www.elsevier.nlrlocatereconbase
Monopsonistic competition for the ‘‘best’’
workers
1Fabio Fiorillo, Stefano Santacroce, Stefano Staffolani
) Department of Economics, UniÕersity of Ancona, Piazzale Martelli 8, 60121 Ancona, ItalyReceived 14 July 1998; accepted 5 July 1999
Abstract
In this paper we analyse an economy where firms use labour as the only production factor, with constant return to scale. We suppose that jobs differ in their non-wage characteristics so each firm has monopsonistic power. In particular, we suppose that workers are heterogeneous with respect to their productivity. Then, each firm has incentives to offer higher wages in order to recruit the most productive workers. Competition among firms leads to a symmetric equilibrium wage, which is higher than the reservation wage, and to involuntary unemployment for the less productive workers, who are willing to work at the current wage but are not hired because their productivity is lower than the wage level. If firms have no institutional constraint on paying lower wages for the same job, an endogenous labour market segmentation emerges.q2000 Elsevier Science B.V. All rights
reserved.
Keywords: Human resource; Company Wage Policies; Wage model
1. Introduction
The management of human resources becomes more and more important if Ž workers do not have the same individual characteristics. In the Weiss Weiss,
)Corresponding author. Tel.:q39-071-2207090; E-mail: [email protected]
1
We thank R. Martel, R. Lucchetti, R. Balducci, T. To and the participants at the 12th AIEL
ŽAssociazione Italiana Economisti del Lavoro Conference for their suggestions. All remaining errors.
are ours.
0927-5371r00r$ - see front matterq2000 Elsevier Science B.V. All rights reserved.
Ž .
.
1980 efficiency wage model or in the Manning ‘Company Wage Policies’ model ŽManning, 1993 , workers differ in their productivity level and in their relative. valuation of leisure, respectively, given a scenario where firms must pay the same wage to all workers.
The main assumptions in our model are that workers have the same reservation wage, but they differ in their productivity level.2
Moreover, they earn the same wage if employed in the same firm. There are reasons that can justify the rule of ‘‘one job, one wage’’. As Manning 3 wrote:
. . . seemingly identical workers receive different rewards depending on who
Ž .
they work for . . . many firms both union and non-union seem to have less variation in wages among workers doing a particular job then there are differences in productivity and this seems to be known to the employer . . . It would seem that wages are often attached more to the firm . . . than the individual.
A second group of hypotheses concerns the demand side of our labour market. In this economy firms have a certain monopsonistic power depending upon their job characteristic as in the recent article by Bhaskar and To, 1999, p. 190:
We argue that the appropriate model is monopsonistic competition, where a large number of employers compete for workers and are able to freely enter or exit. However, different jobs have different non-wage characteristics, giving each employer market power in choosing the wage, even though she employs only a small fraction of the work force.
As in their paper we suppose that firms are spatially differentiated,4 thus
workers incur travelling expenses.
The workers and jobs heterogeneity hypothesis leads firms to compete in order to hire the best workers, offering a wage higher than the market clearing one.
Contrary to labour market perfect competition models, firms do not face perfectly elastic labour–supply schedules. Small changes in wages do not produce infinite changes in the available supply of labour. In fact, the presence of travelling expenses makes workers move subject to a cost–benefit calculus.
2
Ž .
In Weiss’s model 1980 workers differ in reservation wage that is positively correlated with
Ž .
productivity. As Rebitzer 1989 notes this assumption holds if ‘‘a large, non-industrial sector of self-employed craftsmen, professional and casual labourers’’ exists. On the contrary Weiss’s model disintegrates.
3
Ž .
Manning, 1993, p. 1. See also the ‘‘gift exchange economy’’ described by Akerlof 1982 and the
Ž .
‘‘social status’’ considerations proposed by Frank 1985 .
4
In order to maximize profits, each firm could offer above-average wages. In this way it raises the dimension of labour force that would like to work for the firm, i.e., it enlarges its basin of attraction, widening the range of the bowl from which the best workers are selected. Under certain conditions, we demonstrate how firms will hire only those applicants whose productivity level is higher than the offered wage.
Following this way, it is possible to calculate a sort of reaction function for each firm, whose wage will depend on the average wage paid in the industry. A symmetrical steady state equilibrium emerges from this framework, characterised by a wage higher than the market clearing level. Let us call this kind of labour market as monopsonistic competition.
