Input–Output Analysis
Input–Output Analysis
Theory of static inputoutput systems
Theory of dynamic inputoutput systems
Classification, aggregation, and computation
The inputoutput method is an adaptation of the neoclassical theory of general equilibrium [seeEconomic Equilibrium] to the empirical study of the quantitative interdependence between interrelated economic activities. It was originally developed to analyze and measure the connections between the various producing and consuming sectors within a national economy, but it has also been applied to the study of smaller economic systems, such as metropolitan areas or even large integrated individual enterprises, and to the analysis of international economic relationships.
In all instances the approach is basically the same: The interdependence of the individual sectors of the given system is described by a set of linear equations. The specific structural characteristics of the system are thus determined by the numerical magnitude of the coefficients of these equations. These coefficients must be determined empirically; in the analysis of the structural characteristics of an entire national economy, they are usually derived from a socalled statistical inputoutput table.
Applications. Application of the inputoutput method in empirical research requires the availability of basic statistical information. By 1963, inputoutput tables had been compiled for more than forty countries. The principal economic applications, as distinct from engineering and businessmanagement applications, have been made in such fields as economic projections of demand, output, employment, and investment for the individual sectors of entire countries and of smaller economic regions (for example, metropolitan areas); study of technological change and its effect on productivity; analysis of the effect of wage, profit, and tax changes on prices; and study of international and interregional economic relationships, utilization of natural resources, and developmental planning.
Some of these applications require construction of special purpose inputoutput models. A great variety of special models is used, for instance, in the analysis of interregional relationships and in the study of problems of developmental planning.
Inputoutput tables
An inputoutput table describes the flow of goods and services between all the individual sectors of a national economy over a stated period of time—say, a year. An example of an inputoutput table depicting a threesector economy is shown in Table 1. The three sectors are agriculture, whose total annual output amounted to 100 bushels of wheat; manufacturing, which produced 50 yards of cloth; and households, which supplied 300 manyears of labor. The nine entries inside the main body of the table show the intersectoral flows. Of the 100 bushels of wheat turned out by agriculture, 25 bushels were used up within the agricultural sector itself, 20 were delivered to and absorbed, as one of its inputs, by manufacturing, and 55 were taken by the household sector. The second and the third rows of the table describe in the same way the allocation of outputs of the two other sectors.
The figures entered in each column of the main body of the table thus describe the input structure
Table 5  Examlpe of an inputoutput table (in physical units)  

Into From  Sector 1: Agriculture  Sector 2: Manufacturing  Sector 3: Households  Total output 
Sector 1: Agriculture  25  20  55  100 bushels of wheat 
Sector 2: Manufacturing  14  6  30  50 yards of cloth 
Sector 3: Households  80  180  40  300 manyears of labor 
of the corresponding sector. In producing 100 bushels of wheat, agriculture absorbed 25 bushels of its own products, 14 yards of manufactured goods, and 80 manyears of labor received from the households. In producing 50 yards of cloth, manufacturing absorbed 20 bushels of wheat, 6 yards of its own products, and 180 manyears of labor. In their turn, the households used their income, which they received for supplying 300 manyears of labor, to pay for 55 bushels of wheat, 30 yards of cloth, and 40 manyears of direct services of labor, which they consumed.
All entries in this table are supposed to represent quantities, or at least physical indexes of quantities, of specific goods or services. A less aggregative, more detailed inputoutput table describing the same national economy in terms of 50, 100, or even 1,000 different sectors would permit a more specific qualitative identification of the individual entries. In a larger table, manufacturing would be represented not by one but by many distinct industrial sectors; its output—and consequently also the inputs of the other sectors—would be described in terms of yards of cotton cloth and tons of paper products, or possibly yards of percale, yards of heavy cotton cloth, tons of newsprint, and tons of writing paper.
Inputoutput tables and income accounts
Although in principle the intersectoral flows, as represented in an inputoutput table, can be thought of as being measured in physical units, in practice most inputoutput tables are constructed in value terms. Table 2 represents a translation of Table 1 into value terms on the assumption that the price of wheat is $2 per bushel, the price of cloth is $5 per yard, and the price of services supplied by the household sector is $1 per manyear. Thus, the values of the total outputs of agriculture, manufacturing, and households are shown in Table 2 as $200 (= 100 x $2), $250 (= 50 x $5). and $300 (=300x$l), respectively. The last row shows the combined value of all outputs absorbed by each
Table 2  Examlpe of an inputoutput table (in dollars)  

Into From  Sector 1: Agriculture  Sector 2: Manufacturing  Sector 3: Households  Total output 
Sector 1: Agriculture  50  40  110  200 
Sector 2: Manufacturing  70  30  150  250 
Sector 3: Households  80  180  40  300 
Total input  200  250  300 
of the three sectors. Such column totals could not have been shown on Table 1, since the physical quantities of different inputs absorbed by each sector cannot be meaningfully added.
