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HB31
.M415
^7
Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
PROGRESSIVE ESTATE TAXATION
Emmanuel
Ivan
Farhi
Werning
Working Paper 06-27
September 25, 2006
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
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LIBRARIES
Progressive Estate Taxation
Emmanuel
Ivan Werning
Farhi
MIT
MIT
ef arhiSmit .edu
iwerningOmit edu
.
September 2006
Abstract
For an economy with altruistic parents facing productivity shocks, the optimal estate
taxation
is
progressive: fortunate parents should face lower net returns
tances. This progressivity reflects optimal
mean
on
their inheri-
reversion in consumption, which ensures
that a long-run steady state exists with bounded inequahty
—avoiding immiseration.
Introduction
Arguably, the biggest risk in
inherit the luck,
life is
good or bad, of
the family one
their parents
is
born
into. In particular,
and ancestors, passed on by the wealth accu-
mulated within their dynasty. This makes them concerned not only with
skills
and earning
from behind the
potential, but also with that of their progenitors.
veil of ignorance, against these risks.
are partly motivated to
on
work because
their children's wellbeing.
The
newborns partly
On
their
They
own uncertain
value insurance,
the other hand, altruistic parents
of the impact their effort can have, through bequests,
intergenerational transmission of welfare determines the
balance between insuring newborns and parental incentives.
One instrument
is
societies use to regulate the degree of this intergenerational transmission
estate taxation. This paper examines the optimal design of the estate tax by characterizing
Pareto
efficient allocations in
an economy featuring the tradeoff between incentives of parents
Warning is grateful to the hospitality of the Federal Reserve Bank of Minneapolis and Harvard University.
This work benefited from useful discussions and comments volunteered by George-Marios Angeletos, Robert
Barro, Peter Diamond, Mikhail Golosov, Jonathan Gruber, Chad Jones, Narayana Kocherlakota, Robert
LucEis, Greg Mankiw, Chris Phelan, Jim Poterba, Emmanuel Saez, Rob Shimer, Aleh Tsyvinski and seminar
*
and conference participants at the University of Chicago, University of Iowa, The Federal Reserve Bank of
Minneapolis, MIT, Harvard, Northwestern, New York University, IDEI (Toulouse), Stanford Institute for
Theoretical Economics (SITE), Society of Economic Dynamics (SED) at Budapest, Minnesota Workshop in
Macroeconomic Theory, NBER Summer Institute and the Texas Monetary Conference at the University of
Austin. All remaining errors are our own.
and insurance of newborns. Our main
result
is
that estate taxation should be progressive:
fortunate parents should face a higher marginal tax rate on their bequests.
We
begin with a two-period Mirrleesian economy with two generations linked by parental
altruism;
we then extend our
economy
analysis to an infinite horizon
similar to Atkeson
and
Lucas (1995) and Albanesi and Sleet (2004). In our simplest economy, a continuum of parents
during the
live
period:
first
and parents are
altruistic
In the second period each
towards this
child.
Parents work, consume and bequeath; children
simply consume.^ Following Mirrlees (1971), parents
and then exert work
output, the product of the two,
Pareto
efficient allocations
For this economy,
if
is
publicly observable.
and derive
efi'ort
We
random productivity draw
are private information; only
study the entire set of constrained
their implications for marginal tax rates.
one assumes that the social welfare criterion coincides with the par-
and
ent's expected utility, then Atkinson
applies: the optimal estate tax
of children,
observe a
first
Both productivity and work
effort.
replaced by a single descendent
is
is
zero.
Stiglitz's (1976) celebrated
That
is,
when no
income should be taxed nonlinearly
uniform-taxation result
is
placed on the welfare
(as in Mirrlees, 1971),
but bequests should
direct weight
go untaxed. This arrangement ensures that the intertemporal consumption choice made by
parents
As a
—trading
off'
their
own consumption
result, the inheritability of welfare across generations is perfect:
rises one-for-one
with that of
its
equality for the children's generation
of no direct concern.
expected
While
utility
is
by manipulating
one
is
undistorted.
a child's consumption
at
any
their children's consumption.
In-
created as a byproduct, since their expected welfare
Indeed, in this economy,
would be higher
this describes
—
parent. In effect, efficiency dictates that altruism be exploited
to provide higher incentives for parents,
is
against their child's consumption
if
parent were not altruistic, the children's
efl&cient allocation.
efficient allocation,
the picture
is
incomplete.
In this
economy
the parent and child are distinct individuals, albeit linked through parental altruism, a form
of externality.
Thus, a complete welfare analysis requires examining the ex-ante
both parents and children. Figure
is
peaked because
1
depicts our economy's Pareto frontier graphically, which
altruistic parent's welfare decreases
erable.
The
Pareto
frontier:
figure.
In this paper
if
the child
allocation discussed in the previous paragraph
is
is
made
relatively too mis-
a particular point lying on the
the peak which maximizes the welfare of parents; marked as point
we explore other
efficient
A
in the
arrangements representing points lying on on
the downward sloping section of the the Pareto frontier, to the right of
Away from
utility of
its
peak.
point A, a role for estate taxation emerges: efficient allocations which
lie
to
the right of the peak can be implemented with a simple tax system that confronts parents
with separate nonlinear schedules for income and estate taxes. Our main result
^
Although some readers have remarked that they
we extend
the time horizon.
find this
assumption reahstic,
it
will
is
that the
be relaxed
when
v„
Figure
1:
Pareto frontier between ex-ante
optimal estate tax schedule
is
utility for parent, Vp,
and
child, Vc
convex: fortunate high-skilled parents face a higher marginal
tax rate on their bequests.
The
intuition for this result
against their parent's luck.
the children's generation
on them
is
The
— which
that progressive estate taxation arises to insure children
progressive estate tax lowers consumption inequality within
is
desirable as long as
in the social welfare criterion
consumption
still
—while
still
some weight, however
is
placed
A
child's
now does
so less
small,
providing incentives to parents.
varies with their parent's, providing
some
incentives, but
than one-for-one, providing some insurance. In other words, consumption mean reverts across
generations,
making the
estate taxes reflects this
inheritability of welfare
mean
is
imperfect.
reversion: fortunate dynasties
The optimal
must
progressivity in
face a lower net return
on
bequests so that they choose a consumption path declining towards the mean.
Our
stark conclusion on the progressivity of estate taxation strongly contrasts with the
theoretical ambiguity in the shape of the optimal
seminal paper showed that
for
bounded
income tax schedule.
A'lirrlees's
(1971)
distributions of skills the optimal marginal income
tax rates are regressive at the top (see also Seade, 1982; Tuomala, 1990; Ebert, 1992). More
recently.
Diamond
(1998) has
unbounded
certain
skill
shown that the opposite
—progressivity at the top —
distributions (see also Saez, 2001).
progressivity of the estate tax do not
is
true with
In contrast, our results
on the
depend on any assumptions regarding the distribution
of skills.
We
then extend our analysis for the two-period setup to an infinite-horizon economy with
non-overlapping generations. This extension
First,
it
is
important
for at least
two reasons.
provides a motivation for studying efficient allocations which do not simply maxi-
mize the expected
utility of
the very
first
generation
—the analogues of point A from Figure
1
for the infinite-horizon
Indeed, these allocations have everyone in distant gener-
economy.
ations converging to misery, with zero consumption.
result
shown by Atkeson and Lucas (1992)
for a taste
the Mirrleesian economy. Loosely speaking,
last generation
on the horizontal
and further to the
left,
axis,
if
This
is
a version of the immiseration
shock economy, which we extend here to
we continue
to plot the expected utility of the
then as we extend the horizon point
to-«^ards misery.
A
moves further
This provides a rationale for focusing on
allocations that place positive weight on future generations, the analogues of the
sloping section of the Pareto frontier, to the right of point
show here by extending the
analysis in Farhi
A
on the
and Werning (2005),
figure.
private one, a steady state exists, misery
is
avoided and there
is
is
downward
However, as we
this result
placing weight on future generations, so that the social discount factor
efficient
is
special,
by
greater than the
social mobility.
Second, the infinite horizon version of our model allows us to make contact with a growing literature on dynamic Mirrleesian models, such as Golosov, Kocherlakota and Tsyvinski
and
(2003), Albanesi
Sleet (2004)
and others
(see further references in Golosov, Tsyvinski
Werning, 2006). In our model each individual
draw and works, and
is
lives for
and
a single period, observes a productivity
then replaced by a single descendant
in the
next period. As usual,
perfect altruism implies that each dynasty behaves as a single infinitely-lived individual, so
our model environment
is
identical to Albanesi
and Sleet (2004). However, our intergener-
ational interpretation of the infinite horizon leads us to study a different planning problem,
one that puts direct weight on the expected
that has a social discount factor that
is
utility of future generations, or equivalently,
higher than the private one.
one
Indeed, to avoid the
immiseration result mentioned above, Albanesi and Sleet impose an ad hoc lower bound on
continuation utility along the equilibrium path; in contrast, our steady-state analysis requires
no such lower imposition.
