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<title>Input–output model</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Input–output model</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the economic model. For the computer interface, see <a href="Input/output" title="Input/output">Input/output</a>.</div>

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<p>In <a href="Economics" title="Economics">economics</a>, an <b>input–output model</b> is a quantitative economic <a href="Mathematical_model" title="Mathematical model">model</a> that represents the interdependencies between different sectors of a national economy or different regional economies.<sup id="cite_ref-ref_1-0" class="reference"><a href="#cite_note-ref-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <a href="Wassily_Leontief" title="Wassily Leontief">Wassily Leontief</a> (1906–1999) is credited with developing this type of analysis and earned the <a href="Nobel_Prize_in_Economics" class="mw-redirect" title="Nobel Prize in Economics">Nobel Prize in Economics</a> for his development of this model.<sup id="cite_ref-ref_1-1" class="reference"><a href="#cite_note-ref-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Origins">Origins</h2></div>

<p><a href="Francois_Quesnay" class="mw-redirect" title="Francois Quesnay">Francois Quesnay</a> had developed a cruder version of this technique called <a href="Tableau_%C3%A9conomique" title="Tableau économique">Tableau économique</a>, and <a href="L%C3%A9on_Walras" title="Léon Walras">Léon Walras</a>'s work <i>Elements of Pure Economics</i> on <a href="General_equilibrium_theory" title="General equilibrium theory">general equilibrium theory</a> also was a forerunner and made a generalization of Leontief's seminal concept.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Alexander_Bogdanov" title="Alexander Bogdanov">Alexander Bogdanov</a> has been credited with originating the concept in a report delivered to the <a href="First_Conference_on_Scientific_Organization_of_Labour" title="First Conference on Scientific Organization of Labour">All Russia Conference on the Scientific Organisation of Labour and Production Processes</a>, in January 1921.<sup id="cite_ref-Belykh_3-0" class="reference"><a href="#cite_note-Belykh-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This approach was also developed by <a href="Lev_Kritzman" title="Lev Kritzman">Lev Kritzman</a>. Thomas Remington, has argued that their work provided a link between Quesnay's tableau économique and the subsequent contributions by <a href="Vladimir_Groman" title="Vladimir Groman">Vladimir Groman</a> and <a href="Vladimir_Bazarov" title="Vladimir Bazarov">Vladimir Bazarov</a> to <a href="Gosplan" title="Gosplan">Gosplan</a>'s method of <a href="Material_balance_planning" title="Material balance planning">material balance planning</a>.<sup id="cite_ref-Belykh_3-1" class="reference"><a href="#cite_note-Belykh-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Wassily Leontief's work in the input–output model was influenced by the works of the classical economists <a href="Karl_Marx" title="Karl Marx">Karl Marx</a> and <a href="Jean_Charles_L%C3%A9onard_de_Sismondi" title="Jean Charles Léonard de Sismondi">Jean Charles Léonard de Sismondi</a>. <a href="Marxian_economics" title="Marxian economics">Marx's economics</a> provided an early outline involving a set of tables where the economy consisted of two interlinked departments.<sup id="cite_ref-Planning_and_the_Real_Origins_of_Input–Output_Analysis,_1984_4-0" class="reference"><a href="#cite_note-Planning_and_the_Real_Origins_of_Input–Output_Analysis,_1984-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>Leontief was the first to use a <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> representation of a national (or regional) economy.

</p>

<div class="mw-heading mw-heading2"><h2 id="Basic_derivation">Basic derivation</h2></div>

<p>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.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Each column of the input–output <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> shows the monetary value of inputs to each sector and each row represents the value of each sector's outputs.

</p><p>Say that we have an economy with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> sectors. Each sector produces <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math></span><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"></span> units of a single homogeneous good. Assume that the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

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<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

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</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span>th sector, in order to produce 1 unit, must use <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation>

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</math></span><img src="./ebea6cd2813c330c798921a2894b358f7b643917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}" loading="lazy"></span> units from sector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>. 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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>th sector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

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</math></span><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"></span>. Then we might write

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<annotation encoding="application/x-tex">{\displaystyle x_{i}=a_{i1}x_{1}+a_{i2}x_{2}+\cdots +a_{in}x_{n}+y_{i},}</annotation>

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</math></span><img src="./5eac179e541ecd191a6e07ac08e8472c9c8d1d2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.933ex; height:2.343ex;" alt="{\displaystyle x_{i}=a_{i1}x_{1}+a_{i2}x_{2}+\cdots +a_{in}x_{n}+y_{i},}" loading="lazy"></span></dd></dl>

