Optimal age-dependent income taxation in a dynamic
extensive model:
The case for negative participation tax to the young people∗
Takao Kataoka† and Yoshihiro Takamatsu‡
July 25, 2017

Abstract
We consider an optimal age-dependent income taxation in a dynamic model where laborleisure choice is extensive margin in each period, each household faces idiosyncratic shocks oflabor productivity and pecuniary cost to work, and there is no insurance market against theidiosyncratic shocks. We show the well known property of optimal participation tax rate in thestatic model continue to hold in our dynamic economy, and participation tax rates for someincome groups with low consumptions are likely negative. In dynamic models, the optimal
participation tax rate depends on age as well as labor income. Our numerical simulations
suggest that negative participation tax should be restricted to young households.
JEL classification: H21, H23, J22
Keywords: Extensive margin; Age-dependent taxation; Labor income tax∗ We are especially grateful to Yukihiro Nishimura for his insightful comments on an earlier version of this paper.
This work was supported by JSPS KAKENHI Grant Numbers 15K17073.
† Faculty of Commerce, Waseda University,
E-mail: kataoka@waseda.jp‡ Department of Economics, Faculty of Humanities and Social Sciences, Shizuoka University,
E-mail: takamatsu.yoshihiro@shizuoka.ac.jp1

1

Introduction

The optimal income taxation literature has developed models to analyze the design of income
tax/transfer programs. Following the seminal paper of Mirrlees (1971), many studies focus exclusively on the intensive labor margin, in other words, households choose hours of work or intensityof work. In that framework, negative marginal tax rates can never be optimal, ruling out in-work
credits.
Much recent literature has emphasized the role of the extensive margin. In the extensive labor
margin, households choose whether or not to participate in labor force with fixed hours of work. In
this setting, it can be optimal to adopt wage subsidies or in-work credits with negative participation
tax for low income households (see e.g., Diamond (1980), Saez (2002), Choné and Laroque (2005,2011) and Jacquet et al. (2013)).
Although these studies have influential findings both theoretical and policy implications, a largepart of the literature is static and does not directly address the intertemporal side of the problem.
There are few studies treat dynamic model with extensive labor margin (see Laroque (2011) and
Choné and Laroque (2017)). They investigate time invariant non-linear income tax/transfer in astationary life-cycle model with perfect insurance market, and shows that a worker faces a negative
participation tax rate if its social weight with respect to lifetime permanent incomes is larger than
the average.1 Under the stationary assumption, the distribution of the productivities and cost to
work are constant overtime and the correlation between current and permanent income becomesa central problem. Because they assume perfect insurance market, personal event, such as illness
or accident, are fully insured by private financial market. The main objective of the government
in their model is the redistribution among households with diﬀerent expected permanent income
at birth, which solely determine their social weight. A perfect correlation between current andpermanent income would make the static and the dynamic model equivalent. The policy instrumentof Laroque (2011) and Choné and Laroque (2017) is age-independent. Thus, in the extensive labor
margin, the design of an optimal tax scheme depending on age or whole history remains an unsettled
question.
Recent dynamic Mirrlees literature (see Kocherlakota (2010)) examines intertemporal shocks
about labor productivities and preferences in the intensive labor margin. With the extensive
margin, Diamond and Mirrlees (1978) and Golosov and Tsyvinski (2006) treat the permanent
disability shocks. It means that once disabled, labor status is no longer a matter of choice. In
our dynamic extensive labor margin with shocks, a household can decide to participate or not in
the labor market in each period. It is important to examine these shocks in the extensive labor
margin since a household who decides not to work at some period for some reasons (e.g., badhealth condition, involuntary unemployment, or lack of ability) may change its labor status andgo back to work in another period. In adopting these settings, we can study a redistributive policy
to avoid ‘poverty trap’.
The results from dynamic Mirrlees literature show that the optimal income tax policy withshocks have to be history-dependent. The history-dependent nature is, however, complicatedand rare in the real world. The implements of the optimal allocation could be combined labor
income tax with other instruments. For examples, Grochulski and Kocherlakota (2010) show that,the optimal allocation can be implemented through a combination of three policy instrumentsthat they term social security systems, namely a tax on current labor income until retirement,
1 Laroque (2011) also examines the usefulness of a complemental positive linear tax on wealth.

2

a retrospective capital income tax only at retirement that is dependent on labor income history,
and a history-contingent payment after retirement. Michau (2014) also show that the optimal
allocation can be implemented by a history-dependent social security system in a life-cycle modelwith both intensive and extensive (retirement) margin and without skill shocks. On the contrary,Weinzierl (2011) examines partial tax reforms to age-dependent labor income taxation, and showsthat there are small welfare losses from implementing not full optimal history-dependent income
taxation, but age-dependent income taxation.2 We adopt latter approach and focus solely on therole of age-dependent income taxation.
Thus, we investigate the age-dependent and age-independent optimal income tax schedule ina dynamic model with extensive labor margin in each period and idiosyncratic shocks of laborproductivity and cost to work without perfect insurance market. The main purpose of the government here is the redistribution among households with diﬀerent history of idiosyncratic shocks.
We derived the following results. First, in the case of age-dependent labor income tax, we derive
the optimal age-specific participation tax rates which is similar to that in static model. On thecontrary to static model, negative participation tax rate for households with lower consumption
apply in each age group. Second, in the case of age-independent labor income tax, we confirm theoptimal participation tax rates are similar to the static model and the stationary life-cycle model.
Finally, we numerically solve the optimal tax policies in some example two periods economies andsuggest that the negative participation tax should be applied to low-productive young households
only.
The remainder of the paper is organized as follows. In Section 2, we present a dynamic modelwith extensive labor margin and idiosyncratic shocks. In Section 3, we investigate the optimal
income tax problem and derive the optimal participation tax rates both age-dependent case andage-independent case. In Section 4, we consider two periods example economies with various
parameter and numerically solves the optimal income tax policies. In Section 5, we present ourconclusions.

