Reverse-Engineering the Philanthropic Discount Rate

by MichaelDickens38 min read9th Jul 2021No comments

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Temporal discountingPatient altruismTiming of philanthropy
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Summary

  • How much of our resources should we spend now, and how much should we invest for the future? The correct balance is largely determined by how much we discount the future. A higher discount rate means we should spend more now; a lower discount rate tells us to spend more later.
  • In a previous essay, I directly estimated the philanthropic discount rate. Alternatively, We can reverse-engineer the philanthropic discount rate from typical investors' discount rates if we know the difference between the two. [More]
  • In theory, people invest differently depending on what discount rate they use. We can estimate the typical discount rate by looking at historical investment performance. But the results vary depending on what data we look at. [More]
  • We can also look at surveys of experts' beliefs on the discount rate, but it's not clear how to interpret their answers. [More]
  • Then we need to know the difference between the typical and philanthropic discount rates. But it's difficult to say to what extent philanthropists and typical investors disagree. [More]
  • Some additional details raise more concerns about the reliability of this methodology. [More]
  • Ultimately, it looks like we cannot effectively reverse-engineer the philanthropic discount rate, even if we spend substantially more effort on the problem. But under some conditions, we prefer to give later as long as we discount at a lower rate than non-philanthropists, which means we don't need to make precise estimates. [More]

Cross-posted to my website.

Introduction

Last time, I attempted to estimate the philanthropic discount rate. We should discount according to the probability that our resources become worthless in the future. I identified three main reasons why this might happen: existential catastrophe, expropriation, and value drift. Then I estimated the probabilities of each of these individually.

As an alternate approach, what if we reverse-engineer the philanthropic discount rate (call it ) from the typical discount rate (call it )?

We can find as a combination of:

  1. the typical discount rate
  2. the difference between and

For example, if we know that and is 0.5 percentage points less than , then .

This approach has some advantages estimating directly. We can only speculate as to the risk of existential catastrophe, expropriation, or value drift. But we can empirically observe by looking at financial markets. Then all we need to know is how philanthropists and ordinary investors differ.

Estimating the typical discount rate

We can estimate most investors' by looking at investment returns. According to standard economic theory, we can determine if we know four things:

  1. The market rate of return
  2. The volatility of the market
  3. People's appetites for risk
  4. How much people consume vs. invest

In this section, I will avoid the mathematical details, but I provide the exact formula in Appendix A.

Let's see if we can use this formula to estimate using historical data.

(Short answer: We can't—it's too hard to pin down the values of the four inputs.)

Deriving from GDP and market investment return

At a national scale, the consumption rate equals the GDP growth rate, and the investment rate represents the aggregate investment return achieved by capital markets. We can use these to estimate .

The Rate of Return on Everything, 1870-2015 (henceforth "RORE") provides equity, bond, and housing returns, as well as GDPs, for 16 countries back as far as 1870.

Table 1 includes:

  • Real GDP growth
  • Mean and standard deviation of log real returns for the weighted combination of all investment assets in the country (weighting determined by RORE)
  • The derived value of for various values of , where is the rate at which marginal utility diminishes with consumption (also known as the rate of relative risk aversion)

Table 1: RORE 1870–2015 by country

=0.5 =1 =1.5 =2 =3
---------- ------- --------- ------------ -------------- ------------ -------------- ------------ ------------
AUS 3.37 5.47 7.23 3.9 2.1 0.5 -1 -3.6
BEL 2.26 5.90 9.11 4.9 3.6 2.6 1.8 0.8
CHE 2.45 5.29 6.82 4.1 2.8 1.7 0.6 -1.1
DEU 3.83 6.14 6.32 4.3 2.3 0.4 -1.3 -4.6
DNK 2.74 6.93 6.43 5.6 4.2 2.9 1.7 -0.5
ESP 3.02 4.10 8.62 2.7 1.1 -0.3 -1.6 -3.5
FIN 3.91 8.03 15.52 6.4 4.1 2.5 1.4 1.1
FRA 2.38 4.48 8.75 3.4 2.1 1 0.1 -1.1
GBR 2.32 4.28 8.02 3.2 2 0.9 0 -1.4
ITA 3.12 4.62 6.72 3.1 1.5 0 -1.4 -3.8
JPN 4.29 5.04 7.45 3 0.8 -1.3 -3.3 -6.7
NLD 3.48 5.29 6.92 3.6 1.8 0.1 -1.4 -4.2
NOR 3.28 6.64 7.26 5.1 3.4 1.8 0.3 -2.1
PRT 2.94 5.04 8.88 3.7 2.1 0.7 -0.4 -2.2
SWE 3.06 6.77 8.62 5.3 3.7 2.3 1 -0.9
USA 3.12 5.51 8.39 4 2.4 0.9 -0.4 -2.4

