2 Input-Output Analysis
The basis of Leontief's analytical system is the input-output table. This table shows how the output of each industry is distributed among other industries and sectors of the economy. At the same time it shows the inputs to each industry from other industries and sectors. A hypothetical input-output or transactions table is illustrated by Table 2-1. (The same applies to Tables 2-2 and 2-3.) The illustrative table is highly simplified, in that only six hypothetical industries are included, but it is realistic in other respects. An actual input-output table may include from 50 to 200 industries, depending upon the degree of aggregation desired. Data were collected by the Bureau of Labor Statistics to make up a 500-industry table in the 1947 study, although the table itself was not published.
Some advantage is gained by disaggregation; that is, by having a detailed breakdown of industries and sectors. If an input-output table is to be used for forecasting, for example, a detailed industrial classification would reveal where bottlenecks might occur during the expansion of production. There are times, however, when it is useful to consolidate the sectors of a large table into a more compact table.1 This is the case when attention is to be focused on one or two particular sectors. As a general rule, however, input-output analysts strive for the maximum amount of disaggregation when constructing a basic transactions table.
'This leads to a number of statistical problems, however, and treatment of these problems is outside the scope of an introduction to input-output analysis. For a discussion of this aspect of the aggregation problem see Walter D. Fisher, "Criteria for Aggregation in Input-Output Analysis," The Review of Economics and Statistics, XL (August 1958), 250-60.
It has been customary in the United States, and in most other countries as well, to value transactions in terms of producers’ prices. Also, in the case of trade activities, outputs are defined as "gross margins" rather than the total value of all transactions—that is, the value of goods handled by trade establishments is not counted. There are a number of technical problems involved in the measurement of gross margins which cannot be discussed here.2 For present purposes it will be convenient to view gross margins as a "mark-up" on the goods handled by trade establishments as a payment for the creation of time-and-place utility.
A prodigious amount of labor is required to construct an input-output table, but once made up it is fairly easy to “read” or interpret. If the reader has no difficulty in understanding the small hypothetical table considered here he will have no difficulty in interpreting the much larger tables that have been published. We will trace through a series of transactions to show the inner workings of the table, but first we will explain its various parts.
Assume that the transactions, recorded in the table are in billions of dollars. Each row (reading from left to right) shows the output sold by each industry or sector along the left-hand side of the table to each industry or sector across the top of the table. Each column (reading from top to bottom) shows the purchases made by each industry or sector along the top of the table from the industries and sectors along the left-hand side. Since this is a square table, there is one row to correspond to each column.
To illustrate, consider the relationship between industry E ((row 5 and column 5) and industry C (row 3 and column 3). To find the share of industry E’s output sold to industry C, read across row 5 until it intersects column 3. We see that industry E sold one billion dollars' worth of goods to industry C during the period covered by the table. To find how much industry E buys from industry C, go over column 5 and read down until this column intersects row 3. We see that industry E bought from industry C products worth five billion dollars. Hence the net transaction between industries C and E during this period is four billion dollars in favor of industry C. There is nothing difficult about reading the table provided we remember the following simple rules:
1. To find the amount of purchases from one industry by another, locate the purchasing industry at the top of the table, then read down the column until you come to the producing industry.
2. To find the amount of sales from one industry to another, locate the selling industry along the left side of the table, then read across the row until you come to the buying industry.
The Make-up of the Table
1. The Processing Sector. The upper left-hand corner of the table has been set off in heavy double lines and labeled the processing sector. This is the sector of an input-output table which contains the industries producing goods and services. Among them we would find agriculture, various manufacturing industries, transportation, communications and other utilities, wholesale and retail trade, the service industries, construction, and as many other industries as are isolated for separate treatment in the table. This is the portion of the hypothetical table that is highly simplified, and in practice we would expect to find this sector expanded to 50 or more industries, thus greatly expanding the size of the entire table.
