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Poverty Measurement An Introduction. Paolo Verme. Basic Concepts. Measuring Welfare. Two Approaches:

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measuring welfare
Measuring Welfare
  • Two Approaches:
  • Welfarist approach. This derives from orthodox economics and the theory of revealed preferences. Individuals are the best judges of their needs and express their needs through consumption. Measuring consumption and consumption choices reveals individual preferences and utility.
  • Non Welfarist approaches. There are two approaches both developed by Sen as a critique to orthodox economics.
    • Basic needs or functionings. Individuals need to achieve a minimum set of basic needs or functionings to be considered as non poor.
    • Capabilities. Individuals need to have the capacity to reach a minimum set of functionings irrespective of whether they actually make use of these functionings or not (freedom of choice).
welfare in orthodox economic theory
Welfare in Orthodox Economic Theory
  • Happiness = Utility = Welfare
  • How to measure utility? Revealed preferences.
  • I = C (present consumption) + S (future consumption)
  • There is no distinction between income, consumption or expenditure
  • The objective is to maximise utility selecting the best consumption bundle under a budget constraint.
welfare according to sen
Welfare According to Sen
  • Poverty is the inability to reach a Minimum Standard of Living.
welfare according to sen1
Welfare According to Sen
  • What is the inability to reach…? It is the lack of capacity to obtain the minimum living standards. However, people are different and have different needs, opportunities, functionalities and capabilities.
  • Needs. Two persons may be have different needs. A person who weighs 150 kgs. has different nutritional needs from a person that weighs 60 kgs.
  • Opportunities. Two persons may have the same, needs and abilities but different opportunities because of discrimination or chance.
  • Functionings. Functionalities is when opportunities meet abilities. It is the capacity to exploit opportunities. One disabled person and one able person have different functionalities vis-à-vis the same opportunity of public transport.
  • Capabilities. Capabilities are the set of functionalities of which a person disposes of. The more the better, even if some functionalities are not used.
  • => How to adjust the welfare measure to individual functionalities is one of the objective of this course (welfare adjustments).
welfare according to sen2
Welfare According to Sen
  • What is a minimum standard of living? It is a concept generally defined by countries and based on normative and positive criteria.
    • Value judgments about what is important for living (normative). Ex: Including or excluding a TV set from the minimum consumption basket.
    • Scientific notions about what is necessary for living (positive). Ex: establishing a minimum amount of daily calories necessary for survival.
    • => How we define these minimum standards is one objective of this course (Poverty Lines).
welfarist or non welfarist approach
Welfarist or non Welfarist Approach?
  • In practice, when we measure poverty with HBSs, we use components of both approaches.
  • We are interested in the notion of minimum standards (basic needs) and will attempt to measure minimum standards with different methodologies.
  • We are also interested in functionings. We will adjust our measure of welfare to individual needs.
  • We recognize that individuals have many functionings and that welfare is multi-dimentional. However, in this course, we limit ourselves to the study of one dimension, income or consumption.
  • We measure welfare with consumption accepting de facto the notion of revealed preferences put forward in orthodox economics.
the components of a poverty analysis
The components of a poverty analysis
  • Who is poor (poverty measurement) => methodologies
  • What are the characteristics of the poor (poverty profile) => statistics
  • Where are the poor (poverty mapping) => spatial analysis
  • Why are the poor poor (causes of poverty) => econometrics
  • What can be done about the poor (pro-poor policies) => PRSP
  • => This course focuses on poverty measurement
tools for poverty measurement
Tools for Poverty Measurement
  • Measurement instrument: Household Budget Survey
  • Welfare measure: Income, consumption, expenditure
  • Welfare adjustments: Prices, Household composition and Economies of scale
  • Welfare threshold: Poverty lines
  • Welfare statistics: Poverty indexes and their decompositions
  • => In the next sections, we explore these tools one by one.
the household budget survey
The Household Budget Survey
  • From Population to Sample:
  • Population
  • Strata
  • Clusters
  • Sample
  • From Sample to Population:
  • Population Weights
from sample to population
From Sample to Population

A Poverty analysis which ignores population weights, is an analysis of the sample, not the population. Even if we calculate only a mean, we need to use weights. Stata provides simple ways to do it with most commands (not all). In stata, population weights are described as ‘pweight’ or ‘fweight’. Remember: A statistics from the sample is only an estimate of the population statistics.

