Discussion Papers In Economics
And Business

Optimal income taxation without commitment:
policy implications of durable goods
Shigeo Morita
Discussion Paper 14-32

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal income taxation without commitment:
policy implications of durable goods
Shigeo Morita
Discussion Paper 14-32

October 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal income taxation without commitment:
policy implications of durable goods ∗Shigeo Morita†
Abstract
This paper examines the design of a tax policy applied to the consumption of durable
goods and labor income. We consider cases wherein the government cannot commit to a
tax policy in the second period. If the type of taxpayers is unrevealed, it is optimal to tax
the durable goods consumption of a high-income earner and subsidize that of a low-income
earner. On the other hand, when the type of taxpayers is revealed, imposing a positive tax
rate on a high-income earner’s durable goods consumption is desirable. This implies thatthe government should design taxes on durable goods consumption to be progressive andsupplement its optimal tax policies.
Keywords: Commitment, Optimal Taxation, Time consistency
Journal of Economic Literature Classification Numbers: D82, H21∗

I would like to thank my supervisor Yukihiro Nishimura for the long discussions, support, and guidance.
I am grateful to David de la Croix, Jean Hindriks, Nobuo Akai, Pierre Pestieau, and Shingo Ishiguro for theirinvaluable comments and suggestions. I am thankful for the helpful comments of the participants at the Welfare Economic Seminar in CORE, Université Catholique de Louvain and the 15th Annual Conference of the
Association for Public Economic Theory. Finally, I would like to extend my indebtedness to my family for theirunderstanding, 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
According to OECD data, durable goods account for 10 percent of the final consumption
expenditure of households in the United States and Japan during the recent decade, implyingthat the tax treatment of durable goods is an important issue for policy makers. For instance,
when people purchase an automobile in Japan, they usually face an automobile acquisition
tax. On the other hand, the federal income tax in the United States is characterized as apreferential tax treatment of housing. In fact, a mortgage interest can be deducted from ataxable income. In public finance literature, the Atkinson-Stiglitz (1976) theorem provides
important implications for both commodity and income taxation under the second-best environment. They reveal that if the utility functions are weakly separable between consumption
and leisure, government expenditure should be financed by taxing labor income and, notcommodities. The other basic message from the standard model is that the optimal-tax-rate
structure is unprogressive. Stiglitz (1987) implied that the marginal tax rate on capital income
tax is zero and does not depend on the income status of taxpayers. These studies reveal that
there is no need for government intervention in the durable goods market.
We should more carefully consider the outcome of these previous studies in terms of thepolicy implications for durable goods. That is, we need to consider the property of durable
goods. Cremer and Gahvari (1995) add the standard non-linear income tax problem to theheterogeneous nature of the preference for housing as well as the quality of housing. Cremer(1998) discusses the tax treatment of a good, wherein the amount of consumption of goos
is chosen before a realized uncertainty; he named it a pre-commit good, which resembles adurable good. Consequently, they demonstrate that government intervention is required in
the choice of these goods under certain conditions. This implies that the Atkinson-Stiglitz
theorem is invalid. Although Naito (1999) revealed that the Atkinson-Stiglitz theorem fails
due to the imperfect substitution in the labor supply through general production technologies,
they indicate another reason why it fails. This paper diﬀers greatly from these studies; we
focus on a time inconsistency problem and the issue originated from it. Since durable goods
can be defined as the goods that individuals consume for multiple time-periods, we extend thestandard model to a dynamic setting. This extension leads to a time inconsistency problem.
Once taxpayers reveal information on their productivity, the government can revise its tax
policy in the next period using that information. Knowing this exploitative behavior on partof the government, taxpayers will change their decision making in the first period. This is
known as a commitment issue under a non-linear income tax problem. In fact, several studies
consider the optimal tax problem without commitment, for instance Roberts (1984), Apps
and Rees (2006), Bisin and Rampini (2006), Brett and Weymark (2008), Krause (2009), Guo
and Krause (2011, 2013), and Berliant and Ledyard (2014). 1 This paper is related to that of1

Roberts (1984) showed that, in an infinite horizon economy, the type of taxpayer was never revealed. On
the other hand, Berliant and Ledyard (2014) indicated the suﬃcient condition that the consumers are separated
in the first period. These studies have mainly made a significant contribution to the framework of the theoreticalanalysis of this topic, while this paper mainly focuses on applying their frameworks to policy implications.

–1–Brett and Weymark (2008), who focus on the saving behavior of taxpayers and discuss the
optimal capital and labor income tax policy. The diﬀerence between previous studies and thispaper is that we focus on the choice of durable goods by taxpayers and examine the optimallabor income and commodity tax system. Therefore, this paper examines the rationale oftaxation on durable goods from the perspective of an optimal non-linear tax problem withoutcommitment. Thereafter, it demonstrates that the choice of durable goods should be distorted
and optimal tax rate should relate to the income status of taxpayers. Moreover, we also findthat a progressive tax structure is required, that is, optimal tax rate for the high-income earnershould be larger than the tax rate for the low-income earner.
The remainder of this paper is organized as follows: Section 2 presents the setting of themodel that analyzes the optimal tax policy. Section 3 examines the eﬀect of commitment,
considering the optimal tax problem with commitment. Section 4 considers the optimal
tax problem without commitment. Section 5 provides a numerical example of the aboveconsidered model.

