Chiappori, Fortin & Lacroix (2002)

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[Journal of Political Economy, 2002, vol. 110, no. 1] 2002 by The University of Chicago. All rights reserved. 0022-3808/2002/11001-0002$10.00

Marriage Market, Divorce Legislation, andHousehold Labor Supply

Pierre-Andre Chiappori

University of Chicago

Bernard Fortin and Guy Lacroix

Universite Laval, Centre de Recherche en Economie et Finance Appliquees, and Centre

Interuniversitaire de Recherche en Analyse des Organisation

This paper provides a theoretical framework for analyzing the impactof the marriage market and divorce legislation on household laborsupply. In our approach, the sex ratio in the marriage market andthe rules governing divorce are examples of distribution factors.These factors are defined as variables that affect the household mem-bers bargaining position but not preferences or the joint budget set.We extend the collective labor supply model developed by Chiappori

to allow for distribution factors. We show that our model imposes newrestrictions on the labor supply functions and eases the identificationof individual preferences and the intrahousehold decision process.The model is estimated using PSID data for the year 1988. Our resultsdo not reject the restrictions imposed by the model. Also, the sex ratioand divorce laws deemed favorable to women are found to affect laborsupply behavior and the decision process in the directions predictedby the theory and to have sizable effects.

This research received financial support from the Fonds pour la Formation de Cher-cheurs et lAide a la Recherche, the Social Sciences and Humanities Research Council ofCanada, the National Science Foundation (grant SBR9729559), and lEcole des HautesEtudes en Sciences Sociales. This paper was partly written while Fortin and Lacroix were

visiting the Departement et Laboratoire dEconomie Theorique et Appliquee, whose hos-pitality and financial support are gratefully acknowledged. We thank Simon Drolet andPhilippe Belley for able research assistance. We are also grateful to Gary Becker, Martin

Browning, Olivier Donni, Chris Flynn, Shoshana Grossbard-Shechtman, Marc Gurgand,James Heckman, Derek Neal, Robert Pollak, Sherwin Rosen, Mark Rosenzweig, WilbertVan der Klaauw, Frances Woolley, a referee, and participants in many seminars for usefulcomments on an earlier version.


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38 journal of political economy

I. IntroductionDoes household behavior depend on the relative bargaining strengthof each spouse? During the last decade, this question has attractedrenewed attention from both empirical and theoretical analysts. On theempirical side, several papers have analyzed the behavioral impact of

variables that may influence the intrahousehold distribution of power.For instance, Thomas (1990) and Browning et al. (1994) have providedevidence that the distribution of total intrahousehold income has asignificant impact on outcomes, thus rejecting the standard incomepooling prediction. More recently, Thomas, Contreras, and Franken-berg (1997), using an Indonesian survey, have shown that the distri-bution of wealth by gender at marriage has a significant impact onchildrens health in those areas in which wealth remains under thecontributors control.1 Duflo (2000) has derived related conclusionsfrom a careful analysis of a reform of the South African social pensionprogram that extended benefits to a large, previously not covered, blackpopulation.2

Relative incomes, however, are not the only possible variables thatmay affect the intrahousehold decision process. It can also depend ona range of variables that change the households environment and, inparticular, the members respective bargaining positions. Factors thataffect opportunities of spouses outside marriage can influence the in-trahousehold balance of power and ultimately the final allocation ofresources, even when the marriage does not actually dissolve (a pointalready emphasized by Haddad and Kanbur [1992]). Variables that

proxy the situation in the marriage market are natural examples of thesefactors. This intuition can be traced back to Becker (1991, chap. 3),who emphasized that the marriage market is an important determinantof intrahousehold utility distribution. In his approach, the state of themarriage market crucially depends on the sex ratio, that is, the relativesupplies of males and females in the marriage market. When the sexratio is favorable to the wifethat is, there is a relative scarcity of

womenthe distribution of gains from marriage will be shifted in herfavor. This may in turn affect intrahousehold decisions. Using U.S. dataat both the household level and the aggregate level, Grossbard-Shecht-man (1993) and Grossbard-Shechtman and Neideffer (1997) found thatan increase in the sex ratio reduces the labor force participation ofmarried women and their hours worked. Angrist (2002) uses data on

1 See also Galasso (1999) for a similar investigation.2 Specifically, Duflo finds that the consequences of this windfall gain on child nutrition

dramatically depend on the gender of the recipient. Using the same database, Bertrand,Mullainathan, and Miller (2001) study the impact on labor supply of younger women

within the household and find again that the new benefits result in a much larger reductionof labor supply when they are received by a woman.


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marriage market 39

immigrants to the United States and similarly finds that higher sex ratiosare associated with lower female labor force participation.Legislation may also play a role in the decision process. Laws gov-

erning divorce influence the ex partners level of income and the as-signment of property rights when a marriage ends. Therefore, they willaffect the spousal relative bargaining positions, at least to the extentthat divorce matters as an outside opportunity. For instance, laws reg-ulating alimony, support orders, and the right to remarry as well asthose defining marital property and its division at divorce may redis-tribute power in a marriage. Therefore, they are likely to influencedistribution and outcomes within the household. In a recent paper, Gray(1998) relates changes in female labor supply to the adoption of uni-lateral-divorce laws3 in many states during the 1970s.4 Using various data

sources and exploiting the legal changes that took place between twoparticular years, he finds a significant impact when marital property lawsare controlled for. In a related way, Rubalcava and Thomas (2000) arguethat variations in Aid to Families with Dependent Children across statesdirectly affect the reservation welfare a spouse may be able to achievein the case of divorce. Finally, the decision process may be influencedby special features of the marriage contracts. Lundberg and Pollak(1994) insist that whether marriage agreements are binding or not is adeterminant of the intrahousehold decision process.5

Together, these empirical investigations very strongly suggest that in-trahousehold bargaining has a significant impact on behavior andshould be analyzed with care. A striking fact, however, is that most ofthese works are not explicitly grounded in a structural model. 6 For that

3 Unilateral-divorce laws specify that either spouse can initiate divorce. By contrast,mutual-consent laws require either the agreement of both spouses or the demonstrationof marital fault.

4 One can argue that divorce laws also affect intrahousehold decisions through theireffects on the probability of divorce. For instance, according to the usual argument,unilateral divorce raises divorce rates by increasing access to divorce. As a result, in uni-lateral-divorce states, married women have more incentives to reduce their marriage-specific investment and to increase their labor supply, given the lack of compensation atdivorce for that type of investment and for reduced marketable human capital (e.g., Peters1986; Parkman 1992). Yet, empirical evidence generally supports the view that divorcelaws do not affect divorce rates. For instance, Peters (1986) and Gray (1998), amongothers, do not reject the hypothesis that the frequency of divorce is similar for householdsfacing unilateral- or mutual-consent divorce laws. According to Friedberg (1998), theadoption of unilateral-divorce laws increased divorce rates, but this effect seems to dis-appear after a decade (Wolfers 2000). These results are in line with the Coase theorem,

which asserts that changes in divorce laws should not affect efficiency in marriage, and

hence the divorce rates, as long as there are symmetry of information and trivial bargainingcosts within marriage (Becker 1991).5 Unfortunately, it is difficult to construct empirical measures of these features.6 Grossbard-Shechtman and Neideffer (1997) have developed a choice-theoretic model

of married womens labor supply in which the reservation wage depends on marriagemarket conditions. However, their empirical analysis is based on a reduced-form modelthat does not take into account the restrictions imposed by their structural model.


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reason, the interpretation of their empirical results is not straightfor-ward. While they certainly suggest that intrahousehold decision makingis more complex than implied by the traditional unitary model, theydo not say much on the true nature of the actual process.

On the theoretical side, it has also become increasingly clear that theconventional unitary model of household behavior, based on the fictionof a single household utility that is maximized under budget constraint,does not provide an adequate context for addressing these issues. Var-ious contributions have tried to introduce alternative frameworks in

which intrahousehold decision processes can be adequately investigated.Manser and Brown (1980) and McElroy and Horney (1981) have pro-posed models based on cooperative game theory. These attempts havebeen generalized by Chiappori (1988), Bourguignon et al. (1993),

Browning and Chiappori (1998), and Chiappori and Ekeland (2001),who have developed a collective framework. In its most general ver-sion, the collective approach relies on the sole assumption that house-hold decisions are Pareto-efficient. It thus nests all models based oncooperative bargaining, at least under symmetric information. It can beproved that this minimal setting is sufficient to generate strong testablerestrictions on behavior. Under additional restrictions, the collectivemodel allows one furthermore to identify the characteristics of the un-derlying structural model (i.e., individual preferences and the decisionprocess) from observed behavior.

While the collective model provides an appealing theoretical frame-work to analyze household behavior, it needs to be generalized to takeinto account variables that, as discussed above, may affect the distri-

bution of intrahousehold power. The first goal of the present paper isto perform this task. The starting point of our analysis is the conceptof distribution factors (Browning and Chiappori 1998). These factorsare defined as variables that can affect the intrahousehold decisionprocess without influencing individual preferences or the joint con-sumption set. The sex ratio is a natural example of a distribution factor.Divorce laws can also be regarded as distribution factors insofar as theyinfluence outcomes only through their impact on spousal bargaining

within marriage. Other examples of distribution factors include specialfeatures of marriage contracts or the share of total nonlabor incomeunder the control of one spouse.7

In this paper, we theoretically investigate and empirically estimate theeffects of distribution factors in the context of a structural microeco-

nomic model of household behavior. The underlying intuition is quite

7 One must reckon that these variables may raise delicate endogeneity problems. Forinstance, variations in nonlabor income over a cross section are likely to be correlated

with other (unobservable) determinants of household decisions (Behrman, Pollak, andTaubman 1995).


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marriage market 41

simple. Whenever the distribution factor under considerationsay, thesex ratiois favorable to one membersay, females are scarcer, whichpresumably increases the wifes bargaining position within the house-holdthe respective weights in the decision process will be shifted inher favor. Standard income effects should, all else equal, lead to a re-duction in female labor supply and an increase in male labor supply.The main purpose of our model is to provide a clean theoretical frame-

work in which this idea can be worked out and to point out the variousrestrictions that an explicit model of the household decision processimposes on behavior. To do so, we extend various versions of the col-lective model by introducing distribution factors. First, we consider themost general collective framework, where each agents utility is allowedto depend on both members consumptions and labor supplies; in other

words, the model allows for intrahousehold externalities of any kind(including public goods). In the absence of distribution factors, resultsby Browning and Chiappori (1998) and Chiappori and Ekeland (2001)imply that a three-commodity model like the one used here cannotgenerate testable restrictions on behavior. We show, however, that in thepresence of at least two distribution factors, the collective model, evenin its most general form, strongly restricts the form of labor supply.8

In its most general version, however, the collective model is notuniquely identified. For that reason, we next concentrate on the par-ticular collective model of labor supply introduced by Chiappori (1992).The identifying assumption here is that household members have ego-tistic or Beckerian caring preferences (Becker 1991). These prefer-ences allow for interdependence of altruistic utility but impose weakseparability between goods consumed by a household member and thoseconsumed by his or her spouse. Efficiency has, in this setting, a verysimple interpretation: household decisions can be modeled as a two-step process, whereby individuals first share their total nonlabor incomeaccording to some sharing rule and then maximize their own utilitiessubject to separate budget constraints. In particular, the intrahouseholddecision process can be fully summarized by the sharing rule. We extendthis model by allowing the sharing rule to depend on the various dis-tribution factors under consideration as well as on wages and nonlaborincome. We show that the main properties of Chiapporis initial modelare preserved. In particular, it is still possible to identify individual pref-erences (up to a translation) and the sharing rule (up to an additive

constant) from the sole observation of labor supply. Furthermore, thenew context allows for a different identification procedure that is both

8A related result was already mentioned in Bourguignon, Browning, and Chiappori(1995), although not in the context of labor supply. For empirical confirmation, see, e.g.,Browning et al. (1994) and Thomas et al. (1997).


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simpler and more robust than before. It follows that the impact ofdistribution factors on behavior (if any) can in this context be given adirect interpretation in terms of intrahousehold transfers, and the welfare con-sequences can readily be assessed.

The presence of distribution factors also generates new testable pre-dictions. For instance, in addition to the general restrictions evokedabove, the theory imposes a close relationship between the effect of anydistribution factor and the impact of cross wages on labor supply. Thesepredictions are very unlikely to be fulfilled unless the model at stake iscorrect, which provides a rather strong test of our approach.

The final contribution of the paper is to estimate and test our col-lective model with the sex ratio and a divorce laws index as distributionfactors. The sex ratio we use is computed by age, race, and state of

residence. Our divorce laws index, which is an indicator of the extentto which the laws are likely to be favorable to women, is also specificto the state of residence. While most papers that have analyzed the

various behavioral effects of divorce laws have focused on one or twoof them, we specifically take into account the four following features:mutual consent versus unilateral divorce, property division, enforcementof support orders, and spousal interest in professional degrees and li-censes. The availability of two distribution factors allows us to test notonly the collective model with private goods but also the general version

with externalities of any kind.Our sample is drawn from wave 23 of the Panel Study of Income

Dynamics (PSID) (1989 interview year) and focuses on couples in whichboth spouses work. We find that both the sex ratio and the divorce laws

affect the spouses labor supply in exactly the manner predicted by thetheory. The parametric constraints associated with both versions of themodel are not statistically rejected. Finally, the parameters of the sharingrule are recovered. According to these parameters, changes in the sexratio and in the divorce laws index have sizable impacts on incometransfers within the households.

The paper is structured as follows. Section II presents our theoreticalframework. Section III discusses the choice of the empirical specificationused for estimation and testing. Section IV describes our data and em-pirical results. Finally, Section V presents conclusions.

II. The Model

A. The Basic Setting

In this section, we develop a collective labor supply model that takesinto account distribution factors. In this framework, the household con-sists of two individuals with distinct utility functions, and the decision


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process, whatever its true nature, leads to Pareto-efficient outcomes.This assumption seems quite natural, given that spouses usually knoweach others preferences pretty well (at least after a certain period oftime) and interact very often. Therefore, they are unlikely to leavePareto-improving decisions unexploited (however, see Udry [1996]).

A General Framework

Formally, let hi and for 2, denote, respectively, member isiC , i p 1,labor supply (with ) and consumption of a private Hicksiani0 h 1composite good whose price is set to unity. We start from the mostgeneral version of the model, in which member is welfare can dependon his or her spouses consumption and labor supply in a very general

way, including, for instance, altruism, public consumption of leisure,positive or negative externalities, and so forth. In this general frame-

work, member is preferences are represented by some utility functionC1, z). Here, z is a K-vector of preference factors,i 1 2 2U (1 h , 1 h , C ,

such as age and education of the two agents. Also, let w1, w2, and ydenote respective wage rates and household nonlabor income. Finally,let s denote an L-vector of distribution factors.

Under the collective framework, intrahousehold decisions are Pareto-efficient. For any given (w1, w2, y, z, s), hence, there exists a weightingfactor m(w1, w2, y, z, s) belonging to [0, 1] such that (h

i, ) solves theiCfollowing program:

1 2max mU (1 m)U (P1)1 2 1 2

{h ,h ,C ,C }

subject to

1 2 1 2w h w h y C C ,1 2

i0 h 1, i p 1, 2,

where the function m is assumed continuously differentiable in its ar-guments. It should thus be clear that the particular location of thesolution on the Pareto frontier depends on all relevant parameters, sincethe value of m depends on w1, w2, y, z, and s. Furthermore, since the

vector of distribution factors, s, appears only in m, a change in s doesnot affect the Pareto frontier but only the final location on it. In the

particular case in which m is assumed to be constant, the collectiveframework corresponds to the unitary model with weakly separablehousehold preferences. In this situation, the distribution factors haveno effect on behavior.

In this general setting with interior solutions assumed, a first testable


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restriction arises on labor supplies. This restriction is given by the fol-lowing result.Proposition 1 (Bourguignon, Browning, and Chiappori 1995). Let

be solutions to program (P1). Thenih(w , w , y, z, s), i p 1, 2,1 21 2

h/s h/sk kp Gk p 2, , L. (R)

1 2h/s h/s1 1

Proof. For any fixed m, h1 and h2, as functions of w1, w2, y, and m, arewell-behaved Marshallian labor supplies. In particular, one gets

i ih(w , w ,y, z, s) p H(w , w ,y, z, m(w , w ,y, z, s)), i p 1,2,1 2 1 2 1 2

so that

i i

h

H

mps m sj j

and

ih/s m/sk k

pi

h/s m/s1 1

is independent of i. Q.E.D.The basic intuition here is that distribution factors affect consumption

and labor supply choices only through the location chosen on the Paretofrontier or, equivalently, through the implicit weighting of each spousesutility. Since this weighting is unidimensional, this implies that the ratiosof the impacts of all distribution factors on the two labor supplies are

equal. It is worth stressing that these restrictions appear only when thereare at least two distribution factors. If it is the case, they provide a testfor Pareto efficiency in a general collective model of labor supply. Recentresults by Chiappori and Ekeland (2001) imply that these conditionsare also sufficient.

Egotistic Preferences

It should, however, be emphasized that this general version of the col-lective model cannot be uniquely identified from the sole knowledgeof labor supplies. There is a continuum of different structural modelsthat are observationally equivalent, that is, that generate identical laborsupply functions. Therefore, in our empirical analysis, we also estimate

and test a model that imposes additional identifying assumptions. Fornow we shall make the following assumption.

Assumption E. Egotistic preferences.Individual utilities have the formz), where is strictly quasi-concave, increasing, and con-i i i i U (1 h, C , U

tinuously differentiable for 2.i p 1,


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According to assumption E, household members have egotistic pref-erences in the sense that the welfare of member i does not depend onthe consumption of member 9 The corresponding model withoutj ( i.distribution factors has been studied by Chiappori (1992). A first result,

which can readily be extended to our framework, is that under as-sumption E, efficiency has a very simple interpretation. Indeed, considerthe household as a two-person economy. From the second fundamental

welfare theorem, any Pareto optimum can be decentralized in an econ-omy of this kind. Specifically, we have the following result.

Proposition 2. Under assumption E, program (P1) is equivalent tothe existence of some function f(w1, w2, y, z, s) such that each memberi ( 2) solves the following program:i p 1,

i i imax U (1 h, C , z) (P2)i i{h,C }

subject to

i i iw h f C ,i

i0 h 1,

where and1 2f p f f p y f.Proof. See Chiappori (1992).The interpretation is that the decision process can always be consid-

ered as a two-stage process: first, nonlabor income is allocated betweenhousehold members, and then each member separately chooses a laborsupply (and private consumption), subject to the corresponding budget

constraint. The function f is called the sharing rule. It describes the waynonlabor income is divided up, as a function of wages, nonlabor income,distribution factors, and other observable characteristics.10

B. Restrictions on Labor Supplies and the Sharing Rule

The collective framework with egotistic preferences imposes certain re-strictions on the labor supply functions. To show this, let us first assumethat the unrestricted labor supply functions hi(w1, w2, y, z, s) are con-tinuously differentiable. From (P2), with interior solutions assumed,these functions can be written as

1 1h p H (w , f(w , w , y, s, z), z) (1a)1 1 2

9 However, our approach can be extended at basically no cost to caring preferences,where each persons utility depends both on his or her subutility index and on his or herspouses (see below).

10 In the presence of household public goods, a sharing rule can still be defined butconditionally on the level of these goods.


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and2 2h p H (w , y f(w , w , y, s, z), z), (1b)2 1 2

where is member is Marshallian labor supply function.iH(7)The particular structure of equations (1a) and (1b) imposes testable

restrictions on labor supply behavior and allows us to recover the partialsof the sharing rule. It is important to note that, in contrast with theprevious result, onedistribution factor is sufficient for these conclusionsto hold. The intuition goes as follows. Consider a change in, say, member1s wage rate. This can have an income effect on his or her spousesbehavior only through its effect on the sharing rule, just as nonlaborincome and the distribution factor. Thus the impact of these variables onthe labor supply behavior of member 1 allows us to estimate the marginal

rate of substitution between w2 and y as well as between s and y in thesharing rule. Technically, it generates two equations involving the cor-responding partials of the sharing rule. The same argument applies tomember 2s behavior, which leads to two other equations. These fourequations allow us to directly identify the four partials of the sharingrule. Finally, cross-derivative constraints on the sharing rule impose re-strictions on the model that can be tested.

To be more precise, using equations (1a) and (1b), define A pand whenever1 1 2 2 1 1 2 2 1 2h /h , B p h /h , C p h /h , D p h /h h 7 h ( 0,w y w y l s y l s y y y 2 1 l l

for Note that all these variables are observable and canl p 1, , L.thus be estimated. Then one has the following results (where the sub-script has been removed for notational convenience).l p 1

Proposition 3. Take any point such that Then the fol-1 2h 7 h ( 0.y ylowing results hold: (i) If there exists exactly one distribution factorsuch that the following conditions are necessary for any pairC ( D,(h1, h2) to be solutions of (P2) for some sharing rule f:

D CDp , (2a)( ) ( )

s D C y D C

D BCp , (2b)( ) ( )

w D C y D C1

D AD

p , (2c)( ) ( )w D C y D C2

CD BCp , (2d)( ) ( )

w D C s D C1


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CD ADp , (2e)( ) ( )

w D C s D C2

BC ADp , (2f)( ) ( )

w D C w D C2 1

BC D C1 1 1h h h 0, (2g)w y( ) ( )1 D C D

and

AD D C2 2 2h h h 0. (2h)w y ( ) ( )2 D C C(ii) Under the assumption that conditions (2a)(2h) hold and for agiven z, the sharing rule is defined up to an additive function k(z)depending only on the preference factors z. The partial derivatives ofthe sharing rule with respect to wages, nonlabor income, and the dis-tribution factor are given by

Df p ,y

D C

CDf p ,s

D

CBC

f p ,w1 D C

ADf p . (3)w2 D C

If there are several distribution factors ( ), an additional setl p 1, , Lof necessary and sufficient conditions are

C Cl 1p , l p 2, , L. (2i)

D Dl 1

Moreover, the partial derivatives of the sharing rule with respect to the

additional distribution factors are given by

C Dl lf p , l p 2, , L. (4)sl D Cl l

Proof. See the Appendix.


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These results suggest three remarks.1. Conditions (2a)(2h) are analogous to Slutsky restrictions in the(general) sense that they provide a set of partial differential equationsand inequalities that must be satisfied by the labor supply functions inorder to be consistent with the collective model. It is important to note,in particular, that these conditions do not rely on any particular assumption on

the functional form of preferences. Of course, the empirical test of thesepredictions is greatly simplified by the use of specific functional forms,as will be the case below. But, in principle, the nature of the restrictionsis nonparametric.11

2. The form of the conditions above is quite different from thoseobtained in Chiappori (1992) for a similar model without distributionfactors. As a matter of fact, the introduction of distribution factors deeply

changes the way the model is identified. In Chiapporis initial contri-bution, identification required second-order derivatives. In our case, tothe contrary, equations (3) and (4) show that the partials of the sharingrule (hence the sharing rule itself, up to an additive constant) can berecovered as functions of the first-order derivatives of the labor supplies(functions A, B, Cl, and Dl). This suggests that the kind of identificationthat may obtain is more robust in this case.12 The same remark appliesto the testable predictions generated by the model, although the orderof derivation must then be increased by one. The conditions aboveinvolve the first derivatives of the functions A, B, Cl, and Dl, and hencethe second derivatives of labor supplies, whereas third derivatives werein general involved in Chiapporis initial model.

Finally, condition (2i) implies that the relative effects of distribution

factors on each labor supply are equal, that is, for1 1 2 2h /h p h /h ,s s s s l 1 l 1since both members of this equation are equal tol p 2, , L, f /f .s sl 1

This conclusion is not surprising since the model at stake, as a particularcase of the general model developed above, must satisfy condition (R)of proposition 1.

C. Caring

The results derived in proposition 2 are based on the assumption thatpreferences are egotistic. However, as shown in Chiappori (1992), theyalso hold in the more general case of caring agents (see Becker 1991),that is, agents whose preferences are represented by a utility function

11 However, a nonparametric estimation procedure requires a detailed modeling of theunobserved heterogeneity (see Blundell et al. 2000).

12 Note, however, that an alternative approach relying on second derivatives can still beused (in the case, e.g., in which for all l). This can be shown to generate identicalC pDl lresults. Intuitively, the second-order conditions in Chiappori (1992) are direct conse-quences of the restrictions in proposition 3.


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that depends on both their egotistic utility and their spouses. Formally,member is utility function can be written as

i i 1 1 1 2 2 2W p W[U (1 h , C , z), U (1 h , C , z)] for i p 1, 2, (5)

where is continuous, increasing, and quasi-concave in egotisticiWutilities and These utility functions impose separability between1 2U U .a members own private goods and his or her spouses. It is clear thatany decision that is Pareto-efficient under caring would also be Pareto-efficient were the agents egotistic. Assume not; then it would be possibleto increase the egotistic utility of a member without decreasing the utilityof the other. But this would increase the caring utility of at least onemember without reducing the caring utility of any member, a contra-diction. In fact, the Pareto frontier of caring agents is a subset of the

Pareto frontier derived by assuming that they are egotistic (Chiappori1992). In Section III, we shall use the results obtained up to now toderive the parametric restrictions imposed by the collective model tothe particular labor supply system considered in our econometric ap-proach and to recover the corresponding sharing rule.

