Problem Set 6 Solutions:
1. Phillips curve
Explain both in words and in equations the bizarre pattern of observations shown
in Figure 13-3 (a very short answer to each is optimal).
The strange pattern is driven by the distinction between the short run and the long run
relationship between inflation and unemployment. In the short run, the downward sloping
Phillips curve holds, so policymakers are able to move along the curve with a tradeoff
between inflation and unemployment:
 t   t 1   (u t  u n )   t
However, this relationship only holds in the short run while consumers’ expectations of
inflation are stable. As policymakers target higher or lower inflation that would
decrease or increase unemployment, consumers’ expectations soon adapt to the policy
changes.  t is not only determined by (ut  un ) , but also by  t 1  Et 1 t . Furthermore,
Aggregate supply shocks from  cause the curve to shift as well (cost-push inflation). In
the long run, there is no tradeoff between unemployment and inflation, as the classical
model holds.
2. Fisher Equation
Irving Fisher is thought to have proposed the “narrow Fisher theory.” This holds
that the nominal interest rate equals a constant plus the rate of inflation. In the
dynamic AD-AS model of Chapter 14, does this hold in:
a. the short run?
No. The “narrow Fisher theory” is closely analogous to the Monetary Policy Rule
equation in the Dynamic AS-AD model:
Narrow Fisher
it   t  
Monetary Policy
it   t      ( t   t* )   Y (Yt  Yt )
In the dynamic model, there can be short run changes in  t or Yt as a result of shocks, or
there may be changes in  t* as a result of monetary policy or Yt as a result of growth.
Any of these changes would lead the dynamic model to differ from the Narrow Fisher
model in the short run.
b. the long run?
Yes. Unlike the short run, in the dynamic model in the long run you have that:
  ( t   t* )   Y (Yt  Yt )  0
So you are left with it   t   , which holds with the Narrow Fisher theory.
3. Economic Growth
Consider a closed full-employment classical economy with an aggregate production
function:
Yt  At K t.25 L.75
t
Assume that labor grows at rate n and that capital does not depreciate. National income is
all paid out to households, which are taxed at rate τ. Households consume 80 percent of
disposable income. All of taxes are spent on consumption.
a. Assume initially that there are no taxes. Derive the per capita production
function, y = f(k), where k = K/L.
Yt At K t.25 L.t75

Lt
Lt
yt  At k t.25
b. Again, with no taxes, assume that technology is constant. Calculate the longrun value of k, y, and per capita consumption (c).
k  Saving  (  n)k
k  (1   )(0.2) Ak .25  (  n)k
Where the 0.2 represents the fraction of disposable income going to savings. Since there
are no taxes and no depreciation, we can say:
k  (0.2) Ak .25  (n)k
0  (0.2) Ak .25  (n)k
(0.2) A
k .75 
n
4
4
 (0.2) A  3
 A 3
k*  

0.11696

n
 
 n 
We can now find the long run value of y:
y t  Ak t.25
.25
 (0.2) A  .75
y*  A
 n 
4
1
4
3
A3
 0.2  3
y*  A    0.5848 1
 n 
n3
And the long run value of c, consumption per worker:
ct  (0.8) yt
4
1
4
3
A3
 0.2  3
c*  (0.8) A    0.4678 1
 n 
n3
c. Now assume that the tax rate goes to 20 percent of GDP. Derive the new longrun equilibrium for k, y, and c. Explain intuitively why things have changed
between (b) and (c).
k  (1   )(0.2) Ak .25  (  n)k
k  (0.8)(0.2) Ak .25  (n)k
0  (0.16) Ak .25  (n)k
(0.16) A
k .75 
n
4
4
 (0.16) A  3
 A 3
k*  

0.08686

n
 
 n 
We can now find the new long run value of y:
y t  Ak t.25
.25
 (0.16) A  .75
y*  A

 n 
4
1
4
3
A3
 0.16  3
y*  A 

0.5429
1
 n 
n3
And the long run value of c, consumption per worker:
ct  (1   )(0.8) yt
4
3
1
 0.16  .3
c*  (0.64) A 

 n 
But if we treat all taxes as going to consumption, then we have:
4
3
1
 0.16  .3
c*  (0.64) A 
 ( ) y *
 n 
4
3
1
1
4
0.16  .3
 0.16  .3
3
c*  (0.64) A 

(
0
.
2
)
A
 n 
 n 
4
3
1
4
A3
 0.16  .3
c*  (0.84) A 

0.45602
1
 n 
n3
The tax lowers disposable income, so there is less income for saving, which causes k* to
decline. The lower steady state level of capital per worker leads to a lower y*. Even if
the taxes are used directly for consumption, consumption per worker in the long run, c*,
still falls in this case, due to the lower level of y*.
4. Dynamic AS-AD
a. Derive the long-run equilibrium for the dynamic AD–AS model. Assume
there are no shocks to demand or supply (εt = υt = 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.
In order to derive the long run values of the endogenous variables in the dynamic AD-AS
model, we must look at the five equations. We assume, by nature of being in the long run,
that there are no shocks (εt = υt = 0) and inflation has stabilized (πt = πt-1). We know
from one of the equations that the expected inflation for the current period is equal to the
inflation in the prior period:
Et 1 t   t 1
So we can replace the expectation of inflation in the Phillips curve with the prior period’s
inflation:
 t  Et 1 t   (Yt  Yt )   t
 t   t 1   (Yt  Yt )   t
And because inflation is stabile:
 t   t   (Yt  Yt )  t
And we know that  t = 0, so:
 t   t   (Yt  Yt )
And since  > 0 we can infer that (Yt  Yt ) = 0, so Yt  Yt .
We can now move to the demand for goods and services:
Yt  Yt   (rt   )   t
Since Yt  Yt and  t = 0, we know that the real interest rate equals the natural interest
rate: rt   .
Now, turning to the Fisher equation, we can show that the nominal interest rate is equal
to the natural interest rate plus the current inflation rate. Since inflation is stabilized, we
know that Et 1 t   t 1  Et  t 1   t and we just demonstrated that rt   , so:
rt  it  Et  t 1
rt  it   t
it  rt   t
it     t
And, plugging these into the monetary policy equation, it must be true that current
inflation equals the target inflation rate:
it   t      ( t   t* )   Y (Yt  Yt )
it  it    ( t   t* )   Y (0)
0    ( t   t* )
 t   t*
So we have:
Yt  Yt
 t   t*
rt  
it     t*
Et  t 1   t*
b. Assume that the central bank lowers its inflation target from π* to π** <
π*. What will the new long-run values of each of the five key variables.
The long run equilibrium we derived above shows that the central bank’s target rate
 t* only influences the nominal variables in the equations, including inflation  t ,
expected inflation Et t 1 , and the nominal interest rate it .
So long run inflation becomes  t   t** . From here we can solve the remaining
equations.
Yt  Yt
 t   t**
rt  
it     t**
Et  t 1   t**
So inflation will go down, expected inflation will go down, and the nominal interest rate
will go down, each by the number of percentage points that the target inflation rate goes
down.
c. Describe the transition path for output, nominal interest rates, and real
interest rates that will be required to move from (a) to (b).
It’s helpful to conceptualize the answer to this question by thinking in terms of time
periods.
Period 1:
Initially output falls, nominal interest rates rise, and real interest rates rise.
To see the change in the nominal and real interest rates, look at the Monetary Policy
equation and the Fisher equation:
it   t      ( t   t* )   Y (Yt  Yt )
rt  it  Et  t 1
The lower 
**
t
will lead to a higher nominal interest rate it . With the higher it , we can
look to the Fisher equation to see that rt will also initially rise, since expected inflation
has not yet changed.
To see the movement of output, we should look at the Dynamic Aggregate Demand and
Dynamic Aggregate Supply Curves.
  
 1 
*
(DAD)
Yt  Yt  
  ( t   t )  
  t
1


1


Y 
Y 


 t   t 1   (Yt  Yt )   t
(DAS)
The change in  to  t** affects the DAD curve, and not the DAS curve. Assuming that
our exogenous terms α, θπ, and θY don’t change, and that there are no shocks in the
transition (εt = υt = 0), we can see that the DAD curve shifts down (to the left) because
the target inflation rate is now lower, and so we will now be below the natural level of
output. As output falls and the DAD curve shifts down, we move along the DAS curve.
With lower output, we see that inflation must fall as well ( t   t 1 ) .
*
t
Period 2 and onward:
After these initial changes in Period 1, we see that in Period 2 the DAS curve shifts down
because expected inflation fell. Et  t 1 ↓. This allows us to move along the new DAD
curve, gradually increasing output Yt . Expected inflation continues to fall and output
continues to climb until a new equilibrium is reached where Yt  Yt at the new target
inflation  t** .
After Period 1, the nominal interest rate and real interest rate gradually decline. The
change in the nominal interest rate can be seen in the Monetary Policy Rule equation, as
inflation falls, the difference between target inflation and actual inflation falls, and the
difference between actual output and the natural level of output falls with each iteration,
until  t   t**  0 and Yt  Y  0 :
it   t     ( t   t* )  Y (Yt  Yt )
The change in the real exchange rate is a bit more complicated. It also trends downward
gradually, but at a slower rate than the nominal interest rate. This is because
rt  it  Et  t 1 and Et  t 1   t , meaning that the real interest rate only declines by:
Δ[   ( t   t* )   Y (Yt  Yt ) ]
while the nominal interest rate declines by:
Δ[  t   ( t   t* )  Y (Yt  Yt ) ]
For pictures diagramming the changes, see pages 430 and 431 in the textbook.