E¢ ciency of Two Sided Investments in an Equilibrium
Unemployment Framework
Lucas Navarroy
October 2010
Abstract
This paper investigates the e¢ ciency of investments by …rms and workers in a matching
model with high- and low-productivity jobs. Search is sector speci…c and random within sectors. Search frictions and ex-post bargaining imply that wage inequality arises as a result of
the di¤erence in investment costs between the sectors. The e¢ ciency properties of the equilibrium are analyzed under the particular division in bargaining proposed by Hosios (1990). The
conclusion is that the equilibrium is ine¢ cient, with a too low fraction of workers and a too
high vacancy-unemployment ratio in the high-productivity sector. The opposite happens in the
low-productivity sector.
1
Introduction
This paper analyzes the e¢ ciency of two sided investments in a labor market model with search
frictions. It presents a model where …rms and workers make ex-ante investments, search for suitable
partners and once they match they bargain over the surplus that results from their investments.
Typically, ex-ante investments and ex-post bargaining imply under-investments. When two
parties commit to invest in advance to make production possible, investments are sunk at the
bargaining stage and then the parties do not get the full marginal bene…ts of their investments. It
is essential for this result that rents arise because of the presence of asset speci…city. In the extreme
case, investments are only productive within the relationship and the outside options are zero.
JEL Classi…cation: C78, J41, J64. Keywords: matching, search frictions, heterogeneity, e¢ ciency.
ILADES-Georgetown University, Universidad Alberto Hurtado. [email protected]. I thank James Albrecht, Susan Vroman and two anonymous referees for their valuable comments. I also thank seminar participants
at Georgetown University, University of Essex, University of Sydney, Universidad de Alicante and University of
Humboldt, Germany. All errors are my own.
y
In a labor market there are a lot of agents on each side of the market taking investments decisions simultaneously and intuitively the degree of asset speci…city is signi…cantly reduced. Indeed,
Cole, Mailath and Postlewaite (2001a, 2001b) and also Felli and Roberts (2002) analyze matching models where competition for partners can prevent holdup problems. From this perspective,
under-investment problems are mitigated in a frictionless labor market. However, in a labor market with search frictions rents from matching take place and they bring back some degree of asset
speci…city to the productive relationship creating again incentives to under-investments (Acemoglu
and Shimer, 1999).
This paper analyzes the e¢ ciency properties of a standard two-sector search and matching
model with two-sided investments. There exist a high- and a low-productivity sector where …rms
and workers participate as a result of their ex-ante allocation decisions. The model generates
a sectorial distribution of …rms and workers. Heterogeneity arises because workers have to pay
for a cost to become high skilled and …rms, correspondingly, need to make higher investments in
order to open a vacancy in the high-productivity sector. It is assumed that a high-productivity
vacancy can only be …lled by a high-skill worker and low-skill workers can only be productive in
the low-productivity sector.1 Also, any …lled vacancy employs one worker to produce one unit of
a good. The number of vacancies in each sector is determined by two free-entry conditions which
drive expected pro…ts from job creation to zero. The distribution of the workforce is the one that
equalizes the expected net income of high- and low-skill workers. The model is related to the two
sector models of Acemoglu (2001) and Albrecht and Vroman (2002) with the main di¤erence being
that the skill composition of the workforce is endogenous.2
The general conclusion of previous research studying two-sided investment in matching models
is that with search frictions both workers and …rms under-invest in human and physical capital,
respectively.3 This is not necessarily what happens in the model presented here. In fact, it is
shown that under the particular division in bargaining satisfying the Hosios condition with costly
investments workers under-invest and …rms tend to over-invest in the high-productivity sector.
This result is explained by the interaction of investment externalities with their general equilibrium
e¤ects. Of particular importance is the endogenous price assumption introduced to the model.
1
It is not considered the possibility of mismatch between high-skill workers and low-productivity …rms previously
analyzed in the literature (McKenna, 1996; Gautier, 2000; Albrecht and Vroman, 2002; Uren 2006; Dolado, Jansen
and Jimeno, 2009). Notwithstanding, the conclusions of the paper are robust to this potential extension for high
enough investment costs in the high-productivity sector . See Blázquez and Jansen (2008) for an e¢ ciency analysis
of Albrecht and Vroman (2002).
2
Aricò (2009) presents a model with endogenous types in both sides of the market and analyzes the e¤ect of subsidies to education and innovation on output and unemployment. Apart from not analyzing the e¢ ciency properties
of his model, the treatment of frictions is quite di¤erent from this paper.
3
Acemoglu (1996), Acemoglu (1997), Masters (1998) and Davis (2001) .
2
Each sector requires capital and labor to produce an intermediate good that is sold in a perfectly
competitive market and used with the other sector’s output to produce a …nal consumption good
(Acemoglu, 2001). As prices depend on the relative supply of the intermediate goods, and therefore
the number of agents in each sector, a higher investment cost of any good reduces its supply
and increases its price, which is the value of output in each productive relationship (i.e. the
match surplus). Given that wages are determined by Nash Bargaining over the match surplus,
the sectorial price di¤erence resulting from di¤erent investment costs explains wage inequality in
the model. These price e¤ects have important implications for the e¢ ciency properties of the
equilibrium. Even when search frictions and rent sharing give incentives to under-investments by
…rms and workers, the relative scarcity of goods produced by the high-productivity sector implies
that the sectorial price di¤erence is higher than it should be. This incentive to invest in the highproductivity sector mitigates under-investment problems. With endogenous prices there is also
more interdependence in the investments decisions in the economy because they do not only rely
on investment costs and matching probabilities but also on their e¤ect on prices.4 .
It is assumed that search is sector speci…c but random within sectors. That is, given the
matching requirements, agents only search for the partners with whom production is possible. This
leads to a labor market with two separate matching functions, one for each sector. Equilibrium in
standard matching models involves trading externalities: a …rm posting a vacancy makes it more
di¢ cult for the other vacancies to meet an unemployed worker (congestion externality) and easier for
the unemployed workers to meet an un…lled vacancy (thick market externality). These externalities
cancel out only when the measure of workers’bargaining power is equal to the elasticity of the
matching function with respect to unemployment (Hosios, 1990; Pissarides, 2000). The e¢ ciency
analysis of this paper is done assuming the Hosios condition holds. This makes it possible to identify
the e¢ ciency e¤ects of investments not related to the standard ones in this type of models. As
previously shown by Acemoglu and Shimer (1999), under the sector speci…c matching assumption
the Hosios condition makes …rms internalize the externalities associated to vacancy creation. An
interesting result of the paper is that there is an externality in the determination of the skill
composition that is not related to the matching assumption. This arises because when workers take
their skill decision they do not consider how a better skill composition increases the average value
of matches in the economy. These investment externalities together with the general equilibrium
e¤ects of the model (in particular the price e¤ects) determine the …nal e¢ ciency properties of the
model.
The paper is structured as follows: The next section describes the general environment. Section
4
Additionally, in line with some empirical literature (Autor, Katz and Kearney, 2005 and 2008) the model with
endogenous prices is consistent with a net supply of skills explanation of wage inequality.
3
3 presents the equilibrium and Section 4 analyzes the e¢ ciency properties of the equilibrium.
Section 5 concludes.
2
Model
2.1
Basic Assumptions
Consider a continuous-time economy populated by a unit mass of workers producing a …nal consumption good Y with two intermediate inputs Yb and Yg according to a CES technology,
)Yg )1= :
Y = ( Yb + (1
(1)
Following Acemoglu (2001), b and g refer to a bad-job and a good-job sector, respectively. Good Y
is sold in a competitive market at a normalized price of 1. It is assumed that < 1 and measures
the share of Yb in the …nal good production function. The …rst order conditions of the aggregate
economy problem lead to the following equilibrium prices for the two intermediate goods,
pb =
1
Yb
pg = (1
Y1
)Yg
(2)
1
Y
1
:
(3)
These two intermediate goods are produced via a Leontie¤ technology using capital and labor.
That is, production of one unit of an intermediate good i = b; g takes place when a …rm with a
vacancy meets a suitable unemployed worker. Then, Yi measures not only aggregate production
but also aggregate employment in sector i.
Firms can only produce in one of the two sectors and need to make ex-ante investments before production takes place. Denote by ki the …rm investment cost in sector i and assume that
investments are generally more costly in the g sector, that is kg kb : Similarly, workers can only
participate in one of the two sectors. Workers producing in sector g must have acquired the necessary skills at a cost c. It is assumed that having incurred the cost c precludes workers from being
able to produce in sector b:5 As a result of the allocation of workers across the sectors there is a
fraction of workers who can only work in sector b: As long as it is more costly for either …rms or
workers to participate in sector g than in sector b, Yb > Yg and therefore pg > pb .
5
We can relax this assumption to allow for high-skill workers to be productive in both sectors. However, even
under a more general assumption about the skill requirements of the di¤erent jobs, it can be shown that for a high
enough price di¤erence between the sectors high-skill workers would never …nd it worthwhile to accept a b sector
o¤er.
4
2.2
Search and Meeting Process
At any moment in time, a job is either …lled or vacant and a worker is either employed or unemployed. Unemployed workers and …rms o¤ering vacancies have to spend resources in order to meet
each other. When a vacant job is …lled with a qualifying unemployed worker, production takes
place and the job generates a rent. As standard in this literature workers and …rms meet according
to a matching function M (u; v) where u is unemployment and v vacancies. This matching function
is twice di¤erentiable, increasing in both arguments and has constant returns to scale.
Given that di¤erent jobs have di¤erent matching requirements …rms and workers could only
search for their potential partners with whom production is possible. This leads to the sector
speci…c search assumption adopted in this section. There are then two matching functions, one for
bad jobs and another for good jobs.
The constant returns to scale assumption implies that M (ui ; vi )=vi = q( i ), where i = vi =ui is
de…ned as the tightness of the labor market in sector i. The function q( i ) represents the ‡ow rate
at which vacancies meet unemployed workers and is decreasing in i . In addition, M (ui ; vi )=ui =
i q( i ) is the ‡ow rate at which unemployed workers meet un…lled vacancies and is increasing in i .
In steady state, unemployment persists because some of the existing jobs break up at the
exogenous rate s providing a ‡ow into unemployment. When a job breaks up, the worker becomes
unemployed and the job becomes an un…lled vacancy.
Given the intermediate production technology, intermediate good production and sectorial employment are equivalent,
Yb =
ub
Yg = 1
(4)
ug :
(5)
The di¤erential costs of operating in either the bad or the good-job sector imply production and
price di¤erences between the sectors. That is, since producing in sector g is more costly than
producing in sector b prices are relatively higher in sector g and also Yb > Yg :
2.3
Bellman Equations and Wage Determination
Denote by JiU the value of unemployment, JiE the value of employment, JiV the value of an un…lled
vacancy and JiF the value of a …lled job in sector i: These values are linked by the following ‡ow
equations:
rJiU = i q( i ) JiE JiU :
(6)
5
Equation (6) states that the ‡ow value of unemployment in any sector is equivalent to the expected
capital gain from …nding a job and realizing a ‡ow value rJiE :6 In a similar way the ‡ow value of
employment in sector i is
rJiE = wi + s(JiU JiE );
(7)
that is, the ‡ow value for an employed worker in sector i equals the wage wi plus the expected
capital loss s JiU JiE :
Similarly, the ‡ow values for a …rm of having a …lled job and a vacancy are
rJiF = pi
wi + s(JiV
rJiV = q( i )(JiF
JiF )
JiV );
(8)
(9)
respectively.
Generalized Nash Bargaining. As standard in this literature, when a matched is formed a
surplus equivalent to pi is realized and the wage is determined by rent sharing. There is a bilateral
negotiation between the parties; a wage is a solution to a generalized Nash Bargaining problem
with threat points equal to the worker’s and the …rm’s respective continuation values. Then,
(1
)(JiE
JiU ) = (JiF
JiV )
(10)
where denotes the exogenous workers’share of the surplus. The wage assumption implies that
the rent implied by the production of a g sector good is transferred in part to the employees via
higher wages.
2.4
Equilibrium conditions
1. Free entry condition. The number of vacancies in any sector is determined by a free entry
condition which drives expected net pro…ts for opening vacancies to zero,
rki = rJiV :
(11)
2. Indi¤erence condition. The equilibrium distribution of workers is determined by a non
arbitrage condition by which workers are indi¤erent between the two sectors. That is, in
equilibrium all the workers have the same expected value of unemployment net of investment
costs,
JgU c = JbU :
(12)
6
It is assumed that the ‡ow income from unemployment is zero. This is a convenient assumption that simpli…es
the calculations. Depending on parameterizations, results would extend though for positive unemployment income
values.
6
Any investment decision taken by any agent in this economy has two e¤ects: Not only it
a¤ects the matching probabilities faced by other agents but also the surplus of the matches
via the e¤ect on prices. Then, an extra worker in sector i increases the matching probability
of i vacancies and then leads to more production and lower prices in that sector.
3. Steady state unemployment. The ‡ow out of unemployment from any sector has to be
equal to the ‡ow back into unemployment to that sector,
q( b ) b ub = s(
ub )
q( g ) g ug = s(1
(13)
ug );
(14)
and the steady state number of unemployed in i is
s
q( b ) b + s
s(1
)
:
q( g ) g + s
ub =
ug =
3
(15)
Equilibrium
In a steady state equilibrium …rms and workers maximize their respective objective functions, given
the matching and separation technologies and all the equilibrium conditions are satis…ed. The
equilibrium is de…ned as a tightness of the labor market i , workforce distribution, value functions
and prices of both goods such that equations (2) to (15) are satis…ed.
Since Yb =
ub and Yg = 1
ug ; the equilibrium intermediate goods prices are
pb =
pg = (1
q( b ) b
q( b ) b + s
)
1
q( b ) b
q( b ) b + s
q( g ) g (1
)
q( g ) g + s
1
+ (1
q( b ) b
q( b ) b + s
)
q( g ) g (1
)
q( g ) g + s
+ (1
)
(1
)=
q( g ) g (1
)
q( g ) g + s
(16)
(1
)=
:
Given that employment Yi increases with i and the size of the workforce in i, we have that pi is
decreasing in i and increasing in j (for j = b; g and j 6= i): Also pg (pb ) is increasing (decreasing)
in .
Using equations (7) to (11), the wage equations are written in the standard way
wi = (pi
rki ) + (1
)rJiU :
(17)
The wage in i is a weighted average of the surplus that the …rm gets (output per worker minus ‡ow
value of capital creation cost) and the ‡ow value of unemployment.
7
The unemployment ‡ow values are given by (6) with (7), (8), (9), and (17) substituted in,
rJiU =
i q( i ) (pi
r+s+
rki )
:
i q( i )
(18)
The equilibrium ‡ow values of vacancies are obtained from (8) and (9). After substitution, the
equilibrium values of the key endogenous variables of the model are obtained from the next three
equations:
rJbV
= rkb = q( b )(1
rJgV
= rkg = q ( g ) (1
rc = q( g )
pg
Dg
g
)
pb
Db
pg
)
Dg
q( b )
b
(19)
(20)
pb
Db
(21)
where
Di = r + s + (1
) q ( i) +
iq ( i) :
Note that rJiV is decreasing in i as expected, and given the price e¤ects, increasing in the other
sector j . Indeed, an increase in j is associated to a greater pi and therefore rJiV (this result is
derived formally in Appendix A). Given the properties of the matching function and the …nal output
technology assumptions an equilibrium always exists. The argument is standard. Prices and the
number of unemployed are uniquely determined by b ; g and : The right hand side in equations
(19) and (20) converges to in…nity as i ! 0 and goes to zero as i ! 1. For any given , the
steady state values of b and g are determined by the intersection of the loci formed by equations
(19) and (20) with (16) substituted in. In Appendix A, it is shown that the loci are upward sloping
and that they intersect only once like in Figure 1. Given the CES …nal output technology, for
certain parameterizations it is also possible to obtain equilibria where only one sector prevails.7 It
follows that, for every , there exists a unique equilibrium pair ( b ; g ). Moreover, given the price
e¤ects associated to changes in ; both loci shift to the right with increases in . This means that
the equilibrium g ( b ) increases (decreases) with and therefore that JgU JbU is monotonically
increasing in : Additionally, given that JgU JbU < 0 for ! 0 and JgU JbU > 0 as ! 1;
there is only one value of (associated to a unique equilibrium pair ( b ; g )) that satis…es (21).
In summary equations (19), (20) and (21) determine the unique triplet ( b ; g ; ) that solves the
model. The solutions for ub and ug are obtained from the steady state equations (15).
7
For an analysis of the existence and uniqueness of the equilibrium in similar models see Cardullo (2009).
8
rkg
θb
θ∗b
rkb
θ∗g
θg
Figure 1
The next section analyzes how investments in the two sides of the market a¤ect the e¢ ciency
properties of the equilibrium. For that reason it is useful to analyze the comparative statics of the
equilibrium with respect to changes in the capital creation cost in the good-job sector (kg ) and the
education cost (c) : The results are summarized below,
Remark 1 In equilibrium,
d g
dkg
< 0;
d b
dkg
< 0;
d
dkg
> 0;
d g
dc
> 0;
d b
dc
< 0;
d
dc
> 0:
See Appendix C for a proof.8 In Figure 1, an increase in kg shifts the good-job-locus equation to
the left. The decrease in g implies a higher pg : The increase in relative prices in sector g is related
to a decrease in b which restores the equilibrium along the bad-job-locus given by (19). Equations
@ i
@ i
(19) and (20) then imply that @rk
< 0 and @rk
< 0. Given that the fall in g is greater than
i
j
the fall in b , the satisfying in addition the non arbitrage condition (21) is greater. Intuitively,
workers …nd it relatively less attractive to participate in sector g when the number of g vacancies
falls.
Regarding the e¤ect of the education cost c, ddc > 0 results from (21). An increase in c requires
a higher di¤erence in the unemployment values between the sectors which happens via the increase
8
It can also be shown that, as expected,
@ i
@r
< 0;
@ i
@s
< 0;
9
@ i
@
< 0:
in pg pb resulting from a higher . The e¤ect of c on labor market tightness is related to ddc > 0.
All else equal, the greater number of low skilled workers resulting from an increase in c leads to a
d
higher pg and a lower pb . It follows that d g > 0 and dd b < 0 to satisfy the respective free-entry
d
conditions. Then dcg > 0 and ddcb < 0.
To conclude this section, note that with = 1=2, kg = kb and c = 0 in equilibrium g = b and
= 1=2; the two sectors are identical and there is no wage inequality because pg = pb . Second, if
kg > kb and/or if c > 0, > 1=2 and there is wage inequality resulting from the fact that pg > pb :
Given rent sharing, wage inequality arises entirely as a result of the di¤erence in investment costs
between the sectors (to either workers or …rms) and their subsequent e¤ect on prices via the product
market. In a way the model provides a net supply of skills argument to wage inequality. The more
it costs to produce in a sector, the lower the supply of goods in that sector, and given the demand
side the higher the prices and wages. Finally, note that if prices were exogenous it would not be
possible to guarantee an equilibrium with an interior solution for . For example, if = 1 (the
intermediate goods are perfect substitutes) pb = and pg = 1
: In this case with no price e¤ects
would be undetermined.
4
E¢ ciency
This section compares the decentralized equilibrium with an allocation that maximizes net total
output. It is assumed that a social planner determines time paths of values ( b ; g ; ) that maximize
the present value of the net total surplus of the economy subject to the same restrictions that govern
private decisions, that is search frictions and the existence of a product market. The planner’s
problem without considering the time dependence of variables is
)
Z 1(
(
ub ) (pb rkb ) + (1
ug ) (pg rkg )
max
e rt dt
(1
) rc
0
g ug rkg
b ub rkb
subject to
:
ub
:
ug
= s(
= s(1
ub )
q( b ) b ub
ug )
q( g ) g ug
The ‡ow of net output of the economy at any point in time consists of the number of workers in
good-jobs (1
ug ) times their net output (pg rkg ), plus the number of workers in bad-jobs
(
ub ) times their net output (pb rkb ) ; minus the ‡ow costs of vacancy creation ( i ui rki ) ; minus
the ‡ow investment cost for workers in the g sector (1
) rc.
