chapter 3 in managerial economic

7/30/2019 Chapter 3 in Managerial Economic

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Chapter 3

Demand Theory

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INTRODUCTION

You would have done demand theory in your diploma course. Your lecturer would have

discussed the theoretical aspect of demand. Now you will be applying this theory to

make decisions in a firm. In managerial economics you have to assume that you are a

manager of a firm.

For your firm to survive or to be established it needs sufficient demand. It cannot survive

if there is no demand even though it has the most efficient production method. As a

manager you have to understand and analyse the forces that determine the demand for

your product. For example how will customers react to an increase in the price of haircut

during festive season and off-festive seasons? How does the recession affect the

demand for your product? Therefore it can be seen that the demand theory is important

for the creation, survival and profitability of your firm.

This chapter will begin by explaining the basic differences between demand function and

demand curve function, followed by market demand curve. Then attention will be given

to different revenue functions. Finally the concept of elasticity and its importance in

decision making will be discuss .The mechanics of demand will also be analyses using

algebraic equation and graph.

3DEMAND THEORY


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Key terms for review:

Determinants of demand

Multiplicative demand function

Total revenue

Average revenue

price elasticity of demand

Cross elasticity

Complements

Luxury goods

Inferior goods

Demand function

Linear demand function

Demand curve

Marginal revenue

Elasticity

Price elasticity

Income elasticity

Substitutes

Basic necessities

Normal goods


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CHAPTER OVERVIEW


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Learning Objectives

After reading this chapter, the students should be able to:

1. Explain a demand function.

2. Derive a demand curve function from the demand function

3. Derive the total, marginal and average revenue function and

describe the relationship between them.

4. Calculate and interpret price, income and cross elasticity.

5. Analyse the importance of elasticity in decision making.


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3.1 THE DEMAND FUNCTION

A textbook definition of effective demand is - demand that is backed by the purchasing

power. This definition can further be analysed as the desire, willingness and ability to buy

a product at a specific price at given point of time.

Demand function refers to the relationship that exists between the quantity demanded

and the determinants of demand. The determinants of demand are the factors that

influence demand. For example, the demand for shoes (Qs) is influenced by price of

shoes (Ps), advertising expenditure (A), income (I) and (T).

The demand function can be written in a functional form as:

Qs = f (Ps, A, I, T ) Eqn 3a

FORMS OF DEMAND FUNCTIONS

There are various forms of demand functions. The specific form of the demand function

is determined empirically. At this stage we shall analyse two forms of demand function.

They are:

a) Linear Demand Function

The determinants of demand for product X in a linear form can be written

as :

Qx =

- 0 Px + 1 Py + 2 I + 3 T Eqn 3b

where Px is price of product X, Py is the price of related product y.

I is income and T is taste,

= constant and includes all the other variables not mentioned

in the demand function.

0 to 3 = coefficients of the demand function.


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The coefficients indicate the marginal impact of each independent

variable on the quantity demanded. For example if the price of X change

by one unit the quantity demanded will change by 0 units.

The sign (+/-) before the coefficient shows the (positive or negative)

relationship between the independent and dependent variable.

b) Multiplicative Demand Function

Q = Px 0 Py

1 I 2 T 3 Eqn 3c

Where = constant

0 to 3 = measures the percentage change in the

dependent variable relative to the percentage

change in the respective independent variable.

The coefficients are replaced by their numerical equivalent after they are

estimated using regression analysis.

Example 1

Suppose the demand function for product X has been estimated and is as follow:

Qx = 1.0 - 2.0Px + 1.5I + 0.8Py + 2.0T

and coefficients are replaced by their numerical equivalents, the positive or

negative signs before the respective coefficient indicates the direction of the relationship

that variable has on Qx. For example, coefficient -2.0 Px , explains that when the price

of X increase by one unit the quantity demanded of X will decrease by 2.0 units.

If we assume Px = 2, I = 4, Py = 2.5 , and T = 2 , The quantity demanded will be:

(Substitute these values into the equation)

Qx = 1.0 - 2.0(2) + 1.5(4) + 0.8(2.5) + 2.0(2) = 9 Units


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Based on the equation we can predict that, at the current values of the independent

variables, quantity demanded of product X will be 9 units, ceteris paribus.

3.2 The Demand Curve

The Law of demand seeks to explain the relationship between quantity demanded and

price, that is quantity and price are inversely related (price increases, quantity demanded

decrease and vice versa). The Law of demand can be depicted in a functional form by a

demand curve function.

A demand curve function shows the relationship between quantity demanded and the

price of the product assuming all the other factors influencing its demand to remain

constant (Ceteris Paribus). The demand curve function is a special sub-case of the

demand function. In other word it is derived from the demand function.

