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http://www.archive.org/details/optimalsavingsdiOOfarh
NBER WORKING PAPER SERIES
OPTIMAL SAVINGS DISTORTIONS WITH RECURSIVE PREFERENCES
Emmanuel
Farhi
Ivan Weming
Working Paper 13720
http://www.nber.Org/papers/w 1 3720
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge,
MA 02 138
January 2008
We thank George-Marios Angeletos, Narayana Kocherlakota, Andy Atkeson and Martin Schneider
for useful
comments and
insightful discussions. All remaining errors are our
own. The views expressed
herein are those of the author(s) and do not necessarily reflect the views of the National Bureau of
Economic Research.
© 2008 by Emmanuel Farhi and Ivan Weming. All rights reserved. Short sections of text, not to exceed
two paragraphs, may be quoted without explicit permission provided that full credit, including © notice,
is
given to the source.
Optimal Savings Distortions with Recursive Preferences
Farhi and Ivan Weming
Emmanuel
NBER Working Paper No.
13720
January 2008
JEL No. HO
ABSTRACT
This paper derives an intertemporal optimality condition for economies with private information, focusing
on a
class
of recursive preferences.
a risk-free asset market,
Our
we
By comparing
it
to the situation
where agents can
recursive preferences are
homogeneous and
satisfy a
balanced growth condition, while allowing
us to separate the role of risk aversion and intertemporal elasticity of substitution.
quantitative exercises that disentangle the respective roles played
optSimal distortions and the implied welfare gains.
Emmanuel
Farhi
Harvard University
Department of Economics
Littauer Center
Cambridge,
MA 02138
and NBER
efarhi@harvard.edu
Ivan
Weming
Department of Economics
MIT
50 Memorial Drive, E5 1-25 la
02142-1347
Cambridge,
MA
and
NBER
iweming@mit.edu
freely save in
derive the optimal savings distortions necessary for constrained optimality.
We perform some
by these two parameters play
in
With
unfettered access to a risk-free asset, agents can perform the following variation to
consumption plans. At any point
their
by one unit and increase
it
can lower their current consumption
in time, individuals
in all future periods
and contingencies by a constant absolute
amount, equal to the net rate of return. At a market equilibrium, individuals find themselves
at
an optimum within
this class of variations.
The corresponding
optimality condition
is
the
familiar intertemporal Euler equation.
Instead, a planner
may
have on work
must consider the response that any change
effort, if
the latter
is
in the
consumption plan
not fully under her control due to private information.
In general, there exists a class of variations available to the planner on the agent's consumption
such that incentives and work
planner finds the
The
effort are preserved.
optimum within
At the
constrained-efficient allocation, the
this class of variations.
variations available to the planner do not always, or even typically, coincide with those
available to agents in a free market equilibrium. Difference in these sets of variations leads to
optimality conditions that are potentially incompatible. Distortions on savings
required to implement the constrained
The first
optimum with an
point emphasized by this paper
is
may
then be
asset market.
that the particular form that the set of allowable
variations for the planner takes, depends critically on preferences.
We
begin by showing that
there exists a particular class of preferences for which the set of variations available to the
planner actually coincides with that available to agents in a free market.
As a
result, the
constrained efficient allocation requires no distortions on agents' savings.
The
preferences
required for this result feature no income effects on work effort and constant absolute risk
aversion.
This particular result demonstrates that the form of the discrepancy between the
constrained-optimum and the market equilibrium
is
likely to
depend, in general, on preference
assumptions.
Next, we propose a class of homogeneous preferences with a balanced growth condition on
work
effort that delivers
consumption
for the
a simple and intuitive class of variations. The allowable variations on
planner in this case are as follows. At any point in time, the planner can
lower the agent's current consumption and increase
by a constant proportional amount.
it
in all future periods
This type of variation
through the asset market, which opens up the possibility
improvements.
The optimal
is
and contingencies
not available to the agent
for the
planner to find Pareto-
savings distortions are dictated by the difference between the
absolute and proportional variations on consumption available to the agent and planner,
respectively.
Proportional changes in consumption leave incentives unaltered precisely because preferences are homogeneous and satisfy a balanced growth condition.
and
plausibility of these variations
is
We believe that the simplicity
a desirable feature of the preferences we propose. They
lead to simple intuitions, transparent theoretical results and a tractable framework for quantitative analysis.
Within
this class of variations the resulting optimality condition
is
extremely simple.
It
requires that the ratio of current utility to lifetime utility always equal the ratio of current
consumption to the expected present discounted value of
simple optimality condition the Golden Ratio.
It
lifetime consumption.
We
term
this
can also be stated as a Modified Inverse
Euler equation in a form that resembles the standard Inverse Euler equation that was derived
as a necessary condition for optimality for the variations considered in Farhi
and Werning
(2006).
These preferences have three advantages.
First,
they are flexible enough to allow us to
study the respective impact of two crucial parameters: the coefficient of relative risk aversion
and the intertemporal
separability of consumption
Second, although they feature non-
elasticity of substitution.
and work
effort,
these preferences call for no savings distortions
- just as the separable preferences studied in the
in the absence of recurring uncertainty
lit-
erature on the Inverse Euler equation. Third, they lead to a very clean separation result for
welfare gains between an idiosyncratic part and an aggregate part.
Towards the end of the paper, we perform some quantitative welfare exercises that compute
the gains from optimal savings distortions.
We
follow Farhi
and Werning
(2006),
where we
developed a new approach to analyze the welfare gains from distorting savings and moving
away from
letting individuals save freely.
consumption and work
effort,
and
attention to the case of geometric
main goal
and the
is
The method
forgoes a complete solution for both
focuses, instead, entirely
on consumption.
We
random walk consumption and constant work
our
restrict
effort.
Our
to isolate and compare the effects that the intertemporal elasticity of substitution
coefficient of relative risk aversion
size of the intertemporal
wedge and the
Thus, although we borrow from Farhi and Werning
welfare gains from optimal distortions.
(2006), the focus in that paper
have on the
was on the generality
in
terms of the stochastic process
the baseline allocation of consumption. Instead, our focus here
is
on a
for
set of stylized baseline
allocations that allow us to clearly separate the impact of different preferences assumptions.
Welfare gains depend crucially on four factors: the concavity of the production function,
the coefficient of relative risk aversion 7, the intertemporal elasticity of substitution p~^ and
the variance of consumption growth a^
As
in Farhi
.
and Werning (2006), we
find that gains are decreasing in the concavity of the
production function. In partial equilibrium with a linear production function, gains can be
extremely large.
By
contrast, for
hypothesis of a geometric
an endowment economy welfare gains are zero under our
random walk consumption
process. For the intermediate case of a
neoclassical production function, welfare gains are greatly mitigated.
The steady state
of the optimal allocation with saving distortions feature lower capital
and
a higher interest rate then the corresponding steady state of the market equilibrium, where
the precautionary savings motive
is
The
at work.
variance of consumption growth and the
coefficient of relative risk aversion control the strength of this
interest rate increase
and the decrease
motive and hence both the
between the baseline steady state and the
in capital
optimal steady state. The intertemporal elasticity of substitution on the other hand controls
the speed of the transition: the higher p~^, the faster the transition, and the higher the welfare
The
gains.
configuration of these three parameters influences greatly the magnitude of the
welfare gains.
2
Constrained Efficiency
In this section
we
present a two period
Free Savings
vs.
economy
to introduce the basic concepts
set the
Against this background, in the next section we turn to an
stage for the rest of the paper.
horizon economy with recursive preferences.
infinite
Consider a simple economy with two periods
but at the beginning of period
values
and
and p{s)
is
—
1
in the second.
Let
=
0, 1.
There
is
no uncertainty at
5
is
si
=
s.
cq
denote consumption in the
a state Si G
the probability of outcome
and consumes and works
(ci(s), Yi(s))
t
t
realized;
we assume S
is finite,
The agent consumes
t
with
=
#5
in the first period
first
period and
denote consumption and output as a function of the realized state in the second
period.
We
adopt a general specification of preferences and denote the agent's
over allocations by f/(co, Ci(-),Fi(-)). Thus,
As
Yi{-) as inputs.
special
benchmark
U
case,
utility functional
takes a scalar qj and two functions Ci(-) and
one can assume the state
si
determines the
worker's productivity and that the worker has an expected utility function n(co,Ci,ei) over
consumption
in
both periods and work
effort 61(5)
=
Yi{s)/s.
