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3
DEWEY
HB31
.M415
Massachusetts
Institute of
Technology
Department of Econonnics
Working Paper Series
INEQUALITY
AND SOCIAL DISCOUNTING
Emmanuel
Farhi
Ivan Warning
Working Paper 05-1
April 27, 2005
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Network Paper Collection at
http://ssrn.com/abstract=712803
Social Science Research
MAY
S 2005
LIBRARIES
Inequality and Social Discounting*
Emmanuel
MIT
Ivan Werning
MIT, NBER and UTDT
[email protected]
[email protected]
Farhi
First Draft: June 2004
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
who
role should estate
We explore
these ques-
experience privately observed taste or
Our positive economy is identical to models with infinite-lived individwhere efficiency requires immiseration: inequality grows without 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
the initial generation's welfare. We study other efficient allocations where the social welfare
criterion values future generations directly, placing a positive weight 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
productivity shocks.
uals
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,
Gary Becker, Olivier Blanchard, Ricardo Caballero, Dean Corbae, Bengt Holmstrom, Narayana
Kocherlakota, Robert Luccis, 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
Freeman and Sadler we thank Dean Corbae for bringing it to our attention.
'For useful discussions and comments
Abhijit Banerjee,
—
Existing normative models of inequality reach an extreme conclusion: inequality should be per-
and
fectly inheritable
without bound, with everyone converging to absolute misery and
rise steadily
a vanishing lucky fraction to
bliss.
This immiseration result
and obtains invariably
tions on preferences (Phelan, 1998);
Thomas and
in partial equilibrium (Green, 1987,
Worrall, 1990), in general equilibrium (Atkeson and Lucas, 1992), and across environ-
ments with moral-hazard regarding work
or productivity (Aiyagari
We
weak assump-
robust; requires very
is
and Alvarez,
effort or
with private information regarding preferences
1995).^
depart minimally from these contributions, adopting the same positive economic models,
but a shghtly different normative criterion. In a generational context, previous work with
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
future generations.
Our
interest here
is
in
exploring a class of Pareto-efficient allocations that take into account the current population along
with unborn future generations.
We
place a positive and vanishing Pareto weight on the expected
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
lower
for
and everyone avoids misery. Indeed, welfare
bility is possible
bound that
is
strictly better
result:
long-run inequality remains
consumption and welfare,
typically remains above an
social
than misery. This outcome holds however small the difference
between social and private discounting, and regardless of whether the source of asymmetric
mation
We
is
begin by modeling a positive economy that
is
is
identical to the taste-shock setup developed
composed
of a
continuum of individuals who
one period and are altruistic towards a single descendant. There
ment
of the only
consumption good
cations. Feasible allocations
constraint in
all
live
a constant aggregate endow-
but experience
— thus ruling out
must be incentive compatible and must
first-best allo-
satisfy the aggregate resource
periods.
only the welfare of the
to that of an
is
in each period. Individuals are ex-ante identical,
idiosyncratic shocks to preferences that are only privately observed
When
infor-
privately observed preferences or productivity shocks.
by Atkeson and Lucas (1992). Each generation
for
mo-
endogenous
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
of the distribution
'Many
is
permanently. As a result, the consumption process inherits a random-walk
cross-sectional inequahty to
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
venient from a practical standpoint. Long-run steady-states often provide a natural
benchmark
it is
also incon-
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.
— but
By
Of
course, the perfect intergenerational transmission of welfare improves parental incentives
at the
expense of exposing newborns to the risk of their parent's luck.
contrast,
no longer
it
remains optimal to link the fortunes of parents and children in our model, but
Rewards and punishments are distributed over
in lock-step.
manner. This creates a mean-reverting tendency
in a front-loaded
random walk
— that
is
strong enough to
cross-sectional distribution for
bound long-run
future descendants, but
all
consumption
in
The
inequality.
result
is
— instead of a
a steady-state
consumption and welfare, with no fraction of the population stuck
at misery.
We also study a repeated
Mirrleesian version of our
economy and derive imphcations
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
is
private — 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
outcome highlights the
do not require any particular asymmetry of information.
fact that our results
More importantly, the Mirrleesian model
locations can be
offers
new
cross-sectional distribution. This
insights into estate taxation.
implemented by combining income and estate taxes.
Specifically,
Feasible
we
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
dynasties, with relatively high levels of current consumption,
must have a declining 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
this
paper
is
to reformulate the social
ideas introduced by Spear
social preferences differ.
Indeed,
we
and Srivastava
are able to reduce
the dynamic program to a one-dimensional state variable, and our analysis and results heavily
exploit the resulting
Bellman
equation.'^
Our dynamic program
Related Literature. Our paper
social planning
most
it
strictly positive
must
future.
efficient
and accurate
4.
closely related to
problem with no discounting of the
planning problem exists then
problem have
is
an
also offers
computational strategy, as illustrated by an example in Section
Phelan (2005), who considered a
He shows
that
solve a static maximization problem,
if
a steady state for the
and that solutions to
this
inequahty and social mobility. Our paper establishes the existence
of a steady-state distribution for the planning problem for any difference in social and private
discounting. Unlike the case with no discounting, there
^We
is
no associated static planning problem
also present a recursive formulation with a two-dimensional state- variable that has a very simple
interpretation and
is
useful for developing an intuitive understanding of our results.
for
economic
steady-state distributions, and as a result, the methods
we develop here
In overlapping-generation models without altruistic links,
all
weight on future generations. Bernheim (1989) was the
efficient place positive direct
out that in the dynastic extension of these models with altruism,
are not attainable by the market.
The
Our work
estate tax
is
negative
first
to point
efficient allocations
(1995) argued that these Pareto efficient allocations
—
it is
a subsidy
equilibria with estate
— so as to internalize the externality
contributes to a large literature on dynamic economies with asymmetric information.
In addition to the
work mentioned above,
this includes recent research
on dynamic optimal taxation
Golosov, Kocherlakota and Tsyvinski, 2003; Albanesi and Sleet, 2004; and Kocherlakota,
2004). This hterature has been handicapped by the immiseration result
steady-state distribution with positive consumption, making
for
Pareto
on future generations.
of giving
(e.g.,
Kaplow
many
and that they can be implemented by market
are natural social objectives
taxation policy.
are very different.
market equilibria that are Pareto
it
and by the non-existence
difficult to
of a
draw long-run conclusions
optimal taxation. Our results provide an encouraging way to overcome this problem.
Our work is
also indirectly related to a
paper by Sleet and Yeltekin (2004), who study an Atkeson-
Lucas environment with a utilitarian planner, who lacks commitment and cares only
generation. In this environment, as in Phelan's,
hold, so the interesting question
is
how
it is
for the current
a foregone conclusion that immiseration will not
to solve for the best subgame-perfect equilibria. Sleet
and
Yeltekin derive first-order conditions from a Lagrangian and use these to numerically simulate the
solution. Interestingly,
it
turns out that the best allocation in their no-commitment environment
is
asymptotically equivalent to the optimal one with commitment but featuring a more patient welfare
criterion.
Thus, our analysis and results provide an indirect, but effective way of characterizing the
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,
problem and draw
its
we develop a
connection to our original formulation.
recursive version of the planning
The
resulting
Bellman equation
is
then put to use in Section 4 to characterize the solution to the social planning problem. Here we
derive our
main
consumption.
some
results
We
on mean-reversion and on the existence of a steady-state distribution
discuss these results in Section 5
related problems
and reformulations. In Section
setup with productivity shocks and focus on
its
and develop intuition
6,
we turn
for
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
infinite-lived agent
is
can be interpreted as a dynasty of individuals
Under
altruistically linked.
this interpretation,
effort choices.
— the differences are only normative.
identical to that of Atkeson-Lucas
who have
finite lives
but are
Atkeson-Lucas and others focus on a particular
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
is
that directly weigh the welfare of
efficient allocations
individuals.
Demography, Preferences and Technology. At any
point in time, our economy
by a continuum of individuals who have identical preferences,
single descendant in the next. Parents
by a
and
all
asymptotically equivalent to postulating preferences for an
is
born
in period
t
live for
populated
is
one period, and are replaced
are altruistic towards their only child
their utility Vt satisfies
=
vt
where
q >
is
the parent's
descendant's utility Vt+i-
and continuously
and w'(oo)
=
0.
own consumption and
We
assume that the
differentiable for positive
The extremes v
=
=
/3
+ pvt+i]
G
,
(0, 1) is
utility function
consumption
u{0) and v
the inverse of the utility function c
identically
Et-i [9tu{ct)
=
u(oo)
or
m~^, or cost function.
This specification of altruism
=
is
M+ — R
>
is
continuous, concave,
may
not be
The
taste shock ^
and independently across individuals and time. For
adopt the normalization that E[^]
:
with the Inada conditions w'(0)
levels,
may
the altruistic weight placed on the
u
simplicity,
finite.
=
oo
We let
c{u) denote
G
distributed
we assume
is
is
finite
and
1.
consistent with individuals having a preference over the entire
future consumption of their dynasty given by
Vt
J2P'^t-l[0t-,sU{(k+s)].
(1)
In each period, a resource constraint limits aggregate consumption to be no greater than
constant aggregate endowment e
>
0.
some
These specifications of individual preferences and technology
are precisely those adopted by Atkeson-Lucas.
Social Welfare.
We
depart from Atkeson-Lucas by assuming that the social welfare criterion can
be represented by preferences given by the
utility function
oo
J2P'^-i[9tu{ct)],
(2)
t=0
with
P>
p.
Thus, social preferences are identical to the individual preferences given by
(1),
except
The final paragraph in Atkeson-Lucas's paper discusses the possible importance of relaxing this assumption.
However, the approach taken in the literature has been to impose an exogenous lower bound on ex-post lifetime
utility, as in Atkeson and Lucas (1993) and Albanesi and Sleet (2004).
for the discount factor.
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
Future generations are
in addition,
If,
they are
must be higher than
To
p.
see this, consider the utilitarian welfare criterion
oo
oo
5]a*E_it;4
where
and
5^
=
/3*
+ (5^~^a +
social preferences are
approaches
(3)
•
(2)
•
J]<5tE_i[^iw(Q)],
+ [3q^~^ + a*. Then
•
more
$=
with
=
the discount factor satisfies
patient. In the limit St+i/St -^ max(/3,tt), so the welfare criterion
max(/3, a).^'^
Atkeson-Lucas' analysis applies to the case with
weight
is
(3)
/?
=
so
/3,
we
focus on the case where enough
placed on future generations to ensure that the long-run social discount factor remains
strictly higher
than the private one, $
>
/3.
Although we adopt the preference
show how our analysis can be
the rest of the paper,
we
welfare criterion (3).
These two specifications are
also briefly
identical for the long-run,
which
is
The
any
finite
horizon but are
our primary concern.
Information and Incentives. Taste shock
their descendants.
adapted to work with the
easily
slightly different for
in (2) directly for
by individuals and
realizations are privately observed
revelation principle then allows us to restrict our attention to
mechanisms
that rely on truthful reports of these shocks. Thus, each dynasty faces a sequence of consumption
functions {q}, where
{0o,9i,
that
(7*
:
...,6't).
maps
A
Ct{9'^)
represents an individual's consumption after reporting the history
dynasty's reporting strategy a
histories of shocks
0* -» 9*.
We
9* into
=
{at}
a current report
9t.
is
a sequence of functions at
Any
strategy
use a* to denote the truth-teUing strategy with
Given an allocation {q}, the
utility
0*"''^
=
9*
9t for all
obtained from any reporting strategy a
=
— O
>
:
a induces a history
cr*(6'*)
9^
of reports
e QK
is
oo
U{{ct},a;P)=J2 Yl P%u{ct{a\e')))Pr{9').
t=Q 6(«ee*+i
^Caplin and Leahy (2005) argue that a similar logic applies to intra-personal discounting within a lifetime. This
is greater than the private one not only across generations, but within generations
leads to a social discount factor that
as well.
^One can
also
adopt the more general welfare criterion E_i ^Zt^o
'^t^t ^°^
some sequence
of positive Pareto weights
{q;}, where the weight at can be interpreted as the probability of being born into generation
veil of ignorance. "In particular,
all i
=
0,1,..
the sequence qq
=
(1
— 0)/{l —
P) and at
= aoP
for
i
>
1
t
for "souls
behind the
delivers St+i/5t
=$
for
}
.
An
allocation {cj}
incentive compatible
is
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
entitlement to expected, discounted
dynasties with the same entitlement v receive the same treatment.
distribution of utiUties v across the population of dynasties: V^(/l)
wiU receive expected discounted
An
allocation
For any given
feasible
utility in the set yl
a sequence of functions
is
that a dynasty with
is
initial
entitlements
incentive compatible for
it is
not exceed the fixed endowment e in
social
allocations.
to
all
is,
ijj
and
then
v.
We
let tp
assume
denote a
the fraction of dynasties
who
M.
v,
where
c\{9^) represents the
consumption
after reporting the sequence of shocks
t
and resources
dynasties;
all
is
We
=
(ii)
9*'
e,
we say
it
delivers expected utility of at
that an allocation {c^
average consumption in the population does
(iii)
periods.
optimum maximizes the average
That
C
each
entitlement v gets at date
initial distribution of
(i)
if:
{cJ'} for
least V to all initial dynasties entitled to v\
A
utility, vq
social welfare function,
the social planning problem given an
weighed by
ip,
initial distribution of
over
all
feasible
entitlements
-(/;
is
maximize
subject to
i;
= U {{c1}
,
>
a* 0)
;
/
Steady States.
Our
focus
and
C/({c^}, a;/?) for all v,
J2 Ct (^*) Pr{e')dw{v) <e
is
on distributions
t
=
0,l,
...
of utility entitlements
(5)
ip
such that the solution
to the planning problem features, in each period, a cross-sectional distribution of continuation
utilities Vt
of
that
is
also distributed according to
consumption to replicate
itself
over time.
these properties a steady state and denote
utility constitutes a state variable
We
also require the cross-sectional distribution
term any
them by
that follows a
We
xp.
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
consumption
in the
is
Atkeson-Lucas case, with
P=
P, the non-existence of a steady state with positive
a consequence of the immiseration result. Starting from any
initial distribution of
entitlements the sequence of distributions converges weakly to the distribution having
at misery
and
full
mass
zero consumption for everyone. This distribution constitutes a trivial steady state,
which does not exhaust the endowment and
is
Pareto dominated by an allocations that improves
everyone's welfare and the social welfare criterion.
In contrast, in our
steady states that exhaust the aggregate endowment in
all
periods.
model we seek
non-trivial
A
3
Bellman Equation
In this section
First,
we study a relaxed
version of the social planning problem. This has two advantages:
— one
— we avoid the need to keep track of the entire population.
because the relaxed problem can be solved by studying a set of subproblems
dynasty with entitlement v
for
each
Second,
each of these subproblems admits a simple recursive formulation, which can be characterized quite
sharply.
Consider a relaxed planning problem where the sequence of resource constraints
by the
(5) is
/oo
oo
5]Q,J]cn^*)Pr(^*)d^(i;)<e5]g„
for
replaced
single intertemporal condition
some
positive sequence {Qt} with Yl't^o Qt
<
(6)
can interpret this problem as representing
c«- Oi^e
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.
Lagrangian argument establishes
the converse: there must exist some 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.
focus on steady states leads naturally to Qt
are only compatible with q
—
$, so
forward.^ Attaching a multiplier A
we adopt
>
=
q^ 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
L = f L^dip{v) where
the Lagrangian
oo
and study the optimization
of
L
subject to v
= U {{c^}
,
a* P)
;
> U{{c^},a;P)
equivalent to the pointwise optimization, for each v, of the subproblem k{v)
V
= U {{c^}
a* P)
,
;
superscript v
Our
first
>
f/({C(},
when denoting
cr;/3).
Where
there
is
no
risk of confusion,
=
for all v.
This
is
supL*' subject to
we henceforth drop the
allocations {c^}.
result characterizes the value function
k{v)
= m&xE\eu{e) -
and shows that
Xc(u{e))
it
satisfies
the Bellman equation
+ pk(w{e))]
(7)
subject to
V
''A simple variational
argument can be used
=
to
distribution with a finite average value for c'{ut).
E[9u{9)
+ pw{9)]
show that Qt+i/Qt
(8)
=P
for
any relaxed problem
at
an invariant
+ pw{0) >
eu{e)
for all ^,^'
e0."
Theorem
1
The value function k{v)
is
9u{e')
+ pw{e')
(9)
continuous, concave, and satisfies the Bellman equation
(V-(9).
The
the latter rules out one-shot deviations from truth-telling, guaranteeing that telling
the truth today
optimal
is
if
the truth
the value function at the continuation
told in future periods.
is
incentive-compatibility condition (4).
full
and an incentive constraint
recursive formulation imposes a promise-keeping constraint (8)
(9). Intuitively,
The
rest
for
utility:
This
is
necessary to satisfy the
implicitly taken care of in (7)
is
by evaluating
any given continuation value w[9), envision the
planner in the next period solving the remaining sequence problem by selecting an entire allocation
that
is
Then k{w{9))
incentive compatible from then on.
Taken together, a
this continuation allocation.
pair u{9)
represents the value to the planner of
and w{9) that
satisfies (8)-(9)
pasted with
the corresponding continuation allocations for each w{9), yields an allocation that satisfies the
incentive-compatibility
(4).
The
full
objective function in (7) then represents the value to the planner
of this allocation.
Among
other things,
We
tained.
Theorem
1
shows that the
respectively. For
any
initial utility
entitlement
=
policy functions (^",y^) by setting Uo{9o)
for
t
>
1
using ut{9')
Our next
in this
is
=
and
g^{9t,v^{9'-'))
result elucidates the connection
way and
Theorem
maximum
2 (a)
Vq,
(7) is at-
an allocation {ut} can then be generated from the
g^{9o,vo)
v^+,{e')
=
and defining
Mt(^*)
and
Wt+i(^*) recursively
g^{9t,Vt{9'-')).
between allocations generated from the policy functions
An
allocation {ut} is optimal for the relaxed problem, given Vq, if
generated by the policy functions {g^,g^) starting at
has limt_»oo
Bellman equation
solutions to the planning problem.
lifetime utility of Vq; (h)
vq,
in the
the policy functions g^{9,v) and g^{9,v) denote the unique solutions for u and w,
let
vo, is incentive compatible,
and only
and
if it
delivers a
an allocation {ut} generated by the policy functions (g^ig^), starting
/?*IE-if((^*~'^)
—
policy functions {g^,g^), starting
and
delivers utility vq; (c)
from
Vq, is incentive
an allocation
compatible
limsupE_i/3*Ui((j*-^(^*-^))
at
{ut.,Vt} generated by the
if
>0
t—»oo
for
all
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
(c) of
the theorem show that the
^We study a Bellman equation
first
case
is
guaranteed
if
is
we can
that also characterizes the problem for g
7^ /? in
no solution
at
problem
all.
is
Parts (b)
verify the limit condition in
Section
5.
The
part (b).
we
latter
will see that
We
automatically satisfied for
is
and state the Bellman
close this section with a detour
associated with maximizing the welfare criterion
to f
=
functions that are
all utility
bounded below and
can be verified in other cases of interest.^
it
U{{dl},(T*;/3)
>
(3).
U{{c^},a]P). Then very
Define k{v)
for the relaxed
=
sup Ylu^o
planning problem
'^*^-i['^t~-^cJ']
subject
similar arguments imply that the value function
k{v) satisfies the Bellman equation
= maxETu -
Xc(u{e))
+
Because this Bellman equation
is
~k{v)
ak(w{e))]
u,w
subject to (8) and
and
analysis
(9).
results that follow
almost identical to the one in
the
(7),
can easily be adapted to this case.
Optimal Inequality
4
In this section
We
we
exploit the connection between the
Bellman equation and the planning problem.
characterize the solution and derive a key equation that illustrates mean-reverting forces in the
dynamics of consumption. The main
enough to imply the existence
of
result of the section
is
to establish that these forces are strong
an invariant distribution with no misery.
sufficient conditions that verify the
requirements of Theorem
Finally,
we provide
and ensure that a solution to the
2,
planning problem exists.
Mean
Reversion
We
now
are
in
a position to study the Bellman equation's optimization problem.
justify the use of first-order conditions with the following
Lemma
1
The value function k{v)
main, with \iiny^yk'{v)
\imy^yk'{v)
Let A
let
=
=
= — oo.
is strictly
If utility is
unbounded
we
begin,
concave and differentiable on the interior of
below, then liiny^yk'{v)
=
1.
do-
its
Otherwise
oo.
k'{v)
be the multiplier on the left-hand side of the promise- keeping constraint
n{9,9') be the multipliers on the incentive constraints (9).
