Building a better damage function

Bob Kopp, Amir Jina, James Rising, and our partners at the Climate Impact Lab, Princeton, and RMS, have a new paper out today.  Our goal was to construct a climate damage function for the USA that is "micro-founded," in the sense that it is built up from causal relationships that are empirically measured using real-world data (if you're feeling skeptical, here's are two videos where Michael Greenstone and I explain why this matters).

Until now, the "damage function" has been a theoretical concept. The idea is that there should be some function that links global temperature changes to overall economic costs, and it was floated in the very earliest economic models of climate change, such as the original DICE model by Nordhaus where, in 1992, he described the idea while outlining his model:

from Nordhaus (1992)

The "extremely elusive" challenge was figuring out what this function should look like, e.g. what should theta_1 and theta_2 be? Should there something steeper than quadratic to capture really catastrophic outcomes? Many strong words have been shared between environmental economists at conferences about the shape and slope of this function, but essentially all discussions have been heuristic or theoretical.  We took a different approach, instead setting out to try and use the best available real world empirical results to figure out what the damage function looks like for the USA.  Here's what we did.

We started out by recognizing that a lot of work has already gone into modeling climate projections for the world, by dozens of teams of climate modelers around the world. So we took advantage of all those gazillions of processor-hours that have already been used and simply took all the CMIP5 models off the shelf, and systematically downscaled them to the county level.

Then we indexed each model against the global mean surface temperature change that it exhibits. Not all models agree on what warming will happen given a certain level of emissions. And even among models that exhibit the global mean temperature change, not all models agree on what will happen for specific locations in the US.  So it's important that we keep track of all possible experiences that the US might have in the future for each possible level of global mean temperature change.  Here's the array of actual warming patterns in maps, where each map is located on the horizontal axis based on the projected warming under RCP8.5 ("business as usual"). As you can see, the US may experience many different types of outcomes for any specific level of global mean warming.

Then, for each possible future warming scenario, we build a projection of what impacts will be in a whole bunch of sectors, using studies that meet a high empirical standard (pretty much the same one Marshall and I used in our conflict meta-analysis, plus a few other additional criteria). This was relying on a lot of previous work done by colleagues here, like Mike/Wolfram/David's crop results and Max's electricity results. We project impacts in agriculture, energy, mortality, labor and crime using empirical response functions. For energy, this got a little fancy because we hooked up the empirical model to NEMS and ran it as a process model. For coastal damages, we partnered with RMS and restructured their coastal cyclone model to take Bob's probabilistic SLR projections and Kerry Emanuel's cyclone projections as inputs---their model is pretty cool since it models thousands of coastal flood scenarios, and explicitly models damages for every building along the Atlantic coast. The energy and coastal models were each big lifts, using process models with empirical calibration, as were the reduced form impacts since we resampled daily weather and statistical uncertainty for each impact in each RCP in each climate model; this amounted to tracking 15 impacts across 3,143 counties across 29,000 possible states of the world for every day during 2000-2099. These are maps of the median scenarios for the different impacts:

Then, within each possible state of the world, we added up the costs across these different sectors to get a total cost. Doing this addition first is important because it accounts for any cross-sector correlations that might emerge in the future due to the spatial correlations in economic activity (across sectors) and their joint spatial correlation with the future climate anomaly (a bad realization for energy, ag, and mortality all might happen at the same time).  We then take these total costs and plot them against the global mean temperature change that was exhibited by the climate model that generated them. There ended up being 116 climate models that we could use, so there are only 116 different global temperature anomalies, but each model generated a whole distribution of possible outcomes due to weather and econometric uncertainty. Plotting these 116 distributions gives us a sense of the joint distribution between overall economic losses and global temperature changes: 

We can then just use normal statistics on these data to describe this joint distribution succinctly, getting out some equations that other folks can plug into their cost calculations or IAMS.  Below is the 5-95th intervals for the probability mass, as well as the median. To our knowledge, this is basically the first micro-founded damage function:

It turns out that Nordhaus was right about the functional form, it is quadratic. In the paper we try a bunch of other forms, but this thing is definitely quadratic. And if you are happy with the conditional average damage, we can get you the thetas: 

E[damage | T] = 0.283 x T + 0.146 x T^2

Now, of course, as we say several times in the paper, this function will change as we learn more about the different parts of the economy that the climate influences (for example, since we submitted the paper, we've learned that sleep is affected by climate). So for any new empirical study, as long as it meets our basic criteria, we can plug it in and crank out a new and updated damage function.

