<<

. 3
( 13)



>>

0.228 [7.48]——— 0.051 [5.64]——— 0.066 [3.55]——— 0.026 [5.03]———
Village mean land p.c.
’0.039 [1.87]— ’0.024 [1.98]——
’0.049 [0.71]
Village mean male adults 0.052 [1.17]
0.397 [6.29]——— 0.188 [9.93]——— 0.178 [4.58]——— 0.095 [8.79]———
Village mean fem adults
’0.619 [7.90]——— ’0.212 [9.14]——— ’0.597 [10.33]——— ’0.152 [10.99]———
Constant
Observations 1125 1125 1125 1125

Absolute value of t statistics in brackets.— signi¬cant at 10%; —— signi¬cant at 5%; ——— signi¬cant at 1%.
Chronic Poverty and All That

Table 2.4. Percentage change in poverty index from marginal change in characteristics

AFT (7) FTA (12) Seq AFT (7)

’0.43 ’0.39 ’0.48
From no education to primary completed
’0.17 ’0.19 ’0.18
Doubling land per capita
’0.05 ’0.08 ’0.09
Reducing distance to town by one kilometre
’0.63 ’1.01 ’0.91
From bad or no road to road accessible for trucks/bus

Source: Calculated from results in Table 2.3.




The most striking insight from the table is that the differences between
the different intertemporal measures of poverty appear relatively small: in
any case, in terms of signi¬cance, the same variables appear to stand out,
with the expected signs: education, land, distance to towns, road access, and
weather variability. Demographic characteristics also matter but not the sex of
the head. Strikingly, even the pro¬le based on the 1994 squared poverty gap
offers broadly a similar set of correlates. Obviously, this does not mean that
the same people are being predicted as being poor across equations.
It is dif¬cult to interpret the differences in the size of the coef¬cients
across equations, as the left-hand side variables are rather different and
most are not directly comparable. To highlight better the different inter-
pretations across the regressions, we can compare the marginal effects rel-
ative to the mean of each left-hand size variable. In other words, we can
establish the percentage change on each poverty measure from a change
in one of the explanatory variables. The relevant marginal effects are not
the coef¬cients given in Table 2.3 as the zeros in the data can be given
direct meaning (a zero squared poverty gap is a zero squared poverty gap,
and not some unobserved negative poverty). The coef¬cients in Table 2.3
give the marginal effects relative to the underlying latent variable of the
statistical model which is assumed to take on negative values. Instead, we
use marginal effects based on the unconditional expected value, evaluated
at the mean of all explanatory variables. Expressing these as a percent-
age of the mean dependent variable for each poverty measure, we obtain
Table 2.4.
These results are suggestive, as there are some interesting differences in
the order of magnitudes of the relative marginal effects. The most striking
differences relate to the infrastructure variables: using the FTA (2.12) mea-
sure (i.e. allowing for compensation over time) suggests that living nearer
to towns or with better roads is associated with considerably lower poverty
than implied by the AFT (2.7). Education improvements are more strongly
related to the AFT measures, especially the measure that effectively only
counts repeated poverty episodes. In short, when using poverty measures over
time, the way aggregation over time is done will affect the characteristics



53
C©sar Calvo and Stefan Dercon

that will be especially highlighted in poverty pro¬les as correlated with lower
poverty. 4



2.9. Conclusions

This chapter has offered a discussion of a number of issues related to measur-
ing poverty over time. It has highlighted some of the key normative decisions
that have to be taken. In particular, we have highlighted the role of compensa-
tion over time (whether poverty spells can be compensated for by non-poverty
spells); the issue of the discount rate (whether each spell should be given an
equal weight); and the issue of the role of persistence (whether repeated spells
should be given a higher weight). We have offered a number of plausible
poverty measures, each with different assumptions regarding these key issues.
We have also shown how these insights can be used to construct a forward-
looking measure of vulnerability. Applying a number of these measures to data
from rural Ethiopia, it is shown that while correlations are high, there would
still be considerable differences in ranking households by poverty according
to different measures, especially those that have different views on the role
of compensation. Turning to a multivariate poverty pro¬le, it was shown that
while similar factors are signi¬cant, their relative importance in identifying
intertemporal poverty is different according to the measure used to summarize
poverty.



Appendix 2.1

A Family of Individual Vulnerability Measures (Based on
Calvo and Dercon, 2006)
Let individual vulnerability (V) be measured by V = v(z,p,y), where z is the poverty
line, and p and y are k-dimensional vectors, containing state-of-the-world prob-
abilities and outcomes, respectively”i.e. pi is the probability of the i-th state
occurring, with outcome yi . We impose yi ≥ 0. It may be easiest to think of
these outcomes as consumption levels in each possible state of the world, espe-
cially if poverty is de¬ned as usual as a shortfall in consumption. We remark
that we mean outcomes after all consumption-smoothing efforts have been deployed.
In other words, their variability across states is taken as a ¬nal word, with no
scope for reducing it further, e.g. by formal insurance, risk sharing, or precautionary
savings.

4
As is well known with poverty pro¬les, these results have only limited policy implica-
tions, as these correlates are not shown to be causal factors, and even if they were, the relative
cost of intervening in terms of infrastructure, land, or education would have to be taken into
account.




54
Chronic Poverty and All That

For each state, de¬ne ˜censored outcome™ ˜i by ˜i ≡ Min(yi ,z), and the ˜rate of
y y
coverage of basic needs™ xi by xi ≡ ˜i /z, so that 0 ¤ xi ¤1. Vectors y and x are de¬ned
˜
y
correspondingly. ei stands for a k-dimensional vector whose elements are 0, except for
the i-th one, which equals 1. We close our notation with vectors y and yc . Their ele-
˜
k
ments are all equal to y and ˜ c , respectively, which in turn are de¬ned by y = i=1 pi ˜i
y ˆ y
and v(z,p,˜) = v(z,p,˜ c ). Note that ˜ c can be written as a function ˜ c (z,p,˜ ) and will
y y y y y
shortly be called the risk-free equivalent to the set of prospects described by (z,p,y),
in the sense that it yields the same degree of vulnerability; y is the expected value
ˆ
of ˜i .
y
We propose eight desiderata. The ¬rst is the FOCUS AXIOM, which imposes v(z,p,y) =
v(z,p,˜). Our measure will thus disregard outcome changes above the poverty line.
y
If vulnerability is understood as a burden caused by the threat of future poverty, it
should not be compensated by simultaneous (ex ante) possibilities of being well off. In
consequence, high vulnerability is not necessarily tantamount to grim overall expected
well-being (as arguably in Ligon and Schechter, 2003), since the ˜promise™ of richness in
some states can raise welfare expectations, with no bearing on vulnerability.
Imagine that a farmer faces two scenarios: rain (no poverty) or drought (poverty).
Does she become less vulnerable if the harvest in the rainy scenario improves? Our
answer is ˜no™. Poverty is as bad a threat as before. It is as likely as before, and it is
potentially as severe as before.
According to this axiom, ˜excess™ outcomes yi ’ z > 0 are ˜wasteful™ and can be
ignored, as far as vulnerability is concerned. Taking this for granted, the remaining
axioms can be presented as follows:
SYMMETRY OVER STATES: v(z,p,˜) = v(z,Bp,B˜), where B is any k — k permutation matrix.
y y
All states receive the same treatment, and the only relevant difference between two
states of the world i and j is the difference in their outcomes (yi , y j ) and probabilities
( pi , p j ).
CONTINUITY AND DIFFERENTIABILITY. Function v(z,p,˜) is continuous and twice-
y
differentiable in y, for tractability and to preclude abrupt reactions to small changes
in outcomes.
SCALE INVARIANCE. v(z,p,˜) = v(Îz,p,Θ) for any Î > 0. Our measure will not depend on
y y
the unit of measure of outcomes.
NORMALIZATION. Min ˜ [v(z,p,˜)] = 0 and Max ˜ [v(z,p,˜)] = 1. We impose closed bound-
y y
y y
aries to facilitate interpretation and comparability.
PROBABILITY-DEPENDENT EFFECT OF OUTCOMES. For “c < ˜i < z and pi pi = 0, /
y
v(z,p,˜)’v(z,p,˜+cei ) = v(z,p ,˜ ) ’ v(z,p ,˜ + cei ) if and only if pi = pi and ˜i = ˜i .
y y y y yy
Should ˜i change, the consequent effect on vulnerability is not allowed to depend on
y
the outcomes or probabilities of other states of the world”for a given pi , the change
in vulnerability depends only on ˜i . 5 In the opposite direction, the effect must be
y


5
A possible counterargument could run: ˜in fact, there could be some relief in considering that
one could have done much better had the odds been more fortunate™ (or to the contrary, ˜one may
rue having missed a better possible outcome, through no fault on one™s own part, and thus one™s
misery will be greater™). We ignore such counterarguments for the sake of tractability. In doing so,
we simply adhere to the common concept of poverty as mere failure to reach a poverty line, with
no regard for ˜subjective™ subtleties.




