What determines demand elasticities and markups in production networks, where firms' demand and pricing are interdependent? I propose competition in markups, an equilibrium concept in which firms choose markups, taking other firms' markups as given. In partial equilibrium with vertical structures, it links upstream demand elasticities to downstream characteristics without backward induction and ties pass-through to the curvature of final demand. In general equilibrium with arbitrary industry networks, it microfounds firms' demand elasticities and markups through their within-industry market shares and four industry-level network statistics---cycle, substitution in production, substitution in consumption, and consumer-price impact. Empirical analysis of Compustat firms from 1997 to 2019 confirms the predicted variation of industry demand elasticities and firm markups. Relative to an exogenous-markup benchmark, endogenous markups dampen the aggregate TFP impact of a productivity shock to an industry's largest firm by 6% for the median industry, and by up to 48% across industries.