POLICY RESEARCH WORKING PAPER 11332
Poverty and Household Size
the relationship between
poverty and household size
Peter Lanjouw should be interpreted wit
Martin Ravallion cautfon. The empirical
relationship is fragile and-
espedail sensitive to
differences in the welfare
indicator used-in measUring-
povefl:
The World Bank
Policy Research Department
Poverty and Human Resources Division
August 1994
I POLICY RESEARCH WORKING PAPER 1332
Summary findings
The widely held view that larger families tend to be By contrast, some measures of child nutritional sratus
poorer in developing countries has influenced research indicate anJelasticiry closer to unity.
and policies. But the basis for this 'stylized fact" is Consideration of the weight attached to child versus
questionable, argue Lanjouw and Ravallion. Widely cited adult welfare may help resolve the nonrobustncss of
evidence of a strong negative correlation between size demographic profiles of poverty. The authors show that
and consumption per person is unconvincing, given that the incidence of severe child stunting is more elastic to
even poor households face economies of size in household size than their Engel curve estimate suggests,
consumption. although the latter is still a fair predictor of child
Lanjouw and Ravallion find that the correlation wasting.
between poverty and household size vanishes in Pakistan A consideration of the purpose of measuring poverty
when the size elasticity of the cost of living is about 0.6. - notably the extent to which it is used to inform
This turns out to be the elasticity implied by a modified policies aimed at promoting child welfare - may go
version of the food-share method of setting scales. some way toward resolving the issue.
This paper --a product of the Poverty and Human Resources Division, Policy Research Department-is part of a larger
effort in the department to assess the robustness of policy conclusions *o measurement assumptions. Copies of the paper
are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Patricia Cook, room
N5-047, extension 33902 (34 pages). August 1994.
The Poficy Research Working Paper Series dissminates the findings of work in progress to encourage the excbange of ideas about
development issues An obective of the series is to get the findings out quickly, even if the prenations ar ls than fslly polished. The
papers cary the names of the athors and should be used and cired accordingly. The fndYngs, interpretations, and condzzsions are the
authors' owm and should not be attributed to the World Bank its Executive Board of Directors, or any of its member countries.
Produced by the Policy Research Disseminarion Center
Poverty and Household Size
Peter Lanjouw and Martin Ravallio&
* Policy Research Deparment, World Bank
These are the views of the authors, and should not be attrbuted to the World Bank. For their
comments on the paper, the authors are gratefil to Jean Draze, Paul Glewwe, -Stephen Howes,
Jean Olson Lanjouw, Michael Lipton, Lant Pritchett, Etienne van de Walle, Dominique van de
Walle, and seminar participants at Yale University. They are particularly grateful to Stephen
Howes for pointing out a computational error in a previous version.
1 Introduction
There is considerable evidence of a strong negative correlation between household size
and consumption (or income) per person in developing countries.' It is often concluded that
people living in larger and (generally) younger households are typically poorer. There has been
much debate on which is the 'cause" and which is the "effect" in this correlation. The position
one takes in that debate can have implications for policy, including the role of population policy
in development, and the scope for fighting poverty using demographically-contngent transfers.
The existence of size economies in household consumption cautions against concluding
that larger families tend to be poorer? Certain goods (water taps, cooking utsils, clothing,
and housing) allow possibilities for sharing such that the cost per person of a given standard of
living is lower when individuals live together than aparL However, while economnes of size may
be common in rich countries3, it is often argued that such economies are a minor issue in poor
countries, since a large share of the budget is devoted to rival goods such as food. Against this,
it can be argued that-even when jointly consumed goods account for a small budget share-the
large household sizes found in many developrg countries allow non-negligible benefits from
sharing the cost of such goods. Possibly the scope for size economies in developing countries
has been understated.
Despite extensive work on welfare measurement in economics, there is still no preferred
method for making inter-personal comparisons across households of different size and/or
composition. Household data on demands and supplies are often used to estimate how
demographic variables influence the cost of a given utility level (on the theory see Deaton and
Muellbauer, 1980). It is now recognized that the empirical le on of such utility-based
methods of welfare measurement ultimately rests on untestable idenifying assumptions.4 In
1
general there will exist more than one possible set of utility functions for household members
which can account for their observable demands and supplies. For example, the interpretation of
demographic differences in household demand behavior as welfare-relevant differences in needs is
problematic; the same demand data may equally well be explained by intra-household
inequalities. The need to distinguish adult from child welfare-and the possibility of a tension
between the two-has also motivated concern about empirical welfare measures used in both
research and policy (Nelson, 1993). And, even without utility-identification problems, there are
grounds for dispute about whether utility' is the appropriate concept for anchoring scales, or
making interpersonal comparisons generally (Sen, 1985).
The choice of a welfare measure, including an equivalence scale, is ultimately based on
value judgements about which differences of opinion must be expected. This alone should make
one cautious about the statements one often hears concerning the relationship between poverty
and household size. However, the way in which the choice of scale alters poverty comparisons
has received very little attention?' For many purposes for which a demographic profile of
povert is required (such as designing a family allowance scheme, or some other form of
"demographic trgeting" such as subsidized family planning services), it is how the measurement
issue affects the povery orrdering of demographic groups that matters most.
This paper argues tests the robustness of statements about the relationship between
poverty and household size. We begin by showing that, for a broad class of poverty measures
and sufficient dispersion in household sizes, the problem can be reduced to that of whedher or not
the value of a size parameter exceeds a unique critical value (section 2). The key question is
then whether or not one believes that the true value of that parameter is above or below this
critical value. We then estimate that critical value for Paldstan (section 3), and compare its value
2
to a range of estimates that may be deemed defensible for Pakistan (section 4). As in other
developing countries, past practices for Pakistan have typically assumed that the cost of a given
level of welfare is directly proportional to household size and (hence) that a per capita
normalization of total expenditures is appropriate. Taking a deliberately eclectic approach, we
test that assumption using various methods. Our conclusions can be found in section 5.
