Conference Paper Service Organizations: Customer Contact and Incentives of Knowledge Managers

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1 econstor Der Open-Access-Pubikationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Pubication Server of the ZBW Leibniz Information Centre for Economics Kirchmaier, Isadora Conference Paper Service Organizations: Customer Contact and Incentives of Knowedge Managers Beiträge zur Jahrestagung des Vereins für Sociapoitik 2014: Evidenzbasierte Wirtschaftspoitik - Session: Industria Organization III, No. C11-V2 Provided in Cooperation with: Verein für Sociapoitik / German Economic Association Suggested Citation: Kirchmaier, Isadora (2014) : Service Organizations: Customer Contact and Incentives of Knowedge Managers, Beiträge zur Jahrestagung des Vereins für Sociapoitik 2014: Evidenzbasierte Wirtschaftspoitik - Session: Industria Organization III, No. C11-V2 This Version is avaiabe at: Nutzungsbedingungen: Die ZBW räumt Ihnen as Nutzerin/Nutzer das unentgetiche, räumich unbeschränkte und zeitich auf die Dauer des Schutzrechts beschränkte einfache Recht ein, das ausgewähte Werk im Rahmen der unter nachzuesenden voständigen Nutzungsbedingungen zu verviefätigen, mit denen die Nutzerin/der Nutzer sich durch die erste Nutzung einverstanden erkärt. Terms of use: The ZBW grants you, the user, the non-excusive right to use the seected work free of charge, territoriay unrestricted and within the time imit of the term of the property rights according to the terms specified at By the first use of the seected work the user agrees and decares to compy with these terms of use. zbw Leibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics

2 Service Organizations: Customer Contact and Incentives of Knowedge Managers Isadora Kirchmaier We anayze the interdependence of human resource management and knowedge management. The service organization is modeed as a queueing network. The optima number of workers in each division, the amount of customer contact and the wage for each manager is determined. We combine three features within the mode. First, each manager may engage in customer contact. We show that athough the fraction of time a worker is busy is increasing in rank of the manager, the customer task acceptance rate is not necessariy monotonic. Second, knowedge management is expicity taken into account. Knowedge acquired by workers depends on the effort of the manager. Third, since this effort is not easiy measurabe, a mora hazard probem might occur. We discuss a bonus contract under different performance evauation schemes. If queueing costs increase we find it might be optima to increase the knowedge and to decrease the number of workers. This impies that decisions are more decentraized. In a numerica exampe we anayze the eimination of midde management. A fattened firm may respond more quicky by pushing decisions downwards. However, we find that the mean response time is higher and the senior manager is more invoved in interna tasks. Subject cassifications: organizationa design; muti-agent mora hazard; queueing network 1 Introduction Organizations provide a framework for empoyees to perform their tasks. The structure determines basic patterns such as who deas with incoming task from cients, who handes a task, which was aready deat with by another empoyee but coud not be soved, and who decides that Address: Bergheimer Str. 58, Heideberg, teephone: , e-mai: i.m.kirchmaier@uniheideberg.de. 1

3 a task cannot be soved at a. It infuences the incentives of each empoyee and thereby the fow of production and communication within an organization. The aim of human resource management (HRM) is to provide an environment so that the objectives of the organizations are accompished through empoyees. The ink between HRM and business outcome is we documented (Koys 2001, Pau and Anantharaman 2003, Purce and Kinnie 2007, Bowen and Ostroff 2004). The specific goa of HRM is achieving high performance through peope (Armstrong 2003). In order to achieve high performance through peope, empoyees abiities, motivations and opportunities have to aign, i.e. they need to have the necessary skis to perform the task, the empoyment contract has to provide adequate incentives and the work environment has to provide possibiities for expression aso if a probem occurs (Boxa and Purce 2003). Due to a shift from manufacturing-oriented to service organizations, customers are a source of production inputs. This introduces uncertainty in the task characteristics and thereby in the sovabiity of the task (Larsson and Bowen 1989). The empoyee in the division who deas with an incoming task may not have the knowedge to sove it. A popuar way to address this ack of knowedge is knowedge management. The manager takes on the job of providing a knowedge-creating and sharing environment for the workers (Gao et a. 2008). Nonaka and Takeuchi (1995) define these managers responsibe for the knowedge management as the knowedge-creating crew. They coach the empoyees thereby producing ong-asting earning, which guides the empoyees to enhance their performance (Redshaw 2000). However, managers differ in their wiingness to coach. It is a time consuming task and might interfere with achieving performance targets (Goeman 2000). HRM provides the wage contract for the managers and thereby infuences the incentive of the managers to coach their subordinates. Since the effort in coaching is not easiy measurabe, it might not be possibe to contingent the contract on that effort and a mora hazard probem occurs. The aim of this paper is to anayze the interdependence of the goas of HRM and the knowedge crew, i.e. the knowedge managers, who act as coaches for the empoyees in their division, taking the connectedness of the divisions through the supervisory structure into account. Especiay, the reward structure, knowedge management, customer task acceptance rate and span of contro of the knowedge managers are discussed. Furthermore, different information structures between HRM and the knowedge managers are considered. Under symmetric information, it is assumed that the amount of coaching is contractabe whie under asymmetric information HRM cannot observe the 2

4 effort of the knowedge manager. We ask the question if it is sti possibe for HRM to provide a bonus contract for the knowedge managers such that it is optima to impement the amount of coaching under symmetric information. Consider a service organization, which consists of different divisions and suppose that the organization generates revenue from soving tasks. The organization is modeed as a queueing network in which the divisions are the nodes and the workers are the servers of each division. The job of the workers is to sove tasks given by their knowedge manager. If the worker is busy, incoming tasks are queued. Tasks arrive due to customer contact of the knowedge manager and aso, depending on the underying supervisory structure, from knowedge managers of other divisions. For each division a knowedge manager is in charge of coordinating the incoming and outgoing tasks. If after processing, a division is not abe to sove a task, the knowedge manager may forward it to other divisions of the organization. Aso, the knowedge manager coaches the workers in his division, e.g. the workers in his span of contro, and thereby infuences the acquired knowedge of the worker. The organization s HRM is in charge of designing a wage contract for each knowedge manager. These contracts provide the incentive structure within the organization and infuence the effort in coaching of the knowedge managers. Additionay, HRM pursues its other goas, determining the optima amount of customer contact of the knowedge manager, i.e. the task acceptance rate, and the optima number of workers in each division. One exampe for such a type of organization is a consuting firm. A consuting firm typicay consists of junior and senior managers where each manager supervises a division. Junior and senior managers receive tasks from customers and aocate them to their workers in the division. Tasks are queued, if a workers are busy. If the division of the junior manager is not abe to sove a given tasks, the junior manager may ask the senior manager for hep. An important part of the job description of each manger is coaching of the workers in his division. In an numerica exampe we consider such a type of organization and discuss the effect of an impementation of midde management. There are severa studies which view organizations as a network of queues. The one cosest to our setup is Beggs (2001). He considers an organization, where depending on their rank, workers may differ in their abiity to sove tasks. A task is queued unti a worker is free to dea with it. If a worker is not abe to sove a task he can send it to a worker on the next division. The wage is 3

5 a function of the contractabe abiity of each worker. The objective is to minimize wage costs but aso to minimize network performance measures such as the average deay of a task. The trade-off between wage costs and deay determines the optima abiity eve and number of workers in each division. Under the assumption that a task can be immediatey deat with Garicano (2000) derives the optima organizationa structure under symmetric information. He shows that there is one division of workers, that speciaizes in production and the other divisions speciaize in supporting that division by attending to unsoved probems. The knowedge of the worker is increasing in the rank of the division. Cavo and Weisz (1978) and Qian (1994) assumes that the owest division is responsibe for production. A the other divisions are supervisors, who invest resources in monitoring immediate subordinates. The output of each division wi be used as an input for the next division. They show that if the divisions have the possibiity to shirk without their supervisor knowing, then it is optima to pay a wage increasing in rank, even if a workers have the same abiities. We provide three main extensions. First, the production process is extended such that each division may accept tasks from customers in addition to unsoved tasks forwarded from subordinates. It provides additiona insight, since we show that the optima task acceptance rate is not necessariy monotonic in rank of the knowedge managers. The task acceptance rate of a knowedge manager depends on the task arriva rate and on the number of unsoved tasks forwarded by direct subordinate knowedge managers. In genera, a knowedge manager of higher rank has a ower task acceptance rate. However, if a knowedge manager of ower rank receives comparaby more unsoved probems it can occur that his task acceptance rate is ower than for the knowedge manager of higher rank. Nevertheess, the traffic intensity, which is the mean number of tasks in service at a worker, is increasing in the rank of the knowedge manager. In the numerica exampe we discuss the case, in which it is optima that a midde manager has a ower task acceptance rate than the senior manager. The midde manager is mainy in charge of deaing with unsoved tasks of junior managers. Second, we take knowedge management expicity into account. The amount of knowedge acquired by workers depends on the effort of the knowedge crew. The knowedge managers are responsibe for the creation and circuation of knowedge in their division. Knowedge creation and coaching induces costs for the knowedge manager. We show that the optima variabe costs of 4

6 knowedge creation per worker incurred by the knowedge manager of each division is increasing in the rank of the manager. Third, since the effort in coaching is not easiy measurabe, we ask the question if it is sti possibe for HRM to provide a bonus contract for the knowedge managers such that it is optima to impement the amount of coaching under symmetric information. Since knowedge creation and coaching induces costs for the knowedge manager, HRM has to take these aspects into account when designing a contract for a knowedge manager. Organizationa design incorporates aso contro toos such as reward structures, task characteristics, and information systems. In a fied study of compensation practices, Eisenhardt (1985) shows that task characteristics and the avaiabe information system to measure outcomes is strongy reated to the choice of the reward structure. If the characteristic of a given task is not known in advance and the outcome is measurabe, then it is more ikey that the reward structure is based on the outcome and not on the behaviour of the manager. We anayze two different types of reward structures for the knowedge managers. Under independent performance evauation ony the own output infuences the payment of the bonus, whie under joint performance evauation the performance of other divisions are reevant as we. We identify conditions, which depend cruciay on the hazard rate of effort, under which a bonus contract exists. The hazard rate of effort gives the change in the cumuative distribution of output resuting from higher effort in reation to the probabiity to receive a bonus. We find that under independent performance evauation the bonus is unambiguousy positive, whie under joint performance evauation instead of a bonus payment a penaty coud occur. A reason is that under joint performance evauation an increase in knowedge of a worker has two effects. First, it increases the expected output of that division. Second, it decreases the output of the divisions, which supervise that division, since ess unsoved tasks are forwarded. If the second effect outweighs the first effect, it is ess ikey that a certain overa output is achieved. Consider the senior knowedge manager, i.e. the manager who is not supervised by any other knowedge managers. If a contract under joint performance evauation exists, it is optima that the senior manager receives a bonus payment and not a penaty, since the second effect does not occur. We find that for the same target output, the wage sensitivy to performance for the senior manager is higher under joint performance evauation than under individua performance evauation. The wage sensitivity to performance is the ratio of the bonus payment to the saary pus bonus payment and is a common measure in 5

