This paper shows that any equilibrium allocation in the cake-eating problem with quasi-geometric discounting is not Pareto efficient. However, efficiency can be established by introducing a planner who controls the initial endowment and makes transfers over time. It is shown than any Pareto efficient allocation can be supported by a perfect equilibrium with transfers.Pareto efficiency

Juan Carlos Chavez-Martin del Campo

English

Research Papers in Economics

Efficiency in the cake-eating problem with quasi-geometric
discounting 
Juan Carlos Chavez-Martin del Campo
Guanajuato School of Economics, Universidad de Guanajuato
Abstract
This paper shows that any equilibrium allocation in the cake-eating problem with
quasi-geometric discounting is not Pareto efficient. However, efficiency can be established
by introducing a planner who controls the initial endowment and makes transfers over time.
It is shown than any Pareto efficient allocation can be supported by a perfect equilibrium
with transfers.
I am greatly indebted to Ravi Kanbur and Ted O´Donoghue for their continued guidance and feedback, and to an anonymous
referee for helpful suggestions. My research has been supported by the Consejo Nacional de Ciencia y Tecnologia
(CONACYT), the Consejo de Ciencia y Tecnologia del Estado de Aguascalientes (CONCYTEA), and the
Ford/MacArthur/Hewlett Program of Graduate Fellowships in the Social Sciences. All remaining errors are my own.
Citation: Chavez-Martin del Campo, Juan Carlos, (2007) "Efficiency in the cake-eating problem with quasi-geometric
discounting." Economics Bulletin, Vol. 4, No. 43 pp. 1-8
Submitted: July 2, 2007.  Accepted: November 14, 2007.
URL: http://economicsbulletin.vanderbilt.edu/2007/volume4/EB-07D90006A.pdf1. Introduction
Economic theories of intertemporal choice generally assume that individuals dis-
count the future exponentially. However, experimental and ¯eld evidence suggests
that many individuals have preferences that reverse as the date of decision making
nears (Lowenstein and Thaler 1989; Ainslie 1992; Angeletos, Laibson, Repetto, To-
bacman, and Weinberg 2001; DellaVigna and Malmendier 2003; Fang and Silverman
2004). This type of behavior has been modeled through quasi-geometric discount
functions that generate time inconsistency. A solution to the decision problem is
usually required to take the form of a subgame perfect equilibrium of a game where
the players are the consumer and his future selves.
The purpose of this note is twofold. First, it studies e±ciency in the cake eating
problem with quasi-geometric discounting without commitment. Second, it investi-
gates the possibility of obtaining e±cient outcomes by introducing a planner who is
able to distribute the initial endowment through a series of transfers over time.
The paper is organized as follows. We describe the model in section 2. In section
3 we introduce the concept of e±ciency used throughout the paper. Sections 4 and
5, respectively, state formal theorems summarizing our ine±ciency result and charac-
terizing the set of e±cient allocations that can be obtained in an equilibrium with
transfers. Section 6 concludes. Section 7 provides proofs of the theorems.
2. The Model
Consider the following economy. Time is discrete and indexed by t = 1;2;:::;T.
There is one agent who lives for T ¸ 3 periods. An initial endowment of one unit of
a consumption good, x, is to be consumed over time.
In period t, preferences over consumption streams x = (x1;:::;xT) 2 RT
+ are
representable by the utility function
Ut(x) = u(xt) + ¯
PT
¿=t+1 ±¿¡tu(x¿)
where (¯;±) 2 (0;1] £ (0;1], and the instantaneous utility function, u : R+ ! R,
satis¯es u0(x) > 0 and u00(x) < 0 over (0;1).
The type of preferences represented by this model incorporates the so-called quasi-
geometric discounting1. The parameter ± is called the standard discount factor and
it represents the long-run, time consistent discounting; the parameter ¯ represents
a preference for immediate grati¯cation and is known as the present-biased factor.
For ¯ = 1 these preferences reduce to exponential discounting. For ¯ < 1, the (¯;±)
formulation implies discount rates that decline as the discounted event is moved
further away in time implying time inconsistency.2
In the present analysis, we assume that the agent is sophisticated in the sense
that she is fully aware of her time inconsistency problem. Similar to Strotz (1956),
Peleg and Yaari (1973), Goldman (1980), Laibson (1997), and O'Donoghue and Ra-
bin (2001) we model this problem by thinking of the agent as consisting of T au-
tonomous selves. For the ensuing games played between selves we consider subgame
1This type of preferences was originally proposed by Phelps and Pollack (1968).
2See Frederick et al. (2002) for a review of the (¯;±) formulation.
1perfect equilibrium as our solution concept. Let ht = (x1;:::;xt¡1) be a feasible
consumption history at time t. Let St represent the set of feasible strategies for self
t. Let S = S1 £ ::: £ ST be the joint strategy space of all selves. A consumption
strategy for self t is a function
st : St ¡!
"
0;1 ¡
t¡1 X
i=1
xi
#
(1)
The allocation functions ct : S ! Rt
+ are given by
c
t(s1;:::;st) = (c
t¡1(s1;:::;st¡1;st(c
t¡1(s1;:::;st¡1))) (2)
with c1 = s1. An equilibrium allocation is a consumption vector x¤ 2 RT
+ supported
by some perfect equilibrium s¤ of the intrapersonal game. Formally,
De¯nition 1 An equilibrium allocation is a consumption vector x¤ 2 RT
+ that sat-
is¯es x¤ = cT(s¤) for some subgame perfect equilibrium s¤ 2 S of the intrapersonal
game.
3. Welfare and time inconsistency
Time inconsistency implies that preferences today con°ict with preferences tomorrow.
Therefore, it is di±cult to evaluate the welfare of a time inconsistent individual since
her di®erent temporal incarnations disagree about what is better for them. The
di±culty with de¯ning a welfare criterion for time inconsistent individuals is well
illustrated by Goldman (1979): \The question of Pareto e±ciency is especially vexing
when the discussion of endogenous preferences is explicit. If we were to identify
players...by both calendar time and the history of actions prior to the times of their
decisions, then Pareto comparisons under alternative histories become impossible
since the set of players changes". He remarks the importance of imposing some
structure to indicate when two possible players are comparable.
We avoid the problem identi¯ed by Goldman (1979) by evaluating each self on her
own terms. This approach, which has been applied by Phelps and Pollak (1968) and
Goldman himself (1979), emphasizes the application of a Pareto criterion to evaluate
equilibrium allocations
De¯nition 2 An equilibrium allocation, x¤, is Pareto e±cient if there is no other
feasible allocation that makes one self better o® without making some other temporal
selves worse o®.
Other authors have proposed alternative welfare criteria. O'Donoghue and Rabin
(1999) advocate maximizing welfare from a \long-run perspective". It involves the
existence of a \...(¯ctitious) period 0 where the person has no decision to make and
weights all future periods equally." This approach incorporates the fact that most
models of present-biased preferences try to capture situations in which people pur-
sue immediate grati¯cation. Other approaches privilege a subset of players in the
2intrapersonal game. For instance, welfare may be evaluated with respect to current
self's perspective. This \dictatorship of the present" approach has been applied by
Cropper and Laibson (1998), and Cropper and Koszegi (2001).3
4. Ine±ciency of the equilibrium allocation
One implication of assuming that the individual is time consistent (¯ = 1) is that
the optimal consumption path from self 1's perspective can be implemented in equi-
librium: his future incarnations will consume and save the amounts he wants them
to. On the other hand, if the individual is time inconsistent (¯ < 1), the equilibrium
allocation may not be e±cient because the strategic interaction of the temporal selves
can originate a coordination failure with a suboptimal outcome as a result.
This conjecture has been explored by Goldman (1979), who shows that no inte-
rior equilibrium allocation other than one which is best for self 1 is Pareto e±cient if
some conditions are satis¯ed. He does not assume separability of the intertemporal
utility function neither quasi-geometric discounting as we do here, however. Under
