3.1. Commodities, preferences, and the decision process

Our model, following BCM, considers the case of an adult couple in a single time period. Labor supply of i, i = m, f , is denoted by hⁱ, with market wage equal to wⁱ. Total time endowment is normalized to one.⁵ A Hicksian composite good C is consumed by the household. This good is used for private (C^m , C^f) and public (K) consumption, with prices set to one. In a very general sense, the notion of public consumption should be understood as any expenditure that increases the utility of both partners, such as expenditures on heating, electricity, housecleaning, among others. Non-labor income is denoted by Y.

Each spouse’s utility can be written as:

Ui=Ui1-hi,Ci,K.     i=m,f

where U ⁱ is strongly quasi-concave, infinitely differentiable, and strictly increasing in all its arguments. For the moment we follow BCM’s conditions, which rule out cases where leisure, individual and public consumption are equal to zero (lim_hi _→1 ∂U ⁱ/∂hⁱ = lim_Ci _→0 ∂U ⁱ/∂Cⁱ=lim_K→0 ∂U ⁱ/∂K=∞for i = m, f ) ); in subsection 3.4. we relax this for leisure and thereby allow for non-participation.

Household decisions are assumed to generate Pareto-efficient out-comes, whatever the mechanism used to reach this agreement. Therefore, there is a function λ such that the household allocation (h^m∗, h^f∗, C^m∗, C^f∗, K^∗) is the solution to the program:

subject to Cm+Cf+K=wmhm+wfhf+Y0≤hi≤1,     i=m,f

The Pareto weight λ ∈ [0, 1] reflects the relative power of m in the household and (1 − λ) that of f ; a larger λ corresponds to a larger weight of m’s preferences in the household allocation problem, favoring the outcomes enjoyed by m. It is assumed that λ = λ(w^m, w^f , Y, z) is a continuously differentiable function of wages, non-labor income, and at least one distribution factor z that affects only the bargaining power rule but not the utilities or the budget.

The bundle (w^f , w^m, Y, z) is assumed to vary within a compact subset K of IR³ ₊ × IR. Moreover, h^m , w^m , C, and K are observed, whereas the individual consumptions C^m and C^f are unobserved. In general, household surveys do not collect information about intra-household allocation of expenditures but about aggregate consumption C. Finally, it is assumed that both partners’ wages are observed, even when a partner does not participate in the labor market (we come back to that in subsection 4.3.1).

3.2. The conditional sharing rule

The solution to the household program (1) can be thought of as a two-stage process in which the couple first agrees on the level of the public expenditure and how to distribute the resulting residual non-labor income between them. Next, conditional on the outcome of the first stage, each member decides, independently of each other, their individual consumption and labor supply.

Formally, let h^i∗ , C^i∗ , for i = m, f , and K^∗, each considered as a function of w^m , w ^f , Y and z, be the solution of program (1). Then a function φⁱ exists such that:

Ci*wm,wf,Y,z=ϕiwm,wf,Y,z+wihi*wm,wf,Y,z,  i=m,f

where ϕ^m and ϕ^f characterize the conditional sharing rule: the portion of non-labor income allocated to each member once spending on the public good has been discounted:

ϕmwm,wf,Y,z+ϕfwm,wf,Y,z=Y-K*wm,wf,Y,z

Note that ϕⁱ can be positive or negative; they could agree to spend beyond their non-labor income on the public good, and transfers between the two are also possible.

Fixing K¯=K*wm,wf,Y,z, the second stage of the household program (1) can be represented as:

with h^i∗ (w^m, w^f , Y, z) and C^i∗ (w^m, w^f , Y, z) as interior solutions to the individual problem (we relax this in subsection 3.4.). The structure of each partners’ labor supplies can be described by:

hm*wm,wf,Y,z=Hmwm,ϕwm,wf,Y,zhf*wm,wf,Y,z=Hfwf,Y-K-ϕwm,wf,Y,z

where ϕ = ϕ^m. When φ is fixed, H^m and H^f are Marshallian labor supply functions. Imposing the condition K^* (w^m, w^f , Y, z)=K, the couple’s labor supplies are:

In this way, i’s labor supply is described as a function of wages, non-labor income, and a distribution factor z as a function ζ(w^m, w^f, Y, K) such that public expenditures are exactly K. Hence, the values of w^m, w^f , and Y are not constrained to ensure that K^∗(w^m, w^f , Y, z)= K; the key role of z is to guarantee that the level of public expenditure is exactly K. This structure generates testable restrictions because the same function φ(w^m, w^f, Y, z) enters each member’s labor supply (see footnote 7).

