Hyperbolic Discounting and State-Dependent Commitment

Takayuki Ogawa
Osaka University of Economics

Hiroaki Ohno
Meiji Gakuin University
Discussion Paper No.20-03
2021 ᖺ 03 ᭶Hyperbolic Discounting and State-Dependent
Commitment∗
Takayuki Ogawa†

Hiroaki Ohno‡

Abstract
In a stochastic economy with uninsurable endowment risk, we establish thecondition under which hyperbolic-discounting consumers commit to a future consumption path by utilizing both illiquid capital and borrowing constraints as acommitment device. There is always the possibility that a state-dependent commitment can be adopted as an equilibrium consumption strategy. On a pathleading to low future endowment, the current self can commit to their own optimal consumption path, which is undesirable for future selves. In contrast, along
the path with a high future endowment, the current self cannot make a commitment and must accept a consumption allocation that future selves will revise. We
also examine the eﬀect of ﬁnancial development on economic growth through consumption commitment. Relaxation of borrowing constraints across stages of lifechanges the current self’s incentive to control future selves’ behavior in diﬀerent
ways, and thus may or may not promote illiquid capital accumulation.
Keywords: Borrowing constraint, Commitment, Endowment risk, Hyperbolic
discounting, Illiquidity.
JEL Classiﬁcation Numbers: E21, E70, G12, G51.

∗

We are grateful to Keiichi Hori, Shinsuke Ikeda, Takeshi Ojima, and seminar participants at
Otaru University of Commerce for their helpful comments. We appreciate ﬁnancial supports from
JSPS KAKENHI Grant Numbers 19K01767 and 20K01669 for Ogawa and 20K01778 for Ohno.
†
Faculty of Economics, Osaka University of Economics, 2-2-8 Osumi, Higashiyodogawa-ku, Osaka,
533-8533, JAPAN. Tel: +81-6-6328-2431. E-mail: tkogawa@osaka-ue.ac.jp.
‡
Department of Economics, Meiji Gakuin University, 1-2-37 Shirokanedai, Minato-ku, Tokyo, 1080071, JAPAN. Tel: +81-3-5421-5342. E-mail: ohno@eco.meijigakuin.ac.jp.

11Introduction

People often face an internal struggle between the present self and future selves. One
cause of such a struggle is a change in preference, which can be modeled by, for example, hyperbolic discounting (Frederick et al. 2002). Numerous studies, such as Strotz(1956), Phelps and Pollak (1968), and Laibson (1997), investigate decision making
under time-inconsistent preferences. However, most of them focus mainly on intertemporal consumption choices, and scarcely treat commitment decisions across stochastic
states of nature. This study contributes to the literature by analyzing the commitment
decision of hyperbolic-discounting individuals in a risky environment and showing thatthe commitment can be state dependent—people may or may not make a consumption
commitment according to states that will arise in the future. On the path along whicha state with a low level of income will be realized, the current self is able to make a commitment that future selves will follow, even though it is desirable for the current young
self but not for future selves. In contrast, along the path with a high level of future
income, the current self is incapable of committing to their own optimal consumption
allocation, and future selves reoptimize ﬂexibly depending on the realized level of income. In other words, there may be a situation where even rational consumers do not
fully utilize commitment devices. We also show that ﬁnancial development changes thepower balance between the current self and future selves when they decide consumption
plans and aﬀects economic growth as well as commitment decisions.
A large body of literature explores decision making of agents with time-inconsistentpreferences and emphasizes the importance of utilizing commitment devices. The ideaof commitment goes back at least to Strotz (1955), who demonstrates that commitment
is useful for avoiding time-inconsistent decision making. With a hyperbolic-discounting
preference, Laibson (1997) ﬁnds that a combination of illiquid assets and borrowing
constraints plays a signiﬁcant role as a commitment device in an economy with no risk
factor. These seminal works have led to an expanding literature on hyperbolic discounting and commitment (see Bryan et al. 2010 for a survey on commitment). For
example, Angeletos et al. (2001) examine a quantitative implication of Laibson (1997).
Compulsory savings through social security systems (Schwarz and Sheshinski 2007) andintroduction of saving ﬂoors (Malin 2008) are also shown to work as commitment devices. Despite these theoretical predictions, experimental studies often report that onlya small fraction of subjects voluntarily utilizes commitment devices (Giné et al. 2010;
Augenblick et al. 2015). Laibson (2015) stresses the psychological or monetary costs
involved in making a commitment as one possible reason. We point out another: thepresence of income risk hinders individuals from making a commitment. Concretely, we
provide the full characterization of equilibrium consumption strategies in a risky situation and show the possibility that individuals cannot make a commitment, particularly
in a highly volatile economy.
Amador et al. (2006) is a notable exception treating the consumption decision of2

