Hi! I’m Nico and I’m on the research team at Founders Pledge. We noticed that the way we compare current to future income benefits is in tension with how we compare income benefits across interventions. However, aligning these two comparisons—choosing the same function for utility from consumption for both—might lead to large changes in our CEAs. So, we are now thinking about how to choose the right approach. Since our framework is based on GiveWell’s, which is used by other organisations, too, I expect that we’re facing the same issues. I’m posting here as a way of thinking out loud and with the hope of getting input from others.

Summary

• Founders Pledge and GiveWell both use different values of $\eta$ (elasticity of marginal utility from consumption) when modelling isoelastic utility from consumption depending on the context. Across interventions, we assume $\eta =1$. Over time within an intervention, we assume $\eta \ge 1.59$. We should choose the same $\eta$ for both models as having different $\eta$ values can lead us to prefer doubling the incomes of richer people relative to poorer people.
• Practically, this inconsistency leads to strange conclusions in existing CEAs. Taking GiveWell's Unlimit Health (deworming) CEA as a stylised example:
  • For two people in Madagascar, we value doubling the income of someone who makes $2,500 30% as much as for someone who makes $500.
  • When the person making $2,500 lives in Côte d’Ivoire, however, we value doubling their income the same (100% as much) as for the person in Madagascar who makes $500.
• Resolving this isn’t straightforward and has large implications for our prioritisation. For example:
  • Using $\eta =1$ everywhere—which implies that income doublings have the same value regardless of absolute income levels—doubles the cost-effectiveness of education and deworming programs and makes economic growth and poverty graduation interventions look substantially better.
  • Using $\eta =1.87$, which is implied by our discount rate, everywhere requires our evaluations to take into account the income levels of recipients and prioritise lower-income regions more. An income doubling in Malawi would be worth roughly 1.9x as much as in Ethiopia, 3.4x as much as in Kenya, 6.4x as much as in Egypt, and 75x as much as in the US. The same is true within countries: in India, an income doubling in Bihar would be worth 3.4x as much as an income doubling in Andhra Pradesh.
• I’m hoping this post will start a conversation around what the right value of $\eta$ is.

Our inconsistent $\eta$ values

We use $\eta =1$ (log-utility) to compare the value of income benefits across people or interventions^[1]. That assumption is convenient because it allows us to disregard absolute income levels: an income doubling is as valuable from $250 to $500 as it is from $2.5k to $5k. Because of that, we can make statements like “the value of a 10% income increase from a deworming program in India equals the value of a 10% income increase from a cash transfer program in Kenya” without knowing the incomes of the recipients.

At the same time, we use $\eta \ge 1.59$ when comparing the value of income benefits in different years within an intervention. A higher $\eta$ reflects more strongly diminishing utility from consumption. Our current choice of $\eta$ assumes that income doublings that occur 10 years from now are worth about 20% less than income doublings today just because the recipients will have higher incomes then. (At $\eta =1$, that 20% discount would be 0%.) We use this higher $\eta$ whenever we apply our default discount rate.

Our discount rate

Both GiveWell and Founders Pledge use a 4% annual discount rate for consumption benefits^[2]; 2.6 of the 4% likely reflect diminishing marginal utility from income doublings. Whenever we estimate the cost-effectiveness of a program that raises future incomes, we apply a 4% annual discount rate. Many of our programs raise incomes decades in the future (e.g., poverty graduation, deworming, and education programs). At those time scales, the discount rate matters. At a 4% rate, an income doubling today is 80% more valuable than an income doubling 15 years from now. 

The 4% discount rate is the sum of 3 components^[3]: temporal uncertainty (1.4%), improving circumstances (1.7%), and compounding non-monetary benefits (0.9%). In this post, I focus only on the latter two. What distinguishes them from temporal uncertainty is that they reflect diminishing marginal utility from income doublings. We only include them in the discount rate because incomes increase over time even absent any intervention.

Improving Circumstances

This component captures the intuition that utility from consumption declines faster than log-utility and that doubling a poor person’s income is more valuable than doubling a rich person’s income (GiveWell 2018a). The discount rate of 1.7% comes from explicitly using $\eta =1.59$ and an annual consumption growth rate of 3% (GiveWell 2018b)^[4]. Because a program recipient will have 3% higher consumption in a year from now absent any intervention, they will value an income doubling 1.7% less.

Compounding non-monetary benefits

This factor represents future non-income benefits caused by higher consumption today. Some examples are reduced stress and improved nutrition whose positive effects compound over time (GiveWell 2020). While I couldn’t find much information about this factor, my best guess is that it effectively reflects diminishing marginal utility of income doublings. 

