14.472 – Spring 2004
Atkinson-Stiglitz
page 1 of 3
August 19, 2004
Notation
xi
y
z
w
u [ x, y ]
consumption in period i
labor supply
net of tax earnings
wage
utility
fi
p
q
number of workers of type i
producer price of good 2
consumer price of good 2
There are two theorems to look at, both involving a lack of taxation of commodities in the
presence of optimal income taxation. One has general nonlinear taxation, while the other has
linear commodity taxation. Both are shown in the two-types model.
A. Full nonlinear taxation
Social welfare maximization assuming full nonlinear taxation:
Maximize x, y
subject to:
∑ fiu i ⎡⎣ x , x , y ⎤⎦
E + ∑ f i ( x + px
i
1
i
2
i
1
i
i
2
− wi y i ) ≤ 0
u i ⎡⎣ x1i , x2i , y i ⎤⎦ ≥ u i ⎡⎣ x1j , x2j , y j w j / wi ⎤⎦ for all i and j
Assume that the only binding constraint is type 1 imitating type 2.
(1)
14.472 – Spring 2004
Atkinson-Stiglitz
page 2 of 3
August 19, 2004
Then we have the FOC for the consumption levels (assuming interior solutions)
(
f1u1x1 ⎡⎣ x11 , x12 , y1 ⎤⎦ − λ ( f1 ) = − µ u1x1 ⎡⎣ x11 , x12 , y1 ⎤⎦
(
)
f1u1x2 ⎡⎣ x11 , x12 , y1 ⎤⎦ − λ ( pf1 ) = − µ u1x2 ⎡⎣ x11 , x12 , y1 ⎤⎦
(2)
)
(
f 2u x21 ⎡⎣ x12 , x22 , y 2 ⎤⎦ − λ ( f 2 ) = µ u1x1 ⎡⎣ x12 , x22 , y 2 w2 / w1 ⎦⎤
(
(3)
)
f 2u x22 ⎡⎣ x12 , x22 , y 2 ⎤⎦ − λ ( pf 2 ) = µ u1x2 ⎡⎣ x12 , x22 , y 2 w2 / w1 ⎦⎤
(4)
)
(5)
Taking ratios, we have
u1x1 ⎡⎣ x11 , x12 , y1 ⎤⎦
u1x2 ⎡⎣ x11 , x12 , y1 ⎤⎦
=p
(6)
f 2u x21 ⎡⎣ x12 , x22 , y 2 ⎤⎦ − µu1x1 ⎡⎣ x12 , x22 , y 2 w2 / w1 ⎤⎦
f 2u x22 ⎡⎣ x12 , x22 , y 2 ⎤⎦ − µu1x2 ⎡⎣ x12 , x22 , y 2 w2 / w1 ⎤⎦
=p
(7)
The first condition is the usual lack of marginal taxation at the top of the earnings distribution.
The second condition involves no taxation if
u x21 ⎡⎣ x12 , x22 , y 2 ⎤⎦
u ⎡⎣ x12 , x22 , y 2 ⎤⎦
2
x2
=
u1x ⎣⎡ x12 , x22 , y 2 w2 / w1 ⎦⎤
1
u ⎡⎣ x12 , x22 , y 2 w2 / w1 ⎤⎦
1
x2
(8)
14.472 – Spring 2004
Atkinson-Stiglitz
page 3 of 3
August 19, 2004
A sufficient condition for the second to give no taxation is separability and the same sub-utility
function, u i = u i ⎡⎣ h [ x1 , x2 ] , y ⎤⎦ , which implies
u xi1 [ x1 , x2 , y ]
u
i
x2
[ x1 , x2 , y ]
=
uhi ⎡⎣ h [ x1 , x2 ] , y ⎤⎦ h1 [ x1 , x2 ]
u ⎡⎣ h [ x1 , x2 ] , y ⎤⎦ h2 [ x1 , x2 ]
i
h
=
h1 [ x1 , x2 ]
h2 [ x1 , x2 ]
(9)
Note that this has assumed that the corner conditions of nonnegative consumption are not
binding.
B. Nonlinear income taxation and linear commodity taxation
The extension to linear consumption taxes follows from this being a more constrained optimum
than with full nonlinear taxation. That is, there is an additional constraint on allowable ( x, y )
vectors. Yet the optimum without this constraint is feasible with the extra constraint.
To proceed with this problem, we could use a mixed direct-indirect utility function:
vi ⎡⎣ y i , z i , q ⎤⎦ =Maximize x
subject to:
u i ⎡⎣ x1i , x2i , y i ⎤⎦
x1i + qx2i = z i
(10)