# (p.178) Appendix B Notes on Sources and Literature

# (p.178) Appendix B Notes on Sources and Literature

This appendix describes the datasets that have been used for particular examples in each chapter, cites that which has been used for the discussion in the text, and provides a guide for further reading. In addition some more recondite supplementary points are mentioned. The arrangement follows the order of the material in the five chapters and Technical Appendix.

# B.1 Chapter 1

For a general discussion of terminology and the approach to inequality you could go to Chapter 1 of Atkinson (1983), Cowell (2008b, 2008c), and Chapter 2 of Thurow (1975); reference may also be made to Bauer and Prest (1973). For a discussion of the relationship between income inequality and broader aspects of economic inequality see Sen (1997). For other surveys of inequality measurement issues see Jenkins and Van Kerm (2008) and for a more technical treatment, Cowell (2000) and Lambert (2001).

## Inequality of What?

This key question is explicitly addressed in Sen (1980, 1992). The issue of the measurability of the income concept is taken up in a very readable contribution by Boulding (1975), as are several other basic questions about the meaning of the subject which were raised by the nine interpretations cited in the text Rein and Miller (1974). For an introduction to the formal analysis of measurability and comparability, see Sen and Foster (1997 [Sen 1973], pp. 43–46), and perhaps then try going on to Sen (1974) which, although harder, is clearly expounded. There are several studies which use an attribute other than income or wealth, and which provide interesting material for comparison: Jencks (1973) puts income inequality in the much wider context of social inequality; Addo (1976) considers international inequality in such things as school enrolment, calorie consumption, energy consumption and numbers of physicians; Alker (1965) discusses a quantification of voting power; Russet (1964) relates inequality in land ownership to political instability. The problem of the size of the cake depending on the way it is cut has long been implicitly recognized (for example, in the optimal taxation literature) but does not feature prominently in the works on inequality measurement. For a general treatment read Tobin (1970), (p.179) reprinted in Phelps (1973). On this see also the Okun (1975, Chapter 4) illustration of ‘leaky bucket’ income transfers.

The issue of rescaling nominal incomes so as to make them comparable across families or households of different types—known in the jargon as ‘equivalization’— and its impact upon measured inequality is discussed in Coulter *et al.* (1992a, 1992b)—see also page 191 below. Alternative approaches to measuring inequality in the presence of household heterogeneity are discussed in Cowell (1980), Ebert (1995, 2004), Glewwe (1991), Jenkins and O'Higgins (1989), Jorgenson and Slesnick (1990). The issues of measuring inequality when the underlying ‘income’ concept is something that is not cardinally measurable—for example measuring the inequality of health status—is discussed in Abul Naga and Yalcin (2008), Allison and Foster (2004).

## Inequality Measurement, Justice, and Poverty

Although inequality is sharply distinct from mobility, inequality measures have been used as a simple device for characterizing income mobility—after covering the material in Chapter 3 you may find it interesting to check Shorrocks (1978). The application of inequality-measurement tools to the analysis of *inequality of opportunity* is addressed in Lefranc *et al*. (2008) and Pistolesi (2009).

On the desirability of equality *per se* see Broome (1988). Some related questions and references are as follows: Why care about inequality? (Milanovic 2007) Does it make people unhappy? (Alesina *et al*. 2004) Why measure inequality? Does it matter? (Bénabou 2000, Elliott 2009, Kaplow 2005) Do inequality measures really measure inequality? (Feldstein 1998)

On some of the classical principles of justice and equality, see Rees (1971), Chapter 7 and Wilson (1966). The idea of basing a model of social justice upon that of economic choice under risk is principally associated with the work of Harsanyi (1953, 1955)—see also Bosmans and Schokkaert (2004), Amiel *et al*. (2009), and Cowell and Schokkaert (2001). Hochman and Rodgers (1969) discuss concern for equality as a consumption externality. A notable landmark in modern thought is Rawls (1971) which, depending on the manner of interpretation of the principles of justice there expounded, implies most specific recommendations for comparing unequal allocations. Bowen (1970) introduces the concept of ‘minimum practicable inequality’, which incorporates the idea of special personal merit in determining a just allocation.

Stark's (1972) approach to an equality index is based on a head-count measure of poverty and is discussed in Chapter 2; Batchelder (1971, p. 30) discusses the ‘poverty gap’ approach to the measurement of poverty. The intuitive relationships between inequality and growth (or contraction) of income are set out in a novel approach by Temkin (1986) and are discussed further by Amiel and Cowell (1994b) and Fields (2007). The link between a measure that captures the depth of poverty and the Gini coefficient of inequality (see Chapter 2) was analysed in a seminal paper by Sen (1976a), which unfortunately the general reader will find quite hard; the huge literature which ensued is surveyed by Foster (1984), Hagenaars (1986), Ravallion (1994), Seidl (1988), and Zheng (1997). The relationship between inequality (p.180) and poverty measures is discussed in some particularly useful papers by Thon (1981, 1983a). An appropriate approach to poverty may require a measure of economic status that is richer than income—see Anand and Sen (2000).

## Inequality and the Social Structure

The question of the relationship between inequality in the whole population and inequality in subgroups of the population with reference to heterogeneity due to age is tackled in Paglin (1975) and in Cowell (1975). The rather technical paper of Champernowne (1974) explores the relationship between measures of inequality as a whole and measures that are related specifically to low incomes, to middle incomes, or to high incomes.

# B.2 Chapter 2

The main examples here are from the tables in *Economic Trends*, November 1987 (based on the Inland Revenue's Survey of Personal Incomes augmented by information from the Family Expenditure Survey), which are reproduced on the website in the file ‘ET income distribution’: the income intervals used are those that were specified in the original tables. If you open this file you will also see exactly how to construct the histogram for yourself: it is well worth running through this as an exercise. The reason for using these data to illustrate the basic tools of inequality analysis is that they are based on reliable data sources, have an appropriate definition of income, and provide a good coverage of the income range providing some detail for both low incomes and high incomes. Unfortunately this useful series has not been maintained: we will get to the issue of what can be done with currently available datasets in Chapter 5.

The example in Fig. 2.9 is taken from the Annual Survey of Hours and Earnings (formerly the New Earnings Survey) data—see file ‘Earnings quantiles’ on the website. The reference to Plato as an early precursor of inequality measurement is to be found in Saunders (1970), pp. 214–15.

