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Institute of
CAPITAL TAXATION
Quantitative Exploration of the Inverse Euler Equation
Emmanuel
Farhi
Ivan Werning
Working Paper
06-1
November 30, 2005
Revised version: April 10, 2009
Room
E52-251
50 Memorial Drive
Cambridge,
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This paper can be downloaded without charge from the
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Social Science Research
j
Capital Taxation*
Quantitative Explorations of the
Inverse Euler Equation
Emmanuel
Farhi
Ivan Werning
Harvard University
ef arhi ©harvard edu
iwerning@mit edu
MIT
.
.
April 10, 2009 (7:11am)
Abstract
This paper studies the efficiency gains from distorting savings in dynamic Mirrleesian
private-information economies.
We
develop a method that pertubs the consumption
process optimally while preserving incentive compatibility. The Inverse Euler equation holds at the
new optimized allocation.
can save freely allows us
investigate
how
to
these gains
compute
Starting from an equilibrium
where agents
the efficiency gains from savings disortions.
depend on
a limited set of features of the economy.
We
We
find an important role for general equilibrium effects. In particular, efficiency gains
are greatly reduced
when, rather than assuming
a fixed interest rate, decreasing re-
turns to capital are incorporated with a neoclassical technology.
ficiency gains for the incomplete
be relatively modest
market model
We compute
in Aiyagari [1994]
the ef-
and find them
to
for the baseline calibration. For higher levels of uncertainty, the
efficiency gains can be sizable, but
we
find that
most
of the
improvements can then
attributed to the relaxation of borrowing constraints, rather than the introduction of
savings distortions.
*
of
Farhi
is
grateful for the hospitality of the University of Chicago.
Harvard University and the Federal Reserve Bank
pleted.
Werning
is
grateful for the hospitality
of Minneapolis during the time this draft
was com-
We thank Florian Scheuer for outstanding research assistance. We thank comments and suggestions
from Daron Acemoglu, Manuel Amador, Marios Angeletos, Patrick Kehoe, Narayana Kocherlakota, Mike
Golosov, Ellen McGrattan, Robert Shimer, Aleh Tsyvinski and seminar participants at the Federal Reserve
Bank of Minnepolis, Chicago, MIT, Harvard, Urbana-Champaign, Carnegie-Mellon, Wharton and the Con-
on the Macroeconomics of Imperfect Risk Sharing at the UC-Santa Barbara, Society for Economic
Dynamics in Vancouver and the NBER Summer institute. Special thanks to Mark Aguiar, Pierre-Olivier
Gourinchas, Fatih Guvenen, Dirk Krueger, Fabrizo Perri, Luigi Pistaferri and Gianluca Violante for enlightening exchanges on the available empirical evidence for consumption and income. All remaining errors are
our own.
ference
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium Member Libraries
http://www.archive.org/details/capitaltaxationqOOfarh
Introduction
1
Recent work has upset a cornerstone result in optimal tax theory. According to Ramsey
models, capital income should eventually go untaxed [Chamley, 1986, Judd, 1985]. In
other words, individuals should be allowed to save freely and without distortions at the
social rate of return to capital. This
on
this issue.
mation
it is
By
contrast, in
important benchmark has dominated formal thinking
economies with idiosyncratic uncertainty and private
infor-
generally suboptimal to allow individuals to save freely: constrained efficient
allocations satisfy
an
Inverse Euler equation, instead of the agent's standard intertempo-
Euler equation [Diamond and Mirrlees, 1977, Rogerson, 1985, Ligon, 1998]. Recently,
ral
extensions of this result have been interpreted as counterarguments to the Chamley-
Judd no-distortion benchmark [Golosov, Kocherlakota, and Tsy vinski, 2003, Albanesi and
Sleet, 2006,
Kocherlakota, 2005, Werning, 2002].
This paper explores the quantitative importance of these arguments. To do so,
from an equilibrium where individuals save
efficiency gains obtained
dress
is
to solve
without distortions, and examine the
from introducing optimal savings
largely unexplored because
—deriving
distortions.
i.i.d.
over time.
1
The issue we ad-
first-order conditions aside
dynamic economies with private information, except
such as shocks that are
cases,
freely,
we start
for
—
it is
some very
difficult
particular
We develop a new approach that sidesteps
these difficulties.
We
lay
down an
infinite-horizon Mirleesian
Agents consume and work
are
assumed additively
economy with
in every period. Preferences
neoclassical technology.
between consumption and work
separable. Agents experience skill shocks that are private infor-
mation, so that feasible allocations must be incentive-compatible.
Constrained
tion
efficient allocations satisfy a
known as the Inverse Euler equation.
simple intertemporal condition for consump-
This condition
standard Euler equation, implying that constrained
tralized in competitive equilibria
return to capital.
Some form
incompatible with the agents'
efficient allocations
where agents save
of savings distortion
is
is
cannot be decen-
freely at the technological rate of
needed.
Equivalently, starting from an incentive compatible allocation obtained from an equi-
librium where agents save
freely, efficiency
saving distortions. In this paper
gains are possible with the introduction of
we are interested
in
computing these
gains.
We do so by
perturbing the consumption assignment and holding the labor assignment unchanged,
while preserving incentive compatibility. The
equation and delivers the same
1
utility
new
allocation satisfies the Inverse Euler
while freeing up resources. The reduction in
Other special cases that have been extensively explored are unemployment and
re-
disability insurance.
sources
is
our measure of efficiency gains. By leaving the labor assignment unchanged,
we focus on
the efficiency gains of introducing savings distortions, without changing the
incentive structure, implicit in the labor assignment. In this way,
between insurance and
the optimal trade-off
we
sidestep resolving
incentives.
There are several advantages to our approach. To begin with, our exercise does not
some components
require specifying
of the economy. In particular,
work
dividual labor assignment or the disutility of
degree to which work effort responds to incentives
portant, since empirical
efficiency gains
knowledge
depend only on the
function
is
of in-
required. In this way, the
not needed. This robustness
is
im-
of these elasticities remains incomplete. Indeed, our
consumption assignment, the
original
consumption and technology. The planning problem
for
is
no knowledge
we
set
utility function
up minimizes resources
over a class of perturbations for consumption. This problem has the advantage of being tractable, even for rich specifications of uncertainty. In our view, having this flexibility is
important for quantitative work. Furthermore, efficiency gains are shaped by
intuitive properties involving
ations,
both partial equilibrium and general equilibrium consider-
such as the variance of consumption growth and
dispersion across agents and
and the concavity
time, the coefficient of relative risk aversion
Finally,
its
of the production function.
our measure of efficiency gains can provide lower and upper bounds on the
efficiency gains obtained
from joint reforms
that introduce savings distortions
full
and change
the assignment of labor.
Two approaches
we pursue
are possible to quantify the
The
both.
first
approach uses directly as
plausible process of individual consumption.
empirical knowledge of consumption risk
as a baseline allocation the
to idiosyncratic skill
Following the
tures geometric
It is
outcome
is
efficiency gains,
a baseline allocation
an empirically
The second approach recognizes
more
that the
limited than that of income risk.
of a competitive
and
economy where agents
It
uses
are subject
shocks and can save freely by accumulating a risk-free bond.
first
approach,
random walk
rithmic. In this case,
magnitude of these
we
analyze a model where the baseline allocation fea-
processes for individual consumption and utility
we obtain closed form solutions. The perturbed
allocation
is
is
loga-
simple.
obtained by multiplying the baseline consumption assignment by a deterministic de-
downward
clining sequence.
The
in the variance of
consumption growth, which indexes the strength
size of this
drift
and the
efficiency gains are increasing
of the precautionary
savings motive. Using a fixed interest rate, the efficiency gains span a wide range, going
from
0%
to
10%, depending primarily on the variance of consumption growth, for which
empirical evidence
We
is
limited.
find that, general equilibrium effects can dramatically mitigate these numbers.
This
because
is
tilting
individual consumption profiles requires decumulating capital.
With a neoclassical technology, as
capital
raising the cost of further decumulation.
ciency gains range from
is
decumulated, the return to capital increases,
With
a capital share of 1/3,
we show
that
effi-
0% to 0.25%, for plausible values of the variance in consumption
growth.
we
adopt the incomplete-market specification from
Turning
to the
second approach,
the seminal
work
of Aiyagari [1994]. In this economy, there
is
no aggregate uncertainty.
Individuals face idiosyncratic labor income risk that they cannot insure.
in a risk-free asset,
but cannot borrow. At a steady
They can save
state equilibrium the interest rate is
constant and equal to the marginal product of capital. Although individual consumption
fluctuates, the cross-sectional distribution of assets
Our baseline allocation from
this
and consumption is
invariant.
We take
steady state equilibrium.
Even with logarithmic utility, taking baseline consumption from this equilibrium model
implies two differences relative to the geometric random-walk case, discussed previously.
On
the one hand, as
fectively in these
is
well known, agents are able to smooth consumption quite
Bewley models. This tends
growth and pushes towards low
a geometric
random walk.
to
efficiency gains.
ef-
minimize the variance of consumption
On
the other hand,
consumption
In particular, expected consumption growth
is
is
not
not equalized
over time or across agents. Indeed, a steady state requires a stable cross-sectional
distri-
bution for consumption, implying some mean-reverting forces. This pushes for greater
efficiency gains, because there are gains
from aligning expected consumption growth
in
individual consumption across agents, without changing the growth in aggregate con-
sumption or
We
capital."
replicate the calibrations in Aiyagari [1994]
librium for a
number
of
and compute the steady
income processes and values
state equi-
for the coefficient of relative risk
we find that efficiency gains are relatively
specifications. Away from this baseline calibration, we
aversion. For Aiyagari's baseline calibration,
small,
below 0.2%
for all utility
find that efficiency gains increase with the coefficient of relative risk aversion
the variance
and with
and persistence of the income process. Efficiency gains can be large
if
one
combines high values of relative risk aversion with large and persistent shocks. However,
for these calibrations
of
we
find that
most
borrowing constraints, rather than
of the efficiency gains are
to the relaxation
to the introduction of savings distortions.
simulations illustrate our more general methodology and provide
2
due
some
These
insights into the
As a result of equalizing expected consumption growth rates, the perturbed allocation will not feature
an invariant distribution for consumption. Instead, the dispersion in the cross sectional distribution of
consumption grows without bound. This is related to the "immiseration" result found in Atkeson and
Lucas [1992].
determinants of the size of efficiency gains from savings distortions.
Related Literature. This paper relates to several strands of literature.
timal taxation literature based
ski,
on models with private information
and Werning, 2006, and the references
for the constrained efficient allocations,
efficiency gains
due
therein].
[2006] for disability insurance
[see
there
the op-
is
Golosov, Tsyvin-
this literature
usually solve
but few undertake a quantitative analysis of the
Two
savings distortions.
to
Papers in
First,
exceptions are Golosov and Tsyvinski
and Shimer and Werning
[2008] for
unemployment
in-
surance. In both cases, the nature of the stochastic process for shocks allows for a low
dimensional recursive formulation that
is
numerically tractable. Golosov and Tsyvinski
[2006] provide a quantitative analysis of disability insurance. Disability
absorbing negative
that can
to the
skill
calibrate their
model and compute
Werning
They consider
worker samples wage
show
that with
utility
a sequential job search model,
from a known
as
an
the welfare gains
and report welfare gains of
[2008] provide a quantitative analysis of
offers
modeled
efficient allocation that satisfies free savings
optimal allocation. They focus on logarithmic
surance.
utility,
They
be reaped by moving from the most
0.5%. Shimer and
they
shock.
is
where
distribution.
unemployment
in-
a risk averse, infinitely lived
Regarding savings distortions,
CARA utility allowing agents to save freely is optimal.
With
CRRA
savings distortions are optimal, but they find that the efficiency gains they pro-
vide are minuscule. As most quantitative exercises to date, both Golosov and Tsyvinski
[2006]
and Shimer and Werning
[2008] are set in partial equilibrium settings
with
linear
technologies.
Second, following the seminal paper by Aiyagari [1994], there
on incomplete-market Bewley economies within the context
is
a vast literature
of the neoclassical
growth
model. These papers emphasize the role of consumers self-smoothing through the precautionary accumulation of risk-free assets. In most positive analyses, government policy
is
either ignored or else a simple transfer
rent policies. In
the
some normative
income tax or
2005].
tures
analyses,
the gap
it
included and calibrated to cur-
some reforms
of the transfer system, such as
of efficiency
used
litera-
of the constrained-inefficiencies in the latter.
in the present
paper
is
often termed constrained-efficiency,
imposes the incentive-compatibility constraints that
asymmetry
Conesa and Krueger,
between the optimal-tax and incomplete-market
by evaluating the importance
because
is
social security, are evaluated numerically [e.g.
Our paper bridges
The notion
and tax system
arise
from the assumed
of information. Within exogenously incomplete-market economies, a distinct
notion of constrained-efficiency has emerged [see Geanakoplos and Polemarchakis, 1985].
The idea
is
roughly whether, taking the available asset structure as given, individuals
could change their trading positions in such a
way
that generates a Pareto
improvement
at the resulting
Krusell,
market-clearing prices. This notion has been applied by Davila, Hong,
and Rios-Rull
[2005] to Aiyagari's [1994] setup.
resulting competitive equilibrium
is inefficient.
They show
we
In this paper
that, in this sense, the
also apply
our method-
ology to examine an efficiency property of the equilibrium in Aiyagari's [1994] model, but
it
should be noted that our notion of constrained
incentive-compatibility,
is
very
efficiency,
which
is
based on preserving
different.
A Two-Period Economy with Linear Technology
2
We
start
with a simple two-period economy with linear technology and then extend the
concepts to an infinite horizon setting with general technologies.
Preferences. There are
symmetric
allocations.
in period
—
t
1.
two periods
t
=
0,1.
