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c∞Dr Oksana Shatalov, Fall 201012.7: Leontief Input-Output ModelEXAMPLE1.LetA=":4:2:3:1#andD="1012#. Solve the matrix equationX°AX=DforX.An Leontief Input-Output Model studies interaction between diÆerent sectors of economics.An assumption that we will make is that everything produced is consumed, and that supplyalways equals demand.EXAMPLE2.A company has two interacting branches,(A)and(B). Branch(A)consumes$0:4of its own output and$0:3of(B)-output for every$1it produces. Branch(B)consumes$0:2of(A)-output and$0:5of its own output per$1of output. How much each branch should produceper month in order to meet exactly a monthly external demand of$50;000for(A)- product and$40;000for(B)-product?≤DeØne your variables:x- the dollar values of outputs of branch (A).y- the dollar values of outputs of branch (B).≤Calculate the total demand for branch (A) and branch (B) (i.e. internal consumption +external demand)
c∞Dr Oksana Shatalov, Fall 20102≤The system can be written as a matrix equation:whereXis the total output matrix,Ais the Input-Output (IO) matrix,Dis the outsidedemand matrix. The productAXis how much gets used internally (internal consumption).≤The matrix equation can be solved as follows (see Example 1):X=AX+D≤In our case, we haveSo, the production schedule isunits of outputs of branch (A), andunits of outputs of branch (B).EXAMPLE3.In Example 2, what is the internal use of production of each brand?Answer:AX=ORyou could just subtract oÆ the DEMAND matrix (external demand) from the TOTAL you justfound:TheLeontief Input-Output Modelcan be described by the equationX=AX+DwhereXis the production matrix (total output),Ais the input-output matrix andDis theconsumer demand matrix. The calculationAX=X°Dis the internal use of the model. Theproduction matrix can be solved by the formula:X= (I°A)°1D: