The Economic and Social Impact of the Sustainable Evolution of the A4 Motorway Brescia - Padua

Abstract

The paper exploits an input-output analysis of a sustainable evolution of the A4 motorway between Brescia and Padua, which includes the improvement of service levels, and therefore also the option of expanding the section three to four lane highway.

Appendices

Appendix 1: Methodology

1.1 A.1.1 The Input-Output Analysis

Miller and Blair (2009) present a summarized version of the input-output table:

		intermediate demand			final demand $D$				output
		1	$\cdots$	n	consumption	investment	public expenditure	exports
productive sector	1	${x}_{11}$	$\cdots$	${x}_{1n}$	${C}_{1}$	${I}_{1}$	${G}_{1}$	${E}_{1}$	${Y}_{1}$
	$\vdots$	$\vdots$	$\cdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$
	n	${x}_{n1}$	$\cdots$	${x}_{nn}$	${C}_{n}$	${I}_{n}$	${G}_{n}$	${E}_{n}$	${Y}_{n}$
added value		${V}_{1}$	$\cdots$	${V}_{n}$	${V}_{C}$	${V}_{I}$	${V}_{G}$	${V}_{E}$	$V$
imports		${M}_{1}$	$\cdots$	${M}_{n}$	${M}_{C}$	${M}_{I}$	${M}_{G}$	${M}_{E}$	$M$
output		${Y}_{1}$	$\cdots$	${Y}_{n}$	$C$	$I$	$G$	$E$	$Y$

The total of what is produced or imported in sector $j$ is equal to the sum of what is used internally (intermediate demand and final internal demand) and exported:

$$\sum_{i=1}^{n}{x}_{ij}+{V}_{j}+{M}_{j}=\sum_{i=1}^{n}{x}_{ji}+{C}_{j}+{I}_{j}+{G}_{j}+{E}_{j},$$

(A.1)

where $\sum_{i=1}^{n}{x}_{ij}$ is the intermediate demand for other goods $i$ used for the production of good $j$ (sector $j$ is a buyer), ${V}_{j}$ the added value at base prices, ${M}_{j}$ the imports of other goods $i$ used in the production of good $j$, $\sum_{i=1}^{n}{x}_{ji}$ the intermediate demand of good $j$ by other sectors $i$ (sector $j$ is a seller), ${C}_{j}$ final consumption expenditure net of imports (i.e. only the consumption of internally produced good $j$ is considered), ${I}_{j}$ gross investments net of imports (i.e. only investments in internally produced goods $j$ are considered), and ${E}_{j}$ exports net of imports (i.e. goods $j$ imported and then re-exported are not considered).²

Given the linear technology hypothesis, the quantity of input $j$ used in each production activity $i$ is proportional to the volume of output $\sum_{j=1}^{n}{x}_{ji}+{V}_{i}+{M}_{i}$ (sector $i$ is a buyer). Thus, the following relationship exists between the values of production and intermediate demand:

$${a}_{ji}=\frac{{x}_{ji}}{\sum_{j=1}^{n}{x}_{ji}+{V}_{i}+{M}_{i}}=\frac{{x}_{ji}}{{Y}_{i}}.$$

(A.2)

The expenditure coefficients (if measured in nominal value) or technical coefficients (if measured in real value) ${a}_{ji}$ express the value of the product of sector $j$ (input) necessary to produce one unit of value of sector $i$ (output). The elements ${a}_{ji}$ can be ordered in the $n\times n$ matrix of technical coefficients ${\text{A}}$:

$$\begin{array}{c}{\text{A}}\\ n\times n\end{array}=\left[\begin{array}{ccc}{a}_{11}& \dots & {a}_{1n}\\ \vdots & {a}_{ji}& \vdots \\ {a}_{n1}& \dots & {a}_{nn}\end{array}\right].$$

(A.3)

Hence, the model in (A.1) can be rewritten as:

$$\sum_{i=1}^{n}{x}_{ij}+{V}_{j}+{M}_{j}=\sum_{i=1}^{n}\left[{a}_{ji}\cdot \left(\sum_{j=1}^{n}{x}_{ji}+{V}_{i}+{M}_{i}\right)\right]+{C}_{j}+{I}_{j}+{G}_{j}+{E}_{j}.$$