Consumption takes place
Agents obtain
v
=
U(c
)
+ pE
[U( Cl
)
-V(m-r 0)].
effective units of labor,
focus on
work occurs only
and
E
is
where
is
an interval of R.
shock. To capture the idea that uncertainty
law of large number holds so
that for
is
(1)
.
the disutility function
from
effec-
the expectations operator.
captured by an individual shock 9 G
is
We
utility
tive units of labor (hereafter: labor for short)
Uncertainty
identical.
in both periods, while
U is the utility function from consumption, V is
where
the
Agents are ex ante
that affects the disutility of
We will sometimes refer to 9 as a skill
idiosyncratic,
we assume
any function / on 0,
E [f]
that a version of
corresponds
to the
average of / across agents.
The
utility
We assume
9
function
U is assumed increasing, concave and continuously differentiable.
that the disutility function
G 0, the function V(-,9)
property that -^-V
disutility
is
V
increasing
(n\; 9) is strictly
is
continuously differentiable and
and convex.
decreasing in
9,
We
also
assume the
that, for
any
single crossing
so that a high shock 9 indicates a low
from work.
Incentive-Compatibility. The shock realizations are private information to the agent, so
we must
ensure that allocations are incentive compatible. By the revelation principle
can consider, without loss of generality, a direct mechanism where agents report
shock realization
of this report.
report
is
9 in
period
f
=
1
and
The agent's strategy
made when
defined by a* (9)
the true shock
=
9 for all 9
is
is 9.
G 0.
are assigned
a
mapping
The truth
consumption and labor
cr
:
—
»
where
telling strategy is
cr{9)
we
their
as a function
denotes the
denoted by
cr* ,
which
.
It is
convenient to change variables and define an allocation by the
with mq
=
ti(co)/ U\(B)
=
triplet
{w 0/
u\,
ri\
}
U(ci(9)). Incentive compatibility requires that truth-telling be
optimal:
a*
eargmax{w o +
iSE[ Ml
(cr(0))-y(n 1
(cr(0));0)]}.
Let 6* be an arbitrary point in 0. The single crossing property implies that incentive
compatibility
equivalent to the condition that n\ be non-decreasing and
is
u 1 (e)^V(n 1 (9),9)=Mn-VM9*),e*)+ /
Je*
Technology.
equal to
t
w and a rate of return on savings equal to q~
^[€(^{6))
c
(2)
We assume that technology is linear with labor productivity in period =
c(w )+/c!
where
-Tj(ni(0);0))<&
36
= U~
l
is
- wmtf)]
l
.
1
The resource constraints are
<k Q
< q-%
(3)
(4)
the inverse of the utility function.
For an allocation to satisfy both resource constraints with equality the required level
of initial capital ko
must be
feo
In
what
follows,
=
we refer to ko
c(«o)
+ <?E
[c(ui(6))
-
wni(0)]
as the cost of the allocation.
Perturbations. Given a utility level v and a non-decreasing labor assignment {n\
tive compatibility
and the requirement that the
of allocations T {{n\
into equation
(1)
}, v).
},
incen-
allocation deliver utility v determines a set
Indeed, this set has a simple structure. Substituting equation
(2)
implies that
+ j6«i(0*) =0 + j6V (n 1 (0*) 0*)-0E
,
Mo
/
'8*
For given v and {n\}, the right hand side
is
fixed.
d6
For any value of uq this equation
can be seen as determining the value of U\{6*). Equation
(2)
then determines the entire
assignment {wi}. Thus, elements of T({rii},v) are uniquely determined by the value
of WoIt is
useful to restate this property as a perturbation.
7
Given a baseline allocation
{mo, wi,«i}
G r
({rci},u),
=
u
for
AgR.
some
any other allocation
uq
-
The reverse
jSA
is
and
=
Uj (0)
also true, for
G T ({ni},u)
{wo, Wi,ni}
w : (0)
+ A for all
9
satisfies
G 0,
any baseline allocation
in T({ni
},
v)
and any
AElRa utility assignment constructed in this way is part of an allocation mT({n\},v).
Note
{tti }.
{{uq
To
that the construction of this perturbation
—
reflect this
j6A,«i
denote
+ A}
|
poses of describing
{ii\),
knowledge
this
A € R}.
all
is
independent of the labor assignment
=
corresponding set of utility assignments by Y( { uq, U\ }, 0)
This notation captures the following point. For the pur-
possible utility assignments consistent with a labor assignment
of the labor assignment itself
placed by knowledge of some baseline
utility
and the
disutility function
assignment.
Free-Savings and Euler equation. The allocation {uQ,U\,ni]
equilibrium with natural borrowing limits
if
and only
if
V
can be
re-
3
is
part of a free-savings
truth telling
and saving zero
is
optimal:
(<7*,0)
€ argmax {u{c (u
(a,k)
)
- k) + jSE \u
l
*•
(c («i
(or
V
(9)))
+ q~
l
k]
-V
/
(n x [a (9)); 6)}
}
J J
(5)
For short,
we
say that the allocation satisfies free-savings. Free-savings implies incentive
compatibility and the Euler equation
tf(c(u
The converse
However,
is
,
(c(u 1 .(e)))]
(6)
not generally true, these two conditions are not sufficient for free-savings.
for a given labor
{uq, mi} are uniquely
assignment
{n-y}
and
utility level v, the utility
assignments
determined by incentive compatibility, the requirement that the
location deliver expected utility v
3
1
))=to- E[U
and the Euler equation. That
is,
al-
there exists a unique
A note on our notation is in order. We model shocks as continuous,
i.e.
is an interval of R, but do not
support on 0. When it does not, our notation still requires defining the
allocation for values of 9 outside the support. Thus, we distinguish allocations that coincide on the support
of 7T, but differ elsewhere. This is absolutely without loss of generality, but plays a role in the definitions of
necessarily
assume
that
n has
full
T({ni},v) and Y({u ,wi},0).
For example, consider the case where the support of n is composed of a finite set of points, so that 6
belongs to a finite set with probability one. This 'finite shock' setting is often adopted as a simplifying
assumption. In this case, one could define the allocation as a finite vector, with elements corresponding
to the points in the support of K. However, with this alternative representation, there can exist two utility
assignments, {u§,u{\ and {uq,u\}, that, together with {«i} are incentive compatible, but are not obtained
from one another through the parallel perturbations we described. To see why this is consistent with the
notation we adopt, note that in this case it is possible to define various labor assignments on
that coincide
on the finite support of 0. For each of these, our representation of all possible utility assignments T {{n\},v)
holds.
8
allocation in
T {{n\],v) that
Euler equation. For any given assignment {ni}
satisfies the
may or may not satisfy free savings. We denote by D(v)
the resulting allocation
labor assignments that are compatible with free savings, given utility
Efficiency and
The Inverse Euler Equation. Consider the allocation
with the minimal cost
-
most
the
efficient allocation in
T
the set of
v.
{uq, U\, n\ } in
({n-i },v).
T {{n\ }, v)
This allocation
is
uniquely determined by the requirement that {uo,Ui,ni} € T({ni},v) and the following
first
order condition
X
1
1
rE
.
w(c(u
Equation
It is
(7) is
fa~
))
W (c.(ui (6)))
the so called Inverse Euler equation.
useful to contrast equation
(7)
with the standard Euler equation
inequality to equation (18) implies that at the
consumption
that equation (7)
is
at
t
+
1 is
(6).
Jensen's
optimum
1
lf(c(uo)<Pq- B[U
as long as
(7)
,
(c(ui(0)))]
(8)
l
uncertain condition on information at
incompatible with equation
(6).
t.
This shows
Thus, the minimal cost allocation
{uo,Ui,tii} in Y({ni},v) cannot allow agents to save freely at the technology's rate of
return, since then equation (6)
would hold as a necessary condition, which is incompatible
with the planner's optimality condition, equation
For any allocation, define t G
U
•
a
,
wedge
Efficiency Gains.
by
(c(u Q ))=q- (l-T)TE[U'(c(u 1 {e)))],
measure of the distortion
intertemporal
1R
(7).
1
in the Euler equation that
or implicit tax
Given a
on savings. 4
utility level v
and
If
is
sometimes referred
equation
(7)
holds then t
a labor assignment {ri\},
>
we
to as the
0.
define the
following cost functions
min
X{{ni},v)=
,u 1/ n 1
{u
,u 1 ,n 1 }sr({ni},u)
E
X ({ni},v)=
The
first
}er({n 1
{M
.
}/
{c{u Q )
+ qE
[c(ui(9))
- wn^B))}
{c(Uo)
+ qlE[c(ui(6))
—wni(6)]}
z')
min
minimization implies that the corresponding allocation
*We do not concern ourselves here with
explicit tax
s.t.
equation
satisfies the
systems that implement
(6)
Inverse Eu-
efficient allocations.
t
\
A(andB
/
/
\A'
c
On the bottom left panel, the labor assignments corresponding to the minimum
of the upper frontier is in D(v). On the top right panel, the labor assignment corresponding to the minimum of the lower frontier is in D(v). On the top left panel, both labor
assignments are in D(v). On the bottom right panel, none of these labor assignments is
Figure
in
1:
DO).
ler equation.
singleton
Given the discussion above, the second minimization takes place over a
— the allocation
is
determined by the constraints. By definition,
Euler equation. Thus, the allocations behind these two costs
it
satisfies the
satisfy, respectively, the In-
verse Euler and standard Euler equations.
The Euler equation acts
fore, the difference
as an additional constraint in the second minimization. There-
between these cost functions
E
X ({ni},v)-x{{ni}
is
positive
{fti}
and captures the
£ D(v),
this coincides
efficiency cost of
f
v)
imposing the Euler equation. Whenever
with the cost of imposing
10
(9)
free savings. This cost difference
can hence also be interpreted as a measure of the efficiency gains from optimal savings
(9)
and
distortions,
First,
is
the
main focus
of the paper.
It
number
has a
our measure of efficiency gains can be computed by a simple perturbation method.
Indeed, consider the allocation {uo,Ui,n-[} G r({ni},i;) that
By
of desirable properties.
definition,
its
cost
is
X
E
({ n i}> v )-
As discussed above,
Euler equation.
allocations in Y({ni},v) can
=
obtained through simple perturbations Y({«0/Wi}/0)
satisfies the
{{wo
—
jSA,
A e R}
+ A}
ii\
be
of the baseline utility assignment {uq, u{) while fixing the labor assignment {ni}. Within
this class of perturbations, there exists a
satisfies the
unique perturbed
Inverse Euler equation. Efficiency gains
X
E
({ni},v)
- x({ni},v) =
Second, given the baseline
utility
c{u
)
-
c(u
assignment
are given
(9)
)
utility
+ qE
[c( Ml
{
by
(0))
-
c{u x )]
work V
is
needed
compute our mea-
in order to
sure of efficiency gains. In other words, one does not need to take a stand on
effort is to
changes in incentives. More generally one does not need
on whether the problem
ard regarding
knowledge
effort, etc.
is
one of private information regarding
This robustness
of these characteristics
is
consumption
is efficient,
skills
how elastic
to take a stand
or of moral haz-
a crucial advantage since current empirical
and parameters
ficiency gains (9) address precisely the question of
of
(10)
.
assignment {uq,Ui}, no knowledge of either the
labor assignment {n\\ or the disutility of
work
Uq, U\ } that
is
limited
and
controversial.
The
ef-
whether the intertemporal allocation
without taking a stand on
how
correctly tradeoff
between
insurance and incentives has been resolved.Third, as this paper will show, the magnitude of the efficiency gains
by some
rather intuitive properties involving both partial equilibrium
(9) is
determined
and general equi-
librium considerations: the relative variance of consumption changes, the coefficient of
relative risk aversion
Finally,
and the concavity
we now show how
(9) is
of the production function.
informative about more encompassing measures of
efficiency gains that also allow for changes in the labor
Decomposition and Bounds. Suppose the economy
ings.
Suppose further
assignment {n\).
is initially
constrained
by
free sav-
that, subject to this restriction, the allocation is efficient.
This
is
5
The robustness properties of the allocations in T ( {n\ }, v), and hence of our measure of efficiency gains,
can be more formally described as follows. Consider the set Q({uq, U\,ri\ }, v) of disutility functions V with
the following properties: V is continuously differentiable; for any 9 6 0, the function V{-,9) is increasing
and convex;
holds.
V has
the single crossing property that
^- V
{n\,9)
is strictly
Theset of allocations T ({n\},v) is the largest of allocations such
(2) and promise keeping (1) hold.
incentive compatibility
11
decreasing in 9 and equation
that for
all
V
(2)
G D.({uo,ui,ni},v),
A in the figure, with labor assignment
represented by point
£
{nf } £ argmin^ ({?7 1 },z;)
s.t.
{n^ £
D(v).
Now consider lifting the restriction of free savings and allowing the optimal savings distortions, as described
by the Inverse Euler equation. The new
efficient allocation is repre-
sented by point C, with labor assignment {nf } £ argminj,^} x({ n l}' v )-
Th e move from
A to C lowers costs by
B
c
xH{nt} v)- X ({n^},v)=xH{nthv)-xH{n!},v)+xH{n } v)-x({n }
/
The
1
term X E ({ n \}> v )
first
~~
E
X
E
{nf} € argminr Ml } X {{ n i}' v )-
({ n \ }' v )
move from
captures the
^ represents
f
l
point
the potential gains from
A
to B,
move from
x{{ n \}> v )> captures the
from allowing optimal savings
point B to point C.
It
v).
where
removing the con-
£ D(v), but maintaining the Euler equation. The second term, X E ({ n i
straint {n\}
/
}'
v)
~
represents the gains obtained
may be sat-
distortions, so that the Inverse Euler equation
isfied.
This decomposition emphasizes that savings distortions
itatively distinct reasons. First, the set of
may be valuable for two qual-
implementable labor assignments
is
enlarged.
