Hyperbolic discounting — The irrational behavior that might be rational after all

When I was in grad school I occasionally overheard people talk about how humans do something called “hyperbolic discounting”. Apparently, hyperbolic discounting was considered irrational under standard economic theory.

I recently decided to learn what hyperbolic discounting was all about, so I set out to write this blog post. I have to admit that hyperbolic discounting has been pretty hard for me to understand, but I think I now finally have a good enough handle on it to write about it. Along the way, I learned something interesting: Hyperbolic discounting might be rational after all.

Rational and irrational discounting

If I offered you $50 now or $100 in 6 months, which would you pick? It’s not crazy to choose the $50 now. One reason is that it’s a safer bet. If you had chosen the delayed $100, there’s a risk that I might forget about the deal when the time came to pay [personal note: I wouldn’t], and you would never get your money. Another reason is that if you invest the $50 now, you might be able to make up some of the remainder in interest.

Valuing immediate money more than future money is a rational behavior known as discounting. Everybody has their own discount factor. Some people might value money in 6 months at, say, 75% of what they’d value it today. Others might value it at 90%.

In the early 1980s, psychologist George Ainslie discovered something peculiar. He found that while a lot of people would prefer $50 immediately rather than $100 in 6 months, they would not prefer $50 in 3 months rather than $100 in 9 months. These two different scenarios are shown in the diagram below, where the green checks indicate the options that people tended to choose.

If you think about it, there’s something inconsistent about this behavior. The two scenarios are actually identical, just shifted by 3 months, and yet the same people behave differently depending on when the scenario would be presented. If we waited 3 months and then asked them again if they would prefer $50 immediately or $100 in 6 months, their original response to Scenario 1 implies they would prefer the fast money (i.e. they would have a high discount rate), but their original response to Scenario 2 implies they would prefer the delayed money (i.e. they would have a low discount rate). In other words, their present self today would make a decision in Scenario 2 that three months from now they will have regretted making. This behavior is time-inconsistent and is therefore considered irrational according to standard economic theory.

The usual way to achieve rational time-consistent discounting is with an exponential discount curve, where the value of receiving something at future time is a fixed fraction of its present value, and where is the constant discount rate.

With an exponential curve, a dollar delayed by six months is always worth the same fixed fraction of a dollar at the baseline date, no matter what the baseline date it is. If you are indifferent between $100 now and $50 in 6 months, you should also be indifferent between $100 in 3 months and $50 in 9 months. The discount rate is constant.

In contrast to an exponential curve, humans tend to show a hyperbolic discount curve, which is considered irrational according to standard economic theory. Confusingly, the hyperbolic discount curve is not defined by one of the hyperbolic functions you may remember from high school. Instead, it is defined as follows, where $ \tau $ is the relative time from now:

Whereas an exponential curve has a constant discount rate, a hyperbolic discount curve has a higher discount rate in the near future and lower discount rate in the distant future. That’s why the participants in Ainslie’s experiment cared more about the delay from 0 to 6 months than about the same delay from 3 to 9 months.

The apparent rationality of an exponential function is often presented either with a hazard rate interpretation or an interest rate interpretation. It turns out, however, that both of these interpretations make implausible assumptions about the world. The rest of this blog post describes how if we make some more plausible assumptions, hyperbolic discount function becomes rational.

Hazard Rate interpretation

According to the hazard-based interpretation of discounting, you should prefer immediate money to future money because there is a risk that the future money will never be delivered. This interpretation is more common in the animal behavior literature. (Pigeons, rats, and monkeys all show hyperbolic discounting.)

Apparent rationality of exponential discounting

Imagine that there could be an event at some time that could cause you to no longer receive your reward. Perhaps the person who owes you the money could die or lose their assets. Assuming a constant hazard rate (i.e. assuming the event is equally likely to happen at any time), the probability that the event has not happened by time $ \tau $ is:

The function is called a “survival” function because it describes the probability that the deal is still “alive” by time . The parameter is the constant hazard rate.

