2
Dynamic Taxation and
Equivalent Tax Systems
2.1 INTRODUCTION
That some taxes may be redundant in the sense that they can be
replicated by other taxes is a well known phenomenon. It was
discovered already by Ramsey (1927) in a static framework
with commodity taxation. If such an equivalence occurs the
"optimal" 1 tax rates would take the form of an equivalence
class. One reason why the question of equivalence is important
is because it leaves the optimal tax structure undetermined. The
aim of this chapter is to explore the exact relations between
different taxes in a dynamic economy by systematically derive
tax equivalence results that have not been derived before in the
literature. Also, it will be shown, the concept of time will
provide a "natural" solution to the indeterminacy problem of
optimal/endogenous tax rates.
At this point it is appropriate to clarify how one may think
of a tax system. A tax system is characterized by the types of
tax instruments available. For example one tax system could be
composed of a flat consumption tax, a progressive capital
income tax, and a lump sum subsidy.2 Another system would
1
Optimal either from a social point of view or from some individual’s point
of view. If they are optimal from the median voter’s point of view we may
refer to them as endogenous tax rates.
2
A tax system which includes a progressive tax would specify the threshold
points between which different rates would be applied.
Dynamic Taxation and Equivalent Tax Systems
17
be a lump sum tax and a flat labour income tax. Within each
system different levels of the rates may be applied, without the
system itself specifying them. In a dynamic economy these rates
can change over time. Thus, within each tax system in a
dynamic economy the tax rates would have a time profile. A
time profile of a certain type of tax instrument, would be
referred to as a sequence of tax rates.
This chapter gives a systematic treatment of equivalence
among linear 3 tax systems in a dynamic economy: an
economy in which individuals act as dynamic optimisers when
choosing consumption, labour supply and savings, and are, as
well as the government, allowed to borrow and lend freely at
the market interest rate.4 Since there is no uncertainty perfect
foresight will be assumed. Generally proportional taxes on
consumption, labour income and capital earnings will be
allowed for.
In order to keep a clear focus on the tax equivalence
problem we study a simple economy and abstract from some
aspects like "age differences", non-linear tax schedules, tax
collection costs, uncertainty and borrowing-lending constraints.
All individuals are assumed to have the same time horizon
as the government (i.e. no "age differences"). The application
that comes into one’s mind is a dynastic economy, i.e. an
economy in which the parents care about their children’s utility.
Presumably the time horizon here is interpreted as infinite. This
type of economy is most common in studies of intertemporal
phenomena such as growth and business cycles, as well as
intertemporal public finance, and most scholars find it a
3
4
In a linear tax system only flat tax rates are imposed.
This framework is standard in many models of dynamic economies.
Examples of applications are optimal savings, equilibrium business cycles
and modern theories of economic growth.
Endogenous Taxation in a Dynamic Economy
18
reasonable representation of a dynamic economy.
The focus is on linear taxes. The intertemporal literature
generally abstracts from non-linear tax schedules,5 mostly
because they impose severe complexities both for the optimum
tax problem and for voting theory (endogenous taxation). Nonlinear consumption taxes are usually assumed away because of
their limited enforceability (consumers could form groups when
doing their purchases).
Tax collection costs and tax evasion are somewhat related.
They certainly can break the equivalence and solve the
indeterminacy problem. If some taxes are more costly to collect
(in some sense) than others, an optimum (or endogenous) tax
framework would suggest to leave this/these commodity/
commodities untaxed. Also, if some tax bases have different
evading characteristics than others this may also break the
eventual equivalence.6 However, the focus of the chapter is an
economy in which taxes can be more or less costlessly
collected, and where evasion is not possible.7
Borrowing-lending constraints may be regarded as market
imperfections. In general, market imperfections (special kinds
of externalities) give rise to the desirability of corrective
taxation (Pigouvian taxation). It should be said, though, that the
results of this chapter do hold in the presence of imperfections
other than borrowing-lending constraints (e.g. imperfect
5
See Chamley (1985 and 1986), Judd (1985a,b and 1987a,b), Lucas (1990).
An exception is Ordover and Phelps (1979) who examine non-linear income
taxation in an overlapping-generations economy where individuals have no
bequest motives.
