Sunday, August 23, 2015

The algebra needed to read about climate change...

This article (at, on a lawsuit from a group of young people demanding that we do what it takes to recover from climate change, looks very interesting. One line seemed either wrong or surprising to me, though.

We must immediately commence carbon emissions reductions of 6% each year until the end of the century. Timing is crucial. If we wait until 2020 to begin emissions reductions the annual requirement is 15% per year.
Starting only 5 years earlier, they are saying that we can do 2/5ths as much reducing each year, for 85 years instead of 80, and get the same result. It seems too dramatic. I want to think about how to analyze it. I don't yet know what assumptions I can make.

  • Should I compare total emissions from now until 2100? (I think so.)
  • Should I assume emissions are growing exponentially from now until 2020 in the 2nd scenario? (I think so.)
  • What else would I need to know? (Are there other factors that make this more complicated?)
This seems like a perfect question for pre-calculus. Too bad I'm not teaching it this semester.

I think I got it. I think this assumes that we are currently increasing our carbon emissions at a rate of about 20% a year.  We are not. It's more like a tenth of that - about 2.5%. (Government source here.)

If you want to do some real math, think about what you would do before continuing.






I figured it like this. I count this year's carbon emissions as 1. If we decrease 6% a year, that means we have 94% of the previous year's emissions. So the total emissions from now until 2100 is
S=1+.94+.94^2+...+.94^84. This simplifies to S = (1-.94^85)/(1-.94). Note that the .94^85 is so close to 0 that we can ignore it. We Get S=1/.06 =  16.666. So the article is saying that for the next 85 years, we can emit 16 times this year's emissions.

If we increase until 2020, we would start with higher emissions, H. 15% decrease per year leaves 85% of the previous year's emissions. Our sum would be
S=H+.85H+.85^2H+...+.85^79H = H(1-.85^79)/(1-.85) = (almost) 1/.15 = 6.666.

16.666 - 6.666 = 10. So somehow we get 10 times this year's emissions within the next 5 years. If our emissions are currently increasing so that our emissions next year is r, then
S = 1 + r + r^2 + ... +r^5 = (1-r^6) / (1-r) = 10. I asked to solve  this and got r = 1.2, for a 20% increase per year.

I asked John Golden to check my work. He used a continuous increase model and got close to 9%, mush lower. But still not low enough to match what's happening.

So it seems that either the article has a typo, or my mathematical model is not including everything it should. Humanity seems to be at a tipping point. Can we change our ways of making decisions, from capitalism to something else, in time to save ourselves from our foolishness? I would like everyone to be able to do this sort of math.


  1. Sue, you did drop the 'H' from your second sum 'S' - you can't subtract the 16.6 and the 6.6 because they are in different "units".

    The proper equation is (sum if we reduce now) = (sum if we reduce after 5 years) + (sum of emissions over the next 5 years). Using N for current emissions rate and H for emissions in 5 years, this is
    (N * 16.58) = (N * r^5 * 6.66) + (N * (r^5-1)/(r-1))
    Solving this numerically for r I get a value of about 1.096, or an assumed 9.6% per year increase.

    Notes: I used 16.58 instead of 16.66 because (.94)^85 is still about ~.005, so it is more accurate. I used five years of increase instead of the six in your equation, thus H = N * r^5 and the 5-year sum is N * (r^5-1)/(r-1).

    While solving numerically by hand (Newton-Raphson) I noticed that the second derivative was large, so the exact solution may be very sensitive to the inputs - i.e. if the 15% was rounded from 14.5, or the 6% was really 6.2% you might get a full percent or more difference in the answer.

    I also note that the final term in each sum is the assumed emissions in the year 2100. The fact that they are so small brings the feasibility of a continuous decrease into question. For 6% reduction, the final term is about 1/200 of current emissions, whereas for 15% reduction it is barely more than a factor of 500,000 below 2020 emissions (which are 58% higher than current using 9.6% growth). These levels would represent a wholesale shift away from carbon emitting, rather than settling at a new, reduced, stable level.

    I suspect that the 6% and 15% numbers come not from a calculation of total carbon emitted into the air, but the results of a more complex model that represents how the atmosphere responds to total airborne carbon, and that such a model would be very non-linear - therefore the increase in needed reduction may not correspond to achieving a fixed value of the integral at all.

  2. Thank you for catching my error. That's two posts in a row with silly errors. (Perhaps an indicator of my stress level, single parenting a 13-year-old.)

    Your answer is very close to John Golden's. I had thought the difference was because he integrated to look at continuous increase, but it's probably too big a difference for that to have made sense.

    I will work through the problem again myself. It's good I was vague when I posted on the blog where this was mentioned. 9% is still more than 3 times what the true increase is. So yeah, probably there probably very non-linear things happening. Unless that article included a typo.

  3. I value the whole discussion. I wish there was more discussion of the modeling so that it didn't seem like our expert's numbers vs your expert's. Discussing the models would lend a lot more weight to the issue, plus raise the value of this kind of thinking.

    Here's the GeoGebra, if anyone wants to play:

    Walt's reasoning looks solid to me.


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