Discussion Papers In Economics
And Business

Time-Inconsistent Discounting and the Friedman Rule:
The Role of Non-Unitary Discounting

Takeo Hori and Koichi Futagami
Discussion Paper 18-04

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANTime-Inconsistent Discounting and the Friedman Rule:
The Role of Non-Unitary Discounting

Takeo Hori and Koichi Futagami
Discussion Paper 18-04

February 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANTime-Inconsistent Discounting and the Friedman
Rule: The Role of Non-Unitary Discounting∗Takeo Hori† and Koichi Futagami‡
February 13, 2018

Abstract
We examine the optimality of the Friedman rule by considering recent developments inbehavioral economics. We construct a simple macroeconomic model where agents discount consumption and leisure at diﬀerent rates. We also consider a standard exponential
discounting model and a hyperbolic discounting model, assuming that the same discounting applies to both consumption and leisure. Money is introduced via a cash-in-advanceconstraint. Although the three models are observationally equivalent, they provide diﬀerent policy implications. The Friedman rule is optimal in the latter two models, whereas it
is not optimal in the first model when agents discount consumption at a higher rate thanleisure.
Keywords: Non-unitary discount rate, Hyperbolic discounting, Exponential discounting,Friedman rule, Optimal inflation
JEL Classification Numbers: E5, E71∗

The authors are grateful to Nobuo Akai, Ken Ariga, Ken-ichi Hashimoto, Shinsuke Ikeda, Ryoichi Imai,
Tatsuro Iwaisako, Kohei Kubota, Noritaka Kudo, Kazuo Mino, Shoko Morimoto, Makoto Saito, Etsuro Shioji,
Katsuya Takii, and seminar participants at Hitotsubashi University, Kobe University, Kyoto University, Kyusyu
University, Osaka University, and SWET for helpful comments. Hori’s research is partially supported by the Japan
Society for the Promotion of Science (JSPS) through Grant-in-Aids for Young Scientists (B) (No. 21830035 andNo. 15K17025). All remaining errors are ours.
†
Department of Industrial Engineering and Economics, School of Engineering, Tokyo Institute of Technology,
2-12-1, Ookayama, Meguro-ku, Tokyo 152-8552, JAPAN. E-mail: hori.t.ag@m.titech.ac.jp‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0034, Japan.
E-mail: futagami@econ.osaka-u.ac.jp

1

1 Introduction
The purpose of this study is to examine how recent evidences from psychology and behavioral
economics aﬀect the optimality of the Friedman rule in a simple macroeconomic model. Since
Friedman (1969), many studies have examined the optimality of the Friedman rule, which sets
the nominal interest rate at zero. Using standard models, Chari and Kehoe (1999) show that theFriedman rule is optimal in the long run.1 In a diﬀerent context, psychology and behavioral economics have revealed that standard economic models often fail to describe people’s preferences
and behaviors. Naturally, a policy recommendation based on a model that cannot accurately
capture people’s behaviors could be misguided. Therefore, when assessing economic policies
such as the Friedman rule, it is quite important to use a model that is consistent with the recent
evidences from psychology and behavioral economics.
For our purpose, we consider the evidence that people use diﬀerent discount rates for different sources of utility (see Subsection 1.1). Motivated by this evidence, Hori and Futagami(2017) develop a non-monetary model in which agents discount their utilities from consumption and leisure at diﬀerent rates; we call this a non-unitary discount rate model. Except forthe diﬀerence in discount rates between consumption and leisure, the model of Hori and Futagami (2017) is standard. The present study introduces money into the model of Hori andFutagami (2017) through a cash-in-advance (CIA) constraint on consumption. In addition, assuming that the same discounting applies to both consumption and leisure, we also consider (i)a standard model with exponential discounting and (ii) a model with time-variable discounting that includes hyperbolic discounting. We then examine how the diﬀerent assumptions ontime-discounting aﬀect the optimality of the Friedman rule.
We first show that in the non-unitary discount rate model, where agents discount consumption and leisure at diﬀerent rates, preferences are time-inconsistent. It is well known thattime-variable discounting, such as hyperbolic discounting, also gives rise to time-inconsistency.
However, the sources of time-inconsistency in the two models are diﬀerent. In the non-unitary
discount rate model, time-inconsistency arises because the marginal rate of substitution between consumption and leisure is time varying. In contrast, in the time-variable discounting
model, the marginal rate of substitution between consumption and leisure is constant over time.
To formally solve our non-unitary discount rate model, we follow the theoretical literatureof time-inconsistency and consider an individual as composed of a sequence of autonomous
decision makers.2 The choices of each decision maker (self) can be considered the outcome of
an intrapersonal game. We show that under log utility, the non-unitary discount rate model becomes observationally equivalent to the standard exponential discounting model and the modelwith time-variable discounting that includes hyperbolic discounting.
In spite of observational equivalence, the three models provide diﬀerent implications for
the Friedman rule. In the standard exponential discounting model and time-variable discounting model, the Friedman rule is optimal. In contrast, in the non-unitary discount model, theFriedman rule is not optimal as long as agents discount the utility of future consumption at ahigher rate than the utility of future leisure. Moreover, we show that the optimal inflation rateincreases as the financial market develops.
The intuition behind this result is as follows. The fact that agents discount their future
consumption more than their future leisure means that today’s self cares less about the future
consumption than future leisure, whereas future selves care about their own consumption as
1