<p>or total output equals intermediate output plus final output. If we let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> be the matrix of coefficients <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation>

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</math></span><img src="./ebea6cd2813c330c798921a2894b358f7b643917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> be the vector of total output, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {y} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation>

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</math></span><img src="./bb25a040b592282dc2a254c3117e792c3c81161f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }" loading="lazy"></span> be the vector of final demand, then our expression for the economy becomes

</p>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =A\mathbf {x} +\mathbf {y} }</annotation>

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</math></span><img src="./13a5ce2ef38155f363faa4d837db975c0cbadf65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.915ex; height:2.509ex;" alt="{\displaystyle \mathbf {x} =A\mathbf {x} +\mathbf {y} }" loading="lazy"></span></dd></dl>

</td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>

<p>which after re-writing becomes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(I-A\right)\mathbf {x} =\mathbf {y} }">

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<annotation encoding="application/x-tex">{\displaystyle \left(I-A\right)\mathbf {x} =\mathbf {y} }</annotation>

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</math></span><img src="./1a2131e9170641638c5f4c842beaf06235873b7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.872ex; height:2.843ex;" alt="{\displaystyle \left(I-A\right)\mathbf {x} =\mathbf {y} }" loading="lazy"></span>. If the matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I-A}">

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<annotation encoding="application/x-tex">{\displaystyle I-A}</annotation>

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</math></span><img src="./9274b91f0029162df383d65384d7748a2ef3d4cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.755ex; height:2.343ex;" alt="{\displaystyle I-A}" loading="lazy"></span> 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 <a href="Minor_(linear_algebra)" title="Minor (linear algebra)">minors</a> of the matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I-A}">

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<annotation encoding="application/x-tex">{\displaystyle I-A}</annotation>

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</math></span><img src="./9274b91f0029162df383d65384d7748a2ef3d4cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.755ex; height:2.343ex;" alt="{\displaystyle I-A}" loading="lazy"></span> are all positive (known as the <a href="Hawkins%E2%80%93Simon_condition" title="Hawkins–Simon condition">Hawkins–Simon condition</a>),<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> the required output vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

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</p>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

<p>Consider an economy with two goods, A and B. The matrix of coefficients and the final demand is given by

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<annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}0.5&0.2\\0.4&0.1\end{bmatrix}}{\text{ and }}\mathbf {y} ={\begin{bmatrix}7\\4\end{bmatrix}}.}</annotation>

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</math></span><img src="./30038ccbf68432c1c9bcab963d69feb33f59d149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.748ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}0.5&0.2\\0.4&0.1\end{bmatrix}}{\text{ and }}\mathbf {y} ={\begin{bmatrix}7\\4\end{bmatrix}}.}" loading="lazy"></span></dd></dl>

<p>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

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\left(I-A\right)^{-1}\mathbf {y} ={\begin{bmatrix}19.19\\12.97\end{bmatrix}}.}</annotation>

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</math></span><img src="./8fca873810067aed45b19a87bddfbb2c02d2ec9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.066ex; height:6.176ex;" alt="{\displaystyle \mathbf {x} =\left(I-A\right)^{-1}\mathbf {y} ={\begin{bmatrix}19.19\\12.97\end{bmatrix}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Further_research">Further research</h3></div>

<p>There is extensive literature on these models. The model has been extended to work with non-linear relationships between sectors.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> 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.

</p>

<div class="mw-heading mw-heading3"><h3 id="Regional_multipliers">Regional multipliers</h3></div>

<p>While national input–output tables are commonly created by countries' statistics agencies, officially published regional input–output tables are rare. Therefore, economists often use <a href="Economic_base_analysis" title="Economic base analysis">location quotients</a> to create regional multipliers starting from national data.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This technique has been criticized because there are several location quotient regionalization techniques, and none are universally superior across all use-cases.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Introducing_transportation">Introducing transportation</h3></div>

<p>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.

</p><p><a href="Walter_Isard" title="Walter Isard">Walter Isard</a> 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.

</p>

<caption><b>Table: Adding Export And Import Transactions</b>

</caption>

<td>Economic Activities

</td>

<td>1

</td>

<td>2

</td>

<td>...

</td>

<td>...

