Poverty Measurement with Ordinal Data

Christopher J. Bennetta and Chrysanthi Hatzimasourab

First Version: September 2011; Revised: November 2012

Abstract

The Foster, Greer, Thorbecke (1984) class nests several of the most widely used indicesin theoretical and empirical work on economic poverty. Use of this general class of indices,however, presupposes a dimension of well-being that, like income, is cardinally measurable.Responding to recent interest in dimensions of well-being where achievements are recordedon an ordinal scale, this paper introduces a general methodology for constructing ordinalindices of poverty and, in particular, shows how this methodology may be applied toconstruct an ordinal analogue of the popular FGT class of indices. The resulting ordinalFGT indices retain the simplicity of the classical FGT indices and also many of theirdesirable features, including additive decomposability. To illustrate their use, we applythe ordinal FGT indices to self-reported data on health status in Canada and the UnitedStates.

JEL classification: I3, I32, D63, O1

Keywords: poverty measurement, ordinal data, FGT poverty indices

a(Corresponding author) Department of Economics, Vanderbilt University, VU Station B #351819,

2301 Vanderbilt Place, Nashville, TN 37235-1819, U.S.A. E-mail: chris.bennett@vanderbilt.edubDepartment of Economics, George Washington University, 2115 G Street, NW Monroe Hall

340, Washington, DC 20052, U.S.A. E-mail: chrysa@gwu.edu∗We are deeply indebted to James Foster and John Weymark for their advice and supportthroughout our work on this project. We also wish to thank Michael Hoy and BuhongZheng for their insightful comments and suggestions. Finally, the second author grate-fully acknowledges the Institute for International Economic Policy of the Elliott Schoolof International Affairs for funding her visit to Vanderbilt University in order to conductthis research.

1 Introduction

In the twenty-five years since it was first introduced, the FGT (Foster, Greer, and

Thorbecke 1984) family of measures has become the most widely used class in empir-

ical work on the measurement of poverty. The attractiveness of the FGT measures stems

largely from their simple structure, their ease of interpretation, and their sound axiomatic

properties. Being defined by two parameters, namely the poverty line z and a scalar mea-

sure of poverty aversion α, each member of the FGT class is easily computed as an average

of the power function defined by α whose argument is the normalized income shortfall

from z. Specific members include the well-known poverty gap, squared poverty gap, and

headcount ratio (i.e., the proportion of the population identified as poor).

Use of the general FGT class of measures presupposes a dimension of well-being

that, like income, is cardinally measurable. Recently, however, considerable interest has

emerged in measures of aggregate deprivation in dimensions of well-being other than in-

come and, in particular, in dimensions of well-being—for example, health, education,

empowerment, and social inclusion—that are often recorded on an ordinal scale.1 Conse-

quently, “a crucial emerging issue is how to measure poverty when data do not have the

characteristics of income, which is typically taken to be cardinal and comparable across

persons ... Must we retreat to the headcount ratio [with ordinal data], or can we continue

to evaluate the depth or distribution of deprivations—key benefits provided by the higher

order FGT measures when the variable is cardinal?”(Foster, Greer, and Thorbecke 2010,

p. 516)

In order to address this issue, this paper introduces a methodology for constructing

ordinal poverty indices from cumulative distributions over the levels of achievement of the

poor. This general approach to the construction of ordinal poverty indices is motivated by

a thought experiment in which an individual is completely unaware of her relative position

in society and draws a level of achievement at random according to the actual distribution

in society and another level of achievement from a reference lottery over the poor states.

The extent of poverty in society is then recorded as the proportion of individuals who

would accept their realized level of poverty drawn from the reference lottery rather than

their draw from the actual distribution in society.

Poverty indices constructed in this manner are completely determined by the cumu-

1Problems surrounding the measurement of poverty with ordinal data are raised, for example, inFoster, Greer, and Thorbecke (2010) and Alkire and Foster (2011a,b). Allison and Foster (2004) werethe first to stress the problems raised by ordinal data in the related context of inequality measurement.See Zheng (2008), Abul Naga and Yalcin (2004), and Madden (2010) for more on the use of ordinal datain this latter context.

