It is well-known that, in static models, minimum wages generate positive worker rents and, consequently, inefficiently low e?ort. We show that this result does not necessarily extend to a dynamic context. The reason is that, in repeated employment relationships, ?rms may exploit workers’ future rents to induce excessively high effort.bonuses; limited liability; minimum wages

Kräkel, Matthias

Schöttner, Anja

English

Research Papers in Economics

Sonderforschungsbereich/Transregio 15  www.sfbtr15.de 
Universität Mannheim  Freie Universität Berlin  Humboldt Universität zu Berlin  Ludwig Maximilians Universität München 
Rheinische Friedrich Wilhelms Universität Bonn  Zentrum für Europäische Wirtschaftsforschung Mannheim 
 
Speaker: Prof. Dr. Urs Schweizer.  Department of Economics  University of Bonn  D 53113 Bonn, 
Phone: +49(0228)739220  Fax: +49(0228)739221 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
*University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany, tel: +49 228 
733914, fax: +49 228 739210, e-mail: m.kraekel@uni-bonn.de  
**University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany, tel: +49 228 
739217, fax: +49 228 739210, e-mail: anja.schoettner@uni-bonn.de 
 
 
 
 
June 2009 
 
 
 
Financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15 is gratefully acknowledged. 
Discussion Paper No. 264 
Minimum Wages and Excessive 
Effort Supply 
 
Matthias Kräkel* 
Anja Schöttner** 
 Minimum Wages and Excessive Eﬀort Supply∗
Matthias Kräkel† Anja Schöttner‡
Abstract
It is well-known that, in static models, minimum wages generate
positive worker rents and, consequently, ineﬃciently low eﬀort. We
show that this result does not necessarily extend to a dynamic context.
The reason is that, in repeated employment relationships, ﬁrms may
exploit workers’ future rents to induce excessively high eﬀort.
Key Words: bonuses; limited liability; minimum wages
JEL Classiﬁcation: D82; D86; J33.
∗Financial support by the Deutsche Forschungsgemeinschaft (DFG), grant SFB/TR
15, is gratefully acknowledged.
†University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany, tel: +49 228
733914, fax: +49 228 739210, e-mail: m.kraekel@uni-bonn.de.
‡University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany, tel: +49 228
739217, fax: +49 228 739210, e-mail: anja.schoettner@uni-bonn.de.
11I n t r o d u c t i o n
As is well-known in economics, minimum wages are ineﬃcient for at least
two reasons: On an aggregate level, they may prevent labor market clearing,
and on a disaggregate level, they can imply ineﬃciently low eﬀort. The latter
problem has been highlighted by contract-theoretic models analyzing moral-
hazard problems under limited liability (see, among many others, Laﬀont and
Martimort, 2002, chapter 4; Schmitz 2005): When agents are protected by
minimum wages or limited liability, they usually earn positive rents under an
incentive contract. These rents raise the principal’s costs of eliciting eﬀort.
Consequently, he optimally induces less than ﬁrst-best eﬀort.
We show that this conclusion may no longer hold in a dynamic setting.
To do so, we also consider a moral-hazard problem under minimum wages,
which leads to positive rents and ineﬃciently low eﬀort in a static model.
However, in a two-period model, the principal optimally uses second-period
rents to generate extra incentives for the agent in the ﬁrst period. This is
achieved by combining a bonus contract with an extension clause that allows
the agent to sign a second-period contract only if he was successful in the
ﬁrst period. When the expected second-period rent is large, the principal
uses the extra incentives to induce more than ﬁrst-best eﬀort.
In practice, this "reversed" ineﬃciency problem of minimum wages (i.e.,
excessively large eﬀorts) should typically apply to low-skilled blue-collar work-
ers. The introduction of minimum wages that are enforced by law (or col-
lective agreements) usually compels ﬁrms to increase wages for unskilled
labor, whereas wages for high-skilled employees are unaﬀected because they
already earn more than the minimum wage. Translating this to our model
means that, for low-skilled workers, the minimum wage constraint is bind-
ing. Hence, blue-collar workers are likely to earn rents in a one-shot game.
In a dynamic environment, ﬁrms then optimally respond by exploiting these
rents, which may result in ineﬃciently high eﬀort.
The paper is related to the contract-theoretic literature on moral hazard
and limited liability, which usually considers a static relationship.1 Two ex-
1Note, however, that there is a rich literature on repeated moral hazard with risk averse
agents. Contrary to our paper, the focus of these models is on consumption smoothing
2ceptions are Ohlendorf and Schmitz (2008) and Kräkel and Schöttner (2008),
who also analyze a two-period principal-agent relationship and show that the
principal employs future rents to generate extra incentives. However, in those
papers, the agent’s eﬀort remains below ﬁrst-best. We obtain a contrary re-
sult by considering a situation where the ﬁrm can replace the worker after
the ﬁrst period.
2T h eM o d e l
A ﬁrm needs one worker to carry out a task in each of two periods. In each
period, the ﬁrm can randomly hire a worker from a pool of homogeneous
agents available on the labor market. Alternatively, in period 2, the ﬁrm
may again employ the worker hired in period 1. However, only one-period
contracts are feasible because the ﬁrm cannot commit not to renegotiate
contractual terms referring to period 2 at the beginning of the second period.
All players are risk neutral. The monetary output of the worker hired in
period  ( =1 2)i s with  ∈ {01}and Pr[ =1 |]=().T h e
variable  denotes the worker’s eﬀort in period ,a n d() is a concave
probability function with 0 ()  0 and 00 ()  0. At the beginning of
 =1 ,t h eﬁrm knows 1. However, due to uncertainty about the future,
2 is still unknown and considered to be the realization of a non-negative
random variable  with commonly known cdf  (·).T h eﬁrm learns 2 at the
beginning of  =2 .
Eﬀort is not observable, but output  is veriﬁable. Hence, at the begin-
ning of period ,t h eﬁrm oﬀers a worker a bonus contract ( ) contingent
on output ,w h e r et h el o wb o n u s is paid to the worker if  =0 ,a n d
the high bonus  if  =1 .T h e ﬁrm’s payment to the worker must be
a tl e a s ta sh i g ha st h em i n i m u mw a g e ,w h i c hi sn o r m a l i z e dt oz e r oi nb o t h
periods. Thus,   ≥ 0. To supplement the period-1 bonus contract, the
ﬁrm announces a probability  ∈ [01] of hiring the period-1 worker again in
period 2, provided that the worker achieved a high output in period 1, i.e.,
if 1 =1 .2
and the renegotiation of long-term contracts.
2Note that such an extension clause is renegotiation-proof: At the beginning of period
3In each period, workers have a reservation value ¯  ≥ 0.E x e r t i n g e ﬀort
 entails cost () with (0) = 0 (0) = 00 (0) = 0 and 0 () 00 ()  0 for
all   0. To guarantee that the ﬁrm is interested in hiring a worker and
implementing eﬃcient eﬀort, we assume that (
 )−(
 )  ¯  ,∀,w i t h
ﬁrst-best eﬀort 
 being deﬁned by 0(
 )=0(
 ). Concavity of the
ﬁrm’s objective function in the second-best case is ensured by the technical
assumptions 000 ()  0 and 000 () ≤ 0.
The timeline for each period  is the following: First, the ﬁrm observes .
Then the ﬁrm oﬀers a bonus contract ( ) supplemented by an extension
probability  if  =1 . The worker accepts or rejects the contract. In case
of acceptance, the worker chooses eﬀort . Finally, output is realized and
payoﬀsa r em a d e .
3S o l u t i o nt ot h eM o d e l
We solve the problem by ﬁrst considering  =2 . If the worker has accepted
the contract (2 2), his expected utility is
2 (2)=2 +( 2 − 2)(2) − (2) (1)
Hence, the worker optimally chooses eﬀort 2 given by
(2 − 2)=
0 (2)
0 (2) (2)
Therefore, the worker’s expected utility is
2 (2)=2 + (2) with (2): =
0 (2)
0 (2)
(2) − (2) (3)
The function (2) denotes the worker’s expected gain from exerting eﬀort
2, i.e., the resulting expected wage increase net of eﬀort cost. (2) is
strictly increasing in 2.
The ﬁrm maximizes 2(2)−2−(2 − 2)(2),t a k i n gi n t oa c c o u n t
the worker’s participation constraint (PC) 2(2) ≥ ¯  and the minimum-
2, the ﬁrm is indiﬀerent between employing the period-1 worker for another period or
hiring a new worker.
4wage condition (MWC) 2 ≥ 0. Thus, using (3), the ﬁrm’s Lagrangian reads
as follows:3
2 (2 2)=2(2) − 2 − (2) − (2)+1 [2 + (2) − ¯ ]+22
Maximization leads to our ﬁrst result:4
Proposition 1 (i) If ¯ (∗
2) with ∗
2 being implicitly deﬁned by 20 (∗
2)−
0 (∗
2) − 0 (∗
2)=0 ,o n l yt h eM W Cw i l lb eb i n d i n ga n dt h eﬁrm induces ∗
2.
(ii) If ¯  ∈
£
(∗
2)
¡

2
¢¤
, both MWC and PC will be binding and the ﬁrm
implements ∗∗
2 with (∗∗
2 )=¯ . (iii) If ¯ 
¡

2
¢
,t h e no n l yt h eP Cw i l l
be binding and the ﬁrm implements 
2 . (iv) We have ∗
2  ∗∗
2  
2 .
The proposition shows that an increasing reservation value ¯  relaxes the
MWC, thus leading to higher implemented eﬀort. The worker earns a positive
rent if and only if case (i) applies. Because ∗
2 and, consequently, (∗
2) is
increasing in 2, case (i) occurs if 2 is suﬃciently large. More precisely,
2 needs to exceed the threshold ˆ  implicitly deﬁned by (∗
2(ˆ )) = ¯ .
Since we are interested in situations where workers may earn rents, we
assume that 2  ˆ  occurs with positive probability. Thus, before un-
certainty about 2 is resolved, the expected rent of the period-2 worker is
¯  := (1 −  (ˆ ))( [2 (∗
2)|ˆ ] − ¯ )  0.
We now turn to the optimal contract for  =1 . The period-1 worker
earns ¯  in the second period if 1 =1and he is hired again in period 2
(which happens with probability ). Thus, a worker’s expected utility from
accepting a contract in  =1is given by
1 (1)=1 +¯  +
¡
1 − 1 + ¯ 
¢
(1) − (1).
Hence, the worker chooses eﬀort according to 1−1+ ¯  = 0 (1)0 (1).
For  =1 ,t h eP Ci s1 (1) ≥ 2¯  and the MWC is 1 1 ≥ 0.I no r d e rt o
rule out extreme cases where, in the absence of a minimum wage in period
3Note that 2 ≥ 0 together with (2 − 2)=0 (2)0 (2) ensures that 2 ≥ 0.
Recall that 2 is a function of (2 2).
4All proofs are relegated to the appendix.
51, the ﬁrm would like to punish the worker for success (i.e., 1  0), let
¯ ¯ . We obtain the following result:
Proposition 2 Assume that ¯ ¯ . In the optimal contract, the ﬁrm sets
 =1 . There exists a cut-oﬀ value ˆ  (
1 ) such that the ﬁrm implements
more than ﬁrst-best eﬀort, 
1 ,i fa n do n l yi f¯ ˆ .
A c c o r d i n gt oP r o p o s i t i o n2 ,t h eﬁrm optimally combines a bonus contract
with an extension clause that guarantees a period-1 worker another contract
in  =2in case of success (i.e., 1 =1 ). Thereby, the ﬁrm can use the
entire expected second-period rent ¯  to generate extra incentives in  =1 .
Formally, this means that the ﬁrm’s cost of inducing a given eﬀort level 1
decreases by (1) ¯  relative to a situation without extension clause (i.e.,
 =0 ). Consequently, the optimal period-1 eﬀort is higher than under  =0
or, equivalently, in a static employment setting. Moreover, if period-1 eﬀort
would already be quite large without extension clause (i.e., ¯ ˆ ), extra
incentives due to contract extension lead to more than eﬃcient eﬀort in
period 1.
Appendix
P r o o fo fP r o p o s i t i o n1 :We have
2
2
= 2
0 (2)
2
2
−1−[
0(2)+
0(2)]
2
2
+1
∙
1+
0(2)
2
2
¸
+2 =0
(4)
and
2
2
= 2
0 (2)
2
2
− [
0(2)+
0(2)]
2
2
+ 1
0(2)
2
2
=0  (5)
Adding both optimality conditions, using that, by (2),
2
2
= −
0 (2)
(2 − 2)00 (2) − 00 (2)
= −
2
2
 0 (6)
yields 1 + 2 =1 . Hence, either (i) only the PC is binding, or (ii) only the
MWC, or (iii) both. In case (i) we have 2 =0and 1 =1 . Inserting into (5)
6shows that the ﬁrm implements ﬁrst-best eﬀort: 20 ¡