Then
U{co,Ci{-),Yi{-))
=
E[u{co,Ci{s),ei{s))].
Technology
is
linear
Co
for
some
q
>
0.
Here,
R=
1/q
+ q^Ci{s)p{s) <q^ei{s)p{s)
is
the rate of return between periods
(1)
and
1.
—
Free Savings
2.1
First-best.
At
The first-best
allocation simply maximizes utility subject only to technology equation (1).
this allocation the first-order conditions for
consumption are given by
Uco{Co,Ci{-),Yi{-))=
/i,
=
9P(s)/i,
^ci(5)(co,Ci(-),Fi(-))
where
/x is
the multiplier on the resource constraint.
The
first-order conditions for
consumption
can be combined into the following generalized Euler equation:
-^E
g^
^ci(5)(co,ci(-),Fi(-))
f/co(co,Ci(-),Fi(-))
In the expected utility case this equation specializes to the familiar Euler equation
1
= RE
"uci(co,ci(s),ei(s))"
(3)
'
_Uco{co,Ci{s),ei{s))_
Competitive equilibrium with free savings.
tains in a free market
The Euler equation equation
economy where individuals have
For example, suppose that agents
live in
(2) also ob-
access to saving at rate of return R.
an incomplete market
setting, facing the
budget
constraints
Co
Then the
savings
fci
All
<
0,
ci{s)
<
Yi{s)
+
(4a)
+
Rki
first-order conditions for the agent's utility
delivers equation equation (2).^
More
(4b)
S.
maximization problem with respect to
Note that the budgets constraints equation (4a)-
equation (4b) imply the resource constraint equation
A general set-up.
Vs G
(1).
under what conditions does equation
generally,
(2)
hold? Consider
the abstract optimization problem of maximizing utility U{co,ci{-),Yi{-)) subject to
(co,Ca(-),Fi(-))e.F
some constraint
for
with J^
^
=
set J-. This nests as special cases
both the
Tfh defined by the resource constraint equation
planning problem
—and the agent's optimization
if we assume that there are some taxes and transfers that
we impose Ci(s) < T{yi{s),s)-\-Yi{s) + Rki in the second period.
Indeed, this result holds more generally, even
are a function of output or the state, so that
(1)
first-best
market setting
in the free
—with T = Tjra defined by the budget constraints equation (4a)-
equation (4b). Suppose that starting from any aUocation
define simple variations that maintain the allocation in
(co-gA,ci(-)
for all
A in neighborhood of A =
0.
ing (increasing) consumption in the
That
is,
+
y(s)/s,
effort
Property equation
is
(5)
is
maintained for
Note that the same output allocation F(s),
states
all
s.
holds for both the first-best planning problem and the agent's op-
More
assume that
Let r
s
E S
=
(2)
By
telling the truth
is
denoted by a*{s)
=
s for all s
{ci{a{s)),Yi{a{s))) in state
s.
E
E
denote the set of
S.
An
agent using strategy
A
E
s
S and
assign con-
loss in generality,
one can
<t
The
states of the world
truth-telling strategy
e S obtains
is
=
{c'({s),Y^{s))
Incentive-compatibility can be expressed as
The second-best planning problem corresponds
(6).
satisfied.
mapping true
all strategies.
U{co,c,{-),Y,{-))>U{co,cU-),Y,^{-))
and equation
an
satisfied at
optimal.
a{s) denote a reporting strategy for the agent,
S. Let
is
it
the revelation principle, the best the
second period accordingly. Without
in the
E
must be
E S from the agent regarding
to request a report r
into reports r
whenever
generally,
Consider next a private-information setting,
observed only by the agent.
sumption and output
(5)
period, while increasing (decreasing) consumption in
first
Second-best with private information.
planner can do
possible to
T:
optimum, then the generalized Euler condition equation
s is
it is
a feasible allocation can be perturbed by decreas-
timization problem in a free-market setting.
where the state
^
A,ri(-))e.F,
parallel across all states s in the second period.
and hence
Y^-)) e
(cq, ci(-),
second-best
VaeE.
to the case where
optimum maximizes
(6)
T = J-sb defined by equation (1)
utility subject to selecting
an allocation
in Tsb-
In this general context, typically property equation (5) with Tsb
however, provides an example where
Proposition
1.
it
Lei ?7(co,Ci(-), Yi(-))
ond argument. Then property equation
The next
fails.
proposition,
holds.
=
U{cq,Ci{-)
— v{Yi{-),-))
where
U
monotone
in its sec-
(5) holds for J^sb for all feasible allocations {co,Ci{-),Yi{-))
^sbProof.
The
result follows
by noting that incentive compatibility equation
if
c{s)
-
v{Y{s),s)
>
c{r)
-
v{Y{r),s)
Vr, s
E
S,
(6)
holds
if
and only
E
which
is
independent of
cq
and invariant to the operation of exchanging
A
c(-) for c(-) 4-
for
D
any A.
If
it is
property equation
without
(5)
holds for
A
all
(not just in a neighborhood around
loss of generality to allow agents to freely save, in
can allow the agent to
select the value for
of preferences identified
A
in this variation.
of the quasi-linear specification c
As a
(co
first
Y^
v{Y\
s) is
is
-I-
A,Yi(-)).
1/g.
The economic
invariant to A.
is
Proposition
the case for
1
(5) is as follows.
freely,
interpretation
(5)
s,
(5) holds.
Any direct mechanism
an ex-post menu
In each state
menu, {d[,Y^). Property equation
guarantee that this
2.2
follows that, for the class
that there are no income effects on work effort.
An equivalent way of postulating property equation
— 9A,ci(r) + A,yi(r)) essentially offers the agent
this
It
period do not then affect the choice between work effort and earnings.
to the loci of points (ci(-)
on
—
R=
then
the sense that the planner
they do not disturb incentive compatibility and property equation
result,
0)
by the proposition, the planner can allow the agent to save
without distortions, at the technological rate of return
Savings from the
A=
in
each state s equal
the agent selects an optimal point
then amounts to assuming that this optimum
then identifies the largest class of preferences that can
all feasible
allocations.
Distorted Savings
From the
previous subsection,
we know that the
variations that result from free savings
do
not generally preserve incentive compatibility. In this situation, what can we say about the
desirability of free savings?
A
Lagrangian approach.
We
approach
The
first is
this question in
two complementary ways.
to attach Lagrange multiplier ii{a)
on the incentive
constraints equation (6), leading to an optimality condition that includes the effect that
may have on
the incentive constraints:
-E/^('^)(-5^^o(co,Ci(a(-)),ri(a(-)))
•
if all
+
5]t/,,(,)(co,Ci(a(-)),Fi(a(-))))
the incentive constraints are slack, so that
this expression boils
down
jj,{a)
to the Euler equation equation (2).
tion equation (2) will typically not hold. Indeed,
if
are nonzero, then one can sign the intertemporal
=
for all
<t
€ S, then
Otherwise, the Euler equa-
one signs the term —qUco{co, Ci(a(-)),
'^ses^ci{s){co,c-i{a{-)),Yi{a{-))) for different strategies
a and
=0.
J
s&s
\
o-ei;
Note that
A
yi(cr(-)))-|-
characterizes which multipliers
wedge required
in the Euler equation.
Another
Feasible variations.
line of attack
to find a different variation, that does pre-
is
serve incentive compatibility, without changing
work
This leads to an intertemporal
effort.
optimality condition that does not involve Lagrange multipliers.
optimality condition with the Euler equation equation
The
idea
is
this
(2).
to find a variation function 5(A,s) on consumption in the second period that
depends on the realized state
s so that:
(co
in a
One can then compare
neighborhood of
A=
0.
+ A,ci(-) +
(5(A,-),ri(-))e^,
(7)
At an optimum we must then have that
t/co(co, ci(-), ei(-))
+ J2 ^ci(«)(^' ci(-), ei(-)) ^^(0, s) =
(8)
•
s&S
For example, with expected
feasible
is
utility
and u(co,ci,ei)
=
u{co,ci)
—
h{ei) a variation that
is
to set 5(A,s) so that
fi{co
where ^(A)
is
+
A,ci{s)
+ 5{A,s))=u{co,Ciis))+A{A)
Vs e 5
(9)
such that
^(A + <5(A,5))p(5) =
0.
(10)
ses
This variation
shifts utility in
a parallel way across states s E S.
