32
Endogenous Taxation in a Dynamic Economy
2.4 TAX EQUIVALENCE RESULTS
What follows are a number of applications of Theorem 1.
Corollary 1 (Non-Equivalence) A system with only labour
income taxation is not generally equivalent to a system with
only consumption taxation.
Proof: Theorem 1 states that any triple of sequences can be
replicated, therefore a system with one of the sequences being
non-zero (e.g. the labour income tax) and the other two
sequences being zero could be replicated. However, the
replicating triple is unique up to only one sequence, say the
labour income tax. Choose the labour income tax sequence to
be zero, then the replicating triple consisting of a consumption
tax and a capital income tax is uniquely determined. Finally it
must be checked whether the capital income tax rate in the
replicating triple always would be zero. Let * denote the
replicating triple (the one with the consumption tax) and take
the original system to be the one with labour taxation.
Equations (13) and (17) become
(13’)
(17’)
Zero capital income tax rate implies ρt* = rt, t = 0,...,T. Then
(13’) and (17’) can only be satisfied in the trivial case: all taxes
zero, i.e. no taxation is equivalent to no taxation. In all other
cases the conditions are violated.
QED
Dynamic Taxation and Equivalent Tax Systems
33
Note that if capital income taxation is allowed in period 0 or if
initial assets a0 are taxed as labour income then the AtkinsonStiglitz equivalence24 occurs: constant tax rates can be
replicated. However, with the introduction of capital income
taxation equivalence can always be guaranteed.
Corollary 2 (Equivalent Tax Systems) A system with labour
income taxation and capital income taxation is equivalent to a
system with consumption expenditure taxation and capital
income taxation.
Proof: Theorem 1 states that a replicating triple of tax
sequences is unique up to exactly one of the sequences. This
implies that a whole sequence may be set to zero, therefore the
real outcomes under system with two tax sequences can always
be obtained under a different system consisting of two other tax
sequences.
QED
The next question is whether the capital income tax could be
made ’redundant’ in the sense that any capital income tax
sequence could be replicated by labour and consumption tax
sequences.
Corollary 3 (Replicable Tax Systems) A system with only
capital income taxation is replicable by a system with labour
income taxation and consumption taxation. The reverse
replicability is not true.
24
The result of Atkinson and Stiglitz (1980) is that a constant tax on labour
income and initial capital endowments is equivalent to a constant tax on
consumption and bequests at the end of the life time. This equivalence does
not generally hold for the standard dynamic economy, and the reason is the
requirement for the tax rates to be constant and that initial capital has to be
taxable.
Endogenous Taxation in a Dynamic Economy
34
Proof: First part same as Corollary 2, second part same as
Corollary 1.
Note that for each period with capital income taxation the
replicating sequences must involve an increasing consumption
tax rate and an increasing subsidy on labour. This follows from
equations (12) and (13). If capital income always is taxed, then
the replicating consumption tax and labour subsidy has to be
ever increasing.
Corollary 4 (Zero-Revenue and Zero-Distortions Tax Systems)
It is always possible to find sequences of tax rates on labour
income, consumption and capital income tax rates, that do not
give rise to distortions or tax receipts. The collection of these
tax sequences is unique up to exactly one of the sequences.
Proof:
Follows directly from Theorem 1.
Theorem 2 (Taxation of Initial Capital) A (less than 100%) tax
on initial capital is replicable by a system with labour income
taxation, consumption expenditure taxation and capital income
taxation, (even if the capital income tax rate is set to zero in
the first period).
Proof: Taxation of initial capital gives rise to no distortions or
tax returns in the future. This system could be represented as
Dynamic Taxation and Equivalent Tax Systems
The replicating triple, {τt*c}Tt=0, {τt*l}Tt=0, {τt*k}Tt=0,
constructed in accordance with (12), (13) and (17)
35
must be
(18)
(19)
(20)
Such a system will give rise to neither intertemporal nor
intratemporal distortions. Labour has to be subsidised at the
same rate as consumption is taxed. The consumption tax and
the labour subsidy have to be ever decreasing over time.
To see what the effective tax on initial capital is, substitute
the replicating sequences (19) and (20) into the budget equation
(6)
(21)
The equivalent tax on initial capital,τk, is then
(22)
By choosing the level of τ0*c any tax on initial capital smaller
than 100% can be replicated, even if τ0*k= 0.
QED
Endogenous Taxation in a Dynamic Economy
36
Theorem 3 (Public Debt Funding) In an economy with labour
income taxation, consumption expenditure taxation and capital
income taxation (even though distortionary), the same real
economic outcome can be obtained regardless whether for a
period public expenditure is (a) purely debt funded, (b) purely
tax funded, or (c) any combination thereof.
