, ' X R~~ -< L ( D .) t Policy, Research, and External Affairs -WORKING PAPERS Development Economics Office of the Vice President The World Benk May 1991 - WPS 685 ! Children and Intra-household Inequality A Theoretical Analysis Ravi Kanbur Structural models of intra-household allocation of resources must be modified to make intra-household allocation nonlinear in total household resources. The Policy, Rcsearch, and Extcemal Affairs Complex distrbutes PRIE Working Papcrs to disseminate the fuiduigs of work iii progfcss and to encourage the cxchangc of ideas among Bank staff and all others intcrested in developmnent issues. These papers canr) the namcs of the authors, reflect only thcir views, and should be used and cited accordingly. 'Ihe findings, intcrprctations, and conclusions arc thc authors' own. They should not be attrbuted lo the World Bank, its Board of Directors, its management, or any of its member countnes. Policy, Research, and External Affairs _1110 _ * I M . S Development Economics WPS 685 This paper - a product of the Research Advisory Staff, Ofiice of the Vice President, Development Economics -- is part ol a larger effort in PRE to understand the design of poverty alleviation policies. Copies are available free lrom the World Bank, 1818 H Stret NW, Washington, DC 20433 Please contact Jane Sweeney, room S3-026, extension 31021 (19 pages, with figures). Arguing that resources within the household are He analyzes the relationship between intra- not allocated according to need, several research- household inequality and total household re- ers have tried to mo l intra-household sources for models of intra-household allocation allocative beha,/ior. Hladdad and Kanbur (1990) that lead to a linear expenditure reduced form. argued that as households become better off, intra-household inequality first increases and He then investigates three structural models: then decreascs. The behavior of intra-houschold inequality as househioid welfarc improves is . Household welfare maximization clearly important for policy, as interventions are otten restricted to the houschold level -- al- * Cooperative bargaining though the objective is to improve the welfare of the least-well-off individual. * A nonicooperative game with children as public goods Kanbur shows here that many of the trac- table derivations of intra-household resource He indicates how these models should be allocation are availablc in what might be called modified to produce reduced forms that are the "linear expenditure systems" framework. better represented in the evidence. Thc PRE Working Paper Series disseminates the findings of work under way in the Bank's Policy, Rcscarch, and Extemal i Affairs Complex. An objectiv'c of the scrics is to get these findings out quickly, even ifprcsentations are less than filly jxl ished. The findings, interprctations, and conclusions in thesc papers do rot neccssarily represent official Bank policy. Produced by the PRE Dissemination Center Contents 1. Introduction 2. Linear expenditure syst:ems and the behavior of intra-household inequality 3. Two simple applications 4. A further application: children as public goods 5. Conclusion: the need for non-linear extensions 1. lr --dustio The issue of intra-household inequality has received increasing attention over the past decade. A number of authors (e.g. Sen, 1984) have argued that resources within the household are not distributed according to need, and this has led to attempts by others to model intra-household allocative behavior (see, for example, the discussion in the recent survey by Behrman and Deolalikar, 1989). The question of what happens to intra- household inequality when total household resources increase has been raised by Haddad and Kanbur (1990c). They argue, on the basis of empirical evidence on calorie adequacy from the Philippines, that as households become better off, intra-household inequality first increases and then decreases - in other words, there appears to be an intra-household Kuznets curve. The behavior of intra-household inequality as the household becomes better off is clearly important for policy, since interventions are often restricted to the household level while the objective is to improve the welfare of the least well off individuals. It is also important as a reduced form test of alternative models of intra-household allocation. It turns out that many of the tractable derivations of the reduced form relationship between intra-household inequality and total household resources, and indeed many of the other tractable implications of intra-household allocation, are only available