Comparing Income Distributions: Statics and Dynamics 1035307324, 9781035307326

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Comparing Income Distributions

Comparing Income Distributions Statics and Dynamics

John Creedy Wellington School of Business and Government, Victoria University of Wellington, New Zealand

Cheltenham, UK • Northampton, MA, USA

© John Creedy 2023

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical or photocopying, recording, or otherwise without the prior permission of the publisher. Published by Edward Elgar Publishing Limited The Lypiatts 15 Lansdown Road Cheltenham Glos GL50 2JA UK Edward Elgar Publishing, Inc. William Pratt House 9 Dewey Court Northampton Massachusetts 01060 USA

A catalogue record for this book is available from the British Library Library of Congress Control Number: 2022952297

This book is available electronically in the Economics subject collection http://dx.doi.org/10.4337/9781035307333

ISBN 978 1 0353 0732 6 (cased) ISBN 978 1 0353 0733 3 (eBook)

EEP BoX

Contents Acknowledgements

ix

1

Introduction 1.1 Chapter Outlines

1 1

2

Alternative Distributions and Metrics 2.1 Value Judgements and Lorenz Curves 2.2 Inequality Measures 2.3 Thirteen Distributions 2.3.1 Adult Equivalence Scales 2.3.2 The Unit of Analysis 2.3.3 The Use of Allocation Rules 2.4 Construction of Distributions 2.5 Inequality Comparisons over Three Years 2.5.1 Comparisons from 2007 to 2010 2.5.2 Comparisons from 2007 to 2011 2.5.3 Adult Equivalence Scales 2.6 Inequality Decompositions 2.7 Conclusions

9 11 14 19 22 23 24 26 29 29 30 35 37 42

3

Interpreting Inequality Measures 3.1 Previous Results for the Gini Measure 3.2 The Atkinson Inequality Measure 3.3 The Abbreviated Welfare Function 3.4 An Equivalent Small Distribution 3.4.1 n-Person Shares 3.4.2 The Leaky Bucket Experiment 3.5 The Pivotal Income 3.6 Conclusions Appendix: Atkinson’s Measure and Aversion

45 47 51 52 53 53 58 61 64 67

4

Inequality-Preserving Changes 4.1 The Gini Measure 4.1.1 Distributions with Equal Means

75 77 78

v

vi

CONTENTS 4.2 4.3

4.1.2 A Common Mean and Variance The Atkinson Measure Conclusions

82 85 87

5

Decomposing Inequality Changes 5.1 Some Preliminaries 5.1.1 Adult Equivalent Scales 5.1.2 Survey Calibration 5.2 Gini Measures for Alternative Distributions 5.3 Inequality and Social Welfare 5.4 Changing Population Structure 5.5 Inequality Decompositions 5.5.1 The Decomposition Method 5.5.2 Contributions to Inequality Changes 5.6 Conclusion

91 95 95 96 97 103 106 111 112 116 118

6

Inequality Over a Long Period 121 6.1 The Measure of Inequality Used 125 6.1.1 Computing the Gini with Grouped Data 128 6.2 Data Sources 129 6.2.1 Types of Income 130 6.2.2 Data Sources Used in Calculations 133 6.3 Empirical Results 135 6.3.1 Gini Indices: All Individuals ‘Total Income’ Before Tax 1935 to 2014 136 6.3.2 Income Data Before and After 1958 139 6.3.3 An Adjustment for the Introduction of PAYE 141 6.3.4 Some Comparisons 144 6.3.5 Gini Indices by Gender and Before and After Tax: 1981–2013 148 6.4 Conclusions 150

7

Regression Models of Mobility 7.1 Some Previous Literature 7.2 The Dataset 7.3 Regression with First-Order Serial Correlation 7.3.1 Specification of the Model 7.3.2 Income Dynamic Estimates 7.4 An Extension: Second-Order Serial Correlation 7.5 Conclusions

155 157 158 160 160 164 168 172

8

Illustrating Differential Growth 8.1 Illustrative Devices for Income Mobility 8.2 The TIM Curve 8.2.1 The TIP Curve 8.2.2 Three Is of Mobility

175 177 178 178 180

CONTENTS 8.3 8.4

vii

The TIM Curve and Galtonian Regression TIM Curves for New Zealand 8.4.1 Sampling Variability 8.4.2 Comparisons with Cumulative Income Growth Profiles 8.5 Conclusions Appendix: Alternative TIM and Income Share Curves

193 195 197

9

Summary Measures of Equalising Mobility 9.1 A Summary Measure and the TIM Curve 9.2 Positional Changes and the TIM Curve 9.3 Illustrative Examples 9.4 Conclusions

203 205 211 213 217

10

Mobility as Positional Change 10.1 Positional Mobility 10.1.1 Maximum Re-ranking Profiles 10.2 Re-ranking Profiles for New Zealand 10.3 Conclusions

219 220 225 226 232

11

Poverty Persistence 11.1 Measures of Pro-Poor Growth 11.1.1 Growth Incidence Curves 11.1.2 An Elasticity Measure of Pro-Poor Growth 11.2 Income Dynamics 11.3 Poverty Persistence 11.4 New Zealand TIM and Poverty Persistence Curves 11.4.1 Pro-Poor TIM Curves 11.4.2 Longitudinal versus Cross-sectional Inequality 11.4.3 Poverty Persistence Curves 11.5 Conclusions

235 238 238 240 242 243 247 248 250 250 255

Bibliography Index

185 188 191

257 271

Acknowledgements This book is based on the results of a programme of income distribution research carried out in recent years. It therefore consists of revised versions of earlier papers. I am grateful for permission to use these papers. Most of the chapters have been co-authored, and I would like to take this opportunity to acknowledge the many valuable contributions of my collaborators. In particular, I would like to mention Norman Gemmell, who is a co-author of six of the chapters and has long been a greatly appreciated source of encouragement and support. Chapter 2 is based on Creedy, J. and Eedrah, J. (2016) Income redistribution and changes in inequality in New Zealand from 2007 to 2011: alternative distributions and value judgements. New Zealand Economic Papers, 50, pp. 129-152. Chapter 3 is based on Creedy, J. (2016) Interpreting inequality measures and changes in inequality. New Zealand Economic Papers, 50, pp. 177-192. The appendix uses material from Creedy, J. (2019) The Atkinson inequality measure and inequality aversion. Victoria University of Wellington Chair of Public Finance Working Paper WP01/209. Chapter 4 is based on Creedy, J. (2017) A note on inequality-preserving distributional changes. New Zealand Economic Papers, 51, pp. 86-95. Chapter 5 is based on Ball, C. and Creedy, J. (2016) Inequality in New Zealand 1983/84 to 2013/14. New Zealand Economic Papers, 50, pp. 323342. Chapter 6 is based on Creedy, J., Gemmell, N. and Nguyen, L. (2018) ix

ACKNOWLEDGEMENTS

x

Income inequality in New Zealand 1935-2014. Australian Economic Review, 51, pp. 21-40. Chapter 7 is based on Creedy, J., Gemmell, N. and Laws, A. (2021) Relative income dynamics of individuals in New Zealand: new regression estimates. New Zealand Economic Papers, 55, pp. 203-220. Chapter 8 is based on part of Creedy, J. and Gemmell, N. (2019) Illustrating income mobility: new measures. Oxford Economic Papers, 71, pp. 733-755. Chapter 9 is based on Creedy, J. and Gemmell, N. (2023) Summary measures of equalising income mobility based on ‘Three "I"s of Mobility’ curves. Journal of Income Distribution (forthcoming). Chapter 10 is based on part of Creedy, J. and Gemmell, N. (2019) Illustrating income mobility: new measures. Oxford Economic Papers, 71, pp. 733-755. Chapter 11 is based on Creedy, J. and Gemmell, N. (2018) Income dynamics, pro-poor mobility and poverty persistence curves. Economic Record, 94, pp. 316-328.

Chapter 1 Introduction This book is concerned with two main features of income distribution comparisons. The …rst …ve chapters (Chapters 2 to 6) examine a range of technical aspects of inequality measurement, including less well-known properties of inequality indices, and the decomposition of inequality changes into component contributions. The next …ve chapters (Chapters 7 to 11) are concerned with various aspects of the graphical display and measurement of income mobility. There is no attempt to offer a comprehensive treatment, or indeed the kind of systematic treatment that would be required of a textbook. Instead it brings together, in much revised form, a number of contributions made over recent years. The book may in some ways be regarded as a sequel to three earlier volumes – Creedy (1985, 1992, 1998) – which also concentrate on the statics and dynamics of income distribution. While the main focus is on methods, illustrative examples are provided using New Zealand data. The remainder of this introduction brie‡y summarises each of the chapters in turn.

1.1

Chapter Outlines

The greater availability of large micro-datasets makes it possible for researchers to examine a wide range of distributions. Chapter 2 sets out the

1

2

CHAPTER 1. INTRODUCTION

range of alternatives, and resulting summary measures, which could be used to compare inequality over time. Using an annual accounting period, alternative welfare metrics and units of analysis are investigated. Thirteen different distributions can be distinguished. In addition, the sensitivity to assumptions about economies of scale within households is examined, and changes in inequality are decomposed into those arising from population and tax structure changes. Illustrations, using New Zealand data for the period 2007 to 2010, show that judgements about changes in inequality depend on the precise choices made. For the same time period, the answer to the question of whether inequality has risen or fallen depends crucially on the combination of welfare metric, income unit, adult equivalence scale and inequality measure used. In empirical studies it is therefore important to explore a range of alternative approaches, providing information for readers to make their own judgements. Chapter 3 is concerned with the problem of how to interpret orders of magnitude of inequality changes. For example, the information that a Gini or Atkinson measure of inequality has increased by a given percentage does not have an obvious intuitive meaning. The chapter explores, mainly in the context of the Atkinson inequality measure, various attempts to interpret orders of magnitude in a transparent way. One suggestion is that the analogy of sharing a cake among a very small number of people provides a useful intuitive description for people who want some idea of what an inequality measure ‘actually means’. In contrast with the Gini measure, for which a simple ‘cake-sharing’ result is available, the Atkinson measure requires a nonlinear equation to be solved. This involves comparisons of ‘excess shares’ – the share obtained by the richer person in excess of the arithmetic mean – for a range of assumptions. The implications for the ‘leaky bucket’ experiments, often used to clarify the nature of inequality aversion, are also examined. An additional approach is to obtain the ‘pivotal income’, above which a small income increase for any individual increases inequality. The properties of

1.1. CHAPTER OUTLINES

3

this measure for the Atkinson index are also explored. Chapter 4 discusses a somewhat neglected, although widely recognised, feature of summary measures of inequality, namely that a given numerical value of a measure can be associated with a range of distributions. Just as different distributions can have the same arithmetic mean, it is possible – given sufficient observations – for higher moments of different distributions to be equal. To explore this aspect, the chapter considers the problem of distributing a …xed amount of money (‘income’) among a given number of people, such that inequality (measured by either the Gini or Atkinson measure) takes a speci…ed value. The issue basically corresponds to the wellknown fact that simultaneous equations admit of many solutions where the number of variables exceeds that of equations (constraints). However, the approach examines cases where there are just one or two degrees of freedom, clarifying the resulting range of distributions. The properties of simultaneous disequalising and equalising transfers are discussed. When examining a time series of annual inequality measures, a complication is that there are many other changes taking place, such as changes in the age composition of the population, labour force participation, and household structure. Chapter 5 shows how survey calibration methods, where sample weights are calculated to ensure that certain aggregates match those obtained from independent data, can be used to examine the contributions to changing inequality of a wide range of variables. The chapter examines approximately 50 population characteristics which, as part of the decomposition method, are assumed to remain constant over the period, so that their separate contributions to inequality can be measured. The chapter provides an empirical analysis of annual income and expenditure inequality in New Zealand over a thirty-year period from the early 1980s. The extent of redistribution through the tax and bene…t system is also explored. Household Economic Survey data are used for each year from 1983/84 to 1997/98 inclusive, 2000/01 and 2003/04, and for each year from 2006/07.

4

CHAPTER 1. INTRODUCTION Chapter 6 explores the challenges involved in extending the time period,

over which annual summary measures are compared, to a longer number of years. In particular, comparisons are complicated by the fact that data coverage and de…nitions used by statistical agencies change over time, so that comparable results are often difficult to obtain. As seen in Chapter 5, New Zealand income inequality indices increased during the late 1980s and early 1990s, but there has been limited change thereafter. Little is known about the levels and changes in inequality over prior decades. Such a long term analysis is restricted by the more limited data availability in earlier decades of the last century. Based on previously unexplored data from Statistics New Zealand Official Yearbooks and Inland Revenue, this chapter reports estimates for the Gini index of income inequality for New Zealand from the middle 1930s to the present. Comparisons with estimates for Australia for the period, 1942 to 2001, reveal some remarkable common features. The vast majority of discussions of inequality, like the …rst …ve substantive chapters in the present book, focus exclusively on ‘static’ comparisons of the distribution of annual incomes in different places or times. Yet it has long been recognised that an important characteristic of individuals’ incomes is that they are subject to systematic variations associated with the life cycle, as well as both exogenous and endogenous variations over time for a large variety of reasons, including health, employment and job changes. Such changes taking place in the distribution of annual incomes over time may give a highly misleading indication of inequality, when measured over a longer accounting period. Furthermore, comparisons are often made of the growth rates from one year to the next in, say, deciles of the relevant distributions. Lower growth rates of lower income deciles, compared with the growth rates of higher deciles, are said to indicate inegalitarian mobility. However, what matters are the relative average growth rates experienced by individuals who were initially in various ranges of the distribution. These rates require longitudinal information, and typically demonstrate higher growth

1.1. CHAPTER OUTLINES

5

rates, on average, for individuals initially in lower deciles compared with those in higher decile groups. The next …ve chapters are concerned with such dynamic aspects of income distribution comparisons. Chapter 7 shows that the pattern of relative income changes, despite being subject to considerable complexity, can be described succinctly using a simple autoregressive stochastic process in which ‘regression towards the mean’ is combined with serial correlation in the stochastic term. The parameters of the model are shown to have convenient interpretations and can be estimated using limited longitudinal data. Using a series of random samples of New Zealand individual taxpayers, with each sample containing income data for the same individuals over three consecutive years, reveals substantial regression towards the mean, combined with negative serial correlation. Remarkable stability in the estimated parameters is observed across the samples over the whole 1997 to 2012 period. These imply that relatively high-income individuals have, on average, lower proportional increases in income from one year to the next compared with those with lower initial incomes, and those with a large increase in one year are more likely to experience a relative decrease the following year. Both of these effects, though combined with a stochastic component that on its own would tend to increase inequality over time, are sufficient in this New Zealand case to ensure that inequality falls as the accounting period over which incomes are measured increases, and there is no systematic tendency for annual inequality to rise. Despite the simplicity of the dynamic process speci…ed, it is nevertheless capable of explaining about 75 per cent of the variation in annual incomes. Chapter 8 introduces a graphical device which illustrates ‘at a glance’ the incidence, intensity and inequality of mobility, considered in terms of differential income growth. The curve, named a ‘Three "I"s of Mobility’, or TIM, curve plots the cumulative proportion of the population (from lowest to highest values of initial income) against the cumulative change in log-incomes per capita over a given period. The advantage is that, like the Lorenz curve

6

CHAPTER 1. INTRODUCTION

in the case of static inequality, the curve is simple to produce, provides convenient comparisons of the different dimensions, and can be suggestive of further analysis. Chapter 9 extends the ‘Three "I"s of Mobility’ framework, discussed in Chapter 8, by proposing associated quantitative summary measures of equalising mobility between two periods. The chapter shows that measures can be based, as in the famous Lorenz curve used to depict cross-sectional inequality, on areas within the diagram. These are area measures of the ‘distance’ from the TIM curve to two alternative curves which depict, in different senses, hypothetical extreme equalising mobility cases. The …rst case involves the equalisation of incomes in the second period, such that all second-period incomes are equal to the actual average, and the average growth rate is equal to the actual average growth over the relevant period. This involves a compression of incomes and no re-ranking. If second-period equality is treated as ‘extreme equalisation’, then any re-ranking of incomes (generated by non-systematic changes) can be regarded as ‘frustrating’ redistribution. The second concept involves a different concept of maximum redistribution, de…ned in terms of the inequality of incomes measured over the two periods combined. This hypothetical extreme involves a combination of differential income growth with maximum re-ranking, whereby second-period incomes are ‘swapped’: the richest person becomes the poorest, and so on. In this case, maximum re-ranking is viewed as a fundamental component of equalising change. The measures are illustrated using a large sample of taxpayers’ incomes in New Zealand, obtained from con…dential unit-record …les. It is suggested that these measures of equalising mobility can usefully augment the visual information provided by the TIM curve concept. Chapter 10 turns to the analysis of income mobility, viewed in terms of re-ranking or positional changes. Individuals can move to higher or lower rank positions, so that the explicit treatment of the direction of change becomes important. In de…ning a re-ranking mobility index, it is therefore

1.1. CHAPTER OUTLINES

7

…rst necessary to decide whether negative re-ranking (dropping down the ranking) is treated symmetrically with positive (upward) movement within the ranking. A second issue concerns the choice of whose mobility is to be included. This chapter proposes a simple illustrative device for positional mobility. The chapter shows how a re-ranking mobility curve, analogous to the TIM curve, can illustrate the incidence, intensity and inequality of positional mobility in the form of re-ranking. This plots the cumulative degree of re-ranking against the cumulative proportion of the population (from lowest to highest incomes). Additionally, since for any given fraction of the population there is a different maximum possible extent of re-ranking, it is useful to consider the cumulative ratio of actual-to-maximum re-ranking against the cumulative proportion of the population. Illustrations of re-ranking are examined based on three panels of New Zealand incomes from 1998 to 2010, revealing a high degree of positional mobility, compared with the maximum possible, among the lowest and highest income individuals. This highlights how some conclusions regarding the extent of re-ranking depend crucially on the re-ranking measure adopted – positive, net or absolute. For example, the highest re-ranking ratios are observed around the 50 to the 70 percentiles for an absolute re-ranking measure but rise steadily towards the 100 percentile when a positive re-ranking ratio is considered. Chapter 11 suggests two new illustrative devices for poverty income dynamics. To examine pro-poor mobility in the form of relative income growth, it suggests that TIM curves can be applied to individuals in poverty. In addition to highlighting the ‘three I’s’ properties (incidence, intensity and inequality) of mobility for alternative poverty de…nitions, this allows the relative mobility of each poverty group to be compared with mobility by the population as a whole. To examine poverty persistence, the chapter suggests that a poverty persistence curve can identify both the extent of movement across a poverty threshold and the particular poor and non-poor incomes for which persistence or movement is prevalent. Applying these concepts to New

8

CHAPTER 1. INTRODUCTION

Zealand income data for individual income taxpayers showed that income dynamics were especially pro-poor during the period 2006 to 2010, with much faster income growth for those on the lowest incomes than those higher up the income distribution. For example, a mobility index based on cumulative income growth rates for those with the lowest …ve per cent of 2006 incomes, is on average around ten times higher than the equivalent index for all taxpayers combined. For the lowest twenty-…ve per cent the equivalent index is around four times higher than mobility across all taxpayers. On poverty persistence, average income growth rates within each percentile of the distribution suggest relatively little movement into poverty, but somewhat more movement out of poverty. However, considering all individuals within each percentile (around 330 individuals per percentile in this case) revealed a relatively mobile population overall, with some individuals observed within all percentiles that are above the 2006 poverty threshold moving into poverty over the …ve year period examined.

Chapter 2 Alternative Distributions and Metrics This aim of this chapter is to explore the use of alternative distributions, and the implications of using different summary measures of inequality, to assess the contribution of the tax and transfer system in reducing inequality. Stress is placed on the role of a range of value judgements, the need to be explicit about them and, for rational policy analysis, the importance of considering the implications of a range of alternative values. The present chapter does not offer new theoretical insights, but it provides an illustration of the need to consider alternative distributions and measures, showing that unequivocal results are seldom available. The context is one – not unusual in economics – in which there is something of a dichotomy between theory and empirical analysis. While considerable attention has been given to inequality in recently policy debates, those reporting empirical evidence often provide only a limited range of results and do not always clarify either the nature of the income concept (including, where relevant, the adult equivalence scales used) or the unit of analysis. In the latter case, the individual is most often used without comment, although it turns out that a number of familiar results regarding comparisons may not necessarily be appropriate: for example, widely used welfare functions can in some circumstances be ‘inequality preferring’. 9

10

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS Alternative approaches are illustrated using New Zealand Household Eco-

nomic Survey data. Results were obtained using the New Zealand Treasury’s microsimulation model, Taxwell.1 The accounting period is thus necessarily a year. Given the use of an annual measure, choices must then be made regarding precisely what is to be measured and the unit of analysis. The former choice concerns what is often referred to as the ‘welfare metric’. For example, this may be pre-tax incomes, wage rates, or a measure of expenditure or consumption. It may be based on individuals’ income, or could assume some form of income sharing within households or families, in which case adult equivalence scales are adopted. Here no attempt is made to allow for the value of leisure.2 A further decision involves the unit of analysis, which could be the family, the household, the individual or the ‘adult equivalent person’: essentially, the choice of unit of analysis determines the weight attached to the welfare metric. Both the welfare metric and the income unit could be arti…cial measures, designed to allow for differences in the composition of households using adult equivalence scales. Ultimately these choices cannot avoid the use of value judgements, so it is important for empirical studies to provide a range of clearly described alternative results, thereby allowing readers to make their own judgements. Indeed, the results presented below demonstrate that the answer to the question of whether inequality has risen or fallen in recent years depends crucially on the combination of welfare metric, income unit and adult equivalence scales used. Section 2.1 begins by brie‡y rehearsing some basic features of inequality comparisons involving Lorenz curves and the value judgements summarised by a type of social welfare function. Section 2.2 describes the inequality measures used, namely the Atkinson and Gini measures, paying particular 1 The

‘well’ in the name of the model comes from Ivan Tuckwell, who contributed greatly to tax and bene…t modelling in the New Zealand Treasury. 2 Furthermore, no attempt is made to allow for changes over time such as the introduction of new commodities, or relative price changes which may have differential impacts on different income groups.

2.1. VALUE JUDGEMENTS AND LORENZ CURVES

11

attention to the value judgements involved. The implications of including zero values in the distributions are also examined brie‡y. Section 2.3 describes the range of distributions examined, distinguished by welfare metric and income unit. The often-neglected value judgements involved in choosing alternative units are discussed. The data and construction of alternative distributions are explained in Section 2.4. Inequality measures for New Zealand in 2007, 2010 and 2011 are compared in Section 2.5. The period from 2007 to 2010 covers years which may be thought to be substantially affected by the global …nancial crises. However, there were few changes in the tax structure. Major reforms took place in 2010, so comparisons involving 2011 are of interest. The sensitivity of results to the assumption regarding economies of scale within households is examined in Section 2.5. A decomposition of inequality changes into population and tax structure changes is presented in Section 2.6. In view of the timing of the tax reforms, the empirical decompositions are examined for the period 2007 to 2011.

2.1

Value Judgements and Lorenz Curves

The most widely-used graphical device used to compare relative inequality (among income units regarded as having no relevant non-income characteristics) is the Lorenz curve. With all incomes ordered from lowest to highest, this plots the cumulative proportion of total income against the corresponding cumulative proportion of people. Hence concern is with relative inequality, so that the units of measurement of incomes (and hence the arithmetic means) are irrelevant. Consider the two Lorenz curves shown in Figure 2.1, where distribution A lies everywhere inside that of distribution B. That is, A’s curve is closer to the upward sloping diagonal line of equality which arises if all incomes are equal. Distribution A is said to ‘Lorenz dominate’ distribution B.

12

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

Figure 2.1: Two Lorenz Curves Further insight was provided by Atkinson (1970), who pointed out that the intuitive judgement that distribution A is unequivocally less unequal than distribution B is consistent with the value judgement expressed by the ‘principle of transfers’.3 This principle is the inequality-disliking value judgement which takes the view (again in the context of homogeneous units) that an income transfer from a richer to a poorer unit, which leaves their relative rank unchanged, reduces inequality. That is, it is possible to move from distribution B to distribution A by a series of transfers, each of which satis…es the principle of transfers. Faced with the desire to compare relative inequality, it is therefore useful to begin simply by examining Lorenz curves to see if this kind of dominance applies. However, in practice – and certainly in the case of the distributions compared in this chapter – such dominance results are rarely available, and it is necessary to make additional value judgements. One way that distributions can be more widely evaluated involves the use 3 The

choice of metric and unit, prerequisites for drawing the Lorenz curve, also involve value judgements, as discussed further below.

2.1. VALUE JUDGEMENTS AND LORENZ CURVES

13

of a social welfare function, expressing explicitly the value judgements imposed in making comparisons. For a distribution  , for  = 1  , suppose the evaluation function – representing the value judgements of an indepenP dent judge – takes the form, =1  ( ), where  ( ) is a function representing the contribution of individual ’s income to  .4 The basic value judgements shared by all judges whose  functions take this form are that evaluations are individualistic, additive and Paretean (such that an improvement for any one unit, with no units being worse off, is judged to increase  ). Furthermore, if  () is concave, so that the slope of the function falls as  increases and for    ,  ( )    ( )  . This additional assumption re‡ects adherence to the principle of transfers (a rank-preserving transfer from  to  must increase  ), where the degree of concavity re‡ects the extent of aversion to inequality. Atkinson (1970) also established that, if the two distributions have the same arithmetic mean income, all functions of this general kind would judge distribution A to be superior to B, in that it gives a higher value of  , as well as being more equal. This result is true irrespective of the precise extent of aversion to inequality.5 If Lorenz dominance is established, all judges who have these basic value judgements would agree about which distribution is preferred to the other. If the arithmetic means of the two distributions differ, welfare (as opposed purely to inequality) comparisons require an explicit trade-off between (loosely speaking) ‘equity and efficiency’. Shorrocks (1983) showed that Atkinson’s result can be extended if, instead of the Lorenz curve, the concept of the Generalised Lorenz curve is used. This plots the product of 4 It

is tempting to think of  () as representing a (cardinal) utility function, assumed to be the same for all individuals. The case where  () =  (implying no aversion to inequality) thus corresponds to the ‘Classical utilitarian’ case. However, it is necessary to think of  () as simply representing the contribution of  to  , re‡ecting the independent judge’s views. 5 For details and elaborations for special cases where further assumptions regarding value judgements can be used to establish dominance results when curves intersect, see Lambert (1993).

14

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

the proportion of total income and the arithmetic mean income against the corresponding proportion of people. Thus, the vertical axis of the Lorenz curve is ‘stretched’ by an amount depending on the arithmetic mean. It is possible to …nd that distribution A is more equal than B, but B is preferred to A if B’s Generalised Lorenz curve is everywhere above that of A. As with the Lorenz curve, dominance results are seldom available, so that more structure needs to be given to the welfare function; that is, more speci…c value judgements need to be speci…ed, leading to particular inequality and welfare measures. These are discussed in the following Section. When making comparisons between pre-tax and post-tax and transfer incomes in any period, only inequality measures are relevant, but when comparing distributions over time, where mean incomes are expected to change, both inequality and welfare measures are of concern.

2.2

Inequality Measures

This section describes the inequality measures used below, paying particular attention to the value judgements associated with each measure. It is also necessary to recognise that some of the distributions examined have zero incomes. The formulae given are for unweighted distributions, but in practice weights are used to deal with both the sample weights (for aggregation to population values) and, in some cases, the unit of analysis. The Atkinson measure, for a relative inequality aversion parameter of , is de…ned as the proportional difference between the arithmetic mean and the ‘equally distributed equivalent’ income. The measure is based on a social welfare function, representing the value judgements of an independent observer, of the form:

1 X 1−   =  =1 1 −  

(2.1)

for  = 0 and  6= 1, and incomes of  , for  = 1  . If  = 1, then

2.2. INEQUALITY MEASURES  =

1 

P

=1

15

log  . The equally distributed equivalent,  , is that income

level which, if obtained by every unit, gives the same ‘total welfare’ as the actual distribution; hence  is the power mean: Ã  =

1 X 1−   =1  

!1(1−) (2.2)

Then for arithmetic mean of ¯, the Atkinson measure,  , is:  = 1 −

 ¯

(2.3)

From the form in (2.1), it is clear that this is a member of the broad class of welfare functions that are individualistic, additive, Paretean, and satisfy the principle of transfers.6 From (2.3),  = ¯ (1 −  ), which expresses the equally distributed equivalent income in terms of ¯ and  . Hence the value of  corresponding to the distribution can be written as 1−   (1 − ) = {¯  (1 −  )}1−  (1 − ). This re‡ects exactly the same ‘trade-off’ between equality, (1 −  ), and mean income as  itself. Hence, the welfare function associated with the Atkinson measure can be expressed in ‘abbreviated’ form as  = ¯ (1 −  ).7 The nature of the trade-off is an important implication of the basic value judgements underlying the use of the Atkinson measure. Any distribution that is concerned with market income and includes nonworkers (and those without other income sources) can have income units with zero income. In these cases, care must be taken in using and interpreting Atkinson inequality measures. To illustrate a difficulty in the presence of zero values, suppose there are  individuals with incomes of [0 1 1  1], so there is only one unit with a zero value and the rest have equal incomes of 6 It

is obviously possible to modify the form of  () to allow, for example, for constant absolute inequality aversion rather than constant relative aversion, but for convenience the latter speci…cation is used here. 7 Instead of writing  , as in (2.1), in terms of all individual incomes, the abbreviated form is expressed in terms of summary measures of the distribution; see Lambert (1993). The abbreviated form is also convenient to avoid negative values of  in cases where   1.

16

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

1 unit, and let  = 05. The equally distributed equivalent income is thus ¡ ¢2 , so that 05 = 1. Hence . The arithmetic mean is ¯ = −1  = −1   for large  inequality is zero. However, for all  ≥ 1, inequality is 1. Zero values thus need to be omitted. In comparing results for different values of relative inequality aversion, it is useful to consider the hypothetical ‘leaky bucket’ experiment suggested by Atkinson when proposing his measure. Consider taking a small amount from person 2, with income of 2 , and transferring part of this to person 1, with 1  2 , so that the effective ‘tax-transfer’ is equalising. The transfer is thought to involve the use of a leaky bucket, so that some income is lost in the process. A judge’s aversion to inequality is re‡ected in the tolerance of leaks. Totally differentiating  in (2.1) with respect to 1 and 2 gives: −  = − 1 1 + 2 2

Transfers which leave  unchanged are thus given by: ¯ µ ¶ 1 1 ¯¯ =− ¯ 2  2

(2.4)

(2.5)

Convert changes to discrete form, and consider taking 1 unit from the richer person, so that ∆2 = −1. The minimum amount that must be given to person 1 is thus:

µ ∆1 =

1 2

¶

(2.6)

Hence, the judge would tolerate a leaking bucket up to a maximum leak of ³ ´ 1 − 21 . The tolerance thus depends on the initial relative incomes of the two individuals and the value of . Figure 2.2 illustrates the leakage from a ‘tax’ of 1 unit that would be tolerated by a judge with varying  values, for three different ratios of 1 to 2 . The values involved in using the Atkinson measure may be compared with the Gini inequality measure. Geometrically, this can be regarded as a ‘distance measure’ of the difference between the Lorenz curve of the distribution

2.2. INEQUALITY MEASURES

17

Figure 2.2: The Leaky Bucket Experiment from the line of equality in Figure 2.1. A commonly used expression for the Gini inequality measure, , for 1  2  3     , is: 2 X 1 − 2 ( + 1 − )    ¯ =1 

=1+

(2.7)

Clearly for the distribution, [0 1 1  1],  tends to zero as  increases, and a Gini value of 1 results from [0 0 0  0 1]. The value judgements associated with the use of the Gini measure are very different from those underlying the Atkinson measure. However, an interpretation in terms of value judgements is discussed below. First, further insight into the Gini can be obtained by de…ning ¯ as an ‘reverse-order-rank-weighted mean’ of  , given by: P =1 ( + 1 − )  ¯ = P  =1 ( + 1 − )

(2.8)

That is, each value is given a weight given by its ‘reverse rank’ (that is, its rank when in descending order – ordered from rich to poor, rather than poor

18

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

to rich). Using

P

=1

 =  ( + 1) 2, it can be seen that: 1 +   ( + 1) ³ ¯ ´ = −  2 ¯

(2.9)

For large samples this reduces to: =1−

¯ ¯

(2.10)

Hence the Gini has some super…cial similarities with the Atkinson measure: both measures can be expressed as the proportional difference between the arithmetic mean income and another type of average, or measure of location. A further similarity arises when the value judgements underlying the Gini measure are considered although, as shown by Sen (1973), they are very different from those associated with the Atkinson measure. Suppose the contribution to  , the evaluation function of the independent judge, for any pair of individual incomes is equal to the smallest income of the pair. It can be shown that the average welfare across all pairs of individuals is ¯ (1 − ), which is of course a welfare function expressed in abbreviated form. Hence the values implicit in the use of the Gini measure contain a ‘maxi-min’ kind of idea, that only the lowest income matters in all pairwise comparisons. The similarity in terms of abbreviated welfare functions means that, for both Atkinson and Gini measures, the form of the trade-off between average income and its inequality is similar for each case, although of course the magnitudes can differ substantially. Consider a ‘social indifference curve’, showing combinations of ¯ and inequality,  , for which  is constant. By differentiating the abbreviated forms, the slope of such an indifference curve is given for each measure by:

¯  ¯ ¯  ¯¯ = ¯    1 − 

(2.11)

This shows that, implicit in the values behind the use of these measures, a proportional change in inequality of ∆  is viewed as being equivalent to a proportional change in ¯ of ∆  (1 −  ).

2.3. THIRTEEN DISTRIBUTIONS

19

In addition, these inequality measures are de…ned only for  ≥ 0. Although any attempt to include negative values in calculating Atkinson measures immediately runs into difficulties, a value of  can mechanically be obtained. Regarding the inclusion of negative values, for example, the distribution [−2 1 1 3] has an arithmetic mean of ¯ = 34 but a value of ¯ = 0. Substitution into (2.9) gives  = 54 = 125.8

2.3

Thirteen Distributions

Crucial choices when measuring inequality concern the nature of the welfare metric and the income unit. This section describes a range of possible distributions. Clearly, the choice depends on the precise question being asked. The distributions examined are listed in Table 2.1, using distinctions between the welfare metric, the unit of analysis and the use of intra-household sharing rules. The …nal column of the table gives the ‘population’ size, where  is the number of households,  is the total number of individuals,  is the number of ‘adult equivalent persons’, and  is the total number of employed individuals. In the table, the …rst eight distributions listed relate to a welfare metric based on some kind of household income measure. Five distributions with metrics based on individual incomes are then listed. The sequence by which the distributions are constructed is described at the end of this subsection. Considering households, the simplest cases are distributions 1 and 2 in Table 2.1 which refer to total household market and disposable income respectively, for each of the  households. While some households may have no market income, the income transfer system ensures that all disposable incomes are positive. The simplest case involving the distribution of individual market incomes is number 9, where the population consists only of those  8 Some

investigators, such as Hyslop and Yahanpath (2005, p. 7), have overlooked the fact that Ginis should not include negative incomes.

20

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

No. 1 2 3 4 5 6 7 8 9 10 11 12 13

Table 2.1: Alternative Distributions Welfare metric Unit Sharing HH market income Household NA HH disposable income Household NA HH market income per AE Household NA HH disposable income per AE Household NA HH market income per AE Individual Equal HH disposable income per AE Individual Equal HH market income per AE Equiv indiv Equal HH disposable income per AE Equiv indiv Equal Individual market income Individual No Individual disposable income Individual No Individual market income Individual Yes Individual disposable income Individual Yes Individual …nal income Individual Yes

Zeros Yes No Yes No Yes No Yes No No No Yes No No

No.             

individuals participating in paid employment or pro…table self-employment, or who receive income from other sources such as rental, interest and capital income.9 This distribution contains no zero values. It may be compared with distribution number 10, that of disposable income for the same population of individuals. In comparing individuals, comparisons of inequality measures usually assume (that is, take the value judgement) that there are no nonincome differences that are relevant: the units are homogeneous. Given that standard inequality measures are designed to deal with homogeneous units, no special problems arise. However, distributions 1 and 2 involve units which are heterogeneous. The nature of households, their composition and the way resources are shared among members, need to be considered explicitly. There is a complex relationship between the distribution of market incomes and inequality in the distribution of resources as more widely perceived. A challenge is to construct distributions which in some way transform household incomes so that 9 Of course, distributions of these sources may be considered separately, but are combined here, as discussed below.

2.3. THIRTEEN DISTRIBUTIONS

21

suitable comparisons can be made. This is where difficulties arise, not only because of the role of value judgements but because comparisons can involve arti…cial income concepts (such as income per adult equivalent person) and arti…cial units (such as the adult equivalent income unit). These aspects are examined in the following two subsections.

Figure 2.3: The Sequence of Distributions Figure 2.3 illustrates the sequence for compiling the various distributions, moving from the distribution of individual market incomes to household incomes and their transformation. The numbers in square brackets within each box refer to the distribution numbers in Table 2.1. For example, [1 to 2] indicates that comparisons involve the movement from pre-tax household income to post tax and transfer household incomes. Distribution 13, which allocates items of government expenditure to individuals, is not included in the …gure in view of the complex nature of the allocation, explained in Section 2.4.

22

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

2.3.1

Adult Equivalence Scales

A common method of dealing with the heterogeneity of households, given only market and disposable incomes, is to make comparisons on the basis, not of actual incomes, but of an arti…cial income construct which re‡ects the differences in the demographic structure of the households. A simple way to obtain an individual-level income is to assume an equal division within the household and divide total income by the number of individuals in the household. But the view is widely taken that not all members of the household have the same consumption needs. Furthermore, there may be economies of scale within a household. The latter can arise because some goods (including some durables and goods like heating and lighting) may be ‘public goods’ within the household and can therefore be consumed simultaneously by several people. In addition, there may be economies from purchasing larger quantities of some goods. Instead of dividing total household income by the number of people in the household (irrespective of their ages or gender), a measure of adult equivalent household size is obtained using a set of adult equivalence scales. However, this approach continues to assume equal sharing, but among adult equivalents. Some people may object to the use of such scales. Those who take this view may, for example, object to treating children in terms merely of a cost or burden faced by parents, rather than as a desired bene…t or advantage. They may consider household structure, fertility decisions, household production and market income as jointly determined. A simple but ‡exible adult equivalence scale is the following. Let  and  denote respectively the number of adults and children in the household, and let  denote the adult equivalent size of the household. Then:  = ( +  )

(2.12)

where  and  are parameters re‡ecting the relative ‘cost’ of a child and economies of scale respectively. This form was introduced by Cutler and

2.3. THIRTEEN DISTRIBUTIONS

23

Katz (1992) and investigated by, for example, Banks and Johnson (1994) and Jenkins and Cowell (1994). Creedy and Sleeman (2005) found that, despite its simplicity, it provided a close …t to 29 alternative sets of equivalence scales. Having obtained the adult equivalent size of each household, it is a simple matter to calculate the total income per adult equivalent person.

2.3.2

The Unit of Analysis

Given the arti…cial welfare metric of income per adult equivalent person, comparisons then depend on the choice of unit of analysis in combination with this metric. The choice is not as straightforward as has often been assumed. In fact, three further pairs of distributions may be considered. First, comparisons can be made using the household as the unit of analysis: this gives distributions 3 and 4 in Table 2.1. Second, perhaps the simplest and most natural choice is to make comparisons using the individual as the unit of analysis: this gives distributions 5 and 6. Third, as …rst suggested by Ebert (1997), the unit of analysis could be the ‘equivalent adult’, giving distributions 7 and 8 in Table 2.1. When using the individual as the unit of analysis, each person ‘counts for one’ irrespective of the household to which they belong. Inequality remains unchanged when one person is replaced by another with the same metric (income per adult equivalent person) but belonging to a different type of household. It thereby satis…es an ‘anonymity principle’. However, it does not necessarily satisfy the ‘principle of transfers’. But if the existence of large economies of scale means that rich large households are highly efficient at generating welfare (in terms of the choice of this metric), it is possible, when using the individual as unit, for evaluations to be inequality-preferring. This was …rst shown by Glewwe (1991) and the welfare aspects were examined by Shorrocks (2004). The third possibility uses the equivalent adult as the income unit. This

24

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

arti…cial income unit is thus combined with its corresponding arti…cial income measure, income per adult equivalent. In this case there are not necessarily integer numbers of equivalent adults (except for single-adult households). Thus the distributions cannot be written simply as vectors. The equivalent adult size must be treated as a household weight in obtaining inequality or other measures. The use of the arti…cial equivalent adult as the unit of analysis means that the income unit and the income concept are treated consistently. Each individual’s contribution to inequality depends on the demographic structure of the household to which that individual belongs. Thus an adult in a oneperson household ‘counts for one’. But an adult counts for ‘less than one’ (has a weight less than 1) when placed in a multi-person household. The use of this income unit is consistent with the principle of transfers. This can be useful because of the general results discussed above linking this value judgement to Lorenz curves. Importantly, reliance on the strong results regarding Lorenz curves can be made only in the case of comparisons using the equivalent person as the unit of analysis. It cannot be assumed that comparisons are insensitive to the choice of income unit. Indeed, it is quite possible for a tax reform to be judged differently, changing inequality and welfare comparisons in opposite directions, when using the individual and the equivalent adult as income units, as shown below. Further examples are given in Decoster and Ooghe (2003) and Creedy and Scutella (2004).

2.3.3

The Use of Allocation Rules

In the previous subsection the welfare metric was based on an assumption of equal sharing among equivalent adults within the household. Further distinctions were then made depending on the choice of income unit. Yet another approach is to begin by using an explicit sharing rule to allocate income to

2.3. THIRTEEN DISTRIBUTIONS

25

individual members of each household. Instead of taking total market income in a household, or individual market income for those with positive values, household income is considered to be shared among all those in the household.10 The particular sharing rule used may be based on special surveys which provide information about income sharing, or it may be rather more ad hoc. Suppose that the allocation rule is based on an additive household size, , de…ned as:11  = 1 + 05 ( − 1) + 03

(2.13)

Hence the …rst adult is given a weight of 1, while all other adults are given a weight of 0.5 and all children are given a weight of 0.3. This type of explicit income-sharing rule is naturally associated with the use of the individual as the income unit. The use of this sharing rule gives rise to distributions 11 and 12 of Table 2.1.12 In addition to comparisons involving market and disposable incomes, ‘…scal incidence’ studies go further and attempt to allocate some components of government expenditure to individuals. In particular, health expenditure can be allocated based on age, gender and summary information about individuals’ use of publicly …nanced health services. Similarly primary, secondary and tertiary education expenditure can be allocated to individuals based on age.13 This gives distribution 13 in Table 2.1. The discussion has so far been in terms of distributions of market and disposable incomes. Some household surveys contain detailed information about household expenditures, and this can be used to compute an addi10 In

the empirical analysis reported below, when this sharing rule is applied, sharing is actually restricted to family members within a household. 11 This formulation actually corresponds to the modi…ed OECD equivalent scale, which does not allow for economies of scale within households. 12 Distribution 11 shares market income. An additional alternative distribution would be to consider individual market incomes as in distribution 9 but with an additional  −  zero values. Comparisons with distribution 12 would then combine the effects of sharing and taxes and transfers. 13 For details of an attribution process, see Aziz et al. (2015).

26

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

tional metric, that of disposable income after the deduction of indirect taxes. However, if – as in New Zealand – there is a broad-based goods and services tax applied at a uniform rate, combined with limited excises (for example, on tobacco, alcohol and petrol), the allocation is straightforward and involves an approximately proportional change. Hence, indirect taxes are ignored in the comparisons reported here.14

2.4

Construction of Distributions

The data used here were obtained from the Household Economic Survey (HES) for the years 2006/07, 2009/10 and 2010/11. Each year’s survey runs from July to June of the next year and contains detailed information about incomes and household characteristics for approximately 8000 individuals, grouped by households. This is sufficient to enable the calculation of market income, welfare bene…ts and direct and indirect taxes. Each individual in the survey is assigned a weight which makes it possible to aggregate from the sample to population values.15 For the purposes of applying the tax and transfer system, ‘Economic Family Units’ (henceforth referred to as ‘families’) were constructed. A family is de…ned as a person, a partner (if relevant), and any children under the age of 19 who is not in full time employment (henceforth referred to as ‘dependent children’). This construction is required to model a signi…cant proportion of the New Zealand tax and transfer system, such as Working for Families and the core bene…ts. From these data, a measure of market income was obtained for each individual. It is possible for some individuals to have negative market incomes, 14 However,

when the incidence of education and health is examined in distribution number 13, Goods and Services Tax is deducted before those components are added. 15 The weights used for this study are taken from Taxwell, rather than Statistics New Zealand’s HES weights.

