Sensitivity analysis

One of the points on which economists generally agree on is that sensitivity analysis is a good thing. Broadly speaking, this means varying the (putatively) crucial parameters of a model and seeing what happens. If the results change a lot, the parameter justifies a closer look.

In the case of the Stern Review of the economics of global warming, sensitivity analysis quickly revelas that the crucial parameter is the pure rate of time preference. This is the extent to which we choose to discount future costs and benefits simply because they are in the future and (if they are far enough in the future) happening to different people and not ourselves. If like Stern, you choose a value near zero (just enough to account for the possibility that there will be no one around in the future, or at least no one in a position to care about our current choices on global warming), you reach the conclusion that immediate action to fix global warming is justified. If, like most of Stern’s critics you choose a rate of pure time preference like 3 per cent, implying that the welfare of people 90 years (roughly three generations) in the future counts for about one-sixteenth as much as the welfare of people alive today, you conclude that we should leave the problem to future generations.

So, responses to a Stern Review provide another kind of sensitivity analysis. If you don’t care (much) about future generations, you shouldn’t do anything (much) about global warming.

88 thoughts on “Sensitivity analysis

  1. Dear Wilful: bushfires and drought have nothing to do with “climate change”, they have ben endemic in Australia for millennia, or how do you explain the excellent rains across southern and eastern Africa over the last year and the long term trend in much of that area for increasing rainfall? More water storage would make desalination unnecessary.

  2. To Roger Jones:

    All scenarios of the World Bank, UN, IPCC and who not have population growth slowing substantially in a few decades from now. Total economic growth would slow too, and probably per capita economic growth would slow with aging. This would reduce the growth rate of energy use; only in a handful of scenarios, this is dominated by a switch to coal as the main energy source.

    And then the spectral window of carbon dioxide saturates.

    So, if you stop looking at the temperature scenarios of the IPCC, and start looking at their first derivatives, you will find that these decades are the ones with the most rapid change.

    You would need to assume massive methane or carbon dioxide releases to reverse this.

    Perhaps my “all” was a bit overdone; “almost all” is more accurate.

  3. Did I say that we would stop all spending on ourselves?

    Estimates have already been run using the Stern approach to intergenerational utility maximising showing that the appropriate savings rate is something absurdly high. True, not 100%, but high enough to be absurd. I believe Toll and others have also raised this objection.

    No other analysis is done using intergenerational utility… though most activity will impact on the currently unborn. The conclusions from intergernational utility maximisation are absurd. There is no reason to switch to this strange, unjustified and generally un-used totally revolutionary policy analysis approach.

    Certainly, switching to this radical new approach does not imply that a person cares more about their children.

    Further, even if you want to use the radical Stern/Quiggin approach, the time value of money should still include an expected cost of capital.

    I think it is dishonest to say that The quote used by Tam (which had obvious errors) is the best that the GW non-activists can do. That is patently untrue and presumably only written for it’s insult value.

  4. I did indeed say that we’d stop all spending. Mea culpa. I’d written with more nuance previously and I assumed that people would understand the reality underlying the dramatic overstatement. Obviously intergenerational-utility-maximising doesn’t imply zero consumption. At a minimum we need to eat! But it does imply implausably high levels of saving & many absurd policy ideas that nobody would accept.

    I made other points besides this one and it’s dispointing that Quiggin chose to ignore the entire arguments because of a rhetorical flourish.

  5. John, I already responded to the claim by Dasgupta about extremely high savings rates, as did Brad DeLong. Dasgupta’s assumption that economic growth arises solely from capital accumulation (no technical progress) is wrong, as is the implicit assumption of an infinite supply of riskless projections yielding 4 per cent returns (we can see this from the fact that the bond rate is 2 per cent).

    Since you didn’t express yourself in terms of parametric choices, but rather with a general claim that maximising intergenerational utility implied zero consumption, I didn’t know whether this was a rhetorical flourish or a formal (wrong) claim.

    As for your claims about strange, unjustified, revolutionary approaches, I suggest you look at the large literature on intergenerational equity, including the Australian Treasury’s paper on this subject. As I pointed out already, the standard assumption in this literature that equity requires constant proportional tax rates makes sense only on the basis of something like Stern’s assumptions.