2. The Payments Sector. On the left-hand side of the table, rows 7 to 11 are set off under the heading payments sector. This sector includes these five rows read all the way across the table. We shall examine each of the five parts of the payments sector in turn.
a. Row 7, gross inventory depletion. By gross inventory depletion we mean the using up of previously accumulated stocks of raw materials, intermediate goods, or finished products. Thus in row 7, column 2, we see that during the period covered by the table industry B used up two billion dollars' worth of the stock it had put into inventory in an earlier period. The amount of inventory depletion in all other industries and sectors can be found by reading down each column until it intersects row 7.
b. Row 8, imports. To find the value of imports purchased by each industry and sector, read down each column until it comes to row 8. This procedure shows, for example, that industry E imported three billion dollars' worth of goods from abroad, while industry D imported nothing.
c. Row 9, payments to government. For simplicity, assume that payments to governments (federal, state, and local) in the form of taxes, represent purchases of government services such as police and fire protection, maintenance of the armed forces, and similar services which most of us take for granted. Although there is no direct correspondence between payments to government and the amount of government services provided to each industry (because, for example, how do you "value" the protection of the Army and Navy?), it will simplify matters if we assume that the figures in row 9 represent the value of government services to each of the industries and other sectors listed across the top of the table.
d. Row 10, depreciation allowances. Reading across row 10 we see the amounts of depreciation allowances set aside by each of the industries listed across the top of the table. These numbers approximate the cost of plant and equipment used up in the production of the goods represented in this table. Note, for example, that industry A (column 1) allowed one billion dollars during the period covered by the table for the depreciation of machinery and other equipment .3
e. Row 11, households. This row represents the wages, salaries, dividends, interest, and similar payments made to households by each of the industries and other sectors listed across the top of the table. We have inserted fairly large figures in this row to indicate in particular the relative importance of payments to labor in our hypothetical economy. Industry A paid out 19 billion dollars in the form of wages, salaries, and other forms of household income; industry B paid out 23 billion dollars, and so on across row 11.
3. The Final Demand Sector. The final demand sector consists of columns 7 through 11 read all the way down the table. The final demand sector is of special importance because it is the autonomous sector-the one in which changes occur which are transmitted throughout the rest of the table. It is here that the transactions which will be discussed presently originate. We will describe each of the parts of this sector briefly.
a. Column 7, gross inventory accumulation. This column shows the amounts of additions to inventories held by each of the industries and sectors along the left-hand side of the table. During any given time period some of the goods produced do not get into the hands of their final consumers. Retailers must stand ready to provide consumers with a variety of goods at all times. Hence they must keep a stock of goods on their shelves. Wholesalers must likewise be ready to ship to retailers upon short notice. And manufacturers will usually have a stock of the goods they produce on hand at any given time. Column 7 shows the amounts of inventories accumulated during the period covered by the table regardless of where those inventories are held, whether at the factory, in warehouses, or in retail establishments.
b. Column 8, exports. This column shows the value of exports from each of the processing industries and other sectors during the period covered by the table. Note that industry A in our hypothetical economy exported five billion dollars' worth of goods while households exported nothing. This would be typical of a national table since residents of one country ordinarily do not sell their labor services in another country. In regional applications, however, households can export labor services across regional boundaries, and it is also fairly common for management and technical consulting services to be exported from one region to another.
c. Column 9, government purchases. Purchases made by all levels of government are given in this column. The entry where the government column and the government row intersect indicates that there are some intragovernmental transactions, just as there are transactions within other industries and sectors included in our table.
d. Column 10, gross private capital formation. This column shows the amount of sales from each industry or sector along the left side of the table to buyers who use their purchases for private capital formation. All entries in the transactions table, except those in column 10, are on current account. Purchases by all buyers for the replacement of or additions to plant and equipment—and any other purchases which are entered on capital account—are summarized by the entries in column 10. Viewed another way, each entry in column 10 can be considered an input from the industry or sector listed at the left of the Gross Private Capital Formation "industry.”
e. Column 11, households. The entries in this column represent purchases of finished goods and services by their ultimate consumers from the industries and other sectors along the left- hand side of the table.
