Income Taxation when Markets are Incomplete



114


M. Tirelli

equilibrium). In Fig. 2(b) below, we plot the aggregate (indirect) utility level
as a function of the return from holding the firm in state 1 (i.e.,
θ1y1), and
fixed
x0, when allocations are feasible in the sense of Diamond. It is im-
mediate that the minimum welfare is obtained at the symmetric allocation,
while the only two CPO are corner solutions. Thus, Fig. 2(b)
shows that
at a minimum welfare point, the symmetric equilibrium is a local CPO-
equilibrium: a Pareto superior equilibrium can be achieved by a Diamond
central planner via discrete portfolios and production reallocations.

- Optimal tax reforms 1. (Interior equilibrium) We start from the interior
equilibrium, and assume that a tax reform, such as the one considered in
Example 1, is introduced:
dt1 = (-1, -5) %. Then a Pareto improvement
is obtained with a
.04% increase in the utility of both agents, and an 4.17%
increase in the share of agent 1 in the firm:

% Ax 0

% Ax ]

% Ax 2

%

% Au

h=1

-7.86

1. 06

1.98

4.17

.04

h=2

7.86

-1. 06

-1.98

-4.17

.04

2. (Corner equilibria) Since the two corner equilibria are symmetric, we
consider only the one in which consumer 1 is the single owner of the firm.
The tax reform
dt1 = (.1, 0) % is weakly Pareto improving:

% Ax 0

% Ax ]

% Ax 2

%

% Au

h=1

-.05

0.5

0

0

.06

h=2

0.5

-0.5

0

0

0

In words, a (global) CPO equilibrium can be improved upon by using pro-
portional income taxes.

Fig. 2.




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