Cahuc, Saint-Martin, and Zylberberg (2001) show numerically that a minimum wage hike can increase both skilled and unskilled employment in a right-to-manage wage bargaining setting. This comment demonstrates that this result crucially depends on an implicitly unrealistic choice for the skilled workers' alternative wage.Minimum wage, Wage bargaining, Employment

JÃ¼rgen Wiemers

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 1 
  
A Comment on "The consequences of the minimum wage when other wages 
are bargained over" 
  
 
 
Jürgen Wiemers  
Institute for Employment Research (IAB) 
Abstract 
Cahuc, Saint-Martin, and Zylberberg (2001) show numerically that a minimum wage hike can increase both skilled 
and unskilled employment in a right-to-manage wage bargaining setting. This comment demonstrates that this result 
crucially depends on an implicitly unrealistic choice for the skilled workers' alternative wage.
 
Citation: Jürgen Wiemers, (2010) ''A Comment on "The consequences of the minimum wage when other wages are bargained over"'', 
Economics Bulletin, Vol. 30 no.1 pp. 374-382. 
Submitted: Nov 17 2009.   Published: January 28, 2010. 
 
     1 Introduction
Several empirical studies (see e.g. Card and Krueger (1995) for the US, Dolado et al.
(1996) for Europe, and Metcalf (2007) for the UK) ￿nd that a minimum wage either
has no adverse e￿ects on employment or, in some cases, even positive e￿ects. These
￿ndings led, in turn, to the development of theoretical models, which question the
idea of an inverse relationship between the minimum wage and employment. Most of
these contributions build on re￿nements of the monopsony model, e.g. through the
introduction of e￿ciency wages (Rebitzer and Taylor, 1995) or search unemployment
(Manning, 1995; Burdett and Mortensen, 1998; Flinn, 2006).
Cahuc, Saint-Martin, and Zylberberg (2001) (henceforth CSZ) argue that wage bar-
gaining provides another channel for a possibly positive e￿ect of a minimum wage on
employment. More precisely, they analyse the impact of a binding minimum wage
in a situation where unions dominated by skilled workers set wages. For the special
cases of a CES technology or insider unions, they show analytically that the relation-
ship between the minimum wage, the bargained wage and employment of skilled and
unskilled workers depends on the magnitude of the elasticity of substitution between
skilled and unskilled labour. In order to demonstrate the empirical relevance of these
theoretical results under more general assumptions, they present a numerical example
using a translog cost function and a non-insider union. They ￿nd that a minimum wage
hike can increase both skilled and unskilled employment for reasonable parameter set-
tings. This comment demonstrates that this result crucially depends on an implicitly
unrealistic choice for the skilled workers’ alternative wage. Using more realistic alter-
native wages, the intriguing result of rising employment even for the unskilled despite
a minimum wage increase vanishes.
2 The CSZ model
For the sake of self-containment and in order to motivate our discussion of the numerical
example, we ￿rst give a short presentation of the CSZ model. The model considers a
monopolist ￿rm producing a single good with two types of labour, i.e. skilled labour lq
and unskilled labour ln. Capital is kept implicit. The ￿rm unilaterally sets employment
levels after the wage bargain, following the ‘right-to-manage’ assumption.
The monopolist ￿rm is assumed to be endowed with a production function F (lq;ln),
which is twice continuously di￿erentiable, increasing, concave and homogeneous of de-
gree  2]0;1[ with respect to skilled and unskilled work. Homogeneity implies that the
cost function C (w;y) = minlq;ln
P
i=q;n wili s.t. F (lq;ln)  y given wages w =(wq;wn)
and output y can be written as C (w;y) = W (w)L with L  y1=, where W (w) can be
interpreted as a wage index which is increasing, concave, and homogeneous of degree
1one with respect to w. The partial wage elasticities, equal to the cost share of the
respective labour type, are de￿ned as
i 
@W
@wi
wi
W
=
wili
WL
; i = q;n; (1)
where Shepard’s Lemma is used on the right hand side. Furthermore, it is assumed
that the monopolist faces a demand with constant price elasticity with absolute value
e > 1. The pro￿t  is therefore  = py   WL, with the price of the good p = Ay 1=e
and shift demand parameter A > 0. Pro￿t maximisation with respect to L gives
L = (W=(A))
 =( ), where  = e=(e   1) > 1 is the mark-up. Labour demands
are given by li = ciW  =( )=wi, i = q;n with c  (=(A))
 =( ), while optimal
pro￿ts amount to  = (1   =)A=( ) (W=)
 =( ).
The wage bargain is modelled by the generalised Nash solution. Only skilled workers
are represented by a trade union. Nonetheless, it is assumed that both wages, wq and
wn, are bargained over. The skilled workers’ contributions to the Nash program is given
by l
q  ( (wq)    (wq)). The parameter  2 [0;1] measures the weight of employment
in the trade union’s objective function, and  (wq) is the utility reached by a skilled
worker if there is no strike. Utility  () is increasing, concave and homogeneous. In
case of a strike, no one gets paid and the represented workers receive the reservation
wage wq. The ￿rm contributes its optimal pro￿t to the Nash program. The Nash
program can therefore be written as
max
wq;wn
N =

l

q  ( (wq)    (wq))
 [
]
1  + (wn   wn); (2)
where  measures the trade union’s bargaining power (0 <  < 1), wn represents the
minimum wage, and   0 is the Kuhn-Tucker multiplier associated with the re-
striction wn  wn. CSZ argue that the minimum wage is binding, if and only if
(=(   )    (s))+(1=   1)=(=   1) > 0. Since this condition is met for the
range of empirical estimates of , , , and the elasticity of substitution between skilled
and unskilled workers,   (dl=ds)  s=l, where l  lq=ln; s  wn=wq, the constraint
wn  wn will be binding, implying that the unskilled wage is exogenous and equal to
the minimum wage. Thus, only the ￿rst order condition @N=@wq = 0 is relevant for
the simulation. It is given by
0 = V (wq)   q (s) + "q (s); (3)
with  
 1
 . Note that V (wq)  0 (wq)wq=( (wq)    (wq)) denotes the skilled
wage elasticity of the utility di￿erence  (wq)    (wq). De￿ning the partial skilled
wage elasticity of pro￿ts q (s)  @
@wq
wq
 =  =(   )q (s) and the partial skilled
wage elasticity of skilled labour demand, "q (s) 
@lq
@wq
wq
lq , (3) implies that the bargained
wage equalises the utility elasticity V (wq) to the sum of the weighted (by  ) skilled
2wage elasticity of pro￿ts and the weighted (by ) skilled wage elasticity of the demand
for skilled workers.
Given this model setup, CSZ derive the following results for the special cases of a CES
technology ( constant) or for an insider union ( = 0):
@wq=@wn S 0 ()  T 1;
@ (wn=wq)=@wn > 0; (4)
@li=@wn

< 0 if   1
R 0 if  > 1
; i = q;n;
i.e. an increase in the minimum wage wn will result in a lower bargained wq, if the
elasticity of substitution  > 1 (vice versa, if  < 1), while the relative wage wn=wq
will always increase after a minimum wage hike. Labour demands for both skill types
will fall, if   1. For  > 1 the employment reaction is theoretically ambiguous. If
the fall in the skilled wage is strong enough, a minimum wage increase can result in
higher employment for both skill types.
3 The CSZ numerical example revisited
Using a numerical example, CSZ try to demonstrate that their theoretical results not
only hold for the special cases of a CES technology or an insider union. Therefore, they
relax both assumptions by employing a translog cost function, which exhibits a non-
constant elasticity of substitution, while at the same time assuming that employment
is a part of the union’s objective function during the bargain (  > 0).
As in CSZ, we assume that workers’ preferences are given by  (wq) = w
q, where
 2]0;1[ denotes the degree of risk aversion. Using this speci￿cation, the ￿rst order
condition (3) can be solved for wq,
wq =

1  

q (s)   "q (s)
 1=
wq: (5)
Assuming a translog cost function amounts to specifying the wage index as
lnW = lnwq + (1   a)lns  
b
2
(lns)
2 ; (6)
where a and b are technology parameters to be calibrated. With the wage index (6),
equation (1) implies q (s) = a + blns. Also, the elasticity of substitution is given by
 (s) = 1 +
b
q (s)
 