3.3. The determination of public expenditures

The efficiency condition for public good expenditures is obtained directly from the first-order conditions for the household program (1). Assuming an interior solution for individual and public consumption, this condition implies that:

∂Um/∂K∂Um/∂C+∂Uf/∂K∂Uf/∂C=1

This condition can be expressed in terms of individual indirect utilities. Let V ⁱ(wⁱ, φⁱ, K) denote the value of the second stage of the household program (2) for member i, that is, the maximum utility that i can achieve given the own wage and conditional on the outcomes (ϕⁱ, K) of the first stage decision. Imposing efficiency, this leads to the following first-stage program:

max ϕm,ϕf,KλVmwm,ϕm,K+1-λVfwf,ϕf,K s.t ϕm+ϕf+K=Y

with first order conditions that imply that:

The ratio ∂Vi/∂K∂Vi/∂ϕi is i’s marginal willingness to pay (MWP) for the public good. Thus, condition (5) states that the individual MWPs must add up to the market price of expenditures on children. From BCM’s Proposition 1 it follows that if i’s preferences are such that both public and private consumption increase with non-labor income (i.e., K and φⁱ are normal “goods”, so i’s MWP is decreasing in K and increasing in φⁱ), a marginal increase in m’s power will increase the household’s expenditures on children if and only if m’s MWP is more sensitive to changes in his income share than that of f , and vice versa. Because a positive transfer from one member to the other decreases the transferer’s MWP for the public good and increases the MWP of the one who receives the transfer, this proposition establishes that there is a particular point where the positive effect on the receiver is sufficient to compensate the reduction to the transferer. Hence, the key property for analyzing changes in the distribution of power within a household is not the magnitude of the MWPs (say, who cares more for children), but how the MWPs respond to changes in individual resources for private consumption.

Intuitively, empowering the female partner comes with a higher fraction of household non-labor income for her. If both private and public goods are normal, she will consume more of all commodities, and, conversely, the male partner will see his share and consumption reduced. The reduction in household expenditures on the public good that comes from the male’s share will be more than compensated by the increase in the females share when the female partner is more sensitive to changes in her share than her partner, that is, when she is willing to spend a larger fraction than her partner on children of the additional monetary unit that comes via her empowerment.

3.4. The participation decision

The next step, and our contribution, is the extension of the framework of BCM, which includes public goods but assumes interior solutions, with the labor force participation decision, thereby allowing for non-participation. For that, the standard reservation wage approach is generalized to a collective model with two adult members such that, at the reservation wage of one household member, not only is that member indifferent between working and not working, but that the other member is indifferent also (^{Blundell et al., 2007}).

To characterize the participation decision of a household member, a procedure similar to the one used by ^{Neary and Roberts (1980)}, who employed the procedure to model household behavior under rationing. Using this procedure, we characterize household behavior in terms of its unconstrained behavior when faced with shadow prices. Our logic follows the steps set out in a collective framework with non-participation and income taxes but without public goods by ^{Donni (2003)}. The reservation wage of i, wⁱ, is defined by

w¯iUhii(1,ϕi,K¯)UCii(1,ϕi,K¯)

where U_x ⁱ stands for the partial derivative of function U ⁱ with respect to variable x = hⁱ, Cⁱ. This equation is the marginal rate of substitution between leisure and private consumption computed along the axis hⁱ = 0 for a given sharing rule φⁱ (and equal to Cⁱ ) and a level of public expenditures equal to K.

