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We propose a long-run risk model with stochastic volatility, a time-varying mean reversion level of volatility, and jumps in the state variables. The special feature of our model is that the jump intensity is not affine in the conditional variance but driven by a separate process. We show that this separation of jump risk from volatility risk is needed to match the empirically weak link between the level and the slope of the implied volatility smile for S&P 500 options.
This paper examines the welfare implications of rising temperatures. Using a standard VAR, we empirically show that a temperature shock has a sizable, negative and statistically significant impact on TFP, output, and labor productivity. We rationalize these findings within a production economy featuring long-run temperature risk. In the model, macro-aggregates drop in response to a temperature shock, consistent with the novel evidence in the data. Such adverse effects are long-lasting. Over a 50-year horizon, a one-standard deviation temperature shock lowers both cumulative output and labor productivity growth by 1.4 percentage points. Based on the model, we also show that temperature risk is associated with non-negligible welfare costs which amount to 18.4% of the agent's lifetime utility and grow exponentially with the size of the impact of temperature on TFP. Finally, we show that faster adaptation to temperature shocks results in lower welfare costs. These welfare benefits become substantially higher in the presence of permanent improvements in the speed of adaptation.