The fundamentals of hyperbolic discounting

A thought experiment to set the stage

Honestly answer these two questions, in order.

Question 1. Would you rather receive $100 today or $110 in one week?

Question 2. Would you rather receive $100 in 52 weeks or $110 in 53 weeks?

If you are like the vast majority of participants in classic behavioral-economics studies, you picked $100 today in question 1, and $110 in 53 weeks in question 2. The temporal distance between the two options is exactly the same (one week), and the implicit return is identical (+10 %). Yet your preference flipped.

This is hyperbolic discounting: the value our brain assigns to a future reward does not decrease linearly, nor even exponentially as classical economic theory assumed. It decays along a hyperbolic curve — very steep in the short term, then almost flat in the long term.

The detour through classical economics (and its failure)

The discounted-utility model (Samuelson, 1937)

Economist Paul Samuelson posited in 1937 that rational individuals discount future rewards according to an exponential function:

V(reward) = M × δ^t

where
  M = reward amount
  δ = discount factor (0 < δ < 1)
  t = delay in periods

This model has a crucial mathematical property: temporal consistency. If you prefer A to B today, you will still prefer A to B tomorrow, in a year, in ten years — because the same exponential function applies everywhere on the time axis.

It is elegant. It is simple. It is wrong as soon as you confront the model with actual humans.

Anomalies pile up (1970-1990)

Three streams of empirical work bring down the exponential model.

Strotz (1955) first theorized temporal inconsistency: if the discount function is not exponential, today's preferences can contradict tomorrow's. He showed mathematically that any non-exponential function produces predictable "preference reversals".

Ainslie (1975) reports a series of experiments on pigeons, rats, then humans, showing that those reversals do indeed occur. His conclusion becomes a reference: "Preferences between two rewards may reverse as the earlier one approaches in time."

Thaler (1981) asked participants what amount they would accept in 1 month, 1 year, 10 years in exchange for $15 today. The implicit annualized discount rates ranged from 345 % in the short term to 19 % in the long term. No exponential function at a fixed rate can produce such a result.

The mathematical formulation of hyperbolic discounting

Mazur's model (1987)

The most widely used expression today comes from James Mazur, who proposed:

V(reward) = M / (1 + k × t)

where
  M = amount
  t = delay
  k = individual-specific hyperbolic discount rate

This equation has a simple property to picture: it decays very fast in the first moments (the present moment is worth far more than one week from now), then very slowly afterwards (one year vs one year + one week become almost equivalent).

The curve, visualized

Perceived value
     |
 100 |•
     | \
  80 |  \
     |   \\
  60 |    \\\\
     |        \\\\\
  40 |             \\\\\\\\\\\\
     |                         \\\\\\\\\\\\\\\\\
  20 |                                          \\\\\\\\\\\\\\\\\\\\\\\\
     |
   0 +------------------------------------------------------> time
     0    1 wk    1 mo            6 mo           1 yr             10 yr

While a classical exponential decreases at the same rate everywhere, the hyperbola collapses almost vertically in the initial phase, then becomes nearly flat. This is exactly what we observe in humans.

The β-δ (quasi-hyperbolic) model — Laibson (1997)

David Laibson, at Harvard, proposed an operational refinement for economists:

V(t=0) = M
V(t>0) = β × δ^t × M

with β ∈ ]0, 1[ ("present bias")
     δ ≈ 0.95 - 1.00 (long-term discount factor)

Laibson's clever idea: separate a large value jump between now and later (the β factor, sometimes called the "present bias"), then a slower and more linear decay afterwards (the classical δ factor).

With this model:

• At t=0: full value (100 %)
• At t=1 (e.g. tomorrow): 75 % × 100 = 75 %
• At t=2 (day after tomorrow): 75 % × 95 = 71 %
• At t=10: 75 % × 0.95^10 ≈ 45 %

The big value jump happens between now and tomorrow. Anything not now is severely discounted from the very first instant — then the decay calms down. This is precisely what clinical studies measure.

Three emblematic preference reversals

The temporal inconsistency predicted by the hyperbolic model shows up constantly in real life. Three canonical examples.

