Inflation targets Consider a central bank that has an inflation target, \(\pi^{*}\). We studied two versions of the Phillips curve in Chapter 9. The general Phillips curve is:

\[
\pi_{t}-\pi_{t}^{e}=-\alpha\left(u_{t}-u_{n}ight)
\]

The first version of the Phillips curve in Chapter 9 was

\[
\pi_{t}-\pi_{t-1}=-\alpha\left(u_{t}-u_{n}ight)
\]

The second version of the Phillips curve in Chapter 9 was

\[
\pi_{t}-\bar{\pi}=-\alpha\left(u_{t}-u_{n}ight)
\]

a. How are the two versions of the Phillips curve different?

b. In either version, in principle, the central bank is able to keep the actual rate of inflation in period \(t\) equal to the target rate of inflation \(\pi^{*}\) in every period. How does the central bank carry out this task?

c. Suppose the expected rate of inflation is anchored (does not move) and equal to the target rate of inflation, that is, \(\bar{\pi}=\pi^{*}\). How does this situation make the central bank's task easier?

d. Suppose the expected rate of inflation is last period's rate of inflation rather than the target rate of inflation. How does this make the central banks task more difficult?

e. Use your answer to parts

(c) and

(d) to answer the question: Why is central bank credibility about the inflation target so useful?

f. In part (b), we asserted that the central bank could always hit its inflation target. Is this likely in practice?

g. One specific problem faced by the central bank is that the natural rate of unemployment is not known with certainty. Suppose the natural rate of unemployment, \(u_{n}\), changes frequently. How will these changes affect the central bank's ability to hit its inflation target? Explain.