Monitoring mechanisms in new product development with risk-averse project manager

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 level of the project manager improves, while the value of monitoring effort shows the opposite feature.

Appendices

Appendix A. Preliminaries on uncertainty theory

Let $\Gamma$ be a nonempty set, and $\mathcal {L}$ a $\sigma$-algebra over $\Gamma$. Each element $\Lambda$ in $\mathcal {L}$ is called an event. Liu (2007) defined an uncertain measure by the following axioms:

Axiom 1

(Normality Axiom) $\mathcal {M}\{\Gamma \}=1$ for the universal set $\Gamma$.

Axiom 2

(Duality Axiom) $\mathcal {M}\{\Lambda \}+\mathcal {M}\{\Lambda ^{c}\}=1$ for any event $\Lambda$.

Axiom 3

(Subadditivity Axiom) For every countable sequence of events $\Lambda _{1},\Lambda _{2},\cdots$, we have

$$\begin{aligned} \mathcal {M}\left\{ \bigcup _{i=1}^{\infty }\Lambda _{i} \right\} \le {\sum _{i=1}^{\infty }}\mathcal {M}\{\Lambda _{i}\}. \end{aligned}$$

The triplet $(\Gamma ,\mathcal {L},\mathcal {M})$ is called an uncertainty space. Furthermore, Liu (2009) defined a product uncertain measure by the fourth axiom:

Axiom 4

(Product Axiom) Let $(\Gamma _{k},\mathcal {L}_{k},\mathcal {M}_{k})$ be uncertainty space for $k=1,2,\cdots$. The product uncertain measure $\mathcal {M}$ is an uncertain measure satisfying

$$\begin{aligned} \mathcal {M}\left\{ \prod _{k=1}^{\infty }\Lambda _{i} \right\} = {\bigwedge _{k=1}^{\infty }}\mathcal {M}\{\Lambda _{k}\} \end{aligned}$$

where $\Lambda _{k}$ are arbitrarily chosen events from $\mathcal {L}_{k}$ for $k=1,2,\cdots$, respectively.

Definition 1

(Liu 2007) An uncertain variable is a measurable function $\xi$ from an uncertainty space $(\Gamma ,\mathcal {L},\mathcal {M})$ to the set of real numbers, i.e., for any Borel set $\mathcal {B}$ of real numbers, the set

$$\begin{aligned}\{\xi \in \mathcal {B}\}=\{\gamma \in \Gamma |\xi (\gamma )\in \mathcal {B}\} \end{aligned}$$

is an event.

Definition 2

(Liu 2007) The uncertainty distribution $\Phi$ of an uncertain variable $\xi$ is defined by

$$\begin{aligned} \Phi (x)=\mathcal {M}\{\xi \le x\} \end{aligned}$$

for any real number $x$.

Definition 3

(Liu 2010c) An uncertainty distribution $\mathrm{\Phi (x)}$ is said to be regular if it is a continuous and strictly increasing function with respect to $x$ at which $0<\Phi (x)<1$, and

$$\begin{aligned} \lim _{x\rightarrow -\infty }\Phi (x)=0,\ \ \lim _{x\rightarrow +\infty }\Phi (x)=1. \end{aligned}$$

Definition 4

(Liu 2010c) Let $\xi$ be an uncertain variable with regular uncertainty distribution $\Phi$. Then the inverse function $\Phi ^{-1}(\alpha )$ is called the inverse uncertainty distribution of $\xi$.

Definition 5

(Liu 2007) Let $\xi$ be an uncertain variable. Then the expected value of $\xi$ is defined by

$$\begin{aligned} E[\xi ]=\int _{0}^{+\infty }\mathcal {M}\{\xi \ge x\}\text{ d }x-\int _{-\infty }^{0}\mathcal {M}\{\xi \le x\}\text{ d }x \end{aligned}$$

provided that at least one of the two integrals is finite. If $\xi$ has an uncertainty distribution $\Phi$, then the expected value may be calculated by

$$\begin{aligned} E[\xi ]=\int _{0}^{+\infty }(1-\Phi (x))\text{ d }x-\int _{-\infty }^{0}\Phi (x)\text{ d }x \end{aligned}$$

or equivalently,

$$\begin{aligned} E[\xi ]=\int _{-\infty }^{+\infty }x\text{ d }\Phi (x). \end{aligned}$$

