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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 Models Section 10.8 Presented by Adam Diehl

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Leontief economic models section 10 8 presented by adam diehl

Leontief Economic ModelsSection 10.8Presented by Adam Diehl

From Elementary Linear Algebra: Applications VersionTenth EditionHoward Anton and Chris Rorres


Wassilly leontief

Wassilly Leontief

Nobel Prize in Economics 1973.

Taught economics at Harvard and New York University.


Economic systems

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

Input-Output Model

  • Example 1 (Anton page 582)


Example 1 continued

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

Solution


Matrices

Matrices

Exchange matrix

Price vector

Find p such that


Conditions

Conditions

Nonnegative entries and column sums of 1 for E.


Key results

Key Results

This equation has nontrivial solutions if

Shown to always be true in Exercise 7.


Theorem 10 8 1

THEOREM 10.8.1

If E is an exchange matrix, then always has a nontrivial solution pwhose entries are nonnegative.


Theorem 10 8 2

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

Production Model

  • The output of each industry is not completely consumed by the industries in the model

  • Some excess remains to meet outside demand


Matrices1

Matrices

Production vector

Demand vector

Consumption matrix


Conditions1

Conditions

Nonnegative entries in all matrices.


Consumption

Consumption

Row i (i=1,2,…,k) is the amount of industry i’s output consumed in the production process.


Surplus

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

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

Example 5 Continued


Solution1

Solution


Productive consumption matrix

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

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

COROLLARY 10.8.4

A consumption matrix is productive if each of its row sums is less than 1.


Corollary 10 8 5

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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