Theory Notes to Accompany Active Decisions

Our model comprises a firm, which sets a default contribution rate for the company savings plan, and the workers who choose their behavior in response. Consider a group of workers with quasi-hyperbolic preferences, so that they have discount function 1, βδ, βδ, ... where 0 < β ≤ 1. For simplicity, we assume that δ = 1, eliminating long run discounting. We also assume that each worker has an exogenously determined and constant optimal savings rate, denoted s, that is known to the worker with certainty but not observed by the firm. The firm does know the probability distribution function for s, denoted fs(s). In our model, the firm acts in period 0 by setting the default contribution rate, denoted d, at which all new employees are automatically enrolled in the company savings plan. In each period thereafter, every worker has the oppotunity to change her savings rate permanently from the default to her optimal level. Should she act, she incurs a stochastic cost ct drawn from a uniform distribution on the interval [c, c̄], where c < c̄. This cost is determined at the beginning of the period and thus is known to the worker at the time of her decision. After each period in which the agent saves at the default d, however, she faces a flow cost L(s, d) ≥ 0; without loss of generality, we assume workers incur this cost at the beginning of the next period.