</td>

<td>Z

</td>

<td>Exports

</td>

<td>Domestic Final Demand

</td>

<td>Total Outputs

</td></tr>

<tr>

<td>1

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<td>2

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<td>...

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<td>...

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<td>Z

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<td>Imports

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr></tbody></table>

<p>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.

</p><p>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 <a href="Industry_classification" title="Industry classification">industry classification</a>, 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.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> 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, <a href="Computable_general_equilibrium" title="Computable general equilibrium">computable general equilibrium</a> models<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup>.

</p><p>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.

</p>

<div class="mw-heading mw-heading3"><h3 id="Technology_Assumptions">Technology Assumptions</h3></div>

<p>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.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Usefulness">Usefulness</h2></div>

<p>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 <a href="Environmentally_extended_input%E2%80%93output_analysis" title="Environmentally extended input–output analysis">environmentally extended input–output analysis</a> (EEIOA). For example, information on fossil fuel inputs to each sector can be used to investigate flows of <a href="Embedded_emissions" title="Embedded emissions">embodied carbon</a> within and between different economies.

</p><p>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 <a href="Regional_Input%E2%80%93Output_Modeling_System" title="Regional Input–Output Modeling System">Regional Input–Output Modeling System</a>. 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.

</p>

<div class="mw-heading mw-heading3"><h3 id="Input–output_and_socialist_planning">Input–output and socialist planning</h3></div>

<p>The input–output model is one of the major conceptual models for a <a href="Socialist" class="mw-redirect" title="Socialist">socialist</a> <a href="Planned_economy" title="Planned economy">planned economy</a>. 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 <a href="Lange_model" title="Lange model">Lange-model socialism</a> and Soviet-style <a href="Material_balance_planning" title="Material balance planning">material balance planning</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

</p><p>In the economy of the <a href="Soviet_Union" title="Soviet Union">Soviet Union</a>, 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 <a href="Analysis_of_Soviet-type_economic_planning" class="mw-redirect" title="Analysis of Soviet-type economic planning">Soviet-type economies</a>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Criticism_of_Input-Output_Models">Criticism of Input-Output Models</h2></div>

<p><a href="The_Australia_Institute" title="The Australia Institute">The Australia Institute</a> 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. <sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> 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. <sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> IO models can be misinterpreted and used to justify projects or policies that are not economically sound. <sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> 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. <sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Measuring_input–output_tables">Measuring input–output tables</h2></div>

<p>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):<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> 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.

</p><p>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 <a href="Benchmarking" title="Benchmarking">benchmarking</a> of <a href="Gross_domestic_product" title="Gross domestic product">gross domestic product</a>. Input–output tables therefore are an instrumental part of <a href="National_accounts" title="National accounts">national accounts</a>. As suggested above, the core input–output table reports only intermediate goods and services that are exchanged among industries. But an array of row <a href="Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vectors</a>, 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.

</p>

<div class="mw-heading mw-heading2"><h2 id="Dynamic_Extensions">Dynamic Extensions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="The_Leontief_IO_model_with_capital_formation_endogenized">The Leontief IO model with capital formation endogenized</h3></div>

<p>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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{I}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{I}}</annotation>

</semantics>

</math></span><img src="./c0977d707252c1ff480cf58fa967d4760d4800ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.221ex; height:3.009ex;" alt="{\displaystyle y^{I}}" loading="lazy"></span>the vector of capital formation, with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{I}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}^{I}}</annotation>

</semantics>

</math></span><img src="./379a8757ecbdaf0ba0debc88788aafad8491fc3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.221ex; height:3.176ex;" alt="{\displaystyle y_{i}^{I}}" loading="lazy"></span> its <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>th element, and by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{ij}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{ij}(t)}</annotation>

</semantics>

</math></span><img src="./fb13eaf2daf7bf79032e40faf3918b3729d4a4bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.149ex; height:3.009ex;" alt="{\displaystyle I_{ij}(t)}" loading="lazy"></span> the amount of capital good <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> (for example, a blade) used in sector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math></span><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"></span> ( for example, wind power generation), for investment at time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}^{I}(t)=\sum _{j}I_{ij}(t)}">

</mrow>

</mrow>

</msubsup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{i}^{I}(t)=\sum _{j}I_{ij}(t)}</annotation>

</semantics>

</math></span><img src="./cc9f487b2065251c8bcea8a27ec3f24ed5e67ce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.86ex; height:5.843ex;" alt="{\displaystyle y_{i}^{I}(t)=\sum _{j}I_{ij}(t)}" loading="lazy"></span>

