A Test of Collective Rationality Within Bigamous Households in Burkina Faso

This paper provides different tests to ascertain efficiency of consumption decisions in households with many decision-makers based on the effect of distribution factors . It also presents a method of determining the number of these decision-makers.
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  A Test of Collective RationalityWithin Bigamous Households in Burkina Faso Anyck Dauphin    Bernard Fortin ✁  Guy Lacroix ✂ Septembre 2003 P RELIMINARY AND INCOMPLETE .N OT TO BE QUOTED WITHOUT PERMISSION . Abstract This paper provides different tests to ascertain efficiency of consumption decisionsin households with many decision-makers. It also presents a method of determining thenumber of these decision-makers. The tests are used to investigate collective rationalitywithin bigamous households living in rural Burkina Faso. The data are consistent with ✄ IDRC, CR´EFA and Universit´e Laval; Phone number: (613) 236-6163; Fax: (613) 567-7748; ☎ CR´EFA, Universit´e Laval and CIRANO; Phone number: (418) 656-5678; Fax: (418) 656-7798; ✆ CR´EFA, Universit´e Laval and CIRANO; Phone number: (418) 656-2024; Fax: (418) 656-7798; 1  collective rationality. Furthermore, we find that the three adults take part in the internaldecision process.KEYWORDS:Intra-household allocation, collective model, polygamy, extended fam-ily, Pareto optimality, African households. JEL numbers: D11, D70 1 Introduction Household collective rationality (CR) has become a very important topic of research over thepast few years. 1 In particular, many have sought to devise ways of testing CR in differentcontexts ( e.g.,  Chiappori (1992), Browning, Bourguignon, Chiappori and Lechene (1994),Udry (1996) and Fortin and Lacroix (1997)). The interest in this topic stems from the factthat ef ciency has always been thought to be an innocuous and natural assumption to makewhen modeling household decisions. Surprisingly, though, Browning and Chiappori (1998)have shown that ef ciency generates testable restrictions on consumption even in very generalsettings that allow for public and private commodities and externalities. Furthermore, whenthere are two potential decision-makers in the household, they have shown that the (Pseudo-)Slutsky matrix is the sum of a symmetric negative semi-de nite matrix and a matrix that has,at most, rank one. They have also shown how this condition can be generalized to householdswith more than two decision-makers. This extension is important since it is likely that in many 1 The notion of “collective rationality” refers to the fact that intra-household decisions yield pareto-ef cientoutcomes. Furthermore, the decision process recognizes that individual preferences may differ across decision-makers. 2  households adult children who live with their parents influence the family decision process.Likewise, polygamous or extended families are quite common in many developing countries.It is thus likely that the decision process may involve more than two decision-makers is suchhouseholds. As a by-product of their analysis, Browning and Chiappori (1998) have also pro-vided a simple test which allows the number of decision-makers in a multi-person householdto be determined.These tests face two limitations, however. First, they cannot be used with cross-sectionaldata that have no variability in regional prices. Second, from their results it is easy to showthat these tests cannot be implemented when the number of observed commodities is less thantwice the number of intra-household decision-makers. In this case, the symmetry plus rank restrictions are always satis ed. This implies, for instance, that the tests do not apply in thestandard labor supply model with one Hicksian consumption good, two leisure commoditiesand two decision-makers.Fortunately,acomplementaryapproachbasedonso-calleddistributionfactors (seeBrown-ing et al. (1994)) provides tests that are less prone to these limitations. 2 These tests can beimplemented with cross-sectional data and, as shown below, only require the number of ob-served commodities to be greater than the number of intra-household decision-makers. Theshare of income accruing to each household member (Browning et al. (1994)) and the state 2 A distributionfactoris avariablethatinfluencesthedecisionprocesswithinthehousehold,butwhichdoesn’tinfluence preferences or the household budget set. 3  of the marriage market (Chiappori, Fortin and Lacroix (2002)) have recently been used asdistribution factors in empirical work. 3 In households with only two decision-makers, Bourguignon, Browning and Chiappori(1995) have shown that the restrictions imposed by distribution factors stem from the fact thatthey affect consumption choices only through their effect on the relative weight of a singleindividual in the household utility function. However, for each additional individual involvedin the decision process there is an associated relative weight. Therefore, the one-dimensionaleffect of distribution factors is lost, so that their result does not extend trivially to the caseof multi-person households. This paper generalizes the distribution factors test to householdswhere there are potentially more than two persons who participate in the decision process. Italso provides a simple method of determining the number of decision-makers when the intra-household consumption decision process is ef cient. The tests are used to investigate whetherthe decision process within bigamous households in rural Burkina Faso is ef cient. Our sur-vey data is consistent with collective rationality. Furthermore, the data indicates that all threespouses influence to some extent the household expenditures. 3 An important class of distribution factors are the “extra-environmental parameters” (EEPs) discussed byMcElroy (1990). 4  2 The Theoretical Framework Consider a household with ✂✁☎✄  members taking part in the decision process (with   ✝✆✞✄  ).Each draws his/her well-being from the consumption of  ✟  market commodities, which can beprivate, public or both. De ne ✠☛✡✌☞✎✍✑✏✓✒✔✍✖✕✗✒✙✘✚✘✛✘✛✒✜✍✣✢✥✤✛✦  as the ✟  -vector representing householdconsumption. 4 All prices are normalized to one. The household budget constraint is thereforegiven by: ✧✦✠✩★✫✪  , where ✧  is a unit vector and ✪  represents the level of household income. 5 Each member ✬  , for ✬✭★✮✄✯✒✙✘✛✘✛✘✚✒✰✱✁✫✄  , has preferences given by a strongly concave and twicecontinuously differentiable utility function ✲✴✳✶✵✷✠✹✸  . Axiom 1  The outcomes of the decision process are (weakly) Pareto-efficient. Axiom 2  Thehousehold’sdecisionprocessdependsonasetof  ✺  variables, ✻✼✡✽☞✿✾❀✏✑✒✰✾❁✕❂✒✙✘✚✘✛✘✛✒✰✾✯❃✥✤✦  ,that are independent of individual preferences and which do not affect the overall household budget constraint. 4 Following convention, we will denote vectors and matrices in boldface characters. Further, the expression ❄❆❅✔❇❉❈✛❊❉❋ denotes the partial derivatives matrix of any vector-valued differentiable function ❇❉❈✛❊❉❋  with respect to ❊  ,whose ●✭❍  thentry is ■❑❏▼▲❈✛❊❉❋❖◆ ■❁✰◗  . 5 This assumes that the household does not produce any of these ❘  goods, or equivalently, that market goodsare perfect substitutes for household production. 5
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