If $\Phi$ is also regular, then

$$\begin{aligned} E[\xi ]=\int _{0}^{1}\Phi ^{-1}(\alpha )\text{ d }\alpha . \end{aligned}$$

Definition 6

(Liu 2007) Let $\xi$ be an uncertain variable with finite expected value $e$. Then the variance of $\xi$ is

$$\begin{aligned} V[\xi ]=E[(\xi -e)^{2}]. \end{aligned}$$

This definition tells us that the variance is just the expected value of $(\xi -e)^{2}$. Since $(\xi -e)^{2}$ is a nonnegative uncertain variable, we also have

$$\begin{aligned} V[\xi ]=\int _{0}^{+\infty }\mathcal {M}\{(\xi -e)^{2}\ge x\}\text{ d }x. \end{aligned}$$

Definition 7

(Liu 2009) The uncertain variables $\xi _{1},\xi _{2},\ldots ,\xi _{m}$ are said to be independent if

$$\begin{aligned} \mathcal {M}\left\{ \bigcap _{i=1}^{m}(\xi _{i}\in \mathcal {B}_{i}) \right\} = {\bigwedge _{i=1}^{m}}\mathcal {M}\{\xi _{i}\in \mathcal {B}_{i}\}, \end{aligned}$$

for any Borel sets $\mathcal {B}_{1},\mathcal {B}_{2},\ldots ,\mathcal {B}_{m}$ of real numbers. More generally, the independence of uncertain vectors was given by Liu (2013).

Theorem 1

(Liu 2010c) Let $\xi _{1},\xi _{2},\ldots ,\xi _{n}$ be independent uncertain variables with regular uncertainty distributions $\Phi _{1},\Phi _{2},\ldots ,\Phi _{n}$ respectively. If the function $f(x_{1},x_{2},\ldots ,x_{n})$ is strictly increasing with respect to $x_{1},x_{2},\ldots ,x_{m}$ and strictly decreasing with respect to $x_{m+1},x_{m+2},\ldots ,x_{n}$, then

$$\begin{aligned} \xi =f(\xi _{1},\xi _{2},\ldots ,\xi _{n}) \end{aligned}$$

is an uncertain variable with an inverse uncertainty distribution

$$\begin{aligned}&\Phi ^{-1}(\alpha )=f(\Phi _{1}^{-1}(\alpha ),\ldots ,\Phi _{m}^{-1}(\alpha ),\\&\quad \Phi _{m+1}^{-1}(1-\alpha ),\ldots ,\Phi _{n}^{-1}(1-\alpha )). \end{aligned}$$

Theorem 2

(Liu 2007) If $\xi$ is an uncertain variable with finite expected value, $a$ and $b$ are real numbers, then

$$\begin{aligned} V[a\xi +b]=a^{2}V[\xi ]. \end{aligned}$$

Theorem 3

(Liu and Ha 2010) Let $\xi$ be an uncertain variable with regular uncertainty distribution $\Phi$, and let $g(x)$ be a strictly monotone (increasing or decreasing) function, then the expected value of $g(\xi )$ is

$$\begin{aligned} E[g(\xi )]={\int _{-\infty }^{+\infty }}g(x)\text{ d }\Phi (x). \end{aligned}$$