A simple example will help to understand our model. Let us suppose that in an economy there are two firms and four workers. Firms have perfect knowledge of workers’ productivity and operate with constant returns to scale. Workers’ reserva-tion wage is Õs1y´, with ´ infinitesimal. Moreover, two of the workers are
‘‘good’’ ones and produce two units of output each day. The other two are ‘‘bad’’ and produce only one unit of output each day. Each firm may pay a wage Õ, or it
can choose to pay a wage w)Õs1y´. If firms pay the reservation wage, all
workers are employed. When firms pay the same wage, their labour force is composed in the same way. If the two firms pay Õ, the profit of each firm is:
Ž . Ž .
p s 1yÕ q 2yÕ s1q2´. When both pay w)Õ the ‘‘bad’’ quality
Õ,Õ
workers will be unemployed, and each firm has a profit:pw,ws2yw. When firm
1 pays w while firm 2 pays Õ, firm 1 can hire the two ‘‘good’’ workers obtaining
Ž .
a profit of: pw,Õs2 2yw , while the other firm employs the bad workers and
Ž .
obtains 2´. The opposite is true when firm 2 pays w and firm 1 pays Õ Table 1 .
Ž . Ž .
In this simple game a la Bertrand , the optimal strategy
`
Õ,Õ is dominated byŽ .
the Nash equilibrium w,w : Therefore, firms will pay a wage which is higher than the reservation wage and a part of the labour force will be unemployed. Note that it is rational for each firm to raise its wage over the wage w paid by the other. So,
Ž .
in this case the wage equals the productivity of the best workers ws2 and profits are zero.
In the next sections we will present a model to clarify the main issues of this introduction. The model is presented in a context of perfect information. This is one of the major simplifications of our model, and it implies that, when unemploy-ment exists, people without jobs are those with a lower productivity level.
Table 1
Firm 2
Õ w
Ž .
Firm 1 Õ 1q2´, 1q2´ 2´, 2 2yw
Ž .
Section 2 will introduce the main characteristics of the model in two different Ž
cases: when firms do not compete for workers monopsony with heterogeneous .
workers and when firms act as monopsonistic competitors for the best workers; in Section 3 we will suggest an alternative functioning of this kind of labour market introducing the possibility of labour market segmentation. Section 4 is devoted to some concluding remarks.
2. The model
In our model, workers are located around a ring, where the whole population Ž .P lives see Fig. 1 . Around this ring there is also a certain endogenous numberŽ .
Ž .
of firms J , which produce an homogeneous item in a perfectly competitive
goods market. So jobs are differentiated by their location.5
Thus, each firm chooses his workers within a limited area of the ring. Let us
Ž .
define Lj as the labour supply attraction basin of the j-th firm.
We suppose that workers are heterogeneous with respect to their productivity Ž .l ; this means that for a given job, there are workers able to produce higher output.6 We assume uniformly distributed productivity, so the number of workers
for each level of productivity is
1
ms P
Ž .
1bya
where b and a are respectively the maximum and the minimum level of productivity level, and P is the population.
Assuming constant return to scale, profits of the j-th firm are given by:
pjs
Ž
1qs lj.
Ž
w ,w n w ,wj. Ž
j.
yn w ,w wŽ
j.
jyFŽ .
2wheresj is an idiosyncratic shock with zero mean, w is the wage paid by firm j,j
w is the outside wage,lis the average productivity of labour, n is employment in the firm and F are exogenous fixed costs.
5
In a more general setting, we could suppose that workers prefer a particular firm in relation to the characteristics of the offered job. Therefore, in our model the spatial localisation is simply a metaphor of job differentiation.
6
Fig. 1. Firms and workers in a ‘‘ring’’ economy.
Employment and average productivity depend both on wage paid by firm j and on wages of other firms. Firm j maximizes its profits choosing the wage wj defined by:
1qs l nqln sn wqn
Ž .
3Ž
j.
wj wj wj jŽ .
The employment at firm level nj is constrained by the labour force that each
Ž Ž ..
firm may find on the market L w ,w .j j
Ž .
All workers are willing to work if the wage R is bigger enough to cover the
Ž . 7
reservation wage plus the travelling expenses arJ , where a is the travelling
Ž . Ž .
cost per unit of distance. So labour force L is less than population P if:
a
w- qR
Ž .
4J
If condition 4 holds, each firm acts as a pure monopsonistic one, since the attraction basins of two contiguous firms do not overlap; in the opposite case each firm will interact with others in a monopsonistic competition framework.
Let us examine the two cases separately.
2.1. Monopsony with heterogenous workers
If condition 4 holds, for some workers it is not convenient to participate in the labour market. In order to calculate the labour force available to the firm,8 that is
the geographic basin of attraction of firm I , let us focus on the worker on point A1 of Fig. 1 for whom it is indifferent to work in firm I or to not participate in the1 labour market.
