Deadline-based incentive contracts in project management with cost salience

Abstract

The contractor’s procrastinating behavior owing to the psychology of cost salience exposes the project manager to the risk of time delay, which brings a significant challenge in project manager’s incentive contract design. This paper considers that a project manager pays a contractor over a menu of deadline-based incentive contracts to conduct a project which consists of two sequential tasks. The contractor is endowed with private cost salience information and unobservable efforts. The subjective assessments about the cost salience 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 the existence of cost salience and information asymmetry on the incentive contract and the project manager’s profit. We confirm that cost salience can impel the project manager to lower both the fixed payment under full information and the penalty/incentive rate under pure moral hazard. Interestingly, we find that moral hazard can weaken the extent of inverse impact caused by the existence of cost salience for the project manager. Our study also shows that, for mitigating the adverse impacts brought by moral hazard, the project manager is more profitable to provide effort incentive when the contractor’s efforts are more productive or the project risk is in a higher level. Finally, other suggestions for mitigating the detrimental impacts brought by adverse selection are provided by numerical experiments.

Appendix: Proofs

Proof of Proposition 1

Based on the expected value criterion, the project manager’s expected profit can be written as

$$\begin{aligned} E[V(T)-W_1(1, T_1)-W_2(1, T_2)]=a-b\sum _{i=1}^2[t_{0i}-t_{1i}e_i]-\sum _{i=1}^2 \left[ \alpha _i-\beta _i(T_i-d_i)\right] . \end{aligned}$$

Because the project manager’s expected profit is decreasing in the fixed payments $\alpha _1$ and $\alpha _2$, at optimality, the individual rationality constraints for the contractor should be binding, i.e., $\text {CE}(1,1)=0$. Thus, we can rewrite the project manager’s problem as

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \max _{(\alpha _i,\beta _i,e_i)} a-b\sum _{i=1}^2[t_{0i}-t_{1i}e_i]-\sum _{i=1}^2 \left[ \alpha _i-\beta _i(T_i-d_i)\right] \\ \text{ subject } \text{ to: }\\ \displaystyle \quad \alpha _{i}-\beta _{i}(T_i-d_i)-\frac{1}{2}\rho \beta _{i}^2 \sigma _{i}^2-\frac{e_i^2}{2}=0, \quad i = 1,2.\\ \end{array} \right. \end{aligned}$$

By substituting the fixed payments into the objective function, we obtain the project manager’s expected profit:

$$\begin{aligned} a-b(t_{01}-t_{11}e_1+t_{02}-t_{12}e_2)-\frac{e_1^2}{2}-\frac{e_2^2}{2} -\frac{1}{2}\rho \beta _{1}^2\sigma _{1}^2-\frac{1}{2}\rho \beta _{2}^2\sigma _{2}^2, \end{aligned}$$

which is decreasing in $\beta _1$ and $\beta _2$ and concave in $e_{1}$ and $e_{2}$. Thus, the project manager would set both $\beta _1$ and $\beta _2$ to zero for getting a maximum profit. Besides, we can yield the contractor’s optimal effort levels $e_1=bt_{11}$ and $e_2=bt_{12}$ by using the first-order condition. Following the determinate individual rationality constraints which are binding, the corresponding optimal fixed payments $\alpha _1$ and $\alpha _2$ for the contractor can be derived immediately. The proof of the proposition is complete. $\square$

Proof of Proposition 2

Under moral hazard, the contractor will choose his efforts $e_{i}$ in task i to maximize his own utility

$$\begin{aligned} \alpha _{i}-\beta _{i}(T_i-d_i)-\frac{1}{2}\rho \beta _{i}^2 \sigma _{i}^2-\frac{{\hat{e}}_i^2}{2}, \end{aligned}$$

which is concave in ${\hat{e}}_{i}$, $i=1,2$. The maximum is completely characterized by the first-order condition and we derive $e_{1}=t_{11}\beta _1$ and $e_{1}=t_{12}\beta _2$. Furthermore, because the project manager’s profit is decreasing in the fixed payments, at optimality, the individual rationality constraints should be binding. Therefore, by substituting $e_{1}$ and $e_{2}$ into the individual rationality constraints and then substituting the fixed payments ($\alpha _1$ and $\alpha _2$) and the effort levels ($e_{1}$ and $e_{2}$) into the objective function, the project manager’s expected profit can be rewritten as

$$\begin{aligned} a-b(t_{01}+t_{02})+\sum _{i=1}^{2}\left[ bt_{1i}^2\beta _i -\frac{1}{2}t_{1i}^2\beta _i^2-\frac{1}{2}\rho \beta _{i}^2\sigma _{i}^2\right] . \end{aligned}$$

By the first-order condition regarding to $\beta _1$ and $\beta _1$, we can obtain $\beta _1=\frac{bt_{11}^2}{t_{11}^2+\rho \sigma _1^2}$ and $\beta _2=\frac{bt_{12}^2}{t_{12}^2+\rho \sigma _2^2}$. Based on the binding individual rationality constraints, the optimal fixed payments ($\alpha _1$ and $\alpha _2$) can be obtained immediately. The proof of the proposition is complete. $\square$

Proof of Proposition 3

Similar to the Proof of Proposition 1. $\square$

Proof of Proposition 4

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

Proof of Proposition 5

Let $\text {CE}(\theta ,\theta )=\text {CE}_1(\theta ,\theta )+\text {CE}_2(\theta ,\theta )$. The incentive compatibility constraints for adverse selection can be written as

$$\begin{aligned} \text {CE}(\theta ,\theta )\geqslant \text {CE}(\theta ,{\tilde{\theta }}),\quad \forall \theta ,{\tilde{\theta }}\in [{\underline{\theta }}, {\overline{\theta }}], \end{aligned}$$

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

$$\begin{aligned} \sum _{i=1}^{2}\left[ \frac{\mathrm {d}\alpha _i(\theta )}{\mathrm {d}\theta }-(t_{0i}-t_{1i}e_i-d_i)\frac{\mathrm {d} \beta _i(\theta )}{\mathrm {d} \theta }-\rho \sigma _{i}^2\beta _i(\theta )\frac{\mathrm {d} \beta _i(\theta )}{\mathrm {d} \theta }\right] =0,\quad \forall \theta \in [{\underline{\theta }},{\overline{\theta }}]. \end{aligned}$$

Differentiating $\text {CE}(\theta ,\theta )$ with respect to $\theta$ yields

$$\begin{aligned} \sum _{i=1}^{2}\left[ \frac{\mathrm {d}\alpha _i(\theta )}{\mathrm {d}\theta } -(t_{0i}-t_{1i}e_i-d_i)\frac{\mathrm {d} \beta _i(\theta )}{\mathrm {d} \theta }-\rho \sigma _{i}^2\beta _i(\theta )\frac{\mathrm {d} \beta _i(\theta )}{\mathrm {d} \theta }\right] -\frac{e_1^2}{2}=-\frac{e_1^2}{2}\leqslant 0. \end{aligned}$$

Thus, the individual rationality constraint is equivalent to

$$\begin{aligned} \text {CE}({\overline{\theta }},{\overline{\theta }})\geqslant 0. \end{aligned}$$

The constraint is binding under the optimal mechanism because the project manager will reap the redundant profit, so that $\text {CE}({\overline{\theta }},{\overline{\theta }})=0$. Because $\text {CE}({\overline{\theta }},{\overline{\theta }})=0$ and $\frac{\mathrm {d}\text {CE}(\theta ,\theta )}{\mathrm {d}\theta }=-e_1^2/2$, we can derive

$$\begin{aligned} \text {CE}(\theta ,\theta )=\text {CE}({\overline{\theta }}, {\overline{\theta }})+\int ^{{\overline{\theta }}}_{\theta } \frac{e_1^2}{2}\mathrm {d}\theta =\frac{e_1^2}{2}({\overline{\theta }}-\theta ). \end{aligned}$$

Combining the definition of $\text {CE}(\theta ,\theta )$ yields

$$\begin{aligned} \sum _{i=1}^{2}\left[ \alpha _{i}-\beta _{i}(T_i-d_i)-\frac{1}{2} \rho \beta _{i}^2\sigma _{i}^2\right] -\frac{\theta e_1^2}{2}-\frac{e_2^2}{2}=\frac{e_1^2}{2}({\overline{\theta }}-\theta ). \end{aligned}$$

By substituting the fixed wages into the objective function, we can derive

$$\begin{aligned} a-b(t_{01}-t_{11}e_1+t_{02}-t_{12}e_2)-\frac{{\overline{\theta }}e_1^2}{2} -\frac{e_2^2}{2}-\frac{1}{2}\rho \beta _{1}^2\sigma _{1}^2-\frac{1}{2}\rho \beta _{2}^2\sigma _{2}^2, \end{aligned}$$

which is decreasing in $\beta _1$ and $\beta _2$ and concave in $e_{1}$ and $e_{2}$. Thus, the project manager would set both $\beta _1$ and $\beta _2$ to zero for getting a maximum profit. Besides, we can yield the contractor’s optimal effort levels $e_1=bt_{11}/{\overline{\theta }}$ and $e_2=bt_{12}$ by using the first-order condition. Following the determinate individual rationality constraints which are binding, the corresponding optimal fixed payments $\alpha _1$ and $\alpha _2$ for the contractor can be derived immediately. The proof of the proposition is complete. $\square$

Proof of Proposition 6

Similar to the Proof of Proposition 5. $\square$

Proof of Proposition 7

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

Proof of Proposition 8

The result is derived directly by comparing the project manager’s profits which are shown in Corollaries 3–6. $\square$