An assumption on the rate of time preference that reflects observations of a bias towards present rewards. With standard exponential discounting the discount rate between any two periods is constant. In contrast, with hyperbolic discounting the discount rate between any two successive periods declines as the time horizon increases, so that the elasticity of intertemporal substitution depends on the timing of the change in the intertemporal price. A frequently used simplification of hyperbolic discounting is quasi-hyperbolic discounting, for which the lifetime utility from consumption is modelled as

with 0 < δ‎ < 1 and 0 < β‎ ≤ 1. These are also known as beta-delta preferences. Here β‎ is the degree of self-control which determines the present bias, and δ‎ is the intertemporal discount factor; β‎ = 1 is the special case of exponential discounting. See also behavioural economics.