Testing the sender: When signaling is not enough

We study signaling in the presence of endogenous information acquisition by the receiver. A firm, after observing the worker’s costly action, may acquire further information by performing a test on the applicant, and then decide to hire him. We consider different models of information acquisition, including the rational inattention, a generalization of the “truth or noise” and a general grading model. We study test effectiveness as function of beliefs generated through signaling by the worker, and provide clear-cut predictions on the complementarity/substitutability between costly information transmission (signaling) and acquisition, and its implications for the equilibrium. We first show that test effectiveness is non-monotone in beliefs. It exhibits increasing regions, where beliefs and test effectiveness act as complements, with higher beliefs inducing more effective tests, and decreasing regions in which higher beliefs crowd out the firm’s information acquisition. We then show that, when beliefs and test effectiveness are complements, the equilibrium involves (at least partial) separation between workers’ types. Since the high type is more willing to face a more exacting test than a low type, he will exert costly effort to improve the firm’s opinion. When beliefs and test effectiveness are substitutes, any signaling attempt by the high type will be mimicked by the low one who benefits more from relaxed standards and indiscriminate hiring, and the only plausible equilibrium involves both types pooling.