable into the form: (A.1) I (y; n) = Si Wg (m, n) I(y g;n^) + i d n , ^ , . . . ,m6j_n&) where y=(y.,, . . . 'Y^ i s t n e d i s t r i b u t i o n of income over the n members of the population and m i s i t s mean. In (A.l) (Shorrocks equation(4)), t o t a l inequality is decomposed into the sum of two terms: a weighted sum of the subgroup inequality indices, I ( y 3 ; n ^ ) , where y g is the vector of incomes of the n^ members of the gth subgroup (1<g<G), and an index of intergroup inequality whose arguments are subgroup mean incomes. This class of ad d i t i v e l y decomposable inequality measures is also shown to s a t i s f y the Pigou-Dalton p r i n c i p l e of transfers (Shorrocks [1980], theorem 3). Further properties which i t may be desirable for the inequality measures to s a t i s f y narrow the class of admissible indices. For example, s a t i s f y i n g the p r i n c i p l e of population r e p l i c a t i o n r e s t r i c t s the form of the weights, Wg(m,n), in the intragroup term (Shorrocks [1980], theorem 4). The assumption of mean independence (income homogeneity) imposes considerable additional structure on the form of the inequality measures (Shorrocks [1980], theorem 5). Shorrocks [1980, p. 622] shows that the class of, "additively decomposable indices s a t i s f y i n g both mean independence and population r e p l i c a t i o n therefore comprise a one parameter family whose members are i d e n t i f i e d by the value of c:" 1 34 (A.2) I c (y) = (i/n) (1/ C(c-1 ))X-[ (y L/m) c -1 ] I 0 (y)=(l/n)^Llog(m/y L ) I n (y) = ( l/n)ZL(y L/m)log(y c/m) ,c*0,1 ,c=0 ,c=1 This family of inequality measures includes the square of the c o e f f i c i e n t of variation (c=2), two indices developed by Theil [1967] from the theory of entropy (c=0 and C=1), and monotonic transformations of the entire Atkinson [1970] family of indices (Shorrocks [1980, p. 622]). As the value of the parameter c decreases, the index becomes increasingly sensitive to inequality lower down in the d i s t r i b u t i o n of income. The family of inequality measures, (A.2) (Shorrocks' equation (31)), i s thus similar to the Atkinson and Kolm-Pollak families of r e l a t i v e and absolute inequality indices, in the sense that these l a t t e r families of indices s a t i s f y the assumptions which Shorrocks has invoked in his axiomatic derivation of the class of a d d i t i v e l y decomposable indices, with the single exception of the summary s t a t i s t i c of the subgroup d i s t r i b u t i o n that i s used to eliminate intragroup inequality. The weights in the intragroup term of the decomposition of the indices (A.2) in the form (A.1) are: (A. 3) w<j (m,n) = (n g/n) (m^  /mf which sum to unity only when c=0 or c=1. Thus the intragroup inequality term i s not in general a weighted average of inequality within groups, as i t is in the decomposition of e t h i c a l indices of inequality. This, Shorrocks argues, may not be regarded as a major handicap, but T h e i l [1967, p. 125] has pointed out a more serious objection. It can be shown that 1-5Lwg i s proportional to the between group term in the corresponding decomposition equation. Thus, apart from the two measures proposed by T h e i l (c=0 and C=1), the 135 decomposition c o e f f i c i e n t s are not independent of the between group contribution. ([1980, p. 624]) Clearly t h i s problem does not arise in the procedure for decomposing inequality proposed by Blackorby, Donaldson, and Auersperg. Decomposing inequality provides a means of quantitatively assessing the contribution of some factor to t o t a l inequality. The decomposition of inequality within and among age-cohorts, for example, determines the proportion of measured inequality attributable to the variation of income with age, and the proportion that is pure interpersonal inequality. As Shorrocks points out, however, there has been some ambiguity in the interpretation of such a procedure. That i s , inequality within groups might be eliminated by assigning each individual his age-cohort mean income, which would eliminate the intragroup term of the decomposition, and leave the intergroup term as the measure of inequality due to the shape of age-income p r o f i l e s . A l t e r n a t i v e l y , mean incomes might be equalized across age-cohorts without changing inequality within cohorts. This would eliminate the intercohort term of the decomposition, but the reduction in inequality i s not simply B (the between group term of the decomposition) because, in general, changing the age group means w i l l also a f f e c t the decomposition c o e f f i c i e n t s and hence the t o t a l within group contribution. Only when these c o e f f i c i e n t s do not depend on the subgroup means w i l l (these alternative procedures) produce the same answer. Of the family of measures (31)(A.2 in thi s appendix), one alone s a t i s f i e s t h i s requirement -- the index I G , for which the corresponding decomposition c o e f f i c i e n t s are the population shares n 3/n. For thi s reason, I 0 i s the most s a t i s f a c t o r y of the decomposable measures, allowing t o t a l inequality to be unambiguously s p l i t into the contribution due to differences between subgroups and the contribution due to inequality within each subgroup in such a way that t o t a l inequality is the sum of these G+1 1 36 c o n t r i b u t i o n s . ( S h o r r o c k s [ 1 9 8 0 , p . 6 2 5 ] ) When t h e d e c o m p o s i t i o n o f i n e q u a l i t y i s c a l c u l a t e d by e l i m i n a t i n g i n t r a g r o u p i n e q u a l i t y f i r s t , t h e o r i g i n a l d i s t r i b u t i o n i s r e p l a c e d by ( m 1 j _ n i , . . . ,^Q]_n&) • T h e r e m a i n i n g i n t e r g r o u p i n e q u a l i t y i s t h e n m e a s u r e d i n t e r m s o f ( m ^ / m ) . I n t r a g r o u p i n e q u a l i t y i s c a l c u l a t e d by s u b t r a c t i n g t h e i n t e r g r o u p t e r m f r o m t o t a l i n e q u a l i t y . T h i s y i e l d s t h e d e c o m p o s i t i o n g i v e n i n ( A . 1 ) . I f i n t e r g r o u p i n e q u a l i t y i s t o be e l i m i n a t e d f i r s t t h e s u b g r o u p d i s t r i b u t i o n s a r e s c a l e d , w i t h o u t c h a n g i n g i n e q u a l i t y w i t h i n t h e g r o u p s , s o t h a t t h e i r m e a n s a r e e q u a l t o t h e mean o f t h e o r i g i n a l d i s t r i b u t i o n . E a c h i n c o m e , yj_ , i s r e p l a c e d by x t = Y i ( m / m g ) . A c c o r d i n g t o S h o r r o c k s , t h i s d e c o m p o s i t i o n w i l l n o t y i e l d t h e same r e s u l t a s ( A . 1 ) , h o w e v e r , " b e c a u s e , i n g e n e r a l , c h a n g i n g t h e a g e g r o u p means w i l l a l s o a f f e c t t h e d e c o m p o s i t i o n c o e f f i c i e n t s a n d h e n c e t h e t o t a l w i t h i n g r o u p c o n t r i b u t i o n " ( S h o r r o c k s [ 1 9 8 0 , p . 