Mutations of the Phillips curve Suppose that the Phillips curve is given by

\[
\pi_{t}=\pi_{t}^{e}+0.1-2 u_{t}
\]

and suppose that \(\theta\) is initially equal to \(O\) and \(\bar{\pi}\) is given and does not change. It could be zero or any positive value. Suppose that the rate of unemployment is initially equal to the natural rate. In year \(t\), the authorities decide to bring the unemployment rate down to 3\% and hold it there forever.

a. Determine the rate of inflation in periods \(t+1, t+2\), \(t+3, t+4, t+5\). How does \(\bar{\pi}\) compare to pibar?

b. Do you believe the answer given in (a)? Why or why not? (Hint: Think about how people are more likely to form expectations of inflation.)

Now suppose that in year \(t+6, \theta\) increases from 0 to 1 . Suppose that the government is still determined to keep \(u\) at \(3 \%\) forever.

c. Why might \(\theta\) increase in this way?

d. What will the inflation rate be in years \(t+6, t+7\), and \(t+8\) ?

e. What happens to inflation when \(\theta=1\) and unemployment is kept below the natural rate of unemployment?

f. What happens to inflation when \(\theta=1\) and unemployment is kept at the natural rate of unemployment?