Collective Female Labour Supply: A Conditional Approach∗

Collective Female Labour Supply:

A Conditional Approach

Olivier Donni^†

Université du Québec à Montréal, CIRPEE and DELTA

Nicolas Moreau^‡

Université Laval and CIRPEE March 6, 2003

1 Introduction

Traditionally, the household, as a whole, is considered as the elementary decision unit ; in particular, consumption and labor supply decisions are modeled as though household members were maximizing a unique utility function under a budget constraint. However, recent dissatisfaction with this so-called unitary model arose in a large part from the weakness of its theoretical foundations. We must admit, at least since Arrow’s famous impossibility theorem, that a household comprising several adult members does not necessarily behave as a single rational agent. Furthermore, the specific restrictions imposed by the unitary model have received little em- pirical support, if any (see Lundberg and Pollak (1996) for a survey these investigations).

For these reasons, Chiappori (1988, 1992) has proposed a model of la- bor supply based upon a collective representation of household behavior.

In this framework, each person is characterized by specific preferences, and

∗We thank François Bourguignon for useful comments and suggestions. We bear the sole responsability for any remaining errors.

†Département de Sciences Économiques, Université du Québec à Montréal, Case Postale 8888, Succursale Centre-Ville, Montréal (Québec), CANADA H3C 3P8, Tél: (514) 987-3000 (# 6816), Email: [email protected].

‡Département d’économique, Université Laval, Québec (Québec), CANADA G1K 7P4, Email: [email protected].

decisions are only assumed to result in Pareto-efficient outcomes. If con- sumption is exclusively private and agents are egoistic, efficiency has a very simple interpretation: household decisions can be modeled as a two step process, whereby individualsfirst share their total nonlabor income accord- ing to some rule and then maximize their own utilities subject to a separate budget constraint. In particular, this setting implies testable restrictions on the spouses’ labor supply under the form of partial differential equations.

Moreover, if these restrictions are satisfied, some elements of the decision process, such as individual preferences and the rule that determines the dis- tribution of nonlabor income within the household, can be retrieved from the observation of both labor supplies. Recently, Donni (2002) has extended this theoretical model to incorporate the possibility of non-participatory de- cisions and non-linear budget sets. Apps and Rees (1997) and Chiappori (1997) have yielded the theoretical basis for introducing household produc- tion. Fong and Zhang (2001) have studied a collective model of labor supply where there are two distinct types of leisure : one type is each person’s in- dependent (or private) leisure, and the other type is spousal (or public) leisure.

The collective approach turns out to be profitable as shown by several re- cent empirical applications. For example, Fortin and Lacroix (1997) closely follow Chiappori’s initial framework. They use a functional form that nests both the unitary and the collective model as particular cases and find, us- ing Canadian data, that the restrictions implied by the unitary setting are strongly rejected, while the collective ones are not. Chiappori et al. (2002) extend the collective models to allow for ‘distribution factors’, defined as being any variable that is exogenous with respect to preferences but may influence the decision process. Using the Panel Study of Income Dynam- ics and choosing the sex ratio and an indicator of the divorce legislation as distribution factors, theyfind that the restrictions implied by the collective model are not rejected.

However, these collective models of labor supply do not account for the fact that, in the majority of developed countries, male labor supply is rigid and largely determined by exogenous constraints. This is a serious prob- lem which has been recently tackled, though. Blundell et al. (2001) have analyzed household labor supply using United Kingdom data from 1978 to 1993 and they emphasize that, in the UK labor market (but this may be true for other countries), when men work, they work nearly always full- time; the wife’s hours of work, on the contrary, are largely dispersed. The theoretical model these authors develop allows for these essential features:

the wife’s labor supply is assumed to be continuous whereas the husband’s

choices are assumed to be discrete (either full-time working or non-working).

They show that the main conclusions which were derived by Chiappori in the initial context are still valid here. One drawback, however, is that such a difference in spouses’ behavior is completely unexplained. Moreover, the result of identifiability and testability given by Blundell et al. (2001) holds only if the husband’s choice between full-time working and non-working is free; in particular, it could be seriously misleading if husband’s unemploy- ment is mistakenly interpreted in empirical studies as the decision of not participating to the labor market.

