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DEPARTMENT OF ECONOMICS

ISSN 1441-5429

DISCUSSION PAPER 25/13

A Dynamic Multidimensional Measure of Poverty

Aaron Nicholas*, Ranjan Ray and Kompal Sinha

Abstract This paper unites two strands of the literature on subgroup decomposable poverty measurement

originating from Foster, Greer and Thorbecke (1984) by incorporating information on both multiple

dimensions and multiple periods. This generalises the Alkire and Foster (2011a) measure into a

dynamic setting. In doing so, it introduces two variants of the ‘transfer’ axiom: one that gives

increasing weight to individuals whose deprivations are concentrated as repeated dimensions in a

specific period (what we term ‘breadth’) versus one that gives increasing weight to individuals

whose deprivations are concentrated as repeated periods in a specific dimension (‘depth’). The

measure is able to differentiate between both aspects of poverty as well as quantify the relative

contribution of each aspect towards overall poverty. This makes it well suited to make comparison

across subgroups when individual longitudinal data is available as well as allowing policy-makers

to quickly identify if the breadth or depth aspect requires more attention since the policy

prescriptions for each may differ. We apply the proposed measure to longitudinal data from China

and Indonesia where we find that the depth aspect of poverty is a large contributor to individual

level poverty. The existing static framework of multidimensional poverty in the literature may

therefore be inappropriate for subgroup comparisons when information over time is available.

Keywords: Multidimensional Poverty and Deprivation; Duration and Dynamics of Poverty;

Dimensional and Dynamic Transfer Axioms; Subgroup Decomposability.

JEL classification: I31, I32

*corresponding author: aaron.nicholas@deakin.edu.au Graduate School of Business Deakin University, Victoria 3170,

Australia

Ranjan Ray Department of Economics Monash University, Victoria 3800 Australia

Kompal Sinha Department of Econometrics & Business Statistics Monash University, Victoria 3800 Australia

© 2013 Aaron Nicholas, Ranjan Ray and Kompal Sinha

All rights reserved. No part of this paper may be reproduced in any form, or stored in a retrieval system, without the prior written

permission of the author.

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1. Introduction

Traditional measures of poverty based on income and expenditure have been extended along two

major directions: the broadening of the measures to incorporate a wider set of dimensions that

together give a more accurate representation of welfare; and the deepening of the measures to

incorporate information that spans over several periods of observations.

Extensions along the first direction, largely influenced by the writings of Sen (1985), move away

from unidimensional measures and into a multidimensional approach based on the individual‟s lack

of access to a wide set of dimensions that include both market and nonmarket goods. Sen (1976)‟s

pioneering contribution introduced the axiomatic approach to the measurement of poverty and

provided the basis for the recent axiomatic approach to the multidimensional measurement of

poverty – examples include Tsui (2002), Bourguignon and Chakravarty (2003), and Alkire and

Foster (2011a) [henceforth AF].1

Extensions along the second direction move away from static measures into a dynamic approach

where repeated observations (or „spells‟) of poverty are treated differently to cases where poverty is

temporary. Examples of such extensions include Foster (2007), Calvo and Dercon (2009), Hojman

and Kast (2009), Duclos et.al. (2010), Hoy and Zheng (2011), Bossert et. al. (2012) and Gradin et

al.(2012).

Despite the usefulness provided by extensions along both directions, the literature has largely

considered both extensions independent of each other, retaining either the unidimensional or static

property of traditional measures. The principal motivation of this paper is to provide a generalised

framework for jointly incorporating both aspects – multidimensionality and dynamics – with a

particular emphasis on the interaction between both aspects and the information gained by

considering them jointly.

Recently, there have been attempts to unite both strands of the literature: Nicholas and Ray (2012)

[NR], and Bossert, Ceriani, Chakravarty and D‟Ambrosio (2012) [BCCD], consider a class of

subgroup decomposable poverty measures based on the „count‟ of an individual‟s deprivations

(Atkinson, 2003). The unique contribution of these papers is to define each deprivation as belonging

to a particular period of time and a particular dimension – an individual‟s poverty score is then

simply a function of the double-sum of these deprivations over time and over dimensions. However,

such a method leads to „path-independence‟ of deprivations, meaning that the measure is invariant

to whether the deprivations were summed first over periods of time or over dimensions. The

1 See, also, Chakravarty and D‟Ambrosio (2006), Bossert et. al. (2007), and Jayaraj and Subramanian (2010) for closely

related work on the measurement of multidimensional deprivation.

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consequence of this is that the measure is unable to differentiate individuals for whom we observe

deprivations in repeated periods of time for specific dimensions from individuals for whom we

observe deprivations over multiple dimensions for specific periods of time. Nor can any of these

two groups of individuals be differentiated from individuals whose deprivations are scattered across

both dimensions and periods of time.

Consider a simple example, where an individual can be deprived in up to three dimensions, and that

we have observations for three periods of time for each dimension. This means an individual can

experience a maximum of nine deprivations. Consider an individual deprived for all three periods in

the first dimension, versus an individual deprived for one period in the first dimension, one period

in the second dimension and one period in the third dimension. While both individuals share the

same count of deprivations (three), the distribution of the deprivations differ, and indeed, we argue

that the first individual should count as more deprived.

In the spirit of this distinction, this paper proposes a measure that quantifies the contribution, to the

overall poverty measure, of the concentration of deprivation in specific dimensions relative to that

of the concentration of deprivation in specific periods of time. The measure of the relative

magnitude of the two contributors to poverty – introduced in this paper as the „Breadth to Depth

ratio‟ – is a useful concept from a policy viewpoint since it allows policy-makers to identify

whether their efforts should be concentrated on specific dimensions of poverty (e.g. access to

hospitals), or on specific periods of time that have been the main cause of poverty (e.g. large

negative macroeconomic shocks such as the Asian or the Global Financial Crisis).

In incorporating these additional properties, we retain the subgroup decomposability property from

Foster, Greer and Thorbecke (1984) [FGT], the dimensional decomposability property from Alkire

and Foster (2011a) and the dynamic decomposability property from Foster (2007).

We follow the framework established in AF by using a „dual-cutoff‟ approach in our axiomatisation

and application whereby non-poor individuals are ignored by the measure and „poor‟ can be defined

by the intersection, intermediate or union methods (Tsui, 2002). Our generalisation of AF allows us

to include new definitions and perspectives on the „transfer‟ axiom.2

Another significant feature of this study is the application of the proposed dynamic

multidimensional poverty measure to the data sets of developing countries. Both the recent

applications of dynamic multidimensional deprivation measures (NR and BCCD) have been on

panel data sets from developed countries. The absence of any similar previous application in a

2 The transfer axiom originated with Pigou (1912), Dalton (1920), with its modern characterisation in Sen (1976).

5

developing country context is largely due to the absence of information on a panel of households in

developing countries for a sufficiently long time period containing information on a reasonably

wide set of dimensions to make such a study of dynamic multidimensional deprivation possible.3

Such panel data sets are rare even in the context of developed countries and, until recently, almost

non-existent in the case of developing countries.4 The recent availability of such information for

China and Indonesia that we utilize in this study makes this study distinctive.5 The time interval

spanned by the data sets of both countries included the Asian Financial Crisis which affected

Indonesia particularly badly, much more than China.

The rest of the paper is organised as follows. The proposed dynamic multidimensional poverty

measure is introduced in Section 2 along with a discussion of its principal properties. The data sets

are described in Section 3 along with some summary features of the data that are relevant for this

study. The empirical results are presented in Section 4 while Section 5 concludes.

2. Analytical Framework

2.1. Notation

Assume we observe, for N individuals in a population of interest, different dimensions of

deprivation and T equally-spaced periods of time. We say that an individual n is deprived in

dimension j at time t when , where { }, { } { }

is individual n‟s achievement in dimension j at time t, and is a cut-off point that determines

whether or not an individual is considered deprived in a particular dimension at a particular time.

For example, in the dimension „health‟, may be the individual‟s Body Mass Index, in which case

would be some threshold below which the individual would be considered underweight and

therefore deprived in the health dimension.

Each individual can be said to have an individual deprivation profile, which is a matrix

.

/ where {(

)

{ } & { }.

3 As BCCD report, the problem of missing information at the household level on material deprivation in several

individual dimensions is present in the data sets of developed countries as well. This forces one to adopt either the

unsatisfactory practice of treating missing information as the household having access to the dimensions concerned as

done in BCCD, or simply not including such households as done in the present study. 4 There is now increasing availability of such panel data in developing countries. Besides the CHNS data set from China

and the IFLS data set from Indonesia that have been used here, an example of such a data set is that contained in the

household surveys conducted by the Research Centre on Rural Economy (RCRE) in Beijing, and used in the above

cited study by Duclos, et al (2010). 5 Mishra and Ray (2012) have recently compared multidimensional deprivation in the static framework between China

and India, but their study was not on panel data and, consequently, did not incorporate any of the dynamic elements of

the present study.

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is a sensitivity parameter along the lines of the poverty measure due to FGT. Call

deprivation inputs. When observed achievement levels are discrete or ordinal, it is common to

restrict such that { }.

The population deprivation profile is a vector ,.... . Define the identification vector

where takes the value 1 if the individual is considered poor, and 0 otherwise.

An individual is considered poor if he has at least different deprivations; this can be based on a

minimum number of periods, or dimensions, or a combination of both. A union method of

identification would set while the intersection method would use . The poverty

index is a function .

