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 [αt,α*]. 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.
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