1.

0 Introduction
Input–output analysis is the name given to an analytical framework developed by Professor
Wassily Leontief in the late 1930s, in recognition of which he received the Nobel Prize in
Economic Science in 1973 (Leontief, 1936, 1941). One often speaks of a Leontief model
when referring to input–output. The term interindustry analysis is also used, since the
fundamental purpose of the input–output framework is to analyse the interdependence of
industries in an economy. Today the basic concepts set forth by Leontief are key components
of many types of economic analysis and, indeed, input– output analysis is one of the most
widely applied methods in economics (Baumol, 2000). This study focuses on the framework
set forth by Leontief and explores the many extensions that have been developed over the last
nearly three quarters of a century. In its most basic form, an input–output model consists of a
system of linear equations, each one of which describes the distribution of an industry’s
product throughout the economy. Most of the extensions to the basic input–output framework
are introduced to incorporate additional detail of economic activity, such as over time or
space, to accommodate limitations of available data or to connect input–output models to
other kinds of economic analysis tools.

The model is widely applied throughout the world; the United Nations has promoted input
output as a practical planning tool for developing countries and has sponsored a standardized
system of economic accounts for constructing input–output tables. Input–output has been also
extended to be part of an integrated framework of employment and social accounting metrics
associated with industrial production and other economic activity, as well as to accommodate
more explicitly such topics as international and interregional flows of products and services
or accounting for energy consumption and environmental pollution associated with
interindustry activity. In this study, we present the foundations of the input–output model as
originally developed by Leontief, as well as the evolution of many methodological extensions
to the basic framework. In addition, we illustrate many of the applications of input–output
and its usefulness for practical policy questions.
The model depicts inter-industry relationships within an economy, showing how output from
one industrial sector may become an input to another industrial sector. In the inter-industry
matrix, column entries typically represent inputs to an industrial sector, while row entries
represent outputs from a given sector. This format therefore shows how dependent each
sector is on every other sector, both as a customer of outputs from other sectors and as a
supplier of inputs. Each column of the input–output matrix shows the monetary value of
inputs to each sector and each row represents the value of each sector's outputs.

1.1 Aim
The aim of this study is to examine the Framework of the Input – Output Theory
1.2 Objectives
1. To examine the basic Framework of the Theory
2. To examine the Theory’s analyses through its various Linear equation
3. To examine the Application and Usefulness of the Theory
4. To examine the assumptions and Limitations of the Theory.

1.3 Input - Output Theory- The Basic Framework

The basic Leontief input–output model is generally constructed from observed economic data
for a specific geographic region (nation, state, county, etc.). One is concerned with the
activity of a group of industries that both produce goods (outputs) and consume goods from
other industries (inputs) in the process of producing each industry’s own output. In practice,
the number of industries considered may vary from only a few to hundreds or even
thousands. For instance, an industrial sector title might read “manufactured products,” or that
same sector might be broken down into many different specific products. The fundamental
information used in input–output analysis concerns the flows of products from each industrial
sector, considered as a producer, to each of the sectors, itself and others, considered as
consumers. This basic information from which an input– output model is developed is
contained in an interindustry transactions table. The rows of such a table describe the
distribution of a producer’s output throughout the economy. The columns describe the
composition of inputs required by a particular industry to produce its output.
FIG 1: Picture Showing the Interindustry transaction table

These interindustry exchanges of goods constitute the shaded portion of the table depicted in
Figure 1. The additional columns, labelled Final Demand, record the sales by each sector to
final markets for their production, such as personal consumption purchases and sales to the
federal government. For example, electricity is sold to businesses in other sectors as an input
to production (an interindustry transaction) and also to residential consumers (a final-demand
sale). The additional rows, labelled Value Added, account for the other (non-industrial) inputs
to production, such as labour, depreciation of capital, indirect business taxes, and imports.
 2.0 Analyses of the Theory through Linear Equations and Matrices (Leontief’s
Matrices)
Input-output analysis is one of a set of related methods which show how the parts of a system
are affected by a change in one part of that system. Input-output analysis specifically shows
how industries are linked together through supplying inputs for the output of an economy.
Suppose there are only two industries producing Coal and Steel. Coal is required to produce
steel and some amount of steel in the form of tools is required to produce coal. Suppose the
input requirements per ton output of the two products are:
INDUSTRY COAL STEEL

