OPHI
Oxford Poverty & Human Development Initiative
Department of International Development
Queen Elizabeth House, University of Oxford
www.ophi.org.uk
IFPRI 2007 Beijing Presentation:
Counting and Multidimensional Poverty
Sabina Alkire & James Foster
OPHI Working Paper #7 (forthcoming)
Towards Ending Extreme Poverty & Hunger
• GHI, and measures of the subjacent, medial and
ultra income poor, rely on aggregate data.
• They extend a current headcount to investigate the
depth of poverty – a welcome conceptual step (if the
data support it – which needs critical consideration).
• This does not allow us to identify the most acutely
poor across dimensions: Who are deprived in all the
dimensions at the same time?
• This paper proposes an identification strategy and
measure that is sensitive to the breadth of poverty
across income, hunger, and potentially other
dimensions (and also satisfies desirable axioms).
Issues for Multidimensional Poverty Meas.
• Poverty of what?
–
–
–
–
–
–
You have chosen Dimensions
You have chosen Variables
Some Variables are Ordinal
You have set Poverty Lines for each dimension
You have set weights for each dimension
You have decided how often to measure poverty.
• Identification : Who is poor?
• Comparison : How do we make an index?
Breadth of poverty across Dimensions
Dimensions
(and indicators
for each D)
Health
Educat’n Income
Nutrition
Empowerment
Individual 1 or
Household 1
Poor
Poor
Poor
Individual 4 or
Household 4
Poor
Poor
Not
Poor
Not
Poor
Poor
Poor
Individual 3 or
Household 3
Not
Poor
Not
Poor
Poor
Individual 2 or
Household 2
Not
Poor
Poor
Not
Poor
Poor
Poor
Poor
Not
Poor
Poor
Multidimensional Poverty Strategy
Two cutoffs
1) Poverty line for each domain
Bourguignon and Chakravarty, JEI, 2003
“a multidimensional approach to poverty defines poverty as a
shortfall from a threshold on each dimension of an individual’s well being.”
2) Cutoff in terms of range of dimensions (number/sum)
Ex: UNICEF, Child Poverty Report, 2003
-Two or more deprivations
Ex: Mack and Lansley, Poor Britain, 1985
-Three or more out of 26
Focus on Pa family–is general (any additive index can be used)
OPTIONS
Weights
Cardinal or ordinal variable
- ordinal can be used with P0 and has good properties
Data
Matrix of well-being scores in D domains for N persons (e.g.
food energy, income/consumption, presence of a
malnourished child in hh, education, and so on)
Persons
13.1 15.2 12.5 20.0


14
7
10
11

y  
Domains
 4
5
1
3 


1
0
0
1



Data
Matrix of well-being scores in several domains for N persons
Persons
13.1 15.2 12.5 20.0


14
7
10
11

y  
 4
5
1
3 


1
0
0
1


Domain specific cutoffs
These
 entries achieve target cutoffs
These entries do not
z
13
12
3
1
Domains
Normalized Gaps
Persons
0
0
0.04
0 


0
0.42
0.17
0.08

g1  
0
0
0.67
0 


0
1
1
0


z
13
12
3
1
Replace
 these entries with 0
Replace these with normalized gap (zj - yji)/zj
Domains
Deprivation Counts
Persons
0

0
0

g 
0

0
Replace these
 entries with 0
Replace these entries with 1
0 1 0

1 1 1
0 1 0

1 1 0
Domains
Identification
Q/Who is poor?
Persons
0

0
0

g 
0

0
Counts

0 1 0

1 1 1
0 1 0

1 1 0
c = (0, 2, 4, 1)
= number of deprivations
Domains
Identification: Union
Q/Who is poor?
A/ Poor if deprived in at least one dimension (ci > 1)
Persons
0

0
0

g 
0

0
Counts

0 1 0

1 1 1
0 1 0

1 1 0
c = (0, 2, 4, 1)
= number of deprivations
Domains
Identification: Union
Q/Who is poor?
A/ Poor if deprived in at least one dimension (ci > 1)
Persons
0

0
0

g 
0

0
Counts

Difficulties
0 1 0

1 1 1
0 1 0

1 1 0
Domains
c = (0, 2, 4, 1)
= number of deprivations
Single deprivation may be due to something other than poverty (UNICEF)
Union approach often predicts very high numbers - political constraints.
Identification
Q/Who is poor?
Persons
0

