J Intell Manuf DOI 10.1007/s10845-014-0993-5
Monitoring mechanisms in new product development with risk-averse project manager Kai Yang · Yanfei Lan · Ruiqing Zhao
Received: 1 September 2014 / Accepted: 27 October 2014 © Springer Science+Business Media New York 2014
Abstract It is necessary for one senior executive (she) to monitor her project manager (he) who conducts early research stage followed by a later development stage in new product development. In this paper, we analyze two monitoring mechanisms: (1) the idea information-based monitoring (IM) mechanism wherein the senior executive engages one supervisor to monitor the project manager’s idea information; (2) the effort-based monitoring (EM) mechanism wherein the senior executive engages another supervisor to monitor the project manager’s effort. Within the framework of uncertainty theory, we first present two classes of bilevel uncertain principal-agent monitoring models, and then derive their respective optimal incentive contracts. We find that the senior executive should set the incentive term as high as possible to motivate each supervisor to monitor the project manager’s idea information and effort no matter how much the design idea value is. We also find that EM mechanism can always dominate IM mechanism when the monitoring costs are equal. Moreover, comparing with a no monitoring scenario, we identify two values of monitoring: the value of monitoring idea information and the value of monitoring effort. Our results show that adopting IM and EM mechanisms can improve the senior executive’s profits obtained in the no monitoring scenario when the revenue uncertainty is sufficiently low. The results also indicate that the value of monitoring idea information decreases as the risk aversion K. Yang · Y. Lan (B) · R. Zhao Institute of Systems Engineering, Tianjin University, Tianjin 300072, China e-mail:
[email protected] K. Yang e-mail:
[email protected] R. Zhao e-mail:
[email protected]
level of the project manager improves, while the value of monitoring effort shows the opposite feature. Keywords New product development · Monitoring mechanism · Information asymmetry · Incentive contract · Uncertainty theory
Introduction In the current highly competitive marketplace, a rapid and efficient new product development (NPD) is an inevitable choice for firms to achieve real success and gain lasting competitive advantage. Motivating and directing the project manager to devise a design idea and invest adequate effort to realize the transformation from the design idea to the new product is an important goal of all the firms. Firms have two main instruments, namely, incentives and monitoring, through which they can control their project managers. Previous theoretical researches in NPD, however, have focused exclusively on the role of incentives in directing the project manager (Xiao and Xu 2012; Chao et al. 2013). Although monitoring mechanism is widely practiced, relatively little is known about it to supervise the project manager. Consequently, in order to enrich the content of monitoring in NPD, we focus on monitoring mechanism, which is more likely to be effective in operating NPD project. In this paper, we investigate how the monitoring mechanisms impact the incentive contracts in NPD, what types of monitoring mechanisms would be more effective, and how the risk aversion level of the project manager affects the values of monitoring. To answer these questions, we study an NPD problem, where a project manager develops a new product for a senior executive by executing two stages: a research stage and a development stage. In the research
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stage, the project manager devises a design idea and subsequently exerts an effort to transform that idea into a product in the development stage. It is likely that the existence of dual information asymmetry between the senior executive and the project manager roots from the fact that the design idea value of the new product in early research stage is unknown to the senior executive and the project manager may exert an unobservable effort in later development stage. In order to ensure that the project manager reveals true design idea value information and invests adequate effort to realize the implementation of the idea, it is inevitable for the senior executive to carry out two monitoring mechanisms: the idea informationbased monitoring (IM) mechanism and the effort–based monitoring (EM) mechanism. Specifically, under IM mechanism, the senior executive engages one supervisor to monitor the project manager’s idea information, while she engages another supervisor to monitor the project manager’s effort under EM mechanism. In addition, due to the variability of technology and market associated with the new product, the design idea value of such new product and the corresponding revenue are assumed to be uncertain variables. Within the framework of uncertainty theory, we first present two classes of bilevel principal-agent monitoring models based on the idea information and the effort, and then derive their respective optimal incentive contracts. Meanwhile, we present and discuss our results under the scenario of no monitoring as a useful benchmark in analyzing the implications and the value of introducing monitoring mechanisms into NPD. By the obtained results, we establish following main findings. First, we find that the senior executive should set the incentive term as high as possible, regardless of the design idea value, to motivate the supervisors to monitor the project manager such that he reveals the true design idea value information and invests adequate effort to realize the implementation of the idea. Second, comparing IM mechanism with EM mechanism, we find that EM mechanism can always dominate IM mechanism when the monitoring costs are equal. In particular, from the senior executives’s perspective, using EM mechanism has higher impacts on improving the project manager’s optimal effort level than that from using IM mechanism. Third, analyzing the no monitoring scenario, we identify two values of monitoring: the value of monitoring idea information and that of monitoring effort. We find that adopting IM and EM mechanisms can improve the senior executive’s profits obtained in the no monitoring scenario when the revenue uncertainty is sufficiently low. The results also suggest that the value of monitoring idea information decreases as the risk aversion level of the project manager improves, while the value of monitoring effort shows opposite feature. The contributions of this study include three aspects. First, we introduce a framework that explicitly allows for simultaneous examination of IM mechanism and EM mechanism in
