Discrete dynamic systems
139
7.
Derive the cobweb system for the price in each of the following de
mand and supply systems, and establish whether the equilibrium price is
(i) stable, (ii) unstable, or (iii) oscillatory.
(i)
q
d
t
=
10
−
3
p
t
q
s
t
=
2
+
p
t
−
1
q
d
t
=
q
s
t
(ii)
q
d
t
=
25
−
4
p
t
q
s
t
=
3
+
4
p
t
−
1
q
d
t
=
q
s
t
(iii)
q
d
t
=
45
−
2
.
5
p
t
q
s
t
=
5
+
7
.
5
p
t
−
1
q
d
t
=
q
s
t
8.
Suppose we have the macroeconomic model
C
t
=
a
+
bY
t
−
1
E
t
=
C
t
+
I
t
+
G
t
Y
t
=
E
t
where
C
and
Y
are endogenous and
I
and
G
are exogenous. Derive the gen
eral solution for
Y
n
. Under what conditions is the equilibrium of income,
Y
∗
, stable?
9.
Verify your results of question 8 by using a spreadsheet and letting
I
=
10,
G
=
20,
a
=
50,
Y
0
=
20, and
b
=
0
.
8 and 1.2, respectively. For what
period does the system converge on
Y
∗
−
Y
0
within 1% deviation from
equilibrium? For the same initial value
Y
0
, is the period longer or shorter
in approaching equilibrium the higher the value of
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 Fall '11
 Dr.Gwartney
 Economics, Supply And Demand, Qd, linear cobweb model

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