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Department of Economics
Working Paper Series
Massachusetts
Institute of
COMMITMENT
vs. FLEXIBILITY
Manuel Amador
Ivan Werning
George-Marios Angeletos
Working Paper 03-40
November] 5, 2003
Room
E52-251
50 Memorial Drive
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at
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Commitment
vs. Flexibility'
Manuel Amador"'
Ivan Werning-^
George-Marios Angeletos^
November
15.
2003
Abstract
This paper studies the optimal trade-off between commitment and
ity in
flexibil-
an intertemporal consumption/savings choice model. Indi\iduals expect
to receive relevant information regarding their
erating a value for flexibility
-
own
but also expect to
generating a value for commitment.
situation
tastes
-
gen-
from temptations
suffer
The model combines
of preferences for flexibihty introduced by
and
-
the representations
Kreps (1979) with
its
recent antithe-
commitment proposed by Gul and Pesendorfer (2002), which nests the
h\T>erbolic discounting model. We set up and solve a mechanism design probsis for
lem that optimizes over the
individual each period.
We
set of
tion takes a simple threshold form
Our
analysis
or the
is
consumption/saving options available to the
characterize the conditions under which the solu-
where minimum sa\dngs
also relevant for other issues
policies are optimal.
such as situations with externalities
problem faced by a "paternahstic" planner, which may be important
thinking about some regulations such as forced
minimum
for
schooling laws.
Introduction
If
people suffer from temptation and self-control problems, what should be done to
help
them? Most
analysis lead to a simple
and extreme conclusion:
take over the individual's choices completely.
'We'd
like to
Diamond and
optimal to
For example, in models with h\-per-
thank comments and suggestions from Daron Acemoglu, Andy Atkcson, Peter
especially Pablo Werning.
'Stanford GSB
tmx, NBER and UTDT
§MIT and
it is
NBER
bolic discounting preferences
it
is
desirable to impose a particular savings plan on
individuals.
Indeed, one
commonly
articulated justification for government involvement in re-
tirement income in modern economies
is
population would save "inadequately"
1977).
the belief that an important fraction of the
if left
to their
own
devices
Diamond,
(e.g.
From the workers perspective most pension systems, pay-as-you-go and
talized systems alike, effectively
of this paper
is
to see
if
such
capi-
impose a minimum saving requirement. One purpose
minimum
saving policies are optimal in a model where
agents suffer from the temptation to "over-consume"
In a series of recent papers Gul and Pesendorfer (2001, 2002a,b) have given prefer-
ences that value commitment an axiomatic foundation and derived a useful representation theorem. In their representation the individual suffers from temptations and
may
exert costly self-control. This formalizes the notion that
as a
way
commitment
is
useful
to avoid temptations that either adversely affect choices or require exert-
ing costly self-control.
On
an axiomatic foundation
the opposite side of the spectrum, Kreps (1979) provided
for preferences for flexibility.
His representation theorem
shows that they can be represented by including taste shocks into an expected
utility
framework.
Our model combines Kreps' with Gul and
Pesendorfer's representations.
Our main
application modifies the intertemporal taste-shock preference specification introduced
by Atkeson and Lucas (1995) to incorporate temptation. In their model the individual
has preferences over random consumption streams. Each period an
is
i.i.d.
taste shock
realized that affects the individual's desire for current consumption. Importantly,
the taste shock at time-t
is
assumed to be private information.
We
modify these
preferences by assuming that agents suffer from the temptation for higher present
consumption. This feature generates a desire
The
for
commitment.
informational asymmetry introduces a trade-off between
flexibility.
Commitment
valued because
it
is
valued because
it
commitment and
reduces temptation while flexibility
allows the use of the valuable private information.
We
solve for the
optimal incentive compatible allocation that trades-off commitment and
One can
interpret our solution as describing the optimal
In addition to
commitment
is
flexibility.
device.
Gul and Pesendorfer's framework, models with time-inconsistent
preferences, as in Strotz (1956), also generate a value for
committment. In particular,
the hyperbolic discounting model has proven useful for studying the effects of a temptation to 'over-consume' as well as the desirability of
committment devices (Phelps
and Pollack, 1968, and Laibson, 1994). As Krusell, Kuruscu and Smith (2002) have
pointed out, however, the temptation framework provided by Gul and Pesendorfer
effectively generalizes the hyperbolic discounting
when
model:
it
results in the limiting case
the agent cannot exert any self-control, giving in fully to his temptations. For
expositional purposes
show that the
We
we
first
treat the hyperbolic discounting case in detail
results extend to
and then
Gul and Pesendorfer's framework.
begin by considering a simple hyperbolic discounting case with two possible
taste shocks.
By
we
solving this case,
on the strength of the temptation
critically
dispersion of the taste shocks.
how the optimal
illustrate
for current
allocation
consumption
depends
relative to the
For the resulting second-best problem there are two
important cases to consider.
For low levels of temptation, relative to the dispersion of the taste shocks,
optimal to separate the high and low taste shock agents.
If
too low, then in order to separate them the principal must
offer
that yield
somewhat
the temptation
too onerous.
to the agent's temptation for higher current consumption. Thus,
single
When
The
not
consumption bundles
both bundles provide more present consumption than their counterparts
best allocation.
is
it is
temptation
is
in the first
strong enough, separating the agents becomes
principal then finds
it
optimal to bunch both agents: she offers a
consumption bundle equal to her optimal uncontingent
allocation. This solution
resolves the average over-consumption issue at the expense of foregoing flexibility.
In this way, the optimal
amount
of flexibility
depends negatively on the strength
of the temptation relative to the dispersion of the taste shocks.
two shocks are simple and
intuitive. Unfortunately,
results are not easily generalized.
examples where 'money burning'
consuming
We
is
These
results with
with more than two shocks, these
show that with three shocks there are robust
optimal:
it
is
optimal to have one of the agents
in the interior of his budget set. Moreover,
any pair of agents. The examples present a wealth of
bunching can occur between
possibilities
with no obvious
discernible pattern.
Fortunately, strong results are obtained in the case with a continuum of taste
shocks.
Our main
result
is
a condition on the distribution of taste shocks that
is
necessary and sufficient for the optimal mechanism to be a simple threshold rule: a
minimum
savings level
imposed, with
is
The optimal minimum savings
level
full flexibility
allowed above this
depends positively on the strength of temptation.
Thus, the main insight from the two type case carries over here:
the strength of temptation and this
We
is
important because
it is
flexibility falls
with
accomplished by increased bunching.
is
extend the model to include heterogeneity
This
tion.
minimum.
in
temptation of current consump-
reasonable to assume that agents suffer from
temptation at varying degrees. Indeed, perhaps some agents do not suffer from temptation at
all.
dividuals that
Allowing for heterogeneity in temptation would imply that those
we observe saving
higher temptation. However,
of a
minimum
The
we
saving policy
rest of the
paper
is
we show
is
more
less are
that the
likely to
main
in-
be the ones suffering from
result regarding the optimality
robust to the introduction of this heterogeneity.
organized as follows. In the remainder of the introduction
briefly discuss the related literature.
Section
1
lays out the basic intertemporal
model using the hyperbolic discounting model. Section
2 analyzes this
model with
two and three taste shocks while Section 3 works with a continuum of shocks. Section 4 extends the analysis to arbitrary finite time horizons
results to the case
and Section 5 extends the
where agents are heterogenous with respect to
Section 6 contains the
more general case with temptation and
their temptation.
self-control
proposed by
Gul and Pesendorfer (2001,2002a,b). Section 7 studies the case where agents discount
exponentially at a different rate than a 'social planner' and preferences are logarithmic. Section 8 diverges to discuss
some
alternative interpretations and applications of
our main results regarding the optimal trade-ofl[ between committment and
The
final
Section concludes.
An
appendix
collects
some
flexibility.
proofs.
Related Literature
At
least since
in the
Ramsey's (1928) moral appeal economists have long been interested
implications
of,
and
justifications for, socially discounting the future at lower
rates than individuals. Recently, Caplin
and Leahy (2001) discuss a motivation
welfare criterion that discounts the future at a lower rate than individuals.
(2002) provides another motivation
and studies implications
of opportunity of a zero social discount rate. In
and agents discount the future exponentially.
for
for
a
Phelan
long-run inequality
both these papers the social planner
Some papers on
social security policies
possible "undersaving" by individuals.
agents
may undersave due
have attempted to take into account the
Diamond
(1977) discussed the case where
OLG
Feldstein (1985) models
to mistakes.
agents that
discount the future at a higher rate than the social planner and studies the optimal
pay-as-you-go system. Laibson (1998) discusses public policies that avoid undersaving
in
hyperbohc discounting models. Imrohoroglu, Imrohoroglu and Joines (2000) use
a model with hyperbolic discounting preferences to perform a quantitative exercise
on the welfare
(2002) use a
effects of
pay-as-you-go social security systems.
Diamond and Koszegi
model with hyperbolic discounting agents to study the policy
endogenous retirement
choices.
effects of
O'Donahue and Rabin (2003) advocate studying pa-
ternalism normatively by modelling the errors or biases agents
may
have and applying
standard public finance analysis.