## Diagrams

One often finds that technical apparatus or analytical results that have become associated with some famous name were introduced years before by someone else in some dusty journals, but were never popularized. So it is with Pen's Parade, set out in Pen (1974), which had been anticipated by Schutz (1951), and only rarely used since—cf. Budd (1970). As we have seen, the Parade is simply related to the cumulative frequency distribution if you turn the piece of paper over once you have drawn the diagram: for more about this concept, and also frequency distributions and histograms, consult a good statistics text such as Berry and Lindgren (1996), Casella and Berger (2002), or Freund and Perles (2007); for an extensive empirical application of Pen's parade see Jenkins and Cowell (1994a). The log-representation of the frequency distribution is referred to by Champernowne (1973, 1974) as the ‘people curve’.

(p.181)
The Lorenz curve originally appeared in Lorenz (1905). Its convex shape (referred to on page 170) needs to be qualified in one very special case: where the mean of the thing that you are charting is itself negative—see page 170 in the Technical Appendix and Amiel *et al*. (1996). For a formal exposition of the Lorenz curve and proof of the assertions made in the text see Levine and Singer (1970) and Gastwirth (1971). Lorenz transformations are used to analyse the impact of income redistributive policies—see Arnold (1990), Fellman (2001), and the references in Question 7 on page 38. On using a transformation of the Lorenz curve to characterize income distributions see Aaberge (2007); see Fellman (1976) and Damjanovic (2005) for general results on the effect of transformations on the Lorenz curve. Lam (1986) discusses the behaviour of the Lorenz curve in the presence of population growth.

The relationship between the Lorenz curve and Pen's parade is also discussed by Alker (1970). The Lorenz curve has further been used as the basis for constructing a segregation index (Duncan and Duncan 1955; Cortese *et al*. 1976). For more on the Lorenz curve see also Blitz and Brittain (1964), Crew (1982), Hainsworth (1964), Koo *et al*. (1981), and Riese (1987).

## Inequality Measures

The famous concentration ratio Gini (1912) also has an obscure precursor. Thirty-six years before Gini's work, Helmert (1876) discussed the ordinally equivalent measure known as Gini's mean difference—for further information see David (1968, 1981). Some care has to be taken when applying the Gini coefficient to indices of data where the number of individuals *n* is relatively small (Allison 1978, Jasso 1979): the problem is essentially whether the term *n*^{2} or *n*[*n* – 1] should appear in the denominator of the definition—see the Technical Appendix page 155. A convenient alternative form of the standard definition is given in Dorfman (1979):

For an exhaustive treatment of the Gini coefficient see Yitzhaki (1998).

The process of rediscovering old implements left lying around in the inequality analyst's toolshed continues unabated, so that often several labels and descriptions exist for essentially the same concept. Hence *M*, the relative mean deviation, used by Schutz (1951), Dalton (1920), and Kuznets (1959), reappears as the maximum equalization percentage, which is exactly 2*M* (United Nations Economic Commission for Europe 1957), and as the ‘standard average difference’ (Francis 1972). Eltetö and Frigyes (1968) produce three measures which are closely related to *M*, and Addo's ‘systemic inequality measure’ is essentially a function of these related measures; see also Kondor (1971). Gini-like inequality indices have been proposed by Basmann and Slottje (1987), Basu (1987), Berrebi and Silber (1987), Chakravarty (1988), and Yitzhaki (1983), and generalizations and extensions of the Gini are discussed by Barrett and Salles (1995), Bossert (1990), Donaldson and Weymark (1980), Kleiber and Kotz (2002), Moyes (2007), Weymark (1981), and Yaari (1988); see also Lin (1990). The Gini coefficient has also been used as the basis for regression analysis
(p.182)
(Schechtman and Yitzhaki 1999) and for constructing indices of relative deprivation (Bishop *et al.* 1991, Chakravarty and Chakraborty 1984, Cowell 2008a, Yitzhaki 1979).

The properties of the more common *ad hoc* inequality measures are discussed at length in Atkinson (1970, pp. 252–57; 1983, pp. 53–58), Champernowne (1974, p. 805), Foster (1985), Jenkins (1991), and Sen and Foster (1997, pp. 24–36). Berrebi and Silber (1987) show that for all symmetric distributions *G* < 0.5: a necessary condition for *G* > 0.5 is that the distribution be skewed to the right. Chakravarty (2001) considers the use of the variance for the decomposition of inequality and Creedy (1977) and Foster and Ok (1999) discuss the properties of the variance of logarithms. The use of the skewness statistic was proposed by Young (1917); this and other statistical moments are considered further by Champernowne (1974); Butler and McDonald (1989) discuss the use of incomplete moments in inequality measurement (the ordinates of the Lorenz curve are simple examples of such incomplete moments—see the expressions on page 114). On the use of the moments of the Lorenz curve as an approach to characterizing inequality see Aaberge (2000). Further details on the use of moments may be found in texts such as Casella and Berger (2002) and Freund (2003). For more on the minimal majority coefficient (sometimes known as the Dauer–Kelsay index of malapportionment) see Alker and Russet (1964), Alker (1965), and Davis (1954, pp. 138–43). Some of the criticisms of Stark's high–low measure were originally raised in Polanyi and Wood (1974). Another such practical measure with a similar flavour is the Wiles (1974) semi-decile ratio: (minimum income of top 5 per cent)/(maximum income of bottom 5 per cent). Like *R, M*, ‘minimal majority’, ‘equal shares’, and ‘high–low’, this measure is insensitive to certain transfers, notably in the middle income ranges (you can redistribute income from a person at the sixth percentile to a person at the ninety-fourth without changing the semi-decile ratio). In my opinion this is a serious weakness, but Wiles recommended the semi-decile ratio as focusing on the essential feature of income inequality.

## Rankings

Wiles and Markowski (1971) argued for a presentation of the facts about inequality that captures the whole distribution, since conventional inequality measures are a type of sophisticated average, and ‘the average is a very uninformative concept’ (1971, p. 351). In this respect^{1} their appeal is similar in spirit to that of Sen and Foster (1997, Chapter 3) who suggest using the Lorenz curve to rank income distributions in a ‘quasi-ordering’—in other words a ranking where the arrangement of some of the items is ambiguous. An alternative approach to this notion of ambiguity is the use of ‘fuzzy’ inequality discussed in Basu (1987) and Ok (1995).

The method of percentiles was used extensively by Lydall (1959) and Polanyi and Wood (1974); for recent applications to trends in the earnings distribution and the structure of wages see Atkinson (2007a) and Harvey and Bernstein (2003). The (p.183) formalization of this approach as a ‘comparative function’ was suggested by Esberger and Malmquist (1972).

