Conglomerate In the conglomerate of Example, sup-pose that $400,000,000 worth of computers...

Conglomerate In the conglomerate of Example, sup-pose that $400,000,000 worth of computers, $200,000,000 worth of semiconductors, and $300,000,000 worth of business forms are produced. How much of each division’s output will be available for consumption?

EXAMPLE Determining Production for a Conglomerate A conglomerate has three divisions, which produce computers, semiconductors, and business forms. For each $1 of output, the computer division needs $.02 worth of computers, $.20 worth of semiconductors, and $.10 worth of business forms. For each $1 of output, the semiconductor division needs $.02 worth of computers, $.01 worth of semiconductors, and $.02 worth of business forms. For each $1 of output, the business forms division requires $.10 worth of computers and $.01 worth of business forms. The conglomerate estimates the sales demand to be $300,000,000 for the computer division, $100,000,000 for the semiconductor di-vision, and $200,000,000 for the business forms division. At what level should each division produce in order to satisfy this demand?

SOLUTION The conglomerate can be viewed as a miniature economy and its sales as the final demand. The input–output matrix for this “economy” is

The final-demand matrix is

where the demand is expressed in hundreds of millions of dollars. By equation (1), the matrix X, giving the desired levels of production for the various divisions, is given by

But

and after computation,

rounded to two decimal places. Therefore,

That is, the computer division should produce $334,000,000, the semiconductor divi-sion $168,000,000, and the business forms division $239,000,000.

Input-output analysis is usually applied to the entire economy of a country having hundreds of industries. The resulting matrix equation (I – A)X = D could be solved by the Gauss-Jordan elimination method. However, it is best to find the inverse of I – A and solve for X as we have done in the examples of this section. Over a short period, D might change, but A is unlikely to change. Therefore, the proper outputs to satisfy the new demand can easily be determined by using the already computed inverse of I – A.

The Closed Leontieff Model The foregoing description of an economy is usually called the open Leontieff model, since it views exports as an activity that takes place external to the economy. However, it is possible to consider exports as yet an-other industry in the economy. Instead of describing exports by a demand column D, we describe it by a column in the input-output matrix. That is, the export column describes how each dollar of exports is divided among the various industries. Since exports are now regarded as another industry, each of the original columns has an additional entry—namely, the amount of output from the export industry (that is, imports) used to produce $1 of goods (of the industry corresponding to the column). If A denotes the expanded input-output matrix and X the production matrix (as be-fore), then AX is the matrix describing the total demand experienced by each of the industries. In order for the economy to function efficiently, the total amount demanded by the various industries should equal the amount produced. That is, the production matrix must satisfy the equation

By studying the solutions to this equation, it is possible to determine the equilibrium states of the economy—that is, the production matrices X for which the amounts produced exactly equal the amounts needed by the various industries. The model just described is called the closed Leontieff model.

We may expand the closed Leontieff model to include the effects of labor and monetary phenomena by considering labor and banking as yet further industries to be incorporated in the input–output matrix.