Inequality of What? Building a Distribution and Summarizing

Inequality

Peter Lanjouw, DECRG

Poverty and Inequality Analysis

Module 5: Inequality and Pro-Poor Growth

Outline

Constructing a measure of welfare– Income/consumption?– Components– Adjustments

Scalar Inequality Measures– Statistical measures– An axiomatic approach

The distribution– Lorenz curves– Stochastic dominance

Alternative measures in practice

Consumption per capita– Value of total consumption (whether or not transacted)

Income per capita– Poor indicator when incomes vary; hard to measure

Consumption per equivalent single adult– Problems with forming scales; composition versus size

economies; intra-household inequality. Food-share ("Engel's Law")

– Sources of noise: other parameters; problem of income elasticity near unity. Identification problems.

Nutritional indicators– "Welfarist" critique (welfare and nutrition are different things);

nutritional requirements/ anthropometric standards.

Recommendations

Use a comprehensive consumption measure, spanning consumption space

Choice between income and consumption is largely driven by the greater likelihood of accuracy of information on consumption.

Recognize the limitations of consumption based measures; look for supplementary measures, especially access to public services, subjective welfare as a clue to measuring objective welfare.

Constructing a Consumption Measure

The consumption measure serves as the foundation upon which much of the subsequent analysis rests.

Principles– Should be comprehensive– Retain transparency and credibility– Goal is to be able to rank individuals credibly in

terms of welfare

Common Steps

Construct a food consumption measure Add basic non-food items (from consumption

module) Add other non-food items (other modules) Add housing expenditures Add use-value of consumer durables

Aggregating Food and Basic Non-Food Expenditures

Food expenditures should include not only purchased items but also consumption out of home production.

Key issue is to distinguish between investments and consumption (avoid double counting).

Health expenditures are usually excluded Two Additional omissions:

– Leisure– Public goods

Imputing Housing and Durables Expenditures

Many surveys ask specifically what an owner-occupier household would pay in rent if it were renting. Where credible this number can be used for home owners.

Elsewhere one can try to predict rent paid based on regression models estimated on subset of renting households.

Purchases of irregular, lumpy items such as consumer durables (tv, car, etc.) cannot be directly added to the consumption definition.

– If supplementary information is available, it is sometimes possible to impute a stream of consumption services from durables ownership

  Aggregate Gini Coefficient

Food Spending 0.359 

Food Spending plusBasic Non-Food Spending 0.420

Food plus Basic Non-FoodSpending Including Energyand Education Spending 0.421

Above With Actual or ImputedWater Expenditures 0.415

Above With Actual or Imputed 0.420Value of Housing Services

 Above With Imputed Value 0.430of Owned Consumer Durables

 Source:

Encuesta de Condiciones de Vida, 1994 (Instituto Nacional de Estatisticos y Censo, Ecuador).

Impact on Inequality of Using Alternative Consumption Aggregations - Ecuador

Adjustments to Consumption Aggregate

Spatial price variation– Cost of living adjustments

Demographic Composition– Equivalence Scales– Economies of Scale

Which unit of analysis?– Household or individual?

Summarizing Inequality in the Consumption Distribution

Some options from basic statistics:

minmax yyrange

2)(1

)( n

i yyn

yVariance

Inequality measurement and normative judgements

All summary measures of inequality embody some normative judgements regarding inequality and the relative importance of different parts of the income distribution.

– Most measures make these judgements implicitly rather than explicitly.

Some measures are clearly unattractive:– Range: reflects well the distance between the poorest and

richest, but is totally unaffected by changes in the distribution of income within these two extremes.

Inequality measurement: Some attractive axioms

The principle of transfers: a transfer from a poorer person to a richer person, all things equal, must result in an increase of inequality (aka Pigou-Dalton principle).

– This property is not satisfied by the range. Scale-neutrality: If we double all incomes, measured

inequality should not change– This property is not satisfied by the variance

Anonymity: If we simply switch location of two individuals in the income distribution, measured inequality should not change.

