The Use of Linear The Use of Linear Systems in Economics:Systems in Economics:

Math 214 PresentationMath 214 PresentationJenn Pope and Reni PaunovaJenn Pope and Reni Paunova

Professor BuckmireProfessor Buckmire

Leontief Input-Output Leontief Input-Output ModelsModels

OutlineOutline

Basics Closed Economy Model

Open Economy Model Linear Algebra

Applications Example

Practical Applications

GoalGoal

What What quantityquantity should each of the should each of the industries in an economy produce, so industries in an economy produce, so that it will be justthat it will be just enoughenough to meet the to meet the

totaltotal demanddemand for that product? for that product?

BasicsBasics

a11 a12 a13 ... a1n

a21 a22 a23 ... a2n

a31 a32 a33 ... a3n

an1 an2 an3 ... ann

Input

I

II

III

N

Output I II III N

CC: consumption matrix: consumption matrix

dd: demand vector: demand vector

xx: production vector: production vector

Closed Leontief ModelClosed Leontief Model

Cx=0Cx=0 Diagonal entries can be >0Diagonal entries can be >0

aij = 1aij = 1

a11 a12 a13 ... a1n

a21 a22 a23 ... a2n

a31 a32 a33 ... a3n

an1 an2 an3 ... ann

C =

Open ModelOpen Model

Final demand and primary inputs Final demand and primary inputs aaijij ≤≤ 1 (j= 1,2,…, 1 (j= 1,2,…, nn)) 1- 1- a aijij=value of the primary inputs needed to =value of the primary inputs needed to

make a unit of the make a unit of the jjth commodity th commodity

a11 a12 a13 ... a1n

a21 a22 a23 ... a2n

a31 a32 a33 ... a3n

an1 an2 an3 ... ann

x = d

Use of Linear AlgebraUse of Linear Algebra

Total Production—Consumption by Industries= Total Production—Consumption by Industries= Outside DemandOutside Demand

X-CX=d =X-CX=d => > (I-C)X=d(I-C)X=d

(I-C) = Leontief Matrix(I-C) = Leontief Matrix

If (I-C) is If (I-C) is invertibleinvertible,,

unique solution: unique solution: x* = (I-C)x* = (I-C)-1-1d d =>production by each sector=>production by each sector

System of EquationsSystem of Equations

xx11 = a = a1111xx11 + a + a1212xx22 + … + a + … + a11nnxxnn + d + d11

xx22 = a = a2121xx11 + a + a2222xx22 + … + a + … + a22nnxxnn + d + d22

……xxnn = a = ann11xx11 + a + an2n2xx22 + … + a + … + annnnxxnn + d + dnn

X= CX + dX= CX + d

Total Total = = Consumption + Outside Consumption + Outside ProductionProduction by Industries by Industries DemandDemand

=> Solve for => Solve for dd

……

(1-a(1-a1111)x)x11 – a – a1212xx22 - … - a - … - a1n1nxxnn = d = d11

-a-a2121xx11 + (1-a + (1-a2222)x)x22 - … - a - … - a2n2nxxnn = d = d22

……

-a-an1n1xx11 – a – an2n2xx22 - … + (1-a - … + (1-annnn)x)xnn = d = dn n

=> MUCH easier with matrices => MUCH easier with matrices 1 a11 a21 ... a1n

a21 1 a22... a2n

... ... ... ...

an1 an2 ... a annx1

x2

...xn

d1

d2

...dn

ExampleExampleEconomy with Labor, Transportation, and Food industriesEconomy 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?How much should each industry produce?

SolutionSolution

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

food.food.

Practical Applications of the Practical Applications of the ModelModel

Any size economy from a business district Any size economy from a business district to the entire worldto the entire world

Most often used for city planning and Most often used for city planning and analysis of our national economyanalysis of our national economy

Government can predict a deeper Government can predict a deeper recession when one industry shrinks recession when one industry shrinks =>subsidize industries=>subsidize industries

Thank you!Thank you!

Questions?Questions?