Table 2: RORE 1870–2015, summary statistics

=0.5 =1 =1.5 =2 =3
median 3.09 5.38 7.74 3.95 2.2 0.9 -0.2 -2.15
mean 3.1 5.6 8.19 4.14 2.5 1.04 -0.24 -2.26

(These s are the median/mean values from Table 1, rather than calculated from the mean/median .)

How should we measure consumption?

In the above tables, I used GDP as the measure of consumption because it's the most well-known metric (and the only consumption metric that comes with the RORE data set). But it's not obvious that we should use GDP. Arguably, we should use net national income, which excludes depreciation of manufactured capital. Or perhaps we should use GDP minus investment.

Subtracting investment from GDP, leaving only the "consumption-y" part, reduces by about 20% (World Bank, 2020). This increases , and the magnitude of the increase depends on .

Table 3: RORE 1870-2015, subtracting 20% from GDP

=0.5 =1 =1.5 =2 =3
median 2.47 5.38 7.74 4.25 2.9 1.65 0.9 -0.3
mean 2.48 5.6 8.19 4.45 3.11 1.98 1.01 -0.42

Correlation between r and g

According to this model, is proportional to for any period , so they should be perfectly correlated. That is to say, people's spending in a particular period depends on how their investments performed. They're willing to spend more when their investments perform better, and vice versa.

In real life, and are only weakly correlated. In the RORE data set, they had a correlation of only 0.14 (Online Appendix Q, Table A.26). So aggregate consumption has little to do with investment return.

Our model predicts perfect correlation between and , but this prediction is not even close to true. That casts doubt on the reliability of this model.

Some other caveats

  • These figures for market return represent gross return, not net. If we used net returns instead, we would get a higher value for (assuming ).
  • Most investors cannot practically invest in a representative sample of the entire market, because doing so requires buying diversified real estate holdings. A less-diversified portfolio would probably not have lower expected return, but it would have greater volatility, resulting in a lower value of (assuming ).

Deriving from endowment portfolios

We can estimate by looking at how universities manage their endowments. This approach has a few advantages:

  • Universities are legally required to publish their financial data.
  • Universities often intend for their endowments to last for centuries.
  • Large universities typically hire teams to manage their endowments, and these teams make intentional, strategic choices about how to spend endowment money over time. (Some of them might even explicitly use the same model as I use to inform their decision-making.)

Let's use the Harvard endowment as an example. Harvard provides annual financial reports going back to fiscal year 2003.

(To get a more complete perspective, we could combine data from multiple universities, but for now I will only look at the Harvard endowment.)

Over 2003-2019, Harvard had an average annual consumption rate of 4.2%. If we define as the consumption rate when excluding gifts, we get an average of 3.2%. (For both and , the median equaled the mean to within 0.1 percentage points). As explained in Appendix A, we can use the consumption rate to derive the discount rate.

See Appendix B for the full table of yearly consumption rates.

I measure both and because it is unclear how we should treat gifts. If we treat gifts simply as part of the portfolio, we get . But arguably, if the endowment receives outside income in the form of gifts, that should not count as part of the portfolio, in which case we can use instead. Ideally, we would treat the gifts as coming from an external asset, determine the value of that asset, and count it as part of the endowment; but we don't have enough data to properly do this.

I calculate the consumption rate (or ) using the expenditures for the current year divided by the value of the endowment at the end of the current year. Arguably, we should use the value of the endowment at the beginning of the year, because the university executives make budget decisions based on the endowment's value at the beginning of the year, not the end. This method would give a median/mean of 4.4% and a median/mean of 3.3%/3.2%, respectively.

The following table gives implied for various values of , using (which approximately equal the long-run market return and standard deviation according to RORE).

