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How many units of each commodity should be produced?

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Suppose the following matrix represents the input-output matrix of a simplified economy that involves just three commodity categories: manufacturing, agriculture, and transportation. How many units of each commodity should be produced to satisfy a demand of 900 units for each commodity?

[0 1/4 1/3]
[1/2 0 1/4]
[1/4 1/4 0]

A) 2259 units of manufacturing, 2520 units of agriculture, and 2097 units of transportation
B) 2520 units of manufacturing, 2097 units of agriculture, and 2259 units of transportation
C) 2241 units of manufacturing, 2538 units of agriculture, and 2097 units of transportation
D) 2259 units of manufacturing, 2097 units of agriculture, and 2241 units of transportation

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Solution suggests a trick that can be used to answer such kind of questions quickly. It also explains in detail how to arrive at the answer. An excerpt from the solution: "Considering that X units of manufacturing, Y units of agriculture, and Z units of transportation should be produced to satisfy a demand of 900 units for each commodity. This gives us following equations based on the given Input-Output matrix."

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Considering that X units of manufacturing, Y units of agriculture, and Z units of transportation should be produced to satisfy a demand of 900 units for each commodity.

This gives us following equations based on the given Input-Output matrix.

X - (Y/4) - (Z/3) = 900 --- (I)
Y - (X/2) - (Z/4) = 900 --- (II)
Z - (X/4) - (Y/4) = 900 --- (III)

Since we need to produce as much to "satisfy the demand" of 900 units, this can also be interpreted as that we need to produce minimum 900 units of each commodity. Which will change above equations to following.

X - (Y/4) - (Z/3) >= 900
Y - (X/2) - (Z/4) >= 900
Z - (X/4) - (Y/4) >= 900

We will ...

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