Appendix B. Proofs of propositions and lemmas

Proof of Lemma 1

For parts (i) and (ii), by Theorems 1 and 2, we firstly simplify $\mathrm MV _\mathrm{N }(x,y)$, which denotes the project manager with idea value $x$ but choosing the pair $(\alpha _\mathrm{N }(y),\beta _\mathrm{N }(y)$, where $x,y\in [\underline{\theta },\overline{\theta }]$ and $x\ne y$,

$$\begin{aligned} \mathrm MV _\mathrm{N }(x,y)\!&= \max \limits _{\hat{e}_\mathrm{N }} \Bigg \{\alpha _\mathrm{N }(y)+\beta _\mathrm{N }(y) (x+v\hat{e}_\mathrm{N })\nonumber \\&\quad -\frac{1}{2}\hat{e}_\mathrm{N }^{2} -\frac{1}{2}\rho \sigma ^{2}\beta _\mathrm{N }^{2}(y)\Bigg \}\nonumber \\&= \alpha _\mathrm{N }(y)+x\beta _\mathrm{N }(y) +\frac{1}{2}(v^{2}-\rho \sigma ^{2})\beta _\mathrm{N }^{2}(y), \end{aligned}$$

where the second equality follows from the first-order condition for the optimal development effort: $e^{*}_\mathrm{N }=v\beta _\mathrm{N }(y)$.

Thus, for any given $x$, the incentive compatibility constraint can be written as

$$\begin{aligned} \begin{array}{ll} \mathrm MV _\mathrm{N }(x)\ge \mathrm MV (x,y), \forall \ y\in [\underline{\theta },\overline{\theta }], \end{array} \end{aligned}$$

which means that $\mathrm MV _\mathrm{N }(x,y)$ obtain its maximal value at $(x,x)$, i.e., a project manager with idea value $x$ has no incentive to pretend to be a that with design idea value $y$, $x\ne y$. Thus, $\mathrm MV _\mathrm{N }(x,y)$ satisfies the first-order condition $\frac{\partial \mathrm MV _\mathrm{N }(x,y)}{\partial y}|_{y=x}=0$ and the second-order condition $\frac{\partial ^{2} \mathrm MV _\mathrm{N }(x,y)}{\partial y^{2}}|_{y=x}<0$. It follows from the first-order condition that

$$\begin{aligned}&\frac{\text{ d }\alpha _\mathrm{N }(x)}{\text{ d }x}+\left( x+\frac{1}{2}(v^{2}-\rho \sigma ^{2})\beta _\mathrm{N }(x)\right) \frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x} =0,\nonumber \\&\quad \forall \ x\in [\underline{\theta },\overline{\theta }]. \end{aligned}$$

(14)

Differentiating (14) with respect to $x$ yields

$$\begin{aligned}&\frac{\text{ d }^{2} \alpha _\mathrm{N }(x)}{\text{ d }x^{2}}+\frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x}+\frac{1}{2}(v^{2}-\rho \sigma ^{2})\left( \frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x}\right) ^{2}\nonumber \\&\quad +\left( x+\frac{1}{2}(v^{2}-\rho \sigma ^{2})\beta _\mathrm{N }(x)\right) \frac{\text{ d }^{2} \beta _\mathrm{N }(x)}{\text{ d }x^{2}}=0, \forall \ x\in [\underline{\theta },\overline{\theta }]. \end{aligned}$$

(15)

It follows from the second-order condition that

$$\begin{aligned}&\frac{\text{ d }^{2} \alpha _\mathrm{N }(x)}{\text{ d }x^{2}}+\frac{1}{2}(v^{2}-\rho \sigma ^{2})\left( \frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x}\right) ^{2}\nonumber \\&\quad +\left( x+\frac{1}{2}(v^{2}-\rho \sigma ^{2})\beta _\mathrm{N }(x)\right) \frac{\text{ d }^{2} \beta _\mathrm{N }(x)}{\text{ d }x^{2}}\le 0, \forall \ x\in [\underline{\theta },\overline{\theta }]. \end{aligned}$$

(16)

Applying (15) to (16) yields

$$\begin{aligned} \frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x}\ge 0, \forall \ x\in [\underline{\theta },\overline{\theta }]. \end{aligned}$$

(17)