2.4. CONSTRUCTION OF DISTRIBUTIONS

27

almost exclusively through negative self-employed income or large capital market losses. As discussed in the previous section, it is invalid to include negative values in the calculation of the Gini and impossible to include in the Atkinson measure, and neither of the reasons for negative income provide a strong indication where they should lie in the income distribution. Hence they were removed from the sample. The New Zealand tax and transfer system was then applied to the data by using Treasury’s non-behavioural microsimulation model, Taxwell. This incorporates the majority of the rules of the tax and transfer system, including: Direct tax and ACC levies; the core and supplementary bene…ts; Independent Earner and Working for Families tax credits; New Zealand Superannuation; the Accommodation Supplement. Of relevance here are earlier policies, such as the Low Income Earner Rebate, the Child Taxpayer Rebate, and Transitional Tax Allowance. Many of these policies need information at the family or household level to calculate entitlements, but all amounts are attributed back to the eligible individual. An assumption is made that the receiver of Working for Families tax credits is the adult rather than the child, and in the case of partnered adults it is attributed to the partner with the least amount of market income, assumed to be the primary caregiver of any dependent children. Thus, processing by Taxwell gives the components of disposable income at the individual level. While Taxwell works with the New Zealand system, many of the same types of policies and issues arise when applying other countries’ systems and it is straightforward to apply these. In the New Zealand tax and transfer system there are two bene…ts – the Student Allowance and the Accommodation Supplement – where a cash payment is tied to a particular choice the person makes (regarding decisions about study and location and type of housing). Both bene…ts are included in the calculation of disposable income. A scheme similar in purpose to the Accommodation Supplement, the Income Related Rental Subsidy, is not a

28

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

cash payment and was instead treated as an ‘in-kind’ payment. This process produces disposable income at an individual level. Those with zero or negative disposable incomes are excluded. This may arise in cases of retired families living entirely from savings, or sole students (who are counted as a family of one) incurring debt. For the present comparisons it is important to maintain a consistent sample across the alternative distributions. If a person or family was marked for deletion, their entire household was deleted. Thus, people and families in these households that did not have invalid data were also deleted, even for distributions that concern themselves with the individual. This ensured that all 13 distributions use an identical dataset. This does not extend to when individuals with zero market incomes were removed for only distributions 9 and 10, as these are valid members of the distributions. As in Aziz et al. (2015), the explicit sharing rule in equation (2.13) was applied only with families, rather than across households. The rationale behind this is that sharing is more likely to occur within a family. Sharing of …nancial resources is not likely to occur to the same extent between families of adults, and is rare across shared living arrangements where the individuals are not related. Similarly, the allocation of health and education expenditure follows Aziz et al. (2015). Education expenditure is based on total government spending on particular types of education. For example, primary and secondary education is decomposed into schooling year or age groups, and those in each category are allocated the appropriate expenditure. Health expenditure is attributed using demographic per capita expenditure pro…les provided by the Ministry of Health. Unlike earlier studies, the Income Related Rental Subsidy is not included in disposable income, but is included in the calculation of …nal income. In contrast, Student Allowance payments are included in the calculation of disposable income. Thus, only cash payments are in disposable income. A second difference is in the calculation of indirect tax which, in view of New

2.5. INEQUALITY COMPARISONS OVER THREE YEARS

29

Zealand’s broad base, is here treated simply as a constant proportion of disposable income, using the tax-inclusive Goods and Services Tax rate. Furthermore, the data were not scaled to …scal aggregates, since relativities of income, taxes and transfers are important to preserve, and it is of less importance to match macroeconomic variables with the national accounts.

2.5

Inequality Comparisons over Three Years

This section compares the alternative distributions and inequality measures. First, the period 2007 to 2010 is chosen as covering the years of the global …nancial crises and a period in which there were few tax and transfer changes. In view of the tax changes made in 2010, additional comparisons are made including 2011. All results in subsections 2.5.1 and 2.5.2 are obtained using, where relevant, adult equivalence scales with  = 05 and  = 08. Sensitivity analyses are reported in subsection 2.5.3. As mentioned earlier, Lorenz dominance results rarely apply in practice, and this is the case here. Of course, even if they were to apply, concern is with the precise extent of redistribution resulting from taxes and transfers and of changes over time, so that the formal inequality measures are needed.

2.5.1

Comparisons from 2007 to 2010

A range of inequality measures for 2007 and 2010 are presented in Table 2.2. A dash (–) in the table in the column relating to Atkinson measures for  = 12 indicates that, in view of the presence of zero values in the distribution, 12 is unity, as discussed above. The table reveals quite substantial differences in the absolute values of inequality, depending on the measure used (the degree of inequality aversion in the case of the Atkinson measures) and the combination of welfare metric and unit of analysis. The implications of the direct tax and transfer system, in reducing inequality when moving from a gross to a net income metric, are shown in

30

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

Table 2.3. These percentage reductions are substantial, but again they vary considerably depending on the comparisons used: for example in 2007 the Gini measure for the comparison between distributions 9 and 10 shows a reduction of 20.9 per cent, whereas the Atkinson measure, for  = 08, falls by 75.3 per cent when comparing distributions 3 and 4. Any comments about the redistributive effects of taxes and transfers must therefore be clear about the precise nature of the comparisons being made. The changes for 2010 are generally slightly higher than for 2007, although otherwise similar. It is also of interest to examine the percentage changes in inequality between the two years. These are shown in Table 2.4. With just two exceptions – for the distribution of individual market income after sharing (number 11) and the Atkinson measure for  = 05 and  = 08 – it could be said that inequality fell from 2007 to 2010. Despite this large degree of agreement among measures and metrics, the extent of the reduction varies substantially. Disposable incomes have generally shown the largest inequality reductions compared with market incomes.

2.5.2

Comparisons from 2007 to 2011

As discussed earlier, it is useful to consider changes in inequality over the longer period 2007 to 2011, in view of the tax changes announced in 2010. The main policy changes made in 2010 concerned a partial shift in the tax mix from personal income taxation towards indirect taxation, with associated adjustments to a number of bene…t levels.16 In particular, the percentage marginal income tax rates, which in 2006/07 were [195 30 39], were changed to [105 175 30 33]. Corresponding thresholds, above which the respective rates applied, were [0 38000 60000] in 2006/07 and became [0 14000 48000 70000] in 2010. The GST rate was increased from 12.5% to 16 The

HES is conducted from July to June, and Taxwell uses this for modelling the April to March tax year. However, when modelling the so called ‘2011 year’ we use the 2010/2011 HES but apply the policies that came into force at October 2010.

2.5. INEQUALITY COMPARISONS OVER THREE YEARS

31

Table 2.2: Inequality Measures: 2007 and 2010 No. Welfare metric Year 2007 1 HH market income () 2 HH disposable income () 3 HH market income per AE () 4 HH disposable income per AE () 5 Market income per AE () 6 Disposable income per AE () 7 Market income per AE ( ) 8 Disposable income per AE ( ) 9 Individual market income ( ) 10 Individual disposable income ( ) 11 Individual market income () 12 Individual disposable income () 13 Individual …nal income () Year 2010 1 HH market income () 2 HH disposable income () 3 HH market income per AE () 4 HH disposable income per AE () 5 Market income per AE () 6 Disposable income per AE () 7 Market income per AE ( ) 8 Disposable income per AE ( ) 9 Individual market income ( ) 10 Individual disposable income ( ) 11 Individual market income () 12 Individual disposable income () 13 Individual …nal income ()

Atkinson for  of: 02 05 08 12

Gini

0.110 0.049 0.106 0.039 0.088 0.034 0.092 0.036 0.094 0.057 0.126 0.065 0.044

0.310 0.122 0.300 0.095 0.247 0.084 0.258 0.088 0.242 0.145 0.334 0.158 0.107

0.628 0.197 0.615 0.152 0.514 0.132 0.535 0.139 0.407 0.243 0.633 0.246 0.165

– 0.308 – 0.237 – 0.201 – 0.213 0.670 0.427 – 0.375 0.236

0.533 0.382 0.523 0.337 0.481 0.317 0.489 0.324 0.502 0.397 0.583 0.438 0.364

0.102 0.042 0.099 0.034 0.087 0.030 0.088 0.031 0.090 0.050 0.125 0.060 0.041

0.289 0.105 0.281 0.082 0.244 0.072 0.247 0.075 0.235 0.127 0.335 0.145 0.099

0.591 0.169 0.580 0.130 0.505 0.113 0.511 0.118 0.398 0.211 0.644 0.226 0.154

– 0.260 – 0.196 – 0.168 – 0.176 0.645 0.357 – 0.331 0.222

0.513 0.356 0.507 0.316 0.476 0.297 0.477 0.302 0.497 0.376 0.580 0.423 0.353

32

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

Table 2.3: Percentage Inequality Reduction from Market to Disposable Income Change Atkinson for  of: Gini 0.2 0.5 0.8 1.2 Reductions for 2007 1 to 2 -55.3 -60.5 -68.6 – -28.3 3 to 4 -63.4 -68.2 -75.3 – -35.5 5 to 6 -61.2 -66.2 -74.3 – -34.1 7 to 8 -61.0 -65.9 -74.0 – -33.7 9 to 10 -39.1 -40.0 -40.4 -36.3 -20.9 11 to 12 -48.3 -52.7 -61.1 – -24.8 Reductions for 2010 1 to 2 -58.6 -63.6 -71.4 – -30.6 3 to 4 -66.2 -70.8 -77.6 – -37.6 5 to 6 -65.8 -70.4 -77.6 – -37.7 7 to 8 -64.9 -69.6 -76.9 – -36.7 9 to 10 -44.8 -46.1 -46.9 -44.7 -24.4 11 to 12 -52.2 -56.7 -64.8 – -27.0

Table 2.4: Percentage Change in Inequality from 2007 to 2010 No. Welfare metric 1 2 3 4 5 6 7 8 9 10 11 12 13

Atkinson 02 05 HH market income () -7.6 -6.7 HH disposable income () -14.5 -14.1 HH market income per AE () -6.6 -6.1 HH disposable income per AE () -13.8 -13.8 Market income per AE () -2.0 -1.4 Disposable income per AE () -13.6 -13.8 Market income per AE ( ) -4.8 -4.2 Disposable income per AE ( ) -14.4 -14.5 Individual market income ( ) -3.9 -2.7 Individual disposable income ( ) -12.9 -12.6 Individual market income () -1.3 0.5 Individual disposable income ( ) -8.7 -7.9 Individual …nal income () -8.2 -7.2

for  of: Gini 08 12 -5.9 – -3.7 -14.0 -15.6 -6.8 -5.7 – -3.1 -14.5 -17.3 -6.3 -1.8 – -1.1 -14.3 -16.3 -6.4 -4.3 – -2.4 -15.1 -17.3 -6.9 -2.2 -3.7 -1.0 -12.9 -16.4 -5.3 1.8 – -0.5 -7.9 -11.6 -3.4 -6.5 -5.8 -3.1

2.5. INEQUALITY COMPARISONS OVER THREE YEARS

33

15%. For further discussion of the tax mix change, see Creedy and Mellish (2011). A range of bene…t abatement thresholds, such as Domestic Purposes Bene…t (DPB), Invalid’s Bene…t (IB), Widow’s Bene…t (WB) were changed from $80 and $180 per week to $100 and $200 per week, with abatement rates of 30% and 70% continuing to apply. The New Zealand Superannuation Non Qualifying Spouse bene…t (NZS NQS) threshold was changed from $80 to $100 per week, with the abatement rate of 70% remaining unchanged. In addition, in 2006/07 there was a Low Income Rebate (a 4.5% tax rebate until $9500 per year where a 1.5% abatement begins). In 2010 an Independent Earner Tax Credit applied, involving a $520 tax credit for income over $24000, abated at 13% after $44000 per year. The Accident Compensation Corporation (ACC) Levy was 1.3% in 2006/07, and 2.04% in October 2010.17 Table 2.5 shows the percentage change over the period for each of the metrics and inequality measures (although distribution number 13 could not be considered for 2011). Unlike comparisons between 2007 and 2010, the direction of change is more ambiguous. The Gini measures show small percentage increases for all distributions except for the disposable income distributions in numbers 10 and 12. There is more ambiguity among the Atkinson measures. Measured inequality in any period is higher, the higher is the degree of relative inequality aversion. But of relevance here is the change in the Atkinson measure between two time periods. It is not necessarily the case that judges will agree about the direction of changes in inequality. If there are equalising changes in the lower ranges of the distribution, more importance will be attached 17 Changes

to Portfolio Investment Entities (PIEs) could not be incorporated into the analysis. There was a temporary additional payment to some bene…t categories to compensate for price rises due to the GST increase. This was paid from October until April 2011 when bene…ts would next be indexed by the CPI. As a compromise, bene…t payments were modelled according to the Taxwell tax year, thus including only half the temporary payment.

34

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

to these by a judge with high inequality aversion, who attaches less importance to high-income changes. Alternatively a judge with lower aversion is more concerned with the changes taking place in higher-income groups. From the evaluation function,  , associated with the Atkinson measure, given in equation (2.1):

 = −  

(2.14)

Hence, the increase in  associated with an increase in  is not only lower for higher incomes, but is lower for higher values of , for a given income. This has the potential to lead to the counter-intuitive result, depending on the precise nature of the distributional changes, whereby a higher aversion implies a decrease in inequality over time, whereas a low aversion implies an increase. Interestingly, in the present context, a judge with a higher degree of aversion to inequality takes the view that there has been a reduction in inequality from 2007 to 2011 for all disposable income distributions (except for individual market incomes in distribution 9, which shows an increase of about 1 per cent). However, a lower  implies a reduction in the cases of distributions of disposable incomes 2, 10 and 12, but an increase for distributions 4, 6 and 8. Both the direction and extent of the measured change in inequality depend on the particular combination of the welfare metric, the unit of analysis and the inequality measure being considered. It is also of interest to consider whether, for each social welfare function, the value of  increases over the period. That is, in those cases where inequality is seen to increase, is this compensated by an increase in real incomes (in the view of the independent judge’s evaluation function)? This question is answered by comparing values of  = ¯ (1 −  ), discussed above, where ¯ is suitably adjusted for in‡ation over the period. Table 2.6 reports percentage changes in the (abbreviated) social welfare function from 2007 to 2011. Generally the changes are positive. Hence, the increase in real

2.5. INEQUALITY COMPARISONS OVER THREE YEARS

35

Table 2.5: Percentage Change in Inequality from 2007 to 2011 No. Welfare metric 1 2 3 4 5 6 7 8 9 10 11 12

02 HH market income () -0.01 HH disposable income () -0.47 HH market income per AE () 0.87 HH disposable income per AE () 1.88 Market income per AE () 4.46 Disposable income per AE () 4.78 Market income per AE ( ) 2.35 Disposable income per AE ( ) 2.85 Individual market income ( ) 0.05 Individual disposable income ( ) -4.32 Individual market income () 1.39 Individual disposable income () -1.01

Atkinson for  of: Gini 05 08 12 -0.69 -1.17 – 0.57 -1.58 -3.12 -6.90 0.07 -0.15 -0.97 – 0.82 0.22 -1.90 -6.55 0.65 3.44 2.40 – 2.40 3.30 1.58 -1.66 2.01 1.39 0.42 – 1.49 1.48 -0.19 -3.60 1.18 0.32 0.83 0.89 0.04 -5.71 -7.31 -11.03 -1.81 2.01 2.60 – 0.54 -1.40 -2.25 -7.04 -0.40

incomes over the period is judged to more than compensate for the increase in inequality, where relevant, although again the percentage changes differ. The exceptions are for distribution number 5 (for Gini and 08 ) and distribution 11 (for 08 ). These cases relate to market rather than disposable incomes, and the distributions contain zero values. Furthermore, the higher values for  = 08 imply that there is greater sensitivity to changes at the lower end of the distribution.

2.5.3

Adult Equivalence Scales

The results presented above are all obtained for a single set of parameters in  = ( +  ) , the expression for adult equivalence scales given in (5.1). Obviously these scales do not affect all the distributions discussed here, but where they are relevant the sensitivity of comparisons to the value of , which re‡ects the extent of economies of scale, was examined. Consider, for example, the distribution of disposable income per adult equivalent, using the individual as unit, and the Atkinson inequality measure. Using  = 05 inequality in 2011 exceeds that in 2007 for all values of , but

36

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

Table 2.6: Percentage Changes in Abbreviated Social Welfare from 2007 to 2011 No. Welfare metric Atkinson for  of: Gini 02 05 08 12 1 HH market income () 2.0 2.3 4.0 – 1.4 2 HH disposable income () 10.0 10.2 10.8 13.3 9.9 3 HH market income per AE () 2.9 3.1 4.6 – 2.1 4 HH disposable income per AE () 10.5 10.6 11.0 12.9 10.2 5 Market income per AE () 1.3 0.5 -0.9 – -0.6 6 Disposable income per AE () 9.5 9.3 9.4 10.1 8.6 7 Market income per AE ( ) 1.4 1.2 1.2 – 0.2 8 Disposable income per AE ( ) 9.5 9.4 9.6 10.6 9.0 9 Individual market income ( ) 3.1 3.0 2.5 1.2 3.0 10 Individual disposable income ( ) 11.4 12.2 13.7 20.2 12.4 11 Individual market income () 1.3 0.5 -3.0 – 0.8 12 Individual disposable income () 9.5 9.7 10.2 14.0 9.8 when  = 12 the order is reversed and the distribution in 2007 is more unequal than in 2011. The case where  = 08 is illustrated in Figure 2.4. The shape of the pro…les indicates that simple assumptions about the way in which changes in the economies of scale parameter affect inequality may be misleading. There is a range over which an increase in  is associated with reduction in inequality, and a range over which an increase in  produces an increase in inequality.18 In addition, the two pro…les intersect, so that for values of  above around 0.45, inequality in 2011 is judged to be higher than in 2007, but for  less than 0.45, the inequality ranking is reversed. A similar kind of sensitivity arises for the distribution of disposable income per adult equivalent person, when the income unit is the equivalent adult. Figures 2.5 and 2.6 show the corresponding pro…les for inequality aversion of  = 02 and  = 08 respectively. For the higher value of  = 12, inequality is judged to be greater in 2007 than in 2011 for all values of . 18 For

discussion of the precise conditions in terms of the relevant joint distributions and the correlation between equivalent income and the number of individuals in the household, see Creedy and Sleeman (2005, pp. 58–60).

2.6. INEQUALITY DECOMPOSITIONS

37

Figure 2.4: Inequality of Disposable Income per Adult Equivalent with Individual as Unit:  = 08 An example involving the distribution of market income per adult equivalent person, with the equivalent adult as the unit of analysis, is illustrated in Figure 2.7. In this case, the distributions contain some zero values, so Atkinson inequality measures are reported only for   1. In this case, with  = 08, inequality falls consistently as  increases (that is, as the extent of scale economies falls), but again the two pro…les intersect. For  less than about 0.65, inequality in 2007 is judged to be greater than in 2011.

2.6

Inequality Decompositions

The previous sections of this chapter have discussed alternative income distribution comparisons, either for a single time period (in moving from market to disposable income), or for two periods. However, changes in measured inequality over time depend on the structure of the population as well as the tax and transfer system. For example, there are systematic variations in incomes over the life cycle, so that a change in the age distribution of the

38

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

Figure 2.5: Inequality of Disposable Income per Adult Equivalent with Equivalent Adult as Unit:  = 02

Figure 2.6: Inequality of Disposable Income per Adult Equivalent with Equivalent Adult as Unit:  = 08

2.6. INEQUALITY DECOMPOSITIONS

39

Figure 2.7: Inequality of Market Income per Adult Equivalent with Equivalent Adult as Unit:  = 08 population could give rise to an observed increase in overall inequality even if the tax system is changed in ways which are designed to make it more redistributive. The fact that the redistributive effect of any tax system cannot be evaluated independently of the population (the pre-tax income distribution) raises the question of how comparisons can be made over time, where typically both the population and the tax structure are different. In …scal incidence studies the question is thus: has the income tax and transfer system become more or less redistributive? The difficulty is therefore to isolate the marginal effect of the tax policy change from that of the population change. The various components are explained as follows. Give two cross-sectional household surveys, let  denote the tax structure for  = 0 1 (an initial period and subsequent period respectively). Similarly let  denote the population in period . For convenience, consider the Gini inequality measure, although the following approach may be used for other summary measures. There are therefore four possible Gini inequality mea-

40

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

sures of both gross market income and disposable income; denote these by  (   ) and  (   ) for   = 0 1 . These four measures can be obtained using each of the combinations of income concept and unit of analysis discussed above. It is of course required that each survey contains enough information about the characteristics of households so that the disposable incomes of each population can be computed for each of the tax structures. The comparisons in previous sections above are of Gini measures, in the case of disposable income in each period, using  (0  0 ) and  (1  1 ). But the …nding that, for example,  (0  0 )   (1  1 ), following a policy change, does not support the inference that the policy reform has reduced inequality. The reduction may have arisen from the population structure changes. The separate effects of tax and population changes can be obtained as follows. Consider the following decomposition:  (1  1 )− (0  0 ) = [ (1  1 )− (0  1 )]+[ (0  1 )− (0  0 )] (2.15) The …rst term in square brackets on the right hand side of (2.15) is the population effect given tax structure 1, and the second term in square brackets is the tax policy effect given initial population 0. However, there is another possible decomposition of the change in inequality, since:  (1  1 )− (0  0 ) = [ (1  0 )− (0  0 )]+[ (1  1 )− (1  0 )] (2.16) The …rst term in square brackets on the right hand side of (2.16) is the population effect given tax structure 0, while the second term is the tax policy effect given population structure 1. Faced with two values for each of the marginal effects, an approach is to obtain the unweighted arithmetic mean: this average is recommended by Shorrocks (2011). Table 2.7 reports the effects of applying the above decomposition to changes between 2007 and 2011. Values are absolute changes, and are the arithmetic means of the relevant components. It is clear from the …nal two

2.6. INEQUALITY DECOMPOSITIONS

41

Table 2.7: Decomposition of Absolute Changes in Inequality of Disposable Income: 2007 to 2011 No. Welfare metric Component: Total Population Tax Decomposition based on Gini 2 HH disposable income () 0.0003 -0.0061 0.0064 4 HH disposable income per AE () 0.0022 -0.0061 0.0083 6 Disposable income per AE () 0.0064 -0.0024 0.0088 8 Disposable income per AE ( ) 0.0038 -0.0044 0.0082 10 Individual disposable income ( ) -0.0072 -0.0126 0.0054 12 Individual disposable income () -0.0017 -0.0085 0.0067 Decomposition based on Atkinson with  = 02 2 HH disposable income () -0.0002 -0.0019 0.0016 4 HH disposable income per AE () 0.0007 -0.0010 0.0018 6 Disposable income per AE () 0.0016 -0.0001 0.0018 8 Disposable income per AE ( ) 0.0010 -0.0007 0.0017 10 Individual disposable income ( ) -0.0025 -0.0039 0.0014 12 Individual disposable income () -0.0007 -0.0027 0.0020 Decomposition based on Atkinson with  = 08 2 HH disposable income () -0.0061 -0.0122 0.0061 4 HH disposable income per AE () -0.0029 -0.0095 0.0066 6 Disposable income per AE () 0.0021 -0.0046 0.0067 8 Disposable income per AE ( ) -0.0003 -0.0066 0.0064 10 Individual disposable income ( ) -0.0177 -0.0228 0.0050 12 Individual disposable income () -0.0055 -0.0129 0.0073 Decomposition based on Atkinson with  = 12 2 HH disposable income () -0.0212 -0.0298 0.0086 4 HH disposable income per AE () -0.0155 -0.0249 0.0094 6 Disposable income per AE () -0.0033 -0.0128 0.0095 8 Disposable income per AE ( ) -0.0077 -0.0168 0.0091 10 Individual disposable income ( ) -0.0471 -0.0535 0.0064 12 Individual disposable income () -0.0264 -0.0361 0.0097

42

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

columns of the table that, for all disposable income distributions and inequality measures, the effect of the tax and transfer changes between 2007 and 2011 was to increase inequality of disposable incomes slightly. However, the population structure changes had the effect, in all cases, of reducing measured inequality. Whether the overall effect was to reduce inequality therefore depends on whether the population component outweighed the tax change effect. For the Atkinson measures this is seen to be more likely for the higher inequality aversion cases, where measures are more sensitive to changes at the lower end of the income distribution. Such decompositions must always be treated with caution. First, the analysis does not consider endogenous labour supply responses to tax changes. On decompositions which allow also for labour supply responses, see Bargain (2012a, 2012b), and Creedy and Hérault (2015). The latter paper also considers the use of money metric utility as the welfare metric in decompositions. In addition, there may be other responses to tax changes which are nevertheless included in population structure changes. These include changes in fertility, household formation, migration and so on. For a more detailed analysis of decompositions, see Chapter 5 below.

2.7

Conclusions

This chapter has emphasised the importance of making a range of value judgements explicit when attempting to measure inequality, and changes in inequality, for any particular population group. Special attention was given to comparisons of alternative distributions and the implications of using different distributions and summary measures. Is is suggested that comparisons are too often based on a limited range of measures which do not provide suf…cient information for readers, whose value judgements may vary widely, to make their own judgements. Using an annual accounting period, alternative welfare metrics and units of analysis were investigated. The implications for

2.7. CONCLUSIONS

43

redistribution, and recent changes in inequality in New Zealand, were illustrated using Gini and Atkinson inequality measures, where the latter also allow for a range of degrees of aversion to inequality. Some of the comparisons involved the use of adult equivalence scales. The use of a welfare metric de…ned in terms of income per adult equivalent (for example, market or disposable income) can be combined with the use of different income units, such as the household, the individual or the equivalent adult. In addition, the sensitivity to assumptions about economies of scale within households was examined. Furthermore, changes in inequality were decomposed into those arising from population and tax structure changes. When considering the period 2007 to 2010 all measures agree that inequality fell, although the extent of the reduction varies. For the period 2007 to 2011 (after the tax reforms of 2010) the answer to the question of whether inequality in New Zealand has risen or fallen was found to depend crucially on the combination of welfare metric, income unit, adult equivalence scale and inequality measure used. When decomposing changes in inequality into tax and population components, it was found that for all disposable income distributions and inequality measures, the effect of the tax and transfer changes between 2007 and 2011 was to increase inequality of disposable incomes slightly. However, the population structure changes had the effect, in all cases, of reducing measured inequality. The overall effect on inequality (depending on whether the population component outweighed the tax change effect) was found to depend on the inequality measure used. It should perhaps not be surprising that such a complex phenomenon as inequality within a heterogeneous population does not allow simple unambiguous comparisons. It is all too easy for researchers, often implicitly, to impose their own value judgements or not to be aware of some of the complexities involved (such as the fact that the use of the individual as unit, when using a measure of income per adult equivalent, can imply a preference

44

CHAPTER 2. ALTERNATIVE DISTRIBUTIONS AND METRICS

for more inequality if household size is strongly positively related to income and there are substantial economies of scale in consumption within households). An aim of the present chapter has thus been to persuade empirical researchers and policy analysts to investigate and report results for a wider range of distributions and comparisons.

Chapter 3 Interpreting Inequality Measures Anyone who reports a summary inequality measure and its changes over time, or who provides a distributional evaluation of a potential policy change in terms of a change in inequality, is sure to be asked the difficult question, ‘what does this mean?’ Is, say, a …ve per cent increase in an inequality measure a major concern for an inequality-averse judge, or may it be regarded as quite small? It is relatively easy to form a view about a change in an aggregate income measure – a simple sum of values. But it is intuitively much more difficult to appreciate orders of magnitude of an inequality measure which is (usually) based on nonlinear transformations of many individual values. Of course, the Lorenz curve provides an extremely convenient graphical device for illustrating aspects of relative inequality.1 Yet in many practical contexts it is necessary to provide a quantitative indication of inequality and its changes. The aim of this chapter is to explore several approaches that have been proposed to help interpret inequality and its changes in the context of the Gini measure. These approaches are applied to the Atkinson inequality measure. 1 Furthermore,

its relationship with value judgements, such as the principle of transfers, has been well explored. The concepts of Lorenz Dominance and Generalised Lorenz Dominance play a central role in inequality and welfare comparisons; see, for example, Lambert (1993).

45

46

CHAPTER 3. INTERPRETING INEQUALITY MEASURES One possible approach, proposed by Subramanian (2002) and extended

by Shorrocks (2005), is to consider an alternative arti…cial distribution that has the same value of the inequality measure as the actual distribution but consists of only two income levels (or, equivalently, income shares), though there may be more than two individuals in the arti…cial distribution. A basic analogy with a ‘cake-cutting exercise’ is therefore involved. Both Subramanian and Shorrocks concentrated mainly on the Gini inequality measure.2 It is necessary to consider in further detail whether Subramanian’s claim, followed by Shorrocks, that ‘the clearest idea we can have of the extent of relative inequality’ is unambiguously provided by the share received by the poorer person (or persons) in a …nite-sample comparison. Does the analogy of sharing a cake among a very small number of people actually provide the kind of intuitive description that is useful for people who want some idea of what an Atkinson inequality measure and its changes ‘actually mean’? The simple observation that a small income increase for a high-income person leads to an increase in inequality, but an increase for a low-income person results in a reduction in inequality, gives rise to the idea of a particular income level that in some sense ‘divides’ the low and high incomes. This idea appears to have given rise to three independent analyses. First, Hoffmann (2001) referred to the dividing line as the ‘relative poverty line’ and explored its value for a range of inequality measures (including the Gini and the generalised entropy class of measures).3 Second, Lambert and Lanza (2006) de…ned the dividing line as a ‘benchmark income or position’, obtaining a wide range of results.4 Third, Corvalan (2014), used the concept of a ‘pivotal’ individual, de…ned as the person such that if an additional small 2 Two-person

cake sharing in the context of the extended Gini measure is examined by Tibiletti and Subramanian (2015). 3 He also reported values of the relative poverty line for …ve regions of Brazil. 4 They also considered the role of the benchmark income in the leaky bucket experiment, in the context of effects on inequality measures, rather than the usual reference to constant ‘social welfare’ evaluations.

3.1. PREVIOUS RESULTS FOR THE GINI MEASURE

47

income increase is given to a poorer individual, the inequality index falls. Corvalan concentrated on the Gini measure of inequality. The additional insights provided by this approach are explored here in the context of the Atkinson index. First, Section 3.1 brie‡y summarises the main results regarding the Gini measure. Section 3.2 provides a brief reminder of the de…nition of the Atkinson measure. The judge’s evaluation function that lies behind the Atkinson measure allows for a well-de…ned trade-off between inequality and total income (or, as it is sometimes expressed, between ‘equity and efficiency’). This trade-off therefore provides an initial indication of what inequality ‘means’: it makes it possible to talk about the extent of total income growth that would be foregone by a judge in order to achieve a given reduction in inequality. This is described in Section 3.3. The Subramanian–Shorrocks approach applied to the Atkinson measure is then discussed in Section 3.4. Examples are given of the variation in the ‘excess share’, for different …nite-sample sizes, as inequality changes. The fact that only two income values (even where   2) are involved means that it is also possible to exploit the ‘leaky bucket’ thought experiment. This is the device Atkinson introduced in order to help clarify the speci…cation of orders of magnitude regarding an ‘inequality aversion’ parameter. The implications of applying this thought experiment to the -person shares are also investigated in Section 3.4. Section 3.5 then turns to the ‘pivotal income’ associated with any degree of inequality aversion and value of the Atkinson inequality measure.

3.1

Previous Results for the Gini Measure

The …rst approach, along ‘cake-cutting’ lines and in the context of the Gini inequality measure, was made by Subramanian (2002). He considered a distribution having the same Gini value as the actual distribution but consisting

48

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

of only two income levels (or, equivalently, income shares). The restriction to just two incomes enables some simple comparisons to be made. He suggested that, ‘our intuitive comprehension of the notion of inequality is sharpest in the context of the canonical two-person cake-sharing problem’ (2002, p. 375). He argued that, ‘the share  of the cake going to the poorer person furnishes just about the clearest idea we can have of the extent of relative inequality’. Subramanian used a de…nition of the Gini measure,  , that is normalised to the range 0 ≤  ≤ 1 for …nite sample sizes, and showed that  = 1 − 2. Shorrocks (2005) subsequently showed that the standard Gini measure, , provides a particularly simple illustration with a more immediate interpretation. The standard Gini measure is ‘replication invariant’, unlike  .5 In the two-person case equality is obtained when each person has 50 per cent of the total income. If the richer person has a share of , the poorer person has a share of 1 − : these shares may also be considered as income levels if total income is normalised to one. Shorrocks showed that  is equal simply to  − 05: he referred to this as the ‘excess share’, since it is the difference between  and the equal share (which he called the ‘fair share’). A complication is that the two-person standard Gini has a maximum value of 0.5, so this case can only be considered where the actual Gini under discussion is less than 0.5 and, even then, the comparison produces a rather wide difference between the two income levels, even for modest Gini values. The approach was extended to allow for more than two individuals in receipt of the lower income value. Shorrocks demonstrated that for a …nite population of  individuals, where total income is normalised to one unit, with the richer person having  and each of the  − 1 poorer people receiving (1 − )  ( − 1), the Gini measure is  =  − 1. Since equality implies shares of 1, the Gini continues to be interpreted as the ‘excess share’. This can be seen as follows: 5 For

a critique of the axiom of replication invariance, see Subramanian (2010).

3.1. PREVIOUS RESULTS FOR THE GINI MEASURE

49

The standard Gini inequality measure for incomes of 1    arranged in ascending order can be written as: 2 X +1 = − 2 ( + 1 − )    ¯ =1 

(3.1)

If one person has income, , and the remaining  − 1 individuals have (1 − )  ( − 1) each, the arithmetic mean is 1 and: " ¶# µ −1 X 1−  + 1 2 − 2 + ( + 1 − ) =   −1 =1 " # −1 +1 2 1−X  + ( + 1) (1 − ) − = −     − 1 =1

(3.2) (3.3)

Using the standard result that the sum of the …rst  integers is  ( + 1) 2, P−1 then =1  =  ( − 1) 2 and substitution and simpli…cation gives the result that:

1 (3.4)  Hence for any given , the income (share) of the richer person is simply =−

 + 1 , and  itself re‡ects the ‘excess share’ of the richer person. This is the result derived by Shorrocks (2005). It is immediately clear that the twoperson case with  = 2 can be applied to provide a simpli…ed interpretation only in the case where   05. Indeed, the standard Gini expression in (3.1) has a maximum value of 0.5 where there are just two individuals. Consider the distribution [0 1], for which the arithmetic mean is 12. Substitution in (3.1) gives  =

3 2

−

2 4

= 12 . Unlike the Atkinson measure, the Gini has a

maximum of 1 only for large . Shorrocks’s approach differs from the earlier analysis by Subramanian (2002), who wrote the Gini as: X +1 2 − ( + 1 − )   − 1  ( − 1) ¯ =1 

 =

(3.5)

This expression gives the same result as the standard formulae in (3.1) for large , but ensures that in the case of  = 2,  has a maximum of 1 for

50

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

the distribution [0 1]. Subramanian showed that, starting from a rank-order welfare function of the form:  =

 X

( + 1 − ) 

(3.6)

=1

The normalised Gini,  , de…ned as the proportional difference between the arithmetic mean, ¯, and the equally distributed equivalent income,  , is given by:

X 2 ( + 1 − )   ( + 1) =1 

 =

(3.7)

This can be contrasted with the Atkinson index, a normalised measure de…ned in the same way but for which, as shown above, the equally distributed equivalent income is a power mean. Indeed,  is a ‘reverse-order-rankweighted’ mean of . Subramanian focused on the share of income received by the poorest person, which he denoted by . Following a similar approach to that used above, it can be shown that: =

1 −  2

(3.8)

Hence,  = 1 − 2. In terms of the Shorrocks concept of excess share,  = 1 − , so that  = 2 − 1, which does not have the convenient interpretation obtained for . Shorrocks mentioned that such shares can be produced for any inequality measure, though none is as simple as the Gini. He compared decile shares and excess shares for a number of countries and inequality measures, including the Atkinson measure (though he did not explore its properties, giving only the formula for the index), and pointed out that in some cases the ranking by inequality measure is not the same as by excess share (for different  values). As mentioned above, the approach adopted by Corvalan (2014) retained the actual distribution but involved the concept of a ‘pivotal’ individual. For the Gini measure, which of course is based on the ranking of individuals’

3.2. THE ATKINSON INEQUALITY MEASURE

51

incomes, Corvalan obtained a simple expression for the rank of the pivotal person. It is thus possible to obtain, given the Gini measure, the percentile above which a (rank preserving) increase in income for any individual produces an increase in inequality (and thus below which an income increase produces a reduction in inequality). For a large population, Corvalan (2014, p. 600) showed that the percentile above which a transfer increases inequality is given simply by 100 (1 + ) 2. Corvalan did not explore the implications for the Atkinson inequality measure.

3.2

The Atkinson Inequality Measure

Consider a population of  individuals with incomes of 1  2    . Suppose an independent judge evaluates income distributions using the following Social Welfare Function:

µ ¶X  1− 1  =  =1 1 − 

(3.9)

where   0 and  6= 1 re‡ects the degree of relative inequality aversion of P the judge. For  = 1,  becomes 1 =1 log  . This welfare function is individualistic, additive, Paretean and satis…es the ‘Principle of Transfers’ (such that a transfer from a richer to a poorer person, which does not affect their relative positions, represents an ‘improvement’). De…ne  as the ‘equally distributed equivalent’ income level. That is,  is the income which, if received by all  individuals, gives the same value of  as the actual distribution. Hence  is the power mean given by:6 (  )1(1−) 1 X 1−  =   =1 

(3.10)

Atkinson’s (1970) inequality measure, , is de…ned for all   0 as: =1− 6 Where

 = 1,  = exp

¡ 1 P 

=1 log 

¢

 ¯

, or the geometric mean.

(3.11)

52

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

where ¯ is arithmetic mean income. Hence inequality is expressed in terms of a ratio of two measures of location (or central tendency), here the ratio of a power mean of order, 1 − , to the arithmetic mean, ¯. An advantage of the Atkinson measure is thus its clear connection to the value judgements involved in the choice of (3.9).7 Nevertheless, interpreting both the size of  and a change in  are far from transparent.

3.3

The Abbreviated Welfare Function

One way to interpret a change in inequality is to refer to the associated abbreviated welfare function which re‡ects the trade-off between total income and its equality. This trade-off is implicit in the value judgements behind the welfare, or evaluation, function itself. For the Atkinson inequality measure, the abbreviated function, denoted  ∗ , is:  ∗ = ¯ (1 − )

(3.12)

Thus  ∗ shows (loosely speaking) the trade-off between ‘equity and efficiency’ that is implicit in  and the inequality measure, , de…ned above, when equity, , is measured by  = 1 − . Hence any change in inequality can be expressed, in social welfare function terms, as equivalent to a particular change in ¯. Totally differentiating (3.12) gives: ¯   (1 − ) ¯    ∗ + = + = ∗  ¯  ¯ (1 − ) and the trade-off (for which

 ∗ ∗

= 0) is such that a 1 percentage change in

equity is equivalent to a 1 percentage change in ¯, and: ¯ µ ¶   ¯  ¯¯  (1 − ) = =− ¯ ¯  ∗ (1 − ) 1−  7 Subramanian

(3.13)

(3.14)

(2002) related the Gini measure to a welfare function based on rank-order weights, using a corresponding equally distributed equivalent income level.

3.4. AN EQUIVALENT SMALL DISTRIBUTION

53

This expression shows that the percentage change in ¯ that is equivalent to a given percentage change in inequality, , is linearly (proportionally) related to the latter, with a slope that depends on  (1 − ). Obviously if  = 05, there is a one-to-one relationship between the percentage changes. ¯ ¯   , and vice versa for With   05, for increasing inequality ¯ ¯ ¯ ∗  

  05. For example, an increase in  from 0.25 to 0.30 represents a 20 per cent increase, which is equivalent in social welfare function terms to a reduction in arithmetic (or total) income of 6.67 per cent. An increase in  from a higher value of 0.30 to 0.36, which is also a 20 per cent increase in the Atkinson measure is, however, equivalent in social welfare function terms to a reduction in ¯ of 8.57 per cent.

3.4

An Equivalent Small Distribution

It is difficult to envisage what a particular value of , and a change in , implies in view of the fact that incomes are usually distributed among a very large number of people. One approach to interpreting orders of magnitude, mentioned in the introduction, is effectively to consider an arti…cial small population having the same inequality measure. The approach asks what a distribution with just two income levels would look like, having the same  value and of course evaluated for the same value of . However, such a division must necessarily polarise the distribution and, as seen below, can therefore provide an exaggerated picture of what an inequality measure ‘means’.

3.4.1

n-Person Shares

Suppose there are just two types of individual and, for convenience only, the total income is normalised to one unit. As relative inequality is of concern, the income values can also be considered as income shares. One person has an income of , and the remaining  − 1 people have an income of

54

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

(1 − )  ( − 1) each. Arithmetic mean income is 1. Thus: " ( ¶1− )#1(1−) µ 1 −  1 =1− 1− + ( − 1)  −1

(3.15)

Given a value of  obtained from an actual income distribution, the aim is to solve (3.15) for . Rearranging this,  is given by the root (or roots) of: µ ¶1− 1−  1− 1−  + ( − 1) (1 − ) −  =0 (3.16)  It may initially seem natural to consider just two individuals in the arti…cial population, producing only a richer and a poorer person and thereby making the properties more transparent. However, as with the standard Gini measure, it is not always possible to solve (3.16), remembering also that feasible solutions require 0    1. Allowing for   2 can enable a feasible solution to be obtained. This aspect can be seen by considering the case where  = 025 for an inequality aversion parameter of  = 02. Figure 3.1 shows the variation in the left hand side of (3.16) as  is varied over the feasible range, for three different values of . Clearly with  = 2 and  = 4 there are insufficient poorer people to generate enough inequality with just the two income levels. When  = 6, a feasible solution can be obtained; the pro…le intersects the horizontal axis for a high value of .8 Consider how the values of  vary as the Atkinson measure increases. Table 3.1 illustrates the effect of increasing  by 20 per cent, from an initial value of  = 025. A dash (–) in the table indicates that for the combination of  and , no feasible solution for  exists: it is not possible to generate the required inequality level with such a small number of individuals. The table shows that for this increase in , the associated increase in the income (share) of the rich person in the small-sample construction varies depending 8 With

just two income shares, there is only one degree of freedom, and hence one equation in one unknown, , to be solved. This is found to admit of just one feasible root (if  is sufficiently large). An attempt to extend the comparison to three income shares would result in more than one feasible solution.

3.4. AN EQUIVALENT SMALL DISTRIBUTION

55

Figure 3.1: Solving for the Two-Income Case: Atkinson Inequality Index of 0.25 and Inequality Aversion of 0.2 on the sample size, , and the degree of inequality aversion. For  = 05 and  = 2, the value of  increases by 3.2 per cent as  increases by 20 per cent. This increases to 6.7 per cent for the same  but for  = 4. For the low inequality aversion parameter of  = 02 in combination with  = 6 and  = 8,  must increase by 6.7 and 7.0 per cent respectively. Yet for  = 09 in combination with  = 6 and  = 8,  must increase by 7.2 and 17.4 per cent respectively. These values of  are in the ranges commonly found for income distributions. The values of  illustrated are around values implied by questionnaires; see, for example, Amiel et al. (1999). Their interpretation is discussed further in Subsection 3.4.2 below. Table 3.1: Top Income, x, for Alternative tion Size  = 025   2 4 6 8 0.2 – – 0.90 0.85 0.5 0.93 0.75 0.66 0.60 0.9 0.84 0.63 0.55 0.46

Inequality Measures and Popula = 030 2 – 0.96 0.87

4 – 0.80 0.68



6 0.96 0.71 0.59

8 0.91 0.65 0.54

56

CHAPTER 3. INTERPRETING INEQUALITY MEASURES Table 3.2 converts the values of  in Table 3.1 into their corresponding

‘excess shares’,  − 1. In the Gini case these would, as Shorrocks (2005) showed, all equal the appropriate standard Gini measure, but this simple property does not carry over to the Atkinson measure. One feature of these values is that, for given , the variation in the excess share as  increases is not monotonic. It increases initially as  increases from  = 2, and then declines slightly after  = 4. Table 3.2: Excess Shares for Alternative Inequality Measures and Population Size  = 025  = 030    2 4 6 8 2 4 6 8 0.2 – – 0.73 0.73 – – 0.79 0.79 0.5 0.43 0.50 0.49 0.48 0.46 0.55 0.54 0.53 0.9 0.34 0.38 0.38 0.34 0.37 0.43 0.42 0.42 However, the excess share decreases monotonically as the degree of inequality aversion, , increases, for which more weight is attached to the lower income range. For a given value of , the richer share, , must decrease as  increases, because it is easier to achieve the given inequality measure, , with a lower share when aversion to inequality is higher. The excess share is by de…nition equal to  − 1 , so this must also fall as  increases, given that such comparisons hold  constant.9 The changes in the excess share vary as  increases from 0.25 to 0.30: it varies from about 8 per cent to over 20 per cent (for the  = 09 combined with  = 8 case). The variation in  as  varies, for given  and , can be examined more formally as follows. Implicit differentiation of (3.16) shows that the slope of this relationship is given by:

¡ ¢−  1−   =  ( − 1) (1 − )− − −

9 Curiously,

(3.17)

Shorrocks reports (2005, Table 3) values of the excess share which, for  = 10, increase as  is increased.

3.4. AN EQUIVALENT SMALL DISTRIBUTION

57

Although this relationship is nonlinear, it can be seen from Figures 3.2 and 3.3 that they are approximately linear. The two …gures show the variation for values of  of 02 and 0.9 respectively. In the case of Figure 3.2 no pro…le is shown for  = 2 because there is no feasible root of (3.16) for all values of  shown.