7
ŽarJ is the travelling expense of the farest worker..
8
For this worker, the following equation must be valid: w1ya2I A1 sR. We
Ž . Ž .
can also write w1yRr a s2I A.1
Ž .
Then it is not difficult to look at 2PI A as the geographic basin of attraction1
of firm I . If the workers are uniformly distributed around the ring, the following1
ratio must be valid: 2I A:11 sM : P, where M are those workers attracted by the1 1
monopsonistic firm I . Substituting 2I A and generalizing for firm j, we obtain1 1 the available labour force for firm j:
wjyR
M wj
Ž
j.
s PŽ .
5a
We suppose that firms have perfect knowledge of workers’ productivity. Thus, for a given level of wage, firm j will employ all workers of its attraction basin with productivity higher than the wage. Clearly the attraction basin is increasing
Ž .
on the wage paid by the firm. Thus, integrating Eq. 5 from w to b, we obtainj
Ž .
the employment in the j-th firm n wj
M w
Ž
.
wyRb j j j
nj
Ž
wj.
sH
m dLsmŽ
byw .j.
Ž .
6P a
wj
Ž . Ž . Ž .
which is increasing in w until wj jF bqR r2 sw see Fig. 2 . It is important
ˆ
j to realize that the wage is not only a variable cost, but also a strategic tool to raise the firm’s average productivity, because output and employment are a function of the wage.Ž . Ž .
The unemployment rate, obtained from Eqs. 5 and 6 :
n w Jj
Ž
j.
bywjus1y s1y
Ž .
7L w Jj
Ž
j.
bya [image:6.595.117.322.405.547.2]is increasing in w because the labour force increases faster than employment.j
Proposition 1 In a monopsonistic economy characterised by workers with equal
reserÕation wage but different labour endownements, inÕoluntary unemployment
exists
Unemployment exists because firms raise the wages above the equilibrium level in order to enlarge their basin of attraction and to hire the most productive workers.
Ž Ž . .
For a given number of firms J w sJ the participation rate is: L w Jj
Ž
j.
wjyRps s J
P a
where LsJM is the total labour force.j
Ž . Ž . Ž . Ž .
Taking into account Eq. 1 , we have: l wj s bqwj r2 , so lwjs1r2. Substituting in lambda, we obtain:
1qsj
nwj
ž
1ys bywj/
snŽ .
8j
Ž . Ž . Ž .
Substituting Eqs. 5 and 6 into Eq. 8 we obtain the optimal wage paid by all firms in a pure monopsonistic framework with heterogeneous workers
1 b 2
U
wj s q R
Ž .
93 1ysj 3
The equilibrium wage is a weighted average of the best workers’ productivity
Ž . Ž . Ž .
and the reservation wage. Substituting Eq. 9 in Eqs. 5 – 7 we may obtain the equilibrium level of the labour force, employment, and unemployment rate.
Table 29 presents some results concerning the effects of parameters on
equilibrium levels.
Some of the relations of Table 2 are obvious, others are more difficult to explain. For example, a positive shock raises labour force more than employment, increasing the unemployment rate; a reduction of transportation costs or a popula-tion growth raises both employment and labour force leaving unchanged the unemployment rate.
Such as in classical monopsony theory, a binding minimum wage may raise
10 Ž .
employment. In our model, a minimum wage Õ raise employment if ÕFw
˜
jsw Ž .x
min w , R
ˆ
j q arJ , where w is the classical condition in monopsonistic labourˆ
j markets, that is the level of wage which maximizes the labour demand function9
All the derivatives hold withsj small enough.
10
Table 2
Derivatives on monopsony
Žbya. a R P s J
Impact of parameter on wages . . q . q .
Impact of parameter on total employment y y y q y q
Impact of parameter on total labour force . y y q q q
Impact of parameter on unemployment rate q . q . q .
Ž . Ž . Ž .
n wj see Fig. 2 , and Rq arJ depends on labour force participation. Given
that wUj Fw , firms operate in an upward sloping labour demand function.
ˆ
j Ž .If the minimum wage is less than the critical value w :
˜
jProposition 2 For a giÕen number of firms, the effects of the introduction of a
minimum wages on a monopsony with heterogeneous workers are:
1. Raising employment at firm level and at aggregate level
2. Enlarging the attraction basin of each firm and raising labour force 3. Raising both the unemployment rate and the participation rate.
Clearly, this analysis is correct when the number of firms is constant, that is when product market is oligopolistic, when entry thresholds exist or generally in the short run.
If we assume free entry condition firms enter until profits are positive and exit
Ž . Ž .
when negative. Substituting Eq. 9 in Eq. 2 we find that the level of profits in a monopsony with heterogeneous workers does not depend on the number of firms
J.