2 The model
We consider a two period economy: t = 1, 2. 2 There exists two types of consumption goods:
the non-durable and durable goods. The durable good is defined as the good consumed overmany periods. In this study, it is assumed that there is no depreciation and no maintenanceinvestment with respect to the durable goods, because whether these settings are included
or not does not depend on the results in this paper. In this economy, there are two kinds ofplayers: taxpayers and the government. To apply the Revelation Principle, we formulate theplanning problem as a game, wherein the strategies is reporting their productivity. The economy comprises two taxpayers that have diﬀerent productivities in the labor market: i = H, L.
This implies that income diﬀerentials among taxpayers exist. The taxpayer whose marginal
productivity is θ H is a high-income earner. On the other hand, the taxpayer whose marginal
productivity is θ L is a low-income earner. Their productivities are private information and are
constant over all periods. In each period, they choose the amounts of the non-durable good,
durable good, and labor supply that maximize their utility functions. We neglect savings byeconomic agents in this model, because the savings decision has no eﬀect on the results inthis paper. The consumption of the non-durable good by type i in period t is denoted by cit ,durable good by type i is denoted by di , and labor supply by type i in period t is denoted by lti .
Therefore, the pre-tax income in period t of the taxpayer whose productivity is θi is definedby yit = θi lti . In this model, their productivities are equal to their wage rates. Then, the budget2

For simplicity, this paper considers a two period model. Even if we extend it to the n-period economy, theresults in this paper are never aﬀected by this treatment. However, when we consider an infinite economy, thecomplete pooling case, mentioned below, cannot be defined.

–2–

constraint in each period can be written as follows:yi1 − T 1 (yit ) = ci1 + (1 + τi )diyi2 − T 2 (yi2 ) = ci2

(1)
(2)

where τi is a linear tax on the durable good and T t (yit ) is an income tax function. In this model,
consumptions of durable goods are generally taxed only at the time of purchase. There is no
essential diﬀerence among timings of taxation to discuss the optimal tax on the durable goodsince taxpayers maximize their utility subject to lifetime budget constraints. We can examine
the detail of optimal policies under the assumption that the government taxes or subsidizes
income and consumption in a nonlinear way, although income dependent commodity taxes
are not used in practice.
It is assumed that the utility function of taxpayers is additive and separable between theirconsumption and labor supply. On the other hand, their preference is not assumed to be
separable in terms of the durable and non-durable goods. Eichenbaum and Hansen (1990)
and Ogaki and Reinhart (1998) found no evidence against strict separability across durable
goods and non-durable goods. In addition, it is assumed that all taxpayers have the same
utility function. Then, the function is given as follows:!
U(ci1 , di , l1i , ci2 , l2i ) =βt−1 {u(cit , di ) − v(lti )}(3)twhere β is a discount factor. This utility function satisfies the following conditions: u′ (·),
−u′′ (·), v′ (·) and v′′ (·) are all positive. The key assumption of the sub-utility function is∂2 uthat ∂c∂d
is also positive. The empirical results in Ogaki and Reinhart (1998) have reported
that the intra-temporal elasticity of substitution between durable and non-durable goods are
significantly diﬀerent from zero and positive. Based on their specification of utility functions,
it justifies the assumption that the second-order partial derivative is positive.
Taxpayers maximize the utility function (3) subject to their budget constraints (1) and(2). Combining with first-order conditions with respect to consumption and labor supply, themarginal tax rate can be defined as follows:vy (lti )θi uc (cit , di )ud (ci1 , di ) + βud (ci2 , di )τi =−1
uc (ci1 , di )MT Rit ≡ T t′ (yit ) = 1 −

(4)
(5)

where uc (·) is the marginal utility with respect to the consumption of the non-durable good,ud (·) is the marginal utility with respect to the consumption of the durable good and vy (·)
is the marginal dis-utility with respect to the labor supply. Thereafter, MT Rit can be seenas the marginal labor income tax rate function, which is one minus the marginal rate of
substitution of labor income for the consumption of the non-durable good in each period. In
contrast, equation (5) is the marginal commodity tax rate. In the absence of the government
intervention in the decision-making, the left-hand side of these equations corresponds to zero.
The optimal tax policy is characterized by the sign of these marginal tax rates.
–3–

3 Benchmark case
To examine the eﬀect of a commitment issue on the optimal tax policy, at the beginning, weanalyze the case that is assumed to be full commitment by the government as a benchmark
case. In this case, suppose that the government can credibly commit to the tax policy. Thismeans that the government ignores the information, of the taxpayer’s productivity, revealedin the first period. The government controls cit , di , and yit for (t = 1, 2; i = H, L) to maximize
the social welfare function. Thereafter, the government’s optimization problem is maximizing the sum of the taxpayer’s utility under the incentive compatibility condition and budget
constraints of the government. Hence, the planning problem in the full commitment case canbe given as follows:!!
max
βt−1 {u(cit , di ) − v(lti )}(6){cit ,di ,yit }i=H,L
t=1,2

i

t

!

