GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.163

A Non-Unitary Discount Rate Model

Koichi Futagami and Takeo Hori

October 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

A Non-Unitary Discount Rate Model ∗Koichi Futagami† and Takeo Hori‡§

Abstract
The standard economic model of intertemporal decision making assumes that a single
discount rate applies equally to discount (dis)utility from all different sources. However, studies such as psychology and behavioral economics have provided evidence
that people might discount (dis)utility from different sources at different rates. This
paper develops a simple model where the agent discounts utility from consumption at
a different rate from disutility of labor supply. We show that in our non-unitary discount rate model, the preferences of the agent are time-inconsistent. The source ofthe time inconsistency is the difference between relative impatience with consumption
and labor supply. It is shown that the policy effects in our model are quite different
from those in the standard model. For example, when the agent discounts utility fromconsumption at a higher rate than the disutility of labor supply, the Friedman rule (thezero nominal interest rate) is no longer optimal. We also make comparisons between
our results and those obtained in a model with a time variable discount rate where thepreferences are time-inconsistent. It is also shown that the policy effects in our model
are quite different from those in a model with a time variable discount rate.
Keywords: Non unitary discount rate, Tax policies, Time-inconsistency, Friedman
rule.
JEL Classification Numbers:D9, E52, H21∗

The authors are grateful to Nobuo Akai, Kenn Ariga, Kyoji Fukao, Mototsugu Fukushige, Ken-ichi
Hashimoto, Shinsuke Ikeda, Ryoichi Imai, Shingo Ishiguro, Tatsuro Iwaisako, Yasushi Iwamoto, Kohei Kubota, Noritaka Kudo, Kazuo Mino, Akira Momota, Shoko Morimoto, Makoto Nirei, Makoto Saito, Etsuro
Shioji, Katsuya Takii, Takayuki Tsuruga, Takashi Unayama and seminar participants at Hitotsubashi University, Kobe University, Kyoto University, Kyusyu University, Osaka University and SWET for helpful comments. This paper is partially supported by the Japan Society for the Promotion of Science (JSPS) through a
Grant-in-Aid for Young Scientists (B) (No. 21800035).
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0034,
Japan. E-mail: futagami@econ.osaka-u.ac.jp‡
Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, 186-8601, Japan.
E-mail: t-hori@econ.hit-u.ac.jp§
Corresponding Author11 Introduction
Until the early 20th century, economists had been greatly concerned with various kindsof psychological and sociological motives that could determine intertemporal choices suchas consumption and saving decisions. Intertemporal choices had been interpreted as thecomposite of many conflicting psychological and sociological motives, such as the bequest
motive and temptations to consume too much today.1 When in 1937 Samuelson proposed
the discounted utility (henceforth, DU) model which was currently accepted as a standard
model, however, many of the concerns about intertemporal choices that had been discussed
until then were summarized by and compressed into a parameter, the discount rate.2, 3 In the
DU model proposed by Samuelson (1937), the intertemporal utility of an agent at time t who
Rs
lives to time s(> t) without uncertainty is specified as Ut = t u(cv )e−ρ(v−t) dv where u(cv ) is
the instantaneous utility from time-v consumption cv and ρ is the subjective discount rate.
We can extend the DU model to cases where an agent derives her utility from more than two
different sources, like consumption and leisure. In such cases, the DU model assumes thata single discount rate is used commonly to discount (dis)utility from all different sources.
However, if each different source of (dis)utility is associated with a particular motiveof intertemporal choice and hence people discount (dis)utility from different sources atdifferent rates, the notion of a unitary discount rate is nonsense. Frederick et al. (2002)
criticize the unitary discount rate assumption of the DU model, by arguing:
When one looks at the behavior of a single individual across different domains, there
is often a wide range of apparent attitudes toward the future. Someone may smoke
heavily, but carefully study the returns of various retirement packages. Another may
squirrel money away while, at the same time, giving little thought to electrical efficiency when purchasing an air conditioner. Someone else may devote two decades of
his life to establishing a career, and then jeopardize this long term investment for some1

The early views of economists about intertemporal choices are well documented by Frederick et al.
(2002).
2
Frederick et al. (2002) provide an excellent review of the historical development of the DU model. The
DU model has been widely accepted until now despite Samuelson’s reservations about its validity.
3
The other factors such as the curvature of the instantaneous utility function also affect intertemporalchoices.

2highly transient pleasure. (Frederick et al. (2002), p.393)

These behaviors of a single person cannot be explained if a single discount rate applies todiscount (dis)utility from all different sources. A person who smokes heavily may discountthe disutility of having poor health in the future at a higher rate. At the same time, hercareful studying the returns of various retirement packages implies that she may discountutility from consumption after retirement at a much lower rate. In fact, there is evidence
that people might discount (dis)utility from different sources at different rates. In Section
2, we present such evidence. Frederick et al. (2002) continue as follows:
Since the DU model assumes a unitary discount rate that applies to all acts of consumption, such intra-individual heterogeneities pose a theoretical challenge. (Frederick etal. (2002), p.394)

Motivated by the above arguments, we present a simple model where a person discounts(dis)utility from different sources at different rates exponentially.
More precisely, we assume that the agent discounts utility from consumption at a different rate from the disutility of supplying labor. When the discount rate for utility fromconsumption is equal to that for the disutility of labor supply, our model reduces to a standard DU model. Therefore, we can easily compare the results obtained in our non-unitary
discount rate model, where people use different discount rates to discount (dis)utility fromdifferent sources, with the results obtained in the standard DU model where people use asingle discount rate to discount (dis)utility from all different sources.
We first show that in our non-unitary discount rate model, the marginal rate of substitution between consumption and labor supply is no longer time-invariant, and hence there
emerges time inconsistency concerning the preferences of agents. When the agent discounts
utility from consumption at a higher (lower) rate than the disutility of labor supply, she attempts to consume more (less) today and supply a larger (smaller) amount of labor today
than she planned in the past.
Studies in behavioral economics suggest that the assumption of time consistency inthe standard DU model is incorrect.4 Authors such as Strotz (1955) and Laibson (1996,
4

As to recent advances in behavioral economics, see Rabin (1998, 2002), Frederick et al. (2002) and3

1997, 1998) show that the problem of time inconsistency emerges if individuals discountfuture utility with a time variable discount function, especially with the “quasi-hyperbolic”
discount function.5 In a model of hyperbolic discounting, as pointed out by O’Donoghue
and Rabin (1999), the agent attempts to experience pleasant things immediately and to
procrastinate regarding unpleasant things. This present-biased preference is the source oftime inconsistency in a model of hyperbolic discounting.
In our non-unitary discount rate model, the difference between the patience with consumption and labor supply is the source of time inconsistency. Because our model assumes
positive discount rates for both consumption and labor supply, the agent is willing to consume much today and to procrastinate regarding supplying labor today. If the discount rate
for consumption is higher than for labor supply, however, the agent tends to be more willing to consume much today than to procrastinate regarding supplying labor today because
she is relatively more impatient with decreases in consumption today than with increases
in labor supply today. This difference in the patience is the source of time inconsistency in
our model.
However, we do not claim that our model substitutes for models of hyperbolic discounting. The hyperbolic discount function is given by v p = V/(1 + kt) where v p is the present(discounted) value of an undiscounted value V, t represents the time distance and k(> 0)
is a constant parameter representing the degree of discounting. As we will see in Section
2, some studies suggest that people use different hyperbolic discount functions (or different
values of k) to discount (dis)utility from different sources. To isolate the roles of differencesof discount functions from the roles of the hyperbolic discount function, we use exponential discount functions in this paper. If we use the hyperbolic discount function, our model
corresponds to the case where people use different values of k to discount (dis)utility fromdifferent sources.
To solve our non-unitary discount rate model formally, we consider the agent as composed of a sequence of autonomous decision makers as in many previous studies.6 We callPesendorfer (2006), for example.
5The “quasi-hyperbolic” discount function used in Laibson (1996, 1997, 1998) and other studies is introduced by Phelps and Pollak (1968) in a model of imperfect intergenerational altruism.
6
See Peleg and Yaari (1973), Goldman (1980), Harris and Laibson (2001) and Luttmer and Mariotti

4the decision maker at time t self t. Then we consider the choices of each decision maker(self) to be the outcome of an intrapersonal game. We show that in our non-unitary discountrate model, the consumption-saving behavior of the agent is affected by consumption taxes
that have no effect on the consumption-saving behavior in the standard DU model.
To examine the welfare effects of taxes, we consider a simple general equilibrium wherelabor is used as the only input in production. We evaluate welfare from the perspective of
all selves and derive policies that maximize the utility levels of all selves. In the standard
DU model, the zero consumption tax rate is optimal. In our non-unitary discount rate
model, however, it is shown that the utility levels of all selves can be improved by a strictly
positive consumption tax (a consumption subsidy) when the agent discounts the utility fromconsumption at a higher (lower) rate than the disutility of labor supply. Furthermore, byintroducing money under the assumption that a fraction of consumption goods must be
financed by cash, we then show that when the agent discounts the utility from consumption
at a higher rate than the disutility of labor supply, the Friedman rule is no longer optimaland development of the financial market (decreases in the fraction of consumption goods
that must be financed by cash) deteriorates the utility levels of all selves.
Laibson (1996, 1997) also provides welfare implications similar to our results in amodel with a time variable discount rate where the problem of time inconsistency exists.
For example, Laibson (1997) shows that development of the financial market may deteriorate welfare. However, his analysis is based on a partial equilibrium model. To emphasize
the importance of our results, we also conduct welfare analysis in a general equilibrium
model where the agent uses a time variable discount function that is applied equally to consumption and labor supply. We show that in the general equilibrium model with a time
variable discount rate, the zero consumption tax rate is optimal although the problem oftime inconsistency exists. This result suggests that the strictly nonzero optimal consumption tax is not a common feature of general equilibrium models where the problem of thetime inconsistency arises. Our results suggests that when the problem of time inconsistency exists in the economy, the optimal policy might be influenced by the sources of time(2003), for example.

5

inconsistency.
The rest of the paper is organized as follows. Section 2 provides evidence that people might discount (dis)utility from different sources at different rates. In Section 3, we
present our non-unitary discount rate model and show how the problem of time inconsistency emerges. Section 4 derives the solution of the intrapersonal game. By considering asimple general equilibrium model, Section 5 examines the effects of taxes on consumptionsaving behavior and utility levels. Section 6 extends our model by introducing money.
Section 7 compares the results obtained in our non-unitary discount rate model with thoseobtained in a model with a time variable discount rate. Concluding remarks are in Section
8.

2 Empirical and Experimental Evidence
As discussed in Introduction, the DU model assumes that a single discount rate applies
equally to all types of goods and all categories of intertemporal choices. The studies such as
psychology and behavioral economics report some empirical experimental observations thatappear to contradict this assumption. Of such observations, we mention the sign effect, themagnitude effect, and the domain effect (or domain independence). What is most relevant
to our model is the domain effect.
The sign effect refers to the finding that gains are discounted at a higher rate than losses.
Loewenstein (1987) asked 30 undergraduates to determine how much you would pay most
now to obtain (avoid losing) four dollars in the five different time delays. He found thaton average, obtaining four dollars was discounted at higher rates than losing four dollars.
Other authors, such as Thaler (1981), Benzion et al. (1989) and Abdellaoui et al. (2009)
also found the sign effects.
Many studies have found that discount rates decrease with magnitudes of outcomes.
More concretely, receiving $1 million is discounted at lower rates than receiving $100.
This is often referred to as the magnitude effect. Many studies have found the magnitude
effects.7
7

see Thaler (1981), Benzion et al. (1989), Raineri and Rachlin (1993), Green, Fristoe, and Myerson

6Many of previous studies were concerned with discount rates related to monetary outcomes. However, recent research has started to study discount rate of non-monetary outcomes. For example, Chapman and her co-authors have studied discount rates for money
and health in a series of articles. Chapman (1996) conducted three experiments and foundthe low correlation between health and money discount rates, which suggested that a person
who exhibited a high discount rate for money did not necessarily exhibit a high discountrate for health. She interpreted her result as showing that contrary to the DU model, people
used the different discount rates for the two domains, money and health. Chapman andElstein (1995) and Chapman et al. (1999) also reported the similar results. The finding thatthe discount rates differ for different domains is referred to as the domain effect (or domain
independence). By using a sample of law students, Lazoro et al. (2001) found that the students did not apply the same discount rate for their choices about money and health. Baker
et al. (2003) showed that both the current and never-before smokers discounted monetaryloss at a higher rate than health losses.
The observations of the domain effects are not confined to money and health. Fuchs(1982) finds no correlations between a standard measure of time discounting (“Would you
choose $1,500 now or $4,000 in five years?”) and other behaviors that one might plausibly
expect to be affected by time discounting (credit card debit, cigarette smoking, and thefrequency of exercise and dental checkups). By using a sample of psychology students who
had previous work experience and were seeking post-graduation jobs, Schoenfelder and
Hantula (2003) found that students in their study used different discount rates to discountfuture salary outcomes and future access to attractive job duties. Loewenstein (1987) foundthat disutility from receiving electric shocks might be negatively discounted while receiving
an amount of money was positively discounted.
Leclerc (1995) showed that money and time/effort were treated differently in decision
making. The domain effect was observed for money and time/effort. Soman (1998) studied
a monetary reward (R) and a loss of time/effort (E). In his experiments, subjects had tochoose whether or not to enter a transaction where they would receive R just after comple-(1994), Myerson and Green (1995), Green, Myerson and McFadden (1997) and Kirby (1997), for example.

7

tion of E. Both R and E would occur at the same time in the future. Subjects who chose toenter this transaction might have evaluated the discounted value of R much more than the
discounted value of E. Soman (1998) observed that many of subjects who chose to enter
the transaction did not actually redeem the required effort E and could not get R. This suggested that after they had decided to enter the transaction, they might have changed theirevaluation and then evaluated R less than the costs of E.
If both R and E are discounted at the same rate (or by the same discount function), it isdifficult to explain the above observation. Consider a person who discounts R (E) by using
a discount function DR (t) > 0 (DE (t) > 0).8 Both R and E will occur after t periods of time.
If she evaluates RDR (t) more than EDE (t), which implies RDR (t) > EDE (t), she chooses
to enter the transaction. When DR (t) is equal to DE (t) for all t ≥ 0, her decision to enter
the transaction apparently implies R > E. This means that she actually redeems E and can
get R. If DR (t) is not equal to DE (t), however, the inequality RDR (t) > EDE (t) does notnecessarily imply R > E. Therefore, she might not redeem E. Soman (1998) interpreted his
results as showing future time/effort was discounted at different speeds from future money.
More specifically, the k parameter of the hyperbolic discount function for effort was foundto be different from that for money.9 Soman (2004) and Zauberman and Lynch (2005) also
showed that people used different discount rates to discount future time and future money.
The final evidence we provide suggests that people might use different discount rates
to discount money- and labor-related (dis)utility. Table 1 is based on micro data from
“Preference and Life Satisfaction Survey” (see Appendix A for details of this survey). Table
1 (a) shows that in the United States, of 6202 respondents in this survey who discountmoney-related utility at positive rates, about 70% of them (4317 respondents) use negative
discount rates to discount the disutility of labor supply. Table 1 (b) provides similar results
for Japan.
The above evidence raises doubts over the assumption of the DU model that a single
discount rate applies equally to discount (dis)utility from all different sources. In the next
section, we provide a model where agents use different discount rates to discount (dis)utility
89

DR (t) (DE (t)) is a decreasing function of t.
Also, see Soman et al. (2005).

8

from different sources.
[Table 1]

3 The Model
We consider an infinitely-lived agent who is endowed with one unit of time that is allocatedto labor or leisure at each moment of time. The preferences of the agent are given by:
Z +∞ no
Ut =u(cv )e−ρc (v−t) − v(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. u(cv ) and v(lv ) represent the instantaneous utility derived from consumption and the instantaneous disutility of labor at time v, respectively. The functions, u(·) andv(·), are twice differentiable and satisfy u0 (·) > 0, u00 (·) < 0, v0 (·) > 0 and v00 (·) > 0. The
parameters ρc and ρl are the subjective discount rates for consumption and labor supply,
respectively. We assume ρc > 0 and ρl > 0 so that we obtain bounded utility (1), althoughsome authors observe negative discount rates as discussed in Introduction. We allow thecase where ρc is not equal to ρl , which means that the agent discounts utility from different
sources at different rates. When ρc is (not) equal to ρl , we call a (non-)unitary discountrate case. When ρc is larger (smaller) than ρl , if the importance that the agent puts on
consumption at different times is compared with that of the disutility of labor at different
times, the agent puts relatively greater (lesser) importance on consumption today than on
future consumption, while the disutility from future labor supply is relatively more (less)important for her than the disutility from labor supply today. In other words, the agent isrelatively more (less) impatient with decreases in consumption today than with increases inlabor supply today.
As discussed in Sections 1 and 2, some studies suggest that people use different hyperbolic discount functions (or different k parameters) to discount (dis)utility from different
sources. However, it should be noted that we use exponential discount functions in (1) to

9isolate the effects of differences of discount functions from those of the hyperbolic discountfunction.
The budget constraint of the agent is given by:ȧv = (1 − τr )rv av + (1 − τw )wv lv − (1 + τc )cv + T v ,(2)

where av denotes the asset holdings at time v and rv (wv ) is the interest rate (the wage rate).
τr , τw , and τc are the interest income tax rate, the labor income tax rate, and the consumption
tax rate, respectively. τr , τw and τc are all assumed to be constant over time. The lump-sumtransfer from the government is denoted by T v . The budget of the government is balanced
at any moment, τr rv av + τw wv lv + τc cv = T v .

