temporal discounting

Why copyrights should be shorter

PNRJ core principles, public policy , , , , , , ,

Jul 3 JDN 2459783

The copyright protection for Mickey Mouse is set to expire in 2024, though a recently-proposed bill that specifically targets large corporations would cause it to end immediately. Steamboat Willie was released in 1928.

This means that Mickey Mouse has been under copyright protection for 94 years, and is scheduled to last 96. Let me remind you that Walt Disney has been dead since 1966. This is, quite frankly, ridiculous. Mickey Mouse should have been released into the public domain decades ago.

Copyright in general has quite a shaky justification, and there are those who argue that it should be eliminated entirely. There’s something profoundly weird—and fundamentally monopolistic—about banning people from copying things.

But clearly we do need some way of ensuring that creators of artistic works can be fairly compensated for their efforts. Copyright is not the only way to do that: A few alternatives that I think are worth considering are expanded crowdfunding (Patreon and Kickstart already support quite a few artists, though most not very much), a large basic income (artists would still create even if they weren’t paid; they really just need money to live on), government grants directly to artists (we have the National Endowment for the Arts, but it doesn’t support very many artists), and some kind of central clearinghouse that surveys consumers about the art they enjoy and then compensates artists according to how much their work is appreciated. But all of these would require substantial changes, and suffer from their own flaws, so for the time being, let’s say we stick with copyright.

Even so, it’s utterly ludicrous that Disney has managed to hold onto the copyright on Mickey Mouse for this long. It makes absolutely no sense from the perspective of supporting artists—indeed, in this case the artist has been dead for over 50 years.

In fact, it wouldn’t even make sense if Walt Disney were still alive. (Not many people live 96 years past their first highly-successful creative work, but it’s at least possible, if you say published as a child and then lived to be a centenarian.) If the goal is to incentivize new creative art, the first few decades—indeed, the first few years—are clearly the most important for doing so.

To show why this is, I need to take a brief detour into finance, and the concept of a net present value.

As the saying goes: Time is money. $1 today is worth more than $1 a year from now. (And if you doubt this, let me remind you of the old joke: “I’ll give you $1 million dollars if you give me $100! Such a deal! Give me the $100 today, and I’ll give you $1 per year for the next million years.”)

The idea of a net present value is to precisely quantify the monetary value of time (or the time value of money), so that we can compare cashflows over time in a directly comparable way.

To compute a net present value, you need a discount rate. At a discount rate of r, an amount of money X that you get 1 year from now is worth X/(1+r). The discount rate should be positive, because money later is worth less than money now; this means that we want X/(1+r) < X, and therefore r > 0.

This is surprisingly hard to get precisely, but relatively easy to ballpark. A good guess is that it’s somewhere close to the prevailing interest rate, or maybe the average return on the stock market. It should definitely be at least the inflation rate. Right now inflation is running a little high (around 8%), so we’d want to use a relatively high discount rate currently, maybe 10% or 12%. But I think in a more typical scenario, something more like 5-6% would be a reasonable guess.

Once you have a discount rate, it’s pretty simple to figure out the net present value: Just add up all the future cashflows, each discounted by that discount rate for the time you have to wait for it.

So for instance if you get $100 per year for the next 5 years, this would be your net present value:

100/(1+r) + 100/(1+r)^2 + 100/(1+r)^3 + 100/(1+r)^4 + 100/(1+r)^5

If you get $50 this year, $60 next year, $70 the year after that, this would be your next present value:

50 + 60/(1+r) + 70/(1+r)^2

If the cashflow is the same X over time for some fixed amount of time T this can be collapsed into a single formula using a geometric series:

X (1 – (1+r)^(-T)) – 1)/r

This is really just a more compact way of adding up, X + X/(1+r) + X/(1+r)^2 + …; here, let’s do that example of $100 per year for 5 years, with r = 10%.

100/1.1 + 100/1.1^2 + 100/1.1^3 + 100/1.1^4 + 100/1.1^5 = $379

100 (1 – 1.1^(-5))/0.1 = $379

See, we get the same answer either way. Notice that this is less than $100 * 5 = $500, which is what we’d get if we had assumed that $1 a year from now is worth the same as $1 today. But it’s not too much less, because it’s only 5 years.

This formula allows us to consider what happens when the time interval becomes extremely long—even infinite. It gives us the power to ask the question, “What is the value of this perpetual cashflow?”

This feels a bit weird for individuals, since of course we die. We can have heirs, but rare indeed is the thousand-year dynasty. (The Imperial House of Japan does appear to have an unbroken hereditary line for the last 2000 years, but they’re basically alone in that.) But governments and corporations don’t have a lifespan, so it makes more sense for them. The US government was here 200 years ago, and may still be here 200 years from now. Oxford was here 900 years ago, and I see no particular reason to think it won’t still be here 900 years from now.

Since r > 0, (1+r)^(-T) gets smaller as T increases. As T approaches infinity, (1+r)^(-T) approaches zero. So for a perpetual cashflow, we can just make this term zero.

Thus, we can actually assess the value of $1 per year for the next million years! It is this:

1 (1-(1+r)^(10^6))/r

which is basically the same as this:

1/r

So if your discount rate is 10%, then $1 per year for 1 million years is worth about as much to you as $1/0.1 = $10 today. If your discount rate is 5%, it would be worth about $1/0.05 = $20 today. And suddenly it makes sense that you’re not willing to pay $100 for this deal.

What if the cashflow is changing? Then this formula won’t work. But if it’s simply a constant rate of growth, we can adjust for that. If the growth rate of the cashflow is g, so that you get X, then X (1+g), then x (1+g)^2, and so on, the formula becomes just a bit more complicated:

X (1-(1+r-g)^(-T))/(r-g)

So for instance if your cashflow grows at 6% per year and your discount rate is 10%, then it’s basically the same as if it didn’t grow at all but your discount rate is 4%. [This is actually an approximation, but it’s a pretty good one.] Let’s call this the effective discount rate.

