Cross-posted to my website.

Summary

• This essay presents a variety of simple models on giving now vs. later for existential risk.
• On the whole, these models do not strongly favor either option. Giving now looks better under certain plausible assumptions, and giving later looks better under others.
  • On the simplest possible model with no movement growth and no external actors, giving later looks better.
  • Higher movement growth/external spending pushes more in favor of giving now.
  • If our efforts can only temporarily reduce x-risk, we should spend a proportion of our budget in each period, rather than spending or saving all of it.
• It has been argued that, because philanthropists are more patient than most actors, they should give later. This argument does not necessarily work for existential risk.
• The probability of extinction has relatively little effect on when to give.

Last updated 2020-09-08.

A simple approach to giving now vs. later

Let's begin by considering a highly simplified model of reality:

• Every year, the world experiences some constant level of positive well-being.^[1]
• And every year, it faces some probability of extinction^[2], which would reduce all future well-being to zero.
• If we spend money on reducing x-risk, we can permanently decrease the probability of extinction.
• If we do not reduce x-risk, no one else will, and the probability will stay the same.

For simplicity, let's say we only have two choices:

1. Give Now: Spend all our money now on reducing existential risk.
2. Give Later: Invest our money for the next 100 years, and then spend all of it on reducing existential risk.

(For our purposes, "giving" and "spending" mean the same thing—using money to do good.)

How do these options compare?

Giving now has the advantage that if we do reduce x-risk now, we get to enjoy a world with reduced risk for an extra century. But giving later allows us to compound our investments for a century, so we can spend much more money on x-risk reduction and therefore reduce it to a lower level.

To compare these options, we need to know the value of spending on x-risk reduction. Let's make an additional assumption: reducing x-risk by a fixed proportion requires exponentially increasing spending. Or in other words, spending on x-risk has logarithmic return.

This means the probability of extinction equals $\frac{p}{1+x}$, where $p$ gives the initial probability and $x$ is how much we've spent in total (see Appendix A for derivation). The scale of $x$ is normalized such that one unit of spending halves x-risk.

Let $r$ be the interest rate. If we give now, we can spend $x$ on reducing x-risk, and if we give later, we can spend $x\cdot (1+r{)}^{100}$.

Thus: if we give now, the x-risk level equals $\frac{p}{1+x}$ from now until the end of time.

In we give later, the x-risk level for the first century equals $p$. Then for all time after the first century, it equals $\frac{p}{1+x\cdot (1+r{)}^{100}}$.

From this, we can calculate the expected utilities of giving now and giving later. See Appendix A for the details of the calculation.

When the interest rate $r$ is high, giving later looks better. Low interest rates make giving now look preferable. The break-even point occurs approximately when $r=p$. When $r<p$, give now. When $r>p$, give later. In other words, give now if and only if the probability of extinction exceeds the interest rate.

So, based on this highly simplified approach, what can we say about whether to give now or later?

To answer that, we just need to know the interest rate and the probability of extinction. Michael Aird created a database of existential risk estimates. My copy of this database converts these estimates into annual probabilities. If we naively take the average of these estimates, we get a 0.23% annual probability of extinction or existential catastrophe^[3], which means we only need to earn higher than a 0.23% real interest rate over the next century for giving later to beat giving now. The highest estimate on the list (at the time of this writing) is 0.71%, so even the most pessimistic outlook only requires that we earn more than 0.71% real interest per year to prefer giving later. (0.71% represents the most pessimistic best guess, rather than the pessimistic end of one person's confidence interval. The high end of a well-informed confidence interval might give a worse probability than 0.71%.)

This model is just meant as an initial attempt; substantial improvements could be made. But under this model, giving later pretty much always beats giving now.

Should we spend our whole budget at once?

Update 2020-09-09: The original text of this section was incorrect. I originally wrote that we always prefer to spend our whole budget at once. While this is true under certain conditions, it is not necessarily true.

If we spend money to directly improve people's well-being (such as by making cash transfers to the developing world), then under certain assumptions, we want to spend the same fraction of our remaining budget in each period, as shown by Ramsey (1928)^[4]. Specifically, we should behave this way when money spent in one period only benefits that period, rather than providing a continuing benefit to all periods. But in the model laid out in the previous section, this assumption doesn't hold. The model assumes that if we reduce x-risk now, this will also reduce x-risk in all future periods. Therefore, we don't need to spend money in every period to continually push down x-risk.

We want to spend our whole budget in one period if and only if the last dollar spent in that period does more good than the first dollar spent in any other period. Under the model described above, this condition holds for most reasonable input parameters. (See Appendix A for the precise conditions where this holds.)

A more accurate model would allow us to spend money at any time, not just at two specific periods (now or 100 years from now). But the two-period model can at least give us a sense of the balance of how valuable it is to spend now versus later.

Relationship to the Ramsey model

Ramsey (1928)^[4:1] proposed what has become the standard economic model for determining optimal spending over time. In this model:

1. Society consumes resources and gains some well-being as a result of this consumption.
2. Due to economic growth, consumption increases over time.
3. Future well-being is considered less valuable than present well-being, according to some discount rate.

One reason to use a positive discount rate is that we might go extinct. If we don't exist, we cannot experience anything of value, so we should discount the future based on this possibility. The Ramsey model assumes that actors can choose the rate of consumption but cannot affect the discount rate. My model of x-risk reduction follows the same outline as Ramsey's model, but instead treats consumption as fixed and allows the discount rate to change via reducing x-risk. Instead of spending money to increase consumption and thus improve well-being, actors can spend money to reduce the discount rate and thus increase the probability of realizing a positive future.

