Household production in a collective model: some new results

Abstract

Household models estimated on labour supplies alone generally assume non-market time to be pure leisure. Previous work on collective household decision-making is extended here by taking domestic work into account in the Chiappori et al. (J Polit Econ 110(1):37–72, 2002) model. Derivatives of the household “sharing rule” can then be estimated in a similar way. Using the 1998 French Time-Use Survey, we compare estimates of labour supply functions assuming first that non-market time is pure leisure and then taking household production into account. The results are similar but more robust when household production is included. Collective rationality is rejected when domestic work is omitted.

Appendices

Appendix 1: Proof of Proposition 1

Recall that ψ = ψ _f . We also have: $\psi _m =\mathit\Pi +y-\psi$

By differentiation of the labour supply equations

$$ \label{eq17} h^f=L^f\left( {\it{w}_f ,\psi \left( {\it{w}_f ,\it{w}_m ,y,...s_l ...;{\rm {\bf z}}} \right);{\rm {\bf z}}} \right) $$

(17)

$$ \label{eq18} h^m=L^m\left( {\it{w}_m ,\mathit\Pi \left( {\it{w}_f ,\it{w}_m ;{\rm {\bf c}},{\rm {\bf z}}} \right)+y-\psi \left( {\it{w}_f ,\it{w}_m ,y,...s_l ...;{\rm {\bf z}}} \right);{\rm {\bf z}}} \right) $$

(18)

we obtain:

$$ \label{eq19} \frac{\partial h^f}{\partial \it{w}_m }=\frac{\partial L^f}{\partial \psi }\frac{\partial \psi }{\partial \it{w}_m } $$

(19)

$$ \label{eq20} \frac{\partial h^m}{\partial \it{w}_f }=\frac{\partial L^m}{\partial \psi _m }\left( {\frac{\partial \mathit\Pi }{\partial \it{w}_f }-\frac{\partial \psi }{\partial \it{w}_f }} \right) $$

(20)

$$ \label{eq21} \frac{\partial h^f}{\partial \it{w}_f }=\frac{\partial L^f}{\partial \it{w}^f}+\frac{\partial L^f}{\partial \psi }\frac{\partial \psi }{\partial \it{w}_f } $$

(21)

$$ \label{eq22} \frac{\partial h^m}{\partial \it{w}_m }=\frac{\partial L^m}{\partial \it{w}^m}+\frac{\partial L^m}{\partial \psi _m }\left( {\frac{\partial \mathit\Pi }{\partial \it{w}_f }-\frac{\partial \psi }{\partial \it{w}_m }} \right) $$

(22)

$$ \label{eq23} \frac{\partial h^f}{\partial y}=\frac{\partial L^f}{\partial \psi }\frac{\partial \psi }{\partial y} $$

(23)

$$ \label{eq24} \frac{\partial h^m}{\partial y}=\frac{\partial L^m}{\partial \psi _m }\left( {1-\frac{\partial \psi }{\partial y}} \right) $$

(24)

$$ \label{eq25} \frac{\partial h^f}{\partial s_l }=\frac{\partial L^f}{\partial \psi }\frac{\partial \psi }{\partial s_l } $$

(25)

$$ \label{eq26} \frac{\partial h^m}{\partial s_l }=\frac{\partial L^m}{\partial \psi _m }\left( {-\frac{\partial \psi }{\partial s_l }} \right) $$

(26)

Note that, with reference to the results in Chiappori et al. (2002), only Eqs. 19 and 20 include a new specific term: $\frac{\partial \mathit\Pi }{\partial \it{w}_f }$

Taking the same notation, we define $A=\frac{h_{\it{w}_m }^f }{h_y^f }$, $B=\frac{h_{\it{w}_f }^m }{h_y^m }$, $C_l =\frac{h_{s_l }^f }{h_y^f }$, $D_l =\frac{h_{s_l }^m }{h_y^m }$. We assume only one distribution factor and suppress the subscripts l and q to simplify the notation. The partial derivatives of the sharing rule with respect to wages, non-labour income and the distribution factor are given by:

$\frac{\partial \psi }{\partial y}=\frac{D}{D-C}; \quad \frac{\partial \psi }{\partial s}=\frac{CD}{D-C}; \quad \frac{\partial \psi }{\partial \it{w}_m }=\frac{AD}{D-C}.$ Only $\frac{\partial \psi }{\partial \it{w}_f }$ is modified. From Hotelling’s lemma, we obtain: $\frac{\partial \mathit\Pi }{\partial \it{w}_f }=-t_f$, and then $\frac{\partial \psi }{\partial \it{w}_f }$ is given by:

$\frac{\partial \psi }{\partial \it{w}_f }=\frac{BC}{D-C}-t_f .$ Note that t _f is fully observed in the data.