We discuss theoretical literature in Subsection 1.2.
See Peleg and Yaari (1973), Goldman (1980), Laibson (1996, 1997), and Luttmer and Mariotti (2003), forexample.
22

much as they do about their leisure. Thus, future selves consume more than that today’s self
prefers. Higher inflation tightens the CIA constraint and reduces the consumption of future
selves, resulting in a positive eﬀect on the welfare of today’s self. Since there is a positive relation between inflation and the nominal interest rate through the Fisher equation, the Friedman
rule is not optimal.
Our results suggest the following points. First, diﬀerent assumptions about time discounting can aﬀect the optimality of the Friedman rule. Thus, incorporating evidences from psychology and behavioral economics is important when assessing various economic policies such as
the Friedman rule. Second, time-inconsistency of preferences does not necessarily aﬀect the
optimality of the Friedman rule. If hyperbolic discounting leads to time-inconsistency, theFriedman rule is optimal. However, if diﬀerences in discount rates lead to time-inconsistency,
the Friedman rule is not optimal. Thus, we need to consider the source of time-inconsistency.
Third, even if two economies are observationally equivalent to each other, the optimal policies
in each economy could diﬀer.

1.1 Relation to Psychology and Behavioral Economics
The literature from psychology and behavioral economics provide evidences supporting ourassumption that people use diﬀerent discount rates for diﬀerent utility sources. Since Hori andFutagami (2017) provide a detailed review of the literature, we present only a short summary
here.3
Frederick et al. (2002) criticize the standard assumption of economics that people apply asingle discount rate to their utility from diﬀerent sources. They argue that a person who smokesheavily may carefully study the returns of various retirement packages. He may discount thedisutility of poor future health at a higher rate. At the same time, a careful study of the returns ofvarious retirement packages suggests a much lower discount rate for the utility of consumption
after retirement.
In fact, recent studies in psychology and behavioral economics provide empirical and experimental evidences that people discount their utility from diﬀerent sources at diﬀerent rates.
The term “domain eﬀect” is used to refer to the finding that the discount rates for the diﬀerent domains may diﬀer. Through a series of experiments, Chapman and her coauthors showedthat people use diﬀerent discount rates for money and health (Chapman, 1996; Chapman et al.,
1999; and Chapman et al., 2001). Using survey data from rural Uganda, Ubfal (2016) foundthat the discount rates for entertainment and cell phone airtime are lower than the discount rate
for money, while those for food are higher than the discount rate for money.
The domain eﬀect has been observed for money and time/eﬀort. Soman (1998) conducted
an experiment to study the discounting of money and time/eﬀort. He found that people discountfuture time/eﬀort at diﬀerent rates from the perspective of future money (see also Soman et al.,2005). Soman (2004) and Zauberman and Lynch (2005) also provided similar results.
These studies motivate us to construct a model where agents use diﬀerent discount rates fortime and consumption. Section 2 presents a model where agents apply diﬀerent discount rates
for consumption and leisure.

3

See the subsection titled “Related Literature from Psychology and Behavioral Economics” in the Introductionof Hori and Futagami (2017).

3

1.2 Relation to Theoretical LiteratureFriedman (1969) states that the rate of nominal interest should be set at zero because the costof printing money is zero. The validity of this simple rule has been debated. Grandmont
and Younes (1973) show its validity by using a simple macroeconomic model with a CIA
constraint. Phelps (1973) argues that if governments must rely on the distortionary taxes tocollect revenue, they must impose the inflation tax to correct the distortion, which could resultin the rule failing to hold. Other arguments attribute the validity of the rule to the shopping timefunction (Woodford, 1990). Mulligan and Sala-i-Martin (1997) summarize these arguments
and further calibrate their model and show that the optimal nominal interest rate should be lessthan 1 percent. Some authors, such as Freeman (1993) and Bhattacharya et al. (2005), useoverlapping generations models to examine the optimality of the Friedman rule.4
One limitation of these studies is that they pay little attention to recent developments in
psychology and behavioral economics. For example, they assume constant discounting, whichis inconsistent with evidences from psychology and behavioral economics. So far, only a few
studies have considered the findings of behavioral economics when examining the optimality of
the Friedman rule. Graham and Snower (2008, 2013) incorporate hyperbolic discounting into
the New Keynesian models with nominal wage rigidity and elastic labor supply. They argue
that agents with hyperbolic discounting attach much less weight to the disutility from futurelabor supply than agents with standard exponential discounting. If nominal wage is sticky,
inflation decreases the future real wage and thus increases future labor demand. Then, agentswith hyperbolic discounting tend to supply more labor since they discount the future disutilityof labor heavily. This yields a positive relation between inflation and labor supply. Graham andSnower (2013) show that because of this positive relation, the Friedman rule is not optimal.
Here, the following point should be noted. In their model, hyperbolic discounting generates
time-inconsistency. However, time-inconsistency itself is not the source of non-optimality of
the Friedman rule. They insist that heavy discounting of the disutility of labor, coupled withnominal wage rigidity, creates a positive correlation between inflation and labor supply, andthis renders the Friedman rule non-optimal.
In contrast to Graham and Snower (2013), we assume flexible prices. We show that without
price stickiness, the Friedman rule is optimal even with hyperbolic discounting. We also showthat if the agents’ discount rate for future consumption is higher than that for future leisure, theFriedman rule is not optimal even without price stickiness. This is because time-inconsistency
causes ineﬃciency with regard to consumption and leisure choice in our model.
Hiraguchi (2016) examines how the temptation utility aﬀects optimality of the Friedman
rule. To this end, he incorporates Gul-Pesendorfer preferences into a search-theoretic model
proposed by Lagos and Wright (2005).5 Hiraguchi (2016) shows that a positive nominal interest
rate reduces temptation, and therefore the Friedman rule is not optimal. Although this is similar
to our result, the mechanism is diﬀerent because Gul-Pesendorfer preferences do not generate
time-inconsistency.
This study is not the first one that assumes diﬀerent discount rates for diﬀerent utility
sources. Banerjee and Mullainathan (2010) consider a simple two-period model consistingof two types of goods: temptation goods that generate utility from today’s consumption, but
not from tomorrow’s consumption, and the usual goods that generate utilities from both today’s
and tomorrow’s consumption. They use their model to explain some of the puzzling behaviors
4

