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Understanding Leontief Input-Output Models in Economics

Explore the application of linear systems in economics using Leontief input-output models. Learn how to determine optimal production quantities for industries to meet total demand, with practical examples and applications. Discover how linear algebra can be used to solve complex economic problems.

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Understanding Leontief Input-Output Models in Economics

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

  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 justenough to meet the totaldemand for that product?

  4. Basics Output I II III N I II III N Input • 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 •  aij≤ 1 (j= 1,2,…, n) • 1-  aij=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)-1d =>production by each sector

  8. 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

  9. … (1-a11)x1 – a12x2 - … - a1nxn = d1 -a21x1 + (1-a22)x2 - … - a2nxn = d2 … -an1x1 – an2x2 - … + (1-ann)xn = dn => MUCH easier with matrices

  10. 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?

  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?

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