It Is Time to Get Some Rest

We introduce the fatigue disutility model. Set in continuous-time, the model captures in a precise way the notion that fatigue accumulates with effort, decays with rest, and produces an increase in marginal instant disutility. The model is meant to be prescriptive, although it rationalizes several anomalies observed in time preference. Under very general conditions the most efficient temporal profile of effort exhibits a high-low-high pattern. We also consider a variant whereby marginal productivity decreases with fatigue, and examine the optimal management of fatigue and productivity under various scenarios. We bring the model to the data and provide a calibration that predicts well the speed profiles observed in swimming competitions.