a. The relation between the deviation of the unemployment rate from the natural rate of unemployment (NRU), inflation, and expected inflation can be obtained by subtracting the natural rate of unemployment from both sides of the Phillips curve equation and rearranging:
πt - πt^e = (m + z) - α(u_t - u^*)
where u_t - u^* is the deviation of the unemployment rate from the natural rate of unemployment.
b. In part a, we assumed that the actual inflation rate equals the expected inflation rate, i.e., πt = πt^e. This condition implies that there are no surprises in the economy that affect the inflation rate, and therefore the economy is in long-run equilibrium. When we derived the natural rate of unemployment using the wage setting and price setting equations, we also assumed that the price level and expected price level are equal in the long run, i.e., P_t = P_t^e. This condition implies that there are no persistent shocks to the economy that affect the price level, and therefore the economy is in a steady state.