COAL 0 3

STEEL 0.1 0

FIG 2: Table Showing the Output of two Industries

Suppose it desired that the Coal industry produce a net output of 200,000 tons of coal and the
Steel industry a net output of 50,000 tons. If the Coal industry just produces 200,000 tons and
the Steel industry produces 50,000 tons some of outputs are used up in producing the other
output. To produce 50,000 tons of Steel requires 3(50,000) =150,000 tons of coal. Likewise
the production of 200,000 tons of coal requires (0.1) (200,000) =20,000 tons of steel. The net
outputs of coal and steel would then be 200,000-150,000=50,000 tons of coal and 50,000-
20,000=30,000 tons of steel. In other words, in order to get net outputs of 50,000 tons of coal
and 30,000 tons of steel it is necessary to produce 200,000 tons of coal and 50,000 tons of
steel. But we want net outputs of 200,000 tons of coal and 50,000 tons of steel. We would at
least have to produce an additional amount to replace the coal and steel used up in producing
200,000 tons of coal and 50,000 tons of steel. Those amounts, as we saw above, are 150,000
tons of coal and 20,000 tons of steel. But in producing these amount we will also use up coal
and steel. In fact, we will use up 3(20,000) =60,000 tons of coal and (0.1) (150,000) =15,000
tons of steel. So we must increase production again to cover these amounts.

We can think of these production increases as Rounds of production. In Round 1 we produce

the net outputs we are trying to achieve. In Round 2 we produce the outputs that were used up
in producing Round 1 outputs. Then in Round 3 we produce the outputs used up in producing
Round 2, and so on. The figures are given in the following table.
 Round Coal Steel

1 200,000 50,000

2 150,000 20,000

3 60,000 15,000

4 45,000 6,000

5 18,000 4,500

6 13,500 1,800

7 5,400 1,350

8 4,050 540

9 1,620 405

FIG 3: Table Showing the Various Rounds of Production of two Industries

The totals for the first nine rounds are 497,570 tons of coal and 99,595 tons of steel. This
approximate how much we must produce to achieve the net outputs we seek. The exact
amounts, found by more advanced methods, are 500,000 tons of coal and 100,000 tons of
steel. We see that 3(100,000) =300,000 tons of coal are used up in producing the 100,000
tons of steel leaving 500,000-300,000=200,000 tons of coal as net output. Also (0.1)
(500,000) =50,000 tons of steel are used up in producing the 500,000 tons of coal leaving us
with 100,000-50,000=50,000 tons of steel as net output.

We find the exact figures by solving two algebraic equations using Simultaneous equation
method to solve it. If x1 is the output of coal and x 2 is the output of steel, then the conditions
that have to be satisfied are:

X1 – 3X2 = 200,000
X2 - 0.1X1 = 50,000.

This is a set of two equations in two unknowns and we can solve it using simple algebra
which is shown below
X1 – 3X2 = 200,000
X2 – 0.1X1 =50,000
Rewrite
X1 - 3X2 = 200,000-----------eqn (1)
-0.1X1 + X2 = 50,000---------eqn (2)
Multiply eqn (1) by 0.1 and eqn (2) by 1
0.1X1 – 0.3X2 = 20,000--------------eqn (3)
-0.1X1 + X2 = 50,000----------------eqn (4)
Add eqn (3) with eqn (4)
0 + 0.7X2 = 70,000
Divide both sides by 0.7
0.7X2÷ 0.7= 70,000÷ 0.7
X2 = 100,000

Substitute X2 = 100,000 into eqn (1)

X1 – 3 (100,000) = 200,000
X1 - 300,000 = 200,000
X1= 200,000 + 300,000
X1 = 500,000

Therefore X1 = 500,000 and X2 = 100,000

A systematic way to solve for any set of net outputs is to find how much coal and steel are
needed to produce a net output of one ton of coal. The equations which have to be satisfied
are:

X1 – 3X2 = 1
X2 - 0.1X1 = 0.

