The natural rate of unemployment

a. The Phillips curve is \(\pi_{t}=\pi_{t}^{e}+(m+z)-\alpha u_{t}\). Rewrite this relation as a relation between the deviation of the unemployment rate from the natural rate, inflation, and expected inflation.

b. We derived the natural rate of unemployment. What condition on the price level and the expected price level was imposed in that derivation? How does it relate to the condition imposed in part (a)?

c. How does the natural rate of unemployment vary with the markup?

d. How does the natural rate of unemployment vary with the catchall term \(z\) ?

e. Identify two important sources of variation in the natural rate of unemployment across countries and across time.