1

lative distribution associated with the reference lottery, thereby enabling entire classes

of indices to be constructed from parametric classes of distributions. In this sense, the

methodology is related to the Atkinson-Kolm-Sen (Atkinson 1970, Kolm 1969, Sen 1973)

methodology, where the specification of a social welfare function (or parametric class of

welfare functions) completely determines the inequality index (or class of indices). The

AKS methodology can also be motivated by a thought experiment, albeit one that asks

what percentage of total income can be discarded without affecting social welfare if income

is equally distributed.

A distinguishing feature shared by all poverty indices constructed from reference lot-

teries is that they are invariant to ordering-preserving transformations applied to the nu-

merical values representing the various levels of achievement. Consequently, a “retreat”

to the headcount ratio with ordinal data is entirely unnecessary since poverty indices con-

structed from reference lotteries not only include the headcount ratio as a special case,

but they can also be made sensitive to the ‘depth’ and ‘distribution’ of poverty that the

headcount ratio ignores.

As a concrete example, we apply our methodology to construct an analogue of the

FGT class of measures for use with ordinal data. In particular, we show that a simple

parametric class of distributions gives a counterpart of the classical FGT class of measures

that retains many of the attractive properties of the classical FGT measures (including,

for example, additive decomposability) and is without the obvious shortcomings inherent

in the application of conventional poverty measures to ordinal data. Furthermore, we

provide an axiomatization of the ordering induced by our ordinal analogue of the FGT

class of measures. This axiomatization is an ordinal counterpart to the axiomatization of

the classical FGT orderings developed by Ebert and Moyes (2002).

In the next section, we outline the construction of ordinal poverty indices using the

concept of a reference lottery, document the basic properties of poverty indices constructed

from reference lotteries, and introduce the parametric class of reference lotteries that

generate the ordinal analogue of the FGT class of indices. In Section 3, we present an

axiomatic characterization of the poverty orderings induced by the ordinal analogues of the

FGT class of indices. Then, in Section 4, we illustrate the application of the ordinal FGT

indices to self-reported health data from the United States and Canada. When applied

to this dataset, these indices suggest that there is unambiguously greater ill-health in the

United States than in Canada for the bottom 20% of their income distributions. Finally,

in Section 5, we present some concluding remarks.

2

2 Measuring Poverty with Ordinal Data

With ordinal data, there are K ordered categories or states of achievement represented

numerically by an ordered set Y = {y1, y2, . . . , yK} in such a way that yi > yj if, and only

if, state i is preferred to state j.2 The observed levels of achievement in a population of size

N are recorded in y ∈ YN and individuals within this population are identified as “poor”

if they fall into one of the k worst states, or equivalently if their level of achievement falls

at or below yk, where yk < yK .

2.1 Poverty Measures as Evaluations of Achievement Lotteries

In order to construct a meaningful measure of poverty for use with data recorded on

an ordinal scale, we consider a thought experiment in which one has the opportunity to

accept a realized level of achievement from the equiprobable lottery Y on y or to decline

this allocation in favor of an alternative allocation drawn independently from a reference

lottery Uα (indexed by α) over the k states of poverty y1, y2, . . . , yk.

When comparing the allocations from these two lotteries, one is certainly better off

accepting one’s realization of the equiprobable draw Y whenever it is above yk and, hence,

out of poverty. Conversely, one will choose to accept the state of poverty generated by

the reference lottery Uα whenever the realization of Y amounts to an even worse state of

poverty.

Ex ante, the probability that one will accept the state of poverty generated by the

reference lottery Uα is equal to the probability that the realization of Y is no larger than

the the realization of Uα, which is given by

πα(y, yk) = P[Y ≤ Uα]. (2.1)

The quantity πα(y, yk) thus tells us the probability that one will be better off facing the

lottery Uα rather than facing an equiprobable draw from the actual distribution in society.

In interpreting the right-hand side of (2.1), note that the statistical independence of Y

and Uα gives

P[Y ≤ Uα] = EY [EUα1(Y ≤ Uα)]

=1

N

N∑i=1

P(yi ≤ Uα),(2.2)

2The number of states can be countably infinite. Our focus on the case where the number of states isfinite is, however, without loss of generality.

