Optimal income taxation and unemployment benefit withinvoluntary unemployment∗
Takao Kataoka† and Yoshihiro Takamatsu‡
February 2019

Abstract
We consider optimal income taxation and search-contingent unemployment benefits
in a model where there are both voluntary and involuntary unemployed households,
where households’ labor productivity and labor disutilities are heterogeneous, andhouseholds choose whether to participate and seek jobs in the labor market. We derive an optimal employment tax rule. We show that employment tax rates dependon the size of the search-contingent unemployment benefit. Our numerical simulations suggest that negative employment tax rates for some income groups with low
consumption are less likely to be negative.
JEL classification: H21, H24, I38, J65
Keywords: Optimal income taxation; Unemployment benefit; Involuntary unemployment; Extensive margin; Employment tax

1

Introduction

Recent literature has emphasized the role of an extensive margin of labor. With anextensive margin of labor, households choose whether to participate in the labor force
with fixed hours of work. In this setting, it can be optimal to adopt wage subsidies orin-work credits with a negative participation tax for low-income households (see, e.g.,
Diamond (1980), Saez (2002), Choné and Laroque (2005, 2011), Jacquet et al. (2013),
and Christiansen (2015)).
The most simple setting for the extensive margin does not consider the demand side ofthe labor market. This means that all households who choose to participate in the labor
force can find employment, and there is no involuntary employment. Some studies develop
optimal income taxation with involuntary employment. First, Boone and Bovenberg (2004,
2006, 2013) consider that households bear a search cost when they enter the labor market∗

We are particularly grateful to Yukihiro Nishimura for his insightful comments and suggestions.
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.jp†1and seek jobs, and households choose work eﬀort as well as search eﬀort with quasi-linear
preferences for leisure. Boone and Bovenberg (2004) show that with exogenous welfare
benefit, the negative marginal tax rate and welfare benefit have a U-shaped relationship.
Additionally, search subsidies (negative employment taxes) are likely to be optimal if thewelfare benefit and the government’s required revenue is low while search costs are high.
Second, Hungerbuhler et al. (2006), Lehmann et al. (2011), and Jacquet et al. (2014)
consider a search-matching framework in which the wage is determined by a bargaining
process. Jacquet et al. (2014) consider wages are determined by Kalai bargaining andshow that this matching environment induces significantly lower employment tax rates
than the usual competitive model with endogenous participation only.
Although these studies have provided influential findings with both theoretical andpolicy implications, a large component of the literature does not focus on the role of wage
subsidies (or in-work credits) and unemployment benefits. In the real world, governments
employ wage subsidies for low-income workers as well as unemployment benefits for bothvoluntary and involuntary unemployed households. Governments often restrict the qualifications for receiving benefits to those who engage in a job search. With involuntary
employment, the search-contingent unemployment benefit could be a more eﬃcient policy
to encourage labor market participation than wage subsidies or in-work credits. A recent
paper by Boadway and Cuﬀ (2018) only investigates the joint design of a piecewise linear income tax/transfer and unemployment benefits with a search-matching framework.
The authors’ model adopts the extensive margin under which households choose whether
to participate in job search and the intensity of that search. The authors find analytically that the optimal employment tax depends on the relative size of the transfer for
the involuntarily unemployed and the voluntarily unemployed. Instead, we adopt a simple
search cost model in which households are heterogeneous about labor productivities, labor
disutilities, and search costs, and the search-contingent unemployment benefit is endogenous. This model is a simple extension of the original extensive margin model used in
Diamond (1980) and Saez (2002) (hereafter referred to as the full employment model) in
which households are heterogeneous about labor productivities and disutilities. We use
this model to consider involuntary unemployment and to compare both a full employment
model and an extensive model with search and involuntary unemployment.
Thus, we examine the relation and the role of wage subsidies and unemployment benefit in the extensive model with search and involuntary unemployment both analyticallyand numerically. We derive the following results. First, in the case of involuntary unemployment, we identify the optimal employment tax rule, which depends on the optimal
search-contingent unemployment benefit. In contrast to the full employment model, negative employment tax rates for households with lower consumption are less likely to occur.
Second, the optimal search-contingent unemployment benefit should be higher when theeﬀect of the benefit on the population of working households is greater. Finally, we numerically solve the optimal tax policies for example economies. We confirm that employment
tax rates and search-contingent unemployment benefit have a positive correlation. Neg2

ative employment tax rates (in-work credits or wage subsidies) are valid only when theemployment probability is close to unity or the search cost is close to zero. This suggeststhat the search-contingent unemployment benefit is a substitute for in-work credits.
The remainder of the paper is organized as follows. In Section 2, we present a modelwith an extensive margin of labor and involuntary unemployment. In Section 3, we investigate the optimal income tax problem with involuntary unemployment and derive optimal
employment tax rates in the case of partial optimal taxation and full optimal taxation. In
the former case, the government only employs labor income tax/transfers. In the latter
case, the government employs labor income tax/transfers as well as unemployment benefit.
In Section 4, we consider an example economy with various parameters and numericallydetermine optimal income tax policies and optimal search-contingent unemployment benefits. In Section 5, we present our conclusions.