55
C©sar Calvo and Stefan Dercon

/
sensitive to the likelihood of that particular state of the world. Note that pi pi = 0
discards ˜impossible™ states ( pi = pi = 0).
¤ ≥
PROBABILITY TRANSFER. For every p j ≥ d > 0, v(z,p +d(ei ’e j ),˜) v(z,p,˜) if ˜i
y y y ˜.
y
¤j

If ˜i is greater than or at least equal to ˜ j , then vulnerability cannot increase as a result
y y
of a probability transfer from state j to state i. Likewise, if ˜i is lower than or at most
y
equal to ˜ j , then vulnerability cannot decrease. Going back to the example of the farmer
y
facing rain and drought, we say that she becomes more vulnerable if a drought becomes
more likely, at the expense of the rainy scenario (or at least, her vulnerability does not
lessen as a result).
RISK SENSITIVITY. v(z,p,˜) > v(z,p,y). Vulnerability would be lower if the expected (cen-
y
sored) outcome ˆ were attained in all states of the world and uncertainty were thus
y
removed. In other words, greater risk raises vulnerability. 6 Thus we link up with our ¬rst
intuition about vulnerability, as a concept aiming to capture the burden of insecurity,
the fact that hardship is also related to fear of future threats.
Alternatively, resorting to the risk-free equivalent ˜ c , the same axiom could be
y
c /ˆ <1. Expected outcome is unevenly and ˜inef¬ciently™ spread across
expressed as ˜ y
y
states of the world, in the sense that a similarly low degree of vulnerability would result
from ˜ c < ˆ being secured in every state. ˜ c /ˆ re¬‚ects this ˜ef¬ciency loss™.
y y yy
CONSTANT RELATIVE RISK SENSITIVITY. For Í > 0, ͘ c (z,p,˜) = ˜ c (z,p,͘). A proportional
y y y y
increase by Í in the outcomes of all possible states of the world leads to a similar pro-
portional increase in the risk-free equivalent ˜ c . While risk sensitivity ensures ˜ c /ˆ < 1,
y yy
we now require this ratio (or ˜ef¬ciency loss™) to remain constant if all state-speci¬c
outcomes increase proportionally.
As compared to the previous axioms, this ¬nal property seems less compelling. Still,
we ¬nd it attractive for its contribution both to narrowing down the families of accept-
able measures to only one, and to securing that risk sensitivities receive an appropriate
treatment. As for this second point, Ligon and Schechter (2003) were the ¬rst to point
out that some existing vulnerability measures hid some awkward assumptions, e.g. risk
sensitivity increasing in initial income, at odds with most empirical ¬ndings on risk
attitudes (e.g. Binswanger 1981).
Needless to say, we are avoiding here terms such as ˜risk aversion™ or ˜utility™. We
intend our choice of language to convey our view of vulnerability as distinct from
expected utility, if only to stress our departure from proposals where vulnerability
boils down to some form of bad ˜overall™ expectations (e.g. Ligon and Schechter,
2003). On the other hand, parallels should be obvious. In fact, the proof of the fol-
lowing theorem heavily draws on results from expected utility theory (mainly Pratt,
1964), necessarily with some departures due to the speci¬c traits of our vulnerability
concept. For this reason and for brevity, it is not provided, but it is available on
request.
THEOREM 1”If all the axioms above are satis¬ed, then

V(·) = 1 ’ E [x· ], with 0 < · < 1. (.24)


6
We implicitly de¬ne the increase in risk as a probability transfer ˜from the middle to the tails™,
in keeping with one of the Rothschild“Stiglitz senses of risk.




56
Chronic Poverty and All That

E is the expected value operator, and we recall xi ≡ ˜i /z is the rate of coverage of basic
y
needs, and 0 ¤ xi ¤ 1. We highlight the simplicity of this single-parameter family of
measures V(·) . 7 Of course, · regulates the strength of risk sensitivity”as · rises to 1, we
approach risk neutrality.
A few remarks are in place. First, for those facing no uncertainty and with known
xi = x— < 1 for all i, V(·) > 0. If vulnerability is about the threat of poverty, certainty of
being poor is but a dominant, irresistible threat. The concept is not con¬ned to those
whom the winds might blow into poverty or out from it. Vulnerability is about risk, but
not only about it.
Second, it is easy to prove that V(·) is equal to the probability of being poor only
if outcomes are expected to be zero in every state of the world where the individual
is poor. If vulnerability were measured as expected FGT0 (as in Chaudhuri and Jalan,
2002), then vulnerability would be overestimated. Ligon and Schechter (2003) have
pointed out the shortcomings of other FGT choices. 8
Finally, V(·) can still be assimilated into the expected-poverty approach to vulnerabil-
ity, provided poverty is measured as in Chakravarty (1983). In some sense, one of the
contributions of this chapter is to identify the Chakravarty poverty index as the best
choice if the poverty analysis moves from static poverty on to vulnerability.


References
Anand, S., and Hanson, K. (1997), ˜Disability-Adjusted Life Years: A Critical Review™,
Journal of Health Economics, 16: 685“702.
Baulch. B., and Hoddinott, J. (2000), ˜Economic Mobility and Poverty Dynamics in
Developing Countries™, introduction to special issue, Journal of Development Studies.
Binswanger, H. P. (1981), ˜Attitudes toward Risk: Theoretical Implications of an Experi-
ment in Rural India™, Economic Journal, 91(364): 867“90.
Bourguignon, F., and Chakravarty, S. R. (2003), ˜The Measurement of Multidimensional
Poverty™, Journal of Economic Inequality, 1(2): 1569“721.
Calvo, C., and Dercon, S. (2006), ˜Vulnerability to Poverty™, mimeo, Oxford University.
Chakravarty, S. R. (1983), ˜A New Index of Poverty™, Mathematical Social Sciences, 6: 307“
13.
Chaudhuri, S., and Jalan, J. (2002), ˜Assessing Household Vulnerability to Poverty
from Cross-sectional Data: A Methodology and Estimates from Indonesia™, Columbia
University Discussion Paper 0102-52.
Dercon, S., and Krishnan, P. (2000), ˜Vulnerability, Poverty and Seasonality in Ethiopia™,
Journal of Development Studies, 36(6): 25“53.
Foster, J. (2007), ˜A Class of Chronic Poverty Measures™, mimeo.
Greer, J., and Thorbecke, E. (1984), ˜A Class of Decomposable Poverty Measures™,
Econometrica, 52(3): 761“6.
and Shorrocks, T. (1991), ˜Subgroup Consistent Poverty Indices™, Econometrica,
59(3): 689“709.
7
For instance, if our last axiom (constant relative risk sensitivity) were replaced by constant
absolute risk sensitivity [Í + ˜ c (z,p,˜) = ˜ c (z,p,˜ + Í), for Í > 0], the less attractive measure V(‚) = 1
y y y y
--E[{e‚(1’x) ’ 1}/{e‚ ’1}], with ‚ > 0, would result.
8
More precisely, we should speak about expected individual poverty, as measured by the function
implicit in the corresponding aggregate FGT index, as in Foster, Greer, and Thorbecke (1984).




57
C©sar Calvo and Stefan Dercon

Jalan, J., and Ravallion, M. (2000), ˜Is Transient Poverty Different? Evidence from Rural
China™, Journal of Development Studies, 36(6): 82“99.
Kanbur, R., and Mukherjee, D. (2006), ˜Premature Mortality and Poverty Measurement™,
mimeo, Cornell University.
Ligon, E., and Schechter, L. (2003), ˜Measuring Vulnerability™, Economic Journal,
113(486): C95“C102.
Pratt, J. W. (1964), ˜Risk Aversion in the Small and in the Large™, Econometrica, 32(1/2):
122“36.
Tsui, K. (2002), ˜Multidimensional Poverty Indices™, Social Choice and Welfare, 19(1): 69“
93.




