Chapter summary

Inequality and poverty provide two alternative perspectives upon the equity of income distribution. Inequality of income means that some households have higher incomes than others. Poverty exists when some households are too poor to achieve an acceptable standard of living. An inequality measure is a means of assigning a single number to the observed income distribution that reflects its degree of inequality. A poverty measure achieves the same for poverty. The starting point of the chapter is a discussion of income. There are two aspects to this: the definition of income and the comparison of income across households with different compositions. If two households differ in their composition, a direct comparison of their income levels will reveal little about the standard of living they achieve. Equivalence scales are used to adjust for composition. We review the use of equivalence scales and some of the issues that they raise. A number of the commonly-used measures of inequality and poverty are discussed and their properties investigated. Importantly, the link is drawn between measures of inequality and the welfare assumptions that are implicit within them.

13.1 Measuring income

A simple definition of income is that it is the additional resources a household receives over a given period of time. Income is a flow, so the period over which measurement takes place must be specified. Evaluating the receipt of resources is the basis of the definition used in the assessment of income for tax purposes and works in a practical setting. What an economist needs in order to understand behaviour, especially when choices are made in advance of income being received, is a forward- looking measure of income. If the flow of income is certain, then there is no distinction between backward- and forward-looking measures. It is when income is uncertain that differences emerge.

The classic backward-looking definition of income was provided by Simons in 1938. This definition is ‘Personal income may be defined as the algebraic sum of (1) the market value of rights exercised in consumption and (2) the change in the value of the store of property rights between the beginning and end of the period in question’. The essential feature of this definition is that it makes an attempt to be inclusive so as to incorporate all income regardless of the source.

A divergence arises between this definition and that used in practice arises through the practical difficulties of assessing some sources of income especially those arising from capital gains. According to the Simons’ definition, the increase in the value of capital assets should be classed as
income. However, if the assets are not liquidated, the capital gain will not be realised during the period in question and will not be received as an income flow.

A different perspective is to view the level of income by the benefits it can deliver. Building upon this idea, Hicks provided what is generally taken as the standard definition of income with uncertainty. This definition states that ‘income is the maximum value which a man can consume during a
week and still expect to be as well-off at the end of the week as he was at the beginning’.

This definition can clearly cope with uncertainty since it operates in terms of expectations. But this advantage is also its major shortcoming when a move is made towards application. Expectations may be ill-defined or even irrational, so evaluation of the expected income flow may be unreasonably
high or low. A literal application of the definition would not count windfall gains, such as unexpected gifts or lottery wins, as income, because they are not expected despite such gains clearly raising the potential level of consumption. For these reasons, the Hicks definition of income is
informative but not perfect.

These alternative definitions of income have highlighted the distinctions between ex ante and ex post measures. Assessments of income for tax purposes use the backward-looking viewpoint and measure income as all relevant payments received over the measurement period. Practical issues limit the extent to which some sources of income can be included, so the definition of income in tax codes does not precisely satisfy any of the formal definitions. This observation just reflects the fact that there is no unambiguously perfect definition of income.

13.2 Equivalence scales

The fact that households differ in size and age distribution means that welfare levels cannot be judged just by looking at income levels. A household of one adult with no children needs less income to achieve a given level of welfare than a household with two adults and one child. Equivalence scales are the tool for adjusting measured incomes into comparable quantities.

Differences between households arise in the number of adults and the number and ages of dependants. These are called demographic variables. The first approach to equivalence scales is based on the concept of **minimum needs**. A bundle of goods and services that is seen as representing the minimum needs for the household is identified. The cost of this bundle for families with different compositions is then calculated and the ratio of these costs for different families provides the equivalence scale.

There are three major shortcomings of this method of computing equivalence scales. Firstly, by focusing on the cost of meeting a minimum set of needs they are inappropriate for applying to incomes above the minimum level. Secondly, they are dependent upon an assessment of what constitutes minimum needs - and this can be contentious. Most importantly, the scales do not take into account the process of optimisation by the households. The consequence of optimisation is that as income rises substitution between goods can take place and the same relativities need no longer apply.

The Engel approach to equivalence scales is based on the hypothesis that the welfare of a household can be measured by the proportion of its income that is spent on food. This is a consequence of Engel’s law which asserts that the share of food in expenditure falls as income rises. If this is
accepted, equivalence scales can be constructed for households of different compositions by calculating the income levels at which their expenditure share on food is equal.

