Discussion Papers In Economics
And Business

Welfare and Tax Policies in a Neoclassical Growth Modelwith Non-unitary Discounting

Ryoji Ohdoi, Koichi Futagami and Takeo Hori
Discussion Paper 15-14

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWelfare and Tax Policies in a Neoclassical Growth Modelwith Non-unitary Discounting

Ryoji Ohdoi, Koichi Futagami and Takeo Hori
Discussion Paper 15-14

May 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWelfare and Tax Policies in a Neoclassical Growth Modelwith Non-unitary Discounting∗
Ryoji Ohdoi†
Tokyo Institute of Technology

Koichi Futagami‡
Osaka University

Takeo Hori§
Aoyama Gakuin University

Abstract
In this paper, we propose a neoclassical growth model with non-unitary discounting, where an individual discounts her future utilities from consumption and leisurediﬀerently. Because this non-unitary discounting induces the individual’s preference
reversals, we regard one individual as being composed of diﬀerent selves. Then we
derive the closed-form solution of the recursive competitive equilibrium in which her
diﬀerent selves behave in a time-consistent way in all periods. With regard to welfare
analysis, we obtain the following three main results. First, the selves in any period
strictly prefer the planning allocation to the laissez-faire allocation if they are given
the same value of a state variable in both situations. Second, the selves in the longrun can prefer the latter to the former allocation if we focus on the overall equilibrium
paths in both situations. Third, a time-consistent tax policy designed by a benevolent
government replicates the planning allocation.
JEL classification: E21; H21; O41
Keywords: Non-unitary discounting; Time-inconsistency; Intrapersonal game; Markovperfect equilibrium; Time-consistent tax policy.

∗

We would like to thank Takao Asano, Shin-ichi Fukuda, Ken-ichi Hashimoto, Tomohiro Hirano, Kat-suhiko Hori, Shinsuke Ikeda, Hirokazu Ishise, Keizo Mizuno, Yoshiyasu Ono, Miho Sunaga, Hajime Tomura,and seminar participants at Kobe University, Kwansei Gakuin University, Okayama University, Osaka University, Tezukayama University, and The University of Tokyo for many helpful comments. Koichi Futagami
gratefully acknowledges financial support from G-COE (Osaka University) ‘Human Behavior and Socioeconomic Dynamics.’ The usual disclaimer applies.
†

Corresponding author: Department of Social Engineering, Tokyo Institute of Technology, 2-12-1, O-

okayama, Meguro-ku, Tokyo 152-8552, Japan; E-mail: ohdoi.r.aa@m.titech.ac.jp‡

Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,

Japan; E-mail: futagami@econ.osaka-u.ac.jp§

College of Economics, Aoyama Gakuin University, 4-4-25, Shibuya, Shibuya-ku, Tokyo 150-8366, Japan;

E-mail: t-hori@aoyamagakuin.jp

1

Introduction
Father: “Could you mow the yard tomorrow instead of playing football? Aftercompleting the job, I will give you $20.”
Son: “Really? I will. Then, I can buy a new computer game!”
Tomorrow has come.
Father: “Why are you going out to play football? Mow the yard! You promised
yesterday, didn’t you?”
Son: “Sorry Dad. I no longer think $20 is enough for the job.”
Why did the boy break his promise? Is it because he is a liar? Of course, there are anumber of possible answers to this question. One possibility, suggested by a large body ofexperimental evidence, is that preference reversals frequently occur over time in people’s
decision-making. As such, it could be that the boy first felt that $20 (or purchasing the
new game) was preferable, but by the following day, preferred his leisure activity.
Although the aforementioned explanation gives one hypothetical answer to the question,
it becomes convincing once we consider the domain eﬀect, or domain independence, often
referred to in experimental psychology literature. The domain eﬀect emerges when the
discount rates (or factors) diﬀer depending on their domains.1 In the above example, thedomain eﬀect emerges if the boy discounts the utilities from the monetary reward ($20)
and from enjoying the leisure activity (football) diﬀerently. For expositional convenience,
let R denote the utility from the monetary reward and F denote the utility from the leisureactivity. We assume R < F . That is, the boy will never mow the yard if he asked to doso right now. Next, suppose that, on the first day, he evaluates the utility from receiving$20 as β1 R, and that from playing football as β2 F , where β1 ∈ (0, 1) and β2 ∈ (0, 1) are
the discount factors specific to the monetary reward and leisure, respectively. Then, if theboy discounts enjoying leisure steeply enough that β1 R > β2 F , he will accept his father’s
job oﬀer on the first day.
Hereafter, we refer to such domain-specific discounting as non-unitary discounting. If
an individual discounts her future utilities in a non-unitary way, this can make her decisions
time inconsistent. There has been a recent upsurge of interest in models of time-inconsistent
1

Chapman (1996), based on experimental studies, notes that the discount rates may be specific tomoney and health. See also Chapman, Nelson, and Hier (1999) and Chapman, Brewer, and Leventhal(2001). On the other hand, Soman (2004) and Zauberman and Lynch (2005) show that diﬀerent discountrates may apply to time and money. Recently, using survey data from Uganda and both hypothetical andincentivized choices over diﬀerent goods, Ubfal (2014) shows that equal discount rates across goods are notsupported empirically. For an excellent discussion on the inconsistency of intertemporal choices due to time
discounting, see Frederick, Loewenstein, and O’Donoghue (2002).

1

preferences, as pioneered by Strotz (1955–56) and Pollak (1968). In this context, theindividual’s decision-making process is formulated as a dynamic non-cooperative gameplayed by her diﬀerent selves, where the current self is aware that her preferences mightchange in future, and makes the current decision taking this into account.2 However, muchof the literature focuses on a class of quasi-hyperbolic discounting, first proposed by Phelps
and Pollak (1968), and made popular by Laibson (1996, 1997).3 Therefore, the purpose ofthis paper is to develop a simple dynamic theory of non-unitary discounting.
We incorporate non-unitary discounting into a discrete-time neoclassical growth modelwith endogenous labor supply. Then, we assume that an individual discounts her oneperiod utility functions of consumption and leisure diﬀerently. Therefore, as the boy doesin the earlier example, the individual changes her mind about the relative importance ofconsumption and leisure as time progresses. There are very few studies on non-unitary
discounting.4 Compared with them, our formulation has such an advantage that the applicability of dynamic programming enables us to derive the closed-form solution of therecursive competitive equilibrium (RCE)Within this framework, we conduct a welfare analysis to consider the central question ofthis paper, namely whether a policy intervention is desirable for the individual. The answer
to this question is no longer straightforward, because as a result of a lack of commitment,each self of the social planner is also involved in strategic interactions with her other selves.
In fact, in their model with quasi-hyperbolic discounting, Krusell, Kuruşçu, and Smith(2002) show that the allocation in the RCE surprisingly attains strictly higher welfare
than that in the planning allocation. More recently, Hiraguchi (2014) extends Krusell et
al. to a general model of non-constant discounting, including the original as a special case,and shows that their result is robust. At the same time, a welfare comparison between
the competitive and planning economies gives rise to the following problem. In order
to correctly identify which achieves higher welfare in each period, we must control the2

For example, see Peleg and Yaari (1973) and Goldman (1980) for the game-theoretic foundations of thesolution concepts in Strotz (1955–56) and Pollak (1968).
3

Applications based on quasi-hyperbolic discounting now cover a broad range of topics, including con-sumption and saving decisions (Laibson, 1996; 1997; Krusell and Smith, 2003), labor supply decisions(DellaVigna and Paserman, 2005), and optimal taxation (Krusell, Kuruşu̧, and Smith, 2002), among others.
4

By formulating a continuous-time model of non-unitary discounting, Futagami and Hori (2010) derive

the equilibrium strategies of an interpersonal game in the same way as Barro (1999). However, they assume
away capital accumulation and, thereby, overlook how the dynamics of state variables aﬀect the equilibrium
paths of the game. In an accompanying paper, Hori and Futagami (2013) incorporate capital accumulation
and a learning-by-doing externality into the model. They then investigate the interactions among development, patience, and saving behaviors. However, although they provide important explanations to some
empirically observed results, they do not perform any welfare analyses.

