Income taxation when markets are incomplete
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reforms; instead, discrete changes in portfolios and/or production alloca-
tions are required to show constrained inefficiency. Both examples do, in
fact, exploit the nonconvexity of the set of centralized attainable allocations
and show that equilibrium allocations may fail to be globally efficient. For
instance, in Dierker et al. (1999) the only interior11, Dreze, equilibrium is at
the minimum aggregate welfare point; yet this is a local CPO equilibrium:
a marginal production change, and a portfolio readjustment, may locally
Pareto improve the allocation, but cannot be supported as a competitive
equilibrium. To put it differently, although the tax reform may be Pareto
improving, it may not survive a successive round of competitive trade. The
only two other equilibria are corners, in which the only firm in the economy
is, respectively, owned by the first and by the second consumer. This exam-
ple is a robust case in which an interior Dreze equilibrium is a local (but
not global) CPO equilibrium. In Example 2 below, first, we show that the
interior equilibrium, despite being a local CPO, is not an IT-CPO; second,
we show that even the two corner equilibria, which are global CPO, fail to be
IT-CPO. To put it differently, tax reforms (as in Definition 1) may have first-
order effects that are strong enough to achieve a local Pareto improvement,
while the same cannot be said of policy reforms in the sense of Diamond.
Example 2. Consider an economy with utilities as in Example 1 above, in
which consumers are partners of the only firm,
Y = {y ∈ R3 : y о = -1, y 1 + y 2 ≤ - y 0}.
This economy has three (no-tax) Dreze equilibria: a symmetric equilibrium
with each consumer holding 1/2 of the firm producing y = (-1, 0.5, 0.5),
and two corner equilibria in which consumer 1 (respectively, consumer 2)
is the single owner of the firm and y = (-1, 1, 0) (resp., y = (-1, 0, 1))
is produced. Figure 2(a) below represents equilibria in the space of second-
period consumption (where, for clarity, we omit present value budget con-
straints).12 Equilibria can be perfectly Pareto ranked: assuming identical
welfare weights, and allowing for date zero lump-sum transfers, the sym-
metric equilibrium is Pareto dominated by either of the two alternative corner
equilibria; the latter two, instead, have the same welfare properties. More-
over, since all equilibria are technologically efficient, this ranking can be
thought to depend only on their risk sharing properties. Corner equilibria
are global CPO (the symmetric equilibrium is a local, but not a global, CPO
11 Here we use the term “interior” referring both to portfolios and allocation: an interior
equilibrium is one in which two (or more) consumers are partners in each firm, and such that
individual consumption is strictly positive.
12 At the first corner equilibrium (θ1 = y1 = 1.0), β2 = λ12 ≤ β1 = λ11 guarantees that
the production plan is a profit maximizer at β . The second corner equilibrium is symmetric
to the first.