A key insight here is that the survival function can also be interpreted as a discount function. If someone says you can receive your reward at time $ \tau $ if you’re willing to wait for it, and if there’s only a 60% chance you’ll actually receive it, you should value that offer at 60% of the current value of the reward. Since the survival function is exponential, and since the survival function is the discount function, your discount function is also exponential, at least according to standard economic theory.

Rationality of hyperbolic discounting

The problem with assuming an exponential survival function is that it assumes you know the hazard rate $ \lambda $. In most situations, you don’t know exactly what the hazard rate is. Instead you have uncertainty around the parameter. Souza (2015) proposes an exponential prior distribution over the hazard rate.

The exponential prior is shown in the graph below. Although this curve looks like a discount function, it is not. It is a distribution over a parameter.

According to Souza, if you average over all the possible exponential survival functions generated by this prior, you get a hyperbolic survival function, or equivalently, a hyperbolic discount function. I’ll try my best to explain:


In the plot below, I’ve drawn 30 exponential survival functions in blue, each with a sampled from the prior distribution defined above. The pink curve is the mean of all of them. Notice that whereas the individual survival curves are all exponential, their mean is hyperbolic, with a distant future characterized by a flat slope and relatively high survival values.

Why do we get a hyperbolic function when we average many exponential functions? To understand why, you first need to know that the instantaneous discount rate of a discount function is the negative of its instantaneous slope divided by its height. This sounds like a mouthful but it actually makes sense if you think about it. At any point in time, the discount rate is the drop in value over one time unit (i.e. the slope), divided by the principal (i.e. the height).

With that in mind, let’s think about time points in the distant future, when almost all of the exponential curves have gone to zero except for the curve with the lowest discount rate. When averaging across these curves in the distant future, the many curves at zero effectively scale down the one remaining curve by the same factor across a broad range of dates. In the figure below, the averaging operation scales down the curve by 86% at two different dates separated by 15 months. When you scale down a curve by a constant factor across a broad range, you scale its height and its slope by the same factor, thus preserving the slope to height ratio (i.e. the discount rate). Therefore, when you average over possible discount functions in the distant future, your resulting discount function is dominated by the discount rate of the curve with the lowest discount rate.

As a consequence, the further you look into the future, the lower your discount rate. This is exactly what hyperbolic discounting is and exactly what humans do. For the choice between $50 now and $100 in 6 months, we apply a heavy discount rate. For the choice between $50 in 3 months and $100 in 9 months, we apply a lighter discount rate.

For a more qualitative intuition, you can think of it this way: When you make a deal with someone, you don’t know what the hazard rate is going to be. But if the deal is still alive after 80 months, then your hazard rate is probably favorable and thus the deal is likely to still be alive after 100 months. You can therefore have a light discount rate between 80 months and 100 months.

Interest Rate interpretation

While hazard functions are one way to explain discounting, another common explanation involves interest rates. A dollar now is worth more than a dollar in a year, because if you take the dollar now and invest it, it will worth more in a year.

Apparent rationality of exponential discounting

If the interest rate is 5%, you should be indifferent between $100 today and $105 in a year, since you could just invest the $100 today and get the same amount in a year. If the interest rate is constant, the value of the dollar will rise exponentially, according to $ v(\tau) = e^{0.05\tau} $, where $ v $ is the value. To rationally maintain indifference between a dollar and its equivalent value in the future after investment, your discount function should decay exponentially, according to $ s(\tau) = e^{-0.05\tau} $.

Rationality of hyperbolic discounting

The problem with this story is that it only works if you assume that the interest rate is constant. In the real world, the interest rate fluctuates.

Before taking on the fluctuating interest rate scenario, let’s first take on a different assumption that is still somewhat simplified. Let’s assume that the interest rate is constant but we don’t know what it is, just as we didn’t know what the hazard rate was in the previous interpretation. With this assumption, the justification for hyperbolic discounting becomes similar to the explanation in the blue and pink plots above. When you do a probability-weighted average over these decaying exponential curves, you get a hyperbolic function.