58
3
A Class of Chronic Poverty Measures
James E. Foster




3.1. Introduction

Traditional measures of poverty based on cross-sections of income (or con-
sumption) data provide important information on the incidence of material
poverty, its depth and distribution across the poor. However, they have little
to say about another important dimension of poverty: its duration. Empirical
evidence suggests that increased time in poverty is associated with a wide
range of detrimental outcomes, especially for children. 1 If so, then this would
provide a strong rationale for using a methodology for evaluating chronic
poverty that explicitly incorporates ˜time in poverty™. This chapter presents
a new class of chronic poverty measures that can account for duration in
poverty as well as the traditional dimensions of incidence, depth, and severity.
There are several methodologies available for measuring chronic poverty
using panel data. Two broad categories may be discerned, each with its own
distinctive strategy for identifying the chronically poor. 2 The components
approach, exempli¬ed by Jalan and Ravallion (1998), constructs an aver-
age or permanent component of income and identi¬es a chronically poor
person as one for whom this component lies below an appropriate poverty
line. 3 Variations in incomes across periods are ignored by this identi¬cation
process and by the subsequent aggregation step when the data are brought
together into an overall measure. The components approach to chronic

1
For example, longer exposure to poverty is associated with: increased stunting, dimin-
ished cognitive abilities, and increased behavioural problems for children (Brooks-Gunn and
Duncan, 1997); worse health status for adults (McDonough and Berglund, 2003); lower levels
of volunteerism when poor children become adults (Lichter, Shanahan, and Gardner, 1999);
and an increased probability of staying poor (Bane and Ellwood, 1986; Stevens, 1994). See
also the conceptual discussions of Yaqub (2003) and Clark and Hulme (2005).
2
This division is due to Yaqub (2000); see also McKay and Lawson (2002).
3
Examples of the components approach can be found in Duncan and Rodgers (1991),
Rodgers and Rodgers (1993), Jalan and Ravallion (1998), and Dercon and Calvo (2006), among
others.




59
James E. Foster

poverty measurement is not especially sensitive to the time a family spends in
poverty and, hence, may not be the best framework for incorporating duration
into poverty measurement.
A second approach to evaluating chronic poverty”called the spells
approach”focuses directly on the period-by-period experiences of poor fam-
ilies, and especially on the time spent in poverty. The identi¬cation of the
chronically poor typically relies on a duration cut-off as well as a poverty line:
Gaiha and Deolalikar (1993), for example, take the set of chronically poor
to be all families that have incomes below the poverty line in at least ¬ve
of the nine years of observations, hence have a duration cut-off of 5/9. As for
the aggregation step, most proponents of the spells approach use a very simple
index of chronic poverty based on the number of chronically poor. 4 While
the number (or percentage) of chronically poor may be an important statistic
to keep in mind, it is a rather crude indicator of overall chronic poverty. In
particular, it ignores the time a chronically poor family spends in poverty and
hence violates a ˜time monotonicity™ property that is especially relevant in
the present context. In addition, other key dimensions of poverty, namely its
depth and distribution, are utterly ignored by the index.
The present chapter adopts the general methodology of the spells approach.
Two distinct cut-offs are used for identifying the chronically poor”one in
income space (the usual poverty line z > 0) and another governing the
percentage of time in poverty (the duration line 0 < ™ ¤ 1). In other words, a
family is considered to be chronically poor if the percentage of time it spends
below the poverty line z is at least the duration cut-off ™. For the aggregation
step, this chapter presents a new class of chronic poverty measures based on
the P· family proposed by Foster, Greer, and Thorbecke (1984), appropriately
adjusted to account for the duration of poverty. All of the measures satisfy
time monotonicity and an array of basic axioms, while certain subfamilies
satisfy the multiperiod analogues of (income) monotonicity and the transfer
principle. Associated measures of transient poverty are de¬ned to account for
poverty that is shorter in duration. Each chronic poverty measure (and its
transient dual) satis¬es decomposability, thus allowing the consistent analysis
of chronic poverty by population subgroup. In particular, pro¬les of chronic
poverty can be constructed to understand the incidence, depth, and severity
of poverty in a way analogous to the standard static case.
The chapter proceeds as follows. Section 3.2 provides a brief overview of
poverty measurement in a static environment to help ground the discussion
of chronic poverty measurement. Section 3.3 introduces time into the analy-
sis. The identi¬cation and aggregation steps are speci¬ed and the new family


4
See for example The Chronic Poverty Report 2004“05, p. 9, which uses a simple headcount.
Duncan, Coe, and Hill (1984) and Gaiha and Deolalikar (1993) use the headcount ratio, or
the percentage of the population that is chronically poor.




60
A Class of Chronic Poverty Measures

of chronic poverty measures is de¬ned. Section 3.4 provides a brief application
of the technology to data from Argentina, while Section 3.5 concludes.


3.2. Traditional Poverty Measurement

Following Sen (1976), poverty measurement can be broken down into two
conceptually distinct steps: ¬rst, the identi¬cation step, which de¬nes the
criteria for determining who is poor and who is not; and, second, the aggrega-
tion step, by which the data on the poor are brought together into an overall
indicator of poverty. The identi¬cation step is typically accomplished by set-
ting a cut-off in income space called the poverty line and evaluating whether a
person™s resources are suf¬cient to achieve this level. There are several varieties
of poverty lines, each with its own information basis and method for updating
over time. Subjective poverty lines consider information from surveys that ask
participants how much it takes to get along. Relative poverty lines depend on
the current income standard in a given society: a common example sets the
poverty line at 50 per cent of the median income. Absolute poverty lines may
be purely arbitrary (such as the US$1 or US$2 per day lines used in World
Bank illustrations) or may be initially derived from consumption studies.
Note that in principle each type of line can be located at the low end or
the high end of conceivable cut-offs (e.g. a relative line at 1 per cent of the
median and an absolute line at US$15 per day); consequently, the use of an
absolute line does not identify a person as being ˜absolutely impoverished™.
Instead, the term ˜absolute™ typically refers to the fact that the poverty line
is to remain ¬xed during the time frame under consideration. In contrast, a
thoroughgoing relative (or subjective) approach will have a different poverty
line at each point in time as the income standards (or norms) change. 5 This
chapter assumes that an absolute poverty line has been selected and that it is
applicable at all time periods under consideration.
The aggregation step is typically accomplished by selecting a poverty index
(or measure). Each index is a method of combining the income data and the
poverty line into an overall indicator of poverty. Formally, it is a function
associating with each income distribution and poverty line a real number,
namely, the measured level of poverty. The simplest and most widely used
measure is the headcount ratio, which is the percentage of a given population
that is poor. It is sometimes helpful to view the headcount ratio as a speci¬c
population average; indeed, if every person identi¬ed as being poor is assigned
a value of ˜1™ while every person outside the set of the poor is assigned a
value of ˜0™, then the headcount ratio is simply the mean of the resulting ˜0“1™
vector.
5
One could also imagine alternative types of hybrid approaches to setting poverty lines
across space and time. See Foster and Sz©kely (2006).




61
James E. Foster

A second method of aggregation is given by the (per capita) poverty gap,
which is the aggregate amount by which poor incomes fall short of the
poverty line, measured in poverty line units, and averaged across the entire
population. It too can be seen as a population average, with those outside
the set of the poor being assigned a value of ˜0™, as before, and those inside
being represented by their normalized shortfall, or the difference between their
income and the poverty line, divided by the poverty line itself. In contrast to
the ˜all or nothing™ approach of the headcount ratio, the poverty gap uses the
normalized shortfall as a continuous measure of individual poverty and views
overall poverty as its average value across society. Consequently, it satis¬es a
standard monotonicity axiom for poverty measures, which requires poverty to
rise when the income of a poor person falls (ceteris paribus). The headcount
ratio does not.
A general method of aggregation suggested by Foster, Greer, and Thorbecke
(1984) proceeds as above, but ¬rst transforms the normalized shortfalls of the
poor by raising them to a non-negative power · to obtain the associated P·
measure. This approach actually includes both of the foregoing measures: P0
is the headcount ratio and P1 is the poverty gap measure. The squared gap mea-
sure P2 from this family takes the square of each normalized shortfall, which
has the effect of diminishing the relative importance of very small shortfalls
and augmenting the effect of larger shortfalls”and hence emphasizing the
conditions of the poorest poor in society. As a simple average of the squared
normalized shortfalls across the population, P2 satis¬es the transfer axiom,
which requires poverty to rise whenever a poor person transfers income to a
richer poor person. All the P· measures satisfy a range of basic axioms as well
as a property linking subgroup and overall poverty levels that has been used
extensively in the empirical literature: Decomposability requires overall poverty
to be the weighted average of subgroup poverty levels, where the weights are
the population shares of the respective subgroups. While there are several
other poverty measures in common use, this chapter will focus on the P· class
in general and the three measures P0 , P1 , and P2 , in particular, in developing
a new class of chronic poverty measures. 6



3.3. The Measurement of Chronic Poverty

A main premiss of chronic poverty evaluation is that poverty repeated over
time has a greater impact than poverty that does not recur. This section
discusses how poverty measurement can be altered to take into account the

6
Other measures can be found in Sen (1976), Clark, Hemming, and Ulph (1981),
Chakravarty (1983), for example; see also Foster and Sen (1997), Zheng (1997), and Foster
(2006).




