This note warns about the careless computation of poverty indexes when the welfare of each household is measured by its net income, since this can be negative. As is illustrated in the case of Mexico, even if only a handful of households report negative incomes, the resulting poverty aggregates, when they go beyond a mere headcount measure, can behave rather badly. The note ends with suggestions on how to deal with the problem.Poverty measures, poverty indexes, negative income, income and expenditure survey, Mexico

Sandoval, Héctor H.

Urzúa, Carlos M.

English

Research Papers in Economics

Revista de Administraci´ on, Finanzas y Econom´ ıa (Journal of Management, Finance and
Economics), vol. 3, n´ um. 1 (2009), pp. 29-36.
Negative Net Incomes and the Measurement
of Poverty: A Note
H´ ector H. Sandoval∗
Carlos M. Urz´ ua ∗∗
Recibido 11 de enero 2008, Aceptado 20 de enero 2008
Abstract
This note warns about the careless computation of poverty indexes when the
welfare of each household is measured by its net income, since this can be
negative. As is illustrated in the case of Mexico, even if only a handful of
households report negative incomes, the resulting poverty aggregates, when
they go beyond a mere headcount measure, can behave rather badly. The note
ends with suggestions on how to deal with the problem.
Resumen
Esta nota advierte sobre el c´ alculo apresurado de ´ ındices de pobreza cuando
el bienestar de cada hogar es medido por su ingreso neto, pues ´ este puede ser
negativo. Como se ilustra en el caso de M´ exico, aun si s´ olo un pu˜ nado de hogares
reportan ingresos negativos, los resultantes agregados de pobreza, cuando van
m´ as all´ a de un mero conteo de cabezas, podr´ ıan comportarse de manera muy
an´ omala. La nota termina con sugerencias acerca de c´ omo tratar el problema.
Clasiﬁcaci´ on JEL: : I32, C81
Keywords: Poverty measures, poverty indexes, negative income, income and expenditure
survey, Mexico
1. Introduction
Poverty measurement is a delicate task. To start with, as ﬁrst noted by Sen
(1976), the researcher has to solve two important problems: How to identify
the poor among a given population, and how to aggregate the corresponding
data into a reasonable index. For instance, rightly or wrongly, in the case of
Mexico the oﬃcial poverty statistics are based on absolute poverty lines, while
the only aggregate poverty measure that is employed is headcount (see Comit´ e
T´ ecnico para la Medici´ on de la Pobreza, 2002). Once those two fundamental
problems mentioned by Sen are solved, there are still other important issues to
be dealt with by the researcher. To give just two examples: Should household
welfare be measured using income or consumption data? Also, should children
∗ Consejo Nacional de Evaluaci´ on de la Pol´ ıtica de Desarrollo Social.
∗∗ Tecnol´ ogico de Monterrey, Campus Ciudad de M´ exico. e-mail: curzua@itesm.mx Tel.:
+52(55)5483 1878.30 Revista de Administraci´ on, Finanzas y Econom´ ıa
and adults needs be treated the same? For instance, in the case of Mexico
the oﬃcial methodology identiﬁes household welfare with net income, while it
ignores the second question.
To those theoretical problems described above, one can add some statistical
issues. Two questions come immediately to mind in particular: How to estimate
in a robust way the standard errors of the resulting poverty measures, and, if
welfare is identiﬁed with income, how to treat the negative net incomes at the
bottom of the distribution. The former question is a bit less ordinary than it
would seem at ﬁrst sight (see, e.g., Urz´ ua et al., 2008, and references therein),
while the latter is certainly easy. But, as this note shows, such an obvious point
can be nevertheless quite important for applied work.
Section 2 presents the algebraic consequences of calculating poverty
measures when some of the net incomes are negative (we shall always assume in
this paper that welfare is identiﬁed with income, and not with consumption).
After that, section 3 illustrates, using several diﬀerent Mexican income and
expenditure surveys, how badly some poverty indexes may 3 behave if the net
incomes of some of the households are actually negative. Finally, the last section
comments on possible ways to deal with the problem.
2. Negative Net Incomes and Poverty Measurement
There could be several reasons for a household to have a net income that is
negative. Aside from the possibility of misreporting, the two most important
factors would be, ﬁrst, that the business of a self-employed member of the
household might have made a loss during the period under consideration, and,
second, that the transfers made by the household to other households might
have exceeded its total income.
But regardless of the reasons that are behind that fact, the point to be
made here is that almost all poverty measures cannot cope with the possibility
of some incomes being negative. In the case of the headcount index the problem
is obviously innocuous, since a household is counted as poor whenever its net
income, be that positive, zero or negative, is below some positivenumber. Going
to the other extreme, there are poverty measures that are, from the start,
undeﬁned when some incomes are negative. This is the case, for instance, of
the Watts index (Watts, 1968), which requires the computation of the logarithm
of each income, so that this number cannot be less or equal than zero.
There are, nevertheless, other indices that could be computed numerically
even if some of the incomes are negative. Yet, those poverty measures could
behave abnormally in such extreme cases. In order to illustrate the point, we
will consider in this paper the popular Foster- Greer-Thorbecke (FGT) class of
poverty measures (Foster et al., 1984):
Pα(z)=
1
n
q 
i=1

