The Discount Rate Nobody Can Pin Down

Every set of payment terms a Canadian firm offers or accepts is an implicit statement about how it values money over time. This article is about what the research says that valuation looks like, and about a methodological result suggesting a large part of what has been attributed to impatience was something else.

The Verdict, Stated First

Five claims, in descending order of confidence.

One. The field's own leading review calls the variation in its central estimate spectacular. That word is in a Journal of Economic Literature abstract, written by three of the discipline's most cited authors on the topic.

Two. A methodologically improved study found no present bias, and then showed what produces it. Andreoni and Sprenger report dynamically consistent preferences, hypothesise that differential risk and transaction costs explain the standard finding, and state that adding the risk back makes the hyperbolic pattern reappear.

Three. A meta-analysis reports no present bias in monetary rewards. The same source records a present-bias factor when real effort rather than money is at stake, which is a boundary rather than a refutation.

Four. The commercially useful number survives all of this. Whatever produces it, firms and individuals behave as though money is very expensive over short horizons, and a citing paper's computation from the review's own table puts the median at 35 percent a year.

Five. Standard trade credit terms price at almost exactly that rate. Declining a 2/10 net 30 discount costs 37.2 percent annualised on our own arithmetic. We do not claim a causal link and we think the coincidence is worth a business owner's attention.

Our Grades For These Claims

Applying the scheme from the first article in this series.

Grade A for the existence of the variation problem, stated in the review's own abstract which we obtained verbatim from the publishing association.

Grade A for the 2012 null result, from an abstract obtained verbatim from the publishing association and a working paper version obtained from the author's own university page.

Grade A for the authors' own explanation of it, quoted verbatim from that working paper.

Grade B for the meta-analytic result, which reaches us through a citing paper and whose central figure our source truncates before stating.

Grade C for the 35 percent figure, which is a citing paper's computation from the review's table rather than a number the review reports, and which we could not verify against the table itself.

Our position: present bias in monetary choices is considerably less secure than its use in policy and business writing implies, and the high short-horizon discount rates are real regardless of what causes them.

A Note On Method

Everything here is verified to August 2026.

We obtained the 2002 review's abstract verbatim from the publishing association and from three further independent sources[1][4]. We did not obtain the review, and in particular we did not obtain its Table 1, which is the source of every discount rate figure discussed below.

We obtained the 2012 paper's abstract verbatim from the publishing association[2] and passages of its working paper version from the lead author's own university page[3][5].

The 35 percent figure is a citing paper's own computation from the review's table, not a figure the review states[6].

The meta-analytic result reaches us through one citing sentence which truncates mid-figure[7], so we report its direction and not its magnitude.

We did not obtain the 1997 paper that gave the model its standard form, nor the meta-analysis, nor any of the corroborating studies.

All arithmetic is ours and uses invented parameters except where a published figure is named.

This article discusses research on intertemporal choice. It is not investment, credit, treasury or capital allocation advice.

The Anomaly, Concretely

What is actually being claimed, stated with numbers because the abstract version is slippery.

A citing source puts it precisely: "someone who prefers $100 today over $102 in 30 days should also prefer $100 in 1 year over $102 in 1 year and 30 days. But this assumption does not always hold... This function accommodates the possibility that someone may plausibly prefer $100 today over $102 in 30 days but also prefer $102 in 1 year and 30 days over $100 in 1 year, a technically inconsistent time preference."[8]

Four observations, ours.

The two choices involve an identical trade: thirty days of waiting for two percent more. Nothing about the trade differs; only when it starts.

Standard discounting says the answer must be the same in both cases, because the ratio between two payoffs thirty days apart is the same wherever you place them on the calendar.

The observed reversal is therefore not a claim that people are impatient, which would be unremarkable. It is a claim that their impatience depends on whether the sooner option is available now.

And that is what makes it a problem rather than a preference. A person who reverses will make plans they later break, not because circumstances changed but because the date arrived.

The Quasi-Hyperbolic Form

The model, which is simpler than its name.

A payoff arriving at delay t is valued at beta times delta to the power t, where delta is ordinary period-by-period discounting and beta is a single extra discount applied to everything that is not available immediately. A payoff available now is valued at full weight.

Reference lists identify the source as Laibson, D. (1997), Golden Eggs and Hyperbolic Discounting, Quarterly Journal of Economics[8]. We did not obtain it and report the functional form from citing descriptions.

Three observations, ours.

The whole model is one extra parameter. Beta equal to one recovers standard exponential discounting; beta below one produces present bias. That parsimony is why it took over.