separability, an e±cient equiibrium allocation arises when ¯ = 1. This equilibrium
allocation coincides with the most preferred allocation from self 1's perspective. For
¯ < 1, preferences are time inconsistent and the equilibrium allocation is always in-
e±cient. Theorem 1 states this result:
Theorem 1 For ¯ < 1, any equilibrium allocation is ine±cient.
What is driving this result? Given the time inconsistency problem, future selves
may allocate resources between consumption and savings di®erently than the pre-
ferences of self 1 would dictate. Therefore, the ¯rst agent's decision to \overconsume"
is his optimal response to this behavior: self 1 is willing to sacri¯ce present consump-
tion if he could prescribe when in the future his extra savings will be consumed.
Since past consumption does not a®ect current satisfaction, the Pareto improvement
follows. More generally, self t may be willing to sacri¯ce current consumption in
order to increase the consumption of some future self if he were certain that future
selves will not distort that decision. Consequently, the Pareto improvement implies
the existence of an in°ection point at which current consumption stops being reduced
vis-a-vis the initial equilibrium.4
To illustrate the ine±ciency result, consider an agent who lives for three pe-
riods, the minimum number of periods to have time inconsistency, and has pre-
ferences represented by a log utility function u(xt) = lnxt. For the sake of sim-
plicity, we assume that ± = 1 and ¯ = 1=2. It is not di±cult to show that the
equilibrium allocation is given by c¤
1 = 3=6, c¤
2 = 2=6 and c¤
3 = 1=6. Notice that
1=18 = argmaxln(3=6 ¡ ¿) + 1=2ln(2=6) + 1=2ln(1=6 + ¿), so self 1 is willing to
3For an analysis of welfare criteria for time inconsistent individuals see Bhattacharya et al. (2004).
4I am very thankful to an anonymous referee for pointing me out the intuition behind the ine±-
ciency result.
3transfer ¿ = 1=18 to self 3 to increase his own welfare. Clearly, this is a Pareto
improvement.
5. Establishing e±ciency through transfers
Assume the existence of a ¯rst-best scenario where a planner has full information
and is able to distribute the endowment of the consumption good through a transfer
scheme ¿ 2 fRT
+ :
PT
t=1 ¿t = 1g. We formally model this setting as a two-stage game
where the players are the planner and the T di®erent selves. In stage 1, the planner
announces how the endowment will be allocated through the transfer scheme. In
stage 2, the di®erent selves play an intrapersonal game. A strategy for self t is a
function
st : St ¡!
"
0;
t X
i=1
¿i ¡
t¡1 X
i=1
xi
#
(3)
We de¯ne equilibria arising in this game as perfect equilibria with transfers.
The questions to be addressed are:
² Is it possible to observe e±cient allocations in a perfect equilibrium with trans-
fers?
² If so, what subset of e±cient allocations can be obtained in equilibrium?
The answer to the ¯rst question is positive. Moreover, it can be shown that the
planner can achieve any Pareto e±cient allocation, x¤, by appropriately distributing
the endowment over time
Theorem 2 Any e±cient allocation x¤ 2 R+ can be supported by a perfect equilib-
rium with transfers.
We prove this theorem by applying the following line of logic. First, notice that,
on the equilibrium path, the agent has no incentive to transfer resources to the fu-
ture even if he actually could choose the point in time at which these resources will
be consumed if transfers coincide with an e±cient allocation: i.e ¿ = x¤ for some
Pareto e±cient allocation x¤ 2 RT
+. From here we obtain the result that the chosen
e±cient allocation arises in equilibrium by doling out the consumption good appro-
priately. Therefore, the planner provides, through the transfer scheme, a mechanism
that makes the individual commit to follow up an optimal consumption path.
6. Concluding remarks
The cake-eating problem o®ers a simple setting to analyze the consequences of time
inconsistent behavior for e±ciency. We have shown that by imposing constraints
on future resource °ows, e±ciency could be established. In fact, most commitmet
devices impose some in°exibility in the way resources will be consumed in the future.
