Income taxation when markets are incomplete
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interest of shareholders, none of its feasible choices will, typically, achieve
shareholders’ unanimous approval. 3
Let β j ∈ RN++ be the evaluation criterion (or state price vector) given to
firm j ∈ J2.An individual optimum of firm j is a vector yj that maximizes
βjyj on Yj. Next, assume that firm j acts in the best interest of shareholders,
and let βj be a (linear) function of its shareholders’ present value vectors,
λh . In principle, we would (at least) like to require that, at equilibrium, βj
is consistent with the market value of the firm, qj , where by consistent we
mean that qj = (1 /βj ) βjyj (see De Marzo (1988)). Consumers’first-order
conditions imply that the firm problem is well defined if ( 1 /β0' ) βj belongs to
( W)⊥ (i.e., to the same subspace to which consumers’ present value vectors
belong).4 Then, any consistent βj leads to a well defined firm’s problem.
Two consistent criteria, βj , have been extensively used in the literature.
One was proposed by Dreze, and the second by Grossman and Hart. In
Dreze (1987) each firm takes production decisions in the best interest of final
shareholders, by evaluating future profits with respect to βj = h∈H θjhλh .
In Grossman and Hart (1979), firms act in the best interest of initial (or
date 0) shareholders, and βj = Yh∈H θjλh. The two criteria mainly differ
in the “timing” at which the evaluation of production projects is made. In
Grossman and Hart projects are chosen when asset markets are still open.
Thus shareholders may always “vote” against a production choice by selling
their shares in the firm. At equilibrium, the no-arbitrage condition ensures
that the market price they are paid matches their private evaluations. On
the contrary, Drèze’s criterion is based on the idea that production plans are
decided after the security markets close. Thus, current (or final) shareholders
rely entirely on the fact that production plans are decided in their interest.5
3 If financial markets are (generically) complete (dim( W) = J = S), consumers agree
on project evaluations: λh = λ ∈ ( W)⊥ for all h, and dim( W)⊥ = 1. Thus, if we assume
that firms act in the best interest of shareholders, it must be that, for given λ, a production
plan yj is chosen such that yj ∈ arg max λyj : yj ∈ Yj for all j. By introducing this
objective function, it is straightforward to show that the corresponding allocation is Pareto
optimal. This objective function is also appropriate in the case of partial spanning. Partial
spanning occurs when financial markets are incomplete, but firms are constrained to propose
projects, whose returns exclusively lie in the span of the marketed securities.
4 A further necessary condition, that is more related to the fact that individual demands
are well behaved, is that, in every state s, (at least) one consumer, h(s), has non-satiated
preferences. This ensures that, in equilibrium, λh(s)(x) > 0, and thus that all firms are
valued, preventing the security matrix from dropping rank.
5 Both criteria can be shown to be shareholder constrained efficient if lump-sum transfers
among shareholders are allowed. Grossman-Hart’s does also require that shareholders have
competitive price perceptions (i.e., shareholders of firm j correctly anticipate the effect of a
change of the production plan of j, on the market value of the firm). Magill and Shafer (1991),
p. 1586, discuss firms’ constrained efficient pricing criteria. For completeness, we say that a
decision yj ofj is shareholder constrained efficient if there does not exist a marginal change