Discussion Papers In Economics
And Business

Optimal Degree of Commitment in a Tax Policy
Yusuke KINAI
Discussion Paper 11-11

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal Degree of Commitment in a Tax Policy
Yusuke KINAI
Discussion Paper 11-11

March 2011

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal Degree of Commitment in a Tax Policy∗
Yusuke KINAI†
Graduate School of Economics, Osaka University

March, 2011

Abstract
In analyzing economic policies, a severe problem is the time-inconsistency problem. In this regard, the choiceof commitment vs. discretion engenders a tradeoff of flexibility and credibility. Therefore, it might be necessary
and acceptable to adopt a discretionary policy to some degree, but past studies do not clarify the degree to whicha government exercises such a discretionary option. This paper clarifies the optimal degree of commitment using
the framework of a repeated game reported by Chari and Kehoe (1990). We point out the relation between the
optimal degree of commitment and the rate of time preference.

Keywords : Commitment vs. Flexibility; Degree of Commitment; Imperfect Public Monitoring;
Repeated Games; Tax Policy.

JEL Classification : E61, E62, H11, H30.

∗ This is a substantially revised version of an earlier paper presented under the same title. I am grateful to Masatsugu Tsuji andKazuo Mino for their invaluable advice and guidance during the process of writing this paper. I also thank Ichiro Obara, Tadashi
Sekiguchi, Takero Doi, Noriyuki Yanagawa, Eiichi Miyagawa, Takashi Oshio, Dan Sasaki, Eiji Tajika, Takashi Komatsubara,
Colin R. McKenzie, Nobuo Akai, Mutsumi Matsumoto, and seminar participants at various seminars and conferences for theircomments. Discussions with Takeki Sunakawa and Yosuke Yasuda are also acknowledged. Of course, the author is responsible
for all remaining errors in this presentation.
† Correspondence to: Yusuke KINAI, Graduate School of Economics, Osaka University, 1-7, Toyonaka, Osaka, 560-0043,
Japan. E-mail address: ykinai@js8.so-net.ne.jp

1 Introduction
Regarding economic policy, an important issue for any government is the decision of when a policy
is to be implemented. For example, as is often the case in the Japanese economy, fiscal policy tends tobe postponed because of pressure from the Liberal Democratic Party (LDP) and other interest groups.
Policy is often altered shortsightedly. Changing a policy before its aim is achieved is often called timeinconsistency. The cause of this problem is related to the degree of a government’s commitment to theoriginal policy. In this paper, we analyze how a government should implement policy management.
Analysis of that problem demands consideration of the time-inconsistency problem, which occurs
under the following situation: Presuming that a government determines a policy at the beginning ofa period. As time passes, the policy, which was originally optimal in the earlier time of the period,
tends not to be optimal any longer. Nevertheless, the government cannot change the policy because of
the commitment that was adopted and announced initially. Consequently, the original policy objective
becomes inconsistent with the current situation.
As one example, in traditional dynamic optimal taxation theory, the ex ante optimal policy is that inwhich the tax rate on capital is zero or extremely low (see Chamley (1986), Judd (1985)). However,this result is based on the assumption that the government can commit to the policy forever; the result
ignores the time-inconsistency problem. This problem is summarized as follows. Even if the government determines a tax structure according to optimal taxation theory, it remains unclear whether thegovernment can continue its implementation exactly as it was determined initially. Therefore, apartfrom the commitment rule, some other rules governing policy changes should be imposed to avoid atime-inconsistency problem.
It is necessary to construct a disciplinary rule in economic policy to avoid such a problem. Broadly
speaking, two methods exist: “rule” and “discretion”1) However, regarding the latter, the following
faults exist. (1) A lag pertains between the point of policy-decision and that of policy-implementation,in addition to a lag between policy-implementation and the point until its effects become apparent.
(2) Because the effect of the policy is uncertain, the policy is not always effective. (3) This mode ofmanagement is devoid of credibility. Therefore, it is apparent that it is necessary to reinforce commitment: to adopt a “rule”. This method also has its faults. A salient fault is that adoption of a “rule”
is based on the assumption that the economic environment does not change much or change quickly,or that the government can control such a transformation. However, it is not actually appropriate for1) the term “discretion” differs from that used in the 1970s in the sense of a quick-fix policy. The term “discretion” used

for this study is a kind of rule based on the assumption that the government cannot commit to the policy, or that the
government uses the database only in the single period as an anchor for a policy decision.

1

us to consider that the government can control the source of uncertainty because the future economic
environment is uncertain.
From the discussion presented above, it is apparently more realistic to adopt the commitment (inother words, rule) despite its faults. However, is it always right? Which management is more desirable, rule or discretion? As described in Taylor (1993), Lohmann (1992) or Amador, Werning andAngeletos (2006), there exists a tradeoff between these two ways of policy management. As explainedin the latter, both “rule” and “discretion” include benefits and drawbacks2) : in the case of a rule, abenefit is that credibility can be established: the reinforcement of a rule can engender higher credibility, although a disadvantage of this option is that a characteristic rigidity of policy-decision can result.
In other words, it is difficult for the government to adapt when a change does occur. However, regarding discretion, the benefit is that flexibility is retained: the government can freely change the policy
according to the circumstances, including the case in which the government coordinates the policy
after the fact, although a disadvantage is decreased credibility and the possibility that frequent policychanges can engender ex ante moral hazard3)
As explained above, because these two ways of management include advantages and disadvantages,
neither an ironclad rule nor a pragmatic discretion is optimal. Therefore, it is considered that, to somedegree, discretionary management is allowed. Moreover, because of asymmetric information or some
contributing factors, the possibility exists that households cannot observe the government’s behavior.
The solution to the time-inconsistency problem therefore remains unclear. In that situation, we should
analyze the extent to which the government should commit to the policy. This paper specificallydescribes such a problem, although it has not been analyzed in past studies. We intend to answer sucha question using the method of a repeated games and numerical methods.
■Motivated Example

It is illustrative to consider the following example4) in which the government

determines taxation to maximize the following function:U(c, l, g) = l + ln(α + c) + ln(α + g),
where c, l, and g respectively denote consumption, labor supply, and public expenditure. The budget constraints of households and the government are written respectively as c = (1 − τ )(1 − l), andg = τ (1 − l), where τ denotes the labor-income tax. Then, the Ramsey rule and the Nash rule aredefined as follows: The Ramsey rule is a policy that the government determines in consideration of thehouseholds’ response function. The Nash rule is a policy that the government re-determines under the2) the term “rule” and “discretion” respectively corresponds to “commitment” and “flexibility”.
3) “Ex ante moral hazard” means that there exist distortions in the households’ ex ante choice because they expect that the

government will change policies after the fact.
4) This example is based on Ljungqvist and Sargent (2004, Ch.22).

