Optimal Taxation Over the Life-Cycle For the Poor and the Richnmathieu/OTPRonline.pdf · The answer...

35
Optimal Taxation Over the Life-Cycle For the Poor and the Rich Nathalie Mathieu-Bolh September 9, 2009 Abstract This paper is a theoretical contribution to the Ramsey optimal taxation literature. In a new life-cycle model, I nd that when individuals face di/erent borrowing con- straints, the progressivity of optimal labor income taxation is limited. Poor and rich individuals generally face two optimal capital income tax regimes involving redistribu- tion across income categories. Optimal capital income taxation generally di/ers from zero even when preferences are additively separable. The conditions for zero optimal capital income taxation are more stringent when borrowing constraints are tight. Op- timal capital income taxation paid by the rich is lower if their weight in the population is larger. JEL: E61, E62, H21, H23 Nathalie Mathieu-Bolh - Department of Economics - University of Vermont - 94, University Place - Old Mill, O¢ ce 231 - Burlington, VT 05405. Email: [email protected]. Tel: 1 (802) 656 0946. I am especially grateful to Roger Gordon. This paper greatly beneted from his detailed feedback. Thanks also goes to the participants of the Econometric Society European Meeting (ESEM 2008). I am also grateful to Frank Caliendo, Julie Cullen, Arnaud Costinot, Margie Flavin, Nora Gordon, Kjetil Storesletten, Irina Telyukova, as well as the seminar participants at UCSD and UVM. I also thank Pierre-Yves Henin and Antoine dAutume for their ongoing support. 1

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Page 1: Optimal Taxation Over the Life-Cycle For the Poor and the Richnmathieu/OTPRonline.pdf · The answer of the past literature is striking: In their seminal papers, Chamley (1985, 1986)

Optimal TaxationOver the Life-Cycle

For the Poor and the Rich

Nathalie Mathieu-Bolh�

September 9, 2009

Abstract

This paper is a theoretical contribution to the Ramsey optimal taxation literature.In a new life-cycle model, I �nd that when individuals face di¤erent borrowing con-straints, the progressivity of optimal labor income taxation is limited. Poor and richindividuals generally face two optimal capital income tax regimes involving redistribu-tion across income categories. Optimal capital income taxation generally di¤ers fromzero even when preferences are additively separable. The conditions for zero optimalcapital income taxation are more stringent when borrowing constraints are tight. Op-timal capital income taxation paid by the rich is lower if their weight in the populationis larger.JEL: E61, E62, H21, H23

�Nathalie Mathieu-Bolh - Department of Economics - University of Vermont - 94, University Place -Old Mill, O¢ ce 231 - Burlington, VT 05405. Email: [email protected]. Tel: 1 (802) 656 0946. I amespecially grateful to Roger Gordon. This paper greatly bene�ted from his detailed feedback. Thanks alsogoes to the participants of the Econometric Society European Meeting (ESEM 2008). I am also gratefulto Frank Caliendo, Julie Cullen, Arnaud Costinot, Margie Flavin, Nora Gordon, Kjetil Storesletten, IrinaTelyukova, as well as the seminar participants at UCSD and UVM. I also thank Pierre-Yves Henin andAntoine d�Autume for their ongoing support.

1

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Introduction

How should a stream of exogenous government expenditure be �nanced using capital and

labor income taxation? The answer of the past literature is striking: In their seminal papers,

Chamley (1985, 1986) and Judd (1985,1987) �nd that capital income taxation should equal

zero in the long run and should not be used for redistribution. My contribution builds

upon two main papers that have challenged these results. In a simple life cycle model with

endogenous labor supply, Erosa and Gervais (2002) show that, as consumption increases with

age, it is generally optimal to use age-dependent capital income taxation in the steady state.

More precisely, they show that capital income taxation di¤ers from zero when the elasticity

of the marginal utility of consumption is not constant in the steady state. With a �at labor

productivity age-pro�le, the limitation to this result is that capital income taxation should

remain equal to zero in the widely used case of additively separable preferences. Conesa et

al. (2007, 2009) quantitatively characterize optimal capital income taxation in a model with

uninsurable income shocks and permanent productivity di¤erences of households. They �nd

that the life-cycle model generates a high positive capital income tax and that insurance and

redistribution are achieved with a progressive labor income tax. When labor is endogenously

supplied and the labor income tax system depends on age, they �nd that optimal capital

income taxation equals zero when preferences are additively separable between consumption

and leisure.

I propose to study Ramsey optimal taxation in a new life-cycle model with endogenous

labor supply. In the proposed framework, individuals have di¤erent income age-pro�les,

borrowing constraints and life expectancies. The focus of this paper is theoretical. I de-

termine the conditions for non zero capital income taxation in the steady state and its

redistribution implications. I study two types of borrowing constraints (from the most to

the least stringent) as well as the role of di¤erent life horizons. My results di¤er from previ-

ous contributions and are summarized as follows. When individuals face di¤erent borrowing

constraints, the progressivity of optimal labor income taxation is limited. Poor and rich

individuals generally face two optimal capital income tax regimes involving redistribution

across income categories. Optimal capital income taxation generally di¤ers from zero even

when preferences are additively separable. The conditions for zero optimal capital income

taxation are more stringent when borrowing constraints are tight. Optimal capital income

taxation paid by the rich is lower if their weight in the population is larger.

2

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In the Yaari (1965)-Blanchard (1985) overlapping generations model, I introduce two

categories of agents, the rich and the poor. The rich are born with no assets on a high

income path and have a long life expectancy. The poor are born with no assets, on a low

income path and face a shorter life expectancy. I study the case of "exogenous" borrowing

constraints as in Deaton (1991). The exogenous borrowing constraint is the most stringent

since it is a no-borrowing limit. This no-borrowing limit has di¤erent implications for the

rich and the poor. With an interest rate superior to the discount rate, the rich individual

wishes to accumulate wealth. Since it is not optimal for the rich individual to borrow, the

exogenous borrowing constraint is non binding. As a result, she consumes a portion of their

lifetime income. By contrast, for the poor individual, desired consumption exceeds current

income. Therefore, she does not accumulate wealth and faces a binding exogenous borrowing

constraint. As a result, the poor consumes her disposable income. The model presupposes

an annuity market like in Yaari (1965) and Blanchard (1985). In case individuals die holding

wealth, their wealth goes to a non pro�t mutual fund. They get a payout during their lifetime

in exchange for returning their wealth to the mutual fund contingent on their death.

A justi�cation for the assumptions of the model regarding the no-borrowing limit can be

found in Aghion et al.(1999). The authors show that the amount lenders are prepared to lend

is capped by an amount proportional to the borrower�s wealth. The existence of borrowing

limits comes from the fact that some borrowers may chose not to repay their loans. In my

model, the poor have zero wealth and therefore face a binding no-borrowing limit. According

to the same logic, the rich who accumulate wealth face a non-binding no-borrowing limit.

In addition, the proposed model is consistent with recent empirical results by Crook and

Hochguertel (2007) about credit constraints in the US showing that wealth reduces the

chance of being credit constrained. The assumption that a rich individual has a longer life

expectancy than a poor individual has been documented by Waldron (2007) who relates

life expectancy with average earnings. In the model, in the absence of income uncertainty,

precautionary saving is ruled out. Feigenbaum (2007) shows that in a �nite horizon general

equilibriummodel, the contribution of precautionary saving to the capital stock is negligeable

in the presence of exogenous borrowing constraints1. However, there is life-cycle saving by

the rich. The labor income age pro�le is hump-shaped, re�ecting the CPS data on labor

1Feigenbaum (2007) also shows that precautionary saving contributes more signi�cantly to the aggregatecapital stock when borrowing constraints are endogenous.

3

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income2. While the increasing part of the income pro�le and the uncertainty about death

increases aggregate consumption, the steep decreasing part of the income pro�le increases

aggregate saving. Considering that the latter e¤ect dominates, accumulated aggregate wealth

compensates for the expected decline in lifetime labor income3.

In this framework, I determine the �second best� (Ramsey) optimal taxation scheme,

assuming commitment by the government to rule out time inconsistency. Age-dependent

labor and capital income taxes are raised to �nance an exogenous public spending sequence.

Once optimal tax rates are determined, the government adjusts the level of debt to satisfy

the period budget constraint. The government chooses the set of labor and capital income

taxes that maximizes a social welfare function taking into consideration the rich and the

poor�s well-being. Because the government cares about the poor, the tax combination that

maximizes e¢ ciency also involves redistribution. I solve the model combining the primal

approach inspired by Atkinson and Stiglitz (1972, 1980) and the two-step method by Calvo

and Obsfeld (1988) for the perpetual youth model. I write the analytical expressions for

optimal labor and capital income taxes to determine in which environment capital income

taxation should be used.

After having considered exogenous borrowing constraints, I determine optimal tax rates

for an alternative scenario in which the poor and the rich face endogenous borrowing con-

straints, described as "natural" debt limits by Aiyagari (1994). In this scenario, individuals

cannot borrow more than their own human wealth. With the interest rate superior to the dis-

count rate, individuals choose not to borrow. Therefore, the exogenous borrowing constraint

does not bind for the poor and the rich. Various scenarios regarding the life horizon are

also studied. First, I determine optimal tax rates in the case when poor and rich individuals

have the same �nite horizon. Second, I consider that the poor and the rich have di¤erent

�nite horizons. I study the e¤ect on optimal capital income taxation of an increase in the life

horizon of the rich relatively to the life horizon of the poor, which results in increasing the

weight of the rich in the population. In addition, I study the case when the life expectancy

of the poor equals zero. In that scenario, the model becomes a simple life cycle model and

the results are consistent with Erosa and Gervais (2002). Finally, I study the case when

the poor and the rich have an in�nite horizon, which provides results consistent with Judd

2Blanchard (1985) introduces a decreasing income function to re�ect retirement.3See Blanchard (1985) and online appendix at http://www.uvm.edu/~nmathieu for a detailed presenta-

tion of the dynamics of aggregate consumption.

4

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(1985).

This paper provides several new theoretical results. When the poor and the rich face

di¤erent borrowing constraints, this tends to limit the progressivity of optimal labor income

taxation across income categories. The conditions for zero capital income taxation are more

restrictive than in previous contributions. When preferences are additively separable between

consumption and leisure, optimal capital income taxation equals zero in the long run only if

poor and rich individuals also have the same marginal utility of consumption and the same

horizon. Poor and rich individuals generally face two optimal capital income tax regimes

involving redistribution across income categories. When the poor and the rich face di¤erent

endogenous borrowing constraints, the conditions for zero optimal capital income taxation

remain more restrictive than in the simple life-cycle model but less restrictive than in the

case with exogenous borrowing constraints. In the case of logarithmic preferences and same

life expectancy, when the di¤erences in income and borrowing constraints result in a lower

consumption age pro�le of the poor compared to the rich, optimal capital income taxation

redistributes to the poor and the young. The larger the weight of the rich in the population,

the lower the optimal capital income tax paid by the rich and the less capital income taxation

redistributes to the poor.

The paper is organized as follows. In the �rst section, I review the literature. In the sec-

ond section, the formal framework is presented. In the third section, I present the theoretical

results.

1 Literature review

The results of the Ramsey benchmark literature are largely determined by the choice of an

in�nite horizon4. In those models, only distortionary tax instruments are available to the

government. In equilibrium, the market economy reaches a second best Pareto optimum.

As a result, capital income taxation cannot be used for improving e¢ ciency. The literature

about optimal taxation has developed in various directions to challenge the result of zero

capital income taxation. This review exclusively focuses on two types of contributions in

the Ramsey optimal taxation literature, simple overlapping generations (OLG) models or

4In�nite horizon can arise due to an in�nite lifetime for the representative agent or an in�nite horizon foraltruistic families.

5

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models incorporating a borrowing constraint.5.

In the dynamic framework of OLG models, we can distinguish between two types of

results. In one (see e.g. Ambler (1999)), derived from the Diamond (1965) model, the market

economy can accumulate too much capital compared to the golden rule. This yields a steady

state equilibrium that is not Pareto optimal. Taxing capital income can therefore improve

e¢ ciency in the �rst place. In the other (see e.g. Erosa and Gervais (2002), Garriga (2003)

or Mathieu-Bolh (2006)), agents work their entire life and save6. As a result, the economy

accumulates capital and reaches a Pareto e¢ cient equilibrium. Capital income taxation

cannot be used to improve e¢ ciency. However, if the government takes into consideration

the well-being of current and future generations, it has a motive to postpone taxation to

the future to some extent. Therefore, using age-dependent taxes, the government should tax

individuals di¤erently according to (the elasticity of) their marginal utility of consumption

in the steady state, which entails redistribution across age cohorts. With a constant labor

productivity age-pro�le, the limitation to this result is that capital income taxation should

remain equal to zero in the widely used case of additively separable preferences (Atkeson et

al. (1999)).

At �rst, the introduction of a borrowing constraint in in�nite horizon models seems to

prove the robustness of the traditional results (Judd (1985) and Mankiw (2000)). Capital

income taxation is undesirable in the long run because it decreases the real wage. Workers

prefer the static distortion of marginal labor income taxes to the cumulative distortion of

the capital income taxes on intertemporal margins. On the contrary, Aiyagari (1995) and

Chamley (2001) �nd that it is optimal to tax capital income in the long run. In both models,

agents are optimizing over an in�nite horizon and face random income shocks. Insurance

markets are incomplete and agents self-insure against those shocks. As a consequence, they

have precautionary saving. If the market economy as a result accumulates too much capital

compared to the golden rule, the optimal capital income tax is positive.

Conesa et al.(2007, 2009) quantitatively characterize capital income taxation in a model

with uninsurable income shocks and permanent productivity di¤erences of households. First,

5Other recent contributions include other features such as human capital or public good provision in theOLG framework. They �nd that capital income taxation is generally positive in the long run (Pirttila andTuomala [2001], Echevaria and Iza [2000], Bovenberg and Van Ewijik [1997]).The positive approach to taxation also suggests that capital income taxation should di¤er from zero when

markets are incomplete (Hubbard and Judd (1986), Imrohoroglu (1998)).6The interest rate is superior to their discount rate.

6

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they �nd that capital income taxation is positive and the high capital income tax relates

to the life-cycle structure of the model. In a life-cycle model with endogenous labor supply,

it is optimal to tax labor at di¤erent ages at di¤erent rates. If the government does not

have access to an age-dependent labor income tax, a positive capital income tax can achieve

the same as a progressive labor income tax. Second, they �nd that market incompleteness

and distributional concerns determine the optimal progressivity of the labor income tax.

However, when labor is endogenously supplied and the labor income tax system depends

on age, they �nd that capital income taxation equals zero when preferences are additively

separable between consumption and leisure, and that redistribution is achieved by labor

income taxation.

Thus, my paper extends the study of Ramsey optimal taxation in life-cycle models with

di¤erences in productivity and incomplete markets in the following way. First, the focus of

the paper is theoretical. Second, contrary to previous contributions individuals with di¤erent

income face di¤erent borrowing constraints and life expectancies. The paper provides new

results that are discussed with respect to two types of borrowing constraints, from the most

to the least stringent. The role of di¤erent life horizons is also studied.

2 The formal framework

2.1 The economy

2.1.1 Population structure

In this model, time is continuous. There are two types of individuals, poor and rich. To

normalize the size of the total population to one, I assume that the cohort size of poor

individuals born each instant equals 1�P+�R

: The poor face a constant rate of death, 1�P. On

a date t; the size of a cohort of poor individuals is 1�P+�R

e� 1�P(t�s)

: Integrating on the date

of birth since �1, the size of the population of the poor on a date t is:Z t

�1

1

�P + �Re� 1�P(t�s)

ds =�P

�P + �R

Each instant, a cohort of size 1�P+�R

of rich individuals is born. The rich face a con-

stant rate of death, 1�R: Therefore, on a date t, the size of a cohort of rich individuals is

1�P+�R

e� 1�R(t�s)

: As a result, the size of the population of the rich is:Z t

�1

1

�P + �Re� 1�R(t�s)

ds =�R

�P + �R

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The size of the entire population is therefore equal to:

�P�P + �R

+�R

�P + �R= 1

By assumption, each cohort is an in�nitely divisible continuum of agents, such that

each cohort of poor agents decreases at a constant rate 1�Pand each cohort of rich agents

decreases at a constant rate 1�R: There is, therefore, no aggregate uncertainty and the size of

the population of the poor and the rich is constant.

