Pricing the future

There was a flurry of triumphant snorts on Friday when some libertarian blogs picked up a post from earlier in the month, which had commented on the recent paper by William Nordhaus on carbon-pricing. The ASI got it from voluntaryXchange, who got it from Newmark's Door, who had spotted the original post at ReasonOnline. The excitement was because Nordhaus, described variously as "perhaps the world's leading expert on the economics of climate change" and "the economic expert on global warming", had done some modelling of the optimal profile of carbon-pricing, and had concluded that, as the ASI put it, "the suggestions of both the Stern Review and Al Gore don't cut it". Rather, Nordhaus estimates that the optimal level in 2010 for the price of carbon is $34 per tonne (tC), equivalent to around 9 cents per gallon of petrol (i.e. bugger all in the grand scheme of things).

Just a couple of points:

1. Apart from the original article (Reason) they conveniently forgot to report that Nordhaus was not advising a static price, but one which should increment in real terms by around 2.5 per cent per year, to $42/tC in 2015, $90/tC in 2050, and $207/tC in 2100. Those later prices would be felt more keenly, if it weren't for the likelihood that they will be dwarfed by higher fossil-fuel prices by then.

2. More importantly, although Nordhaus's paper dresses the analysis up in a great deal of elaborate academic clothing (and very much more substantial clothing than Stern and Gore, it has to be agreed), it all pretty much boils down to the usual culprit - the discount rate. If, like Stern, you believe that it is wrong, on moral grounds, to discount the costs of future disaster, then you choose a very low (almost zero) discount rate and end up with enormous present costs for future risks attributable to global-warming, and therefore a rationale for taking strong, immediate action based on a high present cost of carbon. If, like Nordhaus, you believe that close-to-zero discounting is irrational, you set a modest discount rate which, over the timescales over which the impacts of global-warming might be felt, reduces the present costs of even catastrophic events to quite low values and results in a steady-as-she-goes policy prescription.

The funny thing is that both Stern and Nordhaus can present simple illustrations that demonstrate the irrationality of the opposing perspective. Stern can say, as paraphrased by Nordhaus, that "a positive time discount rate would lead societies to ignore large costs that occur in the distant future." In other words, if one combines economists' willingness to put a value on life with a modest discount rate, one can end up with a low present value for the deaths of even millions of people sufficiently far in the future (and thanks to the wonders of compounding, not that far into the future). It follows that it isn't worth doing much now to avoid large numbers of deaths, even if our actions make those deaths inevitable, provided that those deaths are not too soon.

On the other hand, Nordhaus gives his "wrinkle experiment" illustration of the absurdity of Stern's use of minimal discounting. "Suppose that scientists discover a wrinkle in the climate system that will cause damages equal to 0.1 percent of net consumption starting in 2200 and continuing at that rate forever after", he muses. "How large a one-time investment would be justified today to remove the wrinkle that starts only after two centuries? Using the methodology of the Review, the answer is that we should pay up to 56 percent of one year’s world consumption today to remove the wrinkle. In other words, it is worth a one-time consumption hit of approximately $30,000 billion today to fix a tiny problem that begins in 2200."

So we have two competing approaches, both of which yield absurd results. Should we just choose the version of absurdity that we prefer? Or argue that, in medio veritas, the truth must lie somewhere (but who knows where) between two absurd positions? Or should we stop and consider whether the problem lies with what these two approaches have in common?

What this is telling us is that we have reached the limits of the usefulness of mathematical economics. Though it came to dominate economics over the course of the twentieth century, it was always a dead end. Now we see exactly how sterile and ridiculous is the idea that you can model human action with numbers and formulae. Climate-change theory turns out to be the perfect reductio ad absurdam test of neo-classical, welfare economics. And it fails.

Incidentally, the Nordhaus paper is well worth a read. It may be an economics paper, but many parts of it are eminently readable and entertaining. In particular, Nordhaus's dislike of the British political establishment is palpable, and his shots well-aimed. In his contempt for our "elite", he adopts H.L.Mencken's recommended attitude of journalists to politicians, that of dogs to lamp posts:

"The British government is not infallible in questions of economic and scientific analysis on global warming, any more than it was in its white paper on weapons of mass destruction in Iraq. External review and reproducibility cannot remove all error, but they are essential for ensuring logical reasoning and a respect for opposing arguments."

"The Review takes the lofty vantage point of the world social planner, perhaps stoking the dying embers of the British Empire, in determining the way the world should combat the dangers of global warming. The world, according to Government House utilitarianism, should use the combination of time discounting and consumption elasticity that the Review’s authors find persuasive from their ethical vantage point."

"I have always found the Government House approach misleading in the context of global warming and particularly as it informs the negotiations of policies among sovereign states."