The inputoutput table expressed in value terms can be interpreted as a system of national accounts. The $300 showing the value of services rendered by households during the year obviously represents the annual national income. It equals the total of the income payments (shown in the third row) received by households for services rendered to each sector; it also equals the total value of goods and services (shown in the third column) purchased by households from themselves and from the other sectors. To the extent that the column entries (showing the input structure of each productive sector) cover current expenditures but not purchases made on capital account, the capital expenditures—being paid out of the net income—should be entered in the households’ column.
All figures in Table 2, except the column sums shown in the bottom row, can also be interpreted as physical quantities of the goods or services to which they refer. This requires only that the physical unit in which one measures the entries in each row be redefined as the amount of output of the particular sector that can be purchased for $1 at prices that prevailed during the interval of time for which the table was constructed.
Input coefficients
Let the national economy be subdivided into n + 1 sectors. Sectors 1,..., n are industries—that is, producing sectors—and sector n + 1 is the final demand sector, represented in inputoutput Tables 1 and 2 by households. For purposes of mathematical manipulation, the physical output of sector i is usually represented by x_{i}, and the symbol x_{i j} stands for the amount of the product of sector i absorbed as an input by sector j. The quantity of the product of sector i delivered to the final demand sector, x_{i.n+1}, is usually identified in short as y_{i}
The quantity of the output of sector i absorbed by sector j per unit of j’s total output is represented by the symbol a_{i j} and is called the input coefficient of sector i into sector j. Thus,
A complete set of the input coefficients of all sectors of a given economy arranged in the form of a rectangular table, corresponding to the inputoutput table of the same economy, is called the structural matrix of that economy. Table 3 presents the structural matrix of the economy whose flow
Table 3  Structural matrix corresponding to the inputoutput table of Table 1  

From Into  Sector 1: Agriculture  Sector 2: Manufacturing  Sector 3: Households 
Sector 1: Agriculture  0.25  0.40  0.183 
Sector 2: Manufacturing  0.14  0.12  0.100 
Sector 3: Households  0.80  3.60  0.133 
matrix is shown in Table 1. The flow matrix constitutes the usual, although not necessarily the only possible, source of empirical information on the input structure of the various sectors of an economy. The entries in Table 3 are computed, according to equations (1), from the figures presented in Table 1—for example, a_{11} = 25/100 = 0.25, and a_{12} = 20/50 = 0.40.
In practice, the structural matrices are usually computed from inputoutput tables described in value terms, such as Table 2. In any case, the input coefficients must be interpreted, for analytical purposes described below, as ratios of two quantities measured in physical units. To emphasize this fact, we derived the structural matrix in this example from Table 1, not Table 2.
Theory of static inputoutput systems
The balance between the total output and the combined input uses of the product of each sector, as shown in tables 1 and 2, can be described by the following set of n equations:
A substitution of equations (1) into (2) yields n general equilibrium relationships between the total outputs, x_{1}, x_{2},..., x_{n}, of the producing sectors and the final bill of goods, y_{1}, y_{2},...,y_{n}, absorbed by households, government, and other final users:
If the final demands, y_{l} y_{2},..., y_{n}, that is, the quantities of the different goods absorbed by households and any other sector whose outputs are not represented by the variables appearing on the lefthand side of equations (3), are given, the system can be solved for the total outputs, x_{1},..., x_{2},..., x_{n}.
