Report

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 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. Solution 1 31 2 = 32 3 36 Matrices Exchange matrix Price vector .2 .1 = .4 .5 .4 .4 1 = 2 3 Find p such that = .6 .1 .3 Conditions 0 for = 1,2, … , k 0 for , = 1,2, … , k =1 = 1 for = 1,2, … , k Nonnegative entries and column sums of 1 for E. Key Results = − = This equation has nontrivial solutions if det − = 0 Shown to always be true in Exercise 7. THEOREM 10.8.1 If E is an exchange matrix, then = always has a nontrivial solution p whose 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 1 2 = ⋮ 1 = 2 ⋮ 11 21 = ⋮ 1 12 22 ⋮ 2 ⋯ 1 ⋯ 2 ⋱ ⋮ ⋯ Conditions 0 for = 1,2, … , k 0 for = 1,2, … , k 0 for , = 1,2, … , k Nonnegative entries in all matrices. Consumption 11 1 + 12 2 + ⋯ + 1 21 1 + 22 2 + ⋯ + 2 = ⋮ 1 1 + 2 2 + ⋯ + 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 − = (I − ) = 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 Example 5 Continued 0 .65 = .25 .05 .25 .05 50000 = 25000 0 .55 .10 0 Solution 102,087 = 56,163 28,330 Productive Consumption Matrix If ( − ) is invertible, = (I − )−1 If all entries of (I − )−1 are nonnegative there is a unique nonnegative solution x. Definition: A consumption matrix C is said to be productive if (I − )−1 exists and all entries of (I − )−1 are nonnegative. 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.