3.1

Non-Unitary Discount Rate and Time Inconsistency

This subsection demonstrates that the problem of time inconsistency arises under preferences with non-unitary discount rates, by focusing on the case where τr , τw , τc and T v are
all equal to zero. Before providing a formal solution in the next section, we consider thecase where at time t, the agent chooses the sequence {cv , lv , av }∞v=t without considering thepossibility that she reconsiders her choices at some future time. In other words, when shechooses the sequence {cv , lv , av }∞v=t at time t, she believes that at time v(> t), she will obey
the decision made at time t.
We maximize (1) subject to (2) by setting the present value Hamiltonian as follows:
Hv = u(cv )e−ρc (v−t) − v(lv )e−ρl (v−t) + λv (rv av + wv lv − cv ),where λv is the costate variable associated with the asset holdings and v is larger than t.
From the first-order conditions, we obtain:
v0 (lv ) −(ρl −ρc )(v−t)
e= wv .
u0 (cv )(3)

At time t, the agent plans to consume goods and supply labor according to (3) at time v(> t).
10

If she maximizes her utility once again at time v(> t), however, we obtain:
v0 (lv )= wv .
u0 (cv )(4)

In the unitary discount rate case (ρc = ρl ), (3) is identical to (4). The decision made at timev is consistent with that made at time t(< v). In the non-unitary discount rate case (ρc , ρl ),however, (3) is different from (4). The decisions at different dates are inconsistent. Notethat as shown in the left-hand side of (3), in the non-unitary discount rate case, the marginal
rate of substitution between consumption and labor supply is no longer time-invariant. The
preferences of the agent are time-inconsistent.
[Figure 1]
Figure 1 shows graphically the time inconsistency of the non-unitary discount rate case.
Figure 1 ignores asset holdings for expositional simplicity. In the next section, we present
a formal solution of our model by considering asset holdings. The straight line represents
the budget constraint, cv = wv lv .10 The curved lines are the indifference curves. Panel(a) shows the case where ρc is larger than ρl . When the agent maximizes her utility attime t, the slope of the indifference curve for the time-v(> t) instantaneous utility is given
by v0 (lv )e−(ρl −ρc )(v−t) /u0 (cv ). At point A, (3) holds. At time t, the agent plans to consume
goods and supply labor at point A in a future time v(> t). When the agent maximizes
her utility again at time v, however, the slope of the indifference curve of the time-v instantaneous utility is given by v0 (lv )/u0 (cv ). At point A, v0 (lv )/u0 (cv ) becomes smaller thanv0 (lv )e−(ρl −ρc )(v−t) /u0 (cv ) because ρc is larger than ρl . At time v(> t), the agent wants to consume goods and supply labor at point B where (4) holds, rather than to obey the plan madeat time t (point A). The agent likes to consume more and supply more labor at time v thanshe planned in a past time t(< v). The intuition is as follows: The inequality ρc > ρl suggests that the agent is relatively more impatient with decreases in consumption today thanincreases in labor supply today. Therefore, the agent attempts to consume much today and
10

Please note that we ignore asset holdings for expositional simplicity. Therefore, the budged constraint isgiven by cv = wv lv .

11cares less about the disutility of labor today. At each point of time, therefore, the agent
attempts to consume more today and supply more labor today than she planned in the past.
Panel (b) in Figure 1 presents the case where ρc is smaller than ρl . In this case, theagent is relatively more patient with decreases in consumption today than increases in labor
supply today. The agent cares relatively less about decreases in consumption today andattempts to procrastinate about labor supply today. At each point of time, therefore, theagent attempts to consume less today and supply less labor today (point D) than she plannedin the past (point C).

3.2

Comparison with a Time Variable Discount Rate Model

This subsection observes that the source of the time inconsistency in our non-unitary discount rate model is quite different from that of a model with a time variable discount rate.
Consider the following utility function:
Z +∞
Ut =

{u(cv ) − v(lv )} e−(%·(v−t)+φ(v−t)) dv,(5)t

where % is a positive constant and φ(t) is a function of t. Following Barro (1999), we assumeφ(0) = 0, φ0 (t) ≥ 0, φ00 (t) ≤ 0 and limt→∞ φ0 (t) = 0. In (5), the instantaneous discount rate,% + φ0 (t), varies with time, and the same instantaneous discount rate applies equally to
consumption and labor supply. If v(lv ) is equal to zero for all lv ∈ [0, 1], (5) is equivalent
to the utility function analyzed by Barro (1999). It is well known that when the discountrate is time variable as in (5), the preferences become time inconsistent. As pointed out by
O’Donoghue and Rabin (1999), when the discount rate, % + φ0 (t), decreases with time, thepreferences represented by (5) captures the tendency of the agent to attempt to experiencepleasant things immediately and to procrastinate about unpleasant things. More precisely,at each moment of time, the agent endowed with (5) attempts to consume more today andenjoy more leisure today by procrastinating about labor supply than she planned in the past.
This present-biased preference is the source of the time inconsistency in a model with atime variable discount rate.

12In our non-unitary discount rate model, the difference between patience with consumption and labor supply is the source of time inconsistency. Remember that both u(c) and v(l)are positively discounted, ρc > 0 and ρl > 0. Therefore, the agent is willing to consume
much today and to procrastinate about labor supply today. When ρc > ρl holds, however,
the agent tends to be more willing to consume much today than to procrastinate about labor supply today because she is more impatient with decreases in consumption today thanwith increases in labor supply today. This difference in the patience is the source of time
inconsistency. Our non-unitary discount rate model may be appropriate for describing thesituation where there are (more than) two distinct choice variables that the agent attempts toexperience immediately (or procrastinate about), however, she is more willing to experience
immediately (or procrastinate about) one of them than the other(s).

4 Generalized Euler Equation
This section provides a formal solution of our model by considering asset holdings. Following Peleg and Yaari (1973) and others, we consider the agent as composed of a sequenceof autonomous decision makers who are indexed by time t. We call the decision maker attime t self t. As in Pollak (1968) and others, we consider the choices of each self to be
the outcome of an intrapersonal game. Following Barro (1999), we solve the intrapersonalgame.
In the following analysis, we specify the instantaneous utility functions as:u(cv ) =cv 1−σ,1−σv(lv ) = −andθ(1 − lv )1−γ,
1−γ

where neither σ > 0 nor γ > 0 are equal to one.11 When σ (γ) is equal to one, we assume
the logarithmic utility function u(c) = log c (v(l) = −θ log(1 − l)). A large θ(> 0) means thatagents put relatively large weight on the disutility of labor supply. For analytical simplicity
and to focus on the effects of the non-unitary discount rates, we consider the case where γ
The disutility of labor is often specified as v(lv ) = θlv 1+γ /(1 + γ) where γ ≥ 0. If we use this specificationin our model, however, it becomes difficult to obtain an analytical solution.
1113

is equal to σ. In Appendix C, we examine the general case where γ is different from σ.12
Given future selves’ behaviors and the sequence of {rv , wv }∞v=t , self t chooses ct and lt
that can be considered as constant flows over the infinitesimally short interval [t, t + ∆]. The
objective of self t is then given by:
Z t+∆
Ut =

Z ∞
z(v, t)dv +t

Z ∞z(v, t)dv ≈ [u(cv ) − v(lv )] ∆ +t+∆z(v, t)dv,

(6)

t+∆

where z(v, t) ≡ u(cv )e−ρc (v−t) − v(lv )e−ρl (v−t) . The approximation comes from setting e−ρc (v−t)
and e−ρl (v−t) equal to one in the infinitesimal short interval [t, t + ∆].
Through the choice of ct and lt , self t can influence choices of selves v(≥ t + ∆) byaffecting the asset holdings at+∆ . To derive the optimal choices of self t, we first have toknow the effects of ct and lt on at+∆ , and second have to conjecture the policy functions ofselves v(≥ t + ∆) to know the effects of at+∆ on future selves’ choices.
The budget constraint (2) can be approximated as follows:at+∆ ≈ {1 + (1 − τr )rt ∆}at + {(1 − τw )wt lt − (1 + τc )ct + T t }∆.
In this approximation, we ignore terms involving ∆2 and consider rt and wt to be constantin the infinitesimally short time interval [t, t + ∆]. This equation implies that:∂at+∆= −(1 + τc )∆,
∂ct

and

∂at+∆
= (1 − τw )wt ∆.
∂lt

(7)

More consumption (labor supply) today leads to smaller (larger) asset holdings in the future.
We turn to the policy functions of self v(≥ t + ∆). We conjecture that self v(≥ t + ∆)chooses cv and lv so as to satisfy:
1

1 − lv = (θζv ) σ cv .