The general solution of these equilibrium equations for the unknown x’s in terms of the given y’s can be presented in the following form:
The constant A_{i j} indicates by how much x_{i} would increase if y_{j} were increased by one unit. An increase in y_{i} would affect sector i directly (and also indirectly) if i = j, but even if i ≠ j, sector i is affected indirectly, since it has to provide additional inputs to all other sectors that must contribute directly or indirectly to producing the additional y,_{i}. From the computational point of view, this means that the magnitude of each coefficient A_{i j} in the solution (4) depends, in general, on all the input coefficients appearing on the lefthand side of the system of equilibrium equations, (3). In mathematical language, the matrix
is the inverse of the matrix
The computation involved in finding the solution of (3) is called the inversion of the coefficient matrix. The inverse of the matrix
based on Table 3 is
(Each element of the inverse has been rounded to four decimal places.) When inserted into (4), this yields two equations—namely,
which permit us to determine the total outputs, x_{1}, and x_{2}, of agriculture and manufacturing corresponding to any given combination of the deliveries of their respective products, y_{1} and y_{2,} to the exogenous household sector. For example, setting y_{1}=55 and y_{2}=30, we find that x_{1},=100 and x_{2}=50, which agrees with the figures in Table 1. Only if all the A_{i j} are nonnegative will there necessarily exist a set of positive total outputs for any given set of final deliveries. A sufficient condition for the nonnegativity of the A_{i j} is that in the structural matrix
the sum of the coefficients in each column (or in each row) be not larger than one and that at least one of these column (or row) sums be smaller than one. A national economy whose structural matrix does not satisfy this condition will be unable to sustain itself—that is, the combined input requirements of all sectors in such an economy would exceed the combined productive capabilities of the sectors.
When the structural matrix of a national economy is derived from a set of empirically observed value flows, the condition stated above is generally found to be satisfied.
In applying this criterion to a given structural matrix, it is useful to keep in mind that by doubling the size of the physical unit used in measuring the output of a particular sector, one can double the magnitude of all the technical input coefficients that make up the corresponding row and reduce to onehalf their previous size all entries in the corresponding column.
In an open inputoutput system, households are usually treated as an exogenous sector—that is, total output of households, x_{n+1}, which is total employment, usually does not appear as an unknown variable on the lefthand side of system (3) and on the righthand side of the solution (4). After the outputs of the endogenous sectors have been determined, total employment can be computed from the following equation:
The technical coefficients,a_{n+1,1,} a_{n+1,2,} ..., a_{n+1,n,} are the inputs of labor absorbed by various industries (sectors) per unit of their respective outputs; j/"+, is the total amount of labor directly absorbed by households and other exogenous sectors. The employment equation for the threesector system whose structural matrix is shown in Table 3 is
Households are not always treated as an exogenous sector. In dealing with problems of income generation in its relation to employment, the quantities of consumer goods and services absorbed by households can be considered (in a Keynesian manner) to be structurally dependent on the total level of employment, just as the quantities of coke and ore absorbed by blast furnaces are considered to be structurally related to the amount of pig iron produced by them. With households shifted to the left side of equations (2) and (4), the exogenous final demand appearing on the right side will contain only such items as government purchases, exports, and, in any case, additions to or reductions in stocks of goods—that is, real investment or disinvestment.
When all sectors and all purchases are considered to be endogenous, the inputoutput system is called closed. A static system cannot be truly closed since endogenous explanation of investment or disinvestment requires consideration of structural relationships between inputs and outputs that occur during different periods of time (see “Theory of dynamic inputoutput systems,” below).
Exports and imports
In an inputoutput table of a country or a region that trades across its borders, exports can be entered as positive components and imports as negative components of final demand. If the economy described in Table 1 ceased to be selfsufficient and started, say, to import 20 bushels of wheat and to export 8 yards of cloth, while letting households consume the same amounts of both products as before, a new balance between all inputs and outputs would be established, which is described in Table 4.
The input coefficients of the endogenous sectors, and consequently also the structural matrix of the system and its inverse, remain the same as they were before. To form the new column of final demand, we have to add to the quantity of each good absorbed by households the amount that was exported less the amount that was imported. Defining E_{i}, i=1,...n, as net exports (exports minus imports) of good i, and redefining x_{i,n+1,}i=1,...,n, as final demand for good i by households only, we have
The sectoral outputs can then be derived (see “Theory of static inputoutput systems,” above) from the general solution (4). For our numerical example, we can use equations (5) directly. The total labor requirement of the economy—300 manyears—remains in this particular case unchanged after the economy enters foreign trade, because the total direct and indirect labor content of the 20
Table 4  Inputoutput table of the economy described in Table 1 with foreign trade added (in physical units)  

Into  FINAL DEMAND  
From  Sector 1: Agriculture  Sector 2: Manufacturing  Sector 3: Households  Exports (+) or Imports ()  Total  Total output 
Sector 1; Agriculture  19.04  22.12  55  20  35  76.16 bushels of wheat 
Sector 2: Manufacturing  10.66  6.64  30  +8  38  55.30 yards of cloth 
Sector 3: Households  60.93  199.07  40  40  300 manyears of labor 
bushels of imported wheat happens to be equal to the labor content of the 8 yards of exported cloth.