2

The model

The economy is populated by a government and one unit of continuous households withan indivisible labor supply. Each household is heterogeneous about the two-dimensional
characteristic θ = (w, δ) where w ∈ R+ and δ ∈ R, respectively, represent labor productivity and labor disutility. There are I ∈ N possible levels of labor productivity.
The domain of labor productivity is denoted by Ω = {w1 , w2 , · · · , wI } ∈ RI++ where(w1 < w2 < · · · < wI ). Let F : Ω × R → R+ denote the cumulative probability distribution function (c.d.f) of the population. The c.d.f. is public information.
There are I labor markets for each labor productivity. A household with productivity
wi cannot mimic another productivity wj (i ̸= j). A household chooses labor participation
status s ∈ {0, 1} given its θ. If a household with productivity wi decides to participate(s = 1) in the labor market of productivity wi , that household suﬀers the utility cost of
job search, Ψi , and has the probability of job matches, pi ∈ [0, 1]. If the household pays
the search cost and receives a job oﬀer, then the household works fixed hours and earns wiunits of (before-tax) labor income and suﬀers the utility cost of labor, δ. If the householdpays the search cost but does not find a job, then the household does not get wi anddoes not pay δ. If the household does not participate (s = 0), their labor income is zero,and the household suﬀers no search cost and disutility of labor. The labor productivity is
known to the public if and only if the household works. Note that nonworking households
in this economy are composed of both the involuntary unemployed and the voluntary
unemployed.
There is a divisible consumption good. Let U (c) denote the household’s utility fromconsumption, c. U is a twice continuously diﬀerentiable utility function satisfying U ′ > 0
and U ′′ < 0. All households have no real assets at the beginning of the economy.
The government’s policy is a tax on labor income and unemployment benefits conditional on the job search. Let T (wi ) and −T (0) denote the labor income tax of working
households with productivity wi and the uniform subsistence income for a nonworking
3household, respectively. T : {0} ∪ Ω → R represents a labor income tax schedule. In
addition, the government provides unemployment benefits conditional on a job search, b.
This unemployment benefit is paid only if a household searched for a job in the labormarket and was not employed. We assume that the government can monitor households’
job search activities, but a job oﬀer is not observable. If U (−T (0) + b) − Ψi > U (−T (0)),
some low-productivity workers pretend to be involuntary unemployed. Thus, we assume
the following incentive compatible conditions hold for all i:
U (−T (0) + b) − Ψi ≤ U (−T (0)).

(1)

Given T (0), (1) determines an upper bound of b. The condition (1) also implies no rejectionof a job oﬀer.
Given labor income tax schedule T , search-contingent unemployment benefit b, andindividual characteristic θ, the household chooses labor participation status s ∈ {0, 1} tomaximize the expected utility,s[pi [U (wi − T (wi )) − Ψi − δ] + [1 − pi ][U (−T (0) + b) − Ψi )] + [1 − s]U (−T (0)).
Let Vi denote the diﬀerence of expected utility between a household that participates inlabor market i and a household that does not participate divided by pi as follows:
Vi (pi , T, b) = U (wi − T (wi )) +[1 − pi ]Ψi + U (−T (0))
U (−T (0) + b) −
.
pi
pi

(2)

Suppose a household has a δ smaller than Vi . The household’s expected utility is greaterat s = 1. Thus, the optimal labor participation plan s is 1 (resp. 0) if δ is smaller (resp.
larger) than Vi .
The population of households who participate in labor market i, denoted by ki , is,∫ki =i = 1, 2, · · · , I,dF (θ),(3)θ∈Θiwhere Θi denotes the region of θ defined byΘi ≡ {(w, δ) | w = wi and δ < Vi }.
Thus, the population of households with earning (before-tax) labor income wi , denotedby ni , and the population of nonworking households, denoted by n0 , are,ni = p i × k i ,n0 = 1 −I∑

i = 1, 2, · · · , I,
ni .

i=1

4

The following symmetric relations hold:
∂n0
∂ni
=−
≤ 0,
∂T (wi )∂T (wi )

i = 1, 2, .., I,

∑ ∂ni
∂n0
=−
≥ 0.
∂T (0)∂T (0)I

i=1

Let EU denote the unconditional expected utility as follows:
EU =

I
∑

[1 − ki ]U (−T (0))

i=1

+

I
∑

ki [pi [U (wi − T (wi )) − Ei (δ)] + [1 − pi ]U (−T (0) + b) − Ψi ]

i=1

=

I
∑

[U (−T (0)) + pi ki [Vi − Ei (δ)]] ,

(4)

i=1
∂EU
′
where Ei (δ) denotes E(δ|w = wi and δ ≤ Vi ). We have ∂T(wi ) = −pi ki U for i = 1, . . . , I

where U ′ represents the average marginal utility of consumption over working householdswith wi . The government must finance an exogenous expenditure G. The government’s

budget constraint is given by,I∑

ki [pi T (wi ) − [1 − pi ]b] + n0 T (0) ≥ G.

(5)

i=1

The optimal taxation problem for the government is to find the labor income tax scheduleT and the job search-contingent unemployment benefit b that maximizes the household’s
unconditional expected utility, EU , subject to the budget constraint (5) and IC condition(1).
We consider two types of optimal taxation problems: the full optimal taxation problem
and the partial optimal taxation problem. The former corresponds to the labor income
tax schedule T and the job search-contingent unemployment benefit pair b that exactly
maximize EU subject to (5) and IC condition (1). However, the actual unemployment
benefit system does not necessarily depend on households’ job search activity. Thus, we
also investigate the latter problem, which solves the same maximization problem but withthe additional constraint that the government does not employ the unemployment benefit(b = 0).

3

Employment tax rate

We first consider the case of partial optimal taxation. Then, we examine the full optimal
taxation problem.

5

3.1

Partial optimal taxation

For the partial optimal taxation problem, the government does not employ the unemployment benefit (b = 0). The Lagrangian of the government problem in the case of full
optimal taxation is
LP = EU P + λ

[ I
∑

]
pi ki T (wi ) + n0 T (0) − G ,i=1

where EU P is the unconditional expected utility in the case of partial optimal taxation,and λP is the Lagrange multiplier associated with the government budget constraint in
the case of partial optimal taxation. The first-order conditions of the Lagrangian withrespect to T (wi ) and T (0) are,
[
]
∂LP
∂EU P
∂ki
∂n0
P
=
+λ
pi T (wi ) + pi ki +T (0) = 0,
∂T (wi )∂T (wi )∂T (wi )∂T (wi )
[ I
]
∑ ∂ki
∂EU P
∂n0
∂LP
P
=
+λ
pi T (wi ) + n0 +T (0) = 0.
∂T (0)∂T (0)∂T (0)∂T (0)

for i ̸= 0,

i=1With involuntary unemployment, households with productivity wi that participate in thelabor market and find a job forgo subsistence income −T (0) and pay income tax T (wi ).
Thus, T (wi ) − T (0) is the employment tax for households with productivity wi . We also
define the employment tax rates faced by households with productivity wi in the case ofpartial optimal taxation byτiP =T (wi ) − T (0),wi

for i ̸= 0.

(6)

Using (10), we rewrite the first-order conditions and obtain the optimal employment tax
rule for partial optimal taxation as follows (see Appendix A.1):
1 − giPτiP=,
1 − τiP
ηiP

for i = 1, 2, . . . , I,

(7)

where giP is the average marginal social weight of consumption for the working householdswith productivity wi in the case of partial optimal taxation and expressed in terms of
public funds as follows:
1∂EU P
giP = − P,λ pi ki ∂T (wi )for i = 1, 2, . . . , I,and ηiP is the participation elasticity in the case of partial optimal taxation defined as
ηiP = −

∂ki wi − T (wi ) + T (0),ki∂T (wi )6

for i = 1, 2, . . . , I.

Equation (7) is similar to the standard optimal participation tax rule in the extensive
margin with full employment (see Diamond (1980), Saez (2002), Choné and Laroque (2005,2011), Jacquet et al. (2013), and Christiansen (2015)). We consider the full employment
case as a special case of partial optimal taxation. In the full employment case, there is no
involuntary unemployment and no utility costs for a job search, so pi = 1 and Ψi = 0 for
all i. Thus, the government need not employ the unemployment benefit (b = 0).
In the full employment case, households with productivity wi that participate in labormarkets forgo the subsistence income −T (0) and pay income tax T (wi ). Thus, T (wi ) −T (0) is the participation tax for households with productivity wi . We also define theparticipation tax rates faced by households with productivity wi by
τiE =T (wi ) − T (0),wi

for i ̸= 0.

(8)

Using (8), we rewrite the first-order conditions and derive the optimal participation tax
rule in the case of full employment as follows (See Appendix A.2):
1 − giE
τiE=,
1 − τiE
ηiE

for i = 1, 2, . . . , I,

(9)where ηiE is the participation elasticity in the case of full employment defined as
ηiE = −

∂ni wi − T (wi ) + T (0),ni∂T (wi )for i = 1, 2, . . . , I,

and giE is the average marginal social weight of consumption for the working householdswith productivity wi expressed in terms of public funds in the case of full employment asfollows:
1 ∂EU E
giE = − E,
λ ni ∂T (wi )for i = 1, 2, . . . , I

where EU E is the unconditional expected utility in the case of full employment, and λE
is the Lagrange multiplier associated with the government budget constraint in the caseof full employment. Note that the average marginal social weight of consumption in thecase of full employment, giE , is the same shape as the average marginal social weightof consumption in the case of partial optimal taxation, giP . However, the unconditional
expected utility in the case of full employment, EU E , does not depend on the search cost,Ψ. The arguments of the unconditional expected utility are diﬀerent between the partial
optimal taxation and full employment case. Nevertheless, equations (7) and (9) have thesame pattern. We can apply the same interpretation with both rules. If the right-handside of (7) (resp. (9)) is negative for some working group with lower consumption, thenthe employment (resp. participation) tax rate for such an income group must be negative.

7

3.2

Full optimal taxation

In the full optimal taxation case, there is involuntary unemployment, and the government
employs labor income tax schedule T as well as the job search-contingent unemployment
benefit b. The Lagrangian of the government problem in the case of full optimal taxationis
L = EU + λ
F

F

F

[ I
∑

]
ki [pi T (wi ) − [1 − pi ]b] + n0 T (0) − G

i=1

where EU F is the unconditional expected utility in the case of full optimal taxation, andλF is the Lagrange multiplier associated with the government budget constraint in thecase of full optimal taxation. Assuming the interior solution, the first-order conditions of
the Lagrangian with respect to T (wi ), T (0) and b are
[
]
∂EU F
∂ki
∂n0
∂LF
F
=
+λ
[pi T (wi ) − [1 − pi ]b] + pi ki +T (0) = 0, for i ̸= 0,∂T (wi )∂T (wi )∂T (wi )∂T (wi )
[ I
]
∑ ∂ki
∂LF
∂EU F
∂n0
F
=
+λ
[pi T (wi ) − [1 − pi ]b] + n0 +T (0) = 0,
∂T (0)∂T (0)∂T (0)∂T (0)
i=1
[ I [
]
]
∑ ∂ki
∂LF
∂EU F
∂n
0
=
+ λF
[pi T (wi ) − [1 − pi ]b] − ki [1 − pi ] +T (0) = 0.
∂b
∂b
∂b
∂b
i=1

Using the same method as the previous subsection, we define the employment tax rates
faced by households with productivity wi in the case of full optimal taxation by
τiF =T (wi ) − T (0),wi

for i ̸= 0.

(10)

Using (10), we rewrite the first-order conditions and the optimal employment tax rate rulein the case of full optimal taxation as follows (See Appendix A.3):
[
]
τiF
1
1
∂ki
F
= F 1 − gi −[1 − pi ]b ,pi ki ∂T (wi )1 − τiF
ηi

for i = 1, 2, . . . , I,

(11)where giF is the average marginal social weight of consumption for working households withproductivity wi in the case of full optimal taxation expressed in terms of public funds asfollows:∂EU F
1giF = − F,λ pi ki ∂T (wi )for i = 1, 2, . . . , I,and ηiF is the participation elasticity in the case of full optimal taxation defined as
ηiF = −

∂ki wi − T (wi ) + T (0),ki∂T (wi )8

for i = 1, 2, . . . , I.

Equation (11) is the optimal employment tax rule with involuntary unemployment andsearch-contingent unemployment benefit. Equation (11) shows that the optimal employment tax rate depends on the search-contingent unemployment benefit (the third term ofithe right-hand side of (11)). Since ∂T∂k(wi ) ≤ 0 and b ≥ 0, this term would be non-negativein total. Thus, even when giF > 1 and 1 − giF < 0, the third term cancels it out, and thenegative employment tax rate is less likely to occur.
The search-contingent unemployment benefit is endogenous in full optimal taxation.
We examine the eﬀect of the search-contingent unemployment benefit on optimal employment tax rates. To do so, we rewrite the first-order condition with respect to b (seeAppendix A.4), and we obtain the optimal search-contingent unemployment benefit asfollows:b∗ =gbF − 1 +∑I

∂ki
i=1 ∂b pi [T (wi )−T (0)]∑Ii=1 ki [1−pi ]

hF

,

(12)

where gbF is the average marginal social weight of unemployment benefits for the involuntary unemployed expressed in terms of public funds, and hF is the rate of the increasein the involuntary unemployed due to a marginal increase in unemployment benefit asfollows:
gbF =
F

h =

∂EU F
,
∂b
i=1 ki [1 − pi ]

∑I
F

λ
∑I

1

∂ki
i=1 ∂b [1 − pi ].
∑I
i=1 ki [1 − pi ]The denominator on the right-hand side of (12) shows that when the unemployment benefit
results in a higher number of involuntary unemployed, the benefit should be decreased.
The third term of the numerator on the right-hand side of (12) is positive and shows themarginal increase in tax revenue from an increase in the number of working households
due to the unemployment benefit divided by the number of involuntary unemployed. If
this value is higher, then a larger optimal unemployment benefit is desirable. However,
the total eﬀect of the right-hand side of (12) is ambiguous since the numerator and thedenominator on the right-hand side of (12) could have positive correlations. Numericalsimulations are required to investigate the eﬀect of the search-contingent unemployment
benefit on the optimal employment tax rates.

4

Numerical simulations

The previous section shows that employment tax rates for some working households withlow consumption are less likely to be negative when there is involuntary unemployment
and search-contingent unemployment benefits. This section numerically solves for optimal
tax policies using an example economy. We suggest that there is a positive correlation

9

between employment tax rates for the least productive workers and search-contingent
unemployment benefits, which implies that unemployment benefits in full optimal taxation
are a substitute for in-work credits.

4.1

Parameter setting(1)

Utility function

We assume the utility function U is
U (c) = cα ,where α ∈ R+ determines the curvature of U . We assume that the range of α is α ∈[0.75, 0.99].
(2)

Labor productivity

There are three possible labor productivity levels, (I = 3 and w1 < w2 < w3 ). We
assume the domain of labor productivity (w1 , w2 , w3 ) = (1, 2, 4). This means that thelabor productivity distribution is skewed.
Let q = (q1 , q2 , q3 ) denote the vector of the population where qi denotes that thehousehold population obtains productivity wi . We assume that the range q1 and q3 is∑
q1 ∈ [0.2, 0.45] and q3 ∈ [0.2, 0.45]. Since the total population is unity, 3i qi = 1, therange q2 is q2 ∈ [0.1, 0.6].
(3)Participation ratio at the laissez-faire equilibrium

Let ρ = (ρ1 , ρ2 , ρ3 ) denote the vector of the participation ratio where ρi is the participation
ratio of households who have labor productivity wi when there is no income tax and no
unemployment benefits. We assume ρi ∈ [0.8, 0.99] for all the productivity groups.
(4)

Employment probability

Let p = (p1 , p2 , p3 ) denote the vector of the employment probability where pi is theemployment probability in labor market i. We assume pi ∈ [0.7, 1] for all the productivity
groups.
(5)

Search costLet Ψ = (Ψ1 , Ψ2 , Ψ3 ) denote the vector of the utility cost of job search. We assume theutility cost of job search in labor market i is specified by Ψi = (1−pi )γpi wiα where γ ∈ [0, 1]
is a common parameter, and pi wiα are the expected utility from earnings when there is no
income tax and no unemployment benefits in market i. Our specification assumes thatthe magnitude of the search costs in market i are some ratio of the expected utility from
earnings in the laissez-faire equilibrium. The ratio is the same for all markets. This means
10

that highly productive jobs tend to be more diﬃcult to find. The search cost is higher
when the employment probability in the market is smaller.
(6)

