Discontinuities and Discounting

Summary: Even with a zero rate of pure time preference, and persistent zero real interest rates, we should use discount rates of at least 1% for any real-world project or policy. Health, lives, or utility should be discounted because of the probability that a discontinuity in resources or policy or the state of the world will cause projected future effects to not happen. With policies that provide economic benefits, we must also increase the discount rate to reflect the fact that it will probably be more expensive to purchase lives, health, or utility as overall conditions improve.

This post is meant for economists, effective altruists, and others who are doing cost-benefit calculations over a long time horizon to decide what actions we should be taking now. I am trying to be as plain-language and non-technical as possible, which will involve simplifications of well-studied things. I do assume basic knowledge of cost-benefit analysis and time value of money calculations. Greaves 2017 is an excellent survey article for some of the relevant technical aspects, and this post builds on its foundation. 

Past use of Discount Rates

In the past, cost-benefit analysis was mainly used for investment projects, and was mainly concerned with money flows. Discount rates were based on the agent's opportunity cost of money, which was usually based on a market rate. There were no hard questions or philosophical considerations beyond "Don't waste money." Market rates were never close enough to zero, and time horizons were never long enough, for the considerations in this post to be relevant.

However, in recent years, people have started to use cost-benefit analysis as a decision analysis tool for many more kinds of policies, including those that do not use money at all but involve trade-offs between lives or health or utility today and in the future. This has forced a rethink of how to apply discount rates.

Pure Time Preference

In experiments, tests, and surveys, people always prefer things now to things later even when there is no possibility of trade and investment. This has been called 'pure time preference'. There are many good arguments that society or a government should have a zero rate of pure time preference i.e. it should care just as much about a life today as a life in 100 years.

I don't believe that there's really any such thing as a pure rate of time preference. I think that people discount the future based on uncertainty that the expected benefits will ever actually arrive. Of course, usually they may do this subconsciously, out of evolved heuristics. The monkeys who assumed things would reliably be around in the future died off, and the monkeys that grabbed what they could at each moment reproduced. We observe throughout history that whenever the economic and political situation becomes more stable, people's discount rate goes down and they start caring more about the future. Attempts at measuring time preference typically fail to adequately correct for these uncertainties.

I suspect, but am too lazy to prove, that hyperbolic discounting is actually based on a subconscious assumption of some fixed probability that the person is lying to you about ever delivering future benefits, plus a variable probability over time that they will be unable to even if they intend to. Hyperbolic discounting is entirely rational in any situation other than our weird world with giant reliable financial systems and strong enforcement of contracts, and people's instincts haven't caught up. (If I am right, an experiment that deliberately made the promise more trustworthy would see a lower discount rate, and I would love to see someone try to falsify this, even if I am proven wrong.)

Real Interest Rates

Everything will be discussed in terms of real values, not nominal ones, because tracking inflation is an unnecessary complication.

It makes sense to discount the future based on how much our money would grow if we just put it in the bank. There's no sense implementing a policy that has a 2% return if it costs us 3% to borrow money. For almost all of human history, governments paid a positive real interest rate to borrow. Now they don't, and it may be decades or more before they do again. Similarly, no individual can or will earn a positive real risk-free return on their money for the foreseeable future. A full discussion of this is beyond the scope of this post, but for our purposes it lets us zero out a complication. I'll make the simplifying assumption that governments will keep borrowing more until their borrowing costs increase and they pay zero real interest rates on all their debt, so we can just assume no interest.

Increased Cost of Utility

We don't care about money; we care about what money buys us. This could be lives, or health, or utility. The exchange rate between money and these things changes over time. When society is poor, or unequal, a little money goes a long way. As things get better, it is more expensive to do good.

Discussions of discounting often involve the Ramsey Rule, which is based on increases in wealth and the decreasing marginal utility of consumption, but I think that is both too technically intimidating, and also too simplistic because it just uses averages and does not take inequality into account.  

