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.
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Leontief Economic ModelsSection 10.8Presented by Adam Diehl
From Elementary Linear Algebra: Applications VersionTenth EditionHoward Anton and Chris Rorres
Nobel Prize in Economics 1973.
Taught economics at Harvard and New York University.
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.
Find p such that
Nonnegative entries and column sums of 1 for E.
This equation has nontrivial solutions if
Shown to always be true in Exercise 7.
If E is an exchange matrix, then always has a nontrivial solution pwhose entries are nonnegative.
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.
Nonnegative entries in all matrices.
Row i (i=1,2,…,k) is the amount of industry i’s output consumed in the production process.
Excess production available to satisfy demand is given by
C and d are given and we must find x to satisfy the equation.
x1 = $ output coal-mining
x2 = $ output power-generating
x3 = $ output railroad
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.
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.
A consumption matrix is productive if each of its row sums is less than 1.
A consumption matrix is productive if each of its column sums is less than 1.
(Profitable consumption matrix)
For proof see Exercise 8.