2

assumption that households behave following the Ramsey rule.
As explained in greater detail, households determine their own response function (labor supply function), l(τ ), and the government determines τ ∗ to maximize U(c, l(τ ), g) = l(τ ) + ln(α + (1 − τ )(1 −l(τ ))) + ln(α + τ (1 − l(τ ))), which is the Ramsey rule. In contrast, assume that households determine
their own behavior taking the Ramsey rule above as given, and that the government determines taxation, τ ∗∗ , by maximizing utility: U(c, l(τ ∗ ), g) = l(τ ∗ ) + ln(α + (1 − τ ∗ )(1 − l(τ ∗ ))) + ln(α + τ (1 −l(τ ∗ ))). This is the Nash rule. First, the solutions of the Ramsey rule and the Nash rule do not coincide.
Consequently, the time-inconsistency problem arises. Second, the utility under the Nash rule is higher
than that achieved under the Ramsey rule. In other words, if the government deviates from the policy
and they can cheat the households, then the government could thereby attain a higher level of utility
than that attained under the Ramsey rule.
Next, we can consider the case in which the game described above is repeated. It is worth recallingthat if a household behaves making allowance that the Nash rule is adopted, then the utility is lower
than that attained under the Ramsey rule. Under the assumption that households can observe thebehavior of the government because households can observe cheating or deviation by the government,
they can punish the government in some form. Thereby, the government might seek to avoid sucha punishment and not deviate. However, under the imperfect monitoring case, households cannot
observe the behavior of the government, so the effect of the Nash rule might endure as effective forsome periods. The government might deviate from the policy if the effect of the Nash rule is greaterthan that of the Ramsey rule. Then, another question arises. When or in what circumstances should thegovernment commit to a policy or deviate from it? The analyses described in this paper are designed
to answer that question, with particular emphasis on the relation between the effectiveness of the Nashrule and observability.
■Related Literature

As described below, some studies have examined repeated games to eluci-

date some features of macro-economic policy5) . In the case of monetary policy, for instance, Athey,Atkeson and Kehoe (2005) investigate the monetary policy game under imperfect public monitoringand show that committing to a simple rule is best. Moreover, Rogoff (1985) established the idea ofa “conservative” bank, which engenders the idea of independence of the central bank and showedthat delegation of authority to a more conservative central bank than the government can eliminate an
inflation bias and yield higher utility.
However, regarding fiscal policy, recently, some studies have begun to incorporate information
asymmetry or the existence of an informational incompleteness6) , as in the case of monetary pol5) See also Ljungqvist and Sargent (2004, Ch.22) or Mailath and Samuelson (2006, Ch.6,15) as a textbook.
6) For instance, Sleet (2004) and Albanesi and Sleet (2006) analyze the dynamic optimal taxation problem under a case in3

icy. Some studies analyze the optimal taxation problem in relation to the commitment problem using
the framework of a repeated game.
The seminal work of this field is related to Chari and Kehoe (1990). This work is based on perfectmonitoring, as developed by Abreu (1988). Households might punish the government if the government were to deviate from a policy that is once determined. This mechanism can be regarded as animplicit contract between the government and households. To be more precise, Chari and Kehoe establish a mechanism by which, if the government deviates from the policy, households might punish thegovernment; because of this punishment, it is difficult for the government to deviate from the policy.
Consequently, the government stays the course and remains committed to the policy. Chari and Kehoeemphasize that point and describe the reputation mechanism, which always forces the government tocommit to the original policy, by extending the model of Fischer (1980) into an infinitely repeatedgame. They named such a punishment or reputation mechanism a “sustainable plan”. In other words,
Chari and Kehoe center upon a trigger strategy as a mechanism to retain the commitment7) . Recently,
Phelan and Stacchetti (2001) describe a sustainable plan in a model of dynamic optimal taxation, using
the methods of dynamic programming approach developed by Abreu, Pearce and Stacchetti (1990).
Although some studies including those of Benhabib and Rustichini (1997), Fernández-Villaverde
and Tsyvinski (2002), and Reis (2006) investigate optimal fiscal policy without commitment, to thebest of our knowledge, few studies have specifically addressed the degree of commitment in a tax
policy. This paper specifically addresses the relation between the degree of precision of monitoring
and the degree of commitment, thereby extending the analyses of Chari and Kehoe (1990). With moredetail, the mechanism of creating reputation might be different between the case in which households
can observe government behavior completely, and the case in which households can observe it incompletely. Regarding the former case, if the government deviates from a policy anticipated by households,
then households can sanction the government in some manner, such that the government avoids deviation from a policy to prevent some reprisal. Regarding the latter case, the possibility exists that if
the government deviates from the policy that households expect, they can obtain higher utility becausehouseholds cannot observe the behavior of the government. Consequently, they might be cheated.
Therefore, for the latter case, the possibility exists that discretion obtains a higher utility than a rule

which asymmetric information exists. Another report Kocherlakota (2006) presents a useful survey related to the recent
issue of dynamic optimal taxation.
7) Abreu (1988) call this mechanism “stick and carrot punishment”. See Stokey (1989, 1991), which presents a similar
approach. Moreover, see Ireland (1997) or Kurozumi (2008) as a study which applies the idea of Chari and Kehoe (1990)
to the monetary policy. Recently, Cho and Matsui (2005) analyze this problem under the game in which the moves of thegovernment and households are altered in each period. Moreover, recently, Piguillem and Schneider (2008) investigatesthe optimal taxation problem in which neither government nor households have complete information related to one
another’s state or behavior.

