Bargaining Power and Equilibrium Consumption

After setting s = min(x₃, y₃), the utility maximization problem for consumer 3 can be
rewritten as

max s s.t. (p1 + 1)s = 1

with solution s = 1/(p1 + 1).

Household h’s demand is (α, (1 - α)p₁). Therefore, market clearing for the first
good requires 1/(p1 + 1) = 1 - α. Thus in equilibrium,

p² = (^α/(1 - ^α )^, 1);

C2 = (α, 0),
c 2 = (0 ,α ),
c*₃ = (1 — α, 1 — α ).

Thus a shift of bargaining power from consumer 2 to consumer 1 benefits both members
of the household to the detriment of consumer 3. A reverse shift harms 1 and 2, and
leaves 3 better off.                                                                      g

The examples suggest that comparative statics is sensitive to the degree of substi-
tutability in the economy. Enhanced substitutability appears to mitigate price effects.
Indeed, if in a further variation of Example 1, one assumes linear preferences (perfect
substitutability) for consumer 3, with utility representation u3 (x3 , y3) = x3 + y3, then
the price effect is zero. Moreover, for two-good economies exhibiting CES-utility func-
tions for all individuals with the same elasticity of substitution, the magnitude of the
price effect can be parameterized by the elasticity of substitution in the economy. The
price effect depends negatively on the elasticity of substitution.

The next section will lend additional support to the conclusion that there exists a
negative relationship between substitutability and the price effect. We will examine
societies where all individuals have quasi-linear utilities. In that case, the price effect
is zero. A gain in bargaining power benefits the consumer at the detriment of the
household member who has less power. Other households, however, are not effected
since the price effect is zero. This indicates that sufficient substitutability can com-
pletely eliminate the price effect, confirming the informal conclusion that enhanced
substitutability tends to mitigate price effects.

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