Applied Linear Algebra
Input-Output Analysis
Overview
Students will learn about
Examples of Input-Output Models
Structure of an Input-Output Model
Technological Matrix
The Leontief Inverse
The Open Leontief Model
Economic Examples
How to Solve The Input-Output Problems
Introductory Example
Let’s start with a simple example of an organic farm which produces
two goods, corn and fertilizer:
A ton of Corn is produced using .1 ton of corn (to plant) and .8 ton of
fertilizer;
A ton of Fertilizer is produced using .5 ton of corn and 0 tones of
fertilizer.
We can describe each of the two production processes by pairs of
numbers (a, b), where a represents the corn input and b represents
the fertilizer input. The corn production process is described by the
pair of numbers (0.1, 0.8).
The fertilizer production process is described by the pair of numbers
(0.5, 0).
Introductory Example
The most important question to ask of this model is: What can
produced for consumption?
Corn is used both in the production of corn and in the
production of fertilizer.
Fertilizer is used in the production of corn.
Is there any way of running both processes so as to leave
some corn and some fertilizer for individual consumption?
If so, what combinations of corn and fertilizer for consumption
are feasible?
Introductory Example
Answers to these questions can be found by examining a particular
system of linear equations.
Suppose the two production processes are run so as to produce xC
tons of corn and xF tons of fertilizer.
The amount of corn actually used in the production of corn is 0.1xC –
the amount of corn needed per ton of corn output times the number of
tons to be produced.
Similarly, the amount of corn used in the production of fertilizer is
0.5xF.
Introductory Example
The amount of corn left over for consumption will be the total amount
produced minus the amount used for production of corn and fertilizer:
xC 0.1xC 0.5 x F , 0.9 xC 0.5 x F
The amount of fertilizer needed in production is 0.8xC tons. Thus the
amount left over for consumption is
x F 0.8 xC
Introductory Example
Suppose we want our farm to produce for consumption 4 tons
of corn and 2 tons of fertilizer.
How much total production of corn and fertilizer will be
required? Put another way, how much corn and fertilizer will the
farm have to produce in order to have 4 tons of corn and 2 tons
of fertilizer left over for consumers?
We can answer this question by solving the pair of linear
equations
0.9 xC 0.5 x F 4,
0.8 xC x F 2.
Introductory Example
This system is easily solved. Solve the second equation for xF
in terms of xC:
x F 0.8 xC 2.
Substitute this expression for xF into the first equation:
0.9 xC 0.5(0.8 xC 2) 4
and solve for xC: 0.5xC = 5, so xC = 10.
Finally, substitute xC = 10 back into equation for xF to compute
xF = 0.8 · 10 + 2 = 10.
Input-Output Analysis - Introduction
An input-output model of an economy is designed to
answer the questions
“What should be the level of output for each
industry in an economy in order to meet a vector
of final demands for products and services?”
Input-Output Analysis - Introduction
Production to meet the final demand for an industry’s goods
requires three types of inputs:
Intermediate products produced by the same industry.
Intermediate products produced by other industries.
A set of primary (non-produced) inputs used in the production
of both intermediate and final products (value added).
Input-Output Analysis - Introduction
Likewise the output of an industry is used in three ways:
Some is used as inputs in other producing activities in the
same industry.
Some output is used as inputs in other industries.
The remainder is used to satisfy final demands.
Input-Output Analysis - Introduction
These relationships are the basis of the input-output model.
However, the model has two simplifying basic assumptions.
Each industry produces only one homogeneous commodity.
The combination of inputs (both primary and produced)
required for a unit of production from an industry is fixed. Thus
marginal cost equals average costs and we have constant
returns to scale.
Input-Output Analysis - Introduction
Suppose a national economy is divided into n sectors and x R
n
be a Production Vector (this n sectors are Production Part of
Economy).
Suppose another part of economy only consumes product and
let d be a Final Demand (this is the demand of nonproductive
part of economy).
