*Advanced course « Economics of
Inequality » (Master APE, M2 year)*

Thomas Piketty

Année universitaire 2008-2009

Academic year 2008-2009

**Course Notes E : **

**Optimal redistributive taxation of capital
and capital income**

T. Piketty, « Income Inequality in

__1. Model with capitalists vs
workers: linear capital taxation__

Consider
an infinite-horizon, discrete-time economy with a continuum [0;1] of dynasties.

For
simplicity, assume a two-point distribution of wealth. Dynasties can be of one
of two types: either they own a large capital stock k_{t}^{A},
or they own a low capital stock k_{t}^{B} (k_{t}^{A }>
k_{t}^{B}). The proportion of high-wealth dynasties is exogenous
and equal to λ (and the proportion of low-wealth dynasties is equal to
1-λ), so that the average capital stock in the economy k_{t} is given by:

k_{t}
= λk_{t}^{A }+ (1-λ)k_{t}^{B}

Consider first the case
k_{t}^{B}=0. I.e. low-wealth dynasties have zero wealth
(the “workers”) and therefore zero capital income. Their only income is labor
income, and we assume it is so low that they consume it all (zero savings). High-wealth
dynasties are the only dynasties to own wealth and to save. Assume they
maximize a standard dynastic utility function:

U_{t} =
∑_{t≥0} U(c_{t})/(1+θ)^{t}

(U’(c)>0,
U’’(c)<0)

All dynasties
supply exactly one unit of (homogeneous) labor each period. Output per labor
unit is given by a standard production function f(k_{t}) (f’(k)>0,
f’’(k)<0), where k_{t} is the average capital stock per capita of
the economy at period t. Markets for labor and capital are assumed to be fully
competitive, so that the interest rate r_{t} and wage rate v_{t}
are always equal to the marginal products of capital and labor:

r_{t} =
f’(k_{t})

v_{t} =
f(k_{t}) - r_{t}k_{t}

In such a
dynastic capital accumulation model, it is well-known that the long-run
steady-state interest rate r* and the long-run average capital stock k* are
uniquely determined by the utility function and the technology (irrespective of
initial conditions): in stead-state, r* is necessarily equal to θ, and k*
must be such that:

f’(k*)=r*=θ

I.e. f’(λk_{A})=r*=θ

This
result comes directly from the first-order condition:

U’(c_{t})/
U’(c_{t+1}) = (1+r_{t})/(1+θ)

I.e. if
the interest rate r_{t} is above the rate of time preference θ, then
agents choose to accumulate capital and to postpone their consumption indefinitely
(c_{t}<c_{t+1}<c_{t+2}<…) and
this cannot be a steady-state. Conversely, if the interest rate r_{t} is below
the rate of time preference θ, agents choose to desaccumulate capital (i.e. to borrow) indefinitely
and to consume more today (c_{t}>c_{t+1}>c_{t+2}>…). This
cannot be a steady-state either.

Now assume we introduce linear redistributive
capital taxation into this model. That is, capital income r_{t}k_{t}
of the capitalists is taxed at tax rate τ (so that the post-tax capital
income of the capitalists becomes (1-τ)r_{t}k_{t}),
and the tax revenues are used to finance a wage subsidy s_{t}
(so that the post-transfer labor income of the workers becomes v_{t}+s_{t}).

Note k_{τ}* , k_{Aτ}*=
k_{τ}*/λ and r_{τ}*
the resulting steady-state capital stock and pre-tax interest rate. The Golden
rule of capital accumulation implies that:

_{}

(1- τ) f’(k_{τ}*)= (1-τ)
r_{τ}* = θ

I.e. the
capitalists choose to desaccumulate capital until the point where the net
interest rate is back to its initial level (i.e. the rate of time preference). **In effect, the long-run elasticity of
capital supply is infinite in the infinite-horizon model**: any infinitesimal
change in the net interest rate generates a savings response that is
unsustainable in the long run, unless the net interest rate returns to its
initial level.

The long
run income of the workers y_{τ}* will be equal to:

y_{τ}*
= v_{τ}* + s_{τ}*

with: v_{τ}* = f(k_{τ}*)
- r_{τ}* k_{τ}*

and: s_{τ}* = τ r_{τ}*
k_{τ}*

That is:

y_{τ}*
= f(k_{τ}*) – (1-τ) r_{τ}* k_{τ}* =
f(k_{τ}*) – θk_{τ}*

Question: what is the capital tax rate τ
maximizing workers’ income y_{τ}* = f(k_{τ}*) – θk_{τ}*
?

