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Global non-linear effect of temperature on economic production

Authors:

Abstract

Growing evidence demonstrates that climatic conditions can have a profound impact on the functioning of modern human societies, but effects on economic activity appear inconsistent. Fundamental productive elements of modern economies, such as workers and crops, exhibit highly non-linear responses to local temperature even in wealthy countries. In contrast, aggregate macroeconomic productivity of entire wealthy countries is reported not to respond to temperature, while poor countries respond only linearly. Resolving this conflict between micro and macro observations is critical to understanding the role of wealth in coupled human-natural systems and to anticipating the global impact of climate change. Here we unify these seemingly contradictory results by accounting for non-linearity at the macro scale. We show that overall economic productivity is non-linear in temperature for all countries, with productivity peaking at an annual average temperature of 13 °C and declining strongly at higher temperatures. The relationship is globally generalizable, unchanged since 1960, and apparent for agricultural and non-agricultural activity in both rich and poor countries. These results provide the first evidence that economic activity in all regions is coupled to the global climate and establish a new empirical foundation for modelling economic loss in response to climate change, with important implications. If future adaptation mimics past adaptation, unmitigated warming is expected to reshape the global economy by reducing average global incomes roughly 23% by 2100 and widening global income inequality, relative to scenarios without climate change. In contrast to prior estimates, expected global losses are approximately linear in global mean temperature, with median losses many times larger than leading models indicate.
LETTER doi:10.1038/nature15725
Global non-linear effect of temperature
on economic production
Marshall Burke
1,2
*, Solomon M. Hsiang
3,4
*& Edward Miguel
4,5
Growing evidence demonstrates that climatic conditions can
have a profound impact on the functioning of modern human
societies
1,2
, but effects on economic activity appear inconsistent.
Fundamental productive elements of modern economies, such as
workers and crops, exhibit highly non-linear responses to local
temperature even in wealthy countries
3,4
. In contrast, aggregate
macroeconomic productivity of entire wealthy countries is
reported not to respond to temperature
5
, while poor countries
respond only linearly
5,6
. Resolving this conflict between micro
and macro observations is critical to understanding the role of
wealth in coupled human–natural systems
7,8
and to anticipating
the global impact of climate change
9,10
. Here we unify these see-
mingly contradictory results by accounting for non-linearity at the
macro scale. We show that overall economic productivity is non-
linear in temperature for all countries, with productivity peaking
at an annual average temperature of 13 6C and declining strongly
at higher temperatures. The relationship is globally generalizable,
unchanged since 1960, and apparent for agricultural and non-agri-
cultural activity in both rich and poor countries. These results
provide the first evidence that economic activity in all regions is
coupled to the global climate and establish a new empirical founda-
tion for modelling economic loss in response to climate change
11,12
,
with important implications. If future adaptation mimics past
adaptation, unmitigated warming is expected to reshape the global
economy by reducing average global incomes roughly 23% by 2100
and widening global income inequality, relative to scenarios with-
out climate change. In contrast to prior estimates, expected global
losses are approximately linear in global mean temperature, with
median losses many times larger than leading models indicate.
Economic productivity—the efficiency with which societies trans-
form labour, capital, energy, and other natural resources into new
goods or services—is a key outcome in any society because it has a
direct impact on individual wellbeing. While it is well known that
temperature affects the dynamics of virtually all chemical, biological
and ecological processes, how temperature effects recombine and ag-
gregate within complex human societies to affect overall economic
productivity remains poorly understood. Characterizing this influence
remains a fundamental problem both in the emerging field of coupled
human–natural systems and in economics more broadly, as it has
implications for our understanding of historical patterns of human
development and for how the future economy might respond to a
changing climate.
Prior analyses have identified how specific components of economic
production, such as crop yields, respond to temperature using high-
frequency micro-level data
3,4
. Meanwhile, macro-level analyses have
documented strong correlations between total economic output and
temperature over time
5,6
and across space
13,14
, but it is unknown
whether these results are connected, and if so, how. In particular,
strong responses of output to temperature observed in micro data from
wealthy countries are not apparent in existing macro studies
5
. If
wealthy populations actually are unaffected by temperature, this
could indicate that wealth and human-made capital are substitutes
for natural capital (for example, the composition of the atmosphere)
in economic activity
5,7
. Resolving this apparent discrepancy thus
has central implications for understanding the nature of sustainable
development
7
.
Numerous basic productive components of an economy display a
highly non-linear relationship with daily or hourly temperature
1
. For
example, labour supply
4
, labour productivity
6
, and crop yields
3
all
decline abruptly beyond temperature thresholds located between
20 uC and 30 uC (Fig. 1a–c). However, it is unclear how these abrupt
Mass mi1
Annual average temperature
Slope bi1
Slope bi1
Mass mi2
Slope = mi1bi1 + mi2bi2
Years have
different daily
temperature
distributions
Daily temperature
Daily impact
Annual impact
d
f
e
Application of equation (1)
Change in ln annual
maize yield (×100)
0 10 20 30 40
Temperature during 24 h (°C)
–7.5
–5
–2.5
0
2.5
Change in labour
supplied (min)
Change in labour
supplied (min)
Daily maximum temperature (°C)
403020100
40
0
–40
–80
Change in labour
performance (%)
Wet bulb globe temperature (°C)
20 30
–0.8
–0.6
–0.4
–0.2
0
‘Performance
decrement’
0
–15
–10
–5
c
a
b
Figure 1
|
Highly non-linear micro responses generate smooth and shifted
macro response. ac, Highly non-linear micro-level responses of labour
supply
4
(a), labour performance
6
(b) and crop yield
3
(c) to daily temperature
exposure exhibit similar ‘kinked’ structures between 20 and 30uC. d,e, These
micro-level responses (f
i
(T) in equation (1); d) map onto country-level
distributions of temperatures across different locations and times within that
country (gi(T{T)in equation (1); e). Shifts in country-level distributions
correspond to changes in average annual temperature, altering the fraction of
unit-hours (m
i1
and m
i2
) exposed to different regions of the micro-level
response in d.f, Aggregating daily impacts according to equation (1) maps
annual average temperature to annual output as a non-linear and concave
function that is smootherthan the micro response with a lower optimum (Y(T)
in equation (1)).
00 MONTH 2015 | VOL 000 | NATURE | 1
*These authors contributed equally to this work.
1
Department of Earth System Science, Stanford University, California 94305, USA.
2
Center on Food Security and the Environment, Stanford University, California 94305, USA.
3
Goldman School of Public
Policy, University of California, Berkeley, California 94720, USA.
4
National Bureau of Economic Research.
5
Department of Economics, University of California, Berkeley, California, 94720, USA.
G2015 Macmillan Publishers Limited. All rights reserved
declines at the micro level are reflected in coarser macro-level data.
When production is integrated over large regions (for example, coun-
tries) or long units of time (for example, years), there is a broad
distribution of momentary temperatures to which individual compo-
nents of the economy (for example, crops or workers) are exposed. If
only the hottest locations ormoments cause abruptdeclines in output,
then when combined with many cooler and highly productive
moments they would sum to an aggregate level of output that only
declines modestly when aggregate average temperature increases.