Table 4: Implied δ from Harvard endowment statements

0.5 1 1.5 2 3
3.2 4.7 3.3 1.9 0.7 -1.3
4.4 5.3 4.5 3.8 3.2 2.5

Universities have sources of revenue other than the endowment (most notably including tuition and research funding). A proper analysis would consider all source of revenue and all expenditures. However, as with gifts, we cannot easily model these other revenue sources as capital assets, which makes it harder to deduce using this model. The above analysis treats the endowment as an isolated system that doesn't know anything about the rest of the university's financial activities.

Deriving from public companies' shareholder yield

Let's look at how publicly-traded companies make consumption choices. To simplify, when companies earn income, they can do two things with it: invest it back into the company (via R&D, acquisitions, etc.), or allow shareholders to "consume" it by paying out dividends or buybacks.

In some sense, dividends don't count as consumption, because shareholders might invest the money back into other assets. But it can still be considered consumption from the perspective of the company because the company can no longer invest with that money, and shareholders now have the money to use however they like. We could say that money feeds into shareholders' utility functions, and the company is agnostic as to how shareholders use the money, even if they invest it.

According to Yardeni (2020), "S&P 500 Buybacks & Dividends", the S&P 500 shareholder yield (that is, dividend yield plus buyback yield) from 1998 to 2020 has varied from about 2% to 7%, with an average around 4.5%. (Yardeni does not report exact figures; I'm estimating these based on the graph on page 9.)

Shareholder yield equals dividends + buybacks divided by market capitalization. Is that really the right denominator to use? Market cap represents the discounted present value of the market, not the capital of the underlying companies per se. Arguably, we should use total assets on the denominator instead (calculating shareholder yield as (dividends + buybacks) / assets), or perhaps balance sheet capital (which can be defined in several ways; one such definition is property, plant, and equipment + current assets - debt in current liabilities - cash and short-term investments). Using any of these would give a higher consumption rate.

I don't believe any denominator based on a company's balance sheet properly reflects the value of the capital a company possesses. Companies hold a great deal of value in non-balance-sheet assets such as human capital. In some sectors (such as technology), the majority of a company's assets do not appear on the balance sheet. The market cap of a company represents something like the net value of all its capital, whether physical, human, or otherwise. I'm not entirely convinced that we should use market cap as the definition of "capital", but it seems better than using anything else.

If we assume a 4.5% shareholder yield, along with for the global equities market[1][2], we can construct this table of possible values of depending on :

Table 5: Implied from public companies' shareholder yield

0.5 1 1.5 2 3
4.5 5.2 4.6 4.8 5.7 9.6

The substantially higher volatility of equities (compared to the global market portfolio, which also includes bonds and real estate) results in a higher estimate for .

Deriving from GDP growth

Liu (2012), Inferring the rate of pure time preference under uncertainty, uses similar methodology to estimate as a function of GDP growth, volatility in GDP growth, and the risk-free rate of return, calculated with US data 1889 to 1978 (taken from Mehra & Prescott (1985), The Equity Premium: A Puzzle). They find that, for , "lies within ±1% from zero"; and for , "tends to be negative."[3]

Experts' beliefs about

Drupp et al. (2017), Discounting Disentangled, presents a survey of experts asking their beliefs about the value of the discount rate.

Before proceeding, we should clarify two terms: (1) the utility discount rate and (2) the rate of pure time preference. A pure time preference represents the extent to which we discount future beings merely because they live in the future. Many philosophers argue (and I would agree) that we should not admit any pure time preference. The utility discount rate includes the rate of pure time preference, but it can also account for other reasons to discount the future, such as the possibility of extinction or expropriation.

Unfortunately, Drupp et al. conflate between these two terms. In their survey, they explicitly ask respondents to estimate the "[r]ate of societal pure time preference (or utility discount rate)", as if they mean the same thing. Based on the descriptions in the paper, the authors appear to assume that a pure time preference is the only reason to discount future utility. But it is unclear how survey respondents might interpret the prompt, and they probably interpret it in different ways, which makes it difficult to say how we can use the responses. The modal discount rate was 0%, which at least suggests that many participants interpreted the question as asking for the rate of pure time preference.

Surveyed experts reported a mean discount rate of 1.1% and a median of 0.5%. It's not clear whether we should interpret these as the utility discount rate or as the rate of pure time preference (or perhaps as a weighted average of the two, if some survey respondents used one interpretation and some used the other).

We could also attempt to derive the utility discount rate from experts' beliefs about the social discount rate using this formula: (taken from Gollier et al. (2016), Declining discount rates: Economic justifications and implications for long-run policy, section 2.1). Using the median from RORE and experts' median estimates gives a of 1.0%. But this move seems questionable because we don't know if surveyed experts would agree with this estimate for .

