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Climate change and agriculture: impacts onwheat and corn yields in France
Catherine Benjamin and Ewen Gallic
Faculté des sciences économiques, Université de Rennes 1
Jeudi 28 novembre 2013 - Aber Wrac’h
Introduction Model Results Conclusion
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 2/48
Introduction Model Results Conclusion
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 3/48
Introduction Model Results Conclusion
Climate is changing ;Agriculture as a prime area :
I direct impacts on yields,I consquences on global markets, on prices, on world food
supply, on food security, . . . ;Objective of this study: measuring the impact of weatheron wheat and corn yields in France.
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Introduction Model Results Conclusion
corn wheat
0
20
40
60
80
1960 1970 1980 1990 2000 20101960 1970 1980 1990 2000 2010Year
Per
cent
age China
FranceIndiaUSAArgentinaBrazilIndonesiaMexicoUkraine
Source : FAOSTAT.
Figure 1: Top world producers of wheat and corn, in percentage ofglobal production
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Introduction Model Results Conclusion
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
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Introduction Model Results Conclusion
Yields
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
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Introduction Model Results Conclusion
Yields
A wide literature;With mixed conclusions :
I positive global effects shown for some (Aggarwal andMall 2002; Deschênes and Greenstone 2007),
I negative global effects for others (Parry et al. 2004; Jonesand Thornton 2003) ;
Different aims:I forecasting (Fischer et al. 2005; Deschenes and Kolstad
2011),I determining influent factors of yield variation (Reidsma,
Ewert, and Lansink 2007),I quantification of different effects (Almaraz et al. 2008).
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Introduction Model Results Conclusion
Yields
Yields modelsTwo types of models :
I agronomic based ones (e.g. Tubiello and Fischer (2007)),I regression based ones (e.g. Schlenker and Roberts (2008)).
Strengths Weaknesses
Agronomic • Distribution of weather • Small scale• CO2 Fertilizer effects • A lot of missing effects
Regression • Large scale • Annual data• Ease of calculation • Large dataset needed
Table 1: Strengths and weaknesses of agronomic and regressionbased models
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Introduction Model Results Conclusion
Yields
Model estimatedFrom Deschênes and Greenstone (2007):
yct = αc + γt + X>ctβ +∑
iθifi(W ict) + εct , (1)
yct : yields or profits ;αc : individual specific effects ;γt : year fixed effects ;Xct : observable determinants of farmland values, varyingwith time t ;W ict : annual realization of the i th weather variable ;εct : a residual error.
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Introduction Model Results Conclusion
Yields
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1990 1995 2000 2005 2010Year
Yie
lds
(kh/
ha)
Crop ● ●Corn Wheat
Source : DISAR.
Figure 2: Corn and wheat yields evolution in France (1990–2011).
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Introduction Model Results Conclusion
Yields
7000800090001000011000
Yields
(a) Corn.
4000
6000
8000
Yields
(b) Wheat.Source : DISAR.
Figure 3: Average annual yields (1990–2011, kg ha−1) .
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Introduction Model Results Conclusion
Yields
Yields are observed on a yearly basis, and data are spatiallyaggregated ;Two issues :
I How can one aggregate weather data?I How can one incorporate variability of weather with an-
nual data?
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Introduction Model Results Conclusion
Climate Data
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
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Introduction Model Results Conclusion
Climate Data
Weather Data Description
Average daily temperature and precipitation ;210 weather stations ;Source : NOAA ;From 1990 to 2011.
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Introduction Model Results Conclusion
Climate Data
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0km 100km 200kmN
Figure 4: French weather stations
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Introduction Model Results Conclusion
Data Aggregation
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
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Introduction Model Results Conclusion
Data Aggregation
Intuitive idea
1. Divide France with multiple rectangles ;
2. Estimate temperature and precipitation for each centroidof the rectangles;
3. Averaging for each département.
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Introduction Model Results Conclusion
Data Aggregation
Intuitive idea
1. Divide France with multiple rectangles ;
Figure 5: France divided in multiple rectangles
2. Estimate temperature and precipitation for each centroidof the rectangles;
3. Averaging for each département.
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Introduction Model Results Conclusion
Data Aggregation
Intuitive idea
1. Divide France with multiple rectangles ;2. Estimate temperature and precipitation for each centroid
of the rectangles;
3. Averaging for each département.
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 18/48
Introduction Model Results Conclusion
Data Aggregation
Intuitive idea
1. Divide France with multiple rectangles ;2. Estimate temperature and precipitation for each centroid
of the rectangles;3. Averaging for each département.