Finally, several papers discuss trade-offs similar to those
emphasized here
ious contexts not related to the intertemporal consumption/saving
in var-
problem that
is
our focus. Since Weitzman's (1974) provocative paper there has been great interest in
the efficiency of the price system compared to a
command economy,
see
(1984) and the references therein. In a recent paper, Athey, Atkeson and
Holmstrom
Kehoe (2003)
study a problem of optimal monetary policy that also features a trade-off between
time-consistency and discretion. Sheshinski (2002) models heterogenous agents that
make
choices over a discrete set of alternatives but are subject to
shows that
in
such a setting reducing the set of alternatives
random
may be
errors
and
optimal. Laib-
son (1994, Chapter 3) considers a moral-hazard model with a hyperbolic-discounting
agent and shows that the planner
may reward
the agent for high output by tilting
consumption towards the present.
1
The Basic Model
For reasons of exposition we
inconsistent.
first
study a consumer whose preferences are time-
Following Strotz (1956), Phelps and Pollack (1968), Laibson (1994)
and many others we model the agent
for
subgame
show that
all
perfect equilibria of the
our results go through
in
each period as different selves and solve
game played between
when we
selves.
In section 6
we
use the more general framework pro-
vided by Gul and Pesendorfer (2001,2002a,b) which, in addition, does not require an
game
intrapersonal
Consider
period
t
=
first
interpretation.
the case with two periods of consumption,
from which we evahiate expected
arbitrary finite horizons.
Ed =
so that
make
\
which
Each period agents
more
current consumption
We
the agent at time t}
the available observable variables.
c
and
The
We
denote
utility for selj-1
where
U {)
<
The
N
1.
and
W {)
from periods
W ()
i
=
1,
^ U {),
utihty for self-0 from periods
t
=
(c)
1
— /3
observed privately by
is
first
after conditioning
on
and second period consumption
2 with taste shock 9
is
{k)
and continuously
differentiable"
and
this generality facilitates the extension to
1,
+
Agents have quasi-geometric discounting:
<
taste shock
consumption: higher 9
4.
9U
/?
initial
taste shock 9, normalized
i.i.d.
and saving data
are increasing, concave
notation allows
periods in section
The
and an
k, respectively.
9U (c) + [3W
/3
1,2,
think of the taste shock as a catch-all for the significant
variation one actually observes in consumption
by
an
utility of current
The
valuable.
=
utihty. Section 4 extends the analysis to
receive
marginal
affects the
t
2
W
{k)
self-t
1
and
is
a measure of this disagreement or bias.
in this respect, there is
is
.
discounts the entire future at rate
among
disagreement
On
the different t-selves and
the other hand, there
is
agreement
regarding taste shocks: everyone values the effect of 9 in the same way. Below we often
associate the value of B to the strength of a 'temptation' for current consumption;
thus,
we say that temptation
An
is
stronger
if 13 is
lower.
alternative interpretation to 'hyperbolic' discounting
only periods
1
and
2.
One can simply work with
is
Although
available
if
we consider
the assumption that the correct wel-
fare criterion does not discount future utility at the
both do so exponentially.
is
same
rate as agents do, although
this alternative interpretation
is
available for
'With exponential CARA utility income shocks are equivalent to taste shocks.
^Note that a taste shock for period i = 2 is not included in this expression. However,
only apparent since k cannot depend on 60 and EOo = 1.
its
absence
two-periods we will see that in general
sion of the analysis to
more periods. In
it
does not permit a straightforward exten-
we
section 7
discuss a case in which
it
does
generalize.
We
investigate the optimal allocation from the point of view of self-0 subject to
the constraint that 6
tailoring
is
private information of self-1.
consumption to the taste shock and the
The
self-1
essential tension
's
To
solve the allocation preferred by self-0 with total
income y we now
set
up the
y.
Periods
V2 {y)
eU (c {9)) +
= max /
F (9)
[OU
> 9U
m' (k {9))
+
c{9)
where
flexibility
self-O.
optimal direct truth telling mechanism given
Two
between
constant higher desire for
commitment and
current consumption. This generates a trade-off between
from the point of view of
is
k
(c {9))
(c {9'))
{9)
<
+
W {k
{6))]
+ PW {k {9'))
y for
for all 9,9'
eQ
(1)
6
G
all
dF [9)
the distribution of the taste shocks with support O.
is
This problem maximizes, given total resources
y,
the expected utility from the
point of view of self-0 (henceforth: the principal) subject to the constraint that 9
is
private information of self-1 (henceforth: the agent).
The
incentive compatibility
it
is
in agent-^'s self interest to report truthfully, thus
obtaining the allocation that
is
intended for him.
constraint (1) ensures that
interest rate
is
normalized to zero for simplicity.
The problem above imposes
across ^-agent's
is
ruled out.
agent's types. This choice
First,
may be
it
many
relevance in
possible or
if
In the budget constraints the
a budget constraint for each ^ G 0, so that insurance
The planner cannot
transfer resources across different
was motivated by several considerations.
possible to argue that the case without insurance
situations.
This could be the case
if
pooling risk
is
is
of direct
simply not
insurance contracts are not available because of other considerations
outside the scope of our model.
Second, the cardinality of the taste shocks plays a more important role
ysis
with insurance. The taste shock 9 definitely
in
an anal-
affects ordinal preferences
between
current and future consumption, c and
However, we would
k.
like to
avoid taking a
strong stand on whether or not agents with high taste for current consumption also
have a higher marginal
from
utility
total resources as the expression
9u
+w
implictly
assumes. Focusing on the case without insurance avoids making our analysis depend
strongly on such cardinality assumptions.
Third, without temptation
as Mirrlees (1971) or
(/?
=
1)
incentive constrained insurance problems such
Atkeson and Lucas (1995) are non-trival and the resulting op-
timal allocations are not easily characterized.
the solutions with temptation
(/?
<
more
1)
the optimal allocation without temptation
This would make a comparison with
In contrast, without insurance
difficult.
[j3
=
1) is
straightforward ~ every agent
chooses their tangency point on the budget set - allowing a clearer disentangling of
the effects of introducing temptation.
Finally,
we hope
that studying the case without insurance
may
yield insights into
the case with insurance which we are currently pursuing.
Once the problem above
is
solved the optimal allocation for selj-0 solves a standard
problem:
max {9oU
[cq)
+ (iv2 {yo -
co)}
CO
where
yo, Cq
and
respectively. In
^o represents the initial
t
—
what follows we ignore the
0,
income, consumption and taste shock,
initial
consumption problem and focus on
non-trivial periods.
2
Two Types
In this section
we study the optimal commitment with only two
occuring with probabilities p and
Without temptation,
/3
=
1,
1
—
there
taste shocks, d^
>
6i,
p, respectively.
is
no disagreement between the planner and the
agent and we can implement the ex- ante first-best allocation defined by the solution
to eU' {cfb {9))
/W {kfb {9))
temptation, so that
compatible.
/3
is
Intuitively,
=
close
if
1
and
Cfh (9)
enough to
1,
+
kfb {9)
=
y.
the first-best allocation
the disagreement in preferences
dispersion of taste shocks then, at the
the high shock agent's allocation.
first
For low enough levels of
best, the low
is
is still
incentive
small relative to the
shock agent would not envy
Proposition 1 There
is
exists a
(3*
<
such that for P 6
1
the first-best allocation
[/5*, 1]
implementable.
Proof. At
Define
/3*
=
/5
<
1
1
the incentive constraints are slack at the ex-ante first-best allocation.
to be the value of
equality at the
This result
first
(3
which the incentive constraint of
for
best allocation.
The
on the discrete
relies
when we study a continuum
centive compatible.
result follows.
diflFerence in taste
of shocks in Section
For higher levels of temptation,
lower
i.e.
-I-
k* (6)
=
P >
6i/9h separation
(b) if
P <
9i/9h hunching is optimal,
(c) if
P =
9i/9h separating
First,
p*
P*
> P
first
best allocation
The next proposition
is
he attained with the
—
y for 9
(a) if
Proof.
the
is
not
in-
for &gent-9h
characterizes
Ccises.
Proposition 2 The optimwn can always
ing with equality: c* {9)
3.
/3,
to obtain a higher level of current consumption.
optimal allocations in such
shocks and no longer holds
would take the bundle meant
agent-^;
If oflfered,
holds with
di
optimal,
i.e.
i.e.
We
9h, 9i.
c* {9h)
c{9i)
=
>
budget constraint hold-
have that 9i/9h
c* {9i)
and k
{9h)
=
c{9h) and k{9i)
<
<
k
P* and:
{9i)
k{9h)
and hunching are optimal
follows since
u{c*{eH))-u{c*{ei))
=
Wiy-c*{9i))-W{y~c*{9H))
U'{c{e,)){c{9,)-c{9^))
'W'{y-c {9h))
where
c* is
the
first
(c {9n)
-
c {9i))
U'{c{9,))
'W'{y-c {9^))
in c{0k)
_
9h~-
best allocation.
Now, consider the case where P > P and suppose that
an increase
9i
and a decrease
unchanged increases c{9h)
centive compatible because
+
it
c {9h)
in k {9h) that holds {9i/P)
+
k (9^)
U {c{9h))
< y. Then
U {k (9^))
-\-
k {9h) and the objective function. Such a change
is in-
strictly relaxes the incentive compatibility constraint of
the high type pretending to be a low type and leaves the other incentive compatibility
constraint unchanged.
