Thomas Piketty, Paris School of Economics

Academic year 2011-2012

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 y_L = 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) = y_L

Period 1 savings s = y_L - c₁ (>0)

Period 2 capital income y_k = (1+r)s = c₂

>>> taxing capital income y_k 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(y_L)

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, “The Desirability of Commodity Taxation under Non-Linear Income Taxation and Heterogeneous Tastes”, Journal of Public Economics 2002 [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 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_tk_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_tk_t of the capitalists is taxed at tax rate τ (so that the post-tax capital income of the capitalists becomes (1-τ)r_tk_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%

>>> 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 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

 

References:

R. Lucas, “Supply Side Economics: An Analytical Review”, Oxford Economic Papers 1990 [article in pdf format]

E. Saez, “Optimal Progressive Capital Income Taxes in the Infinite Horizon Model”, NBER Working Paper 2004 [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 w_i can only invest their own wealth k_i=w_i and produce output f(k_i); 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(y_L1,y_L2):

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 France”, mimeo, 2008 [article in pdf format]

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]