It
preserves incentive
compatibility because these parallel shifts cancel each other out on both sides of equation
At an optimum A'{0)
=
so that
uj^o^
d
dA
It
(6).
'
'
'
(11)
Ue,(co,Ci(5))
then follows that
1
=
E ?#^pw.
(12)
^«ci(Co,Ci(s))
which
is
known
as the Inverse Euler equation.
compatible with the Euler equation equation
By
(3),
Jensen's inequality, this condition
except in the special case where there
is inis
no
uncertainty in the marginal rate of substitution ratio Uco{cq, Ci{s))/uci{co-, Ci(s))- Without uncertainty the optimality of no intertemporal distortions follows from Atkinson-Stiglitz's (1976)
result
on uniform taxation, which requires separability between consumption and
assumed
in this case.
effort, as
Logarithmic balanced-growth preferences.
esting special case with several advantages
u(co,Ci)
=
+
log(co)
is
Within
this class of preferences,
an
inter-
the logarithmic balanced growth specification
/31og(ci). In this case the variations induce parallel multiplicative shifts
over second-period consumption:
S{A,s)
for
If
some 5(A).
Intuitively, incentives are
consumption
is
=
6iA)c,is),
provided by proportional rewards and punishments.
down by a constant
scaled up or
(13)
it
does not change the incentives for work
effort.
In this case, unlike the preference class described in Proposition
work
effort are nonzero.
1,
income
effects for
Proportional variations are feasible precisely because of the balanced
growth condition, that implies that income and substitution exactly cancel each other.
This logarithmic case seems economically appealing, because of the primitives and the
simple proportional variations
expected
utility case
permits.
it
condition.
u(c)
It
is
consumption, as
=
c^~"/(l
—
=
u{co)
+ Pu{ci)h{ei)
This class of preferences also
a).
easily verified that
in
simple generalization of this case,
is
to the
where
u(co, Ci, ei)
and where
One
(14)
satisfies
a balanced growth
once again the feasible variations are proportional in
equation (13).
In the next section
we extend
this class to
an
infinite
horizon economy. Preferences that
lead to the feasibiUty of proportional variations turn out to be very tractable. In particular,
they lead to a very simple optimality condition. Within a class of baseline allocations, the
optimum
3
is
easily identified
and
its
welfare improvements quantified.
Recursive Preferences
We now
turn to an infinite horizon and introduce a class of recursive preferences that are
homogeneous
in the
consumption process and separate
elasticity of substitution as in (Epstein
assumed
and
Zin, 1989).
risk aversion
from the intertemporal
Consumption and work-effort are not
to be separable, but satisfy a balanced-growth condition.
For this class of preferences, we provide simple variations on consumption that maintain
incentive compatibility.
affect incentives.
The
variations involve proportional shifts in consumption that do not
Both the homogeneity and the balanced-growth
are crucial for this result.
specification
on preferences
Based on these variations we derive the intertemporal optimaUty condition
the section.
this way,
The condition
is
shown
at the
end of
to be incompatible with allowing agents to freely save. In
an intertemporal wedge on savings
form of distortions on savings are required
some
is
present at the optimal allocation. Thus,
in
any tax implementation of the optimum. In
the next section we explore the welfare gains from adhering to this condition for some simple
cases.
Our
preferences do not satisfy the separability condition required for Atkinson-Stiglitz's
uniform taxation theorem. Despite
this, it is
optimal in the absence of uncertainty to set the
intertemporal distortions to zero. Thus, for these preferences, optimal distortions in savings
arise
from ongoing idiosyncratic uncertainty, just as
in the additively separable expected-utility
case that leads to the Inverse Euler condition.
Moral Hazard
3.1
We
build on the following simple static moral-hazard model. At the beginning of the period,
the agent
s
is
first
exerts effort a, which
is
not observable by the planner.
The
state of nature
then realized from the distribution P{s\a). The planner observes s and gives the agent
consumption
c{s).
The
agent's expected utility
is
given by
E[U{c{s)h{a))\a].
We
suppose the agent's
utility
U{c)
dard balanced- growth assumption,
is
a power function. This specification
for
which income and substitution
equivalent reformulation of the agent's objective
satisfies
the stan-
effects cancel out.
An
is
U{Ch{a))
where
C=
CE[c{s)\a]
=
U-^{E[U{c{s))\a])
represents the certainty-equivalent obtained from the
random consumption
For our dynamic setting, we proceed analogously.
chooses effort at-i, then the state
sumption
c(5*). Effort affects
with
=
/i(0)
1.
5* is
realized
At the
St
and lowers
Preferences are given by the recursion
=
Cis*)h{a{s'-'))
10
start of period
and observed and the planner
the distribution of state
Va{s'-')
c{s).
utility
t
the worker
allocates con-
by a factor
h{at)
<
1
where
C{s')
= CE
[W{c{s')Ms'))
a{s'-'),s'-']
I
(15)
,
represents lifetime-certainty-equivalent consumption, with
CE =
is
(16)
the certainty equivalent function and.
=
W{c,v)
is
R-^¥.R
u-\{l-(3)u{c)
+ (5u{v))
(17)
a time aggregator, mapping current consumption and future utility into a constant-consumption
equivalent.
With
this representation of preferences,
static setting.
By
in the following,
c
=
one can easily see the analogy with the simple
a change of variables, however, the same preferences can be represented
more convenient, way. For any given
effort
plan a
=
{a(s*)},
an allocation
{c(s*)} implies a process for lifetime utility {va{s*)} that solves
Va{s')
=
W{c{s')Xnh{a{s'))va{s'+')\a{s'),s'])
Incentive compatibility of
c,
\/t,sK
(18)
v and a* requires a* to maximize initial lifetime utility
Va'iso)
>
Vo.
Va{So)
(19)
Since preferences are recursive, this implies that a* maximizes continuation utility after any
history
Va'{s')>Va{s*)
Otherwise, a plan that follows a* up to
s*
and
That
after s*
would be preferable to
a*.
\/a,t,s\
(20)
and then switches to the actions prescribed by a
is.
at
Bellman's Principle of Optimality applies to
the agent's dynamic program.
We now
bility.
consider variations in the consumption process that maintain incentive compati-
After history
s'^
the consumption sequence
not affect incentives. At
affected in period r
and
s'^
we
shift
is
just shifted proportionally,
and
this
does
consumption to compensate, so that incentives are not
earlier periods.
The key property we use
is
homogeneity of W(c,
v')
and of CE.
Proposition
2.
Assume u{x) = x^'P/ {I — p) and R{x)
11
=
x^~'' / {1
— j)
with p,j
>0. Suppose
that
c,
V and a* satisfy conditions (18) and (19). Fix a history
A c{s'^)
c{s')
=
for
there exists a
A
=
fort>
{ A'c(s*)
.
Consider the variation:
s^
T and
s* y- s'^
otherwise
c(5*)
Then for any A'
s*
s'^
such that
v and a* satisfy conditions (18) and (19).
c,
Proof. Let v be such that
Va{s^)
=
A'va{s^)
for
i
> T and
5* >-
s'^
Va{s*)
=
Va{s*)
foT
t
>
s*
^
s'^
so that condition equation (18) with c
is
met
T and
for all 5*
with
t
>
t with
s*
^
s'^.
Now
set
A
so
that
va'isn
so that Va'
at
s'^
{s'^)
=
=
Va' {s'^).
W {Ac{s'),A'CE[h{a*{s-'))v,,{s^-^')
Using recursion equation
\
a*{s^),s^])
(18), the inequality
.
equation (20) evaluated
implies
CE
[h{a*{s^))va'{s^^')
> CE
a*{s^),s^]
I
[h{a{s^))va{s^+')
\
a{s^),s^]
,
so that
Va'is^)
=
W{Acis'),A'CE[h{a*{s^))va^{s^^')
a*{sn,s^])
I
>W {Ac{s'),A'CE[hia{s^))va{s^^')
\
a{sn,s^])
=
i).(s^).
Hence, we have that
Va{s'')
a*
is
<
Va'is'')
=
Va'{s'')
for all a.
optimal from period r onward and delivers the same continuation
utility as previously.
For any plan a define an alternative plan a that switches to a* from period r onward:
a(s*)
=
a(5*) for
i
<
r and a{s^)
Va{So)
That
is
is,
<
=
>
t.