"The normatively acceptable real interest rates prescribed by philosophers, economists, or the British government are irrelevant to determining the appropriate discount rate to use in the actual financial and capital markets of the United States, China, Brazil, and the rest of the world. When countries weigh their self-interest in international bargains about emissions reductions and burden sharing, they will look at the actual gains from bargains, and the returns on these relative to other investments, rather than the gains that would come from a theoretical growth model."

If only we had more academics in the UK who would take as robust an attitude to the lazy, patronizing assumptions of our political establishment, rather than trying to embed themselves as part of that establishment. But Oxford and Cambridge were ever thus (ask Adam Smith).

He also provides welcome doses of realism about the mechanisms that would most effectively internalize whatever social-cost of carbon is agreed, and on the appropriateness (or otherwise) of a cap. His numbers may not be any more reliable than Stern, in an area where future uncertainty is so vast, but his rational analysis (where he leaves the numbers and formulae behind) and policy prescriptions are a vast improvement on most of our academic and political establishment.



...come of your comments seem to require a bit of reconsideration.

for example, the idea that mathematics is somehow now discredited because modelling cannot, at this moment, provide exact answers seems to be an unreasonable conclusion.

remember that input factors do become more accurate as research teaches us more about the behavior of the subject we're modelling.

an example is found in oceanography. the accuracy of models that predict the behavior of surface objects are constantly improving. does that mean mathematics as a tool was discredited until the models became more accurate, or did the inputs improve, making the underlying math process look better, or does a combination of both occur?

the answer is both. that tells us that the process of using mathematics to examine potential results remains sound, even as the math improves.

additionally, you do not seem to have considered the possibility that 2200 might be a date that itself is subject to change.

here's an example from a scientific consortium that includes the us national science foundation, the universities of calgary and colorado, the us geological survey, and pennsylvania state university that tells us a rise in sea level of 13 to 20 feet has occurred before-130,000 years ago-from a similar amount of glacial melt as is currently modelled to occur by 2100.

the modelling data was corellated to ice core data. what does that accomplish? ice cores contain air samples trapped at the time the ice froze, and that air can be analyzed for carbon dioxide content.

co2 is a leading indicator of global mean temperature rise, and the ice core record proves today's co2 levels are as high as they were 130,000 years ago.

the combination of the two is a very strong indicator that warming is accelrating at a rate we had not expected, and it is this potential for accelerated warming that also seems to be lacking in your consideration.

I agree with the feeling that neo-classical welfare economics is a sham because of its attempts at basing policy prescription on quantifications of things that cannot or should not be quantified for that purpose.

But I don't think that either of the two examples you quote are absurd enough to make that point. I did not have the time to trace whether either Stern or Nordhaus original examples are vulnerable to my critique, so I am only responding to your formulation.

Using an ordinary discount rate (like the Government's current 3.5 per cent) does not mean that "it isn't worth doing much now to avoid large numbers of deaths". Any sane person applying cost-benefit analysis (okay, contradiction in terms, but nevermind) to a long-term problem would need to consider the specific inflation rates applied to the values of different things. When it comes to avoiding deaths, the respectable hypothesis is that willingness to pay grows in line with income or consumption (or even faster if you believe some academic research). Make that 2.5 per cent a year above the same inflation rate as the one used in defining the 3.5 per cent Government discount rate, and it is now worth doing quite a lot to avoid future deaths: the effective discount rate on life is "only" 1 per cent. Add to the mix the theories in the Green Book about lower discount rates for longer time period and you have reached a world of nearly-no-discounting-of-life from a standard discount rate method. Where is the absurdity?

As for the example attributed to Nordhaus, there seems to be a similar interaction between economic growth and discounting. If you had an investment opportunity that would generate profits of 0.1 per cent of global consumption from 2200 and forever more, guaranteed, what would the shares in that investment be worth today according to a standard financial method? Let's look at it with a dividend growth model. This is a guaranteed opportunity with guaranteed income growth so we will be getting most of our cost of capital covered by the dividend growth element, and we can affort a very high price/earnings multiple. I haven't done the numbers but investing the equivalent of half a year's consumption does not seem manifestly absurd in exchange for a small but guaranteed, growth- and inflation-proofed, permanent income stream.

Now plainly it would be daft to cut consumption by 56 per cent in 2008 on the foot of some global warming theory. But don't accuse the discounting maths of that bit of silliness: the real reason why it would be daft is that there does not exist an opportunity to spend $30 gazillions tomorrow to achieve the 0.1 per cent guaranteed extra consumption from 2200 forever more.

(And your captcha gives me squares too.)

