# Mathematical background¶

This section gives a general overview about the mathematical background of Input-Output calculations. For a full detail account of this matter please see Miller and Blair 2009

Generally, mathematical routines implemented in pymrio follow the equations described below. If, however, a more efficient mechanism was available this was preferred. This was generally the case when numpy broadcasting was available for a specific operation, resulting in a substantial speed up of the calculations. In these cases the original formula remains as comment in the source code.

## Basic MRIO calculations¶

MRIO tables describe the global inter-industries flows within and across countries for $$k$$ countries with a transaction matrix $$Z$$:

$\begin{split}$$Z = \begin{pmatrix} Z_{1,1} & Z_{1,2} & \cdots & Z_{1,k} \\ Z_{2,1} & Z_{2,2} & \cdots & Z_{2,k} \\ \vdots & \vdots & \ddots & \vdots \\ Z_{k,1} & Z_{k,2} & \cdots & Z_{k,k} \end{pmatrix}$$\end{split}$

Each submatrix on the main diagonal ($$Z_{i,i}$$) represents the domestic interactions for each industry $$n$$. The off diagonal matrices ($$Z_{i,j}$$) describe the trade from region $$i$$ to region $$j$$ (with $$i, j = 1, \ldots, k$$) for each industry. Accordingly, global final demand can be represented by

$\begin{split}$$Y = \begin{pmatrix} Y_{1,1} & Y_{1,2} & \cdots & Y_{1,k} \\ Y_{2,1} & Y_{2,2} & \cdots & Y_{2,k} \\ \vdots & \vdots & \ddots & \vdots \\ Y_{k,1} & Y_{k,2} & \cdots & Y_{k,k} \end{pmatrix}$$\end{split}$

with final demand satisfied by domestic production in the main diagonal ($$Y_{i,i}$$) and direct import to final demand from country $$i$$ to $$j$$ by $$Y_{i,j}$$.

The global economy can thus be described by:

$$$x = Ze + Ye$$$

with $$e$$ representing the summation vector (column vector with 1’s of appropriate dimension) and $$x$$ the total industry output.

The direct requirement matrix $$A$$ is given by multiplication of $$Z$$ with the diagonalised and inverted industry output $$x$$:

$$$A = Z\hat{x}^{-1}$$$

Based on the linear economy assumption of the IO model and the classic Leontief demand-style modeling (see Leontief 1970), total industry output $$x$$ can be calculated for any arbitrary vector of final demand $$y$$ by multiplying with the total requirement matrix (Leontief matrix) $$L$$.

$$$x = (\mathrm{I}- A)^{-1}y = Ly$$$

with $$\mathrm{I}$$ defined as the identity matrix with the size of $$A$$.

The global multi regional IO system can be extended with various factors of production $$f_{h,i}$$. These can represent among others value added, employment and social factors ($$h$$, with $$h = 1, \ldots, r$$) per country. The row vectors of factors can be summarised in a factor of production matrix $$F$$:

$\begin{split}$$F = \begin{pmatrix} f_{1,1} & f_{1,2} & \cdots & f_{1,k} \\ f_{2,1} & f_{2,2} & \cdots & f_{2,k} \\ \vdots & \vdots & \ddots & \vdots \\ f_{r,1} & f_{r,2} & \cdots & f_{r,k} \end{pmatrix}$$\end{split}$

with the factor of production coefficients $$S$$ given by

$$$S = F\hat{x}^{-1}$$$

Multipliers (total, direct and indirect, requirement factors for one unit of output) are then obtained by

$$$M = SL$$$

Total requirements (footprints in case of environmental requirements) for any given final demand vector $$y$$ are then given by

$$$D_{cba} = My$$$

Setting the domestically satisfied final demand $$Y_{i,i}$$ to zero ($$Y_{t} = Y - Y_{i,j}\; |\; i = j$$) allows to calculate the factor of production occurring abroad (embodied in imports)

$$$D_{imp} = SLY_{t}$$$

The factors of production occurring domestically to satisfy final demand in other countries is given by:

$$$D_{exp} = S\widehat{LY_{t}e}$$$

with the hat indicating diagonalization of the resulting column-vector of the term underneath.

If the factor of production represent required environmental impacts, these can also occur during the final use phase. In that case $$F_Y$$ describe the impacts associated with final demand (e.g. household emissions).

These need to be added to the total production- and consumption-based accounts to obtain the total impacts per country. Production-based accounts (direct territorial requirements) per region i are therefore given by summing over the stressors per sector (0 … m) plus the stressors occurring due to the final consumption for all final demand categories (0 … w) of that region.

$$$D_{pba}^i = \sum_{s=0}^m F^i_s + \sum_{c=0}^w F^i_{Yc}$$$

Similarly, total requirements (footprints in case of environmental requirements) per region i are given by summing the detailed footprint accounts and adding the aggregated final demand stressors.

$$$D_{cba}^i = \sum_{s=0}^m D_{cba, s}^{i} + \sum_{c=0}^w F_{Yc}^i$$$

Internally, the summation are implemented with the group-by functionality provided by the pandas package.

## Aggregation¶

For the aggregation of the MRIO system the matrix $$B_k$$ defines the aggregation matrix for regions and $$B_n$$ the aggregation matrix for sectors.

$\begin{split}$$B_k = \begin{pmatrix} b_{1,1} & b_{1,2} & \cdots & b_{1,k} \\ b_{2,1} & b_{2,2} & \cdots & b_{2,k} \\ \vdots & \vdots & \ddots & \vdots \\ b_{w,1} & b_{w,2} & \cdots & b_{w,k} \end{pmatrix} B_n = \begin{pmatrix} b_{1,1} & b_{1,2} & \cdots & b_{1,n} \\ b_{2,1} & b_{2,2} & \cdots & b_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ b_{x,1} & b_{x,2} & \cdots & b_{x,n} \end{pmatrix}$$\end{split}$

With $$w$$ and $$x$$ defining the aggregated number of countries and sectors, respectively. Entries $$b$$ are set to 1 if the sector/country of the column belong to the aggregated sector/region in the corresponding row and zero otherwise. The complete aggregation matrix $$B$$ is given by the Kronecker product $$\otimes$$ of $$B_k$$ and $$B_n$$:

$$$B = B_k \otimes B_n$$$

This effectively arranges the sector aggregation matrix $$B_n$$ as defined by the region aggregation matrix $$B_k$$. Thus, for each 0 entry in $$B_k$$ a block $$B_n * 0$$ is inserted in $$B$$ and each 1 corresponds to $$B_n * 1$$ in $$B$$.

The aggregated IO system can then be obtained by

$$$Z_{agg} = BZB^\mathrm{T}$$$

and

$$$Y_{agg} = BY(B_k \otimes \mathrm{I})^\mathrm{T}$$$

with $$\mathrm{I}$$ defined as the identity matrix with the size equal to the number of final demand categories per country.

Factors of production are aggregated by

$$$F_{agg} = FB^\mathrm{T}$$$

and final demand impacts by

$$$F_{Y, agg} = F_Y(B_k \otimes \mathrm{I})^\mathrm{T}$$$