Income Taxation when Markets are Incomplete



Income taxation when markets are incomplete

123


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 Vα denote a generic submatrix
of the Jacobian of
V with respect to ξ . Define

Kα = {u RH : det (DξVα) = 0} = det (DξVα)-1 (0).

This set is closed in RH.26 The set of critical points of the map V, N =
{u RH : det (DξVα) = 0 α} = ∩αKα, is also closed. Therefore its
complement
NR = RH/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(Bc) is closed in Ω*, and its
complement,
Ω** = Ω*\φ(Bc), 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:

(I)

h, rh1 D2uh +Ah - rh2 + rh8Duh = 0

HN

r1

(II)

h, rh2 W + r6 = 0

HJ

r3

(III)

h, -rh1 +rh3WT =0

HN

r2

(VI)

j, Thhrh0θh - Thhλhhrj = 0

J

r6       (7)

(VII)

sS, Thh,jjrh,s + rljλh)Rj = 0

S

( r 8)

(VIII)

sS, Thrh,s + Hrs = 0

S

r7

(IX)

Ilr Il - 1 = 0

1.

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
DzVα ' s.



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