Derive the long-run equilibrium for the dynamic

*AD–AS*model. Assume there are no shocks to demand or supply (*ϵ*=_{t}*v*= 0) and inflation has stabilized (_{t}*π*=_{t}*π*_{t}_{+1}), and then use the five equations to derive the value of each variable in the model. Be sure to show each step you follow.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 equation for goods demand. 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.492

“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.

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, what is the implied sacrifice ratio? Explain.The text analyzes the case of a temporary shock to the demand for goods and services. Suppose, however, that

*ϵ*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? (_{t}*Hint:*It might be helpful to solve for the long-run equilibrium without the assumption that*ϵ*equals zero.) How might the central bank alter its policy rule to deal with this issue?_{t}Suppose a central bank does not satisfy the Taylor principle; that is,

*θ*is less than zero. Use a graph 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?_{π}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

*r*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

*r*varied over time?_{t}

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}_{–1}*π*=_{t}*π*_{t}_{–1}+*η*_{t}_{–1}, where*η*_{t}_{–1}is a random shock. This shock is normally zero, but it deviates from zero when some event beyond past inflation causes expected inflation to change. Similarly,*E*_{t}π_{t}_{+1}=*π*+_{t}*η*._{t}Derive the two equations for dynamic aggregate demand and dynamic aggregate supply 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?

Use the dynamic

*AD–AS*model to solve for inflation as a function of only lagged inflation and the supply and demand shocks. (Assume target inflation is a 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)?_{π}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}