Input-Output(I-O) analysis is another area where matrix algebra becomes handy in the derivation of results. We discuss the process involved in it to start with. I-O analysis is also known as the inter-industry analysis as it explains the interdependence and interrelationship among various industries.

For example, in the two-industry model, coal is an input for
steel industry and steel is an input for coal industry, though both are the
output of respective industries.

**Assumptions **

The economy is divided into finite number of sectors (industries)
on the basis of the following assumptions:

i) Each industry produces only one homogeneous output.

ii) Production of each sector is subject to constant returns
to scale, i.e., two-fold change in every input will result in an exactly
two-fold change in the output.

iii) Input requirement per unit of output in each sector
remains fixed and constant. The level of output in each sector (industry)
uniquely determines the quantity of each input, which is purchased. Moreover,
if 5 men per Rs. 100000 of investment are required at any level of operation,
it is assumed that the same ratio will be required no matter how much the size
of the firm expands or contracts.

iv) The final demand for the commodities is given from outside
the system. The total amount of the primary factor (e.g., labour) is also given.
Presence of these two assumptions makes the system open ended and for this, the
model is called ‘open model’. In contrast to this, in the ‘closed model’, all
the variables are determined within the system.

**Important Key Notes :-**

**Closed Input-Output Model:**
Exogenous sector of the open input-output model is absorbed into the system as
just another industry such that the entire output of each producing sector is
absorbed by other producing sectors as secondary inputs or intermediate
products. Essentially, the household sector is treated as one of the industries
and no portion of the output is sold in the market as final product.

**Endogenous Variable**:
Dependent variable generated within a model and, therefore, its value is
changed (determined) by one of the functional relationships in that model.

**Exogenous Variable:** A
variable whose value is determined from outside a given model.

**Hawkins-Simon Condition:** More
than one unit of a product cannot be used up in the production of every unit of
that product. Condition requires that all principal minors of the technology
matrix must be positive.

**Input Coefficient Matrix:** A matrix
of different secondary inputs required by producing sectors per unit of output.

**Input-Output Model:** An
economic model that represents the interdependencies between different sectors
(industries) of a national economy.

**Input-Output Transaction Matrix:** A matrix
showing the distribution of total output of one industry to all other
industries as inputs and for final demand.

**Model:** A set of equations, functional
relationships and identities that seek to explain some economic phenomenon.

**Open Input-Output Model:** A model
in which the producing sectors interact with household sector of an economy
through their purchase of primary inputs and sales of final products.

**Primary Inputs:** Basic inputs of a
production process such as labour.

**Technology Matrix:** A matrix
obtained by subtracting a given input coefficient matrix from an identity
matrix.