In functional form a demand curve function is written as:

Qx = f (Px) ceteris paribus. Eqn 3d

In a linear form it will be:

Qx = A - 0Px Eqn 3e

A is derived by compressing (adding) all the factors that determine demand except for

price of the product, based on equation Eqn 3b, A is:

A = + 1Py + 2I + 3T

Note: Since traditionally price is being placed on the y-axis and quantity on the x-axis,

the above equation can be rewritten to make Px the subject. But one has to bear in mind

that quantity (Qx) is the dependent variable and price (Px) is the independent variable.

Rewritten in the conventional way (making P the subject):


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Px= A _ 1 Q

0 0

For easy reference we shall assume that A/ 0 is a and 1/ 0 is b. Therefore the

demand curve function will be as follows:

Px = a b Qx Eqn 3f

The demand curve function can be represented graphically in diagram 3.1.

Diagram 3.1 Demand Curve

Example 2

This example will show you how to derive a demand curve function from demand

function.

Based on the estimated demand function from Example 1 where:

Qx = 1.0 - 2.0Px + 1.5I + 0.8Py + 2.0T

Given Px = 2, I = 4, Py = 2.5 , T = 2

Get the value of A:


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A = 1.0 + 1.5 (4) + 0.8 (2.5) + 2.0 (2) = 13

and then by substituting it into the demand curve function:

Qx = 13 - 2.0Px

In the conventional form:

Px = 6.5 0.5Qx

DEMAND CURVE FUNCTION

Diagram 3.2 shows the demand curve:

Diagram 3.2 Demand curve

NOTES

The price is expressed as a linear function of quantity demanded, 6.5 is

the intercept on the vertical axis (a) and 0.5 is the slope term (b). The

negative sign before the slope term or coefficient shows an inverse

relationship between price and quantity. The intercept on the horizontal

axis is 13 (the value of A)


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3.3 THE MARKET DEMAND CURVE

Market demand of a product or service is the sum of individual demand. It shows the

total quantity demanded in the market at various prices. To get the market demand

function we add the individual demand functions horizontally, that is:

Qm = Qa +Qb +Qc +. + Qn Eqn 3g

When P1 = P2 = P3 = = Pn

Example 3

Given individual demand functions as :

Individual 1 Q1 = 10 - 4p

Individual 2 Q2 = 6 - 3p

Individual 3 Q3 = 15 - 0.9p

The market demand curve function is

QM = Q1 + Q2 + Q3

QM = (10 - 4p) + (6 - 3p) + (15 - 0.9p)

QM = 31 - 7.9p or PM = 3.92 - 0.126 QP

The market demand curve can also be determined graphically by adding the

quantity demanded at a given price. For simplicity we assume that there are two

individuals with the following demand curves.


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Diagram 3.3 The market demand curve

Referring to the diagram 3.3 when the price is RM10, Individual 1 buys 3 units and

individual II buys 2 units. The market demand at this price will be 5 units (2+3). At price

RM5 the market demand is (5+6) 11 units.

3.4 TOTAL, MARGINAL AND AVERAGE REVENUE FUNCTIONS

TOTAL REVENUE FUNCTION

The total revenue shows the total dollar sales of a firm at a certain period of time.

Managers are interested in the relationship between price and quantity since it influences

total revenue. When the price changes the total revenue will change. To get the total

revenue function, total quantity demanded (sold) is multiplied by the price of the product

(P x Q).

That is : Px = a b Qx

Total revenue is: TR = P x Q

By substituting P x into the total revenue function we get :


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TR x = (a b Qx) Qx

Therefore the total revenue function is

TRx = a Qx bQx2 Eqn 3h

Example 4

Based on Example 2 :

Px = 6.5 - 0.5Qx

the TR function is equal to

TR = Q (6.5 - 0.5Qx)

= 6.5Q - 0.5Q2

The total revenue function can be represented graphically by plotting quantity against the

total revenue. For example, if the quantity is 2 the total revenue will be: 6.5(2) - 0.5 (2)2

= 9 and at quantity 6.5 and 7 the total revenue is 21.125 and 14 respectively. Using

these values the total revenue curve can be plotted as show in diagram 3.4.

Diagram 3.4 The Total Revenue Curve


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To find the total revenue maximising quantity, differentiate the equation, set it equal to

zero and solve for Q The process is shown below.

TR = 6.5Q - 0.5Q2

TR = 6.5 - Q = 0

Q

Q = 6.5

At quantity 6.5 units the total revenue is 6.5 (6.5) - 0.5 (6.5) 2 which is 21.125.

THE MARGINAL REVENUE FUNCTION

The marginal revenue is defined as the change in the total revenue that results from one

unit change in the quantity demanded. It can be express as a first derivative of total

revenue with respect to Qx

That is:

TRx = a Qx bQx2

MRx = TR = a 2bQx Eqn 3i

Qx

Diagram 3.5 shows a graphical representation of marginal revenue function.