^a(-So)
<
a*(s*) for
Va{So)
=
a dominates a and yields the same
dominated by the recommended action
a*
t
The
result
^a* (^o)
utility as
=
above implies that
Va'{so)-
without the variation, which in turn
which also yields the same
variation. This estabUshes that a* remains incentive compatible.
12
(21)
utility as after the
D
A
Private Information:
3.2
Here we build on Mirrlees'
Dynamic Mirrleesian Economy
static private information model.
the agent privately observes productivity
consumption
gives the agent
At the beginning of the period,
The agent then makes a
0.
c{r) as function of the report.
The
report r and the planner
agent's expected utility
is
¥.[U{c{r)h{r,e))\a].
where r
=
a{6)
is
We
the agent's reporting strategy.
a power function. This specification
which income and substitution
suppose the agent's
utility
U{c)
is
the standard balanced-growth assumption, for
satisfies
effects cancel out.
For our djoiamic setting, we assume the following structure of uncertainty. At the beginning of the period a state
9t is
a
it
realized
finite
and publicly observed by the agent and planner. Then
of values. After observing the shock
to the planner.
by
realized
and observed only by the agent. To simplify we assume that
number
histories
St is
z*
=
We
collect the variables
Sj
and
the agent makes a report
dt
observed by the planner by
zt
=
r^
{st,rt)
9t
take on
regarding
and
their
(s*,r*).
For any reporting strategy a
v,{z\9')
where
Zt+i
We
let
=
=
W {c{z')XE[h{z'^\e,^,)v,{z'-^\e'+')
a* denote the truth- telling strategy
of the next result
Proposition
(c, h,
3.
is
Assume u{x) =
v) satisfying (22)
and
cr^{z*
x^~''/{l
—
=
{ A'c(2*)
c{z*)
Then for any A'
there exists a
on s\ the realization of
We
9t
p)
,
9^)
=
is
9t.
Incentive compatibihty requires
Va.
and R{x)
(23), fix a history z^
c(2*)
=
(22)
,
(23)
in the appendix.
A c{z-')
u{x)
at+,,z\e'])
(5^+1,0-^+1(2;*, 61*+^)).
Va'{zo,9o)>v,{zo^9o)
The proof
\
A
such that
for
=
x^~"'/{l
and consider
2*
=
—
7).
For any
allocation
the following variation:
r
fort>T
and
z* >- z''
otherwise
(c, h,
v) satisfy (22)
and (23)
independent and identically distributed; or
if:
(h)
(a) Conditional
p
—
I
so that
\ogx.
do not impose restrictions on the stochastic process
13
for the observable state St-
Re-
garding the unobservable shock, the requirement in part
and can,
for productivity,
requirement does ensure
in particular,
is
(a)
does not restrict the process
accommodate any degree
What
of persistence.
that the states that affect the evolution of shocks are observable,
that there are no hidden states. Although this implies that the observable state
the sense that Pr(s*+",6I*+"|5*,6'*)
ficient statistic for (s*,6'*), in
allocations typically
depend on the history
may
False past reports
vant.
this
9^
=
s* is
Pr(s*+",6'*+"|s*), optimal
In this way, the history of reports r*
.
a suf-
is rele-
then affect the allocation the agent receives, but do not affect
the planner's capacity to predict the agent's future productivity. This tractability allows us
to find variations that maintain incentive compatibility.
=
In the logarithmic case, p
0,
the crucial property
H/(Ac,
=
Hence, setting A^~^(A')^
reporting strategy.
As a
1 in
result,
that
Ai-^(A')^H/(c,t;').
the variations does not affect the utility delivered by any
no assumption on the structure of uncertainty
is
required.
The Intertemporal Optimality Condition: The Golden Ratio
or The Modified Inverse Euler Equation
3.3
Let us say that an allocation
E Yl'tLo 9*Q
any
AV) =
is
and
is
efficient if
it
minimizes the present value of consumption
delivers a given lifetime utility level in
efficient allocation
an incentive compatible way. Then
cannot be improved by the variations above. That
is,
these variations
cannot reduce the discounted value of consumption.
Fix a node
at all
s'^
.
Increase consumption at s^ proportionally by A, and increase consumption
nodes that follow
it,
s* >- s'^,
proportionally by A'. This variation
propositions above. Indexing the variation by A' and solving for
constant,
we
permitted by the
A = 5(A')
that keeps utility
consider the minimization
min
j
(5(A')c(r )
first-order necessary
and
+
A'
^
q'c{s') Pr[5*|a*, J"]
t>T,S^
\
The
is
J
sufficient condition for optimality is
ct
(1-/3Mq)
Y.7=o'i'^A(^+s]
u{vt)
Thus, optimality requires the ratio of current to lifetime
equated to the ratio of current consumption with
14
(24)
.
j
its
simply
utility
(25)
(1
—
(3)u{cf,) / u{vt)
expected present value
to be
q/ X^^q 9*^Et['^+s]-
Rearranging, the ratio of current consumption and utility must be equated to the ratio of the
present value of consumption with lifetime utility:
Ct
(26)
{1-I3)u{ct)
u{vt)
Both conditions formalize the optimality of a form of consumption smoothing.
We
call
them
the Golden Ratio conditions.
The next
for
result reexpresses the optimality condition
above
in
a way that
is
more
suitable
comparison with the optimality condition - the Euler equation - that results when agents
can save freely at the interest rate q~^
We
.
call this
condition the Modified Inverse Euler
equation.
Proposition
4. Define
ht+iVt+i
Xt+l
(27)
CEt[ht+iVt+iY
At the optimum in (24) the following condition
(a)
holds:
i-p u'{ct)
%.
1
u'{(h+\)
/?
(h) If agents
(28)
'-t+i
can borrow and save freely at the interest rate q
then the allocation must satisfy
^,
the following Euler equation:
x:
'^'
(29)
u'[c,)
Savings will generally be distorted at the optimal allocation, since the Modified Inverse
Euler equation and the Euler equation are incompatible. Thus, in any implementation of the
planner's optimum, agents cannot be allowed to borrow and save freely at the interest rate
l/q.
Suppose that the optimality condition equation
T by solving
for the factor (1
replaced with (1
—
—
(28) holds. Define the intertemporal
r) required so that the Euler equation (29) holds
wedge
when \/q
is
r)/g:
1
-
T
=
E;
„P-7 u'{ct+\
't+\
%
1-p
(30)
u'{ct+x)
u'ict)
so that
r
= —Gov
u'{ct.+1)
u'{c,)
1-p
-X.*+i
u'ict)
Importantly, the intertemporal wedge r
1-7
^t+\
is
(31)
xl-fu'{c,^,)
zero whenever there
is
no uncertainty. For the case
of certainty, Atkinson-Stiglitz's uniform-taxation result requires preferences to be separable
15
between consumption and
However, in our recursive specification preferences are not
leisure.
Interestingly, despite this, the absence of resolution of uncertainty
separable.
between two
periods implies that there should be no intertemporal distortion on savings there.
In other
words, although the separibility conditions required by Atkinson-Stiglitz are violated, their
uniform commodity taxation result holds under certainty with our preferences. Thus, optimal
distortions can be entirely attributed to ongoing idiosyncratic uncertainty, just as in the
additively separable expected-utility case that leads to the Inverse Euler equation (Golosov
et al, 2003).
Note that
on savings
is
=
7
if
positive.
>
0.
We
shall
0,
guaranteeing that the intertemporal distortion
Another interesting case
=
baseline allocation, so that q+i
r
>
one gets that r
1
£t+iCf
when q
is
is
a geometric random walk at the
then follows that
It
study this case in more detail
in the
Vf is
proportional to q, and
next section.
Constant Absolute Risk Aversion Preferences
3.4
In this subsection,
risk aversion the
we show that
for a particular class of preferences
optimal distortion on savings
convenient specification of preferences
is
zero.
with constant absolute
In a static moral-hazard setting, a
is
E[Uic-h{a))\a]
where U{x)
= — e"'*^
is
(32)
exponential. Equivalently, one can express ex ante utility as
CE[c-h{a)\a]
In our
R{x)
—
dynamic
—e'''^
setting,
we
v')
—
u~^ {{1
5.
=
W {c{s')XEK{s'+') - hia{s'))
— P)u{c) + Pu{v')) and CE = R~^ER.
inequalities (19) as before.
Proposition
generalize this specification as follows.