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Introduction Model Results Conclusion
Data Aggregation
Two methods used to estimate the values of temperatureand precipitations at unobserved locations:
I ordinary kriging for temperature (Guillot (2004)),I trivariate thin-plate splines for precipitation (McKenney
et al. (2006)).
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Introduction Model Results Conclusion
Data Aggregation
Temperature
Let Z = (z1, . . . , zn)> be a spatial random variable ;Observations are made at location (s1, . . . , sn) ;Let z(s) be the realization of Z (s) at location s.An estimation of Z (s0), at site s0 is given by :
Z (s0) =n∑
i=1λiZ (si) = λ>Z
with λi providing a minimum variance unbiased estimator.
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Introduction Model Results Conclusion
Data Aggregation
Then, the kriging weights λ must satisfy
E[Zs0 − Zs0
]= 0
minλi
{V[Zs0 − Zs0
],λ ∈ Rn
}.
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Introduction Model Results Conclusion
Data Aggregation
Unbiased estimator, assuming E(Z) = m :
E[Zs0 − Zs0
]= 0
⇔n∑
i=1λiE [Zsi ]− E [Zs0 ] = 0
⇔ mn∑
i=1λi −m = 0
⇔n∑
i=1λi = 1.
⇔ λ1 = 1.
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Introduction Model Results Conclusion
Data Aggregation
Minimum variance : λ solution of[V[Z ] 11> 0
] [λµ
]=[Cov(Zs0 ,Z)
1
].
Details
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Introduction Model Results Conclusion
Data Aggregation
The covariance function is defined as
C (h) = Cov [Z (s),Z (s + h)] ,
where h is the distance between s and s + h ;A parametric form is used :
C (h) = σ2 · ρ(h),
with σ2 the limit of the variogram tending to infinity lagdistance and ρ(·) a covariance function ;
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Introduction Model Results Conclusion
Data Aggregation
The variogram function is given by
γ(h) = 12V [Z (s)− Z (s + h)]
And estimated using
γ(h) = 12nh
nh∑i=1
(Z (si)− Z (si + h))2 ,
where nh is the number of observations separated by h.
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Introduction Model Results Conclusion
Data Aggregation
The spherical covariance function was used :
ρ(h) =
1− 32
hφ
+ 12
(hφ
)3, h < φ
0, h ≤ φ,
where φ is a scale parameter.
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Introduction Model Results Conclusion
Data Aggregation
Precipitation
Let us assume that precipitation depend on both location(s, identified by longitude and latitude) and elevation (q)(as in Boer (2001)) ;Then the measurements can be viewed as
z(si , qi) = f (si , qi) + ε(si , qi), i = 1, . . . , n,
where f (·) is a trivariate function and ε(si , qi) are randomerrors.
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Introduction Model Results Conclusion
Data Aggregation
The trivariate function f (·) can be estimated by minimizingn∑
i=1{z(si , qi)− f (si , qi)}+ λJ2(f ),
where J2(f ) is a penalty functional measuring the smooth-ness of f (·) and λ is a parameter controlling for the trade-off between the fit and the smoothness of f (·). Both canbe estimated.
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 28/48
Introduction Model Results Conclusion
Data Aggregation
The solution is given by
f (s) =4∑
j=1αjφj(s) +
n∑i=1
βiΨ(hi),
where α and β are unknown parameters vectors, φi arelinearly independent polynomials (φ1 = 1, φ2 = lon, φ3 =lat, φ4 = q), Ψ = h, and
hi =√
(lon− loni)2 + (lat− lati)2 + (q − qi)2
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 29/48
Introduction Model Results Conclusion
Including Weather Variability
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
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Introduction Model Results Conclusion
Including Weather Variability
Temperature-related measures
Climate variability is not included because of yields obser-vation frequency ;Growing degree days measure a heat accumulation ;A definition is given by Deschênes and Greenstone (2007)
GDD =
0 si h ≤ 8h − 8 si 8 < h ≤ 3224 si h > 32
,
where h is the daily mean temperature.