4

(commitment) does. Chari and Kehoe emphasized only the former case. The present study investigates both cases by making allowance for the imperfect monitoring case, as developed by Abreu et al.
(1990)8) and shows that the optimal degree of commitment might change depending on the degree of
accuracy of monitoring.
The remainder of this paper is organized as follows. Section 2 presents a description of the model.
Then, section 3 describes analyses of two cases. The first is the case in which households can observe
the government behavior completely. The second is the case in which households can observe it
incompletely. The final section (section 5) is the conclusion.

2 The Model
This section establishes the basic model used for the analyses described in this paper. The structureof our model follows Chari and Kehoe (1990), although our basic model is similar to that of Cooper(1999, Ch.7), which is a variant of Fischer (1980). The model extends the model of Cooper into
an infinitely repeated game. The economy begins in period 0 and continues over time, extending
indefinitely into the future t = 1, 2...∞. All markets in this economy are perfectly competitive. In our
setting, in each game, the model is structured as a Stackelberg game in which the government andhousehold respectively constitute a leader and follower. In what follows, we explain the structure of
the stage game.

2.1 Behavior of Each Agent■Households

This economy includes N identical households that exist for two periods. In our

setting, the single term is divisible into two periods. The second period in t-th period and the first
period in t + 1-th period are not overlapped. Each agent is assumed to obtain utility from consumption
and leisure. Therefore, the utility of an agent alive at the t-th period is written as U(c1t , c2t , n − l2t ),where c1t , c2t , and n − l2t respectively denote the consumption in the first and second stage, and leisurein the second stage. Here, we impose some assumptions on each utility function as follows:(A 1.) U(·) : ℜ+ → ℜ are strictly concave and increasing in each variable.
(A 2.) limc1 →∞ ∂∂U(·)
c = lim U1 (·) = 01c1 →∞∂ U(·)(A 3.) limc2 →∞ ∂ c = lim U2 (·) = 0
2
c →∞
2

8) For studies which analyze the repeated game under public monitoring, see Abreu (1988), Abreu et al. (1990), and Fuden-berg, Levine and Maskin (1994), for instance. The former two studies characterize the equilibrium paths using dynamic
programming. Fudenberg et al. (1994) proved the folk theorem under certain conditions under a public (including imperfect case) monitoring case. Moreover, an earlier report Pearce (1992) presents an excellent survey of repeated games.

5

U(·)
= limn−l2 →∞ U3 (·) = 0.
(A 4.) limn−l2 →∞ ∂∂n−l2Taxes of two kinds exist: a tax on labor income (η ) and a tax on saving (τ ). Households that form at
date t divide endowment (et ) into consumption (c1t ) and saving (st ) in the first period, and consume(c2t ) as after-tax savings and after-tax labor-income in the second period. Consequently, the first-period
and second-period budget constraints are shown respectively as
c1t + st = et , c2t = st Rt (1 − δ ) + l2t (1 − τ ),where st denotes saving. Given history ht , explained later, each household determines its own behavior
by solving the utility maximization problem, as denoted by ft = ( f1t , f2t ) = (st (ht ), l2t (ht )), which isan allocation determined by households.
■Government

The government determines taxation by maximizing the households’ lifetime utility

function. Under this setting, the budget constraint of the government is written as follows (balancedbudget rule):

gt = δt st Rt + τt wt l2t .

(1)

We then assume that the government expenditure gt is treated as given for households. The objective∞function of the government is written as ∑ β t U(c1t , c2t , n −l2t ), where the discount factor is β ∈ (0, 1].
t=0Moreover, we assume that the government expenditure does not affect the households’ utility. The∞ ). We define the history up to t-period
action profile of the government is denoted as ag = ({δt , τt }t=0

as follows.
ht = {as }t−1s=1 : Path of tax rates which happened up to t-period.
ht = (h0 , h1 .., ht ): Path of events up to t-th period
The whole public history is recorded as H =∞∪ht , and the path of events from the initial point up to tt=1

period is denoted as ht = (h1 , h2 ..., ht ). Then, the action of the government at t period can be written as
atg = (δt , τt ), and ht = atg (ht−1 ). Moreover, the government strategy, σ g is a mapping which satisfies
atg = σtg (ht−1 ).
Incidentally, presuming circumstances in which the government must determine whether they continue the policy (in other words, commit to the policy) or change (deviate from) the policy, they signala policy change at the beginning of the time (or each stage-game). Here, we designate such a public
signal of policy-change (ω g ) as follows: For t = 1, 2...∞,

ω g = (ω1g , ..., ωtg ) ∈ Ω,6(2)

where Ω ∈ ℜ: the distributive function of ω is denoted as f (·). In addition, this public signal is
assumed to be observable to all players (i.e. households) in the following subsection 3.2, which corresponds to the perfect monitoring case. In contrast, in subsection 3.3, the signal includes noise suchthat households cannot perfectly observe government behavior, which corresponds to the imperfectmonitoring case.
Here, it is necessary to refer to an allocation rule. An allocation rule (hereinafter, denoted as “α ”)
describes households’ choices in each period, contingent on observation of the profile of government
policy: α maps σtg into ft . Therefore, the decision-making of households at t-th period is a functionof the history of government policy up to t.
■FirmsWe then describe firms’ behavior. We assume that production process is achieved between

the first and second period in each stage game9) , and factor markets are perfectly competitive and thatfirms maximize their profits. Labor and capital stock are used for production. They are functions of
the history ht ; production technology yields constant returns to scale. Therefore, production functions
are expressed as yt = F(kt (ht ), lt (ht )) : ℜ2+ → ℜ+ , where kt , lt , and yt respectively represent capitalstock, labor demand, and output. The problem facing firms is the following.
max yt − Rt kt (ht ) − wt l2t (ht )s.t. yt ≤ F(kt (ht ), l2t (ht ), ht )In fact, maximization yields
Fk (ht ) = Rt (ht ), and Fl (ht ) = wt (ht ).

