The $100 Question
Would you rather have $100 today or $110 tomorrow? Take a moment. Most people take the $100. The bird in the hand. The certain thing. Now try this one: Would you rather have $100 in 30 days or $110 in 31 days? Most people say they'll wait the extra day for the $110. But here's the thing that should stop you cold: it's the same tradeoff. One day of waiting. Ten dollars. That's it.
If you chose differently on those two questions — and statistically, you almost certainly did — then congratulations: you are a time-inconsistent human being. Your preferences reverse depending on how close the decision feels. And that reversal, that wobble in the machinery of your desires, is one of the most consequential mathematical facts about human nature.
Economists have a word for this. Actually, they have several words, because economists love words almost as much as they love models. They call it hyperbolic discounting, and it explains everything from why your gym membership is a waste of money to why we can't agree on climate policy to why Ulysses had himself tied to a mast.
The Rational Discount Rate
Let's start with how economists thought people made decisions about time, back in the days when economists assumed people were rational. (Those were fun days. Everybody had consistent preferences and infinite computational power. It was like a utopia, except it didn't describe any actual humans.)
The standard model is called exponential discounting. The idea is simple: future rewards get multiplied by a constant discount factor for every period you have to wait. If you discount at rate δ, then something worth V₀ right now is worth this much to you at time t:
Exponential Discounting
V = V₀ · e−δt
The discount rate δ is constant — each additional day of waiting costs the same proportional amount.
The beauty of exponential discounting is time-consistency. If you prefer A over B today, you'll still prefer A over B when you re-evaluate the decision next week. Future-you and present-you always agree on the plan. It's as if there's one unified "you" making decisions across all of time, which is a lovely thought if you've never tried to keep a New Year's resolution.
Paul Samuelson proposed this model in 1937, and to his credit, he basically said: "I'm not claiming this is how people actually work, I'm just saying it's mathematically convenient."1 The economics profession heard the first part, ignored the second, and ran with it for about fifty years.
The Hyperbolic Swerve
Here's what actually happens inside your brain when you evaluate a future reward. Psychologist Richard Herrnstein and his colleagues discovered in the 1960s and 70s that both humans and pigeons (always a flattering comparison) discount future rewards not exponentially, but hyperbolically:2
Hyperbolic Discounting
V = V₀ / (1 + kt)
The discount rate decreases over time. The near future is discounted steeply; the distant future barely at all.
- V₀
- The objective value of the reward
- k
- The discounting parameter (higher = more impatient)
- t
- Time until you receive the reward
The crucial difference: with exponential discounting, the rate at which you discount is constant. Waiting from day 30 to day 31 costs you the same fraction of value as waiting from day 0 to day 1. But with hyperbolic discounting, the rate of discounting is enormous for the very near future and gentle for the distant future. Mathematically, the instantaneous discount rate at time t is k/(1+kt), which at t=0 is simply k — potentially huge — but shrinks toward zero as t grows.
This is why the $100-today-vs-$110-tomorrow question feels so different from the $100-in-30-days-vs-$110-in-31-days question. In the first case, you're in the steep part of the curve. The difference between "right now" and "tomorrow" feels enormous. In the second case, you're in the flat part. The difference between "a month from now" and "a month and a day from now" feels like nothing.
And that crossing of the curves? That's where all the trouble lives. When both options are far away, the hyperbolic discounter looks at them calmly and picks the bigger one. But as the smaller-sooner option gets close, the hyperbolic curve shoots up and the preference flips. You, sitting here in January, solemnly swear you'll start exercising in March. March arrives, and somehow the couch wins again. It's not that you're weak-willed. It's that you're literally a different decision-maker when the choice is right here.
The Marshmallow and Its Discontents
In 1972, Walter Mischel sat a bunch of four-year-olds in a room at Stanford's Bing Nursery School and gave them a cruel choice: you can have one marshmallow right now, or, if you wait until I come back, you can have two marshmallows.3
The videos are heartbreaking and hilarious. Kids stare at the marshmallow. They smell it. They lick it. They turn their chairs around so they don't have to look at it. Some eat it before Mischel is even out the door. Others hold out, grimly, for the full fifteen minutes.
The famous follow-up studies found that the kids who waited — the ones who could defer gratification — did better on their SATs, had lower BMIs, handled stress better, and generally turned into the kind of adults who floss regularly and max out their 401(k)s. The marshmallow test became a parable about the virtue of patience.
Like many iconic psychology results, the marshmallow test has had a rough replication journey. A 2018 study by Tyler Watts, Greg Duncan, and Haonan Quan found that once you controlled for the child's socioeconomic background, the predictive power of the marshmallow test shrank dramatically.4 It turns out that a kid from a stable, affluent home has more reason to trust that the second marshmallow will actually appear. The test may measure trust in institutions as much as impulse control. The math of discounting still holds — but the k parameter isn't just about your brain; it's about your world.
Here's what the marshmallow test really shows us mathematically: different people have different values of k. A high-k person has a steep hyperbolic curve — they discount the near future savagely. A low-k person has a flatter curve — they can wait. But everyone's curve is still hyperbolic. Even the patient kids aren't exponential discounters. They're just hyperbolic discounters with a smaller k.
Explore the Curves
Play with the parameters yourself. Adjust the discount rates and see where the exponential and hyperbolic curves diverge — and where preferences reverse.
