( ) Some argue that the hypothesis that the real mterest rate is constant implies that nominal interest rates move one-for-one with actual inflation m the long run that is, that the hypothesis implies that in a regression of i on a constant and the current and many lagged values of tt, the sum of the coefhaents on the inflation variables will be 1. Is this claim correct? (Hint: Suppose that the behavior of actual inflation is given by , = ,-\ + ,, where e is white noise.)

9.5. Policy rules, rational expectations, and regime changes. (See Lucas, 1976, and Sargent, 1983.) Suppose that aggregate supply is given by the Lucas supply curve, Yt =y + , - TTf"), b > 0, and suppose that monetary policy is determined by , = Wf-i + a + e,, where £ is a white-noise disturbance. Assume that private agents do not know the current values of m, or e,; thus -f is the expectation of p, - Pt-i given m, i, £, i,y, i , and Finally, assume that aggregate demand is given by y, = m, - p,.

(a) Find yt in terms of j, m,, and any other variables or parameters that are relevant.

(b) Are mt-i and m, all one needs to know about monetary policy to find yt? Explain intuitively.

(c) Suppose that monetary policy is initially determined as above, with a > 0, and that the monetary authority then announces that it is switching to a new regime where a is zero. Suppose that private agents believe that the probability that the announcement is true is p. What is y, in terms of Wt-i, "it> p, y, b, and the initial value of a

(d) Using these results, describe how an examination of the money-output relationship might be used to measure the credibility of announcements of regime changes.

9.6. Regime changes and the term structure of interest rates. (See Blanchard, 1984; Mankiw and Miron, 1986; and Mankiw, Miron, and Weil, 1987.) Consider an economy where money is neutral. Specifically, assume that tti = Am, and that r is constant at zero. Suppose that the money supply is given by Am, = kAWf-i -I- where f is a white-noise disturbance.

(a) Assume that the rational-expectations theory of the term structure of interest rates holds (see [9.6]). Specifically, assume that the two-period interest rate is given by z, = (i, -i- Etit\i)/2. i/ denotes the nominal interest rate from f to f -H 1; thus, by the Fisher identity, it equals r, + E,lp,+i] - p,.

(i) What is i/ as a function of Am, and fc? (Assume that Am, is known at time r.)

(zi) What is ffif+i as a function of Am, and k?

(zzz) What is the relation between z/ and i,; that is, what is i, as a function of Zfi and fc?

(zv) How would a change in affect the relation between z, and z/ ? Explain intuitively.

(b) Suppose that the two-period rate includes a time-varying term premium: z/ = (z,i -I- £tz/.,)/2 -I- ,, where e is a white-noise disturbance that is independent of £. Consider the OLS regression - = a + b(i - z/) -t e,+i.

(i) Under die ranonal-expectations theory of the term structure (with e, = 0 for all t), what value would one expect for b (Hint: for a univariate OLS regression, the coefficient on the right-hand-slde variable equals the covariance between the right-hand-side and left-hand-side variables divided by the variance of the right-hand-slde variable.)

(ii) Now suppose that has variance . What value would one expect forb?

(izz) How do changes in affect your answer to part (u)? Wliat happens to b as approaches 1?

9.7. (Fischer and Summers, 1989.) Suppose inllation is determined as in Section 9.4. Suppose the government is able to reduce the costs of inflation; that is, suppose it reduces the parameter a in equation (9.9). Is society made better or worse off by this change? Explain mtuitively.

9.8. Solving the dynamic-inconsistency problem through punishment. (Barro and Gordon, 1983b.) Consider a pohcymaker whose objection function is Xr=o )S(yt 12), where a > 0 and 0 < /3 < 1. Vi is determined by the Lucas supply curve, (9.8), each period. Expected inflation is determined as follows. If TT has equaled tt (where - is a parameter) in all previous periods, then " = . If 77 ever differs from 77, then 77" = bja in all subsequent periods.

(a) What is the equilibrium of the model in all subsequent periods if 77 ever differs from 77

(b) Suppose 77 has always been equal to 77, so * = 77. If the monetary authority chooses to depart from 77 = 77, what value of 77 does it choose? What level of its hfetime objective function does it attain under this strategy? If the monetary authority continues to choose 77 = 77 every period, what level of its lifetime objective function does it attain?

(c) For what values of 77 does the monetary authority choose 77 = 77? Are there values of , b, and /3 such that if 77 = 0, the monetary authority chooses 77 = 0

9.9. Other equilibria in the Barro-Gordon modeL Consider the situation described in Problem 9.8. Fmd the parameter values (if any) for which each of the following is an equilibrium:

(a) One-period punishment. 77f equals 77 if 77t 1 = 77f j and equals b/a otherwise; 77 = 77 each period.

(b) Severe punishment. (Abreu, 1988, and Rogoff, 1987.) 77 equals if ,, i = 77f ,, equals > b/a if 77f j = 77 and 77f i * 77, and equals b/a otherwise; T7 = T7 each period.

(c) Repeated discretionary equilibrium, tt = = b/a each period.

9.10. Consider the situation analyzed in Problem 9.8, 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.)

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 flZLi TTt - )-i-

- ,/2]. The policymaker is chosen randomly from a pool of possible policymakers with differing tastes. Specifically, is distributed normally over possible policymakers with mean and variance af > 0. a and b are the same for all possible policymakers.

The policymaker cannot control inflation perfectly. Instead, , = , + e,, where is chosen by the policymaker (taking Trf as given) and where £f is normal with mean zero and variance af > 0. £i, £2, and are independent. The public does not observe , and s, separately, but only ,. Similarly, the public does not observe c.

Finally, assume that n-f is a linear function of iri: -n-f = a + Ptti.

(a) What is the policymakers choice of ttz? What is the resulting expected value of the policymakers second-period objective function, ( 2 - -f) -i- 7 2 - ai7/2, as a function of

(fc>) What is the policymakers choice of tti taking a and p as given and accounting for the impact of tti on

(c) Assuming rational expectations, what is /3? (Hint: use the signal-extraction procedure described in Section 6.3).

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

9.12. The tradeoff between low average inflation and flexibility in response to shocks with delegation of control over monetary policy. (Rogoff, 1985.) Suppose that output is given by = -t- ( - ), and that the social welfare function is yy - /2, where 7 is a random variable with mean and variance a. ir" is determined before is observed; the policymaker, however, chooses TT after is known. Suppose policy is made by someone whose objective function is cyy - /2.

ia) What is the policymakers choice of given , , and ?

ib) What is 77«?

(c) What is the expected value of the true social welfare function, 7 - -2/2?

(d) What value of maximizes expected social welfare? Interpret your result.

9.13. (a) In the model of reputation analyzed in Section 9.5, is social welfare higher

when the policymaker turns out to be a Type 1, or when he or she turns out to be a Type 2?

(b) In the model of delegation analyzed in Section 9.5, suppose that the policymakers preferences are believed to be described by (9.23), with a > a, when TT*" is determined. Is social welfare higher if these are actually the policymakers preferences, or if the policymakers preferences in fact match the social welfare function, (9.9)?

9.14. Money versus interest-rate targeting. (Poole, 1970.) Suppose the economy is described by linear IS and LM curves that are subject to disturbances: = - ai + Ejs, m - p = hy - ki + , where £/5 and Elm are independent, mean-zero shocks with variances afg and o-,, and where a,h.