Advanced Macroeconomic 320
Note #5
1
Almost Complete Macroeconomic Model:
So far we have assumed P to be fixed in the short-run.
Now let’s endogenize P in the ling-run (and endogenize p in the short-run). This
means that now P is not longer assumed to be fixed, and will be determined in the model.
The Phillips curve describes the dynamics of inflation in such a way that
  H (u  u N )   e . And HN<0
The actual rate of inflation  (read as ‘Phi’) is inversely related to the excess
unemployment over the natural rate of unemployment and has an inflationary expectation
as a shift parameter. Note that u-uN measures how much the actual unemployment rate
exceeds the natural rate of unemployment. It is the deflation gap.
Can you draw the Phillips curve?
We can express the above in terms of Y and Ybar(=full employment national income)
instead of u and uN. Then of course, HY >0
  H(
Y Y
Y
)   e . And H’ or HY >0
It says that the actual rate of inflation is positively related to a) the inflation gap (in
percentage terms), and b)the expected inflation rate: the term in the bracket of H
measures how much(by what %) the actual national income Y exceeds Y bar or the
equilibrium national income, and is expressed in percentage terms.
Basically, the Phillips curve indicates the relationship between the actual Y and the longrun equilibrium Y, and its impact on the inflation. We can also think that Y is the
Aggregate Demand and Y bar or the long-run equilibrium national income is the
Aggregate Supply.
I will rewrite the rate of inflation, , into P dot over P.
*What is P dot? It is P’s time derivative, or a change in P over one unit of time period.
p 
dp p

dt t
Advanced Macroeconomic 320
Note #5
2
And thus P dot over P is the percentage rate of inflation, that is, . The Phillips curve can be
rewritten as:
p
Y Y
 H(
) e
P
Y
Now we have 3 equations for the economic system:
AD
Y  C ( y )  I ( r, y )  G : IS or goods market
M
 L(i , y )
: LM or money market
P
AS
p
Y Y
 H(
) e
P
Y
: Phillips Curve ( H  or H y  0 )
We are not ready yet to endogenize the expected rate of inflation or e.
Thus, we may have a simplifying assumptions for now:  e  0 .
This is called Static Expectations (not realistic) Later, we will relax this assumption (in the next
chapter) by specifying the functional form of the expectations, which should be rational, and
thus by endogenizing  e  0 .
Still the key issue is the impact of the fiscal policy on the real national income:
Now solve for dY/dG (or/and dY/dM) by applying Cramer’s rule:

Step 1: Totally differentiating the above equations, we get 3 X3 matrix:
1) dY  C y dY  I r dr  I y dY  d G
2) Lr dr  Ly dY 
1
M
dM  2 dp
p
p
Advanced Macroeconomic 320 Note #5
1
p
1
3)
dp  2 dp  d e  H   dY is simply reduced to;
p
p
Y
1
1
3’) H   dY  dp
p
Y
p
because 2  0 if p is not that large; and
p
d e  0 by assumption

3
Step 2: Rearranging terms into matrix multiplication form:
In order to apply Cramer’s rule. We should have the above total differentials in the form
of matrix multiplication such as:
[coefficient matrix ] [ vector of d endogenous variables]
= [coefficient matrix] [ vector of d exogenous variables]
Here we have to determine,”What are the endogenous and exogenous variables?”
As we have discussed before, it depends on the time perspectives:
In the short-run Y is endogenous and P is fixed and thus exogenous.
In the long-run Y = Yf at long-run equilibrium, and P is flexible and thus endogenous.
How about the new variable P dot?
At any one point of Long-Run equilibrium, p  0 because p does not changes.
Thus p is equal to zero and thus exogenous.
From one LR equilibrium to another, P changes and thus P dot p is endogenous.
However the change in P is so small, and thus the P itself has not changed yet: P is fixed
and exogenous. In the short run, P is fixed, and thus is exogenous. However, (only in
approximation) p  0 , and thus p is endogenous.
Suppose
the diff.
is very small
  0
p

p2
p  0
p1
L-R
p  0
S-R
L-R
1) Short-Run:
One more thing: Choice of Monetary Policy Instrument
Advanced Macroeconomic 320 Note #5
4
still we assume that the monetary policy sets the Money Supply, not the Interest
Rate. This is an assumption, which makes M or MS exogenous.
In this case, the monetary authority control M or the nominal money supply, and allows
the interest rate to be determined endogenously at any rate from the model, M is
exogenous and r or i is endogenous.
Of course, in an alternative monetary policy where the monetary authority is engaged in
the “interest rate pegging” policy, then r is exogenous and M endogenous.
In general, the monetary authority can choose M or i as their target. Whichever
target it chooses, the target variable become exogenous. We will relax the current
assumption later, and show that the two targets do not lead to the same qualitative
result of the long-run equilibrium.
For now, we will start with the first type of monetary policy. Later we will change to the
other monetary policy and examine its implication.
Thus, for now,
endogenous variable ( Y , r, p )
exogenous variable ( G , M , p )
In this case, the result of Step 2 is as follows:

1  C y  I y

Ly

1

 HY

 Ir
Lr
0



0  dY  1

0   dr   0
  
 1  dp  
0
p 

0
1
p
0


0   d G 
M

dM
2

p 


 p   d p 

p2 
Step 3 & 4: Use Cramer’s rule, and LaPlace expansion to get the solution:
dY

dG
1  Ir
0
0
Lr
0
0
0
1
p

1 Cy  I y
 Ir
0
Ly
Lr
0
1
p
H
1
Y
0
Lr
?? > or < 0
Lr (1  C y  I y )  I r L y
Once again, we can rewrite into a slightly familiar form:
Advanced Macroeconomic 320 Note #5
Lr
1

Ly
Lr (1  C y  I y )  I r L y
(1  C y  I y )  I r
Lr
5
The result is the same as before; the sign of dY/dG is ambiguous mainly due to IY term
in the denominator.
* Comment: Be reminded of the Crowding-Out Effect.
The last term of the denominator Ir Ly/Lr measures a chain reaction between goods and
money markets:
Ir Ly/Lr is the same as dI/dr x d L/dY x dr/ dL = dI/dr x dr /dY, that is,
y  md  = L(r, y) r, therefore, md > ms  I  (y*).
This is the Crowding-out effect. By now we are very familiar with this part.
2) Long-Run (Equilibrium)
Recall that in the long-run equilibrium, there would not be any changes: in other words,
Any variable’s dot should be equal to zero ( P dot is equal to zero).
p  0 ; dp  0
Y  Y f ; dY  0
Endogenous ( dr, dp )
Exogenous ( d G , d M , dY, d p )
Because p =0 and dY =0, the above system of equations is simplified as
 I r dr
Lr dr 
 dG
M
dp 
p2
1
dM
p
In this case of Long-run, the result of Step 2 is as follows:
 I r

 Lr

0 
1
 dr  
M 
0
 dp
p 2    
0  

1   dG 
p  d M 
Advanced Macroeconomic 320
Note #5
6
Step 3 & 4: Use Cramer’s rule and LaPlace expansion, we get:
There is no point of talking about dY/dG in the long-run. Why?
Now,
 Lr
L p2
dp

 r
 0
dG  I M
IrM
r
p2
An expansionary fiscal policy is merely inflationary in the long-run.
M
dr
1
p2


 0 ;
dG  I M
 Ir
r
p2
In a word, an expansionary fiscal policy raises the interest rate in the long run.
We can see that when G rises, r goes up with the same Y: the IS curve shifts up.
What would be the short-run comparative statics if we assume that the government sets the
monetary policy in terms of interest rate?
3)Alternative Assumption about Monetary Policies: “Interest Pegging Policy”
Monetary authority may set monetary policy in terms of i or r (interest rate), rather than
in terms of M or MS (nominal quantity of money supply). In the previous version of
macroeconomics, we have learned that they are equivalent. Now we will see that they
differ in terms of stability implications. At least, in terms of stability, the interest pegging
policy seems to be inferior to the monetary policy geared to control the nominal quantity
of money supply.
Of course, there are different views. One of them is by W. Poole’s illustration, which we
will cover later.
Short-run
endogenous variables (Y , M , p )
exogenous variables (G, r, p )
Step 2: Rearranging the already-obtained total differentials, we get:
(1  C y  I y )dy
= dG  I r dr
Advanced Macroeconomic 320 Note #5
1
=
Lr
dy  dM
p
1
1
H  dy
 dp =
p
Y
7
M
dp
p2
p
 dp
p
 Lr dr 
0
Note: the value of P dot /P square would be close to zero.
In the matrix multiplication, we have:

1  C  I
y
y


Ly


1
 H
Y

0
0
1
p
0
0
1
dY

dG
1  Cy  I y
Ly
H
1
Yf



0  dY
 1 I r

0  dM   0  Lr
 


d
p

 


1
0

0
p

0
1

p
0

0  d G 

M 
dr
p2   
 
 p   dp 

p2 
0

0
0
1
p
0
1
p
0

1
(1  C y  I y )
0
0
1
p
Again here, the sign of
dY
dY
is undetermined: Dependency on the magnitude of I y ,
could be of
dG
dG
negative sign or positive sign.
So we have got a similar result as the case where M is exogenous.
Comparison of
dY
in the above two alternative monetary policy setting:
dG
Advanced Macroeconomic 320