2

diﬀerence in the dynamics of the state variables between the two economies. Namely, we
can not evaluate which of the economies performs better if we focus only on their overall
equilibrium paths.5
To overcome this problem, we conduct the welfare comparison in two distinct ways,
namely by considering both on- and oﬀ-paths of the market economy and the planningeconomy. First, we consider the hypothetical situation that, in an arbitrarily given period,a self faces the same value of a state variable in both economies. Our analysis within thisframework shows that the welfare to the self in the social planning case is always strictly
higher than that in the RCE. This means that welfare improvement is always possible fromthe realized allocation in the RCE, which contrasts sharply with the findings of Krusell etal. (2002).
Then, we conduct a welfare comparison between the overall equilibrium paths of thetwo economies. We show that it depends on the relative degree of impatience whether or
not the planning allocation is more Pareto eﬃcient than the allocation in the RCE. The
following two cases arise. If the individual discounts future leisure more steeply than she
does future consumption, the planning allocation is preferable to the laissez-faire allocation
for all selves. However, if the reverse is true, they are not Pareto ranked. We show that
there exists a unique threshold period such that, before this period, any selves strictlyprefer the planning allocation. However, after this period, they strictly prefer the laissezfaire allocation. This means that the allocation in the RCE may achieve a more desirable
outcome than the social planner does for the selves in later periods.
In addition, we incorporate a government’s activity in the baseline model. Assumingthat the government imposes taxes on savings, wages, and interest income, and that it can
not commit to its future policies as the social planner, we show that the time-consistenttax policy replicates the allocation by the social planner. However, this result makes it
diﬃcult to judge the validity of government interventions. If an individual discounts future
leisure more steeply than she does future consumption, the implementation of such a policy
is Pareto improving. However, if the reverse is true, the second result means that the timeconsistent tax policy is preferred by the selves in earlier periods, but not in later periods.
This makes it diﬃcult to resolve the intertemporal conflict for individuals when there areno commitment mechanisms.
As already stated, the most closely related literature to our study is the set of studies
on time-inconsistent preferences resulting from non-geometric discounting. However, ourmodel is also related to a class of preferences exhibiting temptations. Among others,
Banerjee and Mullainathan (2010) consider a two-period, many-good economy, and classify
the goods into two types. The first is a standard good, the consumption of which in both
5

In Section 4.4 (pp. 56) of their paper, Krusell et al. make the same argument.

3

periods attains the individual’s lifetime utility in period 1. The second is a “temptation
good,” the consumption of which in period 2 is not valued in period 1, but becomes toattain utility once period 2 has come. In their two-period model, temptation goods areinterpreted as those with a discount factor of 0. If we set β2 = 0 in the example at the
beginning of the introduction, playing football is a temptation good for the boy. Thus, ourmodel of non-unitary discounting is closely related to their notion of temptation.6
The remainder of this paper is organized as follows. Section 2 sets up the simple
neoclassical growth model with non-unitary discounting, and explains the mechanism of
the time inconsistency in the model. Section 3 formulates the individual’s decision-making
as a dynamic intrapersonal game and characterizes the RCE. Section 4 solves the social
planner’s intrapersonal game, and examines the welfare implications of the RCE. In thissection, we also explain the intuitions behind our results, and discuss how they diﬀer fromthe results in models that use other types of discounting. Then, Section 5 introduces thegovernment’s tax policy, and derives the time-consistent tax policy. Finally, Section 6
concludes the paper.

2

A Macroeconomic Model of Non-unitary Discounting

Time is discrete and indexed by t = 0, 1, 2, . . . For simplicity, we assume there is no uncertainty. There exists a unit mass of homogenous individuals, each of whom is endowedwith one unit of working time in each period. Let ct and lt denote the individual’s consumption and labor supply, respectively. In our model, we assume that in period t ≥ 0,
the individual’s preference is given by the following utility function:
Ut =

∞
∑
[
]
(βc )i uc (ct+i ) + (βl )i ul (lt+i ) ,

(1)

i=0

where uc and ul denote the one-period (dis)utility functions of consumption and labor
supply, respectively. Then, βc ∈ (0, 1) and βl ∈ (0, 1) are the discount factors of consumption and labor supply, respectively. As is clear from (1), if βc = βl , we have thepreferences of standard geometric discounting.7 In contrast, when βc ̸= βl , a problem oftime inconsistency may emerge, as explained in the introduction.
Our purpose is to obtain the implications of non-unitary discounting in a neoclassicalgrowth model with endogenous labor supply. Hereafter, the one-period utility functions,6

In other words, in the model of Banerjee and Mullainathan (2010), time inconsistency occurs. In

contrast, as is well known, Gul and Pesendorfer (2001) propose a utility function (and give its axiomatic
foundations) that exhibits temptation, but that is free from time inconsistency.
7

Exponential discounting, which is the continuous-time counterpart of geometric discounting, was orig-

inally posited by Samuelson (1937).

4

uc and ul , are specified asuc (c) = ln c,ul (l) = ζ ln(1 − l),where ζ > 0 is a constant that governs the weight on the utility from leisure. Assumingthat capital fully depreciates after production in each period, the flow budget constraintof the individual is
kt+1 = rt kt + wt lt − ct ,(2)

where k denotes the individual’s holding of capital stock, and r and w represent the rental
price of capital and the wage rate, respectively.
The production function takes a Cobb–Douglas form, Yt = AKtα L1−α
, where Y , K, andt
L denote the amount of output, demand for capital, and demand for labor, respectively.8
Then, A > 0 is the level of total factor productivity and α ∈ (0, 1) is a constant which
specifies the share of capital income in total output. Let Xt ≡ Kt /Lt denote the ratio of
aggregate demand for capital to that for labor. Then, perfect competition results in
rt = r(Xt ) ≡ AαXtα−1 ,wt = w(Xt ) ≡ A(1 − α)Xtα .

(3)

We first describe the decision-making of the individual who does not consider the possibility of time-inconsistency. Solving the individual’s dynamic optimization problem, andsubstituting (3) and the market clearing conditions, Kt = kt and Lt = lt , into the resulting
equations, we havect /ct−1 = βc αA (kt /lt )−(1−α) ,
t

(4)
α

ζct /(1 − lt ) = (βc /βl ) (1 − α)A (kt /lt ) .

(5)Then, we guess the following saving decision for the individual: kt+1 = sktα lt1−α , where sis the variable still to be solved. Substituting this guess and c = (1 − s)Ak α l1−α into (4),
we obtain s = βc α. On the other hand, by applying this result to (5), we obtain
lt =

1−α
.
1 − α + ζ(1 − βc α) (βl /βc )t(6)Thus, when βl > (<)βc , labor supply decreases (increases) over time and approaches zero(one). This is because when βl > (<)βc , the individual puts more (less) weight on theutility of future leisure than on that of future consumption. In this case, he or she supplies
less (more) labor over time.

8

Thus, in specifying the functional forms, we basically follow Krusell, Kuruşçu, and Smith (2002), who

restrict their attention to the case of log one-period utility functions and a Cobb–Douglas production
technology, with complete depreciation of physical capital after the production in each period.

5

Does the individual follow these decisions after the initial period? The answer is no. If
the individual again solves the problem at time t, he or she does not supply lt , but instead
will supply
l0 =

1−α
.
1 − α + ζ(1 − βc α)For example, consider the case of βl < βc , i.e., the individual puts less weight on future
leisure. Consequently, he or she initially plans to supply more labor in the future periods.
However, when the future period comes, he or she prefers to enjoy the leisure activity, as
the boy did in the introduction, and supplies less labor.

3

Recursive Competitive Equilibrium

As clarified in the previous section, as long as βc ̸= βl , the individual will change hermind about the relative importance between consumption and leisure as time progresses.
Thus, if she could commit to her future decisions, she would do so. However, in our real
economic life, it is sometimes not possible for consumers to commit to their future actions.
In this section, we characterize the recursive competitive equilibrium (RCE) by making the
restriction that commitment can not be achieved.
First, note that the utility function given in (1) is expressed as
Ut = uc (ct ) + ul (lt )[]+ βc uc (ct+1 ) + βc uc (ct+2 ) + (βc )2 uc (ct+3 ) + . . .
[]+ βl ul (lt+1 ) + βl ul (lt+2 ) + (βl )2 ul (lt+3 ) + . . . .
Define Uj,t+1 ≡ uj (jt+1 ) + βj uj (jt+2 ) + (βj )2 uj (jt+3 ) + . . . , where j = c, l. Then, the above
expression gives the following formula:
Ut = uc (ct ) + ul (lt ) + βc Uc,t+1 + βl Ul,t+1 .

