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Donni_O

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• Publié par: Ewlie
• Nombre de lectures: 17
• Langue: English

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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.
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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