(8)where we conjecture that ζv does not depend on the level of asset holdings. This conjecture
As Appendix C shows, when γ is not equal to σ, our non-unitary discount rate model cannot be solved
without extreme assumptions.
1214

turns out to be true. As in Barro (1999), we conjecture that the choices of self t affectthe levels of future consumption but not the shape of the path of future consumption. We
conjecture that the path of future consumption is:
gcv ≡

ċv
1
= {(1 − τr )rv − ωv } .
cv σ

(9)

This specification allows ωt to vary over time. We conjecture that ωt does not depend on
the level of initial assets. We will see that this conjecture also turns out to be true.
By integrating (2) from t + ∆ to +∞ and using (8) and (9), we obtain (µt+∆ + νt+∆ )ct+∆ =
Rv
Rv
R∞
R∞
r
c
r
at+∆ + Wt+∆ where Wt ≡ t {(1 − τw )wv + T v }e− t (1−τ )rs ds dv, µt ≡ t (1 + τc )e t {gs −(1−τ )rs }ds dv
Rv
R∞
c
r
1
and νt ≡ t (1 − τw )wv (θζv ) σ e t {gs −(1−τ )rs }ds dv. Note that at+∆ has no effect on µt+∆ and νt+∆because we conjecture that both ζv and ωv do not depend on at+∆ . We then have:
1
∂ct+∆
=
.
∂at+∆ µt+∆ + νt+∆

(10)

By using the policy functions of future selves, (8) and (9), we rewrite the objectivefunction of self t, (6), as:"#ct 1−σ θ(1 − lt )1−σct+∆ 1−σ −ρc ∆θct+∆ 1−σ −ρl ∆
Ut =
+
∆+
e Φt+∆ +e Ψt+∆ ,
1−σ
1−σ
1−σ
1−σ
where Φt ≡Rv
R ∞ R v (1−σ)gc −ρ duR∞1−σ(1−σ)gcu −ρl }duc}u
σ e t {
t {
e
(θζ
)
dv
and
Ψ
≡
dv. Self t chooses ct
tvtt

and lt so as to maximize this objective function. Note that at+∆ has no effects on Φt+∆ andΨt+∆ because we conjecture that both ζv and ωv do not depend on at+∆ . Then, the first-orderconditions are given by:
ct −σ = (1 + τc )Xt+∆

∂ct+∆
,
∂at+∆

and

θ(1 − lt )−σ = (1 − τw )wt Xt+∆

∂ct+∆
,
∂at+∆

−ρc ∆
1−σ −ρl ∆where Xt+∆ ≡ c−σΦt+∆ + θct+∆e Ψt+∆ . In deriving the first-order conditions, we use
t+∆ e

15

(7). As ∆ approaches zero, the first-order conditions become:ct −σ = (1 + τc )Xt

∂ct
,
∂at

and

θ(1 − lt )−σ = (1 − τw )wt Xt

∂ct
,
∂at

(11)

−σ
where Xt = c−σt Φt + θct Ψt .

From the two conditions of (11), together with (8), we obtain:ζt =(1 + τc ).
(1 − τw )wt(12)Apparently, ζt does not depend on the level of asset holdings. Our conjecture turns out tobe true.
The first condition of (11) and (10) implies:µt + νt = (1 + τc ) (Φt + θΨt ) .

(13)

This equation holds for all t ≥ 0. We differentiate both sides with respect to t, and aftersome manipulations,13 we obtain:ωt =ρc Φt + ρl θΨt,Φt + θΨt(14)

Rv
R ∞ RvR∞cc1−σwhere Φt ≡ t e t {(1−σ)gu −ρc }du dv, Ψt ≡ t (θζv ) σ e t {(1−σ)gu −ρl }du dv, gcu ≡ {(1−τr )ru −ωu }/σand ζv is given by (12). As we conjectured, ωt does not depend on the level of asset holdings.
The behavior of the agent is summarized by:()
1ρc Φt + ρl θΨtċtr=(1 − τ )rt −≡ gct ,ct σ
Φt + θΨt1()
θ(1 + τc ) σ
1 − lt =ct .
(1 − τw )wt

(15)
(16)

Also in the unitary discount rate case (ρc = ρl = ρ), the same equation as (16) is derived.
We call (15) the generalized Euler equation. In the unitary discount rate case, (15) reduces
13

In Appendix B, we present a derivation of (14).

16to the standard Euler equation: ċt /ct = {(1 − τr )rt − ρ}/σ.
In contrast, in the non-unitary discount rate cases (ρc , ρl ), the generalized Euler equation takes a rather different form. In the log-utility case (σ = 1), however, the generalized
Euler equation takes a simple form. When σ is equal to one, we have Φ = 1/ρc andΨ = 1/ρl by definition. The generalized Euler equation reduces to:ċt= (1 − τr )rt − ρ̃,ctwhere ρ̃ ≡ (1 + θ)ρc ρl /(ρl + θρc ). We can derive the same Euler equation by maximizing thefollowing unitary discount rate utility function subject to (2):
Z ∞
Ut =

(u(cv ) − v(lv ))e−ρ̃(v−t) dv.

tWith logarithmic utility functions, the non-unitary discount rate model is observationallyequivalent to a unitary discount rate model in which the discount rate is equal to ρ̃.14 Furthermore, we obtain the following proposition.
Proposition 1
Suppose that the instantaneous utility functions have logarithmic forms. Consider two
agents, one of which has discount rates, ρc1 and ρl1 . The other has ρc2 and ρl2 . If ρc1 ρl1 /(ρl1 +θρc1 ) = ρc2 ρl2 /(ρl2 + θρc2 ) holds, the generalized Euler equations become the same for thetwo agents.
When the utility functions are not logarithmic, the generalized Euler equation takes amore complex forms. Because the generalized Euler equation includes wv , τc and τw (v ≥ t)
through Φt and Ψt , the consumption-saving behavior at time t is influenced by wv , τc and τw .
Remember that in our non-unitary discount rate model, the problem of time inconsistency
arises. Given policy functions of the future selves, self today attempts to affect the future
selves’ behaviors in a preferable manner for self today by controlling the asset holdings left
14

Using logarithmic utility, Pollak (1968), Barro (1999) and many others obtain similar observational
equivalence in models with a time variable discount rate. Karp (2007) considers a more general utility function.

17

to the future selves. Therefore, the saving decision of self today is affected by behaviorsof the future selves. Behaviors of the future selves are influenced by wv , τc and τw (v ≥ t).
Consequently, the consumption-saving behavior of self today is influenced by wv , τc andτw . In the unitary discount rate case, the problem of time inconsistency does not arise.
Therefore, wv , τc and τw have no influence on the consumption-saving behavior of self
today.

5 A Simple General Equilibrium Model
To examine the effects of preference parameters and taxes, we consider a simple generalequilibrium model. Consider a competitive economy where there are identical firms. The
number of firms is normalized to one. The representative firm produces a final good byusing a constant-returns-to-scale technology, Yt = Alt , where Yt is the output level, lt islabor input and A is a positive constant. Through profit maximization, the wage rate wt
becomes equal to A.
The population size is normalized to one. We first consider an economy populated byhomogeneous agents. Subsection 4.3 examines a case of heterogeneous agents. We assume
that the initial asset holdings of the representative agent are zero, a0 = 0. Because theagents are identical and there is no capital, at is constant at zero over time. By using (16)and the goods market equilibrium condition, ct = Alt , we obtain:cE =An θ(1+τc ) o σ1 ,and lE =1 + A A(1−τw )1n θ(1+τc ) o σ1 .

(17)

1 + A A(1−τw )Because σ is strictly positive, we have:
∂cE
∂lE
1 ∂cE
< 0, and
=
< 0,
x
x
∂τ
∂τ
A ∂τ x(18)1−σwhere x = c or w. Because cE is constant, we have ΦE = 1/ρc and ΨE = {θ(1 + τc )/[A(1 − τw )]} σ /ρl .

18

From (15), we obtain the equilibrium interest rate:1

1−σ

1 (1 + θ σ ζ σ )ρc ρl,rE =1 − τr ρl + ρc θ σ1 ζ 1−σσwhere ζ = (1 + τc )/{A(1 − τw )}.

5.1

Consumption-Saving Behavior

By examining the effects on the equilibrium interest rate, we know the effects of the preference parameters and taxes on the consumption-saving behavior. In Figure 2, we depict
a savings curve that represents the relationship between savings and the interest rate. The
equilibrium interest rate is given by rE . Suppose that changes in a parameter strengthen
the saving incentives of each self. For any given interest rate, the savings of each self increases, which results in rightward shifts of the savings curve. The equilibrium interest rate
must decrease from rE to rE0 . If a(n) decrease (increase) in the equilibrium interest rate iscaused by changes in a parameter, therefore, we can conclude that changes in that parameter
positively (negatively) affect the incentive to save.
[Figure 2]
The next proposition summarizes the effects of preference parameters on the equilibrium interest rate by assuming τc = τw = τr = 0.
Proposition 2
1

1−σρl 2 (1 + θ σ ζ σ )
∂rE
=
(i)
> 0,
∂ρc (ρl + ρc θ σ1 ζ 1−σ
σ )2

1

1−σ

1

1−σ

∂rE ρc 2 θ σ ζ σ (1 + θ σ ζ σ )
=
(ii)
> 0,
1
1−σ
∂ρl
(ρl + ρc θ σ ζ σ )2

1−σ∂rE ρc ρl (θζ) σ (ρl − ρc )< (=)(>)0 if and only if ρl < (=)(>)ρc .
=
(iii)
1−σ
1
∂θ
σ(ρl + ρc θ σ ζ σ )2

The first and second parts of Proposition 2 indicate that in an economy with relatively large
discount rates, saving incentive are relatively weak.

