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To understand this tutorial you need to first understand matrix multiplication and matrix inversion.
Here, we look at an application of matrix algebra developed by Wassily Leontief in the middle of this century. He won the Nobel prize in economics in 1973 for this work. The application involves analyzing national and regional economies by looking at how various parts of the economy interrelate.
Here is a simple example, similar to Example 1 in Section 3.4 of Finite Mathematics and Finite Mathematics and Applied Calculus. : Think of the US economy of a country or a region as being composed of various sectors, or groups of one or more industries. For example, consider two specific sectors: the crude petroleum and natural gas sector (Sector 1: crude) and the petroleum refining and related industries (Sector 2: refining). Both produce a commodity: the crude sector produces crude petroleum and natural gas, and the refining sector produces refined petroleum and related products. By one unit of these products product, we shall mean $1 million worth of that product.
To analyse the use of the products of one sector by another, we use an inputoutput table:
To  crude  refining  
From  crude  1,600  72,000 
refining  0  4,000  
Total Output  40,000  100,000 
We read the table as follows:
To use to given data, we obtain the technology matrix by computing how much of each sector went into the production of one unit of each product: we divide each entry by the total for its column:
Technology matrix  =  A  = 

We got these as follows:
a_{11} = units of crude to produce one unit of crude. We are told that 1,600 million units of crude were used to produce 40,000 million units of crude. Thus, to produce one unit of crude, 1,600/40,000 = 0.04 units of crude were used, and so a_{11} = 0.04
a_{12} = units of crude to produce one unit of refined. a_{12} = 72,000/100,000 = 0.72
a_{21} = units of refined to produce one unit of crude. a_{21} = 0/40,000 = 0
a_{22} = units of refined to produce one unit of refined. a_{22} = 4,000/100,000 = 0.04.
We can use the technology matrix to answer some interesting questions. For instance:
Suppose there is an external demand for 30,000 units of crude and 300,000 units of refined products. How much much be produced by each sector in order to meet the demand?
Q Duh! Just produce 30,000 units of crude and 300,000 units of refined. That's a nobrainer, right?
A Wrong! You need to take into account the fact that there are internal usages for these products in order for the sectors to function. That is what the technology matrix is telling us; for instance, to produce one unit of refined product, we need 0.72 units of crude and 0.04 units of refined. Thus, we need to take these internal consumption figures into account when computing the amount that each sector should produce. This is done as follows.
Finding the Production Necessary to Meet a Given Demand 
Let us go back to the original equation X = AX + D and use it to answer a question based on the inputoutput table we have been studying:
To  crude  refining  
From  crude  1,600  72,000 
refining  0  4,000  
Total Output  40,000  100,000 
Notice that production totals were already given in the table, so we know that
X = 

Given the actual production figures above, what was the external demand for the products of each sector? We will answer it in several steps.
Q The correct formula for D is ...
Q The fact that the external demand for crude oil products was negative indicates that:
Let us continue with the issue of crude and refined products. The figures we have been using in the inputoutput table are quite close to the actual figures for the US economy in 1977. In other words, the answer you just obtained above for the demand are quiet accurate, and show that, in 1977, the US was extremely dependent on imported crude oil products. Now suppose that you were a senator in 1977 and wished to propose a program for domestic production of crude products in order to make the US entirely selfsufficient, whle continuing to meet the current demand for refined products. In other words, you will need to answer the following question:
Q The current entry in D for crude products is 33,600, and you wish to adjust production in order to change that figure to zero. How much more crude oil will need to be produced?
A To answer this question, we can use the same formula X = (I  A)^{1}D as before, modified as follows (see the text for the justification):
Let A be as above. Current imports of crude oil amount to 33,600 units, and you wish to adjust production in order to change that figure to zero. How much more crude oil will need to be produced? (answer should be accurate to the nearest 100 units  no commas, please!)
Q How come, if only 33,600 units of crude oil products are bing imported, the economy needs to produce more than that just to avoid having to import any?
A That is because the production of crude oil products actually uses up some additional crude in the process. Here is the matrix (I  A)^{1} in decimal form:
(I  A)^{1}  = 

To learn more about the entries of (I  A)^{1}, consult Section 3.5 of Finite Mathematics or Finite Mathematics and Applied Calculus Also, you should now be ready to try some of the exercises in Section 3.5.
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Last Updated: April, 2006
Copyright © 2000, 2006 Stefan Waner