{yi1 − ci1 − di } ≥ 0(7){yi2 − ci2 } ≥ 0

(8)

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

(9)

s.t.

i

!
t

!
i

Equation (6) is the objective function of the government, which is a utilitarian social welfare
function. Equations (7) and (8) are resource constraints in each period, implying that the
tax revenue in each period is not less than zero. Equation (9) is the incentive compatibility
constraint. The utilitarian government attempts to redistribute income from a high-income
earner to a low-income earner. The redistribution leads a high-income earner pretending to
be a low-income earner. Thus, we consider the case wherein the incentive compatibility constraint for a high-income earner binds throughout the remainder of this paper. It is assumed
that the problem has interior solution for cit , di , and lti , for (t = 1, 2; i = H, L). The solution to
the problem yields the following lemma:
Lemma 1. If the government can commit to the second-period tax policy, then the optimal tax
on the durable good should be zero. Concerning labor income taxes, the high-income earnershould face a zero marginal tax rate on her income, while the low-income earner should betaxed at a positive marginal tax rate.
Since the government can commit to the tax policy, “no-distortion at the top” result remains in each period, implying that a high-income earner should face a zero marginal labor
income tax rate at both periods, while a low-income earners should be assigned a positive
marginal labor income tax rate across time. In other words, the government chooses a contract that, gives the information rent to the high-income earner. This outcome is similar tothe outcome by Stiglitz (1982, 1987). As for the choice of the consumption good, if the–4–preference is separable between labor supply and other consumption goods, the optimal tax
structure requires no taxation of all consumption goods, because the evaluation of consumption does not depend on the productivity; that corresponds to the Atkinson-Stiglitz (1976)
theorem.

4 Non-commitment caseThis section reconsiders the assumption of full commitment. When the assumption is relaxed,
the government is able to exploit the eﬃciency gains in the second period. Once taxpayers
anticipate such exploitation, they adjust their own decision making in the first period. According to how to adjust their decision making, we can consider two possible solutions: thecomplete pooling and complete separation cases. 3 In both these cases, the planning problem
is considered according to the concept of the backward induction.

4.1 Complete pooling caseIn this case, a high-income earner is concerned with the expropriation of her information rentin the second period and avoids reporting her true productivity to the government. This meansthat information with respect to productivity remains completely hidden in the beginning of
the second period. Thereafter, the allocation in the first period is (c¯1 , d̄, y¯1 ), which does notdepend on productivity. In the second period, given as d̄ = (d̄), the government chooses(c2H , c2L , y2H , y2L ) by solving the planning problem as follows:
V2Pool ( d̄) =!

yi{u(ci2 , d̄) − v( 2i )}θ{c2H ,c2L ,y2H ,y2L }
i=H,L
!
s.t.
{yi2 − ci2 } ≥ 0
max

(10)
(11)

i=H,L

yH
yL
u(c2H , d̄) − v( 2H ) ≥ u(c2L , d̄) − v( H2 )
θ
θ

(12)

where V2Pool ( d̄) is the value function of this optimization problem (10) to (12). Equation (11)is the resource constraint in the second period, and equation (12) is the incentive compatibility
constraint for a high-income earner. Since the asymmetric information remains, the problem
corresponds to the standard non-linear income taxation problem. The government knows theoutcome in the second period and solves the planning problem, which can be described as
3

We examine these two cases as non-commitment case, because the number of high-income and low-income
earners is assumed to be one, respectively. If the number of high-income and low-income earners is not one,
we must additionally consider “the partial pooling case,” wherein high-income earners are pooled and the restof them are separated, in the first period. Even if we extend the model to consider the partial pooling case, nonovel result exists.

–5–follows:max{c¯1 ,y¯1 ,d̄}!

y¯1{u(c¯1 , di ) − v( i )} + βV2Pool ( d̄)
θ
i
!
s.t.
{y¯1 − c¯1 − d̄} ≥ 0

(13)
(14)

i=H,L

Equation (14) is the resource constraint in the first period. In this case, the government
does not face the incentive compatibility constraint, since both types of taxpayers are pooled.
In other words, the incentive compatibility constraints for a high-income and a low-income
earners are binding in the period.
4.1.1 Labor income taxation
In the first period, labor income does not depend on the productivity of taxpayers because
their productivity is heterogeneous. The relationship between the labor supply of both typesof taxpayers is summed up as follows:y¯1 = θ H l¯1H = θ L l¯1L =⇒ l¯1H < l¯1L

(15)

where the subscript “bar” denotes the choice in the complete pooling case. To implement thelabor income ȳ1 , the labor supply of a high-income earner should be distorted downwards
and that of a low-income earner should be distorted upward. This implies that a high-income
earner faces a positive marginal income tax rate and a low-income earner faces a negative
marginal income tax rate. On the other hand, since the optimal tax problem in the second
period is identical to that in the static model, the usual pattern of the marginal labor income
tax rate can be obtained in this period. These results lead to lemma 2.
Lemma 2. Under the complete pooling case, the optimal labor income taxation is characterized as follows:
MT R1H > 0, MT R1L < 0
MT R2H = 0, MT R2L > 0
4.1.2 Optimal tax on durable goods
We obtain the optimal tax formula through the optimal condition for durable goods as follows:τH =(1 − ρ)β[ud (c2H , d̄) − ud (c2L , d̄)] > 0
2uc (c̄1 , d̄)(16)(1 + ρ)β[ud (c2H , d̄) − ud (c2L , d̄)] < 0
2uc (c̄1 , d̄)(17)τL = −–6–

where ρ is the Lagrange multiplier for the incentive compatibility constraint. Equation (16) is
the optimal marginal tax rate for the durable good consumption of a high-income earner, andequation (17) is that of a low-income earner. We show the derivation of these equations in∂2 u
the appendix. The derivation also shows that 1 − ρ is positive. Because it is assumed that ∂c∂d
is positive, the solutions in the bracket of these equations are also positive.4 Since a highincome earner receives the information rent, her consumption is higher than a low-income
earner’s consumption. This implies that the marginal utility of durable goods in the second
period for a high-income earner is lower than for a low-income earner. By the definition of
the marginal tax rate for durable goods (equation (5)), we find that the marginal rate of substitution between durable and non-durable goods for a high-income earner is more than one,while that of a low-income earner is less than one. Therefore, a high-income earner should
face a positive tax rate and a low-income earner should face a negative tax rate. This meansthat the government should design taxes on durable goods consumption to be progressive andsupplement its optimal tax policies. Thus, we have the following proposition:
Proposition 1. When the government cannot commit to the second period tax policy, andwhen the taxpayers are completely pooled in the first period, (i) the consumption of the
durable good by a low-income earner should be subsidized and (ii) the consumption by ahigh-income earner should be taxed.

