In economics, an input–output model is a quantitative economic model that represents the interdependencies between different sectors of a national economy or different regional economies.Thijs Ten Raa, Input–Output Economics: Theory and Applications: Featuring Asian Economies, World Scientific, 2009 They are based on are formalized as industry-by-industry or product-by-product Matrices, where rows typically represent outputs from one sector to another, and columns represent inputs. These models are often used by national statistics offices for their system of national accounts, including the quantification of Gross domestic product. Furthermore, they have been used increasingly in recent years to quantify Environmental footprints, enabling a consumption-based accounting of environmental impacts. Because they are linear in nature, computing input-output models is computationally cheap. However, they require large amounts of data and are therefore often subject to delays and sometimes uncertainties. Wassily Leontief (1906–1999) is credited with developing this type of analysis and was awarded the Nobel Prize in Economics for his development of this model.
Origins Francois Quesnay had developed a cruder version of this technique called Tableau économique, and Léon Walras's work Elements of Pure Economics on general equilibrium theory also was a forerunner and made a generalization of Leontief's seminal concept.
Alexander Bogdanov has been credited with originating the concept in a report delivered to the All Russia Conference on the Scientific Organisation of Labour and Production Processes, in January 1921. This approach was also developed by Lev Kritzman. Thomas Remington, has argued that their work provided a link between Quesnay's tableau économique and the subsequent contributions by Vladimir Groman and Vladimir Bazarov to Gosplan's method of material balance planning.
Wassily Leontief's work in the input–output model was influenced by the works of the classical economists Karl Marx and Jean Charles Léonard de Sismondi. Marx's economics provided an early outline involving a set of tables where the economy consisted of two interlinked departments.
Leontief was the first to use a matrix representation of a national (or regional) economy.
Basic derivation The model depicts inter-industry relationships within an economy, showing how output from one industrial sector may become an input to another industrial sector. In the inter-industry matrix, column entries typically represent inputs to an industrial sector, while row entries represent outputs from a given sector. This format, therefore, shows how dependent each sector is on every other sector, both as a customer of outputs from other sectors and as a supplier of inputs. Sectors may also depend internally on a portion of their own production as delineated by the entries of the matrix diagonal. Each column of the input–output matrix shows the value of inputs to each sector and each row represents the value of each sector's outputs, typically in monetary values.
Say that we have an economy with sectors. Each sector produces units of a single homogeneous good. Assume that the th sector, in order to produce 1 unit, must use units from sector . Furthermore, assume that each sector sells some of its output to other sectors (intermediate output) and some of its output to consumers (final output, or final demand). Call final demand in the th sector . Then we might write
or total output equals intermediate output plus final output. If we let be the matrix of coefficients , be the vector of total output, and be the vector of final demand, then our expression for the economy becomes
which after re-writing becomes . If the matrix is invertible then this is a linear system of equations with a unique solution, and so given some final demand vector the required output can be found. Furthermore, if the principal minors of the matrix are all positive (known as the Hawkins–Simon condition), the required output vector is non-negative.
Example
Consider an economy with two sectors, coal and steel, each of which uses both its own output and the other's as inputs to production. The technical coefficient is the amount of good i needed to produce one unit of good j, so each column lists the inputs a sector consumes per unit of its own output:
Reading the columns, producing one unit of coal requires 0.5 units of coal and 0.4 units of steel, while producing one unit of steel requires 0.2 units of coal and 0.1 units of steel. The final demand is the output left for consumers after all inter-industry use. Total output must cover both intermediate use and final demand:
Forming gives
with determinant
Inverting it yields
This is the Leontief inverse. Multiplying by the final demand yields the required gross output:
The result can be checked by subtracting inter-industry consumption from gross output. The intermediate use is
so , as required.
The economy must produce 19.19 units of coal to deliver only 7 to consumers. The difference is the indirect requirement: coal is needed to make steel, and to make the additional coal, and so on through successive rounds of production. The Leontief inverse sums this entire series,
which converges because the economy is productive (the largest eigenvalue of is less than 1).
Further research
There is extensive literature on these models. The model has been extended to work with non-linear relationships between sectors. There is the Hawkins–Simon condition on producibility. There has been research on disaggregation to clustered inter-industry flows, and on the study of constellations of industries. A great deal of empirical work has been done to identify coefficients, and data has been published for national economies as well as for regions, some of which have global coverage. The Leontief system can be extended to a model of general equilibrium; it offers a method of decomposing work done at a macro level.
