Income taxation when markets are incomplete
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The original aspect of our result is the effectiveness of contingent tax re-
forms in modifying consumers’ risk-sharing opportunities, through a change
in the asset span position. The highly simplified representation of our econ-
omy and of the central government, aims precisely to isolate this effect, and,
ultimately, to derive a minimal set of conditions under which this type of tax-
ation leads to well identified welfare effects. Thus, for example, we assume
that a single perishable commodity is traded in the economy. In contrast
with most of the GEI literature, our results are not driven by changes in rela-
tive spot prices (see, for example, Geanakoplos and Polemarchakis (1986),
Geanakoplos et al. (1990), Citanna et al. (1998, 2001)). Asset span varia-
tions are powerful enough to generate changes in individual utilities that
are otherwise typical of spot price, second-order effects (see Section 3.2,
Remark 2).
The transmission mechanism of tax reform is also different from the one
characterizing financial innovation in the sense of Cass and Citanna (1998)
and of Elul (1995, 1999). Precisely, in contrast with Elul (1999), we assume
that the number of assets (or asset markets) is given, and cannot be modified
by the central government. Thus, although tax reforms may tilt the asset
span, they do not change its dimension.
Our representation of the central government is also highly simplified.
A fictitious central planner set ad valorem, anonymous, state-contingent
taxes (and/or subsidies) on individuals’ incomes, subject to a state-by-state
fiscal budget constraint. Less restrictive budgetary rules and/or other typical
motives for public intervention, if introduced, would only strengthen our
results. Similarly, we could, and indeed we do, discuss the case in which
income may be taxed at different rates, depending on its source (endowment
income, returns from different assets, etc.). A necessary condition for our
results to hold is that the number of policy objectives does not exceed the
number of tax instruments: the changes of the utility levels corresponding to
the H consumer types must be less than the tax instruments. This condition
is an incomplete market analogue of that due to Tinbergen (1952).
The approach used to prove our results is comparative statics, and relies
on well-known techniques of differential topology. In particular, the welfare
analysis follows the guidelines traced in Smale (1974), later applied and
extended by Geanakoplos and Polemarchakis (1986) and, more recently, by
Citanna et al. (1998), whose underlying approach is used in this paper.
Some concluding observations may help the reader. First, our contin-
gent taxes are not progressive in income. They are linear income taxes, such
as those typically used to tax individual income from bond holdings. The
assumption that tax reforms may be state-contingent is realistic inasmuch
as it is the decision of implementing a tax scheme contingently on factors
exogenous to the underlined model of the economy, factors which may be