4. Total Gross Output and Total Gross Outlay. The final row and the final column of the table have yet to be explained.
Row 12, total gross outlay, shows the total value of inputs to each of the industries and sectors in each column at the top of the table. The total value of purchases by industry A, for example, is 64 billion dollars, the amount of the entry in row 12, column 1.
The input-output table is essentially a system of double-entry bookkeeping. Within each industry in the processing sector all of the receipts from sales are paid out for goods and services purchased from other industries or sectors. It might help to think of these as payments to factors of production. Some of the receipts are paid to the government in taxes, and some might be added to capital account. But the receipts from all outputs will just balance total outlays for each industry. After taking into account appropriate inventory changes, the total gross output, column 12, of each industry in the processing sector is equal to the total outlays made by that industry. Thus in the hypothetical table, the first six entries in the Total Gross Output column are identical with the first six entries in the Total Gross Outlay row.
This is not true of the totals in the remaining rows and columns, however. We would expect imports and exports to be exactly equal in any given year. Nor are inventory depletions and inventory accumulations likely to be the same during a given time period. Similarly, one would not expect a balance between government purchases and payments to governments, capital spending and depreciation allowances, and payments to and by households in the same year. But the individual differences must "cancel out" when we view the entire economy. As is true of any single processing industry, total outlays must equal total outputs for the economy as a whole. The total of all rows in the payments sector must equal the total of all columns in the final demand sector for the same reason that the Gross National Product computed from the product side must equal Gross National Product computed from factor payments.
One last point may be raised before tracing through a set of transactions. How does the Total Gross Output (or Total Gross Outlay) in the input-output table compare with Gross National Product? They are not the same. The GNP is defined as "the current market value of final goods and services produced in a given year." But even for the same year, GNP will not be the same as the Total Gross Output of an input-output table. In computing GNP every effort is made to eliminate double-counting. But since the input-output table measures all transactions in the economy the value of goods and services produced in a given year is counted more than one time; that is, we deliberately double count.
The objective is different in the two cases. In national-income analysis the object is to measure the final value of goods and services produced by the entire economy in a given year. We obviously wish to count one time only each good and service produced. In the input-output table, however, we wish to account for all transactions. Since some goods will enter into more than one transaction, their value must be counted each time a different transaction takes place. What we have then is an accumulation of value added at each stage of the production process until a good gets into the hands of its final consumer.
Input-output analysis and national-income accounting are not two separate branches of economics, however. As noted in the preface, the 1958 table for the United States has been completely reconciled with our national income and product accounts.
There is nothing rigid about the classifications used in the payments and final demand sectors of the hypothetical transactions table. The industries in the processing sector can be disaggregated to any degree desired—within the limits of data availability. Similarly, the payments and final demand sectors can be split into more rows and columns than those shown in Table 2-1. For example, the import row (and export column) can be disaggregated along geographic lines. Instead of a single government row (and column) there can be three, one each for federal, state, and local governments. And the household row (and column) could be further divided; for example, on the basis of income distribution. The input-output table is a flexible analytical tool. It can be made as detailed or as condensed as necessary for any given purpose. The only limitation is that there must be one row for each column in the processing sector. It is convenient, although not necessary, to have a final demand column for each row in the payments sector.
There is no fixed rule for including. (or excluding) any specific economic activity in the final demand (or payments) sector. Table 2-1 illustrates a relatively "open" input-output model. For some purposes it might be desirable to "close" the system with respect to one or more of the activities in the final demand (payments) sector. Households, for example, can be shifted into the processing sector, and the same is true of any other activity in final demand.4 Similarly, some activities normally included in the processing sector can be shifted to final demand. The construction and maintenance industry can be included in final demand, for example, if one is interested in analyzing the interindustry effects of changes in construction activity. The decision of how "open" or "closed" an input-output table is to be depends largely upon the purpose for which it is to be used. Our hypothetical example illustrates a general-purpose, open, nondynamic input-output system. But it must be emphasized that the basic model can be altered in a number of ways, depending upon the analytical use for which it is intended.