10
The current-value Hamiltonian with multipliers
Hc
Hert = (
(1
ub ) (pb
) rc +
b
rkb ) + (1
b (s(
ub )
and
g
can be expressed as
ug ) (pg
q( b ) b ub ) +
rkg )
g
ub b rkb
(s(1
ug )
ug
g rkg
q( g ) g ug ) :
In what follows there are presented the necessary conditions that solve the optimum control
planner’s problem. First consider the steady state costate equation conditions,9
@Hc
@ub
@Hc
@ug
= r
b
=
(pb
rkb )
b rkb
= r
g
=
(pg
rkg )
g rkg
and
are,10
The necessary conditions for
@Hc
@ b
@Hc
@ g
@Hc
= (pb
@
g
b;
b (s
g
+ q ( b)
(s + q ( g )
=
rkb
q ( b ) (1
( b ))
b
=
rkg
q ( g ) (1
( g ))
g
rkb )
(pg
rkg ) + rc + s (
b
b)
g) :
(22)
(23)
=0
(24)
=0
(25)
g)
= 0:
(26)
0
where ( i ) = qq(( ii)) i < 0 is the elasticity of the matching function for vacancies. The optimality
conditions for labor market tightness indicate that the number of jobs in each sector has to be
determined by equating vacancy creation costs to the average expected value of the matches taking
into account congestion externalities (which reduce matching rates by q ( i ) ( i )).
According to equation (26) the planner sets the skill-composition taking into account the investment costs for workers and the di¤erence in the social value of output associated to the workers’
investment decision. A higher fraction of educated workers means an increase in the net surplus of
the match ((pg rkg ) (pb rkb )) if employed but an increase in the cost of being unemployed if
the match is destroyed ( s ( b
g )) :
The next step is to evaluate the social allocation equations in the decentralized equilibrium.
The Hosios condition is imposed in the two sectors, that is = ( i ). This makes it possible to
9
It is very di¢ cult to obtain analytical solutions for the su¢ cient conditions for a maximization in this problem
as it is done in Acemoglu and Shimer (1999) and Blázquez and Jansen (2003). However, for this case focusing on the
necessary conditions is enough to compare the social allocation with the decentralized equilibrium.
10
It is obtained that price e¤ects cancel out in all the necessary conditions. In fact, given the CRS …nal output
technology, (2) , (3) , (4) and (5) it is shown in Appendix D that
@pb
(
@x
where x = ub ; ug;
ub ) =
@pg
(1
@x
:
11
ug )
identify the model ine¢ ciencies not related to the standard congestion externalities. Then, the
necessary conditions can be rewritten as
rkb =
rkg =
q( b )(1
Db
q( g )(1
Dg
rc = q( g )
g
)
)
pb
pg
pg
Dg
(27)
q( b )
b
pb
+r
Db
pg
Dg
pb
Db
;
where
Di = r + s + (1
)q( i ) +
i q( i ):
It turns out that the …rst two order conditions for e¢ ciency are satis…ed in the decentralized
c
economy if
= ( i ) : However, the expression for @H
in (26) evaluated at the decentralized
@
equilibrium with = ( i ) ; like in the last equation of (27), is negative. This means that, denoting
S to the social allocation, S < . Indeed, note that the equilibrium and social arbitrage conditions
p
pb
di¤er by a term r Dgg
Db : Intuitively, an additional worker in sector g changes total output
Y by pg pb and makes it easier for …rms in that sector to be matched, and harder for …rms in
the low-productivity sector to contact low-skill workers. The change in the ‡ow discounted value
p
pb
of matches associated to entry in sector g is represented by r Dgg
Db : Workers do not consider
this general equilibrium e¤ect of their education decisions.
There are also other general equilibrium e¤ects associated to the education externality. That is,
c
the fact that (24) and (25) but @H
< 0 at the decentralized equilibrium means that the equilibrium
@
S < . Given the comparative statics results for
b and g di¤er from their e¢ cient values with
described in Remark 1, a too high means a too high pg and a too low pb : This price "distortions"
are in turn associated to a too high g and a too low b: The following proposition summarizes
these results.
Proposition 1 If = ( i ), kg > kb and/or c > 0 the decentralized equilibrium exhibits a too high
fraction of low-skill workers, with a too high labor market tightness in the high-productivity sector,
and a too low one in the low-productivity sector. Denoting s to the social allocation
>
S
;
g
>
S
g;
b
<
S
b
It follows directly that the number of high-skill and low-skill unemployed is too low and too high,
respectively,
ug < uSg ; ub > uSb :
12
Note that in the absence of externalities in the determination of worker types, the standard
e¢ ciency result would apply and the decentralized equilibrium would be constrained e¢ cient if
= ( i ). This would not be the case in a model with random matching and kg > kb like
Acemoglu (2001). In such model, it is not possible to make …rms internalize congestion externalities
by imposing a common Hosios’value of ; and as a consequence wages are too high in the good-job
sector and too low in the bad-job sector leading to an ine¢ cient high fraction of bad-job vacancies.11
These e¤ects disappear with a sector-speci…c search assumption like in this paper because job
creation by …rms of any type only create congestion externalities to …rms of the same type and
then a sector-speci…c Hosios condition would give the right incentives to job creation in both sectors.
This result is consistent with Acemoglu and Shimer (1999) and Davis (2001). However, even in an
economy with sector-speci…c search this paper identi…es another kind of ine¢ ciency related to how
worker types are determined. In a model with an endogenous allocation of workers to the sectors,
there is an externality that implies that too few workers allocate to the high productivity sector.
This ine¢ ciency is increasing in the cost of investment for both …rms and workers in the good-jobs
sector. This externality does not obviously arise in the model with an exogenous skill composition
of the workforce.
We can also relate this result to the e¢ ciency analysis of wage inequality. Even if the Hosios
condition does not hold as long as the elasticity of the matching function is the same across the
sectors it will always be the case that the price di¤erence between the sectors is higher than it
should be and then there is too much wage inequality. This too unequal wage distribution does
not maximize social output. A very simple system of taxes and subsidies (which implies a net
subsidy to education) would restore constrained e¢ ciency in the model. It is accepted though that
from an applied point of view the policy maker would need to know the properties of the matching
function in both sectors for intervention purposes. The direction of the ine¢ ciencies arising in the
two sectors when the Hosios condition does not hold are well known. A contribution of this study
is to present an additional force that would a¤ect the e¢ ciency properties of the equilibrium.
11
Due to the random matching assumption present in Acemoglu (2001), any …rm posting a vacancy in any sector
increases labor market tightness making it harder for …rms in both sectors to meet an unemployed worker. Then it
happens that, for a given , the amount of the surplus transferred to workers in Nash Bargaining in bad-job matches
represents less than the value of the congestion externality implied by bad-job creation. The opposite occurs in the
good-job sector, where the wage is higher than the one that makes …rms internalize congestion externalities. Then,
this leads to a too high fraction of bad-job vacancies in the economy.
13
5
Conclusions
This paper presents a two-sector matching model with heterogenous …rms and workers and analyzes
the e¢ ciency properties of the equilibrium under the Hosios’condition. Given that search is sector
speci…c, search externalities are correctly internalized under the Hosios’ condition. However, due
to an education externality, the skill composition is ine¢ ciently biased toward low skilled workers.
This ine¢ ciency translates via price e¤ects into a too high labor market tightness in the high
productivity sector and a too low one for the other sector in equilibrium relative to the planner’s
allocation. The magnitude of these distortions is increasing in the education cost.
References
[1] Acemoglu, Daron (1996) “A Microfoundation for Social Increasing Returns in Human Capital
Accumulation”, Quarterly Journal of Economics, 779-804.
[2] Acemoglu, Daron (1997) “Training and Innovation in an Imperfect Labor Market”, Review of
Economic Studies, 445-464.
[3] Acemoglu, Daron (2001) “Good Jobs versus Bad Jobs”, Journal of Labor Economics, 1-21.
[4] Acemoglu, Daron and Shimer, Robert (1999) “Holdups and E¢ ciency with Search Frictions”
International Economic Review, 827-849.
[5] Albrecht, James and Vroman, Susan (2002) “A Matching Model with Endogenous Skill Requirements”, International Economic Review, 283-305.
[6] Aricò, Fabio (2009) "Both Sides of the Story: Skill-biased Technological Change, Labor Market
Frictions and Endogenous Two-Sided Heterogeneity", CDMA Working Paper 0908.
[7] Autor, David; Katz, Lawrence and Kearney, Melissa (2005) "Rising Wage Inequality: The
Role of Compositions and Prices", NBER Working Paper 11628.
[8] Autor, David; Katz, Lawrence and Kearney, Melissa (2008) "Trends in U.S. Wage Inequality:
Revising the Revisionists", Review of Economics and Statistics, 90, 300-323.
[9] Blázquez, Maite and Jansen, Marcel (2008) “Search, mismatch and unemployment", European
Economic Review, 52(3), 498-526.
[10] Cardullo, Gabrielle (2009) "Equilibrium in Matching Models with Employment Dependent
Productivity", Economics Bulletin, 29(3), 2380-2387.
14
[11] Cahuc, Pierre; Postel-Vinay, Fabien and Robin, Jean-Marc (2006) “Wage Bargaining with
On-the-job search: Theory and Evidence”, Econometrica, 74(2), 323-64.
[12] Cole, Harold; Mailath, George and Postlewaite, Andrew (2001a) "E¢ cient Non-Contractible
Investments in Large Economies", Journal of Economic Theory, 101, 333 373.
[13] Cole, Harold; Mailath, George and Postlewaite, Andrew (2001b), "Investment and Concern
for Relative Position", Review of Economic Design 6, 2001, 241-261.
[14] Davis, Steven (2001) “The Quality Distribution of Jobs and the Structure of Wages in Search
Equilibrium”, NBER Working Paper 8434.
[15] Dolado, Juan; Jansen, Marcel and Jimeno, Juan (2009) “On-the-Job Search in a Matching
Model with Heterogeneous Jobs and Workers”, The Economic Journal, 119, 200-228.
[16] Felli, Leonardo and Roberts, Kevin (2002) "Does Competition Solve the Hold-up Problem?
CEPR Discussion Paper 3535.
[17] Gautier, Pieter (2000) “Search Externalities in a Model with Heterogeneous Jobs and Workers”, Economica 273, 21-40.
[18] Hosios, Arthur (1990) “On the E¢ ciency of Matching and Related Models of Search and
Unemployment”, Review of Economic Studies, 279-298.
[19] Masters, Adrian (1998) “E¢ ciency of Investment in Human and Physical Capital in a Model
of Bilateral Search and Bargaining”, International Economic Review, 477-494.
[20] McKenna, C.J. (1996) “Education and the Distribution of Unemployment ”, European Journal
of Political Economy, 12, 113-132.