Beyond the damage function, there is one other finding which might interest the G-FEED crowd. First, because the South tends to be both hotter, it is disproportionally damaged by nonlinear climate impacts where high temperatures impose higher marginal damages (ag, mortality, energy & labor). Also, along the Gulf and southern Atlantic coast, coastal damages get large. The South also happens to be poorer than the North, which is impacted less heavily (or benefits on net, in many cases). This means that damages are negatively correlated with incomes, so the poor are hit hardest and the rich lose less (or gain). On net, this will increase current patterns of economic inequality (a point the press has emphasized heavily). Here are are whisker plots showing the distribution of total damage for each county, where counties are ordered by their rank in the current income distribution:

Note that nothing about this calculation takes into account the possibility that poor counties have fewer resources with which to cope, this is just about interaction of geography and the structure of the dose-response function.

This widening of inequality probably should matter for all sorts of reasons, including the possibility that it induces strong migration or social conflict (e.g. think about current rural-to-urban migration, the last election, or the Dust Bowl). But it also should matter for thinking about policy design and calculations of the social cost of carbon (SCC).  Pretty much all SCC calculations (e.g. DICE, FUND, PAGE) think about climate damages in welfare terms, but they compute damages for a representative agent that either represents the entire word, or enormous regions (e.g. the USA is one region in FUND). This made sense, since most of the models were primarily designed to think about the inter-temporal mitigation-as-investment problem, so collapsing the problem in the spatial dimension made it tractable in the inter-temporal dimension.  But it makes it really hard, or impossible, to resolve any inequality of damages among contemporary individuals within a region (in the case of FUND) or on the planet (in the case of DICE). Our analysis shows that there are highly unequal impacts within a single country, and this inequality of damages can be systematically incorporated into the damage function above, which as its shown is simply aggregate losses (treating national welfare as equal to average GDP only).  David Anthoff and others have thought about accounting for inequality between the representative agents of different FUND regions, and shown that it matters a lot.  But as far as I know, nobody has accounted for it within a country, and this seems to matter a lot too.

In the online appendix [Section K] (space is short at Science) we show how we can account for both inequality and risk, capturing both in a welfare-based damage function. Using our data and a welfare function that is additive in CRRA utilities, we compute inequality-neutral certainty-equivelent damage functions. These are the income losses that, if shared equally across the entire US population with certainty, would have the same welfare impact as the uncertain and unequal damages that we cover (i.e. shown in the dot-whisker plot above).  Two things to note about this concept. First, this adjustment could theoretically make damages appear smaller if climate changes were sufficiently progressive (i.e. hurting the wealthy and helping the poor). Second, there are two ways to compute this that are not equivelent; one could either compute the (i) inequality in risks borne by different counties or (ii) risks of inequality across counties. We chose to go with the first option, which involves first computing the certainty-equivelent damage for each county, then computing the inequality-neutral equivalent damage for that cross-sectional distribution of risk. (We thought it was a little too difficult for actual people to reasonably imagine all possible unequal states of the future world before integrating out the uncertainty.)