1
^Theorem
To
lemma:
-
Xc'{u{e))]p{9)
-
e\p{e)
The
+ Y^ Oi^ie, e') -
(8)
and
first-order condition for u{6)
^
e'^{e, e')
2 involves various applications of versions of the Principle of Optimality.
<
is
o,
For example, given policy
functions {g'\g^) and an initial value Vo, the individual dynasty faces a recursive dynamic
programming problem
then amount to guessing and verifying a solution to the Bellman
equation of the agent's problem. In particular, that the value function that solves the Bellman equation, with truth
telling, is the identity function. However, one needs to verify that this value function indeed represents the true
value from the msjcimization of the dynasty's sequential problem. This verification is accomplished by part (c) of
with state variable
Theorem
v.
Conditions
(8)
and
(9)
2.
10
with equality
u
if
(9) is interior.
for k{v) derived in
Lemma
1,
The
and must
pk'{w{e))p{9)
-
must be
solution for w{9)
interior, given the
f3\p{9)
+PY1 i^{9, 9')-PY1 ^(^' ^') = 0e'
Using the envelope condition
k'{v)
=
Inada conditions
satisfy the first-order condition
e'
A and adding up across
Y,k'{w{9))p{9)
=
this
9,
becomes
^k'{v).
This key equation can be represented in sequential notation as
=
E,.^[k'{vt+,{9'))]
where {vt}
generated by the policy function g^.
is
CLAR) Markov
Regressive (hereafter:
process.
^k'{vt{9'-')).
Thus, {k'{vt)}
(10)
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)
strictly decreasing. Likewise, using the policy function g^{9,v), conclusions
is
continuous and
about the process {vt}
provide information about the process for consumption.
The
conditional expectation in (10) illustrates that
for the process {k'{vt)}
maximum,
is
toward
zero.
Lemma
/3//3
<
1
so reversion occurs towards an interior utility level
this case
To
derive
is
=
/3
a
CLAR
reversion
By
interior
This feature
contrast, in the Atkeson-Lucas
equation similar to (10) also holds, but the relevant value function in
monotone.
we
see this
CLAR
its
/?,
mean
— away from misery.
key to our results on the existence of invariant distributions.
model with
creates a force for
imphes that the value function k{v) has an
1
write the Bellman equation for the relaxed problem in the Atkeson-Lucas case and
Euler equation. Even though no steady state exists in this model,
if
we assume
a constant relative risk aversion utility function, then the relaxed problem does characterize the
social planning
equation
problem
for
an appropriately chosen value of the price qal- The associated Bellman
is
KAL{v)=mmE[c{u{9))+qALKAL{w{0))]
U,W
subject to (8) and
K'^i [v)
>
0.
(9).
L
/
\
Crucially, here the value function
The envelope and
\
KAiiv)
first-order conditions yield the
is
CLAR
E ^"alH0))p{9) = ^K,{v),
11
/ J
strictly increasing, so that
Euler equation
similar to condition (10)
when qal >
P-^ However, the critical difference
that here {-^^^^(^4)}
is
non-negative process, which implies by the Martingale Convergence Theorem that
almost surely
utilities
g^(9,v)
follows, Vt
^
to
(a.s.)
some
V and
q —
anything other than
K'^j^{vt)
non-monotonicity, which in turn hinges on the lower social discounting
—
Immiseration then
a.s.
>
[3
>
fi.
Economically, the mean-reversion equation implies a form of social mobility. That
of individuals with current welfare above v will eventually
and
rise
is,
descendants
below v* and vice versa. The resulting
,
result pushes the characterization of reversion past the average behavior of the {k[}
process by deriving bounds for
its
evolution. These
an invariant distribution with no mass
and
fall
of famiUes illustrates a strong intergenerational mobility in the model.
fall
Our next
of
a
This highlights the technical importance of our value function's
a.s.
;>
is
must converge
Since incentives must be provided using continuation
finite value.
g'^{0,v), this rules out
^
it
bounds are
We
at misery.
critical for
guaranteeing the existence
summarize the previous
CLAR
equation
state the next result in the following proposition:
Proposition
1
The policy function
satisfies
J2k'{g'"{e,v))p{9)
dee
=
^k'{v).
P
Define the constants
Then
(a) if utility is
7(1
and
-
(b) if utility is
7
=
iP/P)
7
=
iP/P) 2min^(l
max{{l + er.-E{e<9n])/en),
l<n<N
all
9
>
^
i
^„]/^„_i).
unbounded below,
k\v))
-
+
(^1
<
1
bounded below with u{0)
-
E Q. For values of v such that
k'{g-{e, v))
=
<
=
w{e)
= p-^v>v
k'{v)
=
<
-
7(1
k'{v))
+
(1
for values of v such that k'{v)
0,
u{e)
k'{w{e))
for
+ ^n-i - E[^
>
(11)
1,
we have
{p/p)k\v)
1,
the lower
bound in (11) holds;
the
upper bound
in (11) holds for sufficiently high v.
°With logarithmic
Thus,
it is
with qal = P features constant average consumption.
problem that imposes an aggregate resource constraint in every period. With the
utility the partial cqiiilibrinm sohition
also a solution to the
constant relative risk aversion utility function u{c)
has the same
effect.
Indeed, for
ct
<
1
=
c^'" /{I
—
a) there
the appropriate qal satisfies qal
12
is
>
a value of qal, for each value of a, that
P-
Proposition
illustrates
1
a powerful tendency away from misery.
unbounded below, continuation
how much a
Main
it
utility g'^{9, v)
remains bounded even as
supposed to be punished,
is
his child
state the
main
result of this section:
if
The proof
admits an invariant distribution with no misery.
mean- reversion properties discussed
Proposition 2 The
is
^ — oo. Thus, no matter
No Misery
a solution to the relaxed planning problem exists,
conditional-expectation equation (10) and the bounds in Proposition
for the
t"
always somewhat spared.
is
Result: Existence of an Invariant Distribution with
We now
then
parent
For example, with utility
existence of an invariant distribution
Markov process
bounded above, or j
Proposition
in the previous subsection.
g'^ is
{vt} implied by
<
ip*
Thus,
it
^°
with no mass at misery, Tp*{{v})
if either: utility is
unbounded
(a) of
Theorem
2,
leaves
open only two
exists
This situation contrasts strongly with the Atkeson-Lucas case, with
but does not admit a steady
section
we show
We
state,
/?
=
/?,
illustrate this
0,
no solution
(ii)
and everyone ends up at misery. Towards the end of
an
also provides
efficient
method
for explicitly solving
(i)
where a solution
that a solution to the planning problem can be guaranteed so that case
Our Bellman equation
lem.
=
below, utility
possibihties:
the rela:xed problem admits a steady-state invariant distribution with no misery; or
exists.
on the
makes use of both
1.
when combined with part
2,
guaranteed
of this result relies
1.
(i)
this
holds.
the 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^/^ for c > and c{u) = u^/2
for IX > 0. Atkeson-Lucas cover this case for P = $ and show that the optimum involves consumption
inequality growing without bound, leading to immiseration.
Consider the relaxed problem where we ignore the non-negativity constraints on u and w,
k{v)
= ma^E\eu{e) - XuiOf 12 + Mw{e))]
\
U,W
subject to (8) and
(9).
This
is
/ J
,
a linear quadratic dynamic programming problem, so the solution
a quadratic value function with linear policy functions in
g^{9,v)
=
j-{e)v
is
v.
+ ro{0)
For taste shocks with sufficiently small amplitude we can guarantee, by continuity with the deterministic case
=
[vl,vh] with vl
9,
>
that
0.
-y'^{9)
<
1
and
7"'(^)
Moreover, g'^{9,v)
>
>
0,
for u
implying a unique bounded ergodic set for utility
G
[vl-,vh\-
Hence, since the planning problem
is
— which
is
^"When utility is bounded below, we either require that utility be bounded above, or that 7
ensured for a small dispersion of the shocks
as a simple way of ensuring that the ergodic set
from misery. It seems very plausible, however, that these conditions could be dispensed with.
—
13
<
is
1
bounded away
Figure
convex and
original
1:
utility
To
2.
iterated
c{v{l
—
is
=
P)) against c{g"'{9,v){l
reveals a unique,
low values of
u.
p=
0.9,
=
0.975, e
(A)-i
=
0.6, Oh
=
we compute the
1.2, Oi
=
0.75
solution
and p
=
0.5.
for k{v) until convergence.^^
plots the policy function for continuation utility in consumption-equivalent units,
consumption, c{v{l
1
with P
on the Bellman equation
1
Policy function g'^{6,w)
illustrate the numerical value of our recursive formulation,
for the logarithmic case
Figure
2:
turns out to be strictly positive at the steady state, this solution does solves the
problem with non-negativity constraints on
Example
We
Figure
Policy functions g'^{6,v)
v.
— P))
—
/?)),
while Figure 2 does the same for the pohcy function for
against g'^{9,v).
bounded ergodic
Both policy functions are monotonic and smooth. Figure
set for v.
This illustrates the
Note that both policy functions become nearly
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
—
P)) implied
by the solution to the planning problem. ^^ The
'^The details of this numerical exercise where as follows: we solved for u{9) as a function of w {0) using the
and promise-keeping constraints. We then maximized over w{9). We employed a grid for v defined in
terms of equally spaced consumption-equivalent units c((l — (3)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{9). Our iterations were initialized with the value function
corresponding to the feasible plan that features constant consumption:
incentive
v{l
-
,6)
-
\c{v{l
-
,6))
ko{v)
1-13
We stopped
the iterations when ||fcn(u) — A':„_i(w)|| < 10~''-° and verified that the policy functions had also converged.
Note that g"'(0,t') 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.
'^^The invariant distribution was approximated by generated a Monte Carlo simulation for the dynamics of the Vt
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 Vq, the frequencies in a long time-series sample approach the
frequencies of 0. To create the figure we used Matlab's Wavelet Toolbox to approximate the density from the
simulated Monte-Carlo sample.
14
0.61
utility in
Figure
0.62
consumption equivalent units
Steady-State Distributions of DjTiastic Utility
3:
long-run distribution has a smooth bell-curve shape
mean-reverting dynamics of the model, since
The
distribution of taste shocks.
/?.
The degree
cannot be a direct consequence of our two-point
shows the invariant distributions
of inequality appears to decrease with higher values of
by the
intuitively
figure also
it
— a feature that must be due to the smooth,
on the
coefficient
CLAR equation
(10)
/?.
for various values of
This outcome
and the discussion
is
in Section 5
suggested
on features
of the impulse response to shocks. These simulations also support the natural conjecture that as
we approach the Atkeson-Lucas
case,
3 -^
(3,
the resulting sequence of invariant distributions blows
up. since no steady state with positive consumption exists
when $
=
(3.
We now tmn briefly to issues of uniqueness and stability for the invariant distribution guaranteed
by Proposition
of social
2.
This question
is
of economic interest because
it
represents an even stronger notion
mobihty than that implied by the mean-reversion condition
subsection.
That
utiUty level vq,
is,
if
convergence toward the distribution
tp"
(10) discussed in the previous
occurs starting from any
initial
then the fortunes of distant descendants — the distribution of their welfare —
is
independent of the individual's present condition. At the optimum, the past always exerts some
influence
on the present, but
its
influence
is
bounded and
or disadvantages of distant ancestors are eventually
Indeed, under
if
is
wiped
out.
some conditions we can guarantee that the
display this strong notion of social mobility.
process
dies out over time, so that the advantages
compact. This
is
optimum
is
monotone
from any
in v, the invariant distribution ip*
initial distribution ipQ,
15
in our
model does
see this, suppose the ergodic set for the {kf}
guaranteed, for example, by applying Proposition
the policy function g^{9,v)
in the sense that, starting
To
social
1
when 7 <
is
1.
Then,
unique and stable
the sequence of distributions {^t},
generated by
(10) ensures
converges weakly to
g'^,
ip*
w
policy functions for continuation utility
1
This follows since the conditional-expectation equation
.
enough mixing to apply Hopenhayn-Prescott's Theorem. ^^ The monotonicity of the
seems intuitive and plausible, as illustrated by Examples
2.14
and
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.
initial distribution is fast, in
ments of
their
which
of irreducibility,
distribution.
We
is
likely to
it
occurs at the geometric rate P/$. All the require-
verified already for our
hold
if
such as (10), have
Moreover, convergence to this distribution from any
the sense that
theorem have been
Markov processes
irreducible
CLAR condition,
we were
model, except
for the technical condition
assume that the taste shock has a continuous
to
do not pursue this formally other than to note that the forces
for reversion in (10)
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 the relaxed
rithmic utility case
we can do more: we
and
original planning
problems then coincide. For the loga-
characterize transitional dynamics for any initial distribution
of entitlements.
Proposition 3 (a)
If utility is logarithmic, then for
any
there exists a value of e such that the solution to the relaxed
ijj
endowments
first-order stochastic dominance, then the associated
Proof,
its
and original
(h) Suppose two distributions of utility entitlements satisfy
coincide,
(a)
of utility entitlements
initial distribution
ip"'
satisfy ea
Consider the relaxed social planning problem for some A
=
social planning "problem
>
>
e~^.
et-
We
solution also solves the original social planning problem for any distribution
/ k'{v;
—
e)dip{v)
Since utility
is
0.
We
then show that there exists a value for A
unbounded below, we have
k'{vt)
=
Ei_i[l
=
in the sense of
ip^
first
show that
that satisfies
ipQ
e~^ for any initial distribution.
— Xc'{ut)].
Applying the law of iterated
expectations to (10) then yields
E_i[l-Ac'K(^*))]
With logarithmic
t
=
0, 1,
...
Theorem
Now
J
The
2, it
utihty c'{u)
allocation
follows that
is
it
=
c{u), so that
=
0,
and the
k'{v)d'ip{v)
=
imphes jE^iCtdip
incentive compatible by Proposition 2 below,
must
=
-1
A
=
e for all
and applying part
(c)
of
solve the original planning problem.
consider any initial distribution
k'{v; e)dip{v)
J
= (^0^'^ V
ip.
We
argue that
we can
result then follows immediately.
We
find a value of
A
=
e"^ such that
denote the value function for the
Lucas and Prescott (1989).
can be shown that g^ (v, 9) is strictly increasing in v. However, although we know of no counterexample,
we have not found conditions that ensure the monotonicity of g^{6, (O.v)) for 9^6.
^'^See pg. 382-38:i in Stokey,
^'^Indeed,
it
16
=
relaxed problem with A
e
by
^
k{v; e).
The homogeneity
problem implies that
of the sequential
the value function must satisfy
^^^' ^^
Note that
e ^^
— oo,
k'{v
— log(e)/(l — /?);
k'{v] e)dij{v)
defines a unique value of e for
(b)
1) is strictly
log(e);
1
increasing in e and limits to
1
and — oo as
any
initial distribution,
identically at the outset
=
I
k'
(v
-
-^
log(e); 1
j
d^p{v)
*
oo and
=
(12)
initial distribution ^q-
the situation where the planning problem
Then
logarithmic
coincides with the solution to the relaxed problem.
Then our
utility level v* that solves k'{v*)
'0*
can be reinterpreted in
terms of the stability of the cross-sectional distribution in the population.
stable then the cross-sectional distribution of welfare
will eventually settle
The economic
problem
is
down
interest
behind part
attains a distribution of entitlements
creates a Pareto
is
If
Markov process
the
and consumption
in the population
to the steady state.
(b)
is
that
it
implies that the solution to the planning
ex-post Pareto efficient in the following sense.
then implies that there
modified to select
is
dynasties are treated
all initial
=0. With
and started with the
optimum then
the social
is
instead of taking one as given.
previous discussion of convergence to a unique invariant distribution
is
—
a strictly decreasing function.
Perhaps of particular interest
the best
{vt}
e
This part follows immediately from equation (12) which determines e by using the fact that
1) is
utility
^—
respectively. It follows that
/
A;'(-;
" VI^ ^^^^^^ + k(v-
ifj
Consider the optimal allocation that
with some constant endowment level
no alternative allocation that improves the
improvement among individuals
e.
The
proposition
social welfare criterion
in the current generation.
That
is,
the optimal
allocation also solves the original social planning problem modified to require [/({cJ'},cr*;/3)
place of U{{dl}, a*;P)
=
v.
Since this result holds for any distribution of entitlements
for the distribution of continuation utilities implied
in this sense, the optimal allocation
is
by the optimal allocation
ex-post Pareto
in
and
it
>v\n.
also holds
any period. Thus,
efficient.
Sufficient Conditions for Verification
We
conclude this section by describing sufficient conditions for a solution to the planning problem
to exist at the steady state
steps.
First,
we
ip*
identified
by the policy functions
establish that allocations generated
compatible by verifying the condition in part
consumption
is finite
(c)
of
under the invariant distribution
17
in Proposition 2. This involves
two
by the policy functions are indeed incentive
Theorem
ip*.
2.
Second, we verify that average
Lemma
2 The allocation generated from the policy functions {g^,g^), starting from any vq,
guaranteed
bounded below;
is
compatible in the following cases: (a) utility
to be incentive
<
or (d) ^
(c) utility is logarithmic;
1
bounded below, then the
Proof,
(a) If utility is also
utility
unbounded below, but bounded above. Then
is
implies that \imy^_cok'{g^(9,v))
a vl
> —00
such that g'^(6,v)
>
=
It
1.
-
g'"{e,v)
and 9
for all pairs of histories 9
part (b) of
Theorem
2
<
(d) If
7 >
we have
Then
then the bound
If
for all v
by Vh-
We
7 <
1,
-
—
=
A.
Yimt^oo (3^'^-i[vt{9^~^)]
>
<
It
(c)
of
Theorem
2.
one can show that:
log {0/9).
The
incentive constraint
follows that Vt{0^~^)
>
- tA
Vt{e^-^)
- pHA.
=
0.
Since Y\m.t^^l3hA
<
=
0, it
follows
0.
in (11) implies that k' {g^ (^i f^))
have g^{9, v)
now apply
1,
e'
g^{0,v)
{0/ [3)\og{e/e)
then we can define k
< vh ^e
can
1
attained, so there exists
Then
.
that limsupt^^/3*E_i[t;(((j*-i(^*-^)]
immediately.
is
immediately from part
result follows
P'¥._,[vt{a'-\9'-^))] >/3*E_i[i;,(^*-i)]
From
So suppose
(b).
continuous and Proposition
is
,
utility this implies that g^{6,v)
then implies that g'^{e,v)
from part
result follows
k'{g^{6, ))
follows that m.ax^ k' {g'^" (9 v))
d{u{e_))
With logarithmic
is
(b) utility
0.
Using the first-order conditions from the proof of Proposition
(c)
bounded above;
vl-
bounded below, the
(b) If utility is
>
or 7
is
v.
=
It
1
—
(1
—
(3/P)/{l
—
^
1
7),
~ P/P ^^^
follows that the unique ergodic set
the argument in (a) so there exists a vl
the result follows
and define vh by k'{vH)
> —00
is
=
k.
bounded above
such that g^
(9, v)
>
vl-
U
We now find sufficient conditions that guarantee average consumption is finite under the invariant
distribution
ip*
.
the ergodic set for utility v
If
bounded and average consumption
is
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*
9
if
either (a) the ergodic set for v
of
c.
Proof. Part
For part
(a) is
is
bounded; or (b)
utility is
such that c'{u{c))
a convex function
is
immediate since by continuity of the policy functions, consumption
(b), recall
that
J
k'{v)dip*{v)
=
under the invariant distribution
18
ijj*
.
If
is
bounded.
utihty
is
un-
bounded below then
all
solutions for consumption are interior.
corner solutions with g^{9, v)
g'^iO,
v)
is
bounded,
=
l-k'{v)
some
for
for all 6 in this
=
compact
If utility is
bounded below, then
low enough
9 can only occur for
levels of v, so that
Recall that for interior solutions
set.
XE[c'{g^{e,v))]=XE[c'{;u{c{g''ie,v))))]
Applying Jensen's inequality we obtain
c'
The
(u (jE[c{g''{9,v))]dr{v)]) <
result then follows since c!{u{c))
Note that a bounded ergodic
is
set
is
an increasing function of
guaranteed by 7
with sufficiently small amplitude; and condition
aversion utility functions with
The
a
>
<
1,
(b) holds, for
which
and thus has
full
range.
is
ensured for taste shocks
example, for
all
constant relative risk
1.
utility,
For instance, in the case of
average steady state consumption
is
a power function of A,
In fact, in this case the entire solution for consumption
of degree one in the value of the
1-
c.
value of average consumption depends on the value of A.
constant relative risk aversion
endowment
planning problem for any endowment
5
j E[d {u{c{g'^{9,v))))]dr{v) =
is
homogenous
This ensures a steady state solution to the social
e.
level.
Discussion: Mean-Reversion
This section develops an intuitive understanding of the key mean-reversion property discussed pre-
We
viously.
first
revisit the full
derive the impulse response of consumption to a one-time taste shock.
problem with an alternative Bellman equation that
We
then
useful as a source of intuition.
is
Impulse Response
Consider a version of our model where only the
first
generation faces uncertainty.
period, there are two possible values for the taste shock ^0
is
deterministic:
=
I for t
>
We
1.
compare
{9l-,
this to the case
In the
we adopt logarithmic
first
9h}, but thereafter the economy
with no uncertainty in the
This allows us to trace out the consumption response to the taste shock over time.
period.
simplify,
We
9t
^
first
To
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
is
kdet{vi)
= max V"^
19
(logQ
-
Xct]
,
Vi,
the
subject to
<=1
The
associated Bellman equation
is
kdetivt)
= max
ct,vt+l
subject to
Vt
= logQ + /3ft+i. The
(
ct
{k'^eti'^t}}
- Aq + ^A;det(^^i+i)
J
and envelope conditions imply that
first-order
k',Jvt+i)
Condition (13) shows that
log(Q)
\
=
^X^tivt),
(13)
= X~\l-k',Jvt)).
(14)
reverts geometrically towards zero at the rate
a deterministic version of the conditional-expectation equation (10).
translates directly into
back to a
common
consumption by the
discounting coincide, so that
new
level
c{vi/{l
Turning to the
is
In the logarithmic case,
it
Thus, consumption reverts
steady state at the same rate; deviations from the steady-state level of consump-
tion have a half-life of (log2 (/?//?) )~^.
q =
first-order condition (14).
This
/?//?.
/?
=
Note that
in the
Atkeson-Lucas case when social and private
consumption remains perfectly constant
/?,
after the
shock at
its
— /?)).
first
generation at
i
=
0,
the planning problem solves
maxE[^olog(co(^o))
-
AoCo(^o)
+ ^fcdet(yi(^o))l
Uo,Vl
subject to
+ Pv,{eL) >
OLiog{co{eL))
where we omit the other incentive constraint since
is
convex. At the optimum, Cq{6h)
>
eLiog{co{dH))
it is
not binding at the optimum, and the problem
<
Cq(6l), vi{6h)
+ PvjiOH).
viidi),
and
EKet(^l(^))]=0,
implying that Vi{9h)
and equal to A
<
v*
<
Vi{9l) where k'{v*)
=
0.
Note that average consumption
Figure 4 shows the consumption response to a taste shock in the
periods.
That
is
constant
in all periods.
is,
we use
(13)
and
(14) for
t
>
1
first
starting at Vi{9l), v*
period, for subsequent
and
v-[{9h)-
on consumption from the shock dies out over time and consumption returns to a
The
common
effect
steady-
state level. Again, this illustrates that the influence of past fortunes eventually vanishes for distant
descendants.
We
also plot the Atkeson-Lucas case with
has a permanent impact on the consumption of
To provide
incentives for the
first
all
(3
=
$,
where the luck of the
first
generation
descendants.
generation, society rewards the descendants of an individual
20
Figure
Consumption path
4:
reporting a low taste shock.
first is
i
>
1
in response to taste
shock at
^
=
Rewards can take two forms and society makes use
of both.
consumption,
for
exploits differences in preferences:
it
allows an adjustment in the pattern of
a given present value, in the direction preferred by individuals.^'^
Since individuals are more impatient than the planner, this latter form of reward
by
The
standard and involves increased consumption spending, in present-value terms. The second
more subtle and
is
for
tilting the
consumption
profile
toward the present. Similarly, punishments involve
consumption path toward the future.
intensively to provide incentives
returns to a
affecting the
common
first
delivered
tilting the
In both cases, earlier consumption dates are used
— rewards and punishments are front-loaded.
more
Indeed, consumption
steady-state level in the long-run regardless of the initial shock because
consumption of very distant descendants
incentives to the
is
is
not an
efficient
way
for society to
provide
generation.
Another Bellman Equation
Here we develop another Bellman equation that holds
This alternative formulation
is
useful,
for
any value of q not necessarily equal to $.
both as a source of intuition and to motivate our focus on
Consider the following cost minimization problem
oo
^^Some readers may recognize
parents
when conceding
this last
method
some goods more than the parent
wishes,
system of rewards and punishments used by
TV-time. In these instances, the child values
as the time- honored
their child's favorite snack or reducing their
and the parent uses them to provide
21
incentives.
subject to the incentive compatibility constraint
U {{ct}
a* P)
,
;
>
U{{ct},a; P) and
oo
V
= J2p'E.,[9Mct)]
t=o
_-
oo
while delivering utility v and v for the planner and individual, respectively.
must
satisfy the
Then the
value function
Bellman equation
K{v,v)
=
maxE[c(u(^))
u,w,w
+ qK(w (6) ,w
[9))],
subject to
V
=
E[eu{9)
+ Pw{0)]
V
=
E[9u{9)
+ Pw{9)]
and
9u{9)
+ Pw{9) >
This formulation could be used to derive
equation
(7) is
slightly
formulation, however,
The
more convenient
is
that
it
9u{9')
all
+ I3w{9')
for all 9, 9'
Bellman
of our results, although the lower-dimensional
The advantage
for that purpose.
of this cost-minimization
lends itself naturally to economic interpretations.
following story provides a useful reinterpretation
infinite-lived
e 6.
and source
for intuition.
Consider an
household with two members, husband and wife, and assume that consumption
public good
— there
disagree on
how
is
is
a
no intra-period resource allocation problem. However, husband and wife
more
patient, but only the
husband
problem characterizes the constrained Pareto problem
for this
to discount the future.
Suppose the wife
is
can observe and report taste-shock realizations.
Then
this cost-minimization
household, in the sense that the isocost curve K{v,v)
Pareto frontier between husband and wife.
= Kq
The Pareto
represents, given resources Kq, the
frontier
is
non-standard in that
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*)
—
0.
is
bounded,
utility for the wife is attained for
Reductions in the husband's
an
utility to
also decrease utility for the wife, for a given level of resources.
first-order conditions
can be rearranged to deliver
PK2{v,v)=qK2{w{9),w{9))
22
(15)
K\{w{e),w{9))
K2{v,v)
(16)
K2{w{9),w{9))
J3
Condition (15) can then be used to argue that a steady-state requires q
{K2t} would increase without bound; likewise,
if
g
>
=
p. Indeed,
^, then {K2t} decreases
q
ii
toward
<
0, then
zero.
Both
situations clearly do not lend themselves to the existence of an invariant distribution for {v,v).
On
the other hand,
distribution
When
-f^2(f^0)
"f^o)-
g
=
/?
then K2{vt,Vt)
constant along the optimal path and an invariant
is
possible.
is
=
q
if
P,
the state {vt,Vt) moves along a one-dimensional locus given by K2{v,v)
Intuitively, since
no incentives are required
for the wife, she is perfectly insured in
the sense that the marginal cost of delivering welfare to her
Figure
Figure 5 shows that the curve K2{v,v)
=
curves from below, and cuts Ki{v,v)
perfect insurance for the
—
is
held constant across time.
K{v,v)
Isocost curves of
5:
K2{vo,Vo) for continuation
from above.
=
utilities
cuts the isocost
Intuitively, incentives require foregoing
husband and accepting fluctuations
Starting from {v*,v*), rewards can be dehvered in two ways.
in v as rewards
The optimum makes
and punishments.
use of both forms
of rewards, explaining the shape of the schedule for continuation utilities.
The
form of rewarding involves increasing resources K, and can be seen as an upward
first
movement along the diagonal Ki{v,v)
an allocation
of these resources that
movements along the Pareto
frontier,
both forms of rewards and, as a
K\{y,v)
=
and above the
=
is
0.
more
which
result,
However, the husband
to his liking,
at {v*,v*)
is
23
also
rewarded by allowing
which can be represented as
horizontally
{v,v) travels along K2{v,v)
initial isocost curve.
is
flat.
—
The
lateral
solution combines
/C2(fo,t'o) to
the right of
Condition (16)
is
the analog of the conditional-expectation equation (10) obtained from the
one-dimensional Bellman equation.
frontier),
—K1/K2
Here,
it
implies that the slope of the isocost curve (Pareto
reverts geometrically toward 0. Thus, {vt,Vt)
and eventually reverts toward
{y* v*). Intuitively,
husband with
it is
incentives, but
,
efficient to revert
the wife: Patience ensures that the wife has her
moves along K2{v,v)
=
K2{vq,Vq)
the solution deviates from [v* ,v*) to provide the
way
back to this point of ma:x:imum efficiency
for
in the long-run.
Estate Taxation
6
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
is
but are heterogenous
effort
privately observed by the individual
We
distributed across generations and dynasties.
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 no link
be a non-trivial planning problem.
Indeed, in each
period the situation would then be identical to the static, nonlinear income tax problem originally
studied by Mirrlees (1971). Moreover, in the absence of altruism, so that
With
actually coincides with this static solution.
altruism, however,
(5
we
—
Q,
the social
shall see
optimum
below that
it is
always optimal to 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.
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: more
fortunate parents should face a higher average marginal tax rate on their bequests.
reflects the
mean
reversion in consumption explained in the previous section.
A
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
Utility
of this
economy
depends on the
generation
t
is
level of
identical to the canonical optimal taxation setup in Mirrlees (1971).
consumption
c
and work
effort n.
have identical preferences that satisfy
Vt
= Et.Mct)-h[nt)+l3Vt+i],
24
We
assume that individuals
in
,
but
diflFer
An
regarding their productivity in translating work effort into output.
productivity w, exerting work effort n, produces output y
=
wn.
We
individual with
assume that productivity w
is
independently and identically distributed across dynasties and generations. Thus, the productivity
talents of parent
assumption,
if
the
and
child are unrelated
optimum
— innate
skills
are
assumed nonheritable.
features intergenerational transmission of welfare, then
social decision to provide altruistic parents
it
Given
this
represents a
with incentives in this way, and not a mechanical result
originating from the eissumed physical environment.-'^
For convenience, we adopt the power disutility function h{n)
we can
=
rC /^ so that, defining
w~'^
OO
CXD
5]/3*E_i[«(q) -^,/i(y,)]
= X^/?^E_i[r'MQ_i)
t=0
The right-hand
problem
we
—
write total utility over consumption and output as being subject to taste shocks
-^,/i(y,)]
- E_i [^o%o)]
i=l
side of this equation leads to a convenient recursive representation of the planning
in the continuation utility defined
by
Vt
—
^^Q/5*Et_i[u(ct+s-i)
—
Ot+sh{yt+s)\
— where
are abusing notation shghtly by folding the /3~^ into the definition of the utility function u{c).