6
See Boadway et.al. (1994). They assume that consumption taxes cannot be
evaded. Income, on the other hand, may be concealed at some cost, and
therefore some income tax can be evaded.
7
The idea is that collection costs are small enough not to drastically alter the
optimum/endogenous tax formulae.
Dynamic Taxation and Equivalent Tax Systems
19
competition), and production/consumption externalities.
In the concluding section some of the above issues will be
discussed and put in relation to the findings of this chapter. The
chapter is organised as follows:
Section 2.2 clarifies the distinction between static and
dynamic tax theories. It will be shown the equivalence results
that emerge in a standard static optimal commodity taxation
framework.
Section 2.3 treats a "standard" dynamic economy, where
individuals at each date consume one consumption good and
supply labour, and may borrow or lend. The economy is
specified and a general replicability theorem is derived. It states
the conditions under which a linear tax system may be
replicated by other linear tax systems.
In section 2.4 the general tax replicability theorem is
applied to answer specific questions about tax equivalence. One
important result is that a labour income tax is not generally
equivalent to a consumption tax, but when a capital income tax
is introduced the equivalence holds. Furthermore labour income
taxation together with consumption taxation can replicate a
capital income tax, but then the labour tax rate has to be
negative, i.e. labour has to be subsidised. These results, and
others, follow as corollaries of the theorem. Next, two other
cases are studied. They involve more mathematical elaborations
and do not follow directly from the theorem and therefore are
stated as propositions. The first proposition is that taxation of
initial capital can be obtained even without capital income
taxation, if labour income and consumption expenditure taxes
are available. The second proposition is about debt funding and
is in the close spirit of Ricardian Equivalence. It is shown that
the economy can be made public debt neutral in the sense that
public expenditure may be financed solely with public debt for
a period, giving the same real economic outcome as if it had
Endogenous Taxation in a Dynamic Economy
20
been funded by taxes, even when only distortionary taxes are
available.
The case when the same tax rate is applied to all factor
income, (an income-tax system), is studied separately in section
2.5. One result is Theorem 2, which states that under some
circumstances a general tax system can be replicated by an
income-tax system. However, if the time horizon is infinite,
most of these systems must have an income tax asymptotically
converging to 100%, and a consumption subsidy also
converging to 100%.
Throughout section 2.3, 2.4 and 2.5 it has been assumed
that the government can revise tax policy as often as individuals
can make their consumption and labour supply decisions. For
example, if individuals decide every morning and every
afternoon how much to consume and how much to work the
government was assumed to be able to change the tax rates
every morning and afternoon. This might be technically
impossible in practice. Section 2.6 examines tax equivalence
when the timing differs between individuals and the
government. If there is a slight difference in timing, e.g.
individuals can make decisions every period, while the
government only can make decisions every second period, all
tax equivalence results break down and any tax policy is
unique. Immediately it follows that optimal tax policy here is
unique and is no longer a member of an equivalence class. This
is precisely the way the concept of time provides a "natural"
solution to the indeterminacy problem.8
8
It should be noted that this is more than just a "normalisation". If the
timing is the same for individuals as for the government and the prices (and
the taxes) are normalised we get the optimal tax formulae in terms of
quantities. These formulae would not be the same as those we would obtain
for an economy where the timing differs. The latter problem is a more
restricted optimisation problem. The welfare properties of the economy
depend on the time interval during which policy cannot be revised for
Dynamic Taxation and Equivalent Tax Systems
21
Sections 2.8 summaries the findings and gives the motivation
for the framework of tax theory chosen for the remaining
chapters of the thesis.
2.2 STATIC vs DYNAMIC FRAMEWORK
Before carrying on with the dynamic framework it may be
useful to explore some aspects of the static optimum taxation
literature, to see to what extent it could be used for
intertemporal economies and what kind of tax equivalence
results that have been found. There are a number of differences
between the static and the dynamic literature one may think of:
(a) the assumed production constraints differ, (b) individuals’
budget constraints differ, (c) the price normalisation differs, (d)
the concept of time. Each of these differences will be explored
below.
