A growing empirical literature associates climate anomalies with increased risk of violent conflict. This association has been portrayed as a bellwether of future societal instability as the frequency and intensity of extreme weather events are predicted to increase. Informally, many projections allude to an opportunity-cost mechanism, but this mechanism has been formalized only for time-constant income distributions. We extend the classic microeconomic model (Chassang'09) by considering realistic changes in the distribution of climate-dependent agricultural income. Results urge caution regarding naive extrapolation from established patterns: Sustained shifts in the income distribution have different implications than anomalous individual draws. Although war occurs in bad years, conflict may decrease if agents expect more frequent bad years. Unless the model parameters are properly identified, the conflict impact of increased climate variability is thus ambiguous under the opportunity cost mechanism. And since the relation between crop yields and water availability is nonlinear, changes in both the level and the variability of climate affect conflict frequency. This stresses the importance of distinguishing income shocks and climate shocks in empirical work. We identify three measurable statistics of the income distribution that each offer unambiguous qualitative predictions regarding conflict frequency. Since these predictions differ from competing explanations for climate-induced conflict, they may offer a blueprint to empirically distinguish between these mechanisms.