62
A Class of Chronic Poverty Measures

additional dimension of time in poverty. The ¬rst part begins with some
important de¬nitions and notation.


3.3.1. Notation
The basic data are observations of an income (or consumption) variable for a
set {1, . . . ,N} of individuals at several points in time. 7 Let y = (yit ) denote the
matrix of (non-negative) income observations over time, where the typical
entry yit is the income of individual i = 1, 2, . . . ,N in period t = 1, 2, . . . , T. We
adopt the convention that y is a T — N matrix (having height T and length N),
so that each column vector yi lists individual i™s incomes over time, while each
row vector yt gives the distribution of income in period t. It will prove helpful
to use the notation |y| = È t yit to denote the sum of all the entries in a given
matrix, and to de¬ne an analogous notation for vectors (hence |yi | = t yit is
the sum of i™s incomes across all periods while |yt | = i yit is the total income
in period t). It is assumed that incomes have been appropriately transformed
to account for variations across time and household con¬gurations so that
a common poverty line z can be used to establish who is poor in each
period.
It is sometimes useful to express the data in terms of (normalized) shortfalls
rather than incomes. Let g be the associated matrix of normalized gaps, where
the typical element git is zero when the income of person i in period t is z
or higher, while git = (z ’ yit )/z otherwise. Clearly, g is a T — N matrix whose
entries are non-negative numbers less than or equal to one. When an entry
git is equal to zero, this indicates that the person™s income is at least as large
as the poverty line and hence is not in poverty; when an entry is positive,
this indicates that the person™s income falls below the line, with git being a
measure of the extent to which that person is poor. 8 We can similarly de¬ne
the matrix s of squared normalized shortfalls by squaring each entry of g; i.e.
the typical entry of s is sit =(git )2 .
Counting-based approaches to evaluating poverty ignore the extent of the
income gap and instead only take into account whether the gap is positive or
zero. It is therefore helpful to create another matrix h by replacing all positive
entries in g with the number ˜1™. Thus the typical entry hit of h is ˜0™ when the
income of person i in period t is not below z, and ˜1™ when yit is below z. One
statistic of interest in the present context is the duration of person i™s poverty,
or the fraction of time the person is observed to have an income below z.
Denote this by di and note that it can be obtained by summing the entries in
7
The income variable can be any single-dimensional, cardinally meaningful indicator of
well-being. In the present case, where per-period values are not transferable across time, this
may be more consistent with consumption than with income, per se.
8
In this chapter a person with an income of z is not poor; the alternative assumption (that
z itself is a poor income level) could be adopted with a slight change in notation.




63
James E. Foster

hi (the i-th column of h) and dividing by the number of periods; i.e. di = |hi |/T.
In essence the duration is analogous to a headcount ratio, but de¬ned for a
given person over time, not across different people within the same period of
time.
This chapter™s approach to chronic poverty will be based on the percentage
of time a person spends in poverty. Toward this end, it will be useful to derive
matrices from g (and also from s and h) that ignore persons whose duration
in poverty falls short of a given cut-off ™ ≥ 0. Let g(™) (and s(™) and h(™) ) be
the matrix obtained from g (respectively s and h) by replacing the i-th column
with a vector of zeros when di < ™. In other words, the typical entry of g(™),
namely git (™), is de¬ned by g it (™) = git for all i satisfying di ≥ ™ while git (™) = 0 for
all i having di < ™ (with the analogous de¬nition holding for s and h).
As the duration cut-off ™ rises from 0 to 1 the number of non-zero entries
in the associated matrix falls, re¬‚ecting the progressive censoring of data
from persons who are not meeting the poverty duration requirement. The
speci¬cation ™ = 0 would not alter the original matrices at all, so that g(0) = g,
s(0) = s, and h(0) = h. Every poverty observation would be included, regardless
of a person™s duration in poverty. At the other extreme, the cut-off ™ = 1
ensures that any person who was out of poverty for even a single observation
would have a column of zeros; in other words, g(1), s(1), and h(1) consider a
person who fell out of poverty for one period indistinguishable from a person
who was always out of poverty.
As in the static case, the measurement of chronic poverty can be divided
into an identi¬cation step and an aggregation step. There are many potential
strategies for identifying the chronically poor, but all have the effect of select-
ing a set Z of chronically poor persons from {1, . . . ,N}. The aggregation step
takes the set Z as given and associates with the income matrix y an overall
level K (y; Z) of chronic poverty. The resulting functional relationship K is
called an index, or measure, of chronic poverty.


3.3.2. Identifying the chronically poor
What can panel data reveal that cross-sectional observations cannot? By fol-
lowing the same persons over several periods, they can help discern whether
the poverty experienced by a person in a given period is an exceptional
circumstance or the usual state of affairs.
With panel data, there are several income observations linked to each
individual, and this in turn leads to a wide array of potential methods for
deciding when a person is chronically poor. One approach employed by Jalan
and Ravallion (1998) bases membership in Z on a single comparison between
the poverty line z and a composite indicator of the resources an individual
has available through time. The speci¬c income standard employed by Jalan
and Ravallion is Ï(yi ) = |yi |/T, the average or mean income over time; hence



64
A Class of Chronic Poverty Measures

their method identi¬es as chronically poor any person whose mean income is
below the poverty line. As noted above, this approach is not particularly sen-
sitive to the duration of poverty. Nonetheless, it may make good sense when
incomes are perfectly transferable across time and, accordingly, consumption
can be completely smoothed. Anyone with an average income below z would
in the best case be poor for every period; while a person with a mean of z
or above could be out of poverty in every period. However, the assumption
of perfect transferability may be dif¬cult to sustain, particularly for poorer
individuals; and if per-period incomes are even slightly less than perfectly
substitutable, this procedure could easily misidentify persons. 9
At the other extreme is the ˜spells™ approach to identifying the chronically
poor, which bases membership in Z upon the frequency with which one™s
income falls below the poverty line. So, for instance, one might require a
person to be poor 50 per cent of the time or more, before identifying the
person as chronically poor. A higher cut-off (say 70 per cent of the time or
more) would probably lead to a smaller set of persons being identi¬ed as
chronically poor, while a lower cut-off (such as 30 per cent) would probably
expand the set. Note, though, that this approach also contains within it an
implicit assumption”that there is no possibility of transferring income across
periods. Indeed, it is not entirely clear why a person with a tremendous
amount of income (or expenditure) in period one, who is just barely below
the poverty line in the remaining periods, would be considered chronically
poor, as may be required under this approach. Nonetheless, if (1) the poverty
line is considered to be a meaningful dividing line between poor and non-
poor and (2) the observed data on income (or consumption) in each period
faithfully re¬‚ects the constraint facing the person in the given period, then
identifying chronic poverty with suf¬cient time in poverty makes intuitive
sense.
This chapter uses a dual cut-off ˜spells™ approach to identifying the chron-
ically poor: The ¬rst cut-off is the poverty line z > 0 used in determining
whether a person is poor in a given period; the second is the duration line
™ (with 0 < ™ ≥ 1) that speci¬es the minimum fraction of time that must
be spent in poverty in order for a person to be chronically poor. Given the
income matrix y and the poverty line z, the matrix h depicts the poverty
spells for each person, and this in turn yields di , the fraction of time that
person i is observed to have an income below z. Then, given ™, the set of
chronically poor persons is de¬ned to be Z = {i : di ≥ ™}, or the set of all persons
in poverty at least ™ share of the time. Since Z depends on z and ™, the poverty
index can be written as a function K (y; z, ™) of the income matrix and the pair
of parametric cut-offs. The next section constructs several useful functional
forms for K (y; z, ™).

