There is surprisingly scarce evidence regarding the extent to which and how government health expenditure affects health outcomes. Exploiting variation generated by Brazil’s 29th Constitutional Amendment, which mandated minimum thresholds for municipal spending on health, we examine the chain connecting government health spending to health inputs, production and outcomes, with a focus on infant mortality. We find relatively low average elasticities, but relevant heterogeneity in spending returns. Reductions in infant mortality are greater where baseline spending was lower, pointing to concave returns; where investments in infrastructure and personnel were complementary; and particularly where strong institutional and public management capabilities exist.