Income Taxation when Markets are Incomplete

M. Tirelli

It will be clearer as we go along in the paper that our main results hold for
any criterion, β , which is expressed as a functional of consumers’ present
value vectors, λ. For reasons of analytic tractability, we stick with the adop-
tion of a specific criterion, and choose the following modified Grossman-
Hart criterion.⁶

Definition 2 (ModifiedGrossman-Hartcriterion). Forevery (θj, λ,tj) and
all s ∈ S ∪ {0}, β^j = ∑_hj (1 + ts¹) is the pricing criterion of firm
j ∈ J₂.

2.2. The equilibrium with taxes

Gathering the definitions of agents’ actions and markets we obtain a full
description of our economy, E(u, f, η, e, θ, R). We provide the following
definition of equilibrium at the policy instrument vector t^I in T ^I .

Let ω = (η, e, θ, u, f), and let E be the set of economies with typical
element (ω, R).

Definition 3 (E - t^I equilibrium). In an economy (ω, R) in E, an equilib-
rium with tax instruments t^I ∈ T ^I, and firm pricing criteria β, is a pair
((x, θ, y), (q, t)) such that:

(i) ∀h, x^h,θ^h isan individual optimum for consumer h, at (q, y, ω, t, R);
(ii) ∀j, y^j is an individual optimum for firm j, at β^j;

(iii) ∀ j ∈ J_l, Eh θj^h = 0; ∀j ∈ J2, Σh θj^h = 1 ;

(iv) t satisfies fiscal budget balance at t^I.

Observe that, if (i)-(iv) are satisfied, spot markets clear. Moreover, (iv)
can be written differently depending on the type of fiscal policy assumed.
For the equilibrium to be well defined every β ^j must be chosen in the
set of consistent criteria (see Section 2.1.5). Finally, we define a no-tax
equilibrium, an E - t^I equilibrium setting all tax instruments t^I equal to
zero. Observe that our notion of no-tax equilibrium is the one-commodity
analogue of the equilibrium notion in Geanakoplos et al. (1990), when firms’
pricing criteria, β, are defined as in Dreze (1987); it is the one-commodity
dy^j such that y^j + dy^j is optimal for j and (1, Vu^h ) ∙ ∂^χj-dy^j ≥ 0 for all shareholders h,
with strict inequality for some h. Further, we say that β^j is shareholder constrained efficient
for j if, at equilibrium, it supports a decision profile y^j that is shareholder constrained
efficient for j. Assume that lump-sum redistribution is possible at date zero. Then, β^j is
shareholder constrained efficient if and only if ∑ h (1, V u^h ) ∙ ( ∂x^h∕∂y^j ) dy^j ≤ Oforevery
dy^j such that V f^j ∙ dy^j = O.

⁶ This criterion can be shown to be shareholder constrained efficient if: (1) lump-sum
transfers among shareholders are allowed, and (2) shareholders have competitive price per-
ceptions (see the previous footnote).

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