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Lecture 10 S11

# Lecture 10 S11 - Lecture 10 Leontief Input-Output Analysis...

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Lecture 10 Leontief Input-Output Analysis A very important application of matrices and their inverses is found in the branch of applied mathematics called input-output analysis. Wassily Leontief, the primary force behind these new developments, was awarded the Nobel Prize in economics in 1973 because of the significant impact his work had on economic planning for industrialized countries. Among other things, he conducted a comprehensive study of how 500 sectors of the U.S. economy interacted with each other. Of course, large scale computers played an important role in this analysis. Our investigation will be more modest. In fact, we start with an economy comprised of only two industries. From these humble beginnings, ideas and definitions will evolve that can be readily generalized for more realistic economies. Input-output analysis attempts to establish equilibrium conditions under which industries in an economy have just enough output to satisfy each other’s demands in addition to final (outside) demands. Given the internal demands within the industries for each others output, the problem is determine output levels that will meet various levels of final (outside) demands. Two industry model To make the problem concrete, let us start with hypothetical economy with only two industries, electric company E and water company W. Output for both company is measured in dollars. The electric company uses both electricity and water (input) in the production of electricity (output), and the water company uses both electricity and water (input) in the production of water (output). Suppose that the production of each dollar’s worth of electricity requires \$0.30 worth of electricity and \$ 0.10 worth of water, and the production of each dollar’s worth of water requires \$ 0.20 worth of electricity and \$ 0.40 worth of water. If the final result from the outside sector of the economy (the demand from all other users of electricity and water) is d 1 = \$12 million for electricity d 2 = \$8 million for water how much electricity and water should be produced to meet this final demand? To begin, suppose that the electric company produced \$ 12 million worth of electricity and the water company produced \$ 8 million worth of water (the final demand). Then the production processes of the companies would require Electricity Electricity required to required to produce produce electricity water 0.3(12) + 0.2(8) = \$5.2 million of electricity And Water Water required to required to produce produce electricity water

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0.1(12) + 0.4(8) = \$ 4.4 million of water leaving only \$6.8 million of electricity and \$ 3.6 million of water to satisfy the final demand of the outside sector. Thus, to meet the internal demands of both companies and to end up with enough electricity for the final outside demand, both companies must produce more than just the amount demanded by the outside sector. In fact, they must produce exactly enough to meet their own internal demands plus that demanded by the outside sector.
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