Yes, finally the Ofcom has spoken. Not very loudly, it seems. It’s really just a rap on the knuckles for “The Great Global Warming Swindle”, largely because:

“…whilst Ofcom is required by the 2003 Act to set standards to ensure that news programmes are reported with ‘due accuracy’ there is no such requirement for other types of programming, including factual programmes of this type.”

Unbelievable. What planet are they (or rather the legislators responsible for this insanity) on? One that is going to get a hell of a lot warmer, it seems, if we can’t work out how to make rational, science-based decisions. How can the category “factual programmes” even exist without “standards [of] due accuracy”? Has anyone thought about what the word “factual” actually means??

Remind me if I don’t return to this argument later on, but to state the thesis briefly, in complex domains, problems – whether big ones (like GW itself), or small ones, like “Swindle” – almost always have many causes. Dealing just with the immediate cause may be futile. In the case of “Swindle” it may be most effective putting effort into changing the rules of the media game, rather than engaging in trench warfare. Because, if the ultimate arbiter of truth is not factual accuracy then we just end up with a popularity contest. Hey, why not incorporate audience votes in science programmes? Phone-in to vote for your favourite theory of gravity!

Luckily, in the case of “The Great Global Warming Swindle”, the programme:

“…broke rules on impartiality and misrepresented the views of the government’s former chief scientist…” even though it “was ‘on balance’ cleared of ‘materially misleading the audience so as to cause harm or offence’”. (Quotes from the Guardian’s news story on the findings).

But what if they hadn’t broken any rules?

And at least in this case George Monbiot got his retaliation in first, with a comment (and CiF) piece in today’s Guardian, as well as an essay in G2. [Illustrated with the usual photographs, incidentally: someone should devise a market instrument for investors in pictures of power stations, melting ice and – my personal tip – pictures of solar panels and photogenic children in Africa. Oh, sorry, it slipped my mind for the minute that markets are in the dog-house right now.]

George does an excellent job, as usual, in his forensic G2 piece (though there’s a touch of conspiracy theory in his analysis of Channel 4) but in the very last column it all falls to pieces. [See yesterday’s post for my views on conspiracy theories and the need to read the detail – in this case right to the end – to avoid Taleb’s randomness illusion]. Even so, I urge you to read George’s dissection of “Swindle”: you may be surprised. I recollect that I had moreorless bought into the idea (which Monbiot debunks) that Thatcher’s espousal of GW science was partly due to her search for weapons to use against the UK’s coal-mining industry.

Remember, though, that, as well as the particular pathology – in this case the way “Swindle” was given a platform – we also need to look at the underlying causes.

This is where a major problem lies in George’s piece:

“[Channel 4] says [its scheduling of “Swindle” and other programmes] ‘is against the background of the IPCC [Intergovernmental Panel on Climate Change] stating that there is a 90% certainty that the causes of global warming are man-made, it follows that there is a 10% uncertainty. Yet this 10% uncertainty receives a disproportionately small amount of airtime.’ I [George continues] find this argument extraordinary. A 90% level of confidence does not mean that 10% of the evidence suggests that an effect is not occurring — in fact, there is no reliable evidence showing that man-made global warming is not taking place. It is expressed in this way because there is no absolute certainty in science. The ‘very high confidence’ the IPCC expresses in the global warming thesis is the strongest statement any reputable scientist would make about his area of study. It is legitimate and right to stress that there can be no absolute certainty about global warming.” [my italics stress].

90% is not in fact a very high probability when we are discussing scientific findings. In my opinion, it would be more than justified to say that we’re “virtually certain” that “man-made global warming is […] taking place”, and by virtually certain I mean at least 99%. A 99.9% claim would be perfectly reasonable. So why does the IPCC not say this? Saying 90% gives the green light to people like Martin Durkin (the maker of “Swindle”).

I’ve just done a bit of weight-training and consulted the IPCC’s latest massive report (The Fourth Assessment Report, or “AR4″). If we look at Table 1 on pages 120-1 of the Scientific Basis (there are 3 parts to the overall report) we see that, although the IPCC is happy to use the words “virtually certain”, it only does this when a result “can be estimated probabilistically”. For example, a particular set of data may have a definable probability of indicating a trend.

