Flexible Damage Functions

Author

James Rising, Sebastian Cadavid Sanchez, Climate Impact Lab

Published

April 30, 2026

0.1 Energy: Total Country Unconstrained

Flexible damage function parameters at the Country level.

Outcome units: kWh_per_capita — kWh per capita for total, rebased to 2005 baseline; DIAGNOSTIC variant with NO beta constraint applied (allows beta < 0 = concave temperature response). Compare with the constrained total_country version to see where the convexity constraint is binding and how it shifts the regional fit.

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

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 (\(\gamma\))	0.0659
Standard error	1.46e-02
95% CI	[0.0374, 0.0945]
R-squared	0.9084
Observations	3,519,180
Regions	245
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: 0.065944, 245 rows (of 4,655 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	-0.161	0.05301	0.4229	-2.674	0.235	245
beta	0.01022	0.003429	0.02589	-0.04062	0.1522	245
rho	0.02431	0.01166	0.1533	-0.3405	0.4524	245
zeta	0.05992	0.03743	0.05818	0.009076	0.5209	245
eta	0.1107	0.06665	0.11	0.01697	0.9411	245
rsqr1	0.5518	0.5966	0.2115	0.005605	0.8839	245
rsqr2	0.5477	0.5448	0.06525	0.3349	0.7424	245

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)	0	0.0%
T > 20°C (beyond graph)	27	11.0%
T < 0°C (negative crossing)	45	18.4%
Valid crossings (0-20°C)	173	70.6%

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)

6 Convexity Analysis

Figure 5: Beta Sign Distribution (Convexity)

6.1 Convexity by Country

Table 4: Fraction of Convex (beta > 0) Regions by Country (top 20)

country	n_regions	n_convex	frac_convex
ALA	1	1	100.0%
ALB	1	1	100.0%
AND	1	1	100.0%
ARG	1	1	100.0%
ARM	1	1	100.0%
ASM	1	1	100.0%
AUS	1	1	100.0%
AUT	1	1	100.0%
AZE	1	1	100.0%
BDI	1	1	100.0%
BEL	1	1	100.0%
BGR	1	1	100.0%
BHS	1	1	100.0%
BIH	1	1	100.0%
BLM	1	1	100.0%
BLR	1	1	100.0%
BMU	1	1	100.0%
BOL	1	1	100.0%
BRN	1	1	100.0%
BTN	1	1	100.0%

7 R-squared Analysis

7.1 Polynomial Fit Quality (rsqr1)

Figure 7: Regional Fit Quality (R-squared)

7.2 Error Model R-squared (rsqr2) Quantiles

Table 5: rsqr2 Quantiles

0% (Min)	25%	50% (Median)	75%	100% (Max)
0.3349	0.5073	0.5448	0.5929	0.7424

8 Modelled Variance

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

Table 6: Modelled Variance Statistics

Statistic	Value
Modelled variance	0.9812
Sum(η²)	5.9559
Sum(D²) at T=3.0°C	316.0486
N regions	245

9 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 7: Best- and worst-fitting regions

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

Region	α	β	η	R²
AND	-0.992	0.0318	0.214	0.884
LIE	-0.964	0.0343	0.224	0.869
RUS	-1.25	0.0569	0.305	0.846
NGA	0.0191	-0.00506	0.0708	0.00561
YEM	-0.0129	0.00435	0.0749	0.00902
GUY	0.0396	-0.00627	0.0629	0.0219

Figure 8: Fitted polynomial with raw data for worst-fitting regions

10 Regional Parameter Maps

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

10.1 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.

10.2 Beta (Quadratic Coefficient)

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

The no constraint applied (β unconstrained) constraint is enforced for this sector, meaning this is an unconstrained diagnostic run. β can take either sign, letting the data choose convex or concave shape per region..

10.3 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.

11 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 2099: 65,780 rows

Correlation: 0.8531 | RMSE: 1.685973 | Sign agreement: 69.0%