Variations

In this part, we will consider several variations on our initial simple approach and see how they give different results. Unless stated otherwise, these build off the initial model, not off each other.

This list certainly does not include every interesting variation. Future work could extend these to develop more useful models.

Variation: Well-being increases over time

Originally, we assumed that the world experienced the same amount of well-being each year. This makes the math easy, but it's not a realistic assumption. More likely, the world will get better over time as people become better off (and as population grows, depending on your views on population ethics).

Let's take a simple approach to incorporate increasing well-being into our model. Assume global well-being increases logarithmically with global wealth, and that wealth increases exponentially. Thus, well-being increases linearly over time.

This new variation makes giving later look even better than it did in the first model. The break-even interest rate is always less than $p$. (See Appendix B for proof.) This variation relatively favors giving later because:

1. It values future well-being more highly.
2. Giving later does a better job of reducing x-risk in the distant future at the expense of letting it stay high during this century.

Variation: Exogenous reduction in x-risk

Some people believe that, over the next century or so, civilization will collectively recognize the importance of x-risk and begin investing much more in reducing it. How can we incorporate this into our model? (For simplicity, let's return to our initial assumption that well-being is constant across time.)

Suppose one century from now, the world will spend some amount of money on x-risk (call this "exogenous spending"). For simplicity, we assume that this spending occurs entirely at a single moment in time. Thus, we have two options:

1. Spend our budget now on reducing x-risk. Then, one century from now, some amount of exogenous spending will occur.
2. Invest our money for the next 100 years, at which time we spend our money alongside the exogenous spending.

(We could also invest for longer than 100 years. I will address that possibility in the next section.)

This variation makes the model harder to reason about because now we have more free variables. In the initial model, technically we had two free variables: the initial x-risk rate $p$ and the scale factor for spending $x$. The variable $x$ is defined such that one unit of spending equals the halving cost: the cost to halve x-risk from its initial level. (In mathematical terms, x-risk as a function of spending equals $\frac{p}{1+x}$, which gives $\frac{p}{2}$ when $x=1$.) If the halving cost is (say) $1 billion, then we should scale the units of $x$ such that $x=1$ corresponds to $1 billion of spending. But in the initial model, whether to give now or later did not depend on the halving cost; all that mattered was the initial probability $p$. In this new variation, we care about $p$ and the halving cost and the level of exogenous spending, so now we have three free variables instead of one.

More free variables means it's much harder to put precise conditions on when to give now or later. We can start with a simple observation: higher exogenous spending makes giving now look relatively better. (Or, to put it another way, as exogenous spending increases, the break-even interest rate increases as well.) The reason for this is that exogenous spending next century increases undiscounted utility by the same amount whether we give now or later. But if we give now, we're more likely to make it to next century, so this utility increase is worth more.

It's not at all obvious how much exogenous spending to expect relative to our own spending. First of all, who is "we", exactly, and how much money do we have? Then, how much do we expect civilization to spend on x-risk?

In the absence of any clear answer, we can look at a few examples. Suppose the baseline probability of extinction is 0.2% per year, and that we have a budget of $1 billion (which very roughly represents the x-risk budget of the effective altruism community). The following two tables give the break-even interest rate for various levels of exogenous spending. The first table assumes we need to spend $10 million to halve x-risk (that is, one unit of spending is worth $10 million) and the second assumes we need to spend $100 billion. In the first case, we only need to spend 1% of the EA budget to halve x-risk, and in the second case, the entire EA budget only goes 1% of the way to halving x-risk. These show us a spectrum of how the cost to halve x-risk changes things.

Table 1: Break-even real interest rates with $10 million halving cost

Exogenous Spending	Interest Rate
$0	0.20%
$1 billion	0.37%
$10 billion	1.2%
$100 billion	3.2%
$1 trillion	5.5%
$10 trillion	8.0%

Table 2: Break-even real interest rates with $100 billion halving cost

Exogenous Spending	Interest Rate
$0	0.20%
$1 billion	0.20%
$10 billion	0.22%
$100 billion	0.35%
$1 trillion	1.2%
$10 trillion	3.1%
$100 trillion	5.4%
$1 quadrillion	7.9%

We should expect gross world product (GWP) to be much higher in 100 years' time, which means these dollar amounts for exogenous spending will represent a smaller fraction of GWP than they would today. We might reasonably expect real GWP to grow 1.5% per year, which corresponds to about 5x growth in a century. That's why I included up to $1 quadrillion: even though that's a ridiculous amount of money today, society could contribute $1 quadrillion (in 2020 dollars) to x-risk research over the next century by spending something like 5% of GWP per year.

Some observations based on these tables:

• A lower halving cost results in higher break-even interest rates (and thus relatively favors giving now).
• The lower halving cost of $10 million requires something like $100 billion in exogenous spending before giving later looks better. ($10 billion exogenous spending gives a 1.2% break-even interest rate, which seems easily achievable; $1 trillion gives a rate of 5.5%, which might be achievable but seems a bit high.)
• At a halving cost of $100 billion, we require more like $10 trillion in exogenous spending before giving now beats giving later.

Based on my own beliefs about achievable interest rates, halving cost, and how much I expect civilization to spend on x-risk reduction in the future, I weakly lean toward preferring giving later according to this model. I find it somewhat implausible that we could see low enough interest rates, low enough halving cost, or high enough exogenous spending for giving now to look better. However, this is only based on my initial impression. I have not put much thought into the values of the halving cost or exogenous spending.

Variation: Beyond a century

Previously, we assumed we had a binary choice: give now or give a century from now. Let's take the exogenous-spending model from the previous section and extend it to say we can use our money at any point in the future. When is the best time to give?