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Introduction Model Results Conclusion
Including Weather Variability
Precipitation-related measures
To try to incorporate precipitation variability, another setof variables is created :
I # of wet days over a period (month or growing season),I # of consecutive dry days over a period ;
A day is considered wet at a given location if the totalprecipitation exceeds 0.01 inch.
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Introduction Model Results Conclusion
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 33/48
Introduction Model Results Conclusion
Estimation
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 34/48
Introduction Model Results Conclusion
Estimation
AVG
GDDMonths
GDDComplete Growing
season
Figure 5: Different models.
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Introduction Model Results Conclusion
Estimation
Four estimated models:
I pooling
yct = δ +∑
iθi fi(W ict) + εct (2)
I individual fixed effects
yct = αc +∑
iθi fi(W ict) + εct (3)
I time fixed effects
yct = γt +∑
iθi fi(W ict) + εct (4)
I individual and time fixed effects
yct = αc + γt +∑
iθi fi(W ict) + εct (5)
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Introduction Model Results Conclusion
Estimation
Four estimated models:I pooling
yct = δ +∑
iθi fi(W ict) + εct (2)
I individual fixed effects
yct = αc +∑
iθi fi(W ict) + εct (3)
I time fixed effects
yct = γt +∑
iθi fi(W ict) + εct (4)
I individual and time fixed effects
yct = αc + γt +∑
iθi fi(W ict) + εct (5)
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Introduction Model Results Conclusion
Estimation
Four estimated models:I pooling
yct = δ +∑
iθi fi(W ict) + εct (2)
I individual fixed effects
yct = αc +∑
iθi fi(W ict) + εct (3)
I time fixed effects
yct = γt +∑
iθi fi(W ict) + εct (4)
I individual and time fixed effects
yct = αc + γt +∑
iθi fi(W ict) + εct (5)
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Introduction Model Results Conclusion
Estimation
Four estimated models:I pooling
yct = δ +∑
iθi fi(W ict) + εct (2)
I individual fixed effects
yct = αc +∑
iθi fi(W ict) + εct (3)
I time fixed effects
yct = γt +∑
iθi fi(W ict) + εct (4)
I individual and time fixed effects
yct = αc + γt +∑
iθi fi(W ict) + εct (5)
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Introduction Model Results Conclusion
Estimation
Four estimated models:I pooling
yct = δ +∑
iθi fi(W ict) + εct (2)
I individual fixed effects
yct = αc +∑
iθi fi(W ict) + εct (3)
I time fixed effects
yct = γt +∑
iθi fi(W ict) + εct (4)
I individual and time fixed effects
yct = αc + γt +∑
iθi fi(W ict) + εct (5)
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Introduction Model Results Conclusion
Estimation
Only one selected model for each crop :I wheat: individual and time fixed effects, where weather
data is aggregated for each month, and with temperaturein degrees Celsius ;
I corn: individual fixed effects, where weather data is ag-gregated for each month, and with temperature in growing-degree days.