Tracing through a Set of Transactions
Let us now trace through a set of transactions involving one of the hypothetical industries in the processing sector of the input-output table. Consider the sales made by industry C, and the purchases made by the same industry.
The output side. A look at the transactions table indicates that industry C sold seven billion dollars' worth of goods to industry A during the period covered by the table, and it sold two billion dollars' worth to industry B. Intraindustry transactions amounted to eight billion dollars. This means that the firms in industry C purchased from each other goods valued at this amount. Other sales to industries D, E, and F came to one, five, and three billion dollars respectively. This accounts for all transactions within the processing sector of the table.
Additions to inventory in industry C were valued at two billion dollars during the period, and this industry exported three billion dollars' worth of goods to foreign countries. It sold one billion dollars' worth of goods to various government agencies. During the period covered by the table a total of five billion dollars was spent on the finished products of industry C by households. And three billion dollars' worth of the output of this industry was used by its buyers for replacement of or additions to capital equipment. Altogether, the total gross output of industry C was valued at 40 billion dollars in our hypothetical economy.
The input side. Let us look at the purchases made by industry C from the other industries in the table. Purchases from industry A amounted to one billion dollars; from B, seven billion; from D, two billion; and from E and F, one and seven billion respectively. Industry C also used up inventories amounting to one billion, and imported three billion dollars' worth of goods from other countries. It paid taxes of two billion, and set aside one billion in depreciation allowances. Finally, the industry paid out seven billion dollars in wages and salaries. Once again, these individual items must add up to 40 billion dollars-the amount entered in the Total Gross Outlay row.
The interested reader can repeat this process for any industry or sector shown in the table. He will soon develop a facility for following through a set of transactions.
Industries and Sectors
A transactions table consists of a collection of industries and sectors, and it might be helpful to distinguish between these concepts. According to Tiebout, "industries refer to aggregates of firms producing similar products. Sectors refer to the kinds of markets that industries serve."5 This is a useful distinction to keep in mind. When discussing the transactions table, however, we have at times referred to one collection of activities as the processing sector, and we have spoken of the individual activities outside this category as the final demand sector when they are considered collectively. Thus the term sector may be used at times with slightly different meanings, but the meaning which applies in each case should be clear from the context of the discussion.
All firms engaged in producing similar goods, or providing similar services, make up an industry. The concept of the industry is a fuzzy one because of the problem of overlapping. Not many large manufacturing firms, for example, make one product only. The same firm may manufacture automobiles, tractors, refrigerators, deep-freeze units, television sets, and perhaps a wide variety of other products. Generally, however, a firm is classified on the basis of its principal product. If this firm is engaged primarily in the manufacture of automobiles it is included in the automobile industry. If we are interested in analyzing the refrigerator industry, however, we must include in the industry that portion of this firm's activities devoted to the production of refrigerators. A useful method for solving the problem of overlapping in defining an industry has been developed by P. Sargent Florence.6
Consider, for example, the case of four firms manufacturing three products. We will label the firms A, B, C, and D, and the products x, y, and z. The firms may be classified into industries X, Y, and Z. If we arrange the firms and their products as shown in Figure 2-1 we can easily see the principal product of each firm and this will tell us the industry under which that firm should be classified.
Firm A clearly belongs to industry X although it also manufactures smaller quantities of y and z. Firm B belongs to industry Y, and firm C to industry Z. Firm D also belongs to industry X although it makes a wide variety of other products. If we are interested in measuring the total output of industry X it will be necessary to go to all four of the above firms, although only two of them are classified under industry X. The problem of overlapping is primarily a statistical one, encountered when we attempt to measure employment or production in individual industries. It need not trouble us at present, however, since we are only interested in developing the concept of the industry.
A number of industries, different in some respects but similar in others, may be considered collectively as an industry group.