2.2. The breadth versus depth of deprivation

Unlike the simpler unidimensional case in AF, each deprivation input in the dynamic

multidimensional case contributes to both the „breadth‟ of deprivation (number of dimensions

deprived at a particular time t) as well as the „depth‟ of deprivation (number of periods deprived in a

particular dimension j). In applications such as NR and BCCD, an individual‟s deprivation score is

simply the sum of deprivation inputs regardless of which dimension or period they belong to.

Therefore, the poverty index is insensitive to permutations of the individual‟s deprivation profile.

Consider the example below of three individuals with different deprivation profiles with

and .

{

} {

} {

}

A measure that takes as its inputs only the double sum of the deprivation inputs would conclude that

all three individuals are equally deprived since the „count‟ of deprivations would be six in each.

Path-independence with respect to deprivation inputs makes the measure insensitive to any

permutation of the deprivation inputs. However, one may argue that is more deprived than if

we adopt the view that adding one dimension of deprivation to a period that has many dimensions

of deprivation cannot be offset by removing one dimension of deprivation from another period that

has equal or less periods of deprivation.6 This can be seen as analogous to FGT‟s transfer axiom,

but defined over counts of deprivation dimensions rather than income/expenditure.

Similarly is more deprived than if we adopt the view that adding one period of deprivation

to a dimension that has many periods of deprivation cannot be offset by removing one period of

6 can be derived from by adding one deprivation to and removing one deprivation from .

7

deprivation from a dimension that has equal or less periods of deprivation.7 This can be seen as

analogous to FGT‟s transfer axiom, but defined over counts of deprivation periods rather than

income/expenditure. We define both of these properties formally in the next subsection.

Notice that in the dynamic multidimensional case, a permutation of the deprivation inputs could

increase the breadth of deprivation at a particular period while having no effect on the depth in a

particular dimension (e.g. from to ); or increase the depth of deprivation in a particular

dimension, while having no effect on the breadth of deprivation at a particular time period (e.g.

from to ). It is therefore important to distinguish between the two.

To illustrate the importance of our distinction in the policy context, consider another example:

{

} {

} {

}

We argue that and are more deprived than because in the deprivations occur in time

periods and in dimensions that are independent from each other, whereas has deprivation in

three periods for one dimension, and has deprivation in three dimensions for one period. One

may argue it is a subjective matter as to which of the three profiles are the most detrimental to

individual wellbeing; even so, from a policy perspective, and may be given more importance

simply due to the relative ease of targeted poverty-reduction. In , for example, poverty-reduction

policies would target dimension one, while in , the policies would try to understand what was

unique in period one and focus on those factors.

The highlight of a measure that differentiates all three profiles is therefore twofold. Firstly it is able

to better utilise panel data on individuals by allowing for sensitivity of the index towards

deprivation concentrated in certain periods and certain dimensions. This is useful for comparing

different subgroups of a population. Secondly, it is able to highlight whether it is better for poverty-

reduction strategies to be focussed on certain dimensions, certain time periods, or both.

2.3. Dimensional and Dynamic Transfer Axioms

We now state the unique properties of our measure formally by firstly defining two types of

transfers.

Define as the column vector of individual ‟s deprivation profile for period . A dimensional

permutation occurs if there is a rearrangement of any two deprivation inputs and

7 can be derived from by adding one deprivation to and removing one deprivation from .

8

along . Furthermore,

prior to rearrangement and ∑

remains the

same after rearrangement for all .

Define as the row vector of individual ‟s deprivation profile for dimension . A dynamic

permutation occurs if there is a rearrangement of any two deprivation inputs and

along . Furthermore,

prior to rearrangement and ∑

remains the same

after rearrangement for all .

(Axiom 1): Dimensional Transfer

if is obtained from by a dynamic permutation of individual ‟s deprivation

profile in dimension and ∑

∑

where and individual n is

poor.

The Dimensional Transfer axiom requires that the poverty measure should register an increase if

deprivation was transferred to a dimension where there are more periods of deprivation.

(Axiom 2): Dynamic Transfer

if is obtained from by a dimensional permutation of individual ‟s

deprivation profile at period and ∑

∑

where and

individual n is poor.

The Dynamic Transfer axiom requires that the measure should register an increase if deprivation

was transferred to a period where there are more dimensions of deprivation.

2.4. The Proposed Dynamic Measure of Multidimensional Poverty

Consider the following explicit form for :

{

[∑ ((∑ ∑

)

) ] ⁄

[∑

] ⁄

(1a)

where ; , ∑ ∑

.

Like NR and BCCD, the measure continues to be a double-sum across time and dimensions.

However, the double-sum is now defined over a new term,

.

9

{

[ (∑

)

(

∑

)

]

(1b)

is therefore a transformation of deprivation inputs that weights each input according to its

contribution to the breadth (first term) and depth (second term) of deprivation. Increasing the

parameter increases the sensitivity of the measure to both the breadth and depth aspects. This

would therefore give increasing weight to individuals whose deprivations are located along the

same period/dimension relative to those who deprivations are unrelated by time or dimensions. The

parameter gives more weight to either the breadth or depth aspects. For most applications, it

seems appropriate to weight them equally and set ; however, for some applications it may

be useful to increase the weight to either the breadth or depth depending on policy interest. For

example, some policy-makers may not be overly concerned with cases where there is a

concentration of deprivation in a single period of time (breadth) since this is largely retrospective

information (e.g. a financial crisis in the past) with little directions for targeted poverty-reduction.

By setting and both dynamic and dimensional transfer are satisfied. When

only dynamic transfer (sensitivity to depth) is satisfied and when only dimensional transfer

(sensitivity to breadth) is satisfied (proof in Appendix A).

Consistent with the class of subgroup decomposable measures, has a convenient interpretation

as the average poverty score for individuals in the population of interest. Two special cases of the

measure allow unique interpretations: when , gives us the poverty headcount ratio,

while with , , gives us the average share of deprivations for individuals in

the population of interest. Increasing the parameter essentially increases the sensitivity of the

index to the count of deprivations. We discuss the other properties that emerge with different

choices of in subsection 2.7.

NR can be seen as a special case of when and . AK can be seen as a special case

of when while Foster (2007) can be seen as a special case of when . Figure

1 summarises the special cases in the literature that arises from different parametric choices for

.

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Figure 1: Generalisation-tree of the class of subgroup decomposable poverty measures^

^: Note that the strands of literature referred to in the figure do not necessarily represent the origin of the extensions; for

example, multidimensional extensions had appeared as early as Bourguignon and Chakravarty (2003). We simply refer

to measures that are direct special cases of .

2.5. The Breadth-to-Depth ratio

The Breadth-to-Depth ratio (BTD) allows us to identify the relative contribution to the poverty

index of repeated periods in a few dimensions (the depth aspect) versus many dimensions

concentrated in a few periods (the breadth aspect).

The numerator of the BTD is the deprivation index when (dimensional transfer only) while

the denominator is the deprivation index when (dynamic transfer only).

Dimensional and

dynamic

differentiation and

sensitivity

Dimensional and dynamic

combination

𝛀𝜶𝜷𝜹

Nicholas and Ray, 2012

(𝛽 𝑧

Foster, 2007

(𝛽 𝐽 𝛿

Alkire & Foster, 2011a

(𝛽 𝑇 𝛿

Foster, Greer & Thorbecke, 1984

(𝛽 𝑇 𝐽 𝛿

11

∑

(

∑ ∑ 0(

∑

)

1

)

∑

(

∑ ∑ [(

∑

)

]

)

A BTD above 1 indicates that deprivation is more likely to be concentrated within specific periods

of time in the form of multiple dimensions („breadth‟). A ratio below 1 indicates that deprivation is

more likely to be concentrated within specific dimensions in the form of multiple periods („depth‟).

The BTD can therefore serve as an indicator of whether a policy-maker or researcher should be

concentrating their attention on specific periods of time (BTD>1) or concentrating their attention on

specific dimensions (BTD<1).

Note for any individual, the breadth and depth aspects of deprivation can only differ when the count

of deprivations, ∑ ∑

, is greater than zero and less than ( – i.e., the theoretical maximum

number of deprivations. When the count of deprivations is ( , the deprivation profile is „full‟

and therefore both breadth and depth aspects are at their respective maxima. For values of

∑ ∑

that lie between zero and , the differentiation of breadth and depth becomes the

most meaningful and the BDT can take on different values depending on the deprivation profile.

Consider an example where . Imagine that we observe ∑ ∑

. Consider the

following two possible permutations of individual A‟s deprivation profile:

{

} {

}

and show permutations of a deprivation profile that yield the minimum and

maximum value of BTD (respectively) given a fixed deprivation count ∑ ∑

(in this case, 3).

Now consider the poverty index measure, for individual A. Assume that ; .

will yield a BTD of 0.33, while will yield a BTD of 3. This represents the range for

the BTD for all possible permutations of the deprivation profile given that ∑ ∑

. The

larger the BTD range, the more informative the BTD is since it becomes harder to infer the value of

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BTD by looking at ∑ ∑

alone. When , the BTD range is largest for ∑ ∑

.

For the BTD range is maximised for values of ∑ ∑

lying between and (proof in

Appendix B). In applications, the average deprivation count ∑ ∑

can be used as an

indicator of the usefulness of distinguishing depth from breadth. Figure 2 gives us the BTD range

for all possible values of ∑ ∑

based on our example above.

