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

124 3. Preferences and decision making
Becker calls the ‘marriage market’) should matter, at least insofar as the
threat (or the risk) of divorce may play a role in the decision process. Individuals’
income or wealth can also be used as distribution factors. Suppose,
for example, that earned and unearned income is given for any individual
and let Y s denote the total income of person s. Then total household income,
given by Y = Y a + Y b , is all that matters for the budget constraint.
For any given level of Y , the individual contribution of a to total income,
measured for instance by the ratio Y a /Y , can only influence the outcome
through its impact on the decision process; it is thus a distribution factor.
In the collective framework, changes in distribution factors typically lead
to variations in outcomes while the set of efficient allocations remains unchanged;
as such, it provides very useful information on the decision process
actually at stake in the household. For that reason, it is in general crucial
to explicitly take then into account in the formal model. In what follows,
therefore, z denotes a vector of distribution factors.
3.5.4 Modeling efficiency
The basic framework
The characterization of efficient allocations follows the standard approach.
The basic definition is that an allocation is Pareto efficient if making one
person better off makes the other worse off:
Definition 3.1 An allocation ¡ Q, qa , qb¢ is Pareto efficient if any other
allocation (¯q a , ¯q b , ¯Q) that is feasible:
P 0 ¯Q + p 0 ¡ a b
¯q + ¯q ¢ ≤ P 0 Q + p 0 ¡ q a + q b¢
and is such that u a ¡ ¯Q, ¯q a , ¯q b¢ >u a ¡ Q, q a , q b¢ ,itmustbesuchthatu b ¡ ¯Q, ¯q a , ¯q b¢ <
u b ¡ Q, q a , q b¢ (and conversely).
In practice the basic definition is not very tractable and we often use one
of two alternative characterizations. A first characterization is:
Definition 3.2 For any given vector (P, p,x, z) of prices, total expenditure
and distribution factors, an allocation ¡ ¯Q, ¯q a , ¯q b¢ is Pareto efficient if there
exists a feasible ūa , which may depend on (P, p,x, z), such that ¡ ¯Q, ¯q a , ¯q b¢
solves the problem:
max
Q,qa ,qb u b ¡ Q, q a , q b¢
(3.37)
subject to P 0 Q + p 0 ¡ q a + q b¢ ≤ x
and u
(3.38)
a ¡ Q, q a , q b¢ ≥ ū a (P, p,x, z) (3.39)
Thus the Pareto efficient allocation can be derived from maximizing the
utility of one partner holding the utility of the other at a given level: among