11.8. Consider the system given by (11.41). (a) What does the system simplify to when = 1? What are the eigenvalues of the system

11.8. Consider the system given by (11.41).

(a) What does the system simplify to when φπ = 1? What are the eigenvalues of the system in this case? Suppose we look for self-fulfilling movements in ˜y and π of the form πt = λtZ, ˜yt = cλtZ, |λ| ≤ 1. When φπ = 1, for what values of λ and c does such a solution satisfy (11.41)? Thus, what form do the self-fulfilling movements in inflation and output take?

(b) Suppose φπ is slightly (that is, infinitesimally) greater than 1. Are both eigenvalues inside the unit circle? What if φπ is slightly less than 1?

(c) Suppose κ(1 − φπ)/θ = −2(1 + β). What does the system simplify to in this case? What are the eigenvalues of the system in this case? Suppose we look for self-fulfilling movements in ˜y and π of the form πt = λtZ, y˜t = cλtZ, |λ| ≤ 1. When κ(1 − φπ)/θ = −2(1 + β), for what values of

λ and c does such a solution satisfy (11.41)? Thus, what form do the self-fulfilling movements in inflation and output take?