@ Business Mathematics Chapter 1& 2.pptx

BUSINESS MATHEMATICS (5 ECTS)
(ABVM 222)
BY:
BIZUAYEHU A. (M.Sc.)
MAY, 2023
ARBA MINCH, ETHIOPIA
Target Groups: ABVM 2nd year students

Unit One: Equations and Their Applications
1.1 Linear equations
Basic Concepts of Linear Equations and Functions
 An equation is a statement of equality, which shows two mathematical expressions
are equal.
 Equations always involve one or more unknown quantities that need to be solved.
 Among the different types of equations, linear equation is the one that we are
going to deal with in some detail.

Linear equations...
 An equation is a statement of equality of two expressions.
 The two sides of the equality sign are referred to as the left-hand side
(LHS) and the right-hand side (RHS) of the equation.
 Equation: - A mathematical statement which indicates two algebraic
expressions are equal.
Example: Y = 2X + 3
 Algebraic expressions: - A mathematical statement indicating that
numerical quantities are linked by mathematical operations.
Example: X + 2

Linear equations...
 Linear equations: - are equations with a variable & a constant with
degree one.
- Are equations whose terms (the parts separated by +, -, = signs)
- Are a constant, or a constant times one variable to the first power

Linear equations...
 Linear equations: are equations whose terms1 are a constant times
a variable to the first power.
 Accordingly, equations that can be transposed to the form,
a1 x1+ a2 x2+ …+ an xn = c are said to be linear equations.
Where,
• a1, a2, a3, … an and c are constants
• x1, x2, x3, …xn are variables (unknown quantities)
• a1 x1, a2x2, … an xn and c are the terms of the equation (terms of a linear
equation represent the parts separated by plus, minus, and equal signs).

Cont...
 A linear equation may involve two variables, x and y, and constants a,
b, and c in which case the equation relating x and y takes the form,
a x + b y = c
 Note: * No 𝑋2
terms, No X/Y terms, and no XY terms are allowed.
 The following are all examples of linear equations.
2x + 3y = 9, 3x – 9y + z = 23, 4y + 7.5x – 11 = 14
 On the other hand, 4xy + 7x = 8 is not a linear equation because the
tem 4xy is a product of a constant and two variables.
 Likewise, 5𝑿𝟐
+ 3y = 25 is not linear because of the term 5𝑿𝟐
which
is a constant times one variable to the second –power.

Linear equations...
 Example: Assume that Ethiopian Electric Power Corporation Charges
Birr 0.55 per kilowatt-hour consumed and a fixed monthly charge of
Birr 7 for rent of electric meter. If y is the total monthly charge and x is
the amount of kilowatt-hours consumed in a given month, write the
equation for y in terms of x.
 Solution: The total monthly charge will be, 0.55 times the number of
monthly KWh consumption plus Birr 7 for meter rent. Thus, using the
symbols given,
y = 0.55x + 7

 Linear Functions: Functional relationship refers to the case where there is one
and only one corresponding value of the dependent variable for each value of the
independent variable.
 The relationship between x and y as expressed by:
Y = 0.55x + 7
 is called a functional relationship since for each value of x (independent variable),
there is a single corresponding value for y (dependent).
 Thus if we write y as expression involving x and constants x is called the
independent variable, then the value of y depends upon what value we may assign
to x and as a result it is called the dependent variable.
 Therefore, a linear function refers to a linear equation, which does have one
corresponding value of dependent variable for each value of the independent
variable.

Cont...
 The general notion of a linear equation is expressed in a form
Y = mx + b
 Where
• m = slope,
• b = the Y- intercept,
• Y = dependent variable and
• X = independent variable.

Developing equation of a line
 There are at least three ways of developing the equation of a line.
1. The slope-intercept form
2. The slope-point form
3. The two-point form.

1. The slope-intercept form
 Slope is a measure of steepness or inclination of a line and it is
represented by the letter m.
 The slope of a non-vertical line is defined in several ways.
 It is the rise over the run.
 It is the change in y over the change in x.
 Thus, given coordinates of two points (x1, y1) and (x2, y2),
• 𝑆𝑙𝑜𝑝𝑒 = 𝑚 =
𝑅𝑖𝑠𝑒
𝑅𝑢𝑛
=
Δ𝑦
Δ𝑥
=
𝑦2− 𝑦1
𝑥2−𝑥1
, Where, x1 ≠ x2

The slope-intercept form...
 If the value of the slope is positive, the line raises from left to right.
 If the slope is negative, the line falls from left to right.
 If the slope is zero, the line is horizontal.
If the slope is undefined then the line is vertical.

The slope-intercept form...
 The slope-intercept form way of developing the equation of a line
involves the use of the slope & the intercept to formulate the equation.
 Often the slope & the Y-intercept for a specific linear function are
obtained directly from the description of the situation we wish to
model.
Example #1 : Given Slope = 10
Y-intercept = +20, then
 Slope-intercept form: the equation of a line with slope = m and Y-intercept b is
Y = mx + b
Y = 10X + 20

Cont...
 Example #2: Obtain the slope of the straight-line segment
joining the two points (8, -13) and (-2, 5).
 Solution:
m =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
=
5 − (−13)
−2 − 8
=
18
−10
 Therefore, the line that passes through these two points falls
downwards from left to right.
 On the other hand, if the equation of a line is given, then the slope can be
determined more simply. Thus, if a liner equation is written in the form
y = m x + b, “m” is the slope and “b” is often referred as the intercept
term and it is the value at which the straight line intercepts the Y-axis.

2. The slope point form
 In this form, we will be provided with the slope and points on the line say
(x1, y1).
 Then, we determine the intercept from the slope and the given point and
develop the equation.
 Accordingly, the expression we need further is the equation that is true
not only for the point (x1, y1) but also for all other points say (x, y) on the
line.
 Therefore, we have points (x1, y1) and (x, y) with slope m.
 The slope of the line is y - y1 / x - x1 and this is equal for all pair of points
on the line. Thus, we have the following formula for slope.
 point form:
Alternatively, 𝑦 − 𝑦1 = m(𝑥 − 𝑥1)
𝑦 − 𝑦1
𝑥 − 𝑥1
= m

The slope point form...
• Example: Find the equation of a line that has a slope of 3 and that passes
through the point (3, 4).
• Solution: Given m = 3 and (x1, y1) = (3, 4). By substituting these values in
the formula y – y 1 = m (x – x 1)
• We will obtain y – 4 = 3 (x – 3).
• Then, y – 4 = 3 x – 9
y = 3 x – 9 + 4
y = 3 x – 5.
• In another approach, y = m x + b where (x, y) = (3, 4)
y = 3x + b
4 = 3(3) + b
4 = 9 + b
4 – 9 = b = - 5 is the y-intercept.
• Thus, y = 3x – 5 is the equation of the line.

3. The Two – Point Form
 In this case, two points that are on the line are given and completely used to
determine equation of a straight line.
 In doing so, we first compute the slope and then use this value with either
points to generate the equation.
 Taking two points designated by (x1, y1) and (x2, y2) and another point (x, y),
we can develop the expression for the equation of the line asfollows.
𝑦 − 𝑦1
𝑥 − 𝑥1
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1
Therefore, (y – y 1) (x 2 – x 1) = (y 2 – y 1) (x – x 1)
is the expression for the two-point form of generating equation of a straight line.

• Example: A publisher asks a printer for quotations on the cost of
printing 1000 and 2000 copies of a book. The printer quotes
Birr 4500 for 1000 copies and Birr 7500 birr for 2000 copies.
Assume that cost (y) is linearly related to the number of books
printed (x).
a) Write the coordinates of the given points
b) Write the equation of the line

Solution:
 Given the values
x 1 = 1000 Books y1= Birr 4500
x 2 =2000 books y2= Birr 7500
a. Coordinates of the points are:
(x1, y1) and (x2, y2)
• Thus, (1000, 4500) and (2000, 7500)

Cont...
b. To develop the equation of the line, first let’s compute the slope.
m =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
=
7500 − 4500
2000 − 1000
=3000 ÷ 1000 = 3
• Then, consider the formula of two-point form of developing equation of
a line as given by,
𝑦 − 𝑦1
𝑥 − 𝑥1
=
𝑦2 − 𝑦1
𝑥2 − 𝑥1

 We have obtained the value for the slope m = 3 as it’s expressed by
y2 - y1  x2 - x1.
 Subsequently, by substitution this value in the above formula will
result in;
𝑦− 𝑦1
𝑥−𝑥1
= 3
Then, 𝑦 − 𝑦1= 3(𝑥 − 𝑥1)

 In continuation, substitute the value (1000, 4500) in place of
x1 and y2 in the equation y – y1 = 3 (x – x1). As a result, you
will obtain,
y – 4500 = 3 (x – 1000)
y – 4500 = 3x – 3000
y = 3x – 3000 + 4500
y = 3x + 1500……………………… is the equation of the line

1.2. Nonlinear equations
 A linear equation is one related to a straight line,
 For example f(x) = mx+c describes a straight line with slope m and the
linear equation f(x) = 0, involving such an f, is easily solved to give x =
−c/m (as long as m ≠ 0).
 If a function f is not represented by a straight line in this way, we say it is
nonlinear.
• The nonlinear equation f(x) = 0 may have just one solution, like in the linear
case, or it may have no solutions at all, or it may have many solutions.
• For example, if f(x) = 𝑋2− 9 then it is easy to see that there are two solutions
x = −3 and x = 3

Nonlinear equation...
• The nonlinear equation f(x) = 𝑋2
+ 1 has no solutions at all (unless
the application under consideration makes it appropriate to
consider complex numbers).
• More examples of non-linear functions:
• y = 2𝑋2+ 3 (x is squared)
• y = 3𝑋3 + 12 (x is cubed)
• y = 9𝑋0.4
+ 15 (x has the raised exponent 0.4)
• y = 9(1/x) + 4 (x is in the denominator)

Nonlinear equation...
 A system of nonlinear equations is a system of two or more
equations in two or more variables containing at least one
equation that is not linear.
 Basically any function that is not "linear equation", such as
quadratic, circle, reciprocal, exponential, etc.

