Optimal Government Policies
in Models with Heterogeneous Agents
Radim Boh´aˇcek and Michal Kejak∗
CERGE-EI, Prague, Czech Republic
October 15, 2010
Abstract
In this paper we develop a new approach for ﬁnding optimal government policies
in economies with heterogeneous agents. Using the calculus of variations, we present
three classes of equilibrium conditions from government’s and individual agent’s optimization
problems: 1) the ﬁrst order conditions: the government’s Lagrange-Euler
equation and the individual agent’s Euler equation; 2) the stationarity condition on
the distribution function; and, 3) the aggregate market clearing conditions. These
conditions form a system of functional equations which we solve numerically. The
solution takes into account simultaneously the eﬀect of the government policy on
individual allocations and the resulting optimal distribution of agents in the steady
state. This approach is applicable to a wide class of general equilibrium, Bewley
type economies where the government looks for an optimal nonlinear, second-best
ﬁscal or monetary policy. We illustrate it on a steady state Ramsey problem with
heterogeneous agents, ﬁnding the optimal tax schedule.
JEL Keywords: Optimal macroeconomic policy, optimal taxation, computational
techniques, heterogeneous agents, distribution of wealth and income
∗
Contact: CERGE-EI, Politickych veznu 7, 111 21 Prague 1, Czech Republic. Email:
radim.bohacek@cerge-ei.cz, michal.kejak@cerge-ei.cz. First version: September 2002. For helpful comments
we thank Jim Costain, Mark Gertler, Max Gillman, Boyan Jovanovic, Marek Kapiˇcka, Dirk Krueger,
Josep Pijoan-Mas, Thomas Sargent, Harald Uhlig, Gianluca Violante, Galyna Vereshchagina, and the participants
at the CNB/CERGE-EI Macro Workshop 2004, SED 2004 conference, University of Stockholm,
the Macroeconomic Seminar at the Federal Reserve Bank of Minneapolis, SED 2007 conference, Cardiﬀ
Business School, and the Macroeconomic Seminar at the Stern School of Business of NYU. We are especially
grateful to Michele Boldrin, Tim Kehoe and Stan Zin for their support and advice. Anton Tyutin,
Jozef Zubrick´y and Tom´a´s Le´sko provided excellent research assistance. All errors are our own.
1 Introduction
This paper provides a new approach for computing equilibria in which the stationary distribution
of agents is a part of an optimal nonlinear, second-best in a general equilibrium,
Bewley type economy with heterogenous agents. We formulate the optimal government
policy problem as a calculus of variations problem where the government maximizes an
objective functional subject to a system of operator constraints: 1) the ﬁrst order conditions
from the individual agent’s problem; 2) the stationarity condition on the distribution
function; and, 3) the aggregate market clearing conditions. The ﬁrst order necessary conditions
of the government functional problem given by the Euler-Lagrange equation (with
transversality conditions) form altogether a system of functional equations in individual
agents’ and government’s policies and in the distribution function over agents’ individual
state variables. We solve this system numerically by the standard projection method.
It should be emphasized that our approach does not use any additional restrictions or
assumptions on the equilibrium allocations but is strictly derived from the ﬁrst order and
envelope conditions and from the stationarity of the endogenous distribution in the steady
state. Our main contribution is in the formulation of the Euler-Lagrange equation for the
government problem and for the stationary distribution over individual state variables.
In this way, we are able to solve simultaneously for the government optimal policy, for
the optimal individual allocations, and for the (from a government’s point of view) optimal
distribution of agents in the steady state. Additionally, the derived ﬁrst-order transversality
conditions for the boundary agents allow for a qualitative analysis of the shape of the
optimal government policy function. To our knowledge, this paper is the ﬁrst one that
provides a solution method for this kind of problem.
Our approach can be applied to a wide range of government optimal ﬁscal or monetary
policy problems. We illustrate this general methodology on a steady state Ramsey problem
with heterogeneous agents. We recast the original Ramsey (1927) and Lucas (1990) normative
question for an economy with heterogeneous agents: What choice of a tax schedule
will lead to maximal social welfare in the steady state, consistent with given government
consumption and with market determination of quantities and prices? What is the welfare
diﬀerential with respect to social welfare resulting from the existing progressive tax
schedule in the U.S. economy and as well as from the usual ﬂat-tax reform?
The tradeoﬀ between eﬃciency and income or wealth distribution plays a central role
in analyzing tax policies. In dynamic, general equilibrium models with household heterogeneity
from uninsurable idiosyncratic risk, a tax schedule provides incentives for agents to
accumulate wealth. The optimal tax schedule seeks to arrive at such a steady state where
the distribution of agents is optimal with respect to the aggregate welfare in the economy.
2
The steady state wealth distribution is perhaps the most important part of the resulting
equilibrium: it determines equilibrium prices and fractions of poor and rich agents, i.e.
all elements that aﬀect agents’ ability to self-insure against idiosyncratic shocks through
the accumulated buﬀer stock of savings. In our example, we ﬁnd a welfare maximizing
tax schedule on the total income from labor and capital which takes into account simultaneously
its eﬀects on agents’ allocations and on the stationary distribution of agents in
a steady state. Previous models analyzing the eﬀects of government policies in this class
of models were limited to sub-optimal policy reforms exogenously imposed on the model.
Within the context of optimal taxation, several papers have analyzed the steady state implications
(and transition paths) resulting from an ad hoc ﬂat-tax reform or from an ad
hoc removal of double taxation of capital income.
In this paper, we solve for the optimal tax schedule on the total income that maximizes
aggregate welfare in a steady state of a standard neoclassical, general equilibrium,
full information and full commitment economy with heterogeneous agents and incomplete
markets. In order to evaluate the beneﬁts of the optimal tax schedule, we compare the
steady state aggregate levels, welfare, eﬃciency and distribution of resources associated
with this optimal tax schedule to a simulated steady state of the U.S. economy with the
existing progressive tax schedule and to a steady state resulting from a standard ﬂat-tax
reform.
The optimal tax schedule we ﬁnd is a function that is neither progressive nor monotone.
It is a positive, U-shaped function, taxing the lowest income at 45%, decreasing to a
minimum of 19% and rising to 62% at the highest level of total income. It provides
incentives for agents to accumulate assets while preserving the equality measures in the
economy. Its impact on aggregate levels and welfare is large. Compared to the progressive
tax schedule steady state, average welfare increases by 4.4%, capital stock by 49%, output
by 15.8%, and consumption by 5.8%. Relative to the ﬂat-tax steady state, welfare goes up
by 0.8%, capital stock by 15%, output by 4.5%, and consumption by 1.1%. The marginal
tax rate is also a U-shaped function, but almost ﬂat at the low incomes, reaching negative
levels around the average income, and then rising to positive levels.
The eﬃciency and distributional eﬀects of the optimal tax are the main mechanisms
behind these large changes. Related to the eﬃciency are the general equilibrium eﬀects: a
higher stock of capital increases productivity of labor and, therefore, the income of poor
agents. For the steady state distributional eﬀects, the optimal tax schedule concentrates
the agents around the mean at high levels of wealth, something what a social planner with
an access to the ﬁrst-best, lump-sum transfers would do. The high tax rate at low income
levels provides incentives for these agents to save more for precautionary reasons in the long
3
run steady state. The even higher tax rate on high incomes discourages additional savings
by the wealthiest agents. In the middle of the total income levels, the tax rate is lower
than the one found for the ﬂat-tax reform. In this way, the optimal tax schedule solves the
tradeoﬀ between eﬃciency and equality. For comparison, the ad hoc ﬂat tax reform also
increases aggregate levels but does not take into account the distribution of agents. On
the other hand, the progressive tax schedule provides too much short-run insurance at the
cost of the long-run average levels.
Finally, in order to evaluate the short run costs of the optimal tax reform, we compute
transitions from the progressive and the ﬂat-rate tax schedule steady states to the steady
state of the optimal tax schedule. We ﬁnd that a majority of population, 73%, would beneﬁt
from a reform that replaces the progressive tax schedule by the optimal tax schedule. On
the other hand, only one third of agents would support the reform starting from the steady
state with ﬂat tax. These results as well as a detailed eﬃciency and distributional analysis
are described in a great detail in the following sections.
We limit our example to the optimal tax schedule on the total income from labor and
capital that is needed to raise a given fraction of GDP. There are several reasons why we
choose this setup. First, the tax on the total income enables us to study a tax system with
a non-degenerate distribution of agents in a steady state. If the government had an access
to a lump-sum, ﬁrst best taxation the model would collapse to a representative agent one.
Second, to a large extent the current U.S. tax code does not distinguish between the sources
of taxable income. The last reason for a simple tax on the total income is the complexity
of the problem we solve.
By focusing on a steady state analysis and by imposing a single tax rate on labor and
capital income we tried to isolate the shape of the optimal tax schedule on total income. The
closest paper to ours is Conesa and Krueger (2006), who compute the optimal progressivity
of the income tax code in an overlapping generations economy. They search in a class of
monotone tax functions to ﬁnd a welfare maximizing tax schedule. We show in this paper
that limiting the analysis to monotone functions seems rather restrictive with respect to
welfare maximization.
In this paper, we do not address the following important issues related to optimal taxation:
the issue of time-consistency, the issue of the optimal capital income tax rate in the
steady state, and the distortionary eﬀects of taxation on labor supply. Our government
is fully and credibly committed, the tax schedule is constant over time.1
Aiyagari (1995)
showed that for our class of models with incomplete insurance markets and borrowing
constraints, the optimal tax rate on capital income is positive even in the long run (see
1
For the time-consistency problem see Kydland and Prescott (1977) and Klein and Rios-Rull (2004).
4
also a recent paper by Conesa, Kitao, and Krueger (2009)).2
Due to the complexity of our
work, we study the simplest utility maximization problem on the consumption-investment
margin. However, we would like to stress that our methodology can be applied to diﬀerent
aggregate welfare criteria and a wide variety of optimal government policy problems, including
those with endogenous labor supply, separate taxation of labor and capital incomes
as well as public goods, population growth, or a life-cycle earnings process as in Ventura
(1999). Finally, in future research we also plan to analyze a much more diﬃcult problem,
that of a stationary competitive equilibrium which is the limit of the optimal dynamic tax
schedule.
The paper is organized as follows. The following section describes the economy with
heterogeneous agents, deﬁnes the stationary recursive competitive equilibrium and the
stationary Ramsey problem. Section 3 speciﬁes the equilibrium as a system of functional
equations and deﬁnes the operator stationary Ramsey problem. Section 4 formulates the
Ramsey problem by means of the calculus of variations. The ﬁrst-order necessary conditions
for the optimal government policy schedule expressed in the form of a generalized EulerLagrange
equation including the transversality conditions and related analytical results are
described in Section 5. Section 6 illustrates our approach by an example of the optimal
income tax schedule. Section 7 presents the numerical solution and Section 8 concludes.
Appendix contains the proofs and analytical results.
2 The Economy
The economy is populated by a continuum of inﬁnitely lived agents on a unit interval. Each
agent has preferences over consumption in period t ≥ 0, ct, given by a utility function
E
∞
t=0
βt
U(ct), 0 < β < 1,
where U : R+ → R is a twice continuously diﬀerentiable, strictly increasing and strictly
concave function. We assume that the utility function satisﬁes the Inada conditions.
At all t ≥ 0, each agent is identiﬁed by an endogenous state variable, the accumulated
stock of capital, kt ∈ B = [k, ∞) with k = 0, and by an exogenous labor productivity
shock zt ∈ Z = {z1, z2, . . . , zJ }. The shock represents labor eﬃciency units and follows a
ﬁrst-order Markov chain with a transition function Q(z, z ) = Prob(zt+1 = z |zt = z). We
assume that Q is monotone, satisﬁes the Feller property and the mixing condition deﬁned
in Stokey, Lucas, and Prescott (1989). As the labor productivity shock is independent
2
For the optimal capital tax with two types of agents see Chamley (1986) and Judd (1985).
5
across agents there is no uncertainty at the aggregate level. We preserve the heterogeneity
in the economy by assuming incomplete markets.
In each period, agents supply labor and accumulated capital stock to a representative
ﬁrm with a production function F(Kt, Lt), where Kt ∈ B is the aggregate capital stock,
Lt ∈ R+ is the aggregate eﬀective labor. The production function is concave, twice continuously
diﬀerentiable, increasing in both arguments, and displays constant returns to scale.
Proﬁt maximization implies the following factor prices
rt = FK(Kt, Lt) − δ and wt = FL(Kt, Lt), (1)
where δ ∈ (0, 1) is the depreciation rate of capital.
Finally, there is a government that ﬁnances its expenditures by taxing the agents in
the economy. We assume the government is fully committed to a sequence of tax schedules
π = {πt}∞
t=0 to ﬁnance its expenditures. We assume that these expenditures equal a
constant fraction of output, g, in each period, that they are not returned to the agents,
and that the government cannot use the ﬁrst best, lump-sum taxation.3
The policy π is
applied to a broadly deﬁned taxable activity of each agent, xt ∈ R+. We will assume that
xt = x(zt, kt) where x : Z × B → R+ and xz, xk > 0. Thus in each period, the policy
schedule is a function πt : R+ → R, so that an agent with a total income from labor and
capital, yt ∈ R+, yt = y(kt, zt) = rtkt + wtzt, and a taxable activity xt = x(kt, zt) pays
taxes πt(xt)xt and is left with an after-tax income yt − πt(xt)xt. In our example in Section
6, we illustrate this policy by a proportional taxation of this total income from labor and
capital, i.e. when x = y.
The economy’s state is characterized by the sequence of government policies π, and by
a distribution of agents over capital and shock in each period, λ = {λt}∞
t=0. The latter is
in each period a probability measure deﬁned on subsets of the state space, describing the
heterogeneity of agents over their individual state (z, k) ∈ Z × B. Let (B, B) and (Z, Z)
be measurable spaces, where B denotes the Borel sets that are subsets of B and Z is the
set of all subsets of Z.
2.1 Stationary Recursive Competitive Equilibrium
We will analyze the economy in a stationary recursive competitive equilibrium in which
the government policy schedule and the distribution of agents are time-invariant. Given
the equilibrium prices and the time-invariant government policy π, an agent (z, k) solves
3
Our analysis equally applies to the case when government ﬁnances any expenditures {Gt}∞
t=0 and
corresponding revenue neutral reforms.
6
the following dynamic programming problem
v(z, k) = max
c,k+
u(c) + β
z+
v(z+
, k+
) Q(z, z+
) , (2)
subject to a budget constraint
c(z, k) + k+
(z, k) ≤ y + k − π(x(z, k))x(z, k), (3)
with taxable activity x(z, k), total income y = rk + wz, a borrowing constraint,
k+
(z, k) ≥ k. (4)
Deﬁnition 1 (Stationary Recursive Competitive Equilibrium) For a given share
of government expenditures g and a time-invariant government policy schedule π, a stationary
recursive competitive equilibrium is a set of functions (v, c, y, x, k+
), aggregate levels
(K, L), prices (r, w), and a probability measure λ, such that
1. given prices and the government policy, the policy functions solve each agent’s optimization
problem (2);
2. ﬁrms maximize proﬁt (1);
3. the probability measure is time invariant,
λ(z+
, B+
) =
z {(z,k)∈Z×B: k+(z,k)∈B+}
Q(z, z+
) λ(z, k) dk, (5)
for all (z+
, B+
) ∈ Z × B;
4. the aggregate conditions hold,
K =
z
k+
(z, k) λ(z, k) dk, (6)
L =
z
z λ(z, k) dk; (7)
5. the government budget constraint holds at equality,
g =
z
π(x(z, k))x(z, k) λ(z, k) dk /F(K, L); (8)
6. and the allocations are feasible,
z
c(z, k) + k+
(z, k) λ(z, k) dk + gF(K, L) = F(K, L) + (1 − δ)K. (9)
7
Agents have rational expectations, take the behavior of prices as given by a predetermined
function that depends on aggregate variables in equation (1). As we look for optimal
government policy schedule the goal of our paper is to solve the following stationary Ramsey
problem.
Deﬁnition 2 (Stationary Ramsey Problem) A solution to the Ramsey problem for a
stationary economy with heterogeneous agents is a time-invariant government policy schedule
π : R+ → R that maximizes social welfare in the steady state,
max
π
z
v(z, k; π) λ(z, k; π) dk,
consistent with a given government consumption and with allocations satisfying the deﬁnition
of the stationary recursive competitive equilibrium, where v : Z × B → R is the value
function of individual agents and λ : Z × B → [0, 1] is the stationary distribution.
Our notation indicates that the value and distribution functions depend on the government
policy, π, i.e. v(z, k; π) and λ(z, k; π), respectively.
It is easy to show that the solution to the Stationary Ramsey Problem is equivalent to
that of maximizing the average current period utility,
max
π
z
u(c(z, k; π)) λ(z, k; π) dk.
In the following Sections, we will characterize the optimal government policy schedule using
this latter speciﬁcation.4
3 The Operator Stationary Ramsey Problem
Since the problem is to ﬁnd an optimal, welfare maximizing time-invariant function π :
R+ → R, the Stationary Ramsey Problem can be transformed into an operator form. In
order to express the stationary recursive competitive equilibrium in this form, we deﬁne two
operators: an operator on the Euler equation, F, and that on the stationary distribution,
L.
For a given government policy schedule π : R+ → R the Euler equation operator is
deﬁned on the savings function h : Z × B → B, and the stationary distribution operator
is deﬁned on the distribution function λ : Z × B → [0, 1], and the savings function h. We
4
Our analysis of optimal government policies can also be applied to other types of welfare functions.
8
will assume that these functions are square integrable functions on some closed domain5
:
h, λ ∈ L2
(Z × B) where L2
(Z × B) is a Hilbert space with the inner product (u, v) =
Z×B
u(t)v(t)dt. The operator F : C1
(Z×B) ⊂ L2
(Z×B) → C1
(Z×B) ⊂ L2
(Z×B) is the
mapping from a space of continuously diﬀerentiable functions into a space of continuously
diﬀerentiable functions; and the operator L : C1
(Z × B) × C1
(Z × B) → C1
(Z × B) ⊂
L2
(Z × B).
Operator F on the Euler Equation An individual agent’s allocations are characterized
by the Euler equation from the optimization problem (2)-(4). For all (z, k) ∈ Z × B,
u (y − π(x)x + k − k+
) ≥ β
z+
u (y+
− π(x+
)x+
+ k+
− k++
) · (10)
· 1 + y+
k π(x+
) + π (x+
)x+
x+
k Q(z, z+
),
where
y = rk + wz,
x = x(z, k),
y+
= rk+
+ wz+
,
x+
= x(z+
, k+
),
x+
k = xk(z+
, k+
),
y+
k = r
∂k+
∂k
.
Clearly, y − π(x)x is disposable income and {1 + y+
k − [π(x+
) + π (x+
)x+
]x+
k } is the next
period ‘after-policy’ marginal return to capital where π is the marginal government policy
schedule.
The solution of the Euler equation is a time invariant savings function h : Z × B →
B. In the stationary equilibrium, prices, as functions of the aggregate capital K, are
constant. When the government searches for the optimal policy schedule it needs to take
into account the eﬀect of its policy on prices. For this reason it will be advantageous to
introduce aggregate capital, K, as an explicit variable in the speciﬁcation of the equilibrium.
In order to make this eﬀect more transparent, we will write the equilibrium prices as
r(K) and w(K). As the savings function depends on the aggregate capital stock (through
equilibrium prices) and on the government policy, π, we denote k+
= h(z, k; K, π) and
k++
= h(z+
, k+
; K, π) = h(z+
, h(z, k; K, π); K, π). Since K = z kλ(z, k; π) dk, the
optimal savings function is aﬀected through prices by the distribution function λ.
5
In more precise terms we actually assume that the functions are from the subspace W1,2
(Z ×B) which
contains L2
(Z × B)-functions which have weak derivatives of order one.
9
In the text below, we present only the case of the unconstrained agents (i.e. those for
whom the Euler equation holds with equality and k+
(z, k; K, π) > k). The case of the
borrowing constrained agents is discussed in Appendix A.
The operator on Euler equation F is deﬁned by,
F(h; π) ≡ u (c) − β
z+
u (c+
) 1 + y+
k − π(x+
) + π (x+
)x+
x+
k Q(z, z+
), (11)
where
c = y(z, k; K) − π(x(z, k; K))x(z, k; K) + k − h(z, k; K, π),
y(z, k; K) = r (K) k + w (K) z,
c+
= y(z+
, h(z, k; K, π); K) − π(x(z+
, h(z, k; K, π); K)) x(z+
, h(z, k; K, π); K)
+ h(z, k; K, π) − h(z+
, h(z, k; K, π); K, π),
and the terms y+
k = yk(z+
, h(z, k; K, π); K) and x+
k = xk(z+
, h(z, k; K, π); K) are the
marginal eﬀects of individual savings on income y and taxable activity x in the next
period. The operator equation is simply F(h) = 0.
Operator L on the Stationary Distribution In order to guarantee a unique stationary
distribution of agents in the steady state we need to assume that the government
policy function π is such that the resulting individual agents’ behavior does not display
pathological features (for example, that wealthy agents save less than poor agents). The
following assumption is used in virtually all models with heterogeneous agents as a part of
conditions for the existence of a unique equilibrium and is innocuous.
Assumption 1 Assume that for all z ∈ Z and for a given (K, π), the individual savings
function h : Z × B → B is a monotone function of k over the whole interval [k(z), ∞].
Thus for all z ∈ Z and for a given pair (K, π), there exists an inverse function h−1
assigning
a current value of capital k to savings k+
according to k = h−1
(z, k+
; K, π). In Section 6,
we show in Theorem 2 that our example economy is consistent with this assumption.
In the stationary recursive competitive equilibrium, under the Assumption 1 above, the
operator L for equation (5) is
L(λ, h; π) ≡ λ(z+
, k+
; K, π) −
z
λ[z, h−1
(z, k+
; K, π); K, π] Q(z, z+
), (12)
for all (z+
, k+
) ∈ Z × [h(k(z), z), k(z)].
The related operator equation is L(λ, h) = 0.
10
Deﬁnition 3 (Operator Stationary Recursive Competitive Equilibrium) Given
a share of government expenditures g and a time-invariant government policy schedule π,
an operator stationary recursive competitive equilibrium is a set of operators (F, L), prices
(r, w), a savings function h, a probability measure λ, and aggregate levels (K, L) such that
1. given prices and government policy π, the policy function h is the solution to each
agent’s optimization problem in the operator equation
F(h; π) = 0, (13)
2. ﬁrms maximize proﬁt (1),
3. the time-invariant probability measure is the solution to the operator equation
L(h, λ; π) = 0, (14)
4. the capital and labor markets clear, (6)-(7),
5. the government budget constraint (8) holds at equality,
6. and the allocations are feasible, (9).
We can now specify an operator version of the Stationary Ramsey Problem.
Deﬁnition 4 (Operator Stationary Ramsey Problem) A solution to the Operator
Stationary Ramsey Problem for an economy in a stationary recursive competitive equilibrium
in Deﬁnition 1 is a time-invariant government policy π that maximizes social welfare
in the steady state,
arg max
π
z
k
k
W(v (z, k; K, π)) λ(z, k; K, π) dk, (15)
subject to a system of operator equations (13)-(14), consistent with equilibrium prices (1)
and the market clearing conditions (6)-(7) in Deﬁnition 1, where v : Z × B → R is the
value function of individual agents, λ : Z ×B → [0, 1] is the stationary distribution, W is a
positive linear social aggregator function, k is the exogenously given lower bound on capital
holding, and k is the endogenous upper bound on individual savings, i.e. k = h(z, k; K, π).
11
4 Ramsey Problem as Calculus of Variations Problem
The ﬁrst-order conditions for the solution π to the Operator Stationary Ramsey Problem
in Deﬁnition 4 are best formulated in the calculus of variations.6
As it is standard, the
government policy function π and its derivative π are treated as two independent functions.
Additionally, since we are looking for the government policy as a function of individual
activity x rather than capital k, we now reformulate the problem with activity x as an
independent variable.
The social welfare function in equation (15) at the new coordinates x is
z
k
k
W [u (c (z, k; K, π, π ))] λ(z, k; K, π, π ) dk =
z
x(z)
x(z)
W[z, x; K, π(x), π (x)] dx
with
W[z, x; K, π(x), π (x)] ≡ W [u (c (z, k(z, x; K); K, π, π ))] λ [z, k(z, x; K); K, π, π ] kx (z, x; K) ,
(16)
where kx [z, x; K] = [xk(z, k(z, x; K); K)]−1
is the inverse function to the marginal eﬀect of
individual savings on the taxable activity xk.
Before we proceed further, we have to clarify two important aspects of this dynamic
optimal problem. First, observing that K, which determines the equilibrium prices, is one
of the arguments in the objective function above, we have to use the condition on the
aggregate capital stock. Writing the condition properly using the fact that K also depends
on the government policy π and its derivative π , we get at the new coordinates
K =
z
k
k
k λ(z, k; K, π, π ) dk =
z
x(z)
x(z)
K[z, x; K, π(x), π (x)] dx,
where
K[z, x; K, π(x), π (x)] ≡ k(z, x; K) λ [z, k(z, x; K); K, π, π ] kx (z, x; K) .
Second, the bounds on taxable activity, x(z) and x(z), for each z ∈ Z, are endogenous
functions of a chosen government policy. The lower bound x(z) = x(z, k; K) depends
on z, on the exogenously given lower bound on capital, k, and on the equilibrium prices
6
Mirlees (1976) also uses the calculus of variations to derive the ﬁrst-order conditions for the optimum
income tax schedule in an economy with heterogenous agents. However, while his problem is a static
one with an exogenously imposed distribution of agents’ abilities we have a dynamic problem with an
endogenous distribution of agents.
12
determined by K. The upper bound x(z) = x(z, k; K) depends on z, on the endogenous
upper bound of capital, k, and also on K. 7
Finally, we also need to reformulate the side condition of the problem given by the
government budget constraint in equation (8), i.e.
z
k
k
[π(x(z, k; K))x(z, k; K) − g y(z, k; K)] λ(z, k; K, π, π ) dk
=
z
x(z)
x(z)
G[z, x; K, π(x), π (x)] dx = 0, (17)
where
G[z, x; K, π(x), π (x)] ≡ [π(x)x − g y (z, k(z, x; K); K, π, π )] λ [z, k(z, x; K); K, π, π ] kx (z, x; K) .
Deﬁnition 5 (Calculus of Variation Ramsey Problem) The Ramsey problem in the
calculus of variation is formulated as the following generalized isoperimetric problem,
max
z
x(z)
x(z)
W[z, x; K, π(x), π (x)] dx, (18)
subject to
z
x(z)
x(z)
G[z, x; K, π(x), π (x)] dx = 0, (19)
with the deﬁnition of the aggregate capital stock in equation (17), the individual policy
function h given implicitly by the operator Euler equation (13), the distribution function,
λ, given implicitly by the operator equation (14), the endogenously determined bounds of
taxable activity, x(z) and x(z) for all values of z ∈ Z, and the free values of the government
policy at the extreme lower and upper bounds, π(x(z)), and π(x(z)).
Note that since k is endogenous, the endpoint x(z) is equality constrained.
7
Clearly, the maximal interval is [x(z), x(z)] where x(z) is the lower bound of the lowest shock, z, and
x(z) is the upper bound of the highest shock, z. So any taxable activity interval associated with a shock
z ∈ Z is a subinterval of the maximal interval, [x(z), x(z)] ⊂ [x(z), x(z)].
13
5 Necessary Conditions for the Optimal Government
Policy Schedule
In order to derive the ﬁrst-order conditions, we deﬁne the Lagrange function L for the
Calculus of Variations Ramsey Problem in Deﬁnition 5 as
L(z, x) =
⎧
⎪⎨
⎪⎩
0 for x ∈ [x(z), x(z)),
W(z, x) + μG(z, x) for x ∈ [x(z), x(z)],
0 for x ∈ (x(z), x(z)],
(20)
for each z ∈ Z = {z, z}.
Note that the social welfare function in (16) is the sum of integrands W(z, x) =
W[z, x; K, π(x), π (x)] integrated on the intervals [x(z), x(z)] for each z ∈ Z. The same is
true for integrands L(z, x) in equation (20).
