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In [link] above, how much should each person get for his efforts?
We choose the following variables.
As we said earlier, in this model input must equal output. That is, the amount paid by each equals the amount received by each.
Let us say the farmer gets paid dollars. Let us now look at the farmer's expenses. The farmer uses up 40% of his own production, that is, of the dollars he gets paid, he pays himself dollars, he pays dollars to the carpenter, and to the tailor. Since the expenses equal the wages, we get the following equation.
In the same manner, we get
The above system can be written as
This system is often referred to as
Simplification results in
Solving for , , and using the Gauss-Jordan method, we get
Since we are only trying to determine the proportions of the pay, we can choose t to be any value. Suppose we let , then we get
The Open Model
The open model is more realistic, as it deals with the economy where sectors of the economy not only satisfy each others needs, but they also satisfy some outside demands. In this case, the outside demands are put on by the consumer. But the basic assumption is still the same; that is, whatever is produced is consumed.
Let us again look at a very simple scenario. Suppose the economy consists of three people, the farmer , the carpenter , and the tailor . A part of the farmer's production is used by all three, and the rest is used by the consumer. In the same manner, a part of the carpenter's and the tailor's production is used by all three, and rest is used by the consumer.
Let us assume that whatever the farmer produces, 20% is used by him, 15% by the carpenter, 10% by the tailor, and the consumer uses the other 40 billion dollars worth of the food. Ten percent of the carpenter's production is used by him, 25% by the farmer, 5% by the tailor, and 50 billion dollars worth by the consumer. Fifteen percent of the clothing is used by the tailor, 10% by the farmer, 5% by the carpenter, and the remaining 60 billion dollars worth by the consumer. We write the internal consumption in the following table, and express the demand as the matrix D.
| produces | produces | produces | |
| uses | .20 | .25 | .10 |
| uses | .15 | .10 | .05 |
| uses | .10 | .05 | .15 |
The consumer demand for each industry in billions of dollars is given below.
In [link] , what should be, in billions of dollars, the required output by each industry to meet the demand given by the matrix D?
We choose the following variables.
In the closed model, our equation was , that is, the total input equals the total output. This time our equation is similar with the exception of the demand by the consumer.
So our equation for the open model should be , where represents the demand matrix. We express it as follows:
To solve this system, we write it as
where I is a 3 by 3 identity matrix
Therefore, the three industries must produce the following amount of goods in billions of dollars.
We will do one more problem like the one above, except this time we give the amount of internal and external consumption in dollars and ask for the proportion of the amounts consumed by each of the industries. In other words, we ask for the matrix A.
Suppose an economy consists of three industries , , and . Again, each of the industries produces for internal consumption among themselves, as well as, the external demand by the consumer. The following table gives information about the use of each industry's production in dollars.
| Demand | Total | ||||
| 40 | 50 | 60 | 100 | 250 | |
| 30 | 40 | 40 | 110 | 220 | |
| 20 | 30 | 30 | 120 | 200 |
The first row says that of the $250 dollars worth of production by the industry , $40 is used by , $50 is used by , $60 is used by , and the remainder of $100 is used by the consumer. The other rows are described in a similar manner.
Find the proportion of the amounts consumed by each of the industries. In other words, find the matrix A.
Once again, the total input equals the total output.
We are being asked to determine the following:
How much of the production of each of the three industries, , , and is required to produce one unit of ? In the same way, how much of the production of each of the three industries, , , and is required to produce one unit of ? And finally, how much of the production of each of the three industries, , , and is required to produce one unit of ?
Since we are looking for proportions, we need to divide the production of each industry by the total production for each industry.
We analyze as follows:
To produce 250 units of , we need to use 40 units of , 30 units of , and 20 units of .
Therefore, to produce 1 unit of , we need to use 40/250 units of , 30/250 units of , and 20/250 units of .
To produce 220 units of , we need to use 50 units of , 40 units of , and 30 units of .
Therefore, to produce 1 unit of , we need to use 50/220 units of , 40/220 units of , and 30/220 units of .
To produce 200 units of , we need to use 60 units of , 40 units of , and 30 units of .
Therefore, to produce 1 unit of , we need to use 60/200 units of , 40/200 units of , and 30/200 units of .
We obtain the following matrix.
or
Clearly
We summarize as follows:
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