Labor disutility

The labor disutility δi is uniformly distributed over the interval [δ i , δ̄i ] ⊂ R. Thus, theidensity of the labor disutility δi is δ̄ q−δ. Let δi∗ denote the labor disutility, which isiiequal to the diﬀerence of expected utility between participation in the labor market andnonparticipation divided by pi such that δ = Vi when there is no income tax and no
unemployment benefits. With uniform distributions, the population of households thatparticipate in labor market i, ki becomes ki =δ ∗ −δ

qi [δi∗ −δ i ], and the participation ratio, ρi , isδ̄i −δ i

ρi = kqii = δ̄i −δ i .
ii

Let e denote the labor participation elasticity with respect to labor productivity when∂ki /kifor all i. Using the labor
there is no income tax and no unemployment benefits, e = ∂w
i /wi

participation elasticity, e, we obtain the following equations:1α(wi )α,e[]α(wi )α 1∗δ̄i = δi +−1eρi

δ i = δi∗ −where δi∗ = wiα − Ψpii . Given α, e and wi , ρi , pi , and Ψi determine the domains of thelabor disutility for households with productivity wi , δ̄i , and δ i , respectively. We considere ∈ [0.01, 1].

4.2

Benchmark case

Table 1 shows optimal income taxation for full employment, optimal partial taxation, andthe optimal full taxation in the benchmark case where all parameter settings are listed at
the top of the table.
We first focus on full employment (see Table 1 (A)). In this case, the population ofhouseholds who participate in labor market i corresponds to the population of households
who work in labor market i, (i.e., pi = 1). The first part, “No tax,” shows the allocation when no labor income tax is imposed; that is, disposable income yi is equal to laborproductivity wi . The resulting GDP (total production) is 2.07. The last part, “Optimal
income taxation,” is the optimal income taxation for full employment. The redistribution
from more productive workers to less productive workers improves the welfare (EU increases from 1.609 to 1.622) but has a negative impact on GDP (GDP decreases from 2.07
to 1.906). As shown in the previous section, the participation tax rate for some working
1

We rewrite e as follows:e=α(wi )α∂ki ∂Vi wi
qi
wi
=
=
α(wi )α−1.
∂Vi ∂wi kikiδ̄i − δ i
ρi [δ̄i − δ i ]αThus, using the participation ratio, δi∗ − δ i = α(wei )αand δ̄i − δi∗ = [1 − ρi ][δ̄i − δ i ] = α(wei )

11

[

]

1
−1
ρi

.

Table 1: Optimal taxation for the benchmark case
A. Full employment caseParameter setting
α = 0.875, q = (0.3, 0.4, 0.3), ρ = (0.9, 0.9, 0.9), p = (1, 1, 1), e = 0.5
w = (1, 2, 4), Ψ = (0, 0, 0), G = 0iyi
T (wi )
τi
ki
ni
0
0
0
—–
0.1
0.1
EU
No tax
1 1.00
0
0
0.27
0.27
GDP
2 2.00
0
0
0.36
0.36
3 4.00
0
0
0.27
0.27
0 0.207 -0.207—–
0.156 0.156
EU
Optimal income taxation 1 1.253 -0.253 -0.047 0.265 0.265 GDP
2 2.062 -0.062 0.072 0.337 0.337
3 3.502 0.498
0.176 0.242 0.242

1.609
2.07

1.622
1.906

B. Involuntary unemployment caseParameter setting
α = 0.875, q = (0.3, 0.4, 0.3), ρ = (0.9, 0.9, 0.9), p = (0.85, 0.85, 0.85), γ = 0.5, e = 0.5
w = (1, 2, 4), Ψ = (0.064, 0.117, 0.214), G = 0iyi
T (wi )
τi
ki
ni
0
0
0
—–
0.1
0.235
b
0
No tax
1 1.00
0
0
0.27
0.23
EU
1.367
2 2.00000.36 0.306 GDP 1.759
3 4.00000.27 0.229
0 0.249 -0.249—–
0.184 0.306b0Partial optimal taxation
1 1.228 -0.228 0.021 0.255 0.216
EU
1.391
2 1.998 0.002
0.126 0.327 0.278 GDP 1.571
3 3.374 0.626
0.219 0.235
0.2
0 0.232 -0.232—–
0.174 0.298b
0.214
Full optimal taxation
1 1.193 -0.193 0.039 0.259 0.22
EU
1.397
2 1.970.03
0.131
0.33 0.281 GDP 1.587
3 3.357 0.643
0.219 0.237 0.201

group with lower consumption could be negative. This example shows that the participation tax rate for the least productive workers is negative (τ1E = −0.047), and in-work
credits are desirable.
Next, we focus on involuntary unemployment (see Table 1 (B)). Even with no tax, thepopulation of households that participate in labor market i is diﬀerent from the populationof households that work in labor market i due to involuntary unemployment, (i.e., pi < 1).
The second part, “Partial optimal taxation,” shows the case where the government does
not employ the unemployment benefit (b = 0). In this case, the employment tax rate forthe least productive worker is positive (i.e., τ1P = 0.021). The third part, “Full optimal
taxation,” shows the case where the government employs labor income tax as well as thesearch-contingent unemployment benefit. In comparison to partial optimal taxation, theemployment tax rate for the least productive workers is higher (i.e., τ1F = 0.039), andthe search-contingent unemployment benefit is useful (b = 0.214). Full optimal taxation
improves GDP (GDP increases from 1.571 to 1.587) as well as welfare (EU increases
from 1.391 to 1.397). The search-contingent unemployment benefit reduces the uniform
income transfer for a nonworking household (−T (0) decreases from 0.249 to 0.232) so that12

