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.