[Note that our ability to calculate such statistics requires us to make assumptions about randomness – i.e. a bell-shaped curve or Gaussian distribution. This implies that we have a theory about the causes of variation in the data in the first place! For example, if we say we’re 99% certain that the glaciers are melting this finding must have been calculated against a null hypothesis that changes in glacier volume are subject to random fluctuations. This may not be true. There could be reasons we are entirely unaware of for all the world’s glaciers to either melt or grow at the same time (on top of reasons for correlation between glaciers in the same region which have presumably already been taken into account). Such “unknown unknown” correlation would invalidate the null hypothesis and hence the 99% “virtual certainty”. If we’re 99% sure what the data tells us, then surely we must be at least 99% sure of our theoretical understanding. I’m sure Taleb would agree with me! It’s entirely illogical to have more faith in data-driven findings than in any aspect of the underlying theory explaining them! But this is not my main point today.].

No, what baffles me is why the IPCC restricts itself to a maximum of “very high”, that is, 90%, confidence when it comes to “scientific understanding”.

Politics may have played a part in the IPCC process. Some governments may have lobbied for 90% rather than 99% as the maximum possible confidence. But let’s put that to one side. I want to argue that a critical factor is widespread misunderstanding of the scientific process.

Practising scientists often cite the philosopher Karl Popper. They understand that theories can be “falsified”. Some may even have heard of Thomas Kuhn and appreciate that such “falsification” takes place in “scientific revolutions”.

But what happens in such revolutions? In fact, scientific theories are superseded rather than “falsified”. Let’s consider one or two examples very briefly. When Einstein “overturned” Newton’s theory of gravity he didn’t demonstrate that Newton’s equations were wrong. Rather, he showed the limitations of Newton’s theory. Crucial experiments (where the difference was large enough to be measurable) showed that Einstein’s theory made more accurate predictions than Newton’s. In effect, Einstein incorporated Newton’s findings in his own theory of gravity. Albert never said: “Silly old Isaac’s made a mistake there.”

A case closer to the topic in question is the oft-cited theory of the 1970s that we were about to enter a new ice age. Now this theory hasn’t gone away. The Earth would be cooling (though there is debate as to when the next ice age would occur), if it weren’t for global warming. The current theory of global warming includes the ice age cycle as well as all other prior theories for the variation in the Earth’s climate, such as the effect of volcanic eruptions. Quantitative statements about man-made global warming take into account numerous other causes of climate variation.

Now, it’s possible to imagine reasons why the Earth might not warm as much as projected. For example, the solar system could enter some as yet undetected dust cloud. But any quantitative estimates of the effect of such a dust cloud would have to include the effects of man-made GW. And if the planet cooled dramatically as we entered the dust cloud we’d still have to worry about its temperature rising beyond today’s level because of our greenhouse gas emissions when we came out again. Just the same as, if we solve the problem of global warming and get the climate back to something resembling its pre-industrial state, we will – over the longer timescale of millennia rather than decades – need to take account of the Earth’s ice age cycle which was apparently of such concern in the 1970s.

There are examples in science of theories that are (or could be) flat wrong. But these are theories for which there is no evidence or for which the evidence has been misinterpreted due to problems inherent in the data-gathering process. This is most likely when observations are difficult, such as at the frontiers of physics. For example, the infamous string theory could be wrong because it makes no new predictions.

Any replacement for a theory with lots of firm data, such as global warming, would have to provide explanations for all that data. Clearly this is easiest if the new theory explains the old theory as a special case, rather than by invalidating it entirely.  In the history of science theories are almost always shown to be incomplete rather than “wrong”.  In my opinion, Imre Lakatos understands this process most clearly, even though this aspect of his ideas is rarely stressed.

The probability of the theory of global warming actually being wrong is therefore vanishingly small.  Our level of certainty is, in fact, far more than 99%.

So one of the underlying causes of programmes like “Swindle” is that even the scientific establishment is unclear as to the nature of its theory. Even if there are unknown unknowns and the planet does not end up warming over the 21st century and beyond this would not in itself invalidate the theory of global warming.

This entry is intended to serve as an index to the three papers I’ve written on the topic of biofuels, and in particular how to derive a payback period for a biofuel crop. Such a consideration inevitably suggests that growing biofuels on a given plot of land is a bad idea. It’s just a question of how long there’s going to more global warming for if you grow biofuels than if you don’t – centuries in many cases.

A few months ago I outlined the argument in a systematic step-by-step manner in Biofuel Payback Periods (pdf) (5 sides plus footnotes).

My somewhat longer initial essay treatment a year ago, Biofuels Are Not the Answer (pdf) (6 sides plus footnotes), takes a slightly broader view.

I also last year produced a slightly more elaborate critique of the idea that biofuels displace fossil fuel use – The Displacement Fallacy (pdf) (just 1 side).

