Anomalous dry and hot conditions across the US, as well as recent well-publicized research, have amplified the discussion over how human affairs might be affected by changing patters of exposure to extreme heat. You hear the word "non-linear" thrown around a lot in these discussions, so we wanted to clarify some of the things we know (and don't know) about non-linearity in this setting.
For the time-constrained or otherwise impatient, here are the take-homes:
1. Small increases in average temperature can have large effects on extreme heat exposure - so extreme heat exposure tends to increase non-linearly with average temperature.
2. Many economic outcomes we care about (agriculture, other types of economic output) respond negatively to increased exposure to extreme heat, although there is limited evidence that this response is itself non-linear.
3. Because these economic outcomes respond to extreme heat, and extreme heat responds non-linearly to increases in average temperature, then it follows that these outcomes can respond non-linearly to increases in average temperature.
Boiled down even further: extreme heat is bad under current climate, a few degrees of extra warming will increase exposure to extreme heat dramatically in many places, and this could have very large negative impacts on outcomes that folks care about.
Now here's what some of the science says about these points.
1. Extreme heat exposure tends to increase non-linearly with average temperature. This can be pretty easily visualized by looking at some counties in the US. I pulled data for three corn-growing counties in the US (Sioux County in Iowa, Calhoun County in Georgia, and Clay County in Minnesota), and plotted how exposure to extreme heat would change if temperatures increased anywhere between +1C and +7C. Extreme heat as defined here is uses the agronomic notion of "growing degree days", and essentially measures the amount of time an area is exposed to temperatures above a given threshold, set here to 29C. As discussed below, temperatures above 29C have been shown to be particularly harmful for agricultural productivity.
The left panel in each plot shows how much time corn grown in each county spends at different temperatures. The grey line is the average in current climate (the last three decades), and the grey shaded area is the amount of time spent above the magical 29C threshold. The colors show what happens as you increase average temperatures by +2C, +4C, or +6C. Particularly in the places that start out cooler (Clay County in Minnesota), small initial increases in temperature can lead to big increases in exposure to temperatures above 29C.
How much this exposure increases is quantified in the right panel of each plot. The dark line shows the increase in exposure to temperatures above 29C as average temperatures rise up to +7C. The dashed line shows the increase in exposure if every degree of warming was like the first degree of warming - a simple way of looking a looking at linearity. That these lines diverge shows the non-linearity, and as you can read off the right axis, small increases in average temperature mean very large percentage increase in exposure to extreme heat. A +1C warming means anywhere between a 43% (Georgia) and a 61% (Minnesota) increase in exposure to extreme heat under this definition.
2. Many outcomes we care about respond very negatively to extreme heat. This is a very active topic of research, and the other venerable posters on this blog are at the forefront of research on this topic, but let me briefly summarize some findings to date.
Effects of extreme heat on agricultural outcomes are now increasingly well documented. Here's a summary plot from Wolfram and Michael's 2009 PNAS piece, showing how the main US field crops respond to hot temperatures. For both corn and soy, things fall off pretty dramatically above about 29C.
Importantly, though, this is not showing a non-linear response to extreme heat. The paper shows that modeling the yield response to extreme heat linearly does just fine. But this response is still really negative: an extra day spent above 29C (relative to spending it at around 29C, where corn and soy are happy) reduces end of season yield by about 1%. This is huge effect.
David Lobell and colleagues tell a very similar story looking at maize in Africa: yields respond roughly linearly to extreme heat, and the effect size is almost exactly what Wolfram and Michael find in the US.
New findings outside agriculture are eerily similar. Sol shows in a 2010 PNAS paper that non-agricultural economic output in the Caribbean also drops off quickly above 29-30C, and a recent paper by Graff Zivin and Niedell shows a similar dropoff of US labor supply above 29C in industries particularly exposed to climate (construction, agriculture, mining, transportation, etc). Sol has a nice blog post on this.
Again, none of these studies appear to document non-linear responses to extreme heat. Instead, the take-home from all of them is that outcomes respond very negatively (if perhaps linearly) to increased exposure to extreme heat.
3. Outcomes do appear to respond non-linearly to increases in average temperature. This is just a logical extension of the above two points. If extreme heat increases nonlinearly with increases in average temperature, and outcomes respond strongly to changes in extreme heat, then these same outcomes will respond non-linearly to changes in average temperature.
The appropriate adjective to affix to "non-linear" in this setting ("mildly"? "highly"?) is perhaps in the eyes of the beholder. Here is a plot of how Wolfram and Michael's 2009 paper predicts that corn yields will respond to increasing amounts of average warming (I just grabbed the coefficients and standard errors from their appendix Table A5):
The dotted grey line again shows the impact trajectory if every additional degree of warming was like the first +1C of warming - the poor man's linearity. The actual predicted impacts are well below this line starting at about +2C, showing that the responsiveness of US corn yields to temperatures is [fill in preferred adjective] non-linear. I'd imagine the other findings in (2) above would look about the same.
So: increases in average temperature lead to non-linear increases in extreme heat, which will do bad things to outcomes we care about. There isn't a lot of evidence that these outcomes respond non-linearly to extreme heat, but it's not clear how much this matters - a strong negative linear response of these outcomes to extreme heat is enough to generate pretty negative impact projections under future warming.
A final word of caution is probably in order. These empirical studies do a good job of capturing responses to extremes that we've seen in the past. Unavoidably, they do less good of a job imagining how outcomes might respond to future extremes that we haven't yet experienced. It could be that outcomes indeed respond non-linearly to these changes in extremes, instead of responding non-linearly just to changes in averages. If so, this will probably only make a bad story worse.