MIT LIBRARIES
3 9080 02618 3563
llfll
HMH
HI
'
il §llfl$$ii
;
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/inequalitysocial00farh2
OBwev
B31
M415
Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
INEQUALITY, SOCIAL DISCOUNTING
AND ESTATE TAXATION
Emmanuel
Farhi
Ivan Werning
Working Paper 05-1
April 27, 2005
Rev. May 23, 2005
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
2803
http://ssrn.com/abstract=71
at
Inequality, Social Discounting
and Estate Taxation*
Emmanuel
MIT
Ivan Werning
MIT, NBER and UTDT
efarhi@mit.edu
iwerning@mit.edu
Farhi
June 2004
First Draft:
This Version: April 2005
Abstract
To what degree should
societies allow inequality to
be inherited?
What
taxation play in shaping the intergenerational transmission of welfare?
tions
by modeling
altruistically-linked individuals
productivity shocks.
uals
where
Our
positive
economy
efficiency requires immiseration:
is
who
role
should estate
We explore these
ques-
experience privately observed taste or
identical to
models with
inequality grows without
infinite- lived individ-
bound and everyone's
consumption converges to zero. However, under an intergenerational interpretation, previous
work only characterizes a particular set of Pareto-efficient allocations: those that value only
We
where the social welfare
on their welfare so that
the effective social discount rate is lower than the private one. For any such difference in social
and private discounting we find that consumption exhibits mean-reversion and that a steadystate, cross-sectional distribution for consumption and welfare exists, where no one is trapped
at misery. The optimal allocation can then be implemented by a combination of income and
the
initial
generation's welfare.
study other
efficient allocations
criterion values future generations directly, placing a positive weight
estate taxation.
We
find that the optimal estate tax
is
progressive:
fortunate parents face
higher average marginal tax rates on their bequests.
1
Introduction
Societies inevitably choose the inheritability of inequality.
tunity for newborns
and
incentives for altruistic parents
act plays out to determine long-run inequality
Some balance between
is
struck.
We
explore
equality of oppor-
how
and draw some novel implications
for
this balancing
optimal estate
taxation.
we thank Daron Acemoglu, Fernando Alvarez, George-Marios Angeletos,
Blanchard, Ricardo Caballero, Dean Corbae, Mikhail Golosov, Bengt Holrn-
*For useful discussions and comments
Abhijit Banerjee,
Gary Becker,
Olivier
strom, Narayana Kocherlakota, Robert Lucas, Casey Mulligan, Roger Myerson, Chris Phelan, Gilles Saint-Paul,
Nancy Stokey. Jean Tirole and seminar and conference participants at Chicago, Minnesota, MIT and the Texas
Monetary Conference held at the University of Austin in honor of the late Scott Freeman. This work begun motivated by a seminar presentation of Chris Phelan at MIT in May 2004. We also have gained significant insight from
a manuscript by Scott Freeman and Michael Sadler we thank Dean Corbae for bringing it to our attention.
—
Existing normative models of inequality reach an extreme conclusion: inequality should be perfectly inheritable
and
rise steadily
a vanishing lucky fraction to
tions
without bound, with everyone converging to absolute misery and
This immiseration result
bliss.
on preferences (Phelan, 1998); and obtains invariably
Thomas and
or productivity (Aiyagari
and Alvarez,
effort or
1995).
weak assump-
in partial equilibrium (Green, 1987,
Worrall, 1990), in general equilibrium (Atkeson and Lucas, 1992),
ments with moral-hazard regarding work
We
robust; requires very
is
and across environ-
with private information regarding preferences
1
depart minimally from these contributions, adopting the same positive economic models,
but a slightly different normative
work with
criterion. In a generational context, previous
lived agents characterizes the instance
where future generations are not considered
On
only indirectly through the altruism of earlier ones.
(2005) proposes a social planner with equal weights on
infinite-
directly,
but
the opposite side of the spectrum, Phelan
all
Our
future generations.
interest here
is
in
exploring a class of Pareto-efficient allocations that take into account the current population along
We
with unborn future generations.
and vanishing Pareto weight on the expected
place a positive
utility of future generations, this leads effectively to
a social discount rate that
is
lower than the
private one.
This relatively small change produces a drastically different
bounded, a steady-state, cross-sectional distribution exists
bility is possible
lower
and everyone avoids misery. Indeed, welfare
bound that
between
mation
We
social
is
is
strictly better
long-run inequality remains
consumption and welfare, social mo-
typically remains above an endogenous
than misery. This outcome holds however small the difference
and private discounting, and regardless of whether the source
of
asymmetric
infor-
privately observed preferences or productivity shocks.
begin by modeling a positive economy that
Each generation
by Atkeson and Lucas (1992).
live for
for
result:
one period and are
endowment
of the only
altruistic
is
is
identical to the taste-shock setup developed
composed of a continuum
towards a single descendant. There
consumption good
in each period.
is
of individuals
who
a constant aggregate
Individuals are ex-ante identical, but
experience idiosyncratic shocks to preferences that are only privately observed
—thus ruling out
first-
best allocations. Feasible allocations must be incentive compatible and must satisfy the aggregate
resource constraint in
When
all
periods.
only the welfare of the
to that of an
economy with
first
generation
is
considered, the planning problem
infinite-lived individuals.
Intuitively,
is
equivalent
immiseration then results from
the desire to smooth the dynastic consumption path: rewards and punishments, required for incentives, are best delivered
component that
leads cross-sectional inequality to
of the distribution
1
Many
permanently. As a result, the consumption process inherits a random-walk
is
grow endlessly without bound.
consistent with a constant aggregate
find the immiseration result perplexing
and some even
endowment only
find
it
if
Infinite spreading
everyone's consump-
morally questionable, but
it is
also incon-
venient from a practical standpoint. Long-run steady-states often provide a natural benchmark to study dynamic
economies, but such long-run analyses are not possible for private-information economies without a steady-state
distribution with positive consumption. This has impaired the study of long-run implications of optimal taxation,
so
common
in the
Ramsey
taxation literature.
tion eventually converges to zero.
Note, that as a consequence, no steady-state, cross-sectional
distribution with positive consumption exists.
Across generations, this arrangement requires a lock-step link between the welfare of parent and
child.
Of course, the
perfect intergenerational transmission of welfare improves parental incentives
but at the expense of exposing newborns to the risk of their parent's luck. Individuals would value
being insured against the uncertainty of their family's fortune
the biggest risks one faces in
By
contrast,
it
life is
regarding the family one
—
born
is
often recognized that one of
it is
into.
remains optimal to link the fortunes of parents and children in our model, but
no longer in lock-step. Rewards and punishments are distributed over
in a front-loaded
random walk
manner. This creates a mean-reverting tendency
—that
is
all
consumption
in
strong enough to bound long-run inequality.
future descendants, but
The
result
is
—instead of a
a steady-state
consumption and welfare, with no fraction of the population stuck
cross- sectional distribution for
at misery.
We also
study a repeated Mirrleesian version of our economy and derive implications
for
optimal
estate taxation.
In this model, individuals have identical preferences with regard to consumption
and work
but are heterogenous in the productivity of their work
effort,
productivity and work effort
private
is
—only the
resulting output
is
effort.
Information about
publicly observable.
We
show
that the analysis from the taste-shock model carries over to this setup, virtually without change. In
particular, a very similar
Bellman equation characterizes the solution to the
social planning prob-
lem: consumption exhibits mean-reversion and has a steady-state cross-sectional distribution. This
outcome highlights the
fact that our results
More importantly, the
do not require any particular asymmetry of information.
Mirrleesian model offers
locations can be implemented
new
insights into estate taxation.
by combining income and estate
taxes.
Specifically,
we
Feasible
al-
find that a
progressive estate tax, which imposes a higher average marginal tax rate on the bequests of fortu-
nate parents,
is
optimal.
This result reflects the mean-reversion of consumption: more fortunate
must have a declining consumption
dynasties, with relatively high levels of current consumption,
path induced by higher estate tax rates that lower the net rates of return across generations.
Finally,
an important methodological contribution of
planning problem recursively.
In doing so,
(1987) to situations where private
and
we extend
paper
this
ideas introduced by Spear
state variable,
and Srivastava
we
are able to reduce the
and our analysis and
results heavily exploit
social preferences differ.
dynamic program to a one-dimensional
to reformulate the social
is
Indeed,
the resulting Bellman equation.
Related Literature. Our paper
social planning
most
closely related to
problem with no discounting of the
planning problem exists then
problem have
is
it
must solve a
strictly positive inequality
static
and
future.
He shows
that
if
a steady state for the
maximization problem, and that solutions to
social mobility.
of a steady-state distribution for the planning
Phelan (2005), who considered a
problem
discounting. Unlike the case with no discounting, there
steady-state distributions, and as a result, the methods
is
for
Our paper
establishes the existence
any difference
no associated
in social
static planning
we develop here
this
and private
problem
are very different.
for
In overlapping-generation models without altruistic links,
efficient place positive direct
out that
market equilibria that are Pareto
all
weight on future generations. Bernheim (1989) was the
many
dynastic extension of these models with altruism,
in the
Kaplow
are not attainable by the market.
The
estate tax
is
negative
—
to point
efficient allocations
(1995) argued that these Pareto efficient allocations
and that they can be implemented by market
are natural social objectives
taxation policy.
Pareto
first
it is
a subsidy
— so
equilibria with estate
as to internalize the externality of
giving on future generations.
contributes to a large literature on dynamic economies with asymmetric information.
Our work
In addition to the
tion
work mentioned above,
this includes recent research
on dynamic optimal taxa-
Golosov, Kocherlakota and Tsyvinski, 2003; Albanesi and Sleet, 2004; and Kocherlakota,
(e.g.,
2004). This application has been handicapped by the immiseration result
of a steady-state distribution with positive consumption,
clusions for optimal taxation.
Our work
Our
results provide
also indirectly related to Sleet
is
making
it
and by the non-existence
difficult to
draw long-run con-
an encouraging way to overcome
generation. In this environment, as in Phelan's,
Sleet
and Yeltekin derive
for the current
a foregone conclusion that immiseration
will
to solve for the best subgame-perfect equilibria.
from a Lagrangian and use these to numerically
turns out that the best allocation in their no-commitment
asymptotically equivalent to the optimal one with commitment but featuring a more
is
patient welfare criterion.
way
it
it is
how
first-order conditions
simulate the solution. Interestingly,
environment
is
problem.
and Yeltekin (2004), who study an Atkeson-Lucas
environment with a utilitarian planner, who lacks commitment and cares only
not obtain, so that the interesting question
this
of characterizing the
Thus, our own approach and results provide an indirect, but effective
no-commitment problem and of formally establishing that a steady-state
distribution with no one at misery exists.
The
and
rest of the
sets
up the
paper
is
organized as follows. Section 2 introduces the economic environment
social planning problem. In Section 3,
we develop a
recursive version of the planning
The
problem and draw
its
then put to use
Section 4 to characterize the solution to the social planning problem. Here
derive our
main
consumption.
some
in
connection to our original formulation.
results
We
on mean-reversion and on the existence
discuss these results in Section 5
related problems
and reformulations.
In Section
setup with productivity shocks and focus on
its
we turn
Bellman equation
is
we
of a steady-state distribution for
and develop
6,
resulting
intuition for
them by studying
to the canonical optimal-taxation
implications for estate taxes. Section 7 offers
some
conclusions from the analysis. All proofs omitted in the main text are contained in the Appendix.
2
A
Social Insurance
The backbone
of our
Problem
model requires a tradeoff between insurance and
incentives.
This tradeoff
can be due to private information regarding either productivity or preferences.
For purposes of
comparison, we
6,
first
adopt the Atkeson-Lucas taste-shock specification. In Section
we adapt our
arguments to a repeated Mirrleesian model with privately observed productivity shocks. Similar
arguments could be applied to moral-hazard situations with unobservable
Our
An
positive
economy
identical to that of Atkeson-Lucas
is
—the differences are only normative.
can be interpreted as a dynasty of individuals who have
infinite-lived agent
Under
altruistically linked.
effort choices.
this interpretation,
finite lives
but are
Atkeson-Lucas and others focus on a particular
2
set of efficient allocations: those that only directly consider the welfare of the initial generation.
In contrast, our interest here
future generations.
with
lies
This approach
infinitely-lived social planner
who
more patient than
by a continuum of individuals who have
and
point in time, our economy
identical preferences, live for one period,
is
populated
and are replaced
single descendant in the next. Parents born in period t are altruistic towards their only child
their utility v t satisfies
vt
where
ct
>
is
uous derivative
= Ei_i
[9 t u(c t )
+ fivt+i
}
own consumption and P £ (0, 1) is the altruistic weight placed on the
The utility function u(c) is assumed continuous and concave, with a contin-
the parent's
descendant's utility v t +\.
The
all
individuals.
Demography, Preferences and Technology. At any
by a
that directly weigh the welfare of
asymptotically equivalent to postulating preferences for an
is
is
efficient allocations
for all c
taste shock 9
e
6
>
is
satisfying the Inada conditions
lim^o u'(c) = oo and lim^oo
u'(c)
=
0.
distributed identically and independently across individuals and time.
This specification of altruism
is
consistent with individuals having a preference over the entire
future consumption of their dynasty given by
oo
vt
=
YP
J
5
^t-i[e t+ su{ct+s )].
(1)
s=0
In each period, a resource constraint limits aggregate consumption to be no greater than
constant aggregate endowment e
precisely those adopted
Define
U =
>
0.
set of all possible utility values.
be unbounded so that the extremes u
is
These specifications choices preferences and technology are
by Atkeson-Lucas.
u(M+) to be the
cost function c(u)
some
defined on
U
=
u(0) and u
=
lim^oo
u(c)
Note that we allow
may be
as the inverse of the utility function c
=
utility to
finite or infinite.
u~
l
.
The
For simplicity, we
=
< 6^ = 6. We denote the density
< 9<i <
by p(6) and adopt the normalization that E[0] = Yl n=i @nP{9 n = 1- The level of dynastic utility v
then always belongs to the set V = w(M + )/(l — (3) with extremes v = u/(\ — P) and v = u/{\ — /?).
assume
contains a finite
number
of shocks 9
6-y
)
Social Welfare.
We
t
depart from Atkeson-Lucas by assuming that the social welfare criterion can
be represented by preferences given by the
utility function
j^E-itM*)],
(2)
t=o
2
The
final
paragraph
in
Atkeson Lucas (1992) discusses the possible importance of relaxing
this assumption.
with
J3
>
P.
Thus, social preferences are identical to the individual preferences given by
(1),
except
for the discount factor.
Future generations are
This setup puts weight on the welfare of future generations directly.
already indirectly valued through the altruism of the current generation.
also directly included in the welfare function the social discount factor
If,
in addition, they are
must be higher than
To
p.
see this, consider the utilitarian welfare criterion
oo
oo
^a*E_
1 Ut
= ^^E_
4=0
where S t
and
= P + P t_1 a +
+ Pa
l
•
•
•
more
social preferences are
=
rion (3) approaches (2) with
1
'1
+ a Then
1
[^«(c
t
)]
(3)
>
the discount factor satisfies
.
In the limit 8t+i/5 t
patient.
max{/?, a}.
—
>
max{/3, a}, so the welfare
crite-
3 4
'
Atkeson-Lucas' analysis applies to the case with
weight
1
t=0
P =
$, so
we focus on the case where enough
placed on future generations to ensure that the long-run social discount factor remains
is
strictly higher
than the private one,
rest of the paper,
it is
$>
p.
Although we adopt the preference
in (2) directly for the
straightforward to adapt the arguments to the welfare criterion
specifications are slightly different for
any
finite
The two
(3).
horizon but are identical for the long-run, which
is
our primary concern.
Information and Incentives. Taste shock
their descendants.
The
realizations are privately observed
by individuals and
mechanisms
revelation principle then allows us to restrict our attention to
that rely on truthful reports of these shocks. Thus, each dynasty faces a sequence of consumption
functions {q}, where ct (0
(6o, 8i,
that
a
1
.
.
,8 t ).
maps
£
:
.
A
t
)
=
dynasty's reporting strategy a
{a
t
t
histories of shocks 9 into a current report
-* G*.
We
use
ct*
t
} is
a sequence of functions a t
Any
.
strategy
Given an allocation {c
t
},
l
)
=
t
for all 9
the utility obtained from any reporting strategy a
l
e
—
>
:
a induces a history
to denote the truth- telling strategy with a* (9
t
i+1
=
t
represents an individual's consumption after reporting the history 9
of reports
0*.
is
oo
[/({cj.a;/?)^ J2
t
3
=
/?M*(*W)) PK?)-
0<£0t+l
Bernheim (1989) performs similar intergenerational discount factor calculations
Caplin and Leahy (2005) argue that these ideas
deterministic dynastic saving model.
discounting within a lifetime, leading to a social discount factor that
is
in his welfare analysis of a
also apply to intra-personal
greater than the private one not only across
generations, but within generations as well.
4
Onc can
also adopt the
more general welfare
{at}. In particular, the sequence
a =
(1
criterion J^^L
-j3)/(l
- (i) and at =
a E_iut
t
ao/3* for
for
t
>
some sequence of positive Pareto weights
1
delivers S t+ i/5 t
=
ft
for all
t
=
0, 1,
.
.
An
allocation {c t }
is
incentive compatible
if
truth-telling
is
optimal:
U({ct },a*;P)>U({ct },a;P).
(4)
Social Planning Problem. Following Atkeson-Lucas, we identify each dynasty with a number
v,
which we interpret as
that
all
its initial
dynasties with the
entitlement to expected, discounted
same entitlement v
receive the
same treatment.
distribution of utilities v across the population of dynasties: ip(A)
will receive
An
expected discounted
allocation
is
AC
the set
utility in
a sequence of functions
that a dynasty with initial entitlement v gets at date
For any given
is
feasible
v to
initial distribution of
incentive compatible for
(i) it is
if:
entitlements
dynasties entitled to
all initial
exceed the fixed endowment
v;
and
We
v.
then
let ip
denote a
where
0^(0') represents the
A
(ii) it
e,
sequence of shocks
we say that an
delivers
expected
We
optimum maximizes the average
feasible allocations.
ip
That
let e*(ip)
allocation {cvt }
utility of at least
denote the lowest resource
and an endowment
level e
to
is
relevant for
is
level e
/3
=
ip
such
—the
j3.
weighed by
social welfare function (2),
the social planning problem given an
is,
#*.
average consumption in the population does not
(iii)
problem studied by Atkeson and Lucas (1992) which
social
who
consumption
that there exists a feasible allocation that delivers the distribution of utility entitlements
efficiency
assume
the fraction of dynasties
after reporting the
dynasties;
all
e in all periods.
t
v,
and resources
ip
is
We
=
R.
each
{<:£} for
utility, vq
initial distribution of
ip,
over
all
entitlements
maximize
U({cvt },a*,P)diP(v)
subject to v
=
U{{cvt },a*;P) > U{{cv },a;
(3)
t
and
for all v,
/^cf(9')Pr(«')#)<e
J
Our
social planning
e
problem
is
>
and where
J3
Steady States.
on distributions
to the planning
utilities v t
of
that
problem
is
is
= 0,1,...
(5)
non-empty constraint
well defined, with a
are interested in situations with
Our focus
£
l
(3
e
>
set, for all e
>
e*(ip);
we
e*(ip).
of utility entitlements
ip
such that the solution
features, in each period, a cross-sectional distribution of continuation
also distributed according to
consumption to replicate
itself
over time.
these properties a steady state and denote
ip.
We
also require the cross-sectional distribution
term any
them by
utility constitutes a state variable that follows a
We
ip*.
initial distribution of
As we
Markov
shall
process,
entitlements with
demonstrate below, continuation
and steady states are then invariant
distributions of this process.
Note that
in the
Atkeson-Lucas
case,
with
ft
=
j3,
the non-existence of a steady state with
a consequence of the immiseration result: starting from any non-trivial
positive consumption
is
initial distribution ip
and resources
e*(ip)
the sequence of distributions converges weakly to the
7
distribution having
steady states
A
3
ip*
full
mass
at misery,
with zero consumption
that exhaust a strictly positive aggregate
for everyone.
endowment
We
seek non-trivial
e in all periods.
Bellman Equation
In this section
we study a relaxed
version of the social planning problem whose solution coincides
with that of the original problem at steady states. The relaxed problem has two important advan-
problem can be solved by studying a
First, the relaxed
tages.
dynasty with entitlement v
subproblems
set of
—one
—which avoids the need to keep track of the entire population.
for
each
Second,
each of these subproblems admits a simple recursive formulation, which can be characterized quite
sharply.
We
believe that the general approach
we develop
may be
here
useful in other contexts.
Consider the relaxed planning problem where the sequence of resource constraints
by the
replaced
single intertemporal condition
/oo
oo
^Q
for
(5) is
some
t
positive sequence {Qt} with
J>^)Pr(^)^)<eX>,
Ym^o Qt <
°°-
O ne can interpret
(6)
problem as representing
this
a small open economy facing intertemporal prices {Qt}- The relaxed and original versions of the
planning problem are related in that any solution to the former which happens to satisfy the
A
resource constraints in (5) must also be a solution to the latter.
the converse: there must exist
some
Lagrangian argument establishes
positive sequence {Qt} such that the solution to the original
planning problem also solves the relaxed problem. Most importantly, any steady-state solution to
the relaxed problem
Our
is
a steady-state solution to the original one.
are only compatible with q
forward.
Q =
focus on steady states leads naturally to
=
$, so
Attaching a multiplier A
the Lagrangian
L = JL
v
dijj(v)
we adopt
>
l
q
t
for
some
q
>
this value for the relaxed
0.
Indeed, steady states
problem from
to the intertemporal resource constraint (6),
this point
we can form
where
oo
and study the optimization of L subject to v
= U {{$}
equivalent to the pointwise optimization, for each
v
=
it
satisfies
U({tf}, a*;/3)
Theorem
> U{{c
v
t
),
a;
/?).
Our
first result
v, of
,
a*
;
/?)
> U {{cv }
t
,
a;
the subproblem: k(v)
(3)
=
for all v.
sup
characterizes this value function
L
v
This
is
subject to
and shows that
a Bellman equation.
1
The value function k(v)
k(v)
is
continuous, concave, and satisfies the Bellman equation
= maxE[6u(0) -
\c(u(6))
+ 0k(w(9))]
(7)
subject to
=
v
9u(9)
E[9u(6)
+ f3w(9) >
+ f3w(9)]
9u(9')
(8)
+ pw{9')
for
This recursive formulation imposes a promise-keeping constraint
that telling the truth today
necessary to satisfy the
taken care of in
full
is
optimal
the truth
if
is
9'
e 0.
(9)
and an incentive con-
Of
told in future periods.
incentive-compatibility condition
by evaluating the value function
(7)
(S)
9,
out one-shot deviations from truth-telling, guaranteeing
Intuitively, the latter rules
straint (9).
all
(4).
course, this
Intuitively, the rest
at the continuation utility:
is
for
is
implicitly
any given
continuation value w(9), envision the planner in the next period solving the remaining sequence
problem by selecting an entire allocation that
Then k(w(9))
incentive compatible from then on.
is
represents the value to the planner of this continuation allocation. Taken together, a pair u(9) and
w{9) that
yields
satisfies (8)-(9)
an allocation that
pasted with the corresponding continuation allocations for each w(9),
satisfies
the
full
incentive-compatibility
The
(4).
objective function in (7)
then captures the relevant value of allocations constructed in this way.
Among
We
tained.
shows that the maximum in the Bellman equation (7) is atu
w
the policy functions g (9,v) and g (9,v) denote the unique solutions for u and w,
other things,
let
For any
respectively.
Theorem
1
initial utility
entitlement
the policy functions (g u ,g w ) by setting uq{9q)
inductively for t
Our next
in this
is
1
by u
t
(0')
=
g {9 t ,v t {9
t
=
-1
))
result elucidates the connection
way and
Theorem
>
u
Vq,
an allocation {u
t
}
can then be generated from
u
g (9 ,VQ) initially and defining
t- 1
and Vt+^O*) = g w (9 u v t {9
)).
ii t
and v +i(9
(0')
t
t
)
between allocations generated from the policy functions
solutions to the planning problem.
2 (a)
An
allocation
{u t }
is
optimal for the relaxed problem, given
w
u
generated by the policy functions (g ,g
lifetime utility of Vq; (b)
)
vq, if
starting at vq, is incentive compatible,
an allocation {u t } generated by the policy functions
u
(g ,g
and only
if it
and delivers a
w
starting at
),
4-1
= and delivers utility vq; (c) an allocation {u t } generated by the
has lmij-.oo/?' E_i?; t (#
)
u
w
policy functions {g ,g ), starting from v Q is incentive compatible if
vq,
,
t
t
limsup E_ 1 p v (a
t
for
all
-1
t
{9
-1
))
>
reporting strategies a.
Part
(a)
of
Theorem
2 implies that either the solution to the relaxed planning
generated by the policy functions of the Bellman equation, or there
and
part
(c) of
(c).
the theorem show that the
The
can be verified
a
Theorem
first
case
is
guaranteed
if
is
we can
no solution
many
all.
is
Parts (b)
verify the limit condition in
latter is automatically satisfied for all utility functions that are
in
at
problem
bounded below and
other cases of interest.
2 involves various applications of versions of the Principle of Optimality. For example, for any given
policy functions [g u ,g w ) and an initial value vo, the individual dynasty faces a recursive dynamic
programming
The
case with
(3
=
J3
can be studied by the same approach. Recall that the efficiency problem
studied by Atkeson and Lucas (1992) minimizes resources e subject to the sequence of resource
constraints (5) and v
=
U({cvt ),
a*;
> U {{cv }
(3)
t
,
a; (3)
for all v.