6 2 5 ] , e m p h a s i s a d d e d ) . W i t h i n g r o u p i n e q u a l i t y i n t h i s c a s e i s m e a s u r e d i n t h e d i s t r i b u t i o n x M x ^ j , . . . , x n ) w i t h r e s p e c t t o t h e d i s t r i b u t i o n o f t h e p o p u l a t i o n - w i d e mean i n c o m e (m_1_^ ) . T h e p r i m i t i v e s o f t h e w i t h i n g r o u p i n e q u a l i t y i n d i c e s a r e t h e r e f o r e ( x ^ / m ) . B u t (x- L /m) = ( y \ . / m g ) , t h e p r i m i t i v e s o f t h e w i t h i n g r o u p i n d i c e s i n ( A . 1 ) ( w h e r e i n t r a g r o u p i n e q u a l i t y i s e l i m i n a t e d f i r s t ) . T h u s w h i l e S h o r r o c k s ' a r g u m e n t t h a t c h a n g i n g t h e s u b g r o u p means w i l l a f f e c t t h e d e c o m p o s i t i o n c o e f f i c i e n t s i s t r u e i n g e n e r a l , i t i s n o t i n t h i s p a r t i c u l a r c a s e w h e r e i n t e r g r o u p i n e q u a l i t y i s e l i m i n a t e d f r o m t h e o r i g i n a l d i s t r i b u t i o n . T h i s c a n be v e r i f i e d by n o t i n g t h a t i n t e r g r o u p i n e q u a l i t y i s m e a s u r e d i n t e r m s o f 1 37 (y-t / x ) = (m^ /m) which is the same as the intergroup inequality index in (A.1). Shorrocks 1 decomposition of inequality therefore is unique with respect to the order in which i n t r a - and intergroup inequality are eliminated. Blackorby, Donaldson, and Auersperg have addressed the same problem in their discussion of alte r n a t i v e procedures for decomposing inequality. Their decomposition of per capita inequality i s independent of the order in which i n t r a - and intergroup inequality are eliminated, but their decomposition of AKS inequality does not have th i s very desirable property. Fortunately, however, the decomposition of AKS inequality which I suggested in Chapter Two, and employed there and in Chapter Three, i s independent of the order in which i n t r a - and intergroup inequality i s eliminated. Thus, both Blackorby, Donaldson, and Auersperg's [1981] and Shorrocks' [1980] procedures y i e l d unique decompositions of inequality. Members of the family of inequality measures, (A.2), shown by Shorrocks to exhaust the class of ad d i t i v e l y decomposable indices s a t i s f y i n g mean independence and population r e p l i c a t i o n , do not range over a [0,1] interval,.a property often imposed on indices of r e l a t i v e i n e q u a l i t y . 7 6 While the indices (A.2) are bounded below by zero, their upper bound varies widely with the value of c and for c^O they are unbounded above. For positive 7 6Indeed, i t i s d i f f i c u l t to imagine what interpretation can be placed on a mean independent inequality index which i s not confined to a [0,1] range. It i s for thi s reason that Blackorby, Donaldson, and Auersperg [1981] argue that one of their procedures for decomposing AKS indices of inequality, which generates subindices of r e l a t i v e inequality that can take on values outside the [0,1] i n t e r v a l , is unacceptable. 138 values of c, the indices can be normalized to l i e in the [0,1] i n t e r v a l , but they w i l l then f a i l to s a t i s f y the p r i n c i p l e of population r e p l i c a t i o n . For values of c^O the family of a d d i t i v e l y decomposable indices cannot be normalized (Shorrocks [1980, p. 623, n. 7]). E t h i c a l indices of r e l a t i v e inequality, on the other hand, are constructed so as to range over a [0,1] i n t e r v a l , and the Atkinson family of indices, which exhaust the admissible class of mean independent indices in the decomposition approach to the measurement of inequality, s a t i s f y the p r i n c i p l e of population r e p l i c a t i o n . This is further reason to prefer the Blackorby-Donaldson-Auersperg procedure for decomposing inequality to that proposed by Shorrocks. It may prove interesting, nevertheless, to investigate the application of the class of a d d i t i v e l y decomposable inequality measures defined by Shorrocks [1980] to the problem of developing an index of pure interpersonal inequality. Recalling that the d i s t r i b u t i o n in which inequality i s to be measured i s comprised of the annual consumptions of H individuals observed over T y e a r s , 7 7 the indices (A.2) are written as: I, (y) = ( l / H T ) ( l / c ( c - D ) Z Z : [(y u. / m ) c - l ] ,c*0,1 (A.4) L ( y ) = ( l / H T ) I Z l o g ( m / y h J ,c = 0 I 1 (y) = ( 1/HT)X Z (y h t/m)log(y K t/m) ,c=1 where. y h - t i s the income of the hth person in the t t h year. I begin by decomposing I (y), according to (A.1), into inequality 7 7 S i n c e Shorrocks' work and the foregoing discussion of i t are couched in terms of income rather than consumption, I w i l l continue this practice throughout Appendix A in order to minimize confusion. 139 within the income streams of population members and inequality between individuals: (A. 5) I c ( y ) = 21(T/TH)(mw/m)<1 {(1/T) (1/C(C-1 ) ) Z [ (y w t/m L) C-1 ]} + { ( l / H ) ( l / c ( c - l ) ) Z [ ( m h / m ) t - l ] } We may now decompose interpersonal inequality, the second term in (A.5 ) , within and among age-cohorts. This y i e l d s , I c ( y ) = Z { ( 1/TH) (mh/mf d / c ( c - l ) (y ht/mh ) C -1 ]} (A.6) +Z.{(l/H)(m k/m) c (l/c(c-1 ))Z f c[(m h/m u) c-1 ]} + {(1/H) ( l / c ( c - D ) X . n k [ (mk/m)C-1 ]} The decompositions of the indices (A.4) for c=0 and c=1 corresponding to (A.6) are: l 0 ( y ) - [ Z l ( l / T H ) Z : i o g ( n i k / y h t ) ] (A. 7) +t ^ { 1 / H ) h 5 k l°g ( i r ik / mK )^ +[(1/n k )5L log(m/m k)] and, I, (y) = [ 2 I ( l / T H ) ( m L / m ) X ( y k t /m^ ) log(y ht/mK ) ] (A.8) + [ Z (1/H) (m k/mj iZ k(m h/m k)log(m K/m k) ] + [ (l/n k)^(m k/m)log(m k/m) ] In (A.6), (A.7), and (A.8), t o t a l inequality i s decomposed into the sum of three subindices of inequality which measure, respectively: intrapersonal inequality, ; intracohort inequality, 1^ , and intercohort inequality, I R C . Values of the index and subindices of inequality computed for various degrees of inequality aversion are reported in Table 12. The indices (A.2) were also used to calculate annual inequality indices. The weights (A.3) corresponding to annually defined population subgroups were used to calculate a weighted sum of the annual inequality indices which is also reported. "Annual" inequality 1 40 in Table 12 i s thus the intragroup component of a decomposition within and among years of t o t a l inequality in the panel data treated as an income d i s t r i b u t i o n for a single population. Since one of my objectives i s to evaluate the performance of a l t e r n a t i v e procedures in a decomposition approach to the measurement of inequality, I have, where possible, reported normalized versions of the indices in Table 12 to f a c i l i t a t e t h e i r comparison with the Blackorby-Donaldson-Auersperg decomposition indices of r e l a t i v e ( i . e . mean independent) inequality reported in Table 3 of Chapter