Discussion Papers In Economics
And Business

The time consistent public goods provision
Shigeo Morita
Discussion Paper 14-31

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe time consistent public goods provision
Shigeo Morita
Discussion Paper 14-31

October 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe time consistent public goods provision∗
Shigeo Morita†
Abstract
In this study, we reconsider the optimal non-linear tax problem with the public goods
from the perspective of the commitment issue and examine how it aﬀects the condition of
the public goods provision. We show that the Samuelson rule should be modified when thegovernment cannot commit and the skill types of taxpayers are revealed. Even if taxpayers
have the same preference, which is separable and additive with respect to consumption andleisure, the Samuelson rule breaks down. Our analysis focuses on the eﬀect of commitment
issue on the marginal cost of public funds and the level of public goods provision. Ourfindings imply that the level of investment in public goods may be excessive in comparison
to the case where the commitment issue is not considered.
Keywords: Public goods provision, Optimal Taxation, Time consistency
Journal of Economic Literature Classification Numbers: D82, H21∗

I am grateful to my supervisor Yukihiro Nishimura and participants at the Kansai Public Economic Workshop and at the 2014 Autumn Meeting of the Japanese Economic Association. Finally I would like to extend
my indebtedness to my family for their understanding, support and encouragement. All remaining errors are
mine.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail: ngp024ms@student.econ.osaka-u.ac.jp.

1 Introduction
An important role in the public sector is to manage several kinds of services; for instance,
providing health care, police protection, or education. In order to examine the optimal levelof these activities, many researchers analyze the provision of public goods. As a first-best
optimum, the public goods should satisfy the condition that the sum of the marginal utility
for those who benefit from the goods is equal to the marginal cost. This is known as theSamuelson rule. Boadway and Keen (1993) consider another situation; that is, when thegovernment cannot observe the marginal productivity of taxpayers and it is assumed thatthe public goods are financed by non-linear income taxation. They found that if the utility
function of taxpayers is of the weakly separable form, then the Samuelson rule should beapplied in the second-best environment.
Although they use the static model, public goods have the property of durable goods.
Therefore, a dynamic setting is necessary. The traditional dynamic Samuelson rule implies
that the marginal social benefit of public goods should be equal to the marginal cost plus
the static Pigou eﬀect1 and the dynamic eﬃciency eﬀect. The extension leads to topics that
we need to include in our analysis. Batina (1990) considered a time inconsistency. If thegovernment can use public debt policy and maintain the optimal expansion path, that is, thegovernment can commit to its expenditure policy in the future, then the dynamic eﬃciency
eﬀect disappears. He pointed out that such a preliminary finding is unrealistic and showed that
if the government cannot control its allocation over all periods, then the dynamic eﬃciency
eﬀect may decrease the Pigou eﬀect. As for the non-linear income tax model, Aronsson
and Grandlund (2011) focused on the situation where consumers have a time inconsistentpreference for public goods. Assuming a paternalistic government, they showed that thesocial optimal condition of public goods provision should deviate from the Samuelson rule.
This paper diﬀers from these related studies as it focuses on the commitment issue. Previous studies assume full commitment by the government. Then, once taxpayers reveal theirproductivities, the government can reconsider its tax policy in the next period on the basis of
information gained from the previous period. However, anticipating this action by the government, taxpayers adjust their decision making in the first period. This is also known as theratchet eﬀect. Although previous studies of the optimal tax problem without commitment are
discussed in the literature: for instance, Roberts (1984), Apps and Rees (2006), Bisin andRampini (2006), Brett and Weymark (2008), Krause (2009), Guo and Krause (2011, 2013),
and Berliant and Ledyard (2014), no previous studies examined how the commitment issue
aﬀects the provision of public goods. In this study, we utilize two periods and the partial
equilibrium version of the model discussed in Pirttilä and Tuomala (2001). We show thateven if the utility function is satisfied by the condition of separability, then the Samuelson
rule should not be applied when considering the commitment issue. Moreover, we analyze
1

Pigou (1928) pointed out that when the public good is financed by the distortionary tax, the social marginal
cost of public goods is relatively higher than financed by the non-distortionary tax. This is because the deadweight loss is generated by distortionary taxes. The eﬀect is called “Pigou eﬀect.”–1–how the commitment issue aﬀects the marginal cost of public funds (hereafter MCPF).
The study is organized as follows. Section 2 describes the model and considers the benchmark case. Section 3 analyzes the time-consistent public goods provision. As an additional
analysis, we examine how the commitment issue aﬀects the level of public goods provisionand social welfare in the economy. Thereafter, we provide a numerical example in Section 4.
Conclusion is given in Section 5. Proofs of our main results are presented in the Appendix.