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Introduction Model Results Conclusion
Estimation
Temperature effects
Table 2: Summary of effects of temperature on yields∆ yields in responseto an increase of thevariable
Variable
Wheat
↘ Temp. in February↘ Temp. in July↘ Temp. sup. in August↗ Temp. in April∩ Temp. in August (threshold at 21◦C)
Corn ↗ Temp. in April∩ Temp. in June (threshold at 250 GDD)
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Introduction Model Results Conclusion
Estimation
Precipitation effectsTable 3: Summary of effects of precipitation on yields
∆ yields in responseto an increase of thevariable
Variable
Wheat
↘ # of wet days in May↘ Precip. in June∪ Precip. in January (threshold at 110.9 mm)∪ Precip. in July (threshold at 191.82 mm)
∩ # of wet days in February (threshold at 16.3days)
Corn
↗ Precip. in April↗ # of wet days in June↘ Precip. in May∩ Precip. in June (threshold at 99.5 mm)∩ Precip. in August (threshold at 92.2 mm)∪ # of wet days in May (threshold at 16.6 days)∪ # of wet days in July (threshold at 15 jours)
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Introduction Model Results Conclusion
Simulation
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
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Introduction Model Results Conclusion
Simulation
S−10,3+3◦ C
S−10,2+2◦ C
S−10,1
+1◦ C−10% precipitation
S0,3+3◦ C
S0,2+2◦ C
S0,1
+1◦ C
+0% precipitation
Figure 6: Simulated climate scenarios
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 41/48
Introduction Model Results Conclusion
Simulation
Table 4: Yields variations under different climate scenario
+1◦ C +2◦ C +3◦ C
∆ precip. Wheat Corn Wheat Corn Wheat Maïs
0%Mean −7.2% 0.0% −15.6% −3.0% −25.0% −7.5%S.D. (2.0%) (11.6%) (4.8%) (11.8%) (8.4%) (11.9%)Min. −10.7% −19.% −22.8% −22.8% −37.4% −27.9%Max. −5.4% 17.1% −11.56% 13.8% −18.7% 9.1%
−10%
Mean −6.8% −1.3% −15.1% −4.3% −24.5% −8.9%S.D. (2.1%) (11.4%) (4.8%) (11.6%) (8.4%) (11.8%)Min. −10.0% −20.2% −22.1% −23.8% −36.7% −29.0%Max. −5.0% 15.8% −11.4% 12.6% −18.3% 7.8%
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 42/48
Introduction Model Results Conclusion
Simulation
−1.88
1.06
0.61
−1.44
1.86−0.03
−0.14
2.86
2.75
−2.77−3.24
−0.053.54
−2.27
−3.29
−19.180.56
17.091.28
2.34
6.28
−2.81
1.41−0.42
−2.61
0.43
0.21
2.24
−2.31
0.25
0.34
0.73
−1.98
−0.83 1.12
0km 100km 200km
N % change−20 and 00 and 20NA
(a) +0% precipitation, +1◦C.
−2.84
−0.38
−0.49
−2.6
0.15−1.35
−1.41
1.24
0.95
−4.24−4.49
−1.492.17
−3.71
−4.38
−20.17−0.03
15.82−0.41
0.67
4.6
−3.58
−0.17−1.85
−4.09
−1.37
−1.32
0.58
−2.64
−1.14
−1.28
−0.78
−3.18
−2.31 −0.61
0km 100km 200km
N% change
−40 and −20−20 and 00 and 20NA
(b) −10% precipitation, +1◦C.
Figure 7: Average variation of corn yield response to climatechange
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 43/48
Introduction Model Results Conclusion
Simulation
−11.18
−1.6
−6.12
−9.28
−9.33−10.78
−6.25
2.57
−6.32
−14.87−14.13
−3.263.15
−12.74
−15.06
−27.95−8.55
9.14−2.95
−3.69
−7.91
−13.38
−3.46−5.05
−14.33
−6.75
−4.24
−1.36
−13.54
−7.12
−5.26
−4.26
−12.71
−7.55 −6.3
0km 100km 200km
N% change
−40 and −20−20 and 00 and 20NA
(a) +0% precipitation, +3◦C.
−12.15
−3.04
−7.23
−10.45
−11.05−12.1
−7.53
0.94
−8.12
−16.33−15.38
−4.71.78
−14.18
−16.16
−28.94−9.15
7.86−4.64
−5.36
−9.59
−14.15
−5.03−6.48
−15.81
−8.55
−5.76
−3.03
−13.87
−8.51
−6.88
−5.77
−13.91
−9.02 −8.03
0km 100km 200km
N% change
−40 and −20−20 and 00 and 20NA
(b) −10% precipitation, +3◦C.