Key Points
 Notice that non-linear equations do not have the simply nice
y = bx + c kind of formula.
 Non-linear functions, when plotted, will yield bent curves or
waves rather than straight lines.
 Notice that non-linear functions (i.e. curves) don't have a
constant slope. The slope varies along the curve.
 Just keep in mind that not all functions are straight lines.

How do I know that an equation is a linear or nonlinear
equation?
 To determine whether the given equation is linear we have to
determine that a given equation is in the format y = mx + c.
 Where m is the slope, x and y are the variables and c is the y-
intercept.
 For example, y = 2x + 1, here the equation has the highest
degree as one, So it is a linear equation.
 A nonlinear equation will not match this equation.

Cont...
 Note : An equation in which the maximum degree of a
term is 2 or more than two is called nonlinear
equations.
 You can also test an equation is linear or nonlinear by plotting it on the graph.
 If an equation gives a straight line, then that equation is a linear equation.
 Example: y = 2x + 1 is the equation can be represented on the graph as
 Here it represents a straight line so it is a linear equation.

1.3 Economic applications of linear and quadratic models
1.3.1 Functions and curves in Economics
DEFINITION
 A function is a correspondence between a first set, called the domain, and a
second set, called the range, such that each member of the domain
corresponds to exactly one member of the range.
 Consistent with the definition, we will regard a function as a set of ordered pairs,
such that no two pairs have the same first coordinate paired with different second
coordinates.
 The domain is the set of all first coordinates, and the range is the set of all second
coordinates.

Cont...
 A fundamental idea in mathematics and its application is that of a
function; which tells how one quantity depends on others.
 In applications of mathematics, functions are often representatives
of real phenomena or events.
 Functions therefore are models.
 Obtaining a function to act as a model is commonly the key to
understanding business in many areas.

Cont...
 Functions may be represented by formulae.
 There are a number of common ways in which functions are
presented and used.
 We shall consider functions given by formulae, since this provides a
natural context for explaining how a function works.
 If you get a job that pays birr. 700 per hour, the amount of money M (in
Ethiopian birr) that you earn depends on the number of hours (h) that
you work, and the relationship is given by a simple function:
Money = 700 x hours worked
M = 700h birr

Cont...
 The formula M=700h shows that the money M that you
earn depends on the number of hours worked.
 We say that M is a function of h. this context, h is a variable
whose value we may not know until the end of the week.
 Once the value of h is known, the formula M=700h can be
used to calculate the value of M.
 To emphasize that M is a function of h, it is common to
write M=M(h), so that M(h)=700h.
 For example, if you work 30 hours, then the function M (30)
is the money you would earn.
 To calculate the amount earned, you need to replace the
formula by 30.
 Thus, M = M (30) = 700 x 30 = 21,000 birr.

• Note:
1. It is important to remember that h is measured in hours and M is
measured given is Ethiopian birr. The function/formula is not useful
unlessyou stateinwordstheunits you are using.
2. We would also use different letters or symbols for the variables.
Whatever letter/symbol used, it is critical that you explain in
word what they mean

Supply and Demand Functions
 Supply and demand in economics are modeled by increasing
and decreasing functions.
Demand Functions and curves
 The table and graph below show the relationship between
the price per bag of sugar and the quantity q of 5-lb bags that
consumers will demand at that price.

25
20
Demand function: q =D(X)
15
10
5
$1 $2 $3
Price
$4 $5 x
Quantity
(in
millions
)
Figure 1.2 Demand function and curve

Demand Functions and Curves...
 Demand is defined as the quantities of goods consumers are
willing and able to buy at a particular price.
C Notice that this is an if...then statement - if the price is
$1000, then the quantity demanded is 60, and so on. So it is a
function, like y = f(x), with x now being price, and y being
quantity.
C Demand need not be a linear function. It can be a curve or
wavy. But for this example, let us suppose it is linear. Let us
suppose the demand relationship is summarized as:

Qd = -2P + 21
• Notice that P (price) is where x is, and Qd (quantity demand) is
where y is in a usual linear equation. Notice also that it has the
usual formula of a linear equation (Y= mx + b where
Qd=Y, m = -2 and b = 21).
• So let us go through the if. then statements:
• If price is 0, then quantity demanded is 21
• If price is 1, then quantity demand is 19
• If price is 2, then quantity demanded is 17………….
• If price is 11, then quantity demanded is -1 (and we can ignore this and
onwards as "negative" quantities demanded make no sense).

Cont...
• Now, to plot this. For some rather silly reasons, economists decided
that demand & supply diagrams ought to flip the axes around.
• That is, in Demand & Supply diagrams, the x variable (price) is on
the vertical axis and the y variable (Quantity) is on the horizontal
axis. Don't ask why. It is just the convention economists got used to

Cont...
 Because the x-y axes are switched in demand diagrams, then in order to
plot, you have to reverse the reasoning you got used to in high school.
 That is, you need to now start from the vertical axis (price) then find
the corresponding value on the horizontal axis (quantity).
 So if price is 2, find 2 on the vertical axis, then move along to the right
until you find quantity = 17 on the horizontal axis.
 Go through all the prices P = 1 through P = 10 in this fashion, and you'll
have plotted out the demand curve.

Cont...
C Notice that the intercept (21) is no longer on the vertical
axis, but rather on the horizontal axis. Weird, I know. You just
got to get used to it.
C Finally, notice that the slope is no longer merely the
number attached to P in the equation m (that is -2).
C Rather, because the axes are reversed, "Rise over Run" is
now the reciprocal, slope = -1/2.

Shifts in Demand Curves
 Another peculiarity of the reversed axes, is that shifts in the
curves are no longer "up and down", but rather "left and right".
 That is if you increase the intercept (i.e. a positive shock or sudden
increase to demand), you will be shifting the curve to the right
(not "up").
 Whereas a decrease in the intercept (i.e. a negative shock or
sudden decrease in demand) and you will be shifting the curve to
the left (not "down").

Shifts in Demand Curves...
Figure1.4ShiftsinDemandCurves

When does a demand curveshift?
 Anything that causes an increase or decrease in
demand for a good (except for a change in the price of
the good itself) will lead to a shift the demand curve.
Why is all this a shift?
Because all these things increase demand for a good
regardless of the exact price.

Supply Curves
• Supply curves work essentially the same way as demand curves.
• Remember, supply is defined as the quantity that producers and willing to
produce and supply at a particular price.
• Supply is also an if...then statement again.
• If price is 10, then quantity produced is 20.
• If price is 9, then quantity produced is 18, etc.
• Warning! When talking about supply, some people get easily confused and
imagine a reverse causality, and say things like "when supply increases
from 18 to 20, price rises from 9 to 10". This is wrong. The causality of a
supply curve is from price to quantity.

Supply Curves...
 Remember: in deciphering a supply curve, we are talking about how
much producers are willing to supply given a particular price. When
price rises from 9 to 10, then supply rises from 18 to 20. Not the other way
around. Do not confuse yourselves!

Supply Curves...
 Let us suppose the supply relationship is summarized by a linear
function:
Qs = 3P - 2
 Going through the if...then logic:
• If price is 0, then quantity supplied is -2
(negative, ignore)
• If price is 1, then quantity supplied is 1
• If price is 2, then quantity supplied is 4
• If price is 3, then quantity supplied is 7 and so on.

Shifts in Supply Curves
 Shifts in supply curves work on the same rationale as shifts
in demand: that is,
 A positive shock or increase in supply (not related to price)
will shift the supply curve to the right,
 Whereas a negative shock or decrease in supply will shift the
supply curve to the left.