2 The Model
2.1 Environment
Herein, we introduce a model with two time period denoted by t = 1, 22 and a production
technology with constant returns to scale in terms of labor and capital, where wage rates andrates of return correspond to productivity. For simplicity, the interest rate is assumed to beunity. The economy consists of consumers who have two kinds of productivity, θ H and θ L ,where θ H > θ L . A consumer who has a productivity θ H is called a high-type agent, while
another agent is called a low-type agent. For simplicity, it is assumed that the number of eachtype of consumer is unity3 and their productivities are constant over time. Heterogeneity of
productivity between consumers means that income diﬀerentials exist among consumers. In
order to redistribute consumer incomes, the government designs a tax policy and a program
for the provision of public goods. However, the government cannot observe the productivityof consumers. The purpose of the government is to achieve an allocation that maximizes
the social welfare through a redistribution policy that oﬀsets the incentive for consumers tochoose a mimicking strategy. The tax problem discussed in this paper can be regarded as adirect mechanism, whereby consumers report their own productivities to the government. In
this situation, the revelation principle is applied in order to determine the optimal allocation.
We adopt the classical setting of the optimal tax theory proposed by Stiglitz (1982, 1987).

2.2 Consumers
Consumers choose the amount of labor supply and consumption in both periods, where theconsumption of type i agent in period t is defined as cit and the labor supply of type i agent
in period t is defined as lti . Then, the labor income is yit ≡ θi lti . Since consumers live in a
two-periods economy, they can only save in the first period. The amount of savings by thetype i agent can be denoted by si . As with the production technology, the savings technology
2

Even if we extend this model to an economy with n periods, the result does not change. But, under theinfinite-period model, the complete pooling case introduced in Section 3 cannot be defined. We therefore
consider the two-period model, for simplicity.
3
The same result remains in an environment where there is a continuum of consumers. However, we ought
to take into account the partial-pooling case discussed by Apps and Rees (2006) and Krause (2009). Even if thiscase is included in our analysis, the result does not change.

–2–

is linear in terms of the amount of savings. Consumers choose these variables in order tomaximize their utility. It is assumed that all consumers have the same preference with respect
to consumption, the pubic goods, and the labor supply. Moreover, it is also assumed thatthe preference for consumption and the pubic goods is additive and separable for the labor
supply.4 Then, the utility function can be given by!
U i (ci1 , G1 , l1i , ci2 , G2 , l2i ) =βt−1 {u(cit , Gt ) − v(lti )}(1)twhere β ∈ (0, 1) is a discount factor. The sub-utility function u(·) is strictly increasing and aconcave function, and v(·) is strictly increasing and a convex function.

2.3 The government
The purpose of the government is to implement an allocation that maximizes the social welfare through its redistributive policy. Herein, the social welfare function is defined as the sumof consumer payoﬀs. It is assumed that the government cannot observe individual consumer
productivity θi , while the amount of labor income and savings are assumed to be observable.
This allows the government to levy non-linear taxes on these variables. The tax revenue isdivided between the public goods provision and savings by the government, which is denoted by sG . As for the public goods, it has the property of durable goods. Then, it evolves
according to the following diﬀerence equation:
G2 = δG1 + g2

(2)

where δ ∈ (0, 1) is the depreciation factor and g2 is the amount of investment in public
goods in period 2. For simplicity, public goods contribute to the welfare of consumers but
not to the production sector. However, the government can save when consumers use thesame technology. As a result of this saving, the government compensates for the lack ofredistribution between the first and second periods. Thus, the budget constraint in each period
can be formulated as follows:
!
!
!
yi1 −
ci1 − (
si + sG ) − pG1 ≥ 0
(3)
i
!
i

i
!

yi2 + (

i

i

i

G

s +s )−!
ici2 − pg2 ≥ 0(4)

where p is the marginal cost of investment in the public goods. In sum, the government
collects tax revenues through labor and capital income taxes, and uses these taxes to provide
public goods and incur saving in accordance with its redistribution policy.
4

In Boadway and Keen (1993), they found that the Samuelson rule is applied in the second-best environment
if the utility function is ui (v(cit , Gt ), lti ), which is weakly separable. The utility function in this paper is a more
restrictive formulation. It is natural that their result remains the same when the utility function is assumed to bethat shown in equation (1). The details are discussed in our benchmark analysis.

–3–2.4 Benchmark analysis
In order to examine the eﬀect of commitment, we first consider the case where we assume
full commitment by the government. Under this assumption, the government can control
the allocation in both periods through its tax policy in the beginning of period 1, and all
consumers believe that the commitment by the government is credible. Then, the government
chooses c1H , y1H , c1L , y1L , sH , sL , sG , G1 , c2H , y2H , c2L , y2L and g2 , which maximizes
!