Four. Unfortunately, however, although the normalized indices range over a [0,1] i n t e r v a l , i t appears that those indices that can be normalized ( i . e . c>0) generally exhibit too low a degree of inequality aversion to be of much p r a c t i c a l use. An exception is the normalized index in Table 12 for c=1, for which the value of the index of t o t a l inequality in the panel coincides to the t h i r d decimal place with the t o t a l inequality index value in Table 3 for R=0.9. The two indices are repeated together below. From Table 3 from Table 12 R=.9 c=1 I .0185 .0181 I A P .0038 .0037 I A c .0136 .0133 l^c, .001 1 .0012 The two indices produce v i r t u a l l y i d e n t i c a l results in t h i s case, which might be taken as an indication that the two alt e r n a t i v e decomposition procedures are substitutes for one another. But there i s not s u f f i c i e n t evidence to warrant such a 141 conclusion. Furthermore, certain undesirable attributes of the Shorrocks' [1-980] decomposition procedure place i t in an unfavourable l i g h t , compared to the Blackorby-Donaldson-Auersperg procedure, for use in a decomposition approach to the measurement of inequality. It has already been mentioned that the range of the parameter of inequality aversion, c, over which the indices defined by Shorrocks can be normalized, overlaps only very s l i g h t l y with the range of c l i k e l y to be useful in implementing the decomposition approach to the measurement of inequality. Thus, Shorrocks' indices would have to be used almost exclusively in their nonnormalized versions, a disadvantage because of the d i f f i c u l t y of interpreting mean-independent inequality indices that do not range, over a [0,1] i n t e r v a l . 7 8 A close inspection of the results reported in Table 12 w i l l reveal another d i f f i c u l t y with Shorrocks' class of a d d i t i v e l y decomposable inequality indices. As the value of the parameter, c, f a l l s , the degree of inequality aversion exhibited by members of the family of indices (A.2) r i s e s . With i t should r i s e the s o c i a l significance of inequality in the d i s t r i b u t i o n of 7 furthermore, even the normalized versions of Shorrocks' indices do not bear the interpretation a t t r i b u t a b l e to the Blackorby-Donaldson-Auersperg indices of r e l a t i v e inequality: that they measure the proportion of income "wasted" on inequality. 7 9 A l l the results reported in Tables 3 and 12 measure inequality in the same d i s t r i b u t i o n of income which i s characterized by a certain degree of inequality "objectively" or " s t a t i s t i c a l l y " measured. The s o c i a l significance of inequality, however, should r i s e with the degree of inequality aversion, indicating the greater welfare effects of a given amount of dispersion in the d i s t r i b u t i o n of incomes. 1 42 income, 7 9 as is the case with the Blackorby-Donaldson-Auersperg decomposition indices reported in Table 3. Such i s not the case, however, with Shorrocks' class of inequality indices. As the degree of inequality aversion rises ( i . e . as the value of c f a l l s ) , t o t a l measured inequality f i r s t f a l l s , reaches a minimum at approximately c=.5, and increases thereafter. The strange behaviour of these indices with a low degree of inequality aversion results from the extreme importance which they place, for the measurement of inequality, on the upper end of the income d i s t r i b u t i o n . As can be seen from the structure of the indices (A.2), when c i s large incomes below the mean hardly af f e c t the value of the index, while incomes above the mean, p a r t i c u l a r l y the largest incomes, contribute enormously to t o t a l measured inequality. Shorrocks and others have noticed t h i s property of the indices (A.2) with respect to their transfer properties. (T)he square of the c o e f f i c i e n t of variation (corresponding to c=2) gives roughly the same weight to a transfer of $10 from a person with $10,000 to another with $2,000 as a $1 transfer from someone with $100,000 to another with $20,000 . . . (Shorrocks [1980, p. 623]). . .. . This rather perverse result i l l u s t r a t e s why the (square of the) c o e f f i c i e n t of va r i a t i o n is extremely sensitive to changes in the upper t a i l of the d i s t r i b u t i o n . In fact the transfer properties of indices corresponding to c>2 become even stranger. Although they s t i l l s a t i s f y the p r i n c i p l e of transfers, they show l i t t l e concern for equalization except among the very r i c h . This has led Kolm [1976b] and Love and Wolfson [1976] to question whether they should not be eliminated from consideration as inequality measures, as would be the case i f Kolm's "p r i n c i p l e of diminishing transfers" were adopted (Shorrocks [1980, p. 623, n. 8]). Clearly we might well follow Kolm, and Love and Wolfson, and employ only those members of Shorrocks' class of a d d i t i v e l y decomposable indices corresponding to c^2. 1 43 The strange behaviour of these indices c a r r i e s over to the decomposition of t o t a l inequality within and among subgroups. The proportions of t o t a l inequality attributable to the three subindices of intrapersonal, intracohort, and intercohort inequality vary enormously over the range of c. At i t s extreme values (c>5 and c^-3), the r e l a t i v e magnitudes of the three subindices of inequality correspond to the order in which inequality i s eliminated when computing them; that i s , f i r s t intrapersonal, then intracohort, and f i n a l l y intercohort, inequality. For low degrees of inequality aversion, a possible explanation of thi s i s again the perverse behaviour of the indices (A.2) for large values of the parameter c. The computation of subindices involves the substitution of subgroup mean income for the o r i g i n a l group d i s t r i b u t i o n . This naturally reduces dispersion and with i t the opportunity for the indices to take on large values because of a few observations in the upper t a i l of the d i s t r i b u t i o n . The same is l i k e l y true for small values of c also, although in that case i t would be the observations in the lower t a i l of the d i s t r i b u t i o n which contribute to the high degree of measured i n e q u a l i t y . 8 0 In either case, the replacement of actual d i s t r i b u t i o n s by subgroup means removes the influence of observations in one or the other t a i l s of the d i s t r i b u t i o n , thereby causing the subindices of 8 0 F o r c<0, (A.2) can be written, I c ( y ) = ( l / n ) ( l / c ( c - l ) ) Z l [ ( m / y L ) 1 c l - l ] from which i t should be clear that when; the magnitude of c i s large, incomes greater than the mean would have l i t t l e impact on the value of measured inequality while, very small incomes would cause measured inequality to be very large indeed. 