On the other hand, by $\frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x}\ge 0$ and integrating (14) yields

$$\begin{aligned}&\alpha _\mathrm{N }(x)-\alpha _\mathrm{N }(y)\\&\quad =- {\int _{y}^{x}}\left( s+\frac{1}{2}(v^{2}- \rho \sigma ^{2})\beta _\mathrm{N }(s)\right) \frac{\text{ d }\beta _\mathrm{N }(s)}{\text{ d }s}\text{ d }s\\&\quad \ge {\int _{x}^{y}}\left( x+\frac{1}{2}(v^{2}- \rho \sigma ^{2})\beta _\mathrm{N }(s)\right) \frac{\text{ d }\beta _\mathrm{N }(s)}{\text{ d }s}\text{ d }s\\&\quad =x\beta _\mathrm{N }(y)+\frac{1}{2}(v^{2}- \rho \sigma ^{2})\beta _\mathrm{N }^{2}(y)-x \beta _\mathrm{N }(x)\\&\qquad -\frac{1}{2}(v^{2}-\rho \sigma ^{2})\beta _\mathrm{N }^{2}(x) \end{aligned}$$

when $x>y$; and

$$\begin{aligned}&\alpha _\mathrm{N }(y)-\alpha _\mathrm{N }(x)\\&\quad =- {\int _{x}^{y}}\left( s+\frac{1}{2}(v^{2}-\rho \sigma ^{2}) \beta _\mathrm{N }(s)\right) \frac{\text{ d }\beta _\mathrm{N }(s)}{\text{ d }s}\text{ d }s\\&\quad \le {\int _{y}^{x}}\left( s+\frac{1}{2}(v^{2}-\rho \sigma ^{2}) \beta _\mathrm{N }(s)\right) \frac{\text{ d }\beta _\mathrm{N }(s)}{\text{ d }s}\text{ d }s\\&\quad =x\beta _\mathrm{N }(x)+\frac{1}{2}(v^{2}-\rho \sigma ^{2}) \beta _\mathrm{N }^{2}(x)\\&\qquad -\,x\beta _\mathrm{N }(y)-\frac{1}{2}(v^{2} -\rho \sigma ^{2})\beta _\mathrm{N }^{2}(y) \end{aligned}$$

when $y>x$.

By differentiating $\mathrm MV _\mathrm{N }(x)$ with respect to $x$, we have

$$\begin{aligned} \frac{\text{ d }\mathrm MV _\mathrm{N }(x)}{\text{ d }x}&= \frac{\text{ d }\alpha _\mathrm{N }(x)}{\text{ d }x}+\left( x+\frac{1}{2}(v^{2}-\rho \sigma ^{2})\beta _\mathrm{N }(x)\right) \\&\quad \times \frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x} +\beta _\mathrm{N }(x)=\beta _\mathrm{N }(x). \end{aligned}$$

For part (iii), according to $\frac{\text{ d }\mathrm MV _\mathrm{N }(x)}{\text{ d }x}\ge 0$, the participation constraint is equivalent to

$$\begin{aligned} \mathrm MV _\mathrm{N }(\underline{\theta })\ge 0. \end{aligned}$$

(18)

In fact, the senior executive will set the wage for the lowest idea value as low as possible, so that $\mathrm MV _\mathrm{N }(\underline{\theta })=0$. That is the constraint (18) is binding under the optimal wage contract. $\square$

Proof of Lemma 2

Noting with the definition of $\mathrm MV _\mathrm{N }(x)$, we have

$$\begin{aligned}&E\left[ \Pi _{\mathrm{N}}\right] \\&\quad =E\left[ \theta -\mathrm{MV}_{\mathrm{N}}(\theta )+v^{2}\beta _{\mathrm{N}}(\theta )-\frac{1}{2}\left( v^{2}+\rho \sigma ^{2}\right) \beta _{\mathrm{N}}^{2}(\theta )\right] . \end{aligned}$$

Let $G(x)=x-\mathrm{MV}_{\mathrm{N}}(x)+v^{2}\beta _{\mathrm{N}}-\frac{1}{2}\left( v^{2}+\rho \sigma ^{2}\right) \beta _{\mathrm{N}}^{2}(x)$. By differentiating $G(x)$ with respect to $x$,

$$\begin{aligned} \frac{\text{ d }G(x)}{\text{ d }x}=1-\beta _\mathrm{N }(x)+\left( v^{2}-\frac{1}{2}(v^{2} +\rho \sigma ^{2})\beta _\mathrm{N }(x)\right) \frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x}. \end{aligned}$$

