Econometrica, Vol. 72, No. 1 (January, 2004), 119-158
SELF-CONTROL AND THE THEORY OF CONSUMPTION
BY FARUK GUL AND WOLFGANG PESENDORFER1
To study the behavior of agents who are susceptible to temptation in infinite horizon consumption problems under uncertainty, we define and characterize dynamic self-
control (DSC) preferences. DSC preferences are recursive and separable. In economies
with DSC agents, equilibria exist but may be inefficient; in such equilibria, steady state
consumption is independent of initial endowments and increases in self-control. Increasing the preference for commitment while keeping self-control constant increases
the equity premium. Removing nonbinding constraints changes equilibrium allocations
and prices. Debt contracts can be sustained even if the only feasible punishment for
default is the termination of the contract.
KEYWORDS: Self-control, preference for commitment, consumption behavior, timeinconsistency, borrowing constraints, risk premium, sovereign debt.
1. INTRODUCTION
IN EXPERIMENTS, SUBJECTS EXHIBIT a reversal of preferences when choosing
between a smaller, earlier and a larger, later reward (Kirby and Herrnstein
(1995)). The earlier reward is preferred when it offers an immediate payoff
whereas the later reward is preferred when both rewards are received with
delay. This phenomenon is referred to as time inconsistency and has inspired
theoretical work that modifies exponential discounting to allow for disproportionate discounting of the immediate future.2
This paper proposes an alternative approach to incorporate the experimental evidence. We extend our earlier analysis of self-control in two period decision problems (Gul and Pesendorfer (2001)) to an infinite horizon. Our goal is
to reconcile the experimental evidence with tractable, dynamically consisten
preferences and to apply the resulting model to the analysis of problems in
macroeconomics and finance.
As an illustration, consider a consumption-savings problem. A consumer
faces a constant interest rate r and has wealth b at the beginning of period 1.
Each period, the consumer must decide how much to consume of his remaining wealth. Let z(b) denote the corresponding decision problem and let c denote the current consumption choice. Our axioms imply that the consumer has
preferences of the form:
W(z(b)) = max u(c) + v(c) + 8W(z(b')) - v(b)}
ce[O,b]
'We thank Pierre-Olivier Gourinchas, Per Krusell, and Robert Shimer for their suggestions
and advice. We thank four anonymous referees and a co-editor for their comments. This research
was supported by grants SES9911177, SES0236882, SES9905178, and SES0214050 from the National Science Foundation and by the Sloan Foundation.
2See, for example, Strotz (1955), Laibson (1997), O'Donoghue and Rabin (1999), Krusell and
Smith (1999).
119
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120
E GUL AND W. PESENDORFER
where u and v are von Neumann-Morgenstern utility func
(b- c)(l + r) is the wealth in the next period. These preferen
an individual who, every period, is tempted to consume his enti
Were he to do so, the term v(c) - v(b) would drop out. When
less than his endowment, he incurs the disutility of self-contr
The utility function v represents "temptation," that is, the indiv
current consumption. Optimal behavior trades-off the temptatio
with the long-run self-interest of the individual, represented b
main theoretical result of this paper (Theorem 1) is a represent
yielding the utility function W above.
The preferences developed in this paper depend on what the in
sumes and on what he could have consumed. In a simple consum
problem with no liquidity constraints, the maximal temptation v
tation utility of consuming all current wealth. More generally
temptation depends on the set of possible consumption choices f
For example, if there are borrowing constraints, the set of poss
tion choices will reflect these constraints.
To allow a direct dependence of preferences on opportunity sets, we study
preferences over decision problems. Building on work by Kreps and Porteus
(1978), Epstein and Zin (1989), and Brandenburger and Dekel (1993), we develop the appropriate framework to study infinite horizon decision problems.
With the set of decision problems as our domain, we define dynamic self-control
(DSC) preferences and derive a utility function for those preferences. The formula above is a special case of this derived utility function.
In the literature on dynamic inconsistency, nonrecursive preferences of the
form u(cl) + YS 3=2 '-u(c,) are specified for the initial time period (Phelps
and Pollak (1968), Laibson (1997)). In addition, it is assumed that the preferences governing behavior in period t are given by u(c,) + jEt=t+ 3t' -'u(c,,).
Therefore at time t > 1, the individual's period t preferences differ from his
conditional preferences-the preference over continuation plans implied by his
first period preferences and choices prior to period t. This preference change
is taken as a primitive to derive the individual's desire for commitment.
In contrast, we start with preferences that may exhibit a desire for commitment. The description of a "consumption path" includes both the actual consumption and what the individual could have consumed in each period. On
this extended space, preferences are recursive and the conditional preferences
are the same as period t preferences. Our representation theorem shows that
DSC preferences can be interpreted as describing an individual whose temptation utility, v, interferes with his long-run self-interest represented by u + 8W.
The consumer uses self-control to mediate between his temptation utility and
his long-run self-interest.
The recursive structure of our model allows us to apply standard techniques of dynamic programming to find optimal solutions. At the same time,
the model is consistent with the type of preference reversal documented in
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SELF-CONTROL
121
the experimental literature. Consider a consumer who must decide
smaller reward in period 1 and a larger reward in period 2. Since th
ward would lead to a larger consumption in the decision period, th
incurs a self-control cost when opting for the later reward. Now c
situation where the earlier reward is for period t > 1 and the later
t + 1. Since the decision is taken in period 1, the choice does not a
sumption in the decision period and therefore the later reward ca
without incurring a self-control cost. Thus, the agent's behavior a
recursive when we only observe his consumption choices.
Our framework is rich enough to accommodate the infinite horizo
tic dynamic programming problems central to macroeconomics an
economics. Techniques developed in those areas can be applied to D
ences to explore how self-control and preference for commitment
conclusions of standard (macro)economic models. Section 5 contain
tions of DSC preferences to competitive economies. In Section 5.1,
a deterministic exchange economy with DSC preferences. We give
under which a competitive equilibrium exists and find that in gener
itive equilibria are not Pareto efficient. We also examine steady sta
and show that with DSC preferences the steady state distribution o
independent of the initial wealth distribution. In Section 5.2, we o
removing constraints that are not binding given the original equilib
tions and prices, may change these allocations and prices. In Section
alyze a simple stochastic exchange economy and find that (under
assumptions on u and v) increasing the preference for commitme
the predicted premium of risky over safe assets. Section 6 explor
compatible debt contracts. If the only punishment for default is exc
future borrowing, then standard preferences imply that there are
compatible debt contracts (Bulow and Rogoff (1989)). In contrast,
preferences, "natural" debt contracts turn out to be incentive comp
2. STATIONARY SELF-CONTROL PREFERENCES
Consider a decision-maker (DM) who must take an action in every period
t = 1, 2,.... Each action results in a consumption for that period and constrains future actions. The standard approach to this problem is to define preferences for the DM over sequences of consumption realizations. This standard
approach excludes preferences that depend not only on outcomes but also on
what could have been chosen. Such a direct dependence on the opportunity set
is natural in a context where individuals suffer from self-control problems.
To allow for preferences to depend on opportunity sets we make decision
problems the domain of our preferences. Let C denote the compact metric
space of possible consumptions in each period. An infinite-horizon consumption problem (IHCP) is a set of choices, each of which yields a consumption
c E C for the current period and an infinite horizon problem starting next
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122
F GUL AND W. PESENDORFER
period. Choices may yield a random consumption and continuati
Hence, an IHCP is a set of probability measures where each reali
a consumption today and a continuation IHCP.
Let Z be the set of IHCPs. Each z E Z can be identified with
set of probability measures on C x Z. For any compact metri
A(X) denote the set of probability measures on X and let IC(X) d
of nonempty compact subsets of X. We endow A(X) with the we
and note that A(X) is compact and metrizable.3 We endow 1C
Hausdorff topology which implies that /C(X) is a compact metri
domain of preferences Z can be identified with IC(A(C x Z)).
shows that Z is well-defined and a compact metric space.
We use x, y, or z to denote elements of Z. When there is no risk of
we write A instead of A(C x Z). We use JL, v, or r7 to denote el
A lottery that yields the current consumption c and the continua
z with certainty is denoted (c, z).
For a E [0, 1], let a/,t + (1 - a)v E A be the measure that ass
(1 - a)v(A) to each A in the Borel o--algebra of C x Z. Similarly,
:a= {ua/ + (1 - a)v I , E x, v E y} for a E [0, 1] denotes the convex
of x and y.
A preference is denoted >. We make the following familiar ass
preferences.
AXIOM 1 (Preference Relation): >- is a complete and transitive bin
AXIOM 2 (Continuity): The sets {x I x >- z} and {x I z > x} are cl
We say that the function W: Z -> R represents the preference >
iff W(x) > W(y). Axioms 1 and 2 imply that > may be represente
tinuous function W.
A standard DM evaluates a set of options by its best element. Adding options
to the set can never make such a DM worse off. Thus, if x > y, then the best
option in x is preferred to the best option in y and hence
(*) x>y = xUy x.
Axioms 1, 2, together with (*), imply that there
tion U such that W(x) := max,,, U(Iu) represen
3See Parthasarathy (1970).
4See Brown and Pearcy (1995, p. 222).
5This observation is due to Kreps (1979) who provides the
our setting, note that Axiom 2 ensures that the restriction
preference relation on A. Hence there is a continuous utility
relation. For finite x, y we conclude that x > y iff maxx u(t
Porteus argument. To extend the representation to arbitrary
of u in the compact set x. Note that for finite subsets y, y
Continuity (in the Hausdorff topology) then implies x - y.
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SELF-CONTROL
123
and (*) yield a standard DM. In contrast, a DM who is susceptible to
tion may prefer a smaller set of options to a larger set. That is, he m
preference for commitment.
DEFINITION: The preference >- has apreference for commitment at
is x c z such that x >- z.
When {,t} >- {/a, v} - {v} the DM is worse off when v is availab
addition, derives no benefit from the availability of ,It. We interpre
situation where the DM succumbs to the temptation presented by v.
When {I}t >- {t, v} > {v) the DM derives benefit from the availabili
in spite of the presence of v. We interpret this as a situation where
a temptation but the DM exercises costly self-control and choose
presence of v. More generally, we interpret x > x U y >- y as a situat
the DM has self-control. Theorem 3 (Section 3) shows that, within th
of preferences analyzed here, this definition agrees with the everyda
of the term "self-control" as the ability to resist temptation.
DEFINITION: The preference > has self-control at z if there are sub
with x U y = z and x > z > y.
Axiom 3 allows both a preference for commitment and self-contro
AXIOM 3 (Set Betweenness): x > y implies x > x U y > y.
Our next objective is to characterize preferences that have self-co
a stationary, separable representation. A singleton set represents a
where the DM has no choice. Hence, we refer to the restriction of >
gleton sets as the commitment ranking. The following axiom ensures
commitment ranking satisfies von Neumann-Morgenstern's indepen
iom.
AXIOM 4 (Independence): {u} >- {v}, a e (0, 1) implies {axuC + (1
{av + (1 - a)r}.
Recall that (c, z) denotes a lottery that returns the consumption c in the current period and the continuation problem z. Axiom 5 requires that the correlation between the current consumption and the continuation problem does not
affect preferences. The axiom considers two lotteries: pu = (c, z) + (c', z')
returns either the consumption c together with the continuation problem z or
the consumption c' together with the continuation problem z'; v = (c, z') +
(c', z) returns either the consumption c together with the continuation problem z' or the consumption c' together with the continuation problem z. The
axiom requires the DM to be indifferent between {,u} and {v}.