In a recent paper, Donni (2002) treats the rigidity of the husband’s behavior in a quite different way. The starting point of his approach is very simple. For the majority of households, the husband’s hours of work are assumed to be fixed at full-time (defined by legal or socio-cultural norms) while the wife’s hours of work are assumed to freely vary. There is nothing remarkable here except that – and this is in strike contrast with Blundell et al. (2001) – the behavior of the few husbands who do not work is now explained by involuntary unemployment. One important implication is that the employment status of the husband, since it results from demand-side constraints instead of a voluntary choice, is no longer informative about the household decision process and the individual preferences. That is to say, the knowledge of the husband’s labor supply can no longer be used for testing or identifying some structural elements and more information on how the household allocates its resources is necessary to achieve this goal. The basic idea of the approach is then to use the observation of the demand for one commodity, together with the wife’s labor supply, for that purpose. However, the estimation task of this model may become cumbersome because a system of demands must be estimated jointly with the wife’s labor supply.

In the present paper, we consider a simpler approach based on the no- tion of conditional labor supply. The wife’s labor supply is then expressed as a function of her wage, the household income (defined in a specific way) and the demand for one good consumed at the household level. The key- idea is that the conditioning good in the labor supply summarizes all the characteristics of the decision process. We demonstrate that the main prop- erties of Chiappori’s initial model are preserved in this modified setting.

In particular, it is still possible to identify some important elements of the intra-household decision process and generate testable restrictions on house- hold behavior. The present framework turns out to be advantageous for two reasons. First, there is no need to model the determination of the condi- tioning commodity explicitly. Second, there is no need to observe exactly

the variables which affect the decision process (i.e., the distribution factors in the standard terminology). Finally, we estimate and test this collective model using Panel Study of Income Dynamics data for the sole couples in which the husband is working full-time. The wife’s labor supply is condi- tioned on the level of food consumed by the household. All the proofs are collected in the appendices.

2 Theory

2.1 Basic Framework

We consider only the case of married couples (A and B) in a single period setting. The wife’s and husband’s labor supply are respectively denoted by L^B and L^A with market wages w_B and w_A. There are two private commodities: Qand Z with prices set to one. Non-labor income is denoted by y. For convenience, the spouses’ total time endowment is normalized at one.¹ We assume that each household member is characterized by specific preferences. These can be represented by utility functions of the form :

u^I(1−L^I, Q^I, Z^I),

withI =A, B, that are both strongly concave, infinitely differentiable and strictly increasing in all their arguments. The household members are said to be ‘egoistic’ in the sense that their utility only depends on their own con- sumption and leisure. This seems restrictive but all the results immediately extend to the case of ‘altruistic’ agents in a Beckerian sense, with utilities represented by the form :

WI[u^A(1−L^A, Q^A, Z^A), u^B(1−L^B, Q^B, Z^B)],

whereW_I(·) is a strictly increasing function. The crucial hypothesis is the existence of some type of separability in the preferences of the two household members. Finally, we implicitly assume, as in Chiappori (1988, 1992), that there is no public consumption and no domestic production. The budget set is then written as follows :

y+L^B·w_B+L^A·w_A>Q+Z (1)

1This upper bound for members’ hours of work can alternatively be seen as a legal or socio-cultural norm. We do not decide, at this stage, in favour of a particular interpreta- tion.

and

1>L^I >0, Q^I>0, Z^I >0. (2) Let us remark that, in surveys, the consumption is generally recorded at the household level. We thus assume, in what follows, that onlyQ and Z, and notQ^I and Z^I, are observed by the econometrician.

The main originality of the efficiency approach lies in the fact that the household decisions result in Pareto-efficient outcomes and that no addi- tional assumption is made about the process. That is, for any wage-income bundle, the labor-consumption bundle chosen by the household is such that no other bundle in the budget set could make both members better off.

This assumption, even if not formally justified, has a good deal of intuitive appeal. First of all, the household is one of the preeminent examples of a re- peated game. Then, given the symmetry of information, it is plausible that agents find mechanisms to support efficient outcomes since cooperation of- ten emerges as a long term equilibrium of repeated noncooperative relations.

A second point is that axiomatic models of bargaining with symmetric in- formation, such as Nash or Kalai-Smorodinsky bargaining, which have been previously used to analyze negotiation within the household (Manser and Brown (1980) and McElroy and Horney (1981)), assume efficient outcomes.

We suppose that there is no involuntary unemployment, i.e., each person in the household who want, for his/her current market wage, to work has effectively a job. Then, Pareto-efficiency essentially means that there exists a scalar µ such that the household behavior is a solution to the following program :

max

{L^A,Q^A,Z^A,L^B,Q^B,Z^B}(1−µ)·u^B(1−L^B, Q^B, Z^B)+µ·u^A(1−L^A, Q^A, Z^A) (¯P) with respect to the budget set (1) and (2). The parameterµhas an obvious interpretation as a ‘distribution of power’ index. Ifµ= 0, then the household behaves as though the wife always gets her way whereas, ifµ=A,it is as if the husband is the effective dictator. To obtain well-behaved labor supplies and commodity demands, moreover, we assume that the scalarµ∈]0,1[is a single-valued and infinitely differentiable function ofw_B,w_A,yand a vector of specific variablesswhich influence the decision process without affecting preferences or the budget constraint. This is standard in the literature and does not deserve a justification. We then say that a pair of labor supplies, together with a pair of commodity demands, are consistent with collective rationality if they are obtained from a program such as ¯P.

We follow Donni (2002) and admit that, for sociological reasons, the preferences for leisure are weaker for the husband than for the wife while

his market wage is higher. This inclines us to consider only the case where husbands work full-time.² This is consistent with empirical facts since it is often observed, in most countries, that husband’s labor supply is fixed at the current upper bound for the majority of households. Moreover, in order to make things easier, the participatory decisions of the wife are not incorporated in the model. These considerations lead us to assume that:

1> L^B >0 and L^A= 1. (3) That is, the wife’s labor supply is free to vary but the husband’s labor supply isfixed at a upper bound. Of course, this selection rule has to be taken into account in the empirical section of this paper.

2.2 Decentralization and Functional Structure

A well-known result (Chiappori, 1992) is that the decision process, under Pareto-efficiency, can be decentralized. We assume that the selection rule (3) is satisfied by the household. Then, there exists a sharing rule ρ such that the husband is characterized by the following program:

max

{Q^A,Z^A}u^A(0, Q^A, Z^A) s.t. Q^A+Z^A=ρ, and the wife by:

max

{L^B,Q^B,Z^B}u^B(1−L^B, Q^B, Z^B) s.t. Q^B+Z^B =y+wA−ρ+L^B·wB. We may remark that the husband’s wage enters in the household “exoge- nous” income.³ This result has several important consequences. Specifically, it determines the functional structure of labor supplies and commodity de- mands. We have :

Q(wB, wA, y) =ξ^A(ρ) +ξ^B(wB, y+wA−ρ), (4) Z(w_B, w_A, y) =ζ^A(ρ) +ζ^B(w_B, y+w_A−ρ), (5) and

L^B(wB, wA, y) =λ^B(wB, y+wA−ρ), (6)

2Other explanations for the rigidity are possible but it is not important for our purpose.

3The notion of exogeneity refers here to the theoretical model. It is not exactly the same notion as the one used in econometrics.

where the functions ξ^A,ξ^B,ζ^A,ζ^B and λ^B are traditional Marshallian sup- plies and demands. In particular, the wife’s labor supply satisfy Slutsky Positivity :

λ^B₁ −λ^B₂ ·L^B>0.

For convenience, the wife’s share ρ is called the sharing rule. The latter is generally a function of all the exogenous variables w_A, w_B, y and s. Re- markably, the husband’s rationing on the labor market implies that the husband’s wage, contrarily to the wife’s wage, has only an income impact on the household demands.

2.3 Conditional Labor Supply

In the present section, we define the wife’s conditional labor supply whereby the labor supply is expressed as a function of various variables and the level of Z. Conditional demands/supplies are often used in traditional analysis where we assume a single utility function. See for instance Pollak (1969), Chavas (1984), Browning and Meghir (1991) or Browning (1998). In the extended rational setting at stake here we define a somewhat different type of conditional function.⁴ Rewrite the demand for Z as follows :

Z =ζ^A(κ) +ζ^B(w_B,ψ−κ), (7) whereκ=ρand ψ=y+w_A and assume that:

Zy 6=ZwA. (8)

It means, in words, that there is no income pooling in the demand for Z and excludes a rule such as κ = ρ(wB,ψ). Under this assumption, the commodity demand can be locally inverted on ρto yield ρ =κ(w_B,ψ, Z).

