Question: Suppose that the central bank is in a repeated relationship

Suppose that the central bank is in a repeated relationship with the private sector. If the inflation rate is i = i* and output is Y = YT, then suppose the reward each period to the central bank is u1. If consumers anticipate ie = i*, then the one-period reward the central bank receives from fooling consumers with i > ie is u2 and we will have i > ie = i*. When the central bank cannot commit and there is rational expectations, then the reward the central bank gets is u3 < u1 with i = ie = i1 > i* and Y = YT. Now, suppose that the government anticipates that if it deviates from i = i*, then it will totally lose its reputation and consumers will anticipate i = i1 forever after, so that the government will receive a reward of u3 forever after. That is, suppose the central bank has two alternatives, which are (a) set i = i* forever receiving reward u1, and (b) cheat for this period, receiving reward u2 implying that the reward will be u3 in every future period. Suppose that the central bank discounts the future at the rate r.
(a) Show that it is an equilibrium for the government to choose i = i* forever if
u1(1 + r) – u3 > ru2.
(b) Interpret the condition in part (a).
(c) Show that, as r becomes small, the condition in part (a) must hold, and explain this.

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