Whereas the poor hold zero wealth, the rich accumulate wealth. As a result, consumption

by the rich tends to increase with age. Since the length of life is uncertain, the rich leave

unintentional positive bequests. Following Yaari (1965) and Blanchard (1985), I introduce a

perfect annuity market: Each instant a non-pro�t mutual fund collects the �nancial wealth

(WR) from each deceased rich individual. Since a proportion 1�Rof the rich die, the mutual

fund collects 1�RWR from the deceased. At the same time, the mutual fund pays 1

�RWR to

the rich currently alive. The rich get this payout in exchange for returning their wealth to

the mutual fund contingent on their death7.

2.1.2 The behavior of consumers

The individual behavior of the poor and the rich can be described by the following optimiza-

tion program. The utility of an individual of type i (i = P for poor or R for rich) equals

U [Ci (s; z) ; Li (s; z)]: The variable Ci(s; z) denotes consumption on date z of an individual

born on date s: Individuals are endowed with one unit of time, that they share between work

N i (s; z) and leisure, Li (s; z). The objective of an individual is8:

maxCi(s;z);N i(s;z);W (s;z)

Z +1

t

e���+ 1

�i

�(z�t)

U [Ci (s; z) ; Li (s; z)]dz

7By assumption, the unintended bequests do not go to the poor. This assumption is consistent with the�ndings by Gist and Figueiredo (2006). Inheritance size is correlated with net worth.In the scenario when individuals face natural borrowing constraint, they could also leave a negative wealth

to the mutual fund. Then, they would have to pay an insurance premium during their lifetime.8Time until death has an exponential distribution, therefore:

E

Z +1

t

e��(z�t)U [Ci (s; z) ; Li (s; z)]dz

is equivalent to: Z +1

t

e�(�+�i)(z�t)U [Ci (s; z) ; Li (s; z)]dz

8

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st.:_W i (s; z) =

�1

�i+ r̂ (z)

�W i (s; z) + !̂i(z)Ei(z � s)N i (s; z)� Ci (s; z) (1)

and a borrowing constraint9:

W i(s; z) � �Di(s; z) (2)

The rate of time preference is �. Financial wealth W i (s; z), composed of government

bonds Bi (s; z) and shares of capital Ki (s; z) : Shares and bonds are perfect substitutes

and wealth provides an after-tax return equal to r̂(z) + 1�i, where r̂(z) is the after tax rate

(r̂(z) = r(z)(1 � T ik(z)); and1�iW i(s; z) represents the insurance payout. Individuals pay a

proportional labor-income tax T!(z); which is a fraction of their real wage ! (z) : The after

tax real wage, ! (z) (1 � T i!(z)); is simply denoted !̂i(z): It represents the remuneration of

one unit of e¤ective labor. The labor income age-pro�le is given by a function ~N i(z� s) (seesection 2.1.3.):

The borrowing limit is given by Di(s; z): Debt (�W i(s; z)) cannot exceed Di(s; z): De-

pending on how Di(s; z) is further de�ned, the borrowing constraint is more or less stringent.

The program is solved in a standard way. Given �i1 (s; z) ; the multiplier associated with

the budget constraint, and �i2 the multiplier associated with the borrowing constraint, the

optimality conditions are:

�i1 (s; z) = UCi(s;z)

Ei(z � s)!̂i(z) = �UN i(s;z)

UCi(s;z)(3)

_�i1 (s; z) ��� � r̂i (z)

��i1 (s; z) (4)

limz!+1

�i1 (s; z)Wi (s; z) = 0

�i2(Wi(s; z) +Di(s; z)) = 0 (5)

I �rst consider an exogenous borrowing constraint, Di(s; z) = 0: The exogenous borrow-

ing constraint states that individual�s wealth has to be positive or equal to zero. In this

scenario, the poor desires to consume more than her income and therefore does not save. By

assumption, the poor cannot borrow. Therefore, W P (s; z) = 0 and �P2 6= 0: As a result, thepoor consumes her entire disposable income. With the interest rate superior to the discount

9In the presence of a borrowing constraint, a Ponzi-Scheme is excluded. This explains the absence of theusual no-Ponzi game constraint ( lim

z!+1e�

R zt[r(v)(1�Tk(v)+�i]dvW i (s; z) = 0), which would be redundant.

9

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rate, the rich save, therefore WR (s; z) > 0 and �R2 = 0: For the rich, the no-borrowing

constraint is not binding and the Euler equation holds as an equality10.

Alternatively, I consider the endogenous borrowing constraint, Di(s; z) = H i(s; z). The

borrowing constraint states that debt (�W i(s; z)) cannot exceed human wealth, H i(s; z);

which is the present discounted value of future after-tax labor income.

H i(s; z) =

Z +1

t

e�R zt

hr̂i(v)+ 1

�i

idv!̂i(z)Ei (s; z)N i (s; z) dz (6)

Human wealth therefore represents the maximal amount that is feasible to repay at time

t when Ci(s; t) � 0: In the model, the endogenous borrowing constraint is generally more

stringent for the poor than the rich since the poor face a shorter life expectancy and a lower

labor income than the rich: With the interest rate superior to the discount rate, poor and

rich individuals wish to save. As a result, this constraint never binds (it would imply zero

consumption). Then, along an optimal path �i2 6= 0 and the Euler equation holds as an

equality.

2.1.3 The aggregate economy

Aggregate variables are deduced from individual behavior and from the aggregation rules

derived from the population�s structure. For the poor, the link between an individual variable

XP (s; t) and an aggregate variable XP (t) is:

XP (t) =

Z t

�1

1

�P + �Re� 1�P(t�s)

XP (s; t) ds (7)

For the rich , the link between an individual variable XR (s; t) and an aggregate variable

XR (t) is:

XR (t) =

Z t

�1

1

�P + �Re� 1�R(t�s)

XR (s; t) ds (8)

For the entire population, aggregate variable X is:

X(t) = XP (t) +XR (t) (9)

In order to aggregate the model and provide an analytical solution, I need to make

the distribution of labor income across ages explicit, with "relative labor income functions"

10At the individual level, because of the annuity market for life insurance, the consumption-saving in-tertemporal choice does not depend on 1

�R(the rich accumulate wealth at a rate 1

�R+ r̂ (z) and their

discount rate depends on � + 1�R):

10

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assuming exponential functional forms11. The chosen function aims at reproducing the hump

shape of labor income over the life cycle. Since the after-tax real wage is independent of

age, the distribution of labor income is identical to the distribution of e¢ cient labor. The

variable ~N i(s; t) denotes e¢ cient labor on a date t; for an individual born in s:

~N i(s; t) = Ei(s; t)N i(s; t) (10)

The distribution of the e¢ cient labor is given by12:

~N i(s; t) =�aie

�i(t�s) � e�i(t�s)�Ei(t)N i(t) (11)

Aggregate e¢ cient labor is Ei(t)N i(t). The parameters of the exponential functional

forms are chosen to re�ect that the labor income of the poor is lower than labor income

of the rich and that labor income is hump-shaped (see Appendix A). Those functions are

used to derive aggregate labor supply (Appendix A) and the dynamic equation for human

wealth. The equations for aggregate consumption, �nancial and human wealth are presented

in Appendix B.

Firms produce Y (t); using aggregate capital K (t) and e¤ective labor

E(t)N (t) : The production function is �Cobb-Douglas�,

Y (t) = F [K (t) ; E(t)N (t)] = [K (t)]� [E(t)N (t)]1�� : At each instant, �rms maximize

their instantaneous pro�t. Assuming perfect competition, production factors (capital and

e¤ective labor) are paid their marginal product (respectively !(z) and r(z)+d). Investment

is described by: I (t) = _K(t) + dK(t), where d is the rate of capital depreciation (d 2 [0; 1]).

The government �nances exogenous public spending G(t) by means of taxes T (t) and

debt B (t). For any value of the tax rates, the debt level adjusts to satisfy the dynamic

budget constraint:_B (t) = r̂ (t)B (t) +G (t)� T (t) (12)

11See Blanchard (1985). I use slightly di¤erent exponantial functions which enables me to re�ect thehump shape of the labor income function, whereas Blanchard and Fisher assumed a decreasing relative laborincome.12The labor supply choice being endogenous, it implies that e¢ ciency per unit of labor is deduced from:

Ei(s; t) =haie

�i(t�s) � e�i(t�s)iEi(t)N i(t)=N i(s; t)

11

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with:

T (t) = T! (t)! (t)E(t)N (t) + Tk (t) r (t)K (t)

The model is closed with the market clearing condition:

_K (t) =�F

0

K (t)� d�K (t) + F

0

EN (t)E(t)N(t)� C (t)�G (t) (13)

2.2 The government

Once optimal capital and labor income tax rates are chosen to �nance the exogenous gov-

ernment spending. The government cannot use lump-sum taxation but capital and labor

income taxation are not restricted in so far as they are age-dependent. There are no transfers

programs, the redistribution e¤ects are a consequence of the optimal taxation pattern.