To fix ideas, let function f
i
(T) describe the productive contribution
of an individual productive unit in industry i(for example, a firm)
relative to instantaneous (for example, daily) temperature T(Fig. 1d).
For a given country, period, and industry, denote the fraction of unit-
hours spent below the critical temperature threshold as m
i1
and the
fraction above as m
i2
(Fig. 1e). The full distribution of unit-hours
across all temperatures is gi(T{T), centred at average temperature
T. Assume g
i
(.) is mean zero. If productivity loss within a single pro-
ductive unit-hour has limited impact on other units, as suggested by
earlier findings
8,15
, then aggregate production Yis the sum of output
across industries, each integrated over all productive unit-hours in the
country and period:
Y(T)~X
i
Yi(T)~X
ið
?
{?
fi(T):gi(T{T)dT ð1Þ
As Trises and a country warms on average, m
i2
increases gradually for
all productive units (Fig. 1e). This growing number of hours beyond
the temperature threshold imposes gradual but increasing losses on
total output Y(T):
Equation (1) predicts that Y(T)is a smooth concave function
(Fig. 1f) with a derivative that is the average derivative of f
i
(T)
weighted by the number of unit-hours in each industry at each daily
temperature. It also predicts that Y(T)peaks at a temperature lower
than the threshold value in f
i
(T), if the slope of f
i
(T) above the thresh-
old is steeper than minus the slope below the threshold, as suggested by
micro-scale evidence. These predictions differ fundamentally from
notions that macro responses should closely mirror highly non-linear
micro responses
6,16
. Importantly, while aggregate productivity losses
ought to occur contemporaneous with temperature changes, these
changes might also influence the long-run trajectory of an economy’s
output
5,15
. This could occur, for example, if temporary contempor-
aneous losses alter the rate of investment in new productive units,
thereby altering future production. See Supplementary Equations
1–14 for details.
We test these predictions using data on economic production
17
for
166 countries over the period 1960–2010. In an ideal experiment, we
would compare two identical countries, warm the temperature of one
and compare its economic output to the other. In practice, we can
approximate this experiment by comparing a country to itself in years
when it is exposed to warmer- versus cooler-than-average tempera-
tures
18
due to naturally occurring stochastic atmospheric changes.
Heuristically, an economy observed during a cool year is the ‘control’
for that same society observed during a warmer ‘treatment’ year. We
do not compare output across different countries because such com-
parisons are probably confounded, distinguishing our approach from
cross-sectional studies that attribute differences across countries to
their temperatures
13
.
We estimate how economic production changes relative to the pre-
vious year—that is, annual economic growth—to purge the data of
secular factors in each economy that evolve gradually
5
. We deconvolve
economic growth to account for: (1) all constant differences between
countries, for example, culture or history; (2) all common contempor-
aneous shocks, for example, global price changes or technological
innovations; (3) country-specific quadratic trends in growth rates,
which may arise, for example, from changing political institutions or
economic policies; and (4) the possibly non-linear effects of annual
average temperature and rainfall. This approach is more reliable than
only adjusting for observed variables because it accounts for unob-
served time-invariant and time-trending covariates, allows these cov-
ariates to influence different countries in different ways, and
outperforms alternative models along numerous dimensions
15
(see
Supplementary Information). In essence, we analyse whether coun-
try-specific deviations from growth trends are non-linearly related to
country-specific deviations from temperature and precipitation
trends, after accounting for any shocks common to all countries.
We find country-level economic production is smooth, non-linear,
and concave in temperature (Fig. 2a), with a maximum at 13 uC, well
below the threshold values recovered in micro-level analyses and con-
sistent with predictions from equation (1). Cold-country productivity
increases as annual temperature increases, until the optimum.
Productivity declines gradually with further warming, and this decline
accelerates at higher temperatures (Extended Data Fig. 1a–g). This
0 5 10 15 20 25 30
–0.2
–0.1
0
–0.2
–0.1
0
0.1
5 10 15 20 25 30
–0.2
–0.1
0
0.1
51015202530
Global distribution of temperature observations
Global distribution of population
Global distribution of GDP
Annual average temperature (°C)
Annual average
temperature (°C)
Annual average
temperature (°C)
Change in ln(GDP per capita)
Change in ln(GDP per capita)Change in ln(GDP per capita)
ab
Rich
Poor
Rich-country temperatures
Poor-country temperatures
1960–1989 temperatures
1990–2010 temperatures
1960–1989
1990–2010
c
ed
Rich
Poor Rich
Poor
Agricultural GDP Non-agricultural GDP
US
China
Germany
Japan India
Nigeria
Indonesia
Brazil
France
UK
Figure 2
|
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, N56,584). Model includes country fixed effects, flexible
trends, and precipitation controls (see Supplementary Methods). Vertical
lines indicate average temperature for selected countries, although averages
are not used in estimation. Histograms show globaldistribution 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 bbut for early (1960–
1989) and late (1990–2010) subsamples (all countries). d, Same as bbut for
agricultural income. e, Same as bbut for non-agricultural income.
2|NATURE|VOL000|00MONTH2015
RESEARCH LETTER
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result is globally representative and not driven by outliers (Extended
Data Fig. 1h). It is robust to estimation procedures that allow the
response of countries to change as they become richer (Extended
Data Fig. 1i and Supplementary Table 1), use higher-order polyno-
mials or restricted cubic splines to model temperature effects
(Extended Data Fig. 1j–k), exclude countries with few observations,
exclude major oil producers, exclude China and the United States,
account for continent-specific annual economic shocks
19
, weaken
assumptions about trends in growth, account for multiple lags of
growth, and use alternative economic data sources
20
(Extended
Data Table 1).
Accounting for delayed effects of temperature, which might be
important if countries ‘catch up’ after temporary losses, increases stat-
istical uncertainty but does not alter the net negative average effect of
hot temperatures (Extended Data Fig. 2a–c). This ‘no catch up’ beha-
viour is consistent with the observed response to other climatological
disturbances, such as tropical cyclones
15
.
While much of global economic production is clustered near the
estimated temperature optimum (Fig. 2a, black histogram), both rich
and poor countries exhibit similar non-linear responses to temper-
ature (Fig. 2b). Poor tropical countries exhibit larger responses mainly
because they are hotter on average, not because they are poorer
(Extended Data Fig. 1i and Supplementary Table 1). There is suggest-
ive evidence that rich countries might be somewhat less affected by
temperature, as previously hypothesized
5
, but their response is statist-
ically indistinguishable from poor countries at all temperatures
(Extended Data Fig. 2d–f and Extended Data Table 2). Although the
estimated total effect of high temperatures on rich countries is sub-
stantially less certain because there are few hot, rich countries in the
sample, the non-linearity of the rich-country response alone is statist-
ically significant (P,0.1; Extended Data Table 2), and we estimate an
80% likelihood that the marginal effect of warming is negative at high
temperatures in these countries (Extended Data Fig. 2m). Our finding
that rich countries respond non-linearly to temperature is consistent
with recent county-level results in the United States
8
.
Our non-linear results are also consistent with the prior finding of
no linear correlation between temperature and growth in rich coun-
tries
5
. Because the distribution of rich-country temperatures is roughly
symmetrical about the optimum, linear regression recovers no asso-
ciation. Accounting for non-linearity reconciles this earlier result
(Extended Data Fig. 3a and Supplementary Table 3) but reverses
how wealth and technology are understood to mediate economic res-
ponses to temperature.
We do not find that technological advances or the accumulation of
wealth and experience since 1960 has fundamentally altered the rela-
tionship between productivity and temperature. Results using data
from 1960–1989 and 1990–2010 are nearly identical (Fig. 2c). In agree-
ment with recent micro-level evidence
8,21
, substantial observed warm-
ing over the period apparently did not induce notable adaptation.
Consistent with micro-level findings that both agricultural and non-
agricultural labour-related productivity are highly non-linear in instant-
aneous temperature
3,4,6
, we find agricultural and non-agricultural
aggregate production are non-linear in average annual temperature
for both rich and poor countries (Fig. 2d, e and Extended Data Fig.
2g–l). Low temperature has no significant effect on these subsamples,
althoughlimited poor-country exposure to these temperatures severely
limits statistical precision. High temperatures have significant negative
effects in all cases for poor countries, and significant or marginally
significant effects for rich countries (Extended Data Fig. 2p–u).
A global non-linear response of economic production to annual
temperature has important implications for the likely economic
impact of climate change. We find only weak suggestive evidence that
richer populations are less vulnerable to warming, and no evidence
that experience with high temperatures or technological advances
since 1960 have altered the global response to temperature. This
suggests that adaptation to climatic change may be more difficult than
previously believed
9,10
, and that the accumulation of wealth, techno-
logy and experience might not substantially mitigate global economic
losses during this century
8,21
.
We quantify the potential impact of warming on national and global
incomes by combining our estimated non-linear response function
with ‘business as usual’ scenarios (Representative Concentration
Pathway (RCP)8.5) of future warming and different assumptions
regarding future baseline economic and population growth
22
(see
Supplementary Information). This approach assumes future econom-
ies respond to temperature changes similarly to today’s economies—
perhaps a reasonable assumption given the observed lack of adaptation
during our 50-year sample.
In 2100, we estimate that unmitigated climate change will make 77%
of countries poorer in per capita terms than they would be without
climate change. Climate change may make some countries poorer in
the future than they are today, depending on what secular growth
rates are assumed. With high baseline growth and unmitigated
climate change (RCP8.5 and Shared Socio-economic Pathway
(SSP)5; see Supplementary Information), we project that 5% of coun-
tries are poorer in 2100 than today (Fig. 3a), while with low growth,
43% are (SSP3; Fig. 3b).
Differences in the projected impact of warming are mainly a func-
tion of countries’ baseline temperatures, since warming raises produc-
tivity in cool countries (Fig. 4). In particular, Europe could benefit
from increased average temperatures. Because warming harms pro-
ductivity in countries with high average temperatures, incomes in poor
regions are projected to fall relative to a world without climate change
with high confidence (P,0.01), regardless of the statistical approach
used. Models allowing for delayed effects project more negative
impacts in colder wealthy regions; projections assuming rich and poor
countries respond differently (Fig. 2b) are more uncertain because
fewer data are used to estimate each response (Extended Data Fig. 4).
0.1
0.2
0.5
1
2
5
10
20
50
100
200
500
0.1
0.2
0.5
1
2
5
10
20
50
100
200
500
Incomes
today
Incomes in 2100,
no climate change
Incomes in 2100,
with climate change
GDP per capita (US$1,000)
GDP per capita (US$1,000)
aSSP5 SSP3
Incomes
today
Incomes in 2100,
no climate change
Incomes in 2100,
with climate change
0.1
0.2
0.5
1
2
5
10
20
50
100
200
500
b
2010 2050 210020502100
YearYear
2010 2050 210020502100
YearYear
Figure 3
|
Country-level income projections with and without temperature
effects of climate change. a,b, Projections to 2100 for two socioeconomic
scenarios
22
consistent with RCP8.5 ‘business as usual’ climate change: a, SSP5
assumes high baseline growth and fast income convergence; b, SSP3 assumes
low baseline growth and slow convergence. Centre in each panel is 2010,
each line is a projection of national income. Right (grey) are incomes under
baseline SSP assumptions, left (red) are incomes accounting for non-linear
effects of projected warming.
00 MONTH 2015 | VOL 000 | NATURE | 3
LETTER RESEARCH
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The impact of warming on global economic production is a popu-
lation-weighted average of country-level impacts in Fig. 4a. Using our
benchmark model (Fig. 2a), climate change reduces projected global
output by 23% in 2100 (best estimate, SSP5) relative to a world without
climate change, although statistical uncertainty allows for positive
impacts with probability 0.29 (Fig. 5a and Extended Data Table 3).
Estimates vary in magnitude, but not in structure, depending on the
statistical approach (Fig. 5b and Extended Data Table 3). Models with
delayed impacts project larger losses because cold countries gain less,
while differentiated rich–poor models have smaller losses (statistical
uncertainty allows positive outcomes with probability 0.09–0.40).
Models allowing both delayed impacts and differentiated rich–poor
responses (the most flexible approach) project global losses 2.2 times
larger than our benchmark approach. In all cases, the likelihood of
large global losses is substantial: global losses exceed 20% of income
with probability 0.44–0.87 (Extended Data Table 3 and Extended
Data Fig. 5).
Accounting for the global non-linear effect of temperature is crucial
to constructing income projections under climate change because
countries are expected tobecome both warmer and richer in the future.
In a previous analysis in which a linear relationship was assumed and
no significant linear effect was observed in rich countries
5
, it was
hypothesized that countries adapted effectively to temperature as they
became wealthier. Under this hypothesis, the impacts of future warm-
ing should lessen over time as countries become richer. In contrast,
when we account for the non-linear effect of temperature historically,
we find that rich and poor countries behave similarly at similar tem-
peratures, offering little evidence of adaptation. This indicates that we
cannot assume rich countries will be unaffected by future warming,
nor can we assume that the impacts of future warming will attenuate
over time as countries become wealthier. Rather, the impact of addi-
tional warming worsens over time as countries becomes warmer. As a
result, projections using linear and non-linear approaches diverge
substantially—by roughly 50–200% in 2100 (Extended Data Fig. 3c,
d)—highlighting the importance of accounting for this non-linearity
when assessing the impacts of future warming.
Strong negative correlation between baseline income and baseline
temperature indicates that warming may amplify global inequality
because hot, poor countries will probably suffer the largest reduction
in growth (Fig. 5c). In our benchmark estimate, average income in the
poorest 40% of countries declines 75% by 2100 relative to a world
without climate change, while the richest 20% experience slight gains,
since they are generally cooler. Models with delayed impacts do not
project as dramatic differences because colder countries also suffer
large losses (Extended Data Fig. 5).
We use our results to construct an empirical ‘damage function’ that
maps global temperature change to global economic loss by aggreg-
ating country-level projections. Damage functions are widely used in
economic models of global warming, but previously relied on theory
for structure and rough estimates for calibration
11,12
. Using our empir-
ical results, we project changes to global output in 2100 for different
temperature changes (Fig. 5d; see Supplementary Information) and
compare these to previously estimated damage functions
12
.