The
resource constraint requires total consumption not to exceed total output plus
some
fixed
constant endowment
Y,c'i{e')Y>v{e')d^{v)
where individuals are indexed by
<
fY,yt{0')Pr{9')di^{v)
+ e,
their initial utility entitlement v, with distribution
in the
ip
population.
We continue to assume social discounting is lower than private discounting: $ >
problem
is
to choose an allocation {c^{6^),y^{6^)} to
incentive-compatibility constraints
Using the
last expression for
and the resource
maximize average
(3.
The planning
social welfare subject to the
constraints.
the utility function and applying similar reasoning as in the taste-
shock model yields the Bellman equation for the associated relaxed problem
k{v)
= max E\u^ -
Ac(«_)
-
eh{9)
u-,h,w
v
+ Xyih{e)) + ]3kiw{e))]
= E[u_-eh{e) + i3w{e)]
-eh{9)
+ pw{e) >
-eh{6')
+ (3w{e'),
where the function y{h) represents the inverse of the disutihty function, y
=
h~^
.
The arguments
that justify the study of this Bellman equation, are similar to those that underlie Theorems
in the context of the taste-shock
previously,
and imply that an invariant distribution
^^As in the taste shock model, here the case with
Albanesi and Sleet (2004),
1
and
2
model. The results regarding steady states parallel those obtained
who impose an exogenous
/3
=
/?
lower
25
exists
with no immiseration, as in Proposition
leads to immiseration.
bound on dynastic
This case has been studied by
welfare to circumvent immiseration.
Implementation with Income and Estate Taxation
Any
allocation that
incentive compatible
is
taxes on labor income and estates. Here
and
we
feasible
can be implemented by a combination of
describe this implementation, and explore
first
some
features of the optimal estate tax in the next subsection.
For any incentive-compatible and feasible allocation {cj(^*),
In each period, conditional on the history of their
tation along the lines of Kocherlakota (2004).
and any inherited wealth, individuals report
dynasty's reports 9
consume, pay taxes and bequeath wealth subject to the following
ct
where
Rt-i^t
+ bt<
yt{e')
-
Tt{9')
+
we propose an implemen-
y]^{6^)}
-
(1
their current
Tt{9'))Rt-,,tbt-i
=
t
0,l,
the before-tax interest rate across generations, and initially 6_i
is
shock
Of,
produce,
budget constraints
set of
(17)
...
=
0.
Individuals are
subject to two forms of taxation: a labor income tax Tt{9^), and a proportional tax on inherited
wealth Rt-i^th-i at rate at rate Tt{9
Given a tax policy
{Tj"(^'),
)}'^
t'^{9*),
an equihbrium consists of a sequence of interest
yf{9'')},
income and bequests
rates {Rt^t+i}', an allocation for consumption, labor
reporting strategy {crj(^*)} such that:
(i)
{q,
jbt,
taking the sequence of interest rates {Rt^t+i} and the
asset
market clears so that J E^j[b'^{9^)]d(j){v)
equilibrium
For any
is
incentive compatible
if,
—
tax;
policy {Tj,
for all
in addition,
it
interest rates {Rt-i^t}-
r^,
0,1,...
induces truth
6J'(^')};
yt} as given;
We
and a
utility subject to (17),
and
(ii)
the
say that a competitive
telling.
we construct an incentive-compatible
=
u'{c^{9
pRt-i,t
any sequence of
=
t
feasible, incentive-compatible allocation {dl,y^},
competitive equilibrium with no bequests by setting T^{9^)
for
{cj'(^*),
at} maximize dynastic
yt{9^)
—
Ct[9^)
and
))
These choices work because the estate tax ensures
that for any reporting strategy a, the consumption allocation {Cf{a\9*))} satisfies the consumption
Euler equation
«'(c^(a*(^*)))
=
,5i?,,+i J]^x'(c^^l(cT'+l(^^^,+l)))(l
and the labor income tax
allocation
is
and no bequests,
-
rr+i(o-*+^(^',^m))) Pr(^m),
such that the budget constraints are satisfied with this consumption
6J'(^*)
=
0.
Thus, this no-bequest choice
is
optimal for the individual
and labor income y# is mandated given
M* being implemented are invertible, then by the
taxation principle 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 simplj^ choose
a budget-feasible allocation of consumption and labor income, taking as given prices and the tax system.
^^In this formulation, taxes are a function of the entire history of reports,
this history.
However,
if
the labor income histories
j/'
:
6'
26
—
>
regardless of the reporting strategy followed. Since the resulting allocation
by hypothesis,
it
follows that truth teUing
is
budget constraints then ensure that the asset market
As noted above,
in our
economy without
The
optimal.
is
clears.
^^
capital only the after-tax interest rate matters so the
implementation allows any equilibrium before-tax interest rate {Rt^it}.
we
In the next subsection,
set the interest rate to the reciprocal of the social discount factor, Rt-i,t
natural because
it
incentive compatible,
resource constraints together with the
=P
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 (18) 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 /x(^, 6')
represent the multipliers
w_ and w{9) are
=
c'(t/_)-A-A
M'W^))pW-/3Ap(^)-J]M^,^') + E^(^''^) =
e'
and the envelope condition
is k' {v)
=
e'
Together these imply
X.
Y,k'{wi9))p{e)
Using c'(«_)
=
l/u'{c-)
=
A
-I-
k'{v)
^
The
=
left-hand side together with the
equation.
negative
we
=
L'{v),
P
e
arrive at the Modified Inverse Euler equation
ii:;4^M«)-A(i-i).
first
term on the right-hand side
The second term on the right-hand
side
is
novel, since
it is
is
(19)
the standard inverse Euler
zero
when
(3
=
(5
and
is
strictly
when $ > p}^
In our environment, the relevant past history
is
encoded
tax r(0*~\^i) can actually be reexpressed as a function of
in the continuation utility so the estate
Vt{6*'~^)
and
Ot-
Abusing notation we
then denote the estate tax by Tt{v,9t)- Since we focus on the steady-state, invariant distribution,
we
also
drop the time subscripts and write t{v,
9).
^^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.
^^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.
27
The average
estate tax rate t{v)
is
then defined by
l-f(i;)
= 5^(l-r(^,^))p(^)
(20)
e
Using the modified inverse Euler equation (19) we obtain
f{v)
= -A«'(c_(^^))(^-l)
In particular, this impfies that the average estate tax rate
are subsidized.
is
negative, f{v)
<
so that bequests
0,
However, recaU that before-tax interest rates are not uniquely determined
in
our
(18).
With our
particular choice for the before-tax; interest rate, however, the tax rates are pinned
down and
As a consequence, neither
implementation.
acquires a corrective, Pigouvian role.
externalities
are the estate taxes
computed by
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
depends on the choice of the before-tax
be a robust steady-state outcome
In our model
it is
more
in
by Pigou. In our setup without
interest rate.
a version of our economy with capital.
interesting to understand
how
the average tax varies with the history
of past shocks encoded in the promised continuation utility v.
function of consumption, which, in turn,
progressive: the average tax
Proposition 4 In
capital, this result
However, the negative tax on estates would
on transfers
is
for
The average tax
v.
higher.
the repeated Mirrlees economy, the optimal allocation can be
The
is
face lower net rates of return so that their
7
Conclusions
Should privately- felt parental altruism
implications for inequality?
the optimal
The
fortunate
consumption path decreases towards the mean.^°
affect the social contract?
To address
ip* ,
increasing in promised continuation utility v.
progressivity of the estate tax reflects the mean-reversion in consumption.
must
is
implemented by a
combination of income and estate taxes. At a steady-state, invariant distribution
average estate tax f{v) defined by (18) and (20)
an increasing
Thus, estate taxation
an increasing function of
more fortunate parents
is
is
these questions,
If so,
we model a
what are the long-run
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
^*^Farhi,
we
also find that
if
we
value the welfare
good or bad fortune should be tempered. This
Kocherlakota and Werning (2005) explore more general versions of this result and discuss several related
intuitions.
28
produces a steady-state outcome in which welfare and consumption are mean-reverting, long-run
inequality
What
of our
is
bounded, social mobility
is
and misery
possible
is
avoided by everyone.
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
face a higher average marginal tax rate
on
inheritances
is
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.^^
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
its
continuity over the interior of
also
be established as follows. Define the
k*{v)
subject to
at
any
1)
finite
=
bounded, continuity at the extremes can
first-best value function
Then
Then
v.
concavity of the value function k{v) implies
If utility is
= max^^*
^^Q/3*E_i[^4Ui(^*)].
extremes v and
The weak
set.
domain.
its
E.^[etut{e')
k*{v)
is
-
Xc{ut{9'))]
continuous and k{v)
<
k*{v), with equality
continuity of k{v) at finite extremes follows.
Thus, k{v)
is
continuous.
The
constraint (6) with q
—
J3
implies that utihty and continuation utility are well-defined.
Toward a contradiction, suppose
T
lim Y(3'Esetu{ct)
'
^—>oo
T—>oo
^
t=0
is
not defined, for some s
utility is
concave 9u{c)
<
> — 1.
Ac-[-
B
This implies that limy^oo St=o/3*'^^^(-^s^t'"(^)7 0)
for
some A,
T
^/5*max(£;,^tu(Q),0) <
Taking the limit yields lim^^oo X^f=o
/^
^^-iCt
5>
0,
so
it
^^- Since
follows that
T
A^/?%q + S <
=
=
T
/I
J]]
^*E,q
oo. Since there are finitely
-f
many
5
histories 9^
G 0*+^
St=o/5 E_iQ = oo. If there is a non-zero measure of such agents this implies
(6). Thus, for both the relaxed and unrelaxed problems utility and continuation
this implies limj-^oo
a contradiction of
utility are well defined
given the other constraints on the problem.
This
is
recursive formulation below.
'Some progress along these
lines
can be found in Amador, Angeletos and Werning (2005).
29
important
for
our
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[^^i(^)
v
Ou
Define
m=
+ pw{e) >
(e)
maXc>o,6iee(^^(c)
=
—
Xc{u{9))
(21)
E[eu{9)+Pw{e)]
Ou
{9')
and k{v)
Ac)
+ pk{w{e))]
+ pw{9')
=
(22)
— m/{\ —
k{v)
eQ
for all 9,9'
$)
<
(23)
The problem
0.
in (21)
is
equivalent to the following optimization with non-positive objective:
supE[^u(^)
-
Xc{u{9))
-m + pk{w{9))]
(24)
subject to (22) and (23).
Lemma
Proof.
The supremum in
1
If utility is
(21), or equivalently (24), is attained.
bounded the
result follows
and compactness of the constraint
arguments apply when
utility is
immediately by continuity of the objective function
So suppose
set.
utility is
unbounded above and below
only unbounded below or only unbounded above.
We
— similar
first
show
that
lim k{v)
=
and then use
maximum
To
is
without
this result to restrict,
lim
k{v)
= — oo
(25)
i;—»— oo
v—*oo
the optimization within a compact
loss,
set,
ensuring a
attained.
establish these limits, define
oo
h{v] p)
subject to f
=
=
sup
Y^ ^*E_i [9tu{9')
E_i Yl^ol^^^tu{9^)- Since
incentive constraints
it
then the desired limits (25) follow. Since 9u
h{v, P)
<
h{v,
\c{u{9'))
h{v,P).
—
Xc{u)
If
—
- m]
same problem but without the
this corresponds to the
<
follows that k{v)
-
\miy^ooh{v, $)
m<
P)=v- XC{v,
and
(3)
-
(3
=
<$
limy^^oo h{v,/3)
it
= — oo,
follows that
777
^—^,
(26)
where
CO
C{v,l3)
=
miY^(3'E_^c{u{9'))
t=o
subject to f
=
^^g/3*E_i[^iu(^')].
problem, with solution u{9^)
=
Note that C{v,P)
{c')~^{9t'y{v)) for
some
30
is
a standard convex first-best allocation
positive multiplier 7(f), increasing in v
and
=
such that hmij^-ooTlf)
^^^ hm^_oo7
C{v,P)
so that Um^j^-oo h{v,
hm„_^_oo h{v,
Fix a
(3)
/?)
= — oo
= — oo
and
and hm^,^oo
Take any allocation that
V.
bounded
Then
oo-
h{v,
= — oo.
/3)
= — oo,
verifies
the constraints (22) and (23) and
>
Since {9u
—
we can
Xc{u))
is
this defines a closed,
that
we can
with {9u
—
\u{9)\
<
continuous over this restricted
Lemma
\c{u))
—
for
(25),
<
oo
—
m
>
k/p{9).
oo or m
^ — oo,
Thus, there exists an
My^ <
oo such
>
when
Xc{u{9))
either
u
—
My^u-
maximum must be
set.
Since the objective function
attained.
).
some v
k{v)
where the maximization
My^^
*
— — oo
2 The value function k{v) satisfies the Bellman equation (7)-(9
Proof. Suppose that
is
> maxE[^M(^) -
Xc{u{9))
subject to (22) and (23).
k{v)
for all
the
set,
oo be
M^„;.
interval for u{9), for each 9.
maximization to
<
A;
restrict the
concave with the limits
maximization over u{9) so that 9u{9)
strictly concave,
bounded
<
Hence, we can restrict the ma:ximization in (24) to a compact
is
is
let
we can
non-positive,
interval for w{9), for each 9. It follows that there exists
restrict the
restrict the
is
k/{$p{9)). Since k{w{9))
such that we can restrict the maximization to |w(^)|
Similarly,
Using the inequahty (26) this estabhshes
which, in turn, imply the hmits (25).
Then, since the objective
maximization to w{9) such that k{w{9))
this defines a closed,
=
= ^E[c[{cr\9^{v)))],
hm,;_^oo h{v,$)
the corresponding value of (24).
i'^)
Then
+ pk[w{9))]
A>
there exists
such that
>¥.[9u{9)-\c{u{9))^[5k{w{9))]^/l
{u,w) that satisfy (22) and (23). But then by definition
oo
>
k{w{9))
for all allocations
u that yield
w [9]
J];3*E_i[^,M,(^*)
-
Xc{ut{9'))]
and are incentive compatible. Substituting, we
find that
oo
k{v)
> Y^p'e_,[9M0') - HMO'))] + A
t=o
for all incentive-compatible allocations that deliver v, a contradiction
with the definition of k{v).
Namely, that there should be a plan with value arbitrarily close to k{vo).
max„,^ E[^u(^)
-
Xc{u{9))
+ Pk{w{9))]
subject to (22) and (23).
31
We
conclude that k{v)
<
By
and
definition, for every v
and
e
>
there exists a plan {U((^*; v, e)} that
is
incentive compatible
delivers v with value
oo
Y,J5'¥._^[etUt(e';v,e)-Xc{ut{e'-v,e))]
Let {u* {9) w* {6)) e argmax„,^E[6'ti(6')
,
ondutie') =Ut-i{{9i,...,et);w*{eo),€)
-
Xc{u{6))
fori>
+ Pk{w{6))].
>k{v)-£.
Consider the plan Uo{9o)
=
u*{6q)
Then
1.
oo
>
k{v)
5];3'E_i[0,u,(^*)-Ac(«i(^*))]
f=0
oo
=
E_i [eoU*{9o) - Ac(«*(^o)) +^^^*Eo[^i+iMt+i(^*+^)
-
Xc{ut+i{e'+'))]j
t=o
>
Since £
(22)
>
and
was arbitrary
it
Xu{e)
+ (3k(w{e))] - Pe.
follows that k{v)
>
max^^u; E[^ii(^)
—
Xc{u{9))
+
(3k{w{6))] subject to
(23).
both inequalities imply k{v)
Finally, together
to (22)
-
maxE[6'u(^)
and
=
—
maXu^.u,E[^'u(0)
Xc{u{6))
-\-
pk{w{9))\ subject
(23).
Proof of Theorem 2
Suppose the allocation {ut}
Pcirt (a).
incentive compatible
equation
(7),
we
and
is
generated by the policy functions starting from
delivers lifetime utility Vq.
Vq, is
After repeated substitutions of the Bellman
arrive at
T
k{vo)
= J^P'^-ii^Me') -
Xc{u,{0'))]+p^E_,k{vT{9^)).
(27)
t=0
Since k{vo)
is
bounded above
this implies that
oo
k{vo)
so {tit}
is
< ^p'E_,[eMo') -
xc{ut{e'))],
optimal, by definition of k{vo).
Conversely, suppose an allocation {uf}
incentive compatible
and
is
optimal given
vq.
Then, by definition
it
must be
deliver utility vq. Define the continuation utility implicit in the allocation
oo
t=l
and suppose that either uq
(9)
^
(7" {9;
vq) or
wq
g"^ {9; Vq), for
(9) 7^
32
some
9
E Q. Since the
original
plan {ut}
The Bellman equation
incentive compatible, Uo{9) and wo{9) satisfy (22) and (23).
is
then implies that
k{vo)
=
E[g''{e;vo)-Xc{g%9;vo))+Pk{g^{9;vo))]
>
E[uo{9)
>
E_,[uo{9o)
-
+ pk.{wo{9))]
Xc{uo{9))
CXD
-
Xc{uoi9o))]
+J2f^'^-^[Md') ~ HMO'))].
t=i
The
first
inequality follows since uq does not
definition of k{wo{9)).
argument applies
t
>
1.
We
if
maximize
while the second inequahty follows the
(7),
Thus, the allocation {ut} cannot be optimal, a contradiction.
the plan
is
A
similar
not generated by the policy functions after some history
^*
and
conclude that an optimal allocation must be generated from the policy functions.
Pcirt (b). First, suppose an allocation {ut, Vt} generated by the policy functions (^", g^) starting
=
at vo satisfies limt^oo/3*E_ift(^*)
0.
Then, after repeated substitutions of
(8),
we obtain
T
V
= Y, /5'E_i [9Md')] + /3^E_i [vt{9^)]
(28)
.
t=o
Taking the limit we get Vq
Vq.
Next,
we show
Suppose
utility is
Yl't^o /5*E-i[^<^f (^*)] so
=
oo. Since the value function k{v)
maximum, we can bound
<
Similarly,
= — cx),
liminfi^oo/3 E_ift(^*)
utility is
= — oo.
(3
>
implies that
(5
<
av
+ b,
with a
<
0.
Thus,
+ b= -oo
a contradiction since there are feasible plans that yield
conclude that limsupj^^/3*E_it;j(^*)
suppose
Then
we have
t—>oo
(27) implies that k{vo)
We
0-
Vq,
non-constant, concave and reaches an
alimsup^*E_iz;f(^*)
i—»oo
finite values.
is
the value function so that k{v)
limsup/9'E_iA;(z;t(^*))
and then
that the allocation {ut} delivers lifetime utility
unbounded above and limsupj_Qo/^*E-i^<(^*) >
limsupj^Q^/? E_ifi(^')
interior
=
that for any ahocation generated by (p",p"'), starting from finite
<
0.
unbounded below and that liminfj^oo
Using k{v)
<
av
+ b,
with a
liminf /?*E_iA;(z;i(^*))
>
0,
/3*E_ii;i(^*)
<
0.
Then
Then
after
we conclude that
= -oo
t—«x
implying
A;(i;o)
The two
Part
= — oo,
a contradiction. Thus, we must have liminft^oo/5*JE_ii't(^*)
established inequalities imply limt^oo P^E_iVt{9^)
(c).
Suppose limsup^^^ P^E-iVt{a^{9^))
33
>
=
>
0.
0.
for every reporting strategy a.
repeated substitutions of
(9),
implying
T
^""
Therefore, {ut}
Proof of
Part
(a)
is
incentive compatible, since v
Lemma
(Strict Concavity) Let
Theorem
and
is
attainable with truth telling from part (b).
1
functions starting at vq (note:
utility values Va
t=o
Vb,
with
{ut{6'^ vq)
,
Va^
Vb.
is
=
aut{d\va)
+
{I
v^{e')
=
avt{e';va)
+
{l-a)vt{9';vb)
and
required).
Take two
initial
utilities
<(^*)
2 part (b) implies that {ut{9^,Va)}
T
Vt{9\ Vq)} be the plans generated from the policy
Define the average
immediately implies that {uf{9^)} delivers
there exists a finite time
,
no claim of incentive compatibility
-
a)ut{e\vb)
{ut(^*,Vf,)} deliver Va
initial utility v°'
=
aVa
+
(1
and
—
Vb, respectively.
ce)vb-
It
This
also implies that
such that
T
T
so that
ut{9'va)i^u^{9'vb),
for
some history
and
Vb
9^
€ 0'+^. Consider iterating
T
(29)
times on the Bellman equations starting from Va
:
T
k{v,)
= J2p'E.,[9M0';Va)-c{ut{9';Va))]+fE_,k{vT{9^;Va))
Kvb)
=
T
5];5*E_l[^,^i,(^^^;6)-c(«i(^^i;,))]+^Vl^^(t;T(^^;^;6)),
34
.
and averaging we obtain
T
ak{va)
+
J2p'E_i[etunO') -
=
{l-a)k{v,)
[ac{ut{9';Va))
+
(1
-
a)c{ut{9';v,))]]
t=0
+fE^^[ak{vT{e'^;Va))
T
+
-
(1
a)k{vT{9^;Vk))]
4=0
where the
that
inequahty follows from the
strict
we have the
inequality (29),
concavity of the cost function c(u), the fact
strict
and the weak concavity
of the value function k.
inequality follows from iterating on the Bellman equation for
satisfies
is
The
last
weak
since the average plan {u°',v°')
the Bellman equations constraints at every step. This proves that the value function k{v)
strictly concave.
