Frictionless consumer choices and price competition are often associated with competitive markets and vanishing equilibrium profits. We discuss vanishing profits in competitive screening markets like insurance. We assume symmetric firms which exhibit constant returns to scale. Consumer heterogeneity creates the possibility of adverse selection. Firms can offer multiple contracts in equilibrium and (importantly) in any deviation. Nash equilibrium profits vanish if each consumer has a unique optimizing bundle at equilibrium prices or, more generally, if there exists a linear ordering of contracts that dictates the preferences of firms whenever consumers are indifferent between multiple optimal contracts. Of particular interest, equilibrium profits vanish if, for each agent, indifference curves are steeper than iso-profit curves. The results extend to Miyazaki-Wilson-Spence equilibria. We provide examples of economies where there exists an equilibrium with strictly positive profit and show that these examples are robust (hold for an open set of economies).