(3)

2.2 Market Equilibrium
Finally, we formulate equilibrium conditions for each market.
• Resource Constraint:

c1t (ht ) + c2t (ht ) + k(ht ) + gt (ht ) ≤ yt (ht )(4a)• Labor market:
It is assumed that full employment pertains in this economy, meaning that a supply–demandbalance is attained. Denoting ltd as the labor demand, we have the following.
dl2t = l2t.

(4b)k1t = s2t

(4c)

• Capital Market:

9) This assumption means that in each period, saving in the first period is used for production and its output is given as anendowment for the next generation.

7

2.3 Timing of the Stage Game
The timing of the t-th period stage game, given history ht−1 , is the following:
1. The government signals whether it will make a policy change or not.
2. Observing that signal, households determine their own behavior.
a. In the case in which households expect that the government commits to the policy without apolicy change,
households trust the government’s commitment and do not change their behavior. Therefore, thetime-inconsistency problem does not occur
b. In the case in which households expect that the government does not commit and will change
the policy,
households change their own decisions. Under a perfect monitoring case, households can adjust their behavior swiftly, although, under an imperfect monitoring case, it takes for households
some time to adjust expectations. In this case, the effect of a discretionary policy can be transiently greater than that of the commitment policy.
3. The government determines the tax policy. In other words, whether to commit to the policy or not.
4. All markets clear.
Moreover, regarding step 2 explained above, it is noteworthy that if the government can deviate fromthe policy without being noticed by households, the government can attain higher utility. However, thelevel of utility will be lower than when the government does not change the policy if households cometo a realization of policy change.

3 Analysis of Two Cases
This section presents analyses of the model under the two cases: perfect monitoring and imperfectmonitoring. The difference between these two cases is explained as follows (see also Fig. 1.):10)
Case 1. Perfect Monitoring: All players (households) can observe the past action and the signal. In
other words, households can observe the government’s decision completely.
Case 2. Imperfect Monitoring: Only common signals based on the past action are visible. However,10) Regarding the criticism for analysis with a framework of public monitoring, see, for instance, Matsushima (2002, section

5). He claimed that the analysis under public monitoring is not plausible for obtaining the policy implication. However,considering that households can observe the public behavior of the government to some degree using newspapers or theinternet, we can presume that the analysis with a public monitoring framework corresponds to reality.

8

Action(NOT observable)(a1 , a2 , ..., at )(a) Perfect Monitoring

Signal(observable)ω ∈Ωω = (a1 , a2 , ..., at )(b) Imperfect Monitoringω is random variable
Figure 1 Difference between perfect monitoring and imperfect monitoring Case #1.

this signal includes noise; it is not necessarily accurate. Regarded in greater detail, households cannot capture the government’s decision completely. Looking at this case, we consider
the case in which households cannot observe the government’s behavior for some periods.
Subsequently, they finally take notice of a policy change11) .

3.1 Equilibrium Concept
Before entering into analysis, we define the equilibrium concept (sustainable plans) following Chariand Kehoe (1990). An allocation rule ( ft ) describes households’ choices, st (ht ) (saving), and l2t (ht )(labor supply) in each period, contingent upon observed past government policy, given the continuationof the allocation rule after ht−1 .
First, we define the competitive equilibrium12) .
Defnition 1 (competitive equilibrium) We define an equilibrium as a competitive equilibrium if it satisfies the following allocation.
(i) Given the history up to t − 1; ht−1 , the condition for utility maximization, eq. (6) holds.
(ii) The condition for firms’ profit maximization holds.
(iii) Each market equilibrium condition holds: (4a)–(4c) .
(iv) The budget constraint of the government holds: eq.(1).
Here, it is necessary to define some terms. Given continuation of the allocation rule after ht−1 ,the continuation of σtg induces a continuation path (st , l2t ). This continuation path and an allocation
rule then determine a sequence of aggregate actions (Kt , Lt ). An allocation is competitive if and only
if, for all t, the following condition holds. Given the history up to t − 1-th period, the government11) Regarding this point, see section 4.
12) For this definition, see also Chari, Kehoe and Prescott (1989).

9and the allocation rule are competitive, the allocation after t-th period meets the proper definition of
the competitive equilibrium. However, a government strategy σtg is optimal if the continuation of σ gmaximizes the social planner’s objective, given the continuation of the allocation rule after ht−1 andthe condition presented above.
Now, we are ready to state the definition of Sustainable Plans.
Defnition 2 (Sustainable Plan) We define the allocation rule and the government’s strategy, which
satisfy the following condition as Sustainable Plan.
Given the history up to t − 1; ht−1 ,(i) The government strategy σtg is admissible.
(ii) An allocation rule, α , meets the properties of competitive equilibrium.
(iii) The government strategy of σtg meets the property of optimal allocation.
(iv) Each market equilibrium condition: eqs.(4a)-(4c) holds.
The meaning of (i) ∼ (iii) is explainable in a narrative manner. First, the definition of “admissible”∞
is explained as follows: Given the history up to t − 1, ht−1 , then {hs }∞s=t is defined so that {st , lt , ht }t=0

is consistent with the conditions of competitive equilibrium (CE). A government strategy and an allocation rule are a sustainable plan (hereinafter, SP) if, after any history of government policies, (i)the continuation of the government strategy is optimal for the planner given the allocation rule, and(ii) the choices prescribed by the continuation of the allocation rule are optimal for each worker andinvestor, given the government strategy. In other words, an SP specifies behavior for the government and the private sector such that, after every contingency, the government acts optimally given
households’ behavior, and their behavior meets the property of the rational expectations equilibrium
given the government policy. The condition (iii) requires both the private agents’ decision rule and themonetary policy strategy to be sequentially rational. This condition implies the following situation.
Let V (·; σ , α |ht ) denote the payoff of the government, given history ht . Then for all the continuation
government’s strategy, σ ′ satisfies the following constraint of′t
V (kt ; σ , α |ht ) ≥ V (kt ; {σ }t−1
t=0 , σt , α |h ),(5)where “σt′ ” signifies the deviation by the government at t-period. This equation means that the onetime deviation of the government’s strategy never engenders welfare improvement.
Here, regarding the property of SP, we state the following proposition.
∞ is a sustainable plan, then {s , l }∞Proposition 1 Given any history ht−1 , if {st , l2t }t=0s 2s s=t+1 is also asustainable plan.