19

The third part shows that in the unitary discount rate case, θ is irrelevant to the consumptionsaving behavior. In the non-unitary discount rate cases, however, θ affects the consumptionsaving behavior. In the case where ρc is smaller than ρl , θ is negatively related to the incentive to save. An increase in θ indicates that the agent place a relatively large weighton the disutility from labor supply. Because ρc is smaller than ρl , each self does not wantto supply much labor today while she cares relatively less about the disutility from futurelabor supply. When θ increases, self today attempts to increase labor supply of future selves
by reducing the asset holdings left to future selves. Consequently, savings decreases.
The next proposition examines the effects of tax rates and the wage rate (w = A).
Proposition 3
∂rE
> 0.
∂τr
∂rE
(ii)
< (=)(>)0, if and only if (1 − σ)(ρl − ρc ) < (=)(>)0, where x = c or w.
∂τ x∂rE< (=)(>)0, if and only if (1 − σ)(ρl − ρc ) > (=)(<)0.
(iii)
∂A

(i)

(Proof) If we differentiate rE with respect to τr , we obtain ∂rE /∂τr = rE /(1 − τr ) > 0. We
next differentiate rE with respect to x where x = τc , τw or A:
1∂rE ρc ρl (θζ 1−2σ ) σ (1 − σ)(ρl − ρc ) ∂ζ
=
.
1
1−σ
∂x
(1 − τr )(ρl + ρc θ σ ζ σ )2 ∂x
where ∂ζ/∂τc > 0, ∂ζ/∂τw > 0 and ∂ζ/∂A < 0. 
In both the unitary and the non-unitary discount rate cases, τr has the same qualitative effecton rE . In contrast, τc , τw and w(= A) have different effects on rE in the two cases. Whileτc , τw and w(= A) have no effect in the unitary discount rate case, these three variables do
influence the consumption-saving behavior in the non-unitary discount rate cases.
The intuition of the effects of τc is as follows. An increase in τc has two opposing
effects. When ρc is larger than ρl , each self attempts to consume much today, comparedwith future consumption. When τc increases, therefore, self today does not want to decrease20

consumption today while she cares relatively less about decreases in the future consumption
levels. This negatively affects the saving incentives. However, because ρc is larger than ρl ,each self cares relatively less about the disutility of labor supply today, compared with thedisutility of the future labor supply. Faced with an increase in τc , self today attempts todecrease future labor supply more than labor supply today by saving more. This positively
affects the saving incentives. When σ is larger (smaller) than one, the negative (positive)
effects dominate the positive (negative) effects. Consequently, the savings today decrease(increase). When ρc is smaller than ρl , the opposite holds. When σ is smaller (larger) thanone, therefore, the savings today decrease (increase).
An increase in w (a decrease in τw ) increases the incentive of labor supply, and has apositive effect on consumption. Therefore, an increase in w (a decrease in τw ) has effects
similar to a decrease in τc . Then, we can obtain the results in Proposition 3.

5.2

The Welfare Effects of Taxes

We now examine the effects of taxes on welfare. Because the preferences of the agent are
time-inconsistent, the different selves of an agent need not agree on their welfare rankingof the same consumption and labor supply sequences. In this paper, we evaluate welfare
from the perspective of all selves following authors such as Laibson (1996, 1997).15 In
equilibrium, all selves have the same utility level which is given by:
U E = u(cE )/ρc − v(lE )/ρl .

(19)

The interest income tax τr has no effects on utility. By using (17), we differentiate U E withrespect to τ x where x = c or w:!
1 1 − τw ∂cE
∂U E
−σ 1
= cE
−
< (>)0,∂τ x
ρc ρl 1 + τc ∂τ x

15

As pointed out by O’Donoghue and Rabin (1999) and others, welfare comparisons for agents with timeinconsistent preferences are problematic because an agent’s preferences at different times disagree. However,many studies, including Laibson (1996, 1997), Laibson et al. (1998) and İmrohoruğlu et al. (2003), often
make welfare comparisons from the perspective of all selves.

21

if and only if (1 − τw )/(1 + τc ) < (>)ρl /ρc because ∂cE /∂τ x has a negative sign (see (18)).
We then obtain the next proposition.
Proposition 4
The utility levels of all selves are maximized by setting τc = ρc /ρl − 1 (τw = 1 − ρl /ρc ) whenτw (τc ) is equal to zero.
Consider the effects of τc . In the unitary discount rate case, the consumption tax (or subsidy)
decreases the utility level. In contrast, in the non-unitary discount rate cases, the utility
levels of all selves are improved by a consumption tax (subsidy), τc > 0 (τc < 0), when ρc islarger (smaller) than ρl . As discussed in Subsection 2.1, when ρc is larger than ρl , self v(> t)consumes more by supplying more labor than self t prefers. Faced with a consumption tax,
self v(> t) reduces her own consumption and labor supply. Consequently, the consumption
level and labor supply of self v(> t) become close to those favorable for self t. Then, theutility level of self t improves. Because all selves have the same utility level in equilibrium,
the consumption tax can improve the utility of all selves.
Note that Proposition 4 holds even if the utility functions take logarithmic forms (σ =1). As shown in Proposition 1, when σ = 1 holds, the unitary discount rate economy
becomes observationally equivalent to an economy with the non-unitary discount rate. In
the economy with logarithmic utility, the government may misperceive the preferences of
the agent. If the government believes that the agent is endowed with a unitary discount rate
but the agent actually has the non-unitary discount rates, the government cannot implement
policy in an appropriate manner because the effects of taxes on the welfare in these twocases are quite different, as shown in Proposition 4.

5.3

Heterogeneous AgentsThis subsection briefly considers the case of heterogeneous agents, assuming logarithmic
utility functions, σ = 1. The initial asset holdings of all agents are equal to zero. Let ρic
and ρil be the subjective discount rates of agent i. We assume that (1 + θ)ρic ρil /(ρil + θρic ) =(1 + θ)ρcj ρlj /(ρlj + θρcj )(≡ ρ̃) holds for all i and j(, i) and that all agents have the same value22

of θ. Because the behaviors of all agents are observationally equivalent (see Proposition
1) and because the initial asset holdings of all agents are equal to zero, it appears as if theeconomy is populated by identical agents. The equilibrium consumption level and labor
supply of all agents are then given by (17). The equilibrium interest rate is ρ̃. The utility
level of all selves of agent i is: U Ei = u(cE )/ρic − v(lE )/ρil . Assuming τw = 0, we focus
on the effects of τc . Consider a small increase in τc . As is clear from Proposition 4, theutility levels of all selves of agents with ρic /ρil − 1 < (>)τc decrease (improve). The utility
levels of the agents with high ρic /ρil , who are relatively more impatient with decreases in
consumption today, is improved by an increase in τc .

6 An Extension: Monetary Economy
This section extends the basic model by introducing money. As in Section 4, the population
size is normalized to one and we assume that the agents are identical. Subsection 5.1
considers the case of heterogeneous agents.
Let us denote the price level as pt . We assume that a fraction of the purchase of consumption goods must be financed by cash. More precisely, to purchase ct dt units of consumption goods in a time interval of length dt, ηpt ct dt units of cash are needed in the sametime interval. The parameter η ∈ [0, 1] represents the fraction of consumption goods thatmust be purchased by cash. Let us denote the nominal cash holdings of agents at time t
as Mt . When an agent purchases ct units of consumption goods at time t, Mt must satisfy
Mt ≥ ηpt ct , or equivalently:
mv ≥ ηcv ,(20)

where mv ≡ Mv /pv . A larger η means that agents need more cash for purchasing consumption goods. η represents the degree of financial market development. As the financial
market develops, η decreases. The budget constraint is given by:ȧv = rv av − (rv + πv )mv + wv lv − cv + T v ,
23

(21)

where πv ≡ ṗv /pv is the inflation rate, and at is equal to zv + mv where zv represents the asset
holdings other than cash. We assume that at any moment of time, agents can allocate theirportfolio between cash and other assets without any costs.
As for the money-supply behavior of the government, we assume a helicopter drop ofmoney. The monetary authority issues nominal money at a positive and constant growthrate,  ≡ Ṁt /Mt . The newly created money is transferred to agents as lump-sum payments.
The budget constraint of the government is pt T t =  Mt .
As in Section 3, we solve the intrapersonal game. We begin with the effects on at+∆ . The
budged constraint (21) can be approximated as at+∆ ≈ (1+rt ∆)at +{wt lt −ct −(rt +πt )mt +T t }∆because we can ignore terms involving ∆2 and consider rt , wt , and πt to be constant in theinfinitesimally short time interval [t, t + ∆]. This equation implies that:
∂at+∆
= −∆,
∂ct

∂at+∆
= wt ∆
∂lt

and

∂at+∆
= −(rt + πt )∆.
∂mt

(22)

