Hyperbolic Discounting — Present Bias

Discount Parameters

Present bias β 0.70 Long-run discount δ 0.95 Hyperbolic k 0.5 Reward A (now) 10 Reward B (later) 20 Delay t_B (days) 30

β-δ: V(t) = β·δᵗ·u (t>0)

Hyperbolic: V = u/(1+kt)

Exponential: V = u·e^(−rt)

Value A (now): —

Value B (delay): —

Choice: —

Preference reversal at: —

Laibson (1997): β-δ captures the observation that people have extra discounting for immediate vs. delayed — "present bias." β=1: standard exponential (time-consistent). β<1: present-biased. Hyperbolic discounting (Ainslie): discount rate itself declines with delay, causing preference reversal. Key: if you prefer B over A when both are distant, you may flip to A when A is immediate.

Hyperbolic Discounting & Present Bias

Discount Parameters

Discount Functions Compared

Discounted Value of A vs B over Time (advance period)

Preference Reversal Diagram