9
The case of imperfect substitutability is considered by Foster and Santos (2006).




65
James E. Foster

3.3.3. Chronic poverty and aggregation
The ¬rst question that is likely to arise in discussions of chronic poverty is:
How many people in a given population are chronically poor? The answer
comes in the form of the headcount Q(y; z, ™) de¬ned as the number of per-
sons in Z. This statistic is often highlighted in order to convey meaningful
information about the magnitude of the problem; however, when making
comparisons, especially across regions having different population sizes, the
headcount ratio H(y; z, ™) = Q(y; z, ™)/N is commonly used, where N is the
population size of y. The measure H focuses only on the frequency of chronic
poverty in the population and ignores all other aspects of the problem, such
as the average time the chronically poor are in poverty, or the average size of
their normalized shortfalls.
An example will help illustrate these concepts. Consider the income matrix
⎡ ¤
3 9 7 10
⎢7 3 4 8⎥
⎢ ⎥
y=⎢ ⎥
⎣ 9 4 2 12 ¦
832 9

where the poverty line is z = 5 and the duration line is ™ = 0.70. The associated
matrices of normalized poverty gaps, g, and of poverty spells, h, are given by
⎡ ¤
⎡ ¤
1000
0.4 0 0 0
⎢0 1 1 0⎥
⎢0 0.4 0.2 0 ⎥ ⎢ ⎥
⎢ ⎥
h=⎢ ⎥
g=⎢ ⎥
⎣0 1 1 0¦
⎣0 0.2 0.6 0 ¦
0110
0 0.4 0.6 0

Summing the entries of h vertically and dividing by T = 4 yields the dura-
tion vector d = (d1 , d2 , d3 , d4 ) = (0.25, 0.75, 0.75, 0) and hence we see that
Q(y; z, ™) = 2 and so H(y; z, ™) = 0.5; in this population, half of the persons
(namely numbers 2 and 3) are chronically poor.
Now consider a thought experiment in which person 3 in the above
example receives an income of 3 rather than 7 in period 1, and hence the
normalized gap in that period becomes 0.40 and the entry in h becomes one.
Then person 3 would still be chronically poor, but would have a poverty
duration of d3 = 1.0 rather than 0.75. What would happen to H? Clearly, it
would be unchanged even though a chronically poor person has experienced
an increment in the time spent in poverty. In other words, H violates an
intuitive time monotonicity axiom: if the income in a given period falls for a
chronically poor person in such a way that the duration in poverty rises, then
the level of chronic poverty should increase. 10 It can be argued that, while
H conveys meaningful information about one aspect of chronic poverty, and

10
See Foster (2007) for a precise statement of this axiom.




66
A Class of Chronic Poverty Measures

hence is a useful ˜partial index™, it is a bit too crude to be used as an overall
measure. 11
There is a very direct way of transforming H into an index that is sensitive
to changes in the duration of poverty. Consider the matrix h(™) de¬ned above,
which leaves a column unchanged if the person is chronically poor, and
otherwise replaces the column with zeros. Let di (™) = |hi (™)|/T denote the
associated duration level of person i, so that di (™) = di for each chronically
poor person and di (™) = 0 otherwise. Then the average duration among the
chronically poor is given by D(™) = (d1 (™) + · · · + dN (™) )/Q. This is a second par-
tial index that conveys relevant information about chronic poverty, namely,
the fraction of time the average chronically poor person spends in poverty.
Combining the two partial measures yields an overall index that is sensitive
to increments in the time a chronically poor person spends in poverty as
well as to increases in the prevalence of chronic poverty in the population.
De¬ne the duration-adjusted headcount ratio K 0 = H D to be the product of
the original headcount ratio H and the average duration D or, equivalently,
K 0 = (d1 (™) + · · · + dN (™) )/N.
K 0 offers a different interpretation of our thought experiment from the one
provided by H. Return to the original situation in which person 3 is not poor
in period 1. For ™ = 0.70, the relevant h(™) matrix is given by
⎡ ¤
0000
⎢0 1 1 0⎥
⎢ ⎥
h(™) = ⎢ ⎥
⎣0 1 1 0¦
0110

and the respective column averages are given by d1 (™) = d4 (™) = 0 and d2 (™) =
d3 (™) = 0.75. The headcount ratio is H = 0.50 while the mean duration is
D = 0.75 so that the duration-adjusted headcount ratio K 0 is initially 0.375.
Now when person 3™s income in period 1 becomes 3 rather than 7, the fraction
of time spent in poverty rises to 1 for that person, while the mean duration
among all chronically poor rises to 0.875. Consequently, even though H is
unchanged, K 0 rises to about 0.438, with this higher overall level of chronic
poverty being due to person 3™s increased time in poverty.
The above example also shows that K 0 = Ï(h(™) ) = |h(™)|/(T N); in words, K 0
is the mean of the entries in matrix h(™) or, equivalently, the total number
of periods in poverty experienced by the chronically poor, as given by |h(™)|,
divided by the total number of possible periods across all people, or TN. In
the above example, it is easy to see that the mean of the sixteen entries in
h(™) is 6/16 = 0.375, and hence this is the duration-adjusted headcount index
K 0 . Notice that if a chronically poor person were to have an additional period
in poverty, this would raise an entry in the matrix h(™) from zero to one,

11
See the discussion of partial indices in Foster and Sen (1997).




67
James E. Foster

thereby causing the average value K 0 to rise, as noted above. In other words,
K 0 satis¬es the time monotonicity axiom.
There is no doubt that K 0 is less crude than H as an overall measure of
chronic poverty. However, it too may not fully re¬‚ect the actual conditions of
the chronically poor. The matrix h(™), upon which K 0 is based, is unaffected by
changes in incomes (or normalized gaps) that preserve the signs of the entries
of g(™), even if the magnitudes of the entries in g(™) change dramatically.
For example, if the income of person 3 in period 2 were decreased from 4
2
to 2, so that the normalized gap g 3 rose from 0.2 to 0.6, the corresponding
entry in h would obviously be unchanged (namely, h2 = 1), and hence K 0
3
would remain the same. So a chronically poor person is now much poorer in
period 2, and yet this fact goes unnoticed by the duration-adjusted headcount
measure. This is a violation of the (income) monotonicity axiom, which states
that if a chronically poor person™s income is below the poverty line in a given
period, lowering that income further should increase the measured level of
chronic poverty. 12
What is missing from this measure of chronic poverty is information on
the magnitudes of the normalized gaps. Consider the matrix g(™) de¬ned above
whose non-zero entries are the normalized gaps of the chronically poor. The
number of non-zero entries in g(™)”and hence h(™)”is |h(™)|, while the sum of
the non-zero entries in g(™) is |g(™)|. The ratio |g(™)|/|h(™)| indicates the average
size of the normalized gaps across all periods in which the chronically poor
are in poverty. The resulting average gap G(™) = |g(™)|/|h(™)| provides exactly the
type of information that would usefully supplement the adjusted headcount
ratio. De¬ne the duration-adjusted poverty gap index K 1 = K 0 G to be the product
of the duration-adjusted headcount ratio K 0 and the average gap G or, equiv-
alently, K 1 = HDG, the product of the three partial indices that respectively
measure the prevalence, duration, and depth of chronic poverty.
This chronic poverty index provides a third perspective from which to view
our numerical example. Given the duration cut-off ™ = 0.70, the matrix g(™)
associated with the original situation in which person 3 has an income of 4
in period 2 is given by
⎡ ¤
00 0 0
⎢ 0 0.4 0.2 0 ⎥
⎢ ⎥
g(™) = ⎢ ⎥
⎣ 0 0.2 0.6 0 ¦
0 0.4 0.6 0

The respective sum of entries is |g(™)| = 2.4 while the number of periods in
poverty is |h(™)| = 6, and hence the average gap is G = 0.40. Given H = 0.50
and D = 0.75 from before, the resulting level of the duration-adjusted poverty
gap measure is K 1 = 0.15. Now suppose that the period 2 income of person