If self t increases consumption or cash holdings (labor supply), the asset left to self t + ∆then decreases (increases).
We turn to the policy functions of self v(≥ t + ∆). As in Section 3, the choices of selfv(≥ t + ∆) and the path of future consumption are conjectured as follows:
1

1 − lv = (θζ̃v ) σ cv ,g̃cv ≡(23)ċv
1= (rv − ω̃v ).
cv σ

(24)

As in Section 3, we conjecture that ζ̃v and ω̃v do not depend on the level of asset holdings
and that ζ̃v and ω̃v vary over time. In addition, we conjecture that self v(≥ t + ∆) does
not hold more cash than needed for purchasing consumption goods, which means that (20)holds with equality for all v(≥ t + ∆). We will see that our conjectures turn out to be true.
From (20) with equality, (21), (23) and (24), we obtain (µ̃t+∆ + ν̃t+∆ )ct+∆ = at+∆ + Wt+∆
Ru
Ru
R∞
R∞
c
c
1
where µ̃v ≡{1 + η(ru + πu )}e v (g̃s −rs )ds du and ν̃v ≡wu (θζ̃u ) σ e v (g̃s −rs )ds du. Note thatvv

at+∆ has no effects on µ̃t+∆ and ν̃t+∆ because we conjecture that both ζ̃v and ω̃v do not depend24

on at+∆ . We then have:
∂ct+∆
1
=
.
∂at+∆ µ̃t+∆ + ν̃t+∆(25)

The objective function of self t is given by:"#
ct 1−σ θ(1 − lt )1−σct+∆ 1−σ −ρc ∆θct+∆ 1−σ −ρl ∆
Ut =
+
∆+
e Φ̃t+∆ +e Ψ̃t+∆ ,
1−σ
1−σ
1−σ
1−σ
Rv
R∞
R ∞ Rvcc1−σwhere Φ̃t ≡ t e t {(1−σ)g̃u −ρc }du dv and Ψ̃t ≡ t (θζ̃v ) σ e t {(1−σ)g̃u −ρl }du dv. Given the sequenceof {rv , wv , pv , πv }∞v=t , self t maximizes this objective function subject to (22) and mt ≥ ηct .
We set the Lagrangian as follows: Lt = Ut + λt (mt − ηct ) where λt is the Lagrangian
multiplier. Note that at+∆ has no effect on Φ̃t+∆ and Ψ̃t+∆ because we conjecture that both ζ̃v
and ω̃v do not depend on at+∆ . Then, the first-order conditions are given by:
∂ct+∆
−σ
c−σ
t − ct+∆ X̃t+∆∂at+∆!
∆ = ηλt ,

(26)

∂ct+∆
wt ,
∂at+∆

(27)

∂ct+∆
(rt + πt )∆ = λt ,
∂at+∆

(28)

θ(1 − lt )−σ = c−σt+∆ X̃t+∆c−σt+∆ X̃t+∆where X̃t = e−ρc ∆ Φ̃t + θe−ρl ∆ Ψ̃t .

The condition (28) implies λt > 0, which means that self t does not holds more cashthan needed for purchasing consumption goods, or equivalently (20) holds with equality for
self t. Because this applies to self v(≥ t + ∆), our conjecture that (20) holds with equality
for all v(≥ t + ∆) turns out to be true. From (26) and (28), we have:
= c−σ
c−σ
t
t+∆ X̃t+∆∂ct+∆{1 + η(rt + πt )}.
∂at+∆

By using (23), (27), and (29), we obtain:ζ̃t =1 + η(rt + πt )
.
wt

25

(29)

As ∆ approaches zero, we have µ̃t + ν̃t = {1 + η(rt + πt )}(Φ̃t + θΨ̃t ) from (25) and (29). As
in Section 3, by differentiating both sides of this equation with respect to time, we obtain:ω̃t =ρc Φ̃t + ρl θΨ̃t
η(ṙt + π̇t ).
+1 + η(rt + πt )Φ̃t + θΨ̃t

As we conjectured, ζ̃ and ω̃ do not depend on the level of asset holdings.
For analytical simplicity, we proceed by assuming the logarithmic utility functions (σ =1). If σ is not equal to 1, we can obtain the same qualitative results. Because we haveΦ̃t = 1/ρc and Ψ̃t = 1/ρl when σ = 1 holds, the behavior of self t is summarized by:η(ṙt + π̇t )ċt= rt − ρ̃ −,ct
1 + η(rt + πt )
θ{1 + η(rt + πt )}ct ,1 − lt =
wt

(30)
(31)

where ρ̃ ≡ (1 + θ)ρc ρl /(ρl + θρc ).
As in Section 4, we consider the simple general equilibrium. The production technology
is Yt = Alt , where Yt is the output level, lt is labor input and A is a positive constant. Throughprofit maximization, the wage rate wt becomes equal to A. Because there is no capital, we
have at = mt . We focus on the steady state equilibrium where ċt = ṙt = π̇t = 0 holds.
Equation (30) implies rt = ρ̃. Because mt = ηct implies ċt /ct =  − πt , πt is equal to .
Because the nominal interest rate ρ̃ +  cannot be negative,  must be equal to or larger than−ρ̃. By using (31) and ct = Alt , we obtain:
c∗ =

A
,
1 + θ{1 + η(ρ̃ + )}and1 − l∗ =θ{1 + η(ρ̃ + )}.
1 + θ{1 + η(ρ̃ + )}(32)

Apparently, we have ∂c∗ /∂x < 0 and ∂l∗ /∂x < 0 where x =  or η. The utility levels of allselves are given by U ∗ = (ln c∗ )/ρc + {θ ln(1 − l∗ )}/ρl .
To derive  that maximizes U ∗ (henceforth,  ∗ ), we differentiate U ∗ with respect to  byusing (32):
)
(
θη
1
∂U ∗
1
=
−
.
∂
1 + θ{1 + η(ρ̃ + )} ρl {1 + η(ρ̃ + )} ρc
26

(33)

By examining the sign of ∂U ∗ /∂, we obtain the next proposition.
Proposition 5
if ρc ≤ ρl , −ρ̃,
∗
 =




 1 ρc − 1 − ρ̃(> −ρ̃), if ρc > ρl ,η ρlNote that the real interest rate r is equal to ρ̃. When  ∗ is equal to −ρ̃, the nominal interest
rate becomes equal to zero. In the unitary discount rate case and in the non-unitary discountrate case where ρc is smaller than ρl , the Friedman rule is optimal. When ρc is larger thanρl ,  ∗ is larger than −ρ̃. The Friedman rule is not optimal. When ρc > ρl holds, self v(> t)attempts to consume more by supplying more labor than self t(< v) prefers. An increase in 
reduces the future selves’ purchasing power, which causes decreases in c∗ . Furthermore, l∗also falls because of decreases in final goods production. When  increases, consequently,
consumption level and labor supply of self v(> t) become close to those favorable for self t.
Then, the utility level of all selves improves. When ρc > ρl holds, therefore, the monetary
authority can improve the utility levels of all selves by setting the nominal interest rate at astrictly positive level.
Because we consider the case where the unitary discount rate economy becomes observationally equivalent to an economy with the non-unitary discount rate, the monetary
authority possibly misperceives the preferences of the agent. If the monetary authority believes that the agent is endowed with a unitary discount rate and if ρc is actually larger thanρl , the monetary authority cannot implement policy in an appropriate manner.
We next examine the financial market development (decreases in η) by keeping  constant at some level. Because η disappears from U ∗ when  is equal to −ρ̃, we assume  > −ρ̃.
Given (> −ρ̃), we differentiate U ∗ with respect to η by using (32):)(θ(ρ̃ + )
1
∂U ∗
1
=
−
.
∂η
1 + θ{1 + η(ρ̃ + )} ρl {1 + η(ρ̃ + )} ρc(34)

When ρc ≤ ρl , ∂U ∗ /∂η has a negative sign. On the other hand, when ρc > ρl , ∂U ∗ /∂η has a27

positive (negative) sign if and only if η < (>)(ρc − ρl )/{ρl (ρ̃ + )} ≡ η. Because decreases inη represent financial market developments, we obtain the next proposition.
Proposition 6
As the financial market develops,
1. when ρc ≤ ρl holds, the utility levels of all selves increase;
2. when ρc > ρl holds, the utility levels of all selves increase if the financial market isless-developed (η > η), while the utility levels of all selves decrease if the financial
market is well-developed (η < η).

As η decreases, the constraint on consumption purchases (20) becomes loose. This has apositive effect on U ∗ . Because there exists only this positive effect when ρc ≤ ρl holds,
decreases in η improve the utility levels of all selves. When ρc > ρl holds, however, anegative effect is also at work. When ρc > ρl holds, self v(> t) attempts to consume moreby supplying more labor than self t(< v) prefers. As η decreases, the future selves increase
their consumption further, which results in increases in labor supply because of the rise in
the final goods production. The differences between consumption levels (labor supplies) of
the future selves and those favorable for self t(< v) become wider. As a result, a decreasein η negatively affects U ∗ . In an economy with a less-developed (well-developed) financial
market, the positive (negative) effect dominates the negative (positive) effect. The financial
market development improves (diminishes) the utility levels of all selves.

6.1

Heterogeneous Agents

As in Subsection 4.2.1, we assume that (1 + θ)ρic ρil /(ρil + θρic ) = (1 + θ)ρcj ρlj /(ρlj + θρcj )(≡ ρ̃)holds for all i and j(, i) where ρic and ρil are the subjective discount rates of agent i andthat the initial asset holdings of all agents are equal to zero. All agents have the samevalue of θ and the utility functions are logarithmic. Because the behaviors of all agents areobservationally equivalent (see Proposition 1) and the initial asset holdings of all agents areequal to zero, it looks as if the economy is populated by identical agents. We focus on the28steady state equilibrium. The equilibrium consumption level and labor supply of all agentsare then given by (32). The equilibrium interest rate is ρ̃. The utility level of all selves of
agent i is given by U i∗ = u(c∗ )/ρic − v(l∗ )/ρli . We can use Propositions 5 and 6 to evaluate
welfare effects. When (≥ ρ̃) increases, the utility levels of all selves of agents with ρic ≤ ρildecrease. The utility of all selves of agents with ρic > ρil decreases (increases) if ρic /ρil < (>)1 + η( + ρ̃) holds. As η decreases, the utility levels of all selves with ρic ≤ ρil increase. The
utility of all selves with ρic > ρil decreases (increases) if η < (>)ηi ≡ (ρic − ρil )/{ρil (ρ̃ + )}holds. Note that when ρ̃ is kept constant, ηi increases with ρic /ρil . The utility levels of theagents with high ρic /ρil , who are relatively more impatient with decreases in consumption
today, tend to be increased by increases in the inflation rate and to be decreased by thedevelopment of the financial market.

7 Optimal Policy and a Time Variable Discount Rate
Propositions 4, 5 and 6 provide important welfare implications. Laibson (1996, 1997)
provides results similar to Propositions 4, 5 and 6. For example, Laibson (1997) showsthat the development of the financial market may deteriorate welfare in a model with a time
variable discount rate where the problem of time inconsistency arises. However, his analysis
is based on a partial equilibrium model. To emphasize the importance of our results, weconsider a general equilibrium model with a time variable discount rate that is similar to
Barro (1999) by assuming that the agents are identical and the population size is one.
Instead of (1), this section assumes (5). Please note that even if the instantaneous utility
functions have logarithmic forms, Propositions 4, 5 and 6 hold in the non-unitary discountrate model of the previous sections. For simplicity, we assume the logarithmic utility functions: u(c) = ln c and v(l) = −θ ln(1 − l). When θ is equal to zero, (5) becomes exactly thesame as the utility function employed in Barro (1999). The budget constraint is:ȧv = rv av + wv lv − (1 + τc )cv + T v .

(35)

Because we are interested in the optimal consumption tax, the other taxes are omitted and29money is excluded in this section.
It is well known that when the discount rate varies with time as in (5), the problem oftime inconsistency arises. As in Section 3, we consider the agent as composed of a sequenceof autonomous decision makers. If we follow the same procedure as in Section 3, we canderive the behavior of self t, which is summarized by:ċt= rt − ξ,ct
1 + τc1 − lt = θ
ct ,
wt

(36)
(37)

R∞
where ξ ≡ 1/ 0 exp {−(%t + φ(t))} dt.16 Barro (1999) obtains the same Euler equation as(36). Equation (37) is the same as (16) (if we set τw = 0 in (16)). Note that the model withthe time variable discount rate is observationally equivalent to the non-unitary discount rate
model if ξ is equal to ρ̃(≡ (1 + θ)ρc ρl /(ρl + θρc )). The production technology is given by
Yt = Alt , again. Because (37) is exactly the same as (16), the equilibrium consumption level
and labor supply are given by the two equations of (17) again. In equilibrium, all selves
have the same utility level:
U Eξ = (u(cE ) − v(lE ))/ξ.

(38)

In equilibrium, the only difference between the non-unitary discount rate model and themodel with the time variable discount rate is the difference between U E and U Eξ . Let uscompare (38) with (19). In the non-unitary discount rate model, the weight on u(cE ), 1/ρc ,is different from that on v(lE ), 1/ρl . By contrast, in the model with the time variable discountrate, u(cE ) has the same weight as v(lE ).
We now derive the optimal consumption tax in the model with the time variable discountrate by differentiating (38) with respect to τc :
!
∂U Eξ
1
∂cE
1
=
= 0.
1−
c
c
∂τ
cE ξ
1 + τ ∂τc
16

See Appendix D for the derivations of (36) and (37).

30

Because cE is given by the first equation of (17), we have ∂cE /∂τc < 0 (see (18)). The
above equation implies that by setting τc = 0, the utility levels of all selves are maximized.
This result contrasts with Proposition 4, which shows that in the non-unitary discount rate
model, the optimal τc (= ρc /ρl − 1) is strictly not equal to zero. This exercise reveals that thestrictly nonzero optimal consumption tax is not a common feature of general equilibrium
models where the problem of time inconsistency arises.
Note that the equilibrium consumption level and labor supply in the two models areexactly the same, and that in equilibrium, the only difference between the two models is
whether the weights on u(cE ) and v(lE ) are the same or not. When money is introduced in
the same way as in Section 5, we can reasonably conjecture by setting ρc = ρl = ξ in (33)
and (34) that in the model with the time variable discount rate, the zero nominal interest
rate (the Friedman rule) becomes optimal and the development of the financial market (adecrease in η) improves the utility level of all selves.
The analysis in this section provides important policy implications. Even when the
problem of time inconsistency exists in the economy, if it is caused by the time variable
discount rate, the policy maker might not need to take the problem of time inconsistency
into consideration when setting tax rates. However, if the non-unitary discount rates cause
the problem of time inconsistency, the policy maker could not implement policy in an appropriate manner if she does not consider the problem of time inconsistency.

8 Conclusion
The standard DU model assumes that a single discount rate applies equally to discount(dis)utility from all different sources. However, there is some evidence that people might
discount (dis)utility from different sources at different rates. This paper provided a simple model where the agent discounts utility from consumption at a different rate from thedisutility of supplying labor.
We first showed that in our non-unitary discount rate model, the preferences of agentsare time-inconsistent. The difference between patience concerning consumption and labor31

is the source of the time inconsistency. Our non-unitary discount rate model may be appropriate for describing the situation where there are (more than) two distinct choice variables
that the agent attempts to experience immediately (or procrastinate about), however, she
is more willing to experience immediately (or procrastinate about) one of them than theother(s).
In our non-unitary discount rate model, the policy effects on welfare are quite different
from the standard models where a single discount rate applies equally to discount (dis)utility
from all different sources. For example, when the agent discounts utility from consumption
at a higher (lower) rate than the disutility of labor supply, the utility level of agents canimprove by a strictly positive consumption tax (a consumption subsidy). We compared our
results with those obtained in a time variable discount rate model. Although the preferences
are time-inconsistent in both models, the results of welfare analysis are quite different. Our
analysis suggested that our results suggests that when the problem of time inconsistency
exists in the economy, the optimal policy might be influenced by the sources of time inconsistency.
This paper ignored capital accumulation. The introduction of capital accumulation
could affect our results. It is important to examine how our results are affected by the
introduction of capital accumulation and to compare our non-unitary discount rate modelwith a model with a time variable discount rate by considering capital accumulation.

Appendix
A. Preference and Life Satisfaction Survey
Table 1 is based on micro data from “Preference and Life Satisfaction Survey” conducted in
the Global COE Program entitled “Human Behavior and Socioeconomic Dynamics” which
is supported by the Ministry of Education, Culture, Sports, Science and Technology inJapan. This survey is a drop-off style survey that was conducted in February and March
2009. The target populations are individuals who are over 20 years old. Sample in the
United States was selected randomly from households participating in the managed access32

panel of TNS (a formerly National Family Opinion), a global market research company.
Sample in Japan was selected randomly from all over Japan using the Basic Residents Registration System. Cares were take to ensure that the resulting samples were representativeof the total population in both the United State and Japan. Households in samples were
mailed questionnaires and were asked to mail them back. The resulting number of respondents were 10708 in the United States and 6181 in Japan.
The question about the money-related discount rate is “Would you choose to receive$100 in two days or to receive a different amount of money in nine days?” If a respondent
prefers the receipt of $100 in two days to the receipt of (more than) $100 in nine days, we
determine that her money-related discount rate is positive. The question about the laborrelated discount rate is “Would you choose to do 60 minutes of labor this Sunday or to do adifferent minutes of labor next Sunday?” If a respondent prefers doing 60 minutes labor thisSunday to doing (less than) 60 minutes of labor next Sunday, we determine that her laborrelated discount rate is negative. If a respondent gave an answer such as he or she prefers
60 minutes of labor this Sunday to 40 minutes of labor next Sunday, which implies that hisor her labor-related discount rate is negative, but prefers 80 minutes of labor next Sunday
to 60 minutes of labor this Sunday, which implies that his or her labor-related discountrate is positive, we drop him or her from the data because we cannot determine the signof his or her labor-related discount rate. This also applies to the money-related discountrate. We can determine the signs of both the money- and labor-related discount rates of6719 (4942) respondents in the United States (Japan). In the United States (Japan), 6202
(4644) respondents, which amounts to about 92% (94%) of 6719 (4942) respondents, were
found to discount money-related utility at positive rates. Panels (a) and (b) of Table 1 arebased on the data from these 6202 and 4644 respondents, respectively. Table 1 shows that in
the United States (Japan), among the 6202 (4644) respondents who discount money-related
utility at positive rates, about 70% (74%) of them were found to use negative discount rates
to discount the disutility of labor.

