A Note on Collective Models of Labor Supply

with Domestic Production

Olivier Donni

University of Quebec at Montreal,

CIRPEE and DELTA

January 21, 2004

1Introduction

The collective model of labor supply, developed by Chiappori (1988, 1992), is

now a standard tool for analyzing household behavior, and its empirical success

for the last 10 years is considerable. One of the most serious criticisms of this

model, however, concerns the treatment of domestic production. Apps and

Rees (1997) point out that, if domestic production is not taken into account,

a low level of market labor supply will automatically be interpreted as a large

consumption of leisure, whereas it may in fact reﬂect the specialization of one

of the members in domestic production. This may invalidate the well-known

identiﬁcation results for this model.

One answer to these problems is given by Chiappori (1997) who considers a

collective model of domestic and market labor supply. As for the simple model,

each household member is characterized by a system of egoistic preferences

and the decision process results in Pareto-e ﬃcient outcomes. There are one

market good which is bought and one domestic good which is produced from a

technologyusingtimeinput. Thelattercanbeconsumedbyhouseholdmembers

or exchanged on a market. Chiappori then shows that, provided that domestic

and market labor supplies are both observed, the structural model, i.e., the

outcome of the decision process, is fully identiﬁable (except for a constant).

In spite of this important theoretical contribution, empirical estimations of

1collective models accounting for domestic production are surprisingly rare. In

fact, these models are very demanding in terms of data. Time use surveys, even

if they are now broadly available, are generally fragmented and unreliable for

wage and income issues. Also, most authors, in empirical applications, simply

ignore the production sector of households. For example, Chiappori, Fortin and

Lacoix (2001) estimate a simple model of labor supply and completely disre-

1OnenotableexceptionisAppsandRees(1996)withAustraliandata. Theydonotassume

that there is market for the household good.

16

gard the possibility of domestic production. However, this could have serious

repercussions on the interpretation of the results.

Inthepresentnote, weuseChiappori’sframeworkwithmarketabledomestic

production and make the following contributions.

Firstly, we show, using Chiappori, Fortin and Lacoix’s (2001) functional

form as an example, that simple functional forms which are consistent with

the traditional model of labor supply can sometimes be compatible with more

sophisticated models accounting for domestic production. In this case, the es-

timated parameters of the sharing rule can be misleading if, as usually, the

econometrician mistakenly assumes that there is no domestic production. Quite

importantly, however, the direction of the bias can be evaluated. For example,

we conjecture that, in all likelihood, the own-e ﬀect (resp. cross-e ﬀect) of wages

onindividualsharesareover-estimated(resp. under-estimated)whilethee ﬀects

of non-labor incomes are correctly evaluated.

Secondly, if the good that is domestically produced is marketable, market

labor supplies have to satisfy testable restrictions under the form of partial

di ﬀerential equations. The remarkable point is that this result does not rely

on the observation of domestic labor supplies. Moreover, the sharing rule and

the domestic labor supplies are partially identiﬁable from the sole observation

of market labor supply. In othre words, collective models of labor supply which

are consistent with domestic production can be estimated with usual household

surveys. This opens new directions for research.

2Mainresults

2.1 The model

The model is similar to Chiappori’s (1997) and our description will be brief. We

consider a two-person household (say A and B). Each member is characterized

by speciﬁc preferences which can be represented by regular utility functions:

u (L ,C,Z )I I I I

where L denotes member I’s leisure, C denotes his/her consumption of aI I

marketable good and Z denotes his/her consumption of a domestic good (I =I

A,B). This good is produced with the following technology:

A BZ =F(t ,t ),

A Bwhere t denotes member I’s domestic labor and F(·) is concave in t and t .I

There is a market where the domestic good can be exchanged at a constant

2price (in consequence, Z =Z +Z in general). Member I’s wage and incomeA B

2This assumption is usual in the literature since Gronau’s (1977) seminal paper. Even

if the goods trade on outside markets may be imperfect substitutes for domestic goods, our

argument can be seen as an improvement in the usual collective model of labor supply which

simply neglects domestic production.

23are respectively denoted by w and y . We consider the case of cross-sectionalI I

data. The prices of consumption can be nomalized to one.

