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Leontief Economic ModelsSection 10.8
Presented by Adam Diehl
From Elementary Linear Algebra: Applications VersionTenth Edition
Howard Anton and Chris Rorres
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Wassilly Leontief
Nobel Prize in Economics 1973.Taught economics at Harvard and New York University.
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Economic Systems
• Closed or Input/Output Model– Closed system of industries– Output of each industry is consumed by industries
in the model• Open or Production Model– Incorporates outside demand– Some of the output of each industry is used by
other industries in the model and some is left over to satisfy outside demand
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Input-Output Model
• Example 1 (Anton page 582) Work Performed by
Carpenter Electrician Plumber
Days of Work in Home of Carpenter 2 1 6
Days of Work in Home of Electrician
4 5 1
Days of Work in Home of Plumber 4 4 3
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Example 1 Continued
p1 = daily wages of carpenterp2 = daily wages of electricianp3 = daily wages of plumber
Each homeowner should receive that same value in labor that they provide.
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Solution
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Matrices
Exchange matrix Price vector Find p such that
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Conditions
Nonnegative entries and column sums of 1 for E.
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Key Results
This equation has nontrivial solutions if
Shown to always be true in Exercise 7.
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THEOREM 10.8.1
If E is an exchange matrix, then always has a nontrivial solution p whose entries are nonnegative.
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THEOREM 10.8.2
Let E be an exchange matrix such that for some positive integer m all the entries of Em are positive. Then there is exactly one linearly independent solution to , and it may be chosen so that all its entries are positive.
For proof see Theorem 10.5.4 for Markov chains.
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Production Model
• The output of each industry is not completely consumed by the industries in the model
• Some excess remains to meet outside demand
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Matrices
Production vector Demand vector Consumption matrix
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Conditions
Nonnegative entries in all matrices.
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Consumption
Row i (i=1,2,…,k) is the amount of industry i’s output consumed in the production process.
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Surplus
Excess production available to satisfy demand is given by
C and d are given and we must find x to satisfy the equation.
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Example 5 (Anton page 586)
• Three Industries– Coal-mining– Power-generating– Railroad
x1 = $ output coal-miningx2 = $ output power-generating
x3 = $ output railroad
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Example 5 Continued
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Solution
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Productive Consumption Matrix
If is invertible,
If all entries of are nonnegative there is a unique nonnegative solution x.
Definition: A consumption matrix C is said to be productive if exists and all entries of are nonnegative.
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THEOREM 10.8.3
A consumption matrix C is productive if and only if there is some production vector x 0 such that x Cx.
For proof see Exercise 9.
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COROLLARY 10.8.4
A consumption matrix is productive if each of its row sums is less than 1.
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COROLLARY 10.8.5
A consumption matrix is productive if each of its column sums is less than 1.
(Profitable consumption matrix)
For proof see Exercise 8.