2 The critical size elasticity for rank reversal
We consider the class of equivalence scales whereby the money metric of a consumer's
welfare has an elasticity B with respect to household size (which we term the "size elasticity").
The welfare of a typical member of any household is then measured in monetary terms by
guno (001) where x denotes total household consumption expenditure, and n denotes
household size; is6 can be interpreted as the equivalent number of single-persons.
This class of scales has been widely used over many years (Prais and Houthakker, 1955;
Singh 1972; Buhmann et al., 1988; Coulter et al. 1992), though it is clearly quite special. There
are other possibilities, such as normalizing consumptions by 1+0(n-1) (also considered by
Coulter et al., following O'Higgins et al. 1989). Our main results in this section can be shown
to hold for that specification, and (indeed) a more general class of scales, though we will confine
ourselves to the simple iso-elastic scale here. In the empirical work we shall allow for other
differences in household circumstances (such as in demographic composition, and the prices
faced), but for the exposition in this section we shall assume that the households being compared
are homogeneous in other respects. (For example, one is comparing large and small households
amongst those of a given demographic composition or living in a given region.) However, as we
3
shall show for our data, by at least one common method of setting scales the compositional
effects are insignificant, and so the single parameter scale is also defensible empirically.
Consider two possible household sizes, ns ("small") and nL ("large") with nL>ns. (Two
groups is unrestrictive, as we shall only need to make binary comparisons amongst the numerous
demographic groups.) A poverty line is given for small households (say single adults), and this
is denoted zs. The equivalent poverty line for large households is then zL(O)=zCf(nlnS)I. The
distribution functions of household consumption for the two groups are F. and FL, with
corresponding density functions fs and fL.
We shall allow a broad class of poverty measures. In particular, we follow Atkinson
(1987) in considering additive measures of the form (for group i):
PXZ) = fp(X,z,ipi(x)d (i=SL) (1)
0
for which pj(xz)sO, p/xz) >O and p(z)O, with at least one of the latter two conditions holding
with strict inequality (implying that pi'(Z)>O).6 Values of x and z are associated with a measure
of poverty p(x,z), and this function is non-increasing in x and non-decreasing in z. There are
many examples of this class of measures. The widely used head-count index (H) is the
proportion of ihe population for whom the welfare metric is less than the poverty line; H1=FIz),
obtained by setting p(x,z) 1 in equation (1). The poverty gap index is obtained by setting
p(x,z) = 1-xz (so the measure is strictly decreasing and weakly convex in x). The "squared
4
poverty gap" index proposed by Foster, Greer and Thorbecke (1984) has p(x,z.) = (1 -x4z)2, and
this is arn example of a distribution-sensitive (strictly convex) measure.7
Which are poorer, small households or large ones? First we consider the case in which
larger households have higher total consumption, in that Fs(x)>FL(x) for any given value of x.8
Define D(e) - PL-PS- Clearly large households will be less poor if household consumption is a
pure public good; on noting that ZL(O)'=zs, it is readily verified that:'
D(O) = rp(x; Vi(x)-fWxz)]dx < 0 (2)
0
Consider instead the poverty comparison when =1. The answer will depend on how large the
larger household is relative to the smaller one. Let n; be the size of the larger househuld which
equates its poverty with that of the smaller household i.e.,
PL(zpIjn) = f p(' z nI1z5w(x)dx = Ps (3)
0
It is clear that D(1) 0 for all nL>nL. Thus, provided that the larger households are large
enough (in this precise sense), the poverty comparison is clear at the two extremes of the size
elasticity, and there must be at least one "switch point". But we can go further and rule out
multiple switch points by noting that D '(O)>O for all 8. So, for all nL>nL, continuity of D(e)
implies that there must exist a unique 0=6* such that D(O)>O for all 8>9, while D(O)O for all 0 and all nL>ns (since D(O)>O and D'(0)>O for all 0).
Large families will be unambiguously poorer. However, the ranking is ambiguous if there is not
first-order dominance in terms of total household consunptions; there will be some poverty
measures and poverty lines which will rank differently to others at any given combination of 0
and njns. Under certain conditions, the results from the application of stochastic dominance
theory to poverty comparisons can help resolvc the issue. Suppose that, while FL(x) o (4)
0
for all z. Then (using a result from Atkinson, 1987) it can be shown that there exists a unique
0* such that D(0)>O for all 0>O0, while D(0)cO for all 0<0*, provided one restricts attntion to
weakly convex poverty measures (such as all FGT poverty measures for a > 1).
In this framework, the question of whether large households are poorer is thus seen to
depend critically on the extent of dispersion in family sizes and the size elasticity of the
equivalence scale. As we have shown, under certain conditions one can readily establish
6
existence or a single critical value of the size elasticity for which the poverty ranking of
household-size groups switches. We now investigate these issues empirically.
3 Estimating the critical size elasticity for Pakistan
3.1 Data
Our data are from the Pakistan Integrated Household Survey (PIHS) covering 4794
households residing in 300 urban and rural communities between January 1, 1991 and December
31, 1991. The survey was conducted by the Palistan Federal Bureau of Statistics (FBS) in close
collaboration with the World Bank. The format of the survey followed the Living Standard
Measurement Surveys (LSMS), though it also drew on previous FBS surveys. A stratified
sample was taken, based on the 1980 Census sampling frame for Pakistan as a whole, and
household weights were obtained from that sampling frame. A household-level questionnaire was
completed for each household as well as separate questionnaires for adult females and mnales
within each household. A com=mity questionnaire collected prices. We use the data on
expenditures and household demographics for 4763 households (some sample points were
dropped because the information was inconmplete or not internally consistent). Expenditres were
adjusted to reflect both geographic and -urban/rural cost-of-living differences (Lanjouw, 1994).