10

Proof See A.2.

3.2 Benchmark: Case of Perfect Monitoring
This subsection presents analysis of the perfect monitoring scenario as a benchmark. This case isalmost identical to that of Chari and Kehoe (1990). We continue the analysis based on the assumptionthat both players (government and households) can observe each other’s behaviors. Then, we adopt thetrigger strategy. What we should consider is under what mechanism the government will not deviate.
Here, we borrow the idea penalty code proposed in Abreu (1988).
Under this setting, the government solves the following problem.
∞

β t−sU(c1t (ht ), c2t (ht ), n − l2t (ht ))∑{δ ,τ }t=s
max∞

t=s

Incidentally, it is necessary to solve the stage-game in the t-th period. To obtain the Ramsey solution,
we solve the stage game backward.
Second Stage Households solve the following maximization problem given the level of taxation.
max U(e − s, sR(1 − δ ) + l2 w(1 − τ ), n − l2t )From this problem, we can obtain the following first-order conditions (response function):{
Us = −U1 + R(1 − δ )U2 = 0
Ul = (1 − τ )wt U2 −U3 = 0.

(6)First Stage Given s(δ ∗ , τ ∗ ), and n(δ ∗ , τ ∗ ), which are obtained by solving the two first-order conditions presented above, the government solves the following problem.
max U(c1t , c2t , n − l2t )s.t. gt = st (δ )Rt δt + l2t (τ )wt τtand eq. (6)Arranging the first-order condition of this problem yields the following:
U2 sR = λ [sR + δ sk R + τ lk ](7)

U2 l2 = λ [l2 + Rδ sl + τ ll ],(8)∂ s ∂ s ∂ l2 ∂ l2,,,
, and the Lagrange multiplier. We∂δ ∂τ ∂δ ∂τ
designate the solutions as {δ ∗ , τ ∗ }, or {δs∗ , τs∗ }∞s=0 , which denotes the paths of solutions under thewhere sk , sl , lk ,ll , and λ respectively denote

Ramsey rule. The level of utility under the Ramsey rule is denoted as U ∗ (·). Similarly, we can obtain11

the level of utility under the Nash rule and one-shot deviation, which are denoted respectively as U s (·)
and U d (·). As portrayed in the A.1, the following condition holds13) .
Lemma 1U d (·) > U ∗ (·) > U s (·).

Then, presuming a situation in which the government should determine whether to commit to thepolicy or not at s period, the government should determine the option depending on the magnituderelation in the following equation.
∞

∞

t=s

t=s+1

∑ β t−sU ∗ (c1 , c2 , 1 − l1 ) T U d (·) + ∑ β t−s−1U s (·)(9)

The left side of the equation shown above defines the utility obtained by committing to the policy.
However, the first term of the right hand of the equation denotes the utility obtained when the government deviates at s-th period. Moreover, the second term denotes the mean utility that households
obtain, taking the policy change into consideration rationally after the s + 1-th period. From anotherperspective, U s (·) can be interpreted as a result that the government is punished by households.
Using lemma 1, we are now ready to describe the following proposition.
Proposition 2

1. For all s, the following condition holds.
∞β∑ β t−sU ∗ (c1 , c2 , 1 − l1 ) ≥ U d (·) + 1 − β U s (·)

(10)

t=s

This equation can be rewritten asβ[U ∗ (·) −U s (·)] ≥ [U d (·) −U ∗ (·)].
1−β

(11)

The government should commit to the policy when this condition holds.
2. There exists a lower value of β ; β ∈ [0, 1] which satisfies the equation presented above.
Proof See appendix A.3.
This proposition says that no room exists for discretion under this case depending on the interval of

β . Under the perfect monitoring case, the effect of deviation lasts only one period. Under the imperfectmonitoring case, the effect of deviation might last for some periods, depending on the monitoring
ability. In the next subsection, we investigate how this result varies when we incorporate the imperfectmonitoring case.

13) This lemma is only described as a narrative in Cooper (1999, Ch.7); it is not proved formally.

123.3 Case of Imperfect Monitoring
The case in which both players can observe the other’s behavior, as in the previous subsection, isless than realistic. We then consider the following situation.
• Neither government nor households can observe the other’s behavior. In other words, neither
player can observe which action is taken (or not taken) for sure.
• It is assumed that households can obtain information related to government behavior: public
signal.
• However, this public signal includes noise: this signal is not entirely correct. Therefore, we
assume that this is an unobservable stochastic variable: (i.i.d., independent and identically distributed), whose marginal distribution is denoted as f (ωt |σ g ). Moreover, we assume that thesignal has a cumulative distribution function (c.d.f.) ; F(ωt |σ g ). 14)
Regarding the signal, we impose some assumptions:(A 5.) Ω is a finite subset of ℜ.
(A 6.) f (ωt |σ g ) > o holds for all ω ∈ Ω and f (ωt |σ g ) is continuous in σ g .
As in the preceding subsection, we adopt the trigger strategy, which means that the players who deviate are sanctioned.

We should consider how the government should behave sothat it will not deviate from the policy.

In this imperfect monitoring case, we designate apublic signal as ω g , and consider {δt−1 , τt−1 } as a public strategy.

Consequently, the ex-pected payoff of each player (government and households) at the t-th period is written as
Et [U(c1t , c2t , n − l2t )|Ω] = ∑ω ∈Ω f (ω1 , ω2 ....|σ g )U(c1t , c2t , n − l2t ).