If people are using money at all intelligently to purchase utility, they'll be using it to increase the utility of the people who are worst off. This means that if inequality goes down in the future, or if the social safety net improves, the cost of purchasing utility will go up, probably significantly more than the overall economic growth rate. 

Our discount rate for a policy that has economic effects will be, at minimum, the expected annual rate of increase in the cost of purchasing utility. Ideally this would be estimated by forming a historical data set on the cost per QALY achieved by the 10% most effective charities, or by tracking the percentage of the population in various states of extreme distress or subject to easily preventable deaths, but as a kludge we can just double the real per capita rate of economic growth.

Discontinuities

The primary message of this post is that we must discount based on the fact that everything we do can only affect the future with a probability rather than a certainty. There are many things that could cause our actions to have no effect, and each of them causes us to increase the discount rate. I call these discontinuities: anything that causes a break between what we hoped to happen and what actually happened.

Any competent cost-benefit analysis takes into account known unknowns, listing various scenarios and averaging them for an expected value. But I've never seen one that explicitly accounts for the annual tail risks of unknown unknowns, the discontinuities that could scramble plans and predictions by making big changes in the world. These things can usually be safely ignored on a short time horizon, but over long periods they matter a lot more.

I will generally be speaking in terms of an annual discount rate, but as a simple example, imagine that there are three different things that each cause a 10% cumulative chance, between now and the time period we want to affect, of the plan failing. The expected value of the benefits is roughly 70% of what they would be if the plan was guaranteed to do exactly what we projected. (Technically the expected payoff is 0.9^3 ≈ 73% chance of success, but that assumes risk neutrality. A more realistic assumption of risk aversion decreases the expected utility, so I'll cancel them out and just add up failure probabilities for simplicity.) So if the plan gains us 2 utility on a success, our expected value is 1.4 utility.

However, we still spent the resources, meaning that we lost the ability to do anything with them today. This means that, when doing a cost-benefit analysis, we should discount all future benefits by 30%. If this plan gives us at least 30% more benefits than spending money today, we should do it, and if it gives us less, we should not.

In this example, the failure probabilities were fixed, so it is no different than a normal expected-value scenario analysis. But in reality, the chances of these things increase as more time passes. Every year, there is a chance of something scrambling our plans, either directly or by scrambling the entire world.

To gain an intuition for this, imagine that you are a university professor, with a big endowment and research budget, in a random developed country in the year 1920, and that you are implementing a plan to make life better for people in the year 2020. What is the probability that your plan will survive? (Keep in mind that there's a good chance you are Jewish and live in a country in continental Europe.) Then imagine that you live in 1820, or 1720. Consider all of the economic, political, moral, and scientific shifts that have happened in those years. What is the probability that anything you do can even persist at all in a form you intended? Even then, what is the probability that your plan actually made the world better, in a way that the people of the future agree with and appreciate?

So, our annual discount rate must always include the sum total of all the annual probabilities of everything that could possibly happen to make the plan fail or become irrelevant.

As a concrete example, consider a climate policy that costs a billion dollars a year but would, if consistently implemented for a hundred years, prevent a trillion dollars worth of damages in year 100 of the policy. (This is simplistic, but many proposed policies have a similar payoff structure.) The break-even discount rate is between 3 and 4 percent: with a discount rate of 3%, it's a good idea, but with a discount rate of 4%, it's a bad idea.

When determining our discount rate to use, we must add up:

1) the probability that something else will solve climate change in the meantime, perhaps a cheap geoengineering solution. If that happens, we wasted the resources.   

2) the probability that a future administration will overturn the policy. If they do so, the future benefits will never materialize. 

3) the probability that the country ceases to exist (or becomes conquered by a different power that changes the policy). If that happens, we don't get the benefits.

4) the probability of any form of existential risk that would not be affected by the proposed policy.