Consider the relaxed version
of this problem that replaces the sequence of resource constraints with the single intertemporal
constraint (G) for
some sequence {Q
constraints (5)
is
it
t
}-
Then
the solution to this problem satisfies the resource
also a solution to the original problem.
with constant relative risk aversion
this case,
if
utility functions,
Although no steady state
exists in
the relaxed problem with
Q =
characterize the original one, for an appropriately chosen value of qAL
>
0,
q AL
t
not necessarily equal to
P.
we can take the
Since the constraint (6) binds,
the
left
hand
KAL (v) =
inf
side of this inequality. This minimization can then be
is
subject to (8) and
>
t
/?).
The
each v
let
associated Bellman
(10)
>
This problem can be thought of as the limiting version of the
problem and
is
case as
(3
not necessarily 0. Theorems
1
and
2
Bellman equation.
its
Optimal Inequality
In this section
We
U{{c },a;
for
= mmE[c(u{d))+q AL KAL (w(8))]
co and where the discount factor in the objective q AL
also apply to this
4
(9).
= U ({<*}, a*; P) >
done pointwise:
then
KAL (v)
—
to v
J2Zo <&&(&) subject
equation for this problem
A
objective function for the relaxed problem as
we
exploit the connection between the
characterize the solution
and derive a key equation that
dynamics of consumption. The main
enough
to imply the existence of
result of the section
exists.
Mean
Reversion
We
now
are
in a position to
Theorem
2,
1
=
to establish that these forces are strong
Finally,
study the Bellman equation's optimization problem.
The value function k(v)
main, with lim v ^ v k'(v)
mean-reverting forces in the
-co.
is strictly
If utility is
we provide
and ensure that a solution to the planning
justify the use of first-order conditions with the following
Lemma
is
illustrates
an invariant distribution with no misery.
sufficient conditions to verify part (c) of
problem
Bellman equation and the planning problem.
To
begin,
we
lemma:
concave and
differ~entiable
on the interior of
unbounded below, then lim v ^yk'(v)
=
1.
its
do-
Otherwise
problem with state variable vt Conditions (8) and (9) then amount to guessing and verifying a solution to the
Bellman equation of the agent's problem in particular, that the value function that satisfies the Bellman equation,
with truth telling, is the identity function. However, one also needs to verify that this value function represents the
true optimal value for the dynasty from the sequential problem. This verification is accomplished by part (c) of
.
—
Theorem
2.
10
lim v ^y k'(v )
=
Let A
let jjl{9,9')
=
oo.
k'(v)
be the multiplier on the left-hand side of the promise-keeping constraint
be the multipliers on the incentive constraints
-
with equality
if
u
The
(9) is interior.
for k(v) derived in
-
\c' (u(9)))p(6)
Lemma
1,
9Xp(9)
The
(9).
must be
solution for w(9)
-
+p$r p(9, ff)-PY,
e'
—
Using the envelope condition k'(v)
J
k'{g
w
°')
=
Inada conditions
°-
e'
A and adding up across
Y
M
is
0,
interior, given the
satisfy the first-order condition
0Xp{9)
<
9')
9^(9,
and must
pk' [w(9))p(9)
and
first-order condition for u(9)
£
+ ]T 9(i(9, 9') -
(8)
(9,v))p{9)
=
this
9,
becomes
ik\v).
(11)
P
eeo
This key equation can be represented in sequential notation as
E
where {v
t
} is
t
- l [k'{v t+l (9
t
))]=ik'(v
generated by the policy function g w
Regressive (hereafter:
CLAR) Markov
process.
t
t
-1
(9
Thus, {k'(v t )}
.
(12)
))
is
a Conditional Linear Auto
Note that we can translate anything about the
process {k'(vt )} into implications for the process {vt }, since the derivative k'(v)
u
strictly decreasing. Likewise, using the policy function g (9,v), conclusions
is
continuous and
about the process {v
t
}
provide information about the process for consumption.
The
conditional expectation in (12) illustrates that
for the process {k'(v t )}
maximum
at v*
>
toward
zero.
v with k'(v*)
from misery. This feature
is
=
Lemma
0,
1
1
creates a force for
mean
implies that the value function k(v) has an interior
itself
embodies an interesting form of social mobility.
with mobility ensured between them.
social hierarchies,
Descendants of individuals with current welfare above v*
will eventually fall
below
dynasties initially entitled to welfare below v* are guaranteed to access levels above
fall
the value function k(v).
it
is
it.
Similarly
This
rise
important to stress the role played by the non-monotonicity of
Although mean reversion stems from
non-monotonicity of k(v) that ensures that reversion
Atkeson-Lucas case a
is
it.
of families illustrates a strong intergenerational mobility in the model.
In deriving this result,
case
— away
key to our results on the existence of invariant distributions.
can divide the population into two
and
reversion
so reversion occurs towards this interior utility level
Economically, the mean-reversion equation
We
@/0 <
CLAR
11
<
1
in
not toward misery.
equation (12),
By
it is
contrast, in the
may
hold, but the value function in this
misery.
Indeed, the envelope and first-order
equation similar to (12)
monotone and reversion then occurs toward
is
ft/ft
conditions for the Bellman equation (10) yield
qAL
e
which
when
similar to condition (12)
is
KAL (v)
the value function
q AL
>
h
Crucially, unlike the case with
j3.
strictly increasing, so that
is
KAL (v)
>
negative process, which implies by the Martingale Convergence
almost surely
utilities
w
g (9,v)
follows, v t
—
>
some
to
(a.s.)
v and
q —
>
Proposition
mass
unbounded
7(1
7 =
—
—
a.s.
»
Immiseration then
bounds are
v) satisfies the
(l
-
<
1
CLAR
equation (11). In addition:
- *V(0, v)) < 7 (1 -
+ 9 n ^ -E[9\9>
=
(/?//?)
k'(v))
u(6)
=
u
w(9)
>
v
e 6. For values of v such that
bound in (13) holds for
1
unbounded below, continuation
is
(l
-
(13)
£)
max {(1 + 6 n —
E[6
<
6 n ])/9 n }
and
k'(v)
>
1,
we have
= 0/P)k\v)
k'(v)
<
1,
the lower
bound in (13) holds; the upper
sufficiently high v.
illustrates a powerful
a parent
+
9n]/9n _ l }.
k'(w(9))
Proposition
guaranteeing the existence
bounded below, then for low enough values of v such that
is
all 9
critical for
below, then
- fc») +
(P/P) min {1
2<n<N
how much
t)
(3.
6 G 9, where the constants are given by 7
(b) if utility
for
KAL (v
at misery.
The policy function g w (9,
1
if utility is
all
$ >
evolution. These
its
of an invariant distribution with no
for
must converge
it
result pushes the characterization of reversion past the average behavior of the {k't }
process by deriving bounds for
(a)
a non-
is
t )}
here
This highlights the importance of the non-monotonicity of the value
a.s.
function k(v) for our results in the case of
Our next
Theorem that
/?,
Since incentives must be provided using continuation
finite value.
w
g (9,v), this rules out anything other than
^
{KAL {v
Thus,
0.
>
ft
utility
g
tendency away from misery.
w
(9,
v)
remains bounded even as v
supposed to be punished, his child
6
With logarithmic utility qAL = /3 yields a
and a < 1 the appropriate value of q Ah that
,
For example, with utility
is
—
always somewhat spared.
solution with constant average consumption.
yields constant consumption,
12
-00. Thus, no matter
is
strictly
With
above
(3.
u(c.)
=
c
l
~a
/{I
-a)
Main
Result: Existence of an Invariant Distribution with
We now
state the
then
main
result of this section:
if
a solution to the relaxed planning problem exists,
The proof
admits an invariant distribution with no misery.
it
conditional-expectation equation (12) and the bounds in Proposition
mean-reversion properties discussed in the previous subsection.
Proposition 2 The
existence of an invariant distribution
for the Markov process {vt } implied by g
is
bounded above, or 7
Proposition
<
is
guaranteed
ip*
of this result relies on the
Thus,
1.
it
makes use
exists
if either: utility is
Theorem
part (a) of
2,
leaves
This situation contrasts strongly with the Atkeson-Lucas
unbounded
we show that a
open only two
possibilities:
case,
with
/?
=
/?,
(ii)
illustrate this
0,
also provides
an
efficient
method
(i)
no solution
where a solution
solution to the planning problem can be guaranteed so that case
Our Bellman equation
We
=
below, utility
but does not admit a steady state, and everyone ends up at misery. Towards the end of
section
lem.
both
with no mass at misery, ip*({v})
the relaxed problem admits a steady-state invariant distribution with no misery; or
exists.
of
7
1
when combined with
2,
w
No Misery
for explicitly solving the
(i)
this
holds.
planning prob-
with two examples, one analytical and another numerical.
Example 1. Suppose utility is CRRA with a = 1/2, so that u(c) = 4c / 2 for c > and c(u) = u 2 /2
for u > 0. For f3 = (3 Atkeson-Lucas show that the optimum involves consumption inequality
1
growing without bound and leading to immiseration.
Consider the relaxed problem where we ignore the non-negativity constraints on u and w,
k(v)
= maxE[0u(0) -
\u(6)
2
+ J3k(w(6))]
u,w
subject to (8) and
(9).
the value function
is
This
is
a linear-quadratic dynamic programming problem, so
a quadratic and the policy functions are linear in
u
g (6,v)
=
7 ?(0)t;
g"(0 v)
=
7» + 7oW
t
it
follows that
v:
+ 7 «(0)
For taste shocks with sufficiently small amplitude we can guarantee, by continuity with the deterministic case
=
[vl,vh] with v L
convex and
original
7
0,
>
that 7™(0)
0.
<
1
and 7™(0) >
Moreover, g u (6,v)
utility turns
>
for v
implying a unique bounded ergodic set
e
problem with non- negativity constraints on
When
utility is
for utility
[vl,v h ]. Hence, since the planning problem
is
out to be strictly positive at the steady state, this solution does solves the
bounded below, we
u.
either require that utility
for a small dispersion of the shocks, as a simple
It
0,
way
be bounded above, or that 7
of ensuring that the ergodic set
seems very plausible, however, that these conditions could be dispensed with.
13
is
<
1,
which
is
bounded away from
ensured
misery.
Figure
Example
1:
To
2.
w
illustrate the
for the logarithmic case
We
iterated
Figure
c(v(l
—
=
w
against c(g (9,v)(l
reveals a unique,
low values of
is
j3
0.9,
J3
=
0.975, e
=
=
A" 1
=
0.6, 6 h
for k(v) until convergence.
we compute the
1.2, 0,
=
0.75
solution
and p
=
0.5.
8
plots the policy function for continuation utility in consumption-equivalent units,
consumption, c(v(l
1
with
c
Policy function g (9.w)
2:
numerical value of our recursive formulation,
on the Bellman equation
1
(3))
Figure
Policy functions g (6,v)
v.
— /?))
—
against g
(3)),
c
while Figure 2 does the same for the policy function for
Both policy functions are monotonic and smooth. Figure
(9, v).
bounded ergodic
Note that both policy functions become nearly
set for v.
This illustrates the result, discussed immediately after Proposition
1,
flat for
that utility
kept above some endogenous bound.
Figure 3 displays the steady-state, cross-sectional distribution of dynastic utility measured in
—
consumption-equivalent units, c(v(l
/?))
implied by the solution to the planning problem. 9
long-run distribution has a smooth bell-curve shape
mean-reverting dynamics of the model, since
distribution of taste shocks.
8
The
incentive
The
figure also
details of this numerical exercise
and promise- keeping constraints.
— a feature that must be due to the smooth,
cannot be a direct consequence of our two-point
shows the invariant distributions
where as
We
it
The
follows:
we
for various values of
solved for u (0) as a function of
We
then maximized over w(6).
employed a grid
w (9)
using the
for v defined in
terms of equally spaced consumption-equivalent units c((l — 0)v) = {0.01,
,2}. Results with a grid size of 100
and 300 were similar; we report the latter. We used Matlab's splines package to interpolate the value function
.
and used fmincon.m as our optimization routine over w{6). Our
iterations
.
.
were
initialized
with the value function
corresponding to the feasible plan that features constant consumption:
(
,(1
_
p)
k Q (v)
_
Ac(t; (i
_
,
3))
1-0
10~ 10 and verified that the policy functions had also converged.
Note that g
is well within the interior of [.01,2], so that the arbitrary upper and lower bounds from our grid
choice were not found to be binding.
We stopped
the iterations
when
||fc
n (u)
—
fc
n _i(v)||
<
w (9,v)
9
The
invariant distribution
was approximated by generated a Monte Carlo simulation
for the
dynamics of the
1
{v t } process generated by g " with an arbitrary initial value of vq. Since this process converges to a unique invariant
distribution ip* starting from any initial value of vo, the frequencies in a long time-series sample approach the
,
,
frequencies of
ip"
.
