Industrial Production Suppose that the economy of Example experiences a 20% increase in th...

Industrial Production Suppose that the economy of Example experiences a 20% increase in the demand for coal. At what levels should the three industries produce?

EXAMPLE Determining Industrial Production Suppose that an economy is composed of only three industries—coal, steel, and electricity. Each of these industries depends on the others for some of its raw materials. Suppose that to make $1 of coal, it takes no coal, but $.02 of steel and $.01 of electricity; to make $1 of steel, it takes $.15 of coal, $.03 of steel, and $.08 of electricity; and to make $1 of electricity, it takes $.43 of coal, $.20 of steel, and $.05 of electricity. How much should each industry produce to allow for consumption (not used for production) at these levels: $2 billion coal, $1 billion steel, $3 billion electricity?

SOLUTION Put all the data indicating the interdependence of the industries in a matrix. In each industry’s column, put the amount of input from each of the industries needed to produce $1 of output in that particular industry:

This matrix is the input–output matrix corresponding to the economy. Let D denote the final-demand matrix. Then, letting the numbers in D stand for billions of dollars, we have

Suppose that the coal industry produces x billion dollars of output, the steel industry y billion dollars, and the electrical industry z billion dollars. Our problem is to determine the x, y, and z that yield the desired amounts left over from the production process. As an example, consider coal. The amount of coal that can be consumed or exported is just

To determine the amount of coal used in production, refer to the input–output matrix. Production of x billion dollars of coal takes 0 • x billion dollars of coal; production of y billion dollars of steel takes .15y billion dollars of coal; and production of z billion dollars of electricity takes .43z billion dollars of coal. Thus,

This quantity should be recognized as the first entry of a matrix product. Namely, if we let

then

The equations for the amounts of steel and electricity used in production are obtained in a manner similar to the equation for coal. But then, the amount of each output available for purposes other than production is X – AX. That is, we have the matrix equation

To solve this equation for X, proceed as follows. Since IX = X, write the equation in the form

 (1)

So, in other words, X may be found by multiplying D on the left by (I – A)–1. Let us now do the arithmetic.

Applying the Gauss–Jordan method (or using technology), we find that

where all figures are carried to two decimal places. Therefore,

In other words, coal should produce $3.72 billion worth of output, steel $1.78 billion, and electricity $3.35 billion. This output will meet the required final demands from each industry.

The preceding analysis is useful in studying not only entire economies, but also segments of economies, and even individual companies.