View source for Input–output model

== Introduction ==

The **input–output model** is a quantitative economic technique that represents the interdependencies between different sectors of a national economy or different regional economies. This model was developed by the economist [[Wassily Leontief]], who received the Nobel Prize in Economics in 1973 for his development of this method and its application to important economic problems.

== Historical Background ==

The input–output model has its roots in the early 20th century, but it was Wassily Leontief who formalized the method in the 1930s and 1940s. Leontief's work was inspired by the [[physiocrats]]' tableau économique and the [[Marxian economics|Marxian]] reproduction schemes. The model gained prominence during and after World War II, as it was used to analyze the economic impact of war production and post-war reconstruction.

== Theoretical Foundations ==

The input–output model is grounded in the theory of [[general equilibrium]], which posits that supply and demand in an economy are balanced through prices. The model assumes that the economy can be divided into a finite number of sectors, each producing a unique product or service. These sectors are interdependent, meaning the output of one sector is the input of another.

=== Assumptions ===

The input–output model relies on several key assumptions:

* **Linear Production Functions**: Each sector's output is a linear function of its inputs.
* **Fixed Coefficients**: The input coefficients, which represent the amount of input required to produce one unit of output, are constant.
* **No Substitution**: Inputs cannot be substituted for one another.
* **Closed Economy**: The model often assumes no international trade, although extensions can include trade.

== Mathematical Representation ==

The core of the input–output model is the **input–output table**, which captures the flow of goods and services between sectors. The table is typically represented as a matrix, where each element \(a_{ij}\) represents the value of inputs from sector \(i\) required to produce one unit of output in sector \(j\).

=== Input–Output Table ===

The input–output table consists of three main components:

* **Intermediate Consumption**: The upper-left quadrant of the table, showing the inter-sectoral flows of goods and services.
* **Final Demand**: The upper-right quadrant, representing the demand for goods and services by final consumers, government, and for export.
* **Value Added**: The lower-left quadrant, capturing payments to primary factors of production, such as wages and profits.

=== Leontief Inverse ===

A crucial concept in the input–output model is the **Leontief inverse** matrix, denoted as \((I - A)^{-1}\), where \(I\) is the identity matrix and \(A\) is the matrix of technical coefficients. The Leontief inverse captures the total (direct and indirect) requirements of inputs per unit of final demand.

== Applications ==

The input–output model has a wide range of applications in economics and policy analysis:

=== Economic Impact Analysis ===

One of the most common uses of the input–output model is to assess the economic impact of changes in final demand. For example, policymakers might use the model to estimate the effects of a new infrastructure project on employment and output across different sectors.

=== Environmental Economics ===

The model can also be extended to include environmental factors, leading to the development of **environmental input–output models**. These models help in understanding the environmental impacts of economic activities, such as carbon emissions and resource use.

=== Regional Economics ===

In regional economics, the input–output model is used to analyze the economic interactions within a specific region and between regions. This helps in understanding regional economic dependencies and in formulating regional development policies.

== Extensions and Variations ==

Over time, several extensions and variations of the input–output model have been developed to address its limitations and to apply it to more complex economic scenarios.

=== Dynamic Input–Output Models ===

Dynamic input–output models incorporate time as a factor, allowing for the analysis of economic changes over multiple periods. These models are useful for long-term economic planning and forecasting.

=== Multi-Regional Input–Output Models ===

Multi-regional input–output (MRIO) models extend the basic framework to multiple regions, capturing inter-regional trade and economic interactions. MRIO models are particularly useful for analyzing global supply chains and the economic impacts of trade policies.

=== Social Accounting Matrices ===

A **social accounting matrix** (SAM) is an extension of the input–output table that includes additional accounts for institutions such as households, government, and the rest of the world. SAMs provide a more comprehensive picture of the economy and are used in [[computable general equilibrium]] (CGE) models.

== Criticisms and Limitations ==

Despite its widespread use, the input–output model has several criticisms and limitations:

* **Fixed Coefficients**: The assumption of fixed input coefficients is often unrealistic, as it ignores the possibility of technological change and substitution between inputs.
* **Static Nature**: The basic input–output model is static and does not account for changes over time.
* **Data Requirements**: Constructing an input–output table requires detailed and accurate data, which can be difficult and costly to obtain.

== Conclusion ==

The input–output model remains a fundamental tool in economic analysis, offering valuable insights into the interdependencies within an economy. While it has its limitations, ongoing developments and extensions continue to enhance its applicability and relevance in addressing complex economic issues.

<div class='only_on_desktop image-preview'><div class='image-preview-loader'></div></div><div class='only_on_mobile image-preview'><div class='image-preview-loader'></div></div>

== See Also ==

* [[General Equilibrium]]
* [[Environmental Economics]]
* [[Computable General Equilibrium]]

== Categories ==

[[Category:Economics]]
[[Category:Mathematical Economics]]
[[Category:Regional Economics]]