It
follows that
we must have
c {9h)
+k
{9h)
=
y at an optimum.
This also shows that separating
arguments establish parts
(b)
optimal
is
and
proving part
in this case,
(a).
Analogous
(c).
Proposition 2 shows that for /?</?* the resulting non-trivial second-best problem
can be separated into essentially two cases. For intermediate
61/ dh
<
must
offer
P,
levels of
temptation,
i.e.
optimal to separate the agents. In order to separate them the principal
it is
consumption bundles that yield somewhat to the agent's ex-post desire
higher consumption giving
them higher consumption
in the first
period than the
for
first
best.
For higher levels of temptation,
onerous,
U
—
{)
bunching them
W {)
i.e.
/?
<
9[/9h, separating the agents is too
then optimal at the best uncontingent allocation - with
is
=
this implies c(-)
k{-)
=
bunching resolves the disagreement
y/2.
amount
problem at the expense of
flexibility.
In this way, the optimal
depends negatively on the
size of the
disagreement relative to the dispersion of the
taste shocks as
measured by
9i/9h-
Proposition 2 also shows that
c{9)
+
k{6)
=
y.
In this sense,
not required for optimality.
of flexibihty
As
it
is
always optimal to consume
'money burning',
discuss below, with
i.e.
all
setting c(0/j)
+
the resources
k{9h)
more than two types
this
<
is
y, is
not a
foregone conclusion.
Figure
1
below shows a typical case that
=
W ()
c^-"/
{1-a), U
figure
shows consumption
we
(•)
,
and a
=
2, 9^
illustrate these results.
=
1.2, 9i
in the first period, c (0)
also plots the optimal ex-post
consumption
,
for
=
.8,
p
=
set
111 and y
as a function of
both types
We
(i.e.
/3.
=
U (c) =
\.
The
For comparison
the
full flexibility
outcome). Note that these are always higher than the optimal allocation: the principal
does manage to lower consumption in the
first
10
period.
"""---^, ex-ante c^
i--^___
0.54
--..^
1 ""
Q.
E
^"^--^^^
c
o
o
ex-anle
'~"--~-_._
C|_
0.48
\
0.46
0.6
0.8
1
p values
Figure
The
first
Optimal
1:
first
period consumption
figure illustrates Proposition 1
best allocation
is
with two shocks as a function of
2 in the following way.
For high
consumption
in the first period rises as
/3.
the
/3
attainable so the optimal allocation does not vary with
this range. For intermediate
way the
and
(c)
/?
in
/? falls.
In this
principal yields to the agent's desire for higher consumption. For low
enough
(3
P bunching becomes optimal and
=
c{6)
To summarize, with two types we
y/2.
are able to characterize the optimal allocation
which enjoys nice properties. In particular, the budget constraint holds with equality
and we found simple necessary and
sufficient conditions for a
bunching or separating
outcome to be optimal.
Unfortunately, with more than two types extending these conclusions
forward. For example, with three taste shocks, 9^
> 0^ >
Oi-,
it is
simple to construct
robust examples where the optimal solution has the following properties:
get constraint for agent
is
optimal;
and
h,
and 9h
(ii)
it is
is
9m
although
(3
is
satisfied
< 9m/9h
with
remains a
optimal;
(iii)
characterizing the
to
bunching can occur between
show a variety
optimum with more than two
find conditions
optimal allocation
i.e.
/3
9i
>
(i)
the bud-
'money burning'
bunching
m
9m/9h where bunching 9m
and 9m, with 9h
is
separated.
of possibilities that illustrate the difficulties in
Fortunately, with a continuum of types
we
-
sufficient condition for
no longer necessary: there are cases with
The examples seem
section
strict inequality
not straight-
is
types.
more progress can be made.
In the next
on the distribution of 9 which allows us to characterize the
fully.
11
Continuous Distribution of Types
3
Assume that
the distribution of types
= [9,6]. We find it convenient
= U {c) and w = VV (k) and we
u
K {w)
A'
be the inverse functions of
To
to change variables from {c,k) to {u,'w) where
term either pair an
U {c)
and W{k),
and
respectively, so
and
characterize the incentive compatibility constraint (1) in this case consider the
problem faced by agent-^ when confronted with a
V{9)
= m&xl^u(9') +
0'ee
the mechanism
If
C (u)
that C {)
Let
'allocation'.
and convex.
are increasing
(•)
represented by a density / (9) over the interval
is
is
direct
mechanism
{it.
{9)
,
w
[6)):
tu(9')
[P
truth telling then
V (9) =
|u
~u[9)de
+
[9)
+ w (6)
and integrating the
envelope condition we obtain,
^u {9) + w
(see
Milgrom and
[9)
Segal, 2002).
be a non-decreasing function of
=
j^
The
9.
Thus, condition
is
vo (y)
(2),
particular,
it
We now
u
to
and the monotonicity of u are
know that
well
(u, u') also requires
these two conditions are
thus,
= max f
C (u {9)) + K {w [6)) <
convex since the objective function
is
It is
(2)
(2)
Fudenberg and Tirole, 1989).
planner's problem
subject to
+ w{9)
Incentive compatibility of
necessary for incentive compatibility.
also sufficient (e.g.
^u{9)
follows that ih (y)
is
[9u {9)
+ w {9)] f
y and u
is
linear
concave in
{9')
>
u
{9) d9,
(9) for 9'
>
and the constraint
9.
This problem
set is convex.
In
y.
substitute the incentive compatibility constraint (2) into the objective
function and the resource constraint, and integrate the objective function by parts.
This allows us to simplify the problem by dropping the function iu{9), except
its
value at
n
9
:
9.
for
Consequently, the maximization below requires finding a function
^ R and a scalar w representing w
{9).
12
Continuous Distribution of Types
v^ (y)
= max
i
|u(0 + ^ + i A(l - F [9)) -
{y-C {u {9))) + ^u {9) -^u{9)-w-
K-'
u
For any feasible allocation u
bunch some upper
<
{9')
> u (9)
for 9'
9
and u
feasible,
(9)
=
tail
u{9)
it
is
of agents.
is
[9)]
u
^u{9)d9
>
/3)
f
{9)
cW
j^
>9
always feasible to modify the allocation so as to
That
feasible for
we now show that
To gain some
it is
the allocation
is,
any
9.
+ ^w
+w
<
(59
=
u
tail is
(9) for
always
share the ordinal preferences
That
is,
the indifference curves
are equivalent. Informally, these agents can
>
In contrast, agents with 9
their preferences.
{9)
always optimal.
intuition, note that agents with 9
and 9/ [5u
u given by u
Thus bunching the upper
of the planner with a higher taste shock equal to 9/13.
9u
-
Bunching
3.1
9
9 (1
(59
make a
case for
display a blatant over-desire for
current consumption from the principal's point of view, in the sense that there
taste shock that
would
justify these preferences to the planner. Thus,
that these agents are bunched since separating
them
is
it is
is
no
intuitive
tantamount to increasing some
of these agents consumption, yet they are already obviously "over-consuming"
The next
result
shows that bunching goes even further than ^9.
Proposition 3 Define
E
>
9
9
Note that
u* {9)
=
9p
<
j39
u* {9p) for 9
and
>
such that for 9
9p as the lowest value in
9p
9p
<
(i.e.
l39
it
all
13
9p:
-1
<
- P
as long as f
bunchs
>
>
0.
An
optimal allocation u* has
agents above 9p)
>
Proof. The contribution to the objective function from 9
Substituting u
=
Jg
9p
is
1
f {{i-F{e))-e{i-p)f{e})u{9)d9.
du
+ u {9p)
and integrating by parts we obtain,
I\{i-F{9))-9{1-P)f{9))d9
^i{Op)-
Note that,
((l
J'
for all 9
-F
[§))
>
9p so
{1-^3) f
9
optimal to set du
it is
With two types Proposition
9h/9i
<
1//3.
=
(1
- F{9))9
I
or equivalently
0,
2 showed that bunching
is
-
^
u
3.2
To
<0,
9
(9)
=
strictly
the support
is
essentially 9p
unbounded then
9p
may
=
u
{9p)
full flexibility
or
bunch
Assumption
all
,
for
6*
>
9p.
optimal whenever
when
9h/9i
<
9i.
not exist. This occurs, for example,
with the Pareto distribution. One can show that in this case
allow
E 9\9>9
Proposition 3 generalizes this result since with two types
1/P then according to the definition
If
=
(0")) d9
it is
either optimal to
agents depending on the Pareto parameter.
A
<
solve for the optimal allocation for 9
the density / and
/3.