Let u{x)
= — e~^^
and
and consider the recursion
v^{s')
where W{c,
(33)
Assume
The next proposition
u{x)
=
is
—e~^^ and R{x)
16
proved
\
a{s'),s'])
Incentive compatibility requires
in the
= — e~"*'^.
(34)
appendix.
Suppose we have
c,
v and a*
.
satisfying conditions (18)
and
(19).
Fix a history
c(s^)
=
c(5*)
{ c{s^)
+A
for
+
fort
A'
A
there exists a
As above, we say that an
Consider the variation:
.
=
s*
>
5^
T and
s*
s^
>>-
otherwise
c(s*)
Then for any A'
s'^
such that
c,
v
and
a* satisfy conditions (18)
and
(19).
is efficient if it
minimizes the present value of con-
Y,q'c{s')Y>x[s'\a*]
(35)
allocation
sumption
required to deliver a given lifetime utihty level in an incentive compatible way.
efficient allocation
cannot be improved by the variations above.
That
is,
Then any
these variations
cannot reduce the net present value of consumption.
Indexing the variation at any node by A' and solving for
can write the minimization subproblem as
freely at a
market
Proposition
and save
6.
The optimum
in (24) corresponds to the
u'[ct)
tion
CARA
the worker could save and borrow
=
economy where agents can borrow
The following Euler equation
^u'(CE(ct+i
-
holds:
(36)
ht)).
coincide.
Welfare Gains; Quantitative Explorations
In this section,
Section
3.
The
we
investigate the welfare gains from the optimal savings distortions derived in
analysis proceeds along the lines of Farhi
case where the baseline allocation features a geometric
work
and
preferences under consideration, the constrained-optimality condi-
and the Euler equation
4
if
we
interest rate q~^
freely at the interest rate q~^.
Hence, for the
that keeps utility constant
in (24). In this case, the first-order necessary
with the condition obtained
sufficient condition coincides
A
effort
is
constant.
The
settings.
Assumption
The baseline allocation {ct.ht]
geometric random walk Ct+i
(2006).
We focus
on the
random walk consumption process while
analysis in this section covers both to the private-information
and moral-hazard
1.
and Werning
=
is
such that
ht
=
h
is
constant and
Ct
is
a
Qet+i with St+i identically and independently distributed over
time.
17
—
Partial equilibrium
4.1
Let us
first
assume that there
R=
with a gross rate of return
The
if
the baseline allocation
random
also a pure geometric
a pure geometric
random
walk.
Suppose that Assumption
7.
is
then the cost minimizing allocation attainable through our variations
ht is constant,
Proposition
is
q~^.
following proposition shows that
walk and
is
a linear technology to transfer resources from period to period
is
obtained by multiplying {q}
1
holds.
Then
the cost minimizing allocation
{q}
by a deterministic drift g~^:
ct
= ag~%
with
g
=
'
^-')
(qp-'E[s] (E [c^"^])
and a
1
where 3
=
is
such that
q
attainable from the baseline allocation through our varia-
Ct
also follows a geometric
random walk, but with a
stead of E[£:] for the baseline allocation. This
condition
— a necessary and
when
h^
=
Cj.
Note that
>
1
and vice
drift
/3
p =
dh}~f.
ensures that the constrained-optimality
within our class of variations
factor.
It is
This
roles in this
compounded with
effect is
also useful to note that
if
^
>
1.
versa.
Increasing g while maintaining the value of qE[e\
effective discount factor
/3
different drift ^~^E[£:] in-
and h}~P play exactly similar
his constant, h acts as a discount
to produce an effective discount factor
then a
new
sufficient condition for optimality
holds at the optimal allocation
formula:
qg-p^e]
pih}'^.
Hence the optimal allocation
tions
-
(3.
is
exactly equivalent to decreasing the
In other words, the higher g, the lower the effective discount factor
that makes the constrained-optimality condition hold.
Note
also that given qE\£\
elasticity of substitution
discount factor
/3
and
parameter
g,
p.
the intercept
The
a depends only on the intertemporal
risk aversion
parameter 7 only
shifts
the effective
required for the constrained-optimality condition to hold.
Economists are used to thinking of the discount factor as a primitive of the model, and
as the equilibrium interest rate as an outcome. However, contrary to interest rates, discount
factors are not directly observable. In fact,
comes from equilibrium values of
Q,
we
most of the evidence concerning discount factors
interest rates.
Therefore, in the formula for the intercept
prefer to think of the equilibrium interest rate q as the primitive
effective discoimt factor
(3
and to solve
for the
that makes the constrained-optimality condition hold given g and
qne\.
18
We
Intertemporal wedge.
can compute the optimal wedge
in closed form:
— Cov{£,£~^)
^
^
Note that the wedge
and
is
is
E[e]E[£-T]
always positive.
by
entirely determined
that the origin of the wedge
7, that
is
magnitude
Its
example
by the agent's attitude toward
is
independent of p
is
This highlights
risk.
the combination of two factors: the riskiness of tomorrow's
consumption from today's perspective and the agent's
would be no reason to
in this
distort savings
Absent shocks, there
risk aversion.
and the Euler equation would
hold.
Similarly,
if
the
agent were risk neutral, there would be no reason to distort savings and the wedge would also
be
zero.
We
can re-express the wedge using the formalism of cumulants:
generating function of log
(e)
of log(e)
is
moments, as we see from the
notation
is
standard, with
be the moment
:
m{9)
The nth cumulant
m
let
=
given by
=
«;„
=
^^(0).
log E[s']
Cumulants are
=
^2, «3
=
denoting the conditional
mean
of log(£)
first four:
/xi
=
log £;[exp(^ log (e))]
ki
^1, K2
A*3)
1^4
closely related to
=
fJ-4
—
and ^„,
The
3(//2)^-
n >
for
1
,
denoting the nth central conditional moment.
Using this notation we derive a formula that
ties
the wedge to the higher order
moments
or cumulants of log(e):
00
- log(l -
r)
=
m(l)
+ m(-7) - m(l -
7)
= J^ «„/n!
(1
+
(-7)"
-
(1
-
7)")
n=2
In the lognormal case, which
n >
3
log(e):
we explore below, the higher cumulants
and we obtain a closed form
-log(l
-r) =
for the
«;„
of log(£:) are zero for
wedge which depends only on the variance
a"^
of
7a2.
Outside of the lognormal case, higher cumulants are non-zero and higher moments of the
distribution of consumption growth rates affect the wedge. For example,
impact of skewness
K3.
The contribution
negative skewness - K3 - decreases the wedge
Welfare gains.
computed
The
costs k
term to the wedge
of this
if
7
<
1
is
we can analyze the
given by K3
and increases the wedge
Hence
"''
^
if
7
>
1.
and k of the baseline and the optimal allocations are
to be
-
k
=
ac
l-qg-'ne\
19
easily
and
k
^—
=
-
1
qE[e]
Combining these two expressions, we can derive the
cost allowed
relative reduction in expected discounted
by our variations.
Proposition
8.
Suppose that Assumption
holds.
1
Then
the relative expected discounted cost
reduction achieved by going from the baseline allocation to the optimal allocation
k
~k
By
is
_ fl-qg-PE[s]\^^ fl-qg-'E[e]y-^^
1-9E[5]
V
;
V
1-qE[£]
homogeneity, the ratio of the cost of the optimal allocation to the cost of the baseline
does not depend on the current
or in other words, given gE[e], ^
the variations.
It is
consumption
level of
is
c.
Given the cost of the baseline allocation,
a sufficient statistic for the welfare gains attainable through
some comparative
therefore instructive to perform
statics
with respect to
9-
Given qE[e] and
g,
the relative expected cost reduction depends only on the intertemporal
elasticity of substitution
parameter
p.
that given g and gE[£], the intercept
=
At ^
1,
the reduction in cost
This
a direct consequence of the fact noted above
is
a does not depend on
is 0.
This
the risk aversion parameter.
because in this case, the constrained-optimality
is
condition holds at the baseline allocation. Moreover, a Taylor expansion around g
that the cost reduction
When
zero at the
first
k
1
^
order in g and increasing in g
qE[e]
is like
taking the effective discount factor to
0.
reveals
:
Yir^Ekl-
Taking g to
In that case, the optimal allocation for
1 is
In the limit where p goes to
Ct
=
1,
we
for
i
>
1
and
cq
=
cq.
get
,= -E[e]and. = ^3r^,
which
I
,,2
,
g goes to infinity on the other hand, the cost reduction goes to
infinity
A_i =
is
=
is
.-.
exactly the expression derived in Farhi and Werning (2006).
