Socioeconomic spillovers of the 2016–2017 Italian earthquakes: a bi-regional inoperability model

Abstract

In August and October 2016, and January 2017, Central Italy was shaken by four strong earthquakes followed by other earthquake swarms. These disruptive phenomena, besides bringing devastation in the territory directly involved, caused economic blackouts to important transactions among activities, with consequent different reactions in the economic performance of the whole country. Therefore, the overall economic impact of a disaster should encompass the complete representation of phenomenon, and requires an analytical framework to depict the circular flow of income in all its phases. In this perspective, the current study presents an evolution of the inoperability input–output model by introducing a new approach of bi-regional inoperability extended multisectoral model. This allows assessing the intra-regional and the inter-regional effects of the earthquakes in the production processes and in the institutional sectors disposable incomes of two Italian macro areas, the North-Centre and the South-Islands.

Appendices

Appendix 1

See Fig. 11.

Fig. 11

Source: own elaboration

The bi-regional circular flow of income.

Table 3 shows a comparison between the results of the B-IEMM and the B-IIM approach in order to underline advantages and limitations of the two methodologies.

Table 3 Output performance: B-IEMM versus B-IIM (in percentage variation)

Appendix 2: The bi-regional extended multisectorial model approach

Considering an open economic system with b Industries, n Primary factors and h institutional sectors, the main equation of the model can be introduced.

$$\left\{ {\frac{{m^{\rm SI} }}{{m^{\rm NC} }}} \right\} + \left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\} = \left\{ {\frac{{r^{\rm SI} }}{{r^{\rm NC} }}} \right\} + \left\{ {\frac{{f_{d}^{\rm SI} }}{{f_{d}^{\rm NC} }}} \right\}$$

(12)

This structural form better describes the bi-regional dimension of the model where each variables is divided by the two areas, South-Islands (SI) and North-Centre (NC). Equation 12 variables represent the imports vector m, the domestic industry output x, the vector of the total intermediate consumption r and the final demand vector f_d composed by an endogenous and an exogenous part, f_d = f_c + f₀.

$$\left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\} = \left\{ {\frac{{r^{\rm SI} }}{{r^{\rm NC} }}} \right\} + \left\{ {\frac{{f_{c}^{\rm SI} }}{{f_{c}^{\rm NC} }}} \right\} + \left\{ {\frac{{f_{0}^{\rm SI} }}{{f_{0}^{\rm NC} }}} \right\} - \left\{ {\frac{{m^{\rm SI} }}{{m^{\rm NC} }}} \right\}$$

(13)

Equation 14 shows how the intermediate consumption vector r is given by the product of the technical coefficients matrix A[b, b] and the industry output vector x.

$$\left\{ {\frac{{r^{\rm SI} }}{{r^{\rm NC} }}} \right\} = \left\{ {\frac{{A^{{\rm SI},{\rm SI}} }}{{A^{{\rm NC},{\rm SI}} }}|\frac{{A^{{\rm SI},{\rm NC}} }}{{A^{{\rm NC},{\rm NC}} }}} \right\}\left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\}$$

(14)

The net exports vector f can now be defined as,

$$\left\{ {\frac{{f^{\rm SI} }}{{f^{\rm NC} }}} \right\} = \left\{ {\frac{{f_{0}^{\rm SI} }}{{f_{0}^{\rm NC} }}} \right\} - \left\{ {\frac{{m^{\rm SI} }}{{m^{\rm NC} }}} \right\}$$

(15)

Replacing Eqs. 14 and 15 in Eq. 13, the bi-regional extended multisectoral model can also be expressed as follows.

$$\left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\} = \left\{ {\frac{{A^{{\rm SI},{\rm SI}} }}{{A^{{\rm NC},{\rm SI}} }}|\frac{{A^{{\rm SI},{\rm NC}} }}{{A^{{\rm NC},{\rm NC}} }}} \right\}\left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\} + \left\{ {\frac{{f_{c}^{\rm SI} }}{{f_{c}^{\rm NC} }}} \right\} + \left\{ {\frac{{f^{\rm SI} }}{{f^{\rm NC} }}} \right\}$$

(16)

The value added by industry can be defined as

$$\left\{ {\frac{{v_{io}^{\rm SI} }}{{v_{io}^{\rm NC} }}} \right\} = \left\{ {\frac{{L^{\rm SI} }}{0}|\frac{0}{{L^{\rm NC} }}} \right\}\left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\}$$

(17)

with L[b, b] being a diagonal matrix and $l_{j} = 1 - { }\mathop \sum \limits_{i = 1}^{n} a_{ij}$. In order to obtain the value added by its components v_c can be introduced as follows

$$\left\{ {\frac{{v_{c}^{\rm SI} }}{{v_{c}^{\rm NC} }}} \right\} = \left\{ {\frac{{W^{{\rm SI},{\rm SI}} }}{{W^{{\rm NC},{\rm SI}} }}|\frac{{W^{{\rm SI},{\rm NC}} }}{{W^{{\rm NC},{\rm NC}} }}} \right\} \left\{ {\frac{{v_{io}^{\rm SI} }}{{v_{io}^{\rm NC} }}} \right\}$$

(18)

, where W[n, b] represents a matrix of shares of Primary factors.

The value added by Institutional sector vis is given by

$$\left\{ {\frac{{v_{is}^{\rm SI} }}{{v_{is}^{\rm NC} }}} \right\} = \left\{ {\frac{{P^{\rm SI} }}{0}|\frac{0}{{P^{\rm NC} }}} \right\} \left\{ {\frac{{v_{c}^{\rm SI} }}{{v_{c}^{\rm NC} }}} \right\}$$

(19)

with P[h, n] being a shares matrix of the distribution of primary income that contributes in determining the disposable income through the inter-regional and intra-regional transfer flows. Having finalized the first phase regarding the circular flow of income, the disposable income vector can now be reconstructed.

$$\left\{ {\frac{{y^{\rm SI} }}{{y^{\rm NC} }}} \right\} = \left[ {\left\{ {\frac{{I_{1} }}{0}|\frac{0}{{I_{2} }}} \right\} + \left\{ {\frac{{T^{{\rm SI},{\rm SI}} }}{{T^{{\rm NC},{\rm SI}} }}|\frac{{T^{{\rm SI},{\rm NC}} }}{{T^{{\rm NC},{\rm NC}} }}} \right\} \left\{ {\frac{{v_{is}^{\rm SI} }}{{v_{is}^{\rm NC} }}} \right\}} \right]$$

(20)

The matrix T[h, h] represents the shares of the net transfers between the institutional sectors in the secondary distribution of income.

$$\left\{ {\frac{{D^{{\rm SI},{\rm SI}} }}{{D^{{\rm NC},{\rm SI}} }}|\frac{{D^{{\rm SI},{\rm NC}} }}{{D^{{\rm NC},{\rm NC}} }}} \right\} = \left[ {\left\{ {\frac{{I_{1} }}{0}|\frac{0}{{I_{2} }}} \right\} + \left\{ {\frac{{T^{{\rm SI},{\rm SI}} }}{{T^{{\rm NC},{\rm SI}} }}|\frac{{T^{{\rm SI},{\rm NC}} }}{{T^{{\rm NC},{\rm NC}} }}} \right\}} \right]\left\{ {\frac{{P^{\rm SI} }}{0}|\frac{0}{{P^{\rm NC} }}} \right\} \left\{ {\frac{{W^{{\rm SI},{\rm SI}} }}{{W^{{\rm NC},{\rm SI}} }}|\frac{{W^{{\rm SI},{\rm NC}} }}{{W^{{\rm NC},{\rm NC}} }}} \right\} \left\{ {\frac{{L^{\rm SI} }}{0}|\frac{0}{{L^{\rm NC} }}} \right\}$$

(21)