Interestingly, as we show in Theorem 1 below, where we derive the ﬁrst-order conditions,
the relevant Lagrange function that emerges from the solution to the maximization problem
is one amended by a term which captures the eﬀect of the distribution of capital on the
social welfare: L(z, x) = L(z, x)+ΨK(z, x), where Ψ is the marginal eﬀect of the aggregate
capital on social welfare.8
In other words, the term ΨK(z, x) takes into account the eﬀect
of prices (determined by the aggregate capital) on social welfare of agents characterized by
(z, x).
Theorem 1 (First Order Necessary Conditions) Using the modiﬁed Lagrange function
L for the Calculus of Variations Ramsey Problem in Deﬁnition 5,
L(z, x) =
⎧
⎪⎨
⎪⎩
0 for x ∈ [x(z), x(z)),
W(z, x) + μG(z, x) + ΨK(z, x) for x ∈ [x(z), x(z)],
0 for x ∈ (x(z), x(z)],
(21)
for each z ∈ Z = {z, z}, the ﬁrst order necessary conditions for the Ramsey problem are
1. the Euler-Lagrange condition,
z
Lπ(z, x) −
d
dx
Lπ (z, x) = 0; (22)
2. the transversality condition on the free boundary value, π(x(z)), at the equality constrained
endpoint, x(z),
L(z, x) − π (x) −
kx(z, x)
ωπ(z, x)
Lπ (z, x)
x=x(z)
= 0; (23)
8
Note that Ψ is the eﬀect of variation in K on the variation in L, i.e. δL/δK.
14
3. the transversality condition on the free boundary value π(x(z)) at x(z)
Lπ (z, x)
x=x(z)
= 0; (24)
4. and the condition on the Langrange multiplier, μ, at which (19) is satisﬁed.
The marginal eﬀect Ψ of the aggregate capital stock on social welfare is
Ψ ≡ ΨK
z
x(z)
x(z)
LK(z, x)dx + [(L(z, x) − π (x)Lπ (z, x)) xK]x=x(z)
+ [(L(z, x) − π (x)Lπ (z, x)) (xK + xkωK)]x=x(z) ,
with
Ψ−1
K ≡ 1 −
z
x(z)
x(z)
KK(z, x)dx + [(K(z, x) − π (x)Kπ (z, x)) xK]x=x(z) (25)
+ [(K(z, x) − π (x)Kπ (z, x)) (xK + xkωK)]x=x(z) .
Proof For the proof and more detailed speciﬁcations of all terms see the Appendix.
Inspecting the ﬁrst order conditions in Theorem 1, we see that the condition (22) forms a
functional equation in the unknown government policy function, π, with the side conditions
(23)-(24) and the condition on the value of the Lagrangean multiplier, μ. From the setup
of the problem it is clear that the only free boundary values of the government policies are
the values at the lower and upper bounds, x(z) and x(z). Additionally, the more detailed
ﬁrst-order conditions in the Appendix contain savings and distribution functions, h and λ,
and their derivatives with respect to π, π , and K, i.e. hπ, hπ , hK, hπ π , hπ π, λπ, λπ , λK,
λπ π , and λπ π, respectively.
If we knew how agents’ saving policies h and simultaneously how the distribution λ
depend on the government policy schedule, i.e. if we could solve at equilibrium prices
for the optimal policy π which is a function of the distribution and prices which in turn
are determined by h(π, π ) which is itself a function of the optimal policy and prices, the
task of the derivation of the ﬁrst order conditions for this dynamic optimization would be
straightforward. However, not only we have to solve for these functions simultaneously
but also we are in a much more diﬃcult situation since these functions are speciﬁed only
implicitly as operator equations. The proper speciﬁcation of the derivative of the savings
and distribution functions with respect to the government policy functions requires a
generalized concept of derivative, the so called Frechet derivative.
15
5.1 The Eﬀects of Government Policy on Stationary Recursive
Equilibrium
For any government policy schedule π, an agents’ saving policy and the distribution functions
are known only implicitly as a solution to the two operator equations (13)-(14) together
with the aggregate conditions specifying the equilibrium stationary prices. In order
to derive the ﬁrst order conditions in the calculus of variation, we need to specify the
derivatives of the integrand functions, W and G, with respect to the marginal changes in
government policy, π and π . For this purpose, we use the concept of generalized derivatives
on mappings between two Banach spaces (B-spaces), the Fr´echet derivatives.9
Deﬁnition 6 (The Fr´echet Derivative) Given a nonlinear operator N(u) on function
u, its Fr´echet diﬀerential is deﬁned by
NuΔ≡
∂N(u + εΔ)
∂ε
|ε=0,
where Nu is the Fr´echet derivative.
We now formulate functional equations for the unknown F-derivatives hπ, hπ , hK,
hπ π , hπ π, λπ, λπ , λK, λπ π , and λπ π. The following Lemma derives the eﬀects of the
government policy function π on the operator Euler equation by specifying ﬁve unknown
“sensitivity” functions hπ : Z × K −→ R+
, hπ : Z × K −→ R+
, hK : Z × K −→ R+
,
hπ ,π : Z × K −→ R+
, and hπ π : Z × K −→ R+
.
Lemma 1 (The Eﬀects of π, π , and K on the Euler Equation) The total Fderivatives
of the operator Euler equation F with respect to the government policy function
π, to its derivative π , and to the aggregate capital stock K are, respectively,
Fπ = u (c)cπ − β
z+
u (c+
)c+
π R+
+ u (c+
)R+
π Q(z, z+
) = 0, (26)
Fπ = u (c)cπ − β
z+
u (c+
)c+
π R+
+ u (c+
)R+
π Q(z, z+
) = 0, (27)
FK = u (c)cK − β
z+
u (c+
)c+
KR+
+ u (c+
)R+
K Q(z, z+
) = 0, (28)
9
The compliance of the Frechet derivatives (also called the F-derivatives) with the derivations of the
ﬁrst order conditions in the calculus of variation is reﬂected by the fact that the F-diﬀerential is identical
to the variation. Our derivations are more complicated than the standard Fr´echet derivative because our
operators are recursive.
16
and further
Fπ π = u (c) [cπ ]2
+ u (c) cπ π (29)
− β
z+
u c+
c+
π
2
+ u c+
c+
π π R+
+ 2u c+
c+
π R+
π
+ u c+
R+
π π Q(z, z+
) = 0,
and
Fπ π = u (c) cπ cπ + u (c) cπ π (30)
− β
z+
u c+
c+
π c+
π + u c+
c+
π π R+
+ u c+
c+
π R+
π + c+
π R+
π
+ u c+
R+
π π Q(z, z+
) = 0.
Proof For the proof and the deﬁnition of terms see the Appendix.
Similarly, we derive the eﬀects of the government policy on the shape of the distribution
function λ by specifying functional equations which implicitly determine the unknown
“sensitivity” functions λπ : Z × K −→ R+
, λπ : Z × K −→ R+
, λK : Z × K −→ R+
,
λπ π : Z × K −→ R+
and λπ π : Z × K −→ R+
.
Lemma 2 (The Eﬀects of π, π , and K on the Stationary Distribution Function)
The total F-derivative of the operator stationary distribution function L with respect to
the government policy function π, to its derivative π and to the aggregate capital stock K
are, respectively,
Lπ = λπ(z+
, k+
) −
z
λk z, h−1
(z, k+
) h−1
π (z, k+
) + λπ z, h−1
(z, k+
) Q(z, z+
) = 0,
Lπ = λπ (z+
, k+
) −
z
λk z, h−1
(z, k+
) h−1
π (z, k+
) + λπ z, h−1
(z, k+
) Q(z, z+
) = 0,
LK = λK(z+
, k+
) −
z
λk z, h−1
(z, k+
) h−1
K (z, k+
) + λK z, h−1
(z, k+
) Q(z, z+
) = 0,
and further
Lπ π = λπ π (z+
, k+
) −
z
λπ π z, h−1
(z, k+
) + λπ k z, h−1
(z, k+
) h−1
π (z, k+
)
+ λπ k z, h−1
(z, k+
) + λkk z, h−1
(z, k+
) h−1
π (z, k+
) h−1
π (z, k+
)
+ λk z, h−1
(z, k+
) h−1
π π (z, k+
) Q(z, z+
) = 0, (31)
and
Lπ π = λπ π(z+
, k+
) −
z
λπ π z, h−1
(z, k+
) + λπ k z, h−1
(z, k+
) h−1
π (z, k+
)
+ λπk z, h−1
(z, k+
) + λkk z, h−1
(z, k+
) h−1
π (z, k+
) h−1
π (z, k+
)
+ λk z, h−1
(z, k+
) h−1
π π(z, k+
) Q(z, z+
) = 0. (32)
17
Proof For the proof and the deﬁnition of terms see the Appendix.
In this way we obtain functional equations (11), (26)-(30), (12), and (31)-(32) in the
unknown functions h, hπ, hπ , hK, hπ π , hπ π, λ, λπ, λπ , λK, λπ π , and λπ π, respectively.
Finally, by adding the ﬁrst-order conditions from Theorem 1, the problem of ﬁnding the optimal
government policy π is a system of thirteen functional equations in thirteen unknown
functions with two side conditions and one condition on the Lagrange multiplier.
5.2 Analytical Results
Despite the complexity of the problem we are able to derive several analytical results. It
will be useful to compare the optimal government policy and its eﬀects to those of a ﬂat-tax
economy in which the government also consumes a fraction g of the total output. Note
that the term10
gy(z, x) in equation (19) captures the fraction of government expenditures
“related” to an agent (z, x). This fraction of government expenditures gy(z, x) is identical
to the amount of taxes hypothetically paid by the agent (z, x) under a ﬂat tax g on total
income. We will further use this concept as a useful benchmark in the following analysis.
Since the ﬁrst-order conditions in Theorem 1 specify the explicit transversality conditions
on agents with the lowest and the highest taxable activity, they can be used for
qualitative results on the behavior of the policy schedule at these boundary points.
In the following Proposition we relate the tax payment of the “poorest” agent under
the optimal tax policy to the tax payment in the economy with a ﬂat tax g. This poorest
agent is constrained at the lowest value of capital k and is hit by the lowest shock z.
Proposition 1 (Optimal Tax at Lowest Taxable Activity) The transversality condition
in equation (24) in Theorem 1 implies that for an agent with the lowest taxable
activity level x(z), the optimal diﬀerence between taxes paid π(x(z))x(z) and the income
taxes paid under a ﬂat tax regime, gy(z, x(z)), is proportional to the sum of the agent’s
utility and the marginal contribution of his savings to the aggregate welfare,
π(x(z))x(z) − g y(z, x(z)) = −
W [u(c(z, x(z)))] + Ψk
μ
,
where c(z, x(z)) = y(z, x(z)) − π(x(z))x(z) − k is consumption, y(z, x(z)) = rk + wz is
before-tax income, μ < 0 is the shadow price of government expenditures, and Ψ is the
marginal eﬀect of the aggregate capital stock on the aggregate welfare deﬁned in Theorem
1.
Proof See the Appendix.
10
To shorten the notation we write gy(z, x) = gy(z, k(z, x)).
18
The following corollary states the conditions for which the amount of taxes paid by the
poorest agent under the optimal tax policy is larger than that under a ﬂat income tax, i.e.
π(x(z))x(z) > gy(z, x(z)).
Corollary 1 If the savings contribution of the poorest agent to aggregate welfare is nonnegative,
Ψk ≥ 0, then at the lowest taxable activity level x(z) the amount of taxes paid
under the optimal tax policy π(x(z))x(z) is larger than that under a ﬂat income tax,
gy(z, x(z)).
The poorest agent pays more taxes than in the ﬂat-tax regime if one of these three
conditions are satisﬁed: Ψ = 0, or if k = 0, or if simultaneously Ψ > 0 and k > 0). The
Corollary illustrates the incentives to the agent to accumulate more assets.11
We discuss
these incentives in the following Section.
In a similar way, the Proposition below speciﬁes the optimal tax payment of the “richest”
agent relative to his tax burden at the ﬂat tax equal to g. The richest agent’s current
and future savings are at the endogenous upper bound k(z) = k and he receives the highest
shock z.
Proposition 2 (Optimal Tax at Highest Taxable Activity) Assuming π (x(z)) =
δx
δπ
(x(z))
−1
, the transversality condition in equation (23) in Theorem 1 implies that for
an agent with the highest taxable activity x(z), the optimal diﬀerence between taxes paid
π(x(z))x(z) and the taxes paid under the ﬂat tax regime g y(z, x(z)), is proportional to
the sum of the agent’s utility and the marginal contribution of his savings to the aggregate
welfare,
π(x(z))x(z) − g y(z, x(z)) = −
W [u(c(z, x(z)))] + Ψk
μ
,
where c(z, x(z)) = y(z, x(z))−π(x(z))x(z)−k(z) is consumption, y(z) = rk+wz is beforetax
income, μ < 0 is the shadow price of government expenditures, and Ψ is the marginal
eﬀect of the aggregate capital stock on the aggregate welfare deﬁned in Theorem 1.
Proof See the Appendix.
Notice that as ωπ = hπ
1−hk
< 0, δx
δπ
(x(z))
−1
= 1
xk(z,k)ωπ(z,k)
< 0 implies that the assumption
on the slope of the tax schedule at x(z) is satisﬁed whenever it is non-decreasing, i.e.
π (x(z)) ≥ 0, and that Ψ ≥ 0.
The following corollary states the conditions for which the amount of taxes paid by
the richest agent under the optimal tax schedule is larger than that under a ﬂat tax, i.e.
π(x(z))x(z) > gy(z, x(z)).
11
If a substantial debt can be accumulated, k < 0, the ability of the government to tax capital stock
becomes limited. Finally, it is easy to show that the equilibrium might not exist if Ψ < 0.
19
Corollary 2 If the marginal contribution of the aggregate capital stock to the aggregate
welfare is nonnegative Ψ ≥ 0, and k ≤ 0 then at the highest taxable activity level the
optimal tax contribution π(x(z))x(z) is larger than that under a ﬂat tax regime gy(z, x(z)),
π(x(z))x(z) − g y(z, x(z)) = −
W (u(c(z, x(z))) + Ψk
μ
> 0.
Additionally,
π(x(z))x(z) − g y(z, x(z))
π(x(z))x(z) − g y(z, x(z))
≥
W (u(c(z, x(z)))
W (u(c(z, x(z)))
Ψk
W (u(c(z, x(z)))
+ 1
implies that the amount of taxes paid by the richest agent is greater than that of the poorest
agent, i.e.
