Business Mathematics

B.Com. (Hons.) IV Sem.

Course Instructor
Dr. B. B. Mohapatra
Department of Commerce
Maharaja Agrasen College
(University of Delhi), Delhi
 Unit1
• Matrix Representation of Data
ONCE YOU HAVE REPRESENTED THE GIVEN DATA IN THE
QUESTION IN MATRIX FORM, THEN THE TASK INVOLVES
SIMPLE MATRIX MULTIPLICATIONS OR SOLVING OF
SIMULTANEOUS EQUATIONS OR BOTH
Primarily We have to Apply Matrix Algebra esp;
Multiplication, Finding the value of Determinant,
minor, cofactor, ad-joint and Inverse.
Multiplication or Inverse: the rule of thumb
• CRAMER’S RULE
• LEONTIF’S INPUT OUTPUT MODELS
 Input - Output (I-O) Analysis
As we know there are various sectors/industries in the
economy. Each sector produces something by using its own
output and that of others as input. Thus, there is input
and output for all the sectors in the economy.
• Can we put all these information in a matrix framework?

• Why?
• Because by putting it in a matrix framework, we can study
the inter-industry requirements, aggregate output in the
economy and can plan for our economy accordingly.
 Input - Output (I-O) Analysis
Example:
Suppose there are 3 industries in an economy, say; industry
1, industry 2 and industry 3.

Industry 1 produces 60 units of output by using 10 units of

its own output, 5 and 20 units of output of Industry 2
and Industry 3 respectively.

Similarly, Industry 2 produces 60 units of output by using 20

units of its own output, 20 and 10 units of output of
Industry1 and Industry 3 respectively.

Finally, Industry 3 produces 45 units of output by using 15

units of its own output, 30 and 35 units of output of
Industry1 and Industry 2 respectively.

Let us put this in matrix framework as Leontief did.

 Leontief’s I-O Matrix
Consumers (Input)
Producers or
Output/Sales
à Industry 1 Industry 2 Industry 3
Industry 1 10 20 30
Industry 2 5 20 35
Industry 3 20 10 15
 What about Total Output and Input?
Producers or Consumers (Input)
Output Industry 1 Industry 2 Industry 3 Total Output
Industry 1 10 20 30 60
Industry 2 5 20 35 60
Industry 3 20 10 15 45
Total Input
Requirements 35 50 80 165
•The basic assumption: Whatever is produced is consumed
•No need for individual balance but aggregate balance.
•This is a closed model but where do labour, capital and land fit in?
Anyway
•Let’s have a glance over a generalized framework
 Input-Output in a
Generalized Framework

Consumers (Input)
Producers Total
or Output Industry 1 Industry 2 Industry 3 Industry 4 Industry 5 …… Industry n Output

Industry 1 X11 X12 X13 X14 X15 …… X1n X1

Industry 2 X21 X22 X23 X24 X25 …… X2n X2

Industry 3 X31 X32 X33 X34 X35 …… X2n X3

Industry 4 X41 X42 X43 X44 X45 …… X3n X4

Industry 5 X51 X52 X53 X54 X55 …… X4n X5

…… …… …… …… …… …… …… ….. …..

Industry n Xn1 Xn2 Xn3 Xn4 Xn5 Xnn Xn

 Analysis of closed I-O Model
• What Does Each Entry Indicate?
• Starting by Xs…
• Reading across Row
• Reading across Column

How Can we Make use of it?

(for that we need to find out unit level input

requirements and by changing those
requirements we can change the output)
Suppose the inter industry flow of the products
of two industries is given as under:

Consumer’s of Input

Producer of
Output/Sales IND1 IND2 Gross Output

IND1 30 40 120

IND2 20 10 60
Suppose the inter industry flow of the products
of two industries is given as under:

Consumer’s of Input

Producer of
Output/Sales IND1 IND2 Gross Output

IND1 30/120 40/60 120

IND2 20/120 10/60 60

 I-O Coefficient Matrix Or the
Technology Matrix
• aij = xij/xj
• Matrix A: The Technology Matrix
(with the help of our above example)
• Matrix X
• Balancing Equations
• How can we interpret it? (the role of
coefficients)
 The Problems of Closed I-O Model

• What about final demand? Household, government,

exports, etc.
• The Role of Keynesian Economics (we can influence total
output not just through technology matrix but also through
Final demand)
• So we should study input output model by taking the final
demand or consumption into account

Inter industry consumption + final demand = Total Output

 Open I-O Model
Consumers (Input)
Producers Industr Industry Industry Industry Industry Industry Final Total
or Output y1 2 3 4 5 …… n Demand Output

Industry 1 X11 X12 X13 X14 X15 …… X1n C1 X1

Industry 2 X21 X22 X23 X24 X25 …… X2n C2 X2

Industry 3 X31 X32 X33 X34 X35 …… X2n C3 X3

Industry 4 X41 X42 X43 X44 X45 …… X3n C4 X4

Industry 5 X51 X52 X53 X54 X55 …… X4n C5 X5

…… …… …… …… …… …… …… ….. ….. …..