First, observe that if we anticipate no exogenous spending, it doesn't matter when "later" is. The break-even interest rate always approximately equals $p$.^[5] At $r=p$, giving now is as good as giving later, no matter how far in the future you go. As a consequence, if $r>p$, we should postpone our spending indefinitely.

The time horizon only matters when we expect nonzero exogenous spending. In that case, we have essentially three choices:

1. Give now.
2. Give when the exogenous spending occurs.
3. Give in the distant future.^[6]

Under what conditions do we prefer each of these?

To answer this, we need to look at how exogenous spending affects x-risk. Suppose the world spends some amount $y$ on reducing x-risk. This reduces the probability of extinction from its initial level $p$ to a lower level $\frac{p}{1+y}$.

Consider the perspective of a philanthropist 100 years from now. To them, it appears that the baseline probability of extinction equals $\frac{p}{1+y}$ instead of $p$, and also that halving x-risk take a lot more work than it did 100 years ago ($1+y$ times as much work, to be exact). This philanthropist is confronted with exactly the same choice as in our initial simple model with no exogenous spending, except that (a) instead of x-risk $p$, they see x-risk $\frac{p}{1+y}$, and (b) spending is scaled down by a factor of $1+y$ (so for example, if $1+y=100$, then $100 of spending for them would be like $1 of spending today).

Recall that, in our initial model, giving now beats giving later when the interest rate $r$ is less than the probability of extinction, and vice versa. Therefore, giving when exogenous spending occurs is better than giving in the distant future if and only if $r<\frac{p}{1+y}$.

If we derive the formula for the break-even interest rate between options 1 and 2 (see Appendix C), we will see that there is no situation in which we prefer option 2. According to this model, we should never give when the exogenous spending occurs. We should always either give now or in the distant future.[^6] Specifically, we should give now if and only if $r<\frac{p}{1+y}$.^[7] Intuitively, there's nothing special about the time when exogenous spending occurs, so we never want to give at that time.

But if we invest for sufficiently long, we start to run into the limits of our model assumptions. According to this model, we can compound forever at some fixed rate $r$. But if we do this, eventually we will come to control a significant fraction of the world's wealth. Once we reach a certain level of wealth, it will longer make sense to continue postponing our spending. We might not be able to earn enough of an investment return before this point for giving later to beat giving now.

Variation: Including movement growth

The possibility of movement growth substantially affects considerations, so we should include it in our analysis. We might be able to spend money in ways that will encourage others to spend more on x-risk, and their spending might encourage still others, producing a compounding effect. How does this change things?

Let's return to our basic model with no exogenous spending. Suppose that any spending on x-risk also encourages others to spend more. For simplicity, suppose this compounding effect only lasts for some fixed amount of time, say 100 years. (This will limit the scope of our analysis.) Does this change how much we should value giving now vs. later?

Perhaps surprisingly, it hardly changes anything. To see why, assume we live in the world with no movement growth and where, at the market interest rate, we are indifferent between giving now or later. That means we don't care whether we spend one dollar now or $(1+r{)}^{100}$ dollars in a century. Now suppose our donation causes someone else to spend a little money the following year, and that for every dollar we spend, they will spend $\mu$ dollars. That means we cause either $\mu$ dollars of spending next year or $\mu (1+r{)}^{100}$ dollars of spending in 101 years. The comparison between these two options looks nearly identical to the initial comparison between spending $1 now or $(1+r{)}^{100}$ dollars later. The only difference in this new comparison is that we've already spent one dollar (if we choose to give now) or $(1+r{)}^{100}$ dollars (if we give later).

What if, instead of compounding for a century, movement growth continues compounding forever? If the movement growth rate exceeds the interest rate, then clearly we should give now. If not, a higher movement growth rate still pushes toward giving now. It's difficult to say more than that because even calculating the utility of giving now or later at a particular (nonzero) movement growth rate requires computing an infinite sum, and I do not know of any way to evaluate the sum in finitely many steps.

(I wrote a Python script to numerically determine when to give. The program suggests that a movement growth rate greater than 1% usually favors giving now. But I do not entirely trust that the optimization algorithms I used can reliably produce correct answers, so treat this number as uncertain, even given the model assumptions.)

This approach assumes that spending on x-risk reduction contributes to movement growth. Perhaps instead, some types of spending increase movement growth but don't help reduce existential risk, and other types of spending do the opposite. That would add even more complexity to the decision (we now have three choices instead of two), so I will not address it other than to mention it as a possibility.

Variation: Recurring exogenous spending

What if, instead of some exogenous spending occurring at a single point in time, society makes ongoing efforts to reduce x-risk? And it continues these efforts indefinitely, reducing risk closer and closer to zero?

In this model, the long-term future has infinite expected utility.^[8] That means we can no longer compare different choices using ordinary methods, because we cannot say which of two infinities is larger. Unfortunately, with this model variation, we cannot say anything about whether to give now or later. That said, I'm optimistic that we could find a way to compare worlds with nonzero but asymptotically-diminishing discount rates.^[9]

Alternatively, we could assume x-risk will decrease over time and approach some positive minimum rather than zero. In that case, the expected utility of the future would remain finite. It's hard to characterize exactly how this model behaves, because it's much more complicated than the model with point-in-time exogenous spending. Based on some experiments with my Python script (under permanent_reduction_binary_with_extra_exogenous_spending), it appears that this model favors giving now a little more strongly than the point-in-time exogenous spending model, but not overwhelmingly so.

