Math 141-copyright Joe Kahlig, 12C
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Section 5.4: Additional Matrix Applications
Leontief Input-Output Models
An Leontief Input-Output Model studies how sectors of an economy interact with each other. An
assumption that we will make is that all products produced are consumed.
Example: A village has two economies: farming (food) and weaving (cloth). The following information
has been determined for these sectors of the economy.
To produce 1 unit of food requires 0.40 units of food and 0.10 units of cloth.
To produce 1 unit of cloth requires 0.30 units of food and 0.20 units of cloth.
Any extra food or cloth that is produced by the village is exported to a nearby city. If the city demands
3861 units of food and 2754 units of cloth each month, how much food and cloth should be produced
by the village to meet its own need and to supply the city.
x = the total production of food by the village.
y = the total production of cloth by the village.
Math 141-copyright Joe Kahlig, 12C
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Definition: The Leontief Input-Output Model can be described by the equation: X = AX + D
where X is the production matrix, A is the input/output matrix and D is the demand matrix. The
calculation AX is the internal use of the model and the production matrix can be solved by the formula:
Example: An economy for a country has three different sectors: agriculture, energy, and manufacturing. To produce one unit of agriculture the economy uses 0.2 units of agriculture and 0.4 units of
energy. To produce one unit of energy the economy uses 0.2 units of energy and 0.4 units of manufacturing. To produce one unit of manufacturing the economy uses 0.1 units of agriculture, 0.1 units
of energy and 0.3 units of manufacturing.
A) Find the input/output matrix.
B) Suppose the economy is producing 40 units of agriculture, 45 units of energy, and 60 units of
manufacturing. How much of the production is used internally?
units of Agriculture
units of energy
units of manufacturing
C) How much of the production will the country be able to export to another country?
units of Agriculture
units of energy
units of manufacturing
D) What production should the economy have so that 88 units of agriculture, 61 units of energy, and
44 units of manufacturing may be exported to a nearby country?
units of Agriculture
units of energy
units of manufacturing
Math 141-copyright Joe Kahlig, 12C
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closed Leontief Input-Output Models
Example: Consider an isolated village, i.e. there is no exporting of it production. This economy of
the village has three sectors: farming, building, and services. To produce one unit of farming takes
0.5 units of farming, 0.25 units of building and 0.25 units of service. To produce one unit of building
takes 0.4 units of farming, 0.3 units of building and 0.3 units of service. To produce one unit of service
takes 0.4 units of farming, 0.2 units of building and 0.4 units of service.
A) Find the input/output matrix.
B) What are the equilibrium production levels for these three sectors?
Math 141-copyright Joe Kahlig, 12C
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Example: A company makes four products: A, B, C and D. Due to the nature of the products, it takes
one product A and 2 of product B to make a product C. To make a product D requires 2 product
A, 1 product B, and 3 product C. The company has an order for 30 product A’s, 10 product B’s, 20
product C’s and 15 product D’s.
A) How many of each item must be produced so that the order can be met?
B) How many product A’s were used to complete this order?