3.2 Thefragile correlation between poverty and household size
At what size elasticity is consumption per equivalent person orthogonal to household size?
For these data the least squares regression coefficient of log total household expenditure on log
household size is 0.50 (t-ratio=32). At values of 0 above (below) this figure larger households
tend to have lower (higher) consmption per equivalent person. If one controls for differences in
7
household demographic conposition and food prices then the estimate rises to 0.62 (t=37);:0 the
higher value largely reflects the correlation between size and demographic composition.
However consumption per equivalent person is not a measure of poverty as such. Figure
2 gives instead the head-count index of poverty at various values of e for each household size
found in the data." The poverty line used is described in Lanjouw (1994) and was calcuiated on
a per capita basis; we make the normalizing assumption that ihe poverty line pertains to a
household of average size (7.4 persons) (so a household of average size has the same poverty
index for all values of 0). We find that the percent poor generally increases with household size
when 0 = 1.0. However, the correlation vanishes at a size elasticity of about 0.6, and becomes
negative below this figure; at an elasticity of 0.4 there is marked tendency for the poverty rate to
decrease as household size increases.
Given the well-known limitations of the head-count index of poverty, it is of interest to
consider an alternative measure. Figure 3 gives analogous rcsults to Figure 2 for the "squared
poverty gap index" of Foster et al., (1984), discussed in the previous section. The same basic
pattern is evident, and the extent of the reversal in the direction of correlation as one moves from
0 = 1.0 to 0 =0.4 is actually sharper; up to a household size of 10 persons, the squared poverty
gap increases monotonically as size increases when 眅 size elasticity is unity, while it decreases
monotonically at an elasticity of 0.4. The same pattern is obtained for the poverty gap index.
The rank correlation coefficients between three poverty measures and household size are
plotted in Figure 4. The rank correlation coefficient between the head-count index and household
size is zero (0.07) at a size elasticity of 0.6, while for the poverty gap and squared poverty gap
indices the correlation reaches zero at a value of 0 between 0.5 and 0.6.12
8
4 Do households in Pakistan face significant economies of size?
Conr,mon practice in Pakistan and other low-income countries has been to assume that the
size elasticity is unity, and so household expendiures are simply divided by household size."
From the results of the last section it is plain then that larger households will generally be
deemed poorer on average. But is that assumption defensible? Here we consider various
approaches. Our aim is not to come up with a precise answer, but rather to characterize the
range of values that can be supported by various approaches to setting scales when applied to the
Pakistan data. An eclectic approach is warranted, given the aforeinentioned difficulties in
identifying welfare parameters from belhavior. We start with probably the most common method
of estimating equivalence scales, which we then test against some very different alternatives.
4.1 An Engel-curve estimate of the size elasticity
Our first approach is a variation on the well-known Engel (or "iso-prop") method of
estimating equivalence scales whereby the share of spending devoted to food is taken to be an
inverse welfare indicator; the higher the share of non-food spending the better off members of the
household are deemed to be (see Deaton, 1981, and Deaton and Muellbauer, 1986). Later we
comment on the method, after presenting the results.
We follow the common practice of estimating a Lesser-Working model in which the food
share is regressed on the log of expenditure per person and a set of demographic variables.
However we modify the method be adding a parameter for effects of household size
indepeodeatly of these variables. The basic specification thus takes the form:
9
J-1
us1 = ci + Pln(xd/n16) + S *IW relaiv prics + residua (S)
where w, denotes the budget share devoted to food by household i, and 11,, is the proportion of
persons in household i who belong to category j (j=1.... ,J). This specification allows us to
obtain a direct estimate of the size elasticity, by isolating the pure compositional effects in the
demographic variables (the rij 's) from the effect of household size (;).14 By adding the exta
parameter for the size effect of household size, we get our estimate of the size elasticity.
Amongst households of the samre composition-or in the special case where 5j=O for all j-xn0
is the appropriate money metric for ranking households when food share is the welfare indicator.
Under certain assumptions, G in (5) can also be interpreted as the size elasticity of an exact
money metric of utility (Appendix). In esimating equation (5) we also added the (log) food
poverty line (a cluster-specific food price index) and its squared value and regional and
urban/rural dummy variables to pick up differences in relative prices.'5
Table I gives our results. Column 1 is the simple regression of the food share on (log)
household size. There is a tendency for larger households to have higher food sbares, but the
correlation is small (the correlation coefficient is 0.04). When expenditures are added (column
2) the estimated size elasticity of the money metric of welfare is 0.61. The homogeneity
restriction (0=1) is rejected (t=11.4). In column 3 we give the augmented model including both
size and compositional effects, as well as the price index and regional dummy variables. We
obtain a value for 0 of 0.59, with a standard error of 0.044)6 The homogeneity restriction is
again rejected (t=8.8).
10
We also tried the following ae.emative specification for the demographics, giving the
Engel curve (ignoring relative prices and residuals):
J
w = a' + I3ln(xI/n ) + n,, (6)
J-1
in wbich the it,'s are the numbers (rather than proportions) of people in each demographic
group." These reflect differences in both demographic composition (some households are
younger than others) and size (some are simply bigger). Equation (6) can be rewritten as
wi - a* + I1n(xiJn1 ) + (S8AJi (7)
So the size elasticity is now a function of size and demographic composition:
J
3 *0 -( 6*i1/ I3*n, (8)
j-1
The results are given in column 4 of Table 1. At mean points, this specification gives an
elasticity of 0.58, close to our other estimates. Homogeneity is rejected (t=3.4).