PS (26/6/08): Updated Biofuels Are Not the Answer after noticing some broken links in the footnotes. Version 1.1.1, rather than version 1.1, is now referenced.

I was over at Realclimate yesterday, researching something entirely different – but related, obviously – when I can across a post referring to the discounting conundrum. I’m afraid to say I couldn’t resist making a contribution and then another one.

I have to say that the more I think about the issue of GW cost discounting, the more I harden my position. It seems to me that valuing the future less than the present has all the characteristics of a self-fulfilling prediction. We should not discount future capital losses compared to present expenditure.

My parting shot over at Realclimate included the ever so slightly controversial claim that:

“…[the cost of GW] damage will increase with GDP, since economic damage caused by GW will be proportionate to the total GDP whenever it occurs (e.g. consider that Stern discusses insurance losses in terms of total global GDP). [Equivalent funds] I invest will also increase with GDP…”

I fully expected this statement to be attacked, so had to have a riposte ready before posting. [Postscript (a few minutes after writing this post): I was wondering if the lack of activity on the Realclimate thread was because the Moderator had had a Bank Holiday lie-in. It now seems this may well have been the case. There’s now more to read over there].

I was worried that people would claim they could obtain a long-term investment return exceeding GDP. No they can’t. The so-called risk-free rate applies to government bonds. If I could achieve a return greater than the growth in GDP, that would imply that by investing in government bonds and reinvesting the return – in effect doing nothing, by definition, since this is the risk-free rate of return we are talking about – I could, over time, grow my wealth as a share of GDP. That would be nonsense. Merv and Gordo are stupid, but not that stupid.

But I now see (thanks to a presentation [pdf] by Mike Hanemann referred to in a Realclimate post) that it is the other side of the equation – my assertion that the cost of damage will increase with GDP – that is where Stern’s discount rate comes from. The Hanemann presentation notes that:

“Factors underlying the interest rate
• Pure rate of social time preference (ρ)
• How much richer the future generation will be (g)
• If they are richer, how much that lowers the value of money to them (η)
• Interest rate is: r = ρ + ηg
– Nordhaus & Stern both use same η (=1) and similar g (= 1.3%)
– Nordhaus uses ρ = 2.3-3%; Stern ρ = 0.1%
– Result is that for Stern r = 1.4%, while for Nordhaus r = 3.6 – 4.3%. “

[If you’re slightly confused that John Broome in Sci Am claims “Nordhaus discounts at roughly 6 percent a year”, then we’re in the same boat].

Nordhaus’ ρ = 2.3-3% is a self-fulling prediction as I’ve already pointed out. We can hardly avoid future catastrophe by, in effect, simply saying it doesn’t matter much.

My basic problem is with the idea that future generations will be “richer”. What does this actually mean? However rich we are, GW has the potential to destroy fixed proportions of our capital. If Cambridge ever disappears into the North Sea, we will lose a proportion of all the capital ever invested in Cambridge whether this happens in 2008, 2058, 2108 or 2808. Residents in New Orleans lost their houses whether these had all mod cons or hadn’t been modified for 50 years. They lost their TVs whether they were 1960s B&W sets or the latest wide-screen hi-def plasmas. Similarly many people in the Irrawaddy delta have lost everything, regardless of how rich they were. We may all have smart ecohomes in 2108, but we won’t necessarily have the capacity to replace them more quickly (actually our capacity to replace housing stock in the UK is slower than it was 50 years ago). The cost of a catastrophic flood in the East of England will be x thousand houses (and y hectares of farmland lost to the sea etc) whatever their nominal value or indeed how much real costs in terms of time and non-salvageable materials have been invested in them.

The idea that we are 1.3% “richer” each year is a bizarre economic construction. We may just have more sophisticated assets to lose in the future, that’s all.

If the whole planet is rendered uninhabitable by GW, it won’t matter how “rich” we are. Similarly if 10% or 1% is lost or degraded (e.g. due to sea-level rise or desertification) this is 10% or 1%, however rich we are.

Following my previous post, I’m still mulling over the topic of whether we should “Pay Now or Pay More Later” for the costs of climate change. I feel a need to extract the bear bones of the argument questioning whether discount rates apply at all to the “costs” of global warming. The previous post muddies the waters a little by bringing in additional (though valid) arguments, such as that it may be necessary to distinguish between consumption and capital formation.

