a ws is the amount of output from wood industry needed by the steel industry

# A ws is the amount of output from wood industry

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different demand and production levels.awsis the amount of output from wood industry needed by the steelindustry for each dollar of output produced by the steel industry.aijis the amount of output from industryineeded by industryjforeach dollar of output produced by industryj.Ais called the Input-Output matrix. This is usually known.Dis called the demand matrix. This is usually known.Xis the production matrix. It is the unknown that must be solvedfor. It gives the amount that each industry must produce to satisifyall needs.
MatricesInput-Output AnalysisX = A·X + D,Ahas elementsAwsxs= awsorAijxj= aijthe ratioAijxj= aijtypically does not change as the demand andproduction levels change. OnceAis known, it can be reused fordifferent demand and production levels.awsis the amount of output from wood industry needed by the steelindustry for each dollar of output produced by the steel industry.aijis the amount of output from industryineeded by industryjforeach dollar of output produced by industryj.Ais called the Input-Output matrix. This is usually known.Dis called the demand matrix. This is usually known.Xis the production matrix. It is the unknown that must be solvedfor. It gives the amount that each industry must produce to satisifyall needs.must solveX = A·X + Dfor the production levelX
Matrices Input-Output Analysis Must solve X = A · X + D for the production level X X = AX + D (10) X - AX = D (11) I · X - AX = D (12) (I - A) · X = D (13) (I - A) - 1 · (I - A) · X = (I - A) - 1 · D (14) I · X = (I - A) - 1 · D (15) X = (I - A) - 1 · D (16)
MatricesInput-Output Analysis: exampleThree-Sector EconomyIn an economic system, each of three industriesdepends on the others for raw materials.To make \$1 of processed wood requires:\$.30 wood,\$.20 steel\$.10 coal.To make \$1 of processed steel requires:\$.00 wood,\$.30 steel\$.20 coal.To make \$1 of processed coal requires:\$.10 wood,\$.20 steel\$.05 coal.To allow for \$1 consumption in wood, \$4 consumption in steel, and\$2 consumption in coal, what levels of production for wood, steel,and coal are required?
Matrices Input-Output Analysis: solution for example A = 0 . 30 0 0 . 10 0 . 20 0 . 30 0 . 20 0 . 10 0 . 20 0 . 05 , D = 1 4 2 (17) B = (I - A) = 0 . 70 0 - 0 . 10 - 0 . 20 0 . 70 - 0 . 20 - 0 . 10 - 0 . 20 0 . 95 (18) B - 1 = (I - A) - 1 = 1 . 4654 0 . 0469 0 . 1641 0 . 4924 1 . 5358 0 . 3751 0 . 2579 0 . 3283 1 . 1489 (19)
Matrices Input-Output Analysis: solution for example X = (I - A) - 1 D = B - 1 D = 1 . 4654 0 . 0469 0 . 1641 0 . 4924 1 . 5358 0 . 3751 0 . 2579 0 . 3283 1 . 1489 1 4 2 (20) X = 1 . 98 7 . 39 3 . 87 (21)