1. The Use of Linear Systems in Economics: The Use of Linear Systems in Economics: Math 214 Presentation Jenn Pope and Reni Paunova Professor Buckmire Leontief Input-Output Models

2. Outline   Basics   Closed Economy Model   Open Economy Model   Linear Algebra Applications   Example   Practical Applications

3. Goal What quantity should each of the industries in an economy produce, so that it will be just enough to meet the total demand for that product?

4. Basics Input I II III N Output III III N  C: consumption matrix  d: demand vector  x: production vector

5. Closed Leontief Model  Cx=0  Diagonal entries can be >0  aij = 1 C =

6. Open Model  Final demand and primary inputs  a ij ≤ 1 (j= 1,2,…, n)  1-  a ij =value of the primary inputs needed to make a unit of the jth commodity x = d

7. Use of Linear Algebra Total Production—Consumption by Industries= Outside Demand X-CX=d = > (I-C)X=d (I-C) = Leontief Matrix  If (I-C) is invertible, unique solution: x* = (I-C) -1 d =>production by each sector

8. System of Equations x 1 = a 11 x 1 + a 12 x 2 + … + a 1 n x n + d 1 x 2 = a 21 x 1 + a 22 x 2 + … + a 2 n x n + d 2 … x n = a n 1 x 1 + a n2 x 2 + … + a nn x n + d n X= CX + d Total =Consumption + Outside Production by Industries Demand => Solve for d => Solve for d

9. … (1-a 11 )x 1 – a 12 x 2 - … - a 1n x n = d 1 -a 21 x 1 + (1-a 22 )x 2 - … - a 2n x n = d 2 … -a n1 x 1 – a n2 x 2 - … + (1-a nn )x n = d n => MUCH easier with matrices => MUCH easier with matrices

10. Example Economy with Labor, Transportation, and Food industries $1 L requires 40¢ in T and 20¢ in F$1 L requires 40¢ in T and 20¢ in F $1 T requires 50¢ in labor and 30¢ in T$1 T requires 50¢ in labor and 30¢ in T $1 F requires 50¢ in L, 5¢ in T, and 35¢ in F$1 F requires 50¢ in L, 5¢ in T, and 35¢ in F  How much should each industry produce?

11. Solution

12. => the production schedule should be $59,200 labor, $64,800 transportation, and $33,600 food.

13. Practical Applications of the Model  Any size economy from a business district to the entire world  Most often used for city planning and analysis of our national economy  Government can predict a deeper recession when one industry shrinks =>subsidize industries

14. Thank you! Questions?