Note precisely where beta applies. It discounts the future as a block, relative to now, rather than accumulating with delay. The gap between today and tomorrow gets beta; the gap between day 365 and day 366 does not.

And that asymmetry is the entire content of the theory. Everything present bias predicts follows from one option being available immediately and the other not.

The Reversal, Computed

Making it concrete. Our own arithmetic, illustrative parameters.

Take the two choices above. Choice A: $100 today against $102 in 30 days. Choice B: $100 in 365 days against $102 in 395 days.

With a daily delta of 0.9998, which is about 7 percent a year, and varying beta:

At beta = 1.000: waits in both. No reversal, as required.

At 0.990: waits in both. Still no reversal.

At 0.985: takes the money today in A, waits in B. Reversal.

At 0.950, 0.900 and 0.800: same reversal, more strongly.

The threshold is beta below 0.9863 at these parameters.

Two observations.

The reversal appears at a very mild present bias. A beta of 0.986 means the future as a whole is discounted by less than one and a half percent, and that is enough to flip the choice.

Read the 0.95 row plainly. The same person declines to wait a month for two percent when the money is available now, and accepts exactly that trade when both payments are a year out.

We Got It Wrong First

Continuing a practice from the last six articles, because this one failed on the first attempt in an instructive way.

Our first run used a daily delta of 0.999, and produced no reversal at any value of beta. We nearly reported that the effect was harder to generate than expected.

The error was in the parameter, not the model. A delta of 0.999 per day compounds to 0.970 over thirty days, which makes $102 in a month worth $98.99 today. Under those parameters the sooner option wins in both choices for reasons that have nothing to do with present bias, so no reversal is possible.

Three observations, ours.

The demonstration requires a delta patient enough that waiting is worthwhile in the absence of present bias, which is delta above 0.99934 per day, or an annual rate under about 21.4 percent. Our original figure was outside that range.

We caught it because the beta equals one row must show no reversal and the low-beta rows must show one, and a table where nothing changes across beta is a table where beta is not doing anything.

And the general lesson is the one this series keeps arriving at. A demonstration that produces no effect is more often a badly chosen parameter than a real null, which is exactly the reasoning we would criticise if it were used to dismiss a genuine null, so the discipline is to find the error rather than to assume one.

Beta Cancels When Nothing Is Immediate

A property of the model that emerged from doing the arithmetic and which we have not seen stated plainly anywhere. Ours.

In choice B, both payments are in the future. So both are multiplied by beta, and beta cancels from the comparison entirely.

Four observations.

This means present bias has no effect whatever on a choice between two future dates. The model predicts perfectly consistent behaviour once nothing is available now.

Which is why the effect is called present bias rather than impatience. It is a property of the boundary between now and later, not of time generally.

And it generates the commercial prediction directly. A firm choosing between two future courses of action is not subject to this at all. A firm choosing between acting now and acting later is.

That is also, we think, why the remedy that appears in the literature is a commitment device. Moving a decision from now-versus-later into later-versus-later removes beta from the comparison, which is a structural fix rather than an exercise of will.

The Critical Review

The survey the field treats as its stocktake.

Frederick, S., Loewenstein, G., and O'Donoghue, T. (2002), Time Discounting and Time Preference: A Critical Review, Journal of Economic Literature, 40(2), 351–401, June, DOI 10.1257/002205102320161311[1].

Its abstract: "This paper discusses the discounted utility (DU) model: its historical development, underlying assumptions, and 'anomalies', the empirical regularities that are inconsistent with its theoretical predictions. We then summarize the alternate theoretical formulations that have been advanced to address these anomalies. We also review three decades of empirical research on intertemporal choice, and discuss reasons for the spectacular variation in implicit discount rates across studies. Throughout the paper, we stress the importance of distinguishing time preference, per se, from many other considerations that also influence intertemporal choices."[1]

Three observations, ours.

The lead author is the same Shane Frederick whose three-item instrument the fifty-fourth article in this series examined, and who co-authored the paper on non-effects of repeated exposure to it that the fifty-ninth article reported. That is three appearances in this series in three different roles.

The review is a stocktake by proponents, not an attack. These are authors who work in the tradition, which makes the abstract's language more notable rather than less.

And the closing sentence is the one to carry: distinguishing time preference from the many other considerations that also influence intertemporal choices. That is a warning that the measured quantity is a composite, and the 2012 paper below is the strongest demonstration of it.

Spectacular Variation

The phrase, and what it is doing in a peer-reviewed abstract. Ours.

Four observations.