(7)

In what follows, the variables in the next (current) period are represented by those with
(without) a prime. For example, we use k ′ and k rather than kt+1 and kt , respectively.
The individual’s budget constraint is now expressed ask ′ = r(X)k + w(X)l − c,where r(X) and w(X) are given in (3).
Now we are in position to state the individual’s time-consistent decision-making. However, we first need to make a few assumptions. First, we assume the individual, in any
given period, is aware of her preference for change, and makes her current decision taking
this into account. Therefore, her decision-making process is formulated as a dynamic game6

played by her diﬀerent selves. Second, we focus on Markov strategies such that each self
makes a decision based on the values of individual state k and the aggregate capital–laborratio, X. Third, each self rationally perceives the law of motion of X and her next self’s
decisions, given by
X ′ = G(X),k ′′ = g(k ′ , X ′ ),l′ = l(k ′ , X ′ ),where G, g, and l are functions still to be solved. Finally, we seek a stationary equilibrium
in which the functional forms of g, l, and G are time invariant.
Applying the above assumptions to (7), the current self’s problem is given by thefollowing Bellman equation:{
V (k, X) = max
′
k ,l

(
)
ln r(X)k + w(X)l − k ′ + ζ ln(1 − l)()()}+ βc Vc k ′ , G(X) + ζβl Vl k ′ , G(X) ,(8)

where function V is the value function associated with the problem given in (8), and Vc
and Vl are recursively defined by the following functional equations:Vc (k, X) = ln (r(X)k + w(X)l(k, X) − g(k, X)) + βc Vc (g(k, X), G(X)) ,(9)

Vl (k, X) = ln(1 − l(k, X)) + βl Vl (g(k, X), G(X)) .

(10)

We denote the solution to the problem in (8) as ge(k, X) and el(k, X). The stationary
equilibrium requires ge = g and el = l, for all (k, X) in the domains of g and l. Note that(8)–(10) jointly exhibit a recursive formulation of the maximization problem. Thus, we
can solve the problem using a dynamic programming technique.
Then, the stationary RCE is defined as follows:
Definition 1. The stationary RCE is given by the value functions Vj (j = c, l), the current
self ’s decision rules g and l, the law of motion for the aggregate capital–labor ratio G, andthe factor prices r and w, such that
1. Given g(k, X) and l(k, X), Vc (k, X) and Vl (k, X) solve the functional equations (9)and (10), respectively;
2. The current self ’s decision rules are stationary Markov perfect: that is, g(k, X) andl(k, X) solve the problem given in (8);3. The decision rules g(k, X) and l(k, X) are consistent with their aggregate counterparts;
g(K, K/L) = G(K/L)L′ ,l(K, K/L) = L.

4. The factor prices r(X) and w(X) are given by (3).
7

Since r(X) and w(X) are already given, our task is to obtain the functions Vj , g, l, andG that satisfy properties 1–3. Thus, we have the following proposition.
Proposition 1. The RCE is given by
1. Vj (k, X) = aj + bj ln X + dj ln(k + φX), with bj dj and φ given by
bc = −

1−α,(1 − βc )(1 − βc α)11, bl = −,1 − βc1 − βl1−α1+ζ +Ψ
φ=
,
α
1+ζ
dc =

dl =

1
,
1 − βlwhere Ψ is defined as
Ψ≡

βl
βc
+ζ
;
1 − βc1 − βl(11)

2. g(k, X) and l(k, X) are given byg(k, X) =Ψr(X)k,
1+ζ +Ψ

l(k, X) = 1 −r(X)ζ(k + φX);
w(X) 1 + ζ + Ψ(12)

3. G(X) = se AX α , whereΨα.
1+ζ +Ψ

se ≡(13)

Proof. See Appendix A.
Note that in Definition 1 and Proposition 1, k and X are arbitrarily given. This means
the RCE characterizes an individual’s optimal decisions both on and oﬀ the resulting
equilibrium path, which is analogous to a subgame-perfect equilibrium of extensive-form
games. Furthermore, the stationarity of the RCE means that g and l can give any selves’
optimal decisions once we enter the values of (k, X). Thus, the RCE given in Proposition
1 can give an individual’s optimal decision in any state and in any period.
Next, we derive the path of the recursive competitive equilibrium (hereafter, RCEpath). On the RCE path, firms’ aggregate capital (labor) demand K (L) is equal to the
supply k (l). Substituting (3) and X = k/L into (12), we have
L = l(k, k/L) ⇔ L = 1 −αζ(L + φ).
1−α1+ζ +Ψ

Substituting φ in Proposition 1 into the above equation, we obtain the equilibrium employment, as follows:
L = le ≡(1 − α)(1 + ζ + Ψ).
(1 + ζ)[ζ + (1 − α)(1 + Ψ)](14)

Hence, from (13) and (14), we obtain the RCE path.
Lemma 1. Given k0 > 0, the RCE path is given by the sequence {kte }∞t=0 , such thatekt+1
= G(kte /le )le ≡ se A(kte )α (le )1−α .

8Before turning to a welfare analysis in this model, we investigate the characteristics of
the RCE. First, we examine how the RCE path is influenced by changes in βc and βl . From
the definition in (11), Ψ is strictly increasing with respect to both βc and βl . On the other
hand, from (13) and (14), both se and le are strictly increasing functions of Ψ. Therefore,
these properties are summarized in the following lemma.
Lemma 2. On the RCE path, the saving rate and the labor supply are strictly increasing
functions of βj (j = c, l).
Second, we analyze whether the observational equivalence holds across the model with
our non-unitary discounting preferences and those with the standard geometric discounting
preferences. To that end, consider an economy in the same environment as ours, except∑∞ tthat the representative individual’s preference is given by t=0
β [ln ct + ζ ln(1 − lt )]. The
equilibrium path of this economy is given by the sequence {k̂t }∞t=0 , such thatk̂t+1 = βαA(k̂t )α (ˆl)1−α ,
1−α
ˆl =
.
1 − α + ζ(1 − βα)

(15)
(16)

Then, we obtain (15) and (16) by imposing βc = βl = β on (6).
For expositional convenience, let β̂ ≡ Ψ/(1 + ζ + Ψ). From (13) and (15), we find thatthe saving rate becomes the same in these two models if β = β̂. Furthermore, substituting
this result in to (16), we have
ˆl =

1−α
1 − α + ζ(1 − β̂α)=(1 − α)(1 + ζ + Ψ)
(1 − α)(1 + ζ + Ψ) + ζ [(1 + ζ + Ψ) − Ψα]=(1 − α)(1 + ζ + Ψ).
(1 + ζ)[ζ + (1 − α)(1 + Ψ)]

Then, (16) becomes (14) if β = β̂. This establishes the following proposition.
Proposition 2. Given Ψ (i.e., βc and βl ), the RCE path in this model becomes observationally equivalent to that in the standard growth model, with the discount factor given byΨ/(1 + ζ + Ψ).
By incorporating quasi-hyperbolic discounting into a simple neoclassical growth model,
Krusell, Kuruşçu, and Smith (2002) show that the observational equivalence holds between
their model and the standard geometric discounting model.9 Barro (1999) shows the same
property in his continuous-time model of non-constant rate of time preferences. Proposition
2 in our paper states that the observational equivalence holds between our non-unitary
discounting and the standard discounting. At first glance, this result appears to showthat the standard model, or the model of quasi-hyperbolic discounting, can replicate all
9

See Proposition 2 of their paper.

9

our findings. However, this is not correct because, as we show in the next section, ourmodel proposes welfare implications that diﬀer markedly from those of the aforementioned
underlying preferences.

4

Welfare Implications

In this section, we examine the welfare implications of the RCE in our non-unitary discounting model. To this end, we first investigate the planning problem. The social planner’s
preferences are the same as those of the individual in Section 3, and she can not commit
to her future selves’ decisions. As in the case with the RCE, we model this situation as an
intrapersonal game played among diﬀerent selves of the planner. Then, we compare theRCE and the social planning in terms of resulting welfare.

4.1

Social Planning Problem

Note that, in contrast to the market economy in Section 3, the planner can directly aﬀect
the resource constraint by her decisions. This is because she knows that the aggregate
variables must coincide with their corresponding variables for each individual (i.e., (K, L) =(k, l)).
Assume that the current planner expects that if the value of capital is given by k, thenext self’s decisions of savings and labor supply are given by g sp (k) and lsp (k). As in thecase of the RCE, we focus on the case in which the functional forms of g sp and lsp are
stationary. Therefore, the problem of the current planner is given by{
sp ′ }α 1−α′sp ′V sp (k) = max
ln(Ak
l
−
k
)
+
ζ
ln(1
−
l)
+
β
V
(k
)
+
ζβ
V
(k ) ,
c
l
c
l
′
k ,l

(17)

where Vcsp (k) and Vlsp (k) are, respectively, given by the following functional equations:[]Vcsp (k) = ln Ak α (lsp (k))1−α − g sp (k) + βc Vcsp (g sp (k)),(18)

Vlsp (k) = ln(1 − lsp (k)) + βl Vlsp (g sp (k)).