If the imports of good i (that is, the negative E_{i}) happen to exceed the final domestic consumption of that good, the corresponding “net” final demand, y_{i}, will turn out to be negative. As y_{i} diminishes, the total output of all sectors and in particular the total output of sector i must, ceteris paribus, diminish. For some value of y_{i}, the output of sector i will be reduced to zero, which means that the entire direct and indirect demand for that particular commodity will be covered by imports. The industry will then be eliminated from the endogenous part of the inputoutput table. The imports of such goods are called noncompeting, particularly when, as in the case of coffee and certain minerals, even a large increase in domestic demand does not call forth domestic production of the good. The magnitude of domestic demand for a good that can be satisfied without domestic production of the good can be computed in the same way that the total demand for labor is computed from equation (6).
Prices
Prices in an open static inputoutput system are determined from a set of equations that state that the price that each productive sector receives must equal its total payments per unit of output for inputs purchased from itself and from the other industries, plus a “value added” per unit of output, which essentially represents payments made to the exogenous sectors. Defining p_{i} as the price received by industry i and V, as the value added by industry i per unit of output, these equations are
Each equation describes the balance between the price received and payments made by each endogenous sector per unit of its product; V_{i} represents the payments made by sector i, per unit of its product, to all exogenous (that is, final demand) sectors. The V; usually consist of wages, interest on capital and entrepreneurial revenues credited to households, taxes paid to the government, and so on.
The solution of the price equations (9) permits the determination of prices of all products from given values added by each sector. The solution can be written
The constant, A_{i j}, measures the dependence of the price of the product of sector j on the value added by sector i.
The coefficients a_{i j} appearing in each row of the output equations (3) make up the corresponding column of coefficients appearing in the price equations (9); the coefficients A_{ij} appearing in each row of the output solution (4) make up the corresponding column of coefficients in the price solution (10).
Only if all the A_{ij} in the price solution are nonnegative will there necessarily exist positive prices enabling each sector to balance exactly its inputoutput accounts in value terms for any given set of positive values added. Since A_{ji} in the price solution equals A;, in the output solution, this condition is the same as that needed to assure positive outputs for any given set of final demands.
Inserting into (10) the inverse computed for the example used above, we have:
From tables 2 and 3 we can see that in our example the values added (that is, the wages) per unit of output by agriculture and manufacturing amounted to $0.80 and $3.60. According to equations (11), this yields p_{1},= $2 and p_{2} = $5, which are the prices of agricultural and manufactured products used in deriving from Table 1 the value figures presented in Table 2.
The internal consistency of the price and quantity relationships within an open inputoutput system is confirmed by the following identity derived from equations (4) and (9):
On the lefthand side is the sum of the values added paid out by the endogenous sectors to the exogenous sectors of the system; on the righthand side is the sum of the values (quantities times prices) of products delivered by all endogenous sectors to the final (exogenous) demand sector. This identity confirms, in other words, the accounting identity between the national income received and the national income spent, as shown in Table 2.
Theory of dynamic inputoutput systems
Dynamic inputoutput theory grows out of the static theory through consideration of intersectoral dependences involving lags or rates of change of variables over time. Structural relations between stocks and flows of goods constitute the theoretical basis for the inputoutput approach to empirical analysis of the accumulation process and of developmental planning.
The stock of goods produced by sector i that sector j must hold per unit of its full capacity output is called the capital coefficient of good i in sector j and is usually designated by b_{i j}. A column of capital coefficients indicating the stocks of buildings, machinery, inventories of raw materials and spare parts, and other supplies used by a particular sector describes what may be called that sector’s real capital structure. The matrix
describes the real capital structure of a national economy as a whole.