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124
E GUL AND W. PESENDORFER
AXIOM 5 (Separability): {(c, z) + \(c', z')} {(c, z') + {(c', z)}.
Axiom 6 requires preferences to be stationary. Consider the lotterie
(c, y) each leading to the same consumption in the current period. T
requires that {(c, x)} is preferred to {(c, y)} if and only if the contin
problem x is preferred to the continuation problem y.
AXIOM 6 (Stationarity): {(c, x)} > {(c, y)1 iffx > y.
Axiom 7 requires the individual to be indifferent to the timing of r
of uncertainty. In the standard model this indifference is implicit i
sumption that the domain of preferences is the set of lotteries over c
tion paths. The richer domain used in this paper permits agents who
indifferent to the timing of resolution of uncertainty.6 Since our pur
focus on temptation and self-control, we rule out preference for earl
resolution of uncertainty.
To understand Axiom 7, consider the lotteries pL = a(c, x) + (1 -
and v = (c, ax + (1 - a)y). The lottery JL returns the consumption c t
with the continuation problem x with probability a and the consump
with the continuation problem y with probability 1 - a. In contrast, v
c together with the continuation problem ax + (1 - a)y with prob
Hence, Au resolves the uncertainty about x and y in the current period
v resolves this uncertainty in the future. If {(/} {v}, then the DM is in
as to the timing of the resolution of uncertainty.
AXIOM 7 (Indifference to Timing): {a(c, x) + (1 - a)(c, y)} -
(1 - a)y)}.
Axiom 8 requires that two alternatives, v, qr, offer the same tempt
they have the same marginal over current consumption. For any p.L
Z), .'l denotes its marginal on the first coordinate (current consump
pA2 its marginal on the second coordinate (the continuation problem).
AXIOM 8 (Temptation by Immediate Consumption): v1 = rT, {pu} >
{v} and {(p} > {( , r/} > {1r} implies {/, v} - {/, r1}.
To understand Axiom 8, recall that {(Ju > {(,, v} > {v} represents a s
where the DM is tempted by v but uses self-control and chooses /u. Si
6The domain of preferences used here is closely related to the domain of preferenc
Kreps and Porteus (1978). Their "descriptive approach" defines preferences on the fin
analogue of A, that is, lotteries over current consumption and continuation problems f
period. They use this framework to analyze agents that may have a preference for ea
resolution of uncertainty.
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SELF-CONTROL
125
{uL} > {1/, r} >- {rj} means that the DM is tempted by 77 but chooses
both situations lead to the same choice. According to Axiom 8, if vi
the DM is indifferent between the two situations and hence experi
same temptation in both situations.
We call a preference "degenerate" if it never benefits from addit
tions. Thus > is degenerate if for every IHCP x and y, x > x U y.
DEFINITION: The preference >- is nondegenerate if there exists x
that y c x and x > y.
Theorem 1 provides a recursive and separable representation of n
erate preferences that satisfy Axioms 1-8.
THEOREM 1: If the nondegenerate preference >- satisfies Axioms
there is some 8 E (0, 1), continuous functions u, v : C -- R, and a c
function W that represents >- such that
W(z) := max (u(c+ () + W(z'))d(c, z')
z E
-max f v(c)dv(c, z')
for all z E Z. Conversely, for any 5 E (0, 1), continuous u, v : C -X R, there is a
unique continuous function W that satisfies the equation above and the preference
it represents satisfies Axioms 1-8.
PROOF: See Appendix.
Note that if W satisfies the equation in Theorem 1 for some 8, u, v, then the
preference represented by W is continuous only if W is continuous. Therefore,
by Theorem 1 above, there is a unique preference that satisfies Axioms 1-8 for
any 5 E (0, 1) and continuous u, v. Theorem 1 extends our earlier representation for two-period decision problems. To see the relationship to the earlier
work, note that Axioms 4, 6, and 7 imply the following, stronger version of the
independence axiom:7
AXIOM 4*: x > y, a e (0, 1) implies ax + (1 - a)z > ay + (1 -a)z.8
7To see this, note that x > y implies {(c, x)} >- {(c, y)} by Axiom 6. Axiom 4 then implies
{a(c, x) + (1 - a)(c, z) >- {a(c, y) + (1 - a)(c, z)} and by Axiom 7 {(c, ax + (1 - a)z)} >
{(c, ay + (1 - a)z)}. Applying Axiom 6 again then yields the desired conclusion. Note that Axiom 4* applied to singleton sets yields Axiom 4.
8This axiom was first used by Dekel, Lipman, and Rustichini (2001) in their analysis of prefer-
ences over sets of lotteries.
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126
E GUL AND W. PESENDORFER
In Gul and Pesendorfer (2001) we show that Axioms 1-3 and 4* im
the preference can be represented by a utility function W of the for
W(z) = max{U(,L) + V(/u)} - max V(v)
IIEZ
vEZ
where
W,
U,
establish
that
(1)
U(I)
=
an
U
f(u(c)
V() = f v(c)dl(c, z).
To explain the role of the axioms, we outline the main steps in the proo
equation (1). Axiom 5 implies that U is separable, that is,
U(t/)= - (u(c) + W(z'))dt(c, z')
for some u and W. By Axiom 6, W must represent the same preference a
that is, W(x) > W(y) if and only if W(x) > W(y). Axiom 7 implies that
linear, i.e., W(ax + (1 - ay)) = aW(x) + (1 - a)W(y). Since W is linear,
uniqueness of von Neumann-Morgenstern utility functions implies that
3 + 6W for some 3, 6 E R, 5 > 0.
Let x, y be two IHCPs that offer commitment to the consumption c i
first T periods. The IHCP x yields the continuation problem x' in p
T + 1 whereas y yields the continuation problem y' in period T + 1. If W
not constant, we may choose x', y' so that W(x') 7 W(y'). Note that x
as T -> o and hence continuity requires that 6T+I(W(x') - W(y')) -T -+ oo. Clearly, this implies that 6 < 1. We conclude that U(,) = f(u
5W(y))d1u(c, y) for some u: C --- R and 8 E (0, 1).
Axiom 8 considers a situation where W(t{L, v}) = U(/z) + V(Iu) - V(v
W({u, 7r}) = U(Lu) + V(yt) - V(q) and requires that W({,, v}) = W({I-,
when ,tL and v yield the same current consumption. But this implie
V(v) = V(rl) when v and 7r yield the same current consumption and
V(v) = f v(c)dv(c, z) for some v: C -- RI. Nondegeneracy is used to establ
the existence of the alternatives VL, v, r7 with the desired properties. Not
Axiom 8 is only used to establish the particular form of the temptation u
Without it and without the nondegeneracy assumption, we would get an
ogous representation with an unrestricted linear temptation utility V: A
We refer to nondegenerate preferences that satisfy Axioms 1-8 as dyna
self-control (DSC) preferences. If some W of the form given in Theorem
represents the preference relation >-, we refer to the corresponding (u,
as a representation of >- and sometimes as the preference (u, v, 8).
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12
SELF-CONTROL
The next result establishes that the representation provided in The
unique in the sense that (u, v, 8) and (u', v', 8') represent the same
if and only 8 = 8', and u', v' are a common affine transformation of
is,
(u)
for
(u)
(--u)
some
a
>
0,
fu,
/3
E
R.
THEOREM 2: Let (u, v, 5) be
(u', v', 8') also represents > if
such that u' = au + 3u and v'
A DSC preference is nondeg
constant, the DM is indiffer
of v yields the same preferen
3. CHOICE BEHAVIOR AND INTERPRETATION
The function W represents the agent's preference over decision problems. It
is analogous to the indirect utility function in static demand theory or the value
function in a dynamic programming problem. Period 0 choices among IHCP's
maximize W. Hence, in period 0 the agent chooses y over x if
W(y) > W(x).
In period 1, the agent must choose an alternative ,u from a decision problem x. To simplify the notation below, let E [f] denote the expectation of the
(Borel measurable) function f: C x Z ---> R with respect to /z, i.e., E[f] :=
f (c, z)dtL(c, z). We assume that the agent's utility from choosing /L E x is
(2) U(/t) + V(/u) - max
V() = E,[u r7EX
+ v + 8W] - maxE,[v].
r]ex
Note
that
the
agent
chooses
maxE,
[u
+
v
The
indicated
sentation
asse
were
behaving
main
of
pref
problems.
Thi
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128
E GUL AND W. PESENDORFER
from decision problems. In Gul and Pesendorfer (2001) we provid
extension and give conditions that ensure that the DM indeed behav
gested by the representation. Here, we simply assume that the DM be
this way.
Expression (2) extends the agent's utility function to period t > 1 choices in
a dynamically consistent manner. This means that from the perspective of any
period t, choices in periods t' < t were made optimally and choices in periods
t' > t will be made optimally. For example, assume that W({ft}) > W({,u, v})
but E,[u + v + 8W] > E,[u + v + 8W]. This implies that the agent would have
been better-off if alternative v were eliminated from his choice set but never-
theless chooses v from {ju, v} in period 1. This apparent inconsistency is reconciled by allowing the utility to depend on the decision problem. The utility of
choosing jA from {t} is
E,[u + v + 8W] - maxE[v] = E,[u + 8W].
When Ej[v] > E,[v] the utility of choosing A from {,/, v} is
E,[u + v + 8W] - max Ej[v] = E+[u + 8W] - (E,[v] - E,[v])
and the utility of choosing v from {,J, v} is
E,[u + v + 8W] - max E [v] = E,[u + 8W].
nE{IA, v}
We interpret the direct dependence of welfare on the decision problem as capturing an agent who struggles with temptations. An agent facing the IHCP
{[A} is committed to choosing /u and hence experiences no temptation. When
the agent chooses ,u from {,u, v} he must exercise costly self-control because v
presents a temptation. The cost of self-control is given by E,[v] - E,[v]. When
this cost is large enough, the optimal behavior is to succumb to the temptation
presented by v.
An alternative interpretation would be to assert a change in the agent's prefer-
ence between periods. According to this interpretation, each period is occupied
by a distinct decision-maker with an independent utility function governing
choice. The period 0 utility of decision problem x is W(x) and period t > 1
utility of ,L E x is E,[u + v + 8W]. According to this interpretation, welfare in
period t > 1 depends only on the choice ,t and is otherwise independent of the
decision problem.
We adopt the dynamically consistent interpretation for the following reason. The period 0 preference suggests that the agent wishes to avoid temptations even if they do not alter the choice. For example, if E,[u + v + 8W] >
Ej[u + v + 8W] and E [v] > E [v] the agent would be better off if v were eliminated from the set {,u, v} even though he chooses /L from {/z, v} in period 1. We
see no reason why this dependence of welfare on the decision problem should
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SELF-CONTROL
129
suddenly disappear as we change periods. The changing tastes interp
assumes a choice-problem independence of welfare that seems to con
the very premise of a model of self-control.
