Toxic effects of superoxide dismutase (SOD) overexpression are commonly attributed to increased hydrogen peroxide (H₂O₂) production. Still, published experiments yield contradictory evidence on whether SOD overexpression increases or decreases H₂O₂ production. We analyzed this issue using a minimal mathematical model. The most relevant mechanisms of superoxide consumption are treated as pseudo first-order processes, and both superoxide production and the activity of enzymes other than SOD were considered constant. Even within this simple framework, SOD overexpression may increase, hold constant, or decrease H₂O₂ production. At normal SOD levels, the outcome depends on the ratio between the rate of processes that consume superoxide without forming H₂O₂ and the rate of processes that consume superoxide with high (≥ 1) H₂O₂ yield. In cells or cellular compartments where this ratio is exceptionally low (< 1), a modest decrease in H₂O₂ production upon SOD overexpression is expected. Where the ratio is higher than unity, H₂O₂ production should increase, but at most linearly, with SOD activity. The results are consistent with the available experimental observations. According to the minimal model, only where most superoxide is eliminated through H₂O₂-free processes does SOD activity have the moderately large influence on H₂O₂ production observed in some experiments.