(A.4)

Indicating the payments with ${Y}_{j}=\sum_{i=1}^{n}{x}_{ij}+{V}_{j}+{M}_{j}$ and the final demand (sum of final domestic demand and exports) with ${D}_{j}={C}_{j}+{I}_{j}+{G}_{j}+{E}_{j}$, the model in (A.4) can be rewritten as

$${Y}_{j}=\sum_{i=1}^{n}\left({a}_{ji}\cdot {Y}_{i}\right)+{D}_{j},$$

(A.5)

which in vector terms becomes:

$$ \begin{array}{*{20}c} {{\text{Y}} = {\text{A}}\quad {\text{Y}} + {\text{D}}} \\ {n \times 1\quad n \times n\,n \times 1\quad n \times 1} \\ {{\text{Y}} - {\text{A}}\quad {\text{Y}} = {\text{D}}} \\ {n \times 1\quad n \times n\,n \times 1\quad n \times 1} \\ {\left( {{\text{I}} - {\text{A}}} \right)\;\;\;{\text{Y}} = {\text{D}}} \\ {n \times n\;n \times 1\quad n \times 1} \\ {{\text{Y}} = \left( {{\text{I}} - {\text{A}}} \right)^{{ - 1}} \;\;{\text{D}}} \\ {n \times 1\quad n \times n\;\;\;n \times 1} \\ \end{array} , $$

(A.6)

where ${\text{I}}$ is an $n\times n$ identity matrix and ${{\text{L}}=\left({\text{I}}-{\text{A}}\right)}^{-1}$ is the Leontief matrix.

The final consumption of families of product $j$ (${C}_{j}$) depends on their income, which in turn depends on the production of the sector to which this income refers: the final consumption of families ${C}_{j}$ can be considered an endogenous component of the model. The full version of the input-output table is:

		endogenous demand (intermediate + consumption)				other final demand ${D}^{*}$			output
		1	$\cdots$	n	consumption	investment	public expenditure	exports
supply	1	${x}_{11}$	$\cdots$	${x}_{1n}$	${C}_{1}$	${I}_{1}$	${G}_{1}$	${E}_{1}$	${Y}_{1}$
	$\vdots$	$\vdots$	$\cdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$
	n	${x}_{n1}$	$\cdots$	${x}_{nn}$	${C}_{n}$	${I}_{n}$	${G}_{n}$	${E}_{n}$	${Y}_{n}$
income		${R}_{1}$	$\cdots$	${R}_{n}$	${R}_{C}$	-	-	-	$R$
other added value		${V}_{i}^{*}$	$\cdots$	${V}_{n}^{*}$	${V}_{C}^{*}$	${V}_{I}^{*}$	${V}_{G}^{*}$	${V}_{E}^{*}$	${V}^{*}$
imports		${M}_{1}$	$\cdots$	${M}_{n}$	${M}_{C}$	${M}_{I}$	${M}_{G}$	${M}_{E}$	$M$
output		${Y}_{1}$	$\cdots$	${Y}_{n}$	$C$	$I$	$G$	$E$	$Y$

(see Miller and Blair, 2009). In this case, the final (exogenous) demand does not include household consumption ${D}_{j}^{*}={I}_{j}+{G}_{j}+{E}_{j}$, and the added value ${V}_{j}$ must be broken down into labor income ${R}_{j}$³ and other components of the added value ${V}_{j}^{*}$:

$$\sum_{i=1}^{n}{x}_{ij}+{R}_{j}+{V}_{j}^{*}+{M}_{j}=\sum_{i=1}^{n}{x}_{ji}+{C}_{j}+{D}_{j}^{*}.$$

(A.7)

Even in the case of household final consumption and labour income it is possible to calculate coefficients similar to those calculated in (A.2). The household consumption coefficients are:

$${\gamma }_{j}=\frac{{C}_{j}}{\sum_{i=1}^{n}{C}_{i}+{V}_{C}+{M}_{C}}=\frac{{C}_{j}}{C},$$

(A.8)

where $\sum_{i=1}^{n}{C}_{i}$ is the final consumption expenditure of families, and the labor income coefficients are:

$${\rho }_{j}=\frac{{R}_{j}}{\sum_{i=1}^{n}{x}_{ij}+{V}_{j}+{M}_{j}}=\frac{{R}_{j}}{{Y}_{j}},$$

(A.9)

where $\sum_{i=1}^{n}{x}_{ij}$ indicates the quantity of input $i$ used in production activity $j$ (sector $j$ is a buyer).