3

where 1(·) is the indicator function and EY , for example, denotes the mathematical

expectation with respect to the probability distribution of Y . Because the first term in

the sum is the probability that Uα is no smaller than y1, the second term is the probability

that Uα is no smaller than y2, and the N th term is the probability that Uα is no smaller

than yN , we see that πα(y, yk) is merely the average of each individual’s probability

of receiving a higher level of achievement from the lottery Uα. The quantity πα(y, yk),

therefore, may also be interpreted as the proportion of individuals, each of whom in turn

faces the choice between their own realizations from the pair of lotteries Y and Uα, that

would accept their level of achievement generated from the reference lottery Uα.

Clearly, the quantity πα(y, yk) will be equal to zero for any distribution y in which

no individual in y is identified as poor. Indeed, when there are no poor individuals, the

realization of Y must be above yk and, hence, must be above any possible realization

of Uα. Conversely, πα(y, yk) will tend towards one as individual levels of achievement

fall towards the least desirable state y1. Consequently, we may regard the magnitude of

πα(y, yk) as an indicator of the extent of poverty in y (relative to Uα).

As a concrete example, consider the special case in which the reference lottery Uα is

degenerate at yk so that it yields the least deprived level of poverty yk with probability

one. In this case,

πα(y, yk) = P[Y ≤ Uα]

= P[Y ≤ yk],(2.3)

so that πα(y, yk) records the proportion of individuals who would prefer the guaranteed

state of poverty yk to their draw from the prevailing distribution of achievements y. As

a second example, consider the reference lottery Uα that assigns equal probability to the

states of the poor y1, y2, . . . , yk. In this case, πα(y, yk) records the proportion of individuals

that would prefer their random draw from the states of the poor rather than their realized

allocation drawn at random from y.

In general, the formulation in (2.1) provides us with a framework that is particularly

well suited for constructing meaningful poverty indices when the data are ordinal. This

is because the generic index πα(y, yk) (a) has both a simple and appealing interpretation

and (b) it is, by construction, invariant to order preserving transformations of the levels,

which is essential for any measure applied to ordinal data.3

Different reference lotteries over the states of the poor obviously produce different

3Indeed, P[Y ≤ Uα] = P[g(Y ) ≤ g(Uα)] for all strictly positive monotonic transformations g : R → R.

4

poverty indices, but it is not yet clear how the choice of reference lottery ultimately

shapes the index. The following proposition helps to shed light on this issue. Indeed, it

shows precisely how the reference lottery affects the properties of the resulting poverty

index. In our statement of the proposition we denote the collection of poor individuals

by q(y, yk) = {1 ≤ i ≤ N : yi ≤ yk}.

Proposition 2.1. For any reference lottery Uα over the states of the poor y1, y2, . . . , yk,

πα(y, yk) =1

N

∑i∈q(y,yk)

P[yi ≤ Uα]. (2.4)

Proof. It follows from (2.2) that

P[Y ≤ Uα] =1

N

N∑i=1

EUα1(yi ≤ Uα)1(yi ≤ yk) +1

N

N∑i=1

EUα1(yi ≤ Uα)1(yi > yk). (2.5)

Note that 1(yi ≤ Uα)1(yi > yk) = 0 because the support of Uα is restricted to the poor

states (i.e., y1, . . . , yk). Consequently, the second term in the last line of (2.5) is zero.

Hence, the desired result follows from the equivalence

1

N

N∑i=1

EUα1(yi ≤ Uα)1(yi ≤ yk) =1

N

∑i∈q(y,yk)

P[yi ≤ Uα].

The function πiα ≡ P[yi ≤ Uα] in (2.4) is the individual poverty function of the

ith individual. Proposition 2.1 shows that the poverty measure πα(y, yk) is always de-

composable (Foster and Shorrocks 1991, p. 691), with only the poverty functions πiα,

i = 1, . . . , N , influenced by the specification of the reference lottery. Consequently, the

cumulative distribution associated with the reference lottery determines the specification

of the individual poverty functions and, hence, ultimately determines if and how ‘depth’

and ‘distribution’ are accounted for by the aggregate measure.

In short, the proposed methodology amounts to constructing ordinal poverty indices

from cumulative distributions. This approach gives rise to a rather large set of choices, not

unlike the Atkinson-Kolm-Sen methodology that constructs inequality indices from social

welfare functions. In the next subsection, we examine a class of distributions that give rise

to a particularly simple and appealing class of indices that are ordinal analogues of the

classical FGT class. This new class of indices inherits many of the attractive properties

of the classical FGT class, including its simple structure and sound axiomatic properties

5

(Foster, Greer, and Thorbecke 2010). We also develop an axiomatic characterization of

the poverty ordering induced by this class of ordinal indices in Section 3.