Thanks to both of you for pointing out the captcha problem. I've just spent a good part of a day trying and failing to figure out what is causing it. Fortunately, there are alternative modules, so I've now replaced the module we were using with another one that seems to work OK for me. If you still get problems, and it's bad enough that you can't submit a comment, you can contact me by my contact form at I'm sorry we have to use captchas for anonymous users, but we were getting terrible problems with comment spam. You are welcome to register as a user of the site. Authenticated users are not asked to fill out a captcha form with each comment.

Fake consultant,

Sorry for the delay.

I was criticizing mathematical economics specifically, not maths generally, nor its application to climate-change science. Climate-modelling ≠ economics. Oceonography ≠ economics. Atmospheric physics ≠ economics. In these and other areas of science, I support further study to improve our understanding of the relevant systems.

The anonymous commenter after you has given the broader reasons to be critical of the notion of welfare economics (of which environmental economics is a branch). I will respond to that erudite submission separately, but the general point they make about "quantifications of things that cannot or should not be quantified" is the real basis of the argument. I was simply attempting to use a clash of two welfare economists to illustrate the difficulties.

It is not clear to me how you think further studies will help to clarify the appropriate discount rate to use for valuing present costs of future impacts. Even as we refine the science and pin down the risk of those impacts with greater accuracy, a difference of one or two per cent in the discount rate chosen for the purposes of valuing those impacts will have a dramatic impact on the policy prescriptions that follow. As Stern and Nordhaus have demonstrated, it's the difference between valuing current carbon-emissions at $30 or $300 per tonne.

Unfortunately, economics is not amenable to the same approach as the natural sciences. There is a big jump from understanding what might happen to our climate if people behave in a certain way, to the self-referential question of predicting people's behaviour and how incentives might affect that behaviour. The problem with the attitude of a lot of scientists, politicians, journalists, and other intellectuals (in the Hayekian sense) is that they act as though the policy prescriptions flow naturally from their scientific study of climate systems. The questions regarding human behaviour that follow from the science are actually more complex than the climate science itself.

On a couple of points of detail:

1. Is CO2 a "leading indicator of global mean temperature rise"? My understanding is that, historically, it has been a trailing indicator.

2. The date of 2200 is an entirely arbitrary one selected by Nordhaus. His point is only reinforced by consideration of the difficulty of pinning down the date and scale of future impacts with any accuracy at this time. As many of those impacts are dependent substantially on human behaviour, this is inherently unpredictable, even as our scientific understanding of the consequences of human behaviour improves.


Thank you for the expert criticism. The argument against welfare economics in your first paragraph is key. I would not seek to pretend that this illustration, if it stands up, is any more substantial than any other example. The a priori, deductive arguments against welfare economics, or, from my perspective, scientism in economics more generally, are what count.

The importance of examples is only in their ability to provoke consideration of received wisdom. Although economic studies are probably little better than life in general as a laboratory for experiment into the "science" of human action, there may be an aspect of falsifiability - one cannot prove a contention in this way, but one may help to disprove it. If both A and "Not A" can be argued justifiably from the same proposition, the proposition itself must, as Hegel pointed out, be suspect.

Perhaps Stern and Nordhaus's positions are not precisely diametrically opposed, but they are strongly antagonistic, as Nordhaus himself makes clear. Nordhaus feels he has countered Stern by demonstrating the implausible consequences of Stern's chosen discount-rate. But providing a criticism of Stern's position is not the same thing as proving his own position or disproving Stern's criticism of discount rates such as that preferred by Nordhaus. My contention is that both arguments are "right" in their own terms, and therefore demonstrate the weakness of the proposition on which they are based.

The interesting question is whether, as you doubt, their positions are as contradictory as I, following Nordhaus, suggested. It seems to come down to one's understanding of the purpose of discounting and what it encompasses. It is a fair point that the value of life that is relevant is the value at the time of the disaster, not the value now. And it is also reasonable to assume that people will value lives at least in proportion to their ability to pay, so if they are twice as wealthy on average, lives will be worth at least twice as much on average. But that begs the questions: (a) what do we mean by "growth" and "wealth", and (b) do discount rates conventionally take this into account?

I assume neither of us is taking account of inflation as a monetary phenomenon, which does not represent a growth in wealth, in fact marginally (or significantly in hyper-inflation circumstances) the opposite. I will assume we are working in real, not nominal values, to the extent that it is possible to identify real values.

We are talking instead about the likelihood that, through the division of labour and free trade, more people will be able to afford more goods that they desire than they can now. This, indeed, is a significant part of Nordhaus's thinking, accounting for his argument (to simplify) that we should delay any unnecessary action as much as possible, as we will be more able to afford it in the future - a major factor in his low but rising cost of carbon.

It seems that that argument can cut both ways. Yes, things, including lives, will be more valuable in the future, and we ought to calculate our NPVs on the basis of their value at the time. But equally, we will be more able to afford them in the future, so we ought not to sacrifice more than necessary now when our descendants will be more able to make a larger sacrifice. Is this self-cancelling (at least to some extent), from the perspective of establishing a valid discount rate?