7 accounting (Baiman et a. 1995). Within this framework we discuss the reaction of HRM to a change in the intensity of queueing costs. Beggs (2001) shows that if a deay becomes more costy, the abiity of the empoyees increases. We aso find that the knowedge of the workers and the number of workers in each division are substitutes. If deay becomes more costy, there are three effects on the number of workers in each division. First, since workers are abe to sove more probems, ess workers wi be empoyed. Second, if urgency increases more workers coud sove queued tasks more quicky. Third, the effect on the task acceptance rate aso infuences the number of workers. On the one hand, if queueing becomes more costy ess tasks from customers wi be accepted and the number of workers decreases. On the other hand, since the knowedge of workers increases, ess tasks wi be received from direct subordinates and so the task acceptance rate increases and the number of workers increases. We show that if the margina revenue with respect to the expected number of soved tasks is eastic, it is optima to increase the knowedge of the workers and to decrease the span of contro of the knowedge manager. This impies that decisions are more decentraized, since the probabiity to forward a task to the superior knowedge manager decreases. In a numerica exampe we anayze a specific change of the organizationa structure, namey the fattening of the organization. In genera it refers to the eimination of divisions in a firm. We consider three key payers in the creation of knowedge in the organization, the junior manager, the midde manager and the senior manager and discuss the transformation or eimination of midde management. Coombo and Grii (2013) show in an empirica study of Itaian high-tech entrepreneuria ventures that the information overoad probems are key-drivers for the creation of midde management. Consistent with this, we find that an overoad probem for senior managers occurs when midde management is transformed to junior management. Wuf (2012) finds that after fattening, which shoud push decisions downwards, there is more contro and decision making at the top of the organization. In our exampe, if midde management is eiminated or transformed into junior management, the task acceptance rate of the senior manager decines. His division has to sove comparaby more forwarded tasks from subordinate knowedge managers and is therefore more invoved in interna tasks. The eimination or transformation of the midde management resuts in a ess hierarchica structure. A rationa for fattening is that a streamined firm may respond more quicky to customers. However, we find that the mean response time, i.e. the time a task spends 6

8 in the organization, is higher in the fattened organizations. The reason is that in the organization with a midde management, the task acceptance rate of the senior manager is higher. Externa tasks handed directy by the senior manager division are not forwarded to any other manager and therefore are served faster. Given the optima organizationa structure, we characterize the optima wage contracts and find that for a managers the wage sensitivity to performance is higher under joint performance evauation than independent performance evauation. We consider two different types of joint performance evauations. First, the bonus payment depends aso on the output of the subordinate knowedge manager and second the bonus payment depends on the output of a knowedge managers. We find that whie for the first type of performance evauation a bonus contract is optima, under the second type a penaty contract is optima for the junior and midde managers. Wuf (2007) reports that for division managers of ower rank, the bonus payment is typicay inked to performance measures over which the manager has greater contro than the overa performance of the organization. This means for the numerica exampe that the first type of performance evauation is more reevant to junior and midde managers. 2 The mode The mode consists of: (1) an organizationa component, (2) an informationa component, and (3) an economic component. The structure of the organization incorporates besides the supervisory structure and the production process. The informationa component specifies the information asymmetry between HRM and manager. The economic component comprises the objectives of HRM and the incentives for the managers. 2.1 Organizationa structure The organization has L divisions, where L is taken as given. A division is an organizationa unit which consists of one knowedge manager M (the agent) and r workers. Let t be the arriva rate of tasks given by customers for division, which means that on average t tasks arrive per unit of time. The knowedge manager M, to which we wi refer to as the manager, coordinates the arriving tasks and forwards tasks which were processed but unsoved to managers of other divisions. The job of the workers in division is to sove tasks given by their manager M. The 7

9 manager is in charge for the training of the workers in his division. We wi therefore refer to r as the span of contro of M is r. If the manager invests an effort of p [0, 1] in the training of his workers, they are abe to sove a fraction p of the arriving tasks. HRM (the principa) is in charge of designing the payment scheme w of the manager M, decides how many workers are empoyed r in the division, and determines the task acceptance rate t of each manager M. Corporate governance structure. There are two different types of supervision. First, each manager is supervising a division of r workers. Second, a managers, except the head of the organization directy report to a manager of higher rank. Tasks which where processed but coud not be soved are forwarded by the manager to his immediate supervisor manager. We foow Cho (2010) who describes the second supervisory structure by a directed graph. The nodes are divisions and edges are inks between divisions. Let L = {1,..., L} be the set of the divisions. A ink between node and k is denoted by k and is an ordered pair (, k) L L. If the ink k exists, then M is a supervisor of M k, i.e. M k is a subordinate of M. A path from node k to is { 1 2,..., K 1 K } where 1 = k and K =. M contros M k if there is a path from node k to. Let C be the set of divisions that M contros and C d be the set of divisions that M contros directy. If k C and the ink k exists, then M contros directy M k. If such a ink does not exist, but k C then M contros indirecty M k. Division is the head division if M is not controed by any other division, i.e. C = L\ {}. Division is a ow eve division if M is not supervisor to any other division, i.e. C =. The structure of the organization is assumed to have the foowing properties 1. There is a unique head division. 2. Each division, except the head division has ony one supervisor. 3. If manager M supervises M k, then M k cannot supervise M directy or indirecty. The supervisory structure impies a rank of the division. We consider the counting up rank system (Beckmann 1988). The manager of a ow eve division has rank 0. The manager supervising ony the manager of the ow eve division has rank 2 and so on. For a more forma definition and exampes of the impied rank system see Appendix A. Production process. Time is continuous and at each instance consumers send tasks to the divisions. We assume that the manager M chooses the arriva rate of tasks t, given the requirement 8