Therefore, each objectivefunction is shown as the following.
∞

∞

t=s
∞

t=s ω ∈Ω∞

t=s

t=s ω ∈Ω

∑ β t−s Et [U(c1t , c2t , n − l2t )|Ω] = ∑ ∑ f (ω |σ g )β t−sU(c1t , c2t , n − l2t )(12)∑ β t−s Et [U(c1t , c2t , n − l2t )|Ω] = ∑ ∑ f (ω |σ g )β t−sU(c1t , c2t , n − l2t ))(13)14) Strictly speaking, this signal includes signals of two kinds:– ω g : A public signal, which is observable by all players.
– ω p : A private signal, which is different in each household.
However, for simplicity, we ignore the private signal in this paper.

13tHouseholds are aware of being cheated,
For ∆t periods,
and they correct their own behavior.
households are cheated.

s-th period

Figure 2 Imperfect Monitoring Case #1.

Under this setting, eq. (9) under the imperfect monitoring case is rewritten as
the utility without policy-change

z

∞

∑β

}|

t−s

{

[

U(c1 , c2 , 1 − l1 ) T U (·) +
d

t=s

s+∆t

]

∑ ∑β

t−s−1

t=s+1 ω ∈Ω

f (ω |σ )U (·) +gd∞∑ β t−(s+∆t)U s (·). (14)t=s+∆t

The left side of the equation presented above is the same as the perfect monitoring case. The rightside of the equation presented above consists of two terms. The first is the utility obtained using the
Nash rule: one-shot deviation. In other words, this effect is that the government deviates from its
chosen policy at s period. The second term is that of the expected utility, which consists of two factors.
The first factor is the utility for which the Nash rule is effective when households cannot discern thegovernment behavior. The second factor is the effect whereby households expect the government
rationally to adopt the Nash rule. In the case of perfect monitoring, the left side of the equation
presented above is more than, or at least equal to that of the right side, so that the government shouldcommit to the original policy. However, in the imperfect monitoring case, the possibility exists that theright side of the equation presented above is larger than the left-side as long as households are cheatedby the government. If this is the case, then the government should deviate, or take the discretionary
option, rather than commit to the policy. Figure 2 depicts the situation described above.
The equation in which the commitment policy is sustainable under the imperfect monitoring case is
written as15)[]∞
β[U ∗ (·) −U s (·)] ≥ U d (·) −U ∗ (·) + Et ∑ ∑ β t−s−1 [{U d (·)| f (ωt |σ g )}.
1−βω ∈Ω t=s+1(15)

Under such a situation, the distributive function of the public signal is an important consideration.
Here, the following propositions hold:
Proposition 3 For such an interval, β ∈ [β ′ , 1], it is better for the government not to deviate from thepolicy. In other words, the commitment policy is sustainable.
Proof See appendix A.4.
15) We use the fact that the last term of RHS of the equation presented above can be written as14βs
1−β U (·).

U(·)
Discretion

Cheating

ββ′Commitment

βFigure 3 Relation between the value of β and the choice of policy management.

Proposition 4 Comparing β with β ′ , the following equation holds:

β′ > β.
Proof See appendix A.5.
This proposition means that the interval of β is narrower when the commitment policy is sustainable
than in the perfect monitoring case. In other words, this suggests that if private agents are more patientthan in the perfect monitoring case, then the commitment policy is attainable even in the absence of acommitment device and the imperfect monitoring case. Actually, Fig. 3 displays the above-referenced
explanation in intuitive terms. Moreover, if β is less than β ′ , then the possibility exists that it isbetter for the government to deviate from the policy. In other words, the commitment policy is notsustainable.
■Meaning of ∆tHere, we investigate the optimal degree of commitment, first by seeking the valueof ∆t ∗ which satisfies the following equation.
∞

s+∆t

t=0

ω ∈Ω t=s+1

β∑ β t U ∗ (c1t , c2t , n − l2t ) = U d (·) + [ ∑ ∑ f (ω |σ g )β t−s−1U d (·) + 1 − β U s (·)]

15

(16)

Solving this equation, we obtain16)∆t ∗ = logβ (U ∗ (·) −U s (·)).
The discretionary policy is optimal if the government can cheat households for more than ∆t ∗ periods. To summarize, the following can be said.
Proposition 5 The government can cheat households for more than ∆t ∗ period, defined in eq. (16). It
is better for the government to take discretionary management.
Remark. Under the imperfect monitoring case, households are not aware of a policy change (inother words, they are cheated by the government) for ∆t periods. The value of ∆t varies depending on
the accuracy of monitoring, the possibility exists that discretionary policy is optimal. From anotherangle, ∆t can be interpreted as the degree of commitment. The case of ∆t = 1 corresponds to the fullcommitment17) , and ∆t = ∞ means full discretion.
Based on this result, by differentiating ∆t ∗ with respect to β , we obtain the following:
Corollary 1 As the value of β increases, the value of ∆t ∗ decreases.
This corollary means that the degree of commitment is dependent on the value of β . The government
must cheat households longer as households become more patient.

4 Discussion
Some readers might note that the analysis in the previous subsection is a case of information structure, not a case of accuracy of monitoring. Then, in this section, we consider another version ofimperfect monitoring.
We then consider another version of the imperfect monitoring case. We consider the situation described in the preceding subsection.

16) This can be derived as follows:β(U ∗ (·) −U s (·)) = (1 + β s−1 (ω1 + ω2 β + · + ω∆t β ∆t )U d (·)
1−ββ(U ∗ (·) −U s (·)) = (β + β 2 + · · · + β ∆t )U d (·)
1−ββ
1 + β (1 − β ∆t ) d(U ∗ (·) −U s (·)) =
U (·)
1−β
1−β
⇐⇒ ∆t ∗ = logβ (U ∗ (·) −U s (·)).
⇐⇒

17) See the subsection 3.2.