5) the probability of any form of catastrophic risk that reduces the population enough to make the benefits irrelevant.

All of those estimates should be based on an outside view whenever possible. For example, for #3, look at how many nations in the last 200 years have collapsed, or undergone a revolution, or been conquered, and how often they tend to last before that happens. Given that data set, I'd guess there is a roughly 50% chance of the country ceasing to exist in its current form over the next 100 years, so the discount rate, for just point 3, should be about 0.5%. 

Estimates of x-risk add 0.15% for point 4, and probably another 0.15% for point 5. Point 2 probably adds another 1-2% to the discount rate, as does point 1 (I am a tech optimist). So the point estimate is a 3.8% discount rate.

Of course, this assumes that discontinuities result in zero benefit. If they still allow some benefit, the math gets more complicated, but we can approximate that by adjusting the discount rate downward for the policies where a discontinuity still gives some benefit. If there is a 0.5% chance of government change, but we assume a 50% chance that a revolution or invasion does not change the policy, then the adjusted discount rate is 0.25%.

A less controversial policy, with less chance of being made irrelevant by other means, might have a discontinuity-based discount rate as low as 1%, but I think this is as low as we can safely go. Chaos happens, and we are entering an uncertain age.

(Addition March 2022: The damages from carbon emissions that happen today will eventually happen regardless of intervening policy, but could still be made irrelevant by geoengineering or x-risk, so I would use a 1.5% discount rate for the Social Cost of Carbon, even with a zero rate of time preference. By a happy coincidence, this is the utility discount rate used by the standard models.)

Because people don't deal well with very small probabilities, it may be best, when doing the estimation (which will require surveys of experts) to focus on 10- or 25- or even 100-year blocks of time, and then divide.

Then, if the benefit is an economic effect rather than a pure impact on health or lives or utility, we also have to add the expected annual increase in the price of utility, conditional on no discontinuity. (Technically the total discount rate we should use is (1+u)/(1-d) - 1, where u is the growth rate in utility cost and d is the annual chance of a discontinuity. This will always be slightly higher than d + u, which will matter in the long run, but since these things are estimated with high uncertainty the difference is well within the confidence interval so we might as well keep things simple.)

To make an analogy to finance and investing, the discontinuity probability is the risk premium, and the utility price increase is the risk-free rate. We should only invest if our return is higher than both of these put together. In finance, the risk premium of the safest investment is assumed to be zero, and everything else is compared to that, but in this case, there is nothing with a zero risk premium, because we are comparing a certain result today with an uncertain result in the future. Anything we do has risk.

As a second example, consider a private action, perhaps setting up a think tank to research ways to stop a catastrophe, and that nobody else is trying to solve the problem or ever will. The discount rate is based on:

1) the probability that the organization suffers value drift.

2) the probability that the organization is taken over by a malevolent or selfish actor.

3) the probability that an outside force seizes all of its resources.

4) the probability that the government, society, and/or economy collapse for some other reason, making it impossible to continue work, or that another catastrophe makes the work pointless.

All of these risks increase with time, so the further out the benefit is, the less likely it is that the money spent will accomplish anything.

Even in the extreme case where the plan, once set in motion, cannot be disrupted, for example a rocket launch to nudge the trajectory of an asteroid that will hit the Earth in 200 years, we should discount. There are still discontinuities. Every year in the intervening time, there is a probability that people will develop a cheap asteroid deflection system, or that the asteroid will be consumed by mining, or that some other x-risk will make our effort irrelevant.

It should be clear by now that this is not an estimation problem for economic data sets and equations with greek letters. This is something we must think of by looking at historical trends in the appropriate reference class. Look at what has happened to similar plans or organizations in the past, and estimate the annual probability of a discontinuity of policies or resources between now and the time period we wish to affect. 

