Income taxation when markets are incomplete
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as we show in Section 4 below, Theorem 1 holds for a class of economies
that encompasses these two.
Pure exchange GEI. In a pure exchange economy, a CPO is an allocation
that cannot be Pareto improved by a portfolio redistribution, when date zero
lump-sum transfers are also a viable policy instrument. It is well-known
that, in this setting, every competitive GEI entails a CPO allocation. Instead,
Theorem 1 shows that there exists a generic set of economies in which every
pure exchange equilibrium is not an IT-CPO. In other words, our notion of
constrained efficiency (Definition 4) is stronger than the one proposed by
Diamond.
Example 1. Consider an economy with one firm, two consumers, and two
states of uncertainty (J = J1 = 1, H = S = 2). Consumers have different,
quasi-linear, utilities, and identical endowments:
u1 ( x1) = x 01 + 5 д/xj + x 21,
u2 (χ2) = χ2 + X2 + 5^x∣,
e1 =e2 = (3,.5,.5).
The only asset is a riskless bond R = (1, 1). This economy has a unique
equilibrium that coincides with the no-trade equilibrium. Figure 1(a) repre-
sents the equilibrium, in the space of second-period consumption, with the
thick lines being the present value budget constraints (evaluated at the equi-
librium individual state prices), and the dotted line representing the income
transfers line (i.e., the asset span).
- An optimal tax reform. The equilibrium can be locally Pareto improved via
a marginal income tax reform: dt1 = (-1, -5) %; we are taxing the return
from the asset in state 2 more heavily than in state 1. Consumer 1, who has a
particularly strong taste for consumption in state 1, buys the bond issued by
consumer 2, who instead has relatively stronger taste for the good in state 2.
The sacrifice of 2, in terms of second-period consumption, is compensated
by an increase in her consumption at date zero, due to her gains in asset
trade. The summary of results is expressed in percentage change from the
bench mark no-tax CPO equilibrium:
|
% Ax 0 |
% Ax 1 |
% Ax 2 |
% Aθ |
% Au | |
|
h=1 |
-6.71 |
Γ5^^ |
1.45 |
1.52 |
.04 |
|
h=2 |
6.71 |
-1. 5 |
-1.45 |
-1.52 |
.04 |
Figure 1(a) represents the equilibrium in the space of second-period in-
come (i.e., for given equilibrium levels of first-period consumption). The
tax reform tilts the asset span-line clockwise (see Fig. 1(b)); at the new