144 inequality to r e f l e c t more the order in which inequality i s eliminated than the r e l a t i v e contributions of various sources of inequality to the t o t a l . For intermediate values of the degree of inequality aversion, corresponding roughly to -2<c<4, intracohort inequality is the predominant contributor to t o t a l inequality, followed by intrapersonal inequality and f i n a l l y intercohort inequality. In t h i s range, the rank order of the subindices of inequality i s the same as for the subindices of r e l a t i v e inequality, over a l l degrees of inequality aversion, reported in Table 3. of Chapter Four. 8 1 These patterns suggest that, of the class of a d d i t i v e l y decomposable indices defined by Shorrocks [1980], members displaying either high or low degrees of inequality aversion may have to be omitted i f r e l i a b l e estimates of inequality are to be obtained. In conclusion, i t has been shown that the class of a d d i t i v e l y decomposable inequality indices that use subgroup means to eliminate inequality within groups are not well-suited for use in a decomposition approach to the measurement of pure interpersonal inequality. In addition to departing from the practice of the modern theory of employing equally d i s t r i b u t e d equivalents to eliminate inequality, Shorrocks' decomposition indices suffer from several t h e o r e t i c a l drawbacks from which the Blackorby-Donaldson-Auersperg indices are free. The class of indices defined by Shorrocks [1980] do not 'Furthermore, the magnitudes of the proportions of t o t a l inequality attributable to the three sources are very close for .9>R>-1 in Table 3 and 1.5>c>-.5 in Table 12. 1 45 range over a [ 0 , 1 ] i n t e r v a l , which makes them d i f f i c u l t to interpret as mean-independent measures of inequality. While a subset of th i s class of indices i s bounded above, and can therefore be normalized to range over a [ 0 , 1 ] i n t e r v a l , the results of empirically implementing the decomposition approach using Shorrocks' family of indices reveal that the degree of inequality aversion among those indices that can be normalized is too low for them to be of much p r a c t i c a l use. In addition, estimation of the decomposition approach using Shorrocks' indices revealed strong evidence that the indices could not be deemed r e l i a b l e measures of t o t a l or subgroup inequality in other than an intermediate range of the degree of inequality aversion. These shortcomings are s u f f i c i e n t l y serious that the class of a d d i t i v e l y decomposable inequality indices defined by Shorrocks cannot be recommended for use in the decomposition approach to the measurement of pure interpersonal inequality. -Thus, in Chapters Three and Four, the decomposition approach i s developed and implemented in terms of Blackorby, Donaldson, and Auersperg's procedure for decomposing inequality within and among population subgroups. 1 46 TABLE 12 Additively Decomposable Indices of Inequality c = 5 (normalized) c = 4 (normalized) I 3 .4255 . 1051x10- 1 5 .8872 .4642x10" 1 2 XAP 1 .8603 .0571x1 0" 1 5 .3870 . 2025x10" 1 2 JAc 1 .5197 .0466x10" 1 5 .4748 .2484x10" 1 2 .0456 .0014x10" 1 5 .0254 .0133x10" 1 2 annual 3 .4215 .8833 AP X A C annual .3641 .1171 .2299 .0171 .3602 c = 3 (normalized) . 2 7 0 7 x 1 0 " 8 . 0 8 7 1 X 1 0 " 8 . 1 7 0 9 x 1 0 " 8 . 0 1 2 7 X 1 0 " 8 2252 ,0540 , 1 577 ,0135 .2212 c = 2 (normalized) . 1 5 8 5 x 1 0 " " . 0 3 8 0 x 1 0 - * . 1 1 1 0 x 1 0 - * . 0 0 9 5 x 1 0 - " I :AP annual .1986 .0428 . 1 433 .0125 . 1 946 c=1 .5 (normalized) .8889x10-3 . 1 91 8x1 0-3. .0641x10"3 . 0 0 5 6 X 1 0 " 3 . 1 861 .0376 . 1367 .0118 .1820 c=1 (normalized) . 1814x10-1 .0367x10-1 . 1333x10-1 .0115x10- 1 147 TABLE 12 (CON'T) I A C annual . 1841 .0366 . 1 362 .0113 .1780 c=. 5 (normalized) .4629x10-1 .0921x10- 1 .3424x10" 1 .0284x10" 1 1867 0376 1 380 ,0111 ,1826 c=. 25 (normalized) .3503x10"1 .0705x10"1 .2589x10"1 .0208x10"1 1 AP AC RC annual c = 0 . 1 920 .0397 .1413 .0110 . 1879 c=-. 5 .2117 .0483 . 1 527 .01 08 .2075 c = -1 .2486 .0662 .1718 .0106 .2444 c=-1.5 .3151 . 1 030 .201 5 .0106 .3109 • AP AC RC annual c = -2 .4406 . 1834 .2466 .01 06 .4363 c = -3 1.3194 .8871 .421 4 .0109 1 .3149 c = -4 8.2494 7.3821 .8558 .0114 8.2443 c = -5 93.4406 91.3907 2.0375 .0124 93.4269 1 48 APPENDIX B Evaluation Of A Method Of Approximating Long-run Inequality It was seen in Chapter One that measured inequality in the d i s t r i b u t i o n of annual income includes inequality attributable to several d i f f e r e n t sources which should be distinguished from one another. The objective i s to develop an index of pure interpersonal inequality, which requires, inter a l i a , that inequality due to the age-structure of the population be separated from pure interpersonal inequality. While i t has been argued that t h i s requires evaluation of actual l i f e c y c l e income streams, Paglin [1975] has proposed a revised Gini c o e f f i c i e n t of annual income inequality which, "approximates a measure of long-run interfamily inequality" (p. 601). Such a measure, i f reasonably accurate, would provide a r e l i a b l e estimate of long term inequality at s i g n i f i c a n t savings in data c o l l e c t i o n and computation costs. An assessment of the v a l i d i t y of the Paglin-Gini requires a comparison of i t s estimate of long run inequality with our measure of intracohort inequality in the decomposition approach. This is hampered, however, by the fact that Paglin's technique is s p e c i f i c to the Gini c o e f f i c i e n t which i s not included in the set of admissible inequality indices for the decomposition approach. In t h i s appendix I f i r s t describe Paglin's revised Gini c o e f f i c i e n t and then generalize his method to AKS and per capita indices of inequality. I am thus able to compute Paglin-type inequality measures for indices that are admissible under the assumptions of the decomposition 1 49 approach, which may then be compared to the corresponding decomposition approach indices of intracohort inequality. Annual income inequality measures, "combine and hence confuse intrafamily variation of income over the l i f e c y c l e with the more pertinent concept of inter family income variation which underlies our idea of inequality . . . " (Paglin [1975, p. 598], emphasis in o r i g i n a l ) . It i s Paglin's view that the problem i s best thought of as resulting from an inappropriate standard of equality. Annual income equality implies that families not only have equal l i f e t i m e incomes, but also equal annual incomes regardless of age (of family head), a constraint which requires that a l l families have perfectly f l a t age-income p r o f i l e s . Paglin argues that a more reasonable standard of equality for use with annual income data i s equal l i f e t i m e incomes without the added constraint of f l a t age-income p r o f i l e s . This i s made operational by redefining