A. The density f
{9) is differentiable
and
9p
we
require the following condition on
satisfies
J'iO)^
f{e)-
14
2-/3
i-P
<9p.
all 9
for
Assumption
A
places a negative lower
uous and decreasing in
as
—
/3
1
>
the lower
The
/?.
off to
for 9
<
For any density / such that 9f'/f
for
/3
close
enough to
For example,
and
it is
Moreover,
1.
bound
highest lower
bound goes
on the whole support 0, only
bound on the
— oo. Note
of
that
elasticity of
—2
A
is
/ that
attained for
is
/3
contin-
=
does not impose the
and
bound
9p.
is
A is satisfied
assumption A for all
bounded from below assumption
many
densities satisfy
/3.
trivially satisfied for all density functions that are
non-decreasing
also holds for the exponential distribution, the log-normal, Pareto
and
Gamma
distributions for a large subset of their parameters.
Minimum
3.3
Define u* {9)
,
w*
Saving Policies
{9) to
be the unconstrained optimum
{u* (9)
,
w*
(9))
=
r
arg
9
max —u + w
<^
s.t.
Our next
result
shows that under assumption
unconstrained optimum and agents with 9
optimum
the
left
That
for 9p.
of
some point
is,
for agent-^:
C (u) + K [w] <
A
>
agents with 9
9p are
bunched
<
y
9p are offered their
at the unconstrained
the optimal mechanism offers the whole budget line to
(c*,fc*),
given by the ex-post unconstrained
optimum
of the
^p- agent.
Define the Lagrangian function
L{w,u\A)
= ^ui9)+w +
+
1
^j
as:
[ip-l)9f{9)
+ {l-Fi9))]u{9)d9
(^K-'{y-C{u{9))) + ^u{9)-(^^ui9}+w^-J^ ^u{9)d9^ dA
where the function
A
patibility constraint.
is
We
the Lagrange multiplier associated with the incentive comrequire the Lagrange multiplier
Luenberger, 1969, Chapter
Intuitively, the
{9)
A
to be non-decreasing (see
8).
Lagrange multiplier
A
can be thought of as a cumulative distribu-
15
tion function'^. If
of constraints
A
happens to be
differentiable with density A then the
can be incorporated into the Lagrangian as the famihar integral of the
hand
side of each constraint
product of the
left
though
common approach in many
this
is
a
of discontinuity
As we
and
points: 6
(u* (9)
,
would not be
In such cases, working with a density A
A
indeed discontinuous at two
is
9p.
=
lu* (9))
Proposition 4 The
Proof.
A may have points
applications, in general,
shall see, in our case the multiplier
Consider the allocation
and
and the density function X{6). Al-
and these mass points are associated with individual constraints that
are particularly important.
valid.
continuum
Without
{u,
{u* {9p)
,
w) given by (u
w*
{9p)) for 9
allocation {u,
w)
is
loss of generality set
{9)
>
((9)
w (6)) =
{u* {6)
,
w*
[6)) for 9
<
9p
9p.
optimal
A
,
if
=
and only
1.
We
will
if
assumption
construct
A
A
holds.
to
be
left
continuous. Integrating the Lagrangian by parts:
L{w,u\A)
= {^u{9) + w)A
/3
+
^j
i{f3-l)ef{9)-F{9) + A{e))u{9)d9
+
Note that the Lagrangian
is
a
sum
(a-^
J'
(y
-C
on Gateaux
^Except
+ ^u (^)) dA {9)
of integrals of concave functions of to
This implies that the Gateaux differential exists and
Lemma
{u {9)))
differentiability in the
Appendix)
.
[9).
computed
(see the
In particular, at the
proposed
is
easily
for the integrability condition. Also, for notational purposes,
we make A a
function, instead of the usual right-continuous convention for distribution functions.
16
and u
left-continuous
allocation for
dL
lu,
u the Gateaux
{w, u; h^,
=
/7,„|A)
+
+
where x\g
g\
is
differential
r|/i„ (0
y
-^
((/3
is
+
O
-
1)
given by:
A {6}
ef
{9)
(3)
- F (^) + A (^) -
1) /i„ (0)
d^
^
(^-llX[,„,]MA(^)
^ r (I
I
the indicator function over [6'p,^,
i.e.
x\q
=
gi
1 for
6*
e
\9p,6\
and
zero otherwise.
The problem
tion
is
is
convex, the Lagrangian
continuous.
It
follows that
non-decreasing function
A
u
ty,
is
optimal
is
c>L(«;,u;/i„,/i„|A)
is
/ij^
if
and only
a non-decreasing function u
(5) requires
there exists
if
some
=
(4)
<0
(5)
and hu that belongs to the convex cone given by
Condition
alloca-
such that:
5L(u),n;M,u|A)
for all
and the proposed
differentiable
:
that
Q,
^^
R}
A {9) =
(see Luenberger,
Using
0.
this
A'
=
{w, u
Chapter
:
w
E
R
and u
8).
and integrating
(3)
by parts
leads to
dL
{w, u- h^,
/i„|
A)
=
7
K {d)+ Je[
[9)
1
{0)
dK {0)
(6)
where,
1(0)^^1 HP It
1)
Of
{e)
-F{9) + A
follows that condition (4) requires
7
(^)
=
d^
+
for 9
G
(9))
Iy
[9,9p]
(^i
,
i.e.
-
1
j
X[e,^^^dA
where u
is
strictly
increasing. This implies,
A(^)
9
G
{9,9p).
The proposed
=
(l-,5)^/(^)
+
F(^),
(7)
allocation thus determines a unique candidate multiplier
17
A
the separating region
in
to be non-decreasing.
Op)
[9,
and assumption
^4 is
follows that assumption
It
necessary and sufficient for
A
A
{6)
necessary for the proposed
is
solution to be optimal.
We now
A
over the whole range
A
specified
A
prove sufficiency by showing that there exists a non-decrccising multiplier
(^)
=
for
1 in
[9_,
9p) so
is
we only need
satisfies (4)
to specify the value of
A
and
for [9p, 9j
We've
(5).
and we
set
this interval.
The constructed A
To show that A
Op.
such that the proposed Uku
fi
is
is
not continuous,
non-decreasing
it
that remains
all
jump
has an upward
is
at 9
and a jump
show that the jump
to
at
at 9p
upward,
\-[{l-P)9pf{9p) + F{9p)]>0,
which follows from the definition of
9p.
immediate since A jumps from
at
that 7 (^)
<
for all
The proposed
7
<
and that 7
9>9,
1
w and
wherever u
is
u,
see this, note that
^p
if
Otherwise, recall that 9p
9.
which implies
allocation,
=
to
To
7' {Op)
=
{1
-
p) Of
and the Lagrange
increasing. Using (6)
{9)
is
-
=
^ the result
is
the lowest 9 such
{1
-F
(0))
<
0.
multiplier. A, imply that
it
follows that (4)
and
(5)
are satisfied.
The
figure
below
illustrates the
form of the multipher
A {0)
proof of the proposition.
separating with
Figure
2:
full flexibility
The Lagrange
18
multiplier
A {0)
constructed in the
Proposition 4 shows that under assumption
simple.
It
can be implemented by imposing a
minimum
or equivalently, a
A
the optimal allocation
maximum
Such minimum saving
level of savings.
pervasive part of social security systems around the
The next
shows the comparative
result
spect to temptation
bunched
As the temptation
6p decreases).
(i.e.
minimum
/5.
Proof. That
is
Op
=
— C {u {9p))
y
is
i.e.
a
policies are
optimal allocation with
re-
more types
are
decreases,
/?
decreases with
,
in the allocation.
is less flexibility
9p increases with
The minimum savings
j3.
re-
j3.
weakly increasing follows directly from
its definition.
To
see that
decreasing note that Smin solves
9p U' {y
and that
9p,
when
-
Sm\n)
_
..
interior, solves,
^-^
Combining these, we obtain
is
consumption,
In terms of policies, as the disagreement increases the
Proposition 5 The bunching point
Smin
increases,
extremely
vi^orld.
statics of the
savings requirement decreases so there
quirement, Smin
level of current
is
= E[9\9>9p].
E [9\9 >
9p]
U' [y
-
s^in)
/W {smm)
increasing in 6p the result follows from concavity of
U
=
1-
Since
E[9\9>9p]
and W.
Drilling
3.4
In this subsection
we study
cases where assumption
A
does not hold and show that
the allocation described in Proposition 4 can be improved upon by drilling holes in
the separating section where the condition in assumption
Suppose we are
offering the unconstrained
optimum
agents and we consider removing the open interval
found their tangency within the interval
6b.
The
moving
to 9a
critical issue in
to 9a versus
and
falls
Oi,.
will
move
A
is
not
satisfied.
for a closed interval [9a
{9a, Oh).
,
Oi,]
of
Agents that previously
to one of the two extremes, 9a or
evaluating the change in welfare
is
counting
how many
agents
For a small enough interval, welfare rises from those moving
from those moving to
9h.
19
Since the relative measure of agents moving to the right versus the
the slope of the density function this explains
if
/'
>
then upon removing
{0a, Oh)
to the right than the
The proof
undesirable.
is
depends on
assumption A. For example,
more agents would move
As a consequence, such a change
left.
its role in
left
of the next result
formalizes these ideas.
Proposition 6 Suppose an
6
€
[9a,
[^a,^fc]
with db
<
then removing the interior of the set {{u*
9),]
{9))
A
assumption
If the condition in
9p.
,w
allocation {u,w) has {u{9)
{9)
,
w*
=
,w*
does not hold
for 9 G
[9])
(u* {6)
(&„, 9b)}
{9))
for
/o7' 9
G
improves
welfare.