Given the consumption coefficients in (A.8) and the labour income coefficients in (A.9), it is possible to define a $\left(n+1\right)\times \left(n+1\right)$ matrix of technical coefficients which also includes the household “sector”:

$$ \begin{array}{*{20}c} {\overline{{\text{A}}} } \\ {\left( {n + 1} \right) \times \left( {n + 1} \right)} \\ \end{array} = \left[ {\begin{array}{*{20}c} {\text{A}} \hfill & {{{\upgamma}}} \hfill \\ {n \times n} \hfill & {n \times 1} \hfill \\ {{{\uprho}}^{{ \top }} } \hfill & {\rho _{C} } \hfill \\ {1 \times n} \hfill & {} \hfill \\ \end{array} } \right], $$

(A.10)

where $\upgamma$ is the column vector of the consumption coefficients, ${\uprho }^{\top }$ the row vector of the labour income coefficients and ${\rho }_{C}={R}_{C}/C$ is the (scalar) coefficient that refers to “activities of families and partnerships as employers for domestic staff and production of undifferentiated goods and services for own use by families and cohabitations” (NACE item VT). Thus, the model in (A.6) can be rewritten as:

$$\begin{array}{c}\begin{array}{c}\left[\begin{array}{c}\begin{array}{c}{\text{Y}}\\ n\times 1\end{array}\\ R\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\begin{array}{c} = \; \left[ {\begin{array}{*{20}c} {\text{A}} & \upgamma \\ {n \times n} & {n \times 1} \\ {\uprho ^{{ \top }} } & {\rho _{C} } \\ {1 \times n} & {} \\ \end{array} } \right] \\ \left(n+1\right)\times \left(n+1\right)\end{array}\begin{array}{c}\left[\begin{array}{c}\begin{array}{c}{\text{Y}}\\ n\times 1\end{array}\\ R\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\begin{array}{c}+\;\left[\begin{array}{c}\begin{array}{c}{{\text{D}}}^{*}\\ n\times 1\end{array}\\ 0\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\\ \begin{array}{c}\left[\begin{array}{c}{\text{Y}}\\ R\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\begin{array}{c}-\;\left[\begin{array}{cc}{\text{A}}&\upgamma \\ {\uprho }^{\top }& {\rho }_{C}\end{array}\right]\\ \left(n+1\right)\times \left(n+1\right)\end{array}\begin{array}{c}\left[\begin{array}{c}{\text{Y}}\\ R\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\begin{array}{c} = \; \left[\begin{array}{c}{{\text{D}}}^{*}\\ 0\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\\ \begin{array}{c}\left({\text{I}}-\left[\begin{array}{cc}{\text{A}}&\upgamma \\ {\uprho }^{\top }& {\rho }_{C}\end{array}\right]\right)\\ \left(n+1\right)\times \left(n+1\right)\end{array}\begin{array}{c}\left[\begin{array}{c}{\text{Y}}\\ R\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\begin{array}{c}=\;\left[\begin{array}{c}{{\text{D}}}^{*}\\ 0\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\\ \begin{array}{c}\left[\begin{array}{c}{\text{Y}}\\ R\end{array}\right]\\ \left(n+1\right)\times 1\end{array}\begin{array}{c}=\;{\left({\text{I}}-\left[\begin{array}{cc}{\text{A}}&\upgamma \\ {\uprho }^{\top }& {\rho }_{C}\end{array}\right]\right)}^{-1}\\ \left(n+1\right)\times \left(n+1\right)\end{array}\begin{array}{c}\left[\begin{array}{c}{{\text{D}}}^{*}\\ 0\end{array}\right]\\ \left(n+1\right)\times 1\end{array},\end{array}$$

(A.11)

with $R$ wages of the labour employed in the production of goods consumed by families $C$, and where ${\text{I}}$ is a $\left(n+1\right)\times \left(n+1\right)$ identity matrix and ${\bar{\text{L}}} = ({\text{I}} - {\bar{\text{A}}})^{{ - 1}}$ is the complete Leontief matrix, i.e., with endogenous household final consumption.