D. Distribution Factors and Labor Supply: Alternative Explanations

The empirical work below applies the previous results on a specific dataset, using the sex ratio and an index for divorce laws as distributionfactors. While the effects of these variables on the bargaining positionof spouses provide natural explanations for their correlation with laborsupply behavior, they are by no means exclusive. For instance, spatial

variations in the sex ratio (defined as the males/females ratio) couldbe related to labor market considerations (Grossbard-Shechtman 1993).One interpretation is that men will be observed to work longer hoursin states with a low sex ratio because of a relatively strong demand fortheir services. The opposite will be observed for women. Note that thesepredictions run counter to those of the collective model, in which casethe relative scarcity of men should increase their bargaining power andthus their leisure through increased transfers from their spouse. Thetwo theories have opposite empirical predictions, which suggests thatdata should allow us to discriminate between them.

A second explanation involves the demand for labor. Assume thatsome states specialize in male sectors, that is, sectors with a strongerrelative demand for male labor supply. These states will attract relatively

more men through migration. Therefore, they will have high (endog-enous) sex ratios and presumably high male hours of work. Femalehours of work may conceivably be well below the national average insuch states. Conversely, states that concentrate in female sectors willhave low (endogenous) sex ratios and high female hours of work. Note


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that this effect, in contrast with the previous one, goes in the samedirection as the collective explanation. The empirical distinction be-tween them is thus less straightforward but is still not out of reach. First,strong labor demand should translate into high wage rates. Condition-ing the hours equations on the individual wage rates should at leastpartially account for the tight male or female labor markets. A second

way around is to focus on the relation between the sex ratio and thelabor supply of singles. According to the marriage market hypothesis,the sex ratio should have no effect on their labor supply (at least if oneignores its impact on transfers to potential spouses). The labor markethypothesis, to the contrary, predicts that the sex ratio should influencethe labor supply of both singles and couples. This suggests a simple andrather strong test that allows us to discriminate between the two

explanations.Interestingly, a similar analysis can also be conducted with respect to

the correlation between divorce laws and household labor supply. Whilethe impact of these laws on the bargaining power of spouses is likelythe most plausible explanation, alternative theories can be proposed to

justify the correlation. Indeed, a host of socioeconomic or cultural fac-tors may underlie the design of divorce laws (e.g., Ellman and Lohr1998). Such factors may or may not be correlated with spouses laborsupply. As long as the (unobservable) socioeconomic factors that affectdivorce laws and spouses labor supply also influence singles labor sup-ply, we should observe a correlation between the divorce laws and sin-gles labor supply. No correlation should be expected if the collectivemodel is the proper explanation. Just as previously, focusing on the

relation between divorce laws and the labor supply of singles providesa simple test to assess the importance of alternative explanations.

Finally, it should be stressed that the collective model provides strongrestrictions on how distribution factors may affect behavior. Specifically,the conditions in proposition 3 relate the effect of these factors to thatof wages and nonlabor income. While these conditions are direct con-sequences of the collective setting, they have no reason to hold wheneverthe effect under consideration stems from labor market mechanisms.Consequently, they provide a distinct and additional means of testingthe collective explanation. These tests will be carefully considered inthe empirical sections.

III. Parametric Specification of the Model

A. Functional Form of Labor Supplies

In order to estimate and test a collective model of labor supply, we mustfirst specify a functional form for individual labor supply functions. Let


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marriage market 51

us consider the following unrestricted system, where for convenienceand to reflect the empirical analysis, two distribution factors are as-sumed:

1h p f flog w f log w f y0 1 1 2 2 3

f log w log w f s f s f z (6)4 1 2 5 1 6 2 7

and

2h p m m log w m log w m y0 1 1 2 2 3

m log w log w m s m s m z, (7)4 1 2 5 1 6 2 7

where the fis and the mis, for are scalars, and and i p 1, ,6, f m7 7

are K-vectors of parameters.The generalized semilog system (6) and (7) satisfies a number of

desirable properties. First, in its unrestricted form, it does not imposeall the (equality) conditions of the collective model. Therefore, thecollective model yields a set of restrictions that can be empirically tested.Second, as shown below, these restrictions do not impose unrealisticconstraints on behavior. Third, if the collective restrictions are satisfied,it is possible to recover a closed form for the sharing rule (up to anadditive function k(z)) and for the pair of individual indirect utilityfunctions (for any given k(z)). Finally, the fact that equations (6) and(7) are linear in parameters eases the estimation.

Of course, this generalized semilog system also has some limitations.While some restrictions of the unitary model consistent with this system

do not impose unrealistic labor supply behavior, other restrictions doand therefore cannot be tested.13 However, this should not be a seriousproblem since the unitary model of household labor supply has beenrejected in many studies (e.g., Lundberg 1988; Fortin and Lacroix 1997).Second, labor supply curves are either everywhere upward sloping oreverywhere backward bending, though the sign of can changeih/wi

with the level of wj ( ).14 Note, however, that the log form for thej ( i

13 More specifically, the unitary model imposes that labor supplies are independent fromany distribution factorand that theSlutsky matrixof compensated wage effects is symmetricand semidefinite positive. The former constraint requires that f pf p m p m p 0.5 6 5 6These restrictions can be tested. However, the symmetry of the Slutsky matrix requires inaddition either that (i) which implies that each laborf pf p f p m p m p m p 0,2 3 4 1 3 4supply depends only on the own wage rate and on preference factors, or that (ii) f p0

and which implies that labor sup-m , f p m , f p m , f pf pf p m p m p m p 0,0 3 3 7 7 1 2 4 1 2 4plies are the same and depend only on nonlabor income and on preference factors. It isclear that these two cases impose severe constraints on behavior.

14 Using our data set, we tested a more flexible functional form by introducing a second-order polynomial in and y. No coefficients associated with the second-orderlog w , log w ,1 2

variables were found significant (except for the one associated with the cross term inand ).log w log w1 2


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wage rates is likely to reflect more realistic behavior than the linear formthat is frequently used in empirical studies. Thus it allows the effect ofthe wage rate on labor supply to decrease with the level of hours of

work (when the labor supply is upward sloping), which is likely to bethe case.15

The restrictions imposed by the collective model (see proposition 3)to the generalized semilog system can easily be derived. First, using thedefinitions of A, B, Cl, and Dl 2), one gets(l p 1,

f f log w2 4 1A p ,

w f2 3

m m log w1 4 2B p ,

w m1 3

and The conditionC p f/f, C p f/f, D p m /m , D p m /m .1 5 3 2 6 3 1 5 3 2 6 3is satisfied unless It should be stressed that C ( D m /f p m /f.1 1 3 3 5 5

under the collective model, this equation is unlikely to be satisfied. Forone thing, represents the ratio of income effects on labor supplies;m /f3 3this ratio is positive as long as leisure is a normal good for both membersand an increase in y is shared between them. On the other hand,

represents the corresponding ratio of the effects of the distribu-m /f5 5tion factor. Since, by definition, any distribution factor affects the hus-bands and the wifes share of nonlabor income in opposite directions,the ratio must be negative.

Under the assumption that the necessary and sufficient con-C ( D ,1 1ditions take the following form:

m m m4 5 6p p . (8)

f f f4 5 6

Equations (8) summarize the equality restrictions on labor supplyarising from the collective framework. In other words, given our func-tional form, they are equivalent to conditions (2a)(2f) and (2i). Theyactually take a very simple form since only equations (2f) and (2i)impose restrictions.16 This indicates that the functional form under con-sideration fits well the collective model. In practice, equations (8)impose testable cross-equation restrictions in our labor supply system.They require the ratio of the marginal effects of the cross term in

and to be equal to the corresponding ratio of the marginallog w log w1 2effects of each distribution factor on labor supplies. These restrictions

15 It is also worth mentioning that our specification can easily allow for interactionsbetween distribution and preference factors.

16 Equations (2a)(2e) are always satisfied since all partial derivatives in these equationsare zero.


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marriage market 53

stem from the fact that the cross term and the distribution factors enterlabor supply functions only through the same function f. Notice thatthe last equality in (8) holds also when externalities are allowed sinceit corresponds to (R) in proposition 1.