2

The model

The economy has a discrete time index t ∈ {0, 1, 2, · · · , T } and is populated by a governmentand one unit of continuous T + 1 periods-lived households. Each household faces two dimensional
idiosyncratic shock θt = (wt , δt ) at period t, where wt ∈ R+ and δt ∈ R+ respectively represent
labor productivity and pecuniary opportunity cost to work measured in units of consumption goodat that period. There are I ∈ N possible levels of labor productivity. The domain of the laborproductivity is denoted by Ω = {w1 , w2 , · · · , wI } ∈ RI++ . The idiosyncratic shock evolves overtime and let ht denote a history of the shocks up to period t, in other words, ht = (θ0 , θ1 , · · · , θt ) ∈(Ω × R+ )t+1 . Let Fth : (Ω × R+ )t+1 → R+ denote the cumulative probability distribution function(c.d.f) of ht . Ftθ denotes the marginal (conditional or unconditional) c.d.f. of θt . The c.d.f.s are
public information.
There are two commodities at every period: a divisible consumption good and indivisible
labors with various productivities. A household at period t chooses labor supply lt ∈ {0, 1}, after
observing the idiosyncratic shock at the period, (wt , δt ). If she works (lt = 1), she obtains wt unitsof (before tax) labor income and suﬀers disutility amounts to δt unit of consumption. If she does
2 Blomquist and Micheletto (2008) also show that with the intensive margin, age-dependent income taxation is
Pareto improving on an age-independent taxation.

3

not work, on the other hands, her labor income is zero and she suﬀers no disutility. Householdwith productivity wi cannot mimic other productivity wj (i ̸= j). The opportunity cost to work
is private information, while the labor productivity is known to public if and only if she works at
the period. The households and government can access outside financial market with constant realinterest rate r ∈ R. There is no market to insure against the idiosyncratic risk.
The government imposes tax on the labor income. Let yt (wi ) denote the after-tax income, thatis, wi minus labor income tax. yt (0) represents the subsistence income for not-working householdat period t. Thus, yt : {0} ∪ Ω → R represents a disposable income schedule at period t. Given
the disposable income schedule y = (y0 , y1 , · · · , yT ), the households choose a plan of labor supply,l = (l0 , l1 , · · · , lT ), and asset holding, a = (a0 , a1 , · · · , aT ), where lt : (Ω × R+ )t+1 → {0, 1} andat : (Ω × R+ )t+1 → R respectively represent the labor status and the asset holding at the end of
the period, conditional on ht . All the households have (1 + r)a−1 ∈ R unit of real asset at the
beginning of the initial period.
The consumption net of the pecuniary cost to work at period t conditional on ht (hereafter
called net consumption), denoted by ct (ht ), is,ct (ht ) = (1 + r)at−1 (ht−1 ) + yt (wt lt (ht )) − δt lt (ht ) − at (ht ),(1)

and a plan of net consumption is c = (c0 , c1 , . . . , cT ). The expected utility is,∑∫β

U (ct (ht )) dFth (ht ),

t

(2)

ht

t

where β ∈ R+ is time discount factor and U is a twice continuously diﬀerentiable utility function
satisfying U ′ > 0 and U ′′ < 0.3 Given a disposable income schedule y, the household chooses a, cand l to maximize (2) subject to (1) and,aT (hT ) ≥ 0,

for all hT .

(3)Given a choice of asset plan a, net consumption at period t is larger at lt = 1 than that at
lt = 0, if the opportunity cost to work, δt , is smaller than yt (wt ) − yt (0). Thus the optimal labor
supply plan lt is 1 (resp. 0) if δt is smaller (resp. larger) than yt (wt ) − yt (0). Notice that the
optimal choice of labor supply solely depends on θt and yt .
The population of households with (before tax) labor income wi at period t, denoted by nit andthe population of not-working households at period t, denoted by n0t , are,
∫
nit

=

n0t

=

θt ∈Θit1−

I∑

dFtθ (θt ),i = 1, 2, · · · , I,

nit ,

i=1

where Θit denotes the region of θt defined by,Θit ≡ {(wt , δt ) | wt = wi and 0 < δt < yt (wi ) − yt (0)}.
3 It is natural to restrict the consumption to be non-negative. Consumption net of pecuniary opportunity cost,
however, may be negative. We assume the time separable utility function and the common initial asset for notational
simplicity. It is not diﬃcult to generalize these assumptions.

4The working population of households with wi at period t, nit for i ̸= 0, is a non-decreasing
function of yt (wi ) − yt (0) and, the following symmetric relations hold for all t.
∂nit∂yt (wi )∂n0t∂yt (0)

= −

∂nit
∂n0t
=−
≥ 0,
∂yt (0)∂yt (wi )i = 1, 2, .., I,

I
I
∑
∑
∂nit
∂nit
= −
=
≥ 0.
∂yt (0)∂yt (wi )
i=1
i=1

(4)
(5)

The first order condition of the households’ maximization problem with respect to the asset
holding at yields,∫′

U (ct (ht )) = (1 + r)βθ
U ′ (ct+1 (ht , θt+1 )) dFt+1(θt+1 |ht ).

