Academic year 2009-2010
Thomas
Piketty,
Academic
year 2009-2010
Course Notes :
Optimal redistributive taxation of
capital and capital income
Basic result n°1:
without inheritance, and with perfect capital markets, optimal k tax = 0%
Intuition: if 100% of capital accumulation comes from lifecycle savings,
then taxing capital or capital income is equivalent to using differential
commodity taxation (current consumption vs future consumption);
Atkinson-Stiglitz: under fairly general conditions (separable preferences),
differential commodity taxation is undesirable, and the optimal tax structure
should rely entirely on direct taxation of labor income
(to put it differently: if inequality entirely comes from labor income
inequality, then it is useless to tax capital; one should rely entirely on the
redistributive taxation of labor income)
Atkinson-Stiglitz 1976:
Model with two periods t=1 & t=2
Individual i gets labor income yL = vl in period 1 (v = wage
rate, l = labor supply), and chooses how much to consume c1 and c2
Max U(C1,C2) – V(l) under budget constraint: c1
+ c2/(1+r) = yL
Period 1 savings s = yL - c1 (>0)
Period 2 capital income yk = (1+r)s = c2
>>> taxing capital income yk is like taxing period 2
consumption c2
>>> Atkinson-Stiglitz: under separable preferences, there is no
point taxing capital income; it is more efficient to redistribute income by
using solely a labor income tax t(yL)
References:
A.B. Atkinson and J. Stiglitz, “The design of tax structure: direct vs
indirect taxation”, Journal of Public Economics 6 (1976), 55-75
V.
Christiansen, « Which Commodity Taxes Should Supplement the Income
Tax ? », Journal of Public Economics 1984 [article
en format pdf]
E. Saez,
« Direct vs Indirect Tax Instruments for Redistribution :
Short-run vs Long-run », Journal of Public Economics 2004 [article
format pdf]
Basic result n°2:
with infinite-horizon dynasties, optimal linear k tax = 0%, but optimal
progressive k tax > 0%
Simple 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 ktA,
or they own a low capital stock ktB (ktA >
ktB). 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 kt is given by:
kt
= λktA + (1-λ)ktB
Consider first the case
ktB=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:
Ut =
∑t≥0 U(ct)/(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(kt) (f’(k)>0,
f’’(k)<0), where kt 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 rt and wage rate vt
are always equal to the marginal products of capital and labor:
rt =
f’(kt)
vt =
f(kt) - rtkt
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’(λkA)=r*=θ
This
result comes directly from the first-order condition:
U’(ct)/
U’(ct+1) = (1+rt)/(1+θ)
I.e. if
the interest rate rt is above the rate of time preference θ, then
agents choose to accumulate capital and to postpone their consumption
indefinitely (ct<ct+1<ct+2<…)
and this cannot be a steady-state. Conversely, if the interest rate rt is below
the rate of time preference θ, agents choose to desaccumulate capital (i.e. to borrow) indefinitely
and to consume more today (ct>ct+1>ct+2>…).
This cannot be a steady-state either.
Now assume we introduce linear redistributive
capital taxation into this model. That is, capital income rtkt
of the capitalists is taxed at tax rate τ (so that the post-tax
capital income of the capitalists becomes (1-τ)rtkt),
and the tax revenues are used to finance a wage subsidy st
(so that the post-transfer labor income of the workers becomes vt+st).
Note kτ* , kAτ*=
kτ*/λ and rτ*
the resulting steady-state capital stock and pre-tax interest rate. The Golden
rule of capital accumulation implies that:
(1- τ) f’(kτ*)=
(1-τ) rτ* = 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 yτ* will be equal to:
yτ*
= vτ* + sτ*
with: vτ* = f(kτ*)
- rτ* kτ*
and: sτ* = τ rτ*
kτ*
That is:
yτ*
= f(kτ*) – (1-τ) rτ* kτ* =
f(kτ*) – θkτ*
Question: what is the capital tax rate τ
maximizing workers’ income yτ* = f(kτ*) –
θkτ* ?