1.2 A.1.2 Multipliers and Impacts

The Leontief matrices in (A.6) and (A.11) allow the calculation of sector multipliers. Below we denote the elements of the matrix ${\text{L}}$ by ${l}_{ij}$ and the elements of the matrix ${\overline{\text{L}} }$ by ${\overline{l} }_{ij}$ (see Miller and Blair, 2009). Based on the multipliers, it is possible to estimate the direct, indirect, and induced impacts on the economic system due to a change in final demand.

The estimate of direct, indirect, and induced impacts in terms of production (output) is carried out by considering the output multipliers. With reference to the simple model in (A.6) and the complete model in (A.11) the output multipliers are:

$$ \begin{array}{*{20}c} {o\left( j \right) = \sum\limits_{{i = 1}}^{n} {l_{{ij}} } ,} \\ {\bar{o}\left( j \right) = \sum\limits_{{i = 1}}^{{n + 1}} {\bar{l}_{{ij}} } .} \\ \end{array} $$

(A.12)

In sector $j$ an increase in final demand of one euro generates a new output equal to:

• $o\left(j\right)$ when direct and indirect effects are considered,

• $\overline{o }\left(j\right)$ when considering direct, indirect, and induced effects.

As underlined by Miller and Blair (2009), it is interesting to consider the economic impacts of a change in final demand measured not in terms of production, but in terms of income (income) and jobs created (employment). In this regard, multipliers are considered that convert the elements of the matrices ${\text{L}}$ and $\overline{{\text{L}} }$ into the value of employment in euros using coefficients relating to the labour production factor, and these coefficients can be either monetary (income multipliers) or physical (employment multipliers).

The income multipliers weight the elements of the matrices ${\text{L}}$ and $\overline{{\text{L}} }$ considering the income from work. With reference to the simple model in (A.6) and the complete model in (A.11) the income multipliers are:

$$ \begin{array}{*{20}c} {h\left( j \right) = \sum\limits_{{i = 1}}^{n} {\rho _{i} } \cdot l_{{ij}} ,} \\ {\bar{h}\left( j \right) = \sum\limits_{{i = 1}}^{{n + 1}} {\rho _{i} } \cdot \bar{l}_{{ij}} .} \\ \end{array} $$

(A.13)

In sector $j$ an increase in final demand of one euro generates an additional income for families equal to:

• $h\left(j\right)$ when the direct and indirect effects are converted into income,

• $\overline{h }\left(j\right)$ when the direct, indirect, and induced effects are converted into income.

Therefore, to obtain the income the initial income must be multiplied by $1+h\left(j\right)$ or $1+\overline{h }\left(j\right)$.

There is also another type of income multipliers, defined as Type I and Type II multipliers, which are obtained by dividing the multipliers in (A.13) by the initial effect on labour income:

$$ \begin{array}{*{20}c} {h^{I} \left( j \right) = \frac{{\sum\nolimits_{{i = 1}}^{n} {\rho _{i} } \cdot l_{{ij}} }}{{\rho _{j} }} = \frac{{h\left( j \right)}}{{\rho _{j} }},} \\ {h^{{II}} \left( j \right) = \frac{{\sum\nolimits_{{i = 1}}^{{n + 1}} {\rho _{i} } \cdot l_{{ij}}^{*} }}{{\rho _{j} }} = \frac{{\bar{h}\left( j \right)}}{{\rho _{j} }}.} \\ \end{array} $$

(A.14)

In sector $j$ an increase in final demand of ${\rho }_{j}$ euro generates a new income for families of an amount equal to:

• ${h}^{I}\left(j\right)$ when direct and indirect effects are converted into income,

• ${h}^{II}\left(j\right)$ when direct, indirect, and induced effects are converted into income.

Miller and Blair (2009) note that while Type I multipliers likely underestimate economic impact, Type II multipliers likely overestimate it.