Figure 1.6 Shifts in Supply Curves

 (It is particularly important to keep the right/left thing clear for
supply.
 Notice that the left shift represents less supply, yet it "looks"
diagrammatically like the new curve is "above" the old curve.
 Don't let yourself be confused. Left is always less. This is one of
the drawbacks of the flipped axes.)

1.3.2 Modeling of supply and demand analysis, market equilibrium,
national income determination
 In microeconomics, we are concerned with the interaction of supply and
demand. Figure 1.7 shows typical supply and demand curves sketched on
the same diagram.
 Of particular significance is the point of intersection.
 At this point, the market is in equilibrium because the quantity supplied
exactly matches the quantity demanded.
 The corresponding price, P0, and quantity, Q0, are called the equilibrium
price and quantity.

Example
The demand and supply functions of a good are given by
P = −2QD + 50
P = 1/2QS + 25
• where P, QD and QS denote the price, quantity demanded and
quantity supplied respectively.
a. Determine the equilibriumpriceand quantity?
b. Determine the effect on the market equilibrium if the government
decides to impose a fixed tax of birr 5 on each good?

Solution
a. The demand curve has already been sketched in Figure 1.8. For
the supply function
P = 1/2QS + 25
• We have a = 1/2, b = 25, so the line has a slope of 1/2 and an
intercept of 25.
• It, therefore, passes through (0, 25). For a second point, let us choose
QS = 20, say. The corresponding value of P is
P = 1/2(20) + 25 = 35

Cont...
 So the line also passes through (20, 35). The points (0, 25) and (20, 35)
can now be plotted and the supply curve sketched.
 Figure 1.8 shows both the demand and supply
curves sketched on the same diagram.
 The point of intersection has coordinates (10, 30), so the equilibrium
quantity is 10 and the equilibrium price is 30.
 It is possible to calculate these values using algebra.
 In equilibrium, QD = QS. If this common value is denoted by Q, then the
demand and supply equations become
P = −2Q + 50 and P = 1/2Q + 25

 This represents a pair of simultaneous equations for the two
unknown’s P and Q, and so could be solved using the elimination
method described in the previous section. However, this is not
strictly necessary because it follows immediately from the above
equations that
−2Q + 50 = 1/2Q + 25
 Since both sides are equal to P. This can be rearranged to calculate Q:
−21/2Q + 50 = 25 (subtract 1/2Q from both sides)
−21/2Q = −25 (subtract 50 from both sides) Q =
10 (divide both sides by −21/2)

 Finally, P can be found by substituting this value into either of
the original equations. The demand equation gives
P = −2(10) + 50 = 30
 As a check, the supply equation gives
P = 1/2(10) + 25 = 30 ✓

Figure 1.8 Equilibrium price and quantity

b) If the government imposes a fixed tax of birr 5 per good then the
money that the firm actually receives from the sale of each good is the
amount, P that the consumer pays, less the tax, 5: that is, P − 5.
 Mathematically, this problem can be solved by replacing P by P − 5 in
the supply equation to get the new supply equation
P − 5 = 1/2QS + 25
That is, P = 1/2QS + 30
 In equilibrium, QD = QS. Again setting this common value to be Q gives
P = −2Q + 50
P = 1/2Q + 30
Hence −2Q + 50 = 1/2Q + 30

 Which can be solved as before to give Q = 8. Substitution into either of the
above equations gives P = 34. (Check the details.)
 Graphically, the introduction of tax shifts the supply curve upwards by 5
units. Obviously the demand curve is unaltered. The dashed line in Figure 1.8
above shows the new supply curve, from which the new equilibrium
quantity is 8 and equilibrium price is 34.
 Note the effect that government taxation has on the market equilibrium price.
 This has risen to birr 34 and so not all of the tax is passed on to the consumer.
 The consumer pays an additional birr 4 per good. The remaining birr 1 of
tax must, therefore, be paid by the firm.

 Suppose that there are two goods in related markets, which we call
good 1 and good 2. The demand for either good depends on the
prices of both good 1 and good 2. If the corresponding demand
functions are linear then
QD1 = a1 + b1P1 + c1P2
QD2 = a2 + b2P1 + c2P2
 Where Pi and QDi denote the price and demand for the ith good and ai, bi, ci
are parameters.
 For the first equation, a1 > 0 because there is a positive demand when the
prices of both goods are zero.
 Also, b1 < 0 because the demand of a good falls as its price rises.

Cont...
 The sign of c1 depends on the nature of the goods.
 If the goods are substitutable then an increase in the price of good 2
would mean that consumers would switch from good 2 to good 1,
causing QD1 to increase.
 Substitutable goods are therefore characterized by a positive value of c1.
 If the goods are complementary then a rise in the price of either good
would see the demand fall, so c1 is negative. Similar results apply to the
signs of a2, b2 and c2.
 The calculation of the equilibrium price and quantity in a two-
commodity market model is demonstrated in the following example.

Example
The demand and supply functions for two interdependent
commodities are given by
QD1 = 10 − 2P1 + P2
QD2 = 5 + 2P1 − 2P2
QS1 = −3 + 2P1
QS2 = −2 + 3P2
Where QDi, QSi and Pi denote the quantity demanded, quantity supplied
and prices of good i respectively.
Determine the equilibrium price and quantity for this two-commodity model.

Solution
• In equilibrium, we know that the quantity supplied is equal
to the quantity demanded for each good, so that
QD1 = QS1 and
QD2 = QS2
• Let us write these respective common values as Q1 and Q2.
The demand and supply equations for good 1 then become
Q1 = 10 − 2P1 + P2
Q1 = −3 + 2P1
• Hence
10 − 2P1 + P2 = −3 + 2P1

 Since both sides are equal to Q1. It makes sense to tidy this equation up a bit
by collecting all of the unknowns on the left-hand side and putting the
constant terms on to the right hand side:
10 − 4P1 + P2 = −3 (subtract 2P1 from both sides)
−4P1 + P2 = −13 (subtract 10 from both sides)
• We can perform a similar process for good 2. The demand and supply
equations become
Q2 = 5 + 2P1 − 2P2
Q2 = −2 + 3P2
• Because QD2 = QS2 = Q2 in equilibrium. Hence
5 + 2P1 − 2P2 = −2 + 3P2
5 + 2P1 − 5P2 = −2 (subtract 3P2 from both sides)
2P1 − 5P2 = −7 (subtract 5 from both sides)

• We have therefore shown that the equilibrium prices, P1 and P2, satisfy the
simultaneous linear equations
−4P1 + P2 = −13 (1)
2P1 − 5P2 = −7 (2)
• This can be solved by elimination. Following the steps we proceed as follows.
Step 1
• Double equation (2) and add to equation (1) to get
−4P1 + P2 = −13
4P1 − 10P2 = −14 +
−9P2 = −27 (3)
Step 2
• Divide both sides of equation (3) by −9 to get P2 = 3.

Step 3
• If this is substituted into equation (1) then
−4P1 + 3 = −13
−4P1 = −16 (subtract 3 from both sides)
P1 = 4 (divide both sides by −4)
Step 4
• As a check, equation (2) gives 2(4) −
5(3) = −7 ✓
• Hence, P1 = 4 and P2 = 3.

• Finally, the equilibrium quantities can be deduced by substituting
these values back into the original supply equations.
• For good 1,
Q1 = −3 + 2P1 = −3 + 2(4) = 5
• For good 2,
Q2 = −2 + 3P2 = −2 + 3(3) = 7
• As a check, the demand equations also give
Q1 = 10 − 2P1 + P2 = 10 − 2(4) + 3 = 5 ✓
Q2 = 5 + 2P1 − 2P2 = 5 + 2(4) − 2(3) = 7 ✓
In general, with n goods it is necessary to solve n equations in n unknowns.

Quadratic functions
• The simplest non-linear function is known as a quadratic and takes the form:
f (x) = a𝑿𝟐 + bx + c for some parameters a, b and c.
• (In fact, even if the demand function is linear, functions derived from it, such
as total revenue and profit, turn out to be quadratic.
• Consider the elementary equation
𝑋2− 9 = 0
• It is easy to see that the expression on the left-hand side is a special
case of the above with
• a = 1, b = 0 and c = −9.
• To solve this equation, we add 9 to both sides to get 𝑋2 = 9

• So we need to find a number, x, which when multiplied by itself
produces the value 9.
• A moment’s thought should convince you that there are exactly two
numbers that work, namely 3 and −3 because 3 × 3 = 9 and (−3) ×
(−3) = 9
• The equations considered in above is of the special form ax2 + c = 0 in
which the coefficient of x is zero.
• To solve more general quadratic equations, we use a formula that
enables the solutions to be calculated in a few lines of working.
• It can be shown that a𝑋2 + bx + c = 0 has solutions

 The precise number of solutions that an equation can have depends
on whether the number under the square root sign is positive, zero
or negative.
 The number b2
− 4ac is called the discriminant because the sign of
this number discriminates between the three cases that can occur.