For a perpetual cashflow, as long as r > g, this becomes:

X/(r-g)

With this in mind, let’s return to the question of copyright. How long should copyright protection last?

We want it to last long enough for artists to be fairly compensated for their work; but what does “fairly compensated” mean? Well, with the concept of a perpetual net present value in mind, we could quantify this as the majority of all revenue that would be expected to be earned by a perpetual copyright.

I think this is actually quite generous: We’re saying that you should get to keep the copyright long enough to get most of what you’d probably get if we allowed you to own it forever. In some cases this might actually result in a copyright that’s too long; but I don’t see how it could result in it being too short.

Mickey Mouse today earns about $3 million per year. That’s honestly amazing, to continue to rake in that much money after such a long period. But, adjusted for inflation, that’s actually quite a bit less than what he took in just a few years after his first films were released, nominally $1 million per year which comes to more like $19 million per year in today’s money.

This means that our discount rate is larger than our growth rate (r > g) even if r is just inflation; but in fact we should use a discount rate higher than inflation. Let’s use a plausible but slightly conservative discount rate of 5%.

To grow from nominally $1 million to nominally $3 million per year in 94 years means a growth rate of about 1% per year.

So, our effective discount rate is 4%.

Then, a perpetual copyright for Mickey Mouse should be worth approximately:

X/(r-g) = 10^6/(0.04) = $25 million

Yes, that’s right; an unending stream of over $1 million per year ends up being worth about the same as a single payment of $25 million way back in 1928.

But isn’t Mickey Mouse a “fictional billionare”, meaning his total income over his existence has been more than $1 billion? Sure. And indeed, at a discount rate of 5%, $1 billion today is worth about $10 million in 1928. So Mickey is indeed well above that. Even if I use Forbes’ higher estimate that Mickey Mouse has taken in $5.8 billion, that would still only be a net present value of $59 million in 1928.

Remember, time is money. When it takes this long to get a cashflow, it ends up worth substantially less.

So, if we were aiming to let Mickey earn half of his perpetual earnings in net present value, when should we have ended his copyright? By my estimate, when the net present value of earnings exceeded $12.5 million. If we use Forbes’s more generous estimate, when it exceeded $30 million.

So now let’s go back to the formula for a finite time horizon, and try to solve it for T, the time horizon. We want the net present value of the finite horizon to be half that of the infinite horizon:

X (1-(1+r-g)^(-T))/(r-g) = (X/2)/(r-g)

(1+r-g)^(-T) = 1/2

To solve this for T, I’ll need to use a logarithm, the inverse of an exponent.

T = ln(2)/ln(1+r – g)

This is a doubling time, very analogous to a half-life in physics. Since logarithms are very difficult to do by hand, if you don’t have a scientific calculator handy, you can also approximate it by dividing the percentage into 69:

T = 69/(r-g)%

This is because ln(2) = 0.69…, and when r-g is a small percentage, ln(1+r-g) is about the same as r-g.

For an effective discount rate of 4%, this becomes:

T = ln(2)/ln(1.04) = 69/4 = 17

That is, only seventeen years. Even for a hugely successful long-running property like Mickey Mouse (in fact, is there really anything on a par with Mickey Mouse?), the majority of the net present value was earned in less than 20 years.

Indeed, it seems especially sensible in this case, because back then, Walt Disney was still alive! He could actually enjoy the fruits of his labors for that period. Now it’s all going to some faceless shareholders of a massive megacorporation, only a few of which are even Walt Disney’s heirs. Only about 3% of Disney shares are owned by anyone actually in the Disney family.

This gives us an answer to the question, “How long should copyrights last?”: About 20 years.

If we’d used a higher discount rate, it would be even shorter: at 10%, you get only 10 years.

And a lower discount rate simply isn’t plausible; inflation and stock market growth are both too fast for net present value to be discounted much less than 4% or 5%. Maybe you could go as low as 3%, which would be 23 years.

Does this accomplish the goal of copyrights—which, remember, was to fairly compensate artists and incentivize the creation of artistic works? I’d say so. They get half of what they would have gotten if we never released their work into the public domain, and I don’t think I’ve ever met an artist who could honestly say that they’d create something if they could hold onto the rights for 96 years, but not if they could for only 20 years. (Maybe they exist, but if so, they are rare.) Most artists really just want to be credited—not paid, credited—for their work and to make a decent living. 20 years is enough for that.

This means that our current copyright system keeps works out of public domain nearly five times as long as there is any real economic justification for.

Commitment and sophistication

PNRJ cognitive science, heuristics and biases , , , , , , , , , ,

Mar 13 JDN 2459652

One of the central insights of cognitive and behavioral economics is that understanding the limitations of our own rationality can help us devise mechanisms to overcome those limitations—that knowing we are not perfectly rational can make us more rational. The usual term for this is a somewhat vague one: behavioral economists generally call it simply sophistication.

For example, suppose that you are short-sighted and tend to underestimate the importance of the distant future. (This is true of most of us, to greater or lesser extent.)

It’s rational to consider the distant future less important than the present—things change in the meantime, and if we go far enough you may not even be around to see it. In fact, rationality alone doesn’t even say how much you should discount any given distance in the future. But most of us are inconsistent about our attitudes toward the future: We exhibit dynamic inconsistency.

For instance, suppose I ask you today whether you would like $100 today or $102 tomorrow. It is likely you’ll choose $100 today. But if I ask you whether you would like $100 365 days from now or $102 366 days from now, you’ll almost certainly choose the $102.


This means that if I asked you the second question first, then waited a year and asked you the first question, you’d change your mind—that’s inconsistent. Whichever choice is better shouldn’t systematically change over time. (It might happen to change, if your circumstances changed in some unexpected way. But on average it shouldn’t change.) Indeed, waiting a day for an extra $2 is typically going to be worth it; 2% daily interest is pretty hard to beat.

