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138 Analyzing the Model Since a Type2 policymaker always sets inflation to zero, we focus on the behavior of a Type1 policymaker. In the second period, he or she takes irf as given, and therefore chooses ttz to maximize ( 2  trf) . The solution is 772 = b/a. The policymakers firstperiod problem is more complicated, because his or her choice of inflation affects expected inflation in the second period. If the policymaker chooses any value of tti other than zero, the public learns that it is facing a Type1 policymaker, and therefore expects inflation of b/a in the second period. Conditional on \ not equaling 0, the choice of 7 1 has no effect on trf. Thus if the policymaker chooses a nonzero firstperiod inflation rate, he or she chooses it to maximize [  irf)  1/2, and therefore sets tti = b/a. Trf and 2 are then both equal to b/a, and equals y. The value of the objective function for the two periods in this case is thus niNF = (9.16) 2 1 = ai3)b7rf. The Type1 policymakers other possibility is to set m to 0. It turns out that in equilibrium, he or she may randomize between 771 = fo/aand77  O.Thus, let q denote the probability that the Type1 policymaker chooses 771 = 0. Now consider the pubUcs inference problem if it observes zero inflation. It knows that this means either that the policymaker is a Type 2 (which occurs with probabflity 1  p), or that the policymaker is a Type 1 but chose zero inflation (which occurs with probability pq). Thus, by Bayess law, its estimate of the probability that the pohcymaker is a Type 1 is 7 /[(1  p) b qp]. Its expectation of 772 is therefore {qp/[{lp) + qp]}(b/a), which is less than b/a. This analysis implies that the value of the objective function when the pohcymaker chooses 771 = 0 is Wo(q)=b(7rt) + IB 2 {l~p)+qp a (1  p) + qp a qp I ( \ (9.17)  birl. "The key assumpuon is that the two types have different preferences, not that one type always chooses zero inflation. where /3 reflects the importance of the second period in social welfare. A Type2 policymaker, which occurs with probability 1  p, cares only about inflation, and therefore sets inflation to zero in both periods."
or simply <\. (9.19) Thus if the weight on the second period is sufficiently small, the publics uncertainty about the policymakers type has no effects. The second possibility arises when fKo(l) is greater than fliNP In this situation, the Type1 policymaker always chooses it\ = 0: even if the public learns nothing about the policymakers type from observing \ = 0, the cost of revealing that he or she is a Type1 is enough to dissuade the policymaker from choosing positive inflation. Equations (9.16) and (9.17) imply that IFo(l) exceeds ffiNP when { , {>\{\ ) {. (9.20) This condition simplifies to >\~. (9.21) The final possibility arises when (0) > > )(1); the preceding analysis implies that this occurs when 1/2 < /3 < (1/2)[1/(1  p)]. In this case, Type1 pohcymakers would choose zero firstperiod inflation if the public believes they would choose positive inflation, and would choose positive inflation if the public beheves they would choose zero. As a result, the economy can be in equilibrium only if the Type1 policymakers sometimes choose positive inflation and sometimes choose zero. Specifically, q must adjust to the point where the Type1 policymakers are indifferent between 771 = 0 and 771 = b/a. Equating (9.16) and (9.17) and solving for q shows that this requires g = p(2;6l) if<<ilp (922» Note that Woiq) is decreasing in q, the probability that the Type1 pohcymaker chooses zero inflation in the first period: a higher q implies a higher value of Tjf if ttj = 0, and thus a smaller value to the policymaker of choosing 7 1 = 0. The equilibrium of the model can take three possible forms. The first possibility occurs if Ifo(O) is less than PVinf In this case, even if the Type1 policymaker can cause the public to be certain that it is facing a Type2 pohcymaker by setting = 0, he or she will not want to do so. Thus in this case the Type1 policymaker always chooses iri = b/a. Equations (9.16) and (9.17) imply that WoiO) is less than Winf when i? 1 fo2 q /3  <{\ ) , (9.18)
Delegation A second way to overcome the dynamic inconsistency of lowinflation monetary policy is to delegate policy to individuals who do not share the publics For a general value of /8 > 1/2, one can show that the maximum effect occurs at p = (2/3  l)/2/3, and equals [(2/3  \)/2 \ . For /3 < 1/2, there is no effect. Discussion Although this model is highly stylized, the basic idea is simple. The public is unsure about what policies the government will follow in future periods. Under plausible assumptions, the lower the inflation it observes today, the lower its expectations of inflation in future periods. This gives policymakers an incentive to keep inflation low. Because of the simplicity of the central idea, the basic result that uncertainty about policymakers characteristics reduces inflation is highly robust (see, for example, Vickers, 1986; Cukierman and Mehzer, 1986; Rogoff, 1987; and Problem 9.11). This analysis implies that the impact of reputational considerations on inflation is greater when policymakers place more weight on future periods. Specifically, the probabiUty that a Type1 policymaker chooses tti = 0 IS increasing in /3 for 1/2 < /3 < (1/2)[1/(1  p)], and is independent of /3 elsewhere. Similarly, one can show that the impact of the reputational considerations is greater when there are more periods. The model also impUes that the impact on inflation is greater when there is more uncertainty about policymakers characteristics. To see this, consider, for SimpUcity, the case of /3 = 1. If the pohcymakers type is publicly observed, the Type Is always set tti = b/a and the Type 2s always set 7 1 = 0. Under imperfect information, however, the Type Is set tti = 0 with probability q. Thus the uncertainty lowers average firstperiod inflation by pq(b/a). With /3 = 1, (9.21) implies that q = I when p < 1/2; thus for these values of p, the reduction in average firstperiod inflation is pb/a. And (9.22) implies that q = (l  p)/p when p > 1/2; thus for these values, the reduction is (1  p)b/a. The maximum reduction thus occurs at p = 1 /2, and equals b/2a.ln short, the impact of the reputational considerations is greater when the difference between the two types preferred inflation rates is larger (that is, when b/a is larger) and when there is more uncertainty about the policymakers type (that is, when p is closer to 1/2). The idea that reputational considerations cause poUcymakers to pursue less expansionary policies seems not only theoretically robust, but also realistic. Central bankers appear to be very concerned with establishing reputations as being tough on inflation and as being credible. If the public were certain of policymakers preferences and beliefs, there would be no reason for this. Only if the public is uncertain and if expectations matter is this concern appropriate.
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