(Quantitatively it would be captured by $\eta =1.87$ when combined with the improving circumstances component. That comes from solving the last equation in Rethink Priorities’ 2023 report for $\eta$ given $r=0.026$ and $g=0.03$—i.e., assuming that the compounding non-monetary benefits factor also reflects diminishing marginal utility from income doublings. As a result I'm assuming the discount rate reflects $\eta =1.87$ for the remainder of the post.) 

However, I’m more uncertain about this component reflecting diminishing utility because I could only find limited info on it. I’m explaining my reasoning for why I believe it does at the end of the post and would like to learn more about what GiveWell is trying to capture here. If it turns out that this component reflects something else, the argument in this post does not change qualitatively, though the magnitudes of the conclusions would be roughly a third lower.

To recap, we have two different assumptions for how strongly utility from consumption declines^[5]. Under $\eta =1$, which we use for cross-sectional comparisons (i.e., between people/places/interventions), an income doubling has the same value regardless of the beneficiary’s income. Under $\eta =1.87$, an income doubling is worth much less for richer people. For example, doubling the income of the average person in Kenya ($1000 GDP/capita) is worth only about half an income doubling in Malawi (~$500 GDP/capita).^[6] We use the latter framework when comparing benefits within an intervention over time (i.e., longitudinally).

Why it matters

We should use the same $\eta$ for comparisons across time and place. One might argue that the degree of diminishing utility from consumption differs when comparing between places/people vs over time. This would for example be the case if all that matters is one’s relative position in society. However, using different utility functions leads to some unwanted conclusions. We, for example, sometimes value income doublings more for richer people. To avoid that, we need to settle on one single value of $\eta$ for both cross-sectional and longitudinal comparisons. (I’m detailing the argument for this at the end of the post.)

The inconsistent $\eta$ values cause strange conclusions in existing CEAs

Take GiveWell's Unlimit Health CEA, which considers income increases across different geographies and over time within each geography. Unlimit Health runs a deworming program, which in expectation allocates 13% of a new donation to Madagascar and 6% to Côte d’Ivoire.

The CEA roughly assumes that each treated person experiences the same persistent % consumption increases each year over the course of their life. These income benefits are discounted by our default rate, so that an income doubling today is worth 3x as much as the same income doubling in 48 years solely because recipients will be about 4 times as rich. At the same time, current-year income doublings in Madagascar ($500 GDP/capita) and Côte d’Ivoire ($2.5k GDP/capita) are valued the same^[7]. This leads to the following weird conclusion:

• In Madagascar, we value the income doubling of someone who makes $500 about 3x as much as the income doubling of someone who makes $2.5k.
• However, if the person who makes $2.5k lives instead in Côte d’Ivoire, we value their income doubling the same as doubling the income of the $500 earner in Madagascar.

Put another way, if we consistently apply the income doublings framework (log-utility), the CEA would imply a 2.6% annual discount rate for pure time preference: even if the income % increases are as certain to occur in 48 years as they are now, we value an income doubling 3x more today.

Using either $\eta =1$ or $\eta =1.87$ everywhere would substantially impact our prioritisation

We might hope that as long as we settle on either $\eta =1$ or $\eta =1.87$ and use it consistently, our CEAs, or at least the relative ranking of charities, wouldn’t change. Unfortunately, that is not true.

Option 1: Using $\eta =1.87$ also in comparisons across interventions

Using $\eta =1.87$ would require us to start taking into account absolute income levels in our cost-effectiveness analyses. Consider an intervention that, by improving agricultural productivity, raises the incomes of 1M people by 0.1%. Such an intervention in Malawi would be worth 1.9x as much as in Ethiopia, 3.4x more than in Kenya, and 6.4x as much as in Egypt. The same is true within countries: an income doubling in Bihar would be worth 3.4x as much as an income doubling in Andhra Pradesh. Once we consider income doublings in high-income countries, e.g., from increasing frontier growth, those differences become even larger. An income doubling in the US is worth only 5% of an income doubling in Kenya (or about 1.3% of an income doubling in Malawi). Even for our current interventions, the choice of geography would matter more. In the Unlimit Health example above, the Madagascar program would be about 6.8 times as cost-effective as Côte d’Ivoire (up from 2.3 times as cost-effective).