33

B. Derivation of (14)
By definition, we have:µ̇t = {(1 − τr )rt − gct }µt − (1 + τc ),1−γ

ν̇t = {(1 − τr )rt − gct }νt − θ(1 + τc )(θζt ) γ ,Φ̇t = {ρc + (σ − 1)gct }Φt − 1,1−σΨ̇t = {ρl + (σ − 1)gct }Φt − (θζt ) σ .
By differentiating both sides of (13) and using the above four equations, we have:σ(1 + τc )(Φt + θΨt )gct = (1 − τr )rt (µt + νt ) − (1 + τc )(ρc Φt + ρl θΨt ).
In deriving this equation, we use (13). By using (13), we divide the both sides of the above
equation by σ(1 + τc )(Φt + θΨt ):
gct =

(
)
1
ρc Φt + ρl θΨtr(1 − τ )rt −.
σΦt + θΨt

From this equation and (9), we obtain (14).

C. General Case: σ , γThis appendix discusses difficulties that arise when γ is not equal to σ. For simplicity, we
assume that τc , τw and τr are equal to zero.
The objective of self t and the effects of self t’s choices on at+∆ are again given by (6)and (7), respectively. As in Section 3, we conjecture that self v(≥ t + ∆) chooses cv and lv
so as to satisfy:
1σ

1 − lv = (θζ̂v ) γ cv γ .

(39)

We conjecture that ζ̂v does not depend on the level of asset holdings.
The difficult part of the problem arises from the conjecture as to the effects of at+∆ on34

cv where v ≥ t + ∆. We assume that each self has incorrect beliefs about the future selves’
behavior. More precisely, we assume that self t does not know the effects of her choiceson the shape of the path of future consumption. This may be a restrictive assumption. By
proceeding with this assumption, however, we can illustrate the difficulties that arise when
γ is not equal to σ. Self t, however, is assumed to know the effects of her choices on thelevel of ct+∆ . We conjecture that the path of future consumption is:ĝcv ≡ċv
1= (rv − ω̂v ),
cv σ

(40)

As we will see later, ω̂v does depend on the level of asset holdings in this general case. Our
assumption, however, means that self t does not perceive the effects of the level of asset
holdings on ω̂v .
By using (2), (39) and (40), we obtain the effects of at+∆ on ct+∆ :∂ct+∆=∂at+∆ µ̂
where µ̂t ≡

1
σ
γ −1
σ
t+∆ + γ ν̂t+∆ ct+∆

,

(41)

o
R vn σR ∞ R v {ĝc −r }ds
R∞
1
c
s
s
γ e t γ ĝ s −r s ds dv. Note that self t does not
t
e
w
(θ
ζ̂
)
dv
and
ν̂
≡
v
v
t
t
t

perceive the effects of at+∆ on µ̂t+∆ and ν̂t+∆ because ζ̂v does not depend on at+∆ and because
self t does not perceive the effects of at+∆ on ωv . Therefore, the above equation does notinclude ∂µ̂t+∆ /∂at+∆ and ∂ν̂t+∆ /∂at+∆ .
By using the policy functions of future selves, (39) and (40), we rewrite the objectivefunction of self t, (6), as:"σ(1−γ)#
ct 1−σ θ(1 − lt )1−γct+∆ 1−σ −ρc ∆θct+∆ γ −ρl ∆+e Φ̂t+∆ +e Ψ̂t+∆ ,
Ut =
∆+
1−σ
1−γ
1−σ
1−γ

where Φ̂t ≡o
R v n σ(1−γ)
R∞
R ∞ R v (1−σ)ĝc −ρ du
1−γ
c
c}
u
γ e t
γ ĝu −ρl du dv. Self t chooses c
t {
dv
and
Ψ̂
≡
(θ
ζ̂
)
e
t
v
t
t
t

and lt so as to maximize this objective function. Note that self t does not perceive the effects
of at+∆ on Φ̂t+∆ and Ψ̂t+∆ because ζ̂v does not depend on at+∆ and because self t does not35perceive the effects of at+∆ on ωv . As ∆ approaches zero, the first-order conditions become:ct −σ = X̂twhere X̂t = ct −σ Φ̂t + θσcγ tσ(1−γ)
γ −1

∂ct
,
∂at

θ(1 − lt )−γ = wt X̂t

and

∂ct
,
∂at

(42)

Ψ̂t .

By following the procedure that we described in Section 3, we obtain:σ1ζ̂t = ,
wt

and

ω̂t =

ρc ct Φ̂t + ρl θσc γ Ψ̂tγ t
σct Φ̂t + θσ
c γ Ψ̂tγ t.

When γ is equal to σ, ω̂v corresponds to (14). When γ is not equal to σ, ω̂t includes ct .
Because ct does depend on at , ω̂t actually depends on the level of asset holdings. In thissection, however, we assume that self t does not know the effects of at+∆ on ω̂v . Therefore,
under our assumption, there exist intertemporal external effects.
If self t does perceive the dependence of ω̂v on at+∆ (v ≥ t + ∆), we have the following
difficulties: The first difficulty arises from the effects of at+∆ on ct+∆ . Note that both µ̂t+∆and ν̂t+∆ depend on ω̂v , hence on at+∆ , through ĝcv = (rv − ω̂v )/σ (v ≥ t + ∆). The effects
of at+∆ on ct+∆ through µ̂t+∆ and ν̂t+∆ are not included in (41). The next difficulty is causedby the dependence of Φ̂t+∆ and Ψ̂t+∆ on at+∆ . It is apparent that Φ̂t+∆ and Ψ̂t+∆ depend on
at+∆ because Φ̂t+∆ and Ψ̂t+∆ include ω̂v (v ≥ t + ∆). If self t does perceive the dependenceof ω̂v on at+∆ (v ≥ t + ∆), we have to consider the effects of at+∆ on Φ̂t+∆ and Ψ̂t+∆ whenmaximizing Ut . The first-order conditions are no longer given by (42). Because of these
difficulties, the problem becomes intractable.

D. Derivations of (36) and (37)
Using the same procedure as in Section 3, we derive (36) and (37). Again, the effects of ct
and lt on at+∆ are given by the two equations of (7) if we set τw = 0 in (7). As in Section 3,36

the choices of self v(≥ t + ∆) and the path of future consumption are conjectured as:
1

1 − lv = (θχv ) σ cv

and

gcv ≡

ċv
= rv − ξ.
cv

As in Section 3, we conjecture that χv and ξ do not depend on the level of asset holdings.
Because of the logarithmic utility function, we assume that ξ is constant over time. The
effects of at+∆ on ct+∆ are given by:
∂ct+∆
1
=
,
∂at+∆ µt+∆ + νt+∆
Rv c
Rv c
R∞
R∞
1
where µt ≡ t (1 + τc )e t {gs −rs }ds dv and νt ≡ t wv (θχv ) σ e t {gs −rs }ds dv.

The objective function of self t is given by:
Ut = [ln ct + θ ln(1 − lt )] ∆ + (1 + θ)Ωt+∆ ln ct+∆ ,where Ωt ≡R∞
t

exp {−[% · (v − t) + φ(v − t)]} dv. Given the sequence of {rv , wv }∞v=t , self tchooses ct and lt so as to maximize this objective function.
Using the first-order conditions and limiting ∆ to zero, we obtain χt = (1 + τc )/wt andR∞ξ = 1/ 0 exp {−(%t + φ(t))} dt. Then, (36) and (37) are derived.

37

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41

(a) United States
Labor-related discount rate

Money-related discount rate (+)

negative

positive

4317

1885

(70%)

(30%)

(b) Japan
Labor-related discount rate

Money-related discount rate (+)

negative

positive

3426

1218

(74%)

(26%)

Table 1. Differences in Discount Rates

42

(a) ρc > ρlcv
cv = wv lv

B

0

Point A: uv0 (c(lvv)) e−(ρl −ρc )(v−t) = wv
0

Point B: uv0 (c(lvv)) = wv

A

lv

0

(b) ρc < ρlcv
cv = wv lvC0

Point C: uv0 (c(lvv)) e−(ρl −ρc )(v−t) = wv
0

D0

Point D: uv0 (c(lvv)) = wv

lv

Figure 1. Time Inconsistency of Household’s Behavior

43

r

Increases in saving incentives

rE
rE0

savings

0

Figure 2. Saving Incentives and Equilibrium Interest Rate

44