The solution is x1=1.42857 and x2=0.14286. To get the amounts necessary to produce a net
output of 200,000 tons we multiply these figures by 200,000. The outputs required are
285,714 tons of coal and 28,571 tons of steel.

For a net output of one ton of steel the equations to be satisfied are:

X1 – 3X2 = 0
X2 - 0.1X1 = 1.

The solution is X1=4.28571 and X2=1.42857. To get the amounts necessary to produce a net
output of 50,000 tons of steel we multiply these figures by 50,000. We get an output of coal
of 214,286 tons and 71,429 tons of steel. The total outputs required for 200,000 tons of coal
and 50,000 tons of steel are then

285,714 + 214,286 = 500,000 tons of coal and

28,571 + 71,429 = 100,000 tons of steel.

The outputs of coal and steel required to achieve a net output of one ton of coal may be
considered the direct and indirect requirements for one ton of coal. Likewise for the outputs
necessary for a net output of one ton of steel. These can be put together in a table to contrast
them with the direct requirements of production:

Direct Requirements of Production

Industry Coal Steel

Coal 0 3

Steel 0.1 0

Direct and Indirect Requirements of Production

Industry Coal Steel

Coal 1.42857 4.28571

Steel 0.14286 1.42857

The table is the one that gives the most information about how the industries are inter-related.

2.1 Direct and Indirect Requirement of Production

The direct and indirect requirements are usually determined using matrix operations. A
matrix is simply a rectangular array of numbers; i.e., a table. A matrix is characterized by the
number of rows and columns. An n by m matrix, written n×m, is a matrix with n rows and m
columns. A matrix with only one column is also called a column vector. If a matrix has only
one row it is usually called a row vector.
The element of matrix A that is in the i-th row and j-th column is denoted as A i,j. The addition
of two matrices A and B with the same number of rows and columns is defined as the matrix
C such that Ci,j=Ai,j+Bi,j. Subtraction of matrices is defined in an analogous manner.
The definition of multiplication of matrices that is useful is not the multiplication of
corresponding elements. Instead multiplication is defined only for matrices having the proper
number of rows and columns; i.e., the number of rows of the second matrix has to be equal to
the number of columns of the first matrix. If A is an n×m matrix and B is a m×p matrix then
their product AB is defined to be an n×p matrix C such that
Ci,j = ΣAi,kBk,j

where the summation is over k=1 to k=m.

The levels of production of the Coal and Steel industries can be represented as a 2×1 matrix
(a column vector) X where

X = | x1 |
| x2 |

Likewise the levels of net output required to be met, usually called final demands, f 1 and f2
can be represented as a column vector (2×1 matrix) F where

F = | f1 |

| f2 |

The direct requirements per units of output for an economy with two industries can be
represent as a 2×2 matrix A; i.e.,

A = | A1,1 A1,2 |

| A2,1 A2,2 |

For the Coal/Steel economy above

A=| 0 3|

| 0.1 0 |

The amounts of production used up in producing x1 and x2 are equal to

A1,1x1 + A1,2x2
A2,1x1 + A2,2x2

This is just the product of the matrix A and the matrix X; i.e., AX. The production levels are
given by X so the net productions after the amounts used up in production are deducted are
given by

X - AX

We want this to be equal to the required levels F so the equations to be satisfied are, in matrix
form,

X - AX = F

At this point it is necessary to note that there are special type of square matrices, called
identity matrices and denoted as I, that consist of 1's on the diagonal that runs from the upper
left to the lower right and 0's everywhere else. The 2×2 identity matrix is

I=|1 0|

|0 1|

The virtue of the identity matrices is that the product of an identity matrix with any other
matrix for which the product is defined is just the other matrix. In particular, IX is just X. It
turns out that it is often useful to represent a matrix as a product with the identity matrix. The
equations to be satisfied by the levels of output are, in matrix form,