By fixing public expenditures at some arbitrary level K, problem (1) is basically reduced to that considered by ^{Donni (2003)}, who analyzes the participation decision in a framework with only private goods. Let y = Y − K denote the portion of non-labor income not devoted to public expenditures, which could be positive or negative (labor income can also be used for public consumption). Therefore, i’s reservation wage wⁱ is implicitly defined as a function of (w^m, w^f , y):

Without additional assumptions, Equation (6) could have several solutions. Intuitively, there are two reasons that explain why there can be many wage rates for which i is indifferent between working and not working. The first comes from the assumption that the sharing rule ϕⁱ depends on i’s wage, so there could be more than one combination of wⁱ and ϕⁱ at which i is indifferent. The second is related to the possibility that the sharing rule itself may depend on the non-participation of household members. Hence, the uniqueness of a reservation wage for member i has to be explicitly postulated.

A sufficient condition to obtain a unique reservation wage (fixed point) for each member is to define that the function $ⁱ is a contraction mapping (compare with ^{Donni, 2003}):

ASSUMPTION R. For any (w ^m∗ , w ^{f ∗} , y) and (w ^m◦ , w ^{f ◦} , y) ∈ IR ₊ ² × IR, preferences and the sharing rule are such that there is some non-negative real number r < 1 for which the following condition is satisfied:

maxi=m,fw¯iwm*,wf*,y-w¯iwm°,wf°,y|≤r maxi=m,f|wi*-wi°|

Although ^{Donni (2003)} presents this condition in a model with only private consumption, our contribution is the insight that, by fixing K, its applicability extends to the situation with public goods. In particular, this condition does not affect the level of public expenditure; z varies to guarantee that public expenditure is exactly K. Consequently, the distribution factor allows that w^m, w^j, and Y - and thus also wⁱ - can vary freely, whereas K is kept constant. Moreover, the assumption only applies in the neighborhood of the participation frontier; in the interior of other household participation sets the allocation of additional income stemming from the participation of one member could be more complex.

In essence, Assumption R restricts the impact on both individual shares (and hence individual consumption) of a change in one house-hold member’s wage. This amounts to assuming that the Pareto weights are smooth functions of both wages and non-labor income, and therefore that the smoothness of the individual utilities is preserved at the participation frontier of each individual.⁶

Assumption R is not expected to be very restrictive and it simplifies the analysis by allowing us to avoid the need for more restrictive fixed -point theorems to ensure the existence of a well-behaved participation frontier.

Under this assumption, the system of equations w^m and w^f is a contraction with respect to w^m and w^f for any y. Using the Banach contraction principle (^{Green and Heller, 1981}), two corollaries are:

Hence, Assumption R establishes unique reservation wages, not only when both partners participate in the labor market but also in a situation of non-participation.

3.5. Identification

This section discusses the empirical restrictions on each household member’s labor supply implied by the collective setting with children and non-participation, and shows that it is possible to recover the structural model (preferences and the sharing rule) by observing the labor supplies and the household expenditure on children.

Considering the possible combinations of household members’ participation decisions, four sets can be defined. First, the set of w^m, w^f , y for which both household members choose to work defines the Participation set P . Second, f ’s non-participation set, N^f , is formed by the combinations of w^m, w^f , y for which f chooses not to work and m chooses to work. Similarly, third, the combinations for which m chooses not to work and f chooses to work define m’s non-participation set, N ^m. Finally, the non-participation set N , consists of w^m, w^f , y such that both household members choose not to work; this set is not taken into account in identifying individual utilities and the decision process given the lack of information for this purpose - if the hours of work for both partners are zero, the sharing rule cannot be deduced, so individual utilities cannot be recovered.

Therefore, it is assumed that at least one of the partners’ labor supplies is an interior solution to (1). The following theorem establishes the identification and testability results.

THEOREM 1. Let (h ^m , h ^f ) be a pair of labor supplies, satisfying the regularity conditions listed in Lemmas 1-3 (below). Under Assumption R:

The proof of this theorem is developed in the next subsections. First, subsection 3.5.1. identifies the sharing rule in the participation set in which both household members choose to work (P ). Next, subsection 3.5.2. identifies ϕ in the set in which one of the couple does not work (N ^f and N ^m).