16

(17)

s-th periodt
For all periods,
households are note aware whether
the government takes the Nash-rule or not.
Figure 4 Imperfect Monitoring Case #2.

The main difference between the previous subsection (Case 1 of imperfect monitoring) is that weassume the incomplete awareness of government of household behavior forever, although the government is assumed to be aware of household behavior at some point in the previous subsection. Regarding
this point, see Fig. 4.
Then, the expected utility of households and the government are shown respectively as shown below.
∞

∞

t=s
∞

t=s ω ∈Ω∞

t=s

t=s ω ∈Ω

∑ β t−s Et [U(c1t , c2t , n − l2t )|Ω] = ∑ ∑ f (ω |σ g )β t−sU(c1t , c2t , n − l2t )(18a)∑ β t−s Et [U(c1t , c2t , n − l2t )|Ω] = ∑ ∑ f (ω |σ g )β t−sU(c1t , c2t , n − l2t ))(18b)

Under this setting, eq. (9) under the imperfect monitoring case can be rewritten as presented below.
the utility without policy-change

z

∞

∑β

}|

t−s

{

[

U(c1 , c2 , 1 − l1 ) T U (·) +

t=s

d

∞

∑ ∑β

t=s+1 ω ∈Ω

]
t−s−1

f (ω |σ ){U (·) +U (·)}
g

d

s

(19)

The left side of the equation presented above is the same as that in the perfect monitoring case. The
right side of the equation presented above comprises two terms. The first term is the utility obtainedusing the Nash rule: one-shot deviation. In other words, this effect is that the government deviatesfrom its chosen policy at s period. The difference from the previous subsection is the second termof the right side of the equation presented above. The second term is that of the expected utility forwhich the Nash rule is effective when households cannot discern government behavior. In the case of
perfect monitoring, the left-side of the equation presented above is more than, or at least equal to thatof the right-side, so that the government should commit to the original policy. However, in the case ofimperfect monitoring, the possibility exists that the right-side of the equation presented above is larger
than the left-side as long as households are cheated by the government. If such is the case, then thegovernment should deviate, or take the discretionary option, rather than commit to the policy.
The government should take the discretionary option if the RHS of eq. (19) is larger than LHS.
However, another question arises. Under which case should the government deviate? The following
proposition answers such a question.
17

Proposition 6 There exists a lower value, β ′′ ∈ (0, 1) that satisfies the condition for which the commitment solution is sustainable.
Proof See appendix A.6.

.

5 Concluding Remarks
This paper presented analysis of the optimal degree of commitment using the method of a repeatedgame. What we showed can be summarized as follows:
1. In the case of perfect monitoring, the government should not deviate from its policy, dependingon the value of the household’s patience.
2. However, in the case of imperfect monitoring, the possibility exists that the government deviatesfrom its chosen policy if households are more patient than in the perfect monitoring case.
Finally, we conclude this paper by showing the direction of extension. The plausible extension is toconsider private monitoring in which the signals differ among households.18) It is natural to consider
that the signals that the households receive mutually differ. Although the study of private monitoring
is not imported into the analysis of the macro-economic policy, this extension might be important.

Appendix A
A.1 Proof of Lemma 1
In this appendix, we first describe the model of Cooper (1999, Ch.7). Each household lives two
periods and maximizes the following utility: U(c1 , c2 , n − l2 ; gt ). Herein, ct (t = 1, 2), l2 , n − l2 , and e
respectively denote the consumption in t period, labor supply, leisure, and endowment. Moreover, gt
denotes the government expenditure, which does not affect the households’ utility. Actually, U(·) isassumed to be strictly increasing and concave in each variable. As described in section 2, the budget
constraints of the first and second period are written as
c1 + s = e, c2 = sR(1 − δ ) + l2 w(1 − τ ).
However, the government determines the labor-income and capital-income tax so that they maximize
the social welfare to raise the public expenditure; g2 . Then the budget constraint of the government is
the following:

g2 = sRδ + l2 wτ .

18) See, for instance, Matsushima (2004).

18

(20)

First, considering the commitment case, we presume circumstances in which the government determines the tax rates. Subsequently, the government cannot change the policy). Denoting the life-time
utility as V (δ , τ ), the government strategy (or policy) is set as σtg = (δt , τt ) to maximize V (δ , τ ).
max U(e − s, sR(1 − δ ) + l2 w(1 − τ ), n − l2 ; gt )From this, we obtain the following first-order conditions,{
Us = −U1 + R(1 − δ )U2 = 0
Un = (1 − τ )wU2 −U3 = 0,

(21)

Which are interpreted as the households’ response function. Given s(δ , τ )、n(δ , τ ), which is obtainable
by solving these equations, the government solves the following problem.
max U(c1 , c2 , n − l2 ; gt )s.t. gt = s(δ )Rδ + l(τ )wτ
Now, the Lagrange function is the following:
L = U(c1 , c2 , n − l2 , gt ) + λ {gt − sRδ − l2 wτ }= U(e − s(δ , τ ), s(δ , τ )R(1 − δ ) + n(δ , τ )(1 − τ ), L − n(δ , τ ), s(δ , τ )Rδ + n(δ , τ )τ ) + λ {gt − sRδ − l2 wτ }∂L
= −skU1 + (Rsk (1 − δ ) − Rs(τ ) + lk (1 − τ ))U2 − lkU3 + λ {−Rsk δ − Rs(τ ) − lk τ } = 0∂δ∂L= −sl U1 + (Rsn (1 − δ ) − ll (1 − τ ) + l2 (τ ))U2 − ll U3 + λ {−Rsl δ − ll τ − l2 (τ )} = 0,∂τwhere ∂δs ≡ sk , ∂τs ≡ sl , ∂δl2 ≡ lk , and ∂τl2 ≡ ll . Considering eq. (21), from the first-order conditions,
we obtain
U2 sR = λ [sR + δ sk R + τ lk ](22)

U2 l2 = λ [l2 + Rδ sl + τ ll ].