π(x(z))x(z) > π(x(z))x(z).
These Corollaries show that the socially optimal amount of the tax revenues paid by
the ‘boundary’ agents depends only on these agents’ individual characteristics (on their
taxable activity, x, income, y, and consumption, c) and the two aggregate shadow prices:
the shadow price of government spending, μ, and the shadow price of the aggregate capital,
Ψ, both expressed in terms of social welfare.
Together, Corollaries 1 and 2 can say a lot about the shape of the optimal government
policy schedule. If k = 0, then both ends of the tax schedule are above the ﬂat-tax level and
the implied tax schedule is a “U-shape” function. Its bottom lies under the ﬂat-tax level
to clear the government budget constraint proportional to g. This is also the case if Ψ = 0
and if both Ψ > 0 and k > 0. Finally, if Ψ > 0 and k < 0, the optimal tax schedule could
either be “U-shaped” or an increasing function of income. These qualitative assessments
are conﬁrmed by the numerical results in our example in the following Section.
6 An Example: The Optimal Income Tax Schedule
In this section we demonstrate our method by ﬁnding the optimal government policy π
deﬁned as a tax schedule on total income from capital and labor. Therefore, the taxable
activity is
x(z, k; K) = y(z, k; K) = rk + wz,
and the individual budget constraint is
c ≤ (1 − π(x))x + k − k+
.
20
We assume that the borrowing constraint is
k = 0,
and that there are only two levels of shock, Z = {z, z}. The total tax revenues are equal
to a fraction g of the total output.
Thus the Euler equation (10) for a (z, k)-agent’s optimal savings function k+
(z, k) for
all (z, k) ∈ Z × B is now
u (c) ≥ β
z+
u (c+
) (1 − π(x+
) − π (x+
)x+
) r + 1 Q(z, z+
),
where c+
= (1 − π(x+
)) x+
+ k+
− k++
, x+
= rk+
+ wz+
, and k++
= k+
(k+
(z, k), z+
).
Note that for this speciﬁcation xk = yk = r and kx = 1/xk = 1/r.
For this concrete example we ﬁrst analyze the conditions for the existence of the stationary
recursive competitive equilibrium.
6.1 The Existence of Stationary Recursive Competitive Equilib-
rium
Because the tax schedule is an arbitrary function, we must ensure that the ﬁrst order
approach is valid.12
In order to characterize the admissible tax functions and to prove the
Schauder Theorem for economies with distortions, we follow the notation in Stokey, Lucas,
and Prescott (1989), Chapter 18. For each agent (z, k) ∈ B ×Z, denote the after-tax gross
income as
ψ(z, k) ≡ (1 − π(x(z, k))) x(z, k) + k.
Using ψ(z, k), rewrite the Euler equation as
u (ψ(z, k) − k+
(z, k))=β
z+
u (ψ(k+
(z, k), z+
) − k+
(k+
(z, k), z+
))ψ1(k+
(z, k), z+
)Q(z, z+
),
where
ψ1(k+
(z, k), z+
) = 1 − π(x(k+
(z, k), z+
)) − π (x(k+
(z, k), z+
)) x(k+
(z, k), z+
) r + 1
is the marginal after-tax return on an extra unit of investment. In the following Theorem
we establish the validity of the ﬁrst order approach and the existence of the competitive
equilibrium.
12
Again, we analyze the case of borrowing constrained agents in the Appendix.
21
Theorem 2 For a given tax schedule π : R+ → R, if for each (z, k) ∈ B × Z
1. ψ1(z, k) > 0, and
2. ψ is quasi-concave,
then the solution to each agent’s maximization problem and the stationary recursive competitive
equilibrium exist.
Proof See the Appendix.
The following corollary characterizes the set of admissible tax schedules that satisfy
the conditions of Theorem 2. We have earlier introduced the notation for the endogenous
upper bound on capital for any agent, k. Let {w, w} and {r, r} denote some arbitrary,
non-binding lower and upper bounds for equilibrium wage and interest rate, respectively.
Finally, επ
x(x) ≡ π (x)
π(x)
x is the elasticity of the tax rate to the taxable income.
Corollary 3 (Admissible Tax Schedule Functions) Let C2
(R+) be a set of continuously
diﬀerentiable functions from R+ to R. If a tax schedule function π ∈ C2
(R+) belongs
to the set of admissible tax schedules Υ,
Υ = π ∈ C2
(R+) : π(x) (1 + επ
x) < 1 +
1
r
for all x ∈ [rk + wz, rk + wz], then it satisﬁes the conditions of Theorem 2.
The above statement follows directly from the fact that ψ1(z, k) > 0 and that ψ is
quasi-concave. The corollary implies that there exists an upper bound on the marginal tax
rate, π(x) (1 + επ
x). This upper bound is not likely to bind for a very wide range of tax
schedules.13
6.2 The Shape of the Optimal Income Tax Schedule
To obtain the ﬁrst order conditions for the optimal income schedule we plug the examplespeciﬁc
terms into the general conditions in Theorem 1 and Lemmata 1 and 2. These terms
are listed in the Appendix C.
Adapting Propositions 1 and 2 and the related Corollaries 1 and 2 to our example
economy we obtain the following results.
13
As an example, for a realistic equilibrium interest rate r = 0.05, the upper bound on the marginal
tax rate is equal 1/(1 + r) 21. Therefore, for a high level of tax rate, say π(x) = 0.42, the elasticity
of the tax schedule at that level of income would have to be επ
x = 49 in order to violate the admissibility
condition. When numerically solving for the optimal tax schedule in the next Section we do not impose
any of these exogenous bounds but we check the admissibility of the optimal tax schedule ex post.
22
Corollary 4 (Optimal Tax Rates at Lowest and Highest Income) If π is the optimal
tax schedule on the total income, then
1. the optimal tax rate at the lowest income level x(z) is strictly greater than a ﬂat tax
g at that income level,
π(x(z)) > g;
2. and, provided that Ψ ≥ 0, the optimal tax rate at the highest income level x(z) is
strictly greater than the optimal tax rate at the lowest income level,
π(x(z)) > π(x(z)) > g.
Proof The results follow directly from Corollaries 1 and 2.
Thus both the poorest and richest agents pay higher taxes than in a corresponding
ﬂat-tax economy. The government budget constraint and the continuity of the optimal
tax function imply that there must exist a measure of agents at intermediate income levels
who face lower tax rates than the ﬂat tax rate. Thus the optimal tax schedule must be a
“U-shape” function.
We interpret these results from the point of view of the social planner who could use the
ﬁrst best, lump sum taxes. To insure agents against idiosyncratic shocks, he would set such
a tax schedule on each pair of individual states (z, k) to arrive at a stationary equilibrium
in which all agents would accumulate the same amount of capital K and consume the same
amount of goods. In other words, the distribution would consist of a mass of agents at two
points, (z, K) and (z, K). A U-shape transfer (tax) system is the way to induce agents to
arrive at this optimal outcome.
Although our government is constrained from using the ﬁrst best policy, it strives to
accomplish the same outcome. By imposing higher than average taxes on the poorest
and richest agents, the government tries to provide incentives for agents to save towards
the average capital stock. The poorest agents are motivated by high but decreasing tax
schedule, d(π(x)x)
dx
x(z)
< 0. Thus, according to the Euler equation, their expected return
on capital increases. On the other hand, the richest agents are discouraged from further
savings by the increasing tax schedule at high levels of capital. Finally, agents with savings
around the average capital stock are motivated by lower than average taxes to keep their
savings at the current level. The U-shape tax function provides the incentives for agents
to eventually arrive and stay at the individual capital stock k = K.
23
7 Numerical Solution
In this Section we solve for the optimal tax schedule and compare the associated steady
state allocations to those resulting from the existing progressive tax schedule in the U.S.
economy and from the usual ﬂat-tax reform. In order to evaluate the welfare implications
of these tax reforms, we conduct the transition analysis.
The solution to the problem of ﬁnding the optimal policy in the sense of the Operator
Steady State Ramsey Problem in our example leads, as in the general case, to solving
the functional system of the ﬁrst-order conditions given by Theorem 1 together with the
functional equations specifying the stationary equilibrium (expressed by h and λ) and the
sensitivity functions given by F-derivatives hπ, hπ , hK, hπ π , hπ π, λπ, λπ , λK, λπ π , and
λπ π. Thus we obtain the system of thirteen functional equations in thirteen unknown
functions with two side conditions and one condition on the Lagrange multiplier. We
solve this complex functional equation problem by the least squares projection method.
Its application to our problem and the approximation of the optimal tax schedule can be
found in Appendix D.14
7.1 Parameterization
The uninsurable idiosyncratic shock to labor productivity follows a two-state, ﬁrst order
Markov chain. We use the results of Heaton and Lucas (1996) who, using the PSID labor
market data, estimate the household annual labor income process between 1969 and 1984
by a ﬁrst-order autoregression of the form
log(ηt) = ¯η + ρ log(ηt−1) + t,
with ∼ N(0, σ2
). They ﬁnd that ρ = 0.53 and σ2
= 0.063. Tauchen and Hussey (1991)
approximation procedure for a two-state Markov chain implies zL = 0.665, zH = 1.335
and Q(zL, zL) = Q(zH, zH) = 0.74. These values imply an aggregate eﬀective labor supply
equal to one with agents evenly split over the two shocks.15
We set the discount factor at
β = 0.95. The rest of the parameters is taken from Prescott (1986), in particular α = 0.36,
δ = 0.1, and the preference parameter σ = 1.
14
For a more detailed explanation of the use of the projection methods to stationary equilibria in
economies with a continuum of heterogenous agents see Bohacek and Kejak (2002).
15
Similar parameterization is in Storesletten, Telmer, and Yaron (2007) with zL = 0.73, zH = 1.27 and
Q(zL, zL) = Q(zH, zH) = 0.82. Diaz-Jimenez, Quadrini, and Rios-Rull (1997) use zL = 0.5, zH = 3.0 and
Q(zL, zL) = 0.9811, Q(zH , zH) = 0.9261. In future research we plan to add life cycle features to the model
as in Ventura (1999).
24
Finally, for all steady states we consider a Ramsey problem in which government is
required to raise tax revenues equal to 20% of the total output, i.e. g = 0.2.
7.2 The U.S. Progressive Tax Schedule
We model the progressive tax schedule as Ventura (1999), the closest model analyzing a
ﬂat-tax reform in an economy with heterogeneous agents.16
An agent’s budget constraint
is
c + k ≤ rk + zw + k − T,
where T represents the amount of tax paid by the agent according to the progressive
tax schedule. The amount of tax is determined according to which tax bracket the total
taxable income, I = rk + max{0, zw − I∗
}, falls in, with a labor-income tax deductible
amount I∗
≥ 0. There are M brackets with associated tax rates, τm, m = 1, . . . , M,
deﬁned on intervals between brackets’ bounds I0, . . . , IM−1. For M = 5 the tax rates are
τm ∈ {0.15, 0.28, 0.31, 0.36, 0.396} and the tax brackets, expressed as a multiple of the
average income, Im−1 ∈ {0, 0.85, 2.06, 3.24, 5.79}. In addition, capital income, rk, is taxed
at a ﬂat rate τk = 0.25.
For income I ∈ (Im−1 − Im], the total tax is then
T = τ1(I1 − I0) + τ2(I2 − I1) + · · · + τm(I − Im−1) + τkrk.
The government budget constraint is cleared by ﬁnding an equilibrium value of the tax
exemption level I∗
. Aggregate statistics of the steady state are shown in the left column
of Table 1.
7.3 A Flat-Tax Reform
The ﬂat-tax reform consists of replacing the progressive tax schedule with a single ﬂat tax
τ on the total income from labor and capital. The budget constraint of each agent becomes
c + k ≤ (1 − τ)(rk + zw) + k.
Note that the ﬂat tax reform, like in Ventura (1999), does not eliminate taxation of capital
income. We ﬁnd that the equilibrium ﬂat tax rate is τ = 0.254.
INSERT TABLE 1 ABOUT HERE
16
Compared to his model, our agents are inﬁnitely lived, so we omit the life-cycle variables, accidental
bequests, government transfers, and social security tax and beneﬁts. Except for capital depreciation, we
do not consider tax deductions.
25
The middle column in Table 1 describes the steady state results. Relative to the
progressive tax schedule steady state, the ﬂat-tax reform increases the steady state levels
by similar magnitudes found in the literature: capital stock increases by 30%, output by
10.8%, consumption by 4.6%, and welfare by 3.9%. As in Ventura (1999), the ﬂat-tax
reform increases inequality: Gini income coeﬃcients rise from 0.22 to 0.31 before tax and
from 0.21 to 0.32 after tax.17
7.4 The Optimal Tax Schedule
We use our methodology described in the previous Sections to solve for the optimal tax
schedule that maximizes average steady state welfare18
. The right column in Table 1
summarizes the optimal tax schedule steady state. The impact of the optimal tax schedule
is very large. Steady state average welfare increases by 4.4%. Aggregate capital stock rises
by 49%, output by 15.8%, and consumption by 5.8%. Inequality increases too but not as
much as in the ﬂat-tax reform: Gini income coeﬃcients are 0.28 before and 0.27 after tax,
respectively. General equilibrium eﬀects cause the interest rate to drop by almost one half
and the wage to increase due to a higher productivity of labor used in production with
such a high capital stock. Compared to the ﬂat-tax steady state, capital stock increases
by 15%, output by 4.5%, consumption by 1.1%, and welfare by 0.8%.
INSERT FIGURE 2 ABOUT HERE
Figure 2 shows the optimal tax schedule and the marginal tax rate function. The average
tax rate is is a U-shaped function taxing the lowest total income at a 45%, decreasing to
a minimum of 19% and rising to 62% at the highest level of total income. Although the
whole shape of the tax function is important for the resulting allocations, a majority of
agents face the decreasing or the ﬂat part of the tax schedule. The marginal tax rate is also
U-shaped, almost ﬂat and close to zero for low incomes, falling to negative levels around
the average total income and then rising at high income levels. Note that the maximal
marginal rate is 2.5 and that the optimal tax schedule easily satisﬁes the admissibility
condition from Corollary 3 (the interest rate implies an upper bound equal to 19.7).