Proof. In the appendix.
Propositions 6 illustrates by construction
why assumption A
is
necessary for a
simple 'threshold rule' to be optimal and gives some insight into this assumption. Of
course. Proposition 6 only identifies particular
fails.
It
4
We
seems
have not characterized the
likely that
full
improvements whenever assumption
optimum when assumption A does not
'money burning' may be optimal
in
some
A
hold.
cases.
Arbitrary Finite Horizons
We now show that our results extend to arbitrary finite horizons. We confine ourselves
to finite horizons because
with
any mechanism may yield multiple
infinite horizons
equilibria in the resulting
game. These equilibria
that a good equilibrium
is
upon a
deviation.
1996).
We
involve reputation in the sense
sustained by a threat of reverting to a bad equilibria
Some authors have questioned
equilibria in intrapersonal
may
games
(e.g.
the credibility of such reputational
Gul and Pesendorfer, 2002a, and Kocherlakota,
avoid these issues by focusing on finite horizons.
Consider the problem with
N < oo periods
t
=
1,
...,
TV
where the
felicity
function
U (•) in each period. Let 9^ =
^2, •, ^t) denote the history of shocks up to time
A direct mechanism now requires that at time t the agent make reports on the
history of shocks r* = {r\,r2, ,rl). The agent's consumption is allowed to depend
on the whole history of reports: q {r^,r^~^,
r^). A strategy for self-t is a mapping
is
(6'i,
t.
...,
from the history of shocks and past reports into current reports:
Truth
telling requires
R^ {9^9*-'^,
...,9^)
=
9^ for all
20
t
and
/?*
all histories
(6'*,r'~\ ••,'"^)-
9K
We
first
argue that without
nisms that at time
t
optimahty we can
loss in
require only a report
whole history of shocks
This
6*.
on the current shock
r^
and not of the
9t,
the case in Atkeson and Lucas (1995) but in their
is
setup since preferences are time-consistent there
is
mecha-
restrict ourselves to
a single player and the argument
is
straightforward.
we have
In contrast, in the hyperbolic model
A''
players
and the difference
preferences between these selves can be exploited to punish past deviations.
example, an agent at time
t
that
amongst them according to whether there has been a deviation
to choose
past. In particular, she
allocations
For
between allocations can be asked
indifferent
is
in
in the
can 'punish' previous deviating agents by selecting the worst
from their point of view. Otherwise,
if
there have been no past deviations,
she can 'reward' the truth-telling agents by selecting the allocation preferred by them.
Such schemes may make deviations more
and are thus generally
One way
to
desirable.
remove the
the refinement that
costly, relaxing the incentive constraints,
possibility of these
when agents
punishment schemes
are indifferent
to introduce
is
between several allocations choose
the one that maximizes the utility of previous selves. Indeed, Gul and Pesendorfer's
(2001,2002a,b) framework, discussed in Section
control, the hyperbolic
model with
We
punishment schemes.
over which ^-agents are indifferent
of
self-
added refinement.
this
However, with a continuous distribution
to rule out these
dehvers, in the limit without
6,
is
for 9
such a refinement
show that
at
for
is
not necessary
any mechanism the subset
most countable. This implies that the
probability that future selves will find themselves indifferent
is
zero so that the threat
of using indifference to punish past deviations has no deterrent effect.
For any set
A
of pairs
(^i,
w) define the optimal correspondence over x E
M
[x;
A)
=
arg
X
max {xu + w}
{u,w)eA
(we allow the possibility that
Lemma
M
(x;
(Indifference
is
M
{x,
A)
is
countable).
A) has two or more points
empty) then we have the following
For any
A
the subset
X^ C
(set of agents that are indifferent) is at
X
result.
for which
most count-
able.
Proof. The correspondence
M
{x;
A)
is
monotone
21
in the sense that
if
xi
<
X2 and
M{xi;A) and
(ui.wi) e
sense, points at
(u2,u'2)
€ M{x2;A) then ui <
which there are more than a single element
upward 'jumps'.
As with monotonic
functions,
Thus, in an obvious
U2-
M
in
{x;
follows easily that
it
A) represent
M
A) can
(.t;
have at most a countable number of such 'jumps'.
only on the single crossing property of preferences and not on the
This result
relies
linearity in
u and w.
We make
use of this
lemma
again in Section
These considerations lead us to write the problem with
A'^
>
5.
3 remaining periods
recursively as follows.
N
Period Problem
vn
(y)
= max /
c,y
[OU
(c (9))
+ v^-i
{y [9))] clF {9)
J
9U
+ (3vr,-i
{c{9))
[y' [9])
> 9hU
c {9)
where
112 (•)
was defined
<
+
y for
3v^^,
all
^
G
inefficient
we assume, without
for all
< v^-i
is
^Q
9
continuation utility w' {9)
xu' {9)
=
without
[y' [9))
i.e.
t',v-i
optimal
shows up
loss in optimality
in the
because
can be achieved by 'money
i'at-i (y' [9)) for
For the simple recursive representation to obtain
it is
some
y' [6)
critical that,
<
y' {9).
although the
and the agent disagree on the amount of discounting between the current
and next period, they both agree on the
given by uw-i- This
is
utility
obtained from the next period on,
not true in the alternative setup where the principal and the
agent both discount exponentially but with different discount factors.
case separately in Section
For any horizon
A''
this
only required
properties
all
W {)
We
treat this
7.
problem has exactly the same structure as the two-period
problem analyzed previously, with the exception that
We
9,9
loss in optimality, that the
and incentive constraint. This
burning' with the same effect: setting
principal
(ij (/5))
ex-post optimal given the resources available
is
objective function
any
{9)
(9)^
in Section 2.
In the above formulation
mechanism
+ y'
(c
to be increasing
the previous results apply.
Vjsi-i (•)
and concave and
We
sition.
22
summarize
has substituted
since
I'yv-i (•)
W
{).
has these
this result in the next
propo-
Proposition 7 Under assumption A
riods can be
the optimal allocation vnth a horizon of
implemented by imposing a minimum amount of saving St
N peperiod
{yt) in
t.
In Proposition 7 the
minimum
preferences the optimal allocation
and
y' {9, y)
=
y' {9) y.
It
saving
is
is
homogenous
independent of
Proposition 8 Under assumption
the N-period
linearly
a function of resources
follows that the optimal
saving rate for each period that
nism for
is
A
and
U (c) =
yt-
With
in y, so that c {9, y)
CRRA
=
c {9) y
mechanism imposes a minimum
yt-
c^^'^ / {I
—
a) the optimul m.echa-
problem imposes a minim.um saving rate
St
for each, period
t
independent ofyt-
4.1
Hidden Savings
Another property of the optimal allocation
identified in Propositions 4
and
7
is
worth
mentioning. Suppose agents can save, but not borrow, privately behind the planner's
back at the same rate of return as the planner, as
The
in
Cole and Kocherlakota (2001).
possibility of this 'hidden saving' reduces the set of allocations that are incentive
compatible since the agent has a strictly larger set of possible deviations. Importantly,
the mechanism described in Proposition 7 continues to implement the same allocation
when we
allow agents to save privately,
To prove
this claim
and thus remains optimal.
we argue that confronted with the mechanism
7 agents that currently have
no private savings would never
cumulate private savings. To see
this, first
find
in Proposition
optimal to ac-
it
note that by Proposition 7 the optimal
mechanism imposes only a minimum on savings
in each period.
Thus agent-^ always
have the option of saving more observably with the principal than what the allocation
recommends, yet by incentive compatibility the agent chooses not
Next, note that saving privately on his
increasing the
amount
own can be no
better for the agent than
of observable savings with the principal. This
the principal maximizes the agents utility given the resources at
from the point of view of the current
is
to.
self,
its
is
true because
disposal. Thus,
future wealth accumulated by hidden savings
dominated by wealth accumulated with the
23
principal.
It follows
that agents never find
it
optimal to save privately and the mechanism
implements the same allocation when agents can or cannot save
Proposition 9 Under assumption A
the m.echanism described
when agents can save
plern.ents the sam.e allocation
privately.
m
privately.
Heterogeneous Temptation
5
Consider now the case where the level of temptation, measured by
erogeneity in
is
Proposition 7 im-
f3
captures the
commonly
more
Diamond,
If
likely to
is
it
is
the agents that save the least that
be 'undersaving' because of a higher temptation to consume
(e.g.
1977).
the heterogeneity in temptation were due to permanent differences across indi-
viduals then the previous analysis would apply essentially unaltered.
their
random. Het-
held view that the temptation to overconsume
not uniform in the population and that
are
,i5,
/3
at time
they would truthfully report
time
be tailored to their
/3
as described above"*.
it
so that their
To explore other
If
agents
knew
mechanism could
possibilities
we assume
the other extreme, that differences in temptation are purely idiosyncratic, so that
is i.i.d.
across time and individuals. Thus, each period 9
and
(3
are realized together
from a continuous distribution - we do not require independence of 6 and
results.