[21] Pissarides,Christopher (2000) Equilibrium Unemployment Theory, Second Edition, Basil
Blackwell, Oxford.
[22] Uren, Lawrence (2006) “The Allocation of Labor and Endogenous Search Decisions”, Topics
in Macroeconomics, The Berkeley Electronic Press, 6, 1, Article 13.
15
Appendix A. Existence and Uniqueness of the Equilibrium
The equilibrium values of vacancies with the equilibrium
leads to,
q( i )(1
rJiV = rki =
Di
Taking derivatives,
n h 0
q(
)(1
)
pi qq(( ii)) (r + s)
V
i
@rJi
=
@ i
Di2
@rJiV
@ j
=
q( i )(1
Di
value of unemployment substituted in
)
pi :
i
q( i )
+
@pi
@ i Di
o
<0
) @pi
> 0:
@ j
By the implicit function theorem
d
d
d
d
b
=
gB
b
@rJbV
@ g
@rJbV
@ b
=
@pb
=
@ g
=
@rJgV
@ g
@rJgV
@ b
gG
=
=
@pb
@ g
pb
0
Db (q ( b )=q( b ))(r
rkb
q( b )(1
+ s)
q0( b)
(r + s)
) q( b )
pg
0
Dg (q ( g )=q( g ))(r +
=
@pg
@ b
0
rkg
q ( g)
q( g )(1
)
q( g )
(r + s)
s)
q( b )
+
q( g ) +
q( g )
+
>0
@pb
@ b
q( b ) +
@pb
@ b
@pg
@ g
@pg
@ g
=
>0
@pg
:
@ b
Given the CRS in the …nal output technology and that intermediate goods are sold in perfectly
i
competitive markets, it is always the case that @@pji = YYji @p
@ i (see Appendix B below for a proof).
Then, it follows that
d
d
d
d
b
=
gB
b
=
gG
Yb @pb
rkb
q0( b)
@pb
=
(r + s) q( b ) +
<1
Yg @ b
q( b )(1
) q( b )
@ b
rkg
q0( g )
@pg
@pg
Yb
(r + s) q( g ) +
=
> 1;
Yg q( g )(1
) q( g )
@ g
@ g
which imply that
d
d
b
gB
d
d
b
< 0:
gG
Then the good job locus is always steeper than the bad job locus assuring that the equilibrium
is unique. Note also that for the case of perfect substitutes ( = 1) the bad job locus is a horizontal
16
line and the god job locus is a vertical line. In this case there are no price e¤ects and the equilibrium
may not be interior.
Note that as b ! 0 production is concentrated mostly in the good job sector which has g > 0
and …nite. In that case pb ! 0, ub ! . Then …rms will not open vacancies in sector b and only the
good job sector prevails. In that situation pg = (1
)1= and the equilibrium value of vacancies
1=
q( )(1 )(1
)
is obviously rJgV = rkg = r+s+(1g )q( g )+ g q( g ) : The opposite occurs when g ! 0:
Then, for any given the good job locus is always steeper than the bad job locus assuring that
the equilibrium is unique. It is concluded that for every there exists a unique equilibrium pair
( g ; b ) : It is left to show that there is a such that the triplet ( g ; b ; ) satis…es not only the
@rJ V
@rJ V
free entry conditions but also (21). To prove this, consider that @ g > 0 and @ b < 0 which in
turn implies that both the good- and bad-job loci shift to the right with increases in . Indeed,
d
in a version of the model with given, d g > 0 and dd b < 0.12 This means that for higher ,
the equilibrium pair ( g ; b ) involves higher values of g and lower values of b : Then we have that
rJgU rJbU is monotonically increasing in . Finally, given that rJgU rJbU is negative for ! 0
and positive for ! 1 there is one and only one that satis…es the workers’indi¤erence condition.