0
0

g 
0

0
Counts
0 1 0

1 1 1
0 1 0

1 1 0
c = (0, 2, 4, 1)

Domains
Identification: Intersection
Q/Who is poor?
A/ Poor if deprived in all dimensions (ci > 4)
Persons
0

0
0

g 
0

0
Counts
0 1 0

1 1 1
0 1 0

1 1 0
c = (0, 2, 4, 1)

Domains
Identification: Intersection
Q/Who is poor?
A/ Poor if deprived in all dimensions (ci > 4)
Persons
0

0
0

g 
0

0
Counts
0 1 0

1 1 1
0 1 0

1 1 0
Domains
c = (0, 2, 4, 1)

Difficulty: Demanding requirement (especially if J large)
Often identifies a very narrow slice of population
Identification
Q/Who is poor?
Persons
0

0
0

g 
0

0
Counts
0 1 0

1 1 1
0 1 0

1 1 0
c = (0, 2, 4, 1)

Domains
Counting Based Identification
Q/Who is poor?
A/ Fix cutoff k, identify as poor if ci > k
Persons
0

0
0

g 
0

0
Counts
0 1 0

1 1 1
0 1 0

1 1 0
c = (0, 2, 4, 1)

Domains
Counting Based Identification
Q/Who is poor?
A/ Fix cutoff k, identify as poor if ci > k
Persons
0

0
0

g 
0

0
Counts

Example: 2 out of 4
0 1 0

1 1 1
0 1 0

1 1 0
c = (0, 2, 4, 1)
Domains
Counting Based Identification
Q/Who is poor?
A/ Fix cutoff k, identify as poor if ci > k
Persons
0

0
0

g 
0

0
Counts
0 1 0

1 1 1
0 1 0

1 1 0
Domains
c = (0, 2, 4, 1)

Example: 2 out of 4
Note: Especially useful when number of dimensions is large
Union becomes too large, intersection too small
Counting Based Identification
Implementation method: Censor nonpoor data
Persons
0

0
0

g 
0

0
Counts
0 1 0

1 1 1
0 1 0

1 1 0
c = (0, 2, 4, 1)

Domains
Counting Based Identification
Implementation method: Censor nonpoor data
Persons
0

0
0

g (k) 
0

0
Counts
0 1 0

1 1 0
0 1 0

1 1 0
c(k) = (0, 2, 4, 0)

Similarly for y(k), g1(k), etc
Domains
Counting Based Identification
Implementation method: Censor nonpoor data
Persons
0

0
0

g (k) 
0

0
Counts
0 1 0

1 1 0
0 1 0

1 1 0
c(k) = (0, 2, 4, 0)

Similarly for y(k), g1(k), etc
Note: Includes both union and intersection
Next: Turn to aggregation
Domains
Adjusted Headcount Ratio
Return to original matrix
Persons
0

0
0

g (k) 
0

0
Counts
0 1 0

1 1 0
0 1 0

1 1 0
c(k) = (0, 2, 4, 0)

Domains
Adjusted Headcount Ratio
Need to augment information of H
Persons
0

0
0

g (k) 
0

0
Counts
0 1 0

1 1 0
0 1 0

1 1 0
c(k) = (0, 2, 4, 0)

Domains
Adjusted Headcount Ratio
Need to augment information of H
Persons
0

0
0

g (k) 
0

0
0 1 0

1 1 0
0 1 0

1 1 0
Counts
c(k) = (0, 2, 4, 0)
shares of
deprivations (0, 1/2, 1, 0)
Domains
Adjusted Headcount Ratio
Adjusted headcount ratio = D0 = HA = m(g0(k))
Persons
0

0
0

g (k) 
0

0
0 1 0

1 1 0
0 1 0

1 1 0
Counts
c(k) = (0, 2, 4, 0)
shares of
deprivations (0, 1/2, 1, 0)
Average deprivation share among poor
A = 3/4
Domains
Adjusted Headcount Ratio
Adjusted headcount ratio = D0 = HA = m(g0(k)) = 6/16 = .375
Persons
0

0
0

g (k) 
0

0
0 1 0

1 1 0
0 1 0

1 1 0
Counts
c(k) = (0, 2, 4, 0)
shares of
deprivations (0, 1/2, 1, 0)
Average deprivation share among poor
A = 3/4
Domains
Adjusted Headcount Ratio
Adjusted headcount ratio = D0 = HA = m(g0(k)) = 6/16 = .375
Obviously if person 2 has an additional deprivation, D0 rises
Persons
0