in what might be described broadly as the "linear expenditure systems" framework. The objective of this paper is to lay out a generic analysis in this framework, and to show how a number of formulations of intra-household allocation essentially lead to special cases -2- of the framework. This includes (i) household welfare maximization, (ii) intra-household allocation viewed as the outcome of a Nash co-operative bargain and (iii) intra-household allocation as a Nash non-cooperative game, with children as public goods. Each of these structural models has been suggested as an explanation for intra-household allocation. We start, however, by setting out the framework of linear expenditure systems. 2. Linear Expenditure Systems and the Behavior of Intra-household Ineauality We will conduct the discussion in terms of a variable x that depicts total household resources. The index i - 1, 2,..., n will identify each of the n individuals in a household, so that xL is the flow of resources to the n ith individual, and = xi= x. At this level of generality x can hava several 1-1 interpretations. The most convenient way is perhaps to think of it Is some measure of welfare. More concretely, it can be thought of as calorie intake relative to requirement (as in Haddad and Kanbur, 1990a,c). Our focus is on how the allocation xi (i - 1, 2, ..., n) changes with x. In reduced form, we can write (1) x. x (x) ;' 1, 2, ..,n as the functional relationship derived from the structural model of intra- household allocation. A measure of inequality of the intra-household allocation can then be written as -3- (2) I I(xl X), X2(x) . X X.*(X)) Given the reduced form (1), we can therefore derive the relationship between intra-household inequality and total household resources. Consider the following special case of (1): (3) XL -x iF aL(x - Ej); I - 1, 2, .............., n i-1 L > 0 V, E aL - 1 La'. In the next two sections we will discuss structural models that lead to th. reduced form. For now, notice that (3) is nothing but a linear expenditure system for the n "commodities" i - 1, 2, ..., n, with intercepts 9L,total expenditure x, super-numerary expenditure x - E -j, and marginal propensity to spend on commodity i given by x.. How does intra-household inequality behave as total household resources change in this framework? Before answering this question let us rewrite .4- (4) x'(ii - a. r) alx - (XL stg) + 6&x P't + cgj where Ea is PL CL - - 0 The share of xL Ln x, 8L is given by (5) SL , i *nL + PLX-I Thus the squared coefficient of variation of xL, which is the same as the variance cf s&, is given by (6) o.2 o.2 + 2a. x-l + 4xF-2 where o2, o2 and oa are the variancea of the subscripted variables, and .p is the covariance of a and P. -5- We will focus on a! as our measure of intra-household inequality. We are interested in its behavior as a function of x. It is easily shown that this function has a unique minimum at x --¢ gap Of course, the economically relevant range for x is x 2 X-> 0. Thus if ogp a 0 then dX < ° for all x in the relevant range, as shown in figure 1. If a.p < O then there are two cases to consider. If x a x > O i.e. if asp s , then doi > 0 for x > x, as shown in figure 2. But if x- > x i.e. if 0 > xp > X , then a2 follows a U-shape in the relevant range of x >, as shown in figure 3. Translating these conditions on aL and PL into conditions of £L and xi, after making the normalization assumption that ix 1, we get the following, complete, characterization: (7a) o2 _< vX>x (7b) o2- 5g 0,. O v x > xF dx- (7c) ag* < Min (o., 4) -dx- has a U shape in the range x > x Figures 4a and 4b characterize the different ranges of a,; for the cases where o < a2^ and where og > o.. -6- The behavior of intra-household inequality in a linear expenliture system framework depends, therefore, on the pattern of covariance between the parameters xjl and al of the system. But we already have an important result in (7). Notice that in no circumstances can the linear expenditure system generate the invers-' U shape of the Kuznets curve, for which there is some evidence in the data (Haddad and Kanbur, 1990c). This would seem to be a strong argument against models of intra-household allocation that lead to a linear expenditure system as a reduced form. The next two sections consider some applications of this general characterization. 