4.1. Preferences, labor supply, expenditures on children, and the sharing rule

For the empirical illustration of this model, it is important to use a relatively simple parametric specification. Following the semi-log specification popular in empirical work in general (^{Blundell, MaCurdy, and Meghir 2007}), and used in the empirical literature of collective models where both partners work as well (^{Chiappori, Fortin, and Lacroix 2002}), we specify their individual structural labor supply functions as:

Equations (7) and (8) are linear in parameters, which eases the estimation process. Applying Roy’s identity to the underlying indirect utility functions of the ^{Stern (1986)} type,

Vmwm,ϕm=expψ1wmψ1ψ0+ψ1ϕm+ψ2Inwm+ψ3K-ψ2ψ1∫-∞ψ1wmexp⁡(t)tdt

for men (m), (for women (f ) replace ψs with γs) yields the individual labor supply system (7) and (8). In this specification, K is non-separable in the utility function of both members. Note that the efficiency condition (5) for public good expenditures implies the following restriction in parameters:

As in ^{Chiappori, Fortin, and Lacroix (2002)}, the sharing rule is specified as:¹⁰

From the definition of the sharing rule, the expenditure on children has to satisfy the identity K = Y − (ϕ^m + ϕ^f), so that the reduced form is specified as:

Inserting the sharing rule (10) in the structural labor supply functions (7) and (8), the reduced-form functions are:

4.2. Restrictions of the model

In order to focus on labor supplies, the level of public expenditures is fixed to K(w^m, w^f ; Y, z) = K. Hence, using y = Y − K, after rearranging Equation (11) the distribution factor can be expressed as:

Using (14), the reduced-form labor supply functions (12) and (13) can be written as:

The relation between the parameters of the Equations (12)-(13) and the parameters of Equations (15)-(16) is shown in Appendix 1, Table A1.1.

Using Equations (15) and (16), the conditional sharing rule when both partners work, in terms of the household non-labor income devoted to private expenditures and wages, is characterized by the partial derivatives (see footnote 7):

ϕy=A1B4A1B4-B1A4ϕwm=A4B2+A4B4InwfA1B4wm-B1A4wmϕwf=A3B4+A4B4InwmA1B4wf-B1A4wf

Solving this system of differential equations, the conditional sharing rule recovered is:

Table A1.2 in Appendix 1 shows the parameters of the sharing rule (10) and its conditional version (17) in terms of the parameters of the reduced-form labor supply functions.

In addition to the parameter constraints from the efficiency condition for public-good expenditures (9), under a collective approach and with the chosen functional form of the labor supply functions, the ratio of the marginal effects of the interaction between the log wage rates has to be equal to the corresponding ratio of the marginal effects of the distribution factor on labor supplies:

This restriction stems from the fact that the cross term and the distribution factor enter the labor supply functions only through the sharing rule.

Moreover, collective rationality has implications for the ratio of the marginal effects of the expenditures on children (K) on each partner’s labor supply functions:

where the marginal effect of K is the sum of two terms. The first is the marginal effect that corresponds to the individual preferences via a change in the household’s non-labor income (a₁ and b₁), and the second term is the marginal change of K on the sharing rule via the distribution factor (a₅/c₅(1 − c₁) and b₅/c₅(1 − c₁)). Therefore, changes in the expenditures on children only impact individual labor supply functions through income effects, the impact for both partners being equal. Equations (18) and (19) impose testable cross-equation restrictions in the couple’s labor supply functions.

Finally, the parameters of the structural labor supplies (7) and (8) can be expressed in terms of the parameters of their reduced form; they are presented in Appendix 1, Table A1.3.

If the female partner does not work, there is a regime switch in the male partner’s labor supply and the sharing rule, and the parameters change:

To identify the decision process, the model imposes the restrictions that both the male’s labor supply function and the sharing rule have to be continuous along the female’s participation frontier (see subsection 3.5.2.):

Using the partial differential equation of the male’s labor supply in ϕ (see section 3.5.), a relation between s and r is obtained when the female partner does not work:

α~3+rB3+(α~4+rB4)Inwm(α~1+rB1)wf=A3+rB3+(A4+rB4)Inwm(A1+rB1)wf

Using the equalities of the parameters of the sharing rule (17) shown in Appendix 1, Table A1.2, the relation r=sB4Δ is obtained.