measuring welfare with hbss
Measuring Welfare with HBSs
  • HBSs measure income, expenditure and sometimes savings, they do not measure consumption.
  • W=Welfare
  • I=Income
  • S=Savings
  • E=Expenditure
  • C=Consumption
  • I=C+S
  • W=I?
  • C=E?
  • W=C?
  • W=E?
  • W=S?
  • Self-production
  • Self-consumption
  • Durable goods
  • Inter-temporal consumption
how good is income
How Good is Income?
  • It varies according to:
  • Seasons
  • Life-cycle
  • Likely to be poorly reported because:
  • Illicit activities
  • Informal income (tax evasion)
  • Fear
  • Gross Vs. Net income
  • Recall bias
how good is expenditure
How Good is Expenditure?
  • Better than income because:
    • Less vulnerable to seasonality
    • Less vulnerable to life-cycle
    • Closer to the utility that people effectively extract from income
    • Less vulnerable to measurement errors because respondents have less reasons to lie
    • For the poor, by definition, most of income is consumed (little savings, access to credits, little capital goods)
how good is expenditure1
How Good is Expenditure?
  • However:
    • The list of expenditure items is much larger than incomes
    • Households do not report properly certain consumption items such as alcohol, cigarettes, gambling, prostitution, drugs which may consume considerable amounts of income
    • Some consumption items are often neglected such as small, recurrent or irrelevant purchases.
    • Seasonality also affects consumption, especially the structure of consumption
    • Consumption includes durable goods such as a TV set or a stock of flour for the winter
    • Some purchases are very rare such as a flat and last long periods of time
from expenditure to consumption
From Expenditure to Consumption
  • Expenditure and consumption are different:
  • We do not consume everything that we buy. Some purchased goods are wasted before we consume them. Very few HBS measure HH waste. Some other goods are bought and donated to other people. Donations are often but not always measured in HBS. We buy some goods that we may consume in subsequent periods.
  • We consume some of the things that we do not buy. We consume some goods that we produce. Self-consumption is usually measured in HBS. We consume some goods that are donated to us. Donations are often but not always measured in HBS. We consume goods that we bought in previous periods such as durable goods.
from expenditure to consumption1
From Expenditure to Consumption
  • In substance, we measure consumption by measuring expenditure and adjust this measure with self-production, donations, amortization, and other information that may be present in the HBS. In particular, we adjust for:
    • Self-consumption (HH diary)
    • Rents (Estimates)
    • Durable goods (Depreciation)
    • Net donations in kind (Received donations-donations made)
rent imputation
Rent Imputation
  • Rent is part of expenditure and contributes to people’s welfare. Rents are registered for those who pay rent but do not appear for the owners of flats or houses. We need to calculate a fictitious rent for the owners of properities. There are many techniques to do this. One is with a simple econometric procedure as follows:
  • A) Estimate how rents vary according to the property characteristics of tenants:
  • Rent of tenants=a+b*(Properties characteristics)+u
  • B) Predict rent for all:
  • Rent of all=E(a)+E(b)*(Properties characteristics)
  • C) Use the predicted rent for the owners as imputed rent.
imputation of durable goods
Imputation of Durable Goods
  • Durable goods are purchsed occasionally but provide welfare for extended periods of time. We need to spread the cost of th durable good over the life of the good. This is the same process such as amortization of durable goods in companies. In principle we should consider:
    • Change of value of the durable good (depreciation)
    • Opportunity cost of investment (foregone income of alternative investment)
    • Example:
      • I bought a TV set for 250 USD last year and this year is worth 190$.
      • 250 USD in a bank have a return of 10%
      • The total cost of the televisor for this year has been:
    • 250-190=60+(250*0.1)=85 USD
    • Alternatively, we could simply attribute to the purchased good a standard duration in years and divide the cost of the good for its duration taking inflation into account.
    • The problem with such imputation is not about methodology but information. We need to know the purchase cost, inflation, duration of the good, present value and similar information which are not always available in the HBS we dispose of.
estimates of self consumption
Estimates of self-consumption
  • Especially in rural areas, self-consumption can represent an important share of the total HH consumption.
  • First, we need to have a HBS which measures self-consumption.
  • Second, we need to attribute a value to self-consumption via market prices. The issue is what are the relevant market prices to consider given that food prices may be very variable across seasons and locations.
  • Third self-consumption has a cost in terms of agricultural inputs. If we buy seeds or fodder this is accounted for as expenditure and included into consumption. Including into consumption the total market price of grains produced with the seeds bought and consumed or estimatting the value of cattles produced with the fodder bought and consumed would over-estimate total consumption. We add the value of agricultural inputs to the value of the final product.
  • The household that produces goods and services should be considered as an enterprise and treated economically and financially as such. This means that only the value added which is consumed should be considered. The most sophisticated HBS have entire sections dedicated to self-production and self-consumption.
properties of a welfare measure
Properties of a Welfare Measure
  • We have established that consumption may be the best measure of welfare we can use and we have adjusted consumption with the imputation of rents, durable goods and self-consumption.
  • Is household total consumption sufficient to compare individuals, households or larger communities? Not really.
  • A measure of welfare should have the following properties:
    • Horizontally equitable. All equal individuals should be treated equally. But individuals are not equal, they have different needs. Ex: Food and non-food requirements are different for adults and children. We need to ‘equalize’ the welfare measure.
    • Fixed over time and space. A measure of welfare should be comparable across time and space. But prices change over time and space. Ex: Inflation over time and price differentials across regions. We need to adjust the welfare measure to comparable prices.

=> The poverty line or the poverty measure need to be adjusted accordingly.

Adjust the poverty line or the welfare measure? It’s your choice. In this course, we adjust the welfare measure.

adjusting consumption for welfare comparisons1
Adjusting Consumption for Welfare Comparisons
  • Adjust for what?
  • Household size
  • Household composition
  • Household purchasing power
  • Adjust to what?
  • Economies of scale (ES)
  • Adult Equivalent Consumption (AEC)
  • Purchasing Power Parity (PPP)
    • Adjust how?
    • Equivalence Scales
    • Deflators
adjust consumption for hh size and composition
Adjust Consumption for HH Size and Composition
  • Consumption per capita. Consumption is estimated on households but households have different sizes. We need per capita estimates.
  • Economies of scale. Household size has an impact on economies of scales. The more people live under the same roof and share the same resources the more the fixed costs are spread, the more the unit costs are small, the greater is individual welfare.
  • Consumption capacity. Households are composed of different type of members such as adults, children and pensioners with different needs, costs and consumption capacity. A child eat less than an adult. If a child and an adult have the same monetary consumption, the child is better off. These differences in welfare should be taken into account.
  • From HH consumption to Per capita adult equivalent consumption
  • This is done with equivalence scales
equivalence scales
Equivalence Scales
  • Two simple and popular equivalence scales are the Oxford and OECD scales:
  • Oxford:
    • 1 = First adult
    • 0.7 = Other adults (adjust economies of scale)
    • 0.5 = Children (adjust for calories needs)
  • OECD:
    • 1 = First adult
    • 0.5 = Other adults (adjust economies of scale)
    • 0.3 = Children (adjust for calories needs)
equivalence scales1
Equivalence Scales
  • But there are also much more sophisticated scales. A mathematical formula which may capture both adult equivalent consumption and economies of scale is the following.
  • ES = (A+αC)^β
  • A= Number of adults
  • C=Number of children
  • α=Child adult equivalent parameter
  • β=Economies of scale parameter
  • By dividing consumption by ES, we obtain the per capita adult equivalent consumption. With α=1 and β=1, we obtain per capita consumption. However, the general assumption is that α<1 and β<1.
equivalence scales2
Equivalence Scales
  • Note that consumption and, by consequence, poverty measures are very sensitive to the choice of equivalence scales.
  • Note also that equivalence scales are very arbitrary. In fact, there is no scientific ground to argue that the same equivalence scale should be applied universally as suggested by the Oxford and OECD scales for various reasons:
    • A child in a poor country may cost little more than the food required. But a child in a rich country may cost more than an adult.
    • Countries have different population structures. In some countries over 50% of the population is composed of children, in others less than 20%. A small change in the child parameter of the equivalence scale can lead to a substantial increase or decrease in the poverty gap observed between two countries.
    • Economies of scale may be very different across countries. In Africa, people tend to live in extended and elastic families and fixed costs such as heating, water and electricity are very small. In Europe, people live in small and rather established households and fixed costs may be a substantial part of consumption.
poverty lines1
Poverty Lines
  • The monetary value of the minimum set of basic needs necessary to reach a minimum standard of living
  • or, in economic terms:
  • The minimum consumption needed to achieve the minimum level of utility
  • We consider different types of poverty lines
          • Absolute (APL)
          • Relative (RPL)
          • Subjective (SPL)
absolute poverty lines different approaches
Absolute Poverty LinesDifferent Approaches
  • Cost of basic needs (CBN)
    • Least cost approach
    • Expenditure based method
  • Food Energy Intake (FEI)
  • Direct Calories Intake Method (DCI)
  • All PL are composed of a food and a non food component.
  • However, it is the estimation of the food component which makes a difference across methods.
apl cost of basic needs cbn
APL-Cost of Basic Needs (CBN)
  • With a Cost of Basic Needs approach, the standard method is to start with evaluating in monetary terms a minimum amount of food necessary for having a healthy and active life that allows individuals to fully participate in society; we call this the Minimum Food Basket (MFB).
  • The MFB is based on nutritional values. First, the minimum level of energy intake is established.
  • Example. The FAO recommends a level of 2,100 calories/day for an adult in working age.
  • There are at least two different approaches:
    • Least cost approach
    • Expenditure based method
apl cost of basic needs cbn least cost approach
APL-Cost of Basic Needs (CBN)Least Cost Approach
  • With a least-cost approach, we select a number of food baskets that provide the same calories intake and we then select the one that is less costly and use the value of this basket as the poverty line.
  • First, we find a number of products which are traditionally part of the diet of the population we are targeting.
  • Second, we compose an ideal basket of goods selected on the basis of their nutritional values (MFB), in terms of a correct balance between carbohydrates, proteins, vitamins and other nutritional composites to achieve the set number of calories.
  • Third, we evaluate in monetary terms using affordable prices. By multiplying quantities by prices we obtain the value of the MFB. This value can be used as the Food Poverty Line (FPL).