To use an analogy from a more familiar setting, consider a decisio
who must choose between apple and banana in periods 0 and 1. The pr
is
(apple, banana) >- (banana, banana) >- (banana, apple)
> (apple, apple)
Hence, in period 1 the agent always chooses banana. How would the agent rank
(apple, banana) and (banana, banana) in period 1? Advocates of the 'changing
tastes' view could argue that because in period 1 the agent always chooses the
banana, he has no taste for variety in period 1. As a result, (banana, banana) is
preferred to (apple, banana) in period 1. By contrast, the standard interpretation asserts the same ranking of pairs for both periods. After all, the period 0
ranking of pairs suggests that the agent has a preference for variety as long as
banana is chosen in period 1 and period 1 behavior offers no evidence contradicting this preference. We feel similarly about preference for commitment. If
in period 0 the agent chooses y c x over x, then the period 1 choice behavior
provides no evidence that the agent would not prefer commitment to y also in
period 1. Hence, we see no reason to assert a change in preference.9
Our model assumes that period 0 is "special" in the sense that it is a period
prior to the experience of temptation. Choices in all other periods are potentially affected by temptations. This suggests that correct identification of the
parameters of our model requires a correct specification of period 0. However,
immediate temptation together with stationarity allows us to circumvent this
issue and treat any period as "period 0" as long as we hold consumption in
that period fixed. Assume that the agent faces the decision problem x = {/,, v}
where ,u = (c, y), v = (c, z). Hence, the agent must choose between two alternatives that both yield the consumption c in period 1 but differ in the decision
problem starting in period 2.10 The agent chooses (c, x) from z if
u(c) + v(c) + 8W(x) > u(c) + v(c) + 8W(y),
9There is a close connection between our dynamically consistent interpretation and Machina's
(1989) dynamically consistent interpretation of non-expected utility preferences. The "changing
tastes" interpretation is analogous to the consequentialist view in Machina (1989). The changing
tastes interpretation views period t > 0 preferences as situation-independent. The consequentialists in Machina (1989) view preferences to be history independent. In the case of non-expected
utility, the two views imply different behavior. In our case, the distinction is more subtle. While
period t > 0 behavior is the same, the two approaches differ when the welfare implications of
policies are evaluated.
10Recall that (c, x) denotes a lottery that yields the consumption c and the continuation problem x with probability 1.
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130
F GUL AND W. PESENDORFER
which is equivalent to W(x) > W(y). Since W represents the period 0 preference it follows that the period 1 choice among alternatives of the form
(c, x), (c, y) reveals the period 0 preference between x and y. Therefore, we
can interpret period 0 simply as any period in which consumption is unaffected
by the individual's choice.
When a decision problem offers commitment to a particular choice [L, the
agent's utility is W({/}u) = E,[u + 8W]. Therefore, we refer to u + 8W as the
commitment utility of A. We refer to El[v] as the temptation utility of /L. The
DM prefers y c x to x only if max,,x E, [v] > max,, E,[v]. Hence, commitment to y is desirable because x contains alternatives that are more tempting
than the most tempting element of y.
Section 2 provided a definition of self-control based on the period 0 pref-
erence over decision problems. Here we provide an alternative definition
based on choice behavior in period 1. If the agent chooses an alternative /
from x = ({L, v} and E,[v] < Ev[v], it means that he did not choose the most
tempting alternative. In other words, the agent exercised self-control. For any
(Borel measurable) function f:C x Z -R> , let C(z, f) := {t E z I E[f] >
E,[f] for all v E z}. Hence, the set C(z, u + v + 8W) denotes the choices from
z whereas C(z, v) denotes the set of most tempting alternatives in z. The agent
exercises self-control if C(z, v) C(z, u + v+ 8W) = 0. Theorem 3 below shows
that for DSC preferences our definition of self-control based on period 0 preference (given in Section 2) is equivalent to the definition of self-control based
on behavior in period t = 1.
THEOREM 3 (Gul and Pesendorfer (2001)): The DSCpreference (u, v, 8) has
self-control at z if and only if C(z, v) n C(z, a + v + 8W) = 0.
PROOF: See Theorem 2 of Gul and Pesendorfer (2001).
4. PREFERENCE REVERSAL
Experimental evidence on time preference has been the main motivation
for research on preference for commitment. In a typical experiment a subjec
is asked to choose between a smaller, period t reward and a larger, period t +
reward. Subjects tend to resolve this trade-off differently depending on whe
the decision is made. In particular, if the decision is made in period t, then th
smaller period t reward is chosen, whereas the larger period t + 1 reward tend
to be chosen when the decision is made in period t- 1 or earlier. (See, fo
example, Kirby and Herrnstein (1995).) The example below demonstrates that
such a reversal is consistent with DSC preferences.
Suppose there is one good and C = [0,1]. Further assume that u and v are
increasing. To translate the experimental setting into our notation we describ
decision problems in which the agent makes exactly one choice between a pe
riod t reward a and a period t + 1 reward /3 and is otherwise committed to
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SELF-CONTROL
131
a fixed consumption. First, consider the case where this decision is
period t. Let zo denote the IHCP in which the only option is to
each period and let z := {(c + /3, zo)} denote the IHCP in which t
is committed to a "reward" 3 > 0 in the initial period and the consu
thereafter. The decision problem
z := {(c, z), (c + a, zo)}
with 3 > a > 0 describes the period t choice between a reward a in period t
and a reward /3 > a in period t + 1.
Next, consider the case where the decision between the period t reward a
and the period t + 1 reward /3 is made in period t - 1. In that case, the decision
problem is
z' {(c, {(c, z c)}), (c, (c + a, Zo)}).
Note that both choices from z' lead to the consumption c in the decision period. The two alternatives are distinguished by the implied period t decision
problems. The reward a implies the period t decision problem {(c + a, zo)}
whereas the reward 3 implies the period t decision problem {(c, z6)}. Even
though z and z' describe the same intertemporal trade-off, they constitute different decision problems because the decision is made at a different date.
From the decision problem z, the agent chooses the earlier reward a if
u(c + a) + 8W(zo) > u(c) + v(c) - v(c + a) + 8W(zp).
When the agent chooses (c + a, z0) from z, he incurs no self-control cost,
whereas when the agent chooses (c, zp) from z, he incurs a self-control cost
Sa := v(c + a) - v(c). Let Dy := u(c + y) - u(c) be the utility gain associated
with the reward y E {a, /3}. Substituting u(c + /3) + 8W(zo) for W(z,) we can
simplify the above inequality to establish that the DM will choose the smaller,
earlier reward a if Sa > SD3 - Da.
Now consider the decision problem z'. From z' the agent chooses the reward
a if
u(c) + 8u(c + a) + 82W(zo) > u(c) + 8U(c) + 82W(zO).
In this case, the agent experiences no self-control costs with either choice because the decision does not affect consumption in the decision period. Substituting for W(zp), the above inequality simplifies to 0 > 8Dp - D,. We observe
a "preference reversal" if
Sa > D, - D, > 0.
When both of the inequalities above hold, the smaller, earlier reward is chosen
if the reward can be consumed immediately, that is, in the decision-period. But,
if the decision is made earlier, the larger, later reward is chosen.
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132
F. GUL AND W. PESENDORFER
5. MEASURES OF PREFERENCE FOR COMMITMENT AND SELF-CONTROL
In this section, we introduce measures that allow us to compare the prefer-
ence for commitment and the self-control of decision-makers. These measures
are based on similar concepts in Gul and Pesendorfer (2001). The versions
here are weaker to facilitate the analysis of the applications considered in the
next section."1
To distinguish between differences in impatience and differences in preference for commitment (or self-control) we compare agents' behavior in decision
problems that involve no intertemporal trade-offs. The IHCP z is intertemporally inconsequential if every choice tu E z has the same marginal over continuation problems. Recall that for ,u E A(C x Z), u' denotes the marginal on C
(current consumption) and /.2 denotes the marginal on Z (continuation prob-
lem).
DEFINITION: z is intertemporally inconsequential (II) if, for every gu, v c z,
. 2 = V2. Let Zll denote the set of all intertemporally inconsequential IHCPs.
The definition below presents a comparative measure of preference for com-
mitment and of self-control.
DEFINITION: The preference > has more instantaneous preference for commitment [self-control] than >_2 if, for every z E Zn,, >2 has preference for
commitment [self-control] at z implies >1 has preference for commitment
[self-control] at z. The preferences >- and >2 have the same instantaneous
preference for commitment [self-control] if > has more instantaneous preference commitment [self-control] than >_2 and >_2 has more instantaneous preference for commitment [self-control] than >- i.
Our objective is to characterize these measures in terms of the representation (u, v, 8). This characterization assumes that the preferences are regular.
A DSC preference >- with representation (u, v, 8) is regular if v / au + /3 for
any ac, 3 E IR.
Consider two regular preferences i> with representations (u,, vi, 5i), i =
1, 2. Let z = (tu, v} E Z1, be such that E,[ul] > Ej[ul]. Note that this implies
E,[u + 8W] > EV[u + 8W] because ,L and v lead to the same marginal over continuation problems. Therefore, if >_ has no preference for commitment at z,
then it must be that E,[vl] > E1[vl]. Now suppose that there are nonnegative
' These versions are weaker because they compare choices from a restricted set (intertemporally inconsequential) of IHCPs. The restriction necessitates the stronger regularity assumption
below. In the Appendix, we refer to the weaker assumption of Gul and Pesendorfer (2001) as
regular*.
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SELF-CONTROL
133
constants au, ac,, p, f, and constants yu, yv such that
U2 = Cu1 + avVl + Yu,
V2 = I+uul + f3v1 + ?v'
It follows that E,[u2] > EV[u2] and E [v2] > Ej[v2] and hence >2 has no preference for commitment at x. Theorem 4 in Appendix C demonstrates that this
condition is necessary and sufficient for >1 to have more preference for commitment than _2-
An analogous result characterizes our measure of self-control. Consider z =
{/z, v} E Z1, with E,[U1 + v1] > Ej[u1 + vl]. Then, ,u is the optimal choice from z.
If >1 has no self-control at x, then it must be that p/ is at least as tempting as v
and hence E,[v1] > Ej[v1]. Now suppose that there are nonnegative constants
au, cY, /3l, 8 and constants yu, Yv such that
U2 + v2 = a,(ul + + ) + avl1 + Yu,
V2 = 1u(u1 + v1) + P3VV + Yv.
It follows that E,[U2 + v2] > EV[u2 + v2] and E,,[v2] > Ei[v2]. Therefore, >2 has
no self-control at z. Theorem 5 in Appendix C demonstrates that this condition
is necessary and sufficient for >- to have more self-control than >2.
The following corollary analyzes situations where decision-makers either
have the same preference for commitment and differ with respect to their self-
control or have the same self-control and differ with respect to their preference for commitment. Since we only make instantaneous comparisons in this
paper, henceforth, without risk of confusion we say "preference for commitment" and "self-control" rather than "instantaneous preference for commit-
ment" and "instantaneous self-control."
COROLLARY 1: Let ,i, i = 1, 2, be regular DSCpreferences and let (ul, v1, 1 )
be a representation of >1. Then:
(i) >-1 and >_2 have the same preference for commitment and >1 has more
self-control than _2 if and only if there exist y > 1 and 82 E (0, 1) such that
(ul, yv1, 62) represents >2;
(ii) >1 and >2 have the same self-control and >2 has more preference for commitment than 1I if and only if there exist y > 1 and 82 such that (u1 + (1 - y)vi,
yv1, 52) is a representation of >2.
PROOF: See Appendix.
Part (i) of the corollary says that keeping u constant and changing v to yv for
some y > 1 is equivalent to a decrease in self-control without changing preference for commitment. Part (ii) of the corollary says that keeping u + v constant
and changing v to yv for some y > 1 is equivalent to increasing preference for
commitment without changing self-control. We utilize this observation in our
analysis of competitive equilibria.
