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

• • 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 Days of Work in Home of Electrician Days of Work in Home of Plumber 2 4 4 1 5 4 6 1 3

Example 1 Continued

p

1 = daily wages of carpenter

p

2 = daily wages of electrician

p

3 = daily wages of plumber Each homeowner should receive that same value in labor that they provide.

Solution

𝑝 1 𝑝 2 𝑝 3 = 𝑠 31 32 36

Matrices

Exchange matrix 𝐸 = Price vector 𝐩 = .2 .1 .6

.4 .5 .1

.4 .4 .3

𝑝 1 𝑝 2 𝑝 3 Find p such that 𝐸𝐩 = 𝐩

Conditions

𝑝 𝑖 0 for 𝑖 = 1,2, … , k 𝑒 𝑖𝑗 0 for 𝑖, 𝑗 = 1,2, … , k 𝑘 𝑖=1 𝑒 𝑖𝑗 = 1 for 𝑗 = 1,2, … , k Nonnegative entries and column sums of 1 for E.

Key Results

𝐸𝐩 = 𝐩 𝐼 − 𝐸 𝐩 = 𝟎 This equation has nontrivial solutions if det 𝐼 − 𝐸 = 0 Shown to always be true in Exercise 7.

THEOREM 10.8.1

If E is an exchange matrix, then

𝐸𝐩 = 𝐩

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 E m are positive. Then there is exactly one linearly independent solution to

𝐼 − 𝐸 𝐩 = 𝟎

, 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 𝐱 = Demand vector Consumption matrix 𝐝 = 𝐶 = 𝑑 1 𝑑 2 ⋮ 𝑑 𝑛 𝑐 11 𝑐 21 ⋮ 𝑐 𝑛1 𝑥 1 𝑥 2 ⋮ 𝑥 𝑛 𝑐 12 𝑐 22 ⋮ 𝑐 𝑛2 ⋯ ⋯ ⋱ 𝑐 1𝑛 𝑐 2𝑛 ⋮ ⋯ 𝑐 𝑛𝑛

Conditions

𝑥 𝑖 0 for 𝑖 = 1,2, … , k 𝑑 𝑖 0 for 𝑖 = 1,2, … , k 𝑐 𝑖𝑗 0 for 𝑖, 𝑗 = 1,2, … , k Nonnegative entries in all matrices.

Consumption

𝐶𝐱 = 𝑐 11 𝑥 1 𝑐 21 𝑥 1 𝑐 𝑘1 𝑥 1 + 𝑐 12 𝑥 2 + 𝑐 22 𝑥 2 ⋮ + ⋯ + 𝑐 1𝑘 𝑥 𝑘 + ⋯ + 𝑐 2𝑘 𝑥 𝑘 + 𝑐 𝑘2 𝑥 2 + ⋯ + 𝑐 𝑘𝑘 𝑥 𝑘 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 𝐱 − 𝐶𝐱 = 𝐝 (I − 𝐶)𝐱 = 𝐝 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

x

1 = $ output coal-mining

x

2 = $ output power-generating

x

3 = $ output railroad

Example 5 Continued

𝐶 = 0 .65 .55

.25 .05 .10

.25 .05

0 𝐝 = 50000 25000 0

Solution

𝐱 = 102,087 56,163 28,330

Productive Consumption Matrix

If (𝐼 − 𝐶) is invertible, 𝐱 = (I − 𝐶) −1 𝐝 If all entries of (I − 𝐶) −1 are nonnegative there is a unique nonnegative solution x.

Definition: A consumption matrix C is said to be productive if (I − 𝐶) −1 exists and all entries of (I − 𝐶) −1 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.