  • MFB=FPL= ∑ qi*pi
  • where qi= quantity of good i and pi=price of good I
  • The advantage of this approach is that we do not need to know detailed data on household consumption. On the other hand, this approach will not provide a basket of goods which will be necessarily consumed by any household.
apl cost of basic needs cbn expenditure based method
APL-Cost of Basic Needs (CBN)Expenditure Based Method
  • The expenditure based method looks first at consumption patterns in a certain population. Usually, the sample of households that is used for this evaluation is the sample of middle income or poorer households.
  • The food consumed by this population group is included into the basket and the basket is weighted according to the share of different foods consumed by the target population.
  • The basket is then transformed into calories and adjusted so as to reach the minimum amount of calories required.
  • The method ensures that the food basket is relevant to the population we are assessing. However, it does not guarantee the best possible diet to reach the minimum amount of calories.
apl food energy intake
APL-Food Energy Intake
  • The Food Energy Intake method seeks in the data the consumption level at which a person typically attains the minimum food energy intake.
  • First we estimate econometrically how calories intake change as consumption changes:
  • Ln(x)=a+bC+u
  • Where X=Consumption on a food basked; C=Calories obtained from the food basket; u=error term
  • Then the food poverty line can be estimated as:
  • Z=a*+b*R
  • Where Z=PL, a* and b* are the coefficients estimated from the first equation and R is the recommended calories intake.
apl direct calories intake method
APL-Direct Calories Intake Method
  • With the DCI method we take first the quantities consumed by HH.
  • These quantities are transformed into HH calories.
  • HH calories are transformed into calories per capita.
  • Poor HH are those household with calories per capita below a minimum threshold.
  • Problem: two households with the same calories intake may enjoy very different standards of living
relative poverty line
Relative Poverty Line
  • A relative poverty line is not related to absolute income or consumption of the population but is established relatively to the particular distribution of income or consumption observed in a given society.
  • Example. This is the approach followed by the European Union that considers poor all people with per capita income or consumption below 50% of the median income.
  • The problem with a relative poverty line is that there is no fixed benchmark. The PL moves together with the median of the distribution. If all incomes in a distribution increase by 10%, the number of poor below the poverty line does not change. Thus, it is more similar to a measure of distribution than to a measure of poverty. It pinpoints those who are worse off but does not tell us how poor these people really are.
relative poverty line1
Relative Poverty Line
  • Annual Consumption per Capita (Euro)
  • Data Sorted data
  • 10000 4500
  • 9200 6600
  • 13400 7800
  • 4500 9000
  • 7800 9200
  • 15000 10000
  • 9000 10500
  • 14000 12900
  • 12900 13400
  • 10500 14000
  • 6600 15000
  • Median = 10,000
  • Poverty Line = 10,000/2 = 5,000
subjective poverty line
Subjective Poverty Line
  • A subjective poverty line is established by asking people about poverty. This can be done in several ways:
  • Ask people what they think is a minimum amount necessary for living. Answers to this question vary from person to person but one could use an average or median value to estimate a poverty line. The question could be formulated as follows: What do you think is the minimum income necessary to your family every month? And answers could be structured as open ended or multiple choice answers.
  • Alternatively, we can ask people how they would rank themselves on a poverty scale with several steps and estimate overall poverty by taking the mean of the answers. One example is the following: On a scale from 1 to 10 where one is “extremely poor” and 10 is “extremely rich” how would you rank the income status of your household?
poverty lines what is better
Poverty Lines. What is better?
  • Absolute, relative and poverty lines are not substitutes but complement each other and provide different types of information.
  • Absolute poverty lines provide information on poverty as compared to a recognized minimum threshold determined on the basis of normative and positive criteria. Such lines can be used to compare people across space and time. These lines are used to measure real welfare.
  • Relative poverty lines provide information on poverty based on the position of individuals relative to the position of other individuals within the same consumption distribution. Such lines cannot be used to evaluate changes in real welfare, only changes in relative welfare. They can be seen as a measure of distribution.
  • A subjective poverty line provides a picture of self-perceived poverty. This may be very different from absolute or relative poverty but is nevertheless a useful tool for policy. It can be seen as a measure of the feeling of individual deprivation and can used for political purposes.
poverty indexes1
Poverty Indexes
  • In this section we look at three popular measures of poverty:
          • Headcount Index
          • Poverty Gap
          • Severity of Poverty
  • We will see why we need three measures and we will see that these three measures belong to the same class of poverty measures.
  • We will also have a brief look at some other indexes:
          • Watts
          • Atkinson
          • Sen
          • Sen-Shorrocks-Thon
the poverty headcount index
The Poverty Headcount Index