Figure 3.2: Variation in x with A: Epsilon = 0.2

Figure 3.3: Variation in x with A: Epsilon = 0.9

58

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

3.4.2

The Leaky Bucket Experiment

The restriction to two income levels in the …nite population construct means that the ‘leaky bucket’ thought experiment can be applied directly to those two values. In making comparisons between two different values of inequality it is useful to keep in mind what is implied in terms of the willingness to tolerate leaks in making an income transfer between the two incomes, that is from the richer person to each of the other  − 1 people who share a common income level. From (3.9) consider changes in  resulting from a change in the two income levels 1 and 2 , with 1  2 . Totally differentiating gives:  = 1− 1 + ( − 1) 2− 2 Hence, for constant  :

¯ ¶ µ ¶ µ 2 1 2 ¯¯ =− ¯ 1  −1 1

(3.18)

(3.19)

If ∆1 is taken from the richer person, the amount, ∆2 , that needs to be given to each of the  − 1 people with 2 , in order to hold  constant, is given by:

µ ∆2 = ∆1

1 −1

¶µ

2 1

¶

Thus the minimum proportionate increase in 2 is given by: ¶ µ ¶1− µ 1 ∆1 1 ∆2 = 2 1 −1 2

(3.20)

(3.21)

The maximum leak tolerated by the independent judge, expressed as a proportion of the amount taken from 1 , is thus: ∆1 − ( − 1) ∆2 =1− ∆1

µ

2 1

¶

(3.22)

Table 3.3 illustrates the implications of different inequality aversion parameters for the tolerance of leaks in taking a small amount from the richer

3.4. AN EQUIVALENT SMALL DISTRIBUTION

59

person with , and giving equal amounts to each of the poorer individuals such that social welfare is unchanged. All the cases illustrated in the table are for  = 6, and comparisons are given for two inequality values. Hence the top left-hand cell in the main body of the table shows that if  = 025 when  = 02, the judge tolerates a leak of 53.5 per cent of the amount taken from the richer person, with  given by the corresponding value in Table 3.1. Moving along the row, if the associated value of  is 0.9, again giving  = 025, the judge would tolerate a leak of 80.2 per cent of the amount taken from the richer person. Corresponding amounts for the higher inequality value are shown in the right-hand block of the table which, as expected, reveal a greater tolerance for leaks. Table 3.3: Tolerance for Leaky Bucket as Percentage of Amount Taken: n = 6  = 025  = 030  = 02  = 05  = 09  = 02  = 05  = 09 53.5 67.6 80.2 61.2 71.3 83.2 Table 3.4 illustrates a different kind of sensitivity. This shows the values of  that are obtained for the  value corresponding to a speci…ed  and , but for different  values. The bold diagonal values are the assumed Atkinson measures of 0.25 and 0.30. Consider the …rst row, for  = 02. The value of  that is associated with this combination ( = 6,  = 025 and  = 02) is, from Table 3.1, equal to 0.9. If the resulting values of 090 for the richer person and 0105 for each of the poorer people are then used to calculate inequality, based on a different inequality aversion parameter, of  = 05, the Atkinson measure would be 0.55. Similarly if  = 09 were applied to those incomes, an inequality measure of 0.75 would result. Looking along the rows of the table therefore shows that the values of inequality are indeed highly sensitive to variations in , starting from the income shares that are consistent with the relevant value (from the row) of .

60

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

Table 3.4: Atkinson Inequality Measure for given x and Alternative Aversion Coefficients: n = 6  = 02  = 05  = 09  = 02  = 05  = 09  = 02 0.25 0.55 0.75 0.30 0.66 0.87  = 05 0.11 0.25 0.38 0.14 0.30 0.44  = 09 0.07 0.16 0.25 0.09 0.19 0.30 Atkinson’s (1970) initial discussion of the leaky bucket experiment was in the context of …nding a way to obtain a clearer view of the meaning of different values of , using transfers between just two individuals with incomes of 1 and 2 , say. This produces the result that, after taking ∆1 = 1 from person 1 (the richer person), it is necessary to transfer only ∆2 to person 2 (the poorer person) where:

µ ∆2 =

2 1

¶

(3.23)

Comparisons can then be made for various values of the ratio of incomes. Thus if

2 1

= 21 , aversion of  = 02 suggests that a leak of 0.13 is tolerated

(13 per cent of the dollar taken from person 1). This suggests that 0.2 is a relatively low degree of aversion to inequality. Yet, when a distribution is ‘compressed’ into two income levels (shares of 1 unit), the resulting spread of incomes produces a tolerance, as shown in Table 3.3, that is very much greater, for the same aversion coefficient. This is because, as mentioned earlier, the need to generate a given inequality value with just two income values (shares) means that those values must be ‘pushed’ very far apart. The ratio of the two incomes is very much greater than the usual range considered when the interpretation of  is being discussed. For the same case of

2 1

= 12 ,

a high aversion of 1.5 suggests a tolerance for a leak of 35 per cent of the dollar taken from the richer person. This is considerably lower than any of the values shown in Table 3.3. These examples show that the transformation or abbreviation of a distribution to only two income values (with a small number of individuals) means

3.5. THE PIVOTAL INCOME

61

that those incomes have to be polarised to a degree that is unrealistic and well beyond the range that people generally consider when thinking about inequality aversion.

3.5

The Pivotal Income

Consider the concept of the pivotal income, de…ned in the introduction as the income below which a small increase leads to a reduction in inequality. The pivotal income has a particularly simple form for the Gini measure, but concept can also be applied to the Atkinson inequality measure. As mentioned in the introduction, the pivotal income is seen as a dividing line between rich and poor. It may be compared with a measure of location of the distribution, such as the arithmetic mean. The Atkinson measure is differentiable, so that the pivotal income,  ∗ , is given by the solution to  = 0. Thus:  −  =−   ¯ 

1 

1 ¡P

1− =1 

¢+

 ¯ 2

Setting this equal to zero and rearranging gives: P 1− 1 =1  ∗ −  ( ) = ¯

(3.24)

(3.25)

From (3.10) and (3.11): 1 X 1−  = ((1 − ) ¯)1−  =1 

(3.26)

Thus: ∗ = ¯ (1 − )

−1 

(3.27)

This result is in fact a special case of the much more general results obtained for ‘non-positional measures’ by Lambert and Lanza (2006). For example, consider the simple case where  = 05 and  = 03. Sub and a small addition to the income of anyone stitution shows that  ∗ = 143¯

62

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

with an income smaller than 143 times the arithmetic mean produces a reduction in . Furthermore: 1 ∗ = (1 − )−  

(3.28)

so that in this case the pivotal income is 204 times the equally distributed equivalent income. Further examples of the sensitivity of the pivotal income , varies as  is are shown in Figure 3.4. This shows how the ratio,  ∗ ¯ increased, given two Atkinson inequality measures of 06 and 02. Of course, for any given population, the inequality measure increases as  is increased. Hence for a range of values of  that are assumed to be associated with a  must given inequality measure, and thus different populations, the ratio ∗ ¯ be higher for the lower  values.

Figure 3.4: Ratio of Pivotal Income to Arithmetic Mean and Variation in Inequality Aversion The variation in the ratio of the pivotal income to the arithmetic mean, as inequality increases for a given value of inequality aversion, is shown in Figure 3.5 for two different values of . It can be seen that the pivotal

3.5. THE PIVOTAL INCOME

63

Figure 3.5: Variation in Pivotal Income with Inequality income is not very sensitive to variations in inequality for the higher value of inequality aversion, but is much more sensitive for lower . Given the individual data used to compute the Atkinson measure, the location of the pivotal income in terms of the associated percentile can easily be obtained. However, given only the inequality measure, , and the ratio,  = , say, along with the associated degree of inequality aversion, , ∗ ¯ the percentile can be determined only if the form of the income distribution is known. One approach could be to make the convenient simplifying assumption that the form of the distribution is approximated by the lognormal distribution. Suppose that  is lognormally distributed as Λ (|  2 ). In this case it can be shown that:

¶ µ 2  = 1 − exp − 2

(3.29)

so that, given , the variance of logarithms is: 2 = −

2 log (1 − ) 

(3.30)

64

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

Figure 3.6: Proportion Below Pivotal Income and Inequality: Lognormal Distribution By de…nition,  follows the standard Normal distribution, where: =

log (¯ ) −  

(3.31)

and using the fact that, for the lognormal, ¯ = exp ( + 2 2): =

log  + 2 2 

(3.32)

Figure 3.5 showed that there is a convex relationship between pivotal income and inequality, for high values of inequality aversion. However, Figure 3.6 shows that the corresponding relationship between the proportion below the pivotal income and inequality is slightly concave for the case where incomes are assumed to be lognormally distributed.

3.6

Conclusions

This chapter has considered the problem of conveying just what it means to have a particular income inequality value, or a change in that value. One approach is based directly on the trade-off between equity and efficiency

3.6. CONCLUSIONS

65

that is implied by the (abbreviated) welfare function on which the inequality measure is based. For example, given an annual growth rate, it could be said that a judge is prepared to give up a certain number of years of growth to obtain a given percentage reduction in inequality. This relies on an intuitive understanding of a sum of values, compared with a highly diverse set of values of income obtained by many individuals. The second approach takes as its starting point the suggestion that considering the division of a cake between a small number of people provides, ‘just about the clearest idea we can have of the extent of relative inequality’, so that an inequality measure can be reduced to an equivalent expression of two income (share) values. Subramanian (2002), followed by Shorrocks (2005) with a minor modi…cation, showed how a particularly simple summary along such lines is available for the Gini inequality measure (either in its standard form or its normalised form). This chapter has concentrated on the Atkinson inequality measure. It has been shown that no simple interpretation, in terms of excess shares, is available. Furthermore, it is necessary to solve a nonlinear equation in order to calculate the income share values for any given value of Atkinson’s index. Comparisons of ‘excess shares’ obtained for a range of assumptions, and implications for the ‘leaky bucket’ experiment, suggest that, although some interesting insights are available, it is not clear that the approach can provide the kind of information that non-specialists can easily digest in comparing different values of inequality, at least without some discussion, including an indication of the sensitivities involved. An additional, and quite different, kind of insight is provided by the concept of the pivotal income. This was seen to be very easily calculated given the Atkinson measure (and associated degree of inequality aversion). However, it has been shown here that the relationship between the Atkinson measure and the ratio of the pivotal income to arithmetic mean income is highly sensitive to the degree of relative inequality aversion. And for high aversion

66

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

there is little variation in this ratio as the level of inequality increases. In view of the strong interest in inequality and changes in inequality over time, and the widespread reference to inequality measures, both in evaluating policy changes and in making a case for particular types of policy change, there is a strong incentive to try to provide empirical results that provide transparent guidance about orders of magnitude. This is especially challenging, particularly when different types of value judgement are involved and inequality measures are generally obtained as nonlinear functions of individual incomes which reduce values for many heterogeneous individuals to a single dimension. It is suggested that the information provided by converting inequality measures to comparisons involving a simpli…ed cake-cutting exercise, or concentrating on a pivotal income, cannot therefore be expected to provide immediate clear answers to the communication challenge. But with careful use they may provide useful supplementary descriptions. They can augment discussions and interpretations regarding inequality changes, but cannot be expected to solve the difficult communication challenge. Such approaches do at least preserve the basic value judgements behind the inequality measures and allow for the implications of alternative value judgements to be investigated. This is preferred to the use of oversimpli…ed comparisons, and measures used for rhetorical purposes, that actually disguise the implicit value judgements of those reporting results.

APPENDIX

67

Appendix: Atkinson’s Measure and Aversion The Atkinson measure of inequality, , of the distribution, 1    , is expressed as: =1−

 ¯

(3.33)

where ¯ is the arithmetic mean income, and  is the equally distributed equivalent income, de…ned as: à  =

1 X 1−   =1  

!1(1−) (3.34)

The parameter, , is the degree of relative inequality aversion, with  ≥ 0 and  6= 1. When  = 1, the equally distributed equivalent is geometric mean income. Equation (3.34) is based on the associated welfare function, P  = 1 =1 1−  (1 − ), which is Paretean, individualistic, additive, and is concerned with relative rather than absolute inequality. Measured inequality increases with , and, as Atkinson stressed, the ranking of two distributions can change as  changes. However, inequality typically becomes almost unchanged (approaching unity) as  increases beyond about 5, which virtually re‡ects extreme aversion. Hence, it is possible for a change in a tax and transfer system to be judged as inequality increasing or decreasing, depending on the degree of relative inequality aversion. To examine whether precise conditions can be established under which the ranking of two distributions changes, and the rate at which inequality of two distributions converges or diverges, this appendix derives several elasticities. Direct differentiation of  with respect to  is obviously not straightforward. Consider instead, looking at equality,  = 1 − , rather than inequality, so that: =

 ¯

(3.35)

and: log  = log  − log ¯

(3.36)

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CHAPTER 3. INTERPRETING INEQUALITY MEASURES

Letting  = 1 − , differentiation of (3.36) with respect to  gives:  log   log  =  

(3.37)

The change in log-equality as  increases is therefore simply the change in the logarithm of equally distributed equivalent income. Multiplying both sides of (3.37) by , and using the notation,   , to denote the elasticity of  with respect to , gives (since  log  = ):   =  

(3.38)

¢ ¡ P From (3.34), which becomes log  = 1 log 1 =1  : ! Ã  ¢ ¡ P 1  log 1 =1  1  log  1X  = − 2 log  +    =1   

(3.39)

This can be expressed more succinctly as: log   = −1 +  log( 1  

=1

 )

(3.40)

¢ 1− , and again using  log  = : ¢ ¡ P P  log 1 =1   ( =1  ) 1 = P  (3.41)   =1 

Consider, then,  log

Now consider

Hence:

¡ 1 P 

  (  =1  ) . 

=1

In general, for constant, , and variable, :   ( ) =  log  

(3.42)

P   ( =1  ) X  =  log   =1

(3.43)

Substituting this result in (3.41) and writing in elasticity form gives: P P   (   ) log   =1 ¡ 1 P=1 ¢  log( 1   ) =  (3.44)  =1   log  =1 

APPENDIX

69

In general, elasticities of a variable and the logarithm of that variable are related by the simple relationship:  = (log ) log 

(3.45)

Hence:    = (log  ) log   ( = (log  ) −1 + 

) P   (   ) log    =1 ¡ P=1 ¢ log 1 =1 

P

and using the fact that  =  ¯:   =   =

 µ X



P

=1

=1

(3.46) (3.47)

¶ 

log  − (log ¯)

(3.48)

This elasticity has a convenient interpretation. The proportional change in equality, resulting from a proportional change in inequality aversion, is therefore the difference between a weighted average of log-income and the logarithm of ‘equality adjusted’ arithmetic mean income. Furthermore, (3.48) can be converted into an elasticity of  with respect to , as follows:

¶µ ¶ 1−  (3.49)   =   1−  Similarly, the elasticity of  with respect to  is related to  using: µ ¶  (3.50)  = −  1− µ

Consider the special case of  = 0, corresponding to  = 1. In this case P ³  ´ P  log  = 1 =1 log  , which is the logarithm of geometric =1  =1



mean income. Similarly,  is the ratio of the geometric mean income to arithmetic mean, so that log ¯ is also equal to the logarithm of geometric mean. Hence =0 = 0 when  = 1. Alternatively, when  = 1, corresponding to  = 0, substitution into (3.48) gives:  µ ¶ 1 X  log  − (log ¯)  =1 =  =1 ¯

(3.51)

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CHAPTER 3. INTERPRETING INEQUALITY MEASURES

This result shows that =1 is the difference between a share-weighted mean log-income and the logarithm of arithmetic mean income. This is positive, so that  begins positive for low  and becomes negative for   1. This can also be seen by returning to the simple relationship between elasticities, whereby the term,  = 1− is also expressed in terms of  as: ¶ µ 1−   = − (3.52)  Clearly  is negative for   0: inequality is necessarily higher as inequality aversion increases. Hence   0 when   0, and   0 when   1. The typical shapes of the various pro…les can be illustrated by taking a simple numerical example. Suppose there are just 8 individuals, with incomes in ascending order given by: 5, 10, 20, 50, 100, 300, 500, 1000. Figure 3.7 shows how  and  vary as  is increased. Figure 3.8 plots the three elasticities, 1−1− ,  and  1− , as  varies. The more extensive distributions found in practice will nevertheless give rise to similarly shaped smooth pro…les. Suppose  is lognormally distributed as Λ (|   2 ), where  and  2 are respectively the mean and variance of log : for details, see Aitchison and Brown (1957). From the moment generating function, the arithmetic mean is:

¶ µ 2 ¯ = exp  + 2 and the power mean,  , is given by: 1 " ( )# 1− 2 (1 − )2  = exp (1 − )  + 2

(3.53)

(3.54)

Taking logs and subtracting, gives: log

2  =− ¯ 2

(3.55)

Hence, Atkinson’s measure becomes:

¶ µ  2  = 1 − exp − 2

(3.56)

APPENDIX

71

Figure 3.7: Variation in Inequality and Equality with Inequality Aversion

Figure 3.8: Variation in Elasticities with Inequality Aversion

72

CHAPTER 3. INTERPRETING INEQUALITY MEASURES

Differentiating with respect to  gives:

µ ¶  2 2 = exp −  2 2

(3.57)

The elasticity,  , is thus:  =

2 2 exp ( 2 2) − 1

(3.58)

The result in (3.56) shows immediately that, for the lognormal case, a distributional change – involving a change in 2 – must lead to a consistent upward or downward movement of the pro…le of  against . Therefore, intersections cannot occur. An example of inequality and equality pro…les is given in Figure 3.9 for  2 = 04.

Figure 3.9: Variations in Atkinson Measure and Elasticity with Respect to Epsilon This property suggests that there is a need to be concerned about intersecting pro…les, when comparing two distributions, to the extent that they deviate from lognormality. This distribution is known to provide a reasonable

APPENDIX

73

approximation over the complete range of incomes for many empirical distributions. However, particularly when examining net incomes, distributions can have spikes associated with thresholds relating to means-tested bene…ts, as well as those which may be associated with income tax thresholds. For example, in the context of the New Zealand distribution of net incomes, there is a large spike in the lower tail associated with New Zealand Superannuation. The lognormal form is particularly useful when constructing a range of economic models where the distribution is merely one component and it is necessary to be able to describe the distribution succinctly, say for purposes of aggregation. In such cases, deviations such as the spikes discussed here may not be important.

Chapter 4 Inequality-Preserving Changes The increased interest in inequality in recent years has led to considerable attention being given to changes in summary measures of inequality over time. The most widely used measures in the economics literature are the Gini and Atkinson measures, while in more popular public debates attention is most often restricted to the Gini index. The popularity of the Gini index may perhaps be related to the ease with which it can be described using a Lorenz curve diagram, whereas the Atkinson measure requires explicit discussion of value judgements and the concept of inequality aversion. However, following Atkinson’s (1970) contribution, subsequent research has revealed the value judgements that are consistent with the Gini. As explained below, both measures can be expressed as the proportional difference between an equally-distributed equivalent income,  , and the arithmetic mean. For the Atkinson measure, the welfare function used to evaluate the distribution implies that  is a power mean and for the Gini measure, the very different welfare function gives rise to a reverse-rank weighted mean. Judges with a clear understanding and acceptance of the often-implicit value judgements are necessarily indifferent between two distributions with the same arithmetic mean and inequality measure. But should those readers who may not share the respective value judgements – or may not even be aware of them – also be expected to accept that inequality has remained constant if a summary 75

76

CHAPTER 4. INEQUALITY-PRESERVING CHANGES

measure remains unchanged? For example, both Gini and Atkinson measures of annual market and disposable income per adult equivalent person in New Zealand have been relatively stable since the middle 1990s: see Chapter 5 below. The point addressed here is that such stability is consistent with a broad range of distributions. The extent to which distributional changes can be consistent with an unchanged summary inequality measure are not well understood. Are such changes pathological cases, or are they easily generated and thus create a more serious problem? The basic question considered here may be stated more formally as follows. Suppose it is required to distribute a …xed amount of money (for convenience, referred to as ‘income’) among a given number of people, such that the resulting Gini or Atkinson inequality measure takes a speci…ed value. Since these are relative measures, and are thus not affected by equal proportional changes in all incomes, the actual …xed amount is not important and the requirement could just as well be expressed in terms of income shares. Having speci…ed this requirement, the question arises of whether it implies a unique income distribution. Is there more than one allocation that gives rise to the same inequality measure? From one perspective, this is a trivial problem. Imposing one or more moments of a distribution, along with an inequality measure, simply speci…es linear and nonlinear constraints on the values in a distribution. So long as the number of individuals exceeds the number of constraints, there are some degrees of freedom in selecting values: the simultaneous equations do not necessarily have a unique solution. Despite this obvious property of simultaneous equations, the present chapter explores this question more formally, for Gini and Atkinson measures, in very simple contexts where the number of degrees of freedom is deliberately restricted to a minimum. Indeed, situations are explored where only one or two degrees of freedom are available. The results nevertheless are not restricted to the small populations, since it could simply be assumed

4.1. THE GINI MEASURE

77

that the other income levels are held …xed. The fact that incomes must be positive and, in the case of the Gini measure, the ranks of individuals play an important role, means that the choice of individual incomes is further restricted. These restrictions are not modelled explicitly here, but of course they are checked when investigating numerical examples and ultimately limit the range of alternative distributions. Starting from some arbitrary distribution, an inequality measure can be preserved if an equalising transfer in one range of the distribution is appropriately matched by a disequalising transfer elsewhere. The problem is to determine the nature of such transfers if a continuous range of distributions is to be identi…ed. For example, if two measures, the mean and an inequality measure, must be constant, then the consideration of just three individuals (within a larger unspeci…ed and …xed population) provides one degree of freedom and the ability to move between distributions using two income transfers operating in opposite directions. Section 4.1 examines the Gini inequality measure. First, it considers the case where both the Gini and the arithmetic mean are held constant and it is found that a range of distributions can satisfy these constraints. Second, the variance is also held constant, and again a range of such distributions exists. Section 4.2 considers the case of the Atkinson inequality measure, showing that the allocation of a given income among a …xed number of people to achieve a speci…ed Atkinson measure is, like the Gini, not unique. In each case illustrative examples are given of the types of transfer which can lead from one distribution to another having the same inequality.

4.1

The Gini Measure

This section examines the Gini measure, for which a number of different expressions can be found. One of the most frequently used formulae is the following. For a distribution, 1  2     , with arithmetic mean, ¯,

78

CHAPTER 4. INEQUALITY-PRESERVING CHANGES

the standard Gini inequality measure can be written as: 2 X +1 − 2 ( + 1 − )  =   ¯ =1 

The crucial term here is the sum,

P

=1

(4.1)

( + 1 − )  , which, for constant

,  and ¯, must be constant. Writing the reverse-ranked weighted mean as P P ¯ = { =1 ( + 1 − )  }  =1 ( + 1 − ), it is clear that, for large , the . Hence, ¯ is the equally distributed equivalent Gini is effectively 1− ¯ ¯ income for the Gini welfare function. This form of Gini is not scale invariant, though this matters only for small . In this case of  = 2, the maximum value that can be taken by  is 0.5. The following expression for  with  = 2 is given by Shorrocks (2005). An alternative version is given by Subramanian (2002) for the form of Gini that, for small , lies between 0 and 1.

4.1.1

Distributions with Equal Means

Another way of expressing the question raised in the introduction is to ask whether it is possible to have a mean-preserving change in the distribution of , for which the value of  remains unchanged? Would a judge, whose value judgements are characterised by those lying behind the Gini index, actually be indifferent between two (or possibly more) allocations? The answer is particularly simple if  = 2 and there are only two individ denotes the share of income going to the richer person, uals. If ∗ = 2 2¯ then (4.1) becomes  = ∗ −

1 2

and the allocation is:

2 = 2¯  ( + 05)

(4.2)

1 = 2¯  − 2

(4.3)

with:

Clearly, 2  1 and there is only one allocation satisfying the requirement. From this allocation, any transfer which leaves the arithmetic mean

4.1. THE GINI MEASURE

79

unchanged (that is, for which 1 = −2 ) causes  to change. Similarly, any change which leaves  unchanged requires a change in the arithmetic mean income. With two conditions (constant ¯ and ) and only two income levels, there are no degrees of freedom in selecting the allocation. This is of course not surprising with only two individuals and two constraints. Any possibility of having the two conditions satis…ed by more than one allocation requires at least a degree of freedom in the choice of one of the  values. P Suppose  = 3. Let  = =1 ( + 1 − )  , the inverse-rank-weighted sum in (4.1), and let  denote the sum of incomes. Then for unchanged Gini and mean, it is required to have, for given  and : 31 + 22 + 3 = 

(4.4)

1 + 2 + 3 = 

(4.5)

1 =  − 2 − 3

(4.6)

2 = 3 −  − 23

(4.7)

and: From (4.5): and substituting in (4.4):

Thus it is possible to set the value of 3 , the higher income level, and allow the corresponding values of 2 and 1 to be easily obtained by using (4.7) and then (4.6). The above question thus reduces to …nding, for given  and  whether more than one combination of  values can be obtained which preserves the rank order. Suppose  = 10 and  = 6, so that ¯ = 2. Setting 3 = 3 produces 2 = 2 and 1 = 1. However, if 3 = 28, the resulting values of 2 and 1 are 2.4 and 0.8 . Both distributions give values of  of 0.222. For these two distributions [1 2 3] and [08 24 28] the Atkinson values are quite different. For example, for  of 0.2 and 1.2, the Atkinson values for the …rst distribution are 0.018 and 0.110 respectively, while for the second distribution they are 0.022 and 0.152.

80

CHAPTER 4. INEQUALITY-PRESERVING CHANGES It is well known that the Gini measure can be related to an area in the

famous Lorenz curve diagram. With incomes ranked in ascending order, the Lorenz curve plots, starting from the lowest income, the relationship between the cumulative proportion of total income and the proportion of people to whom it is attributed. The Gini measure is twice the area between the diagonal line of equality and the Lorenz curve. As an area measure, the Gini is not concerned with precisely which parts of the distribution contribute most to inequality. The two Lorenz curves are shown in Figure 4.1, where the curve for the second distribution [08 24 28] is the dashed line. This second distribution is closer to the line of equality for the higher part of the distribution but is further from the line of equality for the lower part of the distribution. Since the ranks of the individuals must remain unchanged, the two distributions involve simply moving the two Lorenz curve points (corresponding to the cumulative incomes associated with 1/3 and 2/3rds of the population) up or down somewhat. In each Lorenz curve the area contained by the Lorenz curve and the diagonal is the same. Intersecting Lorenz curves usually motivate the need for an explicit inequality measure, since a basic ‘dominance’ result does not apply, but in this case the Gini measure cannot distinguish between the two distributions. Consider instead the distribution [0 3 3] for which the Lorenz curve follows the diagonal beyond the 2/3rds point and the Gini area is all contained to the left of that point (with  = 0333). Another distribution with the same Gini is obtained simply by reducing inequality at the bottom of the distribution and increasing it at the top end. Thus, to give just one alternative, [02 26 32] is found to have the same Gini value and of course the same arithmetic mean. A way to view the two distributions is to see that, starting from [0 3 3], an equalising transfer of 0.2 is made from person 2 to person 1, and at the same time a disequalising transfer of 0.2 is made from person 2 to person 3. There are two equal transfers from the middle person. The nature of the effective transfers can be seen using the following ap-

4.1. THE GINI MEASURE

81

Figure 4.1: Two Lorenz Curves with Equal Ginis proach, which begins from a given distribution and investigates the changes in 1 , 2 and 3 for which the arithmetic mean and Gini are …xed. The Gini value for  = 3 can be written as: 2 =2− 3

µ

31 + 22 + 3 1 + 2 + 3

¶ (4.8)

Totalling differentiate  with respect to 1 , 2 and 3 , and impose the condition 1 = − (2 + 3 ) to ensure that the arithmetic mean remains unchanged. Then setting  = 0 gives the result, after a little algebra, that: 2 = −2 3

(4.9)

Hence the slope of the relevant constraint is …xed, irrespective of the  values. This is precisely the change illustrated in the …rst example and Figure 4.1, where 3 falls by 0.2 from 3 to 28 and 2 rises by 0.4 from 2 to 24. And of course 1 must change by − (04 − 02) = −02 and thus fall from 1 to 0.8. Importantly, there is thus a range of allocations satisfying the requirement that  = 0222 and ¯ = 2. In each case, any number of new distributions

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CHAPTER 4. INEQUALITY-PRESERVING CHANGES

can be found satisfying the required conditions. In summary, specifying an allocation of a …xed income among a given number of people, where the distribution must have a given Gini value, does not imply a unique allocation. Indeed, a range of distributions is consistent with the speci…ed Gini.

4.1.2

A Common Mean and Variance

In the above example the variance of the …rst distribution is 0667 while for the second distribution it is 0.747. The question arises of whether it is possible to have a change in the distribution which preserves both the mean and the variance, while also holding the Gini value constant. If a judge wishes to allocate a …xed sum of money to achieve both a given Gini and variance, is the resulting distribution unique? In this case, the additional constraint involves a loss of a further degree of freedom, so it is necessary to increase the population size to  = 4. The required conditions, for given values of ,  and , are as follows. For constant Gini (for a given ¯): 41 + 32 + 23 + 4 = 

(4.10)

and for a constant arithmetic mean: 1 + 2 + 3 + 4 = 

(4.11)

Furthermore, a constant variance, for a given arithmetic mean, requires: 21 + 22 + 23 + 24 =  From (4.11): 1 =  −

4 X



(4.12)

(4.13)

=2

and substituting in (4.10): 2 = 4 ( − 3 − 4 ) + 23 + 4 −  = 4 − 23 − 34 − 

(4.14)

4.1. THE GINI MEASURE

83

Substituting (4.14) into (4.13) gives: 1 =  − 3 + 3 + 24

(4.15)

Hence, substituting in (4.12) gives the quadratic:  = [ − 3 − 4 − {4 − 23 − 34 − }]2 + {4 − 23 − 34 − }2 + 23 + 24

(4.16)

For given values of ,  and , and setting a value of 4 , the quadratic (4.16) can be solved for 3 . Extending the requirement to include a …xed third moment would thus be rather cumbersome as it would require the solution to a cubic in addition to a quadratic similar to (4.16). The question then involves determining whether there are two real and distinct roots for 3 and whether those resulting values give rise to corresponding values of 2 and 1 , using (4.13) and (4.14), which are positive and preserve the necessary rank order. For example, suppose that  = 8, so that ¯ = 2, and  = 15, with  = 2154. Setting 4 = 35 it is found in this case that (4.16) has two roots of 2.2 and 2.8. Hence there are two distributions, [02 21 22 35] and [08 09 28 35] that have the same mean, variance and Gini value: in this case,  = 03125. Going from one distribution to the other effectively involves two simultaneous transfers – one being disequalising while the other is equalising, from or towards person 2, depending on which is considered to be an ‘initial distribution’. Consider the slope of the relevant constraint, relating 3 and 4 , by beginning with a particular distribution and investigating changes in  values for which the Gini and …rst two moments are constant. Totally differentiating: µ ¶ 5 1 41 + 32 + 23 + 4 (4.17) = − 4 2 1 + 2 + 3 + 4 and using: 1 = − (2 + 3 + 4 )

(4.18)

84

CHAPTER 4. INEQUALITY-PRESERVING CHANGES

gives: 2 = −23 − 34

(4.19)

Then differentiating 21 + 22 + 23 + 24 totally, setting the result equal to zero and substituting for 1 and 2 using (4.18) and (4.19) gives: 3 − 1 − 2 (2 − 1 ) 3 = 4 4 − 1 − 3 (2 − 1 )

(4.20)

Unlike the previous example, the slope of this constraint is not constant. Hence in general even small discrete changes from a given distribution do not satisfy the additional requirement. Nevertheless the existence of a range of solutions can easily be seen by solving the above simultaneous equations for a different imposed value of 4 . Starting from [08 09 28 35], suppose that there is an equalising transfer of 0.1 from person 4 (the richest individual) to person 3, while there is simultaneously a disequalising transfer of 0.1 from person 1 (the poorest) to person 2. This results in the distribution [07 10 29 34], which ensures that total income is unchanged and is also found to maintain a …xed variance and Gini value. Hence, as before, only one degree of freedom is enough to generate the result that a range of distributions exist such that the …rst two moments and the Gini measure are constant. With a …fth person, giving two degrees of freedom, it is easy to see that a much wider range of distributions can satisfy the three constraints. For example, if a …fth person is added to [02 21 22 35], with 5 = 4, then  = 12,  = 27, ¯ = 24 and  = 375. If one unit is transferred from person 5 to person 4, and at the same time one unit is transferred from person 2 to person 3, resulting in [02 20 23 36 39], the three constraints are satis…ed.

4.2. THE ATKINSON MEASURE

4.2

85

The Atkinson Measure

This section considers the Atkinson inequality measure,  , which is de…ned, for a relative inequality aversion parameter of , as:  = 1 −

 ¯

(4.21)

where  is the equally-distributed equivalent income given by the power mean:

(  =

1 X 1−   =1  

)1(1−) (4.22)

The question arises of whether two or more distributions can be found having the same Atkinson measure. Suppose only the mean is imposed, so that three values provide a necessary degree of freedom, and the sum, , and equally distributed equivalent,  , are given. Using 1 =  − 2 − 3 and the de…nition of the equally distributed equivalent income, the value of 2 , for given values of 3 ,  and  , is given by the root or roots of: ( − 2 − 3 )1− + 1− + 1− − 31− =0 2 3 

(4.23)

It is found that this expression can have no real roots, one root or two real, positive and distinct roots. For example, suppose that  = 6 so that ¯ = 2, and for  = 03,  = 18. Setting 3 = 3, it is found that there are two solutions for 2 , equal to 0.2 and 2.8. However, the symmetry gives rise to corresponding values of 1 of 2.8 and 0.2. Since the ranks are not relevant, having speci…ed the value of 3 , there is thus only one distribution, given by [02 28 30], that is consistent with the imposed value of  and the arithmetic mean, which together give  . Furthermore, using 1 = − (2 + 3 ) and setting the total differential,  , equal to zero gives 2 = −3 (3 − 1 )  (2 − 1 ). Since 2 3 is not constant, this expression cannot be used to examine discrete changes. This contrasts with the Gini case discussed above where only the arithmetic mean and inequality are constant and there are three individuals.

86

CHAPTER 4. INEQUALITY-PRESERVING CHANGES However, simply by setting 3 to alternative values, a range of alternative

distributions clearly exists for which the mean and Atkinson inequality measure are …xed. For example, starting from the above distribution, suppose that person 2 transfers 1.4 to person 3 and 0.4 to person 1. This combines an equalising with a disequalising transfer and results in the distribution [06 10 44], which has the same Atkinson measure as the initial distribution. With one extra individual, and hence an increase in the number of degrees of freedom to two, it is possible to generate an even wider range of possibilities. The equalising transfer in one range of the distribution can be combined with a disequalising transfer in another range of the distribution, involving a different pair of individuals (rather than the two transfers taking place from the middle-income person only). First, for 1  2  3  4 , suppose a small amount − is transferred from 2 to 1 while simultaneously + is transferred from 3 to 4 . Setting the total differential of  equal to zero gives the result that, for an unchanged  : − − + 1 − 2 = − − − 3 − 4

(4.24)

Again, this expresses the tangent to the required constraint imposed on the two changes to ensure that the Atkinson measure is constant. Given the nonlinearity involved, it cannot be used to consider a range of distributions. Consider instead the simple extension of (4.23) to four individuals, giving: + 1− + 1− − 41− =0 ( − 2 − 3 − 4 )1− + 1− 2 3 4 

(4.25)

It is now possible to set 3 and 4 and determine, for given ,  and  , the required values of 1 and 2 . As before, where two real roots for 2 exist, these simply provide interchangeable values for 1 and 2 so that, since the ranks are not relevant, only one distribution satis…es the condition. However, two simultaneous transfers can be examined. Suppose that for  = 4 and  = 03, it is required to have ¯ = 20 and  = 18. First, setting

4.3. CONCLUSIONS

87

4 = 4 and 3 = 25 gives the result that 2 = 135 and 1 = 015. Then suppose that there is a disequalising transfer at the top end of the distribution such that 4 becomes 4.2 and 3 becomes 2.3. Solving (4.25) gives new values for 2 = 13 and 1 = 02, for which the same conditions are satis…ed for  and  . Hence an equalising transfer at the bottom end of the distribution of 005 from 2 to 1 can be combined with a disequalising transfer at the top end, of 0.2 from 3 to 4 , to maintain the Atkinson inequality measure. For this case of − = 005, the expression in (4.24) gives a value of + = 043, showing the extent of the nonlinearity of the constraint, since for the discrete change examined, it is necessary only to transfer 02 from 3 to 4 .

4.3

Conclusions

This chapter has considered the problem of distributing a …xed amount of money (referred to as ‘income’) among a given number of people, such that inequality, as measured by either the Gini or Atkinson measure, takes a speci…ed value. In particular, the question arises of whether a unique distribution is implied. In general, it is trivially obvious that, given a set of constraints (in the form of moments of the distribution), a range of solutions to the implied simultaneous equations can generally be found. However, this chapter has explored the characteristics of the two inequality measures for cases where only one or two degrees of freedom are available. The examination of small populations is not a restriction, since the income values other than the small number being considered may simply be regarded as being …xed. The measures differ in that the inequality constraint for the Gini involves a linear combination of income values, involving the rank positions, whereas for the Atkinson measure a nonlinear constraint is imposed, involving the degree of relative inequality aversion. For both inequality measures, the intuition was con…rmed that only one degree of freedom is necessary to generate a range of possible distributions,

88

CHAPTER 4. INEQUALITY-PRESERVING CHANGES

for cases where the mean and inequality are constrained and, in the case of the Gini measure, where the …rst two moments and inequality are constrained. For the Gini measure it was found that the choice of one income level (the degree of freedom) can give rise to two different distributions giving the same Gini value (and moments): that is, the simultaneous equations can generate two distinct solutions, satisfying the further requirements that values are non-negative and preserve the rank order. Furthermore, a range of distributions is obtained simply by varying the income level imposed. In the case of the Atkinson measure, when one of the income levels is imposed, the simultaneous equations can give rise to only one feasible solution: where two solutions are generated, they imply the same distribution since the rank order of individuals is not important. Nevertheless, a range of distributions is consistent with a …xed Atkinson measure, obtained by varying the imposed income level. In both Gini and Atkinson cases, the variations are identi…ed with simultaneous equalising and disequalising transfers. For a sufficiently large population, it should in general be possible to …nd multiple allocations for which any number of moments of the distribution, along with an inequality measure, are equal. To give just one example using numerical methods, the distributions [1 2 4 5 10] and [08 26 63 52 10] have the same Gini and …rst three moments. The second can be obtained from the …rst by using two disequalising transfers and one equalising transfer. Small variations in the top income (of 10) generate two solutions for each imposed value of 5 . The important question arises of whether this feature matters for the practical measurement of inequality. Any judge who holds the value judgements implicit in one of these measures will not of course be worried by the fact that a ‘stable’ value of inequality is consistent with quite substantial changes in the precise nature of the frequency distribution of income. The judge is by de…nition indifferent to all such distributions. Hence, the main implications concern the reporting of inequality measures and inferences – particularly policy inferences – drawn from them, when it is recognised that

4.3. CONCLUSIONS

89

many readers are unlikely to share those precise value judgements. Unfortunately, inequality measures continue to be frequently reported with no clari…cation of the value judgements involved. The Gini, for example, is often described simply as being ‘widely used’, as if that were sufficient justi…cation for the reader to accept the results without question. It was mentioned in the introduction that in New Zealand, and despite a substantial increase in media interest in inequality, Gini measures of income and expenditure have remained relatively stable since the mid-1990s. Ball and Creedy (2016) explicitly discuss the value judgements and the nature of the social evaluation function behind the Gini index. They also refer to sensitivity analyses which show that a similar pattern holds for a range of unreported Atkinson inequality measures (for different degrees of inequality aversion). Nevertheless, views regarding inequality are so complex, and unlikely to be capable of being summarised in a simple functional form, that widely used summary measures such as the Gini cannot be expected to be universally accepted, particularly as the basis of policy recommendations. Those who argue that inequality has increased or decreased will not necessarily be persuaded by a time series of a summary inequality measure. In particular, any discussion of inequality usually reveals that there are many non-income differences among individuals and households that judges regard as relevant. Like any summary measure, a single statistic involves a large loss of information. The results presented here suggest at least a further need for caution in making inferences about inequality based on such summary measures.

Chapter 5 Decomposing Inequality Changes Comparisons of inequality measures over time generally show annual inequality, where each year’s value is obtained independently. However, attitudes towards changes in inequality are likely to depend on perceived reasons for the change. Inequality may change as a result of deliberate government policy, involving for example changes to income taxation and welfare bene…ts. Measured inequality may also be affected by a wide range of other population changes which may not be closely related to policy variables. They may reinforce or work against policy aims, making ex post evaluation of the efficacy of policies difficult. Such changes may include, for example, changes in the age distribution of the population, resulting from the demographic transition towards an older age structure, and changes in the structure of households, along with changing patterns of labour force participation. This motivates the present attempt to disentangle the contribution of such changes to overall inequality changes. The type of question considered is thus: to what extent can the observed change in inequality over the period be attributed to the changing structure of households over time? This chapter explores the use of survey calibration, combined with decomposition, to examine contributions to annual inequality of income and expenditure in New Zealand over a thirty-year period from the early 1980s. 91

92

CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

This involves the calculation of sample weights to ensure that a range of aggregates, obtained when grossing-up from survey to population levels, match speci…ed values obtained from independent data. Measures of inequality and the extent of redistribution through the tax and bene…t system are therefore obtained which hold a large number of demographic characteristics of the population constant over the period, thereby isolating their in‡uence. Calibration is then combined with decomposition methods to examine the separate contribution of particular types of demographic change. The data are obtained from the Household Economic Survey for each year from 1983/84 to 1997/98 inclusive, and for each year from 2006/07. Statistics New Zealand did not carry out annual surveys from 1998 to 2006, so that comparable information for 2000/01 and 2003/04 only are available during this period.1 Access to these datasets, providing considerable information about the circumstances of individuals and families, has been extremely limited. Hence, despite the large amount of interest in inequality in recent policy debates, few studies are available allowing for consistent comparisons.2 Furthermore, previous studies have used sample weights provided by Statistics New Zealand, but these were obtained only from 1987 (and are calibrated to population age and gender distributions only). Hence values for earlier years effectively assume equal weights for all individuals and households.3 Com1 In

various tables and diagrams below, linear interpolation is used to provide values for missing years. 2 The main analyses of redistribution, …scal incidence and inequality have been carried out within government departments. In particular, see Statistics New Zealand (1999) and work within the Ministry of Social Development reported by Perry (2014). The New Zealand Treasury carries out regular …scal incidence studies. An earlier description within the Department of Labour is Dixon (1996). Summary results for disposable incomes in selected years (1981/82, 1985/86 and 1989/90) using the Jensen (1988) scales are reported in Saunders (1994). Decompositions by income source for 1983/84, 1991/92 and 1995/95, using gross (pre tax) incomes are reported by Podder and Chatterjee (2002), using household income per person as the income measure, with individual as unit of analysis. The selection of years gives a misleading picture of the inequality changes over the 1980s, as seen below. 3 For a description of the integrated weighting used by Statistics New Zealand,

5.0. INTRODUCTION

93

parisons are therefore made here using both Statistics New Zealand weights and those obtained using a much wider range of calibration values. Section 5.1 brie‡y describes the income and expenditure measures and unit of analysis for which results are reported. The changes over time were explored using a variety of inequality measures and income units. However, because of space limitations, results are presented using only Gini measures of income and expenditure per adult equivalent person, with the individual as unit of analysis. Although some year-to-year variations differ, the main comparisons over the period are not substantially affected by the choice of unit or inequality measure. Section 5.1 also describes the approach to calibration of the HES, as this is a crucial component of the analysis. Section 5.2 reports basic results regarding the inequality of market and disposable income per adult equivalent person using the Gini inequality measure. The inequality measure is directly associated with an explicit statement of value judgements summarised in a so-called ‘social welfare function’ (an evaluation function representing the values of an independent judge). Section 5.3 therefore examines variations in ‘social welfare’ over time and its decomposition into changes in arithmetic mean income and equality (de…ned as one minus the inequality measure). Major changes in the population structure took place over the period investigated. This motivates the examination in Section 5.4 of changing inequality over time, under the assumption of an unchanged population structure, de…ned by almost …fty demographic and labour force characteristics. Section 5.5 attempts to disentangle the contributions to changing inequality of particular components of population change. An analysis of this kind is subject to limitations. First, it is not possible, when dealing with a thirty-year period involving substantial changes to the surveys as well as the bene…t system, to apply the full administrative details see Statistics New Zealand (2001).

94

CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

of the tax and bene…t system of one year to a sample survey of all other years.4 This means that the precise role of the tax structure cannot be isolated from other uncalibrated changes (such as changes in the occupational structure, the nature of age-income pro…les and differential wage growth). Second, it must also be acknowledged that tax and bene…t changes may themselves lead to changes that are effectively being treated here as exogenous. For example, to attribute a component of the observed change in inequality to changes in fertility patterns, or changes in the structure of households (such as the growth of single-parent households), neglects the possibility that such demographic changes may themselves have been partly in‡uenced by changes in government tax and bene…t policies. While reference may not therefore strictly be made to ‘ultimate causes’, since not all changes are truly exogenous, it is nevertheless useful to be able to isolate a number of crucial components.5 Such results can usefully also point to the need for further investigations of a different kind. Furthermore, in examining income and expenditure distributions, no allowance is made for leisure (and thus endogenous labour supply variations) or for other changes, such as the introduction of new goods or changes in the quality of goods and services over time, which may contribute to wellbeing.6 4 However,

this can be done for more recent selected years in New Zealand; see Chapter 2. Adam and Browne (2010) looked at the redistributive effect of taxes in the UK over a thirty year period, but this kind of exercise is not possible for NZ in view of the large changes in eligibility rules and their enforcement, the types of bene…t in existence, and the information needed to apply the rules. 5 Of course, the recognition that, for particular purposes, a variable is being treated as exogenous is extremely common in both empirical and theoretical economic analyses. In any analysis of redistribution, it must be recognised that the pre-tax distribution is not itself exogenous. 6 For a decomposition analysis allowing for leisure see Creedy and Hérault (2015).

5.1. SOME PRELIMINARIES

5.1

95

Some Preliminaries

Results are reported using household income per adult equivalent person, both before income tax and transfers (referred to as market income) and after tax and cash transfers (referred to as disposable income). Results are also shown for expenditure. The unit of analysis – effectively the weight attached to each income per adult equivalent – is the individual.7 In all cases the accounting period is the year. Subsection 5.1.1 describes the adult equivalence scales used. In view of the extensive use made of survey calibration to allow for changing population structures, subsection 5.1.2 brie‡y describes the approach used.

