Bargaining Power and Equilibrium Consumption

11

^λh = ^αh ~j~_τ = (1 ^{- α}h ) 77 .

Uh1            Uh2

∂ uh1   ∂ uh2

∂ kk    ∂ k^k    p^k, ^{k 1},...,'   ¹ ’

^xh1      ^xh2

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):

Since U_{h 1} and U_h2 are independent of α_h and xh ₁ + xh2 does not depend on α_h either,
assertion (iii) follows. Using again the fact that variations in α_h 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(x¹_h1) + (1 - αh)ln Uh2(x²_h2)

s.t. x¹_h1 p1 + x²_h2 = w_h¹ p1 + w_h²

The first-order conditions amount to:

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