Second, for any labor assignment, perturbing the consumption assignment to satisfy the
Inverse Euler equation, as opposed to the Euler equation, reduces costs. To the best of our
knowledge,
Note
this
decomposition
is
novel.
that the first source of efficiency gains
may
not be present.
This
the case
is
whenever {nf} £ D(v), so that points A and B coincide. These cases are shown in
the top left and bottom left panels of the figure. Indeed, in numerical simulations we
found that
this is the typical case for
particular,
we adopted
and V(n;6)
=
oc(n/6) 7
standard
iso-elastic utility
.
As
for n,
we
and
utility functions
and distributions
disutility functions
U(c)
=
c
l
~a
/ (\
n. In
—
a)
tried various classes of continuous distribution,
including uniform, log-normal, exponential, and Pareto. For a given specification and
parameters,
that
we
first
computed
minimized cost subject
keeping. 6
We
the analogue of point B,
6
that
planning problem
and promise
then verified that this allocation was compatible with free savings, so that
(5) directly.
distribution parameters for these classes. In
first
a
to the Euler equation, incentive compatibility
{wf } £ D(v), by verifying condition
the
by solving
We tried a wide range of preference and
all cases,
source of efficiency gains was found to be
This can also be thought of as a
"first
we found
that {nf} £ D(v), so that
nil.
order approach" for solving point A. In the next step,
A and B coincide, verifying the validity of the first order approach.
12
we
check
Of course,
is
where
there are examples
source of gains
this
is
One such example
positive.
the case with finite shocks. However, our numerical findings with standard continu-
ous distributions, where the gains are
suggest that conclusions based on finite shock
nil,
examples, should be interpreted with caution. More
where
cisely the situations
of these scenarios. In
paper and
since
X
({ n i }>
v)
first
~ X({ n
i
is
needed
to
is
not the focus of this
this issue further.
source, the second source of efficiency gains
}'
v)
understand pre-
nonzero, to determine the plausibility
case, this first source of efficiency gains
we will not explore
In contrast to the
E
any
this source of gains is
work
>
as l° n g as
is
always present,
consumption is uncertain, so that the Inverse
and Euler equation and the Euler equation are incompatible.
This two-period, linear technology example
C
and
in
is
can be computed numerically. However, in the
environments with concave technologies and an
modeled
C
B and
persistent shocks.
is
shows
x({ n i}' v )
3
that
is
In this section,
However, note
knowledge
3.1
We
are interested
with productivity
i.i.d.
or fully
that
< ^({n?},*) -*({nf}}v) < * E ({nf },v) - X ({nf },v).
of the vertical distance between the
two cost functions x E {{ n i }> v
Horizon
we lay down our
general environment.
tions that preserve incentive compatibility.
to
infinite horizon,
we
informative about the second source of efficiency gains.
Infinite
method
paper
not tractable, except for some special cases such as
**({*?},*) -x({nf},v)
This
rest of the
B
dynamic extensions, computing the analogues
as a stochastic process. For these
of points A,
In particular, points A,
tractable.
compute
We then describe a class perturba-
These perturbations serve as the basis for our
efficiency gains.
The Environment
cast our
model within
a general Mirrleesian
closest to Golosov, Kocherlakota,
Euler equation
[e.g.
Diamond and Mirrlees,
where agents' privately observed
Preferences.
and Tsyvinski
Our economy
is
skills
dynamic economy. Our formulation
[2003].
This paper obtains the Inverse
1977] in a general
dynamic Mirrlees economy,
evolve as a stochastic process.
populated by a continuum of agent types indexed by
distributed according to the measure
is
ip.
i
G
I
Preferences generalize those used in Section 2
13
and
are
summarized by
the expected discounted utility
oo
t=Q
E
!
is
the expectations operator for type
i.
Additive separability between consumption and leisure
we adopt because
fective units of labor.
for
each individual
distributed across
is
captured by an individual specific shock
an interval of the
is
We sometimes
9\ is
all
for
We
agents.
an agent of type
them
pectation notation E'|/(^
of
/
number holds
00
tation
we
will
that result
of
assume
we
l'
It is
l,t
=
l
(9 Q , 9\,
co
.
0\),
,
.
.
1'
i
i' t
i
} /
of/ given history
so that for any function
/ on
0°°, E'
[/]
9
l,t
~l
f
G
.
that a version of the
law
corresponds to the average
i.
we will have for types
that skills follow a
i
G
I,
Markov
note that in our numerical implemenprocess.
We
from a market equilibrium where agents save in a
tp
9
and independently
f
)]
economy, agent types are then
measure
by
t
I
denote the integral f f(9
)dn(6 00 ) using the ex-1
or simply E [/]. Similarly, we write E [f(9 '°°)\9
or
,
,co
across agents with type
To preview the use
G
i
As in Section 2, all uncertainty is idiosyncratic and we assume
of large
The stochastic process
as skill shocks.
corresponding to the law of the stochastic pro-
for the conditional expectation
[f],
affect the disutility of ef-
denote the history up to period
00
G 0, where
9\
i.
Given any function / on
simply E[_ 1
These shocks
identically distributed within each type
and by n the probability measure on
6\
real line.
refer to
l
cess
a feature of preferences that
required for the arguments leading to the Inverse Euler equation. 7
it is
Idiosyncratic uncertainty
as in Section 2,
is
initial asset
will consider allocations
holdings together with
For
this
kind
initial skill.
The
riskless asset.
captures the joint distribution of these two variables.
convenient to change variables, translating consumption allocation into
assignments {u
i,t
l
t
(9
)} /
where
centive constraints linear
u\(9
1,t
)
=
U(c\(9
l,t
)).
This change of variable will
and render the planning problem
convex. Then, agents of type
v
l
i
=
G
I
that
we will
with allocation {u\} and {n\} obtain
f
E<
X>
f=0
i4(0tt)
-
utility
make
in-
introduce shortly
utility
(11)
V(nf(0''');0i)
Information and Incentives. The shock realizations are private information to the agent.
7
The intertemporal additive separability
additive separability of
some general
work
effort
is
of
consumption also plays a
completely immaterial:
disutility function V({nt}).
14
we
role.
However, the intertemporal
could replace YT=Q
f
/5
E[V(rct;#t)] with
We
invoke the revelation principle to derive the incentive constraints by considering a
direct
mechanism. Agents are allocated consumption and labor as a function of the
The agent's strategy determines
history of reports.
of the history,
f
cf*(0
{0 hi )}-
{cr\
oo
•
-
1
Technology. Let
and
all
i
is,
=
letting c
LI
-1
N =
t
.
.
.
I.
denote the inverse of the
Q=
0, 1,
G
Q and Nt represent aggregate capital, labor and consumption for period
respectively. That
J
E
is
where Kt denotes aggregate
of degree one, concave
N+
t
<
(1
by
is strictly
we
—
capital.
it
will
prove conve-
which represents the aggregate amount
t
also
satisfies
impose Kt >
=
<
and
q
0.
<
N
t,
t
t
)
The function F(K, N)
and continuously
concave and
1)K, S
et
- S)K + F(K
cases are of particular interest.
(q~ l
#
being economized in every period. The resource constraints are then
%! + Q + e
In this case,
dip
c(u\)
nj
utility function,
In order to facilitate our efficiency gains calculations,
of resources that
F(K, N)
f
f E'
nient to index the the resource constraints
Two
(12)
t=o
for all reporting strategies {&[}
=
Yj^WM *^)) - V{n\{^{e^));e\)}
>
V{n\{d^);e\)}
f=0
t
that truth-telling,
=9\, be optimal:
-')
£/5'E''[i4(0*)
for
a report for each period as a function
The incentive compatibility constraint requires
oo
t,
entire
1.
is
=
0,l,...
assumed
differentiable, increasing in
The
be homogenous
K and
N.
the neoclassical growth model,
first is
Inada conditions
Fj<(0,
The second case has
One
to
(13)
interpretation
N)
=
oo
and
F^(oo,
N) =
linear technology F(K,
is
that output
is
where
N)
0.
=
linear in labor
with productivity normalized to one, and a linear storage technology with safe gross rate
l
of return q~
wide budget
is
set for a partial
and unit wage. Under
negative
up
Another interpretation
available.
we
and allows us
to
15
1
economy-
+r=
q~ l
avoid corners by allowing capital to be
value of future output,
the natural borrowing limit
that this represents the
equilibrium analysis, with constant interest rate
either interpretation,
to the present
is
K
> —
s
M+s- This represents
summarize the constraints on the economy
t
+\
J2T=i 9
by
the single present-value condition
00
00
-|
E<?'Q< j>'( Nt-«t) + -Ko9
f=0
We
Moral Hazard Extension.
t=o
could extend our framework to incorporate moral hazard
in addition to private information.
unobservable action
For example, each period agents could choose an
that creates additively separable disutility. Effective labor, n\,
e\
function of the history of effort and shocks n\
over
skill
may be
shocks
depends on the
type of model,
all
pursue in further
this
a
In addition, the distribution
detail the notation required to formalize this
our subsequent analysis extends
hybrid model like
).
,
is
.
may also be interpreted this way.
cal applications
a
will not
ft{e
1,f
affected by the sequence of effort, so that the distribution of 6 l,t
ht ~ l
effort choices e
Although we
=
l,t
one seems more
such a
to
This
realistic
setting.
Indeed, our numeri-
important to keep in mind, since
is
than a model focusing exclusively on
private information, as in the standard Mirrlees model.
Feasibility.
An
allocation {u\,n\,Kt,et}
(11)— (13) hold. That
must be
and
utility profile
{v
1
feasible allocations that deliver utility v
is,
incentive compatible
and resource
l
to
agent of type
agents,
which adds further
i
G
I,
feasible.
Free Savings. For the purposes of this paper, an important benchmark
agents can save, and perhaps also borrow,
conditions
} is feasible if
where
the case
is
This increases the choices available to
freely.
restrictions relative to the incentive compatibility constraints.
In this scenario, the government enforces labor and taxes as a function of the history of reports, but does not control consumption directly. Disposable after-tax income
is
Wtn\(a l,t (9 ht ))
rowing
—
Tl(cr
l,t
(6
8
ht
)).
Agents face the following sequence of budget and bor-
constraints:
4(0^)
+4 +1
z>
(<9
)
<
1
wtrbip
.i +1
with
l
a Q given.
borrowing
We
A
1
^)) +
(1
+ r )4(^
_1
t
(14a)
)
fl|
+1 (#
z '
(ub)
f
)
to
be tighter than the natural
limits.
i
maximize
special case of interest
expressed through
T/'"
1 '
(^)>^ +1 (^(^))
allow the borrowing limits
Agents with type
8
'*^)) - Tlia
its effect
is
utility
E^o^'KC^'^^'O) -
where the dependence
on the history of labor n
l '
of the tax
l,t
t
(9
function.
16
).
That
is,
"'
V(nj(0-
on any history
when
7](0
!
'
)
f
(0
f' f
));0|)]
of reports 9
= T '"(n
l
f
z
t
1,t
t
l,t
(9'' ))
,
by
can be
for
some
\
choosing a reporting, consumption and saving strategy {u^c],^^} subject to the se-
quence of constraints
{u\,n\,Kt,et}
is
optimum
the
{crl,c\,a t+1 } for
i
rt
f'
as given.
}
A
feasible allocation
there exist taxes {T\}, such that
truth telling
)
t
t
if
t
with wages and interest rates given by
= FK (K ,N - 5 satisfies
9
the utility assignment u (0''*) = I7(c|(# )).
At
fied.
agent of type
l
t )
t
ates
and {n\,T\,Wt,r
a'
part of a free-savings equilibrium
w = FN (K ,N and
t
taking
(14),
crj(0
f' f
)
=
and gener-
0}
f
a free-savings equilibrium, the incentive compatibility constraints (12) are satis-
The consumption-savings choices
In particular, a necessary condition
of agents
if a\
+l {6
ht
>
)
a
l
t+1 (6
1 '
restrictions.
the intertemporal Euler condition
is
Lr(c(uS)).>0(l
with equality
impose additional further
t
).
+ r t+1 )Ei
Note
W(c(u\ +l ))
(15)
the borrowing limits a t+1 (9 1,t ) are
l
that
if
equal to the natural borrowing limits, then the Euler equation (15) always holds with an
equality.
We say
Efficiency.
by
that the allocation {u\,n\,Kt,et)
the alternative {u t ,n t/
l
either v
period
1
>
l
K
t/
et }
and {v
1
v
}, if
1
>
v
1
,
and
Kq
utility profile
<
Kq, e t
v for a set of agent types of positive measure, Kq
1
{v
< S for
< Kq or
t
1
} is
periods
all
et
dominated
<
St
for
t
and
some
t.
We say that a feasible allocation is efficient if it is not dominated by any feasible allocation. We say that an allocation is conditionally efficient if it is not dominated by a feasible
with the same labor allocation n\ = h\.
As explained in Section 2,
allocations that are part of a free-savings equilibrium are not
conditionally efficient. Conditionally efficient allocations satisfy a
the Inverse Euler equation,
which
is
3.2
and insurance.
order condition,
inconsistent with the Euler equation.
of a free-savings equilibrium therefore acts as a constraint
incentives
first
Efficiency gains can be reaped
Being part
on the optimal provision
by departing from
of
free-savings.
Incentive Compatible Perturbations
In this section,
we
develop a class of perturbations of the allocation of consumption that
preserve incentive compatibility.
We
then introduce a concept of efficiency, A-efficiency,
that corresponds to the optimal use of these perturbations.
enough
Our
perturbation set
is
large
to ensure that every A-efficient allocation satisfies the Inverse Euler equation.
9
Note that individual asset holdings are not part of this definition. This is convenient because
holdings and taxes are indeterminate due to the usual Ricardian equivalence argument.