βt−1

t

!

yi
{u(cit , Gt ) − v( ti )}
θ
i

(5)

subject to equation (2), (3), (4), and
!
t

!
yH
yL
βt−1 {u(ctH , Gt ) − v( tH )} ≥βt−1 {u(ctL , Gt ) − v( Ht )}
θ
θ
t

(6)

Equation (5) is a utilitarian social welfare function, which is the objective function of the government. Equation (6) is the incentive compatibility constraint for a high-type agent. It meansthat the government faces the constraint that induces a high-type agent to choose his/her own
labor supply and consumption, rather than the labor supply and consumption assigned to a
low-type agent. Suppose that uict is the marginal utility with respect to consumption in periodt for a type i agent and uGi t is the marginal utility with respect to the public good in the periodt for a type i agent. In this case, the social optimal condition with respect to the public goodis given by:!
i

MRS Gi 1 ,ci + δβ
1!
i

MRS Gi 2 ,ci = p
2

(7)

ui

where MRS Gi ,ci ≡ uGi t is the marginal rate of substitution of the public good for private conctt tsumption for a type i agent in period t. Equation (7) implies that the Samuelson rule should
be applied; that is, the social marginal benefit of increasing a unit of the public good over thetwo periods should be equal to the producer prices of the public goods related to the aggregate amount of private goods. This is because the utility function is of the separable form andall agents have the same preference; this assumption means that agents evaluate the marginal
benefit in the same manner. In other words, the relation between the agent’s evaluations isˆ G ,cL = MRS L L
MRS
t t
Gt ,c
t

(8)

where the superscript hat denotes the choice by a high-type agent who chooses a mimicking
strategy. For a high-type agent, the marginal benefit of choosing the truth-telling strategy isequal to that of choosing the mimicking strategy. When equation (8) holds, an increase in thelevel of the public goods never aﬀects the incentive compatibility constraint. Moreover, in the–4–benchmark case, the rule for the social optimal allocation should be constant over periods,
since the government can commit to the second-period policy in the first period. Therefore,
the government does not need to deviate from the Samuelson rule in both periods. 5

3 The time consistent public goods rule
In the beginning of the second period, the government can try to exploit the eﬃciency gainin the second period by using information gathered in the first period. This implies that the
optimal tax problem with multiple-periods involves time inconsistency. Being aware of thisrelation, agents (in particular, a high-type agent) also adjust their own decision-making in thefirst period. This is known as a commitment issue. When the assumption of full commitment
is relaxed, we can consider two cases with respect to how to adjust agent behavior. These
cases are candidates for solving the non-commitment case and they called the complete pooling case and complete separation case. 6 Consumers adjusting their behavior aﬀects both tax
revenues and provision of public goods because the tax revenues finance the public goods.
The main purpose of this section is to examine how the commitment issue aﬀects the optimal
condition of public goods provision.

3.1 The complete pooling case
Herein, a high-type agent decreases his/her own labor supply in order to benefit from theinformation rent that he/she receives in the second period. Consequently, the government
cannot observe which agent is a high-type agent at the beginning of the second period. Thismeans that the tax contract assigned in the first period does not depend on the type of agents:(c̄1 , ȳ1 , s̄). In the second period, given as v̄ = (Ḡ1 , s̄, sG ), the government chooses c2H , y2H , c2L ,y2L , and g2 , which maximizes!

yi{u(ci2 , G2 ) − v( 2i )}
θ
i

(9)

subjet to the budget constraint in the second period that corresponds to equation (4) and
y2H
y2L
H
L
u(c2 , G2 ) − v( H ) ≥ u(c2 , G2 ) − v( H )
θ
θ

(10)

Equation (9) is the social welfare function in the second period and equation (10) is the
incentive compatibility constraint in the same period. On the basis of the results in the second
5

It is a rule issue; that is, the government should follow the Samuelson rule in the second-best environment,
is obtained in the first-best environment. Therefore, the amount of public goods in the second-best environment
is not necessarily equal to the level of public goods in the first-best environment.
6
Which case is the solution to the non-commitment case depends on the shape of the function and the rangeof the exogenous variables. The details are analyzed using numerical examples in Section 4.

–5–period, the government chooses c̄1 , ȳ1 , s̄, sG , and Ḡ1 , in order to maximize!
y¯1{u(c¯1 , Ḡ1 ) − v( i )} + βV2Pool (v̄)
θ
i

(11)

subject to!
i{y¯1 − c¯1 − s̄} − sG ≥ pḠ1(12)where V2Pool (v̄) is the value function in the second period.
Herein, it is easy to show that the dynamic formulation of the Samuelson rule in thebenchmark case should be applied as the social optimal condition of the public goods provision in the complete pooling case. In the first period, the incentive compatibility constraints
are binding for both type agents. Therefore, the government faces no incentive compatibility constraints. In the second period, the planning problem corresponds to the standard
non-linear income tax problem. In terms of the evaluation of public goods between a hightype agent and a low-type agent, the outcome in this period conforms with the benchmark
case. Therefore, the social optimal condition of public goods provision in the benchmark
case remains the same as that in the complete pooling case.

3.2 The complete separation case
We consider the case in which the tax system separates a high-type agent from a low-typeagent in the first period; that is, the asymmetric information can be resolved in the second
period. At an optimum, the government can utilize the lump-sum tax policy in order toredistribute income between agents. We turn to the planning problem in this case. Given asv = (G1 , sH , sL , sG ), the government chooses c2H , c2L , y2H , y2L , and g2 , in order to maximize!

yi{u(ci2 , G2 ) − v( 2i )}
θ
i

(13)

subject to equation (4). Let V2S ep (v) be the value function in the second period of the complete
separation case. With this in mind, the government chooses c1H , c1L , y1H , y1L , and G1 to maximize!

yi{u(ci1 , G1 ) − v( 1i )} + βV2S ep (v)θisubject to equation (3) andyH (v)yHu(c1H , G1 ) − v( 1H ) + β{u(c2H (v), G2 (v)) − v( 2 H )}
θ
θ
L
L
y
y
1
2 (v)
L
L
≥ u(c1 , G1 ) − v( H ) + β{u(c2 (v), G2 (v)) − v( H )}
θ
θ
–6–

(14)

(15)

where c2H (v) and y2H (v) are the first-best contracts for a high type agent, and c2L (v) and y2L (v)are the first-best contracts for a low-type agent. Equation (15) is the incentive compatibility
constraint for a high-type agents. The left-hand side is the utility for a high-type agent thatchooses the truth-telling strategy and the right-hand side is the utility received when he/shechooses the mimicking strategy; that is, the second period payoﬀ received by a high-typeagent in the complete pooling case. Suppose that φ is the Lagrange multiplier for the incentive
compatibility constraint, λt is the Lagrange multiplier for the resource constraint in period t,
and v̂y2 is the marginal dis-utility with respect to labor income in period 2 for an agent who
mimics. Combining these factors with the optimal conditions allows us to obtain the social
optimal condition with respect to the public good:!
λ2 !
[MRS Gi 1 ,ci + δβ
MRS Gi 2 ,ci ] + ΨG = p
(16)
1
2
λ1 i
i

where

dc2H
dy2H
dc2L
dy2L
φ H
dg2
L
H
H
L
ΨG ≡ [{uG2 − uG2 }{δ +} + {uc2− v y2} − {uc− v̂y}]λ1 "!!!!!!!!!!!!!!!!!!!!!#$!!!!!!!!!!!!!!!!!!!!!%
dG1 "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!%
dG1
dG1
dG1
dG1
The direct eﬀect

The indirect eﬀect

The derivation of equation (16) is presented in Appendix A. This equation contains two additional eﬀects.
Firstly, it relates to the durability of public goods. Since the government takes into account the outcome in the second period and public goods have the property of durable goods,
increasing a unit of the public goods in the first period influences the incentive compatibility
constraint. The eﬀect is represented by ΨG , and is named “the durable eﬀect.” Furthermore,
we can classify this eﬀect into two terms; the direct eﬀect and indirect eﬀect. The first term,ΨG , indicates the direct eﬀect. Since the public goods provision contributes to the welfare
of agents, an additional public goods provision directly aﬀects payoﬀs both agents, the agent
who chooses the true-telling strategy and the agent who chooses the mimicking strategy. The
indirect eﬀect appears since the endogenous variables chosen in the second period depend on
the public good provision in the first period. This also aﬀects the incentive compatibility constraint. These are illustrated within the second and third brackets of ΨG . A critical questionis whether the direct eﬀect exceeds or falls the indirect eﬀect. To see the diﬀerence between
these eﬀects, suppose that the preference is additive and separable between private consumption and the public good. Under this assumption, the direct eﬀect disappears because themarginal utility of the public good between agents is identical. In contrast, the indirect eﬀect
is negative, as shown in Appendix B. The sign of the indirect eﬀect implies that an incremental unit of the public good restricts the incentive compatibility constraint. Specifically, theadditional public good provision decreases the payoﬀ of agent who chooses the truth-telling
strategy, but increases the payoﬀ of agent who chooses the mimicking strategy.
The second novel eﬀect is a consequence of production ineﬃciency. The engine of thiseﬀect is the ratio of the shadow price of public fund across time periods. The first-order–7–

condition with respect to the government’s saving reveals the following:−λ1 + Ψ sG + βλ2 = 0
whereΨ sG ≡ βφ[{uGH2 − uGL 2 }{δ +

dcH
dyH
dcL
dyL
dg2
} + {ucH2 2 − vyH2 2 } − {ucL 2 − v̂y 2 }]
dsG
dsG
dsG
dsG
dsG