We also found that the demographic composition effects on the Engel curve are only
significant if the homogeneity restrictions (0 = 1 in equation (5), or O0= I in (6)) are imposed
(Table 2). Once relaxed, the equivalence scale implied by the Engel curve is approximated well
by nO with no adjustment for demographic composition. This suggests that the "compositional
effects' in the past scales obtained by this method may actually be due to omitted variable bias
associated with a data-inconsistent homogeneity restriction, given that size is correlated with
composition (larger households tend to be younger). Our modified version of Engel's method
generates a simple iso-elastic scale, as postulated in section 2.
11
4.2 Linitations of Engel's method
Recalling the discussion in section 3.2, it is evident that at a size elasticity around 0.6 the
correlation between size and poverty vanishes. However, before one accepts that conclusion, one
should reflect on how strong the assumptions are which underlie it. Two problems stand out:
i) The method is only valid under rather special assumptions about the properties of the
consumer's cost function (Appendix). The appeal of these assumptions is questionable.'8 For
example, the Appendix shows that if the size elasticity is not independent of utility then one can
frnd that the true size elasticity is unidentified. A similar problem arises if prices are not
independent of household size. It can be readily shown that if larger househoids can buy cheaper
food through bulk discounts and that the price elasticity of demand for food is less than unity
(both are surely plausible assumptions), then our Engel method will have underesfimated the true
size elasticity. Similarly, the existence of public goods within households also leads one to
question the Engel method. Suppose that a household is exactly compensated for an increase in
its size (holding composition and other relevant variables constant). Individuals may still want to
alter their demand behavior-for example, public goods will be cheaper per person, and so there
may be a substitution in favor of such goods, away from goods such as food. If this effect is
strong enough, then food share will fall as size increases, holding utility constant, and the above
method will again underestimate the true size elasticity of welfare.
ni) Inira-household inequalities are often obscure in the models of consumer behavior
used to (inter alia) calibrate scales (Nelson, 1993). Even if one agreed that food share was a
valid indicator of average welfare within a household, there may be better indicators for specific
sub-groups, such as adults or children, and those indicators may respond differently to household
size. The fact that children consume food-a private good-more intensively than adults suggests
12
that a money metric of child welfare may have a higher size elasticity. At one extreme, consider
the Rothbarth method of setting scales, whereby one uses consumption of "adult goods" as the
welfare indicator. Following Deaton and Muellbauer (1986) let adult welfare be measured by
total non-food spending. Then it can be readily verified that size elasticity implied by equation
(5) is PO/(wj 3 -1). The food Engel curves in Table 1 imply an even lower size elasticity; for
example, at mean non-food spending the size elasticity implied by the first regression in Table 1
is 0.1! By contrast, a higher elasticity than 0.6 may be defensible when the scale is anchored to
child welfare. The literature already contains some suggestive evidence. Higher child mortality
rates have been found in households with lower consumption or income per person, and sibling
crowding is thought to be a causative factor (see the survey in Lipton and Ravalion, 1994). And
there is some evidence of discrimination against children (particularly females) in large and poor
households (Dreze and Sen, 1989, Chapter 4; Nag, 1991). Such studies are suggestive, though
inconclusive for the present purposes since they have not tested homogeneity in total expenditure
and size (assuming instead that it is expenditure per capita that matters).
In view of these probkmns, we shall test our Engel curve estimate against two rather
different welfare measures.
4.3 Publc versus private goods within households
Is a size elasticity much below unity believable for a country in which the bulk of
expenditures goes on food items, which are widely perceived to represent private goods? The
consumption by one person of a certain quantity of food precludes the consumption by another of
that quantity, and to maintain living standards the quantity available of such goods must rise
concomitantly with increasing household size.9 While private goods do not permit economies
13
in consumption, the degree to which such economies exist, and their impact on welfare, will be a
function not only of the proportion of private good consumption but also of household size, for
this is what determines the cost saving to individuals from collective consumption.
Suppose tat p is the proportion of household expenditure x which consists of purely
private goods (such as food), with 1- p being allocated to pure public goods (such as a water
tap).'1 Then the monetary measure of average welfare is:
xno = pxln + (l-p)x (9)
This collapses to x/n when there is only one person in the household or only private goods are
consumed. As n increase or p declines, per capita expenditure becomes a less accurate welfare
measure. On solving equation (9) for 0 one obtains:
e In(l-p +pfn) (10)
Inn
Average household size in Pakistan is 7.44 persons. By invering equation (10) numerically one
finds that a size elasticity of 0.59 (the lowest esimate obtained in the last sub-section) is implied
by a budget share on private goods of 0.80; a size elasticity of 0.61 (our highest estimate) is
implied by a share of 0.82.
Thus our Engel curve estimates of the size elasticity is consistent witi what one would
expect at the average household size if about 80% of spending is on private goods within
households, and the rest is public goods. The average food share is 51%, though clearly many
non-food goods (such as clothing) also have rival consumption. Even so, we would conjecture
that a budget share devoted to public goods of 20% is high for poor households m a country such
14
as Pakistan. Without better dara on the actual private-public split of spending, this approach to
the issue must remain somewhat inconclusive.
4.4 Child welfare and household size
Is our Engel curve-based estimate of the size elasticity also a good basis for predicting
how child nutritional status varies with household size? We can offer a simple test. We regress
anthropometric indicators of child nutritional status on In(4n ) -using the estimate of 0 =0.59
from section 4.1-and the log of household size, as well as a number of other variables typically
deemed to be important determinants of child nutritional status (including female literacy and
food prices). If size is significant independently of ln(xJn癕) then the latter is not the right
money metric of economic welfare for predicting child nutritional status. Thus the test equation
takes the form (ignoring the error term and other determiants of child nutrition for expositional
purposes):
nuaiional stus, = yhxin,) + Blnn (11)
yin(x,In, )
where e" =O-6ay is the size elasticity appropriate to a money metric of child nutritional status.