(6)

θt+1

Given y, (1), (3) with equality, and the Euler conditions (6) uniquely determine the optimal asset
plan.4 Let EU denote the maximized expected utility.
The government must finance an exogenous (presented valued) expenditure G. The government’s (presented valued) budget constraint is given by,
T
∑

1
(1
+
r)t
t=0

(
−n0t yt (0) +

I
∑

(
i

)
nt wi − yt (wi )

)
≥ G.

(7)

i=1

The optimal taxation problem of the government is then to find the disposable income schedule ythat maximizes the household’s maximized expected utility, EU , subject to the budget constraint(7).
We consider two kinds of optimal taxation problems, namely optimal age-dependent taxation
problem and optimal age-independent taxation problem. The former corresponds to the disposable
income schedule which exactly maximizes EU subject to (7). Actual income tax system does notnecessary depend on age in most countries, however. Thus we also investigate the latter problem
which solves the same maximization problem with the additional constraint that disposable income
schedule to be age-independent (y0 = y1 = · · · = yT ).
Notice that the optimal age-dependent taxation is not unique due to the Ricardian equivalence. Suppose, for example, y ′ = (y0′ , y1′ , · · · , yT′ ) is an optimal age-dependent disposable income∑
schedule. For any sequence (α0 , α1 , · · · , αT ) ∈ RT +1 satisfying t (1 + r)−t αt = 0, define anotherdisposable income schedule ỹ by,ỹt (wi ) = yt′ (wi ) + αt ,for all i, t.

Then ỹ is optimal as well since it does not aﬀect the budget constrains of the households or thegovernment. Thus, we normalize the optimal age-dependent tax to be y0 (0) = y1 (0) = · · · = yT (0).
This normalization does not aﬀect the conclusions.

4 The optimal labor supply plan, equation (1) at the last period, and equation (3) with equality uniquely solve cT
as a function of aT −1 . The obtained cT , equation (6), equation (1) and the optimal labor supply rule at t = T − 1
uniquely solve at−1 as a function of at−2 . Repeating this procedure yields the optimal asset plan.

5

3

Optimal participation tax

It is well known that the optimal taxation schedule in a static model with extensive margin likely
exhibits the property that participation tax rates for some low income households are negative(see Saez (2002)). Our first task here is to examine if this property continue to hold in the bothof dynamic settings.

3.1

The case of age-dependent taxation

First, we examine the case of age-dependent taxation. The Lagrangian of the government problem
in the case of age-dependent taxation is,
LD= EU + λ

D

[ T
∑

1
(1 + r)tt=0(−n0t yt (0) +

I
∑

nit

(

)
w − yt (w )
i

)

i

]
−G ,

i=1

where λD is the Lagrange multiplier associated with the government budget constraint in the caseof age-dependent taxation. The first order conditions of the Lagrangian with respect to yt (wi ) and∑yt (0) are, using n0t = 1 − i nit ,
[
)]
(
∂LD
∂EU
1
∂nit
D
i
i
i
=
+λ
(w − yt (w ) + yt (0))= 0, for i ̸= 0,−nt +∂yt (wi )∂yt (wi )(1 + r)t∂yt (wi )
[
(
))]
I (
∑
)
∂nit ( i
∂EU
1
∂LD
D
0
i
=
+λ
−nt −
w − yt (w ) + yt (0)= 0.
∂yt (0)∂yt (0)(1 + r)t∂yt (wi )i=1
The expected utility is defined on the net consumption plan c which depends on a, l and y.
Suppose the disposable income of working households with productivity wi at period t increases
by marginal 1 unit. By the chain rule, the eﬀect on the expected utility is the sum of (i) the direct
eﬀect, that is, the increase of net consumption at period t for the case of δt ≤ yt (wi ) − yt (0) givena and l, (ii) the eﬀect via the change of lt for the case of δt close to yt (wi ) − yt (0), and (iii) theeﬀect via the change of asset plan a. The second eﬀect, however, vanishes because the marginal
households switching their labor status are indiﬀerent between work and not-work. The last eﬀect
also vanishes because of the envelope theorem. Thus, the partial derivatives of EU with respect
to the disposable incomes are,∂EU∂yt (wi )∂EU∂yt (0)

∫
=

ht−1

∫
=

∫

βt
β

t

θt ∈ΘithU ′ (ct (ht )) dFtθ (θt | ht−1 ) dFt−1(ht−1 ),

I ∫∑ht−1 i=1

θt ∈Θ̄itfor i ̸= 0,h
U ′ (ct (ht )) dFtθ (θt | ht−1 ) dFt−1(ht−1 ),

where Θ̄it denotes the region of θt defined by,Θ̄it ≡ {(wt , δt ) | wt = wi and δt ≥ yt (wi ) − yt (0)}.

