European Journal of Operational Research 207 (2010) 878–885

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European Journal of Operational Research

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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 a b s t r a c t

Article history:Received 23 April 2008Accepted 17 May 2010Available online 25 May 2010

Keywords:Organization theoryProductionOrganizational behaviorIncentivesIndividual motivation

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.05.023

* Corresponding author. Tel.: +39 011 6705753; faxE-mail address: merlone@econ.unito.it (U. Merlon

This paper introduces and analyzes a model of supervised work group where subordinates decide how toexert their effort in complementary tasks while the supervisors decide incentives. Incentives may be acombination of individual and group-based ones. The optimality of incentives is analyzed when consid-ering two different cost functions for subordinates. The two cost functions describe different individualmotivations; comparing the resulting effort allocations and production optimality, we can relate themto different organizational theories. Our results provide a measure of how motivation among subordi-nates may affect production and incentives. Furthermore, the optimal incentives schemes are examinedin terms of Adams’ equity theory.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

According to Boudreau et al. (2003), the fields of operationsmanagement and human resources management are intimately re-lated, yet they maintain distinct perspectives.

Among the examples which try to merge these perspectives,Bordoloi and Matsuo (2001) applied control theory to deal withworkforce planning taking into account also worker learning andcontrols for the risk. Another example is Gendron (2005), wherea store scheduling problem with constraints deriving from unionrepresentatives’ and human resources personnel’s was approachedand solved.

Recently, Boudreau et al. (2003) examine how human consider-ations affect classical operation management. In their conclusion,they highlight the research challenges and opportunities of bring-ing the human resources and operation management together.

In this paper, we explore this line of research and try to inte-grate human considerations in optimal incentive problems. TheMoral Hazard literature approaches multi-agent relationships indifferent ways. For example, the joint production models provideinteresting insights in terms of income distribution amongthe agents, see for instance Alchian and Demsetz (1972) andHolmström (1982). Another relevant aspect is the comparison be-tween centralized and decentralized structures as far as contract-ing goes. For example, the literature provides conditions underwhich the delegation of the supervisory task, i.e. decentralizing,

ll rights reserved.

: +39 011 6705783.e).

is beneficial; to obtain a first analysis of the advantages and disad-vantages of delegation the reader may refer to Macho-Stadler andPérez-Castrillo (1997).

One aspect usually neglected in the economic literature is therole of individual motivation; while, in psychology, motivation isa concept that has been discussed extensively. According to Spec-tor (2003) work motivation theories are most typically concernedwith the reasons, rather than ability, that some people performtheir jobs better than others. Steers and Black list the stages theevolution of management thought concerning employee motiva-tion has passed through. They are the traditional model, the humanrelations approach and, more recently, the human resources mod-el. In particular, ‘‘this newer approach also assumes that differentemployees want different rewards from their jobs, that manyemployees sincerely want to contribute, and that employees byand large have the capacity to exercise a great deal of self-directionand self-control at work” (Steers and Black, 1994, p. 139).

To this extent the case of Kyocera Corporation is striking. As itconcerns the reward system, Kyocera’s founder Kazuo Inamori, ina booklet describing his philosophy, writes ‘‘We don’t think interms of individual rewards. We don’t buy individuals’ loyalty withmonetary incentives or titles. Rather, we believe that individualswho are endowed with superior capabilities should contributetheir capabilities for the good of the entire group.” (for a discussionof Kyocera’s organizational culture the reader may refer to Bylinsky(1990)). This example is contrasted by pay incentives used at Lin-coln Electric, where most employees are paid on a piece-rate sys-tem (the Lincoln Electric Company has been described in severalcase studies by business schools, see for instance Fast and Berg(1975)). Several comparisons between incentive programs have

1 We recall that Rþþ is the set of positive real numbers; the case C = 0 is trivial.2 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).

A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885 879

been presented in the literature. Among others, Weiss (1987) pro-vides an empirical comparison between individual wage incentivesand group incentives examining the effects in terms of motivationand quit rates.

Following the joint production approach, we consider a modu-lar model of hierarchical organization. Specifically, we concentratemainly on pyramidal structures. This particular structure is wide-spread 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 ofthe different approaches to pyramidal structures see Merlone,submitted for publication). In our model, the organization consistsof two heterogeneous agents interacting in a supervised workgroup with a Cobb–Douglas production function. In the literature,the distinction between team and work group is hazy, neverthelesswe will follow Spector (2003). As a consequence, we will refer to asupervised work group as, in our case, we consider interchangeablesubordinates.