The optimal tax function τ is strictly positive and very nonlinear. Both results are
diﬀerent from Mirrlees (1971) static model with ﬁxed distribution of skills, where a welfare
17
Elimination of capital income tax in Lucas (1990) increases capital stock by 30-34% and consumption
by 6.7%. A ﬂat-rate reform with heterogeneous agents in Ventura (1999) increases the total capital stock
by one third, output by 15%. Without a well calibrated life-cycle earnings process we are not able to
match well the inequality coeﬃcients, especially that of wealth.
18
The residual errors of the functional equations associated to the main approximated functions are
reported in Table ??.
26
maximizing tax schedule is close to a linear, non-decreasing function. In his model, the
marginal tax rate is between zero and one, and zero at both ends of the distribution.
Compared to our model with insurance and savings incentives, Mirrlees (1971) results
follow from labor incentives related to the distribution of skills and consumption-leisure
preferences.19
We want to emphasize that our stationary distribution is endogenous and there are no
restrictions on the optimal tax schedule to be positive or to be of any particular shape.
Conesa and Krueger (2006), also in a general equilibrium framework but with added life
cycle features, searched for the optimal progressivity of a tax schedule, limiting their class
of tax schedules to monotone functions as in Gouveia and Strauss (1994).20
In this class of
functions, the optimal tax schedule is basically a ﬂat tax with a ﬁxed deduction, delivering
a welfare gain of 1.7% compared to the existing tax progressive tax in the United States.
The class of monotone functions seems rather restrictive for the optimal tax schedule. Our
class of admissible functions includes all progressive tax schedules but these were found
signiﬁcantly inferior with respect to the welfare criterion.
7.5 The Tradeoﬀ Between Eﬃciency and Distribution
Apart from the general equilibrium eﬀects, the huge welfare impact of the optimal tax
schedule arises from the distributional eﬀects. The stationary distributions of capital in
the three steady states are shown in Figure 3. Although both the ﬂat and the optimal
tax schedules increase the aggregate levels, the diﬀerence between them is that the ad
hoc ﬂat tax schedule does not take into account the distribution of agents. The ﬂat tax
reform helps more the agents with high incomes: the mean wealth increases much more
than the median so that the median/mean ratio falls to 0.77. In the ﬂat-tax steady state
the aggregate levels increase but from “the optimal distribution” point of view the mass
of agents moves too much to the left while wealthy agents emerge at the right tail of the
distribution. The progressive tax schedule has the lowest inequality measures because the
high taxes on rich agents narrows the distribution towards the mean. However, the low tax
19
Building on the Mirrlees (1971) and Mirrlees (1976) seminal work, Kocherlakota (2005), Golosov,
Kocherlakota, and Tsyvinski (2003) or Albanesi and Sleet (2003) study optimal social planner policies
with asymmetric information. In such an environment, positive capital income taxes are optimal despite
the associated eﬃciency loss and informational frictions are necessary for a characterization of the optimal
policies.
20
The simple search method cannot be used for computation of a stationary competitive equilibrium
which is the limit of the optimal dynamic tax schedule. We show in Bohacek and Kejak (2004) that under
some parameterization the ﬁrst order conditions of the general dynamic Ramsey problem can simplify to
the ﬁrst order conditions of the steady state Ramsey problem analyzed in this paper.
27
rates on low incomes do not provide incentives for the poor households to save and move
to higher income levels. In other words, it provides too much short-run insurance at the
cost of the long-run average levels.
INSERT FIGURE 3 ABOUT HERE
This is exactly what the optimal income tax schedule improves. The main mechanism
behind the large increase in the aggregate levels is the incentive eﬀect of the optimal tax
schedule. The U-shaped function in the top panel of Figure 2 eﬀectively concentrates the
agents around the mean, something what a social planner with an access to lump-sum
transfers would do.21
The high tax rate at low income levels provides incentives for these
agents to save more and move to higher income levels. On the other hand, the even higher
tax rate on high income discourages further savings by the wealthiest agents. In the middle
of the total income levels, the tax rate is lower than that found for the ﬂat-tax reform.
The optimal tax schedule preserves the median/mean wealth ratio of the progressive tax
schedule by increasing the median by 47% and the mean by 49%. The support of the
invariant distribution becomes wider but inequality measures do not increase as much as
in the ﬂat-tax reform.22
To further analyze the tradeoﬀ between eﬃciency and distribution, we adopt the approach
in Domeij and Heathcote (2004) to distinguish the eﬃciency gain from distributional
gains. The eﬃciency gain for an individual agent is the percentage of the original consumption
that would allow the agent to consume the same fraction of the aggregate consumption
after the reform as he or she was consuming in the original steady state. In the case of
logarithmic utility, the gain is the same for all agents (see Domeij and Heathcote (2004)
for a simple proof and other details). The distributional gain is the diﬀerence between the
individual welfare gain and the eﬃciency gain.23
Table 1 displays the average eﬃciency and distributional gains of the optimal steady
state relative to the other two steady states. It is apparent that the steady state associated
with the optimal, U-shaped tax function is welfare and eﬃciency superior to the other
21
In many countries, marginal taxes are favorable to middle income groups. In practice, high rates
on rich can break large fortunes while on poor they provide a ﬂoor for poverty. The result is a more
equal distribution. Saez (2002) studies the optimal progressivity of capital income tax in a partial equilibrium
model with exogenous labor and distribution. He ﬁnds that a progressive tax is a powerful tool to
redistribute accumulated wealth.
22
Table 1 also shows the fraction of agents constrained in their borrowing: only 1.16% of agents is
constrained in the progressive tax schedule steady state. The ﬂat tax schedule increases this number to
1.88%, while the optimal tax steady state it is 1.42 (Domeij and Heathcote (2004) obtained similar results).
23
The individual welfare gain is the percentage of the original consumption level that would make an
agent as well oﬀ as in the optimal tax steady state.
28
two steady states: both average welfare and eﬃciency measures are positive, and naturally
greater for the comparison with the progressive steady state. As it was noted before, the
optimal tax schedule obtains an average distributional loss relative to the progressive tax
(–0.57%) but a gain relative to the ﬂat tax steady state (2.86%).
INSERT FIGURE 4 ABOUT HERE
The individual gains, for agents with high and low labor productivity shocks, are shown
in Figure 4. They are monotonically decreasing functions for all agents at all asset and labor
income levels (with some exceptions). Most of the asset-poor agents have both welfare and
distributional gains while the rich have losses relative to the steady state with the ﬂat tax
schedule. There are two forces present: ﬁrst is the tax rate (especially for the rich agents in
the ﬂat tax steady state) and general equilibrium eﬀects. The huge welfare gains (5-20%)
for poor agents are mostly due to the higher wage in the optimal steady state. Note that
the big eﬃciency gain from the optimal tax schedule is not suﬃcient to compensate all
agents for the more unequal distribution (compared to the progressive tax steady state,
an agent with a low productivity shock has always a distributional loss in Figure 4, top
panel).
INSERT TABLE 2 ABOUT HERE
Table 2 shows the distribution of resources for quintiles of the wealth distribution.
Because of the high tax rate on incomes in the bottom quintile, these agents in the optimal
tax schedule steady state consume 6.5% less than those of the progressive tax schedule.
However, in all the other quintiles the optimal tax schedule steady state, agents consume
on average more than in the other two steady states. Dividing these levels by the average
consumption in each steady state, we can calculate average quintile consumption relative
to the steady state average. Under the optimal tax schedule, the bottom quintile consumes
73% of the average consumption, in the ﬂat tax it is 77%, in the progressive it is 82%. This
shows that the savings incentives of the optimal tax schedule overweight the insurance
aspects (i.e., redistribution) in both the progressive and the ﬂat tax schedules.
The distribution of capital reveals that the incentives contained in the optimal-tax
schedule move the distribution to higher capital levels. The poorest quintile owns on
average 17% more assets than in the progressive steady state. This increase is even larger
for the other quintiles (40% on the top). Again, the ﬂat-rate steady state leads to lower
level of savings by the bottom two quintiles. These levels are reﬂected in the shares of the
total capital stock. For all steady states the bottom quintile owns only around 5% of the
total stock while the top quintile around one third (43% in the ﬂat-tax steady state).
29
The investment-to-income ratios reveal the agents in the bottom quintile of the optimal
schedule invest much more than similar agents in the other two steady states. Agents in
the optimal tax schedule steady state invest 30% of their income, more than those in the
ﬂat-tax (27%) and progressive (22%) steady states. The investment is also more evenly
distributed over the quintiles. Note also that the ﬂat-rate tax schedule favors capital
accumulation by the top quintile.
The income and after-tax income distribution show the diﬀerences between the three
tax schedules. The progressive tax helps the bottom quintile while the ﬂat tax helps the
top quintile. The U-shape of the optimal tax provides the right incentives at the cost of
lowest after-tax income for the poor agents. Finally, the optimal tax actually equalizes
tax contribution share of total tax revenues across the quintiles. Both the ﬂat-tax and
progressive-tax steady states put more relative burden on the higher income quintiles.
INSERT FIGURE 5 ABOUT HERE
Finally, Figure 5 shows the sensitivity functions hπ and λπ. The top panel shows the
eﬀect of a change in the optimal tax schedule on the savings decision of agents. For the
low shock it is close to zero, for the high shock it is negative and monotonically decreasing.
The bottom panel displays the same eﬀects on the probability density function of the
stationary distribution λ, again for each shock. We know from the stationarity condition
of the distribution that the integral of these functions must be zero.24
7.6 Transition to the Optimal Tax Schedule Steady State
Pure welfare steady-state comparisons could be misleading because tax changes imply
substantial redistribution in the short run. In Domeij and Heathcote (2004) model of
capital tax cuts, the expected discounted present value of welfare losses during transition
are so large that they overturn the steady state welfare improvement. The short-run cost in
the form of higher labor taxes is too heavy a price to pay for all except for the wealth-richest
households.25
INSERT TABLE 3 ABOUT HERE
24
Our numerical solution is only very close to zero due to approximation errors.
25
This is similar to Garcia-Mila, Marcet, and Ventura (1995) and Auerbach and Kotlikoﬀ (1987) who
ﬁnd that reducing capital income taxation shifts the tax burden away from households who receive a large
fraction of their income from capital and towards those who receive a disproportionate fraction from labor.
Transition costs in Lucas (1990) reduce the welfare gains from zero capital tax reform to 0.75-1.25 percent
of average consumption in the initial steady state.
30
Table 3 shows the results for our tax reform experiment. It compares the expected
present discounted value from an unanticipated optimal tax reform of the progressive and
ﬂat-tax steady state. In each case the optimal tax schedule is imposed on the stationary
distribution of the initial steady state.26
We guess a suﬃciently large number of convergence
periods and iterate on paths of equilibrium interest rates and wages to clear markets in
each period of the transition, returning possible excess tax revenues to all agents in each
period. The convergence is relatively fast lasting around thirty periods.
INSERT FIGURE 6 ABOUT HERE
Contrary to Domeij and Heathcote (2004), we ﬁnd that the reform makes both the
mean and the median agents in the progressive tax schedule economy better oﬀ. Their
welfare gains are positive but smaller than in the pure steady-state comparison (3.44% and
3.86%, respectively, measured as per period consumption transfers as a percentage of the
initial steady state average consumption). The top panel in Figure 6 shows the expected
present discounted values in the progressive-rate steady state and at the moment of the
unanticipated reform to the optimal tax schedule. While 73% of the population is better
oﬀ from the reform, it is not Pareto improving as the poorest 27% of all households are
worse oﬀ (they are hit by high tax rates from the optimal schedule).
On the other hand, a transition from the ﬂat-tax steady state would not be supported
by the mean nor by the median agent (they loose 1.81% and 1.97%, respectively). The poor
and now also the wealthy, for whom the tax increases dramatically, are worse oﬀ during
the transition. The bottom panel in Figure 6 shows the expected present discounted values
of the ﬂat-rate steady state and of the transition to the optimal tax schedule. Political
support is not suﬃcient, equal only to 33% of the population. We do not know whether
an optimal transition would be welfare improving from this steady state.
As usual, this transition exercise shows that a tax reform is not Pareto improving for
all agents. However, the gains from the optimal tax reform of the existing progressive tax
schedule are so large that they are supported by majority of agents despite their transitional
costs. Conesa and Krueger (2006) also ﬁnd that a majority of the population would beneﬁt
from their optimal tax reform. However, in their case the poor and rich beneﬁt, while it is
the middle class (38%) who would be against the reform.
INSERT FIGURE 7 ABOUT HERE
26
Of course, it is not the optimal transition to the optimal tax schedule steady state. An optimal reform
would implement a time-speciﬁc optimal tax schedule at each period of the transition
31
Finally, Figure 7 shows the eﬃciency and distributional individual gains from transi-
tion.27
Relative to the steady state analysis, the averages for the progressive steady state
reform decline: while the average welfare and eﬃciency gains remain still positive the distributional
loss reaches negative 7%. A reform from the ﬂat rate steady state delivers average
welfare and eﬃciency losses but improves the distribution. Note that due to sizeable general
equilibrium eﬀects, the functions for poor agents are still positive and monotonically
declining.
8 Conclusions
Quah (2003) shows that average levels are of the ﬁrst order importance for economic
growth and welfare, much more important than inequality. Government policies focusing
on aggregate levels, including obviously optimal ﬁscal policy and taxation, are essential.
However, it is the distribution of agents that delivers these aggregate levels. This paper
shows that it is crucial to think of policies that target the distribution of agents. Only in
this way the high aggregate levels and welfare improvements can be achieved.
To our knowledge, this paper is the ﬁrst one that provides a solution method for such
optimal government policies in heterogeneous agent economies. We think of these policies
as optimal because they take into account their eﬀects on the distribution of agents. As
an example, we ﬁnd the optimal tax schedule for a steady state Ramsey problem in an
economy with heterogeneous agents. The optimal tax schedule is U-shaped, it increases
all aggregate levels by providing the right incentives for the agents to accumulate high
aggregate levels but not at the cost of increased inequality. Welfare gain in the steady
state is large: it is positive for both mean and median agent as well as in a transition
following an unanticipated optimal tax reform of the progressive tax schedule steady state.
The approach developed in this paper can be applied to any optimal government policy.