10 of HRM. More formay, suppose that potentia tasks arrive at each division with a given rate T. M accepts the potentia task with probabiity λ and rejects with probabiity 1 λ. This decision is independent for successive customers and of the number of tasks in the system. By choosing λ, M makes sure that tasks arrive at each node according to independent Poisson processes with rate t = λ T to which we wi refer as the task acceptance rate of M. Since each division has a finite number of workers tasks might not be immediatey deat with. If a workers are busy the incoming tasks are queued. M distributes the queued tasks eveny to the queues of the workers in his span of contro. The queue at each worker can be of infinite ength and tasks are processed on a first-come-first-serve basis and the service time foows an exponentia distribution. For simpicity assume that on average, if the queue is busy, one task can be handed by each worker per unit of time, i.e. the service rate at each server is µ = 1. Independent of other tasks, each arriving task is associated with a difficuty eve P which foows a uniform distribution over [0, 1]. The workers in each division have to acquire a certain knowedge measured by p [0, 1], which is the probabiity that a task is soved. The output of division, i, is the number of soved tasks of division. Suppose the ink k exists. If workers in division k cannot sove a task, which happens with probabiity 1 p k, the manager M k forwards the task to its immediate supervisor M who forwards the task to an ide worker or queues it. The probabiity to forward a processed task to another division is independent of its past history and independent of a other tasks. The supervisory structure determines if and to whom unsoved tasks can be passed on. In queueing networks these reationships are captured by the baance equations which state that in the ong-run the rate of fow of tasks out of a division has to equa the rate of fow of tasks into the division. L a = t + a j p j j=1 = 1,..., L where a is the effective arriva rate at division to which we wi refer as the task arriva rate of M, and p j is the proportion of unsoved tasks passed on from division j to division with p = 0, i.e. unsoved tasks are ony passed on to other ayers. Since we consider hierarchica organizations, where division j is aowed to pass ony to one other division so p j = 1 p j if the ink j exits 9

11 and p j = 0 otherwise. The baance equations in matrix notation are a = t + P a (1) where P is the routing matrix and P ij gives the proportion of tasks forwarded from division j to division i. Since the communication structure is uni-directiona P wi be ower trianguar, with P ii = 0. Equation 1 can be soved for a a = P t (BE) where P := ( I P ) 1 and I is the identity matrix. Properties of P are discussed in appendix B. Since P is aso ower trianguar, the baance equation for division can be written as L a = P k t k = t + P k t k (2) k=1 k:k C In order for the queueing probem to be we defined it has to hod that the net infow of each division is ess than the tota service rate ρ = a r < 1 where ρ is the traffic intensity of division. It is mean number of tasks in service at a worker in division and it is equa to the percentage of time a worker is busy. For such queueing networks, Jackson (1963) showed that the joint probabiity distribution for the number of tasks at each division is the product of the margina probabiity distributions at each division. So the performance of each division can be anaysed independenty as a M/M/1 queue for each worker, i.e. a singe queue at each worker where externa arriving tasks foow a Poisson distribution and task service time foows an exponentia distribution. An exampe is given in Figure 1. There are two divisions, where division 1 has four workers, r 1 = 4, and division 2 has two workers, r 2 = 2. The manager of division 2, M 2, is supervisor of the manager of division 1, M 1. M 1 has task acceptance rate of t 1 and M 2 has a task acceptance rate 10

12 1 a 1 r 1 a 1 r 1 2 a 2 r 2 1 (1 p 2 ) a 2 a 1 M 1 t 1 a 1 r 1 a 1 r 1 3 a 1 M 1 p 1 a 1 (1 p 1 ) a 1 a 2 M 2 t 2 a 2 r 2 2 a 2 M 2 p 2 a 2 C i 1 C i 2 4 Division 1, r 1 = 4 Division 2, r 2 = 2 Figure 1: A queuing network with 4 workers (queues) in division 1 and 2 workers (queues) in division 2. For = 1, 2, M is the manager from division. Each division soves on average p a tasks per unit of time. M 2 accepts t 2 tasks from cients and (1 p 1 ) a 1 tasks from the subordinate manager M 1. On average (1 p 2 ) a 2 tasks remain unsoved. of t 2 and receives (1 p 1 ) a 1 tasks from M 1. The baance equations are a 1 = t 1 a 2 = t 2 + (1 p 1 ) a 1 Each worker in division 1 receives task with rate a 1 4. After the workers have deat with the task M 1 separates the tasks into soved cases, the output of division 1, i 1, and unsoved cases which are forwarded to M 2. On average division 1 soves p 1 a 1 cases. Each worker in division 2 receives tasks with rate a 2 2. The average output of division 2 is p 2 a 2 and (1 p 2 ) a 2 tasks remain unsoved. 2.2 Information Structure The effort choice p of M infuences the probabiity of soved tasks. HRM cannot observe this effort choice but ony the number of soved tasks. HRM designs the wage contracts for the managers of the divisions, w, which depends on the number of soved tasks i. 2.3 Economic Structure Performance Measures. Suppose that the organization creates revenue by soving tasks. Thus, one possibe performance measure is the average number of soved tasks of each division. 11

13 Suppose that for each competed task the firm receives a gain of H. For the network of queues considered it hods that under the equiibrium distribution, the externa departure of tasks aso foow independent Poisson processes (Jackson 1963). So the externa departure (output) foows aso a Poisson process with rate θ = p a. The average number of soved tasks of each division is θ to which we wi refer as the throughput of division. Suppose that HRM evauates the throughput of each division by a revenue function H (θ ), which is assumed to be stricty increasing and stricty concave, H θ > 0 and H θθ < 0. H θ denotes the first derivative and H θθ the second derivative with respect to θ. Since divisions with a higher rank might charge more for their soved probems, H is assumed to be nondecreasing in. Another performance measure is the mean number of tasks pending in the organization. Tasks might not be immediatey deat with and are queued. If it is costy to have tasks queued, it is in the interest of the organization to keep the queues short. Reasons why it is costy for the organization coud be storage costs or just it does not want to keep customers waiting too ong for service. The mean number of tasks pending in the division is the mean number of tota tasks in division minus the tasks in service Q = L Q = =1 L =1 a 2 r a Cost Structure. The workers in division receive a fixed wage of c W > 0. We assume that workers in a division of higher rank have a higher fixed wage, i.e. if M has a higher rank than M k then c W > c W k. The managers are assumed to be risk neutra and receive payment scheme, w, designed by HRM. Additionay, the manager has a utiity oss due to knowedge management. We assume that the oss is additive in the effort of the manager of creating knowedge and the number of workers in his division. If the manager invests an effort of p (0, 1) in knowedge creation and training of the workers in his division, they are abe to sove a fraction of p of the arriving tasks and he has a disutiiy of G (p ) = g (p ) + F. G is increasing in p and stricty convex, g p > 0 and g pp > 0. g p denotes the first derivative and g pp the second derivative with respect to p. If there are r workers in his span of contro an additiona oss of c M r occurs, with c M > 0. Objective of HRM. The objective of HRM is to maximize a function comprised of the key 12

14 performance indicators for each division a2 K = H (θ ) β E [w (I) p] c W r a r where the random variabe I = (I 1,..., I L ) is the output of a divisions. Each division generates costs, the wage of the manager w, the wage for each worker c W, and queueing costs βq. HRM maximizes the sum of profits of each division. It is assumed that the average number of tasks in the queue affects the organization vaue ineary by a factor of β (0, 1). If β is high, then a ong queue is more costy for the organization. 2.4 Probem Formuation HRM considers the foowing maximization probem max w,t,p,r L =1 K s.t. E [w (I) t] G (p ) c M r 0 for = 1,..., L (IR) { [ p arg max E w (I) p ] G (p ) c M } p r for = 1,..., L (IC) w (i) 0 for = 1,..., L and i a = P t (LL) (BE) p [0, 1] for = 1,..., L a < r for = 1,..., L r 0 for = 1,..., L where i = (i 1,..., i L ) is the observed output of a divisions. In section 3 the optima organizationa structure is determined, when the effort of the manager is contractabe. In that case HRM offers a wage w such that it is individua rationa for M to accept the contract, i.e. condition (IR) is binding. Since HRM can observe the effort eve condition it does not need to take into account the incentive compatibe condition (IC). We characterize the optima task acceptance rate t, the optima proportion of soved tasks p and the optima number of workers in each division r. In section 4, under the assumption that the effort of the manager 13

15 is not observabe, a bonus contract is characterized. If the effort of the manager is not observabe, HRM has to take condition (IC) into account as we. It is shown that under some conditions the managers M wi accept the contract and the effort eve under fu information can be impemented. 3 Anaysis: The Organizationa Structure In this section the optima organizationa form is determined under the assumption that the effort of the manager is contractabe. In that case it is optima for HRM to offer the manager an expected wage equa to his reservation utiity. Since p is contractabe, HRM does not need to take the incentives of the manager into account. This means that condition IR is binding and condition IC can be omitted. HRM soves the foowing simpified optimization probem: max t,p,r L ( ) H (θ ) β a2 G (p ) c M r c W r r a =1 s.t. a = P t (BE) p [0, 1] for = 1,..., L r 0 and a < r for = 1,..., L t 0 for = 1,..., L The foowing first order conditions are derived in Appendix E.1. The first order conditions for the optima task acceptance rate t is H θ (θ ) p = β 2a r (a ) 2 (r a ) 2 for = 1,..., L 1 (3) At the optimum it has to hod that the margina gain from soving a task has to be equa to the margina oss of an additiona task pending at division. The right hand side is positive since at the optimum it hods that r > a. The margina effect on other divisions, a k t, does not pay a roe due to the hierarchica structure of the organization. The size of t however, does pay a roe on other divisions through a k = m=1 P kmt m. 14