The current inputs and capital stocks required to produce the output of a particular industry might have to be utilized during the period in which the output is produced, or they might have to be made available and used, at least in part, one or more periods before that. An analytically general and at the same time realistic description of dynamic inputoutput relationships can be given if separate variables are used to designate the flows of inputs and of outputs absorbed or produced by the same industry in different years. The balance between the output and the available capacity of sector i in a particular year t can, for example, be described by a linear differential equation involving
structural interrelationships between the inputs and the outputs of the various sectors and the rates of change of the inputs and outputs. The equation is
where x_{i}(t) is the output and x_{i}(t) the rate of change of output of sector i at time t. If the time path of all final demands and the levels of all outputs at an initial point of time are assumed to be given, a system of n such linear differential equations, one for each sector, can be solved for all the n outputs. The solution gives the level of each output, Xi(t), at any point of time—that is, for any t. Although this approach to the study of dynamic inputoutput relationships offers certain theoretical advantages, most empirical work in the field is conducted in terms of discrete period analysis based on systems of difference equations of the following kind:
(Superscripts indicate the time period to which the variables refer.) The first n + 1 terms of this equation are identical to the lefthand side of the corresponding equations in the static system (3). The next n terms represent the deliveries from sector i to itself and to all other sectors in response to needs for additional productive capacities, which in turn depend on the differences between current and future outputs. These changes in outputs multiplied by the appropriate capital coefficients, i.e., productive stock required per unit of additional output, give the magnitude of the deliveries on the capital account.
In a static formulation investment in additional productive capacity is treated as a component of the given final demand, but in a dynamic analysis investment must be explained and cannot be considered as having been fixed beforehand. Hence the final demand for the product of industry i in period t, y(^{t})_{1} now comprises deliveries to households, government, and so on, but no additions to the stock of productive capital.
Equation (14) is a basic building block that can be used to construct a system describing intertemporal inputoutput relationships between the different sectors of a particular economy over an interval of time containing any number of years. The set of six equations (15) spans the intersectoral relationships within a threesector economy, of which only two are endogenous, over a period of three years.
Even if the final deliveries of both goods in each of the three years—that is, all the y’s —are considered given, the six equations would still contain eight unknowns. In the last two equations, which describe the inputoutput balance of both industries in the third year, the amounts allocated to investment are seen to depend on the output levels of the next, that is, the fourth year. Two of the x’s must also be given before one can proceed to solve the system of six equations for the remaining six unknowns. Thus, for example, if the outputs of both sectors in the first year, i.e., and are
given, the system can be solved for the outputs of the second, third, and fourth years as functions of the six y’s, that is, as functions of the annual deliveries of both goods over the period of the first three years.
Instead of being anchored in the first year and solved for the next three years, the system can be used in reverse; that is, after having fixed the output of both endogenous sectors in the last (fourth) year, the system can be solved so as to display the
dependence of production on the final consumption levels over the period of the first three years.
The numerical example of a threesector economy presented above can now be extended to demonsrate the solution of a dynamic inputoutput system. The flow coefficients shown in Table 3 must first be supplemented by a corresponding matrix of capital coefficients. Let it be
The entries in the first column show that 0.20 units of agricultural goods and 0.01 units of manufactured goods would have to be added to the capital stocks held by agriculture if the productive capacity of that sector were to be stepped up so as to enable it to increase its annual production by one unit. The two figures in the second column supply analogous information on the capital structure of manufacturing.
With the appropriate number inserted for all the a’s and b’s, system (15) takes on the form shown in (16).
The terms containing and are transferred to the righthand side in the last two equations, because in the general solution shown below these outputs will be considered given.
For purposes of computation, the fourth year’s total outputs of both sectors were assumed to equal those entered in the last column of Table 1, that is, bushels of wheat, and yards of cloth. The solution for the remaining unknown outputs is given in equations (17).
Equations (17) represent a general numerical solution of the dynamic inputoutput system (16) in the same sense in which the inversion of the flow coefficient matrix incorporated in (5) yields a general solution of the original static system. These six equations describe explicitly the dependence of the total outputs of both industries in the first, second, and third years on the levels of final deliveries of both products in the first, second, and third years. To compute the sequence of annual outputs corresponding to any given sequence of annual deliveries to final demand, one has only to assign appropriate numerical magnitudes to all the y’s on the righthand side of each equation, perform the necessary multiplications, and sum up the results for each line.
As a simple check on the internal consistency of this general solution, the amounts of 55 bushels of wheat and 30 yards of cloth actually allocated to households in Table 1 can be substituted, respectively, for y_{1} and y_{2} in each of the six equations. After the performance of appropriate multiplications and additions the result would show that in this particular case the total output of wheat would be maintained at a constant level of 100 bushels and the total annual output of manufactured products at a constant level of 50 yards of cloth throughout the entire period. Since from the first year on nothing would have been added to, or subtracted from, its productive stocks, the economy would in this particular case maintain itself without expansion or contraction in either sector.
