How serious is the neglect of intrahousehold inequality ?

IPolIcy, Planning,nd Research] WORKING PAPERS Poverty and Inequality Office of the Vice President Development Economics The World Bank November 1989 WPS 296 How Serious is the Neglect of Intrahousehold Inequality? Lawrence Haddad and Ravi Kanbur Ignoring intrahousehold inequality can lead to considerable underestimates of the true levels of poverty and inequality. But the estimated patterns of poverty and inequality across key socioeconomic groups are not affected dramatica'lly. The Policy. Planning, and Research Complex distributes PPR Working Papers to dtsseminate the findings of work in progress and to encourage the exchange of ideas among Bank staff and aD others interested in development issues. These papers carry the names of the authors, reflect only theur views, and should be used and cited accordingly. The findings, interpretauons, and conclusions a* the authors' own. TIhey should not be autnbuted to the World Bank, its Board of Durctors, ius management, or any of its member countries.  Plc,Planning, and Research Poverty and lneqiuality Haddad and Kanbur developed a framework for * Patterns of inequality revealed by house- assessing the consequences of ignoring in- hold level data are somewhat different from t-ahousehold inequality in the measurement and pattems revealed by individual level data, but analysis of poverty and inequality. the differences seem not to be dramai . To confirm these results, the exercise should be re- After applying this framework to data for peated with data from other countries. the Philippines - based mainly on relative caloric intake in households - they concluded Haddad and Kanbur's conclusions are likely that: to be of interest to those considerinig the costly task of surveys focused on intrahousehold e The result of neglecting intrahousehold ine- pattems in developing countries. Unless poli- quality will probably be considerable understate- cymakers are interested primarily in more ment of the levels of poverty and inequality. accurate measurement of levels of inequality and With the Philippine data, measured levels of poverty, the exercise may not be cost-efficient. inequality and poverty were off 30 percent as a result of ignoring intrahousehold variation. This paper is a product of Lhe Office of the Research Administrator. Copies are available free from the World Bank, 1818 H Street NW, Washington DC 20433. Please contact Jane Sweeney, room S3-026, extension 31021 (39 pages with figures and tables). i The PPR Working Paper Series disseminates the findings of work under way in the Bank's Policy, Planning, and Research Complex. An objective of the series is to get these findings out quickly, even if presentations arc less than fully polished. ,'he findings, interpretations, and conclusions in these papers do not necessarily repres-nt official policy of the Bank. Produced at the PPR Dissemination Center  How Serious is the Neglect of Intrahousehold Inequality? by Lawrence Haddad ard Ravi Kanbur Table of Contents 1. Introduction 1 2. A Theoretical Analysis 3 3. An Empirical Analysis 14 3.1 The Data Set and the Variables 1 4 3.2 Measurement and Decomposition of Inequality 1 9 3.3 Measurement and Decomposition of Poverty 28 4. Conclusion 36 References 38 * The Authors would like to thank seminar participants at Stockholm, Leicester, Essex, Bristol, and Warwick for their helpful comments.  1 1. Introduction In the measurenent of inequality and poverty, the significance of intra-household inequality clearly depends on the objective of the exercise. In the growing literature on this subject, the reason for investigating intra-household inequality is that the ultimate object of concern for economic policy is the well-being of individuals. Yet most policy, and most policy analysis, has until recently equated the well-being of individuals with the average (adult-equivalent) well-being of the household to which they belong. The assumption is thus that within a household resources are divided according to need. If this were true, then policy could concentrate on increasing the resources of poor households without getting enmeshed in an intra-household policy that may be difficult to design and even more difficult to execute. However, a growing body of empirical literature has benun to question whether resources within a household are indeed distributed according to need (see Sen, 1984; Harris, 1986; Behrman, 1987). The natural corollary is thus that conventional results on the extent and pattern of inequality and poverty as revealed by household level resouroes have to be re- examined. There is, however, little in the way of quantification of how much of a difference the existence of intra-household inequality wculd make to conventional measures of