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134
E GUL AND W. PESENDORFER
6. COMPETITIVE ECONOMIES
In this section, we present examples of competitive economies with c
sumers who have DSC preferences. There are n consumers, i E {1, ..., n}
with a DSC preference (Ui, vi, 5i). We assume that each preference is re
that is, vi is not an affine transformation of ui. This ensures that we can
the results of Corollary 1 when using the comparative measures of prefe
for commitment and self-control.
We also assume the functions ui and vi are strictly increasing; vi is convex and
continuously differentiable; ui + vi is concave and continuously differentiable.
The curvature assumptions imply that the temptation utility is risk neutral or
risk loving and the commitment utility is risk neutral or risk averse. These assumptions ensure that the maximization problems below have concave objective functions. In terms of behavior, a risk loving temptation utility implies that
the consumer is tempted by instantaneous gambles, that is, gambles that yield
consumption in the current period. Since ui + vi is assumed to be concave, this
temptation does not lead to risk loving behavior.
There are L physical goods, indexed by I E {1,..., L}; the consumption in
each period is contained in the compact set C where C := {c E IRL+ 0 cl < k}
for some k > 0.
6.1. Deterministic Exchange Economies
We first consider a deterministic exchange economy with complete mar-
kets. Consumers take prices as given and must choose a consumption vector each period subject to an intertemporal budget constraint. Consumer i
has endowment ti = (wci, ..., i, .. .). Endowments are bounded away from
zero, that is, there is an s > 0 such that (it > e for all i, t, 1. We denote with
w) = (Cw1, ..., (w,) the vector of endowments.
The sequence p = (p, ..., Pt,...) E (RfIL+)? denotes the prices. For a given
price p we now define the decision problem of a consumer. Let b > 0 denote the consumer's wealth at the start of period t. The consumer must choose
(c, b') in the budget set
Bt(p, b) := {(c, b') I cp + b' = b, c E C, b' >0}.
The corresponding IHCP is denoted by X,(p, b) and defined recursively as
xt(p, b) = {(c, xt+,(p, b')) I (c, b') E B(p, b)}.
For a consumer with DSC preference (ui, vi, 8i), the utility of the IHCP
Xt(p, b) is:
WI(xt(p, b))
= max {ui(c)+ vi(c) + 8AiW(x+l(p, b')) - max vi(c)
(c,b')EBt(p,b)
(c,b')cBt(p,b)
= max
{(ui(c) +vi(c) + 5iW(xt+(p,b'))} - max vi(c).
(c,b')Bt( p,b) {c\ptc<b}
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SELF-CONTROL
135
Hence, given the prices p and wealth b1, the consumer will choose
(ct, bt+1) that solves
(3) max ui(ct) + vi(ct) + jSiW(xt+1(p, btr+)).
(Ct,bt+l )EBt(p,bt)
Let ci = (c1i, ..., cti,...) denote consumer i's consumption choices. Note that
when a consumer chooses a feasible ci, the corresponding b2, b3, ..., are determined uniquely. We say that ci is optimal for consumer i at prices p and
wealth b1 if ci solves (3). The vector c = (c1, ..., c,) denotes an allocation for
the economy.
For the special case of one physical good (L = 1) the following first order
condition must hold for all t > 1 at an interior solution of (3):
(4) u,(c,) + v(ct) = 5p (u'(ct,+l) + v'(ct+l) - v,(b,+l/p,+l)).
Pt+1i
To get an intuition for equation (4) suppose the consumer reduces consumption in period t by a small amount to finance a corresponding increase in consumption in period t + 1. The left-hand side represents the (temptation and
commitment) utility loss due to the reduction in period t consumption. The
term (5pt/(pt+l))(u'(ct+l) + v'(ct+l)) represents the (temptation and commitment) utility gain due to the increase in period t + 1 consumption. The reduction in period t consumption increases the wealth in period t + 1. Therefore, the most tempting consumption choice in period t + 1 changes as a result
of the reduction in period t consumption. The term (6pt/(pt+l))v (bt+l/pt+l)
represents the increase in the temptation utility of the most tempting choice
in period t + 1. This term has a negative sign since it increases the cost of selfcontrol. It is this last term that distinguishes DSC preferences from standard
time separable utility functions.
DEFINITION: The pair (p, c) is an equilibrium for the economy ((ui, vi, 8i),
1i)i= if (i) for all i, ci is optimal a
~n
PROPOSITION 1: An equilibrium for th
PROOF: See Appendix.
To examine the welfare properties of
ate notion of efficiency. Recall that a
consumption but also on the possible
fore, the definition of admissible int
the definition of Pareto efficiency m
of consumption but also the ways in
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136
E GUL AND W. PESENDORFER
set of feasible choices for consumers. If, for example, the social plan
impose arbitrary restrictions on choice sets, then he can improve wel
ply by restricting the consumers' choice sets to the singleton set containin
the equilibrium allocation.
We assume that the planner may re-allocate resources or change pr
cannot put additional restrictions on the feasible choices of consumer
the consumers' decision problem in period t must be of the form Xt(p
some price vector p and some wealth b. Since we permit a limited set
ventions for the planner we obtain a weak notion of Pareto efficiency
show that competitive equilibria with a representative DSC consumer
even this weak notion of efficiency.
DEFINITION: The pair (p, c) is admissible if, for all i, ci is optimal at
wealth p . ci, and if E 1 ci< =o1 oi. An admissible pair (p, c) is Par
mal if there is no admissible (p', c') with Wi(xl(p', p' . c')) > WJ(xl(p,
for all i.
The following example demonstrates that a competitive equilibrium
be Pareto optimal.
EXAMPLE: Consider an economy with two physical goods and a rep
tive consumer. The utility function is given by u(c1, c2) = log c1 - Acl
v(c1, c2) = Acl. The consumption set is C = [0, 2]2, the endowment is
every period, and A E (0, 1). Consider a policy where the planner con
(and destroys) 8 > 0 units of good 1 in every period. For E small, thi
increases the consumer's welfare. More detailed calculations can be found in
Appendix D. Here we provide the intuition. Recall that the cost of self-control
in period 1 is the equilibrium value of the temptation utility A(1 - 8) minus
the maximal temptation utility. The maximal temptation utility is achieved by
converting all wealth into current consumption of good 1. In equilibrium, the
reduction in the endowment of good 1 has two effects: first it decreases the
equilibrium consumption of good 1, and second it increases the price of good 1.
The first effect reduces welfare while the second effect increases welfare by re-
ducing the number of units of good 1 the consumer can afford in period 1 and
hence reducing the cost of self-control. For the particular functional forms we
have chosen, the second effect dominates the first. Note that the government
could also levy a tax on good 1 and use the proceeds to buy (and destroy)
good 1. This policy is equivalent to the confiscation policy and hence the example shows that such a tax policy may be welfare improving.
6.2. Steady State Equilibria
Next, we examine steady state equilibria of a particular class of deterministic
exchange economies. We assume that there is one physical good, i.e., L = 1,
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SELF-CONTROL
137
and that consumers share a common discount factor 8. In addition,
sumers have the same preference for commitment but may differ in
control. By Corollary 1 this implies that there is (u, v, 8) and (A1, ...
Ai > 0 such that i's preferences are given by (u, Aiv, 6). If Ai > Aj,
more self-control than i.
In a steady state equilibrium each consumer's consumption is constant over
time and prices are characterized by a fixed interest rate r. Clearly, for an
economy to have a steady state equilibrium, the aggregate endowment must
be constant over time. Let ct E C be the aggregate endowment in any period and let pr := (1, (1 + r)-',..., (1 + r)-, ...) be the price sequence corresponding to the interest rate r. The interest rate r and the consumption vector
(cl, ..., c1) E C' are a steady state equilibrium for the aggregate endowment
c. e C if (i) I=' ci = ~o and (ii) (pr, c) where ci = (ci, ..., c, ...) is an equilibrium for the economy with endowments c.
Proposition 2 shows that to each aggregate endowment there is a unique
steady state equilibrium. Hence, steady state allocations are uniquely determined by the aggregate resources and the DSC preferences.
PROPOSITION 2: Suppose that u + A,v is strictly concave for all i and
limc,,(u'(c) + v'(c)) -- oo. Then, there is a unique steady state equilibrium
(r, cl, . . cI) for every aggregate endowment Z. Moreover, ci > cj iff Ai < Aj, that
is, consumer i's steady state consumption is higher than consumer j's if and only
if i has more self-control than j.
PROOF: See Appendix.
Note that the conditions of Proposition 2 guarantee an interior solution to
the consumer's maximization problem. Therefore, equation (4) implies that if
(r, cl, ... cl) is a steady state equilibrium, then
(4') (6(1 + r) - 1)(u'(ci) + v'(ci)) = 8(1 + r)v.(ci(1 + r)/r),
where ci(1 + r)/r is the (constant) wealth of consumer i in terms of consump-
tion. Since v' > 0 we must have 8(1 + r) > 1. The concavity of ui + vi and
the convexity of vi imply that to each interest rate r with 8(1 + r) > 1 there is
a unique ci consistent with equation (4'). The uniqueness of the steady state
then follows from the fact that a higher interest rate implies a higher steady
state consumption for all consumers.
For standard consumers with vi = 0 the steady state interest rate satisfies
8(1 + r) = 1. In that case, any consumption ci satisfies equation (4') and hence
there are many possible steady state equilibria for any aggregate endowment.
By contrast, aggregate endowment and the distribution of preferences uniquely
determine the steady state in an exchange economy with DSC preferences.
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138 F GUL AND W. PESENDORFER
As an example, let u(c) = logc and v(c) = c. In tha
1 +Aic = (1 + r).
Setting yi = 1/Ai, the unique solution to these equations and the aggregate
resource constraint is
cE
and
1+r= - .
Hence, a mean preserving spread in the yi
changed but result in greater inequality in
an increase in 7o, the aggregate endowmen
but lead to an increase in the steady state in
DSC preferences have the feature that the
tween consumption in period t and t + 1 de
in period t + 1 and hence on the consumer'
DSC preferences have similar implications
functions introduced by Uzawa (1968) and
models, the marginal rate of substitution b
pend on the consumption in all periods t' >
in period t + 1.
6.3. Borrowing Constraints
In this section we illustrate the effect of
rium consumption and prices. Consider th
istic exchange economy with one physical g
identical preferences (u, Av, 8) where
Ac2 Ac2
u(c) =c- -22
v(c) 2'
=
with
8
>
a,
ment
is
3
every
(Olt
=
21 if t even.
3
A
-2
E
(0
per
=
|
2
When there are no borrowing constraints this economy has a unique equili
rium in which both consumers consume 3/2 in all but the first period. To s
this, let Ct denote the maximally feasible consumption of consumer i in p
riod t. The first order condition for consumer i's maximization problem (eq
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139
SELF-CONTROL
tion (4), substituting cit+1 for bit+l/pt+l) implies
(4") 1= P 8(1-Acit+).
Pt+l
In equilibrium, the first order condition (4") can only be satisfied fo
sumers if C1t+1 = c2t+1 for all t > 1. Since cit is equal to the consum
in terms of period t consumption, it follows that both consumers
the same wealth and the same consumption in all but the first perio
consumption is such that the wealth of both individuals is the sam
in period 2. Equilibrium therefore predicts consumption smoothin
consumers. The equilibrium prices satisfy
Pt
2
Pt+i
1
+
-
v/(1
-
Now assume that consum
maximum amount each c
s denote a consumer's sa
budget set is denoted Bi
B,(p,
s)
=
{(c,
s')
I
sp
The first order conditi
given by (4"). The borro
sumption cit. Suppose co
endowment. Then it = 3
Pt 1
pt+l 8(1-3A)'
then this choice satisfies the first order condition (4"). Hence, ci = oi
price sequence p with pt = (8(1 - 3A))'-1 is an equilibrium of the ec
with borrowing constraints. Note that in equilibrium the borrowing co
is not binding.
This example illustrates the consumers' desire to smooth the maximally
feasible consumption. Smoothing of the maximally feasible consumption
is achieved by refraining from consumption smoothing. Since v is convex,
smoothing temptation economizes on the cost of self-control. Note that we
have chosen a linear u + v and hence have eliminated the consumer's incentive to smooth actual consumption. In general, both motives for consumption
smoothing are present when consumer's have DSC preferences.
The interest rate r = (pt/pt+l) - 1 in the equilibrium with the borrowing
constraint is lower than the interest rate when there is no borrowing constraint.
To see why this is the case, note that in the equilibrium with no borrowing
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140
E GUL AND W. PESENDORFER
constraints the maximally feasible consumption b,+l is greater than
consumers. The introduction of a borrowing constraint therefore re
and hence the cost of self-control. Equation (4") then implies that p,/
adjust downwards to offset this effect.
In the equilibrium above, consumption tracks income because borro
portunities are much greater in low income periods than in high inc
ods. The individual refrains from shifting funds to low income perio
increasing the cost of self-control. With a different borrowing cons
could obtain the opposite result: that is, consumption could be larg
ods where the individual has low endowment. For example, if the i
may borrow 1 unit independent of his future endowment, then eq
consumption in periods of high endowment could be lower than in
low endowment.
6.4. Stochastic, Representative Agent Economy
In this subsection, we analyze a simple example of a stochastic represen-
tative consumer economy (Lucas (1978)). There is one consumer who owns
a productive asset that yields a dividend in each period. Dividends are identically and independently distributed across time and denoted by the random
variable d. Let D denote the realized value of d. We assume that d has mean 1,
variance o-2, and support [0, 2].
Let s denote the asset holdings of the representative consumer at the beginning of the current period. Then, the consumer observes D, the realization
of dividends in the current period, and chooses consumption c E C := [0, 4].
Given the price of the asset p, the choice of c determines s', the consumer's
holding of the asset for next period. The price of consumption is normalized
to 1. The price of the productive asset depends on the dividend realization and
is described by the function p: [0, 2] - IR+. Let
B(s, D) := {(c, s') I p(D)s + Ds = p(D)s' + c}
denote the consumer's budget set given the asset holding s and the realization
of the dividend D.
It is easy to see that the decision problem of the representative consumer defines an IHCP for each initial asset holding s and each initial value of the dividend D. Let y(s, D) denote this IHCP. Similarly, let y(s, d) denote the IHCP
confronting the representative consumer before the current period dividend is
realized.
The utility of y(s, D) for a consumer with the DSC preference (u, v, 8) sat-
isfies
W(y(s,D)) = (c,s')EB(s,D)
max {u(c) + v(c) + 8W(y(s', d))}
-v([D + p(D)]s).
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141
SELF-CONTROL
Hence, a consumer with the DSC preference (u, v, 8) chooses consum
solve
(5) max
u(c) + v(c) + 8W(y(s', d)).
(c,s')EB(s,D)
The first order condition for a solution to (5) is
(6) p(D)(u'(ct) + v'(ct)) = 8E[(u'(ct+1) + v'(ct+))(p(d) + d))]
- 8E[v'(st+(p(d) + d))(p(d) + d)].
In a competitive equilibrium ct = D, st+1 = 1, and Ct+l = d.
To illustrate the effect of temptation on asset prices we consider the follow
ing example:
Ac2
Ac2
1
u(c)=c- 2 v(c) - 0< A <.
2
2'
2
By
Corollary
1,
a
preference
for
co
A = 0 this consumer is a standard risk neutral decision-maker. For A > 0 the
consumer remains risk neutral with respect to instantaneous consumption gambles, that is, gambles that replace current consumption c with a lottery of consumptions with mean c. However, the consumer is risk averse with respect to
investments that affect future wealth. To illustrate this, we compute the equity
premium, that is, the difference in the expected return between a risky and a
risk-free asset, for this economy.
Equation (6) applied to this example together with the equilibrium conditions yield a constant price p that satisfies
p = 8(1 + p - 2Ap - Ap2 - A(r2 + 1)).
Let 1/(1 + r) denote the price of a risk-free asset that pays off one unit of
consumption in the next period, irrespective of the dividend realization. In
equilibrium (see Appendix D for details) the risk-free rate must satisfy
= 8(1-A(p + 1)).
l+r
Let R be the expected return on the productive asset, that is,
-Rd+p- l+p
P _ P
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142
F GUL AND W PESENDORFER
Appropriate substitution (see Appendix D) yields the following exp
the equity premium:
R - r =A2 (+r)
p
The equity premium is positive for A > 0 and increasing in A (see Appendix D
for detailed calculations).
The utility of an individual with DSC preferences depends both on actual
and maximally feasible consumption (wealth). When u + v is concave and v
is convex, there are two sources of risk aversion. First, as with standard concave utility functions, the consumer is averse to consumption risk. Second, the
consumer is averse to risk in wealth because the cost of self-control is convex.
Our example illustrates the second source of risk aversion. In the example,
the consumer is risk neutral with respect to consumption (u + v is linear) but
risk averse with respect to realizations of future wealth. Since investment in a
risky asset implies that future wealth is uncertain, risky assets must offer a risk
premium.
When choosing among lotteries that promise immediate (consumption) rewards the DSC consumer exhibits less risk aversion than when choosing among
assets that promise risky future returns. In the former case, temptation costs
are sunk and hence do not affect the consumer's choice whereas in the latter
case temptation costs add to the consumer's risk aversion.
7. SUSTAINABLE DEBT
In this section we show that for an agent with DSC preferences, incentive
compatible debt contracts are feasible even in an environment where the only
punishment for default is exclusion from future borrowing. Hence, the agent
may save funds at the market interest rate even after default. This restriction
on feasible punishments after default is particularly relevant in the case of sovereign debt. With standard preferences (i.e. no preference for commitment),
Bulow and Rogoff (1989) show that in this environment there is no incentive
compatible contract that allows the individual to borrow.
We assume that there are no investment opportunities that offer commitment. Thus, the consumer always has the option of exchanging his savings for
current consumption. This assumption may be justified by allowing collateralized loans after default. To see how this works, suppose that in period 1 the
consumer invests in a contract that offers a return of 1 unit of consumption
in period 3. In period 2 the consumer is able to use that contract as collateral
for a loan on current consumption. Under this hypothesis there are incentive
compatible contracts that allow individuals with DSC preferences to borrow.
In the following we provide an example of such a contract.
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143
SELF-CONTROL
Let wo = (0, 0, 4, 0, 0, 4, 0, 0,4,...) be the agent's endowment. T
borrows and lends at a fixed interest rate r. A generic debt contrac
noted (/3, 32, 33) with the understanding that 3j is the required ou
balance at the end of any period 3t + j. If, at the end of any period, t
balance is not at the required level, then he is excluded from borrow
future periods. The individual may always invest funds at a rate r >
after default.
The contract (/31, /32, /3) is incentive compatible if for any feasible de
in period t, there is an optimal plan in which the consumer does no
Simple calculations establish
31 = (1 +r)3 + C3n+l,
32 = (1 + r)1 + C3n+2,
P3 =( + r)/2+ c3 -4,
for n = 1, 2,.... Since consumption is nonnegative, the above equations imply
that if there is any borrowing (i.e., /3j > 0 for some j), then /32 > /31 > /3
Hence, borrowing occurs if and only if /2 > 0 and the maximal level of debt
is /2. Finally, the above three equations imply
C3n+2
C3n+l
1
1
(7) c3+2
+ c3+l +c3n=4- (l+r)- P1
(7) (1 + r)2 (1 + r) (1 + r)2
for all n > 1. Under any contract with /2 > 0, a standard
ence for commitment has an incentive to default. To see
of repaying the debt, the individual "deposits" (c3n+l/(l
into a savings account. By (7) this is feasible and yields
units of consumption for period 3 whenever /2 > 0. This
case of the argument given by Bulow and Rogoff (1989
without direct penalties there can be no borrowing.
However, if the agent has a preference for commitmen
patible borrowing is possible. Let (u, v, 8) be the agent's u
u(c) =
2c if c<1,
1+c ifc> 1,
v(c) = Ac.
Assume that 6 is close enough to 1, so that
1 - 82 1 - 54 282 _ 1
a:= 53 + 64 252 -+ 5
Let A := a/(1 - a) and A := (262 - 1)/(1 + S).
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144
E GUL AND W. PESENDORFER
PROPOSITION 3: If A E (A, A), then for all r < (1- 8)/8, the debt co
P3i = 1, 32 = 2 + r, /3 = 0 is incentive compatible.
PROOF: First, consider the optimal program for an agent who can
row starting in period 3 (the period when he will be tempted to defau
that it cannot be optimal to consume more than 1 in periods other th
see this recall that (1 + r)5 < 1 and consumption beyond 1 has a margi
ity of 1 in each period. Hence the incentive to avoid the self-control p
period 3t makes it optimal to consume no more than 1 in all periods b
the other hand, consuming less than 1 in any period cannot be optim
pose that the individual consumes less than 1 in period 5. Consider an
in period 5 consumption financed by a decrease in period 3 consump
marginal utility of consumption in period 5 is 2, and discounting by two p
yields 262. The marginal utility of consumption in period 3 is 1. In add
transfer of resources from period 3 to period 5 increases self-control
periods 3 and 4. The marginal increase in self-control costs is A(1 +
A < A, the increase of period 5 consumption increases payoff. A simil
ment holds for period 4 and hence the individual will consume exactly
in periods 3t + 1, 3t + 2, t > 1. Therefore, the optimal utility starting
period 3t is
1 -A 1- A ( A 282
l 1-_83 5- l+rr -(+r)2 - + l +r
Now, consider the consumer who does not default. In period 3t + 2, the continuation utility of the plan is
W3 = 3 [5 - (1 + A)(1 + r) - (1 + A)(1 + r)2 + 2 + 22].
Straightforward calculations, using the facts r < (1 - 8)/8 and A > A establish
that W3 > Wd3. Therefore, the individual has no incentive to default in any
period 3t + 2. But this is the period with the highest incentive to default. Hence,
the debt contract is incentive compatible. Q.E.D.
The idea behind Proposition 3 extends to other utility functions:
to the savings program, the borrowing program leads to lower self-c
in periods 3t + 1. The reason is that an agent who enters period 5 w
(1 + r),31 is extended additional credit equal to 2 -(1 + r)f31 > 0. Th
are not available to the agent in period 4 and therefore he does not
self-control costs associated with "transferring" them to period 5. U
ings, "credit-worthiness" is not an asset that can be used as collate
fore, the commitment offered by the debt contract cannot be un
open market.
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SELF-CONTROL
145
8. CONCLUSION
The starting point of the time-inconsistency literature is a nonrecursive pref-
erence 1I over consumption streams {ct}tC. Then, it is assumed that the preference ST over consumption streams starting at time r is the same as _i. Since
>i is not recursive this implies that the conditional preference relation induced
by -1 on consumption streams starting at date r is not the same as >-. That is,
there exists some {c1, ..., c _i, C, ...} and {cl, ..., c_1C, . ...} such that
{C1l, C.. 1, CT, ...} I 1 {ClI, , C. i , CT 1 .} and
{C,T ...} >T {CT ...
This "reversal" of preference is called time inconsistency.
The time inconsistency approach postulates that at each T the agent is a
different "player" and behaves in a manner that maximizes >- given the predicted behavior of his subsequent selves. The predicted consumption paths are
typically the subgame perfect Nash outcomes of the resulting game.12
The multiplicity of subgame perfect Nash equilibria implies that a given decision problem may not have a unique payoff associated with it. From a gametheoretic perspective, this is not surprising; we almost never expect all equilibria of a given game to yield the same payoff for a particular agent. In a
multi-person context, subgame perfect Nash equilibrium is meant to capture a
rest point of the player's expectations and strategizing. Then, the multiplicity
of such equilibria is a reflection of the fact that no single person controls the
underlying forces that might lead to the particular rest point.
But this multiplicity is more difficult to understand within the context of a
single person decision problem, even if the person in question is dynamically
inconsistent. In that case, the game-theoretic argument for multiplicity loses
much of its force since it should be straightforward to renegotiate one's self
out of an unattractive continuation equilibrium. And, foresight of this renegotiation would lead to the unraveling of the original plan.13 More generally,
the notion of (subgame perfect) Nash equilibrium is a tool for the analysis of
noncooperative behavior, and its appropriateness often rests on the implicit assumption of "independent" behavior and absence of communication. Therefore, analyzing the interaction between the agent at time r and his slightly
modified self at time T + 1 as the Nash equilibrium of a game may not be appropriate. 4
'2As an alternative, O'Donoghue and Rabin (1999) advocate "naive" behavior as a solution
concept to this game. Naive agents incorrectly predict future behavior to maximize current period
preferences.
"3The argument that renegotiation is particularly plausible in time-inconsistent decision prob-
lems is due to Kocherlakota (1996).
'4For a related critique of the use of Nash equilibrium when modeling a (different) departure
from fully rational behavior, see Piccione and Rubinstein (1997).
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146
E GUL AND W. PESENDORFER
In contrast, the approach of Gul and Pesendorfer (2001) and the cu
paper is to take a single preference, not over consumption, but over a
decision problems and to permit a strict preference for a smaller set o
In our approach, the preference for commitment arises from a desire
temptation rather than from a change in preference. As demonstrated
orem 2, multiplicity does not arise in the preference for commitment
Every (u, v, 8) corresponds to a unique preference over decision probl
all optimal plans yield the same payoff.
Within the time-inconsistency approach, the only formulation of sel
entails using multiplicity of equilibria to support outcomes in w
decision-maker sustains a desirable plan by threatening himself with
sirable behavior after a deviation. Self-control is therefore not a pro
the agent's preference but of the selected equilibrium and a theory
control requires a theory of equilibrium selection. By contrast, the pr
for commitment approach identifies self-control as a property of the
preference. This allows us to identify parameters of the agent's utility
that measure the amount of self-control this agent has.
Dept. of Economics, Princeton University, Princeton, NJ 08540
fgul princeton. edu; www.princeton.edu/fgul
and
Dept. of Economics, Princeton University, Princeton, NJ 08540
pesendor@princeton.edu; www.princeton. edulpesendor.
Manuscript received November, 2000; final revision received February, 2003.
APPENDIX A: INFINITE HORIZON CONSUMPTION PROBLEMS
Our treatment of dynamic decision problems is similar to the "descriptive approach" in Kreps
and Porteus (1978) extended to an infinite horizon.
Let X be any metric space. The set C(X) of all nonempty compact subsets of X (endowed with
the Hausdorff metric) is itself a metric space. If X is compact, then so is C(X) (see Brown and
Pearcy (1995, p. 222)). Let A(X) denote the set of all measures on the Borel a-algebra of X. We
endow A(X) with the weak topology. If X is compact, then A(X) is also compact and metrizable
(with the Prohorov metric) (see Parthasarathy (1970)). For any metric space X, we use B(X) to
denote the Borel ao-algebra of X.
Given any sequence of metric spaces X,, we endow x~ X, with the product topology.
This topology is also metrizable and x'lX, is compact if each Xt is compact (Royden (1968,
pp. 152, 166)).
Let C denote the compact metric space of possible consumptions in each period. Let Z1 :=
/C(J(C)). An element of Z1 is a one period consumption problem. Each choice /li e zl E Z1
is a probability measure on C. For t > 1, define Zt inductively as Z := :C(J(C x Zt-_)). Thus,
each Zt in Z, is a t period consumption problem. An element l,t E zt is a probability measure on
(C, Zt,_), that is, a probability measure on consumption in the current period and t - 1 period
consumption problems.
Let Z* := x lZt. The set of infinite horizon consumption problems (IHCP) are those elements of Z* that are consistent, that is, for z {zt},l E Z* the t- 1 period consumption problem
induced by z, is equal to Zt-_. To be more precise, let Gl : C x Z1 -- C be given by
GI(c, zl) :=c
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SELF-CONTROL
147
and let F1: A(C x Z1) - A(C), F1 : IC(A(C x Z1)) -> kC(A(C)) be defined as follow
F, (z2)(E) :=/ 2(G1 (E)),
Fi(z2) := {F1(U2) I p2 E Z2},
for E in the Borel r-algebra_of C. Thus, F1 (,2) is the probability measure over curre
tion induced by gu2 E z2 and F1(z2) is the one period decision problem induced by z2
inductively, we define G,: C x Z, -- C x Z,_1 by
Gt, ) (c, :=(cF,_(z,))
and Ft: A(C x Z,) - A(C x Zt_1), Ft: KC(A(C x Z,)) -*K C(A(C x Z,-)) by
Ft (t+l)(E) := art+l(G tl(E)),
F,t(zt+i) :- {Ft(Pt+) I= Fa, t+l = zt+},
for E E 3(C x Zt-1). Then, F(zt) is the t - 1 period decision problem induced by z,. Finally,
define
Z := {{z,}tl E Z* I z,t = Ft- (Z,) Vt > 1}
to be the set of all IHCP's.
LEMMA 1: Let X, Y be compact metric spaces and g : X -+ Y be a continuous function. Then,
(i) g : KC(X) - KI(Y) defined by g(A) = {g(a) I a E A} is also continuous. (ii) If g is a bijection,
then it is a homeomorphism.
PROOF: Part (i) follows from exercise X, p. 222 in Brown and Pearcy (1995). Part (ii) follows
from Theorem 8.37, p. 207 in Brown and Pearcy (1995). Q.E.D.
Note that F1 is continuous and hence by Lemma l(i), so is Fl. Then, an inductive
establishes that Ft, Ft are continuous for all t > 1. It follows that Z, the set of all
compact.
As described, every IHCP is an infinite sequence of finite decision problems. Alternatively,
an IHCP can also be viewed as a set of options, each of which results in a probability measures
over current consumption and IHCP's that describe the consumer's situation next period. Hence,
the recursive view identifies an IHCP as an element of KC(A(C x Z)). Indeed, there is a natural
mapping from Z to KC(A(C x Z)).
THEOREM Al: There exists a homeomorphism f: Z --> C(A(C x Z)).
To illustrate how f identifies recursive IHCP's with IHCP's, consider the problem {zt}rl E Z
and assume that each choice is deterministic; that is, each 1t E Zt yields a (certain) period 1
consumption, ct, and a (certain) continuation problem Zt_1 Consider a sequence {/rt}il such
that Irt E Zt and ,t-_ = Ft- (Lt ) for all t. Let (ct, z_ i) denote the element of C x Zt-1 that occurs
with probability 1 according to iL. Since {r1t}?1 is consistent (t-l1 = Ft_ 1(lt)) it follows that
ct = cl for all t; that is, the period 1 consumption induced by At is the same as the period 1
consumption induced by /1. Moreover, the sequence of continuation problems z' := {Zt-_l}t2 is
itself consistent. Hence, we can identify {,rt}ti? with the unique element in A(C x Z) that puts
probability 1 on (c, z'). Repeating this procedure for every consistent sequence {ut} such that
/,t E Zt for all t yields f(z).
The construction of the set of IHCP's shares the following common structure with several
recent contributions. Let Y0 be a set with some property a and let C be an operator that associates with each set that has property a a new set that also has property a. Let Y1 := C(Y0) and
Yt+l := C(Yo x Yt) for t > 1 and let Y* = xo,Yt. The goal is to identify a subset Y of Y* that is
homeomorphic to the set C(Y0 x Y*).
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F GUL AND W. PESENDORFER
148
In Mertens and Zamir (1985), Epstein and Zin (1989), and Brandenburger and Dekel (1993),
Y( is a topological space with some property (for example, a Polish space in Brandenburger and
Dekel) and C(Y)) is a set of measures on Y(. In Epstein and Wang (1996), Y( is compact Hausdorff and C(Y() are preferences over a set of functions from Y0 to [0, 1]. In Epstein and Peters
(1999) C(Y() is a set of upper hemi-continuous functions on Y( and in Mariotti and Piccione
(1999), C(Y)) is the set of all nonempty compact subsets of the compact set Y(). In our case, Yo
is a compact metric space and C(Y() is the set of all nonempty, compact subsets of probability
measures on the Borel -c-algebra of Yo.
DEFINITION: Let X be a compact metric space. Let A" E 1C(X) for all n. The closed limit
inferior of the sequence A" (denoted LA") is the set of all a e X such that a = lima" for some
sequence {a"} such that a" E A" for every n. The closed limit superior of A' (denoted LA") is the
set of all a E X such that a = lim a", for some subsequence {a" } such that a"' e A"' for every nji.
The set LA" is the pointwise limit of A" if L A" = LA" = L A".
LEMMA 2: Let X be a compact metric space. The sequence A" E IC(X) converges to A iff
A =LA".
PROOF: The lemma follows from exercise X, p. 132 in Brown and Pearcy (1995)
DEFINITION: Let Y' := A(C) and for t > 1 let Y, := (C, Z1,.... Z,_ ). The sequence o
ability measures {/I, } x7, Y, is Kolmogorov consistent if margc z ....z, ,,+l =- it for a
Let Iyk denote the set of all Kolmogorov consistent sequences in x,, Yt.
LEMMA 3: For every {f,} G yk' there exists a unique Lx cE A(C x Z* ) such that margc p. =
marg ...., = Pit for all t > 1. The mapping i : ykc - A(C x Z*) that associates this tx w
corresponding {/, } is a homeomorphism.
PROOF: The first assertion is the Kolmogorov's Existence Theorem (Dellacherie and
(1978), p. 73). Since every compact space is complete and separable, the second assertion f
from Lemma 1 in Brandenburger and Dekel (1993). QE.D.
DEFINITION: Let D, := {(zl,...,zt) e xt=,Z, zk = Fk(Zk+l) Vk
{{,} ) xt A,(C) x (C x Z,) F,(l,c+ ) = u, Vt > 1}, and Yc := {{ft} E yk
Vt> 1}.
LEMMA 4: For every {/1,} E M' there exists a unique {fi,} E Y' such that il = il and
margc. zt, f , = Lt for all t > 2. The mapping ) : M' - Y' that associates this {,t } with the corresponding {ft } is a homeomorphism.
PROOF: Let m( be the identity function on C and let ml be the identity function on C x Zi.