(a) The Production Constraint
The production constraint in the static literature takes the form9
for an n-good economy. It is assumed that f is homogenous of
degree one in all its arguments, i.e. production constraint obeys
constant returns to scale. By this assumption one avoids
example if the government cannot commit to future tax policy the time
interval may act as a partial precommitment technology. We shall further
explore this issue in chapter 4.
9
Diamond and Mirrlees (1971a,b), Stiglitz and Dasgupta (1971), Diamond
(1975), Deaton (1979), Samuelson (1986). Stiglitz and Dasgupta also study
a special case when there is decreasing returns to scale and the implication
for the desirability of production efficiency. This case may be interpreted as
it there was a fixed factor (capital) which was untaxed.
Endogenous Taxation in a Dynamic Economy
22
Pigouvian elements of taxation, and one can more clearly focus
on the inefficiency aspects. One may ask to what extent y may
be treated as a vector of dated goods, say an economy with one
consumption good ct and one type of labour supply lt at each
date t. The question is can we interpret y=(ct,lt,...,cT,lT) as the
production vector above?10 The production constraint for a
dynamic economy (in discrete time) is usually written as a
difference equation in the economy’s capital stock kt, which is
the accumulated savings of the economy in real terms (i.e. in
terms of the consumption good)
(1)
Take this constraint at a particular time T (slightly rearranged)
(2)
Then at T-1
(3)
Use (3) to substitute for kT in (2) to obtain
(2’)
Then by recursive substitution
(4)
The consumption possibility at time T (i.e. kT+1+cT) is a
function, φ, of the consumption and labour supply at all
10
T.
Assuming that we are at date t and view the economy from date t to date
Dynamic Taxation and Equivalent Tax Systems
23
previous dates and the capital stock at the time at which we
consider the economy. It turns out that if f(k,l) is homogenous
of degree one in (k,l) then φ is homogenous of degree one in all
its arguments (i.e. including kt). Thus, a dynamic production
constraint in intertemporal form differs from the production
constraint used in the static optimal taxation literature by
inclusion of kt as a factor of production. The static literature
abstracts from capital.
(b) Individuals’ Budgets
The budget constraint for an individual in the static literature is
qx=0, where q is a vector of consumer prices and x is a vector
of consumption goods (factors of supply, like labour, take on
negative values).11 There are no endowments since the
Euclidian product is zero. We cannot take an element of x to be
interpreted as capital since the latter is fixed in supply.12
(c) Price Normalisation
When carrying out the optimisation over the tax rates two
normalisations can be made, one for consumers and one for
producers. Usually q1=p1=1, i.e. all consumer goods have a
price in terms of good one, and similarly for all producer
goods.13 This also implies that one good may be untaxed. This
gives one tax equivalence result. If one were to interpret this
result in an intertemporal manner, we could say that one dated
11
Diamond and Mirrlees (1971a,b), Stiglitz and Dasgupta (1971), Diamond
(1975), Deaton (1979), Samuelson (1986). Other authors, e.g. Dixit (1970)
allow qx=I, where I is a transfer to individuals resulting from redistributive
taxation, still there are no endowments.
12
To state it in a different way, the elements of x are flow variables only,
while capital is a state variable.
13
Diamond and Mirrlees (1971a,b), and Stiglitz and Dasgupta (1971).
Endogenous Taxation in a Dynamic Economy
24
good may have a zero tax (e.g. labour income at year 2001, or
VAT at year 1999) but this is as much as can be said from a
static treatment. In the dynamic literature the price vector
usually is already normalised, all prices are in terms of the
consumption good. This is clear from the individuals’ budget
equations
where a is the value of assets in consumption units. The prices
r and w are the returns to savings and labour supply and are
stated in terms of consumption units.
(d) The Concept of Time
The concept of time is probably the most important difference.
In the static literature optimal tax formulas are often stated in
terms of elasticities. However, these elasticities in a dynamic
economy would change over time, and therefore the optimal tax
formulas (even if formulated on the same commodity) would
change over time. Take for example a government who at time
t chooses taxes for commodities consumed (and labour
supplied) at a later date (say time t+h). This government would
find these taxes sub optimal when it reviews its policy at a date
later than t. This is the time-inconsistency problem.14 An
extreme example of how elasticities change over time is that of
savings. Savings is highly elastic in the long run but inelastic
in the short.15
2.3 A DYNAMIC EQUIVALENCE THEOREM
14
Kydland and Prescott (1977) and (1980).