Within the ﬁeld of optimal taxation, in our future research we plan to study the optimal tax
schedule with elastic labor supply and realistic life-cycle income proﬁles. An endogenous
labor-leisure decision might aﬀect the shape of the optimal tax schedule, the aggregate
labor supply and the distribution of labor hours. We would also like to explore diﬀerent
(Rawlsian) welfare functions. Another topic that has received a lot of attention is the
optimal capital taxation in models with heterogeneous agents (see Aiyagari (1995) for the
initial contribution). Finally, we plan to use this methodology to analyze optimal dynamic
taxation.
27
These gains are deﬁned in the same way as in the steady state. A gain from transition is a constant,
per-period percentage of consumption in the original steady state that equalizes its corresponding expected
present discounted value from the whole transition. For details, see in Domeij and Heathcote (2004).
32
A Appendix: Analysis of the Borrowing Constrained
Agents
In general, for all z ∈ Z there exists a current minimal accumulated asset level k(z)
above which agents are not borrowing constrained. For these agents, i.e. for those with
(k, z) ∈ [k, k(z)) × Z and the next period savings k+
equal to k, the Euler equation is
satisﬁed in the form of inequality
u (c) > β
z+
u (c+
) 1 + y+
k
− π(x+
) + π (x+
) x+
x+
k Q(z, z+
),
where
c = (1 − π(x)) x + k − k,
c+
= (1 − π(x+
) )x+
+ k − h(z+
, k; K, π, π ),
y+
k
= r (K) ,
x+
= x(z+
, k; K)),
x+
k = xk(z+
, k; K))
implying h = k.
Taking this into account we can deﬁne the extended Euler equation operator,
F(h) ≡
F(h) for (z, k) ∈ Z × [k(z), k(z)],
h for (z, k) ∈ Z × [k, k(z)),
and thus the operator equation in the form F(h) = 0 determines the savings function with
the segment of constrained savings, h.
For the sake of brevity we present here only the eﬀect of the borrowing constrained
agents at the lowest shock, z, with the next period capital k. The stationarity of the
distribution functions implies
λ(z+
, k) =
k(z)
k
λ(z, k) Q(z, z+
) dk +
z=z
λ(z, h−1
(z, k+
; K, π, π )) Q(z, z+
),
where λ(z+
, k) is the mass of agents with the next period capital k and the next-period
shock z+
. Let us note such amended distribution function by λ.
Clearly, it means that the amended distribution function has a discontinuity at λ(·, k)
in the sense that
λ(z, k) > lim
k↓k
λ(z, k),
for all z ∈ Z. However, since we assume here that functions are integrable in the Lebesgue
sense, it follows that
k(z)
k
λ(z, k)dk =
k(z)
k
λ(z, k)dk,
for any z ∈ Z and the distribution functions λ and λ are equivalent. So we can simply
consider only the distribution function λ given by L in (12) and h given by (11).
33
B Appendix: Proofs
B.1 Proof of Theorem 1
For the ﬁrst order conditions for the Ramsey problem in (18)-(19) we deﬁne
J (ε) =
x(z)
x(z)−εδx
W z, x; K, π (x) , π (x) + μG z, x; K, π (x) , π (x) dx
+
z∈Z\{z,z}
x(z)
x(z)
W z, x; K, π (x) , π (x) + μG z, x; K, π (x) , π (x) dx
+
x(z)+εδx
x(z)
W z, x; K, π (x) , π (x) + μG z, x; K, π (x) , π (x) dx,
where
π (x) ≡ π∗
(x) + εδπ (x) ,
π (x) ≡ π∗
(x) + εδπ (x) ,
K ≡ K∗
+ εδK.
The dependence of the bounds on the value of shocks z ∈ Z makes our problem a little
harder than the standard calculus of variation problem. However, as Theorem 1 states we
construct the variations (the perturbation of functions from the optimum) being zero at
all (interior) bounds – see Figure 8. Therefore, only the values of the government policy
at the boundaries of the maximal interval, π(x(z)) and π(x(z)), are free while all other
interior bounds are ﬁxed. Then the condition
J (0) = lim
ε→0
dJ (ε)
dε
= 0
gives us the ﬁrst-order conditions.
J (0) =
z∈Z
x(z)
x(z)
{{Wπ [z, x; K∗
, π∗
(x) , π∗
(x)] + μGπ [z, x; K∗
, π∗
(x) , π∗
(x)]} δπ (x)
+ {Wπ [z, x; K∗
, π∗
(x) , π∗
(x)] + μGπ [z, x; K∗
, π∗
(x) , π∗
(x)]} δπ (x)
+ {WK [z, x; K∗
, π∗
(x) , π∗
(x)] + μGK [z, x; K∗
, π∗
(x) , π∗
(x)]} δK} dx
− W [z, x; K∗
, π∗
(x) , π∗
(x)] |x(z) + μG [z, x; K∗
, π∗
(x) , π∗
(x)] |x(z) (−δx)
+ W [z, x; K∗
, π∗
(x) , π∗
(x)] |x(z) + μG [z, x; K∗
, π∗
(x) , π∗
(x)] |x(z) δx,
where the last two lines come from the fact that the lower and upper bounds are free.
INSERT FIGURE 8 ABOUT HERE
Using the following notation
W (z, x) ≡ W [z, x; K∗
, π∗
(x) , π∗
(x)]
34
and
L (z, x) ≡ W (z, x) + μG (z, x) ,
integration by parts delivers
x(z)
x(z)
Lπ (z, x) δπ (x) dx = [Lπ (z, x) δπ (x)]
x(z)
x(z) −
x(z)
x(z)
d
dx
Lπ (z, x) δπ (x) dx.
Thus we can rewrite the formula above in a more compact form as
J (0) =
z∈Z
x(z)
x(z)
Lπ (z, x) −
d
dx
Lπ (z, x) δπ (x) + LK (z, x) δK dx (33)
+ [Lπ (z, x) δπ (x)]
x(z)
x(z) − L (z, x) |x(z) (−δx) + L (z, x) |x(z)δx.
At the free upper bound, the variation at the end-value of the policy function, δπ, can be
expressed as
δπ ≡ π (x + δx) − π∗
(x) = π (x) + π∗
(x) δx − π∗
(x) ,
and
δπ (x) = π (x) − π∗
(x) δx.
This implies that
δπ (x) = δπ − π∗
(x) δx, (34)
i.e. the variance of the policy function at the upper bound can be expressed as a function
of the variance at the end-value of policy function, δπ, and the variance at the end-value
of the taxable activity value, δx.
We can similarly specify the variation of the start-value of the policy function,
δπ (x) = −δπ + π∗
(x) δx. (35)
The situation is more complicated at the equality constrained endpoint x (z), because
the upper bound for capital, k, is implicitly given by the saving function, k = h z, k; K, π .
The total variance diﬀerential is
δk hk z, k; K, π − 1 + hK z, k; K, π δK + hπ z, k; K, π δπ = 0
and thus
δk = 1 − hk z, k; K, π
−1
hK z, k; K, π δK + hπ z, k; K, π δπ . (36)
As x = x z, k (z; K, π) ; K , we determine the variation
δx = xk z, k (z; K, π) ; K δk + xK z, k (z; K, π) ; K δK,
where we can further substitute for δk from (36)
δx = xk z, k; K ωK + xK z, k; K δK + xk z, k; K ωπδπ, (37)
35
where
ωK ≡
δk
δK
=
hK z, k; K, π
1 − hk z, k; K, π
,
ωπ ≡
δk
δπ
=
hπ z, k; K, π
1 − hk z, k; K, π
.
Going back to (33) and using (34) and (35) we obtain
J (0) =
z∈Z
x(z)
x(z)
Lπ (z, x) −
d
dx
Lπ (z, x) δπ (x) + LK (z, x) δK dx (38)
+ Lπ (z, x) |x(z)δπ − [π (x) Lπ (z, x) − L (z, x)]x(z) δx
+ Lπ (z, x) |x(z)δπ − [π (x) Lπ (z, x) − L (z, x)]x(z) δx.
Since the upper bound is equality constrained, δx is not independent and we need to
use (37) to obtain
J (0) =
z∈Z
x(z)
x(z)
Lπ (z, x) −
d
dx
Lπ (z, x) δπ (x) + LK (z, x) δK dx (39)
+ Lπ (z, x) |x(z)δπ + [L (z, x) − π (x) Lπ (z, x)]x(z) xKδK
+ Lπ (z, x) |x(z)δπ + [L (z, x) − π (x) Lπ (z, x)]x(z) {(xK + xkωK) δK + xkωπδπ} .
For the variation of the aggregate capital δK we use equation (17)
δK =
z∈Z
x(z)
x(z)
{Kπ (z, x) δπ (x) + Kπ (z, x) δπ (x) + KK (z, x) δK} dx
− K (z, x) |x(z) (−δx) + K (z, x) |x(z)δx,
again using integration by parts and the conditions for the free boundary points,
δK =
z∈Z
x(z)
x(z)
Kπ (z, x) −
d
dx
Kπ (z, x) δπ (x) + KK (z, x) δK dx
+ Kπ (z, x) |x(z)δπ + [K (z, x) − π (x) Kπ (z, x)]x(z) xKδK
+ Kπ (z, x) |x(z)δπ + [K (z, x) − π (x) Kπ (z, x)]x(z) {(xK + xkωK) δK + xkωπδπ} .
The variation of the aggregate capital is
δK = ΨK
z∈Z
x(z)
x(z)
Kπ (z, x) −
d
dx
Kπ (z, x) δπ (x) dx
+ Kπ (z, x) |x(z)δπ + Kπ (z, x) |x(z) + [K (z, x) − π (x) Kπ (z, x)]x(z) xkωπ δπ ,
where
Ψ−1
K ≡ 1 −
z∈Z
x(z)
x(z)
KK (z, x) dx − [K (z, x) − π (x) Kπ (z, x)]x(z) xK
− [K (z, x) − π (x) Kπ (z, x)]x(z) (xK + xkωK) .
36
Substituting the formula for δK into (39) we get
δJ =
z∈Z
x(z)
x(z)
Lπ (z, x) + ΨKπ (z, x) −
d
dx
(Lπ (z, x) + ΨKπ (z, x)) δπ (x) dx
+ [Lπ (z, x) + ΨKπ (z, x)] |x(z)δπ + [Lπ (z, x) + ΨKπ (z, x)] |x(z)
+ [L (z, x) + ΨK (z, x) − π (x) (Lπ (z, x) + ΨKπ (z, x))]x(z) xkωπ δπ,
with
Ψ ≡
δL
δK
= ΨK
z∈Z
x(z)
x(z)
LK (z, x) dx + [L (z, x) − π (x) Lπ (z, x)]x(z) xK
+ [L (z, x) − π (x) Lπ (z, x)]x(z) (xK + xkωK) .
Now, in order to get FOCs we assign δJ = 0. Since the ﬁrst term is zero for any δπ (x)
for all x, the following terms must be zero
z∈Z
Lπ (z, x) + ΨKπ (z, x) −
d
dx
(Lπ (z, x) + ΨKπ (z, x)) = 0,
[Lπ (z, x) + ΨKπ (z, x)] |x(z) = 0,
[L (z, x) + ΨK (z, x)] |x(z) +
1
xkωπ
− π (x) [Lπ (z, x) + ΨKπ (z, x)]x(z) = 0.
Note that
δx
δπ
−1
=
1
xkωK
.
If we denote the modiﬁed Lagrange function by L(z, x) ≡ L (z, x) + ΨK (z, x) , then
clearly the derived FOCs are those in (22)-(25). Q.E.D.
B.1.1 Deﬁnition of Terms in Theorem 1
Recall that according to (16)
W [z, x; K, π (x) , π (x)] = W [u (c (z, k(z, x; K); K, π, π ))] λ [z, k(z, x; K); K, π, π ] kx (z, x; K) ,
then
Wπ (z, x) δπ (x) = lim
ε→0
dW [z, x; K, π (x) , π (x)]
dε
,
with
c(z, k(z, x; K); K, π, π ) = y(z, k(z, x; K); K) − π(x)x + k(z, x; K) − h(z, k(z, x; K); K, π, π ),
y(z, k(z, x; K); K) = r (K) k(z, x; K) + w (K) z,
and π (x) ≡ π (x) + εδπ (x) .
37
To simplify notation we will omit the obvious arguments, writing c =
c (z, k(z, x; K)) , cπ = cπ (z, k(z, x; K)) , λ = λ (z, k(z, x; K)) and so on. Thus,
Wπ = {W [u (c)] u (c) cπλ + W [u (c)] λπ} kx,
where cπ = −x − hπ, and the Frechet derivatives of the savings policy and the distribution
function, hπ and λπ, are given by Lemmas 1 and 2, respectively.
Similarly,
Wπ = {W [u (c)] u (c) cπ λ + W [u (c)] λπ } kx, (40)
where cπ = −hπ , for the Frechet derivatives hπ and λπ . Finally, for hK and λK,
WK = {W [u (c)] u (c) cKλ + W [u (c)] [λK + λkkK]} kx + W [u (c)] λkxK,
where
rK = F11 K, L ,
wK = F21 K, L ,
cK = rKk + wKz + [1 + r − hk] kK − hK.
For the equation
d
dx
Wπ (z, x) = Wπ π (z, x) π (x) + Wπ π (z, x) π (x) + Wπ x (z, x) ,
we obtain, using W = W [u (c)], W = W [u (c)] u (c), and W = W [u (c)] [u (c)]2
+
W [u (c)] u (c),
Wπ π (z, x) = W [cπ ]2
λ + W (cπ π λ + 2cπ λπ ) + Wλπ π kx,
Wπ π (z, x) = {W cπ cπλ + W [cπ πλ + cπ λπ + cπλπ ] + Wλπ π} kx,
Wπ x (z, x) = {W cπ cxλ + W [cπ xλ + cπ λx + cxλπ ] + Wλπ x} kx + {W cπ λ + Wλπ } kxx,
with cπ π = −hπ π , cπ π = −hπ π, cπ x = −hπ kkx(z, x), and λπ x = λπ kkx(z, x).
