Bargaining Power and Equilibrium Consumption

Therefore, in equilibrium, the aggregate consumption for household h is (χ,y) = (δ, 2).
The associated individual shares are

σ

x 1 = -----x = σ ;

σ+τ

τ

x ₂ = -----x = τ ;

σ+τ

_ σ*    _ 1   σ*

y¹     *-* I *^y y     O *-* I — * ;

σ + τ*    2 σ* + τ*

_*                 _λ-*

τ        1τ

y² = σ* + τ* y = 2 σ* + τ* ’

As a function of α, consumer 1 achieves

u 1 = -1 μ 1 )^{1 -γ1} μ σ*+τ* )1^- 1

γ1             ^α

= const ₁ ∙ α ¹ -----------------—

1 - γ2 - (γ1 - γ2)α

which is increasing in α. Consumer 2 achieves

1-γ2

τγ2

^const2 ∙ (¹ -^α )^γ 2 μ -—τ¹—^α—r )
1 - γ1 - (γ2 - γ1)(1 - α)

which is decreasing in α. Hence a shift of bargaining power from consumer 2 to con-
sumer 1 benefits consumer 1 and harms consumer 2, who ends up consuming less of
both commodities.

□

Proof of Proposition 3

Good ` serves as a nume´raire so that the price system assumes the form (p1 , . . . , p`-1, 1).

We consider the first-order conditions of maximizing S_h in household h:⁵

1 ∂ uh1

^αh~jJ ~0~^{-     λ}h pk = 0^, k = ^1, ■ ■ ■ ^, ' ^{- 1};

Uh1 ∂x_h1

^αh~r_τ     ^λh = 0;

Uh1

^{(1 - α}h )— ∙ -—k    ^λh pk = 0^, k = ^1, ■ ■ ■ ^, ' ^- 1;

Uh2 ∂x^kh2

^{(1 - α}h)~j~_τ--^λh = 0■

Uh2

⁵ Note that our assumption of sufficient endowments with the num´eraire good in all households
allows us to work with the entire set of first-order conditions.

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