Hyperbolic Discounting

Hyperbolic Discounting

Hyperbolic discounting is a departure from exponential time preferences in which the period immediately ahead is discounted more sharply than any pair of equally-spaced future periods. The leading parametric form is the quasi-hyperbolic or $(\beta ,\delta )$ model of Laibson (1997):

$U_t=u(c_t)+\beta {\sum }_{k=1}^{\infty }{\delta }^{k}u(c_{t+k}),0<\beta <1,0<\delta <1.$

The standard exponential model is the special case $\beta =1$. The extra parameter $\beta$ applies a one-time “haircut” to all future utilities relative to the present, regardless of how distant they are. This single discontinuity at $k=0$ is enough to break the rational benchmark in two important ways.

Time inconsistency. Because $\beta$ attaches only to the boundary between “now” and “later,” the agent’s ranking of two future dated rewards changes as the earlier date approaches. Concretely: people prefer $$100$ today to $$110$ in $10$ days (present boundary active, $\beta$ applies to the $$110$), but prefer $$110$ in $50$ days to $$100$ in $40$ days (both rewards are “later,” $\beta$ cancels on both sides). The preferred choice reverses as the earlier date moves into the present. An exponential discounter is immune to this — her ranking depends only on the relative delay, not on which date is “now.”

Two behavioral types. Behavior under hyperbolic discounting depends crucially on whether the agent is aware of her own inconsistency (O’Donoghue and Rabin, $1999$):

• Naïfs believe their future selves will follow through on plans made today. They repeatedly plan to save/exercise/quit “starting next week” and repeatedly fail, without learning.
• Sophisticates correctly anticipate that their future selves will also present-bias. They either (a) play a subgame-perfect equilibrium against their future selves, or (b) use commitment devices — binding arrangements made today to constrain tomorrow’s choices (automatic $401$(k) contributions, Christmas-club accounts, forfeiting deposits to StickK).

Household-finance relevance. Present bias is the leading behavioral explanation for under-saving for retirement, credit-card borrowing alongside liquid assets (8.10.5. Household Liabilities Puzzles), procrastination on $401$(k) enrollment, and the dramatic success of default-based nudges such as auto-enrollment and Thaler & Benartzi’s Save More Tomorrow. It is a psychology-of-preferences bias in the lecture’s taxonomy, sitting alongside prospect theory and ambiguity aversion as one of the three core departures from Expected Utility.

Intuition

Most dynamic models in economics assume exponential discounting: lifetime utility is a geometric sum
$U_t=u(c_t)+\delta u(c_{t+1})+{\delta }^{2}u(c_{t+2})+\cdots$
with a single per-period discount factor $\delta \in (0,1)$. This form is mathematically convenient and delivers time-consistent plans, but it fails to describe how real people trade consumption across time. Two stylized observations from the lab are incompatible with any single $\delta$:

• People prefer $$100$ today to $$110$ in $10$ days.
• The same people prefer $$110$ in $50$ days to $$100$ in $40$ days.

Both choices involve the same $10$-day delay for the same $$10$ premium. An exponential discounter ranks them identically; a real subject reverses her preference depending on whether the earlier reward is now or later. Something about the present boundary is special, and hyperbolic discounting is the minimal modification that captures it.

Exponential discounting has one clean property: a reward delayed by one day always costs the same fraction of its value, no matter whether the delay is from today to tomorrow or from day $100$ to day $101$. This “stationarity” is what makes exponential agents dynamically consistent — a plan made today about future consumption is still optimal when that future arrives. Real people are not stationary. A reward moved from “today” to “tomorrow” feels much more painful than the same reward moved from “day $100$” to “day $101$,” even though the delay is the same. Something special happens at the boundary between now and not now.

The quasi-hyperbolic model captures this with the minimum possible machinery: a single extra parameter $\beta <1$ that sits in front of the entire future. It keeps the standard geometric decay ${\delta }^k$ for comparisons between two future periods (so a day-$100$-versus-day-$101$ tradeoff looks normal) but adds a one-shot penalty for any delay past the present. The central tension the model formalizes is between a farsighted long-run self, who correctly weights distant periods against each other, and an impulsive short-run self, who overweights whichever period happens to be “now.” Every period the short-run self takes control, and the long-run self’s plans are overwritten. Whether this leads to ruinous procrastination or to elegant self-binding depends entirely on whether the agent knows which self is in charge.

Properties

($(\beta ,\delta )$ Formulation) A single extra parameter captures present bias.

