Hyperbolic Discounting: Why the Wait Feels Longer When It Starts Now

Everyone discounts the future to some degree; the bias is in the shape of the discount, not its existence. Standard economic models assume exponential discounting: every month of delay shrinks a reward's value by the same fixed fraction, so a month is a month whether it starts today or in a year. Under that rule, preferences between two dated rewards never change as time passes. Observed behavior fits a different curve, one that drops steeply over the first days and weeks of delay and then flattens out.

A steep-then-flat curve has one consequence that an exponential curve cannot produce: two value curves can cross. Seen from a year away, the larger, later reward sits above the smaller, sooner one, because at long delays the curve is nearly flat and the extra amount dominates. As the sooner reward comes within days, its value shoots up the steep part of the curve and overtakes the later one. The same preferences, applied at two moments, give two answers; economists call this time inconsistency, and the switch is the preference reversal the demo looks for.

Laibson's 1997 model splits the curve into two parts: a one-time penalty on everything that is not immediate, and ordinary exponential discounting after that. The split makes the whole bias a matter of now versus later, which is how it tends to feel: the present has a pull no other moment has, and a moment loses that pull as soon as it stops being the present.

George Ainslie set out the idea in a 1975 review in Psychological Bulletin titled "Specious reward." Drawing on animal experiments in which pigeons chose between a small, immediate food reward and a larger, delayed one, he argued that choice between rewards at different delays followed a hyperbolic curve rather than an exponential one, that such curves cross, and that the crossing explained impulsiveness: a preference for the larger, later reward that holds at a distance and collapses as the smaller reward comes close. The same shape, he argued, explains why people and animals bind their own future choices.

Richard Thaler published the first human evidence in Economics Letters in 1981, under the title "Some empirical evidence on dynamic inconsistency." Respondents to a questionnaire were asked what amount, received after a delay, would make them indifferent to receiving a smaller amount now. For 15 dollars now, the median answers were 20 dollars in one month, 50 dollars in one year and 100 dollars in ten years. The implied annual discount rates fall as the delay grows: 345 percent over a month, 120 percent over a year, 19 percent over ten years. Thaler also reported that rates were lower for larger amounts and much lower for losses than for gains.

Kirby and Herrnstein's 1995 study in Psychological Science is the one the demo reproduces. Participants chose between a smaller, earlier reward and a larger, later one, with real money or goods, and the experimenters moved both delays closer in steps to see whether the preference would switch. It did: 34 of 36 participants reversed from the larger, later reward to the smaller, earlier one as both rewards approached. The authors concluded that the assumption of stable preference between two rewards separated by a fixed interval, which they called the stationarity axiom, does not describe human choice.

David Laibson's 1997 paper in the Quarterly Journal of Economics, "Golden eggs and hyperbolic discounting," brought the curve into formal economics with a model in which consumers who know their own future impatience use illiquid assets, such as housing and retirement accounts, as commitment devices.

The pattern appears whenever a cost or reward is near and the alternative is far. The rows below are illustrations drawn from the mechanism, not findings from the studies cited on this page.

The curve does not flatten with willpower, so the useful counters change when a decision is made or what it costs to reverse it.

• Decide from a distance. Choices about next month are made with a flatter curve than choices about today, which is why they are usually better. Make the decision early and in writing.
• Close the option to switch. Automatic savings transfers, deadlines with penalties and appointments with other people all turn a decision made from a distance into one that costs something to undo, which is the commitment Ainslie and Laibson describe.
• Move the far reward closer. A long goal with no near milestones is a flat curve against a steep one; a weekly check-in or a small immediate reward for each step puts something on the steep part of the curve on the right side.
• Add delay to impulses. A fixed waiting period before a purchase or a reply pushes the immediate option off the steep part of the curve, where it was winning, onto the flatter part, where the comparison is fairer.
• Watch for the magnitude effect. Thaler found that small amounts were discounted more steeply than large ones, so a string of small impulsive choices can cost more than one large decision that would have been made carefully.

Hyperbolic discounting is a statement about the shape of the discount curve. Several neighbors concern how people misjudge their future selves in other ways.

• It is not projection bias, the tendency to assume that future tastes will match present ones, as when shopping hungry fills the cart. Projection bias is a mistake about what the future self will want; hyperbolic discounting is a predictable change in how much the present self cares about the future one.
• It is not restraint bias, the overestimate of one's own ability to resist a temptation when it arrives. Restraint bias is a belief about self-control; hyperbolic discounting is the reason that belief is so often wrong.
• It is not the empathy gap, in which a person in a calm state underestimates how a hot state, such as hunger or anger, will change their choices. The empathy gap is about emotional states; the discounting curve applies even to a calm person choosing between two sums of money.