X - AX = F
which can be expressed as
IX - AX = F

The advantage of this second representation is that it can be factored; i.e.,

IX - AX = (I-A)X
so the matrix equations to be satisfied is
(I-A)X = F

For the Coal/Steel economy above

I-A = | 1 -3 |

| -0.1 1 |

The solution to this equation would involve carrying out some algebraic operation so we end
up with X being equal to some matrix. Generally the problem we face is that we have a
matrix equation BX=C and we want to end up with X=something. Suppose that we could find
a special matrix D such that DB=I. Then we could multiply both sides of the matrix equality
BX=C to get DBX=DC. Since DB=I this means we would have IX=DC, but IX is the same
as X so we have the solution to the equations as X=DC. The special matrix D such that DB=I
is called the inverse of B and it is denoted as B -1. So the solution to the matrix equation
BX=C is X=B-1C. The only problem in finding a solution to BX=C is finding B-1.

At this point we do not know even if such a matrix exists. It turns out that there is a simple
test to determine whether an inverse exists. The test is based upon the determinant of the
matrix. If the determinant is not equal to zero then an inverse exists and if the determinant is
equal to zero then an inverse does not exist. For a 2×2 matrix B the determinant of B is

det(B) = B1,1B2,2 - B1,2B2,1

For a 2×2 matrix the determination of the inverse is very simple but it is not as simple for
higher order matrices. to get the inverse of a 2×2 matrix B we just interchange B 1,1 and B2,2
and change the signs of B1,2 and B2,1 and then divide all of the elements of the resulting matrix
by det(B); i.e.,

B-1 = | B2,2/det(B) -B1,2/det(B) |

| -B2,1/det(B) B1,1/det(B) |

For the Coal/Steel economy above the (I-A) was

I-A = | 1 -3 |

| -0.1 1 |

so it determinant is (1)(1)-(-3)(-0.1)=1-0.3=0.7. Thus it does have an inverse and that inverse

is

(I-A)-1 = (1/0.7)| 1 3|

| 0.1 1|

or, carrying out the indicated division,

(I-A)-1 = | 1.4286 4.2857 |

| 0.1429 1.4286 |

When we multiply this matrix times the vector of final demands F, where
F = | 200,000 |

| 50,000 |

we get the solution

X = | 500,000 |

| 100,000 |

In conclusion, the direct and indirect requirements per unit of final demands are given by the
columns of the inverse of the matrix I-A; i.e., (I-A)-1.

For the Coal/Steel economy the inverse of (I-A)

(I-A)-1 = | 1.4286 4.2857 |

| 0.1429 1.4286 |

tells us that for each ton of final demand for coal the economy has to produce 1.4286 tons of
coal and 0.1429 tons of steel. For each ton of final demand for steel the economy has to
produce 4.2857 tons of coal and 1.4286 tons of steel.

2.2 Interregional and International Interactions Using the Input – Output Theory
Input-Output Analysis arose to deal with the problem of interindustry demand, but the same
method can be used to show how changes in one region affect the economies of regions
linked to it. Suppose we have information on how changes in production in Santa Clara and
Santa Cruz Counties affect the demand for each other's output. (Santa Clara County is
essentially the famed Silicon Valley and Santa Cruz County is a county to the south of it over
the Santa Cruz Mountains and on Monterey Bay of the Pacific Ocean.) If production in Santa
Clara County increases there will be more income not only for residents of Santa Clara
County but also for the residents of Santa Cruz County because some of the jobs in Santa
Clara County will go to Santa Cruz County residents. The residents of both counties will
decide how much of their income they will spend, where, and for what. Some of that
spending will be in the two counties and be for goods and services that are produced locally.
Likewise when production in Santa Cruz County increases some of the jobs will go to Santa
Clara County residents and some of these will spend their income in Santa Cruz County as
well as in Santa Clara County. Suppose we have that information in matrix form:

County of Production

County of
Santa Clara Santa Cruz
Residence

Santa Clara 0.5 0.1

Santa Cruz 0.2 0.4

This is like a matrix of marginal propensities to consume in macroeconomic theory. In

macroeconomic theory if income is spent for products outside of the economy it is considered
a leakage. In this regional setting some of these leakages leak back into the economy.

The matrix above corresponds to the matrix A in input-output analysis.

In macroeconomic theory the income multiplier k is equal to:

1/(1-c)

where c is the marginal propensity to consume.