In this paper, we assume that individuals with different moti-vation may have different cost functions and analyze the incentiveproblem the supervisor faces, in order to maximize her own profit.In particular, we analyze how the optimal solution for the super-visor varies when considering two different cost functions for sub-ordinates. Cost function plays an important role for the agent.While in Dal Forno and Merlone (2007) only piecewise constantcost function was considered, in this paper we also considerstrictly convex costs. Usually economics assumes that costs areincreasing and marginally increasing; the other kind of cost func-tion 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 adiscussion about antecedents of discretionary effort and its conse-quences on performance the reader may refer to Bailey et al.(2001) and Sutton (2007)). In this sense, the two different costfunctions can be interpreted in terms of motivation; while thepiecewise constant cost function may be appropriate when subor-dinates have high self-efficacy and are highly motivated, the othercost function seems to be more appropriate for individuals whoare mainly interested in monetary incentives. It is well known(Zhou, 2002) that, with regard to the principal-agent theory, ingeneral it is difficult to derive analytical solutions; therefore, evenin scholarly contributions, strong assumptions are usually re-quired. This tradeoff between analytical tractability and extensivesimplification is acknowledged by other authors approachingorganization complexity (see Ethiraj and Levinthal, 2009). Never-theless, the two different cost functions we analyze can be relatedto different cultures in the organizational structure we consider.As a consequence, by comparing the production outputs underthe two different cost function assumptions, our analysis allowsus to compare productivity under different organizational culturesand to measure what the cost of the culture in terms of produc-tion is. The two different cost functions, and the related organiza-tional cultures, are particularly interesting when consideringchanges in competition induced by emergent countries such asChina (for a discussion on the role of culture in the economic styleof China, the reader may refer to Herrmann-Pillath (2005) and toLum (2003), for an analysis of labor conditions in the samecountry).

The structure of the paper is the following. In Section 2, wepresent the theoretical model. Sections 3 and 4 summarize andanalyze the optimal incentive problem with the two different costfunctions in question, describing how these are related to the sub-ordinates’ motivation. In Section 5, optimal incentive schemes andoutputs are compared and the results are interpreted in terms oforganizational culture consequences on productivity. Finally, Sec-tion 6 is devoted to conclusions and further research.

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 asupervisor (acting as principal) and two subordinates (acting asagents) cooperate. Agent i allocates his effort li with the partner,and the effort ui with the supervisor. The joint production functionfor agents 1 and 2 is C(u1 + u2)a(l1 + l2)b, where C 2 Rþþ is a con-stant factor,1 and a, b 2 (0,1) are, respectively, the output elasticitywith respect to the joint effort with the supervisor and with thepartner. As a consequence, the agents have to decide both howmuch effort to exert, and how to partition it in the two comple-mentary tasks.2 Agents bear a cost for effort: agent i’s cost functionci : R2

þ ! Rþ will be denoted with ci(ui, li); cost functions are privateinformation. Furthermore, each agent can observe the level effort hispartner provides with him, but not the one which is provided withthe supervisor. Conversely, the supervisor can only observe the jointoutput and the effort each agent provides with her. The supervisor’sprofit is a share c 2 (0,1) of the supervised work group productionminus the incentives she pays to her subordinates. In the following,we assume that the output is sold on market at unitary price and theproduction and sharing constants C and c are such that Cc = 1; thisis not restrictive, it simplifies the notation, and allows us to simplyconsider monetary payoffs. Finally, agents’ retribution consists of afixed wage w > 0 plus a performance-contingent reward; we assumethat the fixed wage is sufficient to meet basic needs, in terms of thehierarchy of needs theory (Maslow, 1970), physiological needs andneeds of safety; in economic terms we say that the participation con-straint is met.

Proposition 1. The gross production (u1 + u2)a(l1 + l2)b is maximizedif and only if the aggregate efforts are allocated proportionally to theoutput elasticities.

Proof. The result follows from combining and rearranging the firstorder conditions of the problem

maxu1 ;u2 ;l1 ;l2

u1 þ u2ð Þaðl1 þ l2Þb: ð1Þ

In fact, from

aðu1 þ u2Þa�1ðl1 þ l2Þb ¼ 0;

bðu1 þ u2Þaðl1 þ l2Þb�1 ¼ 0;

(ð2Þ

it followsu1 þ u2

l1 þ l2¼ a

b: � ð3Þ

Condition (3) is necessary in order to maximize the production ofthe supervised group. When either as the result of misalignedincentives or as lack of coordination between subordinates this con-dition is not met, then the effort allocation is not efficient. The per-formance-contingent reward is a linear incentive bt on the jointoutput of the team and a linear incentive bi on the effort each agentexerts with the supervisor. Therefore, the problem can be formal-ized as a bilevel programming problem:

maxbt ;b1 ;b2

ð1� 2btÞðu1 þ u2Þaðl1 þ l2Þb � b1u1 � b2u2; ð4Þ

such that, given the incentives bt, b1, b2, subordinates solve

maxu1 ;l1

wþ btðu1 þ u2Þaðl1 þ l2Þb þ b1u1 � c1ðu1; l1Þ; ð5Þ

maxu2 ;l2

wþ btðu1 þ u2Þaðl1 þ l2Þb þ b2u2 � c2ðu2; l2Þ;

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 restric-tive. Examining the form of the problem, it is rather immediateto predict the behavior of rational supervisor and subordinates inthis interaction situation. It is a finite dynamic game with perfectinformation, with supervisor moving first, and then subordinates,acting simultaneously after observing the incentive. This gamehas a proper subgame starting from the information set of the sub-ordinates. Therefore, there exists a set of subgame perfect Nashequilibria which equals the set of Nash equilibria that can be de-rived by backward induction. In fact, given any feasible incentivesscheme (bt,b1,b2), the subordinates will play a Nash equilibrium3

(ui, li) of the subgame. Knowing the fact that subordinates will playa Nash equilibrium in the subgame, the supervisor will maximizeher profit by choosing the optimal incentive scheme. In the followingsections, we will analyze how different assumptions on the subordi-nates’ cost functions may determine the optimal incentive scheme.