15
See Summers (1981).
Dynamic Taxation and Equivalent Tax Systems
25
This section formalises the standard dynamic economic
framework. The assumptions are carefully stated and the general
replication theorem is derived. It should be pointed out that
everywhere individual heterogeneity is allowed (as long as
consumers have the same time horizon). They may or may not
differ in time preferences, productivity, capital endowments,
time endowments and/or intertemporal substitution. The results
are derived by elaborating the budget constraints without
explicitly incorporating the utility functions of the individuals
and the production functions of the firms.
A1. Individuals’ Budget Constraints
At time t, individual i supplies lti units of labour and consume
cti units of the only consumption good. She is assumed to live
for T periods (possibly infinite) and can invest in both capital,
kt, and government bonds, bt, each yielding the same return, rt.
The sum of the individual holdings of capital and bonds is
referred to as the individual’s assets, ati.16 For each supplied
unit of labour, lti, the individual receives the wage rate wti
(possibly individual specific). The prices rt and wti are real, i.e.
stated in terms of the only consumption good. Allow for the
possibility of taxing labour income, consumption expenditure,
and capital income with respective tax rates τtl, τtc, and τtk. The
individual’s asset equation becomes
(5)
16
We may think of the individual as maximising an intertemporal utility
function, U i(c0i,l0i,c1i,l1i,...,cTi,lTi), when choosing the optimal quantities of
consumption and labour. The exact form of U is not crucial for the results
in this paper, since the results can be proven by only looking at the
individuals’ budget sets.
Endogenous Taxation in a Dynamic Economy
26
which may also be written on intertemporal form17
(6)
where
and
.
The economy is assumed to be endowed with some initial
capital k0 > 0.
A2. Government’s Budget Constraint
The government is allowed to borrow and lend freely at the
market rate of interest and has some exogenous (or endogenous)
expenditure sequence {gt}Tt=0. The expenditure sequence may be
thought of as being used either for government consumption,
for provision of public goods or for lump sum redistribution.18
The governments budget equation takes the form of a difference
equation in total outstanding debt, bt,
(7)
where the bars denote aggregate quantities. If individuals differ
in skills the aggregate labour supply, lt, is defined in efficiency
17
Equation (6) is obtained by iterating (5) and use either aTi+1 = 0 if T is
l im
i
i
i
finite or Tl→i m
∞ PT aT+1 = 0 if T is infinite. Note that aT+1 > 0 (or T→∞ PT aT+1 >
0) cannot be optimal since consumption could then be increased at previous
i
dates, without decreasing consumption at other dates. Furthermore aT+1
<0
l im
i
( or T→∞ PT aT+1 <0) cannot be a market equilibrium.
18
The choice of (and use of) gt will not be explicitly modelled since it will
not generally affect the results. The only way in which it could change the
results is if gt was tax dependent. This can be ruled out if we assume that gt
depends on only real variables (such as consumption, production, marginal
utilities etc.)
Dynamic Taxation and Equivalent Tax Systems
units: lt ≡
27
, where wt is the average wage rate in the
economy.
Given the above assumptions two more relations are obtained.
In the aggregate the sum of capital and bonds equals total
assets, i.e. at=kt+bt. Then the aggregate of equation (5) and
equation (7) are consistent with the resource constraint in the
economy19
(8)
where the prices {rt}Tt=0 and {wt}Tt=0 could be taken as general
equilibrium prices (a closed economy) as well as being
exogenously given (an open economy).
Before carrying on with the analysis a formalisation of the
concepts tax system and tax equivalence is needed.
Definition 1 (Tax system) A tax system is completely described
by the types of tax instruments available.
The definition implies that a tax system does not specify the
levels of the tax rates or the time profile of these levels.
Definition 2 (Replication) A tax system Y is said to be
19
This is a different way of stating Walras’ law (which says that if all agents
satisfy their budget constraints and if n-1 markets clear then the nth market
also clears). As pointed out by Diamond and Mirrlees (1971a,b) we may
reverse the argument: when all markets clear and all but one agent satisfy
their budgets then the last agent satisfies his budget. This relationship has
also been noted by Chamley (1986).
