dsc3707-ass3-sem1-2022
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School
University of South Africa *
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Course
3707
Subject
Mathematics
Date
Nov 24, 2024
Type
Pages
3
Uploaded by ChancellorFang8501
DSC3707/Ass3/S1/2022
Assignment 3
Mathematical Modelling
DSC3707
Semester 1, 2022
Department of Decision Sciences
Important Information:
This is an online module.
All study material will be posted on
my
Unisa.
Please activate your
my
Life email address and ensure you have regular access to the
my
Unisa module site DSC3707-22-S1.
This document contains assignment 3.
ASSIGNMENT 03
DSC3707
DUE DATE: 19 April 2022
We encourage the use of a computer package to check your answers. Assignment 03 covers chapters 23-28
in the textbook. This assignment contributes 30% towards your semester mark and 6% towards the final
mark.
Question 1
Determine the general solution to the recurrence equation
1.1
y
t
-
7
y
t
-
1
+ 12
y
t
-
2
= 0
1.2
y
t
+ 6
y
t
-
1
+ 9
y
t
-
2
= 0
1.3
y
t
-
2
y
t
-
1
+ 4
y
t
-
2
= 0
Question 2
Determine the specific solution to the recurrence equation with general solution
y
t
=
A
(
-
1)
t
+
B
4
t
that satisfies the intital conditions
y
0
= 0 and
y
1
= 10.
Question 3
Income from an investment can be described by the second-order recurrence
y
t
-
5
y
t
-
1
-
14
y
t
-
2
= 18
.
3.1 Determine the general solution to the recurrence equation.
3.2 Determine the specific solution that satisfies the initial conditions
y
0
= 1,
y
1
=
-
1.
Question 4
Consider a simplified national economy as explained in Chapter 23, where the quantities Investment (I),
Production (Q), Income (Y) and Consumption (C) play and important role. Suppose that consumption this
year satisfies the recurrence relation
C
t
=
2
3
Y
t
+
1
2
C
t
-
1
where
Y
t
denotes the national income in year
t
.
Assume also that the relationship between next year’s
income and current investment is given by
Y
t
+1
=
k
2
I
t
,
for some positive constant
k
. Assume that the usual equilibrium conditions
Q
t
=
Y
t
and
Y
t
=
C
t
+
I
t
hold.
4.1 Derive a second-order recurrence for
Y
t
.
2
DSC3707/Ass3/S1
4.2 Show that the model leads to ”business cycles” when 17
-
12
√
2
< k <
17 + 12
√
2.
Question 5
The inverse demand function for a good is
p
D
(
q
) =
192
q
2
+ 4
q
+ 3
and the equilibrium price is
p
*
= 4. Determine the equilibrium quantity and the consumer surplus.
Question 6
The elasticity of demand for a good is
(
p
) = 1
-
1
p
+ 1
.
Given that
q
= 8 when
p
= 1, determine the demand function
q
D
(
p
).
Question 7
Suppose that consumer demand depends upon the price-trend according to the formula
q
= 54
-
20
p
-
7
dp
dt
+
d
2
p
dt
2
.
If the supply function is
q
S
(
p
) =
-
6 + 10
p
, write down the condition for equilibrium and determine the
equilibrium price
p
(
t
) when
p
(0) = 13 and
p
0
(0) = 6.
3
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