Of course producers themselves create additional Intermediate
Demand for goods they need for their own production.
Input-Output Analysis - Introduction
So, the Leontief Model is
amount final
int ermediate
produced demand
x demand d
Input-Output Analysis - Introduction
Example: Suppose the economy consists of three sectors with
unit consumption vectors c1 , c2 , c3 :
Unit consumption
From Manufact. Agricult. Services.
Manufact. 0.5 0.4 0.2
Agricult. 0.2 0.3 0.1
Services. 0.1 0.1 0.3
c1 c2 c3
Input-Output Analysis - Introduction
If sectors decide to produce of x1 , x2 , x3output
x1c1
correspondently, then is intermediate demand for
manufacturing (similarly for other sectors).
So total Intermediate Demand is
Dem x1c1 x 2 c 2 x3 c3 Cx
Production equation is
x Cx d x Cx d , I C x d
Let d 50 30 20 , then we have equation
Input-Output Analysis - Introduction
Solution
0.5 0.4 0.2 50
0.2 0.7 0.1 x 30
0.1 0.1 0.7 20
5 4 2 500 1 0 0 226.1 226.1
2 7 1 300 0 1 0 118.5 118.5
1 1 7 200 0 0 1 77.8 77.8
Structure of an Input-Output Model
Since an input-output model normally encompasses a large
number of industries, its framework is of necessity rather
involved.
To simplify the problem, the following assumptions are as a
rule adopted:
(1) each industry produces only one homogeneous commodity
(2) each industry uses a fixed input ratio (or factor combination)
for the production of its output; and
(3) production in every industry is subject to constant returns to
scale, so that a k-fold change in every input will result in an
exactly k-fold change in the output.
Structure of an Input-Output Model
From these assumptions we see that, in order to produce each
unit of the j-th commodity, the input need for the i-th commodity
must be a fixed amount, which we shall denote by a ij .
Specifically, the production of each unit of the j-th commodity
will require a1 j (amount) of the first commodity, a 2 j of the
second commodity,..., and a nj of the n-th commodity.
Structure of an Input-Output Model
For our purposes, we may assume prices to be given and,
thus, adopt "a dollar's worth" of each commodity as its unit.
Then the statement a 32 0.35 will mean that 35 cents' worth of
the third commodity is required as an input for producing a
dollar's worth of the second commodity.
The symbol a ij will be referred to as an input coefficient.
Structure of an Input-Output Model
For an n-industry economy, the input coefficients can be
arranged into a matrix A a ij
Each column specifies the input requirements for the
production of one unit of the output of a particular industry.
The second column, for example, states that to produce a unit
(a dollar's worth) of commodity II, the inputs needed are: a12
units of commodity I, a 22 units of commodity II, etc.
If no industry uses its own product as an input, then the
elements in the principal diagonal of matrix A will all be zero.
Structure of an Input-Output Model
Input-coefficient matrix
a11 a12 a13 a1n
a a 22 a 23 a 2 n
21
a 31 a 32 a 33 an3
a n1 an2 an3 a nn
The Open Model
If, besides the industries, the model also contains an "open"
sector (say, Households) which exogenously determines a final
demand for the product of each industry and which supplies a
primary input (say, labor service) not produced by the
industries themselves, the model is an open model.
In view of the presence of the open sector, the sum of the
elements in each column of the input-coefficient matrix (or
input matrix , for short) must be less than 1.