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NPD. Result analysis demonstrates the necessity of considering both two mechanisms. Second, we explicitly derive the conditions where monitoring can improve the senior executive’s profits obtained in the no monitoring scenario which allows the senior executive in an effort to improve her profits. Finally, we clearly reveal how the risk aversion level of the project manager impacts the values of monitoring. Indeed, it is this change in the level of risk aversion that proves to be most useful in operating NPD project for the senior executive. The paper is organized as follows. In “Literature review” section, we briefly review related literature. In the “Research setting” section, we describe the essential features of research setting. In “Scenario without monitoring” section, we present the optimal menu of wage contracts under the no monitoring scenario as a benchmark. In “Scenarios with monitoring” section, we present and discuss optimal wage contracts across the two monitoring mechanisms, namely, an idea IM mechanism and an EM mechanism. We demonstrate the value of monitoring idea information and the value of monitoring effort in Values of monitoring section. In “Conclusion and future research” section, we summarize our research conclusions and outline directions for future researches. Preliminaries on uncertainty theory and proofs of all propositions and lemmas are relegated to the appendix for clarity of presentation. Literature review Our paper is related to two streams of literature: the operations management literature on NPD and the economics literature on principal agent problems. New product development has attracted interest from both researchers and practitioners of operations management (see Krishnan and Ulrich (2001), Buyukozkan et al. (2004) and Jiao et al. (2007) for excellent reviews). Recently, there is a small group of papers that study issues of incentives for NPD. For example, Mihm (2010) considered an internal costgaming issue in an NPD project and showed how incentive schemes could provide incentives for engineers to improve cost compliance of the project. Xiao and Xu (2012) investigated the impact of royalty revision on incentives and profits in a two-stage NPD. Chao et al. (2013) developed a principalagent model to study how the firm should design incentive mechanisms for the project manager in two-stage NPD with a stage-gate process. Yang et al. (2014) presented four classes of uncertain principal-agent models to examine the impact of risk attitude on incentives and performances in a twostage NPD under dual information asymmetry. In contrast to the aforementioned papers, we treat the scenario of pure incentive as the benchmark to evaluate the performance of monitoring mechanisms. Specifically, we focus on both IM and EM mechanisms in NPD. Further, we identify which monitoring mechanism is able to dominate the other.
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The economics literature on principal agent problems is vast (see Holmstrom (1982), Demski and Sappington (1984), Mookherjee (1984), Itoh (1991), McAfee and McMillan (1991) and Che and Yoo (2001)). These papers characterized the incomplete information as random variables based on probability theory. However, probability theory is no longer applicable in NPD, because the probability distribution cannot be estimated from the frequency due to the lack of historical data. In order to deal with this kind of incomplete information, uncertainty theory was founded by Liu (2007). Since the pioneer work of Liu (2007), the uncertainty theory has become a powerful mathematical tool to handle various issues including uncertain inference (Liu 2010a; Gao et al. 2010), uncertain risk analysis (Liu 2010b), uncertain finance (Liu 2009; Peng and Yao 2011), uncertain logic (Li and Liu 2009a, b; Li and Wong 2010; Liu 2011), uncertain optimal control (Zhu 2010), uncertain differential game (Yang and Gao 2013, 2014) and uncertain solid transportation problem (Yang et al. 2015). Liu (2010c) provided excellent reviews on uncertainty theory. The notion uncertain random variable was first introduced by Liu (2013) based on uncertainty theory (Liu 2007), and then studied by a number of researchers in the literature (Liu 2014; Sheng and Yao 2014). Recently, several scholars have used uncertainty theory to develop the principal agent theory. For example, Mu et al. (2013) presented an uncertain contract model for the rural migrant worker’s employment problem to maximize the enterprise’s expected utility. Wang et al. (2013) addressed an uncertain price discrimination model in labor market to maximize the employer’s expected welfare. Wu et al. (2013) discussed an uncertain principal-agent problem with multi-dimensional incomplete information between a principal and an agent based on the critical value criterion. Wang et al. (2014) and Feng et al. (2014) incorporated both adverse selection and moral hazard in an uncertain principal-agent problem. Our work builds on these literature and extends them by developing two classes of bilevel uncertain principal-agent monitoring models, in which there exist two principal-agent relationships.
The research setting We consider a NPD problem where a project manager (he) researches and develops a kind of new product for a senior executive (she). The executing process for the new product is divided into two stages: a research stage and a development stage. In the research stage, the project manager devises the new product design idea and subsequently exerts effort to realize the transformation from the design idea to the new product in the development stage. Reasonably assume that the project manager has private information about the value of the design idea, and his effort is unobservable for the senior
executive. Therefore, it is inevitable for the senior executive to design monitoring mechanisms by engaging two types of supervisors to monitor the project manager’s idea information in the research stage and effort in the development stage, respectively. Specifically, one supervisor provides incentives to ensure that the project manager reveals the true value of the design idea, and the other guarantees the project manager to make the greatest effort to develop the product. Under such consideration, we describe several important features of our research setting.
Notations and assumptions Although the senior executive does not know the value of the design idea, she can give a subjective assessment, which is characterized as an uncertain variable θ . Denote its uncertainty distribution by (x) with x ∈ [θ , θ ], where 0 ≤ θ < θ ≤ ∞. Let φ(x) = d(x) dx and then impose the inverse hazard rate (IHR) h(x) = (1 − (x))/φ(x) as a decreasing function of x. This monotonicity condition is commonly adopted in asymmetric information literature (Mu et al. 2013; Wang et al. 2013). Most parametric single-peak distributions have a decreasing IHR, such as linear, zigzag, normal and lognormal uncertain distributions (see Liu (2007) for details). Once the project manager learns the value of the design idea x in the end of research stage, he will invest effort e with cost C(e) = 21 e2 to transform the design idea into the new product in the development stage. The assumption of a quadratic cost function is in accordance with the practical fact, which has been used in Xiao and Xu (2012) and Chao et al. (2013), and implies increasing marginal cost of development effort. In addition, this assumption makes analytical solutions accessible, thus allows us to study the significant impact of risk attitude in NPD field. After the project is completed, the senior executive will subsequently receive a revenue which is driven by three factors. First, the value of design idea x resulting from the research stage determines the revenue of the product—the higher the x is, the higher the revenue will be for the product. Second, the project manager’s effort e will lead to a deterministic increase in the product revenue. Third, the revenue for the product is influenced by an uncertain component ε with mean 0 and variance σ 2 which captures the market uncertainty. To ensure meaningful results, we require that σ 2 ≤ v 2 /ρ which is maintained throughout the analyses of the models and guarantees that the revenue generated by the product is nonnegative. Given the value of design idea x and the project manager’s effort e, the total revenue collected from developing the product is determined by the following additive form: R(x, e) = x + ve + ε, where v (v > 0) measures the effectiveness of the project manager’s effort in enhancing the revenue of the product.
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E [ N (αN (θ ), βN (θ ))]
The incentive contracts Assume that incentive contracts are restricted to be linear which allows us to obtain closed-form solutions and insights pertaining to use of monitoring mechanisms in an NPD setting. We specify that the senior executive engages supervisor i to monitor the project manager by writing a linear contract Wi = ai + bi R, i ∈ {I, E}, where ai is referred to as a fixed payment and the slope coefficient bi (0 ≤ bi ≤ 1) captures the incentive term. Since the revenue is uncertain, the incentive pay is contracted on the revenue R. The subscript “I” stands for “idea information-based monitoring” and “E” stands for “effort-based monitoring”. Turning to the relationship between supervisor i and the project manager, supervisor i offers a menu of contracts to ensure that the project manager truthfully discloses idea value information and invests adequate effort to realize the implementation of the idea. For tractability, we also restrict the focus of our analysis on linear menu contracts of the form: Si (x, R) = αi (x) + βi (x)R, i ∈ {I, E}, where αi (x) denotes the project manager’s payment contingent on his report design idea value x and 0 ≤ βi (x) ≤ 1 for any x ∈ [θ , θ ]. It can be seen that this menu of contracts is based on the project manager’s reported idea value and the revenue generated by the new product, which has been used in recent works that consider incentives in NPD (Xiao and Xu 2012; Chao et al. 2013). In particular, under the no monitoring scenario, the senior executive directly offers a menu of contracts (αN (·), βN (·)) to the project manager without engaging supervisors I and E, in which the subscript “N” stands for “no monitoring”. The utility functions To capture the different risk preference of the three parties, we assume that the senior executive and supervisors I and E are risk-neutral and the project manager is risk-averse. The senior executive’s utility function
= E [R − SN ] = E [(1 − βN (θ ))(θ + veN ) − αN (θ )] . (3) The supervisor’s utility function The expected utilities of supervisors I and E are given by E [πI (αI (x), βI (x))] = E [WI − CI ] = E [aI − αI + (bI − βI )(θ + veI ) − CI ]
(4)
and E [πE (αE (x), βE (x), eE )] = E [WE − CE ] = E [aE − αE + (bE − βE )(θ + veE ) − CE ] ,
(5)
where CI and CE are the fixed monitoring costs, respectively. The project manager’s utility function In reality, the project manager is usually risk averse under revenue uncertainty. That is, the project manager avoids risk, is afraid of risk, or is sensitive to risk. The risk-averse contractor is trying to maximize his utility other than his expected profit. To approach this reality, we assume that the project manager makes a trade-off between the mean and the variance of his uncertain profit, i.e., reporting the value of the design idea y when the true value of the design idea is x and exerting the effort e to maximize the following utility function: MVi (x, y) = E [Si (y, R) − C(ei )|x] 1 − ρV [Si (y, R) − C(ei )|x] , i ∈ {I, E, N}, 2 (6) where ρ > 0 is the project manager’s risk aversion level. The larger the risk aversion level x is, the more conservative his behavior will be. In particular, MVi (x) = MVi (x, x) is the project manager’s utility from exerting optimal effort and truthfully reporting the value of the design idea. Having described the key elements of our research setting, we now turn our attention to analysis of the problem.
Under IM mechanism and EM mechanism, the expected utilities of the senior executive respectively are given by Scenario without monitoring
E [ I (aI , bI )] = E [R − WI ] = E [(1−bI )(θ +veI )−aI ] (1) and E [ E (aE , bE )] = E [R − WE ] = E [(1 − bE )(θ + veE ) − aE ] .