Euler at the baseline.
in the interesting case
Given the importance of ^f we now investigate
,
where the Euler equation holds
20
its
main determinants
at the baseline allocation.
That the
.
Euler equation holds at the basehne means that
-7
which can be re-expressed as
The
fiq-^h'-P^. [£-^] (E [e^-^]
effective discount factor
/3,
=
/3
/3
Knowing
^
7-£
=
1
=
(38)
can then be determined:
^
(5h}
)
5(E[£^])-^(E[ei-^])^
the sufficient statistic g for the welfare gains in formula (37) can be derived
using the formula in Proposition
(7).
Proposition
holds
9. //
Assumption
1
and
the Euler equation holds at the baseline allocation,
then
J
g=lE[e]E[e-^]{E[s'-''])-y.
The optimal change
wedge
in drift
g
is
reflects the strength of the
for inceased
more agents
r:
=
g
{1
—
t)~^^'^.
The
precautionary savings motive: the higher the wedge, the
The higher the intertemporal
larger the gains from frontloading consumption.
substitution, the
wedge
positively related to the
elasticity of
are willing to accept reductions in consumption in the future
consumption today. The two
combined determine the optimal change
effects
in
drift g.
When
e
is
~
lognormally distributed loge
N{fi,a'^), then the
wedge
from the baseline allocation g and the welfare gains can be computed
and the variance
CoroUciry
1.
cr^
of
in
r,
the change in drift
terms of the mean
/x
consumption growth:
Suppose that £
is
lognormally distributed loge
~
N{^,al), then r and g are
given by
T
=
I
—
r
^. ,^
E[e]E [e-f]
,
=
—
1
exp
^
L
— 7(T.ej ~
70",
i
'
e
and
=
exp
(
As we already
discussed, the
magnitude of the shocks
cr^.
When shocks are lognormal,
wedge
is
~
—a^
\P
I
V
—
1 H
<T^.
P
increasing in the degree of risk aversion 7
Moreover, 7 and
a'^
affect the
wedge
in a
in the
complementary way.
the formula takes the remarkably simple form r
21
and
= 1 —exp [— 7(t^]
.
The
crucial
parameter g
associated with
is
-cr^.
The higher the variance
and
of the shocks,
the higher risk aversion, the higher the required change in drift g between the baseline and
p~\ the
the optimum. Similarly, the higher the intertemporal elasticity of substitution
higher
9-
Intuitively, this
different dates
can be seen by taking the limit as p goes to
become
perfect substitutes.
Note however
the limit,
since the required change in drift g goes to
)~^
that in this case, the intercept a converges to 1 — g (E [£""'•']
E [e^~^].
when p goes
Intuitively,
it is
to 0,
best to deliver
lim -
is
it
optimal to front-load consumption more and more.
The
is
non
trivial.
~
+ -~
1 -
+ (l-27)f
-
l-q 3Xp
'p
+
1
Indeed,
we
have:
7cr;
qe^'
i_
gains are increasing in the intertemporal elasticity of substitution p~^: intuitively, as
consumption
at different dates
become more
substitutable,
it
becomes
easier to
compensate
the agent for a decrease in the drift in consumption in order to lower his exposure to
fact,
we can
k
this
risk.
In
derive a simple formula for small a^:
'-1 +
From
In
in the first period so that agents are entirely
cost reduction
1-- qexp p
=
0,
consumption
all
shielded from consumption risk.
The
so that consumption at
The Euler equation and the optimality condition
are incompatible in the limit where p goes to
infinity.
0,
formula
it
77^^-4
p
apparent that at the
is
first
(39)
'
qei^)^
(1
relevant order, risk aversion
and the
in-
2
tertemporal elasticity of substitution enter the formula for the gains only through ^.
Quantitative exploration.
Figure
1
and 2 plot the reciprocal of the
relative cost reduction
using equation equation (37) as a measure of the relative welfare gains as a function of a|.
The
figures use
0.007.
The
an
emxpirically relevant range for
value of gE[e]
In figure
1,
cr^
which
is
taken to vary between
and
set to 0.97.
is
the intertemporal elasticity of substitution p~^
is
set to 1
and the
different
curves correspond to different values of the relative risk aversion coefficient 7 ranging from
1
to 3 in increments of 0.5.
The
gains are increasing in 7;
10%
Increasing 7 by
is
exactly
equivalent to increasing a^ by 10%.
In Figure 2, the relative risk aversion coefficient
7
is
set to 1,
and the
different curves
correspond to different values of the intertemporal elasticity of substitution p"^ ranging from
0.5 to 1 in increments of 0.1.
equivalent to increasing
a'^
The
gains are increasing in
by 5%.
22
p~\
Increasing p~^ by
10%
is
roughly
Figure
and
1:
Welfare gains as a function of a^. Baseline consumption
The Euler equation
a ranging from 1 to 3.
ht is constant.
values of
Two
lessons
holds.
emerge from our simple
The
is
a geometric random walk
correspond to different
different curves
First, welfare gains
exercise.
range from small to
potentially large. Second, they depend a lot on three parameters of the model: 7, p
and
The
play an
coefficient of relative risk aversion
7 and the variance of consumption growth
cr^
a'^.
especially important role over the range consistent with the available empirical evidence con-
cerning these two parameters.
but
its
The intertemporal
influence over the empirically relevant range
because the range for this parameter
than 7 and a^ as can be seen from
4.2
Up
to
elasticity of substitution p^^
is
is
somewhat
less
is
important,
dramatic. This
is
both
smaller and because p~^ enters with a smaller power
(39).
General equilibrium
now we have
of the results
restricted the analysis to partial equilibrium. Alternatively, one can think
we have derived
of return to capital. In Farhi
so far as applying to
an economy facing some given constant rate
and Werning (2006), we argue that neglecting general equiHbrium
effects magnifies the welfare gains
from reforming the consumption allocation. Here we explore
the joint influence of risk aversion and the intertemporal elasticity of substitution on general
equilibrium welfare gains.
Planning problem.
problem,
it
allocations
is
q
Consider a baseline allocation {ct,ht}.
useful to introduce the following notation:
let
In order to setup the planning
T{{ct,ht},A_i) be the
attainable through our variations from the baseline allocation {A_iCi,
that the shifted allocation {A-iCt,ht}
utility increased
is
incentive compatible
by a multiplicative factor A_i to the agent.
23
and
/i^}.
set of
Note
delivers a value lifetime
In general equilibrium, the
Figure
2:
Welfare gains as a function of
walk and h-t is constant. The
from 0.5 to 0.9.
cr?
when
baseline consumption is a geometric random
correspond to different values of p~^ ranging
different curves
planning problem can be set-up as
W{Ko) =
max
vq
(40)
{ct,Kt^i}
subject to
ht {{1
Vt
i-p
r,i.-Tni-'
- ,3)cl-' +
13
for f
{E[vl--;'])
=
0,1,..
{ci}GT({Q,M,A_i),
Kt+i
+ E[q] <
F{Kt, Nt)
+
(1
-
S)Kt for
t
=
0, 1,
...
Ko = Ko.
Necessary and
c?
sufficient conditions for this
course,
allocation
Et
FK{Kt.Nt)
+ {l-5)
and A_i
is
the
maximand
in
(
=
E[ct],
where the superscript
i
-^Kt+i
for i
=
0, 1,
[Uviri])'-'
we have W{Kq) — A-it-F where
W
is
the welfare achieved at the baseUne
40).
Note that we can always decompose q
Cf
are:
Vt+l
=
(ih'-"
Of
problem
=
(^^Ct
with the property that
stands for idiosyncratic.
deterministic parallel shifts in consumption,
we have
24
that
E[cJ]
=
1
and
Since our variations allow for
{q} G T({ct,
/?t}.
A_i)
for
some
A_i
if
and only
The
if
{5j}
G T({cJ,
ht},
A_i).
analysis of this planning problem
tackled in
is
full
generality in Farhi
where we also explore non geometric random walk baseline
(2006),
and Werning
allocations:
we provide
cases where (40) can be separated into two different planning problems, one involving only
the idiosyncratic part of the allocation
instead,
we focus on the
cj
and the other only the aggregate part
Ct-
special case were the baseline allocation features geometric
walk consumption with constant
Here
random
ht-
Geometric random walk with constant hf
Suppose that the baseline allocation features
geometric random walk consumption with constant ht and constant aggregate consumption:
=
Ct+i
where
St+i
Ct£t+i
and
ht
=
h,
independently and identically distributed across agents and time and with
is
E[£i+i] =1. In other words.