the standard of equality as income equality within age-cohorts, thus allowing income variation over the l i f e c y c l e to be excluded from contributing to measured annual income inequality. Annual income inequality measured with respect to th i s revised standard of equality should then more clo s e l y approximate pure inter family inequality. Paglin has applied his technique to the Gini c o e f f i c i e n t as follows. He estimates the mean age-income p r o f i l e of the population from annual cross-sectional data in order to construct a Lorenz curve of inequality in the d i s t r i b u t i o n of cohort mean incomes. This new standard of equality, which Paglin c a l l s a P-reference l i n e , replaces the t r a d i t i o n a l 45° l i n e of equality in a Lorenz diagram. The actual d i s t r i b u t i o n of income 1 50 is represented by the usual Lorenz curve showing the share of t o t a l income accruing to the poorest x per cent (0<X<100) of the population. The situation is i l l u s t r a t e d in Figure IV (Paglin's Figure 1B, p. 599). The t r a d i t i o n a l (Lorenz-) Gini c o e f f i c i e n t i s equal to the ra t i o of the area between the diagonal and the Lorenz curve, to the area below the 45° li n e of equality. In Figure IV i t can be seen to include inequality due to interfamily income differences and to intrafamily income variation over the l i f e c y c l e . The l a t t e r source of inequality is represented by the shaded area between the P-reference li n e and the diagonal, and can be measured by a Gini concentration r a t i o which Paglin c a l l s the age-Gini. The area between the P-line and the L - l i n e represents interfamily income inequality. It too can be measured by a Gini c o e f f i c i e n t , l a b e l l e d the Paglin-Gini, which i s most e a s i l y calculated as the difference between the Lorenz-Gini and the age-Gini. Paglin's method for decomposing annual income inequality i s summarized in the following three steps: (1) Calculate the mean age-income p r o f i l e of the population, assign each individual his cohort mean income, and compute inequality in that d i s t r i b u t i o n with respect to a reference d i s t r i b u t i o n in which everyone receives the population-wide mean income. This y i e l d s a measure of age-related inequality,. (2) Compute inequality in the actual d i s t r i b u t i o n of annual income with respect to the usual standard of equality. (3) Subtract age-related from t o t a l annual income inequality, the difference being non-age-related (long-run, interfamily) inequality. I w i l l now apply t h i s technique to AKS and per capita inequality 151 i n d i c e s . 8 2 The population set is assumed to consist of H individuals, each belonging to one of K age-cohorts having n^ members (1<k<K). The d i s t r i b u t i o n of mean cohort incomes i s denoted H!= ( m i ' • • • ' m K l n K ^ ' w n e r e m k is the mean income of cohort k and 1 „ is a unit vector of dimension n, . I wish to measure inequality in the d i s t r i b u t i o n m with respect to an equal d i s t r i b u t i o n of the population-wide mean income, (mJ_ H). 8 3 This requires the existence of a s o c i a l evaluation function, LI W:R"->R1, assumed to be continuous, increasing, and S-concave, whose image i s , (B.1) w=W(y) where y=(y^, . . . is a d i s t r i b u t i o n of income among the H members of the population. I begin, following Atkinson [1970], by defining the equally d i s t r i b u t e d equivalent of m: (B.2) W(plH)=W(m) The properties of W(.) ensure that p i s unique and well-defined for every vector m, so that p may be written e x p l i c i t l y as, (B.3) p=P(m) A per capita index of age-related inequality i s defined as the difference between the mean of m (=m, the mean of (mJ_H)) and i t s equally d i s t r i b u t e d equivalent, p, defined in (B.3): (B.4) A^=m-p 2Since Paglin's work and the foregoing discussion of i t are couched in terms of income rather than consumption, I w i l l continue this practice throughout Appendix B in order to minimize confusion. 3 I t w i l l be important in what follows that the means of m and (mJ_H) are equal. 1 52 The corresponding AKS index i s defined as A divided by m: (B.5) IA=(m-p)/m=1-p/m Indices of t o t a l annual inequality are constructed from the mean and equally d i s t r i b u t e d equivalent of the actual d i s t r i b u t i o n of income. The l a t t e r i s i m p l i c i t l y defined by, (B.6) W(sj_H)=W(y) The properties of W(.) ensure that s is uniquely determined for every y, allowing s to be defined e x p l i c i t l y as, (B.7) s=S(y) The per capita and AKS indices of t o t a l annual income inequality are given by, (B.8) A=m-s (B.9) I=(m-s)/m=1-s/m Paglin-inequality, "the long-run or l i f e t i m e degree of inequality in the economic system" (Paglin [1975, p. 601]), corresponding to the inequality measured by the Paglin-Gini, i s equal to the difference between t o t a l and age-related annual income inequality. That i s , (B.10) Ap=A-AA=(m-s)-(m-p)=p-s (B.1 1 ) Ip=I-I A = (l-s/m)-(1-p/m) = (p-s)/m (B.5) and (B.11) are the components of an additive decomposition of t o t a l r e l a t i v e inequality, (B.9), into age-related inequality and what I w i l l c a l l Paglin-inequality. I have computed these indices for three values of the free parameter of the Atkinson family of inequality indices. The results are presented in Table 13. I wish to compare this approximation of interfamily l i f e t i m e income inequality with a decomposition approach index of intracohort inequality. Paglin 1 53 has argued that, even i f actual l i f e c y c l e income data were available, equality of l i f e c y c l e incomes i s not a reasonable standard of equality when there i s real growth over time because, "there w i l l be very large differences in l i f e t i m e incomes of older workers and young workers entering the labor force . . . and there is no p r a c t i c a l r e d i s t r i b u t i o n scheme which would enable the older workers to approach the probable l i f e t i m e incomes of the younger" [1975, p. 602]. This problem can be overcome, however, by decomposing interpersonal l i f e t i m e inequality within and among age-cohorts. Lifetime inequality i s measured by the intracohort component of t o t a l inequality in the decomposition approach. The empirical results in Table 13 reveal that the decomposition approach index of intracohort inequality r e f l e c t s a lower degree of long-run inequality than does the index of Paglin-inequality. It appears that t h i s i s due to the i n a b i l i t y of the Paglin-inequality index to account for the e f f e c t s of income mobility. Mobility reduces the dispersion of l i f e t i m e incomes much below the annual income estimates . . . . While the P(aglin)-Gini (Paglin-inequality index) adjusts for average age-related inequality i t also f a i l s to catch the accompanying intracohort mobility. U n t i l we are