2.2 A Parametric Class of Reference Lotteries

In this section we examine the parametric class Uα, α ≥ 0, of reference lotteries whose

corresponding probability distributions are given by

P[Uα ≥ yj] =

(k − j + 1

k

)α

, 1 ≤ j ≤ k, α > 0. (2.6)

When α = 0, the lottery Uα guarantees the least deprived state of poverty yk. Conse-

quently, this lottery when evaluated in (2.1) gives rise to the poverty index (2.3), which

is nothing other than the classical headcount ratio. When α = 1, the lottery is equally

weighted over the poor states. Hence, the index πα(y, yk) records the proportion of people

in society that would prefer an equiprobable draw from the states of the poor rather than

an allocation drawn at random from y. When α is chosen to be greater than 1, increased

probability is placed on the poorest of the poor states. The corresponding indices, there-

fore, become relatively more sensitive to the ‘depth’ of poverty experienced by individuals

in the population. In the limit, as α tends to ∞, the lottery is degenerate at y1, implying

that the corresponding poverty index will be sensitive only to changes in the proportion

of individuals experiencing the worst state of poverty.

It follows from Proposition 2.1 that substitution of this parametric class of lotteries

into (2.1) gives the class of ordinal poverty indices

πα(y, yk) =k∑

j=1

pj

(k − j + 1

k

)α

, α > 0, (2.7)

where pj is the proportion of the population y in the jth state. The class of indices

generated by (2.6) is an ordinal analogue to the classical FGT indices in that members

of this class are also given by average power functions of normalized gaps, albeit with

normalized gaps in levels replaced by normalized gaps in ranks.

To further elucidate this close connection to the classical FGT indices, let GY de-

note the cumulative distribution function that assigns equal probability to the potential

achievement levels in Y. The cumulative distribution GY(·) is a convenient mathematical

device that maps a given level of achievement yi ∈ Y to its corresponding (normalized)

achievement rank GY(yi) ∈ { 1K, 2K, . . . , 1}. Thus, for example, GY(yK) = 1 is the highest

6

achievement rank and GY(yj) = j/K is the achievement rank of an individual in the

jth state of achievement. With the distribution of (normalized) achievement ranks and

poverty rank cut-off computed as

x ≡(GY(y1), GY(y2), . . . , GY(yN)) ∈ [0, 1]N

and

z ≡ GY(yk+1) =k + 1

K,

respectively, the indices πα(y, yk), α > 0, which operate on the levels, are equivalent to

the indices

Π̃α(x; z) =1

N

N∑j=1

(z − xj

z −GY(y1)

)α

1(xj < z), α > 0, (2.8)

which operate on the (normalized) ranks.4 Furthermore, the expression in (2.8) is ordinally

equivalent to

Πα(x; z) =1

N

N∑j=1

(z − xj

z

)α

1(xj < z), α > 0. (2.9)

The alternative representation of πα in (2.9), which is formulated in terms of (nor-

malized) ranks, is identical to the computational formula for the classical FGT class of

indices. We exploit this alternative representation in the next section, where we provide

an axiomatic characterization of the ordinal FGT class of indices.

3 An Axiomatic Characterization of the Poverty Or-

dering induced by Πα

This section supplements our earlier construction of the ordinal FGT indices with an

axiomatic characterization of their induced poverty orderings. Our approach mirrors

Ebert and Moyes’s (2002) characterization of the poverty orderings induced by the classical

FGT class, albeit with their axioms suitably translated for when the data are ordinal.

Let <z (indexed by z) denote a complete, reflexive, and transitive binary relation

on the set of all possible distributions of (normalized) achievement ranks [0, 1]N . The

statement x <z y is interpreted as saying that x exhibits at least as much poverty as y.

4Note that we adopt the weak definition of the poor (Donaldson and Weymark 1986) here under whichthe poor consists of all individuals with endowments less than z = k+1

K . The presence of the term GY(y1)is due to the discreteness of the possible levels of achievement and would not appear with a continuumof levels as in the classical FGT class.

7

The corresponding asymmetric and symmetric factors of <z are ≻z and ∼z, respectively.

