INPUT – OUTPUT ANALYSIS

Introduction

Input-output analysis tries to establish equilibrium conditions under which industries in

an economy have just enough output to satisfy each others demands in addition to final

(outside) demands. Given the internal demand within the industries for each others

output, the problem is to determine output levels will meet various levels of final

(outside) demand

Two industry model

We start with an hypothetical economy with only 2 industries, electric company E and

water company W. Output for both companies is measured in dollars. The electric

company uses both electricity and water (inputs) in the production of electricity (output);

and the water company uses both electricity and water (input) in the production of water

(output).

Suppose the production of each dollar’s worth of electricity required $0.30 worth of

electricity and $0.10 worth of water, and the production o each dollar’s worth of water

requires $0.20 worth of electricity and $0.40 worth of water.

If the final demand from the outside sector of the economy (the demand from all other

users of electricity and water) is

Million for electricity

Million for water

How much electricity and water should be produced to meet this final demand.

Suppose that the electric company produces $12 million worth of electricity and the

water company produces $8 million worth of water (the final demand).

Then the production processes of the companies would require.

Million of electricity

and

Million of water

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leaving only $6.8 million of electricity and $ 3.6 million of water to satisfy the final

demand of the outside sector.

Thus to meet the final demands of both companies and to end up with enough electricity

and water for the final outside demand, both companies must produce more than just the

amount demanded by the outside sector. In fact, they must produce exactly enough to

meet their own internal demand plus that demanded by the outside sector.

Basic input-output problem

Given the internal demand of each industry’s output, we wish to determine output levels

for the various industries that will meet a given final (outside) level of demand as well as

the internal demand

If Total output from Electricity Company

Total output from Water Company

Then the internal demands are: -

- Internal demand for electricity

- Internal demand for water

Combining the internal demand with final demand produces the following systems of

equations

Total Internal Final

Output demand demand

……………………………….. (1)

In matrix form,

………………………………………………………………. (2)

Where Final demand matrix

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Output matrix

- Technology matrix

The technology matrix is the heart of input-output analysis. The elements in the

technology matrix are determined as follows

Remark

Labeling the rows and columns of the technology matrix with the first letter of each

industry is an important part of the process. The same order must be used for columns as

for rows, and that same order must be used for the entries of D (the final demand matrix)

and the entries of X (the output matrix).

Now our problem is to solve (2) for X

X MX D

…………………….. (3), Assuming has an inverse.

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Where

We now find ,

and

=

Then we have

…………………………………… (4)

Therefore, the electric company must have an output of $22 million and the water

company must have an output of $17million so that each company can meet both internal

and final demands.

Check

We use equation (2) to check our work.

Suppose in the original problem that the projected final demand 5 years from now are

and . Determine each companies output for this project

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Summary

Given two industries and with

Technology output Final demand

Matrix matrix matrix

, , …………………. (1)

Where is the input required from to produce a dollars worth of output , the

solution to the input – output matrix equation

Total output internal demand final demand

X = + D ………….. (2)

is

…………………………………………………….. (3)

Assuming that has an inverse

Three industry model

Equations (2) and (3) in the solution to a two- industry input-output problem are the same

for a three industry economy, a four industry economy or an economy with n industries.

The steps from equation 2 to equation 3 hold for arbitrary matrices as long as the matrices

have the correct sizes and exists.

Example 1

An economy is based on three sectors agriculture (A), Energy (E) and manufacturing (M).

Production of a dollars worth of Agriculture requires an input of $0.20 from agriculture

sector and $0.40 from energy sector. Production of a dollars worth of energy requires an

input of $0.20 from energy sector and $0.40 from manufacturing sector. Production of a

dollars worth of manufacturing requires an input of $0.10 from agriculture sector, $0.10

from energy sector and $0.30 from manufacturing sector.

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Find the output from each sector that is needed to satisfy a final demand of $20 billion for

agriculture $10 billion fro energy and $30 billion for manufacturing.

Solution

Since this is a three industry problem, the technology matrix will be a matrix, and

the output and final demand matrices will be column matrices. We form the

matrices as follows

Technology matrix Final demand matrix output Matrix

,

and

Thus, the output matrix X is given by

An output of $33b for agriculture, $37b for energy and $64b for manufacturing will meet

the given final demands.

Problem 1

An economy is based on 3 sectors coal, oil and transportation. Production of a dollars

worth of coal requires an input of $0.20 from coal sector and $0.40 from the

transportation sector. Production of a dollars worth of oil requires an input of $0.10 from

oil sector and $0.20 from the transportation sector. Production of a dollar’s worth of

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transportation requires an input of $0.40 from the coal sector, $0.20 from oil sector and

$0.20 from the transportation sector.

a) Find the technology matrix M

b) Find

c) Find the output from each sector that is needed to satisfy a final demand of $30

billion for coal, $10 billion for oil, and 20 billion for transportation.

Problem 2

The

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