Getting from the typical discount rate to the philanthropic discount rate

In the previous part, we attempted to estimate using several methods. These methods did not produce consistent answers. But if we did manage to determine the value of , how to we then derive the philanthropic discount rate ?

Most people behave as though they have a positive pure time preference. If we know the average rate of pure time preference (call it ), we can calculate as . This gives us the "patient" discount rate, where we only discount due to empirical factors such as the probability of extinction, not due to a pure time preference.

But well-informed philanthropists might substantially disagree with most people about things like the probability of extinction. In that case, we might estimate a different discount rate.

If we assume factual agreement + pure time preference

We can break down the discount rate into the patient discount rate plus the rate of pure time preference. The patient discount rate tells us the extent to which we should discount based on empirical factors such as the probability of extinction or the probability that we lose access to our funds. If we assume everyone agrees on the empirical discount rate, then we can determine the value of the philanthropic discount rate (which equals the patient discount rate) by taking the typical discount rate and subtracting the rate of pure time preference.

What is the rate of pure time preference?

Unfortunately, research on the rate of pure time preference has produced mixed results.

Discounting Disentangled, discussed previously, gave expert opinion on the discount rate, but it did not clearly distinguish between the utility discount rate and the rate of pure time preference.

Frederick (2003), Measuring Intergenerational Time Preference: Are Future Lives Valued Less?, reviewed prior surveys that attempted to measure pure time preference. Then it conducted a series of new surveys, and found that people's reported pure time preference varied depending on the exact question asked. Overall, average answers varied from 0% to 6% depending on survey wording. So it's not clear whether most people exhibit a pure time preference at all, much less what rate they do exhibit. If most people do exhibit a 0% rate, that means the philanthropic discount rate equals the typical discount rate. But the two rates might differ by as much as six percentage points.

For more on the problems with observing pure time preference, see Frederick et al. (2002), Time Discounting and Time Preference: A Critical Review.

Non-factual disagreements other than pure time preference

Philanthropists and typical investors might discount at different rates for reasons other than pure time preference, even if they agree about all empirical facts. Philanthropists might naturally differ from typical investors for two main reasons:

  1. Typical investors don't care about value drift.
  2. Typical investors do care about dying.

Value drift occurs when an altruist becomes less altruistic over time. If I stop spending money to do good, then from an altruistic perspective, my money is being wasted. So I should discount my future spending based on the probability that this will happen. But self-interested investors shouldn't behave this way. Any money that I freely spend must be in accordance with my values at the time (assuming I'm behaving rationally), so I still consider it valuable from a self-interested perspective.

But unlike altruistic investors, typical investors don't care as much about what happens to their money after they die. Many investors do want to ensure their children or grandchildren can lead good lives, but it often makes sense for them to discount the future based on the probability that they die before then. Whereas for an altruist, death does not much diminish the value of money. So this gives a reason why typical investors might discount the future when altruists don't.

If we don't assume factual agreement

When I directly estimated the philanthropic discount rate, I identified three key reasons to discount: extinction/global catastrophe, expropriation, and value drift. As discussed in the previous section, typical investors don't care about value drift in the same way. In addition, many people in the effective altruism community have fairly unusual beliefs that might affect how they discount the future: namely, they pay more attention to existential risks. Many effective altruists assign much higher probabilities to existential catastrophes than most people do. However:

  1. It's possible that most sophisticated investors do generally agree with relatively pessimistic assessments of existential risk.
  2. Perhaps most effective altruists aren't particularly pessimistic about x-risk, and only differentially prioritize it because they care more about the long-run future.

So effective altruists might discount the future more heavily based on their beliefs about x-risk, but this is not obviously the case.

Disagreements about x-risk probably don't substantially affect the overall discount rate. Even a fairly pessimistic 1% annual probability of existential catastrophe does not matter much in comparison to a 10% value drift rate, or a 2% probability of dying.[4]

Additional considerations

Rate of risk aversion ()

As discussed previously, if we want to infer the discount rate from the market rate of return, we need to know how risk-averse most investors are. I presented estimates of the discount rate given various assumptions about risk aversion. But exactly how risk-averse are most investors?

Unfortunately, as with all the other variables we tried to estimate in this essay, we cannot reliably pin down the rate of risk aversion . Different methodologies give different results.

