Bargaining Power and Equilibrium Consumption

7 Appendix

Proof of Proposition 1

Suppose that the household’s budget constraint is never binding as hypothesized.
For every α ∈ (0, 1), we can choose an e(α) > 0 so that the local comparative statics
prevail in the open neighborhood N(α) ≡ (α — e(α),α + e(α)). Each set C(α) =
N(α) ∩ [α*, α*] is relatively open in the interval [α*, α*]. The family C(α), α ∈ [α*, α*],
is an open covering of the compact set [α_t,α*]. It has a finite subcovering. Let us fix
a minimal finite subcovering C(α_k), k = 1, . . . , K. Without loss of generality, assume
α₁ < α₂ < . . . < α_K . We claim that:

(A) If α_t < α ₁, then α_t ∈ C(α ₁).

(B) If α_κ < α*, then α* ∈ C(α_κ).

(C) For each k ≤ K — 1, there exists βk with αk < βk < αk+1 and βk ∈ C(αk) ∩
C (α_k+1).

To show (A), suppose it were false, i.e. α_t < α ₁ and α_t ∈ C(α ₁). Then there
exists k > 1 with α_t ∈ C(α_k) and, consequently, C(α ₁) ⊂ C(α_k), contradicting the
minimality of the covering. Claims (B) and (C) are shown by similar reasoning.

Now fix β ₁,..., β_κ-1 according to (C) and let us go from α_t to α* taking small

steps, namely

from α_t to α ₁, from α ₁ to β ₁,

from β₁ to α₂ , from α₂ to β₂ ,

... ...         ... ...

from βK_-1 to α_κ, and α_κ to α*.

During each step, either the utilities remain unchanged or consumer 1’s utility goes
up and consumer 2’s utility goes down. Hence the assertion.

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