The
progressivity of estate taxes extends to this infinite horizon setup: fortunate parents
face a higher average marginal tax rate on their bequests. Indeed, the average marginal estate
tax rate formula
is
the
same
as in the two-period economy.
the two-period and infinite horizon economies
in the latter.
We
estate rate that
in
is
is
The main
difference
that tax implementations are
between
more involved
adapt Kocherlakota's (2004) implementation, which yields a marginal tax
zero,
on average,
for all parents
when only
the
first
generation
is
weighed
the welfare criterion, the analogue of point A.
Throughout
this paper,
we study an economy without
tion equals aggregate produced output plus an
capital,
endowment.
where aggregate consump-
This no-aggregate-savings
as-
sumption allows us to focus on redistribution within generations and abstract from transfers
across generations. Unfortunately,
tion, only its shape. Farhi,
it
does not allow us to pin down the level of estate taxa-
Kocherlakota and Werning (2005) extend this model among several
—
dimensions
—including capital accumulation,
and show that our main
Our work
on progressive estate taxation
result
relates to a
elements and general
life-cycle
number
is
skill
processes
insensitive to this assumption.
of recent papers that have explored the implications of
including future generations in the social welfare criterion. Phelan (2005) considered a social
planning problem that weighted
the future at
generations equally, which
all
studied
This
is
equivalent to a social discount
than one and higher than the private one. Sleet and Yeltekin (2005) have
is less
how such a
higher social discount factor
may
arise
commitment. None of these papers consider implications
We
equivalent to not discounting
Farhi and Werning (2005) considered intermediate cases, where future
all.
generations receive a geometrically declining weight.
factor that
is
organized the rest of the paper
model environment and Section
to an infinite horizon.
The main
for estate taxation.
the following way. Section
1
describes the two period
2 introduces the associated planning problem.
economy
results for this two-period
in
from a utilitarian planner without
are in Section
results for that
In Section 4
3.
economy
Our main
we describe the extension
are contained in Section
5.
We
use
Section 6 for concluding remarks.
A Two
Parent and Child:
1
There are two periods labelled
single child at
an allocation
t
is
=
1.
=
0, 1.
The parent
The parent works and consumes,
is
and output, and
altruistic
is
Ci
assumed
is
replaced by a
,
where
cq
and
yo represents the
towards the child
=E
ti(co)
over wq and
is
and
represents the child's consumption.
(3
<
- hi
=
increasing, concave
increasing, convex
and
—
Wo
The
1.
vi
utility function u{c)
=
^
while the child only consumes. Thus,
,
where the expectations
is
during
,
i;o
The
lives
a triplet of functions {co{wo) Ci{wo) yoiuJo))
parent's consumption
The parent
t
Economy
Period
)
+
(3vi
child's utility
li(ci)
and
(1)
is
simply
(2)
differentiable; the disutility function h{n)
differentiable.
Substituting equation (2) into equation (1) yields the alternative expression for the parent's
utility:
?;o
=E
u(co)
+
Pu{ci)
- h[
—
Wo
As
(3)
usual, the parent's expected utility can be reinterpreted as that of a fictitious dynasty that
lives for
two periods and discounts at rate
/5.
Following (Atkeson and Lucas, 1992) and others, we abstract from capital accumulation
to concentrate on the distributional assignment of goods across agents within a period,
not over time.
An
allocation
not greater than the
sum
is
resource feasible
aggregate consumption in both periods
if
roo
Co{wo)dF{wo) <
/
eo
+
JO
We
yoiwo)dF{wo)
/
(4)
-JO
ci{wo)dF{wQ)
is
is
endowments and production;
of
POO
Productivity
and
<
ei
(5)
private information so incentives need to be provided for truthful revelation.
say that an allocation
is
incentive compatible
if
the parent finds
optimal to reveal her
it
shock truthfully:
u{co{wo))
+ Pu{c,{wo)) -
h
(y^^)
>
u{co{w))
+
pu{c,{w))
-
h
(y^)
(6)
for all productivity realizations wq-
2
Social Welfare
To study
all
and
Efficient Allocations
constrained efficient allocations for the two-period economy
useful to
it is
work
with the general welfare criterion
W = Vo + aEvi,
which places some weight a >
on the expected
trace out the entire Pareto frontier, since the latter
(7)
utility of children.
is
As we vary a we can
convex, as illustrated in Figure
1.
Substituting equjitions (2) and (3) into equation (7) implies the alternative expression
W = E[u{co) + {P + a)u{ci)-h{yo/wo)].
Thus, the social welfare function
discount factor
/3
=P+a
that
is
is
equivalent to the parent's preferences but with a social
higher than the private one as long as
The planning problem maximizes the
feasible
welfare criterion
and incentive compatible. Formally, the problem
a >
0.
W over allocations that are resource
is
/>oo
max
co,ci,yo
/
[u{co{wo))
+
Puiciiwi))
-
h{yo{wo)/wo)]dF{wo)
7q
subject to the resource constraints in equations (4)-(5) and the incentive compatibility con-
/
straints in equation (6).
useful to divide the planning
It is
problem into two
stages. In the first stage the planner
chooses the profile of output yo{wo) and a schedule of incentives A(zi;o), which
The key
Ci(uio).
imposed
feature
is
equal to
+
Pu{ci{wo)) up to a constant. In the second stage, the
how
best to provide the incentives A{wq), using cq{wo) and
utihty from consumption u{c{){wo))
planner solves the subproblem of
is
that the second stage involves no incentive constraints, these are
by introducing
in the first stage. Formally,
A
and
U_
the
full
problem can be written
as
subject to
max
/
co,ci,yo,A,C/
7o
A{w) +
U_
=
[u{co{wo))+ Pu{ci{wi))
u{co{'Wo))
+
- h{yo{wo)/wo)]dF{wo)
Pu{ci{wo)), the resource constraints in equations (4)-(5)
and the incentive compatibility constraints A{wo) — h{y{wo) wq) > A{w)
all
wq. Note that the incentive constraint does not involve
For our purposes,
it
suffices to focus
Cq, Ci
or
U_;
only
A
on the second stage that takes
—
h{y{w)/wo)
A
and
and
for
t/o-
yo as given,
which allows us to drop the incentive constraint:
/>oo
max
[u{cq{wo))
u{co{wo))
+ /3u{ci{wi))
/
co,ci,UJq
subject to A{wq)
+ [/ =
+
Pu{ci{wi))]dF{wo)
and the resource constraints
in equations (4)-
(5).
It is
convenient to rewrite this problem by changing variables, from consumption to utility
=
assignments Uo{'w)
and the constraints
u{co{w)) and Ui{w)
strictly convex.
=
u{ci{w)), since then the objective
After substituting Uq{wq)
=
A{wo)
+
U_
the problem becomes
/•CX)
max/
[U
+ 0-(3)Ui{w,)]dF{wo)
subject to
/•oo
/
/>oo
C{A{wo) +
U- f3Ur{w,))dF{wo)
<
Cq
C{U,{wo))dF{wo) <
ei
Jo
+
/
Jo
yo{wo)dF{wo)
'0
It is
easy to see that both resource constraints must bind at an optimum.
is
then linear
— PUi{wi)
out
The Main
3
In this section
We
section.
We
we
Result: Progressive Estate Taxation
main
derive two
show that
first
relies
laid out in the previous
on bequests must be progressive.
implicit marginal tax rates
then provide a simple tax implementation that
income and
economy
results for the two-period
on two separate schedules
for labor
estates.
Implicit Marginal Taxes
3.1
For any allocation and constant
R>
we can
define the associated marginal tax rates t{wo)
solving the Euler equation
Below, the constant
R plays
no savings technology,
unimportant
this value
is
not uniquely pinned
anything that follows.
for
levels for the tax,
The
the role of the pre-tax gross interest rate. Since our
but they do not
Different values of
first-order condition for Ui{wo),
At
is
which
strictly positive lagrange multiplier
this equation
Uo{iL>o)
+
it
in equilibrium
R
—
it is
completely
are associated with different
affect its shape.
is
P-P + l3XoC'{Uo{wo))
where
down
economy has
follows that Uo{wo)
necessary and sufficient for optimality,
=
is
XiC'{Ui{wo)).
on the resource constraint
and Ui{wo) move
in the
same
for period
t.
direction with Wq.
/3Ui{wo) must be increasing, in order to provide incentives,
it
From
Since
follows that both
Uo{wo) and Ui{wq) are increasing; hence, both consumptions Coiwo) and Ci{wq) are increasing
in Wq.
Using the fact that C{u)
rearranging
is
the inverse of u(c), so that C'{Ut{wo))
=
I / u' {ci{iuo))
,
and
we obtain
\ u'{co{wo)) \ u'{ci{wo))
Ao
J
From
the
follows
first
order condition for U_
we normalize so that
Our
first result,
R=
it
J
follows that I/Aq
^^
u'ic^iwo))-
=
J^{l/u'{co{w)))dF{w). For what
j3,
can be viewed as simply restating
Aq/Ai.
derived from ecjuation (9)
when
J3
=
the celebrated Atkinson-Stiglitz uniform taxation result for our economy.