6We can rewrite the first order conditions in the age-dependent taxation as follows.
()
nit 1 − gti =)∂nit ( iw − yt (wi ) + yt (0) ,i∂yt (w )for i ̸= 0,

(8)

I
∑
(
)
n0t 1 − gt0 = −)∂nit ( iw − yt (wi ) + yt (0) ,
i)
∂y
(w
t
i=1

(9)

where gti is the average marginal social weight of period t consumption for the households obtaining
disposable income yt (wi ) at the period, expressed in terms of public funds, that is,(1 + r)t β tgti =nit λD
gt0 =(1 + r)t β tn0t λD

∫
∫

∫

ht−1

θt ∈ΘithU ′ (ct (ht )) dFtθ (θt | ht−1 ) dFt−1(ht−1 ),

I ∫∑ht−1 i=1

for i ̸= 0,h
U ′ (ct (ht )) dFtθ (θt | ht−1 ) dFt−1(ht−1 ).

θt ∈Θ̄it

Adding (8) for all i ̸= 0, and (9), we have,1=I∑

nit gti .

(10)

i=0

The intuition behind (10) is clear. Suppose the tax at period t decreases by (1 + r)t unit for all thelabor income. This does not aﬀect the households’ labor-leisure choices and requires 1 unit of thegovernment’s budget. This tax cut increases all the households’ expost utility by their marginalutility of period t consumption. (10) shows the cost must be balanced with the benefit at the
optimal taxation schedule.
Households with productivity wi at period t forgo the subsistence income yt (0) and pay income
tax wi − yt (wi ). Thus, wi − yt (wi ) + yt (0) is participation tax for households with productivity wi
at period t. We also define the participation tax rates faced by the households with productivity
wi at age t by,
τt (wi ) =wi − yt (wi ) + yt (0),wi

for i ̸= 0.

(11)

Using (11), we can rewrite (8) as an age-specific inverse elasticity rule which is the same shape instatic model (see in Saez (2002)),
τt (wi )1 − gti=,i
1 − τt (w )ηtifor i = 1, 2, · · · , I, and t = 0, 1, . . . , T,

(12)

where ηti is the ‘age-specific participation elasticity’ defined as,
ηti =

dnit
yt (wi ) − yt (0)> 0,d(yt (wi ) − yt (w0 ))nit

for i = 1, 2, · · · , I, and t = 0, 1, . . . , T.

Since the weighted average of gti is unity within each period or age group, the right hand side of(12) is likely negative for some working group with lower consumption at that period or age. The
participation tax rate for such an income group must be negative.
The participation tax rules both age-dependent case and static case depend on average marginal
social weight and participation elasticity, so that it seems to have same interpretation. There are,7

however, a diﬀerence with age-dependent case and static one. In the static model, the averagemarginal social weight depend on consumptions at only that period. In the age-dependent case,the history of idiosyncratic shocks have an impact on consumption pattern and saving opportunity.
Thus, the average marginal social weight depends on consumptions that take account of the idiosyncratic shock with respect to productivity and cost to work until that period as well as savings
just prior to that period. In this sense, the age-specific tax rules have more information of household history with productivity. As in the following simulation, age-dependent taxation change
their schedule dramatically because elder people’s marginal social weight diﬀer from young’s one.

3.2

The case of age-independent taxation

Next, we consider the case of age-independent taxation. It appears that this model setting issimilar to that of Laroque (2011) and Choné and Laroque (2017). The main diﬀerence is, however,that we allow for idiosyncratic shocks about productivities and preferences which is not insured in
the outside financial market. In this sense, we incorporate idiosyncratic shocks into the model of
Laroque (2011) and Choné and Laroque (2017) without complementary taxes on wealth.
In the case of age-independent taxation, the government’s policy instrument is restricted by
y0 = y1 = · · · = yT . The Lagrangian of the government’s problem is,
L = EU + λ
I

I

[ T
∑

1
(1
+
r)t
t=0

(
−n0t y(0) +

I
∑

nit

(

)
w − y(w )
i

)

i

]
−G ,

i=1

where λI is the Lagrange multiplier associated with government budget constraint in the case ofage-independent taxation. The first order conditions of the Lagrangian with respect to y(wi ) and∑y(0) are, using n0t = 1 − i nit ,
)]
(
T [
∑
∂LI
1
∂EU
∂nit
I
i
i
i
=
+
λ
(w
−
y(w
)
+
y(0))
= 0, for i ̸= 0,−n+t∂y(wi )∂y(wi )
(1 + r)t∂y(wi )
t=0
[
(
))]
T
I (
∑
∑
)
∂EU
1
∂LI
∂nit ( i
I
0
i
=
+λ
nt +
w − y(w ) + y(0)
= 0.
∂y(0)
∂y(0)
(1 + r)t∂y(wi )
t=0
i=1
Similar to the above discussions, the partial derivatives of EU with respect to the disposable
incomes are,
∫
∫
T
∑
∂EU
t
h
=
β
U ′ (ct (ht )) dFtθ (θt | ht−1 ) dFt−1(ht−1 ),
for i ̸= 0,∂y(wi )iht−1 θt ∈Θ
t=0
∫
T
I ∫
∑
∑
∂EU
h
=
βt
U ′ (ct (ht )) dFtθ (θt | ht−1 ) dFt−1(ht−1 ).
∂y(0)
i
h
θ
∈
Θ̄
t−1 i=1t
t=0
The first order conditions in the age-independent taxation problem with respect to y(wi ) andy(0) can be rewritten as follows.
()
n̄i 1 − ḡ i =)∂ n̄i ( iw − y(wi ) + y(0) ,i∂y(w )

I∑()
n̄0 1 − ḡ 0 = −for i ̸= 0,)∂ n̄i ( iw − y(wi ) + y(0) ,i∂y(w )
i=1

8

(13)
(14)

where n̄i and ḡ i are defined as,n̄i ≡ḡ i ≡ḡ 0 ≡

T∑nit,(1 + r)tt=0

for all i = 0, 1, · · · , I,∫T∑
(1 + r)t β t
t=0nit λI

t=0n0t λI

∫

ht−1

∫
T
∑
(1 + r)t β tθt ∈ΘihU ′ (ct (ht )) dFtθ (θt | ht−1 ) dFt−1(ht−1 ),

I ∫∑ht−1 i=1

θt ∈Θ̄ifor i ̸= 0,h
U ′ (ct (ht )) dFtθ (θt | ht−1 ) dFt−1(ht−1 ),

for i = 0.

Adding (13) for all i and (14), the right hand side vanishes,I∑n̄i (ḡ i − 1) = 0.

(15)

i=0

This means a weighted average of ḡ i is unity.
We also define the age-independent participation tax rates faced by the households with productivity wi by,τ̄ (wi ) =wi − y(wi ) + y(0),wifor i = 1, 2, · · · , I.

(16)

Using (16), we can rewrite (13) as an age-independent inverse elasticity rule which is the same
form in age-dependent taxation and static model,τ̄ (wi )1 − ḡ i=,
1 − τ̄ (wi )η̄ i

for i = 1, 2, · · · , I,

(17)

where η̄ i is the ‘life-cycle participation elasticity’ defined as,η̄ i ≡1dn̄iy(wi ) − y(0) ∑∂nity(wi ) − y(0)
=
i
i
i
t
n̄
d(y(w ) − y(0))
n̄(1 + r) ∂y(wi )tfor i ̸= 0.

Since the weighted average of ḡ i is unity, the right hand side of (17) is likely negative for someworking group with lower consumption. The age-independent participation tax rate for such an
income group must be also negative.
Laroque (2011) examines the optimal income taxation in a stationary life-cycle model withan extensive margin and shows that optimal participation tax rate depends on social weight withpermanent income rather than current income. In their life-cycle model, they focus exclusively on
stationary economy with perfect insurance market, so that at given time there are same characteristics cohort and omit the age. The key factor is the correlation between current and permanent
income, which is fixed and determined at birth. In this sense, they need not to consider households
history about productivities and costs to work. In our model, social weight depends on realized
history of uninsurable idiosyncratic shocks. Thus, the implementation of optimal allocation require more information. Nevertheless, the age-independent participation tax rule with uninsurable
idiosyncratic shocks, equation (17), can be described as the same form of Laroque (2011).

9

4

Numerical simulations in an example economy

The previous section shows that participation tax rates for some working households with small
consumptions are likely negative. In a static setting, less productive households tend to obtainlower consumption and negative participation tax. In our dynamic setting, the link between income
and consumption is not periodwise. This section numerically solves the optimal tax policies in someexample two periods economies and suggests that the negative participation tax should be applied
to low-productive young households only.

4.1

Parameter setting(1)

Utility function and interest rate

All the households are two periods lived (T = 1). Hereafter, we refer to the households at period
0 as young and the households at period 1 as old. In this economy, one period corresponds to 25
years, the both of annual time discount rate and real interest rate are 2%, and the initial asset
holding is zero, that is, β = (1.02)−25 ≈ 0.61, 1 + r = 1.0225 ≈ 1.64, and (1 − r)a−1 = 0.
We assume the utility function U is quadratic in this section, and we normalize U (0) = 0 andU ′ (0) = 1, that is,{
U (c) =1
−αc2 + c for c ≤ 2α
14αotherwise,

where α ∈ R+ is the parameter determines the curvature of U . Marginal utility of consumption
becomes zero when the consumption exceeds the satiation level 1/(2α). We normalize the averagelabor productivity of young households to be unity, as mentioned details below. We consider threepossible values for α, α = 0.1, 0.05, or 0.025. Corresponding satiation levels are 5, 10, or 20 times
large of the average labor productivity for young household.
(2)

Labor productivity

There are three possible labor productivity level, (I = 3 and w1 < w2 < w3 ). We assume themiddle productivity corresponds to the average, w2 = 1 and consider three cases for the domainof labor productivity, namely Ω = (w1 , w2 , w3 ) = (0.5, 1, 1.5), (0.5, 1, 2), and (0.5, 1, 3). Thus, thehighest productivity w3 determines the skewness of the labor productivity distribution.
At period 0, q0i ∈ [0, 1] unit of the young households obtain productivity wi . Since the total
population and the average productivity are unity, we must have,3∑

q0i = 1,i=1
3∑q0i wi = 1.

i=1Let σ denotes the standard deviation of the productivity, in other words,
σ2 =

3
∑

q0i (wi − w̄)2 = q01 (w1 − 1)2 + q03 (w3 − 1).

i=1

The parameter σ determines the dispersion of the labor productivity distribution. Given (w1 , w2 , w3 )10and σ, the above three conditions determine the initial population distribution. We consider threecases for the standard deviation, namely σ = 1/4, 1/3, or 1/2.
We assume the stochastic process of the labor productivity follows a Markov process independent of the path of the cost to work δt . An old household with productivity wi at period 0 obtain∑productivity wj at period 1 with transition probability π i,j ∈ [0, 1] ( j π i,j = 1 for all i). We
consider three cases for the transition probabilities, namely,
{
π

i,j

=

1/3

if i = j

1/3

if i ̸= j

{
,

0.6

if i = j0.2

if i ̸= j

{
, or

0.9

if i = j0.05

if i ̸= j.

Let q1j denotes the population of old households with productivity wj at period 1, defined by∑
q1j = i q0i π i,j .
(3)

The cost to work

At period 0, the cost to work of a young household with productivity wi is uniformly distributediover the interval [δ i0 , δ 0 ] ⊂ R+ . Thus, working populations and the participation elasticities of theyoung households are,ni0 = q0i
η0i =y0 (wi ) − y0 (0) − δ i0i
δ 0 − δ i0,

if y0 (wi ) − y0 (0) ∈ [δ i0 , δ̄0i ],y0 (wi ) − y0 (0).
y0 (wi ) − y0 (0) − δ i0

If we set δ i0 = 0, the participation elasticity of the young households is unity.5
We assume 90% of young households will work for all the productivity groups, if no tax is
imposed,6 so that,i0.1δ i0 + 0.9δ 0 = wi , for i = 1, 2, 3.

(18)

We also assume the coeﬃcient of variation of the cost to work is the same across the age andproductivity groups. Let v denote the coeﬃcient of variation of the cost to work. This assumption
implies,7i

δ 0 − δ i0i
δ 0 + δ i0

=

√

3v,

for i = 1, 2, 3.

(19)

Given the coeﬃcient of variation, (18) and (19) determine the domains of the cost to work forithe young household, δ 0 and δ i0 . We consider three cases for the coeﬃcient of variation, namely
v = 0.1, 0.3, or 0.5.8i

The cost to work of old households with productivity wi is distributed over the interval [δ i1 , δ 1 ].
Consider an old household at period 1 with a history of productivity (w0 = wj , w1 = wi ) and
5 It seems that this participation elasticity is a bit high. Saez (2002) adopt the participation elasticity for lower
income earners equal to 0, 0.5, and 1.
6 The labor force participation rates of young male in OECD countries average are about 86%, while by age 25
to 64 total are around 75%. See e.g., in OECD.Stat.
7 Note that δ i is uniformly distributed over [δ i , δ i ]. The expected value of δ i , E(δ i ) is E(δ i ) = (δ i +δ i )/2 and the
0
0 0
0
0
0
0
√0 i
i
i
variance of δ0i is (δ 0 − δ i0 )2 /12. Thus, the coeﬃcientof variation v is, by the definition, v = (δ 0 − δ i0 )/( 3(δ 0 + δ i0 )).
√8 δ i become negative if we set v larger than 1/ 3 ≈ 0.58.
011

experienced the cost to work δ0 at period 0. There are two possible cases for the determinationof δ1 . The probability p have influence on the stochastic process of the cost to work δt . Withjprobability p ∈ [0, 1], the relative position of δ0 in [δ j0 , δ 0 ] ‘survives’ at period 1, that is,
δ1 =(δ̄1i − δ i1 )(δ0 − δ j0 )j(δ 0 − δ j0 )+ δ i1 .

With probability 1 − p, the relative position of δ0 ‘expires’ and the household draws a new δ fromthe uniform distribution over the interval [δ j1 , δ̄1j ]. This means the distribution of δ1 of all the oldhouseholds with productivity wi is uniform over [δ i1 , δ̄1i ]. We consider 3 cases for p, namely p = 0,0.5, or 1.
Working populations and the participation elasticities of the old households are,y1 (wi ) − y1 (0) − δ i1,
δ̄1i − δ i1
y1 (wi ) − y1 (0)η1i =.
y1 (wi ) − y1 (0) − δ i1

ni1 = q1i

Let ρ denotes the work participation ratio of old households. We consider five cases for thework participation ratio of old households, if there is no tax, namely ρ = 0.1, 0.2, 0.3, 0.4, or 0.5 forall the productivity groups.9 The coeﬃcient of variation of δ1 is equal to v. By the same argumentsiiof δ i0 and δ 0 , these conditions determine δ i1 and δ 1 .
Therefore, we consider 3,645 cases for the parameter combinations.10 Under the specificationsof quadratic preference and uniform distribution of δ, the first order conditions of the optimal
asset holdings (6) at t = 0 are piecewise linear, and we derive a closed form solution of the optimal
asset holdings. Substituting these and the labor supply plans, the government’s objective function
becomes piecewise polynomial. We numerically solved the optimal income tax policy for all thecases.

4.2

Results(1)

The benchmark case

Table 1 shows the optimal age-dependent taxation and the optimal age-independent taxation in
the benchmark case where all the parameters are set at the middle choice. The first part, ‘notax’, shows the allocation when no labor income tax is imposed, that is, the disposable income
yti is equal to the labor productivity wi . The working population of young and old are 0.9 and
0.3. The resulting GDP (total production of the two generations) is 1.23. The middle part is the
age-independent optimal taxation where the disposable income schedule is the same across thediﬀerent age groups. The redistribution from the more productive workers to the less productive
ones improves the welfare (EU increases from 0.33068 to 0.330734) while it has a negative impact
on the GDP (y decreases from 1.23 to 1.2182). As shown in the previous section, the participation
tax rate for the least productive workers is negative (τ01 = τ11 = −0.0025). T represents the total
amount of redistribution. It is rather small, 0.3% of GDP, in this example. The last part shows the
9 The labor force participation rates of over age 65 male in OECD countries average are around 20% and overage 65 total are around 14%. The rates by age 55 to 64 male are around 70% and by age 55 to 64 total are around60%. In our two periods model, the old households is age 55 to 80. The labor force participation rates of old arearound 35%. See e.g., in OECD.Stat.
10 3 for α, 3 for w, 3 for σ, 3 for π, 3 for v, 3 for p, and 5 for ρ.

