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 .
Ž .
P lives see Fig. 1 . Around this ring there is also a certain endogenous number 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 L
as the labour supply attraction basin of the j-th firm.
j
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 m s
P 1
Ž .
b y a 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:
p s 1 q s l w ,w n w ,w y n w ,w w y F 2
Ž . Ž
. Ž
. Ž .
Ž .
j j
j j
j j
where s is an idiosyncratic shock with zero mean, w is the wage paid by firm j,
j j
w is the outside wage, l is 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
In the famous article ‘‘Labour contract as a partial gift exchange’’ George Akerlof reported the Homans study on cash posters productivity. He showed that the performance of cash posters varied
between 306 and 439 cards per hour, and, even if output was easily observable, all cash posters gain the same wage. Empirical evidence of different productivity levels between workers that are paid the
same wage is a common phenomenon and is well documented in the literature of human resources management.
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 w
j
defined by: 1 q s
l n q ln s n w q n
3
Ž .
Ž .
j w
w w
j
j j
j
Ž . The employment at firm level n
is constrained by the labour force that each
j
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 - q R
4
Ž .
J 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 A
1
of Fig. 1 for whom it is indifferent to work in firm I or to not participate in the
1
labour market.
7
Ž .
a r J is the travelling expense of the farest worker.
8
It is reasonable to imagine that the distance between two contiguous firms is constant.
For this worker, the following equation must be valid: w y a 2I A s R. We
1 1
Ž . Ž .
can also write w y R r a s 2I A.
1 1
Ž .
Then it is not difficult to look at 2 P I A as the geographic basin of attraction
1
of firm I . If the workers are uniformly distributed around the ring, the following
1
ratio must be valid: 2I A:1 s M : P, where M are those workers attracted by the
1 1
1
monopsonistic firm I . Substituting 2I A and generalizing for firm j, we obtain
1 1
the available labour force for firm j: w y R
j
M w s P
5
Ž .
Ž .
j j
a 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 obtain
j
Ž .
the employment in the j-th firm n w
j
M w w y R
Ž .
b
j j
j
n w s
m d L s m
b y w . 6
Ž .
Ž .
Ž .
H
j j
j
P a
w
j
Ž . Ž .
Ž .
which is increasing in w until w F b q R r 2 s w see Fig. 2 . It is important
ˆ
j j
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 J b y w
Ž .
j j
j
u s 1 y s 1 y
7
Ž .
L w J b y a
Ž .
j j
is increasing in w because the labour force increases faster than employment.
j
Fig. 2. The demand for labour of a monopsonistic firm.
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 s J the participation rate is:
L w J w y R
Ž .
j j
j
p s s
J P
a where L s JM is the total labour force.
j
Ž . Ž
. Ž
. Ž . Taking into account Eq. 1 , we have: l w s b q w r 2 , so l s 1r2.
j j
w
j
Substituting in lambda, we obtain: 1 q s
j
n b y w
s n 8
Ž .
w j
j
ž
1 y s
j
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
w s q
R 9
Ž .
j
3 1 y s 3
j
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 2
9
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 Õ F w s
˜
j
w Ž
.x min w , R q arJ , where w is the classical condition in monopsonistic labour
ˆ ˆ
j j
markets, that is the level of wage which maximizes the labour demand function
9
All the derivatives hold with s small enough.
j 10
The most complete analysis of the effect on the economic system of minimum wage law is presented in Card and Krueger, 1995. They ‘‘ . . . present a new body of evidence showing that recent
minimum wage increases have not had the negative effects predicted by the textbook model. Some of the new evidence points toward a positive effect of the minimum wage on employment . . . ’’, page 2.
The most recent review can be found in Boal and Ransom, 1997. See also Rebitzer and Taylor, 1993.
Table 2 Derivatives on monopsony
Ž .
by a 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 w
see Fig. 2 , and R q arJ depends on labour force participation. Given
j
that w
U
F w , firms operate in an upward sloping labour demand function.
ˆ
j j
Ž . If the minimum wage is less than the critical value w :
˜
j
Proposition 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 b y R
Ž .
p s m
y F
j
27 a
Proposition 3 If we suppose free entry of firms, monopsony cannot exist because:
Ø If p - 0, all the firms exit. Ž .