I may be missing a connection, but this seems to me to be a different issue to people's time preference, which is the factor that I understood discounting was traditionally taking into account. I mean by that, very roughly speaking, the economic equivalent of the notion that a bird in the hand is worth two in the bush. The question is whether a discount rate that took account of time preference, and the swings and roundabouts of the likelihood of future growth in terms of present value of future goods, would be a mere 1%, or something more like the Government's 3.5%.

This still seems to me to come down to a judgment call, and one, moreover, that is largely determined by one's view of the desirability of the outcome. If the preferred discount rate is 1% because to use a higher rate would result in unacceptable disregard for the future, then there really isn't much point calculating the NPV of future events based on that discount rate - you have already made the judgment which that calculation is supposed to inform, in order to be able to make the calculation. The same goes for the selection of a higher discount rate that takes an equally (but differently) morally-justifiable stance (that one ought to put a more certain threat to quality-of-life today ahead of a less certain threat to quality-of-life tomorrow). The discount rate is chosen because one has already made a judgement about the relative values of today and tomorrow. It cannot then be used to justify that judgement.


In a way, this is about looking at two sides of two coins. We have two costs in the distant future - one extreme event, and one small but sustained impact. The Stern approach of a minimal discount rate yields a rational result when looking at the extreme event but an irrational result when looking at the small but sustained impact. The Nordhaus approach of a commercial discount rate yields a rational result when looking at the small but sustained impact, but an irrational result when looking at the extreme event. By refocusing on Stern's approach to the extreme event ("a world of nearly-no-discounting-of-life"), and Nordhaus's approach to the small but sustained impact ("a very high price/earnings multiple"), you can argue with justification that their positions are not absurd.

But the other side of the coin does not cease to exist because we are not looking at it. The range of impacts we face do not fall so conveniently under the headings of "extreme" or "small but sustained". If we are going to use the approach of calculating NPVs to inform policy decisions, we are going to have to choose a figure that will apply to all impacts, small or extreme, short or sustained, and everything in between. If we choose a low discount-rate, we will have to consider the other side of the small-but-sustained coin, where the action that is mandated today is excessive relative to the avoided harm. If we choose a higher discount-rate, we will have to consider the other side of the extreme coin, where deaths sufficiently distant in the future will be excessively devalued in today's terms. From what I can see, your justification of the two approaches is a fair defence of the light sides of the coins, but they do not refute the unavoidable reality of the dark sides of both coins, as highlighted by each protagonist when criticizing the other perspective.

Discounting is a useful financial tool for indicating whether an investment is likely to be worthwhile over a medium-term (say ten- to twenty-year) period. With sufficient expertise and insight, future values can be estimated over such a period with a reasonable hope of being approximately accurate, and the impact of discounting will not make the accuracy of such judgments irrelevant over such a limited period (though it is getting that way on a 20-year DCF analysis). But, in my opinion, discounting and NPVs are wholly inappropriate tools to use for estimating long-term values over many decades or centuries, where future values are increasingly uncertain as the period extends, and yet the effect of variations in those values is diminished by the compound effect of discounting over such a long period. If you have to use this approach to estimate NPVs, then of course you try to tailor the discount rate to achieve what appears to be a rational outcome according to your moral preferences and assumptions about the future. But you cannot then use calculations based on the discount-rate derived in that manner in order to justify arguments about the right moral course of action based on the scale of future costs - that is simply circular reasoning. In which case, what exactly did the mathematical part of this process gain you?

For such uncertain and long-term impacts, a different approach is needed. Unfortunately, mathematical economics needs some method to calculate future values. Any alternative mathematical approach will suffer from similar problems. Mathematical economics simply does not provide a useful tool to derive policy prescriptions for an issue like AGW. I believe this is symptomatic of a broader malaise with the approach, and a broader ineffectiveness for policy guidance, but I raise it only as an unusually accessible example of its failings, not as a falsification of the principle of mathematical economics. That comes from the sort of deductive analysis of the nature of value that you describe. But my hope is that, as an illustration of the weakness of the approach, it may cause one or two people to consider whether something that is so abstracted from reality, and that can provide such different answers from small, subjective changes in conditions, really provides a sound basis for policy prescriptions.

You sound like an economist. If so (or even if not), if you get the chance, read Nordhaus's section on Stern and discounting (section IX, pp.137-167), which explains the issues much better than I can. For instance, I have simplified by referring only to the discount rate, when Nordhaus is at pains to emphasise that this isn't just a question of the discount rate, but also of the utility function - the other aspect of the social welfare function. The discussion of the complementarity of the discount rate and the utility function addresses some of your arguments with regard to the justifiability of a 1% (net) discount rate.