Workers’ Behavior and Job Quality : howcan Taxation alleviate a Holdup Problem?∗

Guillaume Wilemme

Sciences Po, Paris

November 2013

PRELIMINARY

Abstract

This paper investigates the strategy of job seekers in a segmented labor mar-ket with search frictions. Two inefficiencies of the decentralized equilibrium arisefrom the entry decision of firms and the workers’ behavior in the laissez-faire econ-omy. The social optimum can be achieved with appropriate tax instruments. Aproportional wage tax enables us to achieve the Hosios-Pissarides condition, mean-ing an efficient job creation. The second source of inefficiency is a form of holdupproblem. Workers incur the entire cost of finding more productive jobs, but theyhave to share the benefit with the recruiting firms because of the Nash bargain ofwages. A proportional tax on production alleviates this imperfection by makingjob seekers more selective. The optimal tax system is regressive : it is efficient toredistribute from low-productive to high-productive jobs in order to give workersthe right incentives.

JEL Code: H21, H23, J24, J38, J64

Keywords: holdup, job quality, optimal taxation, search strategy, segmentedlabor market

∗This paper stems from my master degree dissertation at ENSAE ParisTech. Etienne Lehmannsupervised me throughout this period and after. I cannot be thankful enough for his patience and hisguidance. I am also grateful to Alexandre De Cornière and Etienne Wasmer for their helpful comments.

1 Introduction

Search and matching models with bargained wage exhibit market failures because an

agent does not internalize her impact on the meeting probabilities of the others in general

(Diamond, 1982). Traditionally, the search e�ort is supposed exogenous or undirected

towards di�erent types of jobs at best. Under this assumption and assuming a constant

return-to-scale matching technology with identical workers and �rms, the constrained

Pareto optimum can be decentralized through a particular bargaining rule (Hosios, 1990;

Pissarides, 2000) or through a proportional labor income taxation scheme when the bar-

gaining power departs from the Hosios-Pissarides condition (Boone and Bovenberg, 2002).

In this paper, I relax the �rst hypothesis and allow workers to modulate their search ef-

forts according to the prospected segment of the labor market. The search strategy has

now an impact on the match quality and so on the job creation and the aggregate produc-

tion. This paper aims at examining this new source of ine�ciency and at characterizing

the optimal policy. I provide an argument for a regressive taxation, abstracting from the

inequality concern and the labor participation concern.

I model a labor market with search and matching in the Diamond-Mortensen-Pissarides

tradition with horizontal heterogeneity à la Salop (Marimon and Zilibotti, 1999; Decreuse,

2008). This provides a tractable framework for considering di�erentiation in match qual-

ity and mismatch. Workers are unequally productive according to the type of employer.

Contrary to a standard stochastic job matchings model, they observe productivity before

meeting a �rm. Hence, job seekers can adjust the search e�ort for each type of prospected

jobs, possibly zero in some cases. Even though productivity is observed, they have an

incentive to search for less productive jobs because this increases the overall probability

of leaving unemployment. The entry decision of �rms leads to an ine�cient quantity of

jobs created in general (Pissarides, 2000). One more vacancy on the labor market has two

e�ects : �nding a partner becomes more likely for workers and less likely for the others

employers. An optimal level of job creation is achieved when workers and �rms are paid

at their marginal contribution. The search behavior of workers leads to an ine�cient

quality of jobs created. Job seekers are not enough selective given the entry decision of

�rms. Whereas the �rst imperfection disappears when the parameters satisfy the Hosios-

Pissarides condition, the second one cannot be eliminated by this type of constraint.

Therefore, the decentralized equilibrium is never optimal in the absence of taxation.

My �rst contribution is to identify the source of the ine�cient search behavior to

a holdup problem, as �rst described by Grout (1984). The choice of a search strategy

is formally equivalent to an investment decision made by workers and held up by �rms

because of the Nash bargain. Search frictions in the labor market lead to a bilateral

2

monopoly situation when an employer and an employee meet. The surplus of a match

is split according to a Nash bargain. Consequently, the bene�t from �nding better jobs

is shared with the �rms. The holdup arises because the cost of this type of investment

is sunk before any meeting, then it is fully incurred by workers. Being more selective is

costly because it tends to reduce the overall probability to �nd a job and to increase the

total search cost according to a triangular trade-o�.

My second contribution consists in exhibiting a simple optimal �scal scheme. On one

hand, a linear taxation on wages makes the participation decision of �rms optimal given

the workers' search behavior. When the tax is progressive, a wage increase implies a higher

payment to the �scal authority. Workers are then induced to bargain less aggressively

their wage, the e�ective worker's share of the surplus reduces (Pissarides, 1985, 1998;

Lockwood and Manning, 1993). Thus, the bargaining rule can be adjusted to ful�ll the

Hosios-Pissarides condition. On the other hand, the search strategy is made e�cient

by redistributing from low-productive matches to high-productive matches, so that job

seekers have an incentive to be more selective. More precisely, the net-of-tax wage must be

proportional to productivity at a rate equal to one. Workers and social planner are then

reduced to solve the same optimization problem when choosing this variable : maximizing

the value of unemployment. In absence of taxation, the slope of the wage equation as

a function of productivity is equal to the bargaining power of workers. The role of the

e�cient tax system is to disentangle these two properties of the bargaining power : the

worker's share of the surplus and the slope of the wage equation. A proportional tax on

production is a well-suited instrument for this purpose in complement of a labor income

tax.

I also investigate other self-�nancing policies. A subsidy for job creation can only

handle the job creation ine�ciency. Unemployment bene�ts a�ects the search strategy of

job seekers but not their selectivity between di�erent jobs. By studying a general income

taxation setting, I show that the wage tax must be linear and that the policy maker needs

a tax on production to decentralize the social optimum.

I extend the model to alternative search technologies. Very few speci�cation is nec-

essary about the impact of the workers' behavior on the quality of jobs, the probability

to match or the search cost. The solution may not be interior and the objective function

may not be di�erentiable.

The optimal taxation scheme I exhibit modi�es the matching strategy. In a random

search framework, the wage tax for job creation to be e�cient (i.e. the Hosios-Pissarides

condition) also impacts on the matching decision. Thus, a tax on production of opposite

sign is needed to make the selectivity of job seekers unchanged even in this case.

Lastly, the optimal policy can also be implemented for other holdup problems. When

3

�rms' investment are made before hiring workers, the holdup is reversed. Restoring

e�ciency needs now to make the pro�t function proportional to productivity with a

coe�cient equal to 1.

Literature Review. The motivation of this paper relies on the existence of market

ine�ciencies in presence of search frictions and heterogeneity. Shimer and Smith (2001)

deal with heterogeneous agents who choose a search intensity and a matching strategy.

Because of this heterogeneity, it is not possible to compensate the positive and the nega-

tive externalities induced by the search e�orts. The high-productive agents do not search

enough contrary to the low-productive who search too much. Decreuse (2008) �nds the

same kind of result in a model where agents can choose to which labor market segments

to participate. The more the visited locations are, the more the probability to �nd a job

but the lower the expected quality of a job. At equilibrium, workers apply to too many

sub-markets. The decrease in the quality of jobs reduces the entry of �rms. Workers do

not internalize this composition externality.