z − yi
z
α
,α ≥ 0,
where q is the number of poor people in a population of size n, and where the
i−th member has an income yi which is less or equal than the poverty line
z. The headcount index mentioned earlier is obtained when α = 0, while theNegative Net Incomes 31
(relative) poverty-gap measure is found when α =1 :
P1(z)=
1
n
q 
i=1

z − yi
z

.
Already in this case, if yi < 0f o rs o m ei, then that particular poverty-gap ratio
would be (incorrectly) found to be greater than one. This anomalous behavior
is exacerbated as α is further increased, as, for instance, in the case of the
squared poverty-gap index given by:
P2(z)=
1
n
q 
i=1

z − yi
z
2
.
But the consequences of having negative incomes become really worrisome when
one compares diﬀerent FGT measures. If yi ≥ 0 for all i,t h e n
P0(z) ≥ P1(z) ≥ P2(z)) ≥ P3(z) ≥ ... (1)
(the equalities will hold if yi = 0 for all households). This is so simply because

z − yi
z
α
≥

z − yi
z
β
if 0 ≤ α<β .
However if yi < 0 for at least some i, it could be the case, as an example in the
next section will illustrate, that some of the inequalities given in (1) above are
reversed.
Furthermore, as it will be also exempliﬁed in the next section, if some
of the incomes are negative, it could be even the case that Pα(z1) >P α(z2)
when z1 <z 2. That is, even though one would expect as a rule that a poverty
index would decrease as the poverty line is lowered, in some rare instances the
opposite could happen if there were enough negative net incomes. This is so
because if y<0, then (z1 − y)/z1 would be greater than (z2 − y)/z2.
3. Examples Using Mexican Income and Expenditure Surveys
Before going through the particular examples, it is worth to describe very brieﬂy
the way in which the oﬃcial Mexican poverty statistics are produced. For that
end, the authorities use income and expenditure surveys (known as ENIGHs,
by their acronym in Spanish), which are mostly made each even year. Given
a particular survey, the monetary and non-monetary incomes are ﬁrst added-
up for each household (the latter income includes self-consumption, inkind
payments, imputed rents and gifts). After calculating total current income,
net income is obtained by subtracting the transfers made by the household to
other households. Next, expansion factors are used to represent the entire set of
Mexican households or, which is more common, the entire Mexican population
(this is the alternative that we will follow here). Finally, the headcount ratio is
used to report poverty incidence according to three diﬀerent poverty deﬁnitions.
These are: “food poverty”, when net income is too low to cover basic food32 Revista de Administraci´ on, Finanzas y Econom´ ıa
necessities; “capabilities poverty”, when net income is insuﬃcient to buy basic
food, education and health necessities; and “assets poverty”, when net income
is too low to cover basic food, education, health, dressing, housing and public
transportation necessities.
Returning to the purpose of this note, one might now wonder whether
or not there are households that indeed report negative net incomes in the
ENIGHs. For the authorities this issue seems to be irrelevant since their only
interest is on poverty incidence. Thus, one has to go back to each of the surveys
to identify those outliers. Using the ENIGHs corresponding to the years 1992,
1994, 1996, 1998, 2000, 2002, 2004 and 2005, Table 1 presents the corresponding
results.
Table 1
Population with Negative Net Incomes
Year Observations Expanded
(Households) (Individuals)
1992 2 21,660
1994 10 13,352
1996 7 10,983
1998 20 129,961
2000 4 24,568
2002 12 34,477
2004 38 139,134
2005 9 19,227
Source: Own calculations based on the ENIGHs.
As can be appreciated from the table, even though there are a relatively small
number of households reporting negative net incomes in each ENIGH, once the
observations are expanded to cover the entire Mexican population, the numbers
become sizable. For instance, in the years 1998 and 2004 more than one percent
of the population was supposed to have had negative incomes. Taking those
observations at their face value, Figure 1 presents three diﬀerent measures of
food poverty in Mexico from 1992 to 2005: the headcount ratio (which coincides
with the oﬃcial estimates), the poverty gap ratio and the squared poverty gap
ratio. As can be seen there, we are now able to exemplify with real data the
reversal of some of the inequalities in (1) above.
How can we avoid that anomaly? In the last section of this note we present