In Discounting Disentangled, surveyed experts were asked to estimate the value of . They gave a mean of 1.35 and a median of 1.0 (a value of 1.0 corresponds to a logarithmic utility function), but their answers showed high variance.

Studies of the relationship between income and happiness tend to find a roughly logarithmic relationship; for example, Stevenson & Wolfers (2013), Subjective Well-Being and Income: Is There Any Evidence of Satiation? (DOI: 10.1257/aer.103.3.598).[5]

Gordon Irlam's Estimating the Coefficient of Relative Risk Aversion for Consumption summarizes a range of attempts to determine the value of . Estimates vary greatly—Irlam claims that you could justifiably use a value anywhere between 1 and 4.

Why is the risk-free rate so low?

As of July 2021, real Treasury yields are negative: -1.60% for 5-year bonds and -0.21% for 30-year bonds. How do we square this with a positive time preference? Why would anyone with a positive discount rate ever invest in bonds?

Weil (1989), The equity premium puzzle and the risk-free rate puzzle, describes this issue as the risk-free rate puzzle. More recent publications have attempted to resolve the puzzle, for example Maki & Sonoda (2010), A solution to the equity premium and riskfree rate puzzles: an empirical investigation using Japanese data, which explains low risk-free rates as a product of trading costs. The existence of this puzzle somewhat invalidates the approach I used to derive the discount rate based on historical investment performance.

Conclusion

To summarize what we've learned so far:

In theory, we can reverse-engineer the philanthropic discount rate by estimating the typical discount rate and then subtracting the pure time preference.

We can determine the typical discount rate from market data in various ways: by looking at broad market returns, behavior of endowment portfolios, public companies' shareholder yield, or GDP growth. Or we can simply survey experts and ask them what they think.

But when we attempt to reverse-engineer the philanthropic discount rate, we run into some problems:

  1. Different methods to estimate the typical discount rate disagree with each other and have wide margins of error.
  2. All of these methods require that we can accurately estimate how risk-averse people are, which we can't. Attempts to estimate people's risk aversion have produced inconsistent results.
  3. We don't know what rate of pure time preference most people use.
  4. Philanthropists and typical investors might disagree about the discount rate for reasons other than pure time preference, and it's hard to say how much they disagree.
  5. Even if we can accurately estimate all of these parameters, this whole exercise relies on a theoretical model that doesn't match reality—for example, it can't explain why the risk-free rate is as low as it is.

Could we fix all of these problems? Perhaps. But some of these are open research areas that have already received substantial attention from academics, and still remain mysterious. It's hard to say how much additional effort we would need to get a more realistic model with more accurate estimates.

In conclusion, it doesn't look like we can do a good job of reverse-engineering the philanthropic discount rate.

Under some circumstances, we don't care about the philanthropic discount rate; all we care about is the relationship between the philanthropic discount rate and the typical discount rate . If patient philanthropists want to fund a particular cause that's also funded by other actors, and if , then the cause may be over-funded according to the philanthropists' discount rate. In that case, the philanthropists should invest all their money to give later. Sometimes we might have good reason to believe , even if we don't know the exact difference. This greatly simplifies our decision, and we don't need to precisely determine the philanthropic discount rate. For more on this, see Philip Trammell's working paper Dynamic Public Good Provision under Time Preference Heterogeneity[6] (especially section 3, "Interactions between patient and impatient funders"), and his 80,000 Hours interview.

Appendix

Appendix A: Optimal consumption under uncertainty

The Ramsey model defines the relationship between consumption and investment. Traditionally, this model assumes a fixed investment rate. In real life, nearly everyone invests at least some of their money in risky assets like stocks, and even "risk-free" assets like short-term Treasury bills still carry some degree of volatility. If we want to estimate the discount rate using real financial data, our model needs to incorporate uncertainty.

Levhari and Srinivasan (1969)[7] provide a solution. First, some definitions[8]:

  • Let be consumption at time .
  • Let be the discount rate.
  • Let be the rate at which marginal utility diminishes with consumption.
  • Let be the investment return at . follows a normal distribution parameterized by mean and standard deviation (which is to say follows a lognormal distribution).
  • Let be the value of capital at . Capital grows according to .
  • Let be a utility function of consumption. Suppose we have an isoelastic utility function, that is,
  • Our goal is to maximize total expected discounted utility (We discount by instead of because this model follows discrete time, not continuous. In the limiting case as the length of a time step approaches zero, these two discount factors are equal.)

