Loading in 5 sec....

Leontief Economic Models Section 10.8 Presented by Adam DiehlPowerPoint Presentation

Leontief Economic Models Section 10.8 Presented by Adam Diehl

Download Presentation

Leontief Economic Models Section 10.8 Presented by Adam Diehl

Loading in 2 Seconds...

- 143 Views
- Uploaded on
- Presentation posted in: General

Download Presentation
## PowerPoint Slideshow about 'Leontief Economic Models Section 10.8 Presented by Adam Diehl ' - adora

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Leontief Economic ModelsSection 10.8Presented by Adam Diehl

From Elementary Linear Algebra: Applications VersionTenth EditionHoward 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)

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.

Conditions

Nonnegative entries and column sums of 1 for E.

THEOREM 10.8.1

If E is an exchange matrix, then always has a nontrivial solution pwhose 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

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-mining

x2 = $ output power-generating

x3 = $ output railroad

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 thatx 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.