For t > 2, define mt : C x Z, -- C x (x= Zk) as follows: m,(c, z,) = (c, zl ..) iff = c, = ,,
and zk- = Fh_I(zk) for all k = 2,.... t. Note that m, is one-to-one for all t. Also, m,(C x Z,) =
C x Dt. Let 7r( and 7r- be the identity mappings on C and C x Z, respectively. For t > 2, let
7Tr (c,..., zt) = (c, z,) for all (c, ..., z,) E C x Dt. Clearly, -r, is continuous for all t. Since C x D,
is a compact set, 7TT can be extended to all of C x (x = Zk ) in a continuous manner. Hence, rr, is a
continuous function on C x (x = ZA ) and its restriction to C x D, is the inverse of m, : C x Z, -
m,(C x Z,). Since F( is continuous for all t, so is m,.
For {,} E M', define {,lt} by /L(E) := t(m '(E)) for all E E 1(C x ( x=Zk). Clearly, the
{fit} defined in this manner is the unique element in Y' such that fL = -L and margc. z, r , = -,
for all t > 1. Define ({zu,}) to be this unique {/it} and note that 0 is one-to-one. Pick any {fI,}
in y'. Define {ti,} as follows: tl = /i and /,(E) := /t(7r'-ll(E)) for all E C B(C x Z,t-). Note
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SELF-CONTROL
149
that 0 ({/t }) = i,; hence, 4 is a bijection. Observe that the tth element of 0({f t}) depe
on 1t. Hence, without risk of confusion we write Ot(l.t ) to denote this element. Note tha
continuous real-valued function h on C x ( x= Zk) and h on C x Z,, f hdi,(t,) = f
and f hdt1(,t) = f h o rt,dlt. Hence, the continuity of 4( and 0-1 follows from the co
of m, and 7Tt for all t. Q.E.D.
LEMMA 5: i(Y') = { Z | (C X Z*)I(C = )}.
PROOF: Let Ft = C x D, xk,k+l Zk for all t > 1 and I = - ({fit}). Observe that /(Ft) =
D,) = 1 Vt if {t,} e yc. Hence it(C x Z) = /(nli F,) = lim t(Ft) = 1. Conversely, if /(C
1, then I(Ft) = 1 Vt and hence there is a corresponding {f,} E yc. Q.E.D.
LEMMA 6: Let :(z) := {{ftt} c MC I At E z, Vt > 1}. Then 5: Z -* kC(M") is a homeomorphism
and {1t,} E e(z) iff tt E zt for all t.
PROOF: Step 1-Let z ={Zt} E Z, IT, e z,. Then, there exists {v,} e ((z) such that vT = A,.
Proof of Step 1: Let V, = /t,. For k = 1, ..., t - 1, define VT,k, inductively as _k :=
F,-k+l (L,-k+l ). Similarly, define V,+k, for k > 1, inductively by picking any V,+k E FT+k (Z,+k - )
Z,+k. The {vt} constructed in this fashion has the desired properties, concluding the proof of
Step 1.
By Step 1, S(z) =: 0. To see that e(z) is compact, take any sequence {Jt} E $(z) for z =
(Z, Z2, ...). We can use the diagonal method to find a subsequence {t} I such that Ij' converges
to some /t for every t. Since each zt is compact, ,t e z, for all t. Then, the continuity of Ft for all t
implies {/tt} is in M' and therefore in ((z). So ((z) is compact. Suppose z 7 z' for some z, z' e Z.
Without loss of generality, assume there exists some T and T, such that / c z,\z'. By Step 1,
we obtain {v,} E e(z) such that v, = t,. Clearly, {v,} c (z)\:(z'). Therefore, e is one-to-one.
Take any z E IC(MC). Define z, := {,l/t, =/ , for some {it,} E z}. Let z = (zI, Z2,.... Since z is
compact we have z E Z. Clearly, z c :(z). Using the compactness of z again and the continuity
of each F, we have ((z) c z. So s is onto.
To prove that s is continuous, let zn converge to z. By Lemma 2, this is equivalent to Lz = z
for all t. We need to show that s(z") converges to ((z). Take any convergent sequence {,t)}
such that {[tnL} Ec (z") for all n. Then, limttn E zt for all t and therefore lim{,/'t} e (z). Let
{,it} E ((z). Since Lz- = zt, there exists l'n converging to tt such that 7, zn for all n. By Step 1,
for each 1T we can construct {vt}(T) e MC such that v,() = ,. Since F, is continuous for all t,
k (r) converges to /k for all k < r. Consequently, {vt}(n) converges to {fit} as n goes to infinity.
Hence, L (zn) = e(z). Again by Lemma 2, this implies S(z") converges to e(z) and hence s is
continuous.
To conclude the proof recall that Z is compact and : is a continuous bijection. It follows from
Lemma 1(ii) that s is a homeomorphism. Q.E.D.
PROOF OF THEOREM Al: Note that by Lemmas 3-5, f, o ) is a
A(C x Z). Hence, by Lemma l(i), the function : JC(MC) -1 J
gr o O)(A) for all A E AC(MA) is also a homeomorphism. Then, by
homeomorphism from Z to IC(A(C x Z)). Q.E.D.
APPENDIX B: PROOF OF THEOREM 1
LEMMA 7 (A Fixed-Point Theorem): If B is a closed subset of a Banach space with norm II1
and T: B - B is a contraction mapping (i.e., for some integer m and scalar a E (0, 1), IIT'(W
Tm(W') i < all W - W' l for all W, W' B), then there is a unique W* e B such that T(W*) =
PROOF: See Bertsekas and Shreve (1978, p. 55) who note that the theorem in Ortega an
Rheinbolt (1970) can be generalized to Banach spaces. Q.E.D.
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E GUL AND W PESENDORFER
150
LEMMA 8: Let u: C - R, v: C --- be continuous and 8 E (0, 1). There is a uniqu
function W: Z -- I such that
(B.1) W(z) = max
{(u(c) + v(c)
+ iW(z'))d(c, z')} - max v(c)dv(c, z'
t~EZ
FEZ
for all z E Z.
PROOF: Let W be the Banach space of all continuous, real-valued functions on Z (endowed
with the sup norm). The operator T: W --> W, where
TW(z) maxlf((c)
+ (c)+SW (z'))d)
(c, ')EZ
-max| v(c)dv(c, z')
ZE[z
J
is
well-defined
and
that T(W) = W. Hence, W satisfies (B.1). Q.E.D.
By Lemma 8, for any continuous u, v, 8, there exists a uniqu
It is straightforward to verify that Axioms 1-8 hold for any bi
tinuous function W that satisfies (B.1).
In the remainder of the proof we show that if > is nondege
then the desired representation exists. It is easy to show that if
it also satisfies the following stronger version of the indepen
AXIOM 4*: x > y, a E (0, 1) implies ax + (1 - a)z > ay +
Theorem 1 of Gul and Pesendorfer (2001) establishes that >only if there exist linear and continuous functions U, V such th
(B.2) W(z) := max{U(,r) + V()} - maxV(v)
icz
vez
represents
>8
E
(0,
1)
suc
U(I)
=
/(u
V(,u)=
vf
LEMMA
9:
T
8W(z))dg
(c,
PROOF:
Ste
W(z))dli(c,
z
Proof
that
of
U()
Ste
=
therefore,
U(A)
u(c,
Setting
Step
2:
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=
f
u
uf
z)dr
u
:=
Ther
151
SELF-CONTROL
Proof of Step 2: Define K := maxz W(z), k := minz W(z), K := maxz W(z), k := m
By nondegeneracy, U is not constant. It follows that W is not constant. Axioms 6 im
W(x) > W(y) if and only if W(x) > W(y). Hence, W is not constant. Therefore, K
To establish the desired conclusion we will show that
Kk -Kk K - k
(B.3) W(z):=
K-)- k)
- K
+ kW(
(K - k)(K
for all z E Z. For any z E Z there exists a unique a such that
(B.4) W(z) = aK+ (1 - a)k.
Let z* maximize W and z* minimize it. It follows that
W({a(c, z*) + (1 -a)(c, z*)}) = aW({(c, z*)}) + (1 -a)W({(c, z,)})
= u(c) + a + (1 - a)k
= u(c)+ W(z)
= W({(c, z)}).
(The first equality follows from linearity of W, the second equality follows from (B.2) and Step 1,
the third equality follows from (B.4), and the last equality follows from (B.2).) By Axiom 7 we
conclude that
W({(c, z)}) = W({(c, az* + (1 - a)z)})
and by Axiom 6
W(z) = W(az* + (1 - a)z).
Linearity of W together with the fact that W(x) > W(y) iff W(x) > W(y) implies
(B.5) W(z) = aK + (1 - a)k.
Solving (B.5) for a, substituting the result into (B.4) and rearranging terms then yields (B
proves Step 2.
Step 3: 6 < 1.
Proof of Step 3: Let zc be the unique z e Z with the property that zc = {(c, zC)}. Let z be
such that W(z) 0 W(zC). Let y1 = {(c, z)} and define y" inductively as y" = {(c, y"-1)}. Then y"
converges to zc. Hence, by continuity W(y") - W(zc) must converge to zero. But
W(y") - W(zc) = 6"(W(z) - W(zC)).
Since W(z) - W(zc) 0 it follows that 8 < 1. Q.E.D.
Let U' = U - (y/(1 - )) and W' W - (y/1 - S)). The
with
W'(z) := max{U'(,/) + V(/u)} - maxV(v).
jiez
VEZ
Moreover,
U'()
=
W'
f(u(c)
Therefore,
LEMMA
V(,u).
repres
+
without
10:
Then,
l
Assume
there
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is
152
F GUL AND W. PESENDORFER
PROOF: Fix &2 E A(Z) and define V* : A(C) -- R by
V**(l) := V(.l x 2).
Take any v e A and let ,u be the product measure vI x u'. By continuity, there e
enough so that
U(av + (1 - a)/+) + V(ap + (1 - a)/.) < U() + V(y),
U(aL + (1 - a)/) + V(a/- + (1 - a)) < U(q) + V(yi),
V(av + (1 - a)m/) > V(y),
V(a/t + (1 - a})) > V(y).
Axiom 8 implies that W({av + ( - a)u,, W}) (= W({(a + (1 - a)t, i}). It now follows from (B.2)
that V(av + (1 - a)pL) = V(arL + (1 - a)/L). Since V is linear, we have V(v) = V(ui) = V*(vl).
Since V is continuous and linear, so is V*. Hence there is a v : C - R such that V*(vl) =
f v(c)dv' (c) as desired. Q.E.D.
To complete the proof, we show that the co
generacy U is not constant. If V is constant,
assume that neither U nor V are constant.
Suppose V = aU + f/ for some a, /3 c R. Since V is not constant we conclude a -A 0. If a > 0,
replace V with V'= 0. Then, W(z) := max,,z{U(4r) + V'(1)} - max,, V'(v) and the conclusion
of Lemma 10 holds. Nondegeneracy rules out a < -1. If a E (-1, 0) or if V is not an affine transformation of U, then V is not a positive affine transformation of U + V. Hence, the preferences
represented by V and U + V are different and nontrivial (i.e. neither V nor U + V is constant).
Therefore, there exists v, v such that either U(v) + V(v) > U(v) + V(v) and V(v) < U(v) or
U(v) + V(v) > U(v) + V(v) and U(v) < U(v). In either case, since neither U nor V is constant,
we can use the linearity of U and V to find - close to v and u close to v for which all of the above
inequalities are strict and apply Lemma 10. Q.E.D.
