Incentives and individual motivation in
European Journal of Operational Research 207 (2010) 878–885
Contents lists available at ScienceDirect
European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Decision Support
Incentives and individual motivation in supervised work groups
Arianna Dal Forno, Ugo Merlone *
Department of Statistics and Applied Mathematics, University of Torino, Corso Unione Sovietica 218 bis, Torino I-10134, Italy
a r t i c l e
i n f o
Article history:
Received 23 April 2008
Accepted 17 May 2010
Available online 25 May 2010
Keywords:
Organization theory
Production
Organizational behavior
Incentives
Individual motivation
a b s t r a c t
This paper introduces and analyzes a model of supervised work group where subordinates decide how to
exert their effort in complementary tasks while the supervisors decide incentives. Incentives may be a
combination of individual and group-based ones. The optimality of incentives is analyzed when considering two different cost functions for subordinates. The two cost functions describe different individual
motivations; comparing the resulting effort allocations and production optimality, we can relate them
to different organizational theories. Our results provide a measure of how motivation among subordinates may affect production and incentives. Furthermore, the optimal incentives schemes are examined
in terms of Adams’ equity theory.
Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction
According to Boudreau et al. (2003), the fields of operations
management and human resources management are intimately related, yet they maintain distinct perspectives.
Among the examples which try to merge these perspectives,
Bordoloi and Matsuo (2001) applied control theory to deal with
workforce planning taking into account also worker learning and
controls for the risk. Another example is Gendron (2005), where
a store scheduling problem with constraints deriving from union
representatives’ and human resources personnel’s was approached
and solved.
Recently, Boudreau et al. (2003) examine how human considerations affect classical operation management. In their conclusion,
they highlight the research challenges and opportunities of bringing the human resources and operation management together.
In this paper, we explore this line of research and try to integrate human considerations in optimal incentive problems. The
Moral Hazard literature approaches multi-agent relationships in
different ways. For example, the joint production models provide
interesting insights in terms of income distribution among
the agents, see for instance Alchian and Demsetz (1972) and
Holmström (1982). Another relevant aspect is the comparison between centralized and decentralized structures as far as contracting goes. For example, the literature provides conditions under
which the delegation of the supervisory task, i.e. decentralizing,
* Corresponding author. Tel.: +39 011 6705753; fax: +39 011 6705783.
E-mail address: merlone@econ.unito.it (U. Merlone).
0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2010.05.023
is beneficial; to obtain a first analysis of the advantages and disadvantages of delegation the reader may refer to Macho-Stadler and
Pérez-Castrillo (1997).
One aspect usually neglected in the economic literature is the
role of individual motivation; while, in psychology, motivation is
a concept that has been discussed extensively. According to Spector (2003) work motivation theories are most typically concerned
with the reasons, rather than ability, that some people perform
their jobs better than others. Steers and Black list the stages the
evolution of management thought concerning employee motivation has passed through. They are the traditional model, the human
relations approach and, more recently, the human resources model. In particular, ‘‘this newer approach also assumes that different
employees want different rewards from their jobs, that many
employees sincerely want to contribute, and that employees by
and large have the capacity to exercise a great deal of self-direction
and self-control at work” (Steers and Black, 1994, p. 139).
To this extent the case of Kyocera Corporation is striking. As it
concerns the reward system, Kyocera’s founder Kazuo Inamori, in
a booklet describing his philosophy, writes ‘‘We don’t think in
terms of individual rewards. We don’t buy individuals’ loyalty with
monetary incentives or titles. Rather, we believe that individuals
who are endowed with superior capabilities should contribute
their capabilities for the good of the entire group.” (for a discussion
of Kyocera’s organizational culture the reader may refer to Bylinsky
(1990)). This example is contrasted by pay incentives used at Lincoln Electric, where most employees are paid on a piece-rate system (the Lincoln Electric Company has been described in several
case studies by business schools, see for instance Fast and Berg
(1975)). Several comparisons between incentive programs have
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
been presented in the literature. Among others, Weiss (1987) provides an empirical comparison between individual wage incentives
and group incentives examining the effects in terms of motivation
and quit rates.
Following the joint production approach, we consider a modular model of hierarchical organization. Specifically, we concentrate
mainly on pyramidal structures. This particular structure is widespread and, consequently, both the economic (see Beckmann,
1988, for a formal analysis) and simulative literature (for instance,
see Glance and Huberman, 1994) find interest (for an analysis of
the different approaches to pyramidal structures see Merlone,
submitted for publication). In our model, the organization consists
of two heterogeneous agents interacting in a supervised work
group with a Cobb–Douglas production function. In the literature,
the distinction between team and work group is hazy, nevertheless
we will follow Spector (2003). As a consequence, we will refer to a
supervised work group as, in our case, we consider interchangeable
subordinates.
In this paper, we assume that individuals with different motivation may have different cost functions and analyze the incentive
problem the supervisor faces, in order to maximize her own profit.
In particular, we analyze how the optimal solution for the supervisor varies when considering two different cost functions for subordinates. Cost function plays an important role for the agent.
While in Dal Forno and Merlone (2007) only piecewise constant
cost function was considered, in this paper we also consider
strictly convex costs. Usually economics assumes that costs are
increasing and marginally increasing; the other kind of cost function in some cases may model more realistically some situations.
We refer in particular to situations like those described in Smith
(1977) where employees expend their discretionary effort (for a
discussion about antecedents of discretionary effort and its consequences on performance the reader may refer to Bailey et al.
(2001) and Sutton (2007)). In this sense, the two different cost
functions can be interpreted in terms of motivation; while the
piecewise constant cost function may be appropriate when subordinates have high self-efficacy and are highly motivated, the other
cost function seems to be more appropriate for individuals who
are mainly interested in monetary incentives. It is well known
(Zhou, 2002) that, with regard to the principal-agent theory, in
general it is difficult to derive analytical solutions; therefore, even
in scholarly contributions, strong assumptions are usually required. This tradeoff between analytical tractability and extensive
simplification is acknowledged by other authors approaching
organization complexity (see Ethiraj and Levinthal, 2009). Nevertheless, the two different cost functions we analyze can be related
to different cultures in the organizational structure we consider.
As a consequence, by comparing the production outputs under
the two different cost function assumptions, our analysis allows
us to compare productivity under different organizational cultures
and to measure what the cost of the culture in terms of production is. The two different cost functions, and the related organizational cultures, are particularly interesting when considering
changes in competition induced by emergent countries such as
China (for a discussion on the role of culture in the economic style
of China, the reader may refer to Herrmann-Pillath (2005) and to
Lum (2003), for an analysis of labor conditions in the same
country).
The structure of the paper is the following. In Section 2, we
present the theoretical model. Sections 3 and 4 summarize and
analyze the optimal incentive problem with the two different cost
functions in question, describing how these are related to the subordinates’ motivation. In Section 5, optimal incentive schemes and
outputs are compared and the results are interpreted in terms of
organizational culture consequences on productivity. Finally, Section 6 is devoted to conclusions and further research.
879
2. The model
As in Dal Forno and Merlone (2007) and Dal Forno and Merlone
(2009), we consider a model of supervised work group in which a
supervisor (acting as principal) and two subordinates (acting as
agents) cooperate. Agent i allocates his effort li with the partner,
and the effort ui with the supervisor. The joint production function
for agents 1 and 2 is C(u1 + u2)a(l1 + l2)b, where C 2 Rþþ is a constant factor,1 and a, b 2 (0, 1) are, respectively, the output elasticity
with respect to the joint effort with the supervisor and with the
partner. As a consequence, the agents have to decide both how
much effort to exert, and how to partition it in the two complementary tasks.2 Agents bear a cost for effort: agent i’s cost function
ci : R2þ ! Rþ will be denoted with ci(ui, li); cost functions are private
information. Furthermore, each agent can observe the level effort his
partner provides with him, but not the one which is provided with
the supervisor. Conversely, the supervisor can only observe the joint
output and the effort each agent provides with her. The supervisor’s
profit is a share c 2 (0, 1) of the supervised work group production
minus the incentives she pays to her subordinates. In the following,
we assume that the output is sold on market at unitary price and the
production and sharing constants C and c are such that Cc = 1; this
is not restrictive, it simplifies the notation, and allows us to simply
consider monetary payoffs. Finally, agents’ retribution consists of a
fixed wage w > 0 plus a performance-contingent reward; we assume
that the fixed wage is sufficient to meet basic needs, in terms of the
hierarchy of needs theory (Maslow, 1970), physiological needs and
needs of safety; in economic terms we say that the participation constraint is met.
Proposition 1. The gross production (u1 + u2)a(l1 + l2)b is maximized
if and only if the aggregate efforts are allocated proportionally to the
output elasticities.
Proof. The result follows from combining and rearranging the first
order conditions of the problem
max ðu1 þ u2 Þa ðl1 þ l2 Þb :
u1 ;u2 ;l1 ;l2
ð1Þ
In fact, from
(
aðu1 þ u2 Þa1 ðl1 þ l2 Þb ¼ 0;
bðu1 þ u2 Þa ðl1 þ l2 Þb1 ¼ 0;
ð2Þ
it follows
u1 þ u2 a
¼ :
l1 þ l2
b
ð3Þ
Condition (3) is necessary in order to maximize the production of
the supervised group. When either as the result of misaligned
incentives or as lack of coordination between subordinates this condition is not met, then the effort allocation is not efficient. The performance-contingent reward is a linear incentive bt on the joint
output of the team and a linear incentive bi on the effort each agent
exerts with the supervisor. Therefore, the problem can be formalized as a bilevel programming problem:
max ð1 2bt Þðu1 þ u2 Þa ðl1 þ l2 Þb b1 u1 b2 u2 ;
bt ;b1 ;b2
ð4Þ
such that, given the incentives bt, b1, b2, subordinates solve
max
w þ bt ðu1 þ u2 Þa ðl1 þ l2 Þb þ b1 u1 c1 ðu1 ; l1 Þ;
max
w þ bt ðu1 þ u2 Þa ðl1 þ l2 Þb þ b2 u2 c2 ðu2 ; l2 Þ;
u1 ;l1
u2 ;l2
1
ð5Þ
We recall that Rþþ is the set of positive real numbers; the case C = 0 is trivial.
From the functional form of the production function it is immediate to observe
that the two tasks are not additive; for a discussion the reader may refer to Spector
(2003).
2
880
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
where we have defined:
i: index of subordinate, i = 1, 2;
ui: effort subordinate i exerts in the task with the supervisor;
li: effort subordinate i exerts in the task with the colleague;
a: output elasticity with respect to the joint effort with the
supervisor;
b: output elasticity with respect to the joint effort between
subordinate;
ci(ui, li): cost function for subordinate i;
w: fixed wage;
bt: linear incentive on the joint output of the team;
bi: linear incentive on the effort subordinate i exerts with the
supervisor.
For the sake of simplicity we assume w = 0, this is not restrictive. Examining the form of the problem, it is rather immediate
to predict the behavior of rational supervisor and subordinates in
this interaction situation. It is a finite dynamic game with perfect
information, with supervisor moving first, and then subordinates,
acting simultaneously after observing the incentive. This game
has a proper subgame starting from the information set of the subordinates. Therefore, there exists a set of subgame perfect Nash
equilibria which equals the set of Nash equilibria that can be derived by backward induction. In fact, given any feasible incentives
scheme (bt, b1, b2), the subordinates will play a Nash equilibrium3
(ui, li) of the subgame. Knowing the fact that subordinates will play
a Nash equilibrium in the subgame, the supervisor will maximize
her profit by choosing the optimal incentive scheme. In the following
sections, we will analyze how different assumptions on the subordinates’ cost functions may determine the optimal incentive scheme.
3. Piecewise constant costs
In the literature, some factor or factors that keep agents from
working infinitely hard are usually considered. For example, Wageman and Baker (1997) propose several mechanisms considering
effort becoming increasingly either unproductive or unpleasant.
In this paper, as common, we assume that the unpleasantness of
the work increases with respect to the effort. Different cost functions may be considered and the functional form may reflect different underlying assumptions. While economics usually considers
mainly rational agents, other approaches to work group dynamics
take into account other aspects, such as norms and team commitment. Industrial and organizational psychology have proposed different theories of motivation to explain individual behavior in the
organization (for a first survey the reader may refer to Spector
(2003)). Nevertheless, several other aspects underlie group and
team behavior; the role of norms is well documented empirically
(see Coch and French, 1948 and Roy, 1952, for example) and it is
commonly assumed, for work groups, to dictate how much each
person will produce (see Spector, 2003). A first cost function we
consider is, as in Dal Forno and Merlone (2007), the following:
ci ðui ; li Þ ¼
0
if ui þ li 6 ci ;
þ1 if ui þ li > ci :
ð6Þ
First observe that this cost function is non-decreasing with respect
to the aggregate effort. It assumes that each subordinate has a physical capacity ci under which effort has zero cost, or, alternatively,
that at some exertion level the effort becomes unpleasant enough
to lead the individual to conclude that it is not worth working
any harder independently of the reward. In this case, we assume
that each individual knows his individual capacity and uses it with3
In the next sections, we will examine also the equilibrium selection problem.
out goldbricking. This kind of cost function is not completely new to
the principal-agent literature; for instance, Holmström and Milgrom (1991) assume that incentives are necessary to encourage
agents to work beyond a limit they would take pleasure in working.
In our case we assume that this limit coincides with their physical
capacity. Finally, observe that this assumption can also be interpreted in terms of the self-efficacy theory (Bandura, 1982), assuming that both subordinates have high self-efficacy and are motivated
to put in as much effort as they can.
3.1. The agents’ problem
Starting from the subordinates’ information set and assuming
coordination and commitment between agents, problem (5) reduces to:
max ðu1 þ u2 Þa ðl1 þ l2 Þb
sub ui þ li 6 ci
u1 ;u2 ;l1 ;l2
ð7Þ
for all i = 1, 2.
Corollary 2. Consider the production function (u1 + u2)a(l1 + l2)b, with
piecewise constant cost functions (6). Then, any effort allocation such
that
(
a ð
u1 þ u2 ¼ aþb
c1 þ c2 Þ;
ð8Þ
b
l1 þ l2 ¼ aþb
ðc1 þ c2 Þ;
maximizes the production.
Proof. The result follows trivially from first order conditions of
problem (7), assuming the constraint on individual aggregate effort
given by capacities. h
Therefore, there is a continuum of solutions to the considered
problem (7), nevertheless, a rather natural effort allocation is the
one that can be interpreted as focal in the sense of Schelling (1960):
ðui ; li Þ ¼
a b
c;
c
aþb i aþb i
ð9Þ
for all i = 1, 2.
Effort allocations such that (8) holds, will be called efficient, in
fact they satisfy also condition (3) which is necessary to the production maximization.
3.2. The supervisor’s problem
The supervisor’s problem is to design a linear compensation
scheme for the subordinates that induces them to use their capacity to maximize the team output. The supervisor can observe the
efforts ui the subordinates exert with her and the team output.
Each subordinate’s compensation is si = biui + bt(u1 + u2)a(l1 + l2)b,
where, we recall, bi is the incentive given to subordinate i for his
individual effort with supervisor, and bt is the incentive given to
them for the team output. We assume that the supervisor declares
the incentives and then the subordinates decide their efforts in
order to maximize their wage. The supervisor has to solve the
following problem:
max ð1 2bt Þðu1 þ u2 Þa ðl1 þ l2 Þb b1 u1 b2 u2 :
bt ;b1 ;b2
ð10Þ
When considering fully rational agents the solution is obvious.
Since any individual incentive given to agents gives a suboptimal
effort allocation, and null team output incentive makes for subordinates any allocation optimal, the optimal solution is
8
>
< bt ¼ e > 0;
b1 ¼ 0;
>
:
b2 ¼ 0:
ð11Þ
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
Some comments about this incentive scheme are in order. While it
seems to be appropriate for agents with the same capacities or a
low number of repetitions, for agents with different capacities
interacting over a long period of time some problems may arise.
For example, when considering equity theory (Adams, 1965), it is
rather evident that different capacity individuals will find themselves in inequitable situations, and will experience dissatisfaction
and emotional tension, that they will be motivated to reduce. In order to improve the distributive justice (see Spector, 2003, for a discussion about equity theory and distributive justice) a different
incentive scheme may be more appropriate in the long run.
4. Strictly convex costs
Following Holmström and Milgrom (1991), we suppose the effort in the two tasks is perfectly substitutable in the agent’s cost
function. Formally we assume that
ci ðui ; li Þ ¼ ci ðui þ li Þ:
ð12Þ
As it is common, ci is assumed differentiable, increasing and marginally increasing. As mentioned above, while this cost function is
closer to the classical modeling of behavior of the rational individual, we may assume this kind of cost function when agents’ motivation consists only in economic reward and neither social norms nor
other motivational theories determine individual effort. It is not difficult to assume that this kind of cost function may be realistic in
organizations where individual objective consists only in the maximization of the individual payoff. Another interpretation of this
cost function is as the result of high-powered incentives. According
to Steers and Black (1994), individual incentives may, at times, lead
to employees competing with one another; in our case we assume
that the internal competition, driven by high-powered incentives,
may result as this individual perception of cost of effort (for a discussion on high-powered incentives, see Encinosa et al., 2007). As
a consequence, we can think that this kind of cost function is the result of a different culture when compared to the piecewise constant
case. In this case also, we first analyze the agents’ problem and then
the supervisor’s one. Finally, we observe that some of the results we
present in the following can be generalized to more general cost
functions.
4.1. The agents’ problem
In this case, it would be rather unrealistic to assume commitment and coordination between subordinates, rather we will assume that agents solve problem (5) with cost function (12)
max
u1 ;l1
max
u2 ;l2
a
b
a
b
bt ðu1 þ u2 Þ ðl1 þ l2 Þ þ b1 u1 c1 ðu1 þ l1 Þ;
bt ðu1 þ u2 Þ ðl1 þ l2 Þ þ b2 u2 c2 ðu2 þ l2 Þ:
ð13Þ
As a consequence, given the incentives (bt, b1, b2), subordinates find
an effort allocation which is a Nash equilibrium, i.e., u1 ; l1 ; u2 ; l2
must be such that
u1 ; l1
¼ arg max bt u1 þ
a
u2
l1 þ
b
l2
þ b1 u1 c1 ðu1 þ l1 Þ
ð14Þ
and
a
b
u2 ; l2 ¼ arg max bt u1 þ u2 l1 þ l2 þ b2 u2 c2 ðu2 þ l2 Þ:
ð15Þ
While for the piecewise constant cost function it was immediate to
derive the effort allocation, in this case some preliminary considerations are in order.
881
Lemma 3. Given an incentive scheme (bt, b1, b2), if subordinates’ cost
functions have form (12), then an optimal effort allocation
u1 ; l1 ; u2 ; l2 must be such that
c01 u1 þ l1 ¼ c02 u2 þ l2 :
ð16Þ
b1 ¼ b2 :
ð17Þ
Furthermore, optimal incentives must be such that individual incentives are identical, that is
Proof. Since we assume that the effort allocation constitutes a
Nash equilibrium, first order conditions can be derived from (14)
and (15); they are:
8
abt
>
>
>
>
>
< bbt
>
> abt
>
>
>
:
bbt
a1
b
l1 þ l2 þ b1 c01 u1 þ l1 ¼ 0;
b1
u1 þ u2 l1 þ l2
c01 u1 þ l1 ¼ 0;
a1
b
u1 þ u2
l1 þ l2 þ b2 c02 u2 þ l2 ¼ 0;
a
b1
u1 þ u2 l1 þ l2
c02 u2 þ l2 ¼ 0:
u1 þ u2
a
ð18Þ
Combining the second and the fourth equation it is immediate to
obtain
c01 u1 þ l1 ¼ c02 u2 þ l2 :
ð19Þ
b1 c01 u1 þ l1 ¼ b2 c02 u2 þ l2 :
ð20Þ
b1 ¼ b2 :
ð21Þ
For the second part, combining the first and the third equation in
(18) it is immediate to obtain
Finally, by Eq. (19) we obtain
Following this result, incentives schemes will be written using the
shortened notation (bt, b1, b2) = (bt, b).
As mentioned in Section 1, it is well known that principalagents models often require strong assumptions to derive analytical results. This model is not an exception; in the following, as it is
common (see for instance, Holmström and Milgrom, 1987; Schattler and Sung, 1997; Encinosa et al., 2007), we assume a quadratic
cost function
ci ðui ; li Þ ¼ di ðui þ li Þ2 ;
ð22Þ
where di, i = 1, 2 are positive constants.
We also assume a + b = 1.
In this case, a result concerning proportionality of allocated efforts can be proved, as well.
Theorem 4. Given an incentive scheme (bt, b), with b – 0, there exists
a unique equilibrium effort allocation u1 ; l1 ; u2 ; l2 such that
(
u1 ¼ q u1 þ l1 ¼ qk1 ; l1 ¼ ð1 qÞ u1 þ l1 ¼ ð1 qÞk1 ;
u2 ¼ q u2 þ l2 ¼ qk2 ; l2 ¼ ð1 qÞ u2 þ l2 ¼ ð1 qÞk2 ;
where
k1
:¼
u1
þ
l1 ; k2
:¼
u2
þ
l2
ð23Þ
and 0 < q < 1.
Proof. First we will show that such an effort allocation is an equilibrium allocation. Since the objective functions are concave, first
order conditions are also sufficient and the optimal solution must
satisfy (18):
8
a1 k þ k a1 ð1 qÞ1a k þ k 1a þ b 2d k ¼ 0;
>
1 1
>
1
2
1
2
> abt q
>
>
< ð1 aÞb qa k þ k a ð1 qÞa k þ k a 2d k ¼ 0;
t
1
1
2
1
2
1
a1
1a
1a
>
> abt qa1 k1 þ k2
ð1
q
Þ
k
þ
k
þ
b
2d
2 k2 ¼ 0;
>
1
2
>
>
:
a
a
a
ð1 aÞbt qa k1 þ k2 ð1 qÞ k1 þ k2
2d2 k2 ¼ 0:
ð24Þ
882
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
Combining the second and the fourth equation it is immediate to
obtain d1 k1 ¼ d2 k2 , that is
k2 ¼
d1
k ;
d2 1
ð25Þ
u1 þ u2 ¼
8
1a
a1
>
þ b 2d1 k1 ¼ 0;
>
< abt q ð1 qÞ
ð1 aÞbt qa ð1 qÞa 2d1 k1 ¼ 0;
>
>
: k ¼ d1 k :
ð26Þ
a
1a
ðl1 þ l2 Þ;
aa bt
;
2d1 ð1 aÞa1
Putting together the first two equations and rearranging
analogously, we find
8
a
qÞ1a
>
abt ð1q1
ð1 aÞbt ð1qqÞa þ b ¼ 0;
>
a
>
<
k2 ¼ u2 þ l2 ¼
ð1aÞbt qa ð1qÞa
2d1
k
> 1
¼
>
>
: k ¼ d1 k :
2
d2 1
ð27Þ
;
Now consider function f ðqÞ ¼ abt
ð1qÞa
q1a
qa
ð1 aÞbt ð1qÞa þ b; it is
ð34Þ
which, substituted in the first equation of (33), yields
k1 ¼ u1 þ l1 ¼
1
d2
ð33Þ
Combining the first and the second equation of (33) it is immediate
to obtain
so we can discard the third equation as redundant.
By simple algebra we obtain
2
8
>
abt ðu1 þ u2 Þa1 ðl1 þ l2 Þ1a 2d1 ðu1 þ l1 Þ ¼ 0;
>
>
>
< bb ðu þ u Þa ðl þ l Þa 2d ðu þ l Þ ¼ 0
t
1
2
1
2
1
1
1
a1
1a
>
2d2 ðu2 þ l2 Þ ¼ 0;
> abt ðu1 þ u2 Þ ðl1 þ l2 Þ
>
>
:
bbt ðu1 þ u2 Þa ðl1 þ l2 Þa 2d2 ðu2 þ l2 Þ ¼ 0:
ð35Þ
aa bt
:
2d2 ð1 aÞa1
ð36Þ
The focal allocation can be found, as before, as
(
k2 ;
l2
u1 ¼ ak1 ; l1 ¼ ð1 aÞk1 ;
u2
¼a
ð37Þ
¼ ð1 aÞk2 :
easy to prove that there exists a unique q* 2 ]0, 1[ such that
abt
ð1 q Þ1a
q1a
qa
ð1 aÞbt
þ b ¼ 0;
ð1 q Þa
ð28Þ
since function f is continuous in ]0, 1[. Furthermore, it holds
8
a
qÞ1a
>
abt ð1q1
ð1 aÞbt ð1qqÞ1b þ b ¼ þ1;
a
< lim
q!0
qÞ1a
>
: lim abt ð11
ð1 aÞbt
a
q
q!1
qa
ð1qÞa
ð29Þ
þ b ¼ 1;
Corollary 6. The fraction q of effort made with the supervisor is
increasing with respect to the individual incentive b and its derivative
is
and
f 0 ðqÞ ¼ abt bð1qÞ
bbt
b1 1a
q ð1aÞqa ð1qÞb
ðq1a Þ2
að1qÞ1b qa1 þð1bÞqa ð1qÞb
½ð1qÞ1b 2
ð30Þ
< 0:
Thus, since, as b – 0, q* is unique, there exists the unique effort
allocation
8
aþ1
a
ð1aÞbt q
ð1aÞbt q ð1q Þ1a
>
;
>
> u1 ¼ 2d1 ð1q Þa ; l1 ¼
2d1
>
<
aþ1
a
1
a
aÞbt q
qÞ
; l2 ¼ ð1aÞbt q2dð1
;
u2 ¼ ð1
2d2 ð1q Þa
2
>
>
>
a
a
>
ð1aÞbt q
: k ¼ ð1aÞbt q ;
k2 ¼ 2d ð1q Þa :
1
2d ð1q Þa
1
q0 ðbÞ ¼
q2a ð1 qÞ1þa
:
að1 aÞbt
ð38Þ
Proof. Eq. (28) can be written as
f ðq; bÞ ¼ 0;
ð39Þ
where
ð31Þ
2
Theorem 4 does not claim that, with positive individual incentives, all effort allocation equilibria are characterized by efforts
being proportionally allocated between the two tasks. Rather, it
claims that among the proportional effort allocations, i.e. those satisfying condition (3), one and only one is an equilibrium.
In the following corollary, we analyze the effort allocation with
null individual incentives.
Corollary 5. Given an incentive scheme (bt,0), if subordinates’ cost
functions have form (22), then any allocation such that ak1 ¼
ab
a
t
ð1 aÞk2 is an equilibrium allocation, where k1 :¼
;
2ð1aÞa1 d1
aa bt
k2 :¼
a1 . Furthermore, there exists a focal effort allocation
2ð1aÞ
Remarkably, effort allocation (37) is the analogous of the focal
allocation (9) we found for piecewise constant costs in Corollary
2. In this case also, given the aggregate efforts the subordinates
put forth, the production is maximized. Furthermore, we observe
that q defines what fraction of the agents’ efforts is made with
the supervisor. By the implicit function theorem we can prove that,
as expected, this fraction increases as b increases. In fact, a greater
individual incentive reallocates a greater effort with the supervisor.
f ðq; bÞ :¼ abt
ð1 qÞ1a
q
1a
ð1 aÞbt
qa
þ b;
ð1 qÞa
ð40Þ
implicitly defines q as a function of b. By the implicit function
theorem
q0 ðbÞ ¼
of =ob
¼
of =oq
1
að1aÞbt
q2a ð1qÞ1þa
¼
q2a ð1 qÞ1þa
:
að1 aÞbt
ð41Þ
As both 0 < a < 1 and 0 < q < 1 the thesis follows. h
This result proves that when individual incentives are both positive, the effort allocation no longer satisfies condition (3) and has
several implications when considering how the supervisor must
determine incentives optimally.
4.2. The supervisor’s problem
d2
8
aþ1
aa bt
< u1 ¼ a ab1t ; l1 ¼
;
2ð1aÞ
d
2ð1aÞa2 d
1
1
aa bt
: u ¼ aaþ1 bt ; l ¼
:
2
2
2ð1aÞa1 d
2ð1aÞa2 d
2
ð32Þ
2
Proof. Under the assumptions b = 0 and a + b = 1, conditions (18)
become
As for the piecewise constant function, the supervisor’s problem
is to determine the incentive scheme maximizing her own payoff
as in (10).
Before investigating whether individual incentive b increases
supervisor’s profit, we characterize the optimal incentives when
b = 0. To this extent, we consider the aggregate profit
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
u1 þ u2
a
1a
l1 þ l2
2
d1 u1 þ l1
2
d2 u2 þ l2 :
ð42Þ
Theorem 7. With incentive schemes (bt, b) where b = 0, the aggregate
profit (42) is maximized if and only if it is completely shared between
the two agents; by contrast, in order to maximize supervisor’s profit,
half of the aggregate profit must be given to the supervisor and the
remaining equally shared between the agents.
Proof. To maximize the aggregate profit the following program
must be solved:
max
bt 612
u1
þ
a
u2
l1
þ
1a
l2
d1 u1
þ
2
l1
d2 u2
þ
2
l2 :
aaþ1 bt d1 þ d2
max
bt 612
2ð1 aÞa1 d1 d2
d1
aa b
t
2ð1 aÞa1 d1
!2
#a "
aa bt
d1 þ d2
a2 d d
1 2
2ð1 aÞ
aa b
t
d2
2ð1 aÞa1 d2
!2
ð44Þ
;
that is
max
bt 612
"
aa bt
d1 þ d2 a
aa bt
a ð1 aÞ1a
a1 d d
1 2
2ð1 aÞ
2ð1 aÞa1
#2
d1 þ d2
:
d1 d2
ð45Þ
2
Since 0 < a < 1 and dd11þd
> 0, the original program (43) is equivalent
d2
to
2
max bt
bt 612
bt
;
2
Now recall that in Corollary 6 we proved that q is a function of b. As
a consequence, we can consider the supervisor’s profit4 as a function of the individual incentive b
pðbÞ ¼ ð1 2bt Þq2a ðbÞð1 qðbÞÞ12a bqaþ1 ðbÞð1 qðbÞÞa :
ð52Þ
Now derive p with respect to b using expression (38) we found in
Corollary 6, substitute b = 0 and recall that q(0) = a,
1 2bt a
1 2bt aaþ1
a ð1 aÞa ¼ a
:
bt
bt
ð1 aÞa
ð53Þ
Since bt < 1/2, the supervisor’s profit increases as b increases. As a
consequence, incentive schemes (bt, b) where b = 0 cannot be
optimal. h
The consequences of this theorem are striking. In fact, this result proves that, with quadratic costs (22), the individual incentives are necessary. Nevertheless, by Corollary 6, it follows that
individual incentives increase the fraction of effort made with
the supervisor. Therefore, with positive individual incentives the
effort allocation does not satisfy condition (3), and gross production is not maximized. In the following section, we will relate the
assumption underlying the different cost functions we have considered to the managerial literature.
5. Comparing the two cost functions
maxð1 2bt Þ u1 þ u2
bt 612
that is
bt 612
ð51Þ
ð46Þ
and the optimal solution is bt = 1/2, that is, all the profit is shared
between the two agents.
For the second part of the theorem, let us consider the
supervisor’s profit (10) when the incentives are null
maxð1 2bt Þ
ð1 2bt Þq2a ð1 qÞ12a bqaþ1 ð1 qÞa :
p0 ð0Þ ¼ a
#1a
ð50Þ
That is, the supervisor’s profit is proportional to
ð43Þ
Taking into account the focal effort allocation, as given by Eq. (32),
the program (43) becomes
"
i
ð1 aÞbt d1 þ d2 h
ð1 2bt Þq2a ð1 qÞ12a bqaþ1 ð1 qÞa :
2
d1 d2
883
a
1a
l1 þ l2
ð47Þ
;
aa bt
d1 þ d2 a
a ð1 aÞ1a :
2ð1 aÞa1 d1 d2
This problem, since 0 < a < 1 and
d1 þd2
d1 d2
maxð1 2bt Þbt ;
bt 612
ð48Þ
> 0, is equivalent to
ð49Þ
and the optimal solution is bt = 1/4.
This proves the thesis. h
This result shows that the agents are the residual claimant (see
for instance, Varian, 1992) to the output produced if and only if the
supervisor distributes it completely to the agents. Hence, having
the supervisor in charge to decide how to share the output does
not maximize the work group production.
Finally, we prove that, under realistic conditions, non-null individual incentive is necessary for supervisor’s profit maximization.
Theorem 8. When bt < 1/2, incentive schemes which maximize the
supervisor’s profit are such that individual incentives are non-null, i.e.,
b > 0.
Proof. Substituting the effort allocations (23) described in Theorem 4 in the objective function of the supervisor’s problem (10),
we obtain
In Section 3, we found that the optimal incentive scheme resulted in null individual incentives, and that any efficient allocation (8), maximizes the production. Furthermore, in Section 4 we
found that, also with quadratic cost functions, when individual
incentives are null the resultant effort allocation (32) is optimal.
Putting together these results with Theorem 8 thesis, it results that
given the aggregate efforts exerted by subordinates, the production
is not maximized. In other words, assuming that the resulting
capacities k1 and k2 for subordinates with quadratic cost functions
are the actual capacity constraints for piecewise constant cost
functions, then the production would be higher since in this case
the effort allocation would be efficient. Note that in this case we
consider gross production because, otherwise, the different incentives for quadratic cost functions would make a comparison between net productions trivial. This result can be interpreted as a
further example of ‘‘systems that pay off for one behavior even
though the rewarder hopes dearly for another” (Kerr, 1975); the
reason here is that the members of the work group possess different goals and motivation. The contrast is even starker when we assume that the whole incentive mechanism was designed by top
management in order to introduce pay incentives for the supervisor and delegating her the authority to motivate subordinates.
When the two different cost functions are interpreted in terms
of motivation, these results may provide some interesting insights.
Within industrial and organizational psychology the study of employee motivation represents one of the most important topics in
the discipline (Jex, 2002). A variety of theories of human motivation have been developed over the years (for a review the reader
may consult Spector, 2003); we may assume that the two cost
4
For the sake of brevity, the proportionality factor is omitted.
884
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
functions we examined are the result of different motivations the
subordinates in the work group share. While the quadratic cost
function seems to be appropriate for individuals who are interested just in monetary incentives, the piecewise constant cost
function, as we mentioned in Section 3, may be appropriate when
subordinates have high self-efficacy and believe they are capable of
accomplishing tasks and motivated to put forth effort (Bandura,
1982).
In this sense, the fact that with the quadratic costs individual
incentives are necessary, together with the fact that optimal production is not achieved, may highlight some limitation of the
scientific management approach, which, among other aspects,
emphasized the provision of economic reward for high performance (Daft and Noe, 2001).
When considering that, according to Weick (1969), any system
consists of several causal relationships, some direct and others inverse, we may assume that, at least in part, the organization may
have some effects on the individual motivation of the employees.
This aspect is related to Theory X/Theory Y (McGregor, 1960); in
fact, according to this theory, the organization management approach determines how subordinates behave and is determined
by the belief of supervisors about their subordinates. According
to Theory X, the worker is viewed as being indifferent to the organization’s needs, lazy and unmotivated; in this case one approach
might be incentivizing the effort, and in this case, the quadratic
cost function would be appropriate. On the other hand, according
to Theory Y, subordinates are capable and not inherently unmotivated or unresponsive to organizational needs, rather it is the
responsibility of the manager to create organizational conditions
so that subordinates can achieve organizational goals through
achieving their own goals; in this case, assuming that individual
motivation does not consist of economic rewards, the piecewise
constant cost function can be appropriate.
Another interpretation of the differences between the two cost
functions, may be found taking into account organizational commitment. Among the different definitions of commitment, Mowday
et al. (1979) view organizational commitment as consisting of three
components: (1) an acceptance of the organizational goals; (2) a
willingness to work hard for the organization; and (3) the desire
to stay with the organization. We can therefore consider the piecewise constant functions appropriate for describing individuals
highly committed to the organization, since they tend to use their
whole capacity. In fact, quadratic costs, may describe a lower commitment situation. On the other hand, the research on withdrawal
behaviors relates to employees not being at work when scheduled
or needed (Spector, 2003); therefore, we can assume the piecewise
constant cost functions realistic in organizations where these counterproductive behaviors do not occur, while by contrast, the quadratic cost functions may be more appropriately assumed in
organization where employee attendance is incentivized.
A last possible interpretation we present, relates to the work of
authors like Herman (1973) and Smith (1977), and, more recently,
Sutton (2007) who examined the relationship between work attitude and such work-related behavior as job performance; in particular Sutton (2007) interprets impaired organizational performance
as the result of dysfunctional leaders. In this case, piecewise constant and quadratic cost functions, might describe different work
attitudes, the second being the result of dysfunctional leadership;
in fact we found that with the latter cost function the effort is
not allocated efficiently.
Finally, for both cost functions, the optimal incentives are independent of agent capacity. This result is quite interesting since, given the fact that individual incentives are identical, subordinates
with different capacities may perceive inequity in the sense of
Adams (1965). With piecewise constant cost functions, when individuals compare ratio of outcomes to input, those with higher
capacity may experience underpayment while those with lower
capacity on the contrary, may experience overpayment. Vice versa,
as it concerns the quadratic cost function this phenomenon may be
not so evident as the individual incentives, at least partially compensate the supervisor exerting higher effort.
6. Conclusion and further research
In this paper, a simple interaction scheme has been proposed
and analyzed. The subordinates are called to allocate their effort
in two interdependent tasks. This interdependence is twofold:
firstly, in each of the two tasks the performance depends upon
the efforts of both subordinates; secondly, the overall performance
depends on both tasks. Given the tasks and the incentive mechanism, the optimal incentives depend on the cost functions of
subordinates.
When considering the analytical results we obtain from the human resources management perspective several insights may be
provided.
In fact, interpreting the cost functions in terms of subordinate
motivation, it emerges that when individuals are not sensitive to
monetary incentives, and rather share a norm according to which
they use their capacities, then there exists an incentive scheme
such that the effort is efficiently allocated. By converse, when individuals are interested only in gain-sharing, no such a scheme
exists.
In Wageman and Baker (1997) it was shown that the interaction
between task and reward interdependence made it difficult to provide effective guidance in solving organizational design problems;
in this paper we fixed the task interdependence and analyzed the
interaction between reward interdependence and individual motivation. In our case simple pay practices, such as delegating the
manager to decide incentives, may be ineffective and, under some
conditions, even counterproductive. In fact, the outcome is that
having this sort of self-managed work group with a gain-sharing
incentive scheme has several drawbacks; in our example the efficiency of the effort allocation is contingent to the motivation of
the subordinates. In terms of incentive design, we found that, in
the case of quadratic costs, the supervisor incentive should be kept
separated from the work group profit. In fact, when the supervisor
is in charge of deciding how to share the output, the work group
aggregate profit is not maximized. Furthermore, also incentives
schemes that may be optimal in the one-shot iteration may exhibit
drawbacks when the interaction is repeated over time.
We provided several interpretations of the two different cost
functions. Independently of the one we choose, what is relevant
here is both the effort allocation efficiency and the cost of incentives which are peculiar to each of them: clearly, a situation with
piecewise constant costs is preferable both in terms of efficiency
and cost of incentives. Furthermore, the difference in terms of production and cost of incentives may provide a quantitative measure
of the differences between organizational cultures.
Finally, Kidwell and Bennett (1993) suggest that the individual
predisposition to withhold effort may be the result of the combination of the environment, the organization and the group; in this paper we assume that the diverse propensity to withhold effort is
formalized by the two cost functions we consider. When considering this propensity as opposite to exerting the discretionary effort,
we show that when the discretionary effort is not exerted, the
introduction of individual incentives does not allow for the efficient allocation of effort. This way we can give a formal example
of the ‘‘damage done” by dysfunctional leaders according to Sutton
(2007).
In further research, we will analyze some dynamical aspects
of the interaction. For example it is evident that, while having
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
piecewise constant function increases the production and reduces
the amount of paid incentives, at the same time it may lead individuals with higher capacity to experience underpayment inequity.
It would be interesting to examine the effort allocation dynamics
when individuals try to reduce the perceived inequity, and also
to understand whether, in this case, individual incentives may become useful to limit the perception.
Furthermore, it would also be interesting to allow several subordinates to interact in a supervised group, extending these results
both analytically and experimentally.
References
Adams, J.S., 1965. Inequity in social exchange. In: Berkowitz, L. (Ed.), Advances in
Experimental Social Psychology. Academic Press, New York, NY, pp. 276–299.
Alchian, A.A., Demsetz, H., 1972. Production, information costs, and economic
organization. American Economic Review 62, 777–795.
Bailey, T., Berg, P., Sandy, C., 2001. The effect of high-performance work practices on
employee earnings in the steel apparel and medical electronics and imaging
industries. Industrial and Labor Relations Review 54 (2A), 525–543.
Bandura, A., 1982. Self-efficacy mechanism in human agency. American
Psychologist 37, 122–147.
Beckmann, M.J., 1988. Tinbergen Lectures on Organization Theory, second ed.
Springer, Berlin.
Bordoloi, S.K., Matsuo, H., 2001. Human resource planning in knowledge-intensive
operations: A model for learning with stochastic turnover, European Journal of
Operational Research.
Boudreau, J., Hopp, W., McClain, J.O., Thomas, L.J., 2003. On the interface between
operations and human resources management. Manufacturing & Service
Operations Management 5 (3), 179–202.
Bylinsky, G., 1990. The hottest high-tech company in Japan. Fortune 121 (1), 82–88.
Coch, L., French Jr., J.R.P., 1948. Overcoming resistance to change. Human Relations
1, 512–532.
Daft, R.L., Noe, R.A., 2001. Organizational Behavior. South-Western Publishing,
Mason, OH.
Dal Forno, A., Merlone, U., 2007. Incentives in supervised teams: An experimental
and computational approach. Journal of Social Complexity 3 (1), 37–52.
Dal Forno, A., Merlone, U., 2009. Individual incentives in supervised work groups.
from human subject experiments to agent based simulation. International
Journal Internet Enterprise Management 6 (1), 4–22.
Encinosa, W.E., Gaynor, M., Rebitzer, J.B., 2007. The sociology of groups and the
economics of incentives: Theory and evidences on compensation systems.
Journal of Economic Behavior & Organization 62, 187–214.
Ethiraj, S.K., Levinthal, D., 2009. Hoping for a to z while rewarding only a: Complex
organizations and multiple goals. Organization Science 20, 4–21.
Fast, N., Berg, N., 1975. The Lincoln Electric Company. Harvard Business School Case,
#376-028.
Gendron, B., 2005. Scheduling employees in Quebec’s liquor stores with integer
programming. Interfaces 35, 402–410.
Glance, N.S., Huberman, B.A., 1994. Social dilemmas and fluid organizations. In:
Carley, K.M., Prietula, M.J. (Eds.), Computational Organization Theory, Lawrence
Erlbaum Associates, Hillsdale, NJ, pp. 217–239.
885
Herman, J.B., 1973. Are situational contingencies limiting job attitude–job
performance relationships? Organizational Behavior and Human Performance
10, 208–224.
Herrmann-Pillath, C., 2005. Culture, economic style and the nature of the Chinese
economic system. Contribution to the China Roundtable of The International
Economic Association.
Holmström, B., 1982. Moral hazard in teams. Bell Journal Economics 13 (2), 324–
340.
Holmström, B., Milgrom, P., 1987. Aggregation and linearity in prevision of intertemporal incentives. Econometrica 7 (2), 302–328.
Holmström, B., Milgrom, P., 1991. Multitask principal-agent analyses: Incentive
contracts, asset ownership, and job design. Journal of Law Economics &
Organization 7, 24–52.
Jex, S.M., 2002. Organizational Psychology. Wiley, New York, NY.
Kerr, S., 1975. On the folly of rewarding a while hoping for b. Academy of
Management Journal 18 (4), 769–783.
Kidwell, R.E., Bennett, N., 1993. Employee propensity to withhold effort: A
conceptual model to intersect three avenues of research. Academy of
Management Review 18 (3), 429–456.
Lum, T., 2003. Workplace codes of conduct in China and related labor conditions.
Key Workplace Documents, Federal Pubblications, Cornell University, ILR
School.
Macho-Stadler, I., Pérez-Castrillo, D., 1997. An Introduction to the Economics of
Information. Oxford University Press, Oxford.
Maslow, A.H., 1970. Motivation and Personality, second ed. Harper & Row, New
York, NY.
McGregor, D.M., 1960. The Human Size of Enterprise. McGraw-Hill, New York, NY.
Merlone, U., submitted for publication. Organizations as allocators of human and
material resources.
Mowday, R.T., Steers, R.M., Porter, L., 1979. The measurement of organizational
commitment. Journal of Vocational Behavior 14.
Roy, D., 1952. Quota restriction and goldbricking in a machine shop. The American
Journal of Sociology 57 (5), 427–442.
Schattler, H., Sung, J., 1997. On optimal sharing rules in discrete- and continuoustime principal-agent problems with exponential utility. Journal of Economic
Dynamics & Control 21, 551–574.
Schelling, T.C., 1960. The Strategy of Conflict. Harvard University Press, Cambridge,
MA.
Smith, F.J., 1977. Work attitudes as predictors of attendance on a specific day.
Journal of Applied Psychology 62 (1), 16–19.
Spector, P.E., 2003. Industrial and Organizational Psychology, third ed. Wiley, New
York, NY.
Steers, R.M., Black, J.S., 1994. Organizational Behavior, fifth ed. Harper Collins, New
York, NY.
Sutton, R., 2007. The No Asshole Rule. Sphere, London.
Varian, H.R., 1992. Microeconomic Analysis. Norton & Company, New York, NY.
Wageman, R., Baker, G., 1997. Incentives and cooperation: The joint effects of task
and
Contents lists available at ScienceDirect
European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Decision Support
Incentives and individual motivation in supervised work groups
Arianna Dal Forno, Ugo Merlone *
Department of Statistics and Applied Mathematics, University of Torino, Corso Unione Sovietica 218 bis, Torino I-10134, Italy
a r t i c l e
i n f o
Article history:
Received 23 April 2008
Accepted 17 May 2010
Available online 25 May 2010
Keywords:
Organization theory
Production
Organizational behavior
Incentives
Individual motivation
a b s t r a c t
This paper introduces and analyzes a model of supervised work group where subordinates decide how to
exert their effort in complementary tasks while the supervisors decide incentives. Incentives may be a
combination of individual and group-based ones. The optimality of incentives is analyzed when considering two different cost functions for subordinates. The two cost functions describe different individual
motivations; comparing the resulting effort allocations and production optimality, we can relate them
to different organizational theories. Our results provide a measure of how motivation among subordinates may affect production and incentives. Furthermore, the optimal incentives schemes are examined
in terms of Adams’ equity theory.
Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction
According to Boudreau et al. (2003), the fields of operations
management and human resources management are intimately related, yet they maintain distinct perspectives.
Among the examples which try to merge these perspectives,
Bordoloi and Matsuo (2001) applied control theory to deal with
workforce planning taking into account also worker learning and
controls for the risk. Another example is Gendron (2005), where
a store scheduling problem with constraints deriving from union
representatives’ and human resources personnel’s was approached
and solved.
Recently, Boudreau et al. (2003) examine how human considerations affect classical operation management. In their conclusion,
they highlight the research challenges and opportunities of bringing the human resources and operation management together.
In this paper, we explore this line of research and try to integrate human considerations in optimal incentive problems. The
Moral Hazard literature approaches multi-agent relationships in
different ways. For example, the joint production models provide
interesting insights in terms of income distribution among
the agents, see for instance Alchian and Demsetz (1972) and
Holmström (1982). Another relevant aspect is the comparison between centralized and decentralized structures as far as contracting goes. For example, the literature provides conditions under
which the delegation of the supervisory task, i.e. decentralizing,
* Corresponding author. Tel.: +39 011 6705753; fax: +39 011 6705783.
E-mail address: merlone@econ.unito.it (U. Merlone).
0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2010.05.023
is beneficial; to obtain a first analysis of the advantages and disadvantages of delegation the reader may refer to Macho-Stadler and
Pérez-Castrillo (1997).
One aspect usually neglected in the economic literature is the
role of individual motivation; while, in psychology, motivation is
a concept that has been discussed extensively. According to Spector (2003) work motivation theories are most typically concerned
with the reasons, rather than ability, that some people perform
their jobs better than others. Steers and Black list the stages the
evolution of management thought concerning employee motivation has passed through. They are the traditional model, the human
relations approach and, more recently, the human resources model. In particular, ‘‘this newer approach also assumes that different
employees want different rewards from their jobs, that many
employees sincerely want to contribute, and that employees by
and large have the capacity to exercise a great deal of self-direction
and self-control at work” (Steers and Black, 1994, p. 139).
To this extent the case of Kyocera Corporation is striking. As it
concerns the reward system, Kyocera’s founder Kazuo Inamori, in
a booklet describing his philosophy, writes ‘‘We don’t think in
terms of individual rewards. We don’t buy individuals’ loyalty with
monetary incentives or titles. Rather, we believe that individuals
who are endowed with superior capabilities should contribute
their capabilities for the good of the entire group.” (for a discussion
of Kyocera’s organizational culture the reader may refer to Bylinsky
(1990)). This example is contrasted by pay incentives used at Lincoln Electric, where most employees are paid on a piece-rate system (the Lincoln Electric Company has been described in several
case studies by business schools, see for instance Fast and Berg
(1975)). Several comparisons between incentive programs have
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
been presented in the literature. Among others, Weiss (1987) provides an empirical comparison between individual wage incentives
and group incentives examining the effects in terms of motivation
and quit rates.
Following the joint production approach, we consider a modular model of hierarchical organization. Specifically, we concentrate
mainly on pyramidal structures. This particular structure is widespread and, consequently, both the economic (see Beckmann,
1988, for a formal analysis) and simulative literature (for instance,
see Glance and Huberman, 1994) find interest (for an analysis of
the different approaches to pyramidal structures see Merlone,
submitted for publication). In our model, the organization consists
of two heterogeneous agents interacting in a supervised work
group with a Cobb–Douglas production function. In the literature,
the distinction between team and work group is hazy, nevertheless
we will follow Spector (2003). As a consequence, we will refer to a
supervised work group as, in our case, we consider interchangeable
subordinates.
In this paper, we assume that individuals with different motivation may have different cost functions and analyze the incentive
problem the supervisor faces, in order to maximize her own profit.
In particular, we analyze how the optimal solution for the supervisor varies when considering two different cost functions for subordinates. Cost function plays an important role for the agent.
While in Dal Forno and Merlone (2007) only piecewise constant
cost function was considered, in this paper we also consider
strictly convex costs. Usually economics assumes that costs are
increasing and marginally increasing; the other kind of cost function in some cases may model more realistically some situations.
We refer in particular to situations like those described in Smith
(1977) where employees expend their discretionary effort (for a
discussion about antecedents of discretionary effort and its consequences on performance the reader may refer to Bailey et al.
(2001) and Sutton (2007)). In this sense, the two different cost
functions can be interpreted in terms of motivation; while the
piecewise constant cost function may be appropriate when subordinates have high self-efficacy and are highly motivated, the other
cost function seems to be more appropriate for individuals who
are mainly interested in monetary incentives. It is well known
(Zhou, 2002) that, with regard to the principal-agent theory, in
general it is difficult to derive analytical solutions; therefore, even
in scholarly contributions, strong assumptions are usually required. This tradeoff between analytical tractability and extensive
simplification is acknowledged by other authors approaching
organization complexity (see Ethiraj and Levinthal, 2009). Nevertheless, the two different cost functions we analyze can be related
to different cultures in the organizational structure we consider.
As a consequence, by comparing the production outputs under
the two different cost function assumptions, our analysis allows
us to compare productivity under different organizational cultures
and to measure what the cost of the culture in terms of production is. The two different cost functions, and the related organizational cultures, are particularly interesting when considering
changes in competition induced by emergent countries such as
China (for a discussion on the role of culture in the economic style
of China, the reader may refer to Herrmann-Pillath (2005) and to
Lum (2003), for an analysis of labor conditions in the same
country).
The structure of the paper is the following. In Section 2, we
present the theoretical model. Sections 3 and 4 summarize and
analyze the optimal incentive problem with the two different cost
functions in question, describing how these are related to the subordinates’ motivation. In Section 5, optimal incentive schemes and
outputs are compared and the results are interpreted in terms of
organizational culture consequences on productivity. Finally, Section 6 is devoted to conclusions and further research.
879
2. The model
As in Dal Forno and Merlone (2007) and Dal Forno and Merlone
(2009), we consider a model of supervised work group in which a
supervisor (acting as principal) and two subordinates (acting as
agents) cooperate. Agent i allocates his effort li with the partner,
and the effort ui with the supervisor. The joint production function
for agents 1 and 2 is C(u1 + u2)a(l1 + l2)b, where C 2 Rþþ is a constant factor,1 and a, b 2 (0, 1) are, respectively, the output elasticity
with respect to the joint effort with the supervisor and with the
partner. As a consequence, the agents have to decide both how
much effort to exert, and how to partition it in the two complementary tasks.2 Agents bear a cost for effort: agent i’s cost function
ci : R2þ ! Rþ will be denoted with ci(ui, li); cost functions are private
information. Furthermore, each agent can observe the level effort his
partner provides with him, but not the one which is provided with
the supervisor. Conversely, the supervisor can only observe the joint
output and the effort each agent provides with her. The supervisor’s
profit is a share c 2 (0, 1) of the supervised work group production
minus the incentives she pays to her subordinates. In the following,
we assume that the output is sold on market at unitary price and the
production and sharing constants C and c are such that Cc = 1; this
is not restrictive, it simplifies the notation, and allows us to simply
consider monetary payoffs. Finally, agents’ retribution consists of a
fixed wage w > 0 plus a performance-contingent reward; we assume
that the fixed wage is sufficient to meet basic needs, in terms of the
hierarchy of needs theory (Maslow, 1970), physiological needs and
needs of safety; in economic terms we say that the participation constraint is met.
Proposition 1. The gross production (u1 + u2)a(l1 + l2)b is maximized
if and only if the aggregate efforts are allocated proportionally to the
output elasticities.
Proof. The result follows from combining and rearranging the first
order conditions of the problem
max ðu1 þ u2 Þa ðl1 þ l2 Þb :
u1 ;u2 ;l1 ;l2
ð1Þ
In fact, from
(
aðu1 þ u2 Þa1 ðl1 þ l2 Þb ¼ 0;
bðu1 þ u2 Þa ðl1 þ l2 Þb1 ¼ 0;
ð2Þ
it follows
u1 þ u2 a
¼ :
l1 þ l2
b
ð3Þ
Condition (3) is necessary in order to maximize the production of
the supervised group. When either as the result of misaligned
incentives or as lack of coordination between subordinates this condition is not met, then the effort allocation is not efficient. The performance-contingent reward is a linear incentive bt on the joint
output of the team and a linear incentive bi on the effort each agent
exerts with the supervisor. Therefore, the problem can be formalized as a bilevel programming problem:
max ð1 2bt Þðu1 þ u2 Þa ðl1 þ l2 Þb b1 u1 b2 u2 ;
bt ;b1 ;b2
ð4Þ
such that, given the incentives bt, b1, b2, subordinates solve
max
w þ bt ðu1 þ u2 Þa ðl1 þ l2 Þb þ b1 u1 c1 ðu1 ; l1 Þ;
max
w þ bt ðu1 þ u2 Þa ðl1 þ l2 Þb þ b2 u2 c2 ðu2 ; l2 Þ;
u1 ;l1
u2 ;l2
1
ð5Þ
We recall that Rþþ is the set of positive real numbers; the case C = 0 is trivial.
From the functional form of the production function it is immediate to observe
that the two tasks are not additive; for a discussion the reader may refer to Spector
(2003).
2
880
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
where we have defined:
i: index of subordinate, i = 1, 2;
ui: effort subordinate i exerts in the task with the supervisor;
li: effort subordinate i exerts in the task with the colleague;
a: output elasticity with respect to the joint effort with the
supervisor;
b: output elasticity with respect to the joint effort between
subordinate;
ci(ui, li): cost function for subordinate i;
w: fixed wage;
bt: linear incentive on the joint output of the team;
bi: linear incentive on the effort subordinate i exerts with the
supervisor.
For the sake of simplicity we assume w = 0, this is not restrictive. Examining the form of the problem, it is rather immediate
to predict the behavior of rational supervisor and subordinates in
this interaction situation. It is a finite dynamic game with perfect
information, with supervisor moving first, and then subordinates,
acting simultaneously after observing the incentive. This game
has a proper subgame starting from the information set of the subordinates. Therefore, there exists a set of subgame perfect Nash
equilibria which equals the set of Nash equilibria that can be derived by backward induction. In fact, given any feasible incentives
scheme (bt, b1, b2), the subordinates will play a Nash equilibrium3
(ui, li) of the subgame. Knowing the fact that subordinates will play
a Nash equilibrium in the subgame, the supervisor will maximize
her profit by choosing the optimal incentive scheme. In the following
sections, we will analyze how different assumptions on the subordinates’ cost functions may determine the optimal incentive scheme.
3. Piecewise constant costs
In the literature, some factor or factors that keep agents from
working infinitely hard are usually considered. For example, Wageman and Baker (1997) propose several mechanisms considering
effort becoming increasingly either unproductive or unpleasant.
In this paper, as common, we assume that the unpleasantness of
the work increases with respect to the effort. Different cost functions may be considered and the functional form may reflect different underlying assumptions. While economics usually considers
mainly rational agents, other approaches to work group dynamics
take into account other aspects, such as norms and team commitment. Industrial and organizational psychology have proposed different theories of motivation to explain individual behavior in the
organization (for a first survey the reader may refer to Spector
(2003)). Nevertheless, several other aspects underlie group and
team behavior; the role of norms is well documented empirically
(see Coch and French, 1948 and Roy, 1952, for example) and it is
commonly assumed, for work groups, to dictate how much each
person will produce (see Spector, 2003). A first cost function we
consider is, as in Dal Forno and Merlone (2007), the following:
ci ðui ; li Þ ¼
0
if ui þ li 6 ci ;
þ1 if ui þ li > ci :
ð6Þ
First observe that this cost function is non-decreasing with respect
to the aggregate effort. It assumes that each subordinate has a physical capacity ci under which effort has zero cost, or, alternatively,
that at some exertion level the effort becomes unpleasant enough
to lead the individual to conclude that it is not worth working
any harder independently of the reward. In this case, we assume
that each individual knows his individual capacity and uses it with3
In the next sections, we will examine also the equilibrium selection problem.
out goldbricking. This kind of cost function is not completely new to
the principal-agent literature; for instance, Holmström and Milgrom (1991) assume that incentives are necessary to encourage
agents to work beyond a limit they would take pleasure in working.
In our case we assume that this limit coincides with their physical
capacity. Finally, observe that this assumption can also be interpreted in terms of the self-efficacy theory (Bandura, 1982), assuming that both subordinates have high self-efficacy and are motivated
to put in as much effort as they can.
3.1. The agents’ problem
Starting from the subordinates’ information set and assuming
coordination and commitment between agents, problem (5) reduces to:
max ðu1 þ u2 Þa ðl1 þ l2 Þb
sub ui þ li 6 ci
u1 ;u2 ;l1 ;l2
ð7Þ
for all i = 1, 2.
Corollary 2. Consider the production function (u1 + u2)a(l1 + l2)b, with
piecewise constant cost functions (6). Then, any effort allocation such
that
(
a ð
u1 þ u2 ¼ aþb
c1 þ c2 Þ;
ð8Þ
b
l1 þ l2 ¼ aþb
ðc1 þ c2 Þ;
maximizes the production.
Proof. The result follows trivially from first order conditions of
problem (7), assuming the constraint on individual aggregate effort
given by capacities. h
Therefore, there is a continuum of solutions to the considered
problem (7), nevertheless, a rather natural effort allocation is the
one that can be interpreted as focal in the sense of Schelling (1960):
ðui ; li Þ ¼
a b
c;
c
aþb i aþb i
ð9Þ
for all i = 1, 2.
Effort allocations such that (8) holds, will be called efficient, in
fact they satisfy also condition (3) which is necessary to the production maximization.
3.2. The supervisor’s problem
The supervisor’s problem is to design a linear compensation
scheme for the subordinates that induces them to use their capacity to maximize the team output. The supervisor can observe the
efforts ui the subordinates exert with her and the team output.
Each subordinate’s compensation is si = biui + bt(u1 + u2)a(l1 + l2)b,
where, we recall, bi is the incentive given to subordinate i for his
individual effort with supervisor, and bt is the incentive given to
them for the team output. We assume that the supervisor declares
the incentives and then the subordinates decide their efforts in
order to maximize their wage. The supervisor has to solve the
following problem:
max ð1 2bt Þðu1 þ u2 Þa ðl1 þ l2 Þb b1 u1 b2 u2 :
bt ;b1 ;b2
ð10Þ
When considering fully rational agents the solution is obvious.
Since any individual incentive given to agents gives a suboptimal
effort allocation, and null team output incentive makes for subordinates any allocation optimal, the optimal solution is
8
>
< bt ¼ e > 0;
b1 ¼ 0;
>
:
b2 ¼ 0:
ð11Þ
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
Some comments about this incentive scheme are in order. While it
seems to be appropriate for agents with the same capacities or a
low number of repetitions, for agents with different capacities
interacting over a long period of time some problems may arise.
For example, when considering equity theory (Adams, 1965), it is
rather evident that different capacity individuals will find themselves in inequitable situations, and will experience dissatisfaction
and emotional tension, that they will be motivated to reduce. In order to improve the distributive justice (see Spector, 2003, for a discussion about equity theory and distributive justice) a different
incentive scheme may be more appropriate in the long run.
4. Strictly convex costs
Following Holmström and Milgrom (1991), we suppose the effort in the two tasks is perfectly substitutable in the agent’s cost
function. Formally we assume that
ci ðui ; li Þ ¼ ci ðui þ li Þ:
ð12Þ
As it is common, ci is assumed differentiable, increasing and marginally increasing. As mentioned above, while this cost function is
closer to the classical modeling of behavior of the rational individual, we may assume this kind of cost function when agents’ motivation consists only in economic reward and neither social norms nor
other motivational theories determine individual effort. It is not difficult to assume that this kind of cost function may be realistic in
organizations where individual objective consists only in the maximization of the individual payoff. Another interpretation of this
cost function is as the result of high-powered incentives. According
to Steers and Black (1994), individual incentives may, at times, lead
to employees competing with one another; in our case we assume
that the internal competition, driven by high-powered incentives,
may result as this individual perception of cost of effort (for a discussion on high-powered incentives, see Encinosa et al., 2007). As
a consequence, we can think that this kind of cost function is the result of a different culture when compared to the piecewise constant
case. In this case also, we first analyze the agents’ problem and then
the supervisor’s one. Finally, we observe that some of the results we
present in the following can be generalized to more general cost
functions.
4.1. The agents’ problem
In this case, it would be rather unrealistic to assume commitment and coordination between subordinates, rather we will assume that agents solve problem (5) with cost function (12)
max
u1 ;l1
max
u2 ;l2
a
b
a
b
bt ðu1 þ u2 Þ ðl1 þ l2 Þ þ b1 u1 c1 ðu1 þ l1 Þ;
bt ðu1 þ u2 Þ ðl1 þ l2 Þ þ b2 u2 c2 ðu2 þ l2 Þ:
ð13Þ
As a consequence, given the incentives (bt, b1, b2), subordinates find
an effort allocation which is a Nash equilibrium, i.e., u1 ; l1 ; u2 ; l2
must be such that
u1 ; l1
¼ arg max bt u1 þ
a
u2
l1 þ
b
l2
þ b1 u1 c1 ðu1 þ l1 Þ
ð14Þ
and
a
b
u2 ; l2 ¼ arg max bt u1 þ u2 l1 þ l2 þ b2 u2 c2 ðu2 þ l2 Þ:
ð15Þ
While for the piecewise constant cost function it was immediate to
derive the effort allocation, in this case some preliminary considerations are in order.
881
Lemma 3. Given an incentive scheme (bt, b1, b2), if subordinates’ cost
functions have form (12), then an optimal effort allocation
u1 ; l1 ; u2 ; l2 must be such that
c01 u1 þ l1 ¼ c02 u2 þ l2 :
ð16Þ
b1 ¼ b2 :
ð17Þ
Furthermore, optimal incentives must be such that individual incentives are identical, that is
Proof. Since we assume that the effort allocation constitutes a
Nash equilibrium, first order conditions can be derived from (14)
and (15); they are:
8
abt
>
>
>
>
>
< bbt
>
> abt
>
>
>
:
bbt
a1
b
l1 þ l2 þ b1 c01 u1 þ l1 ¼ 0;
b1
u1 þ u2 l1 þ l2
c01 u1 þ l1 ¼ 0;
a1
b
u1 þ u2
l1 þ l2 þ b2 c02 u2 þ l2 ¼ 0;
a
b1
u1 þ u2 l1 þ l2
c02 u2 þ l2 ¼ 0:
u1 þ u2
a
ð18Þ
Combining the second and the fourth equation it is immediate to
obtain
c01 u1 þ l1 ¼ c02 u2 þ l2 :
ð19Þ
b1 c01 u1 þ l1 ¼ b2 c02 u2 þ l2 :
ð20Þ
b1 ¼ b2 :
ð21Þ
For the second part, combining the first and the third equation in
(18) it is immediate to obtain
Finally, by Eq. (19) we obtain
Following this result, incentives schemes will be written using the
shortened notation (bt, b1, b2) = (bt, b).
As mentioned in Section 1, it is well known that principalagents models often require strong assumptions to derive analytical results. This model is not an exception; in the following, as it is
common (see for instance, Holmström and Milgrom, 1987; Schattler and Sung, 1997; Encinosa et al., 2007), we assume a quadratic
cost function
ci ðui ; li Þ ¼ di ðui þ li Þ2 ;
ð22Þ
where di, i = 1, 2 are positive constants.
We also assume a + b = 1.
In this case, a result concerning proportionality of allocated efforts can be proved, as well.
Theorem 4. Given an incentive scheme (bt, b), with b – 0, there exists
a unique equilibrium effort allocation u1 ; l1 ; u2 ; l2 such that
(
u1 ¼ q u1 þ l1 ¼ qk1 ; l1 ¼ ð1 qÞ u1 þ l1 ¼ ð1 qÞk1 ;
u2 ¼ q u2 þ l2 ¼ qk2 ; l2 ¼ ð1 qÞ u2 þ l2 ¼ ð1 qÞk2 ;
where
k1
:¼
u1
þ
l1 ; k2
:¼
u2
þ
l2
ð23Þ
and 0 < q < 1.
Proof. First we will show that such an effort allocation is an equilibrium allocation. Since the objective functions are concave, first
order conditions are also sufficient and the optimal solution must
satisfy (18):
8
a1 k þ k a1 ð1 qÞ1a k þ k 1a þ b 2d k ¼ 0;
>
1 1
>
1
2
1
2
> abt q
>
>
< ð1 aÞb qa k þ k a ð1 qÞa k þ k a 2d k ¼ 0;
t
1
1
2
1
2
1
a1
1a
1a
>
> abt qa1 k1 þ k2
ð1
q
Þ
k
þ
k
þ
b
2d
2 k2 ¼ 0;
>
1
2
>
>
:
a
a
a
ð1 aÞbt qa k1 þ k2 ð1 qÞ k1 þ k2
2d2 k2 ¼ 0:
ð24Þ
882
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
Combining the second and the fourth equation it is immediate to
obtain d1 k1 ¼ d2 k2 , that is
k2 ¼
d1
k ;
d2 1
ð25Þ
u1 þ u2 ¼
8
1a
a1
>
þ b 2d1 k1 ¼ 0;
>
< abt q ð1 qÞ
ð1 aÞbt qa ð1 qÞa 2d1 k1 ¼ 0;
>
>
: k ¼ d1 k :
ð26Þ
a
1a
ðl1 þ l2 Þ;
aa bt
;
2d1 ð1 aÞa1
Putting together the first two equations and rearranging
analogously, we find
8
a
qÞ1a
>
abt ð1q1
ð1 aÞbt ð1qqÞa þ b ¼ 0;
>
a
>
<
k2 ¼ u2 þ l2 ¼
ð1aÞbt qa ð1qÞa
2d1
k
> 1
¼
>
>
: k ¼ d1 k :
2
d2 1
ð27Þ
;
Now consider function f ðqÞ ¼ abt
ð1qÞa
q1a
qa
ð1 aÞbt ð1qÞa þ b; it is
ð34Þ
which, substituted in the first equation of (33), yields
k1 ¼ u1 þ l1 ¼
1
d2
ð33Þ
Combining the first and the second equation of (33) it is immediate
to obtain
so we can discard the third equation as redundant.
By simple algebra we obtain
2
8
>
abt ðu1 þ u2 Þa1 ðl1 þ l2 Þ1a 2d1 ðu1 þ l1 Þ ¼ 0;
>
>
>
< bb ðu þ u Þa ðl þ l Þa 2d ðu þ l Þ ¼ 0
t
1
2
1
2
1
1
1
a1
1a
>
2d2 ðu2 þ l2 Þ ¼ 0;
> abt ðu1 þ u2 Þ ðl1 þ l2 Þ
>
>
:
bbt ðu1 þ u2 Þa ðl1 þ l2 Þa 2d2 ðu2 þ l2 Þ ¼ 0:
ð35Þ
aa bt
:
2d2 ð1 aÞa1
ð36Þ
The focal allocation can be found, as before, as
(
k2 ;
l2
u1 ¼ ak1 ; l1 ¼ ð1 aÞk1 ;
u2
¼a
ð37Þ
¼ ð1 aÞk2 :
easy to prove that there exists a unique q* 2 ]0, 1[ such that
abt
ð1 q Þ1a
q1a
qa
ð1 aÞbt
þ b ¼ 0;
ð1 q Þa
ð28Þ
since function f is continuous in ]0, 1[. Furthermore, it holds
8
a
qÞ1a
>
abt ð1q1
ð1 aÞbt ð1qqÞ1b þ b ¼ þ1;
a
< lim
q!0
qÞ1a
>
: lim abt ð11
ð1 aÞbt
a
q
q!1
qa
ð1qÞa
ð29Þ
þ b ¼ 1;
Corollary 6. The fraction q of effort made with the supervisor is
increasing with respect to the individual incentive b and its derivative
is
and
f 0 ðqÞ ¼ abt bð1qÞ
bbt
b1 1a
q ð1aÞqa ð1qÞb
ðq1a Þ2
að1qÞ1b qa1 þð1bÞqa ð1qÞb
½ð1qÞ1b 2
ð30Þ
< 0:
Thus, since, as b – 0, q* is unique, there exists the unique effort
allocation
8
aþ1
a
ð1aÞbt q
ð1aÞbt q ð1q Þ1a
>
;
>
> u1 ¼ 2d1 ð1q Þa ; l1 ¼
2d1
>
<
aþ1
a
1
a
aÞbt q
qÞ
; l2 ¼ ð1aÞbt q2dð1
;
u2 ¼ ð1
2d2 ð1q Þa
2
>
>
>
a
a
>
ð1aÞbt q
: k ¼ ð1aÞbt q ;
k2 ¼ 2d ð1q Þa :
1
2d ð1q Þa
1
q0 ðbÞ ¼
q2a ð1 qÞ1þa
:
að1 aÞbt
ð38Þ
Proof. Eq. (28) can be written as
f ðq; bÞ ¼ 0;
ð39Þ
where
ð31Þ
2
Theorem 4 does not claim that, with positive individual incentives, all effort allocation equilibria are characterized by efforts
being proportionally allocated between the two tasks. Rather, it
claims that among the proportional effort allocations, i.e. those satisfying condition (3), one and only one is an equilibrium.
In the following corollary, we analyze the effort allocation with
null individual incentives.
Corollary 5. Given an incentive scheme (bt,0), if subordinates’ cost
functions have form (22), then any allocation such that ak1 ¼
ab
a
t
ð1 aÞk2 is an equilibrium allocation, where k1 :¼
;
2ð1aÞa1 d1
aa bt
k2 :¼
a1 . Furthermore, there exists a focal effort allocation
2ð1aÞ
Remarkably, effort allocation (37) is the analogous of the focal
allocation (9) we found for piecewise constant costs in Corollary
2. In this case also, given the aggregate efforts the subordinates
put forth, the production is maximized. Furthermore, we observe
that q defines what fraction of the agents’ efforts is made with
the supervisor. By the implicit function theorem we can prove that,
as expected, this fraction increases as b increases. In fact, a greater
individual incentive reallocates a greater effort with the supervisor.
f ðq; bÞ :¼ abt
ð1 qÞ1a
q
1a
ð1 aÞbt
qa
þ b;
ð1 qÞa
ð40Þ
implicitly defines q as a function of b. By the implicit function
theorem
q0 ðbÞ ¼
of =ob
¼
of =oq
1
að1aÞbt
q2a ð1qÞ1þa
¼
q2a ð1 qÞ1þa
:
að1 aÞbt
ð41Þ
As both 0 < a < 1 and 0 < q < 1 the thesis follows. h
This result proves that when individual incentives are both positive, the effort allocation no longer satisfies condition (3) and has
several implications when considering how the supervisor must
determine incentives optimally.
4.2. The supervisor’s problem
d2
8
aþ1
aa bt
< u1 ¼ a ab1t ; l1 ¼
;
2ð1aÞ
d
2ð1aÞa2 d
1
1
aa bt
: u ¼ aaþ1 bt ; l ¼
:
2
2
2ð1aÞa1 d
2ð1aÞa2 d
2
ð32Þ
2
Proof. Under the assumptions b = 0 and a + b = 1, conditions (18)
become
As for the piecewise constant function, the supervisor’s problem
is to determine the incentive scheme maximizing her own payoff
as in (10).
Before investigating whether individual incentive b increases
supervisor’s profit, we characterize the optimal incentives when
b = 0. To this extent, we consider the aggregate profit
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
u1 þ u2
a
1a
l1 þ l2
2
d1 u1 þ l1
2
d2 u2 þ l2 :
ð42Þ
Theorem 7. With incentive schemes (bt, b) where b = 0, the aggregate
profit (42) is maximized if and only if it is completely shared between
the two agents; by contrast, in order to maximize supervisor’s profit,
half of the aggregate profit must be given to the supervisor and the
remaining equally shared between the agents.
Proof. To maximize the aggregate profit the following program
must be solved:
max
bt 612
u1
þ
a
u2
l1
þ
1a
l2
d1 u1
þ
2
l1
d2 u2
þ
2
l2 :
aaþ1 bt d1 þ d2
max
bt 612
2ð1 aÞa1 d1 d2
d1
aa b
t
2ð1 aÞa1 d1
!2
#a "
aa bt
d1 þ d2
a2 d d
1 2
2ð1 aÞ
aa b
t
d2
2ð1 aÞa1 d2
!2
ð44Þ
;
that is
max
bt 612
"
aa bt
d1 þ d2 a
aa bt
a ð1 aÞ1a
a1 d d
1 2
2ð1 aÞ
2ð1 aÞa1
#2
d1 þ d2
:
d1 d2
ð45Þ
2
Since 0 < a < 1 and dd11þd
> 0, the original program (43) is equivalent
d2
to
2
max bt
bt 612
bt
;
2
Now recall that in Corollary 6 we proved that q is a function of b. As
a consequence, we can consider the supervisor’s profit4 as a function of the individual incentive b
pðbÞ ¼ ð1 2bt Þq2a ðbÞð1 qðbÞÞ12a bqaþ1 ðbÞð1 qðbÞÞa :
ð52Þ
Now derive p with respect to b using expression (38) we found in
Corollary 6, substitute b = 0 and recall that q(0) = a,
1 2bt a
1 2bt aaþ1
a ð1 aÞa ¼ a
:
bt
bt
ð1 aÞa
ð53Þ
Since bt < 1/2, the supervisor’s profit increases as b increases. As a
consequence, incentive schemes (bt, b) where b = 0 cannot be
optimal. h
The consequences of this theorem are striking. In fact, this result proves that, with quadratic costs (22), the individual incentives are necessary. Nevertheless, by Corollary 6, it follows that
individual incentives increase the fraction of effort made with
the supervisor. Therefore, with positive individual incentives the
effort allocation does not satisfy condition (3), and gross production is not maximized. In the following section, we will relate the
assumption underlying the different cost functions we have considered to the managerial literature.
5. Comparing the two cost functions
maxð1 2bt Þ u1 þ u2
bt 612
that is
bt 612
ð51Þ
ð46Þ
and the optimal solution is bt = 1/2, that is, all the profit is shared
between the two agents.
For the second part of the theorem, let us consider the
supervisor’s profit (10) when the incentives are null
maxð1 2bt Þ
ð1 2bt Þq2a ð1 qÞ12a bqaþ1 ð1 qÞa :
p0 ð0Þ ¼ a
#1a
ð50Þ
That is, the supervisor’s profit is proportional to
ð43Þ
Taking into account the focal effort allocation, as given by Eq. (32),
the program (43) becomes
"
i
ð1 aÞbt d1 þ d2 h
ð1 2bt Þq2a ð1 qÞ12a bqaþ1 ð1 qÞa :
2
d1 d2
883
a
1a
l1 þ l2
ð47Þ
;
aa bt
d1 þ d2 a
a ð1 aÞ1a :
2ð1 aÞa1 d1 d2
This problem, since 0 < a < 1 and
d1 þd2
d1 d2
maxð1 2bt Þbt ;
bt 612
ð48Þ
> 0, is equivalent to
ð49Þ
and the optimal solution is bt = 1/4.
This proves the thesis. h
This result shows that the agents are the residual claimant (see
for instance, Varian, 1992) to the output produced if and only if the
supervisor distributes it completely to the agents. Hence, having
the supervisor in charge to decide how to share the output does
not maximize the work group production.
Finally, we prove that, under realistic conditions, non-null individual incentive is necessary for supervisor’s profit maximization.
Theorem 8. When bt < 1/2, incentive schemes which maximize the
supervisor’s profit are such that individual incentives are non-null, i.e.,
b > 0.
Proof. Substituting the effort allocations (23) described in Theorem 4 in the objective function of the supervisor’s problem (10),
we obtain
In Section 3, we found that the optimal incentive scheme resulted in null individual incentives, and that any efficient allocation (8), maximizes the production. Furthermore, in Section 4 we
found that, also with quadratic cost functions, when individual
incentives are null the resultant effort allocation (32) is optimal.
Putting together these results with Theorem 8 thesis, it results that
given the aggregate efforts exerted by subordinates, the production
is not maximized. In other words, assuming that the resulting
capacities k1 and k2 for subordinates with quadratic cost functions
are the actual capacity constraints for piecewise constant cost
functions, then the production would be higher since in this case
the effort allocation would be efficient. Note that in this case we
consider gross production because, otherwise, the different incentives for quadratic cost functions would make a comparison between net productions trivial. This result can be interpreted as a
further example of ‘‘systems that pay off for one behavior even
though the rewarder hopes dearly for another” (Kerr, 1975); the
reason here is that the members of the work group possess different goals and motivation. The contrast is even starker when we assume that the whole incentive mechanism was designed by top
management in order to introduce pay incentives for the supervisor and delegating her the authority to motivate subordinates.
When the two different cost functions are interpreted in terms
of motivation, these results may provide some interesting insights.
Within industrial and organizational psychology the study of employee motivation represents one of the most important topics in
the discipline (Jex, 2002). A variety of theories of human motivation have been developed over the years (for a review the reader
may consult Spector, 2003); we may assume that the two cost
4
For the sake of brevity, the proportionality factor is omitted.
884
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
functions we examined are the result of different motivations the
subordinates in the work group share. While the quadratic cost
function seems to be appropriate for individuals who are interested just in monetary incentives, the piecewise constant cost
function, as we mentioned in Section 3, may be appropriate when
subordinates have high self-efficacy and believe they are capable of
accomplishing tasks and motivated to put forth effort (Bandura,
1982).
