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We here describe the data and the statistical framework. The Supplementary Information provides mathematical and computational details, additional results (including decompositions and summary statistics), and robustness checks.

Global climate model simulations.

For the medium-run estimation procedure, we use population-weighted temperature and precipitation for each country from 17 global climate models from the Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report with available simulations for each of the four Representative Concentration Pathways³⁰: BCC-CSM1-1, CCSM4, CESM1-CAM5, FIO-ESM, GFDL-CM3, GFDL-ESM2G, GISS-E2-H, GISS-E2-R, HadGEM2-AO, IPSL-CM5A-LR, IPSL-CM5A-MR, MIROC5, MIROC-ESM, MIROC-ESM-CHEM, MRI-CGCM3, NorESM1-M and NorESM1-ME. For the short-run estimation, we do not want to conflate intermodel variability with interannual variability. We therefore use population-weighted temperature and precipitation from five simulations of the NOAA-GFDL CM2.5 model following the RCP8.5 pathway³¹.

World Bank data set.

Initial GDP per capita comes from the World Bank’s purchasing power parity-adjusted, constant-dollars data set for the year 2010. That data set is in year 2005 dollars, which we rescale to year 2000 dollars using the current-dollars data set to match the units in the historical regressions following ref. 5. Year 2010 population also comes from a World Bank data set. In the global analyses, the initial log GDP per capita ln(y_r0) is 9.0642.

Impacts framework.

We seek distributions for the coefficients ψ in the following two relations:

where I_r^M and I_r^S give the medium- and short-run changes in growth rates in region r due to time t global mean surface temperature T_t^g and conditional on the log change in per-capita economic output (that is, in per-capita GDP) y_rt between the initial time and time t. ΔT_t^g is the change in global mean surface temperature between times t − 1 and t, y_rt is per-capita GDP in region r at time t, and y_r0 is per-capita GDP in region r in the initial period. Note that these equations describe future impacts. They are not regression equations for application to past data: as described below, we follow the fixed-effects specifications in ref. 5 when estimating historical relationships. We project a region’s impacts as a function of global mean surface temperature rather than of regional climate because we aim to produce an impact function that will be useful for climate–economy integrated assessment models, which often simulate only a single global temperature index. At a region’s initial GDP per capita y_r0, the coefficients ψ_{r, T}^M and ψ_{r, T}^S give the effect of, respectively, a 1 °C increase in decadal global mean temperature on medium-run growth and a 1 °C increase in global mean temperature on the variance of short-run growth (shown in Fig. 2). The coefficients ψ_{r, Ty}^M and ψ_{r, Ty}^S describe how temperature interacts with an e-fold (≈2.7-fold) increase in per-capita GDP.

Calculating probability distributions for ψ_r^j.

To obtain probability distributions for each vector of coefficients ψ_r^j in equations (1) and (2), we use the law of conditional probability:

We adapt ref. 5 to obtain central estimates and standard errors for the historical relationship between the climate and the economy (see Supplementary Information). These parameters define the probability p(ω^j) of any sampled set of historical relationships defined by the vector ω^j. Combining this probability p(ω^j) with the conditional probability p(ψ_r^j|ω^j) (described below) allows us to calculate an unconditioned distribution for ψ_r^j, which includes the economic uncertainty about historical climate–economy relationships via p(ω^j) and also scientific uncertainty about how future global mean surface temperature relates to country-level climatic outcomes via p(ψ_r^j|ω^j).

Figure 1 outlines how we calculate the conditional probability p(ψ_r^j|ω^j) by combining state-of-the-art physical climate simulations and socioeconomic projections to account for the spatially heterogeneous implications of global temperature change and for uncertainty about those implications. We begin with a sampled vector ω^j (box A) from historical climate–economy relationships, simulations of temperature and precipitation from physical climate models (box B), and population and GDP projections from the recently developed Shared Socioeconomic Pathways (SSPs; box C; ref. 32). All results in the main text use SSP2, which is the scenario of ‘middle’ challenges. Combining these country-level socioeconomic projections with the country-level climatic projections and each sampled ω^j yields projected impacts for each country (box D) at each decade (in the medium-run analysis) or each year (in the short-run analysis).

The medium-run and short-run specifications calculate these future country-level impacts differently. When estimating medium-run impacts from changing average weather outcomes, we convert each sampled vector ω^j into a sampled impacts trajectory by multiplying ω^j by each time step’s changes in average temperature and precipitation and by their interactions with log GDP per capita. Medium-run impacts arise from the change in global mean surface temperature at each time step rather than the absolute value of temperature because we assume that impacts over this time frame arise primarily from further changes in average climate rather than from changes that have already happened and may have triggered adaptation.

In contrast, when estimating short-run impacts from changing the variability of the weather, we use forecast errors in place of actual changes in temperature and precipitation: we match the panel estimation framework from ref. 5 by assuming that agents are harmed by unexpected weather shocks. This approach differs from literature that assesses whether the climate becomes more variable as global temperature increases^{33, 34, 35, 36}. To separate uncertainty about future warming from weather that is surprising conditional on global warming, we assume that agents correctly anticipate the next year’s global mean surface temperature. Agents then use a straightforward forecasting rule: a linear projection of time t + 1 country-level temperature (or precipitation) on time t country-level temperature (or precipitation) and time t + 1 global mean surface temperature. More complex forecasting methods exist in both the economics and climate science literatures, but the chosen rule is a reasonable heuristic for ordinary agents. In the Supplementary Information we assess robustness to the choice of forecasting rule. Within each simulation of CM2.5, agents estimate the linear projection’s coefficients via an ordinary least squares regression of historical country-level temperature (or precipitation) on lagged country-level temperature (or precipitation) and on contemporaneous global mean surface temperature. Agents use data from all previous years to construct forecasts for the next year.

The final steps in calculating the conditional probability p(ψ_r^j|ω^j) are to convert these country-level impacts into regional impacts and then estimate regional impacts as a function of global mean surface temperature. When the regions of interest encompass more than one country, we aggregate the country-level impacts via a social welfare function that can exhibit aversion to unequal consumption over space (box E). This approach seeks the impacts that affect a regional representative agent in the same way as the combination of its heterogeneous country-level impacts. We employ the same type of power utility social welfare function used to aggregate consumption over time in standard integrated assessment models, where the parameter η controls the degree of inequality aversion^{26, 27}. We vary the parameter between 0 (no inequality aversion) and 4 (high inequality aversion). When aggregating outcomes over time, standard integrated assessment models use values between 1 and 2 (refs 8, 37, 38), and values between 2 and 4 have also been recommended as reasonable^{39, 40}.

We estimate the parameter vector ψ_r^j that best fits a sampled ω^j by combining the simulated regional impacts with the same global climate models’ simulations of global mean surface temperature (box F). The coefficients and standard errors produced by this estimation define a distribution for each region’s desired impact coefficient ψ_r^j (box G), which provides us with the probability p(ψ_r^j|ω^j) of any given value of ψ_r^j given a sampled value of ω^j.

The conditional probability p(ψ_r^j|ω^j) captures uncertainty about how global warming affects country-level temperature and precipitation. This climatic uncertainty arises from variation across physical climate models and across emission scenarios; it does not include uncertainty generated by biases common to all physical climate models^URL: