Income taxation when markets are incomplete
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multipliers (a, μ). Then, the above technique is used to show that tI = 0
would typically not satisfy the first-order (necessary) conditions for a local
maximum of W with Lagrange multipliers r = (μ, a), except for the trivial
case r = 0.8
In showing that (1) has no solution, we have to check that the number of
policy objectives (the utility levels) do not exceed the number of instruments
(tax/transfers).9 Then we prove that the (local) property of G is generic: we
show that our result holds for a set of economies, Ω**, that is (nonempty)
open and dense in Ω*.10
Observe that the submersion property of G is a sufficient condition for
proving the statement of Theorem 1. If DG is locally onto, then the subspace
spanned by the rows of DG is equivalent to Rn+H , therefore including
vectors whose first n entries are zero with the remaining H either positive or
negative. This implies that there also exist directions dtI' which are socially
undesirable,
Dξ ' Fdξ ' + Dtι ' FdtI ' = 0,
Dξ' Vdξ' + Dtι' VdtI' ≤ 0.
This directly implies the following.
Corollary 1. In the context of Theorem 1, for almost every regular economy
ω in Ω **, there exist tax reforms which can achieve any direction of utility
changes in an open neighborhood of the no-tax equilibrium allocations.
This generalization of Theorem 1 should be read as a warning to policy
makers. The effects of our income tax reforms on private risk-sharing op-
portunities may be socially undesirable. This should even increase the level
of concern if interpreted in the context of our simple economy. In contrast
8 Here, one has to be a little careful. The latter planner’s problem is compact but fails to
be convex. The non-convexity of the set of centralized attainable allocations implies that
the constrained Pareto frontier need not be convex either. Then, the set of maxima obtained
from the Lagrangian fails to provide a complete representation of the constrained Pareto
frontier (e.g., any concave segment of the frontier is not attainable from the Lagrangian
maximization). Yet, the proof of our main result shows that the first-order conditions of the
above problem, evaluated at the no-tax equilibrium, do typically fail to hold. Therefore,
competitive equilibria are, typically, not extrema (neither maxima nor minima nor saddle
points) of that problem, and r = 0.
9 Observe that the number of equations in the new system rDG = 0 is equal to the number
of columns of DG, and thus to the number of original unknowns (z', t). The unknowns of
rDG = 0 are the components of r whose number is equal to the number of equations in
G = 0.
10 These steps follow the submersion approach, originally due to Smale (1974). Geanako-
plos and Polemarchakis (1986) first suggested its application to prove constrained subopti-
mality of equilibria when asset markets are incomplete.