You have a perfect foresight equilibrium when everyone knows the whole future and plans for it optimally. Set expected inflation equal to actual inflation, since there are no random variables in this equation. With RE and no random variables, unemployment always equals its natural rate, yet inflation is well above zero due to the temptation to use policy to reduce unemployment. We'll try it out on a simple example. Consider an economy described by 5 equations.
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.