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We provide evidence that industries' supply curves are convex. To guide our empirical analysis, we develop a model in which capacity constraints at the firm level generate supply curves that are convex in logs 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, including that responses to shocks are state dependent and that the Phillips curve is convex.