Now, suppose you have some option to make a commitment, something that will bind you to your earlier decision. It could be some sort of punishment for deviating from your earlier choice, some sort of reward for keeping to the path, or, in the most extreme example, a mechanism that simply won’t let you change your mind. (The literally classic example of this is Odysseus having his crew tie him to the mast so he can listen to the Sirens.)

If you didn’t know that your behavior was inconsistent, you’d never want to make such a commitment. You don’t expect to change your mind, and if you do change your mind, it would be because your circumstances changed in some unexpected way—in which case changing your mind would be the right thing to do. And if your behavior wasn’t inconsistent, this reasoning would be quite correct: No point in committing when you have less information.

But if you know that your behavior is inconsistent, you can sometimes improve the outcome for yourself by making a commitment. You can force your own behavior into consistency, even though you will later be tempted to deviate from your plan.

Yet there is a piece missing from this account, often not clearly enough stated: Why should we trust the version of you that has a year to plan over the version of you that is making the decision today? What’s the difference between those two versions of you that makes them inconsistent, and why is one more trustworthy than the other?

The biggest difference is emotional. You don’t really feel $100 a year from now, so you can do the math and see that 2% daily interest is pretty darn good. But $100 today makes you feel something—excitement over what you might buy, or relief over a bill you can now pay. (Actually that’s one of the few times when it would be rational to take $100 today: If otherwise you’re going to miss a deadline and pay a late fee.) And that feeling about $102 tomorrow just isn’t as strong.

We tend to think that our emotional selves and our rational selves are in conflict, and so we expect to be more rational when we are less emotional. There is some truth to this—strong emotions can cloud our judgments and make us behave rashly.

Yet this is only one side of the story. We also need emotions to be rational. There is a condition known as flat affect, often a symptom of various neurological disorders, in which emotional reactions are greatly blunted or even non-existent. People with flat affect aren’t more rational—they just do less. In the worst cases, they completely lose their ability to be motivated to do things and become outright inert, known as abulia.

Emotional judgments are often less accurate than thoughtfully reasoned arguments, but they are also much faster—and that’s why we have them. In many contexts, particularly when survival is at stake, doing something pretty well right away is often far better than waiting long enough to be sure you’ll get the right answer. Running away from a loud sound that turns out to be nothing is a lot better than waiting to carefully determine whether that sound was really a tiger—and finding that it was.

With this in mind, the cases where we should expected commitment to be effective are those that are unfamiliar, not only on an individual level, but in an evolutionary sense. I have no doubt that experienced stock traders can develop certain intuitions that make them better at understanding financial markets than randomly chosen people—but they still systematically underperform simple mathematical models, likely because finance is just so weird from an evolutionary perspective. So when deciding whether to accept some amount of money m1 at time t1 and some other amount of money m2 at time t2, your best bet is really to just do the math.

But this may not be the case for many other types of decisions. Sometimes how you feel in the moment really is the right signal to follow. Committing to work at your job every day may seem responsible, ethical, rational—but if you hate your job when you’re actually doing it, maybe it really isn’t how you should be spending your life. Buying a long-term gym membership to pressure yourself to exercise may seem like a good idea, but if you’re miserable every time you actually go to the gym, maybe you really need to be finding a better way to integrate exercise into your lifestyle.

There are no easy answers here. We can think of ourselves as really being made of two (if not more) individuals: A cold, calculating planner who looks far into the future, and a heated, emotional experiencer who lives in the moment. There’s a tendency to assume that the planner is our “true self”, the one we should always listen to, but this is wrong; we are both of those people, and a life well-lived requires finding the right balance between their conflicting desires.

Hyperbolic discounting: Why we procrastinate

PNRJ cognitive science, critique of neoclassical economics, heuristics and biases , , , ,

Mar 25 JDN 2458203

Lately I’ve been so occupied by Trump and politics and various ideas from environmentalists that I haven’t really written much about the cognitive economics that was originally planned to be the core of this blog. So, I thought that this week I would take a step out of the political fray and go back to those core topics.

Why do we procrastinate? Why do we overeat? Why do we fail to exercise? It’s quite mysterious, from the perspective of neoclassical economic theory. We know these things are bad for us in the long run, and yet we do them anyway.

The reason has to do with the way our brains deal with time. We value the future less than the present—but that’s not actually the problem. The problem is that we do so inconsistently.

A perfectly-rational neoclassical agent would use time-consistent discounting; what this means is that the effect of a given time interval won’t change or vary based on the stakes. If having $100 in 2019 is as good as having $110 in 2020, then having $1000 in 2019 is as good as having $1100 in 2020; and if I ask you in 2019, you’ll still agree that having $100 in 2019 is as good as having $1100 in 2020. A perfectly-rational individual would have a certain discount rate (in this case, 10% per year), and would apply it consistently at all times on all things.

This is of course not how human beings behave at all.

A much more likely pattern is that you would agree, in 2018, that having $100 in 2019 is as good as having $110 in 2020 (a discount rate of 10%). But then if I wait until 2019, and then offer you the choice between $100 immediately and $120 in a year, you’ll probably take the $100 immediately—even though a year ago, you told me you wouldn’t. Your discount rate rose from 10% to at least 20% in the intervening time.

The leading model in cognitive economics right now to explain this is called hyperbolic discounting. The precise functional form of a hyperbola has been called into question by recent research, but the general pattern is definitely right: We act as though time matters a great deal when discussing time intervals that are close to us, but treat time as unimportant when discussing time intervals that are far away.

How does this explain procrastination and other failures of self-control over time? Let’s try an example.

Let’s say that you have a project you need to finish by the end of the day Friday, which has a benefit to you, received on Saturday, that I will arbitrarily scale at 1000 utilons.

Then, let’s say it’s Monday. You have five days to work on it, and each day of work costs you 100 utilons. If you work all five days, the project will get done.

If you skip a day of work, you will need to work so much harder that one of the following days your cost of work will be 300 utilons instead of 100. If you skip two days, you’ll have to pay 300 utilons twice. And if you skip three or more days, the project will not be finished and it will all be for naught.

If you don’t discount time at all (which, over a week, is probably close to optimal), the answer is obvious: Work all five days. Pay 100+100+100+100+100 = 500, receive 1000. Net benefit: 500.

But even if you discount time, as long as you do so consistently, you still wouldn’t procrastinate.

Let’s say your discount rate is extremely high (maybe you’re dying or something), so that each day is only worth 80% as much as the previous. Benefit that’s worth 1 on Monday is worth 0.8 if it comes on Tuesday, 0.64 if it comes on Wednesday, 0.512 if it comes on Thursday, 0.4096 if it comes on Friday,a and 0.32768 if it comes on Saturday. Then instead of paying 100+100+100+100+100 to get 1000, you’re paying 100+80+64+51+41=336 to get 328. It’s not worth doing the project; you should just enjoy your last few days on Earth. That’s not procrastinating; that’s rationally choosing not to undertake a project that isn’t worthwhile under your circumstances.

Procrastinating would look more like this: You skip the first two days, then work 100 the third day, then work 300 each of the last two days, finishing the project. If you didn’t discount at all, you would pay 100+300+300=700 to get 1000, so your net benefit has been reduced to 300.

There’s no consistent discount rate that would make this rational. If it was worth giving up 200 on Thursday and Friday to get 100 on Monday and Tuesday, you must be discounting at least 26% per day. But if you’re discounting that much, you shouldn’t bother with the project at all.

There is however an inconsistent discounting by which it makes perfect sense. Suppose that instead of consistently discounting some percentage each day, psychologically it feels like this: The value is the inverse of the length of time (that’s what it means to be hyperbolic). So the same amount of benefit on Monday which is worth 1 is only worth 1/2 if it comes on Tuesday, 1/3 if on Wednesday, 1/4 if on Thursday, and 1/5 if on Friday.

So, when thinking about your weekly schedule, you realize that by pushing back Monday’s work to Thursday, you can gain 100 today at a cost of only 200/4 = 50, since Thursday is 4 days away. And by pushing back Tuesday’s work to Friday, you can gain 100/2=50 today at a cost of only 200/5=40. So now it makes perfect sense to have fun on Monday and Tuesday, start working on Wednesday, and cram the biggest work into Thursday and Friday. And yes, it still makes sense to do the project, because 1000/6 = 166 is more than the 100/3+200/4+200/5 = 123 it will cost to do the work.

But now think about what happens when you come to Wednesday. The work today costs 100. The work on Thursday costs 200/2 = 100. The work on Friday costs 200/3 = 66. The benefit of completing the project will be 1000/4 = 250. So you are paying 100+100+66=266 to get a benefit of only 250. It’s not worth it anymore! You’ve changed your mind. So you don’t work Wednesday.

At that point, it’s too late, so you don’t work Thursday, you don’t work Friday, and the project doesn’t get done. You have procrastinated away the benefits you could have gotten from doing this project. If only you could have done the work on Monday and Tuesday, then on Wednesday it would have been worthwhile to continue: 100/1+100/2+100/3 = 183 is less than the benefit of 250.

What went wrong? The key event was the preference reversal: While on Monday you preferred having fun on Monday and working on Thursday to working on both days, when the time came you changed your mind. Someone with time-consistent discounting would never do that; they would either prefer one or the other, and never change their mind.

One way to think about this is to imagine future versions of yourself as different people, who agree with you on most things, but not on everything. They’re like friends or family; you want the best for them, but you don’t always see eye-to-eye.

Generally we find that our future selves are less rational about choices than we are. To be clear, this doesn’t mean that we’re all declining in rationality over time. Rather, it comes from the fact that future decisions are inherently closer to our future selves than they are to our current selves, and the closer a decision gets the more likely we are to use irrational time discounting.

This is why it’s useful to plan and make commitments. If starting on Monday you committed yourself to working every single day, you’d get the project done on time and everything would work out fine. Better yet, if you committed yourself last week to starting work on Monday, you wouldn’t even feel conflicted; you would be entirely willing to pay a cost of 100/8+100/9+100/10+100/11+100/12=51 to get a benefit of 1000/13=77. So you could set up some sort of scheme where you tell your friends ahead of time that you can’t go out that week, or you turn off access to social media sites (there are apps that will do this for you), or you set up a donation to an “anti-charity” you don’t like that will trigger if you fail to complete the project on time (there are websites to do that for you).

There is even a simpler way: Make a promise to yourself. This one can be tricky to follow through on, but if you can train yourself to do it, it is extraordinarily powerful and doesn’t come with the additional costs that a lot of other commitment devices involve. If you can really make yourself feel as bad about breaking a promise to yourself as you would about breaking a promise to someone else, then you can dramatically increase your own self-control with very little cost. The challenge lies in actually cultivating that sort of attitude, and then in following through with making only promises you can keep and actually keeping them. This, too, can be a delicate balance; it is dangerous to over-commit to promises to yourself and feel too much pain when you fail to meet them.
But given the strong correlations between self-control and long-term success, trying to train yourself to be a little better at it can provide enormous benefits.
If you ever get around to it, that is.

Why New Year’s resolutions fail

PNRJ cognitive science, heuristics and biases, Uncategorized , , , , , , , , , ,

Jan 1, JDN 2457755

Last week’s post was on Christmas, so by construction this week’s post will be on New Year’s Day.

It is a tradition in many cultures, especially in the US and Europe, to start every new year with a New Year’s resolution, a promise to ourselves to change our behavior in some positive way.

Yet, over 80% of these resolutions fail. Why is this?

If we are honest, most of us would agree that there is something about our own behavior that could stand to be improved. So why do we so rarely succeed in actually making such improvements?

One possibility, which I’m guessing most neoclassical economists would favor, is to say that we don’t actually want to. We may pretend that we do in order to appease others, but ultimately our rational optimization has already chosen that we won’t actually bear the cost to make the improvement.