Figure 2: Breadth-to-Depth Ratio Range as a function of Count of Deprivation Inputs

2.6. Dimensional and Dynamic Decomposability

One of the attractive features of path-independence is the ability to decompose the measure

according to the individual contribution of each dimension, or the individual contribution of each

time period. This is useful when utilised jointly with the BDT, since it allows one to identify the

dimensions associated with the most depth of deprivation, or the periods associated with the most

breadth of deprivation.

While NR and BCCD have dimensional and dynamic decomposability, decomposition across time

in their measure is equivalent to calculating a static multidimensional measure (as per AF) for each

time period separately. The decomposition therefore only uses information on deprivations

associated with the particular period of time. Similarly, decomposition across dimensions is

equivalent to calculating a dynamic unidimensional measure (as per Foster, 2007) for each

dimension separately. The decomposition therefore only uses information on deprivations

associated with the particular dimension. Our measure has the advantage of allowing the

decompositions to be sensitive to information outside the period or dimension being decomposed

since each deprivation input is transformed to be a function of the entire deprivation profile.

0

0.5

1

1.5

2

2.5

3

3.5

1 2 3 4 5 6 7 8 9

Bre

adth

-to

-De

pth

Rat

io

Count of Deprivation Inputs

Max BTD

Min BTD

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(Axiom 3): Dimensional Decomposability

(

) (

) (

), where

are exogenously

imposed dimensional weights, and (

) is the deprivation index calculated using only the

row vector of each individual‟s deprivation profile.

Dimensional decomposability is useful since it allows the policy maker to calculate the percentage

contribution of each dimension towards the overall deprivation index.

(Axiom 4): Dynamic Decomposability

(

) (

) (

), where

are exogenously

imposed dynamic weights, and (

) is the deprivation index calculated using only the

column vector of each individual‟s deprivation profile.

Dynamic decomposability allows the policy maker to calculate the percentage contribution of each

period towards the overall deprivation index.

Recall that the parameter governs the sensitivity of the index to the count of deprivations. If

, then is insensitive to the order of summation of

(i.e. we have path-independence

with respect to

). We therefore get:

[∑(∑ ∑

)

] ⁄ ∑([∑(∑

)

] ⁄ )

⁄

∑([∑(∑

)

] ⁄ )

⁄

The second and third terms are simply the definitions of dimensional and dynamic decomposability

respectively, proving that satisfies both properties.

When 0, gives us the headcount ratio of poverty: dimensional and dynamic

decompositions are then meaningless since an increase in the number of deprivation inputs has no

effect on the measure if the person is already classified as poor When 1, if decomposition is

desirable it seems reasonable to assume that the decomposition should be made as if 1. This is

because is simply a positive monotonic transformation of each individual‟s poverty score – while

it increases the weight given to certain individuals relative to other individuals, it does not change

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the weights given to certain dimensions (periods) relative to other dimensions (periods).8 Increased

weight to dimensions that have more periods/dimensions of deprivation associated with them is

already captured by . Since dimensional/dynamic decomposability is satisfied for any value of

we recommend using for the purpose of decomposition according to dimensions.

2.7. Other Properties

also satisfies the following properties, which are based on AF but extended to the dynamic

framework. The difference with the following over the axioms in NR is the compatibility with the

poverty focus axiom (i.e. allowing the possibility of non-union identification).

(Axiom 5): Subgroup Decomposability

Define and as two subgroups of the population such that . Decomposability

requires

and holds for any number of subgroups where

is the deprivation index calculated over only a particular subgroup.

The axiom allows deprivation to be decomposed into a subgroup‟s percentage contribution towards

aggregate level deprivation.

(Axiom 6): Replication Invariance

Define a single replication of as ; of as ; of as .

Replication invariance implies for any number of replications of N, and

so long as the number of replications are the same for N, and .

This ensures that larger populations do not automatically count as more deprived.

(Axiom 7): Symmetry (anonymity)

where is any permutation of the vector .

Therefore, all individuals are identical except for their deprivation profiles.

(Axiom 8): Normalisation and Nontriviality

achieves at least two distinct values: a minimum value of 0 and a maximum value of 1.

8 leads to two properties (Deprivation Transfer and Deprivation Transfer Sensitivity) that are only useful in the

context of subgroup comparisons. We discuss these in subsection 2.7: „Other Properties‟.

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(Axiom 9): Poverty Focus

if is obtained from by having a non-poor individual experience an

achievement increase in any deprivation.

Therefore, if an individual is not considered poor, then improvements in that individual‟s

achievements are irrelevant to the measure, similar to a Rawlsian social welfare function.

(Axiom 10): Deprivation Focus

if is obtained from by an increase in the achievement in a deprivation

where an individual is not considered deprived.

Therefore, if an individual is not deprived in a particular period and dimension, then improvements

in that particular deprivation input are irrelevant to the measure.

Proposition 1: satisfies the following core properties: Subgroup Decomposability,

Replication Invariance, Symmetry, Nontriviality, Normalisation, Poverty Focus, Deprivation Focus.

Proof: Since individual scores are simply summed and averaged over individuals, always

satisfies Subgroup Decomposability, Replication Invariance, Symmetry, Normalisation and

Nontriviality. It also satisfies Poverty Focus based on the cutoff and Deprivation Focus based on

the cutoff

Define a deprivation decrement as a decrease in any one of an individual‟s deprivation inputs.

(Axiom 11): Deprivation Monotonicity

if is obtained from by a deprivation decrement among the poor.

Proposition 2: The poverty index satisfies the core properties (axioms 5 to 10), and in

addition, Dynamic Transfer, Dimensional Transfer, Dynamic Decomposability, Dimensional

Decomposability, Deprivation Monotonicity when .

Proof: Dynamic and Dimensional Transfer have been proved in subsection 2.3. Dynamic

Decomposability, Dimensional Decomposability have been proved in subsection 2.6. Deprivation

Monotonicity is proven in Appendix C.

Define an averaging of deprivations as a transfer of deprivations between A and B such that

[ + ] and

[ + ] where . More

16

generally, let be a bistochastic matrix whose elements are unity for every non poor person in

. Then an averaging of achievements can be represented as: .

(Axiom 12): Deprivation Transfer9

where is obtained from by an averaging of deprivations among any two

poor individuals A and B where ≠ ; and is derived by a deprivation decrement of .

A reduction in inequality between poor individuals must reduce poverty.

(Axiom 13): Deprivation Transfer Sensitivity10

where is obtained from by an averaging of deprivations

between A and B, while is obtained by an averaging of deprivations between C and D.

≠ are the deprivation profiles of person A, B, C and D, all four whom which are

poor. is derived by a deprivation decrement of , is derived by a deprivation decrement of

, is derived by a deprivation decrement of

That is, a reduction in inequality between poor individuals must reduce poverty by more than an

equivalent reduction in inequality between relatively less poor individuals.

Table 1: Parameter choices and corresponding properties

>0 >0 >0 ≥1

>0 >0 >1 >2

=1 =0 >0, <1 >0, <1

Dimensional

Transfer √ √ √

Dynamic Transfer √ √ √

Deprivation

Transfer √ √

Deprivation

Transfer

Sensitivity

√

Increasing above 1 allows Deprivation Transfer to be satisfied and increasing it above 2 allows

Deprivation Transfer Sensitivity to be satisfied in addition (proof in Appendix C). However, both of

these come at the cost of Dynamic and Dimensional Decomposability. However, as previously

9 Unlike AF, the Transfer axiom is here defined over deprivation inputs rather than achievements. This distinction is

useful, since it allows a variant of the Transfer axiom to be satisfied even when (which is common in most

multidimensional measurement applications). When changes in achievements have no effect on deprivation

inputs unless they bring the individual above/below a deprivation threshold. 10

While not explicitly stated in AF, Deprivation Transfer Sensitivity is satisfied in their measure when their parameter

.

17

mentioned, since simply changes inter- rather than intra-individual weights, in cases where

it would be appropriate to do dynamic and dimensional decomposition as if

Table 1 summarises key properties that are satisfied with various combinations of the parameters.

We discuss the use of alternate weighting schemes and other extensions to the measure in Appendix

D.

3. Data sets and summary features

In keeping with the motivation of this study to apply the proposed new dynamic measure of

multidimensional poverty in the context of developing countries and show their empirical

usefulness, we have chosen panel data sets from China and Indonesia over roughly

contemporaneous time periods during the 1990s and the first decade of the 21st century. The

empirical evidence is of particular interest since, besides their developmental focus, they cover

periods that include the Asian Financial Crisis and (for China) the global financial crisis. We do not

compare multidimensional poverty between China and Indonesia due to differences between the

two countries in the dimensions on which the required information is available, as well as

differences in the representativeness of the two data sets.

In many cases, the variable denoting deprivation takes the binary form. In other cases, the available

information is cardinal so that one has information on the magnitude of the shortfall from the

deprivation cut-off. The latter allows the measure to better distinguish between deprived individuals

based on the magnitude of their deprivation. We consider both types of data in the application.

