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

4. The collective model: a formal analysis 173
In other words, the structures ¡ va ,vb ,ρ ¢ and ¡ va ε ,vb ¢
ε,ρε , although different,
generate the same collective indirect utilities. It follows that the welfare
conclusions reached by the two structures are always identical. For instance,
if a given reform is found to increase his welfare and decrease her welfare
when the evaluation is made using the first structure, using the second
instead will lead to the same conclusion. We say that different structures
that generate the same collective indirect utilities are welfare equivalent.
The notion of welfare equivalence plays an important role, notably in
the discussion of identification in Chapter 5. In many situations, welfare
equivalent structures are hard to empirically distinguish; in some cases, only
the collective indirect utilities can actually be recovered. The key remark
is that as far are welfare judgment are concerned, identifying collective
indirect utilities is sufficient.
4.4 Application: labor supply with private
consumption
4.4.1 The general setting
An example that has been widely analyzed in the literature concerns labor
supply. In the most stripped down model without household production,
labor supply is modelled as a trade off between leisure and consumption:
people derive utility from leisure, but also from the consumption purchased
with labor income. In a couple, however, an additional issue is the division
of labor and of labor income: who works how much, and how is the resulting
income distributed between members? As we now see, the collective
approach provides a simple but powerful way of analyzing these questions.
Let l s denote member s’s leisure (with 0 ≤ l s ≤ 1) andq s the consumption
by s of a private Hicksian composite good whose price is set to
unity. We start from the most general version of the model, in which member
s’s welfare can depend on his or her spouse’s consumption and labor
supply in a very general way, including for instance altruism, public consumption
of leisure, positive or negative externalities, etc. In this general
framework, member s’s preferences are represented by a utility function
U s ¡ l a ,q a ,l b ,q b¢ .Letw a , w b , y denote respectively real wage rates and
household non-labor income. Finally, let z denote a K-vector of distribution
factors. The efficiency assumption generates the program:
max
{la ,lb ,qa ,qb μU
}
a + U b
subject to q a + q b + w a l a + w b l b ≤ w a + w b + y
0 ≤ l s ≤ 1, s = a, b (4.46)