(2)
In particular, under the scenario of no monitoring, the expected utility of the senior executive is given by
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To explore the influence of monitoring mechanisms, as a benchmark we first examine the scenario of no monitoring. In this scenario, the senior executive directly offers a menu of contracts to the project manager, while the project manager-endowed with private information about the design idea value-decides whether or not to accept a contract and, if so, how much effort to exert. More specifically, we assume the following sequence of events: (1) The project manager
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privately learns the design idea value; (2) The senior executive offers a menu of contracts (αN (·), βN (·)); (3) The project manager decides whether or not to participate (work for the senior executive); (4) Under the signed contract, the project manager exerts unobservable effort; and (5) The senior executive pays based on the realized revenue value. Our problem is one of adverse selection followed by moral hazard. Adverse selection arises from the project managers private information regarding the idea value and moral hazard arises because the senior executive cannot observe the project manager’s effort. In this setting, we present the senior executive’s contract design problem as the following optimization model: ⎧ max E [ N (αN (θ ), βN (θ ))] ⎪ ⎪ ⎪ (αN (·),βN (·)) ⎪ ⎪ ⎪ ⎪ ⎪ subject to: ⎪ ⎨ MVN (x) ≥ MVN (x, y), ∀x, y ∈ [θ, θ ] (IC1-N) ⎪ ⎪ ⎪ ⎪ eN = arg max MVN (x), ∀x ∈ [θ, θ ] (IC2-N) ⎪ ⎪ eˆN ≥0 ⎪ ⎪ ⎪ ⎩ (IR-N) MVN (x) ≥ 0, ∀x ∈ [θ, θ ]. (7)
Lemma 2 Under the scenario of no monitoring, the senior executive’s expected profit can be written as θ x − MVN (x) + v 2 βN E [ N ] = Z
The objective function of Model (7) is equal to the senior executive’s expected revenue generated from the new product minus the wage paid to the project manager. Constraint (IC1-N) is the incentive compatibility constraint for adverse selection, which guarantees the project manager to report his true idea value. Constraint (IC2-N) is the incentive compatibility constraint for moral hazard, which ensures that the project manager chooses his optimal effort level. Constraint (IR-N) is the individual rationality constraint, which reflects the fact that the project manager will participate in the contract by ensuring that his utility is at least as high as his reservation utility. We assume that the project manager has a zero reservation profit. This assumption is common in contract design problem and has been used in both the economic literature (e.g., Che (1993), Myerson (1981)) and the operations management literature (e.g., Corbett et al. (2004), Lim (2001)). Before we derive the optimal menu of contracts under the scenario of no monitoring, we present two lemmas that simplify our problem. The first lemma simplifies the expressions for the constraints (IC1-N), (IC2-N) and (IR-N). The second one simplifies the expressions for the senior executive’s expected profit.
The problem is a deterministic optimal control problem with one decision variable, which is easily solved by applying Pontryagin maximum principle (Casas et al. 2001). The following proposition characterizes the optimal menu of contracts offered by the senior executive under the scenario of no monitoring.
Lemma 1 Under the scenario of no monitoring, a menu of contracts satisfies the constraints (IC1-N), (IC2-N) and (IRN) if and only if (i) dMVdxN (x) = βN (x), ∀x ∈ [θ, θ ]; N (x) ≥ 0, ∀x ∈ [θ , θ ]; (ii) dβdx (iii) MVN (θ) = 0.
θ
−
1 2 v + ρσ 2 βN2 (x) φ(x)dx. 2
(8)
Lemma 1 provides conditions under which the constraints (IC1-N), (IC2-N) and (IR-N) are satisfied while Lemma 2 provides a convenient way to express the senior executive’s expected profit. From Lemmas 1 and 2, we can rewrite the senior executive’s optimization problem (7) as ⎧
θ ⎪ max N = θ x − MVN (x) + v 2 βN ⎪ ⎪ β (·) ⎪ N ⎪ ⎪ ⎪ ⎪ − 21 v 2 + ρσ 2 βN2 (x) φ(x)dx ⎪ ⎪ ⎪ ⎪ ⎨ subject to: (9) ⎪ dMVN (x) = β (x), ∀x ∈ [θ, θ ] ⎪ ⎪ N ⎪ dx ⎪ ⎪ ⎪ d β (x) ⎪ N ⎪ ⎪ ⎪ dx ≥ 0, ∀x ∈ [θ, θ ] ⎪ ⎩ MVN (θ) = 0.
Proposition 1 Under the scenario of no monitoring, the optimal menu of contracts is given by the incentive term + 2 v − h(x) ∗ βN (x) = v 2 + ρσ 2 and the fixed payment x 1 2 αN∗ (x) = v − ρσ 2 (βN∗ (x))2 . βN∗ (s)ds − xβN∗ (x) − 2 θ The corresponding optimal effort level for the project man ager is given by e∗N =
v v 2 −h(x) v 2 +ρσ 2
+
.
As shown in Proposition 1, one particularly useful outcome of our solution is the straightforward manner in which the expressions h(x) and ρσ 2 exist in the incentive term. Firstly, h(x) is linked to adverse selection which results from the project manager’s private knowledge about the idea value of the new product. To facilitate further discussion, we define (x) = v 2 − h(x). Note that (x) is nondecreasing with respect to x. If (θ) < 0, then βN∗ (x) is equal to 0 for any x ∈ [θ , θ ]. Therefore, to avoid unnecessary expositional complication, we assume that (θ) > 0 to exclude that trivial case. In the case (θ ) < 0, there exists a cutoff level ˆ the senior executive θˆ such that (θˆ ) = 0. When x < θ,
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should propose a fixed payment contract βN∗ (x) = 0 . That is, the senior executive worries that the project manager will take advantage of the idea value to shift risk, so she sets the incentive term to zero to prevent such action. Secondly, ρσ 2 is linked to moral hazard which is directly related to market risk experienced by both the senior executive and the project manager. In the presence of moral hazard, the senior executive should bear partial risk and set an appropriate incentive term to motivate the risk-averse project manager. We observe that the incentive term of the optimal menu of contracts is decreasing in the risk aversion level of the project manager ρ because as the project manager becomes more risk-averse, the senior executive should induce the project manager to exert less effort. According to Proposition 1, we compute the senior executive’s expected profit and the project manager’s utility in the following corollary. Corollary 1 Under the scenario of no monitoring, the senior executive’s expected profit is given by
2 + 2 E v − h(θ )
N = E[θ ] + 2(v 2 + ρσ 2 ) and the project manager’s utility is given by + x 2 v − h(s) MVN (x) = ds. v 2 + ρσ 2 θ Corollary 1 can be interpreted as follows. First, in the presence of asymmetric idea information, the term
x v 2 −h(s) + ds is called the project manager’s information θ v 2 +ρσ 2 rent because the senior executive has to give the project manager a piece of the pie in the form of information rent to induce him to reveal the true idea value. Furthermore, the project manager’s information rent increases in x. That is, the project manager with higher idea value earns larger information rents than that with lower idea value. Second, Corollary 1 reveals that adverse selection and moral hazard lead to loss in the senior executive’s expected profit. In short, the results of this scenario serve as a benchmark for deriving results for the alternative scenarios studied. In this way, we can investigate the impact of monitoring mechanisms on the performances in an NPD setting.
Scenarios with monitoring In this section, we analyze the following two monitoring mechanisms: (1) an idea information-based monitoring mechanism wherein the senior executive engages supervisor I to monitor the project manager’s idea information, and (2) an effort-based monitoring mechanism wherein the senior
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executive engages supervisor E to monitor the project manager’s effort. In the following subsections, we will study each of these two mechanisms separately. Idea information-based monitoring mechanism (IM) Under IM mechanism, the senior executive contracts with supervisor I to monitor the project manager’s idea information. More specifically, we assume the following sequence of events: (1) The senior executive offers an incentive contract (aI , bI ) to supervisor I; (2) Supervisor I decides whether or not to participate; (3) Under the signed contract, the project manager and supervisor I simultaneously learn the idea value; (4) Supervisor I offers a contract (αI (x), βI (x)) to the project manager; (5) The project manager decides whether or not to participate; (6) Under the signed contract, the project manager exerts unobservable effort; and (7) The senior executive and supervisor I pay based on the realized revenue value. In this setting, there exist two principal-agent relationships: the relationship between the senior executive and supervisor I as well as the relationship between supervisor I and the project manager. Under this consideration, the senior executive has to take into account supervisor I and the project manager’s optimal decisions in the second level when she designs the contract to maximize her expected profit in the first level. As a consequence, we formulate the senior executive’s contract design problem as the following bilevel principal-agent monitoring idea information model: ⎧ max E [ I (aI , bI )] ⎪ ⎪ ⎪ (aI ,bI ) ⎪ ⎪ ⎪ ⎪ ⎪ subject to: ⎪ ⎪ ⎪ ⎪ ⎪ (IR1-I) E πI (αI∗ (x), βI∗ (x)) ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ∗ ⎪ αI (x), βI (x) are derived from ⎪ ⎪ ⎨ solving the problem PI (10) ⎧ ⎪ ⎪ ⎪ max E (α (x), β (x))] [π ⎪ I I I ⎪ ⎪ ⎪ (αI (x),βI (x)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ subject to: ⎪ ⎪ ⎪ PI : ⎪ ⎪ ⎪ ⎪ eI = arg max MVI (x) (IC-I) ⎪ ⎪ ⎪ ⎪ ⎪ eˆI ≥0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ MVI (x) ≥ 0. (IR2-I) The objective function of Model (10) is equal to the senior executive’s expected revenue generated from the new product minus the wage paid to supervisor I. The individual rationality constraints (IR1-I) and (IR1-I) ensure that supervisor I and the project manager are always guaranteed at least the reservation utility 0. The incentive compatibility constraint (IC-I) guarantees the project manager to choose his optimal effort level. We solve the problem by using backward induction approach from the second level supervisor I offers in which an optimal incentive contract αI∗ , βI∗ to induce the project
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manager to invest optimal effort level eI∗ . The following lemma presents the analysis of supervisor I and the project manager’s optimal decisions in the second level. Lemma 3 Given the incentive contract (aI , bI ), the optimal response αI∗ , βI∗ made by supervisor I is given by βI∗ (x) =
v2
bI v 2 , + ρσ 2
1 αI∗ (x) = −xβI∗ (x) − (v 2 − ρσ 2 )(βI∗ (x))2 . 2 The corresponding optimal effort level for the project man3 Iv ager is given by eI∗ = v 2b+ρσ 2.
Having the information about the strategies αI∗ , βI∗ of supervisor I, the senior executive would use it to maximize her own expected profit. The following proposition characterizes the optimal contracts offered by the senior executive and supervisor I under IM mechanism. Proposition mechanism, the optimal incentive 2 Under IM contract aI∗ (x), bI∗ (x) offered by the senior executive is given by bI∗ = 1, aI∗ =
−2E[θ ]v 2 − v 4 + CI . 2(v 2 + ρσ 2 )
∗ ∗ Given the optimal incentive a I , bI , the corre ∗contract sponding optimal contract αI (x), βI∗ (x) offered by supervisor I is given by v2 βI∗ (x) = 2 , v + ρσ 2 1 αI∗ (x) = −xβI∗ (x) − (v 2 − ρσ 2 )(βI∗ (x))2 2 and the corresponding optimal effort level for the project v3 manager is given by eI∗ = v 2 +ρσ 2. Proposition 2 gives a closed form of the optimal incentive contract mechanism aI∗ , bI∗ . It is worth noting that the optimal incentive term bI∗ is equal to 1 for any x ∈ [θ, θ ]. This can be explained as follows: when supervisor I is risk neutral, the senior executive should set the incentive term as high as possible to motivate supervisor I to monitor the project manager’s idea information no matter how much the design idea value is. The main purpose of the fixed payment aI∗ is to make sure that supervisor I would accept the optimal incentive contract. Proposition 2 shows that the optimal incentive term is greater than that in the benchmark setting of no monitoring; that is βI∗ (x) ≥ βN∗ (x) for any x ∈ [θ , θ ] which is shown in Fig. 1. This is intuitive because the senior executive should set a higher incentive term to prevent the project manager from shirking. Proposition 2 also shows that the optimal effort
No monitoring Monitoring idea information Monnitoring effort
0
Fig. 1 Optimal incentive terms under different scenarios
level for the project manager is greater than its optimal value ∗. obtained in the scenario of no monitoring; that is eI∗ ≥ eN This indicates that from the senior executives’s perspective, using IM can improve the project manager’s optimal effort level. On the basis of Proposition 2, we compute the senior executive’s expected profit and the project manager’s utility in the following corollary. Corollary 2 Under IM mechanism, the senior executive’s expected profit is given by
I = E[θ ] +
2(v 2
v4 − CI , + ρσ 2 )
supervisor I’s expected profit is given by πI = 0 and the project manager’s utility is given by MVI (x) = 0. Comparing with Corollary 1, Corollary 2 reveals that the project manager no longer obtains information rent under IM mechanism. Effort-based monitoring mechanism (EM) Under EM mechanism, the senior executive contracts with supervisor E to monitor the project manager’s effort. More specifically, we assume the following sequence of events: (1) The senior executive offers an incentive contract (aE , bE ) to supervisor E; (2) Supervisor E decides whether or not to participate; (3) Under the signed contract, the project manager privately learns the idea value; (4) Supervisor E offers a menu of contracts (αE (·), βE (·), eE ) to the project manager; (5) The project manager decides whether or not to participate; (6) Under the signed contract, the project manager exerts unobservable effort; and (7) The senior executive and supervisor E pay based on the realized revenue value.