All of the firms which specialize in the manufacture of cotton yarn, for example, make up one industry; firms which make the yarn into cloth make up another industry; and firms which dye or otherwise finish the cloth make up a third.7
A similar distinction may be made in the case of firms specializing in various stages of the production of woolen or synthetic cloth. Each group of firms constitutes a separate industry, but all of them together are members of the textile industry group. In 1945 a Standard Industrial Classification was prepared by several government agencies and published by the Bureau of the Budget. According to this classification (abbreviated as SIC) there are 20 major manufacturing industry groups.8
The operating unit of American industry is the establishment. In general, an establishment consists of a single plant or factory.9 A small firm might operate a single establishment. Larger firms, however, are often made up of two or more establishments. As corporations in this country have increased in size there has been a trend toward decentralization in decision-making. Broad policy is determined by the officers of the firm. But day-to-day management decisions are made at the level of the establishment. The establishment is also the basic unit for analytical purposes since data reported in the Census of Manufactures are based upon the establishment rather than the firm or the plant. Establishments are classified on the basis of their primary or principal products.
The classification of industries and sectors in an input-output table raises a number of technical problems which cannot be discussed here.10 The aggregation problem—or the "index number problem" as it has been known in the past—is as old as the science of economic statistics. For present purposes we will assume that the industries in our hypothetical economy are classified on the basis of their principal products, and that within any industry the products are relatively homogeneous.
Direct Purchases and Technical Coefficients
After an input-output table has been constructed for a given year, a table of input or technical coefficients can be developed from it. By a technical coefficient we mean the amount of inputs required from each industry to produce one dollar's worth of the output of a given industry. Technical coefficients are calculated for processing sector industries only, and may be expressed either in monetary or physical terms. Our hypothetical table is expressed in cents per dollar of direct purchases.
Two steps are involved in the calculation of technical coefficients: Gross output is adjusted by subtracting inventory depletion during the period covered by the table to obtain adjusted gross output. Since gross outlays in the processing sector are identical with gross outputs in this sector, adjusted gross outputs in our hypothetical economy can be computed by subtracting the entries in row 7 from the entries of row 12 of Table 2-1. The results can then be entered as a new row at the bottom of the table. The second step in the calculation of technical coefficients consists of dividing all the entries in each industry's column by the adjusted gross output for that industry.
For example, the adjusted gross output for industry A is equal to 63 (total gross outlay minus gross inventory depletion). To compute the coefficients for column 1, each entry in this column is divided by 63, which gives the entries in column 1 of Table 2-2. Similarly, the adjusted gross output for industry B is 57, and this divided into each entry in column 2 of Table 2-1 gives column 2 of Table 2-2, and so on throughout the remainder of the table.
A specific illustration may make the meaning of Table 2-2 somewhat clearer. From it we see that each dollar's worth of production in industry A will require direct purchases from other industries as follows:
Intraindustry transactions of |
16c |
Purchases by industry A from industry B of |
8c |
Purchases by industry A from industry C of |
11c |
Purchases by industry A from industry D of |
17c |
Purchases by industry A from industry E of |
6c |
Purchases by industry A from industry F of |
3c |
Total direct purchases |
61c |
If the technical coefficients remain constant from year to year, or if they can be adjusted on the basis of new information, we can calculate the amount of direct purchases required from each industry along the left-hand side of Table 2-2, as a result of an increase (or decrease) in the output of one or more of the industries listed at the top of the table. If, for example, the output of industry B were increased $100 (assuming constant technical coefficients), the direct inputs of industry B (purchases from other industries) would be increased by the following amounts:
Inputs from industry |
would be increased by |
A |
$26 |
B (intraindustry) |
7 |
C |
4 |
D |
2 |
E |
— |
F |
11 |
The total increase in the value of direct inputs due to an
increase of $100 in the output of industry B amounts to $50. |
If the input coefficients are relatively stable or if they can be adjusted on the basis of new information, the usefulness of the table of direct coefficients is apparent. By making use of such a table, the management of a typical firm in industry B could tell in advance how much it would have to buy directly from each of its supplying industries when it adds to its own total production.
Stability Conditions for the Table of Technical Coefficients
The table of direct coefficients by itself is of limited usefulness because it shows only the "first-round" effects of a change in the output of one industry on the industries from which it purchases inputs. This table forms the basis, however, for a general solution of an input-output problem which will be discussed in the next section. Because of this it is important that the table of direct coefficients meet certain stability conditions. These are that: (a) at least one column in the table add up to less than unity, and (b) that no column in the table add to more than unity. The mathematical proof of these conditions is quite complex, and no attempt will be made to demonstrate these propositions here.11 When the table is expressed in monetary terms, as is Table 2-2, it is intuitively clear that an industry cannot pay more for its inputs than it receives from the sale of its output. Also, the steps described above for computing input coefficients in the open, static model show that these conditions will be met if in each column the Sum of entries in the payments rows (less the inventory row) is greater than inventory depletion. In practice, these entries are relatively large and the stability conditions are safely met.
Direct and Indirect Purchases
Table 2-2 shows the direct purchases that will be made by a given industry from all other industries within ' the processing sector for each dollar's worth of current output. But this does not represent the total addition to output resulting from additional sales to the final demand sector. An increase in final demand for the products of an industry within the processing sector (coming from households, for example) will lead to both direct and indirect increases in the output of all industries in the processing sector. If, for example, there is an increase in final demand for the products of industry A, there will be direct increases in purchases from industries B, C, and so on. But in addition, when industry B sells more of its output to industry A, B's demand for the products of industries C, D, etc., will likewise increase. And these effects will spread throughout the processing sector.
An integral part of input-output analysis is the construction of a table which shows the direct and indirect effects of changes in final demand. It shows the total expansion of output in all industries as a result of the delivery of one dollar's worth of output outside the processing sector by each industry. A "delivery outside the processing sector" means a sale to households, investors, foreign buyers, a government agency, or any other buyer included in the final demand sector.
There are various methods for computing the combined direct and indirect effects. One is an iterative or step-by-step method which will be illustrated. No attempt will be made, however, to go through all the calculations required to construct a table by this method even for our simple hypothetical example.
Let us assume a one-dollar increase in the demand for the products of industry A. This will increase intraindustry transactions by 16c (see row 1, column 1, of Table 2-2). Thus the gross output of industry A will increase at least $1.16. But when the output of industry A increases, the firms in this industry will step up their purchases from industry B. Sales from industry B to industry A will go up an additional 9c ($1.16 X .08) as a result of the increased activity in industry A. Similarly, sales from industry C to industry A will increase 13c ($1.16 x .11), and so on down column 1 of Table 2-2.
But the indirect effects do not stop here. When industry B expands its production because of an increase in final demand for the products of industry A, the increased demand thus generated will be felt by all other industries in the processing sector which sell to industry B. We could repeat the calculations made above to include each industry in the processing sector, then by adding up all the figures a table would gradually be built up which would show the total requirements, direct and indirect, resulting from the delivery of one dollar's worth of the products of each industry in the processing sector to the final demand sector.
Fortunately for the development of input-output economics there is an alternative method which can be used with high-speed electronic computing equipment to arrive at the same results. In technical terms this method involves taking the difference between an identity matrix and the input coefficient matrix (Table 2-2), and from this computing a transposed inverse matrix.12 This table, on page 26, shows the total requirements, direct and indirect, per dollar of delivery outside the processing sector.
Table 2-3 contains some "rounding error." In computing the inverse, and in other computations to be discussed later, all figures were carried to six decimal places. To simplify the exposition, however, all numbers have been rounded off to the nearest cent.
Table 2-3
Direct & Indirect Requirements per Dollar of Final Demand*
|
|
A |
B |
C |
D |
E |
F |
A |
$1.38 |
.25 |
.28 |
.41 |
.27 |
.23 |
B |
.45 |
1.21 |
.16 |
.19 |
.12 |
.24 |
C |
.27 |
.38 |
1.38 |
.23 |
.17 |
.39 |
D |
.35 |
.25 |
.25 |
1.53 |
.65 |
.41 |
E |
.35 |
.26 |
.31 |
.39 |
1.28 |
.25 |
F |
.38 |
.35 |
.22 |
.30 |
.21 |
1.32 |
What does Table 2-3 show? In Table 2-2, we saw that each dollar's worth of production in industry A required 16c of intraindustry transactions. But it will be recalled that these were direct purchases only. Table 2-3 shows that total intraindustry transactions will rise an additional 22c-to a total of 38c-for each dollar's worth of industry A's products delivered to the final demand sector. This is because when industry A's output rises it must buy more from B, C, and the others in the table. When B sells more to A it must buy more from A, C, etc. The same holds true for all the industries in our hypothetical economy. Thus Table 2-3 shows the total dollar production directly and indirectly required from the industry at the top for each dollar of delivery to final demand by the industry at the left. Each time A sells an additional dollar's worth of goods to households, government, or some other component of final demand, B's output goes up 25c, C's output increases 28c, and so on across the first row of Table 2-3. All other rows in this table are read in the same way.
In one respect the hypothetical example is not very realistic. Most of the transactions in Table 2-3 are quite large relative to an increase of one dollar in sales to final demand by the industry at the left-hand side of the table. This is because small numbers, and few zeros, were used in the hypothetical transactions table. As a result, the ratio of interindustry transactions to final demand is quite high. An actual input-output model will have smaller values in its counterpart of Table 2-3, and there will be much greater variation throughout the table than there is in our hypothetical example.
An actual table of direct and indirect requirements shows, for that the output of the agricultural sector depends upon the demand for processed foods, tobacco, textiles, leather products, and chemicals. Thus there will be fairly large entries in the cells where the agriculture column intersects the rows of these sectors. Most apparel products are sold directly to consumers, however, and the entries in the apparel column will be small.13 In brief, some industries in the processing sector will show relatively large interindustry transactions. Such industries exhibit strong interdependence. Other industries use relatively few raw materials or intermediate products, but they may have substantial labor inputs. If households are not included in the processing sector—customarily they are not—such an industry will exhibit weak interdependence.
Stability Conditions for the Table of Direct and Indirect Coefficients
In an earlier section the stability conditions for the table of direct coefficients were given, and it was noted that in practice these conditions will generally be met. There is a fundamental condition that must also be met by the table of direct and indirect requirements (Table 2-3) known as the "Hawkins-Simon condition."14 The mathematical proof of this condition given by Hawkins and Simon is much too complex to be discussed here, but its meaning can be made intuitively clear. Basically, the Hawkins-Simon condition states that there can be no negative entries in the table of direct and indirect requirements."15 What would a negative entry in Table 2-3 mean? In essence it would mean that each time the industry with a negative entry expanded its sales to final demand, its direct and indirect input requirements would decline. Carried to the extreme this would mean that the more this industry expanded its output the less it would have to buy from other industries. This is clearly a logical contradiction and an economic absurdity.
The Hawkins-Simon condition is an important one. The appearance of one or more negative entries in a table of direct and indirect requirements per dollar of sales to final demand is a signal that something has gone wrong. There could have been a mistake in the construction of the transactions table, or computing errors in deriving the table of direct input coefficients. It is necessary then to go back, locate the cause of an obvious economic contradiction, and make the necessary adjustments or corrections.
Conclusions
1Each row of Table 2-3 shows the output directly and indirectly required from each sector at the top of the table to support the delivery of $1.00 to final demand by the sector at the left of each row. Each column shows the output required for a single sector (directly and indirectly) to support $1.00 of delivery to final demand by each of the processing sectors.
Table 2-3 is a general solution of the hypothetical input-output system. It illustrates the principle of economic interdependence. The table can be used to show how a change in demand for the output of one sector stimulates production in other sectors. It shows the end result after all of the "feedback effects" have worked themselves out. The model illustrated here is a static one. No effort has been made to introduce the time lags that would be involved in achieving the equilibrium results given in Table 2-3. The dynamics of input-output analysis will be discussed briefly in Chapters 5 and 7.
Once a general solution or table of direct and indirect coefficients has been obtained, the input-output model can be used for a variety of analytical purposes. Some of the major uses will be discussed in the following chapter.
References
CHENERY, HOLLIS B. and PAUL G. CLARK, Interindustry Economics (New York: John Wiley & Sons, Inc., 1959), pp. 13-65.
EVANS, W. DUANE and MARVIN HOFFENBERG, "The Interindustry Relations Study for 1947," The Review of Economics and Statistics, XXXIV (May 1952), 97-142.
LEONTIEF, WASSILY, et al., Studies in the Structure of the American Economy (New York: Oxford University Press, 1953).
National Bureau of Economic Research, Input-Output Analysis: An Appraisal (Princeton: Princeton University Press, 1955).
STONE, RICHARD, Input-Output and National Accounts (Paris: Organization for European Economic Co-Operation, June 1961), pp. 21-31.
Endnotes
1This leads to a number of statistical problems, however, and treatment of these problems is outside the scope of an introduction to input-output analysis. For a discussion of this aspect of the aggregation problem see Walter D. Fisher, “Criteria for Aggregation in Input-Output Analysis,” The Review of Economics and Statistics, XL (August 1958), 250-60.
2For a discussion of this and other statistical problems involved in the construction of a transactions table see W. Duane Evans and Marvin Hoffenberg, "The Interindustry Relations Study for 1947,” The Review of Economics and Statistics, XXXIV (May 1952),97-142. See also The 1947 Interindustry Relations Study, Industry Reports: General Explanations, U.S. Department of Labor, Bureau of Labor Statistics, Report 9 (March 1953) and Industry Reports: Manufacturing Methodology, BLS Report No. 10, idem (March 1953).
3An input-output table is compiled for a given time period. In practice this is usually a calendar year. There is no reason, however, why the period could not be either longer or shorter than a year.
4An illustration is given in Chapter 3.
5Charles M. Tiebout, The Community Economic Base Study (New York: Committee for Economic Development, December 1962), p. 29.
61nvestment Location, and Size of Plant (Cambridge: The University Press, 1948), p. 3.
7All of these operations may be carried on by a single firm in one or more plants. If this is the case we say it is an integrated firm, and we refer to this form of integration as vertical integration to distinguish it from the horizontal integration characteristic of many multiplant firms, such as chain stores, which specialize in one phase of economic activity.
8The 20 major industry groups are referred to as the two-digit classification. There is a further breakdown into three-digit and four-digit classifications. An example of the two-digit classification is number 22, Textile Mill Products. Under this, one three-digit classification is number 225, Knitting Mills. As part of the latter we find Full-fashioned Hosiery Mills (number 2251).
9In some cases an establishment may consist of more than one plant if these are engaged in the same kind of activity and are located within the same state.
10See for example Mathilda Holzman, "Problems of Classification and Aggregation," in Wassily Leontief, et al., Studies in the Structure of the American Economy (New York: Oxford University Press, 1952), pp. 326-59, and Richard Stone, Input-Output and National Accounts, OEEC (1961), p. 101-12.
11 For a proof in the case where all technical coefficients are positive see Robert Solow, "On the Structure of Linear Models," Econometrica, XX (January 1952), 29-46. See also Carl F. Christ, "A Review of Input-Output Analysis" in Input Output Analysis: An Appraisal (Princeton: Princeton University Press, 1955), pp. 148-49.
12The meaning of these terms and an illustrative computation are given in Chapter 7.
13See Evans and Hoffenberg, op. cit., p. 140.
14 David Hawkins and H. A. Simon, "Some Conditions of Macroeconomic Stability," Econometrica, 17 (July-October 1949), 245-48.
15See William J. Baumol, Economic Theory and Operations Analysis (Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1961), pp. 306-8.