"Spectacular variation in implicit discount rates across studies" is not hedged language, and we verified the phrase verbatim across four independent reproductions of the abstract[9]. Reviews normally say heterogeneity, or considerable variation, and this one did not.

The claim is about the field's central estimated parameter. Discount rate is to intertemporal choice what effect size is to everything else in this series, and the review is saying nobody agrees what it is.

Note that the abstract promises to discuss the reasons for the variation rather than to resolve it, and we did not obtain those reasons, which is a gap we would rather name than write around.

And a paper's decision to foreground this in its abstract is a signal about what its authors thought most needed saying after three decades of work.

The Measured Quantity Is A Composite

The review's closing warning, developed, because everything after it in this article depends on taking it seriously. Ours.

The abstract ends by stressing "the importance of distinguishing time preference, per se, from many other considerations that also influence intertemporal choices."[1]

Four observations.

An elicited discount rate is whatever makes the observed choices consistent, which means it absorbs every reason a person might prefer sooner money, not only impatience.

Some of those reasons are entirely rational. A future payment may not arrive, may cost something to collect, may be worth less if inflation runs, and may be needed less if circumstances improve. All four push toward taking money now and none is a time preference.

Which means the spectacular variation and the composite warning are the same observation stated twice. A quantity that absorbs many different considerations will vary with whichever of them a given study happens to leave in.

And it is why the 2012 challenge below is a test of the warning rather than a separate finding. The authors removed two of the composite's components and watched the parameter change, which is what the review predicted would happen if the parameter was a composite.

Forty-Three Studies, Thirty-Five Percent

A number derived from the review's table by someone else, which we report with care because we could not check it.

An academic paper states: "Frederick, Loewenstein, and O'Donoghue (2002, table 1) survey 43 such studies. The estimates are quite dispersed, but the majority of them points to high discount rates, much higher than prevailing market interest rates. We compute the mean estimate in each of the studies covered by the survey, and then compute the median value of these means. We obtain an annual discount rate of 35%."[6]

Four observations, ours.

This is not a figure the review reports. It is a computation performed by a citing paper on the review's table, and it is described transparently: mean within each study, then median across studies.

We did not obtain the table and cannot verify the computation. We grade the figure C for that reason and use it as an order of magnitude rather than an estimate.

The qualitative claim around it is the more reliable part: the majority of estimates point to discount rates much higher than prevailing market interest rates. That is a statement about direction and it is what the commercial sections below rest on.

And the same source names the limitation immediately: these studies "use a single rate to exponentially discount future utility", which is the assumption the whole present-bias literature exists to challenge[6].

What Thirty-Five Percent Prices

Making the figure concrete. Our own arithmetic, applying that rate to invented amounts.

A promised $1,000, discounted at 35 percent a year, is worth:

In 30 days: $975.64. In 90 days: $928.67. In 180 days: $862.43.

In one year: $740.74. In two: $548.70. In three: $406.44.

Three observations.

At that rate a dollar three years out is worth forty cents. A firm applying it rejects nearly every investment with a slow payback, however sound the investment is.

The one-year figure is the useful anchor. Money a year away is worth about three quarters of money now, on the empirical rate, against roughly ninety-five cents at a plausible cost of capital.

And that gap is the practical content of the whole literature. The behavioural rate and the financial rate differ by a factor of several, and decisions made on the first will look indefensible under the second.

The Trade Credit Coincidence

An observation we found while computing the above, which we think is worth a business owner's attention. Our own arithmetic, standard trade credit formula.

Terms of 2/10 net 30 offer a two percent discount for paying twenty days early. Declining it means borrowing at:

2/10 net 30: 37.2 percent annualised.

1/10 net 30: 18.4 percent. 2/10 net 60: 14.9 percent. 3/10 net 30: 56.4 percent. 1/15 net 45: 12.3 percent.

Three observations.

The most common term in commerce, 2/10 net 30, annualises to 37.2 percent. The citing paper's median-of-means from the research literature is 35 percent.

Those two numbers were produced by completely unrelated processes. One is a convention of trade, the other a summary of laboratory and field estimates, and they land within about two points of each other.

The variation across terms is also instructive. Extending the net period from 30 to 60 days more than halves the implied rate, from 37.2 to 14.9 percent, because the borrowing period doubles while the discount stays fixed.

What We Do And Do Not Claim About It

Because a coincidence between two numbers is the kind of thing this series has criticised others for over-reading. Ours.

Four points.

We do not claim a causal link. Trade credit terms long predate the research literature, the 35 percent figure is one citing paper's computation from a table we could not obtain, and two numbers landing near each other is weak evidence of anything.