Introducing D[h, h] as the product of the previous structural matrices and substituting it in the disposable income Eq. 20, the vector y can also be expressed as

$$\left\{ {\frac{{y^{\rm SI} }}{{y^{\rm NC} }}} \right\} = \left\{ {\frac{{D^{{\rm SI},{\rm SI}} }}{{D^{{\rm NC},{\rm SI}} }}|\frac{{D^{{\rm SI},{\rm NC}} }}{{D^{{\rm NC},{\rm NC}} }}} \right\} \left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\}$$

(22)

The closing of the circular flow of income loop has been obtained through the construction of the endogenous final demand vector f_c,

$$\left\{ {\frac{{f_{c}^{\rm SI} }}{{f_{c}^{\rm NC} }}} \right\} = \left\{ {\frac{{G^{{\rm SI},{\rm SI}} }}{{G^{{\rm NC},{\rm SI}} }}|\frac{{G^{{\rm SI},{\rm NC}} }}{{G^{{\rm NC},{\rm NC}} }}} \right\} \left\{ {\frac{{y^{\rm SI} }}{{y^{\rm NC} }}} \right\}$$

(23)

, where G can be decomposed in

$$\left\{ {\frac{{G^{{\rm SI},{\rm SI}} }}{{G^{{\rm NC},{\rm SI}} }}|\frac{{G^{{\rm SI},{\rm NC}} }}{{G^{{\rm NC},{\rm NC}} }}} \right\} = \left\{ {\frac{{F^{{\rm SI},{\rm SI}} }}{{F^{{\rm NC},{\rm SI}} }}|\frac{{F^{{\rm SI},{\rm NC}} }}{{F^{{\rm NC},{\rm NC}} }}} \right\}\left\{ {\frac{{K^{{\rm SI},{\rm SI}} }}{{K^{{\rm NC},{\rm SI}} }}|\frac{{K^{{\rm SI},{\rm NC}} }}{{K^{{\rm NC},{\rm NC}} }}} \right\}$$

(24)

Here, the matrix F = F1C, is composed by F1[b, h] which transforms the consumption by Institutional sector into consumption by I–O, meanwhile each element of the diagonal matrix C[h, h] represents the propensity of consumption by Institutional sector.

$$\left\{ {\frac{{F^{{\rm SI},{\rm SI}} }}{{F^{{\rm NC},{\rm SI}} }}|\frac{{F^{{\rm SI},{\rm NC}} }}{{F^{{\rm NC},{\rm NC}} }}} \right\} = \left\{ {\frac{{F1^{{\rm SI},{\rm SI}} }}{{F1^{{\rm NC},{\rm SI}} }}|\frac{{F1^{{\rm SI},{\rm NC}} }}{{F1^{{\rm NC},{\rm NC}} }}} \right\}\left\{ {\frac{{C^{\rm SI} }}{0}|\frac{0}{{C^{\rm NC} }}} \right\}$$

(25)

The matrix K can be rewritten as K1s(I−C), where K1[b, h] represents the matrix that transforms the gross investment by Institutional sector into I–O, the scalar s resumes the "active saving" and I−C captures the saving propensity by Institutional sector.

$$\left\{ {\frac{{K^{{\rm SI},{\rm SI}} }}{{K^{{\rm NC},{\rm SI}} }}|\frac{{K^{{\rm SI},{\rm NC}} }}{{K^{{\rm NC},{\rm NC}} }}} \right\} = \left\{ {\frac{{K1^{{\rm SI},{\rm SI}} }}{{K1^{{\rm NC},{\rm SI}} }}|\frac{{K1^{{\rm SI},{\rm NC}} }}{{K1^{{\rm NC},{\rm NC}} }}} \right\}s\left[ {\left\{ {\frac{{I_{1} }}{0}|\frac{0}{{I_{2} }}} \right\} - \left\{ {\frac{{C^{\rm SI} }}{0}|\frac{0}{{C^{\rm NC} }}} \right\} } \right]$$

(26)

After making the required substitutions, the endogenous final demand formation vector f_c can therefore be expressed as

$$\left\{ {\frac{{f_{c}^{\rm SI} }}{{f_{c}^{\rm NC} }}} \right\} = \left\{ {\frac{{G^{{\rm SI},{\rm SI}} }}{{G^{{\rm NC},{\rm SI}} }}|\frac{{G^{{\rm SI},{\rm NC}} }}{{G^{{\rm NC},{\rm NC}} }}} \right\}\left\{ {\frac{{D^{{\rm SI},{\rm SI}} }}{{D^{{\rm NC},{\rm SI}} }}|\frac{{D^{{\rm SI},{\rm NC}} }}{{D^{{\rm NC},{\rm NC}} }}} \right\} \left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\}$$

(27)

By defining and replacing E as E = GD in Eq. 27, and substituting in Eq. 16, the structural form of the bi-regional extended multisectoral model can be obtained.

$$\left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\} = \left\{ {\frac{{A^{{\rm SI},{\rm SI}} }}{{A^{{\rm NC},{\rm SI}} }}|\frac{{A^{{\rm SI},{\rm NC}} }}{{A^{{\rm NC},{\rm NC}} }}} \right\}\left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\} + \left\{ {\frac{{E^{{\rm SI},{\rm SI}} }}{{E^{{\rm NC},{\rm SI}} }}|\frac{{E^{{\rm SI},{\rm NC}} }}{{E^{{\rm NC},{\rm NC}} }}} \right\} \left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\}\left\{ {\frac{{f^{\rm SI} }}{{f^{\rm NC} }}} \right\}$$

(28)

Alternatively, Eq. 28, can be defined in its reduced form.

$$\left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\} = \left[ {\left\{ {\frac{{I_{1} }}{0}|\frac{0}{{I_{2} }}} \right\} - \left\{ {\frac{{A^{{\rm SI},{\rm SI}} }}{{A^{{\rm NC},{\rm SI}} }}|\frac{{A^{{\rm SI},{\rm NC}} }}{{A^{{\rm NC},{\rm NC}} }}} \right\} - \left\{ {\frac{{E^{{\rm SI},{\rm SI}} }}{{E^{{\rm NC},{\rm SI}} }}|\frac{{E^{{\rm SI},{\rm NC}} }}{{E^{{\rm NC},{\rm NC}} }}} \right\}} \right]^{ - 1} \left\{ {\frac{{f^{\rm SI} }}{{f^{\rm NC} }}} \right\}$$

(29)

The model for the disposable income, at this point, can be solved.

$$\left\{ {\frac{{D^{{\rm SI},{\rm SI}} }}{{D^{{\rm NC},{\rm SI}} }}|\frac{{D^{{\rm SI},{\rm NC}} }}{{D^{{\rm NC},{\rm NC}} }}} \right\} \left\{ {\frac{{x^{\rm SI} }}{{x^{\rm NC} }}} \right\} = \left\{ {\frac{{D^{{\rm SI},{\rm SI}} }}{{D^{{\rm NC},{\rm SI}} }}|\frac{{D^{{\rm SI},{\rm NC}} }}{{D^{{\rm NC},{\rm NC}} }}} \right\} \left[ {\left\{ {\frac{{I_{1} }}{0}|\frac{0}{{I_{2} }}} \right\} - \left\{ {\frac{{A^{{\rm SI},{\rm SI}} }}{{A^{{\rm NC},{\rm SI}} }}|\frac{{A^{{\rm SI},{\rm NC}} }}{{A^{{\rm NC},{\rm NC}} }}} \right\} - \left\{ {\frac{{E^{{\rm SI},{\rm SI}} }}{{E^{{\rm NC},{\rm SI}} }}|\frac{{E^{{\rm SI},{\rm NC}} }}{{E^{{\rm NC},{\rm NC}} }}} \right\}} \right]^{ - 1} \left\{ {\frac{{f^{\rm SI} }}{{f^{\rm NC} }}} \right\}$$

(30)