Commonly used functions are within our estimated uncertainty, but
differ in two important respects.
Europe North America Central and East Asia
Oceania Latin America Middle East/North Africa
Southeast Asia Sub-Saharan Africa South Asia
–100
–75
–50
–25
0
25
50
75
2020 2040 2060 2080 2100 2020 2040 2060 2080 2100
Year
2020 2040 2060 2080 2100
–100
–75
–50
–25
0
25
50
75
Percentage change in GDP per capita
–100
–75
–50
–25
0
25
50
75
–100 –50 0 50 100
Percentage change in
GDP per capita
a
b
Figure 4
|
Projected effect of temperature changes on regional economies.
a,b, Change in GDP per capita (RCP8.5, SSP5) relative to projection using
constant 1980–2010 average temperatures. a, Country-level estimates in 2100.
b, Effects over time for nine regions. Black lines are projections using point
estimates. Red shaded area is 95% confidence interval, colour saturation
indicates estimated likelihood an income trajectory passes through a value
27
.
Base maps by ESRI.
–75
–50
–25
0
25
50
2020 2040 2060 2080 2100
Year
Percentage change in
GDP per capita
Percentage change in
GDP per capita
Percentage change in
GDP per capita
Percentage change in
GDP per capita
–75
–50
–25
0
25
50
2020 2040 2060 2080 2100
Year
–75
–50
–25
0
25
2020 2040 2060 2080 2100
Year
–75
–50
–25
0
25
50
12345
Temperature change (°C)
Richest 20% in 2010
Poorest 20% in 2010
20th–40th percentile
60th–80th percentile
40th–60th percentile
DICE 2010
FUND 3.8 PAGE09
IAMs
This
study
ab
cd
Pooled response, short-run effect
Differentiated response, long-run effect
Differentiated response, short-run effect
Pooled response, long-run effect
Figure 5
|
Global damage estimates arising from non-linear effects of
temperature. a, Change in global GDP by 2100 using benchmark model
(Fig. 2a). Calculation and display are the same as Fig. 4. b, Same as
a(point estimate only) comparing approaches to estimating temperature
effects (pooled/differentiated: rich and poor countries assumed to respond
identically/differently, respectively; short run/long run: effects account for 1 or
5 years of temperature, respectively; see Supplementary Methods). c, Mean
impacts by 2010 income quintile (benchmark model). d, Projected income
loss in 2100 (SSP5) for different levels of global mean temperature increase,
relative to pre-industrial temperatures. Solid lines marked as in b. Blue
shaded areas are interquartile range and 5th–95th percentile estimates. Dashed
lines show corresponding damages from major integrated assessment
models (IAMs)
12
.
4|NATURE|VOL000|00MONTH2015
RESEARCH LETTER
G2015 Macmillan Publishers Limited. All rights reserved
First, our projected global losses are roughly linear—and slightly
concave—in temperature, not quadratic or exponential as previously
theorized. Approximate linearity results from the broad distribution of
temperature exposure within and across countries, which causes the
country-weighted average derivative of the productivity function in
Fig. 2a to change little as countries warm and prevents abrupt transi-
tions in global output even though the contribution of individual
productive units are highly non-linear (see Fig. 1). Global losses are
slightly concave in global temperature because the effect of compound-
ing negative growth declines mechanically over time (Extended Data
Fig. 6e and Supplementary Information). These properties are inde-
pendent of the growth scenario and response function (Extended
Data Fig. 6a).
Second, the slope of the damage function is large even for slight
warming, generating expected costs of climate change 2.5–100 times
larger than prior estimates for 2 uC warming, and at least 2.5 times
larger for higher temperatures(Extended Data Fig. 6b–d). Notably, our
estimates are based only on temperature effects (or effects for which
historical temperature has been a proxy), and so do not include other
potential sources of economic loss associated with climate change,
such as tropical cyclones
15
or sea-level rise
23
, included in previous
damage estimates.
If societies continue to function as they have in the recent past,
climate change is expected to reshape the global economy by substan-
tially reducing global economic output and possibly amplifying exist-
ing global economic inequalities, relative to a world without climate
change. Adaptations such as unprecedented innovation
24
or defensive
investments
25
might reduce these effects, but social conflict
2
or dis-
rupted trade
26
—either from political restrictions or correlated losses
around the world—could exacerbate them.
Online Content Methods, along with any additional Extended Data display items
and SourceData, are available in theonline version of the paper;references unique
to these sections appear only in the online paper.
Received 3 April; accepted 15 September 2015.
Published online 21 October 2015.
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Supplementary Information is available in the online version of the paper.
Acknowledgements We thank D. Anthoff, M. Auffhammer, V. Bosetti, M. P. Burke,
T. Carleton, M. Dell, L. Goulder, S. Heft-Neal, B. Jones, R. Kopp, D. Lobell, F. Moore,
J. Rising, M. Tavoni, andseminar participants at Berkeley,Harvard, Princeton, Stanford
universities, Institute for the Study of Labor, and the World Bank for useful comments.
Author Contributions M.B. and S.M.H. conceived of and designed the study; M.B. and
S.M.H. collected and analysed the data; M.B., S.M.H. and E.M. wrote the paper.
Author Information Replication data have been deposited at the Stanford Digital
Repository (http://purl.stanford.edu/wb587wt4560). Reprints and permissions
information is available at www.nature.com/reprints. The authors declare no
competing financial interests. Readers are welcome to comment on the online version
of the paper. Correspondence and requests for materials should be addressed to M.B.
(mburke@stanford.edu).
00 MONTH 2015 | VOL 000 | NATURE | 5
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RESEARCH LETTER
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Extended Data Figure 1
|
Understanding the non-linear response function.
a, Response function from Fig. 2a. bf, The global non-linear response reflects
changing marginal effects of temperature at different mean temperatures. Plots
represent selected country-specific relationships between temperature and
growth over the sample period, after accounting for the controls in
Supplementary Equation (15); dots are annual observations for each country,
dark line the estimated linear relationship, grey area the 95% confidence
interval. g, Percentage point effect of uniform 1uC warming on country-level
growth rates, as estimated using the global relationship shown in a. A value of
21 indicates that a country growing at 3% yr
21
during the baseline period is
projected to grow at 2% yr
21
with 11uC warming. ppt, percentage point.
h, Dots represent estimated marginal effects for each country from separate
linear time-series regressions (analogous to slopes of lines in bf), and grey
lines the 95% confidence interval on each. The dark black line plots the
derivative LY
LTof the estimated global response function in Fig. 2a. i, Global
non-linearity is driven by differences in average temperature, not income. Blue
dots (point estimates) and lines (95% confidence interval) show marginal
effects of temperature on growth evaluated at different averagetemperatures, as
estimated from a model that interactscountry annual temperature with country
average temperature (see Supplementary Equation (17); LY
LTit
~^
b1z^
b2:!
TiÞ.
Orange dots and lines show equivalent estimates from a model that includes an
interaction between annual temperature and average GDP. Point estimates are
similar across the two models, indicating that the non-linear response is not
simply due to hot countries being poorer on average. jk, More flexible
functional forms yield similar non-linear global response functions. j, Higher-
order polynomialsin temperature, up to order 7. k, Restricted cubic splines with
up to 7 knots. Solid black line in both plots is quadratic polynomial shown in
a. Base maps by ESRI.
LETTER RESEARCH
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GDP/capita
time
growth in
GDP/cap
time
t-1
t
t+1
t+2
one-year
growth eect
persistent
growth eect
level eect
one-year
growth eect
persistent
growth eect
level eect
-.03
-.02
-.01
0
.01
.02
-.03
-.02
-.01
0
.01
.02
0 10 20 30 0 10 20 30
a
b
c
0 lags 1 lag
3 lags 5 lags
dy/dTdy/dT
temperature (C) temperature (C)
t-1
t
t+1
t+2
0.02
0.00
0.02
0.04
0.02
0.00
0.02
0.04
0
0.04
0.02
0.00
0.02
0.04
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Growth,
rich countries
Ag growth,
rich countries
Non-ag growth,
rich countries
Growth,
poor countries
Ag growth,
poor countries
Non-ag growth,
poor countries
Growth,
rich minus poor
Ag growth,
rich minus poor
Non-ag growth,
rich minus poor
de
jkl
ghi
fGrowth,
rich countries
Ag growth,
rich countries
Non-ag growth,
rich countries
Growth,
poor countries
Ag growth,
poor countries
Non-ag growth,
poor countries
Growth,
rich minus poor
Ag growth,
rich minus poor
Non-ag growth,
rich minus poor
mn
st u
pqr
o
p-valuep-value
p-value
dy/dTdy/dT
dy/dT
10 20 0 10 20 0 10 20 0 10 20 0 10 20 0 10 2030 30
temperature (C) temperature (C)
RESEARCH LETTER
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Extended Data Figure 2
|
Growth versus level effects, and comparison of
rich and poor responses. a, Evolution of GDP per capita given a temperature
shock in year t. Black line shows a level effect, with GDP per capita returning to
its original trajectory immediately after the shock. Red line shows a 1-year
growth effect, and blue line a multi-year growth effect. b, Corresponding
pattern in the growth in per-capita GDP. Level effects imply a slower-than-
average growth rate in year tbut higher-than-average rate in t11. Growth
effects imply lower rates in year tand then average rates thereafter (for a 1-year
shock) or lower rates thereafter (if a 1-year shock has persistent effects on
growth). c, Cumulative marginal effect of temperature on growth as additional
lags are included; solid line indicates the sum of the contemporaneous and
lagged marginal effects at a given temperature level, and the blue areas its
95% confidence interval. dl, Testing the null that slopes of rich- and poor-
country response functions are zero, or the same as one another, for quadratic
response functions shown in Fig. 2. Black lines show the point estimate for
the marginal effect of temperature on rich-country production for different
initial temperatures (blue shading is 95% confidence interval) (d,g,j), the
marginal effect poor-country production for different initial temperatures
(e,h,k), and the estimated difference between the marginal effect on rich- and
poor-country production compared at each initial temperature (f,i,l).
df, Effects on economy-wide per-capita growth (corresponding to Fig. 2b).
gi, Agricultural growth. jl, Non-agricultural growth. mu, Corresponding
Pvalues. Each point represents the Pvalue on the test of the null hypothesis that
the slope of the rich-country response is zero at a given temperature (m,p,s),
that the slope of the poor-country response is zero (n,q,t), or that rich-
and poor-country responses are equal (o,r,u) for overall growth, agricultural
growth, or non-agricultural growth, respectively. mu, Red lines at the bottom
of each plot indicate P50.10 and P50.05.
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0 5 10 15 20 25 30
0.15
0.10
0.05
0.00
0.05
temperature
growth rate of GDP/capita
2020 2040 2060 2080 2100
60
40
20
0
20
40
year
change in global GDP/cap (%)
2020 2040 2060 2080 2100
60
40
20
0
20
40
year
change in global GDP/cap (%)
100 50 0 50 100
100
80
60
40
20
0
Change in GDP/cap in 2100 (%) from zero lag models
Change in GDP/cap in 2100 (%) from 5 lag models
SAsia
SSA
EU
MENA
LA
OCEA
SEAsia
NA
EAsia
SAsia
SSA
EU
MENA
LA
OCEA
SEAsia
NA
EAsia
BHM baseline
BHM w/ DJO sample
BHM w/ DJO sample + FE
DJO quadratic
DJO + trends
DJO + trends + yearFE
BHM, 0lag
DJO, 0lag
DJO, 5lag
BHM, 5lag
ab
cd
BHM w/ DJO sample
DJO quadratic
BHM baseline
DJO + trends + yearFE
DJO + trends
BHM w/ DJO sample + FE
Extended Data Figure 3
|
Comparison of our resultsand those of Dell, Jones
and Olken
5
.a, Allowing for non-linearity in the original Dell, Jones and Olken
(DJO)
5
data/analysis indicates a similar temperature–growth relationship as in
our results (BHM) under various choices about data sample and model
specification (coefficients in Supplementary Table 3). b, Projections of future
global impacts on per-capita GDP (RCP8.5, SSP5) using the re-estimated non-
linear DJO response functions in aagain provide similar estimates to our
baseline BHM projection (shown in blue, and here using the sample of
countries with .20 years of data to match the DJO preferred sample).
c, Projected global impacts differ substantially between DJO and BHMif DJO’s
original linear results are used to project impacts. Lines show projected change
in global GDP per capita by 0- and 5-lag pooled non-linear models in BHM
(blue), and 0- and 5-lag linear models in DJO (orange). d, Projected regional
impacts also differ strongly between BHM’s non-linear and DJO’s linear
approach. Plot shows projected impacts on GDP per capita in 2100 by region,
for the 0-lag model (x-axis) and 5-lag model (y-axis), with BHM estimates
in blue and DJO estimates in orange. See Supplementary Discussion for
additional detail.
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Extended Data Figure 4
|
Projected impact of climate change (RCP8.5,
SSP5) on regional per capita GDP by 2100, relative to a world without
climate change, under the four alternative historical response functions.
Pooled short-run (SR) response (column 1), pooled long-run (LR) response
(column 2), differentiated SR response (column 3), differentiated LR response
(column 4). Shading is as in Fig.5a. CEAsia, Central and East Asia; Lamer, Latin
America; MENA, Middle East/North Africa; NAmer, North America; Ocea,
Oceania; SAsia, South Asia; SEAsia, South-east Asia; SSA, sub-Saharan Africa.
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Extended Data Figure 5
|
Projected impact of climate change (RCP8.5) by
2100 relative to a world without climate change, for different historical
response functions and different future socioeconomic scenarios. ap, The
first three columns show impacts on global per-capita GDP (analogous to
Fig. 5a), for the three different underlying socioeconomic scenarios and four
different response functions shown in Fig. 5b. Last column (d,h,l,p) shows
impact on per capita GDP by baseline income quintile (as in Fig. 5c), for SSP5
and the different response functions. Colours correspond to the income
quintiles as labelled in d. Globally aggregated impact projections are more
sensitive to choice of responsefunction than projected socioeconomic scenario,
with response functionsthat allow for accumulating effects of temperature (LR)
showing more negative global impacts but less inequality in these impacts.
