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Applications of Matrices Leontif Input-Output Models Models for Simple Economies Suppose we consider a simple economy with a lumber industry and a power industry. Suppose further that production of 10 units of power require 4 units of power and 25 units of power require 5 units of lumber. 10 units of lumber require 1 unit of lumber and 25 units of lumber require 5 units of power. If surplus of 30 units of lumber and 70 units of power are desired, find the gross production of each industry. Models for Simple Economies: Creating a Technology Matrix In the previous problem we have Outputs the following information Inputs Power Lumber (converting all numbers to percentages) Power 0.4 0.2 Lumber 0.2 0.1 Models for Simple Economies: Creating a Technology Matrix We can convert Outputs this information to Inputs Power Lumber the following Power 0.4 0.2 technology matrix Lumber 0.2 0.1 or Leontif matrix .4 .2 A .2 .1 Models for Simple Economies: The gross production matrix The gross production matrix for the economy can be represented by the column matrix x1 X x2 Where x1 is the gross production of power and x2 is the gross production of lumber. Models for Simple Economies: The technological equation X (or IX where I is the identity matrix) is the amount of production that is desired. AX is the amount that is actually produced. So IX-AX=(I-A)X is the amount of surpluses, D, (also called final demands) Models for Simple Economies: The technology equation ( I A) X D Is called the technology equation of the Leontif equation. Models for Simple Economies Original Question : If surplus of 30 70 units of lumber and D 70 units of power 30 are desired, find the gross production of each industry. Find X Models for Simple Economies To find X, take the inverse of (I-A) if it exists ( I A) X D 1 X ( I A) D Models for Simple Economies 1 0 .4 .2 .6 .2 I A 0 1 .2 .1 .2 .9 Finding the inverse of I-A .6 .2 1 0 1 .333 1.667 0 1 .333 1.667 0 .2 .9 0 1 .2 .9 0 1 0 .833 .333 1 1 .333 1.667 0 1 0 1.8 0.4 0 1 0.4 1.2 0 1 0.4 1.2 Models for Simple Economies Then the inverse of I-A 1 1 .8 .4 is .4 1 .2 ( I A) To find the amount to produce for the given 1 amount of surplus, we X ( I A) D must multiply the inverse of I-A and D 1.8 .4 70 138 .4 1.2 30 64 X Hence the gross production are : Lumber : 64 units Power : 138 units Models for Simple Economies: Another Example The economy of a developing nation is based on agricultural products, steel, and coal. An input of 1 ton of agricultural products requires an input of 0.1 ton of agricultural products, 0.02 ton of steel, and 0.05 ton of coal. An output of 1 ton of steel requires an input of 0.01 ton of agricultural products, 0.13 tons of steel, and 0.18 tons of coal. An output of 1 ton of coal requires an input of 0.01 ton of agricultural products, .2 tons of steel, and 0.05 ton of coal. Find the necessary gross productions to provide surpluses of 2350 tons of agricultural products, 4552 tons of steel, and 911 tons of coal. What is the technology matrix? Models for Simple Economies: Another Example Output 0 .1 .01 .01 Input Agriculture Steel Coal A .02 .13 0.2 Agriculture 0.1 0.01 0.01 .05 .18 .05 Steel 0.02 0.13 0.2 Coal 0.05 0.18 0.05 Models for Simple Economics: Another Example What is the surplus matrix? Find the technological equation. What is (I-A)-1? What is the production matrix? Models for Simple Economics: Another Example What is the surplus matrix? 2350 D 4552 911 Models for Simple Economics: Another Example Find the technological equation. ( I A) X D 1 0 0 0.1 .01 .01 2350 0 1 0 .02 .13 .20 X 4552 0 0 1 .05 .18 .05 991 Models for Simple Economies: Another Example What is (I-A)-1? 1.11 .016 .015 1 ( I A) .041 1.20 .254 .066 .229 1.10 Models for Simple Economics: Another Example What is the production matrix? Thus in our to have the 2700 given surplus in this 1 problem 2700 units of X ( I A) D 5800 2200 agriculture, 5800 units of steel, and 2200 units of coal must be produced. Closed vs Open Leontif Models In both of the previous examples some amount of surplus in production. That is 0 D 0 0 These models are called Open Leontif Models. Closed vs Open Leontif Models If a company want to find out the amount of production need with no surplus, that is 0 D 0 0 Then the model is called a Closed Leontif Model. Closed Leontif Models What is the technological equation for a closed leontif model? Closed Leontif Models What is the technological equation for a closed leontif model? ( I A) X 0 Where 0 is actually a column matrix of all zeros.