Regional multipliers While national input–output tables are commonly created by countries' statistics agencies, officially published regional input–output tables are rare. Therefore, economists often use location quotients to create regional multipliers starting from national data. This technique has been criticized because there are several location quotient regionalization techniques, and none are universally superior across all use-cases.Lehtonen, Olli & Tykkyläinen, Markku. "Estimating Regional Input Coefficients and Multipliers: Is the Choice of a Non-Survey Technique a Gamble?", 16 July 2007. Retrieved 29 May 2019.
Introducing transportation
Transportation is implicit in the notion of inter-industry flows. It is explicitly recognized when transportation is identified as an industry – how much is purchased from transportation in order to produce. But this is not very satisfactory because transportation requirements differ, depending on industry locations and capacity constraints on regional production. Also, the receiver of goods generally pays freight cost, and often transportation data are lost because transportation costs are treated as part of the cost of the goods.
Walter Isard and his student, Leon Moses, were quick to see the spatial economy and transportation implications of input–output, and began work in this area in the 1950s developing a concept of interregional input–output. Take a one region versus the world case. We wish to know something about inter-regional commodity flows, so introduce a column into the table headed "exports" and we introduce an "import" row.
+Table: Adding Export And Import TransactionsEconomic Activities 1 2 ... ... Z Exports Domestic Final Demand Total Outputs1 2 ... ... ZImports
A more satisfactory way to proceed would be to tie regions together at the industry level. That is, we could identify both intra-region inter-industry transactions and inter-region inter-industry transactions. The problem here is that the table grows quickly.
Input–output is conceptually simple. Its extension to a model of equilibrium in the national economy has been done successfully using high-quality data. One who wishes to work with input–output systems must deal with industry classification, data estimation, and inverting very large, often ill-conditioned matrices. The quality of the data and matrices of the input-output model can be improved by modelling activities with digital twins and solving the problem of optimizing management decisions. Moreover, changes in relative prices are not readily handled by this modelling approach alone. Input–output accounts are part and parcel to a more flexible form of modelling, computable general equilibrium models.
Two additional difficulties are of interest in transportation work. There is the question of substituting one input for another, and there is the question about the stability of coefficients as production increases or decreases. These are intertwined questions. They have to do with the nature of regional production functions.
Technology Assumptions To construct input-output tables from supply and use tables, four principal assumptions can be applied. The choice depends on whether product-by-product or industry-by-industry input-output tables are to be established.ابونوری, اسمعیل, فرهادی, & عزیزاله. (2017). آزمون فروض تکنولوژی در محاسبه جدول داده ستانده متقارن ایران: یک رهیافت اقتصاد سنجی. پژوهشهای اقتصادی ایران, 21(69), 117-145.Eurostat manual of supply, use and input-output tables, 2008, Eurostat. Office for Official Publications of the European Communities, (P.24)
Usefulness and applications
Because the input–output model is fundamentally linear in nature, it lends itself to rapid computation as well as flexibility in computing the effects of changes in demand.
Environmental analyses Input–output models for different regions can also be linked together to investigate the effects of inter-regional trade, and additional columns can be added to the table to perform environmentally extended input–output analysis (EEIOA). For example, information on fossil fuel inputs to each sector can be used to investigate flows of embodied carbon within and between different economies. Since most input-output models are quantified in monetary units, environmental intensities are typically expressed in damage per dollar, meaning that every dollar spent in an industry is assumed to have the same impact. This can lead to bias in the estimations, since price does not always reflect environmental damage (e.g., in Co2 or water use) well. This is the case for wine for example - while prices might differ greatly between different bottles of wine, it is unlikely that a wine that costs 200$ per bottle will have a twenty-fold effect of a bottle that only costs 10$. To circumvent this problem, physical input-output tables can be constructed. These are based on physical quantities rather than prices and are thus better suited for environmental accounting.
National accounts The structure of the input–output model has been incorporated into national accounting in many developed countries, and as such can be used to calculate important measures such as national GDP. Input–output economics has been used to study regional economies within a nation, and as a tool for national and regional economic planning. A main use of input–output analysis is to measure the economic impacts of events as well as public investments or programs as shown by IMPLAN and Regional Input–Output Modeling System. It is also used to identify economically related industry clusters and also so-called "key" or "target" industries (industries that are most likely to enhance the internal coherence of a specified economy). By linking industrial output to satellite accounts articulating energy use, effluent production, space needs, and so on, input–output analysts have extended the approaches application to a wide variety of uses.