Leontief Economic Models Section 10 8 Presented by
Leontief Economic Models Section 10. 8 Presented by Adam Diehl From Elementary Linear Algebra: Applications Version Tenth Edition Howard 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) 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
Example 1 Continued p 1 = daily wages of carpenter p 2 = daily wages of electrician p 3 = daily wages of plumber Each homeowner should receive that same value in labor that they provide.
Solution •
Matrices •
Conditions •
Key Results •
THEOREM 10. 8. 1 •
THEOREM 10. 8. 2 •
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 •
Conditions •
Consumption •
Surplus •
Example 5 (Anton page 586) • Three Industries – Coal-mining – Power-generating – Railroad x 1 = $ output coal-mining x 2 = $ output power-generating x 3 = $ output railroad
Example 5 Continued •
Solution •
Productive Consumption Matrix •
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.
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.
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