12

Table 1: Optimal taxation of the benchmark caseParameter setting
α = 0.05, w = (0.5, 1, 2), π i,i = 0.6, π i,j = 0.2 (j ̸= i), p = 0.5, σ = 1/3, v = 0.3, ρ = 0.3
i
y0i
τ0i
ni0
y1i
τ1i
ni1
0
0
—–
0.1
0
—–
0.7
EU
0.33068
No tax
1 0.5000.133 0.50
0
0.078
GDP
1.23
2 1.00
0
0.7
1.00
0
0.153
T
0
3 2.0000.067 2.00
00.069 T /GDP 0
Age0 0.003—–
0.104 0.003
—–
0.703
EU
0.330734
independent 1 0.504 -0.0025 0.134 0.504 -0.0025 0.078
GDP
1.2182
2 1.001 0.0027 0.697 1.001 0.0027 0.152
T
0.0036
optimal
3 1.973 0.0153 0.065 1.973 0.0153 0.066 T /GDP 0.0030
tax
Age0 0.003—–
0.102 0.003
—–
0.706
EU
0.330739
dependent
1 0.506 -0.0056 0.134 0.502 0.0026 0.077
GDP
1.2173
optimal2 1.001 0.0015 0.698 0.995 0.0074 0.150
T
0.0043tax
3 1.973 0.0150 0.065 1.972 0.0155 0.066 T /GDP 0.0035

optimal age-dependent tax, where the participation tax rate for the least productive young worker
is negative but the one for the least productive old worker is positive (i.e., τ01 < 0 and τ11 ≥ 0). In
comparison to the age-independent taxation, the optimal age-dependent tax improves the welfare(EU increases from 0.330734 to 0.330739) since a fine-tuned income tax/transfer is available. It
results in larger redistribution (T increases from 0.0036 to 0.0043).
Remember that equation (12) holds within each age group in the age-dependent taxation.
Thus, the participation tax rate of the least productive age t household is negative if and only if its
marginal social weight is larger than the average of age t households. This negative participation
tax, however, is less likely to happen for old households. The reason is as follows. A large part ofold households is retiring because of high utility cost to work and the retiree’s social weight is likely
higher than the working olds who happens to get low utility cost δ. Therefore, it is likely that themarginal social weight of the least productive worker is lower than the average of old households,
and negative participation tax for old may not be optimal.
(2)

Sensitivity analysis

To see the robustness of this property, we conducted the same calculation for all the 3,645 parameter combinations. Figure 1 shows the average optimal participation tax rates (PTR) with eachproductivity level in the age-independent taxation, age-dependent taxation at period 0 and period
1 for all cases. The average optimal participation tax rates in figure 1 have same patterns withbenchmark case. Furthermore, we calculate the first and third quantile of the optimal participation
tax rates in order to see the statical dispersion of the result. One can confirm that in many cases,
optimal participation tax rates are similar with the benchmark case. Indeed, these statistics of
optimal age-dependent participation tax rates for the least productive working old household are
positive, while the corresponding ones for least productive working young household are negative.
Next, we count the frequency of signs of τ 1 in the optimal age-independent taxation, τ01 and τ11
in the optimal age-dependent taxation. The results are shown in Table 2. The first part of Table 2reports the optimal age-independent participation tax rate for the least productive workers, τ 1 isnegative for 3,605 out 3,645 cases of parameter combinations (about 99%). For these cases, 2,652
cases (about 74%) have the same property of the benchmark case (i.e., τ01 < 0 and τ11 ≥ 0).
The rest of Table 2 reports the relationships of the sign patterns of τ 1 , τ01 , and τ11 with various
13Figure 1: Optimal average participation tax rate
Age−independent tax

Age−dependent tax at period 0

Age−dependent tax at period 1

0.04

0.03

PTR

0.02

0.01

0.00

−0.01
w1
w2
w3
w1
w2
w3
w1
w2
w3
(Notes) This figure shows the average optimal participation tax rates (PTR) with each productivity level (see colorlines). The black lines show the range of first and third quantile of the optimal PTR with each productivity level.

parameters. We first pay attention to the parameters Ω, σ, and π ii aﬀecting the productivity distribution and its stochastic process. If, other things being equal, the sign pattern of the middle case(i.e., τ01 < 0 and τ11 ≥ 0) likely maintains when w3 , σ, or π ii is small. The negative participation
tax for the worker with the lowest income should be restricted to the young, if the productivity
distribution is less skewed, the productivity is less dispersed, or social mobility is high. We especially take notice of the relation between the level of average optimal participation tax rate for the
least productive households and the transition probabilities about productivities, π ii (see figure2). Figure 2 shows that in the case of high social mobility about productivity (i.e., π ii is 0.333 inthis case), the optimal participation tax for the least productive working old households would bepositive.
Finally, we focus on the parameters v, p, and ρ aﬀecting the labor disutility distribution and its
stochastic process. If, other things being equal, the sign pattern of the middle case likely maintains
if v is large, or p is middle or large case, or ρ is small. They suggest the negative participation
tax should be restricted to the young if the cost of work is dispersed, persistency of the preference
shock is middle or large, or work participation ratio of the old is not large. In general, a large partof old is retiring because of high utility cost. Their marginal social weight tends to be larger thanthose who get low utility cost and participating in the workhorse. Therefore, the marginal social
weight of working old of the least income tends to be lower than the average of marginal social
weight of the old. Figure 3 plots the relation between the level of average optimal participation
tax rate for the least productive households and the work participation ratio of the old household,ρ. It shows that the optimal age-dependent participation tax rate for old would be positive when
their work participation ratio is low. The negative participation tax rate for old is inappropriate
for the case that the working population of old is small. It is true even when the participation
elasticity of old is high since the elasticity does not aﬀect the sign of participation tax rate but

