A Continuous-Time Version of the Principal–Agent Problem

Abstract

This paper describes a new continuous-time principal–agent model, in which the output is a diffusion process with drift determined by the agent’s unobserved effort. The risk-averse agent receives consumption continuously. The optimal contract, based on the agent’s continuation value as a state variable, is computed by a new method using a differential equation. During employment, the output path stochastically drives the agent’s continuation value until it reaches a point that triggers retirement, quitting, replacement, or promotion. The paper explores how the dynamics of the agent’s wages and effort, as well as the optimal mix of short-term and long-term incentives, depend on the contractual environment. The understanding of dynamic incentives is central in economics. How do companies motivate their workers through piece rates, bonuses, and promotions? How is income inequality connected with productivity, investment, and economic growth? How do financial contracts and capital structure give incentives to the managers of a corporation? The methods and results of this paper provide important insights to many such questions. This paper introduces a continuous-time principal–agent model that focuses on the dynamic properties of optimal incentive provision. We identify factors that make the agent’s wages increase or decrease over time. We examine the degree to which current and future outcomes motivate the agent. We provide conditions under which the agent eventually reaches retirement in the optimal contract. We also investigate how the costs of creating incentives and the dynamic properties of the optimal contract depend on the contractual environment: the agent’s outside options, the difficulty of replacing the agent, and the opportunities for promotion. Our new dynamic insights are possible due to the technical advantages of continuous-time methods over the traditional discrete-time ones. Continuous time leads to a much simpler computational procedure to find the optimal contract by solving an ordinary differential equation. This equation highlights the factors that determine optimal consumption and effort. The dynamics of the agent’s career path are naturally described by the drift and volatility of the agent’s pay-offs. The geometry of solutions to the differential equation allows for easy comparisons to see how the agent’s wages, effort, and incentives depend on the contractual environment. Finally, continuous time highlights many essential features of the optimal contract, including the agent’s eventual retirement. In our benchmark model a risk-averse agent is tied to a risk-neutral principal forever after employment starts. The agent influences output by his continuous unobservable effort input. The principal sees only the output: a Brownian motion with a drift that depends on the agent’s effort. The agent dislikes effort and enjoys consumption. We assume that the agent’s utility function has the income effect, that is, as the agent’s income increases it becomes costlier to compensate him for the effort. Also, we assume that the agent’s utility of consumption is bounded from below.