able to modify our s t a t i c inequality c o e f f i c i e n t s by an index of mobility or c o l l e c t more longitudinal household income data for an extended period of time, our estimate of inequality of l i f e t i m e incomes (or the more d i f f i c u l t trend in the inequality of l i f e t i m e income) w i l l remain crude (Paglin [1977, p. 527]). In the decomposition approach, the three components of t o t a l inequality measure inequality a r i s i n g from different sources. This decomposition of inequality i s not e n t i r e l y clear cut, depending somewhat on the number of years of data that are used 154 in the computation of the indices. Intrapersonal inequality, for example, picks up the effect of income variation over time. When there are only a few years of data t h i s index largely r e f l e c t s the impact of income mobility although in small part i t also accounts for the effect of individuals being at d i f f e r e n t stages of the l i f e c y c l e . The l a t t e r i s true because the slope of a segment of a l i f e c y c l e income p r o f i l e depends on an individual's age. Early in the working l i f e i t may r i s e quite steeply while l a t e r , though higher, i t generally l e v e l s off subst a n t i a l l y . Where the slope is greater, intrapersonal inequality w i l l be greater even i f there i s zero income mobility ( i . e . variation around the l i f e c y c l e income path). The interpretation of intercohort inequality also changes with the number of years of data. When few years of data are used to compute indices in the decomposition approach, intercohort inequality predominantly r e f l e c t s the stage-of-l i f e c y c l e e f f e c t . A young cohort w i l l have a representative income which i s less than that of a middle-aged cohort precisely because of the t y p i c a l shape of income p r o f i l e s . Where the horizon of the data set i s s u f f i c i e n t to cover a large part of the l i f e c y c l e of each i n d i v i d u a l , representative l i f e c y c l e income (the equally d i s t r i b u t e d equivalent of an individual's income stream) i s a reasonable summary s t a t i s t i c of l i f e c y c l e income, and intercohort inequality r e f l e c t s interpersonal (representative) income differences a t t r i b u t a b l e to economic growth. The interpretations of these inequality indices are pertinent to the comparison of them with the Paglin-inequality 1 55 indices. With ten years of data the index of intrapersonal inequality may be regarded as predominantly r e f l e c t i n g the effects of income mobility, and, to a much lesser extent, stage-o f - l i f e c y c l e e f f e c t s which are primarily captured in the intercohort term. The empirical difference between the Paglin-inequality index and the decomposition approach index of intracohort inequality is due to. the i n a b i l i t y of the former to account for the effects on long-term inequality of income mobility. Thus i t might be expected that the sum of intrapersonal and intracohort inequality would approximate Paglin inequality. This i s , in fact, borne out by the results reported in Table 13. For each value of the free parameter, the sum of intrapersonal and intracohort inequality l i e s within the range of the ten annual Paglin-inequality indices. Furthermore, intercohort inequality should be approximately equal to the index of age-related inequality, with a tendency to be s l i g h t l y less than that index because some part of the inequality due to the shape of l i f e c y c l e income p r o f i l e s i s captured by the intrapersonal inequality index, as argued above. 8 4 This hypothesis i s confirmed for low degrees of inequality aversion. In support of his suggested procedure for measuring long-run inequality, Paglin argued that economic growth renders l i f e t i m e income equality an unreasonable standard against which to measure inequality in the d i s t r i b u t i o n of l i f e t i m e income. Paglin also c r i t i c i z e d the use of age-specific Gini c o e f f i c i e n t s "This would also explain why the intrapersonal inequality index values tend to l i e toward the upper end of the range of the annual Paglin-inequality indices. 156 (and, by implication, any other inequality indices) on the grounds that, the empirical c o e f f i c i e n t s available are not r e a l l y s p e c i f i c by age of family head but in fact represent broad age groups. This introduces spurious income variance by not f u l l y eliminating the effect of the age income p r o f i l e . However, even i f we had t r u l y age-s p e c i f i c G i n i , we would have the problem of weighting and combining fifty-some measures into one c o e f f i c i e n t (Paglin [1975, p. 602]). Paglin's f i r s t point i s , in fact, most applicable to his own procedure for measuring long-term inequality and, as has been seen, represents no real d i f f i c u l t y for the use of age-specific inequality measures. Furthermore, the "problem of weighting and combining fifty-some measures into one c o e f f i c i e n t " i s f u l l y resolved in the procedure for decomposing inequality within and among population subgroups suggested by Blackorby, Donaldson, and Auersperg [1981]. To investigate the performance of age-s p e c i f i c annual income inequality indices v i s - a - v i s the generalized Paglin procedure and the decomposition approach (involving the use of data covering more than a single year), I have estimated annual income inequality and decomposed i t within and among age-cohorts using the Blackorby-Donaldson-Auersperg procedure. The results are reported in Table 14. Comparing the results with those of the generalized Paglin procedure reported in Table 13, i t can be seen that intracohort inequality i s very close to Paglin inequality, especially for low degrees of r e l a t i v e inequality a v e r s i o n . 8 5 When R=-5, however, there i s 5For R=.5, Paglin-inequality is never more than .002 less than intracohort inequality in the same year. The corresponding figure i s .005 for R=-.5 (except in the l a s t year when Paglin-inequality exceeds intracohort inequality by .009). 157 s i g n i f i c a n t d i s p a r i t y between indices of Paglin-inequality and intracohort inequality. This appears to be due to the extreme annual variation in age-related inequality at higher values of r e l a t i v e inequality aversion, re s u l t i n g in similar i n s t a b i l i t y of Paglin-inequality over time. The pattern of annual intracohort inequality exhibits a high degree of s t a b i l i t y at a l l values of the free parameter, R. This I take as p a r t i a l evidence of the superiority of decomposing annual income inequality within and among age cohorts to approximate long-run inequality rather than using the generalized Paglin technique developed herein. In conclusion, where panel data are not avai l a b l e , the Blackorby-Donaldson-Auersperg procedure for decomposing inequality can be used to compute an estimate of. long-run inequality. It appears from the empirical work to offer more r e l i a b l e estimates than the generalized Paglin procedure, p a r t i c u l a r l y at high degrees of r e l a t i v e inequality aversion. Both, however, are incapable of accounting for the e f f e c t s of income mobility on measured inequality, which can be accomplished only when panel data are available. In t h i s case the decomposition approach i s the best method of measuring l i f e t i m e income inequality. 