Proposition
t{wq)
=
1.
When
(5
in equation (8)
=
(5
and
the optimal allocation implies a zero marginal estate tax rate:
the marginal rate of substitution u' [ci{wq)) / u' {cq{wq))
8
is
equated
across
all dynasties, i.e.
Atkinson and
for
all
wq.
showed
Stiglitz (1976)
separable from work
effort,
that, provided preferences over
a group of goods
is
then consumption within this group should not be distorted. In
other words, the imphed marginal taxes for these goods should be equahzed to avoid distorting
their relative
to
consumption
—uniform taxation
consumption at both dates,
equalized across agents
>
/3
zero.
and implies that the
Ci,
equation
/?
ratio of marginal utilities
is
zero.'
implies that the ratio of marginal utilities
(9)
there must be
not equalized across agents:
cannot be
In our context, this result applies
—the estate tax can be normalized to
In contrast, whenever
is
and
cq
optimal.
is
some
distortion, so the marginal estate tax
Indeed, since consumption increases with productivity estate taxation must
be progressive.
Proposition
When $ > 8
2.
estate tax: r{wo)
^
for
all
the optimal allocation implies a nonzero
and progressive marginal
wq and t{wo)
R=
is
increasing in wq.
For
Q the marginal tax
rate is
r{wo)
=
-0lf3-l)u'{co{w,)) n^u'{co{w))-'dF{w)]
and Co{wo), Ci{wo) and yo(^o)
We
fl^^e
increasing in wq.
emphasize that the interesting implication
with productivity:
taxation
is
overall level of estate tax cannot
(10)
progressive.
for the
tax rate here
is
that
3.2
down
A
increases
Without an aggregate savings technology the
be uniquely pinned down,
it is
completely irrelevant. Farhi,
Kocherlakota and Werning (2005) extends the analysis to an economy with
pins
it
capital,
which
the level of estate taxation.
Simple Tax Implementation
We next show that we can implement efficient allocations,
and the progressive
implicit marginal
tax rates that go with them, with a simple tax system. In our implementation, the government
We
confronts parents with two separate schedules: an income tax and an estate tax.
an allocation
is
say that
implementahle by non-linear income and estate taxation Tf{yo), T2 and T^{b)
'
if,
for all Wo, the allocation {co{wo) Ci{wq) yo{wo)) solves
,
,
max
{u{co)
+ Pu{ci) -
h{yo/wo)}
co,ci,yo
^ One difference is that Atkinson
and Stiglitz (1976) assume a linear technological transformation between goods, whereas we assume no possible transformation. Their result on uniform taxation implies that
marginal rates of substitution are equalized across agents and that they are all equal to the marginal rate of
transformation.
Our
result only
emphasizes the former.
subject to
+ h=yo-T\h)-Tf{yo),
co
c,
Furthermore, without
=
bi
+
(y2
Rh,
to change things so that
It is trivial
bi
=
- T2)/R
+ y2-Tl
is
it
the child that pays the estate tax at
we can assume that
loss of generality
2/2
— T^ —
0.
To
i
=
1.
see this, define
then
C + b^ = yo-T\k-{y2-T2)/R)-Tf{yo)-T^{yo)
=.y,-f\k)-fy{yo)
where f^yo)
Our next
= Tf (yo) + T|(yo)
= T\k -
and f'ih)
(yz
-
T2)/R).
result establishes formally that efficient aUocations can
separate nonlinear income and estate taxation.
The
idea
is
be implemented with
to define T^{b) so that
1
=
The proof then
—
t{w)
exploits the fact that marginal tax rates are progressive to ensure that the
bequest problem faced by the parent
Proposition
1
3.
convex.
is
Suppose cq{wq), Ci{wo),
yo{'Wo)
and t{wo) are increasing functions.
Then
there exists tax functions T'^{y) andT''{b) that implements this allocation, with T'^{b) convex.
Proof.
Use the generalized inverse
of Ci{w),
where possible
flat
portions of Ci{w) define dis-
continuous jumps, to define
-'
""'-'^
<'^'
l-.((c,)-(o))
and normalize so that r^(0)
function T^{b)
is
=
Note that by the monotonicity of t{w) and
=
Co{wo)
+
R-^Ci{wo)
income /
Ty{yo)
let
be:
We now
the
+ T\ci{wo))
can express this in terms of output y by using the inverse of yo{wo): P{y)
Then we
Co{uj),
convex. Next define net income
I{wo)
We
0.
=
yo
—
(co(/), Ci(/))
/^(yo)- Finally, let the
=
=
I{yo ^(y))-
consumption allocation as a function of net
{coiI-\l)),c^{I-\l))).
show that the constructed tax
functions, T'^{y)
10
and T^{b), implement the
alloca-
/
tion.
For any given net income / the consumer solves the subproblem:
V{I)
subject to Co
+
constraint set
R~^ci
is
+ T''(c]) <
convex, since
=
This problem
/.
convex.
T'' is
1
It
pu{ci)}
convex, the objective
is
is
concave and the
follows that the first-order condition
+ T'"{b)
Combining equation
sufficient for optimality.
+
max{u(co)
u'{cfy)
(8)
and equation
conditions for optimality are satisfied by Co(/),Co(/) for
(11)
follows that these
Hence V(I)
/.
all
it
=
u{cq{I))
+
/3u(co(/)).
Next, consider the worker's maximization over
j/o
given by
max{ViI{y))-h{y/wo)}.
y
We
need to show that
yo(^f^o)
solves this problem,
plemented since consumption would be given by
Ci{wq).
Co(/(j/o('"-'o)))
Now, from the previous paragraph and our
yo{wo) e argmax{y(/(y))
which implies that the allocation
definitions
=
it
Co(rfo)
and
is
im-
Ci(/(yo(ii'o)))
=
follows that
- h{y;wo)}
y
O yoiwo)
e argmax{u(co(/(y)))
-l-/3«(ci(/(y)))
- h{y/wo)}
y
•^ Wo e argmax{u(co(i(j))
w
Thus, the
first
line follows
from the
{cQ{wo),Ci{wQ),yo{wQ))
3.3
is
Pu{ci{w))
which
last,
compatibility of the allocation, equation
-I-
(6).
is
-
h{yo{iu) wq)}
guaranteed by the assumed incentive
Hence, yoiwo)
is
optimal and
it
follows that
D
implemented by the constructed tax functions.
Discussion
Without estate taxation there
of parents
and child move
in
is
perfect inheritability of welfare. In particular, consumption
tandem, one-for-one. This situation
is
only optimal
when
the
children are not considered independently in the welfare criterion, so that insuring them,
against the risk of their parent's fortune
In contrast,
still
when insurance
is
is
not valued.
provided to the children's generation their consumption
varies with their parent's, but less than one-for-one.
of welfare
is
imperfect: consumption
mean
The
intergenerational transmission
reverts across generations.
11
The
progressivity of
the estate tax schedule reflects this
mean
returns on bequests in order to give
present, that
is,
reversion. Fortunate parents
them
incentives to
tilt
their
must
face a lower net
consumption towards the
towards themselves. Likewise poorer parents need to face higher net returns
so that their consumption slopes upward. This explains the progressivity of estate taxes.
Another intuition
is
based on the interpretation of altruism as a form of externality. In
the presence of externalities', some form of corrective Pigouvian taxes are generally desirable.
Think
of a -parental bequest as a
consumption good with a positive externality to the
then the Pigouvian logic implies that we should subsidize bequests. Since expected
our concern, and utility
tion.
concave, this externality
Thus, the subsidy rate should be highest
lowest
no
is
—
capital, the
Pigouvian
level of taxation turns
However, the relative tax conclusion
estates.
None
—or equivalently,
poor parents. Optimal estate taxation
for
of these
ity or effort
greatest for children with low
is
utility is
consump-
the negative tax should be
thus progressive. Since our economy has
is
out to be irrelevant
in this
child;
—we may tax or subsidize
argument remains robust.
arguments require the private-information structure. However,
if
productiv-
were observable, then the first-best allocation would be achievable. Consumption
and wealth would then be equated across parents. Although one can
still
think of a progres-
sive estate tax in this situation for out-of-equilibrium levels of parental wealth,
irrelevant given the lack of parental inequality. In'this sense, our results rely
between redistributive and corrective motives
for taxation (see also
it
becomes
on an interaction
Amador, Angeletos and
Werning, 2005).
A
4
Economy with
Mirrleesian
We now turn
Sleet (2004).
to a repeated version of this
t
has ex-ante welfare
=
vt
where
/3
<
1 is
the coefficient of altruism.