1   q (s)
: (7)
3In order to replicate the results in CSZ, we use the same calibration for the parameters
of the model. Thus, we choose the elasticity of substitution  = 1:2, the trade union
bargaining power  = 0:4, the mark-up  = 1:3, and we assume the technology to
be homogenous of degree  = 0:65. The benchmark relative wage is set to s = 0:5,
while the proportion of unskilled workers is ln=(lq + ln) = 0:67.1 The skilled workers’
reservation wage wq as well as the demand shift parameter A can both be set to an
arbitrary value, since they only determine the level of wages and employment, but
not the variation with respect to s. As in CSZ, we choose unity for both parameters.
With these parameter values, we can calibrate the technology parameters a = 0:53 and
b = 0:05 using (7) and the relation of skilled to unskilled workers,
lq
ln
=
sq (s)
1   q (s)
: (8)
Finally, a choice for the weight of employment in the trade union’s utility function, ,
is needed in order to simulate the reaction of the bargained wage wq with respect to
a change in the relative wage s.2 Since there is practically no empirical information
on the magnitude of , CSZ arbitrarily set a relatively low value,  = 0:15, while
remarking that the positive relationship between an increase in the minimum wage
and employment can also be found for higher values of , if the mark-up increases as
well.
Figure 1 reproduces the CSZ wage and employment reactions to a change in the relative
wage s, giving the levels instead of the changes relative to the benchmark. As can be
seen from the upper left panel, the CSZ calibration results in a skilled wage of 15:4
in the benchmark (s = 0:5). Since the benchmark reservation wage is wq = 1, this
implies a ratio of the alternative wage to the bargained wage (henceforth ￿wage ratio￿)
of wq=wq = 1=15:4 = 0:065, which is unrealistically low. Even under the extreme
assumption that workers necessarily become unemployed in the case of disagreement,
1Apparently, there is a typo on p. 348 in CSZ. The sentence ￿The corresponding proportion of
skilled workers in total employment is about 2/3￿ (emphasis by author) should read ￿...proportion
of unskilled workers...￿, since 2/3 is approximately the correct share for unskilled workers using the
de￿nitions of skilled and unskilled labour in Sneessens and Shadman-Metha (1995). The simulation
results of CSZ are obviously based on the correct proportion, ln=(lq +ln) = 0:67, since we are able to
replicate their results using this setting. Thus, CSZ assume 2/3 of all workers to receive the minimum
wage in the benchmark. Nevertheless, this rather high share of minimum wage receivers is not crucial
for the model’s ability to produce positive employment e￿ects after an increase in the minimum wage.
For example, by slightly increasing the bargaining power to  = 0:5 and at the same time reducing
the weight of employment in the trade union’s utility function to  = 0:08, the positive employment
e￿ect for both types of labour can be recovered using ln=(lq + ln) = 0:33 in the benchmark.
2Employing the relative wage s as the exogenous parameter instead of directly using the minimum
wage wn simpli￿es the simulation, since the expression (5) can be used to compute wq for a given s.
Using wn would require solving the ￿rst order condition (3) for each value of wn numerically for wq.
Following the latter approach does not change the simulation results qualitatively, but quantitatively
we ￿nd small di￿erences to the former approach.
4workers would still receive unemployment bene￿ts and/or other transfers. For example,
the OECD (2004) reports gross replacement ratios for 21 countries in 2003 varying
between 0:08 and 0:53, with an (unweighted) average of 0:3, and thus markedly higher
than the benchmark wage ratio assumed in CSZ. Allowing for the possibility to obtain
a job elsewhere will typically lead to an expected wage ratio at least an order of
magnitude higher than in the CSZ calibration. 3
Figure 1: Reaction of wages and employment when the relative wage s rises
s
0.5 0.6 0.7 0.8 0.9 1.0
10
11
12
13
14
15
Skilled wage
s
0.5 0.6 0.7 0.8 0.9 1.0
7.8
8.0
8.2
8.4
8.6
8.8
9.0
9.2
Unskilled wage
s
0.5 0.6 0.7 0.8 0.9 1.0
0.0008
0.0009
0.0010
0.0011
0.0012
0.0013
0.0014
0.0015
Skilled employment
s
0.5 0.6 0.7 0.8 0.9 1.0
0.00138
0.00140
0.00142
0.00144
0.00146
0.00148
0.00150
0.00152
0.00154
Unskilled employment
Replication of CSZ simulation, variables in levels.
A closer inspection of the ￿rst order condition (3) reveals why the choice of the bench-
mark wage ratio is crucial for a positive employment e￿ect of a minimum wage to
occur. Given the speci￿cation for workers’ preferences  (wq) = w
q, from the de￿ni-
tion of V (wq) we get V (wq) = w
q=
 