Note #5
 Lr
dY
When M is exogenous,


dG  Lr (1  C y  I y )  I r L y
When r is exogenous,
In both cases, the sign of
8
1
1  Cy  I y  Ir
Ly
Lr
dY
1

dG 1  C y  I y
dY
is unclear.
dG
However, apparently there is a very big difference between the two multipliers. What is it?
In the case of “interest pegging policy” there is no crowding out to the government fiscal policy
multiplier. Some may think,”This is better as there is no offsetting effect from the money
market”. Not really as we will show below.
Although it is not so apparent right now, there is another difference between the two. And it
concerns the qualitative nature of equilibrium or dynamic stability. In a word, when r is
exogenously controlled by the monetary authority, the long-run equilibrium is a unstable one.
There is no such instability characteristics in the case where the monetary authority controls the
money supply.
In fact, there are two different perspectives or theories to this point:
(1) Stability (Dynamic); and
(2) William Poole’s analysis
Please, keep the above question- “which monetary policy is superior?” or “do different monetary
policies matter?” – in your mind as we are making a deep exploration of the issue of stability.
(1) Dynamic Stability Analysis
*Some Background Information for Dynamic Stability: Condition for Stability
The Correspondence Theorem’ says that if and only if there is a dynamic stability (=
tendency to go back to equilibrium), then there is a good comparative static, or
comparative static puts on ‘right’ signs.
This means that dynamic stability is a general guarantee of goodness of comparative
static. We can check dynamic stability instead of the sign of comparative static.
How do we check ‘stability’ in the given model? Recall that we have AS-AD model
where one focus variables are P and Y.
In terms of P,
follows:
dp
 0 is a requirement for the dynamic stability. It can be illustrated as
dp
Advanced Macroeconomic 320
Note #5
9
ES
P
For stability
P   P  0
P*
For stability
P   P  0
P
ED
Q
The same graph can be translated into a different dimension of P dot and P.
P
Convergence to  p *
P  0
0
P
p*
P
P  0
So the Stability requires
dp
 0 .
dp
Advanced Macroeconomic 320
Note #5
You can show that in a system with a unstable equilibrium,
10
dp
dp
 0 or
 0
dp
dp
Advanced Macroeconomic 320 Note #5
11
Now then, what would be the sign of d P / d P in the above two alternative monetary policy
settings?
Let’s get it by Cramer’s rule respectively:
(1) When M :
(2) when



1  C y  I y  I r 0  dY  1

  
L
L
0


  dr   0
y
r

 H
 1  dp  
0


0
p
 Yf

dP

dP
=
1  Cy  I y
 Ir
Ly
Lr
H
Yf
0
1  Cy  I y
 Ir
Ly
H
Yf
Lr
0
(
0
1
p
0


0   d G 
M 
dM
p2   
 
 p   d p 

p2 
r:




0  dY  I I r
1  C y  I y 0
 
1

  
L
0

y

 dM   0  Lr
p

  dp  

H

1
0 0


0
 Y f

p
0
M
 2
P
1 Cy  I y
Ly
H
Yf
0
0
0
1

P
I H'
M
)( 1) 23 ( Y )
P2
Yf
I H'
1
( )( 1) 33{(1  CY  I Y ) LY  Y }
p
Yf
 or  0
dP

dP
0
1
p
0
M
 2
P
0
0
1  CY  I Y
Ir
0
LY
H'
Yf
 Lr
0
1
p
0
=
M
(1) 2  3 (0  0)
P2
=
0 at all times.
0  dG 
M  
 dr
p2   
0   dP 
This means
dP 
 0 , and
dP
dP 
 0 at any time.
if r , exogenous, there is no possibility of stability as
dP
if M is exogenous, there is a possibility of stability with
In general, the correspondence theorem would tell us that comparative statics would be of wrong
signs.
We can deduct that in general the interest pegging monetary policy may be more prone to
instability.
Advanced Macroeconomic 320
Note #5
12
(2) However, there is a different view: William Poole’s Analysis: (Reference at Section 2.7 of
W.Scarth’s book, Chap. 2. pp34-36.);
The optimal monetary policy depends on the specific nature of uncertainty or Shocks in
the economy.
i.)
LR- Agg.
Demandii.)
When the economy is subject to good market shocks: (IS shifts around), then M is
better.
LM ( if M )
r
IS 2
IS 2
IS1
Y1
IS
IS
IS1
Y1
Y2
Relatively stable
iii.)
Y2
Relatively unstable
When an economy is subject to stochastic monetary shocks (LM shifts around),
then r is better.
LM1
LM
LM
LM 2
( if M )
( if r )
r
IS
IS
Y1
Y2
Advanced Macroeconomic 320
Note #5
13
In the short-run, financial asset markets shows significant fluctuations in the national income
while the aggregate expenditure (AE = C + I + G + X-M: the basis of IS curve) are quite
predictable.
 It would be best when “monetary authority should try to peg interest rates in the short-run and
the money supply in the long-run.”