The previous paragraph assumed that the interest rate was constant but unknown. In the real world, the interest rate is known but fluctuates over time. Farmer and Geanakoplos (2009) showed that if you assume that interest rate fluctuations follow a geometric random walk, hyperbolic discounting becomes optimal, at least asymptotically as . In the near future, you know the interest rate with reasonable certainty and should therefore discount with an exponential curve. But as you look further into the future, your uncertainty about the interest rate increases and you should therefore discount with a hyperbolic curve.

Is the geometric random walk a process that was cherry picked by the authors to produce this outcome? Not really. Newell and Pizer (2003) studied US bond rates in the 19th and 20th century and found that the geometric random walk provided a better fit than any of the other interest rate models tested.


When interpreting discounting as a survival function, a hyperbolic discounting function is rational if you introduce uncertainty into the hazard parameter via an exponential prior (Souza, 2015). When interpreting the discount rate as an interest rate, a hyperbolic discounting function is asymptotically rational if you introduce uncertainty in the interest rate via a geometric random walk (Farmer and Geanakoplos, 2009).


Religions as firms

I recently came across a magazine that helps pastors manage the financial and operational challenges of church management. The magazine is called Church Executive.

Readers concerned about seasonal effects on tithing can learn how to “sustain generosity” during the weaker summer months. Technology like push notifications and text messages is encouraged as a way to remind people to tithe. There is also some emphasis on messaging, as pastors are told to “make sure your generosity-focused sermons are hitting home with your audience”.

Churches need money to stay active, and it’s natural that pastors would want to maintain a healthy cash flow. But the brazen language of Church Executive reminded me of the language of profit-maximizing firms. This got me thinking: What are the other ways in which religions act like a business?

This post is my attempt to understand religions as if they were businesses. This isn’t a perfect metaphor. Most religious leaders are motivated by genuine beliefs, and few are motivated primarily by profit. But it can still be instructive to view religions through the lens of business and economics, if only as an exercise. After working through this myself, I feel like I have a better understanding of why religions act the way they do.


As with any business, one of the most pressing concerns of a religion is competition. According to sociologist Carl Bankston, the set of religions can be described as a marketplace of competing firms that vie for customers. Religious consumers can leave one church to go to another. To hedge their bets on the afterlife, some consumers may even belong to several churches simultaneously, in a strategy that has been described as “portfolio diversification”.

One way that a religion can ward off competitors is to prohibit its members from following them. The Bible is insistent on this point, with 26 separate verses banning idolatry. Other religions have been able to eliminate competition entirely by forming state-sponsored monopolies.


Just like a business, religions need to determine how to price their product. According to economists Laurence Iannaccone and Feler Bose, the optimal pricing strategy for a religion depends on whether it is proselytizing or non-proselytizing.

Non-proselytizing religions like Judaism and Hinduism earn much of their income from membership fees. While exceptions are often made for people who are too poor to pay, and while donations are still accepted, the explicit nature of the membership fees help these religions avoid having too many free riders.

Proselytizing religions like Christianity are different. Because of their strong emphasis on growth, they are willing to forgo explicit membership fees and instead rely more on donations that are up to the member’s discretion. Large donations from wealthy individuals can cross-subsidize the membership of those who make smaller donations. Even free riders who make no donations at all may be worthwhile, since they may attract more members in the future.

Surge Pricing

Like Uber, some religions raise the price during periods of peak demand. While attendance at Jewish synagogue for a regular Shabbat service is normally free, attendance during one of the High Holidays typically requires a payment for seating, in part to ensure space for everyone.

Surge pricing makes sense for non-proselytizing religions such as Judaism, but it does not make sense for proselytizing religions such as Christianity, which views the higher demand during peak season as an opportunity to convert newcomers and to reactivate lapsed members. Thus, Christian churches tend to expand seating and schedule extra services during Christmas and Easter, rather than charging fees.

Product Quality

Just as business consumers will pay higher prices for better products, consumers of polytheistic religions will pay higher “prices” for gods with more wide-ranging powers. Even today, some American megachurches have found success with the prosperity gospel, which emphasizes that God can make you wealthy.

