REVIEW OF ECONOMIC PERSPECTIVES – NÁRODOHOSPODÁŘSKÝ OBZOR,
VOL. 13, ISSUE 1, 2013, pp. 30–42, DOI: 10.2478/v10135-012-0014-6
Job Differentiation vs. Unemployment
Samir Amine1
Abstract: We use a matching model in which the horizontal job differentiation results
from the rationale response of firms to the state of the labor market. We show that a
decrease in the labor market tightness gives firms an incentive to raise the
differentiation degree of jobs. Comparative statics suggests that an increase in
unemployment benefits and in the minimum wage improves productivity of skilled
workers by making jobs more differentiated, and leads to a raise in unemployment rate.
Keywords: Job Differentiation, Productivity, Matching, Unemployment
Classification JEL: J64, J65
Introduction
In context of economic crisis, unemployment and the deteriorating situation of unskilled
workers have become major priorities of the labor market institutions in several
developed countries. In response, they try to reform their public policies such as
minimum wage and income redistribution system. This paper aims to study the effects
of the minimum wage and unemployment benefits in a matching model with an original
presentation of interactions between unemployment and job differentiation.
In the literature, other search models with ex-ante heterogeneous workers have been
proposed (Marimon and Zilibotti, 1999; Gautier and Teulings, 2004; Nickell, 2004). In
these models, job differentiation is generally regarded as an exogenous parameter. In the
same mind, other authors, mainly Acemoglu (1999), Gautier (1999), Albrecht and
Vroman (2002), have attempted to endogenize the skill requirements. However, their
models focus on vertical differentiation and consider that job productivity (when filled
by a skilled worker) does not depend on the state of the labor market. In other words,
job differentiation remains essentially exogenous (Mortensen and Pissarides, 1999).
The main contribution and originality of this paper focus on two points. Firstly, we
argue that horizontal jobs differentiation is an endogenous variable which results from
the rationale response of firms to the state of the labor market. In other words, we will
show that firms will be encouraged to offer more adapted jobs to the abilities of skilled
workers until unemployment (labor market state) would facilitate their recruitment.
Secondly, we study this horizontal differentiation by considering that there is a skill bias
in favor of skilled workers.
1
Université du Québec en Outaouais and CIRANO, 283 boul. Alexandre-Taché, C.P. 1250,
succursale Hull, Gatineau, Québec, Canada, J8X 3X7, and CIRANO, Montréal, Canada, email :
samir.amine@uqo.ca
Volume 13, Issue 1, 2013
31
We use a matching model (Pissarides, 2000) in which the horizontal job differentiation
is represented as a point of a line segment. Thus, the degree of adequacy is measured by
the distance between the two ends of the segment and the point. Intuitively, we assume
that an increase in this degree of differentiation raises the output of well-suited workers
and lowers the output of ill-suited workers. When entering the labor market, firms
decide on this degree by maximizing the value of a vacancy. The hiring process
between workers and firms is formalized by the usual matching function (Petrongolo
and Pissarides, 2001).
We show that a decrease in the labor market tightness gives the firms an incentive to
raise the differentiation degree of jobs. In this framework, we study the effects of
unemployment benefits and minimum wage on productivity and unemployment. We
obtain that an increase in unemployment benefits improves the productivity of skilled
workers by making jobs more differentiated, and leads to a raise in unemployment rate.
Comparative statics also suggests that the minimum wage has the same effects on the
model variables.
The rest of paper is organized as follows: The model is presented in Section 1. Then,
solving of the model and the definition of its equilibrium are discussed in Section 2. We
define and study the comparative statics properties of the model in Section 3.
The Model
Consider an economy populated by N heterogeneous workers and by M homogeneous
firms. Workers who are horizontally differentiated by their type of qualification are
divided into two categories: λS the proportion of skilled workers, and (1-λS) the
proportion of unskilled workers. We assume that each employee can only apply for one
job per period, and that each firm offers a single job. In contrast to workers, firms in this
economy are identical and their number is exogenous.
1.1 Job Differentiation
Each job is characterized by a degree of adequacy to both types of workers. This
formalization is the major element of our framework. A given profile corresponds to a
point of a line segment whose length is normalized to unity.
Figure 1: The degree of adequacy of a job
The distance, αୗ measures the adequacy of job to the qualifications of skilled workers,
while 1 െ ߙௌ ൌ ߙேௌis the adequacy to unskilled workers. When αୗ, the job corresponds
REVIEW OF ECONOMIC PERSPECTIVES
32
perfectly to skilled workers and their productivity reaches the maximum, normalized at
β, whereas that of unskilled is equal to its minimal value µ (0 <µ ≤ 1).
Figure 2: The productivity of skilled worker
Figure 3: The productivity of unskilled worker
When ߙௌ= 1, the productivity of unskilled is equal to unity while that of skilled is equal
to its minimal value βµ. Formally, we assume that worker productivity ‫ݕ‬௜with i={s; ns}
is an increasing function f of the degree of adequacy α୧ to job.
( )SS fy α= (1) and ( )NSNS fy α= (2)
Considering that job is biased towards skilled workers, their productivity can be
rewritten:
( )SS fy αβ= (3)
with β>1 which represents the level of qualification (skill bias) of skilled workers
relative to unskilled. If job is horizontally undifferentiated, when ߙேௌ ൌ ߙௌ ൌ 1/2,
skilled workers have higher productivity than the unskilled.
( )NSNS fy α= < ( )SS fy αβ= (4)
This skill bias can be explained through the complementarity relation between
technology and qualification. Skilled workers are more able to adapt their abilities
(namely in computing) to all types of jobs. This increasing function ݂ሺߙ௜ሻ is assumed
strictly concave ሺ݂ᇱ
൐ 0; ݂ᇱᇱ
൏ 0ሻ.
Volume 13, Issue 1, 2013
33
Figure 4: Adequacy and productivity
Moreover, each type of job corresponds to the productivity of a skilled or unskilled
worker. At this productivity yi, the inverse function of ݂ሺߙ௜ሻ, combines a degree of
adequacy to skilled and unskilled workers:






= −
β
α S
S
y
f 1 (5) and ( )NSNS yf 1−
=α (6)
Given that ߙேௌ ൅ ߙௌ ൌ 1, we deduce the degree of adequacy of unskilled workers:














−= −
β
S
NS
y
ffy 1
1 (7)
We note g(.) the function thus obtained:






=
β
S
NS
y
gy (8) and ( )NSS ygy β= (9)
REVIEW OF ECONOMIC PERSPECTIVES
34
Taking into account hypotheses of the function ݂ሺߙ௜ሻ , we verify that substitution
relationship (of productivities), represented by g(.), is decreasing and concave ሺ݃ᇱ
൏
0; ݃ᇱᇱ
൏ 0ሻ.
Figure 5: Substitution relationship
Increasing ‫ݕ‬ௌ requires that job is better suited to the characteristics of skilled workers
and less suited to those of unskilled‫ݕ‬ேௌwhose productivity necessarily decreases. For
‫ݕ‬ௌ ൐ ‫ݕ‬ேௌ the slop e݃ሺ‫ݕ‬ௌሻ has a value greater than unity and increases with‫ݕ‬ௌ.
1.2 Hiring Process
There are frictions in the labor market that stops instantaneous matching of unemployed
workers (skilled or unskilled) and vacant jobs. In order to give solid microeconomic
foundations to meeting process between workers and firms, we use here the "urn-ball
model" (Petrongolo and Pissarides, 2001). According to this model and the "job search
theory" (McKenna, 1985), we assume that an unemployed worker meets one firm likely
in each period. This firm is taken at random from all the firms. We also consider that
workers orient correctly their job search to the extent that they meet the firms offering
more or less differentiated jobs in one point which represents the whole labor market
(Albrecht and al., 2003). The "firm-employee" match is done at this point which
includesሺߣௌ ܰሻ skilled workers, ሺ1 െ ߣௌሻ ܰ unskilled and M firms. Let θ, represents the
labor market tightness (θ = M / N). Given that each worker sends a single application,
the probability that the firm doesn’t meet a skilled worker is given by:
SN
M
λ






−
1
1 (10)
Volume 13, Issue 1, 2013
35
Considering that jobs (M) and workers (N) approaching to infinity, the probability to fill
a job by a skilled worker is:
θ
λS
eqS
−
−= 1 (11)
This probability is a decreasing function of θ. Owing to the congestion effect, a rise in
the number of vacant jobs has a negative impact on the probability of filling a job. In the
same way, increasing ߣௌ causes a rise in this probability owing to the opportunity effect.
Moreover, the probability to meet an unskilled worker is given by the following
equation:
( )








