Therefore:
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λh = αh ~j~τ = (1 - αh ) 77 .
Uh1 Uh2
∂ uh1 ∂ uh2
∂ kk ∂ kk pk, k 1,...,' 1 ’
xh1 xh2
(21)
(22)
Because of differentiability and strict concavity, the demand of household h for
commodities k = 1, . . . , ` - 1 is independent of the bargaining power αh and 1 - αh
of individual h1 and h2, respectively. Hence, by the budget constraint and budget
exhaustion also the aggregate household demand for commodity ` is independent of
αh. Therefore, the market clearing price system (p1, . . . ,p`-1, 1) does not depend on
internal bargaining power of households and, hence, changes of bargaining power in
household h have no effect on other households. This establishes assertions (i) and (ii).
In contrast to all other goods, a shift of power in household h affects the distribution
of the num´eraire good in household h, as we shall establish next. Using the notation
for equilibrium values we obtain from equation (21):
αh
u h I + x h 1
1 - αh
U h 2 + x h 2
(23)
Since Uh 1 and Uh2 are independent of αh and xh 1 + xh2 does not depend on αh either,
assertion (iii) follows. Using again the fact that variations in αh have no effect on
aggregate excess demand, we conclude that if households are completely homogeneous
with respect to Uhi and wh , then a market equilibrium does not exhibit any positive
net trades. Therefore, x'h 1 + x'h2 = w'h and via equation (23) we obtain the expressions
in (iv).
□
Proof of Proposition 4
We normalize prices by p2 = 1. Then the problem of household h is given by:
max ln Sh = αh ln Uh1(x1h1) + (1 - αh)ln Uh2(x2h2)
s.t. x1h1 p1 + x2h2 = wh1 p1 + wh2
The first-order conditions amount to:
1
αh Uh,( x h 1)
Uh01(x1h1) - λhp1 = 0;
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