2.2.1 Objective

In a �nite horizon model, the government has an incentive to levy a one-time tax on the

wealth of agents alive. By doing that, the government could eliminate distortions and reach

a �rst best Pareto optimum. However, such a policy would have two major inconveniences.

First it could be considered unfair that generations currently alive pay all taxes while the

future generations bene�t from the elimination of taxation. So, it is reasonable to assume

that the government sets an objective that takes into consideration the well-being of current

generations as well as future generations. As a consequence, I distinguish between two

elements in the government�s objective. The �rst element takes into consideration the well-

being of agents from each cohort born in s < t (t is the initial date of the �scal plan). It

is equal to the integral between �1 and t of the individual well-being of each cohort alive

on date t. The second element takes into consideration the well being of an agent from each

cohort born in s > t: It is equal to the integral from t to +1 of the individual well-being of

each cohort to be born after date t: The extent to which the government wishes to redistribute

welfare from current to future generations is given by the intergenerational discount factor,

�: Second, the policy would be time inconsistent. In the social welfare presented below, I

eliminate time inconsistency related to social preferences13. The well-being of agents alive

and to be born is discounted at the same discount rate � and with respect to their date of

birth. Agents alive and to be born are therefore treated symmetrically in the government�s

13See Calvo and Obstfeld [1988] for more details. This is only a necessary (not a su¢ cient) condition fortime consistency. This is why I have presuposed that the governement is committed to its policy.

12

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criteria. Furthermore, I assume that the government puts the same weight on the well-being

of the poor and the rich, therefore the government�s objective is:

SW1(t) =

�Z t

�1e��(s�t)IWR

1 (s; t) ds+

Z +1

t

e��(s�t)IWR1 (s; s) ds

�+

�Z t

�1e��(s�t)IW P (s; t)ds+

Z +1

t

e��(s�t)IW P (s; s)ds

�with individual well-being functions (IW ) simply re�ecting the intertemporal utility of each

category of agents.

The well-being function for an individual alive on date t is:

IW i(s; t) =

Z +1

t

e���+ 1

�i

�(z�s)

U�Ci(s; z); Li(s; z)

�dz

For an individual to be "born" on a date s > t; the well-being function is:

IW i(s; s) =

Z +1

s

e���+ 1

�i

�(z�s)

U�Ci(s; z); Li(s; z)

�dz

2.2.2 The primal approach

According to the primal approach, I reformulate the government�s problem of choosing op-

timal taxes as a problem of choosing optimal allocations, called the "Ramsey problem". In

the usual formulation of the problem, the optimal allocations satisfy the "implementability"

constraint and the "feasibility" constraint. In this paper, the optimal allocations also need to

satisfy the additional borrowing constraint. The implementability constraints are simply the

intertemporal budget constraints of respectively the poor and the rich. Each intertemporal

budget constraint is re-written using the optimality conditions of respectively the poor and

the rich to replace after-tax prices by allocations. In addition, for each category of individ-

uals, I distinguish between two types of implementability constraints. The �rst is faced by

an individual born before t, the initial date of the plan. It takes into consideration W (s; t);

the wealth accumulated by the individual since birth. The second is faced by an individual

born after the initial date of the plan. It takes into consideration W (s; s) wealth at the time

of his birth, which by assumption equals zero. The feasibility constraint is the aggregate

market clearing condition. The government budget constraint is satis�ed by Walras law.

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De�nition An allocation ffC(s; z); N(s; z)g+1s=�1; K(z)g+1z=t ; is"implementable" for the government if:

1)�fCi(s; z); N i(s; z);W i(s; z)g+1s=�1

+1z=t

satis�es the borrowing constraint:

W i(s; z) � �Di (s; z)

2)�fCi(s; z); N i(s; z);W i(s; z)g+1s=�1

+1z=t

satis�es the implementability constraint:

For an agent born on a date s < t :Z +1

t

e���+ 1

�i

�(z�s) �

UCi(s;z)Ci (s; z) + UN i(s;z)N

i (s; z)�dz = UCi(s;t)W

i (s; t)

For an agent born on a date s > t :Z +1

s

e���+ 1

�i

�(z�s) �

UCi(s;z)Ci (s; z) + UN i(s;z)N

i (s; z)�dz = UCi(s;s)W

i (s; s)

if W i(s; z) > �Di (s; z) :

If W P (s; z) = 0; the poor�s implementability constraint is replaced by the following static

constraint14:

CP (z � n; z) = �UNP (z�n;z)

UCP (z�n;z)NP (z � n; z)

3) ffC(s; z); N(s; z)g+1s=�1; K(z)g+1z=t satis�es the feasibility constraint:

_K (z) =�F

0

K (t)� d�K (t) + F

0

EN (t)E(t)N(t)� C (z)�G (z)

For convenience, using the primal approach, individual well-being can be re-written to

include the implementability constraint. Given e�(�+1�R)(z�s)

�i(s); the multiplier associated

with the implementability constraint of the generation born at time s; the individual well-

being function for an individual of type i born on date s < t becomes15:

IW i(s; t) =

Z +1

t

e���+ 1

�i

�(z�s) �

�(Ci (s; z) ; Li (s; z))�dz

14In the case of a no borrowing constraint, the constraint is binding only for the poor. This constraintdenotes the fact that the poor consume their disposable income: CP (z�n; z) = !̂P (z)EP (z�s)NP (z�n; z)and at the optimum: !̂P (z)EP (z � s) = �UNP (z�n;z)

UCP (z�n;z):

15I give a general expression of the allocations and tax rates on all dates, including the initial date t: Ihave incorporated the term e

(�� 1�i)(z�t)

UCi(s;t)Wi(s; t) in �: This term re�ects the fact that agents born

prior to t have accumulated wealth. The state of the economy follows a continuous time Markov process,with a unique and invariant probability density. Consequently, the allocation rules on all dates after t aretime invariant. On a date t, the allocation rules include terms relative to the �nancial wealth W i(s; t) ofagents born prior to that date. After date t; the �rst order conditions do not include this term.

14

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where :

�(Ci (s; z) ; Li (s; z)) =

264 (U(Ci (s; z) ; Li (s; z))

+�i(s)

UCi(s;z)C

i (s; z) + UN i(s;z)Ni (s; z)

�e(��1�i)(z�t)

UCi(s;t)Wi(s; t)

! 375For an individual to be born on a date s > t the individual well-being function is:

IW i(s; s) =

Z +1

s

e���+ 1

�i

�(z�s) �

�(Ci (s; z) ; Li (s; z))�dz

where : �(Ci (s; z) ; Li (s; z)) =�

(U(Ci (s; z) ; Li (s; z))+�i(s)

�UCi(s;z)C

i (s; z) + UN i(s;z)Ni (s; z)

� �since W i(s; s) = 0:

Using the de�nitions of the individual welfare constraints, and switching integrals, I

rewrite the government�s objective as:

SW (t) =

Z +1

t

e��(z�t)

(Z z

s=�1e�(z�s)

"e���+ 1

�R

�(z�s)

��CR (s; z) ; LR (s; z)

�+e

�(�+ 1�P)(z�s)

��CP (s; z); LP (s; z)

�#ds

)dz

for z = [t; +1[ and � � � < � + 1�R:

To simplify and make the discussion more intuitive, I set t = 0 and think in terms of age

n = z � s and time z. Then, the government�s objective becomes:

SW (0) =

Z +1

0

e��z

(Z +1

0

e(���)n

"e� 1�Rn�(CR (z � n; z) ; LR (z � n; z))

+e� 1�P�n��CP (z � n; z); LP (z � n; z)

� # dn) dz(14)

2.2.3 Solution

Solution to the Ramsey Problem:

The government maximizes the social welfare criteria (14) subject to the borrowing con-

straint and the feasibility constraints (the implementability constraint is included in the

objective), given K(0) and with K(z) � 0 for all z: The problem can be divided into 2

sub-problems. First, the government solves a static problem: On a date z = [0;+1); thegovernment optimally allocates a level of aggregate consumption and labor between indi-

viduals of all ages (alive and to be born), in order to maximize their instantaneous utility.

Second, the government solves a dynamic problem: The government chooses an optimal al-

location path fC (z) ; N (z) ; K(z)g+1z=0 , that maximize its objective subject to the feasibility

15

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constraints (given that on all dates z > 0; the aggregate consumption and leisure allocations

are optimally spread across ages).

The static problem:

V (C(z); L(z)) =

maxfC(z�n;z);N(z�n;z)g+1n=0

R +10

"e� 1�Rn�(CR (z � n; z) ; LR (z � n; z) )

+e� 1�P�n�(CP (z � n; z) ; LP (z � n; z))

#e(���)ndn

st.:

C (z) =

Z +1

0

�1

�P + �Re� 1�RnCR (z � n; z) +

1

�P + �Re� 1�PnCP (z � n; z)

�dn (15)

N (z) =

Z +1

0

�1

�P + �Re� 1�RnNR (z � n; z) +

1

�P + �Re� 1�PnNP (z � n; z)

�dn (16)

W i(z � n; z) � �Di(s; z) (17)

The static problem is solved in a standard way. Given 1(z), 2(z); i3(z) the multipliers

associated respectively with (15), (16), and (17) the optimality conditions for all n and z > 0

are: �e� 1�Rn�CR(z�n;z) + e

� 1�Pn�CP (z�n;z)

�e(���)n=

1(z)

�P+�R

�e� 1�Rn+ e

� 1�Pn�

(18)�e� 1�Rn�NR(z�n;z) + e

� 1�Pn�NP (z�n;z)

�e(���)n= � 2(z)

�P+�R

�e� 1�Rn+ e

� 1�Pn�

(19)

i3(z)(Wi(s; z) +Di(s; z)) = 0 (20)

The dynamic problem :

maxffC(z);N(z)g;K(z)g+1z=0

Z +1

z=0

e��zV (C(z); L(z))dz

st.:_K (z) = (F

0

K (z)� d)K (z) + F0

EN (z)E(z)N(z)� C (z)�G (z) (21)

with K(0) given and K(z) � 0 for all z:I use the result by Benveniste and Scheinkman (1979). The V (�) function is assumed

to be strictly concave and can be considered as the value function of a dynamic problem in

which z represents time. V (�) is continuously di¤erentiable in C(z) and N(z), and at theoptimum:

VC(z) = �CR(z;z) + �CP (z;z)

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VN(z) = �NR(z;z) + �NP (z;z)

for z > 0.

Given (z) the multiplier associated with the feasibility constraint, the necessary condi-

tions of the dynamic problem are therefore (for z > 0):

VC(z) = (z)

VN(z) = � (z)E(z)!(z)

_ (z) = (�� r (z)) (z) (22)

The implementability constraint incorporates the transversality condition for each indi-

vidual. I have introduced the implementability constraint into the government�s objective.

Therefore, the transversality condition is veri�ed at the aggregate level16.

Optimal tax rates:

Once the government problem is solved, it is possible to �nd the expressions for the

optimal capital and labor income tax rates after date zero. Optimal capital income taxation

for the rich (TRk ) is deduced from the di¤erence between the rich individual�s marginal rates

of substitution (MRS) and the social MRS between current and future consumption:

� � r̂R(z � n; z)� (�� r (z)) =_UCR(z�n;z)UCR(z�n;z)

�_VC(z�n;z)VC(z�n;z)

(26)

When the poor face an endogenous borrowing constraint, optimal capital income taxation

for the poor is deduced from the di¤erence between the poor individual�s MRS and the social

MRS between current and future consumption:

� � r̂P (z � n; z)� (�� r (z)) =_UCP (z�n;z)UCP (z�n;z)

�_VC(z�n;z)VC(z�n;z)

16When the borrowing constraint is exogenous, P3 (z) 6= 0 and R3 (z) = 0. The static problem can berewritten as: �

e� 1�R

n�CR(z�n;z) + e

� 1�P

nUCP (z�n;z)

�e(���)n =

1(z)

�P + �R

�e� 1�R

n+ e

� 1�P

�n�

(23)

�e� 1�R

n�NR(z�n;z) + e

� 1�P

nUNP (z�n;z)

�e(���)n = � 2(z)

�P + �R

�e� 1�R

n+ e

� 1�P

n�

(24)

CP (z � n; z) = �UNP (z�n;z)

UCP (z�n;z)NP (z � n; z) WR > 0 (25)

When the borrowing constraint is endogenous, i3(z) = 0:

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Optimal labor income taxes (T P! and TR! ) are deduced from the ratio of individual and

social MRS between consumption and labor:

T P! (z � n; z) = 1�

UNP (z�n;z)UCP (z�n;z)

=EP (n)

VN(z�n;z)VC(z�n;z)

=E(n)(27)

TR! (z � n; z) = 1�

UNR(z�n;z)UCR(z�n;z)

=ER(n)

VN(z�n;z)VC(z�n;z)

=E(n)(28)

3 Results

The focus of this discussion is the steady state. In a �nite horizon framework, in the steady

state, the government allocates aggregate consumption and labor optimally between individ-

uals of all ages n: Since choosing optimal allocations is equivalent to choosing optimal tax

rates on capital and labor income, the expressions below provide steady state age-dependent

tax rates.

3.1 Redistributive labor income taxation

The labor income taxes are written in the steady state:

T P! (n) = 1�UNP (n)

UCP (n)

=EP (n)

(n)�NR(n)

+(1� (n))�NP (n)

(n)�CR(n)

+(1� (n))�CP (n)

=E(n)(29)

TR! (n) = 1�UNR(n)

UCR(n)

=ER(n)

(n)�NR(n)

+(1� (n))�NP (n)

(n)�CR(n)

+(1� (n))�CP (n)

=E(n)(30)

Proposition 1: When the poor and the rich face di¤erent borrowing constraints, this tends

to limit the progressivity of optimal labor income taxation across income categories.

Proof

From (29) and (30), it is straightforward that if:

UNP (n)

UCP (n)=EP (n) >

UNR(n)

UCR(n)=ER(n)

or equivalently if:UNP (n)

UNR(n)

ER(n)

EP (n)>UCP (n)UCR(n)

(31)

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then:

T P! (n) < TR! (n)

If (31) holds, labor income taxation is progressive across income categories. Consumption

is a function of human and �nancial wealth. Two main elements in�uence human wealth:

lifetime expectancies and income pro�les. As a consequence, other things equal, the rich

have a larger human wealth than the poor. Asset accumulation is in�uenced by borrowing

constraints. Therefore, given some income pro�les for the poor and the rich, the more

stringent the borrowing constraint on the poor compared to the rich, the larger the di¤erence

between their consumptions. As a result, the largerUCP (n)

UCR(n)

and the less likely (31) holds.

If (31) does not hold, labor income taxation is not progressive across income categories. If

age-dependent labor income taxation does not redistribute to the poor, then age-dependent

capital income taxation can be used to redistribute to the poor.

3.2 Non zero capital income taxation and redistribution conse-quences

When the borrowing constraint is exogenous and binding for the poor, from (26), steady

state capital income taxation is deduced from:

r(n)� r̂R(n) =@U

CR(n)@n

UCR(n)

�@@n [ (n)�CR(n)+(1� (n))UCP (n)] (n)�

CR(n)+(1� (n))U

CP (n)

(32)

where@U

CR(n)@n

UCR(n)

is the individual MRS between current and future consumption; and@ (n)@n ( (n)�CR(n)�(1� (n))�CP (n)) (n)�

CR(n)+(1� (n))�

CP (n)is the social MRS between current and future consumption.

The weights of �CR(n) and �CP (n) in the government�s optimality conditions are denoted

(n) and 1 � (n): They depend on the respective horizons 1�Pand 1

�Rof the poor and

the rich17. Note that if the poor and the rich have the same horizon, �R = �P . Then,

(n) = 1� (n) = 50%:

The variable �CR(n) is re-expressed as follows:

�CR(n) =�1 + �R + �R�C

R(n)�UCR(n)

with �CR(n) the general equilibrium elasticity of the rich:

�CR(n) =

UCR(n)CR(n)CR (n) + ULR(n)CR(n)L

R (n)

UCR(n)

17 (n) = e� 1�R

n

e� 1�R

n+e

� 1�P

nand 1� (n) = e

� 1�R

n

e� 1�R

n+e

� 1�P

n

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Expression (32) has several ingredients indicating why the social MRS generally di¤ers

from the private MRS between current and future consumption and capital income taxation

may di¤er from zero. These ingredients are the poor and the rich individuals�marginal

utilities of consumption, UCP (n) and UCR(n), their horizons, �P and �R; which determine

the weights (n) and 1� (n) and the general equilibrium elasticity �CR(n).

The following proposition and its corollary clarify the conditions for non zero capital

income taxation.

Proposition 2: When the poor face a binding exogenous borrowing constraint, optimal

age-dependent capital income taxation di¤ers from zero, if one of the following su¢ cient

(but not necessary) conditions is veri�ed:

P1: UCP (n) 6= UCR(n).

P2: �P 6= �R and preferences are non logarithmic.

P3: Preferences are not additively separable between consumption and leisure.

Proof

- To show that (P1) is su¢ cient for optimal capital income taxation to be di¤erent from

zero, I show that (32) di¤ers from zero when (P1) is veri�ed but (P2) and (P3) are not. I

set UCP (n) 6= UCR(n); �P = �R; and logarithmic preferences (�CR(n) is constant and equal

to �1): Condition (P3) is not veri�ed since logarithmic preferences are additively separablebetween consumption and leisure. Equation (32) simpli�es and capital income taxation is

deduced from:

r(n)� r̂R(n) =@U

CR(n)@n

UCR(n)

�@U

CR(n)@n

+@U

CP (n)@n

UCR(n)

+UCP (n)

(33)

This equation shows that private MRS between current and future consumption is di¤er-

ent from the social MRS. Optimal capital income taxation is therefore di¤erent from zero.

The Proof that (P1) is su¢ cient for optimal capital income taxation to be di¤erent from

zero also proves that (P2), and non additively separable preferences (P3), are non necessary

to obtain TRk (n) 6= 0:- To show that (P2) is a su¢ cient for optimal capital income taxation to be di¤erent

from zero, I show that (32) di¤ers from zero when (P2) is veri�ed but (P1) and (P3) are

not. Setting �P 6= �R and non logarithmic preferences, UCP (n) = UCR(n); and additively

separable preferences (�CR(n) constant and equals ��CR 6= �1), equation (32) simpli�es and

20

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capital income taxation is deduced from:

r(n)� r̂R(n) =@ (n)@n

�R�1��CR

� (n)�R(1��CR )+1

(34)

This equation shows that optimal capital income taxation di¤ers from zero. The social

MRS depends on the poor and the rich respective horizons, which determine (n). However,

the di¤erent horizons of the poor and the rich matter only when the elasticity of substitution

of consumption di¤ers from one.

The proof that (P2) is su¢ cient for optimal capital income taxation to be di¤erent from

zero also proves that (P1) and (P3) are not necessary to obtain Tk(n) 6= 0:- To show that (P3) is a su¢ cient condition for non-zero optimal capital income taxation,

I show that (32) di¤ers from zero when (P3) is veri�ed but (P1) and (P2) are not. Setting

non additively separable preferences (�CR(n) non constant), UCP (n) = UCR(n) and �P = �R;

equation (32) simpli�es and capital income taxation is deduced from:

r(n)� r̂R(n) =�R @�C

R(n)

@n

�R(1+�CR(n))+2

(35)

Optimal capital income taxation is therefore di¤erent from zero. It depends on the form of

the rich individual�s preferences (which determine �CR(n)). Because of the binding exogenous

borrowing constraint, the form of the preferences of the poor is irrelevant since they cannot

reallocate consumption across periods.

The proof that (P3) is su¢ cient for optimal capital income taxation to be di¤erent from

zero also proves that (P1) and (P2) are not necessary to obtain Tk(n) 6= 0:

Corollary: When the poor face a binding exogenous borrowing constraint, optimal capital

income taxation equals zero if condition (C1) or (C2) is veri�ed:

C1: UCP (n) = UCR(n); preferences are separable additively between consumption and

leisure (�CR 6= 1) and �P = �R.

C2: UCP (n) = UCR(n) and preferences are logarithmic.

Proof

- The proof that optimal capital income taxation equals zero if (C1) is veri�ed can be

directly deduced from either (33), (34) or (35). According to equation (35), capital income

taxation di¤ers from zero when UCP (n) = UCR(n) and �P = �R; as long as preferences are

not additively separable between consumption and leisure. It is therefore straightforward

that with additively separable preferences, capital income taxation equals zero. In that case,

21

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the model is reduced to a life cycle model in which agents di¤er only by their age, labor

choices and preferences. According to equation (35), if preferences are additively separable,

then �CR(n) is constant, @

@n�C

R(n) = 0 and capital income taxation equals zero.

- The proof that optimal capital income taxation equals zero if (C2) is veri�ed can

be deduced from (34). According to (34), capital income taxation di¤ers from zero when

�P 6= �R; UCP (n) = UCR(n); and �CR(n) constant but di¤erent from �1 (non logarithmic

preferences). If preferences are logarithmic (�CR= 1); the numerator of (34) is zero. In that

case, capital income taxation is therefore equal to zero.

I now focus on the sign of capital income taxation which will determine redistribution

across income categories.

Proposition 3: When the poor face a binding exogenous borrowing constraint, only the rich

face capital income taxation. If the private MRS between current and future consumption is

larger than the social MRS, capital income taxation is positive and redistributes to the poor.

Proof

Since the poor individual�s desired consumption exceeds their current income, the poor

do not save. Since they cannot borrow, they consume their entire income each instant. As a

result the interest rate does not in�uence their consumption saving decision, it is therefore

not optimal to tax or subsidize the poor through capital income taxation. From (32), it is

straightforward that:

If:@U

CR(n)

@n

UCR(n)>

@@n

� (n)�CR(n) + (1� (n))�CP (n)

� (n)�CR(n) + (1� (n))�CP (n)

then:

TRk (n) > 0

TRk (n) is negative otherwise.

As a result, when the poor face a binding no-borrowing constraint, there are two optimal

capital income tax regimes, one for the poor, one for the rich. When a positive tax rate

is faced by the rich and no taxation is faced by the poor, the rich accumulate capital at

a slower pace. As a result, their marginal utility of consumption is increased. The capital

income tax scheme redistributes from rich to poor (and from poor to rich otherwise). At the

same time, with a positive capital income tax, labor income is taxed less overall to �nance

exogenous government expenditure. The redistributive e¤ect toward the poor is attenuated

22

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by a general equilibrium e¤ect. Positive capital income taxation limits capital accumulation,

decreasing output and the marginal product of labor in equilibrium.

The conditions for zero capital income taxation are therefore more restrictive than in the

simple life-cycle model. Proposition 2 contrasts with Erosa and Gervais (2002) for whom non

additively separable preferences is necessary to obtain Tk(n) 6= 0: Its corollary also contrastswith Erosa and Gervais (2002) for whom additively separable preferences alone are su¢ cient

to obtain Tk(n) = 0: The reason is that contrary to Erosa and Gervais (2002), the proposed

model takes into consideration di¤erent productivity pro�les and incomplete markets and

horizons. In the case when preferences are additively separable, because of di¤erent income

pro�les and borrowing constraints there is no reason to expect that UCP (n) = UCR(n) or

that �P = �R: In that case, capital income taxation di¤ers from zero in the long term.

3.3 Endogenous versus exogenous borrowing constraints

When the borrowing constraint is endogenous for the poor and the rich, optimal capital

income taxes for the poor and the rich are respectively deduced from:

r(n)� r̂P (n) =

� @UCP (n)@n

UCP (n)

��

@@n [ (n)�CR(n)+(1� (n))�CP (n)] (n)�

CR(n)+(1� (n))�

CP (n)

(36)

r(n)� r̂R(n) =

� @UCR(n)@n

UCR(n)

��

@@n [ (n)�CR(n)+(1� (n))�CP (n)] (n)�

CR(n)+(1� (n))�

CP (n)

(37)

�CP (n) is rewritten as follows:

�CP (n) =�1 + �P + �P �C

P (n)�UCP (n)

with �P the shadow price, and �CP (n) the general equilibrium elasticity:

�CP (n) =

UCP (n)CP (n)CP (n) + ULP (n)CP (n)L

P (n)

UCP (n)

Proposition 4: When the borrowing constraint is endogenous, optimal age-dependent cap-

ital income taxation di¤ers from zero, if one of the following su¢ cient (but not necessary)

conditions is veri�ed:

P1, P2 or P4: preferences are non additively separable and �R @@n�C

R(n) 6= �P @@n�C

P (n)

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Conditions (P1) and (P2) are similar to the case with exogenous borrowing constraints.