Following the basic idea of the collective approach, we simply assume that

the decision process, whatever its true nature, always generates Pareto-e ﬃcient

outcomes. From the Theorems of Welfare Economics (or the Principle of Sep-

aration between consumption and production), the allocation problem can be

decentralized. Firstly, household members determines domestic labor supply in

order to maximize proﬁt: © ª

A BΠ(w ,w )=max F(t ,t ) −w t −w tA B A B A B

t ,tA B

where Π(w ,w ) is a normalized proﬁt function, and they agree with a distri-A B

bution of total income:

Ψ =y +y + Π(w ,w )+Tw +Tw .A B A B A BP

Secondly, each household member receives a share ρ ,with ρ = Ψ, of totalI I I

income Ψ and independently maximizes his/her utility with his/her personal

budget constraint:

max u (L ,C,Z )I I I I

L ,C ,ZI I I

subject to

w L +C +Z = ρ (w ,w ,y ,y ).I I I I A B A BI

In other words, the function ρ can be seen as the natural generalzation of theI

sharing rule in the case of household production. The leisure demands have the

following structure:

I I IL =L (w , ρ (w ,w ,y ,y )).I A B A B

Moreover, the domestic labor supplies, which result from proﬁt maximization,

do not depend on incomes. In particular, they have the following structure:

I It =t (w ,w ),A B

A Bwith t = t by symmetry. Using these expressions yields the market laborw wB A

supplies:

I I Ih =T −L (w , ρ ) −t (w ,w ) (1)I A BI

Inthefollowing,weassumethatonlythemarketlaborsuppliesareobservedand

try to determine what can be sait about the leisure demands and the domestic

labor supplies.

3We suppose that there are two sources of income: one for each household member. How-

ever, this assumption can be replaced by the existence of one distribution factor.

32.2 An Informal Look at the Problem

Tobeginwith,itisinterestingtoinvestigatewhatisactuallyestimatedwhenthe

econometrician mistakenly does not account for domestic production. Consider,

for example, the functional form used by Chiappori, Fortin and Lacroix (2001)

and given by

I

h =a +b logw +c logw +d logw logw +e Y +f Y , (2)I I A I B I A B I A I B

where a ,...,f are parameters and Y =y +Tw.Itcanbeshownthat,ifaI I I I I

constraint is applied to parameters, i.e.,

d dA B

= ,

e −f e −fA A B B

this form is compatible with a simple model of labor supply where the sharing

4rule is given by

I I I I I Iρ = π logw + π logw + π logw logw + π Y + π Y (3)A B A B A B1 2 3 4 5

and Marshallian labor supplies by

I Ih = α + β logw + γ ρ .I II I

However, the functional forms (2) for market labor supplies are also compatible

with a more general model accounting for domestic production. Suppose, for

example, that the proﬁt function is given by

Π = −(logw +logw ). (4)A B

Then, from the Hotelling Lemma, the domestic labor supplies are given by

1It = .

wI

Finally, it is quite easy to show that the reduced form of the market labor sup-

plies can be written under the form (2). To do that, we assume that the sharing

ruleisasin(3)andtheMarshalliandemandsforleisurehavethefollowingform:

1I IL = α + β logw − + γ ρ .I II I

wI

In conclusion, for some functional forms at least, the market labor supplies

resulting from the traditional collective model can be seen as resulting from a

more general model with domestic production.

Consider now a more general problem and assume that there exist someP P

Ifunctions H and ϕ ,with ϕ = Y , such that market labor supplies canII I

be written as follows:

I I I IT −h =H (w , ϕ )=L (w , ρ )+t (w ,w ) (5)I I A BI I

4In Chiappori, Fortin and Lacroix’s speciﬁcation, the individual shares add up to total

non-labor income. We choose another normalization to be consistent with our theory.

46

I Iwhere ϕ andH respectivelydenotethesharingruleandtheMarshallianleisure

demand when the econometrician assumes there is no domestic production. In

this case, the econometrician possibly estimates the structural components of a

false model. The question is: what are the scale and the direction of the errors

he makes? To begin with, if we di ﬀerentiate (5) with respect to y ,itiseasytoJ

show that

I Iϕ = ρ , (6)y yJ J

and

I IH =L . (7)ϕ ρ

Inwords,theincome-e ﬀectsofsharesandtheEngelcurves,estimatedwithusual

techniques, are also correctly estimated even if there is domestic production.

