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) + 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 [
αt*]. It has a finite subcovering. Let us fix
a minimal finite subcovering
C(αk), k = 1, . . . , K. Without loss of generality, assume
α1 < α2 < . . . < αK . We claim that:

(A) If αt < α 1, then αt C(α 1).

(B) If ακ < α*, then α* C(ακ).

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

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

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

steps, namely

from αt to α 1, from α 1 to β 1,

from β1 to α2 , from α2 to β2 ,

... ...         ... ...

from βK-1 to ακ, and ακ 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.

27



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