The same analytical procedure can be used to construct and solve an open dynamic inputoutput system incorporating structural change. Both the analytical approach and the numerical manipulations remain essentially the same; only the magnitude of the a’s and b’s inserted in each equation would have to be distinguished by appropriate time subscripts, thus permitting the numerical values of these flow and capital coefficients to change from year to year.
Since no outputs can be negative, only those sequences and combinations of final deliveries that turn out to require nonnegative total outputs in all sectors for all years can in fact be realized within the framework of a particular dynamic structure. The presence of many negative constants in the general solution of the type presented above indicates how narrow the range of alternative developmental paths open to a particular economy might actually be.
The major deficiency of the simple inputoutput approach to the description of dynamic processes presented here is its inability to handle situations in which one or more industries operate over significantly long periods of time under conditions of excess capacity. Stocks of fixed capital invested in one sector cannot as a rule be dismantled and shifted to use in another sector. Thus idle capacities—that is, excess stocks—are bound to appear whenever the rate of output in a particular industry goes down from one year to another.
To take account of idle stocks within the framework of a dynamic inputoutput system, the artificial concept of capacityholding or stockholding activity has to be introduced. For example, whenever the output of the manufacturing sector goes down from one year to the next by, say, one hundred units, the dummy activity of “holding idle manufacturing capacity” is assumed to increase by the same amount. Since the capital coefficients of that new activity are by definition identical to those of the manufacturing industry itself, the total stocks held by the manufacturing sector remain unchanged despite the fact that its annual output has been reduced.
The introduction of this new analytical device transforms the simple dynamic inputoutput scheme described above into a much more complex linear programming model [seeprogramming].
A linear programming formulation can also be appropriately used in the analysis of dynamic (and also static) inputoutput systems in which two or more alternative techniques of production, each described by a different column of input coefficients, are available for some sectors or in which exogenously fixed supplies of several different primary factors of production impose limitations on the attainable combinations of total outputs.
Any linear programming solution of a dynamic inputoutput system will contain as a rule as many nonzero variables as there are equations in the system; the objective function essentially determines which variables should be reduced to zero and thus be eliminated from the set of balance equations so as to make the number of remaining variables equal to the number of equations. The simpler nonmaximizing types of solutions reduce the number of unknowns through deliberate decisions based on direct empirical evidence or pragmatic assumptions.
Classification, aggregation, and computation
The greater is the number of sectors included in an inputoutput table, the more detailed can be the statement of the final results in analytical applications. The majority of inputoutput tables now in use contain from 10 to 100 sectors; however, tables with several hundred sectors have been constructed and used. As better statistical information becomes available, the trend toward larger, more detailed tables becomes more pronounced.
The classification of industries in inputoutput analysis is guided by consideration of technological homogeneity, and the classification of households by structural similarity of expenditure patterns. The problem of aggregation arises when the size of an inputoutput matrix is reduced through combination of some of its columns and of the corresponding rows. The relationship between the properties of the aggregated and of the nonaggregated matrix depends upon the position within the latter of the input columns that are consolidated. Under certain ideal conditions, the consolidated inverse of the original matrix is identical to the inverse of the consolidated matrix. When these conditions are not fully but approximately satisfied, the forementioned identity is of course only approximately realized. [seeAggregation.]
Most applications of the inputoutput method require numerical solutions of large systems of linear equations, inversions of large matrices containing up to several hundred rows and columns, and computationally very similar solutions of large linear programming problems.
An example of the special computational procedures is the iterative procedure used to invert the matrix (I –A ), where A is the structural matrix of coefficients aij and I is the identity matrix, that is,
The inverse of (I —A ), denoted by (I —A )^{1}, can be written as the sum of an infinite series of increasing powers of A, namely,
(I  A)^{1} = I + A + A^{2} + A^{3} + ...
This series is convergent if the structural matrix A satisfies the conditions stated in the section “Theory of static inputoutput systems”—that is, if the national economy described by A is capable of being selfsustaining. A matrix of this kind also possesses another property that is very useful in numerical inputoutput computations: small variations (caused, for example, by observational errors) in the magnitude of its elements can cause only small changes in any element of (I —A )^{1}.
Wassily Leontief
[The inputoutput method is used widely in the development and empirical application of many areas of economics. To gain an appreciation and understanding of its uses, the reader should consultEconomic Growth,article onMathematical Theory; International Trade, article onmathematical Theory; Spatial Economics, article OnThe General Equilibrium Approach. For a general discussion of production relations, which are the major ingredients of inputoutput analysis, seeProduction.]