The variable Ψ sG indicates how an additional saving by the government aﬀects the incentive
compatibility constraint. When the disutility function of eﬀort is iso-elastic, the sign of Ψ sG
is negative. The proof is shown in Appendix C. The negative sign implies that the production
eﬃciency theorem by Diamond and Mirrlees (1971) breaks down, since the marginal social
value of public funds is not equal to the marginal rate of transformation of commodities
between the first and second periods. Therefore, as the shadow price of pubic funds is higher
in the second period than that in the first period, the discount rate of the social marginal
benefit is greater than unity. Hereinafter, we call this eﬀect “the public saving eﬀect” and thisfeature allow us to propose the following.
Proposition 1. Suppose that the government cannot commit to the second period tax policy.
When all agents are completely separate in the first period (even if the preference satisfies
the separability condition and is constant for all agents), then the dynamic formulation of theSamuelson rule should be modified.
It is more interesting for us to examine the interaction with MCPF. Rearranging equation(16) yields the following:!
βλ2
S MB1 +ΨG+(− 1)δMRS Gi 2 ,ci = p"#$%2
βλ2 + Ψ sGithe durable eﬀect "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!#$!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!%
the public saving eﬀect&&
where S MB1 ≡ i MRS Gi ,ci + δ i MRS Gi ,ci is the social marginal benefit of an increase in
1 1
2 2
the level of public goods in the first period. As ΨG is negative, the durable eﬀect means that
MCPF is greater than unity. To begin with, an increase in a unit of the public goods provision
requires additional tax revenues. Then, if the government utilizes distortionary taxes, MCPF
is greater than unity because of the Pigou eﬀect. In contrast, an additional public goods
provision aﬀects MCPF through the demand for taxed goods. The durable eﬀect is diﬀerent
from this response, because increasing a unit of the public goods provision influences thewelfare level through the incentive compatibility constraint. As for the public saving eﬀect, it
implies that MCPF is less than unity because Ψ sG < 0. As λ1 is greater than zero, the sign ofthis term is positive. The weight for the marginal rate of the substitution of public goods forprivate consumption in the second period is higher than that in the first period. This means it
has the opposite eﬀect on MCPF compared with the durable eﬀect. Therefore, if the durable
eﬀect dominates the public saving eﬀect at the social optimum, then MCPF is greater thanunity and vice versa.
–8–

4 Numerical examples
Herein, we provide numerical examples in order to answer the following questions:, (i) Whichcase of social welfare is greater, the complete separation case or the complete pooling case;and (ii) How diﬀerent is the social optimal public goods provision through the social optimal
condition with respect to the pubic goods discussed in the previous sections?

4.1 Specification and Parameterization
It is assumed that utility function is represented by the the constant relative risk aversion(CRRA) function that is satisfied by separability,!
tβt−1 {

yi
1(cit · Gi )1−σ − ( ti )γ }
1−σ
θ

(17)

where σ and γ are non-negative. It is also assumed that σ = 1.0 and γ = 2.0. The otherpreference parameter (a discount factor) is β = 0.38, the wage rate of a high-type agent isθ H = 2.0, and θ L is equal to unity. Concerning a setting of the public goods, the marginal
cost of investment in the public goods is unity and the depreciation factor is 0.5. 7 Table 1
summarizes the results of our simulation.

4.2 The level of social welfare
The level of social welfare in each case is shown in the last line of Table 1. The highest
level of social welfare can be achieved in the benchmark case. This outcome corresponds
to previous studies by Bisin and Rampini (2006) and Guo and Krause (2013). The complete separation case is ranked second and the complete pooling case is ranked third. The
asymmetric information remains in the second period of the complete pooling case, while it
resolves in the complete separation case. The diﬀerence of information structures between
the two cases can enhance the social welfare level in the complete separation case.

4.3 The level of public goods provisions
In order to analyze the level of public goods determined by the social optimal condition
discussed in the previous sections, we focus on the value of public goods provided in the first
period. In fact, the level of public goods is highest in the benchmark case and, lowest in thecomplete pooling case. Since the Samuelson rule should be applied in the benchmark and7This specification and parameterization, as outlined below, refer to Golosov et al. (2006) and Guo andKrause (2013). Their simulations are based on the optimal non-linear income tax problem without the public
goods provision. They examine the two-period case and the infinite-horizon case. In the former case, the levelof social welfare in the complete separation case is larger than that in the complete pooling case; the same result
as we find in this paper.