2
¢
= 0 ¡

2
¢
.F r o m
the binding PC and the non-binding MWC 2  0 we obtain 
¡

2
¢
 ¯ .
In case (ii), 1 =0 , 2 =1and 2 =0 . Inserting into (5) gives
2
0 (2) − 
0 (2) − 
0 (2)=0  (7)
Obviously, the solution to (7), ∗
2,s a t i s ﬁes ∗
2  
2 . The non-binding PC
yields (∗
2)  ¯ . Finally, in case (iii), 1 2  0 and 2 =0 .F r o m t h e
binding PC optimal eﬀort in this scenario, ∗∗
2 , is characterized by (∗∗
2 )=¯ .
Solving (5) for the multiplier 1 yields
1 =1−
20 (∗∗
2 ) − 0 (∗∗
2 )
0 (∗∗
2 )
 (8)
1  1 implies that 20 (∗∗
2 ) − 0 (∗∗
2 )  0 and, hence, ∗∗
2  
2 .F r o m( 8 )
and 1  0 we obtain
2
0 (
∗∗
2 ) − 
0 (
∗∗
2 ) − 
0 (
∗∗
2 )  0 (9)
Using that 000  0 and 000  0, it is straightforward to verify that (·) is a
convex function. Thus 2(·)−(·)−(·) is concave in eﬀort. Consequently,
comparison of (7) and (9) gives ∗∗
2  ∗
2.
Proof of Proposition 2: To make the ﬁrm’s problems for the two periods easily
comparable and to be able to apply Proposition 1, we state the optimization
program in a general form that incorporates both periods  =1 2.I np e r i o d
,t h eﬁrm’s problem is:
max

() −  − ( − )()
s.t.  −  =
0 ()
0 ()
− ¯ 
 +
¡
 −  + ¯ 
¢
() − () ≥ ¯ 
  ≥ 0
 =0 if  =2 
7Inserting for − and using the deﬁnition for (·) from (3), this problem
simpliﬁes to
max

£
 + ¯ 
¤
() −  − [()+()]
s.t.  =m a x
½
¯  − () ¯  −
0 ()
0 ()
0
¾
 =0 if  =2 
where the expression for  follows from the PC and the MWC for  and
, respectively. Note that
¯  − ()  ¯  −
0 ()
0 ()
⇔ ¯  − ¯   [() − 1]
0 ()
0 ()
− ()
is true since the right-hand side of the last inequality is negative and ¯  ¯ 
by assumption. Hence,  ≥ 0 is satisﬁed and  =m a x {¯  − ()0}.
The ﬁrm’s problem can then be transformed to:
max

( £
 + ¯ 
¤
() − [()+()] if () ≥ ¯ 
£
 + ¯ 
¤
() − [¯  + ()] otherwise
s.t.  =0if  =2 .T h u s , t h e ﬁrm sets  =1in  =1 . To see that
the ﬁrm may induce 1  
1 , assume for a moment that ¯  =0 . Then,
the optimal period-1 contract is equivalent to the optimal period-2 contract.
Hence, by Proposition 1, the ﬁrm would implement 1 = 
1 if ¯  (
1 ).
Consequently, because ¯ 0, there is a critical value ˆ  such that 1  
1
for all ¯ ˆ .M o r e o v e r ,ˆ  (
1 ).
References
Kräkel, M. and A. Schöttner (2008): Relative Performance Pay, Bonuses
and Job-Promotion Tournaments. Bonn ECON Papers No. 16/2008.
Laﬀont, J.-J. and D. Martimort (2002): T h eT h e o r yo fI n c e n t i v e s .P r i n c e -
ton University Press.
8Ohlendorf, S. and P.W. Schmitz (2008): Repeated Moral Hazard, Limited
Liability, and Renegotiation. CEPR Discussion paper No. 6725.
Schmitz, P.W. (2005): Workplace Surveillance, Privacy Protection, and
Eﬃciency Wages, Labour Economics 12, 727—738.
9

Minimum Wages and Excessive Effort Supply

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