According to (17),
G [z, x; K, π (x) , π (x)] = [π(x)x − g y(z, k(z, x; K); K)] λ [z, k(z, x; K); K, π, π ] kx (z, x; K) ,
and therefore,
Gπ (z, x) = {xλ + [π(x)x − g y] λπ} kx,
Gπ (z, x) = [π(x)x − g y] λπ kx, (41)
GK (z, x) = {−g [rKk + rkK + wKz] λ + [π(x)x − g y] [λK + λkkK]} kx + [π(x)x − g y] λ kxK,
and
d
dx
Gπ (z, x) = Gπ π (z, x) π (x) + Gπ π (z, x) π (x) + Gπ x (z, x) ,
Gπ π (z, x) = {[π (x) x − gy] λπ π } kx,
Gπ π (z, x) = {xλπ + [π (x) x − g] λπ π} kx,
Gπ x (z, x) = {[π (x) x − grkx] λπ + [π (x) x − gy] λπ x} kx + [π (x) x − gy] λπ kxx.
38
Lastly, according to (17)
K [z, x; K, π (x) , π (x)] = k(z, x; K) λ [z, k(z, x; K); K, π, π ] kx (z, x; K) ,
and thus
Kπ (z, x) = kλπkx,
Kπ (z, x) = kλπ kx, (42)
KK (z, x) = {kKλ + k [λkkK + λK]} kx + k λ kxK,
and
d
dx
Kπ (z, x) = Kπ π (z, x) π (x) + Kπ π (z, x) π (x) + Kπ x (z, x) ,
Kπ π (z, x) = kλπ π kx,
Kπ π (z, x) = kλπ πkx,
Kπ x (z, x) = k2
x + kkxx λπ + kλπ xkx.
B.2 Proof of Lemma 1
From equation (16) the Euler equation operator with the variation π ≡ π + εδπ is
F(h, π) (z, k; K, π(x), π (x)) ≡ u (c (z, k; K, π, π ))
− β
z+
u c+
z+
, h (z, k; K, π, π ) ; K, π, π R+
z+
, h (z, k; K, π, π ) ; K, π, π Q(z, z+
),
and
Fπ (z, k) δπ = lim
ε→0
dF [z, k; K, π (x) , π (x)]
dε
,
Using abbreviated notation x = x(z, k), c = c(z, k), h = h(z, k), h+
= h(z+
, h), x+
=
x(z+
, h), y+
= y(z+
, h), π+
= π(z+
, h), c+
= c+
(z, z+
, k), and R+
= R+
(z, z+
, k),
Fπ (z, k) = u (c) cπ − β
z+
u c+
c+
π R+
+ u (c+
)R+
π Q(z, z+
) = 0,
where
c = y − ˜π(x)x + k − h,
y = rk + wz,
c+
= y+
− ˜π(x+
)x+
+ h − h+
,
R+
= 1 + y+
k − π x+
+ π x+
x+
x+
k .
Terms for Fπ above and in equation (26) are
cπ = −x − hπ,
c+
π = 1 + r − h+
k − π +
x+
+ π+
x+
k hπ − h+
π − x+
,
R+
π = rhkπ − 1 + 2π x+
+ π x+
x+
x+
k hπ x+
k − π +
x+
+ π+
x+
kkhπ.
39
For
Fπ (z, k) = u (c) cπ − β
z+
u c+
c+
π R+
+ u c+
R+
π Q(z, z+
) = 0,
we use terms
cπ = −hπ ,
c+
π = 1 + r − h+
k − π +
x + π+
x+
k hπ − h+
π ,
R+
π = rhkπ − x+
+ 2π x+
+ π x+
x+
x+
k hπ x+
k − π +
x+
+ π+
x+
kkhπ .
For
FK (z, k) = u (c)cK − β
z+
u c+
c+
KR+
+ u c+
R+
K Q(z, z+
) = 0,
the terms are, as well as for equation (28),
cK = rKk + wKz − [π (x)x + π (x)] xK − hK,
c+
K = rKh + wKz+
− π +
x+
+ π+
x+
K + x+
k hK − h+
K,
R+
K = rKhk + rhkK − 2π x+
+ π x+
x+
x+
k x+
K + x+
k hK − π +
x+
+ π+
x+
kK + x+
kkhK .
For
Fπ π (z, k) = u (c) c2
π + u (c) cπ π
−β
z+
u c+
c+
π
2
+ u c+
c+
π π R+
+ 2u c+
c+
π R+
π + u c+
R+
π π Q(z, z+
) = 0,
we use, as well in equation (27), the terms
cπ π = −hπ π ,
c+
π π = 1 + r − h+
kπ + h+
kkhπ hπ + 1 + r − h+
k hπ π − h+
π khπ + h+
π π
− x+
+ π x+
x+
+ 2π x+
x+
k hπ x+
k hπ
− π x+
x+
+ π x+
x+
kk (hπ )2
+ x+
k hπ π ,
R+
π π = rhkπ π − 3 x+
k
2
+ x+
x+
kk hπ − 3π x+
+ π x+
x+
x+
k
3
(hπ )2
− 2π x+
+ π x+
x+
3x+
k x+
kk (hπ )2
+ x+
k
2
hπ π
− π x+
x+
+ π x+
x+
kkk (hπ )2
+ x+
kkhπ π .
Finally,
Fπ π (z, k) = u (c) cπ cπ + u (c) cπ π − β
z+
u c+
c+
π c+
π + u c+
c+
π π R+
+ u c+
c+
π R+
π + c+
π R+
π + u c+
R+
π π Q(z, z+
) = 0.
40
Terms for Fπ π above and in equation (27) are
cπ π = −hπ π,
c+
π π = − h+
kπ + h+
kkhπ hπ + 1 + r − h+
k hπ π − h+
π khπ + h+
π π
− 1 + π x+
x+
+ 2π x+
x+
k hπ x+
k hπ
− π x+
x+
+ π x+
x+
kkhπ hπ + x+
k hπ π ,
R+
π π = rhkπ π − x+
k
2
+ x+
x+
kk hπ − 3π x+
+ π x+
x+
x+
k
3
hπ hπ
− 2π x+
+ π x+
x+
3x+
k x+
kkhπ hπ + x+
k
2
hπ π
− π x+
x+
+ π x+
x+
kkkhπ hπ + x+
kkhπ π .
Q.E.D.
B.3 Proof of Lemma 2
Using equation (16), the stationary distribution operator is
L(h, λ, π) ≡ λ(z+
, k+
; K, π, π ) −
z
λ[z, h−1
(z, k+
; K, π, π ); K, π, π ] Q(z, z+
),
and so
Lπ z+
, k+
δπ = lim
ε→0
dL [z+
, k+
; K, π (x) , π (x)]
dε
,
with π ≡ π + εδπ. Therefore, abbreviating for (K, π, π ),
Lπ z+
, k+
= λπ z+
, k+
−
z
λπ z, h−1
z, k+
+ λk z, h−1
z, k+
h−1
π z, k+
= 0
with h−1
π (z, k+
) = hπ (z, h−1
(z, k+
)) /hk (z, h−1
(z, k+
)) .
Similarly, we can derive the total F-derivative of the Euler equation operator with
respect to the derivative of government policy function,
Lπ z+
, k+
= λπ z+
, k+
−
z
λπ z, h−1
z, k+
+ λk z, h−1
z, k+
h−1
π z, k+
= 0
with h−1
π (z, k+
) = hπ (z, h−1
(z, k+
)) /hk (z, h−1
(z, k+
)) , and
LK z+
, k+
= λK z+
, k+
−
z
λK z, h−1
z, k+
+ λk z, h−1
z, k+
h−1
K z, k+
= 0,
with h−1
K (z, k+
) = hK (z, h−1
(z, k+
)) /hk (z, h−1
(z, k+
)) .
Further, we can derive the following total F-derivative of the Euler equation operator
Lπ π z+
, k+
= λπ π z+
, k+
−
z
λπ π z, h−1
z, k+
+ λπ k z, h−1
z, k+
h−1
π z, k+
+ λπ k z, h−1
z, k+
+ λkk z, h−1
z, k+
h−1
π z, k+ 2
+ λk z, h−1
z, k+
h−1
π π z, k+
= 0
41
where h−1
π π (z, k+
) = hπ π (z, h−1
(z, k+
)) /hkk (z, h−1
(z, k+
)) .
Finally,
Lπ π z+
, k+
= λπ π z+
, k+
−
z
λπ π z, h−1
z, k+
+ λπ k z, h−1
z, k+
h−1
π z, k+
+ λπk z, h−1
z, k+
+ λkk z, h−1
z, k+
h−1
π z, k+ 2
+ λk z, h−1
z, k+
h−1
π π z, k+
= 0
where h−1
π π(z, k+
) = hπ π (z, h−1
(z, k+
)) hkk (z, h−1
(z, k+
)) .
Q.E.D.
B.4 Proof of Proposition 1
Using the deﬁnition for L in equation (21), the boundary ﬁrst-order condition in (24) can
be expressed as
Lπ (z, x(z)) = Wπ (z, x(z)) + μGπ (z, x(z)) + ΨKπ (z, x(z)) = 0.
The term Wπ in (40) evaluated at (z, x(z)) gives
Wπ (z, x(z)) = {W [u (c (z, k))] u (c (z, k)) cπ (z, k) λ (z, k) (43)
+ W [u (c (z, k))] λπ (z, k)} kx (z, x(z)) .
From cπ = −hπ it follows that
cπ (z, k) = −hπ (z, k) = 0,
since the ‘variation’ of the savings policy function related to the lowest shock, h (z, ·), with
respect to the slope of the government policy function, hπ , at (z, k) is clearly zero. Note
that the saving function of the borrowing-constrained agents is ﬂat and equal to zero. It
implies that the ﬁrst term in (43) is also equal to zero. Using Gπ , and Kπ from (41) and
(42), respectively, we get
W [u (c (z, k))] + μ [π(x(z))x(z) − g y (z, k)] + Ψk = 0,
where we used the fact that both λπ (z, k) and kx (z, x(z)) are non-zero. The result of the
Proposition follows. Q.E.D.
B.5 Proof of Proposition 2
We see that terms L and Lπ appear in the boundary ﬁrst-order condition in (23). Using
the deﬁnition of L in (21), we get
L (z, x(z)) = W u c z, k + μ π(x(z))x(z) − g y z, k + Ψk λ z, k = 0,
42
where we have used (20), (17), and (17) for W, G, and K, respectively, and also the fact that
at the upper endogenous limit on capital, k, λ z, k = 0. Thus the ﬁrst-order boundary
condition (23) becomes
π (x(z)) −
kx(z, x(z))
ωπ(z, x(z))
Lπ (z, x(z)) = 0.
If we assume that π (x(z)) − kx(z,x(z))
ωπ(z,x(z))
> 0, then the condition above is satisﬁed if the
second term Lπ (z, x(z) is equal to zero. By the inspection of Wπ in (40) we see that since
λ z, k = 0 then again the ﬁrst term in (40) is zero and
W u c z, k + μ π(x(z))x(z) − g y z, k + Ψk = 0,
where we again used the fact that both λπ z, k and kx (z, x(z)) are non-zero. This implies
the result of the Proposition. Q.E.D.
B.6 Proof of Theorem 2
We need to show that the ﬁrst order approach to each agent’s maximization problem
is valid. First, agents maximize over a quasi-convex set: Ψ = {x ∈ B : 0 ≤ x ≤
ψ(k, z) for all (k, z) ∈ B × Z}. If the function ψ is increasing and quasi-concave, then
the set Ψ is quasi-convex. Further, we need to satisfy Assumptions 18.1 in Stokey, Lucas,
and Prescott (1989), particularly that (i) β ∈ (0, 1); (ii) utility function u : R+ → R is
twice continuously diﬀerentiable, strictly increasing and strictly concave function; (iii) for
some ¯k(z) > 0, ψ(k, z)−k is strictly positive on [0, ¯k(z)) and strictly negative for k > ¯k(z),
where the value ¯k, the maximum sustainable capital stock out of after-tax income for any
agent, is deﬁned as ¯k = max{¯k(z1), . . . , ¯k(zJ )}; and, (iv) given the tax-schedule function
the right-hand side of the Euler equation is strictly positive
β
z+
u (ψ(k+
(k, z), z+
) − k+
(k+
(k, z), z+
))ψ1(k+
(k, z), z+
) Q(z, z+
) > 0,
where
ψ1(k+
(k, z), z+
)Q(z, z+
) = 1 − τ(y(k+
(k, z), z+
)) − τ (y(k+
(k, z), z+
)) y(k+
(k, z), z+
) r+1.
It can be easily checked that the assumptions (i)-(iii) are satisﬁed from our previous
assumptions and the model. The assumption (iv) follows directly from the fact that ψ is
increasing in k, i.e. ψ1 > 0.
The other assumptions needed for proving the existence of a stationary recursive competitive
equilibrium (see Assumption 18.2 in Stokey, Lucas, and Prescott (1989)) are satisﬁed
: (i) the equilibrium marginal return on capital for any k ∈ B is ﬁnite (in our case
the interest rate r); and (ii) that limc→0 u (c) = ∞.
Then to prove the Schauder’s Theorem, let C(B, Z) be the set of continuous bounded
functions h : B × Z → B and deﬁne a subset F = {h ∈ C(B, Z)} where the function h
satisﬁes 0 ≤ h(k, z) ≤ ψ(k, z), all (k, z) ∈ B ×Z, and h and ψ −h are nondecreasing. Note
43
that B × Z is a bounded subset of R2
and that the family of functions F is nonempty,
closed, bounded, and convex. Deﬁne an operator T on F
u (ψ(k, z) − (Th)(k, z)) = β
z+
u (ψ((Th)(k, z), z+
) − h[(Th)(k, z), z+
])
· [(1 − τ(y(·)) − τ (y(·))y(·)) r + 1] Q(z, z+
),
where y(·) = y((Th)(k, z), z+
).
Then it is easy to prove that T is well deﬁned, continuous and that T : F → F. From the
conditions on function h and ﬁnite return on capital, it follows that F is an equicontinuous
family. That the operator T has a ﬁxed point in F follows from the Schauder’s Theorem
(see e.g. Theorem 17.4 in Stokey, Lucas, and Prescott (1989)).
The existence of the stationary recursive competitive equilibrium is standard from the
monotonicity, Feller and mixing property of Q and the non-decreasing policy functions (see
Chapter 12 in Stokey, Lucas, and Prescott (1989)). Q.E.D.
