All variables are logs of their level values. Assume that all expectations are rational - i.e., with 0 mean error. Then, conjecture the form of the solution. Then Substitute in for the expectations. The price level is directly proportional to the money supply, a positive function of investment shocks, a negative function of money demand shocks, and a negative function of unexpected money supply increases.
Economies are always exceptionally at their optimum.
Insofar as policy makes prediction more difficult, discretionary policy is itself a major cause of non-optimalities. For mathematical simplicity more than anything else, the probability that you will be able to change your price today is independent of how long since you were last allowed to change your price. Consider the case of an individual price-setter with a fixed capacity. Assume further that you anticipate no changes in AD, and all other producers are producing at capacity. So long as your impact on the overall array of prices is negligible, you will want to set your price so that you too produce at full capacity.