Suggested Solution for Review Question for Eco 3320
1. Suppose that the following system of equations describe the economy
AD
Y  C ( y )  I (r , )  G : IS or goods market
M
 L(i , y )
: LM or money market
P
And P or the price level is fixed.
1) Get the total differentials of the above system of equations.
2) Get the comparative statics of dY / dG, dY/dM, dr/dG, and dr/dM.
2. From the above system of equation, now change/modify the investment equation into
I= I(r,Y) where IY >0
1)
2)
3)
4)
Discuss the economics meaning of IY >0.
Get the total differentials of the above system of equations.
Get the comparative statics of dY / dG.
Discuss the possible signs of dY/dG. Relate the positive and negative signs of dY/dG
respectively to the different slopes of all possible IS and LM curves. And Illustrate
dY/dG (changes in Y in response to a change in G) for all the possible cases of ISLM cuves.
The next question is what we will cover on October 12, Wednesday: I really would like you
to try this question up to Sub-question 3) below. We have done all the basis for this question,
and thus this question is a kind of application, except for two points(price stability issue of
dP dot/dP; and perhaps LaPlace expansion for the determinant of 3 by 3 matrix). On October
12, I will demonstrate them on the projector screen from my note.
You will first get the total differentials of three equations.
Next, in arranging the components for matrix format, we have two different sets of
circumstances depending on a given assumption:
i)
if we deal with the short-run, we must regard P dot as endogenous. And thus we
must put P dot on the left side of the equal sign;
if we deal with the long-run, we must regard P as endogenous.
For now, we will consider only the short-run for dY/dG.
ii)
If the monetary authority set ‘M’, then we must put ‘M’ or dM on the left side of
the equal sign.
If the monetary authority sets ‘r’ as the target, then we put ‘r’(interest rate) or dr
on the left-side.
3. Suppose that the following system of equations describe the economy
AD
Y  C ( y )  I ( r, y )  G : IS or goods market
M
 L(i , y )
: LM or money market
P
AS
Y  Yf
p
 H(
)e
P
Yf
: Phillips Curve ( H  or H y  0 ) and assume
e is equal to zero for now as we have not endogenized the expectations yet.
We are thinking in terms of Short-run as well as Long-run. G is always exogenous.
M or r could be exogenous as the target of monetary policy depending on their
contents.
So far we have just assumed that M is exogenous and r or i is endogenous. We will
not do so anymore. Your first question is,’Is there any difference/”. Yes, you will
see it in this question.
For the first time, we will make a distinction between two alternative cases where
the monetary authority use different policy instruments for the monetary policy:
First, “Money Supply is the policy instrument or target(Money Supply
Targeting)” - If the monetary authority set ‘M’, then we must put ‘M’ or dM
on the left side of the equal sign;
Second, “Interest Rate Pegging”- If the monetary authority sets ‘r’ as the
target, then we put ‘r’(interest rate) or dr on the left-side
1) Get the total differentials for all three above equations (by now you have done so many
times for the first two equation; the only new one is the third. For the third equation,
there are only P dot, P, and Y.

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
1
M
dM  2 dp
p
p
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
2) Lr dr  Ly dY 
2) Suppose that we are looking at the economy for the short-run. What does this mean in
terms of which variables will be endogenous and which will be exogenous? Especially
pay attention to P dot and P.
Answer:
i)In ‘Money Supply Targeting’
Short-run - endogenous variables(
Long-run - endogenous variables(
); and exogenous variables (
); and exogenous variables (
)
)
); and exogenous variables (
); and exogenous variables (
)
)
ii)In ‘Interest Rate Pegging’
Short-run - endogenous variables(
Long-run - endogenous variables(
3) For this question only, let’s focus on above case i): the short-run only (which one, out of
P dot and P, is endogenous?); the monetary authority is using money supply as the
monetary instrument, and thus sets a certain target of money supply.
Rearrange the terms in the total differentials of the above three equations from your
answer to question 1). And make it into a matrix product form of coefficient matrix and
vector(coefficient matrix x endogenous variable vector = coefficient matrix x
exogenous variable vector).
Now get the comparative statics of dY/dG for short-run when the monetary authority sets
M as the target of monetary policy.
Short-Run, and 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:
Lr