If 𝒃𝟐 − 4ac > 0 then there are two solutions
and
If 𝒃𝟐
− 4ac = 0 then there is one solution
If 𝒃𝟐
− 4ac < 0 then there are nosolutions because does not exist.

Solve the following quadratic equations
(a) 2𝑥2
+ 9x + 5 = 0
(b) 𝑥2
− 4x + 4 = 0
(c) 3𝑥2
− 5x + 6 = 0

National income determination
 Macroeconomics is concerned with the analysis of economic theory
and policy at a national level.
 Initially we assume that the economy is divided into two sectors,
households and firms.
 Firms use resources such as land, capital, labour and raw materials to
produce goods and services. These resources are known as factors of
production and are taken to belong to households.

Cont...
 National income represents the flow of income from firms to
households given as payment for these factors.
 Households can then spend this money in one of two ways.
 Income can be used for the consumption of goods produced by
firms or it can be put into savings.
 Consumption, C, and savings, S, are therefore functions of
income, Y: that is,
C = f (Y)
S = g(Y)

• For some appropriate consumption function, f, and savings function, g.
• Moreover, C and S are normally expected to increase as income rises, so f and g are both
increasing functions. We begin by analyzing the consumption function. As usual we need to
quantify the precise relationship between C and Y.
• If this relationship is linear then a graph of a typical consumption function is shown in Figure
1.9. It is clear from this graph that if
C = aY + b
Then a > 0 and b > 0.
Figure 1.9

• The intercept b is the level of consumption when there is no income
(that is, when Y = 0) and is known as autonomous consumption.
• The slope, a, is the change in C brought about by a 1-unit increase in
Y and is known as the marginal propensity to consume (MPC). As
previously noted, income is used up in consumption and savings so
that
Y = C + S

 It follows that only a proportion of the 1 unit increase in income is
consumed; the rest goes into savings.
 Hence the slope, a, is generally smaller than 1: that is, a < 1. It is
standard practice in mathematics to collapse the two separate
inequalities a > 0 and a < 1 into the single inequality
0 < a < 1
• The relation
Y = C + S
• enables the precise form of the savings function to be determined from
any given consumption function.

Example
 Sketch a graph of the consumption function
C = 0.6Y + 10
 Determine the corresponding savings function and sketch
its graph.
Solution
• The graph of the consumption function C = 0.6Y + 10
has intercept 10 and slope 0.6. It passes through (0, 10).
• For a second point, let us choose Y = 40, which gives C = 34.
• Hence the line also passes through (40, 34).
• The consumption function is sketched in Figure 1.10.

• To find the savings function we use the relation
Y = C + S
• This gives,
S = Y − C (subtract C from both sides)
= Y − (0.6Y + 10) (substitute C)
= Y − 0.6Y − 10 (multiply out the brackets)
= 0.4Y − 10 (collect terms)
• The savings function is also linear. Its graph has intercept −10
and slope 0.4. This is sketched in Figure 1.11 using the fact
that it passes through (0, −10) and (25, 0).

• For the general consumption function
C = aY + b
• We have
S = Y − C
= Y − (aY + b) (substitute C)
= Y − aY − b (multiply out the brackets)
= (1 − a) Y − b (take out a common factor of Y)
• The slope of the savings function is called the marginal
propensity to save (MPS) and is given by 1 − a: that is,
• MPS = 1 − a = 1 − MPC

The result, MPC + MPS = 1, is always true, even if the consumption
function is nonlinear.
• The simplest model of the national economy is illustrated in Figure 1.12,
which shows the circular flow of income and expenditure.
• This is fairly crude, since it fails to take into account government activity
or foreign trade.
• In this diagram investment, I, is an injection into the circular flow in the
form of spending on capital goods.

Figure:1.12 Model of the national economy

1.3.3 Modeling of revenue, cost and profit
 Linear Cost – Output Relations Analysis: In order to grasp the concept of
linear cost output relations, let us consider the relationship among different
types of cost on the following a coordinate plane.
 Definitions: Cost is resource sacrificed to produce a given good or render
service.
 Different classification of costs based on different basis for classification is
possible but for our purpose here let’s define fixed cost, Variable cost and the
sum of the two totals cost as hereunder.

Costs...
 Fixed cost is a cost component that does not change with the
number of units produced whereas
 Variable cost is a cost component that varies with change in
number of units produced.
 Then at each level of production, total cost is the summation of
fixed cost and variable cost.
 Marginal cost is the additional cost incurred in producing one
more unit of output.

• Illustration: Assume that total manufacturing cost and the number
of units produced are linearly related. The total cost originates from
the fixed cost line because of zero level of production the total cost
will be equal to the fixed cost (see the above figure (Fig 1.2.1)).
Accordingly,
- Fixed costs (FC) = AD = BE = CF
- The segment BG is the Total Cost (TC) of producing AB units of outputs.
- The segment CI is the TC of producing AC units of outputs.
- The segment AD is the TC of producing zero units of outputs.
- The ratio

Break – Even Analysis:
Definition: Breakeven point is the level of sales at which profit is zero.
According to this definition, at breakeven point sales are equal to fixed cost
plus variable cost. This concept is further explained by the following equation:
[Break even sales (BS) = Fixed cost (FC) + variable cost (VC)]
Breakeven sales= Selling price (SP)*Quantity (Q)
VC= VC per unit*Quantity (Q)
SP*Q=FC+ VC/unit*Q
SP*Q-VC/unit*Q=FC
Q(SP/unit-VC/unit)=FC
Q=FC/Sp/unit-Vc/unit- is quantity to be produced or sold at
breakeven point

Break – Even Analysis...
Important linear cost – output expressions (equations):
 Total cost (TC) =VC+ FC
 Revenue (R) = SPQ
 Average Revenue (AR) = R ÷Q = PQ ÷Q = SP
 Average Variable Cost (AVC) = VQ ÷ Q = VC = Slope (m)
 Average Fixed Cost (AFC) = FC ÷ Q
 Average Cost = C ÷Q = AVC + AFC
 Profit ( p ) = R – TC

Example: A book company produces children’s books. One time
fixed costs for Little Home are $12,838 that includes fees to the
author, the printer, and for the building. Variable costs amount to
$14.50 per book the books are then sold to bookstores around the
country at $39.00 each. How many books must be printed and
sold to break-even?

Solution:
Given, V = $14.50
FC = $12,838
Sp = $39
Let Q = the number of books printed and sold
Thus, C = VQ + FC
TC = 14.5Q + 12,838 is the cost equation.
The revenue (R) is also given by,
R = SP Q
= 39Q

 To obtain the quantity of books to be printed and sold to
break-even, you need to equate the R and C equations.
39Q = 14.5Q + 12,838
39Q – 14.5Q = 12838
24.5Q = 12838
Q = 12838/24.5
Q = 524 books must be printed and sold to break – even.

Quiz (5%)
The demand and supply functions for two interdependent commodities are given by
QD1 = 40 − 5P1 − P2
QD2 = 50 − 2P1 − 4P2
QS1 = −3 + 4P1
QS2 = −7 + 3P2
Where QDi, QSi and Pi denote the quantity demanded, quantity supplied and price of
good i respectively.
Determine the equilibrium price and quantity for this two-commodity model. Are these
goods substitutable or complementary?

Objectives
After going through this unit, you will be able to:
 Define a sequence;
 Explain how sequences and series are related;
 Discuss arithmetic and geometric progressions; and
 Describe certain applications of sequences and series in Economics.
Unit Two
2. Sequences, Series and Mathematics of Finance

2.1 Sequence and Series
2.1.1 Sequence
 Definition: A sequence of numbers is an endless succession of numbers
placed in a certain order.
 The numbers in the sequence are called terms of the sequences.
 The terms of the sequence are got by certain well defined rules, which will be
clear to you very soon.

Examples; Sequences
i. 3, 7, 11, 15, …
In this sequence each term is obtained by adding 4 to the previous term.
So the next term would be 19.
ii. 4, 9, 16, 25, . . .
This sequence can be rewritten as 22, 32, 42, 52, . . . The next term is 62,
or 36. The dots (. . .) indicate that the sequence continues indefinitely – it
is an infinite sequence.

