Close to zero?

by John Q on December 5, 2006

In yet another round of the controversy over discounting in the Stern Report, Megan McArdle refers to Stern’s use of “a zero or very-near-zero discount rate”. Similarly Bjorn Lomborg refers to the discount rate as “extremely low” and Arnold Kling complains says that it’s a below-market rate.

So what is the discount rate we are talking about? Stern doesn’t pick a fixed rate but rather picks parameters that determine the discount rate in a given projection. The relevant parameters are the pure rate of time preference (delta) which Stern sets equal to 0.1 and the intertemporal elasticity of substitution (eta) which Stern sets equal to 1. The important parameter is eta, which reflects the fact that since people in the future will mostly be richer than us, additional consumption in the future is worth less than additional consumption now.

Given eta = 1, the discount rate is equal to the rate of growth of consumption per person, plus delta which is 0.1. A reasonable estimate for the growth rate is 2 per cent, so Stern would have a real discount rate of 2.1 per cent. Allowing for 2.5 per cent inflation that’s equal to a nominal rate of 4.6 per cent. The US 10-year bond rate, probably the most directly comparable market rate, is currently 4.44 per cent; a bit above its long-run average in real terms. So, Stern’s approach produces a discount rate a little above the real bond rate.

Arguments about discounting are unlikely to be settled any time soon. There’s a strong case for using bond rates as the basis for discounting the future. There are also strong arguments against, largely depending on how you adjust for risk. But to refer to the US bond rate as “near-zero” or “extremely low” seems implausible, and to say it’s below-market is a contradiction in terms. It seems as if these writers have confused the discount rate with the rate of pure time preferences.



dsquared 12.05.06 at 6:26 pm

I would argue for a very much lower eta (or possibly higher, it’s late), on the basis that although future generations of us will probably be richer than us, the costs of global warming will fall on future generations of Bangladeshis and Pacificc Islanders, who will for the most part be poorer than us now.


Shelby 12.05.06 at 7:57 pm

Do you mean “Allowing for 2.5 per cent inflation“?


David Wright 12.05.06 at 8:07 pm

The best review of these issues I have seen is Nordhaus’s commentary on the Stern report. Nordhaus has run similiar utilitiarian optimizations and understands the issues very well.

There are a number of ways to set the values of the delta and eta parameters that determine the relative weights of utility among different generations. One way is to fit both delta and eta to emperically from observed investment patterns. “Utilitiarian fundamentalists,” on the other hand, argue that we must choose delta = 0 (or very close), on ethical grounds; Stern falls into this school. As far as I know, no one argues that there is an a priori correct choice for eta, which makes it rather strange that the Stern report just picked eta = 1 out of a hat and didn’t bother to investigate the consequences of other choices.

These choices make a difference. With delta = 0 and eta = 1, Nordhaus was able to replicate Stern’s result that the immediate imposition of a large CO2 tax is optimal. He also showed that if both delta and eta are fit emperically, you get delta = 3%/year (must larger than Stern’s value), and an optimal tax that is currently very small and only gets big in 50 years or so. He also does the interesting exercise of accepting Stern’s a priori delta = 0 value, but then fitting eta to observed investment patterns, which gives eta = 2.2, and again results in an optimal CO2 tax that is currently very small and only gets big in 50 years or so.


Jamie 12.05.06 at 9:53 pm

For those of us who are not economists, can David Wright or DD or JQ clarify ‘delta’?

As I understand it, positive delta means that future generations don’t matter as much as, well, we do. Also, previous generations mattered more, though we can’t help them now so it doesn’t matter that they mattered more. Delta isn’t the economist’s ordinary discount rate, which makes perfectly good sense because of the real rate of interest. It’s the “pure rate of time preference”.

As I said, I’m not an economist, and I have a bad feeling that I’ve misunderstood. What would it even mean to determine delta empirically? Wouldn’t that be trying to test empirically whether average utilitarianism or total utilitarianism is correct?