–9–

Table 1: Results of simulation
Benchmark case
High
Low
i
c1
0.893466 0.443849
yi1
1.88829
1.20472is0.553725 0.104287
i
c2
0.948375 0.787473
yi2
1.65153
1.02771
Payoﬀ-0.846492 -0.945162
G1
1.97365g20.725424sG-0.875972
S MB1
1.00
λ2
1.00
λ1
ΨG
Social welfare-1.79164

Pooling caseHighLow0.590996 0.590996
1.99574
1.99574
0.533473 0.533473
1.62404
1.265821.69052
1.21287-1.53149 -2.13712
1.319171.50384
0.423365
1.00
1.00-3.66859

Separation caseHighLow
1.01529
0.26941.88046
1.39809
0.0397036 0.105052
0.970125 0.970125
1.66288
1.36307-0.836572 -1.90097
1.54457
1.534990.304534
1.252211.69018-1.51776-2.73754

complete pooling cases, the diﬀerence in the public goods levels between the cases depends
on the values of the endogenous variables. On the other hand, the level of public goods in thecomplete separation case is ranked second. A key variable to discuss the diﬀerence between
the benchmark case and the complete separation case is ΨG . Since the value of this variablein the complete separation case is negative, the sum of the marginal rates of substitution
should be greater than the producer price of the public good in terms of the private good. By
comparing the Samuelson rule in this case with the benchmark case, we find that the level of
public goods in the complete separation case is lower than in the benchmark case.

5 Conclusion
In this study, we reconsider the optimal non-linear tax problem with the public goods provision from the perspective of the commitment issue. If taxpayers have the same preferences
and the preferences are separable consumption and leisure, the Samuelson rule does not need
to be modified, as discussed in previous studies: Christiansen (1981) and Boadway and Keen
(1993). However, the outcome is altered dramatically when the commitment issue is considered. We show that the Samuelson rule should be modified when the taxpayers are completely
separated by tax policy, even if the utility function satisfies the separability condition. Moreover, we point out through numerical examples that the eﬀects of the commitment issue also
aﬀect the level of public goods.

– 10 –

AppendicesA. Proof of equation (16)Consider the second-period planning problem in the complete separation case. The Lagrangian can be written as follows:
L2 =

!

!
yi
{u(ci2 , δG1 + g2 ) − v( 2i )} + λ2 [ {yi2 + si − ci2 } + sG − pg2 ]
θ
i
i

(18)

The first order conditions are∂L2= ucH2 − λ2 = 0∂c2H∂L2= ucL2 − λ2 = 0∂c2L∂L2= −vyH2 + λ2 = 0
H
∂y2
∂L2
= −vyL2 + λ2 = 0
L∂y2∂L2 ! i=uG2 − λ2 p = 0
∂g2
i

(19)
(20)
(21)
(22)
(23)

∂L2 ! i={y2 + si − ci2 } + sG − pg2 = 0
∂λ2
i

(24)

∂u i
dv 1
i
i
where the notations on derivatives are denoted by uict = ∂ci (ct , G t ), vyt = dli θi and uG t =∂u(ci , Gt ). From the envelope theorem, we find
∂Gt t

t

t

!
dV2S ep
=δ
uGi 2
dG1
i

dV2S ep
= λ2
dsG

(25)
(26)

Next, we turn to the first period planning problem. The Lagrangian in the first period is
L1 =

!

!
yi1
i
{u(c1 , G1 ) − v( i )} + βV2S ep (v) + λ1 [ {yi1 − ci1 − si } − sG − pG1 ]θiiHyH (v)y+φ[u(c1H , G1 ) − v( 1H ) + β{u(c2H (v), G2 (v)) − v( 2 H )}θθy1L
y2L (v)
LL−u(c1 , G1 ) + v( H ) − β{u(c2 (v), G2 (v)) − v( H )}]θθ– 11 –(27)

The optimality conditions with respect to the private consumption of each consumer, thepublic goods and saving are as follows:∂L1= (1 + φ)ucH2 − λ1 = 0∂c1H∂L1= (1 − φ)ucL2 − λ1 = 0
L∂c1
dV2S ep∂L ! i=uG + β− λ1 p
∂G1
dG1
i

dc2H
dyH
dcL
dyL
dG2
dG2
− vyH2 2 + uGH2
} − {ucL 2 − v̂y 2 + uGL}] = 0
dG1
dG1
dG1
dG1
dG1
dG1
dV2sep
∂L1
=β
− λ1
∂sG
dsG
dcH
dyH
dcL
dyL
dG2
dG2
+βφ[{ucH2 2 − vyH2 2 + uGH2
} − {ucL 2 − v̂y 2 + uGL}] = 0
dsG
dsG
dsG
dsG
dsG
dsG

