Minimum wage increases can lead to wage reductions by imperfectly competitive firms

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Minimum wage increases can lead to wage reductions by
imperfectly competitive firms
Ken Clarka,b,*, Leo Kaasc, Paul Maddena
aSchool of Social Sciences, University of Manchester, Oxford Rd, Manchester, M13 9PL, UK
bIZA, Bonn, Germany
cUniversity of Konstanz, Konstanz, Germany
Abstract
In a model with imperfect competition and multiple equilibria we show how an increase in the minimum wage
can lead firms to reduce wages (and employment). We find some empirical support for this in the Card–Krueger
minimum wage data.
Keywords: Minimum wages; Oligopoly; Oligopsony
JEL classification: D43; E24; J48
1. Introduction
In a competitive labour market, a minimum wage increase raises wages to the new minimum wage
level if they were initially below that level, with no change otherwise; the effect on employment can only
be a reduction. Although monopsony can reverse the employment effect (e.g. Manning, 2003), the wage
predictions are unchanged. Recently, Bhaskar and To (2003) have shown how oligopsony may also
change the wage prediction, with firms who originally set wages above the new minimum wage further
* Corresponding author. School of Social Sciences, University of Manchester, Oxford Rd, Manchester, M13 9PL, UK. Tel.:
+44 161 275 3679; fax: +44 161 275 4812.
E mail address: ken.clark@manchester.ac.uk (K. Clark).
First publ. in: Economics Letters 91 (2006), 2, pp. 287-292
Konstanzer Online-Publikations-System (KOPS)
URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-88643
URL: http://kops.ub.uni-konstanz.de/volltexte/2009/8864/

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increasing their wage. We expand on these possibilities by showing that in a model of oligopsony where
the firms are also oligopolists, minimum wage increases can induce a reduction of wages (along with a
reduction of employment). We present some evidence of this phenomenon in the Card and Krueger
(1995) data on the New Jersey fast-food sector.
2. The model
We combine features of Dixon (1992) and Kaas and Madden (2004) in a 2-stage game, with wages
and employment determined at stage I and prices and output at stage II. There are two firms, duopsonists
in the labour market at stage I and duopolists in the stage II output market.1The production function of
firm i is yi=liwhere yiis output and liis labour employed. At wage w the upward sloping labour supply
is S(w), and the downward sloping inverse output demand is p=P(Y) where p is output price and Y
aggregate quantity. The revenue function R(Y)=P(Y)Y is increasing (so demand is elastic) and concave.
The unique Walrasian equilibrium wage wWEis defined by w(=p)=P(S(w)).
Strategic interaction between firms under laissez-faire is as follows.
2.1. Stage I
Firms choose wages wiz0 and labour demands Ji. If w1=w2=w we assume that initially half of the
supply S(w) offer to work at each firm, any unsuccessful offers being diverted to the other firm, implying
for i=1,2 and jpi;2
li¼ min Ji;max
1
2S w
ð Þ;S w
ð Þ
Jj
???
:
?
ð1Þ
If w1Nw2then workers S(w1) offer themselves to firm 1 and those with the lowest reservation wage
are hired first,3leaving a residual of max[S(w2) min(J1,S(w1)),0] for firm 2, giving
l1¼ min J1;S w1
ðÞðÞ; l2¼ min J2;max S w2
ðÞ
l1;0
ðÞ½?:
ð2Þ
2.2. Stage II
Firms choose prices p1, p2for the sale of up to the output levels yi=li. Since demand is elastic, this
stage II (bBertrand–EdgeworthQ) subgame always has a unique Nash equilibrium, with market-clearing
prices p1=p2=P(l1+l2) — see Madden (1998).
Thus subgame perfect equilibria of the two-stage game reduce to the Nash equilibria of the
simultaneous move game where firm i chooses (wi, Ji) and, with lidefined by Eqs. (1) and (2), payoffs
1For instance, a town in New Jersey has 2 fast food outlets that are the sole employers of the town’s teenage labour supply
(duopsony) and the only suppliers of fast food to the town (duopoly). Generalizing the results to the oligopoly case is
straightforward but tedious.