</p><p>We assume that it takes one year for investment in plant and equipment to become productive capacity. Denoting by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{ij}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{ij}(t)}</annotation>

</semantics>

</math></span><img src="./16f9da247d8b245a066c98863c79118e345daa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.099ex; height:3.009ex;" alt="{\displaystyle K_{ij}(t)}" loading="lazy"></span> the stock of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> at the beginning of time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>, and by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \in (0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta \in (0,1]}</annotation>

</semantics>

</math></span><img src="./d0590fccb82a0ae667ed13dd0bbf0dff27ff2d9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.8ex; height:2.843ex;" alt="{\displaystyle \delta \in (0,1]}" loading="lazy"></span> the rate of depreciation, we then have:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{ij}(t+1)=I_{ij}(t)+(1-\delta _{ij})K_{ij}(t)}</annotation>

</semantics>

</math></span><img src="./b8045d9d22e1b4df100fda306fb2e6b7321e9943.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.611ex; height:3.009ex;" alt="{\displaystyle K_{ij}(t+1)=I_{ij}(t)+(1-\delta _{ij})K_{ij}(t)}" loading="lazy"></span>

</p>

</td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}K_{ij}(t)}</annotation>

</semantics>

</math></span><img src="./e7c3d2badd316f54ee865fe3d22183c12d8d4472.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.609ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}K_{ij}(t)}" loading="lazy"></span> refers to the amount of capital stock that is used up in year <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}_{j}(t)}</annotation>

</semantics>

</math></span><img src="./cdceb517e28f74bbe7ad3d5b3acef0047fa20457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.888ex; height:3.009ex;" alt="{\displaystyle {\bar {x}}_{j}(t)}" loading="lazy"></span> the productive capacity in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span>, and assume the following proportionalty between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{ij}(t)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{ij}(t)}</annotation>

</semantics>

</math></span><img src="./16f9da247d8b245a066c98863c79118e345daa67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.099ex; height:3.009ex;" alt="{\displaystyle K_{ij}(t)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}_{j}(t)}">

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}_{j}(t)}</annotation>

</semantics>

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{ij}(t)=b_{ij}{\bar {x}}_{j}(t)}</annotation>

</semantics>

</math></span><img src="./07f843a72cc6644e8c8ffd3cc95b2505a6f935e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.561ex; height:3.009ex;" alt="{\displaystyle K_{ij}(t)=b_{ij}{\bar {x}}_{j}(t)}" loading="lazy"></span>

</p>

</td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table>

<p>The matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=[b_{ij}]}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B=[b_{ij}]}</annotation>

</semantics>

</math></span><img src="./03108cc0b865727a8a3e54f8a8ba0e4ebc0016f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.631ex; height:3.009ex;" alt="{\displaystyle B=[b_{ij}]}" loading="lazy"></span> is called the capital coefficient matrix.

From (<b><a href="#math_2">2</a></b>) and (<b><a href="#math_3">3</a></b>), we obtain the following expression for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{I}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{I}}</annotation>

</semantics>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{I}(t)=B{\bar {x}}(t+1)+(\delta -I){\bar {x}}(t)}</annotation>

</semantics>

</math></span><img src="./4ff5acd5908d53d8b92cb54354802c7b47612f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.403ex; height:3.176ex;" alt="{\displaystyle y^{I}(t)=B{\bar {x}}(t+1)+(\delta -I){\bar {x}}(t)}" loading="lazy"></span>

</p><p>Assuming that the productive capacity is always fully utilized, we obtain the following expression for (<b><a href="#math_1">1</a></b>) with endogenized capital formation:

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)=Ax(t)+Bx(t+1)+(\delta -I)Bx(t)+y^{o}(t),}</annotation>

</semantics>

</math></span><img src="./874b29165a95398a976c829661162787ad88d913.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.164ex; height:2.843ex;" alt="{\displaystyle x(t)=Ax(t)+Bx(t+1)+(\delta -I)Bx(t)+y^{o}(t),}" loading="lazy"></span>

</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{o}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{o}}</annotation>

</semantics>

</math></span><img src="./715c273872badc7349d859db5c7953bf5f0dda67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.19ex; height:2.676ex;" alt="{\displaystyle y^{o}}" loading="lazy"></span> stands for the items of final demand other than <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{I}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{I}}</annotation>

</semantics>

</p><p>Rearranged, we have

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Bx(t+1)&=(I-A+(I-\delta )B)x(t)-y^{o}(t)\\&=(I-{\bar {A}}+B)x(t)-y^{o}(t)\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./71245af85a53e32443098f14fcc7b47d4541bcc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.292ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}Bx(t+1)&=(I-A+(I-\delta )B)x(t)-y^{o}(t)\\&=(I-{\bar {A}}+B)x(t)-y^{o}(t)\end{aligned}}}" loading="lazy"></span>

</p><p>wehere <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {A}}=A+\delta B}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {A}}=A+\delta B}</annotation>

</semantics>

</math></span><img src="./9c110e7bea9d5b746e2e49a9425b6829a018b005.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.27ex; height:2.843ex;" alt="{\displaystyle {\bar {A}}=A+\delta B}" loading="lazy"></span>.

</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t+1)}</annotation>

</semantics>

</math></span><img src="./fa60578cbf90491ff3848900cbcf7951e16efccb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.981ex; height:2.843ex;" alt="{\displaystyle x(t+1)}" loading="lazy"></span> for given <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

</semantics>

</math></span><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{o}(t)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y^{o}(t)}</annotation>

</semantics>

</math></span><img src="./6f06a54c2d9adc9f0552ed279c812556c0c3a580.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.839ex; height:2.843ex;" alt="{\displaystyle y^{o}(t)}" loading="lazy"></span>:

<msup>

</mrow>

</msup>

<mover>

</mover>

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<msup>

</mrow>

</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x(t+1)=[I+B^{-1}(I-{\bar {A}})]x(t)-B^{-1}y^{o}(t)}</annotation>

</semantics>

</math></span><img src="./c0481b66d84f2f1b67f974879535db7c29af1dfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.834ex; height:3.176ex;" alt="{\displaystyle x(t+1)=[I+B^{-1}(I-{\bar {A}})]x(t)-B^{-1}y^{o}(t)}" loading="lazy"></span>

</p><p>This is the <b>Leontief dynamic forward-looking model</b><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>

</p><p>A caveat to this model is that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

This is because some products, such as energy items, are not used as capital goods, and the corresponding rows of the matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

This fact has prompted some researchers to consolidate the sectors until the non-singularity of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> is achieved, at the cost of sector resolution.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Apart from this feature, many studies have found that the outcomes obtained for this forward-looking model invariably lead to unrealistic and widely fluctuating results that lack economic interpretation.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> This has resulted in a gradual decline in interest in the model after the 1970s, although there is a recent increase in interest within the context of disaster analysis.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Input–output_analysis_versus_consistency_analysis">Input–output analysis versus consistency analysis</h2></div>

<p>Despite the clear ability of the input–output model to depict and analyze the dependence of one industry or sector on another, Leontief and others never managed to introduce the full spectrum of dependency relations in a market economy. In 2003, Mohammad Gani, a pupil of Leontief, introduced consistency analysis in his book <i>Foundations of Economic Science</i>, which formally looks exactly like the input–output table but explores the dependency relations in terms of payments and intermediation relations. Consistency analysis explores the consistency of plans of buyers and sellers by decomposing the input–output table into four matrices, each for a different kind of means of payment. It integrates micro and macroeconomics into one model and deals with money in a value-free manner. It deals with the flow of funds via the movement of goods.

</p>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">However, CGE models rely on economic production functions, such as CES functions, and are not suited to representing actual technologies in detail, whereas in IO there is no limit in resolution. Furthermore, the use of CES functions results in using a number of separability assumptions which can have strong implications on the assumed technology.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </span>

</li>

</ol></div></div>

<ul><li><a href="Anthropogenic_metabolism" title="Anthropogenic metabolism">Anthropogenic metabolism</a></li>

<li><a href="Computable_general_equilibrium" title="Computable general equilibrium">Computable general equilibrium</a></li>

<li><a href="Economic_base_analysis" title="Economic base analysis">Economic base analysis</a></li>

<li><a href="Economic_planning" title="Economic planning">Economic planning</a></li>

<li><a href="EIOLCA" class="mw-redirect" title="EIOLCA">EIOLCA</a></li>

<li><a href="Environmentally_extended_input%E2%80%93output_analysis" title="Environmentally extended input–output analysis">Environmentally extended input–output analysis</a></li>

<li><a href="Fiscal_multiplier" title="Fiscal multiplier">Fiscal multiplier</a></li>