Figure 8: Average variation of corn yield response to climatechange
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 44/48
Introduction Model Results Conclusion
Simulation
−5.53−5.41
−6.21
−6.34
−9.72
−8.45−7.48
−7.08
−9.18
−6.02
−6.5−7.92
−10.72
−7.31−7.58
−8.43
−8.52
−6.33−8.53
−8.47
−5.56
−7.44
−6.6−6.4
−7.95
−6.85
−5.59
−5.89
−6.9
−5.75
−7.85
−6.2
−5.98−5.61
−8.96 −8.9
−7.41
0km 100km 200km
N % change−20 and 0NA
(a) +0% precipitation, +1◦C.
−5 −5.3
−5.54
−5.8
−9.42
−7.96−7.07
−6.64
−8.86
−5.53
−5.88−7.94
−9.99
−6.84−7.13
−7.91
−7.96
−5.69−7.87
−7.98
−5.11
−6.93
−6.21−6.1
−7.65
−6.06
−5.4
−5.37
−6.26
−5.32
−7.18
−5.63
−5.69−5.22
−8.67 −8.27
−6.93
0km 100km 200km
N % change−20 and 0NA
(b) −10% precipitation, +1◦C.
Figure 9: Average variation of wheat yield response to climatechange
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 45/48
Introduction Model Results Conclusion
Simulation
−18.88−18.66
−21.87
−21.2
−32.67
−28.63−26.91
−23.63
−31.38
−20.39
−23.67−26.4
−37.44
−25.75−25.21
−30.15
−30.17
−25.91−28.61
−29.72
−19.34
−25.02
−22.67−22.2
−26.44
−23.4
−18.82
−19.96
−23.29
−19.26
−27.64
−21.4
−20.06−18.87
−30.95 −31.2
−27.27
0km 100km 200km
N % change−40 and −20−20 and 0NA
(a) +0% precipitation, +3◦C.
−18.35−18.55
−21.21
−20.67
−32.37
−28.15−26.5
−23.19
−31.07
−19.91
−23.05−26.42
−36.7
−25.28−24.76
−29.62−29.61
−25.27−27.95
−29.22
−18.89
−24.51
−22.28−21.9
−26.14
−22.62
−18.63
−19.44
−22.66
−18.83
−26.97
−20.83
−19.78−18.48
−30.65 −30.56
−26.79
0km 100km 200km
N % change−40 and −20−20 and 0NA
(b) −10% precipitation, +3◦C.
Figure 10: Average variation of wheat yield response to climatechange
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 46/48
Introduction Model Results Conclusion
1 Introduction
2 ModelYieldsClimate DataData AggregationIncluding Weather Variability
3 ResultsEstimationSimulation
4 Conclusion
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 47/48
Introduction Model Results Conclusion
Agriculture is vulnerable to climate fluctuations ;We estimated the variation of yields in France using panelmodels ;Mixed results:
I globally negative effects for wheat,I standard deviation too large for corn ;
Some limits to keep in mind:I producers can’t adapt by changing their crops,I CO2 mitigation effects are not taken into account,I neither is irrigation
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Références I
Aggarwal, P.K. and R.K. Mall (2002). “Climate Change and Rice Yields inDiverse Agro-Environments of India. II. Effect of Uncertainties in Scenariosand Crop Models on Impact Assessment”. English. In: Climatic Change52.3, pp. 331–343. issn: 0165-0009. doi: 10.1023/A:1013714506779.Almaraz, Juan Jose et al. (2008). “Climate change, weather variability andcorn yield at a higher latitude locale: Southwestern Quebec”. In: Climaticchange 88.2, pp. 187–197.Boer, E. (2001). “Kriging and thin plate splines for mapping climate vari-ables”. In: International Journal of Applied Earth Observation and Geoin-formation 3.2, pp. 146–154. issn: 03032434.Deschênes, Olivier and Michael Greenstone (2007). “The economic impactsof climate change: evidence from agricultural output and random fluctu-ations in weather”. In: The American Economic Review 97.1, pp. 354–385.Deschenes, Olivier and Charles Kolstad (2011). “Economic impacts of cli-mate change on California agriculture”. In: Climatic Change 109.1, pp. 365–386.