hyperbolic-discounting individuals in a stochastic economy.1 In the presence of taste
shocks consisting of private information on future selves, the current self encountersa conﬂicting choice between commitment and ﬂexibility—namely, either removing all
ex-post choices from future selves or leaving ﬂexible options open to them. Taking into
account an incentive compatibility condition for future selves, the current self makes
a take-it-or-leave-it oﬀer to future selves. When the time-inconsistency in preferenceis slight, the current self can compel future selves to accept the current self’s optimal consumption plan. However, as the time-inconsistency strengthens, the current
self cannot help incorporating the future selves’ tastes into a consumption plan so as
to make future selves reveal their true preferences. While Amador et al. clarify how
incentive-compatible consumption plans relate to the degree of hyperbolic discounting,
we explicitly introduce borrowing constraints and illiquid capital into a stochastic dynamic optimization model as a commitment device. This enables us to associate thefeasibility conditions of commitment with the degree of borrowing constraints and to
examine the eﬀect of ﬁnancial development on economic growth through the accumulation of illiquid capital.
The relationship between borrowing constraints and economic growth is an open
issue that attracts a lot of interest.2 For instance, as demonstrated by Jappelli andPagano (1994, 1999), relaxing borrowing limits leads to encouragement of consumption
and reduction of capital accumulation, with long-term negative eﬀects. In a generalequilibrium model with lenders and constrained borrowers, Biederman (2000) ﬁnds
that a relaxed constraint on borrowing has a positive or negative welfare eﬀect. Constantinides et al. (2002) indicate that the presence of borrowing constraints forces adecrease in demand for risky assets, which explains the empirical fact of an abnormallyhigh equity premium. Kiyotaki and Moore (1997) build a model in which the borrowing constraint on productive investors causes ineﬃcient credit allocation and accounts
for a prolonged economic slump through degradation of collateral values.3 A strand ofresearch analyzes the growth eﬀect of hyperbolic discounting. A crucial conclusion isthat the neoclassical growth model has a similar structure between standard exponential and hyperbolic discounting (Barro 1999; Krusell et al. 2002; Krusell and Smith
2003; Strulik 2015). However, no such study takes into account commitment devices.
In this paper, we provide a new insight by paying attention to the role of borrowing
constraints and illiquid capital as a commitment device.
In our model, a consumer encounters diﬀerent borrowing constraints in diﬀerent
stages of life. With the borrowing constraint relaxed, the current self gains stronger
commitment power: they can accumulate more illiquid physical capital, thereby controlling the behavior of future selves. This enhances the welfare of the current self
but worsens that of future selves. On the other hand, as the borrowing constraints fu1

Ambrus and Egorov (2013) extensively reexamine the results of Amador et al. (2006).
Aiyagari (1994) and Huggett (1997) investigate the property of a neoclassical growth model with
uninsured idiosyncratic income risk and borrowing constraints. See Townsend (2010) and Lerner andTufano (2011) for surveys regarding ﬁnancial innovations, economic growth, and welfare.
3
See Gregorio (1996) and Kitaura (2012) for a model incorporating human capital accumulation.
2

3

ture selves encounter are relaxed, it becomes diﬃcult for the current self to control the
consumption decisions of future selves, therefore bringing less incentive to the current
self to accumulate illiquid capital. This is harmful to the current self but beneﬁcial to
future selves. Thus, the total eﬀect of ﬁnancial development on economic growth andwelfare is generally ambiguous. In contrast to previous studies, with time-consistentpreferences, our results reﬂect the current self’s incentive to manage time-inconsistentbehavior through accumulating illiquid capital.
The rest of the paper proceeds as follows. Section 2 presents the basic structureof the model. Section 3 solves the optimization problem of a consumer who has timeinconsistent multiple selves inside. Section 4 compares welfare among some consumption strategies elaborated by the current self and determines a subgame-perfect equilibrium in a game played by multiple selves. Section 5 discusses the eﬀect of ﬁnancial
development on capital accumulation and welfare. Section 6 concludes the paper.

2

The Model

Using an overlapping-generations model with uninsured endowment risk, we study thecondition under which hyperbolic-discounting consumers can commit to a future consumption path with the help of illiquid assets and borrowing constraints. Individuals
live in one of three periods: young, middle-aged, and old. The population size of eachgeneration is normalized to unity.
The generation born in period t receives a non-stochastic endowment, eyt (> 0), when
young and a stochastic endowment, emi
t+1 (> 0), when middle-aged:embwith probability π,
t+1
emi
=
t+1
mg
et+1 with probability 1 − π,mgwhere embt+1 < et+1 and 0 < π < 1. The risk on endowment is uninsurable.
Being endowed with eyt , the young take out short-term loan byt+1 , which is repaid inyy
the next period, and invest in illiquid physical capital zt+2(≥ 0), which yields Azt+2
units
of output two periods later and cannot be liquidated before maturity. For analytical
simplicity, we omit the young’s consumption. The ﬂow budget equation when young isthen given byyzt+2= ety + byt+1 .

(1)

Given the gross loan rate of interest r and the endowment realized at the beginning of
period t+1, the middle-aged repay the existing loan and roll over loans for consumption
cmi
t+1 :
y
mi
mi
cmi
t+1 + rbt+1 = et+1 + bt+2

for i = g, b,(2)

where bmi
t+2 denotes the amount of new loans. The old then obtain the return on physicalcapital investment and consume the rest after repaying the loan:ymicoi
t+2 + rbt+2 = Azt+2

4

for i = g, b,

(3)

mi
mi
oi
where coit+2 represents consumption when old. bt+2 , ct+1 , and ct+2 are state-dependent
in the absence of insurance markets.
Suppose a small open economy in which the loan interest rate r is constant overtime and individuals are indiﬀerent between lending in the international loan market
and investing physical capital by themselves:

Assumption 1: r2 = A.
Following Phelps and Pollak (1968), Pollak (1968), and Laibson (1997), we modela hyperbolic-discounting consumer as a sequence of diﬀerent selves with diﬀerent preferences. The expected utility for the young self is

 mg


og
ob
(4)
π ln cmb
t+1 + δ ln ct+2 + (1 − π) ln ct+1 + δ ln ct+2 ,where 0 < δ < 1. As time passes, the discount factor declines, and an individual
becomes more impatient to consume now rather than in the future. The utility for the
middle-aged self is then characterized byoi
ln cmit+1 + βδ ln ct+2

with 0 < β < 1.

