Anthony F. Shorrocks

THE CLASS OF ADDITIVELY DECOMPOSABLE INEQUALITY MEASURES

This paper considers a wide class of inequality indices and identifies those which are additively decomposable. The sub-class of mean independent, additively decomposable measures turns out to be a single parameter family which includes the squared coefficient of variation and the two Theils entropy formulas.

Inequality Decomposition by Factor Components

This paper disaggregates the income of individuals or households into different factor components, such as earnings, investment income, and transfer payments, and considers how to assess the contributions of these sources to total income inequality. In the approach adopted, a number of basic principles of decomposition are proposed and their implications for the assignment of component contributions are examined.

Inequality Decomposition by Population Subgroups

This paper examines the implications of imposing a weak aggregation condition on inequality indices, so that the overall inequality value can be computed from information concerning the size, mean, and inequality value of each population subgroup. It is shown that such decomposable inequality measures must be monotonic transformations of additively decomposable indices. The general functional form of decomposable indices is derived without assuming that the measures are differentiable. The analysis is suitable for extension to the many other kinds of indices for which a similar relationship between the overall index value and subaggregates is desirable.

Income inequality and income mobility

The usual indices of inequality are derived from observations on income, wealth etc. corresponding to a particular point or period of time, It has been frequently argued that inequality values by themselves do not accurately reflect the differences between individuals, since the true situation depends to a large extent on how the relative positions of individuals vary over time. Thus, it has been argued, “static” measures of inequality should be supplemented by “dynamic” measures of changes through time, which we shall call measures of mobility. Studies which have proposed ways of quantifying these dynamic changes broadly fall into two categories: those which use elementary statistics, such as the correlation coefficient; and those which make more sophisticated suggestions based on transition matrices and other simple stochastic specifications of dynamic processes. Shorrocks [9] provides a number of references and discusses some of the issues involved in deriving an index of mobility from transition matrices. Particular consideration is given to the interval of time between observations, since a relationship is expected between the amount of observed movement and the length of time over which movement can take place; in a short space of time there is little opportunity for movement, even if the society is inherently very mobile. These earlier attempts to define an index of mobility are mainly concerned with stock variables, interpreted in a wide sense to include social status and occupation as well as wealth and the assets of firms. Once attention is turned to flow variables, such as income, it becomes apparent that there is another important consideration. Observed variations in income depend not only on the interval between observations, but also on the length of the accounting period chosen for incomes. Data availability and custom dictate that the period selected is normally one year, although shorter intervals, a week or a month, are occasionally used. If the accounting period were extended from, say, one month to one year, variations in monthly incomes (previously classified as dynamic changes) become subsumed within the annual income figure. Some of the dynamic changes are therefore incorporated in the static inequality value, and the distinction between the static and dynamic aspects becomes very blurred. Similarly, as we pass from annual to lifetime income inequality, intra-lifetime income mobility is lost in the process of aggregation. However, the effects of income variations over time do not disappear altogether: they are reflected in the changes recorded in the inequality value. Those occupying the highest and lowest positions in the income hierarchy rarely remain there forever. So the aggregation of incomes over time tends to improve the relative position of those temporarily found at the bottom of the distribution, and the situation of those at the top tends to deteriorate. For this reason it is commonly supposed that inequality falls as the accounting period is lengthened. Empirical confirmation of this relationship requires longitudinal income data samples, of which very few exist. However, the little evidence available agrees with expectati0ns.l For example, Soltow [l0] traced the annual incomes of a sample of Norwegians over the period 1928-l960. The Gini coefficient for the 33 years combined was 0.134 compared to an average value of 0.183 for the separate years. Using US data, Kohen et al. [3] found that the Gini coefficient for family income and earnings of young men (aged 16-24) fell by 4.7-7.4 “/,, when cumulated over two years, and by 9.2-10.8 % when cumulated over three. For middle-aged men (4559 years old), aggregating incomes over two years caused the Gini to decline by about 4 %.” There are reasonable grounds, therefore, for supposing that the existence of mobility causes inequality to decline as the accounting interval grows. Furthermore, intuition suggests that the extent to which inequality declines will be directly related to the frequency and magnitude of relative income variations. If the income structure exhibits little mobility, relative incomes will be left more or less unaltered over time and there will be no pronounced egalitarian trend as the measurement period increases. In contrast, inequality may be expected to decrease significantly in a very (income) mobile society. The main purpose is to exploit this relationship between mobility and inequality, to derive an index of mobility for flow variables. In essence, mobility is measured by the extent to which the income distribution is equalized as the accounting period is extended. Defining Mobility as the complement of rigidity, as much as we define equality as the complement of inequality. For inequality measures with the desirable properties.