Substitute this into the labor supply to obtain the conditional labor supply, denoted by

L^B(w_B,ψ, Z) =λ^B(w_B,ψ−κ(w_B,ψ, Z)),

where κ is implicitly defined by (7). This is what we call the ‘conditional’

sharing rule. A key remark, at this stage, is that the distribution of ex- ogenous income between non-labor income and husband’s wage – or, more generally, the distribution factors – does not enter in the conditional la- bor supply. In other words, the information concerning the distribution of exogenous income is completely summarized by the conditioning commodity.

4This concept is not completely new. It is taken from a working paper by Bourguignon et alii (1995) in the demand context. See also Donni (2002) for another application.

There are two distinct advantages to modelling a conditional labor supply instead of a direct one.

1. There is no need to model the determination of the conditioning com- modity explicitly. Indeed, the conditional supply approach does not require an explicit structural model for the conditioning commodity at all. In practice, it turns out to be very useful because the estimation of labor supply models is generally very expensive in computer-time.

2. There is no need to observe exactly the distribution of income between household members. This is particularly compelling since, in empirical work, such information is generally not very reliable. Moreover, the effect of any distribution factor (which is unoberved or unknown) is incorporated in the conditioning commodity.

One important remark is that the conditioning commodity must be con- sumed in an optimum way (defined by (7)). This contrasts in the unitary framework with the more traditional conditional demands of Pollak (1969).

Be that as it may, the attractiveness of this approach largely depends on the properties of the conditional labor supply. More specifically, the underlying framework is desired to be testable and identifiable from the ob- servation of one conditional labor supply. To investigate these questions, we assume that the conditional labor supply exists and is three times continu- ously differentiable in a neighborhood P. We now introduce some pieces of notation :

α(w_B,ψ, Z) =−L^B_ψ

L^B_Z and β(w_B,ψ, Z) = −L^B_Z

α_ψ·L^B_Z−α_Z·L^B_ψ. It is stated in the discussion of the first proposition that the function α corresponds to the slope of the husband’s demand for Z. It can also be shown that the function β is the inverse of the derivative of this slope.

Assume now that the conditional functions at stake satisfy some regularity conditions.

Assumption R The wife’s conditional labor supply is such that L^B_Z 6= 0, α_Z 6= 0 and α_ψ·L^B_Z 6=α_Z·L^B_ψ. for any(w_B,ψ, Z)∈int(P).

We can give an interpretation of thefirst condition in terms of absence of income pooling. It states that the distribution of resources within the house- hold does matter – and that this distribution is conveniently represented by the level of Z – to explain the wife’s labor supply or, more generally, there exists at least one distribution factor with an impact on the wife’s be- havior. Otherwise we would necessarily haveL^B_Z = 0. The second condition means that the husband is affected by the household consumption of Z (at the second order at least). The third condition is more complex.

The next result is the basis of the conditional approach. It states that (some elements of) the wife’s preferences and the sharing rule can be re- trieved from the sole observation of the wife’s conditional labor supply.

Proposition 1 There exists a wife’s conditional labor supplyL^B(wB, Z,ψ).

Assume R. Then, the conditional sharing rule can be retrieved on P up to a constant ². Specifically, the derivatives of the sharing rule on int(P) are given by

κ_wB =α_wB ·β, κ_Z =α_Z·β, and κ_ψ =α_ψ·β.

Moreover, for each choice of², the wife’s preferences between total consump- tion and leisure (i.e., the marginal rate of substitution) are uniquely defined.

Finally, the individual commodity demands can be retrieved up to a constant.

The complete proof is given in Appendix 1. We briefly give thefirst step of the argument here. We may remark that, by definition, the slope of the husband’s demand forZ is given by the increase in Z due to a variation of one unity ofκkeepingψ−κandw_Bconstant. A variation of one unity ofψ such that L^B and wB remain constant provides the slope of the husband’s demand sinceL^B depends only on ψ−κ and w_B. Consequently, we apply the Implicit Function Theorem to L^B(w_B, Z,ψ) and differentiate Z with respect toψ. This yields the slope of the husband’s demand:

ζ^A₁ =−L^B_ψ L^B_Z.

The other structural elements of the decision process, even if the reasoning is less intuitive, can be retrieved similarly.

The conditional approach has two drawbacks for identification purpose.