(28)
(29)
(30)

+βφ[{ucH2(31)

By the definition of public goods between the first period and the second period (equation(2)), it is easy to show that the marginal eﬀect of public goods in the first period on in thedg2
dG2
second period is dG= δ + dG
. Using equations (19), (20), (24), (26), (28), and (29), equation11(30) can be simplified to equation (16). !
B. Proof of the sign of ΨG∂2 u
We consider the sign of ΨG . It is assumed that ∂c∂G
is zero. Let A be the Hessian withrespect to equation (19) to (24). We can rewrite these equations as a matrix formation,⎞⎛ H ⎞ ⎛
0 ⎟⎟⎜⎜⎜ dc2 ⎟⎟⎟ ⎜⎜⎜⎟⎟⎟⎜⎜⎜ dcL ⎟⎟⎟ ⎜⎜⎜
0
⎟⎟⎟⎟
⎜
⎜⎜⎜
2 ⎟
⎟⎟⎟
⎜⎜⎜ dyH ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜
0⎟⎟ dG1A ⎜⎜⎜⎜ 2L ⎟⎟⎟⎟ = ⎜⎜⎜⎜⎜⎜⎜ dy2 ⎟⎟⎟ ⎜⎜⎜ & 0 ⎟⎟⎟⎟⎟⎜⎜⎜ dg ⎟⎟⎟ ⎜⎜⎜ δ ui ⎟⎟⎟2 ⎟i g,g ⎟
⎜⎝
⎠ ⎜⎝
⎠
dλ2
0
where

⎞
⎛ H
0
0
0
0 −1 ⎟⎟⎟⎜⎜⎜ uc,c
⎟
⎜⎜⎜
L
0
0
0 −1 ⎟⎟⎟⎟⎜⎜⎜ 0 uc,c
⎟
H
⎜⎜⎜ 0
0
0
1 ⎟⎟⎟⎟
0 −vy,y⎜⎟A ≡ ⎜⎜⎜
L
0
1 ⎟⎟⎟⎟⎟
00 −vy,y
⎜⎜⎜ 0
&
⎟⎟
⎜⎜⎜
i
0
0
0
⎟⎟⎠
⎜⎜⎝ 0
i ug,g −p ⎟
−1 −1
1
1
−p
0

The determinant of A is

H L H L 2
|A| = −uc,cuc,c vy,y vy,y p +

!
i! j

j i
uic,c uc,c
vy,y

!
i

– 12 –

uig,g −!
i! jjuic,c viy,y vy,y!
iuig,g < 0(32)

It implies that the inverse matrix of |A| exist and Cramer’s theorem is available. Then, we
obtain&δ i uig,g L H L
dc2H=
puc,c vy,y vy,y < 0
(33)
dG1
|A|
&
δ i uig,g H L L
dy2H=
puc,c uc,c vy,y > 0
(34)
dG1
|A|
&
δ i uig,g H H L
dc2L=−
puc,c vy,y vy,y > 0
(35)
dG1
|A|
&
δ i uig,g H L H
dy2L=−
puc,c uc,c vy,y < 0
(36)
dG1
|A|
&
δ i uig,g H H L
dg2
H L
L L
=
vy,y {uc,c uc,c − uc,cvy,y − uc,cvy,y } > 0
(37)
dG1
|A|
These equations imply that ΨG is negative.
C. Proof of the sign of Ψ sG
We show that Ψ sG is negative. Suppose that the dis-utility is an iso-elastic utility functionyiyi
of work eﬀort, that is, v( θti ) = νκ ( θti )ν , κ > 0, ν ≥ 1. We can rewrite equations (19) to (24) as amatrix formation,⎛ H ⎞ ⎛ ⎞⎜⎜⎜ dc2 ⎟⎟⎟ ⎜⎜⎜ 0 ⎟⎟⎟⎜⎜⎜ dcL ⎟⎟⎟ ⎜⎜⎜ 0 ⎟⎟⎟
⎜ ⎟
⎜⎜⎜
2 ⎟
⎜⎜⎜ dyH ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ 0 ⎟⎟⎟⎟⎟A ⎜⎜⎜⎜ 2L ⎟⎟⎟⎟ = ⎜⎜⎜⎜ ⎟⎟⎟⎟ dsG⎜⎜⎜ dy2 ⎟⎟⎟ ⎜⎜⎜ 0 ⎟⎟⎟⎜⎜⎜ dg ⎟⎟⎟ ⎜⎜⎜ 0 ⎟⎟⎟2 ⎟⎜⎝⎠ ⎜⎝ ⎟⎠1
dλ2