It follows from $\sigma ^{2}\le \frac{v^{2}}{\rho }$ and $0\le \beta _\mathrm{N }(x)\le 1$ in Assumption that $\frac{\text{ d }G(x)}{\text{ d }x}\ge 0$. According to Theorem 3, the senior executive’s expected profit can be written as

$$\begin{aligned} E\left[ \Pi _{\mathrm{N}}\right]&= \int _{\underline{\theta }}^{\overline{\theta }}\left( x-\mathrm{MV}_{\mathrm{N}}(x)+v^{2}\beta _{\mathrm{N}}\right. \nonumber \\&\quad \left. -\,\frac{1}{2}\left( v^{2}+\rho \sigma ^{2}\right) \beta _{\mathrm{N}}^{2}(x)\right) \phi (x)\text{ d }x. \end{aligned}$$

$\square$

Proof of Proposition  1

We relax the senior executive’s problem by ignoring the constraint that $\frac{\text{ d }\beta _\mathrm{N }(x)}{\text{ d }x}\ge 0$. The Hamiltonian of the senior executive’s relaxed optimal control problem is

$$\begin{aligned}&H\left( \beta _\mathrm{N },\mathrm MV _\mathrm{N },\lambda ,x\right) \\&\quad =\left( x-\mathrm{MV}_{\mathrm{N}}+v^{2}\beta _{\mathrm{N}}-\frac{1}{2}\left( v^{2}+\rho \sigma ^{2}\right) \beta _{\mathrm{N}}^{2}\right) \phi (x)+\lambda \beta _\mathrm{N }. \end{aligned}$$

According to Pontryagin maximum principle, the necessary conditions are: (i) $\frac{\partial H}{\partial \beta }=0$, (ii) $\frac{\partial H}{\partial \mathrm MV _\mathrm{N }}=-\frac{\text{ d }\lambda }{\text{ d }x}$ and $\lambda (\overline{\theta })=0$, (iii) $\frac{\partial H}{\partial \lambda }=\frac{\text{ d }\mathrm MV _\mathrm{N }(x)}{\text{ d }x}$ and $\mathrm MV _\mathrm{N }(\underline{\theta })=0$. From the transversality condition, we know $\lambda (x)=\Phi (x)-1$. The optimal $\beta _\mathrm{N }^{*}(x)$ follows directly from condition (i)

$$\begin{aligned} \left( v^{2}-\left( v^{2}+\rho \sigma ^{2}\right) \beta _{\mathrm{N}}^{*}(x)\right) \phi (x)+\Phi (x)-1=0. \end{aligned}$$

By simple calculation and the assumption $\beta _{\mathrm{N}}(x)\ge 0$, we can obtain

$$\begin{aligned} \beta _{\mathrm{N}}^{*}(x)=\frac{\left( v^{2}-h(x)\right) ^{+}}{v^{2}+\rho \sigma ^{2}}. \end{aligned}$$

It is straightforward to verify that $\frac{\text{ d }\beta _{\mathrm{N}}^{*}(x)}{\text{ d }x}\ge 0$, $x\in [\underline{\theta },\overline{\theta }]$. Following the determinate optimal incentive coefficient $\beta _{\mathrm{N}}^{*}(x)$, the optimal fixed payment $\alpha _{\mathrm{N}}^{*}(x)$ and the optimal effort level $e^{*}_{\mathrm{N}}$ for the project manager can be obtained immediately. Therefore, the proof of the proposition is complete. $\square$

