**Wassily Leontief Input Output Analysis, Input Output Model**

Any account of Leontief as an economist must begin with his work on input-output. The input-output model is a way of representing the interdependence between the various sectors of the economy. Sector A supplies inputs (raw materials, semifinished goods etc.) to sectors B, C and D and itself receives inputs from sectors X, Y and Z. X, Y and Z may themselves either ship inputs to B, C and D, or receive inputs from them. The whole economy can thus be represented as a complex inter-relationship of sectors. An input-output table is a statistical representation of these inter-sectoral relationships. We illustrate this in figure 1 for a simple economy comprising three sectors.

The three-by-three array or matrix within the double lines shows the shipments made between the sectors. The first element (X11) expresses the output of sector 1 which is used as an input into sector 1 itself (seed planted to grow more corn, for example). X12 indicates the quantity of output of sector 1 supplied for use within sector 2, and so on. Y, is the net output of sector 1, going to final demand (consumption, investment etc.). X1 is the gross output (net output plus intermediate inputs) of sector 1. V1 is the value added within the first sector-the sum of wages and other factor incomes.

Interpreting the table in this way, we identify the destination of all output produced by sector 1 by reading along the first row. Reading down the first column we identify the value of inputs used in sector 1 - both intermediate inputs and primary factors. The first row and the first column must therefore have the same sum, as the revenue from the output of sector 1 is exhaustively divided among intermediate inputs and primary factors. The first of these relationships is the more important; the first row of the table can be written out in equation form as follows:

Gross output Intermediate inputs Final demand

X1 = X 11 + X12 + X13 + Y1

The input-output table for an economy is by itself merely an accounting framework, a means of representing the inter-industry flows in the economy in a particular period. In the above account we have implicitly assumed that all flows are measured in monetary units, and this is the normal practice with input-output tables. We could alternatively have described the flows in physical units-so many tons of coal used as an input in the production of so many tons of steel. This is indeed the way in which Leontief himself prefers to think of the input-output system, and the foundation of the input-output model - rather than the accounting framework for inter-sectoral flows which we have described -is precisely the assumption that there is a direct, technologically-determined relationship between the output of one sector and the inputs from other sectors necessary to produce that output. Each ton of steel-it is assumed - requires so many tons of coal as an input in its production. (Or, holding prices constant, each dollar's worth of steel requires so many cents worth of coal to produce it).

Doubling the output of steel will require twice as much coal as an input. It is this special assumption about the nature of technological relationships in an economy which turns the accounting framework we have described above into a model suitable for analysis and projection of the economy.

Let us define aij as the input of i necessary to produce a unit of j. (Its value can be calculated by using the formula aij = Xij/Xj). Then the equation above can be written:

X1 = a11X1+al2X2 + a13X3 + Yl

The assumption of fixed input coefficients is equivalent to assuming that the aijs are constant and independent of the output levels. If we make this assumption we can then use the equation to calculate the gross output of sector 1 required to meet a specified level of final demand (Y1) and also to supply sectors 2 and 3 with their required inputs. However inputs required for sectors 2 and 3 depend on those sectors' gross output levels {X2 and X3), and to the extent that they produce inputs for sector 1, their gross output levels depend themselves on the output level of sector 1, which we are trying to determine. In other words, the whole set of input-output equations must be solved simultaneously.

There are now straightforward techniques developed by Leontief for solving the simultaneous system of equations and computing the total gross output required in each sector to produce any given bill of final goods (see appendix). But whereas Leontief and his collaborators in the 1930s had to make the computations painfully and largely by hand, modern computer technology makes the solution simple.

The input-output model also allows the investigator quickly to calculate the consequences of a change in the structure of final demand. We can illustrate this by reference to an example which Leontief himself has considered on several occasions. Suppose the government reduces military spending as a result of a disarmament agreement. The immediate impact will fall on the armaments industry, but that sector's suppliers will also be affected indirectly, as will its suppliers' suppliers, and so on. The input-output model enables the investigator to calculate the total impact, direct and indirect, of such a change on the output of all the sectors of the economy. The consequences for employment can be calculated if output per man in each sector is known. A similar set of calculations will reveal the consequences of replacing military spending by alternative forms of government spending or by increased private final demand.