w
q   w
q

. Thus, the elasticity V (wq) is strictly
decreasing in wq, where limwq!wq V (wq) = 1 and limwq!1 V (wq) = . Figure 2 plots
V (wq) for the CSZ values wq = 1 and  = 0:9. As mentioned above, the CSZ calibra-
tion sets the benchmark value of wq to 15:4. At that wage, the slope of V (wq) is small,
3For example, assuming that a) unemployment bene￿ts are indexed to the gross wage with a
replacement rate of 0:3, b) the unemployment rate amounts to u = 0:1, and c) skilled workers can
obtain the same wage elsewhere, the expected wage ratio is given by (u  0:3  wq + (1   u)  wq)=wq =
0:93.
5since the elasticity is already close to its lower limit  = 0:9 (V (15:4) = 0:98). Starting
from this benchmark and using the CSZ calibration, an increase in the minimum wage,
which is re￿ected in an increase in the relative wage s, leads to an increase of the
term q (s)   "q (s) in the ￿rst order condition. In order for (3) to be satis￿ed, this
must be accompanied by an increase in V (wq) by the same amount. Because of the
small slope of V (wq) at the benchmark value for wq, this increase in V (wq) requires
a relatively strong fall in the skilled wage. 4 For the CSZ calibration, the drop in wq
proves to be su￿ciently high to produce positive employment e￿ects for the skilled as
well as for the unskilled workers. Put di￿erently, the simulated positive employment
reactions in CSZ hinge crucially on an unrealistically low wage elasticity of trade union
utility in the initial situation.
Figure 2: Plot of the trade unions’ elasticity of utility V (wq).
w
q
0 5 10 15 20
V w
q
0
1
2
3
4
5
Note that there is no compelling reason for choosing  arbitrarily, since the ￿rst order
condition (3) can be employed to calibrate the parameter. Since the reservation wage
wq is the numeraire in the model, the benchmark value for wq should be chosen in a
way to re￿ect empirical wage ratios. Employing the average gross replacement rate
of 0:3 in 2003 from OECD (2004) gives a benchmark value of wq = 3:33. Given the
other CSZ parameter settings, this implies a benchmark value of  = 0:38. Using this
calibration, the demand for both types of work decreases as a result of an increase in
the minimum wage (see Figure 3).5 This holds a fortiori, if we consider the possibility
that skilled workers can obtain a comparable wage elsewhere, since this will result in
4Inverting u = V (wq), we get wq = V  1 (u). The elasticity of this inverse function with respect
to u is given by wq (u) 
 
dV  1 (u)=du
 
u=V  1 (u)

. In the benchmark, wq (0:98) =  12:5.
5Calibrating  has no e￿ect on the technology parameters a and b, since  does not occur in the
equations (7) and (8), which are used for calibrating a and b. Also, second order conditions for a
maximum of (2) are satis￿ed at all values of s using  = 0:38 for the benchmark calibration.
We tried to recover the positive employment e￿ects reported in CSZ by varying the bargaining
power , the mark-up ; the skilled workers degree of risk aversion , and the benchmark elasticity
6an expected wage ratio close to one and ￿ because of the steepness of V (wq) close to
wq ￿ to a correspondingly small decrease in wq.
Figure 3: Reaction of wages and employment when the relative wage s rises
s
0.5 0.6 0.7 0.8 0.9 1.0
3.05
3.10
3.15
3.20
3.25
3.30
Skilled wage
s
0.5 0.6 0.7 0.8 0.9 1.0
1.8
2.0
2.2
2.4
2.6
2.8
3.0
Unskilled wage
s
0.5 0.6 0.7 0.8 0.9 1.0
0.0144
0.0146
0.0148
0.0150
0.0152
0.0154
0.0156
Skilled employment
s
0.5 0.6 0.7 0.8 0.9 1.0
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
Unskilled employment
Simulation based on OECD (2004) gross replacement rates, variables in levels.
4 Concluding remarks
We have shown that the model proposed in Cahuc, Saint-Martin, and Zylberberg (2001)
is not able to produce a positive employment e￿ect for an increase in the minimum
wage, especially for unskilled workers, if more realistic settings for the ratio of the
skilled alternative wage to the bargained skilled wage are used in the calibration. The
question remains whether it is possible to change some elements of the CSZ model in a
way which allows for positive employment e￿ects to occur using reasonable parameter
settings. We leave this question to future research.
of substitution , re￿ecting the wide range of estimates for these parameters. While variations in ,
, and  do not change the employment reactions qualitatively, a c.p. increase in  to approximately
1:3 and higher leads to an increase in the demand for skilled labour ￿ at the cost of an even stronger
fall in the demand for unskilled labour and overall employment.
7References
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mum wage when other wages are bargained over. European Economic Review 45 (2),
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Card, D. and A. Krueger (1995). Myth and Measurement: The new economics of
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Dolado, D., F. Kramarz, S. Machin, A. Manning, D. Margolis, and C. Teulings (1996).
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Flinn, C. J. (2006). Minimum Wage E￿ects On Labor Market Outcomes Under Search,
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Manning, A. (1995). How Do We Know that Real Wages Are too High? Quarterly
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A Comment on "The consequences of the minimum wage when other wages are bargained over"

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