<li><a href="Gross_output" title="Gross output">Gross output</a></li>

<li><a href="Linear_programming" title="Linear programming">Linear programming</a></li>

<li><a href="Industrial_metabolism" title="Industrial metabolism">Industrial metabolism</a></li>

<li><a href="Industrial_organization" title="Industrial organization">Industrial organization</a></li>

<li><a href="IPO_model" title="IPO model">IPO model</a></li>

<li><a href="Material_balance_planning" title="Material balance planning">Material balance planning</a></li>

<li><a href="Material_flow_analysis" title="Material flow analysis">Material flow analysis</a></li>

<li><a href="Net_output" title="Net output">Net output</a></li>

<li><a href="Shift-share_analysis" title="Shift-share analysis">Shift-share analysis</a></li>

<li><a href="Social_metabolism" title="Social metabolism">Social metabolism</a></li>

<li><a href="Socialist_economics" title="Socialist economics">Socialist economics</a></li>

<li><a href="Urban_metabolism" title="Urban metabolism">Urban metabolism</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-ref-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-ref_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ref_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Thijs Ten Raa, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=nu0FAvNiFhYC">Input–Output Economics: Theory and Applications: Featuring Asian Economies</a></i>, World Scientific, 2009</span>

</li>

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</style><cite id="CITEREFWalras1874" class="citation book cs1">Walras, L. (1874). <a rel="nofollow" class="external text" href="https://archive.org/details/lmentsdconomiep02walrgoog"><i>Éléments d'économie politique pure, ou théorie de la richesse sociale</i></a> [<i>Elements of Pure Economics, or The Theory of Social Wealth</i>]. L. Corbaz.</cite></span>

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<li id="cite_note-Belykh-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Belykh_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Belykh_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFBelykh1989" class="citation journal cs1">Belykh, A. A. (July 1989). "A Note on the Origins of Input–Output Analysis and the Contribution of the Early Soviet Economists: Chayanov, Bogdanov and Kritsman". <i>Soviet Studies</i>. <b>41</b> (3): <span class="nowrap">426–</span>429. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F09668138908411823">10.1080/09668138908411823</a>.</cite></span>

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<li id="cite_note-Planning_and_the_Real_Origins_of_Input–Output_Analysis,_1984-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Planning_and_the_Real_Origins_of_Input–Output_Analysis,_1984_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFClark1984" class="citation journal cs1">Clark, D. L. (1984). "Planning and the Real Origins of Input–Output Analysis". <i>Journal of Contemporary Asia</i>. <b>14</b> (4): <span class="nowrap">408–</span>429. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00472338485390301">10.1080/00472338485390301</a>.</cite></span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://bloomingtontutors.com/blog/how-to-understand-and-solve-leontief-input-output-model-technology-matrix-problems">"How to understand and solve Leontief input-output model (technology matrix) problems"</a>. Bloomington Tutors.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFNikaido1970" class="citation book cs1">Nikaido, H. (1970). <i>Introduction to Sets and Mappings in Modern Economics</i>. New York: Elsevier. pp. <span class="nowrap">13–</span>19. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-10038-5</bdi>.</cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFSandberg1973" class="citation journal cs1">Sandberg, I. W. (1973). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1914043">"A Nonlinear Input-Output Model of a Multisectored Economy"</a></span>. <i>Econometrica</i>. <b>41</b> (6): <span class="nowrap">1167–</span>1182. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1914043">10.2307/1914043</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0012-9682">0012-9682</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1914043">1914043</a>.</cite></span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFFleggWebberElliott1995" class="citation journal cs1">Flegg, A. T.; Webber, C. D.; Elliott, M. V. (1995). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/00343409512331349173?journalCode=cres20">"On the Appropriate Use of Location Quotients in Generating Regional Input–Output Tables"</a></span>. <i>Regional Studies</i>. <b>29</b> (6): <span class="nowrap">547–</span>561. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995RegSt..29..547F">1995RegSt..29..547F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00343409512331349173">10.1080/00343409512331349173</a>.</cite></span>

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Lehtonen, Olli & Tykkyläinen, Markku. <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/254336057_Estimating_Regional_Input_Coefficients_and_Multipliers_Is_the_Choice_of_a_Non-Survey_Technique_a_Gamble">"Estimating Regional Input Coefficients and Multipliers: Is the Choice of a Non-Survey Technique a Gamble?"</a>, 16 July 2007. Retrieved 29 May 2019.</span>