Assumption
holds and E[e] =1.
1
Define
l3e
Proposition 10. Suppose
=
f3{E [e^-^])^ and
that
= Ctc\ where Ct and A_i
CRRA preferences:
Assumption
4 = h'-^P,.
holds and E[e] =1.
1
The solution
to
(40)
is
are the solutions of the standard neoclassical growth model with
Ct
1
-
= (l-4)
^
P
max
^ {Ct,Kt+i}
TPl^
(41)
subject to
Kt+i
+ Ct<
F{Kt, Nt)
+
(1
-
5)Kt
for t
=
0, 1,
...
Ko = Ko.
The property
relies crucially
we saw
that the idiosyncratic component of the baseline allocation
on the assumption of geometric random walk with constant
above, the planner only wants to affect the drift of {c]}, which
case of an
is
endowment economy where
1
=
is
ht-
Intuitively, as
impossible in the
E[cJ].
In the case where the baseline allocation
is
a geometric random walk with constant
can therefore
restrict
welfare gains
come from modifying the aggregate component
our attention to the aggregate part of the allocation:
Euler equation at the baseline.
already optimal
Suppose that
we
the potential
of the allocation.
in addition, the baseline allocation repre-
sents a steady state where the Euler equation holds.
25
all
ht,
p-i
p-i
0.5
=
p-'
0.75
=
l
6WpE
SWge
fss
SWpE
5Wge
rss
5WpE
SWge
rss
0.09%
0.34%
1.07%
0.10%
0.69%
6.53%
12.33%
0.38%
0.79%
3.82%
5.28%
7.03%
0.10%
0.37%
0.75%
3.82%
4.55%
3
3.82%
4.55%
5.28%
1.56%
2
2.02%
3.62%
7=1
7 =
7 =
=
5.15%
9.85%
Table
Let qss
=
—
(1
we can
that case,
S
+
5.28%
Welfare Gains.
1:
FpciKss, Nss))~
5.28%
be the inverse of the steady state
interest rate.
In
derive as above an expression for P^:
4 = qss (E [s-^])-' E [e^-]
(E [s]r'
=
qss (E
[e])'
J^_
E E
[e]
That the baseline allocation
is
[£-T]
a steady state implies in particular that E[£] =1.
We
can
therefore simplify the formula for $^:
Pe
The optimal
rate qss
When
terms of
e is
//
lognormally distributed logs
and
E [g^-^]
E [e] E [e-r]
'
= 4-
a^.
We
~
N{fj,,a'^),
then we can compute P^ and qss in
get the remarkably simple formula:
qss
is,
qss
allocation will eventually reach a steady state where the inverse of the interest
given by qss
is
=
=
Pe
=
qss exp (-70"e)
Equation (42) shows that the new interest rate
is
Kss < Kss) by
The
a factor given by exp
(70-^).
(42)
higher than the initial interest rate (that
higher risk aversion and the variance
of consumption growth, the higher the increase in steady state interest rates,
and the higher
the reduction in steady state capital stock. Because the baseline allocation has no trend, the
intertemporal elasticity of substitution does not affect the level of the
The only thing our
variations allow in this case
is
7 and the variance
As we
of
consumption growth
is
a'^
by the
controlled only by the relative risk
a'^.
just discussed, the coefficient of relative risk aversion
sumption growth
interest rate q^g.
to correct the externality created
precautionary savings motive, the intensity of which
aversion
new
7 and the variance of con-
control the decrease in capital between the baseline steady state
and the
optimal steady state. The intertemporal elasticity of substitution, on the other hand, controls
the speed of the transition: the higher p~^, the faster the transition, and the higher the welfare
gains.
26
We now
F{K,N) = K^N^-"" +
function
of
compute the welfare gains
consumption growth
computations: a^
rss
=
Qss
~
^
~
=
(1
-
in general equilibrium for the neoclassical
5)K.
at the highest
We
a =
set
0.36, S
end of the values we used
3.07%.
—
1,
2
—
0.5, 0.75
and
our partial equilibrium
fixed at rss, the welfare gains in general equilibrium
and
SWpE
SWge and
if
— and three different
the interest rate were
the interest rate fss at the
state for the optimal allocation.
important general lesson from this exercise, as pointed out
is
1
For each configuration of these
3.
parameters, we report the welfare gains in partial equilibrium
(2006),
in
set the variance
We take the initial interest rate at the baseline allocation to be
We perform the computations of welfare gains for three different
values for the relative risk aversion coefficient 7
An
We
.09.
0.007.
values of the intertemporal elasticity of substitution p~^
new steady
=
production
in Farhi
that taking into account the concavity of the production function
into account general equilibrium effects
and Werning
— that
is,
taking
— greatly mitigates the welfare gains. This because
of the consumption process —the optimal policy
is
in general equilibrium, reducing the drift
under partial equilibrium
—yields lower and lower gains as consumption and capital go down
over time and the equilibrium interest rate increases.
As a consequence,
it is
optimal to reduce
the drift differential. Eventually, under the optimal allocation, the drift differential goes to
and the economy reaches the new steady
state with a higher interest rate
and a lower
capital
stock.
Even though the
partial equilibrium welfare gains can
equilibrium welfare gains never go above 0.79%.
The
7
=
3.
the general
=
1
and the highest value
of the
For those parameter values, the new interest rate
substantially higher than the initial interest rate: rss
this large difference in interest rates
,
highest gains are reached for the highest
value of the intertemporal elasticity of substitution p~^
relative risk aversion coefficient
be as high as 12.33%
and therefore
=
5.28% whereas rss
=
is
3.07%. Despite
in steady state capital stocks, the general
equilibrium welfare gains are moderate at 0.79%.
5
Conclusion
This paper studied constrained
focused on
how
efficient allocations in private
information economies.
We
the optimal savings distortions featured in those allocations depend on indi-
viduals' preferences.
We
introduced a recursive class of preferences that allowed a separation
of risk aversion from intertemporal substitution,
and derived general
results
on the nature of
optimal distortions.
We
then performed a quantitative investigation
consumption
allocations.
We
for
a class of geometric random walk
showed that savings distortion depend only on
27
risk aversion
and
.
the variance of the shocks to consumption. However, the welfare gains from these distortions
depend on both parameters, although we found greater
The purpose
of the quantitative exercise
was to
limited in terms of the consumption allocations
we undertake a comprehensive exploration
it
sensitivity to risk aversion.
illustrate the role preferences,
but
it
was
In Farhi and Werning (2006)
considered.
of savings distortions and welfare gains for general
consumption processes.
Appendix
Proof of Proposition 3
Part
(a).
The proof
parallels the proof of Proposition 2 closely. First note that since prefer-
ences are recursive an incentive compatible allocation satisfies
yz\e\a,
Va>{z\9')>v,{z\9')
(43)
so that truth-telling maximizes continuation utility after any history of reports.
Let
v„{z\
e*)
=
A'v^iz*, e')
for
t
>
r and
2*
^
z"
Va{z\
e^)
=
Va{z\e')
iox
t
>
T and
0*
^
F
so that condition equation (22) with c
met
is
for all 0^
with
t
>
r with
= w{^c{z^),^'<CE[h{z^^\er+iW{z^^\d^+')
v,.{z\e^)
I
So that
ity
Va'^z^.O'')
=
Va'{z'^,9'^) for all
equation (43) evaluated at
z'^
9'^.
6^
^
6'^
I
(22), the inequal-
a,V,^^e^
CE[h{z^+\er+,)v^{z^+\e^+')
I
i;<,.(^^r)
=
solve
(j,Vi,r,^i)
Using the recursion equation
>
^^"""^
A
implies
CE[A'{s^+')h{z^+\er+i)v,^{z^^\e-^')
for all histories
Let
.
and reporting plans
a.
at+uz\0^]
Hence,
W{Ac{z^),Cn^'{s^^')h{z^+\er+,)v,4z^'-\e^^')
I
<l,^^r])
>W{Ac{sn,A'is^-'')CE[h{z-+\er+x)v,{z^^\e^+')\at+,,z\e^])=v,{z\9^).