Proof of Lemma 3

Substituting $E\left[ \pi _\mathrm{I }(\alpha _\mathrm{I }(x),\beta _\mathrm{I }(x))\right]$ and $\mathrm MV _\mathrm{I }(x)$, supervisor I’s problem $P_{I}$ can be expressed as:

$$\begin{aligned} \left\{ \begin{array}{ll} \max \limits _{\left( \alpha _\mathrm{I }(x), \beta _\mathrm{I }(x)\right) }a_\mathrm{I }- \alpha _\mathrm{I }(x)+(b_\mathrm{I }-\beta _\mathrm{I }(x)) (x+ve_\mathrm{I })-C_\mathrm{I }\\ \text{ subject } \text{ to: }\\ \quad \quad e_\mathrm{I }=\arg \max \limits _{\hat{e}_\mathrm{I }\ge 0} \big (\alpha _\mathrm{I }(x)+\beta _\mathrm{I }(x) (x+v\hat{e}_\mathrm{I })-\frac{1}{2}\hat{e}_\mathrm{I }^{2}\\ \qquad \quad -\frac{1}{2}\rho \sigma ^{2}\beta _\mathrm{I }^{2}(x)\big )\\ \quad \quad \alpha _\mathrm{I }(x)+\beta _\mathrm{I }(x)(x+ve_\mathrm{I }) -\frac{1}{2}e_\mathrm{I }^{2}-\frac{1}{2}\rho \sigma ^{2}\beta _\mathrm{I }^{2}(x)\ge 0. \end{array} \right. \end{aligned}$$

(19)

To solve Model (19), we employ the two-step optimization method, which is described as follows.

Step 1: The project manager will choose effort $e_\mathrm{I }$ to maximize his certainty equivalent, that is the same as maximizing

$$\begin{aligned} \mathrm MV _\mathrm{I }(x)=\alpha _\mathrm{I }(x) +\beta _\mathrm{I }(x)(x+ve_\mathrm{I }) -\frac{1}{2}e_\mathrm{I }^{2}-\frac{1}{2}\rho \sigma ^{2}\beta _\mathrm{I }^{2}(x) \end{aligned}$$

which is concave in $e_\mathrm{I }$. The maximum is completely characterized by the first-order condition

$$\begin{aligned} v\beta _\mathrm{I }(x)-e_\mathrm{I }=0, \end{aligned}$$

which implies

$$\begin{aligned} e^{*}_\mathrm{I }=v\beta _\mathrm{I }(x). \end{aligned}$$

Substituting $e_\mathrm{I }$ into Model (19) yields

$$\begin{aligned} \left\{ \begin{array}{l} \max \limits _{\left( \alpha _\mathrm{I }(x), \beta _\mathrm{I }(x)\right) }a_\mathrm{I } -\alpha _\mathrm{I }(x)+(b_\mathrm{I }- \beta _\mathrm{I }(x))(x+v^{2}\beta _\mathrm{I }(x))-C_{I}\\ \text{ subject } \text{ to: }\\ \quad \quad \alpha _\mathrm{I }(x)+x\beta _\mathrm{I }(x)+ \frac{1}{2}(v^{2}-\rho \sigma ^{2})\beta _\mathrm{I }^{2}(x)\ge 0. \end{array} \right. \end{aligned}$$

(20)

Step 2: At optimality, the IR condition is binding. If it were not, we could replace $\alpha _\mathrm{I }(x)$ with $\alpha _\mathrm{I }'(x)<\alpha _\mathrm{I }(x)$. Since the objective function is decreasing in $\alpha _\mathrm{I }(x)$, $\alpha _\mathrm{I }'(x)$ improves the objective function. After substituting the fixed payment into the objective function of Model (20), Model (20) reduces to:

$$\begin{aligned} \max \limits _{\beta _\mathrm{I }(x)} \pi _\mathrm{I }=a_\mathrm{I }+b_\mathrm{I }x- \frac{1}{2}(v^{2}+\rho \sigma ^{2})\beta _\mathrm{I }^{2}(x)+ b_\mathrm{I }v^{2}\beta _\mathrm{I }(x)-C_{I} \end{aligned}$$

By the first-order condition, we can obtain

$$\begin{aligned} \beta ^{*}_\mathrm{I }(x)=\frac{b_\mathrm{I }v^{2}}{v^{2}+\rho \sigma ^{2}}. \end{aligned}$$

Following the determinate optimal incentive coefficient $\beta ^{*}_\mathrm{I }(x)$, the optimal fixed payment $\alpha ^{*}_\mathrm{I }(x)$ and the optimal effort level $e^{*}_\mathrm{I }$ for the project manager can be obtained immediately. The proof of the proposition is complete. $\square$

