In
economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interac ...
, an input–output model is a quantitative economic
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
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 Wassily Leontief (1906–1999) is credited with developing this type of analysis and earned 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
Marie-Esprit-Léon Walras (; 16 December 1834 – 5 January 1910) was a French mathematical economics, mathematical economist and Georgist. He formulated the Marginalism, marginal theory of value (independently of William Stanley Jevons and Carl ...
's work ''Elements of Pure Economics'' on
general equilibrium theory
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
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
The State Planning Committee, commonly known as Gosplan ( ), was the agency responsible for economic planning, central economic planning in the Soviet Union. Established in 1921 and remaining in existence until the dissolution of the Soviet Unio ...
'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
Karl Marx (; 5 May 1818 – 14 March 1883) was a German philosopher, political theorist, economist, journalist, and revolutionary socialist. He is best-known for the 1848 pamphlet '' The Communist Manifesto'' (written with Friedrich Engels) ...
and
Jean Charles Léonard de Sismondi
Jean Charles Léonard de Sismondi, also known as Jean Charles Leonard Simonde de Sismondi (; 9 May 1773 – 25 June 1842), whose real surname was Simonde, was a Swiss historian and political economist, who is best known for his works on French ...
.
Karl Marx
Karl Marx (; 5 May 1818 – 14 March 1883) was a German philosopher, political theorist, economist, journalist, and revolutionary socialist. He is best-known for the 1848 pamphlet '' The Communist Manifesto'' (written with Friedrich Engels) ...
'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
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
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
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
shows the monetary value of inputs to each sector and each row represents the value of each sector's outputs.
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 goods, A and B. The matrix of coefficients and the final demand is given by
:
Intuitively, this corresponds to finding the amount of output each sector should produce given that we want 7 units of good A and 4 units of good B. Then solving the system of linear equations derived above gives us
:
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 the national economy as well as for regions. 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.
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.
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
Industry classification or industry taxonomy is a type of economic taxonomy that classifies companies, organizations and traders into industrial groupings based on similar production processes, similar products, or similar behavior in financial m ...
, 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.
Usefulness
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. 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.
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.
Input–output and socialist planning
The input–output model is one of the major conceptual models for a
socialist
Socialism is an economic ideology, economic and political philosophy encompassing diverse Economic system, economic and social systems characterised by social ownership of the means of production, as opposed to private ownership. It describes ...
planned economy
A planned economy is a type of economic system where investment, production and the allocation of capital goods takes place according to economy-wide economic plans and production plans. A planned economy may use centralized, decentralized, ...
. This model involves the direct determination of physical quantities to be produced in each industry, which are used to formulate a consistent economic plan of resource allocation. This method of planning is contrasted with price-directed
Lange-model socialism and Soviet-style
material balance planning.
In the economy of the
Soviet Union
The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet ...
, planning was conducted using the method of material balances up until the country's dissolution. The method of material balances was first developed in the 1930s during the Soviet Union's rapid industrialization drive. Input–output planning was never adopted because the material balance system had become entrenched in the Soviet economy, and input–output planning was shunned for ideological reasons. As a result, the benefits of consistent and detailed planning through input–output analysis were never realized in the
Soviet-type economies
Soviet-type economic planning (STP) is the specific model of centralized planning employed by Marxist–Leninist socialist states modeled on the economy of the Soviet Union.
The post-''perestroika'' analysis of the system of the Soviet econ ...
.
Criticism of Input-Output Models
The Australia Institute critiques input-output (IO) models for their biases and limitations in assessing the economic impacts of projects and policies. There are limitations and biases inherent in IO models, citing concerns that they are "biased" and "abused" by organizations like the Australian Bureau of Statistics and the Productivity Commission. For instance, the institute's research points out that IO models often assume fixed prices and don't account for resource constraints, which can lead to unrealistic and inflated economic impact estimates. IO models can be misinterpreted and used to justify projects or policies that are not economically sound. The Australia Institute suggests that more robust and comprehensive economic analysis methods should be used to assess economic impacts, rather than relying solely on IO models.
Measuring input–output tables
The mathematics of input–output economics is straightforward, but the data requirements are enormous because the expenditures and revenues of each branch of economic activity have to be represented. As a result, not all countries collect the required data and data quality varies, even though a set of standards for the data's collection has been set out by the United Nations through its System of National Accounts (SNA): the most recent standard is the 2008 SNA. Because the data collection and preparation process for the input–output accounts is necessarily labor and computer intensive, input–output tables are often published long after the year in which the data were collected—typically as much as 5–7 years after. Moreover, the economic "snapshot" that the benchmark version of the tables provides of the economy's cross-section is typically taken only once every few years, at best.
However, many developed countries estimate input–output accounts annually and with much greater recency. This is because while most uses of the input–output analysis focus on the matrix set of inter-industry exchanges, the actual focus of the analysis from the perspective of most national statistical agencies is the
benchmarking
Benchmarking is the practice of comparing business processes and performance metrics to industry bests and best practices from other companies. Dimensions typically measured are Project management triangle, quality, time and cost.
Benchmarking is ...
of
gross domestic product
Gross domestic product (GDP) is a monetary measure of the total market value of all the final goods and services produced and rendered in a specific time period by a country or countries. GDP is often used to measure the economic performanc ...
. Input–output tables therefore are an instrumental part of
national accounts
National accounts or national account systems (NAS) are the implementation of complete and consistent accounting Scientific technique, techniques for measuring the economic activity of a nation. These include detailed underlying measures that ...
. As suggested above, the core input–output table reports only intermediate goods and services that are exchanged among industries. But an array of row
vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
s, typically aligned at the bottom of this matrix, record non-industrial inputs by industry like payments for labor; indirect business taxes; dividends, interest, and rents; capital consumption allowances (depreciation); other property-type income (like profits); and purchases from foreign suppliers (imports). At a national level, although excluding the imports, when summed this is called "gross product originating" or "gross domestic product by industry." Another array of column vectors is called "final demand" or "gross product consumed." This displays columns of spending by households, governments, changes in industry stocks, and industries on investment, as well as net exports. (See also Gross domestic product.) In any case, by employing the results of an economic census which asks for the sales, payrolls, and material/equipment/service input of each establishment, statistical agencies back into estimates of industry-level profits and investments using the input–output matrix as a sort of double-accounting framework.
Dynamic Extensions
The Leontief IO model with capital formation endogenized
The IO model discussed above is static because it does not describe the evolution of the economy over time: it does not include different time periods.
Dynamic Leontief models are obtained by endogenizing the formation of capital stock over time.
Denote by
the vector of capital formation, with
its
th element, and by
the amount of capital good
(for example, a blade) used in sector
( for example, wind power generation), for investment at time
. We then have
We assume that it takes one year for investment in plant and equipment to become productive capacity. Denoting by
the stock of
at the beginning of time
, and by
the rate of depreciation, we then have:
Here,
refers to the amount of capital stock that is used up in year
.
Denote by
the productive capacity in
, and assume the following proportionalty between
and
:
The matrix