3. Piecewise constant costs

In the literature, some factor or factors that keep agents fromworking infinitely hard are usually considered. For example, Wag-eman and Baker (1997) propose several mechanisms consideringeffort becoming increasingly either unproductive or unpleasant.In this paper, as common, we assume that the unpleasantness ofthe work increases with respect to the effort. Different cost func-tions may be considered and the functional form may reflect differ-ent underlying assumptions. While economics usually considersmainly rational agents, other approaches to work group dynamicstake into account other aspects, such as norms and team commit-ment. Industrial and organizational psychology have proposed dif-ferent theories of motivation to explain individual behavior in theorganization (for a first survey the reader may refer to Spector(2003)). Nevertheless, several other aspects underlie group andteam behavior; the role of norms is well documented empirically(see Coch and French, 1948 and Roy, 1952, for example) and it iscommonly assumed, for work groups, to dictate how much eachperson will produce (see Spector, 2003). A first cost function weconsider is, as in Dal Forno and Merlone (2007), the following:

ciðui; liÞ ¼0 if ui þ li 6 �ci;

þ1 if ui þ li > �ci:

�ð6Þ

First observe that this cost function is non-decreasing with respectto the aggregate effort. It assumes that each subordinate has a phys-ical capacity �ci under which effort has zero cost, or, alternatively,that at some exertion level the effort becomes unpleasant enoughto lead the individual to conclude that it is not worth workingany harder independently of the reward. In this case, we assumethat each individual knows his individual capacity and uses it with-

3 In the next sections, we will examine also the equilibrium selection problem.

out goldbricking. This kind of cost function is not completely new tothe principal-agent literature; for instance, Holmström and Mil-grom (1991) assume that incentives are necessary to encourageagents to work beyond a limit they would take pleasure in working.In our case we assume that this limit coincides with their physicalcapacity. Finally, observe that this assumption can also be inter-preted in terms of the self-efficacy theory (Bandura, 1982), assum-ing that both subordinates have high self-efficacy and are motivatedto put in as much effort as they can.

3.1. The agents’ problem

Starting from the subordinates’ information set and assumingcoordination and commitment between agents, problem (5) re-duces to:

maxu1 ;u2 ;l1 ;l2

ðu1 þ u2Þaðl1 þ l2Þb sub ui þ li 6 �ci ð7Þ

for all i = 1,2.

Corollary 2. Consider the production function (u1 + u2)a(l1 + l2)b, withpiecewise constant cost functions (6). Then, any effort allocation suchthat

u1 þ u2 ¼ aaþb

�c1 þ �c2ð Þ;l1 þ l2 ¼ b

aþb�c1 þ �c2ð Þ;

(ð8Þ

maximizes the production.

Proof. The result follows trivially from first order conditions ofproblem (7), assuming the constraint on individual aggregate effortgiven by capacities. h

Therefore, there is a continuum of solutions to the consideredproblem (7), nevertheless, a rather natural effort allocation is theone that can be interpreted as focal in the sense of Schelling (1960):

ðui; liÞ ¼a

aþ b�ci;

baþ b

�ci

� �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 pro-duction maximization.

3.2. The supervisor’s problem

The supervisor’s problem is to design a linear compensationscheme for the subordinates that induces them to use their capac-ity to maximize the team output. The supervisor can observe theefforts 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 hisindividual effort with supervisor, and bt is the incentive given tothem for the team output. We assume that the supervisor declaresthe incentives and then the subordinates decide their efforts inorder to maximize their wage. The supervisor has to solve thefollowing problem:

maxbt ;b1 ;b2

ð1� 2btÞðu1 þ u2Þaðl1 þ l2Þb � b1u1 � b2u2: ð10Þ

When considering fully rational agents the solution is obvious.Since any individual incentive given to agents gives a suboptimaleffort allocation, and null team output incentive makes for subordi-nates any allocation optimal, the optimal solution is

bt ¼ e > 0;b1 ¼ 0;b2 ¼ 0:

8><>: ð11Þ

A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885 881

Some comments about this incentive scheme are in order. While itseems to be appropriate for agents with the same capacities or alow number of repetitions, for agents with different capacitiesinteracting over a long period of time some problems may arise.For example, when considering equity theory (Adams, 1965), it israther evident that different capacity individuals will find them-selves in inequitable situations, and will experience dissatisfactionand emotional tension, that they will be motivated to reduce. In or-der to improve the distributive justice (see Spector, 2003, for a dis-cussion about equity theory and distributive justice) a differentincentive scheme may be more appropriate in the long run.