Option 2: Using $\eta =1$ also in comparisons across time

$\eta =1$ implies removing the 2.6 percentage points from the discount rate that are due to diminishing marginal utility from income doublings (i.e., reducing the discount rate to 1.4%^[8]). This reduction makes categories of interventions that raise incomes over long time horizons look substantially better. For example:

• Human capital: The cost-effectiveness of GAIN and others would roughly double.
• Economic growth: If effects persist over 40 years, programs would become about 50% more cost-effective.
• Deworming: Unlimit Health’s rating would almost double from 12.1 to 23.2^[9].
• Poverty graduation: Our rating of Bandhan would increase by ~20%.

Next steps

We should be consistent in our choice of $\eta$ and use the same value for comparisons across places/interventions and over time. That being said, I’m not sure what the right choice of $\eta$ is. Since even small changes in the discount rate can change the relative cost-effectiveness of different interventions, we want to make sure we get it right. I’ve briefly looked into what others have said about $\eta$:

• GiveWell provides some intuition that $\eta =1.59$ is ballpark right.
• Open Philanthropy noted that $\eta =1$ might accord better with life satisfaction data. However, they also say that conclusion is contested.
• SoGive provides a table with empirical estimates of $\eta$.

Through this post, I’m hoping to get input from others on how we should best choose $\eta$.

Appendix

The "compounding non-monetary benefits" factor likely reflects diminishing marginal utility from income doublings

My best guess is that the “Compounding non-monetary benefits” component of GiveWell’s discount rate effectively captures diminishing marginal utility from consumption doublings. However, I’m uncertain about that because there is not much publicly available reasoning about it. Here I’m outlining how I’ve come to that conclusion.

First, I’ve found the following info on this factor:

• “There are non-monetary returns not captured in our cost-effectiveness analysis which likely compound over time and are causally intertwined with consumption. These include reduced stress and improved nutrition.” (GiveWell 2020)
• “Conceptually, it's trying to capture the dynamics behind poverty traps. e.g., an argument for increasing consumption now being more valuable than in the future is that it releases credit constraints, allows additional spending on things like nutrition, and those investments pay off down the line. This parameter was originally 1.9%, and meant to capture the benefits of being able to get returns on capital. But then I think that argument got weaker based on long term fade out evidence from GiveDirectly's program. I'd wanted to cut it to 0, but Caitlin made the case that (a) there may still be other unmeasured non-monetary benefits that compound over time (b) getting money earlier might still allow consumption smoothing. So we halved it as a kind of compromise position.” (Email from James Snowden to Rethink Priorities, in A review of GiveWell’s discount rate — Rethink Priorities) 

My first intuition was that this factor captures the idea that giving a person income today is better than giving it to them one year from now because the effects from better nutrition can accumulate for one additional year overall (or because it allows for consumption smoothing over more periods in total). However, I’m thinking that that scenario wouldn’t require a discount. Presumably, the beneficiaries of a program also change over time, and the average age of a target group will stay roughly the same, so that the non-income benefits accumulate for the same number of years on average. (As a stylised example: a program that gives income transfers to all people over 18 in a community will still do so in 10 years from now rather than restricting it to people over 28).

As a result, I’m thinking that the discount rate is trying to capture some (non-age) change in the characteristics of recipients, so that income doublings a year from now are worth less than an income doubling today because there will be fewer future non-income benefits that accumulate. If those changes are independent of absolute income levels, I think they also shouldn’t require a discount. Say a person gets a certain utility benefit from doubling their consumption today, and an additional x% of that utility benefit from future non-income benefits. If that x% is the same across absolute levels of income, there is no need for a discount rate because the increase in utility would be the same (consumption doublings have the same value because of log-utility and so the x% share of that utility benefit is also equal).^[10]

My best guess for the “Compounding non-monetary benefits” component is then that it reflects some change in the characteristics of recipients that is captured by an increase in income; in other words: that it reflects a poverty trap dynamic where the sum of future non-income benefits is higher at lower absolute income levels. For example, wealthier people might face fewer credit constraints/poverty traps (perhaps because they have collateral), have other options to smooth consumption and guard against income shocks, benefit less in relative terms from reduced stress and improved nutrition, etc. However, such a discount implies that the marginal utility from consumption doublings is lower at higher income levels—i.e., diminishes more strongly than log-utility. Together with the improving circumstances discount, we can model it as an $\eta$ of 1.87 (by backing out $\eta$ from a discount rate of 2.6% and a consumption growth rate of 3% in a modified Ramsey equation).

Since that conclusion required some interpretation based on the info that I could find, I’m overall uncertain about it. It could be that this component reflects something else entirely—I would be interested in learning more about what GiveWell is capturing here.

Using different $\eta$ values for comparisons across time and place can lead us to prefer doubling the incomes of richer relative to poorer people

There are different ways in which we could specify when to use log-utility vs $\eta$ > 1. I’m outlining two options here that are similar to how we currently implicitly do it and explain why either of them will sometimes lead us to prefer doubling the incomes of richer relative to poorer people. (As Karthik points out in the comments, at other times, we will prefer doubling the incomes of poorer relative to richer people.)

Option 1: Use log-utility across people, but $\eta >1$ for each individual person (over time)

Suppose that for any comparison across different people we use log-utility but that for each single person we use $\eta >1$ (for example to make comparisons across time or different states of the world). This has the following issue of preferring rich people’s income doublings.

Consider three people, A, B, and C, with respective incomes of 5, 10, and 10 USD. Suppose we compare doubling the income of A twice (from 5 -> 10 and 10 -> 20) to doubling the incomes of both B and C (10 -> 20 each). The income doubling from 5 -> 10 for A has the same value as doubling the income from 10 -> 20 for B because we use log-utility across people. Doubling A's income from 10 -> 20 is less valuable than doubling it from 5 -> 10 because we use $\eta >1$ for each individual. Since the other three income doublings are all worth the same (A: 5 -> 10, B and C: 10 -> 20), we prefer doubling the incomes of both B and C compared to doubling A’s income twice. Now, since the income doubling for A from 10 -> 20 has the same value as that for B from 10 -> 20, it must be that we prefer doubling C’s income from 10 -> 20 over doubling A’s income from 5 -> 10. But that means we prefer doubling the richer person’s income.

Option 2: Use log-utility across countries, but $\eta >1$ within each country

Another distinction might be to use log-utility for comparisons across countries (or another geography), but to use $\eta >1$ for any comparison within a country (e.g., across people or time). In order to make sense of across-country comparisons, we need to specify which incomes in each country are worth the same across countries. Say that, for example, doubling the income of the average earner in a country is worth the same across countries. Now, doubling the mean income in the US (~$50k) is worth the same as doubling the income of the average earner in Madagascar (~$500). However, within Madagascar, doubling the income of a richer person is worth less than doubling the income of a poorer person, so doubling the income of a person making $1000 is worth less than doubling the income of a person making $500. Since doubling the income of a person making $500 in Madagascar was worth the same as doubling the income of a person making $50k in the US, we have that doubling the income of a person making $1000 (in Madagascar) is worth less than doubling the income of a person making $50k (in the US). So we are again preferring doubling the income of a richer person relative to a poorer person.

1. ^{^}

   This follows from using a constant moral weight for income doublings, regardless of the initial income level. See, e.g., 2023 GiveWell cost-effectiveness analysis – version 4 (public).

2. ^{^}

   Other organisations also use this rate, likely following GiveWell’s choice. Rethink Priorities notes in their 2023 review: "Some organizations defer to GiveWell’s discounting practices. For example, Charity Entrepreneurship stated in an email that it 'defer[s] to GiveWell’s 4%'." According to the report, CE-incubated charities also use the 4%. Open Philanthropy as far as I can tell does not use it.

3. ^{^}

   GiveWell provides information on their discount rate in the following three documents: Discount rate 2020_JS, CAM Discount Rate for 2018 Giving Season, and Discount rate 2018.

4. ^{^}

   One way to derive the discount rate from the elasticity of marginal utility from consumption and the consumption growth rate is to use a modified Ramsey equation. See, e.g., Rethink Priorities' 2023 review of GiveWell's discount rate (p. 48).

5. ^{^}

   Rethink Priorities pointed out this inconsistency with regards to the Improving Circumstances component in their 2023 review of GiveWell’s discount rate. See, e.g., the discussion on p. 21–22 of their report.

6. ^{^}

   55% to be precise. The inverse or $r_{givewell}$ as derived in Rethink Priorities’ 2023 report on the last page when using $\eta =1.87$ and $g=1$.

7. ^{^}

   While the incomes of program recipients might be more similar than the difference in GDP/capita would suggest, let’s assume for the sake of illustration that the deworming benefits accrue to people with the average income in each country (or that the difference in the average income of the recipients in the two countries is 5x).

8. ^{^}

   Under log-utility ($\eta =1$) we would not need any discounts for diminishing marginal utility from income doublings (since there are none). For a derivation, see Rethink Priorities’ 2023 report.

9. ^{^}

   This comes from changing cell B16 in the Deworm the World sheet from 4% to 1.4%, and recording the change in B198 in the Unlimit Health sheet.

10. ^{^}

    However, if those future non-income benefits aren’t captured anywhere else, it might make sense to increase the moral weight of an income doubling to systematically account for them.