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

140 3. Preferences and decision making
is that at the Nash bargaining equilibrium ¡ ū a , ū b¢ , ū a is increasing in Ta
and decreasing in Tb (while, obviously, ū b is decreasing in Ta and increasing
in Tb). Hence, any change that increases a member’s threat point without
changing the Pareto frontier (the typical impact of a distribution factor)
will ameliorate this member’s situation.
Finally, the symmetry axiom can be relaxed. Then the general form is
a straightforward generalization of the previous one: instead of maximizing
the sum of log surpluses, one maximizes a weighted sum of the form
γ a log (u a − T a )+γ b log ¡ u b − T b¢ . In this form, the weights γ s introduce
an asymmetry between the members’ situations.
Kalai-Smorodinsky
An alternative concept has been proposed by Kalai and Smorodinsky (1975).
It relies on the following, monotonicity property. Consider two bargaining
problems such that (i) the range of individually rational payoffs thatplayer
a can get is the same in the two problems, and (ii) for any given, individually
rational utility level for player a, the maximum utility that player b can
achieve (given the Pareto frontier) is never smaller in the second problem
than in the first. Then player b should do at least as well in the second
problemthaninthefirst. In other words, if one enlarges the Pareto set by
inflating b’s opportunities while keeping a’s constant, this change cannot
harm b.
Kalai and Smorodinsky prove that there exists a unique bargaining solution
that satisfies all the previous axioms except for independence, which is
replaced with monotonicity. Moreover, the solution has an interesting interpretation.
Define the aspiration level As of player s as the maximum utility
he/she can get that is compatible with feasibility and the partner’s individual
rationality constraint; this corresponds to the point on the Pareto
frontier that leaves the partner, say s0 , at their threat point utility T s0.
Define,now,theideal point as the point ¡ Aa ,Ab¢ ; obviously, this point
lies outside of the Pareto frontier. The solution, now, is to chose a point
U = ¡ ua ,ub¢ on the Pareto frontier so that
ua − T a
ub − T b = Aa − T a
Ab − T b
In words, the bargaining is here influenced, in addition to the threat points,
by the players’ aspirations about what they might receive within marriage.
The surplus share received by player s, u s − T s , is directly proportional to
the maximum gain s could aspire to, A s − T s .
Non cooperative foundations
Finally, an on going research agenda, initially proposed by Nash, is to provide
noncooperative foundations to the bargaining solutions derived from
axioms. The most influential framework is the model of Rubinstein (1982),