Endogenous Taxation in a Dynamic Economy
28
replicable by another tax system X if for all (possible) tax
sequences in Y there can be found tax sequences in X so that
both systems would give rise to the same budget sets (and
thereby same optimal choices) for the individuals, and yield the
same tax receipts for the government. The sequences in X
satisfying this condition are referred to as replicating
sequences.
Definition 3 (Tax Equivalence) A tax system X is said to be
equivalent to another tax system Y if X is replicable by Y, and
Y is replicable by X.
Note that saying that X is replicable by Y is not saying that Y
is replicable by X. For equivalence to hold it must be explicitly
checked in each case whether the replicability works in both
ways. One could also think of some cases where tax sequences
in X can be found for particular tax sequences in Y.20
What follows is a general replicability theorem for
consumption, labour income and capital income taxation. The
proof of replicability is constructed by considering two different
collections of tax sequences. Conditions are derived under
which the individuals’ intertemporal budget constraints are the
same under both collections of tax sequences. Under these
circumstances the quantities chosen, {ct}Tt=0 and {lt}Tt=0, will be
the same under both collections of sequences for all individuals.
Then, both collections of sequences will yield the same tax
20
Consider the following example: Assume two tax systems, X and Y.
Suppose that for constant (not time-varying) tax rates in Y, there can be
found tax sequences in X that give rice to the same individual choice and the
same tax receipts. But it may be the case that for tax rates in Y that are
moving over time no such sequences in X can be found. For an explicit case
see the comment to Corollary 1.
Dynamic Taxation and Equivalent Tax Systems
29
receipts. Note that this is true regardless whether the producer
prices are fixed or endogenous. Henceforth we will speak of
real outcomes as shorthand for the individual’s choice and the
total of tax receipts.
Theorem 1 Assume A1 and A2 and consider any triple of tax
sequences {τtc}Tt=0, {τtl}Tt=0, {τtk}Tt=0. The real outcome in the
economy can be obtained by another triple, which would be
unique only up to exactly one of its sequences. Once one of the
sequences in the replicating triple has been determined, the
replicating triple becomes unique.
Proof: Suppose there is a given triple of tax sequences {τtc}Tt=0,
{τtl}Tt=0, {τtk}Tt=0. The question is whether the real outcome can
be obtained by another triple, say {τt*c}Tt=0, {τt*l}Tt=0, {τt*k}Tt=0,
and the degree of uniqueness of these sequences. For the
sequences to generate the same choices they must give rise to
the same intertemporal budget constraints [equation (6)]. The
intertemporal budget constraint for the new sequences
becomes21
(9)
where
.
For consumption prices being equal under both triples the
following is required
21
Superscripts denoting individuals are omitted from now on.
Endogenous Taxation in a Dynamic Economy
30
(10)
Similarly, for prices of labour to be equal the following is
required
(11)
Combining (10) and (11) gives22
(12)
That is, once a sequence {τt*l}Tt=0 (or {τt*c}Tt=0) is chosen the
sequence {τt*c}Tt=0 (or {τt*l}Tt=0) is uniquely determined.
Shifting (10) forward and combining with (10) gives23
(13)
That is, once the sequence {τt*c}Tt=0 is chosen, the sequence of
capital tax rates is determined up to τ0*k. Use (13) to obtain the
relation
(14)
22
Condition (12) will ensure that the individuals’ intratemporal optimality
conditions are the same under both collection of tax sequences (i.e. yielding
the same marginal rate of substitution between consumption and leisure at
each point in time).
23
Condition (13) ensures that the individuals’ intertemporal optimality
conditions are the same under both collections of sequences, (i.e. yielding the
same Euler equations for the individuals).
31
Dynamic Taxation and Equivalent Tax Systems
or equivalently by using also (12)
(15)
Substitute these two expressions for Pt* into (9) and premultiply
by
to obtain
(16)
This is exactly the budget equation for the original triple if
(17)
Thus the only requirements on the new triple are (12), (13) and
(17). They define the new sequences up to exactly one of the
sequences, i.e. once {τt*c}Tt=0 (or {τt*l}Tt=0 or {τt*k}Tt=0) is chosen
the remaining sequences are uniquely determined.
QED