We do not claim the terms were set to exploit a bias. Nothing we obtained bears on why 2/10 net 30 became standard, and a supplier's pricing of early payment reflects its own funding costs and credit risk.

What we do claim is arithmetic and checkable. A firm that routinely declines a 2/10 net 30 discount is borrowing at over 37 percent a year, and that is a fact about its own behaviour regardless of any psychology.

And that is the useful form. The behavioural literature suggests firms will decline such discounts more often than a cost-of-capital calculation warrants, and the test is not psychological. It is a line item in the payables ledger.

The Challenge

The paper that reopened the question, published in a top general economics journal.

Andreoni, J., and Sprenger, C. (2012), Estimating Time Preferences from Convex Budgets, American Economic Review, 102(7), 3333–3356, December[2].

Its abstract opens by restating the problem: "Experimentally elicited discount rates are frequently higher than what seems reasonable for economic decision-making. Such high rates are often attributed to present-biased discounting. A well-known bias of standard measurements is the assumption of linear consumption utility."[2]

And the methodological trap it identifies: "Attempting to correct this bias using measures of risk aversion to identify concavity, researchers find reasonable discounting but at the cost of exceptionally high utility function curvature."[2]

Three observations, ours.

The first sentence concedes the empirical fact this article has been reporting. Elicited rates are implausibly high, and the dispute is about what that means rather than whether it is observed.

The second sentence names an assumption almost nobody thinks about. Standard measurement assumes utility is linear in the amounts offered, so a choice between $100 now and $102 later is treated as a choice between one hundred and one hundred and two utils. If utility is concave, that is wrong, and the error loads onto the estimated discount rate.

And the third sentence describes a trade-off the field had been stuck in: correcting for curvature using risk aversion measures produced sensible discount rates and absurd curvature. Neither parameterisation was believable.

Surprisingly, Dynamically Consistent

The result.

"We present a new methodology for identifying time preferences, both discounting and curvature, from simple allocation decisions. We find reasonable levels of both discounting and curvature and, surprisingly, dynamically consistent time preferences."[2]

A working paper version states the estimates directly: "We estimate annual discount rates substantially lower than normally obtained, dynamically consistent discounting, and limited though significant utility function curvature."[4]

Four observations, ours.

Dynamically consistent means no present bias. Not a small one; the paper reports consistency, which is beta equal to one in the terms of the model above.

And the discount rates were substantially lower than normally obtained, which addresses the spectacular-variation problem from the other end: a better method produced a more plausible number.

The word "surprisingly" is the authors' own, in the abstract. They did not expect this and say so.

The method matters and is stated in one line: the choices are allocations across two dates rather than binary picks, which lets discounting and curvature be identified together instead of one being assumed.

And They Explain What Produces It

The part that lifts this from a null result to a structural reappraisal, and it is the authors interrogating their own finding.

From their working paper: "Our results raise several important questions for future research. First, why did we find no evidence of present bias or hyperbolic discounting? One hypothesis is that this may be the result of measures we took to equate transaction costs of sooner and later payments and to increase confidence of receiving future payments. This interpretation suggests that some of the behavior attributed to present bias in the literature may actually be an artifact of differential risk or transactions costs over sooner and later payments."[3]

Four observations, ours.

The mechanism is not psychological at all. A payment promised in a month is less certain than cash in hand, and collecting it costs something. A rational person discounts it for both reasons and looks impatient.

The design fix follows directly. Equalise the transaction costs and raise confidence in the future payment, and the apparent bias goes with them.

The word "artifact" is theirs, and it is a strong claim about a large literature, stated as a hypothesis rather than a conclusion.

And this is the eleventh instance in this series of the pattern the fiftieth article counted. A famous result turns out to be reproducible from something other than its stated mechanism, and here the reproduction is by the people who found the null.

Adding The Risk Back

The confirmation, and it is the strongest form such a hypothesis can take.

The same paper states: "Indeed, in Andreoni and Sprenger (2009) we show that when differential payment risk is exogenously added back into the decision environment, a hyperbolic pattern of discounting appears."[5]

Four observations, ours.

This is removal and restoration. Take out the differential risk and the effect disappears; put it back and the effect returns. That is a far stronger test than either half alone.

It is also the same structure as the fifty-first article's second experiment, where a failed replication was followed by a study identifying what had produced the original result. Both go beyond reporting a null to demonstrating a cause.

We did not obtain the 2009 paper and report this sentence describing it.

And note the honesty of publishing it. The authors could have reported a clean null and left the interpretation open, and instead identified what they think produced decades of contrary findings.