    Finally, my statement that the FT letter quoted by Tam was the best the do-nothing lobby can do was a bit unfair, but a lot of the criticism of Stern has not risen above this standard.

  6. From memory, Sir Partha estimated a savings rate of 97.5% (it might have been 98.5%). Why would anyone believe we’d have any technical progress, if we were saving that amount of GDP? Indeed, the social rate of 0.1% (representing the probability of extinction) would have to increase because we’d all be extinct (from starvation).

    The biggest problem the Greenists have is their continual over-reaching.

  7. Sinclair said: The biggest problem the Greenists have is their continual over-reaching.
    Exactly. My point about Stern’s discounting appears not to have been understood. Because of the 0.1 social discount rate, damages that would be only 0.5 per cent of GDP in 2100, because of its much higher level then even if the World economy grows at only 2.5 per cent a year, have the high present value of 5 per cent of today’s GDP. Sir Nicolas Stern would have been less disingenuous if he had stated simply that his estimate of costs of not preventing climate change damages would be just 0.5 per cent of likely global GDP in 2100. Our successors would be able to make appropriate adaptation to that level of damage with some ease, whereas his Review asks us to give up one per cent of our much lower level of income now to save half that proportion for our descendants.

  8. Dear Tam: al the experts disagree with you.

    And this thread isn’t about whether the weather will change, that is (for the purposes of these arguments) a given. The question is what would the costs be. It’s so easy to point to costs already being incurred.

  9. Wilful: “The question is what would the costs be”. such as? falling prices of oil as the northern hemisphere warms up? there has been NO detectable warming in the tropics, and only a small proportion of the total global population lives in the Southern. Can I suggest you start the process for acquiring a des res. dacha somewhere north of Moscow?

  10. John Q:
    It is the authority of the argument that counts, not the authority of the one that makes the argument.

    In your particular case, I would be extra careful with saying “take a look at my CV” as they might just do that. Not that this matters.

    I pity the students of the London School of Economics for two reasons, and of U Queensland for one.

    First, despite having at least 50 person years of technical support at his disposal, Nick Stern produced a report that has so many technical flaws that it would fail as a master’s thesis in economics. Someone like that is not fit to teach economics. I indeed think that it is nonsense to grant a PhD for life.

    Second, the technical errors aside, Nick Stern placed political expedience above intellectual honesty — by a selective review of the evidence, taking an extreme position on ethical matters, both without alerting the reader to this, and by refusing to do sensitivity analyses. You seem to applaud this attitude, but I think it disqualifies you and him as honest brokers of scientific knowledge. You abandoned your position as policy analysts, and adopted the position of policy advocates. Advocates have a useful role in society, but they do not belong at university.

    With professors like that, I would think twice before hiring students from LSE or UQ.

  11. Richard Tol said:

    “And then the spectral window of carbon dioxide saturates.”

    The spectral window for carbon dioxide is already saturated, which is why the response is logarithmic and not linear or higher (water vapor response to a linear increase in other forcings).

  12. Richard,

    I find this kidn of argument pretty disappointing from you. For a start, it seems strange to say that I applaud Stern’s failure to do sensitivity analysis when commenting on a post that begins “One of the points on which economists generally agree on is that sensitivity analysis is a good thing. ”

    Next, given that you’re using emotive criticisms of opponents to push your case, you’re hardly in a position to accuse others of acting as advocates.

    Third, you’re labelling as “extreme” a view which, as you conceded in an earlier post, has a strong body of support, but which you don’t happen to like. Then you use advocacy of “extreme” positions as evidence of bias. In effect, anyone who disagrees with you cannot have a legitimate position if you argue this way

    Finally, you haven’t responded at all to my substantive point that the approach proposed by Stern yields a discount rate very close to the relevant market rate for evaluating riskless, namely the real bond rate, while the approach you propose is well above the market rate. And as Brad DeLong and Tyler Cowen have pointed out, you can’t fix this hole in your argument by appealing to uncertainty – in this context, uncertainty implies a need for more action, not less.

  13. John:

    The empirical evidence is that people and their governments have a pure rate of time preference between 2 and 4% per year (at least in rich countries); not 0.1%. Saying that Stern’s discount rate is close to the market is just hogwash.

    More generally, you came out strongly supporting the Stern Review — although you did say that perhaps he should have done some more sensitivity analysis.

  14. Chris: large swathes of tropical South America and Africa have no data in your chart, while for the Northern Territory your claimed anomaly is 2C. But the BoM shows for about half the NT +0 to +0.5, and for the rest less than +0.15, all within the SEE.

  15. Richard, you still haven’t responded to my point – if the pure rate of time preference is between 2 and 4 per cent (and the elasticity of intertemporal substitution is nonzero) how can the real bond rate be between 1 and 2 per cent as it has been for most of the last century?

    Also, can you point to a summary of the literature supporting your claim?

  16. “large swathes of tropical South America and Africa have no data in your chart”

    “large” meaning 5 dots out of 50 odd in tropical South America and 1 dot out of 80 odd in tropical Africa.

    “while for the Northern Territory your claimed anomaly is 2C”

    NCDC’s chart shows 1C actually.

    “But the BoM shows for about half the NT +0 to +0.5, and for the rest less than +0.15”

    Slowy getting less inaccurate, but still wrong. BoM’s map show +0.5 to +1.0/century for about 40% of NT, +1.0 to +1.5/century for about 50% of NT and +1.5 to 2.0/centrury for about 10% of NT.

    I can see the level of self-delusion it takes to be a global warming denialist (as in “there has been NO detectable warming in the tropics”). You have to believe that 6 out of 130 means “large”, you have to read dots meaning 1C as 2C and you have to carelessly read contour maps.

  17. John — my earlier post did not get through

    I suggest you start with Newell and Pizer, 2003, JEEM and Evans and Sezer, 2004, Appl Econ Let, and trace the substantial empirical literature on discount rates from there

    you may trace it all the way to Aristotle’s tirade against usury — which he may not have written had the discount rate been lower


  18. Richard, I know the Newell and Pizer paper, but it doesn’t have any reference to the pure rate of time preference. It uses the real bond rate as the riskless discount rate, just as I’ve suggested.

    They use a 4 per cent rate for illustrative purposes, but both the current real bond rate (about 2 per cent) and the historical average for the last 100 years or so (between 1 and 3 per cent, depending on details of the calculation) are lower than this. Assuming that rising incomes account for a discount rate of at least 2 per cent, it’s hard to see any evidence of pure time preference here.

  19. I was involved in the preparation of the intergenerational review done by Treasury and I can assure you that it does not suggest that intergenerational utility maximising is an appropriate public policy approach.

    Also, the relevant capital return is not the risk-free rate, but the expected rate. These aren’t the same thing.

  20. Then why the concern about the possibility that tax rates might be significantly higher in 2050 – why not just borrow overseas to finance public expenditure now and stick the grandkids with the bill?

    It struck me at the time that the authors of the intergenerational report didn’t really understand the assumptions implicit in their analysis (for example, much the same people were advocating an 8 per cent real discount rate), and your comment suggests that this view was right.

  21. John:

    The pure rate of time preference is the money discount rate minus the rate of risk aversion times the per capita consumption growth.

  22. To restate my point, since the real bond rate is around 2 per cent, the coefficient of risk aversion is 1 and per capita consumption growth is 2 per cent, the implied pure rate of time preference is close to zero. So Stern’s assumptions match market data. I’ve made this point quite a few times, without so far seeing a response from you.

  23. Sure, John, the pure rate of time preference is zero — this is quite a stunning discovery: 3000 years of economic theory and data proved wrong! — forget about Krugman, I will nominate you for the Nobel Prize from now on

  24. Thanks for this kind nomination, Richard. Unfortunately, it appears that Ramsey, Pigou, Solow, Sen and others have beaten me to this discovery, but I’m glad that you agree (as shown by your substantive non-response) that the case for a positive rate doesn’t stand up well to scrutiny.

  25. Richard Tol (13th): “The pure rate of time preference is the money discount rate minus the rate of risk aversion times the per capita consumption growth�.
    JQ (14th): To restate my point, since the real bond rate is around 2 per cent, the coefficient of risk aversion is 1 and per capita consumption growth is 2 per cent, the implied pure rate of time preference is close to zero. So Stern’s assumptions match market data. I’ve made this point quite a few times, without so far seeing a response from you.
    Truly JQ as inventor of new new math doth deserve a Nobel.
    He accepts Tol’s definition that prtp = (mdr – rar)*percapconsgrowth, and then provides data, such that apparently prtp = (2-1)*2 = 0; to lesser mortals, (2-1)*2 = 2, which is not “close to zeroâ€?. In fact JQ’s data and Tol’s algebra result in no difference between the prtp and the real bond rate as the correct discount rate of 2%.