We compute these adjusted damages for a range of parameters that separately describe risk aversion and inequality aversion, since these are value judgements and we don't have strong priors on what the right number ought to be. Below is a graph of what happens to the damage function as you raise both these parameters above one (values of one just give you back the original damage function, which is the dashed line below). Each colored band is for a single inequality aversion value, where the top edge is for risk aversion = 8 and the lower edge is risk aversion = 2:

National inequality-neutral certainty-equivalent loss equal in value to direct damages under different assumptions regarding coefficients of inequality aversion and risk aversion. Shaded regions span results for risk aversion values between 2 and 8 (lower and upper bounds).  Dashed line is same as median curve above.

What we see is that adjustment for inequality starts to matter a lot pretty quickly, more so than risk aversion, but the two actually interact to create huge welfare losses as temperatures start to get high. For a sense of scale, note that in the original DICE model, Nordhaus defined "catastrophic outcomes" as possible events that might lower incomes by 20%.

Bob, David Anthoff and I have debated a bit what the right values for these parameters are, and I'll be the first to say I don't know what they should be. There are several estimates out there, but I think we really don't talk about inequality aversion much so there's not a ton to draw on. But, just like the discount rate (which has received a lot of attention/thought/debate), these ethical parameters have a huge influence on how we think about these damages. And looking at this figure, my guess is that inequality aversion may be just as influential on the SCC as the discount rate---especially once we start having global estimates with this kind of spatial resolution.  I think this is one of the most important directions for research to go: figuring out how we are supposed to value the inequality caused by climate change and accounting for it appropriately in the SCC. 

Drink more coffee and run faster! Get fit in only 9 seconds! A rant on NYTimes exercise coverage.

As a fairly half-assed exerciser, I am always on the lookout for quick/easy/delicious ways to get in shape without having to really do anything.  Thus a lot of the exercise articles on NYTimes's "Well" blog are irresistible clickbait for me -- as they are for the presumably millions of other NYT-reading half-assed exercisers who ensure that these articles consistently top of the NYT most-read lists.  Drink more caffeine and run faster!  Take a hot bath and run faster!  Get all the exercise you need in only seven minutes!  Scratch that -- only four!!  Scratch that -- only one!!!

image stolen from NYT article

Here's one from this week:  Hot weather workout?  Try a hot bath beforehand.  Triple clickbait for me, as this week is really hot in California, I was hoping to get in some runs, and I certainly enjoy hot baths (+long walks on the beach, white wine, etc).  Original study here (can't tell if that's paywalled), published in the Journal of Strength and Conditioning Research.  This study takes n=9 (!) people, 8 dudes and 1 woman, and first has them run 5k on a treadmill in a lab where they cranked the temperature to 90F.  Then all subjects underwent about a week of heat acclimation, where they pedaled a stationary bike in the hot lab for five 90-minute sessions.  Then, finally, they ran the 5k again in the hot lab. Low and behold, times had improved!  The now-heat acclimated participants shaved an average of about a minute and a half off their 5k times (~ 6% reduction) when running in hot conditions for the second time.

But wait a sec.  You took some mildly fit runners (5k time of 25min is not exactly East African pace), had them put in almost 8 hours of training, and then tested to see whether they got faster??  As a lazy, mildly-fit runner myself, 8 hours of training constitutes a good month for me, so I would be PISSED if I didn't get faster with that much training.  Now, you might say, they were pedaling a bike and not running, and this is indeed what the paper tries to argue in some hard-to-understand phrasing ("Cycling training controlled for performance that could arise from increased training volume were participants to acclimate through running").  But to me it seems pretty unlikely that a lot of biking is going to give you no running benefit at all, and some quick googling found about 1000 blog posts by runner/bikers that claim that it does (and also 1000 that claim that is doesn't, go figure).  In any case, clearly this is not anywhere close to the optimal research design you'd want for figuring out the causal effect of training-when-hot on performing-when-hot.   Strangely, the "hot bath" part does not appear in the paper at all, but just in an off-hand comment by a research quoted by the NYT.  So that's weird too.

