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**1. ** Derive the long-run equilibrium for the dynamic *AD*–*AS* model. Assume there are no shocks to demand or supply (*ε _{t}* =

**2. ** Suppose the monetary-policy rule has the wrong natural rate of interest. That is, the central bank follows this rule:

where *ρ*′ does not equal *ρ*, the natural rate of interest in the goods demand equation. The rest of the dynamic *AD*–*AS* model is the same as in the chapter. Solve for the long-run equilibrium under this policy rule. Explain in words the intuition behind your solution.

**3. ** “If a central bank wants to achieve lower nominal interest rates, it has to raise the nominal interest rate.” Explain in what way this statement makes sense.

**4. ** The *sacrifice ratio* is the accumulated loss in output that results when the central bank lowers its target for inflation by 1 percentage point. For the parameters used in the text simulation (see the FYI box), what is the implied sacrifice ratio? Explain.

**5. ** The text analyzes the case of a temporary shock to the demand for goods and services. Suppose, however, that *ε _{t}* were to increase permanently. What would happen to the economy over time? In particular, would the inflation rate return to its target in the long run? Why or why not? (

**6. ** Suppose a central bank does not satisfy the Taylor principle; in particular, assume that *θ _{π}* is slightly less than zero, so the nominal interest rate rises less than one-for-one with inflation. Use a graph similar to Figure 15-13 to analyze the impact of a supply shock. Does this analysis contradict or reinforce the Taylor principle as a guideline for the design of monetary policy?

**7. ** The text assumes that the natural rate of interest *ρ* is a constant parameter. Suppose instead that it varies over time, so now it has to be written as *ρ _{t}*.

How would this change affect the equations for dynamic aggregate demand and dynamic aggregate supply?

How would a shock to

*ρ*affect output, inflation, the nominal interest rate, and the real interest rate?_{t}Can you see any practical difficulties that a central bank might face if

*ρ*varied over time?_{t}

**8. ** Suppose that people’s expectations of inflation are subject to random shocks. That is, instead of being merely adaptive, expected inflation in period *t*, as seen in period *t* − 1, is *E _{t}*

Derive both the dynamic aggregate demand (

*DAD*) equation and the dynamic aggregate supply (*DAS*) equation in this slightly more general model.Suppose that the economy experiences an

*inflation scare*. That is, in period*t*, for some reason people come to believe that inflation in period*t*+ 1 is going to be higher, so*η*is greater than zero (for this period only). What happens to the_{t}*DAD*and*DAS*curves in period*t*? What happens to output, inflation, and nominal and real interest rates in that period? Explain.What happens to the

*DAD*and*DAS*curves in period*t*+ 1? What happens to output, inflation, and nominal and real interest rates in that period? Explain.What happens to the economy in subsequent periods?

In what sense are inflation scares self-fulfilling?

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**9. ** Use the dynamic *AD*–*AS* model to solve for inflation as a function of only lagged inflation and supply and demand shocks. (Assume target inflation is constant.)

According to the equation you have derived, does inflation return to its target after a shock? Explain. (

*Hint*: Look at the coefficient on lagged inflation.)Suppose the central bank does not respond to changes in output but only to changes in inflation, so that

*θ*= 0. How, if at all, would this fact change your answer to part (a)?_{Y}Suppose the central bank does not respond to changes in inflation but only to changes in output, so that

*θ*= 0. How, if at all, would this fact change your answer to part (a)?_{π}Suppose the central bank does not follow the Taylor principle but instead raises the nominal interest rate only 0.8 percentage point for each percentage-point increase in inflation. In this case, what is

*θ*? How does a shock to demand or supply influence the path of inflation?_{π}

1 John B. Taylor, “Discretion Versus Policy Rules in Practice,” *Carnegie-Rochester Conference Series on Public Policy* 39 (1993): 195–214.

2 These estimates are derived from Table VI of Richard Clarida, Jordi Gali, and Mark Gertler, “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory,” *Quarterly Journal of Economics* 115, no. 1 (February 2000): 147–180.

3 For a brief introduction to this topic, see Argia Sbordone, Andrea Tambalotti, Krishna Rao, and Kieran Walsh, “Policy Analysis Using DSGE Models: An Introduction,” *Federal Reserve Bank of New York Economic Policy Review* 16, no. 2 (2010): 23–43. An important early paper in the development of DSGE models is Julio Rotemberg and Michael Woodford, “An Optimization-Based Econometric Framework for the Evaluation of Monetary Policy,” *NBER Macroeconomics Annual* 12 (1997): 297–346. A good textbook introduction to this literature is Jordi Galí, *Monetary Policy, Inflation, and the Business Cycle* (Princeton, NJ: Princeton University Press, 2008).