SUBSTITUTIONS AMONG TAXES: IMPLICATIONS FOR
WELFARE LOSS
The third main question of distorting taxation is that of tax reform: What is
the implication on social welfare of substituting one set of taxes for another
while holding revenue constant? This tax substitution experiment is perhaps
the most compelling of all second-best exercises within the pure allocational
theory of taxation, if only because governments occasionally engage in such
tax substitutions. We continue to assume a one-consumer equivalent econ-
omy with linear technology.
As long as the tax changes are ``small,'' the expressions for marginal loss
and total tax revenue are all that are needed to determine the eYciency
implications for any given equal-revenue substitution among taxes. Begin
with the total diVerential of dead-weight loss, Eq. (13.11), with respect to all
the taxes:
dL
P
N
k1
P
N
i1
t
i
M
ik
dt
k
(13:58)
(there is no need to assume an untaxed good in this exercise). Equation
(13.58) is an appropriate measure of marginal loss for any given change in
the vector of tax rates. The reason why is that a substitution can always be
viewed as a multistep series of individual loss experiments, in which one tax is
reduced and the revenue returned to the government lump sum, after which a
second tax is imposed, with its revenue returned to the consumer lump sum,
and so on, for any number of tax changes. Because M
ik
M
ki
, the order of
substitution is irrelevant.
Next, add the values of dt
k
that hold revenue constant. These can be
determined by totally diVerentiating the tax revenue equation:
dT
P
N
k1
M
k
P
N
i1
t
i
M
ik
dt
k
(13:59)
Setting dT 0, Eq. (13.59) determines all possible tax substitutions that keep
revenue unchanged. Once the appropriate values for dt
k
have been deter-
mined from Eq. (13.59), they can be substituted back into Eq. (13.58) to
determine the resulting change in dead-weight loss.
When only two taxes change, Eq. (13.59) describes the exact relation-
ship between the two changes necessary to hold revenue constant. Suppose,
decreasing-cost industries with proWt constraints are virtually identical to the optimal tax rule, as
we shall discover in Chapter 21. One common example is the U.S. postal service, which has long
used the IER to justify its policy of covering its cost increases primarily by increasing rates on Wrst-
class mail (relatively inelastic demands) rather than on the other classes of mail such as parcel post
(relatively elastic demands). We will return to the postal service example in Chapter 21.
13. THE SECOND-BEST THEORY OF TAXATION IN ONE-CONSUMER ECONOMIES 439
for example, t
j
and t
k
are to be changed, dt
i
0, for i 6 j, k. From Eq.
(13.59),
dT 0 M
k
P
N
i1
t
i
M
ik
dt
k
M
j
P
N
i1
t
i
M
ij
dt
j
(13:60)
or
dt
k
dt
j
qT
qt
j
qT
qt
k
(13:61)
As expected, the two rates must change in direct ratio to the marginal
changes in tax revenue with respect to each of the taxes. Presumably, one tax
is increased and the other is decreased. Notice also that the relevant marginal
revenue changes are the changes at the compensated equilibria, not the actual
equilibria. This is consistent with the deWnition of loss in terms of compen-
sated equilibria.
To complete the analysis, the marginal loss with respect to changes in t
j
and t
k
is
dL
P
N
i1
t
i
M
ik
dt
k
P
N
i1
t
i
M
ij
dt
j
(13:62)
from Eq. (13.58). Substituting in the equal-revenue constraint, Eq. (13.61),
and recalling that qL/qt
k
P
N
i1
t
i
M
ik
yields:
dL
qL
qt
k
qT
qt
j
qT
qt
k
0
B
B
@
1
C
C
A
dt
j
qL
qt
j
dt
j
2
6
6
4
3
7
7
5
(13:63)
Rearranging terms:
dL
dt
j
qL
qt
k
qT
2t
j
qT
qt
k
0
B
B
@
1
C
C
A
qL
qt
j
2
6
6
4
3
7
7
5
qL
qt
k
dt
k
dt
j
RR
qL
qt
j
(13:64)
Equation (13.64) gives an entirely plausible result. The change in loss from
increasing one tax (say, t
j
) and lowering another tax (say, t
k
) to keep total
tax revenue constant is a linear combination of the marginal losses
from changing t
k
and t
j
individually. The marginal loss for the revenue-
compensating tax, t
k
, is weighted by the amount that t
k
must be changed
440 SUBSTITUTIONS AMONG TAXES: IMPLICATIONS FOR WELFARE LOSS
Get Public Finance, 2nd Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
