Exponential discounting: V = R · e^(−r·t) — consistent over time; no preference reversals.
Hyperbolic discounting: V = R / (1 + k·t) — decreases rapidly in the short run, slowly in the long run.
The key consequence: preference reversals. At time 0, you may prefer the larger later reward B. But as A's delivery approaches, hyperbolic discounting inflates its present value, causing you to switch preference — even though nothing fundamental changed. This explains procrastination, addiction, and self-control failures.
The crossover point marks when present-biased preferences take over. Exponential discounters are time-consistent; hyperbolic discounters are not.