Cours avancé « Economie des inégalités » (Master APE, année M2)

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


Thomas Piketty

Année universitaire 2009-2010

Academic year 2009-2010


Course Notes F :

Optimal redistributive taxation of capital and capital income



E. Saez, “Optimal Progressive Capital Income Taxes in the Infinite Horizon Model”, NBER Working Paper 2004

T. Piketty, « Income Inequality in France, 1901-1998 », CEPR Working Paper n°2876 (July 2001) (Appendix, pp.30-32)


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 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:



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* , k*= 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%


>>> 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 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 2. In the absence of taxation, all long-run steady-state wealth distributions (kA , kB) (kA > kB) are characterized by  the following condition:

(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






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