# B.3 Chapter 3

The dataset used for the example on page 72 is given in the file ‘LIS comparison’ on the website. The artificial data used for the example in Tables 3.3 and 3.4 are in the file ‘East West’.

## Social Welfare Functions

The traditional view of social welfare functions is admirably and concisely expounded in Graaff (1957). One of the principal difficulties with these functions, as with the physical universe, is—where do they come from? On this technically difficult question, see Boadway and Bruce (1984, Chapter 5), Gaertner (2006), and Sen (1970, 1977). If you are sceptical about the practical usefulness of SWFs you may wish to note some other areas of applied economics where SWFs similar to those discussed in the text have been employed. They are introduced to derive interpersonal weights in applications of cost-benefit analysis, and in particular into project appraisal in developing countries—see Layard (1994), Little and Mirrlees (1974, Chapter 3), and Salanié (2000, Chapters 1, 2). Applications of SWF analysis include taxation design (Atkinson and Stiglitz 1980, Salanié 2003, Tuomala 1990), the evaluation of the effects of regional policy (Brown 1972, pp. 81–84), the impact of tax legislation (Mera 1969), and measures of national income and product (Sen 1976b).

As we noted when considering the basis for concern with inequality (pages 12 and 179) there is a connection between inequality and risk. This connection was made explicit in Atkinson's seminal article (Atkinson 1970) where the analogy between risk aversion and inequality aversion was also noted. However, can we just read across from private preferences on risk to social preferences on inequality? Amiel *et al*. (2008) show that the phenomenon of preference reversals may apply to social choice amongst distributions in a manner that is similar to that observed in personal choice amongst lotteries. However, experimental evidence suggests that individuals' attitude to inequality (their degree of inequality aversion *ε*) is sharply distinguished from their attitude to risk as reflected in their measured risk aversion—Kroll and Davidovitz (2003), Carlsson *et al*. (2005); estimates of inequality aversion have been made using classroom experiments (Amiel *et al*. 1999) and from representative sample survey evidence (Pirttilä and Uusitalo 2010). Estimates of inequality aversion across country (based on data from the World Bank's World Development Report) are discussed in Lambert *et al*. (2003); an interesting study on changes over time in attitudes to inequality in one country is to be found in Grosfeld and Senik (2010). If we were to interpret *U* as individual utility derived from income we would then interpret *ε* as the elasticity of marginal utility of income, and then one could perhaps estimate this elasticity directly from surveys of subjective happiness: this is done in Layard *et al*. (2008). Cowell and Gardiner (2000) survey methods for estimating this elasticity and HM Treasury (2003), page 94, provides a nice example of how
(p.184)
such estimates can be used to underpin policy making. Ebert and Welsch (2009) examine the extent to which conventional inequality measures can be used to represent rankings of income distributions as reflected in survey data on subjective well-being.

The dominance criterion associated with quantile ranking (or Parade ranking) on page 33 and used in Theorem 1 is known as *first-order dominance*. The concept of *second-order dominance* refers to the ranking by generalized Lorenz curves used in Theorem 3 (the shares dominance used in Theorem 2 can be seen as a special case of second-order dominance for a set of distributions that all have the same mean). First-order dominance, principles of social welfare, and Theorem 1 are discussed in Saposnik (1981, 1983). The proofs of Theorems 2 and 4, using slightly more restrictive assumptions than necessary, were established in Atkinson (1970) who drew heavily on an analogy involving probability theory; versions of these two theorems requiring weaker assumptions but rather sophisticated mathematics are found in Dasgupta *et al*. (1973), Kolm (1969), and Sen and Foster (1997, pp. 49–58). In fact a lot of this work was anticipated by Hardy *et al.* (1934, 1952); Marshall and Olkin (1979) develop this approach and cover in detail relationships involving Lorenz curves, generalized Lorenz curves, and concave functions: readers who are happy with an undiluted mathematical presentation may find this the most useful single reference on this part of the subject (see also Arnold 1987).

Shorrocks (1983) introduced the concept of the generalized Lorenz curve and proved Theorem 3. As a neat logical extension of the idea Moyes (1989) showed that if you take income and transform it by some function *ϕ* (for example by using a tax function, as in the exercises on page 38) then the generalized Lorenz ordering of distributions is preserved if and only if *ϕ* is concave—see also page 181 above. Iritani and Kuga (1983) and Thistle (1989a, 1989b) discuss the interrelations between the Lorenz curve, the generalized Lorenz curve, and the distribution function. A further discussion and overview of these topics is to be found in Lambert (2001).

Where Lorenz curves intersect we know that unambiguous inequality comparisons cannot be made without some further restriction, such as imposing a specific inequality measure. However, it is also possible to use the concept of *third-order dominance* discussed in Atkinson (2008) and Davies and Hoy (1995). For corresponding results concerning generalized Lorenz curves see Dardanoni and Lambert (1988).

## SWF-Based Inequality Measures

For the relationship of SWFs to inequality measurement, either in general form, or the specific type mentioned here, see Atkinson (1974, p. 63; 1983, pp. 56–57), Blackorby and Donaldson (1978, 1980), Champernowne and Cowell (1998), Dagum (1990), Dahlby (1987), Schwartz and Winship (1980), Sen (1992), and Sen and Foster (1997). The formal relationships between inequality and social welfare are discussed in Ebert (1987) and Dutta and Esteban (1992). For a general discussion of characterizing social welfare orderings in terms of degrees of inequality aversion see Bosmans (2007a). The association of the Rawls (1971) concept of justice (where society gives priority to improving the position of the least advantaged person) with a social welfare function exhibiting extreme inequality aversion is discussed
(p.185)
in Arrow (1973), Hammond (1975), Sen (1974, pp. 395–98), and Bosmans (2007b). Lambert (1980) provides an extension of the Atkinson approach using utility shares rather than income shares. Inequality measures of the type first suggested by Dalton (1920) are further discussed by Aigner and Heins (1967) and Bentzel (1970). Kolm (1976a) suggests a measure based on an alternative to assumption 5, namely constant absolute inequality aversion (see page 165 above), so that as we increase a person's income *y* by one unit (pound, dollar, etc.) his welfare weight *U*′ drops by *κ*% where *κ* is the constant amount of absolute inequality aversion: this approach leads to an inequality measure which does not satisfy the principle of scale independence. He also suggests a measure generalizing both this and Atkinson's measure. See also Bossert and Pfingsten (1990) and Yoshida (1991). The implications of using absolute rather than relative measures in analysing world income distribution are examined in Atkinson and Brandolini (2009a). The SWF method is interpreted by Meade (1976, Chapter 7 and appendix) in a more blatantly utilitarian fashion; his measure of ‘proportionate distributional waste’ is based on an estimation of individual utility functions. Ebert (1999) suggests a decomposable inequality measure that is a kind of ‘inverse’ of the Atkinson formula.