To
create the figure
we used Matlab's Wavelet Toolbox
simulated Monte-Carlo sample.
14
to
approximate the density from the
0.61
0.6
utility in
Figure
P.
The degree
intuitively
3:
0.62
consumption equivalent
units
Steady-State Distributions of Dynastic Utility
of inequality appears to decrease with higher values of
by the
coefficient
CLAR equation
on the
These simulations
of the impulse response to shocks.
we approach the Atkeson-Lucas
(12)
case,
/3
—
> /3,
/?.
This outcome
and the discussion
is
in Section 5
suggested
on features
also support the natural conjecture that as
the resulting sequence of invariant distributions blows
up, since no steady state with positive consumption exists
when $
=
P-
We now turn briefly to issues of uniqueness and stability for the invariant distribution guaranteed
by Proposition
2.
This question
is
of
level vq,
That
is, if
convergence toward the distribution
then the fortunes of distant descendants
of the individual's present condition.
present, but its influence
is
Indeed, under some conditions
it
represents an even stronger notion
(12) discussed in the previous
occurs starting from any
i/j*
—the distribution of
exerts
some
wiped
we can guarantee
To
that the social
optimum
generated by g w converges weakly to
,
(12) ensures
influence
on the
initial distribution ip Q
ip*
in
our model does
see this, suppose the ergodic set for the {A;^}
compact. This is guaranteed, for example, by applying Proposition
the policy function g w (9,v) is monotone in v, the invariant distribution ip*
from any
independent
out.
is
in the sense that, starting
is
initial utility
dies out over time, so that the advantages or disadvantages
display this strong notion of social mobility.
process
—
their welfare
At the optimum, the past always
bounded and
of distant ancestors are eventually
if
interest because
by the mean- reversion condition
of social mobility than that implied
subsection.
economic
,
1
when 7 <
is
1.
Then,
unique and stable
the sequence of distributions
{ip t },
This follows since the conditional-expectation equation
enough mixing to apply Hopenhayn-Prescott's Theorem. 10 The monotonicity of the
'See pg. 382-383 in Stokey
and Lucas with Prescott (1989).
15
w
policy functions for continuation utility
1
and
seems intuitive and plausible, as illustrated by Examples
11
2.
Another approach suggests uniqueness and convergence without relying on monotonicity. Grunwald,
Hyndman, Tedesco and Tweedie
(1999)
show that one-dimensional,
with the Feller property that are bounded below and satisfy a
a unique and stable invariant distribution.
the sense that
initial distribution is fast, in
ments of
their
theorem have been
of irreducibility, which
distribution.
We do
is
hold
if
condition, such as (12), have
Moreover, convergence to this distribution from any
occurs at the geometric rate @/$. All the require-
it
verified already for our
likely to
CLAR
Markov processes
irreducible
we were
to
model, except for the technical condition
assume that the taste shock has a continuous
not pursue this formally other than to note that the forces for reversion in (12)
could be further exploited to establish uniqueness and convergence.
Our
focus on steady states, where the distribution of utility entitlements replicates itself over
time, has exploited the fact that the relaxed and original planning problems
we can do more and
for the logarithmic utility case
Proposition 3
there exists
is
an endowment
level e*(ip)
initial distribution of utility entitlements ip
such that the solution to the original social planning problem
generated by the policy functions (g u ,g w ) from the relaxed problem with
monotone
e* is
a
e*(i;
)<e*(i;
One
increasing, in that
coincide. However,
characterize transitional dynamics.
any
// utility is logarithmic, then for
must
a
if ip
'
-< ip
b
Q =
t
ft
in the sense of first- order stochastic
.
The function
dominance then
b
).
interesting application of this result
modified to select the best
to the situation where the planning problem
is
initial distribution
instead of taking one as given.
ip,
Then
dynasties are treated identically and started with identical utility level v* solving k'(v*)
is
all initial
=
0.
The
optimal allocation then evolves according to the dynamics implied by the policy function g w (6,v).
The
cross-sectional distribution of welfare will spread out
from
its initially
egalitarian condition as
dynasties experience varying luck in the realization of their shocks.
By
convergence to a unique invariant distribution ip* of the Markov
implied by the policy function g w (9, v) takes on additional economic meaning. It implies
applying Proposition
process {v t }
3,
the stability of the cross-sectional distributions of welfare and consumption in the population. That
if
is,
the
Markov process {v
t
}
generated by g w
welfare and consumption eventually settle
As mentioned
for
any
(ip,e) as
in Section 3, for
any
down
stable, then the cross-sectional distributions of
is
to the steady state.
utility function specification
the solution to a relaxed problem with some sequence of prices {Qt}, that are not
necessarily exponential. Proposition 3 identifies the distributions
More
lead to exponential prices in the logarithmic case.
pair (ip,e),
one can characterize the solution
we can show
that
Q =
t
ft
+
A/5* for
and endowment
pairs
(ip, e)
generally, with logarithmic utility for
some constant
A.
The
that
any
entire optimal allocation
can then be characterized by the policy functions from a non-stationary Bellman equation. Since
n Indeed,
can be shown that g w (9,v) is strictly increasing in v. However, although we know
of no counterexample, we have not found conditions that ensure the monotonicity of g w {9, v) for all 6^0.
for the general case
it
16
prices are asymptotically exponential, in that lim t _ 00
l
P~ Qt
=
follows that long-run
1, it
dynamics
are always dominated by the policy functions (g u ,g w ) from the relaxed problem with exponential
prices
Q =
t
that
/?*
we have
characterized.
Sufficient Conditions for Verification
We
conclude this section by describing sufficient conditions
to exist at the steady state
we
First,
steps.
ip*
identified
consumption
Lemma
is
by the policy functions
establish that allocations generated
compatible by verifying the condition
finite
for
in part (c) of
under the invariant distribution
in Proposition 2.
This involves two
by the policy functions are indeed incentive
Theorem
2.
Second,
we
verify that average
ip*.
2 The allocation generated from the policy functions (g u ,g w ), starting from any
guaranteed to be incentive compatible in the following cases: (a)
is
a solution to the planning problem
bounded below;
(c) utility is logarithmic;
<
or (d) 7
1
or 7
>
utility is
bounded above;
vq,
is
(b) utility
0.
We now find sufficient conditions that guarantee average consumption is finite under the invariant
distribution
is
ip*
.
If
the ergodic set for utility v
bounded and average consumption
is
is
bounded away from the extremes, then consumption
trivially finite.
Even when a bounded ergodic
set for utility
v cannot be ensured, finite average consumption can be guaranteed for a large class of utility
functions.
Lemma
3 Average consumption
is finite
under the invariant distribution
ip*
^Tc(g u (8,v))p(9)diP*(v)<cx>
e
if either (a) the
of
ergodic set for v
is
bounded; or (b)
such that d{u{c))
is
a convex function
c.
Note that a bounded ergodic
set
is
guaranteed by 7
with sufficiently small amplitude; and condition
aversion utility functions with
The
a>
<
(b) holds, for
1,
which
and thus has
full
range.
for
ensured for taste shocks
all
constant relative risk
1.
utility,
A.
For instance, in the case of
average steady state consumption
is
a power function of
In fact, in this case the entire solution for consumption
of degree one in the value of the
planning problem
is
example, for
value of average consumption depends on the value of
constant relative risk aversion
5
utility is
endowment
any endowment
e.
is
A,
homogenous
This ensures a steady state solution to the social
level.
Discussion: Mean-Reversion
This section develops an intuitive understanding of the key mean-reversion property discussed previously.
We
first
derive the impulse response of consumption to a one-time taste shock.
17
We
then
revisit
the
problem with an alternative Bellman equation that
full
useful as a source of intuition.
is
Impulse Response
Consider a version of our model where only the
period, there are
is
two possible values
deterministic:
=
>
1 for t
We
1.
compare
generation faces uncertainty.
shock 6
G {0l,9h}, but
this to the case
In the
thereafter the
first
economy
with no uncertainty in the
first
To
This allows us to trace out the consumption response to the taste shock over time.
period.
simplify,
We
6t
for the taste
first
we adopt logarithmic
utility.
begin by studying a subproblem of the deterministic planning problem from the second
generation onward, that
planning problem
is
for
=
t
1,2,
—
For a given, promised continuation utility
the
Vj,
is
oo
fcdetfa)
= maxV^'"
1
7T
t=\
{a}
(logQ
-
Ac*)
v
,
'
subject to
oo
4=1
The
associated Bellman equation
is
= max
kdet(v t )
Ct,Vt+l
subject to vt
=
log
Ct
+
pv t+ j. The
(log(ct )
- Aq + (3kdet (vt+1
)
\
I
first-order
*4etK+l)
ct
and envelope conditions imply that
=
(14)
!*£*(*)>
= A-^l-fc^M).
(15)
Condition (14) shows that {k'det (v t )} reverts geometrically towards zero at the rate (3/$. This
is
In the logarithmic case,
it
a deterministic version of the conditional-expectation equation
translates directly into
back to a
common
consumption by the
discounting coincide, so that
level c t
=
c{y-y/{\
Turning to the
Thus, consumption reverts
first-order condition (15).
steady state at the same rate; deviations from the steady-state level of consump-
tion have a half-life of (log 2 (/3 P))~
new
(12).
—
first
P=
l
$,
Note that
in the
Atkeson-Lucas case when social and private
consumption remains perfectly constant
/?)).
generation at
t
maxE[0 o log
=
0,
the planning problem solves
-
(c o (0))
A o cq(0)
+ pk det
co,vi
(
Vl {8))}
subject to
0l log (c (9 L ))
+ p Vl (9 L >
)
6 L log (c
18
(M) + Pv
1
(9 H ).
after the
shock at
its
where we omit the other incentive constraint since
>
convex. At the optimum, c {6 H )
is
it is
c (9 L ), Vi(8 H )
not binding at the optimum, and the problem
<
Vi(6 L ),
and
E[k'det (v 1 (6))]=0,
implying that Vi(6 H )
and equal to A
-1
<
<
v*
=
Vi(9 L ) where k'{v*)
Note that average consumption
0.
That
constant
in all periods.
Figure 4 shows the consumption response to a taste shock in the
periods.
is
is,
we use
and
(14)
on consumption from the shock
(15) for t
>
1
dies out over time
first
starting at v 1 (9 L ), v*
period, for subsequent
The
and v±(6 H ).
effect
and consumption returns to a common steady-
state level. Again, this illustrates that the influence of past fortunes eventually vanishes for distant
We
descendants.
also plot the Atkeson-Lucas case with
has a permanent impact on the consumption of
Figure
To provide
4:
Consumption path
incentives for the
reporting a low taste shock.
first is
>
=
(3,
consumption,
for a
where the luck of the
first
1 in
response to taste shock at
t
=
generation, society rewards the descendants of an individual
Rewards can take two forms and society makes use
exploits differences in preferences:
it
of both.
in present- value terms.
tilting the
given present value, in the direction preferred by individuals.
12
consumption
profile
Somc readers may recognize this
when conceding their child's
The second
12
is
toward the present. Similarly, punishments involve
method
The
allows an adjustment in the pattern of
Since individuals are more impatient than the planner, this latter form of reward
by
generation
descendants.
standard and involves increased consumption spending,
more subtle and
is
first
for t
all
(3
delivered
tilting the
as the time-honored system of rewards and punishments used by
snack or reducing their TV-time. In these instances, the child values
some goods more than the parent wishes, and the parent uses them to provide incentives.
parents
last
favorite
19
consumption path toward the future.
intensively to provide incentives
common
returns to a
affecting the
both
In
cases, earlier
—rewards and punishments are front-loaded.
first
Indeed, consumption
steady-state level in the long-run regardless of the initial shock because
consumption of very distant descendants
incentives to the
consumption dates are used more
not an efficient
is
way
for society to provide
generation.
Another Bellman Equation
Here we develop another Bellman equation that holds
This alternative formulation
q
=
is
useful,
for
any value of
q not necessarily equal to
J3.
both as a source of intuition and to motivate our focus on
P-
Consider the following cost minimization problem
oo
*(«,«)=
mm ^g^c^PrC*
(C '
}
),
gt
t=0
subject to the incentive compatibility constraint
1
U ({ct}
,
a*
;
(3)
>
U({ct},a;
(5)
and
oo
4=0
oo
t=0
that
delivering utility v
is,
and v
for the planner
and
individual, respectively.
Then
the value
function must satisfy the Bellman equation
K{v,v)
= maxE\c(u(9))+qK(w(9),w{9))],
u,w,u>
-*
subject to
v
=
E[6u(6)
+ J3w(6)]
v
=
E[6u(6)
+ 0w(6)]
and
6u{9)
+ f3w{9) >
9u{6')
This formulation could be used to derive
equation
formulation, however,
The
more convenient
(7) is slightly
is
that
it
all
+ 0w{6')
for all 9, 9'
e 0.
of our results, although the lower-dimensional
for that purpose.
The advantage
of this cost-minimization
lends itself naturally to economic interpretations.
following story provides a useful reinterpretation and source for intuition.
infinite-lived
Bellman
Consider an
household with two members, husband and wife, and assume that consumption
a public good
—there
is
is
no intra-period resource allocation problem. However, husband and wife
20
disagree on
how
Suppose the wife
to discount the future.
more
patient, but only the
husband
problem characterizes the constrained Pareto problem
for this
is
can observe and report taste-shock realizations.
Then
this cost-minimization
The Pareto
Pareto frontier between husband and wife.
K
=
household, in the sense that the isocost curve K(v, v)
K
represents, given resources
frontier
is
non-standard in that
the
,
not
it is
everywhere decreasing and does not represent the usual transfer of private goods between two
agents.
Instead,
it
arises
from differences in preferences that generate a disagreement about the
optimal consumption path for the only public good. Since disagreement on preferences
the Pareto frontier
is
non-monotone and the highest possible
interior utility level for the
the
left
The
of this point
must
=
husband, where Ki(v*,v*)
utility for the wife
attained for an
Reductions in the husband's
0.
utility to
also decrease utility for the wife, for a given level of resources.
first-order conditions
can be rearranged to deliver
PK (v,v)=qK
2
K
K
2
1
(v,v)
2
(v,v)
=
"
E
p
(w(9),w(9))
(16)
Kx (w{e),w(0))'
K (w(e),w(6))
(17)
2
Condition (16) can then be used to argue that a steady-state requires q
{K2t }
is
bounded,
is
would increase without bound;
likewise,
q
if
>
{K2t }
$, then
=
$. Indeed,
if
q
<
J3,
decreases toward zero.
then
Both
situations clearly do not lend themselves to the existence of an invariant distribution for (v,v).
On
the other hand,
distribution
When
K
2
q
is
=
if
q
=
(3
K
(v t ,Vt)
2
is
constant along the optimal path and an invariant
possible.
(3,
the state (v t ,v t ) moves along a one-dimensional locus given by
Intuitively, since
(vq,vq).
then
no incentives are required
for the wife, she
the sense that the marginal cost of delivering welfare to her
Figure 5 shows that the curve
K
2
(v,v)
=
curves from below, and cuts Ki(v,v)
perfect insurance for the
=
K
2
from above.
2
(v,v)
=
perfectly insured in
held constant across time.
is
continuation
(v ,i>o) for
is
K
utilities cuts
the isocost
Intuitively, incentives require foregoing
husband and accepting fluctuations
Starting from (v* v*), rewards can be delivered in two ways.
,
in v as
rewards and punishments.
The optimum makes
use of both forms
of rewards, explaining the shape of the schedule for continuation utilities.
The
first
form of rewarding involves increasing resources K, and can be seen as an upward
movement along the diagonal Ki(v,v)
allocation of these resources that
is
along the Pareto frontier, which at
rewards and, as a
above the
more
0.
However, the husband
is
also
rewarded by allowing an
to his liking, which can be represented as lateral
(v*, v*) is horizontally flat.
result, (v,v) travels
initial isocost curve.
=
along
K
2
(v,v)
=
Note that punishments
section of the Pareto frontier. Thus, ex-ante efficiency
13
K
2
The
(v ,v
will
)
movements
solution combines both forms of
to the right of K\{y,v)
and
push the agent on the upward sloping
demands ex-post
inefficiency.
13
Returning to the analogy in footnote 12: parents often complain that the punishments they choose to
their children hurt them more than they do their kids.
21
=
inflict,
on
Figure
Condition
(17) is the
—Ki/K2
and eventually
husband with
Isocost curves of
Here,
it
v)
implies that the slope of the isocost curve (Pareto
reverts geometrically toward
0.
Thus, (vt ,vt ) moves along K2(v,v)
=
K2(vq,vo)
reverts toward (v* v*). Intuitively, the solution deviates from (v*, v*) to provide the
,
incentives,
but
it is
efficient to revert
the wife: Patience ensures that the wife has her
6
K(v,
analog of the conditional-expectation equation (12) obtained from the
one-dimensional Bellman equation.
frontier),
5:
way
back to this point of
maximum
efficiency for
in the long-run.
Estate Taxation
We now turn
to a
repeated Mirrleesian economy and study optimal taxation. In this version of our
model, individuals have identical preferences over consumption and work
regarding their labor productivity, which
distributed across generations
is
effort
but are heterogenous
privately observed by the individual
and dynasties.
We
and independently
continue to focus on the case where the social
welfare criterion discounts the future at a lower rate than individuals.
Unlike the taste-shock model, here even
between parent and
child, there
would
still
if
we were
to restrict allocations to feature
be a non-trivial planning problem.
period the situation would then be identical to the
static,
always optimal
to
With
link
Indeed, in each
nonlinear income tax problem originally
studied by Mirrlees (1971). Moreover, in the absence of altruism, so that
actually coincides with this static solution.
no
altruism, however,
(3
we
=
0,
the social
shall see
optimum
below that
it is
link welfare across generations within a dynasty to enhance incentives for parents.
Despite differences between the Mirrleesian economy and our taste-shock model, our previous
analysis
can
be adapted virtually without change.
22
In particular, a recursive representation can
be derived, and the Bellman equation can be used to characterize the solution and to establish
that a steady-state, invariant distribution exists. This highlights the fact that our model requires
asymmetric information, but not any particular form of
We
some
it.
focus on an implementation of the allocation that uses income and estate taxes, and derive
interesting results for the latter.
We
find that estate taxation should be progressive:
fortunate parents should face a higher average marginal tax rate on their bequests.
mean
reflects the
reversion in consumption explained in the previous section.
A
more
This result
higher estate tax
ensures that the fortunate face a lower net rate of return across generations, and that consequently
their
consumption path decreases over time toward the mean.
Repeated Mirrlees: Productivity Shocks
Each period
economy
of this
depends on the
Utility
generation
t
is
identical to the canonical optimal taxation setup in Mirrlees (1971).
level of
consumption c and work
effort n.
t
differ
t
t
t
in
],
regarding their productivity in translating work effort into output.
productivity w, exerting work effort n, produces output y
is
assume that individuals
have identical preferences that satisfy
V =E _ 1 [u(c )-h(n )+pVt+1
but
We
=
wn.
We
An
individual with
assume that productivity
w
independently and identically distributed across dynasties and generations. Thus, the produc-
tivity talents of parent
assumption,
if
the
and
optimum
child are unrelated
skills
are
represents a
assumed physical environment. 14
=
n 1 /j so
that, defining 9
=
iy~ 7
,
write total utility over consumption and output as being subject to taste shocks
oo
oo
Y^pE^u^-eMyt)}
t=0
The
it
this
with incentives in this way, and not a mechanical result
For convenience, we adopt the power disutility function h(n)
we can
assumed nonheritable. Given
features intergenerational transmission of welfare, then
social decision to provide altruistic parents
originating from the
—innate
=^^E_ [A(c -i)-^%
1
i
i )]
-E_x[0 o %o)]
t=l
right-hand side of this equation leads to a convenient recursive representation of the planning
problem
in the continuation utility defined
are abusing notation slightly by folding the
The
by v
_1
/?
t
=
S
YlT=o0 E(-i[«(ct+ s -i) — 6t+sh(yt+ s )] (where we
into the definition of the utility function u(c)).
resource constraint requires total consumption not to exceed total output plus
some
fixed
constant endowment
f
14
As
Y^
in the taste
Albanesi and
Sleet.
cV et
t(
)
Pr (^)
(tyiv)
< f J2
v et
t(
)
Pr (#
£
)
d ^( v )
+
e
i
=
0, 1,
.
.
= j3 leads to immiseration. This case has been studied by
shock model, here the case with
who impose an exogenous lower bound on dynastic welfare to circumvent immiseration.
(2004),
23
where individuals are indexed by
their initial utility entitlement v, with distribution
in the
ip
population.
We
continue to assume social discounting
problem
is
t
is
>
lower than private discounting:
t
to choose an allocation {c^(9 ),y^(9 )} to
maximize average
(3.
The planning
social welfare subject to the
incentive-compatibility constraints and the resource constraints.
Using the
the utility function and applying similar reasoning as in the taste-
last expression for
shock model yields the Bellman equation
= max
k(v)
for the associated relaxed
E[w_ - Xc(u_) - 8h{9)
+
Xy(h{8))
U-,h,w
=
v
-8h{d)
E[u_ - 8h(6)
+ (3w{6) >
problem
+ J3k(w(8))]
+ (3w(6)}
-9h{9')
where the function y(h) represents the inverse of the
+
Pw(8'),
disutility function, y
=
h~ l
that justify the study of this Bellman equation, are similar to those that underlie
in the context of the taste-shock model.
previously,
The
.
The arguments
Theorems
1
and
2
results regarding steady states parallel those obtained
and imply that an invariant distribution
exists
with no immiseration, as in Proposition
2.
Implementation with Income and Estate Taxation
Any
allocation that
is
incentive compatible
taxes on labor income and estates. Here
and
we
can be implemented by a combination of
feasible
describe this implementation, and explore
first
some
features of the optimal estate tax in the next subsection.
For any incentive-compatible and feasible allocation {cvt {9
tation along the lines of Kocherlakota (2004).
t
i
),
In each period, conditional on the history of their
and any inherited wealth, individuals report
dynasty's reports 6
we propose an implemen-
y^{9 )}
shock 9
their current
t
,
produce,
consume, pay taxes and bequeath wealth subject to the following set of budget constraints
ct
where
Rt-i,t
+b <
t
l
y t (9
)
-T
t
(0<)
+
(1
-
n^R^A-i
i? t _ 1}4 6 t _i at rate 7^(0*).
Given a tax policy {Tl'(9
{R
t
,t+i}]
an allocation
for
t
t
)
)
T (9
t
),
(18)
.
=
0.
Individuals are
and a proportional tax on inherited
r^fl*), yl{9 )}
)
an equilibrium consists of a sequence of interest rates
consumption, labor income and bequests {cv {9
strategy {cr^(9 )} such that:
.
ly
t
t
15
= 0, 1,
the before-tax interest rate across generations, and initially 6_i
is
subject to two forms of taxation: a labor income tax
wealth
t
(i)
{c t
,
bt
,
cr t
}
maximize dynastic
t
)
)
^(O
1
)};
and a reporting
utility subject to (18),
taking the
In this formulation, taxes are a function of the entire history of reports, and labor income y t is mandated given
However, if the labor income histories y ( 0* —» R* being implemented are invertible, then by the
this history.
:
we can rewrite T and r as functions of this history of labor income and avoid having to mandate
labor income. Under this arrangement, individuals do not make reports on their shocks, but instead simply choose
taxation principle
a budget-feasible allocation of consumption and labor income, taking as given prices and the tax system.
24
{R
sequence of interest rates
clears so that
fE_i[b^(9
incentive compatible
if,
t
)}
t^
and the tax policy {Tt r t yt }
t+ i}
d(p(v)
,
=
in addition,
for all i
it
=
0, 1,
induces truth
For any feasible, incentive-compatible allocation
.
We
.
.
,
for
any reporting strategy
b^(9
t
)
=
satisfies
cr,
1
is
we construct an incentive-compatible
)
=
—
4
yt(#
)
ct (9
t
)
and
v
u'(c t {8
t
{i? t _i it }.
say that a competitive equilibrium
{c", y^},
PH -i,t
any sequence of interest rates
the asset market
(ii)
telling.
competitive equilibrium with no bequests by setting T^{9
for
and
as given;
))
These choices work because the estate tax ensures that
the resulting consumption allocation {c"(cr f (#'))} with no bequests
the consumption Euler equation
U'(c?(aW)-)=/?iWE u
'(^
Ot+i
The
labor income tax
is
such that the budget constraints are satisfied with this consumption
cation and no bequests. Thus, this no-bequest choice
is
optimal for the individual regardless of the
reporting strategy followed. Since the resulting allocation
follows that truth telling
is
optimal.
then ensure that the asset market
As noted above,
in our
The resource
clears.
is
incentive compatible,
economy without
capital only the after-tax interest rate matters so the
In the next subsection,
set the interest rate to the reciprocal of the social discount factor, Rt-i,t
it
it
constraints together with the budget constraints
t
natural because
by hypothesis,
16
implementation allows any equilibrium before-tax interest rate {i2 _ 1)t }.
we
allo-
— $~
l
-
This choice
is
represents the interest rate that would prevail at the steady state in a version of
our economy with capital.
Optimal Progressive Estate Taxation
In this subsection
we
derive an important intertemporal condition that
must be
satisfied
by the
optimal allocation. This condition has interesting implications for the optimal estate tax, computed
using (19) at the optimal allocation.
Let A be the multiplier on the promise-keeping constraint and
on the incentive constraints. Then the first-order conditions
for
let n(6, 9')
represent the multipliers
u_ and w(9) are
c'(u_)-A-A =
0V(w(9))p(9)-p\p(e)-J2M6') + 'E ti0',O) =
l
16
Since the consumption Euler equation holds with equality, the
same
estate tax can be used to implement alloca-
tions with any other bequest plan with income taxes that are consistent with the budget constraints.
25
and the envelope condition
is k'
—
(v)
Y
k'{w{e)) P {0)
J
Using c'(w_)
=
l/u'(c_)
=
A
negative
arrive at the Modified Inverse Euler equation
first
(3
>
side
tax r(^
,6 t )
,
is
novel, since
is
the standard inverse Euler
zero
it is
when
/?
=
(3
and
is
strictly
Xl
(3.
In our environment, the relevant past history
f_1
V^
term on the right-hand side
The second term on the right-hand
when
ik'(v),
JY"W)
left-hand side together with the
equation.
=
p
+ k'(v) we
u'(c_)
The
Together these imply
A.
is
encoded
in the continuation utility so the estate
can actually be reexpressed as a function of
t_1
t> t
(0
)
and
8t
Abusing notation we
.
then denote the estate tax by T t (v,9 t ). Since we focus on the steady-state, invariant distribution,
we
also drop the time subscripts
The average
and write t(v,9).
estate tax rate f(v)
1
is
then defined by
- f(v) =
£
(1
-
t(v,6)) P (6)
(21)
e
Using the modified inverse Euler equation (20) we obtain
f(v)
= -AV(c>))(J-l)
In particular, this implies that the average estate tax rate
However,
are subsidized.
implementation.
is
negative, f(v)
recall that before-tax interest rates are not
As a consequence, neither
are the estate taxes
<
0,
so that bequests
uniquely determined in our
computed by
(19).
particular choice for the before-tax interest rate, however, the tax rates are pinned
acquires a corrective, Pigouvian role.
externalities
down and
Differences in discounting can be interpreted as a form of
from future consumption, and the negative average tax can then be seen as a way
of countering these externalities as prescribed
by Pigou. In our setup without
depends on the choice of the before-tax interest
be a robust steady-state outcome
In our
With our
model
of past shocks
it is
more
encoded
rate.
in a version of our
in the
promised continuation
function of consumption, which, in turn,
is
progressive: the average tax on transfers for
However, the negative tax on estates would
economy with
interesting to understand
capital, this result
how
capital.
the average tax varies with the history
utility v.
The average tax
an increasing function of
v.
more fortunate parents
higher.
is
is
an increasing
Thus, estate taxation
is
^This equation can also be derived from an elementary variation argument. This is done in Farhi, Kocherlakota
and Werning (2005), who also show that this equation, and its implications for estate taxation, generalize to an
economy with capital and an arbitrary process for skills.
26
Proposition 4 In
economy, the optimal allocation -can
the repeated Mirrlees
be
implemented by a
combination of income and estate taxes. At a steady-state, invariant distribution
average estate tax f(v) defined by (19) and (21)
The
is
ip*
the optimal
increasing in promised continuation utility v.
progressivity of the estate tax reflects the mean-reversion in consumption.
must face lower net
,
The fortunate
consumption path decreases towards the mean. 18
rates of return so that their
Conclusions
7
Should privately-felt parental altruism
implications for inequality?
affect the social contract?
To address these
questions,
what are the long-run
If so,
we modeled a
central tension in society:
the tradeoff between ensuring equality of opportunity for newborns and providing incentives for
altruistic parents.
Our model's answer
is
that society should indeed exploit altruism to motivate parents, linking
the welfare of children to that of their parents. However,
of future generations directly, the inheritability of
we
also find that
if
we value the welfare
good or bad fortune should be tempered. This
produces a steady-state outcome in which welfare and consumption are mean-reverting, long-run
inequality
What
of our
is
bounded, social mobility
is
possible
and misery
avoided by everyone.
is
instruments should society use to implement such allocations? For a Mirrleesian version
model we
progressive:
find
an important role
for the estate tax.
more fortunate parents should
The optimal tax on
inheritances
is
face a higher average marginal tax rate on their bequests.
This result illustrates an interesting way in which the conflict between corrective and redistributive
taxation
is
optimally resolved. Further examination of other situations with similar conflicts remains
an interesting direction
for future work.
19
Appendix
Proof of Theorem
Weak
1
concavity of the value function k{v) follows because the relaxed sequence problem has a
concave objective and a convex constraint
continuity over the interior of
its
also
its
domain.
be established as follows. Define the
k*(v)
set.
The weak concavity
If utility is
of the value function k{v) implies
bounded, continuity at the extremes can
first-best value function
= max^T/^E-x^u^) -
Ac(u
t
(0*))
]
4=0
subject to v
at
any
finite
= X^o^'^-il^*^*)]- Then
extremes v and
18
Farhi, Kocherlakota
19
Some
v.
Then
k*(v)
is
continuous and k(v)
<
k*(v),
continuity of k(v) at finite extremes follows.
and Werning (2005) explore more general versions
of this result
Thus, k(v)
and discuss other
progress along these lines can be found in Amador, Angeletos and Werning (2005).
27
with equality
is
intuitions.
continuous.
The
constraint (6) with q
Toward a
= $
and continuation
implies that utility
utility are well-defined.
contradiction, suppose
T
]T/?'£a
lim
T—>oo
t
u(ct)
t=o
is
not defined, for some
utility is
concave 9u(c)
s
> — 1.
This implies that limx^oo ^] t=0
< Ac + B
for
B>
some A,
so
it
^2(3 m&x{Es 9 u{c
t
t
t
),0}
^
t=0 $*
=
E_iCf
i
A^(?E
t
),
=
0}
oo. Since
a
ct
+B
4=0
t=0
limit yields lini7'_ 00
l
T
< ^ ]P f? E s q + S <
t=0
max{£'s 6 u(c
follows that
T
T
J
Taking the
0,
i
/?
oo. Since there are finitely
many
s
histories
S+1
£
X^t=o^*^-i c< = °°- ^ there is a non-zero measure of such agents this implies
a contradiction of (6). Thus, for both the relaxed and unrelaxed problems utility and continuation
this implies limj^oo
utility are well defined given the other constraints
on the problem.
This
is
important
for
our
recursive formulation below.
We
next prove two lemmas that imply the rest of the theorem.
Consider the optimization
problem on the right hand side of the Bellman equation:
supE[0u(0)
-
\c(u{9))
+ J3k(w{6))]
(22)
u,w
v
6u
Define
m
= max c >
Oi e e
(0)
e(0w(c)
=
+ pw(6) >
—
Ac)
E[6u(6)
Qu
and k
+ pw(6')
(0')
(v)
+ 0w(6j\
=
k (v)
(23)
e
for all 0, 6'
— m/(l —
$)
<
(24)
0.
The problem
in (22)
is
equivalent to the following optimization with non-positive objective:
supE[0u(0) -Xc(u(9))
-m + pk{w{9))}
(25)
subject to (23) and (24).
Lemma
Proof.
A.l The supremum
If utility is
and compactness
in (22), or equivalently (25), is attained.
bounded the
result follows
of the constraint set.
arguments apply when
utility
is
immediately by continuity of the objective function
So suppose
utility
is
unbounded above and below
only unbounded below or only unbounded above.
We
— similar
first
show
that
lim k(v)
v—*oo
and then use
this result to restrict,
maximum
attained.
is
without
=
lim k(v)
v—>— oo
loss,
= -co
the optimization within a compact
28
(26)
set,
ensuring a
To
establish these limits, define
oo
= sup J2 P
h(v; 0)
l
-
E-i [QMS*)
m
-
Ac(u(0*))
t=o
=
subject to v
t
E_i Ylt^oP ^tu(9
incentive constraints
it
t
Since this corresponds to the same problem but without the
)-
<
follows that k(v)
h(v,J3).
—
then the desired limits (26) follow. Since 9u
h(v,P)
<
h(v,0)
—
Xc(u)
=
lim,,-^ h(v,
If
<
ra
and
<
ft
lim„__ 00 h(v,
ft it
= — oo,
J3)
follows that
—
777
- XC(v,P) -
v
=
J3)
(27)
where
oo
C(u,/?)
=
V^'E_ic(u(0'))
inf
i=0
=
subject to v
t
t
Yl'tLoft '^-i[6t'u(9
problem, with solution «(#')
such that lim„__ 00 ^(v)
=
=
Note that C(v,/3)
)]1
and lim _ 00 7
i;
C(v,P)
Fix a
=
this defines a closed,
bounded
>
we can
Xc(u))
is
this defines a closed,
that
we can
in turn, imply the limits (26).
interval for w(9), for each
restrict the
bounded
restrict the
is
M
<
|if(0)|
9.
VyW
It
with (9u
—
Xc(u))
interval for u(9), for each
maximization to
\u(9)\
<
M
v>u
9.
continuous over this restricted
Lemma
set,
the
A. 2 The value function k(v)
Proof. Suppose that
for
concave with the limits
is
when
> maxE[0u(0) -
Xc(u(9))
u
—
>
Thus, there exists an
set.
<
>
k/p(9).
v>w
00
m
—
00 or u
—
M
00 such
ViU
<
>
—00,
Since the objective function
be attained.
Bellman equation
Xc(u{9))
29
—
either
(7)-(9).
some v
k(v)
M
.
maximum must
satisfies the
(26),
.
Hence, we can restrict the maximization in (25) to a compact
is
00 be
restrict the
follows that there exists
— —00
>
we can
non-positive,
maximization over u{9) so that 9u(9)
strictly concave,
<
the constraints (23) and (24) and let k
k/(j3p(9)). Since k{w{9))
such that we can restrict the maximization to
—
Using the inequality (27) this establishes
Then, since the objective
maximization to w{6) such that k(w(9))
Since (9u
(6 1 (v)))~
1
the corresponding value of (25).
Similarly,
1
/3)
verifies
and
Then
00.
= ±-E[cfe)1-/3
T
Take any allocation that
v.
(v)
first-best allocation
positive multiplier j(v), increasing in v
= —00 and lim^oo h(v, (3) = —00.
= —00 and hm ,_ 00 h(v,$) = —00, which,
so that lim^^.oo h(v,
linv^-oo h(v,$)
some
,
(c')~ (9 t y(v)) for
a standard convex
is
+ J3k(w(9))]
where the maximization
Then
subject to (23) and (24).
is
k(v)
>
E[6u{9)
-
there exists
A>
such that
+ J3k(w(6))] + A
Xc(u{9))
(u,w) that satisfy (23) and (24). But then by definition
for all
> Y^P^-ilOMO*) -
k{w(6))
for all allocations
u that
yield
w
(9)
Ac(«t(^))]
and are incentive compatible. Substituting, we
find that
oo
k(v)
> J^p'E^leMe^-xc^tie
1
))]
a contradiction with the definition of k(v).
for all incentive-compatible allocations that deliver v,
Namely, that there should be a plan with value arbitrarily close to
maxu w E[6u(9) -
Xc(u(9))
,
By
and
+ J3k(w(9))]
definition, for every v
delivers v
and e >
subject to (23
+a
)
and
there exists a plan
k(vo).
We
conclude that k(v)
<
(24).
{u^Q
1
;
v, e)}
that
is
incentive compatible
with value
oo
]T ft E_j [OMe v,e)-X
1
c(u 4 (^; v,e))]
;
>
k(v)
-
e.
t=o
Let (u*{6),w*(9)) E Sirgmax UiW E[9u(9)
andut(0')
=
ut-i((0 u
.
.
.
-
+ J3k{w(9))].
Xc(u{9))
,9 t );w*(6 ),e) for t
>
Consider the plan u (9Q )
=
u*(9
)
Then
1.
oo
= E_xk^(0o)-Ac(u*(0o))+^^^E o [0 m ^ +1 (0 i+1 )-Ac(^ +1 (0 t+1 ))]
t=0
>
Since e
(23)
>
and
maxE[0u(fl)-Au(0)+/3/c(w(#))l
was arbitrary
follows that fc(u)
>
ma,x u<w E[9u(9)
—
Xc(u(9))
+ J3k(w(9))}
subject to
(24).
Finally, together
to (23)
it
- /k
and
both inequalities imply k(v)
= max U}W E[9u(9) —
Xc(u(6))
+ j3k(w(9))}
subject
(24).
Proof of Theorem 2
Part
(a).
Suppose the allocation {u
incentive compatible
and
t
}
is
generated by the policy functions starting from v
delivers lifetime utility v
30
.
,
is
After repeated substitutions of the Bellman
equation
( / )
we
,
arrive at
T
= Y,P ^-i{8Me )-\c{u
t
k(v Q )
t
t
t
{d
))}
+
E- k(v T {e T )).
(3
(28)
1
t=o
Since k(v
is
)
bounded above
this implies that
oo
k(v
<
)
YtP'E-iWMe 1
)
HMO
1
))},
4=0
so {ut }
optimal, by definition of k(vo).
is
Conversely, suppose an allocation {u t }
is
optimal given v
Then, by definition
.
it
must be
incentive compatible and deliver utility vq. Define the continuation utility implicit in the allocation
w
= J^p
(9 Q )
t
-1
E o [9
t
u t (0
t
)}-
t=i
and suppose that
plan {u t }
u
either
u
^g
(9)
(9;
u
incentive compatible,
is
v
(9)
)
w (9) ^ g w (9; v for some 9 £ 0.
and w (9) satisfy (23) and (24). The
or
),
Since the original
Bellman equation
then implies that
k(v
E[g u (9;v )-Xc(g u (9;v ))+Pk(g w (9;v
=
)
> E[u
- Xc(u
(9)
))}
+ Pk(w Q (9))]
(9))
oo
> E_j [uo(6
)
- \c(u
+ X>* E-i [MO') ~ HMO'))}
(9 ))]
t=i
The
argument applies
>
1.
We
Part
if
(7),
while the second inequality follows the
Thus, the allocation {u t } cannot be optimal, a contradiction.
definition of k(wo(6)).
t
does not maximize
inequality follows since u
first
the plan
A
similar
not generated by the policy functions after some history 9
is
l
and
conclude that an optimal allocation must be generated from the policy functions.
(b). First, suppose an allocation {u t v t ] generated by the policy functions (g
u
,
at Vq satisfies lim t _ oo
t
/?
E_iu
t
t
(0
)
=
Then,
0.
after repeated substitutions of (8),
,
g
w starting
)
we obtain
T
r
= Y,P ^-A e tMO
t
t
)\
+ P T E^[vT (9
T
(29)
)}.
t=0
Taking the limit we get v
v
Next,
.
Hm -,ooP
t
t
we show
^~iVt{9
Suppose
lim supj^oo
t
)
/5
E_iv
'^
,
-i[OtM0
t
)]
so that the allocation
that for any allocation generated by (g ,g
=
(9
w
{u t } delivers lifetime
starting from finite v
),
,
utility
we have
0.
unbounded above and limsup^^
t
t
t
Yl'tLoP
u
utility is
(
=
)
=
oo.
|t
/?'
E_it> t (6
Since the value function k(v)
31
is
)
>
0.
Then
>P
implies that
non-constant, concave and reaches
an
interior
maximum, we can bound
t
<alimsup 5 E_
limsup/J'E-ifc^fl*))
/
(28) implies that k(v
We
finite values.
Similarly,
=
)
liminf t _ 0O /3*E_ 1 'U f (<9
<
)
with a
b,
<
0.
Thus,
+ 6 = -oo
u i (^)
/3'
E_iut (0') <
0.
unbounded below and that liminf
utility is
=
+
-co, a contradiction since there are feasible plans that yield
conclude that lim sup^,^
suppose
1
av
t—too
t—>oo
and then
<
the value function so that k(v)
<
-co. Using k(v)
+
av
b,
with a
>
0,
i
_ 00 /3
t
<
E_it't(^
)
<
>
0.
0.
Then
Then
after
we conclude that
= -oo
liminf/^E.i/cfWfl'))
t—>oo
implying k(v
The two
Part
= — oo,
)
a contradiction. Thus,
we must have liminf
established inequalities imply lim^oo/?
(c).
t
Suppose limsup 4 _^ 0O
repeated substitutions of
E_if
/3
t
(a'(^
4
E_
|t
>
t
))
u t (6
1
for
)
=
i
t
_too
/?
E_iU£ (^
t
)
0.
every reporting strategy a.
(9),
T
T
v>Y,P' E -i [°^t (*W)] + f E-iVr {° {0
T
))
t=o
implying
T
v
Therefore, {u t }
Proof of
Part
(a)
is
>
incentive compatible, since v
Lemma
is
u, (^(fl*))]
t
•
attainable with truth telling from part (b).
1
t
t
(Strict Concavity) Let {u t (9 ,Vo),vt (9 ,v )} be the plans generated from the policy
no claim of incentive compatibility
functions starting at vq (note:
utility values v a
and
Vb,
^
with v a
v?(9
2 part (b) implies that
immediately implies that {u°(9
there exists a finite time
T
t
Define the average
v^.
<(0')
Theorem
]T 0- E_i [0
lim inf
l
)
= au
=
t
{9\v a )
avti^Va)
{«j(#\fa )} and
+
(\
is
required).
-
a^O ,v
1
b)
+ (1-0^(9*;^)
{«((#*,
i> b
a
)} deliver
= av a +
va and v b respectively. This
,
(1
—
a)v b
such that
t
u
l
t
(9
;
va
)}y^J2
i=0
t=o
32
initial
utilities
)} delivers initial utility v
]T 0* E_i [6
Take two
p*
E_! [OMB*; v b )}
.
.
It also
implies that
so that
ut
for
some
and
history
i+1
e
6>*
t
(9 ;v a
)^u
T
Consider iterating
.
t
(6 -v b ),
t
(30)
times on the Bellman equations starting from v a
Vf,:
T
= £]/?*£_! [0^(0*;^) -c(u
k{va )
(0';
t
Ua ))]
+^ T E_
1
fe(u T (0
T
;
Ua ))
4=0
T
= £)i&'E_ 1 pt w
fc(v 6 )
t
(e*;«i,)-c(Ti t (fi
r
:r
;^))]+ & E_ 1 A:(wr^ ;«6)),
t
)
t=o
and averaging we obtain
ak(v a )
+ (l-a)k(v b
^^E.!^^ )- [ac(^(0
=
)
4
i
;t; Q ))
+ (l- a )c( Ut (^;^))]]
4=0
+£ T E_x [ak(v T (9 T
;
v a ))
+
(1
- a )^ r (# r
«>))]
;
T
J^'E^u?^) -Qc^f^*))]
<
+
T E_ k(v$(9 T
l
))
t=0
where the
that
from the
strict inequality follows
we have the
inequality
(3(1),
concavity of the cost function c
strict
and the weak concavity
of the value function k.
inequality follows from iterating on the Bellman equation for v
satisfies the
is
Bellman equations constraints at every
The
,
the fact
last
weak
a
a
since the average plan (u ,v )
This proves that the value function k(v)
step.
strictly concave.
(b) (Differentiability) Since the value function k(v)
is,
a
(it)
there
is
concave,
it is
sub-differentiable
at least one sub-gradient at every point v. Differentiability can then
is
—that
be established by
the following variational envelope arguments.
Suppose
first
that utility
is
unbounded below. Fix an
interior value v
for initial utility.
For a
neighborhood around vo define the test function
W(v)
Since
W(v)
is
= E [9(g u (9, v + (v-
(see
v Q ))
-
Xc(g
u
(9,
v
)
+
(v
-
v
the value of a feasible allocation in the neighborhood of
with equality at v
Theorem
)
.
Since W'(vq) exists
Theorem
4.10, in Stokey,
k'(v
Finally, since c'(u)
>
this
by the upper bound k(v)
<
)
=
+ 0k (g™ (6, v
i>
it
follows that
W'(v Q
)
=1-
k'(v)
+ m/(l —
<
J3)
33
))]
W(v) <
k(v),
by application of the Benveniste-Scheinkman
follows,
Lucas and Prescott, 1989), that k'(v
shows that
h(v,0)
it
))
)
also exists
and
A E[c'(u*(9))}.
1.
The
limit
lim^-oo
(31)
k'(v)
=
1
is
introduced in the proof of Theorem
inherited
1,
since
Urn,,—.*, Jj;/l(u,/?)
The
it is
=
1.
= — oo
limit lim„_^ c k'(v)
inherited by the upper
since. lim^oo
=
-^h(v,P)
long as g u (9,v
)
>
bound
Then
+ m/(l — $)
h(v,P)
9
0, for all
loss in generality
utility function, so
,
interior
t;
t
t
any
oo.
=
Otherwise
1,
))
the plan {{v/va)u t }
is
incentive compatible
as
is
required.
set.
>
for all reporting strategies
a solution {u t } to the planner's sequence problem
2,
utility
are possible here
a different envelope argument
bounded below, then limsup^^ E-iP v (a(9
Theorem
<
suppose that the
G 0. However, corner solutions with g u (9,vo)
t
that, applying
u
if
introduced in the proof of Theorem
provide one that exploits the homogeneity of the constraint
If utility is
= — oo,
the argument above establishes differentiability at a point v
even with Inada assumption on the
We
<
bounded below, and without
is
zero.
is
k(v)
from lim^^ k(v)
-oo.
Next, suppose utility
of zero consumption
follows immediately
is
a
so
ensured. Then, for
and attains value v
for the agent.
In
addition the test function
W(v)
= J2~PtE
9t
-
f=0
satisfies
W(v) <
W(vq)
k(v),
Scheinkman Theorem, that
The proof of lim^ s
we show
=
= — oo
and
fc(i>o)
k'(vo) exists
k'(v)
that lim^^^ k'(v)
=
t
t
{9
)-\c(-u
v
\v
is
differentiate.
and equals W'(v
the
is
-u
same
t
{9
t
)
follows
It
from the Benveniste-
).
as in the case with utility
unbounded below.
Finally,
problem
oo. Consider the deterministic planning
oo
k(v)
= meix2_]p{ u ~
t
Ac(u t ))
t=o
subject to v
=
P tu t- Note
that k(v)
is
plans are trivially incentive compatible,
it
Yl'tLo
must have limv -, u
k'(v)
=
CLAR
with \im v -> u tf(v)
follows that k(v)
<
=
oo. Since deterministic
k(v), with equality at v.
Then we
oo to avoid a contradiction.
Proof of Proposition
The
differentiable
1
equation was shown in the main text, so we focus here on the bounds. Consider the
program
maxVpn {0 n un
ii.
in
<
-c(u n +(3k(w n )}
)
n
V
= Y^Pn(9nUn + P^n)
n
9 n un
This problem and
constraint on u.
its
But
of neighboring agents
+ Pwn >
9 n u n+1
+ Pw n+X
for
notation require some discussion.
this notation allows us to consider
is
n
=
We
1, 2,
.
.
,
.
K-
1,
do not incorporate the monotonicity
bunching
in the following way.
bunched, then we group these agents under a single index and
34
If
any
let
set
pn be
Likewise
the total probability of this group.
this group,
which
is
what
is
let 9 n
represent the conditional average of
within
relevant for the promise-keeping constraint and the objective function.
Let 9 n be the taste shock of the highest agent in the group. The incentive constraint must rule the
highest agent in each group from deviating and taking the allocation of the group above him.
Of
course, every combination of
bunched agents leads to a
We
program.
different
study
all
of
them. The optimal allocation of our problem must solve one of these programs, although not necessarily the
condition
The
one that yields the highest value, since
is
may
not be feasible
if
the monotonicity
violated.
first-order conditions are
Pn{6n
-
-
Xc'{u n )
+ Qn
A0 n }
Pn0k'(w n - PX} +
=
where, by the Envelope theorem, A
Consider
first
case with utility
tion holds with equality.
n
1
'
=
-
n _ 1/u n _ 1
Mn-:)
<
=
so that the
first
order condition for consump-
the first-order conditions for consumption,
we
get
=l-k'{v)
\E[c'(u(6))}
first-order conditions for
-
n
k'(v).
unbounded below,
Summing
\i
/?(//„
)
The
one
this
imply
1
OiK
Xd( Ul )
XE[c'(u e )]
1-A
0i pi
h
h
e1
This implies
^<^-(i-a£,
Using
PPI
p
we
get
p
fc'M >
p
-^"-4
8
x)
Ok-i /%-i
6K
Pk
=
+
n
=
01
01
K
^ (uk)
,
we
>
0i
Pk
P
"0i
P
A
+£
J
XE[d(u g )]
1-A
K
K
K
1 ~ X
>
-(1-A);
~ Ok-i
35
N
get
This implies
V-K-l
,//
k'{v)
7T~
P
Similarly, writing the first-order conditions for
(1
1
l
TK
7
K-1
—
—
1
1
0i
J
Using
k (w K )
we
= —A + - —
P Pk
get
k'
(%) <
_I^ +
k'(y)
e;-o;
P
_
A
A)
7
&k-i
we have
Wi,
Qx
1
+
(1
@K-\
wk < w n <
For any n,
1
A
-r
1
ft
+I
PY 0i
k'(wn )
<
P
P
1
®K-\
(l-fc») +
l-|P
> l-k'(gw
+
1
arrive at the expression in the text
Turning to the bounded
w
we take the worst
most unfavorable to each bound, noting that
satisfied
+
K
TK-\
@K-\
@K-1
(9,v))
when A >
implies k'(w(9))
1
with
=
utility case,
/i n
=
=
all
eK
Qk-\
& K-\
—
case scenario:
k'{y)
>
-
k'(v))
+i-L
we choose the subproblem
0.
the first-order conditions and constraints are
w (6) = P~
l
v
>
v.