The formal proof of this point is given below (see the proof of Proposition 1).

The underlying intuition is that incomes do not enter the proﬁtfunctionand

the domestic labor supplies. Also, the sharing rule for both models is computed

inthesameway.

Now if we di ﬀerentiate (5) with respect to w , use (7) and rearrange, weJ

obtain:

ItI I wJϕ = ρ + , (8)w wJ J IL ρ

with J = I.Thecros-eﬀects of wages on shares, estimated with usual tech-

niques,aregenerallynotcorrectifdomesticproductionisnottakenintoaccount.

However, we conjecture that

ItwJ > 0

IL ρ

if leisure is normal and domestic labor of household members are gross sub-

5stitute. It means, if we accept these assumptions, that the cross-e ﬀects of

wages are in general over-estimated when the econometrician neglects domestic

production. One remarkable point is that only the slope, and not the level of

domestic labor supplies, is relevant to measure the derivatives of the sharing

rules. The bias is small when domestic labor supplies are relatively unsensitive

to wages and/or when leisure demands are very sensitive to income shares. In

the particular case where the production function is additive, i.e.,

Z =f (t )+f (t ),A A B B

the bias vanishes. P P

IFinally, if we di ﬀerentiate the adding up restriction ϕ = Y withII

respect to w and use (8), we obtain:I

ItwJ I Jϕ =T − ρ − ,w wJ J IL ρ

5The ﬁrst statement is uncontroversial but the second one is more questionable. We base

our conjecture on empirical evidence, given by Leibowitz (1974), that husbands’ and wives’

times are strongly, or perfectly substituable. Similarly, Hill and Juster (1985) underline that

a higher value of wage increases spouse’s domestic labor. This e ﬀect is of small amplitude.

56

6

6

6

6

P P

INow, we di ﬀerentiate the adding up restriction ρ = Ψ = Π+ Y and usesII

Hotelling Lemma, we obtain:

I J Jρ =T −t − ρ .w wJ J

All in all, these relations give: µ ¶JtwI I I Iϕ = ρ + t − . (9)w wI I JL ρ

The sign of the second term in the right-hand-side is clearly undetermined and

it depends on the level of domestic labor supply. This sign is negative (at least

if our conjecture is accepted) if member J does not participate to domestic

production.

One ﬁnal interpretation of (6)-(9) is that, if the traditional sharing rule ϕ

can be identifed and the domestic labor supplies are oberved, than the sharing

rule ρ, which is consitent with domestic production, can be identiﬁed as well.

This result does not suppose that returns to scale are constant.

2.3 Main Result

Itispossibletoshowthatsomestructuralcomponentsofmodelcanberetrieved

fromthesoleobservationofthemarketlaborsupply. Weintroducethefollowing

deﬁnitions:

A B A B A B A Bh h −h h h h −h hy y y y y y y yB A A B B A A BA= − ,B = −

B B A Ah −h h −hy y y yA B B A

whereweassumethatdenominatorsarediﬀerent from zero, and the following

regularity condition.

Condition R The market labor supplies are such that:

B B A A A B A Bh =h ,h=h ,hh =h h ,A=0,B=0y yy y y y y y y y A BA B B A B A A B

almost everywhere.

The main result is then formally stated in the next proposition.

Proposition 1 Assume collective rationality with domestic production. Then,

under Condition R,

1. The market labor supply have to satisfy testable constraints under the form

of partial di ﬀerential equations;

2. The share and the domestic labor supply of member A (resp. B)canbe

identiﬁed up to a function of w (resp. B).A

6The complete proof of this Proposition is given in Appendix. We will only

sketch the basic steps here. The critical point is that the proﬁt function and

the domestic labor supplies are not observed by the econometrician. However,

the theory says that these functions do not depend on incomes. Thus, using

the result of Chiappori (1997), it is possible to retrieve the e ﬀect of incomes on

individual shares and the e ﬀect of these shares on leisure demand. The next

step is to identify the cross-e ﬀect of wages. To do so, we again use the fact that

the domestic labor supply doe not depend on incomes. The partner’s wage has

only an income e ﬀect through the sharing rule. At this stage, several remarks

are in order.