12
See Foster (2007) for a rigorous statement of this axiom.




68
A Class of Chronic Poverty Measures

3 falls from 4 to 2. Clearly H and D are unaffected by this change, and so
K 0 would likewise be unchanged. However, the average gap G would rise to
about 0.47, and hence the duration-adjusted gap would now be K 1 = 0.175,
re¬‚ecting the worsened circumstances for person 3. K 1 rises as a result of the
income decrement since it satis¬es the monotonicity axiom.
The duration-adjusted gap measure has a simple expression as the mean of
the entries of the matrix g(™), so that K 1 = Ï(g(™) ) = |g(™)|/(T N). In words, K 1
is the sum of the normalized shortfalls experienced by the chronically poor,
or |g(™)|, divided by T N, which is the maximum value this sum can take. 13
While K 1 is sensitive to magnitude of the income shortfalls of the chroni-
cally poor, the speci¬c way the gaps are combined ensures that a given sized
income decrement has the same effect on overall poverty whether the gap is
large or small. One could argue that a loss in income would have a greater
effect the larger the gap, in which case the square of the normalized gaps,
rather than the gaps themselves, could be used. For example, suppose that the
initial level of income is 4 and the poverty line is 5, so that the normalized
gap is 0.20 and the squared (normalized) gap is 0.04. Decreasing the income
by one unit will increase the squared gap to 0.16, an increase of 0.12. Now
suppose that the initial level of income is 2, so that the normalized gap is 0.60
and the squared gap is 0.36. The unit decrement would raise the squared gap
to 0.64, which represents a much larger increase of 0.28. Using squared gaps,
rather than the gaps themselves, places greater weight on larger shortfalls.
Consider the matrix s(™) whose non-zero entries are the squared normal-
ized gaps of the chronically poor. The number of non-zero entries is |h(™)|
so that the average squared gap over these periods of poverty is given by
S(™) = |s(™)|/|h(™)|. If this partial index is used instead of G(™) to supplement
the duration-adjusted headcount ratio, the resulting chronic poverty index
would place greater weight on larger shortfalls. The resulting duration-adjusted
FGT measure K 2 = K 0 S is a chronic poverty analogue of the usual FGT index
P2 (just as K 0 and K 1 respectively correspond to P0 and P1 of the same
class). K 2 has a straightforward expression as the product of partial indices
K 2 = HDS and as the mean of the entries of the matrix s(™) of squared gaps
K 2 = Ï(s(™) ) = |s(™)|/(TN). It is the sum of the squared (normalized) gaps of the
chronically poor, divided by the maximum value this sum can take.
Referring once again to the numerical example, the matrix of squared gaps
is given by
⎡ ¤
00 0 0
⎢ 0 0.16 0.04 0 ⎥
⎢ ⎥
s(™) = ⎢ ⎥
⎣ 0 0.04 0.36 0 ¦
0 0.16 0.36 0


TN is the value of |g(™)| that would arise in the extreme case where all incomes were 0.
13




69
James E. Foster

and hence K 2 = Ï(s(™) ) =(1.12)/16 = 0.07. Now recall that the income of
3
person 2 in period 3 is y2 = 4, so that a unit decrement in income causes
the squared normalized gap to rise from 0.04 to 0.16, and raising K 2 by
3
about 0.008. In contrast, a unit decrement from y3 = 3 raises the squared
normalized gap from 0.16 to 0.36, and lifting K 2 by about 0.013. With K 2 ,
the impact of a unit decrement is larger for lower incomes than for higher
incomes.
Analogous reasoning demonstrates that K 2 is sensitive to the distribution
of income among the poor. Let i and j be two chronically poor persons with
income vectors yi and y j . Suppose that their income vectors are replaced with
yi = Îyi + (1 ’ Î)y j and y j = (1 ’ Î)yi + Îy j , respectively, for some Î µ (0,1/2].
This represents a uniform ˜smoothing™ of the incomes of persons i and j,
with the value Î = 1/2 yielding the limiting case where yi = y j = (yi + y j )/2
is a simple average of the two vectors. A transformation of this type is a
multidimensional analogue of a progressive transfer (among the poor) and
it is easy to show that K 2 will not rise; indeed, if their associate normalized
gap distributions gi and g j were not initially identical, K 2 would fall as result
of the progressive transfer. 14 In the numerical example, if the income vectors
of persons 2 and 3 are replaced by the average vector (with Î = 1/2), then
K 2 falls from 0.07 to about 0.54. In contrast, this smoothing of incomes
affects neither the average duration of poverty, nor the average shortfall
among the chronically poor, and hence K 0 and K 1 are entirely unaffected. The
property requiring such a transformation to lower chronic poverty is called
the transfer axiom. 15 The measure K 2 satis¬es this axiom while K 0 and K 1 both
violate it.
The general approach to constructing chronic poverty measures can be
applied to obtain analogues of all of the indices in the FGT class. For any · ≥
0 let g · (™) be the matrix whose entries are the · powers of normalized gaps for
the chronically poor (and zeros for those who are not chronically poor). 16 The
duration-adjusted P· measures are the general class of chronic poverty measures
de¬ned by K · (y; z, ™) = Ï(g · (™) ) = |g · (™)|/(TN); in other words, K · is the sum
of the · power of the (normalized) gaps of the chronically poor, divided by
the maximum value that this sum could take. It can be shown that time
monotonicity is satis¬ed by all K · ; monotonicity is satis¬ed by K · for · > 0;
and the transfer axiom is satis¬ed by K · for · >1.
The K · measures satisfy a wide range of general properties for chronic
poverty measures, some of which are direct analogues of the static poverty

14
See Kolm (1977) and Tsui (2002). The condition g i and g j rules out the case mentioned
by Tsui (2002) where the two chronically poor persons are poor in the same periods and have
the same incomes below the poverty line.
15
See Foster (2007) for a rigorous de¬nition of this axiom.
16
For · = 0, the entries of the matrix are more precisely de¬ned as the limit of the entries
of g · (™) as · tends to 0.




70
A Class of Chronic Poverty Measures

axioms and others more explicitly account for the time element in chronic
poverty. 17 One axiom that deserves special mention due to its importance in
empirical work is the following:
Decomposability. For any distributions x and y we have
N(x) N(y)
K (x, y; z,™) = K (x; z, ™) + K (y; z, ™).
N(x, y) N(x, y)
According to this property, when a distribution is broken down into two sub-
populations, the overall chronic poverty level can be expressed as a weighted
average of subgroup chronic poverty levels, with the weights being the respec-
tive subgroup population shares. 18 All the measures in the K · class satisfy
decomposability and hence are well suited for the analysis of chronic poverty
by population subgroup.
For each measure of chronic poverty K · , a corresponding measure of tran-
sient poverty can be de¬ned to account for spells of poverty among those
who are not chronically poor. Applying K · to a cut-off of ™ = 0 removes all
the restrictions concerning the duration of poverty. The resulting quantity
K · (y; z, 0) = Ï(g· ) takes into account every spell of poverty for all persons,
including those who are not chronically poor, and, in fact, is the overall (or
average) level of measured poverty (using P· ) across the T periods. In con-
trast, K · (y;z,™) limits consideration to the shortfalls of the chronically poor,
and hence the difference R· (y; z, ™) = K · (y; z, 0) ’K · (y; z, ™) is an intuitive
measure of transient poverty. The shares of overall poverty that are chronic
K · (y; z, ™)/K · (y; z, 0) and transient R· (y; z, ™)/K · (y; z, 0) are helpful tools for
understanding the nature of poverty over time, and are used in the empirical
application below.