BIBLIOGRAPHY
Almon, Clopper Jr. 1966 The American Economy to 1975: An Interindustry Forecast. New York: Harper.
Artle, Roland 1959 Studies in the Structure of the Stockholm Economy: Towards a Framework for Projecting Metropolitan Community Development. Stockholm School of Economics, Business Research Institute.
Belgium, Institut National De Statistique 1964–1965 Tableau “entriessorties” de la Belgique en 1959. 7 vols. Brussels: The Institute.
Berman, Barbara R.; Chinitz, Benjamin; and Hoover, Edgar M. 1960 Projection of a Metropolis: Technical Supplement to the New York Metropolitan Region Study. Cambridge, Mass.: Harvard Univ. Press.
Bruno, Michael 1962 Interdependence, Resource Use and Structural Change in Israel. Jerusalem: Bank of Israel.
Carter, A. P. 1967 Changes in the Structure of the American Economy: 194719581962. Review of Economics and Statistics 49: May.
Chenery, Hollis B.; and Clark, Paul G. 1959 Interindustry Economics. New York: Wiley.
Conference On InterIndustrial Relations, DrieberGen, Netherlands,1950 1953 InputOutput Relations: Proceedings. Edited by the Netherlands Economic Institute. Leiden (Netherlands): Kroese. → This conference, held in September 1950, was the first international conference on the subject.
Conference On Research In Income And Wealth 1955 InputOutput Analysis: An Appraisal. Studies in Income and Wealth, Vol. 18. Princeton Univ. Press.
Cornfield, Jerome; Evans W. Duane; and Hoffenberg, Marvin 1947 Full Employment Patterns, 1950. Monthly Labor Review 64:163190, 420432.
France, Institut National De La Statistique Et Des Etudes Economiques 1966 Methodes de la comptabilite nationale—cadres et definitions de la base 1959. Paris: The Institute. → Contains interindustry relations tables for 1959.
France, Institut National De La Statistique Et Des Etudes Economiques,Etudes de comptabilite nationale [1965]: No. 8. → A special number of “Etudes et conjoncture.” Devoted to the discussion of inputoutput tables and their applications by the French government in economic planning.
Great Britain, Central Statistical Office 1961 InputOutput Tables for the United Kingdom, 1954. London: H.M. Stationery Office.
Hatanaka, Michio 1960 The Workability of InputOutput Analysis. Ludwigshafen am Rhein (Germany): Fachverlag fµr Wirtschaftstheorie und Okonometrie.
Hirsch, Werner Z. 1959 Interindustry Relations of a Metropolitan Area. Review of Economics and Statistics 41:360369.
Hoglund, Bengt; and Werin, Lars 1964 The Production System of the Swedish Economy: An InputOutput Study. Stockholm: Almqvist & Wiksell.
IfoInstitut Fur Wirtschaftsforschung 1964 InputOutput Tabellen fµr die westdeutsche Volkswirtschaft mit vorldµfigen Ergebnissen fµr 1961. 2 vols. Munich: The Institute.
International Conference On InputOutput Techniques, Geneva, 19611963 Structural Interdependence and Economic Development: Proceedings. Edited by Tibor Barna. New York: St. Martins. → This conference, held under the auspices of the United Nations, was the third international conference on the subject.
Italy, Istituto Centrale Di Statistica 1965 Primi studi sulle interdipendenze settoriali dell’ economia italiana. Note e relazioni, No. 27. Rome: The Institute. → Contains an inputoutput table of Italy for the year 1959.
Japan, Gyosei Kanricho, Tokei Kijunkyoku 1963 Report on the Compilation of Japan’s Interindustry Relations Table for 1960. Tokyo: The Agency. → In Japanese, with an English summary.
Japan, Keizai Kikakucho 1965 Medium Term Economic Plan 19641968. Tokyo: The Agency. → Contains inputoutput tables for Japan for 1950, 1960, and projections to 1968.
Johansen, Leif 1960 A Multisectoral Study of Economic Growth. Amsterdam: North Holland Publishing.
Kossov, V. V. 1964 Mezhotraslevoi balans (Interindustry Balance). Akademiia Nauk Sssr, Tsentral’ nyi Ekonomikomatematicheskii Institut. Moscow: Nauka. → A textbook for inputoutput analysis.
Leontief, Wassily (1941) 1951 The Structure of the American Economy; 1919–1939: An Empirical Application of Equilibrium Analysis. 2d ed. New York: Oxford Univ. Press.