Inequality measurement and normative judgements

Gini Coefficient:

– Attaches more weight to transfers affecting middle income classes

i

ii Ynrn

Gini ]2/)1([22

Inequality measurement and normative judgements

Coefficient of variation: – Attaches equal weights to all income levels– No less arbitrary than other judgements.

Standard deviation of logarithems:– Is more sensitive to transfers in the lower income

brackets. Bottom line: The degree of inequality cannot in

general be measured without introducing social judgements.

The Atkinson class of inequality measures

Atkinson (1970) introduces the notion of ‘equally distributed equivalent’ income, YEDE.

This represents the level of income per head which, if equally shared, would generate the same level of social welfare as the observed distribution.

This then defines a measure of inequality of the following form:

IA = 1- (YEDE/μ)

The Atkinson class of inequality measures

A low value of YEDE relative to μ implies that if incomes were equally distributed the same level of social welfare could be achieved with much lower average income.

So IA would be large. Everything hinges on the degree of inequality aversion

in the social welfare function. With no aversion, there is no welfare gain from

redistribution so YEDE is equal to μ and IA = 0.

The Atkinson class of inequality measures

Atkinson proposes the following form for his inequality measure:

1

1

1)(1 ii

iA f

Y

YI

The Atkinson class of inequality measures

A key role here is played by the distributional parameter ε. In calculating IA you need to explicitly specify a value for ε.

When ε=0 there is no social concern about inequality and so IA = 0 (even if the distribution is “objectively” unequal).

When ε=∞ there is infinite weight to the poorer members of the population (“Rawls”)

How to choose ε?

Selecting ε (Okun’s leaky bucket)

Okun (1975) asks the hypothetical question:“Consider a transfer of $4000 per head from the top 5

percent giving, in principle, $1000 to each of the bottom 20 percent. But some of the money “leaks” away in the process. How much leakage would you be willing to accept before abandoning the proposed redistribution?”

Selecting ε (Okun’s leaky bucket)

Atkinson shows that the answer to this question can be used to back out a value of ε.

g represents the ratio of the average income of the “transferee” to the “recipient”.

x represents proportion of transfer received by the recipient.

gx

1

Selecting ε (Okun’s leaky bucket)

Suppose a hypothetical transfer from someone with twice the income level of the proposed recipient (g=2)

If you will not tolerate less than half of the transfer going to the poor (x=0.5) then ε=1.

If you will accept up to 75% leakage, then x=0.25 and ε=2.

Recommendations

No inequality measure is purely ‘statistical’: each embodies judgements about inequality at different points on the income scale.

To explore the robustness of conclusions: Option 1: measure inequality using a variety of

inequality measures (not just Gini). Option 2: employ the Atkinson measure with multiple

values of ε. Option 3: look directly at Lorenz Curves, apply

Stochastic Dominance results.

The Lorenz Curve

To compare inequality in two distributions:– Plot the % share of total income received by the poorest nth

percentile population in the population, in turn for each n and each consumption distribution.

– The greater the area between the Lorenz curve and the hypotenuse the greater is inequality.

Second Order Stochastic Dominance (Atkinson 1970):– If Lorenz curves for two distributions do not intersect, then they

can be ranked irrespective of which measure of inequality is the focus of attention.

– If the Lorenz curves intersect, different summary measures of inequality can be found that will rank the distributions differently.

Useful Readings

Atkinson, A. B. (1983) The Economics of Inequality 2nd Edition (Oxford: Clarendon Press). Atkinson, A. B. (1983) Social Justice and Public Policy (Cambridge, Mass: MIT Press). Atkinson, A.B. and Bourguignon, F. (eds) (2000) Handbook of Income Distribution Vo1. 1 (North Holland: Elsevier Science Publishers).

Deaton, A. and Zaidi, S. (1999) ‘Guidelines for Constructing Consumption Aggregates for Use as a Money-Metric Welfare Measure’, LSMS working paper.

Foster, J. and Sen, A. (1997) On Economic Inequality: Expanded Edition with Substantial Annexe (Oxford: Oxford University Press).