I identify one of the two sources of ine�ciency as a form of holdup. The optimal policy

I study can be used in others articles. Acemoglu and Shimer (1999) proposes a model with

�rms' capital investment. When the investment is exogenous, the equilibrium is socially

optimal under the Hosios-Pissarides condition. The ine�ciency due to the holdup arises in

the other case and the Hosios-Pissarides condition is not su�cient anymore. If wages are

posted rather than Nash-bargained and if workers can direct their search, the equilibrium

turns out to be e�cient. Davis (2001) studies investments before matching in both side

of the market. In particular, he looks at the e�cient bargaining that decentralizes the

social optimum according to which side of the market incurs the holdup. He highlights

the di�erent roles of the bargaining power impacting on e�ciency.

I suggest a public policy that alleviates the ine�ciencies of the private agents' equilib-

rium. Boone and Bovenberg (2002) analyzes the optimal �nancing of public expenditures

with a labor income taxation. Contrary to us, they only account for the job creation inef-

�ciency. Jobs do not vary on productivity, so their model is absent of matching strategy

or choosy search strategy. About optimal taxation on the labor market, Hungerbühler,

Lehmann, Parmentier and Van der Linden (2006) consider a labor market segmented by

job's type. A worker is assigned to a speci�c type and cannot search for jobs of other

types. In this paper, I precisely focus on the consequence of this last hypothesis.

4

2 The search and matching framework

2.1 The plot

I introduce a search and matching model with two-sided heterogeneity. The frame-

work comes from Decreuse (2008). The entire labor market is represented as a circle of

perimeter 1, where any location is a sector characterized by a labor demand and supply.

A measure 1 of in�nitely-lived and risk-neutral workers is uniformly distributed on this

circle. On the contrary, �rms are endogenously distributed. They can have at most one

employee. Workers are more or less productive according to the job sector. There is no

vertical heterogeneity, i.e. no agent is more skilled or uniformly more productive than

another. Workers di�er just in the sectors in which they are productive. Heterogeneity

comes from the match output y(x). It only depends on the distance x between the �rm

and the worker on the circle. The function y is de�ned from [0, 12] into R+. It is di�eren-

tiable and strictly decreasing : workers are less productive when matched with a further

�rm. The match output is also assumed to be nil when the individual and the recruiting

�rm are opposite on the circle : y(12) = 0.

The search strategy of workers consists in choosing a search intensity for each segment

of the labor market. Searching is costly. Exerting an e�ort ξ0 at a distance x from the

worker's location costs Cx(ξ0). A same e�ort may imply di�erent costs according to the

distance x. This accounts for potential abilities of workers to match more easily with

nearby employers for instance. Functions Cx are increasing, convex and twice di�eren-

tiable. The search cost and the marginal search cost are nil for a zero search intensity :

Cx(0) = C ′x(0) = 0. For the sake of simplicity, we assume the symmetric and steady

state. Thus, any unemployed worker chooses the same strategy, an e�ort ξ(x) for each

distance x. The total search e�ort ξT and the total search cost CT (ξ) are de�ned by :

ξT = 2

∫ 12

0

ξ(x)dx CT (ξ) = 2

∫ 12

0

Cx(ξ(x))dx

As a consequence of the symmetry hypothesis, �rms are uniformly distributed.1 A

�rm holding an open vacancy incurs a �ow cost k. Let v to be the measure of vacancies

and u of unemployed workers per location. The meeting technology is supposed to be

identical on each sub-market : M(ξTu, v) is the measure of meetings at a location per

unit of time. For simplicity, This function is a Cobb-Douglas form so the elasticity of the

meeting function relative to the measure of employers η is constant. Assuming constant

returns to scale, this function is fully described by a function m depending on the market

1See Decreuse (2008).

5

tightness θ, i.e. the ratio of the mass of employers to the mass of e�cient job seekers.

θ =v

ξTum(θ) = M(1,

1

θ)

The function m is di�erentiable, strictly increasing and concave from R+ onto R+ :

η = θm′(θ)m(θ)

belongs to [0, 1]. A worker exerting an e�ort ξ(x) for a distance x meets a

�rm at this distance at the Poisson rate 2ξ(x)m(θ). To have equality between the �ow

of workers drawing a meeting with a �rm at distance x and the �ow of �rms drawing a

meeting with a worker at a distance x, the �ow probability for an employer to meet an

unemployed at distance x must be 2ξ(x)ξT

m(θ)θ

given that the search strategy is the same

for each worker. As a consequence, 2ξ(x)ξT

is the measure of meetings characterized by a

distance x and m(θ)θ≡ q(θ) is the rate at which a vacancy is �lled.

The optimal behavior of workers imply that any meeting leads to matching. Indeed,

a worker knows for which meeting matching is rejected by one of the two parties. She

will never exert a strictly positive search e�ort for these meetings. In other words, the

decision to match can be omitted. This is a main point of the choosy search framework

(see Decreuse, 2008). Thus, one can de�ne the expected value of a variable A over the

newly created matches :

A(ξ) =

∫ 12

0

A(x)2ξ(x)

ξTdx

Matches face an exogenous destruction rate s, independent of the job productivity.

There is no on-the-job search. Lastly, time is discounted at a rate r.

2.2 The social optimum

We solve for the inter-temporal problem of the social planner. The unemployment rate

u and the gross output Y are two state variables. For a small time period ∆t, a fraction

s∆t of employed workers loose their job and a fraction ξTm(θ)∆t of unemployed workers

are hired. Thus, the unemployment rate obeys the law of motion: u = s(1−u)−ξTm(θ)u.

To understand the dynamic of the gross output, we can focus on a change from ξ to ξ′.

The distribution of job productivity does not adjust instantaneously. Matches break

up at a rate s and are replaced by matches with mean productivity y(ξ′). The arrival

rate of these newly hired workers is ξ′Tm(θ)u. The law of motion of the gross output

is then : Y = ξTm(θ)uy (ξ) − sY . As agents are risk-neutral, the objective function is

the production Y net of the cost of vacancies kξT θu and of the cost of search activities

CT (ξ)u. The planner's problem is :

maxθ,ξ

∫ ∞0

[Y − kξT θu− CT (ξ)u

]e−rtdt s.t.

{Y = ξTm(θ)uy(ξ)− sYu = s (1− u)− ξTm(θ)u

6

Let H to be the current value Hamiltonian with λ and ψ the multipliers of the two

constraints :

H = Y − kξT θu− CT (ξ)u+ λ[ξTm(θ)uy(ξ)− sY

]+ ψ

[s (1− u)− ξTm(θ)u

]We are interested in a steady-state solution. The costate equation associated to the

dynamic of production (∂H/∂Y = rλ) determines the shadow value of a unit produced

by a job, depending on the discount rate and on the job destruction rate :

λ =1

r + s

We will systematically replace λ by its expression for the next developments. The second

costate equation (∂H/∂u = rψ) de�ned the shadow value of an unemployed worker :

(r + s)ψ = ξTm(θ)

(y(ξ)

r + s− ψ

)− kξT θ − CT (ξ) ⇔ ψ = Ψo(θ, ξ) (1)

At a rate ξTm(θ), an unemployed is matched and the match produces a present-

discounted value y(ξ)/(r + s). This variable ψ is crucial in this model, it corresponds

to the opportunity cost of matching today rather than waiting for another job, possibly

better.

Notice that the Hamiltonian is linear in the unemployment rate, it can be written :

H = γ + ∂H∂uu. Since γ does not depend on θ and ξ, the planner's problem is equivalent

to maximizing the shadow value ψ thanks to the second costate equation :2

maxθ,ξ

{ξTm(θ)

(y(ξ)

r + s− ψ

)− kξT θ − CT (ξ)

}The �rst-order condition relative to the market tightness is :

k

q(θ)= η

(y(ξ)

r + s− ψ

)⇔ θ = Θo (ξ, ψ) (2)

An increase in θ by 1 percent makes �rms wait longer before meeting a worker, leading

to a higher expected vacancy cost ("congestion externality"). The cost increases by

the expected cost of a vacancy times its elasticity with respect to the market tightness

times the number of jobs created per in�nitesimal period at each location : kq(θ)× (1 −

η) × ξTm(θ). On the other hand, the number of matches is greater because workers

match more rapidly ("thick-market externality"). The gains in terms of output writes(y(ξ)r+s− ψ − k

q(θ)

)× ηξTm(θ) : the net production per job times the number of additional

jobs per period, at each location. The optimal market tightness in equation (2) is such

that these negative and positive externalities balance each other at the margin.

2This results is emphasized by Acemoglu and Shimer (1999).

7

The search intensity satis�es the following �rst-order condition :{ξ(x) = 0 if y(x)

r+s− ψ − k

q(θ)≤ 0

C ′x(ξ(x)) = m(θ)(y(x)r+s− ψ − k

q(θ)

)else

}⇔ ξ = Ξo (θ, ψ) (3)

Jobs whose production is not high enough to compensate the value of an unemployed

and the expected cost of the vacancy are not prospected, thus the search e�ort is zero

beyond a threshold distance. Because y(1/2) = 0, this threshold is strictly inferior to 1/2.

When the planner makes job seekers exert a strictly positive e�ort, the search intensity

equalizes the marginal cost and the marginal gain of searching. The more productive is

a match, the more intensive is the search e�ort, meaning ξ(x) is decreasing in x.

De�nition 1 An e�cient allocation is described by a value of an unemployed ψ, a market

tightness θ and a search strategy ξ ful�lling (1), (2) and (3).

The existence of an e�cient allocation is proved as a particular case of a decentralized

equilibrium (see below).

2.3 Private agents' behavior

In this subsection, I consider the equilibrium of private agents taking as given the

(net-of-tax) wage pro�le. The problem can be formulated in terms of Bellman equations.

Let W (x) to be the present-discounted value of a match for a worker, characterized by

a distance x. U is the asset value of unemployment. An individual matches with a �rm

at the rate ξTm(θ), then she expects a capital gain from getting employed W (ξ) − U .

In any case, the job seeker incurs the total search cost CT (ξ). Therefore the return to

unemployment writes :

rU = ξTm(θ)(W (ξ)− U

)− CT (ξ) (4)

When working at a distance x, an employee earns the net-of-tax wage w(x). She may

return to the pool of the unemployed if her match breaks, this occurs at a rate s. The

asset value of a match satis�es :

rW (x) = w(x) + s (U −W (x)) (5)

Equations (4) and (5) provide an expression of the return to unemployment :

rU = ξTm(θ)

(w(ξ)− rUr + s

)− CT (ξ) (6)

8

An unemployed who �nds a job earns a �ow expected wage w(ξ) but renounces to

the �ow return to search rU . This return is lost for the duration of the match. As

time is discounted, the present-discounted cost from leaving unemployment corresponds

to ψ = rUr+s

. We denote J(x) the asset value of employing a worker far from a distance

x, its expectation before meeting a worker is J(ξ). Assuming the free-entry condition for

�rms at any time, there is equality between the expected cost of holding a vacancy and

the expected value of �lling a vacancy :

k

q(θ)= J(ξ) (7)

A match yields a net �ow pro�t π(x) to the employer, with a risk to be broken at a

rate s :

rJ(x) = π (x)− sJ(x) (8)

Assume there is no taxation or, if there is, it does not directly impact on the expected

pro�t : π(ξ) = y(ξ) − w(ξ). From equations (7) and (8), the total expected cost of a

vacancy is equal to the present-discounted value of pro�ts :

k

q(θ)=y(ξ)− w(ξ)

r + s(9)

This equation can be interpreted as a decreasing relation between the market tightness

and the expected wage. The more the expected wage is, the less the �ow pro�ts are and

so less �rms decide to open job vacancies. Combining equations (6) and (9),

(r + s)ψ = ξTm(θ)

(y(ξ)

r + s− ψ

)− kξT θ − CT (ξ)⇔ ψ = Ψe(θ, ξ) (10)

The term in big parentheses is the expected surplus of a match when an employee

and an employer meet. The two negative terms in the right-hand side are the �ow cost

of vacancies per unemployed and the search cost. They are sunk before any meeting.

This equation de�nes how private agents determine the value of unemployment given the

search behaviors and the market tightness.

Searching with the strategy ξ must be the optimal choice of job seekers, from equation

(6), the search strategy is solution to :

maxξ

{ξTm(θ)

(w(ξ)

r + s− ψ

)− CT (ξ)

}The search e�ort then follows :{

ξ(x) = 0 if w(x)r+s− ψ ≤ 0 or π(x) < 0

C ′x(ξ(x)) = m(θ)(w(x)r+s− ψ

)else

(11)

9

For too low-productive matches, the job seekers do not exert any search e�ort. These

are jobs whose discounted labor income w(x)r+s

is below the value of unemployment ψ. If

productivity is high enough, the search intensity is such that the marginal cost and the

marginal bene�t equalize. If the pro�t is nonpositive, the �rm refuses to match so workers

rationally do not apply.

A value of an unemployed ψ, a market tightness θ and a search strategy ξ ful�lling

equations (9), (10) and (11) de�ned an equilibrium with an exogenous wage pro�le. A

wage pro�le makes the decentralized equilibrium e�cient if and only if it satis�es :

y(ξ)− w(ξ)

r + s= η(

y(ξ)

r + s− ψ) (12)

y(x)

r + s=

k

q(θ)+w(x)

r + s(13)

According to the �rst equation, the mean wage must be such that �rms obtain on

expectation a share η of the surplus of a match. The second equation states that produc-

tion must compensate exactly the expected vacancy cost and the labor income, for each

match. This equation is always satis�ed on expectation, this is the free-entry condition.

If this constraint were not ful�lled, there would be ex-ante ine�cient matches, meaning

matches whose output is below the expected vacancy cost plus the labor income.

Proposition 1 A decentralized equilibrium can achieve an e�cient allocation (ψ∗, θ∗,

ξ∗) if the wage pro�le is such that, for any distance x, w(x) = y(x) − κ0 with κ0 =

η(y(ξ∗)− (r + s)ψ∗) > 0.

There is as many e�cient wage pro�les as social optima. In any case, the wage equation

must be linear in productivity with a slope equal to one. This statement mimics a basic

result in microeconomics : e�ciency needs equality between net productivity and wage.

Here, net productivity incorporates the initial vacancy cost. However, there is no reason

for such a wage pro�le to emerge at equilibrium.

2.4 The wage setting with taxation

Assume wage is Nash-bargained with a constant bargaining power of workers β with

0 ≤ β < 1. Production and labor income are taxed at rates τy and τw. For each job,

the �scal authority deducts τyy(x) + τww(x) and repays τyy(ξ) + τww(ξ) so the tax is

self-�nancing. The net �ow pro�t writes :

π(x) = (1− τy)y(x)− (1 + τw)w(x) + τyy(ξ) + τww(ξ) (14)

10

Taxation is neutral on expectation : π(ξ) = y(ξ) − w(ξ). If the tax rates are positive,

jobs whose productivity and wage are lower than their mean values receive a subsidy,

whereas the productive and high-paying jobs are taxed. The surplus of a match de�ned

by a distance x is S(x) = J(x) +W (x)− U . The problem is the following :

maxw(x)

(W (x)− U)β J(x)1−β

The after-tax wage solves the �rst-order equation :

W (x)− U = βS(x) (15)

where β, the worker's share of the expected surplus, satis�es

β =β

1 + τw − βτw

A nonzero tax rate on wages makes the surplus of a match depend on the wage. For

a positive rate, the match incurs a net loss. Any increase in the wage implies paying

more taxes. Therefore, both agents have an incentive to bargain a lower wage. This

is equivalent to a decline in the worker's share of the surplus, thus β is decreasing in

τw.3 The amount of the tax on production is the same whatever the wage, hence it does

not in�uence the surplus sharing and so, the e�ective bargaining power. From equation

(15), it can be derived W (ξ) − U = βS(ξ). This implies that the expected discounted

wage is a weighted average of the expected discounted productivity and the value to be

unemployed :

w(ξ)

r + s= β

y(ξ)

r + s+ (1− β)ψ (16)

Equation (15) leads also to W (x)− W (ξ) = β(S(x)− S(ξ)). After some algebra, we

�nd :

w(x)− w(ξ) =◦β(y(x)− y(ξ)) (17)

where◦β, the slope of the wage curve, ful�lls

◦β =

β(1− τy)1 + τw

A steep wage curve induces a large gap between the high-productive and the low-

productive jobs. The more progressive is the taxation (i.e. the higher τy and τw), the

�atter is the wage curve. In absence of taxation, the bargaining power β plays several

roles. One is the worker's share of the surplus β. The other one is the slope of the

3See Pissarides (1985, 1998) and Boone and Bovenberg (2002).

11

wage equation when de�ning the wage as a function of productivity◦β. Both matters

in terms of e�ciency. The tax in�uences two quantities in the surplus-sharing equation

(15) : the e�ective bargaining power and the surplus of a match. Because any tax levy

is redistributed, the impact of the taxation is neutral on the expected surplus : only the

worker's share of the surplus β is involved in (16). This is not true anymore at a distance

x : the two e�ects combine in (17). The success in decentralizing the social optimum is

due to the disentanglement of these two aspects, thanks to the �scal tools.

Because the taxation is neutral on expectation, the value of an unemployed for private

agents (i.e. function Ψe) is not impacted in equation (10). By incorporating the surplus

sharing rule in the free-entry condition (9), we de�ne the equilibrium market tightness

given the search strategy and the value of an unemployed :

k

q(θ)=(

1− β)( y(ξ)

r + s− ψ

)⇔ θ = Θe

(ξ, ψ, β

)(18)

Firms decide to open new vacancies until the total expected cost of the vacancy equals

their share of the expected surplus of a match. The higher their share (i.e. (1− β)), the

higher the number of vacancy created. From substituting the expected wage in equation

(17) thanks to the free-entry condition (9), it derives the search e�ort :ξ(x) = 0 if

◦βy(x)+(1−

◦β)y(ξ)

r+s− ψ − k

q(θ)≤ 0

C ′x(ξ(x)) = m(θ)

( ◦βy(x)+(1−

◦β)y(ξ)

r+s− ψ − k

q(θ)

)else

⇔ ξ ∈ Ξe(θ, ψ,

◦β)

(19)

There may be several search strategies that satis�es this condition, Ξe(θ, ψ,

◦β)de-

notes the set of possible search e�ort functions ξ.4

De�nition 2 A decentralized equilibrium is described by a value of unemployment ψ, a

market tightness θ and a search strategy ξ such that equations (10), (18) and (19) are

ful�lled, given the tax parameters τy and τw.

In appendix, I show that an equilibrium exists if τw > −1.

4Because of the surplus sharing, both agents agree on the decision to match or not. Consequently,

the condition π(x) < 0 is integrated in◦βy(x)+(1−

◦β)y(ξ)

r+s − ψ − kq(θ) ≤ 0.

12

Proposition 2 An equilibrium satis�es the following properties :

1. The search intensity ξ(x) is continuous and decreasing in the distance x.

2. If τy1−τy + β τw

1+τw≤ 0, workers do not visit some labor markets, meaning there exists

σ such that ξ(x) = 0 for x > σ.

Job seekers search more intensively for high-productive jobs. The condition τy1−τy +

β τw1+τw

≤ 0 is equivalent to β ≤◦β.5 In words, if the taxation is too progressive, job seekers

exerts a nonnegative search e�ort everywhere. The laissez-faire equilibrium (τw = τy = 0)

is a particular case.

3 E�ciency and the role of taxation

3.1 The two sources of ine�ciency

The parameters β and◦β impact on the equilibrium through two di�erent channels.

They correspond each one to a source of ine�ciency.

Proposition 3 We have the following properties :

1. Ψe(θ, ξ) = Ψo(θ, ξ).

2. Θe(ξ, ψ, β) ≤ Θo(ξ, ψ) if and only if 1− β ≤ η.

3. Given θ and ψ, we denote ξe ∈ Ξe(θ, ψ,◦β) and ξo = Ξo(θ, ψ).

• When◦β ≤ 1, ξe(x) ≥ ξo(x) if and only if y(ξe) ≥ y(x). In particular, y(ξe) ≤

y(ξo).

• When◦β ≥ 1, ξe(x) ≥ ξo(x) if and only if y(ξe) ≤ y(x). In particular, y(ξe) ≥

y(ξo).

Given a market tightness and a search strategy, private agents and the social planner

value unemployment the same way meaning Ψe = Ψo. The depart from optimality arises

when the two other equilibrium variables are determined.

On the one hand, �rms create too few vacancies when their share of the surplus is below

the elasticity of the meeting function and reciprocally (when taking the search strategy

and the value of unemployment as exogenous). This source of ine�ciency concerns the

5The proof of the second statement of Proposition 2 is straightforward from (18) and (19). Thiscondition is not necessary.

13

quantity of jobs created. When the worker's share of the surplus β increases, �rms obtain

less pro�ts in �ows taking as given the search strategy ξ and the value of an unemployed

ψ. They open less job vacancies, so the market tightness Θe(ξ, ψ, β) decreases.

On the other hand, the search strategy function chosen by workers is always less steep

than the e�cient one when◦β ≤ 1, given a value of unemployment and a market tightness.