A sequence such as 3, 6, 9, and 12 (stopping after a finite number of terms) is
a finite sequence.
Suppose we write u1 for the first term of a sequence, u2 for the second and so on. There may be a
formula for un, the nth term:
Examples: The nth term of a sequence
i. 4, 9, 16, 25, . . . The formula for the nth term is un = (n +1)2.
ii. un = 2n + 3. The sequence given by this formula is: 5, 7, 9, 11, . . .
iii. un = 2n + n. The sequence is: 3, 6, 11, 20, . . .
Or there may be a formula that enables you to work out the terms of a sequence
from the preceding one(s), called a recurrence relation:

Examples: Recurrence Relations
Suppose we know that: un = un−1 + 7n and u1 = 1.
Then we can work out that u2 = 1 + 7 × 2 = 15, u3 = 15 + 7 × 3 = 36, and so on, to find
the whole sequence: 1, 15, 36, 64, . . .
un = un−1 + un−2, u1 = 1, u2 = 1
The sequence defined by this formula is: 1, 1, 2, 3, 5, 8, 13, . . .

2. Series
 A series is formed when the terms of a sequence are added together.
 The Greek letter (pronounced “sigma”) is used to denote “the sum of”:
 Series is the sum of the terms of a sequence.
 The sum of the first n terms of a sequence is denoted by Sn; i,e
U1 + U2 = 2 + U3 + ……+ Un
𝑛
= ∑ 𝑈𝑟
𝑟=1
(Read as "sums Ur, r=1 to r=n")

Examples: Series
a. In the sequence 3, 6, 9, 12, . . . , the sum of the first five terms is the
series: 3 + 6 + 9 + 12 + 15.
6
b. ∑(2𝑟 + 3) = 5 + 7 + 9 + 11 + 13 + 15
r =1

Examples: Series
𝑟=5
𝑘
1
𝑟2
=
1
25
+
1
36
+
1
49
+
1
64
+
1
81
+
1
100
+ ⋯ +
1
𝑘2

2.2 Arithmetic and Geometric Progressions
2.2.1 Arithmetical sequence and series
1.Arithmetical sequence
 An arithmetic sequence is one in which each term can be obtained by adding a
fixed number (called the common difference) to the previous term.
Examples: Some Arithmetic Sequences
a. 1, 3, 5, 7, . . . The common difference is 2.
b. 13, 7, 1,−5, . . . The common difference is −6.

Arithmetical sequence...
 An arithmetical sequence of n ∈ N real numbers an ∈ R,
(an)n∈N,
 is defined by the property that the difference d between neighbouring elements
in the sequence be constant, i.e., for n > 1
• In an arithmetic sequence with first term a and common difference d, the formula
for the nth term is:
un = a + (n − 1)d
Other letters may be used instead of n and d

Arithmetical sequence...
a.Whatisthe10thtermofthearithmeticsequence5,12,19,...?
Inthissequence a=5 and d = 7. So the 10th term is: 5 + 9 × 7 = 68.
b.Ifanarithmeticsequencehasu10 =24andu11 =27,whatisthefirstterm?
Thecommon difference,d,is3.
Usingtheformulaforthe11thterm:
27 = a + 10 × 3
• Hence the first term, a, is −3.

2. Arithmetic Series
 When the terms in an arithmetic sequence are summed, we obtain an
arithmetic series.
 Suppose we want to find the sum of the first 5 terms of the arithmetic
sequence with first term 3 and common difference 4. We can calculate it
directly:
S5 = 3 + 7 + 11 + 15 + 19 = 55
 But there is a general formula:

 If an arithmetic sequence has first term a and common difference d,
the sum of the first n terms is:
𝑆𝑛 =
𝑛
2
((2𝑎 + 𝑛 − 1 𝑑)
We can check that the formula works:
𝑆5 =
5
2
((2 × 3 + 4 × 4) = 𝟓𝟓

2.2.2 Geometrical sequence and series
1. Geometrical sequence
 A geometric sequence is one in which each term can be obtained
by multiplying the previous term by a fixed number, called the
common ratio.
Examples: Geometric Sequences
a.
1
2
,1, 2, 4, 8, . . . Each term is double the previous one. The common
ratio is 2.
b. 81, 27, 9, 3, 1, . . . The common ratio is
1
3

 In a geometric sequence with first term a and common ratio r, the formula for
the nth term is:
‫ݑ‬𝑛 = 𝒂𝒓𝒏−𝟏
 When the terms in a geometric sequence alternate between positive and
negative, the value of r is negative.
Examples: Consider the geometric sequence with first term 2 and common
ratio 1.1.
a. What is the 10th term?
• Applying the formula, with a = 2 and r = 1.1, ‫ݑ‬𝑛 = 𝑎𝑟𝑛−1
u10 = 𝑎𝑟𝑛−1
= 2 × (1.1)10−1
= 2 × (1.1)9
= 4.7159

b. Which terms of the sequence are greater than 20?
The nthterm is given by un = 𝑎𝑟𝑛−1
= 2 × (1.1)𝑛−1
.
It exceeds 20 if: 2 × (1.1)𝑛−1
> 20
(1.1)𝑛−1> 10
Taking logs of both sides log10 (1.1)𝑛−1
> log1010
(n − 1) log10 1.1 > 1
𝑛 >
1
log10 1.1
+ 1 = 25.2
So, all terms from the 26th onwards are greater than 20.

2. Geometric Series
 Suppose we want to find the sum of the first 10 terms of the
geometric sequence with first term 3 and common ratio 0.5:
S10 = 3 + 1.5 + 0.75 + · · · + 3 ×(0.5)9
For a geometric sequence with first term a and common ratio r, the sum of the
first n terms is: 𝑠𝑛 =
𝑎(1−𝑟𝑛)
1−𝑟
So, the answer is, 𝑠10 =
3(1−0.5)10)
1−𝑟

Consequently,
This result is summarized as follows.

Infinite Series
Given an infinite sequence 𝑎𝑛 , then
𝑛=1
∞
𝑎𝑛=𝑎1+𝑎2+𝑎3+𝐾+𝑎𝑛+𝐾 𝒊𝒔 𝒄𝒂𝒍𝒍𝒆𝒅 an infinite series.
𝑎𝑛=
𝑛
𝑛+1
is the infinite sequence
1
2
,
2
3
,
3
4
,
4
5
,K

Convergent and Divergent Series
 A convergent series is a series whose partial sums tend to a specific number,
also called a limit.
 A divergent series is a series whose partial sums, by contrast, don't
approach a limit.
 Divergent series typically go to ∞, go to −∞, or don't approach one specific
number.
 In short,
@ If the infinite series has a sum or limit, the series is convergent.
@ If the series is not convergent, it is divergent.

1. Arithmetic series since no sum exists, it diverges.
2. Geometric series
 If |r| > 1, diverges
 If |r| < 1, converges since the sum exists

2.3 Interest Rates, Savings and Loans
2.3.1 Interest Rates
 Interest is the compensation one gets for lending a certain asset.
 For instance, suppose that you put some money on a bank account for a year. Then,
the bank can do whatever it wants with that money for a year. To reward you for
that, it pays you some interest.
 The asset being lent out is called the capital. Usually, both the capital and the
interest are expressed in money. However, that is not necessary.
 For instance, a farmer may lend his tractor to a neighbours, and get 10% of the
grain harvested in return. In this course, the capital is always expressed in money,
and in that case it is also called the principal.

Interest Rates...
 In sum, the difference arises because a rational being is assumed to invest/use money
available on productive activity that results in a higher future sum and,
 The difference between the present and future value of money is referred to as the
time value of money.
• The world of finance calculates interest in two different ways:
1. Simple Interest. A simple interest system primarily applies to short-term financial
transactions, with a time frame of less than one year.

 Interest on loans of a year or less is frequently calculated as simple interest, a type of
interest that is charged (or paid) only on the amount borrowed (or invested) and not
on past interest.
 The amount borrowed is called the principal. The rate of interest is given as a
percentage per year, expressed as a decimal. For example, 6% = 0.06 and 11.5% =
0.115. The time the money is earning interest is calculated in years.
 One year’s interest is calculated by multiplying the principal times the interest rate,
or Pr.
 If the time that the money earns interest is other than one year, we multiply the
interest for one year by the number of years or Prt.

Simple Interest
I =Prt
Where
• P is the principal;
• r is the annual interest rate (expressed as a decimal);
• t is the time in years.

• A deposit of P dollars today at a rate of interest r for t years produces
interest of
• I = Prt. The interest, added to the original principal P, gives
P + Prt = P (1 + rt).
• This amount is called the future value of P dollars at an interest rate r
for time t in years.
• When loans are involved, the future value is often called the maturity
value of the loan. This idea is summarized as follows.

Future or Maturity Value for Simple Interest
The future or maturity value A of P birr at a simple interest rate r for t years is
A = P (1 +rt).
2. Compound Interest: A compound interest system primarily applies to long-term
financial transactions, with a time frame of one year or more.
In this system, interest accrues and compounds upon previously earned interest.
• Compound interest = P × 1 + interest rate)
n−1
]
• Where:
p = principal
n = number of compounding periods

• Suppose that someone gives you the option of receiving birr 500 now or birr
500 in 3 years’ time. Which of these alternatives would you accept? Most people
would take the money now, partly because they may have an immediate need
for it, but also because they recognize that birr 500 is worth more today than in
3 years’ time. Even if we ignore the effects of inflation, it is still better to take the
money now, since it can be invested and will increase in value over the 3-year
period. In order to work out this value we need to know the rate of interest and
the basis on which it is calculated.