Discounting Visualizer
Compare exponential and hyperbolic discounting. Watch for where the curves cross — that's preference reversal.
The Ulysses Contract
If you're a hyperbolic discounter — and you are — then you have a problem. The "you" who makes plans is not the same "you" who executes them. Monday-you is going to start that diet. But when Monday arrives, Monday-you has become Today-you, and Today-you wants pancakes.
The ancient Greeks understood this, or at least Homer did. In the Odyssey, Ulysses knows he won't be able to resist the Sirens' song when he hears it. So he does something brilliant: he has his sailors tie him to the mast and fill their own ears with wax. He pre-commits. He takes away his future self's ability to choose badly.5
This idea — the commitment device — is everywhere once you know to look for it. Christmas clubs were savings accounts that literally wouldn't let you withdraw money until December. Gym contracts charge you even when you don't go, which is the point: you're paying your future self a penalty for being lazy. The website StickK.com lets you put money on the line and donate it to a charity you hate if you don't meet your goal.6 The math of these devices is straightforward: they change the payoff structure so that the hyperbolic curve no longer reverses your preference.
David Laibson, the Harvard economist who's done more than anyone to bring hyperbolic discounting into mainstream economics, proposed a clever simplification in 1997. Instead of the full hyperbolic function, he suggested what's called the quasi-hyperbolic or β-δ model:7
Quasi-Hyperbolic (β-δ) Model
V0 = u0 + β · Σ δt ut
β < 1 captures present bias. You discount everything in the future by an extra factor β, on top of the normal exponential discounting δ.
- β
- Present-bias parameter (β = 1 means no bias; β = 0.7 means you value the future at 70% right off the bat)
- δ
- Standard exponential discount factor (patience over time)
The elegance of the β-δ model is that it captures the essential weirdness of hyperbolic discounting — the dramatic overweighting of "right now" — with just one extra parameter. Typical estimates put β around 0.7, meaning people instantly discount the entire future by about 30% just because it's not right this second.
Test Your Own Present Bias
Here's your chance to see how you actually make now-vs-later decisions — and whether a commitment device changes your outcomes.
The Commitment Device Game
Face 8 now-vs-later decisions. See your present bias score, then try again with commitment devices enabled.
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Results
Save More Tomorrow
The most consequential application of hyperbolic discounting might be Richard Thaler and Shlomo Benartzi's "Save More Tomorrow" program.8 The insight is devastatingly simple: people won't save more now, because "now" is the steep part of the curve. But they'll happily agree to save more starting with their next raise, because that's the flat part. It's the $100-today-vs-$110-tomorrow problem in reverse: instead of asking people to sacrifice today, you ask them to sacrifice in the future, when sacrifice feels cheap.
The results were staggering. In the first implementation at a mid-sized manufacturing company, savings rates went from 3.5% to 13.6% over four years. The workers didn't feel any pain — each increase coincided with a raise, so their take-home pay never actually went down. Thaler and Benartzi hadn't changed the economics. They'd changed the timing.
The Key Insight
Hyperbolic discounting isn't a flaw to be corrected — it's a feature to be designed around. The most effective interventions don't fight human nature; they use its own geometry against it. If people overweight the present, then frame the sacrifice as happening in the future. If preferences reverse when temptation arrives, then lock in the choice before it does.
Discounting the Planet
There's one domain where hyperbolic discounting becomes genuinely terrifying, and that's climate policy. When economists model the costs and benefits of reducing carbon emissions, they have to decide: how much do we discount the future? What is a dollar of climate damage in 2100 worth today?
If you use a standard exponential discount rate of 5%, then $1 of damage in 2100 is worth about $0.02 today. At that rate, catastrophic climate change seventy-five years from now is barely worth worrying about. Nicholas Stern's famous 2006 review used a near-zero discount rate and concluded we should spend massively on climate now. William Nordhaus used a higher rate and concluded we should go slow. They weren't really disagreeing about the science. They were disagreeing about the discount rate.
And here's the hyperbolic twist: if people actually discount hyperbolically, then the appropriate social discount rate decreases over time. Damage fifty years from now should be discounted less per year than damage five years from now. Several economists have argued that this is in fact what we should do — use a declining discount rate for long-term policy.9 It would dramatically increase the present value of avoiding climate catastrophe.
The math of procrastination, it turns out, is also the math of survival.
Monday Is Always Tomorrow
Here's the deepest thing about hyperbolic discounting: it means there is no single "you." There's the you who plans and the you who acts, and they have genuinely different preferences. The planner is wise and patient, gazing serenely at the flat part of the curve. The actor is urgent and hungry, trapped in the steep part near t = 0.
We call this "weakness of will," as if it's a moral failing. But the math tells a different story. It's not weakness — it's geometry. The shape of the curve changes depending on where you stand, and at every moment, you stand at the steepest point. The diet starts Monday because Monday is always tomorrow, and tomorrow is always in the flat, easy, abstract part of the curve — until it becomes today.
The good news is that once you see the curve, you can work with it. You can tie yourself to the mast. You can commit your future raises. You can put your alarm clock across the room so you have to get out of bed to turn it off. You can't make the curve exponential — that's not how humans work — but you can be the kind of hyperbolic discounter who knows they're a hyperbolic discounter.
And that, it turns out, makes all the difference.