The impacts of private risk aversion magnitude and moral hazard in R&D project under uncertain environment

Abstract

The R&D project manager tends to misreport risk aversion magnitude and shirk under uncertain environment for acquiring information rent and risk premium, which brings a significant challenge for the firm when designing compensation contracts. We consider an agency problem where a firm employs a manager who has private information about his risk aversion magnitude and unobservable efforts to implement a R&D project through a menu of incentive contracts. Both the subjective assessments about the risk aversion degree and the project variability are characterized as uncertain variables. Within the framework of uncertainty theory and principal-agent theory, we investigate the impacts of information asymmetry on the optimal compensation contracts and the firm’s profits under four information structures. We demonstrate that, counterintuitive as it sounds, the manager’s optimal contract under full information is the same as that under pure adverse selection. Nevertheless, compared to the case under full information, the firm should distort the commission rate upwards under pure moral hazard and dual asymmetric information. We also show that when the manager’s efforts are observable, hidden information about the risk aversion magnitude has no effect on the firm’s profit. However, when unobservable, private risk aversion degree always brings about information rent and induces a loss for the firm’s profit. Finally, our study provides managerial recommendations on mitigating the adverse impacts caused by asymmetric information.

Appendix

Proof of Proposition 1

Noting that because the project manager’s expected profit is decreasing in the fixed payment, the individual rationality constraint for the manager must bind at optimality. Thus, we can substitute it into the objective function of Model (8) and obtain the firm’s expected profit:

$$\begin{aligned} (r_1ve_1-r_2d_0+r_2d_1e_2)-\frac{1}{2}(e_1^2+e_2^2)-\frac{1}{2}xw_1^2(x)\sigma ^2, \end{aligned}$$

which is decreasing in $w_1$ and concave in $e_{1}$ and $e_{2}$. Thus, the firm would set both $w_1$ to zero for getting a maximum profit. Besides, we can yield the manager’s optimal effort levels $e_1=r_1v$ and $e_2=r_2d_1$ by using the first-order condition. Following the determinate individual rationality constraint which is binding, the corresponding optimal fixed payment $w_0$ can be derived immediately. The proof of the proposition is complete. $\square$

Proof of Proposition 2

Based on the incentive compatibility constraint for moral hazard, the manager will choose his optimal efforts in the research and development stages to maximize his own utility:

$$\begin{aligned}&w_0(x)+ w_1(x)(r_1ve_1-r_2d_0+r_2d_1e_2)-\frac{1}{2}(e_1^2+e_2^2)\\&\quad -\,\frac{1}{2}xw_1^2(x)\sigma ^2, \end{aligned}$$

which is concave in $e_1$ and $e_2$. We can derive the optimal effort level $e_{1}^*=r_1v w_1$ and $e_{2}^*=r_2d_1 w_1$ by the first-order condition. Furthermore, as the individual rationality constraint should be binding at optimality, by substituting $e_{1}^*$ and $e_{2}^*$ into the individual rationality constraint and then substituting the fixed wage ($w_0$) and the effort levels ($e_{1}^*$ and $e_{2}^*$) into the objective function, the firm’s expected profit can be rewritten as

$$\begin{aligned}&-\frac{1}{2}\left[ (r_1v)^2+(r_2d_1)^2+x\sigma ^2\right] w_1^2(x)\\&\quad +\left[ (r_1v)^2+(r_2d_1)^2\right] w_1(x)-r_2d_0. \end{aligned}$$

By the first-order condition regarding to $w_1$, we can obtain $w_1(x)=\frac{(r_1v)^2+(r_2d_1)^2}{(r_1v)^2+(r_2d_1)^2+x\sigma ^2}$. Based on the binding individual rationality constraint, the optimal fixed payment $w_0$ can be obtained immediately. The proof of the proposition is complete.$\square$

Proof of Lemma 1

The incentive compatibility constraint for adverse selection can be written as

$$\begin{aligned} \mathrm{CE}(x,x)\geqslant \mathrm{CE}(x,y),\quad \forall x,y \in [0, 1], \end{aligned}$$

which means that $\mathrm{CE}(x,y)$ obtains its maximal value at $\mathrm{CE}(x,x)$, i.e., the manager can obtain his maximal profit $\mathrm{CE}(x,y)$ if and only if $x=y$. Thus, $\mathrm{CE}(x,y)$ satisfies the first-order condition (i.e., local incentive compatibility constraint) $\frac{\partial \mathrm{CE}(x,y)}{\partial y}\bigm |_{y=x}=0$ and the second-order condition $\frac{\partial ^{2} \mathrm{CE}(x,y)}{\partial y^{2}}\bigm |_{y=x}\leqslant 0$. It follows from the first-order condition that

$$\begin{aligned} \frac{\mathrm{d}w_0(x)}{\mathrm{d}x}-\left[ r_1ve_1-r_2d_0+r_2d_1e_2-xw_1(x)\sigma ^2\right] \frac{\mathrm{d} w_1(x)}{\mathrm{d} x}=0.\nonumber \\ \end{aligned}$$

(11)

Differentiating the first-order condition (11) with respect to x yields

$$\begin{aligned}&\frac{\mathrm{d}^2w_0(x)}{\mathrm{d}x^2}-[r_1ve_1-r_2d_0+r_2d_1e_2-x\sigma ^2w_1(x)]\frac{\mathrm{d}^2 w_1(x)}{\mathrm{d} x^2}\nonumber \\&\quad -\,x\sigma ^2\left( \frac{\mathrm{d}w_0(x)}{\mathrm{d}x}\right) ^2-\sigma ^2w_1(x)\frac{\mathrm{d} w_1(x)}{\mathrm{d} x}=0. \end{aligned}$$

(12)

It follows from the second-order condition that

$$\begin{aligned}&\frac{\mathrm{d}^2w_0(x)}{\mathrm{d}x^2}-[r_1ve_1-r_2d_0+r_2d_1e_2-x\sigma ^2w_1(x)]\frac{\mathrm{d}^2 w_1(x)}{\mathrm{d} x^2}\nonumber \\&\quad -\,x\sigma ^2\left( \frac{\mathrm{d}w_0(x)}{\mathrm{d}x}\right) ^2\leqslant 0. \end{aligned}$$

(13)

On the basis of (12) and (13), we gain the monotonicity condition

$$\begin{aligned} \frac{\mathrm{d} w_1(x)}{\mathrm{d} x}\leqslant 0,\quad \forall x\in [0,1]. \end{aligned}$$

(14)

Suppose, next, that both the local incentive compatibility and monotonicity conditions hold. Then it must be the case that all the manager’s incentive compatibility conditions hold. To see this result, without loss of generality, suppose that $x>y$. By integrating the local incentive compatibility condition (11) and using the monotonicity condition (14), we can obtain

$$\begin{aligned}&w_{0}(x)-w_{0}(y)\\&\quad =-\int ^{x}_{y}(r_1ve_1-r_2d_0+r_2d_1e_2-\,sw_1(s)\sigma ^2)\frac{\mathrm{d}w_1(s)}{\mathrm{d} s}\mathrm{d} s \nonumber \\&\quad \geqslant (w_1(y)-w_1(x))(r_1ve_1-r_2d_0+r_2d_1e_2)\\&\qquad -\,\frac{1}{2}x(w_1^2(y)-w_1^2(x))\sigma ^2. \end{aligned}$$

That is to say

$$\begin{aligned} \mathrm{CE}(x,x)\geqslant \mathrm{CE}(x,y), \quad \forall x, y \in [0,1]. \end{aligned}$$

