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Input and Output Analysis -The Leontief Models

The outside demand contains all external demand factors. Industries in the United States normally have some demand from the states and some demand from foreign countries that wish to import the goods.
             2.2 The Closed Leontief Model.
             The closed Leontief model assumes that there is no external demand. This will cause all production to stay within the economy. Everything that is produced will also be consumed within the model. Since the total consumption within the system for any given industry is found in the total of the coefficients of the columns, each column will add to 1 in a closed model.
             The problem that needs to be solved is to be able to satisfy the demands of the industries involved. Let  be the production vector. This is a vector that gives all the production necessary to satisfy a certain demand. Since this is a closed model, the production should also be equal to the demand vector. So under the closed model, the equation looks like this:.
             Where  is the  consumption matrix and  is both the  production vector that meets the demands of each industry and the  demand vector. This is true because all consumption stays within the model. In order to solve for an  vector that makes the equality true, some algebra can be done first:.
             The solution  that will solve this is an eigenvector.
             2.3 The Relation to a Markov matrix.
             In order to solve  we are really looking for a vector  that is not affected when multiplied by matrix . Then the system has reached a steady state. In order for the economy to balance (consumption equal to production), it needs to be guaranteed that some steady state exists. There are special types of matrices that will guarantee that we can find a solution x. These matrices are called Markov matrices.
             There are two main properties that define a Markov matrix:.
             1. Each entry in the matrix is nonnegative.
             2. Each column in the matrix adds to 1.
             Based on these properties, it can be said that the closed Leontief model is a Markov matrix.


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