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In this setting, we develop a bilevel principal-agent monitoring effort model where the senior executive has to take into account supervisor E and the project manager’s optimal decisions in the second level when she designs the wage contract to maximize his expected profit in the first level. As a consequence, we build the senior executive’s contract design problem as the following bilevel programming model: ⎧ max E [ E (aE , bE )] ⎪ ⎪ (aE ,bE ) ⎪ ⎪ ⎪ ⎪ ⎪ subject to: ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (IR1-E) E πE αE∗ (θ ), βE∗ (θ ), eE∗ ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎪ αE∗ (θ ), βE∗ (θ ) and eE∗ are derived from ⎪ ⎪ ⎪ ⎪ ⎨ solving the problem PE ⎧ ⎪ max E [πE (αE (θ ), βE (θ ), eE )] ⎪ ⎪ ⎪ ⎪ ⎪ (αE (·),βE (·),eE ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ subject to: ⎪ ⎪ ⎪ ⎪ PE : ⎪ MVE (x) ≥ MVE (x, y), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∀x, y ∈ [θ, θ ] (IC-E) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ MVE (x) ≥ 0, ∀x ∈ [θ, θ ]. (IR2-E) (11) The objective function of Model (11) is equal to the senior executive’s expected revenue generated from the new product minus the wage paid to supervisor E. The individual rationality constraints (IR1-E) and (IR2-E) ensure that supervisor E and the project manager are always guaranteed at least the reservation utility 0. The incentive compatibility constraint (IC-E) guarantees the project manager to report his true idea value. As the mentioned method under the scenario of monitoring idea information, we solve the problem by using backward induction approach from the second level in which supervisor E offers optimal wage contract αE∗ , βE∗ , eE∗ to induce the project manager to reveal the true idea value. The following lemma presents the analysis of supervisor E and the project manager’s optimal decisions in the second level. Lemma 4 Given the incentive contract (aE , bE ), the optimal response (αE∗ (·), βE∗ (·), e∗E ) made by supervisor E is given by βE∗ (x) = 0, 1 αE∗ (x) = v 2 bE2 , 2 e∗E = vbE . Having the information about the strategies αE∗ , βE∗ , e∗E of supervisor E, the senior executive would use it to maximize her own expected profit. The following proposition characterizes the optimal contracts offered by the senior executive and supervisor E under EM mechanism.
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Proposition EM mechanism, the optimal incentive 3 Under contract aE∗ , bE∗ offered by the senior executive are given by bE∗ = 1, 1 aE∗ = −E[θ ] − v 2 + CE . 2
Given the optimal incentive contract a E∗ , bE∗ , the cor- responding optimal menu of contracts αE∗ (·), βE∗ (·), eE∗ offered by supervisor E is given by βE∗ (x) = 0,
1 αE∗ (x) = −E[θ ] − v 2 , 2 eE∗ = v. Proposition 3 provides us with form of the optimal a closed incentive contract mechanism aE∗ , bE∗ . Note that the senior executive offers the same incentive term obtained in Proposition 2 to supervisor E. This suggests that when supervisor E is risk neutral, the senior executive should set the incentive term as high as possible to motivate supervisor E to monitor the project manager’s effort no matter how much the design idea value is. Comparing with Proposition 2, the term ρσ 2 does not exist in the fixed payment aE∗ , because supervisor E can monitor the project manager’s effort level. In other words, moral hazard is not an issue, since we do not have resource constraints here. As shown in Proposition 3, the optimal incentive term βE∗ (x) is equal to 0 for any x ∈ [θ, θ ] which is shown in Figure 1. In other words, supervisor E should propose a fixed payment contract (βE∗ (x) = 0) even if he can monitor the risk averse project manager’s effort. It is because that supervisor E worries that the project manager will take advantage of the idea value to shift risk, so he sets the incentive term to 0 to prevent such action. Comparing with Propositions 1 and 2, Proposition 3 also shows that the optimal effort level for the project manager is the greatest among these three sce∗ . This implies, from the senior narios; that is eE∗ ≥ eI∗ ≥ eN executives’s perspective, using the mechanism of monitoring effort has highest impact on improving the project manager’s optimal effort level. By Proposition 3, we compute the senior executive’s expected profit and the project manager’s utility in the following corollary. Corollary 3 Under EM mechanism, the senior executive’s expected profit is given by 1
E = E[θ ] + v 2 − CE , 2 supervisor E’s expected profit is given by πE = 0
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and the project manager’s utility is given by
VI VE
Similar to Corollary 2, Corollary 3 shows that the project manager can no longer obtain information rent under EM mechanism.
Values of monitoring In this section, we are in a position to examine the values of monitoring. In particular, we identify two values: the value of monitoring idea information and the value of monitoring effort. Specifically, we characterize the value of monitoring idea information and the value of monitoring effort as a function of one key variable, namely, the risk aversion level of the project manager ρ.
Value of monitoring
MVE (x) = 0.
ρ
0
Fig. 2 Impact of the risk aversion level of the project manager on the value of monitoring
project manager is so risk-averse that the senior executive is not willing to monitor the idea information.
Value of monitoring idea information Value of monitoring effort For the senior executive, we examine the value of monitoring idea information, which is defined to be the difference between the senior executive’s expected profit with and without monitoring the project manager’s idea information. The expression for the value of monitoring idea information is given by
2 + 2 v4 − E v − h(θ ) , (12) VI = I − N = 2(v 2 + ρσ 2 )
For the senior executive, we study the value of monitoring effort, which is defined to be the difference between the senior executive’s expected profit with and without monitoring the project manager’s effort. The expression for the value of monitoring effort is given by
2 + 2 E v − h(θ ) 1 , (13) VE = E − N = v 2 − 2 2(v 2 + ρσ 2 )
where N and I are given in Corollaries 1 and 2, respectively.
where N and E are given in Corollaries 1 and 3, respectively.
Proposition 4 The value of monitoring the project manager’s idea information satisfies the following conditions:
Proposition 5 The value of monitoring the project manager’s effort satisfies the following conditions:
(i) VI > 0, if σ 2 < v 4 −E
v 4 −E
v 2 −h(θ)
v 2 −h(θ)
+ 2
if σ 2 ≥ 2ρCI (ii) VI is decreasing in ρ.
+ 2
−2v 2 CI
2ρCI −2v 2 CI
and VI ≤ 0,
;
Proposition 4 shows that the value of monitoring idea information is positive when the revenue uncertainty (σ 2 ) is sufficiently low. In other words, when the outcome is stable, the senior executive can always benefit if she engages supervisor I to monitor the project manager’s idea information. On the other hand, monitoring the project manager’s idea information becomes less important for the senior executive. Just as shown in Proposition 4, the value of monitoring idea information decreases as the risk aversion level of the project manager (ρ) improves which is shown in Fig. 2. The intuition behind this finding is that, for a big value of ρ, the
(i) VE > 0, if σ 2
x. By differentiating MVN (x) with respect to x, we have dMVN (x) dαN (x) 1 = + x + (v 2 − ρσ 2 )βN (x) dx dx 2 dβN (x) + βN (x) = βN (x). × dx N (x) For part (iii), according to dMV ≥ 0, the participation dx constraint is equivalent to
(18)
In fact, the senior executive will set the wage for the lowest idea value as low as possible, so that MVN (θ ) = 0. That is the constraint (18) is binding under the optimal wage contract.
Proof of Lemma 2. Noting with the definition of MVN (x), we have E [ N ]
1 2 v + ρσ 2 βN2 (θ ) . = E θ − MVN (θ ) + v 2 βN (θ ) − 2 Let G(x) = x − MVN (x) + v 2 βN − 21 v 2 + ρσ 2 βN2 (x). By differentiating G(x) with respect to x,
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2
θ
when x > y; and
MVN (θ ) ≥ 0.
It follows from σ 2 ≤ vρ and 0 ≤ βN (x) ≤ 1 in Assumption that dG(x) dx ≥ 0. According to Theorem 3, the senior executive’s expected profit can be written as θ x − MVN (x) + v 2 βN E [ N ] =
According to Pontryagin maximum principle, the neces∂H sary conditions are: (i) ∂∂βH = 0, (ii) ∂MV = − ddλx and N N (x) λ(θ) = 0, (iii) ∂∂λH = dMV and MVN (θ) = 0. From the dx transversality condition, we know λ(x) = (x) − 1. The optimal βN∗ (x) follows directly from condition (i) v 2 − v 2 + ρσ 2 βN∗ (x) φ(x) + (x) − 1 = 0. By simple calculation and the assumption βN (x) ≥ 0, we can obtain + 2 v − h(x) ∗ . βN (x) = v 2 + ρσ 2 dβ ∗ (x) It is straightforward to verify that dNx ≥ 0, x ∈ [θ, θ ]. Following the determinate optimal incentive coeffi∗ (x) and the opticient βN∗ (x), the optimal fixed payment αN ∗ mal effort level eN for the project manager can be obtained immediately. Therefore, the proof of the proposition is complete.
