Signalling with career concerns

Consider an agent (manager, artist, etc.) who has imperfect private information about his productivity. At the beginning of his career (period 1, short run"), the agent chooses among publicly observable actions that generate imperfect signals of his productivity. The actions can be ranked according to the informativeness of the signals they generate. The market observes the agent’s action and the signal generated by it, and pays a wage equal to his expected productivity. In period 2 (the long run"), the agent chooses between a constant payoff and a wage proportional to his true productivity, and the game ends. We show that in any equilibrium where not all types of the agent choose the same action, the average productivity of an agent choosing a less informative action is greater. However, the types choosing that action are not uniformly higher. In particular, we derive conditions for the existence of a tripartite equilibrium where low and high types pool on a less informative action while medium (on average, lower) types choose to send a more informative signal.