4.3. Stochastic specification

For household t, starting from Equations (11)-(13) and (20), the complete system of equations to be estimated, is:

where X _tl is a vector of exogenous variables. A stochastic model is obtained through the inclusion of the error terms on the right-hand side of each equation, where the vector of errors (ε_tp, ε_tnp, ε_tf , ε_tK ) follows a joint normal distribution with a covariance matrix:

The stochastic model is a type 4 Tobit model (^{Amemiya 1985}) or switching regression model (^{Maddala 1983}), with simultaneity. The log-likelihood function of the econometric model is specified in Appendix 2.

6.1. Reduced form model parameters

Tables 2, 3 and 4 and Appendix 1, Table A4.1 show the parameter estimates of the unrestricted model (24)-(25), which assumes that the male’s labor supply function is continuous along the female’s participation frontier, and the associated collective version, which imposes the restrictions (18)-(19) in the estimation process. Two versions are estimated, one using the age-to-age sex ratio variable as a distribution factor (columns labeled (age)), and the other using the 2-year-band sex ratio variable (columns labeled (2yr)).

Using the log-likelihood values for each model it is possible to construct LR statistics to test the collective restrictions (18)-(19). In the version employing the age-to-age (2-year-band) sex ratio the test statistic of 1.79 (4.55) is smaller than the critical value of χ² _0.05 = 5.99. Hence, for both sex ratios, the collective model cannot be rejected, a finding that is consistent with the hypothesis that the presence of children in a household generates non-separabilities in individual consumption. Others that have not explicitly considered this aspect have usually rejected the collective rationality when analyzing a household with children (see ^{Fortin and Lacroix, 1997}; ^{Donni, 2007}).

Table 2 presents the estimates of the parameters of expenditures on children. The magnitudes of the coefficients are very similar in the unrestricted and the collective versions. The marginal effect of a change in the male’s wage rate on the expenditures on children is (c₂ + c₄ ln w^f)/w^m, so for all specifications and everything else being equal, an increase in the male’s wage rate implies an increase in the money spent on children if the female’s wage is more than MXN $8 (that is if w^f > exp(−c₂/c₄)), which is the case for the large majority of the sample. In both versions of the unrestricted model, at the mean wage rate of both parents, a MXN $1 increase in the male’s wage rate (equivalent to an annual increase of MXN $2,408 in labor income at the mean hours worked by men) increases the annual expenditure on children by approximately MXN $61.

The marginal effect of the female’s wage rate is determined by (c₃ + c₄ ln w^m)/w^f , and is positive if the male’s wage is larger than MXN $23 using the age-to-age sex ratio as distribution factor, and $26 with the 2-year-band, which is the case for just over half of the sample. In the unrestricted model with the age-to-age sex ratio as distribution factor and at the mean wage rate of both parents, a MXN $1 increase in the mother’s wage (equivalent to an annual increase of MXN $308 in her labor income, at the mean hours worked by women) increases the annual expenditure on children by approximately MXN $5 (approximately $2 with the 2-year-band). The non-labor income is not statistically significant at conventional levels.

The age-to-age sex ratio has a negative and statistically significant effect on expenditures on children; for example, a one-standard deviation increase in the age-to-age sex ratio (0.07 points) reduces the annual amount spent on children by approximately MXN $646 in the unrestricted model.

Because an increase in the sex ratio is related to an increase in the bargaining power of the female partner (and a corresponding decrease in that of the male partner), this result suggests that fathers care more about their children than mothers (although, under the proposed specification, the adequate indicator of parents’ preferences regarding children is their marginal willingness to pay, whose estimated values are shown later). These results reject the implication of the unitary approach that no distribution factor is associated with intrahousehold allocations.

Most other control variables are statistically significant at conventional levels. As expected, the presence of a larger number of children under 15 increases the expenditure on them. However, if a child under five is present, all else equal, the expenditures are reduced. Children under five contribute to higher expenditures through the total number of children, but an autonomous correction is made since there are no school expenditures for them. Parental education has a positive effect on the expenditures on children, especially the female’s; while an additional year in the male’s education increases the annual amount spent on children by approximately MXN $215, that same factor in the female’s education increases the expenditure by MXN $300.

The estimates of the reduced-form female household member’s labor supply function are shown in Table 3. The own-wage effect of female labor supply is determined by (b₃ + b₄ ln w^m )/w^f , and is positive at male hourly wage rates inferior to MXN $9 but the nega-tive backward bending effect dominates for higher male wage rates. Therefore, if the husband earns more than MXN $9 per hour, a higher potential wage for the woman does not result in a greater labor supply for her; only if the man earns less than MXN $9 the wife is inclined to work more hours. The cross-wage effect of female labor supply, (b₂ + b₄ ln w^f )/w ^m , is positive for female wage rates less than MXN $62 in the model with the age-to-age sex ratio as a distribution factor (and less than MXN $59 using the 2-year-band). Thus, for the most relevant female wage range, all other factors being equal, women who participate work more if the husband has a higher wage, while for those women who do not work the probability of starting to participate increases with the wage of their partner. In sum, the own-wage income effect tends to dominate the substitution effect for very small values of the male wage rate, while a woman tends to increase her working hours upon a wage increase of her partner within a wide range of her own wage rate.

The parameter of the sex ratio variable in the couple’s reduced-form labor supply functions is the result of two effects, one of the sharing rule and the other of the expenditures on children (see the structural labor supply Functions (7) and (8)). Interestingly, the effect of both sex ratios on the female’s labor supply is positive, but imprecisely determined, in both the unrestricted and collective model. In the collective version, the magnitude of both sex ratios is smaller and better determined: the age-to-age sex ratio parameter passes from a p-value of 60% in the unrestricted model to 21% in the collective one, while the corresponding value for the 2-year-band falls from 53% to 33 percent.

With respect to the control variables, the female household member’s age and education have a significantly positive effect on her labor supply. As expected, an increase in the number of children, other factors being equal, is accompanied by a decrease in her number of hours worked; the presence of a pre-school child also reduces the number of hours worked.

Table 4 reports the estimates of the parameters of the reduced-form male labor supply function. In a working couple, the own-wage effect of the labor supply, (a₂ + a₄ ln w^f )/w^m, is always negative and the cross-wage effect, (a₃ + a₄ ln w^m)/w^f , is positive for a wide range of male wage rates. The former indicates a backward bending of the male labor supply, and the latter suggests that men tend to increase working hours upon a wage increase of their partner. Evidence of similar male labor supply behavior has been found for the Netherlands by ^{Bloemen (2010)} and ^{Kapteyn, Kooreman, and van Soest (1990)} when male and female labor supply is estimated simultaneously.

Comparing the unrestricted with the collective model, there is a change of sign in the effect of both sex ratios on the male labor supply; it passes from a negative effect to a positive. The constraints (18) and (19) imposed by the collective model seem to be restrictive regarding the influence of distribution factors on the male’s hours worked. Nevertheless, only in the unrestricted model with the 2-year-band sex ratio the distribution factor is statistically significant at the 5% level. Regarding the control variables, only the male’s education is significant, with a positive sign, in the male’s labor supply.

The parameter estimate of s, associated with (22), the assumption of a regime switch in the male’s labor supply and its continuity along the female participation frontier, is negative but is not estimated precisely. ^{Bloemen (2010)}, under a similar logic of the parametric specification for a sample with all possible combinations of working and non-working partners in the Netherlands, has found that the corresponding parameter for a working husband and a non-working wife is statistically significant, whereas the parameter associated with a working wife and a non-working husband is not significantly different from zero. This unsatisfactory result does not constitute a rejection of the collective approach but is instead a rejection of the auxiliary assumptions of a continuous regime switch of the male labor supply function due to a change in the females participation decision. Female non-participation in the labor market affects the working hours of her partner via her potential wage and the correlation between them (ρ_npf ≈ −0.53, see Table A4.1), but a non-working female partner does not involve a continuous shift in the male labor supply. The reason for the rejection of a regime switch may be that the female reservation wage tends to show little variation and is only captured by the correlation coefficient.

With respect to the nuisance parameters (Table A4.1), all the standard deviations of the dependent variables are estimated pre-cisely. In addition, the only correlations that are statistically significant at the 10% level are those between the female’s participation equation and the male’s labor supply when she does not work (negative), and the female’s participation equation and the expenditures on children (positive). These findings suggest that unobserved variables that influence women’s decision to participate in the labor market are negatively correlated with those that influence men’s hours worked, and positively with the expenditures on children.

