Income Taxation when Markets are Incomplete

M. Tirelli

where y = ∑_j y^j. Since we are considering economies in Ψ *, under the
assumption that H ≤ S - J , Theorem 2 implies that r ⁸ = 0, and thus r = 0.
System (6) has no solution.

(2.1) r1 = 0, for some h, implies that either r_j⁴ ∈ (W)⊥ or r_j⁴ = 0 for all j :
Suppose not, so that r_j⁴ ∈ ( W) / {0} for some j, and rh_l = 0 for some
h; the latter implies r_h² = r_h⁸ Du^h by (I), and r⁶ = 0 by (II) and consumers’
foc’s; next, postmultiplying (V)j by r_j⁴, and using strict quasi-convexity of
f ^j , yields a contradiction, and so r_j⁴ = 0.

(2.2) r¹ = 0 implies r = 0:

Now r1 = 0 implies that r_j⁴ ∈ (W)⊥ or r_j⁴ = 0 for all j, by (2.1).
Also r_h¹ = 0 for all h, and the latter, implies r_h³ = 0 for all h, by (III)
(and the full rank property of W). Then, (III) and (I), respectively, imply
ɪj∈ Ji θj ^rjs = 0, r2 s = rhD_su^h for all h, s. Using these results, and con-
sumers’ foc’s (λ^h ∙ R^j = qj), in (VII) yields (9) in (2.0) above. Therefore,
r = 0.

(2.3) r_h¹ = 0 for some h and r⁴ = 0 implies r = 0:

Firstly, r_h¹ = 0, for some h, implies r_h² = r_h⁸Du^h by (I), and r⁶ = 0,
r_h² W = 0 for all h, by (II). Using r_j⁴ = 0, for all j, in (III) implies that
either rh_l = rh WT or (rh, r^) = 0 for all h. Postmultiplying (I) by r_h¹ T,
and using the latter result, yields r_h³W^TD²u^hWr_h^3T = 0 for all h. By strict
quasi-concavity of u^h , this holds iff r_h³ = 0 for all h, and hence r ¹ = 0.
Finally, we can apply (2.0).

We can finally gather the results in Cases (a) and (b), respectively in
Parts 1, 2 above, and conclude the proof by applying Sard’s theorem: there
exists a dense set of economies, Ω** in Ω*, such that, for every ω in Ω**,
G (ω) is a submersion (see, e.g., Guillemin and Pollack (1974), p. 62). □

Appendix

Lemma 3. The natural projection φ : E → Ω is proper

Proof. Let C be a compact and nonempty subset of Ω. We have to show
that, for each converging sequence (ω^r)_r=₁,₂,..., (ω^r) → ω *, ω^r,ω * ∈ C ,the
sequence φ^-1 (ω^r') = (ξ^r, ω^r) converges in E (i.e., (ξ^r, ω^r) → (ξ*, ω*) ∈
E). First, note that every parameter vector (ω^r) ∈ C has boundaries. Second,
(by Assumption 2(2)) there are subsequences such that y^r is defined on a
compact set when ω^r ∈ C; also, by continuity of f, f (y^r) → |f (y^*)| < ∞,
i.e., (5) of the equilibrium first-order conditions (2) is satisfied. Further,
since f is twice continuously differentiable, Df(y^r) → |Df(y^*)| < ∞

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