The distribution of wealth

This chapter is concerned with the distribution of personal wealth, which usually refers to the material assets that can be sold in the marketpace, although on occasion pension rights are also included. We summarise the available evidence on wealth distribution for a number of countries. This confirms the well known fact that wealth is more unequally distributed than income, and points to a long term downward trend in wealth inequality over most of the twentieth century. We also review the various theories that help account for these feature. Lifecycle accumulation is one popular explanation of wealth differences, but inheritance is also widely recognised as playing a major role, especially at the upper end of the wealth range. A recurrent theme in work on wealth distribution is the relative importance of these two sources of wealth differences. We discuss the results of studies that assess the contributions of inheritance and lifecycle factors, and give attention also to a variety of related issues, such as the link between wealth status across generations, and the possible motives for leaving bequests.

The Impact of Income Components on the Distribution of Family Incomes

Recent studies attempt to quantify the extent to which income inequality is due to income components, such as earnings, investment income and transfer income. This paper examines the fundamental issues and problems involved in assigning inequality contributions to components. Different decompositions are applied to U.S. family income data. A wide range of contributions can be obtained, even when naturally derived decomposition rules are used. Some absurd results obtained serve to warn against indiscriminate use of decomposition formula.

Poverty Orderings and Welfare Dominance

This paper examines the partial orderings of discrete distributions derived from various poverty indices and sets of welfare functions. The poverty ordering with respect to some indexP is the ordering obtained whenP ranks consistently over a range of admissible poverty lines. The poverty orderings derived from the headcount ratio, the per-capita income gap and another “distribution-sensitive” index are characterized in some detail when the poverty standard is allowed to take any positive value, and these orderings are shown to coincide with the natural interpretation of first, second and third degree “welfare dominance”, respectively. Additional results are then obtained for the situation in which the admissible poverty lines cannot exceed some finite upper bound.

Revisiting the Sen Poverty Index

A modification of the official Sen poverty index yields an index which is not only continous in individual incomes and consistent with the transfer axiom, but also admits a geometric interpretation in terms of the area beneath a graph called the inverse generalized Lorenz curve for normalized poverty gaps by Jenkins and Lambert (1993) or the poverty gap profile by Shorrocks (1994).

The Age-Wealth Relationship: A Cross-Section and Cohort Analysis

THE age profile of individual asset holdings is frequently supposed to follow a hump pattern, increasing during the working lifetime and declining in later years. The theoretical explanation of such a relationship is firmly established, since it is a characteristic feature of life-cycle saving models. However, the empirical evidence has never been critically examined. A number of studies based on sample survey information have been regarded as confirming this hump pattern, but on closer examination the evidence is far from conclusive. Moreover the survey results conflict fundamentally with alternative estimates of the age-wealth relationship derived from estate tax data. This paper begins in section II by examining this basic conflict. In the following section the relevance of cross-section studies is questioned, and a cohort is identified whose lifetime wealth characteristics can be studied. The sequence of observed wealth distributions for this cohort is obtained by selecting successive ten years age groups from the Estate duty statistics at intervals of a decade. Section IV is devoted to an analysis of the variation in the composition of any cohort as it ages. These composition changes arise from the fact that wealthier individuals have a lower mortality rate and therefore tend to become a larger proportion of the surviving cohort independently of accumulation behaviour. In these circumstances the characteristics of the representative individual will not necessarily correspond to the representative behaviour of the group. It is, however, possible to correct for this change in composition so that the empirical estimates can be compared with the predictions of economic theory.

A Decomposition Analysis of Regional Poverty in Russia

The paper applies a new decomposition technique to the study of variations in poverty across the regions of Russia. The procedure, which is based on the Shapley value in cooperative game theory, allows the deviation in regional poverty levels from the all-Russia average to be attributed to three proximate sources: per capita income, inequality, and local prices. Contrary to expectation, regional poverty variations turn out to be due more to differences in inequality across regions than to differences in real income per capita. However, when real income per capita is split into nominal income and price components, differences in nominal incomes emerge as more important than either inequality or price effects for the majority of regions.