Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

5. Empirical issues for the collective model 213
clothing tend to be very colinear and we have to treat clothing as an
assignable good.
Two remarks are in order at that point. First, the identifiability result
just presented is, by nature, non parametric, in the sense that it does not
rely on the choice of a specific functional form for either preferences or
Pareto weights. 7 Under an explicitly parametric approach, stronger identification
results may obtain; for instance, it may be the case that one
exclusive good only is sufficient to identify all the relevant parameters.
Clearly, these additional properties are due to the specific functional form
under consideration. Second, the result is generic, in the sense that it holds
for ‘almost all’ structures. An interesting remark is that (non-generic) exceptions
include the case in which Pareto weights are constant; in such a
case, the collective indirect utilities are not identifiable in general. 8 To see
why, simply note that, in that case, the household maximizes a collective
utility of the form:
U ¡ q a , q b , Q ¢ = μu a (q a , Q)+u b ¡ q b , Q ¢
(5.19)
under budget constraint; remember that here μ is a constant. Standard
results in consumer theory guarantee that we can recover U from observed
(household) demand. However, for any given U there exists a continuum of
u a and u b such that (5.19) is satisfied. For instance, take any such u a and
u b that are strongly increasing and concave, pick up any smooth function
φ, and define ū a and ū b by:
ū a (q a , Q) = u a (q a , Q)+εφ (Q)
ū b ¡ q b , Q ¢ = u b ¡ q b , Q ¢ − μεφ (Q)
Then μū a +ū b = U and (5.19) is satisfied; moreover, on any compact set,
ū a and ū b are concave and increasing for ε small enough.
Ironically, the case of a constant Pareto weight corresponds to the Samuelson
justification of the unitary setting, in which a single, price-independent
welfare index is maximized. From an identification viewpoint, adopting a
unitary framework is thus a very inappropriate choice, since it rules out the
identification of individual welfares.
Our general conclusion is that welfare relevant structure is indeed identifiable
in general, provided that one can observe one exclusive consumption
7 This notion of ‘non parametric’, which is used for instance by econometricians,
should be carefully distinguished from the perspective based on revealed preferences -
which, unfortunately, is also often called ‘non parametric’. In a nutshell, the revealed
preferences approach does not require the observability of a demand function, butonly
of a finite number of points; it then describes relationship that must be satisfied for the
points to be compatible with the model under consideration. This view will be described
in Subsection 5.3.4.
8 This case is ‘non generic’ in the sense that in the set of continuous functions, constant
functions are non-generic.