(5)

Here, β measures the degree of time inconsistency in preference. The optimal consumption path for the middle-aged self deviates from that for the young self. As pointed out
by Laibson (1997), however, the young manage to control the middle-aged’s behavior
by making use of illiquid assets and borrowing constraints.
Due to imperfect information in the loan market, lenders are assumed to require
suﬃcient collateral. As a result, the maximum amount the young can borrow is constrained at a fraction φy (> 0) of the present value of expected income obtained in thenext period:miyy Et [et+1 ]bt+1 ≤ φ

r

emb
t+1
with 0 ≤ φ <.
Et [emit+1 ]y

The second inequality in the second equation ensures that the repayment rbyt+1 is lessthan the ﬂow income emi
t+1 even if state b arises. Using (1), we can rewrite the borrowing
constraint asyzt+2≤ z̄t+2 ≡ eyt + φy

Et [emi
t+1 ]
,
r

(6)

where z̄t+2 is an upper bound of physical capital investment consisting of ﬂow income
eyt and the borrowing φy Et [emit+1 ]/r when young. Similarly, the borrowing constraint the
middle-aged face ismbmi
t+2 ≤ φy
Azt+2y
= φm rzt+2rwith 0 ≤ φm < 1,

where the equality in the ﬁrst equation comes from assumption 1.
5

(7)

3

Optimization

We consider a subgame-perfect equilibrium in a game played among multiple selves.
A consumer is said to be sophisticated if the young self elaborates a time-consistentconsumption plan that accurately predicts the reaction of the middle-aged self. We ﬁrst
derive the middle-aged self’s reaction function and then solve the utility maximization
problem for the sophisticated young self. (In what follows, we abbreviate the time
subscript to simplify exposition.)

3.1

The middle-aged self

Let us analyze the middle-aged’s behavior. After the endowment shock is realized, themiddle-aged who are in state i = b, g maximize their own utility (5) subject to thebudget equations (1)–(3) and the borrowing constraint (7):
 mi

mi
y
y
max
ln
e
+
b
−
r(z
−
e
)
+ βδ ln(Az y − rbmi ) + μmi (φm rz y − bmi ),mib

where μmi is a Lagrange multiplier for (7). The ﬁrst-order necessary conditions are1
emi + bmi − r(z y − ey )βδr−μmi (φm rz y − bmi ) = 0,

Az y − rbmi
μmi ≥ 0,− μmi = 0,bmi ≤ φm rz y .

Two cases may arise, depending on whether the borrowing constraint is binding: either
bmi = φm rz y or bmi < φm rz y .
In the case of bmi < φm rz y , we obtain μmi = 0 and
b

mi

βδ
rW i ,= rz −
1 + βδ
y

c

mi

rW i
,
=
1 + βδβδAW ic =,1 + βδ
oi(8)where W i represents the present value of lifetime income when state i arises:
W i ≡ ey +

emi
,
r

where W b < W g .

The consumption allocation (8), which is optimal for the middle-aged, is achievable aslong as the middle-aged’s borrowing constraint is not strictly binding:Condition 1a: bmi < φm rz y⇔

0 ≤ zy <βδW i.
(1 − φm )(1 + βδ)The other case is the one where the borrowing amount reaches the ceiling:μmi ≥ 0,bmi = φm rz y ,6which determines the consumption allocation as
coi = (1 − φm )Az y .
cmi = r W i − (1 − φm )z y ,(9)This implies that the middle-aged cannot smooth consumption intertemporally:
Condition 1b:

βδr
1
≥ oi
mi
c
c

⇔

βδW i
≤ zy .
(1 − φm )(1 + βδ)

The presence of borrowing ceilings prevents a consumer from increasing consumption
when middle-aged. While this is harmful to the impatient middle-aged, it is beneﬁcial
to the patient young. By adjusting the level of illiquid physical capital, the young may
actually avoid this unpleasant situation for themselves.
Keeping in mind that conditions 1a and 1b depend on the realized state i = b, g, we
can summarize the results in the following lemma:
Lemma 1 The middle-aged’s reaction is a function of illiquid capital left by the young.
b(a) Full ﬂexibility: If 0 ≤ z y < (1−φβδWm )(1+βδ) , then the middle-aged’s borrowing
constraint is not binding in both states b and g and consumption is given by (8).
bgy< (1−φβδW(b) State-dependent commitment: If (1−φβδWm )(1+βδ) ≤ z
m )(1+βδ) , then the
middle-aged’s borrowing constraint is binding only in state b. Consumption instate g is given by (8), in which i = g, whereas consumption in state b is given by(9), in which i = b.
gy(c) Full commitment: If (1−φβδWm )(1+βδ) ≤ z , then the middle-aged’s borrowing constraint is binding in both states b and g and consumption is given by (9).

A smaller z y brings lower income when old, so that the middle-aged have less incentive toborrow and can ﬂexibly choose their own consumption allocation, not being restrictedby the borrowing limit. This case is called full ﬂexibility. However, as z y increases,
so does the incentive to borrow, and the amount of borrowing eventually reaches thelimit. The situation occurs ﬁrst in state b, where the ﬂow income when middle-aged,
emi , is lower. The middle-aged self is deprived of a ﬂexible choice, whereas the young
self can commit to the consumption path they desire. Hence, the decision on whether
to commit depends on the realized state. We call this type of consumption schedule
state-dependent commitment. With a suﬃciently large z y , the young can fully commit to their own optimal consumption allocation in both states b and g, whereas the
middle-aged have no option but to follow the young’s decision. This case is called fullcommitment.

3.2

The young self

We next derive the level of illiquid capital, which determines an equilibrium consumption path from lemma 1. Obviously, the young want to commit to the path thatmaximizes their own utility by suﬃciently accumulating capital. However, we showthat they may or may not be able to aﬀord to do so.
7

3.2.1

Full commitment

In the case of full commitment presented in lemma 1c, the young can control the middleaged’s behavior in both states of nature so as to maximize the young’s utility (4) subject
to the young’s borrowing constraint (6) and the middle-aged’s reaction function (9):max
π ln r W b − (1 − φm )z y + (1 − π) ln r [W g − (1 − φm )z y ]yz+ δ ln(1 − φm )Az y + μy (z̄ − z y ),where μy is a Lagrange multiplier for (6). The ﬁrst-order necessary conditions areπ(1 − φm )
δ(1 − π)(1 − φm )+ y − μy = 0,
+ g
(10)
−
b
m
y
m
y
W − (1 − φ )z
W − (1 − φ )zz
μy (z̄ − z y ) = 0, μy ≥ 0, z y ≤ z̄.
(11)
In the case of z y < z̄ or μy = 0, the level of capital is calculated as 4zF C ≡

1
2(1 − φm )(1 + δ)−

(1 − π + δ)W b + (π + δ)W g(12)[(1 − π + δ)W b + (π + δ)W g ]2 − 4δ(1 + δ)W b W g.

This provides the following Euler equation:πδr
1−π+ mb = oi
for i = g, b.
mgccc
From the viewpoint of the young self, they can smooth consumption over time andacross states of nature and also has commitment power in both state b and state g.
Full commitment is thus desirable for the young self but not for the middle-aged self.
From the viewpoint of the middle-aged self, they have no ﬂexible option for diversifying
the endowment risk intertemporally—in fact, consumption when old is the same acrossstates, at cob = cog = (1 − φm )Az F C , while consumption cmi is volatile in response to
the realized level of endowment. This strategy is feasible as long as the middle-aged’s
borrowing constraint is binding in both states of nature and the young’s is not:Condition 2a:βδW g≤ z F C < z̄,(1 − φm )(1 + βδ)

where the ﬁrst equality is from lemma 1c and the second equality means that the
young’s borrowing constraint is not binding.
Let us now turn to the case of z y = z̄ or μy ≥ 0. The borrowing ceiling hampers
the young’s intertemporal smoothing of consumption, although commitment remains
possible in both states of nature:z y = z̄,
4

1−π
π
δr
+ mb ≤ oi .
mg
c
c
c

See appendix A for the derivation.

8

Full commitment
Constrained

Unconstrained

z̄
0

βδW g
(1−φm )(1+βδ)zF C

Figure 1: Full commitment.

While both the young and the middle-aged face each borrowing ceiling, this situation
has a diﬀerent meaning between the two groups. As seen in the second equation above,for hyperbolic-discounting individuals, the patient young self desires to shift income tothe third period of life; in contrast, the impatient middle-aged self wants to decrease
the old’s consumption, because condition 1b holds now:
1
βδr
> oi
mi
c
c

for i = g, b.

These conditions are equivalent to
Condition 2b:βδW g≤ z̄ ≤ z F C ,m(1 − φ )(1 + βδ)

where the ﬁrst inequality comes from substituting z y = z̄ into the condition in lemma
1c and the second inequality implies the binding borrowing constraint of the young.
The results are summarized in the following lemma and illustrated in ﬁgure 1:g

FC
Lemma 2 (Full commitment) If (1−φβδWfor the z F C in (12), the youngm )(1+βδ) ≤ z
can make a full commitment by which they can commit to the consumption path in bothstates of nature. Moreover,(a) Unconstrained: Under condition 2a, the young are unconstrained by the borrowing limit in the ﬁrst stage of life and can smooth consumption over time andacross states of nature.
(b) Constrained: Under condition 2b, the young are constrained by the borrowing
limit in the ﬁrst stage of life and are restricted to smoothing consumption intertemporally.
3.2.2

State-dependent commitment

Lemma 2 implies that there are some regions where full commitment is infeasible.
Then, as in the case of Lemma 1b, the young reluctantly devise state-dependent
commitment—that is, they make a commitment in state b, but they allow the middleaged to have a ﬂexible choice in state g.
9In state g, the young have no choice but to follow the impatient middle-aged’s decision, although they are willing to maximize their own utility. From (8), the consumption
allocation in state g satisﬁes the following Euler equation:
βδr
.
cog
By contrast, the consumption decision in state b is in the patient young’s hand. The
young choose the z y that maximizes their own utility (4) subject to their borrowing
constraint (6) and the middle-aged’s reaction function (9), in which i = b:max
π ln r W b − (1 − φm )z y + δ ln(1 − φm )Az y + μy (z̄ − z y ).
y
1

cmg

=

z

The ﬁrst-order necessary conditions are (11) andδ
1 − φm−− μy = 0.
−π
W b − (1 − φm )z y z yIn the case of z y < z̄ or μy = 0, the level of capital is obtained byz SDC ≡



δW b
FC
,
<
z
(1 − φm )(1 + δ)

(13)which is smaller than the z F C in (12) and gives the Euler equation,
1
cmb
t+1

=

δr
.
cob
t+2

In comparison with full commitment, the young here cannot diversify the endowment
risk across states of nature. In addition, the discount factor in the Euler equation diﬀers
across states—it is βδ in state g and δ in state b. This strategy is possible if
Condition 3a: z SDC <βδW g,(1 − φm )(1 + βδ)z SDC < z̄.

The ﬁrst condition means that the middle-aged’s borrowing constraint is binding instate b but not in state g. We obtain it by substituting the z SDC in (13) into theb
SDCcondition in lemma 1b and noting that (1−φβδWis always fulﬁlled withm )(1+βδ) < z
β < 1. The second condition is the non-binding borrowing constraint of the young.
In the case of z y = z̄ or μy ≥ 0, the presence of a borrowing ceiling disturbs theintertemporal consumption smoothing of the young in state b:δr1≤ oi .
cmbc
This is the case where the young’s borrowing amount reaches the ceiling and, simultaneously, the condition in lemma 1b is satisﬁed. In this case, either of the following two
conditions is required:
Condition 3b:

βδW b
βδW g
SDC
<
≤
z̄
≤
z
,
(1 − φm )(1 + βδ)(1 − φm )(1 + βδ)
10State-dependent commitment
Constrained

Unconstrained

z̄
0

βδW b
(1−φm )(1+βδ)βδW g(1−φm )(1+βδ)δW b(1−φm )(1+δ)


SDC

=z

g

Figure 2a: State-dependent commitment for z SDC < (1−φβδWm )(1+βδ) .
State-dependent commitment
Constrained

z̄
0

βδW b

(1−φm )(1+βδ)βδW g(1−φm )(1+βδ)δW b(1−φm )(1+δ)

SDC



=z

g

SDC
Figure 2b: State-dependent commitment for (1−φβδW.
m )(1+βδ) ≤ z

Condition 3c:

βδW b
βδW g
≤
z̄
<
≤ z SDC .
(1 − φm )(1 + βδ)(1 − φm )(1 + βδ)

Therefore, we conclude that:
Lemma 3 (State-dependent commitment) The young can make a state-dependent
commitment by which they can commit to the consumption path in state b but not instate g, and furthermore:(a) Unconstrained: Under condition 3a, the young are unconstrained by the borrowing limit in the second stage of life and can smooth consumption over time instate b.
(b) Constrained: Under either condition 3b or condition 3c, the young are constrained by the borrowing limit in the second stage of life and restricted to smoothing consumption intertemporally in state b.
g

Figure 2a describes the case where z SDC < (1−φβδWm )(1+βδ) , whereas ﬁgure 2b illustratesg
SDC
that for (1−φβδW.
m )(1+βδ) ≤ z
3.2.3

Full ﬂexibility

In the region where both full and state-dependent commitment are impossible, theyoung are governed by the middle-aged’s decisions in both states of nature. In other11

words, the middle-aged have a fully ﬂexible option in every situation. The consumption
allocation is thus determined by (8), satisfying the following Euler equations with thelower discount factor, βδ:
1
βδr
=
cmi
coi

for i = g, b.

This case is possible for any z y and z̄ such that
Condition 4: 0 ≤ z y ≤ z̄ < z F F ,where z F F ≡βδW b(< z SDC < z F C ).
(1 − φm )(1 + βδ)(14)Here, z F F is smaller than the z F C in (12) as well as the z SDC in (13). This condition
comes from the young’s borrowing constraint (6) and the condition in lemma 1a. The
amount of capital investment is indeterminate within the range of condition 4 but it
does not aﬀect the consumption allocation the middle-aged will decide on.5 It is formally
noted that:
Lemma 4 (Full ﬂexibility) As long as the level of capital is under that in condition
4, the young cannot help taking the full-ﬂexibility strategy by which the consumption
allocation in both states of nature is ﬂexibly determined by the middle-aged.

4

Welfare and Equilibrium

As shown in the previous section, there are potentially three kinds of consumption
strategies relevant here: full commitment, state-dependent commitment, and full ﬂexibility. When the region within which each strategy is feasible overlaps, the young adopt
a welfare-maximizing strategy, and a subgame-perfect equilibrium is determined.
Compare the welfare attained by each strategy in the relevant region. It is intuitivethat:
Lemma 5 (Welfare comparison) In terms of the young’s welfare, full commitment
is superior and full ﬂexibility is inferior to the other two strategies regardless of whether
the young’s borrowing constraint is binding.
Proof: See appendix B.
It is straightforward that full commitment (full ﬂexibility) is the best (worst) strategy
considering the young self’s utility. Furthermore, this lemma states that this welfare
ranking holds even when the young’s intertemporal consumption choices are constrainedby borrowing limits.
5

The indeterminacy of equilibrium capital is due to the linear production function and free international lending and borrowing. Mundell (1957) points out that international specialization patterns of
production become indeterminate in the presence of international capital movement. Ono and Shibata(2010) show that this property also holds in a dynamic-optimization setting.

12Given the welfare ranking in this lemma, cases are divided into three types according
to the degree of time-inconsistency caused by hyperbolic discounting. Figure 3 classiﬁes
the strategy adopted in each case and in each region. The horizontal axis z̄ stands forthe maximum amount the young can invest in capital.
Note that the z F C in (12) is independent of β. Figure 3a illustrates the case with alower degree of time-inconsistency in preference, that is, a β close to unity:6


δW b
βδW g
FC
Condition 5a:<<z.
(1 − φm )(1 + δ)
(1 − φm )(1 + βδ)
In the stochastic environment, there is a possibility that the full-commitment equilibrium disappears and the state-dependent commitment is instead employed. Being
endowed with higher income in state g, Wtg , as well as the larger β (condition 5a), themiddle-aged have less incentive to borrow and are beyond the control of the young. In
contrast, if the income in state b, Wtb , is low, or if the young are capable of suﬃciently
investing in illiquid capital due to the higher borrowing capacity z̄, then the young
can commit to the desired consumption path in state b (see the region where z F F ≤ z̄in ﬁgure 3a). What is worse, if the higher income is also received in state b or if theborrowing limit the young face is severe, the middle-aged’s behavior is completely outof control, and full ﬂexibility is a unique equilibrium strategy (see the region where0 ≤ z̄ < z F F in ﬁgure 3a). This establishes the following proposition:
Proposition 1 Under condition 5a, there is no full-commitment equilibrium; instead,the state-dependent commitment is an equilibrium strategy under z F F ≤ z̄ as well as
condition 5a.
Figure 3b considers a higher degree of time-inconsistency measured by a lower β:
Condition 5b:

δW b
βδW g
<
≤ zF C .
(1 − φm )(1 + δ)
(1 − φm )(1 + βδ)Every type of equilibrium, including the full-commitment strategy, emerges. The lower
β raises the borrowing demand of the impatient middle-aged and makes it easier forthe young to adopt the full-commitment strategy.
Figure 3c treats a signiﬁcantly higher degree of time-inconsistency, such that
Condition 5c:



βδW g
δW b
FC
,
≤
<
z
(1 − φm )(1 + βδ)(1 − φm )(1 + δ)

The lower β needs more illiquid assets to control the middle-aged and eliminates theregion for state-dependent commitment unconstrained by the young’s borrowing ceiling.
As β approaches zero, full commitment is a unique equilibrium strategy even if theyoung’s borrowing constraint is extremely tight, that is, z̄ → 0.
The conditions for state-dependent commitment to arise, presented in ﬁgure 3, areincluded in:
6bFC
Remember that applying (12) and (13) to lemma 1 requires z SDC ≡ (1−φδW.
m )(1+δ) < z

13

State-dependent commitment
Full ﬂexibility
Constrained

Unconstrained

z̄
0

βδW b

(1−φm )(1+βδ)


FF

=z

δW b
(1−φm )(1+δ)


SDC

βδW g

zF C

(1−φm )(1+βδ)=zgFigure 3a: z F C < (1−φβδWm )(1+βδ) (condition 5a).

Full commitment
State-dependent commitment
Full ﬂexibility

Constrained
Constrained

Unconstrained

Unconstrained

z̄
0

βδW b

(1−φm )(1+βδ)

FF



=z

δW b

(1−φm )(1+δ)

SDC



βδW g

(1−φm )(1+βδ)

zF C

=z

g

b

βδW
FC
Figure 3b: (1−φδW(condition 5b).
m )(1+δ) < (1−φm )(1+βδ) ≤ z

Full commitment

State-dependent commitment
Full ﬂexibility

Constrained

Unconstrained

Constrained

z̄
0

βδW b
(1−φm )(1+βδ)


FF

βδW g
(1−φm )(1+βδ)

=z

δW b
(1−φm )(1+δ)


SDC

zF C

=z

g

b

δW
Figure 3c: (1−φβδWm )(1+βδ) ≤ (1−φm )(1+δ) (condition 5c).

14

Proposition 2 State-dependent commitment is an equilibrium strategy if any of thefollowing conditions are fulﬁlled:b(a) condition 5a and (1−φβδWm )(1+βδ) ≤ z̄.
b

g

b

g

βδW
(b) condition 5b and (1−φβδWm )(1+βδ) ≤ z̄ < (1−φm )(1+βδ) .
βδW(c) condition 5c and (1−φβδWm )(1+βδ) ≤ z̄ < (1−φm )(1+βδ) .

Keeping the average income E[emi ] constant, an increased income volatility measured
by the diﬀerence W g − W b raises the income in state g but decreases it in state b. Thiswidens the region within which state-dependent commitment is adopted; in other words,
the presence of income risk prevents the current self from making a commitment for all
possible states in the future. The result is consistent with the ﬁndings of experimental
studies reporting that the voluntary demand for commitment devices is much smaller
than that estimated in theoretical predictions (Giné et al. 2010; Augenblick et al.
2015; Laibson 2015). Even if individuals have access to commitment devices, theycannot eﬀectively utilize them to accomplish full commitment; furthermore, there aresome stochastic situations where commitment is impossible in some states of nature(propositions 1 and 2).

5

Financial Development

This section investigates an impact of ﬁnancial development on consumption decisions
and capital accumulation.

5.1

Insurance markets

The purpose of this study is to clarify the commitment strategy employed in a risky
situation. What would the consumption decision be if there were no endowment risk,or equally, if an insurance market were complete? If the risk were perfectly diversiﬁed
either within or among countries, it would hold that:W b = W g = E[W i ],z F C = z SDC =δE[W i ],(1 − φm )(1 + δ)from (12) and (13). Neither condition 5a nor 5b is fulﬁlled, but condition 5c remains,thereby transforming ﬁgure 3c to ﬁgure 4. Instead of causing the state-dependent
commitment equilibrium to vanish, this leads the range of full ﬂexibility and full commitment to widen.
Corollary 1 An economy with complete insurance markets has a two-alternative strategy: full commitment is an equilibrium strategy if z̄ ≥ z F F ; otherwise, full ﬂexibility is
an equilibrium strategy.
15

Full commitment
Full ﬂexibility

Constrained

Unconstrainedz̄
0βδE[W i ](1−φm )(1+βδ)FF

δE[W i ](1−φm )(1+δ)


FC
SDC

=z

=z

=z

Figure 4: A non-stochastic economy.

Laibson (1997) emphasizes an important role of combining illiquid assets and borrowing constraints as a commitment device in a deterministic world. He imposes assumption A1 in his paper (p. 452) to concentrate on the unconstrained full commitment
equilibrium—the condition z F C = z SDC < z̄ in our context. More precisely, he assumes
an increasing path of ﬂow income, which brings a strong borrowing motive in the current
self and makes the borrowing constraint bind the future self. This study complements
his contribution by taking account of not only income risk but also the entire parameter
range and showing the possibility for state-dependent commitment to arise.

5.2

Borrowing constraints

It is controversial whether the ﬁnancial innovation that relaxes borrowing limits encourages economic growth and enhances welfare (e.g., Jappelli and Pagano 1994, 1999;
Biederman 2000). Our model contains the two borrowing constraints a consumer encounters in diﬀerent stages of life, which aﬀect capital accumulation in contrary directions.
The level of illiquid capital under condition 5b, for example, is illustrated in ﬁgure
7
5. A rise in the maximum loan-to-value ratio for the young, φy , raises z̄ and promotes
capital accumulation z y . It strengthens the young’s commitment power, so that the
young’s welfare improves and the middle-aged’s worsens. Conversely, an increase in the
maximum loan-to-value ratio for the middle-aged, φm , raises the boundary conditions
uniformly for a given z̄ and discontinuously reduces capital z y . This is because arelaxation of the middle-aged’s borrowing constraint makes it diﬃcult to adopt thecommitment strategy for the young, and hence the young need less illiquid capital as acommitment device. Welfare is also inﬂuenced inversely.
Corollary 2 A relaxation of borrowing constraints has an ambiguous eﬀect on economic growth and welfare.
7
In ﬁgure 5, the shadowed area, 0 ≤ z̄ < z F F , represents the indeterminate level of capital shownin lemma 4.

16

zy

Full commitment
UnconstrainedzF C
ConstrainedβδW g(1−φm )(1+βδ)State-dependent commitment
Unconstrainedz SDC
Constrained

Full ﬂexibilityzF Fz̄ = e
0

βδW b
βδW g
δW b
(1−φm )(1+βδ) (1−φm )(1+δ) (1−φm )(1+βδ)



=z


FF



=z


SDC

z

FC

Figure 5: Borrowing constraints and illiquid capital.

17

y

mi ]

+ φy E[er(a) A rise in φy boosts capital accumulation. It is beneﬁcial to the young self andharmful to the middle-aged self.
(b) A rise in φm depresses capital accumulation. It is harmful to the young self andbeneﬁcial to the middle-aged self.
The present relation between borrowing constraints and economic growth diﬀers
from that in the previous literature with time-consistent preferences. In an overlappinggenerations model such as those by Jappelli and Pagano (1994, 1999) and Biederman(2000), the relaxation of borrowing constraints alters intergenerational income distribution through an increased or decreased savings motive in the existing generation andthus aﬀects the welfare of future generations. However, in our model, internal struggle
within a single person matters. The presence of borrowing constraints gives the young
self an incentive to manage the behavior of the middle-aged self and changes the demand for illiquid capital as a commitment device. Thus, incompleteness in the ﬁnancial
market aﬀects diﬀerent selves in diﬀerent ways and may or may not promote capital
accumulation through an increased or decreased commitment motive in the young self.

6

Conclusion

Using an overlapping-generations model with endowment risk, we clarify how hyperbolicdiscounting consumers utilize illiquid capital and borrowing constraints as a commitment device. There is always the possibility that the current self can commit to an
optimal path only in some state of nature—which is called the state-dependent commitment. This occurs notably in a highly volatile economy. This result is a problem
speciﬁc to a stochastic world, which is rarely treated in the literature. We also provide a detailed welfare comparison and full characterization of equilibrium strategies,
including full commitment and full ﬂexibility.
In the presence of hyperbolic-discounting individuals, the ﬁnancial developmentthat relaxes borrowing constraints has two opposite eﬀects on capital accumulation. It
directly induces capital formation by the patient current self; however, it simultaneously
allows the impatient future self to make a ﬂexible consumption decision and lowers thecurrent self’s motive for accumulating capital as a commitment device. In general, thetotal eﬀect on economic growth and welfare remains ambiguous.
Recently, humanity has been exposed to undiversiﬁed risks such as the global ﬁnancial crisis of 2007-2008 and coronavirus disease 2019. In times of increased risk, ourmodel predicts that people are more likely to select state-dependent commitment—ahyperbolic-discounting individual executes a predetermined plan on the path toward
economic contraction, but leaves a ﬂexible choice for the expansionary path in the
future.
This study provides some directions for future research. First, an extension of ourmodel to a dynamic stochastic general equilibrium setting can generate implications
on asset pricing, such as the term structure of interest rates and the excess returns of18

holding illiquid assets. Second, introduction of a heterogeneous degree of borrowing
limits and time discounting can enrich the quantitative and qualitative results of themodel. Third, taking costs of self-control into account may alter the welfare eﬀects ofvarious economic policies, as pointed out in, for example, Krusell et al. (2010) and
Laibson (2015).

19

F (z y )

zF C
Wb
1−φm

0

Wg
1−φm

zy

Figure A.1: The determination of z F C .

Appendices
Appendix A: The derivation of (12)This appendix derives (12) in the text. We can reduce (10) in which μy = 0 to
F (z y ) ≡ (z y )2 −(1 − π + δ)W b + (π + δ)W g y
δW b W g+z= 0,(1 − φm )(1 + δ)
(1 − φm )2 (1 + δ)

which satisﬁes
δW b W g> 0,
F (0) =
(1 − φm )2 (1 + δ)WbπW b (W g − W b )
< 0,
F
=
−
1 − φm(1 − φm )2 (1 + δ)Wg
(1 − π)W g (W g − W b )F=> 0.
1 − φm(1 − φm )2 (1 + δ)
Figure A.1 depicts the shape of F (z y ). Consumption in the full-commitment case isgiven by (9), and we ﬁnd that:cmi ≥ 0 if z y ≤Wi.
1 − φmSince we obtain cmb < 0 for the larger root of F (z y ) = 0, z y is uniquely determined by
the smaller root, which is (12).
20Appendix B: The proof of lemma 5
For analytical reasons, the level of capital in each state of nature is temporally denotedby z i (i = b, g). Using the consumption function (9), we can express the young’s lifetime
utility U as a function of z b and z g :
U = U z b , z g ≡ πub + (1 − π)ug ,ui ≡ ln r[W i − (1 − φm )z i ] + δ ln(1 − φm )Az i .
Diﬀerentiating them with the level of capital yields
dub
dug
dU
=
π
+
(1
−
π)
,
dz y
dz b
dz g 
(1 − φm )(1 + δ)
δW i
dui
i
=
−z .
dz i
[W i − (1 − φm )z i ]z i (1 − φm )(1 + δ)

(A.1)

Compare ﬁrst the lifetime utility attained by each strategy when the young’s borrowing is unconstrained. The maximum utility is achieved by full commitment at
z b = z g = z F C , which is in (12), and requires
dU
= 0,
dz y

dub
< 0,
dz b

dug
> 0,
dz g

where the ﬁrst condition is equivalent to (10) with μyt = 0.
Utility under state-dependent commitment can be expressed by z b = z SDC in (13)gand z g = (1−φβδWm )(1+βδ) , which provides consumption equal to (8) in state g:rW g,
1 + βδβδAW g.
cog = (1 − φm )Az g ] =1 + βδ

cmg = r[W g − (1 − φm )z g ] =Evaluating (A.1) by these capital levels (or consumption levels) generates
dU
> 0,
dz y

dub
= 0,
dz b

dug
(1 − φm )(1 + βδ)(1 − β)
=
> 0.
dz g
βW g

Unless β = 1, the total marginal utility of capital is positive, reﬂecting the fact that theconsumption allocation in state g is governed by the middle-aged and is not optimalfor the young. The level of lifetime utility is hence shown to be lower than that of fullcommitment: FC FC
βδW g
SDC
U z
,
<
U
z ,z
.
(A.2)
(1 − φm )(1 + βδ)






unconstrained
unconstrained state-dependent commitment

21

full commitment

i

The utility under the full-ﬂexibility strategy corresponds to the case of z i = (1−φβδWm )(1+βδ) ,satisfying
dui(1 − φm )(1 + βδ)(1 − β)
=
> 0.
dz i
βW i

dU
> 0,
dz y

As long as β = 1, the total marginal utility of capital is higher than that of statedependent commitment because the young are under the control of the middle-aged inboth states of nature. Therefore, the welfare ranking of these two strategies is 8




βδW g
βδW g
FF
SDC
U z ,
< U z
,
.
(A.3)
(1 − φm )(1 + βδ)(1 − φm )(1 + βδ)






full ﬂexibility

unconstrained state-dependent commitment

By the same logic, we can evaluate the utility attained by state-dependent commitment and full commitment when hitting the young’s borrowing limit. First, theconsumption obtained under constrained state-dependent commitment equals that ung
der z b = z̄ and z g = (1−φβδW
From lemma 3, state-dependent commitment ism )(1+βδ) .
feasible if z̄ lies above the z F F in (14). Thus, we have
dub
dub
≥
> 0,
dz F F
dz̄

dU
dU
≥
> 0,
dz F Fdz̄

with equality when z̄ = z F F . Consequently, even if the young’s borrowing constraint isbinding in the state-dependent commitment strategy, it is superior to the full-ﬂexibility
strategy in terms of the young’s welfare:




βδW g
βδW g
FF
U z ,
≤ U z̄,,(A.4)(1 − φm )(1 + βδ)(1 − φm )(1 + βδ)






full ﬂexibility

constrained state-dependent commitment

with equality when z̄ = z F F .
Full commitment constrained by the borrowing ceiling exists if both condition 2b
and condition 5b (or 5c) are fulﬁlled. As seen in ﬁgure 3b, conditions 2b and 5b equalz SDC <

βδW g
≤ z̄.
(1 − φm )(1 + βδ)This indicates a utility higher than that in unconstrained state-dependent commitment:


βδW g
SDC
,
< U (z̄, z̄) .
(A.5)
U z
  
(1 − φm )(1 + βδ)
constrained
unconstrained state-dependent commitment

full commitment

8
It is worth noting that for a time-consistent consumer with β = 1, the inequalities in (A.2) and(A.3) are not preserved and are replaced by equality, implying that the three strategies all ensure
optimality.

22

In contrast, under conditions 2b and 5c, unconstrained state-dependent commitment isinfeasible (see ﬁgure 3c), and hence, all we have to prove is that constrained full commitment is more desirable than full ﬂexibility. The existence condition of constrained
full commitment is
zF F <βδW g≤ z̄.
(1 − φm )(1 + βδ)This immediately means that constrained full commitment is welfare-superior to full
ﬂexibility:


βδW g
FF
U z ,
< U (z̄, z̄) .
(A.6)
  
(1 − φm )(1 + βδ)
constrained
full commitment

full ﬂexibility

The magnitude relationship from (A.2) to (A.6) proves lemma 5.

23

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