3. Two simple a2plications 3.1 Household Welfare Maximization Suppose that the intra-household allocaticn (1) is the result of the maximization of a household welfare function: (8) max W(xC, x2, ...I,Xa) x1 I2 .., X,, n S. t. X 3 X L-1 If the welfare function were specialized to the following case of the Stone- Geary utility function, -7 (9) W(X, X2, .... ,X) a ln (x - xj) ; x > 0 VI L- 1. then it is well known that the optimal solution to (8) is exactly as given by (3). Hence the behavior of intra-household inequality is determined by the pattern oA the minimum consumption levels il and the weights at given to tne individuals in the welfare function. If, for example, Xi and aL are negatively correlated then case (7c) obtains and intra-household inequality follows a U shape, not the inverse-U shape of the Kuznets curve that is found empirically by Haddad and Kanbur (1990c). If, on the ot;er hand, all aL are the same, then case (7a) obtains and inequality decreases continously. If a;. is positive, case (7c) may still ebtain, rovided the covariance between ixand a is not too high. If a;. is high enough, in particular if it is higher than a!, then intra-household inequality will decrease as the household's resources increase. 3.2 Two Person Nash Co-operative Bgrgaining Haddad and Kanbur (1990b) have investigated the behavior of intra- household inequality for a two person household where allocations are determined as outcomes to a Nash bargain. As is well known, if we specify the total resources being bargained over as x, the threat points of the two individuals as xl, and x2, and their "bargaining strength" parameters as a,, -8- and aa (a, + a2 - 1), then the Nash solution to the co-operative bargain iB given as the solution to the following problem: (10) Max (X1 - jI) 1 (X2 - X1, X2 s.t. x1 + x2 - x But a logarithmic transform of the maximand Li (10) gives us (9) for n - 2 and hence (3) as the solution with n - 2. We are back, therefore, to the results in (7) for tbq case of n - 2, which we have already discussee in section 3.1. The two person Nash bargaining model with fixed xl, ij, a- and a2 leads to one of the three patterns for intra-household inequality, shown in (7a), (7b) and I/c), as total household resources increase. In the symmetric bargaining model, with al - a2. we have a- - o2 0 and therefore case (7a) -- inequality decreases continuously. Since none of these outcomes is like the Kuznets curve observed in the data (Haddad and Kanbur, 1990c), this model will have to be modified. Haddad and Kanbur (1990b) consider endogenizing x, and xi as x changes and find that, under certain conditions, intra-household inequality does indeed follow a Kuznets curve. 4. A Further ARDlication: Children as Public Goods The two person Nash bargaining framework is clearly inappropriate when there are children involved, unless we assume that the welfare of children is subsumed under the objective function of one of the two players. When -9- children are present, one way of modelling their role in the household is as public goods, from which dults gst utility but towards the maintenance of which each adult makes a voluntary contribution. This perspective, which is reflecte6, for example, in the policy debate on whe-her welfare payments intended for children should be given through the father or the mother, leads to a model of non-cooperative Nash equilibrium between the players, in a game over contributions for child up-keep. What are the implications oL this mo -l for intra-household inequality? Consider the case where there are two adults, indexed 1 and 2, and a child, indexed 3. The consumptions of the three indivi, '..is are x , x2, and X3. The two adults Lave individual resources Yi and Y2, which they decide to split between own consumptions, xL, and contributions to the child's consumption, c.. Clearly, Cl + C2 - X3. The child's consumption is a public goocA, i.e. X3 enters both adults' utility functions (along with their own respective zonsumptions). Each adult decides on his or her contribution child consumption conditional upon the oth,r adults' contribution. Let adult i's utility function be given by (11) UL = yLln (xL - m) + (1 - y1)ln(c, + C2 - X3) i - 1,2 This is a Stone-Geary from with minimum consumptions for the adult ar.A m3 for the child. The weights on the contribution of own consumption and child - 10 - consumption (after logarithmic transformation) are y1 and (1 - yi). For simplicity, we assuAme that neither adult's own consumption affects the other's utility. The two adults solve the problems: (12) Max Y1ln (xl - mi) + (1 - YL)ln(ci + cj -3) xi., CL s.t. XI + CL yJ i - 1,2 ; j 1 1 if i - 2 2 if i - 1 This leads to the following solutions for i 1, 2 (13a) xi =m + Yi [Y1 - (mF + h3 - c2)] (13b) cl = [m3 - c21 + (1 - yl)171 - (El + E3 - c2)] (14a) X2 - F2 + y2 1Y2 - (A2 + E3 - C1) (14b) C2 a PT3 - Cl] + (1 - Y2)[Y2 - (O2 i o3 - cl)] A number of authors have considered the utility function (11) for public goods games (e.g. Ulph, 1988 and Woolley, 1988) and have derived the Nash equilibrium solution to cl and c2. Solving for cl and c2 simultaneously in (13b) and (14b), we get the following as interior solutions to the Nash game: - 11 - (15) Cl. = (1 - Y)V - yl(l - Y2)Y2 + Y1(l y2)x3 - (1 - YORxl + yl(l - Y2)R2 (15) ~~~~~~~~~~~1 - YlY2 (16) c = (1 - Y2)72 - Y2(' - Y071 + Y2(l - Y1)g3 - (1 - Y2)R2 + Y2(l - YOKI 1 - Y1Y2 Using these in (13a) and (14a), and noting that X3 C1 + C2, we have a complete characterization of consumption allocation in the interior Nash equilibrium: (17) - + Y2) + (1 - y1)H1 - y1l - Y2)ffi2 - y1(l -Y2)ffi =2(l - Y1)(Y1 + y2) + (1 - y2)fi2 iy2(1 - yd)iF - y2(l - Y053 ,,(1 -y1)(l - Y2)(Yl + Y2) + [y1(l - Y2) + y2(l - y1)]fi - (1 - y1)(l - y2)(il mii) 1 - Y1Y2 It will be seen that the equations in (17) are in fact in the form of a linear expenditure system. If we define the following: (18) x Y1 + Y2 a Y,( - Y2) a2 Y2(1 - YI) (1C3 - Yl) (I - Y2) al =1 -YlY2 ; 1-Y1Y2 ; 3= 1-YIY2 61 k2 - M2 R3 31 -)l - X23 then it is seen that (17) is nothing other than (3) for i - 1, 2, 3. Thus the reduced form of the children as a public goods model of intra-household - 12 - allocation, where preferences of adults as between own consumption and child's consumption are given by Stone-Geary utility functions, leads once again to the linear expenditure system. Equations (17) and (18) can be used to discuss intra-household inequality in this model. Notice first of all that the allocation depends only on x - y, + Y2, i.e. on total household resources. The division of income does not matter. This is a strong result which has immediate policy implications. It suggests that, in this framework, the policy debate on targeting of child payments to the mother or the father is irrelevant - the Dublic goods game serves as a perfect aggregator and what matters is the total level of resources. As discussed by Bergstrom and Varian (1985), the above is a general result for public goods, and does not depend on the Stone-Geary utility function. However, what the linear expenditure specialization allows us to do is to consider explicitly the behavior of inequality as a function of total household resources. We know from the discussion in section 3 that only one of three outcomes is possible - inequality always increasing, inequality always decreasing, or a U shape where inequality first decreases and then increases. To the extent that the empirical evidence points to an inverse - U shape where inequality first increases and then decreases, this is an argument against the children as a public goods model - at least against the interior Nash equilibrium solution of this model. Consider now possible corner solutions. Following Woolley (1988) consider the case where one adult, say adult 2, is constrained to set c2 - 0. Using this in (13) and (14) we get the following allocation as solution: - 13 - (19a) x1 i,m + Yf 17Y - (mi + mg)] (19b) X2 72 (19c) X3 -m3 + (1 - Y1)[y7 - (mix + 53)] It is seen immediately that the distribution of total household resources between the adults now does affect the intra-household allocation. Most particularly, increases in y, increase xl and X3, but increases in Y2 only increase x2. If we think of individual 2 as being the male adult, this model does provide a rationalization of targeting extra resources to the female, since at least some of these will get to the child. The allocation in (19) allows a richer variety of shapes in the relationship between intra-household inequality and total household resources. but it depends on how exactly the increase in resources is divided between y1 and y2. If increments are distributed in constant proportion, so that y1 - ax and y2 - (1 - 8)x, then we get the allocation (20) xI = (1 - Y053. - y1mi + y18x X2 = (1 - 8)x X3 - Ylm3 - (1 - YO)Hl + a(' - YO)X - 14 - This is, of course, another linear expenditure system, so that a similar characterization to the one in (7) obtains. The only chance for a variation is if after a certain level of x the interior solution comes into play. This switch in regimes between two different linear expenditure systems can lead to a richer pattern of behavior. If the increment in x comes solely from Y2 (say, male income) then the behavior depends on the position of x2 relative to x1 and X3. If x2 is already greater than xl and X3, the inequality will increase inexorably. Consider now the case where x increases through increases in y1 (female income). Unit increments in this are divided according as y, to x1 and (1 - y1) to X3. In this case both x, and X3 will increase relative to x2. Thus inequality will decrease. It should then be clear that if initial increments to household resources go to the male, and subsequent increments to the female, then we will indeed get the Kuznets relationship of inequality first increasing and then decreasing as total household resources increase. 