Math 70 - Manyo
4.7 - page 1 of 5
4.7 – Leontief Input – Output Analysis
Read pages 245- 253
Homework: page251 1, 2, 3, 4, 7, 15, 17, 19, 29, 31
Q1: Suppose that a hypothetical economy depends on only two industries, the electric company E and the water
company W. Output for each company is measured in dollars.
The electric company uses (input) both electricity and water in the production (output) of electricity.
The water company uses (input) both electricity and water in the production (output) of water.
Suppose that:
The production of each dollar's worth of electricity requires
$0.30 worth of electricity and $0.10 worth of water.
The production of each dollar's worth of water requires
$0.20 worth of electricity and $0.40 worth of water.
If the final demand from all other users of electricity and water is
d 1 = $12 million for electricity
d 2 = $8 million for water
Internal demands
of the industries
Final demands from
the outside sectors
of the economy.
how much electricity and water should be produced ?
Basic Input-Output Problem:
Given the internal demands within the industries for each other's output, determine the total output levels for
each industry that will meet the given final (outside) level of demand as well as the internal demand.
A. How much electricity and water will the companies use to produce $12 million of electricity and
$8 million of water?
Math 70 - Manyo
4.7 - page 2 of 5
Let x1 = the total output of the electric company
x 2 = the total output of the water company
Q2: A. What are the total output equations of this model ?
B. The matrix of the coefficients above is called the technology matrix M. Find M.
E
E
output
input from E
to produce $1
of electricity
input from E
to produce $1
of water
input from W
to produce $1
of electricity
input from W
to produce $1
of water
input
W
W
C. The model in part A has equation 𝑋 = 𝑀𝑋 + 𝐷, where 𝐷 is the demand matrix and 𝑋 is the ouput matrix.
Rewrite this equation inserting all matrices.
D. To produce $1 worth of electricy we need _______of electricity and ________ of water.
To produce $1 worth of water we need _______of electricity and ________ of water.
E. Solve the matrix equation for 𝑋:
𝑋 = 𝑀𝑋 + 𝐷
Math 70 - Manyo
4.7 - page 3 of 5
F. Find the solution of the input-output problem.
Solution of a Two – Industry, Input-Output Problem
Given two industries C1 and C 2 with
Technology
Matrix
C1
Total Output
Matrix
Final Demand
Matrix
C2
x 
d 
X   1
D   1
 x2 
d 2 
where ai j is the input required from C i to produce a dollar's worth of output for C j .
M 
C1  a11 a12 
C2 a 21 a 22 
The solution to the input-output matrix equation
Total output = Internal Demand + Final Demand
X = MX +
D
is
X  ( I  M ) 1 D assuming I  M  has an inverse.
Math 70 - Manyo
4.7 - page 4 of 5
Q3: A large energy company produces electricity, natural gas, and oil.
The production of a dollar's worth of electricity requires inputs of $0.30 from electricity, $0.10 from natural
gas and $0.20 from oil. The production of a dollar's worth of natural gas requires inputs of $0.30 from
electricity, $0.10 from natural gas and $0.20 from oil. Production of a dollar's worth of oil requires inputs of
$0.10 from each sector. Find the output for each sector that is needed to satisfy a final demand of $25
billion for electricity, $15 billion for natural gas and $20 billion for oil.
Math 70 - Manyo
4.7 - page 5 of 5
Q3: A large energy company produces electricity, natural gas, and oil.
The production of a dollar's worth of electricity requires inputs of $0.30 from electricity, $0.10 from
natural gas and $0.20 from oil. The production of a dollar's worth of natural gas requires inputs of $0.30 from
electricity, $0.10 from natural gas and $0.20 from oil. Production of a dollar's worth of oil requires inputs of
$0.10 from each sector. Find the output for each sector that is needed to satisfy a final demand of $25 billion
for electricity, $15 billion for natural gas and $20 billion for oil.
Let x1 = the total output of the electric company (𝐸)
x 2 = the total output of the natural gas company (𝐺)
𝑥3 = the total output of the oil company (𝑂)
The total amount of electricity needed is the sum of amounts of electricity needed to produce electricity,
natural gas and oil (internal demand) plus the final (external) demand of electricity.
𝑥1 =. 30𝑥1 + .30𝑥2 + .10𝑥3 + 25
The total amount of natural gas needed is the sum of amounts of natural gas needed to produce electricity,
natural gas and oil (internal demand) plus the final (external) demand of natural gas.
𝑥2 =. 10𝑥1 + .10𝑥2 + .10𝑥3 + 15
The total amount of oil needed is the sum of amounts of oil needed to produce electricity, natural gas and oil
(internal demand) plus the final (external) demand of oil.
𝑥3 =. 20𝑥1 + .20𝑥2 + .10𝑥3 + 20
Using the technology matrix 𝑀, the final demand matrix 𝐷 and total demand matrix 𝑋, we get
𝐸
𝐺 𝑂
𝑥1
𝐸 . 3 . 3 . 1 𝑥1
25
𝑥2 = 𝐺 . 1 . 1 . 1 𝑥2 + 15 or 𝑋 = 𝑀𝑋 + 𝐷, with solution 𝑋 = 𝐼 − 𝑀
𝑥3
𝑂 . 2 . 2 . 1 𝑥3
20
1 0
𝐼−𝑀 = 0 1
0 0
0
.3
0 − .1
1
.2
.3 .1
.7 −.3 −.1
.9 −.1
. 1 . 1 = −.1
.2 .1
−.2 −.2
.9
Using a calculator, find
𝐼−𝑀
−1
𝐼−𝑀
−1
1.58 0.58 0.24
= 0.22 1.22 1.16
.4
.4
1.2
1.58 0.58 0.24
𝐷 = 0.22 1.22 1.16
.4
.4
1.2
25
53
15 = 27
20
40
That is, the total output of electricity is $53 billion dollars
the total output of natural gas is $27 billion dollars
the total output of oil is $40 billion dollars
−1
𝐷