We add to this regression a number of other variables for child i or its household.
We consider two widely used indicators, namely stunting (as indicated by child height-for-
age, relative to international standards), and wasting (as indicated by weight-for-height). The
former is generally interpreted as an indicator of "long-term" nutritional status, while the latter
better reflects current status. Following convention, a z-score of -2 or lower is taken to indicate
"undernutrition" (though we consider a lower cut-off point later). The z-score is calculated as
15
the difference between each child's height and the median height of that child's reference age
group, expressed as a proportion of the standard deviation of the reference group.
The probit estimates are given in Tables 3 and 4 for stunting and wasting respectively. In
each case we first give the "unconditional" probit of the probability of undernutrition regressed
on household size, followed by various augmented models.
We find that the incidence of stuntig tends to be higher in larger households (column 1
of Table 3), while wasting tends to be lower (column I of Table 4), though in the latter case the
effect is not statistically significant. When we add consumption per equivalent person (using our
Engel curve estimate of the size elasticity) we find that household size is still significant in
explaining stunting, indicating a higher size elasticity for this welfare indicator than implied by
the Engel curve method; indeed, the size elasticity for a money metric of stunting is not
significantly different from 1.0. When we add the rest of the variables to the stunting model,
household size becomes insignificant controlling for our Engel curve based estimate of
consumption per equivalent person; the size effect on stundng is attrbutable to other household
chancteristics correlated with size. Size is insignificant in dil of the models of wasting (Table 4).
We also examined the incidence of 'severe" stunting and wasting, by setting the cut-off
point at 2.5 standard deviations below the median. The results for wasting were very similar,
and so are not reported. However, there is a notable difference in the stnting equation (Table
5). For the augmented model- of stunting (column 3), we can now reject the Engel curve estimate
of the size elasticity in favor of an elasticity of unity (t= 1.41)-all our results for the incidence
of more severe stunting suggest a positive correlation with household size. This was still not true
for wasting.
16
It thus appears that our Engel curve elasticity of about 0.6 is defensible for calibrating a
money metric of child wasting. However, for child stunting, a stronger case can be made for
using an elasticity of unity, particularly if one focuses on the incidence of more severe stunting.
s Conclusions
One of the "stylized facts" about poverty in developing countries is that large families
tend to be poorer, and some effort has gone into explaining why this might be so, and what
implications it has for policy. However, the basis for this stylized fact is questionable. Widely
cited evidence of a strong negative correlation between size and consumption per person is
unconvincing, given that even poor households face economies of size.
We have characterized and estimated the critical value of the household-size elasticity of
the cost of living at which the relationship between poverty and size switches sign. For Pakistan,
the positive correlation between poverty incidence and household size drops rapidly at size
elasticities below 0.7, and vanishes at 0.6 (between 0.5 and 0.6 for a distribution-sensitive
poverty measure). Recognizing the uncertainties of welfare measurement, we have made an
eclectic assessment of what size elasticity might be defensible. An elasticity of 0.6 is implied by
a modified version of Engel's method of setting scales. However, such an approach to welfare
measurement has a number of liritatis; the true value of the size elasticity cannot be inferred
from demand behavior without some (potentially strong) identifying assumptions. When we
consider instead the allocation of expenditLres between public and private goods, it appears that
the share of jointly consumed goods in the budgets of even large poor households would need to
be fairly high (around 20%) to justify a size elasticitr as low as 0.6. There may also be a
tension over household size between adult and child welfare, which can only be exposed by more
17
direct evidence on child welfare. We have shown that the incidence of child stunting is more
elastic to household size than our Engel curve estimate suggests, though the latter is still a fair
predictor of child wasting.
It is plain from these results that empirical statements about the relationship between
poverty and household size should be interpreted with considerable caution. The empirical
relationship is quite fragile, being particularly sensitive to differences in the assumed size
elasticity. Furthermore, the different welfare measures examined here suggest sufficiently
different elasticities to be consistent with either the conventional view that larger households tend
to be poorer, or that household size and poverty are roughly orthogonal or even negatively
correlated. The differences do appear to bear some relationship to the weight one attaches to
child versus adult welfare; at the two extreres considered here, the Rothbarth method based on
non-food spending as a measure of adult welfare suggests that small households tend to be poorer
while the anthropometric indicator of severe child stunting implies that it is larger households
who tend to be poorer. This suggests that a consideration of the purpose of poverty
measurement-notably the extent to which it is used to inform policies aimed at promoting child
welfare-may go some way toward resolving the issue.
18
Appendix: Identifying assumptions for our Engel curve estimate of the size elasticity
Under what conditions can the parameter we have added to the Engel curve, namely e in
equation (5), be interpreted as the size elasticity of money metric utility? Let adults maximize
utility which depends on the household's consumption of composite food and non-food goods and
on household size (other variables such as demographic composition can be readily introduced).