In this sense, the fact that with the quadratic costs individual
incentives are necessary, together with the fact that optimal production is not achieved, may highlight some limitation of the
scientific management approach, which, among other aspects,
emphasized the provision of economic reward for high performance (Daft and Noe, 2001).
When considering that, according to Weick (1969), any system
consists of several causal relationships, some direct and others inverse, we may assume that, at least in part, the organization may
have some effects on the individual motivation of the employees.
This aspect is related to Theory X/Theory Y (McGregor, 1960); in
fact, according to this theory, the organization management approach determines how subordinates behave and is determined
by the belief of supervisors about their subordinates. According
to Theory X, the worker is viewed as being indifferent to the organization’s needs, lazy and unmotivated; in this case one approach
might be incentivizing the effort, and in this case, the quadratic
cost function would be appropriate. On the other hand, according
to Theory Y, subordinates are capable and not inherently unmotivated or unresponsive to organizational needs, rather it is the
responsibility of the manager to create organizational conditions
so that subordinates can achieve organizational goals through
achieving their own goals; in this case, assuming that individual
motivation does not consist of economic rewards, the piecewise
constant cost function can be appropriate.
Another interpretation of the differences between the two cost
functions, may be found taking into account organizational commitment. Among the different definitions of commitment, Mowday
et al. (1979) view organizational commitment as consisting of three
components: (1) an acceptance of the organizational goals; (2) a
willingness to work hard for the organization; and (3) the desire
to stay with the organization. We can therefore consider the piecewise constant functions appropriate for describing individuals
highly committed to the organization, since they tend to use their
whole capacity. In fact, quadratic costs, may describe a lower commitment situation. On the other hand, the research on withdrawal
behaviors relates to employees not being at work when scheduled
or needed (Spector, 2003); therefore, we can assume the piecewise
constant cost functions realistic in organizations where these counterproductive behaviors do not occur, while by contrast, the quadratic cost functions may be more appropriately assumed in
organization where employee attendance is incentivized.
A last possible interpretation we present, relates to the work of
authors like Herman (1973) and Smith (1977), and, more recently,
Sutton (2007) who examined the relationship between work attitude and such work-related behavior as job performance; in particular Sutton (2007) interprets impaired organizational performance
as the result of dysfunctional leaders. In this case, piecewise constant and quadratic cost functions, might describe different work
attitudes, the second being the result of dysfunctional leadership;
in fact we found that with the latter cost function the effort is
not allocated efficiently.
Finally, for both cost functions, the optimal incentives are independent of agent capacity. This result is quite interesting since, given the fact that individual incentives are identical, subordinates
with different capacities may perceive inequity in the sense of
Adams (1965). With piecewise constant cost functions, when individuals compare ratio of outcomes to input, those with higher
capacity may experience underpayment while those with lower
capacity on the contrary, may experience overpayment. Vice versa,
as it concerns the quadratic cost function this phenomenon may be
not so evident as the individual incentives, at least partially compensate the supervisor exerting higher effort.
6. Conclusion and further research
In this paper, a simple interaction scheme has been proposed
and analyzed. The subordinates are called to allocate their effort
in two interdependent tasks. This interdependence is twofold:
firstly, in each of the two tasks the performance depends upon
the efforts of both subordinates; secondly, the overall performance
depends on both tasks. Given the tasks and the incentive mechanism, the optimal incentives depend on the cost functions of
subordinates.
When considering the analytical results we obtain from the human resources management perspective several insights may be
provided.
In fact, interpreting the cost functions in terms of subordinate
motivation, it emerges that when individuals are not sensitive to
monetary incentives, and rather share a norm according to which
they use their capacities, then there exists an incentive scheme
such that the effort is efficiently allocated. By converse, when individuals are interested only in gain-sharing, no such a scheme
exists.
In Wageman and Baker (1997) it was shown that the interaction
between task and reward interdependence made it difficult to provide effective guidance in solving organizational design problems;
in this paper we fixed the task interdependence and analyzed the
interaction between reward interdependence and individual motivation. In our case simple pay practices, such as delegating the
manager to decide incentives, may be ineffective and, under some
conditions, even counterproductive. In fact, the outcome is that
having this sort of self-managed work group with a gain-sharing
incentive scheme has several drawbacks; in our example the efficiency of the effort allocation is contingent to the motivation of
the subordinates. In terms of incentive design, we found that, in
the case of quadratic costs, the supervisor incentive should be kept
separated from the work group profit. In fact, when the supervisor
is in charge of deciding how to share the output, the work group
aggregate profit is not maximized. Furthermore, also incentives
schemes that may be optimal in the one-shot iteration may exhibit
drawbacks when the interaction is repeated over time.
We provided several interpretations of the two different cost
functions. Independently of the one we choose, what is relevant
here is both the effort allocation efficiency and the cost of incentives which are peculiar to each of them: clearly, a situation with
piecewise constant costs is preferable both in terms of efficiency
and cost of incentives. Furthermore, the difference in terms of production and cost of incentives may provide a quantitative measure
of the differences between organizational cultures.
Finally, Kidwell and Bennett (1993) suggest that the individual
predisposition to withhold effort may be the result of the combination of the environment, the organization and the group; in this paper we assume that the diverse propensity to withhold effort is
formalized by the two cost functions we consider. When considering this propensity as opposite to exerting the discretionary effort,
we show that when the discretionary effort is not exerted, the
introduction of individual incentives does not allow for the efficient allocation of effort. This way we can give a formal example
of the ‘‘damage done” by dysfunctional leaders according to Sutton
(2007).
In further research, we will analyze some dynamical aspects
of the interaction. For example it is evident that, while having
A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885
piecewise constant function increases the production and reduces
the amount of paid incentives, at the same time it may lead individuals with higher capacity to experience underpayment inequity.
It would be interesting to examine the effort allocation dynamics
when individuals try to reduce the perceived inequity, and also
to understand whether, in this case, individual incentives may become useful to limit the perception.
Furthermore, it would also be interesting to allow several subordinates to interact in a supervised group, extending these results
both analytically and experimentally.
References
Adams, J.S., 1965. Inequity in social exchange. In: Berkowitz, L. (Ed.), Advances in
Experimental Social Psychology. Academic Press, New York, NY, pp. 276–299.
Alchian, A.A., Demsetz, H., 1972. Production, information costs, and economic
organization. American Economic Review 62, 777–795.
Bailey, T., Berg, P., Sandy, C., 2001. The effect of high-performance work practices on
employee earnings in the steel apparel and medical electronics and imaging
industries. Industrial and Labor Relations Review 54 (2A), 525–543.
Bandura, A., 1982. Self-efficacy mechanism in human agency. American
Psychologist 37, 122–147.
Beckmann, M.J., 1988. Tinbergen Lectures on Organization Theory, second ed.
Springer, Berlin.
Bordoloi, S.K., Matsuo, H., 2001. Human resource planning in knowledge-intensive
operations: A model for learning with stochastic turnover, European Journal of
Operational Research.
Boudreau, J., Hopp, W., McClain, J.O., Thomas, L.J., 2003. On the interface between
operations and human resources management. Manufacturing & Service
Operations Management 5 (3), 179–202.
Bylinsky, G., 1990. The hottest high-tech company in Japan. Fortune 121 (1), 82–88.
Coch, L., French Jr., J.R.P., 1948. Overcoming resistance to change. Human Relations
1, 512–532.
Daft, R.L., Noe, R.A., 2001. Organizational Behavior. South-Western Publishing,
Mason, OH.
Dal Forno, A., Merlone, U., 2007. Incentives in supervised teams: An experimental
and computational approach. Journal of Social Complexity 3 (1), 37–52.
Dal Forno, A., Merlone, U., 2009. Individual incentives in supervised work groups.
from human subject experiments to agent based simulation. International
Journal Internet Enterprise Management 6 (1), 4–22.
Encinosa, W.E., Gaynor, M., Rebitzer, J.B., 2007. The sociology of groups and the
economics of incentives: Theory and evidences on compensation systems.
Journal of Economic Behavior & Organization 62, 187–214.
Ethiraj, S.K., Levinthal, D., 2009. Hoping for a to z while rewarding only a: Complex
organizations and multiple goals. Organization Science 20, 4–21.
Fast, N., Berg, N., 1975. The Lincoln Electric Company. Harvard Business School Case,
#376-028.
Gendron, B., 2005. Scheduling employees in Quebec’s liquor stores with integer
programming. Interfaces 35, 402–410.
Glance, N.S., Huberman, B.A., 1994. Social dilemmas and fluid organizations. In:
Carley, K.M., Prietula, M.J. (Eds.), Computational Organization Theory, Lawrence
Erlbaum Associates, Hillsdale, NJ, pp. 217–239.
885
Herman, J.B., 1973. Are situational contingencies limiting job attitude–job
performance relationships? Organizational Behavior and Human Performance
10, 208–224.
Herrmann-Pillath, C., 2005. Culture, economic style and the nature of the Chinese
economic system. Contribution to the China Roundtable of The International
Economic Association.
Holmström, B., 1982. Moral hazard in teams. Bell Journal Economics 13 (2), 324–
340.
Holmström, B., Milgrom, P., 1987. Aggregation and linearity in prevision of intertemporal incentives. Econometrica 7 (2), 302–328.
Holmström, B., Milgrom, P., 1991. Multitask principal-agent analyses: Incentive
contracts, asset ownership, and job design. Journal of Law Economics &
Organization 7, 24–52.
Jex, S.M., 2002. Organizational Psychology. Wiley, New York, NY.
Kerr, S., 1975. On the folly of rewarding a while hoping for b. Academy of
Management Journal 18 (4), 769–783.
Kidwell, R.E., Bennett, N., 1993. Employee propensity to withhold effort: A
conceptual model to intersect three avenues of research. Academy of
Management Review 18 (3), 429–456.
Lum, T., 2003. Workplace codes of conduct in China and related labor conditions.
Key Workplace Documents, Federal Pubblications, Cornell University, ILR
School.
Macho-Stadler, I., Pérez-Castrillo, D., 1997. An Introduction to the Economics of
Information. Oxford University Press, Oxford.
Maslow, A.H., 1970. Motivation and Personality, second ed. Harper & Row, New
York, NY.
McGregor, D.M., 1960. The Human Size of Enterprise. McGraw-Hill, New York, NY.
Merlone, U., submitted for publication. Organizations as allocators of human and
material resources.
Mowday, R.T., Steers, R.M., Porter, L., 1979. The measurement of organizational
commitment. Journal of Vocational Behavior 14.
Roy, D., 1952. Quota restriction and goldbricking in a machine shop. The American
Journal of Sociology 57 (5), 427–442.
Schattler, H., Sung, J., 1997. On optimal sharing rules in discrete- and continuoustime principal-agent problems with exponential utility. Journal of Economic
Dynamics & Control 21, 551–574.
Schelling, T.C., 1960. The Strategy of Conflict. Harvard University Press, Cambridge,
MA.
Smith, F.J., 1977. Work attitudes as predictors of attendance on a specific day.
Journal of Applied Psychology 62 (1), 16–19.
Spector, P.E., 2003. Industrial and Organizational Psychology, third ed. Wiley, New
York, NY.
Steers, R.M., Black, J.S., 1994. Organizational Behavior, fifth ed. Harper Collins, New
York, NY.
Sutton, R., 2007. The No Asshole Rule. Sphere, London.
Varian, H.R., 1992. Microeconomic Analysis. Norton & Company, New York, NY.
Wageman, R., Baker, G., 1997. Incentives and cooperation: The joint effects of task
and