158 Population share 159 T A B L E 13 T o t a l , A g e - R e l a t e d , a n d P a g l i n I n e q u a l i t y I n d i c e s R = . 5 R = - . 5 R=-5 YEAR I I I 1968 . 0 9 1 7 .0081 . 0 8 3 6 . 2 6 1 6 .0231 . 2 3 8 5 . 6 9 6 4 . 1 364 . 5 6 0 0 1969 . 0 8 8 9 . 0 0 7 8 . 0 8 1 0 . 2 5 5 4 . 0 2 1 8 . 2 3 3 7 . 6 9 0 8 . 1 1 32 . 5 7 7 7 1970 .0871 . 0 0 8 2 . 0 7 9 0 .2441 . 0 2 2 5 . 2 2 1 6 . 6 4 4 4 . 0 7 7 6 . 5 6 6 8 1971 . 0 8 6 0 . 0 0 9 5 . 0 7 6 5 . 2 4 2 7 .0261 .21 66 . 7 9 2 9 . 0 8 7 9 . 7 0 5 0 1 972 . 0 8 6 9 .0071 . 0 7 9 8 .251 4 . 0 1 9 8 .231 6 . 7 3 5 0 . 0 6 9 5 . 6 6 5 5 1973 . 0 8 1 8 . 0 0 6 5 . 0 7 5 3 . 2 3 0 0 . 0 1 8 2 . 2 1 1 8 .6531 . 0 6 6 6 . 5 8 6 5 1 974 . 0 9 0 2 . 0 0 7 7 . 0 8 2 5 . 2 5 2 8 . 0 2 1 7 .231 2 . 8 6 6 4 . 0 8 0 8 . 7 8 5 6 1975 . 0 9 1 2 . 0 0 6 8 . 0 8 4 4 . 2 5 0 5 . 0 1 9 4 .231 1 . 6 5 5 2 .071 1 . 5 8 4 0 1976 . 0 9 2 5 . 0 0 6 0 . 0 8 6 6 . 2 6 0 4 .01-69 . 2 4 3 6 . 8 1 0 6 . 0 6 6 5 .7441 1 977 . 0 8 6 5 . 0 0 5 2 . 0 8 1 3 . 2 5 4 9 .01 52 . 2 3 9 7 . 8 1 2 9 . 0 6 7 9 . 7 4 5 0 1 60 T A B L E 14 I n t r a - a n d I n t e r - • A g e - C o h o r t I n e q u a l i t y I n d i c e s R = . 5 R = - . 5 R=-5 YEAR I ne I IAc I IAC *Rc 1968 . 0 9 1 5 , . 0 8 4 5 . 0 0 7 0 . 2 6 1 8 . 2 3 9 7 . 0 2 2 0 . 6 9 6 4 .5571 . 1 3 9 3 1969 . 0 8 8 7 . 0 8 2 3 . 0 0 6 5 . 2 5 5 6 . 2 3 6 9 . 0 1 8 7 . 6 9 0 9 . 5 4 9 7 .1412 1 970 . 0 8 7 0 .0801 . 0 0 6 9 . 2 4 4 2 . 2 2 6 0 . 0 1 8 2 . 6 4 4 4 .5271 .1173 1971 . 0 8 5 8 . 0 7 7 8 .0081 . 2 4 2 8 . 2 1 9 9 . 0 2 2 8 . 7 9 2 9 .5361 . 2 5 6 8 1 972 . 0 8 6 7 . 0 8 0 6 . 0 0 6 2 . 2 5 1 6 . 2 3 2 2 . 0 1 9 3 . 7 3 5 0 . 5 5 5 9 .1791 1 973 .081 6 . 0 7 6 3 . 0 0 5 3 .2301 .21 42 . 0 1 5 9 .6531 .5191 . 1 340 1974 . 0 9 0 0 . 0 8 3 3 . 0 0 6 7 . 2 5 3 0 .231 5 . 0 2 1 5 . 8 6 6 4 . 5 2 8 3 .3381 1975 . 0 9 1 0 .0851 . 0 0 5 9 . 2 5 0 6 . 2 3 1 7 . . 0 1 8 9 . 6 5 5 2 . 5 2 5 5 . 1 2 9 7 1976 . 0 9 2 4 .0871 . 0 0 5 3 . 2 6 0 6 . 2 4 3 0 . 0 1 7 6 . 8 1 0 6 . 5 6 0 6 . 2 5 0 0 1 977 . 0 8 6 3 .081 1 . 0 0 5 2 .2551 . 2 3 0 3 . 0 2 4 8 . 8 1 2 9 . 5 5 7 3 . 2 5 5 6 161 A P P E N D I X C D a t a T h e f o l l o w i n g v a r i a b l e s , d r a w n f r o m t h e e l e v e n y e a r f a m i l y t a p e o f t h e P a n e l S t u d y on I n c o m e D y n a m i c s , w e r e u s e d i n t h e c o n s t r u c t i o n o f t h e c o n s u m p t i o n v a r i a b l e . 1 ASFS=Amount S a v e d on F o o d S t a m p s F U M Y = F a m i l y U n i t Money I n c o m e HV=House V a l u e ' O R P A H = O t h e r R e t i r e m e n t ( i n c o m e ) , P e n s i o n s , a n d A n n u i t i e s : H e a d O R P A O = O t h e r R e t i r e m e n t ( i n c o m e ) , P e n s i o n s , a n d A n n u i t i e s : O t h e r s R M P = R e m a i n i n g M o r t g a g e P r i n c i p a l R V F H = R e n t a l V a l u e o f F r e e H o u s i n g T R H S = T R a n s f e r s : H e a d & S p o u s e T R O = T R a n s f e r s : O t h e r s T X H S = T a X e s : H e a d & S p o u s e T X O = T a X e s : O t h e r s T X = T a X e s : t o t a l h o u s e h o l d I b e g a n by c a l c u l a t i n g h o u s e h o l d t o t a l s f o r t h o s e v a r i a b l e s t h a t a r e d i s a g g r e g a t e d by H e a d a n d O t h e r s (ORPA) o r by H e a d & S p o u s e a n d O t h e r s ( T R , T X ) . T h u s , ( C . 1 ) ORPA = ORPAH + ORPAO ( 1 9 7 4 - 7 7 ) 1 T h e e l e v e n y e a r t a p e i s d a t e d 1 9 6 8 - 1 9 7 8 . S i n c e d a t a a r e c o l l e c t e d f o r t h e p r e v i o u s y e a r , t h i s c o r r e s p o n d s t o t h e c a l e n d a r y e a r s 1 9 6 7 - 7 7 . D a t a on t r a n s f e r s , T R , w e r e not. c o l l e c t e d f o r 1 9 6 7 , s o I h a v e u s e d o n l y t h e l a s t t e n y e a r s o f d a t a , 1 9 6 8 - 7 7 . 1 62 (C.2) TR = TRHS + TRO (1968-77) (C.3) TX = TXHS + TXO (1969-77) I then calculated the imputed rental value of owner-occupied housing, ROOH, according to the formula prescribed in the Panel  Study on Income Dynamics: Since RMP is missing for the years 1972-74, I estimated i t s value by linear interpolation. There were two other cases of missing data: ORPA for 1968 and ASFS for 1973. These were both assigned zero values for a l l households. Two further adjustments were necessary before I could compute estimates of gross income, net income, and consumption. F i r s t , the income variable on the data tape, FUMY, is defined to include t o t a l transfers, TR. Since these are not a part of gross income, I calculated Family Unit Money Income from Market Sources as, (C.5) FUMYMS = FUMY - TR (1968-77) I then calculated gross income, YG, as, (C.6) YG = FUMYMS + RVFH (1968-77) Second, in the data set, TR i s defined to include ORPA. In keeping with my d i s t i n c t i o n between interpersonal and intertemporal r e d i s t r i b u t i o n , I wished to calculate net income, YN, as gross income adjusted for public transfers and private interpersonal r e d i s t r i b u t i o n . This required that ORPA, which represents private intertemporal r e d i s t r i b u t i o n , be excluded from TR in the computation of net income: (C.7) YN = YG + (TR - ORPA) - TX + ASFS (1968-77) F i n a l l y , I calculated consumption, C, as, (C.4) ROOH = .06(HV - RMP) (1968-77) 163 (C.8) .' C = YN + ORPA + ROOH (1968-77) This t o t a l household nominal consumption variable was then deflated by the U. S. Consumer Price Index (1975=100) (International Monetary Fund [1980, p.343]). The ca l c u l a t i o n of Family Adult Equivalent Consumption and the grouping of families into age-cohorts required the following demographic variables from the Panel Study on Income Dynamics: AFH=Age of Family Head NFU=Number in the Family Unit NCFU=Number of Children in the Family Unit A complete l i s t i n g of the number, location, width, name, and year of the variables drawn from the data tape follows: 1968 449 821 5 HV 451 827 5 RMP 457 843 4 RVFH 510 976 4 ASFS 525 1020 5 TRHS 527 1 026 5 TRO 529 1 032. 