The same result holds with several distribution factors. This is a straightforward result from Chiappori et al. (2002). Note also that when there is more than one distribution factor, testable restrictions similar to those presented in Chiappori et al. (2002) can be derived from the model.

Finally, with no domestic production, the model simplifies to $\mathit\Pi$ = 0, and thus $\frac{\partial \mathit\Pi }{\partial \it{w}_f }=0$, and ψ is now simply non-labour income. In this case, Eq. 20 reduces to:

$$ \frac{\partial h^m}{\partial \it{w}_f }=\frac{\partial L^m}{\partial \psi _m }\left( {-\frac{\partial \psi }{\partial \it{w}_f }} \right) $$

and (4^′) to:

$$ \frac{\partial h^m}{\partial \it{w}_m }=\frac{\partial L^m}{\partial \it{w}^m}+\frac{\partial L^m}{\partial \psi _m }\left( {-\frac{\partial \psi }{\partial \it{w}_m }} \right) $$

Then, $\frac{\partial \psi }{\partial \it{w}_f }$ reduces to:

$$ \frac{\partial \psi }{\partial \it{w}_f }=\frac{BC}{D-C} $$

And thus the formulas in Chiappori et al. (2002) are found as a special case of the more general model developed here.

Appendix 2: Description of domestic tasks

Domestic activities include all activities around:

• food and drink: preparation (cutting, cooking, making jam), presentation (laying the table), kitchen and food clean-up (washing up)

• housework: interior cleaning, clothes activities (laundry, mending, sewing, knitting, repairing and maintaining textiles), storing interior household items and tidying

• interior maintenance and repair of house and vehicles: repairing, water and heating upkeep

• household management: financial (bills, count,...)

• shopping

• childcare: physical and medical care, reading, talking with and listening to children, homework help, picking up/dropping off children, playing and leisure with children

• care for household adults

• care for animals and pets

• lawn, garden and houseplants

Appendix 3: Computation of the male’s share

In Table 4, the derivatives of the sharing rule have been computed directly calculating the derivatives of $\mathit\Pi +y-\psi$ using formulas symmetrical to those which appear in Section 3.3.

The formulas in Appendix Appendix 1: Proof of Proposition 11 show that the derivatives of the sharing rules are not symmetrical for the man and the woman, because of the term t _f in the derivative with respect to $\it{w}_{f}$. It thus must be checked whether the results are the same when computing directly the derivatives of the male’s share. In this latter case, ψ represents now the man’s share and $\mathit\Pi +y-\psi$ the woman’s share.

As the reduced forms of labour supply are identical, we expect the derivatives with respect to the male and female wages to have an opposite sign and to be about the same absolute value in Table 3. The same should hold for the derivative with respect to the sex-ratio (the coefficient is in fact the exact opposite, see Table 3). The derivative with respect to non-labour income is the complement to 1 of the coefficient computed in Table 3; indeed, we exchange ψ and $\mathit\Pi +y-\psi$, so that we exchange and as $\mathit\Pi$ does not depend on y.

Table 4 presents the results. When comparing with the results of model 2b in Table 3, it can be seen that the derivatives relative to wages show, as expected, an opposite sign and a similar value: an increase in either the male or the female wage should have an exact opposite effect on the male and the female income share. The same expected result is observed for the sex ratio, where the parameters obtained in the two models are exactly opposite and both significant. As expected also the coefficient found for non labour income is the complement to 1 for the coefficient found in the case of the female’s share: when non labour income increases, say by one euro, then it was found (Table 3) that the female share increased by about 62 cents. Here, it is found that the male share does increase in that case by 1–62 cents = 38 cents. For this coefficient, χ ² tests show that on one hand, we cannot reject the null-hypothesis, but on the other hand, the hypothesis that it equals to 1 can be rejected (these are the exact symmetries of the results found in Table 3).