14

Table 2: The frequency of sign patterns of participation tax by age

Total Case40

α = 0.1
α = 0.05
α = 0.025Ω = (0.5, 1, 1.5)
Ω = (0.5, 1, 2)Ω = (0.5, 1, 3)
σ = 1/4
σ = 1/3
σ = 1/2
π ii = 1/3
π ii = 0.6
π ii = 0.9v = 0.1
v = 0.3
v = 0.5p=0
p = 0.5
p=1
ρ = 0.1
ρ = 0.2
ρ = 0.3
ρ = 0.4
ρ = 0.5

17
12
11
0
9
31
40
0
0
40
0
0
21
12
7
0
18
12
3
20
17
0
0

Age-independent taxation
τ1 ≥ 0Age-dependent taxation
τ01 ≥ 0,
τ01 ≥ 0,
τ01 < 0,
τ01 < 0,
τ11 ≥ 0
τ11 < 0
τ11 ≥ 0
τ11 < 0
0
0
40
0
3,605
parameter settings
0
0
17
0
1,198
0
0
12
0
1,203
0
0
11
0
1,204
0
0
0
0
1,215
0
0
9
0
1,206
0
0
31
0
1,184
0
0
40
0
1,175
0
0
0
0
1,215
0
0
0
0
1,215
0
0
40
0
1,175
0
0
0
0
1,215
0
0
0
0
1,215
0
0
21
0
1,194
0
0
12
0
1,203
0
0
7
0
1,208
0
0
0
0
1,215
0
0
18
0
1,197
0
0
12
0
1,193
0
0
3
0
726
0
0
20
0
709
0
0
17
0
712
0
0
0
0
729
0
0
0
0
729

Age-independent taxation
τ1 < 0Age-dependent taxation
τ01 ≥ 0, τ01 ≥ 0, τ01 < 0,
τ01 < 0,
τ11 ≥ 0
τ11 < 0
τ11 ≥ 0
τ11 < 0
1
4
2,652
948
0
0
1
0
0
1
1
0
0
1
0
0
1
0
0
1
0
0
0
1
0
0
0

0
0
4
1
0
3
3
0
1
4
0
0
4
0
0
2
1
1
1
1
0
1
1

933
875
844
948
918
786
913
907
832
1,007
953
692
801
877
974
672
1,008
972
635
637
637
532
211

Figure 2: Optimal average participation tax rates with π ii
Average PTR for the least productive worker

0.004

0.000

Tax Systemage−independent tax τ1−0.004age−dependent tax τ10
age−dependent tax τ11−0.008−0.012
1/3

0.6

0.9

πii

only the size of it.

15

265
328
355
266
288
394
258
308
382
163
262
523
388
326
234
540
188
220
90
70
75
197
516

Average PTR for the least productive worker

Figure 3: Optimal average participation tax rates with ρ

0.000

Tax Systemage−independent tax τ1−0.005age−dependent tax τ10
age−dependent tax τ11−0.010

0.1

5

0.2

0.3

ρ

0.4

0.5

Conclusion

In the present paper, we investigated optimal age-dependent income taxation in a dynamic modelwith extensive labor margin in each period and uninsurable idiosyncratic shocks of labor productivity and cost to work. When the government can only employ age-independent labor income
tax, the optimal participation tax rates are similar to the static model and the stationary life-cycle
model. The age-independent participation tax rate for low consumption group must be negative.
When the government can employ age-dependent labor income tax, the optimal participation tax
rates are age-specific. Negative participation tax rate for low consumption households apply ineach age group. The age-specific participation tax rules depend on the average marginal social
weight that take account of the idiosyncratic shock with respect to productivity and cost to work
until that period. On the contrary, participation tax rates in the static model depend on averagemarginal social weight at only that period. In this sense, the age-specific tax rules have more
information of household history with productivity.
In our example economy, we simulate optimal participation tax rate both age-independent
taxation and age-dependent taxation. We calculate the optimal participation tax rate under various
parameter specifications and found that well known property of negative participation tax rate forlow consumption households are valid in the age-independent case and for only young in the agedependent case. In our dynamic setting, the link between the income and consumption is notperiodwise. Young households tend to work much harder than old households in some reasons(e.g., to raise expense for children or health conditions). Thus, the marginal social weight of theleast productive working young households is likely higher than the average of young households.
On the contrary, a large part of old households is retiring and their social weight is likely higher
than the working olds. Therefore, it is likely that the marginal social weight of the least productive
working old is lower than the average of old households, and negative participation tax for old may
not be optimal even when the participation elasticity of old is high. In-work policy that targets
young people who want to work, have greater benefit. This suggest that in-work credits or wage

16

subsidies for young people are desirable when the extensive labor margin is important.

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17