I

Input–Output Analysis

Wassily Leontief

Input–output analysis is a practical extension ofthe classical theory of general interdependencewhich views the whole economy of a region, acountry and even of the entire world as a singlesystem and sets out to describe and to interpret itsoperation in terms of directly observable basicstructural relationships.

Wassily Leontief, a Russian-born Americaneconomist, started the construction of the firstinput–output tables of the American economywhen he joined the faculty at Harvard Universityin 1932. These tables, for the years 1919 and1929, were published together with the formula-tion of a corresponding mathematical model andnumerical computation based on it in 1936 and1937. Thus from the very outset the newmethodology – for the development of whichLeontief was awarded 40 years later a Nobelprize – emphasized the importance of closemutual alignment of systematic fact finding andtheoretical formulation.

In the late Twenties Leontief spent 3 years atthe Institute for the World Economy at the

University of Kiel (Germany) on derivation ofstatistical supply and demand curves. That earlyexperience with curve fitting taught him not torely on indirect statistical inference as a substitutefor painstaking direct factual inquiry.

With its emphasis on disaggregation permit-ting detailed quantitative description of the struc-tural properties of all component parts of a giveneconomic system the input–output analysismoved in a direction directly opposite to that ofthe highly aggregative approach that began,approximately at the same time, to dominate fun-damental economic research under the powerfulinfluence of the Keynesian paradigm presented inKeynes’s General Theory. Hand-in-hand with adisaggregated data base went an equallydisaggregated theoretical model, the empiricalimplementation of which involved numericalcomputations exceeding in their complexity andscale anything that had been carried out up to thattime along these lines in economics or any othersocial science.

The limited capabilities of the Wilbur linearanalog computer used in the first large scale com-putation forced Leontief to scale down his prob-lem by neglecting some of the detail contained inthe disaggregated data base. Subsequent rounds ofcomputation were carried out at first on HowardAiken’s, Mark I and Mark II computers, and lateron the early electronic machines. Thirty yearslater the race between the economists and statisti-cians compiling more and more detailed factualinformation, and engineers constructing more and

This chapter was originally published in The NewPalgrave: A Dictionary of Economics, 1st edition, 1987.Edited by John Eatwell, Murray Milgate and PeterNewman

# The Author(s) 1987Palgrave Macmillan (ed.), The New Palgrave Dictionary of Economics,DOI 10.1057/978-1-349-95121-5_1072-1

more powerful machines, was won hands downby the latter.

A standard input–output table contains squarearrays of figures arranged in chess-board fashion.Each row and the corresponding column bears thename of one particular sector, say, steel industry,automobile industry, electric power utilities,advertising services, and so on. Each individualentry represents the amount (which can, of course,be zero) of the commodity or service produced bythe sector – identified by the name of the row inwhich it appears – that has been delivered to thesector named at the head of the column in whichthat entry is placed. The small schematicinput–output table presented below (Table 1)describes intersectoral transactions between thethree sectors of the elementary economydescribed by it.

Examining these figures, one finds that to pro-duce one bushel of wheat, agriculture requires0.25 bushels of wheat (seed), 0.14 tons of steeland 0.80 man years of labour. A similar set oftechnical coefficients – 0.40 units of agriculturaland 0.12 of manufactured products – describe theinput requirements for production of one yard ofcloth. Listed column by column these sets oftechnical input coefficients represent the struc-tural matrix at the producing part of the giveneconomy.

While the figures in Table 2 were derived fromthe input–output table (Table 1), estimates of themagnitudes of the technical coefficients could be,and in some instances actually are obtaineddirectly from technical, engineering data sources.

The structural matrix of an economy provides abasis for determination of total sectoral output aswell as magnitude of inter-sectoral transactions thatwould enable the producing sectors to deliver tohouseholds and to other so-called final users a spec-ified ‘bill of goods’. Considering the vector of finaldemand, consisting of 55 bushels of wheat and30 yards of cloth, as given, the following set ofbalanced equations can be used to determine thetotal amounts of wheat (x1), of cloth (x2), as well asthe total amount of labour (L) needed to balanceunder these particular technological conditions theoutputs and inputs of both producing sectors,

1� 0:25ð Þx� 0:14x2 ¼ y1�0:40x1 þ 1� 0:12ð Þx2 ¼ y2

(1)

The general solution of these two equations:

1:457y1 þ 0:662y2 ¼ x10:232y1 þ 1:242y2 ¼ x2

(2)

permits us to compute the total levels of output ofwheat, xi and cloth, x2 required directly and indi-rectly to satisfy any given vector (y1, y2) of ‘finaldemand’.

An increase in the final deliveries of agricul-tural products, y1 by one unit would for instancerequire a rise of total agricultural output, x1,by 1.1457 units, 0.1457 of which will haveto be used to satisfy the additional input require-ments of the agricultural and manufacturingsectors.

Formulated in short-hand matrix notation, thebalance equations (1), describing the relationshipbetween the column vector of final demand, y, andthe column vector, x, of total outputs of all pro-ducing sectors can be written as:

I � Að Þx ¼ y (3)

where A represents the upper, square part of thestructural matrix (Fig. 2) describing the materialinput requirements of all producing sectors, x isthe column vector of total outputs and y, the col-umn vector of final deliveries of both goods. Thegeneral solution of that linear equation is,

x ¼ I � Að Þ�1y (4)

where (I�A)�1 represents the so-called inverse ofmatrix (I�A).

Total labour requirement can be computed in aseparate step,

L ¼ l0x ¼ l0 I � Að Þ�1y (5)

where l0 is a row vector of technical labour coef-ficient representing the technologically deter-mined amounts of labour that each industryemploys per unit of its total output.

2 Input–Output Analysis

The same set, A, of structural coefficients thatcontrols the physical flows, determines also therelationship between the prices of goods and ser-vices produced by different industries and the‘value added’ payments (expressed in the mone-tary units) made by each industry per unit of itsoutput. These include wages, profits, taxes, etc. Inshort, all payments other than those made forgoods and services purchased from other produc-ing sectors.

This set of value added–price equations, (oftenreferred to as a ‘dual’ to set (3) of physicalinput–output relationships) can be formulated asfollows,

I � A0ð ÞP ¼ V (6)

and its solution for the unknown prices as,

P ¼ I � A0ð Þ�1V (7)

where P is the column vector of prices of allsectoral outputs and V is the given column vectorof values added (per unit of their respective out-puts), in different sectors.

In the schematic input–output table (Fig. 1)considered above all amounts entered along aparticular row are measured in the same appropri-ately selected physical unit, for instance,wheat – in bushels; cloth – in yards; labour – inman years. No column totals are entered, since

adding amounts measured in incomparable phys-ical units would make no sense. In most publishedinput–output tables, all transactions are measuredhowever in value terms – usually in ‘base year’prices. Since these are assumed to satisfy theprice-value added equations describedabove – each column total, including the valueadded per unit of total output, must naturally beequal to the total output figures entered at the endof the corresponding row.

Value figures entered along a particular rowcan however also be interpreted as representingphysical amounts of the good in question, pro-vided the physical unit in which they are measuredis implicitly defined as the quantity of that goodpurchasable for, say, one dollar.

In the case of a table some rows of which arepresented in conventional physical amounts, saykwh of electric power, or tons of copper, whilesome other rows – in monetary units, appropriate‘equilibrium prices’ can be computed throughsolution of the corresponding ‘dual’ Eq. (7).

To do so it would suffice to re-define the phys-ical unit of the products of each sector as theamount purchasable for, say, one dollar, or someother monetary unit, at the price actually used indetermination of the value figures entered on thebase year table. These prices might of course bedifferent from the equilibrium prices.

From the outset the development ofinput–output analysis was marked by a succession

Input–Output Analysis, Table 1

Agriculture Manufacturing Households Total

Agriculture 25 20 55 100 bushels

Manufacturing 14 6 30 50 yards of cloth

Households 80 180 – 260 man-years

Input–Output Analysis, Table 2

Sector 1 Sector 2

Sector 1 0.25 0.40

Sector 2 0.14 0.12

Household 0.80 3.60

Input–Output Analysis 3

of empirical applications. In Leontief’s early vol-ume, The Structure of American Economy,1919–1929 (1941), this was the computation ofthe effects of changes in the input structure ofdifferent industries on levels of output and pricesof their products, and in particular on the ‘standardof living’ of households.

