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Welcome

Use of mathematical

tools in business

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Prepared for

Farzana Lalarukh

Assistant professor, Department of Finance

Faculty of Business Studies

University of Dhaka

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We are

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Agenda

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Measuring GDP in the product approach using matrix

• GDP is the market value of all final goods and services made within the borders of a country in a year.

GDP can be determined in three ways,

1) the product (or output) approach,

2) the income approach, and

3) The expenditure approach.

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Assumptions• There are only two firms in Bangladesh• A firm consists of three lands• Paddy and wheat are produced by two firms

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Production in each land

Firm-1 Firm-2

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Measuring procedure

• Total production from all firms is calculated

• GDP=Total production Market Price

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Summation Total Production

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So the GDP is 40+60+64=164 taka

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Case Analysis Using Inverse MatrixTo produce PRAN MANGO JUICE it is necessary to use 8 units of vitamin A, 14 units of vitamin B, and 13 units

of vitamin C. One tube of mixture P contains 1 unit of A, 2 units of B

and 3 units of C. One tube of mixture Q contains 2 units of A, 3 units of

B, and 2 units of C. One tube of mixture R contains one unit of A, 2 units of

B, and 2 units of C.

How many tubes of each type of mixture should be used to produce the juice?

Tabulate the data as follows

VITAMIN

P Q R REQUIREMENT

A 1 2 1 8

B 2 3 2 14

C 3 2 2 13

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Let x tubes of P, y tubes of Q and Z tubes of R be used to produce the juice. Then

x+2y+z=8

2x+3y+2z=14

3x+2y+2z=13

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Writing the equations in the matrix form

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We can solve the problem by the inverse of a matrix

Let A =

Here the determinant of A │A│ =

= 2=4-5 = 1

223

232

121

223

232

121

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M =

=

32

21

22

11

23

1223

21

23

11

22

1223

32

23

22

22

23

Here the minor matrix,

101

412

522

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Here Adjoint Matrix,

A =

A-1 = .AdjA

=

145

012

122

A

1

145

012

122

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z

y

x

=

145

012

122

×

13

14

8

=

3

2

1

=

135640

01416

132816

Putting the value in the equation

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Hence 1 tube of mixture P, 2 tubes of mixture Q, 3 tubes of mixture R ,

should be used to produce the juice.

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Hence 1 tube of mixture P, 2 tubes of mixture Q, 3 tubes of mixture R ,

should be used to produce the juice.

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What is Cost Function

Cost function is a mathematical system relating a firms total cost to its output.

TOTAL COST=

(VARIAVLE COST x OUTPUT)

(+) FIXED COST

It is also a relationship between activity and its cost.

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Who use it

Manufacturer company, Service company such as gas, electricity, Oil company, Many other small organizations.

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Case analysis

General Electronics Company(GEC) charges its

customers according to their usage of electricity

and follows a fixed cost of 3.35 taka and 3.380

paisa per unit for the first 350 unit,5.634 paisa per

unit in excess of 350.

The requirements are finding out cost function,

graph, total and average cost of using 200 units.

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Case analysis (cont’d)

For first 350 units

C(k)=3.35+0.0338k

Total cost of using 200units:

C(200)=3.35+0.0338(200)

=10.11 taka

Average cost=

=0.05055

In excess of 350 units

C(350)=3.35+0.0338(350)

C=15.18

SO,

C(k)=15.18+0.5634(k-350)

200

11.10

Graph

Break even analysis

• Break even point is the point where total revenue is equal to the total cost, that is, profit=0

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Case analysis-2SoundTech, a manufacturer of compact disc (CDs) has a fixed cost of 1000 Taka variable cost of 7 Taka Selling price is 12 Taka per CD. A) find the revenue, cost and profit function? a)R(q) =12Q C(q) =7Q+10000 P(q) =12Q-(7Q+10000) = 5Q-10000 b) What is the profit if 2800 CDs are made and sold? b)P(2800) =5(2800)-10000 =4000

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c) What is the profit if 1000 CDs are made and sold? c) P(1000) =5(1000)-10000 = -5000 d) At What no. Of CDs made and sold will manufacturer

break even? d) Break even point: 5Q-10000=0 Q=2000 CDs

e) At what sales volume (revenue) will break even occurs?

e) Break even dollar volume of sales: R(2000) =12(2000) = 24000 Taka

Graph

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Time value of money

• Present value is the current worth of a future sum of money given a specified rate of return.

• Future value is the value of an asset or cash at a specified date in the future.

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Annuity• An annuity is an amount of money that occurs

(received or paid) in equal amounts at equally spaced time intervals.

• These occur so frequently in business that special calculation methods are generally used.

Example of present value

• Sanjib is planning to buy a car next year that will cost taka 40,000 at that time. If the interest rate is 13% how much should he set aside now and to deposit in a bank in order to pay for the purchase?

• So he should set aside now and to deposit in a bank taka 35,398.23 in order to pay for the purchase.

Example of Future Value • Shamim places taka

10,000 in a savings account @ 13% interest compounded annually. How much will he get at the end of 5 years?

• He will get taka 11592.74 at the end of 5 years.

Parts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

(Ordinary Annuity)EndEnd of

Period 1EndEnd of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

EndEnd ofPeriod 3

Parts of an AnnuityParts of an Annuity

0 1 2 3

$100 $100 $100

(Annuity Due)BeginningBeginning of

Period 1BeginningBeginning of

Period 2

Today EqualEqual Cash Flows Each 1 Period Apart

BeginningBeginning ofPeriod 3

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Present Value of Annuity • Selim took a loan of taka 15,00,000 from Brac

Bank at 15% interest to be repaid in equal yearly installment over next 20 years.

• What will be his annual payment at the beginning of each year?

• What will be his annual payment at the end of each year?

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FindingsMatrix is used to calculate large amount of data

in a simple and smarter way.Cost function and break even point analysis help

to find marginal cost, marginal revenue and optimum level of production.

Interest on loan/deposit, amount of annuity etc are easily calculated by using time value of money concept.

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ANY QUEARY?

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THANKS TO ALL