On the other hand, if $x<y$, we can also obtain

$$\begin{aligned}&w_{0}(y)-w_{0}(x)\\&\quad =-\int ^{y}_{x}(r_1ve_1-r_2d_0+r_2d_1e_2-sw_1(s)\sigma ^2)\frac{\mathrm{d}w_1(s)}{\mathrm{d} s}\mathrm{d} s \nonumber \\&\quad \leqslant (w_1(x)-w_1(y))(r_1ve_1-r_2d_0+r_2d_1e_2)\nonumber \\&\qquad -\,\frac{1}{2}x(w_1^2(x)-w_1^2(y))\sigma ^2. \end{aligned}$$

That is to say

$$\begin{aligned} \mathrm{CE}(x,x)\geqslant \mathrm{CE}(x,y), \quad \forall x, y \in [0,1]. \end{aligned}$$

This result establishes the equivalence between the monotonicity condition together with the local incentive compatibility condition and the full set of the manager’s incentive constraints.

Differentiating $\mathrm{CE}(x,x)$ with respect to x yields

$$\begin{aligned} \frac{\mathrm{d} \mathrm{CE}(x,x)}{\mathrm{d} x}= & {} \frac{\mathrm{d}w_0(x)}{\mathrm{d}x}-[r_1ve_1-r_2d_0+r_2d_1e_2\\&-\,xw_1(x)\sigma ^2]\frac{\mathrm{d} w_1(x)}{\mathrm{d} x}-\frac{1}{2}xw_1^2(x)\sigma ^2\nonumber \\= & {} -\frac{1}{2}xw_1^2(x)\sigma ^2\leqslant 0. \end{aligned}$$

The individual rationality constraint is equivalent to

$$\begin{aligned} \mathrm{CE}(1,1)\geqslant 0. \end{aligned}$$

The constraint is binding under the optimal mechanism because the firm will reap the redundant profit, so that $\mathrm{CE}(1,1)=0$.$\square$

Proof of Lemma 2

Because

$$\begin{aligned} \frac{\mathrm{d} \mathrm{CE}(x,x)}{\mathrm{d}x}=-\frac{1}{2}w_1^2(x)\sigma ^2, \end{aligned}$$

we can derive

$$\begin{aligned} \mathrm{CE}(x,x)=\mathrm{CE}(1,1)-\int ^{1}_{x}-\frac{1}{2}w_1^2(t)\sigma ^2\mathrm{d}t. \end{aligned}$$

Combining the definition of CE(x, x) in equation (2) yields

$$\begin{aligned} w_0(x)= & {} \int ^{1}_{x}-\frac{1}{2}w_1^2(t)\sigma ^2\mathrm{d}t-(r_1ve_1-r_2d_0+r_2d_1e_2)w_1(x)\\&+\,\frac{1}{2}(e_1^2+e_2^2)+\frac{1}{2}xw_1^2(x)\sigma ^2. \end{aligned}$$

By substituting the fixed wage into the objective function, we can derive the firm’s expected profit

$$\begin{aligned} E[Q-W(x,Q)]= & {} r_1ve_1+r_2d_1e_2-r_2d_0-\frac{1}{2}(e_1^2+e_2^2)\\&+\,\frac{1}{2}\sigma ^2\int _0^1(h(x)+x) w_1^2(x)f(x)\mathrm{d}x. \end{aligned}$$

$\square$

Proof of Proposition 3

We can use the first-order condition $\frac{\partial E[Q-W]}{\partial e_{1}}=0$ and $\frac{\partial E[Q-W]}{\partial e_{2}}=0$ to yield the first-best effort levels: $e_{1}^{A}=r_1v$ and $e_{2}^{A}=r_2d_1$. Substituting it into the objective function and ignore the monotonicity constraint in Lemma 1, the firm’s problem can be rewritten as unconstrained optimization problem:

$$\begin{aligned}&\mathop {\max }\limits _{0\leqslant w_1(x)\leqslant 1} r_1ve_1+r_2d_1e_2-r_2d_0 -\frac{1}{2}(e_1^2+e_2^2)\\&\quad +\,\frac{1}{2}\sigma ^2\int _0^1(h(x)+x) w_1^2(x)f(x)\mathrm{d}x, \end{aligned}$$

which is decreasing in $w_1(x)$. Consequently, the firm’s optimization problem can be obtained as $w_1(x)=0$. The corresponding optimal fixed payment $w_0(x)$ for the manager can be obtained immediately. The proof of the proposition is complete.$\square$

Proof of Lemma 3

By using the first-order condition $\frac{\partial \mathrm{CE}(x,x)}{\partial e_{1}}=0$ and $\frac{\partial \mathrm{CE}(x,x)}{\partial e_{2}}=0$, the manager selects his optimal effort levels: $e_{1}=r_1vw_1(x)$ and $e_{2}=r_2d_1w_1(x)$. When the manager selects the contract $(w_0(x), w_1(x))$, his expected utility

$$\begin{aligned}&w_0(x)-r_2d_0w_1(x)+\frac{1}{2}\left[ (r_1v)^2+(r_2d_1)^2\right] w_1^2(x)\\&\quad -\,\frac{1}{2}x\sigma ^2w_1^2(x). \end{aligned}$$

Similarly, if the manager selects the contract $(w_0(y),w_1(y))$, his expected utility

$$\begin{aligned}&w_0(y)-r_2d_0w_1(y)+\frac{1}{2}\left[ (r_1v)^2+(r_2d_1)^2\right] w_1^2(y)\\&\quad -\,\frac{1}{2}x\sigma ^2w_1^2(y). \end{aligned}$$

The rest of the proof is similar to the Proof of Lemma 1. Therefore, the proof of the lemma is complete.$\square$

Proof of Lemma 4

Similar to the Proof of Lemma 2.$\square$

Proof of Proposition 4

Through the similar method used in the Proof of Proposition 3, the firm’s problem can be rewritten as

$$\begin{aligned}&\mathop {\max }\limits _{0\leqslant w_1(x)\leqslant 1}-r_2d_0+\int _0^1\left\{ \left[ (r_1v)^2+(r_2d_1)^2\right] w_1(x)\right. \\&\left. \quad -\,\frac{1}{2}\left[ (r_1v)^2+(r_2d_1)^2+(x+h(x))\sigma ^2\right] w_1^2(x)\right\} f(x)\mathrm{d}x. \end{aligned}$$

The first-order variation and the second-order variation of the firm’s expected profit are derived as

$$\begin{aligned}&\int _0^1\left\{ \left[ (r_1v)^2+(r_2d_1)^2\right] -\left[ (r_1v)^2+(r_2d_1)^2\right. \right. \\&\quad \left. \left. +\,(x+h(x))\sigma ^2\right] w_1(x)\right\} f(x)(\delta w_1(x))\mathrm{d}x \end{aligned}$$

and

$$\begin{aligned} -\int _0^1\left[ (r_1v)^2+(r_2d_1)^2+(x+h(x))\sigma ^2\right] f(x)[\delta w_1(x)]^2\mathrm{d}x. \end{aligned}$$

Following the determinate optimal incentive commission rate $w_1(x)$, the corresponding optimal fixed payment $w_0(x)$, the optimal effort levels $\mathrm{e}^{\mathrm{D}}_1$ and $\mathrm{e}^{\mathrm{D}}_2$ can be obtained immediately. The proof of the proposition is complete.$\square$

Proof of Proposition 5

The result is derived directly by comparing the project manager’s profits which are shown in Propositions 1–4.$\square$

Proof of Proposition 6

The result is derived directly by comparing the project manager’s profits which are shown in Corollaries 1–4.$\square$

Proof of Proposition 7

Similar to the Proof of Proposition 6.$\square$

Proof of Proposition 8

Similar to the Proof of Proposition 6.$\square$