Example 5

Based on Example 4:

TR = 6.5Q - 0.5Q2

The marginal revenue function is

MR = TR = 6.5 - Q

Q

We studied

optimization in chapter2. Make a point to

revise the chapter if

ou find difficulties


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The marginal revenue slope downwards and the intercept is at 6.5. The

marginal curve is shown in Diagram 3.5

MR/p

6.5 P = 6.5 -0.5Q

MR = 6.5 Q

or a 2bQx

0 6.5 Qtty

Diagram 3.5 The Marginal Revenue Curve

The marginal revenue is zero at quantity 6.5.

MR = 6.5 - Q = 0

Q = 6.5

THE AVERAGE REVENUE FUNCTION

Average revenue is the revenue earned per unit of output sold. It is calculated by

dividing total revenue by the quantity.

TRx = a Qx bQx

2

ARx = TR = a bQx Eqn 3j

Qx


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NOTES

The average revenue function is also the demand curve function.

Example 6

The average revenue function can be computed from the above total revenue

function.

Given:

TR = 6.5Q - 0.5Q2

AR = TR = 6.5 - 0.5Q

Qx

The average revenue function is similar to the demand curve function

P = 6.5 - 0.5Q.

The relationship between demand curve, marginal revenue and the total

revenue curves.

The relationship between demand curve, marginal revenue and the total revenue

curve functions can be analyse diagrammatically.


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Diagram 3.5 The relationship between Total Revenue and Marginal Revenue Curve.

Based on diagram 3.5 first we begin by comparing the similarities and differencesbetween the average /demand curve and marginal revenue functions.

a. AR and MR has the same intercept, this indicates that both curves

must begin from the same point on Y axis.

b. The slope of marginal revenue curve is twice (2 b Qx) of the

demand curve (b Qx )

Next we shall look at the relationship between marginal revenue and total revenue.

a. When marginal revenue is zero the total revenue is at its

maximum. (refer to quantity Q*)

b. When marginal revenue is positive the total revenue is increasing

(refer to quantity O- Q*)

c. When marginal revenue is negative the total revenue is decreasing

(refer to quantity after Q*)

Again, section

2.2 previously

explained this

relationship


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QUESTIONS

1. Given the demand curve function for caps as:

Q = 3200 - 12P

Q - Quantity demand for caps

P - Price of caps.

a) Rewrite the demand curve function in the conventional way.

b) How many caps will be sold at RM10 each?

c) At what price would sales equal zero?

2. There are three individuals with different demand curve function wanting to buy

badges (the quantity is in hundreds). Their demand curve function for each

individual is as follows.

Individual I PI = 5 - 5 Q1

Individual II PII = 4 - 3.5Q

Individual III PIII = 8 - 1.5Q

a) What is the market demand curve function?

b) How many badges will be bought in the market at price RM2 and

how will this be distributed among the individuals?

3. Suppose the demand function for special Sukom files is

Qd = 2500 - 6P

a) Derive the total revenue function?

b) At what output is the total revenue maximum?

c) Derive the marginal revenue function?


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d) At what output is marginal revenue zero?

e) What can you conclude from the answers in (b) and (d)

4. A firm selling hot-dogs estimates the demand function for hot-dogs as

Qh = 150.3 + 10.2 A + 9.6 Py - 15.4 Ph.

Where Qh is the quantity of hot dogs, A is advertising expenditure, Py is the price

of hot dog competitor and Ph is the price of hot dog

Given A is RM100; Py is RM2; and Ph is RM18.00.

a) Determine the demand curve function and plot the demand curve

using any three price-quantity combinations.

b) Derive the marginal revenue function.

SUGGESTED SOLUTIONS

1. a) Q = 3200 - 12P

P = 3200 - Q

12

P = 266.67 - 0.083 Q

b) When P = 10

Q = 3200 - 12 (10)

= 3080 units

c) When Q = 0

P = 266.67 - 0.083 (0)

P = 266.67

2. a) Market demand is equal QI + QII + QIII


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(make Q the subject)

PI = 5 - 5 Q1

Q1 = 5 - P

5

Q1 = 1.0 - 0.20P

PII = 4 - 3.5Q

Q11 = 4 P

3.5

QII = 1.14 - 0.29P

PIII = 8 - 1.5Q

QIII = 8 - P

1.5

QIII = 5.33 - 0.67P

QM = QI + QII + QIII

QM = 7.47 - 1.16P

PM = 6.44 - 0.86Q

b) When P = 2

QM = 7.47 - 1.16 (2)

= 5.15

Each Individual will buy:

QI = 1-0.2(2) = 0.6QII = 1.14 - 0.29(2) = 0.56

QIII = 5.33 - 0.67(2) = 3.99

3. a) P = 416.67 - 0.17Q

TR = PQ

TR = 416.67Q - 0.17Q2

b) TR maximising quantity.


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TR = 416.67 - 0.34Q

Q

Q = 1225.5

c) MR = TR = 416.67 - 0.334Q

Q

d) MR = 0

416.67 - 0.34 Q = 0

Q = 1225.5

e) When marginal revenue is zero the total revenue is maximum.