B. Sharing Rule

If the restrictions (8) are satisfied, the partials of f are given by

f m3 4f p ,y

D

m m4 4f p f, f p f,s 5 s 61 2D D

f m m log w4 1 4 2f p ,w1 D w1

m f f log w4 2 4 1f p , (9)w2 D w2

where D p f m f m .3 4 4 3Solving this system of four differential equations, one obtains the

sharing rule equation

1f p (m f log w f m log w f m log w log w1 4 1 2 4 2 4 4 1 2

D

f m y m f s m f s ) k(z). (10)3 4 4 5 1 4 6 2

In equation (10), the function k(z) is not identifiable, since the variablez affects both the sharing rule and the preferences. This reflects thefact that, for any given z, the sharing rule can be recovered up to anadditive constant for each individual.

C. Individual Labor Supplies

It is also possible to recover the individual labor supply functions as-sociated with this setting. Since they must have a functional form con-sistent with equations (1a) and (1b), it is clear, from equations (6), (7),and (10), that they take the following semilog form:

1h p a log w a f a (z) (11)1 1 2 3

and

2h p b log w b (y f) b (z). (12)1 2 2 3


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Using the expressions for the partials of the restricted system (1a)and (1b) with respect to (w1, w2, y) and the partials of f given by (9),one easily recovers the following parameters: a p (f m f m )/m ,1 1 4 4 1 4

and The functions a3(z)a p D/m , b p (f m f m )/f, b p D/f.2 4 1 4 2 2 4 4 2 4and b3(z) are not identifiable since they depend on k(z) in equation(10).17

Slutsky conditions on compensated individual labor supplies (see [2g]and [2h] in proposition 3) are given by

a b1 1 a h 0, b h 0.2 1 2 2

w w1 2

These conditions are verified for each observation in the empirical anal-

ysis. Global conditions for these inequalities are a 0, a 0, b 0,1 2 1and b 0.2

D. Indirect Utility Functions

It can be shown (Stern 1986) that the indirect utility functions consistentwith the labor supply functions (11) and (12) must have the followingform:

exp(a w )2 11v (w , f , z) p [a f a (z) a log w ]1 1 2 1 3 1 1[ ]a2a w2 1

a exp(t)1 dt( )a t2

and

exp(b w )2 22v (w , f , z) p [b f b (z) b log w ]2 2 2 2 3 1 2[ ]b2b w2 2

b exp(t)1 dt.( )

b t2

It is easy to show that Roys identity applied to each of these indirectutility functions (or the corresponding expenditure functions) yields

the individual labor supply system (11) and (12). These functions canbe used to perform intrahousehold welfare analysis of changes in ex-ogenous variables.

17 Identification of these functions would require additional identifying restrictions. Forinstance, it obtains whenever a variable in z affects preferences but not the sharing rule.


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marriage market 55

IV. Data and Empirical ResultsA. Data

The data we use in this study are taken from the University of MichiganPanel Study of Income Dynamics for the year 1988 (interview year 1989).Our sample consists of 1,618 households in which both spouses havepositive hours of work and are between 30 and 60 years of age. 18 Thislatter restriction was used in order to eliminate as much as possible full-time students and retired individuals and to reduce cohort effects. Re-moving couples in which spouses are aged less than 30 increases theproportion of stable households, for which the hypothesis of efficiencyin the intrahousehold decision process is more likely to be satisfied.

The dependent variables, male and female annual hours of work, are

defined as total hours of work on all jobs during 1988. The measure ofthe wage rate is the average hourly earnings, defined by dividing totallabor income over annual hours of work. Nonlabor income includes,among other things, imputed income from all household net assets19

and is net of total household savings.20 This variable is treated as anendogenous variable in the empirical section. It should be stressed thatthe PSID provides information on net assets at the beginning of 1984and 1989. Therefore, our measure of savings is the annual averagechange in total net household assets over this period (expressed in 1988dollars). In order to reduce measurement errors on this variable, wefurther restricted our sample to households with stable couples over the198489 period.

Table 1 presents descriptive statistics for our sample. Panels AC re-

port statistics on individual households, whereas panel D focuses onvarious aspects of the marriage market. According to the data in panel

18 Conditioning the sample on working spouses may induce a selectivity bias, especiallyin the case of females. We ignore this bias in the analysis. The basic reason is that sucha correction requires an extension of the collective model to corner solutions, a task thatis beyond the scope of this paper. The reader is referred to Donni (2002) for recent workin this direction. See also Blundell et al. (2000) for an investigation of the related (butdifferent) problem of discrete labor supply decisions. There is some evidence that theselectivity bias is not likely to be a problem, though. For instance, using PSID data, andon the basis of a standard recursive labor supply model, Mroz (1987) could not reject thehypothesis of no selectivity bias in the womens labor supply equation.

19We use a nominal interest rate of 12 percent. We also experimented with nominalinterest rates of 8 percent and 10 percent, but this did not significantly affect the results.

20 Removing household savings from the measure of nonlabor income is consistent withan intertemporally separable life cycle model involving a two-stage budgeting process. In

the first stage, the couple optimally allocates life cycle wealth over each period in orderto determine the vector of period-specific levels of nonlabor income net of savings. Ateach period, nonlabor income net of savings plus total household wage income is equalto the level of household consumption expenditures (thisrepresents period-specifichouse-hold budget constraints). The second stage corresponds to period-specific Pareto-efficientallocations of goods and labor supplies (see Blundell and Walker [1986] for a discussionof a life cycle, two-stage budgeting process in the case of a one-individual household).


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TABLE 1Descriptive Statistics

MeanStandardDeviation Minimum Maximum

A. Women

Hours of work 1,741 570.69 500 5,120Log wage 2.04 .69 2.30 4.07

Age 37.77 7.74 21 60Schooling* 5.17 1.48 1 8

White .75Black .23

B. Men

Hours of work 2,235 635.48 88 5,824Log wage 2.47 .61 .26 4.61

Age 40.62 7.82 30 60Schooling* 5.25 1.64 1 8White .75Black .23

C. Family Characteristics

Children 6 .33 .62 0 3Children 717 .97 1.07 0 6Nonlabor income 8,068.80 24,197.42 113,984.76 344,804.75

D. Marriage Market

Sex ratio:White .49 .01 .46 .57Black .46 .02 .41 .56

Divorce laws:Property division

(communityp1) .04 0 1

Mutual/unilateral(mutualp1) .22 0 1

Enforcement (courtpaymentp1) .75 0 1

Spousal interest (degree asassetp1) .54 0 1

Divorce laws index 2.48 .88 1 4

* The education variables follow the 1989 coding. Thus, e.g., a value of 4 corresponds to 12 grades and no furthertraining, whereas a value of 5 corresponds to 12 grades plus nonacademic training.

B, men work, on average, more yearly hours than women and earn asomewhat higher hourly wage rate. Men are also nearly three years olderthan their spouses, on average; they both have similar schooling levels.The distribution by race is identical among men and women. A closelook at the data reveals that there are very few interracial marriages in

our sample.Panel C reports the average number of preschoolers and school-age

children per household as well as household nonlabor income. Thesevariables are all treated as endogenous in the empirical work. Althoughthere is mixed evidence concerning the endogeneity of number of chil-


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marriage market 57

dren in womens labor supply (e.g., Mroz 1987), we deem it preferableto instrument these variables. The average nonlabor income per house-hold is approximately $8,000. Its variance is large essentially because

younger households tend to have negative assets (mortgage), whereasolder households have (on average) positive assets.

Our sex ratio index is computed at the state level using data fromthe Census of Population and Housingof 1990. It corresponds to the num-ber of males that are of the same age and same race as the husband ofeach household over the corresponding number of males and females.

We experimented with various definitions of the sex ratio: means of sexratios using the number of females who are two years younger than thehusband or on the basis of individuals who are at most two or five years

younger than the husband of each household. The results were veryrobust to the definition used.21 Our sex ratio index is computed underthe assumption that the relevant marriage market is limited to onesown race. As shown in table 1, the mean sex ratio is slightly higher for

whites than for blacks, but that for blacks has a larger variance that isobservable both statewise and agewise.

The model was also estimated using sex ratios computed at the countylevel. The county of residence reported in the PSID was matched tocounty-level data on male and female populations from the 5 PercentPublic Use Microdata Sample of the 1990 census. Unfortunately, manycases turned out to have too few observations to compute meaningfulsex ratios. Sex ratios for blacks were particularly prone to measurementerrors. We thus used state sex ratios as instruments for county sex ratios.