−=
−
−−
θ
λ
θ
λ SS
eeqNS
1
1 (12)
The impact of θ is identical on this probability due to the same effect. However, any
increase in ߣௌ, causes a decrease of this probability. Finally, the probability to meet a
skilled or unskilled is given by the sum of: ‫ݍ‬ ൌ ‫ݍ‬ௌ ൅ ‫ݍ‬ேௌ
θ
1
1
−
−= eq (13)
This probability does not depend on the proportion of skilled workers but of the labor
market tightness whose impact is negative (the congestion effect).
1.3 Utilities, Profits
In accordance with traditional matching models, wages given to workers result from a
bargaining process according to their bargaining power. This wage is noted wi with
i={s; ns}. In this model the utility of employees is represented only by their wages and
that of the unemployed corresponds to their unemployment benefits denoted b. For
firms, entering the labor market and creating a vacancy impose a cost noted c. We
consider the profit PT of a firm whose job is filled by a skilled or unskilled T={s; ns}.
We obtain:
TTT wyP −= (14)
Sharing the total surplus associated with a job is done accordingly to Nash's generalized
rule, depending on the bargaining power of employees and firms. We denote σ (0<σ<1)
the bargaining power of workers. We obtain:
( )bybw TT −=− σ (15)
For σ =1employees capture all the surplus created by the current job; ‫ݕ‬் ൌ ‫ݓ‬். The
firm's profit can then be rewritten:
( )( )byP TT −−= σ1 (16)
REVIEW OF ECONOMIC PERSPECTIVES
36
The value of a vacant job is denoted by F given by:
NSNSSS PqPqcF ++−= (17)
This value decreases with the cost c and increases with the probability of meeting an
unemployed. However, under the assumption of labor market free-entry, the value of a
vacancy is equal to zero (F=0) and we have:
NSNSSS PqPqc += (18)
2. Solving the model
Solving the model consists of establishing interactions at the stationary equilibrium,
between labor market tightness and Job differentiation.
2.1 Optimal job differentiation
When entering the labor market, the firm decides on the degree of differentiation.
Formally, productivity of a skilled worker ‫ݕ‬ௌ is obtained by maximizing the value of the
vacancy under the constraint imposed by the substitution relationship:
axΜ NSNSSS PqPqcF ++−= s.c 





=
β
S
NS
y
gy
The first order condition satisfies (appendix A):
0
1
'. =





+
ββ
S
NSS
y
gqq (19)
Taking into account the concavity of g(.), we easily obtain the following result
(appendix A):
Proposition 1: In the optimum of profits, productivity ሺ‫ݕ‬ௌሻ is an increasing function of
the ratioሺ‫ݍ‬ௌ/‫ݍ‬ேௌሻ.
This result is very intuitive and is interpreted in the following way. An increase in the
probability to meet a skilled worker (increase inሺ‫ݍ‬ௌ/‫ݍ‬ேௌሻ) has the effect of facilitating
its recruitment. Therefore, firms adapt their job-technology to the labor market state by
creating jobs more differentiated in favor of skilled workers. This improvement of the
matching quality leads to an increase in productivity. The concavity of g(.) ensures that
the first order condition is satisfied (Amine andLages,2010, 2011).
2.2 Labor market tightness and unemployment
In this section, we establish the interactions between the labor market tightness
(unemployment) and productivity. The ratioሺ‫ݍ‬ௌ/‫ݍ‬ሻcan be written as follows:








q
qS =
θ
θ
λ
1
1
1
−
−
−
−
e
e
S
(20)
Volume 13, Issue 1, 2013
37
The derivative of this ratio with respect to θ depends on the sign of the following
expression:
θ
λ
θ
λ
θ
λ S
S
e
e
S
−
−
−1 _
θ
θ
θ
1
1
1
1
−
−
−
e
e (21)
To obtain the sign of this expression, we consider the following function: ߝሺ‫ݔ‬ሻis strictly
increasing for ‫ݔ‬ ൐ 0. (appendix B)
( ) x
x
xe
e
x −
−
−
=
1
ε (22)
Knowing thatߣௌ ൏ 1, we deduce that the expression (21) is negative. Therefore, the
derivative of the ratio ሺ‫ݍ‬ௌ/‫ݍ‬ሻ with respect to θ is negative. Given the proposition1, we
can then state the following result:
Proposition 2: In a matching model with heterogeneous workers, an increase in the
labor market tightness reduces the job differentiation.
The result is interpreted with great simplicity when the ratio ሺ‫ݕ‬ௌ/ߠሻisgreater than unity.
Indeed, any increase in the labor market tightness (i.e. decrease in unemployment)
produces a sufficient decrease in the probability to meet a skilled worker relative to
unskilled. Considering the proposition 1, firms react by reducing the differentiation of
jobs, thus deteriorating the productivity. Therefore, proposition 2 implies a positive
relationship between unemployment and job differentiation.
2.3 Equilibrium
The equilibrium of the labor market can be defined as a coupleሺ‫ݕ‬ௌ; ߠሻ which satisfies
the following expressions:
( ) ( ) ( )( ) 01 =−+−−+− byqbyqc NSNSSSσ (23)
0
1
'. =





+
ββ
S
NSS
y
gqq (19)
This equilibrium is obtained from equations (16), (18) and (19). Once the labor market
tightness is determined, we derive the equilibrium unemployment noted U:
(24)
The equilibrium unemployment rate, denoted u, is then given by:
qu θ−= 1 (25)
( )( ) ( )MqqNNU NSSSS +−−+= λλ 1
REVIEW OF ECONOMIC PERSPECTIVES
38
3. Comparative Statics
In this section, we study the effects of the unemployment benefits and the minimum
wage on variables of the model and particularly on job differentiation.
3.1 Unemployment benefits effects
In order to deduce the unemployment benefits effect, we totally differentiate the
equilibrium equation (23) according to this parameter. We obtain:
( ) ( ) ( ) 011 =−−
















−





∂
∂
+−
∂
∂
− qdbdb
y
g
q
by
q SNS
S
S
σθ
βθθ
σ (26)
Given that
డ௤ೄ
డఏ
൏ 0,the expression ( ) 0<
















−





∂
∂
+−
∂
∂
b
y
g
q
by
q SNS
S
S
βθθ
is negative. We
obtain the effect of b on θ:
( )
0<
















−





∂
∂
+−
∂
∂
=
b
y
g
q
by
q
q
db
d
SNS
S
S
βθθ
θ
(27)
Using the propositions 1 and 2, we establish the effect on productivity ‫ݕ‬ௌ while the
impact on unskilled productivity is given by the substitution relationship (equation (8)).
Concerning the effect on wages is given by differentiation of the equation (15):
0)1( >−+= dbdydw SS σσ (28)
?)1( =−+= dbdydw NSNS σσ (29)
Given that
డ௤
డఏ
൏ 0,the impact on the unemployment rate is deduced as follows:
0>
∂
∂
−= dq
q
qddu
θ
θθ (30)
The increase in unemployment benefits reduces the expected profits of firms and leads
to the decrease in tightness θ by reducing the creation of vacancies. Consequently,
unemployment rate rises as well as the probability to meet a skilled worker. Therefore
and according to the proposition 1 and 2, firms react face to this labor market state by
adapting their technology and by creating more differentiated jobs in favor to skilled
workers. The matching quality is thus improved and the productivity of skilled workers
rises, while that of unskilled is reduced. Moreover, the positive impact on the wages of
skilled workers is explained partly by the increased productivity, and secondly by the
increase in utility of the unemployed b. On the contrary, the effect on the wages of
unskilled remains undetermined because we have two opposite effects.
Volume 13, Issue 1, 2013
39
Table 1: Unemployment benefits effects
θ Sy NSy q u Sw NSw
b - + - + + + ?
3.2 Minimum wage
Keeping the same analytical framework, we study here the effects of minimum wage on
variables of the model. We note m the minimum wage paid to unskilled workers.
Indeed, introducing this public policy does not imply any changes in the functions of
productivity of the two workers, or in the meeting process presented in the previous
sections. Nevertheless, the profit of a job filled by an unskilled worker can be rewritten
as follows:
myP NSNS −= (31)
While the profit of a job filled by a skilled always depends on the bargaining power of
workers and on their unemployment benefits:
( )( )byP SS −−= σ1 (32)
As previously, the optimal differentiation of jobs is obtained by maximizing the value of
the vacancy under the constraint of the substitution relationship. Consequently, the two
main model results are verified with the introduction of the minimum wage. At the
equilibrium, the coupleሺ‫ݕ‬ௌ; ߠሻverifiesthe following two equations:
( ) ( ) ( )( ) 01 =−+−−+− myqbyqc NSNSSSσ (33)
0
1
'.)1( =





+−
ββ
σ S
NSS
y
gqq (34)
To deduce the impacts of a minimum wage on the equilibrium variables, we
( ) ( ) ( ) ( ) 011 =−−





−
∂
∂
+−
∂
∂
− dmqdmy
q
by
q
NSNS
NS
S
S
σθ
θθ
σ (35)
Given that, 0,0,0 >><
∂
∂
σ
θ
m
qS
,
REVIEW OF ECONOMIC PERSPECTIVES
40
the expression ( ) ( ) 0<





−
∂
∂
+−
∂
∂
my
q
by
q
NS
NS
S
S
θθ
is negative. We obtain the effect of m
on θ:
( ) ( )
0<