4.2 Complete separation case
Suppose that the information with respect to the productivity of taxpayers is completely revealed in the beginning of period 2. Using the information gleaned in the first period, thegovernment is free to re-optimize its tax policy in the second period. In this case, the social
optimal allocation in the second period can be achieved through a personalized lump-sum tax
policy. Given as d = (d H , d L ), the government chooses (c2H , c2L , y2H , y2L ) through solving thesocial planning problem as follows:
V2S ep (d) =max{c2H ,c2L ,y2H ,y2L }

!

i=H,L

yi2
i
i
{u(c2 , d ) − v( i )}
θ(18)

{yi2 − ci2 } ≥ 0

(19)

s.t.

!

i=H,L

where V2S ep (d) is the value function in the second period. Equation (18) is the objective function, and equation (19) is the resource constraint in the second period. Since the governmentis able to identify taxpayers, the incentive compatibility constraints need not be added to4

If the preference between durable and non-durable goods is satisfied with the separability, the bracket
in the left hand side of these equations disappears. Therefore, the government should not tax durable goods
consumption because the additional consumption of durable goods does not aﬀect the incentive compatibility
constraints through the marginal utility of consumption.

–7–the planning problem. Similarly, with the complete pooling case, the government chooses
c1H ,c1L ,y1H ,y1L , d H , and d L in anticipation of the outcome in period 2 as follows:max{c1H ,c1L ,y1H ,y1L ,d H ,d L }!

yi{u(ci1 , di ) − v( 1i )} + βV2S ep (d)
θ
i
!
s.t.
{yi1 − ci1 − di } ≥ 0

(20)
(21)

i=H,L

yH
yH (d)u(c1H , d H ) − v( 1H ) + β{u(c2H (d), d H ) − v( 2 H )}
(22)
θ
θ
yL
yL (d)≥ u(c1L , d L ) − v( H1 ) + β{u(c2L (d), d L ) − v( 2 H )}θθHH
where c2 (d) and y2 (d) are the first-best contract for a high-income earner, and c2L (d) and y2L (d)also are the first-best contract assigned to a low-income earner. Equation (21) is the resource
constraints in period 1, and equation (22) is the incentive compatibility constraint for a highincome earner. The left hand side shows the payoﬀ when a high-income earner chooses thetruth-telling strategy. In contrast, the right hand side is the payoﬀ that a high-income earnerchooses the mimicking strategy, that is, the payoﬀ that is realized in the complete poolingcase.
4.2.1 Labor income taxation
Clearly, the pattern of the marginal labor income tax rate in the first period is similar as ina static model. On the other hand, due to the first-best nature in the second period, the zeromarginal labor income tax rates for both types is desirable. To sum up, we have lemma 3.
Lemma 3. Under the complete separation case, optimal labor income taxation is characterized as follows:
MT R1H = 0, MT R1L > 0
MT R2H = 0, MT R2L = 0
4.2.2 Optimal tax on durable goods
The social optimal condition with respect to the durable good consumption by each skill typeof taxpayer contains an additional term. Let Γi be the indirect eﬀect of the durable good
consumption of a taxpayer whose productivity is θi , on the incentive compatibility constraint
through the amount of consumption and labor income in the second period. These are given
as follows:yH 1 dyH
dcL
yL 1 dyL
dcH
ΓH = [{uc (c2H , d H ) 2H − vy ( 2H ) H 2H } − {uc (c2L , d L ) H2 − vy ( H2 ) H H2 }]dd
θ θ dddd
θ θ dd
H
H
H
L
dc
y
dy
dc
y2L 1 dy2L
1 2
2
2
2
L
H
H
L
L
Γ = [{uc (c2 , d ) L − vy ( H ) H L } − {uc (c2 , d ) L − vy ( H ) H L }]dd
θ θ dddd
θ θ dd–8–

The sign of Γi is shown in the appendix. The results in the appendix indicate that the firstbracket of ΓH is negative, while the second bracket is positive. This means that the eﬀectof durable good consumption on the payoﬀ of the taxpayer who chooses the truth-telling
strategy has a negative sign, but the eﬀect of durable good consumption on the payoﬀ of
the mimicker has a positive sign. Thereafter, the sign of ΓH is negative, which implies thatthe additional durable goods consumption restricts the incentive compatibility constraint. On
the other hand, we find that the first bracket of ΓL is positive, while the second bracket isambiguous because the additional consumption of durable goods reduces not only the utility
from consumption that the mimicker receives but also the disutility the mimicker receives
from her labor supply. In total, the sign ΓL cannot be determined. Based on these variables,
we can find the optimal tax rate for durable goods for each type as follows:ψΓH
1 + ψ uc (c1H , d H )ψΓLτL = −
1 − ψ uc (c1L , d L )τH = −(23)(24)

where ψ is the Lagrange multiplier for the incentive compatibility constraint. The derivationof these equations also is in the appendix. Equation (23) is the optimal tax rate for the
durable good consumption of a high-income earner, while equation (24) is that of a lowincome earner. Clearly, the signs of these tax rates depend on Γi . Then, a high-income earnershould face a positive tax rate due to ΓH < 0. On the other hand, whether the government
should tax a low-income earner’s durable good consumption is ambiguous. These results
allow us to obtain the following proposition:
Proposition 2. When the government cannot commit to the second period tax policy, andwhen skill types of taxpayers are completely separated in the first period, the consumption ofdurable goods by a high-income earner should be taxed.

5 Numerical illustration
This section provides numerical examples to examine (i) whether the social welfare in thecomplete separation casce is higher than in the complete pooling case and (ii) how the diﬀerent optimal tax rates on these two cases are determined. Table 1 summarizes the results ofthis simulation. Note that uit is the payoﬀ of the taxpayer whose marginal productivity is θi in"period t. Then, t uit is a total payoﬀ of the taxpayer whose marginal productivity is θi , and" ii,t ut is the social welfare.

–9–5.1 Parameterization
It is assumed that the utility function (3) is represented by the constant relative risk aversion(CRRA) function satisfied with separability5 as follows:!
tβt−1 {

yi
1(cit · di )1−σ − ( ti )γ }
1−σ
θ

(25)

where σ and γ are non-negative and named as preference parameters. It is assumed that
σ is 0.5 and γ is 2.0. Similarly in the second section in this paper, there exist two typesof taxpayers: a high-income earner and a low-income earner. They have the same utility
function (25) and discount factor, the latter is assumed to be 0.38. To simplify, the marginal
productivity for a low-income earner is assumed to be unity, while the marginal productivity
for a high-income earner is 1.2.

5.2 Simulation results
5.2.1 Social welfare
The value of social welfare in each case is reported in the last line. Clearly, the social welfare
in the full commitment case is the highest in these cases. Table 1 also reports that the social
welfare in the complete separation case is ranked second, while the worst welfare level is in
the complete pooling case. The diﬀerence of information structure between these cases in thesecond period plays an important role in determining the welfare level. These results imply
that the complete separation case should be chosen as the equilibrium of the non-commitmentcase under this parameterization.
5.2.2 Payoﬀ of taxpayers
To begin with, income redistribution is the basic purpose of tax policy. It is important for us
to focus on not only the social welfare level but also the payoﬀ of each taxpayer. According
to Table 1, in the second period, the payoﬀ that both taxpayers receive in the complete separation case is higher than in the complete pooling case because the asymmetric information is
resolved in the beginning of the second period. In contrast, the payoﬀ that they receive in thefirst period of the complete separation case is lower as compared with the complete poolingcase. As for a high-income earner, the taxpayer’s labor income in the complete pooling caseis lower than in the complete separation case, since a high-income earner avoids choosing
the true-telling strategy. This implies that the disutility of eﬀort that a high-income earnerreceives in the complete pooling case is lower than in the complete separation case. On the5This specification and parameterization as below refer to Bisin and Rampini (2006), Golosov et al. (2006)
and Guo and Krause (2013). Their simulations examine the two-period and infinite-horizon cases. In the former
case, social welfare in the complete separation case is higher than in the complete pooling case, as are similar
with the results of this paper.

– 10 –

ci1
yi1
di
ci2
yi2
τi
ui1
ui
"2 i
u
" t tii,t utTable 1: Result of simulation
Benchmark case
Pooling case
High
Low
High
Low
0.553202 0.0991258 0.380688
0.380688
1.307480.59824
0.978895
0.978895
0.851039
0.40235
0.598208
0.598208
1.45678 0.0884616
3.09468
0.488211
0.881102 0.664142
2.300641.28225000.325864-0.0897649
0.185137 0.0415248 0.288981 -0.00381286
1.68778 -0.063765 -0.954434-0.563329
0.826493 0.0172941 -0.0737039-0.217878
0.843794-0.291585

Separation caseHigh
Low0.605436
0.106646
1.43699
0.808085
0.933872
0.5991191.260450.810311.316370.754386
0.00376926 -0.635543
0.069875-0.147458
0.966530.824418
0.437156
0.165821
0.60298

other hand, Table 1 indicates that a low-income earner’s consumption in the complete poolingcase is higher than in the complete separation case. This implies that the utility a high-income
earner receives from an amount of consumption in the complete pooling case is higher thanin the complete separation case. Comparing the total payoﬀ of both taxpayers in the complete
pooling case with the complete separation case, we find that the complete separation case isa desirable outcome.
5.2.3 Optimal tax rate
The sixth line in Table 1 shows the optimal tax rate for durable goods consumption. In both
non-commitment cases, the simulation result means that the optimal tax rate for durable good
consumption should be progressive.

6 Conclusion
An analysis based on the optimal tax theory is necessary because the tax treatment on durable
goods is an important issue for policy makers. We focus on the commitment issue in the
public finance literature. In both the complete pooling and complete separation case, weshow the rationale for taxation on durable goods. Moreover, this result means that the tax
treatment for durable goods should be separated from the commodity tax for other goods.
This supports a real estate acquisition tax or an automobile acquisition tax, which diﬀers
in terms of the value added tax (VAT). On the other hand, the tax rate should depend on
the wage rate of taxpayers, since the incentive for choosing the labor supply diﬀers withincome status. In fact, we show that the durable goods consumption by a high-income earner– 11 –

should be taxed in both cases, while the durable goods consumption by a low-income earnershould be subsidized, at least in the complete pooling case. This provides an implication to
the diﬀerential tax treatment of consumer durables. To sum up, although the consumptionof durable goods tends to be subsidized, the results in this paper imply that subsidizing ahigh-income earner’s consumption is unnecessary.

Appendix
A. Proof of Lemma 1
Let L be the Lagrangean for the optimization problem (6) to (9). It can be described asfollows:
!
!
!
!
yi
L=
βt−1
{u(cit , d1i ) − v( ti )} + λ1 [ {yi1 − ci1 − d1i }] + λ2 [ {yi2 − ci2 }]
(26)
θ
t
i
i
i
!
!
yL
yH
+φ[ βt−1 {u(ctH , d1H ) − v( tH )} −βt−1 {u(ctL , d1L ) − v( Ht )}]θθttwhere λt ≥ 0 is the Lagrange multiplier for the resource constraints in period t and φ ≥ 0 is
the Lagrange multiplier for the incentive compatibility constraint.
The first-order conditions with respect to each variable are as follows:∂L= (1 + φ)uc (ctH , d H ) − λt = 0 ∀t
∂ctH
!
∂L
=
(1
+
φ)
βt−1 ud (ctH , d H ) − λ1 = 0
∂d H
t
ytH 1
∂L
=
−(1
+
φ)v
(
∀t
y H ) H + λt = 0
θ θ
∂ytH
∂L
= (1 − φ)uc (ctL , d L ) − λt = 0 ∀t
∂ctL
!
∂L
=
(1
−
φ)
βt−1 ud (ctL , d L ) − λ1 = 0
∂d L
t

ytL 1
ytL 1
∂L
)
+
φv
(
∀t
=
−v
(
y L
y H ) H + λt = 0
θ θL
θ θ
∂ytL
∂L ! i={y1 − ci1 − di } = 0∂λ1i∂L ! i={y2 − ci2 } = 0
∂λ2
i

yH
yL
∂L ! t−1=
β [{u(ctH , d H ) − v( tH )} − {u(ctL , d L ) − v( Ht )}] = 0∂φθθt– 12 –

(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)

Dividing (29) by (27) implies the optimal condition for labor income and non-durable goods
as follows:1=

vyHt
θ H ucHt(36)∀t

The marginal income tax rate for a low-income earner follows from dividing (32) by (30) andarranging it as follows:yL

1=vy ( θtL )θ L uc (ctL , d L )yLvy ( θtL )yLvy ( θHt )+ φ[ L−]θ uc (ctL , d L ) θ H uc (ctL , d L )

∀t

(37)

The term in the brackets on the right hand side is positive because, by the shape of the utility
function, it satisfies the Spence-Mirrlees condition, which is the indiﬀerence curve of thehigh-income earner at any point in (cit − yit ) space is flatter than that of the low-income earner.
Thus, the marginal labor income tax rate for a low-income earner is positive.
On the other hand, combining equations (27) and (28) leads to the optimal condition withrespect to the non-durable and the durable goods for the high-income earner as follows:" t−1
β ud (ctH , d H )
1= t
(38)
uc (c1H , d H )Thereafter, the marginal commodity tax rate for high-income earner should be zero. Since
taxpayers have the same preference, the marginal commodity tax rate for the low-income
earner is similar. Substituting (30) into (31) yields the optimal condition with respect tonon-durable and durable goods for the low-income earner as follows:" t−1
β ud (ctL , d L )
(39)
1= t
uc (c1L , d L )!

B. Proof of Lemma 2
Consider the optimization problem (10) to (12). The Lagrangean can be given as follows:

L2Pool =

!
yi
yH
yL
{u(ci2 , d̄) − v( 2i )} + η2 [ {yi2 − ci2 }] + ρ[u(c2H , d̄) − v( 2H ) − u(c2L , d̄) + v( H2 )] (40)
θ
θ
θ
t
i

!

where ηt ≥ 0 is the Lagrange multiplier for the resource constraints in period t, and ρ ≥ 0 is
the Lagrange multiplier for the incentive compatibility constraints. The first order conditions– 13 –

can be written as follows:∂L2Pool= (1 + ρ)uc (c2H , d̄) − η2 = 0∂c2H∂L2Pool
y2H 1
=
−(1
+
ρ)v
(
) + η2 = 0
y
θH θH
∂y2H

∂L2Pool= (1 − ρ)uc (c2L , d̄) − η2 = 0∂c2L∂L2Pooly2L 1
y2L 1= −vy ( L ) L + ρvy ( H ) H + η2 = 0
θ θ
θ θ
∂y2L

(41)
(42)
(43)
(44)

It is easy to show that the optimal conditions between non-durable goods consumption andlabor income are (36) and (37). Then, the marginal income tax rate for the high-income
earner is zero, while the marginal income tax rate for the low-income earner is positive.
Next, we consider the optimization problem (13) to (14). Let L1Pool be the Lagrangean forthis problem.
!
!
ȳ1
L1Pool ={u(c̄1 , d̄) − v( i )} + βV2Pool + η1 [ {ȳ1 − c̄1 − d̄}]
(45)
θ
i
i
The first order conditions with respect to the non-durable goods, the durable goods and labor
income are as follows:∂L1Pool= 2uc (c̄1 , d̄) − 2η1 = 0∂c¯1∂L1Pool∂V2Pool= 2ud (c̄1 , d̄) + β− 2η1 = 0∂d̄∂d̄
! ȳ1 1
∂L1Pool=−vy ( i ) i + 2η1 = 0
∂y¯1
θ θ
i

