We characterize competitive equilibrium in markets (financial etc.) where price taking Bayesian decision makers screen to accept or reject applicants. Unlike signaling models, equilibrium fails to resolve imperfect information. In classical statistics terminology, some qualified applicants are rejected (type I error) and some unqualified applicants are accepted (type II error). We report three new results: i. optimal firm behavior is deduced to be a Bayesian variant of the Neyman-Pearson theorem; ii. competitive equilibrium entails screening if and only if (net of screening costs) the cost of type II errors exceed the cost of type I errors, i.e. contrary to signaling (where buyers identify more qualified applicants who self screen to differentiate themselves e.g. Stiglitz 1975), price taking firms screen to avoid lower quality sellers; iii. equilibrium groups the least attractive applicants into a single high risk assignment pool. Depending on costs of screening, the unique equilibrium may involve complete pooling (all applicants trade at one price) or partial separation (there are m separate pools with successive pools supported by a single (rising) price and a subset of agents of different screen levels trading at that price). A screening equilibrium has and the mth secondary market entails no screening, as the most adversely selected agents are assigned to the high risk pool. Screening induces market segmentation. Invariably secondary markets contain individuals who with better or different screening mechanisms could be accepted in the primary market. What roles traits such as ethnicity, gender, and race might assume in such decision making is relegated to subsequent research to explore the statistical theory of discrimination