An ingenious way of extending dominance results to cases where individuals differ in their needs as well as their incomes is the concept known as sequential dominance (Atkinson and Bourguignon 1982, 1987). Further discussion of multidimensional aspects of inequality are to be found in Diez *et al*. (2007), Maasoumi (1986, 1989), Rietveld (1990), Savaglio (2006), and Weymark (2006); multidimensional inequality indices are discussed by Tsui (1995).

## Inequality and Information Theory

The types of permissible ‘distance’ function, and their relationship with inequality are discussed in Cowell and Kuga (1981); Love and Wolfson (1976) refer to a similar concept as the ‘strength-of-transfer effect’. The special relationship of the Herfindahl index and the Theil index to the strong principle of transfers was first examined in Kuga (1973). Krishnan (1981) (see also reply by Allison 1981) discusses the use of the Theil index as a measure of inequality interpreted in terms of average distance. Kuga (1980) shows the empirical similarities of the Theil index and the Gini coefficient, using simulations.

The Herfindahl (1950) index (closely related to *c*^{2}, or to Francis' standard average square difference) was originally suggested as a measure of concentration of individual firms—see Rosenbluth (1955). Several other inequality measures can be used in this way, notably other members of the *E*_{θ} family. The variable corresponding to income *y* may then be taken to be a firm's sales. However, one needs to be careful about this analogy since inequality among persons and concentration among firms are rather different concepts in several important ways: (i) the definition of a firm is often unclear, particularly for small production units; (ii) in measuring concentration we may not be very worried about the presence of tiny sales shares of many small firms, whereas in measuring inequality we may be considerably perturbed by tiny incomes received by a lot of people—see Hannah and Kay (1977). The relationship between the generalized entropy measures and the Lorenz curves is examined further
(p.186)
in Rohde (2008) and the problem of capturing Lorenz orderings by a small number of inequality measures is considered by Shorrocks and Slottje (2002).

A reworking of the information theory analogy leads us to a closely related class of measures that satisfy the strong principle of transfers, but where the average of the distance of actual incomes from inequality is found by using population shares rather than income shares as weights, thus:

*θ*= 0 (equivalently Atkinson index with

*ε*= 1).

## Building an Inequality Measure

The social value judgements implied by the use of the various *ad hoc* inequality measures in Chapters 2 and 3 are analysed in Kondor (1975) who extends the discussion in the works of Atkinson, Champernowne, and Sen cited in the notes to Chapter 2. The question of what happens to inequality measures when all incomes are increased or when the population is replicated or merged with another population is discussed in Aboudi *et al*. (2010), Frosini (1985), Eichhorn and Gehrig (1982), Kolm (1976a, 1976b), and Salas (1998). Shorrocks and Foster (1987) examine the issue of an inequality measure's sensitivity to transfers in different parts of the distribution and Barrett and Salles (1998) discuss classes of inequality measures characterized by their behaviour under income transfers; Lambert and Lanza (2006) analyse the effect on inequality of changing isolated incomes. The Atkinson and generalized entropy families are examples of the application of the concept of the quasi-linear mean, which is discussed in Hardy *et al*. (1934, 1952) and Chew (1983).

Distributional principles that can be applied when households are not homogeneous are discussed in Ebert (2007) and Shorrocks (2004). The axiomatic approach to inequality measurement discussed on page 66 is not of course restricted to the generalized entropy family; with a suitable choice of axiom the approach can be extended to pretty well any inequality measure you like: for example see Thon's (1982) axiomatization of the Gini coefficient, or Foster (1983) on the Theil index. The validity of standard axioms when viewed in the light of people's perceptions of inequality is examined in Amiel and Cowell (1992, 1994a, 1999) and Cowell (1985a); for a discussion and survey of this type of approach see Amiel (1999) and Kampelmann (2009). The problematic cases highlighted in the examples on page 38 and 65 are based on Cowell (1988a). Ebert (1988) discusses the principles on which a generalized type of the relative mean deviation may be based and Ebert (2009) (p.187) addresses ways of axiomatizing inequality that will be consistent with the apparently heterodox views illustrated in Question 4 on page 75.

The normative significance of decomposition is addressed by Kanbur (2006). Examples of approaches to inequality measurement that explicitly use criteria that may conflict with decomposability include basing social welfare on income satisfaction in terms of ranks in the distribution (Hempenius 1984), the use of income gaps (Preston 2007), the use of reference incomes to capture the idea of individual ‘complaints’ about income distribution (Cowell and Ebert 2004, Devooght 2003, Temkin 1993)— see also the discussion on page 195.

# B.4 Chapter 4

## The Idea of a Model

For an excellent coverage of the use of functional forms in modelling income distributions see Kleiber and Kotz (2003).

## The Lognormal Distribution

Most texts on introductory statistical theory give a good account of the normal distribution—for example Berry and Lindgren (1996), Casella and Berger (2002), or Freund and Perles (2007). The standard reference on the lognormal and its properties (Aitchison and Brown 1957) also contains a succinct account of a simple type of random process theory of income development. A summary of several such theories can be found in Bronfenbrenner (1971) and in Brown (1976). On some of the properties of the lognormal Lorenz curve, see also Aitchison and Brown (1954).

## The Pareto Distribution

An excellent introduction to Pareto's law is provided by Persky (1992). Pareto's original work can be consulted in Pareto (1896, 1965, 2001) or in Pareto (1972), which deals in passing with some of Pareto's late views on the law of income distribution; the development of Pareto's thought on inequality is discussed in Maccabelli (2009). Tawney (1964) argues forcefully against the strict interpretation of Pareto's law:

It implies a misunderstanding of the nature of economic laws in general, and of Pareto's laws in particular, at which no one, it is probable, would have been more amused than Pareto himself, and which, indeed, he expressly repudiated in a subsequent work. It is to believe in economic Fundamentalism, with the New Testament left out, and the Books of Leviticus and Deuteronomy inflated to unconscionable proportions by the addition of new and appalling chapters. It is to dance naked, and roll on the ground, and cut oneself with knives, in honour of the mysteries of Mumbo Jumbo.