4. Strictly convex costs

Following Holmström and Milgrom (1991), we suppose the ef-fort in the two tasks is perfectly substitutable in the agent’s costfunction. Formally we assume that

ciðui; liÞ ¼ ciðui þ liÞ: ð12Þ

As it is common, ci is assumed differentiable, increasing and mar-ginally increasing. As mentioned above, while this cost function iscloser to the classical modeling of behavior of the rational individ-ual, we may assume this kind of cost function when agents’ motiva-tion consists only in economic reward and neither social norms norother motivational theories determine individual effort. It is not dif-ficult to assume that this kind of cost function may be realistic inorganizations where individual objective consists only in the max-imization of the individual payoff. Another interpretation of thiscost function is as the result of high-powered incentives. Accordingto Steers and Black (1994), individual incentives may, at times, leadto employees competing with one another; in our case we assumethat the internal competition, driven by high-powered incentives,may result as this individual perception of cost of effort (for a dis-cussion on high-powered incentives, see Encinosa et al., 2007). Asa consequence, we can think that this kind of cost function is the re-sult of a different culture when compared to the piecewise constantcase. In this case also, we first analyze the agents’ problem and thenthe supervisor’s one. Finally, we observe that some of the results wepresent in the following can be generalized to more general costfunctions.

4.1. The agents’ problem

In this case, it would be rather unrealistic to assume commit-ment and coordination between subordinates, rather we will as-sume that agents solve problem (5) with cost function (12)

maxu1 ;l1

btðu1 þ u2Þaðl1 þ l2Þb þ b1u1 � c1ðu1 þ l1Þ;

maxu2 ;l2

btðu1 þ u2Þaðl1 þ l2Þb þ b2u2 � c2ðu2 þ l2Þ:ð13Þ

As a consequence, given the incentives (bt,b1,b2), subordinates findan effort allocation which is a Nash equilibrium, i.e., u�1; l

�1;u

�2; l�2

� �must be such that

u�1; l�1

� �¼ arg max bt u1 þ u�2

� �a l1 þ l�2� �b þ b1u1 � c1ðu1 þ l1Þ ð14Þ

and

u�2; l�2

� �¼ arg max bt u�1 þ u2

� �a l�1 þ l2� �b þ b2u2 � c2ðu2 þ l2Þ: ð15Þ

While for the piecewise constant cost function it was immediate toderive the effort allocation, in this case some preliminary consider-ations are in order.

Lemma 3. Given an incentive scheme (bt,b1,b2), if subordinates’ costfunctions have form (12), then an optimal effort allocationu�1; l

�1;u

�2; l�2

� �must be such that

c01 u�1 þ l�1� �

¼ c02 u�2 þ l�2� �

: ð16Þ

Furthermore, optimal incentives must be such that individual incen-tives are identical, that is

b1 ¼ b2: ð17Þ

Proof. Since we assume that the effort allocation constitutes aNash equilibrium, first order conditions can be derived from (14)and (15); they are:

abt u�1 þ u�2� �a�1 l�1 þ l�2

� �b þ b1 � c01 u�1 þ l�1� �

¼ 0;

bbt u�1 þ u�2� �a l�1 þ l�2

� �b�1 � c01 u�1 þ l�1� �

¼ 0;

abt u�1 þ u�2� �a�1 l�1 þ l�2

� �b þ b2 � c02 u�2 þ l�2� �

¼ 0;

bbt u�1 þ u�2� �a l�1 þ l�2

� �b�1 � c02 u�2 þ l�2� �

¼ 0:

8>>>>><>>>>>:

ð18Þ

Combining the second and the fourth equation it is immediate toobtain

c01 u�1 þ l�1� �

¼ c02 u�2 þ l�2� �

: ð19Þ

For the second part, combining the first and the third equation in(18) it is immediate to obtain

b1 � c01 u�1 þ l�1� �

¼ b2 � c02 u�2 þ l�2� �

: ð20Þ

Finally, by Eq. (19) we obtain

b1 ¼ b2: � ð21Þ

Following this result, incentives schemes will be written using theshortened notation (bt,b1,b2) = (bt,b).

As mentioned in Section 1, it is well known that principal-agents models often require strong assumptions to derive analyti-cal results. This model is not an exception; in the following, as it iscommon (see for instance, Holmström and Milgrom, 1987; Schat-tler and Sung, 1997; Encinosa et al., 2007), we assume a quadraticcost function

ciðui; liÞ ¼ diðui þ liÞ2; ð22Þwhere di, i = 1,2 are positive constants.

We also assume a + b = 1.In this case, a result concerning proportionality of allocated ef-

forts can be proved, as well.

Theorem 4. Given an incentive scheme (bt,b), with b – 0, there existsa unique equilibrium effort allocation u�1; l

�1;u

�2; l�2

� �such that

u�1 ¼ q u�1 þ l�1� �

¼ qk�1; l�1 ¼ ð1� qÞ u�1 þ l�1� �

¼ ð1� qÞk�1;u�2 ¼ q u�2 þ l�2

� �¼ qk�2; l�2 ¼ ð1� qÞ u�2 þ l�2

� �¼ ð1� qÞk�2;

(

ð23Þwhere k�1 :¼ u�1 þ l�1; k

�2 :¼ u�2 þ l�2 and 0 < q < 1.

