Bargaining Power and Equilibrium Consumption

equilibrium price system is

p = (2(1 - α) ^,1);

the equilibrium consumption bundles are c0 = (α, 1 /2), cɜ = (1 — α, 1 /2).

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Now we are prepared to consider the case of three individuals, labelled i = 1, 2, 3.
Consumers 1 and 2 form the two-person household h. In this household, consumer
1 has bargaining power α and consumer 2 has bargaining power 1 - α. Consumer 3
constitutes the single household k . We are going to scrutinize several representative
examples which are almost exhaustive in that they exhibit three possible allocative
responses to a shift of bargaining power within the two-person household:

(a) Only one member is affected.

(b) The two members are affected in opposite ways.

(c) Both members are affected in the same way.

The examples differ only in individual consumer preferences. The analysis suggests
that less substitutability leads to more drastic price effects. We start with the following
example of case (a).

Example 1.

Here consumer 1 benefits from more bargaining power, to the detriment of consumer
3 while consumer 2 is unaffected. Household h is endowed with ω_h = (1, 0). Its two
members, i = 1, 2 have utility representations

u1(x1, y1) = x1 and u2(x2, y2) = y2.

The household maximizes

_{Sh =} u₁αu1₂-α ₌ x₁αy₂1-α _{, 0 <} α _< 1.

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