(19)

Then, we arrive at the following lemma.
Lemma 3. The Markov-perfect equilibrium of the intrapersonal game among the diﬀerent
selves of the social planner is given by lsp (k) = lsp and g sp (k) = ssp Ak α (lsp )1−α , wheressp = βc α,
lsp =

(20)

1−α
.
1 − α + ζ(1 − βc α)Proof. See Appendix B.

10

(21)

Once the social planner’s time-consistent decision rule is given by (g sp (k), lsp ), thespoutcome path of the social planning is given by the sequence {ktsp }∞t=0 , such that kt+1 =g sp (ktsp ), with historically given k0 . Since βl does not aﬀect ssp or lsp , we can immediately
show the following proposition.
Proposition 3. Under the assumptions of log one-period utility and the Cobb–Douglas
production function, the social planner’s time-consistent decision-making is intrinsicallythe same as those who discount their future consumption and leisure equally.
As stated explicitly in this proposition, this result owes much to the specifications of
the functional forms. Nevertheless, we think that this proposition is noteworthy, because in
the RCE with the same specifications, this result does not hold but only the observational
equivalence holds.
The intuition behind this proposition is explained as follows. First, note that in any
given period, the current self can influence the next self’s decisions only by choosing k ′ , thevalue of which corresponds to the decision node used in game theory. This is a common
characteristic in models of intrapersonal games. However, as Lemma 3 shows, the valueof lsp does not depend on capital in this model.10 Furthermore, the planner perceives
this result. Then, the current planner gives up trying to control her next self’s labor
supply, and instead tries to manipulate her next self’s savings by choosing her own saving
decision. Since the model exhibits standard geometric discounting when we focus only
on the utility from consumption, such decision-making induces the standard result forintertemporal optimization. This is the reason why the saving rate in our non-unitary
discounting model becomes the same as that in the standard model when we consider
the social planner’s decision-making. Finally, since the saving behaviors are the same
between these two models, the amount of physical capital in all periods is the same as well.
Therefore, the constant value of the labor supply in our model is essentially determined by
the same value as that in the standard discounting model.

4.2

Welfare Comparison: Preliminary Results

Having characterized the decision-making by diﬀerent selves of the social planner, we now
conduct a welfare comparison between the RCE and the social planning. In doing so, we
first consider a hypothetical situation in which, for an arbitrarily given period, a self inboth economies faces the same value of the state variable, k. Of course, since se ̸= ssp andle ̸= lsp , the economies experience diﬀerent dynamic equilibrium paths, even if their initialvalues of physical capital are equal (see Lemma 1 and Lemma 3). Nevertheless, to consider
such a situation is important for two reasons. First, as is well known, in models with
10

This result is directly the result of the specifications of the functional forms.

11preference reversals, we must consider the welfare of the selves in arbitrary periods, notonly in the initial period. Second, in order to correctly identify which allocation achieves
higher welfare, the RCE or the social planning, we must impose a control on the diﬀerencein the state variable kt , because otherwise the diﬀerence in the state would itself bringabout a diﬀerence in welfare.
An individual’s welfare with her state given by k is V sp (k) in the social planning,whereas it is V e (k) in the RCE, where V e (k) is defined as
V e (k) ≡ V (k, k/le ).
Briefly, V e (k) is the welfare evaluated when the market equilibrium conditions (K = k, L =le ) are imposed.
Our analysis essentially follows that of Krusell, Kuruşçu, and Smith (2002), and proceeds by taking the following three steps. First, we define the following welfare evaluation
function, W :W (s, l, k) ≡1ζln(1 − s) +ln(1 − l)1 − βc1 − βlα1+
ln k +[βc α ln s + ln A + (1 − α) ln l] .
1 − βc α(1 − βc )(1 − βc α)(22)The derivation of W (s, l, k) is given in Appendix C. Intuitively, W (s, l, k) is obtained byevaluating the individual’s utility after imposing c = (1 − s)Ak α l1−α and k ′ = sAk α l1−α .
Thus, by its definition, the following identity holds:
V x (k) ≡ W (sx , lx , k),x = e, sp.

Second, we show that there exists a unique (s∗ , l∗ ) that maximizes W (s, l, k). Since
this function in (22) is strictly concave in (s, l), the necessary and suﬃcient conditions for(s∗ , l∗ ) are given by
1
1
βc α
1
=
,
1 − βc 1 − s(1 − βc )(1 − βc α) s

ζ
1
1−α
1
=
.
1 − βl 1 − l(1 − βc )(1 − βc α) l

Therefore, we haves∗ = βc α,
l∗ =

(23)

(1 − α)(1 − βl ).
(1 − α)(1 − βl ) + ζ(1 − βc )(1 − βc α)(24)Finally, we complete the analysis by checking which of V e (k) and V sp (k) is nearer toW (s∗ , l∗ , k). We can show the following lemma.
Lemma 4. ssp (≡ s∗ ) ⋛ se and l∗ ⋛ lsp ⋛ le if and only if βc ⋛ βl .

12(a) Case of βc > βl(b) Case of βl > βc

Figure 1: Contours of W and the relationship among (s∗ , l∗ ), (se , le ), and (ssp , lsp )Proof. Recall that if βc = βl , (se , le ) = (ssp , lsp ) = (s∗ , l∗ ). From (13), (20), (23), and theresult obtained in Lemma 2 that se is increasing in βl , we can show that ssp ≡ s∗ ⋛ se ⇔βc ⋛ βl . On the other hand, from (14), (21), (24) and the result in Lemma 2 that le isincreasing in βl , we can show that l∗ ⋛ lsp ⋛ le ⇔ βc ⋛ βl .
Having obtained Lemma 4, we graphically examine which achieves the higher welfare
in the (s, l)-plane. In panels (a) and (b) of Figure 1, (s∗ , l∗ ) is located at point O, andthe closed curves represent the indiﬀerence curves. The value of the welfare evaluation
function W (s, l, k) increases as the curves approach O. At any point on the curve passing
through point A (B), V e (k) (V sp (k)) is achieved. From Lemma 4, the distance from point
A to point O is necessarily farther than that from point B to point O, as long as βc ̸= βl .
Furthermore, from this lemma, we can show that points A and B are located in the same
quadrant of the coordinate plane, with its origin given by point O. This means that whenβc ̸= βl , the indiﬀerence curve passing through point A is always located outside the curve
passing through point B. Therefore, we have the following proposition.
Proposition 4. Suppose that βc ̸= βl . Then, V e (k) < V sp (k) holds for all k > 0.
Consider an economy that starts in a given period t with its state given by k. Proposition 4 shows that as long as βc ̸= βl , the social planning always achieves higher welfare
than the RCE, given any t and k. This result contrasts sharply with the findings of Krusell,
Kuruşçu, and Smith (2002) who show that the RCE always performs better than the planning economy in their quasi-hyperbolic discounting model. Recently, Hiraguchi (2014)
extended the work of Krusell et al. to a more general model of non-constant discount13

ing, including the original work as a special case, and showed that their result is robust.
Therefore, the findings of Hiraguchi (2014) make our results stand out even further.
The intuition behind this result is explained as follows. We simplify the discussion byfocusing on the case of βl > βc , which induces the situation in which the current self wants
her future selves to enjoy leisure more than they actually do. Note that this gives thecurrent self an incentive to save more amount of the goods. Note too that this strategic
aspect potentially works in both the RCE and the social planning. However, in the latter
economy, such a strategic interaction has no impact on the resulting allocation, since theplanner perceives that she can not manipulate her future selves’ labor supply, as explainedin the previous subsection. Accordingly, both ssp and lsp are free from this intrapersonal
strategic interaction. In contrast, in the RCE, a next self’s labor supply decision is l(k ′ , X ′ ),which can be predicted by the current self. Then, motivated by a desire for self-control,
the current self’s saving rate is excessively high.

4.3

Welfare Comparison of Equilibrium Paths

As already noted, the RCE and the social planning generate diﬀerent equilibrium sequencesof physical capital. On the equilibrium path, {ktx }∞t=0 , for x = e, sp, the welfare of a self inperiod t is given by V x (ktx ). Since k0e = k0sp = k0 , Proposition 4 indicates that the welfareof the initial self is always higher in the social planning than it is in the RCE. Then, does
this result apply to the other selves?
First, we consider the case of βc > βl , where se < ssp and le < lsp are satisfied. Sincex )α (lx )1−α is given by
the solution of the diﬀerence equation ktx = sx A(kt−1}{]
1 − αt [ x
x 1−α
x
t
ln s A(l ),kt = exp α ln k0 +
1−α

(25)

it follows thatkte < ktsp

∀t = 1, 2, . . .

On the other hand, from Proposition 4, we have already shown that V e (k) < V sp (k), forall k. Furthermore, we can readily show the following lemma.
Lemma 5. V x (k) is strictly increasing in k for both x = e and x = sp.
Proof. From the identity V x (k) ≡ W (sx , lx , k) and the definition of W in (22), V x (k) is
expressed asαln k + Ax ,1 − βc α
where Ax is a collection of other terms, independent of k. The above equation shows thatV x (k) =V x (k) is strictly increasing in k.
Therefore, we have the following proposition, which shows that the allocation on the
RCE path is Pareto dominated by that of the social planning.
14

Proposition 5. Suppose that βc > βl . Then, V e (kte ) < V sp (ktsp )∀t = 0, 1, 2 . . .

Next, consider the case of βl > βc . From the foregoing analysis, the following inequality
then holds:
kte > ktsp

∀t = 1, 2, . . .

Since V e (k) < V sp (k) holds for all k in this case, the welfare comparison is not straightforward. Therefore, we first focus on the welfare comparison of the steady states.
When the saving rate and labor supply are constant over time, the steady-state valueof physical capital is given by1

k = Kss (s, l) ≡ (sA) 1−α l.

(26)Thus, the steady state of k in the RCE is Kss (se , le ), while that in the social planning is
Kss (ssp , lsp ). On the other hand, we can define the function, Wss (s, l), as follows:
Wss (s, l) ≡ W (s, l, Kss (s, l))
[
]
α
1
1
ζ
=
ln (1 − s)s 1−α A 1−α l +ln(1 − l).
1 − βc1 − βlFrom the definition of Wss (s, l) and the identity V x (k) ≡ W (sx , lx , k), we can verify thatV x (Kss (sx , lx )) ≡ Wss (sx , lx ).
∗ ), where
By a simple calculation, we find that Wss (s, l) is maximized at (s∗ss , lsss∗ss = α,∗lss=

1 − βl.
1 − βl + ζ(1 − βc )Thus, we have the following lemma.
Lemma 6. Given βc , there exists a unique β l ∈ (βc , 1), such that ssp < se < s∗ss and∗ if and only if β < β < β .
lsp < le < lsscll

Proof. From the proof of Lemma 4, we have ssp < se and lsp < le ⇔ βl > βc . Our∗ hold. Fromremaining task is to derive the condition under which se < s∗ss and le < lssthe definition in (13), se converges to α(= s∗ss ) as βl → 1. Since se is increasing in βl ,se < s∗ss , for all βl ∈ (0, 1). On the other hand, from the definition of le in (12), we have
∗ , we havethat le converges to 1/(1 + ζ) as βl = 1. Finally, from the above definition of lss∗ = 1/(1 + ζ) when β = β , and l∗ → 0 when β → 1. Since le is increasing in β , there
lss
c
l
l
l
ss
∗ if and only if β < β . These results show
exists a unique β l ∈ (βc , 1), such that le < lssllthat this lemma is true.
The previous lemma shows that V e (Kss (se , le )) > V sp (Kss (ssp , lsp )). In other words,
the RCE achieves a higher steady-state welfare than the social planning does if βc < βl < β l .
On the other hand, it is ambiguous which achieves higher welfare if βl > β l . Therefore, we
15

focus on the former situation. Since V e (k0 ) < V sp (k0 ), there exists at least one T , suchthat V e (kte ) > V sp (ktsp ) if t ≥ T . Furthermore, in Appendix D, we show that this period
T is unique. Therefore, we have the following proposition.
Proposition 6. Suppose that βc < βl < β l . Then, there exists a unique T ∗ > 0, such thatV e (kte ) > V sp (ktsp ) if and only if t ≥ T ∗ .
Proof. See Appendix D.
Proposition 6 states that the RCE allocation can achieve a more desirable outcome
than the social planner allocation for the selves in later periods. The intuition behind thisresult is explained as follows. As already stated, when βl > βc in the RCE environment,each self can not help saving excessively. Indeed, as shown in Proposition 4, this induces
a welfare loss to herself. However, such a decision by the current self is favorable for her
future selves, because their assets increase. Therefore, in terms of a welfare comparison
between the equilibrium paths, the RCE can be more desirable in the long run.
Based upon the above result, can we conclude that in the long run, the selves in alaissez-faire environment (surprisingly) do better jobs than the social planner? We must
be cautious when answering this question, because as mentioned in Section 4.2, we must
control for the diﬀerence in the state variable before discussing the eﬃciency of the equilibrium. To clarify this point, we focus on the case of βc < βl < β l . Then, from Propositions
4 and 6, there exists a period t ≥ T ∗ , such that the following two inequalities are satisfied
simultaneously:
V sp (ktsp ) < V e (kte ) < V sp (kte ).
The first inequality comes from Proposition 6, which shows that the RCE is more desirable
than the social planning for the self in this period, because she has greater assets. On theother hand, the second inequality is obtained from Proposition 4, and shows that, given kte ,the self strictly prefers the allocation by the social planner to the allocation on the RCE
path. Thus, the RCE is suboptimal, and welfare improvement is always possible fromits realized allocation. This finding is the motivation for our interest in the possibility of
government intervention, which we examine in the next section.

5

Analysis of Tax Policies

Motivated by the results in the previous section, we now introduce the government’s activity
into the competitive economy examined in Section 3. We assume that the government
imposes taxes on individuals’ wage income, interest income, and savings. Our goal is todesign an optimal tax policy that is time consistent. However, before we do that, we mustfirst qualitatively examine how the RCE is aﬀected by the government taxes, when all tax
16

rates are constant over time. This will provide us with a benchmark for the main analysis.
Then, we examine the optimal policy when the current government seeks to maximize thecurrent self’s utility, but is not able to commit to its policies in future periods.

5.1

The RCE with a Time-invariant Tax PolicyLet τr ∈ (0, 1), τw ∈ (0, 1), and τi ≥ 0 denote the rates of interest income, wage income,and savings. The budget constraint of the individual now becomes(1 + τi )k ′ = (1 − τr )r(X)k + (1 − τw )w(X)l − c.
We further assume that there is no government expenditure and its budget must be balanced in each period. Consequently, the government’s budget constraint is given byτr r(X)K + τw w(X)L + τi K ′ = 0.

(27)

From (27), we can readily find that one of the tax rates is determined by the other two rates.
Thus, we choose τw and τi as independent variables, which are denoted by τ = (τw , τi ).
As in the case of the laissez-faire environment in Section 3, each self rationally perceives
the law of motion for the aggregate demand of the capital–labor ratio, given by
X ′ = G(X, τ ).
We guess that the labor supply on the RCE path is constant over time, based on the
result in Section 3. In that case, on the RCE path, the labor supply in any period is givenby the constant, le . Therefore, K ′ = G(X, τ )L′ = G(X, τ )L. This assumption is verified
to be true when we solve for the equilibrium. Then, from Eq. (27), τr is given byτr = τr (X, τ ) ≡ −τw w(X) + τi G(X, τ )
.
r(X)X

(28)

Next, we define the following new functions:r̂(X, τ ) = (1 − τr (X, τ ))r(X),ŵ(X, τ ) = (1 − τw )w(X),and a new variable:k̂ ′ = (1 + τi )k ′ .
We assume that the current self expects that its future selves’ savings and labor supply
are given by k̂ ′′ = g(k ′ , X ′ , τ ) and l′ = l(k ′ , X ′ , τ ), respectively, where g governs the futuresavings, including tax.
The current self’s problem can be stated as follows:{ ()
V (k, X, τ ) = max ln r̂(X, τ )k + ŵ(X, τ )l − k̂ ′ + ζ ln(1 − l)k̂′ ,l()()}+ βc Vc (1 + τi )−1 k̂ ′ , G(X, τ ), τ +ζβl Vl (1 + τi )−1 k̂ ′ , G(X, τ ), τ
, (29)
17where the functions Vc and Vl satisfy the following functional equations:Vc (k, X, τ ) = ln (r̂(X, τ )k + ŵ(X, τ )l(k, X, τ ) − g(k, X, τ ))()+ βc Vc (1 + τi )−1 g(k, X, τ ), G(X, τ ), τ ,()
Vl (k, X, τ ) = ln(1 − l(k, X, τ )) + βl Vl (1 + τi )−1 g(k, X, τ ), G(X, τ ), τ .