Proof of Lemma 3. Substituting E [πI (αI (x), βI (x))] and MVI (x), supervisor I’s problem PI can be expressed as: ⎧ max aI − αI (x) + (bI − βI (x))(x + veI ) − CI ⎪ ⎪ (αI (x),β ⎪ I (x)) ⎪ ⎪ ⎪ ⎪ ⎪ subject to: ⎪ ⎨ eI = arg max αI (x) + βI (x)(x + v eˆI ) − 21 eˆI2 ⎪ e ˆ ≥0 I ⎪ ⎪ ⎪ ⎪ 1 2 β 2 (x) ⎪ − ρσ ⎪ I 2 ⎪ ⎪ ⎩ αI (x) + βI (x)(x + veI ) − 21 eI2 − 21 ρσ 2 βI2 (x) ≥ 0. (19)
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To solve Model (19), we employ the two-step optimization method, which is described as follows. Step 1: The project manager will choose effort eI to maximize his certainty equivalent, that is the same as maximizing 1 1 MVI (x) = αI (x) + βI (x)(x + veI ) − eI2 − ρσ 2 βI2 (x) 2 2 which is concave in eI . The maximum is completely characterized by the first-order condition vβI (x) − eI = 0, which implies eI∗ = vβI (x). Substituting eI into Model (19) yields ⎧ max aI −αI (x)+(bI −βI (x))(x +v 2 βI (x))−C I ⎪ ⎪ ⎨ (αI (x),βI (x)) (20) subject to: ⎪ ⎪ ⎩ 1 2 2 2 αI (x) + xβI (x) + 2 (v − ρσ )βI (x) ≥ 0. Step 2: At optimality, the IR condition is binding. If it were not, we could replace αI (x) with αI (x) < αI (x). Since the objective function is decreasing in αI (x), αI (x) improves the objective function. After substituting the fixed payment into the objective function of Model (20), Model (20) reduces to: 1 max πI = aI + bI x − (v 2 + ρσ 2 )βI2 (x) + bI v 2 βI (x) − C I βI (x) 2 By the first-order condition, we can obtain βI∗ (x) =
v2
bI v 2 . + ρσ 2
Following the determinate optimal incentive coefficient βI∗ (x), the optimal fixed payment αI∗ (x) and the optimal effort level eI∗ for the project manager can be obtained immediately. The proof of the proposition is complete.
Proof of Proposition 2. For the given (αI∗ , βI∗ ), Model (10) can be rewritten as ⎧ max I = −aI + (1 − bI )(E[θ ] + v 2 βI∗ (x)) ⎪ ⎪ ⎨ (aI ,bI ) (21) subject to: ⎪ ⎪ ⎩ aI −αI∗ (x)+(bI −βI∗ (x))(E[θ ]+v 2 βI∗ (x))−C I ≥ 0. At optimality, the IR condition is binding. If it were not, we could replace aI (x) with aI (x) < aI (x). Since the objective function is decreasing in aI (x), aI (x) improves the objective function. After substituting the fixed payment into the objective function of Model (21), the problem reduces to 2 1 2 bI v 2 2 max I = E[θ ] − (v + ρσ ) 2 bI 2 v + ρσ 2 +
bI v 4 − CI . v 2 + ρσ 2
By the first-order condition, we can obtain bI∗ = 1. Following the determinate optimal incentive coefficient bI∗ , the optimal fixed payment aI∗ can be obtained immediately. The proof of the proposition is complete.
Proof of Lemma 4. By the methods used in the proof of Lemmas 1 and 2, we can rewrite supervisor E’s problem PE as ⎧
θ ⎪ max E = θ aE − MVE (x) + xbE + 21 v 2 bE2 ⎪ ⎪ βE (·) ⎪ ⎪ ⎪ ⎪ ⎪ − 21 ρσ 2 βE2 (x) − C E φ(x)dx ⎪ ⎪ ⎪ ⎪ ⎨ subject to: (22) ⎪ dMVE (x) = β (x), ∀x ∈ [θ , θ ] ⎪ ⎪ E ⎪ dx ⎪ ⎪ ⎪ dβE (x) ≥ 0, ∀x ∈ [θ, θ ] ⎪ ⎪ ⎪ ⎪ dx ⎪ ⎩ MVE (θ ) = 0. We relax supervisor E’s problem by ignoring the constraint that dβdEx(x) ≥ 0. The Hamiltonian of supervisor E’s relaxed optimal control problem is 1 H (βE , MVE , λ, x) = aE − MVE + xbE + v 2 bE2 2 1 − ρσ 2 βE2 − C E φ(x) + λ(x)βE . 2 By Pontryagin maximum principle, the necessary condi∂H tions are: (i) ∂∂βH = 0, (ii) ∂MV = − ddλx and λ(θ) = 0, E E (x) (iii) ∂∂λH = dMV and MVE (θ ) = 0. Condition (ii) dx λ d implies dx = φ(x). Integration of this expression gives λ(x) = (x) − 1. The optimal βE∗ (x) follows directly from condition (i) −ρσ 2 βE∗ (x)φ(x) + (x) − 1 = 0. By simple calculation and the assumption βE (x) ≥ 0, we can obtain βE∗ (x) = 0. dβ ∗ (x) It is straightforward to verify that dEx ≥ 0, x ∈ [θ, θ ]. Following the determinate optimal incentive coefficient βE (x), the optimal fixed payment αE (x) for the project manager can be obtained immediately. The proof of the proposition is complete.
Proof of Proposition 3. For the given (αE∗ , βE∗ ), Model (11) can be written as ⎧ max E = −aE + (1 − bE )(E[θ ] + v 2 bE ) ⎪ ⎪ ⎨ (aE ,bE ) (23) subject to: ⎪ ⎪ ⎩ aE + E[θ ]bE + 21 v 2 bE2 − C E ≥ 0.
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At optimality, the IR condition is binding. If it were not, we could replace aE (x) with aE (x) < aE (x). Since the objective function is decreasing in aE (x), aE (x) improves the objective function. After substituting the fixed payment into the objective function of Model (23), the problem reduces to 1 max E = E[θ ] + v 2 bE − v 2 bE2 − C E , bE 2 By the first-order condition, we can obtain bE∗ = 1. Following the determinate optimal incentive coefficient bE∗ , the optimal fixed payment aE∗ can be obtained immediately. The proof of the proposition is complete.
Proof of Proposition 4. The results follow directly from Corollaries 1 and 2.
Proof of Proposition 5 The results follow directly from Corollaries 1 and 3.
Proof of Proposition 6. The results follow directly from Propositions 4 and 5.
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