3.1 China

The Chinese data came from the China Health and Nutrition Survey (CHNS). This is an ongoing

international project between the Carolina Population Center at the University of North Carolina at

Chapel Hill and the National Institute of Nutrition and Food Safety at the Chinese Centre for

Disease Control and Prevention. This project was designed to examine the effects of health,

nutrition and family planning policies and programs implemented by the national and local

governments and to see how the social and economic transformation is affecting the health and

nutritional status of the population. A detailed description of the CHNS data base has been

presented in Popkin, Du, Zhai and Zhang (2010). The surveys took place over a three day period

using a multi-stage, random cluster process to draw a sample of over 4000 households in nine

provinces that vary substantially in geography, economic development, public resources and health

indicators. We converted household level information to the individual level by assuming that the

18

household‟s access to a facility such as drinking water or electricity is the same for all individuals in

that household. In a departure from previous applications on CHNS data, we supplemented the

individual and household level information with the community level information available in the

CHNS community identifiers. The community questionnaire (filled out for each of the primary

sampling units) collected information from a knowledgeable respondent on community

infrastructure (water, transport, electricity, communications, and so on), services (family planning,

health facilities, retail outlets), population, prevailing wages, and related variables.

The dimensions at the household, individual and community levels considered in this study are

described in Appendix E1. Appendix E2 describes the three different samples used for China.

Appendix E2 also provides the cut offs used in the quantitative dimensions (years of education,

BMI, BP, for both countries and, additionally for Indonesia, expenditure) in defining the “poor”.

While the CHNS data set is longitudinal, there was a trade-off between the length of the time

interval and the number of dimensions on which information is available for the panel of

individuals. The first two samples consider different combinations of time and dimensions through

the use of exclusively „qualitative‟ dimensions where an individual is either „poor‟ or „not poor‟ in a

dimension – there is no information on gradations of poverty.11

In contrast, sample three contains

only dimensions that are „quantitative‟, meaning there is information on the size of the poverty gap.

This means that setting in will now have an effect on the measure.

For the analysis, we focus initially on sample one given the breadth of dimensions it covers:

Appendix E3 presents the summary statistics, year and dimension wise, of the poverty rates in

China for the panel of individuals in sample one. Only individuals aged 18 years and above in the

first year of the panel were included in construction of the balanced panel. The sample statistics for

the panel in the other two samples have not been presented here for space reasons, but are available

on request. A couple of features are worth noting: (a) in case of dimensions that are common to the

three samples or panels, there is hardly any difference in the summary statistics between the three

samples; (b) while some dimensions such as electricity, drink water, radio/TV recorded large

improvements over the period, this is not consistently true; for example, the proportion of

individuals with abnormal blood pressure increased sharply while morbidity (in the form of illness)

and income poverty did not show a pronounced trend in any direction; (c) the third panel shows that

there was a reduction in income poverty in China during 2006-9 that not only confirms the widely

11

Note that though in case of some of these dimensions, for example, BMI, blood pressure and years of schooling, the

quantitative information is available, we converted them into qualitative dimensions for consistency with the others,

such as access to toilet, fuel, etc. However, the quantitative information was used in the third sample which considered

the depth in the poverty in these dimensions and their sensitivity to the weights attached to the depth via the parameter,

α.

19

held view that the global financial crisis bypassed China but that the country witnessed a growth in

income in sharp contrast to the experience of US and the European countries.

3.2 Indonesia

The Indonesian data came from the Indonesia Family Life Survey (IFLS). The Indonesian Family

Life Survey (IFLS) is a longitudinal survey in Indonesia conducted over 1993-2007.12

The sample

is representative of about 83% of the Indonesian population and contains over 30,000 individuals

living in 13 of the 27 provinces in the country. The first wave of the IFLS (IFLS1) was conducted

in 1993/94 by RAND in collaboration with Lembaga Demografi, University of Indonesia. IFLS2

and IFLS2+ were conducted in 1997 and 1998, respectively, by RAND in collaboration with UCLA

and Lembaga Demografi, University of Indonesia. IFLS2+ covered a 25% sub-sample of the IFLS

households. IFLS3, which was fielded in 2000 and covered the full sample, was conducted by

RAND in collaboration with the Population Research Center, University of Gadjah Mada. The

fourth wave of the IFLS (IFLS4), fielded in 2007/2008 covering the full sample, was conducted by

RAND, the Center for Population and Policy Studies (CPPS) of the University of Gadjah Mada and

Survey METRE. The present study considered all the four waves of the IFLS.

Like the third sample for China, the Indonesian data provided information on the quantitative

dimensions and so allowed a closer examination of the sensitivity of the poverty estimates to the

poverty gap sensitivity parameter, α. The quantitative dimensions that were considered in this study

are (i) body mass index (BMI), (ii) blood pressure (BP), (iii) years of education of each individual,

and (iv) per capita household expenditure. A balanced panel dataset of individuals aged 18 years

and above in the first year with information available in all the four dimensions was created. An

individual is considered „poor‟ on BMI if her/his BMI is less than 18.5; deprivation on account of

BP is defined as having a diastolic measure not within the 60-90mm (millimetres of mercury) range

and the systolic measure not within the 90-140 mm range.13

The poverty cut-off for education is

having completed primary education14

. The poverty cut-off for expenditure is half sample median.

Three of these, namely, BMI, education and income/expenditure were considered by AF in their

calculations on the IFLS data set.

Table E4 presents the summary statistics, year and dimension wise, on the poverty rates in

Indonesia. The two health dimensions provide mixed messages, with BMI showing steady

improvement, and BP showing no clear pattern. It is worth noting that in 1997, around the time of

12

See http://www.rand.org/labor/FLS/IFLS.html for a detailed description of the IFLS data set. 13

This information is taken from the blood pressure chart at http://www.vaughns-1-pagers.com/medicine/blood-

pressure.htm 14

Primary Education in Indonesia is 6 years.

20

the Asian Financial crisis, everyone in the sample had a BP outside the normal range. There was a

sharp improvement over the next three years, but there has been a sharp increase in the proportion

of individuals outside the normal range since then. In contrast, the non-health dimensions generally

agree that there has been a steady improvement in Indonesia in the deprivation scenario over the

decade, 1997-2007, considered here. Note, however, that even by the end of the sample period,

there was considerable deprivation in Indonesia with respect to primary education with nearly 40 %

of individuals in 2007 not having completed primary education.

4. Results

This section provides empirical evidence on the usefulness and interpretation of the proposed

measure, , through the use of its properties of dynamic, dimensional and subgroup

decomposition as well as reporting the sensitivity of the decomposition to changes in , which is

the parameter that governs sensitivity towards the breadth and depth of deprivation. In the case of

China, the results are reported for two samples, namely, the qualitative (Sample 1) and the

quantitative (Sample 3), while for Indonesia only the quantitative information was used. In all our

calculations we assume (that is, the breadth and depth aspects are given equal weight) and

that (in order to preserve subgroup decomposability). The results presented in Tables 2-7

consider three models each. Model 1 is the base case where (this leads to the measure used in

NR). Model 2 sets , therefore satisfying Dimensional Transfer and Dynamic Transfer

properties. Model 3 increases to 2, thereby giving additional weight to deprivations repeated in

the same dimension or period of time.

4.1. Multidimensional Poverty in China

The results for the qualitative sample for China (sample one) are presented in Tables 2 and 3, while

those for the quantitative sample are presented in tables 4 and 5. In the quantitative sample we set

to take advantage of the additional information.

We turn firstly to the results on subgroup decompositions (Tables 2 & 4). We define subgroups

according to three different groupings: 1) male/female; 2) province; 3) rural/urban. The reported

calculations present three numbers: the first is the percentage share of overall poverty contributed

by each population subgroup; the second calculation is the poverty index calculated for each

subgroup; and lastly, the rank of each subgroup (from least to most deprived) according to .

Note that the percentage share calculation is sensitive to the size of the subgroup while is not,

21

by virtue of being an average across individuals (as per the replication invariance axiom).

However, is not comparable across models because of changes in the choice parameters.

We can see that on average, rural deprivation exceeds urban deprivation, and females are more

deprived than males. In terms of provinces, the Guizhou province consistently comes up as the most

deprived from both subgroup contribution as well as . The ranking of the other provinces vary

across the two tables due to differences in the choice of dimensions and the use of quantitative

information in sample three. While increasing generally increases the weight given to subgroups

who are more deprived, the magnitude of the effect is considerably different when comparing the

qualitative application (Table 2) to the quantitative one (Table 4). While the deprivation ranking

remains the same in the former, the rankings for Jiangsu, Shandong and Hubei differ as increases

in the latter. Indeed, the deterioration of the Jiangsu province as increases suggests a stronger

concentration of dimensions along the breadth/depth aspects, even though the count of deprivations

may be smaller than Shandong or Hubei.

We also include BTD calculations for each subgroup which is useful on two fronts: it allows us to

quickly identify whether differences between groups can be attributable to differences in

depth/breadth concentration; and it allows us to identify at the overall level whether we should be

paying more attention to individual dimensions (through dimensional decomposition) or to

individual periods (through dynamic decomposition).15

Since the BTD calculations presented in Tables 2 and 4 are all less than 1, it suggests that the

breadth aspect is less significant than the depth aspect for China; i.e. it is more likely to observe

deprivations occurring over multiple periods of time in the same dimension rather than over

multiple dimensions in the same period of time. The reciprocal of the BTD in the data suggests that

the depth aspect contributes around twice the percentage to deprivation relative to the breadth

aspect when .

Like the subgroup poverty scores, the BTD changes considerably more in the quantitative

application. Recall that the Jiangsu province experienced a decline in rankings as was increased

in Table 4. The BTD for Jiangsu in fact reveals that it has the highest depth concentration (i.e.

lowest BTD) among the provinces, which explains the driving factor behind its high sensitivity to

changes to .