(46)
(47)
(48)

Dividing (48) by (46) yields the following:!
ivy ( ȳθ1i )θi uc (c̄1 , d̄)

=2

(49)

Since the utility function satisfies with the Spence-Mirrlees condition, we can obtain thefollowing:1−1−vy ( θȳH1 )θ H uc (c̄1 , d̄)vy ( θȳL1 )θ L uc (c̄1 , d̄1 )– 14 –

>0

(50)

<0

(51)

These expressions mean that the marginal income tax rate for a high-income earner is positive, and the marginal income tax rate for a low-income earner is negative. !
C. Proof of Proposition 1
Consider the optimization problem (10) to (12) again. By the envelop theorem, we obtainthe following:∂V2Pool∂d̄= (1 + ρ)ud (c2H , d̄) + (1 − ρ)ud (c2L , d̄)(52)

Substituting into (47) yields the following:1
ud (c̄1 , d̄) + β[(1 + ρ)ud (c2H , d̄) + (1 − ρ)ud (c2L , d̄)] = uc (c̄1 , d̄)2(53)

Rearranging equation (53) and subtracting it in equation (5) give equations (16) and (17)respectively.
Subtracting equation (41) from equation (43), we find that c2H > c2L . Since it is assumed∂2 uthat ∂c∂d
is positive, [uc (c2H , d̄) − uc (c2L , d̄)] is also positive. This implies that equation (17) is
negative. Because η2 is positive, it follows from (43) that (1 − ρ) is positive. Then, equation(16) is positive. Therefore, Proposition 1 can be established. !
D. Proof of Lemma 3
We consider the optimization problem (18) to (19). Let LS2 ep be the Lagrangean for thisproblem. It can be formulated as follows:
LS2 ep =

!
!
yi2
i
i
{u(c2 , d ) − v( i )} + ζ2 [ {yi2 − ci2 }]
θ
i
i

(54)

where ζt is the Lagrange multiplier for the resource constraints in the period t. The first orderconditions with respect to endogenous variables are∂LS2 ep= uc (c2H , d H ) − ζ2 = 0∂c2H∂LS2 ep
y2H 1
=
−v
(
y H ) H + ζ2 = 0θ θ∂y2H∂LS2 ep= uc (c2L , d L ) − ζ2 = 0L∂c2∂LS2 epy2L 1= −vy ( L ) L + ζ2 = 0
θ θ
∂y2L
S ep
!
∂L2
=
{yi2 − ci2 } = 0∂ζ2i– 15 –

(55)
(56)
(57)
(58)
(59)

Substituting (55) into (56), and (57) by (58), the marginal labor income tax rate for both
taxpayers are determined as follows:yivy ( θ2i )θi uc (ci2 , di )= 1 ∀i(60)

This expression implies that the marginal labor income tax rate should be zero.
Next, we consider the optimization problem in the first period. The Lagrangean for thisproblem can be formulated as follows:!
!
yi
LS1 ep ={u(ci1 , di ) − v( 1i )} + βV2S ep + ζ1 [ {yi1 − ci1 − di }]
(61)
θ
i
i
yH
yH (d)+ψ[u(c1H , d H ) − v( 1H ) + β{u(c2H (d), d H ) − v( 2 H )}
θ
θ
L
L
y
y (d)
−u(c1L , d L ) + v( H1 ) − β{u(c2L (d), d L ) − v( 2 H )}]θθwhere ψ is the Lagrange multiplier for the incentive compatible constraints. Combining thiswith the first-order conditions with respect to this problem, it is easy to demonstrate that thepattern of the marginal income tax rate is also the similar in the second period in the complete
pooling case. !
E. Proof of Proposition 2
We consider the optimization problem (20) to (22) again. The first-order condition withrespect to durable goods for both the high-income and low-income earner can be derived andrearranged as follows:∂V2S ep∂LS1 epHH= (1 + ψ)ud (c1 , d ) + β{+ ψud (c2H , d H )} + βψΓH − ζ 1 = 0
H
H
∂d
∂d
∂LS1 ep∂V2S epLL= (1 − ψ)ud (c1 , d ) + β{− ψud (c2L , d L )} + βψΓL − ζ1 = 0
∂d L
∂d L

(62)
(63)

where
dc2H
y2H 1 dy2H
dc2L
y2L 1 dy2L
L
L
−
v
(
)
}
−
{u
(c
,
d
)
−
v
(
)
}]
y
c
y
2
dd H
θ H θ H dd H
dd H
θ H θ H dd H
y2H 1 dy2H
dc2L
y2L 1 dy2L
dc2H
L
H
H
L
L
Γ ≡ [{uc (c2 , d ) L − vy ( H ) H L } − {uc (c2 , d ) L − vy ( H ) H L }]dd
θ θ dddd
θ θ ddWe turn to the optimization problem (18) to (19). Applying the Envelop theorem to thisproblem yields the following:ΓH ≡ [{uc (c2H , d H )∂V2S ep= ud (c2H , d H )H∂d∂V2S ep= ud (c2L , d L )L∂d– 16 –(64)(65)

