Signaling through tests

A firm (sender), privately informed about product quality, chooses a public test. Tests vary in informativeness and return a binary result. The market (receiver) forms interim beliefs based on test informativeness and posteriors based on test results. We show that standard single-crossing does not hold everywhere. A more informative test is less costly to the high type, who fails it less often, while better interim beliefs may benefit either the high or the low type depending on the prior. When a firm's expected quality is low, an increase in interim beliefs makes the market more sensitive to test results. Then the high type has more incentives to choose a more informative test and separation occurs. When a firm's expected quality is high, a further increase in interim beliefs makes the market less sensitive to test results. In this case, the unique equilibrium is pooling with both types choosing a test with intermediate level of informativeness.