The quasi-hyperbolic utility function is
$U_t(c_t,c_{t+1},c_{t+2},\dots )=u(c_t)+\beta \delta u(c_{t+1})+\beta {\delta }^{2}u(c_{t+2})+\beta {\delta }^{3}u(c_{t+3})+\cdots$
with $\beta \in (0,1]$ and $\delta \in (0,1)$. The ratio of weights on adjacent future periods is
$\frac{\beta {\delta }^{k+1}}{\beta {\delta }^k}=\delta ,$
exactly the exponential case. But the ratio of the weight on period $t$ to the weight on period $t+1$ is
$\frac{1}{\beta \delta }>\frac{1}{\delta },$
strictly larger than exponential whenever $\beta <1$. Only the first step — from $t$ to $t+1$ — is discounted more heavily than exponential; all later steps are exponential. This is why the model is called quasi-hyperbolic: it is a discrete two-parameter approximation to the fully hyperbolic form $D(k)=1/(1+\alpha k)$ studied in psychology (Ainslie $1975$), chosen because $(\beta ,\delta )$ is analytically far more tractable.

(Preference Reversal) The same ranking flips as the earlier date approaches.

Consider a choice between (A) $$100$ at delay $k$ and (B) $$110$ at delay $k+10$. A quasi-hyperbolic agent with $\beta <1$ picks differently at $k=0$ than at $k=40$:

• At $k=0$: weights are $u(100)$ vs. $\beta {\delta }^{10}u(110)$. The present reward avoids the $\beta$ penalty; the delayed reward pays it. For $\beta$ small enough (e.g. $\beta =0.7$, $\delta =0.99$), $(A)$ wins.
• At $k=40$: weights are $\beta {\delta }^{40}u(100)$ vs. $\beta {\delta }^{50}u(110)$. Both rewards carry the same $\beta$; it cancels. The comparison reduces to $u(100)$ vs. ${\delta }^{10}u(110)$, which for reasonable $\delta$ favors $(B)$.

The ranking has reversed even though nothing about the choice itself changed — only the agent’s position in time relative to the earlier date. An exponential agent faces the same comparison at both points in time and picks the same option. Preference reversal is the behavioral fingerprint of hyperbolic discounting and the empirical basis on which the model is identified in the lab and field.

(Naïfs) Naïfs repeatedly plan and fail.

A naïf (O’Donoghue and Rabin, $1999$) believes her future selves will follow the plan she makes today. Formally, at every date $t$ she solves her optimization problem assuming ${\beta }_{\text{future}}=1$ even though her own current $\beta <1$. This produces a characteristic failure pattern: the agent genuinely intends to start saving, exercising, or studying next week, believes sincerely she will do so, and is surprised when next week she again postpones. Naïfs never learn, because each postponement is followed by a fresh plan that is equally unrealistic. The cost of being naïve grows with the complexity of the task — trivial decisions recover quickly, but a naïf who needs to choose a retirement plan may procrastinate for decades. Empirically, $401$(k) enrollment is the canonical example: large majorities of workers say they intend to enroll “soon” but never do until the default changes (Madrian & Shea, $2001$).

(Sophisticates) Sophisticates use commitment devices.

A sophisticate correctly anticipates that her future self will also present-bias, so she solves a dynamic game in which each period’s self plays against the next. The resulting subgame-perfect equilibrium is typically inefficient relative to the long-run self’s optimum, but it is internally consistent. A sophisticate who foresees that her period-$t+1$ self will overspend uses a commitment device — an action today that binds tomorrow — to move the equilibrium closer to the long-run optimum. Examples include automatic payroll deduction for retirement savings, Christmas-club accounts that refuse withdrawals before December, StickK contracts that forfeit money to an unappealing charity on failure, locked-away cash, and deliberately not keeping tempting food in the house. Each of these makes sense only for an agent who simultaneously wants the future outcome and knows she will not choose it in the moment — a signature diagnostic for sophisticated present bias.

(Household-Finance Implications) Present bias explains under-saving and over-borrowing.

The same $\beta <1$ that makes agents procrastinate on retirement enrollment also makes them over-weight the immediate pleasure of a consumption purchase against its distant financing cost. Together, these forces generate the twin household puzzles documented in 8.10.5. Household Liabilities Puzzles: households save too little for retirement and hold high-interest credit-card debt alongside liquid assets. Laibson ($1997$) shows that a quasi-hyperbolic household will rationally hold an illiquid asset (home equity, retirement account) precisely because illiquidity functions as a commitment device against the future self — the household chooses assets whose withdrawal cost exceeds the immediate temptation to spend. The illiquidity itself has positive welfare value, which is impossible in any exponential model.

(Save More Tomorrow) Defaults and pre-commitment exploit the inconsistency.

Thaler & Benartzi ($2004$) designed Save More Tomorrow (SMarT), a commitment device that asks workers to pre-commit today to increasing their $401$(k) contribution rate at the time of future pay raises. Because the increase is a future decision, today’s present-biased self is happy to agree to it ($\beta$ affects only the immediate choice, not the distant one). When the pay raise arrives, the default is to increase the contribution, and present-biased inertia works for saving rather than against it. SMarT boosted average contribution rates from $3.5\%$ to $13.6\%$ over four pay raises at one firm. The program works because it correctly identifies that hyperbolic discounters will under-save when asked to act now but will agree to act later — the exact pattern the $(\beta ,\delta )$ model predicts.