A further alternative is to select for attention a set of goods that are consumed only by adults, termed ‘adult goods’, and such that the expenditure upon them can be treated as a measure of welfare. Typical examples of such goods that have been used in practice are tobacco and alcohol. If these goods have the property that changes in household composition only affect their demand via an income effect (so changes in household composition do not cause substitution between commodities), then the extra income required to keep their consumption constant, when household composition changes, can be used to construct an equivalence scale.

The methods described so far have attempted to derive the equivalence scale from an observable proxy for welfare. A general approach which can, in principal, overcome the problems identified in the previous methods is illustrated. The outer indifference curve represents the consumption levels of these two goods necessary for a family of composition d 2 to obtain welfare level U* and the inner indifference curve the consumption requirements for a family with composition d 1 to obtain the same utility.

The construction of an equivalence scale from preferences makes two further issues become apparent. Firstly, the minimum needs and budget share approaches do not take account of how changes in family structure may shift the indifference map. With the utility approach it then becomes cheaper to attain each indifference curve, so that value of the equivalence scale falls as family size increases. The second problem centres around the use of a household utility function. Many economists would argue that a household utility function cannot exist; instead they would observe that households are composed of individuals with individual preferences. Under the latter interpretation, the construction of a household utility function suffers from the difficulties of preference aggregation identified by Arrow’s impossibility theorem. Among the solutions to this problem now being investigated is to look within the functioning of the household and to model its decisions as the outcome of an efficient resource allocation process.

13.3 Statistical inequality measures

The existence of inequality is easily perceived: differences in living standards between the rich and poor are only too obvious both across countries and, sometimes to a surprising extent, within countries. The substantive economic questions about inequality arise when we try to construct a quantitative measure of inequality. Without a quantitative measure it is not possible to provide a precise answer to questions about inequality.

The simplest conceivable measure, the range calculates inequality as being the difference between the highest and lowest incomes expressed as a proportion of total income. Any index that has
this property of independence is called a **relative index**. The range takes no account of the dispersion of the income distribution between the highest and the lowest incomes, so it is not sensitive to any features of the income distribution between these extremes. An ideal measure should possess more sensitivity to the value of intermediate incomes than the range.

The **relative mean deviation**, D, takes account of the deviation of each income level from the mean so that it is dependent upon intermediate incomes. It does this by calculating the absolute value of the deviation of each income level from the mean and then summing. This summation process gives equal weight to deviations both above and below the mean and implies that D is linear in the size of deviations.

This line of reasoning is captured by the **Pigou-Dalton principle of transfers**. The basis of this principle is the requirement that any transfer from a poor household to a rich one must increase inequality regardless of where the two households are located in the income distribution. Any
inequality measure that satisfies this principle is said to be sensitive to transfers. The Pigou-Dalton principle is generally viewed as a feature that any acceptable measure of inequality should possess and is therefore expected in an inequality measure.

Pigou-Dalton principle of transfers: The inequality index must decrease if there is a transfer of income from a richer household to a poorer household which preserves the ranking of the two households in the income distribution and leaves total income unchanged.

Before moving on to further inequality measures, it is worth describing the Lorenz curve which is a helpful graphical device for presenting a summary representation of an income. Although not strictly an inequality measure as defined above, Lorenz curves are considered because of their use in illustrating inequality and the central role they play in the motivation of other inequality indices.

The Lorenz curve is constructed by arranging the population in order of increasing income and then graphing the proportion of income going to each proportion of the population. The graph of the Lorenz curve therefore has the proportion of population on the horizontal axis and the proportion
of income on the vertical axis. If all households in the population had identical incomes the Lorenz curve would be the diagonal line connecting the points (0,0) and (1,1). If there is any degree of inequality, the ordering in which the households are taken ensures that the Lorenz curve lies below the diagonal since, for example, the poorest half of the population must have less than half total income.