3
2
Ž
byR.
pjs m yF
27 a
Proposition 3 If we suppose free entry of firms, monopsony cannot exist because:
Ø Ifp-0, all the firms exit. Ž .
Ø Ifp)0, firms enter up to Eq. 4 does not hold, thus monopsony change in a monopsonistic competition labour market.
2.2. Monopsonistic competition with heterogeneous workers
Ž .
Assuming that Eq. 4 does not hold, then
a
w) qR
Ž .
10Ž .
where J w indicates that now the number of firms is endogenous to the model. Let us calculate the labour force available to the j-th firm which pays a wage of
w when the outside wage paid by the other firms is w.j
It is important to realise that, when a firm pays its workers more than the other firms, it faces a labour supply function that is different from that one calculated when it pays its workers less than other firms.
Ž .
Paying a wage above the average wj)w , a firm attracts a greater number of
workers and recruits those workers whose individual productivity is higher than the wage.11 So, a higher wage leads to higher average productivity of workers
employed.
The employment in a firm that pays wj)w is indicated by the area B of Fig. 3.
We can write:12
1 wjyw
L w ,wj
Ž
j.
sž
q/
PŽ .
11J w
Ž .
aŽ .
The first term in Eq. 11 represents the share of labour force available to the firm when wages are the same for the whole economy and the second term is the way in which the firm could increase this share. A worker will decide to move to
Ž .
another firm only if the net-income wjyw covers, at least, his increased travelling expenses.
Ž .
Integrating Eq. 11 we obtain the firm employment
w ,w
Ž
.
b j
n
˜
jŽ
w ,wj.
sH
m dLŽ .
12P wj
Ž
Given the distribution of labour force productivity, firms the black points on .
the ring in Fig. 1 compete on the labour market to hire the best workers offering wages which are higher than the reservation wage. In this case, the average
˜
Ž . Ž .productivity of the j-th firm is simplyls bqwj r2 .
On the contrary, if firm j pays wj-w, it loses some of its more productive
workers who are hired by firms paying higher wages. However, it can use its monopsonistic power over those workers with productivity between w and w.j Employment is now given by area C of Fig. 3. In fact low productive workers cannot be employed by high wage firms but only by low wage ones and so they have to decide if work or not, having no possibility to decide where to work.
11 Ž .
The same hypothesis has been proposed by Manning 1993 .
12
Following the same procedure used for monopsony, there is a person on point B of Fig. 1 who is indifferent to work in firm I or in firm I . For this worker, the following equation must be valid:1 2
Ž . Ž .
wjaI Bs1 wyaI B. We can also write w2 jywr a sI ByI B; the second term is equivalent to:1 2
Ž .
Fig. 3. Employment in firm J with monopsonistic competition.
It is not difficult to show that employment in firm j is:
n
ˆ
jŽ
w ,wj.
sn1 w ,wŽ
j.
qn2 w ,wŽ
j.
Ž .
13w
L w ,w
Ž
.
M wŽ
.
b j j j j
n
ˆ
jŽ
w ,wj.
sH
m dLqH
m dLŽ .
14P P
w wj
Ž Ž ..
where the first integral n1 w ,wj gives the number of workers with productivity
Ž Ž ..
higher than w, while the second n2 w ,wj represents those workers with productivity lower than w, who have to choose between being hired at a lower wage than the average wage, or being unemployed. The average productivity is:
n1 w ,w
Ž
j.
bqw n2 w ,wŽ
j.
wqwjˆ
ls q
n w ,w
ˆ
Ž
j.
2 n w ,wˆ
Ž
j.
2Ž .
In the symmetric case, all firms have the same shock sjss ;j . Substituting
˜
in equation lambda the definitions of n, and
˜
l we obtain the reaction function ofŽ . Ž U.
firm j when outside wages w are smaller than the equilibrium wage w ;
ˆ
Proposition 4 There is a unique fixed point for reaction functions, this fixed point
is the Nash-symmetric equilibrium wage wU which is stable13
Ž .
In the short-run J w sJ; the equilibrium wage is:
b 2a
U
w s y
1ys J
Let us analyse this equilibrium. The employment of each firm is:
m 2a sb
n
˜
snˆ
snž
y/
Ž .
15J J 1ys
Ž . ŽŽ .
Thus, a common positive shock s)0 raises the equilibrium wage dwrds
. ŽŽ . .
)0 but reduces firm employment d nrds -0 . This result depends on the competition between firms. They try to hire more workers by raising their wage, but, given the symmetric behavior, workers with a productivity which is lower than the new wage will be fired, so that the employment level decreases.
According to Bhaskar and To, this situation may be defined ‘‘Oligopsony’’ in the labour market.
Proposition 5 In an oligopsonistic labour market, the unemployment rate is
decreasing in job differentiation and it is increasing in theÕariance of workers’
productiÕity.
The complete derivatives of variables with respect to the parameters are shown in Table 3.14
Until now we have considered a given, exogenous number of firms. However, contrary to the previous analysis, we could endogenise the number of firms. This will modify the previously results.
For the sake of simplicity we set ss0. Perfect competition in the product Ž .15
market sets profits to zero; consequently we obtain from Eq. 2 the endogenous number of firms as a function of the wage:
2
m
Ž
byw.
J w
Ž .
sŽ .
162 F
13
Appendix A will demonstrate this proposition.
14
All the derivatives hold withsj small enough.
15
Table 3
Derivatives on oligopsony
Žbya. J a P s
Impact of parameter on wages . q y . q
Impact of parameter on firm employment y y q q y
Impact of parameter on total employment y y q q y
Impact of parameter on unemployment rate q q y . q
This enables us to find an equilibrium value of w under the zero profit assumption. We obtain:
1r3
aF
U
w sby2
ž /
Ž .
172m
We may calculate JU, the number of firms compatible with the free entry condition.
1r3 2
a
U
J s
ž
2 m/
Ž .
18F
Ž .
From Eq. 15 it is easy to calculate employment in each firm:
1r3
2m
U 2r3
n s
ž /
FŽ .
19a
The unemployment rate is simply given by:
U 2r3 1r3
Lyn 2 aF
U
u s s1y
ž
/ ž /
Ž .
20L bya P
Unemployment is completely involuntary: the unemployed would like to work for current wages, but they are not hired because wages are an instrument used by firms in order to select the best workers. As we mentioned above, in this model, as in those with adverse selection, even in a setting where unemployment is high firms do not cut wages.
Ž .
A wider range of productivity bya — i.e., stronger heterogeneity—
wors-ens the economy’s employment performance. Moreover, a reduction in travelling
Ž .
expenses a, or, more generally, a reduction in job differentiation increases the unemployment rate because the power of wages, as a tool to enlarge the attraction basin, becomes stronger.
So, according to this framework, countries with more homogeneous workers and less job differentiation should have lower wages and a lower unemployment rate.
Ž Ž .. number of firms. In the long run, substituting the number of firms Eq. 16 in Eq. Ž15 , we obtain.
2 F
ns
byw
that is an upward sloping relation between employment at firm level and wages. Ž The total employment and the unemployment rate are respectively: nJs by
. Ž . Ž . Ž .
w r bya P and us1y byw r bya .
Proposition 6 In the long run, the effects of minimum wages on a monopsonistic
competition economy with heterogeneous workers are:
1. Raising employment at firm level.
Ž .
2. Reducing the number of firm, because of Eq. 16 .
3. Reducing total employment, because the reduction of the number of firms J is greater than the increase in firm employment.
4. Raising the unemployment rate.
3. Unemployment vs. market segmentation
There could be some social norms, or institutional rules which forbid defini-tively firms to pay a wage lower than the equilibrium wage wU of the previous section. In this case the dynamics of the labour market would immediately end, and the previous results could be seen as a steady state solution.
In the absence of these norms, an intriguing question arises: why don’t new firms enter the market exploiting the unemployed workers and paying them a lower wage?
In other words, we wonder why people do not move towards a secondary sector characterised by lower wages and lower productivity. If this possibility exists and it is economically profitable, the labour market will be segmented in two or more sub-markets.
Let us label each segment of the market with i, ranking the segments from the most to the least productive. Segment i hires a certain number of workers, paying a wage of w .i 16
Each segment chooses the wage and, consequently, the number of workers, by
Ž .
using the same reasoning of the most productive segment is1 . The only difference is that the new firms know that the range of productivity is no longer between a and b, since the upper segment of firms has already hired those with
16
Ž .
Ž . Ž . productivity between wi and b. Therefore, we should integrate Eq. 12 or Eq. Ž13 between w. iq1 and w , where wi iq1 is the optimum wage rate resulting from
Ž .
profit maximization of the iq1 th segment. This behavior continues until condition 10 holds.
In fact, as workers with higher productivity are employed in firms of a higher segment, workers disposable for lower segments decrease. The firm’s attractive power becomes weaker and weaker because of decreasing wage. Monopsonistic competition is a feasible market structure only if all workers are able to reach any firm to offer their skills. In fact, it is possible that some workers have to bear transportation costs higher than the benefits of wage. Moreover, firms face another constraint on the supply side, since the demand of labour must be less than its supply.
