1. Leontief Economic Models Section 10.8 Presented by Adam Diehl
From Elementary Linear Algebra: Applications Version Tenth Edition Howard Anton and Chris Rorres

2. Wassilly Leontief
Nobel Prize in Economics Taught economics at Harvard and New York University.

3. Economic Systems Closed or Input/Output Model Open or Production 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

4. Input-Output Model Example 1 (Anton page 582) 2 1 6 4 5 3
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
Days of Work in Home of Plumber 3

5. 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.

6. Solution
𝑝 1 𝑝 2 𝑝 3 =𝑠

7. Matrices
Exchange matrix 𝐸= Price vector 𝐩= 𝑝 1 𝑝 2 𝑝 3 Find p such that 𝐸𝐩=𝐩

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

9. 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.

10. THEOREM
If E is an exchange matrix, then 𝐸𝐩=𝐩 always has a nontrivial solution p whose entries are nonnegative.

11. THEOREM
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 for Markov chains.

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

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

14. 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.

15. Consumption
𝐶𝐱= 𝑐 11 𝑥 1 + 𝑐 12 𝑥 2 +…+ 𝑐 1𝑘 𝑥 𝑘 𝑐 21 𝑥 1 + 𝑐 22 𝑥 2 +…+ 𝑐 2𝑘 𝑥 𝑘 ⋮ 𝑐 𝑘1 𝑥 1 + 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.

16. 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.

17. Example 5 (Anton page 586) Three Industries Coal-mining
Power-generating
Railroad
x1 = $ output coal-mining
x2 = $ output power-generating
x3 = $ output railroad

18. Example 5 Continued
𝐶= 𝐝=

19. Solution
𝐱= 102,087 56,163 28,330

20. 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.

21. THEOREM
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.

22. COROLLARY
A consumption matrix is productive if each of its row sums is less than 1.

23. COROLLARY
A consumption matrix is productive if each of its column sums is less than 1. (Profitable consumption matrix) For proof see Exercise 8.