Symbolically, this fact may be stated thus
n
a
i 1
ij 1 ( j 1,2, , n)
The Open Model
If industry I is to produce an output just sufficient to meet the
input requirements of the industries as well as the final
demand of the open sector, its output level must satisfy the
following equation:
x1 a11 x1 a12 x 2 a1n x n d 1
or
(1 a11 ) x1 a12 x 2 a1n x n d 1
where d 1 denotes the final demand for its output and a1 j x j
represents the input demand from the j-ih industry
The Open Model
For the entire see of n industries, the "correct" output levels
can therefore be summarized by the following system of linear
equations:
(1 a11 ) x1 a12 x 2 a1n x n d 1
a 21 x1 (1 a 22 ) x 2 a 2 n x n d 2
..........................................................
a n1 x1 a n 2 x 2 (1 a nn ) x n d n
The Open Model
In matrix notation, this may be written as
(1 a11 ) a12 a1n x1 d1
a (1 a ) a x d
21 22 2n 2 2
an1 an 2 (1 ann ) xn d n
The Open Model
This system can also be written as ( I A) x d , where x and d
are, respectively, the variable vector and the final-
demand (constant-term) vector.
The matrix I-A is called the technology matrix, and we may
denote it by T. Thus the system can also be written as Tx=d.
As long as T is nonsingular — and there is no a priori reason
why it should not be — we shall be able to find its inverse , and
obtain the unique solution of the system from the equation
x T 1 d ( I A) 1 d
Example
For purposes of illustration, suppose that there are only three
industries in the economy and that the input-coefficient matrix
is as follows:
a11 a12 a13 0.2 0.3 0.2
a a 22 a 23 0.4 0.1 0.2
21
a31 a 32 a 33 0.1 0.3 0.2
Note that in each column sum is less than 1, as it should be.
Further, if we denote by a oj the dollar amount of the primary
input used in producing a dollar's worth of the j-th commodity,
we can write: a 01 0.3, a 02 0.3 and a 03 0.4 .
Example
With the matrix A above, the open input-output system can be
expressed as follows:
0.8 0.3 0.2 x1 d1
0.4 0.9 0.2 x d
2 2
0.1 0.3 0.8 x3 d3
By keeping the vector d in parametric form, our solution will
appear as a " formula" into which we can feed various specific
vectors to obtain various corresponding specific solutions.
Example
By inverting the technology matrix, the solution can be found,
approximately (because of rounding of decimal figures), to be:
x1 0.66 0.30 0.24 d1
x T 1d 1 0.34 0.62 0.24 d
2 0.384 2
x3 0.21 0.27 0.60 d3
Example
If the specific final-demand vector (say, the final-output target
of a development program) happens to be
10
d 5 in billions of $,
6
then the following specific solution values will emerge
1
x1 0.6610 0.305 0.246 9.54 24.84
0.384 0.384
7.94 7.05
x2 20.68 x3 18.36
0.384 0.384
Example
An important question now arises.
Would the amount required be consistent with what is available in
the economy? The required primary input may be calculated as:
a x 0.3(24.84) 0.3(20.68) 0.4(18.36) $21.00
j 1
0j j
Therefore, the specific final demand will be feasible if and only if
the available amount of the primary input is at least $21 billion.
If the amount available falls short, then that particular production
target will, of course, have to be revised downward accordingly.
How to Solve The Input-Output Problems
Question 1:
How much output is available for final demand given the total
output level?
Answer.
In this case the vector x is assumed to be known and we
should calculate the unknown vector d. The matrix equation
x=Ax + d immediately gives d=x – Ax and the right-hand side
is easily evaluated to get d.
How to Solve The Input-Output Problems
Example. The output levels of machinery, electricity and oil of
a small country (Georgia) are 3000, 5000 and 2000
respectively.
Each unit of machinery requires inputs of 0.3 units of electricity
and 0.3 units of oil
Each unit of electricity requires inputs of 0.1 units of machinery
and 0.2 units of oil
Each unit of oil requires inputs of 0.1 units of electricity and 0.2
units of machinery
Determine the machinery, electricity and oil available for export.
