Quasi-hyperbolic discounting
Also known as: Beta-delta model, Beta-delta discounting
A tractable model with one extra kink that captures present bias in a single parameter.
What it means
Quasi-hyperbolic discounting approximates true hyperbolic curves with a simpler two-parameter form: a present-bias factor beta that applies to all future periods uniformly, and a standard exponential factor delta applied period by period. The beta term creates a single sharp discontinuity between now and later — everything beyond the present is shrunk by an extra constant — which reproduces the preference reversals of hyperbolic discounting while remaining mathematically convenient. When beta equals one the model collapses to exponential discounting, so beta cleanly indexes the degree of present bias. Its tractability made it the workhorse of behavioral economics for analyzing saving, procrastination, and self-control, including the distinction between sophisticated and naive agents. The form is often credited to Phelps and Pollak and was brought to behavioral prominence by Laibson.
Examples
With beta below one, you'd take $100 today over $110 tomorrow, yet prefer $110 in 31 days to $100 in 30 — the extra kink between now and later flips the choice.
On Sunday, Monday's run and its payoff sit equally far off, so you commit. By Monday the effort is now while the benefit is 'later' and takes the extra beta hit — so you skip.
A saver who knows his beta is below one locks his money in a notice account, because he expects next month's self to shrink next month's saving the same way and spend it instead.
First described in Phelps & Pollak (1968); applied by Laibson (1997).