With the onset of World War II, attention wascentred on the transition from peacetime to a wareconomy. In particular, on the effects of changesin the level and composition of final demand onthe intersectoral distribution of output andemployment. The first official US input–outputtable – for the year 1939, compiled for the USBureau of Labor Statistics, provided a basis forpreparation of a detailed multisectoral projectionof postwar production and employment levels.Correctly predicting serious steel shortages,instead of large surpluses anticipated by leadingeconomic and industry experts, that report gainedwider interest in the new approach not only ingovernment circles, but among large industrialcorporations as well. The Western Electric Com-pany (the manufacturing arm of A.T.&T.) havingsuccessfully employed input–output analysis toanticipate impending shortages of lead, one of itsprincipal raw materials, even produced an educa-tional film describing the methodology used.

In one of the early applications of the samemodelling technique to what later on becameknown as operations research the smallinput–output team organized – under the nameProject Scoop – by the US Air Force constructeda detailed structural matrix of its far-flungmaterialprocurement and training operations. It was not asquare, but rather a rectangular matrix showingfor some sectors not one but several input vectorscorresponding to two or more alternative technol-ogies that could be used to produce a particularweapon or to provide a particular type of pilottraining. Confronted with the problem of optimalchoice between alternative ‘cooking recipes’, DrGeorge Dantzig, a young mathematician on theProject’s staff, invented the still very widely usedSimplex method of linear programming, whichconsists of a series of inversion of structuralinput–output matrices with sequential substitutionat alternative vectors of technical coefficients.

Not unlike research conducted in modernnatural sciences, input–output analysis wasfrom the outset most successfully conductedby closely coordinated teams rather than indi-vidual investigators. The first of such academicresearch groups was the Harvard EconomicResearch Project directed by Leontief over aperiod of nearly 30 years. Another centre wasorganized by Richard Stone in the Departmentof Applied Economics at the University of Cam-bridge. He was responsible for formal incorpo-ration of input–output tables in the UnitedNations system of national accounts designedby him.

Many of the young foreign economists whocame for completion of their graduate or postgrad-uate studies to the United States spent from a fewmonths up to several years at the HERP and afterreturning home introduced input–output analysisnot only as a subject of academic instruction andresearch but also as a new field of governmentalstatistics.

In Norway, Canada, Japan and in many othercountries governmental planning agencies andcentral statistical offices compile nationalinput–output tables and carry out practical appli-cations of input–output analysis, but also engagein fundamental methodological research. InSoviet Russia this was the first non-marxist, math-ematical approach to economics adapted, on therecommendation of Oscar Lange, after World WarII as a subject of academic instruction and as a toolof economic planning.

The first International Conference onInput–Output Analysis organized by ProfessorTinbergen was held in Dreibergen, Holland in1950; the eighth has been held in Japan in 1986.Proceedings of these and of other similar scientificmeetings published in book form provide a goodaccount of the current state of the art in the generalfield of input–output analysis and its variousapplications.

One of the fundamental theoretical questionsthat came up in connection with the earlyinput–output computations concerned the condi-tions under which none of the elements of theinverse (I � A) � 1 can be negative. The answerto it was provided by Herbert Simon – the future

4 Input–Output Analysis

Nobel prizewinner – and David Hawkins, a phi-losopher, in the form of the following theorem:

The necessary and sufficient conditions forsome of the elements of (I � A)�1 to be positive,and all to be non-negative, are:

1� a11j j > 0,

1� a11ð Þ �a12�a21 1� a22ð Þ

����

����> 0, . . .

1� a11ð Þ �a12��� �a1n�a21 1� a22ð Þ� � � �a2n⋮ ⋮ ⋮

�an1 �an2� � � 1� annð Þ

��������

��������

> 0:

(8)

If these conditions are satisfied for any particularnumbering of sectors it will necessarily be satis-fied for any other numbering sequence too. Theeconomic interpretation of this theorem is that fora system, in which each sector functions byabsorbing directly or indirectly outputs of someother sectors, to be able not only to sustain itselfbut also to make some positive deliveries to finaldemand, each one of the smaller and smallersub-systems contained within it has to be capableof sustaining itself and yielding a surplus deliver-able to outside users as well.

An example of a system unable to sustain itselfin this sense could be an economy so badly dam-aged by some natural catastrophe or war that onlyexternal assistance, taking the form of an importsurplus, could prevent it from complete collapse.Exports are entered in a standard input–outputtable and in the corresponding set of balanceequations, as positive and exports as negativecomponents of the final bill of goods. The nega-tive elements of the inverse (I � A) � 1 multipliedinto such negative components of the vector y offinal demand would yield in this case positive totaloutputs x.

