Income taxation when markets are incomplete
115
Theorem 1 establishes a set of sufficient conditions for marginal, per-
sonal income tax reforms to provide social insurance. In particular, the
following requirements must be met: markets are sufficiently incomplete,
and consumers sufficiently heterogeneous, 2 ≤ H ≤ S - J; there are
enough policy instruments; the initial economy is chosen in a generic set in
which interiority conditions hold (see Theorem 2 in Section 4). The special
case of a representative consumer, who is also the single owner of the only
firm in the economy (H = J = 1), is analyzed in Diamond and Mirrlees
(1992). Diamond and Mirrlees show that there exist economies where ad
valorem contingent taxes on dividends can be used to implement a first best
equilibrium.13 Yet, even in this simple case the type of economy for which
optimality is achieved is that peculiar type in which consumers do not want
to redistribute income across states. In other words, economies in which the
initial endowment distribution, or the type of consumer preference, is such
that the restrictions arising from market incompleteness to consumers are
non-binding.14
For a better understanding of our results, we go back to our economy.
Local perturbations of the tax system act directly on the asset span, tilting
the subspace of feasible income transfers away from its original position. In
doing so, the consumer may transfer income in a direction that is orthogonal
to the initial asset span. Yet, unlike in the case of fixed payoff structures,15 or
in that of simple linear technologies (e.g., Example 2), the final asset span
crucially depends on firms’ behavior. It is far from obvious that production
decisions do not offset the effects of tax reform, thereby neutralizing the
planner’s policy.16 To complete our analysis of Theorem 1 we need to bring
into the picture production decisions and firms’ state prices, β. Our explana-
tion of Theorem 1 is based on the idea that firms, unlike the central planner,
take the asset span as given. First, observe that, at t1 = 0, firm j ’s pricing
criterion reduces to βj,0 = h θjhλh ; hence, the initial no-tax equilibrium
allocation is still optimal for firm j if the tax reform does not have a direct
13 I thank Herakles Polemarchakis for pointing out this reference to me.
14 A further peculiar situation is represented by the case of generically (or effectively)
complete markets. When the number of consumer types satisfies 1 ≤ H ≤ J , it is always
possible to find asset structures that would allow consumers to effectively achieve a full
efficient diversification of market risk.
15 See also the literature on real indeterminacy (e.g., Geanakoplos and Mas-Colell (1989),
Balasko and Cass (1989)).
16 Remember that after each policy reform re-allocations of commodity and assets are
completely decentralised (i.e., achieved competitively by private agents through trade on the
existent commodity and asset markets).