We
continue to assume that
For any set
A
/3
of available pairs [u,
arg
<
1 for
j3
[3
for
our
simplicity.
w) agents with
{6, /3)
maximize
their utility:
max </^
—u + w
{u,w)eA yjj
Note that
this arg
implies that
max
set
is
we can without
identical for all types with the
loss in optimality
same
ratio
x
=
9/ ^ which
assume the allocation depend only on
X.
To
see this note that the allocation
a given x only
the
lemma
if
the x-agent
in section 4
is
may depend on
indifferent
showed that the
(5.
and
/3
independently
amongst several pairs of
set of
x
for
u,
w.
for
However,
which agents are indifferent
is
course, if agents can only report at f = 1 then one cannot costlessly obtain truthful! reports
However, with large enough
it is likely that the cost of revealing /3 would be small.
*0f
on
9
N
24
of
measure
As a consequence, allowing the
zero.
allocation to
depend on
6 or
/3
independently, in addition to x, cannot improve the objective function. Without loss
in optimality
The
we
limit ourselves to allocations that are functions of
objective function
is
then:
E [eu (x) + w {x)] =E[E [Ou (x) + w (x)| x]] =
where a
of X
and
(x)
=E
F (x)
{6\x) and / (x)
be
its
is
f
[a (x)
the density over x. Let
u{x)
+ w (x)] / (x) dx
X=
[x, x]
be the support
cumulative distribution.
Define Xp as the lowest value such that for x
E [a (x)| X >
modify our previous assumption
Assumption A. For x e
A
"
that
2-«'(x)
xf'ix)^
/(x)
Xp
in the following way.
we have
[x, Xp],
>
x]
X
We
x only^.
-
Note that without heterogeneity a
l-a(x)/x
=
(x)
/3x so that
assumptions
A
and
A
are
equivalent in this case.
Heterogenous Temptation Problem
max
{a
(x)
u
+ ly (x) =
xu
(x)
/
[x)
+ w (x)} dF (x)
<-)M) J
subject to
xu
(x)
c{u{x))
u{x)
+ w (x) +
+ k{w{x)) <
> u (y)
for all
x
u
(x)
dx
1
>
y
^Given /3 < 1 a simpler argument is available. The planner can simply instruct agents with given
X to choose the element of the arg max with the lowest u, since the planner has a strict preference
for the lowest u element. The argument in the text is similar to the one used to extend the analysis
to arbitrary horizons and can be applied to the case where /? > 1 is allowed.
25
The proof
of the next result closely follows the proof of proposition 4.
Proposition 10 Under assumption A agents with x < Xp
optimum and agents with x > Xp
strained
are offered their uncon-
are bunched at the unconstrained
optimum
for agent Xp.
Commitment with
6
Self Control
Gul and Pesendorfer (2001,2002a,b) introduced an axiomatic foundation
We
ences for commitment.
review their setup and representation result briefly in
how we apply
general terms and then describe
In their static formulation the primitive
P {A)
choices, with utility function
maXagyip(a)
this case
then
P
if
is
some
for
a set
A
is
for prefer-
it
is
to our framework.
a preferences ordering over sets of
over choice sets A. In the classical case
utility function
P (A) =
p defined directly over actions. Note that
in
reduced to A' without removing the best element, a* from A,
In this sense, committment, a preference for smaller sets,
not altered.
is
not valued.
To model a preference
a set A' that
is
a
strict
for
commitment they assume a consumer may
subset
.4, i.e.
P {A') > P (.4)
such preferences can be represented by two
by the
and A'
utility functions
C
U
A.
and
strictly prefer
They show
V
that
over choices a
relation:
P {A) = max {p (a) + t{a)} — max
a€.4
One can
think of
when choosing a
t
(a)
t
(a)
ae.4
— maXag^ t (o)
as the cost of self-control suffered
instead of argmax^g^i
(a).
dynamic
In a
by an agent
setting recursive prefer-
ences with temptation can be represented similarly (Gul and Pessendorfer (2002a,b).
In our framework the action
and
k.
is
a choice for current consumption and savings, c
In order to nest the hyperbolic preferences
and Smith (2001) and
model we follow
use:
(c,
k)
=
On
t (c,
k)
=
(p
p
(c)
[On
26
+ w {k)
(c)
+ i5w (A:))
Krusell,
Kuruscu
>
where the parameter
captures the lack of
no self-control and yields
—*
selects
oo the agent has
difference
is
whatever
is
best for his previous 'selves'
objective function
{9u{e)
maximizes
(i.e.
converge
that in the limit
oo we obtain a tie-breaking criteria that whenever an agent
The
/
—^
As
fully to his temptation. His preferences essentially
by the hyperboHc model. The only
to those implied
as
self control.
is
indifferent he
t).
is:
+ w{e))f{9)de +
+ pwie))f{9)de
{9u{9)
(p
Je
Je,
-(p f [9u{9)+l3w{9))f{9)d9
Je,
where
of that
{u,
is
w)
is
the allocation for the "temptation agent" and
chosen by the "self-control agent"
a support larger than
9
Given a set of offered
that maximizes 9u
+
=
given by
(m,
w)
f3w with
=
<
9/3/ 13
=
(1
+
4>/3)
/ (1
is
the allocation
9.
pairs, the "self-control agent" will
/3
w)
convenient to define everything for
where 9
[9, 9]
choose an allocation that maximizes 9u
will
It is
.
{u,
+
0), while the
+ (3w.
the "self-control" agents choose from the same set
choose an allocation
"temptation agent"
Since both the "temptation" and
follows that,
it
u{9)=u{9l3/P),
(8)
so that the "self-control agent" 9 acts as a "temptation-agent" with a lower taste
shock.
Substituting (8) into the objective function
(1
The
+
/"
(9u
0)
first
+ /3io
(913 /l3\)
f
{9)
d9
obtain,
-
4>
f {9u
{9)
+ (3w {9)) f (9) d9.
term can be shown to equal,
(1
Note that h
represent
(^/3//3)
we
[9)
its
=
+
(/.)
{9u {9)
13/ /3 /
JiJ3/0
f (9p//3] ^/l3
+
f3iu {9))
f (9P/I3) P/pd9.
^
is
the density of the
corresponding distribution function.
27
'
random
variable 9P/^; let
H {9)
The
objective function can be written as,
3
+
(1
<^)
§
r§
rS-e
r
+
{9u (9)
/3w (9))
hi9)d9-d
{9u (9)
+
Biv {9))
f
(9) d9.
Substituting in the incentive constraint,
9u
(9)
+
/3w {9)
=
u{9)de
I
+
vo,
Je
where vq
{l
=
{9J3/P)u{9)
+ (3w{§)
parts,
we
obtain:
+ (P)^Jf il-H{9))u{9)de-(l>J^ {l-F{9))uie)d9+(^il +
where we are taking both
h
and integrating by
{9)
=
0, for all 9
>
intervals of integration as being
and /
^/3//5
(^)
=
for all 9
<
9.
<P)p/l3-cP^vo.
from ^ to ^ by letting
In addition
we
require
u to be
non-decreasing and the budget constraint:
vo+ ( u
- 9u {9) <
(9^ d9
(3K~' {y
Definition. Let 9p be the lowest value such that for 9
y_7(l +
Assumption A. For
9
€
(l
We now
strained
for 9p.
0)
^
(1
[9,9p],
-
^
m -<P{1-F
we have
Denote
to
>
9p,
{9))] d9
<
that
+ ,/.)(/3//3)'/(^/3//3)-0/W>O
show that the optimal allocation
optimum and
- C {u {9)))
is
to offer
bunch types higher than
this allocation
by {w*,u) as before.
28
all
types below 9p their uncon-
9p at the unconstrained
optimum
The Lagrangian
L=
is
jU{l + <l>)^{l-H{9))-<l>{l-F{9))-il-A{e))\u{e)d9
+
{/3K-'
/
{y-C {u (9)))+ 9ii)
dk
/3
+ l(l+</>)^-0
where
A
is
a non-decreasing Lagrange multipUer for the budget constraint, normahzed
=
so that A(^)
1.
Proposition 11 The
Proof.
We
allocation
(l-A(0)No
allocation {w*,u)
is
optimal
if
and only
if
assumption
follow the proof of Proposition 4 as closely as possible.
we
A
holds.
At the proposed
have:
dLiw,u-h^Ji^\A)= l^{{l + <P)^{l-H{9))-cl){l-F{9))-{l-A{9))]h^{9)d9
+
Equation
(^P
l^^[^f^-^]\\9..e]hudA
(5) requires
A{9)
—
{1
+
4>)
/),„!
A)
=
7
(/3//9
—
+
1).
;i
+ 0)|-(/.-(l-A(^))
Using this and integrating
L,
by
(3)
parts leads to
dL {w, u-
/i„,
[9)
hu {9)+
7
(0) dh^, {9)
(9)
where,
^{9)^
It
Nil + 4>)t [I- H {9))- <t^{l-F [9))- {I- K{9))\d9+e,j'^(^~^
follows that (4) requires
7
(^i)
=
for 9
G
29
9,9r,
,
i.e.
where u
is
strictly increasing.
This
turn requires,'
in
A
6
G
=
(^)
-
1
+
(1
0)
(1
-
+
// [6))
I
- F
(1
(10)
(9))
A
GiA'en the proposed allocation, this defines a unique multiplier
[9,9p).
separating region
non-decreasing.
