Corollary 1 (Power Illusion)
(i) If αh1 = α1 for all h and α2 > maxh=1 {α%}, then UU111 < U721.
(ii) If αh1 = α1 for all h and α2 < minh=1 {α2h}, then U11 > U721.
The corollary illustrates that a member of a sociological group is better off if he has
the highest internal bargaining power even if the level of his power is much smaller
than in another economy where all individuals of the group have the same bargaining
power, that is α1 > αɪ. The underlying intuition runs as follows: Diversity across
households opens trade opportunities. The gains from trade will, as a rule, accrue
primarily to the members of a sociological group who have relatively higher bargaining
power than other members of the group. The absolute level of bargaining power is
not important. When, however, the bargaining power of other individuals in the same
sociological group is enhanced as well and all individuals of the sociological group enjoy
an identical level in bargaining power, the original gain is totally eroded.4
5.3 An Example
To illustrate the preceding proposition by solving explicitly for the market equilibria,
we consider again a society with n > 1 identical two-member households.
To simplify notation, we use the symbols xh1 and xh2 to denote quantities of the
first good consumed by household member h1 and h2, respectively. The symbols yh1
and yh2 denote quantities of the second commodity consumed by household member
h1 and h2, respectively.
Each household h is endowed with ωh = (1, 2). The two members of household h
have utility representations
Uh1(xh1, yh1) = xh1 and
Uh2(xh2, yh2) = yh2.
The household h maximizes
Sh = Uhα1 Uh12-α = xhα1 yh1-2 α or ln Sh = α ln xh1 + (1 - α) ln yh2.
4When separate sphere consumption does not apply in the strict way postulated above, only partial
erosion will occur, e.g. when all individuals have Cobb-Douglas utility functions.
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