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

256 6. Uncertainty and Dynamics in the Collective model
The sharing rule as a risk sharing mechanism
We now further specify the model by assuming that prices do not vary:
ps = p, s =1,...,S
Let V X denote the indirect utility of agent X. For any ex post efficient
allocation, let ρ X s denote the total expenditure of agent X in state s:
ρ X s = X
i
pic X s,i
Here as above, ρ X is the sharing rule that governs the allocation of household
resources between members. Obviously, we have that ρ a s + ρ b s = y a s +
y b s = ys. Ifwedenoteρ s = ρ a s,thenρ b s = ys − ρ s. Program (6.16) becomes:
W (y1, ..., yS; μ) = max
ρ 1 ,...,ρ S
X
πsV a (ρs)+μ X
πsV b (ys − ρs) (6.19)
s
In particular, in the absence of price fluctuations, the risk sharing problem
is one-dimensional: agents transfer one ‘commodity’ (here dollars) across
states, since they are able to trade it for others commodities on markets
once the state of the world has been realized, in an ex post efficient manner.
When is a unitary representation acceptable?
The value of the previous program, W (y1, ..., yS; μ), describes the household’s
attitude towards risk. For instance, an income profile (y1, ..., yS) is
preferred over some alternative (y0 1, ..., y0 S ) if and only if W (y1, ..., yS; μ) ≥
W (y0 1,...,y0 S ; μ). Note, however, that preferences in general depend on the
Pareto weight μ. That is, it is usually the case that profile (y1, ..., yS) may
be preferred over (y0 1, ..., y0 S ) for some values of μ but not for others. In that
sense, W cannot be seen as a unitary household utility: the ranking over
income profiles induced by W varies with the intrahousehold distribution
of powers (as summarized by μ), which in turns depends on other aspects
(ex ante distributions, individual reservation utilities,...).
A natural question is whether exceptions can be found, in which the
household’s preferences over income profiles would not depend on the member’s
respective powers. A simple example can convince us that, indeed,
such exceptions exist. Assume, for instance, that both VNM utilities are
logarithmic:
V a (x) =V b (x) =logx
Then (6.19) can be written as:
X
πs log (ρs)+μ X
πs log (ys − ρs) (6.20)
max
ρ 1 ,...,ρ S
s
s
s