dsc3707-ass1-sem1-2022
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University of South Africa *
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Course
3707
Subject
Mathematics
Date
Nov 24, 2024
Type
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3
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DSC3707/Ass1/S1/2022
Assignment 1
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 1.
ASSIGNMENT 01
DSC3707
DUE DATE: 7 MARCH 2022
We encourage the use of a computer package to check your answers. Assignment 01 covers chapters 1-7 in
the textbook. This assignment contributes 35% towards your semester mark and 7% towards the final mark.
Question 1
(a) Suppose the market for a commodity is governed by supply and demand sets defined as follows. The
supply set
S
is the set of pairs (
q, p
) for which
252
p
-
q
= 8400
,
and the demand set
D
is the set of pairs (
q, p
) for which
84
p
+
q
= 16800
.
Sketch
S
and
D
and determine the equilibrium set
E
=
S
∩
D
, the supply and demand functions
q
S
, q
D
, and the inverse supply and demand functions
p
S
, p
D
.
(b) Suppose that the government decides to impose and excise tax of
T
on each unit of the commodity in
(a). What price will the consumers end up paying for each unit of the commodity?
(c) Find a formula for the amount of money the government obtains from taxing the commodity in the
manner described in (b). Determine this quantity explicitly when
T
= 4.
Question 2
Suppose that the supply and demand sets for a particular market are
S
=
{
(
q,p
)
|
3
p
-
q
= 5
}
,
D
=
{
(
q,p
)
|
p
+
q
2
+ 2
q
= 9
}
,
where
q
is the quantity of units sold, and
p
is the price in rands per unit.
(a) Sketch
S
and
D
and determine the equilibrium set
E
=
S
∩
D
. Comment briefly on the interpretation
of the results.
(b) Suppose the government imposes an excise tax of R2 on each unit sold. What is the new equilibrium
set?
Question 3
Suppose that the demand function for a good is
q
D
(
p
) = 5
-
p
ln(
p
)
,
p
≥
0
where
p
is the price in rands per unit and
q
the number of units sold.
2
DSC3707/Ass1/S1
(a) Sketch the demand function on the interval 0
≤
p
≤
10.
(b) Determine the gradient of the demand function.
(c) For which value of
p
will there be no demand?
Question 4
You plan to invest an amount
C
of capital. Every year the current amount will earn interest at
r
% per year,
compounded annually.
You will also add an amount 0
,
1
C
at the end of every year.
Set up a recurrence
relation for
y
t
, the amount you have after
t
years. Find an expression for
y
t
and determine when you will
have 10
C
capital.
Question 5
Suppose that the population of a country has a growth rate of 4% per year and an attrition due to emigration
limited to 0
,
5 million per year. Let
y
t
be the population after
t
years. If the current population (at
t
= 0) is
30 million, what will the population be in
t
years? Formulate and solve a recurrence equation for
y
t
. When
will the population be doubled?
Question 6
Suppose that the supply and demand sets for a certain good are
S
=
{
(
q,p
)
|
2
p
-
3
q
= 12
}
,
D
=
{
(
q,p
)
|
2
p
+
q
= 30
}
,
and suppliers operate according to the cobweb model That is, if
p
t
and
q
t
are (respectively) the price and
quantity in year
t
, then
p
t
=
p
D
(
q
t
) and
q
t
=
q
S
(
p
t
-
1
). Suppose also that the initial price is
p
0
= 10.
(a) Find an expression for
p
t
.
(b) How does
p
t
behave as
t
tends to infinity?
Question 7
Suppose you invest R50 000 in a special savings account where, for the first ten years, interest of 6% is
paid annually at the end of each year and, thereafter, interest is continuously compounded at an annual
equivalent rate of 7%. How much money do you have in the account after 14 years if you remove no money
from it during that period?
3
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