We define <z on all of [0, 1]N rather than on { 1K, 2K, . . . , K−1

K, 1}N for simplicity. Such

idealizations are standard in axiomatic analyzes with discrete variables, e.g., in consumer

theory where it is typically assumed that commodities are perfectly divisible even when

they are not.

The first two axioms are suitably reformulated statements of standard properties.

CONTinuity: <z is continuous on [0, 1]N .

FOCus: Let x ∈ [0, 1]N and suppose that xi ≥ z. Then,

x ∼z (x1, . . . ,xi−1,xi + c,xi+1, . . . ,xN)

for all constants c such that z < xi + c ≤ 1.

The focus axiom states that only the ranks of poor individuals play a role in deter-

mining the ordering of two distributions. Our next axiom is a separability axiom.

INDependence: Let x1,x2 ∈ [0, 1]N satisfy x1 ∼z x2 with x1

i = x2i for some 1 ≤ i ≤ N .

Then, for every γ ∈ [0, 1],

(x11, . . . ,x

1i−1, γ,x

1i+1 . . . ,x

1N) ∼z (x

21, . . . ,x

2i−1, γ,x

2i+1 . . . ,x

2N).

The independence axiom implies that the poverty ordering of achievement ranks for

any subgroup of individuals can be derived without reference to the ranks in which the

rest of the the population find themselves.

SYMMetry: For all x = (x1, . . . ,xN) ∈ [0, 1]N and any permutation π of {1, 2, . . . , N},(x1, . . . ,xN) ∼z (xπ(1), . . . ,xπ(N)).

MONotonicity: Let x ∈ [0, 1]N be such that xi < z ≤ 1. Then,

x ≻z (x1, . . . ,xi−1,xi + c,xi+1, . . . ,xN)

for all constants c > 0 satisfying xi + c ≤ 1.

Symmetry says that individual identities play no role in determining the intensity

of poverty, whereas Monotonicity says that an increase in a poor person’s rank should

decrease the overall poverty level; see, e.g., Zheng (1997). Lastly, we impose two invariance

8

axioms.

SCALE Invariance: For all x1,x2 ∈ [0, 1]N and all 0 < λ ≤ 1, x1 ∼z x2 implies

λx1 ∼λz λx2.

TRANSlation Invariance: For all x1,x2 ∈ [0, 1]N and all γ ∈ R such that x1 +

γ1N ,x2 + γ1N ∈ [0, 1]N and z + γ ∈ [0, 1],

x1 ∼z x2 implies x1 + γ1N ∼z+γ x2 + γ1N ,

where 1N is an n× 1 vector of ones.

The axioms stated above are ordinal analogues of the axioms for cardinally measurable

attributes used in Ebert and Moyes (2002). Taken together, they impose sufficient struc-

ture to characterize the representation of <z. Specifically, one can establish the following

result:

Proposition 3.1. The poverty ordering <z satisfies CONT, FOC, MON, IND, SYMM,

SCALE, and TRANS if and only if it is represented by

Πα(x; z) =1

N

N∑i=1

a(z)(z − xi

)α1(xi < z), for all x ∈ [0, 1]N and all α > 0. (3.1)

Proof. The proof is identical to the proof of Theorem 1 in Ebert and Moyes (2002) after

substituting levels for ranks.

As in the case of the classical FGT index, one can easily verify that α > 0 in Propo-

sition 3.1 above must be replaced by α > 1 if we impose the additional requirement

that <z satisfy the following transfer axiom, which states that the overall poverty level

should decrease when a poor individual’s shift upwards in rank if offset by a less deprived

individual’s equal downward shift in rank:

Transfer Suppose that 0 < xi < xj < z ≤ 1. Then,

x ≻z (x1, . . . ,xi−1,xi − c,xi+1, . . . ,xj−1,xj + c,xj+1, . . . ,xN)

for all c > 0 satisfying xi − c ≥ 0 and xj + c ≤ 1.

In summary, the ordinal FGT index is sensitive to ‘depth’ for α > 0, and sensitive to

both ‘depth’ and the ‘distribution’ of achievement ranks for α > 1.

9

4 Empirical Illustration

We now illustrate the ordinal FGT indices using self-reported health statuses in Canada

and the United States from the Joint Canada/United States Survey of Health (JCUSH).