Tom T. 12.05.06 at 10:57 pm

JQ, I just have a couple of questions, and pardon me if they’re ignorant. I’m genuinely trying to understand.

1. I thought that an eta = 1 does NOT reflect “the fact that … additional consumption in the future is worth less than additional consumption now” but rather corresponds to an assumption that consumption by individuals living in the future is fully interchangeable with consumption by individuals living now. To value consumption in the future less than consumption today would require a value of eta greater than 1. Isn’t that part of Dasgupta’s point?

2. What about Dasgupta’s statement that given Stern’s parameters, “It is an easy calculation to show that the current generation in that model economy ought to save a full 97.5% of its GDP for the future!” How does that reconcile with your conclusion that Stern’s parameters are market-reasonable?


Ragout 12.06.06 at 12:58 am

In general, this debate is really making me sympathetic to the people on an earlier Crooked Timber thread who were saying that economists are trying to slip in ideology under the guise of science.

If I understand Nordhaus, one way to think about what the economists are doing is to say: we observe the the real interest rate, and we want to attribute some of it to other observables like the growth rate, and some of it to different components of the utility function (delta and eta). Nordhaus wants to do the partitioning by assuming a particular utility function, and Stern wants to do it by introspection and moral argument.

Well that’s all fine, but why should I think that these economists’ intuitions about eta, delta, and the mathematical form of utility functions are any better than my intuition? They have an intuition about delta and eta, and I have an intuition about whether $100 billion a year seems reasonable compared to the size of the problem, national income, and the amount the government spends on other things.

And even if the Stern or Nordhaus are exactly right, where does this get us? Any given policy or project is going to have to be examined on a case-by-case basis. For example, a gas tax effects not only global warming but also congestion, pollution, and how often we need to go to war in the Middle East. Any particular project to fight global warming will have costs, benefits, and risks of its own. Personally, I favor filling up an ocean or two with ping pong balls, in order to raise the reflectivity of the earth. But this is just as much because I want to sail the ping pong seas, as because I want to fight global warming.


a 12.06.06 at 1:14 am

Have not been following this, but usually the bond rate is only used for discounting when talking about nominal amounts. If you want to discount “real” amounts, one should discount using the bond rate minus the inflation rate.


Alex Gregory 12.06.06 at 4:15 am

I too, am somewhat ignorant about this.

Along with Shelby, could you explain what the additional 2.5% is being added for? Is it inflation?

Further, I don’t quite understand why utilitarian “fundamentalists” commit themselves to the value of delta being 0. Can’t a higher delta than this be justified simply because of additional risk? The Stern report does not allow for all risks to future generations, and there is a possibility, however small, that there will be less people about the future than we’d predict. Given that, shouldn’t we set delta to higher than 0 on the grounds that we at least know for certain that the current generation exists? Or have I hugely misunderstood what delta represents?


John Quiggin 12.06.06 at 6:26 am

D’oh! It is of course 2.5 per cent inflation. fixed now, I hope.

I’ll repeat my promise of a bigger piece, coming Real Soon Now, in which All Will Be Explained.


aaron 12.06.06 at 8:53 am

What would be wrong with using the expected real gdp growth rate?


A-ro 12.06.06 at 9:01 am

That is some nice blogging.


aaron 12.06.06 at 9:08 am

I liked Lomborg’s justification for using delta 0 in the Copenhagen Concensus. He basically said that Global Warming is so unimportant that it’s not worth arguing over.


Walt 12.06.06 at 9:55 am

ragout: Your ping pong seas made me laugh very hard.


dv 12.06.06 at 10:40 am

Could somebody explain what it means to fit delta empirically? In particular, it seems we should distinguish between intrapersonal time preference, and time preferences that have an interpersonal dimension: while it seems perfectly fine for me to decide that I care less about my own future consumption than about my own present consumption, it seems somewhat less fine for me to decide that I care more about my present consumption than about someone else’s (future generations’) consumption because they live in the future. So what are empirical measures of time preferences across generations, and how do such empirical findings bear on the normative question what we owe to future generations? Or am I just misunderstanding what is going on when people talk about ’empirically’ determining the rate of delta?


