9.11. More on solving the dynamic-inconsistency problem through reputation. (This is based on Cukierman and Meltzer, 1986.) Consider a policymaker who is in office for two periods and whose objective function is

can/2]. The policymaker is chosen randomly from a pool of possible policymakers with differing tastes. Specifically, c is distributed normally over possible policymakers with mean c and variance o> 0. a and b are the same for all possible policymakers. Er, The policymaker cannot control inflation perfectly. Instead, where is chosen by the policymaker (taking as given) and where , is normal with mean zero and variance > 0. E1, E2, and c are independent. The public does not observe, and , separately, but only . Similarly, the public does not observe

c. Finally, assume that is a linear function of : = a + B.

(a) What is the policymaker's choice of 2? What is the resulting expected value of the policymaker's second-period objective function, b(2-2)+ Cm2 am/2, as a function of ?

(b) What is the policymaker's choice of taking a and as given and ac- counting for the impact of on?

(c) Assuming rational expectations, what is ? (Hint: use the signal-extraction procedure described in Section 6.3).

(d) Explain intuitively why the policymaker chooses a lower value of * in the first period than in the second.

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