inequality and poverty. Is the understatement (if any), likely to be large? Even if the understatement of the levels of inequality and poverty is large, are the patterns of inequality and poverty grossly different when one takes account of intra-household inequality? An answer to the latter  2 question is inportant since policy design (e.g. directing resouroes to particular regions, crcp groups etc.) often relies on the pattern of poverty and inequality (see, for example, the use by Anand (1983) of inequality and poverty deccmposition to analyse the efficacy of various policies in Malaysia). The object of this paper is to present a framework in which these questions can be addressed, and then to apply this framework, to a data set from the Philippines on intra-household inequality in nutritional status. Our empirical conclusions are likely to be of interest to those who are considering undertaking the costly task of an intra-household focused survey in developing countries. These oonclusions can be stated very crudely but sinply as follows: (i) The neglect of intra-household inequality is likely to lead to a considerable understatement of the levels of inequality and poverty. (ii) However, while the patterns of inequality revealed by household level data are somewhat different to those revealed by individual level data, these differences can be argued to be not dramatic. The plan of the paper is as follows. The next section develops an analytical framework for assessing the inpact of intra-household inequality on the levels of inequality and poverty. Section 3 applies this framework after introducing our data set. Section 4 concludes the paper.  3 2. A Theoretical Analysis We suppose that the object of interest is the well-being of individuals, which is measured by some agreed standard (consumption, nutrition etc.) and denoted y. It is assumed that all relevant corrections and adjustments have been made and incorporated into y (e.g. price differences, needs differences etc.) so that it really does represent the variable on which social welfare is define;. Now let x be the average of y within a household. Thus the distribution of individuals by x would ignore intra-household inequality and it is the difference between this distribution and the distribution of y that lies at the heart of the analysis in this paper. Denote the conditional density of y given x as a(y| x). This captures inequality within a household whose average standard of living is x. If p(x) is the marginal density of x in the population, then the density of y in the population, f (y), is clearly (1) f(y) = la(yi x) p(x) dx where the integration is over the permissible range of x (perhaps non-negative). Notioe that by definition (2) E(yJ x) = Iya(yj x) dy = x  4 where E represents the expectation operator. Hbnce, (3) E(y) = Iyf(y) dy = llya(y x) p(x) dydx = IE(yI x) p(x) dx = E(x) Thus the mean of y is the same as the mean of x. In fact, it can be shon that the distribution of y is a mean preserving spread of the distribution of x. To see this, consider a convex function h(.). Note that (4) Efh(y)) = Ih(y) f(y) dy = I[Jh(y) a(yl x) dyj p(x) dx > Ih(x) p(x) dx by Jensen's inequality = E{h(x)} What (4) tells us is that the expectation of all convex functions is greater under the distribution of y than under the distribution of x. It therefore follows (see Rothschild and Stiglitz, 1970) that f(y) is a mean preserving spread of p(x), which is a fairly obvious result. Since f(y) is a mean preserving spread of p(x) , it follows from Atkinson (1970) that the Lorenz curve of y will be unambiguously below the Lorenz curve of x on a Lorenz diagram. This is the sense in which inequality will always be understated by using only the household level information. The "Lorenz class" of measures (see Anand, 1983) will always be lower for x than for y - for  5 exanple, the Gini coefficient or the Theil index will always be understated. To illustrate the nature of the discrepancy, consider as a measure of inequality the coefficient of variation. Since the means of y and x are the same, in this case we might as well use the variance. Triting V(y) as the variance of y , V(x) as the variance of x and V(yl x) as the variance of y conditional on x (i.e. the variance of well-being within a household whose average well-being is x), we know fran the analysis of variance that (5) V(y) = N(y x) p(x) dx + V(x) Thus the degree of discrepancy depends on what V(y | x) looks like for different values of x . In effect, the right hand side of (5) deconposes the inequality of y into an intra-household coxponent and an inter-household conponent. The size of the intra-household component - the discrepancy between V(y) and V(x) - is an empirical matter and in the following section we provide quantification of