An
Utility
=
rC j^ with 7
>
1
Vt
Et-i[u{ct)
tion satisfies the Inada conditions u'{0)
function h{n)
economy with an
Horizon
infinite horizon, as in
Albanesi and
work and receive a random productivity draw. An individual
All generations
born into generation
Infinite
solving
-
h{nt)
+
Pi't+i],
We assume that the utility function over consump= oo and u'{oo) = 0. We adopt a power disutihty
to ensure that the planning
individual with productivity w, exerting
work
problem
effort n,
is
convex.
produces output y
= w
n.
can then be written as
00
Vt
=
Y.l^' ^*-i [^(^+«) - ^t+sKvt+s)]
s=0
12
(12)
where
=
6t
and hence
t
is
=
0,1
.
.
w^'^ can be interpreted as a taste shock to producing output.
is
9t,
independently and identically distributed across dynasties and generations
With innate
.
Productivity Wt,
assumed nonheritable, intergenerational transmission of welfare
talents
not mechanical linked through the environment but
may
arise to provide incentives for
altruistic parents.
Since productivity shocks are assumed to be privately observed by individuals and their
descendants each dynasty faces a sequence of consumption functions {q}, where
sents an individual's consumption after reporting the history 9*
reporting strategy a
shocks
We
=
{at}
into a current report
9^
a sequence of functions
is
Any strategy a
9t.
utility
U{{ct},a;P)^Yl
An
allocation
{q}
is
E
{9o, 9i,
0*+^ -^
:
Q
.
.
,
.
that
induces a history of reports
use a* to denote the truth-telling strategy with at
Given an allocation {q}, the
at
=
{9^)
=
9t
for all
6**
A
9t).
maps
o"*
:
repre-
Ct{9'')
dynasty's
histories of
G'"*"^
—<
0'+^
G 0'+^
obtained from any reporting strategy
a
is
P'H^t{'^'iS')))-(^Myi'^\0')))]Fii9').
incentive compatible
if
truth-telling
is
optimal, so that
U{{c,},a*;P)>U{{c,],a-f3)
(13)
for all strategies a.
We
identify dynasties
An
population.
allocation
yl{9^) represents the
at date
t
their initial utility entitlement vq with distribution
a sequence of functions {ct,yt} for each
is
consumption and income that a dynasty with
after reporting the
entitlements
compatible
to v\ and
by
'ip
and resources
for all dynasties;
(iii)
sequence of shocks
e,
we say that an
(ii) it
Consider the
all
sum
and
entitlement v gets
initial distribution of
feasible
if:
(i) it is
incentive
periods:
j Y,yl{9')?v{9')diP{v)
/
1
\
1
Y^at¥._,Vt=[l-^—)v, +
^
t=o
The
first
t
=
0,l,...
of expected utilities weighted by geometric Pareto weights at
oo
p.
is
Cj(^*)
delivers expected utility of v to all initial dynasties entitled
lY,c\{9')Y>x{9')di,{v)<e+
>
allocation (cj"}
in the
average consumption in the population does not exceed the fixed endowment e
plus income generated in
with P
initial
For any given
9^.
where
v,
ip
term
is
P
y/
(14)
=
°°
-^—T^'¥..,[u{c,)-9th[yt)].
P
P i=o
exogenously given, since we take as given a distribution
13
/3*
(15)
for the
entitlements Vq- Thus, the welfare criterion
initial utility
is
given by
so
J2P'^-iH^)-^tHyt)]
(16)
Future generations are already indirectly valued through the altruism of the current generation.
If,
factor
in addition,
they are also directly included
must be higher than the
When P =
/3,
in the welfare function
the social discount
more
private one (see Faiiii and Werning, 2005, for
details).
the planning problem seeks the lowest constant resource level e to ensure that
there exists a feasible allocation that delivers the distribution of utility entitlements
precisely the efficiency problem studied in Albanesi
and
WTien
Sleet (2004).
,5
>
/3
we
the social optimum as maximizing the average social welfare function (16), weighed by
all feasible
allocations.
entitlements
ip
That
is,
This
ip.
is
define
over
ib,
the social planning problem given an initial distribution of
and an endowment
level e is to
maximize
Ju{{cn,a*,,d)d^{v)
(17)
subject to the the resource constraints (14), as well as the promise keeping and incentive
constraints: v
and
=
and U{{c^},a*\p)
[/({cJ'},cr*;/3)
>
U{{c^},a;l3) for
all initial
entitlements v
strategies a.
We
are interested in distributions of utility entitlements
ip
such that the solution to the
planning problem features, in each period, a cross-sectional distribution of continuation
ties vt
that
is
also distributed according to
of consumption
and income to
ijj.
We also
require the cross-sectional distribution
replicate itself over time.
We
entitlements with these properties a steady state and denote
Werning
(2005),
we approach
term any
them by
Markov
initial distribution of
ip*.
Follo^vang Farhi
the planning problem by studying a relaxed version of
solutions to both problems coincide for steady state distributions
characterize.
The relaxed problem has continuation
utili-
which
ib*,
is all
it.
and
The
we seek
utility as a state variable that follows
process. Steady states are then invariant distributions of this
Markov
to
a
process.
Define the relaxed planning problem to be equivalent to the social planning problem except
that the sequence of resource constraints (14)
r
•^
is
replaced by the single intertemporal condition
^
1
t=o
1
e'
P
Letting A be the multiplier for this intertemporal resource constraint we form the Lagrangian
14
L = J L" dib{v) where
^''
=
E
E-^'("(<(^*))
t=o
- ^<(^*) - ^Myti^')) + %(^'))
and study the maximization of L subject to v
For any endo-Rinent
cr.
maximizing
this
The
t/"({c«)},
>
o-*; /?)
= snpU
BeUman
where v
some key properties
JTie value
— — oo,-
it is
-\-
Xy{h{e))
-F
3k{w{e))]
(21)
= E[u{e) - dh{e) -F pw{9)]
(22)
is strictly
=
w and
e.o'ee.
(23)
The next lemma
u.
concave and continuously differentiable on {v,v)
unbounded below on both sides
A:'(r)
for
forau
of the value function k{v).
function k{v)
the derivative has lim^.^j,
1
and
limi^,^f k'{v)
limj^,^t^A;(i')
=
lim,.— d
^(i.')
= — oo:
and
Our
first
= — oo.
Steady States aiid Progressive Taxation
5
We
are interested in steadj- state distributions v' that have
result is that this is not possible
l3
is
equation
Denote by g^{v,6) and g^{v,9) the optimal policy function
1.
is
(20)
u{d)-dh{d)+pw{e)>u{e')-eh{e')+dw{9')
Lemma
Maximizing L
-^
V
characterizes
and
of the subproblem:
i".
= maxE[u(^) - Ac(«(^)) - 9h{9)
-
L^({cr},cr;/3) for all v
U{{c{i4)},a;,3) for aU a.
continuous, concave, and satisfies the
subject to
>
the component planning problem k{v) defined by equation (20)
\'alue function of
^(lO
L'({(^'},o"*;,i3)
equi-valent to solving the relaxed problem.
is
k{v)
=
(19)
a unique positive multiplier A(e) so that the
level e. there exists
Lagrangian
=
equi%'alent to the point^ise optimization, for each
subject to V
Pr(^*)
e«
=
,3.
We
when
The
at miserj- v.
future generations are not weighed directly, so that
then show that, in contrast, whenever 3
with no mass at misery.
no mass
>
i3
efficient allocation displays
generations that keeps inequaUty bounded.
a steady state distribution exists
a form of
The mean reversion
inverse Euler equation which implies that estate taxation
15
is
is
mean
rex^ersion across
characterized
progressive.
by a modified
An
5.1
For
/3
=
Immiseration Result
we have
P,
distribution
ip
to modify our definition for the Social Planning problem.
problem
of initial welfare entitlements, the planning
is
to
For any
minimize the net
resources required to deliver the utility entitlements in an incentive compatible way:
inf e
(24)
subject to,
j Yl^dl{0')
-
<
yl{e'))di>{v)
e
(25)
e«
[/({cj'},a;/?)
t/({c;'},a*;/3)
.
From
paper.
this
We
>
=
i;
ally
for
C/({cj^},a;;3) for ally
Proposition
andd
(27)
program, we can define an invariant distribution exactly as
are interested in steady state distributions
first result is
(26)
that this
4.
is
basically not possible
when
(i
Suppose that limu^ooSupc"(u)/c'(ii)
invariant distribution
ip*
-0*
=
<
without
full
in Section 4 of the
mass
at misery.
Our
j3.
Then
oo.
if
§ =
P, there exists
no
without full mass at misery.
