Income Taxation when Markets are Incomplete

M. Tirelli

extended system of equilibrium first-order conditions, and let F(ξ; ω) = 0
denote such a system (see Section 4 for the precise definitions). The num-
ber of equations in the system is n, equal to the sum of the number n ^, of
endogenous variables and the number S of policy instruments. The no-tax
equilibrium set is

E = {(ξ,ω) : F(ξ; ω) = 0, t^, = 0},

and, when restricted to Ω*, it is a manifold. Next, define the function V :
E → RH such that V (ξ; ω) = (u¹ (x¹ ), ... , uH(xH)) is the vector of
utility functions. Further, let

To prove Theorem 1 it suffices to show that there exists a marginal tax change
dt^I , in an open neighborhood of t^I = 0, such that the new equilibrium
(ξ + dξ) is Pareto superior, that is, Dξ V = {du¹, ... , duH) ^ 0. This can
be done by proving that G behaves locally as a submersion, i.e.,

_gg ⁽ _ξ ' _t'∙,Λ-( DF ( ξ ' ∙t' ; ω )ʌ

DG(.^ξ,t ∙ ^ω = (dv ( ξ.^,t ; ω ) )^_j,ω) ∈ E

has full row rank. In fact, observe that, when the subspace spanned by the
rows of DG is equivalent to R^n+H, it includes vectors whose first n entries
are zero and the remaining H positive:

D_ξ , Fdξ ^, + D_tI, Fdt^{1 ,} = 0,

D_ξ, Vdξ^, + D_tI, VdtI^, > 0

for some tax reform, dt^', and corresponding changes in the endogenous
equilibrium variables dξ^, .⁷

An equivalent, but instructive, way to describe our proof is as follows.
Assume that G is a submersion at a no-tax equilibrium, for an economy ω.
Then, rDG (ξ^,, t', ω) = 0 if and only if r = 0, i.e., the following system
has no solution,

This has a natural interpretation in terms of standard welfare analysis. As-
sume that a planner chooses (ξ^,, t') such that first-order (necessary) con-
ditions for the optimality of the equilibrium allocation hold: the welfare
function we have in mind is W = ɪ^h ^ah^^h {^χh} + μF (ξ^,, t' ; ω), with

⁷ Constrained suboptimality is equivalent to the existence of a feasible direction of tax
reforms in the sense of Guesnerie (1977).

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