Or take the one-minute-of-exercise-is-all-you-need study (original study; NYT clickbait).   This study takes n=25 "sedentary" men and divides them into three groups:  a control group (n=6), a group that does moderate-intensity stationary-bike pedaling for 45min at a time (n=10), and a group that does high-intensity ("all out") stationary-bike pedaling for 3x 20 seconds (n=9).  Only 60 total seconds!  This happens three times a week for 12 weeks, after which researchers compare various measures of fitness, including how much oxygen you can take up at your peak (i.e. VO2-max).  Both the moderate (VO2 = 3.2) and intense groups (VO2=3.0) had improved significantly upon the control group (VO2 = 2.5), but the post-training VO2max levels in the moderate and intense groups were not statistically different from each other.  Hence the paper's exciting title:  "Twelve Weeks of Sprint Interval Training Improves Indices of Cardiometabolic Health Similar to Traditional Endurance Training despite a Five-Fold Lower Exercise Volume and Time Commitment", and the NYT clickbait translation:  "1 Minute of All-Out Exercise May Have Benefits of 45 Minutes of Moderate Exertion".

But wait a sec.  Sample size is n=19 in these treatment groups combined!  A quick calculation suggests that, at this sample size, the study is DRAMATICALLY underpowered to find an effect. That is, the study has a high chance of failing to find an effect when there actually is one.  I calculate that to detect a significant difference in means of 0.2 between these two groups (on a control group standard deviation of 0.7, given in the table), the study would have needed 388 participants, or about 20x what they had! (This assumes the researchers would want an 80% chance of correctly rejecting the null that the two groups had the same mean;  in Stata, if I did it right:   power twomeans 3.2 3.0, sd(0.7)).  Even reliably detecting a 20% increase in VO2 max between the two treated groups would have needed 46 participants, more than twice the number they had.  Put more simply:  with sample sizes this small, you are very unlikely to find significant differences between two groups, even when differences actually exist.  So maybe the moderate exercise group actually did do better, or maybe they didn't, but either way this study can't tell us.  The same thing appears to be true with the 4-min-exercise paper (n=26) -- it's way underpowered.  And I haven't looked systematically, but my guess is this is true of a lot of the studies they cover that find no effect.  Andrew Gelman is always grumping about studies with large effects, but we should probably be just as cautious believing small-N studies that find no effect.

So should the NYT stop covering these underpowered or poorly designed studies?  There's not a lot at stake here I guess, so one reasonable reaction is, "who gives a sh*t"?  But surely this sort of coverage  crowds out coverage of other higher-quality science, such as why your roller-bag wheels wobble so much when you're running to catch a flight, or why eggs are egg-shaped.  And cutting down on this crappy exercise coverage will save me the roughly 20min/week of self-loathing I feel when I click on yet another too-good-to-be-true NYT exercise article, look up the article it actually references, and find my self not very convinced.  Twenty minutes I could have spent exercising!!  That's like 20 1-minute workouts...

Trump's climate gift to Russia

Trump's recent announcement that the US would withdraw from the Paris Accords was hailed as a monumental political, environmental, and economic mistake.  But given all the theater surrounding the announcement, others also saw it as an effort to distract the public from the ongoing investigation of the Administration's ties to Russia.

It's hard to see how this latter claim could actually be evaluated.  But it got me thinking:  what are the benefits to Russia of the US withdrawing from the Paris accords?  Was the US withdrawal a climate gift to Russia?

Now, I'm guessing Trump has not read our paper showing that warming temperatures will have unequal economic effects around the world (unlike Obama, to repeat my shameless self promotion from last week).  In that paper, and consistent with a huge microeconomic literature, we see clear evidence in the historical data that cold high-latitude countries tend to experience higher GDP growth when temperatures warm, with the reverse being true in most of the rest of the world where average temperatures are already warmer (the US included).  Basically, if you're currently cold, you do better when it warms up;  if you're already warm, additional warming hurts you.  Pretty intuitive, and also shows up very clearly in the half century of data we have on economic growth from around the world.