 Let us begin by assuming that the birr 500 is invested for 3 years at 10% interest
compounded annually. What exactly do we mean by ‘10% interest compounded
annually’? Well, at the end of each year, the interest is calculated and is added on to
the amount currently invested. If the original amount is birr 500 then after 1 year the
interest is 10% of birr 500, which is
10
100
×500 = 50 Birr
• So, the amount rises by birr 50 to 550.

 What happens to this amount at the end of the second year? Is the interest
also birr 50?
 This would actually be the case with simple interest, when the amount of
interest received is the same for all years. However, with compound interest,
we get ‘interest on the interest’.
 Nearly all financial investments use compound rather than simple interest,
because investors need to be rewarded for not taking the interest payment
out of the fund each year.

Compound interest...
 Under annual compounding the interest tained at the end of the second year is 10% of
the amount invested at the start of that year. This not only consists of the original 500,
but also the 50 already received as interest on the first year’s investment.
 Consequently, we get an additional.
10
100
×550 = 55 Birr, , raising the sum to birr 605.
 Finally, at the end of the third year, the interest is 60.5 birr.
 So, the investment is birr 665.50.
 You are therefore 165.50 birr better off by taking the birr 500 now and investing it for 3
years.

 The calculations are summarized in Table 2.1.
End of year Interest (ETH birr) Investment (ETH birr)
1 50 550
2 55 605
3 60.5 665.5
 The calculations in Table 2.1 were performed by finding the interest earned
each year and adding it on to the amount accumulated at the beginning of the
year.
 This approach is rather laborious, particularly if the money is invested over a
long period of time.

 What is really needed is a method of calculating the investment after, say, 10
years without having to determine the amount for the 9 intermediate years.
This can be done using the scale factor approach.
 To illustrate this, let us return to the problem of investing 500 at 10% interest
compounded annually.
 The original sum of money is called the principal and is denoted by P, and the
final sum is called the future value and is denoted by S (A).
 The scale factor associated with an increase of 10% is
1 +
10
100
= 1.1

 So at the end of 1 year the total amount invested is P (1.1).
 After 2 years we get P (1.1) × (1.1) = P (1.1)2and
 after 3 years the future value is S = P (1.1)2
× (1.1) = P (1.1)3
 Setting P = 500, we see that
 S = 500 (1.1)3
= birr 665.50 which is, of course, the same as the amount
calculated previously.
 In general, if the interest rate is r% compounded annually then the scale
factor is 1 +
r
100
so after n years, S = P (1 +
r
100
)3
• Given the values of r, P and n it is simple to evaluate S using the power xy
key on a calculator.

 In general, to find a formula for compound interest, first suppose that P birr is
deposited at a rate of interest r per year.
 The amount on deposit at the end of the first year is found by the simple interest
formula, with t = 1.
A = P (1 + rx1) = P(1 + r)
 If the deposit earns compound interest, the interest earned during the second year
is paid on the total amount on deposit at the end of the first year.
 Using the formula, A = P(1 + rt) again, with P replaced by P(1 + r) and t = 1, gives
the total amount on deposit at the end of the second year.
 A= [P(1 + r)] (1 + r x 1) = P (1 + r)2
• In the same way, the total amount on deposit at the end of the third year is
A= 𝐏 (𝟏 + 𝐫)𝟑

 Generalizing, if P is the initial deposit, in t years the total amount on
deposit is A= P (1 + r)3
called the compound amount.
NOTE: Compare this formula for compound interest with the formula for simple interest.
Compound interest A= 𝑃 (1 + 𝑟)𝑡
Simple interest A = P(1 + rt)
 The important distinction between the two formulas is that in the compound
interest formula, the number of years, t, is an exponent, so that money
grows much more rapidly when interest is compounded.

 Interest can be compounded more than once per year.
 Common compounding periods include
• semiannually (two periods per year),
• quarterly (four periods per year),
• monthly (twelve periods per year), or
• daily (usually 365 periods per year).
 The interest rate per period, i, is found by dividing the annual interest rate, r,
by the number of compounding periods, m, per year.
 To find the total number of compounding periods, n, we multiply the number
of years, t, by the number of compounding periods per year, m.

The following formula can be derived in the same way as the
previous formula.
Compound Amount
A = P (1 + i)n
Where 𝐢 =
𝐫
𝐦
and 𝐧 = mt
A is the future (maturity) value;
P is the principal;
r is the annual interest rate;
m is the number of compounding periods per year;
t is the number of years;
n is the number of compounding periods;
i is the interest rate per period.

Examples: If you save birr 500 at a fixed interest rate of 6% paid annually:
i. How much will you have after 10 years?
Using the formula above, A= 𝑃 (1 + 𝑟)𝑡
y10 = 500 (1 + 0.06)10
y10 = 500 (1.06)10
= birr 895.42.
ii. How long will you have to wait to double your initial investment?
The initial amount will have doubled when:
500 (1.06)𝑡
= 1000
(1.06)𝑡
= 2
Taking logs of both sides:
tlog10 1.06 = log10 2
t =
log10 2
log10 1.06
= 11.8957
So, you will have to wait 12 years.

Examples 2: You invest £1000 for two years in the bank, which pays interest at an
equivalent annual rate of 8%.
a) How much will you have at the end of one year if the bank pays interest annually?
You will have: 1000 × 1.08 = 1080 birr.
b) How much will you have at the end of one year if the bank pays interest quarterly?
• Using the formula above with m = 4, you will have 1000 × (1.02)4
= 1082.43 birr.
• Note that you are better off (for a given equivalent annual rate) if the interval of
compounding is shorter.
c) How much will you have at the end of 5 years if the bank pays interest monthly?
Using the formula with m = 12 and t = 5, you will have:
10000 × (1 +
0.08
12
)5×12
= 1489.85 birr

 From this example, you can see that if the bank pays interest quarterly and the
equivalent annual rate is 8%, then your investment grows by 8.243% in one year.
 This rate is known as the Annual Percentage Rate, or APR; it is the rate of interest
that, if compounded annually, gives the same yield.
 Banks often describe their savings accounts in terms of the APR, so that customers do
not need to do calculations involving the interval of compounding.

Examples: Annual Percentage Rate
 If a bank pays interest monthly at an equivalent annual rate of 6%, what is the APR?
 If you invested an amount P for t years, you would have
P(1 +
1 + 0.06
12
)12𝑡
Whereas if you invested at interest rate i compounded annually, you would have;
P (1 + i)t
• The yield is the same if:
P (1 + i)t
= P(1 +
1 + 0.06
12
)12𝑡
(1 + 𝑖) = (1 +
1 + 0.06
12
)12
i = 0.0617 So the APR is 6.17%
• Note: When we say, for example, “the annual interest rate is 3%” or “the interest rate
is 3% per annum” we normally mean the equivalent annual rate, not the APR.

2.3.2 Regular Savings
 Suppose that you invest an amount A at the beginning of every year, at a fixed
interest rate i (compounded annually).
 At the end of t years, the amount you invested at the beginning of the first year
will be worth 𝐀(𝟏 + 𝐢)𝐭, the amount you invested in the second year will be worth
𝐀(𝟏 + 𝐢)𝐭−𝟏
, and so on.
 The total amount that you will have at the end of t years is:
𝑆𝑡 = 𝐴(1 + 𝑖)𝑡 + 𝐴(1 + 𝑖)𝑡−1 + 𝐴(1 + 𝑖)𝑡−2 + ⋯ + 𝐴(1 + 𝑖)2 + 𝐴(1 + 𝑖)
= 𝐴(1 + 𝑖) + 𝐴(1 + 𝑖)2 + 𝐴(1 + 𝑖)3 + ⋯ + 𝐴(1 + 𝑖)𝑡−1 + 𝐴(1 + 𝑖)𝑡

 This is the sum of the first t terms of a geometric sequence with first term A(1
+ i), and common ratio (1 + i). We can use the formula from section 3.2.2
 The sum is:
=
𝐴(1+𝑖)(1− 1+𝑖 𝑡)
1−(1+𝑖)
=
𝑨(𝟏+𝒊)
𝒊
( 𝟏 + 𝒊 𝒕-1)
 So, for example, if you saved birr 200 at the beginning of each year for 10 years, at 5%
interest, then you would accumulate
200(1.05)
0.05
( 1.05 10
-1)=2641.36 birr.