a fine-tuned income tax/transfer is available.2

4.3

Sensitivity analysis: Monte Carlo simulations

To investigate the robustness of this property, we conducted a Monte Carlo simulation from10,000 samples where all the parameters (i.e., α, q, ρ, p, γ, and e) are uniformly distributed
over the specified range. Figure 1 shows the average optimal participation tax rate for eachproductivity level in the full employment case, the average optimal employment tax rate
for each productivity level for partial optimal taxation and full optimal taxation, and theaverage optimal unemployment benefit. Average optimal employment (participation) tax
rates and optimal unemployment benefits in Figure 1 have almost the same patterns as thebenchmark case. Additionally, we calculate the first and third quartiles of these cases todetermine the statistical dispersion of the result. The results show that, in many cases, theoptimal employment (participation) tax rates are similar to the benchmark case. Thesestatistics for optimal participation tax rates for the least productive working households
are negative while those of optimal employment tax rates for the least productive working
households are positive, and the statistics are higher for full optimal taxation. We also
examine the distribution of optimal employment tax rates for the least productive working
households for both partial optimal taxation and full optimal taxation, τ1P and τ1F (seeFigure 2). The distribution of employment tax rates for full optimal taxation is moredispersed and has a thicker right tail than that of partial optimal taxation.3
Next, we examine the relationships between the optimal employment tax rate of theleast productive working households and the optimal unemployment benefit with various
employment probabilities and search costs. To do so, we derive several fitted curves from10,000 samples. We first focus on the employment probabilities (See Figure 3 (A)). Whenthe employment probability is equal to unity, all households that participate in the labormarket find a job, and there is no involuntary unemployment. From equation (11) above,
the higher the p1 , the closer to zero is the third term on the right-hand side of (11), andthe optimal employment tax rate rule becomes similar in form to the optimal participation
tax rule. Figure 3 (A) indicates that this intuition is valid. If the employment probability
p1 is higher, then the optimal employment tax rate for full optimal taxation is close tozero or negative. On the other hand, parameter p1 does not seem to aﬀect the level of
unemployment benefit. Second, we examine the eﬀect of search costs (see Figure 3 (B)).
If the utility cost of job search Ψ1 is smaller, then the unemployment benefit is rapidly
decreasing and close to zero. Following this move in unemployment benefits, the optimal
employment tax rate for full optimal taxation is also decreasing. This implies a positive
correlation between unemployment benefits and the optimal employment tax rate for full
optimal taxation.
2

In this example, the diﬀerence in welfare between full optimal taxation and partial optimal taxation,
expressed in terms of public funds, amounts to 0.467% of GDP.
3
We count the sign pattern about τ1P and τ1F . In our Monte Carlo simulation, 78.5 % of the samples
show that τ1F is greater than τ1P with the same parameter settings.

13

Figure 1: Optimal average PTR, ETR, and UBPartial optimal
taxationFull employment case

Full optimal
taxation

Unemployment
benefit

PTR, ETR and UB

0.3

0.2

0.1

0.0

w1

w2

w3

w1

w2

w3

w1

w2

w3

w1 ~ w3(Notes) This figure shows the average optimal participation tax rate (PTR) for each productivity level(red line), the average optimal employment tax rate (ETR) for each productivity level (green and bluelines), and the average optimal unemployment benefit (UB) for all productivity levels (see black diamondpoint). The black lines show the range for the first and third quartiles of the optimal PTR, ETR for eachproductivity level, and UB for all productivity levels. The cross points show the medians.

Finally, we examine the relationships between the optimal employment tax rates ofvarious types of working households, uniform income transfer for nonworking households,
and the optimal unemployment benefit (see Figure 4). Note again that these policy instruments are all endogenous. Figure 4 shows that the optimal employment tax rates
for various type of working households and the optimal search-contingent unemployment
benefit have positive correlations, and the optimal uniform income transfer for nonworking
households and the optimal search-contingent unemployment benefit have negative correlations. Particularly, the slope of the fitted curve of the employment tax rate is steepest
for the least productive working households so that when the optimal unemployment benefit is small and close to zero, the optimal employment tax rates for the least productive
working households should also be small. Thus, in-work credits (negative employment tax
rates) for the least productive working households and the search-contingent unemployment benefit are substitutes. This suggests that when there are fewer search frictions in
the labor market or the employment probability is high, it is more desirable to employ
in-work credits than the search-contingent unemployment benefit. In contrast, when there
are many search frictions and households cannot easily obtain a job oﬀer, the government
should employ the search-contingent unemployment benefit rather than in-work credits as
14

Figure 2: Histogram of τ1P and τ1F
ETR of Full optimal taxation, τF1

150

ETR of Partial optimal taxation, τP1

100

50

0
−0.1

0.0

τ1

0.1

0.2

(Notes) This figure shows the histogram of the optimal employment tax rate for the lowest productivity
level.

a redistribution policy. This is because there is a need to raise incentives among households
to participate and search in the labor market.