Due to its complexity, I will not address this variation in detail. Still, this seems like a fruitful subject for future research, because the real world will probably contain recurring exogenous spending.

Variation: Temporary benefits only

What if spending on existential risk reduction does not permanently reduce x-risk? What if the benefits only last for, say, 100 years, and then x-risk reverts to its initial level?

Let's assume the following:

1. We can either spend our budget now or in 100 years.
2. Our spending only reduces x-risk for 100 years, after which the benefit disappears.
3. In 100 years' time, some exogenous spending will occur, and this spending will re-occur every 100 years so that it has a persistent effect.

Under this altered model, efforts to reduce x-risk look far less valuable in general. With the permanent-benefits assumption, the value of the future can grow arbitrarily large as the probability of extinction approaches zero. But when we can only reduce x-risk temporarily, we can at best reduce the risk to (near) zero during a single century, after which it will increase again.

In general, giving now beats giving later more often under the temporary-benefits model than with permanent benefits. In either case, by giving later, you can reduce x-risk by a larger amount because you have more money. But in the permanent-benefits model, this larger x-risk reduction stays in place forever, whereas in the temporary-benefits case, it only lasts for a century.

As described in the variation with exogenous spending, we should consider the scale our budget relative to the cost to halve x-risk. With temporary benefits to spending, the break-even interest rate depends on the relative size of our budget. Increasing the budget pushes in favor of giving now. For example:

1. When our budget equals the halving cost, the break-even interest rate is 0.4%.
2. When our budget is 2.6x the halving cost, the break-even interest rate is 0.9%.
3. When our budget exceeds 2.7x the cost to halve x-risk, giving now beats giving later for any interest rate.
4. Similarly, with a small budget but sufficiently large exogenous spending, giving now looks preferable to giving later for any interest rate.

Let's look at why giving now always beats giving later when the budget is greater than 2.7x the halving cost. Recall that this model requires us to spend all our money either now or next century. (The requirement to spend all our money at once doesn't make much sense for this model variation, but let's accept it for now.) So no matter how much interest we earn, the best we can do is reduce the probability of extinction to (arbitrarily close to) zero during the next century; we can't do anything to reduce x-risk two, three, four or more centuries from now. This puts an upper limit on how much good we can do by giving later, even with infinite spending. At a certain point, higher interest rates don't produce meaningful marginal benefits.

It seems likely that the "effective altruist budget" is insufficient to halve existential risk. Therefore, if we assume no exogenous spending, we should prefer giving later to giving now. But it also seems likely that x-risk spending will increase in the future. Thus, if we adopt the temporary-benefits model, our belief about whether to give now or later almost entirely depends on how much exogenous spending to expect.

Should we spend our whole budget at once?

In the previous section, I assumed that we had to spend all our money either now or in a century. But this assumption doesn't make much sense.

When we have the ability to permanently reduce x-risk, we usually want to use our entire budget in a single lump sum, either now or at some indefinite point in the future (at least according to the given model assumptions). But recall that this result depends on a specific condition: the last dollar spent in one period must do more good than the first dollar spent in any other period. This can happen in one of two ways:

1. Spending has linear or increasing marginal utility.
2. Spending has diminishing marginal utility, but it diminishes slowly enough that we still prefer to spend our entire budget in one period.

In the temporary-benefits model, utility of spending diminishes much more rapidly than in the permanent-benefits model because reducing x-risk only linearly increases the expected length of the future, rather than exponentially increasing it. And reducing x-risk becomes more costly over time. Therefore, each marginal dollar does much less than the last to increase the expected length of the future.

So we prefer to spread our spending over time (see Appendix D for proof). In this case, our simple binary choice of give now vs. give a century from now doesn't reflect how we should actually behave. Instead, at each point in time, we should determine the optimal proportion of our budget to use, and then invest the rest for later.

It is much harder to determine exactly how much to spend in each century, so I will not attempt to do so in this essay. This would likely make a fruitful subject for future discussion, particularly because this model seems perhaps more realistic than any of the others in this essay.

Assumptions

The models discussed in this essay make a lot of assumptions. I already laid out some of these explicitly when first describing my basic approach. In this section, I will address some implicit premises. I cannot comprehensively cover every assumption made, but we should review a few of the most important ones and discuss their implications.

Ramsey (1928)^[4:2] gives a longer list of assumptions made by models similar to the ones in this essay.

Future happiness outweighs suffering

This model assumes that, conditional on non-extinction, the world will experience positive well-being in perpetuity. But the future might not consist of positive experiences on balance. It looks more likely than not that the world currently experiences net negative utility thanks to wild animal suffering, and presumably has done so for as long as sentient life has existed. So the assumption of positive well-being requires either that the world is in fact net positive currently, or that we will improve things in the future such that it becomes net positive.

Alternatively, even if we expect future happiness to outweigh suffering, we could claim that suffering matters more.

How exactly an extinction event affects the well-being of sentient life depends on the nature of the event. If it wipes out humanity while leaving most animals alive, that would end human-caused animal suffering, but allow wild animal suffering to persist. A catastrophe that wipes out all life (such as a massive asteroid impact, or perhaps a misaligned AI) would reduce utility to zero.

We cannot give to any other cause

As discussed previously, the models outlined in this essay are based on the Ramsey model, but has some key differences. Most significantly, the traditional Ramsey model treats the discount rate as fixed, and decision-maker must decide how much to spend on directly improving well-being at each point in time. The model in this essay treats well-being as fixed, but allows the discount rate (that is, the probability of extinction) to vary.

