Income Taxation when Markets are Incomplete

Income taxation when markets are incomplete

(for any finite B). The asset market clearing condition, together with the
no-short sell conditions, imply that θ^r → |θ*| < ∞ is a subsequence
of (θ^r). Then, Assumption 2(2), and the spot market clearing conditions,
imply that individual demands have subsequences (x^r) → |x*| < ∞ such
that (2) holds. Since the utility function is twice continuously differentiable,
Du (x^r) → |Du (x*) | (for any finite A). This also implies that (λ^r) →
|λ*| < ∞, and thus (1) is satisfied. The last condition - together with the
fact that θ^r is bounded - implies that β^r also converges to a finite bounded
entity. Then, note that, by definition of T', (t^{1 r}) is always bounded. The
above facts imply that for every firm the Lagrange multiplier υ^r is such that
υ^r → |υ*| < ∞ and (4) holds. Moreover, the above facts, and (3) and (7)
respectively, also imply that there exist subsequences q^r → |q^*| < ∞, and
(t0^r) → 110| < ∞. This completes the proof. □

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