It still seems to me that the principle of internalisation of costs – which I thought was established in the mainstream as the way to deal with the global warming problem – and the idea that the polluter pays – which again I thought was not seriously doubted as the appropriate principle – lead to a need to consider carbon emissions made now as a debt owed by the emitter to the global population, not as a cost that might be incurred in the future which doesn’t matter much because we’ll all be rich.

If we do this, then it follows that we have to be temporally impartial. If I have a debt of the form that a cost will be occurred (global warming related losses in this case) at some unknown point in the future, then I can amortise the cost of that debt over time – i.e. pay interest – or, if I’m sensible, insure myself – but not reduce it simply because the risk has not yet materialised. To assume otherwise is to be like the Lloyds of London Names who were wiped out when they were hit with insurance losses due to asbestosis. Note also that the businesses the Names kindly insured may have made a profit for their shareholders, but only because the shareholders avoided paying the costs in damage to health that came to light later. The asbestos businesses are analogous to today’s carbon emitters.

It also follows from the debt idea that we must be utilitarian and not prioritarian. (This is quite apart from the fact that there is no justification for the assumption that we will all be richer in the future). Consider buying a PC. By (say) 1995, it was quite clear that we all had a choice between buying a PC for (say) $1000 or putting the $1000 in the bank and buying a much, much better PC for (say) $1400 ($1000 + compounded interest) in 2000. We all struggled with this choice. It generated a million letters to PC magazines asking “Should I buy now or wait?”. The point is that we didn’t say to ourselves: “I won’t forgo a PC now because in the future I’ll be able to afford PC’s easily”. This argument is analogous to that of the prioritarians. It has zero logic. Sorry, negative logic. I can only afford more computers in the future if I don’t spend all my money on computers now, not if I do! Duh!

The problem is that, in dreaming up their daft arguments for “prioritarianism” and “temporal partiality”, the economists have failed to think about how we attribute the costs of carbon emissions, either to mankind as a whole (and I’ve resisted the temptation in pointing out that the cost of species and ecosystem loss is not only infinite – extinction is forever – but that rich people, as we will supposedly all be in the future, put a higher relative value on the environment than do poor people, an argument that could actually make the discount rate negative), individual nations (which is particularly dumb, as the global community is going to hold the historical polluters to their share of the costs one way or another) or individuals. We don’t have to actually answer the question (not easy – it’s going to lead to a lot of argument over the coming decades!), but, as soon as we ask ourselves: “Who’s going to pay for the damage caused by these emissions?” it becomes obvious that the discount rate should be zero. Anyone who argues that we can simply reduce the debt by 1.4% a year or 6% a year or whatever daft figure they dream up is simply spouting tosh.

Ever since the Stern Review report came out last year I’ve been baffled by the debate on discount rates. Something bothers me, but until now I haven’t been able to put my finger on it.

Effect on GDP of catastrophic events due to global warming

My gut feel is that the costs of catastrophic events have somehow been underestimated.

What I’m worried about in particular is the effect on cities (aka concentrations of fixed capital) of rising sea levels exacerbated by storms (though the loss of other resources – farmland, roads etc – is of course important too, and could cumulatively be even more significant). Some time ago my finger in the air estimate was that the loss of capital due to the forced evacuation of a great city – Shanghai was the one I came up with – would be around $1trn a pop, at today’s prices. My point is: how could the global economy keep growing if such losses become a regular occurrence?

Looking back at the Stern Report I see that there is a section on this very point – section 5.4 Impacts of extreme events (p.149ff in my CUP print copy). “Costs of extreme weather alone could reach 0.5 – 1% of GDP”, it says – currently they are 0.2% or $60bn p.a. on average. In 2005 the loss was $200bn, largely due to Katrina. Stern also notes that “Miami alone has $900bn worth of total capital stock at risk.” (p.150). On the other hand “developed economies… invest a considerable amount in fixed capital each year (20% of GDP or $5.5trn…)” of which $1.5trn goes into construction. Now I am able to quantify my qualms. Bear with me here: I freely admit I’m in a bit of a qualm quantification quandary!

1. GDP includes both consumption and capital formation. Now, it is the capital formation that allows us to grow GDP. To put it simply, it is all the new factories we build this year that allow us to consume more next year. If we stop forming capital (i.e. net investment goes to zero), then GDP growth will also fall to zero. We can’t (say) provide more transport without building more transport infrastructure, factories to produce vehicles etc.