Of course, not all religious consumers will prefer the cheap promises of the prosperity gospel. For many religions, product quality is defined primarily by community, a sense of meaning, and in some cases the promise of an afterlife.

Software Updates

A good business should be constantly updating its product to fix bugs and to respond to changes in consumer preference or government regulation. Some religions do the same thing, via the process of continuous revelation from their deity. Perhaps no church exemplifies this better than the Church of Latter Day Saints.

For most of the history of the Mormon Church, individuals of African descent were prohibited from serving as priests. By the 1960s, as civil rights protests against the church received media attention, the policy became increasingly untenable. On June 1, 1978, Mormon leaders reported that God had instructed them to update the policy and allow black priests. This event was known as the 1978 Revelation on the Priesthood.

In the late 19th Century, when the Mormon Church was under intense pressure from the US Government regarding polygamy, the Church president claimed to receive a revelation from Jesus Christ asking him to prohibit it. This revelation, known as the 1890 Revelation, overturned the previous 1843 Revelation which allowed polygamy.

While frequent updates usually make sense in business, they don’t always make sense in religion. Most religions have a fairly static doctrine, as the prospect of future updates undermines the authority of current doctrine.

Growth and marketing

Instead of focusing only on immediate profitability, many businesses invest in user growth. As mentioned earlier, many religions are willing to cross-subsidize participation from new members, especially young members, with older members bearing most of the costs.

Christianity’s concept of a heaven and hell encouraged its members to convert their friends and family. In some ways, this is reminiscent of viral marketing.

International expansion

Facebook and Netflix both experienced rapid adoption, starting with a U.S. audience. But as U.S. growth began to slow down, both companies needed to look towards international expansion.

A similar thing happened with the Mormon church. By the 20th century, U.S. growth was driven only by increasing family sizes, so the church turned towards international expansion.

The graph below shows similar US and international growth curves for Netflix and the Church of Latter Day Saints.[1,2,3,4]


Like any company, most religions try to maintain a good brand. But unlike businesses, most religions do not have brand protection, and thus their brands can be co-opted by other religions. Marketing from Mormons and from Jehovah’s Witnesses tends to emphasize the good brand of Jesus Christ, even though most mainstream Christians regard these churches as heretical.

One of the most interesting risks to brands is genericide, in which a popular trademark becomes synonymous with the general class of product, thereby diluting its distinctive meaning. Famous examples of generic trademarks include Kleenex and Band-Aid. Amazingly, genericide can also happen to religious deities. The ancient Near East god El began as a distinct god with followers, but gradually became a generic name for “God” and eventually merged with the Hebrew god Yahweh.

Mergers and spin-offs

In business, companies can spin off other companies or merge with other companies. But with rare exceptions, religions only seem to have spin-offs. Why do religions hardly ever merge with other religions? My guess is that since there is no protection for religious intellectual property, religions can acquire the intellectual property of another religion without requiring a merger. Religions can simply copy each other’s ideas.

Another reason that religious mergers are rare is that religions are strongly tied to personal identity and tap into tribal thinking. When WhatsApp was acquired, its leadership was happy to adopt new identities as Facebook employees. But it is far less likely that members of, say, the Syriac Catholic Church would ever tolerate merging into the rival Syriac Maronite Church, even if it might provide them with economies of scale and more political power.

On Twitter, I asked why there are so few religious mergers and got lots of interesting responses. People pointed out that reconciliation of doctrine could undermine the authority of the leaders, and that there is little benefit from economies of scale. Others noted that religious mergers aren’t that rare. Hinduism and Judaism may have began as mergers of smaller religions, many Christian traditions involve mergers with religions they replaced, and that even today Hinduism continues to be a merging of various sects.

It’s worth repeating that economic explanations aren’t always great at describing the conscious motivations of religious individuals, who generally have sincere beliefs. Nevertheless, economic reasoning does a decent job of predicting the behavior of systems, and it’s been pretty interesting to learn how religion is no exception.


Part 2: A bipartisan list of people who argue in good faith

In Part 1, I posted a bipartisan list of people who are bad for America. Those people present news stories that cherry pick the worst actions from the other side so that they can get higher TV ratings and more social media points.