Références IIFischer, Günther et al. (2005). “Socio-economic and climate change im-pacts on agriculture: an integrated assessment, 1990-2080”. In: Philosoph-ical Transactions of the Royal Society B: Biological Sciences 360.1463,pp. 2067–2083. doi: 10.1098/rstb.2005.1744.Guillot, Gilles (2004). Introductin à la geostatistique. Institut NationalAgronomique de Paris-Grignon.Jones, Peter G and Philip K Thornton (2003). “The potential impacts ofclimate change on maize production in Africa and Latin America in 2055”.In: Global Environmental Change 13.1, pp. 51 –59. issn: 0959-3780. doi:10.1016/S0959-3780(02)00090-0.McKenney, Daniel W et al. (2006). “The development of 1901–2000 his-torical monthly climate models for Canada and the United States”. In:Agricultural and Forest Meteorology 138.1, pp. 69–81.Parry, M.L et al. (2004). “Effects of climate change on global food pro-duction under SRES emissions and socio-economic scenarios”. In: GlobalEnvironmental Change 14.1. Climate Change, pp. 53 –67. issn: 0959-3780.doi: 10.1016/j.gloenvcha.2003.10.008.
Références III
Reidsma, Pytrik, Frank Ewert, and Alfons Oude Lansink (2007). “Analysisof farm performance in Europe under different climatic and managementconditions to improve understanding of adaptive capacity”. English. In:Climatic Change 84.3-4, pp. 403–422. issn: 0165-0009. doi: 10.1007/s10584-007-9242-7.Schlenker, Wolfram and Michael J Roberts (2008). Estimating the impactof climate change on crop yields: The importance of nonlinear temperatureeffects. Tech. rep. National Bureau of Economic Research.Tubiello, Francesco N. and Günther Fischer (2007). “Reducing climatechange impacts on agriculture: Global and regional effects of mitigation,2000–2080”. In: Technological Forecasting and Social Change 74.7. Green-house Gases - Integrated Assessment, pp. 1030 –1056. issn: 0040-1625.doi: 10.1016/j.techfore.2006.05.027.
References
Ordinary Kriging: minimum varianceMinimum variance is obtained by solving the followin program:minλ
{V[Zs0 − Zs0
],λ ∈ Rn
}s.c. λ1 = 1
V[Zs0 − Zs0
]= V [Zs0 ] + V
[Zs0
]− 2Cov(Zs0 , Zs0)
= V [Zs0 ] + V[λ>Z
]− 2Cov
(Zs0 ,λ
>Z)
= V [Zs0 ] + λ>V[Z ]λ− 2λ>Cov (Zs0 ,Z) .
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 51/48
References
Ordinary Kriging: minimum variance II
The program can be rewritten as
⇔ minλ
L =V [Zs0 ] + λ>V[Z ]λ
− 2λCov (Zs0 ,Z)− 2µ(λ>1− 1),
with µ the Lagrange multiplier.
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 52/48
References
Ordinary Kriging: minimum variance III
The first-order conditions are∂L∂λ
= 0∂L∂µ
= 0
⇔
2V[Z ]λ− 2Cov(Zs0 ,Z) + 2µ1 = 02λ1− 2 = 0.
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 53/48
References
Ordinary Kriging: minimum variance IV
The kriging weights (λ) are solution of the following system[V[Z ] 11> 0
] [λµ
]=[Cov(Zs0 ,Z)
1
],
i.e.
V(Z)λ = Cov(Zs0 ,Z)− µ1⇔ λ = [V(Z)]−1 (Cov(Zs0 ,Z)− µ1) .
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 54/48
References
Ordinary Kriging: minimum variance V
Then, the variance can be written as
V[Zs0 − λ>Z
]=V[Zs0 ] + λ> (Cov(Zs0 ,Z)− µ1)
− 2λ>Cov(Zs0 ,Z).
Return to Temperature
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References
Estimation of J2(f )
The parameter J2(f ) can be estimated with
J2(f ) =∫ +∞
−∞
∫ +∞
−∞
(∂2f∂x2
)2
+ 2(∂2f∂x∂y
)2
+(∂2f∂y2
)2 dxdy
Return to Precipitation
Ewen Gallic Climate change and agriculture: impacts on wheat and corn yields in France - 56/48