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 z1σ z2τ 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+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 u1αu1-α requires

x1 =

--------∙ x,
σ
+ τ

τ
_____________ , ιγ

x2=

^x,
σ
+ τ

σ*

y1 =

σ* + τ* ^ y,
τ
*

y2=

σ* + τ* ^ y

where

= αγ1,
τ
= (1 - α)γ2,

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

Moreover, at the maximum,

_                     _ _*                         _*

uαu1 -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).

28



More intriguing information

1. The name is absent
2. Neighborhood Effects, Public Housing and Unemployment in France
3. The name is absent
4. The name is absent
5. The name is absent
6. Database Search Strategies for Proteomic Data Sets Generated by Electron Capture Dissociation Mass Spectrometry
7. AGRICULTURAL TRADE LIBERALIZATION UNDER NAFTA: REPORTING ON THE REPORT CARD
8. The name is absent
9. Input-Output Analysis, Linear Programming and Modified Multipliers
10. AN ANALYTICAL METHOD TO CALCULATE THE ERGODIC AND DIFFERENCE MATRICES OF THE DISCOUNTED MARKOV DECISION PROCESSES