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80
60
40
20
0
20
40
12345
0.0
0.2
0.4
0.6
0.8
1.0
12345
0.0
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1.0
12345
0.0
0.2
0.4
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1.0
DICE
FUND
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IAM estimate/ our estimateIAM estimate/ our estimateIAM estimate/ our estimate
Change in global GDP/capita (%)
temperature change (C) relative to pre-industrial
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0.01
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Change in annual growth rate (δ)
Income relative to baseline after 50 yrs
Warming hot countries
Economic
“damage”
relative to
baseline
ba
c
d
e
Pooled response, short-run eect
Dierentiated response, long-run eect
Dierentiated response, short-run eect
Pooled response, long-run eect
Extended Data Figure 6
|
Estimated damages at different levels of
temperature increase by socioeconomic scenario and assumed response
function, and comparison of these results to damage functions in IAMs.
a, Percentage loss of global GDP in 2100 under different levels of global
temperature increase, relative to a world in which temperatures remained at
pre-industrial levels (as in Fig. 5d). Colours indicated in figure represent
different historical response functions (as in Fig. 5b). Line type indicates the
underlying assumed socioeconomic scenario: dash indicates ‘base’ (United
Nations medium variant population projections, future growth rates are
country-average rates observed 1980–2010), dots indicate SSP3, solid lines
indicate SSP5. bd, The ratio of estimated damages from each IAM using data
from ref. 12 (shown in Fig. 5d) to damages in a. Colours as in afor results from
this study; IAM results are fixed across scenarios and response functions.
Temperature increase is in uC by 2100, relative to pre-industrial levels.
e, Explanation for why economic damage function is concave: increasingly
negative growth effects have diminishing cumulative impact in absolute
levels over finite periods (see Supplementary Discussion). Red curve is e
df
after f550 years.
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Extended Data Table 1
|
Regression estimates for global sample, main estimate and robustness
Unless otherwise indicated, all models include country fixed effects, year fixed effects, and quadratic country time trends, with errors clustered at the country level. Temperature is measuredin uCandprecipitation
in metres. Columns: (1) main specification, (2) as in column 1 but excluding countries with fewer than 20 years of growth data, (3) as in column 1 but dropping large oil exporting countries, (4) as in column 1 but
dropping United States and China, (5) as in column 1 but adding continent-by-year fixed effects, (6) as in column 1 but adding continent-by-year fixed effects and dropping country time trends, (7) as in column 1
but dropping year fixed effects, (8) as in column 1 but only linear time trend, (9–10) as in column 1 but adding 1 or 3 lags of per capita growth (that is, lagging the dependent variable), (11) as in column 1 but using
growth data from Penn World Tables. Asterisks indicate statistical significance at the 1% (***), 5% (**), and 10% (*) levels.
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Extended Data Table 2
|
Comparing temperature effects on per-capita growth in rich versus poor countries
Unless otherwise indicated, all models include a quadratic in precipitation, country fixed effects, year fixed effects, and quadratic country time trends, with errors clustered at the country level. Temperature is
measured in uC. Columns: (1) main specification with the climate variables interacted with an indicator for whether a country is poor, (2) as in column 1 but allowing the year fixed effects to differ across rich and
poor countries, (3) as in column 1 but restricting sample to countries with at least 20 observations, (4) as in column 1 but restricting sample to countries with at least 20 observations and allowing year fixed effects
to differ across rich and poor countries, (5) as in column 1 but adding continent-by-year fixed effects, (6) as in column 1 but adding continent-by-yearfixedeffectsanddroppingcountrytimetrends(asinref.5).The
estimated linear and quadratic effects in poor countries (and their standard errors) are given in the bottom rows of the table, along with the estimatedtemperatureoptimaforrichandforpoorcountries.Asterisks
indicate statistical significance at the 1% (***), 5% (**), and 10% (*) levels for coefficients in the main part of the table.
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Extended Data Table 3
|
Projected impacts of climate change on global GDP per capita by 2100 under RCP8.5, relative to a world without
climate change
Estimates are from four different response functions (as in Fig. 5b) estimating how growth responds to temperature, and three different scenarios of how future populations and incomes will evolve without climate
change. The left column indicates the historical response function on which the projections are based, the second column describes the statistic being reported (either the point estimate, a given percentile in the
bootstrapped distribution of projections, or the percentage of total runs projecting impacts more negative than zero, 210%, or 220%), and the last three columns give percentage impacts for three different future
scenarios: ‘base’ (United Nations medium variant population projections, future growth rates without climate change equal to country-average rates observed 1980–2010), SSP3, and SSP5.
RESEARCH LETTER
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Supplementary Materials
This Supplementary Materials Appendix contains fourteen sections, with additional treatment of the
following topics:
Framework
A.1 Aggregation across productive units (Equation 1)
A.2 Deriving Figure 1
A.3 Growth eects
Data and methods
B.1 Data
B.2 Empirical approach
Robustness and generalizability of the main result
C.1 Robustness to model specification, samples, and sources of data
C.2 Growth versus level eects
C.3 Studying heterogeneous responses
C.4 Comparison with Dell Jones Olken 2012
Climate projections and their uncertainty
D.1 Building impact projections
D.2 Projected impacts and robustness to alternative specifications
D.3 Constructing the damage function
D.4 Shape of the damage function
D.5 Damage function uncertainty
A Framework
A.1 Aggregation across productive units (Equation 1)
Numerous micro-econometric studies have identified the eects of momentary temperature on small
units of analysis, such as the eect of variation in hourly temperature on county-level agricultural
yields1, 2 or the eect of variation in daily temperatures on the productivity of individual workers3, 4 .
In these cases, the eect of temperature on these scales is highly nonlinear (Figure 1), which has lead
some researchers to suggest that the macroeconomy should exhibit similarly nonlinear responses5, 6 .
However, the units analyzed in micro-econometric studies are vanishingly small, in terms of economic
value, relative to the scale of a macroeconomy. For example, the output of an individual worker is
negligible relative to the output of an entire country. Further, the periods of time analyzed in micro-
econometric studies are typically short, ranging from an hour to a day in most cases and sometimes
weeks or months—periods that are much shorter than the annual timescale over which macroeconomic
data are typically aggregated. Here we use a simple model to consider how highly nonlinear changes
In some cases,1, 7 prior micro studies recover the net eect of a short-lived (e.g. daily) temperature event using
annual data. In these cases, the “daily response” is actually the cumulative response to a daily event, integrated over a
period substantially longer than a day. This approach is particularly important in cases where short-lived events have
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in productivity on short time scales and across many small elements in the economy might aggregate
and be reflected in macroeconomic responses over longer periods of time.
We partition a macroeconomy into “industries” indexed by i, with all individual units of production
within each industry assumed to respond identically to temperature. Each industry could thus be
highly specific—for example, one industry could be all maize farms using a given technology to produce
a specific variety of maize. Production in each industry occurs at numerous small locations in space,
indexed by `; countries, indexed by L, are large collections of locations. The incremental moments
in time that micro studies have analyzed (e.g. hours) are indexed by tand longer periods of time
composed of many sequential moments (e.g. years) are indexed by .