In an attempt to reconcile at least to someextent the so-called fixed coefficient assumptionof linear input–output models with the neoclassi-cal production functions allowing for input sub-stitution, Kenneth Arrow, Tjalling Koopmansand Paul Samuelson provided independentlyfrom each other three different proofs of the‘non-substitution theorem’. They considered a

multisectoral economy in which each productivesector operates on the basis of a neoclassical pro-duction function and all sectors use the samesingle primary factors of production, say labour.The input combinations used by different sectorsare chosen so as to minimize the total amount oflabour that has to be employed by that economy inorder to enable it to deliver to final users anexogenously specified bill of goods. Thenon-substitution theorem states that the combina-tion of the relative amounts of different inputschosen in each sector will be independent of thecomposition of the final bill of goods. That meansthat even if the structure of final demand changesall producing sectors will behave as if they wereoperating on the basis of fixed coefficients ofproduction.

Restrictive assumptions – particularly thosepostulating invariability of production functionsthat control the operations of all sectors – deprivethe non-substitution theorem of much of its practi-cal significance. However, it calls attention to thedifference between the ways in which the termstechnology, and technological change, are used inneoclassical and in input–output theory. Ininput–output modelling the technology used inany particular sector is described as a given columnvector of coefficients, and a change in any elementof that vector is called technological change. Inneoclassical modelling the state of the technologyemployed by a particular sector is described by amuch more general – and because of that muchmore complex – kind of functional relationship thatin input–output analysis would have to be viewedas a set of many (strictly speaking, infinitely many)different technologies, each described by a differ-ent column vector of input coefficients. While pro-viding a convenient basis for deductive reasoningthe neoclassical terminology makes the task ofactual observation of the technological structureof a particular economy and empirical descriptionof processes of technological change extremely,not to say, prohibitively difficult.

Since direct observation of a set of isoquants ishardly ever possible, empirical implementation ofstandard neoclassical models involves nearlyexclusive reliance on more and more sophisti-cated methods of indirect statistical inference.

Input–Output Analysis 5

Neither of the two definitions of technology andtechnological change can be said to bemore correctthan the other. The employment of the simplerdefinition however permitted input–output analysisto advance in the direction of systematic detailedfactual inquiry, while reliance on a definition, muchless serviceable for purposes of empirical descrip-tion but much richer in its theoretical implications,propelled neoclassical economics towards con-struction of elaborate theoretical models erectedon a narrow, fragile data base or even on quitearbitrary, purely theoretical assumptions.

In static input–output models, additions to thestocks of building, machinery, and other kinds ofproductive stocks are treated as a component partof the final demand vector, entered in the right-hand side of the balance Eq. (6). In the followingformulation of a simple dynamic model theseterms are transferred to its left-hand sides anddescribed explicitly as serving technologicallydetermined capacity expansion required for arise in the level of output.

I � Að ÞXt � B Xtþ1 � Xtð Þ ¼ Yt (9)

B is a square matrix of technical capital coeffi-cients each column of which consists of stock-flow ratios, describing the stocks of products ofdifferent industries which the sector in questionmust have on hand per unit of its capacity output.

If the time unit in terms of which the process isobserved and described is relatively long, say,covering a 5 or even 10 year period, the stocksmight be engaged in production in the same timeperiod during which they have been produced. Inthis case, the second term on the left-hand sidewould be B(Xt� Xt�1). Current inputs required formaintenance of the existing capital stock have ofcourse to be accounted for by the appropriateelements of the A matrix.

While bringing to the fore the crucial role that acomplete set of capital coefficients has to play – inaddition to a complete set of current inputcoefficients – in the detailed description of thestructural framework of a given economy, such aset of difference equations is too rigid a tool to beused to describe and project the actual process ofeconomic development and change.

More effective, because more flexible, is anapproach which takes the form of a step-bystepconstruction of complete input–output tables ofthe economy for successive periods of time, eachbased on the knowledge of its state in the previousperiod, of anticipated changes in the final bill ofgoods and expected technological changes.

