Consider the situation analyzed in Problem 11.16, but assume that there is only some finite number of periods rather than an infinite number. What is the unique equilibrium? (Hint: Reason backward from the last period.)

Sol:

Consider the situation in the last period, denoted T. The policymaker's choice of p has no effect on next period's expected inflation; there is no next period. Thus the policymaker's problem in the final period is to take expected inflation as given and choose p in order to maximize the period T objective function. From previous analysis in the solution to Problem 11.16, we know that the policymaker's choice of inflation in this type of situation is p_T = b/a. Since the public knows how the policymaker behaves, expected inflation also equals b/a and thus output equals y n

Now consider the situation in period T - 1. The important point is that the policymaker knows her choice of pT-1 will have no bearing on what happens the next and final period. Regardless of the level of p she chooses in period T - 1, expected inflation next period will be b/a, as described above. Since the policymaker's problem has no impact on the future, she chooses p, taking pe as given, in order to maximize the period T - 1 objective function. Again, the optimal choice is p_T-1 = b/a. The public knows this and so p_T-1^e = b/a and thus output in period T - 1 equals yⁿ.

Working backward, the same thing happens each period. The policymaker knows that expected inflation the following period will be b/a regardless of what she does this period. Thus she acts to maximize the one-period objective function and chooses p = b/a, which results in output equal to the natural rate. Therefore the unique equilibrium for all periods is

Sol:

The politician faces the following problem, where E is defined as the probability of being reelected:

for t = 1, 2. Substituting equation (2) evaluated at t = 2 into the probability that the politician is reelected yields

Since f(·) ³ 0, the derivative given in equation (8) is greater than or equal to zero. Thus picking a higher value for first-period unemployment can never reduce the probability of being reelected and might increase it. Thus it is optimal to pick the highest feasible level of unemployment in the first period, u_H.

Intuitively, since only second-period inflation and unemployment determine the probability of being reelected, the politician wants to face the best possible inflation-unemployment tradeoff in period 2. From equation (2), we can see that is accomplished by having the lowest possible inflation rate in the previous period, period 1. That, in turn, is accomplished by having the highest possible unemployment rate in period 1.

Problem 16