(30)
(31)

The RCE with government taxes is defined in the same manner as in Definition 1.
Thus, we have the following lemma.
Lemma 7. The RCE with government taxes is given by
1. Vj (k, X, τ ) = aj + bj ln X + dj ln(k + φX), with bj and dj given by the same values
as in Proposition 1, but with φ now given by(1 − τw )(1 − α),α + τw (1 − α) − set (τ )φ = φet (τ ) ≡where
set (τ ) ≡Ψ[α + (1 − α)τw ].
Ψ + (1 + ζ)(1 + τi )2. g(k, X, τ ) and l(k, X, τ ) are given byg(k, X, τ ) =r̂(X, τ )(k + φX),
1+ζ +Ψ

l(k, X, τ ) = 1 −ζr̂(X, τ )(k + φX),
1 + ζ + Ψ ŵ(X, τ )respectively, where the value of Ψ is same as in Proposition 1;
3. G(X, τ ) = set (τ )AX α .
Proof. See Appendix E.
In the same way as in Section 3, the labor supply on the RCE path is given bylet (τ ) =(1 − τw )(1 − α)(1 + ζ + Ψ).
(1 + ζ)[ζ(1 + set (τ )τi ) + (1 − τw )(1 − α)(1 + Ψ)](32)

The derivation of this equation is given in Appendix E.

5.2

A Time-consistent Tax Policy

In this section, we consider the case in which the government can not commit to its policies
at future dates. The timing of events is as follows:
1. Observing kt , the government sets (τwt , τit ) so as to maximize the individual’s utility;
2. Given (rt , wt ) and (τwt , τit , τrt ), the individual and the firms make their decisions so
as to maximize their own objectives;
3. All markets clear. Thus, lt , ct , and the pair of factor prices (rt , wt ) are determined;18

4. The values of τrt and kt+1 are determined from the budget constraints of the individual and the government.
Following Krusell, Kuruşçu, and Smith (2002), we obtain the time-consistent optimal
tax policy using the one-shot deviation method. Suppose that the governments at all
future dates set their taxes as τ = (τ w , τ i ). Then, suppose that the current governmenteunilaterally deviates from τ and sets τe = (eτw , τei ). Let G(X,τe, τ ) denote the law of motionof X, which diﬀers from G(X, τ ) obtained in Lemma 7 owing to the current government’seone-period deviation. By definition, G(X,τ , τ ) ≡ G(X, τ ) (i.e., if τe = τ , they are thesame function).
First, we consider the current individual’s problem:{ ()
Ve (k, X, τe, τ ) = max ln r̂(X, τe)k + ŵ(X, τe)l − k̂ ′ + ζ ln(1 − l)k̂′ ,l

(
)
)}
(
e
e
+βc Vc (1 + τei )−1 k̂ ′ , G(X,τe, τ ), τ + ζβl Vl (1 + τei )−1 k̂ ′ , G(X,τe, τ ), τ.
(33)
The functions Vc and Vl satisfy (30) and (31), respectively. Therefore, these functions are
already given. Furthermore, a comparison between the right-hand-sides in (29) and (33)shows that Ve (k, X, τ , τ ) ≡ V (k, X, τ ). From (29) and (33), we find that the individual’s
optimal decision-making is qualitatively the same as that in Section 5.1. This is simply
because each individual makes her decision taking the factor prices and the taxes as given.
Next, we consider the current government’s decision-making. The underlying diﬀerence
between the decision of the government and that of the individual lies in the fact that the
government recognizes that it can aﬀect the market-clearing labor supply. In what follows,
we let elet (eτ , τ ) denote the market-clearing labor supply. Note that, owing to the current
government’s deviation, elet (eτ , τ ) is a diﬀerent function to let (τ ) given in (32).
We can now define the time-consistent tax policy.
Definition 2. The sequence {τt }∞t=0 , with τt = τ ∀t = 0, 1, 2 . . ., is the time-consistent tax
policy if∀k > 0, ∀t = 0, 1, 2 . . . ,()τ = arg max Ve k, k/elet (eτt , τ ), τet , τ .
τetThat is, the sequence of tax rates {τt }∞t=0 , with τt = τ∀t = 0, 1, 2 . . ., is the time-

consistent tax policy if any selves of the government can not obtain a strictly positive
welfare gain by their unilateral one-shot deviation from τ . Solving the above maximization
problem, we obtain the following proposition.
Proposition 7. The time-consistent tax policy τ = (τ w , τ i ) is given by()ζ
1 − βc βlτ w = 0, τ i =−1 .
1+ζ
βc 1 − βl
Accordingly, τ r is given by −βc τ i .
19

Proof. See Appendix F.
Since both the saving tax rate and the wage income tax rate turn out to be constantover time under the time-consistent tax policy, the saving rate and labor supply are givenby the pair (set (τ ), let (τ )). The following results follow as a corollary of Proposition 7.
Corollary 1. In the RCE with a time-consistent tax policy τ ,1. (set (τ ), let (τ )) = (ssp , lsp );2. τ i ⋚ 0 if and only if βc ⋛ βl .
Proof. Property 1 is shown to be true in the proof of Proposition 7 (see Appendix F).
Property 2 of this corollary is straightforward from Proposition 7.
The first property means that under the time-consistent tax policy, τ , the allocationby the social planner is replicated on the RCE path. Then, the second property showsthat the individual’s savings must be subsidized (taxed) when βc > (<)βl . This result is
intuitive given that, in the RCE, each individual’s saving rate is excessively low (high) if
she discounts future consumption at a lower (higher) rate than she does future leisure.
Finally, we must discuss whether the government can move the equilibrium in thecorrect direction by implementing such a time-consistent policy. If βc > βl , the answer
is undoubtedly yes, because Proposition 5 shows that the realized allocation by the social
planner attains higher welfare for all selves of the individual than does the laissez-faire
allocation. Thus, the implementation of the time-consistent tax policy improves the welfare
of all selves (i.e., the policy is Pareto improving).
However, if βl > βc , the answer is less clear, because the realized allocation by theplanner and that in the RCE are not Pareto ranked. As in Section 4.3, we focus on thecase of βc < βl < β l . Then, from Proposition 6, we find that the time-consistent tax policyin Proposition 7 is desirable for the selves of the individual in earlier periods, but not inlater periods. This is because, as explained in Section 4.3, the taxation on their earlier
selves’ savings reduces their assets, which, in turn, reduces their welfare.
Then, is the laissez-faire allocation really favorable for the future selves? The laissezfaire allocation in the RCE, described in Section 3, is no longer time consistent once we also
consider the benevolent government’s decisions. To see why, first note that the laissez-faire
allocation in the RCE is feasible by setting τt = (0, 0)∀t ≥ 0. However, from Proposition
4, such a policy is undoubtedly time inconsistent. Since V sp (k) > V e (k), for all k > 0,a self in any period strictly prefers the allocation designed by the social planner once she
actually faces a decision-making opportunity. Then, it is optimal for the government inthis period to impose a tax on her savings.

20

6Concluding Remarks

We propose a simple neoclassical growth model in which an individual’s non-unitary discounting induces preference reversals. We first characterize the recursive competitive equilibrium, and then show that the equilibrium allocation in our model is observationallyequivalent to the allocations obtained in standard geometric discounting models and quasihyperbolic discounting models. However, this does not mean that the findings in our model
have been already indicated. Rather, from the normative point of view, we derive the following new results which are summarized as follows.
First, a self in any period strictly prefers the planning allocation to the laissez-faire allocation, provided the state variable has the same value. Therefore, our welfare properties
diﬀer from those of previous studies, but the observational equivalence across the models
means we must be cautious when judging the performance of the market mechanism. Second, if we focus on the overall equilibrium paths of the planning and competitive economies,
the following two cases arise. If the individual discounts future leisure more steeply thanshe does future consumption, the planning allocation dominates the laissez-faire allocation in the Pareto sense. However, if the reverse is true, the selves in the earlier periods
strictly prefer the planning allocation, but in later periods, strictly prefer the laissez-faire
allocation. Thus, a conflict arises among the diﬀerent selves of the individual. Third, thetime-consistent tax policy designed by a benevolent government replicates the planning
allocation. This means that the time-consistent tax policy is desirable for the selves of
the individual in earlier periods, but not so in later periods. Thus, when there are nocommitment mechanisms, it is diﬃcult to resolve this conflict.
Our approach using non-unitary discounting is flexible enough to allow for some extensions. First, it will be fruitful to consider other goods, the future consumption of which
potentially exhibit non-unitary discounting. For example, the relationship between durable
and nondurable goods, or that between a consumption good and the individual’s healthstatus could be characterized in this way. Second, introducing heterogeneity among individuals could provide additional insight into the welfare implications presented here. For
instance, one type of individual might discount future leisure more steeply, while anotherdiscounts future consumption more steeply.11 If these two types of individuals coexist, ourresult on the desirability of a policy intervention will change. Introducing and analyzing
such aspects are left for future research.

11

In their continuous-time model without capital accumulation, Futagami and Hori (2010) show that the

behaviors of all such agents can be observationally equivalent.

21

7

Appendices

Appendix A

Proof of Proposition 1

To avoid confusion, we let r(X) = r, w(X) = w, and G(X) = G. The first-order conditionsof the problem in (8) with respect to k ′ and l are respectively given by∂V (k ′ , G)
1
∂Vc (k ′ , G)
= βc
+ ζβl
,
′
′
rk + wl − k
∂k
∂k ′
w
ζ
=
.
′
rk + wl − k
1−l

(A.1)
(A.2)

The proof follows using the method of “guess and verify” for the value functions, Vj (k, X)(j = c, l), and that for the law of motion of the aggregate capital–labor ratio, G(X).
Specifically, we make the following guess:Vj (k, X) = aj + bj ln X + dc ln(k + φX) ∀j = c, l,G(X) = sAX α .

Using the above guess for Vj (k, X), we can rewrite (A.1) as follows:()
k ′ + φX ′ = Ψ(dc , dl ) rk + wl − k ′ ,(A.3)

where the function Ψ(dc , dl ) is defined as12
Ψ(dc , dl ) ≡ βc dc + ζβl dl .

(A.4)

Substituting (A.2) and X ′ = G(X) = sAX α into (A.3) yields13
ζk + w [(1 + ζ)l − 1] + ζφG = Ψw(1 − l).
The above equation provides l as a function of k and X:
ζrk + w + φG
1+ζ +Ψ
wζr=1−(k + Λ(φ, s)X),
1+ζ +Ψwl = l(k, X) ≡ 1 −where function Λ(φ, s) is defined asΛ(φ, s) ≡w + φG.
rX

Since (r, w) are given in (3) and G is given by sAX α , Λ is reduced toΛ(φ, s) =1 − α + φs.
α12Hereafter, its arguments are omitted.

13

We use the fact that k′ = rk + ζ −1 w [(1 + ζ)l − 1] holds from (A.2).

22

(A.5)

Substituting (A.5) into (A.2) yields
c = c(k, X) ≡r
(k + ΛX).
1+ζ +Ψ

(A.6)

Then, substituting (A.5) and (A.6) into the budget constraint and rearranging the terms,
we obtaing(k, X) = rk + wl(k, X) − c(k, X)=

Ψr(k + ΛX) − φG(X).
1+ζ +Ψ

(A.7)

Thus, once the values of Ψ and Λ are determined, the functional forms of l(k, X) andg(k, X) are determined accordingly. Since Ψ and Λ are functions of dj , φ, and s, we now
consider derivations of Vj (k, X) and G(X).
Note that from (A.6) and (A.7), it follows that g(k, X)+φG(X) = Ψc(k, X). Therefore,
from the functional equation, (9), we obtain the following relationship:ac + bc ln X + dc ln(k + φX)= ac βc + (1 + βc dc ) ln c(k, X) + βc bc ln G(X) + βc dc ln Ψ.
From (A.6) and the guess for G(X), a comparison of the coeﬃcients of both sides leads to
bc = −(1 + βc dc )(1 − α) + βc bc α,

dc = 1 + βc dc ,Λ(φ, s) = φ,which results in
bc = −

1−α,(1 − βc α)(1 − βc )

dc =

1
,
1 − βc

φ=

1−α
.
α−s

(A.8)

Following the same procedure, we can obtain bl and dl , as follows:
bl = −

1,
1 − βldl =1.
1 − βl(A.9)

The value of s is determined from the consistency conditions, g(K, K/L) = G(K/L)L′and l(K, K/L) = L. Lastly, we guess that, in equilibrium, the labor supply is constantover time (i.e., L = L′ ). Then, from (A.7), (A.8), and G(X) = sAX α , the consistency
condition is rewritten assL =Ψα(L + φ) − (φs) ⇔1+ζ +Ψ

(
)
(
)
1−α
1−α
αΨ
L+
s= L+
.
α−s
α−s 1+ζ +Ψ

The latter equation has two solutions, namely Ψα/(1 + ζ + Ψ) and α + (1 − α)/L. However,
we can readily find that the second solution must be ruled out.14 Therefore, the value of
14

To see why, suppose otherwise. Then, φ = −L holds. However, if this equation holds, (A.6) becomesc(k, k/L) =r(k/L)(k − l(k/L)) = 0.
1+ζ +Ψ

That is, the amount of consumption becomes zero.

23

s in the recursive equilibrium is given by the first solution:s = se ≡Ψα.
1+ζ +Ψ

Once s is determined, from (A.8), φ is determined asφ=1−α1+ζ +Ψ
.
α
1+ζ

□

Thus, (A.4)–(A.10) jointly show Proposition 1.

Appendix B(A.10)

Proof of Lemma 3

The first-order conditions of the problem in (17) with respect to k ′ and l are respectively
given by∂Vlsp (k ′ )∂Vcsp (k ′ )
+
β
ζ
,
l
Ak α l1−α − k
∂k ′
∂k ′
(1 − α)Ak α l−α
1
= 0.
−ζ
α
1−α
′
Ak l
−k
1−l
1

= βc
′

(B.1)
(B.2)

We make the following guess for Vjsp :Vjsp (k) = aj + dj ln k.
From (B.1), we obtaing sp (k) =Ψ(dc , dl )Ak α l1−α ,1 + Ψ(dc , dl )(B.3)

where the Ψ(·, ·) is defined as in (A.4). Hereafter, we omit the arguments. From (B.2) and(B.3), we obtainlsp (k) = lsp ,where lsp =(1 − α)(1 + Ψ).
ζ + (1 − α)(1 + Ψ)

(B.4)

Since Ψ is unknown, we substitute (B.3), (B.4), and the guess for Vjsp into (18) and (19):
(
)
[
(
)]
1
Ψ
α sp 1−αα sp 1−αAk (l )+ βc ac + dc ln
Ak (l ),
ac + dc ln k = ln
1+Ψ
1+Ψ
[
(
)]
Ψ
sp
α sp 1−αal + dl ln k = ln(1 − l ) + βl al + dl ln
Ak (l ).
1+ΨSince the coeﬃcients of both sides must be equal, we can show that
dc =

α
,
1 − βc αdl = 0.

Substituting this result into the definition of Ψ, we have Ψ = βc α/(1 − βc α). Finally,
substituting the obtained value of Ψ into (B.3) and (B.4), we obtaing sp (k) = βc αAk α (lsp )1−α ,

lsp =

1−α
.
1 − α + ζ(1 − βc α)□This shows Lemma 3.
24

Appendix C

Derivation of (22)

If the saving rate and labor supply are determined such that they are constant over time,
we calculate the utility of a self in period t as follows.
∞
∑
{
}
α 1−α
Ut =
(βc )i ln[(1 − s)Akt+il] + (βl )i ζ ln(1 − l)
i=0

=

]
ζ
1 [
ln(1 − s) + ln(Al1−α ) +ln(1 − l)1 − βc1 − βl∞∑+(βc )i α ln kt+i .

(C.1)

i=0

From (25), we have ln kt+i = αi ln k + [(1 − αi )/(1 − α)] ln(sAl1−α ), where k is the value ofkt , historically given for self t. Substituting this result into the last term in (C.1) yields
Ut =

1
ζ
α
βc α ln s + ln A + (1 − α) ln lln(1 − s) +ln(1 − l) +ln k +
1 − βc1 − βl1 − βc α(1 − βc )(1 − βc α)= W (s, l, k).
□Appendix D

Proof of Proposition 6

Let qt denote qt ≡ ln kt . Then, from (25),xqtx = αt q0 + (1 − αt )qss,

x = e, sp,(D.1)

x is given by q x = ln K (sx , lx ), from (26). Substituting (D.1) into V x (k), we
where qss
ss
ss

obtain V x (ktx ) ≡ V x (t), where V : Z+ → R is given by15
V x (t) ≡αx[αt q0 + (1 − αt )qss
] + Ax .
1 − βc α

From Proposition 4, we know that V e (0) ≡ V e (k0 ) < V sp (k0 ) ≡ V sp (0) always holds.
Moreover, since we are considering the case of βc < βl < β l , V e (T ) > V sp (T ) as T → ∞.
Finally, subtracting V e (t) from V sp (t) yields
V e (t) − V sp (t) =e − q sp )α(1 − αt )(qssss+ Ae − Asp .
1 − βc α

Since the value of αt decreases as t increases, V e (t′ ) − V sp (t′ ) > V e (t) − V sp (t), for all t′ > t,e − q sp > 0. Note that this condition is automatically satisfied for the case of β > β .
if qssssclThus, there exists a unique T ∗ > 0, such that V e (t) > V sp (t) if and only if t ≥ T ∗ . This□

proves Proposition 6.
15

Z+ = {0, 1, 2, . . .} represents the set of all non-negative integers.

