Optimal Dynamic Taxation
75
It is instructive to concentrate on iso-elastic utility, since then
we have two regimes for the after-tax interest rate: ρ = 0 and ρ
= fk. We will see later on that the dynamic behaviour of the
economy does not fundamentally change for the more general
case. Also, for expositional simplicity we keep τl constant in the
graphs. We will see later on that when τl is changing over time
(according to the optimal policy), the fundamental dynamic
patterns do not change. We shall consider the following timing:
at t0 Chamley’s policy is implemented, and at t1 the capitalincome tax is set to zero.
We may think of three initial values of the capital stock when
Chamley’s policy is implemented: lower, equal to, or greater
than its steady-state value. That is, either k(t0) < k*, or k(t0) = k*,
or k(t0) > k*. Assume first that the initial value of the capital
stock is equal to its long-rung value. When ρ = 0 both the slope
and the level of the line q̇ = 0 goes to plus infinity, (we may
think of this as when the line disappears to the left), and we are
left with Figure 3.6, below.
We then have three possibilities for the individual’s
behaviour. The individual may choose initial consumption and
initial labour supply so that initial marginal utility is
(I) equal to steady-state marginal utility, i.e. q(t0) = q*,
(II) greater than the steady-state level, i.e. q(t0) > q*,
(III) smaller than the steady-state level, i.e. q(t0) < q*,
and then adjusting according to the first-order differential
equation q̇ = θq.
Graphically the three possibilities are depicted in Figure 3.7,
below.
Endogenous Taxation in a Dynamic Economy
76
Figure 3.6
Figure 3.7
It is clear that (I) cannot be optimal, when the regime switches
to a zero capital income tax the economy cannot turn back to
its steady state. The economy would accumulate capital forever
and consumption would go to zero. On the same grounds we
Optimal Dynamic Taxation
77
can rule out (II). We are left with (III) as the only possibility.
In fact the initial value of q is such that the economy is
guaranteed to join the unique converging trajectory X, exactly
at the date of the regime switch t1. This path requires capital
decumulation.24 See Figure 3.8, below.
Figure 3.8
It is clear that the analysis above does not fundamentally
change when there is no regime switch but a gradual adjustment
of ρ towards fk.25 Still the economy would decumulate capital
for a period and join the new converging trajectory associated
with the continuous adjustment of ρ towards fk.
24
We could think of a fourth possibility as well: q(t0) is very low. Then the
trajectory would not cross the k̇ = 0 line, but continue left upwards toward
the q-axis. In this case marginal utility goes to infinity and the economy’s
capital stock goes to zero. This would happen if the ρ = 0 regime lasts
forever.
25
When form of the utility function is more general than the iso-elastic form.
Endogenous Taxation in a Dynamic Economy
78
It is also clear that if the size of economy’s initial capital stock
is different from its steady-state value, the same analysis apply.
Suppose the initial capital stock is smaller than its steady state
value. We have three possibilities analogously to the case when
k0=k*: The individual may choose initial marginal utility
(I′) on trajectory X,
(II′) above trajectory X,
(III′) below trajectory X.
Case (I′) and (II′) may be ruled out on the same grounds as
above. We are left with (III′), which at t1 joins X. See figure
3.9.
Figure 3.9
Finally suppose that the economy’s capital stock is greater than
its steady state value. Again we have three possibilities. The
individual may choose a q on trajectory Y, and then follow the
differential equation q̇ = θq. But then the economy cannot
follow trajectory Y, since q grows faster when ρ = 0 than when
ρ = fk. Since q grows faster, consumption will decrease faster
and k will not decumulate as quickly as intended on trajectory
Optimal Dynamic Taxation
79
Y. Clearly this implies that the economy will go on a trajectory
above, and after some time cross the k̇ = 0 line and behave as
in the case (I) above. By the same reason we may rule out any
q(t0) above the trajectory and we are left with case (III′′),
depicted in Figure 3.10. As above the economy will decumulate
capital.
Figure 3.10
The length of the confiscatory regime (t1-t0) depends on the
economy’s funding requirement. If the funding requirement is
large relative to the capital stock k(t0) then the co-state q(t0)
would take on a lower value and join X at a later date. So, the
greater the distance t1-t0 the smaller the q(t0). Smaller q(t0)
is associated with faster capital decumulation.