2.3.3 Paying Back a Loan
 If you borrow an amount L, to be paid back in annual repayments over t years, and the
interest rate is i, how much do you need to repay each year?
 Let the annual repayment be y. At the end of the first year, interest will have been
added to the loan. After repaying y you will owe:
X1 = L(1 + i) – y
• At the end of two years you will owe:
X2 = (L(1 + i) − y) (1 + i) − y
= L(1 + i)2
− y(1 + i) − y
• At the end of three years: X3 = L(1 + i)3
− y(1 + i)2
− y(1 + i) – y
and at the end of t years: Xt = L(1 + i)𝑡
− y(1 + i)𝑡−1
− y(1 + i)𝑡−2
− . . . − y

 But if you are to pay off the loan in t years, Xt must be zero:
= 𝐿(1 + 𝑖)𝑡 = 𝑦(1 + 𝑖)𝑡−1 + 𝑦(1 + 𝑖)𝑡−2 + ⋯ + 𝑦
• The right-hand side of this equation is the sum of t terms of a geometric
sequence with first term y and common ratio (1 + i) (in reverse order).
𝐿(1 + 𝑖)𝑡 =
𝑦(1 + 𝑖)𝑡
− 1)
𝑖
𝑦 =
𝐿𝑖(1 + 𝑖)𝑡
(1 + 𝑖)𝑡−1)
• This is the amount that you need to repay each year.

2.4 Compounding and Discounting
• Investors are investing money and getting returns in the future in terms of
money. This is called cash flow.
• When cash comes to us it is called a cash inflow and is considered a positive
cash flow, and
• When we pay out cash it is called a cash outflow and is considered a negative
cash flow.

 Let P be the principal, or the total amount of money borrowed (say from a
bank as a loan) or invested (say in a bank as deposit) by an individual. Let
the interest be r (r is expressed as a percentage).
 Let t be the period after which the loan is to be repaid, or the invested amount
matures.
 Then if P is invested at a simple interest of r percent per annum for a
period of t years, the interest charge In is given by:
In = P× r×t .

 The amount A owed at the end of t years is the sum of the principal
borrowed and the interest charged:
A = P + In = P + Prt = P (1 + rt).
 Suppose you invest birr 100 in 2016. If this investment earned simple
interest at the rate of 10% per year, the future value of your investment
would be birr 100 plus birr 10 per year for every year that the amount was
invested at 10% per year.
 If you invested birr 100 for 4 years you would have earned birr 140 at the
end of 4 years.

 In general, if you invest birr P (P stands for principal) at 100r% per year for t
years, at the end of t years you will get an amount
A = P + Pr t = P(1+ rt) as we have seen.
 Compound interest is more interesting!
 When investments are made at compound interest rates, the investment earns
“interest on interest”.
 In other words, interest is paid on interest that has been earned in previous
periods.

In the above example, after the first year, amount would be birr 100 + birr 10
(interest =10% of 100, =10) = birr 110 at the beginning of the second period,
this birr 110 is invested and the interest now is 10% of 110, that is birr 11. So
after the second year, the amount goes up to birr 110+ birr 11 = birr 121.
 In general, if a principal P is invested at 100r% compound interest for t years,
the amount obtained at the end of t years is
A = (P (1+ r)t

 Notice the difference with the simple interest formula. There, the term within
parentheses was 1+rt, that is t is multiplied by r and added to 1.
 In the case of compound interest, t appears not as a multiplicative term but as
an exponent, so that (1+r) is raised to the power t.
 We can restate the above by calculating the future value of a single cash flow
 compounded annually as follows:
 Let C0 be the initial cash flow or investment
• r be stated annual rate of interest or return
 t be the life of investment
 Ct be the value of C0 at end of t years.
Then, Ct = C0 (1 + r)t

 This is the equation for compounding, which converts a present cash value to the
future cash value.
 (1 + r)t is called the future value compound factor, denoted by FVCFr,t, where
the subscripts r and t have the meaning mentioned above.
 Thus, Ct = CoFVCFr, t
 Now we turn to the inverse process, that is, we want to find out, if the future cash
flow is of a given amount, what would be the value of the cash flow today. This is
called discounting.
 We begin by considering a single period.

 To convert future cash value into present value, we use the procedure of
discounting.
 To discount a future cash flow to the present, simply rearrange the equation
for compounding, to get,
‫ܥ‬0 =
Ct
(1 + r)t
 Thus discounting is the opposite of compounding.
 In the above equation,
1
(1+r)t " is called the present value discount factor,
denoted by PVDFr,t.
 The discount factor is simply the reciprocal of compound factor.

 Discounting multiple cash flows is simple: we can discount each individual cash
flow and then add the present values (PV).
 The general case is present below [the cash flows (C1, C2, C3…Ct) are unequal and
uneven each year]:
 ܲ‫ݒ‬0 = (
𝐶1
(1+𝑟)
+
𝐶2
(1+𝑟)2 +
𝐶3
(1+𝑟)3 + ⋯ +
𝐶𝑡
1+𝑟 𝑡)
 Suppose r is the interest rate, t the number of years, as before, but now suppose
it is compounded m times a year, that is, m is the number of compounding
periods in a year.

 To find out the relevant formula for compounding in such cases, we find it
is equal to
𝐶𝑡 = C (1 +
r
i
)𝑚𝑡
• For the case of discounting, if there is non-annual discounting the formula is:
C𝑜 = (
𝐶𝑡
(1+
𝑟
𝑖
)𝑚𝑡

2.5 Present Value and Investment appraisal
 Would you prefer to receive (a) a gift of birr 1000 today, or (b) a gift of birr 1500 in
one year’s time?
 Suppose money can be invested at 10% compound interest, compounded annually.
Then birr. 100 could be invested and be worth birr. 110 in one year’s time.
 Put another way, the value of birr 110 in one year time is exactly the same as birr
100 now (if the investment rate is 10% per annum).
 Similarly, birr 100 now has the same value as birr. 100(1.1)2 = birr. 121 in two
years’ time, assuming the investment rate is 10%.
 This demonstrates the concept of the present value of a future sum.
 To state the above ideas more precisely, if the current investment rate is 10%, then

 The present value of birr. 110 in one year’s time is birr.
110
1.1
= Birr100.
 Similarly, the present value of birr 121 in two years’ time is birr.
121
1.12 = Birr100
and so on.
 The investment rate used is sometimes referred to as the discount rate.

Present Value Formula
 The present value of an amount birr A, payable in t years time, subject to a
discount rate of 100i% is given by:
P=
𝐴
(1+i)t
Where,
• P = present value
• A = amount, payable in t year’s time
• i = discount rate (as a proportion)
• t = number of time periods (normally years)

 The present value is also known as the Present Discounted Value ; payments
received in the future are worth less – we “discount” them at the interest rate i.
 Example: Present Value and Investment
 The prize in a lottery is birr 5000, but the prize will be paid in two years’ time. A
friend of yours has the winning ticket. How much would you be prepared to pay to
buy the ticket, if you are able to borrow and save at an interest rate of 5%? The
present value of the ticket is:
P=
5000
(1.05)2 = 4535.15 birr
• This is the maximum amount you should pay.

 An investment opportunity promises you a payment of birr 1000 at the end of each of the
next 10 years, and a capital sum of birr 5000 at the end of the 11th year, for an initial outlay
of birr 10000. If the interest rate is 4%, should you take it?
 We can calculate the present value of the investment opportunity by adding up the present
values of all the amounts paid out and received:
 ܲ = −10000 +
1000
(1.04)
+
1000
(1.04)2 +
1000
(1.04)3 + ⋯ +
1000
(1.04)10 +
5000
1.04 11
 In the middle of this expression we have (again) a geometric series.
 The first term is
1000
1.04
and the common ratio is
1
1.04
Using the formula from section
2.2.2(2)

• ܲ = −10000 +
1000
1.04
(1−(
1
1.04
)
10
)
1−
1
1.04
+
5000
1.04 11
= −10000 + (
(1−(
1
1.04
)
10
)
0.04
+ 3247.90
= −10000 + 25000 (1 − ( 1
1.04)
10
) + 3247.90
= −10000 + 8110.90 + 3247.90 =−10000 + 11358.80 = 1358.8 𝑏𝑖𝑟𝑟
• The net present value (NPV) of an investment project is defined as the
PV of the future returns minus the cost of setting up the project.

2.6 Annuities and Perpetuities
 Annuity refers to a sequence or series of equal periodic payments, deposits,
withdrawals, or receipts made at equal intervals for a specified number of periods.
 For instance,
• regular deposits to a saving account,
• monthly expenditures for car rent,
• insurance,
• house rent expenses, and
• periodic payments to a person from a retirement plan fund are some of the
particular examples of annuity.

Annuity...
 Payments of any type are considered as annuities if all of the following conditions
are present:
1. The periodic payments are equal in amount
2. The time between payments is constant such as a year, half a year, a quarter of a
year, a month etc.
3. The interest rate per period remains constant.
4. The interest is compounded at the end of every time.