Paul Litvak 12.06.06 at 10:56 am

Short preamble on who I am:
2nd year Graduate Student in Psychology and Behavioral Economics at Carnegie Mellon University

As a student of someone who is one of the foremost experts on time preference, I have to say that this debate about delta seems silly to me. Not to sound like a psychologist clucking at economists, but there is extensive body of research showing that our concept of time preference (at least in the discounted utiity sense that people are using here) utterly and completely incoheres with people’s actual choices. There is a really huge literature on this, and I am somewhat loathe to go into every example, but here are a few good references on this:
Ainslie, G. (1975). Specious reward: A behavioral theory of impulsiveness and impulse control. Psychological Bulletin, 82, 463-509.
Frederick, S., Loewenstein, G. and O’Donoghue, T. (2002). Time Discounting and Time Preference A Critical Review. Journal of Economic Literature. 40(2), 351-401.
Samuel M. McClure, David I. Laibson, George Loewenstein, and Jonathan D. Cohen. (2004). Separate Neural Systems Value Immediate and Delayed Monetary Rewards
Science 306: 503-507
Read, D. (2001). Is time-discounting hyperbolic or subadditive? Journal of Risk and Uncertainty, 23, 5-32

The most disturbing of these papers is Read’s paper, which seems to demonstrate that subdividing intervals of time does wacky things to even hyperbolic discounting. (Btw, if we were hyperbolic discounters, then I think we would discount the difference between what happens in 2100 versus 2101 very little, but comparing 2006 to 7 we care very much. so i think the implication for stern’s model is that the short term ills of global warming (10-20 years?) are to be weighted much more strongly than other 10 year differences way out there. but ymmv depending on choice of parameters, and being more a psychologist than an economist, i am by no means an expert in every detail of these models.)

So I guess my question for Stern is, why are people sticking with a DU model? Is there a reason why a beta-delta model is dispreferred? is it just economists sticking to their disciplinary guns?


Love those rank dependent models, prof. quiggin :-)


jet 12.06.06 at 12:25 pm

Aaron, you misunderstand the context of Lomborg’s quote:

I liked Lomborg’s justification for using delta 0 in the Copenhagen Concensus. He basically said that Global Warming is so unimportant that it’s not worth arguing over.

The context was to rate the best ways to spend money in order to save the most human lives. Compared to the other issues on the list, Global Warming can fairly be argued to be unimportant.


Daniel 12.06.06 at 12:33 pm

Because this is a normative rather than descriptive exercise; the question that Stern is trying to answer is the relative weight that we should give to future generations. While the joys and pains of future generations seem very nebulous and far off to us, they will seem very present indeed to them.

An analogy might be that we all (more or less, when talking in ethical terms) agree that the geographical discount rate ought to be zero – that it is arbitrary and wrong for me to discount the utility of other people based on how far away they live from London (we could even make assumptions about a racial discount rate, whereby I assigned a lower weighting to other people’s welfare based on their genetic distance from red-haired Welshmen). This is based on a fundamental ethical principle of equality. Given that, it seems hard to justify “pure time preference” because there is also a fundamental equality at work there; future generations will be people just like me. The only justification for a positive delta is if we are allowing for the possibility of extinction or some such.

So, hyperbolic discounting doesn’t really enter into this discussion – I don’t think anyone is prepared to defend it as a sensible basis for planning, or as a social welfare function, are they?


Adamsmithee 12.06.06 at 1:22 pm

I’m with Tom T., I can’t get the Quiggin line that the Stern discount rate is effectively the growth rate plus 0.1 to reconcile with the Dasgupta critique –would love to know how that works…


Aaron_M 12.06.06 at 1:30 pm

Here are three reasons to discount the welfare of future generations:

1) Risk of extinction
2) Accounting for the declining marginal utility of benefits we provide to future generations because they are richer
3) Reasonable expectations upon any generation.