We may think of two paths not covered by the Figures 3.5-3.8.
First, if the confiscatory regime is permanent, i.e. t1-t0 → ∞,
then q(t0) takes on a value so small that the trajectory never
crosses the k̇ = 0 line and therefore continues to infinity and the
capital stock goes to zero. Second, if the funding requirement
Endogenous Taxation in a Dynamic Economy
80
is small and k(t0) > k*, we have the possibility that q(t0) is large
enough for the economy not to decumulate capital below k*.
The two cases are depicted below in Figure 3.11, as trajectories
IV and V respectively.
Figure 3.11
We have drawn the graphs for a constant labour income tax τl.
When τl changes over time we have to imagine the k̇ = 0 line
"moving" over time (at least between t0 and t1) in Figures 3.7,
3.9-3.11. For example, when τl is increasing over time [as for
the utility function in (36)] the k̇ = 0 line will move outwards as
time goes on. This is so since an (uncompensated) increase in
the labour income tax decreases labour supply and at least not
decreases consumption, if consumption and leisure are
complements. This would make k̇ < 0. To "restore" k̇ = 0 the
level of k has to be higher (or alternatively the level of q has to
be greater). If this is the case, trajectory III will as before be
below the k̇ = 0 line, and towards t1 "chase" the line and cross
it. Clearly, the fundamental pattern of trajectory III does not
change, and on the same basis as before we can always rule out
Optimal Dynamic Taxation
81
the trajectories I and II.
From the graphical analysis we have found a common
characteristic of the economy under the confiscatory regime:
Remark Regardless the initial value of the capital stock when
Chamley’s policy is implemented, the economy always
decumulates capital in the beginning of the regime, and at least
after some time (if the confiscatory regime is long enough) the
pre-tax interest rate becomes greater than the rate of time
preference.
This characteristic will be useful to bear in mind when we in
the next part will study the feedback equilibrium of an optimaltax economy.
To conclude, in this section we have reviewed Chamley’s result,
and at the same time derived additional results regarding the
labour income. We have also studied the dynamic behaviour of
the economy under Chamley’s fiscal policy. The solution was
derived under the assumption that the government can
(credibly) commit to all future taxes. This solution is usually
labelled as second best. We have an economy in which no
lump-sum instruments are available, and we want to find the
optimum combination of distortionary tax instruments so as to
fund an exogenously given public expenditure sequence.26
Because capital income is fixed in the short run, capital
income taxation is like lump-sum taxation. However, capitalincome taxes distort the intertemporal price between
commodities at different dates. So, by the choice of utility
function, it is optimal in an economy without capital to tax
26
Or without loss of generality endogenous public expenditure.
Endogenous Taxation in a Dynamic Economy
82
consumption at different dates at uniform rate, i.e. a zero capital
income tax.27 In the long run the latter effect becomes
predominant, and therefore we find the capital income tax zero
in the long run.28
It is therefore clear that, if the government is allowed to
reoptimise, the government would always find capital inelastic,
and therefore find it optimal to tax it at the maximum rate. This
is true regardless weather the optimal long-run capital income
tax is zero or not.29
The time-inconsistency problem is quite severe, despite this
the open-loop solution is useful as a bench mark at which we
may evaluate optimal policy under different institutional
settings. We shall proceed with the analysis by assuming that
the government cannot commit to future taxes, and evaluate the
optimal policy resulting from this institutional setting. We may
suspect that if commitment is not possible, the government will
find it optimal to always confiscate capital income. We will see
that this intuition is true in a more formalised treatment of
differential games in the next part.
27
Atkinson and Stiglitz (1972) prove that when utility is additively separable
in commodities, these commodities should be taxed at a uniform rate.
28
As Lucas (1990, pp. 299-300) argues: "One principle is that factors of
production in inelastic supply - factors whose income is a pure rent - should
be taxed at confiscatory rates [...] A second principle in Ramsey’s analysis
is that goods that appear symmetrically in consumer preferences should be
taxed at the same rate [...] In my formulation there is but one tax rate
applied to income from old and new capital alike, so these two principles
cannot simultaneously be obeyed."
29
That is, regardless weather utility is separable in commodities at different
dates. Perhaps the most important question may not be whether the optimal
capital-income tax is zero or not, but how different institutional arrangements
may cause a deviation from the second-best optimal policy.