Let the minimum cost to household i of utility u be
In(c) = a, + OIn(n) + O(p1) + u.p, (Al)
where p, is a price index for food facing household i, 4i is a function (defined below), ,B and 0
are parameters to be estimated. (This type of cost function is a familiar one in the literature on
utility-consistent demand functions; for a discussion see Deaton and Muellbauer, 1980.) Taking
the derivative of (Al) w.r.t. ln(p), and eliminating u by inverting the cost function at the utility
maximum one obtains the demand function for food:
i = '(p)p,-P4&P)+Pln(xJn) (A2)
We need to postulate a form for 4. If we assume that
4() = a0+a1Jn(p)+a2[ln(P)t (A)
then terms in the log price and its squared value appear on the right hand side of (5). Under
these assumptions, the value of e estimated from the Engel curve specification in equation (5)
can be interpreted as the size elasticity of the cost of utility. Amongst households who face the
same prices (p=p for all i) and do not differ in other relevant respects, such as demographic
composition and tastes (in particular a,=a for all i) we have:
19
WI = Pa+$(p) + ppau (A4)
ln(x,ni) = a ++(p)+p0U1 (AS)
i.e., under these conditions, both ln(x,/nv4) and food share will be affme transforms of utility,
and hence valid utility functions in their own right (the only difference being that ln(x,ln1) is a
money metric of utility).
Of course, many of these assumptions are rather special, and relaxing one or more of
them may make it imnpossible to identify the size elasticity from observed behavior. For
example, suppose we relax the implicit assumption in (Al) that the size elasticity is independent
of utility.2' There are many ways of doing so, and some are not empirically distinguishable
from equation (5). For example, suppose that, instead of (A1), the cost-function takes the form:
ln(c,) - a+Oln(nj)+p)nsu.[n(n1)]Jpi (A6)
(for which Al is the restricted form in which y-0). It is readily verified that this yields exactly
the same Engel curve as we have estimated, and so is empirically indistinguishable. Yet the true
size elasticity becomes 6a+yIn(n1)]Y1f.p?P which is not estimable.
20
Notes
1. This pattern has been found in innumerable household surveys spanming Asia, Africa and
Latin America; for surveys see Visaria (1980, section 4), Sundrum (1990, chapter 2). and Lipton
and Ravallion (1994, section 4.2).
2. This is not the only reason. Larger households in developing countries tend also to have
more children, who (it is often argued) can achieve a given level of welfare at lower expenditures.
This is often built into demographic equivalence scales (which convert any demographic composition
for a household into an equivalent number of adults); for a survey see Browning (1992). Later we
shall argue that, by at least one common method of setting scales, these demographic compositional
effects are more plausibly attributed to economies of size. There are other reasons why a greater
household size may make at least some members better off; for example, it may make for a more
secure and easily supervised labor force for own-production activities, or it may offer greater
security in old age; in both cases the benefits are presumably appropriated largely by the household
decision maker(s). (Some demographers have stressed such arguments; see, for example, Caldwell,
1976.) Of course, these arguments do not imply that a vnical member of a large household will be
better off; the children may be worse off.
3. See Lazear and Michael (1980) and Nelson (1988), both for the U.S.
4. See Nicholson ;1976), Pollak and Wales (1979), Deaton and Muellbauer (1986), Blundell and
Lewbel (1991) and Browiing (1992).
5. It is known that the cardinal value of a poverty measure can be sensttive to the choice of
equivalence scale (Coulter et al., 1992). Our concern here is with the effect on the poverty ranking.
The question we pose here is quite similar to Atkinson (1992), though both our method, and the
empirical setting, are quite different. Atkinson asks how far one can go in ranking households
(defined in tenns of their demographic composition) in terms of poverty without specifying the
precise form of the underlying welfare function; only a few (seemingly mild) assumptions are made.
Our approach puts more strmcture on the parametric form of the scale. (Atkinson, 1992, comments
on the existence of this approach, but does not explore it further.) Some discussions of the poverty
impact of family allowance schemes have recognized that the answer may depend critically on the
properties of the equivalence scale used (see, for example, Jarvis and Micklewright, 1994).
6. Additive measures satisfy sub-group consistency, as defined by Foster and Shorrocks (1991).
This requires that when poverty increases in any sub-group of the population without a decrease in
poverty elsewhere, then aggregate poverty must also increase.
7. The general class of Foster-Greer-Thorbecke measures is obutined when
p(x,z) ( (-4) (a20). Other distribution-sensitive measures include p(x,t) = log(zJx), as
proposed by Watts (1968), and the Clark et al. (1981) measure p(x,z) = (1 -(xIz)0)/ (psi) .
8. Empirically one typically finds that total consumption tends to be higher for larger
households, even though consumption per person is lower; see, for example, Visaria (1980) and
Sundrus (1990, chapter 2).
21
9. This step ures a result proved in Atkinson (1987).
10. Specifically, the regression includes the proportions of persons in various demographic groups
(adults under 60, children, infants) and a price index for food given by the food component of the
poverty line used below.
11. The smallest sample size is for single-person households (n=59), and all other sample sizes
are 100 or higher. The "14" persons category includes all households over 14.
12. Of course higher values of e are needed to statistically reject the null hypothesis that poverty
and household size are independent. Against the alternadve hypothesis of a positive relationship, the
critical values needed at the 5% level are (rounding up to the first decimal place) 0.7, 0.7 and 0.6
for the head-count index, poverty gap index and squared poverty gap respectively.
13. A number of studies for developing countries have incorporated differences in child costs
though the scales are typically linear (or approximately so) in the number of children.
14. It is more common not to normalize the demographic variables this way, and use instead the
numbers of persons in household i who belong to category j. We discuss this specification below.
15. We also tested a specification in which the size elasticity is a linear function of (log)
household size and (log) household consumption. A joint F-test convincingly rejected this in favor
of a constant elasticity (F=0.68).
16. The standard error for 9 is obtained from a first order Taylor series expansion of 9 around
the estimated parameters.
17. Deaton and Muellbauer (1986) use this specification, though they imposed the restriction that
6=1 .
18. The classic critique of the Engel method of identifying scales is that even when exactly
compensated for an extra child, the parents will presumably still what to spend relatively more on
food, which is consumed intensively by children (Nicholson, 1976; Deaton and Muellbauer, 1986).
Here we focus instead on the problems in using the method to estimate the size elasticity. Chaudjiuri
and Ravallion (1994) examine the performance of various indicators-including food share-in
predicting chronic poverty using panel data from rural India. Food share does not perform well.
However, this is a different point to that raised here, since Chaudhuri and Ravallion were solely
concerned with how well a static indicator predicts poverty at other dates.
19. Large households can, however, be better placed than small households to take advantage of
bulk purchase discounts - particularly with respect to perishable food items, see Nelson (1988).
20. We are grateful to Jean Dreze for help on how to illustrate the foilowing point.
21. This is sometimes termed the "independence of base" condition. It is often imposed, though
it has been rejected in the few times it has been tested (Blundell and Lewbel, 1991; Dickens et al.,
1993Y.
22
Rererences
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(1992) 'Measuring poverty and differences in farnily composition',
Economica 59.
Blundell, Richard and Lewbel, Arthur (1991) 'The information content of equivalence scales',
Journal of Econometrics 50: 49-68.
Browning, Martin (1992) 'Children and household economic behavior', Journal of Economic
Literature 30: 1434-1475.
Buhmann, B., L. Rainwater, G. Schmaus, and T. Smeeding (1988) 'Equivalence scales, well-
being, inequality, aixd poverty: sensitivity estimates across ten countries using the
Luxembourg Income Study (US) database', Review of Income and Wealth 34:
115-142.
Caldwell, John C: (1976) 'Toward a restatement of demographic transition theory',
opulation and Develonment Review 2: 321-366.
Chaudhuri, Shubham, and Martin Ravallion (1994) 'How well do static indicators identify the
chronically poor?', Journal of Public Economics, 53: 367-94.
Clark; S., R. Hemming and D. Ulph (1981) 'On indices for the measurement of poverty', 'Me
Economic Journal 91: 515-526.
Coulter, F.A.E., F.A. Cowell, and S.P. Jenkins (1992) 'Equivalence scale relativities and
the extent of inequality and poverty'. The Economic Joumal 102: 1067-1082.
Deaton, Angus (1981) 'Three essays on a Sri Lanka household survey', LSMS Working Paper
No 11, The World Bank, Washington D.C.
23
Deaton, Angus, and John Muellbauer (1980) Economics and Consumer Behaviour (Cambridge:
Cambridge University Press).
_ _ _ and (1986) 'On measuring child costs: with applications to
poor countries', Journal of Political Economy 94:720-744.
Dickens, Richard, Vaniessa Fry, and Panos Pashardes (1993) 'Non-linearities and equivalence
scales', The Economic Journal (Conference Volume) 103: 359-368.
Dreze, Jean and Amartya Sen (1989) Hunger and Public Action, Oxford: Oxford University
Press.
Foster, James., J. Greer, and E. Thorbecke (1984) 'A class of decomposable poverty measures',
Econometrica 52:761-765.
Jarvis, Sarah J., and John Micklewright (1994) 'The targeting of family allowance in Hungary',
in D. van de Walle and K. Nead (eds) Public Spending and the Poor: Theory and
Evidence, World Bank, Washington DC (in press).
Lanjouw, Peter (1994) 'A poverty profile for Pakistan', mimeo, Poverty and Human Resources
Division, World Bank.
Lazear, E. and R. Michael (1980) 'Family size and the distribution of per capita income',
American Economic Review 70: 91-107.
Lipton, MichaeL and Martin Ravallion (1994) 'Poverty and policy', in Jere Behrnmn and T.N.
Srinivasan (eds) Handbook of Develonment Economics. Volume 3 Amsterdam:
North-Holland.
Nag, Moni (1991) 'Sex preference in Bangladesh, India and Paldstan, and its effect on
fertlity', Working Paper 29, The Population Council, New York.
24
Nelson, Julie A. (1988) 'Household economies of scale in consumption: theory and evidence',
Econometrica 56: 1301-1314.
(1993) 'Household equivalence scales: theory versus rolicy?', Joumal of
Labor Economics 11: 471-493.
Nicholson, J. Leonard (1976) 'Appraisal of different methods of estimating equivalence scales
and their results', Review of Income and Wealth 22: 216-221.
O'Higgins, M., G. Schmaus, and G. Stephenson (1989) 'Income distribution and redistribution: a
microdata analysis for seven countries', Review of Income and Wealth 35: 107-
131.
PoUak, R.A., and T.J. Wales (1979) 'Welfare comparisons and equivalence scales', American
Economic Review 69:216-221.
Prais, S.J., and H.S. Houtakker (1955) The Analysis of Family Budgets Cambridge: Cambridge
University Press.
Ravallion, Martin (1994), Poverty Comparisons, Fundamentals in Pure and Applied Economics
Voltume 56, Chur, Switzerland: Harwood Academic Press.
Sen, Amartya (1985) Commodities and Capabilities. Amsterdam: North-Holland.
Singh, Balvir (1972) 'On the determinants of economies of scale in household consumption',
International Economic Review 13:257-270.
Sundrum, R.M. (1990) Income Distribution in Less Developed Countries. London: Routledge.
Visaria, Pravin (1980) 'Poverty and living standards in Asia', Living Stndards Measument
Study Worling Paper No.2, World Bank, Washington DC.
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Understimdin2 Poverty, New York: Basic Books.