(P/P)k'(v). Since the problem
is
strictly convex, this represents
and u(9)
=
k'(0~ v)
note that
1
1
(i
P
is
1
k'{v)
Oi
> P
that
Ok-i
we obtain
+
P
To
eK
1
+
1
P
After rearranging,
P 6k-i
1
n
<
ex
~6k
I
k'(v)
®k-\
'K-l
for every
K
u and
The
first-order condition for
the unique solution. Recall that in the arguments above establishing the lower bound involved no
assumption on interior solutions
did require u{9)
>u
for all 9,
for u, so this holds for all v.
which must be true
for
The upper bound, on the other hand,
high enough
v, i.e. for
low enough
k'(v).
Proof of Proposition 2
Consider
first
the case with utility unbounded below. Since the derivative k'(v)
strictly decreasing,
we can
x
<
1
if
utility
is
continuous and
define the transition function
Q(x,9)
for all
is
= k'{g w ((k')-
1
{x),9))
unbounded below. For any probability
distribution
probability distribution defined by
tq(v)(A) =
/ l {Q{Xt o )eA)
36
dn(x)dp(6)
fj,,
let Tq(/i)
be the
any Borel
for
set A. Define
+ 7g + ---+73
rq
TQ n
=
'
For example, Tq
with k'(v
=
)
the empirical average of {k'(v t )}?=1 over
{5x ) is
iTL
The
x.
n
following
lemma
histories of length
all
n
starting
establishes the existence of an invariant distribution
by
considering the limits of {Tq iTI }.
Lemma
A. 3
If utility is
that converges weakly,
Proof. The bounds
unbounded below, then for each x <
in distribution, to
i.e.
an invariant distribution on (—00,
(13) derived in Proposition
lim<2(z,0)
=
lim
imply that
1
k'(g
w
{9,v))
{Tq^ u )(5 x )}
there exists a subsequence
1
for all 9
=
0/J3
<
1)
under Q.
6
1.
l)x0-> (—00, 1) to a continuous
transition function Q(x,9): (-00, 1] x0-t (— 00, 1), with Q(l,8) = P//3 and Q(x,6) = Q(x,8),
for all x € (—00, 1). It follows that TU maps probability distributions over — 00, 1] to probability
We
first
extend the continuous transition function Q(x,
6)
(—00,
:
(
distributions over (—00,
1),
and Tq(S x )
=
Tq(5 x ),
for all
x € (—00,
1).
We next show that the sequence {Tq u (S x )} is tight, in that for any e >
set A e such that TU (5 X )(/LJ > 1 — e, for all n. The expected value of
is
simply E-Wivtie*'
1
))]
with x
=
k'(v
)
<
1.
Recall that
E-i^Mfl*
there exists a compact
the distribution T?(6 X )
-1
))]
=
1
(P / J3)
k' {v
)
-»
0.
This implies that
min{0,fc'(^o)}
< E.!^'^^*- 1 ))]
I ?3(<y (-00, -A)(-A) +
for all
A>
0.
Rearranging,
1
isw(-°o.-a)<
-^
tight.
is
Tightness implies that there exists a subsequence T^ ^
weakly to
some
distribution n. Since Q(x,6)
But the
Tq(tt).
linearity of
Tq
is
and
since 0(n)
Recall that
(-00,
for
1).
—
>
00 we must have Tq{tv)
Tq maps
it
continuous in
~
=
x
)-Tq(§ x
)
tv
= Tq (tt)
follows that
7r
that converges weakly,
x,
then
Tq{Tq^^{5 x ))
+ TQMn)V x
^y-
i.e.
in dis-
converges
)
n.
probability distributions over (-00,
This implies that
such distributions,
n(<5 x )
implies that
T^)+1 (5
TQ(TQ M n)(8*)) =
°-'>
1
which implies that (T^(<5 X )}, and therefore {TQ n (5 x )},
tribution, to
- T?(5x )(-oo, -A))l
(1
1]
to probability distributions over
has no probability mass at {1}. Since Tq and Tq coincide
=
Tq(tt), so that
under Q.
37
n
is
an invariant distribution on (-00,
1)
The argument
for
the case with utility bounded below
function Q(x,8) as above, but for
is
unbounded above but 7 <
v.
Define the utility level v
\im v ^y g
vl
>
bounded above then
is
w
(0,
=
v)
let
argument
is
w
is
>
w
we must have that
v
continuous with Q(x,9)
then a simple modification of the one above for
playing the role of
1
If utility
00 for the ergodic set for
minimum
of the policy
continuous over this compact
is
w
of g over v e
minimum
v^ by the
Finally, the transition function
v.
<
Next, define vl to be the
1.
Define the transition
can take on any real value.
k'(v)
defined since g
is
In both cases, since g
v.
=
v by k'(vo)
function g w over v e [fo,^], which
utility
now
since
very similar.
then there exists an upper bound v H
1
>
i£E,
all
is
[vq, v),
this
<
which
minimum
k'(vi)
<
is
is
above misery:
rest of the
unbounded below, with
utility
If
defined since
The
00.
set.
k'iyLj
(things are actually slightly simpler here, since no continuous extension of
Q
required).
is
Proof of Proposition 3
Consider indexing the relaxed planning problem by
by setting A
=
e
_1
,
with associated value func-
We first show that if an initial distribution ip satisfies the condition J k'(v; e) dip(v) = 0,
the solution to the relaxed problem and original problem coincide. We then show that for any
tion k(v;
then
e
e).
initial distribution
Since utility
is
there exists a value for e that satisfies this condition.
unbounded below, we have
k'{v t e)
;
=E
t
_i
[l
—
v
\c!(u t (#*))]
.
Applying the law of
iterated expectations to (12) then yields
t
E_ 1 [l-Ac'«(^)] = ^0fc'( U ;e
With
logarithmic utility c'(u)
for allt
of
0, 1,
Theorem
Now
/
=
.
.
2, it
.
The
=
allocation
follows that
it
k'(v; e) dtp(v)
=
0.
Note that
— — 00.
>
must
k'(v
It
— j^
= implies /E_i[q] dtp = A -1 = e
Lemma 2 below, and applying part (c)
k'(v;e)dip(v)
by
solve the original planning problem.
The homogeneity
=
J
incentive compatible
is
consider any initial distribution
k(v; e)
e
c{u), so that
—
tp.
We
argue that
+
k (v
find a value of A
=
e
_1
such that
problem implies that
of the sequential
-r log(e)
we can
-
log(e); 1) is strictly increasing in e
——
and
log(e); 1
limits to 1 as e
—
00,
and to —00
as
follows that
J
k'(v ]e )d^(v)
defines a unique value of e* for
any
immediately by using the fact that
=
1
J
k (v
- ^-log(e);l) #(u) =
initial distribution
k'(-;l)
is
ip.
The monotonicity
a strictly decreasing function.
38
of e*(ip) then follows
Lemma
Proof of
(a) If utility is also
2
bounded below, then the
unbounded below, but bounded above. Then k'(g w (9, •))
w
w
that
k'(g (9,v)) = 1. It follows that ma,x k'
lim^-oo
-co such
(g
v
that g
w (9,v)
(b) If utility
is
>
With logarithmic
bounded below, the
w (9,v)
-
g
9*~ l
for all pairs of histories
w (9,v)
and 9
<
l~ l
.
u
g (9,v)
-
(9 v))
2
7 >
>
1
))]
1
^*-
)]
=
A.
<
of
Theorem
2.
one can show that:
log (9/9).
follows that
It
^E-!^*-
t
1
1,
(c)
9'
u
g (9,v)
we have lim _ 00 /5* E_ 1 [u
limsup^^E-iMff'-
(d) If
implies
attained, so there exists a vl
is
,
1
The
v^
incentive constraint
'1
1' 1
1
)
> v^O
)
- tA
Then
1
Theorem
—
(9/(3) log(0/0)
/^[^(cr'- ^part (b) of
utility is
continuous and Proposition
immediately from part
result follows
utility this implies that
then implies that g
So suppose
(b).
vl-
d(u(6))
that
is
Using the first-order conditions from the proof of Proposition
(c)
From
from part
result follows
>
t
(0'
-1
)]
=
1
)]
-PtA.
Since
0.
lim^^ (3 tA =
l
0, it
follows
0.
then the bound in (13) implies that
k' (g
w
(9,
>
v))
1
- (3/(3 and
the result follows
then we can define n = 1 — (1 — /3/$)/(l — 7), and define vh by k'iyn) = «•
Then for all v < v # we have g w (9,v) < v. It follows that the unique ergodic set is bounded above
v) >
We can now apply the argument in (a) so there exists a,
> —00 such that w
immediately.
If
7 <
1,
by vh-
vl
Proof of
Part (a)
is
(b), recall
then
all
c
Lemma
g
J
k'(v) dip*(v)
=
under the invariant distribution
solutions for consumption are interior.
with g (9,v)
=
for all 9 in this
for
some
compact
If utility is
9 can only occur for low
set.
enough
ip*.
is
bounded. For part
If utility is
unbounded below
bounded below, then corner
c
levels of v, so that g (9,v)
Recall that for interior solutions
l-k'(v)
= XE[c'{g u (9,v))]=XE[c'(u(c(g u (9,v))))
Applying Jensen's inequality we obtain
c
The
u
vl-
3
immediate since by continuity of the policy functions, consumption
that
(9,
u
E[c{g (9,v))] dP(v)X\ < JE[c'(u(c(g^9,v)))))
result then follows since c'(u(c))
is
an increasing function of
39
c.
#» =
1.
is
solutions
bounded,
References
[1]
[2]
[3]
Aiyagari, Rao, and Fernando Alvarez (1995), "Stationary
Private Information: A Tale of Kings and Slaves," mimeo.
[5]
Amador, Manuel, and George-Marios Angletos, and Ivan Werning
Bernheim, B. Douglas (1989)
mimeo.
"Intergenerational Altruism, Dynastic Equilibria
Studies,
and Social
v56, nl (Jan), pp. 119-128.
Caplin, Andrew, and John Leahy (2004) "The Social Discount Rate," Journal of Political
Economy, vll2, n6 (December): 1257-68.
Emmanuel, Narayana Kocherlakota and Ivan Werning (2005) work
[7]
Farhi,
[8]
Green, Edward
tual
[9]
(2005), "Correc-
Atkeson, Andrew, and Robert E. Lucas Jr. (1992) "On Efficient Distribution with
Private Information," Review of Economic Studies, v59, n3 (July): 427-53.
Welfare," Review of Economic
[6]
with
Albanesi, Stefania, and Christoper Sleet (2004), "Dynamic Optimal Taxation with
Private Information," mimeo.
tive vs. Redistributive Taxation, "work in progress,
[4]
Efficient Distributions
J.
in progress.
(1987) "Lending and the Smoothing of Uninsurable Income, "in Contrac-
Arrangements for Intertemporal Trade: 3-25.
Grunwald, G.K., R.J. Hyndman, L.M. Tedesco, and R.L. Tweedie (2000) "NonGaussian Conditional Linear AR(1) Models," Australian and
New
Zealand Journal of Statistics,
42(4), 479-495.
[10]
[11]
Golosov, Mikhail, Narayana Kocherlakota, and Aleh Tsyvinski (2003) "Optimal
Indirect and Capital Taxation," Review of Economic Studies, v70, n3 (July): 569-87.
Kaplow, Louis (1995) "A Note on
Subsidizing Gifts," Journal of Public Economics, v58, n3
(Nov): 469-77.
[12]
[13]
Kocherlakota, Narayana (2004) "Zero Expected Wealth Taxes:
Dynamic Optimal Taxation," mimeo.
Mirrlees,
tion,"
[14]
[15]
[16]
[17]
[18]
James (1971) "An
Exploration in the Theory of
A
Mirrlees
Approach to
Optimum Income Taxa-
Review of Economic Studies, v38, n2: 175-208.
Phelan, Christopher (1998) "On the Long Run Implications
ard," Journal of Economic Theory, v79, n2 (April 1998): 174-91.
of
Repeated Moral Haz-
Phelan, Christopher (2005) "Opportunity and Social Mobility," Federal Reserve Bank
Minneapolis, Research Department Staff Report 323 (March).
Sleet, Christopher,
and Sevin Yeltekin (2004)
of
"Credible Social Insurance," mimeo.
Spear, Stephen E., and Sanjay Srivastava (1987) "On Repeated Moral Hazard with
Discounting," Review of Economic Studies, v54, n4 (October): 599-617.
Stokey,
Nancy
L.,
and Robert
E. Lucas, Jr., with
"Recursive Methods in Economic Dynamics," Harvard Press.
40
Edward
C. Prescott (1989)
[19]
Thomas, Jonathan, and Tim Worrall (1990) "Income
formation:
An Example of a
Fluctuation and Asymmetric InRepeated Principal- Agent Problem," Journal of Economic Theory
'
Vol 51(2), pp 367-390.
41
U 8 3 G
025
Date Due
Lib-26-67
MIT LIBRARIES
3 9080 02618 3563