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MSc. Econometrics
Input - Output analysis
Professors
Cristoph Hanck
Martin Arnold
Jens Klenke
Students
Tom Kniepkamp tom.kniepkamp@tu-dortmund.de 229134
Camilo Santa Cruz camilo.santacruz@tu-dortmund.de 235633
Tanzer Siddique tanzer.siddique@tu-dortmund.de 230764
Input Output Models
Column {data-width=1000}
Applications
The intermediate purchases matrix
The matrix of intermediate uses (intermediate purchases) illustrates the volume of goods and services consumed by sector (j) that are produced by sector (i). The matrix form of this indicator is shown below. The reading by columns of this matrix indicates the purchases that a sector (j) makes to a sector (i). The reading of this matrix by rows, for its part, determines the sales that a sector (i) makes to a certain sector (j).
\[\begin{align} Z = \begin{bmatrix} z_{11} & z_{12} & z_{13}\\ z_{21} & z_{22} & z_{23}\\ z_{31} & z_{32} & z_{33} \end{bmatrix} \end{align}\]
The technical coefficients
The technical coefficients matrix describes the share of costs for goods, services and primary inputs to the gross production or output. This is obtained by dividing every element of the matriz Z (intermediate uses) with the corresponding output.
\[\begin{align} A = Z \hat{x}^{-1} = \begin{bmatrix} z_{11} & z_{12} & z_{13}\\ z_{21} & z_{22} & z_{23}\\ z_{31} & z_{32} & z_{33} \end{bmatrix} \begin{bmatrix} \frac{1}{x_1} & 0 & 0\\ 0 & \frac{1}{x_2} & 0\\ 0 & 0 & \frac{1}{x_3} \end{bmatrix} \end{align}\]
\[\begin{align} A = \begin{bmatrix} \frac{z_{11}}{x_1} & \frac{z_{21}}{x_2} & \frac{z_{13}}{x_3}\\ \frac{z_{21}}{x_1} & \frac{z_{22}}{x_2} & \frac{z_{23}}{x_3}\\ \frac{z_{31}}{x_1} & \frac{z_{32}}{x_2} & \frac{z_{33}}{x_3} \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{bmatrix} \end{align}\]
The backward and forward linkages
In its simplest form, a measure of the strength of the backward linkage of sector j – the amount by which sector (j) production is dependent on interindustry inputs is given by the sum of the elements in the jth column of the direct input coefficients matrix:
\[\begin{align}\BL{(d)}_j=\sum_{i=1}^{n}a_{ij}\end{align}\]
The forward linkages (also known as multiplier) for sector (j) is defined as the total value of production in all sectors of the economy that is necessary in order to satisfy a dollar’s worth of final demand for sector j’s output. This can be estimate as the column sum of the Leontief matrix.
\[\begin{align}\BL{(t)}_j=\sum_{i=1}^{n}l_{ij}\end{align}\]
Forward Linkages by its part, illustrates about how an increase in output of certain sectors will encourage an increase in the output of other sectors. This linkage analysis indicates how to use the input as intermediate consumption and focused on input structure. Analogous to linkages on the demand side (backward linkages), this is constructed by the row sum of the distribution coefficient matrices and the Ghosh matrix.
\[\begin{align}FL{(d)}_i=\sum_{j=1}^{n}d_{ij}\end{align}\]
\begin{TFL{(t)}i={j=1}^{n}g_{ij}\end{align}
IOM Models
Production function
Production function with two inputs.
Let \(x\) and \(y\) be the two input of an economy and \(A\) the total factor productivity (the parameter of growth which is not determined by the inputs alone, i.e. level of technology). Then at fixed level of output \(Q\), we can describe this relationship as:
\[\begin{align} Q = A x^{\alpha} y^{1- \alpha},\, \alpha \in (0, 1) \end{align}\]
where \(\alpha\) determines the elasticity of both input factors where a higher value of alpha shows that the production is more elastic with respect to \(x\) than to \(y\).