Examples

The classic preference reversal

An experimental subject with quasi-hyperbolic preferences $\beta =0.7$, $\delta \approx 1$ (so the only relevant discounting is the present-future boundary) is asked two questions.

Question 1. “Do you prefer $$100$ today or $$110$ in one week?”

• Weight on today’s $$100$: $u(100)=100$ (linear utility for simplicity).
• Weight on next week’s $$110$: $\beta \cdot u(110)=0.7\cdot 110=77$.
• Subject picks $$100$ today.

Question 2. “Do you prefer $$100$ in $52$ weeks or $$110$ in $53$ weeks?”

• Weight on $$100$ in year: $\beta \cdot u(100)=0.7\cdot 100=70$.
• Weight on $$110$ in year-plus-one-week: $\beta \cdot u(110)=0.7\cdot 110=77$.
• The $\beta$ cancels — both rewards are in the “future” region. Subject picks $$110$.

Now fast-forward $52$ weeks. The two rewards the subject previously preferred in ranking $B>A$ have become today’s $$100$ vs. next week’s $$110$ — the exact Question 1 setup. She now prefers $A>B$, reversing her stated plan. An exponential discounter would be inconsistent to do this; a quasi-hyperbolic one cannot do otherwise.

A naïf procrastinating on $401$(k) enrollment

A new employee with $\beta =0.6,\delta =0.97$ is offered a $401$(k) match worth $$3,000$ per year in foregone wages. Enrolling takes $30$ minutes and costs a small amount of present leisure (utility cost $$20$). The net present value of enrolling is unambiguously enormous — $$3,000$ per year for decades against $$20$ today.

The naïf’s calculation at date $t$ (using her own $\beta$ for today and assuming $\beta =1$ for the future):

• Enroll today: $-20+\beta \delta \cdot 3000=-20+0.6\cdot 0.97\cdot 3000=1726$.
• Enroll next week (as planned): $0+\beta \delta \cdot (-20+\delta \cdot 3000)=0.6\cdot 0.97\cdot (-20+2910)=1682$.

The naïf plans to enroll next week, believing she will follow through. The two options are within rounding distance, and the trivial $$20$ cost tips it slightly to postpone. When next week arrives, her position is identical: she is again looking at an immediate $$20$ cost versus a future $$3,000$ stream, and she again postpones by \44. She repeats this every week for years, losing \$$3{,}000 per year of match the entire time. The key failure is not that the decision is hard — it is that she never corrects the belief that next week’s self will behave differently from this week’s self. A sophisticate running the same calculation would recognize the pattern after one failure and sign up for automatic enrollment instead.

Commitment via Save More Tomorrow

Shlomo Benartzi and Richard Thaler piloted SMarT at a mid-sized manufacturing firm in $1998$. At enrollment, employees with $\beta$-$\delta$ preferences were asked not “increase your contribution rate now” (which their present self would reject) but “pre-commit to increase your contribution rate by three percentage points at each of your next four pay raises.”

• At enrollment ($t=0$): the decision is purely about future consumption. The agent’s $\beta$ sits in front of every term in the calculation, so it cancels — she evaluates the choice as if exponential. She agrees.
• At pay raise ($t=\text{raise date}$): the increase is now default. To reject it, the present-biased self would need to actively call HR and opt out. The utility cost of opting out is small but non-zero, and $\beta$-discounting now weighs a small present cost (opting out) against a tiny present benefit (slightly higher paycheck). Inertia wins.
• Result: average savings rates rose from $3.5\%$ to $13.6\%$ over $40$ months. Workers who used SMarT ended up with four times the retirement savings of workers who refused it.

The mechanism is subtle: SMarT does not fix the hyperbolic agent. It routes the present bias through a channel where it supports the long-run goal instead of opposing it. Present-bias works against saving when the choice is “save now”; it works for saving when the choice is “un-save what you already committed to.” Same bias, opposite direction, because the timing of the decision changed.

See Also

• 8.10. Behavioral Household Finance — parent hub.
• 8.10.1. Stock Market Non-Participation — another puzzle where procrastination plausibly plays a role alongside frictions.
• 8.10.5. Household Liabilities Puzzles — credit-card debt with liquid assets, the canonical application of $(\beta ,\delta )$ preferences.
• 8.1. Principles of Behavioral Finance — the framework for classifying hyperbolic discounting as a psychology-of-preferences deviation.
• 8.5.1. Prospect Theory — the other major departure from Expected Utility in the same taxonomy.