9.8. Solving the dynamic-inconsistency problem through punishment. (Barro and Gordon, 1983b.) Consider a policymaker whose objection function is By a12), where a > 0 and 0 < < 1. y, is determined by the Lucas supply curve, (9.8), each period. Expected inflation is determined as follows. If has equaled (where is a parameter) in all previous periods, then If ever differs from, then = b/a in all subsequent periods. .

(a) What is the equilibrium of the model in all subsequent periods if ever differs from ?

(b) Suppose has always been equal to , so. If the monetary author- ity chooses to depart from, what value of does it choose? What level of its lifetime objective function does it attain under this strategy? If the monetary authority continues to choose every period, what level of its lifetime objective function does it attain?

(c) For what values of does the monetary authority choose 7 = ? Are there values of

a, b, and such that if = 0, the monetary authority chooses π =0?