Therefore, two constraints must apply for market segmentation:
Ž .
1. wi) arJ qR
2. L Jj iGÝin Ji j i
The first constraint states that all workers are willing to participate to labour force; as soon as it does not apply market structure changes from monopsonistic competition to pure monopsony. The second states that the demand for labour is less than its supply.17 Let us define iU as the last segment for which both
ŽU .
conditions hold. In this case the existence of another segment i q1 depends on the possibility to have a pure monopsonistic segment for the workers with the lowest productivity.
As we show in Appendix B, when the first condition does not hold anymore, monopsony profits are negative. Therefore, we could exclude the existence of a pure monopsony segment. While, if the second constraint is binding, we have no unemployment and in the last segment the wage is set equal to the lower bound productivity a.
The relative strength of two constraints depends on the value of the parameters. As we prove in Appendix B:
Proposition 7 With free entry in the long run, a monopsonistic segment cannot
exist; the inÕoluntary unemployment is a possible result eÕen with the existence of
secondary labour market.
This situation could not necessarily be considered as ‘‘marginal’’: there may be very few segments, particularly when the variance of the productivity distribution is low; in this case the share of workers who cannot be employed may be large. The unemployment is involuntary since workers are willing to work both at current and at lower wages, but there is no incentive for firms to segment the labour market.
17
The number of segments characterised by monopsonistic competition is
1r3
1 1 byR U byR 2 1
i-y q J s
ž
m/
yŽ .
212 2 a 2 aF 2
where JU is the number of firms in the long-run in each segment, which is a U Ž Ž 2 . .1r3
constant J s 2 a rF m . So, the number of segments is increasing with the variance of productivity and decreasing with transportation and fixed costs.
It is easy to demonstrate that in the long-run equilibrium each segment has the same number of firms JU and employs the same number of workers nUs
ŽŽ . Ž ..1r3 2r3
2m r a F , while a representative firm of the i-th segment pays a wage ŽŽ . Ž ..1r3
wisby2 i aF r2m , which is decreasing with the number of segments. The unemployment rate is
U U 2r3 1r3
n J 2 aF
us1yi U Us1yi
ž
/ ž /
L J bya P
Ž .
Substituting Eq. 21 , we obtain that
2r3 1r3
1 2 aF ayR
uG
ž
/ ž /
y2 bya P bya
Ž Ž ..
which reverts some of the results of Table 2 see Eq. 20 since the unemploy-ment increases with transportation costs and with fixed costs, and decreases with population. Moreover, it decreases with the difference between the lower bound of productivity and reservation wage. Finally, the effects of the productivity gap is ambiguous.18
Proposition 8 Labour market segmentation does not eliminate unemployment, but
it reÕerses the relation between the unemployment rate and the parameters.
Ž For instance if the parameter a decreases, unemployment rate increases see
[image:15.595.59.256.209.296.2].
Table 2 , but there exists a threshold level of a that leads a new segment to enter
Ž .
the market and to decrease unemployment see Fig. 4 . The same discontinuity effects characterise the other parameters.
What happens if a minimum wage is set by government in an economy as described above? We may distinguish two different effects:
Ø effects on the lowest productivity segment
Ø effects on the number of segments
Both effects reduce the employment, because of Proposition 6.
Fig. 4. Unemployment and transportation costs.
Proposition 9 With free entry of firms, market segmentation does not modify the
negatiÕe impact of minimum wage on labour market.
4. Concluding remarks
In this paper we present different labour market structures, pure monopsony, Ž
oligopsony and monopsonistic competition, which depend as in Bhaskar and To, .
1999 on free entry, on the number of firms and finally on wages. The new feature of this model compared with the work of Bhaskar and To is that we can characterise this unemployment as involuntary. In fact unemployment exists since firms use wages in order to select workers; in this way the less productive workers will not be hired even if they are willing to work for wages below the equilibrium level. Thus, involuntary unemployment is a crucial result of our model. The device used to achieve this result is an explicit consideration of workers’ heterogeneity. A recent literature has been concerned with the introduction of some form of
Ž .
heterogeneity in order to better justify several otherwise meaningless behavioural theories. The increasing importance attached to the management of human re-sources has emphasised the heterogeneity of workers’ characteristics. The Weiss
Ž . Ž .
efficiency wage model 1980 and Manning’s ‘Company Wage Policies’ 1993 , for example, could hardly be explained without an agent heterogeneity assump-tion.