4. a) Qh = A - 15.4 Ph

A = 150.3 + 9.6 (2) + 10.2 (100)

= 150.3 + 19.2 + 1020

= 1189.5

The demand curve function is:

Qh = 1189.5 - 15.4 Ph

Written in the convention form:

Ph = 77.24 - 0.065 Qh

To plot the demand curve we can take any three values of Q.

Price RM Quantity Points

77.24

70.84

61.24

0

100

250

A

B

C


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Price

77.24 A

Demand curve.

70.84 B

61.24 C

D

100 250 Quantity

b) TR = 77.24 Q - 0.065 Q2

MR = TR

Q

= 77.24 - 0.13 Q

3.5 ELASTICITY

Elasticity is a measure of responsiveness of the quantity demanded when there is a

change in its determinants

You would have learnt about concept of elasticity and how to calculate it in the

economics course at diploma level. You would have used a standard formula to calculate

elasticity. For example, the price elasticity is calculated by using following formula: (basic

formula)

EP = percentage change in quantity demanded

percentage change in price

or EP = OD - ND X OP

OP-NP OQ


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Where OD is original demand, ND is new demand, OP is original price and NP is

new price. This method of measuring elasticity is quite sensitive to which value is

chosen as old and which is chosen as the new.

However, in this course you are introduced to two other approaches to compute

elasticity. They are arc price elasticity (where the basic formula is the same) and

point price elasticity. Arc elasticity measures the elasticity over a range of price

and takes into consideration averages. The point elasticity is used to measure a

very small or infinitesimally small change in the price.

PRICE ELASTICITY OF DEMAND (Ex)

Price elasticity of demand measures the responsiveness of quantity demanded

when there is a change in its price. Basically elasticity can be divided into three

broad categories and two extreme cases. Price elasticity of demand is seen a

positive integer

(Note: When elasticity is interpreted the negative sign can be ignored for normal

goods)

a) Elastic (E > 1). Consumers are responsive to the change in price.

b) Inelastic (E < 1). Consumers are not very responsive to the change

in price.

c) Unitary elasticity (E = 1) Consumer are proportionately responsive to

the change in price.

There are two extreme cases.

a) Perfectly elastic (E =

) consumers are very responsive to the changein price. When there is a slight increase in price, the quantity demanded

will fall to zero. The demand curve is a horizontal straight line. When

there is a slight decrease in the price, the consumers will buy up all that

available in the market i.e. price elasticity is infinity.


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b) Perfectly inelastic (E = 0) consumers are not responsive to change in

price. The demand curve is a vertical straight line. Quantity demanded

remain unchanged when the price changes.

MEASUREMENT OF PRICE ELASTICITY

The price elasticity is measured by dividing the percentage change in quantity

demanded by the percentage change in price.

Arc Price Elasticity: As mention earlier the basic formula is the same, however,

the method of calculating the percentage differs. It takes into consideration the

average of the old quantity and the new quantity on the denominator and the

average of old price and new price on the numerator The formula is:

Q2 - Q1 X P2 + P1

P2 - P1 Q2 + Q1

Where: Q1 is original quantity demanded,

Q2 is new quantity demanded,

P1 is original price and

P2 is new price

Point Elasticity: Point elasticity calculates elasticity for an infinitesimally small change in

the independent variable (price). The formula is:

Ep = Qx x P x

P x Q x

The first term is the partial derivative of the demand function in terms of Px.

Px and Qx is the price and the quantity demanded of the product X respectively.

Based on Eqn 3.1 the demand function is

eqn 3k


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Qx = - 0 Px + 1 Py + 2 I + 3 T

The point price elasticity based on the above formula will be;

Ex = - 0 X Px /Qx

Example 7

Based on Example 1 where :

Qx = 1.0 - 2.0Px + 1.5I + 0.8Py + 2.0T

Given Px = RM2 Py = RM2.5 I = 4 and T = 2

Qx = 9 units

The point elasticity can be calculated by differentiating the equation in terms of Px and

then multiplying by the ratio of Px to Qx.

Qx = - 2.0 Qx = 9 Px = 2

Px

The point price elasticity is

Ex = - 2.0 x 2/9

= - 0.44 (inelastic)

When the price changes by one percent the quantity demanded will change by 0.44

percent.


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ELASTICITY ALONG A DEMAND CURVE

In a linear demand curve function the elasticity along the demand curve varies from zero

to infinity. This is because QX / PX is constant but Px / Qx will increase as price

increase. (You may want to check it out yourselves. The elasticity at different points

along the demand curve can be determined by calculating the quantity demanded at

different price level and then substituting it into the elasticity formula). Diagram 3.6

shows elasticity along a demand curve at different points.

Price

E =

E > 1

E = 1 (Mid - point)

E < 1

E = 0 Quantity

Diagram 3.6 Elasticity along the Demand Curve

THE IMPORTANCE OF PRICE ELASTICITY

The concept of elasticity is useful in decision-making. It helps managers to understand

the relationship between elasticity, total revenue and marginal revenue. It assist the

manager in terms of pricing decision that is, whether to change or maintain the price of

the product when the objective is to maximise revenue.

