**4. IO modelling framework**

#### **4.1. Key definitions and assumptions**

A regional or national economic system could be depicted by a capital flow table between all the sectors of the economy that is the base IO table. This table is the sum of all the business sectors that constitute the gross national product [27]. Even though some business sectors' total capital output may be lower to the overall value added; however, for simplification reasons, it is assumed that each business sectors' total income output represents the weight of this sector to a given economic system. In Ref. [27], it is noted that this is one kind of balance of the economic system, which can be called a balance between final output values and value added [27].

Mathematically, the IO table is depicted as a matrix where the rows and columns represent different business sectors. Each cell of the matrix provides the income (production and consumption) between different business sectors for a given time window, usually, annually. Construction business sector, for instance, would create income to the fabricated metal business sector. Obviously, it is too costly to collect and store all the transaction between all economic sectors; therefore, in the most of the developed world states, it happens periodically between 3 and 5 years. For the years with no actual data, the IO table figures are given by a time series regression analysis, based on the last available actual data.

Reviewing the economic impact of air transports in a region, the analysis framework provides results for three distinguished causalities, measuring the changes caused not just to air transport business but also to the maintenance and the new investments delivered to improve air transport infrastructures. As a result, even for the same demand level, the IO analysis may provide different results over time, resulting in high variations in income generated in various business sectors, depending on the capitals spent for construction, operation and maintenance of infrastructures, according to Ref. [28].

According to Ref. [29], it is assumed that the economy can be categorised into *n* sectors. If the total output is denoted by *xi* and by *f i* of the total final demand for sector *i*'s product, then the simple equation accounting for the way in which sector *i* distributes its product through sales to other sectors and to final demand is

$$X\_{i} = z\_{i1} + \dots + z\_{ij} + \dots + z\_{in} + f\_{i} = \sum\_{j=1}^{n} z\_{ij} + f\_{i} \tag{1}$$

In IO analysis, the fundamental assumption is that the flows of sector *i* to *j* depend on the total output of sector *j*. The *zij* terms represent interindustry sales by sector *i* (also known as intermediate sales) to all sectors *j* (including itself, when *j* = *i*) [29].

#### **4.2. Technological coefficients**

The fundamental step of the IO analysis is to convert the interindustry transaction table into the direct purchase coefficient table. Based on the above-mentioned assumptions of the IO table that the flows of sector *i* to *j* depend on the total output of sector *j*, the technical coefficient can be derived by dividing the flows between the business sector *i* to business sector *j* (*zij*) with a overall output of business sector *j* (*Xj* ). The function that depicts the total technical coefficient table is [29].

$$\mathbf{c}\_{\parallel} = \frac{\mathbf{z}\_{\dot{\eta}}}{X\_{\parallel}} \tag{2}$$

where *cij* is the IO coefficient defined as direct input coefficient. The *cij* determines the flows between the business sector's output and its inputs and is defined as follows [29]:

$$z\_{\parallel} = \frac{c\_{\parallel}}{X\_{\parallel}} \tag{3}$$

IO table describes the national economy as a system that depicts the flows between m business sectors. The input output economic system equilibrium equation of this model can be written in matrix as Ref. [29] defines:

$$(I - A)^{-1}X = \; F \tag{4}$$

where *I* is the *m* × *m* matrix, *X* is the nonnegative vector of the total output of the business sectors, *F* is the nonnegative vector of the final demand, *A* is the *n* × *n* nonnegative matrix of technological coefficients or the input-output matrix and *m* is the number of business sectors in which (*I*−*A*) −1 is the multiplier or Leontief inverse matrix [29].

The matrix (*I*−*A*)−1 is the inverse of (*I*−*A*) and is defined as the Leontief matrix. The solution of Eq. (4) has a meaning if and only if (*I*−*A*) is a non-singular M-matrix. Indeed, the theory of M-matrices implies that a nonnegative solution *x* exists corresponding to each nonnegative *m*.

#### **4.3. Modelling regionalisation**

**4. IO modelling framework**

**4.1. Key definitions and assumptions**

150 Mobilities, Tourism and Travel Behavior - Contexts and Boundaries

A regional or national economic system could be depicted by a capital flow table between all the sectors of the economy that is the base IO table. This table is the sum of all the business sectors that constitute the gross national product [27]. Even though some business sectors' total capital output may be lower to the overall value added; however, for simplification reasons, it is assumed that each business sectors' total income output represents the weight of this sector to a given economic system. In Ref. [27], it is noted that this is one kind of balance of the economic system, which can be called a balance between final output values and value added [27]. Mathematically, the IO table is depicted as a matrix where the rows and columns represent different business sectors. Each cell of the matrix provides the income (production and consumption) between different business sectors for a given time window, usually, annually. Construction business sector, for instance, would create income to the fabricated metal business sector. Obviously, it is too costly to collect and store all the transaction between all economic sectors; therefore, in the most of the developed world states, it happens periodically between 3 and 5 years. For the years with no actual data, the IO table figures are given by a

Reviewing the economic impact of air transports in a region, the analysis framework provides results for three distinguished causalities, measuring the changes caused not just to air transport business but also to the maintenance and the new investments delivered to improve air transport infrastructures. As a result, even for the same demand level, the IO analysis may provide different results over time, resulting in high variations in income generated in various business sectors, depending on the capitals spent for construction, operation and mainte-

According to Ref. [29], it is assumed that the economy can be categorised into *n* sectors. If the

simple equation accounting for the way in which sector *i* distributes its product through sales

In IO analysis, the fundamental assumption is that the flows of sector *i* to *j* depend on the

The fundamental step of the IO analysis is to convert the interindustry transaction table into the direct purchase coefficient table. Based on the above-mentioned assumptions of the IO table that the flows of sector *i* to *j* depend on the total output of sector *j*, the technical

*<sup>i</sup>* = ∑ *j*=1 *n zij* + *f*

terms represent interindustry sales by sector *i* (also known as

of the total final demand for sector *i*'s product, then the

*<sup>i</sup>* (1)

time series regression analysis, based on the last available actual data.

 and by *f i*

intermediate sales) to all sectors *j* (including itself, when *j* = *i*) [29].

nance of infrastructures, according to Ref. [28].

*Xi* = *zi*<sup>1</sup> + ..+*zij* + …+*zin* + *f*

total output is denoted by *xi*

total output of sector *j*. The *zij*

**4.2. Technological coefficients**

to other sectors and to final demand is

The approach of a regional IO modelling framework tries to adapt the national input-output tables with the use of location quotients derived from differences of regional and national employment and production patterns. A simple location quotient for each regional economic sector can be defined as:

$$SLQ^{\prime}\_{\!\!\!\!} = \frac{Q^{\prime}\_{\!\!\!\!\!\!}}{T^{\prime}} \frac{\frac{Q^{\prime}}{T^{\prime}}}{Q^{\prime \prime}\_{\!\!\!\!\!\!\!\!\!}} \tag{5}$$

where *Qir* is the total output of business sector *i* in region *r*, *QiN* is the total output of business sector *i* in nation *Ν*, *T <sup>r</sup>* is the sum of all business sectors in region *r* and *TN* is the sum of all business sectors in nation *Ν*.

The SLQ depicts the region's capability in producing the output of a business sector. Referring to Eq. (5), an SLQ of less than 1.0 means that the output of regional business sector i represents a small proportion of the total gross output. Thus, if *SLQi* is less than 1.0, the region imports some of the output of the sector *i* from elsewhere in the nation *N* as in Ref. [30] is analysed. Similarly, if SLQi is greater than 1, the region exports some of the output of its sector to the rest of the nation *N*.
