The β–δ–Δ Sweet Spot


David Laibson and Peter Maxted. In Preparation. “The β–δ–Δ Sweet Spot”.


When agents have present-biased discount functions and are partially or fully sophisticated, equilibria in infinite-horizon problems are non-unique. In addition, the unique equilibrium selected by backward induction features strategic motives that induce pathological properties, including policy function discontinuities and non-monotonicities. Harris and Laibson (2013) show that continuous-time methods can be used to eliminate pathological equilibria. This paper proposes a discrete-time alternative in consumption models that is simple and computationally tractable: use monthly (or shorter) period lengths, which, when combined with calibrated levels of background noise, effectively eliminates the strategic mechanisms that plague present-biased models. The numerical solution to a discrete-time model with monthly periods approximates its well-behaved continuous-time counterpart. Accordingly, economists can work in discrete time while gaining the tractability afforded by continuous time. This paper formalizes these methods and provides numerical examples.