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accounted for in the definition of the state space (e.g., aggregate supply
shocks, financial crises, etc.). Secondly, our model does not endogenously
provide an explanation of why markets are incomplete. Extensions in this di-
rection, although potentially interesting, are beyond the scope of this paper.
Moreover, at this stage, our positive analysis of tax reform does not provide
any specific recipe for policy intervention. Yet, our results do indicate and
analyze interesting effects that income taxes may exhibit in incomplete mar-
ket economies. Finally, our analysis is local. We do not derive an optimal
fiscal policy rule: a mapping from the equilibrium set to the set of feasible
tax rates which a central government should optimally enforce. We instead
trace the direction of (robust) welfare-improving tax reforms.
Our work is organized as follows. In Section 2 we present the benchmark
economy, and give a notion of competitive equilibrium. In Section 3, we state
and discuss our main results (Theorem 1 and Corollary 1). There, we also
discuss and compare our notion of constrained optimality with the standard
ones used in the GEI literature; this is done by mean of a few examples, both
for a pure exchange and for a production economy. The proofs of our results
are in Section 4, which also includes other technical results such as the
(local) existence and regularity of equilibria (Theorem 2 and Corollary 2).
2. The model
2.1. The general framework
2.1.1. Private agents and commodities There are two dates indexed by 0
and 1. At date 1 there is a finite number S of possible states of the world,
s = 1,... ,S, and S = {1,... ,S}. We also use the convention of labeling
date 0 as s = 0. There is a single perishable good that can be used both for
consumption and investment at date 0, and for consumption at date 1. We
denote by N = S + 1 the number of contingent goods in the economy.
There are H ≥ 2 consumers, H = {1,... ,h,... ,H}. Every consumer
h is endowed with a vector (e0h,e1h) ∈ RN++ of contingent goods, where
e1h = e1h, ... ,eSh denotes date 1 endowments. For simplicity, we assume
that each individual consumption set coincides with the nonnegative orthant
of the commodity space, RN+ , and we denote a typical element by (x0,x1).
The utility function uh : RN+ → R represents the preference ordering of con-
sumer h over his consumption set. Lastly, some rather strong, but standard,
assumptions on preferences and endowments are introduced.
Assumption 1. For every h in H:
(1) uh is twice continuously differentiable, and differentiably strictly in-
creasing (Duh(x) ^ 0 ∀x ∈ R+++)1 ;