Here in Part 2, I post a list of people who don’t do that, at least for the most part. This isn’t a list of centrists. If anything, it is a more politically diverse list than the list in Part 1. This is a list of people who usually make good-faith attempts to persuade others about their point of view.

  • Megan McArdle (Twitter, Bloomberg) – Moderately libertarian ideas presented to a diverse audience
  • Noah Smith (Twitter, Bloomberg) – Center-left economics
  • Ross Douthat (Twitter, NYT) – Social conservatism presented to a left-of-center audience
  • Noam Chomsky (Website)
  • Conor Friedersdorf (The Atlantic)
  • Ben Sasse — Has the third-most conservative voting record in the Senate but never caricatures the other side and is very concerned about filter bubbles.
  • Julia Galef (Twitter) – Has some great advice for understanding the other side
  • Nicky Case (Twitter)
  • Fareed Zakaria (Washington Post) – Center-left foreign policy
  • Eli Lake (Twitter, Bloomberg) – Hawkish foreign policy
  • Kevin Drum (Mother Jones) – Center-left blogger who writes in good faith
  • John Carl Baker (Twitter) – One of the few modern socialists I have found who avoids in-group snark.
  • Michael Dougherty (Twitter, The Week)
  • Reihan Salam (Twitter, NRO)
  • Avik Roy (Twitter, NRO) – Conservative health care
  • Ezra Klein (Vox, early days at the American Prospect) – While at the American Prospect, Ezra did an amazing job trying to persuade people about the benefits of Obamacare. Vox, the explainer site that he started, sometimes slips into red meat clickbait. But to its credit, Vox has managed to reach a wide audience with mostly explainer content.

Reading the people on this list with an open mind will broaden your worldview.


Part 1: A bipartisan list of people who are bad for America

Imagine that alien researchers visited America to learn about our political culture. If they wrote a report to send back to their planet, I imagine it would look something like this:

Americans have split themselves up into two opposing political tribes. Most people who associate with these groups are well-intentioned, but occasionally some members of a tribe do something bad or say something dumb. Whenever this happens, members of the opposite side feel good about themselves.

Certain writers and media personalities have learned to exploit this fact for personal gain. They have found that they can maximize their TV ratings and social media points by writing news stories that either cherry pick the worst actions of the other side or which interpret the other side’s actions in the least charitable way possible. As a result, news readers have developed increasingly distorted beliefs about their political opponents. The civic culture of the society is broken.

Below is a bipartisan list of people who are stoking partisan outrage for personal gain. Some of them do it for retweets, some of them do it for TV ratings, and some of them – still culpable – do it because they have entered a filter bubble themselves, fueling their own distorted and harmful sense of mission.

  • Sean Davis (The Federalist) His Twitter account is deliberately uncharitable.
  • Sopan Deb (New York Times) During the presidential campaign, his Twitter feed was nonstop, “look what this stupid Trump supporter said”.
  • Stephen Miller (The Wilderness, ex-NRO)
  • Chris Cillizza (The Washington Post)
  • Sean Hannity (Fox News)
  • Tucker Carlson (Fox News)
  • Samantha Bee I know, she’s a comedian. I like jokes. But given how many people get their news from selectively edited comedy shows, it’s fair to say that comedians bear some responsibility.
  • John Oliver (HBO) It pains me to include him on this list, since he is funny and since his show also includes some constructive policy explainers. But much of the content is selectively edited clips that paint a very distorted picture of the other side.
  • Rachel Maddow (MSNBC)
  • Shaun King (Facebook personality)
  • Greg Gutfeld (Fox News comedian)

It doesn’t matter if some of the people on this list do accurate reporting. What matters is that their reporting is selective. It doesn’t matter if some of the people on this list support some good policy ideas. What matters is that listening to them will destroy your brain’s ability to understand where the other side is coming from. And it doesn’t matter if one side is more filter-bubbled than the other. Both sides are badly filter-bubbled. Avoiding the people in this list is a good place to start.