our preferred method for correcting for those negative incomes, but here
we simply solve the problem by replacing each negative income with a zero.
By doing so, we continue to obtain in particular the oﬃcial poverty incidence
statistics. The contrasting, well-behaved Figure 2 presents the results thus
obtained.
An even more anomalous behavior is illustrated in Figure 3, which shows,
in the years 1996, 2004 and 2005, how the squared poverty gaps could become
larger as the poverty lines become smaller! Once again, the purported paradox
could be solved by replacing the negative net incomes with zero values.Negative Net Incomes 33
There is still another lesson to be drawn from this exercise. Until
recently, most studies that reported poverty measures tended to do so without
any reference on the precision of those statistics. Fortunately, that practice
has become less frequent nowadays, especially after Kakwani (1993) showed,
by means of the time honored delta-method, how to derive the approximate
sampling variance for the class of FGT poverty indices. The approximation is:
Figure 1
Unadjusted Food Poverty in Mexico, 1992-2005
Figure 2
Food Poverty in Mexico, 1992-200534 Revista de Administraci´ on, Finanzas y Econom´ ıa
Figure 3
Unadjusted Squared Poverty Gaps in Mexico, 1992-2005
var(Pα) ≈
P2α − P 2
α
n
.
Note, however, that Kakwani’s approximation may not be always suitable,
since it implicitlyassumes a normal distribution for the poverty measures, which
actually range only from 0 to 1. Furthermore, if some of the net incomes are
negative, then, as we have illustrated before, the FGT measures, and hence its
approximate sampling variances and corresponding conﬁdence intervals, could
behave rather badly. Using once again real data, Figure 4 illustrates this point.
As can be appreciated in the ﬁgure, the normal approximation would lead us to
have, in the years 1996 and 2005, negative bounds for the conﬁdence intervals
of squared poverty gaps! It is worth to mention in passing that, aside from the
possible corrections for negative net incomes, the best way to solve this last
conundrum is to use resampling methods instead of normal approximations
(see, e.g., Deaton, 1977, and Urz´ ua et al., 2008).
4. Conclusions
Throughout this paper we have shown the anomalous implications that negative
net incomes might have in the measurement of poverty. The way in which we
solved those anomalies was to set the negative incomes equal to zero. Although
this procedure has the advantage of rendering the same poverty incidence
statistics as the oﬃcial ones, it certainly contaminates the poverty measures
that go beyond a mere headcount ratio.
What to do then? Clearly, the authorities will have to change their own
methodology to account for those outliers. One possibility to do so, the one
that we recommend most, is to identify household welfare not with income, but
with consumption. This methodological change is also proposed, for even more
important reasons, by Ruiz-Castillo (2005).Negative Net Incomes 35
Barring such a replacement of income with consumption, we would
recommend to explore as an alternative procedure the replacement of net
incomes with net expenditures in the case of households that report negative
values. We believe that Table 2 provides good reasons in favor of such a
procedure. As can be appreciated from that table, most of the households
that report negative net incomes tend to have expenditure levels that would
correspond to the case of much higher income levels. In all the ENIGHs
considered here, the mean of net expenditures of those households is
substantially higher than the urban (food) poverty line that is given in the
last column. Furthermore, in the years 1996, 2000, 2004 and 2005, all the
households had expenditure levels above the poverty line. Hardly an evidence
of penury.
Figure 4
Conﬁdence Intervals for Unadjusted Food Poverty Gaps
Table 2
Net Expenditures of Households with Negative Incomes
Year Observations Mean Min Max Poverty Line
1992* 2 $1,221 $169 $1,250 $168
1994 10 $6,632 $44 $9,292 $193
1996 7 $3,175 $1,349 $8,255 $389
1998 20 $5,163 $102 $13,867 $524
2000 4 $3,352 $2,070 $4,064 $627
2002 12 $3,187 $264 $10,147 $672
2004 38 $6,309 $807 $140,358 $740
2005 9 $10,584 $4,065 $83,783 $791
Source: Own estimates based on the ENIGHs.
*Old pesos converted to new pesos.36 Revista de Administraci´ on, Finanzas y Econom´ ıa
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Negative Net Incomes and the Measurement of Poverty: A Note

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