In these surveys, approximately 3,500 Canadian and 5,200 U.S. residents rated their

individual health as either poor, fair, good, very good, or excellent.5 Due to the complex

sampling design and over-sampling of certain populations, sampling weights have been

appended to the survey data by the Centers for Disease Control and Prevention and

Statistics Canada to render the samples representative of their respective populations.

We use these sampling weights in our subsequent analysis.

We apply the ordinal FGT indices to examine health deprivation or health poverty, as

well as to examine health poverty when the population is decomposed by income quintiles.

We begin by considering the headcount ratios (α = 0) in each country and at various cut-

offs.6 As can be seen in Table 1, more U.S. residents as a proportion of the population

report their health as being less than or equal to poor, fair, or good, than is the case in

Canada. On the other hand, Canadians are less likely than U.S residents to rate their

health status as excellent rather than very good.

For α = 1, the ordinal FGT indices suggest that health status in the U.S. is worse

than in Canada for every cutoff.7 Perhaps more interestingly, the decomposition by

income quintiles demonstrates that the greatest contribution to the disparity between

the two countries occurs at the lowest income quintile. In other words, the disparity

in health statuses between the two countries is greatest at the bottom income quintile

where the self-reported health statuses of income poor U.S. residents are being compared

to self-reported health statuses of income poor Canadians. The α = 1 case provides

us with more insight into the distribution of the poor than the headcount ratios do by

themselves. Such insight may be helpful to policymakers when designing and targeting

their health-care policies.

These data can also be used to illustrate the simple interpretation of the ordinal FGT

indices provided above. For example, if we focus on the first income quintile and a cutoff

of 2, we observe that the FGT indices when α = 1 are 0.165 in the U.S. and 0.102 in

5The survey participants were asked “In general, would you say your health is: poor, fair, good, verygood, or excellent?” See Allison and Foster (2004), and references therein, for a discussion and a literaturereview of the role of self reported health data as a predictor of mortality and overall health.

6A similar analysis using the headcount ratio to look at poverty with self-reported health data wasperformed in Allison and Foster (2004).

7The ordinal FGT index is unchanged at the first cut-off y1 because πα(y, y1) is independent of αsince πα(y, y1) = P[Y ≤ y1].

10

Table 1: Health Poverty Estimates for Canada and the United States.

Index Country Income Quintile Cut-Off1 2 3 4 5

Headcount, α = 0

USA

All 0.037 0.136 0.398 0.732 1.0001 0.087 0.243 0.532 0.790 1.0002 0.052 0.202 0.497 0.783 1.0003 0.014 0.096 0.398 0.750 1.0004 0.012 0.057 0.288 0.684 1.0005 0.008 0.056 0.247 0.644 1.000

CAN

All 0.032 0.111 0.384 0.757 1.0001 0.051 0.154 0.495 0.820 1.0002 0.054 0.170 0.467 0.810 1.0003 0.032 0.104 0.362 0.756 1.0004 0.013 0.084 0.337 0.746 1.0005 0.005 0.042 0.249 0.651 1.000

Ordinal FGT α = 1

USA

All 0.037 0.087 0.191 0.326 0.4611 0.087 0.165 0.287 0.413 0.5302 0.052 0.127 0.251 0.384 0.5073 0.014 0.055 0.170 0.315 0.4524 0.012 0.034 0.119 0.260 0.4095 0.008 0.032 0.104 0.238 0.391

CAN

All 0.032 0.072 0.176 0.321 0.4561 0.051 0.102 0.233 0.380 0.5042 0.054 0.112 0.230 0.375 0.5003 0.032 0.068 0.166 0.314 0.4514 0.013 0.049 0.145 0.295 0.4365 0.005 0.023 0.098 0.236 0.389

Notes: Headcount (α = 0) and Ordinal FGT measure (α = 1) decomposed by incomequintiles. Estimates based on 2,960 and 3,815 Canadian and U.S. respondents from the2003 Joint Canada/United States Survey of Health.

Canada. Consequently, we have that 165 out of every 1,000 U.S. residents would prefer

an equiprobable lottery from the two lowest states of health rather than draw their health

status from the actual distribution of health in society. In contrast, only 102 out of every

1,000 Canadian residents would prefer the equiprobable two-state lottery over the random

draw from the societal distribution in Canada.