I think this is actually quite rare. I’ve seen too many people with resolutions they didn’t share with anyone, for example, to think that it’s all about social pressure. And I’ve seen far too many people try very hard to achieve their resolutions, day after day, and yet still fail.

Sometimes we make resolutions that are not entirely within our control, such as “get a better job” or “find a girlfriend” (last year I made a resolution to publish a work of commercial fiction or a peer-reviewed article—and alas, failed at that task, unless I somehow manage it in the next few days). Such resolutions may actually be unwise to make in the first place, as it can feel like breaking a promise to yourself when you’ve actually done all you possibly could.

So let’s set those aside and talk only about things we should be in control over, like “lose weight” or “save more money”. Even these kinds of resolutions typically fail; why? What is this “weakness of will”? How is it possible to really want something that you are in full control over, and yet still fail to accomplish it?

Well, first of all, I should be clear what I mean by “in full control over”. In some sense you’re not in full control, which is exactly the problem. Your conscious mind is not actually an absolute tyrant over your entire body; you’re more like an elected president who has to deal with a legislature in order to enact policy.

You do have a great deal of power over your own behavior, and you can learn to improve this control (much as real executive power in presidential democracies has expanded over the last century!); but there are fundamental limits to just how well you can actually consciously will your body to do anything, limits imposed by billions of years of evolution that established most of the traits of your body and nervous system millions of generations before there even was such a thing as rational conscious reasoning.

One thing that makes a surprisingly large difference lies in whether your goals are reduced to specific, actionable objectives. “Lose weight” is almost guaranteed to fail. “Lose 30 pounds” is still unlikely to succeed. “Work out for 2 hours per week,” on the other hand, might have a chance. “Save money” is never going to make it, but “move to a smaller apartment and set aside $200 per month” just might.

I think the government metaphor is helpful here; if you President of the United States and you want something done, do you state some vague, broad goal like “Improve the economy”? No, you make a specific, actionable demand that allows you to enforce compliance, like “increase infrastructure spending by 24% over the next 5 years”. Even then it is possible to fail if you can’t push it through the legislature (in the metaphor, the “legislature” is your habits, instincts and other subconscious processes), but you’re much more likely to succeed if you have a detailed plan.

Another technique that helps is to visualize the benefits of succeeding and the costs of failing, and keep these in your mind. This counteracts the tendency for the costs of succeeding and the benefits of giving up to be more salient—losing 30 pounds sounds nice in theory, but that treadmill is so much work right now!

This salience effect has a lot to do with the fact that human beings are terrible at dealing with the future.

Rationally, we are supposed to use exponential discounting; each successive moment is supposed to be worth less to us than the previous by a fixed proportion, say 5% per year. This is actually a mathematical theorem; if you don’t discount this way, your decisions will be systematically irrational.

And yet… we don’t discount that way. Some behavioral economists argue that we use hyperbolic discounting, in which instead of discounting time by a fixed proportion, we use a different formula that drops off too quickly early on and not quickly enough later on.

But I am increasingly convinced that human beings don’t actually use discounting at all. We have a series of rough-and-ready heuristics for making future judgments, which can sort of act like discounting, but require far less computation than actually calculating a proper discount rate. (Recent empirical evidence seems to be tilting this direction.)

In any case, whatever we do is clearly not a proper rational discount rate. And this means that our behavior can be time-inconsistent; a choice that seems rational at one time can not seem rational at a later time. When we’re planning out our year and saying we will hit the treadmill more, it seems like a good idea; but when we actually get to the gym and feel our legs ache as we start running, we begin to regret our decision.

The challenge, really, is determining which “version” of us is correct! A priori, we don’t actually know whether the view of our distant self contemplating the future or the view of our current self making the choice in the moment is the right one. Actually, when I frame it this way, it almost seems like the self that’s closer to the choice should have better information—and yet typically we think the exact opposite, that it is our past self making plans that really knows what’s best for us.

So where does that come from? Why do we think, at least in most cases, that the “me” which makes a plan a year in advance is the smart one, and the “me” that actually decides in the moment is untrustworthy.

Kahneman has a good explanation for this, in his model of System 1 and System 2. System 1 is simple and fast, but often gets the wrong answer. System 2 usually gets the right answer, but it is complex and slow. When we are making plans, we have a lot of time to think, and we can afford to expend the extra effort to engage the full power of System 2. But when we are living in the moment, choosing what to do right now, we don’t have that luxury of time, and we are forced to fall back on System 1. System 1 is easier—but it’s also much more likely to be wrong.

How, then, do we resolve this conflict? Commitment. (Perhaps that’s why it’s called a New Year’s resolution!)

We make promises to ourselves, commitments that we will feel bad about not following through.

If we rationally discounted, this would be a baffling thing to do; we’re just imposing costs on ourselves for no reason. But because we don’t discount rationally, commitments allow us to change the calculation for our future selves.

This brings me to one last strategy to use when making your resolutions: Include punishment.

“I will work out at least 2 hours per week, and if I don’t, I’m not allowed to watch TV all weekend.” Now that is a resolution you are actually likely to keep.

To see why, consider the decision problem for your System 2 self today versus your System 1 self throughout the year.

Your System 2 self has done the cost-benefit analysis and ruled that working out 2 hours per week is worthwhile for its health benefits.

If you left it at that, your System 1 self would each day find an excuse to procrastinate the workouts, because at least from where they’re sitting, working out for 2 hours looks a lot more painful than the marginal loss in health from missing just this one week. And of course this will keep happening, week after week—and then 52 go by and you’ve had few if any workouts.

But by adding the punishment of “no TV”, you have imposed an additional cost on your System 1 self, something that they care about. Suddenly the calculation changes; it’s not just 2 hours of workout weighed against vague long-run health benefits, but 2 hours of workout weighed against no TV all weekend. That punishment is surely too much to bear; so you’d best do the workout after all.