In Part 2, I’ll post a bipartisan list of people who argue in good faith.


Three questions for social scientists: Internet virtue edition

This isn’t news to anybody, but the internet is changing our culture. Recently, I’ve been thinking about how it has changed our moral culture, and I realized that most of our beliefs on this topic are weirdly in tension with one another. Below are three questions that I feel are very much unresolved. I don’t have any good answers to them, and so I think they might be good topics for social science research.

1. Moral Substitution and Moral Licensing versus Moral Contagion

When people do the Ice Bucket Challenge or put a Pride symbol on their profile avatar, they are sometimes accused of virtue signalling, a derogatory term akin to moral grandstanding. Virtue signallers are said to care more about showcasing their virtue than about creating real change.

Virtue signalling is bad, allegedly, for two reasons: First, instead of performing truly impactful moral acts, virtue signallers spend more time performing easy and symbolic acts. This could be called moral substitution. Second, after doing something good, people often feel like they’ve earned enough virtue points that they can get away with doing something bad. This well-studied phenomenon is called moral licensing.

While there are some clear ways that virtue signalling can be bad, there is another way in which it is good. Doing good things makes other people more likely to do good things. This process, known as moral contagion, was famously demonstrated in the Milgram experiments. Participants in those experiments who saw other participants behave morally were dramatically more likely to behave morally as well.

If the social science research is right, then we can conclude that putting a Pride symbol on your avatar make you behave worse (via moral licensing and moral substitution), but it makes other people behave better (via moral contagion). This leaves a couple of open questions:

First, how do the pros and cons balance out? Perhaps your Pride avatar is net positive if you have a large audience on Facebook, but net negative if you have a small audience. And second, how does moral contagion work with symbolic acts? Does the Pride avatar just make other people add Pride symbols to their avatars? Or does it make them behave more ethically in real and impactful ways?

We are beginning to get some quantitative answers to these questions. Clever research from Linda Skitka and others has shown that committing a moral act makes you about 40% less likely to commit another moral act later in the day, whereas hearing about someone else’s moral act makes you about 25% more likely to commit a moral act later in the day, although the latter finding fell short of statistical significance. More research is needed though, particularly when it comes to social media and symbolic virtue signalling.

2. Slacktivism versus Violent Revolution

This question is more for political scientists.

Many people are concerned that the internet encourages slacktivism, a phenomenon closely related to moral substitution. It’s easier to slap a Pride symbol on your Facebook than to engage in real activism. In this way, the internet is really a tool of the already powerful.

On the other hand, some people are concerned that the internet cultivates violent radicalism. Online filter bubbles create anger and online networks create alliances, ultimately leading to violent rhetoric and homegrown terrorism. Many observers already sense the undercurrents of violent revolution.

How can we be worried that the internet is causing both slacktivism and violent radicalism? One possibility is that we only need to worry about slacktivism, and that the violent rhetoric isn’t actually violent – it’s just rhetoric. But the other possibility is that the internet has made slacktivists out of people who otherwise wouldn’t be doing anything at all, and it has made violent radicals out of people who would otherwise be mere activists. I’m not sure what the answer is, but it would be useful to understand this more.

3. Political Correctness: Overton Windows versus Wolf Crying

Perhaps because of filter bubbles on both sides of the political spectrum, the term “political correctness” is back with a vengeance. Leaving aside the question of whether political correctness is good or bad, it would be interesting to understand whether it is effective. On the one hand, political correctness may help define an Overton Window, setting useful bounds around opinions that can be aired in polite company. But on the other hand, if the enforcers squeeze the boundaries too much, imposing stricter and stricter controls on the range of acceptable discourse, they risk undermining their own credibility by “crying wolf”. For what it’s worth, many internet trolls credit their success to a perception that so-called social justice warriors overplayed their cards. I’m not sure how much to believe them, but it seems possible.

Just in terms of effectiveness, is there a point at which political correctness starts to backfire? And more broadly, what is the optimal level of political correctness for a society? Neither of these questions seems easy to answer, but I would love to learn more.