Income Taxation when Markets are Incomplete



Income taxation when markets are incomplete

119


instrument vector be tI = t1. Adding the government budget constraints to
F, we obtain a new system of equations which we denote by F : Ξ × Ψ
Rn, with n = n' + 5 and Ξ = Ξ' × TI Rn'+s. Thus F = 0 represents the
complete system of first-order conditions of the
E - tI .

We endow the space of functions Y × U with the C2 compact-open
(weak) topology. For simplicity, let Y
× U denote the topological space
Y
× U, C2 (Y) × C2 (U). We also assume that the space of endowments has
the usual (Euclidean) topology.

We are now ready to establish local existence ofan interior equilibrium.

Theorem 2 (Local existence and regularity ofan E - tI, tI = 0). Consider
an economy E (ω) with fixed (uh, fj)
h,j. Further, let Assumptions 1, 2 hold
and H
S J .If t1 = 0, there exists a set of economies Ψ * Ψ, open
and of full measure, such that, for every ψ
Ψ *, there exists a (locally
isolated) E
tI equilibrium with the following properties:

(i) the equilibrium is regular;

(ii) W has full rank;

(iii) θjh = 0 for all h, j;

(iv) the following matrix has full rank H:

^ λ1 (z1 - y 1 /H) ∙∙∙ λS (zS - ys/H)

NZ =          :                  :

(3)


..

vλH (zH - y 1 /H) ∙∙∙ λs (Zs - ys/H)

where z = x e, ys = ∑j j2 yS.

Proof. Generic existence and regularity of EtI at t I = 0 substantially
follow from Theorem 2 in Geanakoplos et al. (1990). Generic properties (i)
through (iii) are standard23
,24 while (iv) coincides with the property in item
3 of Lemma 1 of Citanna et al. (2001) in a pure exchange GEI (
J1 = 0). To
23
Geanakoplos et al. (1986, 1989, 1990) point out that there are two sources of non-
differentiability of security demand functions (
θjh ) which follow from the no-short sales
assumption. The first occurs whenever
θjh = 0, ρjh = 0 (i.e., the constraint is binding, but
this has “no cost” for the consumer). Yet, one can show that such a case in not generic in the
endowments. The most problematic source of non-differentiability occurs when
θjh = 0 for
some
(h, j) and rank(W)<J. However, assuming that (H + J)N, they show that: (i)
there exists a generic set of economies for which equilibria are of full rank; (ii) in restricting
attention to such economies, typically equilibria are such that the no-short sale constraints
do not bind. The latter implies that
θjh0 for all h and all j J2 .

24

24 To show regularity we can endow the parameter space, Ψ, with the usual topology: let
O RN(H+J) × RH+ J represent the class of (non-empty and) open subsets of RN(H+J) ×
RHJ ; a subset Ψ Ψ is open if and only if Φ = OΨ for some O O.



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