2Alternatively phrased, we are assuming here uniform rationing at symmetric wages.
3This assumption (innocuous here) is the efficient rationing rule of the Bertrand Edgeworth literature.
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are:
piw1;J1;w2;J2
ð
w1=w2=wWEwith J1, J2zS(wWE) and p1=p2=0 is an equilibrium of this game; if firm 1 undercuts
the Walrasian wage, firm 2 takes the whole market (because of J2zS(wWE) and Eq. (2)), leaving firm 1
with zero profits, and if firm 1 raises the wage, aggregate output can only increase (again because of
J2zS(wWE) and Eq. (2)) and its price falls, ensuring a loss for firm 1. The Appendix shows that this
equilibrium isunique;
Þ ¼ P l1þ l2
ðÞ
wi
½?li:
ð3Þ
Theorem 1. Under laissez-faire the unique equilibrium wage, employment and profit levels are
Walrasian w1¼ w2¼ wWE;l1¼ l2¼1
J2zS(wWE).
2S wWE
ðÞ;p1¼ p2¼ 0
??, supported by any labour demands J1,
The excess demands for labour (JiNli) neutralise the output market power of firms. For instance a
deviation from equilibrium in only J1has no effect on aggregate output (even if J1=0 this remains at
S(wWE)) and so has no effect on output price (which is always P(S(wWE)) ).
With a legally binding minimum wage wMINN0, the game is the same except for the stage I restriction
to wizwMIN. If wMIN=wWEthe equilibrium of Theorem 1 remains, but there is another equilibrium.
Suppose w1=w2=wMINand firms set labour demands which produce unemployment (J1+J2bS(w));
profits are pi=[P(J1+J2)
wMIN], which is concave in Ji.Maximizing with respect to Jiproduces
candidate new equilibrium labour demands J1=J2=J where Y=2J and,
wMIN¼ w Y
The marginal revenue curve w(Y) slopes down as in Fig. 1. In fact Y is the Cournot–Nash
aggregate output level when firms face constant marginal costs w=w
assumed unemployment whenever wNw shown in Fig. 1, and firms earn positive profits
(=(P(Y)
wMIN)Y/2= PV(Y)Y2/4N0.
It turns out that setting the minimum wage and hiring Y/2 workers is indeed equilibrium behaviour
provided that the minimum wage exceeds a certain threshold level. Without the minimum wage, firms
ð Þ ¼ PV Y
ð ÞY=2 þ P Y
ð Þð4Þ
1. The candidate generates the
w, p
Y
ψ (Y)
S(w)
P(Y)
w
WE
w ˜
w
Fig. 1. Equilibrium wages and output.
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would undercut while hiring the same number of workers. The minimum wage removes this option to
undercut, thus sustaining the new equilibrium. In the Appendix we prove
Theorem 2. There exists a wage w ˜ a(w,wWE) such that
(a) For wMINbw ˜, the unique equilibrium is that of Theorem 1.
(b) For w ˜ VwMINVwWEthere are two equilibria, the equilibrium of Theorem 1 and a minimum wage,
positive profits, unemployment equilibrium with w1=w2=wMINand J1¼ J2¼1
¼ wMIN:
(c) For wMINNwWEthe unique equilibrium is the minimum wage, positive profits, unemployment
equilibrium of (b) above.
2Y where w Y
ð Þ
One can argue for the positive profits at the new equilibrium as a selection mechanism — the new
minimum wage equilibrium Pareto dominates the original. The announcement of the new minimum
wage may also make the new equilibrium focal. Thus a minimum wage increase may lead to lower
wages and employment at firms that were originally paying wages above the minimum wage.
3. Empirical evidence
We analysed Card and Krueger’s (1995) bnatural experimentQ data on fast food restaurants in
Pennsylvania and New Jersey in 1992. In April 1992 the New Jersey minimum wage rose from $4.25 to
$5.05 while the minimum wage in Pennsylvania remained at $4.25. Card and Krueger examined the
employment impact of the wage hike, however they also collected data on the starting wage for
restaurant employees.
