Income Taxation when Markets are Incomplete

Income taxation when markets are incomplete

instrument vector be t^I = 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 - t^I .

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 - t^I, t^I = 0). Consider
an economy E (ω) with fixed (u^h, f^j)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 — t^I equilibrium with the following properties:

(i) the equilibrium is regular;

(ii) W has full rank;

(iii) θ_j^h = 0 for all h, j;

(iv) the following matrix has full rank H:

^ ^λ1 (^z1 ^{- y} 1 /H) ∙∙∙ ^λS (^zS ^- ys/H)

NZ =          :                  :

..

v^λH (zH ^{- y} 1 /H) ∙∙∙ ^λs (Zs ^{- y}s/H)

where z = x — e, ys = ∑_j∈ j2 yS.

Proof. Generic existence and regularity of E — t^I at t ^I = 0 substantially
follow from Theorem 2 in Geanakoplos et al. (1990). Generic properties (i)
through (iii) are standard^23,24 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 (θ_j^h ) which follow from the no-short sales
assumption. The first occurs whenever θ_j^h = 0, ρ_j^h = 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 θ_j^h = 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 θ_j^h ≥ 0 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 ⊂ R^N(H+J) × R^H₊ ^J represent the class of (non-empty and) open subsets of R^N(H+J) ×
RHJ ; a subset Ψ ⊂ Ψ is open if and only if Φ = O ∩ Ψ for some O ∈ O.

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