5.1.1

Adult Equivalent Scales

The adult equivalence scale applied here is a two-parameter scale, allowing simply for a weight attached to children and economies of scale. Let  and  denote respectively the number of adults and children in the household, and let  denote the adult equivalent size of the household. Then:  = ( +  )

(5.1)

where  and  are parameters re‡ecting the relative ‘cost’ of a child and economies of scale respectively. This form was introduced by Cutler and Katz (1992) and investigated by, for example, Banks and Johnson (1994) and Jenkins and Cowell (1994). An advantage of this form is that it allows sensitivity analyses to be carried out easily, where the parameters have clear interpretations. Creedy and Sleeman (2005) found that, despite its simplicity, it provided a close …t to 29 alternative sets of equivalence scales. Benchmark values of  = 05 and  = 08 are used here. 7 Podder

and Chatterjee (2002, p. 11) argue that the ‘appropriate weight must be the number of members in the family’. However, the option of using the number of equivalent adults is discussed by Glewwe (1991), Decoster and Ooghe (2003), Shorrocks (2004) and Creedy and Scutella (2004). See also Creedy (2013) and Chapter 2 above.

96

5.1.2

CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

Survey Calibration

The calibration problem can be stated as follows, following the standard approach of Deville and Särndal (1992). For each of  = 1   individuals in a sample survey, information is available about  variables. These are placed in the vector:

⎡

1

⎢ ⎢  ⎢ ⎢  = ⎢ ⎢  ⎢ ⎢  ⎣ 

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(5.2)

These vectors contain only the variables of interest for the calibration exercise, rather than all measured variables. Many of the elements of  are likely to be 01 variables. For example  = 1 if the th individual is in a P particular age group, and zero otherwise. The sum  =1  therefore gives the number of individuals in the sample who are in the th age group. Let the initial or ‘design’ sample weights be denoted  for  = 1   These weights can be used to produce estimated population totals, b  | , based on the sample, given by the -element vector: b | =

 X

 

(5.3)

=1

Suppose that another set of population totals,  , is available. As in the present context, these may relate to another year, or they may relate to other extraneous sources. The problem is to compute new or ‘calibration’ weights,   for  = 1   which are as close as possible to the initial weights,   while satisfying the set of  calibration equations:  =

 X

 

(5.4)

=1

In specifying a criterion by which to judge the closeness of the two sets of weights, denote the distance between  and  as  (   ). The aggregate

5.2. GINI MEASURES FOR ALTERNATIVE DISTRIBUTIONS

97

distance between the design and calibrated weights is: =

 X

 (   )

(5.5)

=1

The problem is to minimise (5.5) subject to (5.4). The Lagrangian is: Ã !    X X X = (5.6)  (   ) +   −   =1

=1

=1

where  for  = 1   are the Lagrange multipliers, and  represents the th element of the vector of known population aggregates,  . The …rst-order conditions generally give rise to nonlinear simultaneous equations which can be solved using numerical methods.8

5.2

Gini Measures for Alternative Distributions

The Gini coefficients from 1984 to 2013 are shown in Table 5.1 for market and disposable income, and expenditure. In each case results are reported using the calibrated weights and the sample weights provided by Statistics New Zealand: before 1987 these were the same for all sample individuals but afterwards the weights ensure that the grossed-up population age and gender distributions match demographic data. The calibrated weights were obtained using a total of 47 calibration values: these cover the age and gender distributions, labour force participation by age and gender, household type, number of individuals per household and housing tenure. It is not possible to include calibration values based on bene…t totals, such as the number of individuals or households receiving a range of social bene…t payments.9 8 For

further discussion of calibration in the New Zealand context see Creedy and Tuckwell (2004). 9 In Section 5.5 below, sample weights are obtained for each year using the calibration totals for all other years. This is straightforward for aggregates such as the number of people in an age and gender group, but the nature of bene…ts

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CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

Table 5.1: Gini Inequality Measures 1984 to 2013 Market income Disposable Year Calibrated StatsNZ Calibrated 1984 0.3915 0.3909 0.2586 1985 0.4025 0.4030 0.2651 1986 0.4056 0.3990 0.2638 1987 0.4062 0.4157 0.2703 1988 0.4136 0.4152 0.2632 1989 0.4503 0.4455 0.2886 1990 0.4633 0.4645 0.3025 1991 0.4592 0.4734 0.3068 1992 0.4670 0.4808 0.2961 1993 0.5015 0.5013 0.3337 1994 0.4957 0.5001 0.3117 1995 0.4956 0.4943 0.3196 1996 0.4971 0.4926 0.3302 1997 0.4901 0.4958 0.3261 1998 0.4850 0.4767 0.3180 1999 0.4922 0.4797 0.3279 2000 0.4994 0.4827 0.3377 2001 0.5067 0.4858 0.3476 2002 0.4942 0.4789 0.3384 2003 0.4817 0.4720 0.3292 2004 0.4692 0.4652 0.3200 2005 0.4635 0.4615 0.3156 2006 0.4579 0.4579 0.3112 2007 0.4523 0.4542 0.3068 2008 0.4597 0.4602 0.3164 2009 0.4508 0.4590 0.3155 2010 0.4438 0.4609 0.3034 2011 0.4763 0.4890 0.3328 2012 0.4545 0.4621 0.3078 2013 0.4638 0.4698 0.3180

income Expenditure StatsNZ Calibrated StatsNZ 0.2594 0.3066 0.3070 0.2688 0.3146 0.3180 0.2613 0.3137 0.3177 0.2774 0.3177 0.3235 0.2670 0.3195 0.3210 0.2906 0.3457 0.3452 0.3043 0.3395 0.3452 0.3198 0.3270 0.3395 0.3094 0.3273 0.3368 0.3340 0.3501 0.3579 0.3215 0.3471 0.3504 0.3271 0.3398 0.3448 0.3360 0.3555 0.3573 0.3352 0.3507 0.3608 0.3291 0.3256 0.3305 0.3322 0.3221 0.3213 0.3354 0.3187 0.3121 0.3386 0.3153 0.3029 0.3339 0.3209 0.3116 0.3292 0.3265 0.3203 0.3245 0.3322 0.3290 0.3199 0.3315 0.3268 0.3154 0.3308 0.3246 0.3108 0.3302 0.3224 0.3247 0.3185 0.3147 0.3243 0.3069 0.3070 0.3177 0.2953 0.2993 0.3439 0.2989 0.3008 0.3167 0.3026 0.3024 0.3289 0.3062 0.3039

5.2. GINI MEASURES FOR ALTERNATIVE DISTRIBUTIONS

99

For market incomes per adult equivalent, the main differences are a somewhat higher Gini value for the calibrated weights in 2001, and a more rapid subsequent decline, along with a lower value in the early 1990s compared with the use of Statistics New Zealand weights. The Gini values of disposable income are generally slightly higher when using Statistics New Zealand weights rather than the calibrated weights (except for 2001).10 Further investigation of the year for which the two sets of weights give the greatest difference showed that in 2001 the calibration weights increased for a substantial number of households with high incomes (over $150,000). Although it is hard to identify causes of this temporary increase, this is discussed further below. Nevertheless the general patterns are broadly similar. The Gini measure of market incomes saw a steady rise through the second half of the 1980s to the early 1990s, from about 0.4 to around 0.5. Subsequently there has been a steady though less marked decline, with the exception of ‘spikes’ around 2001 and 2011. For disposable incomes, the systematic increase in the Gini measure does not appear to have started until the late 1980s, rising from about 0.27 to about 0.33 in the mid-1990s. The pro…le displays some variability subsequently, although generally it has been declining slightly. In the case of expenditure per adult equivalent person, the general movement is one of increasing inequality from 1984 until the mid-1990s, followed by declining inequality. Unlike the cases of market and disposable incomes, the Gini inequality of expenditure had, by 2010, returned to being slightly less than the 1984 value. Generally the inequality of expenditure is higher than that of disposable income until the late 1990s, after which the two are similar. This result contrasts with some other studies which …nd that inchanges over time. New bene…ts are introduced, while others are abolished, while the regulations regarding bene…t status and eligibility can change in signi…cant ways over time. 10 The distributions of market income per adult equivalent obviously have a number of zero observations. Negative values were excluded, since the Gini is de…ned only for non-negative incomes. Zero and negative values of disposable incomes were excluded. These deletions involved a very small number of households.

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equality of expenditure is less than that of disposable income, and this is attributed to some form of consumption smoothing (whereby lower-income households have a higher tendency to dissave, while higher-income households save). However, the comparison is not unequivocal. For simplicity, let  ,  and  denote household ’s savings, disposable income and expenditure respectively in a given period. Let  be an independently distributed random term, re‡ecting preference heterogeneity and other characteristics that are independent of income. Suppose:  = − +  + 

(5.7)

 = (1 − )  +  − 

(5.8)

2 = 2 + (1 − )2 2

(5.9)

Then: Taking variances gives:

The variance of expenditure is higher than that of disposable income, such that  2  2 , if: 1 − (1 − )2 

 2 2

(5.10)

Hence, although (5.7) suggests a tendency for savings to be positively related to disposable income, expenditure inequality can exceed that of disposable income if the variance of the random term is sufficiently high in relation to that of disposable income. Regressions of (5.7) were carried out for each year over the period. While the estimates of  were positive, the estimated values of 2 were found to be from 1.2 to 2.5 times the variance of disposable income,  2 , until the late 1990s when the ratio dropped to around 0.5. Yet the left hand side of (5.10) was generally around 0.3 to 0.6 during the 1980s and 1990s, after which it rose to around 0.8. Hence some form of consumption smoothing is indeed consistent with the inequality of expenditure exceeding that of disposable income, as a result of the high dispersion of  .

5.2. GINI MEASURES FOR ALTERNATIVE DISTRIBUTIONS

101

Figure 5.1: Gini Inequality and Tax Changes 1984 to 2013 The pro…les using calibrated weights are shown in Figure 5.1, which also indicates tax and other changes made over the period: for discussion of the policy changes during the 1980s and early 1990s, see Evans et al. (1996). It appears that the 1980s reforms – involving cuts in the top income tax rate along with bene…t cuts and the ending of centralised wage setting – are associated with increasing inequality. The spikes in the market and disposable income pro…les from 2000 may also be associated with changes in top income tax rates. In the …rst case of an increase from 33 to 39 per announced in 2000 but effective in 2001), the anticipation of the rate increase could have led to a certain amount of income shifting into the year before the increase. Much of the shifting is likely to have been by those in higher-income groups, and hence this contributes to the sudden increase in inequality, followed by a reduction. In the case of the 2010 reduction in the top rate, the opposite incentive effect operated. For direct evidence of the extent of income shifting

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CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

in anticipation of the tax changes, see Claus et al. (2012). To the extent that high-income earners were able to delay income receipt until after the tax reduction took effect, this would contribute to a dip in inequality followed by an increase in the Gini value immediately after the fall in the top rate. Some of the variability during the 2000s may thus perhaps be attributed to income shifting over time resulting from anticipated income tax changes.

Figure 5.2: Redistribution based on Current-Year Sample Weights The redistributive effect of the direct tax and transfer system in each year is shown in Figure 5.2, which reports the difference between the Gini measures of market income and disposable income: this is often referred to as the Reynolds-Smolenski index. Redistribution depends both on the nature of the pre-tax and transfer income distribution and the tax structure, and care must be taken in interpreting these measures because the distribution of market income is not in fact exogenously given independently of the tax structure. Redistribution increased in the period during which the Gini increased in the late 1980s and early 1990s, after which it declined. The use of calibrated

5.3. INEQUALITY AND SOCIAL WELFARE

103

weights produces a higher degree of redistribution than those provided by Statistics New Zealand. The diagram shows that the substantial reduction in the top rate of income tax in 1987 was associated with a slight reduction in the redistributive effect of taxes and transfers. However the further reduction in 1988 was not associated with a reduction. Furthermore, the introduction of a new top rate in 2001 was not associated with an increase in the redistributive effect of direct taxes and transfers. Similarly, the introduction of the bene…t payment, Working for Families, in 2005 was not associated with an increase in the redistributive effect of taxes. However, a small increase in redistribution arose with the tax mix change in 2010.

5.3

Inequality and Social Welfare

The Gini inequality measure is associated with particular value judgements of an independent judge. The judge is regarded as evaluating alternative distributions using an evaluation function, usually referred to as a ‘social welfare function’, expressed in terms of the individual incomes. For the Gini measure this function takes the form of a weighted sum of incomes, where weights depend on the reverse rank order, with incomes ranked from lowest to highest. Thus the lowest income is given a weight of  (the number of people in the population) and the highest income is given a weight of 1. The Gini evaluation function is therefore not individualistic: each individual’s contribution to ‘social welfare’ depends on his or her position in the distribution. The Gini inequality measure is one of a class of measures de…ned in terms of the proportion difference between the arithmetic mean and the ‘equally distributed equivalent’ value, where the latter is that income which, if obtained by everyone, would give the same ‘social welfare’ as the actual distribution. Importantly, the social welfare function,  , can be expressed in terms of the

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arithmetic mean and inequality as:  = ¯ (1 − )

(5.11)

where  denotes the Gini measure and ¯ is arithmetic mean. This is often referred to as the ‘abbreviated welfare function’, compared with the welfare function expressed in terms of individual values. It shows the trade-off between equity (measured by 1 − ) and efficiency (re‡ected in ¯) made by a judge relying on the Gini inequality measure.11 For an extensive treatment of these aspects, see Lambert (1993). Figure 5.3 shows the pro…le of (5.11) along with the Gini measure of disposable income per adult equivalent person (with individual as unit). In calculating  , the real value of the arithmetic mean, ¯, has been adjusted by dividing by its 1984 value. The positive growth in incomes during the late 1980s ensured that  rose over the period. However the post 1991 recession caused  to fall brie‡y. The small ‘dip’ around 2010 is associated with the temporary increase in the Gini measure and the anticipation effects of tax rate reductions, discussed above. A clearer idea of the contributions of growth and changing inequality may be obtained by decomposing the changes in  , using: ¯   (1 − )  = +  ¯ 1−

(5.12)

The change in  , along with the two components of the change in mean income and of equity, is shown in Figure 5.4. The positive growth of  from the middle 1980s until 1992 to 1994 is closely associated with growth in ¯, 11 As

discussed in Chapter 3, for the Gini the equally distributed equivalent,  , is a ‘reverse rank weighted’ mean and, since  = 1 −  ¯  , rearrangement gives,  = ¯ (1 − ). Hence ¯ (1 − ) can serve as the abbreviated welfare function, giving the same ranking of distributions as the actual welfare function, P ( + 1 − )  , and the same trade-off between ‘equity and efficiency’. See also =1 Subramanian (2002), who uses a form of the Gini that is not replication-invariant. The Gini measure shares this abbreviated form with the Atkinson measure, for which the equally distributed equivalent income is a power mean.

5.3. INEQUALITY AND SOCIAL WELFARE

Figure 5.3: Inequality and the Social Welfare Function

Figure 5.4: Components of Changes in Social Welfare Evaluation

105

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despite the reduction in equity, 1 − , in the late 1980s. The direction of change in  has been dominated by the changes in ¯, except for the short period around 1992 and 2010, when inequality increased temporarily. Judges whose values are similar to those re‡ected in the use of the Gini inequality measure would regard the reduction in equity in the late 1980s, following the period of reforms, as being more than compensated by the associated positive growth in real incomes over the period.

5.4

Changing Population Structure

The comparisons in the previous section involve inequality changes over time, based on cross-sectional Household Economic Survey data. These inequality changes are in‡uenced by a range of factors associated with the structure of the population, which are expected to change over the relevant period. For example, there are systematic income changes over the life cycle, so that a change in the age structure of the population may be expected to affect crosssectional comparisons even if no changes in age-income pro…les take place. Table 5.2 presents information about the changing age structure over the period. For example, there has been a substantial increase in female labour force participation as well as an increase in participation among older males over the period. The decline in labour force participation among those aged 15 to 29 re‡ects, among other things, a substantial increase in participation in higher education. These types of change are also stressed by Statistics New Zealand (1999). Table 5.3 presents information about changes in a range of other demographic characteristics of households and individuals. Substantial changes have also taken place in the structure of households. The main change is that the period from the middle 1980s to 2002 saw a steady decline in the proportion of couple parents, along with an increase in the proportion of single-person households. Following an increase during the 1980s, the proportion of sole-parent households has remained steady.

5.4. CHANGING POPULATION STRUCTURE

Table 5.2: Population Structure Male Female Year 15-29 30-44 45-59 60+ 15-29 30-44 45-59 60+ Unit (000) (000) (000) (000) (000) (000) (000) (000) 1984 347 308 207 49 269 226 135 27 1985 353 315 211 50 273 233 137 23 1986 346 319 213 51 272 237 141 22 1987 347 325 216 51 275 243 142 22 1988 333 328 213 47 270 247 150 20 1989 308 323 207 41 249 245 149 19 1990 295 327 205 40 243 252 151 18 1991 280 328 207 39 238 258 155 19 1992 264 330 209 38 227 261 160 18 1993 263 333 217 38 223 264 169 18 1994 267 341 226 44 229 270 177 22 1995 281 352 240 47 239 279 192 23 1996 286 364 251 52 248 288 205 24 1997 282 369 261 54 245 298 213 28 1998 275 369 268 51 235 299 219 30 1999 259 365 273 56 229 302 229 33 2000 259 371 282 61 228 313 236 32 2001 258 375 290 68 227 318 248 41 2002 264 379 299 79 229 325 256 48 2003 266 384 313 81 230 331 270 50 2004 276 391 322 86 236 337 283 55 2005 279 400 335 94 244 344 300 63 2006 290 398 348 100 252 349 312 68 2007 295 394 356 112 260 347 317 73 2008 299 388 360 118 263 339 328 82 2009 294 377 363 128 256 342 336 91 2010 281 372 360 135 244 331 341 100 2011 290 371 365 146 247 325 348 109 2012 290 366 372 153 251 323 352 116 2013 291 365 369 154 248 320 353 122

107

108

Year (Unit) 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013

CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

Table 5.3: Household Characteristics Housing Parents No. in Rent Own Single Couple Sole Couple Other House (000) (000) (000) (000) (000) (000) (000) 290 755 197 242 79 394 133 3.15 290 772 200 245 82 399 135 3.12 291 790 204 250 85 404 138 3.07 295 803 210 255 90 402 141 3.04 298 817 216 261 94 399 144 3.00 301 831 223 267 99 396 148 2.97 305 844 229 273 104 392 151 2.96 308 859 236 279 108 388 155 3.00 321 866 240 285 110 387 165 2.98 334 874 244 291 111 386 175 2.96 347 881 248 297 112 384 186 2.95 361 887 252 304 114 382 197 2.94 374 894 257 310 115 379 208 2.94 386 897 266 314 117 376 210 2.95 398 901 277 318 120 372 213 2.94 410 904 287 322 122 368 215 2.92 423 906 297 326 124 363 218 2.90 433 911 308 330 127 359 221 2.89 443 923 312 337 128 365 223 2.89 453 935 316 344 130 372 226 2.90 463 947 320 352 131 379 228 2.90 473 960 324 359 133 386 231 2.89 483 971 328 366 135 392 233 2.88 495 982 333 372 137 398 237 2.86 507 992 339 377 139 405 240 2.85 518 998 342 382 140 409 243 2.85 526 1002 345 385 141 412 245 2.86 536 1007 348 388 143 416 247 2.86 545 1010 351 391 144 419 249 2.85 555 1014 354 395 145 423 252 2.85

5.4. CHANGING POPULATION STRUCTURE

109

In addition, there has been a steady decline in the number of persons per household and a decline in the proportion of owner-occupied households. It is useful to know the extent to which the pattern of changes in measured crosssectional inequality over the period has been in‡uenced by these demographic and labour force changes. The effects of the demographic changes can be examined by extending the use of survey calibration which, as discussed above, is used to produce weights for ‘grossing up’ from the sample survey to population values. By applying a set of calibration variables for one year to a sample survey that was obtained for another year, thereby obtaining a new set of survey weights, it is possible to make inequality comparisons which hold those population characteristics constant. The comparisons may thus be said to be, to some extent, ceterus paribus comparisons. It allows a range of ‘what if’ questions to be asked, such as: ‘what happens to measured changes in inequality if year 2’s population structure (described by aggregate age distribution and employment participation rates, and so on) were the same as year 1’s?’ Figure 5.5 show the variation in the Gini measure of inequality of market and disposable income, and consumption expenditure, with unchanged population structures over the whole period. In all cases the measures are again of household values per adult equivalent person, using the individual as unit of analysis. For each measure, inequality is shown separately ‘as if’ the population in each year were similar to the 1984 and 2013 populations for the 47 calibration variables used (involving the age/gender structure, labour force participation, household size and structure, and type of housing tenure). The differences between the two pro…les (1984 and 2013 structures) are greater for the inequality of household market income per adult equivalent, compared with disposable income. However, in each case the use of 2013 calibration values produces higher inequality in all years, except for several of the later years in the case of disposable income. If the population structure (as re‡ected in the calibration values) had remained as it was in 1984 the Gini

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CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

Figure 5.5: Inequality 1984 to 2013: Fixed Calibration Weights measure of market income per adult equivalent person would not have risen by as much as shown by the standard annual measures. Hence changing population structure has contributed to increasing inequality, particularly over the 1980s and 1990s. The differences for the distribution of expenditure are less clear, as the pro…les intersect several times over the period. Figure 5.6 shows the redistributive effect of direct taxes and transfers (the Reynolds-Smolensky measure) when initial and …nal year calibrated weights are, in turn, held constant throughout the thirty-year period. The pro…les consider the question of what the redistributive effect would look like if the population structure, as re‡ected in the 47 calibration variables, were to have remained constant at its 1984 or 2013 structure. It is clear that there is consistently more redistribution with the …nal-year calibrated weights. The

5.5. INEQUALITY DECOMPOSITIONS

111

Figure 5.6: Redistribution based on Initial and Final Year Calibrated Weights main difference between this and the earlier results is the ‘dip’ in the redistributive effect of taxes in the early 1990s, revealed using each of the sets of calibration weights.

5.5

Inequality Decompositions

The time pro…les discussed in the previous section can give an initial impression of the importance of a number of changes in the population structure (the calibration values) relative to the remaining factors such as changes in taxes and bene…ts and other non-calibrated variables. The question arises of the extent to which the differences can be attributed to particular calibration variables. For example, there was a drop in the proportion of households consisting of couple parents during a period when inequality was found to increase but, without further analysis, the extent to which the changes in household composition contributed to inequality changes is not clear. The present section therefore investigates the use of a more detailed decomposi-

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CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

tion of inequality changes. The approach is described in the following subsection. Results are presented in subsection 5.5.2.

5.5.1

The Decomposition Method

Suppose for simplicity that there is just one calibration variable, , and comparisons over only two periods, 1 and 2, are involved, so that  takes values 1 and 2 for the two periods and the samples are denoted 1 and 2 . The variable, , may of course be a vector, representing a set of calibration variables relating to a particular feature of the economy (such as numbers in a range of age and gender groups). The change in inequality ∆, is expressed as: ∆ =  (2 | 2 ) −  (1 | 1 )

(5.13)

The change in inequality arises from changes in the sample and changes in the calibration variable, , which in‡uence the weights applied to each sample. The separate effects of these two factors can be obtained as follows. Consider the following decomposition: ∆ = [ (2 | 2 ) −  (2 | 1 )] + [ (2 | 1 )] −  (1 | 1 )]

(5.14)

The …rst term in square brackets on the right hand side of (5.14) is the effect of changing the sample given the value of the calibration variable in period 2, and the second term in square brackets is the effect of changing the calibration variable, for the sample in period 1. However, there is another possible decomposition of the change in inequality, since: ∆ = [ (1 | 2 ) −  (1 | 1 )] + [ (2 | 2 ) −  (1 | 2 )]

(5.15)

The …rst term in square brackets on the right hand side of (5.15) is the sample effect given 1 , while the second term is the effect of changing the calibration variable for sample 2. Faced with two values for each of these effects, and since there is no special reason to select one as more important than the other,

5.5. INEQUALITY DECOMPOSITIONS

113

an approach is to obtain the unweighted arithmetic mean. This average is recommended by Shorrocks (2011), who links it to the Shapley Value, familiar from game theory. For further discussion of the decomposition and applications allowing for labour supply responses to tax changes, see Bargain (2012b) and Creedy and Hérault (2015). In this example there are essentially just two things that in‡uence inequality, the calibration variable,  , and the (combined) group of factors contained in the sample,  , for each period. This gives rise to two alternative ways of decomposing the change, as seen in (5.14) and (5.15). Suppose there are three contributions to inequality change, which gives rise to 3! = 6 alternative decompositions. For example, there may be labour force calibration totals represented by the vector,  , for  = 1 2, where the elements of  give the total number of individuals in the labour force in each of a set of age groups. Furthermore, household characteristics in each period are represented by the vector  , whose elements give the total number in each of a set of household types. With these two sets of calibration variables, along with the sample,  , there are 3! = 6 different decompositions of the change in the inequality measure between the two periods. These are, in no special order: ∆ =

 (2  2 | 2 ) −  (1  2 | 2 ) + (1  2 | 2 ) −  (1  1 | 2 ) + (1  1 | 2 ) −  (1  1 | 1 )

∆ =

(5.16)

 (2  2 | 2 ) −  (1  2 | 2 ) + (1  2 | 2 ) −  (1  2 | 1 ) + (1  2 | 1 ) −  (1  1 | 1 )

(5.17)

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CHAPTER 5. DECOMPOSING INEQUALITY CHANGES ∆ =

 (2  2 | 2 ) −  (2  1 | 2 ) + (2  1 | 2 ) −  (2  1 | 1 ) + (2  1 | 1 ) −  (1  1 | 1 )

∆ =

(5.18)

 (2  2 | 2 ) −  (2  1 | 2 ) + (2  1 | 2 ) −  (1  1 | 2 ) + (1  1 | 2 ) −  (1  1 | 1 )

∆ =

(5.19)

 (2  2 | 2 ) −  (2  2 | 1 ) + (2  2 | 1 ) −  (1  2 | 1 ) + (1  2 | 1 ) −  (1  1 | 1 )

∆ =

(5.20)

 (2  2 | 2 ) −  (2  2 | 1 ) + (2  2 | 1 ) −  (2  1 | 1 ) + (2  1 | 1 ) −  (1  1 | 1 )

(5.21)

In the …rst decomposition above, the …rst term,  (2  2 | 2 )− (1  2 | 2 ), measures the change in  that is attributable to the change in the labour market structure, for given household structure and sample data for year 2. The second term,  (1  2 | 2 ) −  (1  1 | 2 ), measures the change in  that arises from changing only the household structure, for the given labour structure in period 1 and the survey dataset from period 2. The …nal term,  (1  1 | 2 ) −  (1  1 | 1 ), measures the effect on ∆ of changing only the survey dataset, holding both the labour market and household structures constant at their values in year 1. The subsequent decompositions give alternative changes attributable to each of the three components, but for different combinations of given values of the other components. Since no component of the change in  can be viewed as being any more important than other components, an approach to providing a summary of the decompositions is simply to obtain the arithmetic means.

5.5. INEQUALITY DECOMPOSITIONS

115

Hence, consider the arithmetic mean contribution of changing household structure to inequality over the period, which may be denoted, ∆ | . This is obtained as an arithmetic mean of the relevant changes from the above six decompositions. Thus, taking the appropriate term from each decomposition above: ∆ | =

1 [ (1  2 | 2 ) −  (1  1 | 2 ) + 6  (1  2 | 1 ) −  (1  1 | 1 ) +  (2  2 | 2 ) −  (2  1 | 2 ) +  (2  2 | 2 ) −  (2  1 | 2 ) +  (1  2 | 1 ) −  (1  1 | 1 ) +  (2  2 | 1 ) −  (2  1 | 1 )]

(5.22)

Importantly, the terms  (1  2 | 1 ) −  (1  1 | 1 ) and  (2  2 | 2 ) −  (2  1 | 2 ) appear twice in (5.22). But all six terms must be averaged in order to ensure that the three component averages, ∆ | , ∆ | and ∆ | , add to the actual change in inequality, ∆, such that: ∆ = ∆ | + ∆ | + ∆ |

(5.23)

The proportional contributions are thus obtained as ∆ | ∆, and so on. The number of possible decompositions increases rapidly as the number of sets of calibration variables increases. For example, three sets of calibration variables, along with different samples, gives rise to 4! = 24 decompositions. Hence, although there are only four components to the decomposition, each is obtained as the arithmetic mean of 24 separate values where, again, some of the values are necessarily duplicated in obtaining the different decompositions. In the present application, six different sets of components are relevant. These are: age/gender; housing tenure; household type; occupancy rate;

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CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

labour force participation by age and gender, and …nally the sample. Hence 6! = 720 different compositions are needed and the components reported below are respective arithmetic means. It is therefore desirable to carry out these decompositions in the most efficient manner, since all 720 alternatives need to be evaluated. There are many known algorithms for iterating over permutations, but signi…cant speed improvements can be obtained by using results from previous iterations which start with a subsequence of the current iteration. For instance, knowing the combination (1,2,3,4,5,6) means that it is necessary only to obtain the last two calculations for the combination (1,2,3,4,6,5). Taking this optimisation into account, iteration is over the permutations in lexicographic order: for example, see Knuth (2005).

5.5.2

Contributions to Inequality Changes

The decomposition method described above was applied to changes in Gini measures of market and disposable income per adult equivalent for …ve separate periods, chosen to re‡ect different economic conditions and policies. The period 1984 to 2013 was thus divided into changes for 1984 to 1990; from 1990 to 1996; from 1996 to 2001; from 2001 to 2007; and 2007 to 2013. Table 5.4 shows the components of the absolute change in the Gini measure over the period 1984 to 1990. Clearly, for market incomes labour force participation and household type make small contributions to the increase in the Gini over the period, but the main contribution is in the sample which contains all other factors that cannot be measured separately. Regarding the inequality of disposable incomes over this period, the separate contributions of household type and labour force participation are negligible.12 12 The

components for the whole period are not equal to the sums of the separate components for subperiods. This can easily be seen by taking the simple case of subsection 5.5.1 and writing down the relevant decompositions for two-subperiods, 1 to 2 and 2 to 3 and the complete period 1 to 3 . However, the total change in the Gini measure over the whole period 1984 to 2013 must equal the sum of the separate year-to-year changes.

5.5. INEQUALITY DECOMPOSITIONS

117

Table 5.4: Decomposition Analysis Period Age/Sex Tenure Unit Disposable income 1984-1990 0.14 1990-1996 0.34 1996-2001 0.12 2001-2007 0.01 2007-2013 -0.02 1984-2013 -0.33 Market income 1984-1990 0.12 1990-1996 0.27 1996-2001 0.45 2001-2007 0.20 2007-2013 0.55 1984-2013 0.40

Hh Labour Occupancy Sample Total Type Force (Other) 100 × (Change in Gini)

-0.03 -0.04 -0.04 -0.01 -0.01 -0.03

0.19 0.04 0.28 0.02 0.00 0.43

0.17 -0.03 0.06 -0.40 0.29 0.42

0.01 0.05 -0.06 -0.06 0.10 0.37

3.87 2.47 1.31 -3.60 0.78 5.09

4.36 2.83 1.67 -4.03 1.13 5.95

-0.06 0.16 0.16 0.05 0.10 0.37

0.60 0.11 0.60 -0.02 0.00 0.98

1.48 -0.08 -0.44 -1.65 0.38 -0.18

0.15 0.02 -0.24 -0.05 0.14 0.78

4.85 2.85 0.46 -3.18 -0.24 5.38

7.13 3.33 0.99 -4.65 0.92 7.73

The components for the period 1990 to 1996, indicate that for both market and disposable incomes only the age/gender composition of the population is noteworthy. When considering changes over the period 1996 to 2001, the separate components take on more importance for market incomes, with labour force participation changes and changes in the number of people per household actually contributing to reduce inequality. However, these contributions have much less importance for changes in inequality of disposable incomes. When inequality fell slightly over the period 2001 to 2007, only changes in labour force participation is identi…ed as contributing to the fall for market incomes, and this component becomes negligible for disposable incomes. However, the change from 2007 to 2013 in the Gini measure of market income per adult equivalent can be largely attributed to changes in the age structure of the population and changes in labour force participation. Yet only the latter remains noteworthy when considering changes in inequality

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CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

of disposable incomes. In each case the changing sample characteristics contributed most to the change in inequality. This contains all those changes not captured by the other calibration components. As mentioned earlier, the separate contributions of the tax and transfer system cannot be isolated from other changes such as occupational change, age-income pro…les and differential wage growth: for example the 2000 tax and transfer system could not be applied to the 1984 HES sample.

5.6

Conclusions

This chapter has reported the results of an extensive analysis of annual income and expenditure inequality in New Zealand over a thirty-year period from the early 1980s. The extent of redistribution through the tax and bene…t system was also explored by comparing the inequality of market incomes with that of disposable incomes. Household Economic Survey data were used for each year from 1983/84 to 1997/98 inclusive, 2000/01 and 2003/04 , and for each year from 2006/07. The analysis has been conducted with the aim of obtaining a more detailed description of changing inequality over time and its components. It also demonstrates the difficulty of attributing precise causes to the distributional changes. A distinguishing feature of the analysis is that survey calibration methods were used, by imposing independently obtained population totals for 47 population characteristics, covering demographic and labour force information for each year. Results were compared with the use of the HES weights provided by Statistics New Zealand, which are based on calibration using a much more limited set of population totals. The results indicate an increase in the inequality of market and disposable income per adult equivalent person (using the individual as the unit of analysis) from the late 1980s to the early 1990s. Subsequently, inequality

5.6. CONCLUSIONS

119

has – with some variability – remained either constant or has fallen slightly. Comparisons with tax policy changes over the period suggest that some of the variability (particularly around the 2001 and 2010 policy changes) may be attributed to income shifting between time periods in anticipation of changes in the income tax structure. The use of survey calibration methods also makes it possible to examine changes in inequality using the ‘as if’ assumption that the structure of the population (as described by the 47 calibration totals) remains constant over the period. The variation in inequality can then be attributed to changes in the nature of the sample rather than those features of population and labour force structure that are held constant. With the weights adjusted to ensure that the calibration totals remain constant over time, the pro…les of income inequality display different absolute values and somewhat different patterns, being higher in the earlier years when the calibration values for later years were used. Furthermore, with a constant demographic and labour force structure, the inequality of expenditure, though subject to year-to-year variations, displayed a ‘‡atter’ pro…le over the period. The inequality differences obtained using different calibration totals suggested that it would be useful to examine the contribution to changing inequality of particular components of the demographic and labour force change. A decomposition method was used involving …ve sets of variables (age/gender structure, labour force participation, household type, housing tenure type, and occupancy rate) along with the sample itself. These six components involved the use of 720 separate decompositions of each inequality change considered: arithmetic mean contributions to the overall inequality change were thus reported. The separate ‘non sample’ components were found to make a larger contribution to changes in the Gini measure of market income, though for disposable incomes those components were found to be relatively small, except in the cases where the Gini changed by very little or decreased.

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CHAPTER 5. DECOMPOSING INEQUALITY CHANGES

Although it has been possible to examine the substantial differences between the distributions of market and disposable incomes in each year, re‡ecting the redistributive role of taxes and transfers, it has not been possible to isolate the role of changes in the tax structure from a range of other sample characteristics which may be expected to in‡uence the distribution of income. The present chapter has aimed to provide a more detailed description of the New Zealand income distribution and changes over the last thirty years, but interesting questions about the precise causes of those changes remain a challenge for future research.

Chapter 6 Inequality Over a Long Period This chapter examines long-period changes in annual income inequality of individuals in New Zealand. The aim is to provide descriptive measures over the period 1935 to 2013 using a range of data sources. No detailed attempt is made to disentangle the relative contributions of a number of possible causes of changes. Rather it is hoped that information about orders of magnitude over a long period can contribute to the wider debate about inequality. One of the paradoxes of recent years is that the huge increase in media interest in income inequality in New Zealand follows a period from the early to mid-1990s during which, unlike in several other countries, there has been no trend increase in standard annual income inequality measures. Indeed, the term ‘paradox’ was used by Wilkinson and Jeram (2016) in the title of their wide-ranging discussion of inequality and associated policy issues in New Zealand. Evidence of increases in some measures of inequality in other countries such as the United States, United Kingdom and Australia have been documented by, for example, Chetty et al. (2014), Burkhauser et al. (2012, 2016) and Atkinson and Leigh (2007). Many commentators simply refer to ‘increasing inequality’ without making any reference to empirical studies. Information is nevertheless available using data on household income from the early 1980s to 2013, including work within the Ministry of Social Development reported by Perry (2014), and research within the Treasury, by 121

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Creedy and Eedrah (2016) and Ball and Creedy (2016). The latter, largely reproduced in the previous chapter, reported Gini inequality measures of annual income per adult equivalent person and showed that inequality rose during the late 1980s until the early 1990s, after which it remained steady. There are also various publications by Statistics New Zealand, such as (1999). On the period 1987 to 1998, see Crawford and Johnston (2004). Summary results for disposable incomes in selected years (1981–1982, 1985– 1986 and 1989–1990) are reported in Saunders (1994). Decompositions by income source for 1983–1984, 1991–1992 and 1995–1995, using gross (pretax) incomes are reported by Podder and Chatterjee (2002), using household income per person as the income measure, with individual as unit of analysis: however, the selection of years gives a misleading picture of the inequality changes over the 1980s. On the period 1981 to 1996, see also O’Dea (2000), Easton (2013, 2014). Easton (2015a, 2015b) studies the share of top incomes from the 1930s, but mainly from the 1970s. Studies considering macroeconomic in‡uences on the income distribution in New Zealand include Bakker and Creedy (1999) and Martin (2002). On gender differences over the decade, 1998 to 2008, see Papps (2010). On wage inequalities from 1984 to 1998, see Dixon (1996). For many years the stability of the personal income distribution, along with that of labour’s share in total income, was regarded as a ‘stylised fact’. Indeed Pareto (1909) went so far as to regard the distribution as incapable of change. It is possible that many references to increasing inequality have in mind increasing shares of top individual incomes, which was the focus of studies for Australia and New Zealand by Atkinson and Leigh (2007, 2008); however, these studies also showed stability from the middle 1990s. More recently, Wilkinson and Jeram (2016) examine top income shares, suggesting that they have become more stable in New Zealand unlike, for example, the United States where they have continued to grow. The existence of different views about how inequality is perceived raises the important question of how it can

6.0. INTRODUCTION

123

be measured. This chapter concentrates on the Gini measure, as discussed in Section 6.1, clarifying the nature of the value judgements implicit in using this overall measure, and explaining its calculation when faced with grouped income distribution data. Before computing summary measures, any study of inequality must …rst make decisions regarding three ‘Ws’ of inequality: ‘what’, ‘when’ and ‘whose’. The …rst (what) concerns that usually referred to as the ‘welfare metric’ and involves a decision of whether, for example, to consider earned gross income, or total net of tax and transfer income, or consumption (and the form of any adult equivalence scale, if any, to use). The second (when) involves the accounting period over which the metric is calculated. In the case of, say, disposable income, interest may be on a short period, such as a week, or at the other extreme, the lifetime of members of a cohort. The choice of a longer period introduces complexities arising from relative income mobility within the cohort. In New Zealand, much emphasis was placed on the role of mobility, and a longer accounting period, by Barker (1996). The relationships between income mobility characteristics, annual inequality and inequality using a longer accounting period are far from straightforward: see Creedy (1997). The third (whose) concerns the population group and the income unit: the latter may be the individual, household or ‘equivalent adult’. For example, using annual household income per adult equivalent person as accounting period and welfare metric, the choice of unit of analysis is not straightforward: see Chapter 2. Although these choices all involve value judgements, in practice the choice is often strongly in‡uenced by data availability. However, it is important to recognise that the alternatives can produce results suggesting inequality changes in opposite directions or quite different orders of magnitude. Data limitations mean that the concentration here is on incomes of individuals, rather than households. The Household Economic Survey goes back only to 1973, and the income concept used in surveys before 1983 makes it difficult to produce a consistent series

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CHAPTER 6. INEQUALITY OVER A LONG PERIOD

over the life of the survey. These issues are compounded when attempting to take a longer-term perspective, as here. For example, if it is desired to measure the annual taxable income of individuals, changes in the tax structure over time, in‡uencing the components that are included in ‘taxable income’ (such as certain bene…t payments, ‘fringe bene…ts’ or other components of ‘comprehensive income’) can have a substantial effect on measured income and even on the timing of income receipts, and thus on comparisons. The measurement of changes in inequality over a long period must therefore necessarily be accompanied by the important caveat that many other changes are taking place that may affect results but which may not be regarded as fundamentally re‡ecting inequality changes. An obvious quali…cation is that cross-sectional annual inequality is in‡uenced by changes in the age composition of the population group considered, in view of the fact that there are systematic variations in income over the life cycle. After discussing the Gini measure in Section 6.1, Section 6.2 describes the sources of income data and the de…nition of each series. Section 6.3 reports Gini inequality measures for individual before-tax incomes from 1935–2014. Comparisons are made with results reported for individuals by Easton (2013, 2014), who estimates Gini coefficients for the New Zealand adult population using census data at 5–10 yearly intervals from 1926 to 2016. Comparisons are also made with similar estimates for Australia over 1942– 2001 by Leigh (2005). The characteristics of Australian inequality are also examined by Gaston and Rajaguru (2009) and Saunders (1994). Consistent data on before-tax and after-tax incomes from 1981 allow inequality indices to be calculated for 1981–2014 for all individuals and separately for male and female groups. In some ways a better post-tax income measure would be after-tax-and-transfers income, since most social welfare transfers (including, for example, New Zealand’s family tax credits), are analogous to negative income taxes. However, consistent annual data on social transfers received

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125

by individuals are not available for the extended periods covered here and hence are not considered below. Chapter 5 provides comparisons of Gini coefficients for market and post-tax-and-transfer incomes for 1984–2007. A particularly important income measurement issue arises with the introduction in the tax year 1957–58 of the Pay as You Earn (PAYE) system of tax withholding. This signi…cantly affected the coverage of income data. This is discussed in Section 4, which describes the resulting adjustment made to the Gini index for the period 1935–1957 associated with missing data on low income earners (tax non-…lers) before the introduction of PAYE. In summary, …rst, the long-run Gini income inequality estimates reveal a number of features of inequality in New Zealand not previously recognised. For example, after adjustments to render Gini coefficients more comparable across pre- and post-PAYE periods, there is clear evidence of a steady downward trend in inequality after World War II until the 1970s, prior to the familiar upward trend from the late 1980s. Second, identi…able ‘shocks’ to the New Zealand economy can have dramatic effects on year-to-year changes in the Gini inequality measure. Third, comparisons with Australia suggest quite remarkable similarities in both long-run trends, and some short-term changes, in their respective Gini indices. Finally, examining gender differences in income inequality from the 1980s onwards reveals that the substantial increase in inequality observed for all individuals combined in the 1990s was much more prevalent for males than for females, and female income inequality declined especially during the 2000s.

6.1

The Measure of Inequality Used

The highly in‡uential paper by Atkinson (1970) stressed that value judgements are inevitably involved in constructing inequality measures. There is thus a need for an inequality measure to be linked explicitly to value judgements, which may be summarised in a ‘social welfare function’–essentially a

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function showing how an income distribution is evaluated. He showed that, where the income distribution is evaluated using an additive and individualistic social welfare function (where the weight attached to each individual’s income is a concave function of income, thereby satisfying the ‘principle of transfers’ – a basic value judgement that takes the view that a transfer from a richer to a poorer person, which leaves their relative rankings in the distribution unchanged, represents an improvement), it is possible to de…ne an inequality measure based on the proportional difference between arithmetic mean income and an ‘equally distributed equivalent’ income. The latter is de…ned as the value that, if obtained by everyone, produces the same ‘social welfare’ as the actual distribution. The degree of concavity of the weighting function measures the extent of ‘relative inequality aversion’ of the judge whose value judgements are represented by the social welfare function. The equally distributed equivalent income is a ‘power mean’ in this case. The use of an explicit evaluation function not only links inequality to value judgements, but also implies a clear trade-off between total income and its inequality (often referred to as a trade-off between ‘equity and efficiency’). Subsequent research showed that the Gini inequality measure–previously rationalised in terms of areas in the famous Lorenz curve diagram–can also be derived in the same way but with a different form of social evaluation function, in which the rank order of each individual plays an important role. The income of the richest person is given the lowest weight while the poorest person is given the highest weight. In the Gini case, as shown below, the equally distributed income is a reverse-rank-order weighted mean of incomes, rather than the ‘power mean’ of Atkinson’s measure. The welfare function, for incomes  , for  = 1   and income ranked in ascending (strictly, non-decreasing) order, is written as: =

 X

( + 1 − ) 

(6.1)

=1

It can be shown that the equally distributed equivalent income is given by

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127

the reverse-rank weighted mean: X 2 ( + 1 − )   =  ( + 1) =1 

(6.2)

Therefore giving the Gini measure as: =1−

 ¯

(6.3)

where ¯ is arithmetic mean income. Those whose value judgements do not agree with the expression in (6.1) are not likely to be comfortable with the use of the Gini as a summary measure. Its use in popular debates is often ‘justi…ed’ merely on the vague grounds that it is ‘well known’, but it is important that the associated value judgements are clear. Both Gini and Atkinson measures, by giving relatively little weight to the highest incomes in the overall evaluation, are not very sensitive to changes in top incomes. Stability shown by overall inequality measures may, therefore, be quite consistent with higher top income shares. Furthermore, standard inequality measures, such as the Atkinson and Gini measures, can take unchanged values for what may otherwise be judged to be widely different income distributions: for further details, see Chapter 4. However, evaluations based on top incomes, by ignoring the form of much of the distribution, have considerably less clear-cut rationales in terms of the implicit value judgements involved. The present chapter is restricted to using the standard Gini inequality measure. Chapter 5 examined a range of Atkinson measures (for different degrees of inequality aversion) as well as Gini measures, and found that they produced similar variations over time for household income per adult equivalent. However, the computation is less straightforward than suggested above because grouped data, rather than individual observations, must be used. While there are many alternative ways of expressing the Gini measure for individual data, the following covariance form is particularly useful. This

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is given by: =

2 (  ()) ¯

(6.4)

where  () is the cumulative distribution function, and (  ()) is the covariance between  and  (). In samples with individual observations,  ( ) is calculated simply as .

6.1.1

Computing the Gini with Grouped Data

This subsection describes two methods to calculate Gini with grouped data. Results from the two methods are the same. Where grouped data are available, the term is now the arithmetic mean income in group , where there are  groups rather than individuals. In many cases where grouped data are available, it is necessary to use class midpoints, which is complicated by the fact that the highest income group is often open-ended. However, in the present case the data available give total income and the number of individuals within each income group, allowing arithmetic mean income within each income group to be calculated. Furthermore, the potential downward bias arising from using grouped data is very small where the number of groups is sufficiently large, above about 15: see, for example, simulations reported by Lerman and Yitzhaki (1989) and van Ourti and Clarke (2009). This requirement is also satis…ed in the present case, as explained below The number of observations in group  is  ( ) and thus the ‘weight’ attached to each mean is:  ( )  ( ) =  = P  =1  ( )

(6.5)

Two methods are available, as follows. For the …rst method, if each observaP tion has a weight,  , with =1  = 1, the cumulative proportion,  ( ), is obtained, where 0 = 0, as:  X + ˆ ( ) =  2 =0 −1

(6.6)

6.2. DATA SOURCES

129

The Gini coefficient is thus:  ³ ´ 2X ¯ ˆ  ( − ¯)  ( ) −  = ¯ =1

(6.7)

Here ¯ and ¯ are the weighted means of  and ˆ ( ) respectively, so that P P ¯ = =1   and ¯ = =1  ˆ ( ). The second method uses the (integer) frequencies,  ( ), de…ned above. De…ne  as follows. For  = 1, and  = 1     1 : 1 =  + 1 − 

(6.8)

and for  = 2     , and  = 1   ( ):  =  + 1 −

−1 X

 ( ) − 

(6.9)

=1

Then:

⎛ ⎞  ( )  X X 1 2 − 2 =1+  ⎝  ⎠   ¯ =1 =1

(6.10)

It is also possible to obtain a convenient expression for the standard error of the Gini measure, when using grouped data. These were computed in all cases, but when reporting results below, the standard errors are not included because they are so small, because the results are dominated by the very large number of observations involved.

6.2

Data Sources

Statistics New Zealand Official Yearbooks (NZOYB) are available in digitised form for 1893 to 2013 from the Statistics New Zealand website. Inevitably the type of information provided, and de…nitions of reported items, vary over that period. Of particular interest are Inland Revenue based data on incomes for individual taxpayers available annually (with a few exceptions). Prior to the 1930s, NZOYB income data are recorded for individuals and companies combined and hence are unsuitable for income inequality measurement.

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Thereafter, reports for various years provide data for ‘total income’ (and other income categories) and the number of ‘returns’ (taxpayers) across income bands. For the 1930s to 1950s income bands are typically reported in $100 to $1,000 bands up to around $20,000+ or $40,000+ giving around 15– 25 income groups. These increase to up to 36 income groups in the 1960s and 1970s. Inland Revenue data for 1981 onwards are mostly in $1,000 income bands yielding from 51 (1981) to 171 (1999 onwards) income groups for Gini index calculations. Subject to some changes in de…nition, described below, this source provides suitable data from 1935 to 1983. Income data by income band using more consistent de…nitions are available from the Inland Revenue Department (IRD) for 1981–2014. These two series, supplemented by income data sourced from the separate Statistics New Zealand (SNZ) Report on Incomes and Income Tax for some missing years in the 1970s, form the basis for the Gini indices reported below. An important difference, discussed further below, between these data and data analysed by Easton (2013, 2014), is that Easton’s data refer to all adults rather than taxpayers or tax …lers, but Easton’s inequality estimates are only available for census years.

6.2.1

Types of Income

Statistics New Zealand have collected income data over many decades, for a variety of de…nitions. The main income types reported in the NZOYB are: ‘returnable income’, ‘total income’, ‘assessable income’, and ‘analysis total income’. These are de…ned as follows: Returnable Income: Returnable income ‘comprises assessable income plus proprietary income and the classes of non-assessable income. In addition to the proprietary income which is included in returnable income, certain classes of non-assessable income are taken into account in determining the amount of tax payable on the balance of the assessable income. The

6.2. DATA SOURCES

131

classes concerned mainly comprise dividends from companies trading in New Zealand, interest on New Zealand Government securities issued free of tax, and interest on company debentures issued free of tax or with a ‡oating rate of interest. Company dividends, or proprietary income in lieu of company dividends, are actually by far the largest source of non-assessable income.’ See Statistics New Zealand Official Year Book (NZOYB, 1961). Assessable Income: ‘The broad principle adopted in calculating the assessable income is that any expenditure or loss exclusively incurred in the production of assessable income for any year may be deducted from the total income from any assessable source for that year. Depreciation is allowed, varying rates for different classes of assets being …xed. The assessable income is approximately equivalent to the net pro…t as determined by the normal commercial accounting systems. It is, on the whole, rather higher than the commercial net pro…t, since certain types of expenditure which are regarded as a revenue charge in commercial accounts are not permissible deductions from income for income-tax purposes. Where the operations of a source of income which would be assessable for income tax have resulted in a loss for the year, the loss may be set off against assessable pro…ts from other sources (if any) or, in default thereof, may be set off against assessable pro…ts in the three following years. The Land and Income Tax Amendment Act 1953 extends this period to six years, but does not apply to losses incurred before the income year 1949–50. Capital pro…ts are not assessable and capital losses are not deductible.’ (NZOYB, 1954). Non-assessable Income: ‘Certain types of non-assessable income, including war pensions and social security monetary bene…ts, are excluded from the returns, and are therefore completely omitted from these statistics. The social security universal superannuation bene…t became part of the assessable income from 1 October 1951. The coverage of the returns is also incomplete in one other respect.’ (NZOYB, 1954). Total Income: Total income corresponds in concept with the return-

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able income that was used in NZOYB up to 1960 (Statistics New Zealand Official Year Book, 1961). ‘Total income is used in the sense that it is the total of the various component items of income. It does not include income which is exempt from taxation, such as social security bene…ts (other than universal superannuation), the …rst $24 of income from interest in 1959–60 (raised to $60 from 1960–61), war pensions, workers compensation payments, and certain other types of income.’ (Statistics New Zealand Official Year Book,1961). Because total income is returnable income and the term total income is used in later NZOYB, for convenience, the term total income is used in later sections instead of returnable income. Analysis Total Income: Analysis total income is the estimation of total income. ‘The data required for statistical compilation of incomes do not become available for a lengthy period after the end of the income year. This, coupled with the necessary time taken by the statistical processing, means an unduly long time lag before results of the compilation are available. To meet this situation provisional estimates are made on a sample basis.’ (NZOYB. 1976). Analysis total income is used for those years in the 1970s when the actual total income is not available. The primary purpose of the data collection was tax assessment and for this reason de…nitions often relate to tax properties, such as income required to be included in a tax return (‘returnable’) even if not liable to tax, or income that is liable (‘assessable’) or non-liable (non-assessable) for income tax. In addition, the complete collection of income from tax records can take some years. This means that NZOYBs for year t typically report incomes for a number of previous years up to t–2 or earlier. For several years in the 1970s and early-1980s, provisional estimates obtained before complete income records were available were reported in the relevant NZOYB, but subsequent ‘…nal’ records appear not to have been published. For those years, equivalent data were obtained from the SNZ Report on Incomes and Income Tax. As

6.2. DATA SOURCES

133

far as can be ascertained these reports used identical income de…nitions to those in the NZOYB. The relevant terms are de…ned as follows: Returnable income (to 1960): Assessable income+proprietory income+nonassessable income. Here, ‘proprietary income’ refers to income from ownership of assets (such as interest and dividends arising from company share ownership) that are allocable to individuals. Total income (from 1961): Returnable income. The term ‘total’ income corresponds with the ‘returnable’ income that was used in previous Yearbooks: ‘Total income ... does not include income that is exempt from income tax such as social security bene…ts (except for universal superannuation), the …rst £12 of income from interest, war pensions, workers’ compensation payments, and certain other types of income’ (NZOYB, 1961). Analysis total income: Provisional estimate of total income. This is based on income estimated before more complete ‘…nal’ data are available. For some years in the 1970s no …nal income was reported. After 1980, the income data used for Gini calculation are based on an IRD de…nition of ‘taxable income’. That is, income and tax information derived from IR3 tax returns, personal tax summaries and employer PAYE information. IR3 is the individual tax return used by IRD. Completion of this return is required if the individual has income other than salary, wage, interest or dividends. Taxable income for individuals is income on which their personal income tax is assessed for the March year. It is shown below that the Gini estimates based on data from both NZOYB and IRD sources for the overlapping years 1981–83 show that ‘taxable’ and ‘total’ income data yield almost identical Gini values.

6.2.2

Data Sources Used in Calculations

Two further major data consistency issues must be dealt with prior to Gini calculations. First, as with most individual income datasets, a decision is

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required regarding the treatment of negative and zero incomes. Second, there is a problem of how to compare Ginis obtained from data covering almost all taxpayers after PAYE was introduced in 1957–1958, with measures based on pre-1957 income data that were collected only for tax …lers. On the treatment of negative and zero incomes, the Gini measure is not de…ned for negative values. In response, some investigators reset negative values to zero and include those in calculations. However, for the Gini calculations below, only individuals with positive income are considered, thereby ignoring those with zero or negative incomes (losses), for two reasons. First, especially for early decades of NZOYB income data, it is not clear whether some negative incomes have been treated as zeros or ignored, while for some years the lowest income category appears explicitly not to include incomes below $1 or £1. Therefore, for consistency, the calculations below ignore zeros in all years where identi…able. Some yearbooks record the lowest income categories as ‘losses’ ‘nil’ and ‘$1–199’, while others record only ‘losses’ and ‘$1–199’, or simply ‘under $300’. For estimation purposes, data from NZOYB grouping incomes as $0–$199 are treated as equivalent to $1– $199; see, for example, NZOYB (1969) and NZOYB (1970). Examining the recorded values for incomes and numbers of individuals for those latter cases suggests that differences are typographical rather than substantive. Second, Gini estimates for incomes of an ‘adult population’ (such as Easton’s (2013) census-based values), might attempt to include all adult individuals for comparative purposes regardless of income. However, in the present case concern is with, and data are available for, income recipients. In combination with negative and zero values arising for a variety of reasons (including spurious/statistical) and which are difficult to interpret, it is preferable for comparative purposes to omit zeros. An important potential inconsistency that would arise from inclusion of zero incomes within the pre-PAYE (pre1958) period data, is that this would include zero incomes recorded for those …ling tax returns despite having no taxable income (for whatever reasons),

6.3. EMPIRICAL RESULTS Table 6.1: Data Sources Used Source Type Coverage NZOBY Total inc All

135

Period 1935-1972 (excl 1942-45) 1970-1977 NZOBY

Tax Before tax

1976 1978-1983 1981-2014

Before tax

Total inc All (provisional) SNZ Income Total inc All Report IRD Taxable All: male income and female 1981-2013

Before tax

Before and after tax

while omitting an unknown number of other zero incomes for those not …ling tax returns. Table 6.1 shows the data sources used in calculations. The total income data for the periods 1935 to 1941 and 1946 to 1972 are from NZOYB, and record before-tax incomes only. The data for the period 1942 to 1945 are unavailable due to difficulties collecting income data under war conditions (NZOYB 1947–1949). As mentioned above, ‘…nal’ total incomes are not available for most years during 1970–1977. For this reason, SNZ provisional estimates (‘analysis total income’) are used, and the relevant Gini series is separately identi…ed. The taxable income data for the period 1981 to 2014, available from IRD, enable Gini indices to be estimated for both before-tax and after-tax income and for male and female groups separately. Discussion of how Gini measures based on tax-…ler-only data prior to 1957, and PAYE data from 1958, may be compared follows discussion of the results for each separate series.

6.3

Empirical Results

This section presents the empirical results, beginning in Subsection 6.3.1, which reports inequality measures of total income before tax over the pe-

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riod 1935 to 2014 for all individuals. Subsection 6.3.2 discusses the problem raised by the introduction of PAYE in 1958 and Subsection 6.3.3 proposes an adjustment method to deal with this problem. Comparisons with earlier census-based measures are made in Subsection 6.3.4. Subsection 6.3.5 reports Gini inequality measures by gender for both before-tax and after-tax incomes over the period 1981 to 2013.

6.3.1

Gini Indices: All Individuals ‘Total Income’ Before Tax 1935 to 2014

The Gini indices for 1935 to 2014 are shown in Figure 6.1. Again, this …gure does not show con…dence intervals around these Gini estimates because in all cases the standard errors are very small, at less than 1 per cent of the Gini measure for each year. Inequality is seen to be relatively stable from the early 1960s to the late 1980s, after which it rose to 0.47 in 1994. The pattern from the early 1980s is similar to that found for household incomes per adult equivalent person, reported in Perry (2014) and Chapter 5 above. It is hard to escape the view that the increase was associated with the reforms that took place during the 1980s, discussed by, for example, Evans et al. (1996). Relevant changes included the gradual ‘‡attening’ of the marginal income tax rate structure, with the top marginal rate falling from 66, to 48 and then to 33 per cent, along with bene…t reductions and the end to centralised wage setting. The reforms were in‡uenced by severe macroeconomic pressures, as well as a need for structural/microeconomic reforms, and were followed by improved growth. The ‘trade-off’ between equity and efficiency implicit in the social welfare function generating the Gini measure suggests that the growth in average real incomes outweighed the increase in inequality: this is demonstrated for household incomes in the previous chapter. In addition, fringe bene…ts tax was introduced in 1985 such that, along with the major reductions in the top rate of income tax, the strong incentive to divert income into non-taxable forms prior to the reforms was substantially

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removed or reduced by those reforms. It might be expected, therefore, that from the mid-1980s, an increasing amount of income of top earners would be recorded in official data that previously may well have been ‘hidden’. This would very likely contribute to the observed increase in income inequality but how far this is genuine, as opposed to being due to data coverage limitation, is hard to judge without more information. After 1994 the Gini is relatively constant again, except for a spike in the tax year 1999–2000. This spike is associated with the major income tax changes that raised the top marginal rate from 33 per cent to 39 per cent in 2000. This led to a certain amount of income shifting after the announcement of the change; see Claus, Creedy and Teng (2012) and Carey et al. (2015). Both periods of relative stability in the Gini measure nevertheless witnessed other substantial changes, for example, in the structure of industry and in labour force participation (particularly of women) and participation in tertiary education. Interpreting the Gini indices prior to 1981 requires some caution. First, for the period 1981 to 1983, the IRD and NZOYB total income series reveal almost identical Ginis: they are indistinguishable in Figure 6.1. However, comparing ‘total income’ and ‘provisional total income’ series suggests that the latter may not be a good proxy for the former. For the four years of overlap, 1970 to 1972 and 1976, the provisional series underestimates the …nal series by around 3 to 4 per cent (or 1 to 1.5 percentage points) over the period 1970 to 1972, but appears to overestimate the …nal income-based Gini by around 3 per cent in 1976. Considering only the NZOYB total income series, the …rst year of full PAYE implementation in 1959 is a crucial year, indicating that the Gini peaks at 0.425. From 1959, the data suggest a steady decline in the Gini index to 0.387 in 1967, followed by a slight rise to 0.409 in 1972. Thereafter, the observable pattern is generally one of decline again until 1978, although there are various gaps in the series. In addition, the provisional-based series

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Figure 6.1: Gini Measures for All Individuals: Total Income Before Tax 1935 to 2014 suggests a more volatile pattern with a strong spike in the Gini in 1974. This may re‡ect genuine inequality changes, since it coincides with the major global oil shock that substantively affected the New Zealand economy in the early-to-mid 1970s. The sudden large rise in the NZOYB total income Gini from 1957 to 1959 can be attributed to the move to PAYE-based data, which included almost all income taxpayers. This contrasts with the tax-…ler-based data that excluded almost all of the lowest incomes in view of the tax-free threshold that existed. The pre-1959 Gini coefficients inevitably appear much lower because they capture inequality among tax …lers not taxpayers. Figure 6.1 shows a particularly large spike in the Gini coefficient in 1951, to around 0.36, from values around 0.3 in the years before and after. This appears to be associated with particularly rapid increases in national income associated with a huge temporary rise in wool prices in 1950–1951, a point noted also for Australian inequality indices by Leigh (2005). According to SNZ yearbook

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139

data, wool prices between 1948–1949 and 1951–1952 were (in pence per pound of wool): 25.81, 37.98, 87.84, 40.19 respectively. Similarly gross national income over 1947–1948 to 1951–1952 was (£m): 410, 419, 481, 607, 617; that is, substantially higher growth from 1949–1950 to 1950–1951 and from 1950– 1951 to 1951–1952. This spike in wool prices and farmers’ incomes would undoubtedly have had a large and disproportionate impact on income levels within the income distribution in 1951, but especially generating temporarily high incomes for wool farmers and related activities. The following subsection explores a method of adjusting the pre-1959 Gini indices to make them more consistent with the PAYE-based versions.

6.3.2

Income Data Before and After 1958

Prior to PAYE, taxpayers with incomes below £300 per year were generally exempt from income tax and not required to …le a tax return. Figure 6.2, which shows the ‘number of taxpayers’ recorded in the NZOYB data for 1955 to 1961, demonstrates that numbers increased substantially following the introduction of PAYE. NZOYB refer to these data as ‘numbers of returns’ both before and after 1958. These are treated here as numbers of recorded taxpayers. The year numbers (for example, 1958) refer to the …scal years (1957–1958). PAYE was …rst introduced in 1957–1958 but appears not to have been fully implemented until 1958–1959; see Goldsmith (2008, pp. 229– 30) and Vosslamber (2012). In addition, various changes to tax settings including exemptions in the 1958 budget would have substantively affected the number of individuals legally liable to income tax in 1958 and 1959. It can be seen that before 1958 the number of recorded taxpayers is between 600,000 and 700,000. Those numbers increased signi…cantly to over 800,000 in 1958, when PAYE was introduced, and increased again to around 1,050,000 in 1959. Importantly, the data show that the substantial increase in the number

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Figure 6.2: Number of Taxpayers, 1955–1961 of recorded taxpayers in 1958 and 1959 was nearly all for taxpayers with income less than £400. This can be seen in Figure 6.3, which shows income data by income band (up to £3,000) for 5 years from 1956 to 1960. Higher income bands are not shown in the …gure as their distributions are almost identical across years and they account for only around 1 to 2 per cent of the taxpayer population. The numbers in income bands equal or greater than £400 are very similar in all 5 years. However, there was a large increase in the number of people with income less than £400, from around 15,000 in 1956 and 1957 to approximately 100,000 in 1958, and 300,000 in 1959. Hence the trend for total recorded taxpayers is almost identical to the trend for recorded taxpayers with income less than £400. In short, the introduction of PAYE provided data on a large number of income earners not previously captured in income data because they did not need to …le tax returns. The following subsection, based on these data, explores a method of adjusting the pre-1958 Gini indices (for …lers) to make them more comparable to post-1958 indices (for PAYE taxpayers).

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141

Figure 6.3: Number of Taxpayers by Income Group, 1956–1960

6.3.3

An Adjustment for the Introduction of PAYE

As discussed above, the full effect of PAYE (at least on recoded income taxpayers) appears to be from 1959, with data for the number of income earners with income less than £400 signi…cantly underestimated by the …ler-based data before 1959. Two steps are used to adjust the Gini indices. First, Ginis for 1957 and 1958 are adjusted, using the 1959 Gini index as benchmark. From Figure 6.3, the data for taxpayers (before and after 1958) with income greater than or equal to £400 is likely to be a better representation of income earners above or equal to £400 in those years. Thus, the Gini for the sample of taxpayers with income greater than or equal to £400 is calculated for 3 years: 1957, 1958 and 1959. It is assumed that the rate of change in Gini for the sample of taxpayers with income greater than or equal to £400 is the same as for the total population from 1957 to 1959. Using this assumption, the adjusted Gini for the years 1957 and 1958 can be estimated.

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This can be stated more formally as follows. It is assumed that: 1958 () 1959 () 1957 () = = 1957 ( > 400) 1958 ( > 400) 1959 ( > 400)

(6.11)

Here  is the Gini index for year , and  () represents the Gini for all incomes. From equation (6.11), the adjusted Gini index for 1958 is: 1958 () =

1958 ( > 400) 1959 () 1959 ( > 400)

(6.12)

and similarly for 1957. Second, the adjusted Gini index calculated in the …rst step for 1957 is used as a benchmark to estimate the adjusted Gini indices for the years before 1957. For 1935 to 1957 it is assumed that the rate of change in the Gini index estimated from those submitting returns in (NZOYB data) is the same as rate of change in the Gini index for (unobserved) income earners equivalent to those captured later in PAYE-based data. That is:  ( )  () = −1 () −1 ( )

(6.13)

Using the growth rates from (6.13), the adjusted Gini index for each year from 1935 to 1957 is computed as:  () =

 ( ) 1957 () 1957 ( )

(6.14)

Hence, re-arranging (6.12) as:

the 1959 data yield:

1959 () 1958 () = 1958 ( > 400) 1959 ( > 400)

(6.15)

1959 () = 160 1959 ( > 400)

(6.16)

which, together with an estimate of 1958 ( > 400) = 0285, gives 1958 () = 0455. That is, slightly higher than the value of 1959 () = 0425. A similar process for 1957 yields 1957 () = 0458. For years before 1957, values

6.3. EMPIRICAL RESULTS

143

Figure 6.4: Adjusted Gini for All Individual Total Income Before Tax, 1935– 2014 for 1957 () are obtained using (6.14). The results obtained after making these adjustments for all years 1935–1958 are shown in Figure 6.4. An alternative approach to the adjustment would be to apply the form in equation (6.12) to all years from 1935–1958. However, examination of the income distributions for early years reveals that the vast bulk of the distribution (around 50–75 per cent) of all income earners during the 1930s and 1940s had incomes below £400; with the percentage generally falling over this period as general income growth occurred. As a result, these early distributions generate unreliable Gini estimates for the sub-sample with  ≥ $400 in (6.12). By the 1950s this proportion had dropped to 30 per cent in 1950 and to 12 per cent in 1957; such that there can be more con…dence in the annual  ( ≥ 400) obtained from these larger sub-sample sizes. This adjusted series suggests a substantial fall in income inequality during the 1950s from a peak of around 0.566 in 1951, having risen after World War II from 0.445 in 1947. It is likely that the peak in 1951 is associated with the

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Korean War, including its effect on commodity prices as discussed earlier. Clearly the further back in time the adjusted Gini series is extrapolated the more uncertain the accuracy of the method used here becomes–relying on the measured year-to-year changes in the Gini for tax …lers being an accurate representation of the equivalent changes for non-…lers. But, based on these adjustments, the Gini indices for 1935–1941 shown in Figure 6.4 suggest some variability in the index around 0.45.

6.3.4

Some Comparisons

Having obtained an adjusted annual series of Gini coefficients for taxpayers for 1935–2014 in Figure 6.4, it is interesting to compare these with alternative New Zealand estimates based on census data, and also with comparable estimates for Australia. The former are reported in Easton (2013, 2014) for 1926–2014 using census data for all adults. The latter are reported by Leigh (2005) for Australia based on income tax returns adjusted to account for non-taxpayers in the population, and cover the period 1942–2001. Before comparing the two New Zealand Gini estimates, a number of differences in data coverage between the New Zealand census and NZOYB/IRD data must …rst be mentioned. In addition to the difficulties discussed above, associated with estimating the numbers and incomes of taxpayers as opposed to tax …lers before 1958, the data used in the present chapter involve two substantive differences from census data. Census data on adults include all those not in the labour force and hence include many individuals with zero or very low (non-wage) incomes. Where NZOYB taxpayer data report zero or negative incomes, as mentioned above, these were excluded from the Gini estimates reported here. The issue is further complicated in recent censuses by the decision of Statistics New Zealand to change the way those on low incomes are reported in the 2013 census. Easton (2014, p. 14), quoting SNZ, reports: ‘Non-response rates were lower in

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2013 because we have allocated people who selected “no source of income”, but did not answer the total income question, to the “zero income” category. This edit was not carried out in previous years’. It can be expected that both of these differences serve to increase Gini values obtained from census data relative to those calculated from taxpayer-based data. Figure 6.5 compares the Gini estimates from Figure 6.4 with Easton’s census-based values for 1936 to 2014. These are, as expected, substantially higher for the census-based series, at least until the 1980s or 1990s. There is also an important change in the income concept used. In particular, social security bene…ts were included in Easton’s ‘market income’ series from 1981. This inclusion substantially reduces the value of his Gini estimates from that date. Though the census-based series is noticeably higher than the taxpayerbased series, except for 1950–1951 when the taxpayer-based series displays the unusual spike discussed earlier, the pattern of a decline in the Gini from the early 1950s is con…rmed by both series. However, it is less dramatic in the census data, which is also affected much more strongly by the 1981 change of de…nition to include social security bene…ts. It is also of interest that the census-based series, while demonstrating the familiar increase in the Gini from the mid-1980s, suggests that Gini estimates for the mid-1990s to 2013 are not very different from the comparable estimate for 1981. Figure 6.6 compares the annual New Zealand Gini series with Leigh’s (2005) series for Australia over the period 1942–2001. The Australian Ginis relate to pre-tax incomes for adult males based on data for male taxpayers, together with an adjustment for imputed incomes of non-taxpayers. As a result the absolute levels of the Gini coefficients are not comparable across the two countries but it is interesting to compare their trends over time. The similarity in the long-term pattern of the Gini time-series in the two countries is quite remarkable, notwithstanding very different economic con-

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Figure 6.5: Comparing Adult-based and Taxpayer-based Gini Estimates ditions and policies in the two countries over this extended period. Both countries display the same tendency for inequality to fall during the immediate post-World War II decades and both subsequently experience an increase in inequality from the 1980s. The New Zealand upturn appears somewhat later than in Australia and is more rapid, taking less than a decade, compared with the more steady increase in Australia over two decades from the late 1970s. Some support for the trends, and shorter-term movements, in the Gini in Figure 6.6 are provided by estimates of top income shares in New Zealand by Atkinson and Leigh (2008) for 1921–2005. These reveal very similar patterns over the shared 1942–2005 period, including the spike in 1950–1951, turbulence in the 1970s and the rapid upturn in the mid/late 1980s. Interestingly, a number of short-term movements in the Gini are also common to both countries, such as the sharp spike in 1950–1951 associated with the wool price shock, as mentioned also by Leigh (2005, p. S64) for

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147

Figure 6.6: Comparison of Australia and New Zealand Gini Estimates Australia in 1950–1951. The year labels used in Figure 6.5 for Australia are shifted by one year compared to the labels used by Leigh (2005) to account for the fact that Leigh used the label ‘1950’ for 1950–1951, whereas ‘1951’ is used here for 1950–1951. The greater volatility of the Gini estimates at the time of the global oil shocks in the early-to-middle 1970s is also re‡ected in the data for both countries. However, unlike New Zealand, Australian data do not reveal an obvious spike in 2000 when, as argued above, New Zealand incomes were probably subject to substantial responses to the 2000–2001 tax reforms. Unfortunately, the end of the Australian series in 2001 means that it is not possible to con…rm whether the rise in Australian inequality was largely halted by 2000, as in New Zealand. Evidence for top income shares in Australia from Atkinson and Leigh (2008), however, suggests that the share of the top 5 per cent and 10 per cent remained fairly static from 2000 to 2005 (the …nal year of their series). Other inequality indices from Leigh (2005), such as the 90:50 and 75:25 ratios, also suggest some ‘‡attening’ of

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inequality from the mid-to-late 1990s to 2001. An interesting question arising from this Australia–New Zealand comparison, though beyond the scope of this chapter, is how far these similar inequality patterns represent common responses to global factors or merely responses to similar circumstances in the two countries (for example, through their highly integrated labour market). Clearly, at least some speci…c year-to-year Gini changes in Australia and New Zealand coincide with global changes such as the Korean War and 1970s oil shocks, as noted earlier. However, some authors of studies of inequality in other countries have argued, or speculated, that the nature of post-World War II global technological change (for example, skill biases), and increased globalisation, may be responsible for some of the observed increase in inequality in those countries. As in Australia and New Zealand, these increases often appear from the 1980s onwards. However, addressing this question for Australia and New Zealand requires much more sophisticated analysis than the present data permit. Nevertheless, the previously noted ‡attening of the Gini in New Zealand from the early 1990s appears at variance with a number of other major economies, suggesting that global factors are likely, at most, to be only one part of a complex causal process.

6.3.5

Gini Indices by Gender and Before and After Tax: 1981—2013

Inland Revenue income data available from 1981 allow decompositions of taxpayers’ incomes by gender, and the calculation of Gini measures for aftertax incomes. Before-tax and after-tax Gini indices are shown in Figure 6.7 for males and in Figure 6.8 for females. Figure 6.9 then compares the before-tax Ginis for males and females with the equivalent for all individuals. Figures 6.7 and 6.8 reveal some interesting differences. Regarding gender, it is clear that the substantial increase in inequality observed for all individuals combined in the 1990s was much more prevalent for males than

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149

for females. For example, the male before-tax Gini rises persistently from 0.34 in 1985 to 0.48 by 1995, but the female index rises only from 0.39 to 0.43 over the same period. Female inequality, having been higher than for males before the mid-1980s, became lower than for males from around 1991. However, much caution needs to be exercised in interpreting both the level of, and trends in, male and female Gini indices. In particular, the data in the …gures cover a period of increasing female participation in formal incomeearning employment. Furthermore, participation also involves greater use of part-time work by females, but less so over time. These are likely to have affected changes in measured income inequality, but to an unknown extent. Both Figures 6.7 and 6.8 also indicate that, for each gender, the beforetax and after-tax Gini indices become much more similar during the period approximately from the late 1980s to 2000. This perhaps re‡ects the reduced degree of rate progression of the income tax in the mid-to-late 1980s when the number of marginal tax rates was reduced to three in 1988, and the top marginal rate was reduced to 33 per cent by 1990. The subsequent increase in the top rate of income tax in 2000 from 33 to 39 per cent, and reductions in some lower marginal tax rates, resulted in a more progressive income tax from that year. This seems to be re‡ected in a widening of the gap between the pre- and post-tax Gini indices thereafter in Figures 6.7 and 6.8. Indeed, comparing the two …gures suggests that during the 2000s the tax system generally reduced female inequality further as the decade progressed, whereas for males the effect was more ‘one-off’ immediately following the 2000 reforms. When the Gini for all individuals is included in Figure 6.9 this largely tracks the female Gini, and lies above both male and female Ginis, until the substantial late-1980s increase for males. Since the Gini for all individuals includes both within-gender and between-gender inequality dimensions it is possible for the latter to dominate such that the ‘all individuals’ Gini need not lie within the range of values for the two genders separately. After

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Figure 6.7: Male Gini, Total Income, 1981–2013 a transition, the Gini for all individuals largely tracks the male Gini from around 1993 to the end of the period of available data in 2013. For the period 1998 to 2008, Papps (2010) uses a decomposition of the variance of logarithms to examine within- and between-gender effects, and found evidence that male and female distributions are converging.

6.4

Conclusions

This chapter has sought to extend previously available estimates of Gini income inequality indices for New Zealand individuals from the early 1980s to cover a much longer period. Based on income data collected from various sources such as SNZ yearbooks and Inland Revenue, Gini indices were reported for various de…nitions of income earners’ or taxpayers’ total income before tax for the years 1935 to 2014. For the more recent period, covering 1981 to 2014, separate male and female Gini indices were reported and also comparable indices based on both before-tax and after-tax incomes, revealing changes in the redistributive properties of the personal income tax.

6.4. CONCLUSIONS

Figure 6.8: Female Gini, Total Income, 1981–2013

Figure 6.9: Gini Before Tax, Total Income, 1981–2013

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The evidence suggests that the introduction of the PAYE system in 1958 led to substantial changes in the coverage of income data available to SNZ, and hence to estimates of Gini indices after 1958. However, examination of income data around 1958 suggests that the change in data coverage applied almost entirely to those with incomes previously below the tax …ling threshold, most of whom become recorded in the SNZ data only after 1958. The article proposed a method of controlling for this change in coverage. This resulted in the Gini index showing substantial decline during the 1950s, rather than the apparent rapid rise from 1957 to 1959 suggested by indices based on the raw recorded income data. It is worth repeating here that any empirical study of income inequality is severely constrained by available data, which are usually collected for tax administrative purposes. In the present case the analysis is necessarily restricted to individuals and annual incomes. Where the desired welfare metric is a longer-period measure of income, the relationship between annual and longer measures of inequality are complicated by the possibility of changing patterns of relative income mobility. Furthermore, as in the case of the PAYE introduction considered here, variations in the nature of the population group considered mean that published distributions cannot necessarily be taken at face value. An important problem is that it is almost impossible to obtain a consistent measure of ‘income’. The de…nition of what is included in taxable income varies over time, involving, for example, variations in the administration of welfare bene…ts and whether they are included in taxable income. And, as mentioned above, measured income is substantially affected by regulations regarding the taxation of fringe bene…ts. The introduction of fringe bene…ts taxation in the mid-1980s is likely to have increased measured annual income inequality, without necessarily affecting the more fundamental dimensions of inequality. One particular component of taxable income is also worth bearing in

6.4. CONCLUSIONS

153

mind–the relatively generous taxable and universal New Zealand Superannuation (NZS) introduced in 1977 (the age of eligibility for NZS has changed, for both males and females, over time). This gives rise to a mode in the total annual income distribution corresponding to those in receipt of NZS. An increase in the extent of superannuation payments (also partly associated with an ageing population) may increase measured annual taxable income inequality, but lifetime inequality may move in a quite different manner. When examining long-period movements in an aggregative measure, such as income inequality, there is always a danger of committing the post hoc ergo propter hoc fallacy: attributing causality to changes purely because they precede the change. Any analysis of the underlying causes of changing inequality is severely handicapped by the fact that the generation of the income distribution is affected by numerous complex interacting factors, many of which are central to the functioning of the economy. This article has offered only an empirical description of income inequality changes over a long period, derived from available data and with some important caveats. However, hopefully this can contribute to future analysis and more informed public discussions of policy preferences and prescriptions on this important topic.

Chapter 7 Regression Models of Mobility A great deal of attention is currently being paid to establishing the extent of inequality in New Zealand and whether it has risen in recent years. This mostly relates to static, usually annual, measures of income or wealth inequality, such as Gini coefficients or income percentile ratios. Equally important, but much less explored in the New Zealand context, are the dynamics of the income distribution over time. By contrast, the international literature on measuring income inequality over longer and shorter time periods, or ‘income mobility’, has expanded considerably in recent years. As discussed further below, these studies typically involve non-parametric methods such as calculation of summary income mobility indices or income inequality indices constructed from multi-year accounting periods. Recent applications to New Zealand include Creedy et al. (2018) and Creedy and Gemmell (2018a, 2018b). Though less commonly applied in more recent years, parametric methods such as income dynamic regression models provide an alternative approach and have the advantage of capturing a variety of systematic and stochastic processes contributing to changes in income inequality over time, based on a small number of estimated parameters. This chapter pursues the parametric approach for New Zealand, reporting new estimates of simple regression models of income dynamics. Multi-year income data are obtained from a special New Zealand anonymised individual 155

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income dataset compiled from Inland Revenue department (IRD) taxpayer data. Despite the considerable complexity of the underlying process of income change from year to year for individuals, useful summary information can be obtained using simple speci…cations involving variations of the Galtonian model of ‘regression towards the mean’, based on a few regression parameters. In addition to providing succinct descriptive information which can usefully supplement cross-sectional inequality measures, such models are useful when it is required to include income dynamics in larger economic models. For example, when estimating taxpayers’ responses to changes in marginal tax rates, using the concept of the elasticity of taxable income (Feldstein, 1995, 1999) it is useful to be able separately to capture non-tax related changes in incomes during periods when taxes remain unchanged, so providing a counterfactual against which behavioural responses to tax reforms can be isolated; see, for example, Carey et al. (2015). Other contexts include the construction of life-cycle simulation models designed to examine, for example, superannuation schemes or tax incidence over the lifetime; recent New Zealand examples include Ball (2014) and Ball and Creedy (2014). In addition, descriptive information about mobility is important when examining inequality. Attitudes towards changing crosssectional inequality may well be in‡uenced by the extent and nature of mobility between periods, which in‡uences inequality when measured over longer accounting periods. Section 7.1 …rst brie‡y discusses earlier literature. Section 7.2 then introduces the panel data used in the empirical exercises. Section 7.3 therefore presents results for the basic Galton (1889) ‘regression to the mean’ model combined with …rst-order serial correlation – sometimes described as the ‘workhorse’ of simple dynamic models. Section 7.4 extends the model to examine whether a second-order process is supported by the data.

7.1. SOME PREVIOUS LITERATURE

7.1

157

Some Previous Literature

In recent years the literature on changes in income inequality, previously focused mainly on changes in cross-sectional income distributions, has developed a variety of summary, non-parametric measures of income mobility. Thus, following early work by Shorrocks (1978a, 1978b, 1993) on summary mobility indices, Fields and Ok (1996), Fields (2000) and others proposed various conceptual, non-parametric measures suitable for empirical analysis. These include transition matrices, and ‘growth incidence curves’; see, for example, Ravallion and Chen (2003), Jenkins and Van Kerm (2006, 2016), Van Kerm (2009), Bourguignon (2011), Jäntti and Jenkins (2015), Fields et al. (2015). Previous regression-based approaches include the mobility measures proposed by Trede (1998) based on quantile regressions and applied to German and United States data, and analyses of dynamic panel regression coefficients obtained using Mexican data by Duval-Hernández et al. (2017). The latter paper, for example, focuses on estimating income convergence characteristics for panels of individual incomes, effectively capturing regression towards the mean properties. As they pointed out for Mexico, and is reported below for New Zealand, such systematic convergence tendencies serve to counteract other disequilising effects within the overall income dynamic process such that cross-sectional inequality measures may be observed to rise or fall. These alternative measures provide helpful insights into observed income dynamic patterns. However, they typically do not provide guidance on the extent to which income movements re‡ect underlying deterministic factors as opposed to apparently random, idiosyncratic or transitory movements. An exception is Duval-Hernández et al. (2017), who provide con…dence intervals around their estimated convergence parameters, though they do not report how far the estimated systematic components ‘explain’ observed income movements. By contrast, income dynamic regressions, such as those

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explored in this chapter, identify the extent to which overall movement in individuals’ incomes over time can be accounted for by, for example, systematic regression towards the mean and serial correlation in individual incomes. For New Zealand, limited information about income dynamics and mobility has previously been available. An early analysis of longitudinal data, obtained from the Personal Incomes Survey for eight consecutive years 1980 to 1987 and all age groups combined, is contained in Smith and Templeton (1990). They provide a number of tables of transition matrices between income quintiles. The …rst published regression estimates for New Zealand, using IRD data, were obtained by Creedy (1996) for a range of cohorts over the years 1991 to 1993, and used to produce lifetime income simulations for males and females. Transition matrices were also reported by Hyslop (2000) using IRD data for 1994 to 1997. Crawford (2009) used the Linked Income Supplement of the Household Labour Force Survey to examine transition matrices for hourly earnings over 1997 to 2004. The same data source was used by Ballantyne et al. (2003) to examine child poverty dynamics. Transition matrices, obtained using SoFIE data from 2002 to 2010, were also reported by Carter et al. (2014) and Laws (2014). Illustrations of several income mobility measures for individuals income taxpayers in New Zealand were provided by Creedy and Gemmell (2018a).

7.2

The Dataset

The data used throughout this chapter are unpublished longitudinal panel microdata from Inland Revenue, compiled from anonymous individual tax return information and/or employer records for the period 1994 to 2012.1 The dataset includes detailed information on taxable income and its com1 In

principle, tax return data are available from 1981, but PAYE information for non-…lers (a large fraction of total taxpayers) is only available from 1994. The data are discussed in more detail in Laws (2014).

7.2. THE DATASET

159

ponent sources such as wages and salaries, taxable bene…ts, interest income and dividends. The panel was selected randomly based on taxpayers’ IRD numbers: the …nal two digits of an individual’s IRD number were used for selection. Examining income dynamics using tax administration data has several advantages over other sources of income data, in particular by reducing measurement error in reported incomes (especially for PAYE taxpayers) and increasing the size of samples it is possible to analyse. However, a disadvantage is that very little information – other than taxable income – is available for each individual. Hence the analysis cannot consider a range of demographic or other variables (such as education, occupation and location) which may be relevant. Nevertheless, as is shown below, a substantial proportion (over three quarters) of the variation in individuals’ incomes in any year can be ‘explained’ by incomes in the preceding two or three years. It is also useful to compare the present estimates with previous results using similar approaches. For the regression analyses, this requires separate panels for each regression with the number of taxpayers in each case depending on the start year and the number of annual lags included. In particular, three or four years of data for the same individuals are required when the number of lags used is two or three, respectively. In this case, a 2 per cent sample of NZ taxpayers was constructed, based on the …rst year of data required for each income dynamic regression. To be included in the analysis an individual must either have had PAYE earnings (which include taxable welfare bene…ts) or have …led a tax return. The sample does not capture individuals with zero income or people receiving only interest, dividends or portfolio investment entity (PIE) incomes that are fully taxed at source, unless they …led a tax return. The data are trimmed to exclude very low taxable incomes (below $5,000). Thus, for regressions based on earlier years, the number of available observations tends to be smaller, due to smaller numbers of taxpayers. As shown in Table 7.1 in Subsection 7.3.2, these range from nearly 41,000 individuals

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for regressions based on data for the three years (two lags), 1995 to 1997, to over 52,000 for regressions based on the three years, 2010 to 2012. Slightly smaller sample sizes are available when four years are used, mainly due to greater attrition over four, compared to three, years; see Table 7.2 in Section 7.4 for details.

7.3

Regression with First-Order Serial Correlation

This section presents the basic Galton (1889) model and reports estimates for New Zealand. In its simplest form this model contains a single parameter governing the systematic process of relative income change – regression towards the mean – combined with an independent stochastic term covering idiosyncratic movements. It was …rst applied by Galton in the context of the heights of fathers and sons, explaining the absence of a tendency for the distribution of heights to become more dispersed over time. The disequalising idiosyncratic process is eventually counteracted by the regression tendency to stabilise the distribution. When a …rst-order process for the stochastic term is introduced, this simple property is seen to disappear.

7.3.1

Speci…cation of the Model

Let  denote an individual’s income in period , and if  denotes geometric mean income, then  = log (  ) is the logarithm of the ratio of the individual’s income to the geometric mean (that is, their ‘relative income’). A simple, but convenient, statistical description of the process of relative income change is the Galtonian model:  = −1 + 

(7.1)

where  re‡ects ‘regression towards the mean’ and  is a random term. In his pioneering analysis of income dynamics, Gibrat (1931) introduced the special

7.3. REGRESSION WITH FIRST-ORDER SERIAL CORRELATION 161 case where  = 1: on this process and resulting equilibrium distributions, see Aitchison and Brown (1954, 1957) and Brown (1976). The basic Galton process was used to examine early UK longitudinal data by Hart (1976a, 1976b). If the variance of  is constant at 2 , it is easily seen that the ¡ ¢ variance of logarithms of income in period , 2 , stabilises at 2  1 −  2 .

Figure 7.1: Regression to the Mean It is important in what follows to stress that   0, otherwise there can be systematic movements away from the (geometric) mean. Consider Figure 7.1, which plots  = log (  ) against −1 = log ( −1 ). The top righthand segment (the positive orthant) refers to incomes that are initially (that is, in period  − 1) above the geometric mean income. Conversely the bottom left-hand segment refers to incomes that, in  − 1, are below the geometric mean. The Gibrat process is described by the 45 degree line AB, where the only changes are ‘random’. These random changes can of course place some individuals from both positive and negative orthants in the top left- and bottom right-hand segments, implying that those individuals move across

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the geometric mean. A systematic equalising tendency – movement towards the origin (where  = 1 so that  = 0) – is shown by the straight line , which has a slope of less than 45 degrees. The extent to which this line is ‡atter than a 45 degree line re‡ects the strength of the systematic equalising tendency. A process in which there is a systematic tendency for all individuals (in the absence of an idiosyncratic component) to cross the geometric mean, whereby   0, can be ruled out as economically implausible and hence highly unlikely to be observed empirically. The basic Galton model can be extended, following Creedy (1974), by adding a …rst-order autoregressive process to the  , whereby:  = −1 + 

(7.2)

and the  are independently distributed. Combining the two equations gives:  = ( + ) −1 − −2 + 

(7.3)

 = −1 + −2 + 

(7.4)

Rewrite this as: In this case the variation in  2 over time is considerably more complex than the basic Galton case. On the coefficient of variation of multi-period incomes with the Gibrat process ( = 1) and …rst-order correlation in the  , see Creedy (1977). For the extension to the Galton model combined with serial correlation, see Atoda and Tachibanaki (1991). The parameters  and  can be estimated using income data for a constant sample of individuals over three consecutive years. However,  and  cannot be identi…ed. Writing  = − and substituting into  =  +  gives  as the root of the quadratic  +  − 2 = 0. But from empirical values, this generally has two real roots, one of which is negative and the other is positive and close to unity. If the alternative route is taken of substituting  = − into  =  + , exactly the same quadratic is obtained in terms of .

7.3. REGRESSION WITH FIRST-ORDER SERIAL CORRELATION 163 The problem is essentially that a model which has a high degree of positive autocorrelation in the  and a huge degree of regression towards the mean is observationally equivalent to one in which there is a small amount of negative serial correlation in the  and a ‘reasonable’ amount of regression. The contrast, along with the above reasoning regarding the implausible interpretation associated with negative values of , suggests that it is reasonable to adopt the assumption that the positive root is interpreted as . Consider rewriting (7.3) as:  − −1 = ( +  −  − 1) −1 +  (−1 − −2 ) + 

(7.5)

Hence, the proportional change in relative income from −1 to  can be said to depend on (the logarithm of) the individual’s relative income in  − 1 and the previous proportional change from  − 2 to  − 1. Hence, for a given previous proportional change, if −1  −1 and  +  (1 − )  1 the proportional change is positive. The process is thus ‘equalising’ if  +  (1 − )  1. In addition, if the previous income change is positive, the next proportional change is more likely to be negative, depending of course on relative income at  − 1. Of course, the same interpretation follows for the alternative a priori assumption that  is negative and  is positive and close to one, but as pointed out above, this makes no sense as it implies systematic movements across the geometric mean by all individuals. Figure 7.2 shows the systematic changes implied by (7.5), that is, ignoring the stochastic term, , for the combination of  = 09 and  = −015. The pro…les show the proportional change in  from period  − 1 to  (measured on vertical axis), for variations in the ratio of  in period  − 1, for alternative assumptions about the proportional change ∆ = −1 − −2 . For ease of interpretation, it is assumed that geometric mean income remains unchanged. Hence if the previous income change is zero and income in  − 1 is equal to the geometric mean , the proportional change in income from −1 to  is zero, that is, the same as the change in the geometric mean.

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Figure 7.2: Proportional Income Changes Depending on Initial Income and Previous Change

7.3.2

Income Dynamic Estimates

The database used in this section to examine income dynamics was introduced in Section 7.2. As noted there, some individuals have extremely low taxable incomes which may not be representative of regular or substantive earnings. For example, the panel dataset in principle includes students or others who engage in minimal part time work to supplement other nontaxable incomes, those with small amounts of interest on savings deposits, or election day workers who work for a single day once every three years. To avoid possible distortions from their inclusion, individuals were excluded from the following regressions if their recorded taxable income was less than $5,000 in each of the three years used in the relevant regression in Table 7.1.2 2 To

check the impact of income limit restrictions on the regression results, comparable results were also obtained for alternative restrictions on both the maximum and minimum income allowable. The former was set $500,000 in 2012, with previous years’ values adjusted for in‡ation using the New Zealand Consumer Price

7.3. REGRESSION WITH FIRST-ORDER SERIAL CORRELATION 165 Table 7.1: Regressions with First-Order Serial Correlation  1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Coefficient on: −1 −2 0.71745 0.17389 0.71196 0.17514 0.72383 0.18702 0.72436 0.19016 0.68618 0.16356 0.73517 0.15291 0.73048 0.15409 0.74517 0.14780 0.73519 0.16505 0.74942 0.14747 0.72184 0.16972 0.72471 0.15334 0.71772 0.15952 0.71828 0.17215 0.71705 0.17034 0.73784 0.16637

-values (−1 ) (−2 ) 143.74 35.50 148.69 36.86 143.62 37.92 140.77 36.59 146.47 34.34 162.73 35.64 154.53 33.18 161.58 32.93 160.14 36.71 161.93 32.42 159.60 38.13 158.20 34.37 157.00 36.07 160.17 39.52 168.04 40.52 167.97 39.07

2 0.739 0.746 0.747 0.735 0.731 0.758 0.753 0.756 0.756 0.750 0.749 0.735 0.727 0.732 0.747 0.752

 0.9088 0.9054 0.9258 0.9290 0.8734 0.9043 0.9014 0.9079 0.9155 0.9112 0.9086 0.8959 0.8958 0.9079 0.9052 0.9189

 -0.1913 -0.1934 -0.2020 -0.2047 -0.1873 -0.1691 -0.1709 -0.1628 -0.1803 -0.1618 -0.1868 -0.1712 -0.1781 -0.1896 -0.1882 -0.1811

 40,940 41,610 41,884 42,270 42,856 43,839 44,713 45,775 46,870 47,753 48,631 49,411 50,058 50,876 51,736 52,444

Table 7.1 reports a series of regression results from applying ordinary least squares to equation (7.3), for all working age individuals for whom information was available over the three years. For example, where  = 1997 (the …rst row of the table), this means that three consecutive years, 1995 ( − 2), 1996 ( − 1) and 1997 () were used. Two columns on the right-hand side of Table 7.1 convert the coefficients into the corresponding values of  and  for each ‘year’ regression. The sample size, , for each regression is shown in the far right-hand column and can be seen to increase over time, ranging from 40,940 individuals for the ‘1997’ regression in Table 7.1 to 52,444 for the ‘2012’ regression. Index time series. This gave a maximum criteria in 1994 of $325,300. The alternative minimum requirement was set at one quarter of the gross annual rate of superannuation in each year. In this case, taxable income had to exceed $5,059 in 2012 and the threshold diminished to $3,001 in 1994. See Laws (2014, pp. 44-45) for further details. Regression results from this exercise were very similar to those shown here and are therefore not reported.

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The results show a relatively stable value of  over the period, around 087 to 092. Estimates of  also appear fairly stable with values around −016 to −020, with a slight increase in the value of  over time (that is, a slight reduction in the absolute value). The time pro…le of  and  values are discussed in Section 7.4, and displayed in Figure 7.4, where they are compared with outcomes from a second-order autoregressive speci…cation.

Figure 7.3: Income Convergence Properties: Alternative  and  Values To interpret these values, reference may be made to Figure 7.2 above, which shows the systematic income changes, in different parts of the distribution, for the illustrative combination of  = 09 and  = −015. Furthermore, from equation (7.5) it can be seen that, in the absence of serial correlation ( = 0), the rate of convergence of  is given by ∆ =  −−1 = ( − 1) −1 . Hence, these estimated values of  imply relatively strong regression toward geometric mean income levels. Similarly, estimated values of  imply a strong negative autoregressive process. The convergence properties are illustrated in Figure 7.3. Taking rep-

7.3. REGRESSION WITH FIRST-ORDER SERIAL CORRELATION 167 resentative ‘benchmark’ values of  = 090 and  = −018 from Table 7.1, the …gure illustrates pro…les for  over sixteen years for an individual with initial income three times the geometric mean: that is,   = 3;  = log (  ) ≈ 11. Three further  pro…les are also shown for alternative values of  = 080 and  = −010 −025. Similar pro…les for initial  values below the mean (not shown) are symmetrical; in that case  converges on zero from below. The benchmark pro…le, shown by the unbroken line in Figure 7.3, indicates a half-life of only two years, where half of the difference between the initial value of  , and  = 0, has been eliminated. The convergence process is almost complete after seven years. An autoregressive process of  = −010 generates a longer convergence process since, for example, an earlier increase (relative to the geometric mean) is more likely to be followed by a smaller relative reduction. Nevertheless, convergence is almost complete after sixteen years with a half-life of only three years. This outcome is reversed when  = −025 with faster convergence than in the benchmark case. Conversely, allowing for stronger regression to the mean properties with  = 08 (and benchmark  = −018), can be seen to generate faster convergence towards  = 0. In that case, convergence is almost complete after only …ve years (though some oscillation due to negative serial correlation can be seen to persist thereafter) and a half-life of less than two years. It is interesting to compare the parameter estimates in Table 7.1 with earlier regression results for New Zealand. Following a similar approach, Creedy (1996) obtained comparable estimates of  and  using information on taxable incomes in 1991, 1992 and 1993, for males and females in various age groups. Although there were slight differences among age groups, and the regression term, , was a little lower for females compared with males, Creedy (1996) reports similar values of  to those obtained here, despite the calendar time differences involved. For example, Creedy (1996, Tables 1 and 2) reported values for …fty-two age/gender groups between age 18 and 69

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years. For male age groups, values of  were typically found in the range 0.85 to 0.95, and similarly for females (though with generally slightly lower age-equivalent values). Values for  were also similar to those obtained here but with a wider range across the age and gender groups, typically around −009 to −025.

7.4

An Extension: Second-Order Serial Correlation

Suppose the Galtonian process of regression towards the geometric mean, in (7.1), is combined with a second-order autoregressive form for the  , as follows:  = −1 + −2 + 

(7.6)

Then lagging (7.1) and multiplying by the corresponding terms in (7.6) gives: −1 − −2 = −1

(7.7)

−2 − −3 = −2

(7.8)

and:

Hence substitution produces:  = ( + ) −1 + ( − ) −2 − −3 + 

(7.9)

Given information about the incomes of a constant sample of individuals in four consecutive years, it is thus possible to estimate the parameters ,  and , where:  = −1 + −2 + −3 + 

(7.10)

=+

(7.11)

 =  − 

(7.12)

with:

7.4. AN EXTENSION: SECOND-ORDER SERIAL CORRELATION 169  = −

(7.13)

As before, the question arises of whether the basic parameters, ,  and  can be recovered from the estimates of ,  and . Writing  =  −  and  = − ( − ), substitution into (7.12) gives, − ( − ) −  ( − ) = , and hence the following cubic in : ¢ ¡ ( + ) + 2 −   − 22 + 3 = 0

(7.14)

This particular cubic produces only one real root, which is negative. Having solved for , (7.11) can be used to get  and (7.12) can be used to get . The fact that only one real root is produced for all cases with the present data makes it easier to solve (7.14), as follows. Consider the cubic:

and let: and:

0 3 + 1 2 + 2  + 3 = 0

(7.15)

¶ µ 21 0  = 2 − 30

(7.16)

µ ¶ 2 1 231 −  = 3 + 0 2720 30

(7.17)

If  = −43 −272 is negative, there is only one real root. Letting  = 1 30 n n ¡ − ¢05 o13 ¡ − ¢05 o13   and  = − 2 + 108 with  = − 2 − 108 , the root is given by  +  − . As with the …rst-order serial correlation case, it is possible to rearrange the expression in (7.9), so that the proportional change in relative income from  − 1 to  is a function of the (logarithm of) relative income in  − 1, −1 , and the two preceding proportional changes, −1 −−2 , and −2 −−3 . Equation (7.9) can be rewritten as:  − −1 = { + (1 − ) ( + ) − 1} −1 + { ( + ) − } (−1 − −2 ) + (−2 − −3 ) + 

(7.18)

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For example, consider the case above where, by assumption, income relative to the geometric mean is 05, and −1 − −2 = 010. For  = 09 and  = −015 it can be found that the proportional change in relative income (excluding the stochastic term) is 0.066. If, now, second-order serial correlation is introduced with other parameters unchanged and  = −005, the proportional change becomes 0.077, approximately one percentage point higher than in the …rst-order example. This approach, particularly as the single root of the above cubic equation turns out in practice to be a small negative number, appears at …rst sight to avoid the previous difficulty raised by the inability to identify the parameters  and . However, the coefficient on −1 is unchanged if values of  and  are interchanged, although the coefficients on the past changes are affected. Furthermore, the process turns out to be observationally equivalent to a slight modi…cation, in which the speci…cation has a second-order Galton process with …rst-order serial correlation. Suppose instead that  is thought to be determined by a second-order process rather than the simple Galton process considered earlier:  = −1 + −2 + 

(7.19)

If this process is combined with the …rst-order form for  in (7.2), then by lagging (7.19) and multiplying by : −1 − −2 − −3 = −1

(7.20)

and substituting in (7.19) gives:  = ( + ) −1 + ( − ) −2 − −3 + 

(7.21)

Given estimates of the coefficients ,  and , now obtained from the regression:  = −1 + −2 + −3 + 

(7.22)

the aim is again to recover values of the basic parameters. Since, now  =  + ,  =  −  and  = −, the only term that differs from

7.4. AN EXTENSION: SECOND-ORDER SERIAL CORRELATION 171 the previous case is . Using  =  −  and  = −, substitution gives − −  ( − ) =  which can be rearranged to get the cubic in ,  +  + 2 − 3 = 0. Again, empirical estimates result in only a single real root of this cubic, which is positive and slightly below 1. This approach thus produces a value for  which is negative, which can be ruled out following the argument of Section 7.3, that the systematic component cannot have a tendency for everyone to cross the geometric mean income (in the absence of the idiosyncratic component). Another route to solving the equations is to get the cubic in , given by ( + ) + (2 − )  − 2 2 +  3 = 0, but it produces exactly the same value of  as obtained by the previous route. The speci…cation in the present section is thus observationally equivalent to that in the previous section, but produces individual parameters for which it is hard to give a meaningful economic interpretation. Therefore, the assumption is made that the data-generating process is the previous process presented above. Table 7.2 reports the regression results for equation (7.10). In this case, individuals were included if their income exceeded $5,000 in each year. This was approximately the same value as the mimimum taxable income used to be eligible for inclusion in the sample in Section 7.3.2; namely $5,059 in 2012. The -values are not reported here in view of their magnitude: values for all coefficients on −1 exceed 125, while for −2 all -values exceed 18, and for and −3 all coefficients have -values exceeding 14 (with most exceeding 18). The associated parameter values describing the dynamics are given on the right-hand side of the table. Generally, the introduction of an additional lag produces a slightly lower value of , implying a greater degree of regression towards the (geometric) mean, and very similar , (or slightly lower  in early years regressions). The results from the two speci…cations can be compared in Figure 7.4. The values of  obtained from both speci…cations are relatively stable over the period, except for a dip in 2001. It is possible that this arose from the introduction

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of a higher top marginal income tax rate in that year. The announcement in advance of the tax change allowed individuals to shift some of their income between periods. Table 7.2: Regressions with Second-Order Serial Correlation  1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

7.5

Coefficient −1 −2 0.6804 0.1453 0.6792 0.1320 0.6973 0.1559 0.6907 0.1451 0.6537 0.1279 0.7125 0.1271 0.6961 0.1072 0.7164 0.1163 0.6959 0.1424 0.7146 0.1219 0.6871 0.1530 0.6910 0.1236 0.6809 0.1260 0.6906 0.1296 0.6852 0.1293 0.7035 0.1347

on: −3 0.0847 0.0943 0.0759 0.1015 0.0869 0.0662 0.0997 0.0797 0.0834 0.0809 0.0737 0.0851 0.0901 0.0907 0.0935 0.0849

2 0.748 0.755 0.756 0.745 0.740 0.772 0.766 0.768 0.770 0.762 0.762 0.748 0.738 0.743 0.756 0.760

 0.9333 0.9184 0.9262 0.9230 0.8661 0.8972 0.8887 0.8955 0.8889 0.8910 0.8737 0.8670 0.8585 0.8667 0.8603 0.8731

 -0.2529 -0.2393 -0.2289 -0.2323 -0.2124 -0.1847 -0.1926 -0.1791 -0.1929 -0.1764 -0.1866 -0.1760 -0.1777 -0.1760 -0.1751 -0.1696

 -0.0908 -0.0878 -0.0561 -0.0692 -0.0560 -0.0387 -0.0639 -0.0441 -0.0291 -0.0353 -0.0100 -0.0290 -0.0266 -0.0229 -0.0213 -0.0134

 37,833 38,594 38,971 39,495 39,930 40,499 41,695 42,526 43,591 44,520 45,331 46,092 46,628 47,375 48,494 49,222

Conclusions

In debates on income inequality, considerable emphasis is usually placed on changes in a measure of annual incomes over time. Yet, an important characteristic of individual incomes is that they are not constant: indeed they are subject to considerable variability over time. The pattern of relative income changes, despite being subject to considerable complexity, can be described succinctly using a simple autoregressive stochastic process in which Galtonian regression towards the mean is combined with serial correlation in the stochastic term. The parameters of the model have convenient interpretations

7.5. CONCLUSIONS

173

Figure 7.4: Mobility Estimates: First- and Second-Order Serial Correlation and can be easily estimated using limited longitudinal data. Using a series of random samples of New Zealand individual taxpayers with each sample containing income data for the same individuals over three consecutive years, revealed substantial regression towards the mean combined with negative serial correlation. In addition, remarkable stability in the estimated parameters was observed across the samples over the whole 1997 to 2012 period. These imply that relatively high income individuals have, on average, lower proportional increases in income from one year to the next compared with those with lower incomes, and those with a large increase in one year are more likely to experience a decrease the following year.

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CHAPTER 7. REGRESSION MODELS OF MOBILITY

The present results show that both these effects, though combined with a stochastic component that on its own would tend to increase inequality over time, are sufficient in this New Zealand case to ensure that inequality falls as the accounting period over which incomes are measured increases, and there is no systematic tendency for annual inequality to rise. Despite the simplicity of the dynamic process speci…ed, it is nevertheless capable of explaining about 75 per cent of the variation in annual incomes. For pragmatic reasons, given the available data source, the focus of this chapter has been on the dynamics of taxable income, and relating only to individual taxpayers rather than households. Broader approaches to inequality measurement also need to pay attention to the precise ‘welfare metric’ used, such as broader de…nitions of income, or consumption expenditure. For example, for some purposes it may be desired to allow for imputed rents from owner-occupied housing, or other non-taxable income, especially where the tax regime encourages taxpayers to divert some otherwise taxable income into non-taxable forms. This can affect both static measures of income inequality where diverted income is concentrated among particular types of taxpayer, and measurement of income dynamics where changes in the tax regime encourage taxpayers to alter the composition of their incomes over time.3 These caveats should be borne in mind when interpreting the income dynamic results in this chapter.

3 For

example, Creedy et al. (2018) suggest that the rise in measured inequality of taxable incomes during the 1980s and 1990s may partly re‡ect responses to tax reforms in this period such that previously ‘hidden’ (non-taxable) income earned by higher income taxpayers was more likely to be reported as taxable income when marginal tax rates fell post-reform.

Chapter 8 Illustrating Differential Growth When comparing distributions of non-negative economic variables, such as annual income or consumption, the Lorenz curve is ubiquitous. With individual observations arranged in ascending order, this plots the cumulative proportion of total income against the corresponding cumulative proportion of individuals. A normalised area measure of the distance between the Lorenz curve and the line of equality gives rise to the equally famous Gini inequality measure. Furthermore, the concept of ‘Lorenz dominance’ provides an immediate qualitative comparison between the inequality of two distributions, and this can be given a welfare interpretation when combined with the value judgement summarised by the ‘principle of transfers’. The Lorenz curve thus provides a valuable diagrammatic summary, providing much more information than either the density function or the distribution function alone. Where concern is largely for those below a poverty line, an alternative diagrammatic device involves, for incomes again arranged in ascending order, plotting the cumulative poverty gap per person against the corresponding cumulative proportion of people. This is the TIP curve, named by Jenkins and Lambert (1997) for its ability to indicate the ‘Three Is of Poverty’, namely incidence (the poverty headcount), intensity (the poverty income-gap) and inequality (the within-poor distribution). For those below the poverty line, the curve is a straight line only in situations where all the poor have equal 175

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incomes. As with the Lorenz curve, dominance properties hold. For both Lorenz and TIP curves, comparisons involving intersecting curves lead to the need to impose more structure on evaluations, in the form of explicit value judgements and quantitative inequality and poverty measures. In the context of income mobility, several diagrams have been proposed to capture the key properties of mobility in an easily-perceived way. This is complicated by the variety of de…nitions and interpretations of different mobility concepts. For example, following the taxonomy of mobility concepts developed by Fields (2000), and reviewed recently by Jäntti and Jenkins (2015), mobility has been characterised variously as associated with individual income growth (either absolute or relative); directional or non-directional positional change (re-ranking); income share change; impacts on the inequality of longer-term incomes; and ‘income risk’; see Fields (2000, 2008), Jäntti and Jenkins (2015).1 Most illustrative devices for income mobility have focused on income growth measures. These include Trede (1998), Ravallion and Chen (2003), Van Kerm (2009), Bourguignon (2011) and Jenkins and Van Kerm (2006, 2011, 2016), with recent contributions emphasising the welfare dominance properties of alternative income mobility measures or illustrative devices.2 However, it is suggested below that the ‘three Is’ properties can be translated to the context of income mobility, yet none of the existing approaches 1 Several

authors have pointed to the normative ambiguity associated with (possibly desirable) ‡exibility in short-term income movements versus (undesirable) volatility. Jäntti and Jenkins (2015) treat income risk as one aspect of longer term income inequality which has both permanent predictable and transitory unpredictable components. Creedy and Wilhelm (2002) examine income mobility changes and social welfare in the context of a welfare function de…ned over longerperiod incomes and where individuals have a preference, ceteris paribus, for stable incomes. 2 Palmisano and Peragine (2015) propose a similar welfare framework for analysing growth incidence. They argue that, unlike Bourguignon (2011) and Jenkins and Van Kerm (2011), their framework can incorporate horizontal inequality concerns. See also Palmisano and Van de gaer (2016) who develop a mobility index that is ‘history dependent’ and weights individual income growth rates by their position in the initial income distribution.

8.1. ILLUSTRATIVE DEVICES FOR INCOME MOBILITY

177

focuses speci…cally on these three ‘positive’ properties for mobility measures analogous to the TIP curve for poverty. This chapter addresses that gap by offering new illustrative devices for income mobility. Firstly, a modi…cation of income growth pro…les is proposed to illustrate the three Is of mobility. Like Bourguignon (2011) and Jenkins and Van Kerm (2016) this captures longitudinal dimensions. With individuals ranked in ascending order of initial income, it plots the cumulative proportional income change per capita (not per head of the cumulated sub-group) against the corresponding cumulative proportion of individuals. Since the diagram bears a close resemblance to the TIP curve it is described here as a ‘Three Is of Mobility’, or TIM curve. Secondly, comparable devices capable of illustrating the three Is properties for a positional change measure of income mobility are developed. This …rst considers the cumulative observed re-ranking change across individuals ranked in ascending order of the initial income distribution and, secondly, the ratio of observed re-ranking to the maximum feasible re-ranking for each individual. The former may be called a ‘cumulative re-ranking curve’ and the latter a ‘re-ranking ratio’, or , curve. Some illustrative devices for income mobility are summarised in Section 8.1. The TIM curve concept is introduced in Section 8.2. This device is illustrated in Section 8.4 using a longitudinal sample of individuals from New Zealand.

8.1

Illustrative Devices for Income Mobility

A number of authors have previously sought to illustrate distributional dimensions of income mobility across a population or sample of individuals, such as Van Kerm (2009), Bourguignon (2011) and Jenkins and Van Kerm

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(2016).3 This section outlines these brie‡y. For income growth based mobility measures, Van Kerm (2009) and Jenkins and Van Kerm (2016) suggest using ‘Income Growth Pro…les’ (IGPs).4 In de…ning these, much attention is addressed to the welfare dominance properties of individual income growth where individual utilities are based on incomes in both the initial and terminal periods; see Jenkins and Van Kerm (2016, pp. 681-3). Their objective is to produce summary indices of income growth with consistent welfare foundations that are helpful for normative evaluations of alternative distributions of individual income growth. This is therefore rather different from the positive measurement or description of income mobility properties pursued in the present chapter. Jenkins and Van Kerm, 2016) also produce a cumulative version of the IGP (a CIGP) which is conceptually closest to the TIM curve described below, but the latter illustrates the distributional dimensions of mobility in a visually more straightforward manner.

8.2

The TIM Curve

This section begins by summarising the key aspects of the TIP curve developed by Jenkins and Lambert (1997), in subsection 8.2.1. It is then adapted in the income mobility context in subsection 8.2.2.

8.2.1

The TIP Curve

Jenkins and Lambert (1997) demonstrated that three important dimensions of poverty can be summarised by their TIP curve. These are: the incidence 3 Trede

(1998) proposed an early illustrative device based on quantile regressions, showing pro…les of various quantiles of the conditional income distribution. 4 Similar income growth based devices, but using cross-sectional income distributions, were earlier proposed by Ravallion and Chen (2003) and Son (2004). Bourguignon’s (2011) ‘non-anonymous growth incidence curve’ has similar normative properties to the Jenkins and Van Kerm IGP concept and is longitudinal in nature. See also Grimm (2007).

8.2. THE TIM CURVE

179

of poverty, as captured by the headcount poverty measure; the intensity, as measured by the income gap,  −  , where  is the poverty line; and the inequality of poverty within the poor group, capturing how far the incomes of the poorest differ from those closer to the threshold,  . Let  denote individual ’s income, with  = 1  . Given  , the poverty gaps are de…ned by  ( ) = 0 for  >  and  ( ) =  −  for    . When incomes are ranked in ascending order, the TIP curve P is obtained by plotting 1 =1  ( ) against  , for  = 1 2  . That is, the total cumulative poverty gap per capita is plotted against the associated proportion of people.

Figure 8.1: A TIP Curve A hypothetical example is shown in Figure 8.1. The slope at any point is equal to the average poverty gap, with a steeper slope indicating a larger poverty gap. Flattening of the curve therefore shows the extent to which the average poverty gap falls as income rises towards  . Thus, inequality among the poor is re‡ected in the curvature of the TIP curve. The curve becomes horizontal beyond H, beyond which there is no one in poverty. Poverty is

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CHAPTER 8. ILLUSTRATING DIFFERENTIAL GROWTH

unambiguously higher where a TIP curve lies wholly above and to the left of an alternative TIP curve. The TIP curve provides useful information because, for any poverty metric (such as income or consumption) and poverty threshold de…nition, it captures three separate and important distributional dimensions. However, these properties relate to levels of poverty at a point in time, rather than poverty dynamics over time, such as the extent to which the poor stay poor over different time horizons. This, of course, is captured by longitudinal income mobility analysis.

8.2.2

Three Is of Mobility

The three Is properties of income poverty (or other income thresholds) convey important distributional information. They can be translated to corresponding or analogous properties of income mobility, using the ‘Three Is of Mobility’ (TIM) curve, de…ned here. First, de…ne the logarithm of income,  = log  , for individuals  = 1  . Hence  − −1 is (approximately) person ’s proportional change in income from period  − 1 to . With log inP comes ranked in ascending order, plot 1 =1 ( − −1 ) against  =  , for  = 1  . Thus the TIM curve plots the cumulative proportional income change per capita against the corresponding proportion of individuals. In specifying a TIM curve, the focus is on the mobility of a particular group of low-income individuals, those with incomes below  (), for the proportion, , of the population. In this framework  captures the incidence of the particular low income group of concern, just as the headcount poverty measures the incidence of income poverty. Similarly, intensity and inequality properties of mobility can be de…ned. The intensity of mobility of a particular group of individuals, such as those below (), is measured by the height of the TIM curve at the relevant point, . The inequality property of mobility refers to differences in the income growth rates within the speci…ed group

8.2. THE TIM CURVE

181

below  (), and is re‡ected in the curvature of the relevant portion of the TIM curve. The relevance of all three dimensions of mobility to longer-term inequality suggests that a TIM curve, analogous to the TIP curve, can provide similar insights when ‘mobility as income growth’ is the mobility concept of interest. One difference from the CIGP, but shared with the TIP curve, is that the height is obtained by dividing by  rather than .5 However, this apparently small modi…cation is important, since the properties of the TIM curve can more readily illustrate the three mobility characteristics of interest for any population or speci…ed population sub-group. Identifying the inequality of mobility within a given percentile is less straightforward using the CIGP since this requires a visual comparison of slope changes across percentiles below . The TIM curve can be de…ned more formally as follows, ignoring  subscripts for convenience. Suppose incomes are described by a continuous distribution where  ( ) and  ( ) denote respectively the distribution functions of income and log-income at time , with population size, . For incomes ranked in ascending order, the TIM curve plots the cumulative proportional income changes,  −−1 , per capita, denoted  , against the corresponding proportion of people, , where:  =  (−1 ) 

(8.1)

Thus −1 =  −1 () is the log-income corresponding to the  percentile, and the TIM curve plots  , given by: Z −1 ( − −1 )  (−1 )   = 0

(8.2)

against . 5 Jenkins

and Van Kerm (2016) also construct a CIGP where income changes are de…ned in absolute terms, , rather than growth rates, . See the Appendix to this chapter for equivalent TIM curves based on .

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For very large datasets it is convenient to plot values of the cumulative proportional change corresponding to percentiles,  , for 1 = 001 and  = −1 + 001, for  = 2  100. Thus, obtain the cumulative sum  = P 1 =1 ( − −1 ), where, as above,  is the number of individuals in the  P sample. Hence for  = 2  100:  = −1  + 1 = ( − −1 ). −1 +1 The TIM curve is then plotted using just 100 values. Let  denote the arithmetic mean of log-income (equivalent to the logarithm of the geometric mean,  , of income,  ). Equation (8.2) can be written as: Z −1 © ¡ ¢ª ¡ ¢  = ( −  ) − −1 − −1  (−1 ) +  − −1  (−1 )  0

(8.3) ¡ ¢ The term,  −  is equal to log (  ). Hence ( −  ) − −1 − −1 is the proportional change in relative income. Thus,  consists of the cumulative proportional change in income relative to the geometric mean, plus a component that depends only on the proportional change in geometric mean income. Suppose the proportional change in the geometric mean,  − −1 , is equal to . Furthermore, suppose the proportional change in relative income ¡ ¢ depends on income in  − 1, so that ( −  ) − −1 − −1 can be written as the function,  ∗ (−1 ). Then (8.3) can be expressed as: Z −1  ∗ (−1 )  (−1 ) +   = 0

(8.4)

If all individuals receive exactly the same relative income change, relative positions are unchanged and  ∗ (−1 ) = 0 for all −1 . Hence,  plotted against  is simply a straight line through the origin with a slope of . This means that the extent of differences in proportional income changes over any range of the income distribution can be seen immediately by the extent to which the TIM curve deviates from a straight line, which in turn depends on the properties of ∗ (−1 ). Online Appendix A considers a special case where

8.2. THE TIM CURVE

183

¡ ¢ ( −  ) is a linear function of −1 − −1 , re‡ecting Galtonian regression towards the geometric mean, combined with a random component. A hypothetical example of a TIM curve is shown in Figure 8.2. The particular curve illustrated re‡ects a situation in which relatively lower-income individuals receive proportional income increases which are greater than that of geometric mean income. Hence the TIM curve, OHG, lies wholly above the straight line OG.

Figure 8.2: A TIM Curve If all incomes were to increase by the same proportion, the TIM curve would be the straight line OG. The height, G, indicates the average growth rate of the population as a whole, while the height, H, indicates the average growth rate of those below  (): these heights re‡ect intensity properties of mobility. Furthermore, the inequality of growth rates is re‡ected in the degree of curvature. For example, the curvature of the arc OH relative to the straight line OH indicates that lower-income individuals have higher growth than those individuals to the left of, but closer to, . If concern is for those below a poverty line,  , the corresponding percentile is  =  ( ), where,

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CHAPTER 8. ILLUSTRATING DIFFERENTIAL GROWTH

as de…ned above,  () is the distribution function of . The TIM curve gives an immediate indication of whether income changes have been pro-poor. Suppose interest is focused on those below the  percentile, indicated in Figure 8.2. There is less inequality of mobility – a lower dispersion of income changes – among the group below , shown by the fact that the TIM curve from O to H is closer to a straight line than the complete curve OHG.6 The TIM curve also shows that the income growth of those below  is larger than that of the population as a whole. The average growth rate among the poor – the intensity of their growth – is given by the height H.

Figure 8.3: Example of Differential Mobility Pattern Figure 8.3 illustrates a TIM curve re‡ecting a very different pattern of mobility. In this case the lower-income groups experience smaller proportional increases in income than those with higher incomes. If  is to the right of the intersection (from below) of the TIM curve with the line OG, average 6 There is a potential ambiguity in the use of the term ‘inequality’ here since the TIP curve refers to a cross-sectional distribution whereas the TIM curve refers to income changes. To avoid confusion the latter could be referred to as the ‘interpersonal dispersion’ of mobility.

8.3. THE TIM CURVE AND GALTONIAN REGRESSION

185

growth of those in poverty exceeds average growth for the population as a whole. Yet the convexity of the TIM curve to this point demonstrates that this average pro-poor growth represents quite different experiences among the poor.

8.3

The TIM Curve and Galtonian Regression

It is shown above that the TIM curve can be written as: Z −1  =  ∗ (−1 )  (−1 ) +  0

(8.5)

¡ ¢ where  ∗ (−1 ) = ( −  )− −1 − −1 represents the proportional change in relative income. A simple special case is to suppose that: ¡ ¢ ∗ (−1 ) = − −1 − −1 +  

(8.6)

where  is a stochastic term with expected value of zero. For   0, those with −1  −1 experience a systematic relative reduction in income plus a random proportional change. Conversely, those below the geometric mean experience systematic relative income increases. Hence, letting 1 −  = : ¡ ¢  −  =  −1 − −1 +   (8.7) This is the standard Galtonian ‘regression to the mean’ speci…cation; see Creedy (1985). The extent to which  is less than 1 indicates the degree to which those below the geometric mean experience, on average, a higher relative income increase than those above the geometric mean. If, instead,   0, clearly   1 and there is regression away from the geometric mean. Substituting for  −  from (8.7) into equation (8.2): ∙ ¸ Z −1 ¡ ¢  = ( − 1) −1 − −1  (−1 ) 0 £¡ ¢ ¤ +  − −1  (−1 ) ∙Z −1 ¸ +   (−1 )  0

(8.8)

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CHAPTER 8. ILLUSTRATING DIFFERENTIAL GROWTH

The height of the TIM curve, at any value of , referred to here as the ‘intensity’ of mobility associated with the ‘incidence’, , is thus made up of three components, each contained within square brackets. The …rst term is ( − 1) multiplied by the sum, up to −1 =  −1 (), of the differences between log-income and mean log-income in period  − 1 (or the sum of the logarithms of relative income,   ). The second term is  multiplied by the overall growth rate of (geometric mean) income: this term has a linear pro…le. The third term is the sum of the stochastic terms. For values of −1  −1 (incomes below the geometric mean), the slope of  is positive. A turning point occurs for −1 = −1 , after which the slope is negative, since −1  −1 and   1.  Consider the component of the TIM curve,  , say, that re‡ects only

the systematic component of relative income changes, the regression towards the mean. Then:  

Z = ( − 1)

0

−1

¡ ¢ −1 − −1  (−1 ) 

(8.9)

Furthermore, let 1 () denote the …rst moment distribution function of logincome, the proportion of total log-income obtained by those with log-income below . Hence a graph of 1 () plotted against  () gives the Lorenz curve of log-income, with 1 () ≤  (). Then:   = (1 − ) −1 { (−1 ) − 1 (−1 )} 

(8.10)

Given that  (0) = 1 (0) and  (∞) = 1 (∞), this component of the TIM curve starts and ends at zero. Differentiating: µ ¶   1 (−1 ) = (1 − ) −1 1 −   (−1 )  (−1 )

(8.11)

 The slope of  therefore depends on the degree of regression, 1 − , and

the slope of the Lorenz curve of income in  − 1 at the corresponding value of  =  ( ). Up to the arithmetic mean of log-income, the slope of the

8.3. THE TIM CURVE AND GALTONIAN REGRESSION

187

Lorenz curve, 1  , is less than 1, and above the mean the slope is greater than 1. The curvature, re‡ecting the ‘inequality’ of income growth rates, is given by:

 2 

 (−1 )2

= − (1 − ) −1

2 1 ( −1 )   (−1 )2

(8.12)

More regression towards the mean, resulting from a lower value of , means that the pro…le is concave and deviates further from a straight line. It also lies everywhere above the pro…le obtained from a higher  (re‡ecting less regression to the geometric mean). The maximum height of this component of the TIM curve is obtained by setting (8.11) equal to zero, and recognising 1 (−1 ) , equals the well-known property of a Lorenz curve that its slope,   ( −1 ) 7 1 at the point on the curve corresponding to the mean, −1 . This height is thus equal to:

© ¡ ¢ ¡ ¢ª (1 − ) −1  −1 − 1 −1 

(8.13)

The term in curly brackets is positive, given that the Lorenz curve lies below the diagonal of equality, and hence low  is associated with a higher maximum height of the TIM curve. The term in curly brackets is the maximum vertical distance between the Lorenz curve of log-income and the diagonal of equality. The slope of a ray from the origin to a point on the  component of the TIM curve is:

¶ µ 1 ( −1 )  (1 − ) −1 1 −  ( −1 )

(8.14)

and this of course is always positive. This slope depends on the extent of regression towards the mean, and on the slope of a ray from the origin to the corresponding point on the Lorenz curve of log-income in  − 1. The shape of the TIM curve therefore re‡ects the nature of the Galtonian process of income change. This speci…cation also indicates that care needs to be taken in using the term ‘equalising’ when referring to the mobility process, since a substantial degree of regression (  1) can nevertheless be 7 The

tangent to the Lorenz curve corresponding to −1 is parallel to the 45 degree line of equality.

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associated with an increasing variance of logarithms of income from  − 1 to , depending on the extent of random variations, re‡ected in the variance,  2 , of the  s. Where 2 is the variance of log-income at , it is seen from (8.7) that:  2 =  2 2−1 + 2 

(8.15) ¡ ¢ Hence, inequality in periods  − 1 and  are equal only if  2 = 2 1 −  2 . It is possible to have 2−1   2 and a combination of  and  2 such that the sum of incomes over the two periods is more equal than in both of the individual years. Alternatively, annual inequality can increase and ‘longer period’ inequality can lie between the values for the two separate years. Hence a concave TIM curve, re‡ecting substantial regression – with systematically higher income growth rates for lower-incomes – is not unambiguously associated with a reduction in annual inequality, depending on the extent of random variations (which are re‡ected in a less-smooth TIM curve). Longerperiod inequality is less than the highest annual value, but is not necessarily lower than in all years.

8.4

TIM Curves for New Zealand

This section illustrates the TIM curves described in section 8.2, using data from a 2% random sample of individual New Zealand Inland Revenue personal income taxpayers. Using data for 1998, 2002, 2006 and 2010, three separate panels were obtained for 1998 to 2002, 2002 to 2006 and 2006 to 2010, each …ve-year panel containing incomes for both years for the same taxpayers. To avoid the exercise being contaminated by taxpayers with very low incomes, such as small part-time earnings of children, or small capital incomes of non-earners, individuals with annual incomes less than $1,000 were omitted from the sample. This yielded usable samples of  = 29,405, 31,355 and 32,970 individuals respectively for the three …ve-year panels. In each case

8.4. TIM CURVES FOR NEW ZEALAND

189

individuals were ranked by their initial year incomes, with all of the diagrams below showing cumulative percentiles of the income distribution, , in the relevant initial year (1998, 2002, or 2006) on the horizontal axis. Online Appendix B provides further details on the New Zealand income mobility data. Figure 8.4 shows the three TIM curves. Growth rates shown on the vertical axis are measured over the …ve-year period. The right-hand end of the TIM curve represents the average growth rate over the …ve years across all  individuals. This was similar, at around 15%, for the periods 1998 to 2002 and 2006 to 2010, but was around 20%, over the period 2002 to 2006. All three curves rise steeply at the lowest income percentiles and ‡atten out at higher percentiles, yielding relatively concave pro…les and suggesting greater upward mobility especially among the lower percentiles. These curves combine average growth (across all individuals) and relative growth, but the latter can most easily be seen by normalising each curve using the sample average growth rate. Figure 8.5 shows the normalised TIM curves, which end at a normalised cumulative growth rate of one. These allow the curvature of each pro…le to be more easily compared. The 2002 to 2006 normalised TIM reveals unambiguously lower relative mobility than the other two curves, for all . There appears to be a clear ranking according to the extent of relative mobility, with 1998 to 2002 exceeding that of 2006 to 2010 and the latter exceeding 2002 to 2006 which displays less curvature, implying more equality in income growth, at any selected percentile. Section 8.4.1 examines how far these conclusions hold statistically when allowing for sampling variability. Figure 8.5 has been constructed to illustrate the extent of interpersonal inequality of mobility for the sample as a whole, by comparing the concavity of the three normalised TIM curves to the straight line representing equality of mobility. However, if interest is focused on the inequality of mobility for a particular income group, such as the poorest half of the sample, so that

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CHAPTER 8. ILLUSTRATING DIFFERENTIAL GROWTH

Figure 8.4: Three TIM Curves for New Zealand

Figure 8.5: Normalised TIM Curves for New Zealand

8.4. TIM CURVES FOR NEW ZEALAND

191

 = 05, then each TIM curve can be re-normalised using the average income growth rate for this group. Identifying the underlying in‡uences that might explain the different mobility patterns observed across the three periods in Figure 8.5 is beyond the scope of the present chapter. However it is noteworthy that the period revealing least relative mobility, 2002 to 2006, is also the period with higher average income growth, as seen in Figure 8.4, and without major macroeconomic shocks. The other two periods experienced signi…cant shocks, such as the short New Zealand downturn in 1998 and 1999 and the major global …nancial crisis in 2008 to 2010. Such aggregate shocks may be expected to be associated with greater income volatility at the individual (or …rm) level, and hence contribute to greater measured individual income mobility. Given the prominence of the agricultural sector in the New Zealand economy, the country is also especially sensitive to weather-related impacts on incomes in agriculture and downstream industries. As New Zealand Treasury (2015, p. 20) shows, drought conditions were experienced during 1998, 2008 and 2010, but not in the intervening years. These patterns may have contributed to the greater relative mobility of incomes over 1998-2002 and 2006-2010 but less so in 2002-2006.

8.4.1

Sampling Variability

When comparing Lorenz, concentration or TIP curves for different periods or population groups, it is not usual to examine the statistical properties of these simple graphical devices. Indeed, their attractiveness lies in the fact that they can reveal important features ‘at a glance’ and are very easy to produce. Nevertheless, for present purposes it is useful to know the extent to which such diagrammatic comparisons, based on sample data, are likely to allow robust conclusions regarding mobility characteristics. This section therefore considers the sample variation around each of the TIM curves reported above

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to assess how far the position and shape of each TIM curve may be subject to sampling error. With estimates for each TIM in Figure 8.5 based on around 30,000 individuals, drawn randomly from approximately three million New Zealand income taxpayers, sampling errors can be expected to be small. For each normalised TIM curve, con…dence intervals around the sample TIM curve can be computed using bootstrap methods.8 This generates standard errors and error bands associated with, for example, the 5 and 95 percentile of the distribution of bootstrapped estimates.9 The bootstrap method proceeded as follows. Using the initial income-ranked distribution of  individuals,  random draws were obtained to yield one complete bootstrap replicate of the distribution of individuals with their associated income growth rates. The properties (initial income levels, growth rates, TIM value) of the individual at each of twenty ventiles (0.05, 0.10, 0.15 percentiles, and so on) were obtained for this replicated sample. This process is repeated 100 times with the standard deviation of the generated sampling distribution (the estimated standard errors) and 5 and 95 percentiles at the twenty ventiles being recorded after each additional ten replications. This revealed how far 8 Jenkins and Van Kerm (2016) mainly focus on con…dence intervals around estimated differences between any two periods’ IGPs or CIGPs; see their Figure 2. This re‡ects their primary concern over the robustness of welfare dominance conclusions. For the TIM curves presented here, interest is as much related to the robustness of the shape of the curve (re‡ecting inequality of mobility across the income distribution) as to differences from TIMs for other periods. Bootstrapping is therefore applied at various percentiles for each TIM rather than for inter-period TIM differences, though clearly the latter could also readily be tested. 9 The bootstrapped 5 and 95 percentile band, rather than taking plus or minus two standard errors around the TIM, are preferred here because the distribution of the estimates appears not to be Normal. That is, the 5 and 95 percentiles band is narrower, and slightly asymmetric compared to that obtained using two standard errors, due to a more concentrated, slightly skewed distribution. Also, unlike Jenkins and Van Kerm (2016) who were concerned about sample dependence between their different period IGPs, the independent random sampling of around 1 per cent of New Zealand income taxpayers for each of the sets of samples used here, the probabilities of individuals being repeatedly selected in 1998, 2002 and 2006 are low.

8.4. TIM CURVES FOR NEW ZEALAND

193

the estimates are sensitive to extending the number of replications. It was found that the standard errors and relevant percentiles stabilised well before 100 replications. Figure 8.6 shows the outcome of this process for the 2002-06 and 2006-10 normalised TIMs with their associated 5 and 95 percentile error bands obtained at each ventile.10 Two features are immediately clear from Figure 8.6. Firstly, con…dence intervals for the two TIMs shown are small, such that they do not overlap for any ventile up to around the 80 percentile. Secondly, the error bands generally widen at higher ventiles. This aspect, which arises from the cumulative nature of the TIM with increasing ventiles, is less prominent when con…dence intervals are considered as ratios of the TIM values at each ventile. Nevertheless, some widening remains evident as  increases. For example, based on the 2006-2010 normalised TIM, at  = 005, the standard error is 1.7 per cent of the TIM value. This generally rises with , reaching 2.5 per cent at  = 095. This result is also observed for 2002-2006.

8.4.2

Comparisons with Cumulative Income Growth Pro…les

As mentioned in Section 8.1, the TIM curve is closely related to the CIGP proposed by Jenkins and Van Kerm (2016): the TIM is obtained by dividing cumulative growth rates at each percentile, , by the sample size, , rather than by . It was suggested that this conveniently allowed the inequality of mobility among those below  to be illustrated alongside incidence and intensity properties. Differences between the TIM curve in Figure 8.4 and a CIGP can be seen in Figure 8.7, using data for the 2006 to 2010 panel of taxpayers. In this case, because of the large sample size for New Zealand, compared with 10 Bands

based on plus or minus two standard deviations are slightly larger and, of course, symmetrical around the sample TIM.

