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Federal Reserve Bank of Richmond
Working Paper
Optimal Contracts with Reflection
Borys Grochulski
Yuzhe Zhang

In this paper, we show that whenever the agent's outside option is nonzero, the optimal contract in the continuous-time principal-agent model of Sannikov (2008) is reflective at the lower bound. This means the agent is never terminated or retired after poor performance. Instead, the agent is asked to put zero effort temporarily, which brings his continuation value up. The agent is then asked to resume effort, and the contract continues. We show that a nonzero agent's outside option arises endogenously if the agent is allowed to quit and find a new firm (after a random search time of finite expected duration). In addition, we find new dynamics of the reflection at the lower bound. In the baseline model, the dynamics of the reflection are slow, as in Zhu (2013), i.e., the zero-action is used often. However, if the agent's disutility from the first unit of effort is zero, which is a standard Inada condition, or if his utility of consumption is unbounded below, the reflection becomes fast, i.e., the zero-effort action is used seldom.

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Borys Grochulski & Yuzhe Zhang, Optimal Contracts with Reflection, Federal Reserve Bank of Richmond, Working Paper 16-14, 01 Dec 2016, revised 27 Jan 2017.
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Keywords: dynamic moral hazard; quitting; random search; reflective dynamics; ODE splicing; sticky Brownian motion; fast reflection; instantaneous control
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