3.4. An Empirical Illustration

In this section, the new chronic poverty measures are applied to panel data
from Argentina™s Encuesta Permanente de Hogares carried out by Instituto
Nacional de Estadísticas y Censos (INDEC), covering 2,409 households during
the four waves of October 2001, May 2002, October 2002, and May 2003. The
income variable used is equivalent household income, calculated by dividing
total household income in each period by the number of equivalent adults
(using the equivalent adult scale provided by INDEC). 19 The Instituto also
provides a separate poverty line for each region and each period in order to
17
The ¬rst group of axioms includes anonymity, replication invariance, focus, and subgroup
consistency; the second includes time anonymity and time focus. See Foster (2007).
18
This expression can be generalized to any number of subgroups by repeated application
of the two-subgroup formula.
19
In periods where an income is missing or zero for a given household, the methodology
of Little and Su (1989) is followed, which calculates an imputed income using the household™s
incomes at other dates and other household incomes at the same date. In the few cases where




71
James E. Foster

Table 3.1. Chronic poverty in Argentina: estimated levels for various measures
and durations

Measure

H K0 K1 K2

™ = 0.25 0.61 0.44 0.21 0.124
™ = 0.50 0.50 0.42 0.20 0.122
™ = 0.75 0.40 0.37 0.19 0.116
™ = 1.00 0.27 0.27 0.15 0.096
% Chronica 66.3 83.1 90.7 93.5
% Transient 33.7 16.9 9.3 6.5

a
The duration cut-off is ™ = 0.75. Since there are four periods, ™ = 0.25 yields the same values
. .
as overall poverty, and Percentage Chronic can found by dividing the third row by the ¬rst row.
Percentage Transient is 100 minus Percentage Chronic.


capture spatial variations in the cost of living as well as in¬‚ation over time.
Normalized gaps for a household are found by subtracting the equivalent
income from the appropriate poverty line and dividing this difference by the
same line. Since there are T = 4 periods, there are four relevant duration cut-
offs ™, namely, 0.25, 0.50, 0.75, 1.00. Table 3.1 provides the resulting levels of
chronic poverty for each of the measures H, K 0 , K 1 , and K 2 , at each of the
four duration cut-offs.
Notice that for each index, the measured level of chronic poverty rises as the
duration requirement ™ falls. For ™ = 1, the headcount and duration-adjusted
headcount have the same value since the average duration in this special case
is precisely 1. In general, for ¬xed ™ the value of H is higher than that of K 0 ,
which in turn is higher than K 1 , and so forth, re¬‚ecting the mathematical
properties of the measures. However, it is important to remember that actual
poverty comparisons should only be made using the same poverty measure
and with the same duration cut-off.
The ¬rst row provides the chronic poverty level when the duration in
poverty is at least 0.25, and for K · this is clearly the same as the level when
the minimum duration is 0. This latter level K · (y; z, 0) is precisely the static
poverty level that would arise if the data from all periods were merged into
a single distribution and evaluated by P· , or equivalently, t P· (yt ; z)/T, the
average static poverty level across all periods. Now given a minimum duration
of ™ = 0.75, the difference between this overall poverty level K · (y; z, 0) and the
chronic poverty level K · (y; z, ™) is the level of transient poverty, R· (y; z, ™) as
de¬ned above. 20 The ¬nal two rows provide the percentage of overall poverty
due to chronic poverty and transient poverty respectively for ™ = 0.75 given

the household reports zero income in all four periods, an income equal to the lowest social
welfare transfer was assumed.
20
For H, overall poverty is the limit of H(y; z, ™) as ™ tends to zero; this also equals the value
at ™ = 0.25.




72
A Class of Chronic Poverty Measures

Table 3.2. Chronic poverty pro¬le for Argentina

Measurea

Pop. share (%) K0 % contrib. K1 % contrib. K2 % contrib.

GBA 12 0.18 8.3 0.10 8.0 0.065 8.1
NE 15 0.44 25.2 0.26 26.3 0.174 27.1
NW 22 0.34 28.4 0.19 28.1 0.120 27.4
C 28 0.22 23.5 0.13 24.1 0.085 24.6
MW 11 0.24 10.0 0.13 9.6 0.080 9.1
S 12 0.10 4.6 0.05 4.0 0.029 3.6
Total 100 0.27 100.0 0.15 100.0 0.096 100.0

a
Duration cut-off is ™ = 1.00. Percentages may not sum to 100% due to rounding errors.


each measure. It is interesting to note that as the poverty measure changes
from H to K 0 to K 1 to K 2 , the share of chronic poverty becomes larger
while the share of transient poverty falls. For K 2 the chronic poverty share
is 93.5 per cent, suggesting that the transient poverty spells generally do not
involve large shortfalls.
Table 3.2 provides a pro¬le of chronic poverty in each of the six regions
of Argentina, namely, Greater Buenos Aires (GBA), the North-East (NE), the
North-West (NW), the Center of the country (C), the Midwest (MW), and the
South (S). In this example, chronically poor means poor in all four periods;
hence the duration cut-off is ™ = 1. The ¬rst column in the table gives the pop-
ulation shares of the six regions. The second, fourth, and sixth columns pro-
vide the regional chronic poverty levels for K 0 , K 1 , and K 2 , respectively; while
the third, ¬fth, and seventh columns give the percentage contribution of each
region to total poverty (or the population share times the regional poverty
level over the total poverty). Notice that the North-East has only 15 per cent
of the population, yet accounts for at least 25.2 per cent of the chronic poverty
in Argentina; in contrast, the South has 12 per cent of the total population,
but only contributes 4.6 per cent or less to total chronic poverty.



3.5. Conclusions

This chapter has presented a new family K · of chronic poverty measures based
on the P· measures of Foster, Greer, and Thorbecke (1984). Each measure has
an intuitive interpretation as a ˜duration-adjusted P· measure™ and can be
readily calculated as the mean of a particular matrix. All the measures satisfy
a series of general axioms for chronic poverty measures, and an additional
set of axioms that are applicable to measures based on a spells approach
to chronic poverty. Within the class, there is a range of measures satisfying
a monotonicity axiom, and a smaller subset satisfying a transfer axiom.



73
James E. Foster

In particular, K 2 , the duration-adjusted analogue of the usual P2 measure,
satis¬es all of the properties discussed in this chapter. The usefulness of the
class of chronic poverty measures has been illustrated using panel data from
Argentina.
One powerful criticism of the general framework for evaluation used here is
that, being limited to a single income variable, it cannot utilize data on other
dimensions of relevance to poverty and well-being. The approach is indeed
restrictive, and when panel data containing information on other capabilities
become more widely available, multidimensional measures of chronic poverty
will need to be developed. Of course, there remain serious dif¬culties in
formulating effective multidimensional poverty measures, even in the sta-
tic context. The identi¬cation step for multidimensional poverty depends
crucially on assumptions about substitutability across different dimensions;
and the aggregation step has seemingly endless possibilities for bringing
the various dimensions of deprivation into a single coherent index. The
problem of measuring multidimensional chronic poverty is not likely to be
solved anytime soon.
The measurement approach presented in this chapter is based on a very sim-
ple treatment of time in poverty: an earlier period in poverty is given the same
weight as a later period in poverty. Indeed, such a position could be justi¬ed
using the traditional ˜no discounting™ argument of Ramsey (1928). However,
other plausible alternatives exist. For example, one might argue that greater
weight should be placed on income received in earlier periods, as is typically
done by discounting (see Rodgers and Rodgers, 1993, or Dercon and Calvo,
2006). This could be justi¬ed ex ante where there may be greater perceived
or actual value from receiving income earlier. However, when viewed from an
ex post perspective, there may also be good reason to view earlier incomes as
having less weight (see, for example, the discussion in Ray and Wang, 2001).
An additional problem with discounting (one way or the other) is deciding
what to discount: the income received by the chronically poor person, the
utility level, or some function of the gap. And if the discounting procedure
delivers a different ordering depending on the base period for evaluation, this
could certainly make the procedure less attractive.
There are other aspects of the above approach that might be altered. Some
have argued that continuous spells of poverty should have larger weight in the
evaluation of chronic poverty. As noted by Mckay and Lawson (2002), there
are signi¬cant data problems from trying to implement this using panel data.
Another perspective might consider placing greater weight on observations
from a chronically poor person who is in poverty longer. This could indeed be
done either via a new functional form or through dominance orderings over
time, but this will have to await future work.




74
A Class of Chronic Poverty Measures

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76
4
Measuring Chronic Non-Income
Poverty—
Isabel Günther and Stephan Klasen




4.1. Introduction

In recent years, the research agenda on poverty in developing countries has
moved beyond static assessments of poverty levels to consider dynamic tra-
jectories of well-being over time. The main reason for this shift in emphasis
was the recognition that there is considerable mobility of well-being over
time and that only a share of the poor are affected by persistent (or chronic)
poverty, while a much larger share of the total population experiences tran-
sient poverty, or vulnerability to poverty.
Since the two groups were found to be quite different in terms of their
characteristics and in terms of their needs regarding policy interventions, the
research community has developed two largely distinct research agendas, one
focusing on chronic poverty, the other focusing on vulnerability to poverty.
The research agendas complement one another, with chronic poverty focus-
ing on poverty traps and poverty persistence and vulnerability focusing on
risks and shocks and poverty dynamics.
The distinction between chronic and transient poverty is usually closely
linked to conceptualizing poverty in the monetary dimension. This is largely
related to the fact that the stochastic nature of the income-generating process
is well recognized in economics for decades, going back to Friedman™s Perma-
nent Income Hypothesis, which already made a distinction between perma-
nent and transitory incomes (Friedman, 1957). In line with that hypothesis,
consumption is used as the preferred welfare indicator in many applications
in developing countries as it is believed to be a better re¬‚ection of long-term


We would like to thank Bob Baulch, Stephan Dercon, Michael Grimm, David Hulme,
Stephen Jenkins, John Mickleright, an anonymous referee, and participants at a workshop in
Manchester for very useful comments on a ¬rst draft of this chapter.