Table 1
Starting wages: descriptive statistics
New JerseyPennsylvania
BeforeAfter Before After
At minimum
Mean Wage ($)
(standard dev.)
Above $5.05
Reduce wage
Above $5.05 and reduced to $5.05
Sample size
95 (31%)
4.62
(0.3452)
23 (8%)
19 (6%)
18 (6%)
302
268 (89%)
5.08
(0.1066)
33 (11%)
22 (32%)
4.65
(0.3559)
4 (6%)
27 (40%)
20 (29%)
4.62
(0.3597)
2 (3%)
68
Table 2
Full time equivalent employment
New Jersey Pennsylvania
BeforeAfterBefore After
All Restaurants
Wage Reducers
20.67
27.39
21.88
22.06
23.7021.20
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Table 1 shows that a relatively small proportion of New Jersey restaurants offered starting wages
above the new minimum of $5.05 before the hike. Of the 23 firms in New Jersey that were paying in
excess of $5.05, 18 reduced their starting wage to the new minimum when it came into effect. This is
prima facie evidence in favour of the theoretical result. Further analysis of the firms that reduced starting
wages to $5.05 suggests that this was not a trivial reduction in the majority of cases. The mean reduction
was 27 cents while the modal reduction was 45 cents.
There is evidence that firms that lowered wages also reduced employment. Table 2 shows the change
in full-time equivalent employment for all restaurants in New Jersey and Pennsylvania as well as for
those 18 restaurants in New Jersey that reduced their wage to the new minimum. The second row, where
we consider only the wage reducers in New Jersey, shows a substantial and statistically significant
reduction of around five employees (two-tailed p-value=0.028).
Acknowledgement
We are grateful to an anonymous referee for helpful comments on a previous version of this paper.
Leo Kaas thanks the Austrian Science Fund (FWF) for financial support. The data we use are available
on the World Wide Web. See Chapter 1 of Card and Krueger (1995) for access details.
Appendix A. Proof of Theorem 1
To show uniqueness, consider symmetric wages w1=w2=w. (a) There is no equilibrium with
J1+J2bS(w). If firm 1 reduces w1a little it will still be able to hire the same J1, so aggregate output and
price are unchanged and profits higher. (b) There is no equilibrium with J1+J2zS(w), wNwWE.
Aggregate output would be S(w), its price P(S(w))bP(S(wWE))=wWEso at least one firm (i say) makes
a loss, which Ji=0 improves on. (c) There is no equilibrium with J1+J2zS(w), wbwWE. Here p1
+p2=[P(S(w))
w]S(w). By raising wiand taking the whole market either firm can attain profits close
to p1+p2, which must be an improvement for at least one of them. (d) There is no equilibrium with
J1+J2zS(w), w=wWEand JibS(w) and JibS(w) some i. Note p1=p2=0 in any such equilibrium.
Suppose J2bS(w). If 1 deviates to Jˆ1a(0,S(w)
P(Jˆ1+J2)NP(S(w))Nw. The proof of non-existence of asymmetric wage equilibria is available upon
request.
J2) then it makes strictly positive profits since
5
Appendix B. Proof of Theorem 2
Let p(w)=[P(Y)
profits. Suppose w(Y)=wMINzw, so w1=w2=wMIN, J1=J2=Y/2 is the candidate new equilibrium with
p1=p2=p(wMIN)N0.
Given wi=wMIN, J1=Y/2 is a best response to J2=Y/2-concavity of [P(J1+J2)
that deviations to J1Iˆ[0,S(wMIN)
Y/2] are unprofitable and deviations to J1zS(wMIN)
unchanged at S(wMIN)
Y/2 (from Eq. (1)). Suppose firm 1 raises its wage from the candidate level. We
show that this is beneficial only if there is some wage w1=wMINwhich, with J1=S(w1) is also
w]Y/2 where w=w(Y), and let pˆ(w)=[P(S(w))
w] S(w) denote bwhole marketQ
wMIN]J1ensures
Y/2 leave l1
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