14 - The Leontief Model

14 - The Leontief Model - Math 1b Practical February 25,...

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Math 1b Practical — February 25, 2011 The Leontief input-output model in economics We are given a square matrix C of nonnegative real numbers called the consumption matrix . The rows and columns are indexed by ‘industries’ 1 , 2 ,...,n .T h ee n t r y c ij rep- resents the number of units of material that industry j must purchase from industry i in order to manufacture one unit of j ’s own product. If the economy produces x j units of the product of industry j ,and x is the column vector ( x 1 ,...,x n ), then C x gives the amounts of the various products that are used or internally consumed by the industries. Then x C x =( I C ) x gives the amounts of the various products available for external use. Here is an example I found at http://www.krellinst.org/AiS/textbook/unit7/f77_7.11_leontief.html . The coordinates of the column vector x on the right are the economy’s productions of petroleum, textiles, transportation, and chemicals measured in millions of dollars. We have C x = 880 110 550 825 . 50 and ( I C ) x = 20 190 300 25 . 50 . So e.g. $190 million more dollars’ worth of textiles were produced than consumed in the economy’s production. Unfortunately, it seems that $25.5 million more dollars’ worth of chemicals were consumed than produced. I guess these chemicals had to come from
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14 - The Leontief Model - Math 1b Practical February 25,...

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