25
Table 1: Engel Curve Estimation of the Size Elasticity
Explanatory Demographics as Demographics as
variable proportion of persons numbers of persons
1 2 3 4
1 0.494 1.656 27.148 27.204
(59.13) (48.39) (4.12) (4.13)
Log household -0.122 -0.093 -0.092
expenditre (34.78) (24.03) (24.00)
Log Household Size 0.013 0.074 0.055 0.062
(3.031) (17.66) (11.68) (7.10)
Demorahics
Adults -0.004 -0.001
(between 15-60 years) (0.30) (0.84)
Children -0.007 -0.002
(between 5-15 years) (0.50) (1.23)
Infants 0.001 0.000
(below 5 years) (0-09) (0.04)
Adjusted R2 0.0017 0.207 0.298 0.298
RSS 108.04 85.86 75.80 75.78
Note: Dependent variable is the budget share devoted to food. Absolute t-ratios in parentheses. Quadratic in the log
of the food poverty line and urbanlrural/regional dummy variables were included to pick up diffrrnces in
relative prices. Persons over 59 are excluded from the demographic composition variables to avoid singularity.
26
Table 2: Engel Curves with Homogeneity Inposed
Explanatory Demographics Demographics
variable as proportions as numbers of
of persons persons
1 25.372 27.181
(3.821) (4.121)
Log per capita expenditure -0.080 -0.089
(22.162) (23.968)
Demoaraphics
Adults -0.011 -0.005
(between 15-60 years) (0.867) (5.393)
Children -0.041 -.006
(between 5-15 years) (3.095) (5.219)
Infants -0.021 -O.W03
(below 5 years) (1.296) (1.863)
i2 0.288 0.298
RSS 77.071 75.973
Note: Dependent variable is the budget share devoted to food. Absolute t-ratios in
paretheses. Quadratic in the log of the food poverty line and
urbanfrural/regional dummy variables were included to pick up differences in
relative prices. Persons over 59 are excluded fom the demographic
composition variables to avoid singularity.
27
Table 3: Test of the Engel Size Elastidty in a Model of Child Stunting
Explanatory 1720 observations= 1 and 2168 observations=-
variable
1 2 3
1 -0.371 2.906 90.319
(3.844) C7.811) (1.243)
Log Household Size 0.104 0.167 0.056
(2.393) (3.784) (1.113)
Consumption per -0.367 -0.281
equivalent adult' (9.097) (5.635)
Age of child -0010
(7.846)
Gender of child 0.095
(I =male) (2.304)
MoDter's literacy -0.385
(5.772)
Fadter's litercy -0.058
(1.305)
Proportion of adults -0.132
in household (0.587)
Proportion of children -0.212
in household (1.057)
Proportion of infants -0.650
in household (2.563)
Log Likedihood -2666.215 -2623.931 -2533.721
Note: Absolute t-statistics in parentheses. Dependen variable is a binary variable taking a value of 1 for stunting and
0 for no stuntfing. A z-score of for a child's height for age of -2 or lower indicates stunting. The z-score is
calculated as the difference between each child's height and the median height of that child's reference age
group, expressed as a proportion of the standard deviation of the reference group. Quadratic in the log of the
food poverty line and urban/ruraYregional dummy variables were included to pick up differences in relative
prices.
'Engel curve size elasticity of 0.59
28
Table 4: Test of the Engel Size Elasticity in a Model of Child Wasting
Explanatory 917 observations=l and 2971 observations=0
variable
1 2 3
1 -0.563 1.206 -34.789
(5.341) (2.986) (0.432)
Log Household Size -0.072 -0.044 0.004
(1.512) (0.918) (0.063)
Consumption per -0.197 -0.116
equivalent adult' ( 539) (2.160)
Age of child -0.0016
(1.594)
Gender of child 0.089
(1 =male) (1.968)
Mother's literacy -0.068
(0.944)
Father's litercy -0.073
(1.501)
Proportion of adults 0.192
in household (0.980)
Proportion of childrcn 0.062
in household (0.362)
Proportion of infts 0.126
in household (0.526)
Log Likelihood -2122.687 -2112.305 -2044.522
Note: Absolute t-statistics in parentheses. Dependent variable is a binary variable taking a value of 1 for wasting and
0 for no wasting. A z-score of for a child's weight for height of -2.0 or lower indicates wasting. The z-score
is calculated as the differn ce beween each child's weight and the median weight of that child's reference height
group, expressed as a proportion of the standard deviation of the refrence group. Quadratic in the log of the
food poverty line and urban/rural/regional dummy variables were included to pick up differences in relative
prices.
'Engel curve size elasticity of 0.59
29
Table 5: Test of the Engel Size Elastiity in a Model of Stunting (z < -2.5)
Explanatory 1225 observations-I and 2663 observations=0
variable
1 2 3
1 0.B30 2.421 121.225
(8.291) (6.226) (1.637)
Log Household Size 0.160 0.226 0.156
(3.564) (4.914) (2.985)
Consumption per -0.366 -0.289
equivalent adult' (8.632) (5.566)
Age of child 0.007
(5.379)
Gender of child 0.060
(1 =male) (1.397)
Mother's literacy -0.474
(6.401)
Father's literacy -0.069
(1.495)
Proportion of dults 0.191
in household (0.894)
Proportion of children -0.176
in household (0.929)
Proportion of infants -0.180
in household (0.720)
Log Likelihood -2416.239 -2378.094 -2291.574
Note: Absolute t-statistics in parentheses. Dependent variable is a binary variable taldng a value of 1 for stnting and
0 for no stunting. A z-score of for a child's height for age of -2.5 or lower indicates stunting. The z-score is
calculated as the difference between each child's height and the median height of that child's reference age
group, expressed as a proportion of the stmndard deviation of the reference group. Quadratic in the log of the
food poverty line and urban/ruraUregional dummy variables were included to pick up differences in relative
prices.
Engel curve size elasticity of 0.59
30
Fig.1: Critical size elasticity for rank reversal in poverty
nlL>nlL
0
/~~~~ .-4
/ n^~~~~l