Income Taxation when Markets are Incomplete

Income taxation when markets are incomplete

Proof of Theorem 1. Let Ω * be a subset of Ω restricted to Ψ *, and Ω ** be a
subset of Ω* such that, for every ω ∈ Ω**, G(ω) is a submersion. Applying
Lemma 1 and 2 below we conclude that Ω ** is open and dense in Ω. □

Lemma 1. Ω** ⊂ Ω* is open.

Proof. Recall that ξ = (ξ ', t¹), and let Dξ V^α denote a generic submatrix
of the Jacobian of V with respect to ξ . Define

Kα = {u ∈ RH : det (DξV^α) = 0} = det (DξV^α)^-1 (0).

This set is closed in R^H.²⁶ The set of critical points of the map V, N =
{u ∈ RH : det (DξV^α) = 0 ∀α} = ∩_αK_α, is also closed. Therefore its
complement N_R = R^H/N is open.

Let the function φ : E → Ω be the natural projection of the equilibrium
manifold onto the parameter space. Define

Bc = {(ξ, u,f,ψ) ∈ E : det (D^⁾ = 0 ∀α} ,

the set of critical values of φ . Bc is relatively closed, being the preimage of
{0} under a continuous function. Then, to ensure that the image of Bc under
the mapping φ is closed, it suffices to establish that φ is proper, which is
proved as Lemma 3 in the appendix. Thus, φ(B_c) is closed in Ω*, and its
complement, Ω** = Ω*\φ(B_c), is relatively open. □

Lemma 2. Ω** is dense.

Proof. We divide the proof into two parts: the first considers a pure exchange
economy, Jι = 0, J2 = 0, while the second deals with a production econ-
omy, J2 = 0.

Part 1 (pure exchange GEI) Jι = 0, J2 = 0.

In the case of a pure exchange, J2 = 0, J = J₁, (6) reduces to:

²⁶ This is because it is the preimage of a singleton by a continuous function, in a complete
metric space. Here, the continuous function is the polynomial associated to the determinant
of the continuous linear mappings DzV^α ' s.

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