Applicaition of Cramer’s rule yields,& idc2Hi ug,g L H L=
u v v <0dsG
|A| c,c y,y y,y& idy2Hi ug,g H L L=
u u v >0dsG
|A| c,c c,c y,y& i
dc2Li ug,g H H L=
u u v <0dsG
|A| c,c c,c y,y&idy2Hi ug,g H L H=
u u v >0dsG
|A| c,c c,c y,ydg2
1 H L H L=u u v v p<0
dsG |A| c,c c,c y,y y,y– 13 –

(38)
(39)
(40)
(41)
(42)

From equations (19) and (20), it is easy to show that ucH2 is equal to ucL2 . Moreover, it implies
∂cH

2

∂cL

∂ u
that c2H is also equal to c2L , if ∂c∂G
= 0. Then, we can find that ∂s2G − ∂sG2 is zero. These results
give the following equation:H
L 2
1 1 H L ! i v̂y vy,y (θ )Ψ sG = ( H ) uc,c uc,cug,g [ H L H 2 − 1]θ |A|vy vy,y (θ )i(43)

Since the dis-utility function is iso-elastic, equation (41) can be rewritten asyL θ L1 1 H L ! i
Ψ sG = ( H ) uc,cuc,cug,g [( H2 )( H )ν − 1]
θ |A|
y2 θ
i

(44)

By subtracting equation (21) from equation (22), we find that y2H > y2L . Then, equation (42)has a negative sign and Ψ sG is also negative. !

References[1] Aronsson, Thomas and Grandlund, David. 2011, Public goods and optimal paternalism
under present-biased preferences, Economic Letters, 113, 54-57[2] Apps, P and Rees, R, 2006, Repeated Optimal Nonlinear Income Taxation, mimeo.
[3] Batina, Raymond G., 1990, Public goods and dynamic eﬃciency The Modified Samuelson Rule, Journal of Public Economics, 41, 389-400[4] Berliant, Marcus and Ledyard, John O, 2014, Optimal dynamic nonlinear income taxes
with no commitment, Journal of Public Economic Theory, 16, 196-221[5] Bisin, Alberto, and Rampini, Adriano, 2006, Markets as Beneficial Constraints on the
Government, Journal of Public Economics, 90, 601-629[6] Boadway, Robin, and Keen, Michael, 1993, Public goods, self-selection and optimal
income taxation, International Economic Review, 34, 463-478[7] Brett, Craig, and John Weymark, 2008, Optimal nonlinear taxation of income and savings without commitment, Working paper 08-W05. Vanderbilt University[8] Christiansen, Vidar., 1981, Evaluation of public projects under optimal taxation, Review
of Economic Studies, 48, 447-457[9] Diamond, Peter and Mirrlees, James, 1971, Optimal taxation and public production,
American Economic Review, 61, nos.1 and 3, 8-27, 261-278– 14 –

[10] Golosov, Mikhail, Tsyvinski, Aleh and Werning, Iv́an, 2006, New dynamic public finance: a user’s guide, NBER Macroeconomics Annual, 317-387[11] Guo, Jang-Ting, and Krause, Alan, 2011, Optimal nonlinear income taxation with habit
formation, Journal of Public Economic Theory, 13, 463-480[12] Guo, Jang-Ting, and Krause, Alan, 2013, Optimal nonlinear taxation of income andeducation expenditure, Oxford Economic Paper, 65, 74-95[13] Krause, Alan, 2009, Optimal nonlinear income taxation with learning-by-doing, Journalof Public Economics, 93, 1098-1110[14] Pigou, Arthur C., 1928, A Study in Public Finance, London: Macmillan.
[15] Pirttilä, Jukka, and Tuomala, Matti, 2001, On optimal non-linear taxation and public
good provision in an overlapping generation economy, Journal of Public Economics,
79, 485-501[16] Roberts, Kevin, 1982, The Theoretical Limits to Redistribution, Review of Economic
Studies, 51, 177-195[17] Stiglitz, Joseph E., 1982, “Self-selection and Pareto eﬃcient taxation” Journal of Public
Economics 17, 213-240[18] Stiglitz, Joseph E., 1987, “Pareto eﬃcient and optimal taxation in the new new welfare economics”, In: Auerbach, A. J. and Feldstein, M. (Eds.), Handbook of Public
Economics. Vol.2. North-Holland, Amsterdam, 991-1042– 15 –