Scientific and Moral Uncertainty  

I'm less confident about this, and it doesn't affect the main result, but I suspect that epistemic uncertainty also increases our expected discount rate. Learning that the world doesn't work the way we thought it does is another kind of discontinuity. The annual probability that our theories are wrong, and that our policy is therefore mistaken, should be added to the discount rate.

Given how much our science and understanding of the world has changed and improved in the last hundred years, it seems reasonable to increase the discount rate by another 1% to account for epistemic humility and the potential realization that what we think we know is wrong. 

I think the annual discount makes sense, because long chains of causality introduce more chaos and more opportunities to be wrong. It's kind of like weather forecasting: we can be pretty confident in our ability to predict things a few days out, but after 10 days predictions are basically just swallowed up in chaos. 

Currently, with economic forecasting of government policies, it is rare to go out past 10 years, because of this kind of uncertainty. Assuming no large-scale change in economic conditions, we can predict the short-term effects of a tax or transfer policy fairly well. The Congressional Budget Office can know with decent precision what the effects of a tax cut will be over the next few years, and how that will affect labor supply and similar things. But there are a lot of second order effects and long-term adjustments that they don't know how to deal with, so as we project the changes of a policy more years out into the future we become less and less certain of anything. Typically they do this by increasing the error bars, and just stopping when they get too big.

But if we were doing our best to predict the effect of a large-scale change, because if we don't then people will treat all the side effects as though they did not exist, and while we are uncertain we know they are not zero, it might be more responsible to run the model the best we can, include its probabilistic output in the expected value calculation, and then add in an extra 1% or 2% discount rate on all future effects to account for the extra chaotic uncertainty.

Conclusion

None of this changes the fact that the value of preventing human extinction is huge, at least three orders of magnitude more than the entire annual world GDP. And that is just counting the VSL-based value that currently-existing humans place on preventing their own death; it does not include the value that future people place on existing.

In fact, this discounting strengthens the case for making serious sacrifices to stop x-risks. Anything else we might do to affect the far future is rendered irrelevant by the discounting, including the harmful effects of significantly slowing down economic growth. This is because there will be great unpredictability in human affairs as long as they still exist.

It can be argued that, ironically, the ethical implication of all this consequentialist math is to push people into a precautionary-principle-based deontology where we focus all of our actions on stopping whatever is likely to cause existential or catastrophic risk in the short term.  

However, if we cannot do anything about an x-risk, then we should be doing things to help people now rather than in the far future. The certainty of actually helping now is better than a tiny probability of a plan surviving and doing what we think it will, and this is reflected in the discounting.

Radical Tax Reform

I know that this is probably-impractical utopian thinking, but whenever I discuss related ideas in conversation, people ask me to write it up, so here goes:

The core idea is to get rid of all current taxes, and some existing government power and regulation, and replace them with externality taxes and progressive consumption taxes.

Externality Taxes: The Basics

All significant, measurable externalities should be taxed. Whenever possible, replace existing command-and-control regulations with a measure-and-tax system. Instead of people paying real economic costs for regulatory compliance, the money goes straight into the treasury as a transfer. This increases economic efficiency and will end up with less overall compliance costs for many businesses.

I've estimated that just the basic and obvious externality taxes will raise one to two trillion dollars a year in the USA. This gives us between a quarter and a half of the government's budget. In addition, there are some other externalities to tax:

Corporate Power is Taxed with a Progressive Corporate Income Tax

When considering only economic efficiency, i.e. avoiding the distortionary unfairness of double taxation, the ideal corporate income tax is zero. However, corporate size and market concentration are plausibly negative externalities that should be taxed.

All small businesses in competitive markets will pay no tax. Large companies pay exponentially increasing tax rates based on size and market concentration. If this is calibrated so that this is roughly revenue-neutral with the existing corporate income tax system, it would raise a quarter trillion dollars annually. 

