Income Taxation when Markets are Incomplete

Income taxation when markets are incomplete

(2) u^h is differentiably strictly quasi-concave (rD²u^h (x)r^T < 0 ∀r ∈ R^N,
r = 0, such that Du^h(x)r^T = 0 ∀x ∈ R^N₊₊);

(3) indifference surfaces are contained in the interior of the positive orthant
(the closure of {x ∈ R+++ : u^h(x) ≥ u^h(x')} is contained in R + ∣ ∀x' ∈
R^N++);

(4) endowments are strictly positive (e^h ^ 0).

We denote by U the set of utility functions satisfying Assumption 1(1)
through 1(4).

There are J ≥ 1 assets, J = {1,... ,j,... ,J}. Assets may be charac-
terized either by fixed return (e.g., bonds) or by returns that depend on en-
dogenous variables (e.g., stocks or equities). Precisely, we let J = J1 ∪J2,
and J = J1 + J2, where J1 is the set of J1 assets with fixed return ma-
trix R ∈ R⁺₊ ^J1 (say bonds), possibly empty, and J2 is the set of securities
whose return matrix is endogenous and depends on production activities
(say stocks). Pure exchange economies, with an empty set J2, will also be
considered in the paper.

Production is carried out by J2 competitive firms. The technology of a
typical firm j in J2 is represented by its production possibility set Y^j ⊂ R⁺,
with typical input-output vector y^j = (y^j, y 1). Further, firms are endowed
with a vector η^j in R⁺₊₊ of contingent goods. The properties of the production
technologies are summarized in the following.

Assumption 2. For every firm j in J₂ :

(1) Y^j ⊂ R⁺ is closed and convex, 0 ∈ Y^j, Y^j ∩ R⁺₊ = {0} and R⁺_- ⊂ Y^j;
(2) ( Σ_h e^h + Σj(Y^rj + η^j) ∩ R++ ) is compact, and ∀h, e^h ∈ R +∣ ;

(3) let M^j ⊂ R⁺ be a mj -dimensional subspace with 1 ≤ mj ≤ +; then

Y^j ⊂ M^j is an mj -dimensional manifold with boundary; its bound-
ary, ∂Y^j , is twice continuously differentiable, and differentiably strictly
quasi-convex at each point;

(4) firms’ endowments are strictly positive (η^j ^ 0).

Further, we assume that the technology set Y ^j can be represented by
Y^j ={yj ∈ R⁺ : f^j(y^j + η^j) ≤ 0,η^j ∈ R₊⁺₊}, where f^j : R⁺ → Ris a
differentiable transformation function. We also assume that (-y₀^j ,y₁^j)>0,
and denote the typical element of Y^j by y = (y₀^j ,y₁^j).

Assumption 2 implies that f^j(∙) is twice continuously differentiable,
non-decreasing and strictly quasi-convex, and that it satisfies f ^j (η^j) = 0.
We denote by Y the set of transformation functions satisfying Assump-
tion 2(1) through 2(3). Finally, for notational simplicity, in the rest of the
paper we simply write y^j fory^j +η^j. Firms’ endowments η^j are only needed

¹ Hereafter, we use the convention that >, when applied to vectors, denotes a weak in-
equality (» a strong inequality).

<<   <   3   4   5   6   7   8   9   >   >>