Matrix Application

Quiz-3

Following is the input-output table:

	Agriculture(Sale)	Industry(Sale)	Final Demand
Agriculture(Purchase)	3	4	3
Industry(Purchase)	6	10	4

· Find the disposition of sectors’ outputs.

· Find the transaction matrix.

· Find the direct requirement matrix(input-output co-efficient matrix)

· Find the direct and indirect requirement matrix(input output co-efficient matrix)

· What will be the gross production if the demand of the agriculture and industry sector were 8 and 12 respectively?

Solution:

a. Disposition of output of the sectors’ and final output

Total Input Output Table

	Agriculture(Sale)	Industry(Sale)	Final Demand	Total sales
Agriculture(Purchase)	3	4	3	10
Industry(Purchase)	6	10	4	20
Consumer	1	6
Total Sales	10	20

Interpretation:

Out of the production of 10 of Agriculture Sector:

Agriculture Consumes 3 Units.

Industry Consumes 6 Units.

Consumer Consume 1 Unit.

Out of the Production of 20 of Industry:

Agriculture Consumes 4 units.

Industry consumes 10 units.

Consumer consumes 6 units.

B. Inter Sectoral Input and Output Transaction Matrix

Inter sectoral Input and Output Transaction Table

	Agriculture(Sale)	Industry(Sale)
Agriculture(Purchase)	3	4
Industry(Purchase)	6	10

Inter sectoral Input and Output Transaction Matrix

A =

3 4

6 10

Interpretation- Agriculture Sector

Agriculture Sector sale to Agriculture Sector: 3 units

Agriculture Sector sale to Industry Sector: 6 Units

Interpretation – Industry Sector

Industry Sector Sale to Industry Sector: 10 units

Industry Sector sale to Agriculture sector: 4 units.

C. Inter Sectoral Input and Output Co-efficient

Inter Sectoral Input and Output Co-efficient Table

(Dividing column 1 by 10 and column 2 by 20)

	Agriculture(Sale)	Industry(Sale)
Agriculture(Purchase)	0.3	0.2
Industry(Purchase)	0.6	0.5
Consumer	0.1	0.6

Interpretation – Agriculture Sector

Out of 1 unit total sale (Gross sale): Agriculture sale to Agriculture: 0.3, Agriculture sale to Industry: 0.6,

Interpretation – Industry Sector

Out of 1 unit total sale (Gross sale): Industry sale to Industry: 0.5, Industry sale to Agriculture: 0.2,

D. Sectoral direct output (input-output co-efficient matrix)

Table of Input-Output Coefficient

	Agriculture(Sale)	Industry(Sale)
Agriculture(Purchase)	0.3	0.2
Industry(Purchase)	0.6	0.5

Input-Output Coefficient Matrix

A =

0.3 0.2 0.6 0.5

Interpretation:

Out of every 1 unit of gross output of agriculture sector 0.3 are used by the agriculture sector and 0.4 units by industry.

Out of every 1 unit of gross output industry sector 0.2 are used by the industry sector and 0.5 units by agriculture sector.

D. Direct and indirect output matrix (Input-output coefficient matrix)

Formula:

Intersectoral coefficient x Gross Production + Final output = Gross Output

[Matrix of intersectoral coefficients] x [Matrix of Gross Production] + [Matrix of Final Output] = [Matrix of Gross Production]

Where: A: Matrix of intersectoral coefficients X: Matrix of Gross Production D: Matrix of Final Output

AX + D = X

1 0 0 1

If D =

AX + 1 = X

ð AX –X = -1

ð X – XA = 1

ð X(1-A) = 1

ð X = (1- A)^-1

1 0 0 1

0.3 0.2 0.7 0.2

Now, (1 – A ) = =

0.6 0.5 0.6 0.5

Now, Cofactor Matrix of (1 – A )

0.5 0.6

=

0.2 0.7

Hence, Adjoin matrix of ( 1 – A )

0.5 0.2

=

0.6 0.7

Now, Determinants of ( 1 – A )

D = {0.35 – 0.12} = .23

1

D

Hence, Inverse Matrix of (1 – A )^-1 = ------ x [Adjoin Matrix]

1

0.5 0.2

0.23

=----- x

0.6 0.7

2.17 0.87

=

2.61 3.04

If Industry and Agriculture Sector are 8 and 12 Respectively-

2.17 0.87 8

Then the matrix = X

2.61 3.04 12

27.8

=

57.36

Interpretation:

If agriculture and Industry factor are 8 and 12 Respectively

Then Gross Output of Agriculture 27.8 units

Gross Output of Industry 57.36 units