• For example, someone might pay a fixed sum for a guaranteed pension
payment of birr 14,000 a year for the next 5 years.
• The present value of a steady stream of a fixed return of birr a per year for the
next n years when interest rates are i% will be
• An annuity is a financial asset which pays you an amount A each year for N
years. Using the formula for a geometric series, we can calculate the present
value of an annuity:
ܲ‫ݒ‬0 = (
A
(1 + i)
+
A
(1 + i)2 +
A
(1 + i)3 + ⋯ +
A
1 + i n)

 The present value tells you the price you would be prepared to pay for the asset.
• Example annuity: An annuity will pay birr 2,000 a year for the next 5 years, with the
first payment in 12 months’ time. Capital can be invested elsewhere at an interest rate
of 14%. Is birr 6,000 a reasonable price to pay for this annuity?
Solution
• For this annuity (in birr)
P = 2000(1.14)−1
+2000(1.14)−2
+2000(1.14)−3
+2000(1.14)−4
+ 2000(1.14)−5
In this example the annual payment A = 2,000, i = 0.14 and n = 5.
Therefore, using the annuity formula.
=
𝐴(1−(
1
1+𝑖
)𝑛)
𝑖
=
𝐴(1−(1+𝑖)−𝑛)
𝑖
=
2000(1−(1.14)−5)
0.14
=𝟔,𝟖𝟔𝟔.𝟏𝟔
• The PV of this annuity is greater than its purchase price of birr 6,000 and so it is clearly
a worthwhile investment.

Perpetual annuities
 The PV of the stream of returns from a perpetual annuity is an infinite geometric
progression.
 Whether or not one can find the sum of an infinite geometric progression depends on
whether the progression is convergent or divergent.
• To distinguish a convergent and a divergent geometric series is to look at the value
of the common ratio r.
 If |r| > 1 then successive terms become larger and larger and the series diverges.
 If |r| < 1 then successive terms become smaller and smaller and the series converges.

Perpetual annuities ...
 The absolute value is used because it is possible to have a negative common ratio.
 To find the sum of a convergent geometric series (such as the case of a perpetual
annuity) let us look again at the general formula for the sum of a geometric series:
𝑎(1−𝑟𝑛)
1−𝑟
---------------------------(1)
This can be rewritten as
𝑎
1−𝑟
− (
𝑎
1−𝑟
)𝑟𝑛
 If −1 < r < 1 then 𝑟𝑛 → 0 as n→∞ (i.e. the value of 𝑟𝑛 approaches zero as n
approaches infinity) and so the second term in (1) will disappear and the sum to
infinity will be ‫𝑛ܲܩ‬ =
a
1−r
-------------------------------(2)

• We can now use formula (2) for the frog example. The total distance
jumped is
𝑛=5
∞
5(0.5)𝑛
 In this geometric series r = 0.5 and a = 5. The sum for an infinite number of terms
will thus be
a
1−r
=
5
1−0.5
=
5
0.5
= 𝟏𝟎 𝑚𝑒𝑡𝑟𝑒𝑠

 The PV of a perpetual annuity can also be found using this formula
although care must be taken to include the discounting factor in the initial
term, as explained in the following example.
Example: Perpetual annuities
 What is the PV of an annuity which will pay birr 6 a year ad infinitum,
with the first payment due in 12 months’ time? Assume that capital can be
invested elsewhere at 15%.

Perpetual annuities ...
Solution
ܲ‫ݒ‬ = (
A
(1 + i)
+
A
(1 + i)2 +
A
(1 + i)3 + ⋯ +
A
1 + i n)
ܲ‫ݒ‬ =
6
1.15
+
6
1.152 + ⋯ +
6
1.15n
where n→∞. In this geometric series a = 6 /1.15 and r = 1 /1.15.
This is clearly convergent as |r| < 1. The sum to infinity is therefore
NPV =
𝑎
1 − 𝑟
=
6
1.15
1 −
1
1.15
=
6
1.15(1 −
1
1.15
)
=
6
1.15 − 1
=
6
0.15
= 𝟒𝟎

 A simplified formula for the PV of a perpetual annuity can be derived as certain terms
will always cancel out, as Example above illustrates. Assume that an annuity pays a fixed
return A each year, starting in 12 months’ time, and the opportunity cost of capital is i%.
For this annuity
PV = A(1 + i)−1
+A(1 + i)−2
+ ⋯ + A(1 + i)−𝑛
where n→∞
 In this geometric series the initial value a = A(1 + i)−1and constant ratio r = (1 + i)−1.
 Therefore, using the formula for the sum of an infinite converging geometric series
ܲ‫ݒ‬ =
𝑎
1 − 𝑟
=
A(1 + i)−1
1 − (1 + i)−1 =
𝐴
(1 + 𝑖) 1 − (1 + i)−1
=
𝐴
1 + 𝑖 − 1
=
𝐴
𝑖

 Thus the formula for the PV of a perpetual annuity is
ܲ‫ݒ‬ =
A
i
Reworking Example above using this formula we get
ܲ‫ݒ‬ =
6
0.15
= 40 birr
Which is identical to the answer derived from first principles, although the formula
obviously makes the calculations much easier.

Key notes :
When calculating the time value of money, the difference between an annuity
derivation and perpetuity derivation is related to their distinct time periods.
An annuity is a set payment received for a set period of time.
Perpetuities are set payments received forever—or into perpetuity.
 Valuing an annuity requires compounding the stated interest rate.
Perpetuities are valued using the actual interest rate.

2.7 Mortgage Payments and Amortization
 Mortgages are loans that are used to buy homes and other types of real estate.
 Mortgage payment is an arrangement where by regular payments are made in
order to settle an initial sum of money borrowed from any source of finance.
 Such payments are made until the outstanding debt gets down to zero.
 An individual or a firm, for instance, may borrow a given sum of money from a
bank to construct a building or undertake something else.
 Then the borrower (debtor) may repay the loan by effecting (making) a monthly
payment to the lender (creditor) with the last payment settling the debt totally.

• In mortgage payment, initial sum of money borrowed and regular
payments made to settle the respective debt relate to the idea of present
value of an ordinary annuity.
• Along this line, the expression for mortgage payment computation is
derived from the present value of ordinary annuity formula.
• Our intention in this case is to determine the periodic payments to be
made in order to settle the debt over a specified time period.
ܲ =
A(1 − 1 + i −n
)
i

 Now, we progress to isolate A on one side. It involves solving for R in the above present
value of ordinary annuity formula. Hence, multiply both sides by the interest rate i to
obtain:
ܲ =
A(1 − 1 + i −n
)
i
ܲi = A 1 − 1 + i −n
A = 𝑃
𝑖
1 − 1 + i −n
Further, we divide both sides by 1 − 1 + i −n
and the result will be the mathematical
expression or formula for computing mortgage periodic payments as follows.

• Where, A = Periodic amount of an annuity
• i = Interest per conversion period which is given by r ÷ m
• r = Annual nominal interest rate
• m = Interest or conversion periods per year
• n = the number of annuity payments/deposits (number of compounding periods)
• P = Present value of an ordinary annuity

Example: Mortgage Payments and Amortization
1. Emmanuel purchased a house for Birr 115,000. He made a 20% down payment with the
remaining balance amortized in 30 years mortgage at annual interest rate of 11%
compounded monthly.
a. Find the monthly mortgage payment?
b. Compute the total interest?
2. Assume you borrowed Birr 11,500 from a bank to finance construction of a
swimming pool and agreed to repay the loan in 60 monthly equal installments. If
the interest is 1.5% per month on the unpaid balance,
a. How much is the monthly payment?
b. How much interest will be paid over the term of the loan?

Solution
1. Total cost of purchase = Birr 115,000
 Amount paid at the beginning (Amount of down payment) = 20% of the total
cost
= 0.2 x 115,000 = Birr 23,000
 Amount Unpaid or Mortgage or Outstanding Debt = 115,000 – 23,000
= Birr 92,000
t = 30 years
• j = 11% = 0.11 , m = 12 , i = 0.11 ÷ 12 = 0.00916
• n = t x m = 30 x 12 = 360 months

a. The periodic payment A = ?
A = 𝑃
𝑖
1 − 1 + i −n
A = 92,000
0.00916
1 − 1 + 0.00916 −360
A = 92,000(0.009523233)
A = Birr 876.14
b. Total Interest = (A x n) – P
= 876.14 x 360 – 92,000
= Birr 223,409.49

 Over the 30 years period Emmanuel is going to pay a total interest of Birr
223,409.49, which is well more than double of the initial amount of loan.
 Nonetheless, the high interest can be justified by the fact that value of a real estate
is usually tend to increase overtime.
 Therefore, by the end of the term of the loan the value of the real estate (house)
could be well higher than its purchase cost in addition to owning a house to live in
for the 30 years and more.

Chapter Three
Matrix Algebra and its Applications

@ Business Mathematics Chapter 1& 2.pptx

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