This result extends the immiseration result in Atkeson and Lucas (1992),
endowment economy with
who study an
privately observed taste shocks, instead of the Mirrleesian pro-
duction economy with privately observed productivity shocks studied here. They show that
the cross-sectional distribution of consumption disperses steadily over time, with inequality
As a
growing without bound.
result,
almost everyone converges to the misery, consuming
nothing, while a vanishing fraction tend towards
bliss,
consuming the
entire aggregate en-
dowment. Thus, no steady state distribution with positive consumption
of our
knowledge Proposition 4
is
the
first
on continuation
(3
utility to
=
(3
To the best
formal statement of an analogous result in the
context of a Mirrleesian economy, where private information
Researchers that assume
exists.
is
regarding productivity shocks.
have been typically forced to impose an ad hoc lower bound
avoid misery and ensure that an steady-state distribution exists
(Atkeson and Lucas, 1995; Albanesi and Sleet, 2004).
5.2
Steady States and a Modified Inverse Euler Equation
We now
We first
return to efficient allocations where future generations are given positive weight.
derive an important intertemporal condition that
16
must be
satisfied
by the optimal
allocation.
This condition has interesting impHcations
optimal estate tax, computed
for the
later.
Let A be the multiplier on the promise-keeping constraint and
Then the
multipliers on the incentive constraints.
for u{d)
represent the
first-order conditions for interior solutions
and w{6) are
p{e)
- Xc'{u{6M9) -
Xp{B)
- J2 ^(^' ^') +
6'
-
Pk' {wie))p{e)
pxp{e)
The envelope condition
is k'
{v)
=
-/3J2 a^(^> ^') +
From the
A.
E
E
^(^'' ^)
=
z^^^'- ^)
=
(28)
e'
/?
o
(^9)
6'
8'
CLAR
let ii{9,9')
first-order condition for
w{9) we obtain the
equation
^k'{v)
=
9
This equation encapsulates the mean-reversion force
^k'{v^)
P/P < 1 acts as an
mean reverts back to
(30)
Y.^'i9'"iv,d))p{9).
P
=
in the
model. In sequential notation
Et[k'{vt^,)],
(31)
,
so that
autoregressive coefficient ensuring that over time the derivative
k'{vt)
zero,
mean-reverting force provided by
where the function k{v) finds
/?
>
/3 is
exists, increasing inequality
(Proposition
when
/3
=
/3
no such central
and immiseration ensues and no steady state
exists
resolution of the tradeoff between incentives for altruistic parents and in-
surance for newborns gives
—
In contrast,
4).
The optimal
welfare
maximum. The
crucial for the existence of steady state distributions
with bounded inequality, which we prove below.
tendency
its interior
rise to
in contrast to the case
a less than one-for-one intergenerational transmission of
where P
=
p.
The descendants
of a rich parent are
fortunate than those of a poor parent, but less and less so the more distant
the impact of the
initial
The more weight
less intense is
is
is
more
the descendant:
fortune of dynasties dies out over generations.
put on future generations, the higher
is
the link between the welfare of parents and child.
even the smallest amount of mean-reversion
in
the form oi
P compared to /?, and the
But as we will now show,
P > P
puts enough limits on
the transmission of shocks across generations to prevent the distribution of consumption and
welfare from exploding.
17
,
The
first-order conditions (28)-(29)
=
^k'{w{9))
imply that
and
l-Xc'{u{9))
^k' (v)
=
where u_ should be interpreted as the previous period's assignment of
Substituting relation's in equation (32) into the
tion.
CLAR
~
1
\c' {u_)
utility
(32)
from consump-
equation (30) we arrive at a
Modified Inverse Euler equation
The
left-hand side together with the
The second term on
Euler equation.
and
is
strictly negative
We now show
when
>
(3
first
term on the right-hand side
the right-hand side
is
is
novel, since
the standard inverse
it is
zero
when
(3
=
j5
(5.'^
that a steady state exists whenever the welfare criterion places direct weight
on children so that
(3
>
(5.
The proof
follows Farhi
and Werning (2005) quite
closely,
which
proves such a result for an economy with taste shocks.
Proposition
implied by
misery
v.
(a)
5.
g'^
.
(b)
There exists an invariant distribution
Moreover any invariant distribution
Suppose that limu-*oo sup
away
necessarily has a support bounded
An
c" (u) / c' (u)
bounded away from
<
misery.
It
distribution has consumption
invariant
The
ip* is
then any invariant distribution
when absolute
risk aversion
To
follows directly that the allocation implied
and work
on the force
effort that are
for
mean
see this mean-reversion force
logarithmic utility case, u{c)
^Farhi, Kocherlakota
=
is
bounded, so
the invariant distribution has a compact support, that
log(c).
Then
and Werning (2005) show that
is
by the invariant
bounded above. This ensures that the
also a steady state of the original planning problem, for
result relies heavily
equation (33).
oo,''
oo,
from, v.
invariant distribution always exists, but
that limu^ooSupc"(u)/c'('u)
has a support bounded away from
tp*
<
for the Markov process {vt}
ip*
reversion that
is
some endowment
level
behind equation (31) and
most clearly consider, as an example, the
l/u'{c)
=
c
this equation,
and equation
and
its
(33) can be written
implications for estate taxation,
generahze to an economy with capital and an arbitrary process for skills.
* This is the case for most common preference specifications, such as CARA or
CRRA
utility functions.
Indeed, the proof of this result actually shows that promised continuation utility Vt is bounded for all
realizations of the shocks, starting from any vq in the bounded support. It follows that promised utility t't is
^
bounded
that
is,
for all
reporting strategies. This in turn implies that the proposed allocation
is
incentive compatible,
that the temporary incentive constraints in equation (23) imply equation (13) (see
and Werning, 2005).
18
Theorem
2 in Farhi
.
with sequential time notation as
or simply
= -Q +
Ci+i
where
c
=
A""^ is
1
- -
c
j
+
with Et[e(+i]
et+i
=
average consumption at the steady-state cross-sectional distribution. As the
last expression indicates,
with logarithmic
autoregressive coefficient equal to
/3//3
<
utility,
consumption
itself is
autoregressive with an
1.
Tax Implementation
5.3
Any
(
allocation that
is
incentive compatible and feasible, and has strictly positive consumption,
Here we
can be implemented by a combination of taxes on labor income and estates.
describe this implementation, and explore
some
first
features of the optimal estate tax in the next
subsection.
For any incentive-compatible and feasible allocation
mentation along the
lines of
we propose an
implcr,
Kocherlakota (2004). In each period, conditional on the history
of their dynasty's reports 9^~^
6t,
{cj'(^*), yj'(^*)}
and any inherited wealth, individuals report
their current shock
produce, consume, pay taxes and bequeath wealth subject to the following set of budget
constraints
ct
where
Rt-\,t
is
+ ht<
yt{e')
-
Tt{e')
+
-
(1
Tt{e'))Rt-i,tht-i
t
=
0, 1,
.
the before-tax interest rate across generations, and initially 6_i
are subject to two forms of taxation:
(34)
.
=
Individuals
0.
a labor income tax Tt{9^), and a proportional tax on
inherited wealth Rt^i^th-i at rate Tt{9*).^
Given a tax pohcy
{Tj^(^'), t^{9^), y^{9*)},
an equilibrium consists of a sequence of interest
rates {Rt^t+i}', an allocation for consumption, labor
a reporting strategy
{cr]'{9^)}
such that:
(i)
{q,
income and bequests
bt,
to (34), taking the sequence of interest rates {Rt^t+i}
and
(ii)
the asset market clears so that
/
at}
maximize dynastic
and the tax policy
E_i[6"(6'*)] d(p{v)
{c^{9^), b^{9^)};
=
for
alH
=
{Tt,
0, 1,
and
utility subject
yt} as given;
Tt,
.
.
.
We
say that
^In this formulation, taxes are a function of the entire history of reports, and labor income yt is mandated
Q* ^> E* being implemented are invertible, then
if the labor income histories y'^
given this history. However,
:
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 simply choose a budget-feasible allocation of consumption and labor income, taking as given prices
and the tax system.
19
a competitive equilibrium
is
incentive compatible
if,
For any feasible, incentive-compatible allocation
tion
{cj*, y'^},
it
induces truth
=
-
yt{e')
with strictly positive consump-
interest rates {Rt-i^t}-
These choices work because the estate tax ensures
that for any reporting strategy a, the resulting consumption allocation
bequests
=
6J'(0*)
satisfies the
labor income tax
allocation
less of
is
with no
such that the budget constraints are satisfied with this consumption
this no-bequest choice
is
optimal for the individual regard-
the reporting strategy followed. Since the resulting allocation
it
{cJ'((t'(0*))}
consumption Euler equation
and no bequests. Thus,
by hypothesis,
setting
and
ct{9')
any sequence of
The
telling.
we construct an incentive-compatible competitive equilibrium with no bequests by
Tl'{9')
for
in addition,
follows that truth telling
is
optimal.
The
is
incentive compatible,
resource constraints together with
the budget constraints then ensure that the asset market clears.^
As noted above,
in
our economy without capital only the after-tax interest rate matters
so the implementation allows any equilibrium before-tax interest rate {Rt-i,t}-
subsection,
we
This choice
is
set the interest rate to the reciprocal of the social discount factor, Rt-i,t
natural because
state in a version of our
5.4
In the next
it
=
P~^-
represents the interest rate that would prevail at the steady
economy with
capital.