5 Concluding Remarks

This paper has developed a methodology for constructing poverty indices from cumulative

distributions over the states of the poor and has applied this methodology to construct an

ordinal analogue of the classical FGT class of poverty indices. This new class of ordinal

indices retains many of the attractive properties of the classical FGT class and yet is

without the obvious shortcomings inherent in the application of conventional poverty

measures to ordinal data.

11

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Technical Appendix (Not for publication)

Lemma 5.1. A poverty ordering <z on [0, 1]N satisfies the axioms CONT, FOC, IND,

SYMM, and MON if, and only if, it is represented by

Π(x, z) =∑xi<z

π(xi, z) +∑xi≥z

π(z, z) (5.1)

for some π : [0, 1]2 → R where π(x, z) is continuous and strictly increasing in x for all

x ∈ (0, 1).

Proof. The “if” part is obvious.

IND, SYMM, and FOC imply that the poverty ordering ≽z can be represented by an

additive function Π(x, z) of the form

Π(x, z) =∑xi<z

π(xi, z) +∑xi≥z

π(z, z)

The remaining axioms impose structure on the function π: CONT implies that π is

continuous; and MON implies that π(·, z) is strictly increasing.

The representation of <z in (5.1) allows us to associate each distribution x ∈ [0, 1]N

with an equivalent societal achievement (Ebert and Moyes, p. 462). That is, for a given

distribution of achievement percentiles x ∈ [0, 1]N and a given poverty threshold z ∈ [0, 1],

we define the equivalent societal achievement as

e(x, z) =

π−1(N−1Π(x, z), z) if xi < z for some i ∈ N

z if xi ≥ z for all i ∈ N(5.2)

Lemma 5.2. Suppose that <z satisfies the axioms CONT, FOC, MON, IND, SYMM,

SCALE, and TRANS. Then, for any given distribution of achievement x ∈ [0, 1]N and

any given poverty threshold z ∈ [0, 1],

i. x ∼z (e(x, z), e(x, z), . . . , e(x, z))

ii. e(λx, λz) = λe(x, z) for all λ ∈ (0, 1]

iii. e(x+γ1N , z+γ) = e(x, z)+γ for all γ such that x+γ1N ∈ [0, 1]N and γ+z ∈ (0, 1].

Proof. (i) Obvious from the definition; (ii) By definition,

e(x, z)1N ∼z x, (5.3)

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and e(λx, λz)1N ∼λz λx. Because applying SCALE to (5.3) gives λe(x, z)1N ∼z λx, it

follows from the transitivity of ∼λz that

λe(x, z)1N ∼λz e(λx, λz)1n

The equality λe(x, z) = e(λx, λz) is then an immediate consequence of MON. (iii) Anal-

ogous to (ii).

Proof of Proposition 3.1. The “if” part is obvious.

To prove “only if”, let x ∈ [0, 1]N and z be given, and choose ϵ > 0 such that

0 ≤ z − ϵ ≤ xi ≤ z for all 1 ≤ i ≤ N . Parts (ii.) and (iii.) of Lemma 5.2 give

1

λπ−1

(1

N

N∑i=1

π(λxi, λz), λz

)= π−1

(1

N

N∑i=1

π(xi, z), z

)(5.4)

and

π−1

(1

N

N∑i=1

π(xi + γ, z + γ), z + γ

)− γ = π−1

(1

N

N∑i=1

π(xi, z), z

)(5.5)

for all λ ∈ (0, 1] and all γ such that x+ γ1N ∈ [0, 1]N and γ + z ∈ (0, 1].

Ebert and Moyes (2002) show that the only well-defined solution to the pair of func-

tional equations (5.4) and (5.5), which is also strictly increasing in x, is of the form

π(x, z) = a(z)(z − x)α(z) + b(z) (5.6)

with α(z) > 0, a(z) > 0, and b(z) ∈ R. Furthermore, it is demonstrated in Ebert

and Moyes (2002, p.471) that Part (ii.) of Lemma 5.2 implies that α(z) is functionally

independent of z so that α(z) = α and, hence,

π(x, z) = a(z)(z − x)α + b(z), α > 0, a(z) > 0, b(z) ∈ R (5.7)

The desired result, therefore, is obtained upon setting a(z) =(

1z

)α

and b(z) = 0.

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