The Use of Linear Systems in Economics Leontief

Outline § Basics § Closed Economy Model § Open Economy Model § Linear Algebra

Goal What quantity should each of the industries in an economy produce, so that

Basics I Input I II Output III N II III N § C: consumption

Closed Leontief Model § Cx=0 § Diagonal entries can be >0 § aij =

Open Model § § § Final demand primary inputs aij ≤ 1 (j= 1,

Use of Linear Algebra Total Production—Consumption by Industries= Outside Demand X-CX=d => (I-C)X=d (I-C)

System of Equations x 1 = a 11 x 1 + a 12 x

Example Economy with Labor, Transportation, and Food industries • $1 L requires 40¢ in

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

Practical Applications of the Model § Any size economy from a business district to

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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 just enough to meet the total demand for that product?

Basics I Input I II Output III N II III N § 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 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 Ø If (I-C) is invertible, unique solution: x* = (I-C)-1 d =>production by each sector

System of Equations x 1 = a 11 x 1 + a 12 x 2 + … + a 1 nxn + d 1 x 2 = a 21 x 1 + a 22 x 2 + … + a 2 nxn + d 2 … xn = an 1 x 1 + an 2 x 2 + … + annxn + dn X= CX + Total = Consumption Production by Industries => Solve for d d + Outside Demand

… (1 -a 11)x 1 – a 12 x 2 - … - a 1 nxn = d 1 -a 21 x 1 + (1 -a 22)x 2 - … - a 2 nxn = d 2 … -an 1 x 1 – an 2 x 2 - … + (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 ØHow 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?