Optimal Progressive Estate Taxation
In our environment, the relevant past history
is
encoded
in the
continuation utility so the
estate tax T{9^~^,9t) can actually be reexpressed as a function of Vt{9^~^)
notation we then denote the estate tax by Tt{v,9t).
invariant distribution,
The average
we
also
Since
we
and
dt-
Abusing
focus on the steady-state,
drop the time subscripts and write t{v,9).
estate tax rate f{v)
is
then defined by
l-n^) = E(l-^(^'^)M^)
(36)
^Since the consumption Euler equation holds with equality, the same estate tax can be used to implement
allocations with any other bequest plan with income taxes that are consistent with the budget constraints.
20
Using the modified inverse Euler equation (33) we obtain
fiv)
= -A-i^'(c_(^))(^-l)
In particular, this implies that the average estate tax rate
is
<
negative, f{v)
that
0, so
bequests are subsidized. However, recall that before-tax interest rates are not uniquely de-
termined
in
by
With our
(35).
are pinned
our implementation.
As
computed
a consequence, neither are the estate taxes
particular choice for the before-tax interest rate, however, the tax rates
down and
acquires a corrective, Pigouvian role.
DifTerences in discounting can
be interpreted as a form of externalities from future consumption, and the negative average
tax can then be seen as a way of countering these externalities as prescribed by Pigou.
our setup without capital, this result depends on the choice of the before-tax interest
However, the negative tax on estates would be a robust steady-state outcome
In
rate.
in a version of
our economy with capital.
In our
model
it
is
more
how
interesting to understand
the average tax varies with the
history of past shocks encoded in the promised continuation utility
increasing function of consumption, which, in turn,
taxation
is
The average tax
is
an
an increasing function off. Thus, estate
progressive: the average tax on transfers for
Proposition
more fortunate parents
is
higher.
In the repeated Mirrlees economy, an optimal allocation with strictly positive
6.
consumption can
he
implemented by a combination of income and estate
state, invariant distribution ip*
is
is
v.
,
taxes.
At a steady-
the optim.al average estate tax f{v) defined by (35)
and (36)
increasing in promised continuation utility v.
The
progressivity of the estate tax reflects the mean-reversion in consumption.
The
for-
tunate must face lower net rates of return so that their consumption path decreases towards
the mean.'^
Concluding Remarks
6
When
only the
first
generation's welfare
is
of concern,
we obtain
familiar results that echo
those obtained in intragenerational settings. In particular, in our simple two-period economy
we recover
Atkinson-Stiglitz's uniform-taxation result.
As a consequence, bequests should be
undistorted and the transmission of welfare perfect: consumption of parent and child should
move one-for-one.
In our infinite-horizon model,
^Farhi, Kocherlakota
we prove an immiseration
and Werning (2005) explore more general versions of
intuitions.
21
result that parallels
this result
and discuss other
Atkeson-Lucas': a dynasty's consumption inherits a
random walk
property, inequality grows
without bound and everyone converges to misery.
In contrast,
when
planner values insuring children against the family they are born
and study the
allocations
We
role that estate taxation
find that the estate taS should
can play
section for consumption
Farhi, Kocherlakota
and welfare
a number of issues are
still
result
unexplored.
capital, of
remain open questions
We characterize efficient
in
implementing these allocations.
is
then bounded: a steady-state cross-
and Werning (2005) explore some extensions
—and show that the main
human
into.
exists.
capital accumulation, modeling life-cycle elements
the child's
taken into account, the
be progressive to ensure that consumption and welfare
exhibit mean-reversion across generations. Inequality
generations
is
the expected welfare of future generations
and allowing
—by including physical
skills to
be correlated across
on progressive estate taxation holds. However,
For example, the effects of parental investments in
endogenous and variable
for future research.
22
fertility,
and of intervivo transfers
all
Appendix
A
Proof of
and
Strict concavity
limits of k
bound
and
k' at
/c"'^, for
Lemma
1
differentiability follow
from standard arguments. In order to derive the
the bounds of the domain, we derive a lower bound
which we can
compute the corresponding
easily
limits.
Consider the solution {u^°{9^),y'"°{0^)] to the relaxed planning problem
all
V
<
vo, define {ul'>{d'),y^°{e*)}
and an upper
A;""'"
for
a given
vq.
For
by
ul°{9*)
y^o((?<)
=
u^°(^*) for all
=
y^'<'(^')
for
t
>
alH>
1
Let
oo
/^"'"(^O
= J]/3'^-iK°(0O - Ac«n^*)) +
>^yl°{0')
-
dthiy:°{9'))]
t=o
Since {w^°(^*),y^°(^*)}
fc"""(f), for all V
<
vq.
incentive compatible
is
We
and
delivers welfare level v,
we have
k{v)
>
have
k"'""{v)
1
= l-XE
h'{h{y^°{e°)
+ Vo-v)
Hence
lim
D^ — oo
Since k{v)
>
k'^^"{v), for all
v
<
Vq
k"'""{v)
=
1
and both k and fc™" are concave,
lim k'{v)
D— — oo
<
this implies that
1
»
Next define
oo
k{v)
= sup^p'E.MG') -
Xciu{9'))
+
\y{9')
-
9th{y{9'))]
t=o
s.t.
oo
v
=
Y,l3'E_,[u{0')-d'h{y{9'))\
This corresponds to the relaxed planning problem, but without the incentive constraints.
23
Hence we have k{v)
<
k{v).
Let
m = maxu — \c{u) + Ay — Oh{y)
u,y,9
Then
k{v)
< sup^/3*E_i[u(^') -
Xc{u{9'))
+
\y{9')
/3*^_i [-\c{u{e'))
+
Ay(^0]
-
eth{y{d'))]
+
1
m
4=0
<
i;
+ sup
i
^
+
[
VI
t=o
Hence
if
we
1
m
-p
1
1-/9.
define
C(^;)=inf5^/3*£_i[cK0'))-y(^*)
t=o
s.t
^
=
X;/3'£;_i[u(^*)-^tMy(^*))]
t=0
and
k'^^{v)
= v-
1
+m
C{v)
1
1-/3
1-/?J
we have
<
kiv)
Denote by
{u'-^ {9^
,
v)
y^ {9* v)} the solution
,
,
/c'"^'^(z;)
of the
program defining C. Combining the
order conditions for u{9^) and the envelope theorem,
get
=
C'{v) for
alH>0
——- =
C'{v) for
all t
c'{u^{9\v))
1
we
^
Till
9th'{yC{9^,v))
This implies that
lim
C'{v)
=
•^—00
lim u^{9\v)
> — CX)
=u
im y^{9*,v)
=
lim
=
00
Hence
k'^^'iv)
24
1
>
first
<
Since k
k"^^^
and both
and
A:
k""^^ are concave, this
lim
k'{v)
>
imphes that
1
we already have
Since
k'{v)
<
1
this implies that
lim k'{v)
D— — OO
=
1
»
Note that we always have
lim C'(i')
= +00
v—>v
lim A;"^'(u)
= -oo
v—*v
Since
/c(t')
<
k'^^{v),&nd both k and
k™^
are concave, this implies that
limk'{v)
= — oo
V—fV
Finally, note that
=
^"'"^^(^')
lim
V—V
lim k'^^^iv)
= -oo
V—'V
Hence
lirn
V —*v
B
k{v)
=
lim k{v)
Proof of Proposition 4
In order to characterize the optimal allocation
The Lagrangian theorem guarantees
with
= — oo
v—*v
go
=
1
convenient to study a relaxed problem.
that there exists a unique sequence of multipliers {qt}
on (25) such that solving (24)
t>o
it is
^
is
equivalent to solving the following program:
$*
subject to (26) and (27). Note that this problem
25
is
equivalent to the minimization v by v of
subject to
>
U{{di],a*-I3)
Hence C{v;
{qt})
welfare v to the
is
first
U{{dl},a;p)
a
for all
the least possible cost of an incentive compatible allocation delivering
generation.
It
is
trivial to see that
C{v; {qt})
is
the solution of the
following Bellman equation
C{v; {qt+s}s>i)
=
inf E[c{ue)
-
y{he)
+
qt+iC\we, {-^}s>2)]
(37)
subject to
V
U0
For future use,
6^
when
0100
of multipliers.
Hence there
E[ue
—
9h.g
+
f3we
>
U0:
-
+ (5w0i -
exists
6h0i
is v.
exists an invariant distribution -0*,
Since
9hg\
us denote by g'^{v, 6^) the continuation utility after a history of shock
the initial welfare entitlement
Suppose there
t.
let
+
=
-0 is
and
let {qt}
be the associated sequence
a state variable for (24), this shows that qt+\/qt
<
g
<
such that qt+ijqt
1
time dependence on the sequence {qt}
=
in Ct[v; {qt}),
q for
all
t.
We
is
independent of
can therefore drop the
and simply write C{v) as a shortcut
for
C{v,{q%>,).