This could be written as:

k = (1-c)-1.

With the regional interaction there is a matrix of multipliers and the matrix is equal to:

(I-A)-1.

For the above matrix the matrix I-A is:

County of Production

County of
Santa Clara Santa Cruz
Residence

Santa Clara 0.5 -0.1

Santa Cruz -0.2 0.6

The determinant of I-A is (0.5)(0.6)-(-0.1)(-0.2)=0.30-0.02=0.28.

This means the matrix I-A does have an inverse. Remember that for a 2x2 matrix the inverse
is found by interchanging the diagonal elements and changing the sign of the off-diagonal
elements, then dividing every element by the determinant. This gives:

County of Production
 County of
Santa Clara Santa Cruz
Residence

Santa Clara 2.14286 0.35714

Santa Cruz 0.71429 1.78571

This means that when the demand for Santa Clara County's output increases by $1 the output
in Santa Clara County increases by $2.14 and in Santa Cruz County by $0.71. On the other
hand, if the demand for Santa Cruz County's output increases by $1 then output in Santa Cruz
County increases by $1.79 and in Santa Clara County by $0.36.

The above shows how once the matrix A is known how the inter-relationships between the
parts is determined in the form of the inverse of the (I-A) matrix. So once A is determined the
rest is merely numerical computation. But the matrix A first has to be established. The
derivation of the matrix A involves several economic processes. First, there is the distribution
of income (and jobs) to the sub regions. This is given in the form of a matrix which will be
called the matrix J (for jobs). Suppose J has the following value:

County of Production

County of
Santa Clara Santa Cruz
Residence

Santa Clara 0.75 0.20

Santa Cruz 0.25 0.80

This says that 75% of the jobs and income go to Santa Clara County residents and 25% go to
residents of Santa Cruz County. On the other hand, 20% of the jobs and income in Santa Cruz
County go to Santa Clara County residents and the other 80% to Santa Cruz County residents.
But not all of a dollar of production goes for labor income. Let us say that in both counties
one third of the revenue goes for labor income. This means that the effect of additional
dollars of production would have the following effects on incomes. This is the matrix Y (for
income).

County of Production

County of
Santa Clara Santa Cruz
Residence

Santa Clara 0.250 0.067

Santa Cruz 0.083 0.267

There is also the matrix that tells where people spend their money and how much of it goes
for local production. This is the matrix S (for spending).

County of Residence

County of
Santa Clara Santa Cruz
Spending

Santa Clara 0.80 0.30

Santa Cruz 0.10 0.60

This says that that when Santa Clara residents get another dollar of income 80% is spent in
Santa Clara County and another 10% is spent in Santa Cruz County. On the other hand, when
Santa Cruz residents get another dollar of income 30% is spent in Santa Clara County and
60% at home in Santa Cruz County. In both cases all of the spending goes for goods or
services which are produced in the county of the spending.

Note that the orientation of this table is opposite of the previous tables.
To construct the matrix A we have to follow a dollar of production to its disbursement as
income to the two counties and the allocation of the recipients spending between the two
counties. This illustration is going to leave out several other important economic processes
such how much of labour income goes for taxes, savings and imports. These omissions are to
keep the detail to a minimum.

According to matrix J, when a dollar of production is produced in Santa Clara County, $0.25
goes to Santa Clara County residents who spend 80% of it in Santa Clara County and $0.083
goes to Santa Cruz County residents who spend 30% of it in Santa Clara County. Altogether
then the $1 of production in Santa Clara County leads to (0.25)(.8)+(0.083)(.3)=0.225 of
addition consumer demand in Santa Clara County. This is the element in the first row, first
column of the matrix A. The dollar of additional production also leads to increased demand in
Santa Cruz County; i.e,. (0.25)(0.1)+(0.083)(.6)=0.075. This is the element in the second row,
first column of A.