The BTD being less than 1 suggests that the policy-maker or poverty researcher should be focussed

on certain dimensions where deprivation occurs repeatedly over time. We examine this further by

15

Note that BTD cannot be identified when . Similarly, becomes irrelevant in such a case.

22

considering dimensional decomposition of the measure in Tables 3 & 5. Here the ranks depict

percentage contribution to overall deprivation (from least to most).

Table 3 shows that lack of access to toilet, fuel, drink water, road and transport are the more

significant contributors to deprivation, along with blood pressure not being in the normal range.

These dimensions, taken together, contribute well over 50 % of total deprivation and their shares

increase with . In contrast, access to electricity and health (measured by BMI and frequency of

illness) are success stories in China with relatively minor contributions to deprivation and whose

contributions decline with . On the other hand, Table 5 suggests that the most important dimension

is an individual‟s education. Unlike Table 3, in Table 5 the number of years below completing

primary education is taken into account and we can see the measure suggesting that as much as 95%

of deprivation is coming from the dimension when This may be due to the nature of

education – individuals who have chosen not to complete an education remain „uneducated‟ for the

rest of their lives and are therefore considered to be „deep‟ in deprivation for that dimension.

Tables 3 & 5 show that, under dynamic decomposition, the percentage contribution of each period

of time to overall deprivation in China is nearly uniform, and that this picture is quite robust to β.

The share of period 3 (2000) is noticeably lower in the qualitative sample (Table 3) than in the

quantitative sample (Table 5). The BTD in Tables 2 & 4 suggests the breadth aspect should not

have a strong role in the data – Figure 3 confirms this graphically by considering the relationship

between the time period shares and : as increases we see the shares converging towards 0.20;

i.e. „equal‟ shares among the five periods. This is because, as indicated by the BTD, the overall

measure is dominated by the depth aspect, therefore in relative terms, increasing increasingly

discounts the contribution of the breadth relative to the depth aspect.

4.2. Multidimensional Poverty in Indonesia

The multidimensional poverty estimates of Indonesia are presented in Tables 6 and 7. Unlike in the

case of China, the Indonesian calculations are based on only one sample which contains the

quantitative information on the dimensions (see the bottom half of Appendix Table E2).

As in the case of China, Table 6 presents the subgroup decompositions by male/female and

rural/urban grouping. The rankings of the subgroups are the same as in China and the BTD ratio is

also below 1, suggesting the dominance of the depth aspect. Dimensional decomposition in Table 7

also supports the education dimension (i.e. individuals below primary education) as the primary

contributor to deprivation measured quantitatively, although, increasing from 0 to 1 changes the

23

rank of expenditure from the second least to third least contributor, suggesting that after education,

expenditure exhibits the most deprivations over repeated periods.

In the case of dynamic decomposition (Table 7), we see that the largest share of deprivation is due

to 1997. It is well known that Indonesia was one of the worst affected countries in the 1997 Asian

financial crisis. However, like China we also find that as increases the shares converge towards

0.33 (by virtue of there being only three periods here). This is confirmed by Figure 4 which depicts

the relationship in Indonesia between the various sub periods‟ share of poverty and β. The

discounting of 1997‟s contribution as increases can be interpreted as follows: while 1997 clearly

has the most count of deprivations, because most of these deprivations appear only in 1997, they

have very low depth and since depth contributes more to the overall measure, 1997‟s contribution is

increasingly discounted. This highlights an important property of decomposition when :

information from other periods is used when calculating the contribution of each individual period;

similarly, information from other dimensions is used when calculating the contribution of each

individual dimension.

5. Discussion and Conclusion

There has been a large and rapidly proliferating literature on the multidimensional measurement of

poverty and its application to household survey data. The increasing availability of micro datasets

containing a wealth of household and individual information has helped in the development of

increasingly sophisticated measures that take into account the richness of the available information.

We have proposed a dynamic multidimensional measure of poverty that unites two strands of a

literature that finds its origins in Foster, Greer and Thorbecke (1984). Our measure allows

population subgroups to be compared when information regarding both multiple different

dimensions of deprivation and multiple periods of time are available. Following the axiomatic

framework in Alkire and Foster (2011a) we lay out two unique properties of our measure that are

only identifiable when such panel data is available. We argue that the separate identification and

weighting of the breadth and depth aspects of poverty are useful to both policy-makers and

researchers due to the general idea that a concentration of deprivations (whether defined over time

or over dimensions) should be given as much, if not more, attention than the simple count of

deprivations. Furthermore, policy prescriptions for poverty may vary depending on the type and

location of such concentration of deprivation. In our empirical application for Indonesia, for

example, we found that while the breadth of deprivation present during the financial crisis in 1997

24

had a large effect on poverty, poverty over the 1997-2007 is better explained by long-lasting

deprivation manifest in dimensions such as a lack of education. While such a conclusion is clearly

dependent on our choice of deprivation dimensions, our purpose has been to illustrate the usefulness

of the proposed measure, rather than provide conclusive evidence for China and Indonesia. The

choice of deprivation dimensions, cut-offs, as well as the appropriate weights to attach to them

remains a fertile research agenda beyond the scope of this paper [for a brief discussion, see Alkire

and Foster (2011b)].

The empirical application has highlighted that our measure allocates subgroup, dimensional, and

dynamic shares that are different from other models in the literature that exist as special cases of our

measure. This is especially true in the case where quantitative/cardinal information is available, in

which case increasing the sensitivity of the measure to breadth and depth aspects changes the

measure to the extent of switching the deprivation rank of dimension and subgroup shares. Another

point illustrated by our application has been the importance of the depth aspect relative to the

breadth of poverty in both China and Indonesia, suggesting that for several dimensions, deprivation

remains chronic. While the Alkire and Foster (2011a) method was designed primarily to keep track

of changes to multidimensional poverty over different periods of time by calculating a separate

measure for each period, the importance of the depth component revealed here suggests that when

comparing different groups of individuals, information regarding individual deprivation over time is

crucial in accurately characterising individual poverty.

The generality and flexibility of our measure comes at a cost since any researcher or policy-maker

will have to make choices with regards to multiple parameters. While we have provided a guide

based on the desirable properties of the measure, the degrees of freedom available remains

considerable. However, it is important to highlight that the additional flexibility introduced in our

measure can never be detrimental relative to pre-existing measures that are generalised here. A

notable example is the case of a policy-maker/researcher interested in comparing poverty profiles

over time, rather than over subgroups. If so, the special case of the Alkire and Foster (2011a)

measure is appropriate since the concept of depth is not required. However, further assume the

policy-maker/researcher only has qualitative information on deprivations; he may then wish to set

to allow the measure to be increasingly sensitive to the count of deprivations (as per the

deprivation transfer axiom) since changing would have no effect; indeed, defaulting entirely to

the original Alkire and Foster (2011a) would inappropriately amount to assuming that .

When faced with a large degree of undecidedness with regards to parameter choices, we would

advocate calculating the results for various combinations of the parameters. Indeed, understanding

25

how and why poverty measurement changes with the various parameters is arguably more

informative than poverty conveyed by a single „definitive‟ number.

Table 2: Multidimensional Deprivation and its Subgroup Decomposition for

China based on CHNS Qualitative Sample One

Ωαβδ (Hg) BTD(Hg)

MODEL 1 MODEL 2 MODEL 3

MODEL

1 MODEL 2

MODEL

3

α 0 0 0 0 0 0

β 0 1 2 0 1 2

δ 1 1 1 1 1 1

γ irrelevant 0.5 0.5

Ωαβδ 0.3597 0.2155 0.1552 N/A 0.5704 0.3225

(Subgroup Proportion, Ωαβδ, Rank)

Male (0.4381, 0.3520, 1) (0.4386, 0.2092, 1) (0.4331, 0.1501, 1) N/A 0.5612 0.3134

Female (0.5619, 0.3659, 2) (0.5654, 0.2207, 2) (0.5669, 0.1593, 2) N/A 0.5775 0.3295

Province

Jiangsu (0.1539, 0.3425, 3) (0.1492, 0.1988, 3) (0.1453, 0.1394, 3) N/A 0.5690 0.3230

Shandong (0.1327, 0.4010, 6) (0.1403, 0.2541, 6) (0.1457, 0.1899, 6) N/A 0.5517 0.3008

Henan (0.1063, 0.4003, 5) (0.1114, 0.2514. 5) (0.1151, 0.1871, 5) N/A 0.5636 0.3122

Hubei (0.1621, 0.3655, 4) (0.1633, 0.2205, 4) (0.1637, 0.1592, 4) N/A 0.5718 0.3229

Hunan (0.1176, 0.3195, 2) (0.1095, 0.1783, 2) (0.1047, 0.1228, 2) N/A 0.5687 0.3221

Guangxi (0.1068, 0.2600, 1) (0.0923, 0.1346, 1) (0.0856, 0.0898, 1) N/A 0.5193 0.2689

Guizhou (0.2205, 0.4308, 7) (0.2341, 0.2741, 7) (0.2399, 0.2022, 7) N/A 0.6074 0.3613

Rural (0.8119, 0.3912, 2) (0.8316, 0.2401, 2) (0.8384, 0.1743, 2) N/A 0.5834 0.3342