16 The first order conditions for p is H θ (θ ) a = g p (p ) (4) The margina gain of additiona knowedge in division has to be equa to the margina cost of the manager M of coaching the workers in his span of contro to attain that knowedge. The first order conditions for r is a 2 β (r a ) 2 = c where c = c W + c M. The margina gain of an additiona worker by reducing the task pending at division has to be equa to the margina cost of paying one additiona worker. It can be simpified to r = a c + β c (5) so r > a wi hod. The traffic intensity of each division is ρ = a r = c c + β (6) and the variabe cost of knowedge creation per worker are p g p (p ) r + c M = c3/2 + 2c β c + β + cm Under the assumption that c W and c M are increasing in rank, traffic intensity and the variabe cost of knowedge management per worker are increasing in rank of the manager. This means that on average a worker of a division of higher rank has more tasks in service than a worker of a division of ower rank. A manger of higher rank has higher variabe cost of knowedge management than a manager of ower rank. 15

17 The optima number of externa tasks t can be determined from equation BE t = P 1 a = ( I P ) a Due to the hierarchica structure, the task acceptance rate of division depends ony on divisions directy controed by M. t = a k:k C d (1 p k ) a k (7) Suppose M contros ony M k directy and M k contros ony M m directy, then t < t k a a k < (1 p k ) a k (1 p m) a m M wi have ower task acceptance rate than M k if the difference of the task arriva rate is smaer than the difference of the arriving fraction of unsoved probems. This means that M receives comparaby more unsoved probems and therefore has a ower task acceptance rate. In appendix E the second order conditions are derived. In order for a critica point to be a maximum it has to hod that H θθ(θ )θ /H θ (θ ) > 0.5, which means that the absoute vaue of the easticity of the margina gain of throughput is higher than Comparative Static The parameter β measures the intensity of the queueing costs and c are the tota cost per worker. In this section the effect of β and c on p, r, t and ρ are discussed. Combining equation 3, 4 and 5 gives two equations in p and r H θ (θ ) p = c + 2 β (c ) (8) H θ (θ ) a = g p (p ) (9) with a = r c c +. In order to derive the effects we appy the impicit function theorem on equations β 8 and 9. The derivations are given in Appendix E.2. Suppose that at the optimum the easticity of the margina gain of throughput is higher than 16

18 unity 1 < H θθ (θ ( ) ) θ H θ θ Then it can be shown that p β 0, p c 0, r c 0, r β 0 If the easticity of the margina gain of throughput is equa to 1, then a change in β or c has no effect on p. If H θθ a 2 = g pp, a change in c has no effect on r. If urgency increases, i.e. it becomes more important that tasks are not pending, the manager increases training of the workers in his span of contro. This impies that the probabiity to sove a task directy at each division increases. There are three effects on the number of workers in each division. First, since workers are abe to sove more probems, ess workers wi be empoyed. Second, if urgency increases more workers coud sove queued tasks more quicky. Third, the effect on the task acceptance rate aso infuences the number of workers. Under the assumption that the margina gain of throughput is eastic the span of contro wi decrease. Otherwise it may occur that higher urgency resuts in more workers. If the wage costs of the workers increases again it is optima for the manager to train the workers more. Since workers are more costy and are abe to sove more probems ess workers are empoyed. This impies that in both cases knowedge increases and ess tasks are forwarded to superiors, decisions are more decentraized but divided under ess workers. Athough the effect on p and r depends on the assumption on easticities of margina gain, the effect on traffic intensity is aways unambiguous. From equation 6 ρ c β = 2 β ( c + β ) 2 < 0 ρ β = c 2 ( c c + β ) 2 > 0 If urgency increases traffic intensity wi decrease. It means, that on average a worker has ess tasks in service. Since in this mode it hods that ρ = Q N (see Appendix D) it aso means that the ratio of 17

19 tasks pendiong to tota tasks in the division is decreasing. If the wage cost of the workers increase, traffic intensity wi increase. From equation 7 the effect of urgency on the task acceptance rate can be determined t β = r β c c + β + a β + k:k C d p ( k r ) β a k (1 p k ) k ck β ck + β + a k β The first term, which captures the effect of an increase in urgency on the own task arriva rate, is negative. The second term, which captures the effect on unsoved probems from a other divisions forwarded to M, is positive. This effect is positive since the knowedge in divisions increases and ess tasks are forwarded. Therefore more tasks can be accepted from customers. If the decrease in own task arriva rate outweighs the decrease in unsoved probems then the task acceptance rate has to decrease as we. If the decrease of unsoved probems in other divisions predominates, then task acceptance rate wi be higher. The effect of an increase in tota wage costs of the worker is aso ambiguous t c = r c c c + β + a c The first term is negative whie the second term is positive. The effect of an increase in wage cost of the workers on the task acceptance rate does not depend on the other divisions. Higher wage costs of workers decrease the number of workers so the task has to be distributed to fewer workers. This has a negative effect on task acceptance rate. On the other hand, higher wage cost increase the knowedge of the workers which has a positive on the task arriva rate and so the task acceptance rate can be increased. Suppose that ony in division the tota wage costs of workers increase, then the effect on the task acceptance rate of another division k is zero if is not a direct subordinate of k and otherwise t ( k = p r a c c (1 p ) c ) c c + β + a = p a c c (1 p ) t c So if the task acceptance rate in division decreases then it wi increase in division k. 18