The relationship between price, marginal revenue and total revenue can be summarizedin a single number known as price elasticity of demand. That is:

TR = PQ

MR = dTR

dQ

= P + Q P

Q


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= P 1 + Q Q

P P

Given the elasticity formula as

E = Q - P

P Q

and 1 = Q - Q

E P P

next:

MR = P 1 - 1

E

For normal good the price elasticity is negative, the positive sign in the

above equation becomes negative when multiplied by a negative.

MR = P 1 + 1

E eqn 3l

Bases on the above equation the following relationship can be develop:

a. When price elasticity is greater than one the marginal revenue is

positive (Ex > 1, MR > 0)

b. When price elasticity is less than one the marginal revenue is

negative

(E x < 1, MR < 0)

c. When price elasticity is equal to one the marginal revenue is zero

(E x = 1, MR = 0)


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The relationship can be diagrammatically represented as in diagram 3.7

Price E = 1

E < 1

E > 1

TR

DD

MR = 0

Quantity

MR

Diagram 3.7. Elasticity, Total Revenue and Marginal Revenue

A producer will not sell his output in the region where elasticity is less than one. This is

because the total revenue is decreasing and the marginal revenue is negative as more

units are sold. By reducing the output the producer will be able to increase his profits: as

total revenue will increases and total cost decreases.

DETERMINANTS OF DEMAND ELASTICITY

a) Availability of subst itutes

Substitutes play an important role in determining the elasticity of the product.

Products with substitute have a higher price elasticity compared to products with

poor or no substitutes. For example if the price of coca-cola increases, the

demand for its substitutes will increase. The demand for coca-cola will decreaserapidly with a slight increase in price. Thus, the demand for coca-cola is

relatively elastic.

b) Proportion of income spent on the product.

The demand for the product is said to be elastic if the expenditure on the product

forms a large proportion of the total expenditure For example, students who are


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staying outside campus and are renting rooms, if the rent increases, these

students will find alternative accommodation as a large proportion of their income

is spent on rental. The demand for rented rooms can be considered elastic. On

the other hand, demand is inelastic for products that account for a small

proportion of total expenditure e.g. salt.

c) Length of the period

In the long run demand is more elastic than in the short runs as consumers are

able to adjust to new prices. For example, demand for electricity. In the short

run, when the price of electricity increases, consumers will not reduce their

consumption of electricity. Consumers may still use their electrical appliances.

But given some time (long run), consumers are able to find other energy sources,

i.e. replace electrical appliances with non electrical appliances or to use them

less frequently, reducing the consumption of electricity in the long run.

Consumers are more responsive to the changes in price in the long run.

INCOME ELASTICITY (EI)

Income elasticity measures the responsiveness of demand to the change in income.

MEASUREMENT OF INCOME ELASTICITY

Income elasticity is measured by dividing the percentage change in quantity demanded

by the percentage change in income.

Arc Income Elasticity: The formula is very much like the arc price elasticity. For arc

income elasticity the variable price is changed to income. The formula is:

EI = Q2- Q1 X I2 + I1

I2 - I1 Q2+ Q1

Where: Q1 is old quantity demanded,

Q2 is new quantity demanded,

I1 is original income and

I2 is new income


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Point Income Elasticity: Where the change in income is infinitesimally small. The formula

is:

Ei = Qx x I

I Qx eqn 3m

Based on demand function in equation 3.1:

Qx = - 0 Px + 1 Py + 2 I + 3 T

The income elasticity is:

EI = 2 X I / Q x

Example 8

Based on Example the income elasticity for product X is:

Qx = 1.0 - 2.0Px + 1.5I + 0.8Py + 2.0T


Qx = 9

Qx = 1.5; Qx = 9, I = 4

I

EI = 1.5 x 4/9

EiI = 0.67


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When income increase by one percent the quantity demanded will increase by 0.67

percent. This product can be classified as a basic necessity. It is also recession proof

product since the impact on quantity demand is smaller compare to the fall in the income

level (economic growth).

IMPORTANCE OF INCOME ELASTICITY

As a decision maker income elasticity is important. It shows how the consumers react

towards the quantity demand when there is a change in the income.

Income elasticity is useful to:

a) Determine the type of good.

Income elasticity can either be positive or negative. Positive income elasticity is

associated with normal goods, i.e. when income increases the quantity demanded for

these goods increase. Normal goods can further be divided into basic necessities and

luxury goods. For basic necessities the income elasticity is less than one or close to

zero, whereas for luxury good the income elasticity is greater than one.