Independent Corroboration

Other evidence pointing the same way, from the same paper's discussion.

It records: "Keren and Roelofsma (1995) and Weber and Chapman (2005) find that when applying increasing levels of risk to both present and future payments, present bias decreases substantially."[5]

We obtained neither study and report this characterisation.

Three observations, ours.

The logic is elegant. If present bias comes from the future being riskier than the present, then making the present risky too should shrink it, and two independent studies report exactly that.

These predate the 2012 paper by seven and seventeen years, so the hypothesis is not a post-hoc rescue of an inconvenient null.

And the same discussion describes the design principle: "All payments arrive at student mail boxes and individuals receive minimum payments both sooner and later regardless of experimental choice."[5] Both dates involve a trip to the mailbox and both are guaranteed non-zero, which removes the asymmetry.

The Meta-Analysis

Where the aggregate evidence sits, reported at the strength our source allows.

A citing paper records: "The meta-analysis by Imai, Rutter, and Camerer (2021) finds no evidence of present bias in monetary rewards, while finding a mean bias-corrected present-bias factor" for other reward types, with our source truncating before the value[7].

We did not obtain the meta-analysis and have one sentence about it, cut off mid-figure.

Four observations, ours.

No evidence of present bias in monetary rewards is a meta-analytic statement, not a single study, and it is consistent with the 2012 result rather than an isolated finding.

The second clause is the interesting one and we cannot complete it. The source says a present-bias factor was found once the reward was not money, and the number is exactly where our source stops.

That is the position the forty-third article called this series' most common frustration, and it is worse here because the missing figure is the one that would tell you how large the surviving effect is.

And we would rather report a truncated sentence with the gap marked than paraphrase around it, which is the choice this series has made throughout.

Money Against Effort

The boundary that emerges, and it changes what the finding means. Ours.

The same citing paper explains the methodological reason: "monetary earnings may not adequately capture consumption utility in the absence of liquidity constraints and decision isolation. Augenblick, Niederle, and Sprenger (2015) address this concern with CTB decisions in which individuals allocate real-effort tasks across time."[7]

Four observations, ours.

The objection to money is specific and good. Money can be saved, borrowed against and moved around, so a choice about when to receive it is not a choice about when to consume, and a person with any liquidity can undo the experimenter's timing.

Effort cannot be moved. A task allocated to next Tuesday must be done next Tuesday, which makes the timing binding in a way a cash payment is not.

So the emerging picture is a boundary rather than a refutation. Present bias appears where the thing being scheduled cannot be reallocated by the subject, and disappears where it can.

And that has a direct commercial reading. Your firm's cash decisions are the money case and your firm's work scheduling is the effort case, and on this evidence the second is where the bias lives.

What Actually Survives

Our reading, stated directly.

Five statements.

Elicited short-horizon discount rates are very high, and everyone agrees they are. The dispute is about what produces them, not whether they are observed.

The estimates vary spectacularly across studies, on the field's own review's own word, and we could not obtain the reasons it gives.

Present bias in monetary choices did not survive a better method. A top journal reports dynamically consistent preferences, and a meta-analysis reports no evidence in monetary rewards.

A substantial part of the standard finding may be differential risk and transaction costs. The authors of the null hypothesise it, and report that adding the risk back brings the pattern back.

And present bias with real effort is a live and different question, whose magnitude our source truncates before stating.

Your Payment Terms

The first application, and it does not depend on any of the disputed psychology. Ours, and not credit or treasury advice.

Four points.

Compute the implied rate on every discount you are offered and every one you offer. 2/10 net 30 is 37.2 percent annualised; 2/10 net 60 is 14.9 percent. The same two percent prices completely differently depending on the period it buys.

If you decline early payment discounts, write down the rate you are implicitly paying, and compare it with your actual borrowing cost. A firm on a line of credit at ten percent declining a 37 percent discount is making a mistake that has nothing to do with time preference.

If you offer them, notice what you are selling. A two percent discount for twenty days is expensive funding for you, and lengthening the net period is a cheaper way to achieve the same acceleration.

And the diagnostic is in your own ledger. Count how many available discounts went untaken last year and price them. That is a number, not a judgment about anyone's psychology, which is the form this series has repeatedly found works.

Your Capital Decisions

The second application. Ours, untested, and not capital allocation advice.

Four points.

The empirical literature suggests people behave as though money is far more expensive over short horizons than any financing rate, on the order of a citing paper's 35 percent median against market rates in single digits.

