# Incorporating New Technologies in EEIO Models
## 2 Methodology
### Rebalance Make and Use Tables  
After augmenting the Make and Use tables, as described in the previous sections, the tables do not satisfy the requirement that production equals consumption. Specifically, the total commodity output \(q^{U,1}\), calculated from the Use table, differs from total commodity output \(q^{M,1}\), calculated from the Make table, for some commodities. Similarly, total industry output \(x^{U,1}\), calculated from the Use table, is different from the total industry output \(x^{M,1}\), calculated from the Make table, for some industries.  
We introduce an analytical approach to rebalance the Make and Use tables, following [PERSON] and colleagues [18] but extended for rectangular matrices. It scales the Make and Use tables without modifying production recipes. This is justified by the reasoning that the way in which other industries produce their commodities would not change in the short term due to the introduction of these new industries (For more details about this argument, see [PERSON] et al. [18]). Compared to the frequently used biproportional scaling approach for matrix balancing called RAS, it scales the columns instead of the rows.  
We start from matrices \(U^{1}\), \(V^{1}\) and \(y^{c,1}\), and we calculate the normalized Use table in Equation (19), the normalized Make table (also known as the Market Shares Matrix) in Equation (20) and the total final demands \(f^{c,1}\) and \(f^{I,1}\) in commodity and industry terms in Equations (21) and (22).  
\[U^{1}\_n =U^{1}\cdot\widehat{x^{\prime l,1}}^{-1} \tag{19}\] \[V^{1}\_n =V^{1}\cdot q^{\widehat{M,1}}^{-1}\] (20) \[f^{c,1} =y^{c,1}\cdot\mathbf{1}\] (21) \[f^{I,1} =V^{1}\_n\cdot f^{c,1} \tag{22}\]  
Here, the notation \(\hat{X}\) indicates a square matrix with the elements of vector \(X\) in the main diagonal. Using the unbalanced tables, the direct requirement matrices in commodity terms and industry terms are calculated by Equations (23) and (24), respectively. Similarly, the total requirement matrices (Leontief matrices) in commodity terms and in industry terms are calculated by Equations (25) and (26).  
\[A^{1}_{c\times c} =U^{1}\_n\cdot V^{1}\_n \tag{23}\] \[A^{1}_{l\times l} =V^{1}\_n\cdot U^{1}\_n\] (24) \[L^{1}_{c\times c} =(1-A^{1}_{l\times c\times})^{-1}\] (25) \[L^{1}_{l\times l} =(1-A^{1}_{l\times l})^{-1} \tag{26}\]  
Then, we obtain the total output requirements (including direct and indirect) to satisfy new total demands \(f^{c,1}\) and \(f^{I,1}\) as shown in Equations (27) and (28), respectively.  
\[q^{1} =L^{1}_{c\times c}\cdot f^{c,1} \tag{27}\] \[x^{1} =L^{1}_{l\times l}\cdot f^{I,1} \tag{28}\]With these outputs, we can calculate the scaling multipliers for the Make and Use tables using Equations (29) and (30), respectively.  
\[m^{M} =\widehat{q^{M,1}}^{-1}\cdot\widehat{q}^{1} \tag{29}\] \[m^{U} =\widehat{x^{U,1}}^{-1}\cdot\widehat{x_{1}} \tag{30}\]  
With these scaling multipliers, we obtain a balanced Use table by scaling Use transactions and value added as shown in Equations (31) and (32), respectively. The final demand vector in the balanced Use table remains the same. The balanced Make table is obtained by scaling the Make transactions as shown in Equation (33).  
\[U^{2} =U^{1}\cdot m^{U} \tag{31}\] \[v^{2} =v^{1}\cdot m^{U}\] (32) \[V^{2} =V^{2}\cdot m^{M} \tag{33}\]