This paper is conceived in order to evaluate the effect that differences in both workers’ productivity and job characteristics may generate in a labour market equilibrium.19
Differences in job characteristics make labour supply for an individual firm not perfectly elastic: the wage becomes a choice variable for firms. In this situation,
19
Table 4
Derivatives on monopsonistic competition
Žbya. F a P
Impact of parameter on wages y y y q
Impact of parameter on firm employment y q y q
Impact of parameter on the number of firms y y q q
Impact of parameter on unemployment rate q y y q
firms may compete for hiring the best workers. In fact, each firm has monopsonis-tic power: higher wages lead to a better composition of the labour force.
Without free entry, firms can face two alternative labour market structures, pure monopsony or oligopsony, depending on the number of firms. Entry leads to monopsonistic competition, where firms compete for the best workers, raising the
Ž .
wage above its market clearing equilibrium i.e., above the reservation wage . Monopsonistic competition is justified in a simple game framework where firms take the outside wage as given.
Endogenising the number of firms on a zero-profit condition base, we obtain
Ž .
some interesting results see Table 4 . In particular, an increase in the variance of productivity distribution and a decrease in travelling costs raise the unemployment rate, reversing the results of oligopsony.
Finally we allow for endogenous market segmentation; other firms can enter the market and pay lower wages by hiring the less productive workers.20
In all cases the impact of a minimum wage on the unemployment rate is negative: in pure monopsony a minimum wage reduces unemployment but in-creases the labour force and thus the unemployment rate; in monopsonistic competition, with or without segmentation, a minimum wage reduces employment. Our work represents a first attempt at jointly analysing heterogeneity in both non-wage characteristics and productivity levels of workers.
In our framework, we have considered well-informed firms. They know perfectly the productivity of each worker. In this way we pointed out the role of heterogeneity in explaining the dynamics of the labour market in a nearly ‘‘adverse selection’’ model.
It would be interesting to investigate a framework with imperfect information about individual productivity and the turnover rates of workers. In this scenario, firms would not know workers’ productivity beforehand. Thus, there could be an
20
unemployed worker whose productivity was higher than the prevailing wage. Furthermore, attracting workers to low-wage sector will be more difficult under such framework than under a perfect information one.
Appendix A
Ž .
Case 1: wjGw;wG arJ qR
Ž . Ž .
If firm j pays wages above the average, substituting Eq. 12 in Eq. 8 , the
Ž .
firm chooses the level of wage wj that maximizes its profit under the constraint
Ž . Ž .
L w ,wj Gn w ,w . When the constraint is not binding, we obtain a sort ofj j reaction function:
1 b 2a
wjs
ž
q2 wy/
Ž .
223 1ysj J w
Ž .
where s is an idiosyncratic shock at firm level.21 Imposing symmetry between
j
U ŽŽ . Ž ..
firms, we have the equilibrium wage for the economy: w s br1ys y
ŽŽ2a. Ž ..r J
Ž . Ž .
The relation between w and w is convergent because dwj jrdw s 2r3 .
Ž .
Case 2: wj-w; wG arJ qR
Ž .
If firm j decides to pay wages below the outside wage, its employment 13 is given by
n w ,w
Ž
j.
sn1 w ,wŽ
j.
qn2 w ,wŽ
j.
where:
1 wyw
b j
n1 w ,w
Ž
j.
sH
mž
q/
dLJ a
w
w wjyR n2 w
Ž
j.
sH
m dLa
wj
Ž . Ž .ŽŽ . Ž . Ž .. Ž . Ž
So: n1 w ,wj sm byw 1rJ q wjyw r a and n2 w ,wj sm wjy
.ŽŽ . Ž ..
w wjyR r a
It is now possible to calculate the average productivity of workers employed in the j-th firm:
n1 bqw n2 wjqw
ls q
n 2 n 2
21 Ž .
Eq. 22 is the approximate solution, obtained by imposing second order terms containings to be
Ž .
In order to define the optimal value for the wage w , we substitute the value forj
Ž .
n and lin equation lambda for the sake of simplicity let us set the shock:ss0 :
2 a
(
2 bqRy
Ž
Rqb.
q2 R bŽ
y3w.
q6 bŽ
yw.
Ž
Jyw.
w wj
Ž .
s3
23
Ž .
that is the reaction function of firm j when its own wage is lower than outside wages.
Imposing symmetry between firms, so wjsw, for aFw-b we obtain the
U Ž .
unique equilibrium wage: w sby 2arJ , that is the same obtained in Case 1
for ss0. Let’s analyse the characteristic of this fixed point. Remember that we
Ž . Ž .
are analysing situation in which Rq arJ w-b; differentiating Eq. 23 and
calculating the derivates at the extreme, we obtain:
dwj a 1
<wsRqas1y -1
J
dw J byR
and
dwj a 1
<wsbs y1)y1
dw J byR
So the absolute value of the first derivative is less than one at the extreme. We
Ž .
remind that wG arJ qR, then
dwj dwj a
< <
y1- wsb-0- wsRq -1.
dw dw J
ŽŽŽ .2. Ž 2 ..