In this case, workers tend to search too much on far locations and too little where they are

productive. As a consequence, the expected productivity is smaller than what a planner

would choose, the quality of jobs created is not optimal. The e�ect of a change in◦β on

Ξe(θ, ψ,◦β) is ambiguous because of a triangular trade-o�. Workers face three options :

increasing the expected productivity y(ξ) (and so the expected wage), increasing the total

search e�ort ξT (and so the probability to �nd a job) and reducing the total search cost

CT (ξ). They cannot complete two of these actions without countering the third one. The

speci�cation of the search cost function and the output function matters to study such a

change in the slope of the wage curve◦β.

We are not able to compare the equilibrium values to the e�cient ones, nevertheless

this exercise would have been meaningless as uniqueness is not guaranteed (both for the

decentralized equilibrium and the social optimum).

Proposition 4 A decentralized equilibrium is e�cient if and only if :

β = 1− η◦β = 1

In particular, the laissez-faire equilibrium is ine�cient.

The decentralized equilibrium is not optimal if these two conditions are not ful�lled.6

The laissez-faire equilibrium is then ine�cient for any value of the bargaining power.

Firstly, β = 1 − η is exactly the Hosios-Pissarides condition. It states how to share

the surplus of a match according to the ability of each side of the market to create jobs,

depending on the meeting function. When the elasticity of the matching function η is

high, �rms are more e�cient than workers in job creation. The congestion externality

(negative) they have on the other �rms, which want to �ll their vacancy too, is lower

and the thick-market (positive) externality on job seekers is higher. Symmetrically, when

η is low, job seekers produce better externalities. Because the meeting technology has

constant returns to scale, it is possible to �nd a value of the e�ective bargaining power

that makes the externalities to compensate in both side of the market.

6This proves also the existence of a social optimum.

14

Secondly,◦β = 1 is a common condition to holdup problems. Because of the Nash

bargain, a worker does not fully bene�t from an increase in her productivity as long as

β < 1 in absence of taxation. When comparing two jobs with productivity y and y′

(< y), workers compare w(y) and w(y′). The problem arises because w(y) − w(y′) =

β(y − y′) < y − y′. Facing a trade-o� between job quality and individual cost, workers

and social planner do not make the same decision. The job seekers put less weight on

the gains in productivity leading to an ine�cient search strategy. Since workers exert

search e�orts before meeting a �rm and since it is costly and it impacts on the expected

productivities, there is a form of holdup. Workers are forced to share the bene�ts of their

strategy through the wage bargaining. Referring to the triangular trade-o� commented

above, being selective in job productivity is costly through a direct search cost or an

opportunity cost to reduce the job-�nding rate. In any case, workers fully incurs it.

Figure 1 illustrates the ine�ciency of the laissez-faire equilibrium. For some low-

productive jobs, the output is not high enough to compensate the vacancy cost (the

colored area), hence the suboptimality of the equilibrium. For these jobs, it is optimal

for �rms to accept to match anyway because the vacancy cost is already sunk and �nding

another partner is costly. The free-entry in the demand side implies that only the expected

vacancy cost is compensated. The only way for each match to compensate the vacancy

cost is to make the wage curve parallel to the �rst bisector, i.e. its slope equal to 1. The

holdup disappears because workers fully bene�t from a change in his productivity. This

is shown in �gure 2. With taxation, the slope of the wage curve is◦β.

3.2 Optimal taxation

The optimality conditions on the parameters β and◦β de�nes the optimal tax rates.

Proposition 5 A decentralized equilibrium with taxation is e�cient if and only if the

tax parameters satisfy :

τ ∗w = − 1− β − η(1− β)(1− η)

(20)

τ ∗y = 1− η

(1− β)(1− η)(21)

The lump-sum component of the taxation is always negative :

τ ∗y y(ξ) + τ ∗ww(ξ) < 0

Notice that τ ∗w is positive if and only if 1 − β ≤ η. In this case, the bargaining power

of workers is too high. Setting a positive wage tax reduces the e�ective bargaining power

β, which can be adjusted to its optimal value 1− η. The wage tax is the only tool that

15

Labor income, slope: °

Output, slope 1

k

q( )

y( ) -

r+s

y( ) -

r+s

w( ) -

r+s

y(0) r+s

y( ) r+s

Expected vacancy cost:

Value of unemployment

Figure 1: Production and labor income in the laissez-faire economy

Labor income, slope: °

k

q( )

y( ) -

r+s

w( ) -

r+s

y(0) r+s

y( ) r+s

y( ) -

r+s

Expected vacancy cost:

After-tax output, slope 1- - = y w

Value of unemployment

1 1-

Figure 2: Production and labor income, with optimal taxation

Note : The X-axis is associated to the worker's production (meaning the discounted cumulated pro-ductivity). σ is the distance above which workers do not exert a search e�ort. The labor income is the

discounted cumulated net-of-tax wage, of the form w/(r+s). Its equation is : wr+s−

w(ξ)r+s =

◦β(

yr+s −

y(ξ)r+s

)The after-tax output is the worker's production net of the tax levies and subsidies. For a y-productive

match, it corresponds to (1− τy) yr+s − τw

wr+s + τy

y(ξ)r+s + τw

w(ξ)r+s .

Comment : At a given level of worker's production, the distance between the output and the value ofunemployment represents the surplus of a match. Between the output and the labor income, the distanceis the part of the surplus to the �rm (or discounted pro�t). The colored area represents the range ofoutput which is below the sum of the expected vacancy cost and the value of unemployment.

16

makes the level of job creation optimal, the tax on production is used to compensate

the e�ect of the wage tax on the search behaviors and to make it e�cient. The tax on

production adjusts to ful�ll◦β = 1.

These tax parameters may be negative or positive, depending on the value of the

bargaining power β and the elasticity of the matching function η. Never both can be

simultaneously positive. The wage curve is not enough sloping, so it is necessary to

redistribute from low-productive jobs to high-productive ones with at least one of the

two �scal tools. Consequently, the lump-sum component is negative, it is a �xed tax.7

Unsurprisingly, Boone and Bovenberg (2002) exactly �nd our equation (20). The

source of ine�ciency is the same and a linear income taxation is su�cient to restore

e�ciency. The novelty comes from the second equation who corresponds to the search

behavior of workers.

Corollary 1 When the bargaining power satis�es the Hosios-Pissarides condition (β =

1− η), the optimal taxation is de�ned by :

τ ∗w = 0

τ ∗y = − η

1− η

When the Hosios-Pissarides condition is met, the wage tax must be nil to keep the

quantity of jobs created e�cient. It is worth noting that the tax on production can be

very high in absolute value. For example if β = 1− η = 0.5, then τ ∗y = −100%.

4 The choice of the policy instruments

A priori, the government may use any tax tool at its disposal. The combination of a

linear tax on production and a linear tax on wage is especially relevant. It is a simple

schedule for understanding the solution : the surplus sharing must be adjusted to reach

an optimal level of job creation and the slope of the net wage must be equal to one

for workers to behave like the social planner. Here, I explore two other alternatives :

a subsidy for job creation and a subsidy for unemployed. I also prove that a tax on

production is necessary even if we let the income taxation to be any function.

Subsidy for job creation. In addition to the previous taxation scheme, assume the

government provides a �nancial aid A for each vacancy opened, which is �nanced by a

7Replacing the optimal tax rates and the wage equation, it follows τ∗y y(ξ) + τ∗ww(ξ) =

− η(r+s)(1−β)(1−η)

[η( y(ξ)

r+s − ψ) + (1− β)ψ]< 0.

17

lump-sum tax t on jobs. The free-entry of �rms implies equality between the expected

cost of a vacancy and the expected pro�t :

k

q(θ)− A =

y(ξ)− w(ξ)− tr + s

(9')

The budget constraint is cleared when the present-discounted revenue from a job tr+s

is

equal to the �nancial support A. This is simply redistribution from the �rms with �lled

vacancy to the �rms with opened vacancy. As a consequence, equations (9) and (9') are

identical. Yet, the employers' behavior is impacted because the lump-sum tax is shared

with employees through the Nash bargain :

w(ξ)

r + s= β

(y(ξ)

r + s− A

)+ (1− β)ψ (16')

Thus, workers indirectly pay the subsidy for opening vacancies. The wage equation (17)

is unchanged because the tax is independent of productivity. The search strategy of job

seekers in equation (19) remains the same, as well as the value of unemployment (10).

The equilibrium equation (18) becomes :

k

q(θ)=(

1− β)( y(ξ)

r + s− ψ

)+ βA (18')

A subsidy for job creation �nanced by a lump-sum tax on jobs enables to correct the

ine�ciency of job creation, but it has no impact on the ine�ciency of search behaviors.

If the worker's share of the surplus β is above its e�cient value 1−η, the optimal subsidy

restoring an e�cient level of job creation must be positive and vice versa.

Unemployment bene�ts. Maintaining the self-�nancing taxes on production and la-

bor income, the �scal authority provides unemployment bene�ts b to unemployed worker.

The reservation strategy of workers is modi�ed through the return to unemployment :

rU = b+ ξTm(θ)(W (ξ)− U

)− CT (ξ) (4�)

This policy is �nanced by a lump-sum tax on jobs t as above :

k

q(θ)=y(ξ)− w(ξ)− t

r + s(9�)

The government's budget is balanced if the expected unemployment duration times the

amount of unemployment bene�ts is equal to the present-discounted tax levy of a job,b

ξTm(θ)= t

r+s. The lump-sum tax is shared with the employers so the expected wage

follows :

w(ξ)

r + s= β

(y(ξ)

r + s− b

ξTm(θ)

)+ (1− β)ψ (16�)

18

By redistributing from employed workers to unemployed workers, �rms are induced to

open less vacancies given the search strategy of workers because they incur a share of the

lump-sum tax (this is the opposite mechanism of the subsidy to job creation above) , the

equilibrium condition becomes :

k

q(θ)=(

1− β)( y(ξ)

r + s− b

ξTm(θ)− ψ

)(18�)

When the tax was distributed to the employer (previous policy), workers were com-

pensated by a higher job-�nding rate. This is not the case here, the marginal bene�t

from searching is reduced because of the tax. Search e�orts tends to decrease in response

of this change. The equilibrium equation associated to the search decision is :ξ(x) = 0 if

◦βy(x)+(1−

◦β)y(ξ)

r+s− b

ξTm(θ)− ψ − k

q(θ)≤ 0

C ′x(ξ(x)) = m(θ)

( ◦βy(x)+(1−

◦β)y(ξ)

r+s− b

ξTm(θ)− ψ − k

q(θ)

)else

(19�)

Unemployment bene�ts �nanced by a lump-sum tax on jobs alters both channels of

ine�ciency. It is possible to de�ne an optimal value restoring an e�cient level of job

creation as a subsidy for creation. Nevertheless, unemployment bene�ts do not permit by

itself to alleviate the hold-up problem. Optimality requires that search e�orts increase for

high-type jobs and decrease for low-type jobs whereas a change in unemployment bene�ts

make all the search e�orts simultaneously increase or decrease.

General income taxation function. The use of general income taxation is popular

in the literature since the work of Mirrlees (1971). Assume a more general setting, the

planner can tax production at a constant rate τy and can raise a tax T (w) from each job

providing an after-tax wage w. The income taxation is not imposed to be linear.

In appendix, I show that the optimal taxation scheme requires linearity of the taxation.

The optimal tax parameters are then uniquely de�ned by Proposition 5. Besides, the

government cannot �nd an income taxation function restoring e�ciency by itself. A tax

on production is necessary.

Notice that the optimal tax rates depend on very few variables of the model. They

do not depend on the production function nor on the costs, which is a strong result. A

minimum wage might also be a way to make job seekers more selective but its optimal

value requires the government to know the primitives of the model.

19

5 The search technology and the matching strategy

5.1 Alternative search technology

This subsection aims at analyzing the robustness of the optimal policy to the spec-

i�cation of the search technology. The search cost function was taken continuous and

convex. The optimality condition was then simply a �rst-order condition. Our results

are unchanged for a more general search technology as long as job seekers can be choosy,

meaning they still have the choice not to visit some segments of the labor market at zero

cost. With a mathematical formalism, let S be any subset of the space of integrable

functions from [0, 1/2] into R+. We only impose the following property to the set S :

∀ξ ∈ S,∀x ∈ [0, 1/2],1−x × ξ ∈ S

where 1−x(z) = 1 for z 6= x and 1−x(x) = 0. The search cost function Cx can be

any function from R+ into R+ and can be distance-dependent with Cx(0) = 0. This

hypothesis enables us to omit the matching behavior because a worker will never reject

a job o�er for which she exerts a positive search e�ort. The planner's problem for an

optimal search strategy writes :

maxξ∈S

∫ 12

0

{ξ(x)m(θ)

(y(x)

r + s− ψ − k

q(θ)

)− Cx(ξ(x))

}dx

This is exactly the same problem as workers' under the optimal �scal rules. As a

consequence, if this problem admits a solution then the search strategy chosen by private

agents is e�cient with the optimal tax rates. Actually, the optimal policy does not

decentralize the solution of the planner's problem but the problem itself.

The model of Decreuse (2008) is a particular case of this general setting, in which Sis the set of functions that only take values 0 or 1 and search costs are nil. This model is

tractable because the trade-o� associated to the search strategy is not triangular anymore

and the search strategy simpli�es to a one-dimension parameter.

5.2 The matching strategy

The search technology simpli�es the analysis of the stochastic job matchings model be-

cause it incorporates the matching decision. A matching strategy is a choice of accepting

or rejecting a job o�er once the two parties have met, for all type of employer. When the

search intensity is not directed according to the productivity of the job, Hosios (1990)

shows that the condition for an e�cient job-acceptance strategy is the same as the one

for an e�cient job creation. In our setting, the proof is given by Marimon and Zilibotti

(1999). They study the same framework as us with random search and without search

20

e�orts. As a result, they show that the laissez-faire equilibrium is e�cient if and only if

the bargaining power satis�es the Hosios-Pissarides condition.

Consider a meeting between an employer and an employee separated by a distance x,

even if this event happens with a zero probability. The planner realizes the match if and

only if y(x)r+s≥ ψ. Private agents agree on matching whether w(x)

r+s≥ ψ.8 In the laissez-faire

economy with Nash-bargained wage, this condition is equivalent to the planner's one.