77
Isabel Günther and Stephan Klasen

or permanent incomes. 1 In this sense, low consumption (i.e. consumption
below a poverty line) is seen as a re¬‚ection of a chronic inability to generate
suf¬cient incomes to leave poverty, even though households might temporar-
ily escape income poverty.
But empirically it has been shown that in developing countries also house-
holds™ consumption ¬‚uctuates greatly, in fact often not much less than
income. This could be for three reasons. First, households, particularly poor
households, are not able to smooth their consumption due to a lack of assets
and access to credit and/or insurance markets (e.g. Townsend, 1995; Deaton,
1997). Second, ˜permanent™ incomes of households change as a result of per-
manent shocks affecting the lifetime earning paths of individuals, thus forcing
households to re-optimize their consumption decisions. Last, consumption
(and/or incomes) is measured with error and thus much of the ¬‚uctuation is
spurious and related to these errors. 2
To ¬gure out which households are facing permanently low consumption,
i.e. are chronically poor, and which households are ˜only™ transitory poor and
which households (currently non-poor) are facing a high risk of becoming
poor is thus a very important and at the same time quite dif¬cult task and it
is not surprising that a large literature has dedicated itself to this subject. And
with the help of an increasing number of panel data in developing countries,
dynamic assessments of consumption, i.e. an analysis of chronic poverty as
well as vulnerability to poverty, have indeed become more feasible in an
increasing number of countries, thus underpinning the analysis of poverty
dynamics.
At the same time, this exclusive emphasis on incomes in the assessment
of chronic poverty and vulnerability has clear limitations and shortcomings
(see also Hulme and McKay, 2005), as it is well recognized that income (or
consumption) is an inadequate indicator of well-being. If we conceptualize
well-being from a capability perspective, income is but one (and for some
capabilities a rather poor) means to generate capabilities such as the ability
to be healthy, well educated, integrated, clothed, housed, and the like (see
Sen, 1995, 1999; Klasen, 2000); nor do equal incomes translate into equal
capabilities for different individuals, due to the heterogeneity of people in
translating incomes into well-being. It is therefore preferable to study well-
being outcomes directly (e.g. capabilities or functionings; see Klasen, 2000 3 )

1
There are other reasons to prefer consumption to incomes as a welfare measure in
developing countries. See Deaton (1997) and Deaton and Zaidi (2002).
2
This is a dif¬cult issue to sort out with the type of panel data available for developing
countries which typically have only two or three waves and thus do not allow the applica-
tion of common methods to control for measurement error (such as instrumental variable
techniques).
3
In principle, it is preferable to study capabilities to understand the choices people have
at their disposal. In practice, we usually can only observe functionings and thus most studies
are analysing functionings instead of capabilities (e.g. Klasen, 2000).




78
Measuring Chronic Non-Income Poverty

rather than study a speci¬c well-being input. However, there have been few
attempts to integrate the insights from the static analysis of non-income
dimensions of well-being into a dynamic setting and thus investigate chronic
poverty and vulnerability from a non-income perspective. In addition, apart
from the conceptual advantage of studying chronic poverty from a non-
income perspective, there are several advantages (but also limitations) from
a measurement perspective to studying non-income chronic poverty, which
we discuss in more detail below.
The purpose of this chapter is to try to conceptualize chronic poverty
and hence also poverty dynamics from a non-income perspective and then
illustrate ways to explore this topic empirically. Section 4.2 discusses the
potentials as well as limitations of conceptualizing chronic poverty in a non-
income perspective. Section 4.3 presents a ¬rst approach to empirically mea-
sure chronic non-income poverty, focusing on critical functionings related
to health and education, using a panel survey of Vietnam from 1992/3 and
1997/8. Section 4.4 shows the results of this application. Section 4.5 concludes
with highlighting open issues and suggestions for further research.


4.2. Conceptualizing Chronic Non-Income Poverty

It is clear that in principle it should be useful to study chronic poverty in
non-income dimensions (using for example applications of Sen™s capability
approach) as it would allow us to track well-being outcomes rather than simply
track an important well-being input (income) over time. Thus it would allow
us to measure well-being itself rather than only a proxy of it. The same
theoretical reasoning to prefer non-income to income indicators to measure
well-being as in a static framework certainly applies in a dynamic well-being
framework (see e.g. Sen, 1985). In addition, there are some speci¬c advan-
tages (and limitations) of studying poverty using non-income indicators that
emerge particularly in a dynamic poverty framework.


4.2.1. Potentials
Analysing non-income poverty dynamics would ¬rst of all allow an assess-
ment of the relationship between income and non-income chronic and
transitory poverty. Identifying those households where the two approaches
converge would identify those households who are chronically poor from a
multidimensional perspective and thus possibly most deprived and arguably
most deserving of support. This would enrich our assessment of dynamic well-
being. Conversely, where the two approaches fail to converge in identifying
the chronic poor, we would learn more about the dynamic relationship
between income and non-income poverty. This is directly interesting for



79
Isabel Günther and Stephan Klasen

policy purposes as policy makers are interested in reducing income and non-
income poverty and thus knowing the temporal relationship between the
two, e.g. whether improvements in income will eventually improve health
outcomes (but only with a lag), or vice versa, is critical.
The measurement of non-income poverty dynamics might also shed some
new light on the causes of the less than perfect correlation between income
and non-income dimensions of poverty in a static framework (see e.g. Klasen,
2000). In particular, the lack of correlation at one point in time might be
related more to different dynamics of the two well-being approaches, rather
than the lack of a contemporaneous causal relation between the two. More
precisely, for example in a two-wave panel, static assessments of poverty
in both periods could yield the same result regardless of whether income
and non-income dimensions are used. However, the two (income and non-
income) approaches could also agree in the static assessment of poverty in
the ¬rst period, but differ in the dynamics between the ¬rst and second
periods, suggesting that different drivers affect these dynamics. Similarly, the
two approaches could agree in the static assessment of poverty in the second
period, but differ in the dynamics, and thus would not agree in the static
assessment of poverty in the ¬rst period. Last, the two approaches might also
disagree on classifying households in both periods but agree on the dynam-
ics over time. Thus analysing income and non-income poverty dynamics
simultaneously we are able to separate static and dynamic disagreements in
identifying the poor. 4 If we only examined the two periods separately, we
would either ¬nd a lack of overlap in the ¬rst period or a lack of overlap
in the second period. But we would not be able to tell whether this is due
to different dynamics between two periods or whether there is a permanent
disagreement between the two approaches to identifying the poor.
However, even if it turned out that chronic income and non-income
poverty dynamics are highly correlated, there could still be practical advan-
tages focusing on the measurement of non-income poverty, as many indi-
cators of non-income deprivation (e.g. education or housing) are easier to
measure and less prone to measurement error than income (or consumption)
measures. 5 In fact, at times it may be useful to use non-income measures
of well-being as instruments to correct poorly measured incomes (and or
consumption).
A second measurement advantage is that information on past dynamics
of non-income well-being are often easier to get and more reliable than
information on past income series”even when using cross-sectional surveys.
For example, it is easier to get reliable information about the educational
4
If we had more waves, we could also say more about the temporal relationship between
the two variables by explicitly examining leads and lags.
5
See, for example, Zeller et al. (2006) for an example of a short-cut approach to poverty
measurement using non-income indicators.




80
Measuring Chronic Non-Income Poverty

history of a person than that person™s income history. Moreover, some current
non-income indicators can already provide some information about historical
trends in access to critical functionings. For example, the height of an adult
re¬‚ects past nutritional status and the current grade enrolled for a child
at a certain age reveals important aspects of that child™s past educational
history.
In addition, many capabilities/functionings (e.g. education and health) can
be measured at the individual level while income/consumption poverty can
only be assessed at the household level, due to the presence of household-
speci¬c public goods which are impossible to attribute to individual members

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