However, I do not find his assertion of Pareto's recantation convincing—see Pareto (1972); see also Pigou (1952, pp. 650 ff). Oversimplified interpretations of the law have persisted—Adams (1976) suggested a ‘golden section’ value of $\alpha =2/\left[\sqrt{5}-1\right]$ as a cure for inflation. Van der Wijk's (1939) law is partially discussed in Pen (1974, Chapter 6); in a sense it is a mirror image of the Bonferroni index (Bonferroni 1930) which is formed from an average of ‘lower averages’—see Chakravarty (2007). Several of the other results in the text are formally proved in Chipman (1974). (p.188) Nicholson (1969, pp. 286–92) and Bowman (1945) give a simple account of the use of the Pareto diagram. The discussion of a random process model leading to a Pareto distribution is presented in Champernowne (1953, 1973) and the non-technical reader will find a simple summary in Pen (1971, 1974). The Pareto distribution as an equilibrium distribution of a wealth model is treated in Wold and Whittle (1957) and Champernowne and Cowell (1998), Chapter 10. A recent overview of Pareto-type distributions in economics and finance is provided by Gabaix (2008).

## How Good Are the Functional Forms?

The example of earnings displayed on page 96 can be reproduced from file ‘NES’ on the website; the income example of page 88 is taken from the website file ‘ET income distribution’ again, and the wealth example on page 97 is based on file ‘IR wealth’. Evidence on the suitability of the Pareto and lognormal distributions as approximations to actual distributions of earnings and of income can be found in the Royal Commission on the Distribution of Income and Wealth (1975, Appendix C; 1976, Appendix E).

In discussing the structure of wages in Copenhagen in 1953 Bjerke (1970) showed that the more homogenous the occupation, the more likely it would be that the distribution of earnings within it was lognormal. Weiss (1972) shows the satisfactory nature of the hypothesis of lognormality for graduate scientists' earnings in different areas of employment—particularly for those who were receiving more than $10,000 a year. Hill (1959) shows that merging normal distributions with different variances leads to ‘leptokurtosis’ (more of the population in the ‘tails’ than expected from a normal distribution)—a typical feature of the distribution of the logarithm of income. Other useful references on the lognormal distribution in practice are Fase (1970), Takahashi (1959), and Thatcher (1968). Evidence for lognormality is discussed in the case of India (Rajaraman 1975), Kenya (Kmietowicz and Webley 1975), Iraq (Kmietowicz 1984), and China (Kmietowicz and Ding 1993). Kmietowicz (1984) extends the idea of lognormality of the income distribution to bivariate lognormality of the joint distribution of income and household size. Battistin *et al*. (2009) demonstrate that consumption is ‘more lognormal’ than income and explain the economic reasons for this phenomenon.

Atkinson (1975) and Soltow (1975) produce evidence on the Pareto distribution and the distribution of wealth in the UK and the USA of the 1860s respectively. Klass *et al*. (2006) do this using the Forbes 400; Clementi and Gallegati (2005) examine Pareto's law for Germany, the UK, and the USA. For further evidence on the variability of Pareto's *α* in the USA, see Johnson (1937), a cautious supporter of Pareto. The Paretian property of the tail of the wealth distribution is also demonstrated admirably by the Swedish data examined by Steindl (1965) where *α* is about 1.5 to 1.7.

Some of the less orthodox applications of the Pareto curve are associated with ‘Zipf's law’ (Zipf 1949) which has been fruitfully applied to the distribution of city size (Nitsch 2005). Harold T. Davis, who has become famous for his theory of the French Revolution in terms of the value of Pareto's *α* under Louis XVI, produces further evidence on the Pareto law in terms of the distribution of wealth in the pre- Civil War southern states (wealth measured in terms of number of slaves) and of the
(p.189)
distribution of income in England under William the Conqueror—see Davis (1954). For the latter example (based on the Domesday Book, 1086) the fit is surprisingly good, even though income is measured in ‘acres’—i.e. that area of land which produces 72 bushels of wheat per annum. The population covered includes Cotters, Serfs, Villeins, Sokemen, Freemen, Tenants, Lords and Nobles, Abbots, Bishops, the Bishop of Bayeux, the Count of Mortain, and of course King William himself.

However, Davis's (1941) interpretation of these and other intrinsically interesting historical excursions as evidence for a ‘mathematical theory of history’ seems mildly bizarre: supposedly if *α* is too low or too high a revolution (from the left or the right, respectively) is induced. Although there is clearly a connection between extreme economic inequality and social unrest, seeking the mainspring of the development of civilization in the slope of a line on a double-log graph does not appear to be a rewarding or convincing exercise. There is a similar danger in misinterpreting a dynamic model such as of Champernowne (1953), in which a given pattern of social mobility always produces, eventually, a unique Pareto distribution, independent of the income distribution originally prevailing. Bernadelli (1944) postulates that a revolution having redistribution as an aim will prove futile because of such a mathematical process. Finding the logical and factual holes in this argument is left as an exercise for you.

## Other Distributions

Finally, a mention of other functional forms that have been claimed to fit observed distributions more or less satisfactorily (see the Technical Appendix page 158). Some of these are generalizations of the lognormal or Pareto forms, such as the three- parameter lognormal (Metcalf 1969), or the generalized Pareto–Levy law, which attempts to take account of the lower tail (Arnold 1983, Mandelbrot 1960). Indeed, the formula we have described as the Pareto distribution was only one of many functions suggested by Pareto himself; it may thus be more accurately described as a ‘Pareto type I’ distribution (Hayakawa 1951, Quandt 1966). Champernowne (1952) provides a functional form which is close to the Pareto in the upper tail and which fits income distributions quite well; some technical details on this are discussed in Harrison (1974), with empirical evidence in Thatcher (1968)—see also Harrison (1979, 1981) and Sarabia *et al*. (1999).

