Income Taxation when Markets are Incomplete

Income taxation when markets are incomplete

straightforward. As for Part 1, the proof reduces to showing that system (6)
has independent equations.

Case (a): rh = 0 for all h∖

This is analogous to Case (a) in Part 1, except for Equations (IV), (V),
which can, respectively, be perturbed by dr₀⁴, ((dr_.⁴_,s)s>0 , dr⁵).

Case (b): r_h¹ = 0 for some h

If we allow r_h¹ = 0 (r_j⁴ = 0) for some h (respectively, j), we cannot use
perturbations of utility (resp., transformation) functions for those h (resp.,
j ). We proceed, first, by showing that r_h¹ = 0, r_j⁴ = 0 for (at least) one pair
(h, j ), implies that system (6) has no solution (Fact 1). Then, in Fact 2, we
prove that all other cases have the same implication.

Fact 1.Ifr_h¹ = 0, r_j⁴ = 0 for, say, (h, j) = (H, J), then system (6) has no
solution.

Now r_H¹ = 0, r_J⁴ = 0 imply that we can certainly use perturbations of
utility and transformation functions for H and J , respectively. Consider the
“worst” possible case, in which r_h¹ = 0 for all h = H. Since r_h¹ = 0 implies
r_h² = r_h⁸ Du^h , by (I)h, Equation (II)h becomes redundant when the system
is evaluated at equilibrium. Moreover, we can substitute for r_h² using (I)h to
drop this equation from (6). Then, the following perturbations can be used in
the new system: ((dr_H² _,s)_s^J₌₁ ,dr¹ ,dr_.⁴_,0), respectively, for Equations (II)H,
(III), (IV); drh for some h = H, for (VI); dr⁷ for (VIII); ((dr_h⁸)h<H , dr_H,₀)
for (IX); dA^H for (I)H (given r_H¹ = 0). Lastly, use ((dr_j⁴_,s)s>0, dr_j⁵) for (V)j
if j = J , and dB^j otherwise; the latter implies that we can use (d r_J⁴_,s)s>0
to perturb (VII).

Fact 2. Let r_h¹ = 0 for some h. Then, for r ∈{r : r¹ = 0}∪{r : r⁴ = 0},

system (6) has no solution.

We divide our argument into four steps, (2.0)-(2.3).

(2.0) (r¹, r4) = 0 implies r = 0:

Firstly, r_h¹ = 0 implies r_h² = r_h⁸Du^h by (I), and r⁶ = 0 by (II) and
consumers’ foc’s; also r³ = 0 by (III). Then, consumers’ foc’s (λ^h ∙ Rj = qj)
imply that we can rewrite (VII) as

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