5. Conclusion: The Need for Non-Linear Extensions We have seen that models of intra-household allocation that lead to linear expenditure system allocations imply very specific relationships between intra-household inequality and total household resources. We have derived these relationships and have argued that they can be used as reduced form tests of particular models. Of course, this procedure has well known problems, but it is hoped that this way of proceeding will give guidance on how the structural models should be modified. The essential message is that - 15 - they have to be modified so as to make intra-household allocation non-linear in total household resources: (i) In the household welfare maximization models it means using utility functions that are more general than the Stone-Geary form. It should be noted, however, that most generalizations of demand systems in the income dimension essentially involve introducing the logarithm of income as an independent variable. This monotonic transformation will not of course affect our conclusions on the shaRe of the relationship between intra-household inequality and total household resources. In any event, most attention in generalized demand systems is given to the cross-price effects (e.g. the Almost Ideal Demand System of Deaton and Muellbauer, 1980) across commodities, an issue which is not relevant to the models developed here. (ii) In the Nash co-operative bargaining models, the modifications must involve endogenizing the threat points as total household resources increase. This is done by Haddad and Kanbur (1990b) and they do find that with this modification a Kuznets curve is possible. (iii) In the children as public goods model, departures from the Stone- Geary utility function are likely to make the solution intractable. An alternative is to examine corner solutions of the Nash game, as a way not only of possibly generating the Kuznets curve, but also of rationalizing policy concerns about the need to target incremental resources to female adults. All three of these avenues hold out interesting possibilities for further research. - 16 - References Behrman, J. and A. Deolalikar (1989): "Health and Nutrition", in H. Chenery and T.N. Srinivasan (eds.), Handbook of DeveloDment Economics, North Holland: Amsterdam. Bergstron, T.C. and H.R. Varian (1985): "When are Nash Equilibria Independent of the Distribution of Agents' Characteristics?" Review of Economic Studies. Deaton, A. and J. Muellbauer (1980): "An Almost Ideal Demand System", AmerLjn Economic Review. Haddad, L. and R. Kanbur (1990a): "How Serious is the Neglect of Intra- Household Inequality?" Economic Journal. Haddad, L. and R. Kanbur (1990b): "Are Better Off Households More Unequal or Less Unequal?" PRE Working Paper, No. 373, The World Bank. Haddad, L. and R. Kanbur (1990c): "Is There an Intra-Household Kuznets Curve?" PRE Working Paper, No. 466, The World Bank. Sen, A.K. (1984): "Family and Food: Sex Bias in Poverty." in A.K. Sen, Resources. Values and Develooment, Basil Blackwell: Oxford. - 17 - Ulph, D. (1988): "A General Non-Cooperative Nash Model of Household Consumption Behavior", Mimeographed, University of Bristol. Woolley, F. (1988): nA Non-Cooperative Model of Family Decision Making", Working Paper, No. TIDI/125, London School of Economics. 3.\ psusCi4 _ -_-- - -at - - --4 - - - - - -- v~~~~~~ S pIS Fe l \ I~ _ _ _ _ t __ l _ __ Xo* __ _ _ - - - . 2 - - 4-- . - - 0~~~- " afil ex w!S< |., , I. .I> l.Y V+ 1' 1 wa rop sOn VPP 4 -P 07t0 | Q< lX- to ., I . .. - ,s sG O WN.0 4 0V PRE Working Paper Series Contact fllg Author forpape WPS662 Trends in Social Indicators and Jacques van der Gaag May 1991 B. Rosa Social Sector Financing Elene Makonnen 33751 Pierre Englebert WPS663 Bank Holding Companies: A Better Samuel H. Talley May 1991 Z. Seguis Structure for Conducting Universal 37665 Banking? WPS664 Should Employee Participation Be Barbara W. Lee May 1991 G. Orraca-Tetteh Part of Privatization? 37646 WPS665 Microeconomic Distortions: Static Ramon L6pez May 1991 WDR Office Losses and their Effect on the 31393 Efficiency of Investment WPS666 Agriculture and the Transition to the Karen M. Brooks May 1991 C. 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