C Terms for the Example
The terms from Theorem 1 for Example are those in B.1.1 together with
y(z, k(z, x; K)) = x,
kx(z, x; K) =
1
r (K)
,
kK (z, x; K) = −
wK (K) zr (K) + [x − w (K) z] rK (K)
[r (K)]2 ,
and
kxK(z, x; K) = −
rK
[r (K)]2
.
D Appendix: The Least Squares Projection Method
The optimal income tax policy, π, is a solution of the following system of operator equations:
1. FOC for π given by the Euler-Lagrange condition in (22);
2. the Euler equation (11) capturing the individual optimal behavior h;
3. ﬁve operator equations (26)-(30) for F-derivatives of h based on the Euler equation,
hπ, hπ , hK, hπ π , and hπ π;
4. the operator equation for distribution function, λ, in (12); and
5. ﬁve operator equations (31)-(32) for F-derivatives of λ based on the operator equation
for distribution function: λπ, λπ , λK, λπ π , and λπ π.
44
In order to solve the problem numerically, we ﬁrst approximate all the unknown functions
by combinations of polynomials from a polynomial base. Approximated solutions are
speciﬁed by unknown parameters transforming the original inﬁnitely dimensional problem
into a ﬁnite dimensional one. After substituting the approximated functions into the original
operator equations we construct the residual equations. Ideally, the residual functions
should be uniformly equal to zero. In practical situations, however, this is not achievable
and we limit the problem to a ﬁnite number of conditions, the so called projections,
whose satisfaction guarantees a reasonably good approximation. There are many possibilities
how to deﬁne the projections.28
We have chosen the least squares projection method
for its good convergence properties and advantage in solving systems of nonlinear operator
equations. We search for parameters approximating the functional equations that minimize
the squared residual functions.
As we speciﬁed above, in the system of operator equations given by (11),
(12), (22), (26)-(30), and (31)-(32), there are thirteen unknown classes of functions
{π, h, hπ, hπ , hK, hπ π , hπ π, λ, λπ, λπ , λK, λπ π , λπ π}. Since we assume that
the shocks are discrete, z ∈ Z = {z1, z2, . . . , zJ } and J > 1, we deﬁne
the following family of policy and distribution functions, and their derivatives
{hi
(k), λi
(k), hi
π(k), hi
π (k), hi
K(k), hi
π π (k), hi
π π(k), λi
π(k), λi
π (k), λi
K(k), λi
π π (k), λi
π π(k)}J
i=1,
for each shock value z1, z2, . . . , zJ . We interpret the policy function hi
as the next-period
capital function of an agent who was hit by a shock level zi. Analogously, the distribution
function λi
is the distribution of agents with the shock zi, etc. Similarly, we assign the
Euler and distribution function operators to every shock level, Fi
and Li
, respectively.
We approximate all unknown functions by the orthogonal Chebyshev polynomial base
{Ti(x)}∞
i=0 deﬁned for x ∈ [−1, 1].
As we have to deﬁne our approximation on a ﬁnite interval, we set the highest capital
level to a value k, greater than the endogenous upper bound on the stationary distribution.
Let the interval of approximation be [k, k] and the degrees of approximation for
{hi
(k), hi
π(k), hi
π (k), hi
K(k), hi
π π (k), hi
π π(k), λi
(k), λi
π(k), λi
π (k), λi
K(k), λi
π π (k), λi
π π(k)}
be M, Mπ, Mπ , MK, Mπ π , Mπ π, N, Nπ, Nπ , NK, Nπ π , Nπ π, P ≥ 2, respectively.29
Thus, we obtain
hi
m(k; ai
m) ≡
Mm
j=1
ai
m,jφj(k),
λi
m(k; bi
m) ≡
Nm
j=1
bi
m,jφj(k),
28
For an excellent survey and description of these methods see Chapter 11 in Judd (1998).
29
The details on Chebyshev polynomials can be found in Judd (1992), Judd (1998) or in any book on
numerical mathematics. The linear transformation ξ : [k, k] → [−1, 1] is necessary if we want to use the
Chebyshev polynomials on the proper domain. It is straightforward to show that ξ(k) = 2(k−k)/(k−k)−1.
45
with i ∈ {1, 2, ..., J} and m ∈ {∅, π, π , K, π π , π π}, and
π(x; c) ≡
P
j=1
cjφj(x),
for any k ∈ [k, k] and x ∈ [rk + wz, rk + wz], where φj(k) ≡ Tj−1(ξ(k)), and a’s, b’s, and
c’s are the unknown parameters.
Now we have to deﬁne residual functions as approximations to the original operator
functions (11), (12), (22), (26)-(30), and (31)-(32). Substituting the above approximations
for the unknown functions,
RL
(x; p) =
zj
Lπ h, Λ, π +
d
dx
Lπ h, Λ, π , (44)
RFi
m (k; p) = Fi
m(h, π), (45)
RLi
m (k; p) = Li
m(h, Λ, π), (46)
with i = 1, . . . , J and m ∈ {∅, π, π , K, π π , π π} where
p ≡ (a, aπ, aπ , aK, aπ π , aπ π, b, bπ, bπ , bK, bπ π , bπ π, c),
am ≡ (a1
m, a2
m, ..., aJ
m),
bm ≡ (b1
m, b2
m, ..., bJ
m),
and p is of a size S = J × ( m(Mm + Nm)) + P,
h ≡ (h, hπ, hπ , hK, hπ π , hπ π),
hm ≡ (h1
m, ..., hJ
m),
Λ ≡ (λ, λπ, λπ , λK, λπ π , λπ π),
λm ≡ (λ1
m, ..., λJ
m),
for any i = 1, . . . , J.
The least squares projection method searches for a vector of parameters p that minimizes
the sum of weighted residuals,
J
i=1
k
k m
[RFi
m (k; p)]2
+ [RLi
m (k; p)]2
w(k)dk +
k
k
[RL
(x(k); p)]2
w(k)dk,
with the weighting function given by w(k) ≡ 1 − 2k−k
k−k
2 −1/2
and i = 1, . . . , J. After
approximating the integrals by the Gauss-Chebyshev quadrature, we obtain a minimization
problem
min
p∈RS
ˇk
J
i=1 m
[RFi
m (k; p)]2
+ [RLi
m (k; p)]2
+ [RL
(x(k); p)]2
, (47)
46
with ˇk’s being the zeros of the polynomial φ of a degree greater
than the biggest degree of the polynomial approximations, i.e.
max{M, Mπ, Mπ , MK, Mπ π , Mπ π, N, Nπ, Nπ , NK, Nπ π , Nπ π, P}.
Since the least squares projection method sets up an optimization problem we can use
standard methods of numerical optimization, e.g. the Gauss-Newton or the LevenbergMarquardt
methods. Again, the discussion of these methods is not the aim of our paper.
However, we found that these traditional methods did not work in our high-dimensional
problem mainly due to possible multiple local solutions. We tried several other methods
(simulated annealing or genetic algorithm with quantization, for example) and ﬁnally succeeded
with a genetic algorithm with multiple populations and local search. The used
degrees of polynomial approximation for the optimal individual policy functions h, distribution
functions, λ, the related sensitivity functions hπ, λπ, and the optimal government
policy function, π, where 4, 12, 3, 3, and 4, respectively. The residuals of the related
functional equations were of the order 10−3
or 10−4
with the exception of hπ which was of
the order 10−2
.
47
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49
Steady State Results
Tax Schedule
Average Progressive Flat Tax Optimal
Capital 2.54 3.29 3.80
Output 1.40 1.54 1.62
Consumption 0.86 0.90 0.91
Median Wealth 2.45 2.55 3.60
Median/Mean Wealth Ratio 0.97 0.77 0.95
Interest Rate (%) 9.87 6.77 5.33
Wage 0.894 0.983 1.035
Gini Total Income 0.22 0.31 0.27
Gini Total After Tax Income 0.21 0.32 0.28
Constrained agents (%) 1.16 1.88 1.42
Steady State Welfare Gains from the Optimal Tax Schedule
Welfare Gain (%) 4.39 0.84 —
Eﬃciency Gain (%) 5.67 1.27 —
Distributional Gain (%) –0.57 2.86 —
Notes: Wealth in terms of accumulated capital stock. Constrained
agents is a fraction of agents whose wealth equals the exogenous lower
bound on capital. Welfare measured as consumption level corresponding
to the average utility. Average Welfare Gain measured as percentage of
the average consumption each agent would have to receive in the progressive
and the ﬂat-rate tax steady state so that the average welfare
equals that in the optimal tax steady state. Average Eﬃciency and
Distributional Gains deﬁned in the text.
Table 1: Steady State Results.
Steady State Distribution of Resources
Quintile
Tax Schedule 1st 2nd 3rd 4th 5th
Average Consumption Level
Optimal 0.6673 0.8272 0.9171 1.0076 1.1489
Flat 0.6990 0.8189 0.8904 0.9685 1.1213
Progressive 0.7120 0.8113 0.8642 0.9237 1.0148
Average Asset Level
Optimal 0.9166 2.4051 3.6545 4.9934 7.0381
Flat 0.6336 1.7708 2.8488 4.1933 6.9921
Progressive 0.7705 1.8195 2.5666 3.2509 4.2725
Average Investment/Income Ratio
Optimal 0.1222 0.2148 0.2935 0.3692 0.4643
Flat 0.0869 0.1630 0.2363 0.3197 0.4624
Progressive 0.1058 0.1774 0.2252 0.2592 0.2964
Average Income Level
Optimal 1.0141 1.1520 1.2353 1.3221 1.4620
Flat 0.9644 1.1009 1.1881 1.2883 1.4884
Progressive 0.9158 1.0565 1.1410 1.2337 1.3739
Average After-Tax Income Level
Optimal 0.6996 0.8341 0.9142 0.9964 1.1240
Flat 0.7194 0.8213 0.8863 0.9611 1.1103
Progressive 0.7319 0.8168 0.8644 0.9184 0.9948
Average Tax Contribution Share
Optimal 0.1944 0.1957 0.1995 0.2014 0.2090
Flat 0.1588 0.1824 0.1982 0.2133 0.2473
Progressive 0.1311 0.1702 0.1988 0.2258 0.2741
Table 2: Distribution of Resources in Steady States.
Tax Reform
Transition to the Optimal Tax Schedule Steady State
Transition From Steady State
Progressive Flat Tax
Average
Welfare Gain (%) 3.44 –1.81
Eﬃciency Gain (%) 4.14 –1.35
Distributional Gain (%) –7.29 3.25
Median
Welfare Gain (%) 3.86 –1.97
Political Support
% of Population 72.9 33.2
Notes: Average Welfare Gain from transition is a percentage of the average
consumption each agent would have to receive in each period of
transition so that the average welfare from transition equals in expected
present discounted value that of the initial steady state. Average Eﬃciency
and Distributional Gains from transition deﬁned in the text.
Table 3: Transition to the Optimal Tax Schedule Steady
State.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
k∗kL k(z)
k (k, z)
k (k, z)
k
kH k
Figure 1: Policy functions for the next period capital stock. An example with two productivity
shocks z > z. There is an exogenous lower bound kL and an endogenous upper
bound k∗
< kH. The stationary distribution has a unique ergodic set E = [kL, k∗
]. Agents
with shock z and capital stock k < k(z) are borrowing constrained.
1 1.5 2 2.5 3
0.2
0.3
0.4
0.5
0.6
The Optimal Tax Schedule
I
Average
Total Income
1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
The Optimal Marginal Tax Rate
Total Income
I
Average
Total Income
Figure 2: The Optimal Tax Schedule and The Optimal Marginal Tax.
0 2 4 6 8 10 12 14
0
0.5
1
1.5
2
2.5
3
3.5
x 10
−3 Distribution
λ
Assets
Optimal
Flat
Progressive
Figure 3: Stationary Distribution of Agents Over Assets in the Optimal, Flat, and Progressive
Tax Schedule Steady States.
0 1 2 3 4 5 6 7
−5
0
5
10
15
20
25
Individual Welfare and Distributional Gains in the Optimal Tax Schedule Steady State
Relative to the Progressive Tax Steady State (in %)
Welfare Gain zH
Welfare Gain z
L
Distributional Gain zH
Distributional Gain zL
0 2 4 6 8 10 12 14
−20
−10
0
10
20
30
Individual Welfare and Distributional Gains in the Optimal Tax Schedule Steady State
Relative to the Flat Tax Steady State (in %)
Assets
Welfare Gain zH
Welfare Gain z
L
Distributional Gain zH
Distributional Gain zL
Figure 4: Individual Gains in the Optimal Tax Schedule Steady State.
0 5 10 15 20 25 30
−1.5
−1
−0.5
0
0.5
Dk′
0 5 10 15 20 25 30
−3
−2.5
−2
−1.5
−1
−0.5
0
x 10
−3 Dλ
Assets
Figure 5: Savings and Distribution Sensitivity Functions for High (-) and Low (-.-) Productivity
Shocks.
0 2 4 6 8 10 12 14
−10
−5
0
5
10
Reform of the Progressive Tax Steady State
Value in the Initial Steady State (−) and from Transition (−.−)
0 5 10 15 20 25 30
−10
−5
0
5
10
15
20
Reform of the Flat Tax Steady State
Value in the Initial Steady State (−) and from Transition (−.−)
Assets
Figure 6: Welfare Gains from a Tax Reform. Transition from the Progressive and the Flat
Tax Steady State to the Optimal Tax Schedule Steady State.
0 1 2 3 4 5 6 7 8 9 10
−20
−15
−10
−5
0
5
10
15
Individual Welfare and Distributional Gains from Tax Reform (in %)
From Progressive to Optimal Tax Steady State
Welfare Gain zH
Welfare Gain z
L
Distributional Gain zH
Distributional Gain zL
0 2 4 6 8 10 12
−5
0
5
10
15
Individual Welfare and Distributional Gains from Tax Reform (in %)
From Flat to Optimal Tax Steady State
Assets
Welfare Gain zH
Welfare Gain z
L
Distributional Gain zH
Distributional Gain zL
Figure 7: Individual Gains from a Tax Reform.
x(z)
π∗
(x(z))
π(x − δx)
u
r
r
↔
δx
δπ δπ(x)
u
x(z)
π∗
(x(z))
u
x(z)
π∗
(x(z))
x(z)
π∗
(x(z))
π(x)
π(x + δx)
u
r
r
↔
δx
δπ(x)
↑
↓δπ
π
x
π
π∗
Figure 8: Variations Around Optimal Tax Schedule