I have no idea if a quarter trillion is an appropriate estimate of the annual harm done to our politics and society by concentrations of corporate power. I doubt anyone has estimated this in any sensible way. Probably the easiest way to do it would be a QALY survey. Fairly or not, many people hate big business, and would prefer to live in a world where it did not exist. Suppose that, on average, living in a nation with our amount of corporate power costs everyone one-thousandth of a QALY i.e. 1/20th as much discomfort as mild lower back pain. A third of a billion people, times a half a million dollars per QALY, times 1/1000, equals a sixth of a trillion dollars, which is pretty close to the current tax.

Other Externalities

Basically any problem you have with our society can be thought of as an externality to be taxed. The tax system, rather than the criminal justice system or any other power of government, should be the first tool to reach for to try to fix things. 

Advertising to children can be taxed. Advertising to anyone can be taxed. Political lobbying and large donations can be taxed. Actions that break up or damage families can be taxed. Conspicuous displays of wealth can be taxed. Harming the cultural foundations of our civilization can be taxed. Companies failing to hire in an equitable way can be taxed. Social media hate mobs can be taxed. 

I don't know if it would be best to have each separate tax require an act of Congress, or to give the executive branch broad enabling powers to identify externalities. Probably it would be safest for each one to require separate, narrow, enabling legislation. Congresspeople would propose tax-enabling legislation in response to constituent pressure. The legislation should include a rough estimate of the damage caused by the externality, and exactly what actions will be taxed and at what rate. Then we can have a productive discussion revolving mostly around checking the math and thinking about practicalities and side effects rather than shouting at each other. Once the tax is in place, people can either pay it or change their behavior.

Progressive Consumption Tax

Even if we identified all harms that could be profitably taxed without violating rights or causing too many bad side effects, it probably wouldn't be enough money to fund the government. We need another source of money that is as non-distortionary and politically sustainable as possible.

Whenever you tax something, you destroy it. Income taxes discourage productive work and investment. Wealth taxes, if only placed on wealth that the state can see, generate an economy-destroying incentive to hide money rather than invest it. Consumption taxes, on the other hand, have far fewer problems, and possibly some benefits, like discouraging extravagant lifestyles that fuel envy and social division.

In order to allow progressivity, and to avoid double-taxation of retired people and people who have saved, and to more easily transition from the current system, we implement it as an alteration of the income tax system. You would get taxed based on the money you withdraw from the bank rather than the money that goes from your employer into your bank account. Bank accounts that existed at the time of the tax changeover are exempt, under the assumption that you already paid income tax on them, and all of your income from that date forward is deposited in a new account.

Instead of your employer withholding money from your salary and submitting tax information to the government, your bank does, at the time you withdraw or spend it. (Assuming direct deposit. If you have an actual check you cash, it gets reported by the employer as normal and you are assumed to consume all of it, but you get a tax refund for depositing money in an investment account) All brokerage accounts etc. are considered 'in the bank' and untaxed. Charitable contributions are untaxed. Aside from that, the current tax brackets would be roughly the same, except we add lots of increased tax brackets for people who spend more money. Anyone who lives a lavish millionaire lifestyle pays 70%+ marginal tax rates on that spending, and seven-figure and higher lifestyles can be taxed at 80%+ At the top end, you'd lose almost 90% from each withdrawal, so you pay the government almost ten times as much for your consumption as it costs.

Some people will complain that this is unfair, because rich people can 'evade' almost all of their taxes by living a frugal middle-class lifestyle while their money sits around growing. But this is exactly the kind of behavior we want to encourage; it would increase investment and economic growth. We should think of the consumption tax as a kind of externality tax: Purchasing things takes them away from others, and plausibly harms them by decreasing their perception of relative status, but earning and storing money doesn't hurt anyone (assuming all the harms involved in what you did to earn the money are taxed properly as externalities).

Tax Liquid Wealth that Produces Power? 