The final measure, **the Gini**, has been the subject of extensive attention in discussions of inequality measurement and has been much used in applied economics. The Gini, G, can be expressed by considering all possible pairs of incomes and out of each pair selecting the minimum income level.
Summing the minimum income levels and dividing by H 2 u to ensure a value between 0 and 1. The Gini also satisfies the Pigou-Dalton principle. This can be seen by considering a transfer of income of size ∆M from household i to household j with the households chosen so that M j > M i . From the ranking of incomes this implies j > i. Then as required. The effect of the transfer of income upon the measure depends only on the locations of i and j in the income distribution. It might be expected that an inequality measure should be more sensitive to transfers between households low in the income distribution.

The Gini is equal to the area between the Lorenz curve and the line of equality as a proportion of the area of the triangle beneath the line of equality. This property of the Gini makes it clear that the Gini, in common with R, C and D, can be used to rank distributions when the Lorenz
curves cross since the relevant area is always well defined. Since all these measures provide a stronger ranking of income distributions than the Lorenz curve, they must each impose additional restrictions which allow a comparison to be made between distributions even when their Lorenz
curves cross.

13.4 Inequality and welfare

The analysis of the statistical measures of inequality has made reference to ‘acceptable’ criteria for a measure to possess. To be able to say something is acceptable or not implies that there is some notion of distributive justice or social welfare underlying the judgement. It is then interesting to consider the relationship between inequality measures and welfare.

The first issue to address is the extent to which income distributions can be ranked in terms of welfare with minimal restrictions imposed upon the
social welfare function. Let the level of social welfare be determined by the function W = W (M 1 ,..., M H ). It is assumed that this social welfare function is symmetric and concave. Symmetry means that the level of welfare is unaffected by changing the ordering of the households. Concavity ensures that the indifference curves of the welfare function have the standard shape with mixtures preferred to extremes.

The key theorem relating the ranking of income distributions to social welfare is now given.

Theorem: If the Lorenz curves for two income distributions with the same mean do not cross, every symmetric and concave social welfare function will assign a higher level of welfare to the distribution whose Lorenz curve is closest to the main diagonal.

The proof of this theorem is very straightforward. Hence the marginal social welfare of income is greater for a household lower in the income distribution. If the two Lorenz curves do not cross, the income distribution represented by the inner one (that closest to the main diagonal) can be obtained from that of the outer one by transferring income from richer to poorer households. Since the marginal social welfare of income to the poorer households is never less than that from richer, this transfer must raise welfare as measured by any symmetric and concave social welfare function.

The converse of this theorem is that if the Lorenz curves for two distributions cross, then two symmetric and concave social welfare functions can be found that will rank the two distributions differently. This is because the income distributions of two Lorenz curves that cross are not related by simple transfers from rich to poor. This permits the construction of the two social welfare functions, with different marginal social welfares, that will rank them differently. So, if the Lorenz curves do cross the income distributions cannot be unambiguously ranked without specifying the social welfare function. This shows that the Lorenz curve provides the most complete ranking of income distributions that is possible without assuming a specific social welfare function.

M EDE is called the **equally distributed** equivalent income and is that level of income that if given to all households would generate the same level of social welfare as the initial income distribution. Using M EDE , the **Atkinson measure** of inequality is defined by A = 1 - (M_EDE/ mean).

The flexibility in this measure lies in the freedom of choice of the household utility of income function. Given the assumption of a utilitarian social welfare function, it is the household utility that determines the importance attached to inequality by the measure.

13.5 The poverty line

The essential feature of poverty is the possession of fewer resources than is required to achieve an acceptable standard of living. What constitutes poverty can be understood in the same intuitive way as what constitutes inequality and similar issues about the correct measure arise again once
we attempt to provide a quantification.

The concept of **absolute poverty** assumes that there is some fixed minimum level of consumption (and hence of income) that constitutes poverty and is independent of time or place. Such a minimum level of consumption can be a diet that is just sufficient to maintain health and limited housing and clothing. Under the concept of absolute poverty, if the incomes of all households rise, there will eventually be no poverty. Although a concept of absolute poverty was probably implicit in early studies of poverty the appropriateness of absolute poverty has since generally been rejected.

**Relative poverty** is not a recent concept. Even in 1776 Adam Smith was defining poverty as the lack of necessities, where necessities are defined as ‘what ever the custom of the country renders it indecent for creditable people, even of the lowest order, to be without’. This definition makes it
clear that relative poverty is defined in terms of the standards of a given society at a given time and, as the income of that society rises, so does the level that represents poverty. Operating under a relative standard, it becomes much more difficult to eliminate poverty. Relative poverty has
also been defined in terms of the ability to ‘participate’ in society. Poverty then arises whenever a household possesses insufficient resources to allow it to participate in the customary activities of its society.

13.6 Poverty measures

The poverty line is denoted by the income level z, so that a household with an income level below or equal to z is classed as living in poverty. For a household with income M h , the income gap of household h measures how far their income is below the poverty line. Denoting the income gap for
household h by g h it follows that g h = z  M h . Given the poverty line z and an income distribution {M 1 ,..., M H }, where M 1 >= M 2 >=.. >= M H , the number of households in poverty is denoted by q. The value of q is defined by the facts that the income of household q is on or below the poverty line, so M q >= z, but that of the next household is above M q1 > z.

The simplest measure of poverty is the headcount ratio which determines the extent of poverty by counting the number of households whose incomes are not above the poverty line. Expressing the number as a proportion of the population, the headcount ratio is defined by

E = q/H

The major advantage of the headcount ratio is its simplicity of calculation. A measure that uses only information on how far below the poverty line are the incomes of the poor households is the aggregate poverty gap. This is defined as the simple sum of the income gaps of the households that are in poverty.

The interpretation of this measure is that it is the additional income for the poor that is required to eliminate poverty. It provides some information but is limited by the fact that it is not sensitive to changes in the number in poverty. In addition, the aggregate poverty gap gives equal weight to all income shortfalls regardless of how far they are from the poverty line. It is therefore insensitive to transfers unless the transfer takes one of the households out of poverty.

One direct extension of the aggregate poverty gap is to adjust the measure by taking into account the number in poverty. The income gap ratio does this by calculating the aggregate poverty gap and then dividing by the number in poverty. Finally, the value obtained is divided by the value
of the poverty line, z, to obtain a measure whose value falls between 0 (the absence of poverty) and 1 (all households in poverty have no income)

The income gap ratio has the unfortunate property of being able to report increased poverty when the income of household crosses the poverty line and the number in poverty is reduced.

In 1976 Sen suggested that a poverty measure should have the following properties:

1. Transfers of income between households above the poverty line should not affect the amount of poverty.
2. If a household below the poverty line becomes worse off, poverty should increase.
3. The poverty measure should be anonymous i.e. should not depend on who is poor.
4. A regressive transfer among the poor should raise poverty.

Two further properties were also proposed:
1. The weight given to a household should depend on their ranking among the poor i.e. more weight should be given to those furthest from the poverty line.
2. The measure should reduce to the headcount if all the poor have the same level of income.

One poverty measure that satisfies all of these conditions is the Sen measure:

S = E (1+(1-I)Gp(q/q+1))

where G p is the Gini measure of income inequality among the households below the poverty line. This poverty measure combines a measure of the number in poverty (the headcount ratio), a measure of the shortfall in income (the income gap ratio) and a measure of the distribution of income
between the poor (the Gini).

we will also want the aggregate measure to increase if poverty rises in one of the subgroups and does not fall in any of the others. So, if rural poverty rises while urban poverty remains the same, aggregate poverty must rise. Any measure of poverty that satisfies this condition is
termed subgroup consistent.

A poverty measure will satisfy this sensitivity to transfers if the increase in measured poverty caused by a transfer of income from a poor household to a poor household with a higher income is smaller the larger the income is of the lowest income household.

READING: Hindriks and Myles 2004: Chapter 14: Inequality and Poverty

Chapter 14 Inequality and Poverty

A social welfare function permits the evaluation of economic policies that cause redistribution between consumers - a task that Pareto efficiency can never accomplish... numerous difficulties on the path between individual utility and aggregate social welfare. The essence of these difficulties is that if the individual utility function corresponds with what is theoretically acceptable, then its information content is too limited for social decision making. p317

if the assets are not liquidated, the capital gain will not be realized during the period in question and will not be received as an income flow. For this reason, capital gains are taxed only upon realization. p319

In the words of the economist Gorman, “When you have a wife and a baby, a penny bun costs threepence”. A larger household obviously needs more income to achieve a given level of utility but the question is how much more income? Equivalence scales are the economist’s way of answering this question and provide the means of adjusting measured incomes into comparable quantities. p320