Lemma 2.
Then q>
Suppose there
exists
an invariant distribution
ip*
without full mass at misery.
(3.
Proof.
We
will
make use
of
two possible state variables. The
one: v, promised future utility.
u_. Indeed, from the
first
The
other one
order conditions,
first
utility attained
is
state variable
is
the natural
by the previous generation
easy to see that these two state variables are
it is
related by
d{u_)
The
existence of an invariant distribution
=
^C'{v)
with not mass at misery
ip*{'^')
existence of an invariant distribution '4)*{u-) with no mass at misery.
Let xq
=
ug
+
Pw0. Then we can rewrite the Bellman equation (37) as
C{v)
=
inf E[c{ue)
-
y{hg)
subject to
V
=
E[xe
26
-
9h0]
+
qCiwg)]
is
equivalent to the
xe
—
Ohe
>
—
xgi
+ Pwg =
ue
9hgi
Xe
Hence, given a value x for xg, ug and wg are given by the sub-program
minc(u)
+ qC{w)
subject to
+ (5w =
u
The
solution
is
given by the
first
X
order condition
c'{x-pw) + ^C'{w) =
Using the implicit function theorem, we can then compute
du
lC"{w)
dx
^C"{w)+f5c"{x-Pw)
Hence
du
—
<
dx
<
This in turn implies that there exists
max
M>
\ug'
—
1
such that
ug\
<
M max hg
9,6'
The
first
6
order conditions for ug in in (37) imply that
^C'(^_)
=
E[c'{ug)]
Hence
^d{u_)
=
E[d{ug)\
<
d{ue)
Therefore,
log(^)
+ log(c'(u_))<log(c'M)
and hence
log(^)
+
log(c'(^_))
<
log(c'(n_))
27
+
{
max
^
[Ug
-
U-
which we can rewrite as
log(f)
-.
r-
c"{u) \
(
\rnayiu^[u_,ue}
Hence
c'{u)
<
—
Ug
-
- U-
J
£ 0,
for all ^
log(^)
M max
'^
gi^Q
c"(u) \
(
In order to allow for bunching in (37),
<
ug
—
u..
convenient to consider the following program
it is
-
y'p„{c(ii„)
inf
hgi
y(/i„)
+
gC('u;„)}
n
n
-6nK +
This problem and
,ity
constraint on
any
let
Ur,
its
h.
+
(3Wn
U^+l
+ Pw^ +
But
objective function.
constraint
of the
Of
must
(OT
is
which
U =
1,2,
.
.
.
,
K-
1,
do not incorporate the monotonic-
this notation allows us to consider
bunching
in the following way. If
bunched, then we group these agents under a single index and
pn be the total probability of this group. Likewise
of 9 within this group,
i
We
notation require some discussion.
neighboring agents
set of
> -9nhn+l +
is
what
is
let 9n
represent the conditional average
relevant for the promise- keeping constraint and the
Let 9n be the shock of the highest agent in the group.
rule the highest agent in each
The
incentive
group from deviating and taking the allocation
group above him.
course, every combination of
bunched agents leads
to a different program.
The
op-
timal allocation of our problem must solve one of these programs with a strictly monotone
allocation
—since bunching can be characterized by regrouping agents.
acterize solutions to these
The
first
programs with
order conditions for
y'{hn)
This implies
strict
Thus, below we char-
monotonicity of the solution.
/i„
^n is
=
in particular that at
C'{v)9n
+
9nfin,n+l
~
6'„_i^i„_i,„.
the optimum, for any of these programs (and hence for the
program solved by the true optimal
allocation),
y'ihi)>C'{v)9.
It
is
easy to verify that
C >
C, where
C
28
is
the solution to (37) without the incentive
—
compatibility constraints. Let v be the upper
C
bound
of the
domain
for v. Since
both
C
and
are increasing and convex, and since
\\mC[v)
II
=
and
oo
C'{v)
Jim
—
V
^V
=
oo
=
oo
*oo
we have
=
oo
\imy'{hg{v))
=
hrnC(f)
and
lirnC'(i')
Therefore,
V —>v
and since h0 has
is
decreasing in
=
6,
lini/ie(i;)
this in
limhe{v)
v—*v
—^v
V
But
=>
oo
=
Ofor aU
e
e
e
turn impUes that
log(^)
<
"
''
/
c"{u)
\
lim inf(u0
—
u_)
V-
(^maxue[u_,„^] ^,^^^^
Suppose that q <
functions ue are
all
bounded away from
p.
This implies that for v or equivalently w_ high enough, the policy
such that ug
u.
This
>
u^. This in turn implies that
^p*
necessarily has a support
in turn implies that
C'{v)dij*{v)
=
I
c'{u^)dTp*{u_)
<
oo
Integrating
f^C'iv)
over
V,
we
Since 0* doesn't have
P—
E[C'{we)]
get
I
that
=
q,
a.
full
mass
C'{v)di)*{v)
at misery,
= ^ j C'{v)dr{v)
we have J
C'{v)dip*{v)
>
0.
This
in turn implies
D
contradiction.
29
We
have therefore proved that q
> P
at
^C'iv)
we
see that C'{vt)
any
initial
random
But then from the equation
ip*.
=
E[C'{we)]
By
a positive supermartingale.
is
value vq for v, the sequence of
random
the martingale convergence theorem, for
variables {vt} converges almost surely to a
variable C^°° with
E[Cl°°]
Suppose that there must
<
C'{v).
>
exists a v* such that Pr(C^'?'
We
0).
show that
will
this
is
not
possible.
For any realization 9°° define the set of periods where
e
^
since
we have
is finite,
that with probability one
Pr{#0e{9°°)
Hence there
value,
exists
an event
=
and #0^(0°°)
Since g'^{v,9)
is
6°°
oo for
Hence Pr(C^°° >
0.
0)
=
=
oo for
v,
9
E Q. Hence
all
all
e 0)
^
values of 9 occur infinitely often
=
1.
=
Therefore for
0.
Since
ip*
has
full
ip* is
start with
3.
9
This
an invariant distribution, C'{vt)
full
mass
at zero,
i.e.
5
two lemmas, and then proceed to prove the proposition.
The following inequalities hold
7(1
all
0.
at misery.
Proof of Proposition
Lemma
for
mass
for all
C'{g'^{v,d^)) converges almost surely
all v,
This in turn implies that the stochastic process C'{vt) converges almost surely to
C
We
w*
a contradiction.
^y*,
distributed as C'{vq). This implies that the distribution of C'{vo) has
that
=
oo this implies that g^{w*,6)
incentive constraints are not binding at
for all v.
finite
g'^{v* ,9^{9°°)) converges to a finite value w*.
and i^0g{6°°)
implies that C'{vt) converges in distribution to
is
particular value
such that C'{g'^{v*,0*-{6°°))) converges to a positive and
all
continuous in
E Q. This implies that the
to
some
takes on
as
Then
6
6t
-
k\v))
E Q, where
+
(^1
-
<
1
-
k'{g^\e,v))
<
7(1
-
k'{v))
+
(l
I)
-
|
the constants are given by
j
=
mm{l + 9n-i-E[9\9>9n]/9n-i}.
andj={f3/P)
—
2<n<N
30
(/3//3)
max {(1 +
^„
—
E[^
|
9
<
9n])/9n}
program
Proof. Consider the
max y^pnjun -
Ac(u„)
+
-
Xy{hn)
9nhn
+ Pk{lUn)}
n
= ^pn{Un + PiUn -
V
O-aK)
n
-dnK +
This problem and
ity constraint
any
let
on
Un
But
Un+1
+ PWn+1
is
of 9 within this group, which
constraint
must
U =
1,2,
.
.
.
,
K-
I,
do not incorporate the monotonicbunching
in the following way. If
bunched, then we group these agents under a single index and
pn be the total probability of this group. Likewise
objective function.
We
ioT
this notation allows us to consider
neighboring agents
set of
+
-d^hn+l
notation require some discussion.
its
h.
+ PWn >
what
is
is
let
On represent the conditional average
relevant for the promise-keeping constraint and the
Let 9n be the shock of the highest agent in the group.
rule the highest agent in each group from deviating
The
incentive
and taking the allocation
of the group above him.
Of
bunched agents leads
course, every combination of
to a different program.
The
op-
timal allocation of our problem must solve one of these programs with a strictly monotone
allocation
—since bunching can be characterized by regrouping agents.
acterize solutions to these
The
programs with
monotonicity of the solution.
strict
first-order conditions are
PniX^y'ihn)
-On +
Pr,0k'{vJn)
where, by the envelope condition A
Summing
=
A^„}
/3{fin
the first-order conditions for Wn,
we
=
we
E[k'{g-{v,9))]
The
first-order conditions for
.^
^
n
=
1
+
0„fXn
"
^n-l/^n-l
fin-l)
<
=
k'{v).