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Figure 8.6: TIM Curves and Sampling Variability Jenkins and Van Kerm’s UK data, the CIGP is much smoother (see Jenkins and Van Kerm, 2016, Figure 1), and nicely illustrates the (mostly) declining cumulative growth rates as higher initial income individuals are added to the pro…le; that is, as  increases. However, the CIGP also illustrates that if some of the lowest-income individuals have zero or low income growth, as in this case, the CIGP starts from the origin and rises rapidly before declining as  increases. This makes judgements regarding the inequality of mobility among those below any chosen  more difficult, especially at low percentiles, captured in this case by the shape of the pro…le, relative to the -axis, from the origin to the  percentile of interest. Inequality of mobility is more easily assessed using the TIM curve, from the concavity of the curve relative to a straight line from the origin to the point on the curve at the  percentile of interest. Nevertheless, both curves provide useful illustrations of the di-

8.5. CONCLUSIONS

195

Figure 8.7: TIM Curve and CIGP 2006-2010 versity of growth across the initial income distribution and the strong overall ‘regression to the mean’ characteristic.

8.5

Conclusions

Over two decades ago, Jenkins and Lambert (1997) introduced new insights into the poverty measurement literature by demonstrating that the incidence, intensity and inequality of poverty could be illustrated by their ‘Three Is of Poverty’ (TIP) curve. These aspects are also captured, in speci…c ways, by various summary measures of poverty. This chapter has suggested that, for mobility concepts based on income growth and re-ranking, the same three important dimensions can be translated to the context of mobility – incidence, intensity and inequality – and are easily identi…able using new illustrative devices proposed here. The advantage is that, like the Lorenz curve in the case of static inequality, they are simple to produce, provide convenient compar-

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isons of the different dimensions, and can be suggestive of further analysis. Speci…c summary measures (of inequality, poverty, mobility and so on) are inevitably needed to supplement the diagrams. But the measures necessarily involve a loss of information: for example, a wide range of Lorenz curves are consistent with the same Gini inequality measure. The simplicity, the immediate visibility of essential characteristics, the ability to deal with the detail revealed over the whole range of incomes, and the need to communicate such important concepts to a non-technical audience, explain the popularity of diagrammatic methods. For income mobility measured as relative income growth, based on an analogue of the TIP curve, this chapter has proposed that a ‘Three Is of Mobility’, or TIM, curve can provide a useful means of combining and illustrating these three concepts within a single diagram. This plots the cumulative proportion of the population (from lowest to highest values of initial income) against the cumulative change in log-incomes per capita over a given period. Illustrations were examined based on three panels of New Zealand incomes from 1998 to 2010. These showed that income growth rates within the lower part of the income distribution were quite substantially higher than those observed higher up the income distribution, re‡ecting in part a relatively high degree of regression towards the mean, as also found in Chapter 7.11 These TIM curves provide a convenient counterpoint to evidence from crosssectional distributions over various periods, based on household data from the late 1980s and reported by Perry (2017; chapter D). That suggested similar growth rates across income deciles, or lower (higher) growth for the lowest (highest) deciles: this is a quite different cross-sectional result from the longitudinal evidence here. While users of TIM curves, just like users of Lorenz curves, will typically 11 Van

Kerm (2009) and Jenkins and Van Kerm (2011, 2016) report similar regression to the mean patterns in their income growth pro…les for the UK and a selection of other European countries.

APPENDIX

197

not wish to undertake the cumbersome task of constructing con…dence intervals, the present chapter has shown that 95 per cent con…dence intervals around the TIM curve, for the kind of samples used here, are narrow. Hence qualitative comparisons of mobility using such curves, for different periods or population groups, can be made with reasonable con…dence.

Appendix: Alternative TIM and Income Share Curves This appendix provides additional information on the New Zealand income mobility data, and discusses an alternative TIM curve based on changes in the level of income, , as opposed to income growth rates,  ln(). It then considers how the Fields (2000) notion of ‘income share mobility’ may be illustrated. The empirical analyses throughout the chapter are based on data from a 2 per cent random sample of individual New Zealand Inland Revenue personal income taxpayers, provided by the New Zealand Inland Revenue. It is not publicly available. The data cover three longitudinal panels for the same 29,405 taxpayers over the …ve-year period 1998 to 2002 (4 years of income growth), 31,355 taxpayers for 2002-06 and 32,970 taxpayers for 2006-10. Each dataset was initially speci…ed to eliminate taxpayers with less than $1,000 of income in the initial or terminal years. The data for the 2006 to 2010 panel are illustrated in Figure 8.8, which shows the 2006 taxable income level of the individual at each ventile point (0.05, 0.10, 0.15, and so on) and the average income growth rate of individuals within each ventile group. This reveals, for example that income in the lowest ventile is around $6,500 and rises steadily to around $71,000 at the 18 ventile (the 90 percentile), thereafter rising more rapidly to around $175,000 in the top ventile (the value shown is income at the 99 percentile). The regression to the mean feature of the data can be seen from the

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Figure 8.8: Initial Incomes and Income Growth Rates by Ventile: 2006-10 inverse pattern observed for income growth rates over 2006-10 in Figure 8.8, which generally decline at higher initial income levels. These income levels and growth rates have correlation, and rank correlation, coefficients of −052 and −097 respectively. The income growth rate is especially high in the lowest ventile at around 1.5 (or 150 per cent over the …ve years). However, this is unusual, with most ventile average growth rates ranging from around 02 to −02 (a 20 per cent change over …ve years). In total, nine ventiles display negative average growth rates over the period, re‡ecting in part the impact of the global recession in 2008-10, such that many taxpayers’ 2010 income levels remained below their 2006 counterparts. The phenomenon of negative income growth for the highest ventiles is also observed, however, in the other two panels, 1998-2002 and 2002-06, again capturing systematic regression to the mean aspects, as observed for the three TIM curves shown in the above chapter.

APPENDIX

199

TIMs using Income Changes,  Jenkins and Van Kerm (2016) examined Cumulative Income Growth Pro…les (CIGPs), for income changes measured both as proportional income growth, , where  = ln , and in real income units,  (in January 2008 British pounds). The TIM and normalised TIM curves presented in the chapter are based on , but may be based on changes in income units. Figure 8.9 shows two normalised TIM curves obtained using the 2006-2010 income data: the lower panel is the equivalent of normalised TIM curves in the chapter with  replacing . The upper panel has been normalised by subtracting, rather than dividing by, average income change, . This helps to clarify the concave and convex ranges within the curve, which are less evident in the lower panel. Hence the upper curve ends at zero where the cumulative change per capita equals . The curve also starts at zero. However this is dataspeci…c since the lowest-income ranked individual in this case experienced no income change, hence  = 0. The lower panel of Figure 8.9 reveals that this TIM version is much closer to linear than the normalised TIM based on  between around the 10 to the 95 percentiles. This suggests that much of the observed concavity, indicating highly equalising mobility across the whole sample, is substantially due to similar absolute changes in income, , across much of the income distribution, translating into much higher income growth, , at lower income levels compared with higher income levels, except for the very highest and lowest incomes. The top panel of Figure 8.9 reveals this pattern to be less true when the details of income changes (in $s) can be seen. Nevertheless, the pro…le shows that, except at the lowest and highest incomes, in general income change per capita only varies by around $800 to $1,300 (relative to the average change of approximately $6,500) over 2006-10.12 As with the 12 The

values shown are presented in nominal, rather than real, dollar terms, since these data are not being compared with other years.

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CHAPTER 8. ILLUSTRATING DIFFERENTIAL GROWTH

Figure 8.9: Normalised TIM Curves Based on Income Change (Instead of Proportional Income Change): 2006-2010

APPENDIX

201

Figure 8.10: Changes in Cumulative Income Shares TIM based on , the extent of equalising mobility is strongly in‡uenced by income changes experienced by those on initially lowest (10 per cent), and highest (5 per cent), incomes.

Pro…les of Income Share Changes A mobility concept highlighted by Fields (2000, p. 9) was ‘income share movement’. He suggested that, ‘to the extent that people are relativists in their thinking, what they are much more likely to care about is their income as it compares with that of others. If your income rises by 50 percent but everyone else’s rises by 100 percent, you may feel that you have lost ground. Share-movement measures would say that you have experienced downward income mobility, precisely because your share of the total has fallen’. Though Fields does not suggest a diagrammatic representation of share movement, changes in cumulative income shares over any two periods can be illustrated using Lorenz and concentration curves for incomes in two periods

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ranked by …rst period income. Such curves are more usually presented for comparisons of two income cross-sections, though they can be constructed similarly from longitudinal data. Figure 8.10 shows the equivalent of the difference between such longitudinal-based Lorenz and concentration curves, but using the cumulative change in income share on the vertical axis and cumulative percentiles on the horizontal axis. As with the normalised TIM curves, these pro…les indicate a noticeably greater concavity for the 1998-2002 period than for the two later periods. All three pro…les suggest that the income shares of the poorest 20 per cent of the sample experienced an income share gain, on average, of around 0.05 (5 per cent) over each …ve-year period. The lowest half of the sample experienced around a 0.06 to 0.08 share increase, depending on the time period. Clearly, above-median individuals experienced the same decrease in income share. Since the three pro…les refer to different samples of individuals, and hence are conditional on different initial income distributions, these results cannot simply be aggregated to identify longitudinal income changes over 1998-2010. Nevertheless, they appear to suggest quite signi…cant income share increases over an extended period for individuals initially below median income.

Chapter 9 Summary Measures of Equalising Mobility The value of diagrams to summarise income distribution characteristics is exempli…ed by the famous Lorenz curve, which has become a standard device to illustrate the nature of cross-sectional income distributions. With individual observations arranged in ascending order, the Lorenz curve plots (within a box of unit height and base) the cumulative proportion of total income against the corresponding cumulative proportion of individuals. This provides much more information ‘at a glance’, about relative income inequality, than either the density function or the distribution function alone, and can quickly allow qualitative comparisons between different periods or population groups. It gives rise to the famous Gini inequality summary measure, in terms of the area contained between the Lorenz curve and the diagonal line of equality, expressed as a proportion of the area represented by maximum inequality. A challenge arises in the context of income mobility, where the same individuals are observed in, say, two different years and where the ‘basic data’ are in the form of a joint distribution. Where it is required to illustrate the main characteristic of mobility in a simple diagram, examined in detail in Chapter 8, it is based on a convenient curve that re‡ects several important features of differences in income growth rates, conditional on initial income. With 203

204CHAPTER 9. SUMMARY MEASURES OF EQUALISING MOBILITY individuals ranked in ascending order of initial income, the curve plots the cumulative proportional income change per capita (not, as in other growth curves, per head of the cumulated sub-group), against the corresponding proportion of individuals.1 This diagram enables three characteristics of mobility – incidence, intensity and inequality – to be clearly illustrated: it is referred to as a ‘Three Is of Mobility’, or TIM, curve, following the terminology adopted by Jenkins and Lambert (1997) in the context of cross-sectional poverty. The ‘end value’ of the curve is the average proportional growth rate, and dividing all values by this gives a ‘normalised’ TIM curve, denoted nTIM. If all initial incomes are subject to the same proportional growth rate the TIM curve is a straight line from the origin, at a slope given by the average growth rate. If there is a systematic tendency for mobility to be equalising, over the whole range of incomes, the TIM curve is concave. Comparisons among different periods or population groups can easily be made using nTIM curves. If one (normalised) TIM curve lies above a second curve, it can be said that the …rst displays unequivocally more equalising mobility, in terms of relative income changes. However, it may be desired to provide a quantitative measure of the extent to which such mobility differs between two curves. The value of a scalar summary measure increases in situations where nTIM curves intersect, such that one curve displays more equalising mobility than the other over only a range of the distribution. In the context of inequality comparisons using Lorenz curves, a similar need for a scalar summary measure arises where Lorenz curves intersect or where quantitative comparisons are needed to supplement qualitative comparisons between distributions. The present chapter is concerned with the question of whether an overall summary measure of the equalising extent of mobility can be de…ned, as with 1 For

example, Van Kerm (2009) and Jenkins and Van Kerm (2016) introduce an ‘income growth pro…les’ (IGP), and a cumulative version (CIGP) in which income growth is calculated per head of the cumulated group.

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205

the Gini inequality measure and the Lorenz curve in the cross-sectional case, in terms of the area contained by the nTIM curve and the straight line of equal proportional growth? This requires a statement of what is meant by ‘equalising mobility’ as a benchmark against which to compare the nTIM curve. Two concepts are considered. One is in terms of equalising crosssectional incomes in the second period: in this context the re-ranking of individuals is considered to ‘frustrate’ redistribution, since only a compression of incomes is required (with all individuals moving to the mean). The second concept is in terms of inequality based on a longer accounting period, and here a ‘maximum re-ranking’ standard is relevant, in which individuals ‘swap’ positions and incomes in the second period, whereby the richest becomes the poorest, and so on. Section 9.1 proposes summary measures of equalising and disequalising mobility based on the TIM curve, in terms of only differential income growth and second-period inequality. Section 9.2 considers a longer accounting period and introduces the role of rank-order changes. Illustrations using data from New Zealand’s Inland Revenue department are provided in Section 9.3.

9.1

A Summary Measure and the TIM Curve

In the case of the Lorenz curve, the de…nitions of complete equality and inequality (whereby only one person has a positive income while all others are zero) are straightforward to de…ne and envisage. In the former case, the Lorenz curve corresponds to the diagonal line of equality, and in the latter case it corresponds (for a large enough population) to the base and right hand side of the unit-square box. Any actual curve must lie between these two extremes, and the arrangement of incomes in ascending order ensures that the Lorenz curve is convex. In the context of mobility as differential growth, attempting to de…ne the extremes is not so straightforward. However, one extreme case, that of a relative inequality-preserving mobility process, is

206CHAPTER 9. SUMMARY MEASURES OF EQUALISING MOBILITY simple. It arises when the normalised TIM curve is a straight line. Consider one extreme form of inequality-reducing mobility, where income changes are either zero or positive. If only the poorest person has an income increase, while all other incomes remain unchanged, the normalised TIM curve is simply a horizontal line, after following the vertical axis up to 1. But this is an arbitrary case. Another possibility is the set of proportional income changes which produce equal incomes in the second period, equal to the actual average in that period. In terms of incomes, 2 and 1 , for person  = 1   in periods 2 and 1, for growth rates,  , and second-period arithmetic mean income, ¯2 , this requires 2 = 1 (1 +  ) = ¯2 , or (for strictly positive initial incomes):2

µ

 =

¯2 1

¶ −1

(9.1)

However, these  s produce a TIM curve having an average growth rate that differs from the actual average growth rate, which means that the normalised version adjusts the cumulative growth rates per capita by a different amount from that used to obtain the actual TIM curve. This essentially arises because of the basic property that the average of ratios is not the same as the ratio of averages. Suppose it is required to have equal incomes,  e, in the second period and an average growth rate equal to the actual rate, . Using logarithmic changes to approximate proportional changes: 1X (log 2 − log 1 )  =1 

=

(9.2)

The  e and associated  must now satisfy:

and:

2 If

 = log  e − log 1

(9.3)

1X  =   =1

(9.4)



changes are expressed as log-income-changes, then  = log  ¯ − log 1 . This uses the approximation log(1 +  ) =  .

9.1. A SUMMARY MEASURE AND THE TIM CURVE

207

Substituting (9.3) into (9.4) and equating with (9.2) gives: 1X log 2  =1 

log  e=

(9.5)

Furthermore, substitution into (9.3) gives the set of growth rates needed: ! à  1X log 2 − log 1 (9.6)  =  =1 Figure 9.1 illustrates two normalised TIM curves for a given initial income distribution in period 1. The solid line is the actual nTIM curve, and the higher dashed line is the hypothetical curve which would arise from the application of proportional income changes according to equation (9.6). The question is whether a useful measure of the degree of systematic equalising mobility can be obtained in this diagram. Clearly, the area  (between the nTIM curve and the diagonal line of equal proportional changes) alone does not provide an appropriate measure, since the scope for equalising differential income growth depends on the initial income distribution. The same dynamic process (in terms, say, of the relationship between the proportional growth rate and initial income) gives a smaller area for a relatively more equal distribution than for a more unequal distribution (for the same overall income growth rate). The maximum area is  + , where  is the area between the actual nTIM and the hypothetical ‘fully equalising-mobility’ nTIM. Consider a ‘degree of equalising mobility’ measure,  , de…ned as:  =

 +

(9.7)

The maximum value this can take is 1 while the minimum is 0 (when relative incomes do not change). The example shown is one in which there is systematic second-period-equalising mobility, in that the nTIM curve is everywhere above the diagonal nTIM of equal proportional changes, which is the dominant case in empirical applications.

208CHAPTER 9. SUMMARY MEASURES OF EQUALISING MOBILITY

Figure 9.1: Actual nTIM Curve and nTIM corresponding to Equal SecondPeriod Incomes and an Average Growth Rate Equal to the Actual Rate However, there may be cases where there are sufficiently large disequalising changes, along with other equalising changes in other ranges of the distribution, so that the normalised TIM curve moves below the diagonal (‘equal-proportional change’) line for an initial part of its length. It is important to recognise that the equalising TIM or nTIM curve is not uniquely de…ned, as it depends on the form of the initial distribution. A more equal distribution in period 1 produces a lower and less-concave curve. Nevertheless, the fact that a measure of equalising mobility depends on the initial distribution is not really surprising. For example, it corresponds to the fact that, in a slightly different context (though one involving movement from one distribution to another), income tax progressivity measures depend on the initial or pre-tax income distribution. The characteristics of the two curves, the nTIM and the associated ‘equalising nTIM’ curve that generates complete equality in the second period, are

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209

illustrated in Figure 9.2 for a simple hypothetical example of a population consisting of just 7 individuals. Suppose their incomes in the …rst period, arranged in ascending order, are 100, 200, 500, 1000, 1500, 3000, 4500. The corresponding incomes in the second period, following a mobility process, are 180, 280, 680, 1200, 1600, 3400, 4600. The normalised TIM curve, the solid line in the diagram, demonstrates that the process is equalising. However, the curve is still some distance from the curve associated with a fully equalising process.

Figure 9.2: Hypothetical Example of Normalised TIM Curve and Associated Extreme Equalising Mobility In general, denote the proportions of people and cumulative growth per capita by  and  respectively, for  = 1  , with 1 = 1 = 0 and  =  = 1. The areas may be found using the standard trapezoidal rule for the area, , beneath the piecewise-linear curve: 1X ( − −1 ) ( + −1 ) = 2 =2 

(9.8)

Remembering that 05 – the area below the diagonal line of equal proportional

210CHAPTER 9. SUMMARY MEASURES OF EQUALISING MOBILITY changes – must be subtracted to obtain the relevant areas between the nTIM curve and the diagonal, the area  for this hypothetical case is found to be 0.2076 and the area  +  is 1.576. Hence the degree of equalising mobility,  = 0132. Although systematic inequality-increasing mobility has not been observed in practice, the concept of extreme inequality-increasing mobility is somewhat different. The actual nTIM curve arising from a systematically disequalising process would of course be consistently below the diagonal equalproportional-changes line. One substantial inequality-increasing case arises if only the richest person has a positive income increase, and all other incomes remain constant. The nTIM curve follows the bold line around the base and right-hand side of the unit box. But of course this is not the extreme case, as it could be taken further by allowing for negative income changes. Thus, an extreme inequality-increasing case could be de…ned as a mobility pattern that reduces all but the maximum income to zero, and transfers all income to the initially richest person. In this case, all  = −1, except for the richest person. The corresponding nTIM curve is then uniquely de…ned. It is a downward sloping ‘45 degree’ straight line from the origin until reaching  = 1, when it becomes vertical with a height of +1. This concept is shown in Figure 9.3. The area of the triangle indicated by CDE represents the area representation of the furthest distance from the line of equal proportional changes. This area is equal to 1 (half the base, of 2, multiplied by the height of 1). The area between the actual nTIM curve and the line of equal changes is the area, . Hence in this case, a measure of disequalising mobility,  , is simply given by the area , so that  = . If there are disequalising and equalising ranges so that the nTIM curve crosses the diagonal line one or more times, a measure of disequalising mobility may be obtained by taking  −  .

9.2. POSITIONAL CHANGES AND THE TIM CURVE

211

Figure 9.3: Maximum Inequality-Increasing Income Mobility

9.2

Positional Changes and the TIM Curve

In the previous section, extreme inequality-reducing mobility was considered in terms of the hypothetical pattern of changes,  , giving rise to equal incomes in the second period, and with the same average growth rate as actual incomes. With incomes arranged in ascending order, complete equality can be achieved by a compression of incomes towards the arithmetic mean income. That is, equality can be achieved without any changes in the rank-order of individuals in the income distribution. Changes in the relative positions, or re-ranking, actually ‘frustrate’ an otherwise systematic equalising mobility process. Again there is a corresponding analogy with tax progressivity analyses. A progressive tax structure may, for various reasons, introduce some rank-order changes: for example, it can arise if the tax system allows individual-speci…c changes to the tax function, depending on the non-income characteristics of individuals (such as the number of dependents and associ-

212CHAPTER 9. SUMMARY MEASURES OF EQUALISING MOBILITY ated tax deductions). However, it is possible to consider a process of equalising mobility in which an extreme type of re-ranking takes place along with differential income growth. If equality is viewed not in terms of second-period incomes but in terms of a longer accounting period – incomes measured over the two periods – a further type of maximum equalising mobility can be de…ned. An extreme form of longer-accounting-period equalisation can be achieved by a process involving a complete reordering (maximum re-ranking) of period 2’s incomes. Hence, the poorest in period 2 changes place with the richest, the secondpoorest person changes place with the second-richest person, and so on. This complete reversal of ranks, combined with a ‘swapping of incomes’, with the richest person simply replacing the poorest, and so on across the distribution, produces no change in annual relative income inequality in period 2. Furthermore, the combination of ‘changing places’ and ‘swapping incomes’ in period 2 does not produce complete equality, when incomes are measured over two periods, because of the differential income growth from period 1 to period 2. However, it may be regarded as providing an alternative basis with which to compare the actual normalised TIM curve: it combines maximum re-ranking with a reasonable view of a maximum equalising-growth distribution of income changes. Consider again the simple numerical example above, with just seven individuals. Figure 9.4 shows, in addition to the ‘equalising-period-2 income’ nTIM, a ‘maximum re-ranking combined with income-swapping’ curve. It is clear from the …gure that this new TIM curve represents a distribution of income growth changes that, combined with the greatest degree of re-ranking that arises from swapping period 2 income (from the top to the bottom of the distribution, and so on), produces a greater extent of equalisation of incomes. This is because in this benchmark case, longer-period incomes become significantly more equal, rather than simply equalising period 2’s incomes. This suggests that an additional ‘extreme’ measure of equalising changes can be

9.3. ILLUSTRATIVE EXAMPLES

213

Figure 9.4: Hypothetical Example with Maximum Re-ranking and Period 2 Income Swapping obtained by comparing the area underneath the actual nTIM with that below this new hypothetical nTIM curve. This is illustrated in the following section using a practical example.

9.3

Illustrative Examples

This section illustrates the nTIM curves and associated measures, using Inland Revenue Department (IRD) data from a 2 per cent random sample of New Zealand personal income taxpayers. A con…dential dataset was obtained from IRD, giving incomes of a constant group of individuals in 2006 and 2010. To avoid the exercise being contaminated by taxpayers with very

214CHAPTER 9. SUMMARY MEASURES OF EQUALISING MOBILITY low incomes, such as small part-time earnings of children, or small capital incomes of non-earners, individuals with annual incomes less than $1,000 were omitted from the sample. This yielded a usable sample of 32,970 individuals. Figure 9.5 shows, as a solid line, the normalised TIM curve for income changes between 2006 and 2010. This lies above the straight line of equal proportional changes over the whole of its length, demonstrating a substantial amount of equalising mobility over the period. The dashed line in Figure 9.5 is the normalised TIM curve corresponding to the hypothetical case of equalisation of incomes in the year 2010. This completely smooth curve arises from a compression or squeezing of the distribution towards its mean, and avoids any re-ranking. Although the actual nTIM curve looks quite smooth, this is an artifact of the scale of the diagram: when magni…ed it is a jagged edge, re‡ecting the existence of many individuals experiencing relative income reductions in the lower-income ranges and relative income increases in the higher ranges despite the systematic equalising tendency over the whole range. The combined effect is to produce many rank-order changes and to maintain approximate stability in the inequality of incomes in 2006 and 2010: the variance of logarithms in fact increases very slightly from 0.711 to 0.738. These values necessarily apply to the constant large sample of individuals who were present in both years. In the broader cross-sectional distributions, there are exits and entrants into the population of taxpayers, which also combine to stabilise the distribution of income in any year. Using equation (9.7), the extent of equalising mobility,  , as de…ned above, is found to be 0.325. However, care must be taken in interpreting this value, as discussed further below. For example, it does not mean that, over the …ve-year period, differential income growth has moved the initial (2006) income distribution a third of the way towards equality. Figure 9.6 adds to Figure 9.5 the hypothetical nTIM that arises from the maximum re-ranking, that is ‘reverse ranking’ with income swapping, of period 2 (2010) incomes. This diagram also indicates a number of relevant

9.3. ILLUSTRATIVE EXAMPLES

215

Figure 9.5: Normalised and Equalising TIM Curves: NZ Income Taxpayers 2006 to 2010 areas, re‡ecting the distance between curves using an area measure of distance. The …gure also replaces percentiles on the horizontal axis with the equivalent population proportions, to facilitate measurement comparisons. The relevant areas are given in Table 9.1. Figure 9.6 and the table show, …rstly, that area , under the line of equal proportional changes, is equal to 0.5, being half of the box with sides of unit length. This provides a convenient benchmark against which other areas may be compared. Secondly, it can be seen in Table 9.1 that the area, , beneath the nTIM curve is 1.04. Hence, the area between the nTIM and the line of equal proportional changes, labelled  − , is equal to 0.54. Following the same approach, the area between the ‘income swapping’ nTIM and the line of equal proportional changes, area  − , is equal to 1.66. Furthermore, the area between the ‘equalising nTIM’ and the line of equal proportional changes, area  − , is equal to 3.35.

216CHAPTER 9. SUMMARY MEASURES OF EQUALISING MOBILITY

Figure 9.6: Alternative New Zealand nTIM Curves

Table 9.1: Areas Under nTIM Curves in Figure 9.6 Area under curve:

Area label in Value Value Figure 9.6 (net of EPC) Equal prop. changes (EPC)  0.50 nTIM  1.04 0.54 Equalising nTIM  2.16 1.66 Reverse-rank (RR) nTIM  3.85 3.35 Ratios: (i) (ii)

nTIM RR nTIM – EPC nTIM Equalising nTIM – EPC

− −

0.54 3.35

0.161

− −

0.54 1.66

0.325

9.4. CONCLUSIONS

217

Table 9.1 includes two ratios, both measuring the area beneath the nTIM curve (net of the area ) relative to (i) the (net) area beneath the ‘income swapping/reverse rank’ nTIM curve; and (ii) the area beneath the ‘equalising (second period income) nTIM’ curve. It can be seen that ratio (i) is equal to 0.161 (16.1%), while ratio (ii) is equal to 0.325 (32.5%). These values can be interpreted as follows. The extent to which observed income mobility between 2006 and 2010 is equalising, measured by the area  −  in Figure 9.6, represents about 16 per cent of the maximum that would be achieved if actual 2010 incomes had instead be reallocated such that there was a complete re-ranking of all individuals in the sample. It is also the case that the observed mobility represents around 32 per cent of that which would achieve complete income equality in 2010 purely via income compression towards the mean (that is, with no rank changes). As noted earlier, this is the value of the measure  in equation (9.7). However, since the actual nTIM curve captures a mixture of ‘pure’ income compression between 2006 and 2010, and a substantial degree of re-ranking of individuals’ incomes, comparing the two areas ( −  and  − ) in ratio (ii) is not straightforward. Ratio (i), on the other hand, provides a readily interpretable comparison of the progressivity of actual mobility with the extreme progressive case of income swapping (a given total) to achieve a complete re-ranking of individuals’ incomes, and ‘extreme equalising’ changes when viewed from a longer-accounting-period perspective.

9.4

Conclusions

This chapter has extended the ‘Three Is of Mobility (TIM) Curve’ framework, discussed in Chapter 8. The TIM curve, obtained by plotting the cumulative growth per capita against the corresponding proportion of people (ranked in ascending-income order), provides a convenient illustration of systematic equalising tendencies in differential income growth. While such

218CHAPTER 9. SUMMARY MEASURES OF EQUALISING MOBILITY visual qualitative comparisons (between time periods or geographical regions or demographic groups) are useful, in some applications it is desirable to have quantitative summary measures of equalising mobility between two periods. The present chapter has shown that measures can be based, as in the famous Lorenz curve used to depict cross-sectional inequality, on areas within the diagram. These are area measures of the ‘distance’ from the TIM curve to two alternative curves which depict, in different senses, hypothetical extreme equalising mobility cases. The …rst case involves the equalisation of incomes in the second period, such that all second-period incomes are equal to the actual average, and the average growth rate is equal to the actual average growth over the relevant period. This involves a compression of incomes and no re-ranking. If secondperiod equality is treated as ‘extreme equalisation’, then any re-ranking of incomes (generated by non-systematic changes) can be regarded as ‘frustrating’ redistribution. The second concept involves a different concept of maximum redistribution, de…ned in terms of the inequality of incomes measured over the two periods combined. This hypothetical extreme involves a combination of differential income growth with maximum re-ranking, whereby second-period incomes are ‘swapped’: the richest person becomes the poorest, and so on. In this case, maximum re-ranking is viewed as a fundamental component of equalising change. The measures were illustrated using a large sample of taxpayers’ incomes in New Zealand, obtained from con…dential unit-record …les. It is suggested that these measures of equalising mobility can usefully augment the visual information provided by the TIM curve concept.

Chapter 10 Mobility as Positional Change An alternative class of mobility measures is based on the idea of mobility as positional change, rather than relative income change. Positional mobility has been widely studied using transition matrices, which capture the movement of individuals across percentiles of the income distribution between two time periods. However, as Trede (1998) suggests, ‘even if the number of classes is restricted to as few as …ve classes the transition matrix is not easily comprehended at …rst sight’. In addition, the use of a small number of classes increases the likelihood that potentially important within-class mobility is obscured.1 This chapter focuses on income re-ranking positional change. Individuals can move to higher or lower rank positions, so that the explicit treatment of the direction of change becomes important. When de…ning a re-ranking mobility index, it is therefore …rst necessary to decide whether negative reranking (dropping down the ranking) is treated symmetrically with positive (upward) movement within the ranking. A second issue concerns the choice 1 D’Agostino

and Dardanoni (2009) and Cowell and Flachaire (2016) have sought to rede…ne and clarify various rank-related mobility concepts and measures. Cowell and Flachaire (2016) propose what they term a ‘superclass’ of rank-based mobility measures, in which the evaluation of an individual’s positional ‘status’ can be separated from movements between positions (where measurement of the latter uses distance concepts). Neither study presents graphical devices to illustrate their measures.

219

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CHAPTER 10. MOBILITY AS POSITIONAL CHANGE

of whose mobility is to be included. Section 10.1 proposes a simple illustrative device for positional mobility for which there seems to be no direct equivalent in the literature.

10.1

Positional Mobility

In the following discussion, individuals are ranked in ascending order of initial incomes, 0 , so that ranks  = 1   are for individuals from the lowest to the highest income. The initial period is denoted 0, and initial ranks may be de…ned as 0 = . Consider, as in previous sections, the case where it is desired to measure the extent of mobility of a subset of individuals,  ≤ , with the lowest initial incomes, and let ∆ = 1 − 0 = 1 −  denote the change in the rank order of the person who initially has rank, . Three treatments of re-ranking are possible, all related to how negative, or downward, re-ranking is treated. Firstly, negative re-ranking could be treated symmetrically with positive re-ranking such that positional mobility is de…ned in net terms, that is, positive changes in rank net of any negative changes for those with  ≤ .2 This is referred to as ‘net re-ranking’. Secondly, negative movement in the ranking could be ignored, which simply involves setting ∆ = 0 when ∆  0. This is referred to as ‘positive re-ranking’. Thirdly, re-ranking may be measured in absolute terms in which all re-ranking is measured as a positive value. This is referred to as ‘absolute re-ranking’. As Fields and Ok (1996) and Fields (2000) stress, the appropriate choice among these three measures depends on the question of interest. For example, if interest is focused on those below the poverty line as a group, then it may be desirable to balance any upward mobility by some of those in poverty with downward mobility of others in poverty, in order to gain an indication 2 If

individual changes in rank are simply aggregated to obtain an aggregate mobility index, then a change in rank of 50 places by one individual is treated symmetrically with 50 individuals each changing one ranking place.

10.1. POSITIONAL MOBILITY

221

of the overall experience of the group. This suggests a focus on net mobility in this case. If movement per se is the mobility concept of interest, then a non-directional measure such as absolute re-ranking is more relevant. Positive re-ranking quanti…es only those who are moving up, a common metric when assessing the persistence of low income or poverty status for a sub-set of individuals or households. The three re-ranking indices for individual, , are de…ned formally as follows:  = ∆ 

(10.1)

 = ∆ |

∆ 1 0



 = |∆ | 

(10.2) (10.3)

Cumulated across the  lowest-income individuals in period 0, the corresponding aggregate re-ranking indices are given by: 



() =

 X



 X = (1 − 0 )

=1





() =

 X



 X = (1 − 0 ) for ∆ 1 0

=1



(10.5)

=1

and 

(10.4)

=1

() =

 X =1



=

 X

|1 − 0 | 

(10.6)

=1

The absolute re-ranking case may be thought of as describing the extent of overall positional change within the relevant range of the income distribution. To examine the ‘three Is’ properties similar to the TIM, but based on the indices in (10.4), (10.5) and (10.6), one approach would be to plot the value of the relevant  () index against the cumulative fraction of the population,  = . However, there are two difficulties with the indices in (10.4) to (10.6). Firstly, they are not scale independent, since they depend on the population size: more re-ranking is possible in larger populations. One solution

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would be to scale the three  () indices by . However, as shown below, a slightly different rescaling, by (2)2 , yields normalised values,  (), that lie between zero and one (or zero and two for absolute re-ranking). These may then be plotted against 0 6  6 1. Secondly, an individual’s opportunity for re-ranking is partly determined by their initial position. Those among the lowest ranks have less opportunity to move down, other things equal, than those higher up, and vice versa. It is therefore useful to consider the maximum re-ranking possible for each individual. Actual re-ranking may then be compared with these maximum values for any given . On maximum re-ranking, consider a population of  individuals, each with a different income level. In period 0, they are ranked, 0 = , for  = 1     , representing the lowest to the highest incomes. Two polar cases are the maximum and minimum degrees of mobility possible. The former is de…ned here as a complete ranking reversal, ∆ (max).3 Maximum re-ranking implies:  (max) = ∆ (max) = 1 (max) − 0 =  + 1 − 20 

(10.7)

For large , this is approximated by −20 . Where it is desired to measure the extent of re-ranking of the subset of individuals,  ≤ , with the lowest incomes, the cumulative maximum re-ranking index for the net mobility case,   (max ), is given by:   (max ) =

 X =1

3 An

 (max) =

 X ( + 1 − 20 )

(10.8)

=1

alternative argument is that the relevant comparator should be de…ned as when the change in an individual’s position in the ranking is purely random; see Jäntti and Jenkins (2015, pp. 8-9). That is, ‘maximum’ mobility involves independence from initial positions, rather than complete reversals. In that case, given 0 , maximum mobility requires an actual ordering in period 0 to be compared with a random ordering in period 1. Jäntti and Jenkins reject the use of ‘maximum’ when mobility is based on origin independence because, they suggest, “it is difficult to argue that origin independence represents ‘maximum’ mobility in the literal sense”. Trivially, the minimum degree of re-ranking involves no change in the ranks such that 1 (min) = 0 for all , hence ∆ = 0.

10.1. POSITIONAL MOBILITY Using the sum of an arithmetic progression,

223 P

=1

0 = 1 + 2+ ... +  =

( + 1)2 equation (10.8) becomes: 



 X (max ) = ( + 1 − 20 ) = ( + 1) − ( + 1) =1

= ( − )

(10.9)

Hence, for example, if  = 100, each integer,  = 1  , represents a percentile of the distribution. If interest focuses only on the poorest individual (so that  = 1), maximum net re-ranking is given by   (max 1) = (100 − 1) = 99; when  = 2,   (max 2) = 2(100 − 2) = 196; and so on. More generally, since maximum re-ranking (complete ranking reversal) involves all those below the median individual changing positions with those above the median, it follows from (10.9) that the maximum value of   (max ) as  increases is obtained for  = 2, yielding   (max 2) = (2)2 .4 This measure serves to highlight the scale dependence of both   () and   (max ): larger populations imply larger values of both indices. These could be normalised to create a form of per capita index by dividing by 2 such that, from (10.9), the index would become:   (max )2 = (1−). The maximum value would be reached at  = 05, where the index equals 025. However, to yield an index with a maximum value of 1 at  = 2, it is preferable to divide by (2)2 . That is,  (max ) = 4  (max )2 

(10.10)

where lower-case  denotes this normalised measure. Using (10.9) and remembering that  = :  (max ) = 4(1 − )

(10.11)

A similar exercise for positive re-ranking,   (max ), shows that   (max ) also reaches a maximum as  increases, of   (max 2) = 2 4, since all 4 Strictly,

for small , the median individual is  = (+1)2, and   (max ) is given by ( + 1)( − 1)4.

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individuals below 2 experience positive re-ranking in this maximum case. However, above  = 2, as more above-median individuals are included within , their re-rankings are now given by ∆ = 0; hence, the cumulative index,   (max ) remains unchanged for   2. Thus a similarly rescaled  (max ) may be de…ned analogously to (10.10) to yield a positive re-ranking index where 0 6  (max ) 6 1. Finally, for the absolute re-ranking case in (10.6),   (max ), it can be shown that, as with the other cases, this increases as  increases from  = 1 to  = 2 to reach   (max 2) = (2)2 . However, this represents a point of in‡ection rather than a maximum, since inclusion of the absolute value of above-median individuals’ re-ranking in   (max ), ensures that   (max ) continues to increase for   2, reaching   (max ) = 2 2 at  = . As a result, an absolute re-ranking index  (max ) obtained by rescaling by (2)2 lies between zero and two. Finally, to compare actual and maximum re-ranking mobility, the expressions for actual mobility in (10.4) to (10.6) can be similarly rescaled or normalised by (2)2 to obtain actual aggregate re-ranking mobility expressions,  ,  , and  , given in each case by:  = 42 

(10.12)

Thus, 0 6  ,  6 1 and 0 6  6 2. This suggests a convenient illustrative device for positional mobility, a cumulative re-ranking curve, similar to the TIM curve for relative income mobility, that plots alternative s against . This is explored in the next section using income data for three large longitudinal samples of New Zealand individual taxpayers over 1998 to 2010. First, the next subsection shows (max ) pro…les and introduces an alternative illustration based on the ratio of actual to maximum re-ranking: the re-ranking ratio, .

10.1. POSITIONAL MOBILITY

225

Figure 10.1: Maximum Re-ranking Pro…les

10.1.1

Maximum Re-ranking Pro…les

Pro…les for the three (rescaled) maximum re-ranking cases discussed above,  (max ),  (max ), and  (max ), are plotted against  =  in Figure 10.1. This shows the distinct non-linear shape of the maximum pro…les, whichever de…nition of positional mobility is adopted. As expected, the net re-ranking pro…le displays a parabolic shape which differentiation of (10.11) reveals has a slope of 4(1 − 2), that equals zero at  = 05 (the 50 percentile), thereafter declining symmetrically to a slope of −4 at  = 1. The equivalent positive re-ranking pro…le also reaches a maximum at the 50 percentile but remains constant thereafter, while the absolute re-ranking pro…le displays a sigmoid shape, reaching a local point of in‡ection where  (max 05) = 1 at the 50 percentile, but then rising at an increasing rate till  (max ) = 2 at  = 1. The maximum re-ranking indices in Figure 10.1 are invariant to population size, but they vary with the population percentile of interest, . Thus,

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CHAPTER 10. MOBILITY AS POSITIONAL CHANGE

the scope or opportunity for a given degree of actual re-ranking also varies with . A natural index of interest therefore is the ratio of actual to maximum mobility at each percentile, . This is referred to as the re-ranking ratio, , and can be calculated for net, positive and absolute re-ranking. For example, the net re-ranking case is given by:  =

   =   (max)   (max)

(10.13)

where the numerator and denominator are given respectively by (10.12) and (10.10), or by (10.4) and (10.9). This ratio can also be plotted against  to identify how the extent of mobility changes by cumulative percentile of the population relative to the maximum possible for that percentile. Recognising these differences in maximum re-ranking is important when interpreting differences in actual re-ranking for different values of . In particular, a smaller value of  at  = 01, compared to  at  = 03, for example, may be partly or entirely due the fact that individuals up to  = 01 cannot achieve the higher  observed at  = 03, even in the absence of other constraints on re-ranking mobility. The following section illustrates these re-ranking mobility measures using data on New Zealand taxpayers.

10.2

Re-ranking Pro…les for New Zealand

This section turns to an application of the positional mobility measures described in section 10.1 to the same New Zealand income data, to assess both the extent of observed positional mobility and the incidence, intensity and inequality dimensions. This is illustrated …rst by plotting the re-ranking measures  ,  and  against , analogous to the (max) pro…les in Figure 10.1. These are shown for the 2006 to 2010 period in Figure 10.2. In each case, these pro…les could contain concave, linear or convex segments, re‡ecting the degree of re-ranking being experienced as  is increased to in-

10.2. RE-RANKING PROFILES FOR NEW ZEALAND

227

Figure 10.2: Actual Re-ranking 2006 to 2010: Three Cases clude higher income individuals. A greater amount of re-ranking mobility generates pro…les that are more concave or less convex. To assess the incidence, intensity and inequality aspects of these reranking measures, Figure 10.2 can be interpreted as follows. For a given de…nition of positional mobility (net, positive or absolute re-ranking), select a value of  representing the sub-set of low income individuals of interest (the incidence dimension). The height of the pro…le on the vertical axis at this value of  represents the intensity of re-ranking for this group, namely how much re-ranking they have experienced on average. The section of the pro…le to the right of  becomes irrelevant, equivalent to the ‡at section of the TIP curve. The deviation from linearity (concave or convex) of the re-ranking pro…le, from the origin to its value at the selected , provides a measure of the extent of inequality of mobility within . That is, the actual pro…le may be compared to a straight line from the origin to the relevant value of .

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CHAPTER 10. MOBILITY AS POSITIONAL CHANGE

Figure 10.3: Actual Positive Re-ranking: Three Periods For example, in Figure 10.2 the pro…le for absolute re-ranking appears to be approximately linear over a wide range above the 10 percentile. This suggests that, at least for this sample and measure, the extent of re-ranking is relatively constant across the income distribution. As with the TIM curves in Section 11.4, changes in the incidence, intensity and inequality of positional mobility associated with different time periods can be examined by plotting relevant re-ranking pro…les for the three periods. Figure 10.3 illustrates this for the positive re-ranking measure,  , showing that the characteristics of positive re-ranking mobility across the three periods are very similar, both in terms of levels of  at each value of , and the degree of inequality of mobility (concavity) of each pro…le for any given . To the extent that the pro…les differ, there is some evidence of slightly

10.2. RE-RANKING PROFILES FOR NEW ZEALAND

229

Figure 10.4: Actual Re-ranking Ratios 2006-2010 more re-ranking mobility during the …rst period, 1998-2002, as observed for the income growth based TIM curves. Since the maximum positive reranking,  (max) = 1 for  1 05 (see Figure 10.1), the values of  in Figure 10.3 also reveal the values of the re-ranking ratio for  1 05. While some groups across the sample may experience higher re-ranking in Figure 10.2, their movements are constrained to differing degrees by the maximum re-ranking possible. The differences between the actual , and the equivalent (max) can be identi…ed by considering changes in  as  is increased. Relevant pro…les plotting  against  are shown for the three re-ranking measures in Figure 10.4, where values on the vertical axis are simply ratios of the axis values in Figures 10.2 and 10.1. This indicates that, for all three re-ranking measures in the New Zealand case, the extent of positional mobility relative to the maximum achievable is relatively high for the lowest-income individuals (low ), at around 025−03. This steadily declines as  is increased, to a minimum of approximately 02 at around the 20 percentile. Thereafter, the  rises to around the

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CHAPTER 10. MOBILITY AS POSITIONAL CHANGE

70 percentile, while the  pro…le rises to the 100 percentile. From  it may be inferred that the group experiencing absolute re-ranking that is closest to the maximum achievable is the group between approximately the 50 and 70 percentiles. For positive re-ranking, actual and maximum re-ranking are generally closest for the lower and higher population percentiles, reaching around  = 03 in both cases. The  pro…le is rather different; it continues to decline for   02, though at a somewhat slower rate than for   02. The larger ‡uctuations in the  curve, as  approaches 1, re‡ect the fact that the value of both the actual and maximum net re-ranking measures equal zero at  = 1. Hence the ratio can be quite unstable in the vicinity of  = 1, and is, of course, unde…ned at  = 1. The fact that the  and the  pro…les reach the same value for  = 1 is not coincidental. It has already been shown that   (max) = 2 2, while   (max) = 2 4, at  = 1; that is,   (max) = 2  (max) and hence  (max) = 2 (max). This same relationship holds for the actual measures:  = 2 . This can be seen by noting that, when  = 1:   =

X 1

|1 − 0 | 

(10.14)

However, at  = 1 the sum of positive ranking movements must equal the sum of negative ranking movements, so that:   = 2

X 1

¯ ¯ (1 − 0 ) ¯∆ 1 0  

(10.15)

The term after the summation in (10.15) is simply the positive re-ranking measure,   . Hence both  and  are equal at  = 1. Considering the three pro…les in Figure 10.4, the measure of net movement,  , indicates a persistent downward trend as  approaches 1. This suggests that low-income individuals generally experienced more movement in their income rank (relative to the maximum achievable) over this

10.2. RE-RANKING PROFILES FOR NEW ZEALAND

231

period compared with those on higher incomes. This seems likely to be capturing a re-ranking analogue of the ‘regression to the mean’ in income levels observed above. Figure 10.5 shows pro…les for  for the three periods, equivalent to the three  pro…les in Figure 10.3. This reveals considerable volatility in  over the lowest 5 percentiles, perhaps not surprisingly given the numbers of individuals with low incomes in the initial year and who experience a wide range of income changes over the period.5 Much of this mobility probably re‡ects some low-income individuals, such as secondary earners, moving into employment or from part-time to full-time work, while others remain in their initial employment status. These data also include the self-employed who are known to experience greater income volatility. Above the 5 percentile, the pro…les behave similarly to the  pro…les in Figure 10.3, with generally greater re-ranking as a fraction of the maximum possible in 1998 to 2002. However, for percentiles around the 5 to the 40 , the 2006 to 2010  pro…le is more clearly below the 2002 to 2006 equivalent. This was less obvious for the  pro…les in Figure 10.3. These results therefore generally con…rm that, across most percentiles of the initial income distribution, re-ranking mobility was slightly greater in the earliest period examined, 1998 to 2002, and decreased somewhat in the two subsequent periods. They also demonstrate that, across all three periods, re-ranking is typically around 20-30% of the maximum mobility possible, conditional on an individual’s position in the initial income distribution. It also tends to be highest at both the top and at the bottom of those initial distributions. 5 For

example, in 2006, the 5 percentile income level for the 2006 to 2010 panel is only around $6,600.