When an additional dollar of production takes place in Santa Cruz County the additional
spending in Santa Clara County is (0.063)(0.8)+(0.267)(.3)=0.131, the element of the A
matrix in the first row, second column. The final element is the spending in Santa Cruz
County resulting from an additional dollar of production in Santa Cruz County. This is (0.63)
(0.1)+(0.267)(0.6)=0.167. Thus the A matrix is

County of Production

County of
Santa Clara Santa Cruz
Residence

Santa Clara 0.225 0.131

Santa Cruz 0.075 0.167

The (I-A) is then

County of Production
 County of
Santa Clara Santa Cruz
Residence

Santa Clara 0.775 -0.131

Santa Cruz -0.075 0.834

The determinant of this matrix is 0.6365, so there is an inverse. That inverse is

County of Production

County of
Santa Clara Santa Cruz
Residence

Santa Clara 1.3102 0.2058

Santa Cruz 0.1178 1.2175

This says that when there is an additional dollar of demand in Santa Clara County production
goes up by $1.31 in Santa Clara County and about $0.12 in Santa Cruz County. On the other
hand, when demand in Santa Cruz County increase by one dollar production in Sanat Clara
County increases by about $0.21 and about $1.22 in Santa Cruz County. But these are
increases in sales and production rather than income.

The increases in income are found by determining the proportion of production going to
income (one third in this example) and then distributing it according to the J matrix. The
result of this computation for Santa Clara County income due to an increase in Santa Clara
production is (1/3)(1.3102)(0.75)+(1/3)(0.1178)(0.20)=0.335. The results of the computation
is then

County of Production

County of
Santa Clara Santa Cruz
Residence
 Santa Clara 0.335 0.133

Santa Cruz 0.141 0.342

Therefore Leontief’s input output analysis does not come with great deal of theoretical
baggage that is hard to prove in real life. Of course, it is susceptible to distortions from
measurement error or inaccurate modelling, but its underlying strength lies in being driven by
real data. In such a sense, input-output analysis remains an active branch of economics.
Input-output economics provides us with a powerful economic analysis tool in the form of
input-output analysis.

3.0 Applications of Input-Output Theory in Our Economy

There are Two Applications of the Model which are Open Model and Closed Model.

An open model finds the amount of production needed to satisfy an increase in demand
whereas the closed model deals only with the income of each industry. The closed model
means that all inputs into production are produced and all outputs exist merely to serve as
input. That is to say that all outputs are also used as inputs. Industries produce commodities
using commodities as well as factor inputs. Households produce these factor inputs using
commodities. And as a matter of fact, the Leontief Open Production Model provides us with
a powerful economic analysis tool in the form of input-output analysis. Nowadays, many
people apply the input-output methodology to empirical problems requiring economic
analysis. The real strength of the input-output methodology lay in its practical uses as an
implement of economic analysis.

Because the input–output model is fundamentally linear in nature, it lends itself to rapid
computation as well as flexibility in computing the effects of changes in demand. Input–
output models for different regions can also be linked together to investigate the effects of
inter-regional trade, and additional columns can be added to the table to perform
environmentally extended input-output analysis (EEIOA). For example, information on fossil
fuel inputs to each sector can be used to investigate flows of embodied carbon within and
between different economies.

The structure of the input–output model has been incorporated into national accounting in
many developed countries, and as such can be used to calculate important measures such as
national GDP. Input–output economics has been used to study regional economies within a
nation, and as a tool for national and regional economic planning. A main use of input–output
analysis is to measure the economic impacts of events as well as public investments or
programs as shown by IMPLAN and Regional Input-Output Modelling System. It is also
used to identify economically related industry clusters and also so-called "key" or "target"
industries (industries that are most likely to enhance the internal coherence of a specified
economy). By linking industrial output to satellite accounts articulating energy use, effluent
production, space needs, and so on, input–output analysts have extended the approaches
application to a wide variety of uses.

3.0.1 Input-output and socialist planning

The input-output model is one of the major conceptual models for a socialist planned
economy. This model involves the direct determination of physical quantities to be produced
in each industry, which is used to formulate a consistent economic plan of resource
allocation. This method of planning is contrasted with price-directed Lange-model socialism
and Soviet-style material balance planning.

