Bargaining Power and Equilibrium Consumption

Example 2.

Here consumer 1 benefits from more bargaining power to the detriment of consumer 2.
Consumer 3 either gains or loses. Household h is still endowed with ωh = (1, 0). But
now each member i = 1, 2 has Cobb-Douglas preferences with utility representation

ui(xi, yi) = x_i^γiy_i^1-γi , 0 < γi < 1.

The household maximizes

U au 2^-α = ( X11 y 11^{-γ 1} )^α ( x Y2 y1^-γ2 ¢1^-α
αγ1 (1-α)γ2 α(1-γ1) (1-α)(1-γ2)
= x₁ x₂       y₁        y₂            .

Again, α and 1-α lend themselves as measures of relative bargaining power of consumer
1 and consumer 2, respectively.

Household k has the single member 3, with the same consumer characteristics as before.

We obtain:

Fact 2 A shift of bargaining power from consumer 2 to consumer 1 benefits consumer
1 and harms consumer 2, who ends up consuming less of both commodities.

In Example 2 there is more substitutability in the economy than in Example 1.
Example 3 exhibits less substitutability than Example 1, because the preferences of
consumer 3 will be altered from Cobb-Douglas to Leontieff. It turns out that the lack
of substitution by consumer 3 necessitates a major price adjustment to re-equilibrate
the market after bargaining power within household h has shifted. As a result, we
observe a drastic price effect: When bargaining power within their household changes,
the equilibrium utilities of consumers 1 and 2 are moving in the same direction.

Example 3.

Here a shift of bargaining power from consumer 2 to consumer 1 benefits both con-
sumers to the detriment of consumer 3. This example is identical with Example 1,
except that consumer 3 now has Leontief preferences with utility representation

u3(x3, y3) = min(x3, y3).

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