Firstly, the sharing rule can be recovered up to a constant, as shown in the preceding proposition, from the wife’s conditional labor supply. However, its theoretical interpretation is sometimes unclear because it is expressed

as a function of the level of Z. Secondly, the sharing rule and the other structural elements are identified on the participation set but, obviously, this identification cannot be extended beyond, on the non-participation set.

This is simply a converse of the fact that we need less information to estimate the conditional labor supply. In particular, we do not need information on the level of consumption when the wife does not work nor information on the distribution between husband’s wage and non-labor income.

We next show that the wife’s conditional labor supply has to satisfy some constraints to be consistent with collective rationality. The next proposition considers the case where the conditioning commodity is consumed by both household members.

Proposition 2 There exists a wife’s conditional labor supplyL^B(w_B, Z,ψ).

Assume R. Then, for any (wB, Z,ψ)∈ int(P),

1. L^B_wB −L^B_ψ

µL^B−αwB ·β 1−α_ψ·β

¶

>0.

2. α_wBβ_Z−α_Zβ_wB =α_ψβ_Z−α_Zβ_ψ = 0.

These restrictions provide a joint test of collective rationality under spe- cific assumptions, i.e., private consumption, absence of domestic production and egoistic (or caring) agents. The inequality (1) is due to the standard curvature properties of the traditional model of labor supply (Slutsky Pos- itivity) transposed into the collective approach while the system of partial differential equations (2) results from the separability property of the be- havioral functions. The constraints generated by the conditional model are simpler than the corresponding ones in the unconditional model. Moreover, the straightforwardness of the estimation methods based on a single equa- tion suggests that the wife’s conditional labor supply is a profitable tool to make empirical tests of the collective rationality.

For the sake of completeness, we may now consider the cases where the commodity Z is exclusively consumed by one spouse. This is also very instructive. If the commodity Z is only consumed by A, then the corre- sponding demand isZ =ζ^A(κ). In this case, it is easy, even if quite tedious, to show that R simplifies and becomesL^B_Z 6= 0andL^B_ψ 6= 0. The next propo- sition gives the specific constraints derived from this additional assumption.

Proposition 3 There exists a wife’s conditional labor supplyL^B(w_B, Z,ψ) where Z is exclusively consumed by the husband. Assume R. Then, for any (w_B,ψ, Z)∈ int(P),

1. L^B_wB −L^B·L^B_ψ >0, 2. α_ψ =αw_B = 0.

These constraints are specially simple. Quite surprisingly, the condi- tional labor supply has to satisfy the standard Slutsky Positivity as in the unitary framework. The underlying intuition is that, when the condition- ing commodity is fixed, the husband’s share remains unchanged. Thus, the wife’s wage and the exogenous income have the same effect as in a unitary model of female labor supply. The wife’s conditional labor supply has also to satisfy a functional property which also turns out to be very simple. The intuition is immediate if we recall the interpretation of α: it means that, if the demand for the conditioning commodity is constant, the slope of the individual demand for the husband must not change.

Finally, in the case where the commodity Z is exclusively consumed by the wife, R is never satisfied. Consequently, the structural elements of the model are not identifiable. A very simple constraint can, however, be derived.

Proposition 4 There exists a conditional labor supply L^B(wB, Z,ψ). The conditioning commodity is exclusively consumed by the wife. Then, for any (w_B,ψ, Z)∈ int(P),L^B_ψ = 0.

The intuition is that, when the conditioning commodity is fixed, the wife’s share is necessarily constant and, consequently, the effect ofψon this share is equal to zero.

3 Parametric Specification of the Model

3.1 Quadratic Conditional Labor Supply

We consider a quadratic semi-log functional form for the conditional labor supply and assume that the conditioning commodityZ is consumed by both partners :

L^B = a₀₀+a₀₁·lnw_B+a₀₂·ψ+a₀₃·Z+a₁₁·(lnw_B)²+a₂₂·(ψ)² +a₃₃·(Z)²+a₁₂·lnw_B·ψ+a₁₃·lnw_B·Z+a₂₃·ψ·Z.