Proof: Two different tax regimes τ and τ* give rise to the
same real economic outcome according to Theorem 1 if (12),
(13) and (17) are fulfilled. The evolution of the aggregate of
assets under the two regimes are
(23)
(24)
Define
(25)
Then
(26)
Equation (26) implies
(27)
Furthermore, since the real outcome is the same kt*=kt ∀t, then
(28)
Since {τt*c}Tt=0 is free of choice (according to Theorem 1) then
st is free of choice for t=1,...,T. Thus for any time profile of bt,
Dynamic Taxation and Equivalent Tax Systems
37
taxes may be adjusted so as to give rise to the same real
outcome. Then we may choose bt such that gt=bt-bt-1 for a
period, i.e. pure debt funding. Now it is obvious that we may
also choose partial public debt funding as well.
QED
2.5 INCOME TAXATION
For some reasons it might be convenient to tax income
uniformly, i.e. to tax labour income and capital income at the
same rate: an income tax.25 Could it be possible to replicate
any triple of tax sequences by an income tax sequence?26 It is
not possible to see directly whether it would be possible to
replace any tax system by an income tax system. This is so
because an income tax places additional restrictions on the tax
sequences.
Theorem 4 Assume A1 and A2, and a constant interest rate.
Then, an income tax system, with the possibility of having a tax
rate on capital income which is different from the one on
labour income in period 0, can replicate a tax regime
consisting of a constant labour tax rate τl and a constant
capital tax rate τk and of any sequence of consumption tax rates
{τtc}Tt=0. All but one of the replicating income tax sequences
increase steadily over time (and if T→∞, they converge
asymptotically to 1). The only constant replicating income tax
sequence is the one set equal to the capital income tax in the
original system.
25
It might be difficult or costly for the fiscal authorities to perfectly observe
capital income and labour income distinctly. For example a self employed
could overreport (underreport) his labour supply and thereby transform
capital (labour) income into labour (capital) income.
26
That the converse is true is obvious.
Endogenous Taxation in a Dynamic Economy
38
Proof: Suppose the given tax triple is {τtc}Tt=0, τl, τk. Let
{τt*c}Tt=0 denote the sequence of replicating consumption tax
rates, and τt* the replicating income tax rate. An equivalent tax
system must obey equation (12), (13) and (17), therefore
(29)
(30)
(31)
Use (29) in (30) and (31) then
(32)
(33)
Let rt=r ∀t, and rewrite (32) and (33) as
(34)
Dynamic Taxation and Equivalent Tax Systems
39
(35)
where mt = 1 - τt*, and a = 1 + (1 - τk)r.
There are two steady states to the above difference equation
(36)
The only stable steady state is m*, i.e. when τ*=1. Assuming
T→∞, any mt < 1 -τk converges asymptotically to m*=0, while
any mt > 1 -τk explodes and becomes ill defined.27 To ensure
that m0 < 1 -τk, a different tax on capital income must generally
be used in period 0. The only stable equivalent income tax
system {τt*c}Tt=0, τ* = τk is uniquely described by
(37)
(38)
QED
Stationary replication is restricted to cases where τ* = τk. When
the labour tax rate, and or the capital tax rate are non-constant
only non-stationary replication is possible, i.e when the income
tax rate converges to 1. Suppose τk is given within an optimal
27
If T is finite and mt < 1 -τk, then mt is decreasing in time and may not
reach m*. This implies that τt* is increasing over time. On the other hand, if
m0 > 1 -τk, then m may explode in finite time. This would happen when t = ln(1 - (1-τk)/m0) / ln(1 + (1-τk)r) (if the latter expression is an integer).
Endogenous Taxation in a Dynamic Economy
40
taxation framework, then τk = 0.28 Then τ* = 0, and clearly
there is limited scope for the income tax.
2.6 DIFFERENCES IN TIMING
This section will show how the concept of time solves the
problem of indeterminacy of tax structure. Here will be
supposed that tax rates can be changed less often than
individuals can make their consumption, labour, and savings
decisions. The idea is that an individual may decide how much
to consume and how many hours to work on a daily basis,
while the government can only change the tax rates every
second day or once a week. This will break the previous tax
equivalence results.
Theorem 5 Assume A1 and A2, and that the tax rates have to
be constant over two or more periods, then no replicability or
tax equivalence occurs, i.e. any triple of tax sequences {τtc}Tt=0,
{τtl}Tt=0, {τtk}Tt=0, becomes unique.