−
∂
∂
+−
∂
∂
=
my
q
by
q
q
dm
d
NS
NS
S
S
NS
θθ
θ
(36)
Using the propositions 1 and 2, we establish the effect on productivity ‫ݕ‬ௌ while the
impact on unskilled productivity is given by the substitution relationship (equation (8)).
Concerning the effect on skilled wages is given by differentiation of the equation (15):
0>= SS dydw σ (37)
The impact on the unemployment rate is deduced in the same way as unemployment
benefits. We can then state the following result:
Table 2: Minimum wage effects
θ Sy NSy q u Sw
m - + - + + +
Proposition 3: In a matching model with horizontal jobs differentiation, enhancing the
minimum wage leads to improve productivity of skilled workers but to increase the
unemployment rate.
The increase in the minimum wage paid to unskilled workers reduces incentives to
create jobs since the firms share in the profit decreases. Therefore, the labor market
tightness is reduced and the probability to meet a skilled worker increases. To cope with
this increase in the minimum wage, firms become selective by creating highly
differentiated jobs in favor of skilled workers. The matching quality is therefore
improved thus increasing the productivity and wages of skilled.
Conclusion
Using an original formalization of job differentiation, the model results focus on three
key points. The first result concerns the relationship between differentiation and
unemployment rate. We have showed that firms react face to unemployment by
adapting the characteristics of their job in favor to skilled workers. The second result
concerns the unemployment benefits which accentuate the job differentiation and
increase unemployment rate. The last result shows a negative relationship between
minimum wage and the labor market tightness while providing more differentiated jobs
at the expense of unskilled workers.
Volume 13, Issue 1, 2013
41
References
ACEMOGLU, D. (1999). Changes in unemployment and wage inequality: an
alternative theory and some evidence. American Economic Review. 89. Pp.1259–1278.
AMINE, S., LAGES DOS SANTOS, P. (2010). Technological Choices and
Unemployment Benefits in a Matching Model with Heterogenous Workers. Journal of
Economics. 101 (1). Pp. 1-19.
AMINE, S., LAGES DOS SANTOS, P. (2011). The Influence of Labour Market
Institutions on Job Complexity. Research in Economics.65 (3). Pp. 209-220.
ALBRECHT, J.W., GAUTIER, P.A., VROMAN, S.B. (2003). Matching with Multiple
Applications. Economics Letters. 78 (1). Pp. 67-70.
ALBRECHT, J.W., VROMAN, S.B. (2002). Matching model with endogenous skill
requirements. International Economic Review. 43. Pp.283–305.
GAUTIER, P.A. (1999). Unemployment and search externalities in a model with
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99-075/3.
GAUTIER, P.A., TEULINGS, C. (2004). The right man for the job. Review of
Economic Studies.71, Pp.553-580.
MARIMON, R., ZILIBOTTI, F. (1999). Unemployment vs. mismatch of talents:
reconsidering unemployment benefits. Economic Journal. 109. Pp. 266–291.
MC KENNA, C. (1985). Uncertainty and the labour market, Harvester Press.
MORTENSEN, D.T., PISSARIDES, C. (1999). Unemployment response to ‘‘skill
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NICKELL, S. (2004). Poverty and worklessness in Britain. Economic Journal. 114.
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PETRONGOLO, B., PISSARIDES, C. (2001).Looking into the Black Box: A Survey of
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PISSARIDES, C. (2000). Equilibrium Unemployment Theory, Cambridge: MIT Press.
REVIEW OF ECONOMIC PERSPECTIVES
42
Appendix
The first order condition
axΜ NSNSSS PqPqcF ++−= s.c 





=
β
S
NS
y
gy
Using equation (16), F can be rewritten as follows:
( ) ( ) 















−





+−−+−= b
y
gqbyqcF S
NSSS
β
σ1
The first order condition satisfies:
( ) 0
1
0
1
1 =





′+⇔=













′+−=
∂
∂
ββββ
σ S
NSS
S
NSS
S
y
gqq
y
gqq
y
F
By differentiating this equilibrium expression,
ββββ
1
0
1






′−=⇔=





′+ S
NS
SS
NSS
y
g
q
qy
gqq
We obtain:
S
S
NS
S
dy
y
g
q
q
d 





′′−=





ββ
1
Taking into account the concavity of g(.), we deduce easily the proposition 1.
Study of the function ( ) x
x
xe
e
x −
−
−
=
1
ε
Its derivative is given by: 2
'
)(
)1(
)( x
xx
xe
xee
x −
−−
+−
=ε
This derivative has the same sign as the expression )1( x
xe
+−
−
. The latter term is null
in zero and its derivative )1(
xe−
− is strictly positive for 0>x . The function ( )xε is thus
increasing for 0>x .