The Use of Linear Systems in Economics:

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

Outline • Basics • Closed Economy Model • Open Economy Model • Linear Algebra Applications • Example • Practical Applications

Goal What quantity should each of the industries in an economy produce, so that it will be justenough to meet the totaldemand for that product?

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

Closed Leontief Model • Cx=0 • Diagonal entries can be >0 •  aij = 1 C =

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

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)-1d =>production by each sector

System of Equations x1 = a11x1 + a12x2 + … + a1nxn + d1 x2 = a21x1 + a22x2 + … + a2nxn + d2 … xn = an1x1 + an2x2 + … + annxn + dn X= CX + d Total = Consumption + Outside Production by Industries Demand => Solve for d

… (1-a11)x1 – a12x2 - … - a1nxn = d1 -a21x1 + (1-a22)x2 - … - a2nxn = d2 … -an1x1 – an2x2 - … + (1-ann)xn = dn => MUCH easier with matrices

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

Solution

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

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

Thank you! Questions?

The Use of Linear Systems in Economics: