Should the Timing of Inspections be Predictable?

Abstract

A principal hires an agent to work on a long-term project that culminates in a breakthrough or a breakdown. At each time, the agent privately chooses to work or shirk. Working increases the arrival rate of breakthroughs and decreases the arrival rate of breakdowns. To motivate the agent to work, the principal conducts costly inspections. She fires the agent if shirking is detected. We characterize the principal's optimal inspection policy. Predictable inspections are optimal if work primarily generates breakthroughs. Random inspections are optimal if work primarily prevents breakdowns. Crucially, the agent's actions determine his risk attitude over the timing of punishments.

Figures (4)

• Figure 1: Shirking agent's loss (orange) from a perfect inspection with $\lambda_G > \lambda_B$. In this example, $\lambda_G + r = 2$; $\lambda_B + r = 1$; $U_1 = 1.25$; and $U_0 = 2$.
• Figure 2: Shirking agent's loss (orange) from a perfect inspection with $\lambda_B > \lambda_G$. In this example, $\lambda_G + r = 1$; $\lambda_B + r = 2$; $U_1 = 1.25$; and $U_0 = 2$.
• Figure 3: Agent's loss from an imperfect inspection (orange for shirk-always deviation, green for shirk-work deviation) with $\lambda_B \leq \bar{\lambda}_B$. In this example, $\lambda_G + r = 2$; $\lambda_B + r = 1$; $U_1 = 1.25$; $U_0 = 2$; and $\delta = 3.5$. Here, $\bar{L}_S$ is globally concave because $\delta + \lambda_B \geq \lambda_G \geq \lambda_B$.
• Figure 4: Shirking agent's loss (orange) from an imperfect inspection with $\lambda_B > \bar{\lambda}_B$. In this example, $\lambda_G + r = 1$; $\lambda_B + r = 2$; $U_0 = 2$; $U_1 = 1.25$; and $\delta = 5$.

Theorems & Definitions (18)

• Lemma 1: No inspections
• Theorem 1: Periodic perfect inspections
• Theorem 2: Random perfect inspections
• Theorem 3: Imperfect inspections
• Remark 1: Threshold $\bar{\lambda}_B$
• Theorem 4: Random inspections with recovery
• Lemma 2: Exponential solution
• Lemma 3: Zeros
• Claim 1
• Claim 2