Urban (0.1881, 0.2669, 1) (0.1684, 0.1432, 1) (0.1616, 0.0989, 1) N/A 0.5091 0.2647

26

Table 3: Dynamic and Dimensional Decomposition for China (CHNS Qualitative Sample

One) Model 1 Model 2 Model 3

α 0 0 0

β 0 1 2

δ 1 1 1

γ Irrelevant 0.5 0.5

Dynamic Decomposition

Period Contribution* Rank Contribution* Rank Contribution* Rank

Period 1 (1993) 0.2176 5 0.2141 5 0.2097 5

Period 2 (1997) 0.2025 4 0.2046 4 0.2048 4

Period 3 (2000) 0.1967 3 0.2011 3 0.2031 3

Period 4 (2004) 0.1917 2 0.1921 2 0.1941 2

Period 5 (2006) 0.1915 1 0.1880 1 0.1884 1

Total 1 1 1

Dimensional Decomposition

Contribution** Rank Contribution** Rank Contribution** Rank

No access to toilet 0.1240 11 0.1352 12 0.1437 12

No access to fuel 0.1568 13 0.1699 13 0.1830 13

No access to electricity 0.0017 1 0.0013 1 0.0010 1

No access to drink water 0.1165 10 0.1274 10 0.1373 11

No access to at least one

vehicle type 0.0549 5 0.0525 5 0.0502 6

No access to radio/TV 0.0595 6 0.0547 6 0.0478 5

BMI not in normal range 0.0212 2 0.0181 2 0.0160 2

Illness in the last four

weeks 0.0375 3 0.0250 3 0.0164 3

Blood pressure not normal 0.1339 12 0.1289 11 0.1269 10

Individual below primary

education 0.0826 8 0.0940 9 0.1050 9

No access to road in

community 0.0939 9 0.0867 8 0.0772 8

No access to bus station in

community 0.0776 7 0.0753 7 0.0716 7

No access to school in

community 0.0399 4 0.0313 4 0.0237 4

Total 1 1 1

*Contribution is the ratio ( ) where ∑ ((∑

) )

**Contribution is the ratio ( / ) where ∑ ((∑

) )

27

Table 4: Multidimensional Deprivation and its Subgroup Decomposition for China based on CHNS

Quantitative Sample Three

Ωαβδ (Hg) BTD(Hg)

MODEL 1 MODEL 2 MODEL 3

MODEL

1

MODEL

2

MODEL

3

α 1 1 1 1 1 1

β 0 1 2 0 1 2

δ 1 1 1 1 1 1

γ irrelevant 0.5 0.5

Ωαβδ 0.1396 0.0572 0.0369 N/A 0.4752 0.1987

(Subgroup Proportion, Ωαβδ, Rank)

MALE (0.3713, 0.0980, 1) (0.2812, 0.0305, 1) (0.2348, 0.0164, 1) N/A 0.5106 0.2259

FEMALE (0.6687, 0.1766, 2) (0.7490, 0.0811, 2) (0.7905, 0.0552, 2) N/A 0.4638 0.1917

REGION

Heilongjiang (0.0973, 0.0907, 1) (0.0680, 0.0259, 1) (0.0561, 0.0138, 1) N/A 0.4786 0.2013

Jiangsu (0.1421, 0.1418, 4) (0.1568, 0.0641, 5) (0.1678, 0.0443, 6) N/A 0.4516 0.1818

Shandong (0.1314, 0.1488, 6) (0.1408, 0.0654, 6) (0.1447, 0.0433, 5) N/A 0.4631 0.1897

Henan (0.0913, 0.1590, 7) (0.0991, 0.0708, 7) (0.1030, 0.0474, 7) N/A 0.4684 0.1942

Hubei (0.1255, 0.1430, 5) (0.1256, 0.0588, 4) (0.1256, 0.0379, 4) N/A 0.4824 0.2056

Hunan (0.0975, 0.1130, 2) (0.0798, 0.0379, 2) (0.0693, 0.0212, 2) N/A 0.4928 0.2152

Guangxi (0.0980, 0.1179, 3) (0.0792, 0.0391, 3) (0.0703, 0.0224, 3) N/A 0.4995 0.2115

Guizhou (0.2168, 0.2043, 8) (0.2963, 0.1971, 8) (0.2632, 0.1982, 8) N/A 0.4826 0.2050

RURAL (0.7262, 0.1467, 2) (0.7359, 0.0610, 2) (0.7383, 0.0395, 2) N/A 0.4806 0.2025

URBAN (0.2738, 0.1236, 1) (0.2641, 0.0489, 1) (0.2617, 0.0312, 1) N/A 0.4609 0.1882

28

Table 5: Dynamic and Dimensional Decomposition for China (CHNS Quantitative Sample

Three)

Model 1 Model 2 Model 3

α 1 1 1

β 0 1 2

δ 1 1 1

γ irrelevant 0.5 0.5

Dynamic Decomposition

Period Contribution* Rank Contribution* Rank Contribution* Rank

Period 1 (1997) 0.2160 5 0.2135 5 0.2110 5

Period 2 (2000) 0.2035 3 0.2030 3 0.2037 3

Period 3 (2004) 0.1784 1 0.1785 1 0.1845 1

Period 4 (2006) 0.2088 4 0.2083 4 0.2041 4

Period 5 (2009) 0.1933 2 0.1967 2 0.1966 2

Total 1 1 1

Dimensional Decomposition

Contribution** Rank Contribution** Rank Contribution** Rank

BMI not in normal range 0.0071 1 0.0024 1 0.0010 1

Blood pressure not normal 0.3329 2 0.1351 2 0.0481 2

Individual below primary

education 0.6600 3 0.8625 3 0.9509 3

Total 1 1 1

*Contribution is the ratio ( ) where ∑ ((∑

) )

**Contribution is the ratio ( / ) where ∑ ((∑

) )

29

Table 6: Multidimensional Deprivation and Subgroup Decomposition for Indonesia: IFLS

(Quantitative Sample)

(Proportion, Ωαβδ , Rank) BTD(Hg)

MODEL 1 MODEL 2 MODEL 3

MODEL

1

MODEL

2

MODEL

3

α 1 1 1 1 1 1

β 0 1 2 0 1 2

δ 1 1 1 1 1 1

γ irrelevant 0.5 0.5

Ωαβδ 0.0941 0.0377 0.0236 N/A 0.3512 0.1110

MALE (0.3890, 0.0823, 1) (0.3790, 0.0321, 1) (0.3706, 0.0197, 1) N/A 0.3561 0.1138

FEMALE (0.6110, 0.1035, 2) (0.6211, 0.0421, 2) (0.6296, 0.0268, 2) N/A 0.3484 0.1094

RURAL (0.6127, 0.1193, 2) (0.6422, 0.0501, 2) (0.6543, 0.0320, 2) N/A 0.3401 0.1054

URBAN (0.4018, 0.0731, 1) (0.3740, 0.0273, 1) (0.3640, 0.0167, 1) N/A 0.3583 0.1126

Table 7: Dynamic and dimensional decomposition for Indonesia: IFLS (Quantitative

Sample)

Model 1 Model 2 Model 3

α 1 1 1

β 0 1 2

δ 1 1 1

γ irrelevant 0.5 0.5

Ωαβδ 0.0941 0.0377 0.0236

Dynamic Decomposition

Period Contribution* Rank Contribution* Rank Contribution* Rank

Period 1 (1997) 0.4474 3 0.4285 3 0.4035 3

Period 2 (2000) 0.2783 2 0.2857 1 0.2991 2

Period 3 (2007) 0.2743 1 0.2858 2 0.2969 1

Total 1 1 1

Dimensional Decomposition

Contribution** Rank Contribution** Rank Contribution** Rank

BMI not in normal range 0.0211 1 0.0067 1 0.0022 1

Blood pressure not normal 0.0968 3 0.0259 2 0.0072 2

Individual below primary

education 0.8225 4 0.9368 4 0.9773 4

Income 0.0595 2 0.0306 3 0.0133 3

Total 1 1 1

*Contribution is the ratio ( ) where ∑ ((∑

) )

**Contribution is the ratio ( / ) where ∑ ((∑

) )

30

Fig 3: Variation in the Temporal Shares of Deprivation in China with β (for α=1)

Fig 4: Variation in the Temporal Shares of Deprivation in Indonesia with β (for α=1)

31

REFERENCES

Alkire, S. and Foster, J. (2011a), “Counting and multidimensional poverty measurement”,

Journal of Public Economics, 95, 476-487.

Alkire, S. and Foster, J. (2011b), “Understandings and misunderstandings of multidimensional

poverty measurement”, Journal of Economic Inequality, 9: 289-314.

Atkinson, A. (2003), “Multidimensional deprivation: contrasting social welfare and counting

approaches”, Journal of Economic Inequality, 1, 51-65.

Bossert, W., Chakravarty, S., and C. D‟Ambrosio (2013), “Multidimensional Poverty and Material

Deprivation with Discrete Data”, Review of Income and Wealth 59(1), 29-43.

Bossert, W., Chakravarty, S., and C. D‟Ambrosio (2012), “Poverty and Time”, Journal of

Economic Inequality, 10, 145-162.

Bossert, W., Ceriani, L., Chakravarty, S., and C.D‟Ambrosio (2012), “Inter temporal Material

Deprivation”, mimeographed.

Bossert, W., D‟Ambrosio,C., and V. Peragine (2007), “Deprivation and Social Exclusion”,

Economica, 74 (296), 777-803.

Bourguignon, F. and S. Chakravarty (2003), “The Measurement of Multidimensional Poverty”,

Journal of Economic Inequality, 1, 25-49.

Calvo, C. and S. Dercon (2009), “Chronic poverty and all that: the measurement of poverty over

time” in T.Addison, D.Hulme and R.Kanbur (eds), Poverty Dynamics: interdisciplinary

perspectives, 29-58. Oxford University Press, Oxford.

Chakravarty, S. and C. D‟Ambrosio (2006), “The Measurement of Social Exclusion”, Review of

Income and Wealth, 52, 377-398.

Dalton, H., (1920), “The Measurement of the Inequality of Incomes”, Economic Journal, 30: 348-

61.

Duclos. J, Araar, A. and J. Giles (2010), “Chronic and transient poverty: Measurement and

estimation, with evidence from China”, Journal of Development Economics, 91, 266-277.

Foster, J. (2007), “A class of chronic poverty measures” in T. Addison, D. Hulme and R. Kanbur

(eds), Poverty Dynamics: interdisciplinary perspectives, 59-76. Oxford University Press,

Oxford.

Foster, J., Greer, J. and E. Thorbecke (1984), “A class of decomposable poverty indices”,

Econometrica, 52, 761-766.

Gradin, C., del Rio, C. and O. Canto (2012), “Measuring poverty accounting for time”, Review of

Income and Wealth, 58, 330-354.

Hojman, D. and F. Kast (2009), “On the measurement of poverty dynamics”, Faculty Research

Working Paper Series, RWP 09-35, Harvard Kennedy School.

Hoy, M. and B. Zheng (2011), “Measuring lifetime poverty”, Journal of Economic Theory, 146,

2544-2562.

32

Jayaraj, D. and S. Subramanian (2010), “A Chakravarty-D‟Ambrosio View of Multidimensional

Deprivation: Some Estimates for India”, Economic and Political Weekly, XLV, 53-65.

Kahneman, D, and A. Tversky (1979), "Prospect theory: An analysis of decision under

risk." Econometrica: 263-291.

Mishra, A. and R. Ray (2012), “Multidimensional deprivation in the awakening giants: A

comparison of China and India on micro data” Journal of Asian Economics, 23(4), 454-465.

Nicholas, A. and R. Ray (2012), “Duration and Persistence in Multidimensional Deprivation:

Methodology and Australian Application”, Economic Record, 88, 106-126.

Pigou, A.F., (1912), Wealth and Welfare, Macmillan, London.

Popkin,B.M., Du S., Zhai,F. and Zhang, B. (2010), “Cohort Profile: The China Health and Nutrition

Survey- monitoring and understanding socio-economic and health change in China, 1989-

2011”, International Journal of Epidemiology, 39, 1435-1440.

Sen, A. (1976), “Poverty: an ordinal approach to measurement”, Econometrica, 44, 291-231.

Sen, A. (1985), Commodities and Capabilities. North Holland, Amsterdam.

Tsui, K. (2002),”Multidimensional poverty indices”, Social Choice and Welfare, 19, 69-93.

33

APPENDIX A: Dynamic and dimensional transfer

Proposition: satisfies Dynamic Transfer when , and .

Proof:

We restrict our attention to individual ‟s deprivation profile since that is the only thing that

changes. A dimensional permutation ensures that ∑

remains the same after the permutation,

for all t. It also ensures that ∑ ∑

remains the same after the permutation. Therefore a measure

that is path-independent in terms of (as in NR) will not change with a dimensional

permutation. This means that when , dynamic transfer is clearly not satisfied.

First, define

and

.

Dimensional permutation results in

and

where is any non-negative real

number. We need to show individual deprivation score has increased with the

permutation, which is equivalent to showing

when ∑

∑

where .

∑ ∑

∑ ∑

Consider the first term on the right.

∑ ∑

[ (∑ ∑

)

]

Equation (4b) is positive when . This is a necessary condition for the proof since it ensures

that increases in ∑ ∑

translate into increases in the individual‟s deprivation score.

Now consider the second term on the right of equation (4a). Note, ∑ ∑

∑ ∑

Recall that

[ (∑

)

(

∑

)

]

Since ∑

does not change with dimensional permutation, we can ignore its influence on

and instead define

(∑

)

.

(∑

)

(

∑

)

34

(∑

)

(

∑

)

We now need to show that

is a positive function of where

is any where

but . If so this automatically implies

by virtue of ∑

∑

. However, because appears in the denominator as well it is unclear whether its

influence on

is positive.

Differentiating (4c) by yields:

( ) (

) (

∑

)

,

∑

-

Equation (4d) is positive for any and .

Proposition: satisfies Dimensional Transfer when , and .

Following the proof for dynamic transfer, define

(∑

)

( ) (

) (

∑

)

{

∑

}

where is any

where but . Equation (4e) is positive for any and

.

35

APPENDIX B: Breadth-to-Depth Ratio

Proposition: When , the BTD range is maximised when the deprivation count ∑ ∑

. When the BTD range is maximised at some value of ∑ ∑

lying between and

.

Proof:

We first consider the simple case where and deprivation profiles are therefore square

matrices. For simplicity, we restrict our attention to a single individual‟s deprivation score, .

The BTD range can be represented by the gap between the BTDs obtained from and .

Because we have a square matrix, the deprivation count that maximises BTD for will also

automatically minimise the BTD for since will simply be a 90 degree rotation of the

deprivation profile in . This means that finding the deprivation count that maximises the

BTD for will be sufficient proof for the first part of the proposition.

The theoretically maximum any individual deprivation input can contribute to the numerator

term of the BTD is when the deprivation input belongs to a with ∑

. The theoretically

minimum any individual deprivation input can contribute to the denominator term of the BTD

is when the deprivation input belongs to a j with ∑

; that is,

is the only

deprivation input in . Both these conditions can only be simultaneously satisfied with a deprivation

profile where there is only one column with ∑

and each other column is empty. This is

clearly only satisfied when ∑ ∑

, which therefore gives us the deprivation count for

which achieves a maximum.

Given the square matrix assumption, rotating the profile gives us a profile where there is one row of

∑

and each other row is empty. This is also clearly only possible when ∑ ∑

,

which gives us the deprivation count for which achieves a minimum. Since ∑ ∑

allows and to achieve a maximum and minimum BDT respectively,

∑ ∑

maximises the BDT range.

Now consider the case where we have a non-square matrix, . Since the matrix is no longer

square, the deprivation profiles can no longer be rotated and a given deprivation count that

36

maximises the BTD for will not minimise the BTD for . Similarly, a deprivation

count that minimises the BTD for will not maximise the BTD for .

Note that the BTD converges to 1 when ∑ ∑

and tends towards 1 when ∑ ∑

is close to zero. Therefore deprivation counts that are further from the value that maximises the

BTD for will result in a lower BTD while deprivation counts that are further from the value

that minimises the BTD for will result in a higher BTD. Consider an example when by

some integer a. In such a case, setting ∑ ∑

by some integer b results in being b

counts away from a maximum BTD and being a+b counts away from a minimum BTD.

Whereas setting ∑ ∑

results in achieving a maximum BTD and being a

counts away from a minimum BTD.

Similarly, setting ∑ ∑

by some integer c results in being a+c counts away from a

maximum BTD and being c counts away from a minimum BTD. Whereas setting

∑ ∑

results in achieving a minimum BTD and being a counts away from

a maximum BTD.

For deprivation counts ∑ ∑

that lie between and , however, changing the count trades off

the distance between achieving a minimum versus a maximum .

We therefore conclude that the BTD measure is the most informative for deprivation counts that lie

between and . Simulations with and suggest that the BTD range is in

fact maximised when ∑ ∑

.

Because ∑ ∑

is not a choice variable establishing a general proof for the precise point at

which the BTD range is maximised is beyond the scope of this paper.

37

APPENDIX C: Conditions for Deprivation Monotonicity,

Transfer and Transfer Sensitivity

For the following, we restrict ourselves to individual deprivation score (∑ ∑

)

since is simply the average across all individual. Define define

Deprivation Monotonicity

Deprivation Monotonicity requires that

∑ ∑

∑ ∑

.

∑ ∑

[ (∑ ∑

)

] when

∑ ∑

∑∑

[ (∑

)

(

∑

) (∑

)

(

∑

)] for

when (by definition of

)

Deprivation monotonicity therefore holds for when .

Deprivation Transfer

Transfer requires that

( ) (

) where

is any not equal to

.

( ) (

)

∑ ∑

∑ ∑

( ) (

)

∑ ∑

∑ ∑

(

)

Assume that Monotonicity holds. Then,

∑ ∑

and ∑ ∑

>0.

TERM (1)

∑ ∑

( ) (

) ∑ ∑

( ) (

)

. We know from the proof for dimensional transfer that

( ) (

) for and but only for

that share the same dimension or

same period as . Otherwise,

( ) (

)

TERM (2)

∑ ∑

(

) [ (

∑ ∑

)

] for .

38

Since the Term (2) result holds regardless of the data, setting ensures that the index satisfies

the transfer property.

Deprivation Transfer sensitivity

Transfer requires that

( ) (

)

( ) (

)

∑ ∑

( ) (

)

(∑ ∑

) (

)

∑ ∑

( )

∑ ∑

(

)

∑ ∑

∑ ∑

( ) (

)

Assume that Transfer and Monotonicity holds. Then

∑ ∑

, ∑ ∑

,

(∑ ∑

) (

) >0

TERM (1)

Term (1) is positive if Term (1) for Transfer holds, and zero otherwise.