A firm applying an implicit rate of that size rejects almost every slow-payback investment. On our own arithmetic, a dollar three years out is worth forty cents at 35 percent and around eighty at eight.

The remedy is not exhortation. It is naming the discount rate explicitly in the decision document, because an implicit rate cannot be argued with and a stated one can.

And that connects to the fifty-fifth article's finding. Reading the implied rate off a decision already made is the same move as reading the implied switching cost off a supplier you have not re-tendered, and it works for the same reason.

Commitment Devices

The third application, and the model tells you exactly why it works. Ours.

Four points.

As shown above, beta cancels from any choice between two future dates. Present bias operates only where one option is available now.

So the structural fix is to convert a now-versus-later choice into a later-versus-later one, by deciding in advance and removing the option to revisit.

The commercial forms are ordinary. Pre-authorised transfers, scheduled reviews, standing instructions, and decisions minuted before the date they take effect all have this property, and none requires anyone to be more patient.

And the caveat is the honest one. If the 2012 result holds for money, a firm's cash decisions may not need this at all, and the effort-scheduling case is where the model still predicts a problem.

What To Do

Price every trade discount you decline. 2/10 net 30 is 37.2 percent a year. If that exceeds your borrowing cost, declining it is an error you can see in the ledger.

State the discount rate in your decision documents. An implicit rate cannot be challenged and an explicit one can, and the literature suggests implicit rates run far above financing costs.

Convert now-versus-later into later-versus-later. In the standard model the present-bias parameter cancels entirely from choices between two future dates, which is why pre-commitment works structurally rather than motivationally.

Distinguish money decisions from effort decisions. A meta-analysis reports no present bias in monetary rewards; the surviving evidence concerns rewards a subject cannot reallocate, and work scheduling is that case.

Do not treat present bias in cash choices as settled. A top journal reports dynamically consistent preferences with a better method, and its authors hypothesise the standard finding is an artifact of risk and transaction costs.

Equalise friction across dates before concluding anyone is impatient. Two studies report present bias shrinking substantially when risk is applied to present and future payments alike.

Treat the 35 percent figure as an order of magnitude. It is a citing paper's computation from a table we could not obtain, and the review it draws on describes the underlying variation as spectacular.

Notice that a demonstration producing no effect is usually a parameter error. Ours was, and we describe it in the body rather than reporting the null we first produced.

The Limits Of This Analysis

Several caveats matter. This article discusses research on intertemporal choice and is not investment, credit, treasury or capital allocation advice; the applications are our own reasoning and untested. Everything is verified to August 2026. We did not obtain the 2002 review, only its abstract from four independent sources, and in particular we did not obtain its Table 1, which is the source of every discount rate figure discussed here; we also could not obtain the reasons it gives for the variation it describes, which is a gap sitting exactly where a reader would want an explanation. The 35 percent figure is a citing paper's own computation from that table, described transparently by that paper as a median of within-study means, and we could not verify it; we grade it C and use it as an order of magnitude. We did not obtain the 2012 paper beyond its published abstract and passages of two working paper versions from the lead author's own university pages. We did not obtain the 2009 paper in which the authors report that adding differential risk back produces the hyperbolic pattern, and report one sentence describing it. We did not obtain the meta-analysis, and our single citing sentence about it truncates immediately before the figure that would indicate how large the surviving effect is. We did not obtain the 1997 paper that gave the model its standard form, nor either of the two corroborating studies, nor the real-effort study, and report all from citing descriptions. All arithmetic is ours. The reversal demonstration uses invented parameters and our first attempt used a delta too impatient to permit any reversal at any beta, which is described and corrected in the body. The trade credit computation is the standard formula applied to standard terms, and the proximity between 37.2 and 35 percent is a coincidence we explicitly decline to interpret causally. And the whole picture rests on a literature whose own leading review calls its central estimate spectacularly variable.

Related Reading

Behavioural Finance One Percent Feels Like Five The other axis every decision runs on, and the other half of prospect theory. No. 63 Four Rounds And Seven Hundred Million Trips A famous finding said workers stop early when earnings are good. Eighteen years and four rounds later, a study of every New York taxi trip from 2009 to... Behavioural Finance Three Questions The same lead author, in a different role, on a different instrument.

Frequently Asked Questions

What is present bias?

A single extra discount applied to everything not available immediately. It predicts that someone who takes $100 today over $102 in a month will nonetheless wait the extra month when both payments are a year away. The same trade, opposite choice, depending only on whether one option is available now.

Is it well established?