In real life, actors can spend money either on increasing well-being or on reducing x-risk (or on other things). A more sophisticated model would allow for a three-pronged choice: (1) invest, (2) spend on increasing well-being, or (3) spend on reducing x-risk. (Even more complex models could introduce additional choices, although at a certain point, additional complexity becomes more burdensome than enlightening.) This three-pronged choice approach might tell us something about how spending over time should look across cause areas.

The cost of reducing x-risk grows with inflation

The models we used have a built-in assumption that the cost of reducing x-risk grows over time at the rate of inflation—that's why we calculate investment return using the real interest rate rather than the nominal rate.

The cost of goods grows at the rate of inflation, and x-risk reduction is sort of like a good. But the cost of labor grows with GDP. If x-risk requires employing people, and wages grow approximately with GDP, then we should use the GDP-adjusted interest rate instead of the real rate. In that case, we end up with a lower interest rate, and thus giving now looks relatively more appealing.

We do already assume that money spent on reducing x-risk has logarithmic return, and GDP growth could partially drive this effect. Additionally, costs are driven by supply and demand, so if we don't spend anything on x-risk (and no exogenous spending occurs), demand stays fixed, and the cost does not increase. But if, say, doing x-risk research requires hiring people away from other fields, then our costs will increase over time with wages even if we don't spend any money.

Zero discount rate (other than x-risk)

This model assumes we only discount future well-being in proportion to the probability of extinction. Some altruists might discount the future for other reasons, although these mostly only apply for individuals or organizations, not for society as a whole. We might adopt a pure time preference, where we simply consider the future to be less valuable than present, although most philosophers believe that a pure time preference does not have any reasonable justification.

One source of discounting that does apply at the societal level is the possibility of transformative AI. If a sufficiently advanced artificial intelligence would render our efforts useless, then we should discount future well-being by the probability that a transformative AI emerges. Adding this additional discount makes giving now look relatively more appealing.

Fixed interest rate

So far, we have assumed that investors can earn interest at some risk-free rate. In practice, investors almost always prefer to take on at least some degree of risk in order to increase their expected return. How does this apply to existential risk donors?

The optimal investment strategy depends on whether x-risk reductions are permanent or temporary. Under the permanent-benefits model, spending has linear utility. That means investors want to maximize expected return without regard for risk. (As mentioned previously, this seems implausible.)

Under the temporary-benefits variation, spending has decreasing marginal utility (because, unlike in the permanent-benefits model, reducing x-risk does not exponentially increase the expected length of the future). When we have decreasing marginal utility, we prefer less risky investments (all else equal). In addition:

• If we hold more than enough money to halve x-risk, then we should behave similarly to most investors with regard to our risk appetite.
• If we have less money than the halving cost, then we prefer to take on more risk.

The details of optimal investment strategy are given in Appendix E.

On patience

Philip Trammell has written and spoken on how philanthropists should think about giving now vs. later. His analysis essentially supersedes all previous work on the question among effective altruists. He offers a compelling argument that EAs should give later. I cannot do full justice to the argument, but at a high level, it goes like this (as I interpret it):

1. For society as a whole, there exists some optimal balance of consumption and investment.
2. The optimal rate of consumption depends on how much one discounts future welfare—a higher discount means one should consume more now, and vice versa.
3. Most people are "impatient": they don't consider future welfare as important as welfare today. (They have a pure time preference.)
4. Philanthropists should be "patient". (They should not have a pure time preference.)
5. Therefore, from a philanthropic perspective, society will consume too much and invest too little.
6. Therefore, at the margin, philanthropists maximize long-term welfare by investing all their resources rather than donating now.

To put it slightly differently, most people under-value the future, so patient philanthropists can buy future welfare cheaply (so to speak).

This argument is based on the Ramsey model. It assumes we do good by increasing consumption. This most obviously applies for causes like cash transfers, where altruists give money directly to poor people, who can then spend that money to improve their circumstances. It might more loosely apply to other causes such as factory farming, where charitable giving does not do good by increasing consumption per se, but it does create a direct improvement in someone's life circumstances^[10] (where "someone" could be a chicken or a fish, not just a human). As Trammell's argument goes, we should expect to do more good by increasing consumption in the future rather than in the present.

Altruistic efforts to reduce existential risk do not necessarily work the same way. Reducing x-risk does not directly improve people's well-being; instead, it increases the chance that future people will exist at all. When thinking about giving now vs. later, we cannot use exactly the same approach.^[11]

How does Trammell's argument apply to the models developed in this essay?

If our actions can permanently reduce x-risk, then society should collectively spend all of its money either now or later (under these models). The fact that other actors behave impatiently does not change how philanthropists should use our resources on the margin. (In real life, we can't spend all our money instantly. But there's still a meaningful difference between "you should spend 5% of your altruistic resources per year" and "you should spend as quickly as possible without compromising effectiveness.")

If x-risk reductions only work temporarily, that means we should disperse our resources at some particular rate. And if most actors are impatient, they probably spend too quickly, so patient philanthropists should balance this out by investing all their money to give later.

The field of existential risk might not contain many impatient actors. We do see substantial spending from society at large on some potential existential risks, such as climate change and biosecurity, but most of this funding does not specifically focus on minimizing the chance of extinction. Perhaps patient actors account for most or nearly all of the "pure" x-risk funding, in which case they should give at the optimal rate rather than giving (exclusively) later.

Conclusion

For x-risk-focused philanthropists, how to use altruistic resources over time substantially depends on how spending affects x-risk and movement growth. This essay showed what results we get with various assumptions. To summarize:

• If we can permanently reduce x-risk and we expect only modest exogenous spending or movement growth, then we should give later.
• If instead we expect substantial exogenous spending or movement growth, then we should give now.
• If efforts can only temporarily reduce x-risk, then we should give some proportion of our resources each year. We should determine our spending rate based on the interest rate, exogenous spending, and movement growth. But if most actors in the space are impatient, then we should save all our money to give later.