  26. Tam,

    unfortunately, your sarcasm “Truly JQ as inventor of new new math doth deserve a Nobel” fails immediately.

    “The pure rate of time preference is the money discount rate minus the rate of risk aversion times the per capita consumption growth” means:

    prtp = mdr – rar*percapconsgrowth

    So it’s prtp = 2% – 1*2% = 0

    Of course, as anyone who underatands the concepts knows, it makes no sense to create a variable (mdr – rar), as you have done, since mdr is measured in percentage points (like 2% or 3%), and rar is measured as number, like one or two.

  27. Tam, you are wrong. You are confusing a risk premium with the rate of risk aversion

    The coefficient of relative risk aversion is defined as

    rra = -cu”(c)/u'(c)

    where c is the level of consumption, u'(c) is the first derivative of the utility function, and u”(c) is thr second derivative.

    This is basic stuff.

    The classic reference for why rra is close to one is Ken Arrow (1970), Essays in the Theory of Risk Bearing, Chapter 3.

  28. In addition, your argument falls over on your own logic.

    If, as you say, rar = 1%, then your equation

    prtp = (mdr – rar)*percapconsgrowth

    implies prtp = (2%-1%)*2% = 0.0002, which is, in fact, close to zero.

    But your equation is wrong in any case, as is your measure of rar.

    As I said at 78, the correct equation with the correct measure of rar gives prtp = 0.

  29. Thanks for doing garbage pickup on this, Uncle M, but I’m afraid TimTam is a lost cause when it comes to logical reasoning of any kind.

    And while Richard Tol has the mental equipment, it seems that he’d rather engage in cheap shots like #75 than respond to a substantive point that contradicts his views.

  30. John, I presume you know Tol and his work. Why is he so determined to play the spoiler? Could it be that Stern did not make enough references to Tol’s contributions to the field?

  31. Uncle: I am intrigued by this new algebra and arithmetic (JQ is a lost cause in this area). You said:
    “prtp = (2%-1%)*2% = 0.0002, which is, in fact, close to zero”.

    Let us start with 100, then take 2% of that minus 1% of that, which (as I was no doubt wrongly taught) is 2 minus 1 which means one. Multiply that by 2% of 100, which usually equals 2, and we have 2 times one, which in my day equalled 2 and is not “in fact equal to zero”. If the prtp is an absolute number and not a rate as in your #78, then the equation is comparing mixing aples with oranges. Pity again, Milton, no doubt you are an alumnus of UQ.

  32. Tam,

    You don’t start with 100.

    You start with the real risk free rate of interest, which, as a factual matter, is around 2% per year. You then subtract from that the product of the coefficient of risk aversion, which is close to one, and the rate of growth of consumption which also, as it happens, is around 2% per year.

    There is no mixing of apples and oranges. You can see for yourself the derivation of the equation for prtp in chapter 1 of any asset pricing textbook.

  33. Uncle Milton,

    According to the BBC, Stern quoted my work 63 times.


    I do not believe your numbers one bit. Maybe that is because I’m a reactionary. Your numbers do go against 3000 years of research. So, either you have done something truly spectacular and deserve a Nobel Prize — in which case I suggest that you stop blogging and start drafting a paper — or you’re just completely off.

  34. “I do not believe your numbers one bit. ”

    Richard, I’ve used three numbers – the real bond rate (2 per cent), the coefficient of relative risk aversion (1) and the rate of per capita consumption growth (2 per cent) – and plugged them into a standard formula (given by you in #73) to derive the pure rate of time preference (approximately zero). Which of these numbers don’t you believe?

    If you want to disagree, the most plausible line is to claim that the real bond rate is not the proper rate of discount for riskless flows, either because of tax distortions or because of problems associated with the equity premium and risk-free rate puzzles. A case can be made this way, but it’s uncomfortable if you want to rely on market evidence to attack Stern (as, for example, Nordhaus did).

    Thanks for the advice to publish this. I am working on a paper now which will include this point.

  35. Hi John,

    The real bond rate stikes me as suspect.

    I would not write a paper that includes this point. I would write a paper on this point.


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