A solvable continuous time dynamic principal–agent model

Abstract

I study the provision of incentives in a continuous time dynamic moral hazard model with hidden actions and hidden states. I consider a principal-agent model with linear production and exponential utility, whose explicit solution allows me to show how allocations are distorted for incentive reasons, and how access to hidden savings further alters allocations. I solve the model by applying a stochastic maximum principal, where the co-state variables from the agent's optimization problem become state variables for the principal's problem of choosing an optimal contract. I show that the main effect of moral hazard is a distortion on the effort margin, with a smaller effect on the intertemporal consumption allocation. Access to hidden savings shuts down the intertemporal distortions and increases the effort distortion. I also show how the optimal contracts can be implemented via a constant equity share, a constant flow payment, and a constant tax on savings.

A Solvable Dynamic Principal-Agent Model with Linear Marginal Productivity

“…Sannikov [16] and Meng et al [17] studied the related continuous time models and Sannikov [18] gave an overview of the related literatures. Similar to Williams [6] and Su et al [19], the assets and consumption payments occur continuously throughout the contract. The situation where the agent receives a single payment from principal is considered by Holmstrom and Milgrom [1], Schattler and Sung [20], and Cvitanic et al [21].…”

“…The explicit solutions allow us to derive a simple implementable contract and illustrate the effects of information frictions. In particular, the model studied in this article is an extension of Holmstrom and Milgrom [1] and Williams [6].…”

“…We work with a generalized model where the marginal productivity is linear and decreasing, while the marginal productivity of the model in Williams [6] is a constant. Similar to Holmstrom and Milgrom [1], the optimal contract is also linear in our completely dynamic setting with linear marginal productivity and exponential utility.…”

A Solvable Time-Inconsistent Principal-Agent Problem

“…Since the agent's objective function relies on the consumption process { }, that is, it relies on the history of the whole output, so it is non-Markov (the specific proof see [6]). Thus, the agent optimization problem cannot be solved by the standard dynamic programming principle.…”

“…Reference [4] shows that the first best sharing rule is also linear in output in the continuous-time principal-agent model with exponential utility. Reference [5,6] uses the stochastic maximum principle to extended Holmstrom's model, and discuss the optimal solution of the agent with private information in the continuous-time model. Reference [7,8] uses the forward-backward stochastic differential equations to consider the optimal contract under moral hazard.…”