2. Now, Stern notes that capital formation is around 20% of GDP in developed countries. I see no reason to suppose that this type of GDP (aka economic activity) can displace consumption. What I mean is, the resources that support consumption (private cars, say) cannot be diverted into the production of fixed capital (car factories, say) – at least not easily. To put it another way, if buildings are destroyed in a storm flood, they can either be rebuilt or other buildings can be built elsewhere (as intended before the catastrophic event) – you can’t do both unless you have spare construction resources.

3. So, I suggest, long-term GDP growth depends on the growth rate of capital formation.

4. It follows that average annual losses of fixed capital of Stern’s 0.5% – 1% of GDP will reduce the rate of fixed capital formation – which is only 20% of GDP – by 100%/20% times the proportional GDP loss, that is by 2.5% – 5%.

5. So GDP growth will in the long-run be 2.5% – 5% less than otherwise, not the 0.5% – 1% range Stern refers to.

6. Compound this effect on growth and GDP, even at the lower end (an average reduction in capital formation of 2.5% p.a.), if we allow global warming to happen, will only be around 75% of what it would be otherwise after a mere decade and less than a third after 40 years – so anyone starting out on their working life now will have a retirement living standard 1/3 of what it would otherwise have been. If we allow global warming to happen, that is.

7. Sanity check: refugees from a city rendered uninhabitable will move somewhere else and either:

• be poor (reducing total GDP that way), or,

• if fully insured, push up the price of housing etc, in effect lowering everyone’s standard of living, and again reducing total GDP.

8. I guess ultimately the economy might respond and allocate more resources to construction and other fixed capital formation. But if the losses are scattered moreorless randomly through time and around the world, it may be very inefficient to permanently maintain a larger pool of construction resources (how will the market do this?). One year they’d be needed, another not, the next they’d be needed on the other side of the world. Furthermore, for such a change in resource allocation to have no effect on overall GDP growth rates would imply that capital formation is not at present proceeding at an optimal rate. If it is possible to permanently increase the rate of capital formation without other repercussions for the economy – and note that high rates of capital formation tend to be associated with the accumulation of environmental debt, which is what’s getting us in this mess in the first place! – then there would nevertheless certainly be a permanent reduction in consumption as a percentage of GDP.

Hmm…

Response to “The Ethics of Climate Change”

In the last couple of days, an article, “The Ethics of Climate Change” by John Broome in Scientific American (June 2008 ) has led me to take a somewhat different tack and suggest that economists are making a category error when they model the reduction of carbon emissions as a cost that can, if necessary, be deferred. It is surely not a case of the “sacrifices the present generation will have to make to reduce greenhouse gases” (as John Broome puts it in Scientific American) but one of the environmental debt that will continue to be run up as long as we fail to internalise the costs of our emissions.

It is the science that tells us that the elevated level of greenhouse gases is an example of an environmental debt. For example, because the annual amount of carbon dioxide the ocean can absorb has levelled off (according to the IPCC, and as discussed previously in Uncharted Territory) then, at the margin, an extra tonne of carbon dioxide emitted now by the burning of fossil fuels or land-use change will remain in the atmosphere indefinitely. Each year the extra tonne of atmospheric carbon will cause the Earth to gain more heat than it would have done otherwise, causing environmental or financial damage (or at least increasing the risk of such damage in any given year).

Carbon dioxide emissions effectively incur an (until now unaccounted) interest charge payable in perpetuity in terms of environmental damage. It may make sense to establish an international trade in cuts in emissions compared to notional business as usual levels, so that emission reductions are made most efficiently (or at least more politically expediently than would otherwise be the case), but, when we are trying to determine the optimal global emissions reduction path, it is (IMHO) a mistake to consider emission cuts as a form of expenditure. Overall emission reduction targets must rather be motivated by a consideration of the cost of increasing our collective environmental debt that would otherwise be incurred.

And this isn’t the same thing at all. No, no, no!

The problem cannot be analysed as if it were that of an individual deferring costs over their lifetime, on the grounds (say) that they’ll have more money when their career takes off. To put it bluntly, (and think about the analogy of an individual’s lifetime spending profile) deferring forgoing consumption is simply not postponing a cost (as Stern et al would have us believe) – it’s consuming now and taking on debt. And as we – the post credit crunch generation – know only too well, debts can get out of hand. Am I starting to make sense?