(23)

Next, consider the case without commitment. We consider the following extensive game in which (1)households determine the amount of saving, and (2) the government determines the tax rates, and (3)the government determines the amount of labor supply. The outcome under this case is called called
Kydland–Prescott outcome.
Now, we consider the following two-stage game:
1) The government determines the tax rate.
2) Subsequently, households determine the amount of labor supply.
Under this regime, we solve this game backward. At the second stage of this game, the government sets
the tax rates to maximize V (δ , τ ). Compared with the commitment case, the decision of household
19

s’s saving does not respond to the policy-change. Consequently, the government will impose more
capital-income taxes.

max U(e − s, sR(1 − δ ) + l2 w(1 − τ ), n − l2 ; gt )From this first-order condition, we obtain
Ul = (1 − τ )wU2 −U3 = 0.
Given l = l(τ ) obtained from the equation presented above, the government solves the following
problem.
L = U(c1 , c2 , n − l2 , gt ) + λ {gt − sRδ − l2 wτ }= U(e − s(δ , τ ), s(δ )R(1 − δ ) + n(τ )(1 − τ ), n − l2 (τ )) + λ {g2 − sRδ − l2 wτ }Solving this problem yields the following.
U2 sR = λ [sR + τ lk ](24)

U2 l2 = λ [l2 + τ ll ](25)

Now, we derive U d (·) and U s (·). Presuming circumstances under which the government deviates (i.e.,change the capital-income tax rate) at some point. Under this setting, households solve the following
problem.
U d (·) ≡max{c1t ,c2t ,l2t }U(e − s, sR(1 − δ ) + l2 (1 − τ )w, n − l2 )s.t. c2t = st Rt (1 − δ ) + l2t w(1 − τ )Us = −U1 + R(1 − δ )U2 = 0
By adopting this policy, they can obtain higher utility than that obtained under the Ramsey equilibriumin the short run, which is denoted as U d (·) in this paper.
Moreover, U s (·) is the same as “without commitment” case discussed in the above. Under thissituation, the government has an incentive to impose more taxes on capital income. Therefore, theutility level of this situation is lower than that of “commitment case”.
Under this setting, we have the following lemma 1:
Lemma 1U d (·) > U ∗ (·) > U s (·).

Proof: The second part of the inequality is attributable to the fact that the government has an incentive
to impose more taxes on capital-income after the fact and households decrease the amount of theirsaving expecting such government behavior. Consequently, the level of utility without commitment ishigher than that with commitment.
20

A.2 Proof of Prop.1
Given any history up to t, ht , a sustainable plan attains the outcomes ht = σ (ht−1 ), and (st , l2t ) =ft (ht−1 ). Given these outcomes, we define the competitive outcomes as follows:∞
CEht = {{ht }t=0
, which meets the conditions of the competitive equilibrium.}

Then, the government’s strategy σ given the history ht−1 is also admissible. Given ht−1 ∈ CEht , σ ginduces the continuation outcome path, {st , l2t , ht }, which is a part of competitive equilibrium. Then,α given the history ht−1 also meets the property of competitiveness. Finally, {ss , l2s , hs }∞s=t+1 mustmeet the property of competitive equilibrium.

A.3 Proof of Prop.2
1. As in Chari and Kehoe (1990), the right side of eq. (10) is positive (See Lemma 1) although
the left-side of that equation is increasing concomitantly with increasing β . Therefore, eq. (10)holds.
2. Equation (11) can be rewritten as follows.
U d (·) −U ∗ (·)
β
≥ ∗
1−β
U (·) −U s (·)
As β → 1, the left side of the equation presented above converges to ∞. Therefore, there exists
a lower bound β such that the equation presented above holds.

A.4 Proof of Prop.3
The second term of the right-side of equation, (14) in the case of uniform distribution can be
rewritten as follows.
∞

Et
= Et

β

∑ ∑ β t [U d (·)|π (ωt ) + 1 − β U s (·)]

t=s+1 ω ∈Ω∞β

∑ ∑ β t [U d (·)|π (ωt )] + 1 − β U s (·)

t=s+1 ω ∈Ωβ
U s (·)1−β≥ (left-side of the equation, (14))= [ωs+1 + ... + ωs+1+∆t ]U d (·) +21

Then for β ∈ [β ′ , 1] (β ′ ∈ [0, 1], we havethe utility without policy-change

z

∞

}|

{

∞

β

∑ β t U(c1 , c2 , 1 − l1 ) ≤ U d (·) + Et ∑ ∑ β t [{U d (·)|π (ωt )} + 1 − β U s (·)],

t=0

(26)

ω ∈Ω t=s+1

which means that the discretionary option is better.

A.5 Proof of Prop.5
Comparison of the following two equations shows the following: the first and second equations
respectively represent the conditions for which the commitment policy is sustainable under perfect andimperfect monitoring cases.

β[U ∗ (·) −U s (·)] ≥ U d (·) −U ∗ (·)
1−β

(27a)

[
]
∞
β
[U ∗ (·) −U s (·)] ≥ U d (·) −U ∗ (·) + Et ∑ ∑ β t [{U d (·)| f (ωt σ g )},1−βω ∈Ω t=s+1(27b)

In eq. (27b), the second term of the right side exists, which does not exist in the first equation. Therefore, the second equation is expected to hold under the upper value of β . We have β ′ > β .

A.6 Proof of Prop.6
Compare the following two equations: the first and second equations are those conditions for which
the commitment policy are, respectively, sustainable under perfect and imperfect monitoring cases.

β[U ∗ (·) −U s (·)] ≥ U d (·) −U ∗ (·)
1−β

(28a)

[
]
∞
β
[U ∗ (·) −U s (·)] ≥ U d (·) −U ∗ (·) + Et ∑ ∑ β t [{U d (·)| f (ωt σ g )},1−βω ∈Ω t=s+1(28b)

In eq. (28b), the second term of the right side exists, which does not exist in the first equation. Therefore, the second equation is expected to hold under the upper value of β . Consequently, we haveβ′ > β.

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