Other suggestions are Beta distribution (Slottje 1984, Thurow 1970), the Gamma distribution (Salem and Mount 1974, McDonald and Jensen 1979), the sech^{2}- distribution, which is a special case of the Champernowne (1952) distribution (Fisk 1961), and the Yule distribution (Simon 1955, 1957; Simon and Bonini 1958); see also Campano (1987) and Ortega *et al*. (1991). Evans *et al*. (1993) and Kleiber and Kotz (2003) provide a very useful summary of the mathematical properties of many of the above. The Singh and Maddala (1976) distribution is discussed further in Cramer (1978), Cronin (1979), McDonald and Ransom (1979), Klonner (2000) (first-order dominance), and Wilfling and Krämer (1993) (Lorenz curves); cf also the closely related model by Dagum (1977). A generalized form of the Gamma distribution has been used by Esteban (1986), Kloek and Van Dijk (1978), and Taille (1981). An overview of several of these forms and their interrelationships is given in
(p.190)
McDonald (1984) as part of his discussion of the generalized Beta distribution of the second kind; on this distribution see also Bordley *et al*. (1996), Jenkins (2009), Majumder and Chakravarty (1990), McDonald and Mantrala (1995), Parker (1999), Sarabia *et al*. (2002), Wilfling (1996), and for an implementation with Chinese data see Chotikapanich *et al*. (2007). Alternative approaches to parameterizing the Lorenz curve are discussed in Basmann *et al*. (1990, 1991), and Kakwani and Podder (1973).

Other functional forms based on the exponential distribution are considered in Jasso and Kotz (2007). Some of the Lorenz properties noted for the lognormal and for the Pareto hold for more general functional forms—see Arnold *et al*. (1987) and Taguchi (1968).

# B.5 Chapter 5

## The Data

The UK data used for Fig. 5.1 are from Inland Revenue Statistics (see file ‘IR income’ on the website), and the US data in Table 5.1 from Internal Revenue Service, *Statistics of Income: Individual Tax Returns* (see file ‘IRS Income Distribution’). The UK data used for Figs 5.2–5.7 are taken from the *Households Below Average Income* dataset (HBAI), which is now the principal data source for UK income distribution; summary charts and results are published in Department of Work and Pensions (2009).

For a general introduction to the problem of specifying an income or wealth variable see Atkinson (1983). The quality of the *administrative data* on personal incomes—derived from tax agencies or similar official sources—depends crucially on the type of tax administration and government statistical service for the country in question. On the one hand extremely comprehensive and detailed information about income and wealth (including cross-classifications of these two) is provided, for example, by the Swedish Central Statistical Bureau, on the basis of tax returns. On the other, one must overcome almost insuperable difficulties where the data presentation is messy, incomplete, or designedly misleading. An excellent example of the effort required here is provided by the geometric detective work of Wiles and Markowski (1971) and Wiles (1974) in handling Soviet earnings distribution data. Fortunately for the research worker, some government statistical services modify the raw tax data so as to improve the concept of income and to represent low incomes more satisfactorily. Stark (1972) gives a detailed account of the significance of refinements in the concepts of income using the UK data; for an exhaustive description of these data and their compilation see Stark in Atkinson *et al*. (1978) and for a quick summary, the Royal Commission on the Distribution of Income and Wealth (1975, Appendices F and H). For a discussion of the application of tax data to the analysis of top incomes see Atkinson (2007b). As for *survey data* on incomes, the HBAI in the UK draws on the Family Resources Survey and Family Expenditure Survey—see Frosztega (2000) for a detailed consideration of the underlying income concept: UK datasets are available from the UK Data Archive (http://www.data-archive.ac.uk). Summary charts and results for HBAI are published in Department of Work and Pensions (2009) and Brewer *et al*. (2008) provide a useful critique of this source. On the widely used Current Population Survey (CPS) data (see Question 3 in Chapter 2) in the USA
(p.191)
see Burkhauser *et al*. (2004) and Welniak (2003). A general overview of inequality in the USA is provided in Bryan and Martinez (2008), Reynolds (2006), and Ryscavage (1999). On US data and the quality of sample surveys in particular it is worth checking the two classic references Budd and Radner (1975) and Ferber *et al*. (1969). Since publication of the first edition of this book, large comprehensive datasets of individual incomes have become much more readily available and it is impossible to do justice to them. One that deserves attention from the student of inequality are the early example based on data from the Internal Revenue Service and Survey of Economic Opportunity discussed in Okner (1972, 1975); an extremely useful source of internationally comparable microdata in incomes (and much else) is the Luxembourg Income Study (http://www.lis-project.org). An early and comprehensive source of US longitudinal data is the Panel Study of Income Dynamics (http://psidonline.isr.umich.edu/) described in Hill (1992); more recent European examples of longitudinal data are the British Household Panel Survey (http://www.iser.essex.ac.uk/survey/bhps) and the German Socio-Economic Panel (http://www.diw-berlin.de/de/soep). The classic reference on wealth data in the UK is Atkinson and Harrison (1978) and an important resource for international comparisons of wealth distributions is provided by the Luxembourg Wealth Study (Sierminska *et al*. 2006), OECD (2008) Chapter 10.

A good statement of principles concerning the income concept is provided by the Canberra Group (2001) report. Several writers have tried to combine theoretical sophistication with empirical ingenuity to extend income beyond the conventional definition. Notable among these are the income-cum-wealth analysis of Weisbrod and Hansen (1968), and the discussion by Morgan *et al*. (1962) of the inclusion of the value of leisure time as an income component. An important development for international comparisons is the Human Development Index which has income as just one component (Anand and Sen 2000); Fleurbaey and Gaulier (2009) in similar spirit propose a measure of living standards for international comparisons based on GDP per capita adjusted for personal and social characteristics including inequality; perhaps unsurprisingly the ranking of countries by this measure differs substantially from the conventional GDP ranking. Goodman and Oldfield (2004) contrast income inequality and expenditure inequality in the UK context. Stevenson and Wolfers (2008) examine the way inequality in happiness has changed in the USA.

In Morgan (1962), Morgan *et al*. (1962), and Prest and Stark (1967) the effect of family grouping on measured inequality is considered. For a fuller discussion of making allowance for income sharing within families and the resulting problem of constructing ‘adult equivalence’ scales, consult Abel-Smith and Bagley (1970); the internationally standard pragmatic approach to equivalization is the OECD scale (see, for example, Atkinson *et al*. 1995) although many UK studies use a scale based on McClements (1977); the idea that equivalence scales are revealed by community expenditures is examined in Olken (2005). The relationship between equivalence scales and measured inequality is examined in Buhmann *et al*. (1988), Coulter *et al*. (1992b), and Jenkins and Cowell (1994b): for a survey see Coulter *et al*. (1992a). The fact that averaging incomes over longer periods reduces the resulting inequality statistics emerges convincingly from the work of Hanna *et al*. (1948) and
(p.192)
Benus and Morgan (1975). The key reference on the theoretical and empirical importance of price changes on measured inequality is Muellbauer (1974); see also Crawford and Smith (2002), Hobijn and Lagakos (2005), and Slottje (1987). A further complication which needs to be noted from Metcalf (1969) is that the way in which price changes affect low-income households may depend on household composition; whether there is a male bread-winner present is particularly important. On the effect of non- response on income distribution and inequality refer to Korinek *et al*. (2006).