Although the effects of some important variables are quite precisely measured, the limited number of significant parameters can be explained, at least partially, by the small size of the sample.

6.2. Structural model parameters

The estimates presented in Tables 2, 3 and 4, by use of the expressions in Table A 1.2., enable the recovery of the parameters of the (conditional) sharing rule (10) in the specification equations and (17) when both partners work, as well as the parameter r in Equation (23), which allows a regime switch in the sharing rule if the female partner does not work. The parameters, presented in Table 5, turn out to be not very precisely estimated; the most significant parameter is the one related to non-labor income (both the total in specification (10) and the one that discounts the expenditures on children in specification (17)), with a p-value of approximately 10.3 percent. The parameter of non-labor income is around 0.57, indicating that couples seem to share their non-labor income such that 57% goes to man and the remaining 43% to the woman.

The marginal effect of the male and female wage rate on the sharing rule (10) is (α₂ + α₄ ln w^f )/w^m and (α₃ + α₄ ln w ^m )/w^f , respectively, and similarly for specification (17) using the α˜_i instead of α_i . The estimated parameters of the sharing rule using the age-to-age sex ratio imply that, as long as the female’s hourly wage is less than approximately MXN $67, all other factors being equal, the female partner benefits, in terms of a non-labor income transfer, from an increase in the male’s wage (and for rates less than approximately MXN $74 with the 2-year-band). The female’s share also benefits from increases in her wage within a wide range of the male’s wage rate. By way of illustration, the parameter estimates of the conditional sharing rule Equation (17), with the level of expenditures on children fixed, indicate that in the collective model with the age-to-age sex ratio variable as distribution factor and at the mean wage rate of both parents, a MXN $1 increase in the male’s hourly wage (MXN $2,408 annually at the mean) induces him to transfer an additional MXN $214 to the female partner. Also, an extra MXN $1,367 will be transferred to the female partner when her wage increases MXN $1 (MXN $308 annually at the mean hours worked). Hence, at the mean wage rate of both parents, part of the male’s gain in labor income is transferred to his partner, whereas an increase in the female’s wage dramatically improves her bargaining position; she is able to keep the direct gains and in addition extract a larger portion of household non-labor income devoted to private expenditures.

The estimate of r, associated with the assumption of a regime switch in the sharing rule and its continuity along the female’s participation frontier (eq. (23)), is not significantly different from zero; the estimated values of the sharing rule’s parameters are maintained when the female partner does not work. Also, as mentioned above, in ^{Bloemen (2010)} the corresponding parameter for a working woman with a non-working husband is not significantly different from zero. Although the non-participation of a female partner would have reduced overall household resources, it does not imply a shift in the resources toward her; the female’s bargaining power does not seem to be affected by her non-participation in the labor market. Nevertheless, the male partner’s share decreases if the wage rate of his partner increases, regardless of her labor status: the wage rate of a non-working woman may still function as a threat point.

The reason that the male labor supply and the sharing rule of a working man and his non-working female partner are not significantly different from those of a working couple may be that reservation wages of women tend to be very low and show little variation in the sample used. In this scenario, there is a negligible reduction in overall resources for the household when the woman is not working, so there is no visible response in the male partners hours worked or in the distribution of household non-labor income.

The parameters of the structural individual labor supply Functions (7) and (8) can be computed using the expressions in Table A1.3. In general terms, the parameters in Table 6 are not estimated precisely. The small sample size, together with the low variation in the potential wage, can explain part of this result. Nevertheless, when the marginal willingness to pay for expenditures on children is calculated for each member (M W P ^m = ψ₃/ψ₁ and M W P ^f = γ₃/γ ₁), the male partner seems to care more about the children than the female: an increase of MXN $1 in the male’s share, φ^m , is associated with an increase of MXN $1.3 in the money spent on children; a corresponding increase in the female’s share is associated with a reduction of MXN $0.3. Using the same database but considering only working couples and including home production, ^{Sarmiento (2012)} also found that when time and expenditure on children’s education is evaluated, fathers care more than mothers.