Reversal #1 — The New Year's resolution

On December 31, you decide: "Starting January 1st, I will go to the gym every Monday." From far (1 week), the marginal cost of effort is low and the future reward (health, looks) outweighs it.

On Monday January 7 at 7 p.m., the couch enters the short-time window. Its value explodes in the hyperbolic model (β × value = jump), while health remains months away (nearly flat). Preference reverses. You stay home.

Reversal #2 — The B2B contract that never gets signed

A prospect tells you in March: "Very interesting, we'll sign in June for a September kick-off." Seen from March, signing in June takes 10 minutes and yields (estimated savings of $50k over 12 months). In June, the β jump reappears: "Signing now is immediate; the $50k is 12 months out, almost flat." The sense of opportunity disappears. The deal slips.

Reversal #3 — Retirement savings

Every survey shows working adults declaring they want to increase their saving rate in 6 months. When we come back 6 months later, the rate has almost never gone up. Same temporal asymmetry at play.

What this is NOT (essential distinctions)

Why this matters for sales, business and entrepreneurship

Hyperbolic discounting is not a lab curiosity. It is the structural explanation of phenomena every practitioner encounters.

Case #1 — The "14-day free trial" funnel

Why does a 14-day free trial typically convert 3 to 5 × better than a "6-month ROI" demonstration? Because the trial places the reward (immediate usage gain) inside the β window — the zone of maximum value weighting. The mathematically more profitable 6-month ROI is crushed by the hyperbolic weighting.

Case #2 — The "$1 first month" pricing

The psychological lever is threefold: the immediate commitment cost is minimal (β remains high on the trivial spend), the usage value is immediate, and the future cancel friction is itself hyperbolically discounted. "I'll cancel at month-end if needed" — cancellation, projected 30 days out, feels easy — but almost never gets done.

Case #3 — The "claim now" call-to-actions

The lexical repetition of now, today, immediately mechanically activates the reader's β window. This isn't copywriting cliché — it is neuro-temporal activation.

Case #4 — B2C cart abandonment

E-commerce cart abandonment is massively driven by the hyperbolic asymmetry: the spend is today, the benefit (delivery) is in 3 days. Reducing delivery to 24 h, or offering an immediate gratification ("next-purchase code"), restores perceived temporal coherence.

Case #5 — SaaS and the yearly-discount trap

The "save 2 months" annual offer looks rational. In practice it converts far less than the monthly offer: the yearly spend is immediate (heavy, in β window), while the promised savings are spread over 12 months (flat). The reversal often costs several percentage points of conversion.

The magnitude of the effect

Modern replications (Frederick, Loewenstein & O'Donoghue, 2002; Andreoni & Sprenger, 2012) provide operational bounds:

Same individual, same choice: discount rates 10 to 30 × higher in the short term vs the long term. The amplitude exceeds anything an exponential model can predict.

Key concept in one picture

Classical exponential model (RATIONAL):
   value(now) = value(later) × δ^t
   → constant decay, stable choices

Hyperbolic model (OBSERVED in humans):
   value(now) = HUGE
   value(tomorrow) = suddenly small (β jump)
   value(later than tomorrow) = almost the same regardless of "later"
   → unstable choices, predictable reversals

In summary

• Hyperbolic discounting is the non-linear form by which the human brain devalues future rewards: very rapid drop between now and soon, then near-plateau beyond.
• Mazur's model (V = M / (1 + k·t)) and Laibson's β-δ model (1997) are the two canonical formulations. The second is preferred in applied economics for its readability.
• The experimental observation goes back to Ainslie (1975), Thaler (1981), and has been confirmed hundreds of times since on humans, primates and several animal species.
• Temporal inconsistency (preference reversal depending on proximity) is what radically distinguishes the hyperbolic model from the exponential one — and explains why it is so hard to keep long-term commitments.
• The business applications are massive: free trials, introductory pricing, cart abandonment, churn, B2B sales cycles. Any offer design that ignores hyperbolic discounting leaves conversion on the table.

In Chapter 2, we'll dive into the neurobiological mechanisms that produce this curve: the limbic-system / prefrontal-cortex duality (McClure et al., 2004), the neurotransmitters involved, and modulation by stress, age and culture.