How to Solve The Input-Output Problems
Solution. Let us denote the total output for machinery,
electricity and oil by x1 , x2 , x3 respectively, so that
x1 3000, x2 5000, x3 2000
The matrix of technical coefficients is
0 0.1 0.2
A 0.3 0 0.1
0.3 0.2 0
How to Solve The Input-Output Problems
From the equation d=x – Ax we see that final demand vector is
d1 3000 0 0.1 0.2 3000
d 5000 0.3 0 0.1 5000
2
d3 2000 0.3 0.2 0 2000
3000 900 2100
5000 1100 3900
2000 1900 100
Therefore, the country has 2100, 3900 and 100 units of
machinery, electricity and oil, respectively, available for export.
How to Solve The Input-Output Problems
Question 2:
How much total demand is required to satisfy a given level of
final demand?
Answer.
In this case the vector d is assumed to be known and we need
to calculate the unknown vector x. The matrix equation x=Ax +
d rearranges to give x – Ax=d or (I – A)x=d. This represents a
1
system of linear equations and x ( I A) d .
How to Solve The Input-Output Problems
Example. Given the matrix of technical coefficients
0.3 0.1 0.1
A 0.2 0.2 0.2
0.4 0.2 0.3
for three industries, I1, I2 and I3.
Determine the total outputs to satisfy final demands of 49, 106
and 17 respectively.
How to Solve The Input-Output Problems
Solution. To solve this problem we need to find the inverse of
I – A and then multiply by the final demand vector.
The matrix I – A is
1 0 0 0.3 0.1 0.1 0.7 0.1 0.1
0 1 0 0.2 0.2 0.2 0.2 0.8 0.2
0 0 1 0.4 0.2 0.3 0.4 0.2 0.7
How to Solve The Input-Output Problems
We can find the inverse by calculating its cofactors. So the
inverse is
0.52 0.09 0.10
1 1
( I A) 0.22 0.45 0.16
0.306
0.36 0.18 0.54
So
x1 0.52 0.09 0.10 49 120
x 1 0.22 0.45 0.16 106 200
2 0.306
x3 0.36 0.18 0.54 17 150
How to Solve The Input-Output Problems
Question 3:
What changes need to be made to total output when final
demand changes?
Answer.
In this case we assume that the current total output vector x is
chosen to satisfy some existing final demand vector d, so that
1
x ( I A) d
How to Solve The Input-Output Problems
Suppose that the final demand vector changes by an amount .d
In order to satisfy the new requirements, the total output vector
is then given by
x x ( I A) 1 (d d ) ( I A)1 d ( I A)1 d
So, because of x ( I A)1 d we have x ( I A) 1 d
How to Solve The Input-Output Problems
Example. Consider the following inter-industrial flow table for two
industries, I1 and I2:
Output
I1 I2 Final Demand
Input I1 200 300 500
bbbbbbbb I2 100 100 300
Assuming that the total output is just sufficient to meet the output
and final demand requirements, write down
the total output vector
the matrix of technical coefficients
And calculate the new total output vector needed when the final
demand for I1 rises by 100 units.
How to Solve The Input-Output Problems
Solution. To calculate the current total outputs for I1 and I2 all
we have to do is to add together the numbers along each row
of the table. So
x1 200 300 500 1000
x2 100 100 300 500
The first column of the matrix of technical coefficients
represents the inputs needed to produce 1 unit of I1.
The first column of the inter-industrial flow table represents the
inputs needed to produce current total output of I1 which
equals 1000. So the first coefficient is 200/1000=0.2.
How to Solve The Input-Output Problems
Likewise for other coefficients we define that the matrix of
technical coefficients is
0.2 0.6
A
0.1 0.2
If the demand for I1 rises by 100 units and the demand for I2
remains constant, the vector giving the change in final demand
is
100
d
0
How to Solve The Input-Output Problems
Because of
1 0 0.2 0.6 0.8 0.6
I A
0 1 0.1 0.2 0.1 0.8
and the inverse is 1 0.8 0.6
( I A) 1
0.58 0.1 0.8
1 0.8 0.6 100 138
we define that x
0.58 0.1 0.8 0 17
to the nearest unit.
Note that there is an increase in total output of I2 despite the
fact that the final demand for I2 remains unchanged.
