Lost in Taxation

Abstract

Cost–Benefit Analysis (CBA) was developed to assess the net socioeconomic benefits of a wide variety of projects in many fields. In this context, it is relevant to investigate how this method is actually used for project evaluation, and whether its merits and limitations are properly understood by a wider community of economists. In this study, we showcase a debate that took place in Italy in 2019 about an important high-speed rail project, following the publication of a CBA that received much criticism. To learn from this episode, we find it useful to set up a meta-model of CBA that allows the formalisation of a large number of CBA calculations (including potentially ill-founded calculations) and to verify their validity. With this meta model, we review the criticisms formulated during the 2019 CBA debate focusing on two salient topics; whether CBA should include taxation and whether the Rule-of-Half measure of users’ surplus is valid. Our analysis suggests: (1) That the proposed meta-equation can help in structuring the scientific debate regarding CBA and the relevant economic discussion about a given project; (2) with few exceptions, the criticisms formulated regarding the 2019 CBA on these topics were incorrect, mostly incoherent, also from an axiomatic point of view. This indicates that ill-founded methods are at risk of becoming well-accepted in the larger community of economists, with the risk of lowering the general quality of policy recommendations they can formulate. This underlines the need for economists to revise the misguided views of CBA.

1 Introduction

Cost–Benefit Analysis (CBA) is occasionally at the centre of important public debates covering decades, various topics, and many political contexts. Examples include the third London Airport (Roskill commission back in the 70’s), the Boston Big Dig at the eve of the century and, more recently, the Stuttgart 21 Rail Project in Germany, the new Nantes International Airport in France or, in health policy, the Herceptin debate in New Zealand. In 2019, Italy has witnessed an unprecedented episode of public debate triggered by a CBA, relating to pursuing the construction of an international tunnel for a High-Speed/High-Capacity rail project on the Lyon-Turin corridor. In February 2019, an evaluation report done on this infrastructure was released by a working group of the Ministry of Infrastructure and Transport (MIT). The analysis provided a stark negative result with a net present value (NPV), in one scenario, of minus 8 billion euros.

Quickly, the public debate converged, pointing to claims of inconsistencies and striking paradoxes in the results. More than in other circumstances, many methodological aspects of the study have been heavily debated. The debate entailed numerous claims by academics or experts in the field about whether and how a CBA should be conducted. However, many of the criticisms seemed to be in stark contrast with the well-known results and consolidated methods of CBA and transportation sciences. More so, the debate was so intense that it has likely shaped the future of project evaluation for years, at least in Italy.

Thus, it is relevant for the community of economists to explore the economic arguments introduced in these discussions. This endeavour may result in different outcomes. First, if these criticisms, or a large number of them, are correct, the proposed investigation would provide a dispassionate, scientifically scrutinised record of the study’s inconsistencies. This result would also imply that most of the evaluation procedures used worldwide are wrong and provide misleading policy recommendations–an important message to the international community of evaluators. Another possibility is that some of these criticisms resulted from misunderstandings. In this case, the proposed investigation would be beneficial; as such, a quid-pro-quo could be clarified. It would also be an opportunity to provide a deeper understanding of a proper evaluation method; highlighting for instance the ‘devils in the details’ for more fruitful discussions among scientists. Third, another possibility is that these criticisms would be misguided by acceptable scientific criteria of soundness. Such a result would need to be shared with the community of economists so that future evaluation procedures do not operate with ill-founded approaches endorsed by these criticisms. Additionally, this would provide material for the epistemology of science and the study of the spread or persistence of errors in the scientific community. This investigation complements, in a more detailed and formalised manner, elements provided in recent literature (Massiani and Maltese 2022) showing that CBA methods are at risk of being misunderstood and misinterpreted in public discussions.

Therefore, this study contributes to the discussion on proper evaluation methods and to the progress of knowledge in evaluation in two specific ways. First, it proposes a meta-equation that contains various possible CBA calculations as special cases and allows for formal comparison. Second, using this meta-equation, it systemises and disambiguates the various criticisms related to the 2019 CBA. It examines their validity by focusing on criticisms formulated by academics and experts on the two most heavily discussed issues—the inclusion of taxation impacts and Rule of Half (RoH) computation of users’ benefits.

This paper is structured as follows. Section 1 presents the 2019 CBA results and a synopsis of the criticisms formulated by academics and experts. Section 2 provides a formalisation of a transport project, describes various available evaluation measures, and summarises them in a meta-equation, which allows for comparison among methods and verification of methodological claims. Section 3 systematically uses this meta-equation to review the criticisms of the 2019 CBA on two key topics: RoH and taxation. In the concluding section, we discuss the implications of the analysis.

2 A Highly Discussed Evaluation

We first present the main results of the 2019 CBA and then a synopsis of the criticisms raised, restricted to those formulated by academics.

2.1 A Negative Outcome

In May 2018, a CBA of various important transport projects was undertaken, with its masterpiece being the international tunnel of the Lyon-Turin ‘high-speed’ line project. The results of these studies were published in February 2019 (MIT 2019). This CBA followed a series of other studies on the Lyon-Turin project (for instance CIG 2000; OAFTL 2011). The various studies gave very different results. However, to be fair, they often dealt with different project configurations; for instance, different sections ranging from 60 km to more than 280 km. The results (NPV) of the 2019 study are synthesised in Fig. 1 for two traffic scenarios, ‘realistic’ and ‘high traffic.’

Fig. 1

Outcomes of the 2019 Cost–Benefit Analysis for two scenarios (net present value in bln Euro)

Essentially, both traffic assumptions provide negative Net Present Values. Immediately, comments were formulated on the large social costs corresponding to fuel duties and highway operators’ losses. The item ‘state revenues’ consisted of lost excise rights (fuel tax): fewer vehicle-km implied smaller fuel tax income for the state. The project benefits (such as externalities, road congestion, freight surplus, and passenger surplus) are sizeable; all larger than one billion euros, however not sufficient to compensate for costs. Some paradoxical effects also appeared. Most strikingly, the more the tunnel would be used, the less it would be beneficial for society, at least for freight transportation. Another striking paradox is that even if the tunnel were built costlessly, it would have a negative NPV. This generated scepticism among experts and public discussions (Table 1).

Table 1 Synopsis of criticisms to MIT working group study covered in this article

In the next section, we present the criticisms formulated by experts and academics.

2.2 Summary of Criticisms

We used two sources of information to identify the issues at stake. First, we reviewed claims made by economists in the public debate. Second, we used a compendium published by the Observatory of the Turin-Lyon Rail Project (OAFTL 2019), a national body charged with organising the public consultation with local stakeholders about the project. This source gathers comments that academics and experts, made on the MIT Working Group’s CBA during the weeks following its publication.

Table 1 summarises the issues examined in this study.¹ Table 4 in appendix complements Table 1 with other topics that, because of space limitations, we do not investigate in this paper. Generally, criticisms were related to a number of topics, such as: inconsistency with italian or international guidelines, lack of consideration for wider economic effects, arbitrariness in traffic assumptions, underestimation of benefits, erroneous consideration of taxation, and misuse of the Rule of Half.

The latter two topics are arguably the most discussed. We will focus on them for the remainder of this study.

1. Consideration of taxation

This point was immediately and heavily criticised by many economists who stated that this is inconsistent with the logic of CBA–taxes are a transfer; thus, a reduction in taxes coincides with an identical benefit, and these two terms cancel out. In the 2019 CBA, the loss in tax revenues was included, while no corresponding benefit (less tax paid) was visible. This observation supported the idea that the analysis was distorted. Other economists accepted the view that the item users’ surplus (freight and passenger surplus) could actually include the benefits of lower taxation. However, scepticism was expressed for this solution. For instance, one may have noticed that users’ benefits were smaller than the reduction in taxation; whereas the expectation would be that they were larger because the former includes the latter. Others commented that if distortionary taxation actually exists on fuel (this is apparent in the 2019 CBA input data where taxation on fuel is much higher than the corresponding externalities), lower tax revenues would reduce the distortions and this improvement should be included as a benefit, or reduced cost, in the project evaluation.

2. Use of the rule of half (RoH)

In the 2019 CBA, users’ benefits were computed based on the Rule of Half, a well-known rule in transportation science, whose derivation is presented in Appendix A. It essentially states that the benefit of users transferring from one mode to another can be estimated as half of the product between the change in quantity and change in generalised cost. With $Q_{m}$ being the quantity and $C_{m}$ the generalised cost of mode m, the Rule of Half measure of users’ surplus is $\mathop \sum \limits_{m} \frac{1}{2}\Delta Q_{m} \cdot \Delta C_{m}$, which is sometimes simplified by considering only the most impacted mode (let us assume it is mode 1): $\frac{1}{2}{\Delta }Q_{1} \cdot {\Delta }C_{1}$. This result is sometimes found exotic by experts from other fields: the difference in generalised costs between the origin mode and mode 1, $(C_{m} - C_{1} )$, would be the most obvious measure of transferred users’ benefits; why then does it not appear in this calculation? For this reason, the RoH computation was generally criticised and dismissed in the 2019 debate; as if the $\frac{1}{2}$ fraction in its expression would discard half of the cost reduction.

2.3 The reasonableness of economic claims

Checking the validity of these claims has an epistemological dimension and requires an operational definition of scientific correctness. A frequent view in economics is that there is nothing strictly true or false, nothing correct, or incorrect (Little 1995; O’Donnell 1989). This view results, for instance, from the fact that, in a number of economic investigations, the poperian condition of empirical testability/falsifiability is not fulfilled.

However, whenever possible, the criterion of correctness can refer to the internal consistency or axiomatic consistency of claims. For instance, when a measure, such as the Rule of Half is defined, one can investigate its properties. Another criterion is consistency with observation, which occurs even when epistemology and sociology of science warn that scientists sometimes tend to confound facts with factoids (Caldas 2016; Latour and Wollgar 1986). We are aware of this literature and direct our investigation to elementary facts that are easily accessible to readers.

These two criterium are sufficient for a large part of our assessments. In some cases, more complex assessment criteria must be used. For instance, one criticism could be coherent with mainstream economics yet contested by some economists. In another criticism, we found that the criticized method corresponds to a simplified assumption that is often non-testable, as neither would be an alternative assumption. In this context, we posit that the most scientific solution is to report this situation and discuss its implications.

Once some assessment criteria are provided, the next step is to define a project and formalise its evaluation.

3 Formalisation of a Project and Its Evaluation

In this section, we formalise a meta-model that contains a large variety of possible CBA computations. This model facilitates the comparison between competing CBA calculations and claims. We then formalise the welfare function used in the 2019 CBA, with its main assumptions. That allows for formal verification of claims on its correctness.

3.1 A Project and its Impacts

Economic evaluation usually involves comparing two states of the world, each corresponding to scenarios with or without a project. The analyst desires a comprehensive representation of the economy to quantify ‘all’ the impacts of the project and successively assess them in a welfare function. However, the analyst must make some simplifications or epistemic reductions:

• 1. Generally, one concentrates attention on a limited set of markets, which is commonly labelled partial equilibrium analysis.

• 2. Distinctively, they often have to consider simplification in the functioning of the economy. For instance, in practice, the impact of infrastructure on regional development is sometimes not considered in estimated traffic flows.

In subsequent sections, we describe a typical project. Later, we examine how a welfare function W() can be defined to evaluate such projects accepting the two simplifications above.

3.1.1 What is a Project?

We assume that a project impacts two goods, labelled 0 and 1. We concentrate on the case in which these goods are substitutes, as two competing modes would normally be. These could be two transport modes but also a set of alternative goods proposed to the consumer. In a very general sense, a project involves capital costs and improves the quality of at least one consumption alternative. Thus, it shifts consumption from one alternative to another. Normally, it can also impact markets other than the improved good and its closer substitute. These are usually labelled as general equilibrium effects. The analyst must consider if and how to include these impacts and carefully consider the impact of his assumptions on the evaluation.

The investigated market is usually decomposed into various segments (typically origin–destination pairs, purposes, etc.). For each segment, users face a market price and attributes, sometimes labelled in transportation science as levels of service. For the sake of legibility, we do not use a segment subscript in the following formalisation. This avoids an extra notation and summation across segments. Consumer choice can be analysed using utility functions and, with some assumptions, such as constant marginal utility of income (Delle Site and Salucci 2018), in terms of generalized costs. Since it is more suitable for evaluating the RoH claims, we will use this latest concept.

The generalised cost has two components: an observed part and an unobserved individual part. The alternative assumption, where all components of the generalised costs are perfectly measured and identical for all individuals in a demand segment has little support.² We compare the project and no-project situations as follows.

	No-project	Project
Mode 0	C0 + ε0i	C’0 + ε0i
Mode 1	C1 + ε1i	C’1 + ε1i

It results from these definitions that (${\upvarepsilon }_{1i} - {\upvarepsilon }_{0i}$) is an individual unobserved difference in generalised cost (or specific preference, penalty, or demand shifter) between alternatives 1 and 0.³ We focus on a typical situation in which $C^{\prime}_1 < C_1$, ${{C^{\prime}}}_0{ } \le {\text{C}}_0{ }$ and $\left( {C_1 - C^{\prime}_1} \right) > \left( {C_0 - C^{\prime}_0} \right)$. This latest condition $\left( {C_1 - C^{\prime}_1} \right) > \left( {C_0 - C^{\prime}_0} \right)$ may not hold for all segments. However, our assumption remains valid since, in typical projects, segments that respect this condition dominate others. Referring to a rail project, this condition simply means that a project improving rail can have a positive impact on roads through decongestion; however, it normally improves rail more than roads for many demand segments.

When the project takes place, the number of users is impacted in two ways. First, there is a mode shift of a given number of trips ($Q_{01} )$. Second, there are induced quantities $Q_{z0}$ and $Q_{z1}$. Thus, the quantities in project situations can be expressed as initial (no-project) quantities $\pm$ modal shift + induced quantities. Namely:

$$ Q_{1}^{{\prime}} = Q_{1} + Q_{01} + Q_{z1} $$

(1)

$$ Q_{0}^{{\prime}} = Q_{0} - Q_{01} + Q_{z0} $$

(2)

A situation of interest is when $Q_{01} > 0\,;{\text{Q}}_{z1} \ge 0\,;{ }Q_{z0} \ge 0.$ These changes, observed in the two markets of interest, also impact the other markets. For instance, when there is induced traffic in mode 1, ${\text{Q}}_{z1} >$ 0, the consumption of other goods changes by ${\text{Q}}_{z1} .\left( {{\text{p}}_{1} /{\text{p}}_{{\text{z}}} } \right)$.

For reader convenience, Text box 1 summarises these notations and others that will be useful throughout this analysis.

3.1.2 Project Impacts

The analyst is interested in the various impacts of this project. The project has a capital cost K. This cost can be decomposed into various categories $K_{j}$ whenever necessary. K is also suitable for representing the infrastructure maintenance costs. The project also generates user benefits and can provide net benefits E, to producers through market power. Finally, it can generate other impacts, X (typically externalities); It also affects tax revenues, although the inclusion of this component is controversial. These variables are discounted by a factor ${\updelta }_{{\text{n}}}$ in year n, allowing for whatever discounting mechanism is found appropriate; for instance, but not uniquely: ${\updelta }_{{\text{n}}} = \left( {1 + \frac{discount rate}{{100}}} \right)^{n}$.

Formalising the project impacts in more detail, we discuss the users’ benefits: the project will impact transferred users, additional (induced) users, and users remaining on the same alternative. The proposed notations are $B_{z0}$ and $B_{z1}$ for induced users of modes 0 and 1;$B_{01}$ for transferred users; and $B_{00}$,$B_{11}$ for users staying on their mode. The last two components, $B_{00}$,$B_{11} ,$ have a straightforward calculation. For instance, the benefits of users staying on mode 0 will be the difference in generalised costs multiplied by the number of such users: $B_{00} \left( {{\text{C}}_0{ } - {{C^{\prime}}}_0{ }} \right)$. Other components ($B_{z0}$,$B_{z1}$,$B_{01} )$ do not have such a direct expression. In the literature, two proposals are competing, RoH and, less frequently, logsum. We will focus on RoH since it is most frequently used, can be implemented with minimal data, is recommended in many guidelines and is part of the discussion in the 2019 CBA.

Whenever relevant, we can consider that part of the users’ costs consists of mode-specific taxes $t_{0}$ and $t_{1}$; taxes, $t_{{\text{z}}}$, also exist in other goods. The consumption of other goods is potentially modified if transport expenditure changes. If $t_{{\text{z}}}$ is the tax on each unit of the alternative good, then an extra unit consumption of mode 1, resulting from induced users, will decrease the consumption of other goods by ${\text{p}}_{{1}} /p_{z}$ and the corresponding tax revenues by $\frac{{{\text{p}}_{{1}} }}{{ p_{z} }}$.$t_{{\text{z}}}$. Overall, the project’s impact on tax revenues T_n in year n is derived from the fiscal impact of transferred users and induced users. Starting from the expression T = ${\Delta Q}_{{\text{o}}} t_{0} + {\Delta Q}_{1} t_{1} + {\Delta Q}_{{\text{z}}} t_{z}$ we can write⁴ the following expression (where T has a negative value whenever the project reduces tax revenue):

$$ T_{n} = {\text{Q}}_{{\text{01,n}}} \left( {t_{{1}} - t_{{0}} - \frac{{p_{1} }}{{p_{z} }} t_{z} + \frac{{p_{0} }}{{p_{z} }}t_{z} } \right) + {\text{Q}}_{{\text{z0,n}}} {(}t_{{0}} { - }\left( {p_{0} /p_{z} } \right)t_{z} {\text{) + Q}}_{{\text{z1,n}}} {(}t_{{1}} { - }\left( {p_{1} /p_{z} } \right)t_{z} {),} $$

(3)

and the corresponding discounted value will be:

$$ T = \mathop \sum \limits_{n = 0}^{N} { }\frac{{{\text{Q}}_{{\text{01,n}}} \left( {t_{{1}} - t_{{0}} - \frac{{p_{1} }}{{p_{z} }}t_{z} + \frac{{p_{0} }}{{p_{z} }}t_{z} } \right){{ + Q}}_{{\text{z0,n}}} \left( {t_{{0}} { - }\left( {p_{0} /p_{z} } \right)t_{z} } \right){{ + Q}}_{{\text{z1,n}}} \left( {t_{{1}} { - }\left( {p_{1} /p_{z} } \right)t_{z} } \right)}}{{\delta_{n} }} . $$

(4)

One can easily distinguish two parts in this equation, one corresponding to the tax impact observed in the market for alternatives 0 and 1: $\mathop \sum \nolimits_{{{\text{n}} = 0}}^{{\text{N}}} \frac{{{\text{Q}}_{{\text{01,n}}} \left( {{\text{t}}_{{1}} {{ -t}}_{{0}} } \right)+\left( {{\text{Q}}_{{\text{z0,n}}} {\text{t}}_{{0}} {{ + Q}}_{{\text{z1,n}}} {\text{.t}}_{{1}} } \right)}}{{{\updelta }_{{\text{n}}} }}$; and one that can be observed in other markets: $\mathop \sum \nolimits_{{{\text{n}} = 0}}^{{\text{N}}} { }\frac{{{\text{Q}}_{{\text{01,n}}} \left( { - \frac{{{\text{p}}_{1} }}{{{\text{p}}_{{\text{z}}} }}{ } + { }\frac{{{\text{p}}_{0} }}{{{\text{p}}_{{\text{z}}} }}} \right){.}t_{{\text{z}}} + \left( {{{ -Q}}_{{\text{z0,n}}} {.}\left( {p_{0} /p_{z} } \right)t_{{\text{z}}} {{ -Q}}_{{\text{z1,n}}} \left( {p_{1} /p_{z} } \right)t_{{\text{z}}} } \right)}}{{{\updelta }_{{\text{n}}} }}$. Induced demand reduces the consumption of the alternative basket of goods and thus impacts the corresponding tax revenues. Additionally, a mode shift can also impact taxes on alternative goods if it changes the budget allocated to transport. This distinction would typically be labelled by economists as Partial Equilibrium versus General Equilibrium impacts and corresponds to a more general distinction between the effects that are observable in the model and the effects that are not (at least not directly). This distinction will be useful in future developments. Whatever this distinction, some questions (which turned important in at least one occasion) were whether and how an analyst may want to consider T in a welfare function or may prefer to discard it on the grounds that taxes are just a transfer.

In addition to taxation, we consider other impacts, such as externalities associated with trips: $x_{{m}}$ is the positive externality associated with one use of mode $m$; if mode $m$ has more negative externalities than positive ones, $x_{{\text{m}}} < 0$. This is often considered in road transport. The model generalises to a non-constant marginal externality; however, we focus on a simplified case to avoid cumbersome notations. The impact on externalities for year n has an expression comparable to the impact on taxation.⁵ Generally, the various quoted impacts are compatible with traffic induction; induced traffic actually reduces the budget available for other goods, which generates side effects corresponding to part of what is called ‘GE impacts’.

In this section, we formalise the project and its impacts. We pursue our investigation by examining how an evaluation function can be proposed for such projects.

3.2 Possible Formulations of Project Evaluation Functions

Here, we propose a meta-equation of CBA that is apt to summarising a large number of potential functions. This general expression includes various CBA measures, which assist the analyst in making his assumptions explicit and allows for the comparison of various competing CBA formulations. This formalisation can be used successively to assess the claims made regarding the 2019 CBA.

3.2.1 A Meta-equation for Project Evaluation

In its more general form, a CBA generally makes use of an equation such as:

$$ \begin{aligned} & W\left( B^{PE} ,X^{PE} ,E^{PE} ,T^{PE} ,K_{j}^{PE} , \ldots , K_{J}^{PE} ,B^{GE} , X^{GE} ,E^{GE} ,\right.\\ & \left. T^{GE} ,K_{i}^{GE} , \ldots ,K_{I}^{GE} ; \pi_{b} ,\pi_{x},\pi_{t} ,\pi_{k} ,\omega_{T} ,\omega_{k} ,\gamma_{b} ,\gamma_{x },\gamma_{t} ,\gamma_{ki } , \ldots ,\gamma_{kI } \right) \hfill \\ & =\left( \pi_{b} B^{PE} + \pi_{x} X^{PE} + \pi_{e} E^{PE} +\pi_{T}(1-\omega_{T} )T^{PE} {-}\pi_{k} (1 + \omega_{k} )\left(\mathop \sum \limits_{i} K_{i}^{PE}\right) \right)\\ & + \left( \gamma_{b} B^{GE} \left( {B^{PE} } \right) + \gamma_{x} X^{GE} (X^{GE} ) + \gamma_{e} E^{GE} \left( {E^{PE} } \right) + \gamma_{t} T^{GE} \left( {T^{PE} } \right)\right.\\ & \left.+ \mathop \sum \limits_{j} \gamma_{kj } K_{j}^{GE} \left( {K_{j}^{PE} } \right) \right) \end{aligned} $$

(5)

where W(), is welfare function; ${\text{B}}^{{{\text{PE}}}} ,{\text{X}}^{{{\text{PE}}}} ,{\text{E}}^{{{\text{PE}}}} ,{\text{T}}^{{{\text{PE}}}} ,{\text{K}}_{{\text{j}}}^{{{\text{PE}}}} , \ldots , {\text{K}}_{{\text{J}}}^{{{\text{PE}}}}$ are impacts of the project (user benefits, producers, tax revenues, externalities, and capital cost) on observed markets in the model; ${\text{B}}^{{{\text{GE}}}} ,{\text{X}}^{{{\text{GE}}}} ,{\text{E}}^{{{\text{GE}}}} ,{\text{T}}^{{{\text{GE}}}} ,{\text{K}}_{{\text{j}}}^{{{\text{GE}}}} , \ldots ,{\text{K}}_{{\text{J}}}^{{{\text{GE}}}}$ are other impacts of the project (outside of the observed market or not captured by the modelling approach); ${\uppi }_{b} ,{\uppi }_{x} ,{\uppi }_{t} ,{\text{and }}\pi_{k}$ represent methodological parameters for the first category of impacts; ${\upgamma }_{{\text{b}}} ,{\upgamma }_{{\text{x }}} ,{\upgamma }_{{\text{t}}} ,{\upgamma }_{{\text{kj }}} ,{ } \ldots ,{\upgamma }_{{\text{kJ }}}$ are methodological parameters for other impacts; ${\upomega }_{{\text{T}}} ,{\upomega }_{{\text{k}}}$ are methodological parameters for the cost of public funds. This welfare function has two noticeable features that we discuss below.

3.2.2 Modelled vs Non-modelled Impacts

The proposed meta-equation recognises that the analyst always splits the impacts of a project in two parts. The first corresponds to the impacts that are both included in the modelling approach used and measurable on the investigated markets. The second corresponds to other impacts that are not included in the model by definition.

When performing an assessment, an analyst will generally make two types of simplifications: s/he will concentrate on a limited set of goods and corresponding markets and s/he will consider only a limited set of economic mechanisms. These latest can impact other (unobserved) markets or the observed market in a way that the selected modelling approach does not consider. This latest situation gathers impacts that are labelled in various ways in the literature, such as, General Equilibrium impacts—defined as occurring in other markets; or Wider Impacts—partly occurring in observed markets or in other markets. However, analysts are aware that the model is a simplification; mechanisms that are not embedded in the model could be impacted by the investigated policies. Sometimes, CBA seems to deal with only one market, but this is an elusive qualification: often the boundaries elected for the investigated markets can be set with some discretion. Consistency derives from the fact that for each definition of the realm of the modelled impact, the other, wider, impacts are defined as complementary impacts.

Interestingly, the investigated impacts contain information on other impacts.

3.2.3 Methodological Parameters

A second peculiarity of the master equation is a set of methodological parameters, indicated by Greek letters, $\pi$ and $\gamma$ corresponding to several methodological options open to analysts. We comment on a few methodological parameters below.

• ${\uppi }_{b}$ is typically set to 1. However, analysts may want to include equity considerations and weigh user benefits on distributional grounds by decomposing B across various segments (Florio 2014);

• ${\uppi }_{t}$ is a methodological parameter that triggers the inclusion of PE taxation incidence;

• ${\uppi }_{x}$ relates to other impacts such as externalities. Many reasons could justify that ${\uppi }_{x} \ne 1$, for example, the fact that when an evaluation is made with net of tax prices, an adjustment should be made to make the shadow costs of externalities consistent with the prices used for evaluation (Sugden 1998);

• $\pi_{e}$ represents the weight of the companies’ benefits in the welfare function. An analyst may consider profits to have a value different from that of consumer benefits. Weisman (2016) argues that market regulators give different weights to profits and consumer surpluses in their regulations. Although it is rarely done in CBA, Meta-equation (5) allows for this. The proposed formulation is also compatible with the more common choice, where ${\uppi }_{e}$ = 1.

Some discussion also needs to be spent on $\gamma$ parameters. $\gamma_{b} ,\gamma_{{x{ }}} ,\gamma_{t} ,\gamma_{kj} ,{ } \ldots ,\gamma_{{kJ{ }}}$ are variables that indicate how much wider impacts will be considered in addition to properly defined PE impacts. In this regard, the welfare function W(), defined in Eq. (5), may have undesirable properties when some parameters are set to given values or given combinations of values. When an incoherent or flawed CBA is implemented, Eq. (5) allows a comparison with other appropriate parametrisations.

This presentation may make CBA appear extremely ductile. This is only partly true: what makes CBA appear ductile is that there are incoherent ways of producing a CBA. To be fair, the actual variety of methods is much smaller than the possible number of combinations of methodological parameters; analysts use a reduced number of methodological configurations.

Generally, it appears beneficial to have a general formula to represent various possible computations.

3.2.4 Going Partial

In this context, the majority of CBA applications omit the additional impact section of Eq. (5). Choosing Partial Equilibrium analysis (i.e.: $\gamma_{b} ,\gamma_{{x{ }}} ,\gamma_{t} ,\gamma_{{kj{ }}} ,{ } \ldots ,\gamma_{{kJ{ }}}$ = 0) is frequent in project evaluation (e.g.: Cartenì et al. 2019); however, wider impacts are ignored. To clarify, this does not mean that the analysis has omitted any effects outside of the investigated markets; this means that such effects are considered only as much as they are already visible in the investigated markets ⁶. One reason is that, when adequate methods are used, the investigated markets contain information on the value of the alternative uses of resources.

Choosing a PE analysis may appear inappropriate for analysts with limited knowledge of CBA, yet it elaborates on a rigorous tradition that was nicely summarised by Mishan (2007, p. 37).

Reasons for holding valid this partial equilibrium setting are proposed in different instances in the literature (e.g. Lesourne 1972).

The opposite temptation of expanding Partial Equilibrium to General Equilibrium could also generate distortions: in 1975, Whalley already warned: “The most striking feature (…) is that extended partial equilibrium analysis gives a much poorer prediction of the change in product than simple partial equilibrium analysis.” (Whalley 1975). Additionally, there is no a priori reason that a PE analysis provides less favourable results than a GE analysis: Kanemoto and Mera show that the GE measure is smaller than the PE measure if import demand in one region is more price elastic than in other region, along the entire equilibrium path.

“the prices of goods and services other than transportation services are taken as fixed in the PE measure whereas their changes are fully taken into account in the GE measure. It is shown that the second aspect works in the opposite direction in the MD case: the GE measure tends to be smaller than the PE measure.” (Kanemoto and Mera 1985, pag. 357, “MD” refers to Marshal Dupuit surplus).

It is fair to conclude that Partial Equilibrium analysis is the dominant practice in CBA, that its limitations are less than they appear prima facie. There is no surprise thus, that it is used ubiquitously for infrastructures evaluation including large projects like the channel tunnel (Anguera 2006) or Oresund bridge (Knudsen and Rich 2013).

3.2.5 CBAs Are Parametrizations of a Welfare Function

In this setting, various evaluation methods can be compared, as they refer to various parameterisations of Eq. (5). For instance, the EU Guidelines or the 2019 MIT CBA elected specific values for methodological parameters.

For instance referring to EU Guidelines, notwithstanding the variety of ways these Guidelines are implemented, the elected parameters mostly correspond to: ${\uppi }_{T} = 0$,$\omega_{k} \ge 0$, and $K_{j}^{GE} \left( {K_{j}^{PE} } \right)$ = $K_{j}^{PE}$, $\gamma_{kj} = \left( {{{\varphi }}_{j} - 1} \right)$, where $\varphi_{j}$ refers to the correction factors for expenditure category i. So $\gamma_{{kj{ }}} K_{j}^{GE} \left( {K_{j}^{PE} } \right)$ = (${{\varphi }}_{j}$ − 1)${ }K_{j}^{PE}$. For a railway project, $B^{PE}$ was set equal to $B^{ROH}$ (DG Regio 2014, p. 110).

Referring to the 2019 CBA, we can identify the following parameters:

• $\pi_{T} = 1$; different from the EU Guidelines, it fully considers tax incidence such as, changes in excise right revenues;

• $\omega_{k} { } > 1$; the opportunity cost of funding is applied to the cost of capital expenditure;

• $\omega_{T} = 1$; it does not apply the opportunity costs of funding to the tax incidence;

• $B^{PE}$ = $B^{ROH}$; it estimates users’ benefits using the RoH;

• capital expenditures can be written as $\left( {\omega_{k} \cdot K_{NL}^{PE} + \omega_{L} \cdot {{\varphi }}_{L} \cdot K_{L}^{PE} } \right)$, which applies a factor to the opportunity cost of public funds and selectively, an additional factor to labour-related capital expenditures.

Finally, the welfare function elected for the 2019 MIT CBA can be written as:

$$ W^{CBA} = B^{ROH} + X^{PE} + E^{PE} + T^{PE} {-}\left( {\omega_{k} K_{NL}^{PE} + \varphi_{L} \cdot \omega_{L} \cdot K_{L}^{PE} } \right) $$

(6)

where $K_{L}^{PE}$ relates to labour costs of capital expenditures, and ${\text{K}}_{{{\text{NL}}}}^{{{\text{PE}}}}$ relates to other non-labour capital costs. For practical use, this can be rewritten using a shorter notation for capital cost as $K^{CBA} = \omega_{k} \cdot K_{NL}^{PE} + {{\varphi }}_{{\text{L}}} \cdot {\upomega }_{{\text{L}}} \cdot K_{L}^{PE} .$ Thus, the ${\text{W}}^{{{\text{CBA}}}}$ is expressed as

$$ W^{CBA} = B^{ROH} + X^{PE} + E^{PE} + T^{PE} {-}K^{CBA} , $$

(7)

or, making explicit the first term ${\text{B}}^{{{\text{RoH}}}}$ (see Appendix A):

$$ W^{CBA} = \mathop \sum \limits_{n = 1}^{N} \frac{{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( {Q_{0,n} + Q^{\prime}_{0,n} } \right) \cdot \left( {C_{0,n} - C^{\prime}_{0,n} } \right) + \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( {Q_{1,n} + Q^{\prime}_{1.n} } \right) \cdot \left( {C_{1.n} - C^{\prime}_{1,n} } \right)}}{{\delta_{n} }}+ X^{PE} + E^{PE} + T^{PE} {-} K^{CBA} . $$

(8)

Combining these elements, we obtain a formal expression, Eq. 7, of the welfare function used in the 2019 MIT CBA. Such a parameterisation is not unusual and is similar to Massiani & Maltese (2022), as much as it is documented to the one used the previous CBAs of the Lyon-Turin projects, applied to a variety of project sections, project configurations and traffic assumptions. Notably, focusing on T, the impact on tax incomes, many of these evaluations were including such tax impacts (Debernardi et al. 2011; LTF-RFI 2011; OAFTL 2011; Prud’homme 2014). Based on this formulation, we can review various claims regarding the 2019 Lyon-Turin CBA.

Before doing so, we summarise the content presented thus far. In this section, we provide a simple formalisation of a transport project. We define a family of evaluation functions where practitioners explicitly delimit their analysis. They select investigated markets, which can expand beyond the directly impacted transport markets, and a number of economic mechanisms. Furthermore, they specify how their analysis considers the impacts outside of these markets and the modelled impacts. This can typically be performed using correction factors, which represent GE impacts; or by restricting the analysis to a defined PE setting. This latest choice is consistent with practice, and is supported by a large theoretical body of knowledge. Interestingly, due to consumers and producers’ trade-offs, the PE measure of users’ benefits reflects the missed benefits of alternative consumption or a first-order approximation of them. In this setting, most CBA analysts use the RoH measure of transferred users’ benefits. Using the proposed model, we can use the formal expression of our welfare function to review claims about the 2019 CBA on two key topics: taxation and the RoH.

4 A Review of the Claims on the 2019 MIT CBA

In this section, we use the meta-equation of CBA developed in Sect. 3 to evaluate the claims made about the MIT 2019 CBA as described in Sect. 2, about the RoH and taxation.