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFMasaev2021" class="citation journal cs1">Masaev, S. N. (2021). "Leontev Input-Output Balance Model as a Dynamic System Control Problem". <i>Herald of the Bauman Moscow State Technical University. Series Instrument Engineering</i>. <b>2</b> (135): <span class="nowrap">66–</span>82. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.18698%2F0236-3933-2021-2-66-82">10.18698/0236-3933-2021-2-66-82</a> (inactive 7 July 2025). <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:237889078">237889078</a>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite journal}}</code>: CS1 maint: DOI inactive as of July 2025 (link)</span></span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Shinichiro Nakamura and Yasushi Kondo, Waste Input-Output Analysis: Concepts and Application to Industrial Ecology, Springer, 2009, Section 4.1</span>

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<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">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.</span>

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<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><i>Eurostat manual of supply, use and input-output tables</i>, 2008, Eurostat. Office for Official Publications of the European Communities, (P.24)</span>

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<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><cite id="CITEREFLoucksWhitney1973" class="citation book cs1">Loucks, William Negele; Whitney, William G. (1973). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/comparativeecono0000louc_c2j3"><i>Comparative Economic Systems</i></a></span> (9th ed.). Harper & Row. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/comparativeecono0000louc_c2j3/page/178">178–179</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780060440459</bdi>.</cite></span>

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<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><i>Towards A New Socialism</i>, 1993, by Paul Cockshott and Allin Cottrell. Coronet Books Inc. 978-0851245454. "Planning in the USSR", (P.79)</span>

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<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFCampbell2014" class="citation web cs1">Campbell, Rod (8 January 2014). <a rel="nofollow" class="external text" href="https://australiainstitute.org.au/wp-content/uploads/2020/12/Another-way-to-look-at-the-impact-of-coal.pdf">"Another way to look at the impact of coal"</a> <span class="cs1-format">(PDF)</span>. <i>The Australia Institute</i><span class="reference-accessdate">. Retrieved <span class="nowrap">23 April</span> 2025</span>.</cite></span>

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<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://unstats.un.org/unsd/nationalaccount/sna.asp">About SNA</a>, UN</span>

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<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Wassily Leontief, Dynamic Analysis, Ch.3. In: W. Leontief et al.(eds.) Studies in the Structure of the American Economy. 1953, New York, Oxford University Press, 53–90.</span>

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<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><cite id="CITEREFJorgenson1961" class="citation journal cs1">Jorgenson, Dale W. (February 1961). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://doi.org/10.2307/2295708">"Stability of a Dynamic Input-Output System"</a></span>. <i>The Review of Economic Studies</i>. <b>28</b> (2): <span class="nowrap">105–</span>116. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2295708">10.2307/2295708</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0034-6527">0034-6527</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2295708">2295708</a>.</cite></span>

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<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><cite id="CITEREFTsukui1968" class="citation journal cs1">Tsukui, Jinkichi (1968). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1909611">"Application of a Turnpike Theorem to Planning for Efficient Accumulation: An Example for Japan"</a></span>. <i>Econometrica</i>. <b>36</b> (1): <span class="nowrap">172–</span>186. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1909611">10.2307/1909611</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0012-9682">0012-9682</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1909611">1909611</a>.</cite></span>

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<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Dorfman, Robert, Paul Anthony Samuelson, and Robert M. Solow. <i>Linear programming and economic analysis</i>. RAND Corporation, 1958. Chapter 11. </span>

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<li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Jinkichi Tsukui, (1961) On a Theorem of Relative Stability. International Economic Review, 2, 229–230.</span>

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<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><cite id="CITEREFBródy1995" class="citation journal cs1">Bródy, A. (January 1995). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="http://www.tandfonline.com/doi/full/10.1080/09535319500000022">"Truncation and Spectrum of the Dynamic Inverse"</a></span>. <i>Economic Systems Research</i>. <b>7</b> (3): <span class="nowrap">235–</span>248. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F09535319500000022">10.1080/09535319500000022</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0953-5314">0953-5314</a>.</cite></span>

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<li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><cite id="CITEREFSteengeReyes2020" class="citation journal cs1">Steenge, Albert E.; Reyes, Rachel C. (1 October 2020). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/full/10.1080/09535314.2020.1731682">"Return of the capital coefficients matrix"</a>. <i>Economic Systems Research</i>. <b>32</b> (4): <span class="nowrap">439–</span>450. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F09535314.2020.1731682">10.1080/09535314.2020.1731682</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0953-5314">0953-5314</a>.</cite></span>

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<p>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.