The results were very similar to those reported here.22Four features of divorce laws are considered in the empirical analysis:

mutual consent versus unilateral divorce, property division, enforcementof support orders, and spousal interest in professional degrees and li-

21A very natural question, however, is whether the appropriate sex ratio is measured interms of the marriage market or, alternatively, in terms of the remarriage market. Theissue here boils down to a commitment problem. Under the assumption that couples areable to make an up-front binding commitment at the date of marriage, only the balanceof powers (hence the sex ratio) at that dateshould matter. If, conversely, such commitmentscannot be perfectly enforced, then one should rather consider the current value of thesex ratio. From a theoretical perspective, one can probably prefer the secondinterpretationsince members cannot commit not to divorce. Should the prospects on the remarriagemarket brutally evolve, a renegotiation of the initial contract is difficult to prevent, es-

pecially when, in the new context, remaining married would violate one members indi-vidual rationality constraint. An informal support of this view is provided by the findingof Thomas et al. (1997) that wealth at marriage does not seem to influence the intra-household balance of power in those Indonesian regions in which wealth is traditionallypooled within the household.

22 For the sake of brevity, these results are omitted from this paper, although they areavailable on request.


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censes.

23

As of 1989, most states (42) had adopted unilateral-divorcelaws. Among these states, as many as 24 allowed unilateral divorce onlyafter a lengthy separation that lasted between six months and five years.

We follow Peters (1986) and Gray (1998) and define them as mutual-consent states. Property division refers to state marital property systems,

which can be either community property or common law.24 Courts donot have the same discretion to protect vulnerable parties (usually

women) under common law. Therefore, married womens bargainingpower is likely to be stronger in community property jurisdictions. Fur-thermore, insofar as household assets are disproportionately in the hus-bands name, mutual-consent divorce law also advantages women incommon-law states (which represent 96 percent of our sample) butdisadvantages women in community property states.25 Enforcement ofsupport orders relates to the ability of the state to have payment madedirectly to court officers. Finally, spousal interest in professional degreesrefers to states that treat the value of degrees and licenses as divisibleproperty at divorce. The two latter features are likely to favor women.

Panel D of table 1 reports mean values for all four features. 26 Theseare dummy variables that equal one in cases that are deemed to increase

womens bargaining power. As shown in the table, few households inour sample fall under the community property system, and most are inunilateral states. Likewise, the majority of our households live in statesthat provide direct payments of support orders to the courts, androughly half live in states that treat degrees and licenses as divisibleassets on marital dissolution. Following a simple econometric test dis-

cussed below, we can add up the four dummy variables into a singleindicator that we refer to as divorce laws index. This variable is arough proxy of the extent to which state divorce laws are favorableto women in a bargaining context. In our sample, it ranges betweenone and four, with an average of 2.48. Its large standard error (p

) indicates that some states have few provisions that favor women,0.88whereas others have many.

23 Other features of divorce laws have been considered in preliminary work. Unfortu-nately, none turned out to be statistically significant. A very detailed discussion of statedivorce laws relevant to our sample period can be found in Freed and Walker (1991).

24Arizona, Mississippi, and Nevada are community property states that provide for eq-uitable rather than equal distribution of property on dissolution. They are thus treated

as common-law states.25 This suggests, following Gray (1998), introducing interactive terms between themutual-consent and the community property dummy variables in the equations of themodel. However, these terms were never significant in any equation, presumably becauseof the very small proportion (4 percent) of community property states in our sample.

26 Note that the means represent state averages weighted by the distribution of oursample across the various states.


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marriage market 59

B. ResultsThe parametric form that we estimate was introduced in equations (6)and (7).27 Preference factors include the number of preschool-age chil-dren, the number of school age-children, education, age, dummy re-gional variables, and a race dummy ( if white). This specificationp 1is relatively standard in the labor supply literature (e.g., Mroz 1987). Itmust be stressed that the race dummy controls for the potential cor-relation that may exist between the sex ratio and labor supply that couldarise because of a race effect.28

Before we discuss the results, the issue of endogenous covariates mustbe addressed. Indeed, unobserved individual characteristics may be pos-itively correlated with wages or nonlabor income and hours of work,

thus creating spurious correlation between right-hand-side variables andthe error terms of the hours equations. We thus follow Mroz (1987)and use a second-order polynomial in age and education to instrumentthe wages, the nonlabor income, and the number of preschoolers andschool-age children.29 Other instruments include fathers education, re-ligion, and city size (three dummies). In the unrestricted version, thereare 28 parameters to estimate and over 68 instruments (see tables 2 and3 below for the complete list of instruments).

The various versions of the model are estimated using full-informationgeneralized method of moments (GMM). One advantage of this ap-proach is that it also takes into account heteroskedasticity of unknownform in the errors, which cannot be done using a full-information max-imum likelihood (FIML) method (see Davidson and MacKinnon 1993,

chap. 18). Therefore, in the presence of heteroskedasticity of unknownform, our estimator should be asymptotically more efficient than three-stage least squares (3SLS) or FIML.30

Table 2 provides estimation results. In columns 14, we report theparameter estimates of the unrestricted model in which the distributionfactor reflecting the state divorce laws is first broken down into fourseparate dummy variables. Most parameter estimates are statistically sig-

27We also estimated the model by distinguishing between husbands and wifes nonlaborincome to provide one additional distribution factor. Unfortunately, the parameter esti-mates were never statistically significant when we did so.

28We also estimate the model separately for blacks and whites. The results are quitesimilar but less precise than those reported in this subsection.

29 The estimated coefficients of wages and nonlabor income are relatively insensitive to

the instrumentation of the children variables.30 In the unrestricted form of our model, which is linear in parameters, our estimatoris identicalto theDavidson-MacKinnon H3SLS estimator. Theacronym refers to a modified

version of the conventional 3SLS estimator that attains greater efficiency in the presenceof heteroskedasticity of unknown form. However, our estimator does not correspond tothe H3SLS estimator in the restricted version of the model since the restrictions on theparameters are nonlinear.


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60

TABLE 2GMM Parameter Estimates (Hours/1,000)

UnrestrictedModel withDivorce Law

Dummies

UnrestrictedModel withAggregatedLaw Dummies

General Collec-tive Model

ColMod

Ca

Wives(1)

Husbands(2)

Wives(3)

Husbands(4)

Wives(5)

Husbands(6)

Wives(7)

log qf 1.409(.346)

.810(.321)

1.427(.340)

.756(.323)

1.427(.340)

.760(.322)

.873(.289)

log qh .782(.296)

.597(.287)

.749(.296)

.564(.288)

.748(.296)

.568(.288)

.271(.258)

log qf#log qh .440(.126)

.273(.123)

.433(.125)

.255(.124)

.433(.125)

.257(.123)

.215(.104)

Nonlabor income/1,000 .009(.004)

.006(.004)

.008(.003)

.006(.004)

.008(.003)

.006(.004)

.007(.003)

Sex ratio 1.796(.965)

4.549(1.177)

2.143(.956)

4.379(1.139)

2.283(.700)

4.267(1.024)

2.314(.727)

Divorce laws index .046(.014)

.081(.015)

.044(.012)

.082(.014)

.046(.013)

Divorce laws:Property division

(communityp1).102(.084)

.047(.082)


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61

Mutual/ unilateral(mutualp1)

.117(.050)

.022(.053)

Enforcement (courtpaymentp1)

.050(.036)

.091(.035)

Spousal interest (degree asassetp1)

.003(.029)

.112(.027)

Intercept 1.174(.849)

1.102(.941)

1.326(.832)

1.071(.927)

1.391(.777)

1.134(.883)

2.720(.570)

Children 6 .539(.158)

.126(.112)

.510(.155)

.129(.112)

.512(.155)

.127(.111)

.592(.151)

Children 717 .098(.039)

.036(.038)

.087(.037)

.041(.037)

.087(.037)

.041(.037)

.098(.037)

Education .018(.018)

.036(.012)

.023(.018)

.036(.012)

.022(.018)

.036(.012)

.019(.018)

Age .128(.048)

.064(.042)

.130(.047)

.065(.042)

.131(.046)

.064(.042)

.160(.045)

White .017(.049)

.021(.051)

.010(.049)

.015(.051)

.005(.043)

.011(.048)

.018(.044)

Value of function 22.902 23.473 23.497 26Newey-West test .024 2

Note.Asymptotic standard errors are in parentheses. Instruments: Second-order polynomial in age and education (M-F), fathers educatio(M-F), city size (three dummies), Northeast, North Central, West, Protestant (M-F), Jewish (M-F), Catholic (M-F), sex ratio, and divorce laws. Trule are divided by 1,000 (except the one associated with nonlabor income). Each regression includes three region dummies (Northeast, N


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nificant at conventional levels. In particular, those associated with wagerates, nonlabor income, and sex ratio are statistically significant at the5 percent or 10 percent level. A Hansen test does not reject the validityof the instruments and the overidentifying restrictions. The test statisticof 22.9 is to be compared with the critical value of 2x (40) p 55.7.0.05

The parameter estimates of the unrestricted model provide interest-ing results that are worth mentioning. For instance, according to ourresults, a one-percentage-point increase in the sex ratio reduces wivesannual labor supply by 17.9 hours, whereas it increases husbands laborsupply by 45 hours. These results thus reject an important restrictionof the unitary model according to which no distribution factor influ-ences behavior. It also rejects the simple version of the separate spheresmodel (Lundberg and Pollak 1993), which assumes that the threat point

is not divorce but an uncooperative marriage.31

Further evidence onthis matter is provided by the parameter estimates associated with thestate divorce law variables. Indeed, many of them are statistically sig-nificant and of opposite sign in womens and mens equations. Forinstance, women living in community property states and those livingin mutual-consent states tend to work less than otherwise. On the otherhand, men living in states either that have stronger enforcement lawsor that treat licenses and professional degrees as divisible assets tend to

work more than others. These results are also incompatible with boththe unitary model and the simple version of the separate spheres model.

The parameter estimates of the divorce law dummy variables in eachregression are relatively similar in magnitude. A joint Wald test of equal-ity of coefficients in wives labor supply and of equality of coefficients

in husbands labor supply yields a statistic of 0.88, which is much smallerthan the critical value of We thus add up the dummy 2x (6) p 11.07.0.05

variables into a single indicator and report the estimation results of theunrestricted model that uses this divorce laws index in columns 3 and4 of the table. The results of this model are very similar to those of themodel with divorce law dummies. According to our estimates, a one-percentage-point increase in the index, which reflects the adoption ofa divorce law deemed favorable to women, reduces wives labor supplyby approximately 46 hours, whereas it increases husbands labor supplyby 81 hours over a year.

As discussed above, it can be argued that tests of the unitary or sep-arate spheres models may be biased since the sex ratio and divorce lawsare likely to be correlated with unobserved variables related to the labor

markets. We suggested in Section II a convenient way to discriminate

31 Theoretically, one could also test restrictions of this model (or alternative bargainingmodels) that stem from the particular formulation of the Nash bargaining program. How-ever, these restrictions are likely to be very difficult to derive formally (see McElroy andHorney [1990] and Chiappori [1991] for a recent discussion).


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marriage market 63

TABLE 3Parameter Estimates: Singles (Hours/1,000)

OLS GMM

Women Men Women Men

log q .036(.049)

.040(.048)

.177(.253)

.171(.207)

Nonlabor income (/1,000) .001(.001)

.001(.001)

.001(.004)

.003(.002)

Sex ratio 4.187(2.569)

1.121(2.070)

5.857(2.819)

.695(2.488)

Divorce laws index .018(.039)

.015(.034)

.152(.160)

.025(.118)

Intercept .374(1.243)

1.186(1.020)

.739(1.294)

1.405(1.137)

Education .077

(.020)

.038

(.021)

.095

(.035)

.000

(.045)Age .052(.038)

.015(.030)

.079(.062)

.047(.036)

White .123(.111)

.182(.089)

.111(.166)

.206(.110)

Northeast .083(.104)

.052(.082)

.094(.123)

.114(.111)

North Central .202(.078)

.038(.075)

.193(.081)

.015(.080)

West .243(.101)

.166(.092)

.184(.121)

.146(.117)

Value of function 4.470 9.591Number of observations 572 498 572 498

between the marriage market and the labor market hypotheses, namely,

to analyze the impact of the distribution factors on the labor supply ofsingles: this impact should be zero according to the marriage markethypothesis, whereas in the labor market story, the sex ratio should in-fluence the labor supply of both singles and couples in a similar way.Table 3 reports ordinarly least squares and GMM regression results ofmale and female singles hours of work.32 In both GMM estimations,Hansen tests do not reject the validity of the instruments and the over-identifying restrictions. We find that the sex ratio is statistically signifi-cant only in the GMM regression on the sample of women, but itsparameter estimate has the opposite sign of that of wives. Furthermore,the divorce laws index is not statistically significant in either the maleor female regressions. We conclude that although the sex ratio and thedivorce laws may partially reflect conditions in the labor market, it prob-

ably is not the whole story.The columns associated with the general collective model in table 2(cols. 5 and 6) provide results based on the assumption that the ratios

32We did not use the same age group as the one used for couples (3060) since doingthis severely reduced the sample size and made most coefficients nonsignificant.


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of the effects of the sex ratio to the divorce laws index on labor suppliesare equal. This corresponds to the equality in conditionsm /f p m /f5 5 6 6(8).33 The coefficients are very similar in the unrestricted and the re-stricted versions. Moreover, a Newey-West test does not reject the validityof this restriction. The test statistic is equal to the difference in function

values of the restricted and unrestricted versions ( ) and is muchp 0.024smaller than the relevant critical value of Therefore, our2x (1) p 3.84.0.05results do not reject the general version of the collective model, whichallows for externalities of any kind.

Columns 7 and 8 of table 2 provide results for the collective modelwith caring. The constraints imposed by this more restrictive model boildown to as given by (8), where is the ratiom /f p m /f p m /f, m /f4 4 5 5 6 6 4 4of the effects of the cross-wage variable (in log) on labor supplies. Again,

one cannot reject this joint hypothesis since the Newey-West test statisticis equal to Should the distribution factors reflect 22.58 ! x (2) p 5.99.0.05only labor market mechanisms, there would be no reason to expect thatthese specific restrictions would be satisfied. Interestingly, using Waldtests (test statistics of 4.5 and 4.8, respectively), one rejects the hypothesisthat and where subscript 1 holds for the sex ratioC p D C p D ,1 1 2 2and subscript 2 for the divorce laws index. This provides support forthe theoretical approach we used to derive the restrictions of the model.

Also, Slutsky conditions on the labor supply of women are globally sat-isfied, whereas they are locally satisfied for all men in the sample. Allin all, these tests do not reject the collective model with caring. Column9 of table 2 reports the implicit parameters of womens sharing rule asderived from the restricted parameters of the model with caring and

using equation (10). All parameter estimates of the sharing rule (exceptthat of ) are statistically significant at conventional levels.log wh

In order to gain insight into the interpretation of the parameters ofthe sharing rule, table 4 reports the partial derivatives of the sharingrule along with their standard errors. Column 1 of the table replicatescolumn 9 of table 2. Column 2 reports the partial derivatives themselves.They represent the impact of a marginal change in one variable on thenonlabor income accruing to the wife after sharing. According to ourparameter estimates, a $1.00 increase in the wifes wage rate, qf (whichis equivalent to an annual increase of $1,740 [1988] in her labor income,at the mean of hours worked by women), translates into the transfer ofmore income to her husband. At the sample mean, the transfer amountsto $1,634, although this effect is not precisely estimated. Also, a $1.00

increase in the husbands wage rate, qh (equivalent to an annual increase

33 One must reckon that the test performed is approximative since our divorce lawsindex is a discrete variable. This implies that, strictly speaking, the weighting factor m(w1,w2, y, z, s) is not differentiable in this index, which violates an assumption of our generalmodel.


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TABLE 4Sharing Rule and Elasticities

Variable

Elasticities

Sharing RuleUnrestricted

Model

GeneralCollective

Model

CollectiveModel with

CaringCon

o

Coefficients(1)

f/Variable*(2)

Wives(3)

Husbands(4)

Wives(5)

Husbands(6)

Wives(7)

Husbands(8)

Wives(9)

log qf 56.638(29.524)

1,634.357(1,007.120)

.234(.083)

.073(.037)

.227(.084)

.079(.038)

.235(.084)

.073(.038)

.178(.090)

log qh 25.346(22.543)

600.442(643.569)

.074(.069)

.023(.060)

.103(.071)

.031(.057)

.075(.071)

.022(.057)

log qf#log qh 20.063(10.744)

Nonlabor income .698(.170)

.698(.170)

.040(.017)

.030(.017)

.039(.018)

.028(.018)

.040(.018)

.030(.018)

Sex ratio 216.280(88.221)

2,162.795

(882.210)Divorce laws index 4.310

(1.714)4,309.954

(1,713.692)

Note.Asymptotic standard errors are in parentheses.* The derivatives are computed with respect to the qf and qh, not with respect to log qf and log qh. This figure represents the impact of a one-percentage-point increase in the sex ratio. This figure represents the impact of a one-point increase in the divorce laws index.