It can be shown, if the conditioning commodity is consumed by both spouses, that the coefficients of this functional form have to satisfy the following constraints to be consistent with collective rationality :

a12

a13

= a23

2a33

= 2a22

a23

. (9)

If we impose these constraints, the sharing rule can be retrieved.⁵ 3.2 The conditional sharing rule

Let us define Θ = a₀₃+a₂₃ψ+ 2a₁₂a₃₃ a23

lnw_B + 2a₃₃Z. The conditional sharing rule is quadratic and its derivatives are given by

κ_B= 2a₁₂a₃₃Θ

a23(a03a23−2a02a33)wB

, κ_ψ = a₂₃Θ a03a23−2a02a33

,

κ_Z = 2a33Θ a₀₃a₂₃−2a₀₂a₃₃.

Solving this three differential equation system, one obtains the conditional sharing rule equation:

κ = K₀+K₁·lnw_B+K₂·ψ+K₃·C+K₄·lnw_B·ψ+ (10) K₅·ln (w_B)·Z+K₆·ψ·Z+K₇·(ln (w_B))²+

K8·(ψ)²+K9·(Z)²,

whereK0 is an unidentified constant and where K1 = 2a03a12a33

a₂₃∆ , K2 = a03a23

∆ , K3= 2a03a33

∆ , K4 = 2a12a33

∆ , K5 = 4a²₁₂a²₃₃ a₂₃∆ , K₆ = 2a₂₃a₃₃

∆ , K₇ = 2a²₁₂a²₃₃

a²₂₃∆ , K₈= a²₂₃

2∆, K₉ = 2a²₃₃

∆ ,

with∆= (a₀₃a₂₃−2a₀₂a₃₃).It is also possible to recover the female Mar- shallian labor supply associated with this setting.

5If we assume that the conditioning commodityZⁿis only consumed by the husband, it can be shown that, to be consistent with collective rationality, the coefficients of this functional form have to satisfy the following constraints :

a12

a13

= a02

a03

.

Finally, if the conditioning commodity is exclusively consumed by the wife, the coefficients of this functional form have to satisfy the following constraints :

a02= 0, a22= 0, a12= 0 and a23= 0.

In this case, the regularity conditions are not satisfied and the sharing rule cannot be identified.

3.3 The Marshallian labor supply

The Marshallian labor supply does not depend on the conditioning goodZ and takes the following form:

λ^B =a^∗₀+ µ

a01− ∆ 2a33

K1

¶

ln (wB)+

µ

α7− ∆ 2a33

K7

¶

ln²(wB)− ∆ 2a33

(ψ−ρ), (11) whereα^∗₀=a₀₀−_2a^∆₃₃K₀.The Slutsky condition is then given by

a₀₁−a₁₂a₀₃ a₂₃

1 w_B + 2

µ

a₁₁−a²₁₂a₃₃ a²₂₃

¶ln (w_B) w_B + ∆

2a₃₃λ^B≥0. (12) and must be checked for each observation.

4 Data and Empirical Results

4.1 Data

The data we use are taken from the University of Michigan Panel Study of Income Dynamics (PSID) for the year 1990 (interview years 1990 and 1991).

We select households where both spouses are between 25 and 60 years of age and where annual hours of work is positive for wives and between 1800 and 3700 for husbands. We further restrict our sample to stable couples over the years 1990-1991 as information on periodtin the PSID is often available at t+ 1. All these selection criteria lead us to 1829 observations.

The dependant variable, female annual hours of work, is defined as total hours of work on main jobs during 1990. The measure of the wage rate w_B is the average hourly earnings defined by dividing total labor income on main jobs over annual hours of work on main jobs. The conditioning good Z is the sum of the family’s yearly food expenditures at home and outside of the home. The variable ψ is the sum of male annual earnings on main jobs and household nonlabor income. The latter includes, among other things, imputed income from all net assets. Table 1 includes some descriptive statistics.

4.2 The results

Before discussing the results, we have to address some econometric issues.

First, the labor supply equations are estimated using conventional iterated GMM techniques. This estimation method allows us to account for the

possible endogeneity of some of the covariates and is consistent with het- eroscedasticity of unknown form in the error term.

We allow unobserved female characteristics (gathered in the error term) to be correlated with her wage rate, food expenditures and nonlabor income.

Husband’s labor income and the number of children are supposed to be exogenous and measured without error. To instrument the female wage, the nonlabor income and the food expenditures, we use a third order polynomial in female age, female education and male labor income that we replace with its factors (or components or latent vectors) through a partial least squares analysis. The idea is to use a relatively small number of orthogonal instruments which capture a large portion of the variance of the exogenous variables and of the endogenous ones. Estimations are then more stable.

Second, conditioning the sample on working spouses may induce a selec- tivity bias. We use a standard Heckman’s two steps procedure to account for it. We first estimate a reduced form participation equation and then estimate the labor supply equation including the inverse Mill’s ratio. The second step asymptotic covariance matrix is computed using the results of Newey (1984) and Newey and McFadden (1994). Table 2 reports the second step estimation. The results from the participation equation are shown in Table 5.

Preference factors include the number of children, education, age and a race dummy (=1 if Spanish descent). A Hansen’s test does not reject the validity of the instruments and the over-identifying restrictions. The test statistics of 19.146 and 24.435are to be compared with the critical values of the χ²_0.05(18) = 28.869and of the χ²_0.05(20) = 31.410. Conversely to the unrestricted model, most structural parameters of the restricted model are statistically significant at conventional levels. Moreover, a Newey-West’s test does not reject the validity of the collective restrictions (9) since the difference in function values (=5.289) is less thanχ²_0.05(2) = 5.99. Also, the Slutsky condition (12) is satisfied for 93%of the women in the sample. All in all, these tests do not reject the collective model at stake.

The sharing rule estimates are shown in Table 3. Table 4 reports the parameters estimates of the Marshallian labor supply.

5 Appendix

5.1 Proof of Proposition 1

Differentiating the conditional labor supply with respect toψ and Z gives : L^B_ψ =λ^B₂(1−κ_ψ) and L^B_Z =λ^B₂(−κ_Z)

or

−κ_Z·L^B_ψ = (1−κ_ψ)·L^B_Z. (13) Differentiating the demand for Z with respect to ψand Z indicates:

1 = (ζ^A₁ −ζ^B₂)κ_Z and ζ^A₁ = (ζ^A₁ −ζ^B₂)(1−κ_ψ) or

ζ^A₁ = 1−κ_ψ κZ

(14) Using (13) and (14) yields the husband’s Engel curve :

ζ^A₁ =−L^B_ψ

L^B_Z =α(w_A,ψ, Z).

Differentiating this expression again yields:

ζ^A₁₁·κwB =αwB, ζ^A₁₁·κψ =αψ and ζ^A₁₁·κZ =αZ.

This system of partial differential equations, together with (13) yields a solution forκwB,κ_ψ and κZ. That is,

κ_wB =α_wB ·β, κ_ψ =α_ψ·β, and κ_Z=α_Z·β. (15) The slopes of the labor supply can be obtained by differentiating it with respect toψand wB. We obtain:

L^B_ψ =λ^B₂(1−κ_ψ) and L^B_w

B =λ^B₁ −λ^B₂κ_wB Using (15) yeilds:

λ^B₂ = L^B_ψ

1−α_ψ ·β and λ^B₁ =L^B_wB + αw_B ·β 1−α_ψ·βL^B_ψ.

The slopes of the demand forZ can be retrieved in a similar way. Q.E.D.

5.2 Proof of Proposition 2

1. Since the slopes of the Marshallian labor supply can be retrieved, the Slutsky Positivity implies that:

L^B_wB −L^B_ψ

µL^B−αw_B ·β 1−α_ψ·β

¶

>0.

2. From the Young’s Theorem, the derivatives of the sharing rule have to satisfy a symmetry restriction. Simplifying yields:

β_Z αZ

= β_wB αwB

= β_ψ αψ

.

Q.E.D.

5.3 Proof of Proposition 3

To obtain comparable results, we follow the steps of the identification proof with the demand forZ. Differentiating this demand with respect toψ,wB

and Z yields:

κ_ψ = 0, κ_wB = 0 and κ_Z = 1 ζ^A₁ .

Differentiating the labor supply yields with respect to ψand Z indicates:

L^B_wB =λ^B₁, L^B_ψ =λ^B₂ and L^B_Z =λ^B₂(−κZ).

Consequently, we have:

ζ^A₁ =−L^B_ψ

L^B_Z =α and κ_Z =−L^B_Z

L^B_ψ =α⁻¹.

The constraints result form the Slutsky Positivity and the fact that α is independent ofψ and w_B. Q.E.D.

5.4 Proof of Proposition 4

Let us consider the wife’s conditional labor supply. This one can be slightly simplified when the conditioning commodity is exclusively consumed by the wife. By inversion of the commodity demand, the conditional sharing rule can be written asψ−ρ= ¯κ(w_B, Z). Thus the wife’s conditional labor supply does not depend on the exogenous income. Q.E.D.