5

Conclusion

In this study, we investigated optimal income taxation and optimal unemployment benefits
in a model with an extensive margin of labor and involuntary unemployment. In the case of
involuntary unemployment, the government can employ labor income tax/transfers as well
as a search-contingent unemployment benefit. We derive optimal employment tax rules,
which depend on the average marginal social weight, the participation elasticity, the tax
eﬀect on the participant population in labor markets, and the optimal search-contingent
unemployment benefit. Due to the existence of the search-contingent unemployment benefit, a negative employment tax rate for some working households with low consumption
is less likely to occur. In addition, the optimal search-contingent unemployment benefit
should be higher when the eﬀect of the benefit on the population of working households
is greater.
For our example economy, we simulate an optimal participation tax rate in the caseof full employment and an optimal employment tax rate in the case of involuntary unemployment with both partial optimal taxation and full optimal taxation. We calculate
the optimal participation and employment tax rate under various parameter specifications
and find that the participation tax rates for low consumption households are likely to be
negative in the case of full employment. However, this is not the case for involuntary
unemployment. We confirm that employment tax rates and the search-contingent un15

Figure 3: Fitted curve plot of τ1 and b with search related parametersτF1 , ETR of Full optimal taxation

τP1 , ETR of Partial optimal taxationAb , Unemployment benefit

B
0.4

0.20

0.15

0.3

0.10

0.2

0.05

0.1

0.00

0.0
0.7

0.8

0.9

1.0

0.00

0.05

p1

0.10

Ψ1

0.15

0.20

Figure 4: Fitted curve plot between −T (0), τiF , and b
0.3

0.2

− T (0)
τF1
τF2

0.1

τF3

0.0
0.0

0.2

0.4

0.6

0.8

b

employment benefit have a positive correlation. Only when the employment probability
is close to unity or the search cost is close to zero can negative employment tax rates
be valid. When the search cost is high for households, the role of the search-contingent
unemployment benefit is significant. The unemployment benefit that targets households
who want to work and participate in the labor market have a greater benefit than in-work
credits or wage subsidies when there are serious search frictions. These findings suggestthat the search-contingent unemployment benefit is a substitute for in-work credits.
Although we show that the search-contingent unemployment benefit improves welfare

16with involuntary unemployment, the underlying assumption is that the government cancompletely verify whether households search for a job in labor markets. Previous studies
on unemployment benefit shed light on the problem of monitoring beneficiaries (see, e.g.,
Boadway and Cuﬀ (1999) , Boadway et al. (2003), and Boone and Bovenberg (2013)).
Thus, it is important that future studies clarify the role of in-work credits and unemployment benefits in the case of incomplete monitoring of job searches.

References[1] Boadway, R. and K. Cuﬀ (1999) “Monitoring job search as an instrument For targeting transfers,” International Tax and Public Finance, 6, 317-337.
[2] Boadway, R. and K. Cuﬀ (2018) “Optimal unemployment insurance and redistribution,” Journal of Public Economic Theory, 20, 303-324.
[3] Boadway, R., Cuﬀ, K. and N. Marceau (2003) “Redistribution and employment policies with endogenous unemployment,” Journal of Public Economics, 87, 2407-2430.
[4] Boone, J, and L. Bovenberg (2004) “The optimal taxation of unskilled labor with job
search and social assistance,” Journal of Public Economics, 88, 2227-2258.
[5] Boone, J. and L. Bovenberg (2006) “Optimal welfare and in-work benefits with search
unemployment and observable abilities,” Journal of Economic Theory, 126, 165-193.
[6] Boone, J. and L. Bovenberg (2013) “Optimal taxation and welfare benefits with
monitoring of job search,” International Tax and Public Finance, 20, 268-292.
[7] Choné, P. and G. Laroque (2005) “Optimal incentives for labor force participation,”
Journal of Public Economics, 89, 395-425.
[8] Choné, P. and G. Laroque (2011) “Optimal taxation in the extensive model,” Journalof Economic Theory, 146, 425-453.
[9] Christiansen, V. (2015) “Optimal participation taxes,” Economica, 82, 595-612.
[10] Diamond, P.A. (1980) “Income taxation with fixed hours of work,” Journal of Public
Economics, 13, 101-110.
[11] Diamond, P.A. and E. Sheshinski (1995) “Economic aspects of optimal disability
benefits,” Journal of Public Economics, 57, 1-23.
[12] Hungerbühler, M., Lehmann, E., Parmentier, A. and B. Van der Linden (2006) “Optimal redistributive taxation in a search equilibrium model,” Review of Economic
Studies, 73, 743-767.

17

[13] Jacquet, L., Lehmann, E. and B. Van der Linden (2013) “Optimal redistributive
taxation with both extensive and intensive responses,” Journal of Economic Theory,
148, 1770-1805.
[14] Jacquet, L., Lehmann, E. and B. Van der Linden (2014) “Optimal income taxation
with Kalai wage bargaining and endogenous participation,” Social Choice and Welfare, 42, 381-402.
[15] Laroque, G. (2011) “On income and wealth taxation in a life-cycle model with extensive labour supply,” Economic Journal, 121, F144-F161.
[16] Lehmann, E., Parmentier, A. and B. Van der Linden (2011) “Optimal income taxation with endogenous participation and search unemployment,” Journal of Public
Economics, 95, 1523-1537.
[17] Saez, E. (2002) “Optimal income transfer programs: Intensive versus extensive labor
supply responses,” Quarterly Journal of Economics, 117 (3), 1039-1073.