Proof: Suppose there is a given triple {τtc}Tt=0, {τtl}Tt=0, {τtk}Tt=0.
It remains to show that no other triple can give rise to the same
real outcomes. Theorem will be proven by assuming the least
restrictive case: The tax rates has to be constant over two
periods. If the tax rates has to be constant over two periods the
triple satisfies
28
Chamley (1986) and Judd (1985b).
Dynamic Taxation and Equivalent Tax Systems
41
(39)
for ξ = c, l, k. Assume another triple {τt*c}Tt=0, {τt*l}Tt=0,
{τt*k}Tt=0. First it must satisfy
(40)
for ξ = c, l, k. Next, if * gives rise to the same real outcomes,
it must satisfy (12), (13) and (17) as shown in the proof of
Theorem 1. Equation (13) together with (39) and (40) implies
(41)
i.e.
(42)
Equations (39), (40) and (42) give
Endogenous Taxation in a Dynamic Economy
42
(43)
then
(44)
Next, equation (17) together with (44) implies
(45)
then according to (39) and (40)
(46)
Equation (13) together with (44) gives
(47)
Then (45), (46) and (47) give
(48)
Finally (12) and (48) imply
(49)
By (44), (48) and (49) the triple is unique.
QED
It should be noted that it is required that taxes are constant over
every two periods. It is not enough to set e.g. period-one taxes
equal to period-two taxes and then allow tax changes every
Dynamic Taxation and Equivalent Tax Systems
43
period thereafter. This has no correspondence in the static
literature were normalisation can be obtained by letting two
prices be equal.
2.7 SUMMARY AND CONCLUSIONS
This chapter derived tax equivalence results for dynastic
economies. Individuals receive utility from one consumption
good and disutility from work, they may save or borrow
intertemporally. Flat tax rates are used on consumption
expenditure, labour income and income on savings (capital).
A known dynamic tax equivalence result is due to Atkinson
and Stiglitz (1980), and states that labour income taxation is
equivalent to consumption expenditure taxation if initial capital
is taxed at the same rate as labour, and bequests at the same
rate as consumption, at constant rates. However, if the tax rates
are non-constant over time (out of steady state), this
equivalence will not hold as shown in Corollary 1. However if
a capital income tax is introduced, the general tax equivalence
occurs: a system with consumption expenditure and capital
income taxation is equivalent to a system with labour income
and capital income taxation; this was shown in Corollary 2.
These corollaries (and others) followed from Theorem 1. It
established the exact degree of freedom in a tax system. It
stated that exactly one tax sequence is free of choice, i.e. the
same real economic outcome can be obtained for any values of
the entire tax sequence. That is, the same individual
consumption patterns, work patterns, public expenditure etc.
may be obtained given any time path of (for example) the
consumption expenditure tax.
Another important result is Theorem 3 which is a public
debt neutrality proposition with distortionary taxation. Taking
the result of Theorem 1 it was shown that the same real
Endogenous Taxation in a Dynamic Economy
44
economic outcome could be obtained regardless if public
expenditure for a period was purely public debt funded or
purely tax funded or a combination thereof.29
Section 2.6 introduced a difference in timing. It is plausible
that individuals can revise their consumption and labour-supply
decisions more often than the government can change its fiscal
policy. If there is only a ’slight’30 difference in timing the
equivalence results break down, and any tax policy becomes
unique. In the static optimal taxation literature one may obtain
a unique tax structure by fixing one of the tax rates, which one
is arbitrary. In a dynamic context the concept of time provides
a natural solution to this kind of indeterminacy problem.
Furthermore, it is not enough if only two period’s taxes are set
equal and then allow for tax changes every period, the
difference in timing has to prevail over the entire future.
One may think of some relaxations of the assumptions that
would change the results. One would be tax evasion, since
different taxes may have different evasion characteristics. In this
case the tax equivalence results may break down. Another case
is when individuals ’differ in age’, like in an overlapping
generations economy. In such a case the tax system would have
intergenerational redistributive effects. Finally, there is one
market imperfection that would change the results: borrowing
and lending constraints. If these constraints were binding the
equivalence results would be expected to break down.
29
Taking as given the initial level of public debt and either the condition that
public debt is fully paid back when the economy ends or that the ’No-PonziGame’ condition for the government is fulfilled.
30
Since a discrete time set up was used the result was obtained by letting
individuals revise their policy every period and the government every second
period. A continuous-time version of this economy would generate such a
result for an arbitrarily small difference in timing.
Dynamic Taxation and Equivalent Tax Systems
45
Endogenous Taxation in a Dynamic Economy
46
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