Substituting (64) and (65) into (62) and (63), we obtain equations (23) and (24), respectively.
Since ζ1 is positive, (1 − ψ) also is positive. Then, the optimal tax rate for durable goods
depends on the sign of ΓH and ΓL .
We show that ΓH is negative. Let A be the Hessian with respect to equations (55) to (59).
Then, we can rewrite these in a matrix formation as follows:⎛ H ⎞ ⎛⎞⎜⎜⎜ dc2 ⎟⎟⎟ ⎜⎜⎜ uc,d (c2H , d H ) ⎟⎟⎟⎜⎜⎜ dcL ⎟⎟⎟ ⎜⎜⎜0 ⎟⎟⎟⎟⎟
⎜⎜⎜
⎜
2 ⎟
⎟
⎜
⎟ ⎜
⎟
0 ⎟⎟⎟⎟ dd H
A ⎜⎜⎜⎜ dy2H ⎟⎟⎟⎟ = ⎜⎜⎜⎜
(66)
⎜⎜⎜
⎟
⎜
⎟⎟⎟
L ⎟
⎜
⎟
⎜
0 ⎟⎟
⎜⎝⎜ dy2 ⎟⎠⎟ ⎜⎝⎜
⎠
dζ2
0

where

⎞
⎛
0
0
0 −1 ⎟⎟⎟⎜⎜⎜ uc,c (c2H , d H )⎜⎜⎜⎜0 uc,c (c2L , d L )00 −1 ⎟⎟⎟⎟⎟
⎜⎜⎜
⎟⎟
H
y
0 1 ⎟⎟⎟⎟
00 −vy,y ( θ2H )( θ1H )2A ≡ ⎜⎜⎜⎜⎜
⎟⎟⎟
⎜⎜⎜
yL
⎜⎜⎜
0
0
0 −vy,y ( θ2L )( θ1L )2
1 ⎟⎟⎟⎟
⎠
⎝
−1
−1
1
1 0

Computing the determinant of A implies the following:y2H 1 2HH
L
L
|A| = uc,c (c2 , d )uc,c (c2 , d )vy,y ( H )( H )
θ θ

(67)

y2L
y2H
1
H
H
L
L
−{uc,c (c2 , d ) + uc,c (c2 , d )}vy,y ( H )vy,y ( L )( H L )2 > 0θθ θ θThis implies that the inverse matrix of A exists. The application of the Cramer’s rule yields
the following:
dc2H
uc,d (c2H , d H )y2H 1 2y2L 1 2
y2H
y2L
1 2
L
L
=
[u
(c
,
d
){v
(
)(
)
+
v
(
)(
)
}
−
{v
(
)v
(
c,c 2
y,y H
y,y L
y,y H y,y L )( H L ) }] < 0(68)
H
H
L
dd
|A|
θ θ
θ θ
θ
θ θ θ
H
H
H
dy2
uc,d (c2 , d )y2L 1 2
L
L
=
u
(c
,
d
)v
(
c,c 2
y,y L )( L ) < 0(69)
dd H
|A|
θ θ
H
H
H
H
H
H
dy2
uc,d (c2 , d )y2y2y2L 1 2dc2
LL−={uc,c (c2 , d )vy,y ( H ) − vy,y ( H )vy,y ( L )( L ) } < 0(70)dd H dd H|A|(θ H )2
θ
θ
θ θ
dc2L
uc,d (c2H , d H )
y2H
y2L
1
=
vy,y ( H )vy,y ( L )( H L )2 > 0(71)
H
dd
|A|
θ
θ θ θ
L
H
H
dy2
uc,d (c2 , d )y2H 1 2
L
L
=
u
(c
,
d
)v
(
)( ) < 0(72)
c,c
y,y
2
dd H
|A|
θH θH

These mean that ΓH is negative. Hence, it can be shown that the marginal tax rate of durable
goods for the high-income earner is positive.
– 17 –

As with (66), we diﬀerentiate these equations with respect to durable goods by a lowincome earner and rewrite these in a matrix formation as follows:⎛ H ⎞ ⎛⎞
0 ⎟⎟⎜⎜⎜ dc2 ⎟⎟⎟ ⎜⎜⎜⎜⎜⎜⎜ dcL ⎟⎟⎟⎟ ⎜⎜⎜⎜ uc,d (cL , d L ) ⎟⎟⎟⎟⎟
2 ⎟
2
⎜⎜
⎟⎟
⎟ ⎜⎜
0 ⎟⎟⎟⎟ dd L
A ⎜⎜⎜⎜ dy2H ⎟⎟⎟⎟ = ⎜⎜⎜⎜
(73)
⎜⎜⎜
⎜⎜⎜
⎟⎟⎟
L ⎟
⎟
⎟
dy
0
⎜⎜⎝
⎟⎟⎠
2 ⎟
⎟⎠ ⎜⎜⎝dζ20

Applying Cramer’s rule gives the following:

dc2H uc,d (c2L , d L )
y2H
y2L
=
vy,y ( H )vy,y ( L ) > 0
dd L
|A|
θ
θ
dy2H uc,d (c2L , d L )
y2L
H
H
=
uc,c (c2 , d )vy,y ( L ) < 0
dhL
|A|
θ
L
L
L
H
L
H
dc2
uc,d (c2 , d )
y2
y2
y2
y2L
H
H
H
H
=
[u
(c
,
d
)v
(
)
+
u
(c
,
d
)v
(
)
−
v
(
)v
(
)] < 0
c,c
y,y
c,c
y,y
y,y
y,y
2
2
dd L
|A|
θH
θL
θH
θL
dy2L uc,d (c2L , d L )
y2H
H
H
=
v
(
y,y H )uc,c (c2 , d ) < 0
dd L
|A|
θ

(74)
(75)
(76)
(77)

These imply that ΓL cannot be signed. Therefore, the marginal tax rate of durable goods
for the low-income earner is ambiguous. !

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