Ø If p 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 q R
10
Ž .
J w
Ž .
Ž . 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 w w , a firm attracts a greater number of
j
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 w w is indicated by the area B of Fig. 3.
j
We can write:
12
1 w y w
j
L w ,w s q
P 11
Ž .
Ž .
j j
ž
J w a
Ž .
Ž .
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
w y w covers, at least, his increased
j
travelling expenses. Ž
. Integrating Eq. 11 we obtain the firm employment
w ,w
Ž .
b
j
n w ,w s
m d L
12
Ž .
Ž . ˜
H
j j
P
w
j
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 simply l s b q w r 2 .
j
On the contrary, if firm j pays w - w, it loses some of its more productive
j
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 .
w a I Bs w y aI B. We can also write w y w r a sI ByI B; the second term is equivalent to:
j 1
2 j
1 2
Ž .
2I ByI I , the distance between the two firms I I could be written as 1r J.
1 1 2
1 2
Fig. 3. Employment in firm J with monopsonistic competition.
It is not difficult to show that employment in firm j is: n
w ,w s n1 w ,w q n2 w ,w 13
Ž .
Ž .
Ž .
Ž . ˆ
j j
j j
w
L w ,w M w
Ž .
Ž .
b
j j
j j
n w ,w s
m d L q
m d L
14
Ž .
Ž . ˆ
H H
j j
P P
w w
j
Ž Ž
.. where the first integral n1 w ,w
gives the number of workers with productivity
j
Ž Ž
.. higher than w, while the second
n2 w ,w represents those workers with
j
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 b q w
n2 w ,w w q w
Ž .
Ž .
j j
j
ˆ
l s q
n w ,w 2
n w ,w 2
Ž .
Ž .
ˆ ˆ
j j
Ž .
In the symmetric case, all firms have the same shock s s s ; j . Substituting
j
˜
in equation lambda the definitions of n, and l we obtain the reaction function of
˜
Ž . Ž
U
. firm j when outside wages
w are smaller than the equilibrium wage w
;
ˆ
substituting the definitions of n and l, we obtain the reaction function for outside
ˆ
wage higher than equilibrium one.
Proposition 4 There is a unique fixed point for reaction functions, this fixed point is the Nash-symmetric equilibrium wage w
U
which is stable
13
Ž . In the short-run J w s J; the equilibrium wage is:
b 2 a
U
w s y
1 y s J
Let us analyse this equilibrium. The employment of each firm is: m
2 a s b
n s n s n y
15
Ž . ˜
ˆ
ž
J J
1 y s Ž
. ŽŽ
. Thus, a common positive shock s 0 raises the equilibrium wage
dwrd s .
ŽŽ .
. 0 but reduces firm employment
d nrd s - 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 the Õariance 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 s s 0. 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 b y w
Ž .
J w s 16
Ž . Ž .
2 F
13
Appendix A will demonstrate this proposition.
14
All the derivatives hold with s small enough.
j 15
For w s w.
j
Table 3 Derivatives on oligopsony
Ž .
by a 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
a F
U
w s b y 2 17
Ž .
ž
2 m We may calculate J
U
, the number of firms compatible with the free entry condition.
1r3 2
a
U
J s 2 m
18
Ž .
ž
F Ž
. From Eq. 15 it is easy to calculate employment in each firm:
1r3
2 m
U 2r3
n s F
19
Ž .
ž
a The unemployment rate is simply given by:
U 2r3
1r3
L y n 2
a F
U
u s s 1 y
20
Ž .
ž ž
L b y a
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 b y a — 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.
In a free entry economy, where firms earn no profit, the minimum wage will induce firms to exit and consequently it leads to changes in the endogenous
Ž Ž
.. number of firms. In the long run, substituting the number of firms Eq. 16
in Eq. Ž
. 15 , we obtain
2 F n s
b y w that is an upward sloping relation between employment at firm level and wages.
Ž The total employment and the unemployment rate are respectively: nJ s b y
. Ž .
Ž . Ž
. w r b y a P and u s 1 y b y w r b y a .
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