Answer: τ must be such that f’(kτ*)
= θ, i.e. τ = 0%
Proposition 1: The
capital tax rate τ maximizing long run workers’ welfare is τ = 0%
>>> in effect, even agents with zero
capital loose from capital taxation (no matter how small)
(e.g. the corporate tax is shifted on labor in the very long run)
>>> 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
Simple model with capitalists vs
middle class: progressive capital taxation
Now assume we have ktB>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
(i)
λkA∞ +
(1-λ)kB∞ = 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 kτ .[1]
I.e., the tax is equal to 0 if k<kτ , and the tax is equal
to τ(k-kτ) if k>kτ . Further assume
that the threshold kτ 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 kA∞ < kτ
, 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 kτ 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 kτ (with kτ >
k*), then all long-run steady-state wealth distributions (kA∞
, kB∞) (kA∞ >
kB∞) are characterized by the following two conditions:
(ii)
λkA∞ +
(1-λ)kB∞ = k*
(with k* such that f’(k*)=r*=θ)
(iii)
kB∞ < kA∞
< kτ
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 kB∞
< kA∞ < kτ , or kτ
< kB∞ < kA∞ . Assume
that kτ < kB∞ < kA∞
, and note k∞ the average long run capital stock (k∞ = λkA∞ +
(1-λ)kB∞ ). 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 kτ > k* implies that k∞ > k*
, which in turn implies that r∞ = f’(k∞) <
r* = f’(k*) = θ , which leads to a contradiction. Therefore kB∞
< kA∞ < kτ . 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
References:
R. Lucas,
“Supply Side Economics: An Analytical Review”, Oxford Economic Papers
1990 [article
in pdf format]
Basic result n°3:
with imperfect k markets, optimal k tax > 0%
Pretty obvious: assume there is no capital market at all, i.e. individuals
with wealth wi can only invest their own wealth ki=wi
and produce output f(ki); clearly one could raise aggregate output
by redistributing wealth
References:
C. Chamley, “Capital Income Taxation, Wealth Distribution and Borrowing
Constraints”, Journal of Public Economics 79 (1) (January 2001),
pp.55-70 [article
en format pdf]
Other recent
references on optimal capital taxation
With borrowing constraints, capital taxation as a substitute for dynamic
labor income taxation t(yL1,yL2):
J. Conesa, S. Kitao and D. Krueger, “Taxing Capital? Not a Bad Idea
After All!”, American Economic Review 2009 [article en format pdf]
On capital taxation and governance:
M. Desai, A. Dyck, L. Zingales, « Theft and Taxes », Journal
of Financial Economics, 2006 [article
in pdf format]
R. Morck, “Why Some Double Taxation Might Make Sense: The Special Case
of Inter-Corporate Dividends”, NBER Working Paper 9651 (2003) [article
in pdf format]
R. Morck and B. Yeung, “Dividend Taxation and Corporate Governance”, Journal
of Economic Perspectives 2005 [article
in pdf format]
W. Kopczuk and J. Slemrod, “Putting Firms into Optimal Tax Theory”, American
Economic Review 2006 [article in pdf format]
On informational rationales for taxing capital income:
H. Cremer, P. Pestieau et J.C. Rochet, “Capital Income Taxation when
Inherited Wealth is Not Observable”, Journal of Public Economics 87
(2003), 2475-2490 [article
en format pdf]
R. Gordon, « Taxation of Interest Income », NBER Working Paper
9503 (2003) [article
en format pdf]
On family firms and the inefficiency of inheritance:
M. Bennedsen, F. Peres-Gonzalez, K. Nielsen and D. Wolfenzon, “Inside the
Family Firm: The Role of Families in Succession Decisions and Performance”, Quartely
Journal of Economics 2007 [article
in pdf format]
F. Pérez-Gonzalez, “Inherited Control and Firms’ Performance”, American
Economic Review, 2006 [article
in pdf format]
L. Bach, “Family CEO Successions and Firms
Brankruptcies – Evidence from
V. Grossmann and H. Strulik, “Should Continued
Family Firms Face Lower Taxes Than Other Estates?”, mimeo 2008 [article in pdf format]
On private vs social discounting and negative capital taxation:
E. Farhi and I. Werning, “Progressive Estate Taxation”, mimeo 2008 [article
in pdf format]
E. Farhi and I. Werning, “Inequality and Social Discounting”, Journal
of Political Economy 2007 [article
in pdf format]
[1] A similar result
applies if one replaces the progressive capital tax by a progressive tax on
capital income.