H = q/n

yi= Individual consumption i

z = Poverty line

n = Population

q = Number of individuals below the poverty line

the poverty gap and the severity of poverty index
The Poverty Gap and the Severity of Poverty Index

yi= Individual consumption i

z = Poverty line

n = Population

q = Number of individuals below the poverty line

the common structure of the three poverty indexes
The Common Structure of the Three Poverty Indexes

α=0 => P(0) = Headcount Index

α=1 => P(1) = Poverty Gap Index

α=2 => P(2) = Severity of Poverty Index

α is defined as the poverty aversion parameter. The larger is α the more weight we give to the poorest people. This is a normative choice which may reflect, for example, the weight that governments wish to attribute to poverty in public policy.

a graphical illustration of the three indexes



Poverty Index



0 Consumption z

A Graphical Illustration of the Three Indexes

If all people have zero consumption, all the three indexes are equal to 1. All people reach maximum poverty.

If all people consume the same amount as the poverty line z (or above), the poverty indexes are all equal to zero. There is no poverty.

In between the two extremes:

P0 (headcount index) is constant. Each additional poor add an equal amount of poverty.

P1 (poverty gap) is linear and increasing in poverty. Each additional poor increases poverty proportionally to the level of poverty. The poorest contribute to the index more than the less poor.

P2 (severity of poverty) is exponential and increasing in poverty. Each additional poor increases poverty more than proportionally to the level of poverty. The poorest contribute to the index much more than the less poor.

interpretation of the three indexes
Interpretation of the three Indexes
  • The headcount index (P0) is easily interpreted. It is the percentage of poor people in the population.
  • The poverty gap (P1) index can be interpreted as the cost necessary to eliminate poverty. That is because P1 is the sum of the consumption gap between each poor and the poverty line.
  • More difficult is to interpret the severity of poverty index (P2) and, more in general, all indexes where α>1.
  • Note first, that the greater is α the smaller is the size of the index. Comparing the three indexes between each other, only provides information on the size of poverty aversion parameter, i.e. on the normative judgment made about the weight that we wish to give to poverty.
  • The three indexes cannot be compared between each other, they are only relative to themselves and they are only useful when we observe changes over time or space. They mean little in absolute values.
  • Therefore, the severity of poverty index can only be interpreted as a measure of intensity of poverty. If the index is larger, the intensity of poverty is larger. If it the index is smaller the intensity of poverty is smaller.
watts and atkinson poverty indexes
Watts and Atkinson Poverty Indexes