28
Collecting the inequalities,
we have shown that
<
Vc{z\e^)
Thus, a*
=
Va'{z\e^)
in period
^;^.(^^^*)
r
z\9\a,
for all
optimal from period r onward and delivers the same continuation utility as pre-
is
viously.
For any plan a define an alternative plan a that starts at a and then switches to a* from
period r onward:
result
is
<
at{z'',e*) for t
r and
=
0-4(2^,^*)
a;{z^ ,B^) for
t
>
r.
The
above implies that
t'a(2o,^o)
That
=
at{z'' ,9^)
is,
<
=
Vf,{zo,eo)
a dominates a and
va{zo,6o)
same
yields the
<
=
v^^{zo,9o)
utility as
dominated by the recommended action a* which
v„^{zo,9o)-
(44)
without the variation, which in turn
same
also yields the
utility as after
the
variation. This establishes that a* remains incentive compatible.
Part
(b).
Note that
9*)
=
A'v„{z\9*)
v^{z*, 9')
=
v^{z\
v^{z\
Set
A=
for
i
>
r and
iovt>T and
9^)
2* >-
F
^
z^
z*
(A')"'^ so that
v,{z\9^)
=
h{z\9r)W{Ac{zn,^'CE[v,,{z^+\9^+')
\
<l,^^r])
= h{z^,9r)W{c{z^),Cnv^'{z^^\0^+')\a;^,J\9^])=v,{z\9n.
It
follows
by backward induction that
v„{z*, 9*)
In particular, 1)^(20,^0)
=
=
v^{z\
9^)
for all
s*, t
<
r.
^^(^05^0)) so that the result follows from incentive compatibility of
the original allocation.
Proof of Proposition 5
Let V be such that
Va{s*)
=
Va{s^)
{;^(s*)
=
Va{s*)
+
A'
>
T and
for
i
for
t>T
29
and
s*
y
s'"
s*
^
s^
so that condition equation (18) with c
is
met
for all s*
with
t
>
r with
5* 7^ s'^
.
Now
set
A
so
that
v,,{s^)
so that Va'
at
{s'^)
=
W
-W
=
Va' (s'^).
(c(5*)
+ A,CEK.(s^+i) +
(c(s*)
+
A, A'
A'
-
+ CE[w„.(s^+^) -
Using recursion equation
h{a*{s^))
\
h{a*{s^))
\
a*(5"), s"])
a*(s"),
(18), the inequality
5^)
.
equation (20) evaluated
implies
s"^
-
CE[w„.(s"+i)
h{a*{s^))
>
a*{s^),s^]
CE[va{s^+^)
-
h{a{s^))
\
a{s^),s^],
I
so that
Va'{s^)
=
W
(c(s*)
+
A, A'
>
+
W
-
CE[i;„.(s^+i)
{c{s')
+
A, A'
h{a*{s^))
+
a*{s^), s^\)
\
-
CE[^;,(s^+l)
/i(a(sO)
a{s^),s^\)
|
=
ijais^.
Hence, we have that
Va{s'')
a*
is
<
Va'{s'')
=
Va*{s'')
for all O,
optimal from period r onward and delivers the same continuation
utility as previously.
For any plan a define an alternative plan a that switches to a* from period r onward:
a(s*)
=
a(s*) for
t
<
T and a{s*)
<
l^a(-So)
That
is
is,
=
>
t.
Va{So)
<
a*(s*) for t
Va{So)
=
a dominates a and yields the same
dominated by the recommended action
a*
The
result
fa* (So)
utility as
=
above implies that
Va'{So)-
(45)
without the variation, which in turn
which also yields the same
utility as after
variation. This establishes that a* remains incentive compatible.
Proof of Proposition 6
The equation
that defines
-(1
From
this equation
as a function of A'
/3)«(Q)(e-''^
-
1)
get that at A'
=
0:
-
we
A
dA _
p
=
is:
(3u{CE{vt^^
u{CE{vt+i
30
-
-
h,)){e-P^'
ht))
-
1).
the
At the optimum, we must have that
at
A
=0:
dA _
where r
=
defined by r
is
—
g~^
1
Therefore, the following optimahty condition must hold:
1.
{l-/3Hct)=/3ru{aKivt+,-h,)).
—
Noting that u{c)
—u
(c), this
condition
{l-[i)u'{ct)
which
is
=
equivalent to:
(5ru\€E{vt+i-}H))
the optimality condition in the problem where the agents can borrow and save freely
is
at the interest rate
in the text
is
r.
Transforming these two equivalent conditions into the Euler equation
straightforward.
Consider the constrained
as a function of
A
efficient allocation.
in the following
-A
This defines
—A
We
can rewrite the equation that defines
A
way:
= -log(l + r(l-exp(-pA)))
A > — rA. Now
as a concave function of A. Therefore,
agents the constrained efficient allocation and allowing
strategy but also to borrow and save between history
them
consider giving the
to not only choose a reporting
and subsequent periods. The following
s'^
variations are then available to the agents:
c(s^)
c(s*)
=
{ c{s*)
c(s*)
Since
A > — rA,
+A
for 5*
+ rA
for
=
t>
s^
T and
5* >- 5^
otherwise
whatever reporting strategy the agent chooses when these variations are
permissible, he will always achieve lower utility than under the
were given the variations allowed
is
for the planner.
same reporting strategy
if
he
Since the constrained efficient allocation
incentive compatible, he cannot achieve higher utility than under the constrained efficient
allocation without
history
s'^
,
any additional saving or borrowing. Generalizing that argument to any
this proves the proposition.
31
Proof of Proposition 7
When
consumption
is
a geometric random walk and
constant,
is
ht
it
is
possible to derive
lifetime utility in closed form:
Lemma
1.
Suppose that Assumption
1
holds.
Then
Vt
=
Ahct with
1
i-p
(3
A
i-p
.l-/3/ii-^(E[ffi-^])T
The key
feature that delivers this result
is
the homogeneity of agents' preferences. For a
given h, a proportional shift in consumption today moves consumption in every future period
by a proportional
factor, thereby shifting lifetime utility in
the same multiplicative factor.
Hence
a discount factor.
The constant
disutility
consumption equivalent units by
h on the other hand, acts exactly
consumption equivalent units
utility in
to consumption and to the disutility from effort or work.
settings in section 2.1
It is
and
This
directly proportional
Vt is
reminiscent of the static
is
2.2.
then easy to guess an verify that the solution proposed
the level of utility and
like
in Proposition 7
both preserves
the constrained-optimality condition.
satisfies
Proof of Proposition 10
Before proving this proposition,
Lemma
2.
it is
useful to establish the following
Consider the allocation in Proposition
lifetime utility derived
from
{c\,
h}
:
vl~ Ahc\
10.
with
We
A=
(
can write
^^
'"
Lemma.
Vt
=
Vtvl
where
v\ is the
and
j
iL = (i-A)EA-'f
Let us
now prove Proposition
10.
We
only need to check that
E,
Ph^-p FK{Kt,Nt)
holds.
Decomposing
— r^
Vt+l
zP
c\
^
+ {\-5)
into the product
Ctdj.
(E*[Ci1)
and using
condition as
PeQt+\ FKiKt,Nt) + {l-5)
1
32
i-7n
1
Lemma
2,
we can express
this
This
sical
is
the standard Euler equation that
growth problem
(41).
is
trivially verified
by the solution of the neoclas-
This concludes the proof of Proposition
33
10.
References
Diamond, Peter A. and James A.
Mirrlees, "A Model of Social Insurance With Variable
Retirement," Working papers 210, Massachusetts Institute of Technology, Department of
Economics 1977.
Epstein, Larry, and Stan. Zin, "Substitution, Risk Aversion and the Temporal Behavior
of Consumption and Asset Returns: A Theoretical Framework," Econometrica, 1989, 57,
937-968.
Far hi,
Emmanuel and Ivan Werning,
"Capital Taxation: Quantitative Explorations of
the Inverse Euler Equation," 2006. Mimeo.
Golosov, Mikhail, Narayana Kocherlakota, and Aleh Tsyvinski, "Optimal
and Capital Taxation," Review of Economic Studies, 2003, 70 (3), 569-587.
Ligon, Ethan, "Risk Sharing and Information
Studies, 1998, 65 (4), 847-64.
Rogerson, William
P.,
in Village
Economics," Review of Economic
"Repeated Moral Hazard," Econometrica, 1985, 53
34
Indirect
(1),
69-76.
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