The first approach to equivalence scales is based on the concept of minimum needs. A bundle of goods and services that is seen as representing the minimum needs for the household is identified. The exact bundle will differ between households of varying size but typically involves only very basic commodities. p320

The construction of an equivalence scale from preferences makes two further issues become apparent. Firstly, the minimum needs and budget share approaches do not take account of how changes in family structure may shift the indifference map. For instance, the pleasure of having children may raise the utility obtained from any given consumption plan. With the utility approach it then becomes cheaper to attain each indifference curve, so that value of the equivalence scale falls as family size increases. This conclusion then conflicts with the basic sense that it is more expensive to support a larger family. p324-325

The second problem centres around the use of a household utility function. Many economists would argue that a household utility function cannot exist;
instead they would observe that households are composed of individuals with individual preferences. Under the latter interpretation, the construction of a household utility function suffers from the difficulties of preference aggregation identified by Arrow’s Impossibility Theorem. p325

Probably the simplest conceivable measure, the range calculates inequality as being the difference between the highest and lowest incomes expressed as a proportion of total income. As such, it is a very simple measure to compute.

The definition of the range, R, is R = M^H − M^1 / H u

Any index that has this property of independence is called a relative index p325-326

the range takes no account of the dispersion of the income distribution between the highest and the lowest incomes. Consequently it is not sensitive to any features of the income distribution between these extremes. p326

The relative mean deviation, D, takes account of the deviation of each in- come level from the mean so that it is dependent upon intermediate incomes. It does this by calculating the absolute value of the deviation of each income level from the mean and then summing. p326

Although it does take account of the entire distribution of income, the lin- earity of D has the implication it is insensitive to transfers from richer to poorer households, or vice versa, when the households involved in the transfer remain on the same side of the mean income level. p328

This line of reasoning is enshrined in the Pigou-Dalton Principle of Transfers which is a central concept in the theory of inequality measurement. The basis of this principle is precisely the requirement that any transfer from a poor household to a rich one must increase inequality regardless of where the two households are located in the income distribution.

Definition 5 (Pigou-Dalton Principle of Transfers) The inequality index must decrease if there is a transfer of income from a richer household to a poorer household which preserves the ranking of the two households in the income dis- tribution and leaves total income unchanged. p326

The reason why D is not sensitive to transfers is its linearity in deviations from the mean. The removal of the linearity provides the motivation for con- sidering the coefficient of variation which is defined using the sum of squared deviations. p326

Before moving on to further inequality measures, it is worth describing the Lorenz curve. The Lorenz curve is a helpful graphical device for presenting a summary representation of an income distribution and it has played an important role in the measurement of inequality. Although not strictly an inequality measure as defined above, Lorenz curves are considered because of their use in illustrating inequality and the central role they play in the motivation of other inequality indices.

The Lorenz curve is constructed by arranging the population in order of increasing income and then graphing the proportion of income going to each
proportion of the population. The graph of the Lorenz curve therefore has the proportion of population on the horizontal axis and the proportion of income on the vertical axis. If all households in the population had identical incomes the Lorenz curve would then be the diagonal line connecting the points (0, 0) and (1, 1). If there is any degree of inequality, the ordering in which the house- holds are taken ensures that the Lorenz curve lies below the diagonal since, for example, the poorest half of the population must have less than half total income. p329

The Gini , G, can be expressed by considering all possible pairs of incomes and out of each pair selecting the minimum income level. .. It should be noted that in the construction of this measure, each level of income is compared to itself as well all other income levels.p331

There is an important relationship between the Gini and the Lorenz curve. As shown in Figure 14.6, the Gini is equal to the area between the Lorenz curve and the line of equality as a proportion of the area of the triangle beneath the line of equality. As the area of the box is 1, the Gini is twice the area between the Lorenz curve and the equality line. This definition of the Gini makes it clear that the Gini, in common with R, C and D, can be used to rank distributions when the Lorenz curves cross since the relevant area is always well defined. Since all these measures provide a stronger ranking of income distributions than the Lorenz curve, they must each impose additional restrictions which allow a comparison to be made between distributions even when their Lorenz curves cross. p332

Theorem 10 Consider two distributions of income with the same mean. If the Lorenz curves for these distributions do not cross, every symmetric and concave social welfare function will assign a higher level of welfare to the distribution whose Lorenz curve is closest to the main diagonal. p333