We provide evidence that industries' supply curves are convex. To guide our empirical analysis, we develop a model, in which capacity constraints at the plant level generate convex supply curves at the industry level. The industry's capacity utilization rate is a sufficient statistic for the supply elasticity. Using data on capacity utilization and three different instruments, we estimate the supply curve and find robust evidence for an economically sizable degree of convexity. The nonlinearity we identify has several macroeconomic implications: Responses to shocks are state-dependent, the Phillips curve is convex, and higher welfare costs of business cycles than Lucas (1987).