Watts (1968)Atkinson (1987)

yi= Consumption of individual i

z = Poverty line

n = Population

q = Number of poor

ε= Poverty aversion parameter

sen and sen shorrocks thon indexes
Sen and Sen-Shorrocks-Thon Indexes

Sen (1976) Sen-Shorrocks-Thon (SST)

P0=Poverty Headcount

P1=Poverty gap

G^q=Gini index among the poor

Gz-q^q= Gini index calculated on the poverty gaps

poverty decomposition into sub groups
Poverty Decomposition into Sub-groups

K = Number of sub-groups

k = Sub-groups

P = Poverty index

Pk= Poverty index for group k

p = Population

pk= Population of sub-group k

poverty shares by sub groups
Poverty Shares by Sub-groups

K = Number of sub-groups

k = Sub-groups

P = Poverty index

Pk = Poverty index for group k

p = Population

pk = Population of sub-group k

poverty risk
Poverty Risk

K = Number of sub-groups

k = Sub-groups

P = Poverty index

Pk = Poverty index for group k

p = Population

pk = Population of sub-group k

poverty indexes shares and risk example kosovo

Poverty Headcount

(complete pov. line)

Poverty Headcount

(food pov. line)






























Age 26-65







Age > 65







Poverty Indexes, Shares and Risk: Example (Kosovo)
from units to population
From Units to Population
  • Thus, in order of size we have:
    • Population units (Individuals)
    • Population units (Households)
    • Sample units (Individuals)
    • Sample units (Households)
    • Quantiles (Equal groups of sample units)
    • Sample
    • Population
units and quantiles
Units and Quantiles
  • For the measurement of inequality it is essential to understand the distinction between units of measurement and quantiles.
  • Units of measurement can be individuals, households or larger communities.
  • Quantiles are groups of observations of equal size, i.e. containing the same number of units. If we order the surveyed sample in ascending order of one variable such as income and then we split the sample in groups of equal size we obtain ‘quantiles’. If we order individuals in ascending order of income and then we split the sample into ten groups we talk of deciles. We talk of quartiles if the groups are four and quintiles if the groups are five and so on. Evidently the maximum number of quantiles we can obtain from the sample is equal to the number of units in that sample.
  • Dividing the sample into quantiles is useful because we can then calculate statistics for each quantile and compare these statistics across quantiles. For example, we can calculate the percentage of people with access to basic health services and compare this percentage across income quintiles to see if the poorest groups (quintiles one and two) have more or less access than the richest groups (quintiles four and five) to these basic services.
measures of inequality
Measures of Inequality
  • There is a very wide range of measures of inequality. We focus here on a few basic measures to get acquainted with the practice of inequality measurement.
  • We also focus on discrete measures (with income measured in finite quantities - we use formulas) as opposed to continuous measures (with income measured in infinite quantities comprised in an interval - where integrals are used).
  • We consider the following inequality measures:
          • The variance
          • The coefficient of variation
          • The quintile ratio
          • The Lorenz curve
          • The Gini coefficient
properties of an inequality measure
Properties of an Inequality Measure
  • Mean independence. This means that if all incomes were doubled, the measure would not change.
  • Population size independence. If the population were to change, the measure of inequality should not change, ceteris paribus. This is the problem we had with the variance which changes in size if incomes change in size.
  • Symmetry. If you and I swap incomes, there should be no change in the measure of inequality. This principle says that you can swap income and ranks between different people and that this would leave the inequality measure unchanged. Inequality measures are concerned with incomes, not people.
  • Pigou-Dalton Transfer sensitivity. Under this criterion, the transfer of income from richer to poorer reduces measured inequality. If I take X income from the rich and I give it to the poor the inequality measure should decrease.
  • Decomposability. This means that inequality may be broken down by population sub-groups and that the inequality index may be reconstructed by summing-up the sub-group indexes (additivity principle).
  • Statistical testability. One should be able to test for the significance of changes in the index over time.