[9,
It
9p)
and assumption
A
A
assumption
follows that
necessary and sufficient for
is
A
in the
{9) to
be
necessary for the proposed solution
is
to be optimal.
We now
A
over the whole range
specified
A
prove sufficiency by showing that there exists a non-decreasing multipher
(^)
=
1
A
for
9p) so
[9,
Q.
such that the proposed iv,u
we only need
and
satisfies (4)
to specify the value of
A
for [9p. 9j
(5).
We've
and we
set
in this interval.
A
Note that the constructed
decreasing
all
that remains
to
is
not continuous at
is
show that the jump
To show that A
9p.
at 9p
is
upward which
is
non-
requires:
-{l + cP)^{l-H{9)) + <f>il-F{9))<0
This follows from the definition of 9p
for all 9
9p
=
>9, which
9 then the result
The proposed
7
=
A
is
is
is
= -
{I
+ 6)
the lowest 9 such that 7
9:
3
§
[1
-
H {9)) +
(f>{l
(6*)
- F (9)) <
<
0.
If
immediate.
allocation,
whenever u
The next
that
implies 7' {9p)
>
for 9p
w
and
increasing.
and
u,
Using
multiplier, A,
imply that 7
(6) it follows that (4)
result establishes a connection
and
<
and that
(5) are satisfied.
between assumptions
A
and
A
showing
a weaker requirement.
Proposition 12
condition for assumption
Proof. Let ^
=
$
assumption
// the condition for
-
>
{9, e)
A
holds for
il
+
s)
[p
holds for [9_,6pj3/ (3],
[9, 9p]
then assumption
=
A
{e)
A
is
equivalent to
l^y f
30
(^^
(e) //?)
-/
(0)
>
then the
with
/3
(e)
=
$, {9,e)
and
^' (e)
$,
=
(2
-
/3
=
(1
(^, e)
assumption
+ e)
+ e) / (1 +
(/3
p
-
=
/?)
[ep/p)
/ (1
+
^^
Note that $
e).
+
^
{9, 0)
=
=
0,
(2/3/ (e/3//3)
+ p'f
(op/p) 9/p)
/?'
(e)
e)^ Thus:
((2
-^+
e)
/
(^/5//3)
+
(1
-
/3)
/'
(^/3//5) ^/3//3)
A holding at ^ imphes that (2 - /3) f{9) + {l - P) f'{9)§ > 0. This imphes
for e > 0. So if the condition in assumption A
/(^) + (1 - /3) f'{9)9 >
holds for [9,6pP/P] then $e(^,e)
$
{9, 0)
we have that
0,
if
A
>
for all £
holds for
[9,
>
and 9 G
[§,9p].
Given that
e^P/P], then
/o
so that assumption
A
for ^
G
7
Disagreement on Exponential Discounting
[:^,
^p]
holds.
This section departs a bit from our intra-personal temptation environment.
We now
consider the case where individuals discount the future exponentially but do so at a
different rate
than a
'social planner'.
Caplin and Leahy (2001) and Phelan (2002)
provide motivations for such an assumption. Here we simply explore the implications
of such a difference in discount rates.
It is
important to note that this modification constitutes more than just a de-
parture on the form of discounting.
Our previous
analysis relied on the tensions
within an individual due to temptation. In contrast, in the current case we require
some
paternalistic motivation for the social planner's disagreement with agents.
a consequence, some
analysis.
may view
this case as
more ad hoc and somehow
However, we believe that paternalistic motivations
government
policies. In
may
less
As
worthy
of
be behind several
the next section we discuss other examples of paternalism.
For the two period case the analysis requires absolutely no change, only a different
31
interpretation for
,3.
The
now
the objective function
discounting,
it is
difficulty
is
arises
from an assumed chfference
that the planner and agent will disagree on more than just
on the evaluation of future lifetime
in
and
social
whether we can extend the results to longer horizons.
is
to discount future utility relative to present utility:
more
in private
no longer motivated by time inconsistent preferences. The relevant
question that remains
A
difference in discounting in the incentive constraints versus
difficult since
utility itself.
now
there
is
how much
also disagreement
This makes a recursive formulation
the key simplification was that the same value function appeared
the objective function and in the incentive constraints.
now we
Indeed,
require two value functions, one for the planner, v, and one for the
agent, v^. Fortunately, in the case with logarithmic utility these two value fmictions
are related in a simple way. This allows us to keep track of only one value function,
V,
rendering the analysis tractable.
We
can show that
all
our results go through in
this case.
Consider
for the
first
the situation with three periods. Let the exponential discount factor
agent be given by
planner
The
is
P and
factor for the social
highest utility achievable by the agent in the last two periods
some constant
for the last
B'^.
(y)
=
{I
+
P) log {y)
is
+ B^
For any homogenous mechanism the planner's value function
two periods takes the form:
V2{y)
The important point
ficients.
assume the discount
unity.
vt
for
for simplicity
is
=
2\og {y)
+
B''
that these value functions differ only by constants and coef-
As a consequence the
correct incentive constraint for the
written with either value function. That
9U{c{9))
first
is,
+ pv^ {y{e)) > eu
32
[c
{ey)+(5i4 {y (b))
,
period can be
is
equivalent to,
du
where Ps
=
/3(1
(c [9])
+ kv2 {y {9)) > 0U
+ /3)/2 <
1 is
(c (^~))
+ Psv2
{y [e^)
a fictitious hyperbolic discount factor
three periods to go. Note that the incentive constraint (11) has
all
(n)
,
when
there are
the features of the
hyper bohc discounting case.
Thus, we can we can write the three period problem disagreement on exponential
discounting in the same
way
as the hyperbolic discounting problem.
With k remaining periods the discount
generalizes to any finite horizon.
must be applied
Note that
/3k is
The arguments
factor that
is
decreasing in k and converges to zero (this last feature
the planner not discounting the future at
is
special to
all).
Other Interpretations
8
In this section
we
discuss
how our model can be
reinterpreted for other applications.
Paternalism
8.1
Another interesting application of the model to a paternalistic problem
between schooling and
leisure choice.
In
many
an adult so that we can interpret paternalism
preferences of parents and child.
cases the relevant agent
literally as
Alternatively, other adults
child's preference are given
represents schoohng time and
parameter 9
/
by the
utility function
may be
parent's preferences are given by
6U (s) +
on schooling
W
(s)
,
so that
not yet
is
altruistically
legislation.
9U (/) + /3W (s)where
vs.
s
The
taste
other activities.
The
represents other valuable uses of time.
affects the relative value placed
the choice
a struggle between the
concerned about children without parents and support paternahstic
The
is
more weight
is
given to
schooling time.
The
s
+ <
I
allocation of time
1.
In this example
is
constrained by the time
no insurance
is
possible.
33
endowment normalized
to one,
Externalities
8.2
Another origin
a divergence of preferences between the planner and the agents
for
when consumption
of a
ize the effects of their
To make
good generates positive
The
[6- (c
this precise,
last
()
,
k ()))
suppose there are two goods,
= eU [c {d)) +
m
when the
{k (9))
and
/c,
that the agent with
entire allocation
W
+
fv {9, (c, k)) dF {9) =
Thus, we can represent
{k {9))
which
same
(c [6)
dF {9)
,k{9))
(12)
{1-(3)J
9U (c) +
W
I
{BU
is
+
(c {9))
k.
The
utilitarian
W [k
{9)))
dF {9)
(k) as the relevant utility function for agent-^
from the planner's point of view. Although this
agent-6',
is
is:
W=
to the
c
term captures the externality from the consumption of
welfare criterion
by
Agents do not internal-
externalities.
consumption on other agents but the planner does.
taste shock 9 obtains the following utility
V
is
is
given by the expression in (12),
not the utility actually attained
it is
an interpretation that leads
welfare functional.
Conclusion
9
This paper studied the optimal trade-off between commitment and
intertemporal consumption/saving model without insurance.
flexibility in
an
In our model, agents
expect to receive relevant private information regarding their tastes which creates a
demand
for flexibility.
value commitment.
for flexibility
But they
also expect to suffer
The model combined the
from temptations, and therefore
representation theorems of preferences
introduced by Kreps (1979) with the preferences for commitment pro-
posed by Gul and Pesendorfer (2002).
We
setting
solved for the optimal solution that trades-off
up a mechanism design problem.
We
commitment and
We
by
showed that under certain conditions
the optimal allocation takes the simple threshold form of a
ment.
flexibility
minimum
savings require-
characterized the condition on the distribution of the shocks under which
this result holds,
and showed that
if
this condition
34
is
not satisfied, more complex
mechanisms might be optimal. Future work
with a hope of understanding
The model
who
open
is
how
it
may
on the case with insurance,
will focus
affect the results
to a variety of other interpretations.
cares about an agent but believes the agent
is
obtained here.
A
paternalistic principal
biased on average in his choices
would face a similar trade-off as long as the agent has some private information
regarding his tastes that the planner also cares about.
We
discussed applications to
schooling choices by teenagers and situations with externalities.