0
0

g (k) 
0

0
0 1 0

1 1 0
0 1 0

1 1 0
Counts
c(k) = (0, 2, 4, 0)
shares of
deprivations (0, 1/2, 1, 0)
Average deprivation share among poor
A = 3/4
Domains
Adjusted Headcount Ratio
Adjusted headcount ratio = D0 = HA = m(g0(t)) = 6/16 = .375
Obviously if person 2 has an additional deprivation, D0 rises Dim. Mon.
Persons
0

0
0

g (k) 
0

0
0 1 0

1 1 0
0 1 0

1 1 0
Counts
c(k) = (0, 2, 4, 0)
shares of
deprivations (0, 1/2, 1, 0)
Average deprivation share among poor
A = 3/4
Domains
Adjusted Headcount Ratio
The next measure relates to work on
Amartya Sen’s Development as
freedom approach and can be
interpreted as a Measure of
Unfreedom.
(Pattanaik and Xu 1990)
Adjusted Headcount Ratio
Adjusted headcount = D0 = HA = m(g0(k))
Persons
0

0
0

g (k) 
0

0
0 1 0

1 1 0
0 1 0

1 1 0
Domains
Assume cardinal variables
Q/ What
 happens when a poor person who is deprived in
dimension j becomes even more deprived? (ultra poor)
A/ Nothing. D0 is unchanged. Violates monotonicity.
Adjusted FGT
Consider the matrix of alpha powers of normalized shortfalls
Persons
0
0
0.04 0


0
0.42
0.17
0

g1 (k)  
0
0
0.67 0


0
1
1
0



Domains
Adjusted FGT
Consider the matrix of alpha powers of normalized shortfalls
Persons
0 a
 a
0
a

g (k)  a
0
 a
0

0a
0.04 a
0.42 a
0.17 a
0
a
1a
0.67
1a
a
0 a 
a 
0 
0 a 
a 
0 
Domains
Adjusted FGT
Adjusted FGT is Da = m(ga(t)) for a > 0
Persons
0 a
 a
0
a

g (k)  a
0
 a
0

0a
0.04 a
0.42 a
0.17 a
0a
0.67 a
1a
1a
0 a 
a 
0 
0 a 
a 
0 
Domains
Adjusted Foster Greer Thorbecke
Adjusted FGT is Da = m(ga(t)) for a > 0
Persons
0 a
 a
0
a

g (k)  a
0
 a
0
0a
0.04 a
0.42 a
0.17 a
0a
0.67 a
1a
1a
0 a 
a 
0 
0 a 
a 
0 
Domains
Satisfies numerous properties including
decomposability, and dimension monotonicity,

monotonicity (for a > 0), transfer (for a > 1).
The Measure has good properties
• The adjusted headcount, adjusted poverty gap, and
adjusted FGT measures each satisfy a number of typical
properties of multidimensional poverty measures. In
particular, all satisfy:
• Symmetry,
Scale invariance
Normalization
Replication invariance
Focus (MD)
Weak Monotonicity
Dimensional Monotonicity
• Adjusted FGT satisfy
Decomposability and Strong Monotonicity (for a > 0), and
Transfer (for a > 1),.
Extensions
You can use this counting method of
identifying the poor with any other indices.
Derive censored matrix y*(k)
Replace all nonpoor entries with poverty cutoffs
Apply any multidimensional measure – P(y*(k);z)
e.g. Tsui 2002, Bourguignon & Chakravarty 2003,
Maasoumi Lugo 07,
Extension – weighting dimensions.
You can change the weights on the dimensions –
making some more important than others.
Weighted identification
Weight on income: 50%
Weight on food access, malnourished child in house: 25%
Cutoff k = 0.50 (weighted sum)
Poor if income poor, or suffer two or more deprivations
Cutoff k = 0.60
Poor if income poor and suffer one or more other deprivations
Nolan, Brian and Christopher T. Whelan, Resources, Deprivation and
Poverty, 1996
Weighted aggregation
Summary
• Measuring poverty at the level of the person
or hh can identify the acuteness of poverty.
• This is a simple, intuitive, and easy-to
compute measure of multidimensional
poverty. NGOs already use a form of it.
• It can be disaggregated by region, gender…
• It can be used for targeting purposes, for
monitoring, and for national reports.