Proof. First we will show that such an effort allocation is an equi-librium allocation. Since the objective functions are concave, firstorder conditions are also sufficient and the optimal solution mustsatisfy (18):

abtqa�1 k�1 þ k�2� �a�1ð1� qÞ1�a k�1 þ k�2

� �1�a þ b� 2d1k�1 ¼ 0;

ð1� aÞbtqa k�1 þ k�2� �að1� qÞ�a k�1 þ k�2

� ��a � 2d1k�1 ¼ 0;

abtqa�1 k�1 þ k�2� �a�1ð1� qÞ1�a k�1 þ k�2

� �1�a þ b� 2d2k�2 ¼ 0;

ð1� aÞbtqa k�1 þ k�2� �að1� qÞ�a k�1 þ k�2

� ��a � 2d2k�2 ¼ 0:

8>>>>><>>>>>:

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 toobtain d1k�1 ¼ d2k�2, that is

k�2 ¼d1

d2k�1; ð25Þ

so we can discard the third equation as redundant.By simple algebra we obtain

abtqa�1ð1� qÞ1�a þ b� 2d1k�1 ¼ 0;ð1� aÞbtqað1� qÞ�a � 2d1k�1 ¼ 0;

k�2 ¼d1d2

k�1:

8>><>>: ð26Þ

Putting together the first two equations and rearranging

abtð1�qÞ1�a

q1�a � ð1� aÞbtqa

ð1�qÞa þ b ¼ 0;

k�1 ¼ð1�aÞbtqað1�qÞ�a

2d1;

k�2 ¼d1d2

k�1:

8>>><>>>:

ð27Þ

Now consider function f ðqÞ ¼ abtð1�qÞaq1�a � ð1� aÞbt

qa

ð1�qÞa þ b; it is

easy to prove that there exists a unique q* 2 ]0,1[ such that

abtð1� q�Þ1�a

q�1�a � ð1� aÞbtq�a

ð1� q�Þaþ b ¼ 0; ð28Þ

since function f is continuous in ]0,1[. Furthermore, it holds

limq!0

abtð1�qÞ1�a

q1�a � ð1� aÞbtqa

ð1�qÞ1�b þ b ¼ þ1;

limq!1

abtð1�qÞ1�a

q1�a � ð1� aÞbtqa

ð1�qÞa þ b ¼ �1;

8><>: ð29Þ

and

f 0ðqÞ ¼ abt�bð1�qÞb�1q1�a�ð1�aÞq�að1�qÞb

ðq1�aÞ2�

�bbtað1�qÞ1�bqa�1þð1�bÞqað1�qÞ�b

½ð1�qÞ1�b �2< 0:

ð30Þ

Thus, since, as b – 0, q* is unique, there exists the unique effortallocation

u�1 ¼ð1�aÞbtq�

aþ1

2d1ð1�q�Þa ; l�1 ¼ð1�aÞbtq�

a ð1�q�Þ1�a

2d1;

u�2 ¼ð1�aÞbtq�

aþ1

2d2ð1�q�Þa ; l�2 ¼ð1�aÞbtq�

a ð1�q�Þ1�a

2d2;

k�1 ¼ð1�aÞbtq�

a

2d1ð1�q�Þa ; k�2 ¼ð1�aÞbtq�

a

2d2ð1�q�Þa :

8>>>><>>>>:

� ð31Þ

Theorem 4 does not claim that, with positive individual incen-tives, all effort allocation equilibria are characterized by effortsbeing proportionally allocated between the two tasks. Rather, itclaims that among the proportional effort allocations, i.e. those sat-isfying condition (3), one and only one is an equilibrium.

In the following corollary, we analyze the effort allocation withnull individual incentives.

Corollary 5. Given an incentive scheme (bt,0), if subordinates’ costfunctions have form (22), then any allocation such that ak�1 ¼ð1� aÞk�2 is an equilibrium allocation, where k�1 :¼ aabt

2ð1�aÞa�1d1;

k�2 :¼ aabt

2ð1�aÞa�1d2. Furthermore, there exists a focal effort allocation

u�1 ¼ aaþ1bt

2ð1�aÞa�1d1; l�1 ¼ aabt

2ð1�aÞa�2d1;

u�2 ¼ aaþ1bt

2ð1�aÞa�1d2; l�2 ¼ aabt

2ð1�aÞa�2d2:

8<: ð32Þ

Proof. Under the assumptions b = 0 and a + b = 1, conditions (18)become

abtðu1 þ u2Þa�1ðl1 þ l2Þ1�a � 2d1ðu1 þ l1Þ ¼ 0;bbtðu1 þ u2Þaðl1 þ l2Þ�a � 2d1ðu1 þ l1Þ ¼ 0

abtðu1 þ u2Þa�1ðl1 þ l2Þ1�a � 2d2ðu2 þ l2Þ ¼ 0;bbtðu1 þ u2Þaðl1 þ l2Þ�a � 2d2ðu2 þ l2Þ ¼ 0:

8>>>><>>>>:

ð33Þ

Combining the first and the second equation of (33) it is immediateto obtain

u1 þ u2 ¼a

1� aðl1 þ l2Þ; ð34Þ

which, substituted in the first equation of (33), yields

k�1 ¼ u1 þ l1 ¼aabt

2d1ð1� aÞa�1 ; ð35Þ

analogously, we find

k�2 ¼ u2 þ l2 ¼aabt

2d2ð1� aÞa�1 : ð36Þ

The focal allocation can be found, as before, as

u�1 ¼ ak�1; l�1 ¼ ð1� aÞk�1;u�2 ¼ ak�2; l�2 ¼ ð1� aÞk�2:

(�

ð37Þ

Remarkably, effort allocation (37) is the analogous of the focalallocation (9) we found for piecewise constant costs in Corollary2. In this case also, given the aggregate efforts the subordinatesput forth, the production is maximized. Furthermore, we observethat q defines what fraction of the agents’ efforts is made withthe supervisor. By the implicit function theorem we can prove that,as expected, this fraction increases as b increases. In fact, a greaterindividual incentive reallocates a greater effort with the supervisor.

Corollary 6. The fraction q of effort made with the supervisor isincreasing with respect to the individual incentive b and its derivativeis

q0ðbÞ ¼ q2�að1� qÞ1þa

að1� aÞbt: ð38Þ

Proof. Eq. (28) can be written as

f ðq; bÞ ¼ 0; ð39Þ

where

f ðq; bÞ :¼ abtð1� qÞ1�a

q1�a � ð1� aÞbtqa

ð1� qÞaþ b; ð40Þ

implicitly defines q as a function of b. By the implicit functiontheorem

q0ðbÞ ¼ � of=obof=oq

¼ � 1�að1�aÞbt

q2�að1�qÞ1þa

¼ q2�að1� 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 po-sitive, the effort allocation no longer satisfies condition (3) and hasseveral implications when considering how the supervisor mustdetermine incentives optimally.

4.2. The supervisor’s problem

As for the piecewise constant function, the supervisor’s problemis to determine the incentive scheme maximizing her own payoffas in (10).

Before investigating whether individual incentive b increasessupervisor’s profit, we characterize the optimal incentives whenb = 0. To this extent, we consider the aggregate profit

l of Operational Research 207 (2010) 878–885 883

u�1 þ u�2� �a l�1 þ l�2

� �1�a � d1 u�1 þ l�1� �2 � d2 u�2 þ l�2

� �2: ð42Þ

A. Dal Forno, U. Merlone / European Journa

Theorem 7. With incentive schemes (bt,b) where b = 0, the aggregateprofit (42) is maximized if and only if it is completely shared betweenthe two agents; by contrast, in order to maximize supervisor’s profit,half of the aggregate profit must be given to the supervisor and theremaining equally shared between the agents.

Proof. To maximize the aggregate profit the following programmust be solved:

maxbt6

12

u�1 þ u�2� �a l�1 þ l�2

� �1�a � d1 u�1 þ l�1� �2 � d2 u�2 þ l�2

� �2: ð43Þ

Taking into account the focal effort allocation, as given by Eq. (32),the program (43) becomes

maxbt6

12

aaþ1bt

2ð1� aÞa�1

d1 þ d2

d1d2

" #aaabt

2ð1� aÞa�2

d1 þ d2

d1d2

" #1�a

� d1aabt

2ð1� aÞa�1d1

!2

� d2aabt

2ð1� aÞa�1d2

!2

; ð44Þ

that is

maxbt6

12

aabt

2ð1� aÞa�1

d1 þ d2

d1d2aað1� aÞ1�a � aabt

2ð1� aÞa�1

" #2d1 þ d2

d1d2:

ð45Þ

Since 0 < a < 1 and d1þd2d1d2

> 0, the original program (43) is equivalentto

maxbt6

12

bt �b2

t

2; ð46Þ

and the optimal solution is bt = 1/2, that is, all the profit is sharedbetween the two agents.

For the second part of the theorem, let us consider thesupervisor’s profit (10) when the incentives are null

maxbt6

12

ð1� 2btÞ u�1 þ u�2� �a l�1 þ l�2

� �1�a; ð47Þ

that is

maxbt6

12

ð1� 2btÞaabt

2ð1� aÞa�1

d1 þ d2

d1d2aað1� aÞ1�a

: ð48Þ

This problem, since 0 < a < 1 and d1þd2d1d2

> 0, is equivalent to

maxbt6

12

ð1� 2btÞbt ; ð49Þ

and the optimal solution is bt = 1/4.This proves the thesis. h

This result shows that the agents are the residual claimant (seefor instance, Varian, 1992) to the output produced if and only if thesupervisor distributes it completely to the agents. Hence, havingthe supervisor in charge to decide how to share the output doesnot maximize the work group production.

Finally, we prove that, under realistic conditions, non-null indi-vidual incentive is necessary for supervisor’s profit maximization.

Theorem 8. When bt < 1/2, incentive schemes which maximize thesupervisor’s profit are such that individual incentives are non-null, i.e.,b > 0.

4 For the sake of brevity, the proportionality factor is omitted.

Proof. Substituting the effort allocations (23) described in Theo-rem 4 in the objective function of the supervisor’s problem (10),we obtain

ð1� aÞbt

2d1 þ d2

d1d2ð1� 2btÞq2að1� qÞ1�2a � bqaþ1ð1� qÞ�ah i

:

ð50Þ

That is, the supervisor’s profit is proportional to

ð1� 2btÞq2að1� qÞ1�2a � bqaþ1ð1� qÞ�a: ð51Þ

Now recall that in Corollary 6 we proved that q is a function of b. Asa consequence, we can consider the supervisor’s profit4 as a func-tion of the individual incentive b

pðbÞ ¼ ð1� 2btÞq2aðbÞð1� qðbÞÞ1�2a � bqaþ1ðbÞð1� qðbÞÞ�a:

ð52Þ

Now derive p with respect to b using expression (38) we found inCorollary 6, substitute b = 0 and recall that q(0) = a,

p0ð0Þ ¼ a1� 2bt

btaað1� aÞ�a ¼ a

1� 2bt

bt

aaþ1

ð1� aÞa: ð53Þ

Since bt < 1/2, the supervisor’s profit increases as b increases. As aconsequence, incentive schemes (bt,b) where b = 0 cannot beoptimal. h

The consequences of this theorem are striking. In fact, this re-sult proves that, with quadratic costs (22), the individual incen-tives are necessary. Nevertheless, by Corollary 6, it follows thatindividual incentives increase the fraction of effort made withthe supervisor. Therefore, with positive individual incentives theeffort allocation does not satisfy condition (3), and gross produc-tion is not maximized. In the following section, we will relate theassumption underlying the different cost functions we have con-sidered to the managerial literature.

5. Comparing the two cost functions

In Section 3, we found that the optimal incentive scheme re-sulted in null individual incentives, and that any efficient alloca-tion (8), maximizes the production. Furthermore, in Section 4 wefound that, also with quadratic cost functions, when individualincentives are null the resultant effort allocation (32) is optimal.Putting together these results with Theorem 8 thesis, it results thatgiven the aggregate efforts exerted by subordinates, the productionis not maximized. In other words, assuming that the resultingcapacities k�1 and k�2 for subordinates with quadratic cost functionsare the actual capacity constraints for piecewise constant costfunctions, then the production would be higher since in this casethe effort allocation would be efficient. Note that in this case weconsider gross production because, otherwise, the different incen-tives for quadratic cost functions would make a comparison be-tween net productions trivial. This result can be interpreted as afurther example of ‘‘systems that pay off for one behavior eventhough the rewarder hopes dearly for another” (Kerr, 1975); thereason here is that the members of the work group possess differ-ent goals and motivation. The contrast is even starker when we as-sume that the whole incentive mechanism was designed by topmanagement in order to introduce pay incentives for the supervi-sor and delegating her the authority to motivate subordinates.

When the two different cost functions are interpreted in termsof motivation, these results may provide some interesting insights.Within industrial and organizational psychology the study of em-ployee motivation represents one of the most important topics inthe discipline (Jex, 2002). A variety of theories of human motiva-tion have been developed over the years (for a review the readermay consult Spector, 2003); we may assume that the two cost

884 A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885

functions we examined are the result of different motivations thesubordinates in the work group share. While the quadratic costfunction seems to be appropriate for individuals who are inter-ested just in monetary incentives, the piecewise constant costfunction, as we mentioned in Section 3, may be appropriate whensubordinates have high self-efficacy and believe they are capable ofaccomplishing tasks and motivated to put forth effort (Bandura,1982).

In this sense, the fact that with the quadratic costs individualincentives are necessary, together with the fact that optimal pro-duction is not achieved, may highlight some limitation of thescientific management approach, which, among other aspects,emphasized the provision of economic reward for high perfor-mance (Daft and Noe, 2001).

When considering that, according to Weick (1969), any systemconsists of several causal relationships, some direct and others in-verse, we may assume that, at least in part, the organization mayhave some effects on the individual motivation of the employees.This aspect is related to Theory X/Theory Y (McGregor, 1960); infact, according to this theory, the organization management ap-proach determines how subordinates behave and is determinedby the belief of supervisors about their subordinates. Accordingto Theory X, the worker is viewed as being indifferent to the orga-nization’s needs, lazy and unmotivated; in this case one approachmight be incentivizing the effort, and in this case, the quadraticcost function would be appropriate. On the other hand, accordingto Theory Y, subordinates are capable and not inherently unmoti-vated or unresponsive to organizational needs, rather it is theresponsibility of the manager to create organizational conditionsso that subordinates can achieve organizational goals throughachieving their own goals; in this case, assuming that individualmotivation does not consist of economic rewards, the piecewiseconstant cost function can be appropriate.