On the other hand, negative income elasticity is associated with inferior goods. That is,

when income increases, the spending power increases the demand for an inferior good

will decrease. Consumers will buy better quality goods and reduce their purchase of

inferior quality goods. .

b) Forecasting the future demand

A producer will need to forecast the future demand and be prepared to supply the

amount of quantity demanded. One-way to forecast the demand the product is to

determine the phase of business cycle in the economy, that is in which phase (recession,

expanding, peak and contraction) is the economy. If the economy is expanding or during

economic prosperity, consumers are willing to spent more on luxury goods and the

demand for luxury goods will increase. The rate of increase in demand for these good is

greater than the rate of income (economic growth). The reverse is true during recession;


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the demand for luxury goods is weak. During this time producers should concentrate on

basic necessities. Basic necessities are said to be recession proof as the percentage

decrease in demand is much smaller compare to the percentage decrease in income

The consumers spending on these goods is consistent or the impact of decreasing

income is very small.

c) Promotional strategies

Income elasticity is also useful in promoting or marketing a product. Luxury products

that are consumed by higher income-group are advertised in media that reaches them.

Their promotional strategies would differ in terms of service and are given personal

touch. These products are also be displayed in an exclusive manner Whereas normal

products can be promoted in daily media and be displayed to reach the general public .

CROSS ELASTICITY (Exy)

Cross elasticity measures the responsiveness of the demand of one product (X) when

the price of another changes (Y).

MEASUREMENT OF CROSS ELASTICITY

The cross elasticity is measured by dividing the percentage change in quantity

demanded of one product (X) by the percentage change in the price of another product

(Y).

Arc Cross Elasticity:: The formula is :

Exy = Qx, 2 - Qx,1 X Py,2 + Py,1Py, 2 - Py,1 Qx,2 + Qx,1

Where: Qx,1 is old quantity demanded,

Qx, 2 is new quantity demanded,

Py, 1 is original price of product y and

Py, 2 is new price of product y


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Point Cross price Elasticity: Measure the change in the quantity demand of a product

when there is an infinitesimally small change in price of another product.

The formula is :

Exy = Qx x Py

Py Qx eqn 3 n

Based on equation 3a demand function:

Qx = - 0 Px + 1 Py + 2 I + 3 T

The elasticity is:

Ex = 1 X Py /Qx

IMPORTANCE OF CROSS ELASTICITY

The cross elasticity is used to it determine the relationship between the two goods. That

is

If the cross elasticity is positive the two goods are substitutes, for example Pepsi

and Coke. The higher the value of the cross elasticity the better the substitutes.

If the cross elasticity is negative the two goods are complementary, for example

pen and ink.

If the cross elasticity is zero, it shows that the two products are not related.


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Example 9

From Example 1 above, the cross elasticity for product X is

Qx = 1.0 - 2.0Px + 1.5I + 0.8Py + 2.0T


Qx = 9

The cross elasticity is

Qx = 0.8, Qx = 9, Py = 2.5

Py

Exy = 0.8 x 2.5 / 9

= 0.22

When the price of X changes by one percent the quantity demanded for X will change by

0.22 percent. The two products are said to be poor substitutes


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QUESTIONS

1. The demand function for Rainbow Lemonade has been estimated as

QL = 26 - 20PL + 28 Pc - 17Ac + 20AL + 5I

Where

QL is the quantity of Rainbow Lemonade (in cases)

PL is the price of Rainbow Lemonade in RM

PC is the price of competitor in RM

AL is the advertising expenditure of Rainbow lemonade.

AC is the advertising expenditure of competitor.

I is the income per capita in RM

The current values of the independent value are

AL = RM10, PL = RM 1.50, PC = RM 1.8, I = RM12, AC = RM 15

a) Derive the demand curve function and the total revenue function.

b) At what output is the total revenue maximum? Calculate the price at this

output?

c) If the income decrease by 10% what is the impact on Rainbow Lemonade.

(Calculate income elasticity)?

d) What is the price and cross elasticity? Interpret the elasticity

2. Given the following demand function

Qx = 7 - 2PX + 5Py + 8I

Where Qx and Px is the quantity and price of X respectively PY is the price of

another product; and I is per capita income. Given Px = RM5, Py = 5.50 and I =

4.


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a) Calculate the price elasticity. Interpret the elasticity.

b) Calculate the income elasticity. What can you say about the product?

c) Calculate the cross elasticity. What is the relationship between this two

product?

3. A firm XYZ has written childrens comic books. The estimated total revenue

function for the sales is

TR = 150Q - 20Q2

a) Over what range is demand elastic?

b) Derive the demand curve function.

c) Given the current price is RM50, should the firm change the price to

maximise its total revenue?

4. A consultant estimates the price-quantity relationship for Alice Pizza to be

P = 75 - 25 Q

a) At what output is demand unitary elastic?

b) At what output is marginal revenue zero?

c) At price RM7, what is the price elasticity?

5. The price elasticity for fish is estimated to be -0.6 and income elasticity is 0.7. At

a price of RM8.00 per kg and a per capita income of RM30, 000, the demand for

fish is 25,000 kg.

a) Is fish a basic or Luxury good? Explain.

b) If the per capita increase to RM30, 500, what will the quantitydemanded?

c) If the price of fish increases to RM9.00 when per capita is RM30, 000,

how much should the per capita income change for the quantity

demanded to remain at 25,000 kg?


