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

3. Preferences and decision making 127
between ū a and μ is one-to-one; that is, for any ū a , there exists exactly
one μ that picks up the efficient point providing a with exactly ū a ,and
conversely for any μ there is only one allocation that maximizes (3.40)
under budget constraint, therefore only one corresponding utility level ū a .
We can also understand from Figure 3.5 why the maximization of generalized
Samuelson index W ¡ u a ,u b , P, p,x, z ¢ is equivalent to that of a
linear combination μu a + u b . The maximization of a non linear index W
will select a point where the Pareto frontier is tangent to some indifference
curve of W .If−μ denotes the slope of the corresponding tangent, maximizing
μu a + u b leads to exactly the same point. Replacing W with its linear
equivalent can be done at any point, provided that μ varies adequately;
technically, this simply requires that:
μ = ∂W/∂ua
∂W/∂u b
The main drawback of the generalized index version is that a continuum
of different welfare indices lead to the same choices. Indeed, for any
function F strictly increasing in its first argument, the indices W and
¯W = F (W, P, p,x, z) are empirically indistinguishable. The linear version,
from this perspective, has an obvious advantage in terms of parsimony; in
addition, it has a natural interpretation in terms of distribution of powers
(see below).
Finally, we may briefly discuss two particular cases. One obtains when
the cardinal representations of utilities are concave but not strictly concave.
In that case, the UPF may include ‘flat’ (that is, linear) segments (Figure
3.6). Then Program (3.37) is still equivalent to Program (3.40), but the
relationship between ū a and μ is no longer one-to-one. It is still the case
that for any ū a ,exactlyoneμ picks up the efficient point providing a with
ū a . But the converse property does not hold; that is, to some values of μ are
associated a continuum of utility levels ū a ; graphically, this occurs when −μ
is exactly the slope of a flat portion of the UPF. 13 This case is particularly
relevant for two types of situations, namely transferable utility on the one
hand (then the cardinalization is usually chosen so that the entire UPF is
a straight line) and explicit randomization on the other hand.
The second particular case relates to local non differentiability of utility
functions (Figure 3.7). Then the UPF may exhibit a kink, and the one-toone
relationship breaks down for the opposite reason - namely, many values
of μ are associated with the same ū a .
13However, a strictly quasi concave generalized welfare index would still pick up exactly
one point.