Less than its use implies. A 2012 paper in the American Economic Review reports dynamically consistent time preferences using a better method, and a meta-analysis reportedly finds no evidence of present bias in monetary rewards. The evidence that survives concerns rewards a person cannot reallocate, such as effort.

What produces the standard finding then?

Its own challengers hypothesise differential risk and transaction costs: a future payment is less certain and costs more to collect, so discounting it is rational and looks impatient. They report that adding differential payment risk back into the environment makes the hyperbolic pattern reappear, which is a removal-and-restoration test.

What is the number I should actually use?

None from this literature, and one from your own ledger. A citing paper's computation from the review's table gives a median of 35 percent a year, but the review itself describes the variation across studies as spectacular. The usable figure is the implied rate on discounts your firm declined.

What is the trade credit point?

Declining 2/10 net 30 means borrowing at 37.2 percent annualised on our own arithmetic, which is close to the 35 percent median from the literature. We decline to interpret that causally. The checkable part is that a firm on credit at ten percent declining a 37 percent discount is making a visible error.

Why do commitment devices work?

Because in the standard model the present-bias parameter cancels from any choice between two future dates. Deciding in advance converts a now-versus-later choice into a later-versus-later one, removing the parameter from the comparison. That is structural rather than a matter of willpower.

Did your own arithmetic fail?

Yes, on the first attempt. We used a daily discount factor so impatient that waiting a month for two percent was never worthwhile, which makes a reversal impossible at any level of present bias. The demonstration needs a factor above 0.99934 per day. We describe the error rather than reporting the null we first produced.

