In its “static’ version, Professor Leontief’s input-output analysis deals with this particular question: “What level of output should each of the n industries in an economy produce, in order that it will just be sufficient to satisfy the total demand for that product?” The rationale for the term input-output analysis is quite plain to see. The output of any industry (say, the steel industry) is needed as an input in many other industries, or even for that industry itself; therefore the “correct’ (i.e., shortage-free as well as surplus-free) level of steel output will depend on the input requirements of all the n industries. In turn, the output of many other industries will enter into the steel industry as inputs, and consequently the “correct’ levels of the other products will in turn depend partly upon the input requirements of the steel industry. In view of this inter-industry dependence, any set of “correct” output levels for the n industries must be one that is consistent with all the input requirements in the economy. So, input-output analysis should be of great use in production planning, such as in planning for the economic development of a country or for a program of national defense. Nevertheless, the problem posed in input-output analysis also boils down to one of solving a system of simultaneous equations, and matrix algebra can again be of service.
To simplify the problem, the following assumptions are as a rule adopted:
(1) each industry produces only one homogeneous commodity (broadly interpreted, this does permit the case of two or more jointly produced commodities, provided they are produced in a fixed proportion to one another).
(2) Each industry uses a fixed input ratio (or factor combination) for the production of its output.
(3) Production in every industry is subject to constant returns to scale, so that a k-fold change in every input will result in an exactly k-fold change in the output.
These assumptions are, of course, unrealistic. From these assumptions we see that, in order to produce each unit of the jth commodity, the input need for the ith commodity must be a fixed amount, which we shall denote by aij. Specifically, the production of each unit of the jth commodity will require a1j (amount) of the first commodity, a2j of the second commodity,…, and anj of the nth commodity. (The order of the subscripts in aij is easy to remember: the first subscript refers to the input, and the second to the output, so that aij, indicates how much of the ith commodity is used for the production of each unit of the jth commodity.)
For example, we may assume prices to be given and, thus, adopt “a dollar’s worth’ of each commodity as its unit. Then the statement a32= 0.35 will mean that 35 cents’ worth of the third commodity is required as an input for producing a dollar’s worth of the second commodity. The aij symbol will be referred to as an input coefficient.
For an n-industry economy, the input coefficients can be arranged into a matrix A = [aij], as in Table below, in which each column specifies the input requirements for the production of one unit of the output of a particular industry. The Second column, for example, states that to produce a unit (a dollar’s worth) of commodity II, the inputs needed are: a12 units of commodity I, a22 units of commodity II, etc. If no industry uses its own product as an input, then the elements in the principal diagonal of matrix A will all be zero.
To be Continued….