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SUGGESTED SOLUTIONS

1.a) QL = 26 - 20PL + 28 (1.8) - 17 (15) + 20 (10) +5(12)

QL = 81.4 - 20PL Demand curve function

PL = 4.07 - 0.05 QL

TR = 4.07Q - 0.05QL2 Total Revenue function

b) TR maximising output is

TR = 4.07 - 0.1Q = Q Marginal Revenue function

Q

Q = 40.7

P = 4.07 - 0.05 (40.7) = 2.035

Total revenue maximising quantity is 40.7 cases and the

priceRM2.035.

c) Income elasticity:

QL = 26 20(1.5) + 28 (1.8) - 17 (15) + 20 (10) +5(12)

QL = 51.4

EI = QL x I

I Q L

EI = 5 x 12 = 1.17

51.4


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When income falls by 10% quantity demanded will decrease by

1.17 x 10 = 11.7 %.

d) Price elasticity

Ep = QL x P L

PL Q L

QL = 26 20(1.5) + 28 (1.8) - 17 (15) + 20 (10) +5(12)

QL = 51.4

Ep = -20 X 1.5 = - 0.58

51.4

When price increase by 1% the quantity demanded decreases by

0.58%.

Cross elasticity

ELC = QL x PC

PC Q L

ELC = 28 x 1.8 = 0.98

51.4

When the price of competitor increases by 1% the quantity

demanded for Rainbow Lemonade will increase by 0.98%.

2.a) The quantity demanded is

Qx = 7 - 2 (5) + 5 (5.50) + 8 (4)

Qx = 56.5


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Price elasticity

Ep = Qx x P x

P x Q x

Ep = -2 x 5 = -0.18

56.5

When the price increases by 1% the quantity demanded decrease

by 0.18%.

b) Income elasticity

EI = Qx x I

I Q x

EI = 8 x 4 = 0.57

56.5

When income increases by 1% the quantity demanded increases

by 0.57%. The product is a basic necessity.

c) Cross elasticity

Exy = Qx x Py

Py Q x

Exy = 5 x 5.5

56.5

= 0.49

When the price of Y increases by 1% the quantity demanded for X will

increase by 0.49%. The two products are substitutes.


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3.a) TR maximizing quantity is:

TR = 150Q - 20Q2

TR = 150 - 40Q = 0

Q

Q = 3.75

When the total revenue is at the maximum, elasticity is unitary.

Therefore the demand curve is elastic just before the total revenue

maximizing quantity is achieved. In this case it will be before 3.75

units. The elastic range will be 0 to3.75 units

b) Demand curve function is:

P = TR = 150 - 20 Q

Q

c) To find the total revenue maximising price substitute the totalrevenue quantity calculated in (a) into the demand curve function:

TR = 150 Q - 20 Q2

P = 150 - 20 (3.75) Demand curve function

P = RM 75

The firm should increase the price to maximise to RM75 to

increase total revenue.

4.a) Unitary elasticity (calculate the quantity that maximise total

revenue).

TR = 75 Q - 25 Q2


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TR = 75 - 50 Q = 0

Q

Q = 1.5

When total revenue is maximum when elasticity is unitary at Q =

1.5.

c) MR = 0

MR = 75 - 25 Q = 0

Q = 1.5

c) TR = 75 Q - 25 Q2

P = TR =75 -25Q Demand curve function

Q

Given P = 7 , substitute into the demand curve function

7 = 75 - 25Q

Q = 2.72

Price elasticity

Ep = Qx x P x

P x Q x

1 x 7 =0.103 - demand price elasticity.

25 2.72

When the price increases by 1% quantity demanded will decrease

by 0.103%.


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5. a) Fish is an Basic good because the income elasticity is less than

one (EI.= 0.7)

b) The percentage change is income is

30500 - 30000 X 100 = 1.67%

30000

When the income changes by 1% quantity demanded will change

by 0.7%.If income changes by 1.67 quantity demanded will change

by :

1.67 x 0.7 = 1.17%, that is

The change in quantity is:

1.17 x 25000 = 292.5 units

100

The quantity demanded will be (25000 + 292.5) = 25292.5.5

c) At price RM9.00 the quantity demanded will be.

Percentage change in price of fish

9 - 8 x 100 = 12.5%

8

The percentage change in quantity demanded is:

12.5 x 0.6 = 7.5%

The total quantity is 7.5% x 25000 = 1875.

The percentage change income to offset the decrease in quantity

demanded will be.


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7.5% = 10.7 %

0.7%

The income will increases by 10.7% x 30,000 = 3214.28

SUMMARY

This chapter explained how to derive the demand curve, total marginal and average

revenue curve function from a demand function.

It showed how the above three functions are interrelated.

You will also notice how elasticity is calculated. Here emphasis was given on point

elasticity and it looked at price, income and cross elasticity.

As a manager you will be made aware of the importance of elasticity in decision-

making.