Lr (1  C y  I y )  I r L y
1
(1  C y  I y )  I r
Ly
Lr
4) Now this question only, let’s focus on above case ii): it is short-run; and ii) the monetary
authority adopts interest rate as the monetary policy instrument. Thus it comes up with a
fixed rate of interest as its target, and lets money supply change so as to achieve the target
interest rate.
Now rearrange the terms of the total differentials from your answer to question 1). And
make it into a matrix product form of coefficient matrix and vector(coefficient matrix x
endogenous variable vector = coefficient matrix x exogenous variable vector).
Get the new dY/dG for the short-run when the monetary authority adopts interest rate
pegging policy.
Short-run
endogenous variables (Y , M , p )
exogenous variables (G, r, p )
Step 2: Rearranging the already-obtained total differentials, we get:
= dG  I r dr
(1  C y  I y )dy
Lr
H

dy
1
dy
Y
1
dM
p
=

1
dp =
p
M
dp
p2
p
 dp
p
 Lr dr 
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
1

p
0



1 I
0
r
  dY  


0  dM  0  Lr
 


d
p

 
1 
0

0
p


0  d G 

M 
dr
p2   
 
 p   dp 

p2 
0

0
0
0
1
p
0
0
1
dY

dG
1  Cy  I y
Ly
H
1
Yf
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.
5) Compare dY/dG from 3) and dY/dG from 4). Can you illustrate the two different dY/dG
with the appropriate IS-LM curves?
(Recap) Comparison of


dY
in the above two alternative monetary policy setting:
dG
When M is exogenous,
When r is exogenous,
 Lr
dY


dG  Lr (1  C y  I y )  I r L y
dY
1

dG 1  C y  I y
1
1  Cy  I y  Ir
Ly
Lr
In both cases, the sign of
dY
is unclear.
dG
6) Define the Convergence theorem which links “Dynamic Stability” to the goodness (or
not) of comparative statics. What would be the requirement of dynamic stability for d
Pdot/ d P in this setting?
*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.
dp
 0 is a requirement for the dynamic stability. It can be illustrated as
In terms of P,
dp
follows:
ES
P
For stability
If P> P*, then p should fall:
P   P  0
For stability
If P<P*, then P should rise
P   P  0
P*
P
ED
Q
The same graph can be translated into a different dimension of P dot and P.

P
  0
P

0
P
p*
P
dp
 0 .You can show that in a system with a unstable
dp
dp
dp
 0 or
 0
equilibrium,
dp
dp
So the Stability requires
7) Go back the matrix formats of Questions 3) and 4). Get d Pdot/d P for the two different
cases. Are they different? Check the sign(+, 0, or - ) of d P dot/ dP for the two cases.
Compare the signs of d P dot / d P for the two cases of different monetary policies. For
each case, explain the dynamic stability and the implication of “Convergence Theorem”
for the goodness (or not) of comparative statics.
Let’s get it by Cramer’s rule respectively:
(1) When M :
(2) when



1  C y  I y  I r 0  dY  1

  
Lr 0  dr  0
 Ly
  
 H
 1  dp  
0


0
Y
p
f


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 




0  dY  I I r
1  C y  I y 0
 
1

  
0  dM   0  Lr
 Ly
p



 1  dp  0 0
 H
0
 Y f
p 
0
M
 2
P
1 Cy  I y
Ly
0
0
0
1

P
I H'
M
)( 1) 23 ( Y )
2
P
Yf
I H'
1
( )( 1) 33{(1  CY  I Y ) LY  Y }
p
Yf
 or  0
r:
dP

dP
H
Yf
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)
2
P
=
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.