20 Div. 3 Senior Div. 2 Midde Div. 3 Senior Div. 2 Senior Div. 1 Junior Div. 1 Div. 2 Junior Div. 1 Junior (a) (b) (c) Figure 2: Organization A consists of three divisions, where the manager of each division has a different rank (Figure 2(a)). Organization B consists of three divisions, where division one and two are guided by junior managers and division three is guided by a senior manager (Figure 2(b)). Organization C has two divisions ed by managers of different rank (Figure 2(c)). 3.2 Numerica Exampe In the numerica exampe we discuss the effects of the eimination of the midde management division. We consider three different organizationa structures. Organization A (Figure 2(a)) has one junior manager (division 1), one midde manager (division 2) and one senior manager (division 3). In Organization B (Figure 2(b)) the midde management division is transformed into a junior management division. In Organization C (Figure 2(c)) the midde management division is eiminated. The assumed specific functiona forms for the disutiities of the manager G, the revenue of each division H, and the wage cost of the workers c W are given in Tabe 1. For simpicity, we set the wage costs of the workers to zero so that the tota costs of workers are the disutiity a manager has from coaching, c = c M. Since the optima number of workers in each division depends ony on the tota costs c, the same resuts woud be obtained if a share of c is interpreted as the wage costs. In order to make organization A and C comparabe we chose the parameters such that for β = 0.8, their profits are equa, K A = K C = 0. From the first order conditions it foows that for division with rank s p s = h s x s h s a s = c s + 2 (c s ) β ) β r s = a s (1 + c s 19

21 Tabe 1: Functiona forms and parameter Knowedge management cost of manager M due to coaching G (p ) = g (p ) + F g (p ) = x 2 (p ) 2 due to span of contro c M r Revenue of division H (θ ) = h n (θ ) Intensity of queueing costs β {0.5, 0.8} Wage costs of workers c W = 0 Junior Midde Senior h c M x F At the optimum the gain of HRM is a 2 s K s = H s (θ s ) β G s (p s ) R s (r s ) = h s r s a s ( ( ) ) (h s ) 3 2 n xs (c s c s β) 2 In the simpe exampe, the optima amount of coaching p s is independent of β and c since H θθθ /H θ = 1. The optima task acceptance rate can be derived from the baance equation t s = ( I P s ) a s where P A = 1 p 1 0 0, P B = 0 0 0, P C = p p p 1 1 p 2 0 Let β = 0.8. The resuts are given in Tabe 2 and 3. In the exampe for a three organizations, the span of contro of the managers r and the task arriva rate a are decreasing in the rank of the manager. However, the traffic intensity ρ is increasing. This impies that an worker in a division of higher rank is busy at a higher percentage of time than an worker in a division of ower rank. The effort of the manager in coaching p is increasing in rank which impies that a division of higher rank soves on average more tasks than a division of ower rank. The task acceptance rate can be non monotonic in the rank of the managers. In organization A, the senior manager has a higher task acceptance rate than the midde manager. The midde manager receives comparaby more unsoved tasks than the senior manager and therefore can accept ess tasks from customers in order 20

22 Tabe 2: Resuts for organization A,B and C for β = 0.8 A,B,C A A B C Rank Junior Midde Senior Senior Senior Effort in coaching, p Number of Workers, r Effective arriva rate of tasks, a Task acceptance rate, t Traffic intensity, ρ Throughput, θ Tasks pending, Q Tasks pending and in service, N Variabe costs of knowedge management, g (p ) Costs due to span of contro, c M r Queuing costs, βq Revenue, H (θ ) Profit, K for the baance equation to be fufied. Aso the mean number of tasks pending in each division is non monotonic in rank. The average queue ength of the senior manager is ess than of the midde manager. So the main job of midde manager are soving tasks given by customers and reieving the senior manager from a potentia overfow of unsoved tasks from the junior manager. Eiminating the midde management can resut in organization B or C. If instead of one midde manager two junior manager report to the senior manager (Figure 2(b)) the task arriva rate of the senior manager increases and in order for the baance equation to be fufied, it woud be required that t 2 < 0. This means, that there is an overfow of tasks to the head division. If the division of the midde manager is eiminated without repacement (Figure 2(c)) then the probem of overfow does not occur. However, the task acceptance rate of the senior manager decreases since unsoved tasks are not handed by midde managers first. With respect to the overa gain, HRM is indifferent between organization A or C. However, these two organizations differ in their structure. Athough organization C is fatter and has on average ess tasks pending, the mean response time W, i.e. the time a task spends in the organization, is higher in organization C than in A. The forma definition is given in Appendix D. The reason is that in organization A the task acceptance rate of the senior manager is higher and if a task is directy handed by the senior manager it is served faster. Suppose that a deay becomes ess costy, β = 0.5. The resuts are given in Tabe 4 and 5. 21

