This paper investigates the moral hazard problem in finite horizon with both continuous and lump-sum payments, involving a time-inconsistent sophisticated agent and a standard utility maximizer principal: Building upon the so-called dynamic programming approach in Cvitanić et al. (2018) and the recently available results in Hernández and Possamaï (2023), we present a methodology that covers the previous contracting problem. Our main contribution consists of a characterization of the moral hazard problem faced by the principal. In particular, it shows that under relatively mild technical conditions on the data of the problem, the supremum of the principal's expected utility over a smaller restricted family of contracts is equal to the supremum over all feasible contracts. Nevertheless, this characterization yields, as far as we know, a novel class of control problems that involve the control of a forward Volterra equation via Volterra-type controls, and infinite-dimensional stochastic target constraints. Despite the inherent challenges associated with such a problem, we study the solution under three different specifications of utility functions for both the agent and the principal, and draw qualitative implications from the form of the optimal contract. The general case remains the subject of future research. We illustrate some of our results in the context of a project selection contracting problem between an investor and a time-inconsistent manager.

中文翻译：

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时间不一致契约理论

本文研究了有限范围内连续支付和一次性支付的道德风险问题，涉及时间不一致的复杂代理和标准效用最大化原则：建立在 Cvitanić 等人所谓的动态规划方法的基础上。(2018) 以及 Hernández 和 Possamaï (2023) 最近提供的结果，我们提出了一种涵盖先前承包问题的方法。我们的主要贡献包括对委托人面临的道德风险问题的描述。特别是，它表明，在问题数据相对温和的技术条件下，委托人预期效用在较小的受限合约族上的上界等于所有可行合约上的上界。然而，据我们所知，这种表征产生了一类新颖的控制问题，其中涉及通过 Volterra 型控制和无限维随机目标约束来控制正向 Volterra 方程。尽管存在与此类问题相关的固有挑战，我们还是在代理人和委托人的效用函数的三种不同规范下研究了解决方案，并从最优合同的形式中得出了定性含义。一般情况仍然是未来研究的主题。我们在投资者和时间不一致的管理者之间的项目选择承包问题的背景下说明了我们的一些结果。