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. 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.

  2. 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%.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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?

  8. 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.

  9. 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.

  10. 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.

  11. “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.

  12. 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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