Income Taxation when Markets are Incomplete

M. Tirelli

Matching the equations in system (7) with the unknowns it is immediate
that if H ≤ S the system has equations outnumbering the unknowns (r).²⁷
Therefore, we are left to show that system (7) has independent equations.
The proof can be divided into the following two cases (a) and (b).

Case (a): r_h¹ = 0 for all h∖

The following perturbations can be used: ((dr_.²_,s)_s^J₌₁,dr¹), respectively,
for Equations (II), (III); dr_h³ for h = H, for (VI); dr⁷ for (VIII); dA^h for
(I)h (given r_h¹ = 0) for all h. Finally, use ((dr_H²_,s)s>J,dr_H³ ) for (VII).

Case (b): r_h¹ = 0 for some~h∣

Since, as we are going to show, r_h¹ = 0, for some h, implies that r_h¹ = 0
for all h, we cannot use quadratic perturbations of utilities in the present
case. Yet, our result continues to hold when r¹ = 0, and the u^h are fixed in
U.

Consider system (7). Now r_h¹ = 0, say for consumer h = H, implies that
r_H² - r_H⁸ Du^H = 0. Then, consumer H’s foc’s and (II) imply r⁶ = 0. Since
r⁶ is independent of h, r_h² ∈ {W)⊥ holds for all h such that rh = 0. By (III),
r1 = WrhT, i.e., either r1 ∈ {W) or rh = 0 (and r1 = 0) for all h = H.
Then, post-multiplying (I)_h by r_h¹, and using the latter result, together with
strict quasi-concavity of u^h, we have r_h¹ = 0. This holds for all h.

Also r¹ = 0 implies r³ = 0 by (III). Then, consumers’foc’s ( λ^h ∙ Rj = qj )
imply that we can rewrite (VII) as Y_h rhλ^h_s Y₁j Rs ij = Y_h rhλ^h_sz^h_s for all
s ∈ S, or, using the notation introduced in Theorem 2, as

( r 1⁸, ...,rH ) AZ = 0,

where AZ in (3) reduces to

^{λ λ}1 z1 ∙∙∙ ^λk!

..
..

..

λH H   λH H

∖^λ 1 z 1 ∙ ∙ ∙ ^λSzS

Since we are considering economies in Ψ *, under the assumption that
H ≤ S — J, Theorem 2 implies that r⁸ = (rf, ... , rH ) = 0. This im-
mediately implies r = 0. System (7) has no solution.

Part 2 (production GEI) J₂ = 0.

We prove the result for the case of a pure production economy (with
J1 =0, J2 =0), since then extending the argument to mixed cases is

²⁷ This has a natural interpretation: the number of policy objectives (the utility levels)
should not exceed the number of tax instruments (contingent taxes/transfers). The latter
condition is an incomplete market analogue of the one due to Tinbergen (1952). Thus, for
example, this bound could be dropped by introducing date 0, lump-sum transfers, τ ∈ R^H .

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