We are probably already taxing lobbying and political donations as an externality, and of course they also get taxed as consumption. But potential power is still power, and it may be possible for large concentrations of wealth to exert improper influence, without paying any tax, by merely threatening to spend a lot on lobbying. If this turns out to be a problem, we can have a wealth tax on large pools of liquid assets. Make sure people can escape this by locking their money up in long-term investment projects, or by signing some kind of binding pledge to never spend money on lobbying or influence.

That is basically it. We've switched away from taxes that give bad incentives and toward taxes that give good incentives. In most cases, 'tax evasion' is stopping doing things that harm people. The next two points are details on stopping nonproductive tax evasion under this new system:

Include all Perks and Benefits as Income

In order to prevent powerful executives from evading taxes by disguising consumption as business expenses, we can count absolutely everything they get from their employer as income. The cost of all benefits is taxable income. In addition, something like exclusive use of a company car is income equal to the rental cost of the car. Use of an office counts as income equal to its rental cost.

All business travel is income equal to what it would cost an outsider to participate, e.g. a trip to Davos counts as income equal to the airfare plus the hotel cost plus all expenses plus whatever it would cost an ordinary person to bribe or buy their way in. If the event is so exclusive that there is no observable market for its tickets, this cost is assumed to be in the tens of millions.

In order to enforce this in an incentive-compatible way, the government collects taxes from companies on everything they spend on employees at a rate a little higher than the highest possible marginal tax rate of the recipient. (If a company has any employee who gets over a hundred of million dollars including benefits, it pays a tax on any benefit they get equal to ten times the total cost of the benefit.) Then individuals declare the value of all benefits and perks they received, in order to get a tax refund on them. 

For example, if consuming all your income (after including the benefits) would put you in the 50% tax bracket, and you got $100k worth of perks and benefits, then the company paid a bit more than $100k on your benefits (at a 50% consumption tax, you need to withdraw $200k to purchase $100k of consumption). If your actual consumption only takes you to the 40% bracket, you'd get a tax refund of a bit more than $33k. (with 40% consumption tax, you need to withdraw $166k to end up with $100k, so the proper tax would have been $66k.)

The more benefits you report, the higher your refund is, so you have an incentive to report everything you possibly can, even at the highest bracket, because the company was forced to pay a bit more. If employees collectively report more in benefits than the company reported and paid taxes on, the company pays the difference plus an additional fine.

This would result in total compensation becoming more transparent, and taxed properly, and would probably quickly settle down into everyone being paid in mostly cash, which would end a lot of current economic inefficiencies like health insurance being tied to employment.

Include Imputed Consumption Value of Non-investment Assets  

In order to discourage certain kinds of tax evasion, and to properly measure lifestyles, we also need to count as income the consumption value that you get from owning things like yachts and mansions. Most or all of this would apply only to people in the new lavish-lifestyle tax brackets; it isn't sensible or desirable to try to collect these taxes from people in lower brackets.

The imputed consumption from any non-investment asset you own is added to your income. For most things, this can be the rental market value. If you own a home, the government uses real-estate data to look up how much it would cost to rent a similar home, and increases your income by that amount. Same for cars, yachts, vacation property, etcetera. 

In cases where there are no rental markets, we do our best to estimate its auction value and assume that the consumption value is a fixed percentage of that. For example, someone hanging a Picasso on their wall, or who owns a million dollars' worth of jewelry, would pay a percentage of its value each year.

In order to avoid double-taxation, you subtract from the property value the amount that you paid for it. So if you paid $300k for your house, your imputed income is reduced by the rental value of a $300k house. If your house still has that value, you pay no additional taxes. If you bought it before the changeover, it is assumed that you paid appropriate taxes, and if you bought it after the changeover, you already got taxed on the money that left your bank. But if the house value has gone up since you bought it, you pay based on the difference.

This system taxes speculative profits rather than productive investment. Purchasing something that later rises in value is taxed, in a way that buying stock in a productive enterprise is not. The current system gets things exactly backwards, giving people strong incentives for unproductive real estate and art speculation.