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Leontief Economic Models Section 10.8 Presented by Adam Diehl

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## Leontief Economic Models Section 10.8 Presented by Adam Diehl

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**Leontief Economic ModelsSection 10.8Presented by Adam Diehl**From Elementary Linear Algebra: Applications VersionTenth EditionHoward Anton and Chris Rorres**Wassilly Leontief**Nobel Prize in Economics 1973. Taught economics at Harvard and New York University.**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**Input-Output Model**• Example 1 (Anton page 582)**Example 1 Continued**p1 = daily wages of carpenter p2 = daily wages of electrician p3 = daily wages of plumber Each homeowner should receive that same value in labor that they provide.**Matrices**Exchange matrix Price vector Find p such that**Conditions**Nonnegative entries and column sums of 1 for E.**Key Results**This equation has nontrivial solutions if Shown to always be true in Exercise 7.**THEOREM 10.8.1**If E is an exchange matrix, then always has a nontrivial solution pwhose entries are nonnegative.**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.**Production Model**• The output of each industry is not completely consumed by the industries in the model • Some excess remains to meet outside demand**Matrices**Production vector Demand vector Consumption matrix**Conditions**Nonnegative entries in all matrices.**Consumption**Row i (i=1,2,…,k) is the amount of industry i’s output consumed in the production process.**Surplus**Excess production available to satisfy demand is given by C and d are given and we must find x to satisfy the equation.**Example 5 (Anton page 586)**• Three Industries • Coal-mining • Power-generating • Railroad x1 = $ output coal-mining x2 = $ output power-generating x3 = $ output railroad**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.**THEOREM 10.8.3**A consumption matrix C is productive if and only if there is some production vector x 0 such thatx Cx. For proof see Exercise 9.**COROLLARY 10.8.4**A consumption matrix is productive if each of its row sums is less than 1.**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.