Another interpretation of the differences between the two costfunctions, may be found taking into account organizational com-mitment. Among the different definitions of commitment, Mowdayet al. (1979) view organizational commitment as consisting of threecomponents: (1) an acceptance of the organizational goals; (2) awillingness to work hard for the organization; and (3) the desireto stay with the organization. We can therefore consider the piece-wise constant functions appropriate for describing individualshighly committed to the organization, since they tend to use theirwhole capacity. In fact, quadratic costs, may describe a lower com-mitment situation. On the other hand, the research on withdrawalbehaviors relates to employees not being at work when scheduledor needed (Spector, 2003); therefore, we can assume the piecewiseconstant cost functions realistic in organizations where these coun-terproductive behaviors do not occur, while by contrast, the qua-dratic cost functions may be more appropriately assumed inorganization where employee attendance is incentivized.

A last possible interpretation we present, relates to the work ofauthors like Herman (1973) and Smith (1977), and, more recently,Sutton (2007) who examined the relationship between work atti-tude and such work-related behavior as job performance; in partic-ular Sutton (2007) interprets impaired organizational performanceas the result of dysfunctional leaders. In this case, piecewise con-stant and quadratic cost functions, might describe different workattitudes, the second being the result of dysfunctional leadership;in fact we found that with the latter cost function the effort isnot allocated efficiently.

Finally, for both cost functions, the optimal incentives are inde-pendent of agent capacity. This result is quite interesting since, gi-ven the fact that individual incentives are identical, subordinateswith different capacities may perceive inequity in the sense ofAdams (1965). With piecewise constant cost functions, when indi-viduals compare ratio of outcomes to input, those with higher

capacity may experience underpayment while those with lowercapacity on the contrary, may experience overpayment. Vice versa,as it concerns the quadratic cost function this phenomenon may benot so evident as the individual incentives, at least partially com-pensate the supervisor exerting higher effort.

6. Conclusion and further research

In this paper, a simple interaction scheme has been proposedand analyzed. The subordinates are called to allocate their effortin two interdependent tasks. This interdependence is twofold:firstly, in each of the two tasks the performance depends uponthe efforts of both subordinates; secondly, the overall performancedepends on both tasks. Given the tasks and the incentive mecha-nism, the optimal incentives depend on the cost functions ofsubordinates.

When considering the analytical results we obtain from the hu-man resources management perspective several insights may beprovided.

In fact, interpreting the cost functions in terms of subordinatemotivation, it emerges that when individuals are not sensitive tomonetary incentives, and rather share a norm according to whichthey use their capacities, then there exists an incentive schemesuch that the effort is efficiently allocated. By converse, when indi-viduals are interested only in gain-sharing, no such a schemeexists.

In Wageman and Baker (1997) it was shown that the interactionbetween task and reward interdependence made it difficult to pro-vide effective guidance in solving organizational design problems;in this paper we fixed the task interdependence and analyzed theinteraction between reward interdependence and individual moti-vation. In our case simple pay practices, such as delegating themanager to decide incentives, may be ineffective and, under someconditions, even counterproductive. In fact, the outcome is thathaving this sort of self-managed work group with a gain-sharingincentive scheme has several drawbacks; in our example the effi-ciency of the effort allocation is contingent to the motivation ofthe subordinates. In terms of incentive design, we found that, inthe case of quadratic costs, the supervisor incentive should be keptseparated from the work group profit. In fact, when the supervisoris in charge of deciding how to share the output, the work groupaggregate profit is not maximized. Furthermore, also incentivesschemes that may be optimal in the one-shot iteration may exhibitdrawbacks when the interaction is repeated over time.

We provided several interpretations of the two different costfunctions. Independently of the one we choose, what is relevanthere is both the effort allocation efficiency and the cost of incen-tives which are peculiar to each of them: clearly, a situation withpiecewise constant costs is preferable both in terms of efficiencyand cost of incentives. Furthermore, the difference in terms of pro-duction and cost of incentives may provide a quantitative measureof the differences between organizational cultures.

Finally, Kidwell and Bennett (1993) suggest that the individualpredisposition to withhold effort may be the result of the combina-tion of the environment, the organization and the group; in this pa-per we assume that the diverse propensity to withhold effort isformalized by the two cost functions we consider. When consider-ing this propensity as opposite to exerting the discretionary effort,we show that when the discretionary effort is not exerted, theintroduction of individual incentives does not allow for the effi-cient allocation of effort. This way we can give a formal exampleof the ‘‘damage done” by dysfunctional leaders according to Sutton(2007).

In further research, we will analyze some dynamical aspectsof the interaction. For example it is evident that, while having

A. Dal Forno, U. Merlone / European Journal of Operational Research 207 (2010) 878–885 885

piecewise constant function increases the production and reducesthe amount of paid incentives, at the same time it may lead indi-viduals with higher capacity to experience underpayment inequity.It would be interesting to examine the effort allocation dynamicswhen individuals try to reduce the perceived inequity, and alsoto understand whether, in this case, individual incentives may be-come useful to limit the perception.

Furthermore, it would also be interesting to allow several sub-ordinates to interact in a supervised group, extending these resultsboth analytically and experimentally.

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