Leontief Economic Models Section 10.8 Presented by Adam Diehl

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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

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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.

- 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

- Example 1 (Anton page 582)

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.

Exchange matrix

Price vector

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.

- The output of each industry is not completely consumed by the industries in the model
- Some excess remains to meet outside demand

Production vector

Demand vector

Consumption matrix

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.

- Three Industries
- Coal-mining
- Power-generating
- Railroad
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.