Income taxation when markets are incomplete
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analogue of the equilibrium notion in Grossman-Hart (1979), when β is
the Grossman-Hart criterion; it coincides with a standard notion of pure
exchange GEI equilibrium when J2 = 0.
3. Welfare effects of tax reforms
3.1. Our main results
Definition 4 (Income Tax-Constrained Pareto Optimality, IT-CPO). In ev-
ery economy an E - t I equilibrium allocation of commodities fails to be
IT-CPO if there exists a t1 in TI such that the E — tI equilibrium allocation
is Pareto superior.
For (u, f, R) fixed, let Ψ denote the set of economies, whose typical
element Ψ is denoted (e, η, θ). The set Ω = Ψ × U × Y, with typical
element ω = (ψ,u,f), is the set of economies, parametrized by ψ together
with the preferences and technologies. We are now ready to state our first
result.
Theorem 1. Let Assumptions 1 and 2 hold, and 2 ≤ H ≤ S — J, J ≥ 1.
There exists a generic set of economies Ω ** in the parameter space Ω such
that, for every ω in Ω**, each no-tax, E — tI equilibrium is not an IT-CPO.
Observe that our theorem is stated and applies to both pure exchange
economies and production economies (with or without bonds). As we show
in Section 4 below, the requirement to prove Theorem 1 is that there is at
least one asset (J ≥ 1); this can be either a bond or a stock.
To establish our main result we assume that the initial state coincides with
a no-tax equilibrium (E — tI with tI = 0). Further, we restrict attention
to those economies, parameterized by initial endowments, for which no-
tax equilibria exist and are regular (see Theorem 2, and Corollary 2, in
Section 4). For fixed (u, f, R), the set of regular economies is denoted
by Ψ *. Regularity of equilibria is necessary to carry out local analysis (or
comparative statics), since it ensures (local) differentiability of consumer
demand for commodities and assets, and (local) differentiability of the net-
supply schedules. Moreover, we use the notation Ω * = Ψ *× U × Y the
set of regular economies parameterized by endowments, preferences and
technologies. For every ω ∈ Ω* a regular no-tax E — tI equilibrium exists;
that is, for tI in an open neighborhood of zero, an E — tI equilibrium with
tI = 0 exists (again, see Theorem 2 and Corollary 2).
Referring to Section 4 for a complete proof of Theorem 1, here, we
provide a sketch of its main arguments. Let ξ = (ξ', t1) ∈ Rn × TI be
a vector of endogenous variables and policy instruments appearing in the