4.1 RoH Issues

We start from the RoH computation of transferred users’ benefits, which in year n are, whenever the unobserved component of generalised costs is linearly distributed (or any equivalent assumption; see Appendix A for details):

$$ B_{{01,n}}^{{ROH}} = \left( {\mathop \sum \limits_{i} \left( {C_{{0,n}} + \varepsilon_{{0i,n}} } \right) - \left( {C^{\prime}_{{1.n}} + \varepsilon_{{1i,n}} } \right)~\Big| \ \ \ \ \kern-1em\lower1ex\hbox{$\displaystyle (\varepsilon_{{0i.n}} - \varepsilon_{{1i,n}} )\sim U$}} \right)~ = ~\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} ~Q_{{01,n}} \cdot \left( {\left( {C_{{0,n}} - ~C^{\prime}_{{0,n}} } \right) + ~\left( {C_{{1,n}} - ~C^{\prime}_{{1,n}} } \right)} \right). $$

(9)

The benefits of all users (including induced and existing traffic) for year n are

$$ {\text{B}}_{n}^{{{\text{RoH}}}} = 1/2 \left( {{\text{Q}}_{{0,{\text{n}}}} + {{\text Q^{\prime}}}_{{0,{\text{n}}}} } \right) \cdot \left( {{\text{C}}_{{0,{\text{n}}}} - {{\text C^{\prime}}}_{{0,{\text{n}}}} } \right) + 1/2 \left( {{\text{Q}}_{{1,{\text{n}}}} + {{\text Q^{\prime}}}_{{1,{\text{n}}}} } \right) \cdot \left( {{\text{C}}_{{1,{\text{n}}}} - {{\text C^{\prime}}}_{{1,{\text{n}}}} } \right), $$

(10)

and, for the whole life duration of the project,

$$ {\text{B}}^{{{\text{RoH}}}} = \mathop \sum \limits_{{{\text{n}} = 1}}^{{\text{N}}} \frac{{{{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }}\left( {{\text{Q}}_{{0,{\text{n}}}} + {{\text Q^{\prime}}}_{{0,{\text{n}}}} } \right) \cdot \left( {{\text{C}}_{{0,{\text{n}}}} - {{\text C^{\prime}}}_{{0,{\text{n}}}} } \right) + {{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }}\left( {{\text{Q}}_{{1,{\text{n}}}} + {{\text Q^{\prime}}}_{{1,{\text{n}}}} } \right) \cdot \left( {{\text{C}}_{{1,{\text{n}}}} - {{\text C^{\prime}}}_{{1,{\text{n}}}} } \right)}}{{{\updelta }_{{\text{n}}} }}. $$

(11)

Using these formulas, we can investigate different claims dealing with the use of the RoH, starting from Claims 1, 2 and 3.

Claim 1, 2 and 3: The RoH Estimate of Users’ Surplus Records Only Half of Users’ Generalized Cost Reduction

Claim 1 materialises for instance in: ‘One euro less for highway operators is worth 50 cents as car users saving’ (text 10 in OAFTL 2019). Similar statements are made for Claims 2 and 3.

Claim 2: The RoH estimate of users’ surplus would be half of users’ benefits. Literally,

“the method used for the CBA (…) estimates consumer surplus accounting for excise rights and tolls paid, but divides by 2 the resulting value, by a generalized and uncritical application of the Rule of Half. The value of tolls and excise rights are accounted as half among the benefits… (Text 09 in OAFTL 2019).

Claim 3: Measuring transferred users’ benefits as half of the reduction of Generalized Costs of the destination mode underestimates benefits by 50%.

In contrast, we can observe that the Rule of Half does not halve benefits; the difference in generalised costs across modes, ${ }\left( {{\text{C}}_{{0}} {{ + \varepsilon}}_{{{\text{0i}}}} } \right){ - }\left( {{{C^{\prime}}}_{{1}} {{ + \varepsilon}}_{{{\text{1i}}}} } \right)$, is entirely accounted for in this calculation. The ½ fraction results from a mathematical transformation of $\left( {{\upvarepsilon }_{0i} - {\upvarepsilon }_{1i} } \right)$ and the assumed Uniform distribution (or any equivalent assumption) of this random term, and not from an ungrounded reduction of benefits. The formula does not rely on half of the change in Generalized Costs between modes ${ }\left( {{\text{C}}_{{0}} {{ -C^{\prime}}}_{{1}} } \right)$, but on half of the other quantity: $\left( {{\text{C}}_{{0}} {{ - C^{\prime}}}_{{0}} } \right)+\left( {{\text{C}}_{{1}} {{ - C^{\prime}}}_{{1}} } \right)$. Equation (9) ${{ \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} Q}}_{{01,{\text{n}}}} \cdot \left( {\left( {{\text{C}}_{{\text{0,n}}} {{ -C^{\prime}}}_{{\text{0,n}}} } \right)+\left( {{\text{C}}_{{\text{1,n}}} {{ -C^{\prime}}}_{{\text{1,n}}} } \right)} \right)$ could be modified, leaving out the ½ fraction, only if all individuals were identical to the one that obtains the maximum benefits from the project, but nothing would support this assumption.

Thus, we conclude that Claim 1 and its companions Claims 2 and 3 are analytically inconsistent. We now proceed with another related claim:

Claim 4: The RoH Estimates of Users’ Surplus Record Only Half of the Tax Income Reduction (or Toll Reduction)

This claim is very similar to the previous ones. Literally: ‘[using RoH] implies that reduced State income are accounted for twice compared with consumer savings. A choice far from neutral”.⁷ We can start from Eq. (9):

$$ B_{{01,n}}^{{\text{ROH}}} = \left( {\mathop \sum \limits_{i} \left( {\text C_{{0,n}} + \varepsilon_{{0i,n}} } \right) - \left( {\text C^{\prime}_{{1.n}} + \varepsilon_{{1i,n}} } \right) \Big| \, \kern-.05em\lower1ex\hbox{$\displaystyle \left( {\varepsilon_{{0i.n}} - \varepsilon_{{1i,n}} } \right)\sim \text U$}} \right)~ = ~1/2~\text Q_{{01,n}} \cdot \left( {\left( {\text C_{{0,n}} - ~\text C^{\prime}_{{0,n}} } \right) + ~\left( {\text C_{{1,n}} - ~\text C^{\prime}_{{1,n}} } \right)} \right). $$

A direct observation of this equation shows that the Rule of Half accounts for the entire change in transferred users’ generalised costs. This also holds when one wants to make the tax component of generalised costs explicit, as claim 4 suggests. If, with self-evident notations, ${\text{C}}_{0}$ = ${\text{C}}_{0}^{{{\text{NT}}}}$ + ${\text{t}}_{0}$, ${\text{C}}_{1}$ = ${\text{C}}_{1}^{{{\text{NT}}}} + {\text{t}}_{1}$, then

$$ \text B_{01,\text n} = \mathop \sum \limits_{\text i} \left( {\text C_{0, \text n}^{\text {NT}} + \text t_{0.\text n} + \varepsilon_{0\text {i.n}} } \right) {-}\left( {\text C_{1,\text n}^{\prime \text {NT}} + \text t_{1,\text n} + \varepsilon_{1\text {i,n}} } \right). $$

(12)

If, additionally, $\left( {{\upvarepsilon }_{0i} - {\upvarepsilon }_{1i} } \right)\sim U$ (or any equivalent assumptions), then

$$ {\text{B}}_{{\text{01,n}}}^{{{\text{RoH}}}} = {{ \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \text Q}}_{{\text{01,n}}} \left( {\left( {\left( {{\text{C}}_{{\text{1,n}}}^{{{\text{NT}}}} {{ + {\text{t}}}}_{{\text{1,n}}} } \right){ - }\left( {{\text{C}}_{{\text{1,n}}}^{{\prime {\text{NT}}}} {{ + {\text{t}}}}_{{\text{1,n}}} } \right)} \right){-}\left( {\left( {{\text{C}}_{{\text{0,n}}}^{{{\text{NT}}}} {{ + {\text{t}}}}_{{\text{0,n}}} } \right){{ -({\text{C}}}}_{{\text{0,n}}}^{{\prime {\text{NT}}}} {{ + {\text{t}}}}_{{0,{\text{n}}}} { )}} \right)} \right) = {{ \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \text Q}}_{{\text{01,n}}} \left( {\left( {{\text{C}}_{{\text{1,n}}}^{{{\text{NT}}}} {{ -{\text{C}}}}_{{\text{1,n}}}^{{\prime {\text{NT}}}} } \right){-}\left( {{\text{C}}_{{\text{0,n}}}^{{{\text{NT}}}} {{ -{\text{C}}}}_{{\text{0,n}}}^{{\prime {\text{NT}}}} } \right)} \right){ }{.} $$

(13)

We observe that the benefit of reduced taxation for transferred users is fully accounted for in Eq. (12); hence, also in its RoH measure Eq. (13). Taxes may not appear explicitly in $B_{01}^{RoH}$ but a closer inspection suggests that the benefits of reduced taxation are included in the RoH formula. Therefore, claim 4: ‘The RoH estimates of users’ surplus record only half of the tax income reduction (or toll reduction)’ is analytically incorrect.

Claim 5: Considering Half of Cost Savings Penalises Projects Aimed at Modal Shift

Moving to the next claim, we find the following quotation: ‘the value of excise rights and tolls, accounted for half among benefits, seems compensated subtracting their full value as a cost: if so, this would represent a systematic and ungrounded penalization of whatever modal shift’.⁸ It is already clear at this stage that this assertion is based on a misunderstanding of what the RoH really is.

The RoH does not halve costs; therefore, the claimed penalisation is ungrounded.

Claim 6 and 7: With Unobserved Attributes RoH Underestimates Benefits; With Unobserved Attributes the Larger the Modal Shift, the More RoH Underestimates User Benefits

Literally, ‘the higher the modal shift towards a mode that presents non-measured characteristics (…), the larger the underestimate of the surplus, or users’ benefits’.⁹

Formally, this quotation contains two claims:

Claim 6: RoH underestimates users’ benefits. Formally:

If (${\upvarepsilon }_{{0{\text{i}}}} \ne 0{ }$ or ${\upvarepsilon }_{{1{\text{i}}}}$ ≠ 0), then ${\text{B}}_{01} - B_{01}^{\text{ROH}} > 0$,

Claim 7: RoH underestimation increases when there is a large modal shift. Formally:

If (${\upvarepsilon }_{{0{\text{i}}}} \ne 0{ }$ or ${\upvarepsilon }_{{1{\text{i}}}}$ ≠ 0), the larger ${\text{Q}}_{01}$, the larger (${\text{B}}_{01} - B_{01}^{\text{ROH}}$).

Claim 6 is analytically ungrounded. Moreover, it contradicts the concept of Rule of Half, whose role is to account for unobserved components in generalised costs. If both ${\upvarepsilon }_{0i}$ and ${\upvarepsilon }_{1i}$ are non-zero, this is exactly why one uses the RoH estimate. If they are both zero, there is no reason to use RoH. Instead, we can demonstrate a proposition contrary to Claim 6. If ${\upvarepsilon }_{0i}$ and ${\upvarepsilon }_{1i}$ = 0 $\forall i,$ then

$$ \text B_{{01}} \big| \kern-.05em\lower1ex\hbox{$\displaystyle \left( {\varepsilon_{{0i}} = 0;\varepsilon_{{1i}} = 0} \right)$} ~ = ~\mathop \sum \limits_{i} \left( {\left( {\text C_{0} + \varepsilon_{0i} } \right) - \left( {\text C^{\prime}_{1} + \varepsilon_{{1i}} } \right)\big|\,\kern-.05em\lower1ex\hbox{$\displaystyle \left( {\varepsilon_{{0i}} = 0;\varepsilon_{{1i}} = 0} \right)$}} \right) = \mathop \sum \limits_{i} \left( {\text C_{0} - \text C^{\prime}_{1} } \right). $$

(14)

Under such conditions, where unobserved components are zero, the Rule of Half would be unnecessary. Thus, we can reject claim 6.

Claim 7 is related to the previous one and relies on the following premise; it assumes $\left( {{\text{B}}_{01} - {\text{B}}_{01}^{{{\text{RoH}}}} } \right) > 0$. This premise has limited support. If the distribution of the unobserved component of general costs is not uniform, ${\text{B}}_{01}^{{{\text{RoH}}}}$ may well overestimate the real benefits or, equivalently, the sign of $\left( {{\text{B}}_{01} - {\text{B}}_{01}^{{{\text{RoH}}}} } \right)$ may well be negative. In fact, various comparisons converge, suggesting that RoH provides higher benefits than other methods (Bates 2005; Kohli and Daly 2006; Ma et al. 2015). Thus, we can generally reject Claim 7 together with Claim 6.

Claim 8: The RoH Estimate is Adequate Only for Single Mode Studies

Literally: “This rule (RoH) consists in a simplification typically accepted when the analysis deals only with one component of demand (single mode studies) but it is much discussed when one needs to consider the demand on various modes (car, train, air) with different operating costs and benefits for users (multi modal analysis).¹⁰

However, in single mode studies, there are no transferred users. Therefore, by definition, $B_{01}$ is meaningless. Thus, the benefits to users can be expressed without reference to $\left( {{\upvarepsilon }_{0i} - {\upvarepsilon }_{1i} } \right)$; this term would be undefined. Noting that the presence of $\left( {{\upvarepsilon }_{0i} - {\upvarepsilon }_{1i} } \right)$ was one reason for using RoH, one can conclude that, together with induced traffic, multimodality is one of the motives for using RoH. Thus, claim 8: The RoH estimate is adequate only for single mode studies’ appears ill-founded. In contrast, multimodality and the unobserved component of generalised costs are one of the reasons (together with induced demand) that make RoH useful.

Claim 9: The RoH Estimate of Users’ Surplus is Invalid Because the Linear Assumption Behind is Discussible

It is claimed that the Rule of Half embeds a linear demand curve deriving from the uniform distribution assumption for the unobserved components (with our notations: $\varepsilon_{i1} - \varepsilon_{i0} )$ and that such assumption makes it unsuitable. For instance,

“besides the assumption of linear distribution of Utility” (..) “[answers to a 5500 interviews survey] shows that the stated benefit is much higher than the one computed with the Rule of Half, and that the distribution of utility is absolutely not linear.”¹¹

Several detailed claims can be discerned here:

9.1–RoH in general assumes a linear demand/distribution;

9.2–the 2019 MIT CBA, in particular, assumes linear demand/distribution;

9.3–the real distribution/demand is not linear;

9.4–due to the linear assumption, RoH underestimates the real benefits.

One could be more precise in relation to the first claim. The RoH is valid in a wider set of assumptions than the linear one: whenever the average of the unobserved components is the middle point of the interval between the extreme values of $\left( {\varepsilon_{1i} - \varepsilon_{0i} } \right)$ or, in other terms, when $\mathop \sum \limits_{i = min}^{i = max} \frac{{\left( {\varepsilon_{1i} - \varepsilon_{0i} } \right)}}{N}$  = $\frac{{\left( {\varepsilon_{imin} + \varepsilon_{imax} } \right)}}{2}$. The uniform distribution is only one special case and, as such, should be considered a sufficient condition, but not a necessary one, for the validity of the RoH.

Regarding the second claim, the 2019 MIT CBA does not explicitly refer to the linearity of the demand function. One may think that linearity is the key assumption to the Rule-of-Half, but this would not be rigorous, as illustrated in the previous paragraphs.

Regarding points 9.3 and 9.4, scientists usually use one of two assumptions:

• The linear assumption corresponding to the uniform distribution of the unobserved components (or any equivalent assumption leading to RoH).

• The sigmoid assumption corresponding to various Extreme Value distributions.

The results of transportation science suggest that:

• The choice between these two assumptions has little influence on the results (de Jong et al. 2005; Kohli and Daly 2006; Ma et al. 2015). Generally, the transferred users’ benefits differ by less than 5% between these different computations, and this item represents only a fraction of the project benefits; often a small one whenever the initial users of the mode are a sizeable number in the no-project scenario. In the specific case of Lyon-Turin, even under the extreme hypothesis that the whole user surplus would correspond to transferred users, a 5% difference in users’ surplus for freight and passengers represents, in each scenario, 2.3% and 1.8% of total benefits. After consideration of the surplus of previous users, the real impact of this assumption would only be a fraction of this figure, and the sign of the deviation could not be known with certainty.

• The choice between these two assumptions, linear or sigmoid, can hardly be based on theory. Some economists have occasionally taken position on this; however, they have not converged on a single answer (Allen 2008; Neuburger 1971; Scott 1997).

• Empirical data are of limited use when choosing among these assumptions. Available demand functions are mostly calibrated at a scale—whether national, regional, or sectorial—which is of little use in distinguishing various possible curvature assumptions at the level of a given project. Additionally, in many instances, estimation procedures produce nonlinearities whenever the underlying phenomenon is linear, owing to spurious data or a priori imposition of a functional form (Massiani and Maltese 2019). To be thorough, some procedures have been proposed (Ye et al. 2017) to test the adequacy of the Gumbel distribution. The results obtained suggest that such a distribution is inadequate for the investigated data. Other studies also suggest the choice of Extreme Value distributions has little support (Paleti 2019).

The conclusion on this point is that one may well doubt that the relevant demand functions are linear. Yet there is no firm reason, besides convenience (Cox and Snell 1989), to prefer alternative assumptions, mostly embedded in the logit choice model and the corresponding logsum welfare measure. Additionally, the choice between the various assumptions appears to have a limited foundation and influence on the estimated benefits.

We could now close this section on RoH with claim 10: The RoH would be correct only if taxes were deducted from costs and benefits. This latest claim yet already brings us to the issues of taxation, which deserve a treatment of their own. This will be done in the next section, not without providing a summary of the claims analysed so far (Table 2).

Table 2 Summary of claims regarding rule of half

Our analysis strongly suggests a general misunderstanding of the RoH. Generally, the formulated criticisms were not coherent with the essential features of the RoH. We have now to shift to another series of issues related to taxation.

4.2 Taxation Issues

The taxation issue was the most discussed question during the 2019 CBA debate. The leitmotiv was that CBA as a method should not contain taxation which should be confined to the financial analysis. This point deserves further clarification.

To accomplish this, we proceed in two steps. First, we recall several possible computation methods of the net benefits of a project that differ by their treatment of taxation, some of them providing correct results and others incorrect results (Appendix B provides a detailed exposition). Successively we review the various tax related claims.

Generally, most claims formulated about taxation suggested to simply discard the tax impact from the welfare function. This would change Eq. (7):

$$ W^{{CBA}} = B^{{ROH}} + X^{{PE}} + E^{{PE}} + T^{{PE}} -K^{{CBA}} , $$

(7)

to the following

$$ W_{ROH}^{NT} = B^{ROH} + X^{PE} + E^{PE} {-}K^{CBA} , $$

(15)

a computation where the impact for public finance is discarded and which is occasionally referred to as ‘net-of-tax’ measure (Percoco 2019; Trento and Spaziani 2019).

However, this computation is inconsistent. This result conforms to a recent contribution (Massiani 2021) and is presented in more detail in Appendix B. The important point is that the proposed correction on T would impose an equivalent correction on transferred users’ benefits because they also depend on taxes; users enjoy reduced taxation when they shift to a mode with less taxation. In other words, when the impact of the project on tax income is cancelled from the NPV, a corresponding correction must be made on the transferred users’ surplus, which cancels out with the former and does so exactly unless second-order effects of taxation are considered.

Based on this result and using the proposed meta-equation, we investigate Claims 11 to 18 as well as Claim 10 which covers both RoH and taxation.

Claims 11 and 12: Considering Taxation in CBA is Not Conform to the State of the Art; Considering Taxation in CBA Does Not Conform to Guidelines

We group Claims 11 and 12 as both regard practice and recommendations rather than substance. An illustration of these claims is: ‘In fact, in the Cost–Benefit Analysis technique, taxes are deducted from fuel price, like for all productive factors used in the project, like they distort the market price of the economic asset’.¹²

Claim 11 is incorrect. We find a large variety in the ways in which CBA deals with taxation. CBA are often made with full consideration of taxation. Examples include evaluations made by academics (new tunnel in Antwerp by Proost et al. (2014)), consultants (Rail Baltica by Ernst Young (2017)), and administrative bodies (initial Nantes Airport CBA (commented in Brinke and Faber 2011)). Similarly, the 2011 CBA of the Lyon-Turin (LTF-RFI 2011), to the best of our knowledge, includes tax impacts. To be fair, however, a number of CBAs state that they excluded fuel taxes. Examples include the evaluation of the new Lisbon Airport (NERA Consulting 2007) and the new Trujillo-Caceres Road (INECO, ITM Ingenieria, s.d.). This choice is often implemented with explicit reference to EU guidelines (DG Regio 2014).

To be more complete, analysts often deduct project VAT from the project’s price (e.g. Grignon-Massé 2010), but this does not extend to the general exclusion of taxation. Generally, the claim that the state-of-the-art CBA excludes taxation is, at best, discussible if not misinformed.

Switching to guideline recommendations, a recent publication (Massiani 2021) clarified that a sizeable share of guidelines, the largest one in the investigated sample, mandates the inclusion of taxation; examples are: Spain (Ministerio de Fomento 2010); France (Ministère de l’écologie du développent durable et de l’énergie 2019); HEATCO (COWI A/S 2005); and The World Bank (The World Bank 2005).

The picture emerging from this analysis may be confusing; CBA practitioners adhere to different paradigms and guidelines are not unified. This appeals CBA theorists to investigate this question more thoroughly. However, for the purpose of the present paper, stricto sensu, the important conclusion is that Claims 11 and 12 are far too selective, at best discussible, and substantially misinformed.

Claim 13: Cost–Benefit Analysis Should be Made Net of Taxation

This claim deserves its own treatment. It refers to substance rather than practice. One rationale for this claim can be, for instance: ‘the distortion of these assumptions is obvious (..) as they evaluate the validity of an investment based on the actual system of indirect taxation that is fully based on fuel’.¹³

This point is important, as it is at the fracture between two traditions in CBA: the international development stream (active at the World Bank) and the transport economics stream. The first tradition has gained traction in recent years, especially considering its influence through EU Guidelines.

The most important element here is that no reason emerges to ban taxation from the CBA, as long as its inclusion is performed coherently. The idea that taxes should be excluded from the CBA because they are a transfer is also illogic. If taxes are actually a transfer, it provides no support to exclude them but rather dictates to include them coherently. This was discussed for single mode studies by de Rus (2010, pag. 23). If this coherent approach is implemented, taxes will cancel out by construction, in the computation of the NPV, or affect the latter (rightfully) if second-order effects are considered. Interestingly, one should be aware of the existence of CBAs for taxation (Avi-Yonah and Edrey 2018; Russo 2004; Weisbach et al. 2018). Thus, one may wonder how a CBA of tax reforms can be made if the CBA has to exclude taxation.

To conclude, a CBA including taxation is coherent and fully compatible with taxes being a transfer, as long as the various surpluses are estimated consistently; namely, users and governments (not to mention producers if relevant) net benefits all include tax impacts. The alternative method of excluding taxation is cumbersome and requires taxes to be eliminated in two accounts. Importantly, the method of discarding tax impact for the public sector, without a corresponding correction for users, generates wrong results, such as in some so called ‘net of excise’ computations.

Claims 14 and 15: The 2019 CBA Should Not Consider the Reduced Fuel Tax Revenues Among Costs, and NPV Calculation Should Not Include Tax Income Loss

The two claims can be grouped together. Claim 14 can be illustrated by the view that states taxes should be accounted for as a transfer and doubts the 2019 analysis does this correctly:

“I have some doubts on the method used (…) In a correct Cost-Benefit Analysis that would take the point of view of society as a whole, one has to consider that taxes are collected by the State but are paid by economic agents.”¹⁴

“it has no sense to compensate this [user benefit] with a corresponding, but with opposite sign, variation of the benefits for the State”¹⁵

As analysed, these claims are incorrect. If one wants to eliminate taxation from any expression of a welfare function, the users’ surplus must be adjusted accordingly. The only correct provision is that tax has to be considered consistently, which can be made by adopting the formula ${\text{W}}^{CBA }$ or any equally valid implementation of the meta-CBA equation.

Claim 16: With a Distortive Taxation, Transferring Users to a Mode with Reduced Taxation Should Reduce Distortion

Literally, “a reduction of the fuel taxes paid, thanks to the traffic diverted toward less polluting modes, would imply a reduction in the distortion level.¹⁶” (Text 22 in OAFTL 2019). This means that the opportunity cost of public funds should apply to the reduction in tax revenues. Such a computation would reduce the corresponding welfare loss. The proposed correction would generally benefit the project, whenever this latest reduces tax revenues.

This claim relies on the assumption that the opportunity cost of public funds should apply to tax loss as it does to project expenditures; this position is endorsed by mainstream CBA (Boardman et al. 2010). However, some authors argue that the opportunity cost of public funds should not be applied to reduced tax revenues: ‘under certain frequently made separability assumptions of the utility function, one can disregard excess burden effects of the lost tax revenues associated with public subsidies of, for example, electric vehicles’ (Carlsson and Johansson-Stenman 2003). The question seems to have varying answers depending primarily on the features of consumer preferences.

To conclude, claim 16 appears consistent with the mainstream economic analysis, although it is not universally accepted. Incidentally, this claim also requires that taxes be included in the computation, a position in contrast with other claims, which we investigate in the present section.

Claim 17: The 2019 CBA Method Would Question Sustainable Mobility Policies

A literal statement would be: ‘Whenever one would imagine considering among project costs the reduced incomes (from the state and highway concessionaires), the whole environmental policy for emission reduction and the development of electric vehicles, shared mobility systems, (…) would be rediscussed’.¹⁷ This claim is not scientific in nature. It assumes that the method should be chosen based on its outcome rather than on its internal consistency.¹⁸ Additionally, it neglects that, ceteris paribus, in the method used in the 2019 CBA, as in Eq. (7), the larger the environmental benefits of the project, the higher its net social value.

We can also check a slightly different claim, which follows.

Claim 18: The Project Reduces Externality and Tax Revenues in the Same Measure, thus Reduced Tax Revenues Should be Discarded from the Computation¹⁹

Literally:

“the car traffic that could shift to the rail (…) generates negative externalities for 1,785 million euro, nearly perfectly compensated by excise rights for 1,619 million. But even if compensated, the damage would occur anyway. Instead, if the project were achieved, on the one side the damages would be avoided (and would be, as they are, computed as project benefits). But the corresponding excise lost by the State, accounted as a cost for the project, should not be considered because the losses for the State are actually 0: lower excise rights but also minor health expenditures to bear (…).”²⁰

This is a more complex statement than the others investigated in this study. It relies on the belief that, thanks to the project, the state would benefit from a reduction in externality-related expenditures. One can find some intuition for this reflection. However, it is well-known that public expenditure savings are only partially connected to externality reductions. In some cases, public expenditures are a sizeable part of negative externalities, in others they are a proxy; in many other cases, both are poorly connected. Additionally, in a simplified situation in which a given externality merely consists of a reduced State expenditure, this benefit is already correctly considered in the externality account, so there is no reason to duplicate this benefit by discarding the reduced tax revenues from the computation.

Strictly speaking, the investigated claim can be expressed more formally as ‘If $X^{PE} = T^{PE}$,$W_{ROH}^{NT} {\text{wich is }}B_{ROH} +X^{PE} + E^{PE} {-}{ }K^{CBA}$ is a correct estimate of benefits’. We find no analytical support for this claim. The inclusion or (coherent) exclusion of taxes is always necessary and does not depend on the relative magnitudes of the various items involved.²¹

Claim 10: The RoH Would be Correct Only if Taxes were Deducted from Costs and Benefits

Until now, we have left claim 10 which relates both to RoH and taxation. This claim appears at the conclusion of the following sentence:

“One euro less for highway concessionaire is valued as 50 cents saving for car users, a consequence of a traditional method, theoretically superseded, for computing consumer benefits which assumes that (fiscal) transfers are eliminated from costs and benefits, something that is not done in the MIT analysis”²² (our emphasis).

From the formulation of RoH estimates and from its derivation (Appendix A), the correctness of RoH estimates does not depend on the exclusion of taxation. RoH can be computed with or without taxes; what matters is that other parts of the computation are performed accordingly. Therefore, this claim appears unconvincing.

We have now reviewed various claims formulated regarding the role of taxation in the CBA. Table 3 summarises these findings. It is apparent that most criticisms were ill-founded.

Table 3 Claims on taxation

Our review strongly suggests that the claims expressed about the Rule of Half and taxation in the CBA during the 2019 Italian debate were incorrect, with few exceptions. We can now draw the following conclusions from our research.

5 Conclusions

In this paper, we have examined the criticisms formulated by academics and experts toward the 2019 CBA of the Lyon-Turin High-Speed/High-Capacity Line. In a context that is now less passionate and is disconnected from the pressure of policy discussions, there is an opportunity to reflect on how CBA calculations should be implemented.

To do so, we first proposed a meta-equation of CBA evaluation, which can be used to reduce ambiguity in comparing CBA methods and verifying claims; it can also formalise incoherent methods, tracking sources of inconsistencies. We hope that this formalisation will be useful for a large number of analysts.

We then reviewed the criticisms of the 2019 CBA using this meta-equation. We concentrated on comments from academics and experts collected in a compendium by the Observatory of the Lyon-Turin Rail Corridor, an Italian governmental body in charge of concertation on the Lyon-Turin Project. We concentrated on two salient topics: Rule of Half and the inclusion of taxation and we analysed 18 claims.

We generally find that these claims misunderstand or misrepresent CBA. Two claims partly escape this conclusion.

• One claim that bears no inconsistency but requires assumptions:

  Reduced tax revenues would reduce distortions, and this reduction should be considered among the benefits of a CBA when a project reduces levied taxes. To be more precise, this position is not endorsed by all economists; some instead show that under alternative assumptions on preferences, such impacts should be discarded. Interestingly, we can also note that this claim would also make tax inclusion necessary and would invalidate all other claims where taxes should be excluded. The claim that lower fuel tax revenues should be specifically reflected in a lower distortion may however require further investigation due to its intricacy.

• One claim that is approximate in its premise and whose conclusion is discussible:

  This is related to the linearity assumption behind the Rule-of-Half. To be precise, linearity is a sufficient condition for the validity of Rule-of-Half calculation but not a necessary condition. However, even restricting to the specific assumption of linearity, the conclusion that this assumption invalidates the results of a given study is problematic. Neither this assumption nor the alternative sigmoid assumption embedded in the logsum computation, is generally testable; although isolated proposals have been formulated to do such checks. If the underlying criticism were taken literally, it would mean that any economic calculation based on a reasonable yet untested or untestable assumption should be banned. It would make economic evaluation aporetic and would have a number of epistemological limits (Chapter 6 in Wimsatt 2011). Pragmatically, this would leave the floor to other methods that have no reason to be more rigorous, although their lack of rigor may only be less apparent.

The other claims analysed appear based on a misunderstanding or misrepresentation of what a CBA is, or what it should be. Most typically, the proposed ‘correction’ of discarding tax loss from the NPV calculation would introduce a sizeable inconsistency, sometimes in the order of magnitude of billions of euros, in the appraisal of important infrastructure projects. The proposed modification should indeed dictate a similar correction on consumer surplus so that, in the best case, these changes would exactly compensate one another; in the worst case, the proposed elision of taxation would just make the consideration of second-order effects of taxation impossible.

The issues that we explored also have epistemological and deontological implications. First, it appears possible that a community of experts embraces misguided and mostly incoherent interpretations of a well-established analysis technique. This is in line with the literature on the rise and persistence of errors in the scientific community. For instance, this is analysed in the works by Smaldino and O’Connor (2020) on the use of ill-founded statistical methods, Ioannidis (2012) on the lack of self-correction in science, and the abundant literature on the misuse of significance testing in economics and other disciplines (McShane et al. 2019; Ziliak and McCloskey 2008).

As a community and individual, scientists are committed to diffusing sound knowledge. The 2019 debate contradicts the collaborative and incremental nature of progress in science and arguably in society. If no further discussion takes place after the 2019 CBA episode, one of the rare moments where CBA was in the heart of the public debate (at least in one country), the endorsement of inconsistent claims by a large majority of academics raises concern and could deteriorate the quality of policy evaluation. Occasionally, efforts have been made to bring economists to the same table and to reduce misunderstandings.²³ These isolated efforts and structured exchanges among experts must continue.

We hope the present writing contributes to this aim.²⁴

Appendices

Appendices

1.1 Appendix A: The Rule of Half, Synthetic Presentation

In this appendix, we present concisely the Rule of Half, a well-established measure used in evaluation (Button 2006; Neuburger 1971; Quinet and Vickerman 2004) with well-known properties (Massiani and Maltese 2019). We discuss its area of validity and its merits, compared with alternative measures, which have been repeatedly analysed (Bates 2005; Batley 2008; de Jong et al. 2005; Kohli and Daly 2006; Ma et al. 2015).

1.1.1 Derivation of the Rule of Half

We present a simple analytical derivation of the RoH as a measure of user benefits. The total user benefits of a project can be seen as consisting of several benefits.

• $B_{00}$ for users staying on mode 0: $B_{00} = \mathop \sum \nolimits_{i} \left( {{\text{C}}_{0} + {\upvarepsilon }_{0i} } \right){ }{-}{ }\left( {{{C^{\prime}}}_{0} + {\upvarepsilon }_{0i} } \right)$ = ${\text{Q}}_{00} \left( {{\text{C}}_{0} - {{C^{\prime}}}_{O} } \right)$.

• $B_{11}$ for users staying on mode 1: $B_{11} = \mathop \sum \nolimits_{i} \left( {{\text{C}}_{1} + {\upvarepsilon }_{1i} } \right){ }{-}{ }\left( {{{C^{\prime}}}_{1} + {\upvarepsilon }_{1i} } \right) =$ ${\text{Q}}_{11} \left( {{\text{C}}_{1} - {{C^{\prime}}}_{1} } \right)$.

• $B_{z0}$ for induced demand for mode 0.

• $B_{z1}$ for induced demand for mode 1.

• $B_{01}$ for users switching from 0 to 1.

1.1.2 Transferred Users

This last category ‘$B_{01}$’ deserves specific interest as it concentrates difficulties. Indeed, it is written as (with subscript n for year n; this will be omitted in other equations for the sake of simplicity)

$$ B_{{01,n}} = ~\mathop \sum \limits_{i} (C_{{0,n}} + \varepsilon_{{0i,n}} ){-}(C^{\prime}_{{1,n}} + \varepsilon_{{1i,n}} ) = ~\mathop \sum \limits_{i} \left( {C_{{0,n}} - ~C^{\prime}_{{1,n}} } \right) + (\varepsilon_{{0i,n}} - ~\varepsilon_{{1i,n}} ). $$

(16)

This expression cannot be directly computed due to the presence of $\left( {{\upvarepsilon }_{i0,n} { }{-}{\upvarepsilon }_{i1,n} } \right)$, which is the difference of two unobserved components. To compute $B_{01}$, one needs to substitute these terms with known terms. This is done by assuming a distribution for these terms (Cox and Snell 1989) and using indifference conditions for the marginal individuals. In the resulting formulation, transferred users’ benefits depend only on the change in observed generalised costs on each single mode. A surprising outcome that indeed could (and did) generate misunderstanding.

In detail, the derivation process relies on:

• 1. writing indifference conditions for marginal individuals;

• 2. using the resulting equations to express marginal individuals’ benefits based on an observed variable;

• 3. assume a distribution of the unobserved terms to compute the benefits of intermarginal individuals (individuals whose preferences place them in between the two marginal individuals).

We first consider the individual that is indifferent between the two modes in the original situation (max, since it is the maximum benefit gained from the change), such that

$$C_{0} + \varepsilon_{{0,\max }} = C_{1} + \varepsilon_{{1,\max }} $$

(17)

so that,

$$ C_{0} - C_{1} = \varepsilon_{{1,\max }} - \varepsilon_{{0,\max }} $$

(18)

The corresponding benefit B_max is

$$ {\text{B}}_{\max } = \left( { C_{0} + \varepsilon_{0,max} } \right) - \left( { C^{\prime}_{1} + \varepsilon_{1,max} } \right). $$

(19)

Which, given Eq. 17, provides:

$$ {\text{B}}_{\max } = \left( { C_{1} + \varepsilon_{1,max} } \right) - \left( { C^{\prime}_{1} + \varepsilon_{1,max} } \right) = C_{1} - C^{\prime}_{1} . $$

(20)

Similarly, the individual who is indifferent in the project situation (i = min, since s/he gets no benefit from improvement of mode 1), is such that:

$$ C^{\prime}_{1} + \varepsilon_{1,min} = C^{\prime}_{0} + \varepsilon_{0,min} , $$

(21)

$$ {\text{B}}_{{{\text{min}}}} = \left( { C_{0} + \varepsilon_{0,min} } \right) - \left( {C^{\prime}_{1} + \varepsilon_{1,min} } \right) = \left( { C_{0} + \varepsilon_{0,min} } \right) - \left( {C^{\prime}_{0} + \varepsilon_{0,min} } \right), $$

(22)

$$ {\text{ B}}_{{{\text{min}}}} = {\text{C}}_{{0}} {{ -C^{\prime}}}_{{0}} . $$

(23)

These two expressions, B_min and B_max, provide the maximum and minimum values of individual benefits. To estimate the benefit for intramarginal users, we require an additional assumption relating to the distribution of individual-specific benefits between B_min and B_max. If one assumes that these distribute uniformly on the relevant interval (or any equivalent assumption), the resulting estimate of benefits $B_{01}^{ROH}$ will be

$$ \begin{aligned} \text B_{{01}}^{{\text {RoH}}} = 1/2 \cdot \text Q_{{01}} (\text B_{{\max }} + \text B_{{\min }} ) = 1/2 \cdot \text Q_{{01}} \cdot \left( {\left( {\text C_{1} {-}\text C^{\prime}_{1} } \right) + \left( {\text C_{0} - \text C^{\prime}_{0} } \right)} \right) \hfill \\ \end{aligned} $$

(24)

At this point, we observe that it is possible to suppress the unobserved component of the generalised cost formulation. We also observe that the benefits could be computed based on each mode in isolation,

$$ B_{01}^{RoH} = \left( {\mathop \sum \limits_{i} \left( {C_{0} + \varepsilon_{0i} } \right) - \left( {C^{\prime}_{1} + \varepsilon_{1i} } \right)\Big|\, \kern-.05em\lower1ex\hbox{$\displaystyle { \left( {\varepsilon_{0i} - \varepsilon_{1i} } \right)\sim U}$}} \right) = 1/2Q_{01} \left( {\left( {C_{0} - C^{\prime}_{0} } \right) + \left( {C_{1} - C^{\prime}_{1} } \right)} \right). $$

(25)

Most strikingly, if the improvement on the origin mode were assumed 0, the benefits would only depend on the change in the destination mode; an intriguing result on its own that may generate misunderstandings.

Instead, when $\left( {\varepsilon_{i0,n} - \varepsilon_{i1,n} } \right){ }$ is not distributed uniformly (or with another distribution that makes RoH valid) the approximation in RoH measure will be:

$$ B_{01}^{RoH} - \left( {\mathop \sum \limits_{i} \left( {C_{0} + \varepsilon_{0i} } \right) - \left( {C^{\prime}_{1} + \varepsilon_{1i} } \right)} \right) = \mathop \sum \limits_{i} \left( {\varepsilon_{0i} - \varepsilon_{1i} } \right) - \frac{{\left( {\left( {\varepsilon_{1max} - \varepsilon_{0max} } \right) - (\varepsilon_{1min} - \varepsilon_{0min} )} \right)}}{2}. $$

(26)

The Rule of Half measure will be approximated, and the sign of such approximation will depend on the distribution of $\left( {\varepsilon_{0i} - \varepsilon_{1i} } \right)$ around the midpoint of $\left[ {\varepsilon_{imax} ;\varepsilon_{{imin{ }}} } \right]$.

1.1.3 Complete RoH Formulation

Additionally, one wants to include induced traffic (a full demonstration can be found in Neuburger (1971) or, more recently in (Delle Site and Salucci 2018), in Italian) the RoH can be used to compute ${\text{B}}_{{{\text{z}}1}}^{{{\text{R}}o{\text{H}}}}$ and ${\text{B}}_{{{\text{z}}0}}^{{{\text{R}}o{\text{H}}}}$. Adding as well ${\text{B}}_{11}$ and ${\text{B}}_{00}$, the benefits of users staying in their mode, one obtains the total RoH estimate of users benefits:

$$ \begin{gathered} \text B^{RoH} = \text B_{01}^{ROH} + \text B_{z1}^{ROH} + \text B_{11} + \text B_{z0}^{ROH} + \text B_{00} \hfill \\ = \mathop \sum \limits_{n = 1}^{N} \frac{{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( {\text Q_{0,n} + \text Q^{\prime}_{0,n} } \right) \cdot \left( {\text C_{0,n} - \text C^{\prime}_{0,n} } \right) + \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( { \text Q_{1,n} + \text Q^{\prime}_{1.n} } \right) \cdot \left( {\text C_{1.n} - \text C^{\prime}_{1,n} } \right)}}{{\delta_{n} }} \hfill \\ \end{gathered} $$

(27)

where we explicitly use the notation n corresponding to year; RoH is applied to yearly data and discounted.

1.2 Appendix B: A Coherent Treatment of Taxation

In this appendix, we focus on the tax impacts on transferred users. Indeed, while users staying in their mode simply benefit from the reduction in Generalized Costs, transferred users generate impacts that are more complex: their individual benefit is $\left( {C_{0} + \varepsilon_{0} } \right) - \left( {C^{\prime}_{1} + \varepsilon_{1} } \right)$. We compare various competing measures W, which differ in how they treat taxation.

A first possible measure is the one used in the 2019 CBA, denoted $W^{CBA}$:

$$ W^{CBA} = B^{ROH} + X^{PE} + E^{PE} + T^{PE} {-}K^{CBA}. $$

(28)

However, this computation may not be convincing, and it has been criticised; taxes are a transfer, so they should not appear as a cost in the project NPV. This motivated numerous authors to propose another computation, net-of-tax. This can be achieved in two different ways.

Firstly, by using a consistent net-of-tax computation where taxes are eliminated from T and from B. This measure is labelled ${\text{W}}^{{{\text{NT}},{\text{NT}}}}$ to clarify that taxes T^PE are subtracted both from benefits and from costs.

Second, a measure where ${\text{T}}^{{{\text{PE}}}}$ is discarded only once. This is correspondingly labelled $W_{ROH}^{NT}$, to indicate that taxes are subtracted from only one side of the computation. In summary, the two measures are written as:

$$ W^{NT,NT} = B^{NT} + X^{PE} + E^{PE} {-}K^{CBA} $$

(29)

$$ W_{ROH}^{NT} = B^{ROH} + X^{PE} + E^{PE} {-}K^{CBA} $$

(30)

The first of these two formulas is consistent with the notion of net-of-tax computation; costs and benefits are both net-of-tax.

The comparison between these two measures is detailed in the following paragraphs.

1.2.1 Net-of-Tax Computation

We examine the coherent net-of-tax computations, denoted ${\text{W}}^{{{\text{NT}},{\text{NT}}}} .$ We omit n subscripts and discount for the sake of simplicity. We first decompose the generalised costs of each mode in two parts by singling out the tax component in each mode, for instance: $C_{0} = C_{0}^{NT} + t_{0}$, and $C_{1}^{{\prime}} = C_{1}^{{\prime}NT} + {\text{t}}_{{1}}$, where the NT subscript corresponds to the net-of-tax generalised costs. Based on these definitions, the net-of-tax benefit of a transferred user would be: $\left( {\text C_{0} - {\text{t}}_{0} } \right) - \left( {{\text{C}}_{1}^{{\prime}} - {\text{t}}_{{1}} { }} \right) + \left( {\varepsilon_{0{\text{i}}} - \varepsilon_{1{\text{i}}} } \right)$, and expanding this formulation to all transferred users, the aggregate transferred users’ benefit would be

$$ \text B_{01}^ {\text{ {NT}}} = \mathop \sum \limits_{{\text{i}}} \left( {\left( {\text C_{0} { - }\text t_{0} +\varepsilon_{0i} } \right){ - }\left( {{{\text C^{\prime}}}_{{1}} {{\text -t}}_{{1}} +\varepsilon_{1i} } \right)} \right)=\mathop \sum \limits_{{\text{i}}} \left( {\left( {\text C_{0} +\varepsilon_{0i} } \right){ - }\left( {{{\text C^{\prime}}}_{{1}} +\varepsilon_{1i} { }} \right)+\left( {{\text{t}}_{{1}} { -} \text t_{0} } \right)} \right) $$

(31)

The first term of the latest equation is the (gross of tax) benefit of transferred users, $B_{01}$; while the second term of the latest expression is the individual tax gain multiplied by the number of transferred users.

$$ {\text{B}}_{01}^{\text{NT}} = {\text{B}}_{01} - {\text{Q}}_{{{01}}} \left( {{\text{t}}_{{0}} {\text{ -t}}_{{1}} } \right) $$

(32)

This expression $B_{01}^{NT} { }$ is made of ${\text{B}}_{01}$, the transferred users’ benefit minus the tax impact ${\text{Q}}_{{{01}}} .\left( {{\text{t}}_{0} - {\text{t}}_{{1}} } \right)$, a positive value whenever $t_{0} > {\text{t}}_{{1}}$ (some readers prefer to read ${\text{B}}_{01}$ plus the negative impact ${\text{Q}}_{{{01}}} .\left( {{\text{t}}_{{1}} - t_{0} } \right)$).

For users’ categories, other than transferred users, the net-of-taxes benefit can be computed as:

• ${\text{B}}_{{{11}}}^{NT}$ = $B_{11}$: users staying on their modes do not have such tax benefits, ergo net-of-tax benefits for users staying in the same mode are the same as gross benefits

• ${\text{B}}_{{{\text{z1}}}}^{{{\text{NT}}}}$ = $B_{{{\text{z}}1}}$-${\text{Q}}_{{{\text{z}}1}} {(}\left( {{\text{p}}_{1} /{\text{p}}_{{\text{z}}} } \right){\text{t}}_{{\text{z}}} {{ -t}}_{{1}} {)}$: For induced users, the net-of-tax benefit is different from the gross benefits, and the sign of the difference will depend on the difference in taxation between mode 1 and other goods. For instance, if z is more taxed than mode 1, there will be a tax benefit for induced users. If one wants to compute a net-of-tax computation, this tax benefit has to be deducted from the users’ benefits. In Partial Equilibrium, the impact of generic good taxes $\left( {{\text{p}}_{1} /{\text{p}}_{{\text{z}}} } \right){\text{t}}_{{\text{z}}}$ will be neglected so that the relevant benefit will be ($B_{z1}$ + ${\text{t}}_{{1}} Q_{{{\text{z}}1}}$).

Consequently, if an analyst wants to compute a consistent net-of-tax result, ${\text{W}}^{{{\text{NT}},{\text{NT}}}} ,$ two changes need to be made: $T^{PE}$ should be discarded from the calculation and the net-of-tax users’ benefits, ${\text{B}}^{{{\text{NT}}}} ,$ should be used rather than B. Therefore ${\text{W}}^{{{\text{NT}},{\text{NT}}}}$ should be written as (we neglect subscript n in some equations for simplicity of notations and we use Partial Equilibrium expression of $B_{z0}^{\text{NT}}$ and ${{B}}_{{{\text{z1}}}}^{{{\text{NT}}}}$):

$$ W^{NT,NT} = B^{NT} + X^{PE} + E^{PE} {-} K^{CBA} = \left( {B_{00}^{NT} + {\text{B}}_{{{11}}}^{NT} + B_{z0}^{NT} {{ + B}}_{{{\text{z1}}}}^{{{\text{NT}}}} + B_{01}^{NT} } \right)+ X^{PE} + E^{PE} {-}K^{CBA} $$

(33)

$$ \text W^{\text {{NT,NT}}} = \text B_{00} +\text B_{11} +\left( {\text B_{z0} + {\text{t}}_{{0}} \text Q_{z0} } \right)+\left( {\text B_{z1} {{ + \text t}}_{{1}} \text Q_{z1} } \right)+\left( {\text B_{01} { - }\left( {{\text{t}}_{{0}} {\text {-t}}_{{1}} } \right){\text{.Q}}_{{{01}}} } \right) + \text X^{\text {PE}} + \text E^{\text{PE}} {-} \text K^{\text {CBA}} $$

(34)

$$ \text W^{\text{{NT,NT}}} =\text B_{00} +\text B_{11} +\text B_{z0} + \text B_{z1} + \text B_{01} + \left( {{\text{t}}_{{0}} \text Q_{z0} {\text{{ + t}}_{{1}}} \text Q_{z1} - \left( {{\text{t}}_{{0}} {{\text{ -t}}_{{1}}} } \right){\text{.Q}}_{{{01}}} } \right) + \text X^{\text {PE}} + \text E^{\text {PE}} {-}\text K^{\text {CBA}} $$

(35)

Recalling from Sect. 3 that, $\text T^{\text {PE}} { = - }\left( {{\text{t}}_{0} {-\text {t}}_{{1}} } \right){\text{.Q}}_{{{01}}} +\left( {{\text{t}}_{{0}} {\text{Q}}_{{{\text{z}}0}} {{ + \text t}}_{{1}} {\text{Q}}_{{{\text{z}}1}} } \right)$, we can show that:

$$ W^{NT,NT} =B_{00} +B_{11} +B_{z0} + B_{z1} + B_{01} + T^{PE} + X^{PE} + E^{PE} {-}K^{CBA} = W^{CBA} . $$

(36)

In other words, if an analyst wants to make a net-of-tax estimate, this should impact both the state and users, thus leaving the result unchanged; at least if no second-order effects of taxation are considered. If such effects must be considered, only the inclusion of taxation allows us to do so.

1.2.2 Inconsistent Net-of-Tax Computation

The correct net-of-tax computation can be compared to another candidate measure, where the change in government revenues is discarded and users’ benefits are not corrected accordingly. This has been referred incidentally as a ‘no excise’ (‘senza accise’) calculation and proposed (Cini et al. 2019; Percoco 2019) or implemented (Trento and Spaziani 2019). MIT Working Group (MIT 2018, pag. 30) presented this calculation, stating it was erroneous, at the beginning of the 2019 debate. This so-called net-of-tax calculation can be conveniently written as ${\text{W}}^{NT}$

$$ W^{NT} = W^{CBA} - T^{PE} = {\text{B + }}X^{PE} + E^{PE} {-}K^{CBA} $$

(37)

This is not a proper net-of-tax computation, as it does not recognise that taxes are a transfer.

$$ W^{NT} = W^{NT,NT} - T^{PE} $$

(38)

This measure is distorted; if the project reduces tax revenues, $T^{PE}$ will be negative; thus, ${\text{W}}^{NT}$ will overestimate benefits.

This welfare function ${\text{W}}^{NT}$ would be inconsistent and would not recognise that taxation is a transfer; therefore, it affects both public funds and users. In fact, users’ surplus is defined for a given level of taxation; if taxes are excluded from the computation, the surplus needs to be recalculated or complemented with the change in taxation. In other words, users’ surplus calculated with market price quantities includes the benefits of reduced taxation; correcting for taxation in state revenues actually requires a corresponding correction of users’ benefits.

1.3 Appendix C

The following Table 4 presents additional criticisms made on the 2019 CBA which are not investigated in this paper. A more documented version of this table, including exact quotes, is available as additional material.

Table 4 Other criticisms formulated on the 2019 CBA

1.4 Appendix D: Original quotation of claims in Table 1

i “Il metodo utilizzato per l’Analisi Costi-Benefici del Terzo Valico e della Torino Lione stima il surplus del consumatore conteggiando le accise e i pedaggi pagati, ma divide per due il valore ottenuto, per una generalizzata quanto acritica applicazione della regola delle metà. Il valore delle accise e dei pedaggi, conteggiato a metà tra i benefici, sembra (usiamo questa cautela perché nessuna tabella esplicativa è stata pubblicata) essere compensato sottraendo per intero il loro valore come costo: se così fosse, ci troveremmo di fronte a una strutturale e immotivata penalizzazione di ogni cambio modale”

ii “Ancora, risulta molto discutibile il metodo utilizzato per calcolare i benefici. Un euro in meno per i concessionari autostradali vale solo 50 centesimi come risparmio dell'automobilista.”

iii “In seconda istanza, i benefici sociali prodotti dai consumatori che si spostano dalla gomma al ferro valgono solo per metà (la cosiddetta “regola del mezzo”, ovvero la regola per cui, per ottenere l’incremento di benessere, si stima il triangolo al di sotto della curva di domanda dividendo per la metà il valore ottenuto dal prodotto tra i risparmi unitari e il traffico deviato). Questo implica che i minori introiti dello Stato pesano il doppio rispetto ai risparmi dei consumatori. Dunque, una scelta tutt’altro che neutra”

iv “Il tener conto delle accise applicando la regola della metà comporta che sono sempre considerate integralmente come “costo” e solo al 50% come “beneficio” introducendo una deformazione strutturale del risultato”

v “La diversione modale comporta una riduzione delle accise (e dei pedaggi autostradali) (…) Tali riduzioni sono dei costi nell’ACB. Ma ci sono anche dei benefici… perché i consumatori non pagano, ma il loro beneficio conta la metà”

vi “Il contributo delle minori accise pagate all’aumento del surplus del consumatore – misurato dall’area di un triangolo di base t(qs − tp) e altezza t è, dunque, esattamente pari alla metà delle minori entrate da accise per lo Stato, misurate dall’area del rettangolo di uguale base e uguale altezza. Così facendo, ogni spostamento modale dalla strada alla ferrovia che generi, quindi, una riduzione di consumo di carburante si risolve in una perdita netta di “benessere” per la collettività. Discorso del tutto analogo può essere fatto per i pedaggi autostradali.”

vii “Risulta che quanto maggiore è la diversione della domanda verso un modo di trasporto che presenta caratteristiche non misurate (ad es. il confort del viaggio, la possibilità di lavorare e/o di riposare durante il viaggio, etc.) tanto maggiore è la sottostima del surplus, ovvero dei benefici per gli utenti. Se la regola del mezzo viene applicata ugualmente in analogia ai casi mono-modali, occorre essere consapevoli che è possibile andare incontro a tali sottostime, in alcuni casi in maniera anche molto significativa.”

viii “Il surplus del consumatore, che rappresenta il beneficio diretto per i passeggeri e le merci che utilizzano la nuova opera, è calcolato attraverso la “regola della metà”. Tale regola costituisce una semplificazione che è tipicamente accettata quando l’analisi interessa principalmente una sola componente di domanda (analisi mono-modale), ma è molto controversa quando occorre prendere in considerazione la domanda su diverse modalità di trasporto (auto, treno, aereo) che presentano costi operativi e vantaggi di utilizzo differenti (analisi multi-modale)”

ix “A parte la supposizione che la distribuzione di utilità sia lineare, il problema è che si valuta il differenziale di utilità facendo solo riferimento alla variazione di Cgt “ (…) “L’anno scorso abbiamo realizzato oltre 5.500 interviste a viaggiatori a bordo dei treni alta velocità e le risposte hanno dimostrato che il beneficio dichiarato è molto più alto di quello calcolato con la regola della metà e la distribuzione dell’utilità percepita non è affatto lineare.”

x “Una conseguenza di un metodo tradizionale, e teoricamente superato, di calcolare i benefici dei consumatori che però presuppone che vengano eliminati dai costi e dai benefici i trasferimenti, cosa non fatta nella analisi del MIT.”

xi “Citando un documento di un progetto europeo del 2006, il gruppo di lavoro considera queste voci come costi. Si tratta tuttavia di una prassi non condivisa da tutti. Nel manuale sull’analisi costi-benefici di E. J. Mishan e Euston Quah (pag. 240) le correzioni dell’onere fiscale in eccesso sono considerate un errore. Le linee guida della Commissione europea, poi, chiariscono che le tasse sono trasferimenti e non costi

xii “L'approccio convenzionale dell’analisi costi-benefici e le linee guida comunitarie e nazionali suggeriscono che le tasse vengano escluse dal calcolo, perché costituiscono un trasferimento dal consumatore alle casse dello Stato”

xiii “in realtà nella tecnica dell’Analisi Costi-Beneficio, il prezzo della benzina viene depurato dalle imposte, esattamente come tutti i fattori produttivi utilizzati nel progetto, in quanto distorcono il prezzo di mercato del bene:”

xiv “È evidente la distorsione di queste ipotesi che confondono analisi economica ed analisi finanziaria e valutano la convenienza di un investimento sulla base dell'attuale sistema di tassazione indiretto tutto basato sui carburanti

xv “in questa correzione ad esempio le accise vengono eliminate in quanto sono “trasferimenti””

xvi “La distinzione tra analisi finanziaria ed economica aiuta a comprendere che, all’interno di quest’ultima, la variazione di un prezzo o di una tassa (per esempio, un’accisa) non è una risorsa consumata, e quindi un “costo”, ma viene considerata nel totale del “costo percepito” quale “spia” per “stimare un beneficio”; un indizio, cioè, che, misurando la disponibilità a pagare, ci suggerisce quanto quel viaggio valga per l’utente. Non ha quindi senso “compensare” questa posta con una corrispondente, ma di segno opposto, variazione del “beneficio dello Stato”

xvii “Io ho qualche dubbio sui metodi che Ponti usa nel suo articolo di qualche mese fa (vedremo, quando sarà reso pubblico, se gli stessi metodi sono stati adottati nel rapporto della commissione che ha presieduto. In una corretta analisi costi benefici che prenda come punto di vista quello della collettività bisogna tenere conto che le imposte sono percepite dallo Stato ma pagate dai soggetti economici.”

xviii “Va da sé che, nel momento in cui si punta all’utilizzo di prezzi che si avvicinano al mondo ideale della concorrenza perfetta attraverso l’eliminazione delle distorsioni di mercato (tra cui, appunto, le tasse), è quantomeno bizzarro ritrovare i mancati introiti delle accise quale costo per la collettività. In sostanza, siamo davvero sicuri che ridurre il finanziamento della guerra in Abissinia (una delle ragioni per cui si introdussero quelle accise) sia un danno per la società?”

xix [Considerare le tasse] “infatti, significa auspicare un maggiore consumo di carburante per avere maggiore accise; principio che contrasta con la politica ambientale di riduzione dell’inquinamento, per cui lo svecchiamento del parco auto euro = -1–2-3–4- ad altri valori di C02 in favore di auto elettriche o ibride non sarebbe un obiettivo da perseguire per la perdite delle accise sul carburante” Trento p. 9

xx “tra i costi sono stati inclusi le accise che lo stato perderebbe che come è noto non sono altro che trasferimenti”

xxi “(…) le accise superano in valore le esternalità negative prodotte: una loro riduzione, quindi, dovrebbe ridurre la distorsione attuale e quindi far tendere l’assegnazione delle risorse verso una situazione di maggiore efficienza”

xxii “Se si immaginasse di considerare “costo del progetto” la perdita di ricavi (dello Stato e dei concessionari) si metterebbe in discussione tutta la politica ambientale per la riduzione delle emissioni ed anche lo sviluppo dei veicoli stradali a trazione elettrica, i sistemi condivisi, lo sviluppo dei percorsi ciclo pedonali e così via”

xxiii “Ci guadagnate di più col Tir che con il treno. (…) Se la tassazione supera il danno, il motore a combustione interna diventa il motore della finanza pubblica. Gliene potrebbe derivare, ministro, l’obbligo di riaprire i centri storici al traffico privato, e anzi incentivarlo, o di vietare l’uso delle biciclette. Più macchine più accise; e anche soldi risparmiati in sussidi al trasporto pubblico.”

xxiv “Nel nostro caso, il traffico automobilistico che potrebbe passare al ferro secondo lo scenario “realistico” della Struttura Tecnica genera esternalità negative per 1.785 milioni, quasi perfettamente compensate dalle accise per 1.619 milioni. Ma, anche se compensati, i danni ci sarebbero comunque. Invece, se si realizzasse il progetto, da una parte i danni sarebbero evitati (e sarebbero, come sono, computati come benefici del progetto), ma le corrispondenti accise perdute dallo Stato, che la Struttura Tecnica computa come costi del progetto, non vanno considerate perché le perdite dello Stato sono in realtà pari a zero: minori accise, ma anche minori costi sanitari da sostenere. Dunque, nel metodo ha ragione l’Unione europea”