30/36


of $2,240 in his labor income), translates into the transfer of moreincome to his wife. Indeed, the table shows that, at the mean of thesample, $600 will be transferred to his wife, but again this effect isimprecisely estimated. These results suggest that wives in our samplebehave in a more altruistic manner toward their husbands than theother way around, though the effects are not measured with muchprecision. The next row indicates that a $1.00 increase in householdnonlabor income will increase the wifes nonlabor income by 70 cents.

The next couple of rows report the impact of the distribution factorson the intrahousehold allocation of nonlabor income. As indicated, aone-percentage-point increase in the sex ratio will induce husbands totransfer an additional $2,163 of income to their spouses. Likewise, aone-point increase in the divorce laws index similarly induces husbands

to transfer an additional $4,310 to their wives. Both estimates are sta-tistically significant at conventional levels and provide strong supportto the fact that external factors may have sizable impacts on the intra-household decision process.34

The other columns of table 4 report various labor supply elasticities.In general, these elasticities are comparable to those found in the em-pirical labor supply literature. At the sample mean, womens wage elas-ticities are positive and statistically significant in the unrestricted modeland the two versions of the collective model. They are also very close,

varying between 0.227 and 0.235. Mens wage elasticities are negativebut very small (varying between 0.073 and 0.103) and not statisticallysignificant. Cross-wage elasticities are all negative and statistically sig-

nificant only in the case of husbands labor supply. Moreover, both mensand womens labor supply elasticities with respect to nonlabor incomeare negative and significant at the 5 percent or the 10 percent level.

Columns 9 and 10 of the table report the own-wage elasticities ofindividual labor supplies, conditional on sharing and nonlabor income(f and respectively). These elasticities are derived from equationsy f,(11) and (12) and rely on individual preferences alone since they ignoreany effect wage rates may have on the intrahousehold decision process.Both womens and mens elasticities are significant but smaller than

34 The model was also estimated using a sample that excluded couples with preschoolers.Arguably, young children constitute the most important source of nonseparability inspouses preferences (Lundberg 1988). Consequently, including such families in the sam-ple increases the likelihood of rejecting the collective model with caring. The results based

on the restricted sample are very similar to those obtained using the full sample. Theonly noticeable difference relates to the impact of the distribution factors. An increase inboth the sex ratio and the divorce laws index generates much larger transfers from thehusband to his wife when there are no preschoolers in the household. Presumably, spousesare more responsive to changes in the marriage market in the absence of young children.These results are not reported in the paper for the sake of brevity but are available onrequest.


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marriage market 67

those reported in columns 7 and 8. This simply reflects the fact that,in the latter cases, a marginal increase in either spouses wage ratereduces the share of the nonlabor income, which in turn increases laborsupply through an income effect.

V. Conclusion

This paper has two purposes. We first extend Chiapporis (1992) col-lective model of household labor supply to account for so-called distri-bution factors. The main thrust behind this model is the assumptionthat the household decision process, whatever its true nature, leads toobserved outcomes that are Pareto-efficient. It also assumes that pref-erences are egotistic or caring in the Beckerian (1991) sense. Distri-

bution factors are variables that are thought to affect the internal de-cision process but to have no effect on individual preferences or the

joint consumption set. By introducing distribution factors into themodel, we show that the identification of the structural parameters isgreatly simplified. Furthermore, the introduction of distribution factorsgenerates new testable restrictions. Also, when at least two distributionfactors are assumed, the efficiency assumption can be tested even when

very general preferences with externalities of any kind (including publicgoods) are allowed.

The second goal of the paper is to provide further empirical evidenceon the efficiency assumption as well as on the relevance of distributionfactors to the internal decision process. The two factors we consider arestate-level sex ratios and a compendium of state divorce laws. The em-

pirical analysis is based on household labor supply drawn from the 1989wave of the PSID. The efficiency hypothesis, both in a model with caringpreferences and in one with very general preferences, cannot be statis-tically rejected. Indeed, the nonlinear parametric constraints that derivefrom both models are consistent with the data. Our results thus rejectone important prediction of the unitary model, namely, that distributionfactors are irrelevant to intrahousehold decisions. They are also at odds

with Nash bargaining models that assume that the fallback option isinternal to the household. Quite to the contrary, we provide some sup-port for Beckers (1991) claim that the state of the marriage market isan important determinant of the intrahousehold decision process.

Under the assumptions of efficiency and caring preferences, it canbe shown that the internal decision process may be viewed as a two-step

process: Nonlabor income is first allocated among spouses accordingto a so-called sharing rule that depends on distribution factors and other

variables. Next, spouses choose their labor supply subject to their in-dividual budget constraint. Given that efficiency was not rejected, theparameters of the sharing rule associated with our model can be re-


32/36


covered (up to a constant) and analyzed. It turns out that most param-eters of the sharing rule are significantly different from zero. In partic-ular, we find that a one-percentage-point increase in the proportion ofmales in a population defined by age, race, and jurisdiction induceshusbands in this population to increase their transfer to their wives by$2,163, on average. Likewise, passage of a divorce law that is favorableto women will induce husbands to transfer, on average, an additional$4,310 to their wives. The latter result illustrates the usefulness of thecollective approach in analyzing the consequences of public policies,and in particular divorce legislation, on the allocation of income and

welfare within marriage.We reckon that our empirical analysis is subject to some limitations,

though. Indeed, our estimates are conditioned on a sample of individ-

uals that have chosen to live with a spouse and could suffer from se-lectivity biases as a result. In regions in which the sex ratio is relativelysmall, more low-quality men are likely to marry given the scarcity ofmen in the marriage market. A positive correlation between quality inthe marriage market and quality in the labor market will yield a spuriouscorrelation between the sex ratio and male hours of work. More researchon collective models that endogenize both marital choices and laborsupply is clearly needed.

Finally, our approach assumes that the sex ratio is exogenous. It canbe argued that this variable adjusts across regions to equilibrate themarriage markets (Becker 1991). While we present some evidence thatsuggests otherwise, it would be important to pay more attention to thefactors that explain variations of the sex ratio across regions.

Appendix

Proof of Proposition 3

A. One Distribution Factor

Start from

1 1h p H (w , f(w , w , y, s, z), z)1 1 2

and

2 2h p H (w , y f(w , w , y, s, z), z).2 1 2

Then

1

h fw w2 2Ap p ,1h fy y

2h fw w1 1Bp p ,2h 1 fy y


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marriage market 69

1h fs sCp p ,

1h fy y

and

2h fs sDp p .

2h 1 fy y

Assume that Then the last two equations giveC(D.

Df p ,y

D C

CDf p .s

D C

Then the first two lead to

BCf p ,w1 D C

ADf p .w2 D C

These partials are compatible if and only if they satisfythe usual cross-derivativerestrictions. Hence, the following conditions are necessary and sufficient:

D CDp ,( ) ( )

s D C y D C

D BCp ,( ) ( )w D C y D C1

D ADp ,( ) ( )

w D C y D C2

CD BCp ,( ) ( )

w D C s D C1

CD ADp ,( ) ( )

w D C s D C2

and

BC ADp .( ) ( )w D C w D C2 1

If these equations are fulfilled, then f is defined up to an additive functionk(z) depending only on the preference factors z. The inequalities (2g) and (2h)of proposition 3 follow from standard integrability arguments. Finally, the knowl-


34/36


edge of Marshallian labor supplies allows one to recover preferences for anygiven value of k(z).

B. Several Distribution Factors

If there are several distribution factors, then they can enter labor supply func-tions only through the same function f. This implies that

1 2h f hs s sl l lp p

1 2h f hs s s1 1 1

for all l. Moreover, equations (4), which determine the s, are obtained in thefslsame way as the equation for fs in the case of one distribution factor. Noticefinally that condition (2i) combined with the assumption that impliesC (D1 1that for in equations (4). Q.E.D.C (D, lp 2, , L,l l

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