Leontief, Wassily 1966 InputOutput Economics. New York: Oxford Univ. Press. → Contains 11 essays by the author published over a period of 20 years.
Leontief, Wassily et al. 1953 Studies in the Structure of the American Economy. New York: Oxford Univ. Press.
Manne, A. S. et al. 1965 A Consistency Model of India’s Fourth Plan. Sankhya: The Indian Journal of Statistics Series B [1965]:57144. → Also published by the Mit Center for International Studies and the Planning Unit of the Indian Statistical Institute. Contains an interindustry transactions matrix of India for the fiscal year 19601961 and projections for the year 19701971.
Miernyk, William H. 1965 The Elements of InputOutput Analysis. New York: Random House. → A presentation of the essentials of inputoutput analysis in nonmathematical terms.
Miernyk, William H. et al. 1965 The Impact of Space and Space Related Activities on a Local Economy: A Case Study of Boulder, Colorado. Part 1: The InputOutput Analysis. Springfield, Va.: U.S. Department of Commerce, Clearinghouse for Federal Scientific and Technical Information.
Morgenstern, Oskar (editor) 1954 Economic Activity Analysis. New York: Wiley.
Netherlands (Kingdom) Centraal Bureau Voor De StaTistiek 1963 InputOutput Tables for the Netherlands. Statistical Studies, No. 16. Zeist (Netherlands): Haan. → Contains inputoutput tables of the Netherlands for 19481957.
Paretti, Di V. et al. 1960 Struttura e prospettive dell’ economia energetica italiana. Turin (Italy): Einaudi.
Pisa, Universita, Facolta Di Economia E Commercio 1956 The Structural Interdependence of the Economy: Proceedings of an International Conference on InputOutput Analysis, Varenna 27 June10 July 1954. Edited by Tibor Barna. New York: Wiley. → This conference, held at Varenna, Italy, in June 1954, was the second international conference on the subject.
Research Project On The Structure Of The American Economy 1953 Studies in the Structure of the American Economy: Theoretical and Empirical Explorations in InputOutput Analysis, by Wassily Leontief et al. New York: Oxford Univ. Press.
Riley, Vera; and Allen, Robert L. 1955 Interindustry Economic Studies. Baltimore: Johns Hopkins Press. → An annotated bibliography.
Scientific Conference On Statistical Problems, Budapest, 29611962 InputOutput Tables: Their Compilation and Use. Edited by Otto Lukacs. Budapest: Akademiai Kiado. → Contains a description of the application of inputoutput analysis for planning purposes in the Soviet Union and other socialist countries.
Statistical Office Of The European Economic Communities 1965 Tableaux “entreessorties” pour les pays de la Communaute Economique Europienne (seconds version). Brussels: The Office. → Contains 1959 inputoutput tables for Belgium, France, Germany (Federal Republic including West Berlin), Italy, and the Netherlands.
Stone, Richard 1961 InputOutput and National Accounts. Paris: Organization for European Economic Cooperation.
Taskier, Charlotte E. 1961 InputOutput Bibliography: 19551960. New York: United Nations. → An annotated bibliography.
Taskier, Charlotte E. 1964 InputOutput Bibliography, 19601963. New York: United Nations.
Theil, Henri 1966 Applied Economic Forecasting. Chicago: Rand McNally. → Discusses inputoutput methods and the results of economic predictions based on inputoutput tables of the Netherlands.
Tilanus, Christiaan B. 1965 InputOutput Experiments: The Netherlands 19481961. Rotterdam: Universitaire Pers.
U.S. Bureau Of Labor Statistics 1966 Projections 1970: Interindustry Relationships—Potential Demand —Employment. Bulletin 1536. Washington: Government Printing Office.
U.S. Congress, Joint Economic Committee 1966 New Directions in the Soviet Economy: Studies Prepared for the Subcommittee on Foreign Economic Policy. Part IiA: Economic Performance. 89th Congress, 2d Session. Washington: Government Printing Office. → See especially “The 1959 Soviet InputOutput Table,” by V. G. Treml.
U.S. Department Of Commerce, Office Of Business Economics 1965 The Transaction Table of the 1958 InputOutput Study and Revised Direct and Total Requirements Data. Survey of Current Business 45, no. 9:3349, 56.
Wonnacott, Ronald J. 1961 CanadianAmerican Dependence: An Interindustry Analysis of Production and Prices. Amsterdam: NorthHolland Publishing.
Yamada, Isamu 1961 Theory and Application of Interindustry Analysis. Tokyo: Kinokuniya.
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