11.20. Rational political business cycles. (Alesina and Sachs, 1988.) Suppose the relationship between output and inflation is given by yt = y +b(πt − Et −1πt), where b > 0 and where Et −1 denotes the expectation as of period t − 1. Suppose there are two types of politicians, “liberals” and “conservatives.” Liberals maximize aL yt − π 2 t /2 each period, and conservatives maximize acyt −

π 2 t /2, where aL > aC > 0. Elected leaders stay in office for two periods. In period 0, it is not known who the leader in period 1 will be; it will be a liberal with probability p and a conservative with probability 1− p. In period 1, the identity of the period-2 leader is known.

(a) Given Et −1πt, what value of yt will a liberal leader choose? What value will a conservative leader choose?

(b) What is E0π1? If a liberal is elected, what are π1 and Y1? If a conservative is elected, what are π1 and y1?

(c) If a liberal is elected, what are π2 and y2? If a conservative is elected, what are π2 and y2?