Industry n Xn1 Xn2 Xn3 Xn4 Xn5 Xnn Cn Xn

 Analysis of Open I-O Model
• What Does Each Entry Indicate?
• Starting by Xs…
• Reading across Row
• Reading across Column
How Can we Make use of it?

We can estimate the output when the final

demand changes (Keynesian Economics) by
assuming that the basic structure remains the
same i.e. the industry requirements remain
unchanged.
• How?
 I-O Coefficient Matrix Or the
Technology Matrix
• aij = xij/xj
• Matrix A
• Matrix X
• Matrix D
• Inter industry consumption + final
demand = Total Output
• Balancing Equations
 The Leontif Matrix
qHawkins – Simons Conditions for the Viability
of the System
• The Determinant of Leontief Matrix (I-A)
must be positive
• The Diagonal elements of the of the Leontief
matrix must be positive or the aijs must be
less than 1 for all i=j
qEconomic Implications of Hawkins – Simon
Condition ( )
There exist a solution & No negative output
Thanks to Prof. Leontief

Wassily Leontief
1906-1999
Received Nobel Prize in 1973
 Questions
Q1. For the following Transaction Matrix find
the gross output of each industry for the final
demands of 18 and 44 units respectively
Input to Final
Industry I II Demand
I 16 20 4
II 8 40 32
 Q2. Suppose the inter industry flow of the
products of two industries is given as under:
Consumption
Domestic Gross
Production X Y Demand Output

X 30 40 50 120

Y 20 10 30 60
Determine the Technology matrix and test Hawkins-Simons
conditions for the viability of the system. Compute the equilibrium
level of output of the products when the domestic demand vector
is [80 40] (a column vector).
(in I-O, there are basically three types of matrices; (a) inter
industry transaction matrix, Technology matrix and Leontief
matrix.
Q3. Find the Demand vector which is consistent
with the input output matrix

0.2 0.3 0.2

0.4 0.1 0.2
0.1 0.3 0.2
And output vector X =

25
21
18

Also test the Hawkins Simons conditions for the

viability of the system
 Key Take Away
• Finding Technology Matrix
• Determination of Gross Output
• Finding Final Demand
• Technological Viability
Other Important Issues in I-O Analysis
• Equilibrium Prices
• Equilibrium prices in the economy
• Total cost should be equal to total revenue;
there is no scope for profit here
• Apart from inter industry cost what else
could be the cost?
• The cost of Primary Input say labour, Land
Capital etc..
• How will our Input table look like if we
introduce these primary inputs just take
labour for simplicity
 Open I-O Model with Primary Input Labour
Produce Consumers (Input)

rs or Industry Industry Industry Industry Industry Industry Final Total

Output 1 2 3 4 5 …… n Demand Output
Industry
1 X11 X12 X13 X14 X15 …… X1n C1 X1
Industry
2 X21 X22 X23 X24 X25 …… X2n C2 X2
Industry
3 X31 X32 X33 X34 X35 …… X2n C3 X3
Industry
4 X41 X42 X43 X44 X45 …… X3n C4 X4
Industry
5 X51 X52 X53 X54 X55 …… X4n C5 X5
…… …… …… …… …… …… …… ….. ….. …..
Industry
n Xn1 Xn2 Xn3 Xn4 Xn5 Xnn Cn Xn
Total
Primary
Primary Input
Input Required
Labour L1 L2 L3 L4 L5 …… Ln L
 Question
Q1. A two industry input output relationships
are given below:
Consumption Final Gross
Production I II Demand Output
I 16 20 4 40
II 8 40 32 80
Labour Day 80 120
Using Matrix notations, determine:
Gross output required to satisfy the new final demands of 18
and 44 units
Total labour days required
Equilibrium prices if wage rate is 40 per labour day
 Value Added in I-O Model
• Value Added =
Value of Output – Cost of Material Purchased or Inter-industry
Purchases or in other words whatever you pay to primary inputs
i.e. total primary input requirement x cost per unit of primary
input.
! " #! %!! %$! #!
VA= -
" ! #$ %!$ %$$ #$
=[I-A’]P
Or =[I-A]’P or Money value of Final output or whatever you pay to
primary inputs.
P =[[I-A]-|]’[L]’W (assuming Labour as the only primary input and W
is the wage)
• Nevertheless, By applying the same logic you can find value added
by each primary factor as well.
Question
 Important Issues in I-O Analysis
• Given the inter industry table/ transaction matrix--> You have
to find the technology matrix à or the other way around i.e.
given the technology matrix à find the inter industry table/
transaction matrix

• H-S conditions and its interpretations

• given the technology matrix & Final demand à Find gross output

• Change in Final demandà New Gross output and new transaction

matrix.

• Total labor requirements or labour days required or total input

requirements

• Total value added

• Equilibrium prices in the economy