The threat of antibiotic resistance has generated substantial policy attention toward reducing antibiotic use. However, antibiotic use creates a second, opposing externality: innovation incentives rely on revenues from antibiotic sales. While direct payments for innovation have been proposed to decouple use and revenues, their optimal size and welfare consequences remain poorly understood. This paper introduces a framework for optimal antibiotic policy that accounts for use, resistance, and innovation. We estimate key parameters—the demand for antibiotics, the use-resistance externality, and the effect of use on mortality—using electronic health records data and quasi-experimental variation in antibiotic restrictions. We embed these estimates in a structural model to evaluate optimal demand incentives and innovation prizes. We find that antibiotic use lowers mortality but generates substantial resistance externalities: reducing use by 20% lowers resistance by 60%. However, the innovation externality is large and acts in the opposite direction. We show that, contrary to conventional wisdom, antibiotics may often be underused rather than overused. Jointly optimal prizes and utilization policies decrease mortality in our sample by 7.4% and raise welfare by $56.9 billion per year. Across sensitivity analyses, annualized joint welfare gains range from $12.5 billion to $141.2 billion. More broadly, the results illustrate how policies that target opposing externalities can yield large welfare gains when multiple policy instruments are used in combination.