(b) (Differentiability) Since the value function k{v)
is,
v°'
there
is
concave,
it is
— that
can then be established by
at least one sub-gradient at every point v. Differentiability
is
sub-differentiable
the following variational envelope arguments.
Suppose
first
that utility
neighborhood around
W{v)
Since
W{v)
is
unbounded below. Fix an
E[9{g^{9, vo)
+
{v
vq.
-
Vo))
Since W'{vo) exists
Theorem
(see
4.10, in Stokey,
k'ivo)
Finally, since d{u)
>0
=
shows that
this
Next, suppose utility
-
Xc[g\9,
=
+
{v
-
Vo))
is
it
follows,
+ Pk{g^{9, vo))]
it
W'ivo)
k'{v)
bounded below,
<
=
W{v) <
k{v),
by application of the Benveniste-Scheinkman
-A
1
follows that
k'{vo) also exists
and
E[c'{u*{9))].
(30)
1.
say,
by
>
zero.
Then the argument above
0, for all 9
establishes
E Q. However, corner solutions with
are possible here even with Inada assumption on the utility function, so a different
envelope argument
If utility is
that, applying
any
vo)
Lucas and Prescott, 1989), that
differentiability at a point Vq as long as g'^{9, vq)
g'^{9,Vo)
interior value vq for initial utility. For a
the value of a feasible allocation in the neighborhood of vq
with equality at
Theorem
=
is
Vq define the test function
is
required.
We
provide one that exploits the homogeneity of the constraint
bounded below, then limsupj_^^E_i/3*f((a-(^*)) >
Theorem
2,
for all reporting strategies
a solution {uj} to the planner's sequence problem
interior vq, the plan {{v/vQ)uf}
is
incentive compatible
= V^*E_i
\9t-ut{9')
35
- Xc(-ut{9')]
a
so
ensured. Then, for
and attains value v
addition the test function
W{v)
is
set.
for the agent.
In
W{v) <
satisfies
The Hmit
wise
it is
=
W{vo)
k{v),
Scheinkman Theorem, that
lim.y^yk'{v)
= — oo
inherited by the upper
limy^ooh{v,p)
= -oo
is
differ enti able.
It
follows immediately from lim^^c A;(w)
bound h{v,P) introduced
=
\imy^y^k'{v)
from the Benveniste-
follows
and equals W'{vo).
and lim^^oo §^h{v,P)
we show that
FinaUy,
and
k{vo)
k'{vo) exists
=
= — oo,
in the proof of
if
i;
Theorem
<
oo.
Other-
since both
1,
-oo.
Consider the deterministic planning problem
oo.
oo
k{v)
= uiaxy^P
{ut
—
\c{ut))
t=Q
= ^^0 /5*'"i-
Note that k{v)
is
plans are trivially incentive compatible,
it
subject to v
must have liiny^yk'{v)
=
CLAR
<
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 with liiny^yk^{v)
1
equation was shown in the main text, so we focus here on the bounds. Consider the
program
maxy^ Pn{9nUn "
n
V
9nUn
This problem and
constraint on u.
its
But
of neighboring agents
+ PlUn >
which
is
= y^^PnjOnUn+PWn)
6'„M„+i
+ /3Wn+l
ioT
is
U
=
We
notation require some discussion.
this notation allows us to consider
1,2,
...,
K-
1,
do not incorporate the monotonicity
bunching
in the following way. If
bunched, then we group these agents under a single index and
the total probability of this group.
this group,
+ Pk{Wn)}
c(m„)
'
what
is
Likewise
let
any
let
set
p„ be
On represent the conditional average of 9 within
relevant for the promise-keeping constraint
Let 9n be the taste shock of the highest agent in the group.
The
and the objective function.
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
different
program.
We
study
all of
them. The optimal allocation of our problem must solve one of these programs, although not necessarily the
one that yields the highest value, since this one
condition
violated.
The
is
may
not be feasible
if
the monotonicity
first-order conditions are
Pn{9n
-
Ac' (tin)
PniPk'iWr,)
where, by the Envelope theorem, A
=
-
Xdn}
+ On/J^n "
- PX} + /?(/X„ -
k'{v).
Summing
36
^n-lA^n-l
Ai„_i)
=
=
the first-order conditions for consumption,
we
get
=
\E[c'{u{9))]
The
first-order conditions for n
—
(l-X)+
I
l-k'{v)
imply
^^
^^^("i)
=
^^^^("^)1
<
1^
^
This imphes
Pi
:i-A)^
~
^1
Using
we
get
3
>
A;'K)
l
P
+
1
^1
(^
n
Similarly, writing the first-order conditions for
'1
_
A)
-
^J£zlt^I<^
=
= K, we
A(/(uk)
>
^i"
,//
X
k'{v)
:r-
13
'0^
+^
P Ui
(^iJ
—
1
-
^1
get
XE[c'{ue)]
^ 1-A
This implies
f-i>i^-(l-A)/^
Using
,//
P^
X
,
P
we
P l^K-1
P Pk
get
k'{v) -f
fA'-l
For any n,
1
+
lU/c
<
i^n
1-^
^1
^1
<
u^i,
A:'(i;)
c/A'-1
we have
+i
P
for every
h_}_
h
/5L
e K-l
^K-\
'K-l
(^K-l
n
<
k'{Wn)
<
P
0,
1
37
+
k'{v)
'K-l
^K-1
After rearranging,
we obtain
^1
1
+
:i-k'(v))
l^i
>
l-k'{g''{e,v))
1
> e
(1
-
k'{v))
+
1-1
^A'-l
To
arrive at the expression in the text
that
is
Turning to the bounded utihty
satisfied
w
we take the worst
most unfavorable to each bound, noting that
when A >
implies k'{w{9))
1
with ^„
=
=
k'{f3~^v)
—
case scenario:
>
k'{v)
we choose the subproblem
0.
case, note that all the first-order conditions
=
and u{9)
=
1
((5 / (3)k' {v)
and
w {9) =
P~^v >
Since the problem
.
is
v.
The
and constraints are
first-order condition for
strictly convex, this represents
the unique solution.
Since establishing the lower
for all V.
bound involved no assumption on
The upper bound, on the
high enough
v, i.e. for
low enough
>
other hand, required u{9)
interior solutions for
for all 9,
u
this holds
which must be true
for
k'{v).
Proof of Proposition 2
Since the derivative k'{v)
is
continuous and strictly decreasing, we can define the transition function
=
Q{x,9)
for all
X <
1.
k'{g-{{krHx),0))
For any probabihty distribution
//,
let Tq{ii)
be the probability distribution defined
by
Tq{ij){A)
for
any Borel
For example, TQ^ni^x)
=
Lemma
is
_ Tq + TI +
=
...
+ T^
n
the empirical average of k'{vt) over
all histories
of length
n
starting with
X.
4 For each x <
there exists a subsequence TQ^^(n){5x) that converges weakly
1
distribution) to an invariant distribution
Proof. The bounds (11) derived
on (— oo,
in Proposition 1
limg(x,^)
xj,l
We first
/ ^{Q(x,e)eA}d^i{x)dp{9)
set A. Define
^^'"
k'{vo)
=
=
lim
t;—-0O
1)
:
[— oo, 1) x
6 —
»
k'ig'\9,v))
(— oo,
38
in
imply that
extend the continuous transition function Q{x,
transition function Q{x,9)
(i.e.
under Q.
1),
9)
:
= P/p <
(— oo,
1)
with Q{1,9)
1.
x
—^ (— oo,
= P/P
1) to
a continuous
and Q{x,9)
=
Q{x,9),
for all
X G (—00,
1).
It
follows that
Tq maps
and
=
distributions over (—00,
We next
set
is
1),
Tq{Sj:)
probability distributions over [—00, 1) to probability
Tq{5x), for
show that the sequence {Tq^{5x)}
A^ such that Tq^{6x){Ai;)
>
1
simply E^i[k'{vtie'-^))] with x
=
—
e,
is
for all
k'{vo)
<
all
x G (—00,
tight, in that for
ri.
The expected
1).
any £
>
there exists a compact
value of the distribution T^{Sx)
=
Recall that E_i[k'{vt{9^-^))]
1.
k'{vo){P/PY -^
0.
This imphes that
<
mm{0,k'{vo)}
E^,[k'{vt{e'-'))]
< T^{Sx){-cx,,-A)i-A) + {l-T^{5x){-cx,,-A))l
for all
A>
0.
Rearranging,
T^fe)(-cx..-^)< '-7;\°-^>
which implies that {Tq{6x)}, and therefore {Tq^niSx)},
is
tight.
Tightness implies that there exists a subsequence Tqj^,.{Sx) that converges weakly
some
distribution) to
Since Q{x,9)
tt.
is
(i.e.
in
continuous in x, then Tq{Tq ^,J5x)) converges weakly to
TQ{n). But the linearity of Tq implies that
_T^^''^^\5x)-Tq{5x)
and since 0(n)
Recall that
(—00,
for
1).
^ 00 we must have Tq{7v) =
Tq maps
This implies that
such distribution
it
tt.
probability distributions over [—00, 1) to probability distributions over
tt
=
^^(Tr) has
follows that
n
=
no probability mass
Tq{'k)^ so that
tt is
at {!}. Since
Tq and Tq
coincide
an invariant distribution on (—00,
1)
under Q.
When
utility is
bounded below and
upper bound on the ergodic
set for v.
either utihty
is
bounded above
defined as the minimum of the policy function g^ over the
never zero and continuous on a compact set
g'^ is
then follows along similar
or
7 <
This in turn implies lower bound f l
it
set of values of v
attains a positive
1
>
then there exists an
for the ergodic set
with k'{v)
<
1,
since
minimum. The argument
lines as above.
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2072
40
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