25Appendix E

Proof of Lemma 7

Throughout Section 5.1, we focus on the case of τ being time invariant. Therefore, we omit
the argument τ . Furthermore, for the same reason given in Appendix A, we let r̂(X) = r̂and ŵ(X) = ŵ. The first-order conditions of the problem in (29) with respect to k ′ and l
are respectively given by[]∂V (k ′ , G)
1
∂Vc (k ′ , G)
+ ζβl
=
βc
,
1 + τi∂k ′∂k ′r̂k + ŵl − k̂ ′
ŵ
ζ
=
.
′
1−l
r̂k + ŵl − k̂
1

(E.1)
(E.2)

Next, we define the following new variable:φ̂ = (1 + τi )φ.
We make the same guess for Vj and G as in Appendix A. Using (E.1), (A.3) is now givenby()
k̂ ′ + φ̂X = Ψ r̂k + ŵl − k̂ ′ .

(E.3)

Note that equations (E.2) and (E.3) are equivalent to (A.2) and (A.3), respectively. Therefore, we haveζr̂
(k + Λ̂X),
1 + ζ + Ψ ŵr̂
(k + Λ̂X),
c = c(k, X) ≡1+ζ +Ψ
Ψr̂
(k + Λ̂X) − φ̂G,k̂ ′ = g(k, X) ≡1+ζ +Ψ
l = l(k, X) ≡ 1 −

(E.4)
(E.5)
(E.6)

where
ŵ + φ̂G(X)r̂X(1 − τw )(1 − α) + (1 + τi )φs=.
(1 − τr )αΛ̂ ≡From (E.4)–(E.6), we can verify that k ′ +φX ′ = (1+τi )−1 g(k, X)+φG(X) = Ψc(k, X)/(1+τi ) holds. Therefore, for bj and dj (j = c, l), we can obtain the same values as those in the
RCE. On other hand, from Λ̂ = φ, we haveφ=

(1 − τw )(1 − α).
(1 − τr )α − (1 + τi )s(E.7)

Finally, substituting the definition k ′ = (1+τi )−1 k̂ ′ and (E.6) into the consistency condition,(1 + τi )−1 g(k, k/L) = G(k/L)L, we havesL =Ψα(1 − τr )(L + φ) − φs,(1 + τi )(1 + ζ + Ψ)
26

(E.8)

which results ins=

1 − τrΨα.
1 + τi 1 + ζ + Ψ

On the other hand, from (28) and the guess that G(X) = sAX α , τr is given byτr = −

τw (1 − α) + τi s
.
α

(E.9)

Substituting this into the above equation, we haves = set (τ ) ≡Ψ(α + (1 − α)τw ).
Ψ + (1 + ζ)(1 + τi )(E.10)

Finally, substituting the result back into (E.7) and (E.9), we haveφ = φet (τ ) ≡(1 − τw )(1 − α),α + τw (1 − α) − set (τ )τr = τr (τ ) = −τw (1 − α) + τi set (τ ).
α(E.11)

Thus, (E.4)–(E.6), (E.10), and (E.11) jointly prove Lemma 7.
We can obtain (32) from the consistency condition L = l(k, k/L, τ ). More specifically,
substituting (E.4) into this condition, we first haveL=

(1 − τw )(1 − α)(1 + ζ + Ψ) − ζ(1 − τr )αφ.
(1 − τw )(1 − α)(1 + ζ + Ψ) + ζ(1 − τr )αThen, substituting (E.10) and (E.11) into the above equation, and rearranging the terms,
we obtain
L = let (τ ) =(1 − τw )(1 − α)(1 + ζ + Ψ),
(1 + ζ)[ζ(1 + set (τ )τi ) + (1 − τw )(1 − α)(1 + Ψ)]□

which is equivalent to (32).

Appendix F(E.12)

Proof of Proposition 7

We guess thateG(X,τe, τ ) = seAX α ,where se is still to be solved. Note that by the current government’s deviation, (E.8) is nolonger valid, since L′ = L does not hold in this deviation period.
Since the pair of tax rates after this period is always given by τ , the equilibrium valueof L′ is let (τ ). Therefore, (E.6) and the consistency condition now provide the following
equation, instead of (E.8):
selet (τ ) =Ψα(1 − τer )(L + φ) − φs.
(1 + τei )(1 + ζ + Ψ)

(F.1)

On the other hand, by imposing τ = τe and the labor market equilibrium in the next
period, L′ = let (τ ), on (27), we obtaine ′ = 0 ⇔ τer α + τew (1 − α) + τei selet (τ ) = 0.
τer r(X)K + τew w(X)L + τei GL
27

(F.2)

From (E.7), (F.1), (F.2), and the consistency condition, l(k, k/L) = L, the values of se,
φ, L, and τer are determined as the functions of τe and τ . Their values are denoted byseet (eτ , τ ), φeet (eτ,τ) elet (eτ , τ ), and τer (eτ , τ ), respectively.
Given k and the current government’s tax policy, τe, the market-clearing for the consumption good in this period requiresk ′ = seet (eτ , τ )Ak α (elet (eτ , τ ))1−α ,c = (1 − seet (eτ , τ ))Ak α (elet (eτ , τ ))1−α .

On the other hand, in the next period, given k ′ and τ , it follows thatk ′ + φX ′ = (1 + φet (τ )/let (τ ))k ′ .
Substituting these results into (33), we have
(
)
[(
]
(
)
)
Ve k, k/elet (eτ , τ ), τe, τ = ln 1 − seet (eτ , τ ) Ak α (elet (eτ , τ ))1−α + ζ ln 1 − elet (eτ,τ)[]+ βc (bc + dc ) ln seet (eτ , τ )Ak α (elet (eτ , τ ))1−α+ Ψ ln(1 + φet (τ )/let (τ )) + βc ac + βl al ,where bc and dc are given in (A.8). Note that, from the same equation, bl + dl = 0 is
verified.
From the above equation, we find that τe aﬀects V only through seet and e
let . This meansthat obtaining V by choosing τe can also be achieved directly by choosing seet and e
let . The
first-order conditions for s and l are, respectively, given byβc (bc + dc )
1
=
,
s
1−s
(1 − α)[1 + βc (bc + dc )]
ζ
=
.
l
1−l

(F.3)
(F.4)

Let (s, l) denote the solutions to (F.3) and (F.4). Substituting (A.8) into (F.3) and (F.4),
we can explicitly obtain the values of s and l ass = βc α,

l=

1−α
.
1 − α + ζ(1 − βc α)(F.5)

Then, we can obtain the time-consistent tax policy τ by solving s = seet (τ , τ ) and
l=elet (τ , τ ) for τ . Since seet (τ , τ ) = set (τ ), and elet (τ , τ ) = let (τ ), for all τ , substituting(E.10) and (E.12) (or (32)) into (F.5) yields the following two equations:Ψ(α + (1 − α)τw ),Ψ + (1 + ζ)(1 + τi )1−α(1 − τw )(1 − α)(1 + ζ + Ψ)l = let (τ ) ⇔=.
1 − α + ζ(1 − βc α)(1 + ζ)[ζ(1 + βc ατi ) + (1 − τw )(1 − α)(1 + Ψ)]s = set (τ ) ⇔ βc α =From the former equation, we have
1 + τi =Ψ[1 − βc + τw (1 − α)/α] .
(1 + ζ)βc
28

From the latter equation, we have
1 + τi =[](1 − βc )Ψ τw 1 − βc α (1 − βc )Ψ
+−.
(1 + ζ)βc
βc
α1+ζ

Then, from the above two equations, we have τ w = 0 and τ i =1 − βcΨ − 1. From the(1 + ζ)βc

definition of Ψ in Proposition 1, we then have()ζ
1 − βc βl
τi =
−1 .
1+ζ
βc 1 − βlFinally, substituting τ w = 0 and set (τ ) = βc α into (E.9), we obtain τ r = −βc τ i .

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30