Proof of Proposition 2

For the given $(\alpha ^{*}_\mathrm{I },\beta ^{*}_\mathrm{I })$, Model (10) can be rewritten as

$$\begin{aligned} \left\{ \begin{array}{l} \max \limits _{\left( a_\mathrm{I },b_\mathrm{I }\right) } \Pi _\mathrm{I }=-a_\mathrm{I }+(1-b_\mathrm{I }) (E[\theta ]+v^{2}\beta ^{*}_\mathrm{I }(x))\\ \text{ subject } \text{ to: }\\ \quad \quad a_\mathrm{I }-\alpha ^{*}_\mathrm{I }(x)+(b_\mathrm{I } -\beta ^{*}_\mathrm{I }(x))(E[\theta ]+v^{2}\beta ^{*}_\mathrm{I }(x))-C_{I}\ge 0. \end{array} \right. \end{aligned}$$

(21)

At optimality, the IR condition is binding. If it were not, we could replace $a_\mathrm{I }(x)$ with $a_\mathrm{I }'(x)<a_\mathrm{I }(x)$. Since the objective function is decreasing in $a_\mathrm{I }(x)$, $a_\mathrm{I }'(x)$ improves the objective function. After substituting the fixed payment into the objective function of Model (21), the problem reduces to

$$\begin{aligned} \max _{b_\mathrm{I }}\ \Pi _\mathrm{I }&= E[\theta ] -\frac{1}{2}(v^{2}+\rho \sigma ^{2})\left( \frac{b_\mathrm{I }v^{2}}{v^{2} +\rho \sigma ^{2}}\right) ^{2}\nonumber \\&\quad +\,\frac{b_\mathrm{I }v^{4}}{v^{2}+\rho \sigma ^{2}}-C_{I}. \end{aligned}$$

By the first-order condition, we can obtain

$$\begin{aligned} b^{*}_\mathrm{I }=1. \end{aligned}$$

Following the determinate optimal incentive coefficient $b^{*}_\mathrm{I }$, the optimal fixed payment $a^{*}_\mathrm{I }$ can be obtained immediately. The proof of the proposition is complete. $\square$

Proof of Lemma 4

By the methods used in the proof of Lemmas 1 and 2, we can rewrite supervisor E’s problem $P_{E}$ as

$$\begin{aligned} \left\{ \begin{array}{l} \max \limits _{\beta _\mathrm{E }(\cdot )} \Pi _\mathrm{E }=\int _{\underline{\theta }}^{\overline{\theta }} \big (a_\mathrm{E }-\mathrm MV _\mathrm{E }(x)+xb_\mathrm{E } +\frac{1}{2}v^{2}b^{2}_\mathrm{E }\\ \quad -\frac{1}{2}\rho \sigma ^{2}\beta _\mathrm{E }^{2}(x)-C_{E}\big )\phi (x)\text{ d }x\\ \text{ subject } \text{ to: }\\ \quad \quad {\frac{\text{ d }\mathrm{MV}_{\mathrm{E}}(x)}{\text{ d }x}=\beta _{\mathrm{E}}(x)},\ \forall x\in [\underline{\theta },\overline{\theta }]\\ \quad \quad {\frac{\text{ d }\beta _\mathrm{E }(x)}{\text{ d }x}\ge 0}, \ \forall x\in [\underline{\theta },\overline{\theta }]\\ \quad \quad \mathrm{MV}_{\mathrm{E}}(\underline{\theta })=0. \end{array} \right. \end{aligned}$$

(22)

We relax supervisor E’s problem by ignoring the constraint that $\frac{\text{ d }\beta _\mathrm{E }(x)}{\text{ d }x}\ge 0$. The Hamiltonian of supervisor E’s relaxed optimal control problem is

$$\begin{aligned} H\left( \beta _\mathrm{E },\mathrm MV _\mathrm{E }, \lambda ,x\right)&= \left( a_\mathrm{E }-\mathrm MV _\mathrm{E } +xb_\mathrm{E }+\frac{1}{2}v^{2}b^{2}_\mathrm{E }\right. \\&\quad \left. -\,\frac{1}{2}\rho \sigma ^{2}\beta _\mathrm{E }^{2}-C_{E}\right) \phi (x)+\lambda (x)\beta _\mathrm{E }. \end{aligned}$$

By Pontryagin maximum principle, the necessary conditions are: (i) $\frac{\partial H}{\partial \beta }=0$, (ii) $\frac{\partial H}{\partial \mathrm MV _\mathrm{E }}=-\frac{\text{ d }\lambda }{\text{ d }x}$ and $\lambda (\overline{\theta })=0$, (iii) $\frac{\partial H}{\partial \lambda }=\frac{\text{ d }\mathrm MV _\mathrm{E }(x)}{\text{ d }x}$ and $\mathrm MV _\mathrm{E }(\underline{\theta })=0$. Condition (ii) implies $\frac{\text{ d }\lambda }{\text{ d }x}=\phi (x)$. Integration of this expression gives $\lambda (x)=\Phi (x)-1$. The optimal $\beta _\mathrm{E }^{*}(x)$ follows directly from condition (i)

$$\begin{aligned} -\rho \sigma ^{2}\beta _{\mathrm{E}}^{*}(x)\phi (x)+\Phi (x)-1=0. \end{aligned}$$

By simple calculation and the assumption $\beta _{\mathrm{E}}(x)\ge 0$, we can obtain

$$\begin{aligned} \beta _{\mathrm{E}}^{*}(x)=0. \end{aligned}$$

It is straightforward to verify that $\frac{\text{ d }\beta _{\mathrm{E}}^{*}(x)}{\text{ d }x}\ge 0$, $x\in [\underline{\theta },\overline{\theta }]$. Following the determinate optimal incentive coefficient $\beta _\mathrm{E }(x)$, the optimal fixed payment $\alpha _\mathrm{E }(x)$ for the project manager can be obtained immediately. The proof of the proposition is complete. $\square$

Proof of Proposition  3

For the given $(\alpha ^{*}_\mathrm{E },\beta ^{*}_\mathrm{E })$, Model (11) can be written as

$$\begin{aligned} \left\{ \begin{array}{ll} \max \limits _{\left( a_\mathrm{E },b_\mathrm{E }\right) } \Pi _\mathrm{E }=-a_\mathrm{E }+(1-b_\mathrm{E })(E[\theta ]+v^{2}b_\mathrm{E })\\ \text{ subject } \text{ to: }\\ \quad \quad a_\mathrm{E }+E[\theta ]b_\mathrm{E }+\frac{1}{2}v^{2}b_\mathrm{E }^{2}-C_{E}\ge 0. \end{array} \right. \end{aligned}$$

(23)

At optimality, the IR condition is binding. If it were not, we could replace $a_\mathrm{E }(x)$ with $a_\mathrm{E }'(x)<a_\mathrm{E }(x)$. Since the objective function is decreasing in $a_\mathrm{E }(x)$, $a_\mathrm{E }'(x)$ improves the objective function. After substituting the fixed payment into the objective function of Model (23), the problem reduces to

$$\begin{aligned} \max _{b_\mathrm{E }}\ \Pi _\mathrm{E }=E[\theta ] +v^{2}b_\mathrm{E }-\frac{1}{2}v^{2}b_\mathrm{E }^{2}-C_{E}, \end{aligned}$$

By the first-order condition, we can obtain

$$\begin{aligned} b^{*}_\mathrm{E }=1. \end{aligned}$$

Following the determinate optimal incentive coefficient $b^{*}_\mathrm{E }$, the optimal fixed payment $a^{*}_\mathrm{E }$ can be obtained immediately. The proof of the proposition is complete. $\square$

Proof of Proposition  4

The results follow directly from Corollaries 1 and 2. $\square$

Proof of Proposition  5

The results follow directly from Corollaries 1 and 3. $\square$

Proof of Proposition  6

The results follow directly from Propositions 4 and 5.