Here's the original plot from our paper, with the figure on the left showing the historical relationship between temperature and GDP growth for all countries in the world.  If you're average temperature is below about 13C, historically your economy grows faster when annual temperatures warm.  If your at or above 13C, growth slows as temperatures warm.  The US has a population-weighted annual average temperature of just over 13C.  Russia has a population-weighted average temperature of just under 5C.  Russia is cold!

Figure 2 from Burke, Hsiang, Miguel 2015 Nature. Effect of annual average temperature on economic production. a, Global non-linear relationship between annual average temperature and change in log gross domestic product (GDP) per capita (thick black line, relative to optimum) during 1960–2010 with 90% confidence interval (blue, clustered by country, N= 6,584). Model includes country fixed effects, flexible trends, and precipitation controls. Vertical lines indicate average temperature for selected countries. Histograms show global distribution of temperature exposure (red), population (grey), and income (black). b, Comparing rich (above median, red) and poor (below median, blue) countries. Blue shaded region is 90% confidence interval for poor countries. Histograms show distribution of country–year observations. c, Same as b but for early (1960– 1989) and late (1990–2010) subsamples (all countries). d, Same as b but for agricultural income. e, Same as b but for non-agricultural income.

Last week we calculated the potential harm done to the economy of withdrawing from Paris.  The idea was this:  withdrawing from the Paris accords would make global temperatures rise relative to what they would have been if the US had met its Paris obligations (an additional +0.3C by 2100, according to these guys).  For reasons already stated, warming temperatures are bad for overall economic output in the US. So we can then calculate, what's the difference in output between now and 2100 that would occur in a withdrawal versus a non-withdrawal world?  For the US, the effects were pretty big:  I calculated that, in present value (i.e. discounting future losses at 3%), the US economy would lose about $8 trillion between now and 2100 due to the extra temperature increase induced by withdrawing from Paris.

What about Russia?  Again, Russia is cold, so extra warming is likely to help the Russian economy, all else equal.  You can actually see this really clearly in the Russian data.  Below is the plot of Russian GDP growth rates versus Russian temperatures, using data 1990-2010 (1990 being the first post-Soviet-collapse year that "Russia" shows up in the national accounts data).  Specifically, these are growth deviations from trend versus temperature deviations from trend;  we are detrending the data since you don't want to conflate trends in temperature that could be correlated with other trending factors that also affect growth.

This is just 20 data points, but the estimated effects are HUGE.  Basically, Russian GDP growth is multiple percentage points higher when temperatures warm by a degree C.  And the Russia-specific estimate is even higher than what we would predict the effect would be for Russia using the global response function pictured in blue above.

Anyway...  Basically what I did is to re-do the same calculation we did last week for the US, but now focusing on effects on the Russian economy and calculating what happens to Russian GDP in the scenario where the US withdraws from Paris versus the scenario where the US stays in.  To be conservative, I use estimates from the global response function, not the Russia-specific mega-response just noted.

Here's the main finding:  Trump's decision to withdraw the US from Paris is a $2.2 trillion dollar gift to Russia (paid out over the next 85 years).  Below is the figure showing what happens to Russian GDP under a withdrawal versus a no-withdrawal scenario (left), and the annual gains in GDP in each year (to 2100).  By 2100, Russia is ~10% richer than it would have been otherwise, and the (discounted) sum of these GDP gains is about $2.2 trillion dollars.

Given that there's no evidence that Trump has read our paper, I don't think we can claim that this climatic gift to Russia was purposeful.  But it's sadly ironic that an announcement that might have been meant to distract us from Russian meddling was simultaneously a monumental economic gift to that country.

The cost of Paris withdrawal

Lots of discussion today about the potential ramifications of the US withdrawing from the Paris Accords.  Folks have already done some nice calculations looking at the climate consequences of US withdrawal, but there's a lot of interest in the potential economic consequences and I hadn't seen anyone take a heroic stab at that yet.  So.....

