Bargaining Power and Equilibrium Consumption

(¹ - ^αh) TT 2^ ʌ Uh2(xh2) - ^λh = ⁰’

Uh2 (x_h2 )

Using the budget constraint and the first-order conditions yields

1 ττ' TU ʌ (л      ∖2Jw2(whP1 ^{+ w}h ^- xh 1 P1)      n

^αh Uh ₍(x 1 ) U^{h 1(} x^{h 1})   (^{1   α}h ) U hD^lrD +w²   1¹ n^p p¹   0

Uh1(x_h1)                     U_h2w_hP₁ +w_h - x_h1 P₁

or

₁ ^αh F1 (xh 1) = F2 (w¹ P1 ^{+ w}h - xh 1P1) ∙ P1                 (24)

_{1 - αh} 1    1       2                     1

where F1 ≡ ln Uh1 and F2 ≡ ln Uh2 . F₁⁰ and F₂⁰ are strictly decreasing functions.
Hence, for a given P₁ , a higher (equal) α_h requires a higher (identical) consumption of
good 1 to preserve (24). This shows (i) and (ii).

By the same argument, an increase of α_h raises ceteris paribus the aggregate de-
mand for good 1. Further examination of (24) shows that for fixed bargaining power
parameters, aggregate demand for good 1 is a decreasing function of P1. Consequently,
if only αh is increased, then the equilibrium price Pb1 rises and the equilibrium con-
sumption of all first members except h1 is reduced. Finally, market clearing implies
that the equilibrium consumption of h1 goes up. This shows (iii) and, by symmetry,
(iv).

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