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Optimal Taxation Over the Life-Cycle For the Poor and the Richnmathieu/OTPRonline.pdf · The answer...
Transcript of Optimal Taxation Over the Life-Cycle For the Poor and the Richnmathieu/OTPRonline.pdf · The answer...
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
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
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
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
(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
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
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
7
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
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
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
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
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
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.
13
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
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
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)
16
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:
17
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)
18
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
19
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
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
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
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)
23
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
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
(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
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
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
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
32
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)
33
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
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.
35