In more general terms, the input–output rela-tionship between goods produced and consumedover a sequence of successive years can be for-mally described exactly in the same terms as rela-tionships between different sectors are presentedin an ordinary ‘static’ input–output table for asingle year. The solution of a time-phased systemof linear equations describing the intertemporalbalances of inputs and outputs of goods and ser-vices produced and consumed over a long stretchof successive periods of time can be interpreted asinversion of a large triangular matrix; triangularbecause outputs of 1 year can become inputs inlater years, but not vice versa. The results of thisoperation describing the direct and indirect rela-tionships between all appropriately timed inputsand outputs has been called the ‘dynamic inverse’.Since the sets of flow and capital coefficientscontrolling the input–output balances in succes-sive stretches of such an historical process do nothave to remain the same, both that dynamic matrixand its inverse can accurately represent all kindsof structural change, including elimination of oldand introduction of entirely new goods.

Introduction of capital coefficients permitssubdivision of the value-added term, V, on theright-hand side of the dual system (8) into itstwo parts – the returns on capital and wageincome:

I � A0ð ÞP ¼ lB0Pþ 1w (10)

or, solving for P:

I � A0ð ÞP� lB0P ¼ 1w

l represents the rate of return on invested capitaland w, the wage rate. These equations can be usedfor calculating the ‘trade-off curve’ between realwages (i.e. money wage rate divided by a priceindex) and the rate of return on capital for any

6 Input–Output Analysis

given state of technology. Comparison of suchcurves, each reflecting a different combination ofalternative technologies available in different sec-tors, provides a base for numerical assessment ofthe influence of the distribution of incomebetween the return on capital and wages upontechnological choice.

Practical concerns led quite early to construc-tion of regional input–output tables. The munici-pal government of the city of Stockholm was thefirst to compile a detailed metropolitan table. Thecomplex fact-finding task of putting together adetailed input–output map of a particular regionseemed to have been inspired sometimes by thedesire to assert distinct identity. In Canada,French-speaking economists were the first to con-struct a regional table, that of Quebec. In Belgiumone was compiled for the autonomy-seekingFlemish provinces. In addition to pressing needsof developmental planning, similar considerationsseem to have prompted early compilation ofinput–output tables of many less developedcountries.

The next step was construction ofmultiregional input–output tables and models inwhich intraregional transactions were linked witheach other by interregional flows of goods andservices. While comparison of labour, capitaland natural resource ‘contents’ was the object ofsome of the earliest input–output studies ofdomestic and internationally traded goods, neitherthe theoretical formulation nor the available database are yet sufficiently advanced to permitinput–output modelling of international economictransactional trade to be solidly based on directempirical implementation of the comparative costtheory. In most multiregional inputoutput modelsthe structure of international transactions is con-trolled by sets of empirically determined exportand import coefficients. A large multiregionalinput–output model of the world economyconstructed under the auspices of the UnitedNations was published in 1977. Originallyintended to provide a basis for a set of alternativeprojections of the future growth of eight groups ofdeveloped and seven groups of less developedcountries, this large, highly disaggregated modelwas used in a series of other studies such as the

analysis of economic effects of international armstrade, detailed long-run projections of the produc-tion and consumption of non-ferrous metals in theUnited States and construction of alternativemultiregional scenarios of future exploration ofagricultural and energy resources.

As the range of its practical applications wid-ened, the scope of input–output modelling had tobe broadened, along with the contents of the req-uisite data bases.

Analysis of the petroleum refining industry inthe early Fifties requiredmodelling ofmultiproductprocesses. Thirty years later a similar approach wasemployed to describe within the framework of anational input–output table the generation andelimination of various polluting substances.Modelling devices adapted in description of theallocation of the output of transportation and tradesectors have later on been adapted in modelling theactivities of all service industries. Separation of thedescription of the physical from the price and cost-ing aspects of government operations proved to beuseful in construction and theoretical interpretationof input–output tables of simple, not yet fully mon-etized economies of the less developed economies.Richard Stone offered the conceptual framework ofinput–output analysis for the formal description ofdemographic processes.

To the extent to which it can provide a bridgebetween aggregative analysis and detaileddescription of production and consumption ofspecific goods and services input–output analysishas been incorporated into most of the well-known forecasting econometric models.

The general nature of the approach has madethe development of input–output analysis a cumu-lative process. Each refinement in theoreticalstructure and each addition to or improvement inthe accuracy of factual information incorporatedin its data base potentially improved the perfor-mance of the general model in application to allspecial problems.

See Also

▶Hawkins-Simon Conditions▶Leontief Paradox

Input–Output Analysis 7

Bibliography

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Meyer, U. 1980. Dynamische input–output-modelle.Königstein: Athenaum Ökonomie Verlag.

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8 Input–Output Analysis