(P4) is a new condition. With an endogenous borrowing constraint, the poor can reallocate

consumption across periods. It is optimal for the government to use capital income taxation

to smooth consumption when the poor and the rich do not reallocate consumption across

period in the same way, which depends on �R and �P ; the shadow prices and �CR(n) and

�CP (n); the general equilibrium elasticities.

The proof of this proposition is presented in Appendix C.

Corollary: When the borrowing constraint is endogenous, optimal capital income taxation

equals zero if condition (C1) or (C2) or (C3) is veri�ed.

C3: optimal capital income taxation equals zero if UCP (n) = UCR(n); �P = �R and

�R @@n�C

R(n) = �P @@n�C

P (n)

Compared to the case when the borrowing constraint is exogenous and binding for the

poor, there is therefore an additional case for optimal capital income taxation to be zero. In

this additional scenario, individuals have the same optimal way to reallocate consumption

over time at age n.

The proof of this proposition is presented in Appendix C.

Proposition 5: When the borrowing constraint is endogenous, there are two optimal capital

income tax regimes, one for the poor, one for the rich, when UCP (n) 6= UCR(n). When the

private MRS between current and future consumption is larger than the social MRS, capital

income taxation is positive. When UCP (n) > UCR(n); the rich pay a higher tax rate on

capital income than the poor. Therefore, capital income taxation redistributes to the poor.

The proof of the following propositions is presented in Appendix C.

Proposition 6: When preferences are logarithmic and �P = �R, capital income taxation

redistributes to the poor and to the young if UCP (n) > UCR(n). This result is valid wether

borrowing constraints are exogenous or endogenous.

Proof

When the poor face a binding exogenous borrowing constraint, I set

UCP (n) 6= UCR(n); �P = �R; and logarithmic preferences. Capital income taxation is then

given by (33). In that case, poor and rich individuals di¤er only by their income and access

to credit markets. If, as a result CP (n) < CR(n) then, UCP (n) > UCR(n) and the MRS of

the rich is less negative than the social MRS. From (33), capital income taxation is positive.

24

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Only the rich pay capital income taxation which redistributes to the poor. Moreover, in that

case, the MRS of the rich converges slower toward zero than the social MRS. As a result,

capital income taxation is increasing in age and redistributes form the older rich individuals

toward the younger rich individuals.

This result stays robust in the case of endogenous borrowing constraints. Setting UCP (n) 6=UCR(n); �P = �R; and preferences are logarithmic, capital income taxation for the rich is

given by (33). For the poor, it is given by:

r(n)� r̂P (n) =

@UCP (n)

@n

UCP (n)�

@UCR(n)

@n+

@UCP (n)

@n

UCR(n) + UCP (n)

Therefore, for the rich the result is the same as in the case of an exogenous borrowing

constraint. When UCP (n) > UCR(n); it is optimal to tax capital income for the rich at higher

rate than the poor. Capital income taxation redistributes to the poor and to the younger

rich individuals.

Moreover, if:@U

CR(n)

@n

UCR(n)>

@UCR(n)

@n+

@UCP (n)

@n

UCR(n) + UCP (n)

then:@U

CP (n)

@n

UCP (n)<

@UCR(n)

@n+

@UCP (n)

@n

UCR(n) + UCP (n)

As a consequence:

r(n)� r̂R(n) > 0

r(n)� r̂P (n) < 0

It is optimal to tax the rich capital income and to subsidize the poor capital income.The

MRS of the poor converges faster toward zero than the social MRS, therefore the subsidy

decreases with the age and redistributes from the older poor to the younger poor.

When borrowing constraints are endogenous, the corollary of proposition 4 shows that

there is an additional scenario for capital income taxation to equal zero in the long run.

Therefore the conditions for zero capital income taxation are less restrictive than in the

case of exogenous borrowing constraints. However, the conditions for zero capital income

taxation remain more restrictive than in the simple life-cycle model by Erosa and Gervais

25

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(2002). They are also more restrictive that in the model by Conesa et al. (2007, 2009), which

includes uninsurable income shocks and permanent productivity di¤erences of households.

When preferences are additively separable, the existence of separate life cycles is a key

element explaining why optimal capital income taxation di¤ers from zero. Each life cycle is

characterized by an expected income pro�le, a life expectancy and as a consequence a di¤erent

borrowing constraint. The poor and the rich optimize within their own categories. Therefore

there is no reason to expect that UCP (n) = UCR(n): In that case, capital income taxation

di¤ers from zero in the long term and has redistribution consequences. Even if we assume

the same life expectancies for the poor and the rich, the di¤erences in consumption between

the poor and the rich still re�ect the di¤erences in income and wealth. Wealth accumulation

depends on the access to credit markets. When the borrowing constraint is endogenous, and

UCP (n) > UCR(n); it is optimal to tax the rich capital income and to subsidize the poor

capital income. Social welfare increases because the gains from consumption smoothing and

redistribution are larger than the distortions induced by the capital income tax.

3.4 Di¤erent versus same life horizons

From Proposition 2 and its Corrolary, we learn that there is one case in which individuals�

horizons are irrelevant. When the poor and the rich have logarithmic preferences, their

MRS between current and future consumption are constant and equal to one. If UCP (n) =

UCR(n), they are willing to reallocate consumption over ages the same way. As a result, their

weights in the social welfare function are irrelevant and capital income taxation equals zero

(see (34)). By contrast, if UCP (n) 6= UCR(n) their horizons matter. To have some insight

on their roles for redistribution, I study the case of logarithmic preferences: As shown in

Proposition 6, when individuals have the same �nite horizon and logarithmic preferences,

capital income taxation di¤ers from zero and has redistributive consequences. The magnitude

of the redistribution depends on the di¤erence between UCP (n) 6= UCR(n), which is purely

the result of their di¤erent incomes and borrowing constraints.

The next proposition studies the e¤ect on optimal capital income taxation of an increase

in the life horizon of the rich relatively to the life horizon of the poor.

Proposition 7: When preferences are logarithmic, the larger the weight of the rich in the

population, the lower the capital income tax paid by the rich and the less capital income

taxation redistributes to the poor.

26

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Proof

With logarithmic preferences and �P 6= �R; optimal capital income taxation for the rich

is given by:

r(n)� r̂R(n) =

@UCR(n)

@n

UCR(n)�

@@n

� (n)UCR(n) + (1� (n))UCP (n)

� (n)UCR(n) + (1� (n))UCP (n)

(38)

I study the case when the proportion of the rich in the total population becomes extremely

large:

(n) 7�! 1

then:@@n

� (n)UCR(n) + (1� (n))UCP (n)

� (n)UCR(n) + (1� (n))UCP (n)

7�!@U

CR(n)

@n

UCR(n)

from (38):

TRk (n) 7�! 0

Therefore, the larger (n); the larger the weight of the rich in the government�s objective,

the lower capital income taxation in the steady state. The closer to zero capital income

taxation, the less the redistribution e¤ect to the poor.

Limit case: Erosa and Gervais (2002)

It is possible to retrieve Erosa and Gervais (2002) results by setting (n) = 100%: This

results in eliminating the poor from the economy. The model becomes a continuous time life

cycle model with only rich agents. Capital income taxation is then given by:

r(n)� r̂R(n) =

@UCR(n)

@n

UCR(n)�

@�CR(n)

@n

�CR(n)

which is equivalent to:

r(n)� r̂R(n) =

@UCR(n)

@n

UCR(n)�(1 + �R + �R�C

R(n))@U

CR(n)

@n+ UCR(n)

@�CR(n)

@n

(1 + �R + �R�CR(n))UCR(n)

(39)

Similarly to Erosa and Gervais (2002), when preferences are additively separable,

�CR(n) = �1 and @�C

R(n)

@n= 0: Therefore, (39) equals zero. In the case of one unique

life-cycle, the presence of the borrowing constraint is irrelevant for optimal capital income

taxation. Agents accumulate wealth and their borrowing constraint (exogenous or endoge-

nous) is non binding. Therefore, it has no in�uence on their consumption allocation across

ages.

27

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Same in�nite horizon: Judd 1985

The model enables me to retrieve the results by Judd (1985) in the "workers-capitalist"

framework. When both the poor and the rich have the same in�nite lifetime, 1�P= 0 and

1�R= 0: As a result, (n) = 1� (n) = 50%: The poor and the rich have the same weight in

the social welfare function. In addition, in an in�nite horizon model, �C(n) is by de�nition

constant across ages. As a result, capital income taxation equals zero.