Moreover, it is possible to demonstrate that dwj rd w -0. Then, we
Ž .
demonstrated that the first derivative is always decreasing for Rq arJ Fw-b
ŽŽ . Ž .. U
andy1- dwjr dw -q1. So wsw is an attractive fixed point of 23.
Appendix B
Two constraints must apply for market segmentation:
Ž .
1. wi) arJ qR
2. L Jj iGÝin Ji j i
where i indicates the segment of market.
When the second constraint is binding, the profit is maximized by wU sa. i q1
The number of firms that makes profits equal to zero in the segment where the second constraint is binding is
2
U
w ya P
Ž
i.
U
Ji q1s
Ž U . Ž . U U
since wi ya - wiy1yw it is easy to demonstrate that Ji i q1-J , so the lasti segment of the labour market is characterised by a smaller number of firms. The employment of a single firm of that segment is:
2 F U
ni q1s
U
wi ya
which is greater than nU. Total employment in the last productive market segment i
is lower than that in other segments. Note that it means that there is no involuntary employment.
Until both the constraints are not binding we can derive the wages paid by each segment, the number of employed workers and the whole economy’s unemploy-ment in a recursive way.
For a given number of firm, the long-run symmetric equilibrium wage of i-th segment is:
1r3
2a aF
wisbyi U sby2 i
ž /
Ž .
24J 2m
The number of workers hired by a firm of the i-th segment, is simply the
Ž . Ž .
symmetric equilibrium of Eqs. 12 and 13 , integrated between wiy1 and w :i
m
nis
Ž
wiy1ywi.
J
Since the difference between the wages of two close segments is constant:
1r3
aF
2r3
w yw s2
Ž
iy1 i.
ž /
mn is constant:i
1r3
2m
U 2r3
n s
ž /
FŽ .
25a
Each firm in each segment will hire the same number of workers but workers will have a lower productivity and firms will pay less for them. Substituting Eq. Ž24 in Eq. 10 we can calculate for each segment the number of firms. Ž . ŽJ_s i. under which the first constraint starts being effective:
3a 1q2 i
J_si s a
Ž .
26wiy1yR byR
which is increasing with the number of segments. Until the number of segments
Ž . U
grant that Eq. 10 is valid, J )J_s i; thus, labour market is characterised by
Ž .
monopsonistic competition and wages are set along Eq. 24 .
ŽU .
Ž U . Ž .
U Firms in i q1 segment set their wages integrating Eq. 5 from wi q1 and
wU. For profit maximisation i
1 2
U U
wiq1s wi q R
3 3
Ž U .
Monopsonistic profit of segment i q1 is positive if
3
U 2
Ž
wi yR.
m )F
27 a
But condition 10 does not hold if
JU-J_sU i q1
1r3 2
a 3a
2 m
-ž
F/
wUyRi
then, after some simple manipulations, we obtain:
3
U 2
Ž
wi yR.
m -F
27 a
this proves that when JU
-J_s, there are no rooms for a pure monopsonistic market.22
Finally, substituting the equilibrium values in the constraints we may calculate which one is effective.
1r3
aF
˜
byRG
ž /
Ž
1q2 i.
2m
1r3
P aF
ˆ
1G2 i
ž /
bya 2m
Solving for i:
1r3
byR 2m 1
˜
iF
ž /
y2 aF 2
1r3
i 2m
ˆ
iF
ž /
2m aF
22
An intriguing question arises when JU)J_s iUq1but J_s)JiUq1. In this case if firms pay wages
ˆ
˜
If i-i the second constraint is more effective and market segmentation make
ˆ
˜
unemployment disappear, on the contrary if i)i the first constraint must apply
23
ˆ
˜
and unemployment persists. i-i means:
1r3
1 1 bya 2m
-
ž
byRy/ ž /
2 2 P aF
let us write asRqd)R
1r3
Py1 d 2m
1-
ž
Ž
byR.
ž
/
q/
ž /
P P aF
Ž . Ž . Ž .
for high values of P we have Py1r P ,1 and drP ,0, then
1r3
2m
1-
Ž
byR.
ž /
aFˆ
˜
then i-i for
1r3
aF b)Rq
ž /
2m
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Ž .
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Ž .
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23 ˆ ˜ ˆ ˜
Ž . Ž .