We follow the notation of Deryugina and Hsiang (2014)7and describe capital Kiand labor Li
in each industry as having respective productivities AK
iand AL
ithat are functions of instantaneous
temperature T`texperienced at a location `and time t. The total quantity of capital and labor
allocated to industry icould also potentially change with temperature. The price of a unit of output is
pand is a constant in this stylized production function. For a subunit of the economy at a location
`at time tusing technologies described by i, total production Yi`tis then
Yi`t(T`t)=piAK
i(T`t)Ki`t(T`t)AL
i(T`t)Li`t(T`t)1.(2)
For simplicity, we assume that capital and labor are not rapidly reallocated across locations in response
to temperature changes. Changes in the total allocation of time individuals allocate to labor is known
to change with temperature3, however this response can be easily described by changes to labor
productivity AL
isince we observe empirically that labor is not reallocated across dierent industries
in response to temperature3, 7 . Note that in a competitive equilibrium Ki`t
Li`t=
1, such that capital
labor ratios are fixed and output scales linearly with the total quantity of capital and labor allocated
to i(constant returns to scale).
For notational convenience, we define Ui`t=piK
i`tL1
i`tas a scalar measure of resources applied
to iat location `at time t. We think of Uias describing the number of modular units of production
allocated to industry i(e.g. firms). These assumptions and notations allow us to simplify Equation 2
to
Yi`t(T`t)=AK
i(T`t)AL
i(T`t)1
|{z }
fi(T`t)
piK
i`tL1
i`t
=fi(T`t)Ui`t(3)
where fi(T`t) is a function describing how overall productivity in industry iresponds to instantaneous
temperatures. Note that for clarity, we have assumed here that the economy is additively separable
across industries and locations, with firms behaving as atomistic producers. It is likely that large scale
climatic changes generate emergent impacts on firms beyond what an atomistic firm might experience
in response to an isolated change in their individual climate exposure, since cross-firm spillovers might
be substantial and novel price responses might emerge when climatic events are correlated in either
delayed impacts8,9 .
This assumption is consistent with findings in Deryugina and Hsiang (2014) from the United States.
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time or space. For example, climate-induced interruptions in a firm’s supply chain might amplify the
economic impact of a climatic event that the firm is itself exposed to. If these eects are substantial
and cross national boundaries, then our empirical approach below is likely to understate the overall
economic impact of large scale climatic changes since we focus on country-level changes as if they occur
in isolation.
To form a measure of aggregate output, such as Gross Domestic Product (GDP), we must sum
across all industries iand integrate production across all locations in a country and all moments in
time within the period of observation. Thus total output in country Lduring year is then:
YL=X
i
YiL
=X
iZt2Z`2L
fi(T`t)Ui`td`dt. (4)
The spatial and temporal distribution of units Ui`t, as well as the spatial distribution of atmospheric
temperatures, will determine what temperatures T`tindividual units are exposed to. Within country
Land period , we can integrate the number of points in time when individual productive units
are exposed to a momentary local temperature Ti`tto construct a marginal distribution function
summarizing temperature exposure within industry i. Let the shape of this marginal distribution
function be described by gi(.) which is mean zero and can be shifted by the location parameter ¯
TL,
defined as average temperature in country Lduring period .Thusgi(T¯
TL) looks like a histogram
of the temperatures that units Uiare exposed to within a large region and interval of time. For
simplicity, here we assume gi(.) does not change in shape across countries or years, although the
location parameter ¯
TLmay change. In the real world, the shape of gi(.) may change based on changes
in the within-country and within-year distribution of temperatures that productive units are exposed
to.
We note that gi(.) has two important properties. First, for a single industry, the total quantity or
“mass” of productive units Miis the integral of gi(.) over all possible temperatures
Mi=Z1
1
gi(T¯
TL)dT =Zt2Z`2L
Ui`td`dt. (5)
Second, the shape of gi(.) reflects the distribution of productive units across space and time such that
Zx
1
gi(T¯
TL)dT =Zt2Z`2L
Ui`t1[T`t<x]d`dt (6)
for x2(1,1).
We can now write total production at the aggregate level in terms of average temperature ¯
TL,
Note that this marginal distribution is not a marginal probability distribution because the total number of units at
each temperature are not normalized by the total number of units. i.e. this marginal distribution is more analogous to
a histogram measuring frequencies rather than a histogram measuring probabilities.
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measured at the aggregate level, and gi(.)
Y(¯
TL)=X
i
Yi(¯
TL)
=X
iZt2Z`2L
fi(T`t)Ui`td`dt
=X
iZ1
1
fi(T)gi(T¯
TL)dT (7)
which no longer requires we know detailed information on the spatial and temporal distribution of
Ui`t. If the shape of gi(.) is relatively unchanged across periods , then ¯
TLis a sucient statistic
for temperature exposure at the aggregate level. As shown in Figure 1, changing annual average
temperature ¯
TLshifts the distribution of temperature exposure for individual micro-level units.
Essentially, we have changed variables by collapsing the joint spatial and temporal distribution
of temperatures and micro-level productive units into the marginal distribution gi(.) and a location
parameter ¯
TL, which is a country’s annual average temperature.
A.2 Deriving Figure 1
Figure 1 depicts how the application of Equation 7 (i.e. Equation 1 in the main text) to previously
derived micro-level response functions generates a macro-level response. Prior work1–5 has shown that
basic units of the economy, such as crops and labor, have a response to momentary temperature that
is highly nonlinear and well-approximated by a piecewise-linear function similar to Figure 1d (recall
Figure 1a-c). In general, the productivity of basic units in the economy is either flat or slightly increas-
ing at lower temperatures, and then declines steeply with temperature above a critical temperature
threshold. These responses are the function fi(T) in Equation 7. Thus, to develop a sense of how
macro-level responses to temperature should look, we assume the micro-level function fi(.)ispiecewise
linear with kink at the critical instantaneous temperature T=˜
T:
f(T)=(c1+b1Tif T<˜
T
c2+b2Tif T˜
T(8)
where where slope terms b1and b2and intercept terms c1and c2satisfy
c1+b1˜
T=c2+b2˜
T
b1>0,b
2<0,b2>b
1.(9)
These conditions ensure fi(.) is continuous and non-dierentiable due to a “kink” at the critical
temperature ˜
T, with a downward slope above ˜
Tthat is steeper than the upward slope below ˜
T. Figure
1d displays these properties.
