Common value experimentation

In many economic environments, agents often continue to learn about the same underlying state variable, even if they switch action. For example, a worker’s ability revealed in one job is informative about her productivity in another job. We analyze a general setup of experimentation with common values, and show that the value of experimentation must be equal whenever the agent switches action. In addition to the well-known conditions of value matching (level) and smooth pasting (first derivative), this implies that the second derivatives of the value function must be equal at the switching point. This condition holds generally whenever the stochastic process has continuous increments.