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INPUT OUTPUT MODEL
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INPUT -OUTPUT MODEL
Input-Output Analysis
Input-output analysis is a basic method of quantitative economics that portrays macroeconomic
activity as a system of interrelated goods and services.
Assumptions
• Each industry produces only one homogeneous commodity
• Each industry uses a fixed input ratio (or factor combination) for the production of its output;
and
• 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.
Objectives of Input-Output Model
• Rapid increase the National Development rate is main objective of this model.
• Maximising profit.
• Full employment achieved.
• Maximizing saving and increasing rate of capital formation.
• Received maximum Foreign exchanges.
Input-Output Transaction Table
• The traditional I-O model is estimated with a square accounting framework where the number
of industries is equal to the number of commodities.
𝒅𝟏 = 𝒄𝟏 + 𝑰𝟏 + 𝑮𝟏 + ∆𝑰𝟏 + 𝑬𝒙𝟏
𝒅𝟐 = 𝒄𝟐 + 𝑰𝟐 + 𝑮𝟐 + ∆𝑰𝟐 + 𝑬𝒙𝟐
𝑿𝟏 = 𝒙𝟏𝟏 + 𝒙𝟏𝟐 + 𝒅𝟏
𝒙𝟐 = 𝒙𝟐𝟏 + 𝒙𝟐𝟐 + 𝒅𝟐
Where,
𝒅𝟏 = Final demand of agriculture
𝒅𝟐 = Final demand of industry
𝒄𝟏 = 𝑐𝑜𝑛𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑠
𝑰 = 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑠
𝑮 = 𝐺𝑜𝑣𝑒𝑟𝑛𝑚𝑒𝑛𝑡 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑠
∆𝑰 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑬𝒙𝒑 = 𝑒𝑥𝑝𝑜𝑟𝑡𝑠
𝑰𝒎𝒑 = 𝐼𝑚𝑝𝑜𝑟𝑡𝑠
𝑭𝒅 = 𝑓𝑖𝑛𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑
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𝒙𝟏 = 𝑇𝑜𝑡𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑜𝑓 𝑎𝑔𝑟𝑖𝑐𝑢𝑙𝑡𝑢𝑟𝑒
𝒙𝟐 = 𝑇𝑜𝑡𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 𝑜𝑓 𝑖𝑛𝑑𝑢𝑠𝑡𝑟𝑦
𝑥11 𝑥21 𝑥12
𝒂𝟏𝟏 = , 𝑎21 = , 𝑎12 =
𝑥1 𝑥1 𝑥2
On substitution, we get
𝒙𝟏 − 𝒂𝟏𝟏 𝒙𝟏 − 𝒂𝟏𝟐 𝒙𝟐 = 𝒅𝟏
𝒙𝟐 − 𝒂𝟐𝟏 𝒙𝟏 − 𝒂𝟐𝟐 𝒙𝟐 = 𝒅𝟐
Rearranging,
[(𝟏 − 𝒂𝟏𝟏 )𝒙𝟏 ] − 𝒂𝟏𝟐 𝒙𝟐 = 𝒅𝟏
−𝒂𝟐𝟏 𝒙𝟏 + [(𝟏 − 𝒂𝟐𝟐 )𝒙𝟐 ] = 𝒅𝟐
Input-Output Coefficient Matrix
Hawkins –Simon Condition
• The direct and indirect requirements of input i to produce one unit of input i should be less
than 1 unit.
• If this sum is greater than or equal to $1, therefore, production will not be economically
justifiable.
Symbolically, this fact may be stated thus:
𝑛
∑ 𝑎𝑖𝑗 < 1 (𝑗 = 1, 2, … , 𝑛)
𝑖=1
Generalization in Matrix Form for n Industries:
(𝐼 − 𝐴) × 𝑋 = 𝑑
[(1 − 𝐴)−1 − 1 × (1 − 𝐴)] × 𝑋 = (𝐼 − 𝐴)−1 × 𝑑
𝐼 × 𝑋 = (𝐼 − 𝐴)−1 × 𝑑
𝑋 = (𝐼 − 𝐴)−1 𝑑
Where,
X = vector of output(𝑛 × 1)
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d = vector of final demand(𝑛 × 1)
A = input-output coefficient matrix (𝑛 × 𝑛)
I = identity matrix (𝑛 × 𝑛)
(𝐼 − 𝐴)−1 =Leontief inverse (𝑛 × 𝑛)
Leontief Inverse is the direct and indirect requirement of goods to produce 1 unit of final demand.
Dynamic Input-Output Model
A simple dynamic model has the following form:
𝑋𝑡 (𝐼 − 𝐴) − (𝑋𝑡+1 − 𝑋𝑡 )𝐵 = 𝐹𝑡
where, I is the 𝑛 × 𝑛 identity matrix,
A is the usual Leontief input matrix,
B is the matrix of fixed capital coefficients,
X is the vector of total outputs and
f is the vector of final deliveries, excluding fixed capital investment;
t refers to the time period.
Important Points:
• The field is most identified with the work of Wassily Leontief.
• Input-output model is a system of interrelated goods and services where economic activity
takes place.
• One industry’s output is another’s input.
• The basis of the model is the inter industry transaction table in which the rows provide the
distribution of a producers output and columns provide the inputs required by particular
industry.
• Intermediate demand plus final demand constitutes the total demand.
• 𝑋 = (𝐼 − 𝐴)−1 𝑑
• The sum of the elements in each column of the input-coefficient matrix
• A must be less than 1. This is Hawkins – Simon condition.
• Dynamic I-O model has the form: 𝑋𝑡 (𝐼 − 𝐴) − (𝑋𝑡+1 − 𝑋𝑡 )𝐵 = 𝑓𝑡
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