Hence, the matching strategy is e�cient given a value of an unemployed in absence of

taxation. On the contrary, the optimal taxation schedule distorts the matching decision

of private agents, which writes in this case y(x)r+s≥ ψ + k

q(θ). Private agents would be too

selective in the job-acceptance decision. Anyway, there is no consequences on the social

welfare since search is not random.

For the social planner, the matching decision is di�erent from the decision to exert a

non-zero search e�ort : there are matches that should not be prospected by job seekers but

that should not be rejected in case of meeting. Those are jobs whose productivity is higher

than the return to unemployment but not high enough to compensate also the vacancy

cost. This is due to the traditional externalities. When search e�orts can be directed,

there is an impact on the meeting probability of the other agents. In a random search, the

meeting probability does not depend on the job-acceptance decision.9 As private agents

do not internalize this e�ect, the decision to search on a labor market segment is the

same as the decision to accept a match once a meeting takes place. This point is raised

by Decreuse (2008) when he compares his results with those of Marimon and Zilibotti

(1999). Thus, the holdup ine�ciency disappears in a random search framework.

By making individuals more selective in searching, the optimal taxation makes them

more selective in matching. It cannot disentangle these two aspects.

5.3 Random search

If search is random, the problem is studied in Marimon and Zilibotti (1999). There is

only one source of ine�ciency that is associated to the �rms' entry decision. However,

the government cannot decentralize the social optimum with only the Hosios-Pissarides

condition with taxation (equation (20)) because the wage tax modi�es also the matching

decision, which is initially optimal without taxation (see above). From the wage equations

8If one of the party agrees on matching, the other does so because of the Nash-bargain : S(x) ≥ 0 isequivalent to W (x) ≥ U(x), equivalent also to J(x) ≥ 0.

9The model from which these conclusions are derived is very similar to ours. Consider now ξ(x) asthe probability to accept a job at distance x, suppress the search cost Cx(ξ(x)), impose the constraints0 ≤ ξ(x) ≤ 1 and modify the market tightness θ = v/u. For private agents, the equations barely change,except Ξe that must account for the constraints. However, the objective function of the social planner'sproblem writes :

∫∞0

[Y − kθu] e−rtdt. The total vacancy cost kθu is now independent of the matchingdecision ξ, hence the result.

21

(16) and (17), the labor income writes :

w(x)

r + s=◦βy(x)

r + s+ (β −

◦β)

y(ξ)

r + s+ (1− β)ψ

The e�cient matching decision (y(x)r+s≥ ψ) is decentralized if and only if

◦β = β. This

condition implies that labor income is a weighted average of the production of the worker

and the value of unemployment. This was only true in expectation. In terms of tax rates,

this is equivalent to :

τy = − βτw1 + τw − βτw

(22)

The social optimum is decentralized in this case if the tax system satis�es equations

(20) and (22).

The wage tax has an e�ect on the wage bargain and an e�ect on the selectivity of job

seekers. Formally, it a�ects both β and◦β. In a random search framework, the second

impact is not desirable in terms of e�ciency. The role of the tax on production is to

cancel out the selectivity e�ect of the wage tax. As a consequence, the two tax rates have

an opposite sign.

6 Extension to a general holdup problem

The holdup ine�ciency arises in a wide range of models. An analogous taxation scheme

can be implemented in frameworks in which the holdup is made on any form of invest-

ment. Most works in literature concern the reversed holdup problem : �rms make wrong

investments before going to the labor market.10 Employers need a full bargaining power

to act optimally, meaning β = 0 with our notations (without taxation). Acemoglu (1996),

Masters (1998) and Davis (2001) deal with holdup in both sides of the market. They

show that the wrong behaviors of both �rms and workers does not cancel each other out

but cumulate.

In an extension of our model, assume that �rms must make an investment before

�nding an employee : they invest on a stock of capital k > 0. The price of capital

is normalized to one : the employer incur the cost k until the vacancy is �lled. The

productivity of a match at a distance x writes y(x, k), it is di�erentiable and strictly

increasing and concave in k. Thus, there is a trade-o� between the cost of the invest-

ment and its returns. We denote the expectation of a variable A that depends on x

and k : A(ξ, k) =∫ 1

2

0A(x, k)2ξ(x)

ξTdx. The �rms' problem consists in maximizing the ex-

pected pro�t π(ξ,k)r+s

net of the expected cost of the vacancy kq(θ)

. Under di�erentiability

10See Grout (1984), Acemoglu (2001), Acemoglu and Shimer (1999).

22

assumptions, the �rst-order condition writes :

1

q(θ)=

∂π(ξ,k)∂k

r + s

This formula equalizes the marginal cost of increasing the initial investment to the

marginal gain for the �rm in expectations. The e�cient investment is derived from the

same objective function as the e�cient market tightness and the e�cient search strategy :

it maximizes the value of unemployment ψ. This provides the condition :

1

q(θ)=

∂y(ξ,k)∂k

r + s

With our taxation setting and the Nash bargain, the slope of the �ow pro�t curve as

a function of productivity is equal to (1−β)(1−τy). To make private investment decision

e�cient, one needs :

(1− β)(1− τy) = 1 (23)

Firstly, the problems of right �rms' investment and right search strategy are identical.

The gain functions (pro�t for �rms and net-of-tax wage for workers) must be proportional

to productivity with a coe�cient 1. Secondly, this condition cannot be satis�ed simulta-

neously with β = 1− η and◦β = 1 in general. This result is in line with Davis (2001) who

shows that holdups in both side of the market do not cancel out, the holdup of �rms'

investment is a third source of ine�ciency.

Ful�lling equation (23) and◦β = 1, that are associated to the two holdup problems,

implies an equal sharing of the expected surplus : β = 1/2. As a consequence, e�ciency

is restored in the particular case when the elasticity of the matching function η is equal

to 1/2 because, then, the Hosios-Pissarides condition β = 1 − η is satis�ed. The new

source of ine�ciency highlights another role of the bargaining power, which is associated

to the slope of the �ow pro�t as a function of the productivity. The government needs

another �scal tool to control for this third aspect.

7 Conclusion

This paper examines job seekers' behavior on a labor market with search frictions. By

observing the productivity of jobs, workers can choose the search intensity for each type

of jobs. This has consequences on the total probability to �nd a job and the expected

quality of their future job. A Nash-bargained wage induces workers to share the bene�ts

from being more productive with �rms. However, workers incur the entire cost because

it is sunk when matching. The workers' behavior is then suboptimal comparing to the

23

choice of a social planner : the search e�ort should be higher for high-productive jobs

and vice versa. In addition, another cause of ine�ciency comes from the entry decision

of �rms. Except in a particular case (when the Hosios-Pissarides condition is satis�ed),

the �rms' entry decision is suboptimal given the workers' behavior.

With an appropriate tax schedule, the government is able to make workers solve the

same optimization problem as the social planner. It consists in proportional taxes on

wage and on production with a lump-sum tax, so that the receipts are nil for the �scal

authority. The optimal policy is regressive : there is redistribution from low-productive

to high-productive matches.

This paper is not aimed at recommending a regressive taxation on the labor market.

It sheds light on a harmful consequence of progressive taxation that has been omitted in

the literature.