See also Abel (1987) and Gavhari (1988).
Gul and Pesendorfer (2001) propose a model where individuals face the problem of self-control when they
face temptation goods.
54

of the poor. Hori and Futagami (2017) examine the eﬀects of tax policies in a model where
agents apply diﬀerent discount rates to consumption and leisure, and show that the optimal tax
policies are diﬀerent from those obtained in a model in which agents apply a single discountrate (function) to consumption and leisure. Both studies do not consider the monetary policy.
This study is the first one that examines the optimality of the Friedman rule assuming thatagents discount their utility from consumption and leisure at diﬀerent rates.
Organization of the Paper
In Section 2, we present our basic model. Section 3 derives a solution for the intrapersonal
game. Using a simple general equilibrium model, Section 4 examines the optimal monetary
policy. Section 5 compares the results of our non-unitary discount rate model with those of amodel with a discount rate that varies with time distance. Concluding remarks are presented in
Section 6.

2 The Model
The population size of the economy is one. We consider an infinitely-lived representative agent
endowed with one unit of time at each moment of time that is allocated to labor or leisure. The
preference of the agent is given by∫ +∞ {}Ut =u1 (cv )e−ρc (v−t) + u2 (1 − lv )e−ρl (v−t) dv,(1)t

where cv ≥ 0 is the consumption level at time v and lv ∈ [0, 1] is the time allocated to labor supply at time v. u1 (cv ) and u2 (1 − lv ) represent the instantaneous utility derived from consumption
and leisure at time v, respectively. Functions u1 (·) and u2 (·) are twice diﬀerentiable and satisfyu′i (·) > 0 and u′′i (·) < 0 (i = 1, or 2). Parameters ρc > 0 and ρl > 0 are subjective discountrates for consumption and leisure, respectively. We allow the case where ρc is not equal to ρl ,meaning that the agent discounts utility from diﬀerent sources at diﬀerent rates. When ρc is(not) equal to ρl , we call it a (non-)unitary discount rate case.6 The unitary discount rate casegives a standard exponential discounting model where consumption and leisure are discounted
at the same rate.
We denote the price of consumption good and nominal cash holdings of the agent at time
t as pt and Mt , respectively. A fraction of purchase of the consumption good, η ∈ [0, 1], mustbe financed by cash. As the financial market develops, η decreases. Mt must satisfy the CIA
constraint Mt ≥ ηpt ct , or equivalently
mv ≥ ηcv ,(2)

where mv ≡ Mv /pv . The budget constraint is given byȧv = rv av − (rv + πv )mv + wv lv − cv + T v ,(3)

where rv is the real interest rate, wv is the wage rate, πv ≡ ṗv /pv is the inflation rate, and at isequal to bv +mv , where bv represents asset holdings other than cash. The lump-sum transfer fromgovernment is T v . We assume that at any moment of time, agents can allocate their portfolio
If consumption
is tempting in the spirit of Banerjee and Mullainathan (2010), (1) can be rewritten as Ut =∫ +∞u1 (ct ) + t u2 (1 − lv )e−ρl (v−t) dv.
6

5

between cash and other assets without any costs.7
As for the government’s money supply behavior, we assume a helicopter drop of money.
The monetary authority issues nominal money at a constant growth rate, ϵ ≡ Ṁt /Mt . The newly
created money is transferred to agents in a lump-sum manner, T t = ϵ Mt /pt .

Non-Unitary Discount Rate and Time-Inconsistency
This subsection observes that the preference represented by (1) raises the problem of timeinconsistency. For simplicity, we assume that the CIA constraint (2) binds, which is true inequilibrium, as we will see later. Then, the budget constraint (3) can be written asȧv = rv av + wv lv − {1 + η(rv + πv )}cv + T v .

(4)

Before providing a formal solution in the next section, we consider the case where at timet, the agent chooses sequence {cv , lv , av }∞v=t without considering the possibility that she reconsiders her choices at some future time. As in Hori and Futagami (2017), we first maximize (1)subject to (4) by setting the present value Hamiltonian as Hv = u1 (cv )e−ρc (v−t) +u2 (1−lv )e−ρl (v−t) +ψv [rv av + wv lv − {1 + η(rv + πv )}cv + T v ], where v(≥ t) indicates future selves and ψv is the costatevariable associated with (4). From the first-order conditions, we obtainu′2 (1 − lv ) −(ρl −ρc )(v−t)
wv
e
=
.
′
u1 (cv )1 + η(rv + πv )(5)

At time t, the agent plans to consume goods and supply labor according to (5) at time v(> t).
The left-hand side of (5) shows that as long as ρc , ρl , the marginal rate of substitution between
consumption and leisure is time varying.
However, if she maximizes her utility once again at time v(> t), we obtainu′2 (1 − lv )wv=.
′
u1 (cv )1 + η(rv + πv )(6)

In the unitary discount rate case (ρc = ρl ), (5) becomes exactly the same as (6). However,in the non-unitary discount rate case (ρc , ρl ), (5) is diﬀerent from (6), suggesting that thepreferences of the agent are time-inconsistent. The time-varying marginal rate of substitution
between consumption and leisure in (5) is the source of time-inconsistency.

3 Generalized Euler Equation in a Monetary Economy
To provide a formal solution of our model, we consider the agent as composed of a sequence ofautonomous decision makers indexed by time t, following Peleg and Yaari (1973) and others.8
We call the decision maker at time t as self t. As in Pollak (1968) and others, we consider
the choices of each self to be the outcome of an intrapersonal game. We follow Barro (1999)
in solving the intrapersonal game. One of the advantages of the Barro style solution is thatit allows us to compare the results of our model with a wide range of models where agents
discount their future utility non-exponentially, including hyperbolic discounting. Hori and Fu7

In a model with a CIA constraint and discrete time, Lucas (1982) makes the same assumption. Walsh (2003,
Chapter 3) discusses the roles of this assumption.
8
See footnote 2.

6

tagami (2017) discuss other advantages of the Barro style solution (see Section II of Hori andFutagami (2017) for more details).
In the following analyses, we specify the instantaneous utility functions as
u1 (cv ) = log cv ,andu2 (1 − lv ) = θ log(1 − lv ),(7)

where θ is a positive constant.
Given future selves’ behaviors and the sequence of {rv , wv }∞v=t , self t chooses ct , lt , and mt ,which are considered to be constant over the infinitesimal short interval [t, t + ∆]. The objectiveof self t is approximated as
∫ t+∆
∫ ∞
∫ ∞
Ut =
z(v, t)dv +z(v, t)dv ≈ [u1 (cv ) + u2 (1 − lv )] ∆ +z(v, t)dv,
(8)
t

t+∆

t+∆

where z(v, t) ≡ u1 (cv )e−ρc (v−t) + u2 (1 − lv )e−ρl (v−t) . This approximation comes from taking e−ρc (v−t)
and e−ρl (v−t) as equal to one in the infinitesimally short interval [t, t + ∆].
Through the choices of ct , mt , and lt , self t influences the behaviors of selves v(≥ t + ∆) byaﬀecting the asset holdings at+∆ . To obtain the optimal behavior of self t, we have to first know
the eﬀects of ct , mt , and lt on at+∆ , and then conjecture the policy functions of selves v(≥ t + ∆).
Budget constraint (3) can be approximated as at+∆ ≈ (1+rt ∆)at +{wt lt −ct −(rt +πt )mt +T t }∆because we can ignore the terms involving ∆2 and consider rt , wt , and πt as constant in theinfinitesimally short time interval [t, t + ∆]. This equation implies that
∂at+∆
= −∆,
∂ct

∂at+∆
= wt ∆,
∂lt

and

∂at+∆
= −(rt + πt )∆.
∂mt

(9)

Next, we turn to the policy functions of self v(≥ t + ∆). Following Barro (1999), we
conjecture the choices of self v(≥ t + ∆) and the path of future consumption as, respectively,
1 − lv = θζv · cv ,
ċv
gcv ≡
= rv − ωv .
cv

(10)
(11)

Here, ζv and ωv will be determined later. As in Barro (1999), we conjecture that ζv and ωv do
not depend on the level of asset holdings and that ζv and ωv vary over time. In addition, we
conjecture that self v(≥ t + ∆) does not hold more cash than needed for purchasing consumption
goods, meaning that (2) holds with equality for all v(≥ t + ∆). We see that our conjecture turns
out to be true.
We(2), (10), and∫ (11), to obtain(µt+∆ + κt+∆ )ct+∆ ∫= at+∆ + Wt+∆∫, where∫v∫v∫ ∞integrate (3) usingv∞∞
gcs −r s )ds(gcs −r s )ds(tt
dv, κt ≡ t θζv ·wv e
dv, and Wt ≡ t {wv +T v }e− t rs ds dv.
µt ≡ t {1+η(rv +πv )}e
Here, we use our conjecture that (2) holds with equality for all v(≥ t + ∆). Since we conjecturethat both ζv and ωv do not depend on at+∆ , at+∆ has no eﬀects on µt+∆ and κt+∆ . We then have
∂ct+∆
1
=
.
∂at+∆ µt+∆ + κt+∆(12)

If we use (10) and (11), the objective function of self t, given by (8), can now be rewritten
as
(
)
[
]
(
)
−ρl ∆
−ρc ∆
+ Φt+∆ ,+ θ log ct+∆ ρ−1
Ut = log ct + θ log(1 − lt ) ∆ + log ct+∆ ρ−1
c e
l e
7

}
)
(
)
∫ ∞ {(∫ v∫vc−ρc (v−t)c−ρl (v−t)where Φt ≡ t
g
ds
e
+
θ
log(θζ
)+gds
e
dv. Given the sequence of
v
s
s
t
t
∞
{rv , wv , pv , πv }v=t , self t maximizes this objective function subject to (9), (12), and mt ≥ ηct .
We set the Lagrangian Lt = Ut + λt (mt − ηct ), where λt is the Lagrangian multiplier. Note thatat+∆ has no eﬀect on Φt+∆ because we conjecture that both ζv and ωv do not depend on at+∆ .
Then, the first-order conditions are given by
(
)
1
1
∂ct+∆
−
X∆
∆ = ηλt ,(13)
ct ct+∆ ∂at+∆
1
θ
∂ct+∆
=
X∆
wt ,
(14)
1 − lt ct+∆ ∂at+∆
1
∂ct+∆
X∆
(rt + πt )∆ = λt ,(15)ct+∆ ∂at+∆
−ρc ∆
−ρl ∆
where X∆ = ρ−1
+ ρ−1
.
c e
l θe
Condition (15) implies that λt > 0, which means that (2) holds with equality for self t. Since
this also applies to selves v(≥ t + ∆), our conjecture that (2) holds with equality for all v(≥ t + ∆)turns out to be true.
From (13) and (15), we have1X∆ ∂ct+∆={1 + η(rt + πt )}.
ct ct+∆ ∂at+∆(16)

Using (10), (14), and (16), we obtainζt =1 + η(rt + πt )
.
wt

(17)

ζt does not depend on at . Our conjecture turns out to be true.
From (12) and (16), we have−1
µt + κt = {1 + η(rt + πt )}(ρ−1c + θρl ),(18)

as ∆ approaches to zero. From the definitions of µt and κt as well as (17), we haveµ̇t = {rt − gct }µt − {1 + η(ru + πu )},κ̇t = {rt − gct }κt − θ{1 + η(ru + πu )}.
By diﬀerentiating both sides of (18) with respect to time and using (11) and the above two
equations, we obtainωt =(1 + θ)ρc ρlη(ṙt + π̇t )+.
ρl + θρc
1 + η(rt + πt )As conjectured, ωt does not depend on the level of asset holdings. The behavior of self t can besummarized asη(ṙt + π̇t )ċt= rt − ρ̃ −,ct
1 + η(rt + πt )
θ{1 + η(rt + πt )}1 − lt =
ct ,
wt

8

(19)
(20)

where ρ̃ ≡ (1 + θ)ρc ρl /(ρl + θρc ).

3.1 Observational Equivalence
Consider the following preference with standard exponential discounting:
∫ +∞
Ut =
{u1 (cv ) + u2 (1 − lv )} e−ρ̃(v−t) dv.

(21)

t

In (21), consumption and leisure are discounted at the same rate. If we maximize the above
utility function subject to (2) and (3), we obtain the same equations as (19) and (20). Thus, thebehavior of agents endowed with (1) is observationally equivalent to that of agents endowedwith (21).

3.2 Ineﬃciency
In this subsection, we show that the intrapersonal game generates ineﬃciency. Consider thecase where self s believes that self t(> s) will obey the decision made at time s, as in Section
2. Then, (5) holds, and can written as
1 − lts =θ{1 + η(rt + πt )} −(ρl −ρc )(t−s) s
e
ct ,
wt

(22)

under (7). We use superscript s to indicate that cts and lts are optimal for self s(< t). In the intrapersonal game, self t(> s) follows (20), which is not optimal for self s. From the perspectiveof self s, it is optimal if self t(> s) follows (22). If ρc , ρl , (20) does not correspond to (22).
Thus, the intrapersonal game generates ineﬃciency.9
Note that the disparity between (20) and (22) comes from the term e−(ρl −ρc )(t−s) . This term,
derived from the left-hand side of (5), is the source of time-inconsistency. Thus, time-inconsistency
causes intratemporal ineﬃciency in our model.
To interpret the intratemporal ineﬃciency, we compare (20) with (22). If ρc > (<)ρl holds,
we have ct /(1−lt ) > (<)cts /(1−lts ) for t > s. This equation shows that in the intrapersonal game,self t(> s) consumes more (less) than that self s prefers, given lt . The intuition is as follows.
If ρc > (<)ρl , today’s self cares less (more) about consumption of future selves than leisure of
future selves. However, future selves care about their own consumption as much as they doabout their own leisure. Thus, given lt , future selves consume more (less) than that today’s self
prefers.

4 Optimal Monetary Policy
This section examines the eﬀect of monetary policy on ineﬃciency that is observed in Section
3.2. To characterize an optimal policy, we consider a simple general equilibrium that ensuresuniqueness of the equilibrium and simplifies the expression of welfare.
The population size is normalized to one. The production technology is Yt = Alt , where Yt is
the output level, lt is the labor input, and A is a positive constant. Through profit maximization,
the wage rate wt becomes equal to A. Because there is no capital, we have at = mt . We focus
9

Hori and Futagami (2017) show that the intrapersonal game generates two types of ineﬃciencies, intratemporal and intertemporal. The present study focuses on intratemporal ineﬃciency.

9

on the steady-state equilibrium, where ċt = ṙt = π̇t = 0 holds. Equation (19) implies that rt = ρ̃.
Because mt = ηct implies that ċt /ct = ϵ − πt , the inflation rate πt is equal to ϵ. Thus, the nominalinterest rate is ρ̃ + ϵ. To ensure ρ̃ + ϵ ≥ 0, we assume that ϵ ≥ −ρ̃. From (20), wt = A, rt = ρ̃,πt = ϵ, and ct = Alt , we obtainc∗ =A
θ{1 + η(ρ̃ + ϵ)}
θ{1 + η(ρ̃ + ϵ)} ∗and 1 − l∗ ==c.
1 + θ{1 + η(ρ̃ + ϵ)}
1 + θ{1 + η(ρ̃ + ϵ)}A(23)

We use asterisks to denote equilibrium variables. Since the above two equations determine c∗and l∗ uniquely, the equilibrium is also unique.
By diﬀerentiating the two equations of (23) with respect to ϵ, we obtainθηc∗∂(1 − l∗ )
1 ∂c∗
∂c∗
=−
< 0 and=−> 0.
∂ϵ
1 + θ{1 + η(ρ̃ + ϵ)}
∂ϵ
A ∂ϵ

(24)

A higher inflation reduces the value of real money and hence depresses consumption. It also
decreases the cost of leisure in terms of consumption good, w/[1+θ{1+η(ρ̃+ϵ)}]. Thus, leisureincreases with inflation. We also have ∂c∗ /∂η < 0 and ∂(1 − l∗ )/∂η > 0. As the financial market
develops, the CIA constraint loosens, encouraging consumption and discouraging leisure.
We now discuss our welfare measure. Because of time-inconsistency, diﬀerent selves ofan agent need not agree on the welfare ranking of the same consumption and labor supply
sequences. Nevertheless, many studies such as Laibson (1996, 1997) and Laibson et al. (1998)
compare welfare from the perspective of all selves. Note that in our model, the utility levelsof all selves can be given by U ∗ = (ln c∗ )/ρc + {θ ln(1 − l∗ )}/ρl . Thus, we can easily evaluate
welfare from the perspective of all selves.
To derive the inflation rate ϵ that maximizes U ∗ (henceforth, ϵ ∗ ), we diﬀerentiate U ∗ withrespect to ϵ using (23).
{
}
1
∂U ∗
θη
1
=
−
.
(25)
∂ϵ
1 + θ{1 + η(ρ̃ + ϵ)} ρl {1 + η(ρ̃ + ϵ)} ρc
By examining the sign of ∂U ∗ /∂ϵ, we obtain the next proposition.
Proposition 1
{
∗

ϵ =

−ρ̃,
(
1
η

ρc
−1
ρl

)

if ρc ≤ ρl ,− ρ̃(> −ρ̃), if ρc > ρl ,Note that the real interest rate r is equal to ρ̃. When ϵ ∗ = −ρ̃, the nominal interest rateis equal to zero. If ρc ≤ ρl , a zero nominal interest rate is optimal. Thus, the Friedman rule
is optimal. Note that this case includes the standard exponential discounting model where
consumption and leisure are discounted at the same rate. When ρc > ρl , ϵ ∗ is larger than −ρ̃.
Here, the Friedman rule is not optimal.
To interpret Proposition 1, we approximate the utility of self s as
∫ ∞[]U s ≈ [u1 (c s ) + u2 (1 − l s )] ∆ +u1 (ct )e−ρc (t−s) + u2 (1 − lt )e−ρl (t−s) dt}{ s+∆
1 − ρc ∆1 − ρl ∆∗∗∗∗u1 (c ) +u2 (1 − l ) ,(26)≈ [u1 (c ) + u2 (1 − l )] ∆ +
ρc
ρl
10

where ∆ > 0 is infinitesimally small. The second line uses e−ρx ∆ ≈ 1 − ρ x ∆ (x = c or l). The
first term in the second line represents the utility of self s from self s’s own consumption andleisure at time s. We diﬀerentiate the first term with respect to ϵ:()∂ first term
1
1 ∂c∗
= 1−
.
∂ϵ
1 + η(ρ̃ + ϵ) c∗ ∂ϵThis equation shows that the first term is maximized if ϵ = −ρ, because ∂c∗ /∂ϵ < 0 (see (24)).
A higher inflation has a negative eﬀect on self s’s welfare. This is a standard result. Since selfs can control her own behavior, distorting self s’s own behavior deteriorates self s’s welfare.
The second term in the second line of (26) represents self s’s utility from the consumption
and leisure of future selves. By diﬀerentiating this term with respect to ϵ, we obtain()
1∂ second term
1 − ρc ∆ 1 − ρl ∆
1 ∂c∗
−
=
.
∂ϵ
ρc
ρl 1 + η(ρ̃ + ϵ) c∗ ∂ϵBecause of ∂c∗ /∂ϵ < 0, the second term is maximized if{−ρ̃,if ρc ≤ ρl ,)ϵ = 1 ( ρc 1−ρl ∆− 1 − ρ̃(> −ρ̃), if ρc > ρl ,η ρl 1−ρc ∆When ρc < ρl holds, future selves consume less than that today’s self prefers, as discussed in
Subsection 3.2. An increase in ϵ has a negative eﬀect on the consumption of future selves.
Thus, by keeping ϵ as low as possible (ϵ = −ρ), we can maximize the second term. In contrast
with this case, when ρc > ρl holds, future selves consume more than that today’s self prefers.
Since an increase in ϵ decreases the consumption of future selves, it has a positive eﬀect on thesecond term.
To maximize U s , the eﬀects on the first and second terms must be balanced. Thus, we
obtain Proposition 1.
We can examine how development of the financial market aﬀects the optimal inflation rate.
If ρc > ρl , we have ∂ϵ ∗ /∂η < 0 from Proposition 1. Thus, we obtain the following corollary.
CorollarySuppose that ρc > ρl . Then, as the financial market develops, the optimal inflation rate ϵ ∗increases.
The intuition of Corollary is simple. When ρc > ρl holds, future selves consume more thanthat today’s self prefers. Development of the financial market increases the consumption of
future selves further. Thus, to maximize the welfare of self s, the consumption of future selves
must be depressed through higher inflation. The optimal inflation rate ϵ ∗ increases with thedevelopment of the financial market.
We finally examine the eﬀect of development of the financial market (decreases in η) by
keeping ϵ constant at some level. Given ϵ(> −ρ̃), we diﬀerentiate U ∗ with respect to η by using
(23).
{
}
θ(ρ̃ + ϵ)
1
1
∂U ∗
=
−
.
(27)
∂η
1 + θ{1 + η(ρ̃ + ϵ)} ρl {1 + η(ρ̃ + ϵ)} ρc
When ρc ≤ ρl , ∂U ∗ /∂η has a negative sign. On the other hand, when ρc > ρl , ∂U ∗ /∂η has apositive (negative) sign if and only if η < (>)(ρc − ρl )/{ρl (ρ̃ + ϵ)} ≡ η. Thus, we obtain the11following proposition:
Proposition 2
As the financial market develops (as η decreases),
1. when ρc ≤ ρl holds, the utility levels of all selves increase, and
2. when ρc > ρl holds, the utility levels of all selves increase if the financial market is lessdeveloped (η > η) and decrease if the financial market is well developed (η < η).
The intuition of Proposition 2 is similar to that of Proposition 1. As η decreases, the consumption of today’s self increases, which has a positive eﬀect on U ∗ . When ρc > ρl holds, future
selves consume more than that today’s self prefers. Thus, a decrease in η has a negative eﬀect
on U ∗ because it increases future selves’ consumption. In an economy with a less-developed(well-developed) financial market, the positive (negative) eﬀect dominates the negative (positive) eﬀect. Development of the financial market improves (deteriorates) the utility levels of allselves.
When ρc < ρl holds, future selves consume less than that today’s self prefers. Thus, adecrease in η has a positive eﬀect on U ∗ because it increases future selves’ consumption. Thus,
development of the financial market improves the utility levels of all selves.
When ρc = ρl holds, there is no intratemporal ineﬃciency. Development of the financial
market loosens the CIA constraint, which always has a positive welfare eﬀect.

5 A Discount Rate that Depends on Time DistanceTo emphasize the importance of Propositions 1 and 2, this section derives the optimal policy ina model with a discount rate that varies with time distance, as in the model of Barro (1999).
Instead of (1), this section assumes that the representative agent has the following utility
function:
∫ +∞
Ut =
(28)
{u1 (cv ) + u2 (1 − lv )} e−(ϱ·(v−t)+ϕ(v−t)) dv,t

where u1 (c) = ln c, u2 (1 − l) = θ ln(1 − l), ϱ is a positive constant, and ϕ(v − t) is a function oftime distance v−t. Following Barro (1999), we assume that ϕ(0) = 0, ϕ′ (v−t) ≥ 0, ϕ′′ (v−t) ≤ 0,
and limv−t→∞ ϕ′ (v − t) = 0. In (28), the instantaneous discount rate ϱ + ϕ′ (v − t) varies with time
distance v−t. Since we do not specify the functional form of ϕ(v−t), the following result applies
to a wide range of time-variable discount functions, including hyperbolic discount functions.10
This generality is an advantage of the Barro-style model.
It is well known that when the discount rate varies with time distance as in (28), the problemof time-inconsistency arises. As in Section 3, we consider the agent as composed of a sequenceof autonomous decision makers. The agent is faced with the budget constraint (3) and the CIA
constraint (2). By following the same procedure of Section 3, we derive the behavior of self t

A hyperbolic discount function is given by 1/{1 + f · (v − t))}, where f > 0 is a constant parameter. Thus,
when we define ϕ(v − t), which satisfies e−ϱ·(v−t)+ϕ(v−t) = 1/{1 + f · (v − t))}, we obtain a hyperbolic discounting
function.
10

12

as
ċt
η(ṙt + π̇t )= rt − Ω−1 −,ct
1 + η(rt + πt )
θ{1 + η(rt + πt )}1 − lt =
ct ,
wt

(29)
(30)

