Finite Chapter 3.5

Finite Chapter 3.5 - Math 120 SP08 Notes & Examples for...

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Math 120 SP08 Notes & Examples for Section 3.5: Input-Output Models Brodnick Application of Matrices: Used to analyze economy by considering how sectors interrelate. Consider 2 sectors, Sector 1: Oil Sector, and Sector 2: Electricity Sector Both sectors produce a product and we can measure the products by their dollar value. (Let 1 unit = $1 worth of that product) Consider the following scenario: To produce one unit of oil, the oil sector uses 30¢ worth of oil (to power machinery) and 20¢ of electricity. To produce one unit of electricity, the electricity sector uses 25¢ worth of electricity and 10¢ worth of oil. 10,000 units of oil/year 18,000 units of electricity/year Question : How many units should each sector provide to meet both internal and external demands? i.e., Total Supply = Total Demand Unknown: 1 x = total supply units from oil 2 x = total supply units from electricity So 10000 10 . 0 30 . 0 2 1 1 + + = x x x 18000 25 . 0 20 . 0 2 1 2 + + = x x x System of two linear equations in two unknowns: 18000 25 . 0 2 . 0 10000 1 . 0 3 . 0 2 1 2 2 1 1 + + = + + = x x x x x x Rewrite system in matrix form: + = 18000 10000 25 . 0 2 . 0 1 . 0 3 . 0 2 1 2 1 x x x x In symbols: D AX X + = In our example, using this formula would give the following: y electricit of units oil of units 9 . 28910 8 . 18415 18000 10000 18000 10000 75 . 0 2 . 0 1 . 0 7 . 0 18000 10000 25 . 0 2 . 0 1 . 0 3 . 0 1 0 0 1 101 140 101 40 101 20 101 150 1 1 2 1 = - - = -
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Finite Chapter 3.5 - Math 120 SP08 Notes & Examples for...

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