Do it right, and you will rarely if ever have to impose the punishment. But don’t make it too large, or then it will seem unreasonable and you won’t want to enforce it if you ever actually need to. Your System 1 self will then know this, and treat the punishment as nonexistent. (Formally the equilibrium is not subgame perfect; I am gravely concerned that our nuclear deterrence policy suffers from precisely this flaw.) “If I don’t work out, I’ll kill myself” is a recipe for depression, not healthy exercise habits.

But if you set clear, actionable objectives and sufficient but reasonable punishments, there’s at least a good chance you will actually be in the minority of people who actually succeed in keeping their New Year’s resolution.

And if not, there’s always next year.

What is the price of time?

PNRJ critique of neoclassical economics, heuristics and biases , , , , , , , , , , , , , , ,

JDN 2457562

If they were asked outright, “What is the price of time?” most people would find that it sounds nonsensical, like I’ve asked you “What is the diameter of calculus?” or “What is the electric charge of justice?” (It’s interesting that we generally try to assign meaning to such nonsensical questions, and they often seem strangely profound when we do; a good deal of what passes for “profound wisdom” is really better explained as this sort of reaction to nonsense. Deepak Chopra, for instance.)

But there is actually a quite sensible economic meaning of this question, and answering it turns out to have many important implications for how we should run our countries and how we should live our lives.

What we are really asking for is temporal discounting; we want to know how much more money today is worth compared to tomorrow, and how much more money tomorrow is worth compared to two days from now.

If you say that they are exactly the same, your discount rate (your “price of time”) is zero; if that is indeed how you feel, may I please borrow your entire net wealth at 0% interest for the next thirty years? If you like we can even inflation-index the interest rate so it always produces a real interest rate of zero, thus protecting you from potential inflation risk.
What? You don’t like my deal? You say you need that money sooner? Then your discount rate is not zero. Similarly, it can’t be negative; if you actually valued money tomorrow more than money today, you’d gladly give me my loan.

Money today is worth more to you than money tomorrow—the only question is how much more.

There’s a very simple theorem which says that as long as your temporal discounting doesn’t change over time, so it is dynamically consistent, it must have a very specific form. I don’t normally use math this advanced in my blog, but this one is so elegant I couldn’t resist. I’ll encase it in blockquotes so you can skim over it if you must.

The value of $1 today relative to… today is of course 1; f(0) = 1.

If you are dynamically consistent, at any time t you should discount tomorrow relative to today the same as you discounted today relative to yesterday, so for all t, f(t+1)/f(t) = f(t)/f(t-1)
Thus, f(t+1)/f(t) is independent of t, and therefore equal to some constant, which we can call r:

f(t+1)/f(t) = r, which implies f(t+1) = r f(t).

Starting at f(0) = 1, we have:

f(0) = 1, f(1) = r, f(2) = r^2

We can prove that this pattern continues to hold by mathematical induction.

Suppose the following is true for some integer k; we already know it works for k = 1:

f(k) = r^k

Let t = k:

f(k+1) = r f(k)

Therefore:

f(k+1) = r^(k+1)

Which by induction proves that for all integers n:

f(n) = r^n

The name of the variable doesn’t matter. Therefore:

f(t) = r^t

Whether you agree with me that this is beautiful, or you have no idea what I just said, the take-away is the same: If your discount rate is consistent over time, it must be exponential. There must be some constant number 0 < r < 1 such that each successive time period is worth r times as much as the previous. (You can also generalize this to the case of continuous time, where instead of r^t you get e^(-r t). This requires even more advanced math, so I’ll spare you.)

Most neoclassical economists would stop right there. But there are two very big problems with this argument:

(1) It doesn’t tell us the value r should actually be, only that it should be a constant.

(2) No actual human being thinks of time this way.

There is still ongoing research as to exactly how real human beings discount time, but one thing is quite clear from the experiments: It certainly isn’t exponential.

From about 2000 to 2010, the consensus among cognitive economists was that humans discount time hyperbolically; that is, our discount function looks like this:

f(t) = 1/(1 + r t)

In the 1990s there were a couple of experiments supporting hyperbolic discounting. There is even some theoretical work trying to show that this is actually optimal, given a certain kind of uncertainty about the future, and the argument for exponential discounting relies upon certainty we don’t actually have. Hyperbolic discounting could also result if we were reasoning as though we are given a simple interest rate, rather than a compound interest rate.

But even that doesn’t really seem like humans think, now does it? It’s already weird enough for someone to say “Should I take out this loan at 5%? Well, my discount rate is 7%, so yes.” But I can at least imagine that happening when people are comparing two different interest rates (“Should I pay down my student loans, or my credit cards?”). But I can’t imagine anyone thinking, “Should I take out this loan at 5% APR which I’d need to repay after 5 years? Well, let’s check my discount function, 1/(1+0.05 (5)) = 0.8, multiplied by 1.05^5 = 1.28, the product of which is 1.02, greater than 1, so no, I shouldn’t.” That isn’t how human brains function.

Moreover, recent experiments have shown that people often don’t seem to behave according to what hyperbolic discounting would predict.

Therefore I am very much in the other camp of cognitive economists, who say that we don’t have a well-defined discount function. It’s not exponential, it’s not hyperbolic, it’s not “quasi-hyperbolic” (yes that is a thing); we just don’t have one. We reason about time by simple heuristics. You can’t make a coherent function out of it because human beings… don’t always reason coherently.

Some economists seem to have an incredible amount of trouble accepting that; here we have one from the University of Chicago arguing that hyperbolic discounting can’t possibly exist, because then people could be Dutch-booked out of all their money; but this amounts to saying that human behavior cannot ever be irrational, lest all our money magically disappear. Yes, we know hyperbolic discounting (and heuristics) allow for Dutch-booking; that’s why they’re irrational. If you really want to know the formal assumption this paper makes that is wrong, it assumes that we have complete markets—and yes, complete markets essentially force you to be perfectly rational or die, because the slightest inconsistency in your reasoning results in someone convincing you to bet all your money on a sure loss. Why was it that we wanted complete markets, again? (Oh, yes, the fanciful Arrow-Debreu model, the magical fairy land where everyone is perfectly rational and all markets are complete and we all have perfect information and the same amount of wealth and skills and the same preferences, where everything automatically achieves a perfect equilibrium.)