Conclusion

The standard results of the Ramsey optimal taxation literature, derived in an in�nite horizon

framework have been challenged by recent life-cycle models. I extend this literature by

proposing a model in which agents face di¤erent expected income pro�les, life expectancies

and borrowing constraints. This framework provides new theoretical results. The di¤erences

in borrowing constraints faced by the poor and the rich tend to limit the progressivity of the

labor income tax. Even when preferences are additively separable between consumption and

leisure, optimal capital income taxation generally di¤ers from zero in the long run. Poor and

rich individuals generally face two optimal capital income tax regimes involving redistribution

across income categories. When the poor and the rich face di¤erent endogenous borrowing

constraints, the conditions for zero capital income taxation remain more restrictive than in

the simple life-cycle model but less restrictive than in the case with exogenous borrowing

constraints. The larger the weight of the rich in the population, the lower the capital income

tax paid by the rich and the less capital income taxation redistributes to the poor.

When agents of di¤erent ages and income levels face di¤erent borrowing constraints,

these results suggest that, in addition to a progressive labor income tax schedule, the tax

system should include a progressive capital income tax scheme. Some important elements

of the life cycle to determine optimal taxation have been incorporated in this model and

further research could focus on incorporating other relevant elements such as labor mobility

or endogenous skills acquisition.

28

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31

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Appendix

Appendix A

Parameters of the labor income function:

Therefore the parameters of the income functions verify the following conditions. Recall

that z � s represent age.

In the data, the labor income of the poor is lower than the labor income of the rich.

aP e�P (z�s) � e�P (z�s) < aRe

�R(z�s) � e�R(z�s) for z � s < nold

As a result of the functional forms, labor income decreases and can even be negative for

very old ages (larger than nold). However, this is not a problem to represent the aggregate

behavior of the economy. The parameters of the functional forms are chosen such that the

integrals representing aggregate human wealth to converge. For very old ages, there is only

an in�nitesimal number of agents who are still alive in the model and they do not matter in

the aggregate with proper values of the model�s parameters.

The labor income age pro�le of the poor and the rich is hump shaped. Therefore, there

is a maximum for age z � s = nmax described by:

@

@(z � s)(aP e

�i(z�s) � e�i(z�s)) = 0 for z � s = nmax

and:@2

@(z � s)(aP e

�i(z�s) � e�i(z�s)) < 0 for z � s = nmax

with ai; �i; �i positive coe¢ cients.

Aggregate e¤ective labor supply:

Aggregate e¤ective labor supply for an individual of type i is described by:

~N i (t) =

Z t

�1

1

�P + �Re� 1�i(t�s) ~N i(s; t)

=

Z t

�1

1

�P + �Re� 1�i(t�s) �

aie�i(t�s) � e�i(t�s)

�~N i (t) ds

Therefore, implicitly, ai; �i; �i verify:Z t

�1

1

�P + �Re� 1�i(t�s) �

ai exp�i(t�s)� exp�i(t�s)

�ds =

�i�P + �R

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1

�P + �R

ai

1�i� �i

� 11�i� �i

!=

�i�P + �R

Appendix B

Total aggregate consumption18:

Aggregate consumption is directly deduced from individual consumption and the aggre-

gation rules.

C(t) = CR(t) + CP (t)

If the no borrowing constraint is binding for the poor, then aggregate consumption by

the poor is given by:

CP (t) = !̂P (t)EP (t)NP (t)

Using the intertemporal budget constraint and the Euler equation, individual consump-

tion by the rich can be expressed as a function of human and �nancial wealth.

Aggregate consumption by the rich is:

CR (t) = 'R(t)�HR (t) +WR (t)

�whereWR (t) represents aggregate �nancial wealth and HR (t) ; human wealth and 'R(t) the

average propensity to consume19.

If the poor face a non binding no borrowing constraint, his aggregate consumption func-

tion is expressed in a way similar to the rich.

Aggregate �nancial wealth for an individual of type i is given by:

_W i (t) = r̂i (t)W i (t) + !̂i(t)Ei(t)N i (t)� Ci (t)

Because of the annuity system, the accumulation of aggregate �nancial wealth does not

depend on the occurrence of deaths (Blanchard (1985)).

The hump shape of labor income leads to describe human wealth with two dynamic

equations instead of one (as is the case in Blanchard (1985), when labor income is decreasing):

18The details on the aggregation of human and �nancial wealth are provided online at:http://www.uvm.edu/~nmathieu19With additively separable preferences it is possible to make 'R(t) explicit:

'R(t) = 1=(

Z +1

t

e�R zt

h(1�1=�CR )r̂

R(v)+ 1�R

+�=�CRidvdz)

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H i (t) =1

�P + �R

ai1�i� �i

H i� (t)�

1

�P + �R

11�i� �i

H i� (t)

_H i� (t) =

�r̂(t) +

1

�i� �i

�H i� (t)� !̂i(t)Ei (t)N i(t)

_H� (t) =

�r̂(t) +

1

�i� �i

�H i� (t)� !̂i(t)Ei (t)N i(t)

Appendix C

Proof of proposition 4:

- To show that condition (P1) is su¢ cient but not necessary for capital income taxation

to di¤er from zero for the rich, I show that (37) di¤er from zero when (P1) is veri�ed but

(P2) and (P3) are not. Since condition (P4) includes condition (P3), if (P3) is not veri�ed,

(P4) is not veri�ed either.

- Following the same logic, to show that condition (P2) is su¢ cient but not necessary

for capital income taxation to di¤er from zero for the rich, I show that (37) di¤er from zero

when (P2) is veri�ed but (P1) and (P3) (and therefore (P4)) are not. Setting �P 6= �R and

non logarithmic preferences, UCP (n) = UCR(n); and additively separable preferences (�CR(n)

constant and equals ��CR 6= �1), equation (37) becomes:

@@n (n)

�1 + �R + �R�C

R�� @

@n (n)

�1 + �P + �P�C

P�

(n)�1 + �R + �R�CR

�+ (1� (n))

�1 + �P + �P�CP

� (40)

This expression di¤ers from zero unless �P = �R or �CR= 1:

- To show that conditions (P4) is su¢ cient but not necessary for capital income taxation

to di¤er from zero, I show that (37) di¤ers from zero when (P4) is veri�ed but (P1) and (P2)

are not. Starting from equation (37), the proof that optimal capital income taxation for the

rich di¤ers from zero is straightforward. When preferences are non additively separable and

�R @@n�C

R(n) 6= �P @@n�C

P (n); UCP (n) = UCR(n) and �P = �R; (37) becomes:

�R @@n�C

R(n) � �P @@n�C

P (n)�1 + �R + �R�C

R(n)�+�1 + �P + �P �C

P (n)� (41)

The numerator of this expression di¤ers form zero unless

�R @@n�C

R(n) = �P @@n�C

P (n):

34

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The proofs that optimal capital income taxation for the poor also di¤ers from zero would

be similar, based on (36). Therefore, when optimal capital income taxation di¤ers from zero

for the rich, it also di¤ers from zero for the poor.

Proof of the Corollary of Proposition 4:

- The proof that optimal capital income taxation equals zero if (C1) is veri�ed can be

directly deduced from (40). This expression equals zero if �P = �R:

- The proof that optimal capital income taxation equals zero if (C2) is veri�ed can

be directly deduced from (40). This expression equals zero if preferences are logarithmic

(�CR= 1).

- The proof that optimal capital income taxation equals zero if (C3) is veri�ed can be

directly deduced from (41). This expression equals zero when �R @@n�C

R(n) = �P @@n�C

P (n).

Proof of proposition 5:

If UCP (n) 6= UCR(n); the MRS of the poor di¤ers from the MRS of the rich, and it is

straightforward from (36) and (37) that there are two optimal capital income tax regimes,

one for the poor, one for the rich.

From (37), if:@U

CR(n)

@n

UCR(n)>

@ (n)@n

� (n)�CR(n) � (1� (n))�CP (n)

� (n)�CR(n) + (1� (n))�CP (n)

then:

TRk (n) > 0

(and negative otherwise).

When:

UCP (n) > UCR(n)

then@U

CR(n)

@n

UCR(n)>

@UCP (n)

@n

UCP (n)

Therefore, from (36) and (37):

r(n)� r̂R(n) > r(n)� r̂P (n)

As a result, capital income taxation is higher for the rich than for the poor and capital

income taxation tends to redistribute to the poor.

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