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

108 3. Preferences and decision making
two distinct roles. For example, a father who spends time with his child
contributes to the development of the child (through f j (.)) andmayalso
enjoy spending time with the child (captured by the presence of tb j in U b (.)).
Of course, either of these effects could be negative (although not both).
A standard problem with this approach is that the production function,
despite its conceptual interest, cannot be estimated independently of the
utility function unless the home produced commodities are independently
observable; see Pollak and Wachter (1975) and Gronau (2006). Observability
of outputs may be acceptable for agricultural production, or even for
children’s health or education; it is less likely for, say, cleaning, and almost
impossible for personal caring.
If only inputs are observed and not outputs we may be able to recover
information about the technology if we make auxiliary assumptions such
as constant returns to scale and assumptions on preferences. To illustrate
this, consider two partners who consume one single public good C and one
private good c such that a consumes ca and b consumes cb with preferences
given by us (C, cs ) ,s= a, b. Assume that the private good is purchased in
the market and that the public commodity is produced using only the time
inputs of the two partners. That is,
C = f ¡ t a ,t b¢ . (3.11)
Assuming that both partners participate in the labor market at wages w a
and w b respectively, it can then be shown that for any efficient allocation
the partners will minimize the cost of producing the public commodity in
terms of the forgone private commodity, yielding
¡
a b t ,t ¢
f1
f2 (t a ,t b )
= wa
w b
(3.12)
in any interior solution. If we assume constant returns to scale, we can
write:
C = f ¡ t a ,t b¢ = t b φ (r) (3.13)
for some function φ (r) where r = ta
tb . The condition (3.12) then reduces to:
φ 0 (r)
φ (r) − rφ 0 wa
=
(r) wb (3.14)
The testable implication of this equality is that r only depends on the wage
ratio ω; this can be tested on a data set that reports wages and time spent
on household production. Defining,
h(r) =
φ 0 (r)
φ (r) − rφ 0 (r)
this equation can be re-written as:
φ 0 (r)
φ (r) =
1
r + 1
h(r)
(3.15)