The clearest picture I (read: google) could find on the climate implications was this nice website here from ClimateInteractive.org, where they find that a Paris-minus-US world and a Paris-including US world is the difference between +3.6C of warming and +3.3C of warming by 2100.  There are clearly a lot of assumptions that go into this calculation (e.g., what the hell happens after 2030 when the INDCs run out, what happens if Trump's successor (Kamala Harris?  Zuck?  Steph Curry?) re-signs us up, etc etc), but let's take this calculation as God's truth.  Withdrawal gives the world +0.3C of additional warming.

So I wanted to figure out: what is the cost to the US in terms of additional damages that are wrought by this extra warming that withdrawal would bring about?  A difference of +0.3C might not sound like much, but we've got this paper (cited by Obama, so it must be right) that suggests that changes in temperature can affect the growth rate of GDP in rich and poor countries alike.  So the current administration might be right that meeting the US's Paris commitments would have economic costs, but these need to be weight against the benefits of the reduced warming that we would get.  So what are those benefits?

So I took the basic setup we had in that paper, and ran the world forward to 2100 under +3.3C warming versus +3.6C warming, and I looked at what our results in that paper said would happen to US GDP in those different worlds.  See here for more info on how we do these sorts of calculations in this framework.  Basically, we have a function that tells us how growth rates change as temperatures change, derived from historical data.  Then we walk countries along this function as you crank up the temperature to your desired level. To get GDP in levels, you apply these changes in the growth rate to some baseline growth scenario, which we take off-the-shelf from the Shared Socioeconomic Pathways (SSPs, see here).  We also need population numbers, and take those from the SSPs as well.

Below is what I get when I run the US forward under +3.3C warming versus +3.6C warming.  Under SSP5 (the baseline scenario we use), the US clips along at an average per capita growth rate of above 2%/year.  Once you crank up the temperature by 3.5C or so, historical data tells us that we should shave between about 0.5-1 percentage point off of annual US growth.  So this means by 2100, instead of growing at 2%/year under a no warming scenario, the US would be growing at less than 1.5%/year in this much warmer world.  The effects earlier in the century are smaller, of course.

Our comparison is between +3.3C and +3.6C, and the effects on the growth rate are of course smaller.  But even small effects on the growth rate can add up to big cumulative effects on GDP over time.  The left plot below compares total US GDP in the "no withdrawal" world versus the "withdrawal" world, and the right plot gives you the amount of GDP that's lost in each year from withdrawing.  To be crystal clear, we're again only thinking about the differences brought about by the change in temperature between the two scenarios -- we're not thinking about what it would cost the US to meet its Paris commitments.

If my calculations are right, the numbers are large.  By 2100, withdrawing from Paris makes us (i.e. people in the US) about 5% poorer than we would have been otherwise.  The cumulative US GDP losses over time from withdrawal, discounted back to 2010 at 3%, are also impressive - I calculate them to be $8.2 trillion dollars (right plot above).  That is, withdrawing from the Paris agreement costs the US economy $8.2 trillion dollars in present discounted value.  That is a large pile of money.  Even if I'm off by a factor of 5, we're still talking low trillions.  

And to be clear, my calculations do not take into account many other near-term benefits of reducing our own emissions, such as the 'co-benefits' of better health outcomes from cleaner air.  These could be quite big as well.

The key policy question for the Trump administration is:  do we think the costs of meeting our obligations under Paris are going to run more than $8 trillion?  Put another way, are they going to amount to almost half of current US GDP?

To generate $8 trillion in costs between now and 2030 (after discounting at 3%), annual costs would have to be somewhere around $750 billion.  The compliance cost estimates for the Clean Power Plan that I've seen are about two orders of magnitude smaller than that, so even if the CPP only got us 10% of the way to our Paris commitment (which is very conservative), these costs do not come close to the overall benefits -- even if other reductions are many times as expensive as the CPP.  (Hopefully somebody can correct me if I'm way off on these cost numbers -- definitely not my specialty).

With benefits this big, withdrawal seems like bad policy.