My initial idea was to cut off the calculation at some earlier date, perhaps 2100, but that introduces an arbitrary discontinuity as well as ignoring technological change earlier than that. A better and continuous solution, since the paper is producing a probability distribution rather than a single value, is to have the uncertainty of the costs increase over time. If the estimate, before allowing for technological change, is that cost in year t has a mean of Ct, a standard deviation of Ut, replace that with a mean of Ct, a standard deviation of Ct*e**(A*t), where A is a measure of how fast you believe uncertain technological change increases the uncertainty of the cost.
That would leave the mean of the total cost unchanged but increase the variance. If you think, as I do, that technological change will tend to reduce costs, for example because improved medicine reduces the effect of temperature on mortality, put in an additional factor of e**-(B*t).
How do you pick A and B? For B you could try to see how fast each category of cost calculated for a given amount of warming has decreased over time. Lay et. al., for example, reports the effect of calculating the mortality cost of a given amount of warming using numbers calculated with data from two periods thirty years apart; one could do something similar for other costs.
I don't have a good idea of how to calculate A, save that it should not be zero, as it implicitly is in Rennert.
This is a first pass at the problem. Perhaps commenters here can suggest a better way of incorporating uncertain technological change into a calculation of the social cost of carbon.
One issue I have with using exponential discounting for mu has, much longer horizons that typical adoption periods is that all real world exponentials end out being the early parts of sigmoids (S-curves) as the minor now, but hard to improve aspects/constraints come to dominate how the system behaves.
The challenge then would be picking what the inflection and saturation factors should be, based on plausible technology change...
I can't help with the maths here, far beyond me. However, if you go back to the original models - Special Report on Emissions Scenarios from the 1990s, still online - then the A1 family is based upon this century being very much like last century. About the same levels of economic growth, about the same decrease in energy use per unit of GDP, about the same productivity rises more generally and so on.
It might well be that in some of the work underneath that that the assumptions are explained more explicitly. Basing predictions about future changes upon observed past ones doesn't sound like too bad an idea to me.
It occurs to me that B (or however you model changes in cost responsiveness to climate change) may itself depend on climate change: the more severe the effects of climate change, the more incentive there will be for both private and public entities to find ways to ameliorate its impact.
Of course, some technological means of amelioration will probably be of the "beggar thy neighbor" variety: for a price, I can reduce the impact of climate change on you by shifting the impact to somebody else who's not paying me. The global impact of such efforts may be zero or even net unfavorable. Other technological means will presumably be globally net favorable. I don't know how to encourage the latter at the expense of the former.
Basing your estimate of next century on the same rate of change as last century might make sense but basing it on the same level does not, and in the case of technology that is what the literature, at least the article I am commenting on, is doing.
Sorry, I was unclear. The A1 family of scenarios explores roughly the same rate of change. The differences (A1FI, A1T and so on) tend to be about changes in energy production technologies.
Anyway, my underlying here wasn't to really say that this is the answer. Rather, having a look at the assumptions made in those models might lead to your finding someone has already played with these very assumptions.
It seems as if a basic question is what kind of curve you expect technological change to follow. Long ago, I read an essay by Heinlein that talked about projecting trends, where he said that really daring people might project a continuing linear increase . . . but that technology tended to follow exponential growth curves. The conceptual point about choosing the right curve was valid, but I don't think the exponential was the right function. Growth only is exponential if there are no limits to the resources it depends on; if there are scarcity issues the basic form is a logistic curve, an option that Heinlein never mentions. But it seems as if a lot of technologies go through a period of rapid growth and then approach some sort of limit.
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