Bargaining Power and Equilibrium Consumption

For convenient reference, we state an obvious auxiliary result before proceeding to
the proof of Fact 2.

Lemma 1 Let real numbers σ, τ, z > 0 be given. The solution of the problem

max z₁^σ z₂^τ s.t. z1 ≥ 0, z2 ≥ 0, z1 + z2 = z

is z 1 = ^σ ∙ z, z = = -L- ∙ z, with value

¹ σ+τ ² σ+τ

μ ⅛ )^σ ∙ μ . ) ^τ-σ+τ.

Proof of Fact 2

Let x = x₁ + x₂ and y = y₁ + y₂ denote the total amounts purchased by household
h. By Lemma 1, maximization of the Nash product u₁^αu¹^-^α requires

where

= αγ1,
τ = (1 - α)γ2,

^σ* = ^α(1 - γ 1)^,
τ* = (1 ^{- α})(1 ^- γ2).

Moreover, at the maximum,

_ _ _* _*

u^αu¹ -α = (^σγ ɪ Y ^σ (_IL_ γ y

^{1 2} σ + τ σ + τ σ ^* + τ^* σ ^* + τ^*

with

δ = σ +τ = αγ1 +(1 - α)γ2 = γ2 + α(γ1 - γ2);

1 - δ = σ ^* +τ^* = α(1 - γ1) + (1 - α)(1 - γ2) = 1 - γ2 - α(γ1 - γ2).