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

126 3. Preferences and decision making
This setting generalizes Samuelson because the welfare index W depends
on individual utilities, but also directly on prices, total expenditure and distribution
factors. In other words, the household maximizes an index that is
price (and income) dependent, which distinguishes this setting from a unitary
representation. The surprising property is that under tsrict concavity
one can assume without loss of generality that the index W is indeed linear
(as in (3.40)). We shall come back to this issue below.
Ordinal versus cardinal representation of preferences
It is important to understand what, in the previous discussion, requires a
cardinal representation of preferences, and what can be defined using only
a standard, ordinal representation. The set of Pareto efficient allocations is
an ordinal concept; it is not modified when us is replaced with F (us ) for a
strictly increasing mapping F (.). Under smoothness conditions, the set is
one-dimensional, and therefore can be described by one parameter. However,
the parametrization entails cardinality issues. For instance, a natural
parametrization is through the weight μ. Butμdepends on the particular
cardinal representation that has been adopted for ua and ub :ifus is
replaced with F (us ), then the parameter μ corresponding to a particular
efficient allocation has to be modified accordingly. Moreover, the convexity
properties of the Pareto set are also of a cardinal nature. Assuming
smooth preferences, for any given price-income vector, one can find cardinal
representations of preferences such that the Pareto frontier is convex,
linear or concave. In most of what follows, we adopt the convention of always
using a strictly concave representation of utilities. In this case, the
Pareto set is strictly convex. Indeed, for a given price-income vector, take
two points ¡ ūa , ūb¢ and ¡ u0a ,u0b¢ on the Pareto frontier, and let ¡ Q, qa , qb¢ and ¡ Q0 , q0a , q0b¢ be the corresponding consumption vectors. The vector
¡ 00 00a 00b
Q , q , q ¢ = 1 ¡ a b
Q, q , q
2
¢ + 1 ¡ 0 0a 0b
Q , q , q
2
¢
satisfies the budget constraint, and by strict concavity,
u s ¡ Q 00 , q 00a , q 00b¢ > 1
2 us ¡ Q, q a , q b¢ + 1
2 us ¡ Q 0 , q 0a , q 0b¢ = 1
2 ūs + 1
2 u0s
for s = a, b. We conclude that the point 1
2
¡ ū a , ū b ¢ + 1
2
¡ u 0a ,u 0b ¢ belongs to
the interior of the Pareto set.
Graphically, on Figure 3.4, the Pareto set is indeed strictly convex. We
seethatanypointontheUPFcanbedefined either by its coordinate on the
horizontal axis, here u a , as in program (3.37), or by the negative of the slope
of the Pareto frontier at that point, here μ as in program (3.40). Given that
the UPF is strictly concave there is an increasing correspondence between
ū a and μ: alargerū a (or μ) corresponds to an allocation that is more
favorable to a (hence less favorable for a). Note that the correspondence