Under these conditions, the optimal consumption rate is a constant (call it ). is given by

or

(Levhari and Srinivasan do not provide this exact result, but it easily follows from what they do give.)

Using this equation, we can derive the value of as a function of , and :

If we do not know the value of , we can derive it from the investment rate and consumption growth rate for any particular period.

The consumption growth rate must equal the capital growth rate because consumption is a fixed proportion of capital. By definition, the capital growth rate equals . If we can empirically observe and , we can use these to derive :

In theory, this equation must hold for all times, because the actor will decide how much to grow their consumption () based on how much their portfolio earned during that period (). In practice, it doesn't hold, but we'll get to that later.

Now combine our equations for and . As a simplifying assumption, suppose and . Strictly speaking this is false, but it's true in the limit as time steps become arbitrarily small.

This gives us a new formula for :

We can simplify this a bit further. Suppose we identify the period with median investment return and median consumption growth (these must occur in the same period because is monotonic with ). That means (strictly speaking, , but again, this gives when time steps are infinitesimally small). Now we can simplify the formula to

These equations allow us to derive if we can observe the consumption rate or, failing that, we can derive the optimal consumption rate from the interest rate and consumption growth rate at a particular time .

Appendix B: Details on Harvard endowment annual report

The columns in the table below are defined as follows. All numbers are in millions, except for and , which represent percentages.

  • "Endowment": The total value of the endowment
  • "Gifts": Value of gifts made to the endowment
  • "Return": The investment return of the endowment portfolio
  • "Expenditures": Money withdrawn from the endowment to fund the university's operations
  • "Savings": Any portfolio return that's not used for expenditures (= Return - Expenditures)
  • "": Consumption rate (= Expenditures / Endowment)
  • "": Consumption rate excluding gifts (= (Expenditures - Gifts) / (Endowment - Gifts))
Endowment Gifts Return Expenditures Savings
2003 19294 262 2059 770 1289 4 2.7
2004 22587 257 3800 807 2993 3.6 2.5
2005 25853 285 4044 854 3190 3.3 2.2
2006 29219 273 4113 933 3180 3.2 2.3
2007 34912 277 6499 1043 5456 3 2.2
2008 36926 336 2880 1201 1679 3.3 2.4
2009 26138 194 -9591 1416 -11007 5.4 4.7
2010 27557 240 2630 1320 1310 4.8 4
2011 32012 212 5500 1321 4179 4.1 3.5
2012 30745 226 0 1422 -1422 4.6 3.9
2013 32689 222 3267 1499 1768 4.6 3.9
2014 36429 512 4688 1539 3149 4.2 2.9
2015 37615 338 1956 1594 362 4.2 3.4
2016 35665 491 -625 1706 -2331 4.8 3.5
2017 37096 550 2651 1787 864 4.8 3.4
2018 39233 646 3332 1821 1511 4.6 3
2019 40929 613 2326 1908 418 4.7 3.2

  1. Dimson, Marsh, and Staunton (2020). Summary Edition Credit Suisse Global Investment Returns Yearbook 2020. ↩︎

  2. I don't have data on the global shareholder yield, so I am naively assuming that it equals the yield of the S&P 500. ↩︎

  3. The author distinguishes between risk aversion and intertemporal substitutability of consumption, which I do not, so they present the results a little differently. ↩︎

  4. Technically, the probability of dying varies depending on age. 30-year olds only have about a 0.1% annual probability of death, increasing to 2% by age 70 and 5% by age 80. ↩︎

  5. The paper does not explicitly find a logarithmic relationship; it is simply attempting to argue that income continues to increase happiness on the margin, even at relatively high income levels. But the paper's data is public available. I fitted the data to a hyperbolic curve and found that it best fit at , which is a little less risk-averse than a logarithmic utility function. ↩︎

  6. Accessed 2021-07-09. ↩︎

  7. Levhari and Srinivasan (1960). Optimal Savings Under Uncertainty. ↩︎

  8. My definitions differ from Levhari and Srinivasan in a few ways:

    1. They define as one plus the interest rate, and I define it as the interest rate.
    2. They use a discount factor , and I use a discount rate (such that ).
    3. They refer to the rate of diminishing marginal utility as , and I refer to it as .
    ↩︎

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