Economics Chapter 14 Homework Start With The Phillips Curve Equation Line

Questions for Review

1. The equation for the dynamic aggregate supply curve is:

πt= πt–1 +φ(Yt– Yt) + υt.

Recall that φis a positive parameter that measures how rapidly firms adjust their

prices in response to output fluctuations. When output in the economy rises above its

2. The equation for the dynamic aggregate demand curve is:

The dynamic aggregate demand curve is defined by a given monetary policy rule and

illustrates a negative relationship between the quantity of output demanded and infla-

tion. When inflation changes, the central bank follows its monetary policy rule and

156

CHAPTER 14 A Dynamic Model of Aggregate Demand

and Aggregate Supply

YY

t

Y

tt

y

t

=−+

⎡

⎣

⎢⎤

⎦

⎥−

()

++

⎡

⎣

⎢

⎤

⎦

⎥

αθ

αθππ

αθε

π

1

*.

()

3. The dynamic aggregate demand curve is drawn for a given monetary policy rule. If the

central bank changes the rule by increasing the target inflation rate, then the dynamic

aggregate demand curve will shift to the right. Looking at the equation for the dynamic

aggregate demand curve, an increase in the target inflation rate will increase output

for any given level of the inflation rate. When the central bank increases the target

Chapter 14 A Dynamic Model of Aggregate Demand and Aggregate Supply 157

π

Figure 14–2

Y

DAD

all

t–1

Income, output

Y

4. If the central bank decides to increase the response of interest rates to changes in infla-

tion (the parameter θπ), then the central bank has become less tolerant of inflation. In

this case, any increase in inflation will elicit a larger increase in nominal and real

Problems and Applications

1. The five equations that make up the dynamic aggregate demand–aggregate supply

model can be manipulated to derive long-run values for the variables. In this problem,

it is assumed that there are no shocks to demand or supply and inflation has stabilized.

Since inflation has stabilized, it must be true that inflation in time

t

is equal to infla-

tion in time t– 1(πt= πt–1). We also know that expected inflation is equal to last period’s

inflation, or

E

158 Answers to Textbook Questions and Problems

π

Y

BDAS t–1

DAS t

Figure 14–4

Chapter 14 A Dynamic Model of Aggregate Demand and Aggregate Supply 159

it now follows that the real interest rate equals the natural rate of interest since the

demand shock parameter εtequals zero and

Y

t=

Y

t:

Moving now to the monetary policy rule equation on line 1 below, it must be true

that current inflation equals the target inflation rate so that the third term on the right

zeros out. Likewise, the fourth term on the right side will zero out since output is at the

natural level:

2. If the central bank has the wrong natural rate of interest, then it is using a value ρ’

that is different from the real value ρ. Suppose that the wrong natural rate of interest

is defined as follows:

ρ’= ρ + Δρ

In this case, if Δρ equals zero, then the central bank has the correct natural rate of

interest. If the natural rate of interest is wrong, then the long-run equilibrium values

will change. The five equations that make up the dynamic aggregate demand–

it now follows that the real interest rate equals the natural rate of interest since the

demand shock parameter εtequals zero and

Y

t=

Y

t:

The monetary policy rule equation on line 1 below has the wrong natural rate of

interest ρ’. Substitute in the relationship between the correct and incorrect rates of

natural interest and rearrange terms:

The third term on the right side will zero out since output is at the natural level.

Now, combine the rewritten Fisher equation with the rewritten monetary policy rule

equation above:

Intuitively, if the central bank thinks that the natural rate of interest is higher

than it really is, then it will be setting interest rates higher than they should be set,

and Δρ is greater than zero. The higher interest rates will result in lower demand for

160 Answers to Textbook Questions and Problems

3. “If a central bank wants to achieve lower nominal interest rates, it has to raise the nomi-

nal interest rate.” In long-run equilibrium, the nominal rate of interest is equal to the

4. The sacrifice ratio measures the accumulated loss in output associated with a one-per-

centage-point reduction in the target inflation rate. Graphically, the reduction in the

target inflation rate will shift the dynamic aggregate demand curve down and to the

left, resulting in a short-run equilibrium with a lower level of output and a lower infla-

tion rate. Over time, expected inflation will adjust and the dynamic aggregate supply

Chapter 14 A Dynamic Model of Aggregate Demand and Aggregate Supply 161

π

DAS

t+1

t

DAS

t–1,t

DAD

t,t+1,…

DAS

t–1

Figure 14–5

162 Answers to Textbook Questions and Problems

5. Follow the hint given in the problem and solve for the long-run equilibrium with the new

assumption that the demand shock parameter εtis not zero. Since inflation has stabi-

Turning to the Fisher equation on line 1 below, we can show the nominal interest

rate is equal to the natural interest rate plus the current inflation rate plus a new

r

t

=+

Ê

Ë

Áˆ

¯

˜

re

a:

Chapter 14 A Dynamic Model of Aggregate Demand and Aggregate Supply 163

The final values are as follows:

6. The equation for the dynamic aggregate demand curve is given below:

The parameter θπmeasures the central bank’s responsiveness to changes in the infla-

tion rate. When θπis large, the central bank aggressively responds to changes in the

inflation rate. When θπis small but still positive, the central bank has a weak response

YY

r

tt

t

=

=+

re

a

YY

tt

Y

tt

Y

t

=− +

⎡

⎣

⎢

⎤

⎦

⎥−+

+

⎡

⎣

⎢

⎤

⎦

⎥

αθ

αθππ

αθε

π

1

*.

()()()

a. The equation for dynamic aggregate supply is not affected by this change because its

derivation does not involve the natural rate of interest. The equation for dynamic

8. Suppose people’s expectations of inflation are subject to random shocks so that

a. The dynamic aggregate supply curve equation is derived from the Phillips curve and

the expectations equation. In this case, start with the Phillips curve equation on line 1

below and substitute in for the expected inflation term using the expression above:

164 Answers to Textbook Questions and Problems

π

DAS t

Yall DADt,t+1,…

CDAS t+1

πt+1

Figure 14–6

Chapter 14 A Dynamic Model of Aggregate Demand and Aggregate Supply 165

b. If ηtis greater than zero for one period only, then the dynamic aggregate demand curve

will shift to the right and the dynamic aggregate supply curve will not shift. Note that

the dynamic aggregate supply curve depends on the lagged value of this shock parame-

c. In period

t

+ 1, the dynamic aggregate demand curve will shift back to its original posi-

tion (because ηt+1 is zero), and the dynamic aggregate supply curve will shift to the left

d. In subsequent time periods, the dynamic aggregate supply curve will slowly shift back

to its original position as the lower level of output reduces inflation, and hence expecta-

9. Use the dynamic AD–AS model to solve for inflation as a function of only lagged infla-

tion and the two shocks. Start with the dynamic aggregate supply curve and substitute

a. A supply or demand shock will lead to an increase in current inflation. As the economy

adjusts and returns to long-run equilibrium, the inflation rate will return to its target

166 Answers to Textbook Questions and Problems

b. If the central bank does not respond to changes in output so that θYis zero, then the

economy will still return to its target inflation rate after a supply or demand shock

c. If the central bank does not respond to changes in inflation so that θπis zero, then the

d. The Taylor rule says that a one-percentage-point increase in inflation will increase the

nominal interest rate by 1 + θπpercentage points. If the central bank increases the nomi-