In the economy of the Soviet Union, planning was conducted using the method of material
balances up until the country's dissolution. The method of material balances was first
developed in the 1930s during the Soviet Union's rapid industrialization drive. Input-output
planning was never adopted because the material balance system had become entrenched in
the Soviet economy, and input-output planning was shunned for ideological reasons. As a
result, the benefits of consistent and detailed planning through input-output analysis was
never realized in the Soviet-type economies.

3.1 The Assumptions of the Input – Output Theory

 The characteristic assumption of input-output analysis is really a double one, and it is well to
distinguish its two constituents.

One might argue that the constant ratios of consumption of each good to the output of labor
need not be assumed but can be derived as a result of an optimizing procedure on the part of
consumers. But this, if true, would require a utility function wherein all goods are perfect
complements, so that no good has any marginal utility, except in combination with the right
proportions of all others. The argument must be rejected, because its premise is implausible
and also because its conclusion-fixed ratios of consumption to labour output-is contrary to
fact. If any component of the bill of goods turns out to be negative, this is a sign that the
original set of specified outputs is inconsistent.

It enables one, at best, to tell which combinations of inputs and outputs are possible and
which are efficient in the linear programming sense, i.e. which are on the boundary of the
production possibility range so that a reduction in any input or an increase in any output
requires an increase in some other input or a reduction in some other output, cause they are
logically independent and because they differ in the attacks and defences that can be brought
to bear.

1. The assumption of constant returns to scale: The other, and the more controversial, is the
assumption that no substitution among inputs is possible in the production of any good or
service. Alternative ways of putting the second assumption are: There is only one process
used for the production of each output, or, the level of output of a product determines
uniquely the level of each input required.
2. Exclusion of any optimizing from the supply side, because it excludes all choice about the
proportions in which inputs are to be combined in the production of a given output. With
such a production function, all inputs are perfect complements, the marginal product of every
one of them being zero, except in appropriate combination with all the others. The
assumption of constant returns to scale is contested on the ground that functions more
complex than simple proportions are necessary to describe production processes realistically,
particularly in industries like the railroads and utilities, where at least one large installation
(such as a railroad track, a dam, or a telephone line) must be provided before any output
appears. It is defended chiefly on grounds of simplicity. One need observe a productive
process just once, say at one point in time, to obtain estimates of all the parameters of a
simple-proportion production function, and computations are simpler with this form than with
 almost any other. It is quite possible that analyses based on it will lead to empirically
satisfactory results in some problems of course this remains to be seen; if they do, that will be
a splendid defence. Another defence sometimes offered is the argument that not enough is
known to suggest what type of function should be used if proportions are rejected. The
assumption that there is no substitution among inputs is often attacked because economists
expect to find, and do find, substitution. The idea of a process here is the same as that of a
process or activity in linear or nonlinear programing. A process can be defined in terms of a
particular set of proportions among inputs (if all inputs are doubled, the processing being
used more, but output may not double; if the input proportions are changed, then by
definition a new process is being used). Therefore, within any process there is no substitution
among inputs. This explains why observations in a single year enable one to estimate input-
output coefficients numerically. An obvious possibility to try is the ordinary linear function.
If this is not very helpful, as I suspect it will not be, then observations at several different
times on each industry might suggest possible alternative forms, but there is always the risk
that technological change may have intervened to cloud the picture among inputs when
relative prices change. This assumption too is defended on the general grounds of simplicity:
data gathering and computation are much easier if one can regard an industry as a single
process with fixed technical coefficients. In addition, analyses based on this assumption may
yield empirically satisfactory results in certain problems, but this remains to be seen Some
economists, instead of defending this assumption, have scrapped it and admitted the
possibility that each product may be produced by several different processes. If this
modification alone is made, the assumption of constant returns to scale being retained, then
the open form of input-output analysis turns into linear programing. Substitution among
inputs in the production of a particular output is then explained by shifts from one productive
process
to another.
3. It is that no process produces more than one output or, in other words, that there are no joint
products. Two comments may be made here, this is a kind of reverse of the first. First, if a
process produces two or more outputs in nearly constant proportions, such as hides and meat,
one can define a single new output for the process, consisting of all the original outputs
together, and thus satisfy the assumption. Second, joint products cause no difficulty in linear
programming, because every input is regarded as a negative output, so that there are already
at least two "outputs" of each process anyway, and adding a few more positive ones will not
change the character of the problem.
4. Only current flows of inputs and outputs are important, i.e. that problems of capacity and
capital can safely be ignored. This assumption is not necessarily characteristic of input-output
analysis. In fact, it has been replaced in several studies by suitable dynamic assumptions,
with the aid of which the open input-output model can be made to "foresee" and provide for
the capital requirements associated with a given future time-pattern of final demand. This is
strictly true only if the number of alternative processes for producing a product is finite; if the
number of alternative processes is infinite, and they form smoothly curved production
surfaces, then open input-output analysis becomes the familiar continuous production theory
instead of linear programming

