comment

comment

-

Documents
9 pages
Lire
Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

Description

A Note on Collective Models of Labor Supplywith Domestic ProductionOlivier DonniUniversity of Quebec at Montreal,CIRPEE and DELTAJanuary 21, 20041IntroductionThe collective model of labor supply, developed by Chiappori (1988, 1992), isnow a standard tool for analyzing household behavior, and its empirical successfor the last 10 years is considerable. One of the most serious criticisms of thismodel, however, concerns the treatment of domestic production. Apps andRees (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 largeconsumption of leisure, whereas it may in fact reflect the specialization of oneof the members in domestic production. This may invalidate the well-knownidentification results for this model.One answer to these problems is given by Chiappori (1997) who considers acollective model of domestic and market labor supply. As for the simple model,each household member is characterized by a system of egoistic preferencesand the decision process results in Pareto-e fficient outcomes. There are onemarket good which is bought and one domestic good which is produced from atechnologyusingtimeinput. Thelattercanbeconsumedbyhouseholdmembersor exchanged on a market. Chiappori then shows that, provided that domesticand market labor supplies are both observed, the structural model, i.e., theoutcome of the decision process, is fully identifiable (except for a ...

Sujets

Informations

Publié par
Nombre de visites sur la page 17
Langue English
Signaler un problème

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
References
[1] Apps P.F. and Rees R. (1996), ‘Labour Supply, Household Production and
Intra-family WelfareDistribution’, Journal of Public Economics, vol. 60.pp.
199—219.
[2] Apps P.F. and Rees R. (1997), ‘Collective Labor Supply and Household
Production’, Journal of Political Economy, vol. 105, pp. 178—190.
[3] Chiappori P.A. (1988), ‘Rational Household Labor Supply’, Econometrica,
vol. 56, pp. 63—90.
[4] Chiappori P.A. (1992), ‘Collective Labor Supply and Welfare’, Journal of
Political Economy, vol. 100, pp. 437—467.
[5] Chiappori P.A. (1997), ‘Introducing Household Production in Collective
Models of Labor Supply’, Journal of Political Economy, vol. 105, pp. 191—
209
[6] Chiappori P.A., Fortin B. and Lacroix G. (2001), ‘Marriage Market, Divorce
Legislation, and Household Labor Supply’, Journal of Political Economy,
vol. 110, pp. 37—72.
[7] Gronau R. (1977), ‘Leisure, Home Production and Work – The Theory of
the Allocation of Time Revisited’, Journal of Political Economy, vol. 85,
1099—1123.
[8] Hill M.S. and Juster F.T. (1985), ‘Constraints and Complementarities in
Time Use’. In: Juster F.T. and Sta fford F.P. (eds), Time, Goods, and Well-
being, Ann Arbor: Institute for Social Research, University of Michigan.
[9] LeibowitzA.(1974),‘Productionwithinthehousehold’, American Economic
Review, Papers and Proceedings, vol. 62, pp. 243—250.
9