</p>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li>Dietzenbacher, Erik and Michael L. Lahr, eds. <i>Wassily Leontief and Input–Output Economics</i>. Cambridge University Press, 2004.</li>

<li>Isard, Walter et al. <i>Methods of Regional Analysis: An Introduction to Regional Science.</i> MIT Press 1960.</li>

<li>Isard, Walter and Thomas W. Langford. <i>Regional Input–Output Study: Recollections, Reflections, and Diverse Notes on the Philadelphia Experience.</i> The MIT Press. 1971.</li>

<li>Lahr, Michael L. and Erik Dietzenbacher, eds. <i>Input–Output Analysis: Frontiers and Extensions.</i> Palgrave, 2001.</li>

<li>Leontief, Wassily W. <i>Input–Output Economics.</i> 2nd ed., New York: Oxford University Press, 1986.</li>

<li>Miller, Ronald E. and Peter D. Blair. <i>Input–Output Analysis: Foundations and Extensions.</i> Prentice Hall, 1985.</li>

<li>Miller, Ronald E. and Peter D. Blair. <i>Input–Output Analysis: Foundations and Extensions,</i> 2nd edition. Cambridge University Press, 2009.</li>

<li>Miller, Ronald E., Karen R. Polenske, and Adam Z. Rose, eds. <i>Frontiers of Input–Output Analysis.</i> N.Y.: Oxford UP, 1989.[HB142 F76 1989/ Suzz]</li>

<li>Miernyk, William H. <i>The Elements of Input–Output Anaysis,</i> 1965.<a rel="nofollow" class="external text" href="http://www.rri.wvu.edu/WebBook/Miernykweb/new/index.htm">Web Book-William H. Miernyk</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191126210814/http://www.rri.wvu.edu/WebBook/Miernykweb/new/index.htm">Archived</a> 26 November 2019 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li>

<li>Polenske, Karen. <i>Advances in Input–Output Analysis.</i> 1976.</li>

<li>Pokrovskii, Vladimir N. <a rel="nofollow" class="external text" href="https://www.springer.com/physics/complexity/book/978-94-007-2095-4"><i>Econodynamics. The Theory of Social Production</i></a>, Springer, Dordrecht, Heidelberg et cetera, 2011.</li>

<li>ten Raa, Thijs. <i>The Economics of Input–Output Analysis.</i> Cambridge University Press, 2005.</li>

<li>US Department of Commerce, Bureau of Economic Analysis . <i>Regional multipliers: A user handbook for regional input–output modeling system (RIMS II)</i>. Third edition. Washington, D.C.: U.S. Government Printing Office. 1997.</li>

<li>Eurostat <i>Eurostat manual of supply, use and input-output tables.</i> Office for Official Publications of the European Communities, 2008.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.iioa.org">International Input–Output Association</a></li>

<li><a rel="nofollow" class="external text" href="https://www.bea.gov/industry/io_annual.htm">Input–Output Accounts Data</a>, <a href="Bureau_of_Economic_Analysis" title="Bureau of Economic Analysis">Bureau of Economic Analysis</a></li>

<li><a rel="nofollow" class="external text" href="http://www2.sjsu.edu/faculty/watkins/inputoutput.htm">Input–Output Analysis and Related Methods</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210505101943/https://www.sjsu.edu/faculty/watkins/inputoutput.htm">Archived</a> 5 May 2021 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, San José State University</li>

<li><a rel="nofollow" class="external text" href="http://www.doingbusiness.org/">Doing Business project input/output tables for reforms</a></li>

<li>Energy Economics. Input–Output Analysis: <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=onIhwmbL8CA">Lecture – 6</a> and <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=CTt4y8bokWs">Lecture 7</a> – two introductory videos on Input–Output methodology with a focus on energy economics from <a href="Indian_Institute_of_Technology_Kharagpur" class="mw-redirect" title="Indian Institute of Technology Kharagpur">IIT Kharagpur</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Models">Models</h3></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.remi.com/">REMI (Regional Economic Models, Inc.)</a></li>

<li><a rel="nofollow" class="external text" href="http://implan.com/">IMPLAN (Impact Analysis for Planning)</a></li>

<li><a rel="nofollow" class="external text" href="http://www.redyn.com/">REDYN (Regional Dynamics Model)</a></li></ul>

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