PRACTISE QUESTIONS

Q1. The demand function for TV set is given as follow

Qx = 350 3.7Px + 0.2Y + 4.2 Pz

Where Qx is quantity of TV set demanded per month

Px is the price of TV set

Y is the per capita income

Pz is the price of a competitive brand

a. What is the demand curve function if

Px = RM1200 Y = RM5500 Pz = RM1500.


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b. If the firm wishes to increase its total revenue, should the price be increased

or decreased?

c. At what price and output would total revenue maximize?

Q2. The market demand for good X is

Q = 70 3.5P 0.6M + 4Pz

Where Q is the number of units of good X demanded

P is the price of good X

M is income of buyers

Pz is the price of related good.

a) Assuming P = 10, M= 20, and Pz = 6, calculate Q. hence, compute the price,

income and cross elasticity of demand of good X.

b) Is X a normal or inferior good? Explain.

c) Are X and Z substitute or complements? Explain.

d) Is demand for X elastic or inelastic? Why do you say so?

e) Suppose the producer of X wishes to increase total revenue by changing

price, should he increase or decrease price? State your reason.

Q3. Zaids Frozen Pizza has enjoyed rapid growth in West Malaysia. From the

analysis it was found that the demand curve follows this pattern.

Q = 1000 3000P + 10A

Where Q = quantity demanded

P = product price (in RM)

A = advertising expenditure (in RM)

a) Assume that P = 3 and A = 2000Suppose the firm reduces price. Would this increases total revenue? Explain.

b) Assume that P =4 and A = 2100

Suppose the firm reduces price. Would this increase total revenue? Explain.

Q4. Hardwood cutters offers seasoned, split fireplace logs to consumers in Kampung

Parit, Perak. The company is the low-cost provider of firewood in this market with


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fixed costs of RM10 000 per year, plus variable costs of RM25 for each unit of

firewood. Annual demand for the company is:

P= 225 0.125 Q

Where p is the price of firewood per unit and Q is the number of units of firewood.

a) Determine the marginal revenue and the marginal cost function.

b) Calculate the profit maximizing quantity, price and profit.

c) Calculate the price elasticity at the profit maximizing price.

Q5. Syarikat Alpha-Beta has estimated the demand curve for its product as follows:

Q = 8000 - 5P

where Q is quantity sold per week and P is the price per unit.

a) on the estimated demand curve, write the equation for:

i. Based total revenue

ii. Average revenue

iii. Marginal revenue

b) What is the maximum total revenue per week that Syarikat Alpha-Beta can

obtain from sales of its product?

c) Calculate the arc price elasticity of demand for Syarikat Alpha-Betas product

between Q = 3,000 and Q = 3,200.

Q6. The marketing department for a firm that manufactures vehicles has determined

the following demand function for their vehicles:

QV = 5000 0.6PV + 0.2PC + 0.04Y + 0.02A

Where:

QV = the number of the firms vehicles sold weekly

PV = the price of the firms vehicle

PC = the price of a close competitors vehicle

Y = average household income, and

A = weekly advertising dollars spent


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a) If PV = RM10, 000, PC = RM8000, Y = RM12,000 and A = RM4000, find the

price elasticity of demand.

b) Is demand elastic, unitary elastic, or inelastic? Why?

If the firm decreases price, what will happen to total revenue?

c) Assuming values of the variable is as given in part (a) above, determine the

income elasticity. Interpret your answer.

d) Calculate the cross price elasticity of demand and interpret your answer.

Q7. The following functions describe the demand of 3 small firms: A, B and C that

sell hand phone accessories to the customers.

Firm A : P = 500 0.5QA

Firm B : P = 300 0.25QB

Firm C : P = 200 0.125QC

Where Q is the quantity demanded and P is the price.

a) Calculate the market demand function.

b) Using the market demand function, determine the total revenue maximizing

price and quantity.

c) The industry supply equation is given by QS = -1000 + 10P. Determine the

market equilibrium price and quantity

Q8. The research department of White Pigeon wished to estimate the demand

function for its new product, Cage XP. A demand function had been estimated

on 120 respondents using regression analysis. The demand function is as

follows:

Q = 12,000 8P + 1300A + 5Pc + 2I

Where

Q = Quantity demanded for Cage XP cages

P = Price of Cage XP cages (RM70)

A = Advertising expense, in thousands (RM54)


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Pc= Price of competitor's product (RM80)

I = Average monthly income (RM400)

a) Derive an expression for the firms conventional demand curve for the new

product, Cage XP cages If the firms objective is to maximize total revenue

from the sales of Cage XP cages, at what price should the firm charge?

b) Should White Pigeon Company consider reducing its price in order to

increase its total revenue? Explain.

c) Calculate income elasticity of demand. Is the Cage XP cages a luxury,

inferior or necessity good?

d) Calculate the cross elasticity of demand. Are Cage XP cages and interpret

it.

STUDY NOTES

chapter 3 in managerial economic

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