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I analyze the marginal value of reducing greenhouse gas emissions (the “social cost of carbon”) under uncertainty about warming, under uncertainty about how much warming reduces consumption, and under stochastic shocks to consumption growth. I theoretically demonstrate that each of these sources of uncertainty increases the social cost of carbon under conventional preferences. In a calibrated numerical implementation, uncertainty increases the 200-year social cost of carbon by more than 20%. Uncertainty about the consumption impacts of warming contributes the most to this premium and makes the social cost of carbon sensitive to impacts even after 2400.