There was a very good experiment on this, showing that rather than discount hyperbolically, behavior is better explained by a heuristic that people judge which of two options is better by a weighted sum of the absolute distance in time plus the relative distance in time. Now that sounds like something human beings might actually do. “$100 today or $110 tomorrow? That’s only 1 day away, but it’s also twice as long. I’m not waiting.” “$100 next year, or $110 in a year and a day? It’s only 1 day apart, and it’s only slightly longer, so I’ll wait.”

That might not actually be the precise heuristic we use, but it at least seems like one that people could use.

John Duffy, whom I hope to work with at UCI starting this fall, has been working on another experiment to test a different heuristic, based on the work of Daniel Kahneman, saying essentially that we have a fast, impulsive, System 1 reasoning layer and a slow, deliberative, System 2 reasoning layer; the result is that our judgments combine both “hand to mouth” where our System 1 essentially tries to get everything immediately and spend whatever we can get our hands on, and a more rational assessment by System 2 that might actually resemble an exponential discount rate. In the 5-minute judgment, System 1’s voice is overwhelming; but if we’re already planning a year out, System 1 doesn’t even care anymore and System 2 can take over. This model also has the nice feature of explaining why people with better self-control seem to behave more like they use exponential discounting,[PDF link] and why people do on occasion reason more or less exponentially, while I have literally never heard anyone try to reason hyperbolically, only economic theorists trying to use hyperbolic models to explain behavior.

Another theory is that discounting is “subadditive”, that is, if you break up a long time interval into many short intervals, people will discount it more, because it feels longer that way. Imagine a century. Now imagine a year, another year, another year, all the way up to 100 years. Now imagine a day, another day, another day, all the way up to 365 days for the first year, and then 365 days for the second year, and that on and on up to 100 years. It feels longer, doesn’t it? It is of course exactly the same. This can account for some weird anomalies in choice behavior, but I’m not convinced it’s as good as the two-system model.

Another theory is that we simply have a “present bias”, which we treat as a sort of fixed cost that we incur regardless of what the payments are. I like this because it is so supremely simple, but there’s something very fishy about it, because in this experiment it was just fixed at $4, and that can’t be right. It must be fixed at some proportion of the rewards, or something like that; or else we would always exhibit near-perfect exponential discounting for large amounts of money, which is more expensive to test (quite directly), but still seems rather unlikely.

Why is this important? This post is getting long, so I’ll save it for future posts, but in short, the ways that we value future costs and benefits, both as we actually do, and as we ought to, have far-reaching implications for everything from inflation to saving to environmental sustainability.

The sunk-cost fallacy

PNRJ heuristics and biases , , , , , , , , , , , , ,

JDN 2457075 EST 14:46.

I am back on Eastern Time once again, because we just finished our 3600-km road trek from Long Beach to Ann Arbor. I seem to move an awful lot; this makes me a bit like Schumpeter, who moved an average of every two years his whole adult life. Schumpeter and I have much in common, in fact, though I have no particular interest in horses.

Today’s topic is the sunk-cost fallacy, which was particularly salient as I had to box up all my things for the move. There were many items that I ended up having to throw away because it wasn’t worth moving them—but this was always painful, because I couldn’t help but think of all the work or money I had put into them. I threw away craft projects I had spent hours working on and collections of bottlecaps I had gathered over years—because I couldn’t think of when I’d use them, and ultimately the question isn’t how hard they were to make in the past, it’s what they’ll be useful for in the future. But each time it hurt, like I was giving up a little part of myself.

That’s the sunk-cost fallacy in a nutshell: Instead of considering whether it will be useful to us later and thus worth having around, we naturally tend to consider the effort that went into getting it. Instead of making our decisions based on the future, we make them based on the past.

Come to think of it, the entire Marxist labor theory of value is basically one gigantic sunk-cost fallacy: Instead of caring about the usefulness of a product—the mainstream utility theory of value—we are supposed to care about the labor that went into making it. To see why this is wrong, imagine someone spends 10,000 hours carving meaningless symbols into a rock, and someone else spends 10 minutes working with chemicals but somehow figures out how to cure pancreatic cancer. Which one would you pay more for—particularly if you had pancreatic cancer?

This is one of the most common irrational behaviors humans do, and it’s worth considering why that might be. Most people commit the sunk-cost fallacy on a daily basis, and even those of us who are aware of it will still fall into it if we aren’t careful.

This often seems to come from a fear of being wasteful; I don’t know of any data on this, but my hunch is that the more environmentalist you are, the more often you tend to run into the sunk-cost fallacy. You feel particularly bad wasting things when you are conscious of the damage that waste does to our planetary ecosystem. (Which is not to say that you should not be environmentalist; on the contrary, most of us should be a great deal more environmentalist than we are. The negative externalities of environmental degradation are almost unimaginably enormous—climate change already kills 150,000 people every year and is projected to kill tens if not hundreds of millions people over the 21st century.)

I think sunk-cost fallacy is involved in a lot of labor regulations as well. Most countries have employment protection legislation that makes it difficult to fire people for various reasons, ranging from the basically reasonable (discrimination against women and racial minorities) to the totally absurd (in some countries you can’t even fire people for being incompetent). These sorts of regulations are often quite popular, because people really don’t like the idea of losing their jobs. When faced with the possibility of losing your job, you should be thinking about what your future options are; but many people spend a lot of time thinking about the past effort they put into this one. I think there is also some endowment effect and loss aversion at work as well: You value your job more simply because you already have it, so you don’t want to lose it even for something better.

Yet these regulations are widely regarded by economists as inefficient; and for once I am inclined to agree. While I certainly don’t want people being fired frivolously or for discriminatory reasons, sometimes companies really do need to lay off workers because there simply isn’t enough demand for their products. When a factory closes down, we think about the jobs that are lost—but we don’t think about the better jobs they can now do instead.

I favor a system like what they have in Denmark (I’m popularizing a hashtag about this sort of thing: #Scandinaviaisbetter): We don’t try to protect your job, we try to protect you. Instead of regulations that make it hard to fire people, Denmark has a generous unemployment insurance system, strong social welfare policies, and active labor market policies that help people retrain and find new and better jobs. One thing I think Denmark might want to consider is restrictions on cyclical layoffs—in a recession there is pressure to lay off workers, but that can create a vicious cycle that makes recessions worse. Denmark was hit considerably harder by the Great Recession than France, for example; where France’s unemployment rose from 7.5% to 9.6%, Denmark’s rose from an astonishing 3.1% all the way up to 7.6%.

Then again, sometimes what looks like a sunk-cost fallacy actually isn’t—and I think this gives us insight into how we might have evolved such an apparently silly heuristic in the first place.

Why would you care about what you did in the past when deciding what to do in the future? Well there’s one reason in particular: Credible commitment. There are many cases in life where you’d like to be able to plan to do something in the future, but when the time comes to actually do it you’ll be tempted not to follow through.

This sort of thing happens all the time: When you take out a loan, you plan to pay it back—but when you need to actually make payments it sure would be nice if you didn’t have to. If you’re trying to slim down, you go on a diet—but doesn’t that cookie look delicious? You know you should quit smoking for your health—but what’s one more cigarette, really? When you get married, you promise to be faithful—but then sometimes someone else comes along who seems so enticing! Your term paper is due in two weeks, so you really should get working on it—but your friends are going out for drinks tonight, why not start the paper tomorrow?

Our true long-term interests are often misaligned with our short-term temptations. This often happens because of hyperbolic discounting, which is a bit technical; but the basic idea is that you tend to rate the importance of an event in inverse proportion to its distance in time. That turns out to be irrational, because as you get closer to the event, your valuations will change disproportionately. The optimal rational choice would be exponential discounting, where you value each successive moment a fixed percentage less than the last—since that percentage doesn’t change, your valuations will always stay in line with one another. But basically nobody really uses exponential discounting in real life.

We can see this vividly in experiments: If we ask people whether they would you rather receive $100 today, or $110 a week from now, they often go with $100 today. But if you ask them whether they would rather receive $100 in 52 weeks or $110 in 53 weeks, almost everyone chooses the $110. The value of a week apparently depends on how far away it is! (The $110 is clearly the rational choice by the way. Discounting 10% per week makes no sense at all—unless you literally believe that $1,000 today is as good as $140,000 a year from now.)

To solve this problem, it can be advantageous to make commitments—either enforced by direct measures such as legal penalties, or even simply by making promises that we feel guilty breaking. That’s why cold turkey is often the most effective way to quit a drug. Physiologically that makes no sense, because gradual cessation clearly does reduce withdrawal symptoms. But psychologically it does, because cold turkey allows you to make a hardline commitment to never again touch the stuff. The majority of successful smokers report using cold turkey, though there is still ongoing research on whether properly-orchestrated gradual reduction can be more effective. Likewise, vague notions like “I’ll eat better and exercise more” are virtually useless, while specific prescriptions like “I will do 20 minutes of exercise every day and stop eating red meat” are much more effective—the latter allows you to make a promise to yourself that can be broken, and since you feel bad breaking it you are motivated to keep it.

In the presence of such commitments, the past does matter, at least insofar as you made commitments to yourself or others in the past. If you promised never to smoke another cigarette, or never to cheat on your wife, or never to eat meat again, you actually have a good reason—and a good chance—to never do those things. This is easy to confuse with a sunk cost; when you think about the 20 years you’ve been married or the 10 years you’ve been vegetarian, you might be thinking of the sunk cost you’ve incurred over that time, or you might be thinking of the promises you’ve made and kept to yourself and others. In the former case you are irrationally committing a sunk-cost fallacy; in the latter you are rationally upholding a credible commitment.

This is most likely why we evolved in such a way as to commit sunk-cost fallacies. The ability to enforce commitments on ourselves and others was so important that it was worth it to overcompensate and sometimes let us care about sunk costs. Because commitments and sunk costs are often difficult to distinguish, it would have been more costly to evolve better ways of distinguish them than it was to simply make the mistake.

Perhaps people who are outraged by being laid off aren’t actually committing a sunk-cost fallacy at all; perhaps they are instead assuming the existence of a commitment where none exists. “I gave this company 20 good years, and now they’re getting rid of me?” But the truth is, you gave the company nothing. They never committed to keeping you (unless they signed a contract, but that’s different; if they are violating a contract, of course they should be penalized for that). They made you a trade, and when that trade ceases to be advantageous they will stop making it. Corporations don’t think of themselves as having any moral obligations whatsoever; they exist only to make profit. It is certainly debatable whether it was a good idea to set up corporations in this way; but unless and until we change that system it is important to keep it in mind. You will almost never see a corporation do something out of kindness or moral obligation; that’s simply not how corporations work. At best, they do nice things to enhance their brand reputation (Starbucks, Whole Foods, Microsoft, Disney, Costco). Some don’t even bother doing that, letting people hate as long as they continue to buy (Walmart, BP, DeBeers). Actually the former model seems to be more successful lately, which bodes well for the future; but be careful to recognize that few if any of these corporations are genuinely doing it out of the goodness of their hearts. Human beings are often altruistic; corporations are specifically designed not to be.

And there were some things I did promise myself I would keep—like old photos and notebooks that I want to keep as memories—so those went in boxes. Other things were obviously still useful—clothes, furniture, books. But for the rest? It was painful, but I thought about what I could realistically use them for, and if I couldn’t think of anything, they went into the trash.