Leontief Economic ModelsSection 10.8

Presented by Adam Diehl

From Elementary Linear Algebra: Applications VersionTenth 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 carpenterp2 = daily wages of electricianp3 = daily wages of plumber

Each homeowner should receive that same value in labor that they provide.

Solution

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 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

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-miningx2 = $ output power-generating

x3 = $ output railroad

Example 5 Continued

Solution

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 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.