References

1. Frederick, S., Loewenstein, G., & O'Donoghue, T. (2002). Time Discounting and Time Preference: A Critical Review. Journal of Economic Literature, 40(2), 351–401, June, DOI 10.1257/002205102320161311. Publishing association's own record reproducing the abstract in full: on the paper discussing the discounted utility model, its historical development, underlying assumptions and anomalies, being the empirical regularities inconsistent with its theoretical predictions; on the authors summarizing the alternate theoretical formulations advanced to address these anomalies; on the authors reviewing three decades of empirical research on intertemporal choice and discussing reasons for the spectacular variation in implicit discount rates across studies; and on the authors stressing throughout the importance of distinguishing time preference per se from many other considerations that also influence intertemporal choices. Note: the publishing association's record. We obtained the abstract only; we did not obtain the review, and in particular we did not obtain its Table 1, which is the source of every discount rate figure discussed in this article, nor the reasons it gives for the variation it describes. pubs.aeaweb.org
2. Andreoni, J., & Sprenger, C. (2012). Estimating Time Preferences from Convex Budgets. American Economic Review, 102(7), 3333–3356, December, DOI 10.1257/aer.102.7.3333. Publishing association's own record reproducing the abstract in full: on experimentally elicited discount rates frequently being higher than what seems reasonable for economic decision-making; on such high rates often being attributed to present-biased discounting; on a well-known bias of standard measurements being the assumption of linear consumption utility; on researchers attempting to correct this bias using measures of risk aversion to identify concavity finding reasonable discounting but at the cost of exceptionally high utility function curvature; on the authors presenting a new methodology for identifying time preferences, both discounting and curvature, from simple allocation decisions; and on the authors finding reasonable levels of both discounting and curvature and, surprisingly, dynamically consistent time preferences. Note: the publishing association's record. We obtained the abstract only and not the paper. aeaweb.org
3. Working paper version of the same study, hosted on the lead author's own university department page, reproducing passages of its discussion: on the authors' results raising several important questions for future research; on the first being why they found no evidence of present bias or hyperbolic discounting while so many other studies using cash rewards do find present bias; on one hypothesis being that this may be the result of measures taken to equate transaction costs of sooner and later payments and to increase confidence of receiving future payments; and on that interpretation suggesting some of the behavior attributed to present bias in the literature may actually be an artifact of differential risk or transactions costs over sooner and later payments. Note: a working paper version on the author's own university page. Our source for the authors' explanation of their own null result, which does not appear in the published abstract. econweb.ucsd.edu
4. Economics working paper database record for an earlier version of the same study, reproducing an abstract stating that the authors estimate annual discount rates substantially lower than normally obtained, dynamically consistent discounting, and limited though significant utility function curvature; carried on a page which also identifies Andersen, S., Harrison, G. W., Lau, M. I., and Rutström, E. E. (2008), Eliciting Risk and Time Preferences, Econometrica, 76(3), 583–618; Meier, S., and Sprenger, C. (2009), Present-Biased Preferences and Credit Card Borrowing; and Andersen and colleagues (2014), Discounting behavior: A reconsideration, European Economic Review, 71(C), 15–33. Note: a working paper database record. Used for the earlier abstract's more explicit statement of the estimates, and for independent confirmation of related citations; we obtained none of the works listed. ideas.repec.org
5. A second hosted copy of the 2012 study's working paper version, reproducing further discussion passages: on Keren and Roelofsma (1995) and Weber and Chapman (2005) finding that when applying increasing levels of risk to both present and future payments, present bias decreases substantially; on the authors' experimental methodology being designed to eliminate differential risk, being part of differential transaction costs, between sooner and later payments; on all payments arriving at student mail boxes with individuals receiving minimum payments both sooner and later regardless of experimental choice; and on the authors showing, in Andreoni and Sprenger (2009), that when differential payment risk is exogenously added back into the decision environment a hyperbolic pattern of discounting appears. Note: a second hosted copy of the working paper. Our source for the removal-and-restoration result and for the two corroborating studies, none of which we obtained. web.stanford.edu
6. Academic preprint on macroeconomic modelling, section on the time discount rate, recording that since the 1970s many studies have estimated time discount rates using field and laboratory experiments and real-world behavior; that Frederick, Loewenstein and O'Donoghue (2002, table 1) survey 43 such studies; that the estimates are quite dispersed but the majority point to high discount rates much higher than prevailing market interest rates; that the citing authors compute the mean estimate in each study covered by the survey and then the median value of those means, obtaining an annual discount rate of 35 percent; that one immediate limitation is that those studies use a single rate to exponentially discount future utility; and that exponential discounting does not describe reality well because people seem to choose more impatiently for the present than for the future. Note: an academic preprint. The 35 percent figure is this paper's own computation from a table we did not obtain and could not verify; we grade it C and use it as an order of magnitude only. arxiv.org
7. Academic preprint on temptation and intertemporal choice, section on empirical estimates of present bias, recording that while many studies have used monetary rewards to measure present bias, Andreoni and Sprenger (2012) pioneered the convex time budget methodology which elicits monetary-prize allocations between two time periods at various interest rates, allowing risk preferences and quasi-hyperbolic discounting parameters to be estimated jointly; that monetary earnings may not adequately capture consumption utility in the absence of liquidity constraints and decision isolation; that Augenblick, Niederle and Sprenger (2015) address this concern with convex time budget decisions in which individuals allocate real-effort tasks across time; and that the meta-analysis by Imai, Rutter and Camerer (2021) finds no evidence of present bias in monetary rewards while finding a mean bias-corrected present-bias factor, with our source truncating before the value. Note: an academic preprint. Our only source for the meta-analysis, whose central figure it truncates before stating; we did not obtain the meta-analysis or the real-effort study. arxiv.org
8. Repository page for the 2002 review carrying scholarly citing text, recording the implication that someone who prefers $100 today over $102 in 30 days should also prefer $100 in 1 year over $102 in 1 year and 30 days; that this assumption does not always hold and a hyperbolic discount function is sometimes used to reflect it, citing Frederick and colleagues (2002), Laibson (1997) and Loewenstein and Prelec (1992); and that such a function accommodates the possibility that someone may prefer $100 today over $102 in 30 days but also prefer $102 in 1 year and 30 days over $100 in 1 year, a technically inconsistent time preference. Note: a repository page reproducing text from a citing paper. Our source for the concrete statement of the anomaly and for the attribution of the functional form; we did not obtain the 1997 paper. researchgate.net
9. Bibliographic service record for the 2002 review, confirming the citation as Shane Frederick, George Loewenstein and Ted O'Donoghue, Journal of Economic Literature, volume 40, pages 351–401, 2002, and reproducing the abstract identically to reference 1; together with a separate economics database record giving the citation as Journal of Economic Literature, American Economic Association, volume 40(2), pages 351–401, June. Note: two further independent reproductions of the abstract, used to confirm the phrase "spectacular variation" verbatim. semanticscholar.org

This article discusses research on intertemporal choice and is not investment, credit, treasury or capital allocation advice. The 2002 review was not obtained beyond its abstract, and its Table 1, the source of every discount rate figure here, was not obtained. The 35 percent figure is a citing paper's own computation from that table and could not be verified. The 2012 paper was obtained as an abstract plus working paper passages. The meta-analysis reaches this article through one citing sentence that truncates before its central figure. All arithmetic is the authors' own; the reversal demonstration failed on first attempt due to a parameter error, described and corrected in the body. The proximity between the trade credit rate and the literature median is a coincidence this article declines to interpret causally.