78 C
. Fuest, B. Huber Journal of Public Economics 79 2001 71 –91
domestic production domestic investment bears no direct relation to the wealth of households N E 1 N E domestic savings.
1 1
2 2
The government levies a source tax on capital K at rate t . The marginal
k
productivity condition for capital is then given by f 9 k 5 r 1
t 7
s d
k
where k 5 K N L 1 N L denotes capital per unit of labor. fk is the corre-
s d
1 1
2 2
sponding per capita production function. This condition can be solved for k r 1 t
s d
k
with k9 5 1 f 0 , 0. Using the fact that w 5 f k 2 r 1 t k, one obtains the factor–
s d s
d
k
price frontier w 5 w r 1
t 8
s d
k
with w9 5 2 k , 0. Finally, one can now state the country’s aggregate resource constraint which is
N C 1 N C fk 2 rk N L 1 N L 1 rN E 1 rN E 2 N z 2 N z s
d
1 1
2 2
1 1
2 2
1 1
2 2
1 1 2 2
9 where we have assumed, since we are mainly interested in redistributive tax
policy, that exogenous government spending on goods and services is zero.
3. The government’s problem
In this section, we now turn to the government’s optimal tax problem. We proceed in two steps. We will first discuss the structure of the national planner’s
optimal tax problem in some detail. In the following section, we then derive the optimal tax policy.
In what follows, we assume that the government maximizes a utilitarian social welfare function. This implies that the government wishes to redistribute from
wealthy households of type 2 to households of type 1. The planner’s problem is set up as follows: since the non-linear tax schedule TY , K allows the government to
i i
indirectly control C , Y and K , we pose the problem such that the planner directly
i i
i
sets these variables, and we then discuss the tax policy which is required to achieve this outcome.
As a benchmark, consider first the full information case, where the government
13
can observe each individual’s labor supply L and endowment E . The problem of
i i
1 2
the government is then to maximize social welfare N V ? 1 N V ? subject to
1 2
the aggregate resource constraint 9. The solution to this problem yields a first-best optimum. Marginal utilities of the two types of households are equalized
13
Indeed, it would suffice to identify either E or L .
i i
C . Fuest, B. Huber Journal of Public Economics 79 2001 71 –91
79
1 2
i i
i i
V 5 V . The first-best is also characterized by T 5 T 5 0 such that V 5 V
s d
C C
Y K
C Y
and Z 5 0. To implement the first-best, lump-sum taxes conditioned on E are set
i i
14
which achieve an optimal redistribution from type 2 to type 1 households. In the presence of informational constraints, the first-best optimum can no
longer be attained because it is not incentive-compatible. If the government can only observe Y and rK , person 2 will mimic person 1 if the government tries to
i i
15
implement the first-best. Due to these informational constraints, the government has to solve a second-
best problem. In this setting, redistributive policies are limited by the self-selection constraint which precludes that person i, i 51, 2, can gain by mimicking person j,
j ± i. Denote a mimicker by a bar. If person 2 mimicks person 1, this yields
] ]
Y 2Z
1 2
2
S D
]] V
? 5 U C , s d
1
w ]
] as utility for person 2 where Z 5 Z r E 2 K
. As Z . 0, and since, as a s s
dd
2 2
2 1
2
mimicker, person 2 has to report the same labor income as person 1 Y , the labor
1
supply of person 2 will be less than that of person 1. The opposite applies if person 1 mimics person 2. Thus, a mimicker would choose his labor supply so that, given
16
the amount of shifted income, the observable variables match the required levels. The self-selection constraint for a person of type i requires
]
i i
V C , Y , K , r, w, E V C , Y , K , r, w, E i 5 1, 2, j ± 1.
10 s
d
s d
i i
i i
j j
j i
In the standard optimum income tax model with differing abilities, the well-known monotonicity property rules out that the self-selection constraints of both persons
17
bind. In our model, the appendix derives sufficient conditions which ensure for
fairly general utility functions that an analoguous property holds, implying that only the self-selection constraint for the wealthy individual binds.
14 i
2
Denoting optimal lump-sum taxes by T , one obtains T 5 r E 2 E N N 1 N
. 0 and s
ds s
dd
2 1
1 1
2 1
2
T 5 2 N N T , 0. s
d
2 1
15
This can be seen as follows. Using the tax formula in footnote 12, it is straightforward to check that a first-best optimum implies C 5 C 5 C and Y 5 Y 5 Y. If person 2 the high-wealth individual
1 2
1 2
reveals his true type, utility is thus given by UC, Y w. If person 2 mimics person 1, person 2 again achieves C as level of consumption. Person 2 then reports the same income and pays the same amount
of taxes as person 1 in the first-best. Mimicking also implies that the amount rE 2 K of capital
2 1
income is shifted to labor income. Total utility of a mimicker is thus given by UC, Y 2 Zr E 2 K w. But since Z . 0, this implies that a mimicking strategy strictly dominates truth-telling
2 1
in a first-best optimum. This reflects that the wealthy mimicker gains by supplying less labor while consumption remains constant, as explained in the next paragraph in the text.
16
We owe this explanation to a referee.
17
In the pareto-optimizing framework of Stiglitz 1987, it is possible that the self-selection constraint of either the high-ability or the low-ability household binds.
80 C
. Fuest, B. Huber Journal of Public Economics 79 2001 71 –91
4. Optimal tax policy
