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M. Tirelli
asset span effect: the matrix
dt11
M(y1, 0).
dtS1
induces the same column span of M(y1, 0).17 Theorem 1 shows that, if
this is not the case, (M(y 1, 0)) = (My 1, dt1 ),, then the initial production
decisions are no longer individually optimal at βj,0 (βj,0 ^tt1 □yj) = 0) for
some dt1 and some j .18 Moreover, there is room for a change of the (after
tax) payoff vectors in a direction that was otherwise unfeasible for the private
sector at the initial, no-tax, equilibrium (dt1 □yj ∈ (M(y 1, 0) )⊥).19 Yet, for
production changes not to kill the direct span effect of a tax reform, we need
to account for indirect (or equilibrium) state-price effects (i.e., pecuniary
externalities), that is, changes in β j which may support the new allocation
as an individual optimum for j .20 Our proof provides direct calculations
confirming this intuition. It also shows that the assumption that the planner
computes the equilibrium, accounting for the state-price externalities, is not
essential: the argument in the proof also goes through if the central planner
does not anticipate the effects of a change in individual state prices (λ)
on firms’ state prices (β). Thus, no particular informational advantage is
required for the planner to implement a tax reform. For our theorem to
hold, the planner is “only” required to have statistical information on the
consumer types, firm technologies, state prices, and market structure. That
is, based on a complete, but abstract, representation of the economy, a Pareto
improving tax reform can be designed by computing and comparing different
equilibria. Once a tax reform has been chosen, and the corresponding tax
vector publicly announced, the final allocation is determined competitively,
with each agent truthfully revealing her/his type.
17 In fact, a tax reform characterized by uniform tax changes across states satisfies
βj,0(dt1 □ yj + dyj) = 0, and thus βj,0(dt1 □ yj) = 0, for every individually optimal
initial production allocation yj (i.e., dyj such that V fj(yj') ∙ dyj ≤ 0), that is, the ini-
tial no-tax production plans are individually optimal. Here, and elsewhere, □ denotes the
elementwise vector product: for x,y ∈ RC, x□y = (x 1 y 1, ... , xCyC)'.
18 Ataxreformthatproducesadirectspaneffectimpliesthatdt1 □ yj ∈ (M(0, y 1 )).Then,
βj,0(dt1 □yj + dyj) = 0 for every individually optimal initial production allocation yj
(i.e., dyj such that Dfj(yj') ∙ dyj = 0). This implies that yj need not be ex-post optimal
for j at βj,0.
19 Here, by “feasibility” we are directly referring to financial feasibility. On technological
feasibility, observe that after tax-payoffs of a firm j, (1 + dt1)□yj, may not be in Yj .
20 At the tax equilibrium, arising after a tax reform, the production plan yj' = yj + dyj is
optimal for j at βj = βj, 0 + dβj, 0 if Dfj (yj ' ) is collinear to βj, that is, if the production
choice tilts the gradient of the transformation function away from the original position.