A

Appendix

A.1

Derivation of equation (7)The first-order conditions of the Lagrangian with respect to T (wi ) are
[
]
∂EU P
∂ki
∂n0
P
+λ
pi T (wi ) + pi ki +T (0) = 0.
∂T (wi )∂T (wi )∂T (wi )Note that n0 = 1 −∑n

i=1 ni = 1 −∑ni=1 pi ki . We rewrite the first-order conditions using∂ki∂n0∂T (wi ) = −pi ∂T (wi ) as follows:

[
]
∂EU P
∂ki
∂ki
P
+λ
pi T (wi ) + pi ki − pi
T (0) = 0.
∂T (wi )∂T (wi )∂T (wi )∂ki wi −T (wi )+T (0)∂EU
P
Dividing both sides by λP pi ki and using giP = − λP 1p k ∂T,(wi ) and ηi = − ki∂T (wi )P

i i

we rearrange the above equation as follows:
1 − giP = −ηiP

T (wi ) − T (0).
wi − T (wi ) + T (0))−T (0)
Finally, using τiP = T (wiw
, we obtain the following equation,iτiP
1 − giP=.
1 − τiP
ηiP
This equation corresponds to equation (7).

18

A.2

Derivation of equation (9)The Lagrangian of the government problem in the case of full employment is
L = EU + λ
E

E

E

[ I
∑

]
ni T (wi ) + n0 T (0) − G ,i=1

where λE is the Lagrange multiplier associated with the government budget constraint in
the case of full employment. The first-order conditions of the Lagrangian with respect to
T (wi ) and T (0) are
[
]
∂LE
∂EU E
∂ni
∂n0
E
=
+λ
T (wi ) + ni +T (0) = 0, for i ̸= 0,∂T (wi )∂T (wi )∂T (wi )∂T (wi )
[ I
]
∑ ∂ni
∂LE
∂EU E
∂n0
E
=
+λ
T (wi ) + n0 +T (0) = 0.
∂T (0)∂T (0)∂T (0)∂T (0)i=1

i
We rewrite the first-order conditions with respect to T (wi ) using ∂T∂n(w0 i ) = − ∂T∂n(wi ) asfollows:
[
]
∂ni
∂ni
∂EU E
E
+λ
T (wi ) + ni −T (0) = 0.
∂T (wi )∂T (wi )∂T (wi )∂ni wi −T (wi )+T (0)∂EUE
Dividing both sides by λE ni and using giE = − λE1n ∂T,(wi ) and ηi = − ni∂T (wi )E

iwe rearrange the above equation as follows:
1 − giE = −ηiE

T (wi ) − T (0).
wi − T (wi ) + T (0))−T (0)
Finally, using τiE = T (wiw
, we obtain equation (9).
i

A.3

Derivation of equation (11)The first-order conditions of the Lagrangian with respect to T (wi ) in the full optimal
taxation are
[
]
∂ki
∂EU F
∂n0
F
+λ
[pi T (wi ) − [1 − pi ]b] + pi ki +T (0) = 0.
∂T (wi )∂T (wi )∂T (wi )i
We rewrite the first-order conditions with respect to T (wi ) using ∂T∂n(w0 i ) = −pi ∂T∂k(wi ) asfollows:
[
]
∂EU F
∂ki
∂ki
F
+λ
[pi T (wi ) − [1 − pi ]b] + pi ki − pi
T (0) = 0.
∂T (wi )∂T (wi )∂T (wi )19∂ki wi −T (wi )+T (0)∂EU
F,
Dividing both sides by λF pi ki and using giF = − λF 1p k ∂T(wi ) and ηi = − ki∂T (wi )F

i i

we rearrange the above equation as follows:
1 − giF −1∂ki
T (wi ) − T (0)[1 − pi ]b = −ηiF.
pi ki ∂T (wi )wi − T (wi ) + T (0))−T (0)
Finally, using τiF = T (wiw
, we obtain the following equation,
i

[
]
1
1
τiF
∂ki
F
= F 1 − gi −[1 − pi ]b .
pi ki ∂T (wi )1 − τiF
ηiThis equation corresponds to equation (11).

A.4

Derivation of equation (12)The first-order condition of the Lagrangian with respect to b for full optimal taxation is
[ I [
]
]
∑ ∂ki
∂EU F
∂n0
F
+λ
[pi T (wi ) − [1 − pi ]b] − ki [1 − pi ] +T (0) = 0.
∂b
∂b
∂b
i=1

0
Noting that ∂n∂b = −∑I∂ki
i=1 pi ∂b , we rewrite the first-order condition as follows:

[ I
]
I
I
∑ ∂ki
∑
∑
∂EU F
∂ki
F
+λ
[pi T (wi ) − [1 − pi ]b] −T (0) = 0.
ki [1 − pi ] −
pi
∂b
∂b
∂b
i=1

i=1

Dividing both sides by λF∑I

i=1 ki [1 − pi ], we rearrange the above equation as follows:

∂EU F
−1=−
∑
λF Ii=1 ki [1 − pi ] ∂b
1Finally, using gbF =equation,

λF

i=1

∑I 1
i=1

∂EU F
ki [1−pi ] ∂b

b∗ =

∑
∂ki
i
b Ii=1 ∂k[1 − pi ]i=1 ∂b pi [T (wi ) − T (0)]+ ∑I ∂b.
∑I
i=1 ki [1 − pi ]i=1 ki [1 − pi ]∑Iand hF =gbF − 1 +∑I∂ki [1−p ]i∂b∑i=1, we obtain the following
I
k
[1−p
]
i
i
i=1

∑I

∂ki
i=1 ∂b pi [T (wi )−T (0)]∑Ii=1 ki [1−pi ]hF

This equation corresponds to equation (12).

20.