3.2 Limitations and Criticism of the Input – Output Theory

Major limitations faced by input-output analysis are as follows:
1. Its framework rests on Leontief’s basic assumption of constancy of input co-efficient of
production which was split up above as constant returns of scale and technique of production.
The assumption of constant returns to scale holds good in a stationary economy, while that of
constant technique of production in stationary technology. These assumptions sacrifice
reality. They do not treat the inter-industry analysis dynamically even in the so-called
“dynamic model”. It tells us nothing as to how technical co-efficient would change with
changed conditions. Again, some industries may have identical capital structures some may
have heavy capital requirements while others may use no capital. Such variations in the use
of techniques of production make the assumption of constant coefficients of production
unrealistic.
2. This assumption of fixed co-efficient of production ignores the possibility of factor
substitutions. There is always the possibility of some substitutions even in a short period,
while substitution possibilities are likely to be relatively greater over a longer period.
3. The assumption of linear equations, which relates outputs of one industry to inputs of others,
appears to be unrealistic. Since factors are mostly indivisible, increases in outputs do not
always require proportionate increases in inputs.
4. The rigidity of the input-output model cannot reflect such phenomena as bottlenecks,
increasing costs, etc.
5. The input-output model is severely simplified and restricted as it lays exclusive emphasis on
the production side for the economy. It does not tell us why the inputs and outputs are of a
particular pattern in the economy.
6. Another difficulty arises in the case of “final demand” or “bill of goods”. In this model, the
purchases by the government and consumers are taken as given and treated as a specific bill
of goods. Final demand is regarded as an independent variable. It might, therefore, fail to
utilize all the factors proportionately or need more than their available supply. Assuming of
co-efficiency of production, the analysis is not in a position to solve this difficulty.
7. There is no mechanism for price adjustments in the input-output analysis which makes it
unrealistic. “The analysis of cost-price relations proceeds on the assumption that each
industrial sector adjusts the price of its output by just enough to cover the change in the
case of its primary and intermediate output.”
8. The dynamic input-output analysis involves certain conceptual difficulties. First, the use of
capital in production necessarily leads to stream of output at different points of time being
jointly produced. But the input-output analysis rules out joint production. Second, it
cannot be taken for investment and output will necessarily be non- negative.
9. The input-output model thrives on equations that cannot be easily arrived at. The first
thing is to ascertain the pattern of equations, then to find out the necessary voluminous
data. Equations pre suppose the knowledge of higher mathematics and correct data are not
so easy to ascertain. This makes the input-output model abstract and difficult.

4.0 Conclusion
The Input – Output Theory is said by other scholars to just be a baggage of Linear Equations
and matrices, this is evidently true but it still analyses the interrelationship of industries
through figures and gives facts and Data sheets on the activities that is carried in the
production of goods in industries. The Input –Output Theory is also a model that countries
such as the United States of America, Canada use as a tool for Economic Planning, Regional
Planning, Employment Multipliers and Economic Forecasts.
Reference
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Input – Output Models, Interindustry Structure. URBPL 5/6020, University of Utah Pam
Perlich.
Input – Output Analysis, World Bank. Skopje October 26-28, 2011, ECOMOD.
Miller R.E. & Blair P.D., (2009). Input – Output Analysis. Foundations and Extensions
Second Edition. Cambridge University Press, the Edinburgh Building Cambridge CB2
8RY UK. ISBN – 13 978-0-511-65103-8
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