So what conclusion we draw depends on which model assumptions we find most plausible. I do not have a particularly well-informed opinion on the matter, but I'm inclined to believe that (a) some actions can have long-lasting (if not quite permanent) impacts on x-risk, (b) spending on x-risk substantially contributes to movement growth, and (c) most x-risk donors behave impatiently, not because of a pure time preference but simply because they haven't put much thought into the optimal spending rate. I would also guess that the effect of (b) is larger than that of (c). Based on these assumptions, x-risk donors should spread their spending over time, and probably spend at a relatively high rate. But I only have weak confidence in this position.

As a final observation, notice that in the permanent-benefits model, the probability of extinction appears hardly to matter at all. With no exogenous spending, giving now only beats giving later when x-risk is implausibly high—it must exceed the interest rate, which would require an annual x-risk level of 2% to 5% or so (depending on what you believe the interest rate equals). With high exogenous spending, giving now looks better even when extinction is extremely unlikely. The probability of extinction only matters when the amount of exogenous spending falls in a fairly narrow middle range.

The models in this essay are not intended to accurately reflect reality. I believe better models could be developed. This essay simply provides some initial approaches to the problem of how x-risk donors should allocate their resources across time. Future work could create new and better models.

Source code for all calculations done in this essay is available here.

Appendix

Appendix A: Simple approach to giving now vs. later

Derivation of the probability of extinction $\frac{p}{1+x}$

Let $\delta$ be the probability of extinction. First, the rate of x-risk reduction is proportional to the current level of x-risk. We can mathematically represent this as $\frac{d\delta }{dy}=-\alpha y$ for some constant $\alpha$, where $y$ is work output aimed at reducing x-risk. For simplicity, let's assume $\alpha =1$. The solution to this equation is $\delta =p{e}^{-y}$ for any constant $p$, where $p$ gives the probability of extinction before any work is done.

Second, work output is asymptotically logarithmic with spending: $y=\log (1+x)$. (We add 1 so that zero spending corresponds to zero work output instead of $-\infty$.) Plugging this into the previous equation gives $\delta =p{e}^{-\log (1+x)}$, which we can rewrite as $\delta =\frac{p}{1+x}$. One unit of spending represents how much it costs to halve x-risk from its initial level, because $x=1$ will decrease x-risk from $p$ to $\frac{p}{2}$.

Expected utility of giving now or later

If we do not give any money toward reducing x-risk, the expected utility of the future equals

$$\begin{array}{c}\sum _{t=0}^{\infty }{(1-p)}^{t}w=\frac{w}{p}\end{array}$$

where $w$ is the constant level of well-being experienced by sentient beings during each year.

For simplicity, let $w=1$.

The formulas for expected utility of giving now or later follow easily from the definitions given. These formulas are:

$$\begin{array}{c}\text{Utility Now}=\sum _{t=0}^{\infty }{(1-\frac{p}{1+x})}^t=\frac{1+x}{p}\end{array}$$

$$\begin{array}{c}\text{Utility Later}=\left[\sum _{t=0}^{99}\right(1-p{)}^t]+(1-p{)}^{100}\left(\frac{1+x(1+r{)}^{100}}{p}\right)\end{array}$$

These formulas make use of the identity $\sum _{t=0}^{\infty }(1-a{)}^t=\frac{1}{a}$.

Derivation of the break-even point $r=p$

In the main article, I claimed that the break-even interest rate occurs approximately at $r=p$.

$r=\frac{p}{1-p}$ is the exact break-even interest rate. When $p$ is small, $p\approx \frac{p}{1-p}$. This works even for unreasonably large values of $p$: For example, when $p=1/10$, $\frac{p}{1-p}=1/9$, and surely the annual probability of extinction is less than 10%. More generally, $|p-\frac{p}{1-p}|=\frac{p^2}{1-p}$, which shrinks slightly faster than quadratically as $p$ approaches zero.

Now to prove that $r=\frac{p}{1-p}$ gives the exact break-even interest rate. To do this, we will plug in this value of $r$ into the formula for the utility of giving later, and show that it equals the utility of giving now.

$$\begin{array}{c}\text{Utility Later}=\left[\sum _{t=0}^{99}\right(1-p{)}^t]+(1-p{)}^{100}\left(\frac{1+x(\frac{1}{1-p}{)}^{100}}{p}\right)\end{array}$$

We can rearrange this to

$$\begin{array}{c}\left[\sum _{t=0}^{99}\right(1-p{)}^t]+\frac{1}{p}(1-p{)}^{100}+\frac{x}{p}\end{array}$$

Observe that $\frac{1}{p}=\sum _{t=0}^{\infty }(1-p{)}^t$, and therefore $\frac{1}{p}(1-p{)}^{100}=\sum _{t=100}^{\infty }(1-p{)}^t$. Furthermore, clearly $\sum _{t=0}^{99}(1-p{)}^t+\sum _{t=100}^{\infty }(1-p{)}^t=\sum _{t=0}^{\infty }(1-p{)}^t$. Thus we can rewrite the Utility Later formula as

$$\begin{array}{c}\frac{1}{p}+\frac{x}{p}=\frac{1+x}{p}\end{array}$$

which exactly equals Utility Now. $■$

Observations:

1. This proof works no matter when the "give later" time occurs, whether it's a century from now, a millennium, a year, etc.
2. When $r>\frac{p}{1-p}$, you can achieve arbitrarily high expected utility by waiting sufficiently long before spending.

When to spend all money at once

The model assumes we must spend either now or 100 years from now. Let's generalize this to say we must spend either now or $T$ years from now, for some number $T$. We have a budget $B$ that we can spend. Let $x_1$ be the amount we choose to spend now, and $x_2$ be the amount we spend at time $T$. We can discount $x_2$ at the investment rate of return to get that $x_1+(1+r{)}^T{x}_2=B$. What are the optimal values of $x_1$ and $x_2$?

Total utility is given by

$$\begin{array}{c}U=\sum _{t=0}^T(1-\frac{p}{1+x_1}{)}^t+(1-\frac{p}{1+x_1}{)}^T\sum _{t=0}^{\infty }(1-\frac{p}{1+x_1+x_2}{)}^t\end{array}$$

The formula for the optimal values of $x_i$ is complicated, but we can confirm numerically that for most reasonable parameter values, at $T=100$, we prefer to give all now or all later. (See x-risk-now-later.py, using the function permanent_reduction_continuous.)

We can specify the precise formula when $T=1$. In this case, utility is maximized by $x_1=\sqrt{\frac{p(1+r)(1+B)}{r}}-1$. If $x_1\le 0$, that means we should spend all of our money later and none now, and vice versa if $x_1\ge B$. If $0<x_1<B$, then we should spend some money now and some later.

Appendix B: Model variation where well-being increases over time

Expected utility of giving now or later

For any number $a\ge 0$, $\sum _{t=0}^{\infty }(1-a{)}^t\cdot t=\frac{1-a}{a^2}$. We can use this fact to update the formulas for the expected utility of giving now or later.

$$\begin{array}{c}\text{Utility Now}=\frac{1-\frac{p}{1+x}}{(\frac{p}{1+x}{)}^2}=\frac{1}{p^2}[x^2+(2-p)x+(1-p\left)\right]\end{array}$$

$$\begin{array}{c}\text{Utility Later}=\sum _{t=0}^{99}(1-p{)}^t+(1-p{)}^{100}\frac{1}{p^2}\left[\right(x(1+r{)}^{100}{)}^2+(2-p\left)x\right(1+r{)}^{100}+(1-p)]\end{array}$$

Derivation of the break-even point

I will not precisely derive the break-even interest rate in this case. Instead, I will simply show that the break-even point occurs when $r<\frac{p}{1-p}$. From this, it follows that the break-even point is lower in the increasing-wellbeing model than in the constant-wellbeing model.

Plugging in $r=\frac{p}{1-p}$ to the formula for Utility Later gives

$$\begin{array}{c}\sum _{t=0}^{99}(1-p{)}^t+(1-p{)}^{100}\frac{1}{p^2}\left[\right(x(1-p{)}^{-100}{)}^2+(2-p\left)x\right(1-p{)}^{-100}+(1-p)]\end{array}$$

Rearranging these terms, and using the same trick we used in the previous proof to get rid of the summation term, we get

$$\begin{array}{c}\text{Utility Later}=\frac{1}{p^2}[\frac{x^2}{(1-p{)}^{100}}+(2-p)x+(1-p\left)\right]\end{array}$$

Comparing this to the Utility Now formula, we see that they look nearly the same except that Utility Later has an $x^2$ term, and in its place, Utility Later has $\frac{x^2}{(1-p{)}^{100}}$. The latter term is bigger, so giving later provides greater utility than giving now. $■$

Appendix C: Beyond a century

Option 2 is never best

Let $\hat{p}=\frac{p}{1+y}$. As discussed in the main text, option 3 provides greater utility than the other options when $r>\hat{p}$. However, as discussed previously in this appendix, this is an approximation. The true break-even point is $\frac{\hat{p}}{1-\hat{p}}=\frac{p}{1+y-p}$. Let $r_2=\frac{p}{1+y-p}$. If we show that the break-even interest rate $r_1$ between options 1 and 2 is greater than $r_2$, that means there are two cases:

1. The market interest rate is less than $r_1$, so option 1 wins.
2. The market interest rate is greater than $r_2$, so option 3 wins.

Let $U_n$ be the total utility of option $n$. Let $r$ be the market interest rate. Without loss of generality, suppose that rather than spending either now or in a century, we spend either now or next year (this makes the math simpler.)

Then:

$$\begin{array}{cc}{U}_1& =1+(1-\frac{p}{1+x})\left(\frac{1+y+x}{p}\right)\\ U_2& =1+(1-p)\left(\frac{1+y+x(1+r)}{p}\right)\end{array}$$

The break-even interest rate $r_1$ occurs when $U_1=U_2$. Solving for $r_1$ gives

$$\begin{array}{c}{r}_1=\frac{p(1+y+x)}{(1-p)(1+x)}\end{array}$$

We want to find the conditions under which $r_1\ge r_2$:

$$\begin{array}{c}\frac{p(1+y+x)}{(1-p)(1+x)}\ge \frac{p}{1+y-p}\end{array}$$

Simplifying this inequality gives

$$\begin{array}{c}(2+x+y)y\ge py\end{array}$$

When $y=0$, this inequality holds because both sides equal $0$. When $y>0$, we can simplify the inequality to $2+x+y\ge p$. Recall that $p$ is a probability and therefore cannot be greater than 1, so this inequality is always true. The inequality holds in both conditions on $y$. Therefore, $r_1\ge r_2$.