We consider other and more complex uses of the basic input-output model below. First, however, it is worthwhile considering the antecedents of the model and the way in which Leontief developed it. The notion of interdependence within the economy had been current for nearly two centuries. In 1758 Francois Quesnay, the Physiocratic economist, had published his Tableau Economique, a tabular representation of the relations between the output and expenditures of farmers, landowners and manufacturers (Quesnay, 1972; for a modern examination see Barna, 1975). The data however were fictitious. Marx's model of simple and expanded reproduction in a two-sector model also illustrated the interdependence of department I (producer goods) and department II (consumer goods) (Marx, 1956, Part III) and Walras formulated a mathematical model of interdependent markets or general equilibrium (Walras, 1954, Lesson 20). These last two contributions were made in the second half of the nineteenth century.

Early Soviet work devoted to compiling a balance of the national economy for the year 1923/4 has also been cited as an influence on Leontief's thinking. The work in question was a set of tables indicating the breakdown of production and inter-sectoral flows within industry. The distribution of national income between alternative uses was, however, shown separately. Leontief's review of this work appeared in Germany in October 1925 and, in Russian translation, in the December 1925 issue of Planned Economy, the journal of the State Planning Commission of the USSR (1977a, pp. 3-9). The present-day reader finds it hard to see the seeds of the input-output model in the short review, and Leontief himself disclaims the influence of the Soviet balance, pointing out that he left the USSR at the age of nineteen. However the lucky chance of the review enabled later Soviet writers to claim Soviet origins for input-output, and this gave the technique a degree of respectability in the USSR which it would otherwise have lacked.

Leontief himself attaches more significance to his time in Kiel, particularly to some work he did on demand and supply for particular products. Certainly by the time of his removal to Harvard he had a clear notion of what he wanted to do, and of the data required. However it was nearly ten years before his monograph on the American economy was published, and the first version he developed differed in important respects from the model described above, which appears in the later, expanded, edition of the monograph.

The first version was in many respects more sophisticated than the later one. The economic system it describes embraces the household sector not as a recipient of consumer goods produced to satisfy final demand, as described above, but as an integral part of the model. Households form an additional sector or industry; its inputs are consumer goods and its output is labour supplied to other sectors.6 Investment and savings are similarly incorporated in the system. If a sector is making a net investment, its expenditure exceeds its revenues, while for a sector making net savings revenue exceeds expenditure. A separate saving coefficient is established for each sector, such that total expenditure is equal to total revenue divided by the coefficient. Sectors making a net investment will have a savings coefficient of less than one (expenditure exceeds revenue); sectors making net savings—the household sector, for example - will have a coefficient in excess of one.

The version described in the previous paragraph is a closed model, which integrates all economic activity within a single framework. Leontief's later version, first published in 1944, was an open model which treats final demand as exogenous. This version-in many ways less elegant than the first-has proved more satisfactory for practical work, though dynamic input-output models do treat investment demand as endogenous (see below).

As noted earlier, the crucial assumption which underlies any application of the input-output model in either version, open or closed, is the assumption of fixity of input coefficients or perfect complementarity of inputs. On this point Leontief has consistently argued from the need to make simplifying assumptions in order to implement the model empirically - a preoccupation which underlies all his work. His formulation combines elements of both substitution and complementarity, as the separate outputs produced by a single sector are assumed to be perfect substitutes: it is only between the outputs of any two separate sectors that no substitution is permitted. Moreover he points out that many apparent instances of substitution between inputs are due to changes in the inter-industry composition of output -changes which the input-output model captures without appeal to input substitutability. Ultimately though, as Leontief recognises, the realism of the assumption of fixed input coefficients is tested in practical applications of the model.