the discrepancy for a range of inequality measures, based on a particular data set. So nuch for the measured level of inequality. What about the pattern of inequality? Suppose that our households could be split into two mutually exclusive and exhaustive groups U and R ("urban" and "rural"). A typical investigation of the pattern of inequality involves two questions: (i) Which group has higher inequality? (ii) What a fraction of inequality is accounted for by inequality within and inequality between these two groups? These questions are asked  6 very cx mmonly in inequality analysis (e.g. Theil, 1967; Anand, 1983; Tsakloglou, 1988) and they are inportant for policy design. Wculd the answers differ greatly if we ignored intra-household inequality? Taking the second queastion first, using subscripts U and R in an obvious way we car write: (6) V(y) = )AVu(Y) + ARVR(Y) + lAIVU(Y) - VR(y)1 where i and i are population proportions in the two groups (AU + AR = 1) and p represents mean. The between group compcnamt of overall inequality in (6) is that involving the .1roup means. The between group contribution is defined as (7) i(y) = WYAi[.VU(Y) - i.(y)J2 The within group contribution is simply 1 - C (y). If we did not have individual level data but relied on household means, then (8) V(x) = AuVu(x) + ARVR(x) + AuAR [%(x) - PR(x)2 (9) %(x) R [j.U(x) - VR(x)12 CB (x) e V(x)  7 But with a suitable adaptation of (3) it follows that a (y) = ju(x) and R (y) = R (x). Thus the absolute value of the between group conponent is the same whether y or x is used. Since fran (5) we know that V(y) > V(x), we have the result that (10) C (y) < %(x) Henoe the between group contribution to inequality is overstated and the within group contribution is oorrespondingly understated when intra-household inequality is ignored. While (for ease of exposition) we have derived the result for V(.), it holds true for any measure of inequality where the between group coomponent depends only on group means (for this approach to defining "deccmposability", see Shorrocks, 1980). For exanple, it holds true for the well known Theil index of inequality, which forms the basis of many empirical studies. The extent of over statemet or understatement is an empirical matter, and we shall investigate this in the next section in the oontext of our data set. What of che ranking of groups by inequality? For this, note that (11) VU(y) = NIU(yI x) pU(x) dx + VU(X) (!2) VR(Y) = IVR(y I x) pR(x) dx + VR(x)  8 Fran these: (13) Vu(x) VR(x) = [Vu(y) - VR(Y)J - Vu(yIx) pU(x) dx iVR(y I x) pR(X) dx] Thus we get: (14) ([VU(Y) - VR(y)J I 0 => [Vu(x) - VR(x)I > 0) {(Vu(y) VR(y)I > 1Ivu(y I X) pU (x) dx - NR(y I x pR(x) dx]) Similar results can be derived for other indices such as the Theil index. The general point is that, if intra-household inequality in the two groups are sufficiently similar, the rankings will be preserved. However, if intra-household inequality is very much greater in the group with higher overall inequality, then suppression of this intra-household variation could lead to a ranking reversal. Whether this actually happens or not is e empirical matter, and we will investigate it further in the next section. We turn now to an analysis of poverty. The standard approach in the literature (see Sen, 1976) is to choose a poverty line and then define a poverty index based on the gap between the value of the variable measuring the standard of well being, and i'cs critical value as given by the poverty line.  9 Define a gap function" as hMy, z), where z is the poverty line. Then a general definition of a class of poverty indices (see Atkinson, 1987) is (15) P(y) = Jh(y, z) f(y) dy If we only had information on household averages then we would be forced to use (16) P(x) = Ih(x, z) p(x) dx But (17) P(y) = Ih(y, z) f(y) dy = I[Ih(y, z) a(yI x) dy] p(x) dx > I[h(x, z)] p(x) dx if h is convex in y = P(x) Thus if h(., z) is convex in its first argument, there is definitely an understatement of true poverty by using x . To investigate this further, consider the class of poverty indices recently introduced by Foster, Greer and Thorbecke (FG) in 1984. In terns of (15), their index assumes O ) 0; y < z z (18) h(y, z) = 0 y>z  10 Here, a is an index of poverty aversion. When a = 0 , P beocuuss siiply the standard head count ratio or incidence of poverty measure. When a = 1 , P emphasises the average depth of poverty while with a > 1 , P is sensitive to intra-poor transfers. Notice that with a > I , h(y, z) is convex in y . Thus (17) holds and we can be sure that the FG index on x will understate true poverty. However, tor a < 1 h(y , z) is neither convex nor concave over its whole range so that Jensen's inequality can no longer be used. To investigate this further, consider a = 0 Then (19) PO(y) = IPO(y Ix) p(X) dx < PO(Y I P(X)) according as P (yI x) is {conve ) in x- Thus if the incidence of intra-household poverty is corawve in x and the incidence of intra-household poverty at neon household consumption exceeds the incidence of poverty defined on x , then the latter will underestimate the true incidence of poverty. But if the incidenoe of intra-household poverty is convex in x and the incidence of intra-household poverty at mean household consumption is less than the incidence of poverty defined on x , then the latter will overestimate the true incidence of poverty.  11 Other sufficient conditions can also be derived. It can be shown that if (20) y = h(x, c) where c is randon and h is increasing and quasi-concave in its arguments, then PO(X) will understate true poverty if the poverty line is less than mean y , which is equal to mean x . (Ravallion (1988) derives this result in a different context.) A ncoessary and sufficient condition can be derived if we further specialise to Y =X + E E(E) =0O (21) Var (c) =2 Cov(X, E) =0 Then E(y) = E(x) Var(y) = Var(x) + a2 If we further restrict ourselves to y and x being synmetric distributions (e.g. the normal distribution) then it follows easily that (22) PO(Y) PO(x) according as z e3 p(y)  12 Thus the x indicator overstates poverty if the poverty line exceeds the mean of y - we shall see an espirical verification of this result in our data set. Let us now turn to the difference that can be made to an analysis of poverty patterns across mutually exclusive and exhaustive groups. As before, let these be indexed U and R , with population proportions Ai and i . Wb know that (23) P(Y) = VU(Y) + ARPR(y) and the contribution of region U to poverty, CU(y) is written (24) C (y) = (Y) Similarly: (25) P(x) uPU(x) + kR(P) (26) %(x) =vU() CU ~~~ PRU()(x)) P(y Thus (27) %u(y) - U(x) = Ati9u(x PR(x) I PU(y)_ R) P (y) (X-) P~(x) -PR(  13 We already know that if h is convex in y then PU(Y) > Pu(x) and PR(y) > PR(x) , i.e. true poverty is understated in both groups when measured using x . However, for the measured contributions to poverty to be very different, the degree of understatement has to be greatly different in the two regicns. In other words, intra-household inequality, and its pattern, has to be very different when camparing across the two groups. The same is of course also true when considering poverty ranking reversals. If PJ(y) > PR(y) and the pattern of intra-household inequality is the same or very similar in the two groups then Pu(x) > PR(x) will also hold. Cnly if the patterns are significantly different will ranking reversals take place. once again, whether this happens or not is an empirical matter and we turn now to an investigation of our theoretical framework as applied to a particular dataset.  14 3. An Empirical Analysis 3.1 The Data Set and the Variables Having developed a theoretical framework and scme results on what difference the neglect of intra-household inequality can make to the measurement and deccnposition of inequality and poverty, it is now time to investigate a specific dataset. The data used in this study are described and evaluated fully in Bcuis and Haddad (1989a). They cone from a survey of the predominantly rural southern Philippine providence of Bukidnon. The survey was conducted in four rounds over a sixteen month period in 1984-85, covering 448 households ccmprising 2880 individuals. The only good for whicn we can identify individual consumption is food. Therefore we focus on food, converting dietary intake into calories and standardising by calorie requirements, to give calorie adequacy. Calorie adequacy will be our measure of individual well being. There is now a large and controversial literature on the appropriateness of this variable for welfare and policy analysis. However, recall that our object is to investigate the consequences of neglecting intra-household inequality for the measurement of inequality and poverty. Food consumption is one of the few variables on which intra-household data can be collected and as such, is suited to our analysis. Calorie intakes in our data set represent 24 hour recalls by the mDther, of food eaten by individual family menmers. This  15 information may be subject to a number of errors, both in overall quantity recall and allocative recall. Burke and Pao (1976) review the methodological evaluation literature for large scale surveys of individual diets. Using the evaluation criteria of reliability (small variable errors), validity (small biases), respondent burden, and data oosts, they oompare 24 hour recalls with dietary history, food weighing and inventoty-record methodologies. They conclude that "no one methd was consistently advantageous over all others". 24 hour recalls did well in ternm of light respondent burden and ease of collection, but were biased to an extent primarily dependent on the skill and probing abilities of the enumeration team. The studies reviewed covered mainly developed countries, and we should note that different problems may arise from their application in less developed countries: toddler "snacking" away fran home, for instance (although less varied LDC diets may strengthen the 24 hour recall method). The position with respect to the 24 hour recall method is sufmed up by Chavez and Huenemann (in Sahn et al, 1984): "Because of the short time period, this method [24 hour recallJ is more econcmical than the detailed method and the modified dietary history method. Cne day may or may not represent a 'typical' intake for the individual household. Twenty-four hour intakes of a large sample of households may, hoyever, represent a typical daily intake for the community as a ,whole". We have minimised problems of representativeness by using only four-round averages of calorie intake for each individual in an attenpt to make the dietary snapshots more typical. This technique has been used for a number of years by the USDA in its National Food Consumption Surveys (U5CA, 1988).  16 Concerning measurement errors, two sources of evidence attest to the accuracy of our enumerators' data collection efforts. Firstly, calorie consumption figures calculated from two different methodologies (24 hour recall and food expenditure data) exhibited a high degree of correspondence at the means of the data (Bouis and Haddad, 1989b). Furthermore, the 24 hour recall intakes corresponded closely to a small, overlapping, subsample of food weighings conducted simultaneously (Corpus et al, 1987). The dencminator of the calorie adequacy ratio is calorie requirement. We use orthodox reccuerxuded daily allowance (RDA) calorie figures for a healthy Philippine population with requirements diasaggregated into thirty-two age-gender-pregnancy status categories (details in Bouis and Haddad, 1988). We recognise the limitations of RDA's in the context in which we plan to use them. Firstly, in a normal distribution of healthy individuals in a given age-gernder- pregnancy-activity level group, 50% will have intakes below the RDA. This reflects the construction of RDA's as an average requirenent (Davidson, et al, 1979). Secondly, the requirements take no account of an individual's unrestricted physical activity level. Even the crude classification into limited, moderate, and extrene physical activity is difficult to achieve in the absence of well collected time-activity data. Thirdly, the requirenents take no account of individual adaptation to food availability in the form of activity patterns, longitudinal growth retardation and, to a lesser extent, basal metabolic rate adjustment. These problems are not trivial, but until individual requirements for full functional capacity are available the best we  17 can do is to use the RDA's, and note that they represent "an order of nagnitude" (Achaya, 1983). Our object is to assess the seriousness of neglecting intra- household inequality. In our data set, since we have individual level data we can "pretend" that we do not have this information by taking household averages. However, in the emvirical context we now have a choice of whether to take the mean of individual adeuqacy ratios, or to take the ratio of the within-household mean of individual calorie intakes and individual requirements. There are thus three variables of interest: individual calorie adequacy, 0 , mean individual calorie adequacy within the household, 01, and household calorie adequacy, 02 . Mbre precisely, let C. = calorie intake of individual i R. = calorie requirenent of individual i Oi = i/Ri = calorie adequacy of individual i n = number of individuals in household h 01i = n z oi = mean of individual calorie adequacy within the household, which is assigned to each household member. nh = 1=1 = household calorie adequacy, which is 2 nh iR ili assigried to each household me8lber.  18 Referring to our theoretical discussion, 0 corresponds to y and 01 to x . But in the empirical rontext we typically have to deal not with 01 but with 02 since information is collected at the household level on calorie intake and calorie requirement separately While 01 and 02 will differ, we shall see that the difference, and its empirical effect, is not very great. These three variables are calculated for all 2800 individuals in our sanple. We should note that all individuals within a household will have identical values for 01 . The same is true for 02 . Figure 1 shows the relative frequency plots for the three variables. Nor surprisingly, the range for 0 is the largest of the three. Its distribution is skewed to the right but approaches normality. The plot for 01 is less of a normal approximation than 0 , while the plot for 02 is fairly similar to that of 01 . The mean of 0 over the 2880 individuals in the sanple is 0.877-5, indicating that on average our sample is significantly undernourished. The mean of 01 is by definition the same as the mean of 0 . However, the mean of 02 is 0.88835, an excess of 1.2%, indicating slight negative correlation between calorie intake and calorie requirement. Or real object, however, is to examine and compare measures of inequality and poverty defined over 0 , 01 and 02 . We start with inequality.  