17
asset
we show
Moreover,
that A-efficiency
cepts: A-efficiency coincides
we
A
of the
derive
allocation,
creasing utility
in the case
by A
is
and history
t
to decrease utility at this
l,t
9
where the
node by
(5A
1
and compensate by
6>|
+1
=
j4(6^)
new allocation is preserved. We can represent
- 0A u t+1 (9 t+l = u t+1 {9 t+1 + A for all 9\ +v
i
1
',
i'
i
i'
full set of variations
Total lifetime util-
.
the
new
allocation as
1
',
)
)
This perturbation changes the allocation in periods
The
in-
shifts in utility are involved, incentive
compatibility of the
ffj(0«>)
utility func-
a feasible perturbation, of
in the next period for all realizations of
!
unchanged. Moreover, since only parallel
ity is
properties of A-efficient allocations
CRRA form.
Class of Perturbations. For any period
any baseline
some
we term steady-states,
with constant aggregates, which
is
efficiency are closely related con-
with conditional efficiency on allocations that satisfy some
mild regularity conditions. Finally
tion
and conditional
t
and
t
+
1 after
by allowing perturbations
generalizes this idea
history 9 l,t only.
of this kind at
all
nodes:
=
f
wj(6>
.
for all sequences of
{A
1,t
l
{9
)}
)
f
i4(6>
)
+ A^"
such that u
t
(9
)
- SA
j
f
(0
f' f
)
G li(!R+) and such that the limiting con-
l,f
l
1
)
dition
lim £
T
T—»co
E'[AV'V' T ))] =
for all reporting strategies {c\}. This condition rules out Ponzi-like
By
construction, the agent's expected
utility, for
only changed by a
is
00
Y,^MW
i' t
¥'
t
= EjS E
f
))]
f
[w|((7
f' f
''
l
f
(0
))]
+ A!_
1
follows directly from equation (12) that the baseline allocation {u\}
if
and only
the
if
new
allocation {u\}
is
incentive compatible.
of the initial shifter A'_ 1 determines the lifetime utility of the
baseline. Indeed, for
any fixed
infinite history 9
tuting the deterministic strategy o\(d l,t )
OO
£
t
}
that are zero after
node and
its
—
1,0°
new
is
incentive com-
Note that the value
allocation relative to
its
equation (16) implies that (by substi-
9\)
00
p'filcs'"'*)
Note that the limiting condition
for {A'
(16)
.
t=0
t=0
10
10
z
OO
patible
utility.
A _v
constant
It
any strategy {&]},
schemes in
=
YL faffi*)
+ A -i
is trivially satisfied
some period
T.
v ^' ,co e 0CO
for all variations
with
finite
(
17 )
horizon: sequences
This was the case in the discussion of a perturbation at a single
successors.
18
Thus, ex-post realized
utility is the
same along
all
possible realizations for the shocks. 11
Let Y({u^} / A'_ 1 ) denote the set of utility allocations {u\} that can be generated
these perturbations starting from a baseline allocation {u\} for a given initial
a convex
A v
l
__
This
we show
enough
that these perturbation are rich
to deliver the Inverse Euler
equation. In this sense, they fully capture the characterization of optimality stressed
et
{u\, n\, Kt, et]
dominated by another
for
with
{v
utility profile
1
} is A-efficient if it is
feasible allocation {u\,n t ,K t ,et} such that {u\}
feasible
and not
G Y({u\], A^).
l
that conditional efficiency implies A-efficiency, since both concepts
do not allow
changes in the labor allocation. Under mild regularity conditions, the converse
true.
More
signments
on the
precisely, in
{u\, n\}.
12
Appendix A, we define
the notion of regular utility
and
is
also
labor as-
We then show that A-efficiency coincides with conditional efficiency
class of allocations
ular utility
by
[2003].
al.
An allocation
Note
is
set.
Below,
Golosov
by
with regular
and labor assignment
utility
and labor assignments. Indeed, given
{u\, n\}, the perturbations
the utility assignments {u\} such that {u\,n\}
is
Y({u\},
A _^
l
a reg-
characterize
all
regular and satisfies the incentive com-
patibility constraints (12).
We will shortly present a methodology to compute the efficiency gains from restoring
A-efficiency. We refer the reader to the discussion in Section 2 for an extensive motivation
of this question.
Inverse Euler equation. Building on Section
tion
which
is
Proposition
A-efficient
is
the optimality condition for
1.
A
set of necessary
and
we review briefly the
2,
any
A-efficient allocation.
sufficient conditions for
=
<^^
q
^-- = % Ej
1
s
JE\[c'(u't+1 )}
77
LT'(c(«i))
t
=
1/(1
+r
t
)
n\, Kt, et) to be
—
1
J
U'(c(u\
iTtf
A
-
.
(18)
+l ))_
and
rt
is
an allocation {u\,
given by
c'(u\)
where q
Inverse Euler equa-
== FK (Kt+1
N )-5
)-S
t+l/,N
t
(19)
the technological rate of return.
11
The converse
is nearly true: by taking appropriate expectations of equation (17) one can deduce equabut for a technical caveat involving the possibility of inverting the order of the expectations operator and the infinite sum (which is always possible in a version with finite horizon and
finite). This
caveat is the only difference between equation (16) and equation (17).
12
Regularity is a mild technical assumption which is necessary to derive an Envelope condition crucial
for our proof.
tion (16),
19
A A-efficient allocation cannot allow
return, since then equation (15)
ible
agents to save freely at the technology's rate of
would hold
as a necessary condition,
with the planner's optimality condition, equation
Another implication of the Inverse Euler equation
which
incompat-
is
(18).
(18)
concerns the interest rates that
We refer to allocations {u\,n\,Kt,e }
constant aggregates Q = C ss Nt = Nss K = Kss and e = e ss as steady states.
steady state, the discount factor is constant q = qss = 1/(1 + r ss where r ss =
prevail at steady states for A-efficient allocations.
with
At a
,
,
t
t
t
)
t
Fk(Kss
,
N
—
ss )
variables.
Note
5.
that our notion of a steady state
is
only a condition on aggregate
does not presume that individual consumption
It
an invariant distribution
Suppose the
constant nor that there
of the constant relative risk aversion
is
is
consumption.
for the cross section of
function
utility
is
(CRRA) form,
so
— a) for some (7 > 0. The Inverse Euler equation is then EJ[c(w f+1
=
When a = 1, this implies C +\ = (j5/q ss )Ct, so that aggregate consump(fi/ clss)c{u\)
= ^ ss For a > 1, Jensen's inequality implies that
tion is constant if and only if
1/a
< q ss is inconsistent with a steady state. The argument
Ct, so that
Q+i < (p/q ss
for <7 < 1 is symmetric.
thatll(c)
=
c
1-cr
<7
!
/(l
)
]
CT
.
t
/S
ft
)
Proposition
2.
.
CRRA preferences
Suppose
U(c)
=
c
1_cr
/(l
—
a) with
a >
0.
Then
at a steady
state for a A-efficient allocation
(i)
(ii)
a >
1
then q ss
<
/3
and
r ss
>
ifcr<l
then q ss
>
/S
and
r ss
<
if
-1
/3
6
-1
-
1
—
1
j
This result can be contrasted with the properties of steady state equilibria in incomplete markets models,
return.
where agents
As shown by Aiyagari
lower than the discount rate
4
are allowed to save freely at the technological rate of
[1994], in
1//5
—
such cases, the steady
state interest rate is
always
1.
Efficiency Gains
In this section
we
consider a baseline allocation and consider an improvement on
yields a A-efficient allocation.
We
planning problem. In particular,
that allows us to break
show how
up
these problems
mic case we find
the solution into
may
that
define a metric for efficiency gains from this improve-
ment. This leads us to a planning problem to compute these gains.
to solve this
it
we work with
We
then
show how
a relaxed planning problem
component planning problems. Moreover, we
admit a recursive representation. Indeed, in the logarith-
a closed-form solution.
More
20
generally,
we show
that each
component
planning problem
is
mathematically isomorphic to solving an income fluctuations prob-
lem where At plays the
role of wealth.
Planning Problem
4.1
an allocation {u ,n\,Kt,et} with corresponding
l
If
utility profile
{v
t
then
we
KQ =
}, is
not A-efficient,
can always find an alternative allocation {u\,n\,Kt,et} that leaves
changed, so that v
with
1
at least
one
Kq and
=
l
v for
all
i
£
I,
for
some A >
allocations. This
0.
We
<
un-
<
et
we restrict to cases where et =
0,
but economizes on resources: Kq
the rest of the paper,
strict inequality. In
= AQ
et
between these
l
utility
Kq and
et
then take A as our measure of efficiency gains
measure represents the resources that can be saved
in all
periods in proportion to aggregate consumption.
We now
of
introduce a planning problem that uses this metric to compute the distance
any baseline allocation from the
{u\, n\,
K
t,
0},
which
is
feasible
an alternative allocation
{u\,
A-efficient frontier. For
= for all > 0, we seek to maximize A by finding
AQ} with Kq = Ko,
with
n\,K
t ,
any given baseline allocation
et
t
Kf+ + J W [c(u\))dip + AQ <{l-5)K + F(K N
t/
t
i
t
t )
=
0,l,...
(20)
and
K}GY(K},0).
Let
Q =
J*E
z
[c(u|)]<ii/>
The optimal allocation
denote aggregate consumption under the optimized allocation.
{u\, n\, Kt,
XCt} in
AQ *of aggregate resources in every period.
allocation
from the
A-efficient frontier
program
this
A-efficient
is
Our measure
and saves an amount
of the distance of the baseline
is A.
Relaxed Problem
4.2
We now
construct a relaxed planning problem that replaces the resource constraints with
a single present value condition. This
prices or interest rates,
riod.
problem
which encodes the
The relaxed planning problem can be
planning problems corresponding
is
indexed by a sequence of intertemporal
scarcity of aggregate resources in every pe-
further
decomposed
to the different types
i
G
in a series of
component
/.
Given some positive intertemporal prices {Qt} with the normalization that ]2 t= Q QtQ
1
we
=
replace the sequence of resource constraints (20) with the single present value con-
21
}
.
dition
A=EQ(
which
is
( F(Kt/
N + (1 - S)K
t
)
t
- K t+1 -
J
E
!
[c(flj)]
dip)
summing over
obtained by multiplying equation (20) by Qt and
Formally, the relaxed planning problem seeks the allocation {u\,n ,K ,XCt}
l
t
Y({u\},0) that maximizes A given by equation
The connection with the
the resource constraints
(20).
The converse
is
0, 1,
where
problem
{u\, n\, Kt,
AQ}
is
.
.
{u\}
Suppose
the following.
for the relaxed
problem
adapted from Farhi and Werning [2007] and
theorem proved
is
=
<G
that,
satisfies
Then, this allocation solves the original planning problem.
This relaxed problem approach
to the first welfare
t
t
(21).
original planning
given some {Qt}, the optimal allocation
(21)
in
Atkeson and Lucas
is
related
[1992].
also true. Indeed, the prices {Qt} are
Lagrange multipliers and La-
grangian necessity theorems guarantee the existence of prices {Qt} for the relaxed problem.
The following lemma, which follows immediately from Theorem
Luenberger
Lemma 1.
[1969],
provides one such
Suppose {u\,
n\,
1,
Section 8.3 in
result.
K AQ } solves the planning problem and the resource constraints (20)
t,
hold with equality. Then there exists a sequence of prices {Qt} such that this same allocation solves
the relaxed planning problem.
The relaxed planning problem can be decomposed
and
a series of
problem
component planning problems
for capital
where
qt
=
{FK (K t+l ,Nt+1 )
assingment {u[}. The sub-
sufficient for
+ l-5)
£
=
{K
t
+i}.
an interior optimum, are
0,1,...
Qt+i/Qt- This, together with the normalization that
one-to-one relationship between
for capital {Kt
of equation (21) with respect to
The first-order conditions, which are necessary and
t
subproblem
for the utility
maximizes the right hand side
l=q
into a
Yl'tLo
QfQ = L
implies a
{K t+ i} and {Qt}-
The component planning problem
equation (21) with respect to the
for type
utility
i
G
I
maximizes the right hand side of
assignment {u\} £ Y({u\},0).
The
objective
reduces to minimizing the present value of consumption:
c[ul
t=o
(22)
Qo
Each of these component planning problems can be solved independently. Since both the
objective
and the
constraints are convex
it
follows immediately that, given {qt}, the
22
first
order conditions for optimality at an interior solution in the component planning problem
(22) coincide exactly
with the Inverse Euler equation
=
c'(u\)
^E\[c'(u\ +1 )}
£
(18):
=
0,1,...
A Bellman Equation
4.3
In most situations of interest, the baseline allocation admits a recursive representation for
some endogenous
state variable. This is the case
the baseline allocation
depends on the history
marized by an endogenous
condition
x'
.
Note
state vector
In
i
s\
=
M
is
l
value x Q This
.
(x\, 6[)
there
We
must
is
{6\} is a Markov process and
~l
in a way that can be sum-
= M{x\_ lr 0\) and given initial
to
be independent of
i.
Types
the only thing that distinguishes them.
exist a function u
the hat notation
x\
l,t
assumed
state x\ is a function of the history of
what follows we drop
of the agents.
initial
of shocks d
with law of motion
that the transition matrix
then correspond to different
The endogenous
state x\,
whenever
exogenous shocks
such that
u\ (9
and we stop indexing the
l,t
)
6
=
l,t
.
Defining the
u(s\) for all 6
allocation
by the
denote a baseline allocation by u(st) and use the notation
l,t
.
type
c(st) for
c(u(s t )).
The requirement
tive.
that the baseline allocation be recursive in this
Of course, the endogenous
state
and
economic model generating the baseline
allocation.