Appendix B. Price E¤ects in the Equilibrium Analysis
q0 ( i )
@pi
Yi @pi
@ j =
Yj @ i . Using ( i ) =
q( i ) i < 0 and recalling
q( g ) g (1 )
Yg = q( g ) g +s ; the following results are also useful:
I …rst present a proof of
Yb =
q( b ) b
q( b ) b +s
and
that in equilibrium
1
( b)
(q( b ) b + s) b
1
( g)
= Yg s
(q( g ) g + s) g
@Yg
@Yb @Y
= pb
;
= pg
:
@ b @ g
@ g
@Yb
@ b
@Yg
@ g
@Y
@ b
= Yb s
12
Indeed, following an analysis similar to the one of Appendix C below to sign the expressions, it is easy to show
that
V
V
d
d
g
=
and
d
d
:
b
=
@rJg @rJbV
@ b
@
@rJg @rJbV
@
@ b
@rJgV @rJbV
@ g
@ b
@rJbV @rJgV
@ g
@ b
V
@rJbV @rJg
@ g
@
V
@rJbV @rJg
@
@ g
@rJgV @rJbV
@ g
@ b
@rJbV @rJgV
@ g
@ b
17
>0
< 0:
Then, from (16) we have
@pg
@ g
@pb
@ g
@Yg
+ (1
) Yg
@ g
1
( g)
Yb
= (
1) pg s
(q( g ) g + s) g
Y
@Y
=
(1
) Yb 1 Y
@ g
1
( g)
Yg
Yb
= (1
) pg s
;
(q( g ) g + s) g
Y
Yb
= (1
which con…rms that
)(
@pg
@ g
@pb
@ b
@pg
@ b
and also
@pb
@ b
=
=
2
1) Yg
@pb Yb
@ g Yg
Y1
1)pb s
= (1
(1
@Y
@ g
)Y
as claimed in the paper: In a similar way,
1
( b)
(1
(q( b ) b + s) b
1
( b)
)pb s
(1
(q( b ) b + s) b
= (
1
Yg
Y
Yg
Y
)
)
Yg
;
Yb
@pg Yg
@ b Yb :
Appendix C. Proof of Remark 1
We totally di¤erentiate equations (19), (20) and (21). Given that rJiU =
condition can be conveniently rewritten as
where
=
)
kb
kg b :
+
drkg =
@rJgV
d
@ g
g
+
@rJgV
d
@ b
b
+
@rJgV
d
@
drkb =
@rJbV
d
@ g
g
+
@rJbV
d
@ b
b
+
@rJbV
d
@
drc =
Then,
g
c (1
kg
2
6
A6
4
d g
drkg
d b
drkg
d
drkg
rkg d
1
3
g
1
2
3
2
1
7 6 7
7 = 4 0 5 and A 6
4
5
0
2
6
A=6
4
@rJgV
@ g
@rJbV
@ g
1
rkg
@rJgV
@ b
@rJbV
@ b
1
18
rkb
1
rkb d
d g
drc
d b
drc
d
drc
3
b
+ 0d
3
0
7 6 7
5=4 0 5
1
@rJgV
@
@rJbV
@
0
2
3
7
7:
5
rJiV
i
the non arbitrage
We know from Appendix A that
@rJbV
@
@rJiV
@ i
<0,
@rJiV
@ j
> 0 and it is easy to show that
@rJgV
@
> 0 and
< 0: Given that det A > 0 (see proof below); we have
d g
drkg
=
d b
drkg
@rJgV @rJbV
@ b
@
@rJgV @rJbV
@
@ b
<0
<0
rkb +
@rJbV
@ b
rkg
>0
det A
@rJgV @rJbV
@ b
@
@rJgV @rJbV
@
@ b
det A
@rJgV @rJbV
@
@ g
@rJgV @rJbV
@ g
@
det A
@rJgV @rJbV
@ g @ b
=
@rJgV @rJbV
@ b @ g
det A
it is easy to show that
derivations used to prove that
@rJbV
@ g
1
=
d
drc
rkg
det A
=
d b
drc
d
drkg
@rJbV
@
1
=
d g
drc
rkb
det A
=
d
drkg
To sign
@rJbV
@
1
d b
d gB < 1
@rJgV @rJbV
@
@ g
@rJiV
@ i
>
@rJiV
@ j
d b
d g G > 1:
@rJgV @rJbV ;
,
@ g
@
and
>0
<0
>0
: To sign
The signs
d
drc we use from Appendix A the
d
of drcg and ddrcb depend on the sign
of
and
which according to points 1. and 2. of the
proof of det A > 0 below are positive and negative, respectively.
Proof that det A > 0
!
@rJgV @rJbV
@rJbV @rJgV
det A = rkg
+
1
@
@ b
@
@ b
!
@rJgV @rJbV
@rJbV @rJgV
rkb
1
@
@ g
@
@ g
It can be shown that both terms in parenthesis are positive. To prove this use
and the expressions for
@rJiV
@ i
and
@rJiV
@ j
@pi
@ j
=
Yi @pi
Yj @ i
from Appendix A to obtain the sign of the following objects:
1.
@rJbV
@
@rJgV
@
b
@rJgV
@
@rJbV
@
b
=
q( g )(1
Dg
) @pg q( b )(1
@
19
n h 0
) pb qq(( bb)) (r + s)
Db2
q( b )
io
>0
2.
@rJbV
@
@rJgV
@
g
@rJgV
@
@rJbV
@
q( b )(1
Db
=
g
n h 0
q( )
) pg q( gg) (r + s)
) @pb q( g )(1
@
q( g )
Dg2
io
>0
Appendix D. Price E¤ects in the E¢ ciency Analysis
@pg
@x
b
In footnote 11, I claim that @p
ub ) =
@x (
Recall that Yb =
ub and Yg = 1
@pb
Yb = (
@
= (
(1
ug : Using
ug ) for x = ub ; ug;
@Yg
@Yb
@Y
@ = 1; @ = 1, @
@Yb
Yb
+ pb (1
@
Y
Yg
1) pb (1
)
Y
1) pb
)
= pb
pg we have
@Y
@
Yb
Y
+ pg
and
@pg
Yg = (
@
= (
pg @Yg
pg
@Y
+
(1
)
Yg @
Y
@
Yg
Yb
1) pg
+ (1
)
Y
Y
1)
Similar results can be obtained for x = ub ; ug noting that
instance,
@pb
Yb = (
@ub
@pg
Yg
@ub
@Yi
@ui
=
@Yb
Yb
@Y
+ pb (1
)
@ub
Y
@ub
Yg
= (1
) pb (1
)
Y
Yg
@Y
= pg (1
)
Y
@ub
Yg
= (
1) pb (1
)
:
Y
1) pb
20
pb
1;
@Yj
@ui
= 0,
@Y
@ui
=
pi . For