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CHAPTER 10. MOBILITY AS POSITIONAL CHANGE

Figure 10.5: Positive Re-ranking Ratios: Three Periods

10.3

Conclusions

For mobility measures based on positional changes, or the extent of re-ranking of individuals over a given period, it was shown that an equivalent re-ranking mobility curve can illustrate the incidence, intensity and inequality of positional mobility in the form of re-ranking. This plots the cumulative degree of re-ranking against the cumulative proportion of the population (from lowest to highest incomes). Additionally, since for any given fraction of the population there is a different maximum possible extent of re-ranking, it is useful to consider the cumulative ratio of actual-to-maximum re-ranking against the cumulative proportion of the population. Illustrations of re-ranking were examined based on three panels of New Zealand incomes from 1998 to 2010. Evidence on the extent of re-ranking of individual incomes across a …ve year period also suggested a relatively high degree of positional mobility, compared to the maximum possible, among the lowest and highest income individuals. This highlighted how some conclu-

10.3. CONCLUSIONS

233

sions regarding the extent of re-ranking depend crucially on the re-ranking measure adopted – positive, net or absolute. For example, the highest reranking ratios are observed around the 50 to the 70 percentiles for an absolute re-ranking measure but rise steadily towards the 100 percentile when a positive re-ranking ratio is considered.

Chapter 11 Poverty Persistence In the context of cross-sectional inequality and poverty, diagrams produced from simple data manipulations have proved to be invaluable. Examples are the Lorenz curve, with its associated Gini summary measure of inequality based on areas, and the ‘Three Is of Poverty’ (TIP) curve, proposed by Jenkins and Lambert (1997), with its ability to capture easily the incidence, intensity and inequality of poverty. For incomes ranked in ascending order, the Lorenz curve plots the cumulative proportion of total income against the corresponding cumulative proportion of individuals.1 For those below a poverty line, and again for incomes ranked in ascending order, the TIP curve plots the cumulative poverty (income) gap per person against the corresponding cumulative proportion of people. Both diagrams provide valuable visual information that is not apparent from plots of either the density function or the distribution function alone, and they allow easy comparisons among distributions. Indeed, the detail revealed by the simple diagrams is concealed by standard summary measures: for example very different income distributions can be consistent with the same Gini measure. The Lorenz and TIP curves are necessarily concerned only with cross1 Income

transfers satisfying the Dalton-Pigou ‘principle of transfers’ unambiguously move the Lorenz curve towards the diagonal line of equality. Furthermore, ‘Lorenz dominance’ can be associated with welfare dominance for individualistic, additive welfare functions satisfying the principle of transfers.

235

236

CHAPTER 11. POVERTY PERSISTENCE

sectional distributions. However, the characteristics of income mobility are crucial in in‡uencing inequality, when measured over longer accounting periods, and the extent to which people remain in poverty, combined with the characteristics of movements into and out of poverty. Whether overall income growth reduces a static poverty measure depends on the precise form of the distribution of income changes. The welfare interpretation of mobility is necessarily less clear-cut compared with single-period income comparisons. On the one hand, greater relative income movements can be associated with more opportunities and less rigidity within a population, but on the other hand they can also be associated with greater risks. Jenkins and Van Kerm (2016) suggest an approach to welfare evaluation of income mobility using an Atkinson-Bourguignon social welfare function based on individuals’ utility functions that include both initial and terminal year incomes as arguments. Earlier, Creedy and Wilhelm (2002) examined changes in a social welfare function, de…ned using a multiperiod welfare metric, where individuals (with decreasing marginal utility) have a preference for a steady income stream. This allows consideration of the question of whether increased mobility, and increasing single period inequality, can be consistent with lower multi-period inequality and higher social welfare. Various statistical measures of income mobility and poverty dynamics are available, and of course extensive use of income transition matrices has been made for movements between speci…ed income classes or deciles. In recognising the value of simple illustrative devices, the aim of this chapter is to offer two new diagrams to illustrate pro-poor income mobility and poverty persistence. The …rst diagram adapts a more general income mobility curve, referred to as a ‘Three Is of Mobility’ (TIM) curve, proposed by Creedy and Gemmell (2019). It compares the relative income growth of different poverty sub-groups. The second diagram is a ‘poverty persistence curve’: this focuses on income changes which generate movements out of, and

11.0. INTRODUCTION

237

into, poverty (that is, movements across a designated poverty line which is assumed to remain constant in real terms). Each curve captures their respective aspects of poverty dynamics in ways that allow convenient comparisons between distributions, and both are easy to produce. Regarding simple diagrams, Growth Incidence Curves (CIGs) have been used by Ravallion and Chen (2003). However, their approach uses crosssectional data for two periods, rather than longitudinal information about individuals. The GIC plots the growth rate, between two periods, of each quantile or percentile of the distribution in the initial period. It can display relative growth differences, by subtracting the overall income growth, and can be used to examine whether or not income growth is said to be pro-poor. An important distinction needs to be drawn between such a curve, which is based on the growth of quantiles rather than of individuals’ incomes, and one which uses longitudinal data. Bourguignon (2011) developed a longitudinal version of the GIC which he called a ‘non-anonymous growth incidence curve’. A related approach, proposed by Jenkins and Van Kerm (2016), de…nes Income Growth Pro…les (IGPs) which, in using longitudinal data, are similar to those developed by Bourguignon (2011). The IGP involves plotting a measure of average income growth, (), for the th percentile, against , where in their case () is a conditional expectation-based measure amenable to social welfare comparisons.2 Jenkins and Van Kerm (2016) also propose a cumulative version of the IGP, called the CIGP, in which a measure of average income growth, for those with initial incomes below (), is plotted against . The TIM curve, itself motivated by the TIP curve, is a modi…cation of the CIGP curve. 2 See

also Grimm (2007) and Van Kerm (2009). Jenkins and Van Kerm (2016) examine the welfare-dominance properties of individual income growth based on a social welfare function for which individual utilities are based on incomes in both the initial and …nal periods. That is, their objective is to produce summary indices of income growth with consistent welfare foundations that are helpful for normative evaluations of alternative distributions of individual income growth.

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CHAPTER 11. POVERTY PERSISTENCE

Section 11.1 …rst reviews some of the earlier approaches to the analysis of pro-poor growth. Section 11.2 reviews the TIM curve, examined in further detail in Chapter 8 above, and its application to pro-poor growth comparisons. Section 11.3 then considers the modi…cation leading to the poverty persistence curve, relating to movements across a poverty line. Section 11.4 uses longitudinal income data for New Zealand, from administrative taxpayer records, to construct examples of pro-poor income mobility and poverty persistence curves.

11.1

Measures of Pro-Poor Growth

11.1.1

Growth Incidence Curves

The Growth Incidence Curve, introduced by Ravallion and Chen (2003), refers to the change in percentiles of the income (or other welfare metric) distribution from −1 to ; that is, it is based only on the characteristics of the two relevant cross-sectional distributions. Where  () is the distribution function of income at , the th percentile,  (), is given by:  () = −1 ()

(11.1)

The growth rate,   () of the th percentile is:  () −1 −1 ()

  () =

(11.2)

The GIC curve plots   () against . Since all percentiles are subject to some form of growth, the term ‘incidence’ is perhaps not the most appropriate: rather,   () shows the extent of growth of the th percentile. Ravallion and Chen (2003) show that   () can be linked to the slopes of the two Lorenz curves in  − 1 and . The Lorenz curve is obtained by plotting:

1  () = ¯

Z 0

 −1 ()

 ()

(11.3)

11.1. MEASURES OF PRO-POOR GROWTH where  =  () and ¯ is arithmetic mean income,

239 R∞ 0

 (). The slope,

0

 (), is given by:

 () ¯

0 () =

(11.4)

So that substituting for  () and −1 () in (11.2) gives:   () =

0 () ( + 1) − 1 0−1 () 

where:  =

¯ −1 ¯−1

(11.5)

(11.6)

is the growth rate of mean income. Hence if the Lorenz curve is unchanged,   () =   for all  and all percentiles grow at the same rate. R Let  −1 = 0  −1 () = −1 ( ) denote the headcount poverty measure at  − 1, where  is the constant poverty line. Hence,  −1 is the percentile corresponding to  for distribution −1 (). Ravallion and Chen (2003) go on to measure the pro-poor growth rate, PPG, by the mean growth rate:    =

Z

1  −1

 −1

0

  () 

(11.7)

Pro-poor growth, de…ned in this way, leads to a reduction in the Watts (1968) measure of poverty,  , de…ned in terms of a proportional poverty gap and given by:  =

1  

Z 0

 

µ log

¶    ()

(11.8)

Pro-poor growth therefore involves a change in the income distribution that is sufficient to lower the poverty measure. From (11.7) it is clear that the    measure is directly related to the GIC curve: it is the area under the curve up to,  −1 , the percentile associated with the poverty line. However, the mean growth rate of percentiles below the …xed poverty line,  , is not the growth rate of the mean income of those below  . It is also not the mean growth rate of those individuals who were below  in period  − 1. The GIC is based purely on the two marginal distributions in

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CHAPTER 11. POVERTY PERSISTENCE

 and  − 1: the growth rate of the th percentile,

 () −1 ()

− 1, does not refer

to the growth rate between  and  − 1 of the individual at the th percentile in  − 1.

11.1.2

An Elasticity Measure of Pro-Poor Growth

Instead of de…ning pro-poor growth in terms of the arithmetic mean growth rate of percentiles below a …xed poverty line, Essama-Nssah and Lambert (2006) use the concept of the elasticity of poverty with respect to a change in mean income. Letting  () denote a ‘deprivation’ measure, with  () = 0 for  ≥  and  () is a decreasing convex function of  for    , they consider the set of ‘average deprivation’ poverty measures de…ned by: Z   ()  () (11.9)  = 0

Let  ¯ denote the elasticity of  with respect to changes in arithmetic mean income, ¯. Then: ¯

¯ ¯  = =  ¯  

Z 0



 ()   ()  ¯ 

(11.10)

Furthermore, de…ne  ¯ as the elasticity of  with respect to changes in mean income. The ¯ s describe the individual income dynamics over the period. Then (11.10) can be written, using 0 () = () , as:  Z 1  ¯ = 0 () ¯  ()  0 Is is possible to introduce another elasticity  , so that: Z   ¯ = 0 ()    ¯  () 0

(11.11)

(11.12)

It is important to recognise that Essama-Nssah and Lambert (2006) implicitly assume that the individual income changes are such that there are no movements across the poverty line,  . For the headcount poverty measure, which requires  () = 1 for    , equation (11.11) gives ¯ = 0.

11.1. MEASURES OF PRO-POOR GROWTH

241

However, consider the simple case where  () = 1 −  , for which: µ ¶ ¯  =  ( ) 1 − (11.13)  As above,  ( ) is the headcount poverty measure and ¯ is the arithmetic mean income of those below the poverty line. Using 0 () = − 1 , the elasticity becomes:  ¯

1 =−  ( ) ( − ¯ )

Z



0

 ¯  ()

(11.14)

De…ne ˆ as the weighted average income of those in poverty, with weights R  1 ¯ . Hence ˆ = ( ¯  () and: ) 0   ¯ = −

ˆ  − ¯

The elasticity,  ¯ , can be rewritten as: Z Z ¡ ¢ 1  1  0  ()  () + 0 ()  ¯ − 1  () ¯ =  0  0

(11.15)

(11.16)

The …rst term in (11.16) is the elasticity that would result from uniform income growth of  ¯ = 1 for all . The second term re‡ects the contribution of the deviation from uniform growth. Essama-Nssah and Lambert (2006) de…ne the extent of pro-poor growth, , as: ´ ³ ¯ (11.17)  =   ¯ ¯ =1 − ¯ ¯  ¯ where ¯ ¯ =1 is the elasticity resulting from uniform income growth, that ¯ 

is, the …rst term in (11.16). Hence: Z  ¡ ¢ =− 0 ()  ¯ − 1  () 0

(11.18)

For example, in the above case where  () = 1 −  , it can be seen that: µ ¶ ˆ − ¯  = − ( ) (11.19)  In the case where the poorest of the poor experience relatively larger (and positive) income changes compared with those closer to the poverty line, ˆ  ¯ and  is positive.

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Figure 11.1: A TIM Curve

11.2

Income Dynamics

The ‘Three "I"s of Mobility’, or TIM, curve, was de…ned in Chapter 8. For convenience, Figure 8.2 is reproduced as Figure 11.1. To examine pro-poor mobility, suppose interest is focused on a proportion of the population, for example those below the th percentile, as indicated in Figure 11.1. There is less inequality of income changes among the group below , shown by the fact that the TIM curve from O to H is closer to a straight line than the complete curve OHG. The TIM curve also shows that the average income growth among those below  is larger than that of the population as a whole: compare the slopes of the lines OH and OG. The average growth rate among the poor (the intensity of their growth) is given by the height, H =  . If concern is largely for those below a poverty line,  , the corresponding percentile is  =  ( ), where  () is the distribution function of . For alternative values of , the TIM curve therefore gives an immediate indication of whether income changes have been pro-poor, that is, whether the slope of OH exceed that of OG. From the de…nition of  above and Figure 11.1, it is clear that the slope

11.3. POVERTY PERSISTENCE

243

of OH is given by  , while the slope of OG is simply given by  . Hence a measure of the extent of pro-poor mobility can be obtained as:   =

 

(11.20)

which measures the extent to which the cumulative proportional income growth per capita for those below  exceeds that of the population as a whole, . Having de…ned a fraction of the population below the poverty line,  , if the average income growth of the poor is the same as average income growth across the population (that is, income mobility is neither systematically pro-poor nor pro-rich) then   =  and   = 1.3 In principle, the maximum extent of pro-poor mobility by this measure is in…nite. For example,   −→ ∞ as  −→ 0, where the latter indicates zero income growth across the population on average. Alternatively,   −→ ∞, in the case where there is an in…nitely high rate of income growth for the poorest  of the population, if their initial incomes are zero. On the other hand, maximum pro-rich income mobility implies   −→ −∞.4 Using equation (11.20) and New Zealand income growth data, section 11.4 illustrates values of   for different values of  .

11.3

Poverty Persistence

This section turns to the question of how poverty persistence can be shown diagrammatically, re‡ecting the extent to which upward income mobility between two periods shifts individuals from below, to above, a given income poverty threshold,  . Since the TIM curve illustrates the cumulative extent of mobility for those below  , it can show whether the incomes of an 3 There

may nevertheless be differing degrees of inequality of mobility (as represented by the curvature of the TIM curves up to ) within the poor group compared to that within the population as a whole. 4 An alternative measure of the extent of pro-poor income mobility could be based on the difference in slopes of OH and OG in Figure 11.1. That is:   −  . In this case mobility-neutral growth (neither pro-poor nor pro-rich) implies   = 0.

244

CHAPTER 11. POVERTY PERSISTENCE

initially poor group on average grew sufficiently to escape from poverty, but it cannot directly illustrate the extent to which individuals below  escape from poverty. Suppose …rst that poverty is measured in relative terms. If the TIM curve up to  were to lie below the straight line OG of Figure 11.1, it is clear that income growth for those below  is insufficient to lift this group, in poverty at  − 1, above the poverty line at . That is, had the income growth experienced in aggregate by those below  been redistributed among those individuals to maximise the numbers above  at , there is no reallocation that could have lifted all of them out of poverty. Nevertheless, some individuals within this group in time period, −1, may experience sufficient income growth between  − 1 and  to raise their income levels above  . Assume further that the poverty income threshold is constant  −1

in both years. Let  =

denote individual ’s proportional income

growth between  − 1 and . The condition required for those individuals for whom −1   to move out of poverty is given by:  

 −1

−1

(11.21)

More generally, all individuals can be allocated to one of four groups based on their values of  and −1 , as shown in Table 11.1. Table 11.1: Poverty Persistence Move Persist

In poverty  − 1;   −1 −1  

Out of poverty    −1 − 1; −1  

   −1 − 1; −1  

   −1 − 1; −1  

The groups are separated by values of  and −1 , denoted ∗ and ∗−1 respectively, given by ∗ =

 −1

− 1 and ∗−1 =  . These can be illustrated

by a variant of the TIM curve. As discussed above, the TIM curve plots, for incomes in ascending order, cumulative proportional income changes per

11.3. POVERTY PERSISTENCE

245

Figure 11.2: Income Growth Rates Required to Escape and Avoid Entering Poverty capita against the corresponding proportion of people, . Consider an alternative diagram in which individual income growth rates,  and ∗ (which depends on the proportional difference between −1 and  ) are plotted against , for any given income poverty threshold,  , and associated  . Figure 11.2 plots the values of  ∗ against , for a poverty income threshold,  , such that  = 02: that is, 20 per cent of the population are below the poverty line. Hence, the  ∗ pro…le crosses the -axis at  = 02. This is referred to as a poverty persistence curve, ∗ ( ), which is de…ned for a given value of  . To the left of  = 02, growth rates greater than ∗ from period  − 1 to period  are sufficient to move individuals out of poverty, that is, to an income level which places them above  in the population in time period, . Conversely, for those who are placed to the right of  = 02 in period  − 1, growth rates less than  ∗ are sufficiently negative to move the individual into poverty in period . The pro…le of critical values,  ∗ , approaches an

246

CHAPTER 11. POVERTY PERSISTENCE

asymptote of −10 (or −100 per cent); that is, as incomes become very large relative to  , the required (negative) growth rate to move such individuals into poverty from period  − 1 to period  approaches −100 per cent. Movements across the poverty line can be examined more formally as follows. Dropping individual subscripts, for someone with initial income in period  − 1 of , experiencing a proportional change of , poverty is avoided if:

¶ µ     1+  Hence the income change must be such that:    −1  

(11.22)

(11.23)

Suppose the dynamics can be described by the following function:  =  +  () +  

(11.24)

where  denotes the general growth in incomes,  is a stochastic term distributed as  (0 2 ), and the function  () describes the relative income movements. For example, for the basic Galtonian regression towards the mean model,  () = − (1 − ) log (), where  is the regression coefficient and  is the geometric mean income in the initial period. Hence to avoid poverty, it is required that:  +  () +  

 −1 

(11.25)

or:

 − (1 + ) −  () =  (   ) (11.26)  The probability of avoiding poverty in the second period,  (   ) is thus: Z ∞ Z ( ) ¡ ¢ ¡ ¢ 2  (   ) =  | 0  = 1 −  | 0 2 (11.27) 

( )

−∞

The probability of being in poverty in the second period,  (   ), is: ¯ µ ¶  (   ) ¯¯  (   ) =  (11.28) ¯ 0 1 

11.4. NEW ZEALAND TIM AND POVERTY PERSISTENCE CURVES247 As above,  ( ) is the initial headcount poverty measure, the headcount poverty measure in the second period,  ( ), is a weighted average of individual probabilities, given by: ¯ ¶ Z ∞ µ  (   ) ¯¯   ( ) = ¯ 0 1  ()  0

11.4

(11.29)

New Zealand TIM and Poverty Persistence Curves

This section illustrates pro-poor TIMs and poverty persistence curves using a 2 per cent random sample of individual New Zealand personal income taxpayers, using administrative data from the Inland Revenue Department. The panel dataset contains incomes for both 2006 and 2010 for the same taxpayers. To avoid the exercise being contaminated by taxpayers with very low incomes (such as small part-time earnings of children, or small capital incomes of non-earners), individuals with 2006 or 2010 incomes less than $1,000 per annum were omitted from the sample. This yielded a usable sample of 32,970 individuals. Before examining the resulting diagrams, it should be acknowledged that since these data are based on incomes of individuals, as opposed to households, the notion of a poverty line is less meaningful. Clearly many individuals could, and in the dataset do, experience substantial year-to-year changes in income without this necessarily implying that the households of which they are members move into, or out of, poverty. Nevertheless the dataset has the particular advantage of providing more reliable estimates of individuals’ incomes from matched tax records, and has much wider coverage than more limited longitudinal household data based on survey methods available for New Zealand.5 For present purposes it serves to illustrate the conceptual 5 Statistics

New Zealand collected longitudinal household income data between 2001 and 2010 in the SoFIE dataset, in 8 waves. While SoFIE initially involved around 11,000 households, this declined progressively across waves with an attri-

248

CHAPTER 11. POVERTY PERSISTENCE

Figure 11.3: The New Zealand TIM Curve and Poverty Income Dynamics: 2006 to 2010 aspects of interest.

11.4.1

Pro-Poor TIM Curves

To construct a TIM curve for income growth over 2006-10, individuals were …rst ranked by their incomes in 2006. Cumulative income growth rates for the …ve year period were then obtained as described in section 11.2 and plotted against the cumulative population shares. The curve is therefore constructed using the full sample of 32,970 individuals, rather than based on averages within percentiles. Figure 11.3 shows the resulting TIM curve. The vertical height of the point G in the Figure indicates the average growth rate across the full sample, of around 0.14; that is, a 14 per cent income growth rate over the period 2006 to 2010. The TIM curve is everywhere above the straight line, OG, indicating higher growth rates at lower income levels. Moving from left to right in the Figure, from low to high 2006 incomes, the cumulative growth of the lowertion rate reaching 54 per cent by wave 7; see Statistics New Zealand (2011).

11.4. NEW ZEALAND TIM AND POVERTY PERSISTENCE CURVES249

Figure 11.4: Pro-Poor Mobility: 2006 to 2010 income sub-group remains above the overall average. The slope of a line from the origin to any point on the TIM curve, is steeper than the line from the origin to the end point, G. For example, the slope of line OH, for  = 02 is steeper than the line OK for  = 04, which in turn is steeper than the line OG. Hence the degree of pro-poor mobility in these data would seem to suggest greater pro-poor growth, the lower the threshold adopted to de…ne those considered to be poor in 2006. Figure 11.4 shows the values of   =    , using equation (11.20), for different percentile values of  from 0.05 to 0.5, in steps of 0.05. For  = 005 income growth of the poorest 5 per cent in 2006 is approximately ten times that of the full sample: the process could therefore be described as highly pro-poor. As  rises towards 05, the extent of this pro-poor income growth falls, but remains above one, reaching   = 23 at  = 05. By de…nition,   = 10 at  = 10.

250

11.4.2

CHAPTER 11. POVERTY PERSISTENCE

Longitudinal versus Cross-sectional Inequality

The longitudinal data underlying Figures 11.3 and 11.4 suggest strong propoor growth over the 2006-10 period. As argued earlier, this need not necessarily be re‡ected in cross-sectional inequality evidence, but may be associated with decreasing longer-term inequality as the accounting period is lengthened. This can be illustrated by considering the sum, for each individual, of 2006 and 2010 incomes.6 Figure 11.5 shows several Lorenz and concentration curves, with sample individuals ranked by 2006 income levels. The two cross-section curves for 2006 and 2010 are almost indistinguishable from each other, indicating little difference in a Gini-based cross-sectional measure of inequality for each year. However, a longitudinal perspective reveals a distinct reduction in inequality. Firstly, adding 2006 and 2010 incomes for each individual (ranked by 2006 incomes) the concentration curve for longer-term incomes displays less inequality than either of the cross-sections. Secondly, a concentration curve for 2010 incomes (again, for individuals ranked by 2006 incomes), suggests a further reduction in income inequality. These curves con…rm that substantial reductions in longer-term or longitudinal inequality are consistent with relatively unchanged cross-sectional patterns.

11.4.3

Poverty Persistence Curves

As mentioned above, identifying poverty persistence requires a choice of poverty income threshold,  . For illustrative purposes this is arbitrarily set here at 50 per cent of median income in 2006: thus, where  = $34 087, the poverty line is  = $17 044. In this case it turns out that  = 025 (or 25 per cent). This unusual property results from the fact that the distribution function over the relevant range is approximately a straight line 6 While

this does not sum incomes for all years from 2006 to 2010, it is sufficient to demonstrate the point made above.

11.4. NEW ZEALAND TIM AND POVERTY PERSISTENCE CURVES251

Figure 11.5: Longitudinal and Cross-sectional Inequality through the origin. Figure 11.6 shows both the critical growth rate, ∗ (the dashed curve) and median actual growth rates within each percentile. Given the sample size, there are 329 or 330 individuals in each percentile. As required, the critical growth rate, ∗ , crosses the -axis at  = 025. To the left of that point, any median growth rate for a given percentile which is above the ∗ curve implies that for at least half of the individuals in the percentile, income growth was sufficiently large that their income in 2010 exceeded  . Figure 11.6 shows that this condition is satis…ed for about 10 of the 25 percentiles below  . By contrast, above  there are no percentiles for which median growth is sufficiently negative (that is, lying below the ∗ curve) to push median individuals below  in 2010. This suggests a high degree of poverty avoidance over the 2006 to 2010 period for those not initially in poverty, and somewhat less persistence in poverty for those initially below

252

CHAPTER 11. POVERTY PERSISTENCE

Figure 11.6: Poverty Persistence in New Zealand: 2006 to 2010 the poverty line. However, the median percentile growth rates cannot capture the diversity of experience within each percentile. Figure 11.7 replaces the percentile medians with ‘box plots’ for actual percentile growth rates where each box shows the median growth rate and inter-quartile range. The ‘whiskers’ record the maximum and minimum income growth rate within each percentile. Figure 11.7 reveals a richer pattern of movement into and out of poverty. In particular, the whiskers indicate a wide range of growth rates within each percentile of the initial income distribution, such that, for example, every percentile above  includes at least one person who moved into poverty. Similarly, for those percentiles initially below  = 025, there is evidence of many individuals in almost all the lower percentiles moving out of poverty. There are numerous cases of individuals lying between the median and upper quartile (of the percentile distribution) who are observed to move out of poverty. This re‡ects a high degree of volatility in individual taxpayer

11.4. NEW ZEALAND TIM AND POVERTY PERSISTENCE CURVES253

Figure 11.7: Details of Poverty Persistence in New Zealand: 2006 to 2010 incomes from year to year in New Zealand. This volatility can also be observed when frequencies are inserted into the four-way classi…cation in Table 11.1. These are shown in Table 11.2 for three alternative de…nitions of the poverty line,  , of 033, 050, and 067 of median income. Values in Table 11.2 for   = 050 are for the case shown in Figure 11.7. The table shows that, for the lowest poverty line (33 per cent of median income), 14 per cent of individuals move across the poverty line from 2006 to 2010: 9 per cent move out of poverty by 2010 while 5 per cent move into poverty from above the 2006 poverty line. The extent of poverty persistence is sensitive to the poverty line, rising from only 3 per cent or approximately one-quarter (precisely 312) of those in poverty, (at   = 033) to 14 per cent (with   = 050) and 21 per cent or around two-thirds of those in poverty (at   = 067). Hence in this New Zealand case, those initially more deeply in poverty

254

CHAPTER 11. POVERTY PERSISTENCE Table 11.2: Poverty Persistence: 2006 to 2010 2006 headcount (%) In poverty Not in poverty   = 033 Move 9 5 Persist 3 82   = 050 Move 11 8 Persist 14 67   = 067 Move 12 9 Persist 21 58

– those below one-third of 2006 median income – appear to experience the least persistence of the three poverty group de…nitions. A similar picture emerges for those initially above the poverty line, but moving below it. The percentage moving into poverty is also sensitive to poverty line de…nitions, ranging from 82 per cent for a poverty line of 33 per cent of median income, to 58 per cent when the poverty line is 67 per cent of median income in 2006. These values for ‘movers’ suggest that, at least among individual New Zealand income taxpayers, there is substantial movement into and out of the lowest income levels. Since these lowest income levels are typically a few thousand dollars of annual income, this may capture, inter alia, the effects of moving into, or out of, the labour market including temporary migration spells overseas, and/or part-time employment. Additionally, the inclusion of the self-employed may contribute to the observed mobility into and out of low-income status, because they are known to experience more income volatility.7 7 Excuding

taxpayers with less than $1000 of taxable income does not prevent some of those with zero labour market earnings appearing in the sample, since capital income (such as property income, interest, and dividends) is included within the taxable income de…nition.

11.5. CONCLUSIONS

11.5

255

Conclusions

This chapter has suggested two new illustrative devices for poverty income dynamics. To examine pro-poor mobility in the form of relative income growth, it has proposed using TIM curves applied to individuals in poverty. In addition to highlighting the ‘three I’s’ properties (incidence, intensity and inequality) of mobility for alternative poverty de…nitions, this allows the relative mobility of each poverty group to be compared with mobility by the population as a whole. To examine poverty persistence, the chapter suggested that a poverty persistence curve can identify both the extent of movement across a poverty threshold and the particular poor and non-poor incomes for which persistence or movement is prevalent. Applying these concepts to New Zealand income data for individual income taxpayers showed that income dynamics were especially pro-poor during 2006-10, with much faster income growth for those on the lowest incomes than those higher up the income distribution. For example, a mobility index based on cumulative income growth rates for those with the lowest …ve per cent of 2006 incomes, on average was around ten times higher than the equivalent index for all taxpayers combined. For the lowest twenty-…ve per cent the equivalent index was around four times higher than mobility across all taxpayers. On poverty persistence, average income growth rates within each percentile of the distribution suggested relatively little movement into poverty but somewhat more movement out of poverty. However considering all individuals within each percentile (around 330 individuals per percentile in this case) revealed a relatively mobile population overall, with some individuals observed within all percentiles above the 2006 poverty threshold moving into poverty over the …ve year period examined. These New Zealand …gures are of course purely illustrative, depending

256

CHAPTER 11. POVERTY PERSISTENCE

here on the particular poverty threshold chosen and relating to individual, rather than household, incomes. But, suitably applied to householdor family-level income data, the devices proposed here would seem to offer convenient illustrations of the nature and extent of poverty income dynamics that are easily constructed and interpreted.

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Index abbreviated welfare function 52–5, 104 absolute re-ranking 220–21, 224–6, 228, 230 accounting period 10, 123, 212 additive welfare functions 15, 51, 67 adult equivalence scales 10, 22–4, 35–9, 95 after-tax incomes 124, 135, 148–50 age-related population structure 106, 117–18, 167–8 Aitchison, J. 70, 161 allocation rules 24–6, 28, 78–9, 81 alternative distributions 9–44, 46, 55–6, 60, 86, 97–103 alternative TIM curves 197–202, 216 annual inequality 3–5, 10, 91, 188 arithmetic mean income 13–14, 69–70, 77–9, 93, 127–8, 211 artificial income concepts 21, 22, 24 artificial units 21, 24 assessable income 130–31, 133 assets 133 Atkinson, A.B. 12, 13, 125, 146–7 Atkinson inequality measure 2, 14–16, 42, 46–7, 50–55, 85–7 2007 to 2010 29–30, 33–6

arithmetic mean of 15, 18–19, 52, 62, 67 aversion and 14, 29, 33–4, 42, 60–61, 67–73, 85 Gini measure comparison 18, 75–6, 79, 85, 104, 127 negative values 19, 27 pivotal income 61–3 values of inequality 59 Atkinson-Bourguignon social welfare function 236 Atoda, N. 162 Australia 4, 124, 138, 144–8 autoregressive processes 5, 166–8 Aziz, O.A. 28 Ball, C. 122, 156 Bargain, O. 42, 113 before-tax incomes 124, 135–9, 143, 148–51 benchmark income 46, 167 benefit abatement thresholds 33, 73 benefit system 92, 94, 97, 99, 103, 131, 145 bootstrap methods 192 Bourguignon, F. 177, 237 Brown, J.A.C. 70, 161 calibration variables 109, 112–13, 115, 118 calibration weights 96–7, 99, 101, 102–3, 110, 111 capital income 254 Carey, S. 137, 156

271

272 Carter, K. 158 census data 124, 144–5 Chatterjee, S. 95, 122 Chen, S. 237–9 CIPGs see ‘Cumulative Income Growth Profiles’ Claus, I. 102, 137 complete equality 205, 208, 211 concentration curves 191, 201–2, 250 confidence intervals 157, 193 consumption smoothing 100 convergence process 157, 166–7 Corvalan, A. 46–7, 50–51 Crawford, R. 122, 158 cross-sectional inequality 106, 109, 124, 157, 214, 250–51 cubic equation 169–71 ‘Cumulative Income Growth Profiles’ (CIGPs) 178, 181, 192–5, 199, 201, 204, 237 Cutler, D.M. 22–3, 95 data sources 129–35, 139–41, 155–6 decomposition of inequality 37–8, 91–120, 148, 150 degrees of freedom 54, 76, 77, 79, 84, 86 ‘deprivation’ poverty measures 240 Deville, J.-F. 96 differential growth 175–202 direct taxes 29–30, 102, 110 disequalising mobility 205, 208, 210 disequalising transfers 77, 80, 83–4, 86–7 disposable income 27–30, 32, 34, 41–2, 100 adult equivalence scales 35–6, 38, 104, 109 Gini values of 99, 116 households 19–20, 40 dividends 131, 159 Dixon, S. 122

INDEX Duval-Hernández, R. 157 dynamic processes 5, 155–61, 164–8, 171, 207, 242–3, 248 Easton, B. 122, 124, 130, 144–5 economies of scale 22–3, 36, 95 education expenditure 25, 28 equal means, distributions with 78–82 equalising income mobility 203–18 TIMs based on 201, 208 use of term 187 ‘equalising nTIM’ 215, 217 equalising transfers 77, 80, 83–4, 86–7 equally distributed equivalent income 14–16, 50–51, 62, 68, 78, 85, 103–4, 126 ‘equity and efficiency’ trade-off 47, 52, 104, 126, 136 equivalent adult unit see adult equivalence scales Essama-Nssah, B. 240–41 evaluation function 13, 34, 47, 126 see also social welfare functions Evans, L. 101 ‘excess shares’ 2, 47, 48–50, 56 exogenous variables 94 expenditure inequality 95, 99–100 extreme equalising mobility 209, 212–13, 217 family, definition 26 family sharing rule 28 Fields, G.S. 157, 176, 197, 201, 220 first-order serial correlation 156, 160–68, 170, 173 fiscal incidence studies 25, 39 fringe benefits tax 136, 152 Galtonian regression 160–62, 168, 170, 183, 185–8, 246 Gaston, N. 124 Gemmell, N. 155, 158, 236 gender 117–18, 125, 148–51, 167–8 Generalised Lorenz curve 13–14, 45

INDEX geometric mean income 67, 69, 160–63, 167, 170–71, 182, 185 Gibrat process 161–2 Gibrat, R. 160 GICs see Growth Incidence Curves Gini inequality measure 2, 4, 16–18, 47–52, 77–84, 97–103, 175 absolute changes 116 arithmetic mean 77, 81, 127 Atkinson measure comparison 75–6, 79, 85, 104, 127 by gender 148–51 covariance form 127–8 cross-sectional inequality 250 data for calculations 133–5 evaluation function 126 ‘excess shares’ 56 grouped data comparison 128–9 household surveys 39–40 labour force changes 117 negative incomes and 19, 27 New Zealand 124–5 nTIM curve and 205 pivotal income 61 poverty persistence 235 social welfare function 93 survey calibration methods 109–10 taxable income 133 total income before tax 136–9 value judgements 123 Glewwe, P. 23 Goldsmith, P. 139 grouped data 128–9 Growth Incidence Curves (GICs) 237, 238–40 growth rates 4–5, 189, 208 income mobility 180–81, 183–4 poverty persistence 245–6, 248, 251–2 Hart, P.E. 161 health expenditure 25, 28 Hérault, N. 42, 113 HES see Household Economic Survey Hoffmann, R. 46

273 Household Economic Survey (HES) 26, 30, 92, 106, 118, 123 household income 19–25, 28, 95, 99–100, 109, 247 household structure 91, 94, 106, 108, 113–15 household surveys 39–40, 158 households 10, 23, 111, 247 Hyslop, D. 158 IGPs see Income Growth Profiles income definition problems 152 types of 130–33 income changes, TIMs using 199–202, 242 income convergence process 166–7 income dynamics 155–61, 164–8, 171, 242–3, 248 Income Growth Profiles (IGPs) 178, 192, 204, 237 income mobility 6–7, 123, 157, 176, 177–8, 180–85 equalising 203–18 positional changes 219–33 poverty persistence and 8, 236, 243, 249 regression models 155–74 income poverty, properties of 180 income risk 176 income share mobility 197–202 income-sharing rule 24–5 income shifting 101–2, 137, 172 ‘income swapping’ nTIM 212–14, 215, 217 income taxes 30, 39, 73, 131–3, 139, 215 marginal structure 136–7, 149 top rate 103, 136 income transfers 77, 80, 83, 86, 235 indifference curves 18 indirect taxes 26, 28–9 individual as unit of analysis 23, 95 inequality aversion 29, 42, 47, 60–63, 67–73 inequality changes 91–120, 124

274 inequality decompositions 37–8, 111–18 inequality measures 2, 14–19, 45–73, 91, 135–50 see also Atkinson inequality measure; Gini inequality measure inequality-preserving processes 75–90, 205–6 Inland Revenue Department (IRD) data 129–30, 133, 135, 137, 144, 148, 156, 158, 213 IRD data see Inland Revenue Department data Jäntti, M. 176, 222 Jenkins, S.P. 175–8, 181, 192–6, 199, 204, 222, 235–7 Jensen scales 92 Jeram, J. 121–2 Johnston, G. 122 Katz, L. 23, 95 Knuth, D.E. 116 labour force changes 42, 106, 109, 113, 117 labour market structure 114, 254 Lagrange multipliers 97 Lambert, P. 46, 61, 104, 175, 178, 195, 204, 235, 240–41 Land and Income Tax Amendment Act 1953 131 Lanza, G. 46, 61 Laws, A. 158 ‘leaky bucket’ experiments 2, 16–17, 47, 58–61 Leigh, A. 124, 138, 144–8 life-cycle simulation models 156 longitudinal data 158, 177, 197, 202, 237, 247 longitudinal inequality 250–51 Lorenz curves 5–6, 11–14, 24, 45, 80–81, 201–4 differential growth 186–7

INDEX Gini measure comparison 16–17, 126 longitudinal inequality 250 non-negative economic variables 175 poverty persistence 235, 238 principle of transfers 24 sampling variability 191 TIP curve comparisons 176 Lorenz dominance 11–12, 14, 29, 45, 175, 235 marginal income tax rate 136–7, 149 market income 30, 37, 39, 99, 101–2, 109–10, 145 age/gender and 117–18 households 19–20, 40 individuals 21, 26–7 nonworkers 15 percentage inequality reduction 32 maximum re-ranking 6–7, 205, 212–14, 222–6, 230 Mellish, A. 33 mobility see income mobility n-person shares 53–7 negative income 27–8, 99, 134–5, 144, 198 negative re-ranking 219, 220 net profit 131 New Zealand 4–5, 11, 26–8, 91, 94, 121 data sources 92, 99, 103, 106, 122, 144, 155–6, 159 Gini measure for 124–5, 145–8 nTIM curves for 213, 215–16 re-ranking profiles for 226–32 TIM curves for 188–95, 197, 247–54 non-assessable income 130–31, 133 non-parametric methods 155, 157 normalised TIM (nTIM) curves 189–90, 192–3, 199–200, 202–10, 212–17 NZOYB see Statistics New Zealand Official Yearbooks

INDEX O’Dea, D. 122 Ok, E.A. 157, 220 orders of magnitude 2, 45, 47, 53 Palmisano, F. 176 Papps, K.L. 150 Paretean welfare function 13, 15, 51, 67 Pareto, V. 122 Pay as You Earn (PAYE) system 125, 134, 136–44, 158 Peragine, V. 176 Perry, B. 136, 196 pivotal income 2–3, 61–4 ‘pivotal’ individual 46–7, 50–51 Podder, N. 95, 122 population structure 40, 42, 91, 93, 95, 97, 106–11 positional changes, TIM curve and 211–13 positional mobility 6–7, 219–33 poverty, dimensions of 175–7, 178–80, 183 poverty persistence 7–8, 235–56 poverty persistence curves 245, 247–54 PPG see pro-poor growth principle of transfers 12–13, 15, 23–4, 51, 126, 235 pro-poor growth (PPG) 184–5, 238–42, 250 pro-poor income mobility 236, 249 pro-poor TIM curves 248–9 proprietary income 130–31, 133 provisional total income 137–8 Rajaguru, G. 124 range of distributions 81, 82, 84 rank-based mobility measures 77, 79–80, 126, 211, 219 rank-order changes 211, 220 Ravallion, M. 237–9 re-ranking mobility 6–7, 205, 211–12, 217, 219–32 re-ranking ratio (RRR) curve 177, 224, 226, 229–32

275 redistribution 30, 39, 92, 94, 102–103, 110–11 regression with first-order serial correlation 160–68 regression models 155–74, 183, 246 regression towards the mean 5, 156, 160–61 relative income proportional change 163, 169–70, 182 systematic change 160 relative inequality 11, 45, 203 relative inequality-preserving mobility process 205–6 ‘relative poverty line’ 46 replication invariance 48 returnable income 130–33 ‘reverse-order-rank-weighted’ mean 50, 78 reverse rank order 103, 214, 217 ‘reverse rank’ weighting 17–18, 126, 127 Reynolds-Smolenski index 102, 110 RRR see re-ranking ratio curve sample weights 97, 99 sampling variability 112, 118, 191–4 Särndal, C.-E. 96 Saunders, P. 122, 124 savings–income relation 100 second-order serial correlation 168–73 self-employment 231, 254 sensitivity analyses 29, 36, 59, 62–3 serial correlation 156, 168–73 Shapley Value 113 sharing rule, households 24–5, 28 Shorrocks, A.F. 13, 23, 40, 46–50, 56, 65, 78, 113, 157 Sleeman, C. 95 small-sample construction 53–61 Smith, H. 158 SNZ see Statistics New Zealand social indifference curves 18 social welfare functions 34–6, 51–3, 93, 103–6, 125–6, 236 social welfare transfers 124

276

INDEX

SoFIE dataset 247 standard errors 129, 192–3 static inequality 4, 6, 155 Statistics New Zealand Official Yearbooks (NZOYB) 129, 130, 132–5, 137–8, 139, 142, 144 Statistics New Zealand (SNZ) 92–3, 99, 103, 106, 122, 129–30, 132, 135, 138–9, 144–5, 247 stochastic processes 5, 155, 160, 172, 174 Subramanian, S. 46, 47, 49–50, 52, 65, 78, 104 survey calibration methods 3, 91–2, 95, 96–7, 109 systematic equalising tendency 162, 207

Galtonian regression and 185–8 New Zealand 188–95, 247–54 poverty persistence 7–8, 236–7, 242–4 re-ranking similarities 221, 228 TIP curve 175–7, 178–81, 184, 191, 227, 235, 237 top income shares 122, 127, 146–7 top income tax rates 101–3, 136–7, 149 total income 48, 53, 84, 131–3, 135–9, 143, 150–51 transition matrices 158, 219 Trede, M. 157, 219 two-person cake sharing 46, 48–9

Tachibanaki, T. 162 tax changes 24, 101–4, 113, 136–7, 172 tax progressivity analyses 211–12 tax structures 40, 94, 136–7 tax and transfer system 16, 26–8, 29–30, 33, 42, 67, 118, 124 redistributive effects 30, 39, 102–3, 110 taxable income 124, 133, 135, 152–3, 158–9, 165, 171 taxpayer-based Gini estimates 144–5, 146 Taxwell model 10, 27, 30, 33 Templeton, R. 158 ‘Three Is of Mobility’ curve see TIM curve ‘Three Is of Poverty’ curve see TIP curve Three ‘Ws’ of inequality 123 TIM curve 5–6, 177–85, 197–202, 205–17, 229

value judgements 9–19, 52, 93, 125–6, 127 Atkinson measure consistency 75 Gini measure 75, 123 Lorenz curves 45 Van de gaer, D. 176 Van Kerm, P. 177–8, 181, 192–4, 196, 199, 236–7 variance, Gini measure 82–4 Vosslamber, R. 139

unit of analysis 9–10, 19–20, 23–4, 95

Watts measure of poverty 239 welfare function 13–15, 18, 51–5, 67, 75, 93, 126 income mobility 176, 236 percentage changes 34–6 welfare metric 10, 19, 23, 123 Wilhelm, M. 236 Wilkinson, B. 121–2 zero income 15–16, 28, 37, 99, 134–5, 144–5, 159, 163