XE[y'ih{9))]
Summing up
-
- PX} +
the first-order conditions for hn,
Thus, below we char-
get
1
-
k'{v)
get
=
^k'{v)
imply
.,
9i
/^i
Xy'jh,)
^
XE{y'{hg)]
'
9,
p^
9,
-
9,
31
_
1-A
9,
,
This implies
Pi
Oi
Qi
Using
= -A - - —
k [wi)
we
get
^-^-('-4
>
fc'K)
+
1
/c'(t;)
^1
Similarly, writing the first-order conditions for
(1
_
A)
-
^/f^zi =
Ok
V
n = K, we
(^^)
01
get
_
1
eK
Ok
Pk
Oi
^E[c'(/ie)]
>
+
^1
-A
K
This implies
~
Pk
TK
-(1-A)
K-\
0K-1
Using
P.
ru
we
"~/3
\^U7/-
/V
PfJ-K-l
,
1
7
get
k' [iuk)
<
A-
fK-1.
P
For any n,
1
+
lO/^-
<
zi^„
<
0K--[_
+i
\
Ok
1
_0K-\
0K-\.
<
A;'
eK
1
^A'-l
^A'-l
(u;„)
^1
/3
After rearranging,
P
1
k'{v)
/3
<e
1
^A'
tfi,
1-^'
^1
P
_/5
-A,/''!
7/^
1+
fc'(f)
^A'-l
L
^A'-l
+
we obtain
/5
+
{l-k'{v))
+ l-^>l-k'{g-{e,v))
^1
>e
p[
1
1
+
K-\
0.
7k
^K-\
{l-k\v)) +
P
l
P
D
32
By Lemma
we have
1
=
that \[my^yk'{v)
then using the bounds we obtain that
= ^ <
\imk'ig'"(v,9))
v^v
for all
1,
=
1
limk'{v),
v-*v
Oee.
The
lemma
following
Lemma
We
4.
have
Proof. Consider the
when v goes
describes the behavior of the optimal allocation
g'^{v, 9)
> u and Y\rag'^{y,
6) =
V —^V
u, Ym\g^{v, 9)
V —^v
=
to v.
oo
program
max y^p„{u„ -
Ac(n„)
+
-
\y{hn)
hK + ^K'^n)]
n
n
-9nhn + Un + fiwn > -^„/i„+i
The
first
order condition for u„
<
Since k'{g'^{v,9))
j,
P
we
\\mc'{g'"{v,9))
That
lim5f''(t', ^)
=
is
+ u„+i + /9u'n+i
1-Ac'(u„)
have,ii„
>
u.
=
'^k'{wn).
Y\m.k'{v)
=
1, 2,
.
.
.
,
—^v
or equivalently
=
3,
p
-
/"C
1-Ac'(£f"(t;, 9))
Moreover, since \imk'{v,9)
=
1,
^k'{g'"{v, 9)).
we have
Y\m.g^{y,9)
=
u
00 follows from
m[y'{g\v,9))]
and
Hence
V
=
n
for
=
l~k'{v)
= L
D
Since the derivative k'{v)
is
continuous and strictly decreasing, we can define the transition
function
Q[x,9)
for all
a;
<
^
if
utility is
=
k'\g'^{{kr\x),9))
unbounded below. For any probability
the probability distribution defined by
TQ{yi){A)
for
any Borel
= j
l(c?(.,e)eA}
set A. Define
^^'-
=
n
33
d^{x) dp
{9)
distribution
//,
let Tq{ii)
be
For example, Tq^ni^x)
starting with k'{vo)
is
=
the empirical average of {^'(^<)}"=i
The
x.
distribution by considering the limits of
We
now
are
lemma
following
{7q
O"^^^ ^^^ histories of
length n
establishes the existence of an invariant
„}.
able to prove a proposition that implies the
part Proposition
first
5,
and
describes an algorithm to construct an invariant distribution.
Proposition
i.e.
For each x <
7.
in distribution, to
Proof. For
^
all
I
there exists a subsequence {TQ^^n){^x)} that converges weakly,
an invariant distribution on (—00,
G
l\mQ{x,e)=
lim
next show that the sequence {Tq{5x)}
compact
set
tion T^{5:,)
K^ such
that Tq{5^){K^)
>
—
1
{P/pyk'{vo) ->
0.
is
6)
=
(—00,
:
1)
tight, in that for
k'{vo)
<
-A){-A) +
(1
^ (—00, !)
x
any e
The expected
e, for all n.
simply ¥.^i[k'{vt{d^-^))] with x
is
= ^ <1.
k'{g'"i9,v))
Note that we have a continuous transition function Q{x,
We
under Q.
1)
>
there exists a
value of the distribu-
Recall that E_i[k'{vt{e^'^))]
1.
=
This imphes that
mm{0,k'ivo)}<E^,[k'{vtie'-'))]
<
for all
A>
0.
T5(<5,)(-oo,
-
T^{5^){-cx>,
-A))l
Rearranging,
ra(4)(-co,-A)< '-;;y°-^>
Hence we can
find
^^ >
such that
e
T^((5,)(-oo,-/l,)<^
Define a^ by
1
—
=
Oe
sup
Q{x,6)
xe\A,,i)
Since for
T^{Sx)
all
6
e Q,
-rn-l/
lim k'{v,d)
II—>— 00
= Tq^-' {5,)),
<
|
<
1,
we have
a^
>
0.
In addition, for all
so th8.t
m5,){l - a„
1)
< T^-\5,)i-c^,A,) < -
Since we also have
34
n >
1,
this implies
—
Taking K^
is
[A^, 1
— a.-],
impHes that {7g(5x)}„>i
this
is
tight,
and therefore {7Q('5x)}n>0!
tight.
Tightness imphes that there exists a subsequence Tq
distribution, to
some probability
distribution
on (— cxd,
tt
[S^] that
1).
converges weakly,
Since Q(x, 9)
is
i.e.
in
continuous in x,
then Tq{Tq\5j:)) converges weakly to TQ{'n). But the linearity of Tq impHes that
and since
—
(f){n)
Note that
>
for
oo we must have Tq{tt)
contained in (— oo,|].
5.
We
<
Suppose that lim„^ooSupc"(n)/c'('u)
will
make use
n implies that the support of
5.
We
finally
u_. Indeed, from the
away from
first
The
it
is
prove a
distribution
ip
v.
other one
order conditions,
Then any invariant
oo.
of two possible state variables.
one: v, promised future utility.
The
first
utility attained
is
state variable
is
the natural
by the previous generation
easy to see that these two state variables are
it is
by
1
The
=
This proves the second part of Proposition
necessarily has a support bounded
related
Tq^tt)
tt,
that implies the last part Proposition 5
Lemma
Proof.
D
tt.
any invariant distribution
3
lemma
=
- Xc\u^) =
^k'{v)
existence of an invariant distribution ip*{v) with not mass at misery
existence of an invariant distribution ip*{u_) with no mass at misery.
Let X0
=
uq
+ Pws- Then we
k{v)
=
can rewrite the Bellman equation (21) as
sup E[ub
-
\c{ug)
-
6he
+
'Xy{hg)
subject to
V
xe
—
=
-
E[xg
Ohg
>
+
(5we
ub
35
9he]
xq'
=
—
OHqi
xg
+ Pk{we)]
is
equivalent to the
Hence, given a value x for xq, uq and we are given by the sub-program
—
maxzi
\c{u)
+ pk{vj)
subject to
+ Pw =
u
The
solution
is
given by the
first
X
order condition
=
1-Xc'{u)
^k'{^)
=
Using the implicit function theorem, we can then compute
urf x-u \
,
dx
_lk"{^) +
Xd'{u)
Hence
du
—
<
dx
0<
This
in
turn implies that there exists
M>
maxlufl/
1
such that
—
<
ug\
'
'
M max hg
e
e,0'
Consider the program (C). The
where A
=
k'{v).
first
order condition for h^
is
This implies that
y'{he)>kl-k'Ov)
A
This shows that
lim
y' {hg{v))
=
oo
=r-
and since hg has
is
decreasing in
\irnhi{v)
V—fV
V—>V
9,
lim hg{v)
=
for
36
ah
6^
e
9
=
The
first
order condition(33) implies that
c'{u.)>^c'iue_)-X-\^-l)
which can be rewritten as
L'{u^) + X-\l-^)>c'{ue)
/3
/3
This in turn implies that for
all
9
E
Q
^^]
exp^Mmax/ie max
+ X-\l - i)] >
( L'{u_)
c'{ue)
Since
lim/ie(t;)
=Ofor
all^ e
e
we have
/
M max hg
y
e
lim exp
u--*u
This
in
max
u€luj,ug} C'[U)
turn proves that for u^ high enough,
Hence any invariant distribution
-0*
c"(u)\
——
—
all
1
for all ^
£
the policy functions uq are such that uq
necessarily has
equivalent to saying that any invariant distribution
from
=
)
a'
i^}
support bounded away from
necessarily has a support
Ti.
<
k_.
This
is
bounded away
D
V.
37
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52
1
6 5
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Date Due