International comparisons of datasets on inequality and poverty are provided by Ferreira and Ravallion (2009); an early treatment of the problems of international comparison of data is found in Kuznets (1963, 1966) and Atkinson and Brandolini (2009b) provide an excellent general introduction to issues of data quality. Appropriate price adjustments to incomes can be especially problematic when making international comparisons. A standard approach is to use an index of *Purchasing Power Parity* (PPP) rather than converting incomes at nominal exchange rates. The issues involved in constructing PPP are treated in Heston *et al*. (2001); the method of imputation of PPP can have a substantial impact on estimates of between-country inequality and hence on the picture of global inequality; the topic is treated exhaustively in Anand and Segal (2008), Kravis *et al*. (1978a, 1978b), and Summers and Heston (1988, 1991). The issue of international comparability of income distribution data is one of the main reasons for the existence of the Luxembourg Income Study: see Smeeding *et al*. (1990) for an introduction and a selection of international comparative studies; Lorenz comparisons derived from this data source are in the website file ‘LIS comparison’. On the use of data in OECD countries see Atkinson and Brandolini (2001) and on international comparisons of earnings and income inequality refer to Gottschalk and Smeeding (1997). Atkinson and Micklewright (1992) compare the income distributions in Eastern European economies in the process of transition. Other important international sources for studying inequality are Deininger and Squire (1996) and also UNU-WIDER (2005) which provides Gini indices drawn from a large number of national sources.

Beckerman and Bacon (1970) provide a novel approach to the measurement of world (i.e. inter-country) inequality by constructing their own index of ‘income per head’ for each country from the consumption of certain key commodities. Becker *et al*. (2005) examine the effect on trends in world inequality of trying to take into account people's quality of life.

## Computation of the Inequality Measures

For detail on computation of point estimates of inequality go to the Technical Appendix. For an excellent general text on empirical methods including computation of inequality measures and other welfare indicators see Deaton (1997). For a discussion of how to adapt standard methodology to estimation problems in small areas with few observations see Tarozzi and Deaton (2009).

Decomposition techniques have been widely used to analyse spatial inequality (Shorrocks and Wan 2005) including China (Yu *et al*. 2007) and Euroland (Beblo and Knaus 2001) and for the world as a whole (Novotný 2007). For a systematic analysis of world inequality using (fully decomposable) generalized entropy indices
(p.193)
see Berry *et al*. (1983a, 1983b), Bourguignon and Morrisson (2002), Sala-i-Martin (2006), Ram (1979, 1984, 1987, 1992), and Theil (1979b, 1989); Milanovic and Yitzhaki (2002) use the (not fully decomposable) Gini coefficient.

## Appraising the Calculations

An overview of many of the statistical issues is to be found in Cowell (1999) and Nygård and Sandström (1981, 1985). If you are working with data presented in the conventional grouped form, then the key reference on the computation of the bounds *J*_{L}, *J*_{U} is Gastwirth (1975). Now, in addition to the bounds on inequality measures that we considered in the text, Gastwirth (1975) shows that if one may assume ‘decreasing density’ over a particular income interval (i.e. the frequency curve is sloping downwards to the right in the given income bracket) then one can calculate bounds *j*_{L}′, *j*_{U}′ that are sharper—i.e. the bounds *j*_{L}′, *j*_{U}′ lie within the range of inequality values (*j*_{L}, *j*_{U}) which we computed: the use of these refined bounds leaves the qualitative conclusions unchanged, though the proportional gap is reduced a little. The problem of finding such bounds is considered further in Cowell (1991). The special case of the Gini coefficient is treated in Gastwirth (1972) and McDonald and Ransom (1981); the properties of bounds for grouped data are further discussed in Gastwirth *et al*. (1986); Mehran (1975) shows that you can work out bounds on *G* simply from a set of sample observations on the Lorenz curve without having to know either mean income or the interval boundaries *a*_{1}, *a*_{2},…, *a*_{k+1} and Hagerbaumer (1977) suggests the upper bound of the Gini coefficient as an inequality measure in its own right. In Gastwirth (1972, 1975) there are also some refined procedures for taking into account the open-ended interval forming the top income bracket—an awkward problem if the total amount of income in this interval is unknown. Ogwang (2003) discusses the problem of putting bounds on Gini coefficient when data are sparse. As an alternative to the methods discussed in the Technical Appendix (using the Pareto interpolation, or fitting Paretian density functions), the procedure for interpolating on Lorenz curves introduced by Gastwirth and Glauberman (1976) works quite well.

Cowell and Mehta (1982) investigate a variety of interpolation methods for grouped data and also investigate the robustness of inequality estimates under alternative grouping schemes. Aghevli and Mehran (1981) address the problem of optimal choice of the income interval boundaries used in grouping by considering the set of values {*a*_{1}, *a*_{2},…, *a*_{k}} which will minimize the Gini coefficient; Davies and Shorrocks (1989) refine the technique for larger datasets.

