## PROBLEMS AND APPLICATIONS

1. Derive the long-run equilibrium for the dynamic AD–AS model. Assume there are no shocks to demand or supply (ϵt = vt = 0) and inflation has stabilized (πt = πt+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.

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 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.

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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, 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 ϵt 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? (Hint: It might be helpful to solve for the long-run equilibrium without the assumption that ϵt equals zero.) How might the central bank alter its policy rule to deal with this issue?

6. Suppose a central bank does not satisfy the Taylor principle; that is, θπ 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?

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 ρt.

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

2. How would a shock to rt affect output, inflation, the nominal interest rate, and the real interest rate?

3. Can you see any practical difficulties that a central bank might face if rt varied over time?

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 Et–1πt = πt–1 + ηt–1, where ηt–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, Etπt+1 = πt + ηt.

1. Derive the two equations for dynamic aggregate demand and dynamic aggregate supply in this slightly more general model.

2. 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 ηt is greater than zero (for this period only). What happens to the DAD and DAS curves in period t? What happens to output, inflation, and nominal and real interest rates in that period? Explain.

3. 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.

4. What happens to the economy in subsequent periods?

5. In what sense are inflation scares self-fulfilling?

9. 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.)

1. 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.)

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

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

4. 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 θπ? How does a shock to demand or supply influence the path of inflation?