A
Proof of Proposition 6
Suppose that we are
for 9l
offering a
segment of the budget
between the tangency point
line
and that of Oh, with associated allocation cl and ch- Define the
inchfferent
from the allocation c^ and Ch then
9*
G {9i,9h)
for
9h >
9* that is
Upon
9l-
removing the interval 9 € {9*,9h) types move to Ch and 9 £ {Oi,0*) types move to
Cl allocation.
Let A{9h,9l) be the change in utility for the planner of such a move (normalizing
income to y
=
1
for simplicity)
A{9h,9l)=
/
{9U{c*{9H))
+ W{y-c*{9H))}fi9)d9
{eu[c*[9L))
+ w{y-e{eL))}f{9)d9
6l
Bh
{9U{c*{9))
where the function
c* [9) is
+ W{y-c*{9))}f{9)d9
defined implicitly by
eU'[c*{9)]^l3W'{y-c*{9))
and
6* {9h, 9l)
is
(13)
then defined by
9*
=
{9H,eL)U
{c*
9* {9h, 0l)
{en))
u {c*
+ I3W {y -
{9l))
Notice that A{9l,9l) =0.
35
c*
{9h))
+ m{y- e
{9jy)
,
(14)
The
lemma regarding
following
Lemma. The
A {9h,6l)
partial of
9^
the partial derivative oi A{9h,0l)
n
^n
with respect to 6h can be expressed
,.,U'
ofn
,
[9] 9*) is
Proof.
(c*
We
Ai
as:
89 H
defined by.
= {y-l3){9-9*)9*f[9*)
S[9,9*)
Since U'
used below.
{Oh)) dc* [Oh)
[c*
P
where 5
is
>
{9h))
and
dc-{eH)
>
d0H
0,
f
t^e)
=
then sign (Aj)
9)d9
{t
sign {S{9h,9*))
have
=
{9h, 9l)
[9hU
- [r
+
+
{c*
{9h))
{9h.9j^)
U
(c*
{9U'
I
{e* {9h. 9l)
- [OhU {c*
W [y -
+
(c*
U {c*
{9h))
{9h))
{9h))
{9l))
c* {9h))\
W
+
-
W
+
W
+
W (y -
-
c*
c*
-
{y
[y
{9h))
(^//)
89*
c* [9^))]
-
-
{y
f
c*
f
{9']
W
H
^-^^d9
{9h))} f (9)
89*
c* {9^))}
f
f
[9*]
89 H
{9h)]
Combining terms,
Ai(^;/,eL)
/'"
+ {r
=
[eU'
{c*
{9„))
{9h,Bi^) [U [c* {9l))
Now, from
(14)
-
W
- U
[y
{9h))] f {9) d9)
^K^
W [y -
-
[c* {9h))\
+
c* [9^))
W {y -
we have
9U'\c{9)\-W'{y-c{9)) =
Substituting above
36
/3-1
9U'
[c*
89*
c*
{9^))} f (9*)
89h
Ai(^h,^l)
=
9--^9H)f{0)d9^U'{c*{en))^^^
'O'(0H.eL)
+
we
{e* {9h, 6l) [U (c* {6,))
also have that
9*
{9h,9l'
- U
{c* {9h))]
+ W{y-c*
{9^))
-
W
{y
39*
-
c*
(9^))} f {9*
39 H
from (13)
[U
{c* {9l))
- U
= {W {y -
(c* i9H))]
c* (0^))
-W{y-c* {9h))}
(3
So,
^i{9h,9l)
39*
=
e* f {9*)\[IT {c* {9h))
-U [c* {9^))] 39
.e„
j^" {^^9h - 9^ f {9)d9^U'
Differentiating (14)
we
[U
Using the
{9h))
- U
{c* {9,))]
= -
fact that
9U'
[c* {9)\
-
pW
{c*
^
[U
{c*
3c* {9h)
{c*
{9h))
39 H
obtain:
30*
^
H
{9h))
- U {c*
[9*U' {c* {9h))
{1
-
c* {9))
=
{9,))\
[Oh
-
=
-
9*]
^W {y ~
this
U'
3c*
c* {9h))\
7H)
^^^
imphes
[c*
{9h)]
^-—^
Substituting back the result follows.
From
the
derivatives
lemma we
we
only need to sign
5 {9h, 9*).
Clearly,
5 {9*
,
0*)
also get that
3S{9,9*)
39
= [l-P]e*f{9*)-{l-/3)9f{9)-
/
f{9)d9
Notice that
dS
{9,
9*
=
39
3-S{9,9*)
= -{2-P)fi9)-{l-/3)9f'{9)
id9f
37
=
0.
Taking
Note that 9^5 (0,r)/(ae)^ does not depend on ^*,just on
5.
It
^"gyM <
follows that sz.gn
(
if
and only
if
-
f{0)
That
is, if
A
9-5
holds. Integrating
{9, 9*) /
[d9f
^
'
twice:
red~S[d,9*
ren
S{9h.B*)=
1-/3
'
/
/
/
,
dOdB
d9]
Thns S{9h,0*) <
if >!
holds.
This impUes then that Ai
[9,
9l)
<
for all ^
^{9h,9l)=
so that
^7^
>
-^
and clearly 9i G argmaxe^>9^
=>
A {Oh, 9l)<0
punching holes into any offered interval
The converse
is
also true:
if
A
^i, if
;
for all
In other words
is
holds;
and
if
9h and 9^
assumption
A
holds then
not optimal.
it
is
optimal to remove the whole interval.
In other words,
^o)
A
does not hold for some open interval 9 € {di.O^)
then the previous calculations show that
(6'i,
eissumption
A,{9;9L)d9
/
A {9h^ 9l)
>
e arg
S.t.
max A
{9h, 9l)
ei<9L<9H <
This concludes the proof.
38
02
Lemma on
B
Differentiability
Definition. Given a function
x,h E
spaces. If for
T
lim
Lemma.
called the
it is
X
Q C
and
X
and
Y
are
normed
the Umit:
fl
a—>0
exists then
^^ Y, where
Q.
:
a
Gateaux
[T (x
+
ah)
- T {x)]
differential for
x,h ^
Q,
and
is
denoted by
dT [x]
h).
Let
T{x)= I g{x{e),e)dii{9)
Je
(9,0,/.;,) is
in
any measure space (not necessarily
Q
some space
which
for
T is
and
a G
some
for
[0, e]
is
e
and
(a
minimum
Suppose g
integrability).
continuous in x, for
>
:
Q
^ R^
defined (an arbitrary restriction or perhaps a required
restriction to ensure measurability
gx {-,0) exists
a vector space) and x
i? or
Then
all 9.
requirement
{-,9) is
as long as
x
+
concave and
a/i
G
fi
for
for existence! of course) then:
dT{x;h)= f gAx{9),9)h{9)d^t{9)
Je
if
this expression
Proof.
dT (x;
By
h)
is
well defined.
definition
=
lim
-
a^O a
[T {x
+
ah)
- T (x)]
= hm [ -[g {x {9) + ah {9) ,9)-g{x (9)
= [
,
9)]
dp
{9)
gAx{9),9)h{9)dfL
Je
lim
/
"^0 Je
since for
a <
e
-[g{x{9)
+
ah{9)
,9)
rv
-
g{x{e) ,9)]- gAx{9) ,9)h{9)
we have
-[g{x{9)
a
< -Jg{x
+ ah{9)
(9)
,9)
- g{x{9) ,9)]- g,{x[9)
+ Eh {6) ,e)-g{x [9)
39
,
9)]
-
,9)h{9)
g, [x {9)
,
9)
h
{9)
d^i{9)
by concavity of
g.
+ eh {9)
Since g {x (9)
sumption
it
any function /
chapter
cise 7.26, pg. 192,
,
9)
+
follows that i [g {x (9)
integrable. Since
gx {x {9)
g (x (9)
,6),
h
(9)
eh
and
[9]
g^, (.r
integrable
We
(9)
,9)- g{x
if
SLP] we have that
integrable.
is
\
8,
is
9)
,
,
{9)
,
and only
9)]
are integrable by as-
{9)
-g^
{x {9)
+
eh
h
9)
,
(9) is also
integrable [see Exer-
if |/| is
{x (9)
j^ [g
h
9)
{9) .9)
-
g
(,t
{9)
,
^)]-
then have the conditions for Lebesgue's Domi-
nated Convergence Theorem.
It
follows that:
-
Hm
a
(x {9)
[g
Hm -
'e
Q—»o a
[g
+ ah
+ ah [9]
{x {9)
by continuity oig^ and
(9) ,9)
is
=
h
Xi
—
^
,9)
+ ah
Xo and
non- decreasing functions
,
-
9)]
g {x {9)
,
9)]
follows that
g, {x {9)
-
,
dT {x;
h)
=
.ri
G
E
^
fi.
for
for
[0, e]
,
9) h (9)
Jq
in
some
{9)
g^^
is
d^L
{x {&)
,
=
9) h
((?) (i/i.
ST{xo;h) then the
£
Note that the case where
convex and so
is
a E
h
9)
g, [x {9)
convex and that we are interested
obvious requirement that xq
if
g {x {9)
its definition. It
Remark: Suppose Q
and only
-
>
fl
is
is
satisfied
if
the space of
the space of measurable functions.
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Date Due
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