Consider further that the existing international consensus is to implicitly attribute the global warming debt to individual nation states on the basis of their “historical emissions” through consumption of fossil fuels. (One might argue that fossil fuel producer countries and those that have caused carbon emissions through land‑use change have also incurred a debt). Instead of philosophising we can simply look at how international debt is managed today. Government debt is rarely written off – that of developed countries almost never. It follows that we should be temporally impartial (see the Scientific American article for an explanation) in assessing when to make emission reductions. In other words, it would be very imprudent to apply a discount rate on the basis that the cost of damage caused by today’s emissions will be less in the future. This is why it’s so important that we don’t treat a debt as a cost. If I borrow a dollar today I have to pay back more than a dollar in the future. But if I spend a dollar today that is worth more than to me than if I spend a dollar in future.

The question remains as to whether we should be prioritarian (again, see the Scientific American article). To determine this we must look at the present balance sheet of Earth Enterprises inc., taking into account the net present value of the environmental debt. Taking on the environmental debt results in at least some capital investment. The interest on the environmental debt represents future destruction of capital. How should this destruction be discounted in deciding the optimal path of future carbon emissions? The loss of a city in 50 years time due to rising sea levels and more intense storms will cause the loss of today’s capital represented by the city as well as the investment made over the next 50 years. However many bathrooms there are in each house (the article suggests that we can discount because we’ll have more bathrooms in the future), they’ll all be destroyed. The same proportion of global capital will be lost by the destruction of the city in 50 years as would be lost if it were destroyed tomorrow. As for the temporal consideration, we must simply use present-day costs – a discount rate of zero.

As pointed out in the Stern Review, the cost of – or rather, the environmental debt incurred as a result of – greenhouse gas emissions depends on their level in the atmosphere at the time. That is, because of the limited ability of natural systems to absorb increased emissions and eventual positive carbon cycle feedbacks, the marginal cost of a tonne of carbon emissions increases (in real terms) as atmospheric greenhouse gas levels and the temperature increase. This conclusion suggests two possible international conversations:

• First, responsibility for historical emissions can be quantified if politicians wish to attempt such an exercise, but at a lower cost to present emissions.

• Second, the cost of future emissions will be higher than even present-day emissions. My bank manager would have little sympathy if I asked him to grant me a cheap mortgage simply because I missed the opportunity to borrow before the credit crisis, when money was cheap. Similarly, the fact that carbon emissions were cheap in the past is not a valid argument against them being made very expensive for everybody now.

Conclusion

The conventional economic approach – as espoused by Stern and characterised by a discourse focussing on the “costs” of global warming and, as a consequence, making use of discount rates – is deeply suspect.

An alternative approach which is more in tune with the international consensus perception of global warming (and other environmental externality problems) is to characterise the elevation of levels of greenhouse gases as an environmental debt, attributable to nation states or as may otherwise be politically determined.

Such an approach immediately exposes the fallacy of temporal discounting – international debts are typically never forgiven, or, if they are, this is seen as akin to an act of charity. We must be temporally impartial in assessing the costs of global warming (and other environmental debts).

Neither can we be prioritarian and plead present-day poverty. The debt-based approach also reveals that capital losses cannot be simply equated with the loss of consumption opportunities. The error is akin to a company trying to convince the investment community that a loss of $100m in a financial year is tiny compared to its turnover of $5bn. It is only relevant to compare the scale of the loss with the company’s capital. Additionally, losses of capital will be proportionate to the total capital at the time of the loss – any more optimistic assumption would be reckless.

The correct approach therefore is to use a discount rate of zero for global warming losses, and account for all global warming debts at present-day prices. That is, if the emission of a tonne of carbon now increases the risk of the 50% loss of the fixed capital investment represented by a given city in 50 years time by 0.000001%, then the cost of (=the debt incurred by emitting) that carbon is 0.000001% of 50% of the value of the city now, not 0.000001% of 50% of the value of the city discounted by a factor of the order of some percent raised to the power 50 (i.e. multiplied by for example 98.6% – Stern’s figure – or 94% – Nordhaus’ estimate for each of 50 years – try this one at home to see the effect!). To put it bluntly, if Miami has to be permanently evacuated and 50% of the cumulative fixed capital investment in the city ($900bn at present-day prices) is lost, then it is 50% whether the event happens tomorrow, in 50 years time or in 150 years time. Whenever it happens we still have to build New Miami. If you happen to be responsible for some of the environmental debt that causes this to happen, I hope you’ve invested the carbon wisely!

In translating the risk of future losses – that is, the total environmental debt due to greenhouse gas emissions – into a price of carbon we must also take into account that the warming effect of a tonne of emissions at a given time depends on the total amount of emissions up to that point in time. As in any financial market, the cost of a commodity (in this case the right to emit) depends on the behaviour of all other participants in the market.