Flexible Damage Functions

Author

James Rising, Sebastian Cadavid Sanchez, Climate Impact Lab

Published

September 14, 2026

0.1 Agriculture: Wheat Spring Country Collapsed

Flexible damage function parameters at the Country level.

Change in spring wheat yields (log change in yields) under full adaptation

Outcome units: log_yield_change. Log change in spring wheat yield (kg/Ha, log scale): country-level rows extracted from the 1% winsorization aggregated.nc4 file, mean-collapsed over the MC ensemble (batch x GCM).

\[D_{it} = (\alpha_i T_t + \beta_i T_t^2) \cdot Y_{it}^{\gamma}\]

where \(\gamma\) is the income elasticity (fitted globally), \(\alpha_i\) is the linear coefficient, and \(\beta_i\) is the quadratic coefficient.

1 Global Estimation

1.1 Income Elasticity Estimation

The income elasticity \(\gamma\) is estimated using a fixed-effects regression:

\[y_{it} = \gamma \cdot \log(Y_{it}) + \mu_{g(i,T)} + \nu_t + \varepsilon_{it}\]

where \(\mu_{g(i,T)}\) are region-by-temperature-bin fixed effects and \(\nu_t\) are year fixed effects.

Table 1: Income Elasticity (Gamma) Estimation Results

Statistic	Value
Income elasticity (\(\gamma\))	-0.0011
Standard error	2.94e-03
95% CI	[-0.0069, 0.0046]
R-squared	0.7514
Observations	93,343
Regions	138
Gamma quantiles	19

2 Parameter Distributions

2.1 8-Panel Summary

Row 1: gamma, alpha, beta, rsqr1. Row 2: rho, zeta, eta, rsqr2.

Using median gamma: -0.001128, 138 rows (of 2,622 total)

Figure 1: Distribution of Regional Parameters

2.2 Projection Equation and Parameter Definitions

The estimated parameters are used to project damages via Monte Carlo sampling:

\[D_{it}^k = (\hat{\alpha}_{ik} T_t + \hat{\beta}_{ik} T_t^2) Y_{it}^{\hat{\gamma}_k} + \hat{\theta}_{ik} T_t Y_{it}^{\hat{\gamma}_k} + \hat{\phi}_{it}^k\]

where \(k\) indexes the Monte Carlo draw. The parameters in each row of the output CSV control distinct components of this equation:

gamma: income elasticity \(\hat{\gamma}_k\), one of 19 quantile values drawn from \(N(\hat{\gamma}, SE(\hat{\gamma}))\)
alpha, beta: linear and quadratic temperature coefficients; \(\hat{\alpha}_{ik}\) and \(\hat{\beta}_{ik}\) are drawn from the joint normal defined by the VCV below
sigma11, sigma12, sigma22: variance-covariance matrix of \((\alpha, \beta)\), used for joint uncertainty sampling
rho: correlation between regional and global polynomial residuals \(\rho_i\), used to maintain spatial covariance across regions in Monte Carlo draws
zeta: temperature-dependent error scale \(\zeta_{ik}\); the run-specific error term \(\hat{\theta}_{ik}\) is drawn from \(N(0, \zeta_{ik})\)
eta: residual noise standard deviation \(\eta_{ik}\); the annual noise \(\hat{\phi}_{it}^k\) is drawn from \(N(0, \eta_{ik})\)
rsqr1, rsqr2: polynomial fit quality and error model fit, respectively

2.3 Summary Statistics

Table 2: Regional Parameter Summary

Parameter	Mean	Median	Std	Min	Max	N
alpha	-0.01906	-0.02749	0.04869	-0.2229	0.1542	138
beta	-0.004298	-0.001805	0.01047	-0.08901	0	138
rho	0.04002	0.0306	0.1293	-0.2075	0.9189	138
zeta	0.01241	0.006109	0.02259	0.002483	0.1631	138
eta	0.02408	0.01066	0.04617	0.004393	0.3452	138
rsqr1	0.7138	0.8784	0.2973	0.003252	0.9679	138
rsqr2	0.534	0.5402	0.06921	0.1864	0.6787	138

3 Spaghetti Curves

Regional damage function curves showing D(T) = αT + βT² for sampled regions.

Figure 2: Regional Damage Functions

4 Zero Crossings

The zero crossing (extremum) of the parabola occurs at \(T^* = -\alpha / (2\beta)\).

Table 3: Zero Crossing Statistics

Category	Count	Percentage
β = 0 (no crossing)	31	22.5%
T > 20°C (beyond graph)	2	1.4%
T < 0°C (negative crossing)	82	59.4%
Valid crossings (0-20°C)	23	16.7%

Figure 3: Distribution of Zero Crossing Temperatures

5 Slope Analysis

Maximum slope between 0 and 10°C: \(\frac{dM}{dT} = \alpha + 2\beta T\)

The maximum occurs at either T=0 or T=10 (endpoints of interval).

Figure 4: Distribution of Maximum Slopes (0-10°C)

Convexity analysis omitted (beta constraint active).

6 R-squared Analysis

6.1 Polynomial Fit Quality (rsqr1)

Figure 5: Regional Fit Quality (R-squared)

6.2 Error Model R-squared (rsqr2) Quantiles

Table 4: rsqr2 Quantiles

0% (Min)	25%	50% (Median)	75%	100% (Max)
0.1864	0.5028	0.5402	0.5731	0.6787

7 Modelled Variance

Modelled variance statistic: \(1 - \frac{\sum_i \eta_i^2}{\sum_i D_i^2}\)

Table 5: Modelled Variance Statistics

Statistic	Value
Modelled variance	0.9377
Sum(η²)	0.3720
Sum(D²) at T=3.0°C	5.9684
N regions	138

8 Best- and Worst-Fitting Regions

The 3 worst- and 3 best-fitting regions by R-squared, with raw simulation data overlaid on the fitted polynomial curve. Red rows = worst fits, green rows = best fits.

Table 6: Best- and worst-fitting regions

Top 3 best fit (R²) Top 3 worst fit (R²)

Region	α	β	η	R²
LBY	-0.0596	-0.00189	0.00732	0.968
ISR	-0.0663	-0.00302	0.00968	0.967
KWT	-0.116	-8.62e-4	0.0132	0.966
AGO	0.00554	-9.74e-4	0.00971	0.0124
BLZ	0.0861	-0.0302	0.225	0.0272
SLV	0.0622	-0.00772	0.0899	0.0598

Figure 6: Fitted polynomial with raw data for worst-fitting regions

9 Regional Parameter Maps

Maps of key parameters at the country level. Red = negative (damage increases with T), Blue = positive.

9.1 Alpha (Linear Coefficient)

Alpha (\(\alpha\)) represents the linear sensitivity to temperature. Regions with negative alpha experience damage that increases with the first degree of warming.

9.2 Beta (Quadratic Coefficient)

Beta (\(\beta\)) represents the curvature of the damage function.

The concavity (\(\beta \leq 0\)) constraint is enforced for this sector, meaning optimal temperature exists, damages accelerate beyond it.

9.3 R-squared (Fit Quality)

\(R^2\) measures the polynomial fit quality. Higher values indicate that the quadratic form captures more of the variance in the data.

10 F2: Flex vs Raw Comparison

Comparison of flexible damage function predictions against raw simulation means.

For each scenario (RCP × SSP × Model), we compute at the final year:

Flex predicted: \((\alpha \cdot T + \beta \cdot T^2) \cdot Y^\gamma\)
Raw actual: The outcome variable (y) from the source data

Year 2098: 2,024 rows

Correlation: 0.6437 | RMSE: 0.172115 | Sign agreement: 76.1%