23 Tabe 3: Resuts for organization A,B and C for β = 0.8 Organization A B C Tota number of workers, L =1 r Tota effective arriva rate of tasks, L =1 a Tota task acceptance rate, L =1 t Tota tasks pending, L =1 Q Tota tasks pending and in service, L =1 N Q/N Tota throughput, L =1 θ Response time, W Tota variabe costs of knowedge management, L =1 G (p ) Tota costs due to span of contro L =1 R (r ) Tota queueing costs, L =1 βq Tota revenue, L =1 H (θ ) Tota profit, L =1 K Athough in this exampe the decrease in urgency has no effect on the knowedge of the workers, it resuts in a higher span of contro. This means that the effect of a higher task acceptance rate, which resuts in more workers, outweighs that a decrease in urgency coud impy fewer workers. In organization A, the division of the midde manager has the highest gain, which aso impies that by removing division 2 or transforming it into a division of rank 0 does not resut in higher tota gain. 4 Anaysis: The Incentive Structure Under asymmetric information HRM cannot observe the effort a manager puts into coaching of the workers in his span of contro, but the amount of soved tasks of each division. We consider two different types of performance evauation with bonus scheme: first, under independent performance evauation a bonus is paid, if the division reached the targeted output and second, under joint performance evauation a bonus is paid ony if a divisions have achieved their target. We identify conditions, which depends cruciay on the hazard rate of effort, under which such a bonus contract exist. The conditions are simiar to Kim (1997) and Park (1995) who considered simiar bonus contracts for a singe agent under asymmetric information and imited iabiity. We foow their ines of argument and extend a bonus contract for the mutipe agent case. The hazard rate of effort gives the change in the cumuative distribution of output resuting from higher effort in reation to 22

24 Tabe 4: Resuts for organization A,B and C for β = 0.5 A,B,C A A B C Rank Junior Midde Senior Senior Senior Effort in coaching, p Number of Workers, r Effective arriva rat of tasks, a Task acceptance rate, t Traffic intensity, ρ Throughput, θ Tasks pending, Q Tasks pending and in service, N Variabe costs of knowedge management, g (p ) Costs due to span of contro, c M r Queuing costs, βq Revenue, H (θ ) Profit, K the probabiity to receive a bonus. 4.1 Independent Performance Evauation Under independent performance evauation (IPE) the payment of the bonus ony depends on the output of each division i. Proposition 1 (IPE). If there exists some eve of soved tasks 0 < i θ 1 such that F (i,θ ) g p (p ( ) > ( ( )) G p 1 F i, θ > g p (p ) G ( p ) + c M r (10) where F (i, θ ) is the cumuative distribution of the individua success probabiity ( ) i F i, θ = π (i, θ ) F (i, θ ) i =0 = a π ( i, θ ) < 0 then the foowing incentive scheme induces the first-best effort eve p and the manager receives 23

25 Tabe 5: Resuts for organization A,B and C for β = 0.5 Organization A B C Tota number of workers, L =1 r Tota effective arriva rate of tasks, L =1 a Tota task acceptance rate, L =1 t Tota tasks pending, L =1 Q Tota tasks pending and in service, L =1 N Q/N Tota throughput, L =1 θ Response time, W Tota variabe of knowedge management, L =1 G (p ) Tota costs due to span of contro L =1 R (r ) Tota queueing costs, L =1 βq Tota revenue, L =1 H (θ ) Tota profit, L =1 K his reservation eve of utiity. A + B w (i ) = A for i i ese (11) where A = G (p ) + cm r (1 F (i, θ )) B and B = gp(p ) F (i,θ ) The proof and a foowing proofs are given in Appendix F. The saary A and the bonus B depend ony on the own output of each division. Under symmetric information it hods that H θ (θ ) a = g p (p ), therefore B = g p (p ) a π ( i, θ ) = H θ (θ ) π ( i, ) θ The size of the bonus depends on the ratio of the margina gain of one additiona soved task and the probabiity that the target output is reached. Increasing the target output resuts in a ower bonus since π (θ, i ) is increasing in i for i θ 1 (See Appendix C, remark 1). The saary is A = G (p ) + cm r ( ( 1 F i, θ )) B = G (p ) + cm r H θ (θ ) (1 F (i, θ )) π ( i, θ ) (12) Since the third term is decreasing in i, the saary wi get coser to the expected wage G (p )+cm r 24

26 when the target output is increased. 4.2 Joint Performance Evauation Under joint performance evauation (JPE) the payment of the bonus depends on the output of a divisions, i = (i 1,..., i ). Proposition 2 characterizes the saary A, the bonus B, and i the target output for which a bonus wi be paid. Proposition 2 (JPE). If there exists some eve of soved tasks 0 < i θ 1 such that F (i,θ ) 1 F (i, θ ) > g p (p ( ) ) G p + c M F (i,θ ) F (i, θ ) > g p (p ( ) ) G p + c M F (i, θ ) im p 0 F (i, θ) F (i,θ ) r r if if F (i, θ) < 0 (13) F (i, θ) > 0 (14) g p (p ) G ( p ) (15) where F (i, θ) is the cumuative distribution of the joint success probabiity F ( i, θ ) = F (i, θ) ( ) F i, θ =1 = k: C k θ k π ( i k p, θ ) L k m k ( ) F m i m, θ m a π ( i, θ ) L ( ) F m i m, θ m m then the foowing incentive scheme induces the first-best effort eve p and the manager receives his reservation eve of utiity. A + B w (i ) = A for i i ese (16) where A = G (p ) + cm r (1 F (i, θ )) B and B = gp(p ) F(i,θ ) The size of the bonus B depends on the effect of an increase in coaching on the own success probabiity and on the success probabiities of divisions which supervise division. 25