Eco 320 Topic #4. Dynamics
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Ground Work: Two-Variable 1st-Order Differential Equation
When we are interested only in the qualitative aspect of dynamic system, as opposed to
the quantitative aspect, we can use the “phase-diagram” analysis. The qualitative aspect
concerns 1.) the locate of equilibrium, and 2.) the dynamic stability, or convergence
toward equilibrium.
The general form of the first-order differential equation system is
dx
x 
 x t  x t 1
x  f ( x, y; exogenous variables)
dt
y  g( x , y; exogenous variables)
= “intertemporal ∆”
= “∆ in x during a unit time”
= “time derivative of x”
1.) Locale Equilibrium:
The demarcation line are denoted by x  0 and y  0 .
x  0 means 0  f ( x, y; exogenous variables) → (1)
y  0 means 0  g( x, y; exogenous variables) → (2)
If the specific function form of f (∙) is known, from the above (1) we can solve for y in
terms of x and plot y = h(x) in the x-y plane as the x  0 demarcation line.
In general cases where f (∙) is known, at least we may be able to get the slope of the
dy
| x  0 through the implicit functional rule:
x  0 line by getting
dx
dy
f
|x 0   x → (3)
dx
fy
In the same manner, we can get the y  0 line or the demarcation line for the second
equation:
dy
g
|y 0   x → (4)
dx
gy
At this point of time, we can get the signs for 3) and 4), and draw (roughly) the two
demarcation lines corresponding to 1) and 2) respectively.
The point of the intersection between 1) and 2) is the equilibrium of the systems, which is
characterized by x  0 and y  0 . (Recall that at equilibrium there is no tendency for
changes in any variables).
Eco 320 Topic #4. Dynamics
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For an illustration, let’s assume that: f x  0 ; f y  o ; g x  0 ; g y  0 (all arbitrary)
f  g 
Also  x    x  ( x  0 curve is steeper than y  0 curve), we see the
 f   g 
 y   y 
equilibrium at E.
-
Y
x  0 : Slope 
 fx
fy
y  0 : Slope 
 gx
gy
E
x
2.) Convergence:
a.) Horizontal Trajectories:
The next task is to find out “under-currents” in the above phase-space. To do so,
we have to find to which side of x  0 line is x  0 and to which side of x  0
line is x  0 . The answer to this question is to be found in the sign of

dx
x
|y  0 
.
dx
x
i.)
ii.)
as
Y
x , x ,
y
If
as
too, with its sign changing in the order
of , ,
d x
 0
dx
x  0
and finally
Y
x  0
x  0
x  , x  ,too, with its sign changing from
If
d x
 0
dx
x  0
( x )
x  0 means the movement ← (horizontal leftward) along X
x  0 means the movement → (horizontal rightward) along X
x  0
x  0
x
to
Eco 320 Topic #4. Dynamics
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(Case i)
Y
(Case ii)
x  0
x  0
dx
 0
dx
Y
x  0
x  0
x  0
dx
 0
dx
x  0
x
x
These “under currents” are formally called “streamlines” or “phase trajectories”.
x
x
 f x which is simply the partial derivative of
is nothing but
x
x
x  f ( x, y; exogenous variables) with respect to x.
The value
b.) Vertical Trajectories:
y
 g y , which now is simple the partial derivative of
y
y
y  g( x , y; exogenous variables) with respect to y. Depending on sign of
, we
y
could have two different results:
In the same manner
(Case iii)
(Case iv)
Y
if
dy
 0
dy
Y
 y  0
if
dy
 0
dy
 y  0
y  0
y  0
 y  0
 y  0
x
x
Eco 320 Topic #4. Dynamics
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Again, for illustration, let us assume that we have
x
y
 0 (case ii) and
 0 (case iv)
x
y
in hand.
Then we have,
(E: Equilibrium)
Y
x  0
E
y  0
x
As we have already spotted, “E”, we want to answer the other qualitative question: “Is
“E” a stable equilibrium. If a slight deviation is made from “E”, would any force come to
push back to the point of equilibrium “E”? In the above case, the answer is “yes”.
Eco 320 Topic #4. Dynamics
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Y
Any deviation for E, such
as a, b, c and d, will be
wiped away in due course
because of the under
currents at work.
x  0
d
a
E
y  0
c
b
X
Y
x  0
y  0
Any deviation, such as e,
f, g and h, will also be
eliminated.
E
X
x
y
 0 (case ii) and
 0 (case iii), for instance,
x
y
will lead to a very different implication for convergence:
○ Alternatively, the combination of
Y
x  0
y  0
E
Note that there is only one
narrow path, on which the
system is pushed back to
equilibrium.
All elsewhere, any slight
deviation will lead to a
further deviation (nonconvergence).
X
The chance of following
the narrow path is rather
slim.
Eco 320 Topic #4. Dynamics
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- Example 1.
From (Review) assignment: (#3.) x   y  k
y  x  l
1.) and 2.) at equilibrium: x  0 and y ; these characteristics yield demarcation
lines.
Step 1: x  0 part; x  0 then 0   y  k
y  0 part; y  0 then 0  x  l
Y
y  k
x  l
Y
x  0
or
y=k
x  l
or
y  0
k
x
x
l
Step 2: Now let’s get phase trajectories:
x  0 part
y  0 part
dx d(  y  k)

 1  0
dy
dy
● This means that x’s sign should change in the order of (+), (0), and (-) as
the value of y increases.
Y
( y )
x  0
x  0
k
x  0
X
{
x  0
means
x  0
means
Eco 320 Topic #4. Dynamics
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In the similar manner, we get phase trajectories for y :
dy d(  x  l)

1 0
dx
dx
dy
has a positive sign, an increase in the value of X leads to an
dx
increase in the value of y . This means as x increases from zero to a large
positive value, y ’s sign should go through changes from (-), (0) to (+).
As
1.) “figuring out
the sign of
y ”
Y
y  0
y  0
y  0
(x↑)
x
y  0 means y↓
y  0 means y↑
2.) “figuring
out the
trajectories”
(movements)
Y
y  0
y  0
y  0
x
Eco 320 Topic #4. Dynamics
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Step 3: Combining the two graphs with trajectories.
Y
E
k
l
X
- Example 2.
x   x  y
y  x  l
→ (1.)
→ (2.)
1.) Demarcation Lines for x  0 and y  0 :
From (1.): 0   x  y : x  0
From (2.): 0  x  l : y  0
Y
→
→
x = -y or y = -x → (3.)
x  l → (4.)
3
Y
4
y  0 : x  l
x  0 : y   x
X
l
X
Eco 320 Topic #4. Dynamics
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2.) Streamlines:
Y
3
dx
dx  1  0

as x  , x  ; x changes
in the sequence of
(+),(0), (-).
From (1.):
x  0
x  0
x  0
x  0 : y   x
X
Y
4
y  0
y
From (2.):
y 0
0
X
l
Combining:
Y
“Stable around E!”
E
X
dy
dx
Eco 320 Topic #4. Dynamics
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- Example 3.
Motivation - Why do we check “Convergence” or “Dynamic Stability of the
System”?
In most cases, the so-called “Correspondence Theorem” in math tell us that a
dynamic convergence leads to or guarantees good comparative statics.
eg.
Y
 0 iff there exists Convergence
G
(Exception: see Assignment 1#2)

IS:
y  f {C( y)  I(i, y)  G  y} ; f’ > 0
LM:
i  g{M  L(i, y)} ; g’ < 0
P


1.
IS Curve:
y  0  f {C( y)  I(i, y)  G  y} at Equilibrium.
Cy  I y  1
i
;

y
Ii
Depending on the sign of (C y  I y  1) , we could have an upward or
downward sloping IS curve.
When (C y  I y  1)  0 , then
i)
i
0
y
i

y  0  f {C( y)  I(i, y)  G  y}
y

We have to determine the sign of y to the left and right of the IS curve;

y
 f '(C y  I y  1)
y
Eco 320 Topic #4. Dynamics
By Assumption,

y

0
y
11
C  I 1
i
 y y
y
Ii
(C y  I y  1)  0

y ’s sign changes in the order of +, 0 and -.
i

y0
IS Curve

y0

y0
y
ii) Where (C y  I y  1)  0 , then
C  I 1
i
i
, then
 y y
0
y
Ii
y
y  0 : IS Curve 1
i
y

y
 f '(C y  I y  1)  0
y

y
*In a 2 variable case,
does not mean
y
instability.

y changes its sign in the order of -, 0 and +.
Eco 320 Topic #4. Dynamics
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y  0 : IS Curve 2
i
y  0
y  0
y

2. LM Curve: i  g{M P  L(i, y)} ; g’ < 0

i  0 when

i  0  g{
M
 L(i, y ) or money market is in equilibrium.
P
M
 L(i, y )}
P

L
i
  y  0 Always.
i
Li
i

i  0:
LM Curve
y

We have to determine one sign of i off the LM curve.


i
 g'Li  0 Always. This means that as i↑, i changes its sign from + to 0
i
and finally to -.
Eco 320 Topic #4. Dynamics
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i
i  0
i  0
i  0
y
Let us combine the IS and LM as well as what we have learned from 1 & 2.
Case 1.

i
i0
LM

y  0 IS
y
Case 2.

i0
LM
i

y  0 IS
y
Eco 320 Topic #4. Dynamics
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Case 3.
i
Narrow
Path
leading to
Equilibrium
IS
LM
Saddle Path
y
Case 4.
IS
LM