Option 2 can only be optimal when $r>r_1$ and $r<r_2$, but this can never happen. Therefore, we should always prefer option 1 or option 3. $■$

Appendix D: Model variation where spending only temporarily reduces x-risk

Optimality of spreading spending over time

To show that we should spread out our spending across multiple centuries, it is sufficient to show that utility of spending is concave within each period, holding spending in all other periods fixed.

To make this proof a little easier, let's change the assumptions. Instead of x-risk reduction efforts lasting a century, let's say they only last for a year.

Suppose you spend your budget at year $n+1$. Then total utility equals (using the same variable terminology as in previous appendices):

$$\begin{array}{cc}U& =\sum _{t=0}^n(1-p{)}^t+(1-p{)}^n(1-\frac{p}{1+x})\left(\sum _{t=0}^{\infty }\right(1-p{)}^t)\end{array}$$

Then, taking the second derivative:

$$\begin{array}{c}\frac{{\partial }^{2}U}{\partial x^2}=\frac{-2}{(1+x{)}^3}(1-p{)}^n\end{array}$$

The second derivative is negative for any non-negative value of $x$. Therefore, utility is concave with respect to spending. $■$

Appendix E: Optimal investment where spending temporarily reduces x-risk

Suppose we have some risky investment and want to decide how much leverage to use. We should pick the amount of leverage that maximizes expected utility.

For simplicity, assume x-risk reductions only last for a single year rather than a century. (Or we could say that a single period equals a century rather than a year. This simplification changes the answer somewhat, but it's necessary to make the problem appropriately tractable.) From the utility formula in Appendix D, we can see that utility is a linear function of $\frac{-p}{1+x}$. I do not believe the expected utility of this function has a precise closed-form solution, but it does closely resemble an isoelastic utility function with a relative risk aversion coefficient of 2.

More precisely, we can calculate that the relative risk aversion (RRA) for this utility function equals $\frac{2x}{1+x}$. For small $x$, RRA is close to 0; as $x$ increases, it approaches a maximum of 2.

For an isoelastic utility function with RRA = 2, optimal leverage is given by $\frac{\mu }{2\sigma }$, where $\mu$ and $\sigma$ are the mean and standard deviation of the logarithm of the interest rate $r$ (assuming $r$ follows a log-normal distribution). Merton (1969)^[12] gives the precise conditions for optimality, but we might fairly assume that for large $x$, $\frac{\mu }{2\sigma }$ gives approximately optimal leverage for existential risk-focused philanthropists. For smaller $x$, optimal leverage is higher, but it's hard to specify the precise value.

Ordinary investors tend to have RRAs roughly around 2. That means under the temporary-benefits model and with a relatively low halving cost, philanthropists should behave like ordinary investors with respect to how they manage risk. With a relatively high halving cost, philanthropists should be willing to take on much more risk.

Acknowledgments

Thanks to Aaron Gertler and Mindy McTeigue for providing feedback on drafts of this essay.

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1. For the most part, I use the term "well-being" to refer to the value of the world at a particular point in time, and "utility" to refer to the total value across time of taking a particular action. Utility is equal to the sum of the discounted well-being at each point in time. I use this terminology because "utility" refers to the thing we value, which isn't exactly the same as well-being—it's well-being aggregated over time using a particular method. ↩︎

2. We could also talk about existential catastrophes other than extinction. This essay will focus on extinction because it's a dramatic binary event that's easier to reason about. For simplicity, we can say that either society will be good and have some high level of well-being, or everyone dies and the world has zero well-being. We might also be able to talk about other existential catastrophes in the same way by treating catastrophic outcomes as resulting in zero well-being. Even if it's not actually zero, it's much smaller than the level of well-being in an ideal world, so we can round it off to zero. Some types of bad outcomes might result in substantially negative well-being, but that's out of scope of this essay. ↩︎ ↩︎

3. I am conflating these two concepts for simplicity. See prior footnote.^[2:1] ↩︎

4. Ramsey (1928). A Mathematical Theory of Saving. ↩︎ ↩︎ ↩︎

5. More precisely, it always exactly equals $\frac{p}{1-p}$. ↩︎

6. If we do prefer to give in the distant future, when exactly do we give? Do we just keep postponing our giving indefinitely? This problem comes up in many models like the one in this essay. For more, see Rendall (2018), Discounting and the Paradox of the Infinitely Postponed Splurge. ↩︎

7. This follows from the fact that, with an interest rate above $\frac{p}{1+y}$, giving later can achieve arbitrarily large expected utility by waiting sufficiently long before giving. ↩︎

8. If the probability of extinction at time $t$ equals $\frac{1}{1+at}$ for some constant spending rate $a$, then total utility equals ${\sum }_0^{\infty }\frac{1}{1+at}$. This series diverges. ↩︎

9. Limit-discounted utilitarianism (Jonsson & Voorneveld, 2018) can compare infinite utility streams generated by a finite Markov process. A discount rate asymptotically approaching zero cannot arise from a finite Markov process, but it can still be defined with finitely many (and in fact very few) symbols, so there might be something similar to limit-discounted utilitarianism that can compare such utility streams. ↩︎

10. Some factory farming interventions prevent animals from existing rather than improving their lives, which introduces questions of population ethics. But interventions such as welfare reforms do straightforwardly improve animals' lives, so we can treat this as analogous to consumption. ↩︎

11. Can other causes fit into the standard Ramsey model? Direct cash transfers (via GiveDirectly) fit the model well, because they operate by directly increasing poor people's consumption. Others, like malaria nets or cage-free campaigns, don't seem to fit as cleanly, and we might want to model them differently. I will not address any other causes in this essay, but this raises questions for future research. ↩︎

12. Merton (1969). Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time Case. ↩︎