The employment multipliers weight the elements of the matrices ${\text{L}}$ and $\overline{{\text{L}} }$ considering the number of employed people. In this regard, the European System of National Accounts indicates that work units (AWU) are a better indicator than employed people for understanding the actual dynamics of work. The use of AWUs was necessary as the person can take on one or more work positions based on:

• the activity (sole, main, or secondary),

• the position in the profession (employee or self-employed),

• the duration (continuous or non-continuous),

• the working hours (full-time or part-time).

Thus, the AWU represents the quantity of work performed during the year by a full-time employee, or the equivalent quantity of work performed by part-time workers or workers who carry out double work.⁴

As done in Eqs. (A.2), (A.8), and (A.9) an employment-production ratio can be derived; this employment-production ratio illustrates the number of AWUs (average) per one euro of output produced in each sector $j$, i.e., a row vector ${\upvarepsilon }^{\top }$ of employment-production relations:

$${\varepsilon }_{j}=\frac{{N}_{j}}{\sum_{i=1}^{n}{x}_{ij}+{V}_{j}+{M}_{j}}=\frac{{N}_{j}}{{Y}_{j}}.$$

(A.15)

With reference to the simple model in (A.6) and the complete model in (A.11) the employment multipliers are:

$$ \begin{array}{*{20}c} {e\left( j \right) = \sum\limits_{{i = 1}}^{n} {\varepsilon _{i} } \cdot l_{{ij}} ,} \\ {\bar{e}\left( j \right) = \sum\limits_{{i = 1}}^{{n + 1}} {\varepsilon _{i} } \cdot \bar{l}_{{ij}} .} \\ \end{array} $$

(A.16)

In sector $j$ an increase in final demand of one euro generates additional work units of an amount equal to:

• $e\left(j\right)$ when direct and indirect effects are considered,

• $\overline{e }\left(j\right)$ when considering direct, indirect, and induced effects.

Finally, if the analysis is restricted to the effects on the production system (and therefore on $n$ sectors), the truncated multipliers can be calculated:

$$ \begin{array}{*{20}c} {\bar{o}_{T} \left( j \right) = \sum\limits_{{i = 1}}^{n} {\bar{l}_{{ij}} } ,} \\ {\bar{h}_{T} \left( j \right) = \sum\limits_{{i = 1}}^{n} {\rho _{i} } \cdot \bar{l}_{{ij}} ,} \\ {h_{T}^{{II}} \left( j \right) = \frac{{\sum\nolimits_{{i = 1}}^{n} {\rho _{i} } \cdot l_{{ij}}^{*} }}{{\rho _{j} }} = \frac{{\bar{h}\left( j \right)}}{{\rho _{j} }}} \\ {\bar{e}_{T} \left( j \right) = \sum\limits_{{i = 1}}^{n} {\varepsilon _{i} } \cdot \bar{l}_{{ij}} } \\ \end{array} , $$

(A.17)

(Cassar, 2015).

In the case of output multipliers, since the multiplier $o\left(j\right)$ only captures the direct and indirect impacts and the multiplier ${\overline{o} }_{T}\left(j\right)$ also captures the induced impacts, as suggested by Cassar (2015), the induced impact can be estimated by doing the difference between ${\overline{o} }_{T}\left(j\right)$ and $o\left(j\right)$:

$$ {\text{induced}}\;{\text{impact}} = \overline{o}_{T} (j) - o(j). $$

(A.18)

In the case of income multipliers, the direct impact is captured by the vector $\uprho$ whose elements are ${\rho }_{j}={R}_{j}/{Y}_{j}$, as seen in Eq. (A.9). So as suggested by Cassar (2015) the indirect impact is:

$$ {\text{indirect}}\;{\text{impact}} = h(j) - \rho_{j} $$

(A.19)

and the induced impact is:

$$ {\text{induced}}\;{\text{impact}} = \overline{h}_{T} (j) - h(j). $$

(A.20)

Finally, also in the case of income multipliers Type I and Type II the induced impact can be estimated by making the difference between ${h}^{II}\left(j\right)$ and ${h}^{I}\left(j\right)$:

$$ {\text{induced}}\;{\text{impact}} = h_{T}^{II} (j) - h^{I} (j). $$

(A.21)

Appendix 2: Additional Tables

Table A.1. The planned investments by NACE code.

Table A.2. Output and employment multipliers by NACE code.

Table A.3. The 63 NACE codes and the 10 economic sectors.