An illiquid asset a la Laibson (1997) is a good example of a commitment device that
is used by current selves to limit consumer behavior.
4There are some interesting extensions for the cake-eating problem with quasi-
geometric discounting. For instance, one could introduce imperfect information in
the sense that the planner cannot observe the degree of present-biased preference.
We could also incorporate income shocks into to model. In that setting, the planner
would have to play the roles of insurance provider and commitment enforcer simulta-
neously.
7. Proofs
Proof of Theorem 1. First, notice that if there exist periods j and t, j > t,
such that u0(x¤
t) < ¯±j¡tu0(x¤
j), then the equilibrium allocation is ine±cient. Hence
it su±ces to show that an allocation x 2 RT
+ satisfying
u
0(xT¡2) ¸ max[¯±u
0(xT¡1);¯±
2u
0(xT)] (4)
cannot be an equilibrium allocation.
Let x¤ 2 RT
+ be an equilibrium allocation. Assume, towards a contradiction, that
x¤ satis¯es (4). De¯ne the set © = f° 2 R3 j °T¡2 = 1;°T¡1 · 0;°T · 0;°T¡1 +°T =
¡1g, and the function '(¿) = u(x¤
T¡2 + ¿°T¡2) + ¯
P2
t=1 ±tu(x¤
t + ¿°t). Taking the
second derivative of the function '(¢) we have
'00(¿) = u00(x¤
T¡2 + ¿) + ¯°2
T¡1u00(x¤
T¡1 + °T¡1¿) + ¯°2
Tu00(x¤
T + °T¿)
hence, '(¿) is strictly concave for all ° 2 ©. Evaluating the ¯rst derivative of '(¿)
at ¿ = 0, we have
'
0(0) = u
0(x
¤
T¡2) + °T¡1¯±u
0(x
¤
T¡1) + °T¯±
2u
0(x
¤
T)
> u
0(x
¤
T¡2) ¡ max[¯±u
0(x
¤
T¡1);¯±
2u
0(x
¤
T)]
¸ 0
Where the last inequality follows from the initial hypothesis. This shows that the
optimum is strictly positive on ©: i.e. ¿(°) = argmax¿2R+ '(¿) > 0, for all ° 2 ©.
Since © is a compact set and ¿(°) is a continuous function on © by the Maximum the-
orem, ¿(°) attains its minimum on © by Weierstrass theorem. Let À = min°2© ¿(°),
and take any ¹ ¿ 2 (0;À). To see that this is an optimal deviation, it su±ces to show
that °T¡1 2 (¡1;0), where °T¡1 =
¢s¤
T¡1
¹ ¿ is the relative change in consumption at
T ¡ 1 given the increase in consumption by ¹ ¿ at T ¡ 2.
In period T-1, there is no dynamic inconsistency, so the optimal strategy is ob-
tained by solving
s¤
T¡1 = argmaxu(xT¡1) + ¯±u(xT)
subject to the constraint xT¡1+xT = 1¡
PT¡2
t=1 xt¡¿. From the ¯rst order conditions
we have that
ds¤
T¡1
d¿ 2 (¡1;0). By the Mean Value Theorem, there exists ´ 2 (0; ¹ ¿),
such that
¢s¤
T¡1
¹ ¿ = s0
T¡1j¿=´. The result follows. ¤
Proof of Theorem 2. Let x¤ be a Pareto e±cient allocation and set ¿ = x¤. Let
s¤ be a perfect equilibrium of the intrapersonal game. In order to prove the result, it
5su±ces to show that xt ¸ ¿t for all feasible histories ht. We prove this by induction.
For period T, all resources left are consumed, hence sT ¸ ¿t for all sT 2 ST. Next, take
any history ht and assume that xi ¸ ¿i for all i > t. Assume, towards a contradiction,
that xt < yt for some history ht. This implies that u0(xt) > u0(¿t) ¸ ¯±ju0(¿t+j) ¸
¯±ju0(xt+j) for all j > t since an e±cient allocation satis¯es u0(x¤
t) ¸ ¯±ju0(x¤
t+j) for
all t;j ¸ 1. Clearly, self t can improve his welfare by increasing his own consumption.
Hence, by the one deviation property, this cannot be a perfect equilibrium. ¤
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Choice and procrastination.

Consistent plans.

Hyperbolic discounting and golden eggs.

Intertemporally inconsistent preferences and the rate of consumption.

Is addiction rational? theory and evidence.

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Myopia and inconsistency in dynamic utility maximization.

On second-best national saving and gameequilibrium growth.

On the existence of a consistent course of action when tastes are changing.

Overestimating self-control: Evidence from the health-club industry. mimeo,

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The implications of hyperbolic discounting for project evaluation. World Bank, working paper 9356.

Time discounting and time preference: a critical review.

Time-inconsistency and welfare. NBER working paper series.

Efficiency in the cake-eating problem with quasi-geometric discounting

http://www.accessecon.com/pubs/EB/2007/Volume4/EB-07D90006A.pdf

Efficiency in the cake-eating problem with quasi-geometric discounting

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