As a means of introduction to Leontief's work, lets consider
an economy based on two sectors.
Two Industry Model
Consider an economy that is based on two industries, electricity and steel
production. The Electricity industry uses
30% of the electricity produced and
20% of the steel produced in its production of electricity. The
steel industry
uses 10% of the electricity produced and 40% of the steel produced in
its production of steel. This means that for
every dollar of
electricity produced, $0.30 is used by the electric company and $0.20 is used by
the steel industry.
Also for every dollar of steel produced, $0.10 is used
on electricity and $0.40 is used on steel.
Let matrix M represent the model for the two
industries.
Using
the above information, 
Note:
The data for an industry is entered in columns, not rows.
Let x = total output of electricity and y = total
output of steel,
The internal demand for electricity is 0.30x + 0.10y
The internal demand for steel is 0.20x + 0.40y
Let a = demand for electricity and b = demand for steel
x = internal demand for electricity + demand for electricity = 0.30x + 0.10y + a
y = internal demand for steel + demand for steel = 0.20x + 0.40y + b
Using matrix equations, the economy can be written as X = MX +
D
where: 
Solve
the matrix equation X = MX + D for matrix X:
1. Subtract matrix MX from both sides of the
equation.
X - MX = D
2. Multiply matrix X by the identity Matrix I. Since
IX = X, this step is a
substitution.
IX - MX = D
3. Factor out matrix
X
(I - M)X = D
4. Multiply both sides of the equation by 
Since 
,
then
Example 1:
An economy is based on two industries, electricity and steel
production. The Electricity industry uses
30% of the electricity produced and
20% of the steel produced in its production of electricity. The
steel
industry uses 10% of the electricity produced and 40% of the steel produced in
its production
of steel. The demand for electricity is $32 million and the
demand for steel is $48 million. How much
electricity and steel must be
produced to meet this demand?
Solution:
Let x = total output of electricity and y = total
output of steel,
The internal demand for electricity is 0.30x + 0.10y
The internal demand for steel is 0.20x + 0.40y
Let a = demand for electricity and b = demand for steel
x = internal demand for electricity + demand for electricity = 0.30x + 0.10y + a
y = internal demand for steel + demand for steel = 0.20x + 0.40y + b
Using matrix equations, the economy can be written as X = MX +
D
where: 
Solving
for X,
Using your calculator,
1. store matrix M in matrix A
2. store matrix D in matrix D
3 store the Identity matrix, I, in B
4. subtract Matrix A from Matrix B and store in matrix C
5. find the inverse matrix for matrix C and store in matrix E
6. multiply matrices E and D = X 
Thus, x = total demand for electricity is $60 million
and y = total demand for steel is $100 million.
Checking the solution.
electricity: x = 0.30x + 0.10y + a = 0.30(60) + 0.10(100) +
32 = 60
steel : y = 0.20x + 0.40y + b = 0.20(60) + 0.40(100) + 48 = 100
Larger Economy's
Leontif's Input-Output model applies to larger economy's.
Example 2:
Suppose a small country produces three types of energy; electricity, natural
gas, and coal. Production of
electricity requires 40% of electricity
produced, 20% of natural gas produced and 10% of coal produced.
Production
of natural gas requires 10% of electricity produced, 10% of natural gas
produced and 5% of
coal produced. Production of coal requires 40% of
natural gas produced 20% of coal produced. The
demand for electricity is
$800 million, natural gas is $250 million and coal is $120 million. How much
electricity,
natural gas and coal must be produced to meet this demand?
Solution:
Let x = total output of electricity, y = total
output of natural gas, and z = total output for coal
The internal demand for electricity is 0.40x + 0.10y + 0.00z
The internal demand for natural gas is 0.20x + 0.10y + 0.40z
The internal demand for coal is 0.10 x + 0.05y + 0.20z
Let a = demand for electricity, b = demand for natural gas,
and c = demand for coal
x = internal demand for electricity + demand for electricity = 0.40x + 0.10y +
0.00z + a
y = internal demand for natural gas + demand for natural gas = 0.20x + 0.10y +
0.40z + b
z = internal demand for coal + demand for coal = 0.10x + 0.05y + 0.20z + c
Using matrix equations, the economy can be written as X = MX +
D
where: 
Solving
for X,
Using your calculator,
1. store matrix M in matrix A
2. store matrix D in matrix D
3. store the Identity matrix, I, in B
4. subtract Matrix A from Matrix B and store in matrix C
5. find the inverse matrix for matrix C and store in matrix E
6. multiply matrices E and D = X 
Thus, x = total demand for electricity is $1,462
million, y = total demand for natural gas is $772 million
and z = total demand for coal is $381 million.
Checking the solution.
electricity: 0.40x + 0.10y + 0.00z + a = 0.40(1462) +
0.10(772) + 0(381) + 800 = $1,462 million
natural gas: 0.20x + 0.10y + 0.40z + b = 0.20(1462) + 0.10(772) +
0.40(381) + 250 = $772 million
coal: 0.10x + 0.05y + 0.20z + c = 0.10(1462) + 0.05(772) + 0.20(381)
+ 120 = $381 million
The
linear system is consistent and independent (only one solution).
 |
Test #1 will
cover material from Lessons 1 through 6. |
 |
Project
Click on the light bulb.
It will take you to the project, "Leontif: Input-Output
Model".
|
ASSIGNMENT:
Try the following problems from Section 2-7: 1 through 21 odd
Lesson 5 
Return to Preface