Proof of Theorem 2
A DSC preference > represented by (u, v, 6) has the property
stant. Suppose > has a preference for commitment at some z E Z. T
such that v = aU + /3. Then, we may apply the proof of Theorem 4
to conclude that if (u, v, 6) and (u', v', 8') both represent >, the
v' = av + pv, and therefore W' = a W + 3,, for some a > 0, ,/3 , c
implies that 8' = 6 and u' = au + (1 - 6)3,,. If > has no preference fo
representation of > must be constant. To see this, note that for an
exist z, z' such that z > z'. If v is not constant, then there exist c,
follows from the representation that {(c, z)} > (a(c, z') + (1 - a)(c
ficiently close to 1. Thus, the standard von Neumann-Morgenstern
that u' + 6W' is a positive affine transformation of u + 6W. Then
ensures that 6 = 6' and u' is a positive affine transformation of u
straightforward and therefore omitted. Q.E.D.
APPENDIX C: MEASURES
THEOREM 4: Let >-, i = 1, 2, be regular DSC preferences with representation (ui, vi
?2 has more instantaneous preference for commitment than >_ 1 if and only if there exist a
nonnegative matrix 0' and 3' E R2 such that
ul) 0. l' )+/3.
\v] ) v2
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153
SELF-CONTROL
PROOF: Let >* be a binary relation on A := KC(A(C)). Gul and Pesendorfer (2001
>* satisfies Axioms 1-3 and 4* if and only if there exists (U*, V*) such that
(C.1) W*(A) :=max{U*(7
')+V*(V l)}- maxV*(.L1)
AlcA pilA
represents >*. Then, they de
control as in Section 4, but w
problems. If there is W*, (U*
represents >*. The preference
transformation of U*. Theore
are regular* preferences, the
there exist representations (U
U'*
V*
=
Ca
U2
=aU2
+
+
(1
(1
-
-
a,)V
a)V)
*,
,
for a,, av E [0, 1]. We note th
following binary relation >*
{till A E x}, B := {I ll / E y}, a
since
it
can
be
represented
a
E,1 [u], V*(yL) := E,A [v], and
since > is regular, it follows th
Also, if >*, >* are two prefer
instantaneous preference for
than >_. Therefore we may ap
result of Section 2 (Theorem 3
regularity
THEOREM
of
5:
>-).
Let
Q.E.D.
>-,
i
=
1,
2,
be
I1 has more instantaneous self-co
matrix O' and p1' E R2 such that
U2 + V)2 ) U + V2 )
V12
V11I
PROOF:
We
note
that
Gul
and
Pesendor
control analogous to one in Section 4 but w
tial decision problems. In Theorem 9 they
_* has more self-control than >- if and o
that
U+V* + V = au (U* + V*) + (1 - au)V*,
2* = aV,(U + VI*) + (1 - a)V*,
for some au, a, e [0, 1]. Define >*, Ui, and 1/* for i = 1, 2 as in the proof of Theorem 4. Again, it
is easy to verify that >- has more instantaneous self-control than >2 iff >-7 has more self-control
than >-. Therefore we may apply Theorem 9 in Gul and Pesendorfer (2001) and Theorem 3 of
Section 2 to yield the desired result. Q.E.D.
PROOF OF COROLLARY 1: The "if" parts of both stateme
By Theorem 4, if >l has more preference for commitm
sentation of -i for i = 1, 2, there is a nonsingular, nonneg
(C.2) (U)=
V"I7(u2
V2 +3
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154
E GUL AND W. PESENDORFER
Similarly, by Theorem 5, if >- has more self-control than >2 and (ui, vi, 6) is a representation
-i for i = 1, 2, there is a nonsingular, nonnegative matrix 0' and 3' E R22 such that
U2 + V2) (UlVl + vl 3
V2 1)1 V
Suppose >- and >2 have the same preference for commitment an
>2. Without loss of generality we can choose (u2, v2) such tha
some nonsingular, nonnegative matrix O
(C.3) ( )=0.( .
Similarly, reversing the roles of (ul, vl) and (u2, v2) in (C.2) yields
and / E R2 such that
(C.4) 2) = . ( )+/.
Equation (C.3) implies that / = 0 and O = 0-'. But since both O and 0 a
implies
(1
(C.5) (= ) 0, o= I
0 1
for some a > 0, y > 0. Again, without loss of generality, we assume a = 1. To conclude the pro
we show that y > 1. Since >- has more self-control than >2, the regularity of L2, equations (C
(C.5), and the fact that / = 0 imply that for some nonnegative, nonsingular 6 and 3,
U2 + -V2
VU2 VI=0 u +l + J8
,- 12
Since, >-2 is regular and 0 is nonnegative, we conclude 1 + b = y for some b > 0. Hence, y > 1
as desired.
Suppose >2 has more preference for commitment than >-,, and >- and >_2 have the same
self-control. Following the line of argument above, we obtain (U2, v2, 62), a representation of >2
such that u2 + v2 = u1 + v1 and v2 = yvI for some y > 0. Then, since >_2 has more preference for
commitment than >-, (C.2) implies that there is a nonnegative, nonsingular 0 and /3 such that
( vI
)= U2
+ Ul
+ ( )l-v )+p.
12
'))yv
It
y
follows
>
1
as
from
desired.
the
reg
Q.E.D.
APPENDIX D: COMPETITIVE ECONOMIES
PROOF OF PROPOSITION 1: We first show existence of equilibrium in a "truncated"
in which, from period r on, every consumer is committed to the consumption of woi E C
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SELF-CONTROL
period.
endowment. Let
Let
Xit
=
{((),it,
i+l)}
155
be
the
IHC
{(c, it, b') I (c, b') c B,(p, b)} if t <- 1,
xt(p, b)= ' {(ct, xit+l) I (c, 0) e Bt(p, b)} if t = ,
Xit
Let
v
:
IR+
if
t
--
>
R+
+1.
be
defined
as
v (Pt, b)= max vi(c)
(clpt c<b)
and note that v* is convex since vi is convex. It is straightforward to show that the optimal con-
sumption choices in periods 1 < t < r given the IHCP x\(p, t=1 pt wit) solve the following
maximization problem:
subject
{Ct}
max
t=l
T
k=1
t
Ui(c)
++v(C
k=1
t-
Ct,<
k=l
8-1
EPk
.ik
-
LPk
*C
k=l
Note that the objective funct
consumption choices is comp
Proposition 17.C.1 in Mas-Col
truncated economy.
Normalize
equilibrium
pric
set Ptl = 8t. (Note that in th
all consumers are committed
away from zero it follows t
truncation
r.
This
follows
from
increasing, and since aggrega
Let (p7, c7) denote the equi
Let (p, c) denote the limit o
for
the
economy.
therefore
h
Et
-1
S
therefore,
the
the
set
to
of
Since
show
t-
l
the
it
<
set
of
oc.
of
feasible
continuity
marke
that
ci
sol
Henc
feasible
consump
the
agent's
xi(p, Et=? pt * wt), that is, ci is optimal for i at prices p. Q.E.D.
u
EXAMPLE OF PARETO INEFFICIENCY: The utility function is
u(ctl, ct2) = log ctl - Actl + Ct2
and
v(ctl, Ct2) = Actl.
Let (a, 1) be the endowment in every period and A E (0, ). Normalize prices so that the pric
good 1 in period 1 is equal to 1. Straightforward calculations yield
Pti = St,
Pt2 = at.
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156 F GUL AND W. PESENDORFER
The equilibrium welfare of the representative household as a fun
written as
W = u(a, 1)+v(a, 1) - max v(a, a2)+ SW,
(a , )a2t t )a +ta2 b})
W, = log a + - Ab + 8W,,
where b = 2a/(l - 8) is the equilibrium value of the agent's endowment. H
2/(1 - 8) and therefore dW,/da < 0 at a = 1. It follows that destroying e units
period increases welfare, for e sufficiently small.
PROOF OF PROPOSITION 2: The necessary condition for optimality is
u'(ci,) + A,v'(ci,) = 8(1 + r)(u'(cit) + Aiv'(ci,) - A,v'(bi,+l))
where b/,+ is the period t + 1 wealth of agent i in terms of period t + 1 consumpt
state ci = ci and bi, = ci(l + r)/r, yielding
u'(ci) + Av'(ci) = 6(1 + r)(u'(ci) + Aiv'(ci) - Av( )).
Choose i( such that Ai,, = min Ai. We define r by the equation
8(l+r )-l I = I 0 + , r)(
(D. ) 8(1
- (u()
,v'()) = ( ).
-(1
+ +
r)A,,,
Note that r is well-defined since the
the right-hand side is decreasing. Mo
u;())
Ai(
+ Ai, v;(w)
> v'(O)
and the right-hand side converges to v'(w). Hence, there is a unique r that satisfies the above
equation. Furthermore, r > (1 - 8)/6.
Let r E ((1 - 8)/6, r]. For every r in that range there is a unique Cio(r) < wi that satisfies (D.1)
(i.e., cio(r) such that (D.1) is satisfied when r is replaced with r and w is replaced with cio(r)). To
see this observe that since u' (c) + ,,v (c) -- oc as c -> 0, the left-hand side goes to infinity as
cio 0 . Since v is convex, the right-hand side is bounded. For r < r, we have
5(1l+ r) -l 1 (1 + F) - 1
(+r) (u'() + Ajv'()) < (u'(w) + Ai( v'(wi))
8A(1 + r) 8Ai,,(1 +r)
= V ) (< v ).
Since u' + Aiv' is decreasing and v' is increasing, there is
Define ci(r) for each i by replacing Ai, with A, in (D.1
increasing in r. To see this, note that holding consumptio
is strictly increasing in r while the convexity of v implies
Then, the strict concavity of u + Aiv ensures that consum
equality of the two sides.
Thus, Ei ci(r) is increasing in r with
lim ci(r)=0 and rci(r)>o
Continuity implies that there is a unique r'd such that Zi ci(rd) = w.
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SELF-CONTROL
Finally
we
8(l
need
+
81(1
to
see
for
to
r)
r
+-,
that
if
show
(u'(c,)
Ai
either
i
<
Aj
or
shown
(D.2)
p
in
=
the
8(1
+
ci
>
ci
if
Ai/(ci))
and
j.
Stochastic
As
that
157
ci
<
cj,
Q.E.D.
-
the
Representati
text,
+
p
-
the
equ
2p
-
and hence
P =-12oA
(1 - 6 + 26A - /(1 - )2 + 48A(1 - 8Ao2)).
It can easily be checked that for A < 1/(1 + a 2) the price is strictly positive. Since 0-2 < 1 this
implies that for A c (0, 1/2) the price is strictly positive.
The risk free rate satisfies the following no-arbitrage condition:
u'(ct) + v'(c) = 8(1 + r)E[(u'(ut+1) + v'(ct+l))] - 6(1 + r)E[v'(st+(p(d) + d))].
Using the equilibrium conditions ct = D, ct+l = d, st+l = 1, this implies for our special case that
+r=
+
1
1
(1
Equation
positive
2
-
2
A(p
(D.2)
and
+
and
1))
p
>
increasing
1
+
0
8-
imp
note
t
1 + p 1 (1 + r)(1 + p)(1 - A(p + 1) - p)
R-r=
-
.
p 6(1- A(p + 1)) p
Equation (D.2) and some simplification yields
R -r = Ao-21
p
It remains to show that R - r is increasing in A. Equation (D.2) implies
p(l - 6) = (1 - 2Ap - Ap2 - (a(2 + 1)).
It follows that p is decreasing in A and since p > 0 that 1 - 2Ap - A > 0. We conclude that
a8 (1+?r 8 1 ) (1-2Ap-A)(-8p/dA)+ p +p2 0
A p ~ dA \p(1 - A(p + 1)) 6(p( - A(p + 1)))2
which in turn implies that R - r is increasing in A. Q.E.D.
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