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
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 carpenter
p2 = daily wages of electrician
p3 = daily wages of plumber
Each homeowner should receive that same
value in labor that they provide.
Solution
𝑝1
31
𝑝2 = 𝑠 32
𝑝3
36
Matrices
Exchange matrix
Price vector
.2 .1
𝐸 = .4 .5
.4 .4
𝑝1
𝐩 = 𝑝2
𝑝3
Find p such that
𝐸𝐩 = 𝐩
.6
.1
.3
Conditions
𝑝𝑖
0 for 𝑖 = 1,2, … , k
𝑒𝑖𝑗
0 for 𝑖, 𝑗 = 1,2, … , k
𝑘
𝑖=1
𝑒𝑖𝑗 = 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 Em 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
Demand vector
Consumption matrix
𝑥1
𝑥2
𝐱= ⋮
𝑥𝑛
𝑑1
𝑑
𝐝= 2
⋮
𝑑𝑛
𝑐11
𝑐21
𝐶= ⋮
𝑐𝑛1
𝑐12
𝑐22
⋮
𝑐𝑛2
⋯ 𝑐1𝑛
⋯ 𝑐2𝑛
⋱
⋮
⋯ 𝑐𝑛𝑛
Conditions
𝑥𝑖
0 for 𝑖 = 1,2, … , k
𝑑𝑖
0 for 𝑖 = 1,2, … , k
𝑐𝑖𝑗
0 for 𝑖, 𝑗 = 1,2, … , k
Nonnegative entries in all matrices.
Consumption
𝑐11 𝑥1 + 𝑐12 𝑥2 + ⋯ + 𝑐1𝑘 𝑥𝑘
𝑐21 𝑥1 + 𝑐22 𝑥2 + ⋯ + 𝑐2𝑘 𝑥𝑘
𝐶𝐱 =
⋮
𝑐𝑘1 𝑥1 + 𝑐𝑘2 𝑥2 + ⋯ + 𝑐𝑘𝑘 𝑥𝑘
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
x1 = $ output coal-mining
x2 = $ output power-generating
x3 = $ output railroad
Example 5 Continued
0 .65
𝐶 = .25 .05
.25 .05
50000
𝐝 = 25000
0
.55
.10
0
Solution
102,087
𝐱 = 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.