5 FUMY 532 1 042 5 TX 549 1 077 2 NFU 550 1 070 2 NCFU 1 008 1829 2 AFH 1 1 22 2041 5 HV 1 1 24 2047 5 RMP 1 1 30 2063 •4 RVFH 1 183 21 94 4 ASFS 1208 2244 5 TXHS 1213 2265 5 ORPA 1 220 2284 5 TRHS 1 224 2299 5 TXO 1225 2304 5 TRO 1238 2347 2 NFU 1239 2349 2 AFH 1 242 2354 1 NCFU 1514 2706 5 FUMY 1823 3542 5 HV 1825 3548 5 RMP 1831 3564 4 RVFH 1884 3695 4 ASFS 1910 3750 5 TXHS 1915 3771 5 ORPA 1969 1970 1 64 1922 3790 5 TRHS 1 926 3805 5 TXO 1 927 381 0 5 TRO 1 941 3853 2 N F U 1942 3855 2 A F H 1945 3860 1 N C F U 2226 4226 5 FUMY 2423 4542 5 HV 2425 4548 5 RMP 2431 4564 4 RVFH 2478 4680 4 A S F S 251 1 4751 5 TXHS 251 6 4772 5 ORPA 2523 4791 5 TRHS 2527 4806 5 TXO 2528 481 1 5 TRO 2541 4854 2 N F U 2542 4856 2 A F H 2545 4861 2 N C F U 2852 5253 5 FUMY 3021 5542 5 HV 3025 5553 4 R V F H 3064 5638 5 TXHS 3069 5659 5 ORPA 3076 5678 5 TRHS 3080 5693 5 TXO 3081 5698 5 TRO 3094 5741 2 N F U 3095 5743 2 A F H 3098 5748 2 N C F U 3256 5976 5 FUMY 341 7 6129 5 HV 3421 61 40 4 R V F H 3443 6193 4 A S F S 3476 6261 5 TXHS 3481 6282 5 ORPA 3488 6301 5 TRHS 3492 631 6 5 TXO 3493 6321 5 TRO 3507 6368 2 N F U 3508 6370 2 A F H 351 1 6375 2 N C F U 3676 6618 5 FUMY 3817 6828 6 HV 3821 6841 4 R V F H 3851 6914 4 A S F S 3876 6965 5 TXHS 3881 6986 5 ORPAH 3889 7010 5 TRHS 3893 7025 5 TXO 3899 7050 5 ORPAO 3905 7072 5 TRO -1971 1972 1973 1974 1 65 3920 7123 2 NFU 3921 71 25 2 A F H 3924 71 30 2 N C F U 41 54 7437 5 FUMY 4318 7633 6 HV 4320 7640 5 RMP 4330 7663 4 RVFH 4364 7747 4 A S F S 4390 7836 5 TXHS 4396 7859 5 ORPAH 4404 7895 5 TRHS 4409 791 1 5 TXO 441 3 7931 5 ORPAO 441 9 7961 5 TRO 4435 8016 2 NFU 4436 8018 2 A F H 4439 8023 2 N C F U 5029 8933 5 FUMY 521 7 91 29 6 HV 521 9 9136 5 RMP 5229 9159 4 RVFH 5277 9282 4 A S F S 5301 9367 5 TXHS 5307 9390 5 ORPAH 531 6 9427 5 TRHS 5321 9443 5 TXO 5325 9463 5 ORPAO 5332 9495 5 TRO 5349 9554 2 NFU 5350 9556 2 A F H 5353 9561 2 N C F U 5626 9948 5 FUMY 571 7 10129 6 HV 5719 10136 5 RMP 5727 1 01 54 4 RVFH 5776 1 0274 4 A S F S 5800 1 0359 5 TXHS 5807 10386 5 ORPAH 5815 10419 5 TRHS 5820 1 0435 5 TXO 5825 1 0460 5 ORPAO 5831 1 0487 5 TRO 5849 1 0545 2 NFU 5850 10547 2 A F H 5853 1 0552 2 N C F U 6173 10994 5 FUMY 1975 1 976 1977 166 APPENDIX D Inequality in the Original Sample Measuring inequality in the o r i g i n a l sample raises the question of how to treat families that experience a change of family head. When the new head is of a dif f e r e n t age than the previous one, in which age-cohort should the family be included? For that matter, how i s the family's representative consumption to be calculated: separately for each group of years corresponding to a d i f f e r e n t family head, or over the entire sample period? Although the age-cohort to which a family is assigned depends on the age of i t s head, the family is not defined by i t s head. In the Panel Study on Income Dynamics a family retains i t s identity despite a change of head. In keeping with t h i s , I have calculated each family's representative consumption over the whole sample period, ignoring changes of family head. The d i s t r i b u t i o n of families by age-cohort varies from year to year in the o r i g i n a l sample because of changes in family heads. The choice of which d i s t r i b u t i o n to use to decompose inequality within and among age-cohorts i s a r b i t r a r y , but unimportant to the extent that the results are not sensitive to the decision made. There are, therefore, two comparisons to be made. F i r s t , inequality in the o r i g i n a l sample has been measured and decomposed according to the age-cohort d i s t r i b u t i o n s in di f f e r e n t years. The r e s u l t s for 1968 and 1977 are reported in 167 Table 15, from which i t can be seen that measured inequality is quite insensitive to annual differences in the age-cohort d i s t r i b u t i o n . Second, inequality measured in the o r i g i n a l sample can be compared to the subsample results reported in Chapter Four (see Table 3). Inequality in the o r i g i n a l sample is approximately 20 to 25 per cent greater than in the subsample. The breakdown of t o t a l inequality among the three subindices is very similar in the two samples, however, as is the degree to which annual inequality overstates pure interpersonal inequality. I conclude that the subsample results, while understating the degree of actual inequality, do provide an accurate indication of the differences between the t r a d i t i o n a l and decomposition approaches to the measurement of inequality. 1 68 T A B L E 15 R e l a t i v e I n e q u a l i t y i n t h e O r i g i n a l S a m p l e 1968 A g e -• c o h o r t D i s t r i b u t i o n D E C O M P O S I T I O N I N D I C E S R = . 5 R = - . 5 R=-5 I .1113 .31 23 . 9 5 5 3 . 0 2 6 6 ( 2 3 . 9 ) . 0 7 8 4 ( 2 5 . 1) . 2 2 8 3 ( 2 3 . 9 ) .0821 ( 7 3 . 8 ) . 2 2 3 9 ( 7 1 . 7 ) .6171 ( 6 4 . 6 ) He . 0 0 2 6 ( 2 . 3 ) . 0 1 0 0 ( 3 . 2 ) . 1 1 0 0 ( 1 1 . 5 ) ANNUAL I N D I C E S m i n . 0 9 6 9 .2821 . 8 4 5 4 mean . 1 079 .3021 . 9 2 2 5 max .1211 . 3 3 0 3 . 9 6 4 5 1977 Age-- c o h o r t D i s t r i b u t i o n D E C O M P O S I T I O N I N D I C E S R = . 5 R = - . 5 R=-5 I .1113 . 3 1 2 3 . 9 5 5 4 . 0 2 6 6 ( 2 3 . 9 ) . 0 7 8 4 ( 2 5 . 1 ) . 2 2 8 3 ( 2 3 . 9 ) . 0 8 0 2 ( 7 2 . 1 ) . 2 2 2 7 ( 7 1 . 3 ) . 6 0 8 9 ( 6 3 . 7 ) X R C . 0 0 4 5 ( 4 . 0 ) . 0 1 1 3 ( 3 . 6 ) .1181 ( 1 2 . 4 ) ANNUAL I N D I C E S m i n . 0 9 6 8 .2821 . 8 4 5 4 mean . 1 0 7 9 .3021 . 9 2 2 5 max . 1 2 1 0 . 3 3 0 3 . 9 6 4 5 1 69 APPENDIX E A Proof The proposition to be proved is that the members of a family of s o c i a l evalution (or equally distributed equivalent) funcions r e f l e c t i n g "the same set of et h i c a l judgments" are characterized by the same degree of inequality aversion. The p r o o f 8 7 i s in two parts. L With n people, (E.1 ) h " 1 [ ( l / n ) ? l h ( y L ) ] = g - 1 [ ( l / n ) ^ : g ( y t )] i f and only i f g(.) and h(.) are ca r d i n a l l y equivalent. Proof: ( i ) s u f f i c i e n c y : g(.) and h(.) are ca r d i n a l l y equivalent i f they are unique up to a posi t i v e a f f i n e transformation: g(.)=ah(.)+b a>0 In t h i s case, g- 1(y L)=h- 1[(y L-b)/a] and (E.1) can be written, h- 1 [ d/ n ) T h ( y L ) ]=g-1 [ (l/n)2r(ah( y i. )+b) ] =g- 1 [a( 1/n)£lh(y. )+b] =h" 1[ (a(l/n)Ih(y t)+b-b)/a] =h- 1[(1/n)2:h(y L)] ( i i ) n e c e s s i t y : Let z^=g(y l)/n. Then, 7 I am indebted to David Donaldson for the following proof of this proposition. 1 70 h(y L)/n=h(g- 1(nz l))/n=f(z c) (E.1) can then be written, h - 1 [ 2 l f ( z l ) ] = g - 1 [ 2 l z L ] , or (E.2) k ( Z z = )= Z f (z; ) (E.2) is a Pexider equation whose solution i s , 8 8 f(t)=at+b a>0 Therefore, h(y L)=nf(z t) =n(az L+b) =ag(yu)+nb =ag(y L)+b || II . The Population Substitution P r i n c i p l e implies that the equally d i s t r i b u t e d equivalent functions can be written, s„=S n(y)=g- 1[d/n)^g(y c)] Vn Proof: The Population Substitution P r i n c i p l e implies that S*(.) is a d d i t i v e l y separable: (E.3) S n(y)=g; 1[(1/n)£ g„(y t)] Let m<n. (E.3) can be written (E.4) s"(y)=g- 1[(m/n)g n{g^[( l/m ) tg w(y.)} + ( l / n ) i i g M ( y i ) ] Equating (E.3) and (E.4) and dropping the last m-n common terms y i e l d s , g ;i [ ( l/m )Sg r t(y , ) ]=g^ 1 [ ( l/m ) ig m(y c)] It follows from part I that g„(.) and g m(.) are c a r d i n a l l y equivalent. || Eichhorn[1978] 

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