Bargaining Power and Equilibrium Consumption



5 Comparative Statics Across Households

Until now we have focused primarily on intra-household effects, that is, on the utility
changes in a particular household when bargaining power shifts within that household.
Via a series of examples, we have demonstrated that such a shift of bargaining power
can affect the members of the corresponding two-person household in three different
ways: Only one member is affected; the two members are affected in opposite ways;
both members are affected the same way. We have argued earlier that the above exam-
ples can be readily reinterpreted as instances of widespread shifts of bargaining power
in a replica economy. In the resulting replica economy, the main focus remains on intra-
household effects, on the repercussions on the members of those households in which a
shift in bargaining power has occurred. However, we have also seen that third parties
can be affected. In this section, we redirect our attention to such inter-household or
spill-over effects. We start with a neutrality result that can serve as a benchmark.

5.1 A Neutrality Result

We consider a society with n > 1 identical households. Household h (h = 1, . . . , n)
has members
h1 and h2, called the first member and the second member, respectively.
There are
` goods (` > 1). The consumption of good k (k = 1, . . . , `) by individual
hi (i = 1, 2) is denoted by xkhi. Each household h is endowed with wh = (wh1 , . . . , wh` ).
The two members of household
h have quasi-linear utility representations of the form

Uh 1(Xh 1) = Uh 1 (xh 1,... ,xeh-v) + Xh 1                         (16)

Uh 2( Xh 2) = Uh 2 (x h 2 ,...,Xh-1) + Xh 2                      (17)

where uhi is assumed to be strictly concave, strictly increasing and differentiable.
Household
h maximizes

Sh= Uhα1h Uh12h or lnSh=αhlnUh1+(1 h)lnUh2           (18)

where 0 < αh < 1 is the bargaining power of individual h1 in household h. We denote
equilibrium values by
Xhi and equilibrium utilities by Uhi and Uhi. For the following
we assume that for any array of bargaining power parameters (
α1 , . . . , αn) under con-
sideration, each individual consumes a non-negative amount of the natural num´eraire

19



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