On optimal sharing rules in discrete-and continuous-time principal-agent problems with exponential utility

doi:10.1016/S0165-1889(96)00944-X

Journal of Economic Dynamics and Control

Volume 21, Issues 2–3, February–March 1997, Pages 551-574

https://doi.org/10.1016/S0165-1889(96)00944-X Get rights and content

Abstract

We relate the existing first-order approaches for discrete- and continuous-time problems by considering continuous-time principal-agent problems as limiting cases of multi-period discrete-time formulations. Our multi-period discrete-time formulation gives an intuitive explanation of why the validity of the first-order approach to continuous-time formulation in Schattier and Sung (1993) can be more easily established. We also show that although continuous-time formulations are far easier to deal with, their solutions may not be, in general, as simple as the one in the Brownian model. Under slightly modified assumptions solutions typically depend on the entire history of the process, rather than only on the terminal state alone.

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In that paper the principal and the agent have exponential utility functions and the optimal contract is linear. Their work was generalized and extended by Schättler and Sung [14,15], Sung [16,17], Müller [10,11], and Hellwig and Schmidt [7]. The papers by Williams [19] and Cvitanić, Wan and Zhang [3] use the stochastic maximum principle and forward–backward stochastic differential equations (FBSDEs) to characterize the optimal compensation for more general utility functions, under moral hazard.
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An analysis of airport-airline vertical relationships with risk sharing contracts under asymmetric information structures
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Then Schaettler and Sung (1993) proved the general cases with necessary and sufficient conditions in the second approach case. Also Schaettler and Sung (1997) studied the connection between discrete models and continuous model in the second approach case. Mueller (1996) and Mueller (1998), based on these studies, showed that if the principal can see the agent action, in the first best situation according to his words, the linear optimal contract is structured by the risk aversion parameters of both principal and agent.
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If both parties’ productivities are same, or the diffusion rate of the underlying process is unity, optimal linear sharing rule do not depend on the final state. If their conditions not dependent on final state are symmetric as well, then risk sharing disappears completely. In numerical examples, we illustrate the complex impact of uncertainty increase and end-of-period load factor improvement on the optimal sharing rule, and the relatively simple impact on total utility levels.
On the first-order approach in principal-agent models with hidden borrowing and lending
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We provide sufficient conditions for the validity of the first-order approach for two-period dynamic moral hazard problems where the agent can save and borrow secretly. The first-order approach is valid if the following conditions hold: (i) the agent has non-increasing absolute risk aversion utility (NIARA), (ii) the output technology has monotone likelihood ratios (MLR), and (iii) the distribution function of output is log-convex in effort (LCDF). Moreover, under these three conditions, the optimal contract is monotone in output. We also investigate a few possibilities of relaxing these requirements.
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^∗: The authors thank Kerry Back, Phil Dybvig, and an anonymous referee for helpful comments and suggestions.

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On optimal sharing rules in discrete-and continuous-time principal-agent problems with exponential utility

Abstract

Deterministic and stochastic optimal control

Moral hazard and observablity

Bell Journal of Economics

Aggregation and linearity in the provision of intertemporal incentives

Econometrica

Justifying the first-order approach to principal-agent problems

Econometrica