19 3.2 Measurement and Decomposition of Inequality Figure 2 compares the Lorenz curve of 0 with those of 01 and 02 . We proved in Section 2 that the Lorenz curve of 01 would be uniambiguously closer to the line of perfect equality than the Lorenz curve of and this is shown to be the case in Figure 2a. The same comparison holds for 02 and 0 , and in fact the Lorenz curves of 01 and 02 are almost identical. Table 1 quantifies inequality differences with respect to five commonly used measures of inequality: the coefficient of variation, the log-variance, the Gini coefficient, the Theil index T, Theil's second measure L and the Atkinson equally distributed equivalent measure of inequality with inequality aversion parameter equal to 2. The exact definitions of these measures are to be found in Kanbur (1984). The first point to note is how close the measures based on 01 and 02 are to each other. With this in mind, we concentrate on the differences between 0 and 01 As can be seen, the understatements of inequality when intra-household inequality is suppressed can be very large, ranging from around 60% for the log-variance, the Theil T , and Theil L and the Atkinson measure, to 35% for the Gini and the coefficient of variation. It may be tempting to attribute the difference to "within" household inequality, but such a precise attribution depends on whether or not the measure is "decomposable" in the sense of Shorrocks (1980). Only the two Theil measures satisfy the relevant conditions of strict sub-group decomposability.  Figure 1. Relative Frequenc. :stributions for 0. 0 * and 02 0 - individual level -_11 _NNNN----- 01 - individual means | of 0 within hh C~ ~~2 Inee _: re ;S 2 _ X Oon3<|oeg* ;2: ... ... ... .. .. .. .. . -ratio of I ! * household means U -----N _--__  21 Figure 2. Lorenz Curves for 0, 01, and 02 100 98- 70 X,./ 9wqlative ., 68 -45 50 ....-- --~~~~~~~~~PHI 48 ...PHI1 38- 0 18 20 38 40 58 68 70 80 98 108 ec i iis 1908 80 ..v 68-- 45~~~~~~~~~~~~~-. tIIUlativeX i< 58 --~~~~~~~~~~~~~PHI 40. Pl 12 30 , / ,'- 28.. 10. 0 8 I I 4I I I 0 10 20 30 40 58 60 78 80 90 108 geci le|  22 We turn now to the issue of the pattern of inequality as revealed by the data. it is traditional in inequality and poverty analysis to decompose inequality along key socio-economic dimensions. Thus Anand (1983) provides a profile of inequality in Malayasia along racial lines while Tsakloglou (1988) does the same exercise for Greece along regional lines. The exact nature of the profile depends on the policy question at hand. In the Philippine region of Bukidnon, one of the central issues has been the impact of a move from corn to sugar production on inequality and poverty. Bouis and Haddad (1988) provide a detailed analysis of the nutrition and income effects of the incroduction of sugar can cultivation in the study area. Our object here is more limited - it is to investigate the sensitivity of the pattern of inequality, across the sub-groups identified by Bouis and Haddad (1989a) as being important, to the use of individual or household level data. The first panel of Table 2 shows a decomposition of the Theil T index across three mutually exclusive and exhaustive types of households - corn producers, sugar producers and others. As can be seen, the 2880 individuals in the sample are divided as follows: 1565 in corn producing households, 1082 in sugar producing households and 233 in other households. It is immediately seen that if we compare inequality as measured by the Theil T index defined on 0 (individual level data), inequality among individuals in sugar households is greater than that among individuals in corn households, while inequality among households that grow neither crop is greatest. A shift in favour of sugar, perticularly if this creates landless labourers in the process is therefore worrying from the point of view of inequality. Would this conclusion have been greatly affected if we  Table 1. Inequality Measures for 0 , 01 , ard 02 Variable n mean oCefficient Log Variance Gini TheilT TheilL Atkinson of Variation Coefficient (base e) (base e) Measure (c.=2) 0 2880 .87765 .31419 .10897 .1754 .04873 .05078 .10229 01 2880 .87765 .20386 .04257 .1148 .02059 .02083 .04127 (% of 0) (65) (39) (65) (42) (41) (40)