The third reason recognises the fact that once we get into long-term intergenerational cost/benefit analysis of the climate type saying, as welfare economists generally do, that the efficient thing to do is to maximise total welfare becomes hugely problematic. If costs now entail gains for several future generations and the avoidance of costs now entails costs for several future generations then the fact that there are many more people in the future (i.e. future generations 1+2+3+4+5…) makes the welfare interests of the future much more important than the welfare interests of any single generation (i.e. the current generation).

There is enormous debate over how to model the first and second reasons for discounting, but it seems to me that what is really going on is that welfare economists intuitively reject as unreasonable the demands imposed on any current generation for maximizing intergenerational welfare. They then engage in various kinds of tinkering with how the model deals with 1 & 2 in order to produce a CBA that gives them an answer on “efficiency” that seems reasonable (as opposed to welfare maximising) based on what they estimate current costs and future benefits to be and on how rich they estimate future generations to be. All this to avoid questioning their underlying utilitarian normative assumptions, assumptions that more often than not only serve to simplify analysis or to make the presentation of results seem more scientific (i.e. giving a clear single answer on what is efficient).

As it stands the debate between economists over what is intergenerationally “welfare efficient” when it comes to climate change only gives us a lot of second rate theorising about intergenerational justice discussed under the guised of empirical assessment, which in turn is distracting and confusing.


Paul Litvak 12.06.06 at 2:34 pm

In response to Daniel,

I figured someone was going to make that exact counterargument (normative v descriptive). heres the question though.. where is the intuition for whats normatively correct coming from? doesn’t the debate about whether it’s proper to discount this much or that much come from some intuition? and if all you are doing is appealing to moral intuitions when advocating for this discount rate or another, then it seems that what people do *descriptively* when they are assessing time preference matters. unless you have an a priori standard for discounted utilty which does not appeal to consequences that our intuition tells us is tasteful or distasteful.



MQ 12.06.06 at 4:26 pm

Comment #19 from Aaron strikes me as right on. Economists shouldn’t be trying to roll all this stuff up together to get a single cost/benefit metric which tells us how much we should spend today. That is where you need to discount, and introduce lots of strong assumptions in places where non-technical types can’t question them. Instead, economists should be *unpacking* all the various possible consequences of global warming, presenting them as clearly as possible, and letting a democratic process sum up people’s intuitions about whether we should put in a significant effort today to prevent it.

I actually think that something like this process is happening — the polity is gradually deciding that we have some collective responsibility to spend something today, this conclusion is based on a wide range of evidence gathered from different sources, and people currently feel they want the spending to be fairly limited as yet. The big econ models come in around the edges of the process, as one other form of rhetoric.


dsquared 12.06.06 at 5:02 pm

Paul; my second paragraph wasn’t purely meant to be a joke. Descriptively, people care about people roughly according to the inverse squared distance, and in a way that varies inversely with genetic distance. It seems to me that your argument is too strong; if we’re going to uncritically use revealed preference as a guide to moral intuition, then we’re going to have a racist and nationalist discounting rule AFAICS.


leederick 12.06.06 at 5:04 pm

Isn’t there also a problem that we have no say in whether intervening generations screw up our plans? If we go and prevent global warming in the interests of Generation 3+ down the line, what’s to stop Generation 2 from going on a coal burning spree and making our sacrifice for naught?

It seems to me that it’s just not possible to make these intergenerational transfers, the way you can between people currently existing, because you can only hold out the posibility of a contingent benefit for future generations in return for a real investment. You can’t be certain that the transfer you want will take place.


David Wright 12.06.06 at 6:30 pm

1. Dsquared makes a great argument against choosing ethical parameters that reproduce emperically observed decision-making: it ignores the possibility that current decision-making might be unethical.

Lombard, though, has a good counter-point. He says: okay, use whatever delta and eta you want, but apply them consistently to all questions of sacrifice for the benefit of future generations, not just for CO2 emissions. Then when we are done calculating how much our generation should sacrifice, and we see how much our generation actually will sacrifice, direct those dollars to the projects that buy the most future welfare. Lombard claims that curing third world diseases, per dollar invested, buys more future welfare than reducing CO2 emissions. I don’t know if that’s true, but it is certainly possible.

2. The formula rho = eta (c’/c) + delta gives a discount rate for welfare, not a discount rate for money. If utility is not proportional to money, those aren’t the same. So testing whether a particular choice of eta and delta are emperically reasonable is not as simple as computing rho and comparing it to the interest rate. Fitting eta and delta to observed investment patterns involves seeing how much of their income people set aside in long-term investments and how that savings rate varies with income, not just looking at the rate of return on those investments.


dearieme 12.06.06 at 6:49 pm

Her Majesty’s Government believes that all substantial bequests should be reduced by 40%: that surely implies that one should use a rather large discount rate.


John Quiggin 12.06.06 at 9:32 pm

“The formula rho = eta (c’/c) + delta gives a discount rate for welfare, not a discount rate for money.”
This is incorrect. Delta is the discount rate for utility (welfare). Rho is the discount rate for money.


David Wright 12.07.06 at 12:47 am

JQ: I just worked out the time-variation of the marginal welfare of an extra dollar and I see that you are right. Sorry for the incorrect information.


John Quiggin 12.07.06 at 1:49 am

To explain dw’s last comment for the benefit of earlier commenters such #5, #18, #19, with eta = 1 an 1 per cent increase in income implies a 1 per cent reduction in the marginal utility of income. So if income is growing at x per cent per year (in may example 2 per cent) marginal future income is discount by 2 per cent for each year into the future it accrues.

A useful way to look at this, but also a source of some confusion, is that, with eta =1, a 1 per cent increase in income now has the same value as a 1 per cent increase in income at any point in the future. Since income is growing this means a given (constant dollar) increase in real income now is worth more than the same increase in the future.


aaron 12.07.06 at 8:12 am

Increases in income will decrease the marginal utility of additional income, however they will also decrease the negative utility of costs, making us less risk averse in the future. So while addional income is worth less to us, cost are also worth much less. It is wrong to consider preferences in longterm decision making. Our ability to handle the burder of global warming is what matters, not how much we enjoy it.


Ragout 12.07.06 at 8:45 am

Two comments.

First, if we’re going to compare our discount rules to the “real interest rate,” uncritically or not, issues of loss-aversion and hyperbolic discounting inevitably arise, because of the equity premium puzzle. Should we think of the “real interest rate” as the 2% return to bonds or the 6% return to stocks? I note that Nordhaus seems to favor a return much closer to the stock market return, not the return to T-bills.

Second, if the Stern Report is supposed in influence policy in, for example, the US, I don’t see what sense it makes to say that US interests only get 5% weight, since we’re only 5% of world population. I don’t particularly see this as a moral basis for US policy: in some contexts we call those who weight the interests of foreigners equal to national interests “traitors,” which (rightly) implies moral condemnation.

In practical terms, I don’t think such an analysis is going to be persuasive, unless the assumptions are hidden behind a lot of Greek letters. When I worry about global warming, I’m especially concerned about possible effects on the Gulf Stream, even though it affects only a relatively few Westerners. Maybe Stern would tell me that’s an immoral way of deciding policy, but who wants economists to be the world’s moral philosophers?


John Quiggin 12.07.06 at 2:24 pm

Of course the equity premium puzzle is important, and I will be coming back to this. But it’s just as much of a problem for Nordhaus as for Stern. It means that there is no easy way of picking a ‘market rate’ that matches all observations.

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