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.