∫∞
where Ω ≡ 0 exp {−(ϱt + ϕ(t))} dt > 0 is a positive constant. See the appendix for thederivations of (29) and (30). Equation (30) is the same as (20). If Ω−1 is equal to ρ̃(≡
(1 + θ)ρc ρl /(ρl + θρc )), (29) is exactly the same as (19). The model with a discount rate varying
with time distance is observationally equivalent to the non-unitary discount rate model. Furthermore, from the discussion in Subsection 3.1, the behavior of agents with (1) is observationallyequivalent to those of agents with (21) or (28).
As in Section 4, we focus on the steady-state equilibrium, where ċt = ṙt = π̇t = 0 holds. The
production technology is again given by Yt = Alt . Because of the observational equivalence, theequilibrium consumption level and labor supply are again given by the two equations of (23).
In equilibrium, all selves have the same utility level, UΩ∗ ≡ [log(c∗ ) + θ log(1 − l∗ )]Ω, where c∗and l∗ are given by (23).
To derive the optimal ϵ, we diﬀerentiate UΩ∗ with respect to ϵ as follows:
{
}
∂UΩ∗
1
θη
=
− 1 Ω.
(31)∂ϵ
1 + θ{1 + η(ρ̃ + ϵ)} 1 + η(ρ̃ + ϵ)
The above equation implies that ϵ = −ρ̃ is optimal. In spite of time-inconsistency, the Friedman
rule is optimal in the model with a discount rate that varies with time distance. This result is
sharp contrast with Proposition 1. We next diﬀerentiate UΩ∗ with respect to η:{}∂UΩ∗θ(ρ̃ + ϵ)1=− 1 Ω < 0.
(32)
∂η
1 + θ{1 + η(ρ̃ + ϵ)} 1 + η(ρ̃ + ϵ)
In contrast to the non-unitary discount rate case, development of the financial market always
improves welfare.
Why do we obtain the results that are quite diﬀerent from Propositions 1 and 2? The answer
to this question is simple. If agents are endowed with (28) (or (21)), the same discount function(rate) applies to both consumption and leisure. The ineﬃciency discussed in Subsection 3.2
does not exist under (28) (or (21)). Thus, the Friedman rule is optimal and development offinancial market always improves welfare.
As mentioned above, the behavior of agents with (1) is observationally equivalent to thatof agents with (21) or (28). Nevertheless, the three types of preferences derive quite diﬀerent
optimal levels of inflation and financial development.

6 Conclusion
Motivated by recent evidences from psychology and behavioral economics, this study provides
a simple model where agents discount the utility from consumption at a rate diﬀerent from thatfor the utility of leisure. We show that in our non-unitary discount rate model, the preferences of
agents are time-inconsistent. We examine the optimal monetary policy. If a higher discount rate
is applied for consumption than for labor, the Friedman rule is no longer optimal. A strictly13positive nominal interest rate improves social welfare. In addition, the optimal inflation rateincreases as the financial market develops.
We also consider (i) a model with standard exponential discounting and (ii) a model witha discount rate that varies with time distance. Both cases assume that the same discounting
applies to both consumption and leisure. The behavior of agents in our non-unitary discountrate model is observationally equivalent to that of agents of the two models. Nevertheless, theoptimal monetary policies in the three models are quite diﬀerent. In the standard exponential discounting model and the model with a discount rate that varies with time distance, theFriedman rule is always optimal.

Appendix
Using the same procedure as in Section 3, we derive (29) and (30). Again, the eﬀects of ct , lt ,and mt on at+∆ are given by the three equations of (9). As in Section 3, the choices of selvesv(≥ t + ∆) and the path of future consumption are conjectured as
1 − lv = θχv · cv

and

gcv ≡

ċv
= rv − ξt .
cv

As in Section 3, we conjecture that χv and ξ do not depend on the level of asset holdings. The
eﬀects of at+∆ on ct+∆ are given by
∂ct+∆
1
=
,
∂at+∆ µt+∆ + κt+∆
∫v c
∫v c
∫∞
∫∞
where µt ≡ t {1 + η(rv + πv }e t {gs −rs }ds dv and κt ≡ t θχv · wv e t {gs −rs }ds dv.
The objective function of self t is given by
Ut = [ln ct + θ ln(1 − lt )] ∆ + (1 + θ)e−(ϱ·∆+ϕ(∆)) Ωt+∆ ln ct+∆ + Γt+∆ ,
}
∫v
∫∞{
∫∞
where Ωt ≡ t exp {−[ϱ · (v − t) + ϕ(v − t)]} dv and Γt ≡ t (1 + θ) t gcs ds + θ log(θχv ) e−(ϱ·(v−t)+ϕ(v−t)) dv.
Given the sequence of {rv , wv }∞maximize this objective funcv=t , self t chooses ct and lt so as∫ to∞tion. Note that Ωt is constant over time because we have Ωt = t exp {−[ϱ · (v − t) + ϕ(v − t)]} dv =∫∞exp {−[ϱ · s + ϕ(s)]} ds = Ω0 . We denote this constant level as Ω.
0Using the first-order conditions and limiting ∆ to zero, we obtain χt = [1 + η(rt + πt )]/wt
and ξ = 1/Ω + η(ṙt + π̇t )/{1 + η(rt + πt )}. Then, we derive (29) and (30).

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