For ease of comparability across countries and industries of dierent economic size, we normalize
total production in an industry by the the total mass of productive units Miand focus on Yi
Mi. When
examining data, we implement an analogous normalization by focusing on GDP per capita§. We are
§The standard GDP per capita normalization does not account for capital, however we note that, at least in our
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interested in how aggregate productivity changes with each country’s average annual temperature, so
we dierentiate this normalized measure Yi
Miwith respect to ¯
TLwhile substituting from Equation 7
@
@¯
TLYi
Mi=1
Mi
@Yi
@¯
TL
=1
Mi
@
@¯
TLZ1
1
fi(T)gi(T¯
TL)dT
=1
Mi
@
@¯
TL"Z˜
T
1
fi(T)gi(T¯
TL)dT +Z1
˜
T
fi(T)gi(T¯
TL)dT #(10)
where the final equality is simply separating the integral into a portion below the critical temperature
and a portion above the critical temperature. These integrals appear dicult to dierentiate because
the shape of gi(.) is unknown, but a change of variables clarifies the derivative by making the shift
parameter ¯
TLan argument of fi(.) instead. Define a new variable T0=T¯
TL. We analogously
define ˜
T0such that ˜
T=˜
T0+¯
TL. Substituting T0and ˜
T0into Equation 10 and noting the linearity of
fi(.) within the range of each integral we have
@
@¯
TLYi
Mi=1
Mi
@
@¯
TL"Z˜
T0+¯
TL
1
fi(T0+¯
TL)gi(T0)dT 0+Z1
˜
T0+¯
TL
fi(T0+¯
TL)gi(T0)dT 0#
=1
Mi"Z˜
T0+¯
TL
1
b1gi(T0)dT 0+Z1
˜
T0+¯
TL
b2gi(T0)dT 0#
=b1mi1(¯
TL)+b2mi2(¯
TL) (11)
where mi(¯
TL) is defined as the fraction of productive unit-hours exposed to Tbelow ˜
Tfor a given
national mean temperature ¯
TL. Specifically,
mi1(¯
TL)=R˜
T0+¯
TL
1 gi(T0)dT 0
Mi
=R˜
T
1 gi(T¯
TL)dT
R1
1 gi(T¯
TL)dT
mi2(¯
TL)=1mi1(¯
TL)
and m1and m2are illustrated as two shaded masses in Figure 1e.
Thus, the country level aggregated response Yi
Mican be computed by integrating the weighted
average in Equation 11 with respect to average annual temperature ¯
TL, where the weights mi1and
mi2are determined by the shape of the distribution of productive units gi(.) and its location relative
to the critical temperature ˜
T. This integration is depicted in Figure 1f. The aggregated productivity
function recovered from this integration will be smooth if gi(.) is continuous and contains no point-
masses, even though there is a known sharp kink in the micro-level response function fi(.). The
distribution gi(.) “smoothes” over this kink, causing the the eect of average warming (increasing
¯
TL) to be much less abrupt on the macro-economy than local and instantaneous warming is on micro-
level productive units. A broader distribution function gi(.), either caused by a wider dispersion of
simple model, capital-labor ratios are constant in equilibrium so this normalization would still be a valid approach to
comparing countries of dierent size.
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productive assets across space and/or longer periods of observation, will cause more smoothing.
A key characteristic of interest regarding the shape of Yi
Miis the temperature at which the peak
of the function is located. This turning point occurs at the temperature where the derivative of the
response is zero:
@
@¯
TLYi
Mi=b1mi1(¯
TL)+b2mi2(¯
TL)=0
which implies that the turning point temperature ¯
T
Lhas the property:
mi2(¯
T
L)
mi1(¯
T
L)=b1
b2
<1
where the inequality comes from the conditions in Equation 9. This implies
¯
T
L<˜
T
if gi(.) is roughly symmetric or negatively skewed.Note that the greater the dierence between |b1|
and |b2|, the lower the temperature at which the peak occurs. Also, the greater the dispersion in gi(.),
the lower the value of ¯
T
L.
For multiple sectors, total production Y
M=Pi
Yi
Miwill be concave in annual average temperature
because it is the weighted sum of several concave functions of annual average temperature.
We note that this result reconciles a long-standing debate about whether degree days or seasonal
averages are better measures of exposure to climate change,10 since we have shown that the two
measures are closely related mathematically (and identical under certain assumptions). The kinked
micro-level response reported by degree day studies1, 11 has a direct mapping (Equation 7)tothe
seasonal average response that is smoother and roughly quadratic.10 The quality of fit may dier
between these two approaches depending on spatial and temporal autocorrelation of the outcome
variable within averaging periods and regions, as well as the extent to which gi(.) is actually the same
across dierent units of analysis. Nonetheless, at their core the two approaches are not fundamentally
dierent and may produce findings that are mutually consistent even though the temperature response
functions recovered by the two approaches dier.
A.3 Growth eects
Is it plausible that temporary productivity losses caused by temperature can translate into endur-
ing productivity shocks? That is, could we expect that economic growth is aected by temporary
productivity changes? The micro-level responses to temperature in prior analyses (discussed above)
generally characterize temporary changes in productivity. However, as discussed in detail below, our
main analysis estimates the nonlinear eect of temperature on GDP per capita growth, in part be-
cause log GDP per capita is known to have a unit root and thus requires first dierencing for proper
inference.12 As detailed in Section C.2 below, this transformation of the dependent variable does not
This condition may not hold if gi(.) is strongly positively skewed, depending on the values b1and b2.
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itself imply that output is depressed relative to trend in the long run; getting at this question requires
examining the cumulative eect of lagged independent variables, which we implement in Section C.2.
(As stated in the main text and detailed in Section C.2, our results are somewhat ambiguous as to
whether the nonlinear eect of temperature generates a permanent or temporary loss of output relative
to trend.) In this section we point out how, in a simple model, temporary productivity losses generated
through a mechanism like Equation 2 could generate growth eects if savings rates do not change with
temperature to compensate and oset temporary productivity losses.
Let total productivity be smooth, twice dierentiable, and concave with respect to ¯
T, as derived
above in Section A.1-A.2 (we drop the iLsubscripts here). If year-to-year changes in capital stocks
are modest, then we can apply Taylor’s theorem and linearize output with respect to total productive
units M, with local slope (¯
T):
Y= (¯
T)M(12)
where @2
@¯
T2<0. Thus, the temporary change in temperature aects output similarly to a change in
total factor productivity, akin to the approach in Nordhaus and Boyer (2000)13.
Following seminal work by Solow (1956)14, we assume a fraction of Mdepreciates each period
and a fraction sof output is saved and re-invested in augmenting M. The equation of motion for M
is then @M
@t=sY M
where Mis measured in units of U, which recall accounts for both capital and labor contributions to
production. Substituting from Equation 12:
@M
@t=s (¯
T)MM
=s (¯
T)
| {z }
net growth if >0
M.
Thus the productive stock of a country Mwill grow on net if savings rates, average temperature, and
the initial stock M(which aects ) allow investment to outpace depreciation.k
Importantly, if productivity is reduced in a given period due to temperature, this has an impact on
output in subsequent periods because it reduces investment in M. To see this, note that for a given
initial stock Mt1just prior to a temperature realization ¯
Tt1, then output in the subsequent period
will be
Yt= (¯
Tt)Mt
= (¯
Tt)(Mt1+Mt1!t)
= (¯
Tt)Mt1+s (¯
Tt1)Mt1Mt1.(13)
Dierentiating Equation 13 with respect to the prior year’s temperature ¯
Tt1, we see that current
kHere we assume that the climate does not aect depreciation, although recent evidence suggest this may be an
important direction for future work.15, 16
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