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Table 1: Descriptive Statistics

Mean Median Std. Dev. Min. Max.

Female hours of work 1583 1837 665 4 4160

Female wage rate 11 9 14 2.5 441

Food expenditures 5631 5200 2533 240 20398

Male labor income 34908 34908 24008 750 350000 Nonlabor income 6107 2800 11544 −8500 168600

Table 2: Second Step Labor Supply Estimates Unrestricted Model Collective Model ln(w_B) 2811.558^∗∗∗ 1350.162^∗

(1251.4) (805.9)

ψ −0.010 0.016^∗∗

(0.015) (0.007)

Zⁿ −0.284 −0.593^∗∗∗

(0.216) (0.161)

ln(w_B)·ψ 0.005 −0.004

(0.005) (0.003)

ln(w_B)·Zⁿ 0.035 0.105^∗

(0.083) (0.063) ψ·Zⁿ −2.8E−7 −1.9E−6^∗∗∗

(1.3E−6) (6.2E−7) ln²(w_B) −658.779^∗∗∗ −343.613^∗∗

(218.6) (162)

(ψ)² −6.7E−9 3.4E−8^∗∗∗

(2.4E−8) (1.2E−8)

(Zⁿ)² 9.2E−6 2.6E−5^∗

(1.5E−5) (1.4E−5)

Intercept −437.458 1523.370^∗

(1251.4) (807)

Number of children −21.199 −25.009 (25.585) (25.853)

Education −10.646 −14.047

(10.566) (11.269)

Age 7.582^∗∗ 7.193^∗∗

(3.027) (3.076)

Spanish 109.449^∗ 99.274^∗

(59.752) (60.344) Inverse Mill’s ratio −41.773 −61.711 (138.9) (137.7)

Value of function 19.146 24.435

Newey-West test 5.289

Notes: Asymptotic standard errors in parenthese.

Significance levels of 10, 5 and 1% are noted *, ** and *** respectively.

Table 3: Conditional Sharing Rule Estimates Sharing Rule Parameters ln(wB) −211759

(156938)

ψ 3.796^∗∗∗

(1.164)

Zⁿ −105.024^∗∗

(48.454) ln(w_B)·ψ −0.675

(0.435) ln(w_B)·Zⁿ 18.681^∗∗

(8.856) ψ·Zⁿ −0.0003^∗

(0.0002) ln²(w_B) 18832.63 (23197.9) (ψ)² 6.05E−6^∗∗

(1.97E−6)

(Zⁿ)² 0.005

(0.004)

Notes: Asymptotic standard errors in parenthese are computed with the Delta method.

Significance levels of 10, 5 and 1% are noted *, ** and *** respectively.

Table 4: Marshallian Labor Supply Estimates Parameters Estimates ln(w_B) 2545.332^∗∗∗

(835.8) ln²(wB) −449.905^∗∗∗

(154.6) (ψ−ρ) −0.006^∗∗∗

(0.002)

Notes: Asymptotic standard errors in parenthese are computed with the Delta method.

Significance levels of 10, 5 and 1% are noted *, ** and *** respectively.

Table 5: Participation Equation Estimates

Parameters Estimates Standard Error

Intercept −1.631^∗∗∗ 0.488

Wife’s age 0.104^∗∗∗ 0.024

Wife’s education 0.003 0.030

(Wife’s age)² −0.001^∗∗∗ 0.0003

(Wife’s education)² 0.001 0.001

Wife’s health −0.365^∗∗∗ 0.071

City size 1 0.192^∗∗∗ 0.063

City size 2 0.199^∗∗ 0.087

City size 3 0.149^∗∗ 0.064

Husband’s age −0.019^∗∗∗ 0.005

Husband’s education 0.037^∗∗∗ 0.009

Husband’s white −0.065 0.052

Husband’s Spanish −0.202^∗∗∗ 0.071

Husband’s religion 1 0.213^∗∗∗ 0.057

Husband’s religion 2 −0.043 0.160

Husband’s religion 3 0.124 0.086

Husband’s religion 4 0.179 0.121

Number of children −0.087^∗∗∗ 0.019

Note: Significance levels of 10, 5 and 1% are noted *, ** and *** respectively.