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[16] C.A. Pissarides. Equilibrium Unemployment Theory. the MIT press, 2000.

[17] R. Shimer and L. Smith. Matching, Search, and Heterogeneity. The B.E. Journal

of Macroeconomics, 0(1):5, 2001.

A Proof of the existence of a decentralized equilibrium

Let F to be the function such that F (ψ, θ, ξ) = (F1(θ, ξ), F2(ψ, θ, ξ), F3(ψ, θ, ξ)) and

F1(θ, ξ) =F4(ξ)m(θ)

r + s+ F4(ξ)m(θ)

(F6(ξ)

r + s− k

q(θ)

)− F5(ξ)

r + s+ F4(ξ)m(θ)

F2(ψ, θ, ξ) = q−1

k

(1− β) max{F6(ξ)r+s− ψ, 0

}

[F3(ψ, θ, ξ)](x) = C ′−1

(m(θ) max

{ ◦βy(x) + (1−

◦β)F6(ξ)

r + s− ψ − k

q(θ), 0

})

25

given the auxiliary function :

F4(ξ) = 2

∫ 12

0

ξ(x)dx

F5(ξ) = 2

∫ 12

0

C(ξ(x))dx

F6(ξ) = 2

∫ 12

0

ξ(x)

F4(ξ)y(x)dx

This function is de�ned from C = L1([0, 12], [0, α1]× [0, α2]× [0, α3]) into itself,11 where

L1(K,E) denotes the set of the integrable functions from K to E and

α1 =y(0)

r + sα2 = q−1

(k

(1− β)y(0)r+s

)

α3 = C ′−1

(y(0)

r + s

)The set C belongs to the vectorial space E = L1([0, 1

2],R3) associated with the uniform

norm.12 I impose β < 1 (i.e. τw > −1).

Let show that F is continuous on C. I de�ne ||.||1 the 1-norm of a function f from

[0, 12] to R such that ||f ||1 = 2

∫ 12

0|f(x)|dx. Straightforwardly, F4 is continuous.

Next, the function cost C is continuous. The Heine-Cantor theorem states that C is

uniformly continuous on the compact set [0, α3]. Let ε > 0, there exists ηε such that

∀X, Y ∈ [0, α3], |X − Y | < ηε ⇒ |C(X)− C(Y )| < ε

Let ξ, ξ′ ∈ L1([0, 12], [0, α3]) with ||ξ−ξ′||∞ < ηε. Then, for all x in [0, 1

2], |ξ(x)−ξ′(x)| < ηε

and |C(ξ(x))− C(ξ′(x))| < ε. Hence, |F5(ξ)− F5(ξ′)| < ε. F5 is continuous.

It can be shown the following inequalities :

|F6(ξ)− F6(ξ′)| < y(0)

∥∥∥∥ ξ

||ξ||1− ξ′

||ξ′||1

∥∥∥∥1

< y(0)

(∥∥∥∥ ξ

||ξ||1− ξ′

||ξ||1

∥∥∥∥1

+

∥∥∥∥ ξ′

||ξ||1− ξ′

||ξ′||1

∥∥∥∥1

)< 2

y(0)

||ξ||1||ξ − ξ′||1 < 2

y(0)

||ξ||1||ξ − ξ′||∞

It derives that F6 is continuous at any ξ in L1([0, 12], [0, α3]).

By elementary operations, F1 and F2 are continuous. For all x in [0, 12], the function

x→ [F3(ψ, θ, ξ)](x) is continuous at (ψ, θ, ξ). Because [0, 12] is compact, F3 is continuous.

Thus F is continuous on C.11Abusively I denote f = (ψ, θ, ξ) ∈ C with f(x) = (ψ, θ, ξ(x)) for x in [0, 1

2 ].12The norm of (ψ, θ, ξ) ∈ E is ||(ψ, θ, ξ)||∞ = max

x∈[0, 12 ](|ψ|, |θ|, |ξ(x)|)

26

Let show that F (C) is relatively compact. First, I show that F (C) is equicon-

tinuous at any x0 in [0, 12]. Let f ∈ F (C), then there exists (ψ, θ, ξ) in C such that

f = F (ψ, θ, ξ). For x in [0, 12], ||f(x)− f(x0)||∞ = |[F3(ψ, θ, ξ)](x)− [F3(ψ, θ, ξ)](x0)|. By

continuity of F3(ψ, θ, ξ), it follows :

∀ε > 0, ∃η(ψ, θ, F6(ξ), x0) > 0 /

|x− x0| < η(ψ, θ, F6(ξ), x0)⇒ |[F3(ψ, θ, ξ)](x)− [F3(ψ, θ, ξ)](x0)| < ε

By de�ning η∗(x0) = min0 ≤ ψ ≤ α1

0 ≤ θ ≤ α2

0 ≤ a ≤ y(0)

η(ψ, θ, a, x0), we prove the equicontinuity of F (C).

Now, I prove that S(x) = {[F (ψ, θ, ξ)](x) | (ψ, θ, ξ) ∈ C} is relatively compact for

all x. S(x) is a subset of R3, it is su�cient to show that S(x) is bounded. Clearly,

S(x) ∈ [0, α1]× [0, α2]× [0, α3]. Thus S(x) is relatively compact.

According to the Arzelà-Ascosi theorem, F (C) is relatively compact.

Let show that F has a �xed point. F is continuous from the convex closed set

C into itself and F (C) is relatively compact. The Schauder theorem states that F has a

�xed point.

B General income taxation

In addition to a commodity tax τy, each job providing an after-tax wage w gives T (w)

to the �scal authority. T (.) is the taxation function, its derivative is denoted T ′(.). The

�ow pro�t writes π(x) = y(x) − w(x) − T (w(x)) − τy(y(x) − y(ξ)). The wage bargain

leads to the following wage setting :

w(x)

r + s=

β

1 + (1− β)T ′(w(x))

(y(x)

r + s− T (w(x))− τy(y(x)− y(ξ))

)+

(1− β)(1 + T ′(w(x)))

1 + (1− β)T ′(w(x))ψ

(24)

The problem of optimal taxation consists in �nding T (.) and τy such that :

w(x)

r + s=

y(x)

r + s− η

(y(ξ)

r + s− ψ

)(25)

2

∫ 12

0

T (w(x))ξ(x)

ξTdx = 0 (26)

The �rst equation is the de�nition of the e�cient wage pro�le in Proposition 1. Ac-

cording to the second one, the tax receipts are nil to satisfy the budget constraint of the

government. Suppose T (.) be such that (25) is true. Denote γ = (r + s)ψ, (24) then

writes :

T ′(w) +1

(1− β)(w − γ)T (w) = β

[η + τy(1− η)] (y(ξ)− γ)

(1− β)(w − γ)− 1− β(1− τy)

1− β(27)

27

The solutions of this di�erential equation have the following expression :

T (w) = [η + τy(1− η)](y(ξ)− γ)− 1− β(1− τy)1− β

(w − γ)− κ1

(w − γ)β

1−β(28)

We exclude the possibility of having T (γ) ∈ {−∞,∞} so κ1 = 0. Thus, the labor

income taxation is linear. With (26), one can easily obtain the optimal parameters of

Proposition 5.

28