Bewley economies
In these models, described in
more
At a steady
for
in
A leading
example
Huggett [1993] and Aiyagari
detail in Section 6,
on
restric-
in this paper
each individual
is
income or productivity and saves using a
state, the interest rate
is
[1994].
subject to an
riskless asset.
this asset is constant, so that the agent's solution
can be summarized by a stationary savings
rule.
marized using asset wealth as an endogenous
agent's optimal saving rule.
hardly
law of motion depend on the particular
its
the case of incomplete markets
exogenous Markov process
way is
The baseline
state,
allocation can then be
with law of motion
sum-
M given by the
13
For such baseline allocations
we
can reformulate the component planning problems
recursively as follows.
The idea
utility
is
to take
St
as
an exogenous process and keep track of the additional
lifetime
Af_i previously promised as an endogeneous state variable.
For any date r define the continuation plans {nJ}^L Q with uj(s)
=
n + T (s) and the
t
value function corresponding to the component planning problem starting at date t with
13
Another example are allocations generated by a dynamic contract. The state variable then includes the
promised continuation utility [see Spear and Srivastava, 1987] along with the exogenous state.
23
state
s
£ ^E
Qr+t.
'
K(A_,s;t)
=
inf
[c(Hj)
=
sT
\
s.t.
s]
{uj} G
Y(K},A_).
This value function satisfies the Bellman equation
-
K(A_,s;t) =min[c(w(s)+A_
+ q T E[K{A,s';T +
jSA)
A
The optimization over A
1)
|
s]l.
(23)
one-dimensional and convex and delivers a policy function
is
g(A-,s;r). Combining the necessary and sufficient first-order condition for
A
with the
;T
+
envelope condition yields the Inverse Euler equation in recursive form
c'(u(s)
+ A_-
Sg(A_
i
/
= |E[c
s;T))
/
This condition can be used to compute
An optimal plan for
{<?(/
-
g(A(s
/
{u
T )} by setting w(s
~1
t
With
),s t ;t)
with
t
f
)
initial
}
(u(s
/
)+g(A_,s;T)-^(g(A_
given g(-,
g(-, -;t) for
-,t
+
/
s;T)
/
/
s
l))
\
s]
1).
can then be generated from the sequence of policy functions
=
u(st)
+
A(s f_1 )
condition A_!
a fixed discount factor
q,
=
—
jSA(s
f
)
and using the recursion A(s
t
)
=
0.
the value function
is
independent of time, K(A-,s), and
solves the stationary Bellman equation
K(A_,s) =min[c(u(s)
A
+ A_ - jSA) + qE[K(A,s')
\s}].
(24)
This dynamic program admits an analogy with a consumer's income fluctuation problem
that
is
both convenient and enlightening.
We
transform variables by changing signs and
switch the minimization to a maximization. Let
=
and U(x)
and
satisfies
equation
(24)
—c(—x). Note that the pseudo
A_ = —A^,K{A^;s) = — K( — A_;s)
utility
Inada conditions at the extremes of
its
function
U
is
increasing, concave
domain. 14 Reexpressing the Bellman
using these transformation yields:
K(A-;s) =max\U{-u{s)
+ A- -
jSA)
+
qE[K(A;s') \s]].
A
This reformulation can be read as the problem of a consumer with a constant discount
CRRA U(c) = c 1_(T /(l — a) for a > and
c > 0. Then for a > 1 the function U(x) is proportional to a CRRA with coefficient of relative risk aversion
<7 — cr/((r — 1) and x G (0, oo). For u < 1 the pseudo utility U is "quadratic-like", in that it is proportional
to — (— x)P for some p > 1, and x € (— oo,0].
14
An
important case
is
when
the original utility function
24
is
factor q facing a constant gross interest rate
must decide how much
to save BA;
benefit of this analogy
and used;
sively studied
+r —
B~ l entering the period with pseudo
,
A_, receives a pseudo labor income shock — u(s). The
financial wealth
The
1
models [Aiyagari,
pseudo consumption
that the
is
=
—u(s)
consumer
+ A_ —
BA.
income fluctuations problem has been exten-
at the heart of
it is
then x
is
fictitious
most general equilibrium incomplete market
1994].
With logarithmic utility, the Bellman equation can be simplified considerably. The idea
is
best seen through the analogy, noting that the pseudo utility function
— e~ x
in this case.
It is
known
well
that for a
consumer with
CARA
is
exponential
preferences a one
unit increase in financial wealth, A, results in an increase in pseudo-consumption,
r/(l
+ r) =
1
—
/3
in parallel across all periods
and
that this implies that the value function takes the
states of nature.
form K(A_;s)
It is
x,
of
not hard to see
= e~^~^ A ^k(s).
These
ideas are behind the following result.
Proposition
tion (24)
is
3.
With logarithmic
utility
and constant discount
a,
the value function in equa-
given by
K{A-;s)
=e^-® L -k{s)
t
where function k{s) solves the Bellman equation
k{s)
where
A=
(q/ffi/fa
-
Appendix
This solution
in equation (25),
process for
4.4
The
is
•
(E[fc(s')
|
s]f
(25)
,
"^.
B.
A
can be obtained from k(s) using
m
nearly closed form: one needs only compute k(s) using the recursion
which requires no optimization.
No
simplifications
on the
stochastic
skills are required.
Idiosyncratic Planning
full
1
Ac(s) -^
j
The optimal policy for
Proof. See
1
6)
=
Problem
planning problem maximizes over
also consider a version of the
utility
assignments and
capital. It is useful to
problem that takes the baseline sequence of
25
capital {Kt } as
maximizing A 7 subject to
given. Thus, define the idiosyncratic planning problem as
[E
and {u\} G Y({u\},0). The
consumption that
tion in
Of course,
The
is
i
i
[c{ii t )}dip
+X
I
Ct
efficiency gains
=C
£
t
= 0,1,...
1
A represent the constant proportional reduc-
possible without changing the aggregate sequence of capital.
the total efficiency gains are larger than the idiosyncratic ones:
idiosyncratic planning
problem lends
itself to
A > X
1
.
a similar analysis. In order to avoid
we only sketch the corresponding analysis. We can define a corresponding relaxed problem, given a sequence of prices {Qt}- We can also prove an analogue of Lemma
repetitions,
(1).
This relaxed planning problem can then be decomposed into a series of component
planning problems.
prices {Qt},
we
When
the baseline allocation
is
recursive,
and given
sequence of
a
can study the component planning problems using a Bellman equation
as in equation (23).
The corresponding
first-order conditions also take the
form
of
an
Inverse Euler equation
f
c (ui)
where
qt
=
=
^E\[c'(4 +1 )]
£
=
0,1,...
.
Qt+i/Qt- In other words, the marginal rates of substitution corresponding
to the Inverse Euler
equation
histories in every period.
sarily linked to
E
!
[c'(u\
f
must be equalized across types and
These marginal rates of substitution, however, are not neces-
any technological
cratic efficiency gains
+1 )/ (/3c'(u[))]
rate of transformation as in equation (19).
The
idiosyn-
thus correspond to the gains from equalizing the marginal rate of
substitution across types
and
histories in every period,
without changing the sequence of
capital.
5
Idiosyncratic
In this section,
and Aggregate Gains with Log
we focus on the
problem can be decomposed
gate planning problem.
case of logarithmic
into
utility.
Utility
We first show that the planning
an Idiosyncratic planning problem and a simple Aggre-
We then illustrate this result in the simple benchmark case we the
baseline allocation of consumption
is
a geometric
26
random walk.
.
A Decomposition:
5.1
When utility is
Idiosyncratic and Aggregate
logarithmic, our parallel shifts in utility imply proportional shifts in con-
sumption. This allows us
prove a decomposition
to
a simple Aggregate planning
also
makes
problem involving the idiosyncratic
A using
gains
efficiency gains
A7
.
It
possible to solve the idiosyncratic efficiency gains and the corresponding
it
allocation in closed
consumption. In
form when the baseline allocation
this case, the
A can be solved out almost
lem with
result: the efficiency
optimum
explicitly
in the
recursive and features constant
is
planning problem and the efficiency gains
by combining the solution
of the Idiosyncratic prob-
that of the Aggregate planning problem.
Given A 7 6
allocation
[0,
1),
the Aggregate planning problem seeks to determine the aggregate
{Q, Nt, Kt, AQ}
that
maximizes
=
Ko
A, subject to
Ko,
i
K t+l +Ct + \C <(l-5)K + F(K N
t,
t
t
t
t )
= 0, 1,
.
.
and
00
00
t
zp u(c
t=o
Proposition
4.
= j:p u(c (i-x
t
t
)
t
i
))
t=o
Consider a baseline allocation {u\, n\,K t ,0} such that
fined and finite. Suppose that the
optimum
A7
f
P LI(Ct)
is
well de-
planning problem are such that the
in the Idiosyncratic
resource constraints hold with equality. Let
Yl'tLo
denote the idiosyncratic efficiency gains identified
by the Idiosyncratic problem. Then the total efficiency gains
A
underlying the planning problem
are determined by the Aggregate planning problem.
Proof. See
Appendix
The proof
C. m
of Proposition 4 also establishes that the utility assignment {u\} that solves
the Idiosyncratic planning problem
nal planning
The
problem are related by
efficiency gains
and aggregate
A
-A
and the
u\
—
u\
utility
+ St
where
A can then be decomposed
efficiency gains
X A Aggregate
.
St
=
l
t
U(Ct)
}
that solves the origi-
—
LT(Q(1
—A
7
)).
into idiosyncratic efficiency gains
efficiency gains are simply defined as
XA
X
1
=
7
.
The analysis
knowledge
of
A
of the evolution of the aggregate allocation {Ct,Kt} only requires the
7
,
but can otherwise be conducted separately from the analysis of the
iosyncratic problem.
ministic
The aggregate planning problem
growth model, which, needless
Css
,
Kt
= K ss and
Nt
= N
ss
.
is
Then
id-
simply that of a standard deter-
to say, is straightforward to solve.
suppose that the baseline allocation represents
Q =
assignment {u
a steady state
For example,
with constant aggregates,
the optimized allocation aggregate allocation
27
{Ct,K
t
}
converges to a steady state {C SS ,KSS } such that
Css = F(KSS ) — 5KS s — AC SS We
that the baseline allocation
gregate consumption
Q=C
and labor Nt be constant
Kt
ss
6
+ FK (KSS ,NSS =
)
recursive with state
is
s
1//3
and
6.
and features constant ag-
not necessary for the argument that aggregate capital
It is
.
-
will put that result to use in Section
-
Suppose
I
at the baseline allocation.
The idiosyncratic
we
can then be easily computed. Propositions 2 implies that
can use
efficiency gains
cjt
=
/3
to
com-
pute the idiosyncratic allocation. Proposition 3 then allows us to compute the solution in
closed form. This leads to
A,
where
=
with q
k(s) solves equation (24)
-
(1
=1
ft
f k{s)dtp
We
jS.
can then apply Proposition 4 and com-
pute the efficiency gains A and the corresponding A-efficient allocation by solving the
Aggregate planning problem, a version of the neoclassical growth model. This provides a
complete solution to the planning problem
when
the baseline allocation
is
recursive and
features constant aggregate consumption.
Example: Steady States with Geometric
5.2
Although the main virtue of our approach
line allocations, in this section
tain the
to
we
U(s). Moreover,
_1
E
=
we assume
that
we can flexibly apply it to various base-
begin with a simple and instructive case.
assumption of logarithmic
be a geometric random walk:
is
Random Walk
utility
St+\
=
that log(e)
throughout.
We
£t$t 'with, et i.i.d.
is
We
main-
take the baseline allocation
and
c(s)
=
s,
so that u(s)
=
normally distributed with variance of so that
We also assume that the baseline allocation represents a steady
state with constant aggregates Q = C ss Kt = Kss and Nt = Nss which require E[e] == 1.
We define r ss = Fk(Kss Nss — 5 and q ss = 1/(1 + r ss ). Moreover, we assume that the Eu-1
= exp(oj).
ler equation holds at the baseline allocation, which requires ^ ss = ^Efs
E
[e]
•
[e
]
exp(<r£2 ).
,
,
)
]
Although extremely
pirical
benchmark.
hypothesis
stylized, a
First,
most
random walk
theories
is
an important conceptual and em-
—starting with the simplest permanent income
—predict that consumption should be close
to a
random walk. Second, some
authors have argued that the empirical evidence on income, which
for
consumption, and consumption
component
nious
[e.g.
itself
Storesletten, Telmer,
statistical specification for
The advantage
the intertemporal
is
that
we
wedge and
j5
shows the importance
is
a
major determinant
of a highly persistent
and Yaron, 2004a]. For these reasons, a parsimo-
consumption
may favor a random walk.
obtain closed-form solutions for the optimized allocation,
the efficiency gains.
veals important determinants for the
The transparency
magnitude of efficiency
28
gains.
of the exercise re-
A geometric random
walk however,
special for the following reason.
is
If
we apply
tion 5.1, then the idiosyncratic efficiency gains are zero.
are aggregate. This
Ef
[c'
(u\ +l
)
/
(fie'
is
because
(u\))] are
the sequence of prices
{Q
t
the decomposition of Sec-
The entirety of the efficiency gains
at the baseline allocation, the
marginal rates of substitution
already equalized across types and histories to
}
Q =
given by
QojS
t
1
=
and Qo
(1
—
_1
/3
Therefore,
.
/3)/C ss solve the relaxed
version of the Idiosyncratic problem. These marginal rates of substitution are not equalized,
however,
to the
marginal rate of transformation
—
1
5
+ Fk(Kss Nss ).
,
Therefore, this
section can be seen as an exploration of the determinants of aggregate efficiency gains.
An Example Economy.
random walk
ric
markets. To see
and
disutility
for
this,
consumption
=
They
1
>
/3g
EJ£
-1
].
e f+ i is i.i.d.
Finally,
Skills
with incomplete
over consumption
i+c <
that a
t
evolve as a geometric random
on the economy's
linear savings
budget constraints
q~ 1 a
t
t
>
-\-6 t n t
0.
t
=
0, 1,...
Suppose the
suppose that individuals have no
l
rate of return q~ is
initial assets,
so that
In the competitive equilibrium of this example individuals exert a constant
n ss , satisfying n ss v'(n ss )
no
=
1,
ef-
Individuals can only accumulate a risk-
to the rate of return
face the sequence of
and the borrowing constraint
utility
a geomet-
v(n/9) for some convex function v(n) over work
where
£f+i#t,
a t+
-1
=
paying return q~ l equal
technology.
arises as a competitive equilibrium
can be interpreted as productivity.
walk, so that 6 t+ i
economy where
suppose that individuals have logarithmic
from labor V(n;9)
fort n, so that 6
less asset
Indeed, one can construct an example
and consume
all their
such that
—
ciq
0.
work effort
labor income each period, c t
=
9tn ss ;
assets are accumulated, at remains at zero. This follows because the construction en-
sured that the agent's intertemporal Euler equation holds at the proposed equilibrium
consumption process. The
work
level of
effort n ss is
defined so that the intra-period
consumption-leisure optimality condition holds. Then, since the agent's problem
vex,
follows that this allocation
it
trivially holds,
Although
ric
it is
optimal for individuals. Since the resource constraint
for
very special example economy,
consumption
is
q
linear with a rate of return q~ l
<
1
>
1.
imposes, for a given discount rate
2
exp(cr£
)
<
it
illustrates that a
geomet-
a possible equilibrium outcome.
Partial Equilibrium: Linear Technology.
is
con-
an equilibrium.
this is certainly a
random walk
is
is
We
first
study the case where the technology
This imposes q
/3,
=
q ss
=
/3exp(cf ). Note that
an upper bound on the variance of the shocks
-1
jS
.
Since the idiosyncratic efficiency gains X 1 are zero, the solution of the planning prob-
29
lem can be derived by studying the Aggregate planning problem, which takes
ably simple form. The aggregate consumption sequence
planning problem
is
efficiency gains
that solves the Aggregate
given by
Q=
The
{Q}
a remark-
C ss exp
A and the optimal
(73-g^J
exp(-fcr 2 ).
assignment {u} are then readily computed.
utility
We can also derive the intertemporal wedge t that measures the savings distortions at the
optimal allocation U'(c(u(s )) = /S(l - t^^E [^(c(w(s f+1 )))|s ].
t
Proposition
5.
Suppose that
that the baseline allocation
Q=
C ss
,
f
)
by r
=
=
Then
exp
1
utility
is
logarithmic and that the technology
—
(
the
and
exp(-cr 2 ). The
st
.
Suppose
that the Euler equation holds at
consumption assignment of the solution of the planning problem
i^raof) exp (—tof)
The optimized
linear.
is
random walk with constant aggregate consumption
a geometric
that the shocks e are lognormal with variance of
the baseline.
c(s
is
f
The intertemporal wedge
efficiency gains are given
by A
=
at the optimal allocation
-
1
is
ex P
jjprrr^
given by
is
given
^~6 C^^•
(
allocation has a lower drift than the baseline allocation. Intuitively, our
perturbations based on parallel shifts in utility can be understood as allowing consumers
to
borrow and save with an
artificial
which
idiosyncratic asset, the payoff of
correlated
is
with their baseline idiosyncratic consumption process: they can increase their consumption today
by reducing
their
consumption tomorrow
consumption tomorrow more
consumption
is
low.
in states
in such a
where consumption
The desirable insurance properties
attractive, leading to a front-loading of
is
way that they reduce
their
high than in states where
of these perturbation
make them
consumption. In other words, because our pertur-
bations allow for better insurance, they reduce the benefits of engaging in precautionary
As
savings by accumulating a buffer stock of risk free assets.
front-load consumption,
allocation,
by superimposing
where the variance
in the
a
growth
downward
rate of
drift
a result,
it is
optimal to
exp(— of) on the baseline
consumption of indexes the strength
of the precautionary savings motive at the baseline allocation.
In this example, the Inverse Euler equation provides a rationale for a constant and
positive
to the
wedge r
= 1 — exp (—of)
Chamley-Judd benchmark
in the agent's Euler equation. This
result,
where no such
distortion
is
is
in stark contrast
optimal in the long
run, so that agents are allowed to save freely at the social rate of return.
The efficiency gains
are increasing in of.
gains. For small values of of, the
Note that when of
wedge is given by t ~
gains then takes the form of a simple
of.
0,
there are
The formula
Ramsey formula A ~
30
=
(/3/(l
—
no
efficiency
for the efficiency
2
/3)
)
t 2 /2. At the
other extreme, as of
—
then q
>
the efficiency gains asymptote 100%.
to infinity; in contrast, the cost of the
optimal allocation remains
General Equilibrium: Concave Technologies. In
tion that utility
random walk
= Nss
and Nf
is
logarithmic.
We also assume
,
that the shocks
a linear technology,
surprising, nor
is it
issue arises in the
this section,
e
are lognormally distributed
We
The point
specific to the
Ramsey
may
effects
and
forces
emphasized
a geometric
C ss K — Kss
,
t
that the Euler equation
here.
is
certainly not
Indeed, a similar
literature, the quantitative effects of taxing capital greatly de-
as
it
may,
important to confront
it is
16
made most
that general equilibrium considerations are important can be
We
consider the extreme case of a constant endow-
ment: the economy has no savings technology, so that
the baseline allocation
of intertemporal prices
cjt
is
Q
<
Nt for
A-efficient. This follows since
such that the Inverse Euler equation
endowment
case
is
an extreme example, but
it
t
=
0, 1,
.
.
.
[Huggert,
one finds a sequence
(18) holds.
exchange economy there are no efficiency gains from changing the
the fixed
is
Q =
be greatly reduced. This point
model or
from the following example.
Then
maintain the assump-
depart from the partial equilibrium assumption of
reach meaningful quantitative conclusions.
this issue to
we
that the baseline allocation
pend on the underlying technology. 13 Unsurprising
1993].
that
and consider instead the case of concave accumulation technologies.
argue that the efficiency
clearly
is
finite.
representing a steady state with constant aggregates,
holds at the baseline allocation.
We
The reason
implying that the present value of the baseline consumption allocation goes
1,
>
— — log(/3)
Thus, in
allocation.
17
this
Certainly
serves to illustrate that general
equilibrium considerations are extremely important.
Consider
tion 5.1,
we
now a neoclassical production function F(K, N). Applying the results in Sec-
can decompose the planning problem into a idiosyncratic planning problem
and an aggregate planning problem, with a corresponding decomposition
gains.
A
;
=
We
0.
have already argued that the corresponding
for efficiency
efficiency gains are equal to zero
Therefore, and just as in the partial equilibrium case,
all
the efficiency gains are
aggregate.
The aggregate planning problem
is
extremely simple.
It
seeks to determine the ag-
15
Indeed, Stokey and Rebelo [1995] discuss the effects of capital taxation in representative agent endogenous growth models. They show that the effects on growth depend critically on a number of model
specifications. They then argue in favor of specifications with very small growth effects, suggesting that a
neoclassical growth model with exogenous growth may provide an accurate approximation.
16
A similar point is at the heart of Aiyagari's [1994] paper, which quantified the effects on aggregate
savings of uncertainty with incomplete markets. He showed that for given interest rates the effects could
be enormous, but that the effects were relatively moderate in the resulting equilibrium of the neoclassical
growth model.
17
This point holds more generally in an endowment economy with CRRA utility when the baseline allocation is a geometric random walk.
31
.
gregate allocation {Q, Nt,
K
t ,
ACt} that maximizes the proportional amount of resources
saved in every period A, subject
to the resource constraints
Kt+1 + Q + XC <(l-S)K + F(K Nss
SS
the restriction that the initial capital
allocation
and the requirement
t,
t
is
equal the
initial capital
Kq
—
.
.
Kq of the baseline
^
ss)
=
t=0
a simple modification of the neoclassical
is
= 0, 1,
that
£/5'U(C«)
This problem
t
)
growth model. The solution
=
involves a transition to a steady state with an interest rate equal to fss
which
efficiency gains,
reflect the strength of the
how much lower than fss
This, in turn
is
the interest rate
r ss
=
1
consumption
Suppose one wishes
to
—
The
1.
precautionary savings motive, depend
/
(/3
exp (of) )
depends on the variance of consumption growth
Empirical Evidence.
1//3
at the baseline allocation.
of.
random walk
accept the
What does
as a useful empirical approximation.
specification of
the available empirical
evidence say about the crucial parameter of?
Unfortunately, the direct empirical evidence on the variance of consumption growth
is
very scarce, due
of consumption.
18
to the unavailability of
Moreover,
much
of the variance of
may be measurement error or attributable
manent changes we are interested in here.
There are a few papers
that,
somewhat
the variance of consumption growth.
vide a sense of what
good quality panel data
broad categories
consumption growth
to transitory taste
We briefly review some
PSID
in panel data
shocks unrelated to the per-
tangentially, provide
currently available. Using
is
for
some
direct evidence
of this recent
work
on
to pro-
data, Storesletten, Telmer,
and
Yaron [2004a] find that the variance in the growth rate of the permanent component of
food expenditure
data, but
impute
lies
total
between l%-4%.
Blundell, Pistaferri,
18
face value, their estimates
statistical
19
model
of
consump-
For the United States the PSID provides panel data on food expenditure, and
that food expenditure
See their Table
3,
is
is the most widely used
work by Aguiar and Hurst [2005]
unlikely to be a good proxy for actual consumption.
pg. 708.
See their footnote
19,
pg
21,
which
a vari-
imply enormous amounts of mobility and a very large
source in studies of consumption requiring panel data. However, recent
show
PSID
Krueger and Perri [2004] use the panel
element in the Consumer Expenditure survey to estimate a
At
[2004] use
consumption from food expenditure. Their estimates imply
ance of consumption growth of around 1%. 20
tion.
and Preston
refers to the estimated autocovariances
32
from
their Table VI.
0.25
0.15
0.05
x 10
Figure
2:
Efficiency gains in
x 10
% when baseline consumption is
a geometric
random walk
consumption and the Euler equation holds as function of of. The left panel shows the
gains in partial equilibrium and the right panel shows the gains in general equilibrium.
The bottom dotted line corresponds to /? = .96, the middle dashed line to /3 = .97 and top
for
plain line to
=
j8
.98.
— around 6%-7%—although most of
variance of consumption growth
tributed to a transitory, not permanent, component.
enormous empirical challenges faced
21
this
should be
at-
In general, these studies reveal the
in understanding the statistical properties of house-
hold consumption dynamics from available panel data.
An
interesting indirect source of information
is
the cohort study
by Deaton and Pax-
son [1994]. This paper finds that the cross-sectional inequality of consumption
the cohort ages.
The
rate of increase then provides indirect evidence for of; their point
estimate implies a value of of
ology finds
rises as
much lower
—
0.0069.
However, recent work using a similar method-
estimates [Slesnick and Ulker, 2004, Heathcote, Storesletten, and
Violante, 2004].
Calibration.
21
We now
They specify
a
display the efficiency gains as a function of of.
Markov
We
choose three
transition matrix with 9 bins (corresponding to 9 quantiles) for consumption.
We
thank Fabrizio Perri for providing us with their estimated matrix. Using this matrix we computed that
had an average across bins of 0.0646 (this is for the year
2000, the last in their sample; but the results are similar for other years).
the conditional variance of consumption growth
33
possible discount factors
imposes q
with
cc
=
=
/3
=
0.96, 0.97
and
/Sexp(of). For the neoclassical
0.98.
For the linear technology model,
growth model, we take F(K,N)
this
= K N
a
l
~a
1/3. Figure 2 plots the efficiency gains as a function of exp(u^) for the linear
technology model and for the neoclassical growth model. The figure uses an empirically
relevant range for exp(crj).
Consider
first
the linear technology model. For the parameters under consideration,
the efficiency gains range from minuscule
discount factor
effect of the
that
is,
moving from
this, interpret
/3
=
is
f>
.96 to
/3
-
less
than 0.1% to very large
-
over 10%. The
nearly equivalent to increasing the variance of shocks;
=
.98
has the same effect as doubling
(i\.
To understand
the lower discounting not as a change in the actual subjective discount, but
as calibrating the
model
to a shorter period length.
But then holding the variance of the
innovation between periods constant implies an increase in uncertainty over any fixed
length of time. Clearly, what matters
the
is
amount
of uncertainty per unit of discounted
time.
Consider
now
the neoclassical
growth model. There again, there
ation in the size of the efficiency gains,
the efficiency gains are
ences in interest rates
is
considerable vari-
depending on the parameters. However, note
that
much smaller than in the linear technology model. Large differfss — r ss are necessary to generate substantial efficiency gains: it
takes a difference of around 2%, to get efficiency gains that are bigger than 1%.
Lessons. Three lessons emerge from our simple exercise.
tentially far
from
trivial.
First, efficiency
gains are po-
Second, they are quite sensitive to two parameters available in
our exercise: the variance in the growth rate of consumption and the subjective discount
factor. Third,
We
general equilibrium forces can greatly mitigate them.
conclude
that, in
our view, the available empirical evidence does not provide
liable estimates for the variance of the
permanent component
of
re-
consumption growth.
Thus, for our purposes, attempts to specify the baseline consumption process directly are
impractical. That
nious
is,
even
if
one were willing
statistical specifications for
to
assume the most
consumption, the problem
is
stylized
that the
remains largely unknown. This suggests that a preferable strategy
is
and parsimo-
key parameter
to use
consumption
processes obtained from models that have been successful at matching the available data
on consumption and income.
It
general equilibrium models
crucial to the quantitative assessment of the
is
also follows
from
the efficiency gains.
34
this discussion that
it is
crucial to use
magnitude
of
An Aiyagari Economy
6
We now we take the steady state equilibrium allocation of an incomplete market economy.
We adopt the calibrations from the seminal work of Aiyagari [1994].