For general information on the concept of the standard error see Berry and Lindgren (1996) or Casella and Berger (2002). On the sampling properties of inequality indices generally see Victoria-Feser (1999). Formulas for standard errors of specific inequality measures can be found in the following references: Kendall *et al*. (1994, sec. 10.5) (relative mean deviation, coefficient of variation), David (1968, 1981), Nair (1936) (Gini's mean difference), Gastwirth (1974a) (relative mean deviation), Aitchison and Brown (1957, p. 39) (variance of logarithms). For more detailed analysis of the Gini coefficient see Davidson (2009), Deltas (2003), Gastwirth *et al*. (1986), Giles (2004), Glasser (1962), Lomnicki (1952), Modarres and Gastwirth (2006),
(p.194)
Nygård and Sandström (1989), Ogwang (2000, 2004), and Sandström *et al*. (1985, 1988). Allison (1978) discusses issues of estimation and testing based on microdata using the Gini coefficient, coefficient of variation, and Theil index. The statistical properties of the generalized entropy and related indices are discussed by Cowell (1989) and Thistle (1990). A thorough treatment of statistical testing of Lorenz curves is to be found in Beach and Davidson (1983), Beach and Kaliski (1986), Beach and Richmond (1985), and Davidson and Duclos (2000); for generalized Lorenz estimation refer to Bishop *et al*. (1989), and Bishop *et al*. (1989). See also Hasegawa and Kozumi (2003) for a Bayesian approach to Lorenz estimation and Schluter and Trede (2002) for problems of inference concerning the tails of Lorenz curves. For a treatment of the problem of estimation with complex survey design go to Biewen and Jenkins (2006), Cowell and Jenkins (2003), Binder and Kovacevic (1995), Bhattacharya (2007), and Kovacevic and Binder (1997). Cowell and Victoria-Feser (2003) treat the problem of estimation and inference when the distribution may be censored or truncated and Cowell and Victoria-Feser (2007, 2008) discuss the use of a Pareto tail in a ‘semi- parametric’ approach to estimation from individual data. The effect of truncation bias on inequality judgements is also discussed in Fichtenbaum and Shahidi (1988) and Bishop *et al*. (1994); the issue of whether ‘top-coding’ (censoring) of the CPS data makes a difference to the estimated trends in US income inequality is analysed in Burkhauser *et al*. (2008). So-called ‘bootstrap’ or resampling methods are dealt with by Biewen (2002), Davidson and Flachaire (2007), and Van Kerm (2002)—see also Davison and Hinkley (1997). For an interesting practical example of the problem of ranking distributions by inequality when you take into account sampling error, see Horrace *et al*. (2008).

On the robustness properties of measures in the presence of contamination or outliers see Cowell and Victoria-Feser (1996, 2002, 2006) and for the way inequality measures respond to extreme values go to Cowell and Flachaire (2007). Chesher and Schluter (2002) discuss more generally the way measurement errors affect the comparison of income distributions in welfare terms.

## Fitting Functional Forms

Refer to Chotikapanich and Griffiths (2005) on the problem of how to choose a functional form for your data and to Maddala and Singh (1977) for a general discussion of estimation problems in fitting functional forms. Ogwang and Rao (2000) use hybrid Lorenz curves as a method of fit. If you want to estimate lognormal curves from grouped or ungrouped data, you should refer to Aitchison and Brown (1957, pp. 38–43, 51–54) first. Baxter (1980), Likes (1969), Malik (1970), and Quandt (1966) deal with the estimation of Pareto's *α* for ungrouped data. Now the ordinary least squares method, discussed by Quandt, despite its simplicity has some undesirable statistical properties, as explained in Aigner and Goldberger (1970). In the latter paper you will find a discussion of the difficult problem of providing maximum likelihood estimates for *α* from grouped data. The fact that in estimating a Pareto distribution a curve is fitted to cumulative series which may provide a misleadingly good fit was noted in Johnson (1937), while Champernowne (1956) provided the warning about uncritical use of the correlation coefficient as a criterion of suitability
(p.195)
of fit. The suggestion of using inequality measures as an alternative basis for testing goodness-of-fit was first put forward by Gastwirth and Smith (1972), where they test the hypothesis of lognormality for United States IRS data; see also Gail and Gastwirth (1978b, 1978a). To test for lognormality one may examine whether the skewness and the kurtosis (‘peakedness’) of the observed distribution of the logarithms of incomes are significantly different from those of a normal distribution; for details consult Kendall *et al*. (1999). Hu (1995) discusses the estimation of Gini from grouped data using a variety of specific functional forms.

# B.6 Technical Appendix

For a general technical introduction see Duclos and Araar (2006) and Cowell (2000); functional forms for distributions are discussed in Kleiber and Kotz (2003) and Evans *et al*. (1993).

The formulas in the Technical Appendix for the decomposition of inequality measures are standard—see Bourguignon (1979), Cowell (1980), Das and Parikh (1981, 1982), and Shorrocks (1980).

For a characterization of some general results in decomposition, see Bosmans and Cowell (2010), Chakravarty and Tyagarupananda (1998, 2000), Cowell (2006), Foster and Shneyerov (1999), Kakamu and Fukushige (2009), Toyoda (1980), Shorrocks (1984, 1988), and Zheng (2007). Establishing the main results typically requires the use of functional equations techniques, on which see Aczél (1966). For applications of the decomposition technique, see the references on spatial and world inequality in Chapter 5 (page 192) and also Anand (1983), Borooah *et al*. (1991), Ching (1991), Cowell (1984, 1985b), Frosini (1989), Glewwe (1986), Mookherjee and Shorrocks (1982), and Paul (1999).

Decomposition by income components is discussed by Satchell (1978), Shorrocks (1982), and Theil (1979a). Applications to Australia are to be found in Paul (2004), to New Zealand in Podder and Chatterjee (2002), and to UK in Jenkins (1995). The issues underlying an application of the *Shapley value* to decomposition analysis are examined in Sastre and Trannoy (2002). The use of partitions into subgroups as a method of ‘explaining’ the contributory factors to inequality is dealt with in Cowell and Jenkins (1995) and Elbers *et al*. (2008). Alternative pragmatic approaches to accounting for changes in inequality are provided by Bourguignon *et al*. (2008), Morduch and Sicular (2002), Fields (2003), and Jenkins and Van Kerm (2005); Cowell and Fiorio (2009) reconcile these alternatives with conventional decomposition analysis.

The relationship between decomposition of inequality and the measurement of poverty is examined in Cowell (1988b). As noted in Chapter 3 the decomposition of the Gini coefficient presents serious problems of interpretation. However, Pyatt (1976) tackles this by ‘decomposing’ the Gini coefficient into a component that represents within-group inequality, one that gives between-group inequality, and one that depends on the extent to which income distributions in different groups overlap one another. The properties of the Gini when ‘decomposed’ in this way are further discussed by Lambert and Aronson (1993), Lerman and Yitzhaki (1984, 1989), Yitzhaki and Lerman (1991), and Sastry and Kelkar (1994). Braulke (1983) (p.196) examines the Gini decomposition on the assumption that within-group distributions are Paretian. Silber (1989) discusses the decomposition of the Gini coefficient by subgroups of the population (for the case of non-overlapping partitions) and by income components.

The data in Table A.3 is based on Howes and Lanjouw (1994) and Hussain *et al*. (1994). For recent decomposition analysis of China, see Kanbur and Zhang (1999, 2005), Lin *et al*. (2008), and Sicular *et al*. (2007).

## Notes:

(^{1})
But only in this respect, since they reject the Lorenz curve as an ‘inept choice’, preferring to use histograms instead.