TERM (2)

(∑ ∑

) (

) [ (

∑ ∑

)

] for .

TERM (3)

When does not share the same dimension or period as

then, ∑ ∑

( ) (

)

∑ ∑

( ) (

)

If it shares the same dimension then:

( ) (

) (

∑

)

,

∑

-

( ) (

) for and

( ) (

) for .

Therefore, to ensure Transfer Sensitivity holds, set and .

39

APPENDIX D: Weights and Extensions

Because our measure is a generalisation of the class of measures in NR and BC, it can incorporate

the extensions used there, namely in terms of „persistence‟ (Gradin et al, 2012; Bossert et al 2012)

and „loss-aversion‟ (Hojman and Kast, 2009). Given our focus on the interaction between „depth‟

and „breadth‟ of deprivations, we avoided the use of such extensions in our empirical applications

as they would make it harder to identify the effect of depth versus breadth given the additional

trade-offs introduced with such extensions. However, we briefly describe how these can be

incorporated into our measure – the interested reader is advised to refer to the aforementioned

papers directly for more details regarding their properties.

0∑.(

∑ ∑

)

/

1 ⁄

is a generic weight associated with each

(the transformed deprivation input). These

weights can be defined in a variety of ways, though they clearly change the interpretation of the

measure. „Persistence‟, which gives increasing weight to deprivations that occur in consecutive

periods, for example, can be incorporated by defining ( ) where is the length of

the deprivation spell associated with a particular

. Another way of defining persistence (BCCD)

could be where the weights are increasing in the number of consecutive periods

that an individual experiences breath of deprivation above a predetermined cutoff.

Persistence may not always be relevant to every dimension of deprivation. One can, for example,

imagine that being unemployed for three consecutive periods and then being employed for three

consecutive periods is superior to alternating in and out of employment for six periods since one

incurs an „adjustment cost‟ when changing states. This can be captured with the concept of loss-

aversion (Kahneman and Tversky, 1979; applied to poverty indices in Hojman and Kast, 2009).

Consider, for example:

*

(

∑

∑

) (

)+ ∑

∑

*

(

∑

∑

) (

)+

40

This weighting scheme has two features: 1) it allows a particular period to be weighted heavier if it

is proceeded by more dimensions of deprivation and to be weighted by less if it is proceeded by less

dimensions of deprivation. 2) it captures the concept of loss-aversion by weighting increases in the

breadth of deprivation in the subsequent period more heavily than an equivalent decrease in the

breadth of deprivation in the subsequent period (in absolute terms).

Another potential use of the general weights is to assign different importance to different

dimensions. As suggested by Atkinson (2003), it is not unreasonable in most applications to start

with the case where

; that is, where each dimension is weighted equally. With additional

information, however, a researcher or policy-maker may assign different weights (so long as

∑ ). Reasons behind changing the weights include „double counting‟ in the sense that some

dimensions may essentially be capturing the same aspect of deprivation, which justifies the

discounting of the importance of the associated dimensions; another reason is that individuals may

actually put very low importance on certain dimensions; e.g. in Bossert et al (2013) dimensions are

weighted based on the views of society regarding the importance of those dimensions -- “consensus

weighting”.

41

APPENDIX E1: Description of Dimensions used for analysis

Type Variable Description

Chin

a Indonesia

Multidimensional Deprivation Dimension

Household

Access to

Toilet (D1)

Individual‟s household has access to improved

toilet facility as per UN norms. UN defines

improved facility as having own flush toilet,

own pit toilet, traditional pit toilet, ventilated

improved pit latrine, pit-latrine with slab, flush

toilet, and composting toilet X X

Access to

fuel (D2)

Fuel used for cooking by individual‟s

household is kerosene, electricity, LPG or

biogas X X

Access to

electricity

(D3)

Individual‟s household has access to electricity X X

Access to

drink water

(D4)

Individual‟s household has access to improved

drinking water source. UN defines improved

drinking water source as piped water into

dwelling, plot or yard, public tab/standpipe,

tube well, borehole, protected dug well,

protected spring and rainwater.

X X

Access to

atleast one

vehicle type

(D5)

Individual‟s household owns atleast one of the

following mode of transport: a bicycle,

motorcycle or car.

X X

Access to

radio/TV

(D6)

Individual‟s household owns atleast a radio,

b/w Television or a colour Television X X

Individual

BMI<18.5 or

BMI>30

(D7)

If BMI of the individual is not normal (i.e.,

either greater than 30 or less than 18.5.)

X X

Illness in the

last four

weeks (D8)

whether individual suffered illness in the last

four weeks.

X

Blood

pressure not

normal (D9)

If the blood pressure is normal (i.e., 90=<

systolic< 120 and 60<= diastolic<80)

X X

Individual

atleast

primary

educated

(D10)

Individual is educated upto primary ( year 6). X X

Community

Access to

road (D11)

If the village/neighbourhood has a stone/gravel

or paved roads.

X

Access to

bus/train

If there is a bus stop in the

village/neighbourhood.

X

42

station (D12)

Access to

school (D13)

If there is a primary/middle/upper middle

school in the village/neighbourhood.

X

APPENDIX E2: Description of balanced samples used for analysis

China Analysis: Sample Description

Sample 1 Balanced panel Sample of 1874 individuals over five years (1993-1997-2000-2004-2006)

covering 7 provinces.

Dimensions D1-D2-D3-D4-D5-D6-D7-D8-D9-D10-D11-D12-D13

Description This sample comprises of household, individual and community dimensions.

All dimensions are qualitative.

Sample 2 Balanced panel Sample of 1934 individuals across six years (1991-1993-1997-2000-2004-2006)

covering 8 provinces.

Dimensions D1-D2-D3-D4-D5-D6-D7-D8-D9-D10

Description This sample comprises of household and individual dimensions only. All

dimensions are qualitative.

Sample 3 Balanced panel Sample of 2458 individuals across five years (1997-2000-2004-2006-2009)

Dimensions 1. Years of education of individual (cut off is primary education i.e., 6 years)

2. Individual's Body Mass Index (cut off is 18.5)

3. Individual's Blood Pressure (systolic/diastolic).i.e., more than 120/80 then

high BP.

Description All dimensions in this sample are quantitative

Indonesia Analysis: Sample Description

Balanced

panel Individuals across three years 1997-2000-2007

Dimensions 1. Years of education of individual (cut off is primary education i.e., 6 years)

2. Individual's Body Mass Index (cut off is BMI > 18.5)

3. Individual's Blood Pressure (systolic/diastolic).i.e., more than 120/80 then

high BP.

4. Expenditure. ( cut off is half median)

Description This sample comprises of household and individual dimensions. All dimensions

are quantitative.

43

Appendix E3: Dimension wide poverty rates for China (Sample 1, Qualitative – balanced panel of 6 years and 13 dimensions)

CHNS

wave

No A

cces

s to

Toil

et

No A

cces

s to

fuel

No a

cces

s to

ele

ctri

city

No a

cces

s to

dri

nk w

ater

No a

cces

s to

atl

east

one

veh

icle

type

No a

cces

s to

rad

io/T

V

BM

I not

in n

orm

al r

ange

Illn

ess

in t

he

last

four

wee

ks

Blo

od p

ress

ure

not

norm

al

Indiv

idual

bel

ow

pri

mar

y

educa

ted

No a

cces

s to

road

in

com

munit

y

No a

cces

s to

bus/

trai

n

stat

ion i

n c

om

munit

y

No a

cces

s to

sch

ool

in

com

munit

y

Inco

mea l

ess

than

0.5

*M

edia

n

1991 0.6665 0.9189 0.0523 0.6526 0.1827 0.4823 0.1185 0.2391 0.4726 0.4454 0.8538 0.4885 0.2560 0.2344

1993 0.6573 0.8769 0.0164 0.6660 0.1637 0.4330 0.0944 0.1596 0.5239 0.4228 0.8964 0.4495 0.1837 0.2344

1997 0.6147 0.7414 0.0036 0.5557 0.2376 0.3740 0.0867 0.1585 0.6039 0.3982 0.7542 0.3217 0.1611 0.2488

2000 0.5952 0.6952 0.0092 0.5485 0.2550 0.2950 0.0841 0.1827 0.6485 0.3833 0.8481 0.3638 0.1432 0.2904

2004 0.5428 0.7086 0.0056 0.5141 0.2996 0.1652 0.1067 0.3479 0.6963 0.3366 0.9389 0.3628 0.2365 0.2776

2006 0.4772 0.6321 0.0056 0.4962 0.3258 0.1113 0.1042 0.2986 0.7045 0.4310 0.9656 0.4228 0.2817 0.3074

aIncome is used as a benchmark for comparison and not a dimension in itself.

44

Appendix E4: Dimension wise Poverty rates in Indonesia (for Qualitative Sample)

Indonesi

a

wave

Not

norm

al

BP

BM

I<18.5

Ed

uca

tion

(B

elow

Pri

mary

)

Hou

seh

old

per

cap

ita

exp

end

itu

re <

0.5

*M

edia

n

1997 1.000 0.124 0.579 0.114

2000 0.297 0.119 0.370 0.102

2007 0.443 0.107 0.379 0.094