Answer: τ must be such that f’(k_{τ}*)
= θ, i.e. τ = 0%

__Proposition 1__: The
capital tax rate τ maximizing long run workers’ welfare is τ = 0%

>>> this is the theoretical basis for
the “zero capital tax is socially optimal” result

>>> this result requires three strong
assumptions: infinite elasticity of capital supply; perfect capital markets;
and linear capital taxation

__2. Model with capitalists vs middle
class: progressive capital taxation__

Now assume we have k_{t}^{B}>0.
I.e. all dynasties accumulate capital and save according to the dynastic,
infinite-horizon utility function.

The
important point is that convergence in individual wealth levels does not necessarily
occur in a such a model. In fact, any wealth distribution such that the average
wealth is equal to k* (the “golden rule” capital stock) can be a long-run
steady-state.

__Proposition
2. In the absence of taxation, all
long-run steady-state wealth distributions (k ^{A}_{∞} , k^{B}_{∞})
(k^{A}_{∞}^{ }> k^{B}_{∞})
are characterized by the following
condition: __

(i)
λk^{A}_{∞}^{ }+
(1-λ)k^{B}_{∞} = k*
(with k* such that f’(k*)=r*=θ)

Consider
now the effects of progressive taxation. Assume that individual capital stocks
are taxed each period at a marginal tax rate τ>0 above some capital
stock threshold k_{τ }.[1]
I.e., the tax is equal to 0 if k<k_{τ} , and the tax is equal
to τ(k-k_{τ}) if k>k_{τ} . Further assume that
the threshold k_{τ} is larger than the “golden rule” capital stock
k* (defined by f’(k*)= r*=θ). One can easily show that the only long run
effect of this progressive capital tax is to truncate the distribution of
wealth. That is, the long run distribution
of wealth must be such that k^{A}_{∞} < k_{τ}
, but long run average wealth is unchanged (it is still equal to the “golden
rule” level k*). Note that this truncation result holds no matter how small the
tax rate τ : τ just needs to be strictly positive (say, τ =
0,0001%), and one gets the result according to which individual wealth levels
above the threshold k_{τ} must completely disappear in the long
run. This illustrates how extreme the dynastic model really is.

__Proposition
3.__ With progressive capital taxation at rate τ>0 levied on capital
stocks above some threshold k_{τ} (with k_{τ} >
k*), then all long-run steady-state wealth distributions (k^{A}_{∞}
, k^{B}_{∞}) (k^{A}_{∞}^{ }>
k^{B}_{∞}) are characterized by the following two conditions:

(ii)
λk^{A}_{∞}^{ }+
(1-λ)k^{B}_{∞} = k*
(with k* such that f’(k*)=r*=θ)

(iii)
k^{B}_{∞} < k^{A}_{∞}
< k_{τ}

__Proof__ : In
steady-state, after-tax interest rates faced by both types of dynasties must be
equal to the rate of time preference. This implies that both types of dynasties
must be in the same tax bracket in the long run: either k^{B}_{∞}
< k^{A}_{∞} < k_{τ} , or k_{τ}
< k^{B}_{∞} < k^{A}_{∞} .
Assume that k_{τ} < k^{B}_{∞} < k^{A}_{∞}
, and note k_{∞} the average long run capital stock (k_{∞} = λk^{A}_{∞}^{ }+
(1-λ)k^{B}_{∞} ). The long run before-tax interest
rate r_{∞} is given by r_{∞}
= f’(k_{∞}), and the long run after-tax interest rate (1-τ)r_{∞}
faced by both types of dynasties is such that (1-τ)r_{∞} =
θ. But k_{τ} > k* implies that k_{∞ }> k*
, which in turn implies that r_{∞} = f’(k_{∞}) <
r* = f’(k*) = θ , which leads to a contradiction. Therefore k^{B}_{∞}
< k^{A}_{∞} < k_{τ} . This implies
that the tax does not bind in the long run and that r_{∞} =
θ and k_{∞} = k*, in the same way as in the absence of tax.
CQFD.

**>>> even with infinitely elastic
capital supply and perfect capital markets, there is scope for progressive
capital taxation: as long as some lower wealth individuals can accumulate
capital and compensate for the higher wealth individuals’ desaccumulation,
progressive capital taxation entails no efficiency cost**

[1] A similar result
applies if one replaces the progressive capital tax by a progressive tax on
capital income.