The Aiyagari Economy. Aiyagari
Bewley economy, where
[1994] considered a
tinuum of agents each solve an income fluctuations problem, saving
Efficiency labor
is
an
£f is
i.i.d.
random
standard deviation
labor supply
is
=
cr£
N =
ss
variable
n ss
—
w ss n
t
+c <
t
is
(1
+ rss )a + w
is
not allowed:
t
ss
at
>
given by the net marginal product of
St,
librium requires average assets, under
to
equal the capital stock
as a
Markov process. We
their initial conditions
i
is
ip,
<
/3" 1
1,
agent optimization leads to an
denote by
a function of the state variable
St
To apply
as follows. For
=
nous
state.
so,
which evolves
result
by
ip.
developed in Sec-
,
method with Af_i
{Q}. Using the resource
as the
endogenous
this
for
state variable
consumption
and
St
as the exoge-
in equation (23),
and
we compute an aggregate sequence of consumption
constraints, we can solve for A and a sequence for capital {Kt}
assumes no
ip,
taxation.
It is
straightforward to introduce taxation.
conjecture that since taxation of labor income acts as insurance,
to net
(at, nt)
we seek the appropriate sequence of discount factors {cjt}
{qt} we solve the non-stationary Bellman equation (23) using
Using the underlying policy function
For simplicity
.
distributed according to the invariant distribution
integrating in every period over
22
=
Kss
steady-state equi-
this result,
any given
a policy iteration
A
ip.
take this as our baseline allocation with agents distinguished
Numerical Method. To solve the planning problem we use the
tion 4.2.
w ss =
capital, r ss =
given by the marginal product of labor
is
invariant cross-sectional distribution for
Individual consumption
Agents
0.
—
which we
22
efficiency
nt
r ss
S.
zero and
wss is the steady-state wage.
where
For any interest rate
Fk(Kss,Nss )
mean
budget constraints
the interest rate
)
distributed with
.
The equilibrium steady-state wage
Fn(KSS/ Nss and
£f
)
assumed Normally
In addition borrowing
0, 1,
- p) log(n ss +
'
a t+ i
=
(1
With a continuum of mass one of agents the average
.
face the following sequence of
t
+
plog(nf-i)
Labor income is given by the product
for all
in a risk free asset.
specified as a first-order autoregressive process in logarithms
logOt)
where
a con-
it
income. Lower uncertainty will then only lower the efficiency gains
35
However, we
effectively reduces the variance of shocks
we compute.
N
= Kss
that has Kq
From
.
Section 4.2,
a solution. Otherwise,
Ffc(Xf,
ss
)
—
Calibration.
we
We
FK(K ,Nss ) —
+
new sequence
take a
simulate the
with a share of capital of
CRRA
1
5
t
is
met
this constitutes
of discount factors given
by
=
q'
t
(1
+
S)' 1 and iterate until convergence to a fixed point.
The discount
gari [1994].
=
the condition l/q t
if
so that U(c)
=
c
economy
factor
set to
is
parameter values considered in Aiya-
for all the
/3
=
.96,
the production function
The
0.36, capital depreciation is 0.08.
1-D
7(l —
utility
is
Cobb-Douglas
function
is
assumed
with a G {1,3,5}. Aiyagari argues, based on var-
cr)
ious sources of empirical evidence, for a baseline parametrization with a coefficient of
autocorrelation of p
=
0.6
and
income of 20%. Following
a standard deviation of labor
we also consider different values for the coefficient of relative risk aversion
Aiyagari,
,
the
autocorrelation coefficient p G {0,0.3,0.6,0.9} and the standard deviation of log income,
Std(log(n
))
f
=cr£ G
{0.2,0.4}.
These values are based on the following studies. Kydland [1984] finds that the stan-
dard deviation
cr£
of annual hours
from the PSID and the NLS,
worked from PSID data
Abowd and Card
and
[1987]
around 15%. Using data
is
Abowd and Card
that the standard deviation of percentage changes in real earnings
about 40% and 35% respectively. They report a
about
0.3,
resulting in a estimate of
cr£
first
Some
recent studies, within a
life
PSID
27%
data,
to
The
cently
by
first
Storesletten et
al.
[2004a]
and
Storesletten et
is
al.
due
shocks. This leads to estimates of p around or above 0.9
0.3
gari [1994].
and
0.6,
on
the high
end of the range
The second view, dating back
and developed Guvenen
[2007]
of
to Lillard
and Guvenen
model where agents
values for p and
a life-cycle
aE
.
an alternative approach and
in the
observed
Two views have been
cross-
articu-
cr£ ,
face individual-specific
around
0.8
and 25%
[2004b], posits that the increase
to large
and
we have shown,
a
and Weiss
is
[1979]
and Hause
better explained
profiles.
respectively. Since
for
<j£
[1980],
argues that the increase in cross sec-
income
key object
and persistent income
a range of estimates for
by an
alternative
This view leads to lower
both approaches are based
framework, the estimates are not directly relevant
setup. Indeed, as
of
for
[1996]
parameter values explored by Aiya-
[2009],
tional earnings inequality over time within a cohort
on
are
view, dating back to Deaton and Paxson [1994] and developed most re-
in earning inequality over time within a cohort
between
Heaton and Lucas
by using the increase
sectional inequality of earnings over time within a cohort.
lated.
40%
cycle context, consider
estimate a process for log earnings indirectly,
and annual hours
order serial correlation coefficient p of
of 34%. Using
estimate a range of 0.23 to 0.53 for p and a range of
[1989] find
our analysis
is
for
our
infinite
horizon
the conditional variance
consumption growth which depends on the conditional variance of permanent income
36
1.68
1.66
1.64
1.62
1.58
1.56
1.54
150
Figure
for
Path of aggregate consumption for the baseline and the optimized allocations
1, Std(lqg(ttf) - log(n _i)) = .4 and p = 0.9.
3:
=
cr
t
growth. Life-cycle models incorporate a retirement period, generating a longer horizon
consumption than
for
for labor income. This reduces the
impact of permanent income
shocks on consumption. Taking this into account, estimates for the persistence p and the
standard deviation of labor income uz derived in the context of a life-cycle model would
to
be adjusted downwards in order to be used in our numerical exercises.
We find
that the optimized allocation always features aggregates converging to
have
therefore
Results.
new
steady state values:
of aggregate
consumption
represents a steady state,
for the
Q—
for
C ss K — K ss
,
>
t
,
qt
—
cj ss
>
as
one particular parameter
aggregate consumption
its
optimized allocation
»
is initially
above
is
this level,
tually reaching a new, lower, steady state. For the
t
— cop
>
Figure 3 plots the path
case. Since the baseline allocation
constant. Aggregate
consumption
but declines monotonically, even-
same parameter
values, Figure 4
a typical sample path for individual income, cash in hand, consumption and
timized consumption appears more persistent and displays a
initial
1,
2
and
=
1),
Table
1
measure
includes the idiosyn-
and aggregate components X and X A For references, the tables show the baseline
1
.
and optimized steady
state interest rates r ss
replicates Aiyagari's Table
the
Op-
trend in the
3 collect the results of our simulations. All tables report our
of efficiency gains, A. In the logarithmic utility case (a
23
utility.
periods.
Tables
cratic
downwards
shows
II
r ss
.
The second column, showing
r ss ,
(pg. 678).
In the case of logarithmic utility functions,
more general
and
we provided
a formal proof of this result in Section 5.1. In
CRRA utility function case, we rely solely on our numerical
37
results.
Income
Cash
100
50
Consumption: baseline
Figure
4:
log(«f_i))
The
(solid)
150
hand
in
50
and optimized (dashed)
Utility:
100
baseline (solid) and optimized (dashed)
=
Simulation of a typical individual sample path for a
=
.4andj0
=
150
Std(log(ftf)
1,
—
0.9.
baseline's interest rate
r ss is
decreasing in the size and persistence of the shocks,
Precautionary saving motives are
as well as in the coefficient of relative risk aversion.
stronger and depress the equilibrium interest rate. The interest rates at the optimized
allocation are consistent with the conclusions from Proposition
1/ ft
—
1
when
utility is logarithmic,
that fss increases with
These comparative
<j
and the
and
size
f ss
>
1//S
—
1
when a >
Moreover,
statics for r ss are precisely the reverse of those for r ss
and Euler equations,
cr
and —a,
power coefficients
we
find
This
.
is
perhaps
in the Inverse Euler
respectively.
Efficiency gains are increasing with the variance of the shocks
They
1.
=
and the persistence of the labor income shocks.
not surprising, given the reversal in the sign of the
tence.
In particular, fss
2.
and with
their persis-
also increase with the coefficient of relative risk aversion. Aiyagari argues,
based on various sources of empirical evidence,
of autocorrelation of p
=
preferred specification,,
we
0.6
for a parameterization
with a coefficient
and a standard deviation of labor income
find that efficiency gains are small -
of 20%. For this
below 0.2%
for all three
values for the coefficient of relative risk aversion.
In the logarithmic case, efficiency gains are moderate - always less than 1.3%.
shows that idiosyncratic efficiency gains A'
gains X A Our finding that idiosyncratic gains are
decomposition along the lines of Section
completely dwarf aggregate efficiency
5.1
.
modest could have perhaps been anticipated by our
example, where idiosyncratic gains are zero.
cratic
component require
Our
illustrative
geometric random-walk
Intuitively, efficiency gains
differences in the expected
38
from the idiosyn-
consumption growth
rate across in-
o-
=
1
and Std(logQf) - log(n _!))
f
=
.2
Efficiency Gains
?ss
''ss
Idiosyncratic
4.14%
4.13%
4.09%
3.95%
4.17%
4.17%
4.17%
4.17%
0.0%
0.0%
0.0%
0.2%
p
0.3
0.6
0.9
a
=
1
and Std(log(n
Aggregate
0.0%
0.0%
0.0%
0.0%
)
f
- log(n
t
Borrowing
0.0%
0.0%
0.0%
0.1%
Total
0.0%
0.0%
0.0%
0.2%
-i))
-
-4
Efficiency Gains
1'ss
1'ss
Idiosyncratic
4.06%
3.97%
3.79%
3.38%
4.17%
4.17%
4.17%
4.17%
0.1%
0.2%
0.4%
1.2%
p
0.3
0.6
0.9
Table
dividuals.
.
'
Aggregate
0.0%
0.0%
0.0%
0.1%
Borrowing
0.1%
0.2%
0.3%
1.1%
Total
0.1%
0.2%
0.4%
1.3%
Efficiency Gains for replication of Aiyagari [1994] when
1:
a
=
1.
When individuals smooth their consumption over time effectively the remain-
ing differences are small
— as a
result, so are the efficiency gains. Efficiency
gains from the
aggregate component are directly related to the difference between the equilibrium and
optimal stead-state capital. With logarithmic
tween the equilibrium steady-state
utility this is equivalent, to the difference be-
interest rate
and
l
ft~
—
1,
the interest rate that obtains
with complete markets. Hence, our finding of low aggregate efficiency gains
related to Aiyagari's [1994]
or for
moderate
directly
conclusion: for shocks that are not implausibly large
risk aversion, precautionary savings are small in the aggregate, in that
steady-state capital
our Table
main
is
and
interest rate are close their
complete-markets
levels, as
shown
in
1.
For the range of parameters that
we
cule - less than 0.1%- to very large -
consider, the efficiency gains range
more than 8.4%. However,
from minus-
efficiency gains larger
than 1.3% are only reached for combinations of high values of relative risk aversion (J
greater than 3-
and both
large
and highly persistent shocks - a standard deviation
labor income of 40%, and a mean-reversion coefficient p greater than
The Role
of Borrowing Constraints.
The market arrangement
imposes borrowing constraints that limit the
rally.
/3(1
At the baseline
+ r ss )Ej.
ever, for
low
/
[L7 (c(u[
ability to trade
allocation, the Euler equation holds
+1 ))] holds
levels of assets
when
assets
of
0.6.
in Aiyagari's
economy
consumption intertempo-
with equality U'(c(u\))
=
and current income are high enough. How-
and income, the agent may be borrowing constrained so
39
that
.
<7
=
3 and Std(log(n f )
— log(n
t
_i))
=
.2
Efficiency Gains
ss
Total
4.21%
4.37%
4.45%
4.75%
0.0%
0.0%
0.2%
0.7%
I
SS
4.09%
4.02%
3.88%
3.36%
0.3
0.6
0.9
a
-=
Borrowing
0.0%
0.0%
0.2%
0.7%
3 and Std(log(n f ) - -log(n f _i))
=
.4
Efficiency Gains
I'ss
1~ss
Total
3.77%
3.47%
2.89%
1.47%
4.62%
4.77%
5.03%
5.56%
0.2%
0.5%
1.2%
4.2%
p
0.3
0.6
0.9
Table
Borrowing
0.2%
0.5%
1.2%
3.3%
Efficiency Gains for replication of Aiyagari [1994]when
2:
the Euler conditions holds with strict inequality U'(c(u\))
>
B(l
+
r ss
u
=
3.
l
)Ef [U'(c(u t+1 ))]
In contrast, since our planning problem does not impose arbitrary restrictions
the perturbations,
it
places no such limits
on intertemporal
reallocation of consumption.
Thus, the perturbations can effectively undo limits to borrowing. As a
efficiency gains
we compute can be
on
result, part of the
attributed to the relaxation of borrowing constraints,
not to the introduction of savings distortions.
To get an idea of the efficiency gains that are obtained from the relaxation of borrowing
we performed the following exercise. The basic idea is to use the perturbations to construct a new allocation where the Euler equation always holds with equality.
That is, the new allocation stops short of satisfying the Inverse Euler equation, so it does
constraints,
not introduce positive intertemporal wedges. Instead,
poral
wedges
that
were present due
to
borrowing
it
removes the negative intertem-
constraints.
We compute
the resource
savings from this perturbations as a simple measure of the efficiency gains due to the
relaxation of borrowing constraints.
24
More precisely, we seek to determine
B stands
for
borrowing, that
the unique allocation {u\'
satisfies the
following constraints.
,
n\,
B
Kf, A CSS }, where
First,
we impose
that
the allocation be achievable through parallel perturbations of the baseline allocation, and
deliver the
same
utility
K }eY(K},0).
B
24
Theoretically,
posed
it
isn't
evident that these perturbations actually produce positive efficiency gains, as op-
to negative ones. In our cases, the
measure always came out
40
to
be positive.
N
E
a
=
5
and Std(log(n
f
)
-
=
log(n f _i))
.2
Efficiency Gains
ss
Total
5.34%
5.39%
5.48%
5.72%
0.0%
0.0%
0.2%
1.0%
ss
'
4.01%
3.89%
3.61%
2.66%
0.3
0.6
0.9
a
=
5
and Std(log(n
f )
— log(n
f
Borrowing
0.0%
0.0%
0.2%
1.0%
=
_i))
.4
Efficiency Gains
r ss
r$s
Total
0.6
3.43%
2.90%
1.95%
0.9
-0.16%
5.54%
5.55%
5.58%
5.52%
0.2%
0.7%
2.1%
8.4%
p
0.3
Table
Efficiency Gains for replication of Aiyagari [1994]when
3:
we impose that the
Second,
+J
K?+l
and require
B
l
[c(wj'
)
Pr(^
l
f
)]
dip
+ XB C <
SS
(1
- S)Kf + F{K?,
,
n\,
B
))
=
fS[l-6
K B A B Q}
,
+ FK (K?+1 Nss )]E\[U
,
not satisfy the Inverse Euler equation and
of the efficiency gains deriving
The rightmost column
gains
f
(c(u
A and A
B
and persistence
is
less
l '
t
is
Note
= 0, 1,
•
•
Ko of the baseline
B
+1 ))}.
that this allocation does
therefore not A-efficient.
We adopt X B
drawn by comparing
of shocks,
as our
from the relaxation of borrowing constraints.
and
all utility
XB
that
can be
at-
most important conclu-
the efficiency gains
A B deriving from the relaxation of borrowing
come from
t
)
in each of table reports the efficiency gains
sion of our numerical exercise can be
for all size
ss
KB =
tributed to the relaxation of borrowing constraints. Perhaps the
efficiency gains
5.
can improve on the baseline allocation by insuring that
the Euler equation holds for every agent in every period.
measure
=
we impose that the Euler equation hold for every agent in every period
W{c(u)
allocation {u (
(7
resource constraints hold
that the initial capital be equal the initial capital
allocation. Finally,
The
Borrowing
0.2%
0.7%
2.0%
6.3%
constraints.
specifications,
most
A with
We
the
find that
of the efficiency
the relaxation of borrowing constraints. Indeed, the difference
between
than 0.1% except for the largest and most persistent shocks - a standard
deviation of labor income growth of 40%, and a mean-reversion coefficient p greater than
41
In other words,
0.9.
their
most
of the efficiency
consumption over time by
come from allowing agents
alleviating
to better
smooth
borrowing constraints, rather than by
opti-
mally distorting savings as prescribed by the substitution the Inverse Euler equation to
the Euler equation.
Conclusions
7
The main contribution
role that savings distortions
to evaluate the welfare
paper
of this
is
to
provide a method for evaluating the auxiliary
may play in social
We put it
insurance arrangements.
importance of recent arguments
and
for distorting savings
to
use
capital
accumulation based on the Inverse Euler equation.
We
quantitative explorations of
modate
several extensions
it.
and
The method developed here
it
may be
paper [Farhi and Werning, 2008]
is
flexible
of interest to investigate
the quantitative conclusions found here for the
arate
own
believe that the methodological contribution of this paper transcends our
enough
how
these
to
accom-
may
benchmark Aiyagari economy. In
we pursued
affect
a sep-
such an extension using Epstein-Zin
preferences that separate risk aversion from the intertemporal elasticity of substitution. In
ongoing work,
finite
we study an environment with overlapping-generations instead
of the in-
horizon dynastic setup used here. This introduces intergenerational considerations
that are absent in our setting. In particular,
with an infinitely-lived agent
that the planner chooses a declining sequence for capital. In
model
this feature, in
and of
itself, affects
wards finding lower potential
we have shown
an overlapping-generations
future generations negatively. This points to-
efficiency gains.
However, new opportunities
for intertem-
poral reallocations do arise since consumption can be shifted across generations, for any
given aggregate sequence of consumption. Finally, as discussed in Section
6,
an overlap-
ping generation requires a different calibration of the income process. Despite these
considerations, the basic
methodology and decompositions introduced here
such extensions.
42
new
are useful for
Appendix
A
Proofs for Section 3.2
we
To simplify the arguments,
T >
assume
first
We also assume that V (n; 0)
0.
that the horizon
is finite
with terminal period
continuously differentiable with respect to n and
is
0.
We also assume throughout that shocks are continuous.
Consider an assignment for
U
value
!,t
(0
)
u
conditional on history Q
l
(
e
ut
)
LP
=E
We
and labor {u\,n\}.
utility
l,t
define the continuation
as follows
Kt+*l0
y+ ')-V>i
f'
f+s
»,t
);0JM-s
+B
s=0
Fix a history
(e
tr|''
=
1'3
)
e
{' s
z,f
and consider
except
if
6
{'s
a report
f' f
>z
in
,
G 0. Define the strategy
0J
which case
This strategy coincides with truth-telling except in period
report
can be different from the true shock
6\
For clarity and brevity
we
as follows:
cr|.'
t
after history 9
l,t
where
the
0\.
,s
use the notation
!,s
for crJ (0 ).
We
denote the continua-
t
tion utility after history B hS
for short. Similarly
this
O
Qht-l
denote by
E
i'
[IP (§
^
say that the assignment for
©^
^
utility
continuation utility IP (0 Z/f
s
;6
i' t
)]
or
E
and labor [u
;
0,-^) is
differentiable with respect to the true shock
rVtnie '*)-^
1
1
30
is
(op
under the strategy op, by IP
1 '*
[IP (&'*)
bounded by
9\,
l
t
10*'*]
{ u\,
n\ }
,
!,t
(0''
s
;
&'*)
the expectation of
t.
f
>
and
absolutely continuous in the true shock
and the
^-E
~1
)
which
is
derivative,
§i,t+A
ip
which we denote by
\Qi,t
integrable with respect to
same assignment
the corresponding continuation
that {u}}
or IP
J
del
a function b(9\;
that share the
)
n\ } is regular if for all
,
Consider two regular, incentive compatible, assignments for
and
f 's
(0
continuation utility conditional on the realized history 6 l,t at date
We
Q\,
we
>z
utilities.
for labor [n\ }.
We now show
utility
Q\.
and labor {u\,n\}
Denote by
that there exists
{
IP
}
and
A_j G
1R
{
U
1
}
such
eYKnj}^).
Note we can
restate the result as follows: for all
43
t
with
<
t
< T — 1 and
for all
e Qt there
Qi,t-i
exists
f
A
{d
ht ~
l,t
and consider the
=u
1
,^)
-4
+
where
9
{'
=
t+s
|
i
(e
i' t
w
A
(e
i
it
{u\,n\}
'1
1' 1
)
G
and
x
e
+ a (e^-
f
)
(27)
Using Theorem 4 in Milgrom and Segal
.
-\ety
v
B
(
fiW
«)+^iw E
;
)
(
'-'- 1
l
,
1
6 t+s ) for all s
...,
0J; 0J +1 ,
for [u\, n\).
{w*f , n[} share the
>
L7
and
Using the
/5;,f+l
Z
«
1(^-1^1)
6>
^'
t
"s
fact that
same labor assignment,
=
0'' f
-s
^1
for all s
>
(28)
0.
shocks are continuous and
this implies that there exists
R such that
_ 7Ti
u
fpi,t\
'
(<9
and regularity implies the following Envelope condition
The same expression holds
that
;
l
strategies trJ
1
(fl
= u
(&'*)
1
[2002], Incentive compatibility
LT'fV''-
R such that
G
)
u
Fix a history 9
l
(pi,t
1A
+A
e
ii.t-1
[6
e
n
r
„
5
'<
{
)ey«t=
ei
_ U i(Qut+i\
Qi(0i,t+i\
i,t+A
E
\{e\_ lt e
del
For t = T, the last term on the right hand side vanishes and we have that for all
T~ 1
qi,t
G R such that (27) holds. We proceed by induction on
e ©r+i^ there exists A (0
)
Z
t.
Suppose
holds.
that
t
>
'
and that
Then applying
(28)
and using the
a
we immediately find
for all 6 ht
t+1
G
,
all
6 ht
1
G
A
(6^) G
R
such that
(27)
fact that
0"
gji0M E a
that for
there exists
He
i,t-\1
1 ''-.
& there
5i>
.,?'*)
exists
A
'
Q*'(V) =LT(0
i' f
)+A(0
,;
o,
l,t
(9
1
)
G
R such that
'- 1
The proof follows by induction.
When
the horizon
limit condition. For
is infinite,
the regularity conditions
must be complemented with
example, expanding our Envelope condition over two periods,
find
44
a
we
U
l
=
f
(V- )
IF (V'
+ A (f^
f
)
1
)
+jS
2 /'
ae~r !=(?l
/fi
't+i
E
E
!
?}+!=
de t+i
Consider the sequence which
utility
(T
if)
if
and labor assignment
is
n
{u\, n\}.
)
The
first
1
|(T
!
'
f
>
f+1
)
1
iTiK^'*- ,^) dm
element of the sequence
t+1
\(5i,t-l1 Sifr
i,t+A
!
^
'
J
second element of the sequence
k^- ,^)
is
del.
is
% d
de\
e\=e\
U
E
i
f^i,t+i\
Our proof then carries over if the condition
of iterations goes to infinity
B
f+2
constructed by iterating our Envelope condition for the
u
Similarly, the
! '
^ f+ i
is
kV^^^+1)] dVjK^- ,^)
1
that the limit of these terms
equal to zero
is
added
„ej
when the number
requirements for regularity.
to the
Proof of Proposition 3
With logarithmic
utility the
Bellman equation
K(s,A_) =min[sexp(A_
=
- jSA) + qE[K{s',A)
Substituting that K( A_ ,s)
-
/3)A_)
=
\
min[sexp((l- 6)A_+£(A_ - A))
i
A
k{s) exp((l
is
=
fc(s)exp((l
min[s exp((l
-
— j6)A_)
/3)A_
s]]
+ qE[K(s',A)
s}}
\
gives
+ /S(A_ -
A))
+ qE[k(s') exp((l - /3)A)
and cancelling terms:
fc(s)
=
min[sexp(/S(A_
- A))
+.flE[fc(s')exp((l
-
«)(A
A
=
/
min[sexp(-/3<i)+<7E[/c(s )exp((l
45
-
fi)d)
s]}
\
-
A_))
|
si]
|
s]],
=
+ qE[k(s')
min[sexp(-Bd)
s]
\
exp((l
-
B)d)]
d
where d = A — A_.
Define q(s)
=
qH[k(s
We
f
)
|
can simplify
s]/s
further.
and
=
M{q)
The
one dimensional Bellman equation
this
min[exp(-jSd)
+
#exp((l
-
j8)d)].
first-order conditions gives
=
Bexp(-^d)=q(l-B)exp({l-B)d)
Substituting back into the objective
Mtf)
we find
= i^'expt-M =
T^exp (-/Jlog-L)
^ = B^,
1
(l-B)iis
a constant defined in the obvious
way in
terms of
The operator associated with the Bellman equation
T[k)(s)
where
A =
Bq?
Combining
= sM
(
= {q/Bf/{\ -
q
(29)
_f
that
=
where B
*=log
*M£Ll£ \ = As
1
is
-?
B.
then
(E[fc(s')
|
s])
p
,
1
B) "^.
the Bellman k(s)/s
= M(q) = AqP
with equation
(29) yields the policy
function as a function of K(s). This completes the proof.
C
Proof of Proposition 4
Consider the aggregate allocation {Ct,Nt,Kt,XCt} that solves the Aggregate planning
problem and the
utility
assignment {u\} 6 Y({u\},0) that solve the Idiosyncratic plan-
ning problem as well as the corresponding idiosyncratic efficiency gains X 1
.
Since the resource constraints hold with equality in the Idiosyncratic planning prob-
lem,
we know from Lemma
1
that there exists a sequence of prices {Qt}, such that
!
c (u\)
=
q
JE)[c'(u\ +l )}
46
£
= 0,1,.-.
(30)
N
where
qt
=
Qt+i/Qt- Moreover, the sequence {q
The aggregate
allocation
{Q,
N
Kj,
f/
AQ}
t
} is
given by q
=
PQ/Ct+i.
the necessary and sufficient
satisfies
t
first-
order conditions
U'(C
t
)=p(l-6 + FK (Kt+1 ,Nt+1 ))U
,
(C t+1 )
t
= 0,l,:..
(31)
Define the following sequence {St}
= -U({l-X
5t
We
have Y^L
^S =
t
With
0.
{u\} £ Y({Wf},0) as follows u\
1
)C
t
)
+ U(C
u\
+ St.
The
(31),
c'(u\)
we find
=
=
0,l,...
then define a
utility
assignment
allocation {u\,n t ,K t ,XCt} then satisfies
the constraints of the original planning problem.
equation (30) and equation
we
this choice of {St},
=
t
)
t
l
Moreover using
c'(u\)
=
all
c'{5t)c'{u\),
that
^E\[c'(u't+1 )]
f
=
0,l,...
where
1
Hence the
=
q t {FK (Kt+1 ,
t+1
)
+ l-S)
1
= 0,1,...
allocation {u t ,n\,K t/ XCt} satisfies the sufficient first order conditions in the
planning problem.
l
It
therefore represents the
47
optimum.
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