1 The second statement in Proposition 1 can be interpreted as follows. If

Iρ (w ,w ,y ,y ) is a particular solution for the sharing rule, then the generalA B A B∗

solution is

I I Iρ (w ,w ,y ,y )= ρ (w ,w ,y ,y )+k (w )A B A B A B A B I∗

Iwhere k (w ) is an unknown function. By comparison, in the simple modelI

of labor supply or in the Chiappori’s (1997) model, the undeterminacy of the

sharing rule is a constant.

2 Condition R excludes the hypothesis of income pooling. It also excludes

market labor supplies which are linear in incomes such as in (2). This may

explain why this functional form raises identiﬁcation issues.

3 Even if identiﬁcation is incomplete, the result is very useful specially in view

of policies based on cash subsidies whose e ﬀect is exactly to alter the members’

respective non-labor income. The cross-e ﬀectofwagesonsharesisimmediately

interpretable in terms of welfare. That is, it is possible to say whether an

increase inw (for example) has a positive or a negative impact on memberB’sA

welfare. The own-e ﬀect of wages on shares is not identiﬁable. However, even if

it was possible, this e ﬀect would be useless without the identiﬁcation of utility

functions.

3Apendix

3.1 Proof of Proposition 1

Deﬁne ρ = ρ et Y − ρ = ρ .Ifwediﬀerentiate the market labor supply (1)A B

with respect to y and y , we obtain:A B

A A Ah = −L ρ ,y ρ yA A

A A Ah = −L ρ ,y ρ yB B¡ ¢

B B

h = −L 1 − ρ ,y ρ yA A¡ ¢

B Bh = −L 1 − ρ ,y ρ yB B

76

If Condition R is satisﬁed, Chiappori (1997) shows that this system can be

A A B B A Bsolved with respect to ρ , ρ , ρ , ρ , L et L as a function of w ,w ,A By y y y ρ ρA B A B

y et y . We then adopt the following notations:A B

B B Ah −h hy y yA A A B Aρ = h = − (10)y yA A A B A Bh h −h h Ay y y yB A A B

A A Bh −h hy y yB B B A Aρ = h = − (11)y yA A A B A Bh h −h h By y y yB A A B

B B Ah −h hy y yA A A B Bρ = h = − (12)y yB B A B A Bh h −h h Ay y y yB A A B

A A Bh −h hy y yB A BB Bρ = h = − (13)y yB B A B A Bh h −h h By y y yB A A B

A B A Bh h −h hy y y yA B A A BL = − =A (14)ρ B Bh −hy yA B

A B A Bh h −h hy y y yB B A A BL = − =B (15)ρ A Ah −hy yB A

The cross-derivative restrictions imply that

A ACondition 1 h A =.h A .y yy B y AA B

Consider now the derivatives of the market labor supply (1) with respect to

w (J =I). We obtain:J

A A A Ah = −L ρ −t (16)w ρ w wB B B

B B B Bh = −L ρ −t (17)w ρ w wA A A

Ifwedi ﬀerentiatetheseexpressionswithrespecttoy andy ,anduse(10)-(15),A B

with Condition R, we obtain:

AwA A Bρ = −h (18)w yB AA AyA

BwB B A

ρ = −h (19)w yA BB ByB

The cross-derivative restrictions imply that

A 2 A A ACondition 2 h A −h A A +h A A −h A A =0w y w y y y w yy w y y y B A y B A A y A B AA B A A A A A

A 2 A A A 3 h A −h A A +h A A −h A A =0w y w y y y w yy w y y y B A y B A B y A B BB B A A B A A

Introducing (18) and (19) in (16) and (17) yields the derivatives of the do-

mestic labor supply:

AwA A B At = h −hw y wB A BAyA

BwB B A B

t = h −hw y wA B AByB

8The symmetry implies that

A Bw wB AA A B BCondition 4 h −h =h −h . ky w y wA B B AA By yA B

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